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W/S Pro-independence activists rally to oppose visit of Spanish Prime Minister Pedro Sanchez to Barcelona
M/S Woman behind window
C/U Protester holds sign demanding dialogue
SOT, Berta, Barcelona resident (Spanish): "The president of the Generalitat has repeatedly asked him to meet him (Pedro Sanchez) but Mr. Sanchez has refused and today he has come to take the picture with the injured police, but he has refused to receive our president of the Generalitat (Quim Torra) who is the one who represents the citizens of Catalonia. "
W/S Protesters rallying *CUTAWAY*
SOT, Berta, Barcelona resident (Spanish): "She has systematically refused to meet an unprecedented political crisis and refuses to use the dialogue and only uses repression against our politicians and civil representatives. And police and judicial repression."
C/U Signs demanding dialogue
M/S Police overseeing rally
SOT, Rosa, Barcelona resident (Spanish): "To defend the rights of the Catalans and the Spaniards because what they are doing to us will also be done to them. And it is starting to be demonstrated in demonstrations in Madrid, Valencia and other places in Spain. We fight for our rights and freedoms. We will fight today and always at the expense of anything. "
W/S Police overseeing protesters
W/S Protesters rallying
M/S Woman shows sign reading (Catalan): "Freedom for political prisoners"
M/S Police overseeing protesters, Catalan flag
M/S Man holds sign demanding for dialogue
W/S Police helicopter hovering over city
M/S Man holds sign demanding for dialogue
W/S Protesters clapping
WS Protesters rallying to oppose Sanchez visit
SCRIPT
Thousands of people took to the streets of Barcelona on Monday, to show their discontent about the planned visit of Spanish Prime Minister Pedro Sanchez to the city, after a tense weekend of protests.
Footage shows a heavy police presence as protesters rallied to demand freedom for jailed pro-independence activists.
Activists blamed Sanchez for reportedly refusing to meet Catalan independence leader Quim Torra, who requested a meeting with the Prime Minister to initiate a dialogue on the independence movement.
During Monday's visit, Sanchez was due to meet the police officers who were injured during the last five consecutive evenings of rioting. According to the authorities, 288 police officers were injured and 194 people were arrested over the past weekend.
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TITLE: Elementary proof that $c^n > n^c$ for $n>>1$?
QUESTION [0 upvotes]: I need to prove that for $c>1$ we can find some bound $n_0$ such that $c^n>n^c$ for all $n\in \mathbb N, n > n_0$.
Now all proofs I found so far include more advanced tools like that $f(x) = x/\log x$ grows unbounded, and to prove that we need again more theory. Is there a more elementary proof of this statement that does not get too long? The reason is that I want to show it a student that just started taking math courses at a university.
REPLY [1 votes]: Write $n=(1+m)c$.
$c^n>n^c$ is equivalent to $c^m>m+1$.
Now use the inequality $$
\begin{align*}
c^m\geq c^{\lfloor m\rfloor}&\geq 1+(c-1)\lfloor m\rfloor+(c-1)^2\lfloor m\rfloor(\lfloor m\rfloor-1)/2\\
&\geq 1+(c-1)(m-1)+(c-1)^2(m-1)(m-2)/2\\
&>1+(c-1)(m-1)+2(m-1)\\
&>1+2(m-1)\\
&>1+m,
\end{align*}
$$
Whenever $m>2+\frac{4}{(c-1)^2}$.
REPLY [1 votes]: I need to prove that for $c>1$ we can find some bound $n_0$ such that
$c^n>n^c$ for all $n\in\mathbb{N},n>n_0$
$$\lim_{n\to \infty } \, \frac{c^n}{n^c}=\infty\quad (*)$$
Indeed as $ \lceil c\rceil\ge c$ then $\dfrac{c^n}{n^{\lceil c\rceil }}\le \dfrac{c^n}{n^c}$ and applying L'Hopital rule $\lceil c\rceil $ times as $c>1$ we get $\dfrac{c^n \log^{\lceil c\rceil} c}{\lceil c\rceil !}\to\infty \text{ as } n\to\infty$
Thus if the LHS of the inequality tends to $\infty$ then to a greater extent does the RHS. So $(*)$ is proved.
This means that for any $M>0$ there exists an $\bar n$ such that for $n>\bar n$ it happens that $\dfrac{c^n}{n^c}>M$. I choose $M=1$ so that there exists an $n_0$ such that for $n>n_0$ we have $\dfrac{c^n}{n^c}>1$ that is $c^n>n^c$
QED
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And so the Kristen Stewart soap opera trundles on, turning out to be far more interesting than all of her movie roles put together.
Now Robert Pattinson has allegedly moved out of their LA home after his girlfriend 'completely humiliated' him by having an affair with director Rupert Sanders. There's obvz just not enough room in their mansion for the two of them. He's told her that he 'needs space'. Sources close to the actor say that Robert was the one who actually called for her over the top apology in the first place. ''Robert wanted the world to know that he hadn't done anything wrong and that she had made a fool of him publicly... 'He still can't fathom why she cheated. Kristen is begging for another chance. Robert is undecided.''
Meanwhile another source - according to The Sun - says Rob is both "heartbroken and angry", adding: "I'm not sure they'll be able to recover from this."
What's more, Kristen now apparently wants to write a full grovelling letter of apology to Rupert Sanders' wife - Liberty Ross - which we can guarantee will go down like a lead balloon. Sources say "Kristen is going to do the decent thing and write Liberty a private letter expressing her deepest regret for her actions... Kristen feels she made a terribly naïve ['naive' my HOOP] mistake and will do anything to make amends. The fact that Liberty and Rupert have two kids has now dawned on her and she feels awful."
Want our advice? Just leave it out, Kristen. This is getting faaaar too big for you to handle. And who does your PR? Haven't they heard of crisis management? Say as little as possible, isn't that the golden rule?
Besides, you didn't feel so awful and it wasn't such a 'momentary indescretion' when you took Ross and Sanders' kids for ice cream two months ago. (The Daily Mail report that she'd been doting over the kids during the press tour for Snow White). And those pictures of you cosying up to your director were taken BEFORE you took the kids out out so now we're not sure what to believe at all. Now we think the affair might have been going on longer than you initially made out. Hmmm...
As for how this will affect their forthcoming movie releases? Given the backlash on Twitter, Hollywood bosses have decided to hold off on a Snow White sequel for the time being but as per analyst Paul Dergarabedian, the last installment of Twilight should still do just fine, maybe even better. Which might lead some skeptics to consider that underneath it all, this could be viewed as one big publicity stunt. “Fans think of Bella and Edward and Kristen and Rob like family... There's a soap opera going on off-screen, and people love to follow that."
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Longtime Mesa developer Nicolai Hähnle of AMD has sent out a big patch series today introducing a real runtime linker for the RadeonSI Gallium3D driver and hopefully to be used by the RADV Vulkan driver as well.
This runtime linker changes the way shaders are loaded and sent off to LLVM to the AMDGPU back-end. Rather than a bunch of hard-coding, there's an actual linker in place.
Nicolai Hähnle explained:
Basically, instead of hard-coding that we have a single .text section in the ELF generated by LLVM, we align ourselves more with the ELF standard and actually look at all the sections in the file(s), lay them out in memory, and resolve relocations between them.
There is still hard-coding of ".text" sections for the purpose of gfx9+ merged shaders.
The immediate consequence is that we will be able to emit .rodata in LLVM and emit absolute or relative relocations that will be resolved when shaders are uploaded to the GPU.
As a next step, I want us to explicitly record LDS symbol in the ELF symbol table and have ac_rtld lay out and resolve those symbols at load time. This will allow us to use LDS both for communication between shader parts and for temporary variables used within each part.
With the Mesa 19.1 branching imminent and these patches not yet being reviewed, we will likely only find this work with Mesa 19.2 next quarter but is certainly interesting and at least gives RADV time to potentially hook into this new functionality as well. More details via this patch series of ten patches touching roughly one thousand lines of code.
9 Comments
| 415,426
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May 14, 2004
( Phoenix, AZ ) Robrt Pela reviews Tapestry: A Musical Revue Based on the Music of Carole King.
Additional theater reviews
Robrt Pela's reviews at phoenixnewtimes.com
[ Robrt Pela ]
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Tapestry by Robrt Pela for KJZZ.
Attribution InformationTitle: Tapestry
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This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 2.5 License.
| 195,068
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Dating Question of the Week: "I'm In A Relationship, But Can't Stop Thinking About Another Woman" Dating Question of the WeekThis week’s question comes from a dater named “Young Usher”. He’s currently in a relationship, but met another woman that he can’t seem to stop thinking about. He’s asked our matchmaker, Christen, for her advice on what he should do. What would YOU do? To let us know, follow us on social media and YouTube to leave your comments and thoughts.*And don’t forget to follow Christen on Twitter and Instagram: @isthatchristen* dating, dating advice, matchmaker, matchmaking, relationships, Dating QuestionsChristen TurnerJanuary 28, 2019dating advice, dating, matchmaking, matchmaker, houstontexas, houstonComment Facebook0 Twitter LinkedIn0 Pinterest0 0 Likes
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smeezekitty wrote:I have a new lens setup. I took apart the lens from a broken camcorder and used only the front part. With a bit of cutting and sanding, it fits perfectly in the hole in the front of the case.
It is much more practical than having the lens on a tripod way out in front. I can turn the lens to focus it although it doesn't change it that much -- probably because how far it is from the disc. It only focuses on objects rather close in.
That was a good idea to use another lens and cover the lens hole out right reduces noise and lets the light from the lens enter .
Yes to check the lens focus to the disk put some white paper on the nipkow placement point it at things at different distance and see where it will work and check the image on the paper if its not in focus it never will ...
When trying a lens its some thing that's has to be worked out by testing ... its a 32 line camera so your not going to need long distance focusing you want it to focus close arms length or 2 at most so you can fill the screen.
Try the paper test and see how in focus a object is on that white sheet you might have to adjust the lens distance or the nipkow mounting position .
I tended to do this in my garage with the door open and with sun light the focusing on that white sheet is easy but you want close so a lamp and test card is more wanted here ....we want reflected light ...Troys nipkow camera on this forum he shows this procedure .
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{\bf Problem.} Compute
\[\prod_{k = 1}^{12} \prod_{j = 1}^{10} (e^{2 \pi ji/11} - e^{2 \pi ki/13}).\]
{\bf Level.} Level 2
{\bf Type.} Precalculus
{\bf Solution.} Let
\[P(x) = \prod_{k = 1}^{12} (x - e^{2 \pi ki/13}).\]The roots of this polynomial are $e^{2 \pi ki/13}$ for $1 \le k \le 12.$ They are also roots of $x^{13} - 1 = (x - 1)(x^{12} + x^{11} + x^{10} + \dots + x^2 + x + 1).$ Thus,
\[P(x) = x^{12} + x^{11} + x^{10} + \dots + x^2 + x + 1.\]Now, $e^{2 \pi ji/11},$ for $1 \le j \le 10,$ is a root of $x^{11} - 1 = (x - 1)(x^{10} + x^9 + x^8 + \dots + x^2 + x + 1),$ so $e^{2 \pi ji/11}$ is a root
of
\[x^{10} + x^9 + x^8 + \dots + x^2 + x + 1.\]So, if $x = e^{2 \pi ji/11},$ then
\begin{align*}
P(x) &= x^{12} + x^{11} + x^{10} + \dots + x^2 + x + 1 \\
&= x^2 (x^{10} + x^9 + x^8 + \dots + x^2 + x + 1) + x + 1 \\
&= x + 1.
\end{align*}Hence,
\begin{align*}
\prod_{k = 1}^{12} \prod_{j = 1}^{10} (e^{2 \pi ji/11} - e^{2 \pi ki/13}) &= \prod_{j = 1}^{10} P(e^{2 \pi ji/11}) \\
&= \prod_{j = 1}^{10} (e^{2 \pi ji/11} + 1).
\end{align*}By similar reasoning,
\[Q(x) = \prod_{j = 1}^{10} (x - e^{2 \pi ji/11}) = x^{10} + x^9 + x^8 + \dots + x^2 + x + 1,\]so
\begin{align*}
\prod_{j = 1}^{10} (e^{2 \pi ji/11} + 1) &= \prod_{j = 1}^{10} (-1 - e^{2 \pi ji/11}) \\
&= Q(-1) \\
&= \boxed{1}.
\end{align*}
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\begin{document}
\begin{frontmatter}
\title{Maximal Likely Phase Lines for a Reduced Ice Growth Model\tnoteref{mytitlenote}}
\tnotetext[mytitlenote]{This work was partly supported by the NSF grant 1620449, and NSFC grants 11531006 and 11771449.}
\author[mymainaddress]{Athanasios Tsiairis}
\ead{thtsiairis@hotmail.com}
\author[mymainaddress]{Pingyuan Wei}
\ead{weipingyuan@hust.edu.cn}
\author[mymainaddress]{Ying Chao\corref{mycorrespondingauthor}}
\ead{yingchao1993@hust.edu.cn}
\author[mymainaddress,mysecondaryaddress]{Jinqiao Duan}
\cortext[mycorrespondingauthor]{Corresponding author}
\ead{duan@iit.edu}
\address[mymainaddress]{School of Mathematics and Statistics, \& Center for Mathematical Sciences, Huazhong University of Sciences and Technology, Wuhan 430074, China}
\address[mysecondaryaddress]{Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA}
\begin{abstract}
We study the impact of Brownian noise on transitions between metastable equilibrium states in a stochastic ice sheet model. Two methods to accomplish different objectives are employed. The maximal likely trajectory by maximizing the probability density function and numerically solving the Fokker-Planck equation shows how the system will evolve over time. We have especially studied the maximal likely trajectories starting near the ice-free metastable state, and examined whether they evolve to or near the ice-covered metastable state for certain parameters, in order to gain insights into how the ice sheet formed. Furthermore, for the transition from ice-covered metastable state to the ice-free metastable state, we study the most probable path for various noise parameters via the Onsager-Machlup least action principle. This enables us to predict and visualize the melting process of the ice sheet if such a rare event ever does take place.
\end{abstract}
\begin{keyword}
Maximal likely trajectory; most probable path; Brownian noise; stochastic ice sheet system.
\end{keyword}
\end{frontmatter}
\section{Introduction}
\label{intro}
Tremendous variety of nonlinear complex dynamical systems are subject to noisy perturbations \cite{Ar,Duan}. These noises usually play a pivotal role on setting up the dynamical behavior of the system. It is incredibly important to study the influence of noise on ice sheets as it is closely connected with the livelihood of human beings \cite{Agarwal2018,Thorndike2000}. Indeed, with the increasing global mean temperatures over the last decades, the behaviour of ice sheets, such as those on Greenland and Antarctica, become a major topic in climate research \cite{Mulder2018,Toppaladoddi2017}. Research studies show that the melting of the Greenland Glaciers has caused the increase of the sea level by 0.5 mm per year in the period 2003-2008 \cite{Woot2008} while if the Antarctic glaciers were ever going to melt the sea level would rise by an astonishing 58 meters \cite{Fretwell2013}. All these make the study of the ice sheet more imminent. More results of interactions among atmosphere, ice, land and ocean for feedbacks that are relevant to understandings of climate change could be found in \cite{Ar,Franzke2017}.
In this present paper, we consider an ice sheet model for the development of ice sheets with boundary on the polar sea (such as the Arctic Ocean for the Greenland ice sheet), developed by Weertman \cite{Weertman1964,Weertman1976}, with the accumulation and ablation perturbed by Brownian fluctuations. In the direction of deterministic analysis, Weertman's idea has been further extended and improved through both conceptual and numerical models; see K\"all\'en et al. \cite{Kallen1979} and Oerlemans-Van der Veen \cite{Oerlemans1984}. As noise is present, the stochastic model has two metastable states in certain parameter ranges (see Section 2), which are referred as ``ice-covered" state and ``ice-free" state respectively. Random fluctuations may lead to switching between these two states, and such transitions occur widely in not only climate model, but also biological, chemical, physical, and other systems \cite{Chen2018,Duan,OM}. In order to gain insights into the evolution trajectory of the formation and even melting of ice sheet, it is of interest to study the transitions between these two metastable states. The objective of this present paper is to study the maximal likely trajectories and most probable transition paths for such a stochastic system as time goes on. This offers the following information for the ice sheet: (i) The evolution trajectories from ice-free state, indicating the forming routes of this ice sheet. (ii) The Transition paths from ice-covered state to ice-free state, predicting the melting routes on the assumption that the ice sheet will go completely in certain time interval.
We remark that ``maximal likely trajectory" and ``most probable path" here are two mathematical terms that have essential distinction \cite{Cheng2019,OM}: Firstly, The maximal likely trajectory is an evolution trajectory starting from one initial state with unknown final state, while the most probable path is a continuous transition trajectory from one metastable state to another metastable state. Secondly, the former is determined by maximizing the probability density function at every time instant, while the latter should be understood as the probability maximizer where sample solution paths lie within a tube. Thirdly, the former is obtained via numerically solving an initial value problem, while the latter is calculated by solving the two-point boundary value problem.
This paper is organized as follows. In Section 2, we introduce the stochastic ice sheet model influenced by Brownian motion and present the solutions of the deterministic counterpart. In Section 3, we present the methods used with some preliminary results. We further consider the most probable pathways by simulating exact model solutions under noise and compare these to the maximal likely trajectories. In Section 4, we present final results with varying parameters and noise intensity while also examine examine the transition from one state to another. More specifically we investigate the possibility of a transition from the ice-covered to the ice-free state. Finally, we summarize the above results in Section 5.
\section{Model}
We first introduce the height-mass balance feedback in an ice sheet system influenced by the Brownian motion, and then show most probable transition pathways and maximal likely trajectories for such a system.
The development of ice sheets is governed by nonlinear processes \cite{Ar}. Consider an idealised ice sheet with length $X$. It is natural to choose the coordinates such that the point $x = 0$ corresponds to the boundary with the polar sea; see K\"all\'en et al. \cite{Kallen1979} and Oerlemans-Van der Veen \cite{Oerlemans1984}. Indeed, we can refer to the Arctic Ocean for the Greenland ice sheet as a practical examples.
Note that the ice can be treated as a perfectly plastic material, by horizontal stress balance \cite[Page 277-278]{Ar} in the ice, the height (or thickness) of the ice sheet $h$ satisfies
\begin{equation}\label{Eqn-ice thickness}
h(x,t)=\sqrt{\sigma}\Big(\frac{X(t)}{2}-\big|x-\frac{X(t)}{2}\big| \Big)^{\frac{1}{2}},
\end{equation}
where $\sigma$ is a yield stress parameter. The maximum height of the ice sheet is indicated by $H(t)$ and given by $H(t)=h(X/2,t)=\sqrt{\sigma X(t)/2}$.
\par
On the other hand, by the mass balance for the ice cap, we have following continuity equation
\begin{equation}\label{Eqn-continuity}
\rho_i\frac{\partial h}{\partial t}=P_i-M,
\end{equation}
where the right-hand side is the mass balance for the ice cap (the difference between accumulation $P_i$ and ablation $M$). Indeed, the mass balance depends on the distance from the polar ocean, represented by $r$, and the height of the ice sheet. Note that the ice sheet is located on the north of polar sea, we always consider $r\leqslant 0$. Some experiments also indicated that both accumulation and ablation are influenced by random fluctuations arising from the complex environment in actual situation \cite{Ar,Franzke2017}. We consider a linear relation of the form
\begin{equation}\label{Eqn-mass balance}
P_i(x,t)-M(x,t)=G(x,t)=\rho_i\beta\Big(h(x,t)-\lambda(x-r)+\varepsilon_0{\xi}(t)\Big),
\end{equation}
where $\beta>0$, $\lambda>0$ are constants, ${\xi}(t)$ is a stochastic noise, $\varepsilon_0$ is the strength of the noise.
In this paper, we will consider stochastic noise ${\xi}(t)=\dot{B}_t$ as a Gaussian white noise, which is a special stationary stochastic process, with mean $\mathbb{E}B_t=0$ and covariance $\mathbb{E}(B_tB_s)=\delta(t-s)$, and, formally, can be understood as the ``time derivative" of Brownian motion (also called Wiener process) \cite[Page 51]{Duan}.
\begin{table}
\caption{Parameters of the conceptual ice sheet model \cite{Ar}}
\begin{tabular}{llll}
\hline\noalign{\smallskip}
Parameter & Meaning & Value & Unit\\
\noalign{\smallskip}\hline\noalign{\smallskip}
$\rho$ & ice density & 0.9 & $kg$ $m^{-3}$\\
$\sigma$ & yield stress parameter & 6.25 & $m$\\
$\beta$ & coefficient in the linear mass balance & $10^{-3}$ & $yr^{-1}$\\
\noalign{\smallskip}\hline
\end{tabular}
\label{tbl:table1}
\end{table}
\par
Assume that the snow accumulating on the northern half of the ice sheet flows into the Arctic Ocean or melts close to it. The evolution of the ice sheet is then governed by the mass balance conditions on the southern half of the ice sheet.
\par
Substituting $G(x,t)$ into the continuity equation (\ref{Eqn-continuity}) and integrating over the southern part of the ice sheet, we obtain (with $h=\sqrt{\sigma} (X-x)^{\frac{1}{2}}$ )
\begin{equation}
\int_{X/2}^{X}\frac{\partial h}{\partial t}dx=\sqrt{\frac{\sigma X}{2}}\frac{dX}{dt}=\beta\int_{X/2}^{X}\Big[h(x,t)-\lambda(x-r)+\varepsilon_0{\xi}(t)\Big]dx.
\end{equation}
Hence the length $X$ of the ice sheet can be determined by the following stochastic differential equation (SDE)
\begin{equation}\label{Eqn-L}
\dot{X}=f(X)+\varepsilon g(X)\xi(t),
\end{equation}
where $f(X)
=-{\frac{\beta\lambda}{\sqrt{2\sigma}}} \Big(\frac{3}{4}X^{\frac{3}{2}}-r{X}^{\frac{1}{2}} \Big)+\frac{1}{3}\beta X$, $g(X)={X}^{\frac{1}{2}}$ and $\varepsilon= \frac{\beta\varepsilon_0}{\sqrt{2\sigma}}$.
\par
For the deterministic counterpart
\begin{equation}\label{Eqn-D}
\dot{X}=f(X),
\end{equation}
the vector field $f(X)$ can be rewritten as $-U^{\prime}(X)$, with the potential function $U(X)=\frac{\beta\lambda}{\sqrt{2\sigma}} \Big(\frac{3}{10}X^{\frac{5}{2}}-\frac{2}{3}r{X}^{\frac{3}{2}} \Big)-\frac{1}{6}\beta X^2$; see Fig. \ref{fig_potential}(a).
Since there are two control parameters, $\lambda$ and $r$, it is desirable to consider beyond codimension-one bifurcation events. The parameter space is now represented jointly by the $(r,\lambda)$-plane. A greater perspective to the fold bifurcation is shown in Fig. \ref{fig_potential}(b) in terms of the cusp catastrophe surface. This cusp codimension-two phenomenon is observed because we have independent control of both $r$ and $\lambda$.
\begin{figure}
\centering
\subfigure[]{
\includegraphics[width=2.2in]{potential2.eps}
}
\subfigure[]{
\includegraphics[width=2.2in]{bifurcation2.eps}
}
\caption{(Color online) (a) The potential $U$ as a function of the length $X$ with different parameter $\lambda$; (b) The cusp catastrophe surface for the deterministic ice sheet model in (\ref{Eqn-D}) with $r$ and $\lambda$ as control parameters. Indeed, equation (\ref{Eqn-D}) always has an equilibrium state, 0; when $r=0$, there exists only one additional equilibrium state, $\frac{32\sigma}{81\lambda^2} $ ($\approx2469.1 km$, if $\lambda=0.001$); but for $r<0$, there exists two additional equilibrium state,
$
X_{\pm}=\frac{4}{3}\frac{|r|}{1\mp\sqrt{\Delta}},
$
if the discriminant $\Delta:=1+\frac{27r}{2\sigma}\lambda^2>0$. Note that $0<\Delta<1$, both of $X_{\pm}$ are bigger than 0. Thus, the state $X_{-}$ is an unstable node and the state $X_{+}$ is a stable node.
For $r=-250km$, the reasonable value of $\lambda$ should be small about $0.0014$. The deterministic model with $\lambda=0.001$ has been studied in \cite{Ar}, in which these two states are $X_{-}\approx 63.9 km$ and $X_{+}\approx 1738.6 km$. }
\label{fig_potential}
\end{figure}
\par
Without noise, in certain parameter ranges, equation (\ref{Eqn-D}) has two stable equilibrium states separated by an unstable equilibrium (which is a saddle node). In one of the stable equilibrium states, the length of ice sheet is 0 and corresponds physically to the ice sheet being melted completely. This state is usually referred to as the ice-free state. In the other equilibrium state the length of ice sheet is large and corresponds to the ice sheet being formed. This state is called ice-covered state. For different initial value of the length of ice sheet, the trajectory of $X$ may lie either in the domain of attraction of the ice-free state or in the domain of attraction of the ice-covered state. That is, the ice-free stable state 0 and ice-covered stable state $X_{+}$ are resilient (see Fig. \ref{fig_potential}(a)): the ice length states will locally be attracted to 0 or $X_{+}$, as time increases for the deterministic system. When noise is present, the system (\ref{Eqn-L}) may show switches between state 0 and state $X_{+}$, and then these two states are called metastable states. By passing through the unstable saddle state $X_{-}$, the ice length starting near the ice-free state 0 in interval $(0,X_{-})$ arrives at an ice-covered state (near $X_{+}$).
\par
We now examine these system trajectories or orbits for the stochastic ice sheet model:
(i) How does the system evolve from ice-free situation (near $0$) to ice-covered situation (near $X_+$)? It means that we can try to understand how an ice sheet is formed.
(ii) If the ice sheet melted, how would the system transit from ice-covered situation to ice-free situation? We should also wonder how likely is such a transition for this model?
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\setcounter{equation}{0}
\section{Methods}
Two different methods will be used for te purposes of this paper. I this part they will be reviewed and explained.
In the first subsection, we consider the ice sheet system (\ref{Eqn-L}) with Brownian noise (that is, $\xi(t)=\dot{B}_t$), and use the \emph{maximal likely trajectories} to discuss the evolution of trajectory starting from different initial states. In the second subsection, we focus on transition paths connecting two special fixed states for this system. The so called \emph{most probable path} will be proposed via Onsager-Machlup's function.
\subsection{Maximal Likely Trajectory Based on Fokker-Planck Equation}
We consider the maximal likely trajectory \cite{Duan,Cheng2016} for the stochastic system (\ref{Eqn-L}), starting at an initial state $X_0$. This is reminiscent of studying a deterministic dynamical system by examining the evolution of its trajectory starting from an initial state. Each sample solution path starting at this initial state is a possible outcome of the solution path $X_t$. Then an interesting question to raise is: What is the maximal likely trajectory of $X_t$? To answer this question, we need to decide on the maximal likely position $X_{ml}$ of the system (starting at the initial point $X_0$), at every given future time $t$, and this would be the maximizer for the probability density function $p(X, t)=p(X, t; X_0, 0)$ of solution $L_t$. Indeed, the probability density function $p(X, t)$ is a surface in the $(X, t, p)$-space. At a given time instant $t$, the maximizer $X_{ml}(t)$ for $p(X, t)$ indicates the maximal likely location of this orbit at time $t$. Therefore, $X_{ml}(t)$ follows the top ridge or plateau of the surface in the $(X, t, p)$-space as time goes on, and the trajectory (or orbit) traced out by $X_{ml}(t)$ is called the \emph{maximal likely trajectory} starting at $X_0$. The maximal likely trajectories are also called `\emph{paths of mode}' in climate dynamics and data assimilation \cite{Miller1999,Cheng2019}.
\par
For the stochastic ice sheet system with Brownian noise
\begin{equation}\label{Brownian}
d{X}=f(X)dt+\varepsilon g(X)dB(t),
\end{equation}
the Fokker-Planck equation \cite{Duan} in terms of the probability density function $p(X,t)$ for the solution process $X_t$ given initial condition $X_0$ is
\begin{equation}\label{local_pde}
p_t = -(f(x)p(x,t))_x + \frac{1}{2}\varepsilon^2(g(x))^2p(x,t)_{xx}, \qquad p(x,0) = \delta(x-x_0).
\end{equation}
By numerically solving the Fokker-Planck equation, we can find the maximal likely position $X_{ml}(t)$ as the maximizer of $p(X, t)$ at every given time $t$. \par
We consider the unit of time as ``$kyr$" and the unit of length as ``$km$" in the numerical calculation throughout this paper. In Fig. \ref{fig:Actual}(a)(b), from around 100 stochastic simulations of system (\ref{Brownian}) with $\varepsilon_0=0.01$ and $\varepsilon_0=0.1$, respectively, we observe The maximal likely trajectory (red line) is actually located in the middle of the blue area created from all the possible outcomes. that simulation orbits are more likely to concentrate around the maximal likely trajectory $X_{ml}(t)$ (red line). This result indicates that the orbit according to the Fokker-Planck equation is actually the maximal likely evolution trajectory of the system (\ref{Brownian}).
\par
\begin{figure}
\subfigure[]{
\includegraphics[width=2.3in]{Actual1800_0p01_title.eps}
}
\subfigure[]{
\includegraphics[width=2.3in]{Actual1600_0p1_title.eps}
}
\caption{(Color online) Let $r=-250(km)$ and $\lambda=0.001$. Simulations of system (\ref{Brownian}) (blue lines) and the corresponding maximal likely trajectory (red line) : (a) Initial state $X_0=1800$, noise intensity $\varepsilon_0=0.01$; (b) Initial state $X_0=1600$, noise intensity $\varepsilon_0=0.1$.
}
\label{fig:Actual}
\end{figure}
The state which attracts (or repels) all nearby orbits is referred as a \emph{maximal likely stable (unstable) equilibrium state} \cite{Wang}, which depends on noise intensity $\varepsilon_0$ as well as the ice sheet system parameters $\lambda$, $r$. Fig. \ref{fig:Actual} shows that the maximal likely equilibrium states in these two cases are $1736.8km$ and $1734.7km$, respectively.
We will exhibit the number and value of maximal likely stable equilibrium states for the stochastic ice sheet model more thoroughly in Section 4.
\subsection{Most Probable Paths Using Onsager-Machlup's Method}
As mentioned above, maximal likely trajectories can help us to understand the dynamics of stochastic ice sheet system starting at different initial states. However this doesn't provide any help if we aim for a particular final state. We would like to investigate how we would go from one initial state to a final state of our choice. This is a matter of finding the most probable path for two fixed points. The two fixed points of course have a particular interest as we choose them to be the metastable states. In this subsection, we use Onsager-Machlup's method \cite{OM} to study the most probable transition path connecting initial and final states. From now on we will concentrate exclusively on the transition connecting two metastable states.
Note that system (\ref{Brownian}) is an SDE with multiplicative noise, and the Onsager-Machlup function for SDEs with multiplicative noise can be referred to Bach et al. \cite{Bach1976}. However, it is more convenient to apply Onsager-Machlup's method for an SDE with additive noise numerically \cite{OM}. To this end, we make a transformation: $Z=2\sqrt{X}$ ($X>0$). By It\^o formula, the system (\ref{Brownian}) can be suitably converted to an SDE with additive noise:
\begin{equation}\label{Eqn-Z}
dZ=F(Z)dt+\varepsilon dB_t,
\end{equation}
where $f(Z)=-{\frac{\beta\lambda}{\sqrt{2\sigma}}} \Big(\frac{3}{16}Z^{2}-r \Big)+\frac{1}{6}\beta Z-\frac{\beta^2\varepsilon_0^2}{4\sigma}{Z}^{-1}$ and $\varepsilon= \frac{\beta\varepsilon_0}{\sqrt{2\sigma}}$. Note that, under the variable transformation, $F(Z)=0$ is equivalent to $f(X)-\frac{1}{4}\varepsilon^2=0$. The extra term with respect to $\varepsilon$ appears here.
It means that equilibrium states for the deterministic counterpart of additive noise system (\ref{Eqn-Z}) are affected by the strength of the noise. As we assume that the noise strength is small, we next look into the transitions between $2\sqrt{X_0}$ and $2\sqrt{X_{+}}$ for system (\ref{Eqn-Z}). By inverse transformation, we then have a better understanding for the transitions between the two metastable states of the original system (\ref{Eqn-L}).\\
\par
For $0\leqslant t_0 \leqslant t_1$, let $\mathcal{T}=\{ z\in C([t_0,t_1];\mathbb{R}):z(t_0)=z_0, z(t_1)=z_1\} $ denote the set of trajectories connecting a point $(t_0,z_0)$ on $2\sqrt{L_0}$ and a point $(t_1,z_1)$ on $2\sqrt{L_{+}}$. Consider an infinitesimally small tubular neighborhood of $z$,
$$
K(z,\delta)=\{ {z}^{\prime}\in C([t_0,t_1]:|z-z^{\prime}|\leqslant \delta, \text{for }z \in C([t_0,t_1], \delta>0\}.
$$
If $z\in\mathcal{T}$ is differentiable, then the measure of the transition paths lying in the small tubular neighborhood satisfies
\begin{equation}
\mu(K(z,\delta))\propto C(\delta) \int_{\mathcal{T}}\exp\Big(-\frac{1}{2\varepsilon^2}\int_{t_0}^{t_1}OM(z,\dot{z})dt \Big)d\mu_W[z],
\end{equation}
where $C(\delta)$ is a constant with respect to $\delta$, $\mu_W[z]$ is the Wiener measure \cite{OM,Duan}, symbol $\propto$ denotes the proportionality relation, and $OM(z,\dot{z})$ is the so called Onsager-Machlup function which is given by:
\begin{equation}
OM(z,\dot{z})=\big(\dot{z}-F(z)\big)^2+\varepsilon^2F^{\prime}(z).
\end{equation}
We remark that the probability of a transition event lying within a tube along a smooth path $z\in \mathcal{T}$ can be determined by integrating $\mu_\varepsilon[z]$ over this tube. The \emph{most probable path} or \emph{optimal path} connecting points $(t_0,z_0)$ and $(t_1,z_1)$ is defined as minimizers of the OM functional $I_\varepsilon:\mathcal{A}\to \mathbb{R}$ given by
\begin{align}\label{OM functional}
I_\varepsilon[z]:=\int_{t_0}^{t_1}OM(z,\dot{z})dt,
\end{align}
where $\mathcal{A}=\{ z\in H^1([t_0,t_1];\mathbb{R}):z(t_0)=z_0, z(t_1)=z_1\}$. In analogy to classical mechanics, we also call the OM function the Lagrangian function and the OM functional the action functional \cite{DB}.
\par
Indeed, if we restrict ourselves to twice differentiable functions $z(t)$, the most probable path can be found by variation of the OM functional $I_\varepsilon$. We thus get the Euler-Lagrange equation
\begin{eqnarray}\label{EL}
\frac{d}{dt}\frac{\partial OM(z,\dot{z})}{\partial\dot{z}}=\frac{\partial OM(z,\dot{z})}{\partial z},
\end{eqnarray}
i.e.,
\begin{equation}\label{evolu}
\ddot{z}=F^{\prime}(z)F(z)+\frac{\varepsilon^2}{2}F^{\prime\prime}(z).
\end{equation}
with boundary conditions $z(t_0)=z_0, z(t_1)=z_1$.
Note that the problem of solving (\ref{evolu}) is referred to as the two-point boundary value problem. By Theorem 10 in \cite[Section 8.2.5]{Hasty2000}, if there exists a twice differentiable solution to the Euler-Lagrange equation, then it is indeed a (local) minimizer. As this boundary value problem may not have a solution, the most probable transition pathway may not exist.
\par
We rewrite equation (\ref{evolu}) in Hamiltonian form
\begin{equation}
\label{Hamiltonian}
\begin{split}
\dot{z}=&\Phi+F(z), \\
\dot{\Phi}=&-F(z)\Phi+\frac{\varepsilon^2}{2}F^{\prime\prime}(z),
\end{split}
\end{equation}
with Hamiltonian function $H(z,p)=\frac{\Phi^2}{2}+F(z)\Phi-\frac{\varepsilon^2}{2}F^{\prime\prime}(z)$. Here the momentum variable $\Phi=\dot{z}-F(z)$ measures the deviation from the deterministic flow. The corresponding Freidlin-Wentzell functional \cite{Chen2018} can be expressed in terms of $\Phi$ as $I_{F}=\int_{t_0}^{t_1}\Phi^2(t)dt$. Therefore, as $\varepsilon\to 0$, most probable paths are well approximated by heteroclinic orbits of (\ref{Hamiltonian}) connecting deterministic solutions and minimizers of $I_\varepsilon$ converge uniformly to minimizers of $I_{F}$ \cite{Dykman2001,Chen2018}. \\
\par
In order to illustrate the effectiveness of this method, for a fixed initial point, we choose the final point as that gained by maximal likely trajectory, and choose the transition time as minimal time arriving at the final point. Note that, usually, these two points are not metastable states, we choose them only for the consideration of methodology. Then, we compare the most probable path (red line) for the stochastic system (\ref{Brownian}) connecting these two fixed points (but not necessarily that they are metastable states here) with the corresponding maximal likely trajectory (blue line) in Fig. \ref{Versus}(a) and (b). Some simulations of actual sample paths (green lines) nearby these two trajectories are also presented in the figures. In Fig. \ref{Versus}(a), maximal likely trajectory and most probable path coincide with each other. But, in Fig. \ref{Versus}(b), we see that they have obvious difference.
\begin{figure}
\subfigure[]{
\includegraphics[width=2.15in]{MPPMLT.eps}
}
\subfigure[]{
\includegraphics[width=2.28in]{MLTMPP.eps}
}
\caption{(Online color) The most probable path (red line) versus the maximal likely trajectory (blue line) for (\ref{Eqn-L}): (a) $\varepsilon_0=0.01$, initial point $X(0)=1800$, final point $X(35.42)=1736.8$; (b) $\varepsilon_0=0.1$, initial point $X(0)=65$, final point $X(97.94)=1734.7$.}
\label{Versus}
\end{figure}
\section{Result}
In the following section, we compute the maximal likely evolution trajectories and most probable transition paths, in order to analyze how the ice sheet is formed and how it melts away. The maximal likely trajectories starting from different initial points and most probable transition paths starting from $X_0=0$ and ending at $X_+=1738.6$, are deterministic estimators as time goes on. We will examine maximal likely trajectories and most probable transition paths when system parameters change followed by a brief discussion on general behavior of the system.
\subsection{Maximal Likely Trajectory}
The deterministic dynamical system is bistable in some range of ice sheet parameter space that contains $\lambda=0.001$, $r=-250$.
For simplicity, we firstly consider five maximum likely trajectories with initial points $X_0=1800$, $1600$, $1000$, $100$ and $50$ to both sides of the equilibria in the deterministic dynamical system. Due to the lengthy computation process time is capped to $T=100$.
\begin{figure}
\subfigure[]{
\includegraphics[width=2.3in]{B_MLT_0p01_title.eps}
}
\subfigure[]{
\includegraphics[width=2.23in]{Close_to_free_0p01.eps}
}
\caption{(Color online) Let $\varepsilon_0=0.01$, $r=-250(km)$ and $\lambda=0.001$. Maximal likely evolution trajectories of stochastic system (\ref{Brownian}) starting at various initial concentration $X_0$.
}
\label{fig:MLT}
\end{figure}
Fig. \ref{fig:MLT}(a) shows only one maximal likely equilibrium state for stochastic ice sheet system (\ref{Brownian}) with $\varepsilon_0=0.01$, and the value of this maximal likely stable equilibrium state is $1736.8$ which differs slightly from the deterministic stable state $X_+=1738.6$ due to the effect of noise. It is also noticeable that the other deterministic stable state $X_0=0$ is not a maximal likely equilibrium state. We observe that the maximal likely evolution trajectories starting close to the ice-free state (e.g. the pink line in Fig. \ref{fig:MLT}(a)) will go to or go close to zero firstly, but they will reach the ice-covered state ultimately. This could be understood as a reason why the ice sheet concerned here has been formed in its local environment initially.
As shown in Fig. \ref{fig:MLT}(b), the maximal likely evolution trajectories will reach zero for some time when the values of initial points are smaller than $61.5$.
\begin{figure}
\centering
\includegraphics[width=2.4in]{MLS.eps}
\caption{(Color online) For $\varepsilon_0=0.01$, $r=-250$ and $\lambda=0.001$, the change of maximal likely evolution trajectories of stochastic system (\ref{Brownian}) starting at $X_0=1800$ with respect to different noise intensities. }
\label{fig:diff noise}
\end{figure}
Furthermore we investigate the impacts of different levels noise intensity on the maximal likely trajectory for system (\ref{Brownian}) in figure \ref{fig:diff noise}. For convenience, we consider the initial state $X_0=1800$. The maximal likely stable equilibrium state reduces with the increase of the noise intensity as seen in Fig. \ref{fig:diff noise}. Recall that, in Fig. \ref{fig_potential}, the metastable states depend on model parameters $\lambda$ and $r$. With different model parameters, the maximal likely stable equilibrium state will also change; see Fig. \ref{fig:Rlambda}.
\begin{figure}
\subfigure[]{
\includegraphics[width=2.3in]{lambda0p0012.eps}
}
\subfigure[]{
\includegraphics[width=2.3in]{R-100.eps}
}
\caption{(Color online) Let $\varepsilon_0=0.01$. Maximal likely evolution trajectories of stochastic system (\ref{Brownian}) with different model parameters: (a) $r=-250(km)$ and $\lambda=0.0012$; (b) $r=-100(km)$ and $\lambda=0.001$.
}
\label{fig:Rlambda}
\end{figure}
We end this section with a short summary of key points. First, while the deterministic model has two metastable states, in the stochastic it's reduced to one. In addition this metastable state is dependent both on the parameters and the noise intensity.
\subsection{Most Probable Path}
While in the previous section we saw how the trajectory of the system evolves over time, we are also interested to see the trajectory it would follow to reach a specific end.
For stochastic ice sheet system (\ref{Brownian}), we now examine the most probable transition pathway starting at the ice-covered metastable state $X_+=1738.6$ and ending at the ice-free state $0$. As seen in Fig. \ref{MPP}(a), the most probable ice sheet height $L_{mp}$ decreases slowly at first, but after 40-50 kyrs from the start the melting rate accelerates. After that it continues to decrease on a downward trend and passes the `barrier' $X_-=63.9$ after about 80kyrs, and finally, it reaches the ice-free state. The evolution of such most probable transition pathway depends crucially on the noise intensity $\varepsilon_0$ and the system running time $t_1$ as seen in Fig. \ref{MPP}(a) and (b).
\begin{figure}
\subfigure[]{
\includegraphics[width=2.3in]{Diff_noise.eps}
}
\subfigure[]{
\includegraphics[width=2.3in]{Diff_Time.eps}
}
\caption{(Online color) For system (\ref{Eqn-Z}), set initial value $z(0) =2\sqrt{X_+}$ and final value $z(t_1) =0$. (a) For fixed $t_1=100$, the change of most probable path for original system (\ref{Eqn-L}) with respect to different noise strength $\varepsilon_0$; (b) For fixed $\varepsilon_0=0.01$, the change of most probable path for original system (\ref{Eqn-L}) with respect to different time interval parameters.}
\label{MPP}
\end{figure}
Fig. \ref{MPP}(a) shows the most probable path with different noise intensity whilst Fig. \ref{MPP}(b) with different running time. Both affect the most probable path. It should be noted that this shows the most probable path for this transition to happen, not the likeliness for this event. The transition from the ice-covered to the ice-free state would be a rare event. As we have seen in the previous section, Fig. \ref{fig:MLT}, even when starting from very low height the most probable outcome is to reach the metastable state $X_+$.
\section{Conclusion}
In this work, we have established an ice-sheet model with Brownian noise. It was constructed closely to its deterministic predecessor. We examine the maximal likely trajectories and most probable transition paths, i.e., we visualize the trajectories from different initial state (including ice-free state) to ice-covered state as well as transition pathways from ice-free state to ice covered state. The maximal likely trajectories are calculated via numerically solving the Fokker-Planck equation for the stochastic ice sheet model (\ref{Brownian}). The most probable transition pathways are computed by numerically solving a two-point boundary value problem.
For a stochastic ice-sheet system, we have observed that the maximal likely trajectories starting from near ice-free state would converge to only one maximal likely stable equilibrium state, which could be recognized as a ice-covered state. It shows that there would always be a very thick ice sheet no matter what the initial situation is, and the length of ice sheet would reach definite value (e.g. 1736.8\emph{km} in the case of Fig. \ref{fig:MLT}) after a long time. At the same time, we have also noticed some peculiar or counter-intuitive phenomena. For example, the initial ice cap might melt for some time and then gradually form a big one, if its initial length is small enough (e.g. smaller than 61\emph{km} in the case of Fig. \ref{fig:MLT}). This phenomenon does not occur in the case of deterministic model.
Although it seems that the total melting of the ice-sheet in this model is a very rare event, it is necessary to study the transition from ice-covered state to ice-free state due to this even is more concerned about the people. The method of most probable transition paths by minimizing the Onsager-Machlup action functional could thus be applied. For certain evolution time scale and system parameters, we have indeed observed that the most probable transition pathway exists under Brownian noise. Furthermore, we have characterized the evolution with varying noise parameters. Therefore, we can predict the melting route at a given future time.
The findings in this work may provide helpful insights for further practical research, to verify the change of an ice sheet and so on. And these two methods applied here could be also used in other practical models.
\section*{Acknowledgements}
The authors are grateful to Xiaoli Chen, Xiujun Cheng and Yuanfei Huang for helpful discussions and comments. This work was partly supported by the NSF grant 1620449, and NSFC grants 11531006 and 11771449.
\section*{References}
\bibliography{mybibfile}
\end{document}
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History and past events
History
The fellowship started in 1976 when a group of families meeting with Brook Chapel in Runcorn started holding services in their homes in Frodsham and then holding services in the Frodsham Community Centre. It is still going!
Other pages have more on the history of Christian life on the site of Main Street Community Church.
Some Past Events
2017
We inducted Paul Wintle as our leader and pastor.
2012
- In July we had a visit from a YWAM team they completed their trip by travelling down to London for outreach work as part of the Olympic games.
2011
- Christmas Journey was hosted again on behalf of Frodsham Churches Together. Some photographs from Christmas Journey 2011
- Mike Vickers visited us on Sunday morning 15 May and then at ’Open Door’ on the same afternoon. has highlights of Mike’s cross walk from Glasgow to Chester.
- Mothering Sunday in England is on the fourth Sunday of Lent, 3 April in 2011. For some years we have presented a daffodil flower to the ladies of the congregation. Some photos from the 2011 service included these (two selected from five):
2010
- Freedom in Christ course.
2009
- September 20 to 27, an eco-week where Christian faith met environmental responsibility and climate change.
- The afternoon of June 14 Family Fun Day in Castle Park. The day ended with a special “Songs of Praise”-style event
- At the end of May, Dave Bennett of Bridge-Builder, or Pocket Testament League, led three sessions to help, encourage and challenge us in helping people we know to consider Christianity.
- Just Curious? about the Christian faith?
- Men’s visit to Mitchell Group
- A series on the ten commandments for today
- Topics from a Church family meeting on 25 February 2009
- Our theme for the year is “Keep on keeping on”
2008
- Life ’08 from June 13 to June 22
- In April the church building was used for a two-day Christian RE Resource Exhibition for local teachers
- After many years, from April 7 2008, Vale Royal Citizens Advice Bureau closed the drop-in service for advice held at the church building on Thursday mornings
- In March Easter Journey was in the church building for a second year
- The programme of All-age services for the year was based on the fruit of the spirit
- In January, our theme for the year was announced as “Give me also springs of living water”
2007
- July 15. Public service of re-dedication following the change of name. Also the first public showing of the Prodigal Son paintings in the hall.
- July 1. Name change from Main Street Chapel to Main Street Community Church.
- Easter: the first production of Easter Journey.
- What’s my place in this church? seminar on March 24
- 2007 had a series of all-age services, a detailed teaching programme, a theme: “With God Nothing is Impossible” and a vision statement.
2006
- Pictures from the 2006 Candle-lit carol service
Christmas Journey 2006
… for children from 4 to 8 to enjoy the Christmas Story
The team putting this together was from different churches in the town. The event was supported by Frodsham Churches Together, Wellspring Christian Trust and the Frodsham Christmas Festival.
You can see some photographs from the 2006 event.
- As in many previous years, in December 2006 we held a senior’s Christmas tea.
- In Autumn 2006, Tim Coad joined us as Lead Elder. Read more about Tim and Eunice and Tom.
2005
- During the 2005 Frodsham Christmas Festival, Christmas Journey returned to Frodsham. You can see some photographs from the 2004 and 2005 events.
- November 11 to 13 2005. The Mbale Youth Brass Band visited and performed music, drama and dance. They gave performances at Chester Cathedral and Frodsham Methodist Church on the Saturday and took part in the service at Main Street Community Church on the Sunday. There are some photographs from the weekend.
- Easter 2005. Some members of the 2004 mission team visited Main Street Community Church. There are a few photographs on the Mission 2004 page
2004
Christmas Journey 2004
The team putting this together was from different churches in the town. The event was supported by Frodsham Churches Together, Wellspring Christian Trust and the Frodsham Christmas Festival team. You can see some photographs from the 2004 and 2005 events.
- For two years to November 2004, Philip and Helen Clarke shared their lives with us and ministered to us. Philip modelled the role of lead elder to help the leadership adapt to this different style of church organisation.
- 25 June to 5 July, 2004. We had the privilege of sharing with and learning from a team from Timothy Ministries. We renewed old friendships and made new ones as we shared together the good news of Jesus. There were events for everyone, including baseball, basketball and free food! There are some photographs here.
- In Lent 2004 we worked through Rick Warren’s book The Purpose Driven Life
2002 and earlier
- In January 2002, The Labyrinth visited Main Street Community Church. Creative and reflective, restful and searching, a walk through the labyrinth will refresh your senses and your spirit. Visit the website to experience it on the Internet. The labyrinth website.
- Local Alpha courses
- Thanks to Richard, Marian, Judy, Ron, Tom, Philip and Jean who came from the USA to share in the Main Street Mission from 9 to 19 June, 2000.
- Taking part in the Churches Together in Frodsham Creative Faith exhibition (June 2000)
- Taking part in the Frodsham Millennium Mystery Play (April 2000)
- Life and Death Decisions. Community Bible Studies from the book of Acts. On Thursdays at the Wellspring Centre, Church Street, Frodsham, each week for ten weeks from January to March 2000. 1.30pm to 3pm repeated at 7.30pm to 9pm. Led by Dr Larry Alan Correll.
- Walk Thru the Old Testament. A Friday evening and Saturday overview of the whole of the Old Testament at Main Street Community Church on 4 and 5 February 2000. More information from the Walk Thru the Bible Ministries UK Website.
- On 1st July 1999, Larry and Susan Correll of Timothy Ministries joined us for two years to help with leadership and direction of the fellowship.
- On a weekend in March 1998 the Reverend Richard Russell, formerly lecturer in philosophy at Manchester College, Oxford, then Vicar of Widcombe, Bath, spoke to us about “New Millennium … New Vision?”. The sessions were entitled “Where there is no vision the people perish” and “Living Between Memory and Hope”.
- In October 1997 the Frodsham Churches arranged for the Bible Experience Exhibition to visit Frodsham.
- The World Vision 24/30-hour famine was been tackled by the Youth Fellowship several times.
- Imagine, by Dr Andrew Basden, is an update of Matthew 25 for those in the IT industry.
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TITLE: Cosine of the sum of two solutions of trigonometric equation $a\cos \theta + b\sin \theta = c$
QUESTION [5 upvotes]: Question:
If $\alpha$ and $\beta$ are the solutions of $a\cos \theta + b\sin \theta = c$, then show that:
$$\cos (\alpha + \beta) = \frac{a^2 - b^2}{a^2 + b^2}$$
No idea how to even approach the problem. I tried taking two equations, by substituting $\alpha$ and $\beta$ in place of $\theta$ in the equation and manipulating them, but that didn't get me anywhere. Please help!
REPLY [4 votes]: Here's a picture showing angles $\theta$ (at $P$) and $\phi$ (at $Q$) such that
$$a \cos\theta + b \sin\theta = c = a \cos\phi + b \sin \phi$$
The measure of $\angle PAQ$ is the sum of these angles.
Note that $P$ and $Q$ lie on the circle with diameter $\overline{AB}$, and that the diameter bisects $\angle PAQ$. From here, we have many approaches to the final relation; here's one: Clearly,
$$\cos\frac{\theta+\phi}{2} = \frac{a}{d} \qquad\qquad \sin\frac{\theta+\phi}{2} = \frac{b}{d}$$
so that, by the Double-Angle Formulas,
$$\cos(\theta+\phi) = 2\cos^2\frac{\theta+\phi}{2} - 1 = \frac{2a^2-d^2}{d^2} = \frac{2a^2-(a^2+b^2)}{a^2+b^2} = \frac{a^2-b^2}{a^2+b^2}$$
$$\sin(\theta+\phi) = 2 \sin\frac{\theta+\phi}{2}\cos\frac{\theta+\phi}{2} = \frac{2ab}{d^2} = \frac{2ab}{a^2+b^2} $$
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Amador CountyHome Sitemap Contact
Administration
As directed by the Amador County Board of Supervisors the Office of the County Administrative Officer is responsible for the administration of all County departments and programs including development and maintenance of the County budget, oversight of County departments, staff support to the Board of Supervisors, and management of specific programs as directed by the Board of Supervisors.
| 42,088
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Understanding and managing the risk that third-party service providers or suppliers pose to your operations should be an essential component of any comprehensive cybersecurity risk program. The risk that third-party vendors pose organizations is often not well understood. This leads to organizations exposing themselves to unnecessary risk that is otherwise avoidable.
Third-party entities can pose risks in a variety of ways. From the poor implementation of required security protocols to a lack of in-depth personnel vetting, there are many ways that security vulnerabilities with third-party vendors can translate to a security incident for your organization..
In this article, we’ll outline some third party risk management best practices that you can use to ensure your risk management policy is headed in the right direction. From the beginning, it’s important to understand that third-party risk management should be an extension of your existing risk management efforts, not simply an afterthought. As will become clear, it isn’t enough from a security or a compliance perspective to simply trust that third-party entities that provide IT services are developing and implementing policies and procedures that are consistent with your requirements. Rather, it is ultimately your responsibility to ensure that third-party entities are maintaining their contractual security obligations.
It is important from the beginning to understand what may constitute a third-party. Within a cyber security context, a third-party vendor is an entity that you share network access or information with. This may include a cloud service provider, payment processing provider, or supply chain partner. Each third-party vendor must be accounted for when you are assessing your third-party risk. If there are gaps in your assessment there will naturally be gaps in your security.
Perform Your Due Diligence
One of the best ways you can manage risk stemming from third-party entities is to ensure that you partner with businesses that take cybersecurity seriously. In order to ensure that the third-party entities you share your network or information with have an adequate cybersecurity presence for your security needs, you’ll have to perform a certain level of due diligence. Businesses that partner with, or outsource to, third-parties must ensure that the third-party’s cybersecurity protocols and security controls are sufficient for their security needs. They must also ensure that the third-party provider understands their security needs and security concerns, and can offer assurances that those needs will be met.
Due diligence can take a variety of forms, many of which come down to the specific security and compliance requirements you are bound by. Ultimately, you’ll have to make the final decision whether the third-party provider poses an acceptable level of risk. An important element in that calculation is to have a full understanding of the company you are doing business with. Taking the time to gain a full-field view of a third-party provider is an essential component to reducing risk.
Assess your Third Party Risk ManagementAssess your Third Party Risk Management
Security Assessment and Validation
With any third-party entity you allow network access to or share information with, it is advisable to perform a security assessment. If this can’t be done in-house, consider utilizing a third-party security provider to assess the company. The reason for performing a security assessment is simple; you’ll want to assess for yourself what the third-party provider or vendor’s security risks are. It is one thing for a third-party vendor to make claim regarding their security efforts. It is quite another to see for yourself what security vulnerabilities exist for yourself.
Along with performing a security assessment, you should also validate that the third-party vendor you are working with is properly implementing security protocols over time. In today’s threat landscape, it’s not enough to ensure that security protocols are in place during a single moment in time. Regular validation of third-party vendors is an effective way to help reduce the risk for your organization. Determining how long between validations should be determined with your cybersecurity team or the third-party security provider you work with.
Integrate Risk Management Into the Vendor Selection Process
A recurring theme in our risk management best practices is identifying areas of risk and minimizing them early on, or avoiding them entirely. One way that you can minimize risk is by selecting the right third-party vendor from the beginning. Integrating risk management into the vendor selection process is critical to ensure that your entire vendor network poses a manageable level of risk. During the vendor selection process members of your IT and security teams should be involved. Alongside this, cybersecurity risks posed by third-party vendors should be a central component of a comprehensive risk assessment prior to working with a vendor.
By integrating risk management into the selection process for vendors, you ensure that each relationship you enter into is done with a clear understanding of the risks it poses to your operations and security. With the state of risk that companies face today, every potential area of vulnerability must be identified and addressed. It’s also important to know how to build an effective vulnerability management program. This includes every third-party vendor or provider along with all supply-chain suppliers. Given the scope of this for some organizations, it makes sense to integrate cybersecurity risk assessments into the selection process from the outset. Your company will be sure that they are working with providers that understand your security concerns and will implement adequate safeguards to ensure you have an acceptable level of risk.
Clearly Define Areas of Responsibility
One best practice that is sometimes overlooked to the peril of organizations is clearly defining areas of responsibility when it comes to security. This is especially important from a compliance perspective, as some regulatory authorities have specific regulatory requirements for who is responsible for safeguarding sensitive data in a third-party vendor situation. Every relationship with a third-party provider should have clearly spelled out areas of responsibility. There will most often be areas that you are solely responsible for security, areas where the third-party provider will have sole responsibility, and areas where you have overlapping responsibility. Understanding exactly what your areas of responsibility are, and how shared responsibility will be managed, is critical. Equally important, however, is for the third-party entity to fully understand the security requirements they must implement to ensure they meet their security obligations. If they don’t have a clear understanding of what is required of them from the outset, it is much more likely that security lapses may occur.
The most effective way to ensure that each party understands their security obligations is to utilize ironclad contracts. Make sure your contracts spell out in clear language the security requirements that each party will be required to meet. This will position you favorably if there is ever a dispute over areas of sole or overlapping responsibility. More importantly, however, you may have a compliance requirement to create a clear contract that demarcates security responsibilities.
As with all other aspects of managing third-party cybersecurity risk, involving your IT and security departments into the management processes of understanding and outlining areas of cybersecurity responsibility is strongly encouraged. Not only should your personnel be involved, but the third-party vendor’s IT and security staff should also be engaged in the process. Both parties must fully understand where their responsibilities start, end, and overlap. In addition to this, each party must understand what should occur if there is a data breach, and have a response plan in place that is regularly reviewed.
Stay Vigilant
Navigating 3rd party risk management can be tricky. In today’s cyber landscape, the scope of risk facing companies is large and often underestimated. Once you begin peeling back the layers on your third-party vendor relationships and exposing areas where third-party vendors may pose a significant risk to your information or networks, it can be a daunting process to assess, quantify, and minimize that risk. A mistake many companies make is in thinking that the landscape of risk they face is static rather than dynamic. Due to the fact that the threat landscape is constantly shifting, there are always new areas of risk opening up. What this means is that managing third-party risk from a cybersecurity perspective requires an approach that is constantly vigilant.
While there are many ways organizations do this, you’ll want to set up a system for continuous monitoring and reassessment of approved third-party vendors that you work with. This should be a high priority in any organization that carries risk from third-party vendors. Due to the fact that cybersecurity risk is shared risk, your vendors should also be doing this same thing, hopefully to the same degree. The net effect of this is a stronger communal network where areas of risk overlap. If your entire third-party network exercised the same high degree of cybersecurity awareness and monitoring, the risk level of all entities within your network will be lower.
The Importance of Security Expertise
One of the challenges that many companies face is gaining access to a sufficient level of expertise to accurately assess the risk that third-party providers pose to them. Assessing risk from third-party vendors is a complex process, and within a cybersecurity context requires an in-depth understanding of exactly what vulnerabilities may exist and what efforts need to be taken to minimize the risk those vulnerabilities pose. There are also compliance requirements to consider. Your compliance requirements will be a deciding factor in both the third-party vendors you choose to work with and the security protocols that they are required to put in place. Some regulatory structures have strict requirements regarding the nature of a relationship between a business and third-party vendors, particularly in regards to understanding and defining areas of risk.
The problem for many companies is that they don’t have access to security professionals in-house that can help them identify and mitigate the risk posed by third-party vendors. In many cases, it’s not financially viable to operate third-party risk management internally. Although sometimes third-party risk management aligns completely with internal risk management efforts, more often it is an external process that is managed under a different organizational structure. This can present challenges for both security staff and management for getting an accurate understanding of what risks third-party vendors pose and what steps are being taken to minimize those risks. Also, remember that managing third-party risk is an ongoing process, not a one-off event. This means that there is generally a team tasked specifically with identifying and managing third-party risk.
The solution that some companies choose is to work with a third-party security provider that specializes in third-party risk management. It is particularly helpful to acquire these compliance advisory services when there are compliance considerations that must be accounted for. Even ensuring operations internal to your organization are adhering to regulatory compliance requirements can be a daunting process. Assessing whether compliance requirements have been met by a single third-party vendor can be challenging, and given the fact that many companies work with a variety of third-party vendors, providers, and supply chain partners, working with a security assessor that can focus on ensuring third-party risk is minimized and compliance requirements are met can free up resources for other areas.
Closing Thoughts
In today’s world, the level of risk that companies face is staggering. The toll for a harmful cybersecurity event is just as shocking. Due to this, companies must be proactive in their approach to finding a third-party management for a risk assessment and threat and vulnerability management program. Implementing the best practices we have outlined are important components of a comprehensive risk-management approach to cybersecurity, but there are also many others. Working with a third-party security provider that understands this is one step toward ensuring the security of your network assets and information. If you would like to find out more information about how third-party risk management services can help identify and minimize your risk and provide you the cybersecurity solutions you need, please contact RSI Security today.
Sources
-
- “Understanding the Cybersecurity Threat: The Board’s Role.” In Corporate Governance Advisor, 26:9–17. Aspen Publishers Inc., 2018.
- Preimesberger, Chris. “10 Ways Enterprises Can Limit Third-Party Cyber-Risk.” EWeek, February 8, 2017, 2–2.
-
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\begin{document}
\title[Homologous Non-isotopic Symplectic Tori
in Homotopy $E(1)$'s]{Homologous Non-isotopic
Symplectic Tori\\
in Homotopy Rational Elliptic Surfaces}
\author{Tolga Etg\"u}
\address{Department of Mathematics and Statistics, McMaster University,
Hamilton, Ontario L8S 4K1, Canada}
\email{etgut@math.mcmaster.ca}
\author{B. Doug Park}
\address{Department of Pure Mathematics, University of Waterloo, Waterloo,
Ontario, N2L 3G1, Canada}
\email{bdpark@math.uwaterloo.ca}
\thanks{B.D. Park was partially supported by an NSERC research grant.}
\subjclass[2000]{Primary 57R17, 57R57; Secondary 53D35, 57R95}
\date{May 15, 2003. Revised on \today}
\begin{abstract}
Let $E(1)_K$ denote the homotopy rational elliptic surface corresponding to a
knot $K$ in $S^3$ constructed by
R. Fintushel and R.J. Stern in \cite{fs:knots}. We
construct an infinite family of homologous non-isotopic symplectic
tori representing a primitive $2$-dimensional
homology class in $E(1)_K$ when
$K$\/ is any nontrivial fibred knot in $S^3$.
We also show how these tori can be non-isotopically embedded as
homologous symplectic submanifolds in other symplectic $4$-manifolds.
\end{abstract}
\maketitle
\section{Introduction}
This paper is a continuation of studies initiated in \cite{ep1}
and \cite{ep:k3} regarding infinite families of non-isotopic and symplectic
tori representing the same homology class in a symplectic 4-manifold.
Let $E(1)_K$ denote the closed 4-manifold that is homotopy equivalent
(hence homeomorphic) to the rational elliptic surface $E(1)\cong\cpk \#9\cpkk$
and is obtained by performing knot surgery (as defined in \cite{fs:knots})
on the rational elliptic surface
using a knot $K$\/ in $S^3$. Our main result is the following:
\begin{theorem}\label{theorem:main}
Let $K\subset S^3$ be a nontrivial fibred knot. Then there
exists an infinite family of pairwise non-isotopic symplectic tori
representing the primitive homology class\/ $[F]=[T_m]$ in $E(1)_K$,
where\/ $[F]$ is the homology
class of the fiber in a rational elliptic surface $E(1)\cong \cpk \#
9\cpkk$.
\end{theorem}
Examples of homologous, non-isotopic, symplectic tori were first
constructed in \cite{fs:non-isotopic} and then in \cite{ep1},
\cite{ep:k3}, \cite{vidussi:non-isotopic} and
\cite{vidussi:E(1)_K} (also see \cite{fs:lagrangian} and
\cite{vidussi:lagrangian}). Recall that infinite families of
non-isotopic symplectic tori representing $n[F]\in H_2(E(1)_K)$,
$n\geq 2$, were constructed in \cite{ep1}. The family of tori we
construct in this paper is in some sense the `simplest' example
known so far, when measured in terms of the `geography size' of
the ambient (simply-connected) symplectic 4-manifold, the
divisibility of the homology class represented, and the complexity
of the knotting of the tori. In \cite{vidussi:E(1)_K}, using a
different technique, Vidussi already constructed symplectic tori
representing the same primitive class in $E(1)_K$ for some
particular fibred $K$, namely the trefoil and other fibred knots
that have the trefoil as one of their connected summands.
It should be noted that the
non-existence of such an infinite family of tori in $\cpk$ and
$\cpk \# \cpkk$ is proved by Sikorav in \cite{sikorav} and by Siebert and Tian
in \cite{st}, respectively. It is also conjectured that there is at most
one
symplectic torus (up to isotopy) representing each homology class in
$\cpk \# n \cpkk$ for
$n < 9$.
The proof of Theorem~\ref{theorem:main} is spread out over the next three
sections. We will review the relevant definitions in
Section~\ref{sec:link surgery}. In Section~\ref{sec:generalization},
we will present a
direct generalization in the form of Proposition~\ref{prop:generalization}.
In this introduction and elsewhere in the paper by isotopy we mean smooth
isotopy
and all homology groups have $\zz$ coefficients.
\section{Link Surgery 4-Manifolds}
\label{sec:link surgery}
In this section, first we review the generalization of the
link surgery construction of
Fintushel and Stern \cite{fs:knots} by Vidussi
\cite{vidussi:smooth}, and then give specific link surgeries that
will be used in the following sections.
For an $n$-component link $L\subset S^3$,
choose an ordered homology basis of simple closed curves $\{(\alpha_i, \beta_i)
\}_{i=1}^{n}$ such that
the pair $(\alpha_i, \beta_i )$ lie in the $i$-th boundary
component of the link exterior
and the intersection of
$\alpha_i$ and $\beta_i$ is 1.
Let $X_i$
($i=1,\dots, n$) be a 4-manifold containing a 2-dimensional torus
submanifold $F_i$ of self-intersection $0$. Choose a Cartesian product
decomposition $F_i = C_1^{i} \times C_2^{i}$, where each $C^i_j \cong S^1$
($j=1,2$) is an embedded
circle in $X_i$.
\begin{definition}\label{def:data}
The ordered collection
$$\mathfrak{D} \, =\; \big( \{(\alpha_i, \beta_i)
\}_{i=1}^{n}\: , \: \{(X_i , \hspace{1pt}
F_i= C_1^{i} \times C_2^{i} )\}_{i=1}^{n}\big)$$
is called a \emph{link
surgery gluing data}\/ for an $n$-component link $L$.
We define the \emph{link surgery manifold corresponding to} $\mathfrak{D}$
to be the closed $4$-manifold
\[
L(\mathfrak{D}) \: :=\; [\coprod_{i=1}^{n} X_i\setminus\nu
F_i]\hspace{-20pt}\bigcup_{F_i\times\partial D^2=(S^1\times
\alpha_i)\times\beta_i}\hspace{-20pt} [S^1\times(S^3\setminus \nu
L)]\, ,
\]
where $\nu$\/ denotes the tubular neighbourhoods. Here, the gluing
diffeomorphisms between the boundary 3-tori identify the torus
$F_i = C_1^{i} \times C_2^{i}$\/ of $X_i$\/
with\/ $S^1\times \alpha_i$\/ factorwise, and act as the complex
conjugation on the last remaining\/ $S^1$ factor.
\end{definition}
\begin{remark}
Strictly speaking, the diffeomorphism type of the link surgery manifold
$L(\mathfrak{D})$\/ may possibly
depend on the chosen trivialization of $\,\nu F_i \cong F_i \times D^2$ (the
framing of $F_i$). However, we will suppress this dependence in our notation.
It is well known (see e.g. \cite{gs}) that the diffeomorphism type of
$L(\mathfrak{D})$
is independent of the framing of $F_i$ when $(X_i,F_i) = (E(1), F)$.
\end{remark}
We fix a Cartesian product decomposition\/
$F= C_1 \times C_2$ in $E(1)$.
Let $K$\/ be a knot in $S^3$, and
let $M_K$ denote the 3-manifold that is the result
of the 0-framed surgery on $K$.
Fix a meridian circle $m=\mu(K)$\/ in $M_K$.
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=.5]{nhopf-ab.eps}
\end{center}
\caption{Hopf link $L= A\cup B$}
\label{fig:hopf}
\end{figure}
\begin{definition}
Let\/ $L \subset S^3$ be the Hopf link in Figure~$\ref{fig:hopf}$.
For the link surgery gluing data\/
\begin{eqnarray}\label{eq:data}
\mathfrak{D} \!\!\! &:=& \!\!\!
\big(\{ (\mu(A),\lambda(A)), (\lambda(B),-\mu(B))
\}, \\ && \{ (X_1, F_1=C_1^1\times C_2^1),
(S^1\times M_K, T_m = S^1\times m )\}\big),
\nonumber
\end{eqnarray}
we shall denote\/ $L(\mathfrak{D})$ by $(X_1)_K$.
Here, $\mu(\,\cdot\,)$ and $\lambda(\,\cdot\,)$ denote
the meridian and the longitude
of a knot, respectively.
In particular, when $(X_1, F_1=C_1^1\times C_2^1) = (E(1), F=C_1\times C_2)$,
we denote $L(\mathfrak{D})$\/ by $E(1)_K$. This notation is consistent
with that of Fintushel and Stern in \cite{fs:knots} as
there is a diffeomorphism
between our $L(\mathfrak{D})$ and their fiber sum
$E(1)_K = E(1) \#_{F=T_m}\! (S^1 \times M_K)$.
\end{definition}
Note that there is a canonical framing of $T_m$ in $(S^1\times M_K)$\/ given by
the minimal genus
Seifert surface of the knot $K$. We shall always use this framing to
trivialize $\nu T_m$.
\begin{lemma}\label{lemma:E(1)_K is symplectic}
If\/ $K\subset S^3$ is a fibred knot, then $E(1)_K$ is a symplectic
$4$-manifold.
\end{lemma}
\begin{proof}
This is because there exists a fiber bundle
$\,(S^1\times M_K) \rightarrow T^2$ when $K$\/ is
fibred, so
$(S^1\times M_K)$ admits a symplectic form with respect to which
$T_m$ is a symplectic submanifold (cf.$\;$\cite{thurston}).
Hence we may express $E(1)_K$ as a \emph{symplectic}\/ fiber sum
$E(1)\#_{F=T_m}\! (S^1\times M_K)$ along symplectic submanifolds $F$ and $T_m$
(cf.$\;$\cite{g:sum}).
\end{proof}
\begin{lemma}
The homology class\/ $[F]=[T_m]\in H_2(E(1)_K)$
is primitive.
\end{lemma}
\begin{proof}
Since\/ $[\mu(A)]=[\lambda(B)] \in H_1(S^3\setminus \nu L)$,
we must have\/ $[S^1\times \mu(A)]=[S^1\times \lambda(B)]$\/ in\/
$H_2(S^1\times (S^3\setminus \nu L) )$, and so\/
$[F]=[T_m]$\/ in\/ $H_2(E(1)_K)$.
Let $\Sigma$ denote a Seifert surface of $K$.
Let $\Sigma_K$ denote a closed surface in $E(1)_K$
that is the internal tubular sum of
a punctured section in $[E(1)\setminus \nu F]$ and a punctured surface
$\{ {\rm point} \} \times\Sigma$, glued together along $K$. Then we have
$[\Sigma_K] \cdot [T_m]= \pm 1$.
\end{proof}
\section{Family of Homologous Symplectic Tori in $E(1)_K$}
\label{sec:symplectic family}
Let $T_C := S^1 \times C \subset [S^1 \times (S^3\setminus \nu L)]
\subset E(1)_K$,
where the closed curve $C\subset (S^3\setminus \nu L )$\/
is given by Figure~\ref{fig:doublehopf}.
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=.85]{ndoublehopf.eps}
\end{center}
\caption{3-component link $L_q=A\cup B \cup C\,$ in $S^3$}
\label{fig:doublehopf}
\end{figure}
\begin{lemma}
If\/ $K$ is a fibred knot, then $T_C =S^1\times C$ is a
symplectic submanifold of\/ $E(1)_K$ and we have\/
$[T_C]=[F]\,$ in\/ $H_2(E(1)_K)$.
\end{lemma}
\begin{proof}
It is easy to see that the link exterior
$Y:=(S^3\setminus \nu L)$\/ is diffeomorphic to
$S^1 \times \mathbb{A}$, where $\mathbb{A}\cong S^1 \times [0,1]\,$
is an annulus.
Hence we have
\[
[S^1 \times (S^3\setminus \nu L)] \:\cong\:
[S^1 \times (S^1 \times \mathbb{A})]
\:\cong\: T^3\times [0,1] .
\]
We may assume that the symplectic form on $E(1)_K$ restricts to
\[
\omega \,=\, dx\wedge dy \,+\, r\,dr\wedge d\theta
\]
on $[S^1 \times (S^1\times \mathbb{A})]$,
where $x$\/ and $y$\/ are the angular coordinates on the first and the second
$S^1$ factors respectively, and
$(r,\theta)$ are the polar coordinates on the annulus $\mathbb{A}$.
We can embed the curve $C$\/ inside $(S^1\times \mathbb{A})$ such that
$C$ is transverse to every annulus of the form, $\{{\rm point}\} \times
\mathbb{A}$, and the restriction $dy|_{C}$ never vanishes.
It follows that $\omega |_{T_C} = (dx\wedge dy)|_{T_C} \neq 0$, and
consequently
$T_C$ is a symplectic submanifold of $E(1)_K$.
To determine the homology class of $T_C$, note that\/ $[C]=[\mu
(A)] + q [\mu (B)]$\/ in\/ $H_1(Y)$.
When we glue\/ $[(S^1\times M_K)\setminus \nu T_m]$\/ to\/ $S^1\times
Y$, the homology class $[\mu(B)]$ gets identified with
$[\{{\rm point}\}\times\lambda(K)]
\in H_1((S^1\times M_K)\setminus \nu T_m)$, which is
trivial. Hence by K\"unneth's theorem, $[T_C]=[S^1\times
\mu(A)]\,$ in $H_2([S^1\times Y]\cup [(S^1\times M_K)\setminus \nu
T_m])$. It follows that\/ $[T_C]=[F]\,$ in $H_2(E(1)_K)$.
\end{proof}
\section{Non-Isotopy: Seiberg-Witten Invariants}
\label{sec:sw}
Our strategy is to show that the isotopy types of the tori $\{ T_C
\}_{q\geq 1}$ can be distinguished by comparing the Seiberg-Witten
invariants of the corresponding family of fiber sum 4-manifolds $\{E(1)_K
\#_{T_C=F} E(n)\}_{q\geq 1, n\geq 1}$. Note that there is a
canonical framing of a regular fiber $F$\/ in $E(n)$, coming from
the elliptic fibration $E(n)\rightarrow \mathbb{CP}^1$.
\begin{lemma}
The fiber sum\/ $E(1)_K \#_{T_C=F} E(n)$\/ is diffeomorphic to the
link surgery manifold $L_q(\mathfrak{D}')$, where
\begin{eqnarray}
&& \quad\quad \mathfrak{D}' : = \: \big(
\{(\mu(A),\lambda(A)),(\lambda(B),-\mu(B)),
(\lambda(C),-\mu(C))\}, \\[3pt]
&& \{(E(1),F=C_1\times C_2),
(S^1\times M_K,T_m=S^1 \times\mu(K)),(E(n),F=C_1\times C_2)\}\big).
\nonumber
\end{eqnarray}
\end{lemma}
\begin{proof}
We already observed in the proof of
Lemma~\ref{lemma:E(1)_K is symplectic} that the fiber sum construction
corresponds to this type of link surgery. (See also \cite{ep:k3}.)
\end{proof}
Recall that the Seiberg-Witten invariant\/
$\overline{SW}_{\!\!X}$\/ of a 4-manifold $X$\/
can be thought of as an element of the group ring of $H_2(X)$,
i.e.\/ $\overline{SW}_{\!\!X} \in \zz [ H_2( X ) ]$. If we
write\/ $\overline{SW}_{\!\!X} = \sum_g a_g g \hspace{1pt}$, then
we say that\/ $g\in H_2( X )$\/ is a Seiberg-Witten
\emph{basic class}\/ of $X$\/ if\/ $a_g\neq 0$. Since the
Seiberg-Witten invariant of a 4-manifold is a diffeomorphism
invariant, so are the divisibilities of Seiberg-Witten basic
classes. The Seiberg-Witten invariant of the link surgery manifold
$L_q(\mathfrak{D}')$ is known to be related to the Alexander
polynomial\/ $\Delta_{L_q}$ of the link $L_q$.
\begin{lemma}
$\Delta_{L_q}(x,s,t)= 1-x(st)^q$, where the variables\/ $x$, $s$
and\/ $t$\/ correspond to the components\/ $A$, $B$\/ and\/ $C$\/
respectively.
\end{lemma}
\begin{proof}
This follows readily from the formula in Theorem 1 of \cite{morton}
which gives the multivariable Alexander polynomial
of a braid closure and its axis
in terms of the representation of the braid.
We view $A$\/ as the axis of the closure of a 2-strand braid, $B\cup C$.
See \cite{ep1} for details on a similar computation.
\end{proof}
\begin{theorem}\label{theorem:seiberg-witten}
Let\/ $\iota:[S^1\times(S^3\setminus\nu L_{q})]\rightarrow
L_{q}(\mathfrak{D}')$\/ be the inclusion map. Let\/
$\xi:=\iota_{\ast}[S^1\times\mu(A)],$
$\tau:=\iota_{\ast}[S^1\times\mu(C)] \in
H_2(L_{q}(\mathfrak{D}') ).$ Then\/ $\xi$ and\/ $\tau$ are both primitive
and linearly independent.
The Seiberg-Witten invariant of\/ $L_{q}(\mathfrak{D}')$ is
given by
\begin{equation}\label{eq:sw}
\overline{SW}_{\!L_q(\mathfrak{D}')}\, = \: (\xi^{-1}-\xi)^{n-1}
\cdot \Delta^{\rm sym}_K(\xi^2 \tau^{2q})\hspace{1pt} ,
\end{equation}
where\/ $\Delta^{\rm sym}_K$ is the\/ \emph{symmetrized}
Alexander polynomial of the knot\/ $K$.
\end{theorem}
\begin{proof}
Let\/ $N := (S^3\setminus \nu L_q)$, and let $Z:=
[(S^1\times M_K) \setminus \nu T_m]$. Recall from \cite{doug:pft3}
that we have $\overline{SW}_{\!E(n)\setminus \nu
F}=([F]^{-1}-[F])^{n-1}$, and also
$$
\overline{SW}^{\,\pm}_{\!Z} = \:
\overline{SW}^{\,\pm}_{\!(S^1\times M_K) \setminus \nu T_m} =\:
\frac{\Delta_K^{\rm sym}([T_m]^2)}{[T_m]^{-1}-[T_m]}
\; .$$
From the gluing formulas in \cite{doug:pft3} and \cite{Taubes:T^3},
we may conclude that
$$
\overline{SW}_{\!L_q(\mathfrak{D}')} \,=\:
\overline{SW}_{\!E(1)\setminus \nu F} \cdot
\overline{SW}_{\!E(n)\setminus \nu F} \cdot
\overline{SW}^{\,\pm}_{\!(S^1\times M_K) \setminus \nu T_m} \cdot
\Delta_{L_q}^{\rm sym}(\xi^2,\sigma^2,\tau^2),
$$
where $\sigma:=\iota_{\ast}[S^1\times\mu(B)]$. Note that $\sigma
=1\in \zz[H_2(L_q(\mathfrak{D}'))]$,
since we have\/ $[\mu(B)]=[\lambda(K)]=0\,$ in\/ $H_1(Z)$.
Also note that $\mu(K)$ and $\lambda(B)$ are identified
by the gluing data $\mathfrak{D}'$, and\/
$[\lambda(B)]=[\mu(A)]+q[\mu(C)]\in H_1(N)$. It follows
that\/ $[T_m]=\iota_{\ast}[S^1\times \lambda(B)]=
\xi \tau^q \in \zz[H_2(L_q(\mathfrak{D}'))]$. Thus
we have
$$\Delta_{L_q}^{\rm sym}(\xi^2,\sigma^2,\tau^2) \:=\:
\xi^{-1}\tau^{-q} - \xi \tau^q \:=\: [T_m]^{-1}-[T_m]\, .$$
Hence
\begin{equation}\label{eq:sw almost}
\overline{SW}_{\!L_q(\mathfrak{D}')}
\,=\:([F]^{-1}-[F])^{n-1}\cdot \Delta_K^{\rm sym}(\xi^2\tau^{2q})\hspace{1pt} .
\end{equation}
Note that the fiber $F$\/ in\/ $E(n)$\/ gets identified with\/
$S^1\times \lambda(C)$ by the gluing data $\mathfrak{D}'$,
and we also have\/ $[\lambda(C)]=[\mu(A)]
+q[\mu(B)]=[\mu(A)]\,$ in\/ $H_1([S^1\times N]\cup Z )$.
Therefore we can identify\/ $[F]=\xi\,$ in (\ref{eq:sw almost}),
and we obtain Equation
(\ref{eq:sw}).
Next we show that $\xi$ and $\tau$ are primitive and linearly
independent elements of $H_2(L_{q}(\mathfrak{D}'))$. We can
proceed in two different ways. A Mayer-Vietoris argument,
combined with Freedman's classification theorem
(cf.$\;$\cite{FQ}), shows that $L_q(\mathfrak{D}')$\/ is
homeomorphic to $E(n+1)$. It is not too hard to find two closed
surfaces $R$\/ and $S$\/ in $L_q(\mathfrak{D}')$\/ satisfying
\[
\xi \cdot [S] \;=\; \tau \cdot [R] \;=\; 1 \, ,
\]
and
\[
\xi \cdot [R] \;=\; \tau \cdot [S] \;=\; [R] \cdot [S] \;=\; 0 \, .
\]
For example, we can let $S$\/ be the
internal tubular sum of punctured sections from\/
$[E(1)\setminus \nu F]$\/ and\/ $[E(n)\setminus \nu F]$\/ summands,
together with a suitable punctured surface from the $Z$\/ summand.
Let $R$\/ be
the internal tubular sum of the self-intersection $(-1)$ disks bounding
the circle $C_2$ in
$[E(1)\setminus \nu F]$\/ and\/ $[E(n)\setminus \nu F]$\/ summands,
together with a suitable punctured surface from the $Z$\/ summand.
In $L_q(\mathfrak{D}')$, $S$\/ plays the role of a section in
$E(n+1)$, while $\xi$\/ plays the role of the homology class of
the fiber. Note that we have\/ $[\mu(C)]=[\lambda (A)]-[\mu(B)]
=[\lambda(A)]\in H_1([S^1\times N]\cup Z)$, and
the gluing data $\mathfrak{D}'$ identifies
$\lambda(A)$
with a meridian circle $\mu(F)$ of the fiber $F$\/ in
$\partial [E(1)\setminus \nu F]$. Hence $\tau$ plays the role of the
homology class of the rim torus\/ $C_1 \times \mu (F)$\/ in $E(n+1)$.
$R$\/ plays the role of a self-intersection $(-2)$ sphere transversally
intersecting the above rim torus once. The pairs\/ $(\xi ,[S])$\/
and\/ $(\tau, [R])$\/ form homology bases for two $\bigl(
\begin{smallmatrix}
0 & 1 \\
1 & 0
\end{smallmatrix} \bigr)$ summands in the intersection form of
$L_q(\mathfrak{D}')$.
Alternatively, we can argue more algebraically as follows.
Consider the composition of homomorphisms
\begin{equation}\label{eq:homomorphism}
H_1(N) \longrightarrow H_2(S^1\times N)
\stackrel{\iota_{\ast}\;}{\longrightarrow}
H_2(L_q(\mathfrak{D}')),
\end{equation}
where the first
map is a part of the K\"unneth isomorphism
\begin{equation}\label{eq:Kunneth}
H_1(N) \oplus H_2(N)
\stackrel{\cong}{\longrightarrow} H_2(S^1\times N).
\end{equation}
Note that $H_2(N) \cong \zz\oplus \zz\,$,
as is easily seen from the long exact
sequence of the pair\/ $(N,\partial N)$ as follows.
\[
\begin{array}{ccccccccc}
H^0(N)&\longrightarrow & H^0(\partial N) &\longrightarrow &
H^1(N,\partial N) & \stackrel{0}{\longrightarrow} &
H^1(N) & \longrightarrow & H^1(\partial N) \\
_{||} && _{||} && _{||} & & _{||}
&& _{||}\\[3pt]
\zz & \longrightarrow & \zz^3 & \longrightarrow & H_2(N) &
\stackrel{0}{\longrightarrow}
& \zz^3 &
\longrightarrow & \zz^6
\end{array}
\]
Note that the first map sends the generator $\hspace{1pt} 1\in \zz\,$ to
the diagonal element\/ $(1,1,1)\in \zz^3$, while the last map is injective.
We have also used the Lefschetz duality theorem (for manifolds with boundary)
to identify $H_2(N)\cong
H^1(N,\partial N)$.
Next consider the long exact sequence of the pair
$(L_q(\mathfrak{D}'), S^1\times N)$:
\[
0 = H_3(L_q(\mathfrak{D}')) \longrightarrow
H_3(L_q(\mathfrak{D}'),S^1\times N) \longrightarrow H_2(S^1\times
N) \stackrel{\iota_{\ast}\;}{\longrightarrow}
H_2(L_q(\mathfrak{D}'))
\]
The kernel of the last map $\iota_{\ast}$ is isomorphic to
$H_3(L_q(\mathfrak{D}'),S^1\times N)$. By Lefschetz duality
theorem (for relative manifolds),
$H_3(L_q(\mathfrak{D}'),S^1\times N)$ is in turn
isomorphic to
\[
H^1(L_q(\mathfrak{D}')\setminus (S^1\times N)) \:\cong\;
H^1(E(1)\setminus \nu F) \oplus H^1(E(n)\setminus \nu F) \oplus
H^1(Z).
\]
Since we have $H^1(E(1)\setminus \nu F) = H^1(E(n)\setminus \nu F)=0\,$
and
\begin{equation}\label{eq:z plus z}
H^1(Z)
=H^1(S^1 \times( M_K \setminus \nu m) ) \cong \; \zz\oplus \zz \, ,
\end{equation}
the kernel of $\iota_{\ast}$ is isomorphic to\/ $\zz\oplus\zz\,$.
Finally we observe that only one $\zz$ summand of (\ref{eq:z plus z}) lies
in the kernel of the composition (\ref{eq:homomorphism}). The other
$\zz$ summand belongs to the kernel of
\[
H_2(N) \longrightarrow H_2(S^1\times N)
\stackrel{\iota_{\ast}\;}{\longrightarrow}
H_2(L_q(\mathfrak{D}')),
\]
where the first map is the second part of the K\"unneth
isomorphism (\ref{eq:Kunneth}). We have thus shown that the kernel
of the composition (\ref{eq:homomorphism}) is of rank one. It
follows immediately that $\xi$ and $\tau$ are linearly
independent, since we already have shown that $\sigma$ is trivial.
A more detailed analysis shows that\/ $\{\xi ,\tau\}$\/ can be
extended to a basis of $H_2(L_q(\mathfrak{D}'))$, which we shall
omit. (Also see the proof of Proposition 3.2 in
\cite{McMullen-Taubes} for a similar argument.)
\end{proof}
\begin{corollary}\label{cor:divisibilities}
If\/ $K$ is a nontrivial fibred knot,
then the tori\/ $\{ T_C \}_{q \geq 1}$ are
pairwise non-isotopic inside $E(1)_K$. In fact, there is no
self-diffeomorphism of\/ $E(1)_K$ that maps one element of this
family to another.
\end{corollary}
\begin{proof}
Let's choose $n$\/ to be $2g+1$, where $g$ is the genus of $K$.
Remember that the degree of the symmetrized Alexander polynomial of a fibred
knot is the same as its genus (see e.g. Proposition 8.16 in \cite{bz}).
Since we assume that $K$ is nontrivial, i.e. not the unknot,
$g >0$. A Seiberg-Witten basic class of $L_q(\mathfrak{D}')$ with the
highest divisibility is divisible by $2gq$.
This could be seen by observing that
the highest power of $\tau$ in (\ref{eq:sw}) of
Theorem~\ref{theorem:seiberg-witten} is $2gq$
(hence there cannot be a basic class
with divisibility higher than $2gq$) and our choice of $n=2g+1$\/ ensures
that there is a basic class (namely $\tau^{2gq}$) with this highest possible
divisibility. On the other hand, since the
Seiberg-Witten invariant is a diffeomorphism invariant, so are
the divisibilities of basic classes. Therefore, $L_q(\mathfrak{D}')$
is diffeomorphic to $L_{q'}(\mathfrak{D}')$ if and only if\/ $q=q'$
proving that the tori
in $\{ T_C \}_{q \geq 1}$ are different up to isotopy and in fact even up to
self-diffeomorphisms of\/ $E(1)_K$.
\end{proof}
This concludes the proof of Theorem~\ref{theorem:main}.
\section{Generalization to Other Symplectic 4-Manifolds}
\label{sec:generalization}
When $K$\/ is the unknot, $E(1)_K$ is diffeomorphic to $E(1)$. In
this unknot case, our family of tori $\{T_C\}_{q \geq 1}$ are
easily seen to be all isotopic to one another. The isotopy can
actually be visualized by erasing the $B$\/ component in
Figure~\ref{fig:doublehopf} (This corresponds to filling in $\nu
B$\/ with\/ $(M_K\setminus \nu m)$, which, in the unknot case, is
diffeomorphic to a solid torus\/ $S^1\times D^2$.), and
straightening out the $C$\/ component through the tubular
neighbourhood of $B$, which has now been filled in. Note that the
normal disks of $B$\/ are the Seifert surfaces of the unknot.
Suppose that $K$\/ is not fibred. Then, unlike the fibred case where the degree
of the Alexander polynomial is (the same as the genus hence)
strictly greater than
$0$ unless the knot is the unknot, the Alexander polynomial of $K$\/ might
be constant and the Seiberg-Witten invariant doesn't seem to be
delicate enough to distinguish the tori we constructed. On the
other hand, when $K$ is not fibred and has a non-constant Alexander
polynomial, the tori in our family\/ $\{ T_C \}_{q\geq 1}$ are still pairwise
non-isotopic in $E(1)_K$, but there is no natural symplectic
structure on $E(1)_K$ and we don't know whether $T_C$ is symplectic
with respect to a symplectic structure on $E(1)_K$.
In fact, it is known that $E(1)_K$ doesn't admit any symplectic
structure if the Alexander polynomial of $K$\/ is not monic
\cite{fs:knots}.
On a more positive note, we can easily extend Theorem~\ref{theorem:main}
to $E(n)_K$ $(n\geq 2)$
and more generally to $X_K$, where $X$\/ is a symplectic
4-manifold satisfying
certain topological conditions as in \cite{ep1}.
\begin{proposition}\label{prop:generalization}
Assume that\/ $F$ is a symplectic\/
$2$-torus in a symplectic\/ $4$-manifold\/ $X$.
Suppose that\/ $[F]\in H_2(X)$ is
primitive, $[F]\cdot[F]=0$, and $H^1(X\setminus\nu
F\hspace{1pt})=0$. If $\,b_2^+(X)=1$, then we also assume
that $\,\overline{SW}
_{\! X\setminus \nu F}\neq 0\,$ and is a finite
sum.
Then there exists
an infinite family of pairwise non-isotopic symplectic tori in\/ $X_K$
representing the homology class\/ $[F]\in H_2(X_K)$ for
any nontrivial fibred knot\/ $K\subset S^3$.
\end{proposition}
The divisibility argument
in the proof of Corollary~\ref{cor:divisibilities}
may not work in this general setting, but after observing
that an isotopy between these tori should preserve $\xi$
and $\tau$, one can resort to a homology basis argument
due to Fintushel and Stern which was announced in \cite{fs:ipam}.
It may be possible, as in the rational elliptic surface case,
to show that these non-isotopic tori are inequivalent
under self-diffeomorphisms of $X_K$ once we know the
Seiberg-Witten invariant of\/ $[X \setminus \nu F]$\/ explicitly,
but a general argument
doesn't seem to exist at this moment.
\subsection*{Acknowledgments}
We would like to thank Ronald Fintushel, Sa\v{s}o Strle
and
Stefano Vidussi for their encouragement
and
helpful comments on this and other works of ours.
| 12,450
|
by Nedim Hadrovic
Coalition staff journalist
Newly minted and taking examples from democracies way west of its borders, Kosovo’s Constitution has the ingredients to perform what it endeavors to do. In practice, though, it falls short of ensuring that the country’s many laws promising good governance work on the ground. This is perhaps nowhere more reflected than in Kosovo’s media.
The constitutional guarantee of freedom of expression “without impediment” has not translated into strong protections from political meddling, budget shortfalls and physical attacks on journalists.
“Kosovo’s legislation provides a healthy infrastructure for media,” Albulena Sadiku of the Balkan Investigative Reporting Network (BIRN) tells me, but “the problem lies with the editorial independence of the majority of media outlets, who are by default totally dependent on financial input through advertising funds.”
Although a widespread problem throughout the Balkans, the issue of advertising revenue exercising influence over the editorial integrity of publications is amplified in Kosovo. But it is only one piece making up the larger jigsaw of issues plaguing the integrity of the country’s press.
Playing Favorites
Imer Mushkolaj, Board President of the self-regulatory Kosovo Press Council, points out the close relationships among the country’s media and its major stakeholders, and mentions that this is immediately apparent in the reporting.
If a journalist is writing a comparative economic piece on different public companies, for example, and writes disproportionately about one more than another, Mushkolaj says over time, it becomes obvious “that the journalist is working for that company.”
These two examples may naturally be traced to the general and current problems of low revenue within newsrooms and low wages for journalists, respectively. The problem is that in Kosovo, these challenges practically have become part of the mainstream.
The constitutional guarantee of freedom of expression “without impediment” has not translated into strong protections from political meddling, budget shortfalls and physical attacks on journalists.
This all contributes to “a very low public trust towards media, specifically online media, which are often seen as spin doctors, sensational click-bait and revenge tools” in the wider arena of Kosovo politics, says Faik Ispahiu of KALLXO.com, a joint news project of Internews Kosova and BIRN.
The 2013 Human Rights Report by the US State Department says “growing financial difficulties left the editorial independence and journalistic professionalism of both print and television media vulnerable to outside influence and pressure.”
The dire financial state of Kosovo’s media also means journalists often aren’t paid – and when this happens, either the content suffers or journalists retaliate against the outlet through lawsuits.
Kosovo journalist Naim Krasniqi (left) is confronted and threatened by an employee of a private air conditioning company in the Palace of Justice on June 23, 2016, a few weeks after a Krasniqi wrote an article about alleged corrupt dealings between the Justice Ministry and the company. (Photos: KALLXO)
The Media as a Target
This retaliation can boil over into verbal and physical attacks against bereft journalists – which was exactly what happened to Gazeta Tribuna’s Zekirja Shabani. The former economy editor-in-chief, Shabani announced to the editorial board that he planned to sue the paper for not paying the entire staff for two months. He was called into the newspaper owner’s office, where he was verbally abused, subject to pressure to sign an illegal contract termination, and physically assaulted.
Attacks on journalists and media workers are a widespread problem in the country. To this day, Shabani says he gets calls from people threatening to kill him. “In Kosovo, where newspapers are dependent on politics, criminal and other interest groups, attacks on journalists are evident,” he says.
According to the European Commission, 26 cases of attacks, threats and obstruction of journalists were under investigation in Kosovo in 2015.
The hierarchical power structure in the country is evident in the preferential treatment of cases by police. In a classic example, a journalist who point out abuse of power by a politician’s son was immediately silenced by the police, whereby a parallel case of a media worker who was threatened by a person in the institutional ranks received practically no attention.
Shabani, who is now the head of the Kosovo Journalist Association – making him the target of further threats – recently founded Gazeta Fjala, an online portal that claims to be committed to fair and unbiased reporting.
Founding his own outlet was an act of resilience for Shabani – against the reactions and rebuttals from his colleagues and fellow outlets following his run-in at Tribuna. “It was hard for me to find another job, even though I have more than 10 years experience both as a journalist and editor,” Shabani explains.
Online portals, however, are subject to much criticism when it comes to professionalism, as most exercise “copy-paste” journalism based on incomplete research. Further, almost none of the journalists “employed” by online portals have regular employment contracts. The result is a very shaky system of journalism that is hardly reliable.
An Un-free Press?
Kosovo’s legislative progression in the past decade-and-a-half, since the effective power transfer to the UN Interim Administration Mission in Kosovo in 1999 and then to independence in 2008, has seen promising constitutional changes, many of which promised transparency in many aspects. But the lack of any effective implementation of these laws means this has fallen way short of initial promises.
Media still remains a meddled-with aspect of society. Access to information and transparency thereof is only promised on paper. Adding to this the poor treatment and general state of media workers, and Kosovo finds itself ranked 90th worldwide in terms of press freedom, according to both Reporters Without Borders and Freedom House. Relegated nearly a hundred countries down, alongside many countries that exercise more downright blatant abuses of power, can’t be good for Kosovo.
And although the physical abuse of journalists has waned since the Serb handover in 1999, following brutal attacks by police forces, “this is a problem that still very much exists and is something that is manifested in both verbal and physical forms, which we try and keep track of this as best we can,” says Oliver Vujovic of the South East Europe Media Organisation.
Civil society, as is standard practice in the so-called transitional Balkan states, has a very strong presence and monitors these problems through platforms and tangible, data-based reportage. Sites such as KALLXO.com tally abuse cases against whistleblowers and has received 6,000 cases in the past three years.
But these cases hardly ever are addressed by institutions in Kosovo, with excuses ranging from “we have previous priorities” to “it’s buried in the paperwork.” According to KALLXO’s Ispahiu, “in such a fragile media scene, the government often seems to ignore its legal obligation” to the media.
And this ignorance extends to a lack of prosecution of abuse cases against journalists and media workers. “Our staff has been physically attacked, our cars have been vandalized, our senior staff has been threatened with their lives and lives of their families,” Ispahiu says, “all this because we were reporting facts and the truth.”
If even tech-based initiatives like KALLXO fall on deaf institutional ears, they will just become vacant datasets serving as a mere nod to what technology is capable of in theory, but where exactly it fails in practice. Bringing the potential of NGOs that base their work on tech and new media to the forefront of public debate in Kosovo is key. If it only puts its potential on both sides of the institutional coin, Kosovo might be the first Balkan country to actually see an effective benefit from its large civil society presence..
| 277,823
|
\section{The second variation of $\mW_3$}
\label{sec:stable}
In order to prove Theorem~\ref{thm:stability_theorem}, we first compute the second variation of the $\mV_3$-functional at a volume-normalized shrinking gradient Ricci soliton.
\begin{thm}
\label{thm:V3_second_var}
Let $(M^n,g,e^{-\phi}\dvol)$ be a volume-normalized shrinking gradient Ricci soliton. Then
\begin{equation}
\label{eqn:V3_second_var} \mV_3^{\prime\prime}[\psi,t] \geq \left(\tau^2\cv_{2,\phi}-\frac{\tau}{2}\cv_{1,\phi}\right)\int_M \left[ \tau\left|\nabla\psi_1\right|^2 - \frac{1}{2}\psi_1^2 + \frac{1}{2}\left(c-\frac{t}{\tau}\right)^2\phi_0^2 \right] d\nu ,
\end{equation}
where $\psi_1=\psi_0+c\phi_0$ for $\psi_0=\psi+\frac{nt}{2\tau}$ and $c$ such that $\int\psi_1\phi_0\,d\nu=0$. Moreover, equality holds in~\eqref{eqn:V3_second_var} if and only if $(M^n,g,e^{-\phi}\dvol)$ is isometric to a shrinking Gaussian or $t=0$.
\end{thm}
\begin{proof}
Let $\gamma\colon(-\eps,\eps)\to C^\infty(M)\times(0,\infty)$ be a smooth path with $\gamma(0)=(\phi,\tau)$ and $\frac{\partial\gamma}{\partial s}=(\psi,t)$. Define functions $A,B,C,D\colon (-\eps,\eps)\to\bR$ as follows:
\begin{align*}
A(s) & = -\int_M \left(\tau^3\cv_{3,\phi}-\frac{\tau}{2}\cv_{2,\phi}\right)\psi_0\,d\nu , \\
B(s) & = \int_M \frac{t}{\tau}\left(\tau^2\cv_{2,\phi}\right)\left(\tau\sigma_{1,\phi}-\frac{n}{4}\right)d\nu , \\
C(s) & = \int_M t\left\lp \tau^2\cE_{2,\phi} + \frac{\tau^2}{3}\cB_\phi, \Ric_\phi\right\rp d\nu , \\
D(s) & = \int_M \frac{t}{6}\left\lp 4\tau\cB_\phi + \Ric, \tau\cRic_\phi\right\rp d\nu .
\end{align*}
Since $\left(\mV\left(g,\phi(s),\tau(s)\right)\right)^\prime=\left(A+B+C+D\right)(s)$, it suffices to compute the derivatives of $A,B,C$ and $D$ at $s=0$. This task is made simpler by combining Lemma~\ref{lem:grs_sigmak}, Lemma~\ref{lem:v1_local_all_var}, Lemma~\ref{lem:v2_local_all_var} and Lemma~\ref{lem:v3_local_all_var} to conclude that
\begin{equation}
\label{eqn:grs_var_vk}
\left(\tau^k\cv_{k,\phi}\right)^\prime(0) = \tau^{k-1}\cv_{k-1,\phi}\left(\tau\Delta_\phi + \frac{1}{2}\right)\psi_0 - \frac{t\tau^{k-2}}{2}\cv_{k-1,\phi}\phi_0
\end{equation}
for $k\in\{1,2,3\}$. Furthermore, \cite[Corollary~8.4]{Case2014sd} implies that
\begin{equation}
\label{eqn:grs_var_s1}
\left(\tau\sigma_{1,\phi}-\frac{n}{4}\right)^\prime(0) = \tau\Delta_\phi\psi_0 - \frac{t}{2\tau}\phi_0 + \frac{nt}{4\tau}
\end{equation}
and direct computation yields
\begin{align}
\label{eqn:grs_var_cRic} \left(\tau\cRic_\phi\right)^\prime(0) & = \tau\nabla^2\psi_0 + \frac{t}{2\tau}g , \\
\label{eqn:grs_var_cE2} \left(\tau^2\cE_{2,\phi}\right)^\prime(0) & = -\tau^2\cv_{1,\phi}\nabla^2\psi_0 - \frac{t}{2}\cv_{1,\phi}g , \\
\label{eqn:grs_var_cB} \left(\tau^2\cB_\phi\right)^\prime(0) & = -\frac{t}{2}\nabla^2\phi_0 + \frac{t}{4\tau}g .
\end{align}
By adapting the proof of Corollary~\ref{cor:grs_crit}, we see that $(\phi,\tau)$ is a critical point of $\mV_3$. Hence
\[ A^\prime(0) = -\int_M \left(\tau^3\cv_{3,\phi}-\frac{\tau}{2}\cv_{2,\phi}\right)^\prime(0)\,\psi_0\,d\nu . \]
Therefore, by Lemma~\ref{lem:grs_sigmak} and~\eqref{eqn:grs_var_vk},
\begin{equation}
\label{eqn:Aprime}
A^\prime(0) = \left(\tau^2\cv_{2,\phi}-\frac{\tau}{2}\cv_{1,\phi}\right)\int_M \left[ \tau\lv\nabla\psi_0\rv^2 - \frac{1}{2}\psi_0^2 + \frac{t}{2\tau}\phi_0\psi_0\right]d\nu .
\end{equation}
Using Lemma~\ref{lem:grs_sigmak} and Lemma~\ref{lem:estimates}, we observe that
\begin{multline*}
B^\prime(0) = -\frac{t}{2\tau}\int_M \left(\tau^2\cv_{2,\phi}\right)^\prime(0)\,\phi_0\,d\nu \\ + t\tau\cv_{2,\phi}\int_M \left(\tau\sigma_{1,\phi}-\frac{n}{4}\right)^\prime(0)\,d\nu - \frac{t\tau}{2}\cv_{2,\phi} \int_M \phi_0\,(d\nu)^\prime(0) .
\end{multline*}
From~\eqref{eqn:grs_var_vk} and \eqref{eqn:grs_var_s1} we compute that
\[ B^\prime(0) = \int_M \left[ \frac{t}{2}\left(\tau\cv_{2,\phi}+\frac{1}{2}\cv_{1,\phi}\right)\psi_0\phi_0 + \frac{t^2}{4\tau}\cv_{1,\phi}\phi_0^2 + \frac{nt^2}{4}\cv_{2,\phi} \right]d\nu . \]
Applying Lemma~\ref{lem:grs_sigmak} and Lemma~\ref{lem:estimates} yields
\begin{equation}
\label{eqn:Bprime}
B^\prime(0) \geq \left(\frac{t\tau}{2}\cv_{2,\phi} + \frac{t}{4}\cv_{1,\phi}\right) \int_M \left[ \psi_0\phi_0 + \frac{t}{\tau}\phi_0^2 \right]d\nu
\end{equation}
with equality if and only if $(M^n,g,e^{-\phi}\dvol)$ is isometric to a shrinking Gaussian or $t=0$.
Using first~\eqref{eqn:grs_var_cE2} and~\eqref{eqn:grs_var_cB} and second~\eqref{eqn:sigma1_estimate}, we compute that
\begin{align*}
C^\prime(0) & = -\frac{t}{2\tau}\int_M \tr\left( \tau^2\cv_{1,\phi}\nabla^2\psi_0 + \frac{t}{2}\cv_{1,\phi}g + \frac{t}{6}\nabla^2\phi_0 - \frac{t}{12\tau}g\right)d\nu \\
& = -\frac{t}{2\tau}\int_M \left[ \tau\cv_{1,\phi}\psi_0\phi_0 + \frac{nt}{2}\cv_{1,\phi} + \frac{t}{6\tau}\phi_0^2 - \frac{nt}{12\tau} \right]d\nu .
\end{align*}
Lemma~\ref{lem:grs_sigmak} and Lemma~\ref{lem:estimates} then yield
\begin{equation}
\label{eqn:Cprime}
C^\prime(0) \geq -\frac{t}{2}\cv_{1,\phi}\int_M \left[ \psi_0\phi_0 + \frac{t}{\tau}\phi_0^2 \right]d\nu
\end{equation}
with equality if and only if $(M^n,g,e^{-\phi}\dvol)$ is isometric to a shrinking Gaussian or $t=0$.
From~\eqref{eqn:grs_var_cRic} we compute that
\[ D^\prime(0) = \frac{t}{6}\int_M \left[ \frac{1}{2}\lp\nabla\psi_0,\nabla\phi_0\rp - \tau\lp\nabla^2\psi_0,\nabla^2\phi_0\rp - \frac{t}{2\tau^2}\phi_0^2 + \frac{nt}{4\tau^2} \right] d\nu . \]
Recalling that $\delta_\phi\nabla^2u=d\Delta_\phi u + \Ric_\phi(\nabla u)$ for all $u\in C^\infty(M)$, we compute using Lemma~\ref{lem:estimates} that
\[ \int_M \lp\nabla^2\psi_0,\nabla^2\phi_0\rp\,d\nu = \frac{1}{2\tau}\int_M \lp\nabla\psi_0,\nabla\phi_0\rp\,d\nu . \]
Inserting this into the previous display and using Lemma~\ref{lem:estimates} again yields
\begin{equation}
\label{eqn:Dprime}
D^\prime(0) \geq 0
\end{equation}
with equality if and only if $(M^n,g,e^{-\phi}\dvol)$ is isometric to a shrinking Gaussian or $t=0$.
Now, combining~\eqref{eqn:Aprime}, \eqref{eqn:Bprime}, \eqref{eqn:Cprime} and~\eqref{eqn:Dprime} yields
\[ \mV_3^{\prime\prime}[\psi,\tau] \geq \left(\tau^2\cv_{2,\phi} - \frac{\tau}{2}\cv_{1,\phi}\right)\int_M \left[ \tau\lv\nabla\psi_0\rv^2 - \frac{1}{2}\psi_0^2 + \frac{t}{\tau}\psi_0\phi_0 + \frac{t^2}{2\tau^2}\phi_0^2 \right] d\nu \]
with equality if and only if $(M^n,g,e^{-\phi}\dvol)$ is isometric to a shrinking Gaussian or $t=0$. The final conclusion follows from the observation that
\begin{align*}
\int_M \lv\nabla\psi_0\rv^2\,d\nu & = \int_M \left[ \lv\nabla\psi_1\rv^2 + \frac{c^2}{\tau}\phi_0^2\right] d\nu , \\
\int_M \psi_0^2\,d\nu & = \int_M \left[ \psi_1^2 + c^2\phi_0^2 \right]d\nu . \qedhere
\end{align*}
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:stability_theorem}]
It follows from Theorem~\ref{thm:V3_second_var} and the proof of~\cite[Theorem~9.2]{Case2014sd} that
\[ \mW_3^{\prime\prime}[\psi,\tau] \geq \left(\tau^2\cv_{2,\phi} - \frac{\tau}{2}\cv_{1,\phi} + \frac{1}{8}\right)\int_M \left[ \tau\lv\nabla\psi_1\rv^2 - \frac{1}{2}\psi_1^2 + \frac{1}{2}\left(c-\frac{t}{\tau}\right)^2\phi_0^2 \right] d\nu \]
with equality if and only if $(M^n,g,e^{-\phi}\dvol)$ is isometric to a shrinking Gaussian or $t=0$. By Lemma~\ref{lem:grs_sigmak}, we see that $\tau^2\cv_{2,\phi}-\frac{\tau}{2}\cv_{1,\phi}+\frac{1}{8}>0$. Hence $\mW_3^{\prime\prime}[\psi,\tau]\geq0$ with equality if and only if
\begin{enumerate}
\item $(M^n,g,e^{-\phi}\dvol)$ is isometric to a shrinking Gaussian, $c-\frac{t}{\tau}=0$ and $\int\lv\nabla\psi_1\rv^2\,d\nu=\frac{1}{2\tau}\int\psi_1^2\,d\nu$, or
\item $t=0$, $\int\psi\phi_0\,d\nu=0$ and $\int\lv\nabla\psi\rv^2\,d\nu=\frac{1}{2\tau}\int\psi^2\,d\nu$.
\end{enumerate}
The conclusion follows from a straightforward application of Lemma~\ref{lem:grs_obata}.
\end{proof}
| 170,814
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TITLE: Show that $47$ divides $5^{23}+1$
QUESTION [1 upvotes]: Show that $47$ divides $5^{23}+1$.
My attempt:
Since $47$ is prime and $47$ does not divide $5$, by Fermat's Little Theorem,
$5^{47-1} \equiv 1 \pmod {47}$
$5^{46} \equiv 1 \pmod {47}$
Now I noticed that $\mathbb{Z}_{47}$ was a field. So that means each element in $\mathbb{Z}_{47}$ has an multiplicative inverse in $\mathbb{Z}_{47}$. I went on to proceed to find the inverse of $5$ by the Extended Euclidean Algorithm which gave me $19$.
Now if I multiply both sides by $5^{-1}$ twenty-three times, I can reduce the power of $46$ to $23$,
Now, $(5^{-1} \cdot \ldots \cdot 5^{-1}) 5^{46} \equiv (5^{-1} \cdot \ldots \cdot 5^{-1}) \pmod {47}$
So, $5^{23} \equiv 19^{23} \pmod {47}$
But this didn't help me at all. So without giving the solution can someone give me a hint of a way to proving the above?
REPLY [0 votes]: I think you did all the hard work and got entangled with the easy one. As you said, we're working in a field, so:
== How many solutions on any field with characteristic $\;\neq2\;$ are there to $\;x^2-1=0\;$ ?
== Observe carefully what you wrote (everything's done modulo $\;47\;$):
$$1=5^{46}=\left(5^{23}\right)^2$$
Finish the proof.
| 69,842
|
Hey Now (Mean Muggin) (Edited, Single)
Released: Nov 2004
Label: Sony Urban Music/Columbia
More high-profile than ever, thanks to his hit MTV show Pimp My Ride, L.A. flamespitter Xzibit returns to the rap game with this ferocious single. "Hey Now (Mean Muggin)" flaunts X's signature gruff rhymes, with a banging Dr. Dre beat ideal for streets and clubs alike.
Tracks
Welcome to Rhapsody
You're just minutes away from millions of songs.
| 148,848
|
TITLE: Unequal circles within circle with least possible radius?
QUESTION [7 upvotes]: It is the classical will-my-cables-fit-within-the-tube-problem which lead me to the interest of circle packing. So basically, I have 3 circles where r = 3 and 1 circle where r = 7 and I am trying to find the least r for an outer circles of these 4 smaller circles.
After a couple of hours of thinking and some sketches with a compass I am getting close to the actual result.
But how can I calculate this?
With what formula?
EDIT:
Thanks for the great answer.
And then I come to wonder.
What happens if you add another of the small circles, so you have four circles with r = 3? It is very close to 11.7
REPLY [5 votes]: I finished up the other approach. The arrangement with larger radius $A$ and three circles of smaller radius $B,$ all tangent to an outer circle of radius $R,$ gives a cubic $$ R^3 - (A+2B) R^2 + A B R + A B^2 = 0. $$ For $A=7, B=3$ this gives $R \approx 10.397547282.$ Note that the coefficient of $R^3$ is positive, when $R=0$ the result is positive, but when $R=A$ the result is negative (for $A>B>0$). So there is a negative root, an unsuitable root $0 < R < A,$ and finally the real thing when $R>A.$ Notice that, for $A=B=1,$ we get the correct $R=1+\sqrt 2,$ meaning the centers of the four small circles are on the corners of a square, and the center of the circumscribing circle is at the center of the same square, all very symmetric in that case.
It is possible for $(A,B,R)$ to come out integers, for example $(A=9,B=5,R=15)$ or $(A=32,B=11,R=44).$ These are in the infinite family $$ A = n^3 + 4 n^2 + 4 n = n (n+2)^2, B = n^2 + 3 n + 1, R = n^3 + 5 n^2 + 7 n + 2 = A + B + 1. $$
The $(9,5,15)$ arrangement is especially good for a diagram here, as there are many visible $30^\circ-60^\circ-90^\circ$ right triangles, as well as one with sides $14,11, 5 \sqrt 3$ where I drew a pale green line in pencil.
Umm. It turned out it was possible to solve the Diophantine equation for much larger values; it is obvious that $R | A B^2,$ and a little extra fiddling with unique factorization (I'm taking $\gcd(A,B)=1$) shows that $R | AB,$ so that $AB/R$ is an integer, and the equation becomes
$$ R \cdot (A + 2 B - R) = (R + B) \cdot (AB/R). $$ Here are the first hundred integer solutions
A B R A+2B-R R+B AB/R
9 5 15 4 20 3
25 22 55 14 77 10
32 11 44 10 55 8
75 19 95 18 114 15
128 93 248 66 341 48
144 29 174 28 203 24
147 62 217 54 279 42
245 41 287 40 328 35
363 244 671 180 915 132
384 55 440 54 495 48
400 183 610 156 793 120
405 118 531 110 649 90
507 395 1027 270 1422 195
567 71 639 70 710 63
605 237 869 210 1106 165
784 505 1414 380 1919 280
800 89 890 88 979 80
845 404 1313 340 1717 260
847 190 1045 182 1235 154
867 847 2057 504 2904 357
1089 109 1199 108 1308 99
1183 363 1573 336 1936 273
1296 1043 2682 700 3725 504
1440 131 1572 130 1703 120
1445 906 2567 690 3473 510
1521 278 1807 270 2085 234
1536 755 2416 630 3171 480
1568 435 2030 408 2465 336
1575 596 2235 532 2831 420
1805 1253 3401 910 4654 665
1859 155 2015 154 2170 143
1936 1845 4510 1116 6355 792
2023 895 3043 770 3938 595
2205 1672 4389 1160 6061 840
2352 181 2534 180 2715 168
2400 1477 4220 1134 5697 840
2475 382 2865 374 3247 330
2527 1266 4009 1050 5275 798
2560 597 3184 570 3781 480
2592 1045 3762 920 4807 720
2601 820 3485 756 4305 612
2645 2169 5543 1440 7712 1035
2925 209 3135 208 3344 195
3179 687 3893 660 4580 561
3249 1205 4579 1080 5784 855
3584 239 3824 238 4063 224
3645 3421 8397 2090 11818 1485
3703 2248 6463 1736 8711 1288
3757 502 4267 494 4769 442
3872 1967 6182 1624 8149 1232
3971 1076 5111 1012 6187 836
4205 4188 10121 2460 14309 1740
4335 271 4607 270 4878 255
4375 2871 7975 2142 10846 1575
4693 885 5605 858 6490 741
4704 3905 9940 2574 13845 1848
4761 2233 7337 1890 9570 1449
4851 1555 6531 1430 8086 1155
5103 3590 9693 2590 13283 1890
5184 305 5490 304 5795 288
5408 3249 9386 2520 12635 1872
5415 638 6061 630 6699 570
5600 993 6620 966 7613 840
5625 2888 9025 2376 11913 1800
5733 1364 7161 1300 8525 1092
5760 2513 8616 2170 11129 1680
5808 1745 7678 1620 9423 1320
5819 2130 8165 1914 10295 1518
5887 4411 11629 3080 16040 2233
6137 341 6479 340 6820 323
6144 5707 14048 3510 19755 2496
6727 5340 13795 3612 19135 2604
6875 2807 10025 2464 12832 1925
6877 1945 8947 1820 10892 1495
7200 379 7580 378 7959 360
7200 4939 13470 3608 18409 2640
7497 790 8295 782 9085 714
7569 4510 13079 3510 17589 2610
7623 6383 16203 4186 22586 3003
7744 1227 8998 1200 10225 1056
7840 4059 12628 3330 16687 2520
7935 1684 9683 1620 11367 1380
8019 3592 12123 3080 15715 2376
8064 2155 10344 2030 12499 1680
8112 3115 11570 2772 14685 2184
8125 2634 10975 2418 13609 1950
8379 419 8799 418 9218 399
8649 5489 15469 4158 20958 3069
8993 1353 10373 1326 11726 1173
9248 7085 18530 4888 25615 3536
9251 4491 14471 3762 18962 2871
9477 3437 13257 3094 16694 2457
9583 8835 21793 5460 30628 3885
9680 461 10142 460 10603 440
10051 958 11017 950 11975 874
10240 6061 17632 4730 23693 3520
10571 5510 17081 4510 22591 3410
10625 2036 12725 1972 14761 1700
10647 10256 24999 6160 35255 4368
10816 2605 13546 2480 16151 2080
A B R A+2B-R R+B AB/R
| 47,361
|
TITLE: Writing roots of f(x) as f(a) for some a
QUESTION [0 upvotes]: I was solving a problem when this random thought popped into my head. Suppose you have a function, say $f(x)=x^2-1=(x-1)(x+1)$. The roots of this function are $-1$ and $-1$. We can write these roots as $f(a)$ for some $a$, because solving $f(x)=-1$ gives $x=0$ and solving $f(x)=1$ gives $x=\pm \sqrt{2}$. Therefore, we can write the function as either $f(x)=(x-f(\sqrt{2}))(x-f(0))$ or $f(x)=(x-f(-\sqrt{2}))(x-f(0))$.
My questions:
1. Has any work been done on such representations of f(x)?
2. If not, does this seem like a nice idea or is it just a trivial thing that won't lead anywhere?
3. Can you think of any way to reverse the process? For example, given that $f(x)=(x-f(\sqrt{2}))(x-f(0))$, is it possible to find $f(x)$'s that "work"?
Thanks!
EDIT: Example: For $f(x)=x$ the only root is $x=0$. So we can write it (as we did above) as $f(x)=x-f(0)$.
For the reverse problem, we want to solve $f(x)=x-f(0)$ for $f(x)$. Plugging $x=0$ yields $f(0)=-f(0)$ or $2f(0)=0$ or $f(0)=0$. Therefore $f(x)=x-f(0)=x$. So the only function that "works" is indeed $f(x)=x$.
EDIT (2): Harder example: Solve $f(x)=-(x-f(0))$.
REPLY [0 votes]: HINT
$$\begin{align}
f(x) &= f(0) - x \\
f(1) &= f(0) - 1 \\
f(2) &= f(0) - 2 \\ &= f(1) - 1 \\ &\vdots \\
\implies f(n) &= f(n-1) -1
\end{align}$$
Therefore, $f(x) = -x$.
| 164,960
|
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The profile for Janusz Gol
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| 207,669
|
TITLE: In Luminous Distance calculations, why does the power fall of by a factor of $(1+z)$?
QUESTION [0 upvotes]: This is a follow up to this question here about the $(1+z)$ factor in the Luminous Distance formula. As the universe expands, I understand why the energy falls as the wavelength is stretched (by a factor of $z+1$), but in the Luminous Distance formula
$$L_D=\chi(1+z)$$
we have an additional an additional factor of $z+1$. I've read Dodelson and this article (page 201) and both seem to argue that the frequency of photons crossing the shell is different between emission and observation:
We must take into account the fact that the rate of photon reception
is smaller than the rate of emission by a factor of $a = 1/(1+z)$.
Both authors wave their hands and say 'this is so', but I don't understand the process. The time interval, $\Delta t$, is the same from emission to observation (that is, expansion doesn't change the time interval). The number of photons crossing the entire shell doesn't change (you can't add or remove photons from the expanding shell). So how does the rate of photons crossing the shell (or hitting a detector) change?
REPLY [2 votes]: The additional $1+z$ factor is cosmological time dilation (e.g. see Zhang et al. 2013). The frequency with which photons are received is reduced by this factor.
That is, if a distant source emits $n$ photons per second in its rest frame, then those are received in the current epoch at a rate of $n/(1+z)$. The reason is precisely the same reason that individual photons have frequencies reduced by the same factor.
| 149,620
|
\begin{document}
\maketitle
\begin{abstract}
In \cite{CT20}, we have studied the Boltzmann random triangulation of the disk coupled to an Ising model on its faces with Dobrushin boundary condition at its critical temperature. In this paper, we investigate the phase transition of this model by extending our previous results to arbitrary temperature: We compute the partition function of the model at all temperatures, and derive several critical exponents associated with the infinite perimeter limit. We show that the model has a local limit at any temperature, whose properties depend drastically on the temperature. At high temperatures, the local limit is reminiscent of the uniform infinite half-planar triangulation (UIHPT) decorated with a subcritical percolation. At low temperatures, the local limit develops a bottleneck of finite width due to the energy cost of the main Ising interface between the two spin clusters imposed by the Dobrushin boundary condition. This change can be summarized by a novel order parameter with a nice geometric meaning.
In addition to the phase transition, we also generalize our construction of the local limit from the two-step asymptotic regime used in \cite{CT20} to a more natural diagonal asymptotic regime. We obtain in this regime a scaling limit related to the length of the main Ising interface, which coincides with predictions from the continuum theory of \emph{quantum surfaces} (a.k.a.\ Liouville quantum gravity).
\end{abstract}
\newcommand*{\g}{\Map{g}}
\newcommand*{\B}{\Map{b}}
\section{Introduction}
The two-dimensional Ising model is one of the simplest statistical physics models to exhibit a phase transition. We refer to \cite{MW73} for a comprehensive introduction. The systematic study of the Ising model on random two-dimentional lattices dates back to the pioneer works of Boulatov and Kazakov \cite{Kaz86,BouKaz87}, where they discovered a third order phase transition in the free energy density of the model, and computed the associated critical exponents. In their work, the partition function of the model was computed in the thermodynamic limit using matrix integral methods applied to the so-called \emph{two-matrix model}, see \cite{LZ04} for a mathematical introduction. Since then, this approach has been pursued and further generalized to treat other statistical physics models on random lattices, see e.g.\ \cite{EynOra05,EynBon99}.
In this paper, we will follow a more combinatorial approach to the model originated from a series of works by Tutte (see \cite{Tut95} and the references therein) on the enumeration of various classes of embedded planar graphs known as \emph{planar maps}, which is essentially another name for the random lattices studied in physics. The approach of Tutte utilizes a type of recursive decomposition satisfied by these classes of planar maps to derive a functional equation that characterizes their generating function. This method was later generalized by Bernardi, Bousquet-M\'elou and Schaeffer \cite{BMS02, BBM11,BBM16} to treat two-colored planar maps with a weighting that is equivalent to the Ising model.
From a probabilistic point of view, the aforementioned recursive decomposition can be seen as the operation of removing one edge from an (Ising-decorated) random planar map with a boundary, and observing the resulting changes to the boundary condition. By iterating this operation, one obtains a random process, called the \emph{peeling process}, that explores the random map one face at a time. Ideas of such exploration processes have their roots in the physics literature \cite{Wat95}, and was revisited and popularized by Angel in \cite{Ang02}. The peeling process proves to be a valuable tool for understanding the geometry of random planar maps without Ising model, see \cite{CurPeccot} for a review of recent developments.
In our previous article \cite{CT20}, we extended some enumeration results of Bernardi and Bousquet-M\'elou \cite{BBM11} to study the Ising-decorated random triangulations with Dobrushin boundary condition \emph{at its critical temperature}. We used the peeling process to construct the local limit of the model, and to obtain several scaling limit results concerning the lengths of some Ising interfaces. In this paper, we extend similar results to the model \emph{at any temperature}, and show how the large scale geometry of Ising-decorated random triangulations changes qualitatively at the critical temperature. In particular, our results confirm the physical intuition that, at large scale, Ising-decorated random maps at non-critical temperatures behave like non-decorated random maps.
A similar model of Ising-decorated triangulations (more precisely, a "dual" to ours) has been studied in a recent work of Albenque, M\'enard and Schaeffer \cite{AMS18}. They followed an approach reminiscent of Angel and Schramm in \cite{AS03} to show that the model has a local limit at any temperature, and obtained several properties of the limit such as one-endedness and recurrence for some temperatures. However they studied the model without boundary, and hence did not encounter the geometric consequences of the phase transition. Their model can also be studied with a boundary, and the methods introduced in \cite{CT20} and this article were recently applied to that in \cite{T20} by the second author of this work.
\medskip
We start by recalling some essential definitions from \cite{CT20}.
\begin{figure}[t!]
\centering
\includegraphics[scale=0.9]{Fig1.pdf}
\caption{
(a) A triangulation $\tmap$ of the 7-gon with 19 internal faces. The boundary will no longer be simple if one attaches to $\tmap$ the map inside the bubble to its left.
(b) an Ising-triangulation of the $(3,4)$-gon with 18 monochromatic edges (dashed lines).
}
\label{fig:def-map}
\end{figure}
\paragraph{Planar maps.}
Recall that a (finite) \emph{planar map} is a proper embedding of a finite connected graph into the sphere $\mathbb{S}^2$, viewed up to orientation-preserving homeomorphisms of $\mathbb{S}^2$. Loops and multiple edges are allowed in the graph.
A \emph{rooted map} is a map equipped with a distinguished corner, called the \emph{root corner}.
\begin{center} \vspace{-1.ex}
\emph{All maps in this paper are assumed to be planar and rooted.}
\end{center} \vspace{-1.ex}
In a (rooted planar) map $\map$, the vertex incident to the root corner is called the \emph{root vertex} and denoted by $\rho$. The face incident to the root corner is called the \emph{external face}, and all other faces are \emph{internal faces}. We denote by $F(\map)$ the set of internal faces of $\map$.
A map is a \emph{triangulation of the $\ell$-gon} ($\ell\ge 1$) if its internal faces all have degree three, and the boundary of its external face is a simple closed path (i.e.\ it visits each vertex at most once) of length $\ell$. The number $\ell$ is called the \emph{perimeter} of the triangulation. Figure~\refp{a}{fig:def-map} gives an example of a triangulation of the $7$-gon.
\paragraph{Ising-triangulations of the $(p,q)$-gon.}
We consider the Ising model with spins on the internal \emph{faces} of a triangulation of a polygon. A triangulation together with an Ising spin configuration on it is written as a pair $\bt$, where $\sigma \in \{\+,\<\}^{F(\tmap)}$. Observe that $\sigma$ can also be viewed as a coloring, and by combinatorial convention, we sometimes refer to it as such.
An edge $e$ of $\tmap$ is said to be \emph{monochromatic} if the spins on both sides of $e$ are the same. When $e$ is a boundary edge, this definition requires a boundary condition which specifies a spin outside each boundary edge. By an abuse of notation, we consider the information about the boundary condition to be contained in the coloring $\sigma$, and denote by $m\bt$ the number of monochromatic edges in $\bt$.
In this work we consider the \emph{Dobrushin boundary conditions} under which the spins \emph{outside} the boundary edges are given by a sequence of the form $\+^p \<^q$ ($p$ $\+$'s followed by $q$ $\<$'s, where $p,q\ge 0$ are integers and $p+q\ge 1$ is the perimeter of the triangulation) in the clockwise order from the root edge.
We call a pair $\bt$ with this boundary condition an \emph{Ising-triangulation of the $(p,q)$-gon}, or a \emph{bicolored triangulation of the $(p,q)$-gon}. Figure~\refp{b}{fig:def-map} gives an example in the case $p=3$ and $q=4$. We denote by $\bts_{p,q}$ the set of all Ising-triangulations of the $(p,q)$-gon.
For $\nu>0$, let
\begin{equation*}
z_{p,q}(t,\nu) = \sum_{(\tmap,\sigma)\in \mathcal{BT}_{p,q}} \nu^{m(\tmap,\sigma)} t^{\abs{F(\tmap)}}
\end{equation*}
When $z_{p,q}(t,\nu)<\infty$, we can
define a probability distribution $\prob_{p,q}^{t,\nu}$ on $\mathcal{BT}_{p,q}$ by
\begin{equation*}
\prob_{p,q}^{t,\nu}(\tmap,\sigma) = \frac{ t^{\abs{F(\tmap)}} \nu^{m(\tmap,\sigma)} }{z_{p,q}(t,\nu)}
\end{equation*}
for all $(\tmap,\sigma)\in \mathcal{BT}_{p,q}$. A random variable of law $\prob_{p,q}^{t,\nu}$ will be called a \emph{Boltzmann Ising-triangulation of the $(p,q)$-gon}.
We collect the partition functions $( z_{p,q}(t,\nu) )_{p,q\ge 0}$ into the following generating series:
\begin{equation*}
Z_q(u;t,\nu) = \sum_{p=0}^\infty z_{p,q}(t,\nu)\, u^p
\qtq{and}
Z(u,v;t,\nu) = \sum_{p,q\ge 0} z_{p,q}(t,\nu)\, u^p v^q = \sum_{q=0}^\infty Z_q(u;t,\nu) v^q \,,
\end{equation*}
where by convention $z_{0,0} = 1$.
\paragraph{Phase diagram of the Ising-triangulations.}
The condition $z_{p,q}(t,\nu)<\infty$ does not depend on $(p,q)$:
For any pairs $(p,q),(p',q')\ne (0,0)$, one can construct an annulus of triangles which, when glued around \emph{any} bicolored triangulation of the $(p,q)$-gon, gives a bicolored triangulation of the $(p',q')$-gon. Thus $z_{p,q}(t,\nu)\le C\cdot z_{p',q'}(t,\nu)$, where $C$ is the weight of the annulus.
It has been shown in \cite[Section 12.2]{BBM11} that for all $\nu>1$, the series $t\mapsto z_{1,0}(t,\nu)$ converges at its radius of convergence $t_c(\nu)$. Then the above argument implies that $t_c(\nu)$ is the radius of convergence of $t\mapsto z_{p,q}(t,\nu)$ and we have $z_{p,q}(t_c(\nu),\nu)<\infty$, for all $(p,q)\ne (0,0)$ and $\nu>1$.
In this paper we always restrict ourselves to the case $\nu>1$. (This is called the \emph{ferromagnetic} case since in this case the weight $\nu^{m(\tmap,\sigma)}$ favors neighboring spins to have the same sign.)
We shall call $t_c(\nu)$ the critical line of the Boltzmann Ising-triangulation. It separates the inadmissible region $t>t_c(\nu)$, where the probabilistic model is not well-defined, from the subcritical region $t<t_c(\nu)$, where the probability for a Boltzmann Ising-triangulation to have size $n$ decays exponentially with $n$. (Here the size of an Ising-triangulation is defined as its number of internal faces.) It has also been shown in \cite{BBM11} that the function $t_c(\nu)$ is analytic everywhere on $(1,\infty)$ except at $\nu_c=1+2\sqrt 7$. This further divides the critical line into three phases: the high temperatures $1<\nu<\nu_c$, the critical temperature $\nu=\nu_c$, and the low temperatures $\nu>\nu_c$.
\begin{figure}[h!]
\centering
\includegraphics[scale=1.2]{Fig2.pdf}
\caption{Phase diagram of the Boltzmann Ising-triangulation for $\nu>1$. The critical line $t_c(\nu)$ is divided by $\nu_c=1+2\sqrt7$ into the high temperature, low temperature and critical temperature phases. Although hardly visible in the graph, the third derivative of $t_c(\nu)$ has a discontinuity at $\nu=\nu_c$.}
\label{fig:tc_nu_plot}
\end{figure}
In our previous paper \cite{CT20}, we studied the model at the critical point $(\nu,t)=(\nu_c,t_c(\nu_c))$. Results in \cite{CT20} include an explicit parametrization of $Z(u,v;t_c(\nu_c),\nu_c)$, the asymptotics of $z_{p,q}(t_c(\nu_c),\nu_c)$ when $q\to \infty$ and then $p\to \infty$, the scaling limit of the main interface length and the local limit of the whole triangulation under that asymptotics. In this paper, we will extend this study to the critical line $t=t_c(\nu)$ in order to shed more light on the nature of the phase transition at $\nu=\nu_c$. For this reason we will write throughout this paper
\begin{equation*}
z_{p,q}(\nu)=z_{p,q}(t_c(\nu),\nu) \,, \qquad
Z_q(u;\nu)=Z_q(u;t_c(\nu),\nu)
\qtq{and}
Z(u,v;\nu)=Z(u,v;t_c(\nu),\nu) \,.
\end{equation*}
In \cite{CT20}, we have characterized $Z(u,v;t,\nu)$ as the solution of a functional equation, and solved it in the case of $(\nu,t)=(\nu_c,t_c(\nu))$. In this paper we solve the equation for general $(\nu,t)$ and give the solution in terms of a multivariate \RP:
\begin{theorem}[\RRP\ of $Z(u,v;t,\nu)$]\label{thm:Zparam}
For $\nu>1$, $Z(u,v;t,\nu)$ satisfies the parametric equation
\begin{equation}\label{eq:RP:Z General}
t^2 = \hat T(S,\nu), \qquad
t\cdot u = \hat U(H;S,\nu),\qquad
t\cdot v = \hat U(K;S,\nu) \qtq{and}
Z(u,v;t,\nu) = \hat Z(H,K;S,\nu) \,,
\end{equation}
where $\hat T$, $\hat U$ and $\hat Z$ are rational functions whose explicit expressions are given in Lemma~\ref{lem:RP Z0} and in \cite{CAS2}.
\end{theorem}
To specialize the above \RP\ of $Z(u,v;t,\nu)$ to the critical line $t=t_c(\nu)$, one needs to replace the parameter $S$ by its value $S_c(\nu)$ that parametrizes $t=t_c(\nu)$. It turns out that the function $S_c(\nu)$ itself has \RP s on $(1,\nu_c)$ and $(\nu_c,\infty)$, respectively. More precisely, $S_c(\nu)$ satisfies a parametric equation of the form
\begin{equation*}
\nu=\check \nu(R) \qtq{and} S_c(\nu)=\check S(R) \,,
\end{equation*}
where $\check \nu(R)$ and $\check S(R)$ are piecewise rational functions on the intervals $(R_1,R_c]$ and $[R_c,\infty)$, where the values $R_1,R_c,R_\infty$ correspond to $\nu=1,\nu_c,\infty$ in the sense that $\check \nu(R_1)=1$, $\check \nu(R_c)=\nu_c$ and $\check \nu(R_\infty)=\infty$. The expressions of $\check \nu(R)$, $\check S(R)$ and of $R_1,R_c,R_\infty$ are given in Section~\ref{sec:specialization of RP}. By making the substitution $\nu=\check \nu(R)$ and $S=\check S(R)$ in \eqref{eq:RP:Z General}, we obtain a piecewise \RP\ of $t_c(\nu)$ and $Z(u,v;\nu)$ of the form
\begin{equation*}
t_c(\nu)^2 = \check T(R), \qquad
t_c(\nu)\cdot u = \check U(H,R),\qquad
t_c(\nu)\cdot v = \check U(K,R),\qtq{and}
Z(u,v;\nu) = \check Z(H,K,R) .
\end{equation*}
See Section~\ref{sec:specialization of RP} for more details.
In \cite{CT20}, we computed the asymptotics of $z_{p,q}(t,\nu)$ when $(\nu,t)=(\nu_c,t_c(\nu_c))$ in the limit where $p\to \infty$ after $q\to \infty$. The following theorem extends this result to the whole critical line $t=t_c(\nu)$, and also to the limit where $p,q\to \infty$ at comparable speeds.
\begin{theorem}[Asymptotics of $z_{p,q}(\nu)$]\label{thm:asympt}
For any fixed $\nu>1$ and $0<\lambda_{\min}<\lambda_{\max}<\infty$, we have
\begin{align*}
u_c(\nu)^q \cdot z_{p,q}(\nu) & = \frac{a_p(\nu)}{\Gamma(-\alpha_0)} \cdot q^{-(\alpha_0+1)} + O\m({ q^{-(\alpha_0+1+\delta)} }
&&\text{as } q\to\infty \text{ for each fixed }p\ge 0. \\
u_c(\nu)^p \cdot~~\, a_p(\nu) & = \frac{b (\nu)}{\Gamma(-\alpha_1)} \cdot p^{-(\alpha_1+1)} + O\m({ p^{-(\alpha_1+1+\delta)} }
&&\text{as } p\to\infty. \\
u_c(\nu)^{p+q} \cdot z_{p,q}(\nu) & = \frac{b(\nu)\cdot c(q/p)}{\Gamma(-\alpha_0) \Gamma(-\alpha_1)} \cdot p^{-(\alpha_2+2)} + O\m({ p^{-(\alpha_2+2+\delta)} }
&&\text{as } p,q\to\infty \text{ while }q/p \in [\lambda_{\min}, \lambda_{\max}].
\end{align*}
where the exponents $\alpha_i$, $\delta$ and the scaling function $c(\lambda)$ only depend on the phase of the model, and are given by
\begin{equation*}
\begin{tabular}{|c|c|c|c|c|}
\hline & $\alpha_0$ & $\alpha_1$ & $\alpha_2$ & $\delta$ \\\hline
$\nu>\nu_c$ & $3/2$ & $3/2$ & $3$ & $1/2$ \\\hline
$\nu=\nu_c$ & $4/3$ & $1/3$ & $5/3$ & $1/3$ \\\hline
$\nu\in (1,\nu_c)$ & $3/2$ & $ -1$ & $1/2$ & $1/2$ \\\hline
\end{tabular}
\qquad
c(\lambda) = \begin{cases}
\lambda^{-5/2} & \text{when }\nu>\nu_c \\
\frac43 \int_0^\infty (1+r)^{-7/3}(\lambda+r)^{-7/3} \dd r & \text{when }\nu=\nu_c \\
(1+\lambda)^{-5/2} & \text{when }\nu\in (1,\nu_c)\,.
\end{cases}
\end{equation*}
On the other hand, $u_c(\nu)$, $a_p(\nu)$ (for $p\ge 0$) and $b(\nu)$ are analytic functions of $\nu$ on $(1,\nu_c)$ and $(\nu_c,\infty)$, respectively. And $u_c(\nu)$ is continuous at $\nu=\nu_c$. An explicit parametrization of $u_c(\nu)$ is given in Section~\ref{sec:dom of cvg}. Parametrizations of $b(\nu)$ and of the generating function $A(u;\nu) := \sum_{p} a_p(\nu) u^p$ are explained in Section~\ref{sec:local expansion} and given in \cite{CAS2}.
\end{theorem}
\begin{remark}
The exponents $\alpha_i$ and the scaling function $c(\lambda)$ satisfy a number of consistency relations.
First, one can exchange the roles of $p$ and $q$ in the last asymptotics of Theorem~\ref{thm:asympt}. Since we have $z_{p,q}=z_{q,p}$ for all $p,q$, this implies that $c(\lambda)\lambda^{\alpha_2+2}=c(\lambda^{-1})$ or, in a more symmetric form, $c(\lambda)\lambda^{(\alpha_2+2)/2}=c(\lambda^{-1}) \lambda^{-(\alpha_2+2)/2}$.
By replacing the factor $a_p(\nu)$ in the first asymptotics of Theorem~\ref{thm:asympt} with the dominant term in the second asymptotics, we obtain \emph{heuristically} that
\begin{equation*}
u_c(\nu)^{p+q} \cdot z_{p,q}(\nu) \sim \frac{b(\nu)\cdot (q/p)^{-(\alpha_0+1)}}{\Gamma(-\alpha_0) \Gamma(-\alpha_1)} \cdot p^{-(\alpha_0+\alpha_1+2)}
\qt{when } p,q\to\infty \text{ and }q\gg p \,.
\end{equation*}
This suggests that $\alpha_0+\alpha_1=\alpha_2$ and $c(\lambda)\sim \lambda^{-(\alpha_0+1)}$ when $\lambda \to \infty$. One can verify that both relations are indeed satisfied by $\alpha_i$ and $c(\lambda)$ in all three phases. Notice that thanks to the equation $c(\lambda)\lambda^{\alpha_2+2}=c(\lambda^{-1})$, the asymptotics $c(\lambda)\eqv\lambda \lambda^{-(\alpha_0+1)}$ is equivalent to $c(\lambda)\eqv[0]{\lambda}\lambda^{-(\alpha_1+1)}$.
\end{remark}
The local distance between bicolored triangulations (or actually any maps) is defined by
\begin{equation*}
d\1{loc}(\bt,\bt[']) = 2^{-R}\qtq{where}
R = \sup\Set{r\geq 0}{ \btsq_r=\btsq[']_r }
\end{equation*}
and $\btsq_r$ denotes the ball of radius $r$ around the origin in $\bt$ which takes into account the colors of the faces. The set $\bts$ of (finite) bicolored triangulations of a polygon is a metric space under $d\1{loc}$. Let $\overline{\bts}$ be its Cauchy completion. Recall that an (infinite) graph is \emph{one-ended} if the complement of any finite subgraph has exactly one infinite connected component. It is well known that a one-ended map has either zero or one face of infinite degree \cite{CurPeccot}. We call an element of $\overline{\bts}\setminus \bts$ a \emph{bicolored triangulation of the half plane} if it is one-ended and its external face has infinite degree. Such a triangulation has a proper embedding in the upper half plane without accumulation points and such that the boundary coincides with the real axis, hence the name. We denote by $\bts_\infty$ the set of all bicolored triangulations of the half plane.
Moreover, let $\bts_\infty^{(2)}$ be the set of bicolored triangulations which have an infinite boundary and exactly \emph{two ends} (as defined in \cite[14.2]{topograph}).
We extend the local convergence to arbitrary fixed temperature as follows.
\begin{theorem}[Local limits of Ising-triangulations]~\label{thm:cv}\\
For every $\nu>1,$ there exist probability distributions $\prob_p^\nu$ and $\prob_\infty^\nu$, such that
\begin{equation*}
\prob_{p,q}^\nu\ \cv[] q\ \prob_p^\nu\ \cv[] p\ \prob_\infty^\nu
\end{equation*}
locally in distribution. Moreover, $\prob_p^\nu$ is supported on $\bts_\infty$ for all $\nu>1$, whereas $\prob_\infty^\nu$ is supported on $\bts_\infty$ when $1<\nu\leq\nu_c$ and on $\bts_\infty^{(2)}$ when $\nu>\nu_c$. In addition, for $0<\lambda'\leq 1\leq\lambda<\infty$, we have
\begin{equation*}
\prob_{p,q}^\nu\ \cv[]{p,q}\ \prob_\infty^\nu \qt{when} \quad \frac{q}{p}\in [\lambda',\lambda]
\end{equation*}
locally in distribution.
\end{theorem}
This result partially confirms a conjecture in our previous paper \cite{CT20}, which basically stated that $\prob_{p,q}^{\nu_c} \to \prob_\infty^{\nu_c}$ locally in distribution whenever $p,q\to \infty$.
\paragraph{Peeling process and perimeter processes.}
Similarly as in \cite{CT20}, we will consider a peeling process that explores the main interface $\iroot$, which is imposed by the Dobrushin boundary condition. This exploration process reveals one triangle adjacent to $\iroot$ at each step, and swallows a finite number of other triangles if the revealed triangle separates the unexplored part in two pieces. Formally, we define the \emph{peeling process} as an increasing sequence of \emph{explored maps} $\nseq \emap$. The precise definition of $\emap_n$ will be left to Section~\ref{sec:peeling}.
The peeling process is also encoded by a sequence of \emph{peeling events} $\nseq[1] \Step$ taking values in a countable set of symbols, where $\Step_n$ indicates the position of the triangle revealed at time $n$ relative to the explored map $\emap_{n-1}$. Again, the detailed definition is left to Section~\ref{sec:peeling}.
The law of the sequence $\nseq[1] \Step$ can be written down fairly easily and one can perform explicit computations with it. We denote by $\Prob_{p,q}^\nu\equiv\Prob_{p,q}$ the law of the sequence $\nseq[1] \Step$ under $\prob_{p,q}^\nu$, where the $\nu$ is omitted when it is clear from the context.
We denote by $(P_n,Q_n)$ the boundary condition of the unexplored map at time $n$ and by $(X_n,Y_n)$ its variations, $X_n=P_n-P_0$ and $Y_n=Q_n-Q_0$. Now $(X_n,Y_n)$ is actually a deterministic function of the peeling events $(\Step_k)_{1\le k\le n}$ with a well-defined limit when $p,q\to\infty$. This allows us to define the law of the process $\nseq{X_n,Y}$ under $\Prob\yy=\lim_{p,q\to\infty}\Prob_{p,q}$ despite the fact that $P_n=Q_n=\infty$ almost surely in the limit. Then, under $\Prob\yy$, the process $(X_n,Y_n)_{n\ge 0}$ is a two-dimensional random walk. It was proven in \cite{CT20} for the corresponding expectations of the increments that
\begin{equation}\label{eq:mu}
\EE\yy(X_1)=\EE\yy(Y_1)=\mu:=\frac{1}{4\sqrt 7}>0\qquad \text{when}\quad \nu=\nu_c,
\end{equation}
yielding the interface drifting towards the infinity. Considering the temperature $\nu$ as a variable, this drift actually defines an order parameter, as the following proposition states:
\begin{proposition}[Order parameter]\label{prop:orderparam intro}
Let $\mathcal{O}(\nu):=\EE_\infty^\nu((X_1+Y_1)\id_{|X_1|\vee|Y_1|<\infty})$. Then
\begin{equation*}
\mathcal{O}(\nu)=\begin{cases}
0,\qquad\text{if}\quad 1<\nu<\nu_c \\
f(\nu)\qquad\text{if}\quad \nu\geq\nu_c,
\end{cases}
\end{equation*}
where $f:[\nu_c,\infty)\to\R$ is a continuous, strictly increasing function such that $f(\nu_c)=2\mu>0$ and $\lim_{\nu\nearrow\infty}f(\nu)<\infty$ exists. Moreover, for $1<\nu<\nu_c$, we have the drift condition $\EE_\infty^\nu(X_1)=-\EE_\infty^\nu(Y_1)>0$.
\end{proposition}
The proposition \ref{prop:orderparam intro} has the following geometric interpretation: For $\nu\in (1,\nu_c)$, the peeling process starting from the \< edge on the left of the root $\rho$ drifts to the left. This process also follows the left-most interface starting from $\rho$. This essentially tells that the left-most interface in the local limit stays near the infinite \< boundary segment, hitting it almost surely infinitely many times. Similarly, the right-most interface starting from the right of $\rho$ drifts to the right following the \+ boundary. Since $\EE_\infty(X_1)+\EE_\infty(Y_1)=0$, these two interfaces have the same geometry up to reflection. Thus, iterating the peeling process consistently reveals that the local limit has a percolation-like interface geometry. For $\nu\in [\nu_c,\nu)$, the peeling process explores an interface which drifts to infinity and hits the boundary only finitely many times. The fact that this drift is increasing in $\nu$ means that the lower the temperature is, the faster the interface tends to the infinity. In fact, it is also shown that if $\nu\in (\nu_c,\infty)$, the peeling process approaches a neighborhood of the infinity in a finite time almost surely.
Let $T_m:=\inf\{n\ge 0: \min\{P_n,Q_n\}\leq m\},$ which can be seen as the first hitting time of the interface in a neighborhood of the infinity. For $\nu=\nu_c$, we find an explicit scaling limit of $T_m$ under the diagonal rescaling:
\begin{theorem}\label{thm:scaling}
Let $\nu=\nu_c$. For all $m\in\natural$, the jump time $T_m$ has the following scaling limit:
\begin{equation}\label{eq:Tm scaling}
\forall t>0\,,\qquad \lim_{p,q\to\infty} \Prob_{p,q}\m({\mu T_m>tp} = \int_t^\infty(1+s)^{-7/3}(\lambda+s)^{-7/3}ds
\end{equation}where the limit is taken such that $q/p\to\lambda\in (0,\infty)$. In particular, for $\lambda=1$, \begin{equation*}
\lim_{p,q\to\infty} \Prob_{p,q}\m({T_m>tp} =(1+\mu t)^{-11/3}.
\end{equation*}
\end{theorem}
Since the hitting time $T_m$ is roughly the interface length of a finite Boltzmann Ising-triangulation, we land to the following conjecture:
\begin{conjecture}[Scaling limit of the interface length]
Let $\eta_{p,q}$ be the length of the leftmost interface in $\bt$ under $\prob\pqy$. Then
\begin{equation*}
\prob_{p,q}(\eta_{p,q}>tp/\mu)\cv[]{p,q}\int_{t/E}^\infty(1+s)^{-7/3}(\lambda+s)^{-7/3}ds\qquad\text{while}\quad \frac{q}{p}\to\lambda,
\end{equation*}
where $E$ is the expectation of a random variable under $\prob_\infty$ taking values in $\{0,1,\eta_{k,1},1+\eta_{k,1} : k=0,1,\dots\}$ and given by the contribution of a peeling step in the total interface length.
\end{conjecture}
The idea behind the above conjecture is explained in \cite[Section 6]{CT20} in a similar setting. The main obstacle of the proof for the above conjecture is the fact that the expected value of $\eta_{k,1}$ is hard to estimate. One could find an asymptotic estimate for the volume of a finite Boltzmann Ising-triangulation, but it turns out not to be sufficient. However, the analog of the conjecture could be proven for the model with spins on vertices, or with spins on faces and a general boundary. The former is conducted in \cite{T20}, in which setting $E=1$. The conjecture is also supported by a prediction derived from the \emph{Liouville Quantum Gravity}, seen as a continuum model of \emph{quantum surfaces} studied eg. in \cite{matingoftrees}, which also inspired us to find the correct constant in the scaling limit of Theorem~\ref{thm:scaling}. More discussion about this is given in Section~\ref{sec:onejumpscaling}.
Recall from \cite{CT20} that at the critical temperature, due to the positive drift $\mu>0$, we observe a different interface geometry than in the case of percolation on the UIHPT (see \cite{Ang02} as well as \cite{Ang05,AC13}), where the percolation interface hits the boundary almost surely an infinite number of times.
In this work, we show that in the high temperature
phase ($1<\nu<\nu_c$), the Ising model behaves in the local limit like subcritical face percolation, whereas in low temperatures ($\nu>\nu_c$), the local limit contains a bottleneck separating the \+ and \< regions with positive probability. In particular, in the latter case the local limit is not almost surely one-ended. This property could be viewed as a quantum
version of the 2D Ising model in the ferromagnetic low-temperature regime, corresponding the model on regular
lattices with two "frozen" regions. Both of the cases are predicted in physics literature, though not extensively studied (see \cite{Kaz86,ADJ97}). More about the geometric interpretations is found in Section~\ref{sec:orderparam}.
To understand the phase transition at the critical point in greater detail, one should also consider the so-called near-critical regime. In our context, this means
that we let the temperature tend to the critical one at some rate as the boundary size tends to infinity. Intuitively, one would expect that if $\nu\to\nu_c$ fast along with the perimeter, the geometry corresponds to the model at the critical point, whereas if the convergence is slow, the model lies in the pure gravity universality class. An interesting question is to determine whether this phase transition is "sharp" or contains some intermediate regime, which also exhibits a different geometric behavior. These problems are considered in an upcoming work.
\paragraph{Outline.}
The paper is composed of two parts, which can be read independently of each other.
The first part, which spans Section~\ref{sec:RP of GF}--\ref{sec:coeff asymp}, deals with the enumeration of Ising-decorated triangulations. We start by deriving explicit \RP s of the generating function $Z(u,v;t,\nu)$ and its specialization $Z(u,v;\nu)\equiv Z(u,v;t_c(\nu),\nu)$ on the critical line (Section~\ref{sec:RP of GF}). Using these \RP s, we show that for each $\nu>1$, the bivariate generating function $Z(u,v;\nu)$ has a unique dominant singularity and an analytic continuation on the product of two $\Delta$-domains (Section~\ref{sec:singularity structure}). We then compute the asymptotic expansion of $Z(u,v;\nu)$ at its unique dominant singularity (Section~\ref{sec:local expansion}). Finally, we prove the coefficient asymptotics in Theorem~\ref{thm:asympt} using a generalization of the classical transfer theorem based on double Cauchy integrals (Section~\ref{sec:coeff asymp}).
The second part, which comprises Section~\ref{sec:peeling perim}--\ref{sec:locallimit c diag} and Appendix~\ref{sec:bigjumplemma proof}, tackles the probabilistic analysis of the Ising-triangulations at any fixed temperature $\nu\in(1,\infty)$. It uses the combinatorial results of the first part as an input, and leads to the proofs of Theorems~\ref{thm:cv} and \ref{thm:scaling}. First, we introduce the different versions of the peeling process, which are chosen depending on the temperature $\nu$ and the limit regime. We also study the associated perimeter processes, whose drifts in the limit $p,q\to\infty$ define an order parameter which describes the geometry of the interface in the local limit (Section~\ref{sec:peeling perim}). After that, we provide a general framework for constructing some local limits, which we use to show the local convergence results for $\nu\in(1,\nu_c)$ and $\nu\in(\nu_c,\infty)$, respectively (Section~\ref{sec:locallimits}). Finally, we strengthen the local convergence at $\nu=\nu_c$ (\cite{CT20}) to a regime where $p,q\to\infty$ simultaneously at a comparable rate, as well as prove Theorem~\ref{thm:scaling} (Section~\ref{sec:locallimit c diag}). A central tool for the proofs in this last section is an adaptation of the proof of the one-jump lemma from \cite[Appendix B]{CT20} to the diagonal convergence of the perimeter processes (Appendix~\ref{sec:bigjumplemma proof}).
\paragraph{Acknowledgements.}
The authors would like to thank M. Albenque, J. Bouttier, S. Charbonnier, K. Izyurov, M. Khristoforov, A. Kupiainen and L. Ménard for enlightening discussions. Both authors have been primarily supported by the Academy of Finland via the Centre of Excellence in Analysis and Dynamics Research (project No. 271983), as well as the ERC Advanced Grant 741487 (QFPROBA). The second author also acknowledges support from the Icelandic Research Fund (Grant Number: 185233-051) as well as from the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program « Investissements d’Avenir » (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).
\section{Rational parametrizations of the generating functions.}\label{sec:RP of GF}
The functional equations satisfied by the generating functions $Z_0(u;t,\nu)$ and $Z(u,v;t,\nu)$ were derived in our previous work \cite{CT20}. The result were written in the form of
\begin{equation}\label{eq:E0 and E}
\mc E_0 \mb({ Z_0(u) ,u; t,\nu, z_{1,0}, z_{3,0} } = 0
\qtq{and}
Z(u,v;t,\nu) = \mc E \mb({ Z_0(u), Z_0(v), u,v; t,\nu, z_{1,0}, z_{3,0} }
\end{equation}
where $\mc E_0$ and $\mc E$ are explicit rational functions with coefficients in $\rational$. Let us briefly summarize their derivation:
\begin{enumerate}
\item
We start by expressing the fact that the probabilities of all peeling steps sum to one. This gives two equations (called loop equations or Tutte's equations) with two catalytic variables for $Z(u,v;t,\nu)$. These equations are linear in $Z(u,v;t,\nu)$.
\item
By extracting the coefficients of $[v^0]$ and of $[v^1]$ from these two equations, we obtain four algebraic equations relating the variable $u$ to the series $Z_p(u;t,\nu)$ for $p=0,1,2,3$, whose coefficients are polynomials in $t$, $\nu$ and $z_{1,0}(t,\nu)$, $z_{3,0}(t,\nu)$. These equations are linear in the three variable $Z_1$, $Z_2$ and $Z_3$. After eliminating these variables, we obtain the first equation of \eqref{eq:E0 and E}.
This procedure is essentially equivalent to the method used in \cite[Chapter~8]{EynardBook} to solve Ising model on more general maps.
\item
Using the four algebraic equations found in Step 2, one can also express $Z_1(u;t,\nu)$ as a raitonal function of $Z_0(u;t,\nu)$, $u$, $t$, $\nu$ and $z_{1,0}(t,\nu)$, $z_{3,0}(t,\nu)$. Then, plug this relation into one of the two loop equations, and we obtain the second equation of \eqref{eq:E0 and E}.
\end{enumerate}
In this section, we first solve the equation for $Z_0(u;t,\nu)$ with the help of known \RP s of $z_{1,0}(t,\nu)$ and $z_{3,0}(t,\nu)$. Then, the solution is propagated to $Z(u,v;t,\nu)$ using its rational expression in $Z_0(u;t,\nu)$ and its coefficients. Finally, we specialize the parametrization of $Z(u,v;t,\nu)$ to the critical line $t=t_c(\nu)$ by replacing two parameters ($S,\nu$) with a single parameter $R$.
\subsection{\RRP\ of $Z_0(u;t,\nu)$}
\begin{lemma}\label{lem:RP Z0}
$Z_0(u;t,\nu)$ has the following \RP:
\begin{align}
t^2 &= \hat T(S,\nu) \hspace{5.5mm} :=
\frac{ (S-\nu) (S+\nu-2) (4S^3-S^2-2S+\nu^2-2\nu) }{ 32 (1-\nu^2)^3 S^2 }
\label{eq:RP:T General}\\
tu &= \hat U(H;S,\nu) \, := \ H\cdot
\frac{2(4S^3-S^2-2S +\nu^2-2\nu) -4(S+1)S^2 H +4S^2 H^2 -S H^3}{16(1-\nu^2)^2 S}
\label{eq:RP:U General} \\
Z_0(u;t,\nu) &= \hat Z_0(H;S,\nu) := \ \frac{\hat U(H;S,\nu)}{\hat T(S,\nu)} \cdot
\frac{ (S-\nu)(S+\nu-2) + 2(S-\nu)S H - 2S^2 H^2 + S H^3 }{ 4(1-\nu^2)S H } \ .
\label{eq:RP:Z0 General}
\end{align}
\end{lemma}
\begin{proof}
The following \RP s of $z_{1,0}(t,\nu)$ and $z_{3,0}(t,\nu)$ were obtained in \cite{CT20} by translating a related result from \cite{BBM11}: $t^2 = \hat T(S,\nu)$ and
\begin{align}
t^3 \cdot z_{1,0}(t,\nu) =&\ \hat z_{1,0}(S,\nu) :=
\frac{ (\nu-S)^2 (S+\nu-2) }{ 64 (\nu^2-1)^4 S^2 }
(3S^3 -\nu S^2 -\nu S +\nu^2 -2\nu) \,,
\label{eq:RP:z1 General}\\
t^9 \cdot z_{3,0}(t,\nu) =&\ \hat z_{3,0}(S,\nu) :=
\frac{ (\nu-S)^5 (S+\nu-2)^5}{2^{22} (\nu^2-1)^{12} S^8}
\cdot \big(\, 160S^{10} -128S^9 -16(2\nu^2-4\nu+3)S^8
\label{eq:RP:z3 General}
\\& +\ 32(2\nu^2-4\nu+3)S^7 - 7(16\nu^2-32\nu+27)S^6 - 2(32\nu^2-64\nu+57)S^5 \notag
\\& +\ (32\nu^4-128\nu^3+183\nu^2-110\nu+20)S^4 - 4(7\nu^2-14\nu-2)S^3 \notag
\\& +\ \nu(\nu-2)(9\nu^2-18\nu-20)S^2 + 14\nu^2(\nu-2)^2 S - 3 \nu^3 (\nu-2)^3 \notag
\,\big) \,.
\end{align}
Substituting $u$ by $U/t$, and then $t$, $z_{1,0}$, $z_{3,0}$ by their respective parametrizations in the first equation of \eqref{eq:E0 and E}, we obtain an algebraic equation of the form $\hat{\mc E}_0(Z_0,U;S,\nu) = 0$.
It is straightforward to check that \eqref{eq:RP:U General}--\eqref{eq:RP:Z0 General} cancel the equation, that is, $\hat{\mc E}_0(\hat Z_0(H;S,\nu), \hat U(H;S,\nu); S,\nu)=0$ for all $H$, $S$ and $\nu$. See \cite{CAS2} for the explicit computation.
On the other hand, we know that \eqref{eq:E0 and E} uniquely determines the formal power series $Z_0(u)$, see \cite[Section~3.1]{CT20}. When $H\to 0$, Equations~\eqref{eq:RP:U General}--\eqref{eq:RP:Z0 General} clearly parametrize an analytic function $Z_0(u)$ near $u=0$. Therefore they are indeed a \RP\ of $Z_0(u;t,\nu)$.
\end{proof}
\begin{remark}
The proof of Lemma~\ref{lem:RP Z0} followed a guess-and-check approach. To actully derive the parametrization \eqref{eq:RP:U General}--\eqref{eq:RP:Z0 General}, we first check that the plane curve defined by $\hat{\mc E}_0(Z_0,U;S,\nu)=0$ has zero genus using the command \code{algcurves[genus]} of Maple, so it does have a \RP\ with coefficients in $\rational(S,\nu)$.
Theoretically, one should be able to produce one such parametrization using the Maple command \code{algcurves[parametrization]}. However, the execution takes too much time, presumably due to the presence of two free parameters $(S,\nu)$ in the coefficient ring. Instead, we followed the steps below to find \eqref{eq:RP:U General}--\eqref{eq:RP:Z0 General}:
\begin{enumerate}
\item
Choose a set of values $\mc Q \subset \rational \cap (1,\infty)$ for $\nu$. In practice we used the integers $\mc Q=\{2,3,\ldots,20\}$.
\item
For each $\nu \in \mc Q$, use \code{algcurves[parametrization]} to find a \RP\ for the solution of the equation $\hat{\mc E}_0(Z_0;U;S,\nu)$ in the form of $U=\tilde U_\nu(\tilde H;S)$ and $Z_0=\tilde Z_{0,\nu}(\tilde H;S)$.
\item
Check that for each $\nu \in \mc Q$, there is rational function $\tilde H_{\infty,\nu}(S)$ such that $\tilde H=\tilde H_{\infty,\nu}(S)$ is the unique pole of both $\tilde U_\nu$ and $\tilde Z_{0,\nu}$, and $\tilde H=\infty$ is the unique point on the Riemann sphere such that $\tilde U_\nu(\tilde H)=0$ and $\tilde Z_{0,\nu}(\tilde H)=1$.
We define the renormalized \RP\ by $\hat U_\nu(H;S) = \tilde U_\nu(\tilde H;S)$ and $\hat Z_{0,\nu}(H;S) = \tilde Z_{0,\nu}(\tilde H;S)$, where $H$ is related to $\tilde H$ by the M\"obius transform $H = \Lambda_\nu(S) /(\tilde H_{\infty,\nu}(S)-\tilde H)$, and the scaling factor $\Lambda_\nu(S)$ is to be chosen later.
\item
This M\"obius transform maps $\tilde H = \tilde H_{\infty,\nu}(S)$ to $H=\infty$, and $\tilde H=\infty$ to $H=0$. Since the unique pole of $\tilde U_\nu$ and $\tilde Z_{0,\nu}$ is mapped to $\infty$, both $\hat U_\nu$ and $\hat Z_{0,\nu}$ are polynomials of $H$. We choose $\Lambda_\nu(S)$ heuristically to make their coefficients look as simple as possible (in particular, lower their degrees as rational functions of $S$).
\item
Compute the rational function $\hat U$ (resp.\ $\hat Z_0$) of $\nu$ of lowest degree (defined as the maximum of the degrees of the numerator and the denominator) with coefficients in $\rational(S)[H]$ that interpolates between the functions $(\hat U_\nu)_{\nu \in \mc Q}$ (resp.\ $(\hat Z_{0,\nu})_{\nu \in \mc Q}$). Increase the size of the set $\mc Q$ until the interpolation results $\hat U$ and $\hat Z_0$ stablize.
This gives the parametrization \eqref{eq:RP:U General}--\eqref{eq:RP:Z0 General}.
\end{enumerate}
\end{remark}
\subsection{\RRP\ of $Z(u,v;t,\nu)$.}
We plug the parametrizations \eqref{eq:RP:T General}--\eqref{eq:RP:z3 General} into the second equation of \eqref{eq:E0 and E} to obtain a \RP\ of $Z(u,v;t,\nu)$ of the form
\begin{equation*}
t^2=\hat T(S,\nu) \qquad
tu = \hat U(H;S,\nu) \qquad
tv = \hat U(K;S,\nu) \qtq{and}
Z(u,v;t,\nu) = \hat Z(H,K;S,\nu) \,,
\end{equation*}
where the rational functions $\hat T$ and $\hat U$ are defined in Lemma~\ref{lem:RP Z0}, and the expression of $\hat Z$ is given in \cite{CAS2}.
\subsection{Specialization of $Z(u,v;t,\nu)$ to the critical line $t=t_c(\nu)$.}\label{sec:specialization of RP}
\paragraph{\RRP\ of the critical line.}
Recall that $t_c(\nu)$ is defined as the radius of convergence of the series $z_{1,0}(\adot,\nu)$. The series have nonnegative coefficients, and have a real rational parametrization of the form $t^2=\hat T(S,\nu)$ and $t^3\cdot z_{1,0} = \hat z_{1,0}(S,\nu)$ given by \eqref{eq:RP:T General} and \eqref{eq:RP:z1 General}. As explained in \cite[Appendix~B]{CT20}, the value $S=S_c(\nu)$ that parametrizes the point $t=t_c(\nu)$ is either a zero of $\partial_S \hat T(\adot,\nu)$ or a pole of $\hat z_{1,0}(\adot,\nu)$. More precise calculation (see \cite{CAS2}) using the method of \cite[Appendix~B]{CT20} shows that $S_c(\nu)$ is the largest zero of $\partial_S \hat T(\adot,\nu)$ below $S=\nu$ (which parametrizes $t=0$). The equation $\partial_S \hat T(S,\nu)=0$ factorizes, and $S_c(\nu)$ satisfies
\begin{align}
2 S^3 -3 S^2 - \nu^2 +2 \nu = 0 & \qt{if }\nu \in (1,\nu_c] \,,
\label{eq:def S_c high} \\
3 S^2 -\nu^2 +2 \nu = 0 & \qt{if }\nu \in [\nu_c,\infty) \,,
\label{eq:def S_c low}
\end{align}
where $\nu_c = 1+2\sqrt7$.
It is not hard to check that $S_c(\nu)$ has the following piecewise \RP:
\begin{equation}\label{eq:RP:S_c}
\nu = \check \nu(R) = \begin{cases}
\frac{1}{2}(2-3R+R^3) & \\
\frac{27}{13+2R-2R^2} &
\end{cases}
\quad\tq{and}
S_c(\nu) = \check S(R) = \begin{cases}
\frac{1}{2}(R^2-1) & \qt{for }R \in (R_1,R_c] \\
\frac{3(2R-1)}{13+2R-2R^2} & \qt{for }R \in [R_c,R_\infty)
\end{cases}
\end{equation}
where $R_1=\sqrt3$, $R_c=\sqrt7$, $R_\infty=\frac{1+3\sqrt3}2$ correspond respectively to the coupling constants $\nu=1$, $\nu=\nu_c$ and $\nu=\infty$.
Plugging \eqref{eq:RP:S_c} into $\hat T(S,\nu)$ gives the following piecewise \RP\ of $t_c(\nu)$:
\begin{equation*}
t_c(\nu)^2 = \check T(R) := \begin{cases}
\displaystyle
\frac{3R^2-1}{2 R^3 (4-3R+R^3)^3} & \text{for }R \in (R_1,R_c]
\vspace{1.5ex} \\
\displaystyle
\frac{ (1+R)^2 (13+2R-2R^2)^3 (19-10R-2R^2)
}{ 128 (R-5) (4+R)^3 (7-R+R^2)^3
} & \text{for }R \in [R_c,R_\infty)
\end{cases}
\end{equation*}
\paragraph{\RRP\ of $Z(u,v;t,\nu)$ on the critical line.}
Define $\check U(H;R) = \hat U(H;\hat S(R),\hat \nu(R))$ and $\check Z(H,K;R) = \hat Z(H,K;\hat S(R),\hat \nu(R))$.
Then $Z(u,v;\nu) \equiv Z(u,v;t_c(\nu),\nu)$ has the piecewise \RP:
\begin{equation*}
t_c(\nu) \cdot u = \check U(H;R) \qquad t_c(\nu) \cdot v = \check U(K;R)
\qtq{and} Z(u,v;\nu) = \check Z(H,K;R) \ ,
\end{equation*}
where
\begin{equation}\label{eq:RP:U}
\check U(H;R) := \begin{cases}
\displaystyle
\frac{\mn({3-10 R^2+3 R^4}+\mn({1-R^4}H-2 \mn({1-R^2}H^2 -H^3
}{ R^2 \m({3-R^2}^2 \m({4-3 R+R^3}^2
} H
\hspace{23mm} \qt{for }R \in (R_1,R_c]
\vspace{1.5ex} \\
\displaystyle
-\frac{ (13+2R-2R^2)^2H
}{ 256 (5-R)^2 (4+R)^2 (7-R+R^2)^2}
\bigg( 8(1+R)(5-R) \m({ 19-10R-2R^2-3(1-2R)H }
\vspace{0.5ex} \\
\displaystyle ~\hspace{15.5mm}
+12(1-2R)\m({ 13+2R-2R^2 } H^2 + \m({ 13+2R-2R^2 }^2 H^3 \bigg)
\qt{for }R \in [R_c,R_\infty)
\end{cases}
\end{equation}
whereas $\check Z(H,K;R)$, too long to be written down here, is given in \cite{CAS2}. Since we look for the asymptotics of $z_{p,q}(\nu)$ when $p,q\to\infty$ with fixed values of $\nu$, we will be interested in the singularity behavior of $\check U(H;R)$ and $\check Z(H,K;R)$ at fixed values of $R$. For this reason we introduce the shorthand notations
\begin{equation*}
\check U_R(H) := \check U(H;R)
\qtq{and}
\check Z_R(H,K) := \check Z(H,K;R) \,.
\end{equation*}
\subsection{Domain of convergence of $Z(u,v;\nu)$ and its parametrization.}\label{sec:dom of cvg}
\paragraph{Definition and parametrization of $u_c(\nu)$.}
For all $R\in (R_1,R_\infty)$, let $\check H_c(R)$ be the smallest positive zero of the derivative $\check U_R'$. Using the expression \eqref{eq:RP:U}, it is not hard to find that
\begin{equation}\label{eq:RP:H_c}
\check H_c(R) := \begin{cases}
\displaystyle
\frac{R^2-3}2
& \text{for }R \in (R_1,R_c]
\vspace{1.5ex} \\
\displaystyle
\frac{5+4 R-R^2- \sqrt{3(5-R) (1+R) \left(R^2-7\right)}}{13+2 R-2 R^2}
& \text{for }R \in [R_c,R_\infty)
\end{cases}
\end{equation}
For $\nu>1$, let $u_c(\nu)$ be the function parametrized by $\nu=\check \nu(R)$ and $t_c(\nu)\cdot u_c(\nu) = \check U_R(\check H_c(R))$, where $R\in (R_1,R_\infty)$.
\begin{lemma}\label{lem:dom of cvg}
For all $\nu>1$, the double power series $(u,v)\mapsto Z(u,v;\nu)$ is absolutely convergent \Iff\ $|u|\le u_c(\nu)$ and $|v|\le u_c(\nu)$.
\end{lemma}
\begin{proof}
First, we notice that the proof can be reduced to the problem of estimating the radii of convergence of two univariate power series: it suffices to show that the series $u\mapsto Z(u,0;\nu) \equiv Z(0,u;\nu)$ is divergent when $|u|>u_c(\nu)$, and the series $u\mapsto Z(u,u;\nu)$ is convergent at $u=u_c(\nu)$.
Indeed, since the double power series $Z(u,v;\nu)$ has nonnegative coefficients, the divergence condition implies that $Z(u,v;\nu)$ is divergent when $|u|>u_c(\nu)$ or $|v|>u_c(\nu)$, and the convergence condition implies that $Z(u,v;\nu)$ is absolutely convergent for all $|u|\le u_c(\nu)$ and $|v|\le u_c(\nu)$.
The univariate series $u\mapsto Z(u,0;\nu)$ has nonnegative coefficients and the following \RP:
\begin{equation*}
t_c(\nu) \cdot u = \check U_R(H) \qtq{and}
Z(u,0;\nu) = \check Z_R(H,0) \,.
\end{equation*}
It is not hard to check that this \RP s are real and proper (see \cite[Appendix~B]{CT20} for the definitions and characterizations of these properties), and the parametrization $t_c(\nu)\cdot u = \check U_R(H)$ maps a small interval around $H=0$ increasingly to an interval around $u=0$. Hence the parametrization of the radius of convergence of $u\mapsto Z(u,0;\nu)$ can be determined in the framework of \cite[Proposition~21]{CT20}. More precisely, the radius of convergence $u_c^*(\nu)$ should satisfy $t_c(\nu) u_c^*(\nu) = \check U_R(\check H_c^*(R))$, where $\check H_c^*(R)$ is the smallest positive number that is either a zero of $\check U_R'$, or a pole of $H\mapsto \check Z_R(H,0)$. Comparing this to the definition of $\check H_c(R)$, we see that $\check H_c^*(R) \le \check H_c(R)$, and hence $u_c^*(\nu)\le u_c(\nu)$.\footnote{
Using its explicit expression, one can check that $H\mapsto \check Z_R(H,0)$ has no pole on $[0,\check H_c(R)]$. Hence $\check H_c^*(R) = \check H_c(R)$ and $u_c^*(\nu) = u_c(\nu)$. But this is not necessary for the proof.}
This shows that $u\mapsto Z(u,0;\nu)$ is divergent when $|u|>u_c(\nu)$.
We apply the same argument to the series $u\mapsto Z(u,u;\nu)$, which has the \RP
\begin{equation*}
t_c(\nu) \cdot u = \check U_R(H) \qtq{and}
Z(u,u;\nu) = \check Z_R(H,H) \,.
\end{equation*}
Again, the \RP\ is real and proper. Using its explicit expression, one can check that the rational function $H\mapsto \check Z_R(H,H)$ has no pole on $[0,\check H_c(R)]$. With the same argument as for $u\mapsto Z(u,0;\nu)$, we conclude that $u_c(\nu)$ is the radius of convergence of $u\mapsto Z(u,u;\nu)$ and the series is convergent at $u=u_c(\nu)$ (because $Z(u_c(\nu),u_c(\nu);\nu) = \check Z_R(\check H_c(R),\check H_c(R))$ is finite).
This concludes the proof of the lemma.
The necessary explicit computations in the above proof can be found in \cite{CAS2}.
\end{proof}
\paragraph{Notations:} In the following, we will use the renormalized variables $(x,y)= \m({ \frac u{u_c(\nu)}, \frac v{u_c(\nu)} }$.
A parametrization of the function $(x,y)\mapsto Z(u_cx,u_cy;\nu)$ is given by $x=\check x(H;R)$, $y=\check x(K;R)$ and $\tilde Z(x,y;\nu) = \check Z(H,K;R)$, where $\check x(H;R) \equiv \check x_R(H) := \check U_R(H) /\, \check U_R(\check H_c(R))$ is still a rational function in $H$. In the low temperature regime $\check x(H;R)$ is no longer rational in $R$ due to the square root in \eqref{eq:RP:H_c}. However it remains continuous on $(R_1,R_\infty)$ and smooth away from $R_c$. These regularity properties will be more than sufficient for our purposes.
\paragraph{Definition of holomorphicity and conformal bijections:}
We say that a function is \emph{holomorphic} in a (not necessarily open) domain if it is holomorphic in the interior of the domain and continuous in the whole domain.
This definition is also valid for functions of several complex variables, in which case \emph{holomorphic} means that the function has a multivariate Taylor expansion that is locally convergent. A \emph{conformal bijection} is a bijection which is holomorphic and whose inverse is also holomorphic.
\paragraph{Definition of $\Hdom[0](R)$:}
By \cite[Proposition~21]{CT20}, for each $R\in (R_1,R_\infty)$, the mapping $\check x_R$ induces a conformal bijection
from a compact neighborhood of $H=0$ to the closed unit disk $\cdisk$. We denote by $\cHdom[0](R)$ this neighborhood and by $\Hdom[0](R)$ its interior.
It is not hard to see that $\Hdom[0](R)$ is the connected component of the preimage $\check x_R^{-1}(\disk)$ which contains the origin. This characterization of $\Hdom[0](R)$ will be used in the proof of Lemma~\ref{lem:unique dominant}. Notice that it implies in particular that $\Hdom[0](R)$ is symmetric \wrt\ the real axis.
\section{Dominant singularity structure of $Z(u,v;\nu)$}\label{sec:singularity structure}
In this section, we prove that the bivariate generating function $(x,y)\mapsto Z(u_c(\nu)x,u_c(\nu)y,\nu)$ has a unique dominant singularity at $(x,y)=(1,1)$, and is ``$\Delta$-analytic'' in a sense similar to the one defined in \cite{FS09} for univariate generating functions. Before starting, let us briefly describe the state of the art for the singularity analysis of algebraic generating functions of one or two variables.
For a generating function $F(z)=\sum_{n\ge 0} F_n z^n$ of one complex variable, a dominant singularity of $F$ is by definition a singularity with minimal modulus. Moreover, this minimal modulus is equal to the radius of convergence $\rho$ of the Taylor series $\sum_{n} F_n z^n$, so the dominant singularities of $F$ are simply those on the circle $\Set{z\in \complex}{|z|=\rho}$.
When $F$ is algebraic, it behaves locally near a singularity $z_*$ like $(z-z_*)^{r}$ with some $r\in \rational$. In particular, one can find a disk centered at $z_*$ such that (a branch of) $F$ is analytic in the disk with one ray from $z_*$ to $\infty$ removed.
Since algebraic functions have only finitely many singularities, it follows that any univariate algebraic function $F(z)$ with finite radius of convergence has an analytic continuation in a domain of the form $\bigcap_{i} \m({z_i\cdot \slit}$, where $z_i$ are the dominant singularities of $F$, and $\slit$ is the disk of radius $1+\epsilon>1$ centered at $0$, with the segment $[1,1+\epsilon]$ removed. This ensures that the classical transfer theorem (see \cite[Chapter VI.3]{FS09}) always applies to algebraic functions, and gives coefficient asymptotics of the form $F_n \sim \sum_i c_i\cdot z_i^{-n} \cdot n^{-r_i}$ with $c_i\in \complex$ and $r_i\in \rational$. In particular, when the dominant singularity is unique, the asymptotics has the simple form of $F_n \sim c\cdot z_*^{-n} \cdot n^r$.
When $F(x,y) = \sum_{m,n} F_{m,n} x^m y^n$ is an algebraic function of two complex variables, the situation is much more complicated. First, the singularities of $F(x,y)$ are in general no longer isolated points. Also, the definition of dominant singularities has to be generalized: instead of minimizing $|z|$ in the univariate case, one needs to minimize the product $|x|^\lambda |y|$, where $\lambda=\lim \frac mn$ is defined by the regime of $m,n\to \infty$ in which one looks for the asymptotic of $F_{m,n}$.
The general picture for the singularity analysis of bivariate algebraic functions is still far from being fully understood. The only systematic study we found in the literature concerns the case where $F(x,y)$ is rational or meromorphic. See \cite{ACSVhome} for references. (A non-rational case has also been studied in \cite{Greenwood18}. But it concerns functions of a special form, and does not cover the case we are interested in here.)
When $F(x,y)$ is rational (or of the form studied in \cite{Greenwood18}), the locus of singularities of $F$ is an algebraic sub-variety of $\complex^2$. In that case, sophisticated tools from algebraic geometry can be used to locate the dominant singularities, and to study $F(x,y)$ locally near the dominant singularities.
For the Ising-triangulations, the singularity locus of the generating function $(x,y)\mapsto Z(u_c x,u_c y,\nu)$ is much harder to describe, since it involves describing branch cuts of the function in $\complex^2$. Luckily, the structure of dominant singularities is very simple: regardless of the relative speed at which $p,q\to \infty$, the dominant singularity is always unique and at $(x,y)=(1,1)$. Moreover, the function has an analytic continuation ``beyond the dominant singularity'' in both the $x$ and $y$ coordinates, in the product of two $\Delta$-domains. Proposition~\ref{prop:singularity structure} gives the precise formulation of the above claim.
\paragraph{Notations.}
We denote by $\disk$ the open unit disk in $\complex$ and by $\mathrm{arg}(z) \in (-\pi,\pi]$ the argument of $z \in \complex$. For $\epsilon>0$ and $0\le \theta<\pi/2$, define the $\Delta$-domain
\begin{equation*}
\Ddom = \set{\, z\in (1+\epsilon)\cdot \disk\, }{\, z\ne 1 \text{ and }|\mathrm{arg}(z-1)|>\theta \, }.
\end{equation*}
When $\theta=0$, the above definition gives $\Ddom[0] = (1+\epsilon)\cdot \disk \setminus [1,1+\epsilon)$, which is a disk with a small cut along the real axis. We call this a slit disk, and use the abbreviated notation $\slit \equiv \Ddom[0]$.
We denote by $\partial \Ddom$ and $\cDdom$ be the boundary and the closure of $\Ddom$. When $\theta \in (0,\pi/2)$, these are taken \wrt\ the usual topology of $\complex$. When $\theta=0$ however, we view $\slit$ as a domain in the universal covering space of $\complex\setminus \{1\}$, and define $\partial \slit$ and $\cslit$ \wrt\ that topology. In this way the closed curve $\partial \slit$ will be a nice limit of $\partial \Ddom$ when $\theta \to 0^+$, as illustrated in Figure~\refp{a}{fig:Hdom}.
\begin{figure}
\centering
\includegraphics[scale=0.75]{Fig3.pdf}
\caption{(a) Boundaries of the unit disk $\disk$, the $\Delta$-domain $\Ddom$ and the slit disk $\slit$. For the sake of visibility, $\Ddom$ and $\slit$ are drawn for two different values of $\epsilon$.\\
(b) Boundaries of the domains $\Hdom[0](R)$, $\Hdom[\epsilon,\theta](R)$ and $\Hdom(R)$ defined by the parametrization $\check x_R$ at a non-critical temperature $R\ne R_c$. By definition, $\Hdom[0](R)$ (resp.\ $\Hdom[\epsilon,\theta](R)$ and $\Hdom(R)$) is the connected component of the preimage $\check x_R^{-1}(\disk)$ (resp.\ $\check x_R^{-1}(\Ddom)$ and $\check x_R^{-1}(\slit)$) containing the origin.\\
(c) Boundaries of the domains $\Hdom[0](R)$ and $\Hdom(R)$ defined by $\check x_R$ at the critical temperature $R=R_c$. Notice that at the point $\check H_c(R)$, the curve $\partial \Hdom(R)$ in (b) has a tangent, while the curve $\partial \Hdom(R_c)$ in (c) has two half-tangents at an angle $2\pi/3$.
}
\label{fig:Hdom}
\end{figure}
\begin{proposition}\label{prop:singularity structure}
For all $\nu>1$ and $\theta \in (0,\frac\pi2)$, there exists $\epsilon>0$ such that (an analytic continuation of) the function $(x,y)\mapsto Z(u_c(\nu) x, u_c(\nu) y; \nu)$ is holomorphic in $\cslit \times \cDdom$. Moreover, when $\nu\ge \nu_c$, we can take $\theta=0$, i.e., find $\epsilon>0$ such that the function is holomorphic in $\cslit \times \cslit$.
\end{proposition}
\begin{remark}
As mentioned at the end of the previous section, by ``holomorphic in $\cslit \times \cslit$'', we mean that a function has complex partial derivatives in the interior $\slit \times \slit$ of $\cslit \times \cslit$, and is continuous in $\cslit \times \cslit$.
This will be later used to express the coefficients $z_{p,q}(\nu)$ as double Cauchy integrals on the contour $\partial \slit \times \partial \slit$, so that their asymptotics when $p,q\to\infty$ can be estimated easily.
For this purpose, it is not absolutely necessary to prove the continuity of $(x,y)\mapsto Z(u_c(\nu) x, u_c(\nu) y;\nu)$ on the boundary of $\cslit \times \cslit$ (in particular, at the point $(1,1)$). But not knowing this continuity would require one to approximate the contour $\partial \slit \times \partial \slit$ by a sequence of contours that lie inside $\slit \times \slit$, which complicates a bit the estimation of the double Cauchy integral.
\end{remark}
The rest of this section is devoted to the proof of Proposition~\ref{prop:singularity structure}. To this end, we will construct the desired analytic continuation of $(x,y)\mapsto Z(u_c(\nu) x, u_c(\nu) y;\nu)$ based on the heuristic formula $Z(u_c x, u_c y) = \check Z(\check x^{-1}(x), \check x^{-1}(y))$. The proof comes in two steps: First, we show that for each fixed $R$, the rational function $\check x_R$ defines a conformal bijection from a set $\cHdom(R)$ to $\cslit$ for some $\epsilon>0$. Then, we try to show that for all $\epsilon$ small enough, the rational function $\check Z_R(H,K)$ has no pole, hence is holomorphic, in $\cHdom(R) \times \cHdom(R)$.
It turns out that this is true only when $\nu \ge \nu_c$. When $\nu\in (1,\nu_c)$, one needs to reduce the domain $\cHdom(R) \times \cHdom(R)$ a bit, which corresponds to replacing one factor in the product $\cslit \times \cslit$ by a $\Delta$-domain $\cDdom$ with some opening angle $\theta>0$.
\subsection{The conformal bijection $\check x_R: \cHdom(R) \to \cslit$}
\begin{lemma}[Uniqueness and multiplicity of the critical point of $\check x_R$]
\label{lem:unique dominant}
For all $R\in (R_1,R_\infty)$, $\check H_c(R)$ is the unique zero of the rational function $\check x_R'$ in $\cHdom[0](R)$. It is a simple zero if $R\ne R_c$, and a double zero if $R=R_c$.
\end{lemma}
\begin{proof}
By definition, $\check H_c(R)$ is a zero of $\check x_R'$. One can easily check that it is a simple zero if $R\in (R_1,R_\infty)\setminus \{R_c\}$, and a double zero if $R=R_c$. It remains to show its uniqueness in $\cHdom[0](R)$.
By the definition of $\Hdom[0](R)$, the restriction of $\check x_R$ to this set is a conformal bijection. Therefore the derivative $\check x_R'$ has no zero in $\Hdom[0](R)$. On the other hand, $\check x_R'$ is a polynomial of degree three for all $R\in (R_1,R_\infty)$, so it has three zeros (counted with multiplicity), one of which is $\check H_c(R)$. In the following we show that the two other zeros are not in the set $\partial \Hdom[0](R) \setminus \{ \check H_c(R) \}$, and this will complete the proof.
When $R\in [R_c,R_\infty)$, we check by explicit computation (see \cite{CAS2}) that all three zeros of $\check x_R'$ are on the positive real line. Since $\Hdom[0](R)$ is a topological disk containing $H=0$ and is symmetric with respect to the real axis, its boundary intersects the positive real line only once (at $\check H_c(R)$). Hence $\check x_R'$ has no zero on $\partial \Hdom[0](R) \setminus \{ \check H_c(R) \}$.
Now assume that for some $R_*\in (R_1,R_c)$, the derivative $\check x_{R_*}'$ vanishes at a point $H_*\in \partial \Hdom[0](R_*) \setminus \{ \check H_c(R_*) \}$. Then $H_*$ is a zero of the quadratic polynomial $\chi_{R_*}$ defined by $\chi_R(H) \equiv \chi(H;R) := \frac{\partial_H \check x(H;R)}{H-\check H_c(R)}$. On the other hand, applying the mapping $\check x_{R_*}$ to the inclusion $H_*\in \partial \Hdom[0](R_*) \setminus \{ \check H_c(R_*) \}$ gives that $\check x_{R_*}(H_*) \in \partial \disk \setminus \{1\}$. This is equivalent to $\check x(H_*;R_*) = \frac{s+i}{s-i}$ for some $s \in \real$, since $s\mapsto \frac{s+i}{s-i}$ defines a bijection from $\real$ to $\partial \disk \setminus \{1\}$.
There is one more algebraic constraint that $H_*$ must satisfy:
\begin{itemize}
\item
If $\partial_H \chi(H_*;R_*)=0$, then we have found an algebraic constraint satisfied by $H_*$.
By solving the system $\chi(H_*,R_*)=0$ and $\partial_H \chi(H_*,R_*)=0$ (the first equation is quadratic in $H$, and the second linear in $H$), we find a unique solution for the pair $(H_*,R_*)$ such that $R\in (R_1,R_c)$.
But this pair gives a numerical value $|\check x_{R_*}(H_*)| \ne 1$ (see \cite{CAS2} for the numerical computation), which contradicts the previously established condition $\check x_{R_*}(H_*) \in \partial \disk$. Thus the case $\partial_H \chi(H_*;R_*)=0$ can be excluded.
\item
If $\partial_H \chi(H_*;R_*)\ne 0$, then the equation $\chi(H;R)=0$ defines a smooth implicit function $H=\check H_*(R)$ in a neighborhood of $(H,R)=(H_*,R_*)$. We claim that the derivative $\od{}R \mn|{\check x(\check H_*(R),R)}$ must vanish at $R_*$.
To see why, notice that $\partial_H \chi(H_*;R_*)\ne 0$ \Iff\ $\partial_H^2 \check x(H_*;R_*) \ne 0$. Therefore $\check x(\check H_*(R);R)\ne 0$, $\partial_H \check x(\check H_*(R);R)=0$ and $\partial_H^2 \check x(\check H_*(R);R)\ne 0$ for all $R$ in a neighborhood of $R_*$: the equation on $\partial_H \check x$ is a direct consequence of $\chi(\check H_*(R);R)=0$, while the two inequalities hold near $R=R_*$ by continuity. It follows that there exists a change of variable $H=\check H_R(h)$ which is holomorphic in $h$, and smooth in $R$, such that
\begin{equation*}
\check x(\check H_R(h);R) = \check x(\check H_*(R),R) \cdot (1+h^2)
\end{equation*}
for all $(h,R)$ in a neighborhood of $(0,R_*)$. In the variable $h$, the preimage of the unit disk $\disk$ by the mapping $\check x_R$ is determined simply by the number $|\check x(\check H_*(R),R)|^{-1}$ via the relation
\begin{equation*}
\check x^{-1}_R(\disk) = \Set{\check H_R(h)}{|1+h^2| < |\check x(\check H_*(R),R)|^{-1} } \,.
\end{equation*}
When $R=R_*$, we have $|\check x(\check H_*(R_*),R_*)|^{-1} = 1$, and the set $\Setn{h}{|1+h^2|<1}$ locally looks like the two-sided cone $\Set{h}{|\mathfrak{Re}(h)|<|\mathfrak{Im}(h)|}$, as in the middle picture of Figure~\ref{fig:local-merger}.
By assumption, the point $H_*=\check H_*(R_*)$ is on the boundary of the domain $\Hdom[0](R)$, which is a connected component of $\check x^{-1}_R(\disk)$ (the one that contains $H=0$). Therefore one side of the two-sided cone must belong to $\Hdom[0](R)$.
Assume that the derivative $\od{}R \mn|{\check x(\check H_*(R),R)}$ does not vanish at $R_*$, then we would have $|\check x(\check H_*(R),R)|<1$ either for $R>R_*$ or for $R<R_*$ in a neighborhood of $R_*$. But, as shown in the right picture of Figure~\ref{fig:local-merger}, when $|\check x(\check H_*(R),R)|<1$, the preimage $\check x^{-1}_R(\disk)$ has only one connected component locally at $\check H_*(R)$. This connected component belongs to $\Hdom[0](R)$ due to the continuity of $R\mapsto \check x(H;R)$.
It follows that $H=\check H_*(R)$ (i.e. $h=0$ in the variable $h$) is a zero of the derivative $\check x'_R$ inside $\Hdom[0](R)$. This is a contradiction because $\check x_R$ defines a conformal bijection on $\Hdom[0](R)$.
Thus the derivative $\od{}R \mn|{\check x(\check H_*(R),R)}$ must vanish at $R_*$.
\begin{figure}
\centering
\includegraphics[scale=0.9]{Fig4.pdf}
\caption{Local behavior of the set $\check x_R^{-1}(\disk)$ in the coordinate $h$. The region $\Hdom[0](R)$ is colored in yellow, and the region $\check x^{-1}_R(\disk) \setminus \Hdom[0](R)$ is colored in green.}
\label{fig:local-merger}
\end{figure}
The vanishing of $\od{}R \mn|{\check x(\check H_*(R),R)}$ is equivalent to
\begin{equation*}
\od{}R \log \abs{ \check x(\check H_*(R),R) } \,=\, \od{}R \Re\m({ \log \check x(\check H_*(R),R) } \,=\, \Re \m({ \frac{ \od{}R \check x(\check H_*(R),R) }{ \check x(\check H_*(R),R) } } = 0 \,.
\end{equation*}
Or equivalently, $\od{}R \check x(\check H_*(R),R) = ir\cdot \check x(\check H_*(R),R)$ for some $r\in \real$.
Recall that $\check H_*(R)$ is the implicit function defined by $\chi(H;R)=0$ and $\check H_*(R_*)=H_*$.
Using the chain rule, we get
\begin{equation*}
\left. \od{}R \check x(\check H_*(R),R) \right|_{R=R_*} = \mB({ \partial_R \check x - \frac{\partial_R \raisebox{3pt}{$\chi$}}{\partial_H \raisebox{3pt}{$\chi$}} \cdot \partial_H \check x }(H_*,R_*) \,.
\end{equation*}
\end{itemize}
To summarize, if there exists $R_*\in (R_1,R_c)$ such that $\partial \Hdom[0](R_*) \setminus \{ \check H_c(R_*) \}$ contains a zero $H_*$ of $\check x_{R_*}'$, then the pair $(H_*,R_*)$ satisfies the system of algebraic equations
\begin{equation}\label{eq:no real sol}
\chi(H;R) := \frac{\partial_H \check x(H;R) }{H-\check H_c(R)} = 0
\ ,\qquad
\check x(H;R) = \frac{s+i}{s-i}
\qtq{and}
\partial_R \check x - \frac{\partial_R \raisebox{3pt}{$\chi$}}{ \partial_H \raisebox{3pt}{$\chi$}} \cdot \partial_H \check x = ir\cdot \frac{s+i}{s-i}
\end{equation}
for some $r,s\in \real$. After eliminating $H$, we obtain two polynomial equations relating $R$, $r$ and $s$. Since these variables are all real, the real part and the imaginary part of each equation must both vanish. We check that the resulting system of four polynomial equations has no real solution using a general algorithm \cite{KRS16realroot} implemented in Maple as \code{RootFinding[HasRealRoot]}, see \cite{CAS2}. This proves that $\check H_c(R)$ is the unique zero of $\check x_R'$ in $\partial \Hdom[0](R)$ for all $R\in (R_1,R_c)$, and completes the proof of the lemma.
\end{proof}
\begin{remark}
The second equation in \eqref{eq:no real sol} implies $\check x_R(H) \in \partial \disk$ but does not guarantee $H\in \partial \Hdom[0](R)$, because the mapping $\check x_R$ is not injective on $\complex$. In fact, if one removes the last equation from \eqref{eq:no real sol}, then the system does have a solution $(H,R,s)$ with $R \in (R_1,R_c)$ and $s \in \real$. This solution corresponds to a critical point of $\check x_R$ which is not on $\partial \Hdom[0](R)$, but is mapped to $\partial \disk \setminus \{1\}$ nevertheless.
The purpose of the last equation of \eqref{eq:no real sol} is precisely to avoid this kind of undesired solutions.
Without the last equation, the algebraic system \eqref{eq:no real sol} contains two complex equations with four real unknowns ($\Re(H)$, $\Im(H)$, $R$, $s$). So generically, we do expect it to have a finite number of solutions.
The last equation adds one \emph{complex} equation to the system while introducing only an extra \emph{real} variable. So with it, we expect \eqref{eq:no real sol} to have no solutions in the generic case (which is indeed true in our specific case).
In general, if the mapping $\check x(H)$ depends on $m$ real parameters $(R_1,\ldots,R_m)$ instead of $R$, then provided that $\check x(H)$ has continuous derivatives \wrt\ each of the parameters, one can replace the last equation of \eqref{eq:no real sol} by $m$ complex equations with $m$ extra real variables. Then we would have a system of $m+2$ complex equations with $3+m+m=2m+3$ real variables, which would have no solution in the generic case.
\end{remark}
\paragraph{Definition of $\Hdom(R)$:}
For each $R\in (R_1,R_\infty)$, the above lemma and Proposition~21(iii) of \cite{CT20} imply that there exists $\epsilon>0$ for which $\check x_R$ defines a conformal bijection from a compact set $\cHdom(R) \supset \cHdom[0](R)$ to $\cslit$. For $\theta \in (0,\pi/2)$, let $\cHdom[\epsilon,\theta](R)$ be the preimage of the $\Delta$-domain $\cDdom \subset \cslit$ under this bijection.
We denote by $\partial \Hdom(R)$ and $\Hdom(R)$ the boundary and the interior of $\cHdom(R)$, and similarly for $\cHdom[\epsilon,\theta](R)$.
Notice that the notation $\cHdom(R)$ fits well with the previously defined $\cHdom[0](R)$, since the latter is in bijection with the closed unit disk $\cdisk$, which can be viewed as a special case of the domain $\cslit$ with $\epsilon=0$.
\paragraph{Geometric interpretation of Lemma~\ref{lem:unique dominant}.}
We know that analytic functions preserve angles at non-critical points.
More generally, if $f$ is an analytic function such that $H\in \complex$ is a critical point of multiplicity $n$ (that is, a zero of multiplicity $n$ of $f'$, with $n\ge 0$), then $f$ maps each angle $\theta$ incident to $H$ to an angle $(n+1)\theta$.
Since $\Hdom[0](R)$ is mapped bijectively by $\check x_R$ to the unit disk (whose boundary is smooth everywhere), the boundary of $\Hdom[0](R)$ forms an angle of $\pi/(n+1)$ at each $H\in \partial \Hdom[0](R)$ which is a critical point of multiplicity $n$ of $\check x_R$.
Therefore, Lemma~\ref{lem:unique dominant} tells us that the boundary of $\Hdom[0](R)$ is smooth everywhere except at $H=\check H_c(R)$, where it has two half-tangents forming an angle of $\pi/2$ if $R\ne R_c$, or an angle of $\pi/3$ if $R=R_c$.
This is illustrated by the red curves in Figure~\refp{b}{fig:Hdom}~and~\refp{c}{fig:Hdom}.
For the same reason, the boundary of $\Hdom[\epsilon,\theta](R)$ has also two half-tangents at $H=\check H_c(R)$. They form an angle of $\pi-\theta$ if $R\ne R_c$, and an angle of $\frac23(\pi-\theta)$ if $R=R_c$. (In particular, when $\theta=0$ and $R\ne R_c$, the angle is equal to $\pi$, i.e.\ the two half-tangents become a tangent.) This is illustrated by the blue and cyan curves in Figure~\refp{b}{fig:Hdom}~and~\refp{c}{fig:Hdom}.
From this we deduce the following corollary, which will be used to derive the local expansion of the bivariate function $\check Z(H,K;R)$ at $(H,K) = (\check H_c(R), \check H_c(R))$ at critical and high temperatures.
\begin{corollary}\label{cor:double angle bound}
For all $R\in (R_1,R_\infty)$ and $\theta \in (0,\frac\pi2)$, there exist a neighborhood $\mc N$ of $(\check H_c(R),\check H_c(R))$ and a constant $M_\theta<\infty$ such that
\begin{equation}\label{eq:double angle bound}
\max \m({ |\check H_c(R)-H| , |\check H_c(R)-K| } \le M_\theta \cdot
\abs{ (\check H_c(R)-H) + (\check H_c(R)-K) }
\end{equation}
for all $(H,K) \in \mc N \cap \m({ \cHdom(R) \times \cHdom[\epsilon,\theta](R) }$.
When $R=R_c$, one can take $\theta=0$ so that \eqref{eq:double angle bound} holds for all
$(H,K) \in \mc N \cap \m({ \cHdom(R) \times \cHdom(R) }$.
\end{corollary}
\begin{proof}
For $R \in (R_1,R_\infty) \setminus \{R_c\}$, the boundary of $\cHdom[\epsilon,\theta](R)$ has two half-tangents at $H=\check H_c(R)$, both at an angle of $\frac{\pi-\theta}2$ with the negative real axis. When $\theta=0$, the two half-tangents becomes a tangent that is orthogonal to the real axis. For any $\theta\in (0,\frac\pi2)$, we can choose $\theta_1>\frac\pi2$ and $\theta_2>\frac{\pi-\theta}2$ such that $\theta_1+\theta_2<\pi$. Then there exists a neighborhood $N$ of $(\check H_c(R),\check H_c(R))$ such that $\arg(\check H_c(R)-H) \in (-\theta_1,\theta_1)$ for all $H\in N \cap \cHdom(R)$, and $\arg(\check H_c(R)-K) \in (-\theta_2, \theta_2)$ for all $K\in N \cap \cHdom[\epsilon,\theta](R)$.
In polar coordinates, this means that $\check H_c(R)-H = r_1 e^{i \phi_1}$ and $\check H_c(R)-K = r_2 e^{i \phi_2}$ satisfy $|\phi_1|\le \theta_1$ and $|\phi_2|\le \theta_2$, so that $|\phi_1-\phi_2|\le \theta_1+\theta_2<\pi$. It follows that
\begin{align*}
\abs{ (\check H_c(R)-H) + (\check H_c(R)-K) }^2
&= \abs{r_1 e^{i\phi_1} + r_2 e^{i\phi_2}}^2
= r_1^2 + r_2^2 + 2r_1r_2 \cos(\phi_1-\phi_2)
\\ & \ge
r_1^2 + r_2^2 + 2r_1r_2 \cos(\theta_1+\theta_2)
\\&
= \mb({ r_1-r_2 \cos(\theta_1+\theta_2) }^2 + \mb({ r_2 \sin(\theta_1+\theta_2) }^2 \,.
\end{align*}
This implies that $r_2 = |\check H_c(R)-K| \le \frac{1}{\sin(\theta_1+\theta_2)} \cdot \abs{ (\check H_c(R)-H) + (\check H_c(R)-K) }$, and by symmetry, the inequality \eqref{eq:double angle bound} with $M_\theta = \frac{1}{\sin(\theta_1+\theta_2)}$, for all $(H,K) \in (N\times N) \cap \m({ \cHdom(R) \times \cHdom[\epsilon,\theta](R) }$.
When $R=R_c$, the boundary of $\cHdom(R)$ has two half-tangents at $H=\check H_c(R)$ at an angle of $\frac\pi3$ with the negative real axis. In this case, we can take $\theta_1=\theta_2=\frac{5\pi}{12} >\frac\pi3$ so that $\theta_1+\theta_2<\pi$. Then, the same proof as in the $R\ne R_c$ case shows that there exists a neighborhood $N$ of $\check H_c(R)$ such that \eqref{eq:double angle bound} holds with $M_0 = \frac{1}{\sin(5\pi/6)}$ for all $(H,K)\in (N\times N) \cap \m({ \cHdom(R) \times \cHdom(R) }$.
\end{proof}
\subsection{Holomorphicity of $\check Z$ on $\cHdom(R) \times \cHdom(R)$.}
The previous subsection showed that for $\epsilon>0$ small enough, $\cslit \times \cslit$ is mapped analytically by the inverse function of $(H,K)\mapsto (\check x_R(H),\check x_R(K))$ to the domain $\cHdom(R) \times \cHdom(R)$. Ideally, we want to show that the other part of the rational parametrization $(H,K) \mapsto \check Z_R(H,K)$ does not have poles on $\cHdom(R) \times \cHdom(R)$. Then the formula $Z(u_c x,u_c y) = \check Z_R( (\check x_R)^{-1}(x), (\check x_R)^{-1}(y) )$ would imply that $(x,y)\mapsto Z(u_c x,u_c y)$ has an analytic continuation on $\cslit \times \cslit$.
By continuity, any neighborhood of the compact set $\cHdom[0](R) \times \cHdom[0](R)$ contains $\cHdom(R) \times \cHdom(R)$ for all $\epsilon$ small enough. On the other hand, the poles of $\check Z_R$ form a closed set. Hence to prove that the domain $\cHdom(R) \times \cHdom(R)$ does not contain any poles for $\epsilon$ small enough, it suffices to show that the compact set $\cHdom[0](R) \times \cHdom[0](R)$ does not contain any poles of $\check Z_R$.
It turns out that this is almost the case:
\begin{lemma}\label{lem:poles of Z in H0*H0}
For all $R\in (R_c,R_\infty)$, the rational function $\check Z_R$ has no pole in $\cHdom[0](R) \times \cHdom[0](R)$.\\
\phantom{\textbf{Lemma $16$.} }
For all $R\in (R_1,R_c]$, $(\check H_c(R),\check H_c(R))$ is the only pole of $\check Z_R$ in $\cHdom[0](R) \times \cHdom[0](R)$.
\end{lemma}
\begin{proof}
By definition, a pole of $\check Z_R$ is a zero of the polynomial $D_R$ in the denominator of $\check Z_R(H,K) = \frac{N_R(H,K)}{D_R(H,K)}$, where $N_R$ and $D_R$ are coprime polynomials of $(H,K)$. With an appropriate choice of the constant term $D_R(0,0)$, we can take $N(H,K,R):= N_R(H,K)$ and $D(H,K,R) := D_R(H,K)$ to be polynomial in all three variables $(H,K,R)$. We check by explicit computation (see \cite{CAS2}) that $D(\check H_c(R),\check H_c(R),R) \ne 0$ for all $R\in (R_c,R_\infty)$, and $D(\check H_c(R),\check H_c(R),R) = 0$ for all $R\in (R_1,R_c]$.
Then it remains to show that $D$ does not vanish for any $(H,K) \in \cHdom[0](R) \times \cHdom[0](R) \setminus \{ (\check H_c(R),\check H_c(R)) \}$ and $R\in (R_1,R_\infty)$. For this we use the following lemma, whose proof will be given later:
\begin{lemma}\label{lem:pole system}
If the polynomial $D$ vanishes at a point $(H,K,R)$ such that $(H,K) \in \cHdom[0](R) \times \cHdom[0](R)$ and $R\in (R_1,R_\infty)$, then both $N$ and $\partial_H N \cdot \partial_K D - \partial_K N \cdot \partial_H D$ vanish at $(H,K,R)$.
\end{lemma}
This lemma tells us that the poles of $\check Z_R$ in the physical range of the parameters (that is, for $R\in (R_1,R_\infty)$ and $(H,K) \in \cHdom[0](R) \times \cHdom[0](R)$) satisfy the system of three polynomial equations
\begin{equation}\label{eq:pole system 3}
D \,=\, N \,=\,
\partial_H N \cdot \partial_K D - \partial_K N \cdot \partial_H D \,=\, 0
\end{equation}
instead of just $D=0$. However, it is not easy to verify whether \eqref{eq:pole system 3} has a solution $(H,K,R)$ satisfying $(H,K) \in \cHdom[0](R) \times \cHdom[0](R) \setminus \{ (\check H_c(R),\check H_c(R)) \}$, for two reasons: On the one hand, the solution set of \eqref{eq:pole system 3} contains at least one continuous component: $(H,K,R)=(\check H_c(R), \check H_c(R), R)$ is a solution of \eqref{eq:pole system 3} for all $R\in (R_1,R_c]$. On the other hand, it is not easy to distinguish between points in $\cHdom[0](R)$ from points in the preimage $\check x_R^{-1}(\cdisk)$ which are not in $\cHdom[0](R)$. To mitigate these issues, we construct an auxiliary equation that eliminates some solutions of the system which are known to be outside $\cHdom[0](R) \times \cHdom[0](R) \setminus \{ (\check H_c(R),\check H_c(R)) \}$.
Since $\check x_R$ is a conformal bijection from $\cHdom[0](R)$ to the unit disk, we know that $H=0$ is its unique (simple) zero in $\cHdom[0](R)$. Hence the polynomial $H\mapsto \check U_R(H)/H$ does not vanish on $\cHdom[0](R)$. (Recall that $\check x_R$ is defined as $\check U_R$ divided by a constant that only depends on $R$.) On the other hand, $\check H_c(R)$ is the unique zero of $\check x_R'$ in $\cHdom[0](R)$ by Lemma~\ref{lem:unique dominant}. Thus if $(H,K) \in \cHdom[0](R) \times \cHdom[0](R)$ is different from $(\check H_c(R),\check H_c(R))$, then either $\check U_R'(H)\ne 0$ or $\check U_R'(K)\ne 0$.
Let $\mc{NZ}(H,K,R)= \frac{\check U_R(H)}H \cdot \frac{\check U_R(K)}K \cdot \check U_R'(H)$. Then the above discussion shows that for $R\in (R_1,R_\infty)$ and $(H,K) \in \cHdom[0](R) \times \cHdom[0](R) \setminus \{ (\check H_c(R),\check H_c(R)) \}$, either $\mc{NZ}(H,K,R) \ne 0$, or $\mc{NZ}(K,H,R) \ne 0$.
It follows that if $(H,K)$ is a pole of $\check Z_R$ in $\cHdom[0](R) \times \cHdom[0](R) \setminus \{ (\check H_c(R),\check H_c(R)) \}$, then either $(H,K,R)$ or $(K,H,R)$ is a solution to the system of equations
\begin{equation}\label{eq:pole system 4}
D \,=\, N \,=\,
\partial_H N \cdot \partial_K D - \partial_K N \cdot \partial_H D \,=\, 0
\qtq{and}
X\cdot \mc{NZ} = 1
\end{equation}
where $X\in \complex$ is an auxiliary variable used to express the condition $\mc{NZ}\ne 0$ as an algebraic equation.
A Gr\"obner basis computation (see \cite{CAS2}) shows that this system has no solution with real value of $R$. By contradiction, $\check Z_R$ has no pole in $\cHdom[0](R) \times \cHdom[0](R) \setminus \{(\check H_c(R),\check H_c(R))\}$ for all $R\in (R_1,R_\infty)$. This completes the proof.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:pole system}]
In this proof we fix an $R\in (R_1,R_\infty)$ and drop it from the notations.
Since the double power series $(x,y)\mapsto Z(u_c x,u_c y)$ is absolutely convergent for all $x,y$ in the unit disk $\cdisk$, and $\check x$ is a homeomorphism from $\cHdom[0]$ to $\cdisk$, the rational function $\check Z(H,K) = Z(u_c\cdot \check x(H), u_c\cdot \check x(K))$ is continuous on the compact set $\cHdom[0] \times \cHdom[0]$.
Assume that $D$ vanishes at some $(H,K) \in \cHdom[0] \times \cHdom[0]$. The boundedness of $\check Z$ on $\cHdom[0] \times \cHdom[0]$ implies that $N$ also vanishes at $(H,K)$. If $\partial_H D(H,K) = \partial_K D(H,K) = 0$, then $\partial_H N \cdot \partial_K D - \partial_K N \cdot \partial_H D$ obviously vanishes at $(H,K)$. Otherwise, consider the limit of $\check Z(H+\varepsilon h,K+\varepsilon k)$ when $\varepsilon \to 0^+$, where $h,k \in \complex$. By L'H\^opital's rule, for all $(h,k)$ such that $h \cdot \partial_H D(H,K) + k \cdot \partial_K D(H,K) \ne 0$, we have
\begin{equation}\label{eq:alpha limit}
\lim_{\varepsilon \to 0^+} \check Z(H+ \varepsilon h,K+\varepsilon k) = \frac{
h \cdot \partial_H N(H,K) + k \cdot \partial_K N(H,K)}{
h \cdot \partial_H D(H,K) + k \cdot \partial_K D(H,K)} \,.
\end{equation}
By the continuity of $\check Z$ on $\cHdom[0] \times \cHdom[0]$, the above limit is independent of $(h,k)$ as long as the pair satisfies that $(H+\varepsilon h,K+\varepsilon k) \in \cHdom[0] \times \cHdom[0]$ for all $\epsilon>0$ small enough. From Figure~\ref{fig:Hdom} (or more rigorously the geometric interpretation of Lemma~\ref{lem:unique dominant}), we see that for all $H\in \cHdom[0]$, there exists $h_*\ne 0$ such that $H+\varepsilon h_* \in \cHdom[0]$ for all $\epsilon>0$ small enough. Similarly, there exists $k_*\ne 0$ such that $K+\varepsilon k_* \in \cHdom[0]$ for all $\epsilon>0$ small enough. By taking $(h,k)$ to be equal to $(h_*,0)$, $(0,k_*)$ and $(h_*,k_*)$ in \eqref{eq:alpha limit}, we obtain that
\begin{equation*}
\frac{h_* \partial_H N(H,K)}{h_* \partial_H D(H,K)} \,=\,
\frac{k_* \partial_K N(H,K)}{k_* \partial_K D(H,K)} \,=\,
\frac{h_* \partial_H N(H,K) + k_* \partial_K N(H,K)}{
h_* \partial_H D(H,K) + k_* \partial_K D(H,K)} \,,
\end{equation*}
provided that the denominators of the three fractions are nonzero.
By assumption, $\partial_H D(H,K)$ and $\partial_K D(H,K)$ do not both vanish. It follows that at least two of three fractions have nonzero denominators. From the equality between these two fractions, we deduce that $\partial_H N \cdot \partial_K D - \partial_K N \cdot \partial_H D=0$ at $(H,K)$.
\end{proof}
Now we use the continuity argument mentioned at the beginning of this subsection to deduce the holomorphicity of $\check Z_R$ on $\cHdom \times \cHdom$ (or $\cHdom \times \cHdom[\epsilon,\theta]$, see below) from Lemma~\ref{lem:poles of Z in H0*H0}. The low temperature case is easy, since $\check Z_R$ does not have any pole in $\cHdom[0] \times \cHdom[0]$ for all $R\in (R_c,R_\infty)$. When $R\in (R_1,R_c]$, one has to study the restriction of $\check Z_R$ on $\cHdom \times \cHdom$ more carefully near its the pole $(\check H_c(R), \check H_c(R))$. This is done with the help of Corollary~\ref{cor:double angle bound}.
\begin{lemma}\label{lem:holomorphicity Z}
For all $R\in [R_c,R_\infty)$, there exists $\epsilon>0$ such that $\check Z_R$ is holomorphic in $\cHdom(R) \times \cHdom(R)$.\\
For $R\in (R_1,R_c)$ and $\theta \in (0,\pi/2)$, there exists $\epsilon>0$ such that $\check Z_R$ is holomorphic in $\cHdom(R) \times \cHdom[\epsilon,\theta](R)$.
\end{lemma}
\begin{proof}
As in the previous proof, we fix a value of $R \in (R_1,R_\infty)$ and drop it from the notation.
\paragraph{Low temperatures.}
When $R\in (R_c,R_\infty)$, Lemma~\ref{lem:poles of Z in H0*H0} tells us that $\check Z$ has no pole in $\cHdom[0] \times \cHdom[0]$. Since the set of poles of $\check Z$ is closed, and $\cHdom[0] \times \cHdom[0]$ is compact, there exists a neighborhood of $\cHdom[0] \times \cHdom[0]$ containing no pole of $\check Z$.
By continuity, this neighborhood contains $\cHdom \times \cHdom$ for $\epsilon>0$ small enough. It follows that there exists $\epsilon>0$ such that $\check Z$ is holomorphic in $\cHdom \times \cHdom$.
\paragraph{Critical temperature.}
When $R=R_c$, Lemma~\ref{lem:poles of Z in H0*H0} tells us that $(\check H_c,\check H_c)$ is the only pole of $\check Z$ in $\cHdom[0] \times \cHdom[0]$.
First, let us show that $\check Z$, when restricted to $\cHdom \times \cHdom$, is continuous at $(\check H_c,\check H_c)$. Notice that this statement does not depend on $\epsilon$, since two domains $\cHdom \times \cHdom$ with different values of $\epsilon>0$ are identical when restricted to a small enough neighborhood of $(\check H_c,\check H_c)$.
We have seen in the proof of Lemma~\ref{lem:poles of Z in H0*H0} that the numerator $N$ and the denominator $D$ of $\check Z$ both vanish at $(\check H_c, \check H_c)$. Therefore their Taylor expansions give:
\begin{equation}\label{eq:Zl limit at (Hc,Hc)}
\check Z(\check H_c -h, \check H_c -k) = \frac{
\partial_H N(\check H_c,\check H_c) \cdot (h+k) + O\m({ \max(|h|,|k|)^2 } }{
\partial_H D(\check H_c,\check H_c) \cdot (h+k) + O\m({ \max(|h|,|k|)^2 } }
\qt{as }(h,k)\to (0,0).
\end{equation}
We check explicitly that $\partial_H D(\check H_c, \check H_c) \ne 0$, see \cite{CAS2}. On the other hand, thanks to Corollary~\ref{cor:double angle bound} (the critical case), we have $\max(|h|,|k|) = O\m({|h+k|}$ when $(h,k)\to (0,0)$ in such a way that $(\check H_c-h,\check H_c-k) \in \cHdom \times \cHdom$. Then it follows from \eqref{eq:Zl limit at (Hc,Hc)} that $\check Z(H,K) \to \partial_H N(\check H_c,\check H_c) / \partial_H D(\check H_c,\check H_c)$ when $(H,K)\to (\check H_c,\check H_c)$ in $\cHdom \times \cHdom$. That is, $\check Z$ restricted to $\cHdom \times \cHdom$ is continuous at $(\check H_c,\check H_c)$.
Next, let us show that for some fixed $\epsilon_0>0$, every point $(H,K)\in \cHdom[0] \times \cHdom[0]$ has a neighborhood $\mc V(H,K)$ such that $\check Z$ is holomorphic in $\mc V(H,K) \cap (\cHdom[\epsilon_0] \times \cHdom[\epsilon_0])$. (Recall that this means $\check Z$ is holomorphic in the interior, and continuous in the whole domain).
For $(H,K)=(\check H_c,\check H_c)$, the expansion of the denominator in \eqref{eq:Zl limit at (Hc,Hc)} shows that there exists $\epsilon_0>0$ and a neighborhood $\mc V(\check H_c,\check H_c)$ such that $(\check H_c,\check H_c)$ is the only pole of $\check Z$ in $\mc V(\check H_c,\check H_c) \cap (\cHdom[\epsilon_0] \times \cHdom[\epsilon_0])$. Moreover, the previous paragraph has showed that $\check Z$ is continous at $(\check H_c,\check H_c)$ when restricted to $\cHdom \times \cHdom$. It follows that $\check Z$ is holomorphic in $\mc V(\check H_c,\check H_c) \cap (\cHdom[\epsilon_0] \times \cHdom[\epsilon_0])$.
For $(H,K) \in \cHdom[0] \times \cHdom[0] \setminus \{(\check H_c,\check H_c)\}$, since $(H,K)$ does not belong to the (closed) set of poles of $\check Z$, it has a neighborhood $\mc V(H,K)$ on which $\check Z$ is holomorphic.
By taking the union of all the neighborhoods $\mc V(H,K)$ constructed in the previous paragraph, we see that there is a neighborhood $\mc V$ of the compact set $\cHdom[0] \times \cHdom[0]$ such that $\check Z$ is holomorphic in $\mc V \cap (\cHdom[0] \times \cHdom[0])$. By continuity, $\mc V$ contains $\cHdom \times \cHdom$ for some $\epsilon>0$ small enough. Hence there exists $\epsilon>0$ such that $\check Z$ is holomorphic in $\cHdom \times \cHdom$.
\paragraph{High temperatures.}
When $R\in (R_1,R_c)$, Lemma~\ref{lem:poles of Z in H0*H0} tells us that $(\check H_c,\check H_c)$ is also a pole of $\check Z$. The rest of the proof goes exactly as in the critical case, except that the domain $\cHdom \times \cHdom$ has to be replaced by $\cHdom \times \cHdom[\epsilon,\theta]$ for an arbitrary $\theta \in (0,\pi/2)$ due to the difference between the critical and non-critical cases in Corollary~\ref{cor:double angle bound}.
\end{proof}
\begin{remark}\label{rem:extended holomorphicity Z}
In fact, the above proof shows the holomorphicity of $\check Z_R$ in a larger domain than the one stated in Lemma~\ref{lem:holomorphicity Z}. In particular, one can check that the following statement is true: \emph{for each compact subset $\mathcal K$ of $\Hdom$, there exists a neighborhood $\mathcal V$ of $\cHdom[0]$ such that $\check Z$ is holomorphic in $\mathcal{K\times V}$.}
This remark will be used to show that $x\mapsto A(u_c x)$ is analytic on $\slit$ in Corollary~\ref{cor:A Delta-analytic}.
\end{remark}
\begin{proof}[Proof of Proposition~\ref{prop:singularity structure}]
The proposition follows from Lemma~\ref{lem:holomorphicity Z} and the definition of $\cHdom(R)$:
At critical or low temperatures, the inverse mapping of $(H,K)\mapsto (\check x_R(H), \check x_R(K))$ is holomorphic from $\cslit \times \cslit$ to $\cHdom(R) \times \cHdom(R)$. For $\epsilon>0$ small enough, $(H,K)\mapsto \check Z_R(H,K)$ is holomorphic in $\cHdom(R) \times \cHdom(R)$. Hence their composition defines an analytic continuation of $(x,y)\mapsto \tilde Z(x,y;\nu)$ on $\cslit \times \cslit$.
At high temperatures, it suffices to replace $\cslit \times \cslit$ by $\cslit \times \cDdom$, and $\cHdom(R) \times \cHdom(R)$ by $\cHdom(R) \times \cHdom[\epsilon,\theta](R)$.
\end{proof}
\section{Asymptotic expansions of $Z(u,v;\nu)$ at its dominant singularity}\label{sec:local expansion}
In this section, we establish the asymptotic expansions (Proposition~\ref{prop:local expansion}) of the generating function $Z(u_c x,u_c y)$ at its dominant singularity $(x,y)=(1,1)$. For this we define the function $\mc Z(\mc h,\mc k;\nu)$ by the change of variable
\begin{equation*}
Z(u_c(\nu) x, u_c(\nu) y; \nu) = \mc Z(\mc h,\mc k; \nu) \qtq{with}
\mc h = \m({1-x}^\delta \qtq{and}
\mc k = \m({1-y}^\delta
\end{equation*}
Recall that $\delta = 1/2$ when $\nu\ne \nu_c$ (non-critical case), and $\delta = 1/3$ when $\nu=\nu_c$ (critical case).
The proof relies on Lemma~\ref{lem:unique dominant} (location and multiplicity of the zeros of $\check x_R'$) and Lemma~\ref{lem:poles of Z in H0*H0} (location and multiplicity of the poles of $\check Z_R$) of the previous section, as well as the following property of the rational function $\check Z_R$: for all $H\ne 0$,
\begin{equation}\label{eq:vanishing derivatives}
\partial_K \check Z_R (H,\check H_c(R) )=0
\qt{for all }R\in (R_1, R_\infty), \qtq{and}
\partial_K^2 \check Z_{R_c}(H,\check H_c(R_c))=0 \,.
\end{equation}
These identities can be easily checked using a computer algebra system (see \cite{CAS2}). And we will see that these identities and Lemmas~\ref{lem:unique dominant}~and~\ref{lem:poles of Z in H0*H0} are the only properties of the \RP\ $(\check x_R,\check Z_R)$ used in the proof of Proposition~\ref{prop:local expansion}, in the sense that if $(\check x_R,\check Z_R)$ were a generic element chosen from the space of functions having these properties, then the asymptotic expansions of $Z(u,v;\nu)$ will have the exponents of polynomial decay as given in Proposition~\ref{prop:local expansion}.
From now on we hide the parameter $\nu$ and the corresponding parameter $R$ from the notations.
\begin{lemma}\label{lem:singular expansion}
For $\nu>\nu_c$, $\mc Z(\mc h,\mc k)$ is analytic at $(0,0)$. Its Taylor expansion $\mc Z(\mc h,\mc k) = \sum_{m,n\ge 0} \mc Z_{m,n} \mc h^m \mc k^n$ satisfies $\mc Z_{1,n} = \mc Z_{n,1} = 0$ for all $n \ge 0$ and $\mc Z_{3,3}>0$.
For $\nu \in (1,\nu_c]$, we have a decomposition of the form $\mc Z(\mc h,\mc k) = Q(\mc h,\mc k) + \frac{J(\mc h\mc k)}{\mc D(\mc h,\mc k)}$, where $Q(\mc h,\mc k)$, $J(r)$ and $\mc D(\mc h,\mc k)$ are analytic at the origin. The denominator satisfies $\mc D(0,0) = 0$ and $\partial_{\mc h} \mc D(0,0) = \partial_{\mc k} \mc D(0,0) = 1$, whereas
\begin{equation*}
Q(\mc h,\mc k) = \sum_{m,n\ge 0} Q_{m,n} \mc h^m \mc k^n
\qtq{and}
J(r) = \sum_{l\ge 1} J_l r^l
\end{equation*}
satisfy: If $\nu\in (1,\nu_c)$, then $J_1>0$.\\
~\phantom{satisfy:} If $\nu=\nu_c$, then $Q_{1,n} = Q_{n,1} = Q_{2,n} = Q_{n,2} = 0$ for all $n \ge 0$, $J_1=J_2=0$ and $J_3>0$.
The three nonzero coefficients in the above statements can be computed by:
\begin{align}
\label{eq:Z_3,3}
\mc Z_{3,3} &= \frac{1}{\check x_2^3} \mh({ \check Z_{3,3} -2 \, \frac{\check x_3}{\check x_2}\, \check Z_{2,3} + \m({\frac{\check x_3}{\check x_2}}^2 \check Z_{2,2} }
&&\hspace{-1cm}\text{when }\nu>\nu_c ,
\\
\label{eq:J_1}
J_1 &= \frac1{\check x_2^{1/2}} \lim_{H\to \check H_c} \partial_H \check Z(H,\check H_c)
&&\hspace{-1cm}\text{when }\nu\in (1,\nu_c),
\\
\label{eq:J_3}
J_3 &= \frac{1}{\check x_3^{5/3}} \cdot \frac{4}{13}\lim_{H\to \check H_c} \frac{ \partial_H \partial_K \check Z(H,H) }{ (\check H_c - H)^3 }
&&\hspace{-1cm}\text{when }\nu=\nu_c,
\end{align}
where the numbers $\check x_n$ and $\check Z_{m,n}$ are the coefficients in the Taylor expansions $1-\check x(\check H_c - h) = \sum_{n\ge 2} \check x_n h^n$ and $\check Z(\check H_c-h,\check H_c-k) = \sum \limits_{m,n} \check Z_{m,n} h^m k^n$.
\end{lemma}
\begin{proof}
Recall that $Z$ has the parametrization $x=\check x(H)$, $y=\check x(K)$ and $Z(u_c x, u_c y) = \check Z(H,K)$. The function $h\mapsto \mc h = \mn({ 1-\check x(\check H_c-h) }^\delta$ is analytic and has positive derivative at $h=0$. (The exponent $\delta$ has been chosen for this to be true.)
Let $\psi$ be its inverse function. Then the definition of $\mc Z$ implies that
\begin{equation}\label{eq:Z-tilde = Z-hat(psi)}
\mc Z(\mc h,\mc k) = \check Z \m({\check H_c-\psi(\mc h), \check H_c-\psi(\mc k)} \,.
\end{equation}
The proof will be based on the above formula and uses the following ingredients: The form of the local expansions of $\mc Z$ will follow from whether $(\check H_c,\check H_c)$ is a pole of $\check Z(H,K)$ or not. The vanishing coefficients will be a consequence of the vanishing of $\partial_K \check Z(H,\check H_c)$ and of $\partial_K^2 \check Z(H,\check H_c)$ given in \eqref{eq:vanishing derivatives}. Finally, the non-vanishing of the coefficients $\mc Z_{3,3}$, $J_1$ and $J_3$ will be checked by explicit computation.
\paragraph{Low temperatures ($\nu>\nu_c$).}
By Lemma~\ref{lem:poles of Z in H0*H0}, $(\check H_c,\check H_c)$ is not a pole of $\check Z(H,K)$ when $\nu>\nu_c$. Thus \eqref{eq:Z-tilde = Z-hat(psi)} implies that $\mc Z$ is analytic at $(0,0)$.
By the definition of $\check x_2$ and $\check x_3$, we have $\mb({1-\check x(\check H_c - h)}^{1/2} = \check x_2^{1/2} h \mb({ 1 + \frac{\check x_3}{2 \check x_2} h + O(h^2)}$. Then the Lagrange inversion formula gives,
\begin{equation*}
\psi(\mc h) = \frac{1}{\check x_2^{1/2}} \mc h - \frac{\check x_3}{2 \check x_2^2} \mc h^2 + O(\mc h^3) \,.
\end{equation*}
In particular, $\psi(\mc h) \sim \mathtt{cst}\cdot \mc h$.
Hence \eqref{eq:Z-tilde = Z-hat(psi)} and the fact that $\partial_K \check Z(H,\check H_c)=0$ for all $H$ (Eq.~\eqref{eq:vanishing derivatives}) imply that $\partial_{\mc k} \mc Z(\mc h,0)=0$ for all $\mc h$ close to $0$, that is, $\mc Z_{1,n}=\mc Z_{n,1}=0$ for all $n\ge 0$.
On the other hand, we get the expression \eqref{eq:Z_3,3} of $\mc Z_{3,3}$ by composing the Taylor expansions of $\psi(\mc h)$ and of $\check Z(\check H_c-h,\check H_c-k)$, while taking into account that $\check Z_{1,n}=\check Z_{n,1} = 0$.
By plugging the expressions of $\check x(H)$ and $\check Z(H,K)$ into the relation \eqref{eq:Z_3,3}, one can compute the function $\mc Z_{3,3}(\check \nu(R))$, which gives a parametrization of $\mc Z_{3,3}(\nu)$. The explicit formula, too long to be written down here, is given in \cite{CAS2}. We check in \cite{CAS2} that it is strictly positive for all $R\in (R_c,R_\infty)$.
\paragraph{High temperatures ($1<\nu<\nu_c$).}
When $\nu\in (1,\nu_c)$, Lemma~\ref{lem:poles of Z in H0*H0} tells us that $(\check H_c,\check H_c)$ is a pole of $\check Z(H,K)$. Moreover, this pole is simple in the sense that the denominator $D$ of $\check Z$ satisfies that $D(\check H_c,\check H_c)=0$ and $\partial_H D(\check H_c,\check H_c) = \partial_K D(\check H_c,\check H_c) \ne 0$.
Then it follows from \eqref{eq:Z-tilde = Z-hat(psi)} that $\mc Z = \mc N/\mc D$ for some functions $\mc N(\mc h,\mc k)$ and $\mc D(\mc h,\mc k)$, both analytic at $(0,0)$, such that $\mc D(0,0)=0$ and $\partial_{\mc h} \mc D(0,0)=\partial_{\mc h} \mc D(0,0)=1$.
We will show in Lemma~\ref{lem:singular division} below that there is always a pair of functions $Q(\mc h,\mc k)$ and $J(r)$, both analytic at the origin, such that $\mc N(\mc h,\mc k) = Q(\mc h,\mc k)\cdot \mc D(\mc h,\mc k) + J(\mc h \mc k)$. This implies the decomposition $\mc Z(\mc h,\mc k) = Q(\mc h,\mc k) + \frac{J(\mc h \mc k)}{\mc D(\mc h,\mc k)}$. Notice that $J(0)=0$, because $\mc N(0,0) = \mc D(0,0) = 0$ by the continuity of $\check Z |_{\cHdom[0] \times \cHdom[0]}$ at $(\check H_c,\check H_c)$.
Taking the derivatives of the above decomposition of $\mc Z(\mc h,\mc k)$ at $\mc k=0$ gives
\begin{equation*}
\partial_{\mc h} \mc Z(\mc h,0) = \partial_{\mc h} Q(\mc h,0) \qtq{and}
\partial_{\mc k} \mc Z(\mc h,0) = \partial_{\mc k} Q(\mc h,0) + \frac{J_1 \cdot \mc h}{\mc D(\mc h,0)} \,.
\end{equation*}
For the same reason as when $\nu>\nu_c$, we have $\partial_{\mc k} \mc Z(\mc h,0)=0$ for all $\mc h$ close to $0$. On the other hand, $\mc D(\mc h,0)\sim \mc h$ as $\mc h\to 0$ because $\partial_{\mc h} \mc D(0,0)=1$. Thus the limit $\mc h\to 0$ of the above derivatives gives
\begin{equation*}
\lim_{\mc h\to 0} \partial_{\mc h} \mc Z(\mc h,0) = \partial_{\mc h} Q(0,0) \qtq{and}
\partial_{\mc k} Q(0,0) + J_1 = 0.
\end{equation*}
By symmetry, $\partial_{\mc h} Q(0,0) = \partial_{\mc k} Q(0,0)$, therefore $J_1 = - \lim_{\mc h\to 0} \partial_{\mc h} \mc Z(\mc h,0)$.
After expressing $\mc Z(\mc h,0)$ in terms of $\check Z(H,\check H_c)$ and $\psi(\mc h)$ using \eqref{eq:Z-tilde = Z-hat(psi)}, we obtain the formula \eqref{eq:J_1} for $J_1$.
We check by explicit computation in \cite{CAS2} that $J_1(\nu)$ has the parametrization
\begin{equation*}
J_1 \m({\check \nu(R)} =
\frac{
\sqrt{\m({1+R^2} \m({7-R^2}^3 \m({14R^2 -1 -R^4}^5}
}{ \sqrt2\, (3R^2 -1) \m({29 +75R^2 -17R^4 +R^6}^2 }
\end{equation*}
which is strictly positive for all $R\in (R_1,R_c)$.
\paragraph{Critical temperature ($\nu=\nu_c$).}
When $\nu=\nu_c$, the point $(\check H_c,\check H_c)$ is still a pole of $\check Z(H,K)$ by Lemma~\ref{lem:poles of Z in H0*H0}, and one can check that it is simple in the sense that $\partial_H D(\check H_c,\check H_c) = \partial_K D(\check H_c,\check H_c) \neq 0$. Therefore, the decomposition $\mc Z(\mc h,\mc k) = Q(\mc h,\mc k) + \frac{J(\mc h \mc k)}{\mc D(\mc h,\mc k)}$ remains valid. Contrary to the non-critical case, now we have $\check x_2=0$ and $\delta=1/3$, thus $\psi(\mc h) \sim \check x_3^{-1/3} \mc h$. Together with the fact that $\partial_K \check Z(H,H_c)= \partial_K^2 \check Z(H,H_c)=0$ for all $H$ (Eq.~\eqref{eq:vanishing derivatives}), this implies
$\partial_{\mc k} \mc Z(\mc h,0) = \partial_{\mc k}^2 \mc Z(\mc h,0) = 0$ for all $\mc h$ close to $0$. Plugging in the decomposition $\mc Z(\mc h,\mc k) = Q(\mc h,\mc k) + \frac{J(\mc h \mc k)}{\mc D(\mc h,\mc k)}$, we obtain
\begin{equation*}
\partial_{\mc k} Q(\mc h,0) + \frac{J_1 h}{\mc D(\mc h,0)} = 0
\qtq{and}
\partial_{\mc k}^2 Q(\mc h,0) + \frac{J_2 \mc h^2}{\mc D(\mc h,0)} - J_1 \mc h \cdot \frac{\partial_{\mc k} \mc D(\mc h,0)}{\mc D(\mc h,0)^2} = 0 \,.
\end{equation*}
Since $\partial_{\mc h} \mc D(0,0)=1$ and $\mc D(\mc h,0) \sim \mc h$ as $\mc h\to 0$, the last term in the second equation diverges like $J_1 \mc h^{-1}$ when $\mc h\to 0$, whereas the other two terms are bounded. This implies that $J_1 = 0$.
Plugging $J_1=0$ back into the two equations, we get
\begin{equation*}
\partial_{\mc h} Q(\mc h,0) = 0 \qtq{and}
\partial_{\mc k}^2 Q(\mc h,0) + \frac{J_2 \mc h^2}{\mc D(\mc h,0)} = 0 \,.
\end{equation*}
The first equation translates to $Q_{1,n}=Q_{n,1}=0$ for all $n \ge 0$. Then, $Q_{1,2}=0$ tells us that in the second equation $\partial_{\mc k}^2 Q(\mc h,0) = Q_{0,2} + O(\mc h^2)$, whereas $\frac{J_2 \mc h^2}{\mc D(\mc h,0)} \sim J_2 \mc h$ when $\mc h\to 0$. Therefore we must have $J_2=0$, which in turn implies $\partial_{\mc k}^2 Q(\mc h,0) =0$, that is, $Q_{2,n}=Q_{n,2}=0$ for all $n\ge 0$.
To obtain the formula for $J_3$, we calculate from the decomposition $\mc Z(\mc h,\mc k) = Q(\mc h,\mc k) + \frac{J(\mc h \mc k)}{\mc D(\mc h,\mc k)}$ that
\begin{align}
\partial_{\mc h} \partial_{\mc k} \mc Z(\mc h,\mc h) =
\partial_{\mc h} \partial_{\mc k} Q(\mc h,\mc h)
&+\frac{1}{\mc D(\mc h,\mc h)^2}
\mh({
\m({
\mc D(\mc h,\mc h) - 2\mc h \partial_{\mc h} \mc D(\mc h,\mc h)
} J'(h^2)
- \partial_{\mc h} \partial_{\mc k} \mc D(\mc h,\mc h) J(h^2)
} \notag
\\&+\frac{1}{\mc D(\mc h,\mc h)}
\mh({
J''(h^2)\cdot h^2 +
2 \mB({
\frac{
\partial_{\mc h} \mc D(\mc h,\mc h) }{ \mc D(\mc h,\mc h)
}
}^2 J(h^2)
} \label{eq:dh dk Z-tilde}
\end{align}
When $\mc h\to 0$, we have $\partial_{\mc h} \partial_{\mc k} Q(\mc h,\mc h) = O(h^4)$ because $Q_{1,n} = Q_{n,1} = Q_{2,n} = Q_{n,2} = 0$. Moreover, using
\begin{align*}
\mc D(\mc h,\mc h)
& \sim 2\mc h &
\mc D(\mc h,\mc h) - 2\mc h \partial_{\mc h} \mc D(\mc h,\mc h)
& = O(\mc h^2) &
\partial_{\mc h} \partial_{\mc k} \mc D(\mc h,\mc h)
& = O(1) &
\frac{ \partial_{\mc h}\mc D(\mc h,\mc h) }{ \mc D(\mc h,\mc h) }
& \sim \frac{1}{2\mc h} \\
&\text{and}&
J''(\mc h^2) & \sim 6J_3 \cdot \mc h^2 &
J' (\mc h^2) & \sim 2J_3 \cdot \mc h^4 &
J (\mc h^2) & \sim J_3 \cdot \mc h^6
\end{align*}
we see that the first line of \eqref{eq:dh dk Z-tilde} is a $O(\mc h^4)$, whereas the second line is $\frac{13}{4} J_3 \mc h^3 + O(\mc h^4)$.
Therefore we have $J_3 = \frac{4}{13} \lim_{\mc h \to 0} \mc h^{-3} \partial_{\mc h} \partial_{\mc k} \mc Z(\mc h,\mc h)$. Finally, we obtain the expression \eqref{eq:J_3} of $J_3$ using the relation $\mc Z(\mc h,\mc h) = \check Z( \check H_c-\psi(\mc h), \check H_c-\psi(\mc h))$ and the fact that $\psi(\mc h) \sim \check x_3^{-1/3} \mc h$ when $\nu=\nu_c$.
Numerical computation gives $J_3 = \frac{27}{20} \m({\frac32}^{2/3} > 0$.
\end{proof}
\begin{lemma}[Division by a symmetric Taylor series with no constant term]
\label{lem:singular division}
Let $\mc N(\mc h,\mc k)$ and $\mc D(\mc h,\mc k)$ be two symmetric holomorphic functions defined in a neighborhood of $(0,0)$.
Assume that $(0,0)$ is a simple zero of $\mc D$, that is, $\mc D(0,0) = 0$ and $\partial_{\mc h} \mc D(0,0) = \partial_{\mc k} \mc D(0,0) \ne 0$. Then there is a unique pair of holomorphic functions $Q(\mc h,\mc k)$ and $J(r)$ in neighborhoods of $(0,0)$ and $0$ respectively, such that $Q$ is symmetric and
\begin{equation}\label{eq:singular division}
\mc N(\mc h,\mc k) = Q(\mc h,\mc k)\cdot \mc D(\mc h,\mc k) + J(\mc h\mc k) \,.
\end{equation}
\end{lemma}
\begin{remark}
When $\mc D(0,0)=0$, the ratio $\frac{\mc N(\mc h,\mc k)}{\mc D(\mc h,\mc k)}$ between two Taylor series $\mc N(\mc h,\mc k)$ and $\mc D(\mc h,\mc k)$ does not in general have a Taylor expansion at $(0,0)$.
The above lemma gives a way to decompose the ratio into the sum of a Taylor series $Q(\mc h,\mc k)$ and a singular part $\frac{J(\mc h\mc k)}{\mc D(\mc h,\mc k)}$ whose numerator is determined by an univariate function. The lemma deals with the case where $\mc N(\mc h,\mc k)$ and $\mc D(\mc h,\mc k)$ are symmetric, and the zero of $\mc D(\mc h,\mc k)$ at $(0,0)$ is simple. The following remarks discuss how the lemma would change if one modifies its conditions.
\begin{enumerate}
\item
In \eqref{eq:singular division}, instead of requiring $Q(\mc h,\mc k)$ to be symmetric, we can require the remainder term to not depend on $\mc k$. Then the decomposition would become $\mc N(\mc h,\mc k) = Q(\mc h,\mc k) \cdot \mc D(\mc h,\mc k) + J(\mc h^2)$. Notice that the remainder term does not have any odd power of $\mc h$, which is a constraint due to the symmetry of $\mc N$ and $\mc D$.
Without the assumption that $\mc N$ and $\mc D$ are symmetric, we would have a decomposition $\mc N(\mc h,\mc k) = Q(\mc h,\mc k) \cdot \mc D(\mc h,\mc k) + J(\mc h)$ where the remainder is a general Taylor series $J(\mc h)$. The proof of Lemma~\ref{lem:singular division} can be adapted easily to treat the non-symmetric case.
\item
If $(0,0)$ is a zero of order $n>1$ of $\mc D$ (that is, all the partial derivatives of $\mc D$ up to order $n-1$ vanishes at $(0,0)$, while at least one partial derivative of order $n$ is nonzero), then one can prove a division formula similar to \eqref{eq:singular division}, but with a different remainder term. For example, when $n=2$, the remainder term can be written as $J_1(\mc h\mc k)\cdot (\mc h+\mc k) + J_2(\mc h\mc k)$ if $\partial_{\mc h}^2 \mc D(0,0) \ne 0$, or as $J_3(s+t)$ if $\partial_{\mc h}^2 \mc D(0,0)=0$ but $\partial_{\mc h} \partial_{\mc k} \mc D(0,0) \ne 0$.
\item
As we will see in the proof below, the decomposition \eqref{eq:singular division} can be made in the sense of formal power series without using the convergence of the Taylor series of $\mc N$ and $\mc D$. (In fact this is the easiest way to construct $Q(\mc h,\mc k)$ and $J(r)$.) The decomposition \eqref{eq:singular division} will be used in the proof of Proposition~\ref{prop:local expansion} to establish asymptotics expansions of $\mc Z(\mc h,\mc k) = \frac{\mc N(\mc h,\mc k)}{\mc D(\mc h,\mc k)}$ when $(\mc h,\mc k)\to (0,0)$. For this purpose, it is not necessary to know that the series $Q(\mc h,\mc k)$ and $J(r)$ are convergent. Everything can be done by viewing \eqref{eq:singular division} as an asymptotic expansion with a remainder term $O\m({ \max(\abs{\mc h},\abs{\mc k})^n }$ for an arbitrary $n$. However, we find that presenting $Q(\mc h,\mc k)$ and $J(r)$ as analytic functions is conceptually simpler. For this reason, we will still prove that the series $Q(\mc h,\mc k)$ and $J(r)$ are convergent even if it is not absolutely necessary for the rest of this paper.
\end{enumerate}
\end{remark}
\begin{proof}
The proof comes in two steps: first we construct order by order two series $Q(\mc h,\mc k)$ and $J(r)$ which satisfy \eqref{eq:singular division} in the sense of formal power series, and then we show that these series do converge in a neighborhood of the origin.
We approach the construction of $Q(\mc h,\mc k)$ and $J(r)$ as formal power series as follows: Assume first that $Q(\mc h,\mc k)$ and $J(r)$ are given together with the assumptions of the theorem. In that case, for all $n\ge 0$, let $\mc D_n(s,t) = [\lambda^n] \mc D(\lambda s,\lambda t)$, and similarly define $\mc N_n(s,t)$ and $Q_n(s,t)$. By construction, $\mc D_n$, $\mc N_n$ and $Q_n$ are homogeneous polynomials of degree $n$. The assumptions of the lemma ensure that $\mc D_n$ and $\mc N_n$ are symmetric, $\mc D_0=0$, and $\mc D_1(s,t) = d_{1,0}(s+t)$ where $d_{1,0} := \partial_h \mc D(0,0) \ne 0$. On the other hand, let $J_l = [r^l] J(r)$. Then \eqref{eq:singular division} is equivalent to
\begin{equation}\label{eq:singular division rec}
\mc N_n \,=\, \mc D_1 Q_{n-1} \,+\, \m({ \mc D_2 Q_{n-2} +\cdots+ \mc D_n Q_0 } \,+\, J_l \cdot (st)^l \cdot \id_{n=2l \text{ is even}}
\end{equation}
for all $n \ge 0$.
Let us show that this recursion relation indeed uniquely determines $Q_n$ and $J_l$, such that $Q_n(s,t)$ is a homogeneous polynomial of degree $n$ and $J_l\in \complex$. When $n=0$, \eqref{eq:singular division rec} gives $J_0 = \mc N_0 \in \complex$.
When $n\ge 1$, we assume as induction hypothesis that $Q_m(s,t)$ is a symmetric homogeneous polynomial of degree $m$ for all $m<n$. Then
\eqref{eq:singular division rec} can be written as
\begin{equation*}
\tilde{\mc N}_n = d_{1,0}(s+t) \cdot Q_{n-1} + J_l \cdot (st)^l \cdot \id_{n=2l \text{ is even}} \,,
\end{equation*}
where $\tilde{\mc N}_n := \mc N_n - \m({ \mc D_2 Q_{n-2} +\cdots+ \mc D_n Q_0 }$ is a symmetric homogeneous polynomial of degree $n$.
By the fundamental theorem of symmetric polynomials, a bivariate symmetric polynomial can be written uniquely as a polynomial of the elementary symmetric polynomials $s+t$ and $st$. Isolating the terms of degree zero in $s+t$, we deduce that there is a unique pair $Q_{n-1}(s,t)$ and $\tilde J_n(r)$ such that $Q_{n-1}(s,t)$ is symmetric, and
\begin{equation*}
\tilde{\mc N}_n(s,t) = d_{1,0}(s+t) \cdot Q_{n-1}(s,t) + \tilde J_n(st)\,.
\end{equation*}
Moreover, since $\tilde{\mc N}_n(s,t)$ is homogeneous of degree $n$, the polynomials $Q_{n-1}(s,t)$ and $\tilde J_n(st)$ must be homogeneous of degree $n-1$ and $n$ respectively. When $n$ is odd, this implies $\tilde J_n(st)=0$, and when $n=2l$ is even, we must have $\tilde J_n(st) = J_l \cdot (st)^l$ for some $J_l\in \complex$.
By induction, this completes the construction of $Q_n(s,t)$ and $J_l \in \complex$, such that the series defined by $Q(\lambda s,\lambda t) = \sum_n Q_n(s,t) \lambda^n$ and $J(r) = \sum_l J_l r^l$ satisfy \eqref{eq:singular division} in the sense of formal power series.
Now let us show that the series $J(r)$ has a strictly positive radius of convergence.
Since $\mc D(0,0) = 0$ and $\partial_{\mc h} \mc D(0,0) = \partial_{\mc k} \mc D(0,0) \ne 0$, by the implicit function theorem, the equation $\mc D( \mc{h,\tilde k(h)} )=0$ defines locally a holomorphic function $\mc{\tilde k}$ such that $\mc{\tilde k}(0)=0$ and $\mc{\tilde k}'(0)=-1$.
In particular, $\mc{h\cdot \tilde k(h)}$ has a Taylor expansion with leading term $-\mc h^2$, so the inverse function theorem ensures that there exists a holomorphic function $\varphi$ such that $s^2 = \varphi(s) \cdot \tilde{\mc k}(\varphi(s))$ near $s=0$.
Taking $\mc h=\varphi(s)$ and $\mc k=\tilde{\mc k}(\varphi(s))$ in \eqref{eq:singular division} gives that
\begin{equation*}
\mc N \m({ \varphi(s), \tilde{\mc k}(\varphi(s)) } = J(s^2)
\end{equation*}
in the sense of formal power series.
Since $\mc N$, $\tilde{\mc k}$ and $\varphi$ are all locally holomorphic, the series on both sides have a strictly positive radius of convergence.
It remains to prove that $Q(\mc h,\mc k)$ also converges in a neighborhood of the origin. Even though $\mc D(0,0)=0$, the Taylor series of $\mc D(\mc h,\mc k)$ still has a multiplicative inverse in the space of formal Laurent series $\complex(\!(x)\!)[\![y]\!]$. Therefore we can rearrange Equation \eqref{eq:singular division} to obtain in that space
\begin{equation*}
Q(\mc h,\mc k) = \frac{\mc N(\mc h,\mc k) - J(\mc h\mc k)}{\mc D(\mc h,\mc k)}.
\end{equation*}
The \rhs, which will be denoted by $f(\mc h,\mc k)$ below, is a holomorphic function in a neighborhood of $(0,0)$ outside the zero set of $\mc D(\mc h,\mc k)$.
As seen in the previous paragraph, this zero set is locally the graph of the function $\mc{\tilde k(h) \sim -h}$ when $\mc h\to 0$.
It follows that there exists $\delta>0$ such that $f$ is holomorphic in a neighborhood of $(\cdisk_{3 \delta} \setminus \disk_{2\delta}) \times \cdisk_\delta$, where $\cdisk_{3 \delta} \setminus \disk_{2\delta}$ is the closed annulus of outer and inner radii $3\delta$ and $2\delta$ centered at the origin. The usual Cauchy integral formula for the coefficient of Laurent series gives
\begin{equation*}
Q_{m,n} = \m({ \frac{1}{2\pi i} }^2 \oint_{\partial \disk_\delta} \frac{\dd \mc k}{\mc k^{n+1}} \m({
\oint_{\partial \disk_{3\delta}} \frac{\dd \mc h}{\mc h^{m+1}} f(\mc h,\mc k) -
\oint_{\partial \disk_{2\delta}} \frac{\dd \mc h}{\mc h^{m+1}} f(\mc h,\mc k) }.
\end{equation*}
However, by construction, the Laurent series $\sum_{m \in \integer} Q_{m,n} \mc h^m$ does not contain any negative power of $\mc h$. This implies that the integral over $\partial \disk_{2 \delta}$ in the above formula has zero contribution.
Therefore we have
\begin{equation*}
\abs{Q_{m,n}}
= \abs{ \m({ \frac{1}{2\pi} }^2 \oiint_{\partial \disk_{3\delta} \times \partial \disk_\delta} \frac{\dd \mc h\, \dd \mc k}{\mc h^{m+1} \mc k^{n+1}} f(\mc h,\mc k) }
\le (3 \delta)^{-m} \delta^{-n} \! \cdot \!\! \sup_{\partial \disk_{3\delta} \times \partial \disk_\delta} \!\! |f| \,.
\end{equation*}
It follows that the series $Q(\mc h,\mc k) = \sum Q_{m,n} \mc h^m \mc k^n$ converges in a neighborhood of $(0,0)$.
\end{proof}
\begin{proposition}[Asymptotic expansions of $Z(u,v)$]\label{prop:local expansion}
Let $\epsilon=\epsilon(\nu,\theta)>0$ be a value for which the holomorphicity result of Proposition~\ref{prop:singularity structure} and the bound in Corollary~\ref{cor:double angle bound} hold. Then for $(x,y)$ varying in $\cslit \times \cslit$ (when $\nu\ge \nu_c$) or $\cslit \times \cDdom$ (when $1<\nu<\nu_c$), we have
\begin{align}
\label{eq:asym Z 1}
Z(u_c x,u_c y) &= Z\1{reg}\01(x,y) + A(u_c x)\cdot (1-y)^{\alpha_0} + O\mb({ (1-y)^{\alpha_0 + \delta} } \qt{for }x\ne 1\text{ and}\hspace{-2ex} &&\text{as } y\to 1 \,,
\\
\label{eq:asym Z 2}
A(u_c x) &= A\1{reg}(x) + b\cdot (1-x)^{\alpha_1} + O\mb({ (1-x)^{\alpha_1 + \delta} } &&\text{as } x\to 1 \,,
\\
\label{eq:asym Z diag}
Z(u_c x,u_c y) &= Z\1{reg}\02(x,y) + b\cdot Z\1{hom}(1-x,1-y) + O\m({ \max( \abs{1-x}, \abs{1-y})^{\alpha_2+\delta} }
&&\text{as } (x,y)\to (1,1) \,,
\end{align}
where $b=b(\nu)$ is a number determined by the nonzero constants $\mc Z_{3,3}$, $J_1$ and $J_3$ in Lemma~\ref{lem:singular expansion}, and $Z\1{hom}(s,t)$ is a homogeneous function of order $\alpha_2$ (i.e.\ $Z\1{hom}(\lambda s,\lambda t) = \lambda^{\alpha_2} Z\1{hom}(s,t)$ for all $\lambda>0$) that only depends on the phase of the model. Explicitly:
\begin{equation*}
b(\nu) = \begin{cases}
\mc Z_{3,3}(\nu) & \text{when } \nu > \nu_c \\
J_1(\nu) & \text{when } \nu\in(1,\nu_c) \\
-J_3(\nu_c) & \text{when } \nu = \nu_c
\end{cases}
\qtq{and}
Z\1{hom}(s,t) = \begin{cases}
s^{3/2} t^{3/2} & \text{when } \nu > \nu_c \\
\frac{s^{1/2} t^{1/2}}{s^{1/2}+t^{1/2}} & \text{when } \nu\in(1,\nu_c) \\
\frac{-st }{s^{1/3}+t^{1/3}} & \text{when } \nu = \nu_c \,.
\end{cases}
\end{equation*}
On the other hand, $Z\1{reg}\01(x,y) = Z(u_cx,u_c) - \partial_v Z(u_cx,u_c) \cdot u_c\cdot (1-y)$ is an affine function of $y$ satisfying
\begin{equation}\label{eq:local integrability}
Z(u_cx,u_c) = O(1) \qtq{and}
\partial_v Z(u_cx,u_c) = O((1-x)^{-1/2}) \qt{when } x\to 1\,,
\end{equation}
whereas $A\1{reg}(x)$ is an affine function of $x$, and $Z\1{reg}\02(x,y) = Z\1{reg}\01(x,y) + Z\1{reg}\01(y,x) - P(x,y)$ for some polynomial $P(x,y)$ that is affine in both $x$ and $y$. The functions $A\1{reg}(x)$ and $P(x,y)$ will be given in the proof of the proposition.
\end{proposition}
\begin{remark}\label{rem:local expansion}
For a fixed $x$, \eqref{eq:asym Z 1} is an univariate asymptotic expansion in the variable $y$. It has the form
\begin{center}
(analytic function of $y$ near $y=1$) + \texttt{constant} $\cdot\, (1-y)^{\alpha_0}$ + $o((1-y)^{\alpha_0})$,
\end{center}
which makes it a suitable input to the classical transfer theorem of analytic combinatorics. More precisely, when we extract the coefficient of $[y^q]$ from \eqref{eq:asym Z 1} using contour integrals on $\partial \slit$, the contribution of the first term will be exponentially small in $q$, whereas the contributions of the second and the third terms will be of order $q^{-(\alpha_0+1)}$ and $o(q^{-(\alpha_0+1)})$, respectively. Similar remarks can be made for \eqref{eq:asym Z 2} \wrt\ the variable $x$.
The asymptotic expansion \eqref{eq:asym Z diag} has a form that generalizes \eqref{eq:asym Z 1} and \eqref{eq:asym Z 2} in the bivariate case.
Instead of being analytic \wrt\ $x$ or $y$, the first term $Z\1{reg}\02(x,y)$ is a linear combination of terms of the form $F(x) G(y)$ or $G(x) F(y)$, where $F(x)$ is analytic in a neighborhood of $x=1$, and $G(x)$ is locally integrable on the contour $\partial \slit$ near $x=1$. (The local integrability is a consequence of \eqref{eq:local integrability}.) As we will see in Section~\ref{sec:proof diagonal}, a term of this form will have an exponentially small contribution to the coefficient of $[x^py^q]$ in the diagonal limit where $p,q \to\infty$ and that $q/p$ is bounded away from $0$ and $\infty$. On the other hand, the homogeneous function $Z\1{hom}(1-x,1-y)$ is a generalization of the power functions $(1-y)^{\alpha_0}$ and $(1-x)^{\alpha_1}$ of the univariate case.
Indeed, the only homogeneous functions of order $\alpha$ of one variable $s$ are constant multiples of $s^\alpha$. We will see in Section~\ref{sec:proof diagonal} that the term $Z\1{hom}(1-x,1-y)$ gives the dominant contribution of order $p^{-(\alpha_2+2)}$ to the coefficient of $[x^py^q]$ in the diagonal limit.
\end{remark}
\begin{proof}
First consider the non-critical temperatures $\nu\ne \nu_c$. In this case we have $\delta=1/2$, and the definition of $\mc Z(\mc h,\mc k)$ reads $Z(u_c x,u_c y) = \mc Z \mn({(1-x)^{1/2}, (1-y)^{1/2}}$.
As seen in the proof of Lemma~\ref{lem:singular expansion}, for any $\mc h\ne 0$ close to zero, the function $\mc k\mapsto \mc Z(\mc h,\mc k)$ is analytic at $\mc k=0$ and satisfies $\partial_{\mc k} \mc Z(\mc h,0)=0$. Hence it has a Taylor expansion of the form
\begin{equation*}
\mc Z(\mc h,\mc k) = \mc Z(\mc h,0)
+ \frac12 \partial_{\mc k}^2 \mc Z(\mc h, 0) \cdot \mc k^2
+ \frac16 \partial_{\mc k}^3 \mc Z(\mc h, 0) \cdot \mc k^3 + O(\mc k^4) \,.
\end{equation*}
Plugging $\mc h = (1-x)^{1/2}$ and $\mc k=(1-y)^{1/2}$ into the above formula gives the expansion \eqref{eq:asym Z 1} with $\alpha_0=3/2$, $Z\1{reg}\01(x,y) = \mc Z((1-x)^{1/2},0) + \frac12 \partial_{\mc k}^2 \mc Z((1-x)^{1/2},0)\cdot (1-y)$, and
\begin{equation*}
A(u_cx) = \frac16 \partial_{\mc k}^3 \mc Z((1-x)^{1/2},0) \,.
\end{equation*}
We can identify the coefficients in the affine function $y\mapsto Z\01\1{reg}(x,y)$ as $\mc Z((1-x)^{1/2},0)= Z(u_c x,u_c)$ and $\frac12 \partial_{\mc k}^2 \mc Z((1-x)^{1/2},0) = -u_c \cdot \partial_v Z(u_cx,u_c)$. The first term is continuous at $x=1$, thus of order $O(1)$ when $x\to 1$. For the second asymptotics of \eqref{eq:local integrability}, it suffices to show that $\partial_{\mc k}^2 \mc Z(\mc h,0) = O(\mc h^{-1})$.
\paragraph{Low temperatures ($\nu>\nu_c$).}
In this case, $\mc Z(\mc h,\mc k) = \sum_{m,n} \mc Z_{m,n}\, \mc h^m \mc k^n$ with $\mc Z_{1,n}=\mc Z_{n,1}=0$. Hence
\begin{equation*}
A(u_c x) = \sum_{m\ne 1} \mc Z_{m,3}(1-x)^{m/2} = \mc Z_{0,3} + \mc Z_{2,3}\cdot (1-x) + \mc Z_{3,3} \cdot (1-x)^{3/2} + O\m({ (1-x)^2 } \,,
\end{equation*}
which gives the expansion \eqref{eq:asym Z 2} with $\alpha_1=3/2$, $A\1{reg}(x) = \mc Z_{0,3} + \mc Z_{2,3}\cdot (1-x)$ and $b=\mc Z_{3,3}>0$.
Moreover, since $\mc Z$ is analytic at $(0,0)$, we have obviously $\partial_{\mc k}^2 \mc Z(\mc h,0) = O(1)$, which is also an $O(\mc h^{-1})$.
On the other hand, by regrouping terms in the expansion $\mc Z(\mc h,\mc k) = \sum_{m,n} \mc Z_{m,n}\, \mc h^m \mc k^n$, one can write
\begin{align*}
\mc Z(\mc h,\mc k) = &
\sum_{m\ge 0} \mc Z_{m,0}\, \mc h^m
+ \sum_{n\ge 0} \mc Z_{0,n}\, \mc k^n
- \mc Z_{0,0}
+ \sum_{m\ge 2} \mc Z_{m,2}\, \mc h^m \cdot \mc k^2
+ \sum_{n\ge 2} \mc Z_{2,n}\, \mc k^n \cdot \mc h^2
- \mc Z_{2,2}\, \mc h^2 \mc k^2 \\&
+ \mc Z_{3,3}\, \mc h^3 \mc k^3
+ O\m({ \max(|\mc h|,|\mc k|)^7 } \,.
\end{align*}
After plugging in $\mc h = (1-x)^{1/2}$ and $\mc k=(1-y)^{1/2}$, we can identify the first line on the \rhs\ as $Z\1{reg}\02(x,y) = Z\1{reg}\01(x,y) + Z\1{reg}\01(y,x) - P(x,y)$ with $P(x,y) = \mc Z_{0,0} + \mc Z_{2,0} \cdot (1-x) + \mc Z_{0,2} \cdot (1-y) + \mc Z_{2,2} \cdot (1-x)(1-y)$. The term $\mc Z_{3,3} \, \mc h^3 \, \mc k^3$ becomes $b\cdot (1-x)^{3/2}(1-y)^{3/2}$. Thus we obtain the expansion \eqref{eq:asym Z diag} with $\alpha_2=3$ and $Z\1{hom}(s,t) = s^{3/2}t^{3/2}$.
\paragraph{High temperatures ($1<\nu<\nu_c$).}
In this case, we have $\mc Z(\mc h,\mc k) = Q(\mc h,\mc k) + \frac{J(\mc h\mc k)}{\mc D(\mc h,\mc k)}$. Straightforward computation gives that
\begin{align*}
\partial_{\mc k}^2 \mc Z(\mc h,0) &= \partial_{\mc k}^2 Q(\mc h,0)
+ \frac{2 J_2 \mc h^2}{\mc D(\mc h,0)}
- 2J_1 \mc h \cdot \frac{\partial_{\mc k} \mc D(\mc h,0)}{\mc D(\mc h,0)^2}
\\
\partial_{\mc k}^3 \mc Z(\mc h,0) &= \partial_{\mc k}^3 Q(\mc h,0)
+ \frac{6 J_3 \mc h^3}{\mc D(\mc h,0)}
- 6J_2 \mc h^2 \cdot \frac{\partial_{\mc k} \mc D(\mc h,0)}{\mc D(\mc h,0)^2}
- 3J_1 \mc h \cdot \frac{\partial_{\mc k}^2 \mc D(\mc h,0)}{\mc D(\mc h,0)^2}
+ 6J_1 \mc h \cdot \frac{( \partial_{\mc k} \mc D(\mc h,0) )^2}{\mc D(\mc h,0)^3} \,.
\end{align*}
Using the fact that $Q(\mc h,\mc k)$ is analytic at $(0,0)$, and $\mc D(\mc h,0)\sim \mc h$, $\partial_{\mc k} \mc D(0,0) = 1$ and $\partial_{\mc k}^2 \mc D(\mc h,0) = O(1)$ when $\mc h\to 0$, we see that $\partial_{\mc k}^2 \mc Z(\mc h,0) = O(\mc h^{-1})$, whereas all terms in the expansion of $\partial_{\mc k}^3 \mc Z(\mc h,0)$ are of order $O(\mc h^{-1})$, except the last term, which is asymptotically equivalent to $6J_1 \mc h^{-2}$.
It follows that
\begin{equation*}
A(u_c x) = \frac16 \partial_{\mc k}^3 \mc Z((1-x)^{1/2},0)
= J_1\cdot (1-x)^{-1} + O\m({ (1-x)^{-1/2} }\,,
\end{equation*}
which gives the expansion \eqref{eq:asym Z 2} with $\alpha_1=-1$, $A\1{reg}(x) = 0$ and $b=J_1>0$.
On the other hand, Corollary~\ref{cor:double angle bound} and the relations $\check H_c-H\sim \mathtt{cst} \cdot \mc h$ and $\check H_c-K\sim \mathtt{cst} \cdot \mc k$ imply that $\max(|\mc h|,|\mc k|)$ is bounded by a constant times $|\mc h+\mc k|$ when $(x,y)\to (1,1)$ in $\cslit \times \cDdom$. It follows that
\begin{equation}\label{eq:1/D estimate}
\frac{1}{\mc h+\mc k} = O\m({ \max(|\mc h|,|\mc k|)^{-1} }
\qtq{and}
\frac{1}{\mc D(\mc h,\mc k)} =
\frac{1}{\mc h+\mc k + O\m({(\mc h+\mc k)^2} }
= \frac{1}{\mc h+\mc k} + O(1) \,.
\end{equation}
From these we deduce that $\frac{J(\mc h\mc k)}{\mc D(\mc h,\mc k)} = \frac{J_1 \mc h \mc k}{\mc h + \mc k} + O\m({ \max(|\mc h|,|\mc k|)^2 }$.
Thus we can regroup terms in the decomposition $\mc Z(\mc h,\mc k) = Q(\mc h,\mc k) + \frac{J(\mc h\mc k)}{\mc D(\mc h,\mc k)}$ to get
\begin{equation*}
\mc Z(\mc h,\mc k) =
\sum_{m\ge 0} Q_{m,0} \mc h^m
+ \sum_{n\ge 0} Q_{0,n} \mc k^n - Q_{0,0}
+ \frac{J_1\mc h \mc k}{\mc h + \mc k}
+ O\m({ \max(|\mc h|,|\mc k|)^2 }
\end{equation*}
After plugging in $\mc h = (1-x)^{1/2}$ and $\mc k=(1-y)^{1/2}$, we can identify the first three terms on the \rhs\ as $Z\1{reg}\02(x,y) = Z\1{reg}\01(x,y) + Z\1{reg}\01(y,x) - Q_{0,0}$ up to a term of order $O\m({ \max(|1-x|,|1-y|) }$. The term $\frac{J_1\mc h \mc k}{\mc h + \mc k}$ becomes $b\cdot \frac{(1-x)^{1/2} (1-y)^{1/2}}{(1-x)^{1/2} + (1-y)^{1/2}}$. Thus we obtain \eqref{eq:asym Z diag} with $\alpha_2=1/2$ and $Z\1{hom}(s,t) = \frac{s^{1/2} t^{1/2}}{s^{1/2} + t^{1/2}}$.
\paragraph{Critical temperature ($\nu=\nu_c$).}
At the critical temperature, $\delta=1/3$ and the definition of $\mc Z(\mc h,\mc k)$ reads $Z(u_c x,u_c y) = \mc Z((1-x)^{1/3},(1-y)^{1/3})$. In this case, $\mc k \mapsto \mc Z(\mc h,\mc k)$ has a Taylor expansion of the form
\begin{equation*}
\mc Z(\mc h,\mc k) = \mc Z(\mc h,0)
+ \frac16 \partial_{\mc k}^3 \mc Z(\mc h, 0) \cdot \mc k^3
+ \frac1{24} \partial_{\mc k}^4 \mc Z(\mc h, 0) \cdot \mc k^4 + O(\mc k^5) \,,
\end{equation*}
because $\partial_{\mc k} \mc Z(\mc h,0)=\partial_{\mc k}^2 \mc Z(\mc h,0)=0$.
Plugging $\mc h = (1-x)^{1/3}$ and $\mc k=(1-y)^{1/3}$ into the above formula gives \eqref{eq:asym Z 1} with $\alpha_0=4/3$, $Z\1{reg}\01(x,y) = \mc Z((1-x)^{1/3},0) + \frac16 \partial_{\mc k}^3 \mc Z((1-x)^{1/3},0)\cdot (1-y)$, and
\begin{equation*}
A(u_c x) = \frac1{24} \partial_{\mc k}^4 \mc Z((1-x)^{1/3},0) \,.
\end{equation*}
As in the non-critical case, we identify $\mc Z((1-x)^{1/3},0)= Z(u_c x,u_c)$ and $\frac16 \partial_{\mc k}^3 \mc Z((1-x)^{1/3},0) = -u_c \cdot \partial_v Z(u_cx,u_c)$. The first term is still continuous at $x=1$, thus of order $O(1)$ when $x\to 1$. Let us show that $\partial_{\mc k}^3 \mc Z(\mc h,0)$ is analytic at $\mc h=0$ so that the second term is also continuous.
From the expansion $\mc Z(\mc h,\mc k) = Q(\mc h,\mc k) + \frac{J(\mc h\mc k)}{\mc D(\mc h,\mc k)}$ with $J_1=J_2=0$ and $Q_{1,n}=Q_{2,n}=0$ for all $n$, we obtain
\begin{align*}
\partial_{\mc k}^3 \mc Z(\mc h,0) &= \partial_{\mc k}^3 Q(\mc h,0)
+ \frac{6 J_3 \mc h^3}{\mc D(\mc h,0)}
\\
\frac{1}{24} \partial_{\mc k}^4 \mc Z(\mc h,0) &= Q_{0,4}
+ \sum_{m\ge 3} Q_{m,4} \mc h^m
+ \frac{J_4 \mc h^4}{\mc D(\mc h,0)}
- J_3 \mc h^3 \cdot \frac{\partial_{\mc k} \mc D(\mc h,0)}{\mc D(\mc h,0)^2} \,.
\end{align*}
Recall that $\mc D(\mc h,0)\sim \mc h$ and $\partial_{\mc k} \mc D(0,0)=1$. Then it is not hard to see that $\partial_{\mc k}^3 \mc Z(\mc h,0)$ is analytic at $\mc h=0$.
On the other hand, the second and the third terms in the expansion of $\frac{1}{24} \partial_{\mc k}^4 \mc Z(\mc h,0)$ are $O(\mc h^3)$, whereas the last term is equivalent to $J_3 \mc h$. It follows that
\begin{equation*}
A(u_c x) = \frac1{24} \partial_{\mc k}^4 \mc Z((1-x)^{1/3},0)
= Q_{0,4} - J_3 \cdot (1-x)^{1/3} + O\m({ (1-x)^{2/3} } \,,
\end{equation*}
which gives the expansion \eqref{eq:asym Z 2} with $\alpha_1=1/3$, $A\1{reg}(x) = Q_{0,4}$ and $b=-J_3<0$.
As in the high temperature case, we still have the estimate \eqref{eq:1/D estimate} when $(\mc h,\mc k)\to (0,0)$ such that the corresponding $(x,y)$ varies in $\cslit \times \cslit$.
Moreover, at the critical temperature we have $J_1=J_2=0$ and $Q_{1,n} = Q_{n,1} = Q_{2,n} = Q_{n,2} = 0$ for all $n$. Therefore $\frac{J(\mc h\mc k)}{\mc D(\mc h,\mc k)} = \frac{J_3 (\mc h \mc k)^3}{\mc h + \mc k} + O\m({ \max(|\mc h|,|\mc k|)^6 }$, and we can regroup terms in the decomposition $\mc Z(\mc h,\mc k) = Q(\mc h,\mc k) + \frac{J(\mc h\mc k)}{\mc D(\mc h,\mc k)}$ to get
\begin{align*}
\mc Z(\mc h,\mc k) =&
\sum_{m\ge 0} Q_{m,0} \mc h^m
+ \sum_{n\ge 0} Q_{0,n} \mc k^n
- Q_{0,0}
+ \sum_{m\ge 3} Q_{m,3} \mc h^m \cdot \mc k^3
+ \sum_{n\ge 3} Q_{3,n} \mc k^n \cdot \mc h^3 \\&
+ \frac{J_3(\mc h \mc k)^3}{\mc h + \mc k}
+ O\m({ \max(|\mc h|,|\mc k|)^6 }
\end{align*}
After plugging in $\mc h = (1-x)^{1/3}$ and $\mc k=(1-y)^{1/3}$, we can identify the terms on the first line of the \rhs\ as $Z\1{reg}\02(x,y) = Z\1{reg}\01(x,y) + Z\1{reg}\01(y,x) - P(x,y)$ up to a term of order $O\m({ \max(|1-x|,|1-y|)^2 }$, where $P(x,y) = Q_{0,0} + Q_{3,0}\cdot (1-x) + Q_{0,3}\cdot (1-y)$. The term $\frac{J_3(\mc h \mc k)^3}{\mc h + \mc k}$ becomes $-b\cdot \frac{(1-x)(1-y)}{(1-x)^{1/3} + (1-y)^{1/3}}$. Thus we obtain \eqref{eq:asym Z diag} with $\alpha_2=5/3$ and $Z\1{hom}(s,t) = \frac{-st}{s^{1/3} + t^{1/3}}$.
\end{proof}
\begin{corollary}\label{cor:A Delta-analytic}
The function $x\mapsto A(u_c x)$ has an analytic continuation on $\slit$.
\end{corollary}
\begin{proof}
We have seen in the proof of Proposition~\ref{prop:local expansion} that $A(u_c x) = \frac1{m!} \partial_{\mc k}^m \mc Z((1-x)^\delta,0)$, where $m=\frac1\delta+1$ is equal to $3$ when $\nu\ne \nu_c$, and equal to $4$ when $\nu=\nu_c$. The change of variable $\mc h = (1-x)^\delta$ defines a conformal bijection from $x\in \slit$ to some simply connected domain $\mc U_\epsilon$ whose boundary contains the point $\mc h=0$.
In the proof of Lemma~\ref{lem:singular expansion}, we have shown that the mapping $h\mapsto \mc h = \mn({1-\check x(\check H_c-h)}^\delta$ has an analytic inverse $\psi(\mc h)$ in a neighborhood of $\mc h=0$ such that $\mc Z(\mc h,\mc k) = \check Z(\check H_c -\psi(\mc h),\check H_c -\psi(\mc k))$.
Let $\Psi(\mc h)=\check H_c-\psi(\mc h)$, then $\Psi$ is a local analytic inverse of the mapping $H\mapsto \m({1-\check x(H)}^\delta$, and
\begin{equation}\label{eq:Z-tilde = Z-hat(psi) bis}
\mc Z(\mc h,\mc k) = \check Z(\Psi(\mc h),\Psi(\mc k)) \,.
\end{equation}
By Lemma~\ref{lem:unique dominant}, $\check x$ defines a conformal bijection from $\Hdom$ to $\slit$. On the other hand, $x\mapsto (1-x)^\delta$ is a conformal bijection from $\slit$ to $\mc U_\epsilon$. It follows that $\Psi$ can be extended to a conformal bijection from $\mc U_\epsilon$ to $\Hdom$.
Now fix some $x_*\in \slit$ and the corresponding $\mc h_* = (1-x_*)^\delta \in \mc U_\epsilon$ and $H_* =\Psi(\mc h_*) \in \Hdom$.
Let $\mc K \subset \Hdom$ be a compact neighborhood of $H_*$. According to Remark~\ref{rem:extended holomorphicity Z}, there exists an open set $\mc V$ containing $\cHdom[0]$ such that $\check Z$ is holomorphic in $\mc K\times \mc V$. As $\check H_c\in \mc V$, this implies in particular that $\check Z$ is analytic at $(H_*,\check H_c)$.
Since $\Psi(\mc h_*)=H_*$ and $\Psi(0)=\check H_c$, and we have seen that $\Psi$ is analytic at both $\mc h_*$ and $0$, the relation \eqref{eq:Z-tilde = Z-hat(psi) bis} implies that $\mc Z$ is analytic at $(\mc h_*,0)$. It follows that $A(u_c x) = \frac{1}{m!} \partial_{\mc k}^m \mc Z((1-x)^\delta,0)$ is analytic at $x=x_*$.
\end{proof}
\begin{corollary}
A parametrization of $x\mapsto A(u_c x)$ is given by $x=\check x(H)$ and
\begin{equation*}
\check A(H) = \begin{cases}
\frac{1}{\check x_2^{3/2}} \mB({ \check Z_3(H) - \frac{\check x_3}{\check x_2} \check Z_2(H) } & \text{when }\nu \ne \nu_c \\
\frac{1}{\check x_3^{4/3}} \mB({ \check Z_4(H) - \frac{\check x_4}{\check x_3} \check Z_3(H) } & \text{when }\nu = \nu_c
\end{cases}
\end{equation*}
where $\check x_n$ are defined as in Lemma~\ref{lem:singular expansion}, and $\check Z_n(H)$ are defined by the Taylor expansion $\check Z(H,\check H_c-k) = \sum_n \check Z_n(H) k^n$.
\end{corollary}
\begin{proof}
We have seen in the previous proof that $A(u_c x) = \frac{1}{m!} \partial_{\mc k} \mc Z((1-x)^\delta,0)$ with $m=3$ if $\nu\ne \nu_c$ and $m=4$ if $\nu=\nu_c$. Moreover, $\mc Z$ satisfies $\mc Z(\mc h,\mc k) = \check Z(\check H_c -\psi(\mc h), \check H_c -\psi(\mc k))$, where $\psi(h)$ is the local inverse of $h\mapsto (1-\check x(\check H_c-h))^\delta$. It follows that
\begin{equation}\label{eq:A as Taylor coefficient}
\check A(H) \equiv A(u_c \cdot \check x(H)) = \frac{1}{m!} \partial_{\mc k}^m \check Z\m({ H,\check H_c - \psi(\mc k)}\big|_{\mc k=0 \ .}
\end{equation}
Using the definition of the coefficients $\check x_n$ and the Lagrange inversion formula, it is not hard to obtain that
\begin{equation*}
\psi(\mc k) = \begin{cases}
\frac{1}{\check x_2^{1/2}} \mc k - \frac{\check x_3}{2 \check x_2^2} \mc k^2 + O(\mc k^3) & \text{when }\nu \ne \nu_c \\
\frac{1}{\check x_3^{1/3}} \mc k - \frac{\check x_4}{3 \check x_3^{5/3}} \mc k^2 + O(\mc k^3) & \text{when }\nu = \nu_c \,.
\end{cases}
\end{equation*}
Now plug $k=\psi(\mc k)$ into $\check Z(H,\check H_c-k) = \sum_n \check Z_n(H) k^n$, and compute the Taylor expansion in $\mc k$ while taking into account the fact that $\check Z_1(H)=0$ for all $\nu$ and $\check Z_2(H)=0$ when $\nu=\nu_c$ (see Equation~\eqref{eq:vanishing derivatives}). According to \eqref{eq:A as Taylor coefficient}, $\check A(H)$ is given by the coefficient of $\mc k^m$ in this Taylor expansion. Explicit expansion gives the expressions in the statement of the corollary.
\end{proof}
\section{Coefficient asymptotics of $Z(u,v;\nu)$ --- proof of Theorem~\ref{thm:asympt}}\label{sec:coeff asymp}
Theorem~\ref{thm:asympt} gives the asymptotics of $z_{p,q}$ when $p,q\to \infty$ in two regimes: either $p \to\infty$ after $q \to\infty$, or $p \to\infty$ and $q \to\infty$ simultaneously while $q/p$ stays in some compact interval $[\lambda_{\min},\lambda_{\max}] \subset (0,\infty)$. We will call the first case \emph{two-step asymptotics}, and the second case \emph{diagonal asymptotics}. Let us prove the two cases separately.
\subsection{Two-step asymptotics}\label{sec:proof two-step}
At the critical temperature $\nu=\nu_c$, the two-step asymptotics of $z_{p,q}$ has already been established in \cite{CT20}. The basic idea is to apply the classical transfer theorem \cite[Corollary VI.1]{FS09} to the function $y \mapsto Z(u_c x,u_c y)$ to get the asymptotics of $z_{p,q}$ when $q\to\infty$, and then to the function $x\mapsto A(u_c x)$ to get the asymptotics of $a_p$ when $p\to\infty$.
Proposition~\ref{prop:singularity structure}~and~\ref{prop:local expansion} provide all the necessary input for extending the same schema of proof to non-critical temperatures.
\begin{proof}[Proof of Theorem~\ref{thm:asympt} --- two-step asymptotics]
According to Proposition~\ref{prop:singularity structure}, for any fixed $x\in \slit$, the function $y\mapsto Z(u_c x,u_c y)$ is holomorphic in the $\Delta$-domain $\Ddom$. And \eqref{eq:asym Z 1} of Proposition~\ref{prop:local expansion} states that, as $y \to 1$ in $\Ddom$, the dominant singular term in the asymptotic expansion of $y\mapsto Z(u_c x,u_c y)$ is $A(u_c x) \cdot (1-y)^{\alpha_0}$.
It follows from the transfer theorem that
\begin{equation}\label{eq:asym Z_q}
u_c^q \cdot Z_q(u_c x) \eqv q \frac{A(u_c x)}{\Gamma(-\alpha_0)} \cdot q^{-(\alpha_0+1)}
\end{equation}
(Recall that $Z_q(u)$ is the coefficient of $v^q$ in the generating function $Z(u,v)$.) The above asymptotics is valid for all $x\in \cslit \setminus \{1\}$. It does not always hold at $x=1$ because $A(u_c)=\infty$ in the high temperature regime.
However, if we replace $x$ by $\frac{u_0}{u_c}x$ for some arbitrary $u_0\in (0,u_c)$, then the asymptotics is valid for all $x\in \cslit$. Then, by dividing the asymptotics by the special case of itself at $x=1$, we obtain the convergence
\begin{equation*}
\frac{Z_q(u_0 x)}{Z_q(u_0)} \cv[]q \frac{A(u_0 x)}{A(u_0)}
\end{equation*}
for all $x\in \cslit$. For each $q$, the \lhs\ is the generating function of a nonnegative sequence $\m({ \frac{u_0^p \cdot z_{p,q}}{Z_q(u_0)} }_{p\ge 0}$ which always sums up to $1$ (that is, a probability distribution on $\natural$). According to a general continuity theorem \cite[Theorem IX.1]{FS09}, this implies the convergence of the sequence term by term:
\begin{equation*}
\frac{u_0^p \cdot z_{p,q}}{ Z_q(u_0) } \cv[]q \frac{u_0^p \cdot a_p}{A(u_0)}
\end{equation*}
for all $p\ge 0$. On the other hand, \eqref{eq:asym Z_q} implies that $u_c^q\cdot Z_q(u_0) \eqv q \frac{A(u_0)}{\Gamma(-\alpha_0)} \cdot q^{-(\alpha_0+1)}$. Multiplying this equivalence with the above convergence gives the asymptotics of $z_{p,q}$ when $q\to\infty$ in Theorem~\ref{thm:asympt}.
The asymptotics of $a_p$ in Theorem~\ref{thm:asympt} is a direct consequence of the transfer theorem, given the asymptotic expansion \eqref{eq:asym Z 2} of $x\mapsto A(u_c x)$ in Proposition~\ref{prop:local expansion} and its $\Delta$-analyticity in Corollary~\ref{cor:A Delta-analytic}.
\end{proof}
\subsection{Diagonal asymptotics}\label{sec:proof diagonal}
\newcommand{\Vet}{V_{\epsilon,\theta}}
\newcommand{\Ve}{V_\epsilon}
In the diagonal limit, we have not found a general transfer theorem in the literature that allows one to deduce asymptotics of the coefficients $z_{p,q}$ from asymptotics of the generating function $Z(u_cx,u_cy)$.
However, it turns out that with the ingredients given in Proposition~\ref{prop:singularity structure}~and~\ref{prop:local expansion}, we can generalize the proof of the classical transfer theorem in \cite{FS09} to the diagonal limit in the case of the generating function $Z(u_cx,u_cy)$. Let us first describe (a simplified version of) the proof in \cite{FS09}, before generalizing it to prove the diagonal asymptotics in Theorem~\ref{thm:asympt}:
Given a generating function $F(x) = \sum_n F_n x^n$ with a unique dominant singularity at $x=1$ and an analytic continuation up to the boundary of a $\Delta$-domain $\cDdom$, one first expresses the coefficients of $F(x)$ as contour integrals on the boundary of $\Ddom$
\begin{equation*}
F_n = \frac{1}{2\pi i} \oint_{\partial \Ddom} \frac{F(x)}{x^{n+1}} \dd x \,.
\end{equation*}
Next, one shows that the integral on the circular part of $\Ddom$ is exponentially small in $n$ and therefore
\begin{equation*}
F_n = \frac{1}{2\pi i} \int_{\Vet} \frac{F(x)}{x^{n+1}} \dd x + O \m({ (1+\epsilon)^{-n} }\,,
\end{equation*}
where $\Vet = \partial \Ddom \setminus (1+\epsilon)\cdot \partial \disk$ is the rectilinear part of the contour $\partial \Ddom$ (see Figure~\refp{b}{fig:V-path}). Then one plugs the asymptotic expansion of $F(x)$ when $x\to 1$ into the integral. One shows that any term that is analytic at $x=1$ in the expansion will have an exponentially small contribution, and terms of the order $(1-x)^\alpha$ and $O\m({ (1-x)^\alpha }$ have contributions of the order $n^{-(\alpha+1)}$ and $O\m({ n^{-(\alpha+1)} }$, respectively.
\begin{proof}[Proof of Theorem~\ref{thm:asympt} --- diagonal asymptotics]
By Proposition~\ref{prop:singularity structure}, the function $(x,y)\mapsto Z(u_cx,u_cy)$ is holomorphic in $\cslit \times \cDdom$, and hence we can express the coefficient $[x^py^q] Z(u_cx,u_cy)$ as a double contour integral and deform the contours of integral to the boundary of that domain. This gives
\begin{equation*}
u_c^{p+q} \cdot z_{p,q} =
\m({\frac{1}{2\pi i}}^2 \oiint_{\partial \slit \times \partial \Ddom} \frac{Z(u_cx,u_cy)}{x^{p+1}y^{q+1}} \dd x\dd y \,.
\end{equation*}
First, let us show that the contour integral can be restricted to a neighborhood of the dominant singularity $(x,y)=(1,1)$ with an exponentially small error.
Let $\Ve = \partial \slit \setminus (1+\epsilon)\cdot \partial \disk$ be the rectilinear portion of the contour $\partial \slit$.
It consists of two oriented line segments living in the Riemann sphere with a branch cut along $(1,\infty)$.
Similarly, define $\Vet = \partial \Ddom \setminus (1+\epsilon)\cdot \partial \disk$. The two paths $\Ve$ and $\Vet$ are depicted in Figure~\refp{a--b}{fig:V-path}.
For all $(x,y)\in \partial \slit \times \partial \Ddom$, we have $|x|\ge 1$ and $|y|\ge 1$. Moreover, if $(x,y)\notin \Ve \times \Vet$, then either $|x|=1+\epsilon$ or $|y|=1+\epsilon$. Since $Z(u_cx,u_cy)$ is continuous on $\partial \slit \times \partial \Ddom$, it follows that
\begin{align*}
&\ \abs{
\m({\frac{1}{2\pi i}}^2
\iint_{(\partial \slit \times \partial \Ddom) \setminus (\Ve \times \Vet)}
\frac{Z(u_cx,u_cy)}{x^{p+1}y^{q+1}} \dd x\dd y
}
\\ \le &\
\m({\frac{1}{2\pi}}^2
\sup_{(x,y)\in \partial \slit \times \partial \Ddom}
\hspace{-8mm} \abs{Z(u_cx,u_cy)} \hspace{4mm}
\cdot (1+\epsilon)^{-\min(p,q)}
\ = \ O\m({ (1+\epsilon)^{-\lambda_{\min} p} }
\end{align*}
where we assume \wlg\ $\lambda_{\min}\le 1$, so that $\min(p,q)\ge \lambda_{\min} p$ whenever $q/p \in [\lambda_{\min},\lambda_{\max}]$.
Thus we can forget about the integral outside $\Ve \times \Vet$ with an exponentially small error in the diagonal limit.
Using the expansion \eqref{eq:asym Z diag} in Proposition~\ref{prop:local expansion}, we can decompose the integral on $\Ve \times \Vet$ as
\begin{equation*}
\m({\frac{1}{2\pi i}}^2
\iint_{\Ve \times \Vet}
\frac{Z(u_cx,u_cy)}{x^{p+1}y^{q+1}} \dd x\dd y
\ = \ I\1{reg} + b\cdot I\1{hom} + I\1{rem}
\end{equation*}
where $I\1{reg}$, $I\1{hom}$ and $I\1{rem}$ are defined by replacing $Z(u_cx,u_cy)$ in the integral on the \lhs\ by $Z\02\1{reg}(x,y)$, $Z\1{hom}(1-x,1-y)$ and $O(\max(|1-x|,|1-y|)^{\alpha_2+\delta})$ respectively.
As mentioned in Remark~\ref{rem:local expansion}, $Z\02\1{reg}(x,y)$ is a linear combination of terms of the form $F(x)G(y)$ or $G(x)F(y)$, where $F$ is analytic in a neighborhood of $1$, and $G$ is integrable on $\Ve$ and $\Vet$ for $\epsilon$ small enough.
Consider the component of $I\1{reg}$ corresponding to one such term: the integral factorizes as
\begin{equation}\label{eq:factorized iint}
\iint_{\Ve \times \Vet}
\frac{F(x)G(y)}{x^{p+1}y^{q+1}} \dd x\dd y
\ =\
\m({ \int_{\Ve } \frac{F(x)}{x^{p+1}} \dd x } \cdot
\m({ \int_{\Vet} \frac{G(y)}{y^{q+1}} \dd y }\,.
\end{equation}
Since $F$ is analytic in a neighborhood of $1$, we can deform the contour of integration $\Ve$ in the first factor away from $x=1$, so that it stays away from a disk of radius $r>1$ centered at the origin. It follows that the integral is bounded as an $O(r^{-p})$.
On the other hand, the second integral is bounded by a constant $\int_{\Vet}|G(y)|\dd |y|<\infty$ thanks to the integrability of $G$ on $\Vet$.
Hence the \lhs\ of \eqref{eq:factorized iint} is also an $O(r^{-p})$. Since $I\1{reg}$ is a linear combination of terms of this form, we conclude that there exists $r_*>1$ such that $I\1{reg} = O(r_*^{-p})$ when $p,q\to \infty$ and $\frac qp\in [\lambda_{\min},\lambda_{\max}]$.
\begin{figure}
\centering
\includegraphics[scale=1]{Fig5.pdf}
\caption{The paths of integration $\Ve$, $\Vet$ and $\mc \Ve$, $\mc \Vet$.}
\label{fig:V-path}
\end{figure}
Next, let us prove that $I\1{rem} = O(p^{-\alpha_2+2+\delta})$ in the same limit. Consider the change of variables $s=p(1-x)$ and $t=p(1-y)$, and denote by $\mc \Ve$ and $\mc \Vet$ respectively the images of $\Ve$ and $\Vet$ under this change of variable, as in Figure~\refp{c--d}{fig:V-path}. (Notice that these paths now depend on $p$.)
Then $I\1{rem}$ can be written as
\begin{equation*}
I\1{rem} = \m({ \frac{1}{2\pi i} }^2 \iint_{\mc \Ve \times \mc \Vet} \frac{ O\m({ \max(p^{-1}|s|,p^{-1}|t|)^{\alpha_2+\delta} } }{ (1-p^{-1}s)^{p+1} (1-p^{-1}t)^{q+1} } \frac{\dd s\dd t}{p^2} \,.
\end{equation*}
On the one hand, there exists a constant $C_1$ such that $\abs{ O\m({ \max(p^{-1}|s|,p^{-1}|t|)^{\alpha_2+\delta} } } \le C_1 \cdot p^{-(\alpha_2+\delta)} \cdot \m({ |s| + |t|}^{\alpha_2+\delta}$ for all $(s,t)\in \mc \Ve \times \mc \Vet$. On the other hand, since $|s|\le p\epsilon$ and $|t\cos \theta|\le p\epsilon$ for all $(s,t)\in \mc \Ve \times \mc \Vet$, and $q\ge \lambda_{\min} p$, it is not hard to see that
\begin{equation*}
\abs{ \m({1-\frac sp}^{p+1} }
= \m({ 1+\frac{|s|}{p} }^{p+1}
\ge e^{C_2 |s|}
\qtq{and}
\abs{ \m({1-\frac tp}^{q+1} }
\ge \m({ 1+\frac{|t|\cos \theta}{p} }^{q+1} \ge e^{C_2 |t|}
\end{equation*}
for some constant $C_2>0$ that only depends on $\epsilon$, $\theta$ and $\lambda_{\min}$. It follows that
\begin{equation*}
\abs{I\1{rem}} \le \frac{C_1}{4 \pi^2} \cdot p^{-(\alpha_2+2+\delta)} \iint_{\mc \Ve \times \mc \Vet} \m({|s|+|t|}^{\alpha_2+\delta} e^{-C_2(|s|+|t|)} \dd|s|\,\dd|t| \,.
\end{equation*}
The integral on the \rhs\ is bounded by the constant $4\int_0^\infty \dd r_1 \int_0^\infty \dd r_2 \cdot e^{-C_2(r_1+r_2)}\cdot \m({r_1+r_2}^{\alpha_2+\delta} <\infty$. Hence $I\1{rem} = O\m({ p^{-(\alpha_2+2+\delta)} }$.
To estimate the term $I\1{hom}$, we make the same change of variables as for $I\1{rem}$. Since $Z\1{hom}$ is homogeneous of order $\alpha_2$, we have
\begin{equation*}
I\1{hom} = \m({ \frac{1}{2\pi i} }^2 \iint_{\mc \Ve \times \mc \Vet} \frac{ p^{-\alpha_2} Z\1{hom}(s,t) }{ (1-p^{-1}s)^{p+1} (1-p^{-1}t)^{q+1} } \frac{\dd s\dd t}{p^2} \,
\end{equation*}
Using again the fact that $|s|\le p\epsilon$ and $|t|\cos \theta \le p\epsilon$ for $(s,t)\in \mc \Ve \times \mc \Vet$,
we can expand in the denominator in the integral as $\m({1-p^{-1}s}^{-(p+1)} \m({1-p^{-1}t}^{-(q+1)} \!= \exp \m({s+\frac qpt} \cdot \m({1+O \m({ \max(|p^{-1}s^2|,|p^{-1}t^2| }}$. More precisely, the big-O means that there exists a constant $C_3$ depending only on $\epsilon$, $\theta$ and $\lambda_{\min}$ such that for all $(s,t)\in \mc \Ve \times \mc \Vet$
\begin{equation*}
\abs{ \m({1-\frac sp}^{-(p+1)} \m({1-\frac tp}^{-(q+1)} - e^{s+\frac qpt} } \le C_3 \frac{(|s|+|t|)^2}p \cdot \abs{e^{s+\frac qpt}} \,.
\end{equation*}
Moreover, there exists $C_4>0$ such that $|e^{s+\frac qpt}| \le e^{-C_4(|s|+|t|)}$ for all $(s,t)\in \mc \Ve \times \mc \Vet$.
It follows that
\begin{align*}
&\ \abs{
I\1{hom} - \m({ \frac{1}{2\pi i} }^2 \iint_{\mc \Ve \times \mc \Vet}
p^{-\alpha_2}Z\1{hom} (s,t) \cdot e^{s+\frac qpt} \cdot \frac{\dd s\dd t}{p^2}
}
\\ \le &\
\frac{1}{4\pi^2} \iint_{\mc \Ve \times \mc \Vet} p^{-\alpha_2} |Z\1{hom}(s,t)|\cdot C_3 \frac{(|s|+|t|)^2}p e^{-C_4(|s|+|t|)} \cdot \frac{\dd |s|\, \dd |t|}{p^2}
\\ \le &\
C_5 \cdot p^{-(\alpha_2+3)} \iint_{\mc \Ve \times \mc \Vet} (|s|+|t|)^{\alpha_2+2} e^{-C_4(|s|+|t|)} \dd |s|\, \dd |t| \,.
\end{align*}
where for the last line we used the bound $\abs{Z\1{hom}(s,t)}\le \mathtt{cst}\cdot (|s|+|t|)^{\alpha_2}$, which is a consequence of the fact that $Z\1{hom}(s,t)$ is homogeneous of order $\alpha_2$ and continuous on $\Ve \times \Vet$.
The integral on the last line is bounded by $4 \int_0^\infty \dd r_1 \int_0^\infty \dd r_2 \cdot e^{-C_4(r_1+r_2)}\cdot \m({r_1+r_2}^{\alpha_2+2} <\infty$. Thus we have
\begin{equation*}
I\1{hom} = p^{-(\alpha_2+2)} \cdot \m({ \frac{1}{2\pi i} }^2
\iint_{\mc \Ve \times \mc \Vet} Z\1{hom} (s,t) e^{s+\frac qpt} \dd s\dd t
+ O\m({ p^{-(\alpha_2+3)} } \,.
\end{equation*}
Let $\mc V_\infty$ and $\mc V_{\infty,\theta}$ be obtained by extending the line segments in $\mc \Ve$ and $\mc \Vet$ to rays joining the origin to the infinity. Thanks to the exponentially decaying factor $e^{s+\frac qpt}$ in the above integral, one can replace the domain $\mc \Ve \times \mc \Vet$ of the integral by $\mc V_\infty \times \mc V_{\infty,\theta}$ while committing an error that is exponentially small in $p$ (recall that $\mc \Ve$ and $\mc \Vet$ depend on $p$). Therefore $I\1{hom} = \tilde c(q/p)\cdot p^{-(\alpha_2+2)} + O \m({ p^{-(\alpha_2+3)} }$, where
\begin{equation*}
\tilde c(\lambda) := \m({ \frac{1}{2\pi i} }^2
\iint_{\mc V_\infty \times \mc V_{\infty,\theta}} Z\1{hom} (s,t) e^{s+\lambda t} \dd s\dd t \,.
\end{equation*}
With the previous estimates for $I\1{reg}$ and $I\1{rem}$, we get
\begin{equation*}
u_c^{p+q} z_{p,q} = b\cdot \tilde c(q/p)\cdot p^{-(\alpha_2+2)} + O \m({ p^{-(\alpha_2+2+\delta)} } \,.
\end{equation*}
where the big-O estimate is uniform for all $p,q\to \infty$ such that $q/p\in [\lambda_{\min},\lambda_{\max}]$.
We finish the proof by computing $\tilde c(\lambda)$ or, in the notation of Theorem~\ref{thm:asympt}, $c(\lambda) = \Gamma(-\alpha_0)\Gamma(-\alpha_1) \cdot \tilde c(\lambda)$. Notice that the value of $\tilde c(\lambda)$ does not depend on the angle $\theta$ appearing in the contour of integration $\mc V_{\infty,\theta}$.
\paragraph{Low temperatures.}
When $\nu > \nu_c$, we have $Z\1{hom}(s,t)=s^{3/2}t^{3/2}$ and Proposition~\ref{prop:singularity structure}~and~\ref{prop:local expansion} allow us take $\theta=0$. Then the double integral defining $\tilde c(\lambda)$ factorizes as
\begin{equation*}
\tilde c(\lambda) =
\m({ \frac{1}{2\pi i} \int_{\mc V_\infty} s^{3/2} e^s \dd s }
\cdot \m({ \frac{1}{2\pi i} \int_{\mc V_\infty} t^{3/2} e^{\lambda t} \dd t } \,.
\end{equation*}
After the change of variable $t'=\lambda t$ in the second factor, the formula simplifies to $\tilde c(\lambda) \!=\!
\m({ \frac{1}{2\pi i} \int_{\mc V_\infty}\! s^{3/2} e^s \dd s }^2 \lambda^{-5/2}$.
Since the contour $\mc V_\infty$ lives in the Riemann sphere with a branch cut along $(-\infty,0)$, the function $s^{3/2}$ should be understood as its principal branch \wrt\ this branch cut. Therefore
\begin{align*}
\frac{1}{2\pi i} \int_{\mc V_\infty} s^{3/2} e^s \dd s &\ =
\frac{1}{2\pi i} \int_0^\infty \m({ (-r+i0)^{3/2} - (-r-i0)^{3/2} } e^{-r} \dd r
\\ &\ =
\frac{1}{2\pi i} \int_0^\infty \m({ -ir^{3/2} - ir^{3/2} } e^{-r} \dd r
\ =\ -\frac{\Gamma(5/2)}{\pi} \,.
\end{align*}
Recall that in the low temperature regime, $\alpha_0=\alpha_1=3/2$, and by Euler's reflection formula, $\Gamma(5/2)\Gamma(-3/2)=\pi$. It follows that $c(\lambda) = \Gamma(-3/2)^2\cdot \tilde c(\lambda) = \lambda^{-5/2}$.
\paragraph{High temperatures.}
When $\nu\in (1,\nu_c)$, we have $Z\1{hom}(s,t) = \frac{s^{1/2} t^{1/2}}{s^{1/2} + t^{1/2}}$ and thus
\begin{equation*}
\tilde c(\lambda) =
\m({ \frac{1}{2\pi i} }^2 \int_{\mc V_{\infty,\theta}} \m({
\int_{\mc V_\infty} \frac{s^{1/2}}{s^{1/2} + t^{1/2}} e^s \dd s
} t^{1/2} e^{\lambda t} \dd t \,.
\end{equation*}
The inner integral can be expanded in a similar way as in the low temperature case
\begin{align*}
\int_{\mc V_\infty} \frac{s^{1/2}}{s^{1/2} + t^{1/2}} e^s \dd s
& =
\int_0^\infty \m({ \frac{(-r+i0)^{1/2}}{(-r+i0)^{1/2} + t^{1/2}}
-\frac{(-r-i0)^{1/2}}{(-r-i0)^{1/2} + t^{1/2}} } e^{-r} \dd r
\\ & =
\int_0^\infty \m({ \frac{ i r^{1/2}}{ i r^{1/2} + t^{1/2}}
-\frac{-i r^{1/2}}{-i r^{1/2} + t^{1/2}} } e^{-r} \dd r
= \int_0^\infty \frac{2i r^{1/2} t^{1/2}}{r+t} e^{-r} \dd r \,.
\end{align*}
Plugging the \rhs\ into the expression of $\tilde c(\lambda)$ and changing the order of the integrals on $r$ and on $t$ yield
\begin{equation*}
\tilde c(\lambda) =
\frac1\pi \int_0^\infty \m({
\frac{1}{2\pi i} \int_{\mc V_{\infty,\theta}} \frac{t \cdot e^{\lambda t}}{r+t} \dd t
} r^{1/2} e^{-r} \dd r \,.
\end{equation*}
The function $t\mapsto \frac{t\cdot e^{\lambda t}}{r+t}$ is meromorphic on $\complex$ and has a unique (simple) pole at $t=-r$, with a residue of $-r \cdot e^{-\lambda r}$. By closing the contour $\mc V_{\infty,\theta}$ far from the origin in the direction of the negative real axis, we see that the integral on $t$ is given by $-1$ times the residue. Therefore
\begin{equation*}
\tilde c(\lambda) = \frac{1}{\pi} \int_0^\infty r^{3/2} e^{-(1+\lambda)r} \dd r = \frac{\Gamma(5/2)}{\pi} (1+\lambda)^{-5/2} \,.
\end{equation*}
In the high temperature regime, we have $\alpha_0=3/2$ and $\alpha_1=-1$. Thus $c(\lambda) = \Gamma(-3/2)\Gamma(1) \cdot \tilde c(\lambda) = (1+\lambda)^{-5/2}$.
\paragraph{Critical temperature.}
When $\nu=\nu_c$, we have $Z\1{hom}(s,t) = \frac{-st}{s^{1/3} + t^{1/3}}$ and one can take $\theta = 0$. In the low and high temperature regimes, we have used the relation $\int_{\mc V_\infty} f(x)\dd x = \int_0^\infty \m({f(-r+i0)-f(-r-i0)} \dd r$ to expand integrals on $\mc V_\infty$. By applying this relation to the integral on $s$ and the integral on $t$ simultaneously, we get
\begin{align*}
\tilde c(\lambda) &= \m({ \frac{1}{2\pi i} }^2 \iint_{\mc V_\infty \times \mc V_\infty} \frac{-st}{s^{1/3} + t^{1/3}} e^{s+\lambda t} \dd s \dd t
\\&=
\m({ \frac{1}{2\pi i} }^2 \iint_{(0,\infty)^2}
\m({ \sum_{(\sigma_1,\sigma_2)\in \{-1,+1\}^2}
\frac{\sigma_1\sigma_2}{(-r_1+\sigma_1 \cdot i0)^{1/3} + (-r_2+\sigma_2 \cdot i0)^{1/3}}}
\cdot (-r_1r_2) \cdot e^{-(r_1+\lambda r_2)} \dd r_1 \dd r_2 \,.
\end{align*}
The principal branch of the function $s^{1/3}$ prescribes that $(-r\pm i0)^{1/3} = r^{1/3} e^{\pm i\frac\pi3}$.
One can check by direct computation that
\begin{equation*}
\sum_{(\sigma_1,\sigma_2)\in \{-1,+1\}^2}
\frac{\sigma_1\sigma_2}{(-r_1+\sigma_1 \cdot i0)^{1/3}
+ (-r_2+\sigma_2 \cdot i0)^{1/3}}
= \frac{-3 r_1^{1/3} r_2^{1/3}}{r_1+r_2} \,.
\end{equation*}
Therefore
\begin{equation*}
\tilde c(\lambda) = \m({ \frac{1}{2\pi i} }^2 \cdot 3 \iint_{(0,\infty)^2}
\frac{r_1^{4/3} r_2^{4/3} e^{-(r_1+\lambda r_2)}}{r_1+r_2} \dd r_1 \dd r_2 \,.
\end{equation*}
One can ``factorize'' this double integral using the relation $\frac{1}{r_1+r_2} = \int_0^\infty e^{-r_1 r} e^{-r_2 r} \dd r$\,:
\begin{align*}
\tilde c(\lambda) &= -\frac{3}{4\pi^2} \int_0^\infty
\m({ \int_0^\infty r_1^{4/3} e^{-(1+r)r_1} \dd r_1 }
\m({ \int_0^\infty r_2^{4/3} e^{-(\lambda+r)r_2} \dd r_2 }
\dd r
\\ &= -\frac{3}{4\pi^2} \int_0^\infty
\m({ \Gamma(7/3) \cdot (1+r)^{-7/3} }
\m({ \Gamma(7/3) \cdot (\lambda+r)^{-7/3} } \dd r
\\& = -\m({ \frac{\sqrt 3}{2\pi} \Gamma(7/3) }^2
\int_0^\infty (1+r)^{-7/3} (\lambda+r)^{-7/3} \dd r \,.
\end{align*}
When $\nu=\nu_c$, we have $\alpha_0=4/3$ and $\alpha_1=1/3$. And by Euler's reflection formula, $\Gamma(7/3)\Gamma(-4/3) = \frac{\pi}{\sin(7\pi/3)} = \frac{2\pi}{\sqrt 3}$. It follows that
\begin{equation*}
c(\lambda) = \Gamma(-4/3) \Gamma(-1/3) \cdot \tilde c(\lambda)
= \frac{4}{3} \int_0^\infty (1+r)^{-7/3} (\lambda+r)^{-7/3} \dd r \,.
\qedhere
\end{equation*}
\end{proof}
\section{Peeling processes and perimeter processes}\label{sec:peeling perim}
\newcommand*{\hl}{\\\cline{1-6}}
\tabulinesep=1.2mm
\newcolumntype{L}{>{\!$\displaystyle}l<{$\!}}
\newcolumntype{S}{>{\!$\displaystyle}l<{$\!\!}}
\newcolumntype{R}{>{\!$\displaystyle}r<{$\!}}
\newcolumntype{C}{>{\!$\displaystyle}c<{$\!}}
\newcommand*{\zp}[1]{t\zz{#1}}
\newcommand*{\zzp}[2]{t\zzz{#1}{#2}}
\newcommand*{\zph}[1]{t\zzh{#1}}
\newcommand*{\zzph}[2]{t\zzzh{#1}{#2}}
\newcommand*{\zq}[1]{t\zz[p,q+1]{#1}}
\newcommand*{\zqh}[1]{t\zz[p+1,q]{#1}}
\newcommand*{\zqo}[1]{t\zz[0,q+1]{#1}}
\newcommand*{\zzq}[2]{t\zzz[p,q+1]{#1}{#2}}
\newcommand*{\zzqh}[2]{t\zzz[p+1,q]{#1}{#2}}
\newcommand*{\zzqo}[2]{t\zzz[0,q+1]{#1}{#2}}
\newcommand*{\ap}[1]{\,\frac{a_{#1}}{a_{p}}\,}
\newcommand*{\aph}[1]{\,\frac{a_{#1}}{a_{p+1}}\,}
\newcommand*{\apo}[1]{\,\frac{a_{#1}}{a_0}}
\newcommand{\pq}[1][P_n,Q_n]{\raisebox{-2pt}{$\!_{#1}\!$}}
\begin{table}[t!]
\centering
\begin{tabu}{|L|S|C| |L|S|C| L}
\cline{1-6}
\step &\Prob_{p,q+1}(\Step_1 = \step) &(X_1,Y_1) &
\step &\Prob_{p,q+1}(\Step_1 = \step) &(X_1,Y_1)
\hl \cp & \zp{p+2,q} &(2,-1)
& \cm & \nu \zp{p,q+2} &(0,1)
\hl \lp & \zzp{p+1,q-k}{1,k} &(1,-k-1)
& \lm & \nu \zzp{p,q-k+1}{0,k+1} &(0,-k) & (0\le k \le \frac{q}{2})
\hl \rp & \zzp{k+1,0}{p-k+1,q} &(-k+1,-1)
& \rn & \nu \zzp{k,1}{p-k,q+1} &(-k,0) & (0\le k\le p)
\hl \rp[p+k] & \zzp{p+1,k}{1,q-k} &(-p+1,-k-1)
& \rn[p+k] & \nu \zzp{p,k+1}{0,q-k+1} &(-p,-k)
&(0<k<\frac{q}2)
\hl
\end{tabu}
\caption{Law of the first peeling event $\Step_1$ under $\Prob_{p,q+1}$ and the corresponding $(X_1,Y_1)$, where the peeling is without target. We use the shorthand notations $t=t_c(\nu)$ and $z_{p,q}=z_{p,q}(t,\nu)$.
}\label{tab:prob(p,q)}
\end{table}
We recall first the essentials of the \emph{peeling process} for Ising-triangulations with spins on faces, introduced in \cite{CT20}. The peeling process is the central object both in the construction of the local limits and in the proofs of the local convergences. It can be viewed as a deterministic exploration of a fixed map, driven by a \emph{peeling algorithm} $\mathcal{A}$. The basic definition is identical to that of \cite{CT20}, with the exception that in this work the peeling algorithm $\mathcal{A}$ is defined in a slightly more general way. In particular, the algorithm chooses an edge on the explored boundary instead of a vertex. While the algorithm used in \cite{CT20} still works in the low-temperature regime, we will need different algorithms in high temperatures, as explained in Section~\ref{sec:hightemplimit}. We will also note that, unlike in \cite{CT20}, here the different peeling algorithms result different laws of the peeling process.
Throughout this work we assume the following: if an Ising-triangulation has a bicolored boundary, the algorithm $\mathcal{A}$ chooses an edge at the junction of the $\+$ and $\<$ boundary segments on the boundary of the explored map. This edge may either have spin $\+$ or $\<$. It is easy to see that deletion of the chosen edge and exposure of the adjacent face preserves the Dobrushin boundary condition of the map. Thus, we call such an algorithm $\mathcal{A}$ \emph{Dobrushin-stable}. We make the following convention: if the algorithm always chooses a \< edge to peel, we denote it by $\algo_\<$; otherwise if it always chooses a \+ edge, we denote it by $\algo_\+$; otherwise, the algorithm is "mixed", choosing either type of the edges, and denoted by $\algo_m$.
The choice of the peeling algorithm in each of the temperature regime stems from the different expected interface geometries in the respective regimes. At $\nu=\nu_c$, we already saw in \cite{CT20} that the peeling algorithm $\algo_\<$ is particularly well-suited, due to the fact that we take the limit $q\to\infty$ first, after which there is an infinite \< boundary. For $\nu>\nu_c$, we can still make the same choice. However, for $\nu\in (1,\nu_c)$, we will notice that whether we choose the peeling to explore the left-most or the right-most interface from the root $\rho$, the interface will stay close to the boundary of the half-plane. Thus, in order to explore the local limit by roughly distance layers, we need to combine two different explorations. This leads us to choose a mixed algorithm $\algo_m$. In the first limit $q\to\infty$, however, the simplest choice which works is $\algo_\+$.
When we take the local limits $q\to\infty$ and $p\to\infty$ one-by-one, we always peel from a boundary with more \< edges. This is to ensure the peeling process is compatible with the $q=\infty$ case. In the limit $(p,q)\to\infty$, it is more natural to peel from the boundary which contains the vertex $\rho^\dagger$ opposite to the root and at the junction of the \+ and \< boundaries, which can be seen as a point in the infinity. For this purpose, we introduce the \emph{target} $\rho^\dagger$ for the peeling. See the following subsection for a more precise definition. A summary of the peeling algorithms and the existence of the target is presented in Table~\ref{tab:peelingalgo}.
\begin{table}[b!]
\begin{center}
\begin{tabu}{|L|L| |L|}
\hline \text{Local convergence} & \nu\in(1,\nu_c) & \nu\in[\nu_c,\infty)
\\\hline \prob_{p,q}^\nu\cv{q}\prob_p^\nu & \algo_\+ & \algo_\<
\\\hline \prob_p^\nu\cv{p}\prob_\infty^\nu & \algo_m & \algo_\<
\\\hline \prob_{p,q}^\nu\cv{p,q}\prob_\infty^\nu & (\algo_m,\rho^\dagger) & (\algo_\<,\rho^\dagger)
\\\hline
\end{tabu}
\caption{A summary of the choices of the peeling algorithm in each of the temperature regimes.}\label{tab:peelingalgo}
\end{center}\vspace{-0.7em}
\end{table}
In the following subsection, we define the versions of the peeling process used in this work in the finite setting. After that, we generalize those for infinite Ising triangulations of the half-plane, and study the properties of the associated perimeter processes.
\subsection{Peeling of finite triangulations}\label{sec:peeling}
\paragraph{Peeling along the left-most interface.} Assume that an Ising-triangulation $\bt$ has at least one boundary edge with spin \<. In this case, the peeling algorithm $\algo_\<$ chooses the edge $e$ with spin \< immediately on the left to the origin. We remove $e$ and reveal the internal face $f$ adjacent to it. If $f$ does not exist, then $\tmap$ is the edge map and $\bt$ has a weight 1 or $\nu$. If $f$ exists, let $*\in\{\+,\<\}$ be the spin on $f$ and $v$ be the vertex at the corner of $f$ not adjacent to $e$. Then the possible positions of $v$ are:
\begin{description}[noitemsep]
\item[Event $\CC^*$:] $v$ is not on the boundary of $\tmap$;
\item[Event $\RR^*_k$:] $v$ is at a distance $k$ to the right of $e$ on the boundary of $\tmap$; ($0\le k\le p$);
\item[Event $\LL^*_k$:] $v$ is at a distance $k$ to the left of $e$ on the boundary of $\tmap$. ($0\le k< q$).
\end{description} We also make the identification $\RR^*_{p-k}=\LL^*_{q+k}$ and $\LL^*_{q-k}=\RR^*_{p+k}$, which is useful in the sequel.
Denote $\tilde\steps = \{\cp,\cm\}\cup\{\lp,\lm,\rp,\rn:\ k\ge 0\}$. The peeling process along the leftmost interface $\iroot$ is constructed by iterating this face-revealing operation, yielding an increasing sequence $\nseq \emap$ of \emph{explored maps}. We can use this sequence as the definition of the peeling process. At each time $n$, the explored map $\emap_n$ consists of a subset of faces of $\bt$ containing at least the external face and separated from its complementary set by a simple closed path. We view $\emap_n$ as a bicolored triangulation of a polygon with a special uncolored, not necessarily triangular, internal face called the \emph{hole}. It inherits its root and its boundary condition from $\bt$. The complement of $\emap_n$ is called the \emph{unexplored map at time $n$} and denoted by $\umap_n$. It is a bicolored triangulation of a polygon.
Notice that $\umap_n$ may be the edge map, in which case $\emap_n$ is simply $\bt$ in which an edge is replaced by an uncolored digon. This may, however, only happen at the last step of the peeling process.
To iterate the peeling, one needs a rule that chooses one of the two unexplored regions, when the peeling step separates the unexplored map into two pieces. Here, we assume that the boundary contains no target vertex which determines the unexplored part (this case is treated separately in the sequel). In this case, we choose the unexplored region with greater number of \< boundary edges (in case of a tie, choose the region on the right). This in particular guarantees that when $q=\infty$ and $p<\infty$, we will choose the unbounded region as the next unexplored map.
We apply this rule recursively starting from $\umap_0=\bt$. At each step, the construction depends on the boundary condition of $\umap_n$:
\begin{enumerate}
\item If $\umap_n$ has a bichromatic Dobrushin boundary, let $\rho_n$ be the boundary junction vertex of $\umap_n$ with a \< on its left and a \+ on its right ($\rho_0=\rho$). Then $\umap_{n+1}$ is obtained by revealing the internal face of $\umap_n$ adjacent to the boundary edge on the left of $\rho_n$ and, if necessary, choose one of the two unexplored regions according to the \mbox{previous rule}.
\item If $\umap_n$ has a monochromatic boundary condition of spin \<, then the peeling algorithm $\algo_\<$ chooses the boundary edge with the vertex $\rho_n$ as an endpoint according to some deterministic function of the explored map $\emap_n$, which we specify later in Sections~\ref{sec:lowtemplimit}~and~\ref{sec:hightemplimit}. We then construct $\umap_{n+1}$ from $\umap_n$ and $\rho_n$ in the same way as in the previous case.
\item If $\umap_n$ has a monochromatic boundary condition of spin \+ or has no internal face, then we set $\emap_{n+1}=\bt$ and terminate the peeling process at time $n+1$.
\end{enumerate}
We denote the law of this peeling process by $\Prob_{p,q}^\nu\equiv\Prob_{p,q}$, where on the right we have dropped the dependence of $\nu$ in order to ease the notation and continue to do so in the sequel in the context of the peeling processes, except when the temperature $\nu$ should be emphasized. Let $(P_n,Q_n)$ be the boundary condition of $\umap_n$, and $(X_n,Y_n)=(P_n-P_0,Q_n-Q_0)$. Also, let $\Step_n\in\tilde\steps$ denote the peeling event that occurred when constructing $\umap_n$ from $\umap_{n-1}$. Then the \emph{peeling process} following the left-most interface can also be defined as the random process $\nseq{\Step}$ on $\tilde\steps$, with the law $\Prob_{p,q}$. We view the above quantities as random variables defined on the sample space $\Omega=\bts=\bigcup_{p,q} \bts_{p,q}$. In the sequel, one should understand that any of the sequences $\nseq{\emap}$, $\nseq{\umap}$ and $\nseq{\Step}$ can be viewed as the peeling process, since together with the boundary condition, they contain the same essential information. Table~\ref{tab:prob(p,q)} collects the distribution of the first peeling step $\Step_1$ and the associated perimeter change in the peeling process driven by $\algo_\<$.
\begin{table}[b!]
\begin{center}
\begin{tabu}{|L|S|C| |L|S|C| L}
\cline{1-6}
\step &\Probh_{p+1,q}(\Step_1=\step) &(X_1,Y_1) &
\step &\Probh_{p+1,q}(\Step_1=\step) & (X_1,Y_1)
\hl \cp & \nu \zph{p+2,q} & (1,0)
& \cm & \zph{p,q+2} & (-1,2)
\hl \lp & \nu \zzph{p+1,q-k}{1,k} & (0,-k)
& \lm & \zzph{p,q-k+1}{0,k+1} & (-k+1,-1) &(0\le k\le \frac{q}{2})
\hl \rp & \nu \zzph{k+1,0}{p-k+1,q} & (-k,0)
& \rn & \zzph{k,1}{p-k,q+1} & (-k-1,1) &(0\le k\le p)
\hl \rp[p+k] & \nu \zzph{p+1,k}{1,q-k} & (-p,-k)
& \rn[p+k] & \zzph{p,k+1}{0,q-k+1} & (-p-1,-k+1) &(0<k< \frac{q}{2})
\hl
\end{tabu}
\vspace{3ex}
\begin{tabu}{|L|L| |L|L|}
\hline \step & \Probh_{0,q+1}(\Step_1=\step)& \step & \Probh_{0,q+1}(\Step_1=\step)
\\\hline \cp & \zqo{2,q} &\cm & \nu \zqo{0,q+2}
\\\hline \lp & \zzqo{1,q-k}{1,k} &\lm & \nu \zzqo{0,q-k+1}{0,k+1}
\\\hline \rp & \zzqo{1,k}{1,q-k} &\rn & \nu \zzqo{0,k+1}{0,q-k+1}
\\\hline
\end{tabu}
\caption{Law of the first peeling event $\Step_1$ under $\Probh_{p+1,q}$ and the corresponding $(X_1,Y_1)$, where the peeling is without target. Due to the possibility that there is no $\+$ edge on the boundary, we also present the step probabilities under the law $\Probh_{0,q+1}$. The notational conventions coincide with Table~\ref{tab:prob(p,q)}.}\label{tab:probh(p,q)}
\end{center}\vspace{-0.7em}
\end{table}
\paragraph{Peeling along the right-most interface.} The peeling process along the right-most interface is similar to the previous one, except that the algorithm $\algo_\+$ chooses the \+ edge adjacent to $\rho_n$ if possible. Again, in case there are more than one holes, we fill in the one with less \< edges by an independent Boltzmann Ising-triangulation, and if the hole has a monochromatic \< boundary, the peeling continues on that according to some deterministic function. A small subtlety here is that the distribution of this peeling process differs from the previous one, such that the step distribution involves a spin-flip due to the deleted boundary edge of different spin. In particular, we also need to take into account that the peeling algorithm chooses a \< edge if the unexplored part has a monochromatic boundary. We denote the distribution of this peeling by $\Probh_{p,q}$. For the explicit probabilities of the first peeling step, see Table~\ref{tab:probh(p,q)}.
\paragraph{Peeling with the target $\rho^\dagger$.} Let $\algo$ be any Dobrushin-stable peeling algorithm (in the sense of the previous paragraphs). Considering the local limits when $p,q\to\infty$ simultaneously, it is convenient to define a \emph{peeling process with a target}, where the target is the vertex $\rho^\dagger$ at the junction of the \< and \+ boundaries opposite to $\rho$. The definition of this peeling process is as in the previous paragraphs, except when the peeling step separates the unexplored map into two pieces: in this case, the unexplored part corresponds to the one containing $\rho^\dagger$, and the other one is filled. If $\rho^\dagger$ is contained in both of the separated regions, the one with more \< edges is chosen for the unexplored part.
\subsection{Peeling of infinite triangulations}\label{seq:infinitepeeling} Obtaining the limits of the peeling process for a general temperature $\nu$ is just a straightforward generalization of the analysis in our previous work \cite{CT20}. Indeed, the asymptotics of Theorem~\ref{thm:asympt} give the limit according to the recipe given in \cite{CT20}. The first limit $q\to\infty$ yields exactly the same form for the peeling process, where the step probabilities only depend on $\nu$. Following the notation of \cite{CT20}, let $\Prob_p(\Step_1=\step):=\lim_{q\to\infty}\Prob_{p,q}(\Step_1=\step)$ and $\Prob_\infty(\Step_1=\step):=\lim_{p\to\infty}\Prob_p(\Step_1=\step)$. The quantities after the first limit $q\to\infty$ are collected in Table~\ref{tab:prob(p)}.
\begin{table}[h]
\centering
\begin{tabu}[t]{|L|L|L| |L|L|L| L}
\cline{1-6}
\step & \Prob_p(\Step_1=\step) & (X_1,Y_1) &
\step & \Prob_p(\Step_1=\step) & (X_1,Y_1) &
\hl \cp & t \ap{p+2} u & (2,-1) &
\cm & \frac{\nu t}{u} & (0,1) &
\hl \lp & t \ap{p+1} z_{1,k} u^{k+1} & (1,-k-1) &
\lm & \nu t z_{0,k+1} u^k & (0,-k) & (k\ge 0)
\hl \rp & t z_{k+1,0}\ap{p-k+1} u & (-k+1,-1) &
\rn & \nu t z_{k,1} \ap{p-k} & (-k,0) & (0\le k\le \frac{p}{2})
\hl \rp[p-k] & t z_{p-k+1,0} \ap{k+1} u & (-p+k+1,-1) &
\rn[p-k] & \nu t z_{p-k,1} \ap{k} & (-p+k,0) & (0\le k <\frac{p}{2})
\hl \rp[p+k] & t z_{p+1,k} \ap1 u^{k+1} & (-p+1,-k-1) &
\rn[p+k] & \nu t z_{p,k+1} \ap0 u^k & (-p,-k) & (k>0)
\hl
\end{tabu}
\caption{Law of the first peeling event $\Step_1$ under $\Prob_{p}$ and the corresponding $(X_1,Y_1)$, where the peeling is without target. We use the shorthand notations $t=t_c(\nu)$, $u=u_c(\nu)$, $z_{p,k}=z_{p,k}(t,\nu)$ and $a_p=a_p(\nu)$. Note the cutoff $p/2$ in the finite boundary segment, which is used for the convergence $p\to\infty$ in the $\nu>\nu_c$ regime (see Table~\ref{tab:pinfty}).}\label{tab:prob(p)}
\end{table}
Taking the second limit $p\to\infty$ yields a similar peeling process for all $1<\nu\leq\nu_c$, but for $\nu>\nu_c$ the asymptotics of Theorem~\ref{thm:asympt} yield additional non-trivial peeling events. Indeed, since the perimeter exponents $\alpha_0$ and $\alpha_1$ of $z_{p,k}$ and $a_p$ coincide in that case, the probabilities $\Prob_p(\Step_1=\rp[p\pm k])$ and $\Prob_p(\Step_1=\rn[p\pm k])$ have non-trivial limits when $p\to\infty$. For that reason, when $p=\infty$ or $q=\infty$, we identify $\rho^\dagger$ with $\infty$ and introduce the following additional peeling step events:
\begin{description}[noitemsep]
\item[Event $\RR^*_{\infty-k}$:] $v$ is at a distance $k$ to the right of $\infty$ on the boundary of $\tmap$, viewed from the origin ($0\le k< \infty$);
\item[Event $\LL^*_{\infty-k}$:] $v$ is at a distance $k$ to the left of $\infty$ on the boundary of $\tmap$, viewed from the origin ($0\le k< \infty$).
\end{description}
Let $\steps=\tilde\steps\cup\{\RR^*_{\infty-k}, \LL^*_{\infty-k}:\ *\in\{\+,\<\},\ k\ge 0\}$. Observe that the set $\steps$ in \cite{CT20} corresponds to the set $\tilde\steps$ here. We make the identifications $\RR^*_{\infty-k}=\LL^*_{\infty+k}$ and $\LL^*_{\infty-k}=\RR^*_{\infty+k}$, as well as the convention $\Prob_{p,q}(\Step_1=\RR^*_{\infty\pm k})=\Prob_p(\Step_1=\RR^*_{\infty\pm k})=0$. Thus, the peeling process can always be defined on $\steps$.
We define $\Prob_\infty(\Step_1=\rp[\infty\pm k]):=\lim_{p\to\infty}\Prob_p(\Step_1=\rp[p\pm k])$ and $\Prob_\infty(\Step_1=\rn[\infty\pm k]):=\lim_{p\to\infty}\Prob_p(\Step_1=\rn[p\pm k])$. The events $\rp[\infty\pm k]$ and $\rn[\infty\pm k]$ can be viewed as jumps of the peeling process to the vicinity of $\infty$. This property of infinite jumps results a positive probability of bottlenecks in the local limit when $\nu>\nu_c$. See Section~\ref{sec:lowtemplimit} for a more precise analysis of the local limit structure in the low temperature regime. The peeling step probabilities for $p,q=\infty$ are collected in Table~\ref{tab:pinfty}.
\begin{table}[h]
\centering
\begin{tabu}[t]{|L|L|L||L|L|L| L}
\cline{1-6}
\step & \Prob_\infty(\Step_1=\step)& (X_1,Y_1) &
\step & \Prob_\infty(\Step_1=\step)& (X_1,Y_1) &
\hl \cp & \frac{t}{u} & (2,-1) &
\cm & \frac{\nu t}{u} & (0,1) &
\hl \lp & t u^k z_{1,k} & (1,-k-1) &
\lm & \nu t u^k z_{0,k+1} & (0,-k) & (k\ge 0)
\hl \rp & t u^k z_{k+1,0} & (-k+1,-1) &
\rn & \nu t u^k z_{k,1} & (-k,0) & (k\ge 0)
\hl \rp[\infty-k] & \frac{t a_0}{b}a_{k+1}u^k\id_{\nu>\nu_c} &(-\infty,-1) &
\rn[\infty-k] & \frac{\nu t a_1}{b}a_ku^k\id_{\nu>\nu_c} & (-\infty,0) & (k\ge 0)
\hl \rp[\infty+k] & \frac{t a_1}{b}a_ku^k\id_{\nu>\nu_c} & (-\infty, -k-1) &
\rn[\infty+k] & \frac{\nu t a_0}{b}a_{k+1}u^k\id_{\nu>\nu_c} & (-\infty, -k) & (k>0)
\hl
\end{tabu}
\caption{Law of the first peeling event $\Step_1$ under $\Prob_\infty$ and the corresponding $(X_1,Y_1)$, where the peeling is without target. We have the same shorthand notation as in the previous tables as well as $b=b(\nu)$.}\label{tab:pinfty}
\end{table}
\begin{table}
\begin{center}
\begin{tabu}{|L|L|L| |L|L|L| L}
\cline{1-6}
\step & \Probh_{p+1}(\Step_1=\step) & (X_1,Y_1) &
\step & \Probh_{p+1}(\Step_1=\step) & (X_1,Y_1) &
\phantom{k\le }
\hl \cp & \nu t \aph{p+2} & (1,0)
& \cm & t \aph{p} \frac1{u^2} & (-1,2) &
\hl \lp & \nu t z_{1,k} u^k & (0,-k)
& \lm & t z_{0,k+1} \aph{p} u^{k-1} & (-1,-k+1) & (k\ge 0)
\hl \rp & \nu t z_{k+1,0}\aph{p-k+1} & (-k,0)
& \rn & t z_{k,1} \aph{p-k} \frac1{u} & (-k-1,1) & (0\le k\le p)
\hl \rp[p+k] & \nu t z_{p+1,k} \aph1 u^{k} & (-p,-k)
& \rn[p+k] & t z_{p,k+1} \aph0 u^{k-1} & (-p-1,-k+1) & (k>0)
\hl
\end{tabu}
\vspace{3ex}
\begin{tabu}{|L|L|L||L|L|L| L}
\hline \step & \Probh_\infty(\Step_1=\step)& (X_1,Y_1) &\step & \Probh_\infty(\Step_1=\step) & (X_1,Y_1)
\\\hline \cp & \frac{\nu t}{u} & (1,0) &\cm & \frac{t}{u} & (-1,2)
\\\hline \lp & \nu t u^k z_{1,k} & (0,-k) &\lm & t u^k z_{0,k+1} & (-1,-k+1)
\\\hline \rp & \nu t u^k z_{k+1,0} & (-k,0) &\rn & t u^k z_{k,1} & (-k-1,1)
\\\hline
\end{tabu}\phantom{$k\le$}\raisebox{3.2em}{}~~~
\caption{Laws of $\Step_1$ under $\Probh_{p+1}$ ($p\ge 0$) and $\Probh_\infty$, respectively, obtained by taking two successive limits in Table~\ref{tab:probh(p,q)}. The peeling is without target. Since we only need this distribution in the high-temperature regime $\nu\in(1,\nu_c)$, the bottleneck events are omitted.
}\label{tab:probh(p)}
\end{center}\vspace{-1em}
\end{table}
The proof that $\Prob_p$ defines a probability distribution on $\steps$ goes similarly as in \cite[Lemma 6]{CT20}, as well as that $\Prob_\infty$ is a probability distribution on $\steps$ for $1<\nu<\nu_c$. For $\nu>\nu_c$, the total probability from Table~\ref{tab:pinfty} sums to
\begin{equation*}
t(\nu+1)\left(\frac{Z_0(u)}{u}+Z_1(u)+\frac{\frac{a_0}{u_c}+a_1}{b}\left(A(u)-a_0\right)\right),
\end{equation*} which is shown to be equal to one either by a coefficient extraction argument similar to the one of \cite[Lemma 6]{CT20}, or by a computer algebra calculation.
It follows that $\Prob_p$ and $\Prob_\infty$, respectively, can be extended to the distribution of the peeling process $\nseq{\Step}$, and we have the convergence $\Prob_{p,q}\cv[]q\Prob_p\cv[]p\Prob_\infty$ in distribution, where $\Prob_p$ and $\Prob_\infty$ satisfy the spatial Markov property (see \cite[Proposition 2, Corollary 7]{CT20}). By symmetric arguments, we recall the same properties for the laws $\Probh_p$ and $\Probh_\infty$, which are obtained as the distributional limits of $\Probh_{p,q}$. The explicit laws of the first peeling step are collected in Table~\ref{tab:probh(p)}. The expectations corresponding to $\Prob$ and $\Probh$ are called $\EE$ and $\hat{\EE}$, respectively.
By the diagonal asymptotics part of Theorem~\ref{thm:asympt}, it is also easy to see that convergences $\Prob_{p,q}\cv{p,q}\Prob_\infty$ and $\Probh_{p,q}\cv{p,q}\Probh_\infty$ hold for every appropriate $\nu$. More precisely, since the coefficient function $\lambda\mapsto c(\lambda)$ is continuous on every interval bounded away from zero for every fixed $\nu\in(1,\infty)$, we conclude that $\lim_{p,q\to\infty}\frac{c\left(\frac{q-m}{p-k}\right)}{c\left(\frac{q}{p}\right)}=1$ for any fixed $k,m\in\Z$ when $q/p\in [\lambda_{\min},\lambda_{\max}]$, and the convergence of the one-step peeling transition probabilities follows. The rest is a mutatis mutandis of the proof of the convergence $\Prob_p\cv{p}\Prob_\infty$.
\subsection{Order parameters and connections to pure gravity}\label{sec:orderparam}
We define \begin{equation*}
\mathcal{O}(\nu):=\EE_\infty^\nu((X_1+Y_1)\id_{|X_1|\wedge|Y_1|<\infty})=(\nu+1)t_c(\nu)\left(\frac{Z_0(u_c(\nu))}{u_c(\nu)}-Z'_0(u_c(\nu))-u_c(\nu)Z'_1(u_c(\nu))\right).
\end{equation*}
Above, the cases $|X_1|=\infty$ and $|Y_1|=\infty$ may appear if $\nu>\nu_c$, and the latter only if we consider the peeling with target $\rho^\dagger$. The next proposition states that $\mathcal{O}$ can be viewed as an \emph{order parameter} for the phase transition around $\nu=\nu_c$. Recall from \eqref{eq:mu} the drift at $\nu=\nu_c$.
\begin{proposition}\label{prop:orderparam} The quantity $\mathcal{O}(\nu)$ satisfies
\begin{equation*}
\mathcal{O}(\nu)=\begin{cases}
0,\qquad\text{if}\quad 1<\nu<\nu_c \\
f(\nu)\qquad\text{if}\quad \nu\geq\nu_c,
\end{cases}
\end{equation*}
where $f:[\nu_c,\infty)\to\R$ is a continuous, strictly increasing function such that $f(\nu_c)=2\EE_\infty^{\nu_c}(X_1)=2\EE_\infty^{\nu_c}(Y_1)=2\mu>0$ and $\lim_{\nu\nearrow\infty}f(\nu)=\sqrt{\frac{7}{3}}f(\nu_c)$. Moreover, for $1<\nu<\nu_c$, we have the \emph{drift condition} $\EE_\infty^\nu(X_1)=-\EE_\infty^\nu(Y_1)>0$.
\end{proposition} The proof is a computation by a computer algebra, presented in \cite{CAS2}. Note that $\mathcal{O}$ is discontinuous at $\nu=\nu_c$, and that $\mathcal{O}(\nu)=\EE_\infty(X_1+Y_1)$ for $1<\nu\leq\nu_c$.
Moreover, the above drift condition in this regime shows that the peeling process started from the $\<$ edge next to the origin drifts to the left, swallowing the $\<$ boundary piece by piece. By symmetry, we also obtain $\hat\EE_\infty(Y_1)=\EE_\infty(X_1)$ and $\hat\EE_\infty(X_1)=\EE_\infty(Y_1)$, which in turn yield that the peeling process following the right-most interface drifts to the right. In Section~\ref{sec:hightemplimit}, we will use these properties to modify the peeling algorithm so that the peeling process will explore a neighborhood of the origin in a metric sense, which will be enough to construct the local limit in the high-temperature regime.
\begin{remark}\label{rem:orderparam2}
There is another, and perhaps more natural, order parameter \begin{equation*}
\tilde{\mathcal{O}}(\nu):=\Prob^\nu_\infty(|X_1|\vee|Y_1|=\infty)=\begin{cases}
0 & \text{if}\quad 1<\nu\le\nu_c \\
(\nu+1) t_c(\nu) \m({ \frac{\frac{a_0(\nu)}{u_c(\nu)}+a_1(\nu)}{b(\nu)}
\m({ A(u;\nu)-a_0(\nu) } }
& \text{if}\quad\nu>\nu_c \phantom{\le \nu_c}.
\end{cases}
\end{equation*}
It is easy to see that $\tilde{\mathcal{O}}$ is even continuous at $\nu=\nu_c$. This order parameter is the probability of the occurrence of a finite bottleneck in a single peeling step in the (to-be-constructed) local limit $\prob_\infty^\nu$. It can also be shown to be increasing and have the limit $\frac{\sqrt{3}}{12}$ as $\nu\to\infty$. However, since the order parameter $\mathcal{O}$ encapsulates all what we need in the proofs of the local convergences, $\tilde{\mathcal{O}}$ is not studied further in this work. Its only non-zero occurrence is related to Lemma~\ref{lem:hit 0} in Section~\ref{sec:lowtemplimit}.
\end{remark}
\begin{remark}
In the physics literature, the order parameter for the two-dimensional Ising model is traditionally the magnetization of the Ising field. We do not know the connection of $\mathcal{O}$ or $\tilde{\mathcal{O}}$ to the magnetization. Unlike the magnetization for the Ising model on a regular lattice, $\mathcal{O}$ is discontinuous at the critical temperature. Moreover, it does not tell us about the global geometry of the spin clusters, rather it serves as a "measure" of the interface behaviour in the local limit. An interesting curiosity is that we can show the free energy density per boundary edge has a second order discontinuity, even though it is known that the free energy density per face has a third order discontinuity, telling that the phase transition should be of third order. More precisely, by the work of Boulatov and Kazakov \cite{BouKaz87} or an explicit computation \cite{CAS2}, we have
\begin{equation*}
-\lim_{n\to\infty}\frac{1}{n}\log([t^n]z_{p,q}(t,\nu))=F(\nu)
\end{equation*} where $n$ is the number of interior faces and $F$ has a third order discontinuity at $\nu=\nu_c$. However, we find
\begin{equation*}
-\lim_{q\to\infty}\frac{1}{q}\log(z_{p,q}(\nu))=-\lim_{p,q\to\infty}\frac{1}{q}\log(z_{p,q}(\nu))=\log(u_c(\nu)),
\end{equation*}
which can be shown to have a second order discontinuity at $\nu=\nu_c$.
\end{remark}
\begin{figure}
\centering
\includegraphics[scale=1.1]{Fig6.pdf}
\caption{The graph of the order parameter $\mathcal{O}$.}
\label{fig:orderparam}
\end{figure}
\paragraph{Pure gravity-like behavior and some literature remarks.} It has been conjectured by physicists that the Ising model outside the critical temperature falls within the pure gravity universal class (see \cite{ADJ97}). In particular, in the seminal work of Kazakov \cite{Kaz86}, the fact is justified by computing the zero-temperature and the infinite-temperature limits of the free energy, which both coincide with the ones derived from the one-matrix model. The analysis of our peeling process, and the geometry in the further sections, will give a different perspective to this phenomenon.
First, we note that $\lim_{\nu\searrow 1}\EE_\infty(Y_1)=-\lim_{\nu\searrow 1}\EE_\infty(X_1)=-\frac{1}{2}$. From \cite[Section 3.2]{ACpercopeel}, we check that this coincides with the drift of the perimeter process of an exploration which follows the right-most interface of a finite percolation cluster on the UIHPT decorated with a face percolation configuration with parameter $p=1/2$. This is natural due to the symmetry of the $\+$ and $\<$ spins. We stress that, since percolation on the triangular lattice is \emph{not} self-dual, this falls in the subcritical regime of percolation. Observe also that the geometry of large Boltzmann Ising-triangulations in the high-temperature regime essentially should not depend on the exact value of $\EE_\infty(X_1)=-\EE_\infty(Y_1)$, as long as it is strictly positive and the perimeter exponents $\alpha_0+1$ and $\alpha_2+2$ of the asymptotics of Theorem~\ref{thm:asympt} are equal to $5/2$. Therefore, the geometry of the Ising-decorated random triangulation of the half-plane in the high-temperature regime is similar to the one of the UIHPT decorated with subcritical face percolation. To our knowledge, this phenomenon has never been explicitly written, though intuitively well understood.
In low temperature, we have $\lim_{\nu\to\infty}\mathcal{O}(\nu)=\frac{1}{2\sqrt{3}}$. This, in turn, coincides with the expectation of the number of edges swallowed (both to the right or to the left) after a peeling step of the non-decorated UIHPT of type I. At the level of the peeling process, we find that
\begin{equation*}
\begin{aligned}
&\ \lim_{\nu\to\infty}\Prob_\infty(\Step_1=\cp)
\!\!&=&\ \lim_{\nu\to\infty} \sum_{k=0}^\infty \Prob_\infty(\Step_1=\lp)
\!\!&=&\ \lim_{\nu\to\infty} \sum_{k=0}^\infty \Prob_\infty(\Step_1=\rp)
\\
=&\ \lim_{\nu\to\infty} \sum_{k=0}^\infty \Prob_\infty(\Step_1=\rp[\infty-k])
\!\!&=&\ \lim_{\nu\to\infty} \sum_{k=1}^\infty
\Prob_\infty(\Step_1=\lp[\infty-k])
\!\!&=&\ \lim_{\nu\to\infty} \sum_{k=1}^\infty
\Prob_\infty(\Step_1=\rn[\infty-k])
\ =\ 0
\end{aligned}
\end{equation*}
and
\begin{align*}
\lim_{\nu\to\infty}\Prob_\infty(\Step_1=\cm) &= \frac{1}{\sqrt{3}} &
\lim_{\nu\to\infty}\sum_{k=0}^\infty\Prob_\infty(\Step_1=\lm) \ \ \ \,
&= \frac{1}{2}-\frac{1}{2\sqrt{3}} \\
\lim_{\nu\to\infty}\sum_{k=0}^\infty\Prob_\infty(\Step_1=\rn)
&= \frac{1}{2}-\frac{\sqrt{3}}{4} &
\lim_{\nu\to\infty}\sum_{k=0}^\infty\Prob_\infty(\Step_1=\lm[\infty-k])
&= \frac{\sqrt{3}}{12}\,.
\end{align*}
Since these quantities sum to one, we conclude that either the bottlenecks survive in the zero temperature limit, or the limit does not define a probability distribution. The former follows if we can change the limit and the summation above, and that indeed can be done by the following simple argument: We notice that $\Prob_\infty(\Step_1=\lm[k])\sim\frac{M(\nu)}{k^{5/2}}$ as $k\to\infty$, where
\begin{equation*}
M(\nu)=\lim_{k\to\infty}k^{5/2}\left(\frac{\nu t_c(\nu)}{u_c(\nu)}\frac{a_0(\nu)}{\Gamma(-3/2)}(k+1)^{-5/2}\right)=\frac{\nu t_c(\nu)}{u_c(\nu)}\frac{a_0(\nu)}{\Gamma(-3/2)}.
\end{equation*}
An explicit computation shows that $\lim_{\nu\to\infty}M(\nu)\in (0,\infty)$, so $M(\nu)$ is bounded. Moreover, we can show that $\Prob_\infty(\Step_1=\lm[\infty-k])$ has exactly the same asymptotics as $k\to\infty$. By this asymptotic formula, one can then find a summable majorant for the above series for large enough $\nu$, and therefore the exchange of the limit and the sum follows from the dominated convergence theorem.
Hence, we find a zero temperature limit of the peeling process which shares the behaviour of the peeling process in the low-temperature regime. In that case, the peeling process constructs an infinite triangulation, which consists of two infinite triangulations with the geometry of the UIHPT that are glued together by just one vertex, which can be viewed as a pinch point in the vicinity of both the origin and the infinity. The construction of this local limit is the same as in the upcoming Section~\ref{sec:lowtemplimit}.
The existence of the finite bottlenecks for $\nu>\nu_c$ is well predicted in the physics literature. More precisely: When $\nu=\infty$, the spins are totally aligned. Therefore, for $\nu>\nu_c$, it is predicted that the energy of a spin configuration is proportional to the length of the boundary separating different spin clusters, and hence the minimal energy configurations should be those with minimal spin interface lengths. In our setting, we consider an annealed model where we sample the triangular lattice together with the spin configuration. Hence, a bottleneck in the surface is formed. This is explained eg. in \cite{ADJ97} and \cite{AJL00}. To our knowledge, this is the first time when the existence of the bottlenecks on Ising-decorated random triangulations is shown rigorously.
\section{Local limits and geometry at $\nu\neq\nu_c$}\label{sec:locallimits}
\subsection{Preliminaries: local distance and convergence}
For a map $\map$ and $r\ge 0$, we denote by $[\map]_r$ the \emph{ball of radius $r$} in $\map$, defined as the subgraph of $\map$ consisting of all the \emph{internal} faces which are adjacent to at least one vertex within the graph distance $r-1$ from the origin. By convention, the ball of radius $0$ is just the root vertex. The ball $[\map]_r$ inherits the planar embedding and the root corner of $\map$. Thus $[\map]_r$ is also a map. By extension, if $\sigma$ is a coloring of \emph{some faces} and \emph{some edges} of $\map$, we define the ball of radius $r$ in $(\map,\sigma)$, denoted $[\map,\sigma]_r$, as the map $[\map]_r$ together with the restriction of $\sigma$ to the faces and the edges in $[\map]_r$. In particular, we have $[[\map,\sigma]_{r'}]_r = [\map,\sigma]_r$ for all $r\le r'$.
Also, if an edge $e$ is in the ball of radius $r$ in a bicolored triangulation of a polygon $\bt$, then one can tell whether $e$ is a boundary edge by looking at $\btsq_r$, since only boundary edges are colored.
\newcommand{\CM}{\mathcal{C\hspace{-1pt}M}}
The \emph{local distance} for colored maps is defined in a similar way as for uncolored maps: for colored maps $(\map,\sigma)$ and $(\map',\sigma')$, let
\begin{equation*}
d\1{loc}((\map,\sigma),(\map',\sigma')) = 2^{-R}\qtq{where}
R = \sup\Set{r\geq 0}{ [\map,\sigma]_r=[\map',\sigma']_r }\,.
\end{equation*}
The set $\CM$ of all (finite) colored maps is a metric space under $d\1{loc}$. Let $\overline \CM$ be its Cauchy completion. As was the case with the uncolored maps (see e.g.\ \cite{CurPeccot}), the space $(\overline \CM, d\1{loc})$ is Polish (i.e.\ complete and separable). The elements of $\overline \CM \setminus \CM$ are called \emph{infinite colored maps}. By the construction of the Cauchy completion, each element of $\CM$ can be identified as an increasing sequence of balls $(\bmap_r)_{r\ge 0}$ such that $[\bmap_{r'}]_r = \bmap_r$ for all $r\le r'$. Thus defining an infinite colored map amounts to defining such a sequence.
Moreover, if $(\prob\0n)_{n\ge 0}$ and $\prob\0\infty$ are probability measures on $\overline \CM$, then $\prob\0n$ converges weakly to $\prob\0\infty$ for $d\1{loc}$ if and only if
\begin{equation*}
\prob \0n([\map,\sigma]_r=\bmap) \ \cv[]n\ \prob \0\infty([\map,\sigma]_r=\bmap)
\end{equation*}
for all $r\ge 0$ and all balls $\bmap$ of radius $r$.
When restricted to the bicolored triangulations of the polygon $\bts$, the above definitions construct the corresponding set $\overline \bts \setminus \bts$ of infinite maps. Recall that $\bts_\infty$ is the set of \emph{infinite bicolored triangulation of the half plane}, that is, elements of $\overline \bts \setminus \bts$ which are one-ended and have an external face of infinite degree. Recall also the set $\bts_\infty^{(2)}$, consisting of \emph{two-ended} bicolored triangulations with an infinite boundary.
\subsection{A general algorithm for constructing local limits}\label{sec:generalalgo}
In this section, we provide an algorithm for constructing local limits and proving the local convergence for a generic setup of Boltzmann Ising-triangulations of the disk. The algorithm is already used in our previous work \cite{CT20} in the proof of the local convergence $\prob_{p.q}\cv[]q\prob_p$.
\newcommand{\probl}{\mathds{P}_l}
\newcommand{\problinf}{\mathds{P}_\infty}
\newcommand{\Probl}{\Prob\0l}
\paragraph{Assumptions.} Suppose we are given a family of probability measures $\{\probl:l=1,2,\dots\}$ supported on $\overline{\bts}$, where the index $l$ is either the full perimeter or the length of a finite boundary segment of a bicolored, possibly infinite, Boltzmann Ising-triangulation with Dobrushin boundary conditions. For example in the latter case, we may have $l=q$ if $\probl=\prob_{p,q}^\nu$ and we consider the convergence $q\to\infty$. The point is that $\{\probl:l=1,2,\dots\}$ is assumed to be a one-parameter family. Recall that the peeling process of a fixed triangulation can be viewed as a deterministic sequence $\nseq{\emap_n,\umap}$ of explored and unexplored maps, respectively, driven by a peeling algorithm $\mathcal{A}$. By convention, $\law\0l\nseq{\emap}$ denotes the law of the sequence of the explored maps under $\probl$. \newcommand{\pqn}[1][n]{\tilde p,\tilde q,#1}
\newcommand{\upqn}[1][n]{\umap_{\pqn[#1]}^*}
Let $(\upqn)_{\pqn \ge 0}$ be a family of independent random variables which are also independent of $\nseq \Step$, such that $\upqn$ is a Boltzmann Ising-triangulation of the $(\tilde p,\tilde q)$-gon, where possibly $\tilde p=\infty$ or $\tilde q=\infty$. Consider $\mathbb{Z}$ with its nearest-neighbor graph structure and canonical embedding in $\mathbb{C}$, viewed as an infinite planar map rooted at the corner at $0$ in the lower half plane. Then, the upper half plane is the unexplored map $\law\0l\umap_0$, and $\law\0l\emap_0$ is defined as the deterministic map $\Z$ in which the following holds, depending whether the boundary of length $l$ is monochromatic or not: the monochromatic boundary of length $l$ is contained in $[0,l]$ (if it has spin \+) or in $[-l,0]$ (if it has spin \<), or the bichromatic boundary of length $l=l_1+l_2$ is contained in $[-l_1,l_2]$. Assume that under $\probl$, one can recover the distribution of $\emap_n$ as a deterministic function of $\emap_{n-1}$, $\Step_n$ and $(\upqn)_{\tilde p,\tilde q \ge 0}$. We define $\law\0l \nseq \emap$ by iterating that deterministic function on $\law\0l \emap_0$, $\law\0l \nseq \Step$ and $(\upqn)_{\pqn \ge 0}$. Let $\filtr_n$ be the $\sigma$-algebra generated by $\emap_n$. Then the above construction defines a probability measure on $\filtr_\infty = \sigma(\cup_n \filtr_n)$, which we denote by $\Probl$. Moreover, assume $\Probl\cv[]l\Prob\0\infty$ in distribution with respect to the discrete topology. That is, there exists a distribution $\Prob\0\infty$ such that for any element $\omega$ in the (countable) state space of the sequences $\nseq \Step$ and $(\upqn)_{\pqn \ge 0}$ up to time $n_0<\infty$, we have $\Prob\0l(\omega) \cv[]l \Prob\0\infty(\omega)$.
For the peeling algorithm $\mathcal{A}$, we make two assumptions. First, we assume that the algorithm is \emph{Dobrushin-stable}, in the sense that $\mathcal{A}$ always chooses a boundary edge at the junction of the \< and \+ boundary segments. This choice guarantees that the boundary condition always remains Dobrushin or monochromatic. Second, we assume that $\mathcal{A}$ is \emph{local}, by which we mean the following: If the boundary is bichromatic, $\mathcal{A}$ chooses the boundary edge according to the previous rule such that it is connected to the root $\rho$ via an explored region by the peeling excluding the boundary. On the other hand, if the boundary is monochromatic, $\mathcal{A}$ chooses an edge whose endpoints have a minimal graph distance to the origin, according to some deterministic rule if there are several such choices.
\paragraph{Convergence of the peeling process.}
Since $(\upqn)_{\pqn \ge 0}$ has a fixed distribution and is independent of $\nseq \Step$, it follows that $\law\0l\nseq \Step$ and $(\upqn)_{\pqn \ge 0}$ converge jointly in distribution when $l\to\infty$ with respect to the discrete topology.
However, because $\law\0l \emap_0$ takes a different value for each $l$, the initial condition $\law\0l \emap_0$ cannot converge in the above sense. This is not a problem, since for any positive integer $K$, the restriction of $\law\0l \emap_0$ to a finite interval $[-K,K]$ stabilizes at the value that is equal to the restriction of $\law\0\infty \emap_0$ on $[-K,K]$.
Therefore, let us consider the truncated map $\emapo_n$, obtained by removing from $\emap_n$ all the boundary edges adjacent to the hole. Then the number of the remaining boundary edges is finite and only depends on $(\Step_k)_{k\le n}$. It follows that for each $n$ fixed, $\emapo_n$ is a deterministic function of $(\Step_k)_{k\le n}$, $(\upqn[k])_{\tilde p,\tilde q\ge 0; k\le n}$ and $\emap_0$ restricted to some finite interval $[-K,K]$ where $K$ is determined by $(\Step_1,\dots, \Step_n)$. As the arguments of this function converge jointly in distribution with respect to the discrete topology (under which every function is continuous), the continuous mapping theorem implies that
\begin{equation}\label{eq:peeling cvg}
\Prob\0l(\emapo_n=\bmap) \cv[]l \Prob\0\infty(\emapo_n=\bmap)
\end{equation}
for every bicolored map $\bmap$ and for every integer $n\ge 0$. We can extend this convergence for finite stopping times according to the following proposition, whose proof is a mutatis mutandis of the proof of an analogous statement \cite[Lemma 12]{CT20}.
\begin{prop}[Convergence of the peeling process]\label{lem:stopped peeling}
Let $\filtr^\circ_n$ be the $\sigma$-algebra generated by $\emapo_n$.
If $\theta$ is an $\nseq{\filtr^\circ}$-stopping time that is finite $\Prob\0\infty$-almost surely, then for every bicolored map $\bmap$,
\begin{equation}\label{eq:stopped peeling cvg}
\Prob\0l(\emapo_\theta = \bmap) \cv[]l \Prob\0\infty(\emapo_\theta = \bmap) \,.
\end{equation}
\end{prop}
\paragraph{Construction of $\problinf$.}
Recall that the explored map $\emap_n$ contains an uncolored face with a simple boundary called its hole. The unexplored map $\umap_n$ fills in the hole to give $\bt$. We denote by $\frontier_n$, called the \emph{frontier} at time $n$, the path of edges around the hole in $\emap_n$. For all $r\ge 0$, let $\theta_r = \inf\Set{n\ge 0}{ d_{\emap_n}(\rho,\frontier_n)\ge r}$, where $d_{\emap_n}(\rho,\frontier_n)$ is the minimal graph distance in $\emap_n$ between $\rho$ and vertices on $\frontier_n$. It is clear that this minimum is always attained on the truncated map $\emapo_n$, therefore $d_{\emap_n}(\rho,\frontier_n)$ is $\filtr^\circ_n = \sigma(\emapo_n)$-measurable and $\theta_r$ is an $\nseq{\filtr^\circ}$-stopping time. Expressed in words, $\theta_r$ is the first time $n$ such that all vertices around the hole of $\emap_n$ are at a distance at least $r$ from $\rho$. Since $\bt$ is obtained from $\emap_n$ by filling in the hole, it follows that
\begin{equation*}
\btsq_r\ =\ [\emapo_{\theta_r}]_r
\end{equation*}
for all $r\ge 0$. In particular, the peeling process $\nseq\emap$ eventually explores the entire triangulation $\bt$ if and only if $\theta_r<\infty$ for all $r\ge 0$. A sufficient condition for this is provided by the following lemma.
\begin{lemma}\label{lem:cover r-ball}
If the frontier $\frontier_n$ becomes monochromatic in a finite number of peeling steps $\Prob\0\infty$-almost surely, then $\theta_r$ is almost surely finite for all $r\ge 0$.
\end{lemma}
\begin{proof}
We have $\theta_0=0$. Assume that $\theta_r<\infty$ almost surely for some $r\ge 0$. Then the set $V_r$ of vertices at a graph distance $r$ from the origin in $\tmap$ is $\Prob\0\infty$-almost surely finite. Since by assumption the frontier $\frontier_n$ becomes monochromatic in a finite time $\Prob\0\infty$-almost surely, the spatial Markov property yields that $\frontier_n$ is monochromatic infinitely often.
On the event $\{\theta_{r+1}=\infty\}$ and at the times $n>\theta_r$ such that $\frontier_n$ is monochromatic, the peeling algorithm $\algo$ chooses to peel an edge with an endpoint in $V_r$ by the locality assumption of $\algo$. Since $V_r$ is finite, there exists a $v\in V_r$ at which such peeling steps occur infinitely many times. But each time the vertex $v$ is swallowed with a non-zero probability, as a consequence the transition probabilities of the one-step peeling. Therefore $v$ can remain forever on the frontier only with zero probability. This implies that $\Prob\0\infty(\theta_{r+1}<\infty)=1$. By induction, $\theta_r$ is finite $\Prob\0\infty$-almost surely for all $r\ge 0$.
\end{proof}
We define the infinite Boltzmann Ising-triangulation of law $\problinf$ by the laws of its finite balls $\law\0\infty \btsq_r := \lim\limits_{n \to\infty} \law\0\infty{} [\emap_n]_r$. The external face of $\law\0\infty \bt$ obviously has infinite degree. Moreover, every finite subgraph of $\law\0\infty \bt$ is covered by $\emap_n$ almost surely for some $n<\infty$. If the peeling process only fills finite holes by the family $(\upqn)_{\pqn \ge 0}$, it follows that the complement of a finite subgraph only has one infinite component. That is, $\problinf$ is one-ended, which together with the infinite boundary tells that the local limit is an infinite bicolored triangulation of the half-plane. If the limiting map, however, includes infinite holes to fill in with the peeling, the map has several infinite connected components with positive probability. In the following section, we see a concrete example of that case.
\subsection{The local limit in low temperature $(\nu>\nu_c)$} \label{sec:lowtemplimit}
Throughout this section, fix $\nu\in (\nu_c,\infty)$. For simplicity, let us first consider the case where $q\to\infty$ and $p\to\infty$ separately. In Section~\ref{sec:orderparam}, the order parameter $\mathcal{O}$ told us that for $\nu>\nu_c$, the peeling process has a tendency to drift to infinity. Moreover, from Table~\ref{tab:pinfty} we already read that $X_1=-\infty$ with a positive probability. Thus, we have $\EE_\infty(X_1)=-\infty$. These properties intuitively mean that the left-most interface drifts to infinity much faster than in the critical temperature, in fact even in a finite time almost surely. Thus, the construction of the local limit and the proof of the local convergence follows by choosing $\algo=\algo_{\<}$ and after we verify the assumptions of the algorithm in the previous section. The geometric view is similar to that in the critical temperature \cite{CT20}, with the exception that in this case the interface in a realization of the local limit is contained in a ribbon which is finite. Therefore, the local limit is not one-ended, unlike in $\nu=\nu_c$, and contains a bottleneck between the origin and infinity.
In order to be more precise, let us consider the $\Prob_p$ -stopping time
\begin{equation*}
T_m\ =\ \inf\Set{n\ge 0}{P_n\le m},
\end{equation*}
where $m\ge 0$ is a cutoff. In particular, $T_0$ is the first time that the boundary of the unexplored map becomes monochromatic. Observe that for $p>2m$, we can write $T_m\ =\ \inf\Set{n\ge 0}{\Step_n\in\Set{\rp[p+k+1], \rn[p+k]}{k\geq-m}}$. This extends to $p=\infty$ in a natural way, and thus $T_m$ is also a well-defined stopping time under $\Prob_\infty$.
Following the notation of \cite{CT20}, denote by $\law_{p,q}X$ (resp.\ $\law_p X$ and $\law_\infty X$) a random variable which has the same law as the random variable $X$ under $\Prob_{p,q}$ (resp.\ under $\Prob_p$ and $\Prob_\infty$).
We start by giving an upper bound for the tail distribution of $T_0$, which implies in particular that the process $ \law_p\nseq P$ hits zero almost surely in finite time. In other words, the peeling process swallows the \+ boundary almost surely, exactly as for $\nu=\nu_c$. What makes the low-temperature regime different is that this property actually holds also for $\Prob_\infty$, since by the infinite jumps of the peeling process we may have $T_m<\infty$. Moreover, we can easily find the explicit distribution of $T_m$ under $\Prob_\infty$.
\begin{lemma}[Law of $T_m$, $\nu>\nu_c$]\label{lem:hit 0}
Let $\nu\in(\nu_c,\infty)$.
\begin{enumerate}
\item There exists $\gamma>0$ such that $\Prob_p(T_0> n)\le e^{-\gamma n}$ for all $p\ge 1$. In particular, $T_0$ is finite $\Prob_p$-almost surely.
\item Under $\Prob_\infty$, the stopping time $T_m$ has geometric distribution with parameter
\begin{equation*}
r_m:=\Prob_\infty(P_1\leq m)=\Prob_\infty(T_m=1)
\end{equation*} supported on $\{1,2,\dots\}$. That is,
\begin{equation*}
\Prob_\infty(T_m=n)=(1-r_m)^{n-1}r_m
\end{equation*} for $n=1,2,\dots$. In particular, $T_m$ is finite $\Prob_\infty$-almost surely for all $m\geq 0$.
\end{enumerate}
\end{lemma}
\begin{proof}
Since $\nu>\nu_c$, we have $\Prob_p(\Step_1=\rn[p])\longrightarrow\Prob_\infty(\Step_1=\rn[\infty]):=\tilde{r}\in(0,1)$, which yields
$
\Prob_p(T_0= 1) \ \ge\ \Prob_p(\Step_1=\rn[p]) \ \ge\ r
$
for all $p\ge 1$, for some constant $r\in (0,1)$. It follows by the Markov property and induction that for all $n\ge 0$,
\begin{equation*}
\Prob_p(T_0>n+1) \ = \ \EE_p \mb[{ \Prob_{P_n}(T_0\ne 1) \idd{T_0>n} }
\ \le\ (1-r)^{n+1}\,,
\end{equation*}from which the first claim follows.
For the second claim, the data of Table~\ref{tab:pinfty} for $\nu>\nu_c$ shows that
\begin{align*}
\Prob_\infty(T_m=1) &=\ \Prob_\infty(P_1\leq m)
\\ &=\ \sum_{k=0}^\infty \m({ \Prob_\infty(\Step_1=\rp[\infty+k])
+ \Prob_\infty(\Step_1=\rn[\infty+k]) }
\ +\ \sum_{k=1}^{m-1} \Prob_\infty(\Step_1=\rp[\infty-k])
\ +\ \sum_{k=1}^m \Prob_\infty(\Step_1=\rn[\infty-k])
\\ &=\ t\m({ \frac{a_0}{bu}\m({ \nu(A(u)-a_0)+\sum_{k=2}^ma_ku^k }
+\frac{a_1}{b}\m({ A(u)+\nu\sum_{k=1}^ma_ku^k}
} \ =:\ r_m.
\end{align*}
By the spatial Markov property and induction, \begin{equation*}
\Prob_\infty(T_m>n+1) \ = \ \EE_\infty \mb[{ \Prob_\infty(T_m\ne 1) \idd{T_m>n} }
= (1-r_m)^{n+1}\,
\end{equation*}for all $n\geq 0$, which shows that $T_m$ has geometric distribution with parameter $r_m$.
\end{proof}
\begin{remark}
Observe that by the above proof, $\lim_{m\to\infty}r_m=\tilde{\mathcal{O}}(\nu)$, which was introduced as an order parameter in Remark~\ref{rem:orderparam2}.
\end{remark}
The above lemma entails that $T_m$ can directly, without further conditioning, be regarded as a time of a large jump of the perimeter process. In other words, unlike in \cite{CT20}, a suitably chosen peeling process will explore any finite neighborhood of the origin in a finite time, and thus no gluing argument of locally converging maps is needed. In particular, the general algorithm of Section~\ref{sec:generalalgo} applies. If one wanted to study the local limit via gluing, one could note that conditional on $n<T_m$, the process $P_n$ has a positive drift, a behaviour reflected by the order parameter $\mathcal{O}$.
It is easy to see that the above lemma also holds if we define more generally $T_m:=\inf\{n\ge 0: \min\{P_n,Q_n\}\leq m\}$ and consider the convergence of the peeling process with the target $\rho^\dagger$ under the limit $p,q\to\infty$ while $q/p\in [\lambda',\lambda]$. We omit the details of this here. The stopping time $T_m$ is extensively studied in Section~\ref{sec:onejumpscaling} for $\nu=\nu_c$, and the computation techniques for $\nu>\nu_c$ are similar. The biggest difference compared to the $p=\infty$ case is the fact that in the $p,q<\infty$ case, the triangle revealed at the peeling step realizing $T_m$ must hit the boundary at a distance less than $m+1$ from $\rho^\dagger$. The perimeter variations $(X_1,Y_1)$ will also have a different law, and in particular both $X_1$ and $Y_1$ may have infinite jumps (though not simultaneously).
Recall that in our context of the peeling along the leftmost interface, the peeling algorithm $\algo_\<$ is used to choose the origin $\rho_n$ of the unexplored map $\umap_n$ \emph{when its boundary $\frontier_n$ is monochromatic of spin \<}. (See Section~\ref{sec:peeling}.) Under $\Prob_p$, we can ensure $\theta_r<\infty$ almost surely for all $r\ge 0$ with the following choice of the peeling algorithm $\algo=\algo_\<$: let $\rho_n = \algo(\emap_n)$ be the leftmost vertex on $\frontier_n$ that realizes the minimal distance $d_{\emap_n}(\rho,\frontier_n)$ from the origin.
This algorithm is obviously local. Since $T_0<\infty$ almost surely by Lemma~\ref{lem:hit 0}, Lemma~\ref{lem:cover r-ball} gives $\theta_r<\infty$ almost surely in $\Prob_p$. Moreover, everything in this paragraph clearly also holds after replacing $\Prob_p$ by $\Prob_\infty$.
\begin{proof}[Proof of the convergence $\prob_{p,q}^\nu \protect{\cv q} \prob_p^\nu \protect{\cv p} \prob_\infty^\nu$ for $\nu>\nu_c$]
The $(\filtr^\circ_n)$-stopping time $\theta_r$ is almost surely finite under $\Prob_p$ and $\Prob_\infty$, and $\btsq_r = [\emapo_{\theta_r}]_r$ is a measurable function of $\emapo_{\theta_r}$. Thus, the assumptions of the general algorithm for local convergence hold with the choice $\probl=\prob_{p,q}^\nu$ with $l=q$ in the first limit, and after $\prob_p^\nu$ is defined, also with $\probl=\prob_p^\nu$ with $l=p$ in the second limit. In the first limit, the family $(\upqn)_{\pqn \ge 0}$ consists of independent finite Boltzmann Ising-triangulations, which fill in the finite holes formed in the peeling process (exactly as in $\nu=\nu_c$, see \cite{CT20}). Assuming $\prob_p^\nu$ is defined for all $p\geq 0$, then the family $(\upqn)_{\pqn \ge 0}$ also contains the elements $\umap^*_{\infty,\tilde{q},n}$ with law $\prob_{\tilde{q}}^\nu$, which fill in the hole with infinite \+ boundary after a bottleneck is formed. Putting things together in this order, it follows from Proposition~\ref{lem:stopped peeling} that $\prob_{p,q}^\nu(\btsq_r = \bmap) \cv[]q \prob_p^\nu(\btsq_r = \bmap)\cv[]p\prob_\infty^\nu(\btsq_r = \bmap)$ for all $r\ge 0$ and every ball $\bmap$. This implies the local convergence $\prob_{p,q}^\nu \cv[]q \prob_p^\nu \cv[]p \prob_\infty^\nu$.
\end{proof}
\begin{proof}[Proof of the convergence $\prob_{p,q}^\nu \protect{\cv {p,q}} \prob_\infty^\nu$ while $0<\lambda'\leq\frac{q}{p}\leq\lambda$ for $\nu>\nu_c$]
The assumptions of the general algorithm for local convergence hold with the choice $\probl=\prob_{p,q}^\nu$ with $l=p+q$, where $l\to\infty$ such that $q\in [\lambda' p,\lambda p]$. Since the peeling process with the target $\rho^\dagger$ has the same limit in distribution as the untargeted one, the local limit is indeed $\prob_\infty^\nu$.
\end{proof}
The above constructed local limit $\prob_p^\nu$ is one-ended, since the peeling process only fills in finite holes. By Lemma~\ref{lem:hit 0}, the untargeted peeling process of the local limit $\prob_\infty^\nu$ swallows the infinite \+ boundary $\Prob_\infty$-almost surely in a finite time, resulting a finite bottleneck, after which the peeling process continues to peel the infinite triangulation with infinite \< boundary and finite \+ boundary. Since the latter one is one-ended, it follows that the local limit $\prob_\infty^\nu$ consists of two independent triangulations of laws $\prob^\nu_{\tilde{p}}$ and $\prob^\nu_{\tilde{q}}$, for some $\tilde{p}\ge 0$ and $\tilde{q}\ge 0$, the second one modulo a spin flip, glued together along a finite bottleneck. That is, the local limit $\prob_\infty^\nu$ is two-ended.
\subsection{The local limit in high temperature $(1<\nu<\nu_c)$}\label{sec:hightemplimit}
Throughout this section, fix $\nu\in (1,\nu_c)$. We start by considering first the convergence $\prob_{p,q}^\nu\to\prob_p^\nu$. For that purpose, we choose the peeling algorithm $\algo_\+$, for a reason explained in Lemmas~\ref{lem:integralconvergence}~and~\ref{lem:hit0h}. Again, the only thing to show is that $\Probh_p(T_0<\infty)=1$ for every finite $p\geq 0$. However, due to the fact that $\Probh_p(T_m=1)\sim c_m\cdot p^{-5/2}$, we need a different strategy as in \cite{CT20} to prove that result. At this point, recall the drift of the perimeter processes: $\EE_\infty(X_1)=-\EE_\infty(Y_1)>0$ from Proposition~\ref{prop:orderparam}. This drift is used to estimate the drift of the perimeter process for a large $p<\infty$.
\begin{lemma}\label{lem:integralconvergence}
Let $\nu\in (1,\nu_c)$. Then,
\begin{equation*}
\lim_{p\to\infty}\EE_p(X_1)=\EE_\infty(X_1)\qquad\text{and}\quad\lim_{p\to\infty}\EE_p(Y_1)=\EE_\infty(Y_1).
\end{equation*}
Likewise,\begin{equation*}
\lim_{p\to\infty}\hat{\EE}_p(X_1)=\hat{\EE}_\infty(X_1)\qquad\text{and}\quad\lim_{p\to\infty}\hat{\EE}_p(Y_1)=\hat{\EE}_\infty(Y_1).
\end{equation*}
\end{lemma}
\begin{proof}
For $p>m>1$, we make the decomposition
\begin{equation*}
\EE_p(X_1)=\EE_p\left(X_1\id_{\{X_1\geq -m\}}\right)+\EE_p\left(X_1\id_{\{X_1\leq -p+m\}}\right)+\EE_p\left(X_1\id_{\{X_1\in (-p+m,-m)\}}\right).
\end{equation*} By the convergence of the peeling process,
\begin{equation*}
\EE_p\left(X_1\id_{\{X_1\geq -m\}}\right)\cv[]p\EE_\infty\left(X_1\id_{\{X_1\geq -m\}}\right)\cv[]m\EE_\infty(X_1).
\end{equation*}
For the second term, $\Prob_p(X_1\leq -p+m)=\Prob_p(P_1\leq m)\sim c_m\cdot p^{-5/2}$ as $p\to\infty$ for some constant $c_m>0$, which shows that
\begin{equation*}
\EE_p\left(X_1\id_{\{X_1\leq -p+m\}}\right)=-p\Prob_p(X_1\leq -p+m)+\sum_{k=0}^m k\Prob_p(X_1=k-p)\cv[]p 0.
\end{equation*}
Finally, the third term can be explicitly written using the data of Table~\ref{tab:prob(p)} as
\begin{equation*}
\EE_p\left(X_1\id_{\{X_1\in (-p+m,-m)\}}\right)=-\sum_{k=m+1}^{p-m-1}k\Prob_p(X_1=-k)=-\sum_{k=m+1}^{p-m-1}k\left(tz_{k+2,0}\frac{a_{p-k}}{a_p}u+\nu t z_{k,1}\frac{a_{p-k}}{a_p}\right).
\end{equation*}
By the asymptotics $z_{k+2,0}\frac{a_{p-k}}{a_p}\underset{p\to\infty}\sim z_{k+2,0}u_c^k\underset{k\to\infty}\sim\text{ cst}\cdot k^{-5/2}$ and a similar one for $z_{k,1}\frac{a_{p-k}}{a_p}$, the sum on the right hand side can be approximated by a remainder of a convergent series, and therefore taking the limits $p,m\to\infty$ yields the claim.
The case $\lim_{p\to\infty}\EE_p(Y_1)=\EE_\infty(Y_1)$ is similar, except easier, since it only requires one cutoff at $Y_1=-m$. Indeed, the same asymptotics hold for $Y_1$. The cases $\lim_{p\to\infty}\hat{\EE}_p(X_1)=\hat{\EE}_\infty(X_1)$ and $\lim_{p\to\infty}\hat{\EE}_p(Y_1)=\hat{\EE}_\infty(Y_1)$ follow by symmetry.
\end{proof}
\begin{remark}
By Proposition~\ref{prop:orderparam} and symmetry, we have then
\begin{equation*}
\lim_{p\to\infty}\EE_p(X_1)>0\qquad\text{and}\quad\lim_{p\to\infty}\hat{\EE}_p(X_1)<0,
\end{equation*}
and likewise\begin{equation*}
\lim_{p\to\infty}\EE_p(Y_1)<0\qquad\text{and}\quad\lim_{p\to\infty}\hat{\EE}_p(Y_1)>0.
\end{equation*} This property is the main implication of Lemma~\ref{lem:integralconvergence}, which we keep on using in this section.
\end{remark}
\begin{remark}
In \cite{CT20}, we used the same decomposition to show that $\EE_p(X_1)\cv[]p-\frac{1}{3}\EE_\infty(X_1)<0$ at $\nu=\nu_c$. This blow-up of the probability mass was due to the fact that $\Prob_p(X_1\leq -p+m)\sim c_m\cdot p^{-1}$ at $\nu=\nu_c$. Currently, we do not have an interpretation of this symmetry breaking.
\end{remark}
Under mild conditions, a Markov chain on the positive integers with an asymptotically negative drift is expected to be recurrent. The next lemma verifies this in our case.
\begin{lemma}\label{lem:hit0h}
If $\nu\in(1,\nu_c)$, then $T_0$ is finite $\Probh_p$-almost surely.
\end{lemma}
\begin{proof}
Since $\nseq{P}$ is an irreducible Markov chain on the positive integers, it is enough to show that $\Probh_{p'}(T_{p'}<\infty)=1$ for some $p'>0$. Namely, this means that the chain will return to the finite set $\{0,\dots p'\}$ infinitely many times, and thus there exists a recurrent state.
Observe that by Lemma~\ref{lem:integralconvergence}, there exists an index $p_*>0$ such that $\hat{\EE}_{p'}(X_1)\leq-a$ for some $a>0$ if $p'>p_*$. On the other hand, $\hat{\EE}_{p'}(X_1)\leq\max_{0\leq i\leq p_*}\hat{\EE}_{i}(|X_1|)<\infty$ for $p'\leq p*$. Then, it follows that the set $\{0,\dots,p_*\}$ is actually positive recurrent; see \cite[Theorem 1]{FK04} for a more general statement via Lyapunov functions, in which the Lyapunov function is chosen to be the identical mapping.
\end{proof}
Now, the proof of the local convergence $\prob_{p,q}^\nu\cv{q}\prob_p^\nu$ goes along the same lines as in the case $\nu\geq \nu_c$. Let us proceed to the proof of the convergence $\prob_p^\nu\cv{p}\prob_\infty^\nu$. For this, we cannot just choose the peeling algorithm $\algo_\+$ (or $\algo_\<$, respectively) since the peeling exploration under that algorithm drifts to the right (resp. to the left) in the limit by Lemma~\ref{lem:integralconvergence}. These drifts, however, allow us to construct a mixed peeling algorithm $\algo=\algo_m$ as follows.
\paragraph{Peeling algorithm $\algo_m$.} Recall that for $\nu\in(1,\nu_c)$, we have the drift conditions $\EE_\infty(X_1)>0$ and $\EE_\infty(Y_1)<0$ (together with the symmetric conditions $\hat{\EE}_\infty(X_1)<0$ and $\hat{\EE}_\infty(Y_1)>0$). These conditions allows us to construct the following sequence of stopping times:
Set $X_0=Y_0=0$ and $\tau_0^r=0.$ \begin{itemize}
\item Start peeling with $\algo_\<$ until the time $\tau_1^l:=\inf\{n>0:Y_n<-1\}$, which is almost surely finite under $\Prob_\infty$ due to the drift condition.
\item Proceed peeling with $\algo_\+$ until the time $\tau_1^r:=\inf\{n>\tau_1^l:X_n<-X_{\tau_1^l}-1+\min_{0\leq m\leq\tau_1^l}X_m\}$, which is a.s. finite under $\hat{\Prob}_\infty$, conditional on $\tau_1^l$.
\end{itemize} Repeat inductively for $k\geq 1$:
\begin{itemize}
\item At time $\tau_{k-1}^r$, run peeling with $\algo_\<$ until $\tau_k^l:=\inf\{n>\tau_{k-1}^r:Y_n<-Y_{\tau_{k-1}^r}-1+\min_{\tau_{k-1}^l\leq m\leq\tau_{k-1}^r}Y_m\}$.
\item At time $\tau_k^l$, run peeling with $\algo_\+$ until $\tau_k^r:=\inf\{n>\tau_k^l:X_n<-X_{\tau_k^l}-1+\min_{\tau_{k-1}^r\leq m\leq\tau_k^l}X_m\}$.
\end{itemize}
Obviously, the above constructed $\algo_m$ is a local and a Dobrushin-stable peeling algorithm. We denote the law of this peeling process by $\tilde\Prob_p$ ($p\in\N\cup\infty$). Note that the above stopping times may be infinite if $p<\infty$. Let
\begin{equation*}
\tilde\theta_R:=\inf\{n>\tau_R^l:X_n<-X_{\tau_R^l}-1+\min_{\tau_{R-1}^r\leq m\leq\tau_R^l}X_m\}.
\end{equation*}
The stopping time $\tilde\theta_R$ may be infinite for $p<\infty$, but the drift condition assures that $\tilde\Prob_\infty(\tilde\theta_R<\infty)=1$. It follows that under $\tilde\Prob_\infty$, the peeling process with algorithm $\algo_m$ explores the half-plane by distance layers, in the sense that the finite stopping time $\tilde\theta_R$ is an upper bound for the covering time $\theta_R$ of the ball of radius $R$. Hence, choosing $\algo=\algo_m$ in the general construction of the local limit and $\theta=\theta_R$ in Proposition~\ref{lem:stopped peeling} will give the construction of $\prob_\infty^\nu$ and yield the local convergence $\prob_p^\nu\cv{p}\prob_\infty^\nu$. To be a bit more precise, we still need to verify that $\tilde\Prob_p\to\tilde\Prob_\infty$ weakly. This is shown in the following lemma.
\begin{figure}
\begin{center}
\includegraphics[scale=0.7]{Fig7.pdf}
\end{center}\vspace{-1em}
\caption{Left: a glimpse of a realization of the local limit as $\nu<\nu_c$. In this case, the geometry is reminiscent to subcritical face percolation. Right: a glimpse of a realization of the local limit as $\nu>\nu_c$. In this case, the local limit almost surely contains a finite "bottleneck" between the extremities $\rho$ and $\rho^\dagger,$ being an infinite \emph{two-ended} Ising-decorated random triangulation.}
\end{figure}
\begin{lemma}\label{lem:mixedpeelingconv}
Let $\nu\in(1,\nu_c)$. Then $\tilde\Prob_p\to\tilde\Prob_\infty$ as $p\to\infty$.
\end{lemma}
\begin{proof}
From the construction of $\tilde\Prob_p$ and by the spatial Markov property, for all $n\ge 1$ and all $\step_1,\cdots,\step_n\in \steps$, as well as for all $k\in [1,n]$ and $1\le m_1^l\le m_1^r\le\dots\le m_k^l\le m_k^r\le n$, we have
\begin{align*}
&\tilde\Prob_p(\Step_1=\step_1,\cdots, \Step_n=\step_n, \tau_1^l=m_1^l, \tau_1^r=m_1^r,\dots,\tau_k^l=m_k^l, \tau_k^r=m_k^r)\\ &=\Prob_p(\Step_1=\step_1,\dots,\Step_{m_1^l}=\step_{m_1^l},\tau_1^l=m_1^l)\hat\Prob_{p+x_{m_1^l}}(\Step_1=\step_{m_1^l+1},\dots,\Step_{m_1^r-m_1^l}=\step_{m_1^r},\tau_1^r=m_1^r) \\ &\cdots\Prob_{p+x_{m_{k-1}^r}}(\Step_1=\step_{m_{k-1}^r+1},\dots,\Step_{m_k^l-m_{k-1}^r}=\step_{m_k^l},\tau_k^l=m_k^l)\hat\Prob_{p+x_{m_k^l}}(\Step_1=\step_{m_k^l+1},\dots,\Step_{m_k^r-m_k^l}=\step_{m_k^r},\tau_k^r=m_k^r)\\
&\cdot\Prob_{p+x_{m_k^r}}(\Step_1=\step_{m_k^r+1},\dots,\Step_{n-m_k^r}=\step_n),
\end{align*} where the peeling events $(\step_i)_{1\le i\le n}$ completely determine the perimeter variations $(x_i)_{1\le i\le n}$. By the convergences $\Prob_p\to\Prob_\infty$ and $\hat{\Prob}_p\to\hat{\Prob}_\infty$, and by another application of the spatial Markov property, the right hand side tends to the limit $\tilde\Prob_\infty(\Step_1=\step_1,\cdots, \Step_n=\step_n, \tau_1^l=m_1^l, \tau_1^r=m_1^r,\dots,\tau_k^l=m_k^l, \tau_k^r=m_k^r)$. The claim follows.
\end{proof}
\begin{proof}[Proof of the convergence $\prob_p^\nu \protect{\cv p} \prob_\infty^\nu$ for $1<\nu<\nu_c$]
We write
\begin{equation*}
\prob_p^\nu(\btsq_R=\bmap)=\prob_p^\nu([\emapo_{\tilde\theta_R}]_R=\bmap, \tilde\theta_R<N)+\prob_p^\nu(\btsq_R=\bmap, \tilde\theta_R\geq N),
\end{equation*}
where the last term satisfies
\begin{equation*}
\prob_p^\nu(\btsq_R=\bmap, \tilde\theta_R\geq N)\leq\tilde{\Prob}_p^\nu(\tilde\theta_R\ge N)\cv[]p\tilde{\Prob}_\infty^\nu(\tilde\theta_R\ge N)\cv[]N 0
\end{equation*}
by Lemma~\ref{lem:mixedpeelingconv} and the drift condition. Thus, letting first $p\to\infty$ and then $N\to\infty$ yields the claim.
\end{proof}
\begin{proof}[Proof of the convergence $\prob_{p,q}^\nu \protect{\cv {p,q}} \prob_\infty^\nu$ while $0<\lambda'\leq\frac{q}{p}\leq\lambda$ for $1<\nu<\nu_c$]
It is not hard to see that the counterparts of the above lemmas also hold under the diagonal rescaling; each of the proofs is just a mutatis mutandis of the previous ones. The essential matter is that the peeling processes under $\prob_{p,q}^\nu$ converge towards the peeling processes under $\prob_\infty^\nu$, due to the diagonal asymptotics. Again, we take into account $\rho^\dagger$ as a target.
\end{proof}
\section{The local limit at $\nu=\nu_c$ under diagonal rescaling}\label{sec:locallimit c diag}
Throughout this section, we assume that $\nu=\nu_c$ and $\frac{q}{p}\in[\lambda',\lambda]$ for some $0<\lambda'\leq 1\leq\lambda<\infty$ as $p,q\to\infty$, and study the local limit of $\prob_{p,q}$ in this setting. We find, unsurprisingly, the same local limit $\prob_\infty=\prob_\infty^{\nu_c}$ as discovered in \cite{CT20}. Moreover, we find the scaling limit of the first hitting time of the peeling process in a neighborhood of $\rho^\dagger$. The starting point of our analysis is the diagonal asymptotics (Theorem~\ref{thm:asympt}) \begin{equation*}\label{eq:diagasympt}
z_{p,q}(\nu)\sim
\frac{b\cdot c(q/p)}{\Gamma\left(-\frac{4}{3}\right)\Gamma\left(-\frac{1}{3}\right)}u_c^{-(p+q)}p^{-11/3}\qquad (\nu=\nu_c) .
\end{equation*}
We choose the peeling process with the target $\rho^\dagger$, described in Section~\ref{sec:peeling}: If the peeling step $\step_n$ splits the triangulation into two pieces, we choose the unexplored part $\umap_n$ to be the one containing $\rho^\dagger$. If $\rho^\dagger$ is included in both, we choose the one in the right. This gives rise to a different perimeter variation process $(X_n,Y_n)$, whose law is described in Table~\ref{tab:prob(p,q)diag}.
\begin{table}[t!]
\centering
\begin{tabu}{|L|S|C| |L|S|C| L}
\cline{1-6}
\step &\Prob_{p,q+1}(\Step_1 = \step) &(X_1,Y_1) &
\step &\Prob_{p,q+1}(\Step_1 = \step) &(X_1,Y_1)
\hl \cp & \zp{p+2,q} &(2,-1)
& \cm & \nu \zp{p,q+2} &(0,1)
\hl \lp & \zzp{p+1,q-k}{1,k} &(1,-k-1)
& \lm & \nu \zzp{p,q-k+1}{0,k+1} &(0,-k) & (0\le k \le\theta q)
\hl \rp & \zzp{k+1,0}{p-k+1,q} &(-k+1,-1)
& \rn & \nu \zzp{k,1}{p-k,q+1} &(-k,0) & (0\le k\le \theta p)
\hl \lp[q-k] & \zzp{p+1,k}{1,q-k} &(1,-q+k-1)
& \lm[q-k] & \nu \zzp{p,k+1}{0,q-k+1} &(0,-q+k)
&(0<k<\theta q)
\hl \rp[p-k] & \zzp{p-k+1,0}{k+1,q} &(-p+k+1,-1)
& \rn[p-k] & \nu \zzp{p-k,1}{k,q+1} &(-p+k,0)
&(0\le k<\theta p)
\hl
\end{tabu}
\caption{Law of the first peeling event $\Step_1$ under $\Prob_{p,q+1}$ and the corresponding $(X_1,Y_1)$ under the peeling process of the left-most interface with the target $\rho^\dagger$. In the table, $\theta\in (0,1)$ is an arbitrary cutoff, which roughly measures whether the perimeter process has only small jumps or not. Observe that the last two rows of the table are redundant with the second and the third row, respectively, in order to emphasize the cutoff for taking the limit. Taking the limit $(p,q)\to\infty$ gives the data of Table~\ref{tab:pinfty}.
}\label{tab:prob(p,q)diag}
\end{table}
Accordingly, we define for $m\geq 0$
\begin{equation*}
T_m:=\inf\{n\ge 0: \min\{P_n,Q_n\}\leq m\}.
\end{equation*}
Using the peeling steps, we also see that $T_m=\inf\{n\ge 0: \Step_n\in\{\rp[p-k+1],\rn[p-k], \lp[q-k-1], \lm[q-k]: 0\leq k\leq m\}\}$.
\subsection{The one-jump phenomenon of the perimeter process}\label{sec:onejumpscaling}
\newcommand{\tauxy}{\tau^\epsilon_x}
\newcommand{\barrier}[1][x]{#1 f_\epsilon}
Next, we investigate an analog of the \emph{large jump} phenomenon discovered in \cite{CT20}. For that, fix $\epsilon>0$ and let \begin{equation*}
\barrier[](n) = \mb({ (n+2)(\log(n+2))^{1+\epsilon} }^{3/4}.
\end{equation*}Define the stopping time
\begin{equation*}
\tauxy = \inf\Set{n\ge 0}{\abs{X_n-\mu n} \vee \abs{Y_n-\mu n} > \barrier(n) }\,.
\end{equation*}
where $x>0$.
\begin{lemma}[One jump to zero]\label{lem:one jump diag}
For all $\epsilon>0$ and $0<\lambda'\leq 1\leq\lambda<\infty$,
\begin{equation*}
\lim_{x,m \to\infty} \limsupp \Prob_{p,q} (\tauxy<T_m) = 0\quad\text{while}\quad\frac{q}{p}\in[\lambda',\lambda].
\end{equation*}
\end{lemma}
\begin{figure}
\begin{center}
\includegraphics[scale=0.8]{Fig8.pdf}
\end{center}\vspace{-1em}
\caption{Left: a glimpse of a realization of the local limit at $\nu=\nu_c$ after just $q\to\infty$, which is considered in \cite{CT20}. The interface is still finite. In this work, we bypass this intermediate limit via the diagonal rescaling. Right: a realization of the ribbon containing the interface in the local limit at $\nu=\nu_c$.}
\end{figure}
The proof of Lemma~\ref{lem:one jump diag} is a modification of the proof of the analogous Lemma 10 in \cite{CT20}. The necessary changes are left to Appendix~\ref{sec:bigjumplemma proof}. Next, we prove the main scaling limit result of this article.
\newcommand*{\anom}{\mathcal{E}}
\newcommand*{\nom}{\mathcal{N}}
\begin{proof}[Proof of Theorem~\ref{thm:scaling}]
First, assuming that a scaling limit of $p^{-1}T_m$ exists for every $m\geq 0$, it actually does not depend on $m$. Namely, since $T_0\ge T_m$, the strong Markov property gives
\begin{align}
\Prob_{p,q}(T_0-T_m >\epsilon p) \ &= \ \EE_{p,q}\m[{ \Prob_{P_{T_m},Q_{T_m}}(T_0 >\epsilon p) }
\notag \\ &
\le\ \EE_{p,q}\m[{\sum_{p'=0}^m\Prob_{p',Q_{T_m}}(T_0 >\epsilon p)+\sum_{q'=0}^m\Prob_{P_{T_m},q'}(T_0>\epsilon p)}.\label{eq:Tdifference}
\end{align}
Let $M>0$ be some large constant, and fix $p'\leq m$. We write \begin{align*}
\Prob_{p',Q_{T_m}}(T_0 >\epsilon p)&=\Prob_{p',Q_{T_m}}(T_0 >\epsilon p, \ Q_{T_m}>M)+\Prob_{p',Q_{T_m}}(T_0 >\epsilon p| \ Q_{T_m}\leq M)\Prob_{p',Q_{T_m}}(Q_{T_m}\leq M)\\ &\leq\Prob_{p',Q_{T_m}}(T_0 >\epsilon p, \ Q_{T_m}>M)+\sum_{q'=0}^M\Prob_{p',q'}(T_0>\epsilon p).
\end{align*}
By \cite[Proposition 2]{CT20} (actually, by its analog for the peeling with target), $\Prob_{p',q}\cv[]q\Prob_{p'}$. Therefore, the first term can be bounded from above by $\Prob_{p'}(T_0>\epsilon p)+\epsilon'$ for any $\epsilon'>0$, provided $M$ is large enough. In that case, we obtain
\begin{equation*}
\sum_{p'=0}^m\Prob_{p',Q_{T_m}}(T_0 >\epsilon p)\le \sum_{p'=0}^m\Prob_{p'}(T_0>\epsilon p)+\sum_{p'=0}^m\sum_{q'=0}^M\Prob_{p',q'}(T_0>\epsilon p)+(m+1)\epsilon'.
\end{equation*}It is easy to see that the right hand side converges to zero as $p\to\infty$ and $\epsilon'\to 0$. The second term in Equation~\eqref{eq:Tdifference} is treated similarly, and finally we deduce $\Prob_{p,q}(T_0-T_m >\epsilon p)\cv[]{p,q} 0$.
Let us then proceed to the existence of the scaling limit. First, fix $x>0$, $m\in\natural$ and $\epsilon\in(0,\mu)$. Take $p$ and $q$ large enough such that $\Prob_{p,q}$-almost surely, $\tauxy\le T_m$. Denote $\anom := \{ \tauxy<T_m \}$ and $\nom_n := \{ \tauxy>n \}$. Clearly $\nseq \nom$ is a decreasing sequence, and one can check that
\begin{equation}\label{eq:anomaly inclusion}
\nom_{n+1} \ \subset\ \nom_n\setminus\{T_m=n+1\}
\ \subset\ \nom_{n+1} \cup \anom\,.
\end{equation}
Let $c_m(\lambda):=\lim_{p,q\to\infty} p\cdot\Prob_{p,q}(T_m=1)$, where the limit is taken such that $q/p\to\lambda$. In other words, $q=\lambda p+o(p)$, and from the asymptotics of Theorem~\ref{thm:asympt},
\begin{equation}\label{eq:def c_m}
\Prob_{p,q}(T_m=1)\sim\frac{c_m\left(\frac{q}{p}\right)}{p}
\end{equation}
as $p,q\to\infty$, $q/p\to\lambda$.
On the event $\nom_n$, we have $P_0+\mu n -\barrier(n) \le P_n\le P_0+\mu n + \barrier(n)$ and $Q_0+\mu n -\barrier(n) \le Q_n\le Q_0+\mu n + \barrier(n)$. This, in particular, gives
\begin{equation}\label{eq:lambdaestimate}
\frac{\lambda p+\mu n-x f_\epsilon(n)+o(p)}{p+\mu n+x f_\epsilon(n)}\le\frac{Q_n}{P_n}\le\frac{\lambda p+\mu n+x f_\epsilon(n)+o(p)}{p+\mu n-x f_\epsilon(n)}.
\end{equation}
Denote $\lambda_n:=Q_n/P_n$. Then combining the previous equation with \eqref{eq:def c_m}, we also obtain that for $P_0=p$ and $Q_0=q$ large enough,
\newcommand*{\cmore}[1][p]{ \frac{c_m(\lambda_n) +\epsilon}{#1+\mu n - \barrier(n)} }
\newcommand*{\cless}[1][p]{ \frac{c_m(\lambda_n) -\epsilon}{#1+\mu n + \barrier(n)} }
\begin{equation*}
\cless \id_{\nom_n} \ \le\ \id_{\nom_n} \Prob_{P_n,Q_n}(T_m=1) \ \le\ \cmore \id_{\nom_n} \,.
\end{equation*}
By Markov property, $\Prob_{p,q}(\nom_n\setminus\{T_m=n+1\})
= \Prob_{p,q}(\nom_n) - \EE_{p,q}\m[{ \id_{\nom_n} \Prob_{p,q}(T_m=1) }$. Therefore
\begin{align*}
\m({1-\cmore} \Prob_{p,q}(\nom_n) &\ \le\ \Prob_{p,q}(\nom_n\setminus\{T_m=n+1\})
\\&\ \le\ \m({1-\cless} \Prob_{p,q}(\nom_n)\,.
\end{align*}
Combining these estimates with the two inclusions in \eqref{eq:anomaly inclusion}, we obtain the upper bounds
\begin{equation*}
\Prob_{p,q}(\nom_{n+1})\ \le\ \m({1-\cless} \Prob_{p,q}(\nom_n) \,,
\end{equation*}
and the lower bounds
\begin{align*}
\Prob_{p,q}(\nom_{n+1}\cup\anom)
&\ \ge\ \Prob_{p,q}\m({ (\nom_n\setminus\{T_m=n+1\}) \cup\anom }
\\&\ \ge\ \Prob_{p,q}(\nom_n\setminus\{T_m=n+1\}) + \Prob_{p,q}(\anom\setminus\nom_n)
\\&\ \ge\ \m({1-\cmore} \Prob_{p,q}(\nom_n) + \Prob_{p,q}(\anom\setminus \nom_n)
\\&\ \ge\ \m({1-\cmore} \Prob_{p,q}(\nom_n\cup \anom) \,.
\end{align*}
Then, by iterating the two bounds, we get
\begin{equation*}
\Prob_{p,q}(\nom_N) \le \prod_{n=0}^{N-1} \m({1-\cless} \quad\text{and}\quad
\Prob_{p,q}(\nom_N\cup \anom) \ge \prod_{n=0}^{N-1} \m({1-\cmore}
\end{equation*} for any $N\ge 1$.
Since $\nom_n\subset \{T_m>n\}\subset \nom_n \cup\anom$ up to a $\Prob_{p,q}$-negligible set, the above estimates imply that
\begin{equation*}
\prod_{n=0}^{N-1} \m({1-\cmore} - \Prob_{p,q}(\anom) \ \le\ \Prob_{p,q}(T_m>N)
\ \le\ \prod_{k=0}^{N-1} \m({1-\cless} + \Prob_{p,q}(\anom)\,.
\end{equation*}
From the Taylor series of the logarithm we see that $-x-x^2\le \log(1-x)\le -x$ for all $x\ge 0$. Therefore, for any positive sequence $\nseq x$, we have
\begin{equation*}
\exp\mB({ -\sum_{n=0}^{N-1} x_n -\sum_{n=0}^{N-1} x_n^2}
\ \le\ \prod_{n=0}^{N-1}(1-x_n)\ \le\ \exp\mB({ -\sum_{n=0}^{N-1} x_n }\,.
\end{equation*}
Now, we consider the sum \begin{equation*}\sum_{n=0}^{tp}\frac{c_m(\lambda_n)\pm\epsilon}{p+\mu n\mp x f_\epsilon(n)}.
\end{equation*}
First, by \eqref{eq:lambdaestimate}, we see that $\lambda_n=\frac{\lambda p+\mu n}{p+\mu n}\left(1+o(1)\right)$ where $o(1)$ is uniform over all $n\in[0,tp]$ as $p\to\infty$. Namely,
\begin{equation*}
\abs{1-\frac{\lambda p+\mu n\pm xf_\epsilon(n)+o(p)}{p+\mu n\mp x f_\epsilon(n)}\cdot\frac{p+\mu n}{\lambda p+\mu n}}=\abs{\frac{x f_\epsilon(n)\left(\frac{p+\mu n}{\lambda p+\mu n}+1\right)+o(p)}{p+\mu n\mp x f_\epsilon(n)}},
\end{equation*}
where the right hand side tends to zero uniformly on $n\in[0,tp]$ as $p\to\infty$.
On the other hand, we also have
$\frac{c_m(\lambda_n)\pm \epsilon}{p+\mu n\mp \barrier(n)} = \frac{c_m(\lambda_n)\pm \epsilon}{p+\mu n} (1+o(1))$ uniformly on $[0,tp]$, for any fixed $t>0$. Hence,
\begin{equation*}
\sum_{n=0}^{tp} \frac{c_m(\lambda_n)\pm \epsilon}{p+\mu n\mp \barrier(n)}
\ \cv[]p \int_0^{t} \frac{c_m\left(\frac{\lambda+\mu s}{1+\mu s}\right)\pm\epsilon}{1+\mu s}\dd s=\pm \frac{\epsilon}{\mu}\log(1+\mu t)+\int_0^{t} \frac{c_m\left(\frac{\lambda+\mu s}{1+\mu s}\right)}{1+\mu s}\dd s\,.
\end{equation*}
Above, we also used the fact that $c_m(\lambda)$ is continuous in $\lambda$, which follows directly from its definition and is also seen below via an explicit expression. We also have
\begin{equation*}
\sum_{n=0}^{tp} \left(\frac{c_m(\lambda_n)+\epsilon}{p+\mu n- \barrier(n)}\right)^2 \cv[]p 0
\end{equation*} for all $t>0$. Combining this with the last three displays, we conclude that
\begin{align}\label{eq:Tmscalingbounds}
&(1+\mu t)^{- \frac{\epsilon}\mu}\exp\left(-\int_0^{t} \frac{c_m\left(\frac{\lambda+\mu s}{1+\mu s}\right)}{1+\mu s}\dd s\right) - \limsupp \Prob_{p,q}(\anom)
\ \le\ \liminf_{p,q\to\infty} \Prob_{p,q}(T_m>tp) \notag
\\& \ \le\ \limsup_{p,q\to\infty} \Prob_{p,q}(T_m>tp)
\ \le\ (1+\mu t)^{\frac{\epsilon}\mu}\exp\left(-\int_0^{t} \frac{c_m\left(\frac{\lambda+\mu s}{1+\mu s}\right)}{1+\mu s}\dd s\right)
+ \limsupp \Prob_{p,q}(\anom) \,.
\end{align}
Now take the limit $m,x\to\infty$.
First, using the data of Table~\ref{tab:prob(p,q)diag}, we observe that the sequence $(c_m(\lambda))_{m\ge0}$ is increasing with a finite limit: \begin{align}\label{eq:defcinfty}
c_m(\lambda)\, =&\
\lim_{p,q\to\infty} \m({ p\cdot \Prob_{p,q}(P_1\wedge Q_1\leq m)}
\ =\ \lim_{p,q\to\infty} p\cdot \sum_{k=1}^m
\m({ \Prob_{p,q}(\rp[p-k+1],\rn[p-k])+\Prob_{p,q}(\lp[q-k-1],\lm[q-k]) }
\notag \\
=&\ -\frac43 \frac{t_c}{b \cdot \lambda^{7/3} c(\lambda)}
\sum_{k=1}^m(1+\nu_c)\m({ \frac{a_0}{u_c}+a_1 } a_k u_c^k
\notag \\ \cv[]m &\
-\frac43 \frac{t_c}{b \cdot \lambda^{7/3} c(\lambda)}
(1+\nu_c) \m({ \frac{a_0}{u_c}+a_1 } (A(u_c)-a_0)
\ =:\ c_\infty(\lambda).
\end{align}
Furthermore, we notice that $(1+\nu_c)\left(\frac{a_0}{u_c}+a_1\right)(A(u_c)-a_0)=-b\mu$, a computation already done in the proof of \cite[Proposition 11]{CT20}. This gives $c_\infty(\lambda)=\frac{4}{3}\frac{\mu}{c(\lambda)\lambda^{7/3}}.$
Moreover, in the limit $m,x\to\infty$, the error term $\limsup\Prob_{p,q} (\anom)$ tends to zero due to Lemma~\ref{lem:one jump diag}. The middle terms $\liminf_{p,q\to\infty}\Prob_{p,q}(T_m>tp)$ and $\limsup_{p,q\to\infty}\Prob_{p,q}(T_m >tp)$ do not depend on $m$ due to the convergence $\Prob_{p,q}(T_0-T_m>\epsilon p) \cv[]{p,q} 0$ seen at the beginning of the proof. Thus by sending $m\to\infty$ and $\epsilon\to 0$, the monotone convergence theorem finally yields \begin{equation*}
\lim_{p,q\to\infty} \Prob_{p,q}\m({T_m>tp} = \exp\left(-\int_0^t c_\infty\left(\frac{\lambda+\mu s}{1+\mu s}\right)\frac{\dd s}{1+\mu s}\right).
\end{equation*}
Now recall that
\begin{equation*}c(\lambda)=\frac{4}{3}\int_0^\infty(1+s)^{-7/3}(\lambda+s)^{-7/3}ds.
\end{equation*}
We note first that
\begin{equation*}
c\left(\frac{\lambda+x}{1+x}\right)=\frac{4}{3}(1+x)^{11/3}\int_x^\infty(1+s)^{-7/3}(\lambda+s)^{-7/3} \dd s.
\end{equation*}
This yields
\begin{align*}
& \od{}{x} \m({ \int_0^{\mu^{-1}x} c_\infty \m({ \frac{\lambda+\mu s}{1+\mu s} } \frac{\dd s}{1+\mu s} }
=\frac1\mu \cdot c_\infty\m({ \frac{\lambda+x}{1+x} } \cdot\frac{1}{1+x}
\\ =\ &(1+x)^{-7/3}(\lambda+x)^{-7/3}\left(\int_x^\infty(1+s)^{-7/3}(\lambda+s)^{-7/3}ds\right)^{-1}
=-\od{}{x} \log\int_x^\infty(1+s)^{-7/3}(\lambda+s)^{-7/3}\dd s.
\end{align*}
Finally, integrating this equation on each of the sides gives the claim.
\end{proof}
In order to prove the diagonal local convergence in its full generality as Theorem~\ref{thm:cv} suggests, we also show the following generalized bounds:
\begin{prop}\label{prop:scalingTm}
For all $m\in\natural$, the scaling limit of the jump time $T_m$ has the following bounds:
\begin{equation*}
\forall t>0\,,\qquad \liminf_{p,q\to\infty} \Prob_{p,q}\m({T_m>tp} \ge \exp\left(-\int_0^t\max_{\ell\in[\lambda',\lambda]}c_\infty\left(\frac{\ell+\mu s}{1+\mu s}\right)\cdot\frac{ds}{1+\mu s}\right)
\end{equation*} and
\begin{equation*}
\limsup_{p,q\to\infty} \Prob_{p,q}\m({T_m>tp} \le \exp\left(-\int_0^t\min_{\ell\in[\lambda',\lambda]}c_\infty\left(\frac{\ell+\mu s}{1+\mu s}\right)\cdot\frac{ds}{1+\mu s}\right)
\end{equation*}
where $c_\infty$ is defined as in \eqref{eq:defcinfty} and the limit is taken such that $q/p\in [\lambda',\lambda]$.
\end{prop}
\begin{proof}
We modify the above proof as follows: First, \eqref{eq:lambdaestimate} translates to
\begin{equation*}
\frac{\lambda' p+\mu n-x f_\epsilon(n)}{p+\mu n+x f_\epsilon(n)}\le\frac{Q_n}{P_n}\le\frac{\lambda p+\mu n+x f_\epsilon(n)}{p+\mu n-x f_\epsilon(n)}
\end{equation*}
conditional on $\nom_n$. Then, the identity $Q_n/P_n:=\lambda_n=\frac{\lambda p+\mu n}{p+\mu n}\left(1+o(1)\right)$ is to be replaced by the bounds
\begin{equation*}
\frac{\lambda' p+\mu n}{p+\mu n}\left(1+o(1)\right)\le\lambda_n\le \frac{\lambda p+\mu n}{p+\mu n}\left(1+o(1)\right).
\end{equation*}
Finally, we notice that $\lambda\mapsto c_m(\lambda)$ is a continuous function for every $m\ge 0$ on any compact strictly positive interval, having the limit $c_\infty(\lambda)$ as $m\to\infty$ with the same property. Therefore, we can replace $c_m\left(\frac{\lambda+\mu s}{1+\mu s}\right)$ in \eqref{eq:Tmscalingbounds} by its minimum or maximum over the interval $[\lambda',\lambda]$, respectively, and finally take the limit $m\to\infty$.
\end{proof}
The limit law
\begin{equation}\label{scalinglimitlaw}
\Prob(L>t):=\int_t^\infty(1+x)^{-7/3}(\lambda+x)^{-7/3}dx
\end{equation}
can be interpreted as the law of the quantum length of an interface resulted from the conformal welding of two quantum disks in the Liouville Quantum Gravity of parameter $\gamma=\sqrt{3}$, introduced in the context of the mating of the trees theory in \cite{matingoftrees} and studied in \cite{AHS20}. More precisely, this measure results from a welding of two independent quantum disks of parameter $\gamma=\sqrt{3}$ and weight $2$ along a boundary segment of length $L$. See \cite{matingoftrees,AG19,AHS20} for precise definitions of such quantum disks. In particular, a quantum disk conditioned to have a fixed boundary length is well-defined.
As defined in \cite{AG19}, an $(R,L)$-length quantum disk $(D,x,y)$ is a quantum disk decorated with two marked boundary points $x,y$, which is sampled in the following way: First, a quantum disk $D$ of a fixed boundary length $R+L$ is sampled. Then, conditional on $D$, the boundary point $x$ is sampled from the quantum boundary length measure. Finally, define $y$ to be the boundary point of $D$ such that the counterclockwise boundary arc from $x$ to $y$ has length $R$. By giving the quantum disk an additional weight parameter and setting its value to $2$, the points $x$ and $y$ can in fact be sampled independently from the LQG boundary length measure, as explained in \cite{AHS20}.
For two independent $\sqrt{3}$-quantum disks, there is a natural perimeter measure on $(0,\infty)^2$, given by
\begin{equation}\label{perimeterlaw}
dm(u,v)=u^{-7/3}v^{-7/3}du dv.
\end{equation}
This measure is the Lévy measure of a pair of independent spectrally positive $4/3$-stable Lévy processes, which has a direct connection to the jumps of the boundary length processes of SLE($16/3$). On the other hand, it is known that the typical disks swallowed by the SLE($16/3$) are $\sqrt{3}$-quantum disks; see \cite{MSW20}. This perimeter measure allows us to randomize the boundary arc lengths of the quantum disks as follows.
Due to the convergence $q/p\to\lambda\in(0,\infty)$ (and $p/p\to 1$) in our discrete picture, we consider the measure \eqref{perimeterlaw} conditional on the set $\{(u,v): u=1+L,\ v=\lambda+L,\ L>0\}$, such that the two independent quantum disks have perimeters $(\lambda,L)$ and $(L,1)$, respectively. This gives rise to the law of the segment $L$ as
\begin{equation*}
\Prob(L\in dx)=\mathcal{N}^{-1}(1+x)^{-7/3}(\lambda+x)^{-7/3}dx
\end{equation*}
where $\mathcal{N}$ is a normalizing constant in order to yield a probability distribution. Gluing the two quantum disks along the boundary segment of length $L$ such that the marked boundary points of the two disks coincide to the points $\rho$ and $\rho^\dagger$, respectively, finally yields \eqref{scalinglimitlaw} as the law of the interface length.
The same law of $L$ has been recently derived in \cite[Remark 2.7]{AHS20} as a special case of the general conformal welding of quantum disks. Since the parameters there also match with the expected ones for the universality class of the critical Ising model, this gives some hints that the Ising interfaces should indeed converge towards an SLE$(3)$-curve on a LQG surface, as predicted in the literature. This convergence remains as an important open problem.
\newcommand{\ribbon}{\mathcal{R}}
\newcommand*{\rmap}[1][m]{\mathcal{R}_{#1}}
\newcommand*{\uleft}[1][T_m]{\umap_{#1}}
\newcommand*{\uright}[1][T_m]{\umap^*_{#1}}
\newcommand{\kk}{\mathcal{K}_m}
\newcommand{\pjump}{\mathcal{P}}
\newcommand{\qjump}{\mathcal{Q}}
\newcommand{\pleft}{\mathcal{P}}
\newcommand{\qleft}{\mathcal{Q}}
\newcommand{\pright}{\mathcal{P}^*}
\newcommand{\qright}{\mathcal{Q}^*}
\subsection{The local convergence under the diagonal rescaling}
We recall first the definition of the local limit $\prob_\infty=\prob_\infty^{\nu_c}$ (see \cite{CT20}). The probability measure $\prob_\infty$ is defined as the law of a random triangulation of the half-plane which is obtained as a gluing of three infinite, mutually independent, one-ended triangulations $\law_\infty\umap_\infty$, $\law_\infty\ribbon_\infty$ and $\law_\infty\umap_\infty^*$ along their boundaries, which satisfy the following properties: $\law_\infty\umap_\infty$ has the law $\prob_0$, $\law_\infty\umap_\infty^*$ has the law $\prob_0$ under the inversion of the spins, and $\law_\infty\ribbon_\infty$ is defined as the law of the increasing sequence $(\lim_{n\to\infty}\law_\infty[\emapo_n]_r)_{r\ge 0}$ under $\Prob_\infty$. The fact that the ball $[\emapo_n]_r$ stabilizes in a finite time, and thus the limit $n\to\infty$ is well-defined, follows from the positive drift of the perimeter processes. Observe that the boundary of $\law_\infty\ribbon_\infty$ consists of three arcs: a finite one consisting of edges of $\partial\emapo_0$ only, and two infinite arcs of spins \< and \+, respectively. The gluing is performed along the infinite boundary arcs such that the spins match with the boundary spins of $\umap_\infty$ and $\umap^*_\infty$, respectively.
Then, fix $m\ge 0$, and define $\ribbon_m$ as the union of the explored map $\emapo_{T_m-1}$ and the triangle explored at $T_m$. Now the triple $(\umap_{T_m},\ribbon_m,\umap_{T_m}^*)$ partitions a triangulation under $\prob_{p,q}$, such that $\umap_{T_m}$ and $\umap_{T_m}^*$ correspond to the two parts separated by the triangle at $T_m$. We will reroot the unexplored maps $\uleft$ and $\uright$ at the vertices $\rho_\umap$ and $\rho_{\umap^*}$, which are the unique vertices shared by $\umap_{T_m}$ and $\ribbon_m$, and $\umap_{T_m}^*$ and $\ribbon_m$, respectively. Now the boundary condition of $\uleft$ is denoted by $(\pleft,(\qleft_1,\qleft_2))$. This notation is in line with \cite[Theorem 4]{CT20}. Similarly, the boundary condition of $\uright$ is $((\pright_1,\pright_2),\qright)$. Observe that the condition
\begin{equation*}
\Step_{T_m} \in \{ \rp[P_{T_m-1}+\kk], \rn[P_{T_m-1} +\kk] \}
\end{equation*}
uniquely defines an integer $\kk$, which represents the position relative to $\rho^\dagger$ of the vertex where the triangle revealed at time $T_m$ touches the boundary. Here, we make the convention that $\rp[p+k]=\lp[q-k-1]$ and $\rn[p+k]=\lm[q-k-1]$ for $k\ge 0$. In particular, $|\kk|\leq m$. See also \cite[Figure 12]{CT20} for a similar setting when $q=\infty$.
\newcommand{\PrE}[4]{\prob#1 \mb({ ([#2]_r,[#3]_r,[#4]_r) \in \mathcal{E} }}
\begin{lemma}[Joint convergence before gluing]\label{lem:loc cv on big jump}
Fix $\epsilon,x,m>0$, and let $\mathcal{J} \equiv \mathcal{J}^\epsilon_{x,m} := \{\tauxy = T_m \ge \epsilon p\}$. Then for any $r\ge 0$,
\begin{equation*}\label{eq:loc cv on big jump}
\begin{aligned}
& \limsup_{p,q\to\infty} \abs{ \PrE\pqy{\rmap}{\uleft}{\uright}
- \PrE\yy{\rmap[\infty]}{\uleft[\infty]}{\uright[\infty]} }
\\ \le \ &
\limsup_{p,q\to\infty} \prob_{p,q} (\mathcal{J}^c) + \prob_\infty(\tauxy<\infty)
\end{aligned}
\end{equation*}
where $\mathcal{E}$ is any set of triples of balls.
\end{lemma}
\begin{proof}
The proof is a mutatis mutandis of the proof of \cite[Lemma 14]{CT20}. With the idea of that proof, the only thing one needs to take care of is the fact that the random numbers $\pright_1$, $\pright_2$, $\qleft_1$ and $\qleft_2$ tend to $\infty$ uniformly, and that $\pleft$ and $\qright$ stay bounded, conditional on $\mathcal{J}$. Observe also that the random number $\kk$ is automatically bounded in this setting, so we do not need any condition for $\kk$ on the event $\mathcal{J}$.
Similarly as in \cite{CT20}, we have the lower bounds $\pright_1\geq\mu(\epsilon p-1)-xf_\epsilon(\epsilon p-1)=:\underline{\pright_1}$ and $\pright_2\ge p+\min_{n\ge 0}(\mu n-x f_\epsilon(n))-1-m=:\underline{\pright_2}$ as well as the upper bound $\qright\le m+1$ for the boundary condition of $\uright$.
For completeness and convenience, let us show similar bounds for the boundary condition of $\uleft$. Let $S^\+$ and $S^\<$ be the distances from $\rho$ to $\rho_{\umap^*}$ and $\rho_\umap$ along the boundary, respectively. First, expressing the total perimeter of $\uleft$, the number of edges between $\rho$ and $\rho^\dagger$ clockwise and the number of \+ edges on the boundary of $\uleft$, respectively, we find the equations
\begin{equation*}
\begin{cases}
\qleft_1+\qleft_2+\pleft=Q_{T_m-1}-\kk \\
S^{\<}+\qleft_2+\max\{0,\kk\}=q \\
\pleft=\delta-\min\{0,\kk\}
\end{cases}
\end{equation*}
where $\delta=1$ if $\Step_{T_m} =\rp[P_{T_m-1}+\kk]$, and otherwise $\delta=0$, as well as $S^{\<}$ is the number of $\<$ edges on $\ribbon_m\cap\partial\emapo_0$. The solution of this system of equations is
\begin{equation*}
\begin{cases}
\qleft_1=Y_{T_m-1}+S^{\<}-\delta \\
\qleft_2=q-S^{\<}-\max\{0,\kk\} \\
\pleft=\delta-\min\{0,\kk\}
\end{cases}.
\end{equation*}
We have $S^{\<}\in [0,1-\min_{n\ge 0}(\mu n-x f_\epsilon(n))]$, and the function $n\mapsto\mu n-x f_\epsilon(n)$ is increasing. Therefore, we deduce $\qleft_1\ge\mu(\epsilon p-1)-x f_\epsilon(\epsilon p-1)-2=:\underline{\qleft_1}$ and $\qleft_2\ge q+\min_{n\ge 0}(\mu n-x f_\epsilon(n))-1-m=:\underline{\qleft_2}$, together with $\pleft\le m+1$. The claim follows.
\end{proof}
\newcommand{\pglued}[1]{\prob#1 \mb({ [\mop]_r \in \mathcal{E} }}
\newcommand{\mop}{\tmap \oplus \tmap'}
\begin{proof}[Proof of the convergence $\prob\pqy^{\nu_c} \to \prob\yy^{\nu_c}$.]
The triangulation $\law\pqy \bt$ (respectively, $\law\yy \bt$) can be represented as the gluing the triple $\law\pqy (\rmap,\uleft,\uright)$ (respectively, $\law\yy (\rmap[\infty],\uleft[\infty],\uright[\infty])$) along their boundaries. This is done pairwise between the three components, taking into account that the location of the root vertex changes during this procedure. Given a triangulation $\tmap$ with a simple boundary, and an integer $S$, let us denote by $\overrightarrow \tmap^{S}$ (resp. $\overleftarrow \tmap^{S}$) the map obtained by translating the root vertex of $\tmap$ by a distance $S$ to the right (resp. to the left) along the boundary. Denote by $\rho$ and $\rho'$ the root vertices of two triangulations $\tmap$ and $\tmap'$, respectively, and let $L$ be the number of edges in $\tmap$ and $\tmap'$ which are admissible for the gluing. More precisely, we assume that $L$ is a random variable taking positive integer or infinite values, such that
\begin{equation}\label{eq:gluing length}
\law\pqy L \cv[]{p,q} \infty \text{ in distribution and }\law\yy L = \infty \text{ almost surely.}
\end{equation}
Finally, let $\mop$ be the triangulation obtained by gluing the $L$ boundary edges of $\tmap$ on the right of $\rho$ to the $L$ boundary edges of $\tmap'$ on the left of $\rho'$. The dependence on $L$ is omitted from this notation because the local limit of $\mop$ is not affected by the precise value of $L$, provided that \eqref{eq:gluing length} holds.
Now under $\prob\pqy$, we have
\begin{equation}\label{eq:gluing-of-3}
\bt = \overrightarrow{(\umap\rmap)}^{S^\+ + S^\<} \oplus \uright \qtq{where} \umap\rmap = \uleft \oplus \overleftarrow{(\rmap)}^{S^\<}
\end{equation}
where we recall that $S^\+$ and $S^\<$ are the distances from $\rho$ to $\rho_{\umap^*}$ and $\rho_\umap$ along the boundary, respectively. Similarly, $\law\yy \bt$ can be expressed in terms of $\uleft[\infty]$, $\rmap[\infty]$, $\uright[\infty]$ and $S^\jj$ using gluing and root translation.
On the event $\mathcal{J}$, the perimeter processes $\nseq X$ and $\nseq Y$ stay above $\mu n-xf_\epsilon(n)$ up to the time $\tauxy$. Thus their minima over $[0,\tauxy)$ are reached before the deterministic time $N_{\min} = \sup\Set{n\ge 0}{\mu n-xf_\epsilon(n)\le 0}$, and $S^\+$ and $S^\<$ are measurable functions of the explored map $\emapo_{N_{\min}}$. It follows that $\law\pqy S^\jj$ converges in distribution to $\law\yy S^\jj$ on the event $\mathcal{J}$. Using the relation \eqref{eq:gluing-of-3} together with \cite[Lemmas 15-16]{CT20}, we deduce from Lemma~\ref{lem:loc cv on big jump} that for any $x,m,\epsilon>0$, and for any $r\ge 0$ and any set $\mathcal{E}$ of balls, we have
\begin{equation*}
\limsupp \mb|{\, \prob\pqy ( \btsq_r \in \mathcal{E} )
- \prob\yy( \btsq_r \in \mathcal{E} ) \, } \ \le\
\limsupp \prob\pqy (\mathcal{J}^c) + \prob\yy (\tauxy<\infty) \,.
\end{equation*}
The left hand side does not depend on the parameters $x,m$ and $\epsilon$. Therefore to conclude that $\prob\pqy$ converges locally to $\prob\yy$, it suffices to prove that
$\displaystyle \limsupp \prob\pqy (\mathcal{J}^c) + \prob\yy (\tauxy<\infty)$\, converges to zero when $x,m\to\infty$ and $\epsilon\to 0$. The latter term converges to zero, since if $x\to\infty$, we have $\tauxy \to\infty$ almost surely under $\prob\yy$. For the first term, a union bound gives
\begin{equation*}
\prob\pqy(\mathcal{J}^c) \ \le\ \prob\pqy (\tauxy< T_m) + \prob\pqy(T_m<\epsilon p) \,,
\end{equation*}
where the first term on the right can be bounded using Lemma~\ref{lem:one jump diag}:
\begin{equation*}
\lim_{m,x \to\infty} \limsupp \prob\pqy(\tauxy<T_m) \ =\ 0 .
\end{equation*}
For the last term, we use the lower bound of Proposition~\ref{prop:scalingTm}:
\begin{equation*}
\lim_{\epsilon\to 0}\ \limsupp \prob\pqy(T_m<\epsilon p)\ \leq\
1-\lim_{\epsilon\to 0}\exp\left(-\int_0^\epsilon \max_{\ell\in[\lambda',\lambda]}c_\infty\left(\frac{\ell+\mu s}{1+\mu s}\right)\frac{ds}{1+\mu s}\right) \ =\ 0 \,.\qedhere
\end{equation*}
\end{proof}
\appendix
\section{A one-jump lemma for the process $\law_{p,q}\nseq{X_n,Y}$ at $\nu=\nu_c$}\label{sec:bigjumplemma proof}
\newcommand*{\ea}{\asymp}
\newcommand{\cst}{\mathrm{cst}}
\newcommand{\pp}[2]{\mathfrak{p}_{#1,#2}}
\newcommand{\pxx}[1][k]{\mathfrak{p}^x_{#1}}
\newcommand{\pyy}[1][k]{\mathfrak{p}^y_{#1}}
\newcommand{\Py}{\Prob\py{}}
\newcommand{\PY}{\Prob\yy{}}
\newcommand{\Ey}{\EE\py{}}
\newcommand{\EY}{\EE\yy{}}
The proof is mostly a modification of a similar proof \cite[Appendix C]{CT20}. Here, we need to take care that both $X_n$ and $Y_n$ stay close to their asymptotic mean for $n<T_m$ with high probability, as $p,q\to\infty$ with $q/p\in[\lambda',\lambda]$, where $0<\lambda'\leq 1\leq\lambda<\infty$. We follow the exposition and the notation of \cite{CT20}.
For starters, we write
\begin{equation*}
\pp k{k'} = \Prob_\infty ( -(X_1,Y_1) = (k,k') ) \qtq{and}
\pxx = \Prob_\infty (-X_1=k) \ ,\quad
\pyy = \Prob_\infty (-Y_1=k) .
\end{equation*}
Then, for $k\leq p-2$ and $k'\leq q-1$, the basic relation for the comparison of the laws of the perimeter processes reads
\begin{equation}\label{eq:Doob}
\Prob_{p,q}(-(X_1,Y_1)=(k,k'))=\frac{z_{p-k,q-k'}}{z_{p,q}u_c^{k+k'}}\pp k{k'}=\frac{z_{p-k,q-k'}u_c^{(p+q)-(k+k')}}{z_{p,q}u_c^{p+q}}\pp k{k'},
\end{equation}
as easily verified using the data of Table~\ref{tab:prob(p,q)diag}. Observe that this condition is reminiscent of the Doob $h$-transform of a random walk, ceased to satisfy it since the condition \eqref{eq:Doob} breaks down for $k>p-2$ or $k'>q-1$. We also introduce the following notation: If $A$ and $B$ are two \emph{positive} functions defined on some set $\Lambda$, we say that
\begin{itemize}
\item $A(y) \la B(y)$ for $y \in \Lambda$, \emph{if}
there exists $C>0$ such that $A(y)\le CB(y)$ for all $y \in\Lambda$;
\item $A(y) \ea B(y)$ for $y \in \Lambda$, \emph{if}
$A(y) \la B(y)$ and $B(y) \la A(y)$.
\end{itemize}
We fix a cutoff $\theta \in (0,1)$ and let $p_\theta := \frac2{1-\theta}$ so that $\theta p \le p-2$ and $\theta q \le q-1$ for all $p,q\ge p_\theta$. The following lemma gives estimates for the jump probabilities of the perimeter processes in a single peeling step:
\newcommand{\Xp}{\{-X_1\le p-2\}}
\newcommand{\Yp}{\{-Y_1\le q-1\}}
\newcommand{\ykXp}[1][=k]{\{-Y_1#1\} \cap \Xp}
\newcommand{\xkYp}[1][=k]{\{-X_1#1\} \cap \Yp}
\newcommand{\zfrac}[1][k]{\frac{z_{p-#1,q-k'} u_c^{p+q-(#1+k')}}{z_{p,q} u_c^{p+q}}}
\newcommand{\Ap}{\mathcal{A}_x}
\newcommand{\zg}[1][h]{\{W \ge #1\} \cap \Ap}
\newcommand{\zl}[1][h]{\{W \le #1\} \cap \Ap}
\newcommand{\idzg}[1][h]{\id_{\zg[#1]}}
\newcommand{\idzl}[1][h]{\id_{\zl[#1]}}
\begin{lemma}\label{lem:estimates}Assume throughout the lemma that $q/p$ lies in a fixed compact interval $I\subset\R_+$ such that $0\notin I$. Then the perimeter increments during the first peeling step satisfy the following probability estimates:
\begin{enumerate}[label=(\roman*)]
\item\label{item:estimate infty}
$\pxx \ea \pyy \ea k^{-7/3}$ for $k\ge 1$.
\item\label{item:estimate X}
$\Prob_{p,q}(\xkYp) \ea k^{-7/3}$ and $\Prob_{p,q}(-X_1=p-k) \ea p^{-1} k^{-4/3}$ for all $p,q\ge p_\theta$ and $1\le k\le \theta p$.
\item\label{item:estimate Y}
$\Prob_{p,q}(\ykXp) \ea k^{-7/3}$ and $\Prob_{p,q}(-Y_1=q-k) \ea p^{-1} k^{-4/3}$ for $p,q\ge p_\theta$ and $1\le k\le \theta q$.
\item\label{item:estimate zfrac}
$\abs{\zfrac -1} \la p^{-1} \abs{k} + p^{-1/3}$ and $\abs{\zfrac -1} \la p^{-1} \abs{k'} + p^{-1/3}$ for any $(k,p)$, $(k',q)$ such that $-2\le k \le \theta p$ and $-2\le k' \le \theta q$.
\item\label{item:estimate large jump}
For $p,q\ge p_\theta$, $x\in [1,\theta (p\wedge q)]$ and $m\ge 1$,
\begin{equation*}
\Prob_{p,q}(-X_1 < p-m, \ -Y_1< q-m \text{ and } (-X_1)\vee (-Y_1)\ge x)
\ \la \ x^{-4/3} + p^{-1} m^{-1/3} \,.
\end{equation*}
In the following, let $\Ap=\{ (-X_1) \vee (-Y_1) \le x\}$ and $W$ be either $\mu-X_1$ or $\mu-Y_1$.
\item\label{item:estimate S}
$\Prob_{p,q}(\zg) \la h^{-4/3}$, $\EE_{p,q}[W \idzg] \la h^{-1/3}$ and $\EE_{p,q}[W^2 \idzl] \la h^{2/3}$ for $p,q\ge p_\theta$, $x\in [1,\theta (p\wedge q)]$ and $h\in [1,x]$.
\item\label{item:estimate E}
$|\EE_{p,q}[W \id_{\Ap}]| \la x^{-1/3}$ for $p,q \ge p_\theta$ and $x\in [1,\theta (p\wedge q)]$.
\item\label{item:estimate small jump}
For $p,q\ge p_\theta$, $x\in [1,\theta (p\wedge q)]$ and $\xi \in [2x^{-1},1]$,
\begin{equation*}
\log \m({ \EE_{p,q}[e^{\pm \xi W} \id_{\Ap}] }
\ \la\ x^{-4/3} e^{\xi x}.
\end{equation*}
\end{enumerate}
\end{lemma}
\newcommand{\varsubset}{\subset}
\renewcommand{\subset}{\subseteq}
\newcommand{\absb}[1]{\mb|{#1}}
\newcommand{\absB}[1]{\mB|{#1}}
\newcommand{\absh}[1]{\mh|{#1}}
\newcommand{\absH}[1]{\mH|{#1}}
\begin{proof}
\begin{enumerate}[label=\textbf{(\roman*)},wide=0pt,listparindent=\parindent,]
\item
Proven in \cite{CT20}.
\item
First, since $\Prob_{p,q}(\{-X_1=1\}\cap\Yp)$ has a finite limit as $p,q\to\infty$ while $q/p\in I$, we have $\Prob_{p,q}(\{-X_1=1\}\cap\Yp)\ea 1$.
Then for $2\leq k\leq\theta p$, we write
\begin{equation*}
\Prob_{p,q}(\xkYp)=\frac{z_{p-k,q}}{z_{p,q}u_c^k}\pp k{0}+\frac{z_{p-k,q-1}}{z_{p,q}u_c^k}\pp k{1}.
\end{equation*}
Since $k\leq\theta p$, the asymptotics of Equation~\eqref{eq:diagasympt} yield $\frac{z_{p-k,q}}{z_{p,q}u_c^k}\ea 1$ and $\frac{z_{p-k,q-1}}{z_{p,q}u_c^k}\ea 1$. The first estimate follows then by \textbf{(i)}. For the second estimate, we note that
\begin{equation*}
\Prob_{p,q}(-X_1=p-k)=t_c\frac{z_{k,q-1}z_{p-k+2,0}}{z_{p,q}}+t_c\nu_c\frac{z_{k,q}z_{p-k,1}}{z_{p,q}}\sim C(p,q)\cdot a_k u_c^k\cdot\left((p-k+2)^{-\frac{7}{3}}+(p-k)^{-\frac{7}{3}}\right)\cdot p^{\frac{4}{3}}
\end{equation*}
where $C(p,q)$ is a bounded constant depending on $p,q$ and bounded away from zero. Since $a_k u_c^k\ea k^{-4/3}$ as well as $(p-k+2)^{-7/3}\ea p^{-7/3}\ea (p-k)^{-7/3}$, the desired result follows.
\item
Since $\Prob_{p,q}(\{-Y_1=1\}\cap\Xp)$ has a finite limit, $\Prob_{p,q}(\{-Y_1=1\}\cap\Xp)\ea 1$. Then, assume $2\leq k\leq\theta q$, in which case
\begin{equation*}
\Prob_{p,q}(\ykXp)=\frac{z_{p,q-k}}{z_{p,q}u_c^k}\pp 0{k}+\frac{z_{p+1,q-k}}{z_{p,q}u_c^k}\pp {-1}{k}
\end{equation*}
Since $k\leq\theta q$, the asymptotics of Equation~\eqref{eq:diagasympt} yield $\frac{z_{p,q-k}}{z_{p,q}u_c^k}\ea 1$ and $\frac{z_{p+1,q-k}}{z_{p,q}u_c^k}\ea 1$. The first estimate follows. Secondly, for $k=2,\dots,\theta q$,
\begin{equation*}
\Prob_{p,q}(-Y_1=q-k)=t_c\frac{z_{p+1,k}z_{1,q-k-1}}{z_{p,q}}+t_c\nu_c\frac{z_{p,k}z_{0,q-k+1}}{z_{p,q}}\sim \tilde{C}(p,q)\cdot a_k u_c^k\cdot\left((q-k+1)^{-\frac{7}{3}}+(q-k-1)^{-\frac{7}{3}}\right)\cdot p^{\frac{4}{3}}
\end{equation*}
where $\tilde{C}(p,q)$ is bounded and bounded away from zero. The result follows since $(q-k+1)^{-7/3}\ea q^{-7/3}\ea (q-k-1)^{-7/3}$ and $q\ea p$.
\item
From the asymptotic expansion $z_{p,q}u_c^{p+q}=\frac{b\cdot c(q/p)}{\Gamma(-4/3)\Gamma(-1/3)}p^{-11/3}\left(1+O(p^{-1/3})\right)$, we see that there exist constants $C=C(\theta)$ and $p_0=p_0(\theta)$ such that for all $p,q\geq p_0$, $-2\leq k\leq\theta p$ and $-2\leq k'\leq\theta q$,
\begin{equation*}
\frac{(p-k)^{-11/3}}{p^{-11/3}}\left(1-C p^{-1/3}\right)\leq\frac{z_{p-k,q-k'}}{z_{p,q}u_c^{k+k'}}\leq\frac{(p-k)^{-11/3}}{p^{-11/3}}\left(1+C p^{-1/3}\right).
\end{equation*}
After writing down the Taylor expansions of each of the sides of the inequality, the rest of the proof of the first estimate goes similarly as the proof of a corresponding claim in \cite{CT20}. The second estimate follows after swapping the roles of $p$ and $q$, and $k$ and $k'$, respectively, and noting that $q^{-1} \abs{k'} + q^{-1/3}\ea p^{-1} \abs{k'} + p^{-1/3}.$
\item
We estimate
\begin{align*}
&\ \Prob_{p,q}(-X_1 < p-m, \ -Y_1<q-m \text{ and } (-X_1)\vee (-Y_1)\ge x) \\ \le&\ \Prob_{p,q}\left(\{-X_1\le p-2\}\cap\{\theta q>-Y_1\geq x\}\right)+\Prob_{p,q}\left(\{-Y_1\le q-1\}\cap\{\theta p>-X_1\geq x\}\right) \\ +&\ \Prob_{p,q}(-X_1\in [\theta p,p-m])+\Prob_{p,q}(-Y_1\in[\theta q,q-m]) \\ \la&\ \sum_{k=x}^{\theta q}k^{-7/3}+\sum_{k=x}^{\theta p}k^{-7/3}+\sum_{k=m}^{(1-\theta) p}p^{-1}k^{-4/3}+\sum_{k=m}^{(1-\theta) q}p^{-1}k^{-4/3}\la x^{-4/3}+p^{-1}m^{-1/3},
\end{align*}
where we used the results \textbf{(ii)}-\textbf{(iii)}.
\item
This is a mutatis mutandis of the proof of a corresponding claim in \cite{CT20}, after one notices that the conditions $x\leq \theta q$ and $h\geq 1$ imply $\{h-\mu\leq -Y_1\leq x\}\subset\{1\leq -Y_1\leq \theta q\}$.
\item
For $k\leq p-2$ and $k'\leq q-2$, Equation~\eqref{eq:Doob} gives the estimate \begin{equation*}
\abs{\Prob_{p,q}\left((-X_1,-Y_1)=(k,k')\right)-\pp k{k'}}\leq\pp k{k'}\abs{\frac{z_{p-k,q-k'}}{z_{p,q}u_c^{k+k'}}-1}.
\end{equation*}
If $W=\mu-X_1$, the equation above then yields
\begin{align*}
\abs{\EE_{p,q}[W \id_{\Ap}]-\EE_\infty[W \id_{\Ap}]}
=&\ \abs{\sum_{k=-2}^x (\mu+k) \m({
\Prob_{p,q} \m({ -X_1=k, -Y_1\in [-1,x] }
- \Prob_\infty\m({ -X_1=k, -Y_1\in [-1,x] }
} }
\\ =&\
\abs{\sum_{k=-2}^x \sum_{k'=-1}^x(\mu+k) \m({
\Prob_{p,q} \m({-X_1=k, -Y_1=k'}
- \Prob_\infty\m({-X_1=k, -Y_1=k'}
} }
\\ \le&\
\sum_{k=-2}^x \sum_{k'=-1}^x |\mu+k|\abs{\Prob_{p,q}(-X_1=k, -Y_1=k')-\pp k{k'}}
\\ \le&\
\sum_{k=-2}^x\sum_{k'=-1}^x|\mu+k|\pp k{k'}\abs{\frac{z_{p-k,q-k'}}{z_{p,q}u_c^{k+k'}}-1}
\\ \la&\ \sum_{k=-2}^x|\mu+k|(k+3)^{-7/3}\left(p^{-1}|k|+p^{-1/3}\right)\la p^{-1/3}
\end{align*}
where we used the estimates \textbf{(i)} and \textbf{(iv)}. If $W=\mu-Y_1$, symmetry and the second estimate in \textbf{(iv)}
yield the same asymptotic upper bound. The rest of the proof is a mutatis mutandis of the proof of an analogous claim in \cite{CT20}.
\item This is a mutatis mutandis of the proof of a similar claim in \cite{CT20}. \qedhere
\end{enumerate}
\end{proof}
\newcommand{\tauxx}[1][x]{\tau_{#1}}
Let $\tauxx = \inf \Set{n\ge 0}{|X_n-\mu n| \vee |Y_n-\mu n| > x}$. Then, using the Markov property, we find the following analog of \cite[Lemma 24]{CT20}.
\begin{lemma}\label{lem:cst barrier}
Fix some $\epsilon>0$ and let $x =\chi \mb({ N (\log N)^{1+\epsilon} }^{3/4}$.
Then for any $0<\lambda_{\min}\leq 1\leq\lambda_{\max}<\infty$, $\theta\in(0,\lambda_{\min})$, $p,q\ge \tilde p_\theta:=p_\theta/(1-\theta)$ such that $\frac{q}{p}\in[\lambda_{\min},\lambda_{\max}]$ as well as $m\ge 1$ and $\chi,N \ge 2$ such that $x\in [1,\frac{\theta}{1+\theta} (p\wedge q)]$, we have
\begin{equation*}
\Prob_{p,q} (\tauxx \le N, \tauxx < T_m) \la \frac1{(\log \chi + \log N)^{1+\epsilon/2}} + \frac Np m^{-1/3}\,.
\end{equation*}
\end{lemma}
\begin{proof}
For $n\ge 1$, let $\Delta X_n = X_n-X_{n-1}$ and $\Delta Y_n = Y_n-Y_{n-1}$, and
\begin{equation*}
J_x = \inf\Set{n\ge 1}{ (-\Delta X_n) \vee (-\Delta Y_n) \ge x}.
\end{equation*} Following the corresponding proof in \cite{CT20}, we bound the probability of the event $\{\tauxx \le N, \tauxx < T_m\}$ both in the cases $\{J_x\le \tauxx \}$ (large jump estimate) and $\{\tauxx < J_x\}$ (small jump estimate).
\medskip
\noindent\textbf{Large jump estimate: union bound.} We have
\begin{equation*}
\Prob_{p,q}( \tauxx \le N, \tauxx < T_m \text{ and } J_x \le \tauxx )\le \sum_{n=1}^N \Prob_{p,q}( n \le \tauxx \text{ and } J_x=n < T_m ) \,.
\end{equation*}
If $n\leq\tauxx$, in particular $P_{n-1}\ge p-x$ and $Q_{n-1}\ge q-x$. Let
\begin{equation}\label{eq:defD}
\mathcal{D}:=\bigg\{(p',q') : p'\ge p-x,\ q'\ge q-x,\ \frac{\lambda_{\min}-\theta}{1+\theta}\leq\frac{q'}{p'}\leq(1+\theta)\lambda_{\max}+\theta \bigg\}.
\end{equation}
Then for $i<\tauxx$, we have the estimates
\begin{equation*}
\frac{Q_i}{P_i}-\lambda_{\max}\leq\frac{q+\mu i+x}{p+\mu i-x}-\lambda_{\max}\leq\frac{(q-\lambda_{\max} p)+\mu(1-\lambda_{\max})i+(1+\lambda_{\max})x}{p-x}\leq \frac{(1+\lambda_{\max})x}{p-x}\leq\theta(1+\lambda_{\max})
\end{equation*}
and
\begin{equation*}
\frac{Q_i}{P_i}-\lambda_{\min}\ge\frac{q+\mu i-x}{p+\mu i+x}-\lambda_{\min}\ge\frac{(q-\lambda_{\min} p)+\mu(1-\lambda_{\min})i-(1+\lambda_{\min})x}{p}\ge-\frac{(1+\lambda_{\min})x}{p}\ge-(1+\lambda_{\min})\frac{\theta}{1+\theta},
\end{equation*}
since by assumption, $\lambda_{\min}p\leq q\leq\lambda_{\max} p$ and $1\leq x\leq \frac{\theta p}{1+\theta}$. In particular, this holds for $i=n-1$, and we conclude that $(P_{n-1},Q_{n-1})\in\mathcal{D}$.
On the other hand, $J_x=n<T_m$ immediately implies $P_n>m$, $Q_n>m$ and $ (-\Delta X_n) \vee (-\Delta Y_n) \ge x$. Thus, the Markov property of $\nseq{P_n,Q}$ gives the upper bounds \begin{align*}
\Prob_{p,q}( n \le \tauxx \text{ and } J_x=n < T_m ) &\leq \EE_{p,q}\left(\Prob_{P_{n-1}, Q_{n-1}}\left(P_1>m, \ Q_1>m, \ (-X_1) \vee (-Y_1)\ge x\right)\id_{\{P_{n-1}\geq p-x, Q_{n-1}\geq q-x\}}\right) \\ &\leq \sup_{(p',q')\in\mathcal{D}}\Prob_{p',q'}\left(-X_1<p-m, \ -Y_1<q-m, \ (-X_1) \vee (-Y_1)\ge x\right).
\end{align*}
The assumptions $p,q\geq\tilde{p}_\theta=\frac{p_\theta}{1-\theta}$ and $1\leq x\leq\frac{\theta}{1+\theta}(p\wedge q)$ ensure that, for $p'\geq p-x$ and $q'\geq q-x$, the condition $p'\wedge q'\geq p_\theta$ with $1\leq x\leq \theta(p'\wedge q')$ is satisfied. Hence, by Lemma~\ref{lem:estimates} \textbf{(v)},
\begin{equation}\label{eq:large jump estimate}
\Prob_{p,q}(\tauxx \le N, \tauxx < T_, \text{ and } J_x \le \tauxx )\la N\left(x^{-4/3}+p^{-1}m^{-1/3}\right) = \frac{\chi^{-4/3}}{ (\log N)^{1+\epsilon} } + \frac{N}{p} m^{-1/3}.
\end{equation}
\newcommand{\unit}{\mathbf{e}}
\newcommand{\tauu}{\tau^{\unit}_x}
\noindent\textbf{Small jump estimate: Chernoff bound.} For each of the four unit vectors $\unit \in \integer^2$, define
\begin{equation*}
\tauu = \inf\Set{n\ge 0}{ (\mu n-X_n, \mu n-Y_n)\cdot \unit \ge x } \,,
\end{equation*}
so that $\tauxx = \min_{\unit} \tauu$. We start by estimating
\begin{align}
\Prob_{p,q}( \tauxx \le N, \tauxx < T_m \text{ and } J_x > \tauxx )
\ &\le\ \Prob_{p,q}( \tauxx \le N, \tauxx < J_x ) \notag \\
&\le \sum_{\unit}\Prob_{p,q}\left(\tauxx=\tauu\le N, \tauu < J_x\right) \,. \label{eq:pre Chernoff}
\end{align}
If $\tauu = n < J_x$, then $(\mu n-X_n, \mu n-Y_n) \cdot \unit = \sum_{i=1}^n (\mu -\Delta X_i, \mu -\Delta Y_i) \cdot \unit \ge x$, and $(-\Delta X_i) \vee (-\Delta Y_i) \le x$ for all $i=1,\ldots,n$. Therefore, applying the Chernoff bound,
\begin{align}
\Prob_{p,q}(\tauxx=\tauu \le N, \tauu < J_x) &\leq e^{-\xi x} \sum_{n=1}^N \EE_{p,q} \m[{ \idd{\tauxx=\tauu=n} \prod_{i=1}^n
e^{\xi (\mu-\Delta X_i, \mu-\Delta Y_i) \cdot \unit}
\idd{(-\Delta X_i) \vee (-\Delta Y_i) \le x } }\notag \\
&\leq e^{-\xi x} \sum_{n=1}^N \EE_{p,q} \m[{ \idd{\tauxx=n} \prod_{i=1}^n
e^{\xi (\mu-\Delta X_i, \mu-\Delta Y_i) \cdot \unit}
\idd{(-\Delta X_i) \vee (-\Delta Y_i) \le x } }
\label{eq:stopped Chernoff}
\end{align}
for all $\xi\ge0$.
\newcommand{\phix}[1][p,q]{\varphi^{x,\unit}_{#1}(\xi)}
\newcommand{\phixs}[1][p_*,q_*]{\varphi^{x,\unit}_{#1}(\xi)}
\newcommand{\phixp}[1][p',q']{\varphi^{x,\unit}_{#1}(\xi)}
\newcommand{\phixi}[1][P_i,Q_i]{\varphi^{x,\unit}_{#1}(\xi)}
For $p,q\in \natural \cup \{\infty\}$, let $\phix = \EE_{p,q}[ e^{\xi (\mu-X_1, \mu-Y_1) \cdot \unit} \id_{\Ap}]$, where $\Ap = \{(-X_1) \vee (-Y_1) \le x \}$ was already encountered in Lemma~\ref{lem:estimates}. Since the pair $(X_1,Y_1)$ takes only finitely many values on the event $\Ap$ and $\law_{p,q}(X_1,Y_1) \to \law_\infty(X_1,Y_1)$ in distribution, we have $\phix \to \phix[\infty]$ as $p,q \to\infty$ and $q/p\in I$ for any compact interval $I\subset\R_+$ such that $0\notin I$. Recall the set $\mathcal{D}$ defined by \eqref{eq:defD}. Since the peeling process converges in this set when $p',q'\to\infty$, the function $\phixp$ is continuous in its one-point compactification $\mathcal{D}\cup\{(\infty,\infty)\}$. Hence, there exists a pair $(p_*,q_*)=(p_*(x,\unit,\xi), q_*(x,\unit,\xi))\in \mathcal{D}\cup\{(\infty,\infty)\}$ such that
\begin{equation*}
\phixs = \sup_{(p',q')\in \mathcal{D}\cup\{(\infty,\infty)\}} \phixp.
\end{equation*}
Let $\nseq[1]{\Delta X^*_n, \Delta X^*}$ be a sequence of i.i.d.\ random variables independent of $\nseq{X_n,Y}$ and with the same distribution as $\law_{p_*,q_*}(X_1,Y_1)$. Define
\begin{equation*}
(U_i,V_i) = \begin{cases}
-(\Delta X_i , \Delta Y_i) & \text{if }i \le \tauxx
\\ -(\Delta X_i^*, \Delta Y_i^*) & \text{if }i > \tauxx.
\end{cases}
\end{equation*}
On the event $\{\tauxx=n\}$, the future $(U_i, V_i)_{i>n}$ of the process is an i.i.d.\ sequence independent of the past such that
$\EE_{p,q}[e^{\xi (\mu+U_i, \mu+V_i) \cdot \unit} \idd{U_i \vee V_i \le x }] = \phixs$. Therefore we can continue the bound \eqref{eq:stopped Chernoff} with
\begin{align}
&\ e^{-\xi x} \sum_{n=1}^N \EE_{p,q} \m[{ \idd{\tauxx=n} \prod_{i=1}^n
e^{\xi (\mu+U_i, \mu+V_i) \cdot \unit} \idd{U_i \vee V_i \le x } }
\notag \\ =\ &
e^{-\xi x} \sum_{n=1}^N \mb({\phixs}^{-(N-n)} \EE_{p,q} \m[{
\idd{\tauxx=n} \prod_{i=1}^N
e^{\xi (\mu+U_i, \mu+V_i) \cdot \unit} \idd{U_i \vee V_i \le x } }
\notag \\ \le \ &
e^{-\xi x} \cdot (1\vee \phixs^{-N}) \cdot \EE_{p,q} \m[{ \prod_{i=1}^N
e^{\xi (\mu+U_i, \mu+V_i) \cdot \unit} \idd{U_i \vee V_i \le x } }.
\label{eq:completed Chernoff}
\end{align}
Now $\tauxx$ is a stopping time with respect to the natural filtration $\nseq \filtr$ of the process $\nseq{U_n,V}$. Therefore for all $i\ge 0$,
\begin{equation*}
\EE_{p,q}\Econd{ e^{\xi (\mu+U_{i+1}, \mu+V_{i+1}) \cdot \unit} \idd{U_{i+1} \vee V_{i+1} \le x} }{\filtr_i}
\ =\ \idd{i< \tauxx} \cdot \phixi + \idd{i\ge\tauxx} \cdot \phixs \le \ \phixs \,,
\end{equation*}
where we have the last inequality due to the fact that $(P_i,Q_i)\in \mathcal{D}$ on the event $\{i< \tauxx\}$. By expanding the expectation in \eqref{eq:completed Chernoff} with $N$ successive conditioning, we see that it is bounded by $\phixs^N$. Then, combining \eqref{eq:stopped Chernoff} and \eqref{eq:completed Chernoff} yields
\begin{equation*}
\Prob_{p,q}(\tauxx=\tauu \le N, \tauu < J_x)
\ \le\ e^{-\xi x} (\phixs^N \vee 1) \,.
\end{equation*}
By Lemma~\ref{lem:estimates}\ref{item:estimate small jump}, there exists a constant $C$ such that $\phix \le \exp(C x^{-4/3} e^{\xi x})$ for all $p,q\ge p_\theta$, $x\in [1,\theta (p\wedge q)]$, $\xi \in [2 x^{-1},1]$ and unit vector $\unit \in \integer^2$. Note that we already have seen in the derivation of the large jump estimate that the conditions $p'\ge p-x$ and $q'\ge q-x$ imply $p'\wedge q'\geq p_\theta$ and $1\leq x\leq \theta(p'\wedge q')$. Therefore, we also have $\phixs \le \exp(C x^{-4/3} e^{\xi x})$ by the definition of $\phixs$.
Hence,
\begin{equation*}
\Prob_{p,q}(\tauxx=\tauu \le N, \tauu < J_x)
\ \le\ \exp(-\xi x + C\cdot N x^{-4/3} e^{\xi x}) \,.
\end{equation*}
Plugging this into \eqref{eq:pre Chernoff} and taking $\xi x = c\log \log x$ with $c=1+\epsilon/2$ yields
\begin{equation*}
\Prob_{p,q}( \tauxx \le N, \tauxx < T_m \text{ and } J_x > \tauxx )
\ \le\ 4 \exp(-c\log\log x + C N x^{-4/3} (\log x)^c).
\end{equation*}
Thanks to the relation between $x$ and $N$ given in the assumptions, we have $N x^{-4/3} (\log x)^c \ea \chi^{-4/3} \frac{(\log \chi + \log N)^c}{(\log N)^{1+\epsilon}}$, which is bounded by a constant for $\chi,N\ge 2$. It follows that
\begin{equation*}
\Prob_{p,q}( \tauxx \le N, \tauxx < T_m \text{ and } J_x > \tauxx )
\ \la\ \exp(-c\log\log x) \ea (\log \chi + \log N)^{-c}.
\end{equation*}
By adding the large jump estimate \eqref{eq:large jump estimate} to the above small jump estimate, we conclude that $\Prob_{p,q}( \tauxx \le N, \tauxx < T_m ) \la (\log \chi + \log N)^{-c} + N p^{-1} m^{-1/3}$, where we again use the boundedness of $\chi^{-4/3} \frac{(\log \chi + \log N)^c}{(\log N)^{1+\epsilon}}$.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:one jump diag}]
Let $\Delta_n = |X_n-\mu n| \vee |Y_n-\mu n|$. Recall that $\tauxy = \inf \Set{n\ge 0}{\Delta_n > \barrier(n)}$ where $\barrier[](n) = \mb({ (n+2)(\log(n+2))^{1+\epsilon} }^{3/4}$, and we want to prove that
\begin{equation*}
\lim_{x,m\to\infty} \limsupp \Prob_{p,q}(\tauxy<T_m) =0 \quad\text{while}\quad\frac{q}{p}\in[\lambda',\lambda]\,.
\end{equation*}
Consider the sequences $(N_k)_{k\ge 0}$ and $(x_k)_{k\ge 0}$ defined by $N_0=x_0=0$,
\begin{equation*}
\Delta N_k := N_k - N_{k-1} = 2^k \qtq{and}
\Delta x_k := x_k-x_{k-1}
= \frac{x}3 \mb({ \Delta N_k \m({ \log \Delta N_k }^{1+\epsilon} }^{3/4}.
\end{equation*}
Then we have $N_k = 2^{k+1}-2$ and
\begin{align*}
x_k = \frac{x}3 \sum_{i=1}^k 2^{\frac34 i} \cdot (i \log 2)^{\frac34(1+\epsilon)}
\le \frac{x}3 \cdot \frac{2^{\frac34 (k+1)}}{2^{3/4}-1} (k \log 2)^{\frac34 (1+\epsilon)}
\le x \mb({ 2^k (\log 2^k)^{1+\epsilon} }^{3/4}.
\end{align*}
In other words, $x_k \le \barrier(N_{k-1})$.
\newcommand*{\kex}{K^\epsilon_{x,m}}
Consider the sequence of horizontal segments $I_k = \Set{(n,x_k)}{n\in (N_{k-1}, N_k]}$. Due to the previous inequality, all of these segments are below the curve $\Delta_n = \barrier(n)$. Let $\kex$ be the index $k$ where $\Delta_n$ goes above $I_k$ for the first time up to $T_m$, that is,\
\begin{equation*}
\kex = \inf \Set{k\ge 1}{\exists n \in (N_{k-1},N_k] \text{ s.t.\ } \Delta_n>x_k \text{ and } n<T_m} \,.
\end{equation*}
Then we have $\{\tauxy < T_m\}\subset \{\kex < \infty\}$, and an union bound would allow us to restrict our consideration to the set on the right hand side of the inclusion.
Observe that, for any $n\ge 1$, the conditions $\Delta_{N_{k-1}}\le x_{k-1}$ and $\Delta_{n+N_{k-1}}>x_k$ imply $\tilde \Delta_n := |X_{n+N_{k-1}} - X_{N_{k-1}} -\mu n| \vee |Y_{n+N_{k-1}} - Y_{N_{k-1}} -\mu n| > \Delta x_k$.
Therefore by Markov property of $\law_{p,q} \nseq{X_n,Y}$,
\begin{equation*}
\Prob_{p,q}(\kex=k) \le
\EE_{p,q}\m[{ \Prob_{P_{N_{k-1}},Q_{N_{k-1}}}\mb({ \exists n \in (0,\Delta N_k] \text{ s.t.\ } \Delta_n >\Delta x_k \text{ and } n<T_m } \id_{\{\Delta_{N_{k-1}} \le x_{k-1} \}} }.
\end{equation*}
On the other hand, $\Delta_{N_{k-1}}\le x_{k-1}$ also implies
\begin{equation*}
\frac{Q_{N_{k-1}}}{P_{N_{k-1}}}-\lambda\leq\frac{(q-\lambda p)+\mu(1-\lambda)N_{k-1}+(1+\lambda)x_{k-1}}{p+\mu N_{k-1}-x_{k-1}}\leq\frac{(1+\lambda)x_{k-1}}{p-x_{k-1}}
\end{equation*}
and
\begin{equation*}
\frac{Q_{N_{k-1}}}{P_{N_{k-1}}}-\lambda'\geq\frac{(q-\lambda' p)+\mu(1-\lambda')N_{k-1}-(1+\lambda')x_{k-1}}{p+\mu N_{k-1}+x_{k-1}}\geq -\frac{(1+\lambda')x_{k-1}}{p+x_{k-1}}.
\end{equation*}
We note that $x_{k-1}\le x f_\epsilon(\Lambda p)$, which is of smaller order than $p$.
Let $\lambda_{\min}$ and $\lambda_{\max}$ be positive constants such that $\lambda_{\min}<\lambda'\leq 1\leq\lambda<\lambda_{\max}$. Then for $p$ large enough, $\frac{Q_{N_{k-1}}}{P_{N_{k-1}}}\in[\lambda_{\min},\lambda_{\max}]$. In this case, we obtain
\begin{align*}
&\ \EE_{p,q}\m[{ \Prob_{P_{N_{k-1}},Q_{N_{k-1}}}\mb({ \exists n \in (0,\Delta N_k] \text{ s.t.\ } \Delta_n >\Delta x_k \text{ and } n<T_m } \id_{\{\Delta_{N_{k-1}} \le x_{k-1} \}} }\\ &\le
\sup_{p'\ge p-x_{k-1}, \ q'\ge q-x_{k-1}, \ \frac{q'}{p'}\in[\lambda_{\min},\lambda_{\max}]} \Prob_{p',q'}(\tauxx[\Delta x_k] \le \Delta N_k, \tauxx[\Delta x_k] < T_m)\,.
\end{align*}
Let $k_0=k_0(p,q)$ be the largest $k$ such that $N_k \le \Lambda (p\wedge q)$, where $\Lambda \ge 1$ is some cut-off value that will be sent to infinity after $p$, $x$ and $m$. Explicitly, $k_0=\floor{\log_2\left(\frac{\Lambda}{2}(p\wedge q)+1\right)}$, and $\Delta N_{k_0}=\frac{N_{k_0}}{2}+1=O(p)$.
Then, for any fixed $x$, $m$ and in the limit $p,q\to\infty$, we have $\Delta x_k \le \Delta x_{k_0}\leq\frac{\theta}{1+\theta} (p\wedge q)$ and $p-x_{k-1} \ge p- \barrier(\Lambda (p\wedge q)) >\tilde p_\theta$ as well as $q-x_{k-1} \ge q- \barrier(\Lambda (p\wedge q)) >\tilde p_\theta$ for all $k\le k_0$.
Therefore we can apply Lemma~\ref{lem:cst barrier} to bound the above supremum, and obtain for large enough $p,q$ and $k_0$ that
\begin{align*}
\Prob_{p,q}(\kex \le k_0) & \la \sum_{k=1}^{k_0} \m({
\frac1{ (\log (x/3) + \log(\Delta N_k))^{1+\epsilon/2}} + \frac{\Delta N_k}p m^{-1/3} }
\\ & = \sum_{k=1}^{k_0} \frac1{ (\log(x/3)+k \log 2)^{1+\epsilon/2} } + \frac{N_{k_0}}p m^{-1/3}
\la \frac1{ (\log x)^{\epsilon/2} } + \Lambda m^{-1/3}\,.
\end{align*}
On the other hand, $k_0<\kex <\infty$ implies $T_m > N_{k_0}$. Therefore
\begin{align*}
\Prob_{p,q}(k_0 < \kex < \infty) \ &\le\
\Prob_{p,q}(T_0 > N_{k_0},\ k_0 < \kex < \infty) \\ &=\
\EE_{p,q}\left(\Prob_{P_{N_{k_0}-1},Q_{N_{k_0}-1}}(T_0\neq 1)\id_{(T_0>N_{k_0}-1)}\id_{(k_0 < \kex < \infty)}\right).
\end{align*}
Now $T_0>N_{k_0}-1$ implies $P_{N_{k_0}-1}\ge 1$ and $Q_{N_{k_0}-1}\ge 1$, and together with $k_0<\kex$ also $\Delta_{N_{k_0}-1}\le x_{k_0}$. This yields the estimate
\begin{equation*}
\lambda'-\frac{(1+\lambda')x_{k_0}}{p+x_{k_0}}\leq\frac{Q_{N_{k_0}-1}}{P_{N_{k_0}-1}}\leq\lambda+\frac{(1+\lambda)x_{k_0}}{p-x_{k_0}}.
\end{equation*}
Therefore, for $p$ large enough, we have $0<\lambda_{\min}<\lambda'-\frac{(1+\lambda')x_{k_0}}{p+x_{k_0}}<1<\lambda+\frac{(1+\lambda)x_{k_0}}{p-x_{k_0}}<\lambda_{\max}<\infty$. On the other hand, for $p', q'>0$ such that $q'/p'\in[\lambda_{\min},\lambda_{\max}]$, we have $\Prob_{p',q'}(T_0=1)\sim -\nu_c t_c\frac{4}{3}\frac{a_0 a_1}{b c(q'/p')}(p')^{4/3}(q')^{-7/3}$. Thus, there exist a constant $\delta=\delta(\lambda_{\min},\lambda_{\max})>0$ such that
\begin{equation*}
\Prob_{p',q'}(T_0\ne 1)\leq 1-\frac{\delta}{p'}.
\end{equation*}
We also have the trivial estimate $P_n\le p+2n$. In the end, we conclude \begin{align*}
\EE_{p,q}\left(\Prob_{P_{N_{k_0}-1},Q_{N_{k_0}-1}}(T_0\neq 1)\id_{(T_0>N_{k_0}-1)}\id_{(k_0 < \kex < \infty)}\right)&\leq \prod_{n=0}^{N_{k_0}-1}\left(1-\frac{\delta}{p+2n}\right)\leq\exp\left(-\sum_{n=0}^{N_{k_0}-1}\frac{\delta}{p+2n}\right)\\ &\leq
\exp\left(-\int_0^{N_{k_0}/p}\frac{\delta dx}{1+2x}\right)=\left(1+2\frac{N_{k_0}}{p}\right)^{-\frac{\delta}{2}}\\ &\leq 2^{-\frac{\delta}{2}}\left(\frac{N_{k_0}}{p}\right)^{-\frac{\delta}{2}}.
\end{align*}
Since $N_{k_0}\geq 2\left(2^{k_0-1}-1\right)\geq\frac{\Lambda}{2}(p\wedge q)-1$, it follows that $\left(\frac{N_{k_0}}{p}\right)^{-\frac{\delta}{2}}\la\Lambda^{-\frac{\delta}{2}}$.
We conclude that for every fixed $\Lambda>0$, and uniformly for $x>0$ and $m\ge 1$,
\begin{equation*}
\limsupp \Prob_{p,q}(\tauxy < T_m) \le
\limsupp \Prob_{p,q}(\kex < \infty) \la (\log x)^{-\epsilon/2} + \Lambda m^{-1/3} + \Lambda^{-\frac{\delta}{2}}.
\end{equation*}
Taking the limit $m,x\to\infty$ and then $\Lambda \to\infty$ finishes the proof.
\end{proof}
\bibliographystyle{abbrv}
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\end{document}
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TITLE: The importance and applications of order of a group?
QUESTION [7 upvotes]: Recently, I'm exposed to some exercises and theorems concerning order of a group. For example,
If an abelian group has subgroups of orders $m$ and $n$, respectively, then it has a subgroup whose order is $\operatorname{lcm}(m,n)$.
A simple proof of Sylow theorem for abelian groups
If a finite group $G$ of order $n$ has at most one subgroup of each order $d|n$, then $G$ is cyclic
IMHO, the classic result of this kind is Sylow theorems that appear in most standard textbooks about abstract algebra. As such, I would like to ask about the importance of order of a group in abstract algebra and its applications in other branches of mathemactics.
Thank you for your elaboration!
REPLY [6 votes]: The classification of finite simple groups was one of the great mathematical achievements of the 20th Century. It is also one where a single result on the order of the groups played a key role, namely the Feit–Thompson theorem, or odd order theorem:
Theorem. (Feit-Thompson, 1963) Every group of odd order is soluble*.
The proof is famously long, at 255 pages, and has recently been Coq-verified [1].
The derived subgroup of a soluble group is a proper normal subgroup, and so a soluble group is simple only if it is abelian. Therefore, the Feit-Thompson theorem has the following corollary:
Corollary. Every non-cyclic finite simple group has even order.
There are other results in this vein, with much shorter proofs. For example, Burnside's theorem (Wikipedia contains a proof):
Theorem. (Burnside, 1904) Let $p, q, a, b\in\mathbb{N}$ with $p, q$ primes. Then every group of order $p^aq^b$ is soluble.
Therefore, every non-cyclic finite simple group must have order divisible by three primes. Moreover, at least one of these primes occurs twice in the prime decomposition of the order:
Theorem. (Frobenius, 1893) Groups of square-free order are soluble.
You can find a proof of this theorem on Math.SE here. The answer there links to the article [2], where the theorem is Proposition 17 (page 9). The article also claims that the result is due to Frobenius in [3].
*In American English, solvable.
[1] Gonthier, Georges, et al. "A machine-checked proof of the odd order theorem." International Conference on Interactive Theorem Proving. Springer, Berlin, Heidelberg, 2013.
[2] Ganev, Iordan. "Groups of a Square-Free Order." Rose-Hulman Undergraduate Mathematics Journal 11.1 (2010): 7 (link)
[3] Frobenius, F. G. "Uber auflösbare Gruppen." Sitzungsberichte der Akademie der Wiss. zu Berlin (1893): 337-345.
| 66,161
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But they’re running into some problems.
Posted By Shubhankar Parijat | On 12th, Nov. 2012 Under News
People have been clamouring for more Mega Man action for months- years- now, and Capcom hasn’t complied so far. It seems, however, that they might be beginning to turn around.
The publisher said that it’s “hoping” to bring the first six games of the Mega Man series to the 3DS’ eShop. This isn’t the first time those games are releasing as downloadable purchases- they’re also available via Wii Virtual Console, the first four games via PSN. There’s also a Mega Man Anniversary Collection.
However, Capcom is encountering some problems bringing these games over to the 3DS. They haven’t specified what problems these are exactly, but they’re being worked on. Japan already has four games, Europe has one while North America has neither.
Do you want this to happen? Tell us what you think in the comments section below.
[Nintendo Power (via Destructoid}]
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| 194,049
|
Rome 1960 has drifted well back into the un-remembered games of history
Now, as some may have realised, I'm fairly young.
There are also some older people on this forum.
So where is the cut-off point for a 'Distant Games' - Olympics which have faded into distant history.
1952 - Helsinki, Finland
1956 - Melbourne, Australia
1960 - Rome, Italy
1964 - Tokyo, Japan
1968 - Mexico City, Mexico
1972 - Munich, West Germany
1976 - Montreal, Canada
1980 - Moscow, U.S.S.R
paul92
1984 - Los Angeles, United States
1988 - Seoul, South Korea
1992 - Barcelona, Spain
1996 - Atlanta, United States
2000 - Sydney, Australia
2004 - Athens, Greece
2008 - Beijing, China
2012 - London, United Kingdom
Does the cut-off point vary between people's ages or do we all agree there is a set number of years in which Olympic Games are still recognised as fairly modern/recent.
For myself, it is Moscow which have the 'long forgotten fee' to them.
So copy and paste the latest list below (or above if you're first) and add your name below the Olympics which you consider to be in forgotten history.
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Category
- Door Accessories
- Door Furniture
- Door Security
- Door Systems
- External Doors
- Internal Doors
Product categories
- Bespoke Doors
- Door Accessories
- Door Furniture
- Door Security
- Door Systems
- External Doors
- Internal Doors
- Uncategorized
Category: Door Systems
Door Systems
Pocket doors are a relatively new introduction into the internal door market, but they’re becoming ever more popular as time goes by. Pocket doors are a great solution for open plan spaces and also small rooms, but the key to getting a pocket door right is all about what’s going on behind the plasterboard. Traditionally most people opt for standard hinged swinging doors because it’s the norm, but who says you have to go with the crowd? Hinged doors are sensible choices, but whether you want something more extravagant.
Showing all 3 results
Classic DOUBLE Pocket Door System£510.00 – £569.00Select options
Double pocket doors allow you the freedom to change your room from open plan to cosy in one easy movement without constraining your room layout.
Imagine opening up and closing off different areas of your layout in an instant, with doors that are there when you need them, and slide into a pocket in the wall when you don’t. And because the doors slide into a pocket in the wall, the doors themselves do not take up valuable space in the room when they are open.
The Eclisse double sliding pocket door system can also be installed against an existing masonry wall by simply creating a false stud wall alongside the existing masonry one. All you need to do is to construct the frame in exactly the same way, still applying plasterboard to both sides of the frame for strength. By doing this you will only be adding 100mm thickness to the existing wall. This is nothing compared to the space you will save by installing pocket doors instead of traditional hinged doors!
SIZING
- Not sure which size you need? Have a look at this Size Chart & Technical Spec ECL_Classic_Double_UK_20
- PLEASE NOTE INBUILT TOLERANCE – If setting out your studwork before delivery of your frame please note there is approximately an 8mm tolerance built into the height of these pocket door systems which you will need to take into account. Packers may be required to accommodate for this gap if you are unable to adjust the studwork above the system. Note: Frame should be installed at finished floor level.
- What is FINISHED WALL THICKNESS? see diagram.
- Using other stud work sizes? See our chart Eclisse-Studwork-Chart
Classic SINGLE Pocket Door System£289.99 – £309.99Select options
Single pocket doors offer a space-saving and stylish solution
Sliding pocket doors are a perfect space-saving alternative to a traditional hinged door for downstairs toilets, utility rooms, bathrooms, en-suites, storerooms and pantries, walk-in wardrobes, in fact they are ideal for any room where you wish to maximise the usable floor space in a property.
The Eclisse single sliding pocket door system is quick and easy to install. The beauty of the Eclisse system is that there are kits that will fit 10 standard UK door panel sizes and 3 standard ROI door panel sizes. This means you have the freedom to use our system with an existing standard door panel you may already have, or to match the door you use with other doors in your property.
There is also a fire rated version available.
SIZING
- Not sure which size you need? Have a look at this size chart & technical spec ECL_Classic Single_UK_8_18.
- What is FINISHED WALL THICKNESS?
- Using other stud work sizes? See our chart Eclisse-Studwork-Chart
Internal Oak Nuvu Roomfold Frame Set£249.00 – £299.00Select options
General Information Door Style: Roomfold Kit Range: Internal Roomfold Collection Construction Material: American White Oak (Frame) Finish: Unfinished Texture: Natural Core Type: Solid Engineered Guarantee: 10 Years Conditional This Roomfold Kit door from LPD’s Internal Roomfold collection features a Natural Unfinished surface on top of a Solid Engineered core. Having a room that can transform is a luxury you cant afford to …
Internal Oak Nuvu Roomfold Frame Set
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Hi! My name is Nicole Poirier. You won’t see any fancy theology, philosophy or psychology degrees here yet but psychology and the human and spiritual experience are what make up my world, how I perceive it and my existence in it.
I am a natural empath and clairsentient. These gifts give me the ability to read people very well. I can do this by scanning you and asking you simple questions. By seeing how you answer and what you answer I am able to see qualities in yourself that you, yourself, may be unaware of. I can easily recognize traumas or underlying reasons that may make you act the way you do.
By scanning for negative things I am able to tell what’s going on– almost like a Doberman sensing out someone’s false sense of ego. My keen sense of emotion can tell me what’s really going on with you beneath the surface.
How are you doing emotionally? Balancing and acknowledging emotions can lead to joy, enthusiasm, peace and relaxation! Sometimes all you need is an outsider’s perspective.
I have successfully been present, yes PRESENT, with all my emotions no matter how uncomfortable they can be. I have become more stress resistant in doing so and can show you techniques and coping strategies that come with changing for the better 💃🏽
I have fine-tuned my psychic, empathic and receptive abilities & use them in my writing and coaching and can give you clear guidance and advice from the spiritual realm.
Affirming the life that you want to live is challenging at times but completely do-able and I can help you do it!
Faith is important! I have been highly dedicated to numerology and giving my fears and worries up to my angels. If you are riddled with doubt let me help give you some insight into what you may be missing.
Through my experiences given to you in my writing and with my one on one coaching sessions I hope to help you create a positive change in yourself and the life that you want.
One on One Coaching with Nicole
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For eleven years I was a college professor. Part of my job was to advise new graduate students. I made a point to ask my students why they wanted to go to grad school. It’s a serious decision after all.
That’s why I had to ask myself, “Why do I want to live in an RV?” I keep a running list of reasons and I look at them often. Here are the big ones:
Full-time RVing excites me. I’m talking about that butterfly feeling in my stomach. I get it when I daydream about setting up my RV kitchen, pulling into a new camping spot, or sipping wine with my boyfriend in lounge chairs under the awning. As long as I keep getting that butterfly feeling, my dream will stay alive.
I love to travel. I’m a nomad at heart. I don’t want to wait until I have enough vacation time to ride my bike through Glacier National Park or experience the charm of Savannah. Authentic travel involves adventure. Even using my resourcefulness to fix a broken fridge can feel adventurous.
I want less stress. There’s no doubt that living on the road has its own challenges. But most of my stress stems from my work and keeping up a lifestyle that robs me of my joy. As with most people who reach mid-life, my goals have changed. Instead of working like a dog to gain career recognition, I want to do things that make me smile at the end of the day.
I’m an introvert. Full-time RVing suits many introverts, especially me. I need solitude to recharge. I enjoy extended periods alone, especially when I’m traveling. Other introverts like Chris and Cherie at Technomadia prove that I’m in good company.
I want more work flexibility. I want the freedom to alter my work schedule to fit whatever else I’m doing that week or month (zip-lining for the first time, touring a local brewery). RVing can allow me to work where and when I want to, given certain constraints (e.g., job requirements, finding a site with reliable WiFi).
My RV dream means the world to me. But I know it all won’t be peaches and cream. I have plenty of doubts about living on the road. But that’s for next time.
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\begin{document}
\author[K. Divaani-Aazar, A. Ghanbari Doust, M. Tousi and Hossein Zakeri]
{Kamran Divaani-Aazar, Akram Ghanbari Doust, Massoud Tousi\\ and\\ Hossein Zakeri}
\title[Cohomological dimension and relative Cohen-Maculayness]
{Cohomological dimension and relative Cohen-Maculayness}
\address{K. Divaani-Aazar, Department of Mathematics, Alzahra University, Vanak, Post Code 19834, Tehran, Iran-and-School of Mathematics,
Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran.}
\email{kdivaani@ipm.ir}
\address{A. Ghanbari Doust, Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran.}
\email{fahimeghanbary@yahoo.com}
\address{M. Tousi, Department of Mathematics, Shahid Beheshti University, G.C., Tehran, Iran-and-School of Mathematics, Institute for Research
in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran.}
\email{mtousi@ipm.ir}
\address{H. Zakeri, Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran.}
\email{hoszakeri@gmail.com}
\subjclass[2010]{13C14; 13C05; 13D45.}
\keywords {Arithmetic rank; cohomological dimension; generalized fractions; local cohomology; relative Cohen-Macaulay module; system of parameters.}
\begin{abstract} Let $R$ be a commutative Noetherian (not necessary local) ring with identity and $\fa$ be a proper ideal of $R$. We introduce a notion
of $\fa$-relative system of parameters and characterize them by using the notion of cohomological dimension. Also, we present a criterion of relative
Cohen-Macaulay modules via relative system of parameters.
\end{abstract}
\maketitle
\section{Introduction}
Throughout, the word ring stands for commutative Noetherian rings with identity. Consider the following naturally-raised questions:
\begin{question}\label{1.1} Let $\fa$ be a proper ideal of a ring $R$, $M$ a finitely generated $R$-module and $c=\cd(\fa,M)$. Is there a sequence $x_1,
x_2, \ldots, x_c$ of elements in $\fa$ such that $$\cd\left(\fa,M/\langle x_{1}, x_2, \ldots, x_{i} \rangle M\right)=c-i$$ for every $i=1, 2, \ldots, c$?
If yes, how can we characterize such sequences?
\end{question}
Let $R$ be a ring, $\fa$ an ideal of $R$ and $M$ a finitely generated $R$-module with $M\neq \fa M$. Then $M$ is said to be $\fa$-relative Cohen-Macaulay,
$\fa$-RCM, if $\grade(\fa,M)=\cd(\fa,M)$. This notion was introduced by Majid Rahro Zargar and the fourth author in \cite{RZ2} and its study was continued
in \cite{Ra1}, \cite{Ra2}, \cite{Ra3}, \cite{RZ1} and \cite{CH}. Relative Cohen-Macaulay bigraded modules were already introduced and investigated by
Ahad Rahimi; see \cite{R} and \cite{JR}. Also, the closely related notion of cohomologically complete intersection ideals was examined by Michael
Hellus and Peter Schenzel in \cite{HS}.
\begin{question}\label{1.2} Over a local ring $T,$ a finitely generated $T$-module $N$ is Cohen-Macaulay if and only if every system of parameters of
$N$ is an $N$-regular sequence. Is there an analogue characterization for $\fa$-relative Cohen-Macaulay $R$-modules?
\end{question}
This paper is dealing with the above questions. Although these questions don't look so related in the beginning, surprisingly, they become connected through
a notion of relative system of parameters. Here, we introduce this notion and through investigation of its properties, we answer the above questions.
Let $c:=\cd(\fa,M)$ denote the cohomological dimension of $M$ with respect to $\fa$; i.e. the supermum of the integers $i$ for which $\text{H}_{\fa}^i(M)\neq 0$.
Recall that when $R$ is local with the unique maximal ideal $\fm$ and $\dim_RM=d$, a sequence $x_1, x_2,\ldots, x_d\in \fm$ is called a system of parameters
of $M$ if the $R$-module $M/\langle x_1, x_2,\ldots, x_d\rangle M$ has finite length. This is equivalent to say that $$\Rad\left( \langle x_1, x_2, \ldots,
x_d \rangle +\Ann_RM\right)=\Rad\left(\fm+\Ann_RM \right).$$ We call a sequence $x_{1},x_2, \ldots, x_{c}\in \fa$ an $\fa$-relative system of parameters,
$\fa$-Rs.o.p, of $M$ if $$\Rad\left(\langle x_{1}, x_2, \ldots, x_{c} \rangle+\Ann_{R}M\right)=\Rad\left(\fa+\Ann_{R}M\right).$$
System of parameters appear in many contexts. Especially, Monomial Conjecture on system of parameters of local rings stands for decades until recently
solved by Yves Andr$\acute{e}$; see \cite{An}.
Although over a local ring every finitely generated $R$-module possesses a system of parameters, this is not the case for $\fa$-relative systems of
parameters. It is immediate that $R$ admits an $\fa$-relative system of parameters if and only if $\ara(\fa)=\cd(\fa,R)$. Let $K$ be a field. For a
square-free monomial ideal $\fa$ of a polynomial ring $R=K[x_1,\dots ,x_n]$, it is known that $\cd(\fa,R)=\pd_R\frac{R}{\fa}$; see \cite[Theorem 1]{Ly}.
Characterizing monomial ideals $\fa$ satisfying $\ara(\fa)=\pd_R\frac{R}{\fa}$ has been an active area of research for years; see e.g. \cite{Ba1}, \cite{Ba2}
and \cite{SV}.
Assume that $\fa$ is contained in the Jacobson radical of $R$ and $M$ possesses an $\fa$-Rs.o.p. We prove that a sequence $\underline{x}=x_1,x_2,\ldots,
x_c\in \fa$ is $\fa$-relative system of parameters of $M$ if and only if $$\cd \left(\fa,M/\langle x_1, x_2, \ldots, x_i \rangle M \right)=c-i$$ for every
$1\leq i\leq c$; see Theorem \ref{2.9}. Also, we show that $M$ is $\fa$-relative Cohen-Macaulay if and only if every $\fa$-relative system of parameters of
$M$ is an $M$-regular sequence if and only if there exists an $\fa$-relative system of parameters of $M$ which is an $M$-regular sequence; see Theorem
\ref{3.2}. These two results yields that if $M$ is $\fa$-RCM and $\underline{x}=x_{1}, x_2, \ldots, x_{c}\in \fa$ is an $\fa$-Rs.o.p of $M$, then
$M/\langle x_{1}, x_2, \ldots, x_{i} \rangle M$ is $\fa$-RCM for every $i=1,\ldots,c$; see Corollary \ref{3.4}.
\section{Question 1.1}
Theorem \ref{2.9} is the main result of this paper. To prove it, we need Lemmas \ref{2.4}, \ref{2.5}, \ref{2.6} and \ref{2.8}. We begin by recalling
some needed definitions.
Let $\fa$ be an ideal of $R$ and $M$ a finitely generated $R$-module. Recall that the arithmetic rank of $\fa$, denoted by $\ara\left(\fa\right)$, is
the least number of elements of $R$ required to generate an ideal with the same radical as $\fa$. Among other things, this paper deals with the local
cohomology modules $$\text{H}_{\fa}^{i}\left(M\right):=\varinjlim \limits_{n\in \mathbb{N}} \text{Ext}_R^i\left(R/\fa^n,M\right); \ i\in \mathbb{N}_0.$$
If $\fb$ is another ideal of $R$ such that the ideals $\fa+\Ann_{R}M$ and $\fb+\Ann_{R}M$ have the same radical, then the
Independence Theorem \cite[Theorem 4.2.1]{BS} yields a natural $R$-isomorphism $\text{H}_{\fa}^{i}\left(M\right)\cong \text{H}_{\fb}^{i}\left(M\right)$
for all $i\in \mathbb{N}_0.$ One easily sees that $\cd\left(\fa,M\right)=-\infty$ if and only if $M=\fa M$. On the other hand, \cite[ Corollary 3.3.3]{BS}
implies that $\cd\left(\fa,M\right)\leq \ara\left(\fa\right)$. In the case $\left(R,\fm\right)$ is a local ring, it is known that $\ara\left(\fm\right)=\dim R=
\cd\left(\fm,R\right)$.
\begin{definition}\label{2.1} Let $M$ be a finitely generated $R$-module and $\fa$ an ideal of $R$ with $M\neq \fa M$.
\begin{enumerate}
\item[i)] Let $c=\cd\left(\fa,M\right)$. A sequence $x_{1},x_2, \ldots, x_{c}\in \fa$ is called $\fa$-{\it relative system of parameters}, $\fa$-Rs.o.p, of $M$ if
$$\Rad\left(\langle x_{1}, x_2, \ldots, x_{c}\rangle+\Ann_{R}M\right)=\Rad\left(\fa+\Ann_{R}M\right).$$
\item[ii)] {\it Arithmetic rank} of $\fa$ with respect to $M$, $\ara\left(\fa,M \right)$, is defined as the infimum of the integers $n\in \mathbb{N}_0$ such that
there exist $x_1, x_2, \ldots, x_n\in R$ satisfying $$\Rad\left(\langle x_{1}, x_2, \ldots, x_{n}\rangle +\Ann_{R}M\right)=\Rad\left(\fa+\Ann_{R}M\right).$$
\end{enumerate}
\end{definition}
Clearly if $x_{1},x_2, \ldots, x_{c}\in R$ is an $\fa$-Rs.o.p of $M$, then for all $t_{1},\ldots,t_{c}\in \mathbb{N}$, every permutation of $x_{1}^{t_{1}},\ldots ,
x_{c}^{t_{c}}$ is also an $\fa$-Rs.o.p of $M$. One may easily check that $\cd\left(\fa,M\right)\leq \ara\left(\fa,M \right)$. Obviously, $\ara\left(\fa,R\right)=
\ara\left(\fa\right)$.
Our first result provides a characterization for existence of relative system of parameters. Although it is an easy observation, we include its proof for the
reader's convenience.
\begin{lemma}\label{2.2} Let $M$ be a finitely generated $R$-module and $\fa$ an ideal of $R$ with $M\neq \fa M$. Then $\fa$ contains an $\fa$-Rs.o.p
of $M$ if and only if $\ara\left(\fa,M \right)=\cd\left(\fa,M\right)$.
\end{lemma}
\begin{prf} Set $c:=\cd\left(\fa,M\right)$. Let $x_{1},x_2, \ldots, x_{c}\in \fa$ be an $\fa$-Rs.o.p of $M$. Then $$\Rad\left(\langle x_{1},x_2, \ldots, x_{c}\rangle+\Ann_RM\right)=\Rad\left(\fa+
\Ann_RM\right),$$ and so $\ara\left(\fa,M \right)\leq c \leq \ara\left(\fa,M \right)$. Thus $\ara\left(\fa,M \right)=c$.
Next, suppose that $\ara\left(\fa,M \right)=\cd\left(\fa,M\right)$. Hence, there are $y_{1}, y_2, \ldots, y_{c}$ in $R$ such that $$\Rad\left(\langle y_{1},\ldots,y_{c}\rangle+\Ann_RM\right)=\Rad\left(\fa+
\Ann_RM\right).$$ There is $n\in \mathbb{N}$ such that $y_{i}^{n}\in \fa+\Ann_RM$ for every $1\leq i\leq c$. So for each $1\leq i\leq c$, there are
$z_i\in \fa$ and $w_i\in \Ann_RM$ such that $y_i^n=z_i+w_i$. Now, $$\Rad\left(\langle z_{1},z_{2}, \ldots, z_{c}\rangle+\Ann_RM\right)=
\Rad\left(\fa+\Ann_RM\right),$$ and so $z_{1},z_{2},\ldots, z_{c}\in \fa$ is an $\fa$-Rs.o.p of $M$.
\end{prf}
This note is also concerned with the special case of the notion of generalized fractions. This notion is described as follows: Let $x_{1}, \ldots, x_{n}$
be a sequence of elements of $R$ and $M$ an $R$-module. Set $$U:=\left \{ \left(x_{1}^{\alpha_{1}}, \ldots, x_{n}^{\alpha_{n}}\right)|\alpha_{1}, \ldots,
\alpha_{n} \in \mathbb{N} \right \}.$$ Then $U$ leads to a module of generalized fractions $U^{-n}M$:\\
For every $r,s\in M$ and $\left(x_{1}^{\alpha_{1}}, \ldots , x_{n}^{\alpha_{n}}\right), \left(x_{1}^{\beta_{1}}, \ldots , x_{n}^{\beta_{n}}\right)\in
U$, we write $$\left(r,\left(x_{1}^{\alpha_{1}}, \ldots , x_{n}^{\alpha_{n}}\right)\right)\sim \left(s,\left(x_{1}^{\beta_{1}},
\ldots , x_{n}^{\beta_{n}}\right)\right)$$ if there exist integers $\delta_i \geq \max\lbrace \alpha_{i}, \beta_{i}\rbrace$; $i=1, \ldots, n$
such that $$x_{1}^{\delta_1-\alpha_{1}}\ldots x_{n}^{\delta_n-\alpha_{n}}r-x_{1}^{\delta_1-\beta_{1}}\ldots x_{n}^{\delta_n-\beta_{n}}s\in
\langle x_{1}^{\delta_1}, \ldots,
x_{n-1}^{\delta_{n-1}}\rangle M.$$ It is easy to verify that $\sim$ is an equivalence relation on $M\times U$. Then the equivalence class
of an element $\left(r,\left(x_{1}^{\alpha_{1}}, \ldots , x_{n}^{\alpha_{n}}\right)\right)$ is denoted by $\frac{r}{\left(x_{1}^{\alpha_{1}}, \ldots,
x_{n}^{\alpha_{n}}\right)}$ and we let $U^{-n}M$ stand for the set of all equivalence classes of $\sim$. With naturally defined
sum and scalar multiplication, $U^{-n}M$ forms an $R$-module. It is easy to see that $\frac{r}{\left(x_{1}^{\alpha_{1}}, \ldots,
x_{n}^{\alpha_{n}}\right)}\in U^{-n}M$ is zero if and only if there exists an integer $\delta \geq \max\lbrace \alpha_{1},\ldots ,
\alpha_{n}\rbrace$ such that $x_{1}^{\delta-\alpha_{1}}\ldots x_{n}^{\delta-\alpha_{n}}r\in \langle x_{1}^{\delta}, \ldots, x_{n-1}^{\delta}\rangle M$.
For more details see \cite{SZ}.
The next result is very crucial in this paper and may also have applications in other contexts.
\begin{lemma}\label{2.4} Let $\fa=\langle x_{1},\ldots, x_{d}\rangle$ be an ideal of $R$ and $M$ a finitely generated $R$-module.
Then for every $i=1,\ldots, d,$ one has the following exact sequence $$\text{H}_{\fa}^{d-i}\left(\frac{M}{\langle x_{1},\ldots, x_{i-1},
x_{i}\rangle M}\right)\rightarrow \text{H}_{\fa}^{d-i+1}\left(\frac{M}{\langle x_{1}, \ldots, x_{i-1}\rangle M}\right) \stackrel{x_{i}}
\longrightarrow \text{H}_{\fa}^{d-i+1}\left(\frac{M}{\langle x_{1},\ldots, x_{i-1}\rangle M}\right) \rightarrow 0.$$
In particular, there is an exact sequence $$\text{H}_{\fa}^{d-1}\left(M/x_{1}M\right)\longrightarrow \text{H}_{\fa}^{d}\left(M\right) \stackrel{x_{1}}
\longrightarrow \text{H}_{\fa}^{d}\left(M\right) \longrightarrow 0.$$
\end{lemma}
\begin{prf} We first prove the last assertion. Denote $M/x_{1}M$ by $\overline{M}$ and let $-: M\lo \overline{M}$ be the natural
epimorphism. Set $$U:=\left \{ \left(x_{1}^{\alpha _{1}}, x_{2}^{\alpha _{2}}, \ldots, x_{d}^{\alpha _{d}},1\right)| \ \alpha_{1},\ldots,
\alpha_{d}\in \mathbb{N} \right \}$$ and $$V:=\left \{ \left(x_{2}^{\alpha_{2}}, x_{3}^{\alpha_{3}}, \ldots, x_{d}^{\alpha_{d}}, 1\right)|\
\alpha_{2},\ldots,\alpha_{d} \in\mathbb{N} \right \}.$$
Then, by \cite[Remark 2.2]{KSZ}, $\text{H}_{\fa}^{d}\left(M\right)\cong U^{-d-1}M$ and $\text{H}_{\fa}^{d-1}\left(\overline{M}\right)\cong V^{-d}\overline{M}$.
Define $$\varphi: V^{-d}\overline{M}\lo U^{-d-1}M$$ by $$\varphi\left(\frac{\overline{r}}{\left(x_{2}^{\alpha_{2}}, x_{3}^{\alpha_{3}}, \ldots,
x_{d}^{\alpha_{d}}, 1\right)}\right)
=\frac{r}{\left(x_{1}, x_{2}^{\alpha_{2}}, \ldots, x_{d}^{\alpha_{d}},1\right)}$$ and let $\psi:U^{-d-1}M\lo U^{-d-1}M$ denote the map defined by
multiplication by $x_1$.
It suffices to show that the sequence $$V^{-d}\overline{M}\overset{\varphi}\lo U^{-d-1}M\overset{\psi}\lo U^{-d-1}M\lo 0$$ is exact.
Let $z=\frac{r}{\left(x_{1}^{\alpha_{1}},\ldots,x_{d}^{\alpha_{d}},1\right)} \in U^{-d-1}M $. Then $z=\frac{x_{1}r}{\left(x_{1}^{\alpha_{1}+1},
x_{2}^{\alpha_{2}}, \ldots, x_{d}^{\alpha_{d}},1\right)}$, and so $\psi$ is surjective.
Clearly, $\im \varphi \subseteq \ker \psi$. Now, let $\psi\left(z\right)=0$. Then, there is an integer $\delta \geq \max\lbrace \alpha_{1},\ldots
,\alpha_{d}\rbrace$ such that $x_{1}^{\delta-\alpha_{1}}\ldots x_{d}^{\delta-\alpha_{d}}x_{1}r \in \langle x_{1}^{\delta},\ldots,x_{d}^{\delta}
\rangle M$.
Hence, $$x_{1}^{\delta+1-\alpha_{1}} x_{2}^{\delta-\alpha_{2}}\ldots x_{d}^{\delta-\alpha_{d}}r=\sum \limits_{i=1}^{d} x_{i}^{\delta}r_{i},$$
where $r_1, \ldots, r_d\in M$. This yields
that
$$\begin{array}{ll}
z&=\frac{r}{\left(x_{1}^{\alpha_{1}},\ldots ,x_{d}^{\alpha_{d}},1\right)}\\ &=\frac{x_{1}^{\delta+1-\alpha_{1}}x_{2}^{\delta-\alpha_{2}}\ldots x_{d}^{\delta-\alpha_{d}}r}{\left(x_{1}^{\delta+1},x_{2}^{\delta},\ldots,x_{d}^{\delta},1\right)}\\
&=\underset{i=1}{\overset{d}\sum }\frac{x_{i}^{\delta}r_{i}}{\left(x_{1}^{\delta +1},x_{2}^{\delta}, \ldots,x_{d}^{\delta},1\right)}\\
&=\frac{x_{1}^{\delta}r_{1}}{\left(x_{1}^{\delta+1},x_{2}^{\delta},\ldots,x_{d}^{\delta},1\right)}\\
&=\frac{r_{1}}{\left(x_{1},x_{2}^{\delta},\ldots,x_{d}^{\delta},1\right)}\\
&=\varphi \left(\frac{\overline{r}_{1}}{\left(x_{2}^{\delta}, x_{3}^{\delta}, \ldots, x_{d}^{\delta}, 1\right)}\right).
\end{array}
$$
So, $\ker \psi \subseteq \im \varphi $. This completes the proof of the last assertion.
Now, we show the first assertion. Set $N:=M/\langle x_{1},...,x_{i-1}\rangle M$. Then by using the same argument as above, we have the following exact sequence
$$\text{H}_{\fb}^{d-i}\left(N/x_{i}N\right)\longrightarrow \text{H}_{\fb}^{d-i+1}\left(N\right) \stackrel{x_{i}}\longrightarrow \text{H}_{\fb}^{d-i+1}
\left(N\right) \longrightarrow 0,$$ where $\fb:=\langle x_{i},x_{i+1},\ldots, x_{d}\rangle$. This yields our claim.
\end{prf}
\begin{lemma}\label{2.5} Let $M$ be a finitely generated $R$-module and $\fa$ an ideal of $R$ with $M\neq \fa M$. Let $c:=\cd\left(\fa,M\right)$ and
$\underline{x}=x_{1}, x_2, \ldots, x_{c}\in \fa$ be an $\fa$-Rs.o.p of $M$. Then for every $0\leq i\leq c$, the sequence $x_{i+1}, x_{i+2}, \ldots, x_{c}$
forms an $\fa$-Rs.o.p of $M/\langle x_{1},x_2, \ldots, x_{i}\rangle M$, and so $\cd\left(\fa,M/\langle x_{1},x_2, \ldots, x_{i}\rangle M\right)=c-i$.
\end{lemma}
\begin{prf} We do induction on $i$. The case $i=0$ holds trivially. Next, assume that $i>0$ and the claim holds for $i-1$. Set $\overline{M}:=M/
\langle x_{1},x_2, \ldots, x_{i-1}\rangle M$. Then, by the induction hypothesis, $\cd\left(\fa,\overline{M}\right)=c-i+1$. As $$\Rad\left(\fa+\Ann_RM\right)=
\Rad\left(\langle x_{1}, x_2, \ldots, x_{c} \rangle+\Ann_RM\right),$$ Lemma \ref{2.4} yields the exact sequence: $$\text{H}_{\fa}^{c-i}\left(\overline{M}/x_i
\overline{M}\right)\rightarrow \text{H}_{\fa}^{c-i+1}\left(\overline{M}\right) \stackrel{x_{i}}
\longrightarrow \text{H}_{\fa}^{c-i+1}\left(\overline{M}\right) \rightarrow 0.$$
Since $\text{H}_{\fa}^{c-i+1}\left(\overline{M}\right)$ is $\fa$-torsion and $x_i\in \fa$, each element of $\text{H}_{\fa}^{c-i+1}\left(\overline{M}\right)$
is annihilated by some power of $x_i$. Hence, as $\text{H}_{\fa}^{c-i+1}\left(\overline{M}\right)$ is nonzero, the map
$$\text{H}_{\fa}^{c-i+1}\left(\overline{M}\right) \stackrel{x_{i}}\longrightarrow \text{H}_{\fa}^{c-i+1}\left(\overline{M}\right)$$ is not injective. So, $\text{H}_{\fa}^{c-i}\left(\overline{M}/x_i \overline{M}\right)\neq 0$. Consequently, $$c-i\leq \cd\left(\fa,\overline{M}/x_i \overline{M}\right)=
\cd\left(\fa,M/\langle x_{1},x_2, \ldots, x_{i}\rangle M\right).$$
Now, one has the following display of equalities:
$$\begin{array}{ll}
\Rad\left(\fa+\Ann_R\left(M/\langle x_{1},x_2, \ldots, x_{i}\rangle M\right) \right)&=\Rad\left(\fa+\left(\langle x_{1},x_2, \ldots, x_{i}\rangle+\Ann_RM\right)\right)\\
&=\Rad\left(\fa+\Ann_RM\right)\\
&=\Rad\left(\langle x_{1}, x_2, \ldots, x_{c} \rangle+\Ann_RM\right)\\
&=\Rad\left(\langle x_{i+1}, \ldots, x_{c}\rangle+\left(\langle x_{1},x_2, \ldots, x_{i}\rangle+\Ann_RM\right)\right)\\
&=\Rad\left(\langle x_{i+1}, \ldots, x_{c}\rangle+\Ann_R\left(M/\langle x_{1},x_2, \ldots, x_{i}\rangle M\right)\right).
\end{array}
$$
So, $\ara\left(\fa,M/\langle x_{1},x_2, \ldots, x_{i}\rangle M\right)\leq c-i.$ Thus $$\ara\left(\fa,M/\langle x_{1},x_2, \ldots, x_{i}\rangle M\right)=\cd\left(\fa,M/\langle x_{1},x_2, \ldots, x_{i}\rangle M\right)=c-i,$$ and the sequence $x_{i+1}, x_{i+2}, \ldots, x_{c}$ is an $\fa$-Rs.o.p of $M/\langle x_{1},x_2, \ldots, x_{i}\rangle M$.
\end{prf}
Let $\fa$ be an ideal of $R$ and $M, N$ two finitely generated $R$-modules such that $\Supp_RN\subseteq \Supp_RM$. Then, by \cite[Theorem 2.2]{DNT}, $\cd(\fa,N)\leq \cd(\fa,M)$.
In particular if $\Supp_RN=\Supp_RM$, then $\cd(\fa,N)=\cd(\fa,M)$. In the rest of the paper, we shall use this several times without any further comment.
\begin{lemma}\label{2.6} Let $\fa$ be an ideal of $R$ which is contained in the Jacobson radical of $R$
and $x$ an element of $\fa$. Assume that $M$ is a nonzero finitely generated $R$-module with $\ara\left(\fa,M \right)=1$.
If $\text{H}_{\fa}^{1}\left(M/xM\right)=0$, then $\Rad\left(\fa+\Ann_RM\right)=\Rad\left(\langle x \rangle+\Ann_RM\right)$.
\end{lemma}
\begin{prf} Set $\overline{M}:=M/xM$ and assume that $\text{H}_{\fa}^{1}\left(\overline{M}\right)=0$.
As $\fa$ is contained in the Jacobson radical of $R$, it follows that $\overline{M}\neq\fa
\overline{M}$, and so $\cd\left(\fa,\overline{M}\right)\geq 0$. Since $\text{H}_{\fa}^{1}\left(\overline{M}\right)=0$ and $\ara\left(\fa,M \right)=1$,
one deduces that $\cd\left(\fa,\overline{M}\right)=0$. Set $T:=R/\left(\langle x \rangle+\Ann_RM\right)$. Then $\cd\left(\fa,T\right)=\cd\left(\fa,\overline{M}\right)=0$.
There is $y\in R$ such that $$\Rad\left(\fa+\Ann_RM\right)=\Rad\left(\langle y \rangle+\Ann_RM\right).$$ So, $\text{H}_{\fa}^{i}\left(T\right)=\text{H}_{\langle y \rangle}^{i}\left(T\right)$ for every $i\geq 0$. We may and do choose $y$ in $\fa$. By \cite[Remark 2.2.20]{BS}, there is the following exact
sequence $$0 \longrightarrow \text{H}_{\langle y \rangle}^{0}\left(T\right) \longrightarrow T \longrightarrow T_{y} \longrightarrow \text{H}_{\langle y
\rangle}^{1}\left(T\right) \longrightarrow 0,$$ which implies that the natural map $\theta : T\longrightarrow T_{y}$ is surjective. In particular,
there is $t\in T$ such that $\frac{t}{1_T}=\frac{1_T}{y}$, and so $y^{n}\left(yt-1_T\right)=0_{T}$ for some $n\in \mathbb{N}$. As $yt$ belongs to the Jacobson
radical of $T$, $yt-1_T$ is a unite in $T$, and so it follows that $y^{n}\in \langle x \rangle+\Ann_RM$. Thus, $$\Rad\left(\fa+\Ann_RM\right)=\Rad\left(\langle x \rangle+\Ann_RM\right).$$
\end{prf}
A special case of the next result has already been proved by Michael Hellus; see \cite{He2} and \cite[Remark 1.2]{He1}.
\begin{lemma}\label{2.8} Let $\fa$ be a proper ideal of $R$ and $M$ a nonzero finitely generated $R$-module. Let $n\in \mathbb{N}$ be such that
$\cd\left(\fa,M\right)\leq n $ and $x_1,\ldots, x_n \in \fa$. Consider the following conditions:
\begin{enumerate}
\item[i)] $\Rad\left(\langle x_1 \ldots, x_{n} \rangle+\Ann_RM\right)=\Rad\left(\fa+\Ann_RM\right)$.
\item[ii)] The map $\text{H}_{\fa}^{n-i+1}\left(M/\langle x_1 \ldots, x_{i-1} \rangle M\right)\stackrel{x_{i}}\longrightarrow \text{H}_{\fa}^{n-i+1}
\left(M/\langle x_1 \ldots, x_{i-1} \rangle M\right)$ is surjective for all $i=1,\ldots, n$.
\end{enumerate}
Then i) implies ii). Furthermore if $\fa$ is contained in the Jacobson radical of $R$, then i) and ii) are equivalent.
\end{lemma}
\begin{prf} i)$\Rightarrow$ii) It follows by Lemma \ref{2.4}.
ii)$\Rightarrow$i) We do induction on $n$. Assume that $n=1$. Since
$$\Rad\left(\langle x_1 \rangle+\Ann_RM\right)\subseteq \Rad\left(\fa+\Ann_RM\right),$$ it suffices to show that $$\V\left(\langle x_1 \rangle+\Ann_RM\right)\subseteq \V\left(\fa+\Ann_RM\right).$$ Let $\fp\in \V\left(\langle x_1 \rangle+\Ann_RM\right) $.
Then $\fp\in \Supp_RM$. Set $T:=R/\Ann_RM $. Then $$\cd\left(\fa T,T\right)=\cd\left(\fa,M\right)\leq 1.$$
So, $\text{H}_{\fa}^{1}\left(-\right)$ is a right exact endofunctor on the category of $T$-modules and $T$-homomorphisms. By the assumption, the map
$\text{H}_{\fa}^{1}\left(M\right)\stackrel{x_1}\longrightarrow \text{H}_{\fa}^{1}\left(M\right) $ is surjective. Now, we have the following display of $R$-isomorphisms:
$$\begin{array}{ll}
\text{H}_{\fa}^{1}\left(M\right)\otimes_{R} R/\fp &\cong \text{H}_{\fa T}^{1}\left(M\right) \otimes_{T} T/\fp T \\
&\cong \left(\text{H}_{\fa T}^{1}\left(T\right)\otimes_{T} M\right)\otimes_{T} T/\fp T \\
&\cong \text{H}_{\fa T}^{1}\left(T\right)\otimes_{T} M/\fp M \\
&\cong \text{H}_{\fa T}^{1}\left(M/\fp M\right)\\
&\cong \text{H}_{\fa}^{1}\left(M/\fp M\right).
\end{array}$$
This shows that the natural map $$\text{H}_{\fa}^{1}\left(M/\fp M\right)\stackrel{x_1}\longrightarrow \text{H}_{\fa}^{1}\left(M/\fp M\right)$$
is surjective. But $x_1\in \fp$, and so the above map is zero. Thus, $\text{H}_{\fa}^{1}\left(M/\fp M\right)=0$. Since $\fa$ is contained in the Jacobson
radical of $R$ and $\fp\in \Supp_RM$, it turns out that $M/\fp M \neq \fa \left(M/\fp M\right)$, and so $\Gamma_{\fa}\left(M/\fp M\right)\neq 0$.
One has
$$\Rad\left(\Ann_R\left(M/\fp M\right)\right)=\Rad\left(\fp+\Ann_RM\right)=\fp,$$
and so $\Supp_R\left(M/\fp M\right)=\Supp_R\left(R/\fp\right).$ This implies that $$\cd\left(\fa,R/\fp\right)=\cd\left(\fa,M/\fp M\right)=0.$$ Hence
$\Gamma_{\fa}\left(R/\fp\right)\neq 0$, which implies that $\fa \subseteq \fp$, and so $\fp \in \V\left(\fa+\Ann_RM\right)$.
Next, assume that $n>1$ and the case $n-1$ is settled. Set $\overline{M}:=M/x_1M$. As $$\cd\left(\fa,M/\left(0:_{M}x_1\right)\right)\leq \cd\left(\fa,M\right)\leq n ,$$
applying the functor $\text{H}_{\fa}^{n}\left(-\right)$ on the
exact sequence $$0\longrightarrow M/\left(0:_{M}x_1\right) \stackrel{x_1}\longrightarrow M\longrightarrow \overline{M} \longrightarrow 0$$ yields that $\text{H}_{\fa}^{n}\left(\overline{M}\right)$
is a quotient of $\text{H}_{\fa}^{n}\left(M\right) $. So, the map $\text{H}_{\fa}^{n}\left(\overline{M}\right)\stackrel{x_1}\longrightarrow \text{H}_{\fa}^{n}\left(\overline{M}\right)$ is surjective.
But this map is zero, and so $\text{H}_{\fa}^{n}\left(\overline{M}\right)=0 $. Thus, $\cd\left(\fa,\overline{M}\right)\leq n-1$.
Since $$\text{H}_{\fa}^{n-i+1}\left(\frac{M}{\langle x_1 \ldots, x_{i-1} \rangle M}\right)=\text{H}_{\fa}^{\left(n-1\right)-\left(i-1\right)+1}
\left(\frac{\overline{M}}{\langle x_2, \ldots, x_{i-1}\rangle \overline{M}}\right),$$ by the induction hypothesis
$$\Rad\left(\langle x_2, \ldots, x_n \rangle+\Ann_R\left(\overline{M}\right)\right)=\Rad\left(\fa +\Ann_R\left(\overline{M}\right)\right).$$
Now by the argument given in the second paragraph of the proof of Lemma \ref {2.5}, we deduce that
$$\Rad\left(\langle x_1, \ldots, x_n\rangle+\Ann_R\left(M\right)\right)=\Rad\left(\fa+\Ann_R\left(M\right)\right).$$
\end{prf}
Now, we are ready to present the main result of this paper.
\begin{theorem}\label{2.9} Let $\fa$ be an ideal of $R$ which is contained in the Jacobson radical of $R$ and $M$ a nonzero finitely generated $R$-module.
Assume that $c:=\cd\left(\fa,M\right)=\ara\left(\fa,M \right)$ and $x_1,\ldots, x_{c}\in \fa$. Then the following are equivalent:
\begin{enumerate}
\item[i)] $x_1,\ldots, x_{c}$ is an $\fa$-Rs.o.p of $M$.
\item[ii)] The map $\text{H}_{\fa}^{c-i+1}\left(M/\langle x_1, \ldots, x_{i-1}\rangle M\right) \stackrel{x_{i}}\longrightarrow \text{H}_{\fa}^{c-i+1}
\left(M/\langle x_1,
\ldots, x_{i-1}\rangle M\right)$ is surjective for all $i=1, \ldots, c$.
\item[iii)] $\cd\left(\fa,M/\langle x_{1},x_2, \ldots, x_{i}\rangle M\right)=c-i$ for every $i=1, 2, \ldots, c$.
\end{enumerate}
\end{theorem}
\begin{prf} For $c=0$, there is nothing to prove. So, in the rest of the argument, we assume that $c\geq 1$.
i)$\Leftrightarrow$ii) and i)$\Rightarrow$iii) are immediate by Lemmas \ref{2.8} and \ref{2.5}; respectively.
iii)$\Rightarrow$i) We do induction on $c$. Suppose that $c=1$ and set $\overline{M}:=M/x_1 M$. Then $\ara\left(\fa,M \right)=1$ and $\text{H}_{\fa}^{1}\left(\overline{M}\right)\stackrel{\left(iii\right)}=0$. So, Lemma \ref{2.6}
implies that $$\Rad\left(\fa+\Ann_RM\right)=\Rad\left(\langle x_1 \rangle+\Ann_RM\right).$$ Thus $x_1$ is an $\fa$-Rs.o.p of $M$. Next, suppose
that $c>1$ and the claim holds for $c-1$. One has $\cd\left(\fa,\overline{M}\right)\stackrel{\left(iii\right)}=c-1$ and, for each $2\leq i \leq c$, $$\begin{array}{ll}\cd\left(\fa,\frac{\overline{M}}{\langle x_2, x_3, \ldots, x_{i} \rangle\overline{M}}\right)&=\cd\left(\fa,\frac{M}{\langle x_{1}, x_2,
\ldots, x_{i} \rangle M}\right)\\
&\stackrel{\left(iii\right)}=\cd\left(\fa,M\right)-i\\
&=\cd\left(\fa,M\right)-1-\left(i-1\right)\\
&\stackrel{\left(iii\right)}=\cd\left(\fa,\overline{M}\right)-\left(i-1\right).
\end{array}
$$
Hence by the induction hypothesis, the sequence $x_{2}, x_3, \ldots, x_{c}$ forms an $\fa$-Rs.o.p of $\overline{M}$. Thus
$$\Rad\left(\fa+\Ann_R\overline{M}\right)=\Rad\left(\langle x_2, x_3, \ldots, x_{c} \rangle+\Ann_R\overline{M}\right),$$ which implies that
$$\Rad\left(\fa+\Ann_RM\right)=\Rad\left(\langle x_1, x_{2}, \ldots, x_{c} \rangle+\Ann_RM\right).$$
Therefore, $x_{1},\ldots,x_{c}$ is an $\fa$-Rs.o.p of $M$.
\end{prf}
Next, we record the following immediate conclusion which may be interesting in its own right.
\begin{corollary}\label{2.10}
Let $\left(R,\fm\right)$ be a local ring, $M$ a $d$-dimensional nonzero finitely generated $R$-module and $x_{1},\ldots, x_{d}\in \fm$. Then the following
are equivalent:
\begin{enumerate}
\item[i)] $x_{1},\ldots, x_{d}$ is a system of parameters of $M$.
\item[ii)] The map $\text{H}_{\fm}^{d-i+1}\left(M/\langle x_{1}, \ldots, x_{i-1}\rangle M\right) \stackrel{x_{i}}\longrightarrow \text{H}_{\fm}^{d-i+1}
\left(M/\langle x_{1},
\ldots, x_{i-1}\rangle M\right)$ is surjective for all $i=1,\ldots, d$.
\end{enumerate}
\end{corollary}
Let $\left(R,\fm\right)$ be a local ring. Next, we will mention two results for system of parameters that their
analogues don't hold for relative system of parameters; see Example \ref{2.11}.
First: Every $R$-regular sequence is a part of a system of parameters of $R$.
Second: Let $M$ be a maximal Cohen-Maculay $R$-module and $A$ be a square matrix of size $n$ with entries in $R$. Let $x_1, x_2, \ldots, x_n$ be a system
of parameters of $M$ and $y_1, y_2, \ldots, y_n\in \fm$ be such that $[y_1, y_2, \ldots, y_n]^T=A[x_1, x_2, \ldots, x_n]^T$. Then by \cite[Theorem]{DR}, $y_1,
y_2, \ldots, y_n$ is a system of parameters of $M$ if and only if the map induced by multiplication by det $A$ from $M/\langle x \rangle M$ to $M/\langle y
\rangle M$ is injective.
\begin{example}\label{2.11}Let $K$ be a field, $R=K[[x,z]]$ and $\fa=\langle x \rangle$. Then $\ara\left(\fa\right)=\cd\left(\fa,R\right)=1$. Set $y:=zx$. Then
\begin{enumerate}
\item[i)] One has $y\in \fa\setminus Z_{R}\left(R\right)$ and $x\notin \Rad\left(\langle y \rangle\right)$.
Hence, $\text{H}_{\fa}^{1}\left(\frac{R}{\langle y \rangle}\right)\neq 0$ by Lemma \ref{2.6}. So, by Lemma \ref{2.5}, $y$ is not an $\fa$-Rs.o.p of $R$.
\item[ii)] The natural map $R/\langle x \rangle\overset{z}\lo R/\langle y \rangle$ is injective, $x$ is an $\fa$-Rs.o.p of $R$
and $[y]=[z][x]$, while $y$ is not an $\fa$-Rs.o.p of $R$.
\end{enumerate}
\end{example}
\section{Question 1.2}
Our main result in this section is Theorem \ref{3.2}. To prove it, we need the following lemma.
\begin{lemma}\label{3.1} Let $\fa=\langle x_{1},\ldots,x_{n}\rangle$ be an ideal of $R$ and $M$ a finitely generated $R$-module with $\fa M\neq M$.
Set $g:=\grade\left(\fa,M\right)$. Then
\begin{enumerate}
\item[i)] $\fa$ can be generated by elements $y_{1},\ldots,y_{n}$ such that $y_{i_{1}}, \ldots, y_{i_{h}}$
forms an $M$-regular sequence for all $i_{1},\ldots,i_{h}$ with $1\leq i_{1}<\cdots<i_{h}\leq n, h\leq g$.
\item[ii)] If $\fa$ is contained in the Jacobson radical of $R$ and $g=n$, then $x_{1},\ldots,x_{n}$ forms an $M$-regular sequence.
\end{enumerate}
\end{lemma}
\begin{prf} i) Follows by \cite[Theorem 125 (b)]{Ka}.
ii) Follows by \cite[Theorem 129]{Ka}.
\end{prf}
Here is the right place to bring the following immediate corollary of Lemma \ref{2.2}.
\begin{corollary}\label{2.3} Let $M$ be a finitely generated $R$-module and $\fa$ an ideal of $R$ with $M\neq \fa M$. Then the following are equivalent:
\begin{enumerate}
\item[i)] $M$ is $\fa$-RCM and it possesses an $\fa$-Rs.o.p.
\item[ii)] $\grade\left(\fa,M\right)=\ara\left(\fa,M \right)$.
\end{enumerate}
\end{corollary}
\begin{prf} It is clear by Lemma \ref {2.2} and the inequality $\grade\left(\fa,M\right)\leq \cd\left(\fa,M\right)\leq \ara\left(\fa,M \right)$.
\end{prf}
\begin{theorem}\label{3.2} Let $M$ be a finitely generated $R$-module and $\fa$ an ideal of $R$ with $\ara\left(\fa,M \right)=\cd\left(\fa,M\right)$.
Consider the following conditions:
\begin{enumerate}
\item[i)] $M$ is $\fa$-RCM.
\item[ii)] Every $\fa$-Rs.o.p of $M$ is an $M$-regular sequence.
\item[iii)] There exists an $\fa$-Rs.o.p of $M$ which is an $M$-regular sequence.
\end{enumerate}
Then i) and iii) are equivalent. Furthermore if $\fa$ is contained in the Jacobson radical of $R$, all three conditions are equivalent.
\end{theorem}
\begin{prf} Set $c:=\cd\left(\fa,M\right)$.
i)$\Rightarrow$iii) Let $y_{1}, y_2, \ldots, y_{c}\in \fa$ be an $\fa$-Rs.o.p of $M$ and set $J:=\langle y_{1},y_2, \ldots, y_{c}\rangle$. Then $$\Rad\left(J+\Ann_RM\right)=\Rad\left(\fa+\Ann_RM\right),$$ and so $$\grade\left(J,M\right)=\grade\left(\fa,M\right)=c.$$
By Lemma \ref{3.1} i), there exist $x_{1}, x_2, \ldots, x_{c}\in R$ which forms an $M$-regular sequence and $J=\langle x_{1}, x_2, \ldots, x_{c}
\rangle$. Now, $x_{1}, x_2, \ldots, x_{c}$ is our desired $\fa$-Rs.o.p of
$M$.
iii)$\Rightarrow$i) Let $\underline{z}=z_{1},\ldots,z_{c}\in \fa$ be an $\fa$-Rs.o.p of $M$ which is an $M$-regular sequence. Then $$\ara\left(\fa,M
\right)\leq c\leq \grade\left(\fa,M\right)\leq \ara\left(\fa,M \right).$$ So, $M$ is $\fa$-RCM by Corollary \ref{2.3}.
ii)$\Rightarrow$iii) It is obvious.
i)$\Rightarrow$ii) Let $\underline{z}=z_{1}, z_2, \ldots, z_{c}\in \fa$ be an $\fa$-Rs.o.p of $R$ and set $J:=\langle z_{1},z_2, \ldots, z_{c}\rangle$.
Then
$$\Rad\left(J+\Ann_RM \right)=\Rad\left(\fa+\Ann_RM\right),$$ and so
$$\begin{array}{ll}
\grade \left(J,M \right)&=\grade \left(J+\Ann_RM,M \right)\\
&=\grade \left(\fa + \Ann_RM , M \right)\\
&=\grade \left(\fa,M \right)\\
&=c.
\end{array}$$
Thus Lemma \ref{3.1} ii) yields that $\underline{z}$ is an $M$-regular sequence.
\end{prf}
The following example shows that in Theorem \ref{3.2}, the assumption that $\fa$ is contained in the Jacobson radical of $R$ is necessary.
\begin{example}\label{3.3}
Let $K$ be a field. Consider the ring $R=K[x,y,z]$ and let $\fa=\langle x,y,z \rangle$. It is clear that $R$ is $\fa$-RCM. We can see that $\fa=\langle
y\left(1-x\right), z\left(1-x\right), x\rangle$, so that $y\left(1-x\right), z\left(1-x\right), x$ is an $\fa$-Rs.o.p of $R$. But $y\left(1-x\right),
z\left(1-x\right), x$ is not an $R$-regular sequence.
\end{example}
Next, we record the following corollary of Theorem \ref{3.2}.
\begin{corollary}\label{3.4} Let $\fa$ be an ideal of $R$ which is contained in the Jacobson radical of
$R$. Let $M$ be an $\fa$-RCM $R$-module and $\underline{x}=x_{1}, x_2, \ldots, x_{c}\in \fa$ an $\fa$-Rs.o.p
of $M$. Then $M/\langle x_{1}, x_2, \ldots, x_{i} \rangle M$ is $\fa$-RCM for every $i=1,\ldots,c$.
\end{corollary}
\begin{prf} Set $\overline{M}:=M/\langle x_{1}, x_2, \ldots, x_{i} \rangle M$. By Theorem \ref{3.2}, $x_{1}, x_2, \ldots, x_{c}$ is an $M$-regular sequence,
and so $x_{i+1}, x_{i+2}, \ldots, x_{c}$ is an $\overline{M}$-regular sequence. On the other hand, by Lemma \ref{2.5}, the sequence $x_{i+1},x_{i+2}, \ldots,
x_{c}$ is an $\fa$-Rs.o.p of $\overline{M}$. Applying Theorem \ref{3.2} again implies that $\overline{M}$ is $\fa$-RCM.
\end{prf}
\begin{acknowledgement}
The authors thank Sara Saeedi Madani for introducing them to the references \cite{Ba1}, \cite{Ba2} and \cite{SV}.
\end{acknowledgement}
| 5,944
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TITLE: Induced map on cohomology being zero impies null-homotopic?
QUESTION [6 upvotes]: Let $f:A^\bullet\to B^\bullet$ be a morphism of chain complexes (of any give abelian category). We know that if $f$ is homotopic to the zero map, then $f$ will induce zero map on cohomology. I want to know if the converse true, i.e. if $f$ induces zero map on cohomology, is it true that $f$ is homotopic to zero map?
If not, under what conditions it can be true?
REPLY [2 votes]: Let $\mathcal A$ be an abelian category. Your condition (zero map on cohomology implies null-homotopy) is equivalent to the condition that $\mathcal A$ be semisimple (i.e. any exact triple $0\to X\to Z\to Y\to 0$ in $\mathcal A$ splits, i.e. is isomorphic to $0\to X\to X\oplus Y\to Y\to 0$).
On the one hand, your condition implies all Yoneda Exts are zero, so that the homological dimension of $\mathcal A$ is zero; i.e. $\mathcal A$ is semisimple.
On the other hand, if $\mathcal A$ is semisimple, the functor $h:(K^n,d^n)\mapsto (H^n(K),0)$ which sends a complex $K$ to its complex of cohomology objects with zero differentials in all degrees, induces an equivalence between $D(\mathcal A)$ and the category of cyclic complexes of objects of $\mathcal A$ (complexes with zero differentials in all degrees).
| 30,610
|
Sometimes it’s better to put love into hugs than to put it into words”
– Unknown Author!
- I have been thinking of talking about the Free Hugs Campaign . So finally today it just occurred to my mind to share my views about it. This time when I was in India I came across A.R Rehman song Jiya to Jiya. In the video they have introduced the free hugs campaign ( To See The Video). The free hugs campaign was started way back in 2004 by someone called “Juan Mann”, he is an Australian guy who introduced his free hugs campaign by a band called “ ”.(Click Here To See The Video)
- This guy shared his story on how he felt the need to introduce something that can comfort any person. He once went to a party, totally depressed and heartbroken due to some personal issues. And then a complete stranger in the party gave him a hug that made him feel extremely happy and relaxed. From then onwards he wanted to spread this among strangers. I really appreciate his effort behind commencing something that can cheer up people.
I know many people who even feel reluctant to hug their loved ones. All they do is a handshake!!! You don’t need a reason to hug your loved ones at least!
v A hug can drive away any sorrow.
v A hug has a power to lighten up someone.
v A hug can make your loved one feel wanted and loved.
v A hug can trigger warmth in anyone’s heart.
v A hug can relieve your tension and stress level.
v A hug can literally combat depression .
v A hug can wipe away the tears of a heartbroken person when words of sympathy don’t seem to be enough.
v A hug can ease anyone’s pain.
v In fact, it’s scientifically proved that hugs literally decrease blood pressure.
- I believe that you can only value the importance of a hug when you get deprived of it!!! So from now on don’t hesitate to hug your loved ones whenever you meet them. Don’t miss an opportunity to cheer anyone up by showing their importance in your life. After all, a tender touch is one of the most needed expressions that all of us crave for all the time. If you are a shy person or wait for the other person to come to you and give you a hug then this time surprise your loved ones by hugging them when they least expect it. I am sure they would love it without any doubt you will rejuvenate yourself .
- Personally speaking, I do feel reluctant in hugging strangers. But I really liked the concept and the thought behind the Free Hugs Campaign. But there are a few incidents when I hugged strangers. I guess I have shared that with all of you in . Second incident that pops up to my mind is this old lady I meet very often; usually I see her standing at the corner of our place. Whenever I give her some money or things to eat I usually give her a hug and then she gives a million dollar smile to me. Only these two incidents I can recall where I have hugged a complete stranger.
How about you? Share your story when you hugged a complete stranger.
| 159,929
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David Hunt, AIA, NCARB, LEED AP
It is so rewarding to be part of smart, creative and talented teams, working together to develop elegant and meaningful solutions to our clients' challenges. Seeing our work transform our clients' organizations, to support those they serve, motivates me every day.
David Hunt has 30 years of continuous and dedicated experience in facility management, design, and construction services. David has built a career successfully delivering integrated design, planning, architecture, construction management and general contracting services for clients in the healthcare, education, commercial and civic markets. He has a passion for delivering high-quality services and award-winning projects and values professional integrity, hard work, continuous improvement and collaboration.
David has been active in promoting project delivery and design excellence through the DBIA and AIA, as a jurist of the annual DBIA Western Region Design Awards and membership on the AIA Project Delivery Knowledge Community. He has been a prior speaker at several conferences including the DBIA, AIA, CCFC and Healthcare Design Conference.
| 23,913
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:
Rohinton Mistry
December 22, 2014
November 23, 2016
Nilanjana S Roy
Speaking Volumes: Mr Mistry’s homecoming
October 23, 2010
Nilanjana S Roy
PEN India statement on Rohinton Mistry and Such a Long Journey
October 19, 2010
May 3, 2013
Nilanjana S Roy
The BS column: Rohinton and the Rat Pack
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TEL AVIV (Oct. 13)
Abraham Tzafati, the Iranian-born youth who recently attempted to take his life at a session of the Knesset attended by Premier David Ben Gurion and other members of the Israel Cabinet, was adjudged mentally unsound today by a government psychiatrist. Although the Court decided not to continue with the youth’s projected trial, the defendant was kept under arrest at the suggestion of the Ministry of Justice.
Tzafati made strong protestations against the court’s decision, contending he was in full possession of his mental faculties. He said he intended to commit suicide in the Knesset for the benefit of Israel. Shouting and shedding tears after the court’s ruling was pronounced, Tzafati said the “entire proceedings were a Mapai fabrication.” He collapsed after being removed from the courtroom, and was taken to a government hospital.
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\section{Adjunction between stacks and cocomplete tensor categories} \label{adju}
Every morphism $f : Y \to X$ of schemes (or algebraic stacks) induces a pullback functor $f^* : \Q(X) \to \Q(Y)$, which is a cocontinuous tensor functor. Now we may ask if $f$ can be reconstructed from $f^*$ and if any cocontinuous tensor functor $\Q(Y) \to \Q(X)$ is induced by a morphism. This motivates the definition of \emph{tensorial schemes}, or more generally \emph{tensorial stacks}. We will give a quite conceptual definition of them with the help of an adjunction between stacks and cocomplete tensor categories.
In this section, we assume that all schemes and stacks are defined over some fixed commutative ring $R$ and we work with $R$-linear cocomplete tensor categories. This assumption is not really necessary and in fact one should omit it when one is interested in algebraic geometry over $\F_1$.
\begin{defi}[$\SPEC$ and $\Q$] \label{stacksdef}
\noindent
\begin{enumerate}
\item For us a \textit{stack} is a pseudo-functor $\Sch^{\op} \to \Cat$ which satisfies effective descent with respect to the fpqc topology. Together with natural transformations and modifications, we obtain the $2$-category of stacks, denoted by $\Stack$. We do not require that stacks factor through groupoids $\Gpd$.
\item Let $\C$ be a cocomplete tensor category. Its \emph{spectrum} $\SPEC(\C)$ is a stack defined by
\[\SPEC(\C)(X) = \Hom_{c\otimes}\bigl(\C,\Q(X)\bigr)\]
for schemes $X$. If $X \to Y$ is a morphism of schemes, its pullback functor $\Q(Y) \to \Q(X)$ induces a functor $\SPEC(\C)(Y) \to \SPEC(\C)(X)$. Then $\SPEC(\C)$ is a stack basically because $\Q(-)$ is a stack by descent theory for quasi-coherent modules (\cite[4.23]{Vis05}). This construction provides us with a $2$-functor
\[\SPEC : \Cat_{c\otimes}^{\op} \to \Stack.\]
\item Let $F$ be a stack. The cocomplete tensor category of quasi-coherent modules $\Q(F)$ is defined as $\Hom_{\Stack}(F,\Q(-))$ with pointwise defined tensor products and colimits. This means that a quasi-coherent module $M$ on $F$ is given by functors $M_X : F(X) \to \Q(X)$ for every scheme $X$, which are compatible with base change, which means that there are compatible isomorphisms $f^* \circ M_X \cong M_Y \circ F(f)$ for morphisms $f : Y \to X$. These isomorphisms also belong to the data. We obtain a $2$-functor
\[\Q : \Stack \to \Cat_{c\otimes}^{\op}.\]
\end{enumerate}
\end{defi}
\clearpage
\begin{rem} \noindent
\begin{enumerate}
\item If $F$ is an algebraic stack, then our definition of quasi-coherent modules on $F$ coincides with the usual one (\cite[7.18]{Vis89}).
\item The definition of $\SPEC$ has also appeared in the independent work \cite[Section 2]{Liu12} (including a very detailed proof that $\SPEC(\C)$ is, indeed, a stack). The current proof of \cite[Theorem 3.4]{Liu12} has a serious gap.
\end{enumerate}
\end{rem}
I have learned the following adjunction from David Ben-Zvi. It is a categorification of the well-known adjunction between commutative rings and schemes \cite[Proposition 1.6.3]{EGAI}.
\begin{prop}[Adjunction between $\SPEC$ and $\Q$] \label{adjunction}
If $F$ is a stack and $\C$ is a cocomplete tensor category, then there is a natural equivalence
\[\Hom_{\Stack}(F,\SPEC(C)) \simeq \Hom_{c\otimes}(\C,\Q(F)).\]
Thus, $\SPEC : \Cat_{c\otimes}^{\op} \to \Stack $ is right adjoint to $\Q : \Stack \to \Cat_{c\otimes}^{\op}$.
\end{prop}
\begin{proof}
This is entirely formal. For a stack $F$ and a scheme $X$, the component functor
\[F(X) \to \Hom_{c\otimes}(\Q(F),\Q(X))\]
of the unit $\eta_F : F \to \SPEC(\Q(F))$ is defined to be the obvious evaluation which comes from the definition of $\Q(F)$.
The counit $\varepsilon_\C : \C \to \Q(\SPEC(\C))$ is also given by evaluation, i.e.
\[\varepsilon_\C(T)_X : \Hom_{c\otimes}(\C,\Q(X)) \to \Q(X)\]
evaluates at $T \in \C$.
\end{proof}
\begin{defi}[Tensorial stacks and stacky tensor categories] \label{tensorial-stacky}
A stack $F$ is called \emph{tensorial} if the unit
\[\eta_F : F \to \SPEC(\Q(F))\]
is an equivalence, i.e. for every scheme $Y$ we have an equivalence
\[F(Y) \simeq \Hom_{c\otimes}(\Q(F),\Q(Y)).\]
A cocomplete tensor category $\C$ is called \textit{stacky} if the counit
\[\varepsilon_\C : \C \to \Q(\SPEC(\C))\]
is an equivalence.
\end{defi}
For example, a scheme $X$ is tensorial if for every scheme $Y$ the functor
\[\Hom(Y,X) \to \Hom_{c\otimes}\bigl(\Q(X),\Q(Y)\bigr),~ f \mapsto f^*\]
is an equivalence of categories. This forces $\Hom_{c\otimes}(\Q(X),\Q(Y))$ to be essentially discrete. We will say more about tensoriality in \autoref{tensoriality}.
Every adjunction restricts to an equivalence between its fixed points. Thus:
\begin{prop}
The $2$-functors $\SPEC$ and $\Q$ yield an anti-equivalence of $2$-categories between tensorial stacks and stacky cocomplete tensor categories.
\end{prop}
\begin{rem}[Analogies]
This can be seen as a categorification of the anti-equivalence between affine schemes and commutative rings. It connects algebraic geometry, which is commutative algebra locally, with categorified or global commutative algebra. We will explain later what this means in detail. Notice that there is a striking analogy to functional analysis: We have a functor $C(-,\mathds{C})$ from topological spaces to the dual category of (involutive) $\mathds{C}$-algebras which is left adjoint to the Gelfand spectrum $\Hom(-,\mathds{C})$. The Theorem of Gelfand-Naimark states that this adjunction restricts to an equivalence of categories between compact Hausdorff spaces and the dual category of unital commutative $C^*$-algebras. Back to our setting, we may imagine stacks as $2$-geometric objects whose $2$-regular functions are precisely quasi-coherent modules. Tensorial stacks are those stacks which are determined by their $2$-semiring of $2$-regular functions.
\end{rem}
\begin{rem}[Size issues]
The reader may notice that our definitions of $\Q(F)$ for a general stack $F$ and of $\SPEC(\C)$ for a general cocomplete tensor category $\C$ might cause set-theoretic problems, since they are outside of some Grothendieck universe fixed beforehand. However, the definitions of tensorial stacks and stacky cocomplete tensor categories including their anti-equivalence still make sense.
\end{rem}
\begin{rem}[Stacky tensor categories]
There are lots of examples of cocomplete tensor categories which are not stacky, since there are quite a few properties of quasi-coherent modules which do not hold in all tensor categories. An example is that every epimorphism $\O \to \O$ is already an isomorphism (see \autoref{jim}). Unfortunately we do not know any classification of stacky tensor categories. At least Sch\"appi (\cite{Sch12a}) has proven that the anti-equivalence above restricts to an anti-equivalence between Adams stacks and so-called weakly Tannakian categories. The latter have an intrinsic characterization in characteristic zero (\cite{Sch13}). This is a generalization of the classical Tannaka duality (\cite{Del90}).
\end{rem}
\begin{rem}[Artin's criteria]
The spectrum $\SPEC(\C)$ of a cocomplete tensor category $\C$ should only be considered to be a geometric object when it is an algebraic stack. Artin has found criteria for a stack to be algebraic (\cite[Theorem 5.3]{Art74}). Recently Hall and Rydh have found a refinement of these criteria (\cite[Main Theorem]{Hall13}). Let us write down what they mean for our stack $X=\SPEC(\C)$ defined by $X(Y) = \Hom_{c\otimes/R}(\C,\Q(Y))$. For brevity we will just say algebras when we mean commutative algebras.
Let $R$ be an excellent ring and let $\C$ be a cocomplete $R$-linear tensor category. Then the stack $\SPEC(\C)$ is an algebraic stack, locally of finite presentation over $R$, if and only if the following conditions are satisfied:
\begin{itemize}
\item For every $R$-algebra $A$ the category $\Hom_{c\otimes/R}(\C,\M(A))$ is actually a groupoid.
\item For every directed diagram of $R$-algebras $\{A_i\}$ the canonical functor
\[\colim_i \Hom_{c\otimes/R}(\C,\M(A_i)) \to \Hom_{c\otimes/R}(\C,\M(\colim_i A_i))\]
is an equivalence of groupoids.
\item For homomorphisms $A' \to A \leftarrow B$ of local artinian $R$-algebras of finite type, where $A' \to A$ is surjective and $B \to A$ is an isomorphism on residue fields, the natural functor from $\Hom_{c\otimes}(\C,\M(A' \times_A B))$ to
\[\Hom_{c\otimes}(\C,\M(A')) \times_{\Hom_{c\otimes}(\C,\M(A))} \Hom_{c\otimes}(\C,\M(B))\]
is an equivalence of groupoids.
\item For every complete local noetherian $R$-algebra $(A,\mathfrak{m})$ the canonical functor
\[\Hom_{c\otimes}(\C,\M(A)) \to \lim_n \Hom_{c\otimes}(\C,\M(A/\mathfrak{m}^n))\]
is an equivalence of groupoids.
\item Automorphisms, deformations and obstructions for $\SPEC(\C)$ are bounded, constructible and Zariski-local.
\end{itemize}
It would take too long to elaborate here what the last condition means in terms of $\C$. Let us explain for example what boundedness of deformations and automorphisms means. Let $A$ be a finitely generated $R$-algebra without zero divisors, let $s : \C \to \M(A)$ be an $R$-linear cocontinuous tensor functor. Define a category $\mathsf{Def}_\C(s)$ whose objects are $R$-linear cocontinuous tensor functors $\overline{s} : \C \to \M(A[\e]/\e^2)$ with an isomorphism $\overline{s} \bmod \e \cong s$. The set of isomorphism classes of this category is actually an $A$-module. This is required to be coherent. Besides, the $A$-module of automorphisms in $\mathsf{Def}_\C(s)$ of the trivial extension $s[\e]$ is required to be coherent.
\end{rem}
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Updated: Nov 4, 2021
Debt Relief: What It Means
Estimated read time: 7 minutes
The total household debt in America reached $12.96 trillion, a new record since 2008.
For those whose debt has reached a point where it seems impossible to pay it all off, debt relief programs are an option to help avoid bankruptcy and make payments at an attainable level.
There are a lot of debt relief companies promising to help people get out of debt via debt settlement, a method of negotiating with creditors to reduce the amount you have to repay.
Unfortunately, those who are under the pressure of uncontrolled debt are also vulnerable to scams that present the illusion of resolving their problems, but just leave victims in a worse situation.
For instance, the government sued more than 30 companies in October for a student debt relief scam.
While debt relief companies can help you with negotiating settlements and arranging a payment plan, you need to be careful when deciding which debt relief service to trust with your financial future.
Understanding how debt relief is supposed to work and what are the hallmarks of a scam are key to getting on the right path and avoiding the wrong one.
Here are the things to look for and questions to ask yourself before signing on with one of these companies.
Is the.
Has anyone made consumer complaints.
Know what federal rules apply to debt relief companies.
Whenever you work with a debt settlement company, don't be afraid to ask to read the fine print.
Don't pay money until you understand what the money is for and until you are sure it is legal.
If you find that debt settlement doesn't work for you, there are other options like consumer credit counseling, debt consolidation, or as a last resort, bankruptcy.
Every possible solution has benefits and risks that you need to evaluate and weigh before you decide which path is right for you.
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OpenGL.NET is a C# binding for OpenGL. The wrapping is done by a developed wrapper- generator, which is part of this project. DoxyGen is used to create XML representations of the GL header files.
Tweet this project Short link
Changes: This release included a binding for the NVidia-Cg toolkit and some minor bugfixes.
A software forge that is lightweight and extensible.
Connect Atlassian dev tools (JIRA, Bamboo, Crucbile, FishEye) to IntelliJ IDEA.
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Association Studies of BMI and Type 2 Diabetes in the Neuropeptide Y Pathway
A Possible Role for NPY2R as a Candidate Gene for Type 2 Diabetes in Men
A Possible Role for NPY2R as a Candidate Gene for Type 2 Diabetes in Men
Abstract
The neuropeptide Y (NPY) family of peptides and receptors regulate food intake. Inherited variation in this pathway could influence susceptibility to obesity and its complications, including type 2 diabetes. We genotyped a set of 71 single nucleotide polymorphisms (SNPs) that capture the most common variation in NPY, PPY, PYY, NPY1R, NPY2R, and NPY5R in 2,800 individuals of recent European ancestry drawn from the near extremes of BMI distribution. Five SNPs located upstream of NPY2R were nominally associated with BMI in men (P values = 0.001–0.009, odds ratios [ORs] 1.27–1.34). No association with BMI was observed in women, and no consistent associations were observed for other genes in this pathway. We attempted to replicate the association with BMI in 2,500 men and tested these SNPs for association with type 2 diabetes in 8,000 samples. We observed association with BMI in men in only one replication sample and saw no association in the combined replication samples (P = 0.154, OR = 1.09). Finally, a 9% haplotype was associated with type 2 diabetes in men (P = 1.73 × 10−4, OR = 1.36) and not in women. Variation in this pathway likely does not have a major influence on BMI, although small effects cannot be ruled out; NPY2R should be considered a candidate gene for type 2 diabetes in men.
- CEPH, Centre d'Etude du Polymorphisme Humain
- FHS, Framingham Heart Study
- NHLBI, National Heart, Lung, and Blood Institute
- NPY, neuropeptide Y
- PYY, polypeptide YY
- SNP, single nucleotide polymorphism
Obesity, as measured by BMI, is an important predictor of type 2 diabetes, cardiovascular disease, cancer, and death (1–5). Although environmental factors influence the rising tide of obesity, genetic factors strongly influence obesity; the heritability of BMI within individual populations is ∼30–70% (6–8). Identifying the underlying genetic causes of obesity could provide valuable insights into the pathways that are relevant in patients and help guide the development of more effective preventive measures and therapies. In addition, genes that influence susceptibility to obesity may also contribute to the common sequelae of obesity, such as type 2 diabetes.
Variants in over 120 genes have been reported to be associated with measures of obesity (9), but few of these associations have been repeatedly reproduced with convincing statistical evidence (9) (H.N.L., J.N.H., personal communication). A few exceptions are rare variants in the MC4R receptor in early-onset obesity (10–13) and possibly an association with common variation near INSIG2 (14). As one approach to identify genes contributing to obesity in the general population, we are testing common genetic variation in candidate genes implicated by physiological and genetic studies in humans and model organisms.
Several lines of evidence suggest that the genes encoding the neuropeptide Y (NPY) family of peptides and receptors are good candidates for association studies with obesity (15,16). NPY is a potent stimulus of food intake (15,16), primarily through binding the Y1 and Y5 receptors (16). Two related peptides, pancreatic polypeptide (PP) and polypeptide YY (PYY), inhibit food intake (16). PPY3–36 inhibits food intake by binding to Y2 receptor (16). Suggestive evidence for linkage has been seen for the regions containing NPY (NPY), PYY (PYY), PPY (PP), NPY1R (NPY Y1 receptor), NPY2R (NPY Y2 receptor), and NPY5R (NPY Y5 receptor), although none of these linkages have been consistently reproduced (9). Finally, genetic association studies have implicated these genes in obesity or type 2 diabetes, with P values in the range of 0.001–0.05 (17–23), suggesting that common variation in these genes may be involved in obesity in humans. Larger sample sizes are needed to determine if these results represent true associations.
Because of the considerable biological connection to appetite control and the suggestive genetic data, we sought to comprehensively study a common variation in the genes in the NPY pathway for association with BMI. We selected tag SNPs that capture the majority of common variation in these genes (24–26). We genotyped these tag SNPs in multiple large samples to survey common variation in these genes for association with BMI, and we tested the most associated variants for association with type 2 diabetes.
RESEARCH DESIGN AND METHODS
First, we used reference panels to determine the patterns of linkage disequilibrium and choose tagging SNPs. Second, tag SNPs and multimarker haplotypes comprised of tag SNPs were tested in two screening panels: European-American and Polish subjects. SNPs and haplotypes were tested in the full panels and in men and women separately, based on an a priori hypothesis that effects on BMI or related traits could be sex dimorphic. We tested SNPs and haplotypes for nominal association with BMI in the European-American panel (two-tailed P < 0.05) and then tested these SNPs and haplotypes for replication in the Polish sample (one-tailed P < 0.05). SNPs that met these criteria or comprised haplotypes that met these criteria were genotyped in the replication samples (Framingham unrelated, Scandinavian unrelated, Scandinavian trios, GCI African American, and Maywood African American). In the replication samples, only the specific hypotheses suggested by the screening samples were tested. Specifically, we observed an association in men only, so we only analyzed men from the replication studies. The SNPs carried forward into the BMI replication panels were also tested for association with type 2 diabetes.
Consent.
All subjects gave informed consent, and the project was approved by the institutional review board of Children's Hospital (Boston, MA).
Reference panels.
The European-derived reference sample consists of 93 individuals in 12 multigenerational pedigrees (Centre d'Etude du Polymorphisme Humain [CEPH]) representing 96 independent chromosomes, as previously described (27). The African-American reference panel is comprised of 50 unrelated individuals, as previously described (28).
Screening panels.
The European-American (1,218 case and 624 control subjects) and Polish (700 case and 330 control subjects) panels were obtained from Genomics Collaborative (Table 1), as previously described (14). These individuals were selected from a collection of >60,000 subjects and include healthy control subjects plus patients with osteoarthritis, rheumatoid arthritis, asthma, hypertension, coronary artery disease, myocardial infarction, hyperlipidemia, stroke, type 2 diabetes, or osteoporosis. We determined the BMI distribution in healthy individuals for each decade of life, sex, and country of origin (U.S. or Poland). We designated any subjects from the original set of 60,000 with a BMI between the 90th and 97th percentile of the described distribution as potential obese case subjects and designated any subjects with a BMI between the 5th and 12th percentiles as potential lean control subjects. From this set of obese and lean individuals, a subset of obese and lean individuals was selected by matching for age, sex, and grandparental region of origin. Collectively, these samples will be referred to as the screening panels. There was no significant effect on the odds ratios (ORs) by limiting the analysis to those from the healthy control subjects.
BMI replication panels.
The panel from the Framingham Heart Study (FHS) contains 1,739 unrelated individuals, from which we only analyzed data for the 847 men (Table 1). These individuals are drawn from the offspring cohort, which is the second generation of a longitudinal study of the general population of Framingham, Massachusetts. Height and weight were measured on six separate occasions from 1971 to 1998, as described elsewhere (29,30). We focused on the exam 6 data because BMI was most heritable in this exam cycle (30).
The Scandinavian parent-offspring trios (218 male offspring) are from the Botnia Study conducted in Finland and Sweden; the offspring were ascertained as nondiabetic individuals with waist-to-hip ratios in the upper quintile or lower decile (31). In addition, from this panel and the Scandinavian panels described below, we constructed a separate, nonoverlapping sample of 977 unrelated men to test for association with BMI, consisting of the male control subjects, nondiabetic fathers in parent-offspring trio panels, and a single nondiabetic male sibling from each sibship.
Two African-American panels were studied. The first consists of self-described African-American men born in the U.S. (192 obese and 71 lean). This panel is identical in design to the European-American and Polish case-control panels described above and was also collected by Genomics Collaborative. The second panel consists of African-American individuals from Maywood, Illinois, and contains 866 individuals in nuclear families and sibships and 186 unrelated individuals, as described elsewhere (14). For analytical purposes, we constructed a panel of unrelated men (108 obese and 111 lean) from this sample by taking all the unrelated men and one man from each nuclear family in the top and bottom quartiles of age-adjusted log(BMI).
Type 2 diabetes panels.
The panels used for association studies of type 2 diabetes (Table 2) have been described elsewhere (32,33) and include a panel of 321 Scandinavian trios with offspring with type 2 diabetes, impaired glucose tolerance, or impaired fasting glucose levels (166 male offspring) and 1,189 Scandinavian individuals from discordant sibpairs (280 affected men); these two panels are collectively referred to as Scandinavian related. A case-control study with 942 Scandinavian subjects matched on age, BMI, and geographic region (252 affected men and 254 unaffected men); a Swedish case-control study with 1,028 subjects matched on age and BMI (267 affected men and 267 unaffected men); and a case-control study with 254 subjects from the Saguenay Lac-St. Jean region in Quebec, Canada (70 affected men and 54 unaffected men), were also tested. The European-American and Polish diabetes panels were drawn from the same cohorts as the European-American and Polish BMI panels, as described elsewhere (32,33). The European-American panel includes 2,452 subjects (644 diabetic men and 644 men with normal glucose tolerance). The Polish panel includes 2,018 subjects (422 diabetic men and 422 men with normal glucose tolerance).
Genotyping.
All genotyping was performed using the mass spectrometry–based MassArray platform (Sequenom) (27,34). Primers and probes were designed using SpectroDesigner (Sequenom). Assays were multiplexed (maximum 7-plex) and PCR performed in 6 μl with 5 ng of DNA, 0.6 pmol of each primer, 1.2 nmol of dNTP, and 0.2 units of Taq DNA polymerase (Qiagen) in 1.5× PCR buffer (Qiagen) and 1 mmol/l MgCl2. PCR conditions are as previously described (27). Extra dNTPs were inactivated using 0.3 units of shrimp alkaline phosphatase. Primer extension was performed with 6 pmol of probe for each assay, 5.2 nmol of appropriate termination mix, and 0.64 units of Thermosequenase (Sequenom).
Tag SNP selection.
The European-American CEPH reference panel and African-American reference panels described above were used to assess the patterns of linkage disequilibrium and select tag SNPs because this study began before the release of HapMap (34). The density of coverage and extent of linkage disequilibrium in these samples are similar to those described by HapMap (data not shown).
For each locus, SNPs were chosen from dbSNP and the Celera database to cover the entire gene region, as well as ∼20 kb upstream and 10 kb downstream (Supplementary Tables 1 and 2, which can be viewed in an online appendix at). The density of attempted SNPs was 1 per 1 kb. PPY and PYY lie 10 kb apart, so we characterized these genes together. NPY1R and NPY5R are 14 kb apart, so we also characterized these genes together. After observing an association in NPY2R, we genotyped all reported SNPs from dbSNP. SNPs were included in the analysis of linkage disequilibrium if the allele frequency was >5%, the genotyping success rate was >85%, there was a maximum of one apparent inheritance (Mendelian) error, and the genotypes were in Hardy-Weinberg equilibrium (P > 0.01). The average spacing of working polymorphic SNPs (frequency >1%) is 1.3 kb for NPY, 1.5 kb for PPY and PYY, 560 bp for NPY2R in CEPH, 490 bp for NPY2R in the African-American reference panel, and 1.9 kb for NPY1R and NPY5R. The overall genotype success rate across all panels was 96.4% for polymorphic working markers.
We were unable to analyze the NPY4R/PPYR1 gene because all SNPs tested failed quality control in a manner strongly suggestive of a polymorphic duplication, the presence of which has been confirmed (35,36).
We selected tag SNPs using Tagger (26). Tag SNPs were chosen so that the minimum r2 was >0.8 for all SNPs with frequencies >5%. Additional tag SNPs were genotyped because they were previously selected using the algorithm implemented in Haploview (37). Tag SNPs for all genes were initially genotyped in the screening panels. For NPY2R, SNPs that were predictive of the associated haplotypes (r2 > 0.8) were also genotyped in the screening panels.
To maximize comparability across populations, tag SNPs in NPY2R were chosen for the African-American panels using Tagger by first including all the tag SNPs from the European-American reference panel. Then, we picked extra tag SNPs so that all SNPs >5% frequency in the African-American panel were captured with a minimum r2 > 0.8. We selected all tags SNPs in the region of linkage disequilibrium with the BMI-associated SNPs in either European American or African American subjects and genotyped them in the African-American study samples.
Data analysis.
In the screening panels, all analyses were performed in the total sample and in men and women separately. Because of the design of these panels, BMI was treated as a dichotomous trait, and all SNPs and haplotypes were tested under an allelic model using a χ2 test (1 df). For haplotype analyses, fully phased data were generated using PHASE, version 2.1 (38,39), for each haplotype block, defined using Haploview specifying the “solid spine of LD [linkage disequilibrium]” option (37). Within each block, we tested all haplotypes with frequency >5% for association against all others using a χ2 test (1 df).
To assess significance of our initial results, we permuted the case-control labels within the screening panels 1,000 times. For each permutation, we did Mantel-Haenszel tests of the screening panels for men, women, and the two sexes combined. For the 127 SNPs and haplotypes tested across the six genes, we recorded the best P value for each permutation. We observed an association better than the original result in 221 permutations, giving our data an empirical P value of 0.22 within the screening panels.
For the FHS panel, the unrelated Scandinavian men, Scandinavian trios, and Maywood African-American men, BMI data were available as a continuous measure. To increase the normality of the BMI distributions, we analyzed log(BMI). We created a z score for log(BMI) based on age and, in the FHS cohort, adjusted for smoking status. Specifically, z scores (difference from mean divided by SD) were calculated for each individual based on the means and SDs of the distributions of log(BMI) within each decade of life and sex for each population. In the Scandinavian population, we considered individuals from Botnia, Helsinki, and southern Sweden separately. The z scores were further corrected by regressing against age within each decade, with separate regressions for each sex and geographic population. We performed linear regression to test the association of the age- and smoking-adjusted score and genotype using SAS statistical software (SAS Institute, Cary, NC). We used a multivariate family-based association test using generalized estimating equations, as implemented in PBAT (40), to analyze age-adjusted score as a continuous trait in the Scandinavian trios. To analyze BMI as a dichotomous trait in these samples, we defined the top and bottom quartiles of age-adjusted log(BMI) score as obese and lean and used a χ2 test for association; for the parent-offspring trios, we used the TDTQ4 test (41).
Population stratification.
To assess stratification, we genotyped 128 random SNPs (42,43) in subsamples of the European-American (238 case and 130 control subjects) and Polish (254 case and 114 control subjects) BMI case-control studies. From the 105 SNPs that passed quality control in the European-American panel, we estimate a mean χ2 value of 1.22 and median χ2 value of 0.63 (a mean χ2 of 1.0 and a median χ2 of 0.45 are expected when there is no stratification). Comparing the observed distribution of χ2 values with the distribution expected with no stratification gives a P value = 0.059, suggesting there may be mild stratification in this sample. We saw no evidence for stratification from the 113 SNPs tested in the Polish case-control study (mean χ2 = 0.99 and 0.36; P = 0.50).
RESULTS
To characterize the patterns of common genetic variation in NPY pathway genes, we genotyped 26 SNPs in NPY, 28 SNPs in PPY and PYY, 84 SNPs in NPY2R, and 54 SNPs in NPY1R and NPY5R in a reference sample of 12 European-derived multigenerational pedigrees (CEPH) (27) and 95 SNPs in NPY2R in a panel of 50 unrelated African-American subjects (28) (Supplementary Tables 1 and 2). Using these data, we selected 11 tag SNPs in NPY, 14 tag SNPs in PPY and PYY, 26 tag SNPs in NPY2R, and 26 tag SNPs in NPY1R and NPY5R to capture the underlying common variation; polymorphic missense SNPs were also included.
The tag SNPs were genotyped in the European-American and Polish case-control studies (Table 1, screening panels). Although nominal associations were observed in the NPY, PYY, PPY, NPY1R, and NPY5R genes (Supplementary Tables 3 and 4), only five SNPs in a region upstream of NPY2R showed nominal association with BMI with the same allele in both screening panels (see research design and methods for criteria). The linkage disequilibrium between these five SNPs is high (r2 = 0.46–0.95). The association was observed in men but not in women (Table 3 and Supplementary Table 3). Meta-analysis by Mantel-Haenszel test (44) of the two samples yielded P values between 0.001 and 0.009 and ORs between 1.27 and 1.34. A multimarker haplotype comprised of the associated alleles of these SNPs was similarly associated (frequency 26–32%, P = 0.002, OR = 1.34, and 95% CI 1.11–1.61) (Table 3). We have assessed the significance of the original association by permuting the affected status 1,000 times; we calculated an experiment-wide P value of 0.22 for our data (see research design and methods).
Because we observed mild evidence for stratification in the European-American sample (see research design and methods), we assessed the allele frequencies of one associated SNP, rs11099992, across multiple European populations. Allele frequencies ranged from 0.25 to 0.44, roughly trending from west to east. We rematched our European-American panel along this axis using previously described methods (42) and observed no decrease in association (P = 0.016, OR = 1.33 vs. P = 0.036, OR = 1.28 in the original sample), suggesting the association is not due to stratification in this sample.
Because we observed the strongest potential association for variation upstream of NPY2R and BMI in men, we focused our further replication efforts on this locus. We genotyped the seven SNPs in NPY2R that were associated with BMI in the screening panels or comprise the associated haplotype in a set of 1,739 unrelated individuals from the FHS, 1,018 unrelated Scandinavian men, and 437 Scandinavian parent-offspring trios. Analyzing age-adjusted log(BMI) as a quantitative trait, we found rs11099992 was associated with BMI in the Scandinavian men (one-tailed P = 0.03) but not associated in men from FHS or in the Scandinavian trios (Table 4).
To permit a combined analysis across all of the samples, we examined BMI as a dichotomous trait in the Scandinavian and FHS panels, assigning men in the bottom quartile of age-adjusted log BMI as control subjects and men in the top quartile as case subjects for each panel. The combined association in the European-derived replication samples using a Mantel-Haenszel test (44) does not reach significance for the high-BMI haplotype (one-tailed P = 0.27, OR = 1.06) (Table 5).
Tag SNPs for NPY2R were also tested in two African-American studies. No SNP or haplotype was associated with BMI in these panels (Table 5). However, we are unable to compare these results to the haplotype-based tests in the European-derived samples because the linkage disequilibrium structure is quite different (data not shown).
Because obesity is an important risk factor for type 2 diabetes, we considered the possibility that variation in this gene also influences the risk of type 2 diabetes. We genotyped the same seven SNPs upstream of NPY2R in samples discordant for type 2 diabetes, including >3,000 previously described Scandinavian subjects, 2,400 European Americans, 2,000 Polish subjects, and a panel of 250 French Canadian individuals (32,33) (Table 2). Surprisingly, a 9% haplotype was associated with type 2 diabetes in men only (P = 1.73 × 10−4, OR = 1.36; for women, P = 0.42, OR = 1.07) (Table 6). Through permutation testing (see research design and methods), we obtained a gene-wide corrected P value of 0.02 for this association.
DISCUSSION
We performed an extensive survey of common genetic variation in the peptides and receptors of the NPY pathway for association with BMI. We observed no reproducible association in NPY, PPY, PYY, NPY1R, or NPY5R in two large studies of individuals sampled from the extremes of the BMI distribution, although modest associations at these loci, or associations that are strongly influenced by gene-gene or gene-environment interactions, cannot be ruled out. In the case of NPY2R, we observed an association in both studies between variation upstream of this gene and BMI in men, although we were not able to significantly replicate this result in further panels. Because of our initial result with BMI, we examined the association between variation upstream of NPY2R and type 2 diabetes. Importantly, the association we observed between NPY2R and diabetes is not a replication of our initial BMI result because the phenotype is different and a different haplotype is most strongly associated with diabetes.
Published studies provide a small amount of support for an association between variation upstream of NPY2R and BMI in men. A prior study of NPY2R and BMI in the Pima Indian population (100 obese and 67 lean men) (23) found a trend (P = 0.13, OR = 1.39) toward association with rs2880412 (r2 = 0.86 to rs11099992), although the linkage disequilibrium relationship between these SNPs could be different in the Pima Indian population. A recent study motivated by a prepublication abstract of our results also found an association (P = 0.02, OR = 1.24) with BMI upstream of NPY2R in 6,000 Danish men and women; a stronger result was observed in men compared with women (45). However, this study did not test the SNP or haplotype most strongly associated with type 2 diabetes in our samples. A study by Ma et al. (23) and another by Hung et al. (22) (420 men) have reported that a silent SNP in NPY2R, rs1047214, is associated with BMI in men, although Lavebratt et al. (46) (500 men) observed an association of BMI with the alternate allele. In our samples, this SNP is not significantly associated with BMI (P = 0.657, OR = 0.97) or type 2 diabetes (P = 0.128, OR = 0.93).
We have attempted to estimate the likelihood that the associations with BMI and type 2 diabetes are valid by setting a prior probability for this pathway. We have previously estimated that appropriate prior probabilities for variants in good candidate genes (such as the NPY pathway genes for BMI) range from 0.0003 to 0.003 (47). For type 2 diabetes, these genes would be not considered as strong candidates, giving a range of prior probabilities of 8 × 10−5 (considering these genes as no better than random) to 0.001 (considering these genes as interesting candidates). Using these values and methods described previously to estimate false-positive report probabilities (48), the posterior probabilities that the association with BMI is valid range from 0.02 to 0.19, assuming the OR observed is close to the true genetic effect. These probabilities suggest that this association with BMI is likely spurious. For type 2 diabetes, the posterior probability that this association is valid ranges from 0.04 to 0.87. These probabilities suggest that this association could also be spurious, but NPY2R should be considered a candidate gene to be tested in further samples for association with diabetes in men.
In theory, the different design of our screening and replication samples could have contributed to the inability to replicate an association with BMI. Our screening samples included subjects from the near extremes of the BMI distribution, whereas our replication samples included subjects from the full BMI distribution. The design of the screening sample is very powerful for detecting variants that shift the trait distribution by a constant amount but can also detect variants that influence whether subjects are above or below a threshold (49,50). By contrast, the replication samples are less well powered on a per-sample basis, which we tried to account for by testing several large replication samples. Also, continuous trait data would be much less powerful if the associated variants were to have a threshold effect. To more closely mimic the design of the screening samples, we analyzed the replication samples with BMI as a dichotomous rather than a continuous trait; however, when we compared the top and bottom quartile or reproduced the exact sampling scheme of the screening panels, we still observed no significant association in the combined replication panels (Table 5 and data not shown).
In summary, we have conducted a comprehensive study to look for variants in the NPY pathway that influence BMI and to test for involvement of associated variants with type 2 diabetes. We have presented data that genetic variation in NPY pathway is not an important contributor to BMI, although small effects cannot be excluded. Association of NPY2R with type 2 diabetes in men should be tested in further studies. Our study is the first to observe an association between NPY2R and type 2 diabetes in humans. It has been shown that the diabetic phenotype of the obese ob/ob mouse is reduced when Npy2r has been deleted even though the double-knockout animals show no decrease in body weight (51), lending some biological plausibility to our preliminary finding.
Panels for BMI association studies
Patient samples used to test association to type 2 diabetes
An initial association of NPY2R with obesity in men from the screening panels
Association of NPY2R SNPs to BMI analyzed as a continuous trait
Association of SNPs in NPY2R with BMI in men in all samples
Association of SNPs and haplotypes in NPY2R with type 2 diabetes in male case-control subjects
Acknowledgments
This work was supported by the American Diabetes Association/Smith Family Pinnacle Program Project Grant. The Botnia Study has been funded by grants from the Sigrid Juselius Foundation, the Academy of Finland, the Folkhälsan Research Foundation, the Swedish Research Council, and the Finnish and Swedish Diabetes Research Foundations. X.Z. and R.S.C. are supported in part by a grant, R01HL074166, from the National Heart, Lung, and Blood Institute (NHLBI). The Framingham Heart Study is conducted and supported by the NHLBI in collaboration with Boston University. This manuscript has been reviewed by Boston University and NHLBI for scientific content and consistency of data interpretation with previous Framingham publications, and significant comments have been incorporated before submission for publication.
We thank Mark J. Daly and David Altshuler for insightful discussions and advice about this project and for helpful comments on the manuscript. We thank T. Bersaglieri for genotype data and analysis of PPYR1. We thank J. Butler for technical assistance and members of the laboratories of J.N.H., M.J.D., and D.A. for helpful discussions. We thank the Framingham Heart Study (NHLBI) and the Botnia Study group for providing DNA samples.
Footnotes
Published ahead of print at on 26 February 2007. DOI: 10.2337/db06-1051.
Additional information for this article can be found in an online appendix at.
K.G.A. is currently affiliated with the Biological Samples Platform, Broad Institute of the Massachusetts Institute of Technology and Harvard University, Cambridge, Massachusetts.
J.N.H has received samples from Genomics Collaborative/Seracare LifeSciences as part of a collaboration, and L.C.G. has been a consultant for and served on the advisory boards of Sanofi-Aventis, Bristol-Myers Squibb, and GlaxoSmithKline. 27, 2006.
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\begin{document}
\maketitle
\begin{abstract}
Using the recently developed groupoidal description of Schwinger's picture of Quantum Mechanics, a new approach to Dirac's fundamental question on the role of the Lagrangian in Quantum Mechanics is provided. It is shown that a function $\ell$ on the groupoid of configurations (or kinematical groupoid) of a quantum system determines a state on the von Neumann algebra of the histories of the system. This function, which we call {\itshape q-Lagrangian}, can be described in terms of a new function $\mathcal{L}$ on the Lie algebroid of the theory. When the kinematical groupoid is the pair groupoid of a smooth manifold $M$, the quadratic expansion of $\mathcal{L}$ will reproduce the standard Lagrangians on $TM$ used to describe the classical dynamics of particles.
\end{abstract}
\section{Introduction: The Lagrangian in Quantum Mechanics}
In this letter a new ``classical'' approximation to the dynamics of a quantum system will be discussed. It will take advantage of the recently developed groupoid description of quantum mechanical systems emanating from the seminal work by J. Schwinger (see, for instance \cite{Ci19a} - \cite{Ci20b} and references therein).
There have been many different ways of addressing the relation between the quantum properties of a given system and its emergent classical description. In many cases this task takes the form of a ``quantization'' program, that is, guessing the quantum structure of the system from its classical description; in spite of its interest, the only really meaningful question is indeed the converse one: how does the classical structure emerge from the quantum one?
At the moment, there is not a clear answer to this question beyond the original proclaims of the founding fathers of Quantum Mechanics. N. Bohr coined the ``correspondence principle'', that acknowledges the relation between the quantum properties of a system and its classical description \cite[pages 23-24 and 27-28]{Bo20}: \textit{``The process of radiation cannot be described on the basis of the ordinary theory of electrodynamics, according to which the nature of the radiation emitted by an atom is directly related to the harmonic components occurring in the motion of the system, there is found, nevertheless, to exist a far-reaching correspondence between the various types of possible \underline{transitions} between the stationary states on the one hand and the various harmonic components of the motion on the other hand.... This is equivalent to the statement that, when the quantum numbers are large, the relative probability of a particular \underline{transition} is connected in a simple manner with the amplitude of the corresponding harmonic component in the motion. This peculiar relation suggests a general law for the occurrence of \underline{transitions} between stationary states.'' }
P.A.M. Dirac took a different, and in a sense, much more subtler approach to this question by asking what was the role of the Lagrangian in quantum mechanics \cite{Di33}. He himself provided a meaningful insight by relating the Lagrangian to the generating function of canonical transformations of the theory, hence opening the door to subsequent developments brought by R. Feynman and Schwinger, independently. Each one of them provided different answers to Dirac's conundrum. Feynman kept the classical Lagrangian $L$ but changed in a drastic way the dynamical description of the theory by providing an explicit expression for the transition amplitudes introducing its path integral approach \cite{Fe48}, while Schwinger opted to change the notion of the Lagrangian function, replacing it with an operator-valued distribution $\mathbf{L}$, and giving a quantum variational principle description for the dynamics \cite{Sc91}. In both approaches Bohr's correspondence principle was overridden by the new powerful ideas and relegated to introductory courses on the subject. The saddle point approximation allows to extract from Feynman's path integral approach the classical trajectories of the system, however the problem persists, as the construction of the path integral uses a classical Lagrangian function $L$. Where does this Lagrangian come from?
In this note we will propose a new interpretation for the Lagrangian of a quantum theory as a function $\ell$ defined on the groupoid $K$ describing the kinematics of the quantum system. Such Lagrangian function would allow us to define, on one side, a Feynman-like description of the dynamics of the system by constructing a particular state on a von Neumann algebra of observables of the system and, on the other, a Schwinger-like description of the dynamics of the system by means of a variational principle. In this Letter, we will focus on the implications of the choice of such a q-Lagrangian function $\ell$ regarding the problem discussed above, that is, showing how a natural ``quasi-classical'' description of the dynamics of the theory emerges using the infinitesimal description of the groupoid $K$ supporting the theory, that is, its Lie algebroid, leaving the dynamical interpretation of the q-Lagrangian $\ell$ to future papers.
Lie groups are groupoids with a single-element base space, and their Lie algebroids are nothing else than the Lie algebra of the (Lie) group. Hence, the Lie algebroid description of the dynamics is the natural counterpart of the Lie algebraic description of a dynamics on a group.
The Lie algebroid of a given Lie groupoid can be integrated by using a natural extension of the exponential map, and this map allows us to define a Lagrangian function $\mathcal{L}$ on the Lie algebroid starting from a q-Lagrangian $\ell$ on the Lie groupoid. The function $\mathcal{L}$ will be called the c-Lagrangian of the system and it would provide a natural ``classical description'' of the dynamics of the quantum system.
Lagrangian dynamics on Lie algebroids have been studied in different contexts and their corresponding Euler-Lagrange equations are well understood (see for instance \cite{We96}, \cite{Ma01} and the review \cite{Co06}). In this Letter it will be shown how the second order approximation to such quasi-classical Lagrangian function allows us to reproduce the well-known Lagrangians found in the description of point particles.
\section{Quantum systems and groupoids}
On its most basic terms a quantum system is characterised by the outcomes $x,y,\ldots \in \Omega$ of a family of observables, and by a family of \textit{transitions} $\alpha \colon x \to y$ experienced by the system, whose interpretation is that if the observable $A$ were measured right before the observed transition took place, the outcome would have been $x$, and if measured again right after the transition had taken place, the result would have been $y$. The outcome $x$ of the transition $\alpha \colon x \to y$ will be called its source and the outcome $y$ its target. The natural axioms satisfied by the family of all possible transitions are those of a groupoid (see \cite{Ci19a} for more details). In particular, transitions compose in a natural way: the symbol $\beta \circ \alpha$ denotes the transition resulting from the occurrence of the transition $\beta$ right after the first transition $\alpha$. Two transitions $\alpha$, $\beta$ can be composed only if the target of the first coincides with the source of the second (note the backwards notation for composition): in this case they are said to be composable, and such a composition law is associative. There are unit elements, that is, transitions $1_x\colon x \to x$ such that they do not affect the transition $\alpha \colon x \to y$, when composed on the right, i.e., $\alpha \circ 1_x = \alpha$, or on the left, $1_y \circ \alpha = \alpha$, and whose physical interpretation is that the system remains unchanged during the observation. Finally, the fundamental property that implements Feynmann's principle of microscopic reversibility \cite[page 3]{Fe05}, that is, for any transition $\alpha \colon x \to y$, there is another one, denoted $\alpha^{-1} \colon y \to x$, such that $\alpha^{-1} \circ \alpha = 1_x$ and $\alpha \circ \alpha^{-1} = 1_y$. The collection of all transitions satisfying the previously enumerated properties is called a (algebraic) groupoid $K$ with space of objects (called in what follows ``outcomes'') $\Omega$. The map $s \colon K \to \Omega$ assigning to the transition $\alpha \colon x \to y$ the initial outcome $x$ is called the source map, and the map $t \colon K \to \Omega$ assigning to $\alpha$ its final outcome $y$ is called the target map. In the following, we will always consider $K$ and $\Omega$ to be at least locally-compact, Hausdorff topological spaces for which $s,t$ , as well as $x\mapsto 1_{x}$, $\alpha\mapsto\alpha^{-1}$, and $(\beta,\alpha)\mapsto\beta\circ\alpha$, are continuous maps. When $K$ and $\Omega$ are smooth manifolds and all the maps are smooth (in particular, $s,t$ are smooth submersions), we say that the groupoid is a {\itshape Lie groupoid}.
The previous notions provide a natural mathematical setting to Schwinger's `algebra of selective measurements' \cite{Sc91} introduced to provide an abstract setting to the foundations of atomic physics. Transitions could also be understood in terms of the basic quantum mechanical notion of probability amplitudes because a unitary representation of the given groupoid will associate to them a family of operators directly related to the notion of probability amplitudes or `transition functions' in Schwinger's terminology \cite{Ci19}.
It is also possible to conceive of a groupoid as an abstraction of a certain experimental setting used to describe the properties of a given system. For instance, if we consider a charged particle moving on a certain region where detectors have been placed, the triggering of them will correspond to the possible outcomes of the system and the sequence of such triggerings would be the transitions of the system. Another possible interpretation is offered by earlier descriptions of spectroscopic data. Actually, as A. Connes suggested \cite{Co94}, the Ritz-Rydberg combination principle of frequencies in spectral lines is rightly the composition law of a groupoid (in this case of a simple groupoid of pairs).
In what follows, we will look at the groupoid used to describe a certain quantum system as a kinematical object, which means that transitions and outcomes represent just kinematical information obtained from the system without dynamical content, that is, no specific dynamical law is associated to their description. In this sense, we will say that the groupoid $K$ is a kinematical groupoid. It will also be called the groupoid of ``configurations'' of the system, and it is associated to a specific experimental setting for a (quantum mechanical) system\footnote{Consider, for example, an electron whose motion is analysed through bubble chambers, or through Stern-Gerlach devices or through two-slits walls: these settings result in specific (and different from the others) kinematical groupoids.}.
In order to focus on a specific situation that will be of particular interest for the purposes of the present letter, consider the groupoid of configurations of a point particle moving in a region $\Omega$. The outcomes resulting from determining the position of the particle will provide coordinates $x^k$ for the points of the region $\Omega$ with respect to some reference system and the transitions of the system will be pairs $ \alpha = (y,x)\colon x \to y$ of consecutive detections. In this case, the natural groupoid that should be used to describe the system will be just the collection of all pairs $(y,x)$, $x,y\in \Omega$, with composition law $(z,y) \circ (y,x) = (z,x)$. Note that units $1_x$ are diagonal pairs $(x,x)$, and the inverse of the transition $(y,x)$ is $(x,y)$. The resulting groupoid is called the {\itshape pair groupoid} of $\Omega$ and is denoted by $P(\Omega)$. The region $\Omega$ carries, in general, a natural notion of distance established by the setting prepared by the experimenter. We may even assume that such distance is given by a metric $ds$ on $\Omega$ (determined, for instance, by the presence of a gravitational field). Thus, in what follows, we will assume that $(\Omega, \eta)$ is a Riemannian manifold\footnote{We will not deal in this letter with the relativistic descriptions of particles, and address such a problem to a forthcoming paper.} with metric $\eta = \eta_{kl} \, dx^k\, dx^l$.
We can even consider a slightly more ambitious setting where, together with the position, the spin of the particle can be measured. This would imply that a system of orthogonal vectors $e_a(x)$ will be selected at each point $x$, providing an orientation for the Stern-Gerlach-like apparatus used for the task, and each possible transition would amount to a determination of how these vectors are rotated, that is, the transitions of the system will be determined by linear isometries $T_{yx}$ from the tangent space at the point $x$ to the tangent space at the point $y$ to the manifold $\Omega$, that is $T_{yx}^* \eta_y = \eta_x$, or in local coordinates: $(\eta_y)_{ij} (T_{yx})_k^i (T_{yx})_l^j = (\eta_x)_{kl}$. In classical terms, we may also think that we are following the possible transitions of a rigid body in the domain $\Omega$. The family of all transitions $T_{yx} \colon x \to y$, constitutes a groupoid $R(\Omega)$ with the natural composition law given by composition of linear maps: $(T_{zy} \circ T_{yx})_a^b = (T_{zy})_a^c (T_{yx})_c^b$.
\section{Lagrangians and groupoids}
We will take the point of view that the dynamical description of a quantum system is provided by a particular class of states that capture Dirac's insight into the role of the Lagrangian on quantum theories. The rationale behind this proposal lies in the fact that, given a groupoid $K$ in a large class of groupoids, we may associate a von Neumann algebra $\nu(K)$ to $K$ representing the algebra generated by the (bounded) observables (i.e., self-adjoint elements in $\nu(K)$). This algebra comes as the closure, in an appropriate topology, of a suitable convolution algebra of functions on the groupoid, and will be called the (von Neumann) algebra of the groupoid. In particular, we may follow Connes' theory of integration on groupoids \cite{Co78,Ka82} and define the convolution operation $\star$ on integrable functions on $K$ by setting
$$
(f \star g) (\alpha) = \int_{K^y} f(\gamma) g(\gamma^{-1}\circ \alpha) d \nu^y (\gamma ) \,
$$
where $K^y$ denotes the set of all transitions $\alpha \colon x \to y$, with fixed target $y$, and $\nu^y$ the conditional measure along $K^y$ induced by a measure $\nu$ on $K$. The conjugation operation $f \mapsto f^*$ is defined as $f^*(\alpha) = \overline{f(\alpha^{-1})} \Delta(\alpha^{-1})$ where $\Delta$ is the modular function on $K$, that is $\int f(\alpha) d\nu (\alpha) = \int f(\alpha^{-1}) \Delta (\alpha^{-1}) d\nu (\alpha^{-1})$ for all integrable $f$. Then, $\nu(K)$ is the weak (or strong) closure of the algebra of integrable functions (closed with respect to the convolution product) represented on the Hilbert space $\mathcal{H}=L^{2}(K,\nu)$ by means of the operators $\left(T_{f}(\Psi)\right)(\alpha)\,:=\,\left(\Delta^{\frac{1}{2}}f\right)\star \Psi(\alpha)$ with $\Psi\in\mathcal{H}$.
By a state of the system described by the groupoid $K$ we mean a state of the von Neumann algebra $\nu(K)$ of observables of the system, that is, a positive normalized linear functional $\rho$ on $\nu (K)$. A convenient way of describing states is by means of functions of positive type on the groupoid itself. Suppose that $\varphi$ is a complex-valued function defined on the groupoid $K$ such that
$$
\int_{K} (f^* \star f) (\alpha ) \varphi (\alpha ) d \nu (\alpha) \geq 0 \, ,
$$
for all integrable functions $f$ on $K$. Then, it can be shown that the functional
$$
\rho_\varphi (f) = \int_{K} f(\alpha) \varphi (\alpha) d\nu (\alpha) \, ,
$$
defines a state on the von Neumann algebra $\nu(K)$ provided that the normalization condition $\int_G \varphi (\alpha) d\nu (\alpha) = 1$ holds (see \cite{Ci21} for more details).
A particularly interesting class of states are those defined by functions of positive type satisfying a reproducing property (such states were introduced for the first time in \cite{Ci19c}). It can be shown that if $\varphi$ has the form
$$
\varphi (\alpha ) = \sqrt{p(x) p(y)} e^{ \frac{i}{\hbar} \mathscr{S}(\alpha)} \, ,
$$
for any $\alpha \colon x \to y$, where $p$ is a probability density on $\Omega$, $\hbar$ is a physical constant introduced to make adimensional the argument of the exponential, and the function $\mathscr{S} \colon K \to \mathbb{R}$ satisfies the $\log$-like properties
$$
\mathscr{S}(\alpha \circ \beta ) = \mathscr{S}(\alpha) + \mathscr{S}( \beta ) \, , \qquad \mathscr{S}(\alpha^{-1}) = - \mathscr{S}(\alpha) \,,
$$
then $\varphi$ is of positive type and thus determines a state on $\nu (K)$. We will call such states Dirac-Feynman states, while the function $\mathscr{S}$ will be called an action functional.
A natural way of constructing an action functional $\mathscr{S}$ consists in considering the groupoid of histories associated to a given groupoid. In other words, given the groupoid of configurations $K$ of a quantum system, we may consider the space of absolutely continuous paths $w \colon [t_0,t_1] \to K$. Such paths can be composed in a natural way. If $w' \colon [t_1,t_2] \to K$ is another path, then $w' \circ w \colon [t_0,t_2] \to K$ is the path that takes the values $w(t)$ if $t_0 \leq t \leq t_1$ and $w'(t)$ for $t_1 \leq t \leq t_2$, with matching condition $w (t_1) = w'(t_1) $. Such space of curves can be groupoidified adding the formal inverses of all paths, that is, if $w \colon [t_0,t_1] \to K$ is a path, then $w^{-1}$ is another path, formally described as a map $w^{-1} \colon [t_1,t_0] \to K$, and such that $w^{-1}(s) =w(s)^{-1}$ (note the opposite orientation of the interval $[t_1,t_0]$ with respect to $[t_0,t_1]$). The resulting groupoid is called the groupoid of histories $G(K)$ of the configuration groupoid $K$ and its elements are called histories on the groupoid $K$. Given an integrable function $\ell \colon K \to \mathbb{R}$, we can define the action functional $\mathscr{S}$ on the groupoid of histories $G(K)$ as
$$
\mathscr{S}(w )= \int_{t_0}^{t_1} \ell (w(s)) ds \, ,
$$
for any history $w$ in $G(K)$. Notice that the invariance under the ``time-reversal'' operation $\tau\colon \alpha \mapsto \alpha^{-1}$ of the function $\ell$, a property that will be always assumed in what follows, implies that $\mathscr{S}(w^{-1}) = - \mathscr{S}(w)$.
We can summarise the discussion so far by saying that the choice of a $\tau$-invariant function $\ell$ on the groupoid of configurations $K$ of a quantum system allows to define in a natural way a Dirac-Feynman state on the groupoid of histories $G(K)$. We call the function $\ell (\alpha)$ the {\itshape q-Lagrangian} of the theory in accordance with Dirac's terminology of q-numbers and c-numbers. We must point out at this stage that, even if the q-Lagrangian $\ell$ is an ordinary real-valued function on $K$, it also defines an element of (or is affiliated to) the von Neumann algebra of $K$ when acting on $\mathcal{H}$ by means of the convolution product, hence, it also has a non-ambiguous non-commutative character.
Let us illustrate the previous ideas with a simple example. Consider the groupoid of histories of a quantum system described by the groupoid of pairs of the Euclidean 3-space, $E(\mathbb{R}^3) = \{ (\mathbf{y}, \mathbf{x}) \mid \mathbf{x}, \mathbf{y} \in \mathbb{R}^3\}$. In such case, the groupoid of histories can be identified with the groupoid defined by the family of paths $\gamma \colon [t_0,t_1] \to \mathbb{R}^3$, and a quantum Lagrangian would be given by the function $\ell \colon E(\mathbb{R}^3) \to \mathbb{R}$, $\ell (\mathbf{y}, \mathbf{x}) = \frac{1}{2} || \mathbf{y} - \mathbf{x}||^2$.
\section{The classical Lagrangian}
Once a q-Lagrangian $\ell$ on the kinematical groupoid $K$ is chosen, we will see that its ``infinitesimal counterpart'' on the Lie algebroid of $K$ determines a sort of ``classical Lagrangian'', here also called c-Lagrangian, in a sense that will be clear later.\footnote{We are not applying Schwinger's quantum dynamical principle to the q-Lagrangian $\ell$ in order to describe its associated quantum dynamics. This will certainly constitute the subject of subsequent work.}
As in the case of Lie groups, a Lie groupoid has an infinitesimal description. In the case of a Lie group $G$, its infinitesimal description is given by its Lie algebra $\mathfrak{g}$, defined either as the space of right (or left) invariant vector fields on $G$ or, equivalently, as the space of tangent vectors at the identity element. Because of Lie's third theorem the original Lie group $G$ (actually its universal simply connected covering) can be recovered from its Lie algebra $\mathfrak{g}$, and a natural exponential map $\exp \colon \mathfrak{g} \to G$ exists, that agrees with the standard exponential in the case of groups of matrices. Similar notions hold in the case of Lie groupoids. If $K \rightrightarrows \Omega$ is a Lie groupoid, we can consider the space of right invariant vector fields on $K$. Since the right translation $R_\alpha \colon s^{-1}(y) \to s^{-1}(x)$, $\alpha \colon x \to y$, $R_\alpha (\beta) = \beta \circ \alpha$ maps the space of transitions starting at $y$ into the space of transitions starting at $x$, a right-invariant vector field $X$ must be tangent to the fibres of the source map $s$. Moreover, as in the case of Lie groups, such vector fields $X$ turn to be uniquely determined by its values $X_x$ at the units $1_x$, $x \in \Omega$. Hence we define the Lie algebroid of the Lie groupoid $K$ as the bundle consisting of all $\xi_x$ tangent to the fibres $K_x$ of the source map $s \colon K \to \Omega$ at the outcomes $x \in \Omega$. Denoting such collection of tangent vectors as $A(K)$, we have that for each $x\in \Omega$, $\xi_x \in A(K)_x$, if $\xi_x$ is the tangent vector at $x$ of a curve $\gamma (s) \in K$, $\gamma(0) = x$, whose source is $x$ for all $-\epsilon < s < \epsilon$.
The particular instances of the Riemann groupoid $R(\Omega) $ and the groupid of pairs $P(\Omega)$ provide an excellent illustration of this notion. In the case of the groupoid of pairs $P(\Omega)$, if we fix $x\in \Omega$, then the set of the transitions whose source is $x$ are given by all pairs $(y,x)$, $y \in \Omega$. Hence, the space of tangent vectors at the unit $1_x = (x,x)$ is given by all tangent vectors $\xi_x \in T_x\Omega$, i.e., tangent vectors to curves $(\gamma (s), x)$, $\gamma (0) = x$. Thus, the Lie algebroid $A(P(\Omega))$ can be identified with the tangent bundle $T\Omega$ over $\Omega$ and the anchor map is the identity map $\mu = \mathrm{id}_\Omega \colon T\Omega \to T\Omega$ (see below). In the case of the Riemann groupoid $R(\Omega)$, the space of transitions with fixed source $x$ is given by the collection of all linear maps $T_{yx}$ mapping the linear tangent space $T_x \Omega$ isometrically into $T_y\Omega$. Hence a curve $\gamma (s)$ on $R(\Omega)_x$ can be described as a family of linear maps $T_{x(s)x}$ from $T_x\Omega$ to $T_{x(s)}\Omega$. For small values of $s$, we can select a smooth frame $\{ \sigma^a(s) \}$, $-\epsilon < s < \epsilon$, in such a way that the maps $T_{x(s)x}$ can be identified with a pair of curves $(x(s), R(s))$ with $x(s)$ a curve in $\Omega$ passing through $x$ and $R(s)$ is a curve of orthogonal matrices. Then, the tangent vector $\xi_x$ to such a curve is just a pair consisting of a tangent vector $v_x$ and a skew-symmetric matrix $S_x$ and the Lie algebroid $A(R(\Omega))$ of the Riemann groupoid is a bundle of Lie algebras over the tangent bundle of the manifold $\Omega$.
The space of all cross-sections $\xi \colon \Omega \to A(K)$, $\xi (x) \in A(K)_x$, of the Lie algebroid $A(K)$, carries a natural Lie algebra structure given by $[\xi , \zeta]_x = [X^\xi, X^\zeta](x)$, where $X^\xi$ denotes the right invariant vector field whose values at $x$ are $\xi(x)$ (with $x \in \Omega$), while the bracket on the right hand side of the previous expression is the standard Lie bracket of vector fields. Finally, the Lie algebroid $A(K)$ carries a natural map (the anchor) $\mu \colon A(K) \to T\Omega$, given by the differential of the target map $t \colon K \to \Omega$ acting upon vectors $\xi_x$. The bracket $[\cdot, \cdot ]$ defined before satisfies, in addition to the Jacobi identity, the derivation property
\begin{equation}\label{eq:anchor}
[\xi , f \zeta] = f [\xi , \zeta] + \mu_\xi (f) \zeta \, ,
\end{equation}
where $f$ is an arbitrary smooth function on $\Omega$ and $\mu_\xi := \mu (\xi)$ is a vector field on $\Omega$. As in the case of Lie algebras, a Lie algebroid $A(K)$ can be characterised by families of structure functions obtained as follows. Let $\sigma^a$ be a local family of linearly independent cross-sections of $A(K)$ and $x^k$ local coordinates on $\Omega$. Then, consider the coefficients $C_{ab}^c$ and $\mu_a^k$, defined as
\begin{equation}\label{eq:structure}
[\sigma_a, \sigma_b] = C_{ab}^c \sigma_c \, , \qquad \mu(\sigma_a) = \mu_a^k \frac{\partial }{\partial x^k} \, ,
\end{equation}
that satisfy the non-linear differential equations
$$
\sum_{\mathrm{cyclic\, \, } a,b,c} C_{ad}^e \, C^d_{bc} + \mu_a^k\, \frac{\partial C_{bc}^e}{\partial x^k} = 0 \, , \qquad
\mu_a^k\, \frac{\partial \mu_b^j}{\partial x^k} - \mu_b^k \, \frac{\partial \mu_a^j}{\partial x^k} - C_{ab}^c \, \mu_c^j = 0 \, .
$$
We must point out that all previous notions hold independently whether the bundle $A(K) \to \Omega$ has been constructed starting from a Lie groupoid or not. In fact, we may define a Lie algebroid $A \to \Omega$ as a vector bundle such that its space of cross-sections carries a Lie algebra bracket $[\cdot, \cdot]$, and there is a map $\mu \colon A \to T\Omega$ satisfying (\ref{eq:anchor}).
Although an exponential map can be defined for a general Lie algebroid (more precisely, for its Lie algebra of sections), we concentrate here on a simpler situation where an actual exponential map is constructed for the Lie algebroid $A(K)$ itself. The natural way to do that is by considering an auxiliary $A$-connection $\nabla$. As in the standard calculus with covariant derivatives, given a Lie algebroid $A$, an $A$-connection is a bilinear map $\nabla \colon \Gamma (A) \times \Gamma (A) \to \Gamma (A)$ such that $\nabla_{f\xi} \zeta = f \nabla_\xi \zeta$, and $\nabla_\xi (f \zeta) = f \nabla_\xi \zeta + \mu_\xi (f) \zeta$. If $\sigma_a$ denotes a local basis of cross sections for $A$, the $A$-connection $\nabla$ is characterised by
a family of functions $\Gamma_{ab}^c$, defined as $\nabla_{\sigma_a} \sigma_b = \Gamma_{ab}^c \sigma_c$.
As in the case of standard connections, a full covariant calculus can be developed (see \cite{Cr02} for details).
Given an $A$-connection $\nabla$, we say that the curve $\gamma (s)$ on $A$ such that $\mu (\gamma (s)) = \dot{x}(s)$, with $x(s)$ the projection of the curve on $\Omega$, and $\gamma (0) = x$, is a $\nabla$-geodesic if $\nabla_{\dot{\gamma}} \dot{\gamma} = 0$. Note that the $\nabla$-geodesic $\gamma(s)$ is determined uniquely by its value $x$ at $s = 0$, and the tangent vector $\xi_x = \dot{\gamma}(0)$. Actually if we write the curve $\gamma (s) = \gamma^a (s) \sigma_a (x)$, then the coefficients $\gamma^a$ satisfy the system of differential equations
$$
\frac{d \gamma^a}{ds} + \Gamma_{bc}^a (x(s)) \gamma^b (s) \gamma^b (s) = 0 \, , \qquad \frac{d x^k}{ds} = \mu_a^k (x(s)) \gamma^a (s) \, .
$$
Turning back our attention to the situation where the Lie algebroid $A$ is the Lie algebroid $A(K)$ of a Lie groupoid $K$, fixing an $A(K)$-connection $\nabla$, we define the exponential map $\mathrm{Exp} \colon A(K) \to K$ as $\mathrm{Exp} (x, s\xi_x ) = \gamma_{\xi}(s)$,
where $\gamma (s)$ is the unique $\nabla$-geodesic such that $\gamma (0) = x$ and $\dot{\gamma}(0) = \xi_x$. There is a caveat though with respect to the case of the exponential map for Lie groups. Contrarily to the situation with Lie groups, the exponential map $\mathrm{Exp}$ is not defined for all values of the parameter $s$. Actually, the map $\mathrm{Exp}^\nabla$ is a diffeomorphism on a neighborhood of the space of outcomes (or units) of the theory. Then, the exponential of the tangent vector $\xi_x$ will be given by $\mathrm{Exp}(x, \xi_x) = \gamma_{\xi_x}(1)$, provided that $\xi_x$ lies in the neighborhood where the geodesic $\gamma_\xi(s)$ is defined for $s=1$ (see \cite{Ma05} for details).
Given the above analysis, it makes sense now to consider the c-Lagrangian associated to the q-Lagrangian $\ell$ as the map $\mathcal{L}$ defined on (an open neighbourhood of the zero section of) the Lie algebroid $A(K)$ by
\begin{equation}\label{eq:classicalL}
\mathcal{L} (x, \xi_x) = \ell \left(\mathrm{Exp} (x, \xi_x/c_K ) \right) \, ,
\end{equation}
and $c_K$ is a constant with the dimensions of $\xi_x$. In addition, the constant $c_K$ will fix a radius on $A(K)$ such that for all $\xi_x$ whose ``size'' is smaller than $c_K$ the exponential map would be defined.
If the Lie algebroid $A$ carries a metric along its fibres, say $\eta_x = \eta_{ab}(x) \sigma^a \otimes \sigma^b$, there is a canonical $A$-connection $\nabla^\eta$ associated to it, characterised by the conditions of being torsionless and leaving the metric $\eta$ invariant. Such $A$-connection coincides with the standard Levi-Civita connection in the case of the groupoid of pairs $G(\Omega)$. We will assume that the Lie algebroid $A(K)$ of the groupoid $K$ carries a right-invariant metric $\eta$ and the corresponding Levi-Civita $A$-connection $\nabla^\eta$.
Once a Lagrangian $\mathcal{L}$ is defined on the Lie algebroid $A(K)$ we may describe the dynamics associated to it given by the Euler-Lagrange equations
$$
\frac{d}{dt} \frac{\partial \mathcal{L}}{\partial \xi^a} + \frac{\partial \mathcal{L}}{\partial \xi^c} \, C_{ab}^c\, \xi^b - \mu_a^k \, \frac{\partial \mathcal{L}}{\partial x^k} = 0 \, , \qquad \dot{x}^k = \mu_a^k \, \xi^a \, .
$$
The analysis of this dynamics, introduced by A. Weinstein, has been the subject of intense scrutiny by E. Martinez and others, and has found a wide range of applications (see \cite{Ma01}, \cite{We96} for details). We leave the discussion of the relation between the dynamics of the c-Lagrangian $\mathcal{L}$ and the original q-Lagrangian $\ell$ to forthcoming papers and focus on the structure of the c-Lagrangian in both general and specific instances.
Hence, if the kinematical groupoid of the theory $K$ has a natural scale $c_K$ on it, provided that we consider only dynamics with $|| \xi_x || << c_K$, we can approximate the c-Lagrangian $\mathcal{L}$ by truncating its Taylor expansion around the space of outcomes of the theory. In this sense, we may write the following expansion of $\mathcal{L}$ around the region $\Omega \subset K$
$$
\mathcal{L}(x, \xi_x) = \mathcal{L}(x, 0_x) + \frac{\partial \mathcal{L} }{\partial \xi^a}(x,0_x) \xi^a + \frac{1}{2} \frac{\partial^2 \mathcal{L} }{\partial \xi^a \xi^b} (x,0_x) \xi^a \xi^b + \mathrm{h.o.t.}
$$
Taking into account the structure of the function $\mathcal{L}$, Eq. (\ref{eq:classicalL}), we get
\begin{equation}\label{eq:expansion}
\mathcal{L}(x, \xi_x) = \ell (x, x) + \frac{1}{c_K} \frac{\partial_1 \ell }{\partial x^k}(x,x) \mu_a^k (x) \xi^a + \frac{1}{2c_K^2} \frac{\partial_1^2 \ell} {\partial x^k \partial x^l} (x,x) \mu_a^k \mu_b^l \xi^a \xi^b + O(1/c_K^3) \, ,
\end{equation}
where $ \frac{\partial_1 \ell }{\partial x^k}(x,x)$ denotes the derivative of $\ell$ with respect to the first variable $y$ at the point $(x,x)$.
If we consider the expansion up to second order in (\ref{eq:expansion}), we get a quadratic function $L$, now defined on the whole Lie algebroid $A(K)$
\begin{equation}\label{eq:L}
L(x, \xi_x) = \frac{1}{2} {\eta}_{ab} (x) \xi^a \xi^b + A_a(x) \xi^a - V(x) \,.
\end{equation}
In this expression $\eta= \frac{\partial_1^2 \ell} {\partial x^k \partial x^l} (x,x) \, \mu_a^k \, \mu_b^l \, \sigma^a \otimes \sigma^b$ is a quadratic form along the fibres of the Lie algebroid $A(K)$, the linear term reads $A = A_a\, \sigma^a = \frac{1}{c} \frac{\partial_1 \ell }{\partial x^k}(x,x) \, \mu_a^k \, \sigma^a$, and $V(x) = - \ell (x,x)$. We say that the c-Lagrangian $\mathcal{L}$ (and the q-Lagrangian $\ell$) is regular if the quadratic form $\eta$ above is non-degenerate. In such a case, it defines a metric on the Lie algebroid. Hence, if the q-Lagrangian $\ell$ is regular, there is a natural metric $\eta_\ell$ associated with it that can be used to construct the exponential map that would allow to define the second order approximated Lagrangian $L$.
Clearly, the structure of the second order approximation $L$ is reminiscent of the standard form of a Lagrangian describing the motion of a charged point particle in a gravitational field in the presence of an electromagnetic field (see for instance \cite{Ca95} for a thorough discussion on Feynman's inverse problem and the conditions that guarantee that a Lagrangian takes the previous form).
We end up the discussion of the classical description of the dynamics of a quantum system by briefly analysing a family of q-Lagrangians which are both natural and physically meaningful. We start by considering the groupoid of pairs of a Riemannian manifold $\Omega$ with metric $\eta$, intended to describe the observed transitions of a particle, an electron for instance, in the region $\Omega$. There is a natural function on the pair groupoid $P(\Omega) $ which is the two-point function $\ell (y,x)$ given by
$$
\ell (y,x) = \inf_{\gamma \colon (x,s_0) \to (y,s_1)} \frac{1}{2} \int_{s_0}^{s_1} || \dot{\gamma} (s) ||^2 ds \, ,
$$
where $\gamma \colon (x,s_0) \to (y,s_1)$ is any absolutely continuous curve $\gamma \colon [s_0,s_1] \to \Omega$, such that $\gamma (s_0) = x$ and $\gamma (s_1) = y$. Note that in the particular instance of the $n$-dimensional Euclidean space, then $\ell (y,x) = \frac{1}{2} || y -x ||^2$, as in the example at the end of Sect. 3. The product of the previous Lagrangian times a constant $mc_K^2$ has the physical dimensions of an energy and constitutes the natural candidate for the q-Lagrangian of a free particle moving on the Riemannian manifold $\Omega$.
In this situation, the Levi-Civita connection $\nabla$ associated to the metric $\eta$ is the natural choice to construct the exponential map $\mathrm{Exp} \colon T\Omega \to \Omega \times \Omega$, that is, $\mathrm{Exp}(x, s\xi_x) =(\exp_x(s\xi_x), x)$, where $\exp$ denotes the standard exponential map (see \cite{La85} for details). Note that the c-Lagrangian $\mathcal{L}$ (\ref{eq:classicalL}) is a function on (a tubular neighborhood of the zero section of the) tangent bundle $T\Omega$ while its second order approximation (see Eq. \eqref{eq:L}) is a standard Lagrangian function. A few computations show that the Lagrangian $\mathcal{L}$ associated to the q-Lagrangian $\ell$ has the form
$$
\mathcal{L}(x, v) = m c_K^2 \ell \left(\mathrm{Exp}(x, v / c_K) \right) = \frac{1}{2} m \, \eta_{kl}(x) v^k v^l + O(1/c_K^3) \, ,
$$
which coincides with the standard Lagrangian describing the geodetic motion of a particle on $\Omega$.
\section{Conclusions and discussion}
A new approach to analyse the question raised by Dirac on the role of the Lagrangian function in Quantum Mechanics is presented; it is based on the groupoidal formulation of Schwinger's algebraic description of Quantum Mechanics. It has been shown that a choice of a function $\ell$ on the kinematical groupoid describing a quantum system, called a {\itshape q-Lagrangian}, emerges from a state in a particularly interesting class defined by functions of positive type satisfying a reproducing property. This q-Lagrangian becomes a natural candidate to determine the dynamics of the corresponding quantum system according to Schwinger's variational principle. The function $\ell$, even if it is an ordinary real-valued function on the groupoid of the system, defines an (affiliated) element of the von Neumann algebra of the groupoid, showing in this way its true non-commutative origin in agreement with Schwinger's notion of quantum Lagrangian.
If the kinematical groupoid is a Lie groupoid, one may exploit the associated Lie algebroid to provide an infinitesimal description of the q-Lagrangian that brings in a classical-like flavour. In the particular instance that the q-Lagrangian $\ell$ is regular, there is a canonical exponential map that allows to translate the q-Lagrangian function $\ell$ into a c-Lagrangian $\mathcal{L}$ defined on (a tubular neigborhood of the space of outcomes of) the Lie algebroid of the theory. Hence a classical-like dynamics is provided by the corresponding Euler-Lagrange equations. Such dynamical behaviour would only be defined, in principle, for ``small velocities'' and would only account for a restricted description of the full quantum dynamics. It is shown that the quadratic approximation to the c-Lagrangian $\mathcal{L}$ produces a function on the Lie algebroid whose explicit form is reminiscent of the standard Lagrangian for particles moving on a gravitational background under an electromagnetic field.
In the particular instance of the groupoid of pairs of a Riemann manifold that would provide the natural setting to describe the free motion of a particle on a curved background, the q-Lagrangian $\ell$ is defined to be the natural two-point function on the Riemannian manifold, and the fourth-order approximation of the associated c-Lagrangian is the standard Lagrangian describing the geodetical motion of a classical particle on $\Omega$. Further examples of interest, like the Riemann groupoid suitable to take into account the spin of particles and the corresponding groupoid for a space time $\mathscr{M}$ with Lorentzian metric $\eta$, will be discussed elsewhere.
The analysis of the quantum dynamics determined by a q-Lagrangian $\ell$ will be the subject of forthcoming articles and the relation with Lagrangian dynamics on Lie algebroids will be discussed in detail. We also hope to show that this setting is the appropriate one to address Dirac query that starts from the observation that each solution of Hamilton-Jacobi equation (corresponding to one state of motion in the quantum theory) gives rise to a family of solutions of Hamilton's equations \cite{Di51}, so presumably the family has some deep significance in nature, not yet properly understood. In particular, the problem of reconstructing the quantum dynamics from the classical one will be analysed from the perspective of the integration of the Lie algebroid of a given groupoid \cite{Cr03}. This program can be considered as a natural ``quantization'' program in the groupoidal setting for Quantum Mechanics.
\section*{Acknowledgements}
We acknowledge the support provided by the MINECO research project MTM2017-84098-P and QUITEMAD++, S2018/TCS-A4342, the financial support from the Spanish Ministry of Economy and Competitiveness through the Severo Ochoa Programme for Centres of Excellence in RD(SEV-2015/0554), and the support provided by the Santander/UC3M Excellence Chair Programme 2019/2020. It is a pleasure for us to thank the Istituto Nazionale di Fisica Nucleare (INFN) and the Gruppo Nazionale di Fisica Matematica (INDAM), Italy. L.S. would like to thank the support provided by Italian MIUR through the Ph.D. Fellowship at Dipartimento di Matematica R.Caccioppoli.
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HazMatLab is a Frankfurt-based collective, including the artists Sandra Havlicek, Tina Kohlmann and Katharina Schücke. With an emphasis on process-based working practices, the collective is focused on material research and experimentation, trying to push boundaries in cross-sections between chemistry, biology, art and scientific imagination. All three artists studied at Städelschule in Frankfurt with different professors, and after graduating in 2016 they started collaborating as HazMatLab. After attending a nail art night in their studio, the artists took me through how they transformed an abandoned room in their building into a hidden lab filled with nail polish smells and curious slimes. This lab is where their work takes place: chemical and alchemical experiments with materials from cosmetics, technology or TCM. We talked about their recent solo exhibition “soothing efficacy” at 8. Salon in Hamburg and the importance of collaborative thinking processes in contemporary art practices.
Mine Kaplangı: I would like to start out by asking you for the reference of your collective's name. What does HazMatLab exactly stand for?
Katharina Schücke: Hazmat stands for hazardous material. In 2016 we were writing to nail polish labels abroad, asking them for sponsorship to our sculpture "Slumping No1". The answer we got: we would love to support you - but we cannot ship a large amount of nail polish, because it’s easily inflammable, it’s hazmat! We loved that term for dangerous goods and kept it as our name.
Sandra Havlicek: We follow a tradition we have created years ago, to meet and do our nails regularly. Our nail-sessions are always obsessive evenings. The table is stuffed with nail polish and tools to create the possibility to express any kind of mood on your nail. Through these sessions we were more and more focusing on technical questions –– the tiny space a nail provides, efficient drying, different tools to do a gradient, compatibility of different polish brands, reuse of worn down designs. Having a well-manicured nail was not our priority. We share this experimental approach as we began to expand our research from nail polish to raw and potentially sculptural materials in the beauty industry in general.
MK: When did you start collaborating as a trio?
Tina Kohlmann: Although our individual work is quite different, we liked the idea of doing a show together, developing something new together. The process eventually got so important to us, that we decided to continue collaborating. Our studio in the basement became our lab. It had a big table in the middle, while everything else was built around this working station. We added a cooking station for crystal growing, as well as a coating station and so on. Everything was highly functional. After three month we had to move –– our current lab is smaller and only suitable for some experiments on different crystals and for slime making.
MK: Last year you joined the residency program at IAa Art Space on Jeju Island in South Korea. Did these experiences influence your practice and research?
KS: We came to Jeju Island with the idea of visiting cosmetic labs, learning how they extract the agents from their natural ingredients. We realised how fundamentally cosmetic production is embedded in science and high technology. All processes are kept in secret and unfortunately, we couldn’t even arrange for a site visit.
SH: We specifically wanted to work on Jeju-Island, because the main ingredients of "K-Beauty" – Korean Beauty – have their origin on this island. "K-beauty" is now a worldwide phenomenon and no other beauty market knows how to combine efficiency and fun better. It has opened up a new realm of materials for us. Talking about snail mucus, pig collagen, bee venom, black basalt, starfish and jellyfish extract and also the several steps of cleansing rituals as a method. The insanity "K-Beauty" implies feels familiar and challenging at the same time. We wanted to touch all of these raw materials and study them in our lab.
KS: Jeju is a great island, with legends on their rich natural environment, amazing seafood and sweets. The staff at IAa was supportive, as we were basically busy 24/7. It was a very productive residency. We got cosmetic products like masks, sheet mask, lotions, tools to use for facial cleaning and more. Playing around with it in the studio, we decided to go out in nature to collect the actual ingredients of those products, such as basalt, corals, algae, rice or jellyfish. We learned how to do photogrammetric scans, so we could take all that material as digital 3D data back to Frankfurt. We also worked with a scanner on the highest resolution to be able to enlarge material without a microscope, to study the materials own quality, structure and so on.
TK: In Frankfurt it can be difficult to find the time to develop new ideas. Three people with three different schedules can be hard to combine. In the residency, instead, we really went for it –– it was an intense time. At IAa they have an apartment and a super nice studio set up in the city centre. Until then we worked mostly sculptural, on Jeju though, everything was a bit different and we brought back a lot of data instead of sculptures. It was a new way of working for us, a little bit outside of the box.
MK: I assume most of this research shaped the exhibition "soothing efficacy" that you had recently at 8.Salon in Hamburg?
KS: Yes. „soothing efficacy“ is a term from a Korean Beauty company that describes one of their cosmetic product goals. In the exhibition, we show our intermediate state of research in K-Beauty raw materials in the form of a large-scale photograph titled „Hero Ingredient (Corals)“. The extensive installation „Eco Soul Super Slim“ is a sculptural approach towards methods like soothing, slumping and layering. Taking the title of the exhibition literal, we have developed a recipe for slime to coat ten plinths that are hanging from the ceiling. The slime is dripping down steadily and spreading slowly onto the gallery floor. For the object „Slumping No.1“ we have used another cosmetic product as sculptural material, adding layer after layer of nail polish the object is grown to over 1500 layers so far.
SH: As Tina mentioned earlier, on Jeju we decided to transform our research and the materials we collected into a pile of digital scans that we could work with. We now have a huge archive of different scans. The work "Hero Ingredient (Corals)" is the prototype of a series of six, which shows a collection of raw materials used in K-Beauty. It’s mounted between two large glass panes, like an enlarged microscope slide, it's just pressed together by 4 clips. It took us a while to create this specific format. Although experts advised us against it, this experiment worked out super well! In "soothing efficacy", we show our on-going interest in materials gathered from the beauty industry and materials that don’t want to be formed, like our homemade slime.
© HazMatLab, Above: "Soothing efficacy", installation view, 8.Salon, 2019 // Below: "Hero Ingredient (Corals)", detail, 2019
MK: You are creating spaces that are not familiar to our everyday lives, and although these spaces are structured as installations, there is a performative and dynamic element in them. Like the dripping slime, moving colours in print images and of course the unbalanced position of the nail polish sculptures… How would you describe the relations between the media you choose and the concepts of your works and exhibitions?
SH: The medium plays a very active role in our work. And probably it is the autonomous behaviour of the materials that adds a performative feeling for you to our work.
TK: Slime just has its own mind. We try to control it, but in the end, it just does what it wants –– which is great. A big part of our work is this surprise moment of the unexpected. We expect the unexpected and then see, if we can handle it. I think it also helps not to have a super clear image in your head about what you want to create.
KS: We love to think of the material as the fourth mind in our collaboration. We definitely work with it rather than make something out of it. It’s allowed free bent, it might grow out of shape –– or not. It may overwhelm its origin and provoke with new meaning.
SH: I think the important background is our lab, where we set the structure for experimental strategies from the very beginning. Our lab is the basis: it changes depending on how and where we work, but it's always an empty space, setting up a structure and yet open to any kind of coincidence that will improve the lab. Basically, we set a frame and let the material move within that frame. Then we start again. Until we – the material and us – defined a form.
© HazMatLab, working station, 2016, photo by Wolfgang Günzel
MK: The first time I visited your studio, Donna Haraway's book "Staying with the Trouble Making Kin in the Chthuluce" was placed on your light table. After reading a few chapters I couldn’t stop thinking about how its chapter "Sympoiesis: Symbiogenesis and the Lively Arts of Staying with the Trouble" relates to your art-science practice. How do you think your practice and concept of making art is related to Haraway’s recent studies?
KS: This quote from the book could describe our approach: „A model is like a miniature cosmos, in which a biologically curious Alice in Wonderland can have tea with the Red Queen and ask how this world works, even as she is worked by the complex-enough, simple-enough world." I think we approach her visionary proposal to make kin, to relate symbiotically, her idea of response-ability in our artwork in a way that we often work in an open process.
SH: I think it is HOW we do works connects us to Donna Haraway's theories. She characterises terms like “sympoietical”, “tentacular thinking” and “response-ability”, which make much sense to describe our work. "Sympoietical" –– because we work in a symbiotic relationship with the material. “Tentacular thinking” –– Haraway relates "to the Latin 'tentaculum', meaning 'feeler', and 'tentare', meaning 'to feel' and 'to try'." The three of us work through feeling and trying, the feelers reappear in our work through the nail, the fingertip, the squid, the chicken feet. "Response-ability" –– Haraway contrasts with responsibility: she establishes an ethical principle for those unable to articulate their views through a rational, human voice, like animals, children and plants. Perhaps one can draw a parallel to our way of working with all senses, without a preconceived principle, while the outcome remains open. This is how we create a basis to enable a form of "response-ability". It is kind of a driving force in our artistic practice.
© HazMatLab, Slumping #1, detail, 2016
MK: There is something about your works that makes me smile involuntarily....
TK: ... especially the slime evokes certain emotions, which make it easy to connect with the work. These associations always fascinated us. The slimier, the better. The process of slime making is exhausting and also quite hilarious. I think people get that when they are in an exhibition.
KS: We get extremely excited while experimenting and have lot's of fun. Perhaps it is attached to the material.
SH: Besides the unconventional contents and materials we address, we employ directness. Sculptures combining casts of chicken feet and 99,99% pure bismuth, or a plinth that is overgrown by crystals, the lab, materials like nail polish and green tea mochi, are both - serious and offbeat.
MK: So you don't experiment through a set of rules, such as in science, but you follow the freedom of collective intuition thinking manifesting forms in your practice?
KS: I have the impression that collective work is often misunderstood, because common knowledge regards the artist as a singular brain and author. We were told once that the work of HazMatLab is unsellable, because it lacks this one clear identity. If we have a manifest it’s based on trust: the process comes instinctively.
SH: I totally agree. A collective is a fragile construct: there are more uncontrollable moments than working on your own. The uncertainty, which constitutes what we do, also implies failure.
© HazMatLab, Portrait, 2019, Photo by Jörg Burzinski
MK: What are you working on at the moment?
KS: A new polish mountain is in the making –– for two years we apply layer after layer of nail polish, it’s over 2000 layers now. We experiment with UV light to decrease drying time...because we want to try out what will happen if we work with nail polish on a larger scale.
SH: We’re also continuing our research about new lab-equipment, like cooking tools and professional mixing machines. Producing a larger amount of slime by hand is a lot of fun, but also hard muscle work. We liked the idea that we have machines running all day long producing slime for us 24/7. We want our own slime-factory! We also started to think about spatial questions for our upcoming group show “creating a ‘we’” at Basis Frankfurt. It will take place from August to September 2019, together with the artists Daniel Stubenvoll, Olga Cerkasova and Giuletta Ockenfuss. We plan to set up a spatial dramaturgy, for which we will start experimenting with different light situations.
TK: In China we made a sculpture out of grass jelly, which is a popular dessert with cooling effects and therefore especially nice during a hot summer. We got some of the raw material, dried leaves called "Platostoma palustre" to experiment with it and make another jelly sculpture from it. The first one collapsed after thirty minutes –– it was just too wobbly.
instagram.com/hazmatlab
instagram.com/hazmatlab
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GENEVA, June 11. /TASS/. The launch of national projects in Russia is intended to provide large-scale investments in human development and improving the well-being of citizens, Russian Prime Minister Dmitry Medvedev said on Tuesday at the International Labor Conference.
"Last year we launched 12 national projects, the plan for modernizing and expanding the backbone infrastructure. In fact, this is a large-scale investment in people, in their development," the head of the Russian government said.
The Russian government has published information on its website about all the 12 National Projects, which are going to be implemented up until late 2024. The data were compiled on the basis of documents and certification from the National Projects that were approved at a session of the presidium of the Presidential Council for Strategic Development and National Projects dated December 24, 2018.
The key areas of the country’s strategic development were established by President Vladimir Putin’s order of May 7, 2018. The document outlined 12 priority areas: the Digital Economy, the Ecology, Labor Productivity and Supporting Employment, International Cooperation and Export, Education, Culture, Small and Medium Businesses and Support for Business Initiatives, Healthcare, Demographics, Safe and High-Quality Roads, Housing and Urban Environment and Science. The work on these projects is aimed at "providing breakthrough scientific-technological and socio-economic development for Russia, increasing the standard of living, creating conditions and opportunities for personal fulfillment and unlocking every person’s talent," the government stressed. The budget of the National projects will total 25.7 trillion rubles ($398 bln).
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\begin{document}
\pagenumbering{arabic}
\title[On restricted Verma modules]{On the restricted Verma modules at the
critical level}
\author[]{Tomoyuki Arakawa, Peter Fiebig}
\thanks{T.A. is
partially supported
by the JSPS Grant-in-Aid for Scientific Research (B)
No.\ 20340007. P.F. is supported by a grant of the Landesstiftung Baden--W\"urttemberg }
\begin{abstract} We study the restricted Verma modules of an affine
Kac--Moody algebra at the critical level with a special emphasis on
their Jordan--H\"older multiplicities. Feigin and Frenkel
conjectured a formula for these multiplicities that involves
the periodic Kazhdan--Lusztig polynomials. We prove this conjecture
for all subgeneric blocks and for the case of anti-dominant simple
subquotients.
\end{abstract}
\maketitle
\section{Introduction}
The representation theory of affine Kac--Moody algebras at the
critical level is one of the essential ingredients in the approach towards the geometric
Langlands conjectures proposed by Beilinson and Drinfeld (cf.~
\cite{BeiDri96}). In particular, the correspondence between the center
of the (completed) universal enveloping algebra of an affine
Kac--Moody algebra at the critical level and the geometry of the space
of opers associated with the Langlands dual datum (cf.~\cite{FeiFre92}) is one of the main tools used in the construction of a
part of the Langlands correspondence in \cite{BeiDri96}.
\subsection{The local geometric Langlands conjectures}
In \cite{FreGai06} Frenkel and Gaitsgory formulated the {\em local
geometric Langlands conjectures}, which
relate the critical level representation theory of an affine Kac--Moody
algebra to the geometry of an affine flag manifold. In a
series of subsequent papers, the authors proved parts of these conjectures. In particular, in the paper \cite{FreGai07} a derived
equivalence between a certain category of $D$-modules on the affine
flag variety and a derived version of the affine category $\CO$ at the
critical level was constructed using a localization
functor of Beilinson--Bernstein type. It seems, however, hard to
control the action of this equivalence on the respective hearts of the
triangulated categories, and hence it is not yet possible to
deduce information on the simple critical characters of the category $\CO$
from the Frenkel--Gaitsgory result.
\subsection{The Andersen--Jantzen--Soergel approach}
In this paper we study the critical representation theory by a very
different method that is inspired by results of Jantzen
(cf.~\cite{Jan79}) and Soergel (cf.~\cite{Soe90}) in the case of finite dimensional complex Lie
algebras and of Andersen, Jantzen and Soergel in the case of modular Lie
algebras and quantum groups (cf.~\cite{AJS94}). The analogous results for the
non-critical blocks of $\CO$ for a symmetrizable Kac--Moody algebra
can be found in \cite{Fie06}.
The main idea of
this approach is to first describe the {\em generic} and {\em
subgeneric} blocks of the respective representation theory in as
much detail as possible, and then to deform an arbitrary block
in such a way that it can be viewed as an intersection of generic and
subgeneric blocks. This intersection procedure should then be
described using only some underlying combinatorial datum (like for
example the associated integral Weyl group).
As a result we hope to be able to
construct equivalences between various categories or links to
categories defined in topological terms in the framework of the
geometric Langlands program, and to deduce information on
the respective simple characters. More specifically, we hope to find a
correspondence between certain intersection cohomology sheaves on the
Langlands dual affine flag variety and critical representations.
This paper provides the first steps in the approach described above. Its main result is the calculation of the simple
characters in the subgeneric critical blocks. In order to explain our
result, let us consider the respective categories in some more detail.
\subsection{Critical representations of affine Kac--Moody algebras}
Let us denote by $\hfg$ an affine Kac--Moody algebra and by
$\hfh\subset\hfg$ its Cartan subalgebra (for the specialists it should
be noted here that we consider the central extension of a loop
algebra together with the grading operator). The one-dimensional
center of $\hfg$ acts semisimply on each module in the category $\CO$. Accordingly, each block of the category $\CO$,
i.e.~each of its indecomposable direct summands,
determines a central character. There is one such character, called
the {\em critical character}, which is distinguished in more than one
respect.
In this paper we focus on the following feature of the
critical blocks. Let $\hfhd$ be the dual space of the Cartan
subalgebra and denote by $\delta\in\hfhd$ the smallest positive
imaginary root. Then the corresponding simple highest weight module
$L(\delta)$ is of dimension one, and the tensor product
$\cdot\otimes_\DC L(\delta)$ defines a {\em shift functor} $T$ on
$\CO$ that is an equivalence. Now the critical blocks are exactly
those that are preserved by the functor $T$. This allows us to
consider, for each critical block, the
corresponding graded center $\CA$ (see Section \ref{sec-gradedcenter}).
The graded center is huge and intimately related to the center of the
(completed) universal enveloping algebra at the critical level, which
was determined in the fundamental work of Feigin and Frenkel
(cf.~\cite{FeiFre92}) (conjecturally, $\CA$ is a quotient of the
latter center). We
use the results of Feigin--Frenkel to describe the action of $\CA$ on
the Verma modules contained in the critical blocks.
\subsection{The restricted Verma modules}
Let $\Delta(\lambda)$ be the Verma module
with highest weight $\lambda\in\hfhd$. We define the {\em restricted
Verma module} $\rDelta(\lambda)$ with highest weight $\lambda$ as the quotient of $\Delta(\lambda)$ by
the ideal of $\CA$ generated by the homogeneous
constituents $\CA_n$
with $n\ne 0$. For our approach, the restricted Verma
modules, not the ordinary ones, should be considered as the ``standard
objects'' in the critical blocks.
We denote the irreducible quotient of $\Delta(\lambda)$ by
$L(\lambda)$. Results of Frenkel and
Feigin--Frenkel yield the characters of the restricted Verma modules,
and the knowledge of the character of $L(\lambda)$ for all $\lambda$
is equivalent to the knowledge of the Jordan--Hölder multiplicities
$[\rDelta(\lambda):L(\mu)]$ for all pairs $\lambda,\mu$ of critical
weights.
\subsection{The Feigin--Frenkel conjecture}
Let us choose a critical indecomposable block $\CO_\Lambda$ of $\CO$
and let us identify the index $\Lambda$ with the subset of highest
weights of the simple modules in $\CO_\Lambda$. Since $\Lambda$ is
critical we have $\lambda+\delta\in\Lambda$ if and only if
$\lambda\in\Lambda$.
Let $\hCW$ be the affine Weyl group associated with our data, and
$\CW\subset\hCW$ the finite Weyl group. The {\em integral Weyl group}
$\hCW(\Lambda)$ corresponding to $\Lambda$ is generated by the
reflections with respect to those real roots that satisfy a certain
integrality condition with respect to $\Lambda$. In the critical case,
$\hCW(\Lambda)$ is the affinization of the corresponding {\em finite integral Weyl group} $\CW(\Lambda)\subset\CW$.
We define $\lambda\in\Lambda$ to be {\em dominant} resp.~ {\em
anti-dominant} if it is
dominant resp.~ anti-dominant with respect to the action of
$\CW(\Lambda)$, i.e.~if it is the highest resp.~ smallest element
in its $\CW(\Lambda)$-orbit. We say that $\lambda$ is {\em regular} if it is
regular with respect to $\CW(\Lambda)$ (note that here we only refer to the finite integral Weyl group).
As mentioned before, $\hCW(\Lambda)$ is the affinization of
$\CW(\Lambda)$. In \cite{Lus80} Lusztig associated with a pair
$w,x\in\hCW(\Lambda)$ the {\em periodic polynomial}
$p_{x,w}\in\DZ[v]$ (in Lusztig's paper these polynomials were indexed not by affine Weyl group elements, but by alcoves, see Section \ref{subs-FFconj} for more details). The {\em Feigin--Frenkel conjecture} is the
following.
\begin{conjecture} Let $\Lambda$ be a critical equivalence class.
\begin{enumerate}
\item {\em The restricted linkage principle:} For
$\lambda,\mu\in\Lambda$ we have $[\rDelta(\lambda):L(\mu)]=0$
unless $\lambda$ and $\mu$ are contained in the same
$\hCW(\Lambda)$-orbit.
\item {\em The restricted Verma multiplicities:} Suppose that
$\lambda\in\Lambda$ is regular and dominant. Under some further regularity
conditions on $\Lambda$ (cf.~Section \ref{conj-FFC}), we have
$$
[\rDelta(w.\lambda):L(x.\lambda)]=p_{w,x}(1)
$$
for all $w,x\in\hCW(\Lambda)$.
\end{enumerate}
\end{conjecture}
The Feigin--Frenkel conjecture fits very well into a broader picture
that was anticipated by Lusztig in his ICM address in 1990 in Kyoto
(cf.~\cite{Lus91}). There, Lusztig conjecturally linked the
representation theory of modular Lie algebras, of quantum groups and
of critical level representations of an affine Kac--Moody algebra to
the topology of semi-infinite flag manifolds.
In \cite{AF2} we use the results of the present article in order to prove part (1) of the Feigin-Frenkel conjecture, the restricted linkage principle.
\subsection{Our main result}
Let $\fh\subset\hfh$ be the finite part of the Cartan subalgebra and let us denote by $\lambda\mapsto \ol\lambda$ the corresponding restriction map from $\hfhd$ to $\fhd$.
We call a critical class $\Lambda$ {\em subgeneric}, if its image $\ol\Lambda$ in $\fhd$ contains precisely two elements. Then we can define, following
\cite{AJS94}, a bijection $\alpha\uparrow\cdot
\colon\Lambda\to\Lambda$. Here, $\alpha$ denotes the unique positive
finite root with $\ol\Lambda=\{\ol\lambda,s_\alpha.\ol\lambda\}$.
Here is our result:
\begin{theorem}
\begin{enumerate}
\item If $\nu\in\Lambda$ is
anti-dominant, then we have for all
$\gamma\in\Lambda$
$$
[\rDelta(\gamma):L(\nu)]=
\begin{cases} 1 & \text{ if
$\gamma\in\CW(\Lambda).\nu$},\\
0 & \text{ otherwise}.
\end{cases}
$$
\item If $\Lambda$ is subgeneric, then we have for all
$\gamma,\nu\in\Lambda$
$$
[\rDelta(\gamma):L(\nu)] =
\begin{cases}
1 &\text{ if $\gamma\in\{\nu,\alpha\uparrow\nu\}$},\\
0 &\text{ otherwise}.
\end{cases}
$$
\end{enumerate}
\end{theorem}
The above theorem confirms the Feigin--Frenkel conjecture in the
respective cases. In \cite{AF2} we use it in order to describe the structure of the {\em restricted} category $\rCO_\Lambda$ completely for subgeneric $\Lambda$. As for generic $\Lambda$ the structure of $\rCO_\Lambda$ is easy to determine using the results of Feigin and Frenkel (see also \cite{Hay88,Ku89,Mal90,Mat96}, we completed the first part of the Andersen-Jantzen-Soergel approach towards the description of the category $\CO$ at the critical level.
\subsection{Acknowledgments} Both authors wish to thank the Emmy
Noether Center in Erlangen, where parts of the research for this paper
were done, for its hospitality and
support. The second author wishes to thank Nara Women's University for its hospitality and the Landesstiftung Baden--W\"urttemberg for supporting the project.
\section{Affine Kac--Moody algebras}
In this section we recall the fundamentals of the theory of affine Kac--Moody algebras. The main references are the
textbooks \cite{Kac90} and \cite{MP95}. Our basic data is a finite
dimensional simple complex Lie algebra $\fg$. We denote by
$k\colon \fg\times \fg\to \DC$ its Killing form.
\subsection{The construction of $\hfg$}
From $\fg$ we construct the (untwisted) affine Kac--Moody algebra
$\hfg$ as follows. We first consider the
loop algebra $\fg\otimes_\DC \DC[t,t^{-1}]$ for which the commutator is
the $\DC[t,t^{-1}]$-linear extension of the commutator of $\fg$. That
means that we have $[x\otimes t^n,y\otimes t^m]=[x,y]\otimes t^{m+n}$ for
$x,y\in\fg$ and $m,n\in\DZ$. The loop algebra has an up to isomorphism
unique non-split central extension $\tfg$ of rank one. As a
vector space we have
$\tfg=\fg\otimes_{\DC}\DC[t,t^{-1}]\oplus \DC K$, and the Lie bracket is given by
\begin{align*}
[K,\tfg] &= 0, \\
[x\otimes t^n,y\otimes t^m] &=[x,y]\otimes t^{m+n}+
n\delta_{m,-n}k(x,y) K
\end{align*}
for $x,y\in\fg$, $n,m\in\DZ$ (here $\delta_{a,b}$ denotes the
Kronecker delta). In the last step of the construction we add the
outer derivation operator $[D,\cdot]=t\frac{\partial}{\partial t}$ and get the affine Kac--Moody algebra $\hfg:=\tfg\oplus \DC D$
with the Lie bracket
\begin{align*}
[K,\hfg] & = 0, \\
[D,x\otimes t^n] & = n x\otimes t^n, \\
[x\otimes t^n, y\otimes t^m] & = [x,y]\otimes t^{m+n}+
n\delta_{m,-n}k(x,y) K
\end{align*}
for $x,y\in \fg$, $n,m\in \DZ$.
Let us fix a Borel subalgebra $\fb\subset \fg$ and a Cartan subalgebra
$\fh\subset \fb$. Then the corresponding Borel subalgebra $\hfb$ of $\hfg$ is given by
$$
\hfb := (\fg \otimes_\DC t\DC[t]+\fb\otimes_\DC \DC[t])\oplus \DC
K\oplus\DC D
$$
and the Cartan subalgebra $\hfh\subset \hfb$ is given by
$$
\hfh := \fh\oplus\DC K\oplus \DC D.
$$
\subsection{Affine roots} We denote by $V^\star$ the dual of a vector space $V$ and we write
$\langle\cdot,\cdot\rangle\colon V^\star\times V\to \DC$ for the
canonical pairing.
Let $R \subset \fhd$ be the set of roots of $\fg$ with respect to $\fh$. The projection $\hfh\to \fh$ along the decomposition $\hfh=\fh\oplus \DC K\oplus \DC D$ allows us to embed $\fhd$ inside $\hfhd$. In particular, we can view any $\alpha\in R$ as an element in $\hfhd$. Let us define $\delta,\Lambda_0\in \hfhd$ by
\begin{align*}
\langle\delta,\fh\oplus \DC K\rangle & = \{0\}, \\
\langle\delta,D\rangle & = 1, \\
\langle\Lambda_0,\fh\oplus \DC D\rangle & = \{0\},\\
\langle\Lambda_0,K\rangle & = 1.
\end{align*}
Then we have $\hfhd=\fhd\oplus\DC\Lambda_0\oplus\DC\delta$. The set $\hR\subset \hfhd$ of roots of $\hfg$ with respect to $\hfh$ is
$$
\hR=\{\alpha+n\delta \mid \alpha\in R,n\in\DZ\}\cup\{n\delta\mid n\in \DZ, n\ne 0\}.
$$
For $\alpha\in R$ let us denote by $\fg_\alpha\subset\fg$ the
corresponding root space. The root spaces of $\hfg$ with respect to $\hfh$ are
\begin{align*}
\hfg_{\alpha+n\delta} &= \fg_\alpha\otimes t^n\quad\text{for
$\alpha\in R$, $n\in\DZ$},\\
\hfg_{n\delta} &= \fh\otimes t^n\quad\text{for $n\in\DZ$, $n\ne 0$}.
\end{align*}
The subsets
\begin{align*}
\hR^{\real} &:=\{\alpha+n\delta \mid \alpha\in R, n\in\DZ\},\\
\hR^{\imag} &:= \{n\delta\mid n\in \DZ, n\ne 0\}
\end{align*}
are called the sets of {\em real} roots and of {\em imaginary} roots,
resp.
Let $ R^+\subset R$ be the set of roots of $\fb$ with respect
to $\fh$. Then the set $\hR^+$ of roots of $\hfb$ with respect to $\hfh$ is
$$
\hR^{+}=\{\alpha+n\delta\mid \alpha\in R, n\ge 1\}\cup
R^{+}\cup \{n\delta\mid n\ge 1\}.
$$
Let $\Pi\subset R^+$ be the set of simple roots and denote by
$\gamma\in R^+$ the highest root. Then the set of simple affine roots is
$$
\hPi=\Pi\cup\{-\gamma+\delta\}\subset \hR^+.
$$
For each real root $\alpha\in \hR^{\real}$ the corresponding root space
$\hfg_{\alpha}$ is one-dimensional, and so is the commutator
$[\hfg_{\alpha},\hfg_{-\alpha}]\subset \hfh$. The {\em (affine) coroot
$\alpha^\vee$} associated with $\alpha$ is the unique element in
$[\hfg_{\alpha},\hfg_{-\alpha}]$ on which $\alpha$ takes the value
$2$. Note that $\alpha^\vee$ is contained in $\fh\oplus\DC K$, so we
have $\langle\delta,\alpha^\vee\rangle=0$.
\subsection{The Weyl groups}
For $\alpha\in \hR^{\real}$ we define the reflection $s_\alpha\colon \hfhd\to \hfhd$ by $s_\alpha(\lambda):=\lambda-\langle\lambda,\alpha^\vee\rangle\alpha$.
We denote by $\hCW\subset \GL(\hfhd)$ the affine Weyl group, i.e.~the
subgroup generated by the reflections $s_\alpha$ for
$\alpha\in \hR^+$. The subgroup $\CW\subset\hCW$ generated by the
reflections $s_\alpha$ with $\alpha\in R$ leaves the subset
$\fhd\subset \hfhd$ stable and can be identified with the Weyl
group of $\fg$.
Let $\rho\in\hfhd$ be an element with the property
$\langle \rho,\alpha^\vee\rangle=1$ for each simple affine root $\alpha$. Note that
$\rho$ is only defined up to the addition of a multiple of $\delta$
(the span of the affine coroots is $\fh\oplus\DC K$, and the simple
coroots form a basis in this space). Yet all constructions in the following that
use $\rho$ do not depend on this choice. So let us fix such an
element $\rho$ once and for all.
The {\em dot-action} $\hCW\times\hfhd\to\hfhd$,
$(w,\lambda)\mapsto w.\lambda$, of the affine Weyl group on
$\hfhd$ is obtained by shifting the linear action in such a way that
$-\rho$ becomes a fixed point, i.e.~ it is given by
$$
w.\lambda:=w(\lambda+\rho)-\rho
$$
for $w\in\hCW$ and $\lambda\in \hfhd$. Note that since
$\langle\delta,\alpha^\vee\rangle=0$ we have
$s_\alpha(\delta)=\delta$ for all
$\alpha\in \hR^{\real}$. Hence
$w(\delta)=\delta$ for all $w\in\hCW$ (so the dot-action is
independent of the choice of $\rho$, as we claimed above).
\subsection{The invariant bilinear form}
Denote by $(\cdot,\cdot)\colon \hfg\times\hfg\to\DC$ the form given by
\begin{align*}
(x\otimes t^n, y\otimes t^m) &= \delta_{n,-m} k(x,y), \\
(K, \fg\otimes_\DC \DC[t,t^{-1}]\oplus \DC K) &= \{0\}, \\
(D, \fg\otimes_\DC \DC[t,t^{-1}]\oplus \DC D) &= \{0\}, \\
(K,D) &= 1
\end{align*}
for $x,y\in \fg$, $m,n\in\DZ$. It is non-degenerate, symmetric and {\em invariant}, i.e.~
it satisfies $([x,y],z)=(x,[y,z])$ for all
$x,y,z\in\hfg$. Moreover, it induces a non-degenerate bilinear form on the Cartan
subalgebra $\hfh$ and hence yields an isomorphism $\hfh\stackrel{\sim}\to\hfhd$, which is the direct sum of the isomorphism $\fh\to\fhd$ given by the Killing form $k$ and the isomorphism $\DC K\oplus \DC D\to \DC\Lambda_0\oplus \DC \delta$ that maps $K$ to $\delta$ and $D$ to $\Lambda_0$. We get an induced symmetric
non-degenerate form on the dual $\hfhd$ that is given explicitly by
\begin{align*}
(\alpha,\beta) &= k(\alpha,\beta), \\
(\Lambda_0, \fhd\oplus \DC\Lambda_0) &= \{0\}, \\
(\delta, \fhd\oplus \DC\delta) &= \{0\}, \\
(\Lambda_0,\delta) &= 1
\end{align*}
for $\alpha,\beta\in \fhd$ (here we denote by $k$ also the form on $\fhd$ that is induced by the Killing form).
It is
invariant under the linear action of the affine Weyl group, i.e.~ we have
$$
(w(\lambda),w(\mu))=(\lambda,\mu)
$$
for $w\in\hCW$, $\lambda,\mu\in\hfhd$.
\section{The category $\CO$ for an affine Kac--Moody algebra}
Having recalled the fundamental structural results for an affine
Kac--Moody algebra we now turn to its representation theory. We
restrict ourselves to representations in the affine category $\CO$.
\subsection{The category $\CO$}
Let $M$ be a $\hfg$-module. Its {\em weight space} corresponding to
$\lambda\in\hfhd$ is
$$
M_\lambda:=\{m\in M\mid h.m=\langle\lambda,h\rangle m\text{ for all $h\in \hfh$}\}.
$$
Any non-zero element $m\in M_\lambda$ is said to be {\em of weight $\lambda$}.
We say that $M$ is a {\em weight module} if
$M=\bigoplus_{\lambda\in\hfhd} M_\lambda$.
We say that $M$ is {\em locally $\hfb$-finite} if all finitely generated $\hfb$-submodules of $M$ are finite dimensional.
The {\em affine category $\CO$} is defined as the full subcategory of the category of $\hfg$-modules that consists of all locally $\hfb$-finite weight modules.
\subsection{Highest weight modules}
Our choice of positive roots defines a partial order on $\hfhd$: we set $\nu\varle\nu^\prime$ if and only if
$\nu^\prime-\nu\in\DZ_{\ge 0}\hR^+$.
A {\em highest weight module} of highest weight $\lambda\in \hfhd$ is
a $\hfg$-module $M$ that contains a generator $v\ne 0$ of weight
$\lambda$ such that
$\hfg_\alpha v=0$ for all $\alpha\in\hR^+$. Then $\lambda$ is indeed
the highest weight of $M$, i.e.~ $M_\mu\ne 0$ implies
$\mu\varle\lambda$. Each highest weight module is contained in $\CO$.
For $\lambda\in\hfhd$ denote by $\DC_\lambda$ the one-dimensional
$\hfh$-module corresponding to $\lambda$. We extend the $\hfh$-action
to a $\hfb$-action using the homomorphism $\hfb\to\hfh$ of Lie
algebras that is left inverse to the inclusion $\hfh\subset \hfb$. That
means that $\hfg_\alpha$ acts trivially on $\DC_\lambda$ for all
$\alpha\in\hR^+$. The induced module
$$
\Delta(\lambda):=U(\hfg)\otimes_{U(\hfb)}\DC_\lambda
$$
is called the {\em Verma module} corresponding to $\lambda$. It
contains a unique simple quotient $L(\lambda)$, and both
$\Delta(\lambda)$ and $L(\lambda)$ are highest
weight modules of highest weight $\lambda$.
Moreover, the modules $L(\lambda)$ for $\lambda\in\hfhd$ form a
full set of representatives of the simple isomorphism classes of
$\CO$, i.e.~each simple object in $\CO$ is isomorphic to
$L(\lambda)$ for a unique $\lambda\in\hfhd$.
\subsection{Characters}
Let $\DZ[\hfhd]=\bigoplus_{\lambda\in \hfhd}\DZ e^{\lambda}$ be the group algebra of the additive group $\hfhd$. Let $\widehat{\DZ[\hfhd]}\subset \prod_{\lambda\in\hfhd} \DZ e^{\lambda}$ be the subgroup of elements $(c_\lambda)$ that have the property that there exists a finite set $\{\mu_1,\dots, \mu_n\}\subset\hfhd$ such that $c_\lambda\ne 0$ implies $\lambda\le \mu_i$ for at least one $i$.
Let $\CO^f\subset \CO$ be the full subcategory of modules $M$ that have
the property that the weight spaces $M_\lambda$ are finite dimensional
and such that there exist $\mu_1,\dots, \mu_n\in \hfhd$ such that
$M_\lambda\ne 0$ implies $\lambda\le \mu_i$ for at least one $i$. For
each object $M$ of $\CO^f$ we can then define its {\em character} as
$$
\cha M:=\sum_{\lambda\in \hfhd} (\dim_\DC M_\lambda) e^\lambda\in \widehat{\DZ[\hfhd]}.
$$
The character of a Verma module is easy to calculate. For each $\lambda\in\hfhd$ we have
$$
\cha \Delta(\lambda)=e^{\lambda} \prod_{\alpha\in \hR^+}(1+e^{-\alpha}+e^{-2\alpha}+\dots)^{\dim\fg_\alpha}.
$$
(The above product is well-defined in $\widehat{\DZ[\hfhd]}$.) If $\lambda\in\hfhd$ is non-critical (cf.~ Section \ref{sec-crithyp}), the character of $L(\lambda)$ is known (cf.~\cite{KT00,Fie06}). The principal aim of our research project is to calculate $\cha L(\lambda)$ for the critical highest weights $\lambda$.
\subsection{Multiplicities}
Suppose again that $M$ is an object in $\CO^f$. Then there are well defined numbers $a_\nu\in \DN$ such that
$$
\cha M=\sum_{\nu\in\hfhd} a_\nu \cha L(\nu)
$$
(cf.~\cite{DGK82}). Note that the sum on the right hand side is, in
general, an infinite sum. We define the {\em multiplicity of $L(\nu)$ in $M$} as
$$
[M:L(\nu)]:= a_\nu.
$$
The matrix $[\Delta(\lambda):L(\mu)]$ is invertible, so the problem of calculating $\cha L(\mu)$ for all $\mu\in\hfhd$ is equivalent to the calculation of the multiplicities $[\Delta(\lambda):L(\mu)]$ for all $\lambda,\mu\in\hfhd$.
\subsection{Block decomposition}
Denote by ``$\sim$'' the equivalence relation on $\hfhd$ that is generated
by $\lambda\sim\mu$ if $[\Delta(\lambda):L(\mu)]\ne 0$. For an equivalence class $\Lambda\in \hfhd/_{\textstyle{\sim}}$ we define the
full subcategory $\CO_\Lambda$ of $\CO$ that consists of all objects
$M$ whose irreducible subquotients are isomorphic to $L(\lambda)$ for some $\lambda\in\Lambda$. We have the following decomposition result.
\begin{theorem}[\cite{DGK82,RCW82}] The functor
\begin{align*}
\prod_{\Lambda\in \hfhd/_{\scriptstyle{\sim}}} \CO_\Lambda & \to \CO, \\
(M_\Lambda) & \mapsto \bigoplus_{\Lambda\in\hfhd/_{\scriptstyle{\sim}}} M_\Lambda
\end{align*}
is an equivalence of categories.
\end{theorem}
\subsection{The Kac--Kazhdan theorem}
The following theorem gives an explicit description of the equivalence relation ``$\sim$'' on $\hfhd$. Let us denote by ``$\preceq$'' the partial order on $\hfhd$ generated by
$\nu\preceq\lambda$ if there exist $n\in\DN$ and $\beta\in \hR^+$
such that $2(\lambda+\rho,\beta)=n(\beta,\beta)$ and
$\nu=\lambda-n\beta$. In particular, $\nu\preceq\lambda$ implies
$\nu\varle\lambda$, but the converse is not true.
\begin{theorem}[\cite{KK79}] We have $[\Delta(\lambda):L(\nu)]\ne 0$ if and
only if $\nu\preceq\lambda$.
\end{theorem}
In particular, the equivalence relation ``$\sim$'' is generated by the partial
order relation ``$\preceq$''. For $\Lambda\in\hfhd/_{\textstyle{\sim}}$ set
\begin{align*}
\hR(\Lambda) &:=\{\alpha\in\hR\mid
2(\lambda+\rho,\alpha)\in \DZ(\alpha,\alpha) \text{ for some $\lambda\in\Lambda$}\}\\
&=\{\alpha\in\hR\mid
2(\lambda+\rho,\alpha)\in \DZ(\alpha,\alpha) \text{ for all $\lambda\in\Lambda$}\}
,\\
\hCW(\Lambda) & :=\langle s_{\alpha}\mid \alpha\in
\hR(\Lambda)\cap\hR^{\real}\rangle\subset\hCW.
\end{align*}
If $\Lambda$ is {\em non-critical}, i.e.~ if
$\hR(\Lambda)\subset\hR^{\real}$, then $\Lambda=\hCW(\Lambda).\lambda$
for each $\lambda\in\Lambda$. In this case, the structure of the block
$\CO_{\Lambda}$ can be completely described in terms of the group
$\hCW(\Lambda)$ (which turns out to be a Coxeter group) and the
singularity of its orbit $\Lambda$, cf.~\cite{Fie06}.
\subsection{A duality on $\CO^f$}
We will later need the following duality functor. For convenience
we only define it on the full subcategory $\CO^f$ of $\CO$ that we
defined earlier. All the
modules that we encounter in this article belong to $\CO^f$.
For $M\in\CO^f$ we set $M^\star:=\bigoplus_{\lambda\in\hfhd}
\Hom_\DC(M_\lambda,\DC)$. We endow $M^\star$ with the action of
$\hfg$ given by
$$
(x.\phi)(m)=\phi(-\omega(x).m),
$$
for $x\in\hfg$, $\phi\in M^\star$ and $m\in M$, where
$\omega\colon\hfg\to\hfg$ is the Chevalley-involution, i.e.~the
involution induced on $\hfg$ by the root system automorphism that
sends $\alpha\in \hR$ to $-\alpha\in \hR$ (cf.~\cite[Section 1.3]{Kac90}). Then $M^\star\in\CO^f$ and we
indeed get a duality functor on $\CO^f$. It is exact and maps irreducible modules
to irreducible modules. A quick look at characters shows
the following.
\begin{lemma} For each $\lambda\in\hfhd$ we have
$L(\lambda)^\star=L(\lambda)$.
\end{lemma}
For each $\lambda\in\hfhd$ we denote by
$$
\nabla(\lambda):=\Delta(\lambda)^\star
$$
the dual of the Verma
module with highest weight $\lambda$. By the above
lemma and the exactness of the duality, $\nabla(\lambda)$ and $\Delta(\lambda)$ have the same
Jordan-Hölder multiplicities, and $\nabla(\lambda)$ has a simple socle which is isomorphic to
$L(\lambda)$.
\section{The critical hyperplane}\label{sec-crithyp}
In this section we recall the notion of a critical weight for the
affine Kac--Moody algebra $\hfg$. We introduce a shift functor $T$ on
each of the critical blocks and study the corresponding
{\em graded center} $\CA=\bigoplus_{n\in\DZ}\CA_n$. For a critical weight $\lambda\in\hfhd$ we define the {\em restricted Verma
module $\rDelta(\lambda)$} as the quotient of $\Delta(\lambda)$
by the ideal of $\CA$ generated by $\bigoplus_{n\ne 0}\CA_n$. We state some fundamental
properties of these modules in Theorem \ref{theorem-ResVerma}. The
proof of this theorem is due to Feigin and Frenkel. Then we
recall the {\em Feigin--Frenkel conjecture} on the Jordan-Hölder
multiplicities for restricted Verma modules and, finally, state the
main result of this article in Theorem \ref{theorem-MT}.
\subsection{A shift functor}
The defining relations of $\hfg$ show that the derived Lie algebra of
$\hfg$ coincides with the central extension $\tfg$ of the loop algebra, i.e.~
$$
[\hfg,\hfg]=\fg\otimes\DC[t,t^{-1}]\oplus\DC K.
$$
Hence $[\hfg,\hfg]$ is of
codimension one in $\hfg$, so the quotient $\hfg/[\hfg,\hfg]$ is a
one-dimensional Lie algebra. Each character of $\hfg/[\hfg,\hfg]$ gives rise to a
one dimensional module of $\hfg$. In this way we get the simple
modules $L(\zeta\delta)$ for $\zeta\in\DC$. We have $L(\zeta\delta)\otimes L(\xi\delta)\cong L((\zeta+\xi)\delta)$ for $\zeta,\xi\in \DC$.
Let us define the {\em shift functor}
\begin{align*}
T\colon \hfg\catmod&\to\hfg\catmod,\\
M&\mapsto M\otimes_\DC L(\delta).
\end{align*}
The action of $\hfg$ on the tensor product is the usual one:
$X.(m\otimes l)=X.m\otimes l+m\otimes X.l$ for $X\in\hfg$, $m\in M$
and $l\in L(\delta)$. The functor $T$
is exact and preserves the categories $\CO^f$ and $\CO$. Clearly, it is an equivalence on these categories with inverse $T^{-1}\colon M\mapsto
M\otimes_\DC L(-\delta)$. For $n\in\DZ$ we denote by
$T^n\colon\CO\to\CO$ the $|n|$-fold composition of $T$
or of $T^{-1}$. It is given by the tensor product with the one-dimensional module $L(n\delta)$.
The following lemma is easy to prove (for part (3) use the facts that
$L(\delta)^\star\cong L(\delta)$ and $(M\otimes_\DC
N)^\star=M^\star\otimes_\DC N^\star$).
\begin{lemma}\label{lemma-propT} For each $\lambda\in\hfhd$ we have
\begin{enumerate}
\item $T\Delta(\lambda)\cong \Delta(\lambda+\delta)$,
\item $T L(\lambda)\cong L(\lambda+\delta)$,
\item on $\CO^f$ the functor $T$ commutes with the duality, i.e.~ there is a natural
equivalence
$T\circ (\cdot)^\star\cong (\cdot)^\star\circ T$ of functors.
\end{enumerate}
\end{lemma}
Let us denote by $\lambda\mapsto \ol\lambda$ the linear map $\hfhd\to\fhd$ that is dual to the inclusion $\fh\subset\hfh$.
For a subset $\Lambda$ of $\hfhd$ we denote by $\ol\Lambda$ its image in $\fhd$.
Now we come to the definition of {\em critical equivalence classes}, {\em critical weights} and {\em critical
blocks}.
\begin{lemma}\label{lemma-aequcrit} For an equivalence class $\Lambda\in\hfhd/_{\textstyle{\sim}}$ the following are equivalent.
\begin{enumerate}
\item The functor $T$ maps $\CO_{\Lambda}$ into itself.
\item We have $\lambda+\delta\sim\lambda$ for all $\lambda\in \Lambda$.
\item We have $\DZ\delta\setminus\{0\}=\hR^{\imag}\subset \hR(\Lambda)$.
\item For all $\lambda\in \Lambda$, the restriction of $\lambda$ to the central line
$\DC K\subset\hfh$ coincides with the restriction of $-\rho$, i.e.~ we have $\langle\lambda,K\rangle=\langle-\rho,K\rangle$.
\item The induced dot-action of the affine Weyl group $\hCW$ on $\ol\Lambda$ in
factors over an action of the finite Weyl group.
\end{enumerate}
\end{lemma}
\begin{remark} Note that $w.(\lambda+\xi\delta)=w.\lambda+\xi w(\delta)=w.\lambda+\xi\delta$, so $w.\lambda\equiv w.(\lambda+\xi\delta)\mod\delta$ for all $\lambda$, so the dot-action indeed induces an action of $\hCW$ on the quotient $\hfhd/\DC\delta$ and the statement (5) above makes sense.
\end{remark}
If the above conditions on $\Lambda$ are satisfied, we say that $\Lambda$ is a {\em critical equivalence class}. In this case we call each element $\lambda$ of $\Lambda$ a {\em critical weight}, or {\em of critical level}. We let $\hfhd_{\crit}$ be the set of weights of critical level, i.e.~the union of the critical equivalence classes. Condition (4) above shows that $\hfhd_{\crit}$ is an affine hyperplane in $\hfhd$. It is called the {\em critical hyperplane}.
\begin{proof} From the definition of the blocks, the exactness of $T$ and Lemma \ref{lemma-propT} we deduce the equivalence of (1) and (2).
Note that $(\alpha+n\delta)^\vee=\alpha^\vee+n K^\prime$, where $K^\prime$ is a non-zero multiple of $K$. Hence, $\lambda\in\hfhd$ has the property that $s_{\alpha+n\delta}.\lambda\equiv s_{\alpha+m\delta}.\lambda\mod\delta$ for all $m,n\in\DZ$ if and only if $\langle\lambda+\rho,K\rangle=0$. Hence (4) and (5) are equivalent.
As $(\delta,\delta)=0$ we have $\DZ\delta\setminus\{0\}\subset\hR(\Lambda)$ if and only if $(\lambda+\rho,\delta)=0$ for all $\lambda\in\Lambda$. The latter equation is equivalent to $\langle\lambda+\rho,K\rangle=0$. Hence (3) and (4) are equivalent.
Clearly, (3) implies (2). On the other hand, suppose that (2) holds, but (3) does not hold. Then $\hR^{\imag}\cap\hR(\Lambda)=\emptyset$, and $\Lambda$ is an orbit of $\hCW(\Lambda)$ under the dot-action. So $\lambda,\lambda+\delta\in\Lambda$ implies that $\lambda+\delta$ and $\lambda$ are contained in the same $\hCW$-orbit. The invariance of the bilinear form then yields $(\lambda+\delta+\rho,\lambda+\delta+\rho)=(\lambda+\rho,\lambda+\rho)$ which implies, as $(\delta,\delta)=0$, that $(\lambda+\rho,\delta)=0$, hence $\langle\lambda+\rho,K\rangle=0$, hence (4), which is equivalent to (3). Hence we have a contradiction. So (2) implies (3).
\end{proof}
\subsection{The action of $\hCW$ in the critical hyperplane}
Let us fix a critical equivalence class
$\Lambda\subset\hfhd_{\crit}$. Then we have $\delta\in \hR(\Lambda)$, so
$(\lambda+\rho,\delta)\in\DZ(\delta,\delta)=0$ for some (all)
$\lambda\in\Lambda$. Hence, if $\alpha\in \hR(\Lambda)$, then
either $\alpha=-\delta$ or $\alpha+\delta\in\hR(\Lambda)$.
If we set $R(\Lambda):=R\cap \hR(\Lambda)$,
then
$$
\hR(\Lambda)=\{\alpha+n\delta\mid \alpha\in R(\Lambda),
n\in\DZ\}\cup\{n\delta\mid n\in\DZ,n\ne 0\}.
$$
Moreover, we have $s_{\alpha+n\delta}\in \hCW(\Lambda)$ if and only if
$s_{\alpha}\in\hCW(\Lambda)$. We set $\CW(\Lambda)=\hCW(\Lambda)\cap
\CW$. Then $\hCW(\Lambda)$ is the affinization of $\CW(\Lambda)$.
Let $\alpha\in R(\Lambda)$. We now define a bijection
$\alpha\uparrow\cdot\colon\Lambda\to\Lambda$, following \cite{AJS94}. For
$\lambda\in\Lambda$ let $\alpha\uparrow\lambda$ be the minimal element
in $\{s_{\alpha,n}.\lambda\mid n\in\DZ,s_{\alpha,n}.\lambda\ge \lambda\}$. Note that $\alpha\uparrow\lambda=\lambda$ if $\lambda$ is contained in the reflection hyperplane corresponding to some $s_{\alpha,n}$. We
denote by $\alpha\downarrow\cdot\colon\Lambda\to\Lambda$ the
inverse map.
\begin{definition} We say that
\begin{enumerate}
\item
$\Lambda$ is {\em generic}, if $\ol{\Lambda}$ contains only one element,
\item $\Lambda$ is {\em subgeneric}, if $\ol \Lambda$ contains exactly two elements,
\item $\Lambda$ is {\em regular}, if for some (all) $\nu$ in $\Lambda$ we have that
$w.\nu=\nu$ implies $w=e$ for all $w\in\CW$.
\end{enumerate}
\end{definition}
Note that part (3) of the above definition refers to the action of the
{\em finite} Weyl group only. If $\Lambda$ is subgeneric, then there is a unique finite positive root $\alpha$ such that $\ol\Lambda=\{\ol\lambda,s_\alpha.\ol\lambda\}$. Set $R(\Lambda)^+:=R(\Lambda)\cap R^+$.
\begin{definition} Let $\nu\in \Lambda$. We say that
\begin{enumerate}
\item $\nu$ is {\em dominant}, if $\langle\nu+\rho,\alpha^\vee\rangle\ge 0$ for all $\alpha\in R(\Lambda)^+$.
\item $\nu$ is {\em anti-dominant}, if $\langle\nu+\rho,\alpha^\vee\rangle\le 0$ for all $\alpha\in R(\Lambda)^+$.
\end{enumerate}
\end{definition}
\subsection{The graded center of a critical block}\label{sec-gradedcenter} Since $\Lambda$ is
supposed to be critical, we can consider the functor $T$ as an auto-equivalence on the block $\CO_\Lambda$.
Let $n\in\DZ$ and let $z$ be a natural transformation from the functor $T^n$ on $\CO_\Lambda$ to the identity functor $\id$ on $\CO_\Lambda$. Note that $z$ associates with any $M\in\CO_\Lambda$ a homomorphism $z^M\colon T^nM\to M$ in such a way that for any homomorphism $f\colon M\to N$ in $\CO_\Lambda$ the diagram
\centerline{
\xymatrix{
T^nM\ar[r]^{T^n f} \ar[d]_{z^M}&T^nN\ar[d]^{z^N}\\
M\ar[r]^{f}& N
}
}
\noindent
commutes.
Denote by $\CA_n=\CA_n(\Lambda)$ the complex vector space of all natural transformations $z$ as above such that $z^{T^lM}=T^l z^M\colon T^{n+l}M\to T^l M$ for all $M\in\CO_\Lambda$ and $l\in\DZ$. There is a bilinear map
\begin{align*}
\CA_n\times\CA_m&\to\CA_{m+n}\\
(z_1,z_2)&\mapsto (M\mapsto z_1^M\circ(T^n z_2^M))
\end{align*}
that makes $\CA=\CA(\Lambda):=\bigoplus_{n\in \DZ} \CA_n$ into a graded $\DC$-algebra. It is associative, commutative and unital.
\subsection{Restricted Verma modules}
Let $\lambda\in \Lambda$ and $n\in \DZ$. Each $z\in \CA_n$ defines a homomorphism
$$
z^{\Delta(\lambda)}\colon T^n\Delta(\lambda)\cong\Delta(\lambda+n\delta)\to \Delta(\lambda).
$$
Each such homomorphism is zero if $n>0$. Define $\Delta(\lambda)^-\subset \Delta(\lambda)$ as the submodule generated by the images of all the homomorphisms $z^{\Delta(\lambda)}$ for $z\in \CA_n$ and $n<0$.
\begin{definition} The quotient
$$
\rDelta(\lambda) := \Delta(\lambda)/\Delta(\lambda)^-
$$
is called the {\em restricted Verma module} of highest weight $\lambda$.
\end{definition}
Consider the formal power series
$$
\prod_{j\ge 1}(1-q^j)^{-\rank\,
\fg}=\prod_{j\ge1}(1+q^j+q^{2j}+\dots)^{\rank\,\fg},
$$
and let us define for $n\in\DZ$ the number $p(n)\in\DN$ as the
coefficient of $q^n$ in the above series.
We denote by $\hfn_+:=\bigoplus_{\alpha\in \hR^+}\hfg_\alpha$ the
subalgebra of $\hfg$ corresponding to the positive affine roots. For a
$\hfg$-module $M$ we denote by $M^{\hfn_+}$ the set of $\hfn_+$-invariant
vectors. It is a $\hfh$-submodule of $M$, hence we can also define its
weight spaces $M^{\hfn_+}_\nu$ for $\nu\in\hfhd$.
The following theorem lists the most important properties of the restricted Verma modules. The proofs of the following statements (2), (3) and (4), as well as the main step in the proof of (1), are due to Feigin and Frenkel. We recall the main arguments in Section \ref{sec-FFcenter}.
\begin{theorem} \label{theorem-ResVerma} Let $\lambda\in \hfhd_{\crit}$.
\begin{enumerate}
\item The map $\CA_{n}\to \Hom(T^{n}\Delta(\lambda), \Delta(\lambda))$ is surjective for all $n\in \DZ$.
\item We have $\dim \Hom(T^n\Delta(\lambda), \Delta(\lambda))= p(-n)$ for all $n\in \DZ$.
\item We have
$$
\cha \rDelta(\lambda)=e^{\lambda}\prod_{\alpha\in \hR^{+}\cap \hR^{\real}}(1+e^{-\alpha}+e^{-2\alpha}+\dots).
$$
\item We have $\rDelta(\lambda)^{\hfn_+}_{\lambda-n\delta}=\{0\}$ for $n\ne 0$.
\end{enumerate}
\end{theorem}
\subsection{A conjecture}\label{subs-FFconj} The character of $L(\lambda)$ for a critical
highest weight $\lambda$ is not yet known in general. But we have a
formula for the characters
of the restricted Verma modules and the simple characters can
be calculated once the Jordan--Hölder multiplicites
$[\Delta(\lambda):L(\mu)]$ for critical weights $\lambda$ and $\mu$
are determined. In the following we state a conjecture
that gives a formula for these multiplicities in terms of periodic
Kazhdan--Lusztig polynomials. It is due to Feigin and Frenkel.
Let $\Lambda$ be a critical equivalence class. Then $\hCW(\Lambda)$ is the affinization of the finite Weyl group $\CW(\Lambda)$. So we can think of $\hCW(\Lambda)$ as a group of affine transformations on a vector space. The connected complements of the affine reflection hyperplanes are called alcoves, and the set of alcoves is a principal homogeneous set for the action of $\hCW(\Lambda)$. Let $A_e$ be the unique alcove in the dominant Weyl chamber that contains the origin in its closure. Then the map $w\mapsto A_w:=w(A_e)$ gives a bijection between $\hCW(\Lambda)$ and the set of alcoves. In \cite{Lus80}
Lusztig defined for alcoves $A$ and $B$ a {\em periodic
polynomial} $p_{A,B}\in\DZ[v]$ (we use the normalization and notation of Soergel, cf.~ \cite{Soe97} and \cite{Fie07}). Denote by $w_\Lambda\in\CW(\Lambda)$ the longest element.
\begin{conjecture} \label{conj-FFC} Let $\Lambda$ be a critical equivalence class.
\begin{enumerate}
\item {\em The restricted linkage principle:} For $\lambda,\mu\in
\Lambda$ we have that $[\rDelta(\lambda):L(\mu)]\ne 0$ implies $\mu\in\hCW(\Lambda).\lambda$.
\item {\em The restricted Verma multiplicities:} Let $\lambda\in \Lambda$ be regular and dominant and $w\in \hCW(\Lambda)$. Suppose that for all $x,x^\prime\in \hCW(\Lambda)$ with $p_{A_{w_\Lambda x},A_{w_\Lambda w}}(1)\ne 0$ and $p_{A_{w_\Lambda x^\prime},A_{w_\Lambda w}}(1)\ne 0$ and $x\ne x^\prime$ we have $x.\lambda\ne x^\prime.\lambda$. Then
$$
[\rDelta(w.\lambda):L(x.\lambda)]=p_{A_{w_\Lambda w},A_{w_\Lambda x}}(1)
$$
for all $x\in \hCW(\Lambda)$.
\end{enumerate}
\end{conjecture}
This conjecture is closely related to an anticipated relation between
representations of a small quantum group, the topology of
semi-infinite flag manifolds and the restricted critical level
representations of an affine Kac--Moody algebra, cf.~\cite{Lus91}. We prove part (1) of the above conjecture in \cite{AF2}.
\subsection{The main result}
In the following theorem we summarize the main results of this article.
\begin{theorem}\label{theorem-MT} Suppose that $\Lambda\subset\hfhd$
is a critical equivalence class.
\begin{enumerate}
\item If $\nu\in\Lambda$ is
anti-dominant, then for all
$w\in\CW(\Lambda)$ and $n\ge 0$ we have
$$
[\Delta(w.\nu):L(\nu-n\delta)]=p(n).
$$
\item If $\Lambda$ is subgeneric and $\nu\in\Lambda$ is dominant, then
we have for all $n\ge 0$
$$
[\Delta(\nu):L(\nu-n\delta)] = [\Delta(\alpha\uparrow \nu):L(\nu-n\delta)] = p(n),
$$
where $\alpha$ is the unique positive finite root with $\ol\Lambda=\{\ol\lambda,s_\alpha.\ol\lambda\}$.
\end{enumerate}
\end{theorem}
Let us restate the above results in terms of restricted Verma modules.
\begin{corollary} Let $\Lambda\subset\hfhd$ be a critical equivalence class.
\begin{enumerate}
\item Suppose that $\nu\in \Lambda$ is anti-dominant. Then for any
$\gamma\in\Lambda$ we have
$$
[\rDelta(\gamma):L(\nu)]=
\begin{cases}
1 \text{ if $\gamma\in\CW(\Lambda).\nu$}, \\
0 \text{ else.}
\end{cases}
$$
In particular, Conjecture \ref{conj-FFC} holds for the anti-dominant multiplicities.
\item
If $\Lambda$ is subgeneric, then we have for all $\gamma,\nu\in\Lambda$
$$
[\rDelta(\gamma):L(\nu)]=
\begin{cases}
1 &\text{ if $\nu\in\{\gamma,\alpha\downarrow\gamma\}$},\\
0 &\text{ else,}
\end{cases}
$$
where $\alpha\in R^+$ is such that $\ol\Lambda=\{\ol\lambda,s_{\alpha}.\ol\lambda\}$. In particular, Conjecture \ref{conj-FFC} holds in the subgeneric cases.
\end{enumerate}
\end{corollary}
In the following section we recall the results of
Feigin and Frenkel on the center at the critical level and deduce
Theorem \ref{theorem-ResVerma}. In Section \ref{sec-Proj} we study
the structure of projective objects in a critical block
$\CO_\Lambda$. In particular, we provide some results on the action
of the graded center $\CA$ on a projective object. In Section
\ref{sec-BRST} we introduce the BRST-cohomology functor and recall the
main Theorem of \cite{Ara07}. In Section \ref{sec-proofofMT} we use
the results of Sections \ref{sec-FFcenter}, \ref{sec-Proj} and
\ref{sec-BRST} to give a proof of Theorem \ref{theorem-MT}.
\section{The Feigin--Frenkel center}\label{sec-FFcenter}
In this section we recall the fundamental
results on the Feigin-Frenkel center \cite{FeiFre92}
at the critical level.
The main references are the textbooks
\cite{FreBen04} and \cite{Fre07}.
\subsection{The universal affine vertex algebra
at the critical level}
Set
\begin{align*}
V^{\crit}(\fing) := U(\affg)\*_{U(\fing\*_{\C} \C[t]\+ \C K \+ \C D)}\C_{\crit},
\end{align*}
where $\C_{\crit}$ is
the one-dimensional
representation
of $\fing\*_{\C}\C[t]\+ \C K\+ \C D$
on which
$\fing\*_{\C}\C[t]\+ \C D$
acts trivially and
$K$ acts as multiplication by the critical value $\langle-\rho,K\rangle$.
The space
$V^{\crit}(\fing)$
has a
natural structure of a {\em vertex algebra},
and is called the
{\em universal affine vertex algebra associated with
$\fing$} at the critical level (see e.g., \cite[\S 4.9]{Kac98}).
Let
\begin{align*}
Y(?,z):V^{\crit}(\fing) & \ra \End_{\C} V^{\crit}(\fing)[[z,z\inv]], \\
a& \mapsto a(z) =\sum_{n\in \Z}a_{(n)}z^{-n-1}
\end{align*}
be the state-field correspondence.
The map
$Y(?,z)$
is
uniquely determined by the condition
\begin{align*}
Y((x\* t\inv)\vac,z)=\sum_{n\in \Z}(x\* t^n)z^{-n-1}
\quad \text{for $x\in \fing$,}
\end{align*}
where $\vac$ is the vacuum vector $1\* 1$.
The vertex algebra $\Vcr$
is graded by the Hamiltonian $-D$.
If $a\in \Vcr$ is an eigenvector of $-D$,
its eigenvalue is called the {\em conformal weight}
and is denoted by $\Delta_a$.
We denote by $\partial$
the translation operator. It satisfies
\begin{align*}
Y(\partial a,z)=[\partial,Y(a,z)]=\frac{d}{dz}Y(a,z).
\end{align*}
\subsection{The Feigin-Frenkel center}
\newcommand{\FFC}{\mathfrak{z}(\affg)}
Let $\FFC$ be the center of the
vertex algebra $V^{\crit}(\fing)$:
\begin{align*}
\FFC=\{a\in V^{\crit}(\fing)\mid
[a_{(m)}, v_{(n)}]=0
\text{ for all $v\in V^{\crit}(\fing)$, $m,n\in \Z$}
\}.
\end{align*}
One has
\begin{align*}
\FFC&=\{a\in \Vcri \mid v_{(n)}a=0
\text{ for all $v\in \Vcri$, $n\geq 0$}
\},\\
&=V^{\crit}(\fing)^{G [[t]]},
\end{align*}
where $G$ is the adjoint group of $\fing$
and
$G[[t]]$ is
the $\C[[t]]$-points of $G$.
Let $\{U_p(\fing\*_{\C}\C[t\inv]t\inv)\}$ be the standard filtration of
$U(\fing\*_{\C}\C[t\inv]t\inv)$ and set
\begin{align*}
F_p V^{\crit}(\fing):=U_p(\fing\*_\C{\C}[t\inv]t\inv)\cdot \mathbf{1}
\subset \Vcri.
\end{align*}
This defines a filtration of a vertex algebra.
Let
$\gr\, \Vcri$ be
the associated
graded vertex algebra:
$
\gr\, \Vcri=\bigoplus_{p}F_{p}V^{\crit}(\fing)
/F_{p-1}V^{\crit}(\fing)
$.
It is a commutative vertex algebra
and
one
has
\begin{align*}
\gr\, \Vcri\cong S(\fing[t\inv]t\inv)\cong\C[\fing_{\infty}]
\end{align*}
as differential rings\footnote{A commutative vertex algebra
$V$ is naturally a commutative ring
with a derivation
by the multiplication
$(a,b)\mapsto a_{(-1)}b$
and the derivation $\partial a_{(-n)}=n a_{(-n-1)}$.}
and $G[[t]]$-modules,
where
$\fing_{\infty}$ is the infinite jet scheme of
$\fing$ (cf.\ \cite{EisFre01})
and $\fing$ is identified with $\fing^\star$.
Below we shall identify $\gr\, V^{\crit}(\fing)$ with $\C[\fing_{\infty}]$.
The natural projection $\fing_{\infty}\ra \fing$
gives the embedding $\C[\fing]\hookrightarrow \C[\fing_{\infty}]$.
Let $\{F_p \FFC\}$ be the induced filtration of $\FFC$,
$\gr\, \FFC$ the associated graded vertex algebra.
Certainly, the image of the natural embedding
$
\gr\, \FFC\hookrightarrow \gr\, V^{\crit}(\fing)
=\C[\fing_{\infty}]
$
is contained in
$ \C[\fing_{\infty}]^{G[[t]]}$.
Let $\bar p^{(1)}
,\dots, \bar p^{(\ell)}$,
where $\ell=\rank\, \fing$,
be homogeneous
generators
of
the ring $
\C[\fing]^G\subset \C[\fing_{\infty}]^{G[[t]]}$.
The elements $\bar p^{(i)}_{(-j-1)}:=(\partial)^j \bar p^{(i)
}/j!$ are also $G[[t]]$-invariant
for all $j\geq 0$.
According to \cite{BeiDri96}
(see also \cite{EisFre01}), one has
\begin{align*}
\C[\fing_{\infty}]^{G[[t]]}
=\C[\bar{p}^{(1)}_{(-j-1)},\dots
\bar{p}^{(\ell)}_{(-j-1)}]_{j\geq 0}.
\end{align*}
\begin{theorem}[\cite{FeiFre92}, see also \cite{Fre07}]\label{Th:FF}
The embedding
\begin{align*}
\gr\, \FFC \ra
\C[\fing_{\infty}]^{G[[t]]}
\end{align*}
is an isomorphism.
\end{theorem}
By Theorem \ref{Th:FF}
there exist homogeneous
generators $p^{(1)},\dots,p^{(\ell)}$
of $\FFC$
whose symbols are $\bar{p}^{(1)},\dots, \bar{p}^{(\ell)}$.
Let $d_i=\deg \bar {p}^{(i)}-1$,
so that
$d_1,\dots,d_{\ell}$ are the exponents of $\fing$.
The conformal weight of $p^{(i)}$ is
by definition $d_i+1$.
We write
\begin{align*}
Y(p^{(i)},z)&=p^{(i)}(z)=\sum_{n\in \Z}p^{(i)}_{(n)}z^{-n-1}
=\sum_{n\in \Z}p^{(i)}_{n}z^{-n-d_i-1},
\end{align*}
so that
\begin{align*}
[D, p^{(i)}_n]=n p^{(i)}_n.
\end{align*}
One has
\begin{align*}
[x, p_{(n)}^{(i)}]=0\quad \text{for all $x\in \tfg$.}
\end{align*}
\subsection{The action of the Feigin--Frenkel
center on objects with critical level}
For $k\in\DC$ we denote by $\CO_k$ the category $\CO$ at level $k$, i.e.~ the direct summand of the category $\CO$
on which $K$ acts as multiplication with $k$. In particular, we denote by $\CO_{\crit}$ the category $\CO$ at critical level.
Let $M$ be an object of $\CO_{\crit}$.
Then $M$
is naturally a graded module over the vertex algebra $\Vcri$,
and hence, it is a graded module over its center $\FFC$.
Thus
$M$ can be viewed as a graded module over the
polynomial ring
\begin{align*}
\CZ=\C[p^{(i)}_s; i=1,2,\dots, \ell,\ s\in \Z]
=\bigoplus_{n\in \Z}\CZ_n
\end{align*}
in an obvious manner.
Here
$\CZ_n$ is the subspace of $\CZ$
spanned by elements
$p^{(i_1)}_{n_1}\cdots p^{(i_r)}_{n_r}$
with $n_1+\dots +n_r=n$.
Set
\begin{align*}
\RZm =\C[p^{(i)}_n;i=1,\dots ,\ell, n<0]
=\bigoplus_{n\leq 0}\RZm_n\subset \CZ,
\end{align*}
where $\RZm_n=\RZm\cap \CZ_n$.
\begin{theorem}[{\cite{FreGai06}, see also \cite[Theorem 9.5.3]{Fre07}}]
\label{th:FF-freeness}
For any $\lam\in \affh_{\crit}^{\star}$,
$\Delta(\lam)$ is free over $\RZm$.
Moreover,
the natural map $\CZ_n^-\to\Hom(\Delta(\lambda+n\delta), \Delta(\lambda))$
is a bijection for all $n\leq 0$.
\end{theorem}
We now construct a natural map
\begin{align*}
\CZ_n\ra \CA_n.
\end{align*}
Recall that $L(\delta)$ is one-dimensional. So we can choose a
generator $l$ of $L(\delta)$. This gives us, for any $\hfg$-module
$M$, a map $M\otimes_\DC L(\delta)\to M$, $m\otimes l\mapsto m$. Since
$L(\delta)$ is trivial as a $\tfg$-module, this map is a $\tfg$-module
homomorphism. By iteration we get $\tfg$-module homomorphisms $T^n M=M\otimes_\DC
L(\delta)^{\otimes n}\to M$ for all $n\ge 0$. Using the element
$l^\prime\in L(-\delta)$ that is dual to $l$ with respect to an isomorphism $L(\delta)\otimes_\DC L(-\delta)\to L(0)\cong\DC$ we
analogously get $\tfg$-module homomorphisms $T^{-n} M=M\otimes_\DC L(-\delta)^{\otimes
n}\to M$.
Now suppose that $M$ is contained in a critical block of $\CO$. Let $n\in\DZ$ and $z\in\CZ_n$. Then the composition of the map
$T^n M\to M$ constructed above and the action map $z\colon M\to M$
yields now a {\em $\hfg$-module homomorphism} $T^n M\to M$, as it now also commutes with the action of $D$. This
gives us a natural transformation $T^n \to \id$ and we get an element in
$\CA_n$ that we associate with $z$. Hence we constructed a map $\CZ_n\to
\CA_n$.
\begin{proof}[Proof of Theorem \ref{theorem-ResVerma}]
The action of $\CZ_n$ on $\Delta(\lambda)$ factors over the
action of $\CA_n$. Hence Theorem \ref{th:FF-freeness} implies parts (1) and (2)
of Theorem \ref{theorem-ResVerma}.
It also implies that
$\Delta(\lam)^-$ is spanned by $p^{(i)}_{-n}m$
with $i=1,\dots,\ell$,
$n>0$ and $m\in \Delta(\lam)$.
From the freeness assertion in Theorem \ref{th:FF-freeness} we deduce
part (3) of Theorem \ref{theorem-ResVerma}. Finally, its part (4) is
another Theorem of Feigin and Frenkel \cite{FeiFre90}
which,
for instance,
follows
from
\cite[Proposition 9.5.1]{Fre07}
and Theorem \ref{th:FF-freeness}.
\end{proof}
\section{Projective objects}\label{sec-Proj}
For a general equivalence class $\Lambda\in\hfhd/_{\textstyle{\sim}}$ the block
$\CO_\Lambda$ does not contain
enough projective objects (this includes, for example, all critical
equivalence classes). However, there is a way to overcome this problem
by restricting the set of possible weights for the modules under
consideration. This means that we have to consider the {\em truncated}
subcategories of $\CO$.
\subsection{The truncated categories}
Let us fix a (not necessarily critical) equivalence class $\Lambda\in
\hfhd/_{\textstyle{\sim}}$. We use the following notation:
We write $\{\le \nu\}$ for the set $\{\nu^\prime\in\Lambda\mid
\nu^\prime\le \nu\}$ and use the similar notations $\{<\nu\}$, $\{\ge \nu\}$,
etc.~ for the analogously defined sets. We consider the topology on
$\Lambda$ that is generated by the basic open sets $\{\le\nu\}$ for
$\nu\in\Lambda$. Hence a subset $\CJ$ of $\Lambda$ is open in this
topology if and only if for all $\nu,\nu^\prime\in\Lambda$ with
$\nu^\prime\le \nu$, $\nu\in\CJ$ implies
$\nu^\prime\in\CJ$.
\begin{definition} For an open subset $\CJ\subset\Lambda$ we denote by $\CO_\Lambda^{\CJ}\subset\CO_\Lambda$ the full
subcategory of objects $M$ with the property that each of its subquotients is isomorphic to $L(\lambda)$ for some $\lambda\in \CJ$.
\end{definition}
Note that $\CO_\Lambda^\CJ$ is an abelian category and that
$\Delta(\lambda)$ and $L(\lambda)$ are
contained in $\CO_\Lambda^{\CJ}$ if and only if $\lambda\in\CJ$. We
write $\CO_\Lambda^{\varle\nu}$ instead of $\CO_\Lambda^{\{\varle\nu\}}$.
\subsection{Submodules, quotients and subquotients}
Let $\Lambda\subset\hfhd$ be an equivalence class and $\CJ$ an open
subset of $\Lambda$.
In this section we construct a left adjoint functor $M\mapsto M^{\CJ}$
to the inclusion functor $\CO_{\Lambda}^\CJ\to\CO_\Lambda$. Let us
denote by $\CI=\Lambda\setminus \CJ$ the closed complement of $\CJ$.
\begin{definition} Let $M$ be an object in $\CO_\Lambda$.
\begin{enumerate}
\item We define $M_{\CI}\subset M$ as the submodule generated by the
weight spaces $M_\nu$ with $\nu\in\CI$.
\item We define $M^{\CJ}$ as the quotient of $M$ by the submodule $M_{\CI}$.
\end{enumerate}
\end{definition}
Obviously, the map $M\mapsto M^\CJ$ defines a functor from
$\CO_\Lambda\to\CO_\Lambda^\CJ$ that is left adjoint to the
inclusion. We write $M^{\le\nu}$ instead of $M^{\{\le\nu\}}$ and $M^{<\nu}$
instead of $M^{\{<\nu\}}$, etc. If
$\CI^\prime\subset\CI\subset\Lambda$ are closed subsets, then there is
a natural inclusion $M_{\CI^\prime}\subset M_\CI$.
If $\CJ^\prime\subset\CJ\subset\Lambda$ are open subsets, then there is
a canonical surjective map $M^\CJ\to M^{\CJ^\prime}$.
In addition to the submodules and quotient modules defined above we
will also need the following subquotient modules that are associated
with locally closed subsets. For any subset $\CK$ of $\Lambda$ define
$$
\CK_-:=\bigcup_{\lambda\in \CK} \{\le \lambda\} \text{ and } \CK_+:=\bigcup_{\lambda\in \CK}\{\ge \lambda\}.
$$
Note that $\CK_+$ is closed and $\CK_-$ is open, and $\CK$ is locally closed if $\CK=\CK_-\cap\CK_+$.
\begin{definition} Suppose that $\CK$ is locally closed. We define $M_{[\CK]}$ as the image of the canonical decomposition
$M_{\CK_+}\inj M \sur M^{\CK_-}$.
\end{definition}
If $\CK=\{\lambda,\dots,\mu\}$, we write $M_{[\lambda,\dots,\mu]}$
instead of $M_{[\{\lambda,\dots,\mu\}]}$.
\subsection{Modules admitting a Verma flag}
\begin{definition} We say that $M\in\CO_\Lambda$ {\em admits a Verma flag}
if there is a finite filtration
$$
0=M_0\subset M_1\subset \dots\subset M_n=M
$$
such that for all $i=1,\dots,n$ the quotient $M_i/M_{i-1}$ is isomorphic to a Verma module $\Delta(\mu_i)$ for some $\mu_i\in \hfhd$.
\end{definition}
For each $\mu\in\hfhd$ the multiplicity
$$
[M:\Delta(\mu)]:=\# \{i\mid \mu=\mu_i\}
$$
is independent of the chosen filtration.
It is a well-known fact that $\Ext^1(\Delta(\mu),\Delta(\lambda))\ne 0$ implies that $\lambda>\mu$. Hence we can find a filtration for $M$ as in the definition such that $\mu_i>\mu_j$ implies $i<j$. From this one easily gets the following lemma.
\begin{lemma} \label{lemma-Vermasubquots} Suppose that $M$ admits a Verma flag and suppose that $\CJ\subset\hfhd$ is open, $\CI\subset\hfhd$ is closed and $\CK\subset\hfhd$ is locally closed. Then $M^\CJ$, $M_\CI$ and $M_{[\CK]}$ admit a Verma flag and for the multiplicities we have
\begin{align*}
[M^\CJ:\Delta(\mu)]&=
\begin{cases}
[M:\Delta(\mu)]&\text{ if $\mu\in\CJ$}\\
0&\text{ else,}
\end{cases}
\\
[M_\CI:\Delta(\mu)]&=
\begin{cases}
[M:\Delta(\mu)]&\text{ if $\mu\in\CI$}\\
0&\text{ else,}
\end{cases}
\\
[M_{[\CK]}:\Delta(\mu)]&=
\begin{cases}
[M:\Delta(\mu)]&\text{ if $\mu\in\CK$}\\
0&\text{ else}
\end{cases}
\end{align*}
for all $\mu\in\hfhd$.
\end{lemma}
\subsection{Projective objects}
We say that $\CJ\subset \Lambda$ is {\em bounded} if for any $\lambda\in\CJ$ the set of $\mu\in\CJ$ with $\mu\ge\lambda$ is finite.
\begin{theorem}[\cite{Fie03}, cf.~ also \cite{AF2}] Suppose that $\CJ\subset \Lambda$ is open and bounded and let $\lambda\in \CJ$.
\begin{enumerate}
\item There exists an (up to isomorphism unique) projective
cover $P^{\CJ}(\lambda)$ of $L(\lambda)$ in $\CO^{\CJ}_\Lambda$.
\item If $\CJ^\prime\subset \CJ$ is an open subset and $\lambda\in\CJ^\prime$, then
we have an isomorphism
$(P^{\CJ}(\lambda))^{\CJ^\prime}\cong P^{\CJ^\prime}(\lambda)$.
\item The object $P^{\CJ}(\lambda)$ admits a Verma flag and for each $\gamma\in \Lambda$ we have the BGG-reciprocity formula
$$
\left(P^{\CJ}(\lambda):\Delta(\gamma)\right)=
\begin{cases}
0 & \text{if $\gamma\not\in\CJ$,} \\
[\nabla(\gamma):L(\lambda)] & \text{if $\gamma\in \CJ$.}
\end{cases}
$$
\end{enumerate}
\end{theorem}
By Lemma \ref{lemma-Vermasubquots} and the theorem above, if $\nu,\lambda\in\Lambda$ are such that $\nu\ge
\lambda$, then the module $P^{\CJ}(\lambda)_{[\nu]}$ does not depend on
the open set $\CJ$ as long as $\nu\in\CJ$. We denote this object by
$P(\lambda)_{[\nu]}$.
Let $\CJ\subset\Lambda$ be open and bounded and $\lambda\in \CJ$. Suppose that $\nu\in \CJ$ is a maximal element. Then we can consider $P(\lambda)_{[\nu]}$ as a subspace in $P^{\CJ}(\lambda)$.
We will need the following result later.
\begin{lemma}\label{lemma-HomdualVerma} Under the above assumptions, the restriction map
$$
\Hom(P^{\CJ}(\lambda), \nabla(\nu))
\to \Hom(P(\lambda)_{[\nu]}, \nabla(\nu))
$$
is a bijection.
\end{lemma}
\begin{proof} As a first step we show that the map is
injective. Suppose that $f\in\Hom(P^{\CJ}(\lambda), \nabla(\nu))$
is contained in the kernel. Then $P(\lambda)_{[\nu]}$ is contained
in the kernel of $f$, hence $f$ factors over the map
$P^{\CJ}(\lambda)\to (P^{\CJ}(\lambda))^{\CJ\setminus\{\nu\}}\cong P^{\CJ\setminus\{\nu\}}(\lambda)$. But then it cannot
contain the simple socle $L(\nu)\subset \nabla(\nu)$ in its image,
so $f=0$.
Now we have
\begin{align*}
\dim\Hom(P^{\CJ}(\lambda), \nabla(\nu)) &=
[\nabla(\nu):L(\lambda)]\\
&= (P^{\CJ}(\lambda):\Delta(\nu))\\
&= (P(\lambda)_{[\nu]}:\Delta(\nu))\\
&=\dim \Hom(P(\lambda)_{[\nu]},\nabla(\nu)).
\end{align*}
The last equation holds since $P(\lambda)_{[\nu]}$ is isomorphic to a direct sum of copies of $\Delta(\nu)$ and $\dim\Hom(\Delta(\nu),\nabla(\nu))=1$. Since the map referred to in the lemma is injective and since the dimensions of its source and its image coincide, it is bijective.
\end{proof}
\subsection{The Casimir operator}
We call a $\hfg$-module $M$ {\em smooth} if for all $m\in M$ we have
$\hfg_\alpha.m=0$ for all but a finite number of positive affine roots
$\alpha\in\hR$. In particular, each locally $\hfb$-finite $\hfg$-module is smooth. Recall that on the full subcategory $\hfg\catmod^{sm}$ of
smooth representations there is an endomorphism of the identity
functor, the {\em Casimir operator} $C\colon \id\to \id$ (its
construction can be found, for example, in \cite[Section 2.5]{Kac90}). We will
only need the following property of $C$.
\begin{proposition} Let $\lambda\in\hfhd$. Then $C$ acts on
$\Delta(\lambda)$ as multiplication with the scalar $c_\lambda:=(\lambda+\rho,\lambda+\rho)-(\rho,\rho)\in\DC$.
\end{proposition}
The construction
of $C$ depends on $(\cdot,\cdot)$, hence there is no ambiguity in the
statement of the proposition.
Let $\Lambda\subset\hfhd$ be a (not necessarily critical) equivalence class. For
$\lambda,\mu\in \Lambda$ we have $c_\lambda=c_\mu$, so $\Lambda$ defines a unique scalar $c_\Lambda\in\DC$ such that $C$
acts by multiplication with $c_\Lambda$ on each Verma module in $\CO_\Lambda$.
Now suppose that
$\lambda,\mu\in\Lambda$ are such that they form an atom in the
partially ordered set $\Lambda$, i.e.~ suppose that $\lambda< \mu$
and that there is no $\nu\in\Lambda$ with $\lambda<\nu<\mu$. Moreover, assume that $\ol\lambda\ne\ol\mu$. Then the
object $P^{\varle\mu}(\lambda)$ is an extension of the Verma modules
$\Delta(\lambda)$ and $\Delta(\mu)$, each occurring once (this is, by the BGG-reciprocity, equivalent to $[\Delta(\lambda):L(\lambda)]=[\Delta(\mu):L(\lambda)]=1$, which can be deduced easily from
the analog of Jantzen's sum formula in the Kac-Moody case,
cf.~\cite{KK79}). Hence we have a short exact sequence
$$
0\to \Delta(\mu)\to P^{\varle\mu}(\lambda)\to\Delta(\lambda)\to 0.
$$
\begin{lemma} The endomorphism $C-c_\Lambda\id\colon
P^{\varle\mu}(\lambda)\to P^{\varle\mu}(\lambda)$ is non-zero.
\end{lemma}
\begin{proof} For the proof we use deformation theory,
cf.~\cite{Fie03} and \cite{AF2}. Denote by $A=\DC[[t]]$ the completed polynomial ring in
one variable and by $Q=\Quot A$ its quotient field. Let us fix $\gamma\in \hfhd$ with the property that $(\gamma+\rho,\lambda)\ne(\gamma+\rho,\mu)$. Let $S=S(\hfh)$
be the symmetric algebra over $\hfh$ and consider the algebra map
$\tau\colon S\to A$ that is determined by $\tau(h)=\gamma(h)t$ for all
$h\in\hfh$. This makes $A$, and hence $Q$, into a local $S$-algebra.
In \cite{Fie03} we
constructed the deformed categories $\CO_A$ and $\CO_Q$ as full
subcategories of $\hfg\otimes_\DC A\catmod$ and $\hfg\otimes_\DC
Q\catmod$. We showed that the functors $\otimes_A \DC$ and
$\otimes_A Q$ induce functors $\CO_A\to\CO$ and
$\CO_A\to\CO_Q$. The categories $\CO_A$ and $\CO_Q$ contain {\em deformed Verma modules} $\Delta_A(\nu)$ and $\Delta_Q(\nu)$, resp. On their highest weight spaces the Cartan algebra $\hfh$ acts by the character $\nu+\tau$, which is considered as a linear map from $\hfh$ to $A$ and $Q$, resp. In particular, the Casimir operator $C$ acts on $\Delta_Q(\nu)$ as multiplication with $c_{\nu+\gamma t}=(\nu+\gamma t+\rho,\nu+\gamma t+\rho)-(\rho,\rho)$.
The analogous definition as in the non-deformed case gives
truncated categories $\CO_A^{\varle\mu}$ and $\CO_Q^{\varle\nu}$
for all $\nu\in\hfhd$. For
any $\nu,\nu^\prime\in\hfhd$ there is a projective
object $P^{\varle\nu}_A(\nu^\prime)$ in $\CO_A^{\varle\nu}$ such that
$P^{\varle\nu}_A(\nu^\prime)\otimes_A\DC\cong
P^{\varle\nu}(\nu^\prime)$. Moreover, we have
$$
\Hom(P^{\varle\nu}_A(\nu^\prime_1),
P^{\varle\nu}_A(\nu_2^\prime))\otimes_A\DC=\Hom(P^{\varle\nu}(\nu_1^\prime),
P^{\varle\nu}(\nu_2^\prime)).
$$
The category $\CO_Q$ is semi-simple, i.e.~ each object is isomorphic
to a direct sum of Verma modules $\Delta_Q(\lambda)$. Now each
$P^{\varle\nu}_A(\nu^\prime)\otimes_A Q$ admits a $Q$-deformed Verma flag, hence it splits into a
direct sum of $Q$-deformed Verma modules. The Verma multiplicities of $P^{\varle\nu}_A(\nu^\prime)$, of $P^{\varle\nu}(\nu^\prime)$ and of $P_A^{\varle\nu}(\nu^\prime)\otimes_A Q$ coincide.
Now let $\lambda,\mu\in\Lambda$ be as in the statement of the lemma. Note that $c_{\lambda+\gamma t}\equiv c_{\mu+\gamma t}\mod t$, but our choice of $\gamma$ implies $c_{\lambda+\gamma t}\ne c_{\mu+\gamma t}$. Let us suppose that the action of $C-c_{\Lambda}\id$ on $P^{\varle\mu}(\lambda)$ was
zero.
From the above, our assumptions on $\lambda$ and $\mu$ and the Jantzen sum formula we get an inclusion
$$
P^{\varle\mu}_A(\lambda)\subset P^{\varle\mu}_A(\lambda)\otimes_A Q\cong\Delta_Q(\lambda)\oplus\Delta_Q(\mu).
$$
On each of the modules above the Casimir operator acts. Our assumptions imply that the image of the action of $C-c_\Lambda\id$ on the module on the left is contained in $tP^{\varle\mu}_A(\lambda)$, hence $t^{-1}(C-c_\Lambda\id)$ is a well-defined operator on $P^{\varle\mu}_A(\lambda)$. On the module on the right this operator acts diagonally with eigenvalues in $\DC[[t]]$ which are distinct modulo $t$. Hence $P^{\varle\mu}_A(\lambda)$ decomposes according to the inclusion above, which clearly cannot be the case.
\end{proof}
\begin{lemma}\label{lemma-Casimir} Suppose that
$\lambda,\mu\in\Lambda$ are as above and that
$$
0\to \Delta(\mu)\to M\to X\to 0
$$
is a short exact sequence, where $X$ is a module of highest weight $\lambda$.
If $C$ acts on $M$ as a scalar, then there is a
submodule $Y$ of $M$ with highest weight $\lambda$ that maps surjectively onto $X$.
\end{lemma}
\begin{proof} Since $X$ is a module of highest weight $\lambda$ and since the weights of $M$ are smaller or equal to $\mu$, there is a map $f\colon P^{\varle\mu}(\lambda)\to M$ such that the composition $P^{\varle\mu}(\lambda)\to M\to X$ is surjective. There is a commutative diagram
\centerline{
\xymatrix{
0 \ar[r] & \Delta(\mu) \ar[r]\ar[d]^{f^\prime} & P^{\varle\mu}(\lambda) \ar[r] \ar[d]^f& \Delta(\lambda) \ar[r]\ar@{>>}[d] &0\\
0 \ar[r] & \Delta(\mu) \ar[r] & M \ar[r] & X \ar[r] &0
}
}
\noindent
with exact rows. In order to prove the lemma it is enough to show that $\Delta(\mu)\subset P^{\varle\mu}(\lambda)$ is in the kernel of $f$, i.e.~ that the left vertical map $f^\prime$ is zero. Since any endomorphism of a Verma module is either zero or injective, it suffices to show that $f^\prime$ is not injective.
By assumption, the Casimir element $C$ acts on $M$ by a scalar, which
has to be $c_\Lambda$. So $C-c_\Lambda\id$ acts on $M$ by zero, so
$(C-c_\Lambda\id)P^{\varle\mu}(\lambda)$ is in the kernel of $f$. Since
$C-c_\Lambda\id$ acts by zero on $\Delta(\lambda)$, we have
$(C-c_\Lambda\id)P^{\varle\mu}(\lambda)\subset \Delta(\mu)\subset
P^{\varle\mu}(\lambda)$. By the previous lemma,
$(C-c_\Lambda\id)P^{\varle\mu}(\lambda)$ is non-zero, so the kernel of $f^\prime$ is not trivial, hence $f^\prime$ is not injective, hence it must be zero, which is what we wanted to show.
\end{proof}
\subsection{The action of $\CA$ on projective objects}
Let us fix now a critical equivalence class $\Lambda\subset\hfhd$.
Let $\lambda\in \Lambda$ and $n\ge 0$. In this section we study the
action of $\CA_n$ on
$P^{\varle\lambda}(\lambda-n\delta)$, i.e.~ we want to study the map
$$
\CA_n\to \Hom(T^n P^{\varle\lambda}(\lambda-n\delta), P^{\varle\lambda}(\lambda-n\delta)).
$$
For the ease of notation let us fix an identification
$T^nP^{\varle\lambda}(\lambda-n\delta)\cong P^{\varle\lambda+n\delta}(\lambda)$.
Each map $f\colon P^{\varle\lambda+n\delta}(\lambda)\to
P^{\varle\lambda}(\lambda-n\delta)$ factors over the map
$P^{\varle\lambda+n\delta}(\lambda)\to
P^{\varle\lambda}(\lambda)$. The latter module is isomorphic to
$\Delta(\lambda)$, hence the image of the map $f$ must be contained
in $P(\lambda-n\delta)_{[\lambda]}\subset P^{\varle\lambda}(\lambda-n\delta)$. So the map $f$ induces a unique
map $f^\prime\colon P^{\varle\lambda}(\lambda)\to
P(\lambda-n\delta)_{[\lambda]}$ such that the following diagram
commutes:
\centerline{
\xymatrix{
T^n P^{\varle\lambda}(\lambda-n\delta)\ar[d]\ar[r]^{f}&P^{\varle\lambda}(\lambda-n\delta)\\
P^{\varle\lambda}(\lambda)\ar[r]^{f^\prime}&P(\lambda-n\delta)_{[\lambda]}.\ar[u]
}
}
The next proposition is one of the principal technical ingredients in the proof of our main Theorem.
\begin{proposition}\label{prop-actonproj}
Suppose that $n\ge 0$ is such that $[\Delta(\lambda):L(\lambda-n\delta)]=p(n)$. Then the action map
\begin{align*}
\CA_n&\to \Hom(T^nP^{\varle\lambda}(\lambda-n\delta),
P^{\varle\lambda}(\lambda-n\delta))\\
&\cong\Hom(P^{\varle\lambda}(\lambda), P(\lambda-n\delta)_{[\lambda]})
\end{align*}
is surjective.
\end{proposition}
\begin{proof} Note that $P(\lambda)^{\varle\lambda}\cong\Delta(\lambda)$
and that $P(\lambda-n\delta)_{[\lambda]}$ is isomorphic to a direct
sum of $(P(\lambda-n\delta)_{[\lambda]}:\Delta(\lambda))$-copies of
$\Delta(\lambda)$. By our assumption and the BGG-reciprocity, this number is $p(n)$. Hence the spaces on the right hand side of our map are of dimension $p(n)$.
Now the following Lemma \ref{lemma-actonVerma} shows that the image of the action map
$\CA_n\to \Hom(T^nP^{\varle\lambda}(\lambda-n\delta),
P^{\varle\lambda}(\lambda-n\delta))$ is of dimension $p(n)$. From this we deduce our
claim.
\end{proof}
\subsection{A duality on $\CA$}
Let $\Lambda\subset\hfhd$ again be a critical equivalence class.
In this section we define an algebra involution
$$
\dual\colon\CA\to\CA
$$
which maps $\CA_n$ into $\CA_{-n}$.
Fix $n\in\DZ$ and choose $z\in\CA_n$. We define $\dual z\in\CA_{-n}$
as follows. Let $M\in \CO^f_\Lambda:=\CO^f\cap\CO_\Lambda$ and let $M^\star\in \CO_\Lambda$
be its restricted dual. Then $z$ defines a homomorphism
$z^{M^\star}\colon T^n M^\star\to M^\star$. The dual of this map is a
homomorphism $\left( z^{M^\star}\right)^\star\colon M\to T^n M$.
\begin{definition} For $z\in\CA_n$ and $M\in\CO^f_\Lambda$ define the
map
$$
(\dual z)^M:= T^{-n}\left( z^{M^\star}\right)^\star\colon T^{-n} M\to M.
$$
\end{definition}
One immediately checks that we get a natural transformation
$\dual z\colon T^{-n}|_{\CO_{\Lambda}^f}\to \id_{\CO_{\Lambda}^f}$. As $\CO_\Lambda$ is filtered by the truncated categories, and as each indecomposable projective object in a truncated category is also contained in $\CO_{\Lambda}^f$, this induces a natural transformation $\dual z\colon T^n\to \id$ between the functors on the whole block $\CO_\Lambda$, hence an element in $\CA_{-n}$.
Now we prove the statement that remained open in the proof of Proposition \ref{prop-actonproj}. Fix $\lambda\in\Lambda$ and $n\ge 0$.
\begin{lemma}\label{lemma-actonVerma} For $z\in\CA_n$ the following holds:
$$
(\dual z)^{\Delta(\lambda)}\ne 0 \text{ if and only if } z^{P^{\varle\lambda}(\lambda-n\delta)}\ne 0.
$$
In particular, the image of the map
$\CA_n\to\Hom(T^nP^{\varle\lambda}(\lambda-n\delta),P^{\varle\lambda}(\lambda-n\delta))$
is of dimension $p(n)$.
\end{lemma}
\begin{proof} By the definition of the duality we have
$$
(\dual z)^{\Delta(\lambda)}\ne 0 \text{ if and only if } z^{\nabla(\lambda)}\ne 0.
$$
For each homomorphism $g\colon P^{\varle\lambda}(\lambda-n\delta)\to \nabla(\lambda)$ there is a commutative diagram
\centerline{
\xymatrix{
T^n
P^{\varle\lambda}(\lambda-n\delta)\ar[d]_{T^ng}\ar[rrr]^{z^{P^{\varle\lambda}(\lambda-n\delta)}}&&&
P^{\varle\lambda}(\lambda-n\delta)\ar[d]^{g}\\
T^n\nabla(\lambda)\ar[rrr]^{z^{\nabla(\lambda)}}&&&
\nabla(\lambda).
}
}
The strategy of the proof is the following. Suppose that
$z^{P^{\varle\lambda}(\lambda-n\delta)}\ne 0$. We show that there is
a map $g$ such that the top right composition in the diagram above is
non-zero. From this we deduce that $z^{\nabla(\lambda)}\ne 0$. We show
that $z^{\nabla(\lambda)}\ne 0$ implies
$z^{P^{\varle\lambda}(\lambda-n\delta)}\ne 0$ in a similar way.
So suppose that $z^{P^{\varle \lambda}(\lambda-n\delta)}\ne 0$. We
have already seen that there is a unique map
$b\colon P^{\varle\lambda}(\lambda)\to P(\lambda-n\delta)_{[\lambda]}$ such
that the following diagram commutes:
\centerline{
\xymatrix{
T^n P^{\varle\lambda}(\lambda-n\delta)\ar[d]\ar[rr]^{z^{P^{\varle\lambda}(\lambda-n\delta)}}&&P^{\varle\lambda}(\lambda-n\delta)\\
P^{\varle\lambda}(\lambda)\ar[rr]^b&&P(\lambda-n\delta)_{[\lambda]}.\ar[u]
}
}
\noindent
Now
$P^{\varle\lambda}(\lambda)$ is isomorphic to $\Delta(\lambda)$ and
$P(\lambda-n\delta)_{[\lambda]}$ is a direct sum of various
copies of $\Delta(\lambda)$. Hence we can
find a map $g^\prime\colon P(\lambda-n\delta)_{[\lambda]}\to \nabla(\lambda)$
such that the composition $P^{{\varle\lambda}}(\lambda)\stackrel{b}\to
P(\lambda-n\delta)_{[\lambda]}\stackrel{g^\prime}\to
\nabla(\lambda)$ is non-zero. By
Lemma \ref{lemma-HomdualVerma}, the map $g^\prime$ admits a lift
$g\colon
P^{\varle\lambda}(\lambda-n\delta)\to
\nabla(\lambda)$. Diagrammatically, the situation now looks as follows:
\centerline{
\xymatrix{
T^n P^{\varle\lambda}(\lambda-n\delta))\ar[d]\ar[rr]^{z^{P^{\varle\lambda}(\lambda-n\delta)}}&&P^{\varle\lambda}(\lambda-n\delta)\ar[dr]^{g}&\\
P^{\varle\lambda}(\lambda)\ar[rr]^b&&P(\lambda-n\delta)_{[\lambda]}\ar[u]\ar[r]^{g^\prime}&\nabla(\lambda).
}
}
If we plug in the map $g$ that we just obtained in the first diagram above, then
top right composition is non-zero, hence so is the bottom left composition. In particular,
$z^{\nabla(\lambda)}\ne 0$. This was the first part of the proof.
Now suppose that $z^{\nabla(\lambda)}\ne 0$. Then the image of $z^{\nabla(\lambda)}$ contains the unique simple submodule $L(\lambda)$ of
$\nabla(\lambda)$. Since
$T^n P^{\varle\lambda}(\lambda-n\delta)\cong
P^{{\varle\lambda}+n\delta}(\lambda)$ is a projective cover of
$L(\lambda)$ in $\CO^{{\varle\lambda}+n\delta}$, and since
$T^n \nabla(\lambda)\cong \nabla(\lambda+n\delta)$ is contained in the latter category, we
can find a map $g^\prime\colon T^nP^{\varle\lambda}(\lambda-n\delta)\to
T^n \nabla(\lambda)$ such that
$z^{\nabla(\lambda)}\circ g^\prime$ is non-zero. For $g:=T^{-n}g^\prime$, the bottom left composition in the first diagram in this proof is
non-zero, hence so is the top right composition. In particular, $z^{P^{\varle\lambda}(\lambda-n\delta)}\ne 0$.
The last statement of the lemma follows from the previous result and
Theorem \ref{theorem-ResVerma}.
\end{proof}
\subsection{A variant for the subgeneric cases}
We will also need the following variant of Proposition \ref{prop-actonproj} in the case that $\Lambda$ is
critical and subgeneric. Suppose that $\alpha$ is the positive finite root with $\ol\Lambda=\{\ol\lambda,s_\alpha.\ol\lambda\}$. Fix $\lambda\in\Lambda$ and $n\ge 0$. We study the
action of $\CA_n$ on the projective cover
$P^{\varle\alpha\uparrow\lambda}(\lambda-n\delta)$, i.e.~we now consider the map
$$
\CA_n\to \Hom(T^nP^{\varle\alpha\uparrow\lambda}(\lambda-n\delta),
P^{\varle\alpha\uparrow\lambda}(\lambda-n\delta)).
$$
Again, let us fix an isomorphism
$T^nP^{\varle\alpha\uparrow\lambda}(\lambda-n\delta)\cong
P^{\varle\alpha\uparrow\lambda+n\delta}(\lambda)$.
As before we see that each map $f\colon
P^{\varle\alpha\uparrow\lambda+n\delta}(\lambda)\to
P^{\varle\alpha\uparrow\lambda}(\lambda-n\delta)$ induces a map
$f^\prime\colon P^{\varle\alpha\uparrow\lambda}(\lambda)\to P(\lambda-n\delta)_{[\lambda,\alpha\uparrow\lambda]}
$
such that the following diagram commutes:
\centerline{
\xymatrix{
P^{{\varle\alpha\uparrow\lambda+n\delta}}(\lambda)\ar[d]\ar[r]^{f}&P^{{\varle\alpha\uparrow\lambda}}(\lambda-n\delta)\\
P^{\varle\alpha\uparrow\lambda}(\lambda)\ar[r]^{f^\prime}& P(\lambda-n\delta)_{[\lambda,\alpha\uparrow\lambda]}.\ar[u]
}
}
\noindent
Note that $P^{\varle\alpha\uparrow\lambda}(\lambda)$ is a non-split
extension of the Verma modules $\Delta(\lambda)$ and
$\Delta(\alpha\uparrow\lambda)$, as $\{\lambda,\alpha\uparrow\lambda\}\subset\Lambda$ is an atom. The module
$P(\lambda-n\delta)_{[\lambda,\alpha\uparrow\lambda]}$ is an extension
of $[\Delta(\lambda):L(\lambda-n\delta)]$ many copies of
$\Delta(\lambda)$ and
$[\Delta(\alpha\uparrow\lambda):L(\lambda-n\delta)]$ many copies of
$\Delta(\alpha\uparrow\lambda)$.
Let us assume that
$[\Delta(\alpha\uparrow\lambda):L(\lambda-n\delta)]=[\Delta(\lambda):L(\lambda-n\delta)]=p(n)$.
Then $P(\lambda-n\delta)_{[\lambda,\alpha\uparrow\lambda]}$ is a
direct sum of $p(n)$ non-split extensions of the Verma modules
$\Delta(\lambda)$ and $\Delta(\alpha\uparrow\lambda)$ (the extensions
are non-split by projectivity).
Let us consider the composition
\begin{align*}
\CA_n&\to \Hom(T^nP^{\varle\alpha\uparrow\lambda}(\lambda-n\delta),
P^{\varle\alpha\uparrow\lambda}(\lambda-n\delta))\\
&\to
\Hom(P(\lambda)_{[\alpha\uparrow\lambda]},
P(\lambda-n\delta)_{[\alpha\uparrow\lambda]}),
\end{align*}
where the last map is induced by the functor
$(\cdot)_{[\alpha\uparrow\lambda]}$.
\begin{proposition}\label{prop-actonprojsub}
Suppose that
$[\Delta(\alpha\uparrow\lambda):L(\lambda-n\delta)]=[\Delta(\lambda):L(\lambda-n\delta)]=p(n)$.
Then the composition $\CA_n\to
\Hom(P(\lambda)_{[\alpha\uparrow\lambda]},
P(\lambda-n\delta)_{[\alpha\uparrow\lambda]})$ constructed above is
surjective.
\end{proposition}
\begin{proof} Consider the homomorphisms
\centerline{
\xymatrix{ & \Hom(P(\lambda)_{[\alpha\uparrow\lambda]},
P(\lambda-n\delta)_{[\alpha\uparrow\lambda]})\\
\CA_n \ar[r] & \Hom(P(\lambda)_{[\lambda,\alpha\uparrow\lambda]},
P(\lambda-n\delta)_{[\lambda,\alpha\uparrow\lambda]}) \ar[u]\ar[d]
\\
&\Hom(P(\lambda)_{[\lambda]},
P(\lambda-n\delta)_{[\lambda]}).
}
}
By Proposition \ref{prop-actonproj}, the lower composition is
surjective. But the kernel of the upper composition is contained in
the kernel of the lower composition, as $P(\lambda-n\delta)_{[\alpha\uparrow\lambda,\lambda]}$ is a direct sum of non-split extensions of $\Delta(\alpha\uparrow\lambda)$ and $\Delta(\lambda)$. Since the spaces on the top and the bottom
share the same dimension, also the upper composition is surjective.
\end{proof}
\section{The BRST cohomology}\label{sec-BRST}
To prove Theorem \ref{theorem-MT} we need a result from \cite{Ara07},
which we explain below.
\subsection{The BRST cohomology associated with the
quantized Drinfeld-Sokolov reduction}
Denote by
$\finn_-:=\bigoplus_{\alpha\in R^+}\fing_{-\alpha}$
the nilpotent subalgebra of $\fing$ corresponding to the set of negative
roots.
Let
$\Psi$ be a non-degenerate character of $\finn_-$
in the sense of Kostant \cite{Kos78}, i.e.~
\begin{align*}
\Psi(x)=k(x,e_{prin})
\end{align*}
for some principal nilpotent element $e_{prin}$ of $\fing$ in $\finn_+
:=\bigoplus_{\alpha\in R^+}\fing_{\alpha}$.
We
extend $\Psi$ to the character $\widehat{\Psi}$
of $\finn_-[t,t\inv]:=\finn_-\* \C[t,t\inv]\subset \affg$
by setting
\begin{align*}
\widehat{\Psi}(u\* t^n)=\Psi(u)\delta_{n,0}
\end{align*}
for $u\in \finn_-$, $n\in \Z$.
Let $\C_{\widehat{\Psi}}$ be the
corresponding one-dimensional representation of $\finn_-[t,t\inv]$.
Set
\begin{align*}
H^i(M):=\BRST{i}{M}
\end{align*}
for $M\in \CO_{\crit}$ and $i\in \Z$.
Here,
$M\* \C_{\widehat{\Psi}}$ is considered as an
$\finn_-[t,t\inv]$-module by the tensor product action,
and
$\BRST{\bullet}{M}$
is the semi-infinite $\finn_-[t,t\inv]$-cohomology \cite{Feu84} with
coefficients in
$M\* \C_{\widehat{\Psi}}$.
The cohomology $H^{\bullet}(M)$
is defined by the semi-infinite analogue of the
Chevalley-Eilenberg complex:
Let $\Cl$
be the unital superalgebra
generated by
odd elements $\psi_{\alpha}(n)$
for $\alpha\in R$, $n\in \Z$,
with the relations
\begin{align*}
\psi_{\alpha}(m)\psi_{\beta}(n)+ \psi_{\beta}(n)\psi_{\alpha}(m)
=\delta_{m+n,0}\delta_{\alpha+\beta,0}.
\end{align*}
Let $\semiwedge{\bullet}$ be the irreducible
representation of $\Cl$ generated by the vector $\vac$
such that
\begin{align*}
\psi_{\alpha}(n)\vac=0,\quad \text{if }\alpha+n\delta\in \widehat{R}^+.
\end{align*}
The space $\semiwedge{\bullet}$ is graded by {\em charge}:
$\semiwedge{\bullet}=\bigoplus_{i\in \Z}\semiwedge{i}$,
where
the charges of $\vac$, $\psi_{\alpha}(n)$
and $\psi_{-\alpha}(n)$
for $\alpha\in R^+$, $n\in \Z$,
are $0$,
$1$, and $-1$,
respectively.
Also, we view $\semiwedge{\bullet}$
as an $\affh$-module
on which $h\in \affh$
acts as $h\mathbf{1}=0$ and $[h,\psi_{\alpha}(n)]=
\bra \alpha+n\delta,h\ket \psi_{\alpha}(n)$.
Let
$M$ be an object of $\CO$.
Set
\begin{align*}
C^{\bullet}(M):=M\* \semiwedge{\bullet}=\bigoplus_{i\in \Z}C^i(M),
\quad C^i(M)=M\* \semiwedge{i}.
\end{align*}
Define an odd operator
$Q$ of charge $1$ on $C^{\bullet}(M)$
by
\begin{align*}
Q:=&\sum_{\alpha\in R^-\atop n\in \Z}
(x_{\alpha}\* t^{-n}+\widehat{\Psi}(x_{\alpha}\* t^{-n}))\*
\psi_{\alpha}(n)
\\ &\quad-
\frac{1}{2} \sum_{\alpha,
\beta,\gamma\in R^-\atop
k,l\in \Z}
c_{\alpha,\beta}^{\gamma}
\id_M\* \psi_{-\alpha}(-k)
\psi_{-\beta}(-l)\psi_{\gamma}(k+l),
\end{align*}
where $x_{\alpha}$ is a (fixed) root vector in $\fing_{\alpha}$ for any $\alpha\in R^-$
and
$[x_{\alpha},x_{\beta}]=
\sum_{\gamma}c_{\alpha,\beta}^{\gamma}x_{\gamma}$.
The operator $Q$ is well-defined because $M\in \CO$.
One has
\begin{align*}
Q^2=0.
\end{align*}
Therefore,
$(C^{\bullet}(M),Q)$ is a cochain complex.
The space $H^{\bullet}(M)$ is
by definition
the
cohomology of the complex $(C^{\bullet}(M),Q)$.
Set
$C^{\bullet}(M)_d:=\{c\in C^{\bullet}(M)\mid
(D\* 1+1\* D) c= dc \}$.
One has
$C^{\bullet}(M)=\bigoplus_{d\in \C}C^{\bullet}(M)_d$.
Because the operator $Q$
obviously preserves each subspace $C^{\bullet}(M)_d$,
$H^{\bullet}(M)$ is also graded by the diagonal action of $D$:
\begin{align*}
H^{\bullet}(M)=\bigoplus_{d\in \C}H^{\bullet}(M)_d.
\end{align*}
\subsection{The functor $F$}
Let $p$ be an element of $\FFC$.
For each $n\in \Z$,
the operator $p_{(n)}\* 1$
commutes with the action of $Q$ on $C^{\bullet}(M)$.
Therefore,
for each $i\in \Z$,
$H^{i}(M)$ is
naturally a graded module
over the commutative vertex algebra
$\FFC$,
and thus can be considered as a graded $\mathcal{Z}$-module.
Denote by $F$ the functor
\begin{align*}
\CO_{\crit}\ra \mathcal{Z}\operatorname{-Mod},
\quad M\mapsto H^0(M),
\end{align*}
where $\mathcal{Z}\operatorname{-Mod}$ is the
category of graded $\mathcal{Z}$-modules.
Set
\begin{align*}
\cha_q F(M):=\sum_{d\in \C}q^{-d} \dim_{\C}F(M)_d
\end{align*}
for a finitely generated object $M$
of $\CO_{\crit}$,
where $F(M)_d=H^0(M)_d$.
\begin{theorem}[\cite{Ara07}]\label{Th:main-of-Ara}$ $
\begin{enumerate}
\item One has $H^i(M)=0$ for all $i\ne 0$
and $M\in \CO_{\crit}$.
In particular, the functor $F$ is exact.
\item
Let $\lam\in \affh^{\star}_{\crit}$.
One has the following.
\begin{align*}
&\cha_q F(\Delta(\lam))=q^{-\bra \lam,D\ket}\prod\limits_{j\geq 1}(1-q^j)^{-\rank\, \fing},
\\
&\cha_q F(L(\lam))=\begin{cases}
q^{-\bra \lam,D\ket}&\text{if $\lam$ is
anti-dominant},\\
0&\text{otherwise.}
\end{cases}
\end{align*}
\end{enumerate}
\end{theorem}
\begin{Rem}
In general,
the correspondence $M\mapsto H^0(M)$ defines a functor
from $\CO_k$ to the
category of graded modules over the
$W$-algebra $\mathscr{W}^k(\fing)$
associated with $\fing$ at level $k$,
which coincides \cite{FeiFre92}
with $\FFC$ if the level $k$ is critical.
In \cite{Ara07},
it was proved that
the functor $H^0(?)$ is exact
and
$H^0(L(\lam))$
is zero or irreducible
for any $\lam$ at any level $k$.
\end{Rem}
\section{The proof of the main Theorem}\label{sec-proofofMT}
We have collected all the ingredients for the proof of our main
Theorem \ref{theorem-MT}. We start with claim (1). Let us state it again:
\begin{theorem} Let $\Lambda\subset \hfhd$ be a critical equivalence class and suppose that $\nu\in\Lambda$ is anti-dominant. Then for all $w\in \CW(\Lambda)$ and $n\ge 0$ we have
$$
[\Delta(w.\nu):L(\nu-n\delta)]=p(n).
$$
\end{theorem}
\begin{proof} By Theorem \ref{Th:main-of-Ara}, (1), the functor $F$
is exact. Hence we have
\begin{align*}
\cha_q F(\Delta(w. \nu)) &=\sum_{\gamma\in\Lambda} [\Delta(w .\nu):
L(\gamma)]
\cha_q F(L(\gamma))\\
&=\sum_{\gamma\in\Lambda\atop \text{$\gamma$ anti-dominant}} [\Delta(w .\nu):
L(\gamma)]
\cha_q F(L(\gamma))
\end{align*}
Note that $\gamma\in\Lambda$ is anti-dominant if and only if $\gamma=\nu+r\delta$ for some $r\in\DZ$. Since $\bra w. \nu,D\ket=\bra \nu,D\ket$
for all $w\in \CW$ (as $\langle\alpha,D\rangle=0$ for all finite roots $\alpha$), the claim now follows directly from Theorem \ref{Th:main-of-Ara}, (2).
\end{proof}
It remains to prove part (2) of Theorem \ref{theorem-MT}. Let us
recall the statement:
\begin{theorem} \label{theorem-dommul} Let $\Lambda$ be a subgeneric critical equivalence class and let $\lambda\in \Lambda$ be dominant. Denote by $\alpha$ the positive finite root with $\{\ol\lambda,s_\alpha.\ol\lambda\}$. Then we have, for all $n\ge 0$,
$$
[\Delta(\lambda):L(\lambda-n\delta)]=[\Delta(\alpha\uparrow\lambda):L(\lambda-n\delta)]=p(n).
$$
\end{theorem}
For the proof of the above statement we need the following result.
\begin{lemma}\label{lemma-prim} Suppose that $\lambda$, $\Lambda$ and $\alpha$ are as in Theorem \ref{theorem-dommul}. For any $n\ge 0$ we then have
$$
\dim\Delta(\lambda)_{\lambda-n\delta}^{\hfn_+}=\dim \Delta(\alpha\uparrow\lambda)_{\lambda-n\delta}^{\hfn_+} = p(n).
$$
\end{lemma}
\begin{proof} Note that $\dim\Delta(\lambda)_{\lambda-n\delta}^{\hfn_+}=\dim\Hom(\Delta(\lambda-n\delta),\Delta(\lambda))$, so Theorem \ref{theorem-ResVerma} yields $\dim\Delta(\lambda)_{\lambda-n\delta}^{\hfn_+}=p(n)$.
Let $\gamma\in\Lambda$ be arbitrary. The Jantzen sum formula yields $[\Delta(\gamma):L(\alpha\downarrow\gamma)]=1$ (note that $\alpha\downarrow\gamma$ is maximal in the set $\{\alpha\downarrow\gamma+n\delta\mid [\Delta(\gamma):L(\alpha\downarrow\gamma+n\delta]\ne 0\}$). Hence $\dim\Hom(\Delta(\alpha\downarrow\gamma),\Delta(\gamma))=1$. Since any non-trivial homomorphism between Verma modules is injective, we can view $\Delta(\alpha\downarrow\gamma)$ as a uniquely defined submodule in $\Delta(\gamma)$.
In particular, we have a chain of inclusions
$$
\Delta(\lambda)\subset\Delta(\alpha\uparrow\lambda)\subset \Delta(\alpha\uparrow^2\lambda)\subset \Delta(\alpha\uparrow^3\lambda).
$$
From now on we view each of these modules as a submodule of all the modules appearing on its right (note that $\dim\Hom(\Delta(\lambda), \Delta(\alpha\uparrow^2\lambda))>1$ in general).
By Theorem \ref{th:FF-freeness}, there is an element $z\in \CZ^-$, uniquely defined up to multiplication with a scalar in $\DC^\times$, such that $\Delta(\lambda)=z\Delta(\alpha\uparrow^2 \lambda)$. It is easy to see that this implies $\Delta(\alpha\uparrow\lambda)=z\Delta(\alpha\uparrow^3\lambda)$.
We fix non-zero vectors $v_{\alpha\uparrow^2\lambda}$ and $v_{\alpha\uparrow^3\lambda}$ of highest weight in $\Delta(\alpha\uparrow^2\lambda)$ and $\Delta(\alpha\uparrow^3\lambda)$, resp.
We consider both as elements in the free $\CZ^-$-module $\Delta(\alpha\uparrow^3\lambda)$. As $\CZ^-$ is a graded polynomial ring (in infinitely many variables), as $\CZ^-_0\cong \DC$ and as $v_{\alpha\uparrow^3\lambda}$ and $v_{\alpha\uparrow^2\lambda}$ are not contained in $\left(\bigoplus_{n<0}\CZ^-_n\right)\Delta(\alpha\uparrow^3\lambda)$, we can extend the set $\{v_{\alpha\uparrow^2\lambda},v_{\alpha\uparrow^3\lambda}\}$ to a $\CZ^-$-basis of $\Delta(\alpha\uparrow^3\lambda)$. In particular, if $v\in \Delta(\alpha\uparrow\lambda)=z\Delta(\alpha\uparrow^3\lambda)$ is of the form $\tilde z v_{\alpha\uparrow^2\lambda}$ for some $\tilde z\in\CZ^-$, then $\tilde z$ is divisible by $z$ in $\CZ^-$.
Now let $v\in \Delta(\alpha\uparrow\lambda)^{\hfn_+}_{\lambda-n\delta}$. Then, again by Theorem \ref{th:FF-freeness}, $v=\tilde z v_{\alpha\uparrow^2\lambda}$ for some $\tilde z\in\CZ^-$. As we have observed above, $\tilde z$ is divisible by $z$ in $\CZ^-$, hence $v$ is contained in $\Delta(\lambda)$. Hence
$$
\Delta(\alpha\uparrow\lambda)^{\hfn_+}_{\lambda-n\delta}=\Delta(\lambda)^{\hfn_+}_{\lambda-n\delta},
$$
and we conclude $\dim\Delta(\alpha\uparrow\lambda)^{\hfn_+}_{\lambda-n\delta}=p(n)$ from what we have shown earlier.
\end{proof}
\begin{proof}[Proof of Theorem \ref{theorem-dommul}] Note first that the claimed identities are equivalent to the
identities
\begin{align*}
[\Delta(\lambda):L(\lambda-n\delta)]&=\dim
\Delta(\lambda)^{\hfn_+}_{\lambda-n\delta},\\
[\Delta(\alpha\uparrow\lambda):L(\lambda-n\delta)]&=\dim
\Delta(\alpha\uparrow\lambda)^{\hfn_+}_{\lambda-n\delta}
\end{align*}
as the right hand sides both equal $p(n)$ by Lemma \ref{lemma-prim}. Hence we want to prove that each simple subquotient with highest weight $\lambda-n\delta$ of $\Delta(\lambda)$ or $\Delta(\alpha\uparrow\lambda)$ corresponds to a primitive vector. As $\Delta(\alpha\uparrow\lambda)$ can be considered as a submodule of $\Delta(\alpha\uparrow^2\lambda)$ and as $\alpha\uparrow^2\lambda$ is dominant again, it is enough to prove the above claim for the simple subquotients of $\Delta(\lambda)$, i.e.~ it is enough to prove that $[\Delta(\lambda):L(\lambda-n\delta)]=\dim
\Delta(\lambda)^{\hfn_+}_{\lambda-n\delta}$.
We prove this by induction on the number $n$.
The case $n=0$ is easy to settle: we certainly have $[\Delta(\lambda):L(\lambda)]=1=\dim \Delta(\lambda)^{\hfn^+}_\lambda$. So let us assume that $n>0$ and that
$$
[\Delta(\lambda):L(\lambda-l\delta)]=\dim \Delta(\lambda)^{\hfn^+}_{\lambda-l\delta}
$$
holds for all $l<n$. Let
$$
V(\lambda):= \Delta(\lambda)/\Delta({\alpha\downarrow\lambda})
$$
be the {\em Weyl module} with highest weight $\lambda$ (again we consider $\Delta(\alpha\downarrow\lambda)$ as a uniquely define submodule in $\Delta(\lambda)$. Now we prove the following statement:
\begin{enumerate}
\item
{\em We have $[V(\lambda):L(\lambda-n\delta)]= \dim V(\lambda)^{\hfn_+}_{\lambda-n\delta}$.}
\end{enumerate}
Suppose that $X$ is a submodule of $V(\lambda)$ such
that there exists a surjection $f\colon X\to L(\lambda-n\delta)$. In order
to prove claim (1) it is enough to show that there is a map
$g\colon \Delta(\lambda-n\delta)\to X$ such that the composition
$f\circ g\colon \Delta(\lambda-n\delta)\to X\to L(\lambda-n\delta)$ is surjective.
Now $L(\lambda-n\delta)$ is not a quotient of $V(\lambda)$ (since $n>0$), hence $X$
is a proper submodule of $V(\lambda)$. So its weights are strictly
smaller than $\lambda$, i.e.~ $X$ is an object in
$\CO_\Lambda^{<\lambda}$. Since $P^{<\lambda}(\lambda-n\delta)$ is a
projective cover of $L(\lambda-n\delta)$ in $\CO_\Lambda^{<\lambda}$, there
is a map $h\colon P^{<\lambda}(\lambda-n\delta)\to X$ such that the
composition $P^{<\lambda}(\lambda-n\delta)\stackrel{h}\to
X\stackrel{f}\to L(\lambda-n\delta)$ is surjective. We are going to
show that the map $h$ factors over the quotient map
$P^{<\lambda}(\lambda-n\delta)\to
P^{\varle\lambda-n\delta}(\lambda-n\delta)\cong
\Delta(\lambda-n\delta)$, so that we get an induced map
$\Delta(\lambda-n\delta)\to X$, which we can take as the map $g$ that we
wanted to construct.
By Lemma \ref{lemma-Vermasubquots}, the module $P^{<\lambda}(\lambda-n\delta)$ is an extension of its subquotients $P(\lambda-n\delta)_{[\nu]}$.
Now $X$ is a submodule of
$V(\lambda)$, which is a module of highest weight $\lambda$. So for each $r>0$ and $z\in \CA_r$ we have
$z^{V(\lambda)}=0$, hence $z^X=0$. Using the Propositions \ref{prop-actonproj} and
\ref{prop-actonprojsub} and the assumption on the multiplicities, we
can inductively show that each Verma subquotient of
$P^{<\lambda}(\lambda-n\delta)$ lies in the kernel of $h$ except possibly $P(\lambda-n\delta)_{[\alpha\uparrow\lambda-n\delta]}$ and $P(\lambda-n\delta)_{[\lambda-n\delta]}$.
But since $[X:L(\alpha\uparrow \lambda-n\delta)]=0$ by part (1) of our main Theorem, also $P(\lambda-n\delta)_{[\alpha\uparrow\lambda-n\delta]}$ is contained in the kernel, so we get an induced map
$$
P(\lambda-n\delta)^{\varle\lambda-n\delta}\cong \Delta(\lambda-n\delta)\to X,
$$
which is what we wanted to show. Hence we proved claim (1).
Secondly, we claim:
\begin{enumerate}
\setcounter{enumi}{1}
\item
{\em The map $\Delta(\lambda)^{\hfn_+}_{\lambda-n\delta} \to V(\lambda)^{\hfn^+}_{\lambda-n\delta}$ that is induced by the canonical map $\pi\colon\Delta(\lambda)\to V(\lambda)$, is surjective.}
\end{enumerate}
So let $v\in
V(\lambda)^{\hfn^+}_{\lambda-n\delta}$, $v\ne 0$, and denote by $X$ the
submodule of $V(\lambda)$ which is generated by $v$. Then $X$ is a
highest weight module with highest weight $\lambda-n\delta$ and there
is a short exact sequence
$$
0\to \Delta({\alpha\downarrow\lambda})\to \pi^{-1}(X)\to X\to 0.
$$
Now the Casimir element $C$ acts on $\pi^{-1}(X)$ as a scalar, since
$\pi^{-1}(X)$ is a submodule of a Verma module. Hence we can apply
Lemma \ref{lemma-Casimir} and we deduce that there is a submodule $Y$
of $\pi^{-1}(X)$ of highest weight $\lambda-n\delta$ that maps
surjectively to $X$. In particular, there is a preimage of $v$ in
$\pi^{-1}(X)^{\hfn^+}_{\lambda-n\delta}\subset \Delta(\lambda)^{\hfn_+}_{\lambda-n\delta}$. So we proved part (2).
Our next step is to prove
\begin{enumerate}
\setcounter{enumi}{2}
\item
{\em We have $[\Delta({\alpha\downarrow\lambda}):L(\lambda-n\delta)]= \dim \Delta({\alpha\downarrow\lambda})_{\lambda-n\delta}^{\hfn_+}$.}
\end{enumerate}
Suppose that the claim is wrong, i.e.~ suppose that $[\Delta({\alpha\downarrow\lambda}):L(\lambda-n\delta)]>\dim
\Delta({\alpha\downarrow\lambda})^{\hfn_+}_{\lambda-n\delta}$. Let $f\colon\Delta(\alpha\downarrow\lambda)\to\Delta(\lambda)$ be a non-zero map and let us consider the composition $g\colon \Delta({\alpha\downarrow\lambda})\stackrel{f}\to \Delta(\lambda)\to\rDelta(\lambda)$.
Let $K$ be the kernel and $X$ the image of $g$. Then $X$ is a highest weight module of highest weight $\alpha\downarrow\lambda$, so $[X:L({\alpha\downarrow\lambda})]=1$. As $g$ factors over the map $\Delta({\alpha\downarrow\lambda})\to\rDelta({\alpha\downarrow\lambda})$, part (1) of our main Theorem implies $[X:L({\alpha\downarrow\lambda}-l\delta)]=0$ for all $l>0$. Our induction assumption implies $[\rDelta(\lambda):L(\lambda-l\delta)]=0$ for all $0<l<n$, hence $[X:L(\lambda-l\delta)]=0$ for all $l<n$.
Now the weights of $K$ are strictly smaller than $\alpha\downarrow\lambda$. By the induction assumption and part (1) of the main Theorem, each subquotient of $K$ of type $L(\lambda-n\delta)$ hence corresponds to a singular vector of weight $\lambda-n\delta$.
Together with the assumption $[\Delta({\alpha\downarrow\lambda}):L(\lambda-n\delta)]>\dim
\Delta({\alpha\downarrow\lambda})^{\hfn_+}_{\lambda-n\delta}$ this allows us to deduce $[X:L(\lambda-n\delta)]>0$.
So $\lambda-n\delta$ occurs as a maximal weight in the maximal submodule of $X$.
Hence there is a primitive vector in $X$, hence in
$\rDelta(\lambda)$, of weight $\lambda-n\delta$. But this contradicts
Theorem \ref{theorem-ResVerma}. Hence our assumption $[\Delta({\alpha\downarrow\lambda}):L(\lambda-n\delta)]>\dim
\Delta({\alpha\downarrow\lambda})^{\hfn_+}_{\lambda-n\delta}$ leads to a contradiction, so we proved claim (3).
Now we can finish the proof of the theorem. Recall that we have to show that $[\Delta(\lambda):L(\lambda-n\delta)]=\dim \Delta(\lambda)^{\hfn_+}_{\lambda-n\delta}$ under the assumption that $[\Delta(\lambda):L(\lambda-l\delta)]=\dim \Delta(\lambda)^{\hfn_+}_{\lambda-l\delta}$ for any $l<n$.
We have
\begin{align*}
\dim\Delta(\lambda)_{\lambda-n\delta}^{\hfn_+} &= \dim\Delta({\alpha\downarrow\lambda})_{\lambda-n\delta}^{\hfn_+}+ \dim V(\lambda)_{\lambda-n\delta}^{\hfn_+}\quad\text{ (by $(2)$)}\\
&= [\Delta({\alpha\downarrow\lambda}):L(\lambda-n\delta)]+[V(\lambda):L(\lambda-n\delta)] \quad\text{ (by $(1)$ and $(3)$)}\\
&= [\Delta(\lambda):L(\lambda-n\delta)],
\end{align*}
which is what we wanted to prove.
\end{proof}
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\section{Additional Proofs}
\subsection{Proof of Theorem~\ref{Main_estimation_theorem} \label{sec:Main_estimation_thm_proof}}
For $k = 1$, we immediately have
\begin{eqnarray}\label{k_eq_1_case}
\Exp_{\theta\sim\calP}\Pr_{\theta}\left[d\langle \vone,\theta \rangle^2 > \tau_{1}\right] \le \sup_{v \in \calS^{d-1}}\Exp_{\theta\sim\calP}\left[d\langle v,\theta \rangle^2 > \tau_{1}\right].
\end{eqnarray}
For $k \in \{2,\dots,T+1\}$, we define the action space $\calA = \calS^{d-1}$, and loss function $\calL(\vkplus,\theta) = \I(\langle \vkplus,\theta \rangle \ge \tau_{k+1})$. Applying Proposition~\ref{chi_sq_fano_prop}, then the inequality~\eqref{Chi_Plus_1_Eq}, we have
\begin{multline}\label{chi_sq_main_recursion}
\Exp_{\theta\sim\calP}\Pr_{\theta}\left[d\langle \vkplus,\theta \rangle^2 > \tau_{k}; A_{\theta}^{k}\right] \\
\le
\sup_{u \in \calS^{d-1}}\Pr_{\theta \sim \calP}\left[d\langle u,\theta \rangle^2 \ge \tau_{k}\right] + \sqrt{\Exp_{\theta \sim \calP}\Exp_{\Prit_0}\left[\left(\frac{\mathrm{d}\Prit_\theta(Z_{k} ; A_\theta^{k})}{\mathrm{d}\Prit_0(Z_{k})}\right)^2 \right]\sup_{v \in \calS^{d-1}}\Pr_{\theta \sim \calP}\left[d\langle v,\theta \rangle^2 \ge \tau_{k}\right] }\\
\le
\sup_{u \in \calS^{d-1}}\Pr_{\theta \sim \calP}\left[d\langle u,\theta \rangle^2 \ge \tau_{k}\right] +
e^{\frac{\lambda^2}{2}\sum_{i=1}^{k} \tau_i}\sqrt{\sup_{v \in \calS^{d-1}}\Pr_{\theta \sim \calP}\left[d\langle v,\theta \rangle^2 \ge \tau_{k}\right] }\\
\le 2
e^{\frac{\lambda^2}{2}\sum_{i=1}^k \tau_i}\sqrt{\sup_{v \in \calS^{d-1}}\Pr_{\theta \sim \calP}\left[d\langle v,\theta \rangle^2 \ge \tau_{k}\right] }.
\end{multline}
Hence,
\begin{align*}
\Exp_{\theta \sim \calP}\Pr_{\theta}&\left[\exists k \in [T+1]: \langle \vk , \theta \rangle^2 > \frac{\tau_k}{d}\right]\\
&= \sum_{k=1}^{T+1} \Exp_{\theta \sim \calP}\Pr_{\theta}\left[\left\{\langle \vk , \theta \rangle^2 > \frac{\tau_k}{d} \right\}\cap \left\{\forall j < k, \langle \vj , \theta \rangle^2 \le \frac{\tau_j}{d}\right\}\right ]\\
&= \sum_{k=1}^{T+1} \Exp_{\theta \sim \calP}\Pr_{\theta}\left[\left\{\langle \vk , \theta \rangle^2 > \frac{\tau_k}{d} \right\};A_{\theta}^{k-1}\right]\\
&\overset{(i)}{\le} \sup_{v \in \calS^{d-1}}\Pr_{\theta \sim \calP}[d\langle v, \theta \rangle^2 \ge \tau_1] + \sum_{k=1}^{T} \Exp_{\theta \sim \calP}\Pr_{\theta}\left[\left\{\langle \vkplus , \theta \rangle^2 > \frac{\tau_{k+1}}{d} \right\};A_{\theta}^{k-1}\right]\\
&\overset{(ii)}{\le} \sup_{v \in \calS^{d-1}}\Pr_{\theta \sim \calP}[d\langle v, \theta \rangle^2 \ge \tau_1] + 2\sum_{k=1}^{T}
e^{\frac{\lambda^2}{2}\sum_{i=1}^k \tau_i}\sqrt{\sup_{v \in \calS^{d-1}}\Pr_{\theta \sim \calP}\left[d\langle v,\theta \rangle^2 \ge \tau_{k+1}\right] }.
\end{align*}
where $(i)$ uses Equation~\eqref{k_eq_1_case}, and $(ii)$ uses~\eqref{chi_sq_main_recursion}.
The theorem follows from a union bound by summing up the above display and combining with Equation~\eqref{k_eq_1_case}.
\subsection{Proof of Corollary~\ref{Main_Cor_estimation}\label{Main_Cor_Proof}}
Set $\tau_1 = 2(\sqrt{\log(1/\delta)} + 1)^2$, so that
\begin{eqnarray*}
\frac{1}{2}(\sqrt{\tau_1} - \sqrt{2})^2 = \log(1/\delta).
\end{eqnarray*}
Now, define $\tau_k \ge \sqrt{2}$ via
\begin{eqnarray*}
\frac{1}{2}(\sqrt{\tau_k} - \sqrt{2})^2 = \lambda^2\sum_{i=1}^{k-1}\tau_i + (k-1)\tau_1.
\end{eqnarray*}
Then, using the fact that $\Pr_{\theta \sim \calP}[d\langle v, \theta \rangle^2 \ge \tau] \le \exp(\frac{-1}{2}(\sqrt{\tau} - \sqrt{2})^2)$ from Lemma~\ref{SphereConcentation}, Theorem~\ref{Main_estimation_theorem} implies that
\begin{eqnarray*}
&& \Exp_{\theta \sim \calP}\Pr_{\theta}\left[\exists k \in [T+1]: \langle \vk , \theta \rangle^2 > \frac{\tau_k}{d}\right] \\
&\le& \exp(\frac{-1}{2}(\sqrt{\tau_1} - \sqrt{2})) + \sum_{k=2}^{T+1} \sqrt{\exp(\lambda^2 \sum_{i=1}^{k-1}\tau_i - \frac{1}{2}(\sqrt{\tau_k} - \sqrt{2})^2)}\\
&=& \exp(\frac{-1}{2}(\sqrt{\tau_1} - \sqrt{2})) + \sum_{k=2}^{T+1} \exp(- \frac{k-1}{2}\tau_1)\\
&\le& \exp(\frac{-1}{2}(\sqrt{\tau_1} - \sqrt{2})) + \sum_{k=2}^{T+1} \exp(- \frac{k-1}{2}(\tau_1-\sqrt{2})^2)\\
&=& \delta + \sum_{k=2}^{T+1} \delta^{k-1} \le \frac{2\delta}{1-\delta}.
\end{eqnarray*}
We now bound that rate at which our $\tau_k$ increase. Since $\tau_i$ are non-decreasing, we have
\begin{eqnarray*}
\frac{1}{2}(\sqrt{\tau_k} - \sqrt{2})^2 = \frac{\tau_k}{2}(1 - \sqrt{2/\tau_k}) \ge \frac{\tau_k}{2}(1 - \sqrt{2/\tau_1}).
\end{eqnarray*}
And thus,
\begin{eqnarray*}
\tau_k \le \frac{2}{1 - \sqrt{2/\tau_1}}\cdot(\lambda^2\sum_{i=1}^{k-1}\tau_i + (k-1)\tau_1) \le \frac{2(\lambda^2 + 1)}{1-\sqrt{2/\tau_1}}\sum_{i=1}^{k-1}\tau_i.
\end{eqnarray*}
For ease, set $\alpha = \frac{2(\lambda^2 + 1)}{1-\sqrt{2/\tau_1}}$, and consider the comparison sequence $\tau_1' = \tau_1$, and $\tau_k' = \alpha\sum_{i=1}^{k-1}\tau_i'$. Then $\tau_k' \ge \tau_k$, and moreover, $\tau_k' \ge \alpha \tau_{k-1}' $, which implies that
\begin{eqnarray*}
\sum_{i=1}^{k-1}\tau_i' = \tau_{k-1}'\sum_{i=1}^{k-1}\frac{\tau_i'}{\tau_{k-1}'} \le \tau_{k-1}\sum_{i=0}^{k-2}\alpha^{-i} \le \frac{\tau_{k-1}'}{1-1/\alpha}.
\end{eqnarray*}
Thus, $\tau_k \le \tau_{k}' = \alpha\sum_{i=1}^{k-1}\tau_i' \le \frac{\alpha}{1 - 1/\alpha}\tau_{k-1}' \le \tau_1'\left(\frac{\alpha}{1 - 1/\alpha} \right)^{k-1} = \tau_1 \left(\frac{\alpha}{1 - 1/\alpha} \right)^{k-1}$. Finally, we bound
\begin{eqnarray*}
&& \frac{\alpha}{1 - 1/\alpha} = 2\lambda^2 \cdot \frac{1 + 1/\lambda^2}{(1 - \frac{1 - \sqrt{2/\tau_1}}{2(\lambda^2 + 1)})(1 - \sqrt{2/\tau_1})} \\
&\le& 2\lambda^2 \cdot \frac{1 + 1/\lambda^2}{(1 - 1/2\lambda^2)(1 - \sqrt{1/(1+\sqrt{\log(1/\delta)} )^2})}\\
&=& 2\lambda^2 \cdot \frac{1 + 1/\lambda^2}{(1 - 1/2\lambda^2)(1 - \sqrt{1/(1+ \log(1/\delta)})}\\
&=& 2\lambda^2 \cdot c(\lambda,\delta).
\end{eqnarray*}
which implies that
\begin{eqnarray*}
\tau_k \le \tau_1 \left(2\lambda^2 \cdot c(\lambda,\delta)\right)^{k-1} = 2(\sqrt{\log(1/\delta)} + 1)^2\left(2\lambda^2 \cdot c(\lambda,\delta)\right)^{k-1}.
\end{eqnarray*}
\subsection{Proof of Proposition~\ref{estimation_cor_2}~\label{Est_Cor_proof}}
Fix any $\lambda \ge 1$. Recall the function $c(\lambda,\delta) := (1 + 1/\lambda^2)\left\{(1 - 1/2\lambda^2)(1 - 1/\sqrt{1+\log(1/\delta)}\right\}^{-1}$ from Corollary~\ref{Main_Cor_estimation}. For $\delta \le 1/e$ and $\lambda \ge 1$, we have
\begin{eqnarray*}
c(\lambda,\delta) \le (1 + 1/\lambda^2)\left\{(1 - 1/2\lambda^2)(1 - \sqrt{1+\log(1/\delta)}\right\}^{-1} \le \frac{2}{(1/2)(1 - 1/\sqrt{2})} := c'.
\end{eqnarray*}
Next, fix any $\eta \ge 0$ and choose $\delta$ such that
\begin{eqnarray}\label{delta_cor_eq}
\log (1/\delta) \le \left(\sqrt{\frac{d \eta}{2 \left(2\lambda^2 \cdot c'\right)^{T}} } - 1 \right)^2.
\end{eqnarray}
If we suppose that
\begin{eqnarray}\label{eta_eq}
\sqrt{\frac{d \eta}{2 \left(2\lambda^2 \cdot c'\right)^{T}} } \ge 2.
\end{eqnarray}
This implies that $\delta \le 1/e$, so $c'\ge c(\lambda,\delta)$ and thus
\begin{eqnarray}
\eta \ge (\left(2\lambda^2 \cdot c'\right)^{T}\cdot \frac{2\left(\sqrt{\log(1/\delta)} + 1\right)^2}{d} \ge (\left(2\lambda^2 \cdot c(\lambda,\delta)\right)^{T}\cdot \frac{2\left(\sqrt{\log(1/\delta)} + 1\right)^2}{d}.
\end{eqnarray}
Then
\begin{eqnarray*}
&&\Exp_{\theta \sim \calP}\Pr_{\theta}\left[ \langle \widehat{v}, \theta \rangle^{2} \ge \eta \right] \\
&\le& \Exp_{\theta \sim \calP}\Pr_{\theta}\left[ \langle \widehat{v}, \theta \rangle^{2} \ge (\left(2\lambda^2 \cdot c(\lambda,\delta)\right)^{T}\cdot \frac{2\left(\sqrt{\log(1/\delta)} + 1\right)^2}{d} \right] \\
&\le& \Exp_{\theta \sim \calP}\Pr_{\theta}\left[ \langle \vk, \theta \rangle^{2} \ge (\left(2\lambda^2 \cdot c(\lambda,\delta)\right)^{k-1}\cdot \frac{2\left(\sqrt{\log(1/\delta)} + 1\right)^2}{d} \forall k \in [T+1]\right] \\
&\overset{(i)}{\le}&\frac{2\delta}{1-\delta} \overset{(ii)}{=} \frac{2\delta}{1-1/e},
\end{eqnarray*}
where $(i)$ uses Corollary~\ref{Main_Cor_estimation} and $(ii)$ uses that $\delta \le 1/e$. Moreover, by Equation~\eqref{delta_cor_eq}, we have
\begin{eqnarray*}
\delta = \exp(-\log(1/\delta)) &\le& \exp\left(- \left(\sqrt{\frac{d \eta}{2 \left(2\lambda^2 \cdot c'\right)^{T}} } - 1 \right)^2\right)\\
&\le& \exp\left(- \left(\frac{1}{2}\sqrt{\frac{d \eta}{2 \left(2\lambda^2 \cdot c'\right)^{T}} }\right)^2\right)\\
&=& \exp\left(- \frac{d \eta}{8 \left(2\lambda^2 \cdot c'\right)^{T}} \right).
\end{eqnarray*}
Thus, if $\sqrt{\frac{d \eta}{2 \left(2\lambda^2 \cdot c'\right)^{T}} } \ge 2$, we have
\begin{eqnarray*}
\Exp_{\theta \sim \calP}\Pr_{\theta}\left[ \langle \widehat{v}, \theta \rangle^{2} \ge \eta \right] \le \frac{2}{1-e} \cdot \exp\left(- \frac{d \eta}{8 \left(2\lambda^2 \cdot c'\right)^{T}} \right).
\end{eqnarray*}
On the other hand, if $\sqrt{\frac{d \eta}{2 \left(2\lambda^2 \cdot c'\right)^{T}} } < 1$, then the right hand side of the above display is at least $1$, so the result also holds vacuously.
\subsection{Proof of Lemma~\ref{Detection_Prop}\label{DetecLemProof}}
Let $c(\lambda,\delta)$ be as above, and fix $\delta \le 1/2$ and $\lambda \ge 1$. Then,
\begin{eqnarray*}
\Exp_{\Prit_0}\left[\left(\frac{\rmd\overline{\Prit}[\cdot;\{A_{\theta}^T\}]}{\rmd\Prit_0}\right)^{2}\right] &=& \Exp_{\theta,\theta' \sim \calP}\Exp_{\Qit}\left[\frac{\rmd\Prit_{\theta}[\cdot;A_{\theta}^T]\rmd\Prit_{\theta'}[\cdot;A_{\theta'}^T]}{(\rmd\Qit)^2}\right] \\
&=& \Exp_{\theta,\theta' \sim \calP}e^{\lambda^2\{|\langle \theta , \theta' \rangle| \sum_{i=1}^T \tau_i + \frac{1}{d}(\sum_{i=1}^T \tau_i)^2\}}\\
&=& e^{\frac{\lambda^2}{d}(\sum_{i=1}^T \tau_i)^2} \cdot \Exp_{\theta,\theta' \sim \calP}e^{\lambda^2(\sum_{i=1}^T \tau_i )\{|\langle \theta , \theta' \rangle| \}}\\
&\le& e^{\frac{\lambda^2}{d}(\sum_{i=1}^T \tau_i)^2} \cdot e^{\frac{4\lambda^4}{d}(\sum_{i=1}^T \tau_i )^2 + \lambda^2(\sum_{i=1}^T \tau_i )\sqrt{2/d}}\\
&=& \exp \left\{\left(\frac{2\lambda^2\sqrt{1 + 1/4\lambda^2} }{\sqrt{d}}\left(\sum_{i=1}^T \tau_i \right)\right)^2+ \frac{\sqrt{2}\lambda^2}{\sqrt{d}}\left(\sum_{i=1}^T \tau_i \right)\right\}\\
&\overset{(i)}{\le}& \exp \left\{\frac{(2\sqrt{5/4} + \sqrt{2})\lambda^2}{\sqrt{d}}\cdot \sum_{i=1}^T \tau_i\right\} \le \exp \left\{\frac{\sqrt{7}\lambda^2}{\sqrt{d}}\cdot \sum_{i=1}^T \tau_i\right\},
\end{eqnarray*}
where $(i)$ holds as long as $\frac{(2\sqrt{5/4} + \sqrt{2})\lambda^2}{\sqrt{d}}\cdot \sum_{i=1}^T \tau_i \le 1$ and $\lambda \ge 1$. On the other hand, from Corollary~\ref{Main_Cor_estimation}, we can choose $\tau_1,\dots, \tau_T$ such that, $\Exp_{\theta}\Prit_{\theta}[A_{\theta}^T] \ge \frac{2\delta}{1-\delta}$, and
\begin{eqnarray*}
\sum_{i=1}^T\tau_i \le 4(2\lambda^2 \cdot c(\lambda,\delta))^{T-1} \left(\sqrt{\log(1/\delta)} + 1\right)^2.
\end{eqnarray*}
Then, as long as $4(2\lambda^2 \cdot c(\lambda,\delta))^{T-1} \left(\sqrt{\log(1/\delta)} + 1\right)^2 \le 1$, we have
\begin{eqnarray*}
\Exp_{\Prit_0}\left[\left(\frac{\rmd\overline{\Prit}[\cdot;\{A_{\theta}^T\}]}{\rmd\Prit_0}\right)^{2}\right] &\le& \exp\left\{\frac{4\sqrt{7}\lambda^2}{\sqrt{d}}(2\lambda^2 \cdot c(\lambda,\delta))^{T-1} \left(\sqrt{\log(1/\delta)} + 1\right)^2\right\}\\
&\overset{(i)}{\le}& \exp\left\{\frac{4\sqrt{7}}{\sqrt{d}}(2\lambda^2 \cdot c(\lambda,\delta))^{T} \left(\sqrt{\log(1/\delta)} + 1\right)^2\right\}.
\end{eqnarray*}
where $(i)$ uses that $c(\lambda,\delta) \ge 1$. Since $\delta \le 1/2$ and $\lambda \ge 1$, we have that
\begin{eqnarray*}
c(\lambda,\delta) \le c'' := \max_{\delta \le 1/2,\lambda \ge 1}c(\lambda,\delta) = c(1,1/2) < \infty.
\end{eqnarray*}
We can then bound
\begin{eqnarray*}
\Exp_{\Qit}\left[\left(\frac{\rmd\overline{\Prit}[\cdot;\{A_{\theta}^T\}]}{\rmd\Qit}\right)^{2}\right] \le \exp\left\{\frac{4\sqrt{7}\left(c'' \lambda^2\right)^T}{\sqrt{d}} \cdot \left(\sqrt{\log(1/\delta)} + 1\right)^2\right\}.
\end{eqnarray*}
First, suppose that the quantity in the above exponential is less than $1/2$, then using the inequality $e^{x} - 1 \le 2x$ for $x \le 1/2$, we can bound
\begin{eqnarray}\label{exminusbound}
\Exp_{\Prit_0}\left[\left(\frac{\rmd\overline{\Prit}[\cdot;\{A_{\theta}^T\}]}{\rmd\Prit_0}\right)^{2}\right] \le \exp\left\{\frac{4\left(c'' \lambda^2\right)^T}{\sqrt{d}} \cdot \left(\sqrt{\log(1/\delta)} + 1\right)^2\right\} - 1 \le \frac{8\left(c'' \lambda^2\right)^T}{\sqrt{d}} \cdot \left(\sqrt{\log(1/\delta)} + 1\right)^2.
\end{eqnarray}
Take $\delta = \frac{\left(c'' \lambda^2\right)^T}{\sqrt{d}}$. If $\delta \le 1/2$ and $\frac{8\left(c'' \lambda^2\right)^T}{\sqrt{d}} \cdot \left(\sqrt{\log(1/\delta)} + 1\right)^2 \le 1$, then
\begin{eqnarray*}
\|\Prit_0 - \overline{\Prit}\|_{TV} &\overset{(i)}{\le}& \frac{1}{2}\sqrt{\Exp_{\Qit}\left[\left(\frac{\rmd\overline{\Prit}[\cdot;\{A_{\theta}^T\}]}{\rmd\Qit}\right)^{2}\right] - 1} + \frac{\sqrt{2\Exp_{\theta\sim \calP}\Prit_{\theta}[A_{\theta}^T]} +\Exp_{\theta\sim \calP}\Prit_\theta[A_{\theta}^T]}{2}\\
&\overset{(ii)}{\le}&\left(\sqrt{\log(1/\delta)} + 1\right)\cdot\sqrt{\frac{2\left(\gamma'' \lambda^2\right)^T}{\sqrt{d}} } + \frac{\sqrt{2\Exp_{\theta \sim \calP}\Prit_\theta [A_{\theta}^T]} +\Exp_{\theta \sim \calP}\Prit_\theta[A_{\theta}^T]}{2}\\
&\overset{(iii)}{\le}&\left(\sqrt{\log(1/\delta)} + 1\right)\cdot\sqrt{\frac{2\left(\gamma'' \lambda^2\right)^T}{\sqrt{d}} } + \frac{\sqrt{2(\frac{2\delta}{1-\delta})}+ \frac{2\delta}{1-\delta}}{2}\\
&\le& \left(\sqrt{\log(1/\delta)} + 1\right)\cdot\sqrt{\frac{2\left(\gamma'' \lambda^2\right)^T}{\sqrt{d}} } + 4\sqrt{2\delta} \\
&\le& \sqrt{2}(\log^{1/2}\frac{\sqrt{d}}{\left(\gamma'' \lambda^2\right)^T} + 4)\cdot \sqrt{\frac{\left(\gamma'' \lambda^2\right)^T}{\sqrt{d}}}.
\end{eqnarray*}
where $(i)$ uses Propostion~\ref{truncChi_detec}, $(ii)$ uses Equation~\eqref{exminusbound}, and $(iii)$ uses the fact that $\Exp_{\theta\sim \calP}\Prit_{\theta}[A_{\theta}^T] \le 2\delta/(1-\delta)$. On the other hand, if $\delta \ge 1/2$, or if $\frac{8\left(c'' \lambda^2\right)^T}{\sqrt{d}} \cdot \left(\sqrt{\log(1/\delta)} + 1\right)^2 \ge 1$, then the last line of the above display is at least $1$, so the conclusion remained true because $\|\Prit_0 - \overline{\Prit}\|_{\TV}\le 1$.
\begin{comment}
For every $\delta > 0$, there exists a choice of $\tau_1,\dots,\tau_T$ such that $\Exp_{\theta}\Pr_{\Pr_{\theta}}[A_{\theta}^T] \ge 1 - \frac{2\delta}{1-\delta}$, and
\begin{eqnarray}
\sqrt{\Exp_{\Q}\left[\left(\frac{d\overline{\Pr}[\cdot;\{A_{\theta}^T\}]}{d\Q}\right)^{2}\right] - 1} \le \left(\sqrt{\log(1/\delta)} + 1\right)\cdot\sqrt{\frac{8\left(c_3 \lambda^2\right)^T}{\sqrt{d}} }
\end{eqnarray}
where $c_3 $ is a numerical constant.
\end{comment}
| 149,811
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KNOXVILLE, Tenn. -- Saturday night at halftime, Lane Kiffin changed his clothes, ditching a black sweater in favor of an orange pullover. Presumably the wardrobe change was a superstitious response to an awful offensive half, one that saw the Vols with nine yards total passing until the final two-minute drive. If only Kiffin were less stubborn about his signal-caller. News flash, Kiffin could coach on the sideline in a burka or a Japanese sumo outfit and the result on the field would be the same -- Jonathan Crompton is going to lose the game.
Early in the season Kiffin adopted the coaching cliche, "If you've got two quarterbacks, you've got none." I'd like to advance another version of that cliche: "If your one quarterback is Jonathan Crompton, then you still ain't got one either." Right now, Kiffin's refusal to make a change at quarterback is slowly bleeding his head-coaching legitimacy among the fan base. In his first season Kiffin has struck an iceberg, and he's going down on the S.S. Crompton. So is his team. It's time for a change.
Lane Kiffin has been brash, confident, and quotable. What he hasn't been is a winner. Anywhere. In 25 games as a head coach, Kiffin is now 7-18 (5-15 with the NFL's Oakland Raiders and 2-3 with the Vols). With Georgia, Alabama, and South Carolina in the next three games, it's altogether possible that Kiffin and the Vols are going to be sitting at 2-6 by the time November arrives. Another season of 5-7 or worse looms. And here's the kicker, next year Tennessee is going to be worse. In this day and age if your second season isn't a good one, you're not going to truly succeed as a coach. Pete Carroll, Nick Saban, Bob Stoops, Urban Meyer, Mark Richt -- every single one of those coaches had great second seasons. That's when their teams made a seismic jump.
Kiffin's situation is unique because his team, due to the graduating seniors on the offensive line and at tailback, and the presumed early departure of Eric Berry, is going to be worse next year. He.
"(Kiffin)." At this point in the season, standing at 2-3, what does Jonathan Crompton, a senior who has managed to win one SEC game in which he completed a pass in his career -- Kentucky last year -- actually give you if he plays great? The chance to finish 6-6? And then be gone from football forever?
Meanwhile, you have a redshirt junior in Nick Stephens standing on the sideline. Worst case scenario, Stephens comes in and only wins two games as your starting quarterback. But at least you give him a chance to prove that he can be your guy for a year in 2010.
My point is, I've finally come around to this argument: Stephens can't be worse. He just can't. And now it's time for a change.
Here are other observations:
1. Gene Chizik and Auburn, particularly Gus Malzahn, have blown Tennessee's highest paid coaching staff in the country out of the water this season in terms of performance.
Kiffin defenders scream, "Talent, talent, talent."
That's all well and good, but does anyone really think Auburn has more talent than Tennessee? Last year's Tennessee-Auburn game ended 14-12 and set back offensive football five decades. Now compare the two teams this year. Which looks different, which looks improved?
That's almost entirely a product of coaching, right?
In fact, this game was almost a perfect laboratory for coaching analysis. Compare the products on the field last night. Kiffin, who was supposedly an offensive guru at USC, has not improved Tennessee's offense. Chizik, under the direction of Gus Malzahn, has completely remade Auburn. They're now 5-0, loving football, and have completely bought in to what the coaching staff is selling.
Meanwhile, Kiffin and Tennessee are regressing offensively and defensively.
You can argue talent differential in games against Florida and Georgia, maybe, but you can't argue talent differential in games like Auburn and UCLA. What you can argue is inferior preparation. Again, I'll say what I did after the UCLA game: if Phillip Fulmer is standing on the sideline and makes every play call that Kiffin did, fans are outraged.
That first half of football was unwatchable.
Kiffin bears the blame.
Taking it further, there were two primary rationales to replace Fulmer: a. the team needed to be coached better and, b. the recruiting had suffered.
So far Kiffin's offense and defense look no different than Fulmer's did. So now the rationale for the coaching change boils down completely to recruiting.
Kiffin is recruiting well, but Tennessee has always recruited well. Fulmer had better players than every other team in the SEC during his tenure (using the NFL Draft as the barometer). What have we seen on the field thus far that offers clear evidence that Tennessee is being coached better?
2. Crompton's receivers didn't help him by making catches, but that's partially because they don't believe in him at quarterback.
Of course not.
They're so worried about trying to make a spectacular play for the offense, that they can't make a simple play.
Why?
Because they don't trust Crompton to make plays. Even if they're not saying it out loud, their body language tells the story. Watch how long the receivers take to get up after another failed pass attempt. The downcast head. They're beaten before the ball is snapped.
3. It's time to toss practice out the window when it comes to evaluating quarterbacks.
I said it earlier, but it bears repeating, Jonathan Crompton has won a single SEC game when he completes a pass -- Kentucky. (He also "beat" Vanderbilt as the starter last year, but his only pass, the first of the game, was intercepted.) In fact, Kentucky is the only team from a major conference that Crompton has beaten in his career.
My point, there's enough game experience film to evaluate at this point to make a decision on what the game play is going to be like. Using practice as a proxy for games doesn't make sense anymore. For whatever reason, if the coaching staff is to be believed, Crompton's talents don't translate to Saturday.
So be it.
Nick Stephens deserves his shot to see what he does in games. By all accounts, he tends to do better in games than practice. Give him a chance.
4. This team is divided already. Changing quarterbacks won't make it any worse.
For the first time in two years, the defense buckled in a game. It happened at the end of the first half as Auburn was in the process of running up 49 offensive snaps in the first half.
Forty-nine!
Tennessee had to take two timeouts to rest its defense.
Let me repeat that, Tennessee had to take two timeouts to rest the defense.
I can't imagine any more glaring indictment of the offensive performance than this. It's downright shameful how wasted the Tennessee defense is. And if you don't think those guys on defense are looking out at the offense, watching Crompton give up a field goal to the other team by dropping the ball on the center exchange, for example, you're fooling yourself.
This team is already divided along offense and defense lines. And it's only going to get worse as long as Crompton is in there. At least if a change is made, there's a tangible sign that the offense is willing to try anything to get better.
5. Why the lack of offensive ingenuity?
I want one person to explain to me why Nu'Keese Richardson carries the ball for 40 yards on the first play of the game and we never see him again.
Kiffin went to war for Richardson, brought on the wrath of an entire nation, turned Nu'Keese into a modern day Helen of Troy -- with an apostrophe -- and he can't even use him for more than one direct snap after the first one is hugely successful? That makes zero sense.
If your offense is awful, isn't it the coaching staff's responsibility to find ways to get players chances to make plays?
Putting this into context, the majorettes twirled flaming batons at halftime of the game. It was the most explosive offensive performance on the part of anyone from Tennessee.
6. Late in the third quarter, Lane Kiffin went for it on 4th-and-1 at his own 29.
This play speaks volumes -- Kiffin is even more frustrated than the biggest fan. I get that. But it's also making him choose risky options that offer limited payoffs. Down just 16-6, Kiffin put the game on the line with this decision.
What was the payoff for this ill-advised gamble? Tennessee punted three plays later from its 34. Yep, they risked the entire game for five yards of field position.
The risk-reward ratio when it comes to the Tennessee offense is completely broken. And it's leading the coach to make poor decisions that have limited tactical benefit -- what were the odds that Tennessee was going to score a touchdown on that drive with a first down at its own 30? To me that's clear evidence that a change needs to be made at the quarterback position. Even the head coach is pressing in his play calling due to the offensive woes.
7. It's time someone calls Kiffin on playing Crompton to help recruiting, and I'm going to do it.
Kiffin defenders consider his refusal to bench Crompton to be a strength. I think that's misguided. In fact, I actually think it's unfair to Crompton.
Why?
Because part of the reason Kiffin is leaving Crompton in at quarterback to bear the brunt of the criticism is because he wants to bring in at least one and potentially two stud quarterbacks in this year's recruiting class. That might make sense in the long-term, but it makes Kiffin's defense of Crompton ring hollow in the meantime.
It also makes Crompton a de facto shield for Kiffin. Instead of blaming the coaches, fans blame Crompton. I want the Crompton shield removed.
8. Two plays from last night that sum up the Crompton era.
a. Late in the third quarter, Crompton drops back and completes a pass to Gerald Jones, running a drag route across the field with three defenders in close proximity to him.
Hooray, a completed pass.
Only, the wide receiver on that same side of the field, I believe it was Denarius Moore, was left all alone running down the sideline. The receiver waved his arm, leaped, and the entire crowd in my section screamed for Crompton to uncork his massive arm and heave a pass down the field that would potentially unlock the offensive miasma.
But Crompton never saw his open receiver. He was locked on his primary read.
That's despite Tennessee rolling Crompton out to one side or the other throughout the game so he'd only have to survey one-third of the field.
Yep, even when receivers are wide-open in the one-third of the field that Crompton is facing, he doesn't see them.
b. The trip and fall play to begin the second half.
Coming out of the locker room, the Tennessee defense holds the Auburn offense to a three-and-out. Tennessee takes possession coming off a touchdown drive. Crompton takes the snap, steps back, is tripped by his own man, and falls down in the backfield for a loss of four.
This play might not be Crompton's fault, but no one on the offense and no one in the crowd was surprised that it happened to Crompton. I don't bear him any ill-will, I don't think he's a bad guy, I hope he's a success in something other than football, but it's time for his football career at Tennessee to come to a close.
Kiffin has given him five games. Crompton has proven that's he not up to the task.
Saturday night as I left the stadium a black man in a suit played a mournful dirge on his saxophone. Standing in Phillip Fulmer Way with an open case in front of him, his music soared into the clear night sky as downcast Vol fans passed him.
One fan, an older man in a Vol parka, reached into his wallet and pulled out a dollar bill. "You're the best player I've seen all night," the man said.
The saxophone player nodded as he continued to play. It was October 3rd and already fall seemed like it stretched forward into eternity.
"That damn Crompton," the man said, shaking his head, "that damn Crompton."
...
Clay Travis is the author of three books. His latest, "On Rocky Top: A Front Row Seat to The End of an Era" chronicles the 2008 Tennessee football season and is on sale now.
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TITLE: What does it mean that there is an "isomorphism of homsets" due to an exponential object?
QUESTION [0 upvotes]: $X^Y$ together with a morphism $\bf{ apply}$ $:X^Y\times Y\to X$ is an exponential object of $X$ and $Y$ if for each $Z$ and each morphism $f:Z\times Y \to X$, there is a unique morphism $\lambda f:Z\to X^Y$ such that $\lambda f \times id_Y \circ \bf {apply}$ $ = f$.
It is obvious that for every $f$ there is a $\lambda f$. But two things are not obvious to me:
Is there necessarily for every morphism $\lambda g:Z\to X^Y$ a corresponding morphism $g:Z\times Y \to X$? Couldn't we have a category where there are these $\lambda f:Z\to X^Y$, but also additional extraneous morphisms $\lambda g:Z\to X^Y$ that have no corresponding $g$? Can't we always add such extraneous morphisms?
Assuming that the previous question is clarified, and there is indeed a bijection between these homsets, why does wikipedia call it a "isomorphism" of homsets? An isomorphism is only an isomorphism within a category, and I don't know what category they're talking about.
REPLY [3 votes]: (i) : No : start from $h : Z\to X^Y$; then you have $h\times id_Y : Z\times Y\to X^Y\times Y$, and then you can compose with $\mathbf{apply}$ to get $g:= \mathbf{apply}\circ (h\times id_Y) : Z\times Y\to X$.
Then by uniqueness, $\lambda g = h$.
(ii) : A bijection between sets is an isomorphism in the category of sets $\mathbf{Set}$.
But here it's even lore than that, because the isomorphism $\hom(Z\times Y, X) \cong \hom(Z, X^Y)$ is natural in all three variables, so it is actually an isomorphism between two functors in the category of functors $C^{op}\times C^{op}\times C\to \mathbf{Set}$
| 61,270
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PROUD TO BE AN INDIAN
Once Again, ISA Students at Rashtrapathi Bhavan
Two of our students, MANASVI UDAYA KUMAR (8-A) and THEJALAKSHMI ANIL (8-B), were selected to be a part of the Proud to be an Indian journey organized by Asianet News channel.
The selected students from all the emirates were taken on a road show around three prestigious schools in the UAE, among which Indian School Al Ain was one of them.
The students had the privilege to watch the Republic Day parade in Delhi on 26th January 2017. They also visited Shimla. The students accompanied by Mrs. Safeeda Nazar, got the golden opportunity to meet Indian President, Pranab Mukherjee at the Rashtrapati Bhavan.
| 198,263
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In computer science, particularly the field of databases, the Thomas write rule is a rule in timestamp-based concurrency control. It can be summarized as ignore outdated writes.
It states that, if a more recent transaction has already written the value of an object, then a less recent transaction does not need perform its own write since it will eventually be overwritten by the more recent one.
The Thomas write rule is applied in situations where a predefined logical order is assigned to transactions when they start. For example a transaction might be assigned a monotonically increasing timestamp when it is created. The rule prevents changes in the order in which the transactions are executed from creating different outputs: The outputs will always be consistent with the predefined logical order.
For example consider a database with 3 variables (A, B, C), and two atomic operations C := A (T1), and C := B (T2). Each transaction involves a read (A or B), and a write (C). The only conflict between these transactions is the write on C. The following is one possible schedule for the operations of these transactions:
<math>\begin{bmatrix}
T_1 & T_2 \\
& Read(A) \\
Read(B) & \\
&Write(C) \\
Write(C) & \\
Commit & \\
& Commit \end{bmatrix} \Longleftrightarrow
\begin{bmatrix}
T_1 & T_2 \\
& Read(A) \\
Read(B) & \\
& Write(C) \\
& \\
Commit & \\
& Commit\\
\end{bmatrix}
</math>
If (when the transactions are created) T1 is assigned a timestamp that precedes T2 (i.e., according to the logical order, T1 comes first), then only T2's write should be visible. If, however, T1's write is executed after T2's write, then we need a way to detect this and discard the write.
One practical approach to this is to label each value with a write timestamp (WTS) that indicates the timestamp of the last transaction to modify the value. Enforcing the Thomas write rule only requires checking to see if the write timestamp of the object is greater than the time stamp of the transaction performing a write. If so, the write is discarded
In the example above, if we call TS(T) the timestamp of transaction T, and WTS(O) the write timestamp of object O, then T2's write sets WTS(C) to TS(T2). When T1 tries to write C, it sees that TS(T1) < WTS(C), and discards the write. If a third transaction T3 (with TS(T3) > TS(T2)) were to then write to C, it would get TS(T3) > WTS(C), and the write would be allowed.
| 36,539
|
\begin{document}
\maketitle
\begin{abstract}
\small We investigate the birational section conjecture for curves over function fields of characteristic zero and prove that the conjecture holds over finitely generated fields over $\Q$ if it holds over number fields.
\end{abstract}
\tableofcontents
\section{Introduction and statement of results}
\subsection{The birational section conjecture}
For a smooth, geometrically connected, projective curve $X$ over a characteristic zero field $k$, we define the \emph{absolute Galois group of $X$} to be the group
\[G_X:=\Gal(\overline {k(X)}/k(X))\]
where $k(X)$ denotes the function field of $X$ and $\overline {k(X)}$ is an algebaic closure of $k(X)$. The Galois group $G_X$ fits into an exact sequence
\[\sexact{G_{X_{\bar k}}}{}{G_X}{}{G_k}\]
where $G_k := \Gal(\bar k|k)$, $\bar k$ being the algebraic closure of $k$ in $\overline {k(X)}$, and $G_{X_{\bar k}}:=\Gal (\overline {K(X)}/K(X) \cdot \bar k)$. Let $x \in X(k)$ be a $k$-rational point, and let $\tilde x$ be a valuation of $\overline {k(X)}$ extending the valuation $\nu_x$ of $k(X)$ corresponding to $x$. We will refer to $\tilde x$ as an \emph{extension of $x$ to $\overline {k(X)}$}. The \emph{decomposition group} $D_{\tilde x}$ of $\tilde x$ fits into a commutative diagram of exact sequences
\[\begin{tikzcd}
1 \arrow{r}{} &I_{\tilde x} \arrow{r}{}\arrow[hookrightarrow]{d}{} &D_{\tilde x} \arrow{r}{}\arrow[hookrightarrow]{d}{} &G_{k(x)} \arrow{r}{}\arrow[equals]{d}{} &1\\
1 \arrow{r}{} &G_{X_{\bar k}} \arrow{r}{} &G_X \arrow{r}{} &G_k \arrow{r}{} &1
\end{tikzcd}\]
where $I_{\tilde x}$ is the \emph{inertia group} at $\tilde x$. We will refer to a splitting of the lower sequence in the above diagram as a \emph{section of $G_X$}. A splitting of the upper exact sequence naturally defines a section $s_{\tilde x} : G_k \to G_X$ of $G_X$, with image contained in $D_{\tilde x}$.
\begin{definition}\label{geometricgaloissections}
We say a section $s$ of $G_X$ is \emph{geometric} if its image $s(G_k)$ is contained in a decomposition group $D_{\tilde x}$ for some $k$-rational point $x\in X(k)$ and some extension $\tilde x$ of $x$ to $\overline {k(X)}$. In this case, we say that the section $s$ \emph{arises from the point} $x$.
\end{definition}
The birational analogue of Grothendieck's anabelian section conjecture for \'etale fundamental groups may be stated as follows.
\begin{conjecture*}
Let $k$ be a finitely generated field over $\Q$, and let $X$ be a smooth, projective, geometrically connected curve over $k$. Then every section of $G_X$ is geometric and arises from a unique $k$-rational point $x\in X(k)$.
\end{conjecture*}
We will refer to this as the \emph{birational section conjecture} or \textbf{BSC}. One may consider the statement for more general fields $k$, so we establish the following terminology.
\begin{definition}\label{bscholds}
\begin{enumerate}
\item Let $X$ be a smooth, geometrically connected, projective curve over a field $k$. We say the birational section conjecture (or \textbf{BSC}) \emph{holds for $X$} if every section of $G_X$ is geometric and arises from a unique $k$-rational point $x\in X(k)$.
\item For a field $k$, we say the birational section conjecture (or \textbf{BSC}) \emph{holds over $k$} if the \textbf{BSC} holds for every smooth, geometrically connected, projective curve over $k$.
\end{enumerate}
\end{definition}
\begin{remark}\label{bscuniqueness}
To prove that the \textbf{BSC} holds for $X$ it suffices to prove that every section of $G_X$ arises from a $k$-rational point $x \in X(k)$. This is necessarily the unique such point, since decomposition subgroups of $G_X$ associated to distinct rank $1$ valuations of $\overline {k(X)}$ intersect trivially \cite[Corollary 12.1.3]{CONF} (in loc. cit. $k$ is a global field but the same argument of proof works over any field).
\end{remark}
It is hoped that the birational section conjecture might be used to prove the Grothendieck anabelian section conjecture for \'etale fundamental groups, via the theory of ``cuspidalisation'' of sections of arithmetic fundamental groups \cite{saidicusp}. Conversely, the anabelian section conjecture for $\pi_1$ implies the birational section conjecture, as follows easily from the ``limit argument'' of Akio Tamagawa \cite[Proposition 2.8 (iv)]{tamagawa}. At present the \textbf{BSC} over finitely generated fields over $\Q$, as well as the anabelian section conjecture for $\pi_1$, are quite open. Few examples are known of curves over number fields for which the {\bf BSC} holds. More precisely, with the notations and assumptions in Definition 1.2(i), if $k$ is a number field, $X$ is hyperbolic, $J(k)$ is finite where $J$ is the jacobian of $X$, and the Shafarevich-Tate group of $J$ is finite, then the \textbf{BSC} holds for $X$ (see \cite[Remark 8.9]{stoll} for related examples). Also, some conditional results on the birational section conjecture over number fields of small degree are known (see \cite{hoshi}). Note that a $p$-adic analog of the birational section conjecture holds by \cite{koenigsmann}. To the best of our knowledge no result on the birational section conjecture is known for curves over finitely generated fields over $\Q$ of positive transcendence degree.
\subsection{Statement of the Main Theorems}
In this paper we investigate the birational section conjecture over function fields. We prove that, for a certain class of fields $k$ of characteristic zero, and under the condition of finiteness of certain Shafarevich-Tate groups, proving that the \textbf{BSC} holds over function fields over $k$ reduces to proving that it holds over finite extensions of $k$. This class of fields contains, in particular, the finitely generated extensions of $\Q$, and for such fields we show further that the statement is independent of finiteness of the Shafarevich-Tate groups. The result therefore reduces the \textbf{BSC} over finitely generated extensions of $\Q$ unconditionally to the case of number fields.
Our approach stems from the proof in \cite{saidiSCOFF} of a similar result for the section conjecture for \'etale fundamental groups. Let us start by describing the aforementioned class of fields, which was introduced in \cite[Definition 0.2]{saidiSCOFF}.
\begin{definition}\label{conditions}
For a field $k'$ of characteristic zero, consider the following conditions on $k'$.
\begin{enumerate}
\item The {\bf BSC} holds over $k'$.
\item For every prime integer $\ell$, the $\ell$-cyclotomic character $\chi_{\ell}:G_{k'}\rightarrow\Z^{\times}_{\ell}$ is non-Tate, meaning that any $G_{k'}$-map $\Z_{\ell}(1)\rightarrow T_{\ell} A$, for some abelian variety $A$ over $k'$ and its $\ell$-adic Tate module $T_{\ell}A$, vanishes.
\item Given an abelian variety $A$ over $k'$, any quotient $A(k')\twoheadrightarrow D$ of the group of $k'$-rational points $A(k')$ satisfies the following:
\begin{enumerate}
\item The natural map $D\rightarrow \widehat D$ is injective, where $\widehat D:=\varprojlim_{N \geq 1} D/ND$.
\item The $N$-torsion subgroup $D[N]$ of $D$ is finite for all $N\geq 1$, and the Tate module $TD$ is trivial (cf. Notation).
\end{enumerate}
\item Given a separated, smooth, connected curve $C$ over $k'$ with function field $K=k'(C)$, $K$ admits the structure of a Hausdorff topological field, so that $X(K)$ is compact for any smooth, geometrically connected, projective, hyperbolic curve $X$ over $K$.
\item Given a separated, smooth, and connected (not necessarily projective) curve $C$ over $k'$ with function field $K=k'(C)$ and a finite morphism $\tilde C\rightarrow C$ with $\tilde C$ separated and smooth, then the following holds. If $\tilde C_c(k'(c))\ne\emptyset$ for all closed points $c\in C^{\textup{cl}}$, where $k'(c)$ denotes the residue field at $c$ and $\tilde C_c$ is the scheme-theoretic inverse image of $c$ in $\tilde C$, then $\tilde C(K)\ne\emptyset$.
\end{enumerate}
For a field $k$ of characteristic zero, we say that $k$ \emph{strongly satisfies} one of the above conditions (i), (ii), (iii), (iv) and (v) if this condition is satisfied by any finite extension $k'|k$ of $k$.
\end{definition}
Condition (i) is expected to hold for all finitely generated fields over $\Bbb Q$ by the \textbf{BSC}.
Conditions (ii)-(v) are satisfied by finitely generated fields over $\Q$ (cf. \cite{saidiSCOFF}, discussion after Definition 0.2).
More precisely, (ii) follows from the theory of weights, (iii) follows from the Mordell-Weil and Lang-N\'eron Theorems, and (iv) follows (for the discrete topology) from Mordell's conjecture: Faltings' Theorem and N\'eron's specialisation Theorem. Condition (v) is satisfied by Hilbertian fields (cf. \cite[Lemma 4.1.5]{saidiSCOFF}), in particular (v) holds for finitely generated fields.
\begin{definition}\label{sha}
Let $k$ be a field of characteristic zero and $C$ a smooth, separated, connected curve over $k$ with function field $K$. Let $\mathcal{A}\rightarrow C$ be an abelian scheme with generic fibre $A := \mathcal{A} \times_C K$. For each closed point $c\in C^{\textup{cl}}$ denote by $K_c$ the completion of $K$ at $c$, and write $A_c := A \times_K K_c$.
We define the Shafarevich-Tate group
\[\Sha(\mathcal{A}) := \ker(H^1(G_K,A)\rightarrow\prod_{c\in C^{\textup{cl}}}H^1(G_{K_c},A_c))\]
where the product is taken over all the closed points of $C$.
\end{definition}
We now state our first two main Theorems.
\begin{maintheorem}
Let $k$ be a field of characteristic zero that satisfies conditions (iv) and (v) of Definition \ref{conditions},
and strongly satisfies conditions (i), (ii), and (iii) of Definition \ref{conditions}. Let $C$ be a smooth, separated, connected curve over $k$ with function field $K$. Let $\X\rightarrow C$ be a flat, proper, smooth relative curve, with generic fibre $X:=\X\times_C K$ which is a geometrically connected hyperbolic curve over $K$ such that $X(K)\ne\emptyset$. Denoting by $\mathcal{J}:=\textup{Pic}^0_{\X/C}$ the relative Jacobian of $\X$, assume $T\Sha(\mathcal{J})=0$. Then the {\bf BSC} holds for $X$.
\end{maintheorem}
\begin{maintheorem}
Let $k$, $C$ and $K$ be as in Theorem A, and assume further that $k$ strongly satisfies conditions (iv) and (v) of Definition \ref{conditions}. For any finite extension $L$ of $K$, let $C^L$ denote the normalisation of $C$ in $L$. Assume that for any such finite extension $L$ and any flat, proper, smooth relative curve $\Y \to C^L$ we have $T\Sha(\J_{\!\Y}) = 0$, where $\J_{\!\Y} := \Pic^0_{\Y/C^L}$ is the relative Jacobian of $\Y$. Then the {\bf BSC} holds over all finite extensions of $K$.
\end{maintheorem}
In the context of $\Q$, this means that if the birational section conjecture holds over all fields of some fixed transcendence degree over $\Q$ then it holds over all fields of transcendence degree one higher, provided that finiteness of $\Sha$ holds. By induction this means that, under the assumption on the relevant Shafarevich-Tate groups, if the birational section conjecture holds over number fields then it holds over all finitely generated fields over $\Q$. As a consequence of one of the main results in \cite{saiditamagawa}, asserting the finiteness of $\Sha$ for isotrivial abelian varieties over finitely generated fields, and using Theorem A, we can prove the following.
\begin{maintheorem} Assume that the {\bf BSC} holds over all number fields. Then the {\bf BSC} holds over all finitely generated fields over $\Bbb Q$.
\end{maintheorem}
Thus Theorem C reduces the proof of the birational section conjecture to the case of number fields.
\subsection{Guide to the proof of the Main Theorems}
Our approach to proving the above Theorems is inspired by the method in \cite{saidiSCOFF}, and relies on a local-global argument. This requires studying the properties of sections of absolute Galois groups of curves over local fields of equal characteristic zero and over function fields of curves in characteristic zero. We relate these two settings by investigating ``\'etale abelian sections'', from where arises the constraint on the Shafarevich-Tate groups.
{\it To the proof of Theorems A and B}. With the notations and assumptions in Theorem A, let $s:G_K \to G_X$ be a section. For each closed point $c\in C^{\cl}$, using the fact that $k$ strongly satisfies condition (ii), as well as a specialisation theorem for absolute Galois groups which may be of interest independently of the context of this paper (cf. \S 3.1), we prove that $s$ naturally induces a birational section $\bar s_c:G_{k(c)}\to G_{\X_c}$ of the absolute Galois group $G_{\X_c}$ of the fibre $\X_c$ at $c$. The fact that $k$ strongly satisfies condition (i) implies that the section $\bar s_c$ is geometric and arises from a unique rational point $x_c\in \X_c(k(c))$ (the unicity follows from the fact that $k$ strongly satisfies condition (iii)(a)).
Next, we consider the \'etale abelian section $s^{\ab}:G_K\to \pi_1(X)^{(\ab)}$ induced by $s$, where $\pi_1(X)^{(\ab)}$ is the geometrically abelian fundamental group of $X$, and similarly the \'etale abelian sections $\bar s_c^{\ab}:G_{k(c)}\to \pi_1(\X_c)^{(\ab)}$ induced by $\bar s_c$, $\forall c\in C^{\cl}$ (cf. \S 2.2). Thus, $s^{\ab}$ and $\bar s_c^{\ab}$ can be viewed as elements of $H^1(G_K,TJ)$, and $H^1(G_{k(c)},T\J_c)$, where $J$ and $\J_c$ are the jacobians of $X$ and $\X_c$ respectively, and $T$ indicates their Tate modules (cf. loc. cit.). We have the following commutative diagram of Kummer exact sequences (cf. $\S2.2$)
\[\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=3em, column sep=2em,
text height=2ex, text depth=0.25ex]
{0 & \widehat{J(K)} & H^1(G_K,TJ) & TH^1(G_K,J) & 0\\
0 & \displaystyle\prod_{c\in C^{\cl}}\widehat{J_c(K_c)} & \displaystyle\prod_{c\in C^{\cl}}H^1(G_{K_c},J_c) & \displaystyle\prod_{c\in C^{\cl}}TH^1(G_{K_c},J_c) & 0\\};
\path[->]
(m-1-1) edge (m-1-2);
\path[->]
(m-1-2) edge (m-1-3) edge (m-2-2);
\path[->]
(m-1-3) edge (m-1-4) edge (m-2-3);
\path[font=\scriptsize,->]
(m-1-4) edge (m-1-5) edge node[left]{} (m-2-4);
\path[->]
(m-2-1) edge (m-2-2);
\path[->]
(m-2-2) edge (m-2-3);
\path[->]
(m-2-3) edge (m-2-4);
\path[->]
(m-2-4) edge (m-2-5);
\end{tikzpicture}\]
where $J_c:=J\times _KK_c$, and $K_c$ is the completion of $K$ at $c$. Moreover, $(s_c^{\ab})_{c\in C^{\cl}}=(x_c)_{c\in C} \in \prod_{c\in C^{\cl}}\widehat{J_c(K_c)}$ via the injective maps $\prod_{c\in C^{\cl}}\X_c(k(c))\to \prod_{c\in C^{\cl}}\widehat{\J_c(k(c))}\to \prod_{c\in C^{\cl}}\widehat{J_c(K_c)}$, where the last map is induced by the inflation map, and the first map is injective as $k$ strongly satisfies condition (iii)(a). The kernel of the right vertical map $TH^1(G_K,J)\to \prod_{c\in C^{\cl}}TH^1(G_{K_c},J_c)$ is the Tate module of the Shafarevich-Tate group $\Sha(\mathcal{J})$, which is trivial by our assumption that $\Sha(\mathcal{J})$ is finite. Thus $s^{\ab}\in \widehat{J(K)}$ by the above diagram. We then prove the following
(cf. Lemma 5.1).
\begin{maintheorem} We use the above notations. Assume that $k$ strongly satisfies the conditions (i), (ii), and (iii) of Definition 1.4.
Then the homomorphism $J(K)\rightarrow\widehat{J(K)}$ is injective and $s^{\ab}$ is contained in $J(K)$.
\end{maintheorem}
Furthermore, the natural map $\prod_{c\in C^{\cl}}\X_c(k(c))\to \prod_{c\in C^{\cl}}\J_c(k(c))$ is injective (this follows from condition (iii)(a)), and inside
$\prod_{c\in C^{\cl}}\J_c(k(c))$ the equality
$$\prod_{c\in C^{\cl}}\X_c(k(c))\cap J(K)=X(K)$$
holds (the map $J(K)\to \prod_{c\in C^{\cl}}\J_c(k(c))$ is injective. See Lemma 5.2 and its proof).
Thus $s^{\ab}=x\in X(K)$ arises from a unique rational point $x$ since $s_c=x_c\in \X_c(k(c))$ for all $c\in C^{\cl}$. To finish the proof of Theorem A, using a limit argument due to Tamagawa relying on the fact that $k$ satisfies condition (iv), it suffices to show that every neighbourhood of the section $s$ has a $K$-rational point. This is achieved using the fact that $k$ satisfies condition (v).
The proof of Theorem B follows easily from Theorem A, and standard facts on the behaviour of birational Galois sections with respect to finite base change.
{\it To the proof of Theorem C}. We argue by induction on the transcendence degree and assume that the \textbf{BSC} holds over all finitely generated fields over $\Bbb Q$ with transcendence degree $<n$, where $n\ge 1$ is an integer. To prove that the \textbf{BSC} holds for curves over a finitely generated field $K$ of transcendence degree $n$ we first reduce to the case of the projective line $\Bbb P^1_K$. Given a section $s:G_K\to G_{\Bbb P^1_K}$ we prove that $s$ has a neighbourhood $Y\to \Bbb P^1_K$ with $Y$ hyperbolic and isotrivial. For such a curve it is proven in \cite{saiditamagawa} that $\Sha (\J_Y)$ is finite, where $\J_Y\to C$ is the relative jacobian of a suitable model $\Y\to C$ of $Y$ as in the statement of Theorem A. We can then (after passing to an appropriate finite extension of $K$) apply Theorem A to conclude that the section $s_Y:G_K\to G_Y$ induced by $s$, and a fortiori the section $s$, is geometric.
\subsection{Guide to the paper}
The layout of the paper is as follows. In \S 2 we will recall some properties of \'etale and geometrically abelian fundamental groups and their sections, which will be necessary for the proofs of our main Theorems.
In \S 3 we work in a local setting. We consider a flat, proper, smooth relative curve over the spectrum of a complete discrete valuation ring with residue characteristic zero, and explain how to define a specialisation homomorphism of absolute Galois groups (Theorem \ref{galspec}) using the specialisation homomorphism of fundamental groups of affine curves.
We apply this in \S \ref{ramificationofsections} to study the specialisation of sections, and the phenomenon of ramification of sections.
In \S 4 we return to the global setting of Theorems A and B, and consider curves over function fields of characteristic zero. In \S \ref{specialisationofsections} we explain how to pass to the local setting and apply the results from \S 3. In \S \ref{etaleabeliansections} we consider \'etale abelian sections and show how to apply a local-global principle, during which we encounter the issue of finiteness of the Shafarevich-Tate group (Proposition \ref{sabjkhat}).
In \S 5 we use the results of \S 3 and \S 4 to prove Theorems A and B, and apply these, along with one of the main results of \cite{saiditamagawa}, to prove Theorem C.
\subsection*{Notation}
\begin{itemize}
\item For a scheme $X$, we will denote the set of closed points of $X$ by $X^{\cl}$.
\item Given a ring $R$ and morphisms of schemes $X \to Y$ and $\Spec R \to Y$, we will denote the fibre product $X \times_Y \Spec R$ by $X_R$.
\item For a field $k$ and a given algebraic closure $\bar k$ of $k$, we will denote the absolute Galois group $\Gal(\bar k|k)$ by $G_k$.
\item For an abelian group $A$ and a positive integer $N$, we denote by $A[N]$ the kernel of the homomorphism $N : A \to A$, $a \mapsto N \cdot a$.
\item Given an abelian group $A$, we denote by $\widehat{A}$ the inverse limit $\widehat{A} := \varprojlim_{N \geq 1} A/NA$ and by $TA := \varprojlim_{N \geq 1} A[N]$ the Tate module of $A$.
\item Given an \emph{abelian variety} $B$ over a field $k$, an algebraic closure $\bar k$ of $k$, we will denote by $TB$ the Tate module of the abelian group $B(\bar k)$.
\end{itemize}
\section{Sections of \'etale and geometrically abelian fundamental groups}
\subsection{Sections of \'etale fundamental groups}
Let $k$ be a field of characteristic zero, $X$ a smooth, geometrically connected, projective curve over $k$, and $U\subset X$ an open subset with complement $S := X \setminus U$ (when considering an open $U\subset X$ we assume $U$ non-empty).
Write $K=k(X)$ for the function field of $X$. Let $z$ be a geometric point of $U$ with image in the generic point. Thus $z$ determines algebraic closures $\ob{K}$, and $\bar k$, of $K$ and $k$ respectively, as well as a geometric point of $U_{\bar k}$ that we denote $\bar z$. The \emph{\'etale fundamental group} $\pi_1(U, z)$ of $U$ fits into an exact sequence:
\[\sexact{\pi_1(U_{\bar k},\bar z)}{}{\pi_1(U,z)}{}{G_k=\Gal(\bar k/k)}\]
We will refer to this as the \emph{fundamental exact sequence} of $U$. The fundamental group $\pi_1(U_{\bar k},\bar z)$ will be called the \emph{geometric fundamental group} of $U$.
\begin{definition}\label{universalcover}
A \emph{universal pro-\'etale cover} $\tilde U\rightarrow U$ of $U$ is an inverse system of finite \'etale covers $\{V_i\rightarrow U\}_{i\in I}$, corresponding to open subgroups of
$\pi_1(U,z)$, such that for any \'etale cover $V\rightarrow U$, corresponding to an open subgroup of $\pi_1(U,z)$, there is a $U$-morphism $V_i\rightarrow V$ for some $i\in I$. A (closed) \emph{point} $\tilde x$ of the universal pro-\'etale cover $\tilde U$ is a compatible system of (closed) points $\{x_i\in V_i\}_{i\in I}$.
\end{definition}
Fix a universal pro-\'etale cover $\tilde U \to U$, and let $\{V_i \to U\}_{i\in I}$ be the inverse system of \'etale covers defining it.
\begin{definition}\label{normalisation}
Let $Y_i \to X$ denote the unique extension of $V_i \to U$ to a finite morphism of smooth, connected, projective curves over $k$. The inverse system of morphisms $\{Y_i \to X\}_{i\in I}$ will be called the \emph{normalisation of $X$ in $\tilde U$}, denoted $\tilde X_{\tilde U} \to X$. A (closed) \emph{point} $\tilde x$ of $\tilde X_{\tilde U}$ is a compatible system of (closed) points $\{x_i\in Y_i\}_{i\in I}$.
\end{definition}
\begin{definition}\label{decompinertiapi1}
With the above notation, let $x\in X^{\cl}$ be a closed point and $\tilde x$ a point above $x$ in the normalisation $\tilde X_{\tilde U}$ of $X$ in $\tilde U$. The \emph{decomposition group} $D_{\tilde x}$ of $\tilde x$ is the stabiliser of $\tilde x$ under the action of $\pi_1(U,z)$ (which acts naturally on the points of $\tilde X_{\tilde U}$).
The \emph{inertia group} $I_{\tilde x}$ is the kernel of the natural projection $D_{\tilde x}\twoheadrightarrow G_{k(x)}= \Aut(\bar k/k(x))$, where $k(x)$ is the residue field at $x$.
\end{definition}
It follows immediately from the definition that decomposition and inertia groups for different choices of $\tilde x$ above $x$ are conjugate, that is, for any $\sigma\in\pi_1(U,z)$ we have $I_{\sigma\tilde x}=\sigma I_{\tilde x}\sigma^{-1}$ and $D_{\sigma\tilde x}=\sigma D_{\tilde x}\sigma^{-1}$. The inertia group $I_{\tilde x}$ is trivial if $x \in U$, while if $x \in S$ it is isomorphic as a $G_{k(x)}$-module to the Tate twist $\hat\Z(1)$.
Recall that the curve $U$ is \emph{hyperbolic} if the geometric fundamental group $\pi_1(U_{\bar k}, \bar z)$ is non-abelian. Denoting by $g$ the genus of $X$, this occurs exactly when $2 - 2g - \deg_k S < 0$, where $S$ is regarded as a reduced effective divisor on $X$. We refer to \cite[Lemma 1.5]{hoshimochizuki} for a proof of the following statement.
\begin{lemma}\label{inertiaproperties}
With the above notation, assume further that $U$ is hyperbolic. Then any two inertia subgroups $I_{\tilde x}, I_{\tilde x'} \subset \pi_1(U, z)$ corresponding to distinct points $\tilde x\ne\tilde x'$ of $\tilde X_{\tilde U}$ intersect trivially.
\end{lemma}
Let $x \in X(k)$ be a $k$-rational point of $X$, and let $\tilde x$ be a point above $x$ in the normalisation $\tilde X_{\tilde U}$ of $X$ in $\tilde U$. There is a commutative diagram
of exact sequences
\begin{equation}\label{decompinjectpi1}
\begin{tikzcd}
1 \arrow{r}{} &I_{\tilde x} \arrow{r}{}\arrow[hookrightarrow]{d}{} &D_{\tilde x} \arrow{r}{}\arrow[hookrightarrow]{d}{} &G_{k(x)} \arrow{r}{}\arrow[equals]{d}{} &1\\
1 \arrow{r}{} &\pi_1(U_{\bar k},\bar z) \arrow{r}{} &\pi_1(U,z) \arrow{r}{} &G_{k} \arrow{r}{} &1
\end{tikzcd}
\end{equation}
where the middle vertical map is the natural inclusion, via which a section of the upper exact sequence naturally defines one of the (lower) fundamental exact sequence. When $x \in U$ the inertia group $I_{\tilde x}$ is trivial, hence there is an isomorphism $D_{\tilde x} \simeq G_{k(x)}=G_k$ (induced by the projection $\pi_1(U,z)\twoheadrightarrow G_k$) and thus $D_{\tilde x}$ gives rise naturally to a section of $\pi_1(U, z)$ in this case. Action by $\pi_1(U_{\bar k}, \bar z)$ permutes the points $\tilde x$ of $\tilde U$ above $x$, so $x$ in fact induces a conjugacy class of sections of the fundamental exact sequence.
It is well known that the upper exact sequence in diagram (\ref{decompinjectpi1}) also splits when $x$ is contained in the complement $S = X \setminus U$.
\begin{definition}\label{cuspidal}
A section $s:G_k\rightarrow\pi_1(U,z)$ is called \emph{cuspidal} if it factors through $D_{\tilde x}$ for some (necessarily $k$-rational) point $x \in S$ and some $\tilde x\in\tilde X_{\tilde U}$ above $x$.
\end{definition}
Now let $\xi : \Spec\Omega \to X$ be a geometric point with image the generic point of $X$. The geometric point $\xi$ naturally determines a choice of an algebraic closure $\overline {K}$
of $K$ and, for each open subset $U \subset X$, a geometric point $\xi:\Spec\Omega\to U$ with image the generic point of $U$. The following is well-known.
\begin{lemma}\label{Gallim}
With $X$, $K$ and $\xi$ as above, there is a canonical isomorphism
\[G_X:=\Gal (\overline {K}/K)\simeq\varprojlim_{U\subset X\textup{ open}}\pi_1(U,\xi)\]
where the limit is taken over all the open subsets of $X$, partially ordered by inclusion.
\end{lemma}
In particular, for any open subset $U \subset X$, the fundamental group $\pi_1(U, \xi)$ is naturally a quotient of $G_X$. In fact we have a commutative diagram
\[\begin{tikzcd}
1 \arrow{r}{} &G_{X_{\bar k}} \arrow{r}{}\arrow[twoheadrightarrow]{d}{} &G_X \arrow{r}{}\arrow[twoheadrightarrow]{d}{} &G_k \arrow{r}{}\arrow[equals]{d}{} &1\\
1 \arrow{r}{} &\pi_1(U_{\bar k},\bar\xi) \arrow{r}{} &\pi_1(U,\xi) \arrow{r}{} &G_k \arrow{r}{} &1
\end{tikzcd}\]
where the middle vertical map is the natural projection, via which a section $s : G_k \to G_X$ of $G_X$ naturally induces a section $s_U : G_k \to \pi_1(U, \xi)$
of the projection $\pi_1(U, \xi)\twoheadrightarrow G_k$.
Thus, by Lemma \ref{Gallim}, a section $s$ of $G_X$ determines, and is determined by, a compatible system of sections $s_U : G_k \to \pi_1(U, \xi)$ for the open subsets $U \subset X$.
\subsection{Geometrically abelian fundamental groups}
Let $k$ be a field of characteristic zero, and let $X$ be a smooth, geometrically connected, projective curve over $k$ such that $X(k) \ne \emptyset$. Let $z:\Spec\Omega\rightarrow X$ be a geometric point with value in the generic point, which determines an algebraic closure $\bar k$ of $k$ and a geometric point $\bar z$ of $X_{\bar k}$. Denote by $\pi_1(X_{\bar k}, \bar z)^{\ab}$ the maximal abelian quotient of the geometric fundamental group of $X$. The \emph{geometrically abelian quotient} of $\pi_1(X, z)$, denoted $\pi_1(X, z)^{(\ab)}$, is defined by the pushout diagram
\[\begin{tikzcd}
1 \arrow{r}{} &\pi_1(X_{\bar k},\bar z) \arrow{r}{}\arrow[twoheadrightarrow]{d}{} &\pi_1(X,z) \arrow{r}{}\arrow[twoheadrightarrow]{d}{} &G_k \arrow{r}{}\arrow[equals]{d}{} &1\\
1 \arrow{r}{} &\pi_1(X_{\bar k},\bar z)^{\ab} \arrow{r}{} &\pi_1(X,z)^{(\ab)} \arrow{r}{} &G_k \arrow{r}{} &1
\end{tikzcd}\]
where the upper row is the fundamental exact sequence. By commutativity of this diagram, a section $s : G_k \to \pi_1(X,z)$ of the \'etale fundamental group of $X$ naturally induces a section $s^{\ab} : G_k \to \pi_1(X, z)^{(\ab)}$ of the geometrically abelian quotient, which we will call the \emph{\'etale abelian section} induced by $s$.
Fix a $k$-rational point $x_0\in X(k)$. Let $J$ denote the Jacobian of $X$, and let $\iota:X\rightarrow J$ be the closed immersion mapping $x_0$ to the zero section of $J$. Note that $\iota$ maps an arbitrary $k$-rational point $x \in X(k)$ to the class of the degree zero divisor $[x]-[x_0]$. Moreover, it induces a commutative diagram of exact sequences
\begin{equation}\label{pi1xpi1j}
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\matrix(m)[matrix of math nodes,
row sep=3em, column sep=2.5em,
text height=1.5ex, text depth=0.25ex]
{1 & \pi_1(X_{\bar k},\bar z)^{\ab} & \pi_1(X,z)^{(\ab)} & G_k & 1\\
1 & \pi_1(J_{\bar k},\bar z) & \pi_1(J,z) & G_k & 1\\};
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(m-1-4) edge[double distance=2pt] (m-2-4);
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(m-2-3) edge (m-2-4);
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(m-2-4) edge (m-2-5);
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\end{equation}
where the vertical maps are isomorphisms.
Hence there is an identification of $G_k$-modules $\pi_1(X_{\bar k},\bar z)^{\ab} \simeq \pi_1(J_{\ob{k}}, \bar z) \simeq TJ$, where $TJ$ is the Tate module of $J$ (see Notation). We fix a base point of the torsor of splittings of the upper exact sequence in (\ref{pi1xpi1j}) to be the splitting arising from the rational point $x_0$, and the corresponding base point of the torsor of splittings of the lower exact sequence in (\ref{pi1xpi1j}), and identify the set of $\pi_1(X_{\bar k},\bar z)^{\ab}$-conjugacy classes of sections of the upper exact sequence in (\ref{pi1xpi1j}) with the Galois cohomology group $H^1(G_k, TJ)$. By functoriality of the fundamental group, any $k$-rational point $x\in X(k)$ induces a section $s_x:G_k\to\pi_1(X,z)$, which in turn induces an \'etale abelian section $s_x^{\ab}:G_k\rightarrow\pi_1(X,z)^{(\ab)}$. Thus we have a map $X(k)\to H^1(G_k,TJ)$ defined by $x\mapsto [s_x^{\ab}]$, where $[s_x^{\ab}]$ denotes the $\pi_1(X_{\bar k}, \bar z)^{\ab}$-conjugacy class of sections containing $s_x^{\ab}$. This map factors through $J(k)$, that is, it coincides with the composite map
\[\begin{tikzcd}[column sep=small]X(k)\arrow[hookrightarrow]{r}{\iota} &J(k) \arrow{r}{} &H^1(G_k,TJ).\end{tikzcd}\]
This is due to the isomorphism $\pi_1(X,z)^{(\ab)}\simeq\pi_1(J,z)$, via which $s_x^{\ab}$ corresponds to the section of $\pi_1(J,z)$ induced by functoriality from the $k$-rational point $\iota(x)\in J(k)$.
\begin{lemma}\label{kummerexactseq}
There is an exact sequence
\[\sexactab{\widehat{J(k)}}{}{H^1(G_k,TJ)}{}{TH^1(G_k,J)}\]
where $\widehat{J(k)}:=\varprojlim_N J(k)/NJ(k)$ (see Notation) and $TH^1(G_k,J)$ is the Tate module of the Galois cohomology group $H^1(G_k,J)$.
\end{lemma}
We shall refer to this sequence as the \emph{Kummer exact sequence}. One can easily derive it from the Kummer exact sequences $\begin{tikzcd}[column sep=small]0 \arrow{r}{} &J(\bar k)[N] \arrow{r}{} &J(\bar k) \arrow{r}{N} &J(\bar k) \arrow{r}{} &0,\end{tikzcd}$ $N \geq 0$, during which one sees, in particular, that the map $J(k)\to H^1(G_k,TJ)$ factors through $\widehat{J(k)}$. Thus we have a sequence of maps:
\begin{equation}\label{sabptth}
\begin{tikzcd}[column sep=small]X(k) \arrow[hookrightarrow]{r}{\iota} &J(k) \arrow{r}{} &\widehat{J(k)} \arrow[hookrightarrow]{r}{} &H^1(G_k,TJ).\end{tikzcd}
\end{equation}
\section{Specialisation of sections in a local setting}
\subsection{A specialisation homomorphism for absolute Galois groups}\label{spechom}
Let $R$ be a complete discrete valuation ring with uniformiser $\pi$, field of fractions $K$ and residue field $k:=R/\pi R$ of characteristic zero. Let $X$ be a flat, proper, smooth, geometrically connected \emph{relative curve} over $\Spec R$, and denote by $X_K:=X\times _{\Spec R}\Spec K$ its generic fibre and $X_k:=X\times _{\Spec R}\Spec k$ its special fibre. Fix an algebraic closure $\ob{K}$ of $K$, and denote by $\ob{R}$ the integral closure of $R$ in $\ob{K}$, and by $\bar k$ the residue field of $\ob{R}$, which is an algebraic closure of $k$.
Let $\bar\xi_1:\Spec\Omega_1\rightarrow X_{\ob{K}}$ be a geometric point with image the generic point of $X_{\ob{K}}$, and similarly let $\bar\xi_2:\Spec\Omega_2\rightarrow X_{\bar k}$ be a geometric point with image the generic point of $X_{\bar k}$. These induce geometric points of $X_K$ and $X_k$, which we denote by $\xi_1$ and $\xi_2$ respectively.
\begin{definition}\label{stilde}
For each closed point $x$ of $X_k^{\cl}$, fix a choice of closed point $y\in X_K^{\cl}$ which specialises to $x$ and whose residue field is the unique unramified extension of $K$ whose valuation ring has residue field $k(x)$ (such a point exists since $X \to \Spec R$ is smooth). We define $\tilde S$ to be the set of these chosen closed points $y\in X_K^{\cl}$. Thus, $\tilde S$ is a subset of $X^{\cl}_K$ in bijection with $X_k^{\cl}$.
We denote by $\tilde S_{\ob{K}}$ the inverse image of $\tilde S$ via the map $X_{\ob K}^{\cl}\to X^{\cl}$.
Thus, $\tilde S_{\ob{K}}$ is a subset of $X_{\ob{K}}^{\cl}$ in bijection with $X_{\bar k}^{\cl}$.
\end{definition}
\begin{definition}\label{pi1stilde}
With $\tilde S$ and $\tilde S_{\ob{K}}$ as in Definition \ref{stilde}, we define the group $\pi_1(X_K - \tilde S)$ to be the inverse limit
\[\pi_1(X_K - \tilde S) := \varprojlim_{B\subset\tilde S\textup{ finite}}\pi_1(X_K \setminus B, \xi_1)\]
over the open subsets of $X_K$ whose complements are finite subsets of $\tilde S$, ordered by inclusion. Similarly, we define $\pi_1(X_{\ob{K}} - \tilde S_{\ob{K}}) := \varprojlim_{B\subset\tilde S_{\ob{K}}\textup{ finite}}\pi_1(X_{\ob{K}} \setminus B, \bar\xi_1)$.
\end{definition}
\begin{definition}\label{universalcovertilde}
A \emph{universal pro-\'etale cover} $\tilde X_{\tilde S} \to X_K - \tilde S$ is an inverse system of finite morphisms $\{Y_i\to X_K\}_{i\in I}$, with $Y_i$ smooth, corresponding to open subgroups of
$\pi_1(X_K - \tilde S)$, such that for any finite morphism $Y \to X_K$, with $Y$ smooth, corresponding to an open subgroup of $\pi_1(X_K - \tilde S)$,
there is an $X_K$-morphism $Y_i\to Y$ for some $i\in I$.
A (closed) \emph{point} $\tilde x$ of $\tilde X_{\tilde S}$ is a compatible system of (closed) points $\{y_i \in Y_i\}_{i\in I}$.
\end{definition}
For a universal pro-\'etale cover $\tilde X_{\tilde S} \to X_K - \tilde S$ and any closed point $\tilde x$ of $\tilde X_{\tilde S}$, we define decomposition and inertia subgroups $D_{\tilde x}, I_{\tilde x} \subset \pi_1(X_K - \tilde S)$ exactly as in Definition \ref{decompinertiapi1}, and they satisfy analogous properties.
\begin{theorem}\label{galspec}
There exists a surjective (continuous) homomorphism $\Sp : \pi_1(X_K-\tilde S) \twoheadrightarrow G_{X_k}$, an isomorphism $\ob{\Sp} : \pi_1(X_{\ob{K}} - \tilde S_{\ob{K}}) \simeq G_{X_{\bar k}}$, and a commutative diagram of exact sequences:
\begin{equation}\label{specialisationgalois}
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\matrix(m)[matrix of math nodes,
row sep=3em, column sep=2.5em,
text height=1.5ex, text depth=0.25ex]
{1 & G_{X_{\ob{K}}} & G_{X_K} & G_K & 1\\
1 & \pi_1(X_{\ob{K}}-\tilde S_{\ob{K}}) & \pi_1(X_K-\tilde S) & G_K & 1\\
1 & G_{X_{\bar k}} & G_{X_k} & G_k & 1\\};
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\end{equation}
The homomorphism $\Sp$, resp. $\ob{\Sp}$, is defined only up to conjugation.
\end{theorem}
The homomorphisms $\ob{\Sp}$ and $\Sp$ will be referred to as \emph{specialisation homomorphisms}. For the proof we need the following well-known result.
\begin{lemma}\label{specialisationaffine}
Let $S$ be a divisor on $X$ which is finite \'etale over $R$, and denote $U := X \setminus S$ which is an open sub-scheme of $X$. Then there exists a surjective (continuous) homomorphism $\Sp_U : \pi_1(U_K, \xi_1) \twoheadrightarrow \pi_1(U_k, \xi_2)$ and an isomorphism $\ob{\Sp}_U : \pi_1(U_{\ob{K}}, \bar\xi_1) \simeq \pi_1(U_{\bar k}, \bar\xi_1)$ making the following diagram commutative.
\begin{equation}\label{specialisationaffine2}
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row sep=3em, column sep=2.5em,
text height=1.5ex, text depth=0.25ex]
{1 & \pi_1(U_{\ob{K}},\bar \xi_1) & \pi_1(U_K,\xi_1) & G_K & 1\\
1 & \pi_1(U_{\bar k},\bar \xi_2) & \pi_1(U_k,\xi_2) & G_k & 1\\};
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(m-1-2) edge node[left]{$\ob{\Sp}_U$} node[below,rotate=90]{$\sim$} (m-2-2);
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\end{equation}
The homomorphism $\ob{\Sp}_U$, respectively $\Sp_U$, is defined only up to inner automorphism of $\pi_1(U_{\ob{k}},\bar\xi_2)$, resp. $\pi_1(U,\xi_2)$.
\end{lemma}
The homomorphisms $\ob{\Sp}_U$ and $\Sp_U$ are called \emph{specialisation homomorphisms of fundamental groups}. The homomorphism $\Sp_U$ is defined in a natural way and induces the homomorphism $\ob{\Sp}_U$. For an exposition, in particular the fact that $\ob{\Sp}_U$ is an isomorphism, see \cite{Luminy}.
\begin{proof}[Proof of Theorem \ref{galspec}]
Let $B\subset\tilde S$ be a finite subset of $\tilde S$, viewed as a reduced closed sub-scheme of $X_K$, and let $\B$ denote its schematic closure in $X$. By construction, $\B$ is a divisor on $X$ which is finite \'etale over $R$ such that $\B_K=B$. Denoting $U := X - \B$, by Lemma \ref{specialisationaffine} there exist specialisation homomorphisms $\Sp_U : \pi_1(U_K, \xi_1) \twoheadrightarrow \pi_1(U_k, \xi_2)$ and $\ob{\Sp}_U : \pi_1(U_{\ob{K}}, \bar\xi_1) \simeq \pi_1(U_{\bar k}, \bar\xi_2)$.
Since there exist such homomorphisms for every finite subset of $\tilde S$, we have a compatible system of surjective homomorphisms $\{ \Sp_U \}$, resp. of isomorphisms $\{ \ob{\Sp}_U \}$ (the compatibility follows from the construction of these homomorphisms).
By Lemma \ref{Gallim}, taking inverse limits gives rise respectively to the surjective homomorphism $\Sp : \pi_1(X_K-\tilde S)\twoheadrightarrow G_{X_k}$ and the isomorphism $\ob{\Sp} : \pi_1(X_{\ob{K}} - \tilde S_{\ob{K}}) \simeq G_{X_{\bar k}}$, and moreover $\pi_1(X_K-\tilde S)$ and $\pi_1(X_{\ob{K}} - \tilde S_{\ob{K}})$ are naturally quotients of $G_{X_K}$ and $G_{X_{\ob{K}}}$ respectively. Thus we have the required homomorphisms in diagram (\ref{specialisationgalois}), and this diagram is clearly commutative.
\end{proof}
One could consider the composite homomorphism $G_{X_K} \twoheadrightarrow \pi_1(X_K-\tilde S) \twoheadrightarrow G_{X_k}$, and similarly for $G_{X_{\ob{K}}}$, to be a specialisation homomorphism for absolute Galois groups. However, we will reserve the label `Sp' for the homomorphism $\pi_1(X_K-\tilde S) \twoheadrightarrow G_{X_k}$, since this will be important in the next section.
\subsection{Ramification of sections}\label{ramificationofsections}
We use the notation of \S \ref{spechom}, and assume further that the closed fibre $X_k$ is \emph{hyperbolic}. With $S$ and $U$ as in Lemma \ref{specialisationaffine}, consider again diagram (\ref{specialisationaffine2}), and recall that the kernel of the projection $G_K\twoheadrightarrow G_k$ is the inertia group $I_K$ associated to the discrete valuation on $K$.
\begin{lemma}\label{sppullback}
\begin{enumerate}
\item The projection $\pi_1(U_K, \xi_1) \twoheadrightarrow G_K$ restricts to an isomorphism $\ker(\Sp_U)\simeq I_K$.
\item The right square in diagram (\ref{specialisationaffine2}) is cartesian.
\end{enumerate}
\end{lemma}
\begin{proof}
The isomorphism $\ker(\Sp)\simeq I_K$ follows from a simple diagram chase, and (ii) follows easily from (i).
\end{proof}
Fix universal pro-\'etale covers $\tilde U_K \to U_K$ and $\tilde U_k \to U_k$ (corresponding to the geometric points $\xi_1$ and $\xi_2$ respectively), and let $\tilde X_{\tilde U_K}$ denote the normalisation of $X_K$ in $\tilde U_K$, and likewise $\tilde X_{\tilde U_k}$ the normalisation of $X_k$ in $\tilde U_k$ (see Definitions \ref{universalcover} and \ref{normalisation}).
\begin{lemma}\label{cuspdecomppullback}
Let $x\in S(k)$, and let $y'$ be the unique ($K$-rational) point of $S_K$ which specialises to $x$. Let $\tilde y'$ be a point of $\tilde X_{\tilde U_K}$ above $y'$. There exists a unique $\tilde x$ in $\tilde X_{\tilde U_k}$ above $x$, so that we have the following commutative diagram:
\[\begin{tikzpicture}[descr/.style={fill=white}, baseline=(current bounding box.center)]
\matrix(m)[matrix of math nodes,
row sep=3em, column sep=2.5em,
text height=1.5ex, text depth=0.25ex]
{1 & I_{\tilde y'} & D_{\tilde y'} & G_K & 1\\
1 & I_{\tilde x} & D_{\tilde x} & G_k & 1\\};
\path[->,font=\scriptsize]
(m-1-1) edge node[above]{} (m-1-2);
\path[->,font=\scriptsize]
(m-1-2) edge node[above]{} (m-1-3);
\path[->,font=\scriptsize]
(m-1-2) edge node[left]{$\ob{\Sp}_U$} node[below,rotate=90]{$\sim$} (m-2-2);
\path[->,font=\scriptsize]
(m-1-3) edge node[above]{} (m-1-4);
\path[->>,font=\scriptsize]
(m-1-3) edge node[left]{$\Sp_U$} (m-2-3);
\path[->,font=\scriptsize]
(m-1-4) edge node[above]{} (m-1-5);
\path[->>,font=\scriptsize]
(m-1-4) edge node[left]{} (m-2-4);
\path[->,font=\scriptsize]
(m-2-1) edge node[above]{} (m-2-2);
\path[->,font=\scriptsize]
(m-2-2) edge node[above]{} (m-2-3);
\path[->,font=\scriptsize]
(m-2-3) edge node[above]{} (m-2-4);
\path[->,font=\scriptsize]
(m-2-4) edge node[above]{} (m-2-5);
\end{tikzpicture}\]
where $D_{\tilde y'}$ (resp. $D_{\tilde x}$) is the decomposition group of $\tilde y'$ (resp. $\tilde x$) in $\pi_1(U_K,\xi_1)$ (resp. $\pi_1(U_k,\xi_2)$).
Moreover, the right square in this diagram is cartesian.
\end{lemma}
\begin{proof}
The image of $D_{\tilde y'}$ under $\Sp_U$ is contained in $D_{\tilde x}$ for some $\tilde x$ in $\tilde X_{U_k}$ above $x$, as follows easily from the functoriality of fundamental groups and the specialisation of points on (coverings of) $R$-curves. Moreover, such $\tilde x$ is unique by Lemma \ref{inertiaproperties}, since $\ob{\Sp}_U$ is an isomorphism, which implies the inertia subgroup $I_{\tilde y'}\subset D_{\tilde y'}$ maps isomorphically to $I_{\tilde x}\subset\pi_1(U_{\bar k},\bar\xi_2)$. Commutativity of diagram (\ref{specialisationaffine2}) then implies that $D_{\tilde y'}$ maps surjectively onto $D_{\tilde x}$, whence the above diagram. As in the proof of Lemma \ref{sppullback}, the right square in this diagram is cartesian.
\end{proof}
\begin{corollary}\label{cuspidalevenifyisnotacusp}
With the notation of Lemma \ref{cuspdecomppullback}, for any $K$-rational point $y$ of $U_K$ specialising to $x$ and any point $\tilde y$ of $\tilde U_K$ above $y$, $D_{\tilde y}$ is contained in $D_{\tilde y'}$ for some $\tilde y'$ in $\tilde X_{\tilde U_K}$ above $y'$. In particular, a section $s:G_K\to\pi_1(U_K,\xi_1)$ with image contained in (hence equal to) $D_{\tilde y}$ is cuspidal (Definition \ref{cuspidal}), even though $y\not\in S_K$.
\end{corollary}
\begin{proof}
By specialisation, the point $\tilde y$ determines a point $\tilde x\in \tilde X_{\tilde U_k}$ such that $\Sp_U(D_{\tilde y}) \subset D_{\tilde x}$. The statement then follows from Lemma \ref{cuspdecomppullback} and the universal property of cartesian squares. Note that for a point $\tilde x$ in $\tilde X_{\tilde U_k}$ above $x$ there exists a point $\tilde y'$ of $\tilde X_{\tilde U_K}$ above $y'$ such that the conclusion of Lemma \ref{cuspdecomppullback} holds (follows easily from a limit argument).
\end{proof}
By Lemma \ref{sppullback} (ii), any section of $\pi_1(U_k,\xi_2)$ naturally induces a section of $\pi_1(U_K,\xi_1)$, but the converse is not true in general. Given any section $s:G_K\rightarrow\pi_1(U_K,\xi_1)$, let us define a homomorphism $\varphi_s:G_K\rightarrow\pi_1(U_k,\xi_2)$ by the composition $\varphi_s:=\Sp_U\circ s$.
\begin{equation}\label{defofvarphisandramified}
\begin{tikzpicture}[descr/.style={fill=white}, baseline=(current bounding box.center)]
\matrix(m)[matrix of math nodes,
row sep=4em, column sep=3em,
text height=1.5ex, text depth=0.25ex]
{1 & \pi_1(U_{\ob{K}},\bar\xi_1) & \pi_1(U_K, \xi_1) & G_K & 1\\
1 & \pi_1(U_{\bar k},\bar\xi_2) & \pi_1(U_k, \xi_2) & G_k & 1\\};
\path[->]
(m-1-1) edge (m-1-2);
\path[->,font=\scriptsize]
(m-1-2) edge (m-1-3) edge node[left]{$\ob{\Sp}_U$} node[below,rotate=90]{$\sim$} (m-2-2);
\path[->]
(m-1-3) edge (m-1-4);
\path[->>,font=\scriptsize]
(m-1-3) edge node[left]{$\Sp_U$} (m-2-3);
\path[->,font=\scriptsize]
(m-1-4) edge (m-1-5) edge[out=160,in=20] node[above]{$s$} (m-1-3) edge[out=200,in=60] node[left]{$\varphi_s$} (m-2-3);
\path[->>,font=\scriptsize]
(m-1-4) edge node[left]{} (m-2-4);
\path[->]
(m-2-1) edge (m-2-2);
\path[->]
(m-2-2) edge (m-2-3);
\path[->]
(m-2-3) edge (m-2-4);
\path[->]
(m-2-4) edge (m-2-5);
\end{tikzpicture}
\end{equation}
\begin{definition}\label{ramified}
We say the section $s$ is \emph{unramified} if $\varphi_s(I_K)=1$. Otherwise we say $s$ is \emph{ramified}.
For an unramified section $s:G_K\rightarrow\pi_1(U_K,\xi_1)$ the map $\varphi_s$ factors through the projection $G_K\twoheadrightarrow G_k$ and $s$
induces a section $\bar s:G_k\rightarrow\pi_1(U_k,\xi_1)$. This induced section will be called the \emph{specialisation of} $s$ and denoted $\bar s$.
\end{definition}
We now investigate under what conditions we may conclude that a given section $s : G_K \to \pi_1(U_K, \xi_1)$ has image contained in a decomposition group.
\begin{lemma}\label{unramifiedcuspidal}
Let $s:G_K\rightarrow\pi_1(U_K,\xi_1)$ be an unramified section, and suppose $\bar s(G_k)\subset D_{\tilde x}$ for some $x \in S(k)$ and some $\tilde x$ in $\tilde X_{\tilde U_k}$ above $x$. Then $s(G_K)\subset D_{\tilde y'}$ for a unique $\tilde y'$ in $\tilde X_{\tilde U_K}$ above the unique point $y' \in S(K)$ specialising to $x$.
\end{lemma}
\begin{proof}
There exists a point $\tilde y'$ in $\tilde X_{\tilde U_K}$ above $y'$ such that the image of $D_{\tilde y'} \subset \pi_1(U_K, \xi_1)$ under $\Sp_U$ is contained in $D_{\tilde x}$ (cf. proof of Corollary \ref{cuspidalevenifyisnotacusp}), and this $\tilde y'$ is unique as follows from Lemma \ref{inertiaproperties}. The pullback of $\bar s$ via the natural projection $G_K\twoheadrightarrow G_k$ gives rise to the section $s$, so $s(G_K)$ must be contained in the pullback of $D_{\tilde x}$ via $G_K\twoheadrightarrow G_k$, which is $D_{\tilde y'}$ by Lemma \ref{cuspdecomppullback}.
\end{proof}
\begin{proposition}\label{weightargument}
Assume that $k$ satisfies condition (ii) in Definition \ref{conditions}. Let $s:G_K\rightarrow\pi_1(U_K,\xi_1)$ be a section, and denote $\varphi_s:=\Sp_U\circ s$ as above. If $\varphi_s(I_K)$ is non-trivial then it is contained in the inertia group $I_{\tilde x}$ of a unique point $\tilde x$ of $\tilde X_{\tilde U_k}$ above a point of $S_k$.
\end{proposition}
\begin{proof}
This follows from \cite[Lemma 1.6]{hoshimochizuki}. Indeed, if $\varphi_s(I_K)$ is non-trivial then, by commutativity of diagram (\ref{defofvarphisandramified}), it is contained in $\pi_1(U_{\bar k},\bar\xi_2)$, and it is a procyclic subgroup of $\pi_1(U_{\bar k},\bar\xi_2)$ because $I_K\simeq\hat\Z(1)$ is procyclic. Since $k$ satisfies condition (ii) of Definition \ref{conditions}, the image of $\varphi_s(I_K)$ under the composite $G_k$-homomorphism
\[\begin{tikzcd}[column sep=small]
\pi_1(U_{\bar k},\bar\xi_2) \arrow[twoheadrightarrow]{r}{} &\pi_1(X_{\bar k},\bar\xi_2) \arrow[twoheadrightarrow]{r}{} &\pi_1(X_{\bar k},\bar\xi_2)^{\ab}
\end{tikzcd}\]
is trivial. It then follows from loc. cit. that $\varphi_s(I_K)$ must be contained in an inertia subgroup of $\pi_1(U_{\bar k}, \bar\xi_2)$, which is unique by Lemma \ref{inertiaproperties}.
\end{proof}
\begin{remark}\label{properimpliesunramified}
Under the hypotheses of Proposition \ref{weightargument}, if $S=\emptyset$, so that $U=X$ is proper over $\Spec R$, then it follows from the arguments in the proof of Proposition \ref{weightargument} that any section $s:G_K\rightarrow\pi_1(X_K,\xi_1)$ is necessarily unramified.
\end{remark}
\begin{lemma}\label{ramified=cuspidal}
Assume that $k$ satisfies condition (ii) in Definition \ref{conditions}, and let $s:G_K\to\pi_1(U_K,\xi_1)$ be a section and $\varphi_s:=\Sp_U\circ s$. If $s$ is ramified then $\varphi_s(G_K)\subset D_{\tilde x}$ for a unique $k$-rational point $x$ of $S_k$ and a unique $\tilde x$ in $\tilde X_{\tilde U_k}$ above $x$, and $s(G_K)\subset D_{\tilde y'}$ for a unique point $\tilde y'$ of $\tilde X_{\tilde U_K}$ above the unique $K$-rational point $y'$ of $S_K$ specialising to $x$. In particular, $s$ is cuspidal.
\end{lemma}
\begin{proof}
If $\varphi_s(I_K)$ is non-trivial then, by Proposition \ref{weightargument}, it must be contained in a unique inertia group $I_{\tilde x}$ for some $x\in S_k$ and some $\tilde x\in\tilde X_{\tilde U_k}$ above $x$. Since $\varphi_s(G_K)$ normalises $\varphi_s(I_K)$, for some $\sigma\in G_K$ we have $\varphi_s(I_K)=\varphi_s(\sigma)\cdot \varphi_s(I_K)\cdot \varphi_s(\sigma)^{-1}\subseteq \varphi_s(\sigma)\cdot I_{\tilde x}\cdot \varphi_s(\sigma)^{-1}=I_{\varphi_s(\sigma)\cdot\tilde x}$. But $\varphi_s(I_K)$ is contained in a unique inertia group, so $\varphi_s(\sigma)\cdot\tilde x=\tilde x$ and $\varphi_s(G_K)$ fixes $\tilde x$, i.e. $\varphi_s(G_K)\subseteq D_{\tilde x}$. Moreover, $x$ is necessarily a $k$-rational point since, by commutativity of diagram (\ref{defofvarphisandramified}), $\varphi_s(G_K)$ maps surjectively onto $G_k$. A similar argument to that used in the proof of Corollary 3.8 implies that $s(G_K)\subseteq D_{\tilde y'}$ for some $\tilde y'$ in $\tilde X_{\tilde U_K}$ above the unique point $y'$ of $S_K$ specialising to $x$. Moreover, such point $\tilde y'$ is unique by Lemma \ref{inertiaproperties}.
\end{proof}
Let $\tilde S$ be as in Definition \ref{stilde}. For a section $s:G_K\rightarrow\pi_1(X_K-\tilde S)$ (see Definition \ref{pi1stilde}), we write $\varphi_s:=\Sp\circ s$ for the composition of $s$ with the specialisation homomorphism $\Sp : \pi_1(X_K - \tilde S)\twoheadrightarrow G_{X_k}$ of Theorem \ref{galspec}.
\begin{equation}\label{varphistilde}
\begin{tikzpicture}[descr/.style={fill=white}, baseline=(current bounding box.center)]
\matrix(m)[matrix of math nodes,
row sep=4em, column sep=3em,
text height=1.5ex, text depth=0.25ex]
{\pi_1(X_K-\tilde S) & G_K\\
G_{X_k} & G_k\\};
\path[->]
(m-1-1) edge (m-1-2);
\path[->>,font=\scriptsize]
(m-1-1) edge node[left]{$\Sp$} (m-2-1);
\path[->,font=\scriptsize]
(m-1-2) edge[out=150,in=30] node[above]{$s$} (m-1-1) edge[out=200,in=60] node[left]{$\varphi_s$} (m-2-1);
\path[->>,font=\scriptsize]
(m-1-2) edge node[left]{} (m-2-2);
\path[->,font=\scriptsize]
(m-2-1) edge node[above]{} (m-2-2);
\end{tikzpicture}
\end{equation}
We define the ramification and specialisation of $s$ analogously to Definition \ref{ramified}. For an open subset $U_k \subset X_k$ with complement $S_k$, we will denote by $S_K$ the set of points of $\tilde S$ which specialise to $S_k$, $D$ the schematic closure of $S_K$ in $X$, and $U := X \setminus D$, thus $U_K=X_K\setminus S_K$. We will denote by $s_U : G_K\rightarrow\pi_1(U_K,\xi_1)$ the section of $\pi_1(U_K, \xi_1)$ naturally induced by $s$, and by $\varphi_U$ the composition $\varphi_U := \Sp_U \circ s_U : G_K \to \pi_1(U_k, \xi_2)$ of $s_U$ with the specialisation homomorphism $\Sp_U : \pi_1(U_K,\xi_1) \twoheadrightarrow \pi_1(U_k,\xi_2)$ of Lemma \ref{specialisationaffine}.
Since $s$ induces such a section $s_U : G_K \to \pi_1(U_K, \xi_1)$ for every open subset $U_K\subset X_K$ as above, it determines a compatible system of sections $\{ s_U \}$, parameterised by the open subsets $U\subset X$ as above. Conversely, such a system determines a section of $\pi_1(X_K-\tilde S)$. Similarly, since $s$ induces a homomorphism $\varphi_U : G_K \to \pi_1(U_k, \xi_2)$ for every open subset of $X_k$, it determines a compatible system of homomorphisms $\{ \varphi_U \}$ whose inverse limit $\varprojlim_{U_k \subset X_k \textup{ open}}\varphi_U$ is exactly the homomorphism $\varphi_s$ of diagram (\ref{varphistilde}).
\begin{lemma}\label{toramifiedsubsets}
A section $s:G_K\to\pi_1(X_K-\tilde S)$ is ramified if and only if there is a non empty open subset $U_k\subset X_k$ for which the section $s_U:G_K\to\pi_1(U_K,\xi_1)$ induced as above by $s$ is ramified.
\end{lemma}
\begin{proof}
Since $\varphi_s(I_K) = \varprojlim_{U_k \subset X_k \textup{ open}} \varphi_U(I_K)$ (with surjective transition maps), $\varphi_s(I_K)$ is trivial if and only if $\varphi_U(I_K)$ is trivial for every open subset $U_k \subset X_k$.
\end{proof}
\begin{proposition}\label{ramifiedsubsets}
Assume that $k$ satisfies condition (ii) of Definition \ref{conditions}, and let $s:G_K\to\pi_1(X_K-\tilde S)$ be a section. Let $U'_k \subset U_k \subset X_k$ be any two non-empty open subsets, and let $\tilde X_{\tilde U_k}$, resp. $\tilde X_{\tilde U'_k}$ be the normalisation of $X_k$ in some universal pro-\'etale cover $\tilde U_k \to U_k$, resp. $\tilde U'_k \to U'_k$.
Suppose that $s_U$ is ramified, with $\varphi_U(I_K)$ contained in the inertia subgroup $I_{\tilde x_U} \subset \pi_1(U_k,\xi_2)$ for some $x\in (X_k\setminus U_k)(k)$ and some $\tilde x_U$ in $\tilde X_{\tilde U_k}$ above $x$ (see Lemma \ref{ramified=cuspidal} and its proof). Then $s_{U'}$ is ramified, and $\varphi_{U'}(I_K)$ is contained in the inertia subgroup $I_{\tilde x_{U'}}\subset\pi_1(U'_k,\xi_2)$ of some $\tilde x_{U'}$ in $\tilde X_{\tilde U'_k}$ above the same point $x\in (X_k\setminus U_k)(k)\subset (X_k\setminus U_k')(k)$.
\end{proposition}
\begin{proof}
The image of $\varphi_{U'}(I_K)$ under the homomorphism $\pi_1(U'_k,\xi_2)\twoheadrightarrow\pi_1(U_k,\xi_2)$ coincides with $\varphi_U(I_K)$, which is nontrivial by assumption. Thus $\varphi_{U'}(I_K)$ must also be non-trivial, and therefore it is contained in an inertia subgroup $I_{\tilde z_{U'}} \subset \pi_1(U'_k,\xi_2)$ for some $z\in (X_k\setminus U_k')(k)$ and some $\tilde z_{U'}$ in $\tilde X_{\tilde U'_k}$ above $z$ (see Lemma \ref{ramified=cuspidal} and its proof). Let $\tilde z_U$ be the image of $\tilde z_{U'}$ in $\tilde X_{\tilde U_k}$. Suppose $\tilde z_U \ne \tilde x_U$. If $z \in X_k\setminus U_k$ then, by functoriality, the image of $I_{\tilde z_{U'}}$ under $\pi_1(U'_k,\xi_2)\twoheadrightarrow\pi_1(U_k,\xi_2)$ is the inertia subgroup $I_{\tilde z_U} \subset \pi_1(U_k,\xi_2)$ at $\tilde z_U$, which intersects trivially with $I_{\tilde x_U}$ by Lemma \ref{inertiaproperties}. Meanwhile, if $z \not\in X_k\setminus U_k$ the image of $I_{\tilde z_{U'}}$ in $\pi_1(U_k,\xi_2)$ is trivial. Both of these contradict compatibility of $\varphi_U$ and $\varphi_{U'}$, so we must have $\tilde z_U=\tilde x_U$ and $z=x$.
\end{proof}
The ramification of a section $s:G_K\to\pi_1(X_K-\tilde S)$ is therefore characterised by the ramification of the system of sections $s_U$ it induces. Let us now fix a universal pro-\'etale cover $\tilde X_{\tilde S} \to X_K - \tilde S$ (see Definition \ref{universalcovertilde}), and denote by $\overline {k(X_k)}$ the separable closure of the function field $k(X_k)$ determined by the geometric point $\xi_2$.
\begin{lemma}\label{galptthunramified}
Let $s:G_K\to\pi_1(X_K-\tilde S)$ be an unramified section, and suppose its specialisation $\bar s : G_k \to G_{X_k}$ is geometric with $\bar s(G_k)\subset D_{\tilde x}$ for some $x\in X(k)$ and some extension $\tilde x$ of $x$ to $\overline {k(X_k)}$. Then $s(G_K)\subset D_{\tilde y}$ for a unique $\tilde y$ in $\tilde X_{\tilde S}$ above the unique ($K$-rational) point $y$ of $\tilde S$ specialising to $x$.
\end{lemma}
\begin{proof}
For any open subset $U_k \subset X_k$, the section $s_U$ is unramified by Lemma \ref{toramifiedsubsets}. Choose $U_k$ so that $x \not\in U_k$, and let $\tilde X_{\tilde U_K}$, respectively $\tilde X_{\tilde U_k}$ denote the normalisation of $X_K$, resp. $X_k$ in some universal pro-\'etale cover of $U_K$, resp. $U_k$. By compatibility of the homomorphisms $\varphi_U$, we have $\bar s_U(G_k) \subset D_{\tilde x_U}$ for some $\tilde x_U$ in $\tilde X_{U_k}$ above $x$. Hence, by Lemma \ref{unramifiedcuspidal}, $s_U(G_K) \subset D_{\tilde y_U}$ for some unique $\tilde y_U$ in $\tilde X_{U_K}$ above the unique point $y$ of $S_K$ specialising to $x$. This is also true for every open subset of $X_k$ contained in $U_k$, so taking inverse limits yields the required statement.
\end{proof}
\begin{lemma}\label{galptthramified}
Assume that $k$ satisfies condition (ii) of Definition \ref{conditions}, and let $s:G_K\to\pi_1(X_K-\tilde S)$ be a section. If $s$ is ramified then $\varphi_s(G_K)\subset D_{\tilde x}$ for a unique valuation $\tilde x$ on $\overline {k(X_k)}$ extending a $k$-rational point $x$ of $X_k$, and $s(G_K)\subset D_{\tilde y}$ for a unique $\tilde y$ in $\tilde X_{\tilde S}$ above the unique $K$-rational point $y$ in $\tilde S$ specialising to $x$.
\end{lemma}
\begin{proof}
Lemma \ref{toramifiedsubsets} implies that, for some open subset $U_k \subset X_k$, the section $s_U$ is ramified. Let $\tilde X_{\tilde U_K}$, respectively $\tilde X_{\tilde U_k}$ denote the normalisation of $X_K$, resp. $X_k$ in some universal pro-\'etale cover of $U_K$, resp. $U_k$. By Lemma \ref{ramified=cuspidal} we have $\varphi_U(G_K) \subset D_{\tilde x_U}$ for some $x \in (X_k\setminus U_k)(k)$ and some $\tilde x_U$ in $\tilde X_{\tilde U_k}$ above $x$, and $s_U(G_K) \subset D_{\tilde y_{U}}$ for some unique $\tilde y_U$ in $\tilde X_{\tilde U_K}$ above the unique $K$-rational point $y$ of $\tilde S$ specialising to $x$. By Proposition \ref{ramifiedsubsets}, this is true for every open subset $U'_k\subset X_k$ contained in $U_k$, thus taking inverse limits over the open subsets of $X_k$ yields the required statement (the uniqueness of $\tilde x$ follows from the same argument as in the proof of \cite[Corollary 12.1.3]{CONF}).
\end{proof}
\section{Sections of absolute Galois groups of curves over function fields}
\subsection{Specialisation of sections of absolute Galois groups}\label{specialisationofsections}
Let $k$ be a field of characteristic zero, and let $C$ a smooth, separated, connected curve over $k$ with function field $K$. Let $\X\rightarrow C$ be a flat, proper, smooth relative curve, with generic fibre $X := \X \times_C K$ which is a geometrically connected curve over $K$. For some $c\in C^{\cl}$, let $K_c$ denote the completion of $K$ with respect to the valuation corresponding to $c$, and let $X_c:=X \times_K K_c$ be the base change of $X$ to $K_c$.
Let $\xi$ be a geometric point of $X$ with value in its generic point. This determines an algebraic closure $\overline {K(X)}$ of the function field of $X$ and an algebraic closure $\ob{K}$ of $K$, as well as a geometric point $\bar\xi$ of $X_{\ob{K}}$. Likewise let $\xi_c$ be a geometric point of $X_c$ with value in its generic point, which determines an algebraic closure $\ob{K_c}$ of $K_c$ and a geometric point $\bar\xi_c$ of $X_{\ob{K_c}}$. Fix an embedding $\ob{K}\hookrightarrow \ob{K_c}$. This determines a natural inclusion $i : X(\ob{K}) \hookrightarrow X_c(\ob{K_c})$. Set-theoretically, an element $y : \Spec\ob{K_c} \to X_c$ of $X_c(\ob{K_c})$ maps the unique point of $\Spec\ob{K_c}$ to a closed point of $X_c$. This closed point will be called an \emph{algebraic point} of $X_{K_c}$ if $y \in i(X(\ob{K}))$; otherwise, it will be called a \emph{transcendental point}. Let us denote by $X_c^{\tr}$ the complement in $X_c^{\cl}$ of the set of all algebraic points of $X_c^{\cl}$.
\begin{definition}
With the above notation, define the group $\pi_1(X_c^{\tr})$ to be the inverse limit
\[\pi_1(X_c^{\tr}) := \varprojlim_{U \subset X \textup{ open}} \pi_1(U_c, \xi_c)\]
over all open subsets $U \subset X$, where $U_c$ denotes the base change $U \times_K K_c$.
\end{definition}
Recall there is an exact sequence $1 \to G_{X_{\ob{K}}} \to G_X \to G_K := \Gal(\ob{K}/K) \to 1$, where $G_X$ is the Galois group of $X$ with base point $\xi$ (cf. Lemma \ref{Gallim}).
\begin{lemma}\label{pullbackgx}
We have a commutative diagram of exact sequences
\[\begin{tikzpicture}[descr/.style={fill=white}, baseline=(current bounding box.center)]
\matrix(m)[matrix of math nodes,
row sep=3em, column sep=2.5em,
text height=1.5ex, text depth=0.25ex]
{1 & \pi_1(X_{\ob{K_c}}^{\tr}) & \pi_1(X_c^{\tr}) & G_{K_c} & 1\\
1 & G_{X_{\ob{K}}} & G_X & G_K & 1\\};
\path[->]
(m-1-1) edge (m-1-2);
\path[->]
(m-1-2) edge (m-1-3) edge (m-2-2);
\path[->]
(m-1-3) edge (m-1-4) edge (m-2-3);
\path[->]
(m-1-4) edge (m-1-5) edge (m-2-4);
\path[->]
(m-2-1) edge (m-2-2);
\path[->]
(m-2-2) edge (m-2-3);
\path[->]
(m-2-3) edge (m-2-4);
\path[->]
(m-2-4) edge (m-2-5);
\end{tikzpicture}\]
where $G_{K_c}=\Gal (\ob{K_c}/K_c)$, $\pi_1(X_{\ob{K_c}}^{\tr})$ is defined so that the upper horizontal sequence is exact, the middle vertical map is defined up to conjugation,
the left vertical map is an isomorphism, and the right square is cartesian.
\end{lemma}
\begin{proof}
For each open subset $U \subset X$, functoriality of the fundamental group yields a diagram
\begin{equation}\label{pullbackdiagrampi1u}
\begin{tikzpicture}[descr/.style={fill=white}, baseline=(current bounding box.center)]
\matrix(m)[matrix of math nodes,
row sep=3em, column sep=2.5em,
text height=1.5ex, text depth=0.25ex]
{1 & \pi_1(U_{\ob{K_c}},\bar\xi_c) & \pi_1(U_c,\xi_c) & G_{K_c} & 1\\
1 & \pi_1(U_{\ob{K}},\bar\xi) & \pi_1(U,\xi) & G_K & 1\\};
\path[->]
(m-1-1) edge (m-1-2);
\path[->]
(m-1-2) edge (m-1-3) edge (m-2-2);
\path[->]
(m-1-3) edge (m-1-4) edge (m-2-3);
\path[->]
(m-1-4) edge (m-1-5) edge (m-2-4);
\path[->]
(m-2-1) edge (m-2-2);
\path[->]
(m-2-2) edge (m-2-3);
\path[->]
(m-2-3) edge (m-2-4);
\path[->]
(m-2-4) edge (m-2-5);
\end{tikzpicture}
\end{equation}
where the rows are the fundamental exact sequences, the middle vertical map is defined up to conjugation,
the left vertical map is an isomorphism (see \cite{Luminy}, Th\'eor\`eme 1.6),
and the right square is cartesian (follows as in the proof of Lemma \ref{sppullback}). The statement then follows by taking the projective limit of these diagrams over the open subsets of $X$.
\end{proof}
Denote by $\X_c:=\X\times_C k(c)$ the fibre of $\X$ above $c$.
\begin{lemma}\label{liftingtoalgpts}
For each closed point $x$ of $\X_c^{\cl}$, there exists an algebraic point $y$ of $X_c$ specialising to $x$ whose residue field is the unique unramified extension $L$ of $K_c$ whose valuation ring $\oh_{L}$ has residue field $k(x)$.
\end{lemma}
\begin{proof}
Let us write $X_{c,L} := X_c \times_{K_c} L$ and $\X_{c,k(x)} := \X_c \times_{k(c)} k(x)$. Let $x'$ be a $k(x)$-rational point of $\X_{c,k(x)}$ which maps to $x$ under the projection $\X_{c,k(x)} \to \X_c$, and let $\m_{L}$ denote the maximal ideal of $\oh_{L}$. The set of $L$-rational points of $X_{c,L}$ which specialise to $x'$ is in bijection with $\m_{L}$ \cite[Proposition 10.1.40]{Liu}. Let $F$ be a finite extension of $K$ whose completion at a place above $c$ is $L$. Then an element of $\m_{L} \cap F$ corresponds to an $L$-rational algebraic point $y'$ of $X_{c,L}$ which specialises to $x'$. The image $y$ of $y'$ under the projection $X_{c,L} \to X_c$ is then an $L$-rational algebraic point of $X_c$ ($k(y)=L$) which specialises to $x$.
\end{proof}
\begin{definition}\label{stildekc}
For each closed point $x$ of $\X_c^{\cl}$, fix a choice of algebraic point $y\in X_c^{\cl}$ which specialises to $x$ and whose residue field is the unique unramified extension of $K_c$ whose valuation ring has residue field $k(x)$ (such a point exists by Lemma \ref{liftingtoalgpts}). We define $\tilde S_c$ to be the set of these chosen algebraic points $y\in X_c^{\cl}$. Thus, $\tilde S_c$ is a subset of $X_c^{\cl}$ which consists of algebraic points and which is in bijection with $\X_c^{\cl}$.
We denote by $\tilde S_{c,\ob{K_c}}$ the set of points of $X_c\times _{K_c}\ob {K_c}$ which map to points in $\tilde S_c$.
Thus $\tilde S_{c,\ob{K_c}}$ is a subset of $X_{\ob{K_c}}^{\cl}$ in bijection with $\X_{\ob{k(c)}}^{\cl}$.
\end{definition}
Let $\xi'_c$ be a geometric point of $\X_c$ with value in its generic point, which determines an algebraic closure $\ob{k(c)}$ of $k(c)$ and a geometric point $\bar\xi'_c$ of $\X_{\ob{k(c)}}$. Let $1 \to G_{\X_{\ob{k(c)}}} \to G_{\X_c} \to G_{k(c)} \to 1$ be the exact sequence of the absolute Galois group of $\X_c$ with base point $\xi'_c$.
\begin{theorem}\label{galspeckc}
With $\tilde S_c$ and $\tilde S_{c,\ob{K_c}}$ as in Definition \ref{stildekc}, there exists a surjective homomorphism $\Sp : \pi_1(X_c - \tilde S_c) \twoheadrightarrow G_{\X_c}$, an isomorphism $\ob{\Sp} : \pi_1(X_{\ob{K_c}} - \tilde S_{c,\ob{K_c}}) \simeq G_{\X_{\ob{k(c)}}}$ and a commutative diagram of exact sequences:
\[\begin{tikzpicture}[descr/.style={fill=white}, baseline=(current bounding box.center)]
\matrix(m)[matrix of math nodes,
row sep=3em, column sep=2.5em,
text height=1.5ex, text depth=0.25ex]
{1 & \pi_1(X_{\ob{K_c}}^{\tr}) & \pi_1(X_c^{\tr}) & G_{K_c} & 1\\
1 & \pi_1(X_{\ob{K_c}}-\tilde S_{c,\ob{K_c}}) & \pi_1(X_c-\tilde S_c) & G_{K_c} & 1\\
1 & G_{\X_{\ob{k(c)}}} & G_{\X_c} & G_{k(c)} & 1\\};
\path[->]
(m-1-1) edge (m-1-2);
\path[->]
(m-1-2) edge (m-1-3);
\path[->]
(m-1-3) edge (m-1-4);
\path[->]
(m-1-4) edge (m-1-5);
\path[->>]
(m-1-2) edge (m-2-2);
\path[->>]
(m-1-3) edge (m-2-3);
\path[-]
(m-1-4) edge[double distance=2pt] (m-2-4);
\path[->]
(m-2-1) edge (m-2-2);
\path[->]
(m-2-2) edge (m-2-3);
\path[->]
(m-2-3) edge (m-2-4);
\path[->]
(m-2-4) edge (m-2-5);
\path[->,font=\scriptsize]
(m-2-2) edge node[left]{$\ob{\Sp}$} node[below,rotate=90]{$\sim$} (m-3-2);
\path[->>,font=\scriptsize]
(m-2-3) edge node[left]{$\Sp$} (m-3-3);
\path[->>,font=\scriptsize]
(m-2-4) edge node[left]{} (m-3-4);
\path[->]
(m-3-1) edge (m-3-2);
\path[->]
(m-3-2) edge (m-3-3);
\path[->]
(m-3-3) edge (m-3-4);
\path[->]
(m-3-4) edge (m-3-5);
\end{tikzpicture}\]
where the map $\Sp$ is defined up to conjugation.
\end{theorem}
\begin{proof}
The existence of $\ob{\Sp}$ and $\Sp$ follows from Theorem \ref{galspec}. Since we have chosen $\tilde S_c$ to consist of algebraic points, $\pi_1(X_c-\tilde S_c)$, respectively $\pi_1(X_{\ob{K_c}} - \tilde S_{c,\ob{K_c}})$ is naturally a quotient of $\pi_1(X_c^{\tr})$, resp. $\pi_1(X_{\ob{K_c}}^{\tr})$. Thus we have the required homomorphisms in the diagram, and it is clearly commutative.
\end{proof}
Let $s:G_K\rightarrow G_X$ be a section, and let $s_c:G_{K_c}\rightarrow\pi_1(X_c^{\tr})$ be the section of $\pi_1(X_c^{\tr})$ induced from $s$ (cf. Lemma \ref{pullbackgx}). This naturally induces a section $\tilde s_c:G_{K_c}\rightarrow\pi_1(X_c-\tilde S_c)$ of the projection $\pi_1(X_c-\tilde S_c)\twoheadrightarrow G_{K_c}$.
\begin{theorem}\label{stildeptth}
With the notation of the above paragraph, assume further that $X$ is hyperbolic. Let $\tilde X_{c, \tilde S_c} \to X_c - \tilde S_c$ be a universal pro-\'etale cover (Definition \ref{universalcovertilde}), and denote by $\overline {k(\X_c)}$ the separable closure of the function field of $\X_c$ determined by the geometric point $\xi'_c$.
\begin{enumerate}
\item Suppose $\tilde s_c$ is unramified and induces a section $\bar s_c:G_{k(c)}\rightarrow G_{\X_c}$. If $\bar s_c$ is geometric with $\bar s_c(G_{k(c)})\subset D_{\tilde x}$ for some $x\in\X_c(k(c))$ and some extension $\tilde x$ of $x$ to $\overline {k(\X_c)}$, then $\tilde s_c(G_{K_c})\subset D_{\tilde y}$ for some unique $\tilde y$ in $\tilde X_{c, \tilde S_c}$ above the unique ($K_c$-rational) point $y$ of $\tilde S_c$ specialising to $x$.
\item Assume further that $k$ strongly satisfies condition (ii) of Definition \ref{conditions}. Suppose $\tilde s_c$ is ramified, and denote $\varphi_s:=\Sp\circ\tilde s_c$. Then $\varphi_s(G_{K_c})\subset D_{\tilde x}$ for a unique valuation on $\overline {k(\X_c)}$ extending a $k(c)$-rational point $x$ of $\X_c$, and $\tilde s_c(G_{K_c})\subset D_{\tilde y}$ for some unique $\tilde y$ in $\tilde X_{c, \tilde S_c}$ above the unique ($K_c$-rational) point $y$ of $\tilde S_c$ specialising to $x$.
\end{enumerate}
\end{theorem}
\begin{proof}
Part (i) follows from Lemma \ref{galptthunramified}, and (ii) follows from Lemma \ref{galptthramified}.
\end{proof}
\subsection{\'Etale abelian sections}\label{etaleabeliansections}
We use the notation of \S \ref{specialisationofsections}, and hereafter we assume that $X$ is hyperbolic and that $X(K)\ne\emptyset$.
\begin{lemma}
For each $c \in C^{\cl}$, there is a commutative diagram
\begin{equation}\label{specialisationofset}
\begin{tikzpicture}[descr/.style={fill=white}, baseline=(current bounding box.center)]
\matrix(m)[matrix of math nodes,
row sep=3em, column sep=2.5em,
text height=1.5ex, text depth=0.25ex]
{1 & \pi_1(X_{\ob{K}}, \bar\xi) & \pi_1(X, \xi) & G_K & 1\\
1 & \pi_1(X_{\ob{K}}, \bar\xi) & \pi_1(\X, \xi) & \pi_1(C, \xi) & 1\\
1 & \pi_1(\X_{\ob{k(c)}}, \bar\xi'_c) & \pi_1(\X_c, \xi'_c) & G_{k(c)} & 1\\};
\path[->]
(m-1-1) edge (m-1-2);
\path[->]
(m-1-2) edge (m-1-3);
\path[->]
(m-1-3) edge (m-1-4);
\path[->, font = \scriptsize]
(m-1-4) edge (m-1-5);
\path[-]
(m-1-2) edge[double distance = 2pt] (m-2-2);
\path[->>]
(m-1-3) edge (m-2-3);
\path[->>]
(m-1-4) edge (m-2-4);
\path[->]
(m-2-1) edge (m-2-2);
\path[->]
(m-2-2) edge (m-2-3);
\path[->]
(m-2-3) edge (m-2-4);
\path[->, font = \scriptsize]
(m-2-4) edge (m-2-5);
\path[->]
(m-3-1) edge (m-3-2);
\path[->, font = \scriptsize]
(m-3-2) edge (m-3-3) edge node[below, rotate = 90]{$\sim$} (m-2-2);
\path[->]
(m-3-3) edge (m-3-4);
\path[right hook->]
(m-3-3) edge (m-2-3);
\path[->, font = \scriptsize]
(m-3-4) edge (m-3-5);
\path[right hook->]
(m-3-4) edge (m-2-4);
\end{tikzpicture}
\end{equation}
where the lower vertical homomorphisms are defined up to conjugation, the middle and right of those maps are injective and the left one is an isomorphism.
\end{lemma}
\begin{proof}
This follows from functoriality of the fundamental group and the fundamental exact sequences for $X$ and $\X_c$. Exactness of the middle row follows from \cite[Expos\'e XIII, Proposition 4.3]{SGA}.
\end{proof}
For the remainder of this section we assume that $k$ strongly satisfies the condition (ii) in Definition 1.4.
Let us fix a section $s:G_K\rightarrow G_X$ of $G_X$, and denote by $s^{\et}:G_K\rightarrow\pi_1(X,\xi)$ the section of the \'etale fundamental group of $X$ induced by $s$. By Lemma \ref{pullbackgx}, $s$ pulls back to a section $s_c:G_{K_c}\rightarrow\pi_1(X_c^{\tr})$, which in turn induces a section $s_c^{\et}:G_{K_c}\rightarrow\pi_1(X_c,\xi_c)$. By Proposition \ref{weightargument} (see also Remark \ref{properimpliesunramified}), $s_c^{\et}$ specialises to a section $\bar s_c^{\et}:G_{k(c)}\rightarrow\pi_1(\X_c,\xi'_c)$. By the following, we may also consider $\bar s_c^{\et}$ the specialisation of $s^{\et}$.
\begin{lemma}
The section $s^{\et}$ extends to a section $s_C^{\et} : \pi_1(C, \xi) \to \pi_1(\X, \xi)$ of the projection $\pi_1(\X, \xi) \twoheadrightarrow \pi_1(C, \xi)$ which restricts to the section $\bar s_c^{\et}$ for each $c \in C^{\cl}$.
\end{lemma}
\begin{proof}
The kernel of the homomorphism $G_K\twoheadrightarrow\pi_1(C, \xi)$ is the inertia group $I_C$ normally generated by the inertia subgroups associated to the closed points of $C$. Since $s_c^{\et}$ is unramified for every $c \in C^{\cl}$, the image of each of these inertia groups under the composition $\Sp_X \circ s_c^{\et} : G_{K_c} \to \pi_1(\X_c, \xi'_c)$ of $s_c^{\et}$ with the specialisation homomorphism $\Sp_X : \pi_1(X_c, \xi_c) \twoheadrightarrow \pi_1(\X_c, \xi'_c)$ is trivial, hence the image of $I_C$ under the composite $\begin{tikzcd}[column sep=small]G_C \arrow{r}{s^{\et}} & \pi_1(X, \xi) \arrow[twoheadrightarrow]{r}{} & \pi_1(\X, \xi)\end{tikzcd}$ is trivial. Thus $s^{\et}$ extends to a section $s_C^{\et}:\pi_1(C, \xi) \rightarrow \pi_1(\X, \xi)$, which must restrict to $\bar s_c^{\et}:G_{k(c)}\rightarrow\pi_1(\X_c, \xi'_c)$.
\end{proof}
Let $\J:=\textup{Pic}^0_{\X/C}\rightarrow C$ denote the relative Jacobian of $\X$, and $J:=\J_K$ the Jacobian of $X$. For each closed point $c\in C^{\cl}$, let $J_c:=J_{K_c}$ denote the Jacobian of $X_c$ and $\J_c:=\J_{k(c)}$ that of $\X_c$. The above sections $s^{\et}$, $s_c^{\et}$ and $\bar s_c^{\et}$ induce \'etale abelian sections $s^{\ab}$, $s_c^{\ab}$ and $\bar s_c^{\ab}$ respectively, while diagram (\ref{specialisationofset}) induces a commutative diagram of exact sequences of geometrically abelian fundamental groups
\begin{equation}\label{specialisationofsab}
\begin{tikzpicture}[descr/.style={fill=white}, baseline=(current bounding box.center)]
\matrix(m)[matrix of math nodes,
row sep=3em, column sep=2.5em,
text height=1.5ex, text depth=0.25ex]
{1 & \pi_1(X_{\ob{K}}, \bar\xi)^{\ab} & \pi_1(X, \xi)^{(\ab)} & G_K = G_C & 1\\
1 & \pi_1(X_{\ob{K}}, \bar\xi)^{\ab} & \pi_1(\X, \xi)^{(\ab)} & \pi_1(C, \xi) & 1\\
1 & \pi_1(\X_{\ob{k(c)}}, \bar\xi'_c)^{\ab} & \pi_1(\X_c, \xi'_c)^{(\ab)} & G_{k(c)} & 1\\};
\path[->]
(m-1-1) edge (m-1-2);
\path[->]
(m-1-2) edge (m-1-3);
\path[->]
(m-1-3) edge (m-1-4);
\path[->, font = \scriptsize]
(m-1-4) edge (m-1-5) edge[out = 165, in = 15] node[above]{$s^{\ab}$} (m-1-3);
\path[-]
(m-1-2) edge[double distance = 2pt] (m-2-2);
\path[->>]
(m-1-3) edge (m-2-3);
\path[->>]
(m-1-4) edge (m-2-4);
\path[->]
(m-2-1) edge (m-2-2);
\path[->]
(m-2-2) edge (m-2-3);
\path[->]
(m-2-3) edge (m-2-4);
\path[->, font = \scriptsize]
(m-2-4) edge (m-2-5) edge[out = 165, in = 15] node[above]{$s_C^{\ab}$} (m-2-3);
\path[->]
(m-3-1) edge (m-3-2);
\path[->, font = \scriptsize]
(m-3-2) edge (m-3-3) edge node[below, rotate = 90]{$\sim$} (m-2-2);
\path[->]
(m-3-3) edge (m-3-4);
\path[right hook->]
(m-3-3) edge (m-2-3);
\path[->, font = \scriptsize]
(m-3-4) edge (m-3-5) edge[out = 160, in = 15] node[above]{$\bar s_c^{\ab}$} (m-3-3);
\path[right hook->]
(m-3-4) edge (m-2-4);
\end{tikzpicture}
\end{equation}
where the middle horizontal row is obtained as the push-out of the middle horizontal row in diagram (9) by the projection
$\pi_1(X_{\ob{K}},\bar\xi)\twoheadrightarrow \pi_1(X_{\ob{K}},\bar \xi)^{\ab}$,
and $s_C^{\ab} : \pi_1(C, \xi) \to \pi_1(\X, \xi)^{(\ab)}$ is induced by $s_C^{\et}$. Since $X(K) \ne \emptyset$ by assumption, the \'etale abelian sections $s^{\ab}$, $s_c^{\ab}$, $s_C^{\ab}$ and $\bar s_c^{\ab}$ correspond to elements of the cohomology groups $H^1(G_K,TJ)$, $H^1(G_{K_c},TJ_c)$, $H^1(\pi_1(C, \xi), TJ)$ and $H^1(G_{k(c)},T\J_c)$ respectively, which are related by the following restriction and inflation maps:
\[\begin{tikzpicture}[descr/.style={fill=white}]
\matrix(m)[matrix of math nodes,
row sep=3em, column sep=2.5em,
text height=2ex, text depth=0.25ex]
{H^1(G_K,TJ) & H^1(G_{K_c},TJ_c)\\
H^1(\pi_1(C, \xi), TJ) & H^1(G_{k(c)},T\J_c)\\};
\path[->,font=\scriptsize]
(m-1-1) edge node[above]{$\res_c$} (m-1-2);
\path[->, font = \scriptsize]
(m-2-1) edge node[left]{$\inf_C$} (m-1-1) edge node[above]{$\res_{C, c}$} (m-2-2);
\path[->, font=\scriptsize]
(m-2-2) edge node[right]{$\inf_c$} (m-1-2);
\end{tikzpicture}\]
\begin{lemma}\label{sabresinf}
With the above notation, we have the following.
\begin{enumerate}
\item $\res_c(s^{\ab}) = \inf_c(\bar s_c^{\ab}) = s_c^{\ab}$;
\item $\inf_C(s_C^{\ab}) = s^{\ab}$ and $\res_{C, c}(s_C^{\ab}) = \bar s_c^{\ab}$.
\end{enumerate}
\end{lemma}
\begin{proof}
Part (i) follows from the fact that $s^{\et}$ and $\bar s^{\et}_c$ both pull back to $s^{\et}_c$ - see diagrams (\ref{pullbackdiagrampi1u}) and (\ref{defofvarphisandramified}), considering the case when $U=X$ is projective. Part (ii) follows from diagram (\ref{specialisationofsab}). See also \cite[Lemma 3.4]{saidiSCOFF}.
\end{proof}
The image of $s^{\ab}$ under the diagonal map
\[\prod_{c\in C^{\cl}}\res_c:H^1(G_K,TJ)\longrightarrow\prod_{c\in C^{\cl}}H^1(G_{K_c},TJ_c)\]
is therefore the family $(s_c^{\ab})_{c\in C^{\cl}}$. This diagonal map fits into the following commutative diagram of Kummer exact sequences (see Lemma \ref{kummerexactseq}).
\[\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=3em, column sep=2em,
text height=2ex, text depth=0.25ex]
{0 & \widehat{J(K)} & H^1(G_K,TJ) & TH^1(G_K,J) & 0\\
0 & \displaystyle\prod_{c\in C^{\cl}}\widehat{J_c(K_c)} & \displaystyle\prod_{c\in C^{\cl}}H^1(G_{K_c},TJ_c) & \displaystyle\prod_{c\in C^{\cl}}TH^1(G_{K_c},J_c) & 0\\};
\path[->]
(m-1-1) edge (m-1-2);
\path[->]
(m-1-2) edge (m-1-3) edge (m-2-2);
\path[->]
(m-1-3) edge (m-1-4) edge (m-2-3);
\path[font=\scriptsize,->]
(m-1-4) edge (m-1-5) edge node[left]{} (m-2-4);
\path[->]
(m-2-1) edge (m-2-2);
\path[->]
(m-2-2) edge (m-2-3);
\path[->]
(m-2-3) edge (m-2-4);
\path[->]
(m-2-4) edge (m-2-5);
\end{tikzpicture}\]
Note that the kernel of the right vertical map is the Tate module $T\Sha(\J)$ of the Shafarevich-Tate group $\Sha(\J)$ (Definition \ref{sha}). Commutativity of this diagram immediately implies the following.
\begin{lemma}\label{tshatrivial}
Suppose that, for every $c\in C^{\cl}$, the section $s_c^{\ab}\in H^1(G_{K_c},TJ_c)$ is contained in $\widehat{J_c(K_c)}$. Then if $T\Sha(\J)=0$, the section $s^{\ab}\in H^1(G_K,TJ)$ is contained in $\widehat{J(K)}$.
\end{lemma}
\begin{proposition}\label{sabjkhat}
Assume that $T\Sha(\J)=0$, and that $k$ strongly satisfies conditions (i), (ii) and (iii)(a) of Definition \ref{conditions}. Then we have the following.
\begin{enumerate}
\item For each $c \in C^{\cl}$, $\bar s_c^{\ab}$ is in the image of the injective map $\X_c(k(c)) \hookrightarrow H^1(G_{k(c)},T\J_c)$, and $s_c^{\ab}$ is in the image of $X_c(K_c)\to H^1(G_{K_c},TJ_c)$.
\item $s^{\ab}$ is contained in $\widehat{J(K)}$.
\end{enumerate}
\end{proposition}
\begin{proof}
Let $\tilde s_c : G_{K_c} \to \pi_1(X_c - \tilde S_c)$ denote the section of $\pi_1(X_c - \tilde S_c)$ induced by $s_c$, and write $\varphi_{s_c} := \Sp \circ \tilde s_c$. Let $\tilde X_{c, \tilde S_c} \to X_c - \tilde S_c$ be a universal pro-\'etale cover, and recall $\overline {k(\X_c)}$ the separable closure of the function field of $\X_c$ determined by $\xi'_c$. By Theorem \ref{stildeptth}, for every $c\in C^{\cl}$ we have $\varphi_{s_c}(G_{K_c}) \subset D_{\tilde x}$ for a unique $x \in \X_c(k(c))$ and some unique extension $\tilde x$ of $x$ to $\overline {k(\X_c)}$, and $\tilde s_c(G_{K_c}) \subset D_{\tilde y}$ for some unique $\tilde y$ in $\tilde X_{c, \tilde S_c}$ above the unique ($K_c$-rational) point $y$ of $\tilde S_c$ specialising to $x$. This implies that $s_c^{\et}(G_{K_c})=D_{\tilde y'}$, for some $\tilde y'$ above $y$ in a universal pro-\'etale cover $\tilde X_c \to X_c$, and $\bar s_c^{\et}(G_{k(c)}) = D_{\tilde x'}$ for some $\tilde x'$ above $x$ in a universal pro-\'etale cover $\tilde\X_c \to \X_c$. This means that $s_c^{\et}$, respectively $\bar s_c^{\et}$ arises from $y \in X_c(K_c)$, resp. $x \in \X_c(k(c))$ by functoriality of the fundamental group, which proves (i) (see the discussion before Lemma \ref{kummerexactseq}). The map $\X_c(k(c))\to H^1(G_{k(c)},T\J_c)$ is injective by condition (iii)(a) of Definition \ref{conditions}.
Since the map $X_c(K_c)\to H^1(G_{K_c},TJ_c)$ factors through the inclusion $\widehat{J_c(K_c)} \hookrightarrow H^1(G_{K_c}, TJ_c)$ (see sequence (\ref{sabptth})), part (i) implies in particular that $s_c^{\ab}$ is contained in $\widehat{J_c(K_c)}$, and since this is true for every $c\in C^{\cl}$, Lemma \ref{tshatrivial} implies that $s^{\ab}\in\widehat{J(K)}$, which proves (ii).
\end{proof}
\section{Proof of the Main Theorems}
\subsection{Proof of Theorem A}
Let $k$ be a field of characteristic zero that satisfies conditions (iv) and (v) of Definition \ref{conditions}, and strongly satisfies conditions (i), (ii) and (iii) of Definition \ref{conditions}. Let $C$ be a smooth, separated, connected curve over $k$ with function field $K$. Let $\X\rightarrow C$ be a flat, proper, smooth relative curve whose generic fibre $X:=\X\times_C K$ is geometrically connected and hyperbolic, with $X(K)\ne\emptyset$. Let $\J:=\Pic^0_{\X/C}$ denote the relative Jacobian of $\X$, and $J := \J_K$ the Jacobian of $X$. For a closed point $c \in C^{\cl}$, denote by $\J_c := \J_{k(c)}$ the Jacobian of $\X_c$. Assume that $T\Sha(\J)=0$. We show that the birational section conjecture holds for $X$ (see Definitions \ref{geometricgaloissections} and \ref{bscholds} and Remark \ref{bscuniqueness}).
Let $s:G_K\rightarrow G_X$ be a section. Under our assumptions, the \'etale abelian section $s^{\ab}$ induced by $s$ is contained in $\widehat{J(K)}$, by Proposition \ref{sabjkhat} (ii).
\begin{lemma}\label{sabinjk}
The homomorphism $J(K)\rightarrow\widehat{J(K)}$ is injective and $s^{\ab}$ is contained in $J(K)$.
\end{lemma}
\begin{proof}
There exist $c_1,c_2\in C^{\cl}$ such that the natural specialisation homomorphism $J(K)\rightarrow\J_{c_1}(k(c_1))\times\J_{c_2}(k(c_2))$ is injective \cite[Proposition 2.4]{poonenvoloch}.
Let $\ell $ be a finite extension of $k$ that contains $k(c_1)$ and $k(c_2)$. Then there is an injective homomorphism $\J_{c_i}(k(c_i))\hookrightarrow\J_{c_i}(\ell )$ for each $i=1,2$, hence an injective homomorphism $\J_{c_1}(k(c_1))\times\J_{c_2}(k(c_2))\hookrightarrow\J_{c_1}(\ell)\times\J_{c_2}(\ell)\simeq(\J_{c_1}\times\J_{c_2})(\ell)$. For ease of notation, let us write $\J_{1,2}(\ell):= (\J_{c_1}\times\J_{c_2})(\ell)$. We have a commutative diagram of exact sequences:
\[\begin{tikzpicture}[descr/.style={fill=white}]
\matrix(m)[matrix of math nodes,
row sep=3em, column sep=3em,
text height=2.5ex, text depth=0.25ex]
{0 & J(K) & \J_{1,2}(\ell) & H & 0\\
0 & \widehat{J(K)} & \widehat{\J_{1,2}(\ell)} & \widehat{H} & 0\\};
\path[->]
(m-1-1) edge (m-1-2);
\path[->]
(m-1-2) edge (m-1-3) edge (m-2-2);
\path[->,font=\scriptsize]
(m-1-3) edge (m-1-4) edge node[left]{$\phi$} (m-2-3);
\path[->]
(m-1-4) edge (m-1-5) edge (m-2-4);
\path[->]
(m-2-1) edge (m-2-2);
\path[->,font=\scriptsize]
(m-2-2) edge node[above]{$\psi$} (m-2-3);
\path[->]
(m-2-3) edge (m-2-4);
\path[->]
(m-2-4) edge (m-2-5);
\end{tikzpicture}\]
where $H$ is defined so that the upper horizontal sequence is exact. Exactness of the lower sequence follows easily from condition (iii) (b) of Definition \ref{conditions}, while condition (iii) (a) implies that the middle and right vertical maps are injective. Therefore the left vertical map is also injective, and the equality $J(K)=\phi(\J_{1,2}(\ell))\cap\psi(\widehat{J(K)})$ holds inside $\widehat{\J_{1,2}(\ell)}$.
For each $c_i$, $i=1,2$, the section $s$ induces an element $\bar s_{c_i}^{\ab}\in H^1(G_{k(c_i)},T\J_{c_i})$, which is contained in the image of the map $\X_{c_i}(k(c_i)) \to H^1(G_{k(c_i)}, T\J_{c_i})$ by Proposition \ref{sabjkhat} (i). This map is injective by condition (iii)(a) of Definition \ref{conditions}, so we may consider $\bar s_{c_i}^{\ab}$ to be contained in $\X_{c_i}(k(c_i))$. Then $\bar s_{c_i}^{\ab}$ is contained in $\J_{c_i}(\ell)$ for each $i=1,2$, due to injectivity of the maps $\X_{c_i}(k(c_i))\hookrightarrow\J_{c_i}(k(c_i))\hookrightarrow\J_{c_i}(\ell)$. Thus $(\bar s_{c_1}^{\ab},\bar s_{c_2}^{\ab})$ is contained in $\J_{c_1}(\ell)\times\J_{c_2}(\ell)$, hence in $\phi(\J_{1,2}(\ell))$. By Lemma \ref{sabresinf}, the image of $s^{\ab}\in\widehat{J(K)}$ in $\widehat{\J_{1,2}(\ell)}$ under $\psi$ is the element $(\bar s_{c_1}^{\ab},\bar s_{c_2}^{\ab})$, and since this lies in $\phi(\J_{1,2}(\ell))$ we have $s^{\ab}\in\phi(\J_{1,2}(\ell))\cap\psi(\widehat{J(K)})=J(K)$.
\end{proof}
Fix a $K$-rational point $x_0\in X(K)=\X(C)$ (non-empty by assumption), and let $\iota:\X\rightarrow\J$ denote the closed immersion mapping $x_0$ to the zero section of $\J$.
\begin{lemma}\label{sabinxk}
$s^{\ab}$ is contained in $X(K)$.
\end{lemma}
\begin{proof}
Since $s^{\ab}$ is contained in $J(K) = \J(C)$, it may be regarded as a morphism $s^{\ab} : C \to \J$. By Lemma \ref{sabresinf}, the pullback of this morphism to $\Spec k(c)$ is precisely $\bar s_c^{\ab}$, considered an element of $\J_c(k(c)) \subset H^1(G_{k(c)}, T\J_c)$. But $\bar s_c^{\ab}$ is contained in $\X_c(k(c))$ by Proposition \ref{sabjkhat} (i), hence the morphism $\bar s_c^{\ab} : \Spec k(c) \to \J_c$ must factor through $\X_c$, where $\X_c$ is considered a closed subscheme of $\J_c$ via $\iota$. Thus, for each $c \in C^{\cl}$, the image of $c$ under the morphism $s^{\ab} : C \to \J$ is a closed point of $\X_c$. This implies that $s^{\ab} : C \to \J$ factors through $\iota:\X\rightarrow\J$, thus $s^{\ab}$ is contained in the subset $\iota(X(K))\subseteq J(K)$.
\end{proof}
Let $z$ be the point in $X(K)$ such that $\iota(z)=s^{\ab}$, and for each $c \in C^{\cl}$ let $\bar z_c$ denote its specialisation to $\X_c$. Let $\bar x_c\in\X_c(k(c))$ be the point associated to $\bar s_c^{\ab}$ by Proposition \ref{sabjkhat} (i).
\begin{lemma}\label{zc=xc}
$\bar z_c=\bar x_c$ in $\X_c(k(c))$ for all $c\in C^{\cl}$.
\end{lemma}
\begin{proof}
Lemma \ref{sabresinf} implies that, for each $c \in C^{\cl}$, $\bar s_c^{\ab}$ is the image of both $\bar x_c$ and $\bar z_c$ under the map $\X_c(k(c)) \to H^1(G_{k(c)},T\J_c)$. This map is injective by condition (iii)(a) of Definition \ref{conditions}, hence $\bar z_c=\bar x_c$.
\end{proof}
\begin{proposition}\label{sisgeometric}
$s$ is geometric.
\end{proposition}
\begin{proof}
By the ``limit argument'' of Tamagawa \cite[Proposition 2.8 (iv)]{tamagawa}, and the fact that $k$ satisfies the condition (iv) in Definition 4.1,
it suffices to prove that for any open subgroup $H \subset G_X$ which contains $s(G_K)$, if $Y \to X$ denotes the corresponding finite morphism with $Y$ smooth, we have $Y(K) \ne \emptyset$.
By construction, we have $G_Y = H$, and $s$ defines a section $s_Y : G_K \to G_Y$ of $G_Y$. For $c\in C^{\cl}$, let $K_c$ be the completion of $K$ at $c$, and write $X_c:=X\times_K K_c$ and $Y_c:=Y\times_K K_c$. By Lemma \ref{pullbackgx}, the section $s$ pulls back to a section $s_c:G_{K_c}\rightarrow\pi_1(X_c^{\tr})$, and likewise $s_Y$ pulls back to a section $s_{Y_c}:G_{K_c}\rightarrow\pi_1(Y_c^{\tr})$. We have the following commutative diagram.
\[\begin{tikzpicture}[descr/.style = {fill = white}, baseline = (current bounding box.center)]
\matrix(m)[matrix of math nodes,
row sep=3em, column sep=3em,
text height=2ex, text depth=0.25ex]
{\pi_1(Y_c^{\tr}) & \pi_1(X_c^{\tr}) & G_{K_c}\\
G_Y & G_X & G_K\\};
\path[right hook->]
(m-1-1) edge (m-1-2);
\path[->]
(m-1-1) edge (m-2-1);
\path[->]
(m-1-2) edge (m-1-3);
\path[->]
(m-1-2) edge (m-2-2);
\path[->,font=\scriptsize]
(m-1-3) edge[out=165,in=15] node[above left, near end]{$s_c$} (m-1-2) edge[out=145,in=25] node[below]{$s_{Y_c}$} (m-1-1);
\path[->]
(m-1-3) edge (m-2-3);
\path[right hook->]
(m-2-1) edge (m-2-2);
\path[->]
(m-2-2) edge (m-2-3);
\path[->,font=\scriptsize]
(m-2-3) edge[out=165,in=15] node[above left, near end]{$s$} (m-2-2) edge[out=150,in=25] node[above left,near end]{$s_Y$} (m-2-1);
\end{tikzpicture}\]
Let $\Y$ be the normalisation of $\X$ in the function field of $Y$, and for each $c \in C^{\cl}$ let $\Y_c$ denote the closed fibre of $\Y$ at $c$. After possibly removing finitely many points from $C$, we may assume that $\Y$ is smooth over $C$. Indeed, the closed fibres $\Y_c$ are smooth except possibly for finitely many closed points $c\in C^{\cl}$ \cite[Proposition 10.1.21]{Liu}. So, if necessary, we may replace $C$ by the largest open sub-scheme $C'\subset C$ such that $\Y_c$ is smooth for every $c\in(C')^{\cl}$, and $\Y \to \X$ by the induced map of fibre products $\Y\times_C C' \to \X\times_C C'$.
So we assume that the fibres $\Y_c$ are smooth for all $c\in C^{\cl}$. For each closed point $\bar x_c \in \X_c^{\cl}$, respectively $\bar y_c \in \Y_c^{\cl}$, choose an algebraic point $x_c \in X_c$, resp. $y_c \in Y_c$ specialising to $\bar x_c$, resp. $\bar y_c$ whose residue field is the unique unramified extension of $K_c$ whose valuation ring has residue field $k(\bar x_c)$, resp. $k(\bar y_c)$. Let $\tilde S_c$, respectively $\tilde T_c$ denote the set of these chosen algebraic points of $X_c$, resp. $Y_c$ (see Definition \ref{stildekc}). The groups $\pi_1(Y_c-\tilde T_c)$ and $\pi_1(X_c-\tilde S_c)$ are naturally quotients of $\pi_1(Y_c^{\tr})$ and $\pi_1(X_c^{\tr})$ respectively, hence $s_c$ naturally induces a section $\tilde s_c:G_{K_c}\rightarrow\pi_1(X_c-\tilde S_c)$, and likewise $s_{Y_c}$ induces a section $\tilde s_{Y_c}:G_{K_c}\rightarrow\pi_1(Y_c-\tilde T_c)$. By Theorem \ref{galspeckc}, there exist specialisation homomorphisms $\Sp_X : \pi_1(X_c - \tilde S_c) \twoheadrightarrow G_{\X_c}$ and $\Sp_Y : \pi_1(Y_c - \tilde T_c) \twoheadrightarrow G_{\Y_c}$ and a commutative diagram
\[\begin{tikzpicture}[descr/.style = {fill = white}, baseline = (current bounding box.center)]
\matrix(m)[matrix of math nodes,
column sep=3em,
text height=2ex, text depth=0.25ex]
{\pi_1(Y_c^{\tr}) & \pi_1(X_c^{\tr}) & G_{K_c}\\[2em]
\pi_1(Y_c-\tilde T_c) & & \\[0em]
& & G_{K_c}\\[0em]
& \pi_1(X_c-\tilde S_c) & \\[3.5em]
G_{\Y_c} & G_{\X_c} & G_{k(c)}\\};
\path[right hook->]
(m-1-1) edge (m-1-2);
\path[->>]
(m-1-1) edge (m-2-1);
\path[->]
(m-1-2) edge (m-1-3);
\path[->>]
(m-1-2) edge (m-4-2);
\path[->,font=\scriptsize]
(m-1-3) edge[out=165,in=15] node[above left, near end]{$s_c$} (m-1-2) edge[out=150,in=25] node[below]{$s_{Y_c}$} (m-1-1);
\path[-]
(m-1-3) edge[double distance=2pt] (m-3-3);
\path[font=\scriptsize,->]
(m-2-1) edge[out=0, in=165] (m-3-3);
\path[font=\scriptsize,->>]
(m-2-1) edge node[left]{$\Sp_Y$} (m-5-1);
\path[->]
(m-4-2) edge[out=5, in=205] (m-3-3);
\path[font=\scriptsize,->>]
(m-4-2) edge node[left]{$\Sp_X$} (m-5-2);
\path[->,font=\scriptsize]
(m-3-3) edge[out=187,in=17] node[above left, near end]{$\tilde s_c$} (m-4-2) edge[out=150,in=10] node[above left,near end]{$\tilde s_{Y_c}$} (m-2-1) edge[out=220,in=70] node[below right]{$\varphi_X$} (m-5-2) edge[out=220,in=70] node[below right,near end]{$\varphi_Y$} (m-5-1);
\path[font=\scriptsize,->>]
(m-3-3) edge node[left]{} (m-5-3);
\path[font=\scriptsize,right hook->]
(m-5-1) edge node[above]{} (m-5-2);
\path[->]
(m-5-2) edge (m-5-3);
\end{tikzpicture}\]
where we denote $\varphi_X:=\Sp_X\circ\tilde s_c$ and $\varphi_Y:=\Sp_Y\circ\tilde s_{Y_c}$. By Theorem \ref{stildeptth}, we have $\varphi_Y(G_{K_c}) \subset D_{\tilde y_c} \subset G_{\Y_c}$ for a unique valuation $\tilde y_c$ on $\overline {k(\X_c)}$ extending a unique $k(c)$-rational point $\bar y_c \in \Y_c(k(c))$. By commutativity of the above diagram, this implies that $\varphi_X(G_{K_c}) \subset D_{\tilde y_c} \subset G_{\X_c}$ for the same valuation $\tilde y_c$ on $\overline {k(\X_c)}$, whose restriction to $k(\X_c)$ corresponds to the image $\bar x'_c$ of $\bar y_c$ in $\X_c$. Thus we have found, for every $c\in C^{\cl}$, unique $k(c)$-rational points $\bar y_c\in\Y_c(k(c))$ and $\bar x'_c\in\X_c(k(c))$ such that $\bar y_c$ maps to $\bar x'_c$ via $\Y_c\rightarrow\X_c$. Moreover, $\bar x'_c$ must be the same as the point $\bar x_c$ associated to $\bar s_c^{\ab}$ (see Lemma \ref{zc=xc} and the paragraph before it).
Recall the section $s^{\ab}$ is associated to a $K$-rational point $z$ (see Lemma \ref{sabinxk} and the paragraph after it). View $z\in X(K) = \X(C)$ as a section $z:C\rightarrow\X$, and denote by $\Y_z$ the pullback of the image $z(C)$ via the map $\Y\rightarrow\X$. Then $\Y_z\rightarrow z(C)$ is a finite morphism, and we can assume, after possibly shrinking $C$, that $\Y_z$ is smooth. Since $z$ specialises to $\bar x_c\in\X_c(k(c))$ (Lemma \ref{zc=xc}), $\bar x_c\in z(C)$ and therefore $\bar y_c\in\Y_z(k(c))$ for every $c\in C^{\cl}$. Then condition (v) of Definition \ref{conditions} implies that $\Y_z(K)\ne\emptyset$. Thus $\Y_z(K)\subseteq\Y(K)=Y(K)\ne\emptyset$, which completes the proof of Proposition \ref{sisgeometric}.
\end{proof}
Thus $s(G_k)$ is contained in a decomposition group associated to a $K$-rational point $x\in X(K)$, which is unique (cf. Remark \ref{bscuniqueness}). This concludes the proof of Theorem A.
\subsection{Proof of Theorem B}
In this section we explain how Theorem B is deduced from Theorem A. Let $k$ be a field of characteristic zero that strongly satisfies the conditions of Definition \ref{conditions}. Let $C$ be a smooth, separated, connected curve over $k$ with function field $K$. For any finite extension $L$ of $K$, let $C^L$ denote the normalisation of $C$ in $L$, and for any flat, proper, smooth relative curve $\Y \to C^L$, let $\J_{\!\Y} := \Pic^0_{\Y/C^L}$ denote the relative Jacobian of $\Y$. Assume that for any such finite extension $L$ and any such relative curve $\Y$ we have $T\Sha(\J_{\!\Y})=0$.
We will show that for any finite extension $L$ of $K$ and any smooth, projective, geometrically connected (not necessarily hyperbolic) curve $X$ over $L$, the birational section conjecture holds for $X$.
\begin{proposition}
With the above notation and hypotheses, let $s:G_L\rightarrow G_X$ be a section of $G_X$. Then $s$ is geometric.
\end{proposition}
\begin{proof}
By the Hurwitz formula, we may choose an open subgroup $H\subset G_X$ containing $s(G_L)$ such that, denoting by $Y\rightarrow X$ the corresponding finite morphism with $Y$ smooth, $Y$ is hyperbolic. We have an isomorphism $H\simeq G_Y$, and $s$ naturally defines a section $s_Y:G_L\rightarrow G_Y$ of the natural projection $G_Y\twoheadrightarrow G_L$.
Let $L'|L$ be a finite extension such that $Y(L')\ne\emptyset$, and let $M|L$ be a Galois extension of $L$ containing $L'$. Then $Y_M(M)\ne\emptyset$, and $s_Y$ restricts to a section $s_{Y_M}:G_M\rightarrow G_{Y_M}$ of the absolute Galois group of $Y_M$.
\[\begin{tikzpicture}[descr/.style = {fill = white}, baseline = (current bounding box.center)]
\matrix(m)[matrix of math nodes,
row sep=3em, column sep=3em,
text height=1.5ex, text depth=0.25ex]
{1 & G_{Y_{\ob{L}}} & G_{Y_M} & G_M & 1\\
1 & G_{Y_{\ob{L}}} & G_Y & G_L & 1\\};
\path[->]
(m-1-1) edge (m-1-2);
\path[->,font=\scriptsize]
(m-1-2) edge (m-1-3);
\path[-]
(m-1-2) edge[double distance=2pt] (m-2-2);
\path[->]
(m-1-3) edge (m-1-4);
\path[right hook->,font=\scriptsize]
(m-1-3) edge (m-2-3);
\path[->,font=\scriptsize]
(m-1-4) edge (m-1-5) edge[out=160,in=20] node[above]{$s_{Y_M}$} (m-1-3);
\path[right hook->,font=\scriptsize]
(m-1-4) edge (m-2-4);
\path[->]
(m-2-1) edge (m-2-2);
\path[->]
(m-2-2) edge (m-2-3);
\path[->]
(m-2-3) edge (m-2-4);
\path[->,font=\scriptsize]
(m-2-4) edge (m-2-5) edge[out=160,in=20] node[above]{$s_Y$} (m-2-3);
\end{tikzpicture}\]
Let $C^M$ denote the normalisation of $C$ in $M$, and let $\Y \to C^M$ be a flat and proper model of $Y_M$ over $C^M$. As in the proof of Proposition \ref{sisgeometric}, after possibly removing finitely many closed points from $C^M$ we may assume that the closed fibres $\Y_c:=\Y \times_{C^M} k(c)$ of $\Y$ are smooth for all $c\in (C^M)^{\cl}$. Then $\Y\to C^M$ is a flat, proper, smooth relative curve whose generic fibre $Y_M$ is hyperbolic and has at least one $M$-rational point. Theorem A then implies that $s_{Y_M}(G_M)$ is contained in a decomposition subgroup $D^M_{\tilde y} \subset G_{Y_M}$ for a unique $M$-rational point $y$ of $Y_M$ and some extension $\tilde y$ of $y$ to $\overline {k(X)}$. Note we use a superscript $M$ to emphasise that $D^M_{\tilde y}$ is a subgroup of $G_{Y_M}$.
Since $M|L$ is a Galois extension, $G_M$ is a normal subgroup of $G_L$, hence $s_Y(G_L)$ normalises $s_{Y_M}(G_M)$ in $G_Y$. Therefore, for any $\sigma \in G_L$, $s_{Y_M}(G_M)$ is also contained in $s_Y(\sigma)^{-1}D^M_{\tilde y} s_Y(\sigma) = D^M_{s_Y(\sigma)\cdot\tilde y}$, which implies that $\tilde y=s_Y(\sigma)\cdot\tilde y$ \cite[Corollary 12.1.3]{CONF}. Thus, $s_Y(G_L)$ normalises $D^M_{\tilde y}$ in $G_Y$, so it is contained in the normaliser of $D^M_{\tilde y}$ in $G_Y$, which is precisely $D_{\tilde y} \subset G_Y$. This implies that $s(G_L)$ is contained in the decomposition subgroup $D_{\tilde y} \subset G_X$ of the same valuation $\tilde y$ of $\overline {k(X)}$, whose restriction to $k(X)$ corresponds to the image $x$ of $y$ in $X$. The point $x$ is then necessarily $L$-rational, since $D_{\tilde y}$ must map surjectively to $G_L$.
\end{proof}
This concludes the proof of Theorem B.
\subsection{Proof of Theorem C}
In this section we prove Theorem C. Assume the {\bf BSC} holds over all number fields. We prove that the {\bf BSC} holds over all finitely generated fields over $\Q$ of transcendence degree $n \ge 1$. We argue by induction on $n$ and assume that the {\bf BSC} holds over all finitely generated fields over $\Q$ of transcendence degree $<n$. Let $K$ be a finitely generated field over $\Q$ of transcendence degree $n$ and $k\subset K$ a subfield which is algebraically closed in $K$ over which $K$ has transcendence degree $1$. We show the {\bf BSC} holds over $K$.
It is well-known that in order to prove that the {\bf BSC} holds over $K$ it suffices to prove that the {\bf BSC} holds for the projective line over $K$ (cf. \cite[Lemma 2.1]{saidiBASC}). Thus, we will show the following.
\begin{proposition}
With $K$ and $k$ as above, let $X=\Bbb P^1_K$, and let $s:G_K\to G_X$ a section of the projection $G_X\twoheadrightarrow G_K$. Then $s$ is geometric.
\end{proposition}
\begin{proof}
Let $\ob{K}$ and $\ob{k}$ be the algebraic closures of $K$ and $k$, respectively, induced by the geometric point $\xi$ defining $G_X$. We claim that there exists an open subgroup $H \subset G_X$ containing $s(G_K)$ such that, denoting by $Y \to X$ the corresponding finite morphism with $Y$ smooth, $Y$ is hyperbolic and isotrivial, meaning that $Y_{\ob{K}}$ descends to a smooth curve $Y_{\bar k}$ over $\bar k$.
Indeed, let $U\subset \Bbb P^1_k$ be an open subset, $U_{\bar k}=U\times _k \bar k$, $U_K=U\times _k K$, and $U_{\ob{K}}=U\times _k \ob{K}$. We have a natural commutative diagram of exact sequences
$$
\CD
1@>>> \pi_1(U_{\ob{K}},\bar \xi) @>>> \pi_1(U_K,\xi) @>>> G_{K} @>>> 1\\
@. @VVV @VVV @VVV @. \\
1@>>> \pi_1(U_{\bar k},\bar \xi) @>>> \pi_1(U_k,\xi)@>>> G_{k}@>>> 1\\
\endCD
$$
where the left vertical map is an isomorphism. Let $\tilde \Delta$ be a characteristic open subgroup of $\pi_1(U_{\bar k},\bar \xi)$ corresponding to a finite morphism $Y_{\bar k}\to \Bbb P^1_{\bar k}$ with $Y_{\bar k}$ smooth and hyperbolic. Such a subgroup exists by the Riemann-Hurwitz formula and the fact that $\pi_1(U_{\bar k},\bar \xi)$ is finitely generated. We write $\Delta$ for the corresponding subgroup of $\pi_1(U_{\ob{K}},\bar \xi)$ and $Y_{\ob{K}}\to \Bbb P^1_{\ob{K}}$ the corresponding finite morphism with $Y_{\ob{K}}$ smooth. The section $s$ induces a section $s_U : G_K \to \pi_1(U_{K},\xi)$ of the projection $\pi_1(U_K,\xi) \twoheadrightarrow G_{K}$. Let $\widetilde H=\Delta \cdot s_U(G_K)$ and $H$ the inverse image of $\widetilde H$ in $G_X$. Then $H$ and the corresponding finite morphism $Y\to X$ are as claimed above.
The section $s$ induces a section $s_Y : G_K \to G_Y = H$ of the natural projection $G_Y\twoheadrightarrow G_K$ with $s_Y(G_K) = s(G_K)$, and one easily verifies that the section $s$ is geometric if (and only if) the section $s_Y$ is geometric. Let $C$ be a separated, smooth and connected curve over $k$ with function field $k(C)=K$, and $\Y\to C$ a flat, smooth and proper relative $C$-curve with generic fibre $\Y_K=Y$. Without loss of generality, we can assume that $Y(K)\ne \emptyset$ (cf. proof of Theorem B). Let $\J$ be the relative jacobian of $\Y$. The Shafarevich-Tate group $\Sha (\J)$ is finite by \cite[Theorem 4.1]{saiditamagawa} since $\J_{\ob{K}}$, being the jacobian of $Y_{\ob{K}}$, is isotrivial (i.e. descends to an abelian variety over $\bar k$). Moreover, $k$ strongly satisfies the conditions in Definition \ref{conditions}, since it is finitely generated and by the above induction assumption. Then the section $s_Y$, and a fortiori the section $s$, is geometric by Theorem A.
\end{proof}
This concludes the proof of Theorem C.
\addcontentsline{toc}{section}{References}
| 192,827
|
TITLE: Does a cubic polynomial split in linear factors over the rational numbers?
QUESTION [2 upvotes]: Let $\alpha + \beta + \gamma = a$ , $\alpha \beta + \alpha \gamma + \beta \gamma = b$, $\alpha \beta \gamma = c$, $\alpha^2 \beta + \gamma^2 \alpha + \beta ^2 \gamma = d$, where $\alpha, \beta, \gamma$ are complex numbers and $a,b,c,d$ are rational numbers.
Can it happen, that for example $\alpha$ is a rational number while $\beta, \gamma$ are irrational, or is this impossible? I guess that from the last equation it would follow that $d$ is not rational if only $\alpha$ is rational, but am unable to prove it.
The polynomial I am considering is $f(x) = x^3 - ax^2 + bx -c = (x-\alpha)(x-\beta)(x-\gamma)$ with the constraint that $\alpha^2 \beta + \gamma^2 \alpha + \beta ^2 \gamma = d$ is rational.
REPLY [1 votes]: If I don't misread your equations, and given the title, you are asking if a cubic $x^3 + a x^2 + b x + c$ can have one rational and two irrational roots, if it has rational coefficients. Just consider:
$$(x - 1) (x^2 - 2) = x^3 - x^2 - 2 x + 2$$
with one rational and two irrational roots.
| 118,454
|
Lucy Waverman's Instant Caesar Salad
Quick and easy if not exactly authentic. Replace the garlic with 2 tqblespoons roasted garlic puree, if desired.
Level: Moderate
Yield: 1 cup
Ingredients
- ½ c. mayonnaise
- 3 anchovy fillets, chopped
- 2 tbsp. lemon juice
- 1 tsp. chopped garlic
- ½ tsp. Worcestershire sauce
- ½ c. olive oil
- Freshly ground pepper to taste
- ½ c. grated Parmesan
Directions
- Process in a food processor until smooth. Thin down with drops of warm water if too thick.
More From Recipes
| 116,089
|
TITLE: Strict 2-groups and group objects in $\textbf{Cat}$
QUESTION [1 upvotes]: Group objects in $\textbf{Cat}$ are strict monoidal categories with an antipode functor endofunctor $\text{inv}$ such that the standard diagram of groups (with the appropriate replacements), shown below, commutes.
The following definition is taken from Wikipedia.
A strict 2-group is a strict monoidal category in which every morphism
is invertible and every object has a strict inverse (so that $X\otimes Y$ and $Y\otimes X$
are actually equal to the unit object).
It seems to be folklore that strict 2-groups are precisely group objects in $\textbf{Cat}$.
Indeed, from the diagram above it is clear that $X\otimes \text{inv}(x) = \text{inv}(x)\otimes X = I$.
On the other hand, I can't see why would morphisms be invertible.
Remarkably, this question is avoided in the category of groupoids, i.e. strict 2-groups are clearly group objects in the category of groupoids.
PS: a question very similar to this one has already been posted, but it looks for tensor inverses instead of true inverses.
REPLY [3 votes]: The following proof also works for non-strict 2-groups (here, "2-group" is used in the "categorized group" sense, not the "$p$-group" sense).
Let $C$ be a (not necessarily strict) monoidal category such that there is a functor $\mathrm{inv}:C \to C$ for which the functors $X \mapsto X \otimes \mathrm{inv}(X)$ and $X \mapsto \mathrm{inv}(X) \otimes X$ are naturally isomorphic to the constant functor with value $I$.
Then, for any morphism $f:X \to Y$ in $C$, the isomorphism $f \otimes \mathrm{inv}(f):X \otimes \mathrm{inv}(X) \to Y \otimes \mathrm{inv}(Y)$ may be factorized in at least two ways, namely through $X \otimes \mathrm{inv}(Y)$ as $(f \otimes 1_{\mathrm{inv}(Y)}) \circ (1_X \otimes \mathrm{inv}(f))$, or through $Y \otimes \mathrm{inv}(X)$ as $(1_Y \otimes \mathrm{inv}(f)) \circ (f \otimes 1_{\mathrm{inv}(X)})$. This shows that $f \otimes 1_{\mathrm{inv}(Y)}$ is a split epimorphism, while $f \otimes 1_{\mathrm{inv}(X)}$ is a split monomorphism.
Also, the functors $- \otimes \mathrm{inv}(Y)$ and $- \otimes \mathrm{inv}(X)$ are equivalences, and so they reflect split epimorphisms and split monomorphisms. It follows then that $f$ must be a split epimorphism and a split monomorphism, so it must in fact be an isomorphism.
Hence, $C$ must in fact be a groupoid. $\square$
| 180,311
|
\section{Dynamics}
\label{sec:dynamics}
Consider a flexible hose connected to multiple quadrotor UAVs as shown in Figure~\ref{fig:intro}. In this section, we present the coordinate-free dynamics for this system. We consider the following assumptions before proceeding to derive the dynamics:
\begin{enumerate}[label=A\arabic{*}.]
\item No water/water-flow in the hose and thus also no pressure forces;
\item Hose is modeled as a series of $n$ smaller links
connected by spherical joints;
\item Each link is massless with lumped point-masses at the end with the hose mechanical properties like stiffness and torsional forces ignored.
\item The quadrotors attach to the hose at their respective center-of-masses.
\end{enumerate}
In the following section, we present the notation used to describe the system.
\subsection{Notation}
Dynamics for the model are defined using geometric-representation of the states. Each link is a spherical-joint and is represented using a unit-vector $q\in\mathbb{S}^2:=\{x\in\mathbb{R}^3~|~\|x\|=1\}$. The position of one end of the cable is given in $\mathbb{R}^3$ and finally, the rotation matrix $R\in SO(3):=\{R\in \mathbb{R}^{3\times 3}|R^\top R = 1, det(R)=+1\}$ is used to represent the attitude of the quadrotor.
Let the hose be discretized into $n$ links with the cable joints indexed as $\inds =\{0,1,\hdots,n\}$ as shown in Figure~\ref{fig:intro}. The position of one (starting) end of the hose is given as $x_0\in\mathbb{R}^3$ in the world-frame. The position of the link joints/point-masses is represented by $x_i\in\mathbb{R}^3$, where the link attitude between $x_{i{-}1}$ and $x_i$ is given by $q_i\in \mathbb{S}^2$ and length of this link-segment is $l_i$ \emph{i.e.,} $x_i = x_{i{-}1}{+
}l_iq_i$. Also, $m_i$ is the mass of the lumped point-mass for link $i$. Let the set $\indi\subseteq \inds$ be the set of indices where the cable is attached to the quadrotor and $n_Q=|\indi|$ is the number of quadrotors. For the $j^{th}$ quadrotor, $R_j\in SO(3)$ is the attitude, $m_{Qj},J_j$ is its mass and inertia matrix (in body-frame) and $f_j\in\mathbb{R},M_j\in\mathbb{R}^3$ are the corresponding thrust and moment for all $j\in\indi$. Finally, the configuration space of this system is given as $Q:= \mathbb{R}^3\times (\mathbb{S}^2)^n\times (SO(3))^{n_Q}$. Table~\ref{tab:my_label} lists the various symbols used in this paper.
\subsection{Derivation}
The kinematic relation between the different link positions is given using link attitudes as,
\begin{align}
x_i = x_0 + \sum_{k=1}^i l_k\q_k,& ~\forall~i\in \inds\backslash\{0\},\label{eq:link-positions}
\end{align}
and the corresponding velocities and accelerations are related as,
\begin{align}
v_i = v_0 + \textstyle\sum_{k=1}^i l_k\dq_k,\quad
a_i = a_0 + \textstyle\sum_{k=1}^i l_k\ddot q_k.
\end{align}
Potential energy $\mathcal{U}:TQ\rightarrow \mathbb{R}$ of the system, due to hose and quadrotors' mass is computed as shown below,
\begin{align}
\mathcal{U} = \sum_{i\in \inds}\mbar_ix_i\cdot g\bm{e_3},\label{eq:pe1}
\end{align}
where $\mbar_i = m_i + m_{Qi}\indicate{i}$ is the net-mass at index $i$ and $\indicate{i}:=\indicator{i} = \begin{cases}
1, ~ \textit{if } i\in \indi\\
0, ~ \textit{else}
\end{cases} $ is an indicator function for the set $\indi$.
\begin{figure*}
\normalsize
\begin{tcolorbox}[title={Equations of motion for \currsys}]
\begin{gather}
\dot{x}_0 = v_0,~
\dot{q}_i = \omega_i\times q_i, \label{eq:dyn_pos}\\
\underbrace{\begin{bmatrix}
M_{00}I_3 & -\hatmap{q}_1M_{01} & -\hatmap{q}_2M_{02} & \hdots & -\hatmap{q}_nM_{0n} \\[2ex]
-M_{10}\hatmap{q}_1 & -M_{11}I_3 & M_{12}\hatmap{q}_1\hatmap{q}_2 &{\scriptsize \hdots} & M_{1n}\hatmap{q}_1\hatmap{q}_n\\[2ex]
-M_{20}\hatmap{q}_2 & M_{21}\hatmap{q}_2\hatmap{q}_1 & -M_{22}I_3 & {\scriptsize \hdots} & M_{2n}\hatmap{q}_2\hatmap{q}_n\\[2ex]
\vdots & \vdots & \vdots & {\scriptsize \ddots} & \vdots \\[2ex]
-M_{n0}\hatmap{q}_n & M_{n1}\hatmap{q}_n\hatmap{q}_1 & M_{n2}\hatmap{q}_n\hatmap{q}_2 & {\scriptsize \hdots} & -M_{nn}I_3
\end{bmatrix}}_{=:\mathbb{M}_{\{q_i\}}}
\begin{bmatrix}
\dot{v}_0 \\ \dot{\omega}_1 \\ \dot{\omega}_2 \\ \vdots \\ \dot{\omega}_n
\end{bmatrix} =
{\begin{bmatrix}
\sum\limits_{i=1}^{n}M_{0i}\|\omega_i\|^2q_i + \sum_{k=0}^{n}u_k \\[2ex]
-\sum\limits_{k=1}^{n}(M_{1k}\|\omega_k\|^2\hatmap{q}_1q_k){-}l_1\hatmap{q}_1\sum\limits_{k=1}^{n}u_k \\[2ex]
-\sum\limits_{k=1}^{n}(M_{2k}\|\omega_k\|^2\hatmap{q}_2q_k){-}l_2\hatmap{q}_2\sum\limits_{k=2}^{n}u_k \\[2ex]
\vdots \\[2ex]
-\sum\limits_{k=1}^{n}(M_{nk}\|\omega_k\|^2\hatmap{q}_nq_k){-}l_n\hatmap{q}_nu_n
\end{bmatrix}}, \label{eq:dyn_q}\\
\dot{R}_j = R_j\hatmap{\Omega}_j,~
J_j\dot\Omega_j = M_j-\Omega_j\times J_i\Omega_j, \label{eq:dyn_R}
\end{gather}
$\forall i\in\inds\backslash\{0\}$, $j\in\indi$, $u_i = (- {\mbar_i}g\ez+ f_iR_i\ez\indicate{i} )$.
\end{tcolorbox}
\vspace*{4pt}
\end{figure*}
Kinetic energy $\mathcal{T}:TQ\rightarrow \mathbb{R}$ is similarly given as,
\begin{gather}
\mathcal{T} = \sum_{i\in\inds}\frac{1}{2}\mbar_i\langle v_i,v_i\rangle + \sum_{j\in\indi} \frac{1}{2}\langle\Omega_j,J_j\Omega_j\rangle, \label{eq:ke1}
\end{gather}
where $\Omega_j$ is the angular velocity of the quadrotor $j$ in its body-frame. Dynamics of the system are derived using the Lagrangian method, where Lagrangian $\mathcal{L}:TQ\rightarrow \mathbb{R}$, is given as,
\begin{gather*}
\mathcal{L} = \mathcal{T}-\mathcal{U}.
\end{gather*}
We derive the equations of motion using the Langrange-d'Alembert principle of least action, given below,
\begin{align}
\delta \int_{t_0}^{t_f}\mathcal{L}dt + \int_{t_0}^{t_f}\delta W_e dt = 0, \label{eq:leastaction}
\end{align}
where $\delta W_e$ is the infinitesimal work done by the external forces. $\delta W_e$ can be computed as,
\begin{align}
\delta W_{e} &= \sum_{j\in\indi}\Big(\langle W_{1,j},\hat{M}_j \rangle + \langle W_{2,j}, f_jR_j\bm{e_3}\rangle \Big), \label{eq:dWe}
\end{align}
\begin{gather}
W_{1,j} =R^T_j\delta R_j, \\
W_{2,j} = \delta x_j = \delta x_0+\textstyle\sum_{k=1}^jl_k\delta q_k,
\end{gather} are variational vector fields [\cite{goodarzi2015geometric}] corresponding to quadrotor attitudes and positions. The infinitesimal variations on $q$ and $R$ are expressed as,
\begin{gather*}
\delta q = \hatmap{\xi}q = -\hatmap{q}\xi,~ \xi\in\mathbb{R}^3~\text{s.t.}~\xi\cdot q = 0, \\
\del{\dq} = {-}\hatmap{q}\dot{\xi}{-}\hatmap{\dot{q}}\xi, \\
\delta R = R\hatmap{\eta},\quad \delta\hatmap{\Omega}=\hatmap{(\hatmap{\Omega}\eta)}{+}\hatmap{\dot \eta},\eta\in\mathbb{R}^3,
\end{gather*}
with the constraints $q\cdot\dot{q}=0$ and $q\cdot{\omega}=0$, $\omega$ is the angular velocity of $q$, s.t. $\dot{q} =\omega\times q$. The cross-map is defined as $\hatmap{(\cdot)}:\mathbb{R}^3\rightarrow \mathbf{so(3)}$ \textit{s.t} $\hatmap{x}y=x\times y,\forall x,y\in\mathbb{R}^3$. Similarly, variations on the link positions are given as,
\begin{gather}
\del{x}_i = \del{x}_0 + \sum_{k=1}^i l_k\del{\q}_k = \del{x}_0 {-} \sum_{k=1}^i l_k\hatmap{q}_k\xi_k, \label{eq:delx}\end{gather}\begin{gather}
\del{v}_i = \del{v}_0{+}\sum_{k=1}^i l_k\del{\dq}_k = \del{v}_0{-}\sum_{k=1}^i l_k(\hatmap{q}_k\dot{\xi_k}{+}\hatmap{\dot{q}}_k\xi_k). \label{eq:delv}
\end{gather}
Finally, we obtain the equations of motion for the system by solving \eqref{eq:leastaction}.
\referarxiv{A}
for the detailed derivation. Equations of motion for the {\currsys} are given in \eqref{eq:dyn_pos}-\eqref{eq:dyn_R}. Note the mass-matrix $\mathbb{M}_{\{q_i\}}$ is a function of link attitudes $\{q_i\}=\{q_1,q_2,\hdots q_n \}$ and we use the following notation similar to [\cite{goodarzi2014geometric}]
\begin{gather}
M_{00}{=}\textstyle\sum_{k=0}^n \mbar_k,M_{0i}{=}l_i\textstyle\sum_{k=i}^n \mbar_k, \nonumber \\ M_{i0}{=}M_{0i},M_{ij} =\textstyle\sum_{k=\max\{ij\}}^n \mbar_k l_il_j.\label{eq:mass_var}
\end{gather}
\begin{remark}
\label{remark:inputIndicator}
In \eqref{eq:dyn_q}, note the use of $f_i, R_i$ for $i\notin \indi$, (since $i\notin \indi$ implies no quadrotor is attached at index $i$ and thus cannot have $f_i$ and $R_i$). However, this notation is used for convenience, since $i{\notin }\indi{\implies }\indicate{i}{=}0$ and thus $f_iR_i\ez\indicate{i}=0$, there by ensuring the right inputs to the system.
\end{remark}
\begin{remark}
\label{remark:doua}
Degrees of freedom for the \currsys~ is $DOF=3(n_Q{+}1){+}2n$ where $2n$ corresponds to the link attitudes DOF, $3n_Q$ the rotational DOF of the quadrotors and $3$ for the initial position $x_0$. Similarly, the degrees of actuation is $DOA=4n_Q$ corresponding to the $4$ inputs for each quadrotor. Thus, the degrees of under-actuation are $DOuA=2n+3-n_Q$. For a typical setup we have $n>>n_Q$, \emph{i.e.,} system is highly under-actuated.
\end{remark}
\begin{remark}
\label{remark:tethered}
For a tethered system, we can assume $x_0{\equiv}0$ $\forall~t$, i.e. the system is tethered to origin of the inertial frame, and we can derive the dynamics as earlier. Equations of motion for this system would be same as \eqref{eq:dyn_pos}-\eqref{eq:dyn_R}, without the equation corresponding to $\dot{v}_0$.
\end{remark}
\begin{table}[!t]
\begin{center}
\caption{List of various symbols used in this work. Note: $k{\in}\inds, i{\in}\inds\backslash\{0\}, j{\in}\indi$, WF - World frame, BF-Body-frame, $|\cdot |$ represents cardinality of a set.} \label{tab:my_label}
\begin{tabular}{c|p{4cm}}
Variables & Definition\\
\hline\hline
$n\in\mathbb{R}^+$ & Number of links in the hose. \\
$\mathcal{S}=\{0,1,\hdots,n\}$ & Set containing indices of the hose-segments.\\
$x_k\in \mathbb{R}^3$ & Position of the $k^{th}$ point-mass of the hose in WF.\\
$v_k\in \mathbb{R}^3$ & Velocity of the $k^{th}$ point-mass of the hose in WF.\\
$l_i\in\mathbb{R}^+$ & Length of the $i^{th}$ segment.\\
$m_k\in \mathbb{R}^+$ & Mass of the $k^{th}$ point-mass in the hose-segments.\\
$\q_i\in\mathbb{S}^2$ & Orientation of the $i^{th}$ hose segment in WF. \\
$\omega_i\in T_{q_i}\mathbb{S}^2$ & Angular velocity of the $i^{th}$ hose segment in WF. \\
\hline\hline
$\mathcal{I}\subset \inds$ & Set of indices where the hose is
attached to the quadrotor. \\
$|\indi|=n_Q$& Number of quadrotors.\\
$x_{Qj}\equiv x_j$ & Center-of-mass position of the $j^{th}$ quadrotor in WF. \\
$R_j\in SO(3)$ & Attitude of the $j^{th}$ quadrotor w.r.t. WF.\\
$\Omega_j\in T_{R_j}SO(3)$ & Angular velocity of the $j^{th}$ quadrotor in BF.\\
$m_{Qj}$, $J_{j}$ & Mass \& inertia of the $j^{th}$ quadrotor.\\
$f_j\in\mathbb{R},~M_j\in\mathbb{R}^3$ & Thrust and moment of the $j^{th}$ quadrotor in BF.\\
\hline
\hline
$\indicate{i} :=\indicator{i} = \begin{cases}
1 ~ \textit{if } i\in \indi\\
0 ~ \textit{else}
\end{cases}$ & Indicator function for the set $\indi$.\\
$\mbar_{k} = m_k + m_{Qk}\indicate{k}$ & Net mass at the $k^{th}$ link joint. \\
$u_k = (- {\mbar_k}g\ez+ f_kR_k\ez\indicate{k} )$ & Net force due to thrusters \& gravity.\\
\hline
\end{tabular}
\end{center}
\end{table}
| 120,503
|
Fourth it has melted then, remove from heat.
Give each child a pretzel rod and let them dip it in the chocolate, then roll the rod in sprinkles.
Allow the pretzels to sit upright in a glass to dry, then eat!
Star Garlands
Cut out different sizes and colors of stars (blue, red, white, silver, gold) and glue them onto a red, white or blue ribbon. Use a flat ribbon, and glue stars on both sides. Then, hang the garland around the door, window, or party table!
Fireworks Paintings
Materials
-.
Flag Tag
Materials
- Buckets of sand
- Small American flags
Instructions
- To set up the race, divide the group into 2 teams. For each team, place a bucket of sand filled with small American flags (1 per teammate) on the opposite end of the yard.
- When the race starts, the first child. The first team to finish should win a small prize, like bubbles for everyone.
Play “The Stars and Stripes Forever” and other July 4th music while the race is happening.
Fireworks Balloons
To make these fun, non-flammable fireworks, pull a red, white or blue balloon over a funnel. Pour in red, white and blue confetti until the balloon is a quarter full (while deflated); Then either inflate with a hand pump, or blow up the balloon. Use a sharpened pencil for popping. (Pop where you’re happy to have a lot of confetti!)
Stars and Stripes Windsock
Glue patriotic paper around a frozen juice-concentrate container, making sure that the paper extends a few inches beyond each end of the can. Then, tape or glue crepe paper streamers to the inside of the paper. On the inside of the can, tape fishing line or ribbon on equal sides of the can so that the windsock will hang evenly. Hang from a tree or the front porch for a patriotic welcome.
I hope these ideas help out with your Fourth of July party, and have a great celebration!
| 1,737
|
Jul 30.
Reader Comments
1. lol - July 30, 2008 1:23 PM
oaihoierg
2. Sam - July 30, 2008 1:35 PM
It reminds me of that little alien in the Lost in Space movie (the remake).
3. Julian - July 30, 2008 1:43 PM
"if he is cuddled his limbs become limp" I have the exact opposite reaction.
This thing is pretty cool i guess, though it does look creepy. maybe its the exposed heart that beats, but most likely it's its uncanny resemblance to dennis kucinich.
4. Icon - July 30, 2008 2:08 PM
This is what happens when Dr. Seuss drops too much acid one night and f***s around with electronics.
God help us all. These little bastards will rule the world in 4 days tops.
5. ZenoZTankof - July 30, 2008 3:53 PM
My dog would hump the crap out of that thing.
Then lets see how the little bastard "feels".
6. Zip - July 30, 2008 4:26 PM
pale color of this robot "skin" looks creepy
7. al - July 30, 2008 7:50 PM
Jack Skellington?
8. bcutid - July 30, 2008 8:33 PM
I saw many people are discussing this on the forum of black dating site called *******Blackcentury dot com********。 You may go there to check it if you are interested. Maybe you can meet your life partner there.
9. Gavin - July 30, 2008 9:21 PM
GRRR DAMN YOU GEEKOLOGIE WRITER NO CREDIT FOR ME!!!!!!!!!!!!!!!!!!
10. Paulie Danger - July 30, 2008 11:31 PM
Pale, small, emotional. Oh f*** me. Technology has built us an emo kid.
11. - July 31, 2008 5:35 AM
It's kind of spooky to look at, other people feeling that yeah?
12. absinthe - July 31, 2008 7:56 AM
O...M...G... you linked to The Sun! I had such respect for you Geekologie writer. Please tell me you didnt know it's a tabloid trash-rag for the terminally dim so everything is ok again...
13. Gingersnapz - August 7, 2008 2:10 PM
I don't a doll that loves me
I want a doll that MAKES love to me
14. Sally - July 2, 2009 1:13 PM
I HELD this thing. It was actually really cute. And then it ran out of batteries. It actually started to 'relax' and looked like it was falling asleep. And it blinked.
And i wanna sculpt it.
15. nono - August 24, 2009 12:50 AM
i like the idea, i mean creating, designing and building a robot that can respond to human emotions, but i do not understand why on earth they design such "look" to the robot. Yea, it looks scary a lil bit. LOL..
16. Dru - January 7, 2010 4:33 PM
I thought Furbies were a evil.
And then they made this.
Some humans don't even feel love - why are we trying to make robots that do?
17. Sally - August 3, 2010 3:35 PM
@gw, it 'dies' when the battery runs out. Yeah, I totally saw it in person! As in, I was the person. It was in robot. I got to hold it... It's meant to be like a 3 year old. Not really, it looks more like a nymph of some sort.
| 281,143
|
\begin{document}
\begin{abstract}
We prove that the full twist is a Serre functor in the homotopy category of type A Soergel bimodules.
As a consequence, we relate the top and bottom Hochschild degrees in Khovanov-Rozansky homology,
categorifying a theorem of K\'alm\'an.
\end{abstract}
\maketitle
\setcounter{tocdepth}{1}
\tableofcontents
\section{Introduction}
The category of Soergel bimodules is a categorification of the Hecke algebra. It can be defined for any Coxeter group, but here we focus on type A only, where the corresponding group is $S_n$ and the category of Soergel bimodules will be denoted by $\SBim_n$. Given a braid on $n$ strands, Rouquier \cite{Rouquier} constructed a complex of bimodules in $\SBim_n$ and proved that it is unique up to a canonical homotopy equivalence. Khovanov and Rozansky \cite{Kh07,KR2} used Rouquier complexes to define {\em Khovanov-Rozansky homology} $\HHH$, a categorification of the HOMFLY-PT polynomial.
In recent years, the Rouquier complex for the full twist braid $\FT_n$ has attracted a lot of attention. Elias and the second author \cite{EH2} proved that $\FT_n$ is in the Drinfeld center of the homotopy category of Soergel bimodules $\KC^b(\SBim_n)$. They also computed the Khovanov-Rozansky homology of the full twist \cite{EH1} and the categorified eigenvalues of $\FT_n$ acting on $\KC^b(\SBim_n)$. The work of the first and second author, Negu\cb{t} and Rasmussen \cite{GH,GNR} related $\FT_n$ to a natural line bundle $\CO(1)$ on the isospectral Hilbert scheme $X_n$.
In this paper, we prove that $\FT_n$ (or rather its inverse $\FT_n^{-1}$) acts as a kind of
Serre functor \cite{BK} in $\KC^b(\SBim_n)$. Let $\k$ be a field of characteristic $\neq 2$, and set $R:=\k[x_1,\ldots,x_n]$. We will consider Soergel bimodules over $\k$. Given a complex of free $R$-modules $X$, we denote by $X^{\vee}=\Homc(X,R)$ the dual complex. Note that the cohomology of $X^{\vee}$ and of $X$ are, in general, related by the universal coefficient spectral sequence which can be rather complicated.
\begin{theorem}
\label{thm:introserre}
For any two complexes $A,B\in \KC^b(\SBim_n)$ one has
$$
\Homc(A,B)\simeq \Homc(\FT_n\otimes B,A)^{\vee}=\Homc(B,\FT_n^{-1}\otimes A)^{\vee}.
$$
\end{theorem}
Here $\Homc(-,-)$ denotes the complex of homs; in the category of complexes of Soergel bimodules, $\Homc(A,B)$ is a $\Z\times \Z$-graded complex of $(R,R)$-bimodules. Theorem \ref{thm:introserre} is true whether we regard $\Homc(-,-)$ as complexes of right $R$-modules or left $R$-modules.
\begin{remark}
Typically one states Serre duality in the context of categories which are linear over a field $\k$. The statement above differs from this typical situation in several ways. First, our category is monoidal, and the duality is taken with respect to the ring $R \cong \End(\one)$ instead of a field. Second, the morphism spaces are bimodules over this ring, and one may take the duals with respect to either the left or right actions. Finally, the Serre duality functor itself is tensoring with an object of the category.
\end{remark}
\begin{remark}
In \cite{Bez, MS} it was proven that action of the full twist on the BGG category $\CO$ is the Serre functor (see \S \ref{sec:smod}), which holds in more general types. We expect that our result also generalizes to other types, though we do not consider this here.
\end{remark}
\begin{remark}
Theorem \ref{thm:introserre} can be compared with a result of Haiman \cite{Haiman} which states that the isospectral Hilbert scheme $X_n$ is Gorenstein with the canonical sheaf $\CO(-1)$, so tensor multiplication by $\CO(-1)$ is a Serre functor.
\end{remark}
\subsection{A reformulation}
\label{subsec:introreform}
It is much more convenient to restate Theorem \ref{thm:introserre} in a more canonical form. First, let $\SBim_{1,\ldots,1}\subset \SBim_n$ denote the full subcategory consisting of direct sums of shifted copies of the trivial bimodule $\one= R$. The inclusion $\SBim_{1,\ldots,1} \rightarrow \SBim_n$ has left and right adjoints $\Pi_L,\Pi_R:\SBim_n\rightarrow \SBim_{1,\ldots,1}$ defined as follows. The left adjoint $\Pi_L(M)=\HH_0(M)$ is the quotient of $M$ by the sub-bimodule of commutators $fm-mf$, for all $f\in R$ and all $m\in M$, while the right adjoint $\Pi_R(M)=\HH^0(M)$ is the sub-bimodule consisting of elements $m\in M$ with $fm-mf=0$ for all $f\in R$. When $M$ is a Soergel bimodule, $\HH_0(M)$ and $\HH^0(M)$ are free $R$-modules, hence can be regarded as objects of $\SBim_{1,\ldots,1}$ (see \S \ref{sec:background}).
The additive functors $\HH_0$ and $\HH^0$ can be extended to complexes, and it is not hard to see that
\[
\HH^0(X) \cong \Homc_{R,R}(R,X),\qquad \HH_0(X) \cong \Homc_{R,R}(X,R)^{\vee},
\]
naturally in $X\in \KC^b(\SBim_n)$. The second of these uses properties of Soergel bimodules. Thus, Theorem \ref{thm:introserre} has the following as a special case (set $A=R$ and $B=X$).
\begin{theorem}\label{thm:introReform}
For any complex $X\in \KC^b(\SBim_n)$ we have $\HH^0(X)\simeq \HH_0(\FT\otimes X)$ in $\KC^b(\SBim_{1,\ldots,1})$.
\end{theorem}
Using the rigid monoidal structure on $\KC^b(\SBim_n)$ is not hard to see that in fact Theorem \ref{thm:introserre} is equivalent to Theorem \ref{thm:introReform}. However, the latter is often preferable because the $R$-action is now canonically defined (the left and right $R$-actions on $\HH^0(-)$ and $\HH_0(-)$ coincide). More importantly, the latter theorem generalizes to a relative version, which we discuss next.
For each subset $I\subset \{1,\ldots,n-1\}$, let $\SBim_I\subset \SBim_n$ denote the full monoidal subcategory generated by the Bott-Samelson bimodules $B_s$ with $s\in I$. Alternatively, the subgroup of $S_n$ generated by $s\in I$ is of the form $S_{k_1}\times \cdots\times S_{k_r}\subset S_n$, and we will write
\[
\SBim_I =: \SBim_{k_1,\ldots,k_r}
\]
by abuse.
The inclusion $\SBim_{n-1,1}\rightarrow \SBim_n$ has left and right adjoints $\pi_L,\pi_R:\SBim_n\rightarrow \SBim_{n-1,1}$ defined as follows. With respect to the identification $R=\k[x_1,\ldots,x_n]$, $\pi_L(M)$ and $\pi_R(M)$ are the cokernel and kernel of $x_n\otimes 1 -1\otimes x_n$ acting on $M$, respectively.
\begin{remark}
In the main body of the paper, we write $\pi^+=\pi_L$ and $\pi^-=\pi_R$ because of the interaction of these functors with positive and negative Rouquier complexes.
\end{remark}
\begin{theorem}\label{thm:introrelative}
Let $\CL_n:=\FT_n\otimes \FT_{n-1}\inv$ denote the Rouquier complex of the Jucys-Murphy braid $\sigma_{n-1}\cdots\sigma_2\sigma_1^2\sigma_2\cdots \sigma_{n-1}$. For each complex $X\in \KC^b(\SBim_n)$ we have
\[
\pi_R(X)\simeq \pi_L(\CL_n\otimes X),
\]
naturally in $X$.
\end{theorem}
By analogy with Theorem \ref{thm:introReform}, we may refer to $\CL_n\inv$ as the \emph{relative Serre functor} for $\SBim_n$ relative to $\SBim_{n-1,1}$. In relation to the conjectures in \cite{GNR}, this result is a monoidal, algebraic analogue of a geometric statement regarding the Hilbert scheme of points $\Hilb^n(\C^2)$ relative to the nested Hilbert scheme $\Hilb^{n-1,1}(\C^2)$. More precisely, $\Hilb^{n-1,1}(\C^2)$ yields a smooth correspondence between
$\Hilb^n(\C^2)$ and $\Hilb^{n-1}(\C^2)$, and the analogues of $\pi_R$ and $\pi_L$ differ by the canonical line bundle
on $\Hilb^{n-1,1}(\C^2)$ which was computed e.g. in \cite[Proposition 3.6.4]{Haiman} and corresponds to $\CL_n^{-1}$.
We expect that this statement generalizes to arbitrary Coxeter systems in the following way. Let $(W,S)$ be any finite Coxeter system with longest element $w_0\in W$. After choosing a \emph{realization} $\mathfrak{h}$ of $W$, there is an associated category $\SBim=\SBim(W,\mathfrak{h})$ of Soergel bimodules (or its diagrammatic version; see \cite{EW} and references therein). Given a subset $I\subset S$, we let $\SBim_I\subset \SBim$ denote the full monoidal, idempotent complete, subcategory generated by Bott-Samelson bimodules $B_s$ with $s\in I$. Note that $\SBim_I$ is just the category of Soergel bimodules associated to the \emph{parabolic subgroup} $W^I\subset W$, defined using the given realization $\mathfrak{h}$ of $W$.
Let $\FT:=F_{w_0}^{\otimes 2}$ denote the Rouquier complex for the ``full twist'' in $\KC^b(\SBim)$, and let $\FT_I:=F_{w_I}^{\otimes 2}$, where $w_I$ is the longest element of $W^I\subset W$. Set $\FT_{S/I}:= \FT\otimes \FT_I\inv$. Equivalently, $\FT_{S/I}=F_{v_I\inv}\otimes F_{v_I}$, where $v_I\in W$ denote a shortest length representative of the coset $w_0 W_I$.
\begin{conjecture}\label{conjecture:generalCox}
Let $\pi_L,\pi_R:\SBim\rightarrow \SBim_I$ denote the left and right adjoints to the fully faithful inclusion $\SBim_I\rightarrow \SBim$. Then $\FT_{S/I}$ tensor commutes all with complexes in $\KC^b(\SBim_I)$ up to natural homotopy equivalence, and
\[
\pi_R(X)\simeq \pi_L(\FT_{S/I} \otimes X) \in \KC^b(\SBim_I),
\]
naturally in $X\in \KC^b(\SBim)$.
\end{conjecture}
The results in this paper prove this conjecture in the special case of subgroups $S_r\times (S_1)^{n-r}\subset S_n$.
\subsection{Khovanov-Rozansky homology}
\label{subsec:introKR}
Finally, we apply the above results to relate the ``top" and ``bottom" $a$-degrees in the Khovanov-Rozansky homology.
This categorifies a result of K\'alm\'an \cite{Kalman} relating the ``top" and "bottom" parts of the HOMFLY-PT polynomial.
\begin{theorem}
\label{thm: intro top bottom}
For any braid $\beta$ on $n$ strands one has
$$
\HHH^n(\beta\otimes \FT_n)(-2n)\simeq \HHH^0(\beta).
$$
\end{theorem}
\begin{remark}
For torus knots, Theorem \ref{thm: intro top bottom} was conjectured in \cite{G,ORS,GORS}. It was recently proved by the fourth author \cite{Nak} based on the explicit computation of the Khovanov-Rozansky homology for torus knots \cite{Mellit}.
\end{remark}
We also prove a ``folk result'' relating the Hochschild cohomology (or homology) of $X\in \KC^b(\SBim_n)$ and its dual $X^\vee$.
\begin{theorem}
Let $\widetilde{\HH}^k(M):=\HH^k(M)(-2k)$. Then
\[
\widetilde{\HH}^k(X)\cong \widetilde{\HH}^{n-k}(X^\vee)^\vee
\]
as complexes of $R$-modules, for all $X\in \KC^b(\SBim_n)$.
\end{theorem}
If $\b$ is a braid, let $r(\b)$ denote the reversed braid, defined by $r(\sigma_i^{\pm})=\sigma_i^{\pm}$ for each elementary braid generator $\sigma_i^{\pm}$, and $r(\b \b') = r(\b') r(\b)$ for all $\b,\b'\in \Br_n$. If $L$ is a link which is presented as the closure of a braid $\b$, then the mirror image $\overline{L}$ can be presented as the closure of the reversed inverse braid $r(\b\inv)$. There is an anti-involution of $\SBim_n$ defined by switching the right and left actions of all bimodules, which exchanges the Rouquier complexes for $\b$ and its reverse $r(\b)$. It follows that $\HH^k(\b)\cong \HH^k(r(\b))$. Since the Rouquier complexes satisfy $F(\b\inv) = F(\b)^\vee$, we obtain the following corollary:
\begin{corollary}
We have $\widetilde{\HH}^k(\b)\cong \widetilde{\HH}^{n-k}(\b\inv)^\vee$ as complexes of $R$-modules, for all braids $\b\in \Br_n$. In particular the complexes which compute the Khovanov-Rozansky homologies of $L$ and $\overline{L}$ are graded dual as complexes of free $R$-modules.
\end{corollary}
\subsection{Remark on conventions}
\label{subsec:intro other rings}
In this paper we have made the choice to work with honest Soergel bimodules rather than the diagrammatic version of Elias-Khovanov \cite{EK}, so that we may discuss Hochschild (co)homology. Also, we have chosen to work over an infinite field $\k$ of characteristic $\neq 2$, so that Soergel's results apply. When $\k$ is a more general ring, one can still define $\SBim_n$, but one loses control over the indecomposable objects in $\SBim_n$. Nonetheless, we believe that all of our main results should hold over $\Z$, but where $\KC^b(\SBim_n)$ gets replaced by the homotopy category of complexes of Bott-Samelson bimodules. These two categories are equivalent when Soergel's results apply.
\section*{Acknowledgements}
The authors would like to thank Tam\'as K\'alm\'an, Andrei Negu\cb{t}, Alexei Oblomkov and Jacob Rasmussen for the useful discussions. We also thank American Institute of Mathematics, where a part of this work was done, for hospitality.
E. G.~ was partially supported by the NSF grants DMS-1700814, DMS-1760329, and the Russian Academic Excellence Project 5-100. M.H.~ was supported by NSF grant DMS-1702274 and also partially supported by NSF grants DMS-1664240 and DMS-1255334. A.M.~ was supported by Austrian Science Fund (FWF) projects Y963-N35 and P-31705. K.N.~ was supported by JSPS KAKENHI Grant Number JP19J12350.
\section{Decategorified story}
\subsection{Jones-Ocneanu trace}
Let $\mathbb{H}_n$ be the Hecke algebra for $S_n$. We adopt Soergel's conventions below. We will work over the field $\Q(q)$ or occasionally $\Q(q)\subset \Q(v)$, where $q=v^{-2}$. (The variable $v$ corresponds to the grading downshift endofunctor (1) of $\SBim_n$.)
The algebra $\HM_n$ is formally generated by elements $H_1,\ldots,H_{n-1}$ modulo the braid relations and
\[
(H_i+v)(H_i-v\inv)=0.
\]
Given $w\in S_n$ with a reduced expression $w=s_{i_1}\cdots s_{i_k}$, we define $H_w=H_{i_1}\cdots H_{i_k}$.
The algebra $\mathbb{H}_n$ has two standard bases as a $\Q(v)$-vector space, namely the positive standard basis $\{H_w\}_{w\in S_n}$ and the negative standard basis $\{{H_{w\inv}\inv}\}_{w\in S_n}$.
\begin{remark}
It is also common to express everything above in terms of $T_w:=(-v)^{\ell(w)}H_w$, where $\ell(w)$ is the Bruhat length of $w$.
\end{remark}
Jones and Ocneanu \cite{Jones,HOMFLY} defined a trace function $\Tr \colon \mathbb{H}_n \rightarrow \Q(v)[a]$,
which (up to a normalization factor) agrees with the HOMFLY-PT polynomial. We define Jones-Ocneanu trace in section \ref{subsec:decat decomp} and list some of its most important properties here.
For $x\in \mathbb{H}_n$, $\Tr(x)$ is a polynomial in $a$ of degree at most $n$ with coefficients being rational functions in $v$. Let $\Tr^n(x)$ (resp. $\Tr^0(x)$) be the coefficient of $a^n$ (resp. $a^0$) in $\Tr(x)$.
\begin{lemma}
\label{lem: trace as coefficient at 1}
If we express $x\in \mathbb{H}_n$ in the positive and negative standard bases as
$$
x=\sum_{w\in S_n}\phi_w H_w=\sum_{w\in S_n}\psi_w H_{w\inv}\inv,
$$
then we have $\Tr^n(x)=(1-q)^{-n}\phi_1$ and $\Tr^0(x)=(1-q)^{-n}\psi_1$.
\end{lemma}
Indeed, it follows from Lemma \ref{lemma:ptr decat} below that for $w\neq 1$ one has $\Tr^n(H_w)=0$ and $\Tr^0(H_{w\inv}\inv)=0$, while $\Tr^0(1)=\Tr^n(1)=(1-q)^{-n}$.
Let $\e\colon \mathbb{H}_n \rightarrow \Q(v)$ be the vector space projection $\sum_{w\in S_n}\psi_w H_{w\inv}\inv\mapsto \psi_1$, and let
$(-)^\vee\colon \mathbb{H}_n \rightarrow \mathbb{H}_n$ be the ring anti-automorphism defined by $H_i^\vee=H_{i}\inv$ and $v^\vee=v^{-1}$.
(We remark that $(H_{w\inv}\inv)^{\vee}=H_{w^{-1}}$.)
Define a pairing $\ip{-,-}\colon \mathbb{H}_n\times \mathbb{H}_n\rightarrow \Z[q^{\pm}]$ over $\mathbb{H}_n$ by $\ip{x,y}:=\e(yx^\vee)$.
By the definition, we have $\ip{xz,y} = \ip{x,yz^{\vee}}$. We also have $\e(xy)=\e(yx)$, hence $\ip{zx,y} = \ip{x,z^{\vee}y}$.
\begin{remark}
This is the pairing which is categorified by the hom pairing of Soergel bimodules (see for instance \cite{EK}, modulo conventions).
\end{remark}
\begin{theorem}[\cite{Kalman}]
For all $x\in \mathbb{H}_n$ one has $\ip{x\FT,1}=\ip{1,x}^{\vee}$ and $\Tr^n(x\FT)=\Tr^0(x)$.
\end{theorem}
In \cite{Kalman}, this is proved by the fact that $\phi_{w_0}=\psi_{w_0}$ in the expansions of $x\HT$.
Equivalently, one has $\ip{\HT,H_{w\inv}\inv}=0$ if $w\neq w_0$ and $\ip{\HT,\HT^{-1}}=1$.
It is also known that the basis $\{H_{w\inv}\inv\}_{w\in S_n}$ is the dual basis of $\{H_w\}_{w\in S_n}$ with respect to this pairing.
(In fact, this orthogonality holds for Hecke algebras with any Coxeter group. For more details and a categorified result, see Appendix \ref{app:anytype}.)
\subsection{Partial traces}
\label{subsec:decat decomp}
We define Jones-Ocneanu trace on the Hecke algebra and its ``partial analogues" following \cite{Jones}. The somewhat nonstandard conventions below are chosen to match with the categorical picture, in $\SBim_n$, discussed later.
The algebra $\mathbb{H}_n$ may be regarded as a bimodule over $\mathbb{H}_{n-1}$, and we have an isomorphism
\[
\mathbb{H}_n \ \ \cong \ \ \mathbb{H}_{n-1} \ \oplus \ \mathbb{H}_{n-1}\otimes_{\mathbb{H}_{n-2}}\mathbb{H}_{n-1}
\]
This isomorphism is not canonical, but two natural choices are (recall that $q=v^{-2}$)
\begin{equation}\label{eq:twoisos}
\mathbb{H}_{n-1} \ \oplus \ \mathbb{H}_{n-1}\otimes_{\mathbb{H}_{n-2}}\mathbb{H}_{n-1}\buildrel\Phi^{\pm} \over \longrightarrow \mathbb{H}_n, \qquad\qquad (x,y\otimes z)\mapsto (1-q)x + y H_{n-1}^\pm z.
\end{equation}
With respect to these isomorphisms, the induced projections $\pTr^\pm:\mathbb{H}_n\rightarrow \mathbb{H}_{n-1}$ are characterized by
\[
\pTr^\pm(x) = \frac{1}{1-q}x,\qquad\quad \pTr^{\pm}(xH_{n-1}^{\pm} y) =0
\]
for all $x,y\in \mathbb{H}_{n-1}$. We have a close relationship between $\pTr^\pm$ and the Jones-Ocneanu trace.
\begin{definition}
Let $\pTr:\mathbb{H}_n[a]\rightarrow \mathbb{H}_{n-1}[a]$ be the map defined by $\pTr(x)=\pTr^-(x) + a\pTr^+(x)$.
\end{definition}
\begin{lemma}\label{lemma:ptr decat}
The map $\pTr:\mathbb{H}_n\rightarrow \mathbb{H}_{n-1}$ satisfies
\[
\pTr(x)=\frac{1+a}{1-q} x, \qquad \qquad \pTr(x H_{n-1} y) = -v xy,\qquad\qquad \pTr(x H_{n-1}\inv y) = av\inv xy,
\]
for all $x,y \in \mathbb{H}_{n-1}$. Furthermore, the composition
\[
\mathbb{H}_n[a] \buildrel \pTr \over\longrightarrow \mathbb{H}_{n-1}[a]\buildrel \pTr \over\longrightarrow \cdots \buildrel \pTr \over\longrightarrow \mathbb{H}_{0}[a]=\Q(q)[a]
\]
is the Jones-Ocneanu trace.
\end{lemma}
In particular, we have $(\pTr^-)^n(x)=\Tr^0(x)$ and $(\pTr^+)^n(x)=\Tr^n(x)$ for $x \in \mathbb{H}_n$.
We wish to categorify this story. It will turn out that the functors which categorify $\pTr^\pm$ are related by a relative version of Serre duality.
\section{Background}
\label{sec:background}
In this section we discuss the backround in Soergel bimodules and Khovanov-Rozansky homology. Throughout this paper, let $\k$ be an infinite field of characteristic $\neq 2$. This guarantees that the results of \cite{Soergel} apply, though we will only need these results in type $A$.
\subsection{Soergel bimodules}
\label{subsec:SBim}
Fix an integer $n\geq 1$, and let $R=\k[x_1,\ldots,x_n]$. The ring $R$ is graded such that the variables $x_i$ have degree 2. The notions of an $R$-bimodule and an $R^e$-module will be identified, where $R^e = R\otimes_\k R = \k[x_1,\ldots,x_n,x_1',\ldots,x_n']$.
Let $B_i$ denote the \emph{elementary bimodules} $B_1,\ldots,B_{n-1}$, defined by
$$
B_i=R\otimes_{R^{(i,i+1)}}R(1),
$$
where $R^{(i,i+1)}\subset R$ is the subalgebra of polynomials which are symmetric in $x_i,x_{i+1}$. We may identify $B_i(-1)$ with $\k[x_1,\ldots,x_n,x'_1,\ldots, x'_n]$ modulo the ideal generated by
\[
x_i+x_{i+1}-x'_{i}-x'_{i+1},\qquad \ \ \ x_ix_{i+1}-x'_ix'_{i+1}, \qquad \ \ \ x_j-x'_j\ (j\neq i,i+1).
\]
\begin{definition}
Let $\SBim_n$ denote the full subcategory of graded $(R,R)$-bimodules generated by $R,B_1,\ldots,B_{n-1}$ and closed under direct sums, tensor product $\otimes_R$, gradings shifts $(\pm 1)$, and direct summands (i.e.~retracts). A tensor product of shifts of elementary bimodules $B_i$ is called a \emph{Bott-Samelson bimodule}; by convention, $R$ (the ``empty tensor product'') is also regarded as a Bott-Samelson bimodule.
\end{definition}
\begin{notation}
Henceforth, the tensor product $\otimes_R$ will simply be denoted $\otimes$.
\end{notation}
The category $\SBim_n$ is additive but not abelian. An important result of Soergel \cite{Soergel} states that, up to isomorphism and shift, the indecomposables $B_w$ in $\SBim_n(\k)$ are in one-to-one correspondence with $w\in S_n$. Furthermore, if $s_{i_1}\cdots s_{i_\ell}$ is a reduced expression of $w\in S_n$, then the Bott-Samelson bimodule $B_{s_1}\otimes\cdots \otimes B_{s_\ell}$ has a unique summand isomorphic to $B_w$, and the remaining summands are $B_v$ for elements $v\in S_n$ of shorter length.
The morphisms in $\SBim_n$ are degree preserving $R$-bilinear maps. In this paper we almost exclusively work with space of morphisms of arbitrary degree $\Hom^{\Z}(M,N)=\bigoplus\Hom(M,N(i))$. In fact, $\Hom^{\Z}$ will occur so often that we will simply write $\Hom^\Z=\Hom$ by abuse, and we will write $\Hom^0$ when we wish to emphasize degree zero morphisms. By convention, every arrow $M\rightarrow N$ will be a degree preserving map in whatever category, unless otherwise specified.
Now, the (graded) hom spaces $\Hom(M,N)$ are graded $R$-bimodules, via
\[
f\cdot \phi\cdot g : m\mapsto f\phi(m)g = \phi(fmg)
\]
for all $f,g\in R$, $m\in M$, and $\phi\in \Hom(M,N)$.
Given two complexes $A=(A^\bullet,d_A)$ and $B=(B^\bullet,d_B)$ in $\KC^b(\SBim_n)$, we define a complex $\Homc(A,B)=(C_k,d_{\Homc})$ where
$$
C^k=\prod_{i\in \Z}\Hom(A^i,B^{i+k}),\ \qquad d_{\Homc}(f):=d_B\circ f - (-1)^k f\circ d_A.
$$
\begin{notation}
It would be more precise to write $\Homc^{\Z\times \Z}_{\KC^b(\SBim_n)}(A,B)$ instead of $\Homc(A,B)$.
\end{notation}
\subsection{Hochschild (co)homology}
\label{subsec:hochschild}
If $M$ is a graded $R$-bimodule, the zeroth Hochschild cohomology $\HH^0(M)$ is defined to be the sub-bimodule of $M$ spanned by homogeneous elements $m\in M$ with $x_im=mx_i$ for all $i=1,\ldots,n$. Note that by definition,
\begin{equation}
\label{eq:HH^0 as hom}
\HH^0(M)\cong \Hom(R,M).
\end{equation}
Dually, the zeroth Hochschild homology $\HH_0(M)$ is defined to be the quotient bimodule $M/[R,M]$, that is to say $M$ modulo the $\k$-submodule spanned by commutators $x_im - mx_i$ for all homogeneous elements $m\in M$ and all $i=1,\ldots,n$.
The higher derived functors of $\HH^0$ and $\HH_0$ are denoted by $\HH^k$ and $\HH_k$; they are zero outside the range $0\leq k\leq n$.
Self-duality of the Koszul resolution of $R$ as a graded bimodule implies the following.
\begin{lemma}\label{lemma:H homology is cohomology}
For each graded $R$-bimodule $M$, we have
\[
\HH_{k}(M)\cong \HH^{n-k}(M)(-2n).
\]\qed
\end{lemma}
We will regard $\HH^k(M)$ and $\HH_k(M)$ as graded $R,R$-bimodules on which the left and right $R$-actions coincide. That is to say, $\HH^k$ and $\HH_k$ may be viewed as endofunctors of the category of graded $R$-bimodules.
We have the following ``Markov moves'' for $\HH_k$ and $\HH^k$.
\begin{lemma}\label{lemma:HH markovs}
Let $M\in \SBim_{n-1}$ be given. Then
\begin{subequations}
\begin{equation}\label{eq:trivialMarkovUpper}
\HH^k(M\sqcup \one_1)\cong \Big(\HH^k(M)\sqcup \one_1\Big) \oplus \Big(\HH^{k-1}(M)\sqcup \one_1\Big)(2)
\end{equation}
\begin{equation}\label{eq:interestingMarkovUpper}
\HH^k\Big((M\sqcup \one_1)\otimes B_{n-1}\otimes (N\sqcup \one_1)\Big)\cong \Big(\HH^k(M\otimes N)\sqcup \one_1\Big)(-1) \oplus \Big(\HH^{k-1}(M\otimes N)\sqcup \one_1\Big)(3)
\end{equation}
\begin{equation}\label{eq:trivialMarkovLower}
\HH_k(M\sqcup\one_1)\cong \Big(\HH_k(M)\sqcup \one_1\Big) \oplus \Big(\HH_{k-1}(M)\sqcup \one_1\Big)(-2)
\end{equation}
\begin{equation}\label{eq:interestingMarkovLower}
\HH_k\Big((M\sqcup \one_1)\otimes B_{n-1}\otimes (N\sqcup \one_1)\Big)\cong \Big(\HH_k(M\otimes N)\sqcup \one_1\Big)(1) \oplus \Big(\HH_{k-1}(M\otimes N)\sqcup \one_1\Big)(-3)
\end{equation}
\end{subequations}
\end{lemma}
Here $M\sqcup \one_1 = M[x_n]$ is the induced $R$-bimodule.
\begin{proof}
Standard, see \cite{Kh07} and also Proposition 3.10 in \cite{Hog18-GT}. Note that \eqref{eq:trivialMarkovLower} and \eqref{eq:interestingMarkovLower} follow from \eqref{eq:trivialMarkovUpper} and \eqref{eq:interestingMarkovUpper} using the isomorphism $\HH_k(M)\cong \HH^{n-k}(M)(-2n)$.
\end{proof}
\begin{corollary}\label{co:HHfree}
For each $1\leq k\leq n$ and each $B\in \SBim_n$, the Hochschild cohomology $\HH^k(B)$ is a free $R$-module of finite rank.
\end{corollary}
\begin{proof}
Since summands of free graded finite rank $R$-modules are free of finite rank, it suffices to prove in the case when $B=B_{i_1}\otimes\cdots\otimes B_{i_r}$ is a Bott-Samelson bimodule. We have
\begin{equation}\label{eq: sts}
(B_i\otimes B_{i+1} \otimes B_i) \oplus B_{i+1} \cong (B_{i+1}\otimes B_{i}\otimes B_{i+1})\oplus B_{i},
\end{equation}
\begin{equation}\label{eq: ss}
B_i^{\otimes 2} \cong B_i(1)\oplus B_i(-1),
\end{equation}
so a straightforward induction allows us to reduce to the case when the index $n-1$ appears at most once among the indices $i_j$. Applying \eqref{eq:trivialMarkovUpper} or \eqref{eq:interestingMarkovUpper} we reduce to the statement for $n-1$.
\end{proof}
Thus, we may view $\HH^k$ as an endofunctor of $\SBim_n$: the input is an arbitrary Soergel bimodule and the output is a direct sum of finitely many copies of $\one$ with shifts.
\subsection{Duals}
\label{subsec:duals}
The category $\SBim_n$ has a contravariant functor $(-)^\vee:\SBim_n\rightarrow \SBim_n$ so that $B^\vee$ is the two-sided dual (or biadjoint) to $B$. This functor satisfies $B_i^\vee = B_i$ for all $i$ and $(M\otimes N)^\vee\cong N^\vee\otimes M^\vee$. The duality functor comes from the observation that each bimodule $B_1,\ldots,B_{n-1}$ is a Frobenius algebra object in $\SBim_n$. Precisely, there are canonical chain maps
\[
B_i\otimes B_i\rightarrow B_i(-1),\qquad B_i(-1)\rightarrow R,\qquad R\rightarrow B(1),\qquad B_i(1) \rightarrow B_i\otimes B_i.
\]
The first and and third of these maps give $B_i(-1)$ the structure of an algebra object, and the second and fourth maps give $B_i(1)$ the structure of a coalgebra object. Moreover, the composition of the first two defines a map $B_i \otimes B_i\rightarrow R$, the composition of the last two defines a map $R\rightarrow B_i\otimes B_i$, and these maps realize the fact that $B_i$ is self-dual. In general we have natural isomorphisms:
\begin{subequations}
\begin{equation}\label{duality for SBim}
\Hom(A,B)\cong \Hom(R,B\otimes A^\vee) \cong \Hom(A\otimes B^\vee,R),
\end{equation}
\begin{equation}\label{duality for SBim right}
\Hom(A,B)\cong \Hom(R,A^\vee \otimes B)\cong \Hom(B^\vee\otimes A,R) .
\end{equation}
\end{subequations}
\begin{remark}
\label{rem:duality with R action}
Recall that $\Hom(A,B)$ is a graded $R$-bimodule. The isomorphisms \eqref{duality for SBim} are isomorphisms of graded left $R$-modules, while \eqref{duality for SBim right} are isomorphisms of graded right $R$-modules. In fact we can say more; for instance the right action on $\Hom(A,B)$ can be understood as corresponding to the $R$-action on $\Hom(R,B\otimes A^\vee)$ via ``middle multiplication'' on $B\otimes A^\vee$.
\end{remark}
\begin{remark}
We can also consider the duality isomorphisms for complexes. For each $A,B\in \KC^b(\SBim_n)$ we have
$$
\Homc(A,B) \cong \Homc(\one, B\otimes A^{\vee}) \cong \Homc(A\otimes B^{\vee},\one)
$$
as complexes of $R$-modules (with the left $R$-action on $\Hom(A,B)$), and
\[
\Homc(A,B) \cong \Homc(\one,A^\vee\otimes B)\cong \Homc(B^\vee\otimes A,\one)
\]
as complexes of $R$-modules (with the right $R$-action on $\Homc(A,B)$).
\end{remark}
It can be useful to phrase this categorical duality in terms of the usual duality in the category of $R$-modules. If $M$ is a graded $R$-module, we let $M^\star:=\Hom^\Z_R(M,R)$ denote the graded $R$-module of homs. There is a natural map $M\to (M^{\star})^{\star}$, which is an isomorphism if $M$ is free and finitely generated.
If $B$ is an $R,R$-bimodule, then we can forget the left action, obtaining a dual bimodule $B^\star:=\Hom_{\k\otimes R}(B,R)$, or we can forget the right $R$-action, obtaining a dual bimodule ${}^\star B:=\Hom_{R\otimes \k}(B,R)$
\begin{lemma}\label{lemma:twodualities}
We have natural isomorphisms
\[
B^\star \cong \ B^\vee \ \cong \ {}^\star B
\]
for $B\in \SBim_n$.
\end{lemma}
\begin{proof}
We will define inverse isomorphisms $\Phi:B^\star \leftrightarrow B^\vee\cong \Hom_{\Z\otimes R}(R, B^\vee) :\Psi$. Let $f:B\rightarrow R$ be a morphism of graded right $R$-modules. Define $\Phi(f)$ to be the composition
\[
R \rightarrow B\otimes_R B^\vee \rightarrow R\otimes_R B^{\vee} \cong B^{\vee},
\]
where the first map is given by duality and the second is $f\otimes \Id$.
In the other direction, let $g:R\rightarrow B^\vee$ be a morphism of graded right $R$-modules, and define $\Psi$ to be the composition
\[
B\cong R\otimes_R B \rightarrow B^\vee\otimes_R B \rightarrow R,
\]
where the second map is $g\otimes \Id_{B^\vee}$ and the last map is given by duality. It is an easy exercise to show that $\Psi$ and $\Phi$ are inverse isomorphisms of graded bimodules $B^\star \cong \Hom_{\Z\otimes R}(R, B^\vee)$; they are clearly natural in $B$.
The proof that $B^\vee\cong {}^\star B$ naturally is similar.
\end{proof}
\begin{notation}
Henceforth, if $M$ is a finitely generated free $R$-module, then $M$ will be regarded as an object of $\SBim_n$, and $M^\star$ will be denoted by $M^\vee$.
\end{notation}
The following is standard.
\begin{lemma}\label{lemma:free homs}
For all $M,N\in \SBim_n$ the hom bimodule $\Hom(M,N)$ is free as a left or right $R$-module.
\end{lemma}
\begin{proof}
By Remark \ref{rem:duality with R action}, it suffices to prove the lemma in the special case $M=R$; in this case $\Hom(R,N)=\HH^0(N)$ is free by Corollary \ref{co:HHfree}.
\end{proof}
Now we consider how the duality functor interacts with Hochschild (co)homology.
\begin{lemma}
\label{lem: HH mirror}
For each $B\in \SBim_n$ one has $\HH^k(B^\vee)(-2n)=\HH^{n-k}(B)^{\vee}$.
\end{lemma}
\begin{proof}
It is easy to see from the definition that complexes of $R$-modules computing Hochschild cohomology of $B$ the Hochschild homology of $B^{\vee}$ are dual to each other. Since the cohomology of both complexes are free over $R$, the cohomologies are dual to each other as well, i.e.~$\HH_k(B^\vee)\cong \HH^k(B)^\vee$. The Lemma now follows from Lemma \ref{lemma:H homology is cohomology}.
\end{proof}
The symmetries between $\HH^k$, $\HH^{n-k}$, $\HH_k$, and $\HH_{n-k}$ are quite a bit more attractive (and easy to remember) after a change in normalization.
\begin{definition}\label{def:normalizedAgrading}
Let $\widetilde{\HH}^k(M):=\HH^k(M)(-2k)$ and $\widetilde{\HH}_{k}(M):=\HH_k(M)(2k)$.
\end{definition}
\begin{proposition}\label{prop:HHsymmetries}
If $k+l=n$, then we have
\[
\widetilde{\HH}^k(M)\cong \widetilde{\HH}_{l}(M) \cong \widetilde{\HH}^{l}(M^\vee)^\vee.
\]
for all $M\in \SBim_n$. These are isomorphisms of functors from $\SBim_n$ to the category of finitely generated free graded $R$-modules.
\end{proposition}
\begin{proof}
This is just a restatement of Lemma \ref{lemma:H homology is cohomology} and Lemma \ref{lem: HH mirror}.
\end{proof}
\begin{remark}
When expressing the Poincar\'e series of Khovanov-Rozansky homology, the variable $a=AQ^{-2}$ is often used instead of $A$. This precisely corresponds to replacing $\HH^k$ by $\widetilde{\HH}^k$. (Here, $Q$ denotes the degree in Soergel bimodules and $A$ denotes the usual Hochschild degree.)
\end{remark}
\begin{corollary}\label{cor:HH_0 as hom}
We have $\HH_0(M)\cong \Hom(M,R)^\vee$, natural for $M\in \SBim_n$.
\end{corollary}
\begin{proof}
Indeed
\[
\Hom(M,R)^\vee\cong \Hom(R,M^\vee)^\vee \cong \HH^0(M^\vee)^\vee\cong \HH_0(M).
\]
Each of these isomorphisms is functorial in $M$.
\end{proof}
\subsection{Rouquier complexes}
\label{subsec:rouquier}
Let $\KC^b(\SBim_n)$ denote the homotopy category of bounded complexes of Soergel bimodules, with differentials of degree $+1$.
It inherits all the structures from $\SBim_n$: it is an additive tensor category with (two-sided) duals and shifts $(\pm 1)$.
It is also triangulated, with cohomological shift functors $[\pm 1]$.
We will need special complexes
$$
\rouq_i \ \ := \ \ 0\rightarrow \underline{B_i}\rightarrow {R}(1)\rightarrow 0,
$$
$$
\rouq_i^{-1}\ \ :=\ \ 0 \rightarrow R(-1)\rightarrow \underline{B_i}\rightarrow 0,
$$
where we have underlined terms in degree zero. Note that $\rouq_i=\Cone(b_i)[-1]$ and $\rouq_i\inv = \Cone(b_i^*)$ for some distinguished morphisms $b:B_i\to R(1)$ and $b^*:R(-1)\to B_i$ which are adjoint to each other.
\begin{theorem}[\cite{Rouquier}]
The complexes $\rouq_i$ and $\rouq_i^{-1}$ satisfy the braid relations up to canonical homotopy equivalence:
\begin{enumerate}
\item $\rouq_i\otimes \rouq_{i+1}\otimes \rouq_i\ \simeq\ \rouq_{i+1}\otimes \rouq_i\otimes \rouq_{i+1}$,
\item $ \rouq_{i}\otimes \rouq_{i}^{-1} \ \simeq \ \one\ \simeq \ \rouq_i^{-1}\otimes \rouq_i$,
\item $\rouq_i\otimes \rouq_j \ \simeq \ \rouq_j\otimes \rouq_i$ if $|i-j|\ge 2$.
\end{enumerate}
\end{theorem}
As a corollary, for any braid $\beta$ expressed as a product of the standard generators $\sigma_i^\pm$ we can define a complex $F(\beta)$ as a tensor product of complexes of the form $\rouq_i^{\pm 1}$; the resulting complex depends only on $\beta$ and not the expression as a product of generators, up to coherent homotopy equivalence. Since $\rouq_i^\vee\simeq \rouq_i^{-1}$, we have
$$
F(\beta)^{\vee}\simeq F(\beta^{-1})\simeq F(\beta)^{-1}
$$
for all $\beta$.
\begin{definition}\label{def:hull}
Given a collection of complexes $X_1,\ldots,X_r\in\KC^b(\SBim_n)$, define the \emph{graded triangulated hull} $\ip{X_1,\ldots,X_r}$ to be the smallest full triangulated subcategory of $\KC^b(\SBim_n)$ containing $X_1,\ldots,X_r$ and closed under grading shifts. We also refer to $\ip{X_1,\ldots,X_r}$ as the \emph{span} of $\{X_i\}$.
\end{definition}
\begin{proposition}\label{prop:rouquier generation}
For each $w\in S_n$ let $F_w$ and $F_w\inv$ denote the Rouquier complexes associated to the positive braid lift of a chosen reduced expression for $w$. Then $\{F_w\}_{w\in S_n}$ and $\{F_w\inv\}_{w\in S_n}$ span $\KC^b(\SBim_n)$.
\end{proposition}
\subsection{Triply graded Khovanov-Rozansky homology}
\label{subsec:KR homology}
Every additive functor can be extended term-wise to complexes. In particular, for a complex $C=(C^\bullet,d_C)$ and $j\in \Z$ we can define Hochschild cohomology $\HH^{j}(C)$ as the complex
\[
\cdots\rightarrow \HH^j(C^k) \rightarrow \HH^j(C^{k+1})\rightarrow \cdots
\]
whose differential is just the functor $\HH^j$ applied to $d_C$. If $C\simeq D$ then $\HH^j(C)\simeq \HH^j(D)$ as complexes of $R$-modules. Each of the natural isomorphisms in \S \ref{subsec:hochschild} extends without trouble to the corresponding categories of complexes. In particular if $X\in \KC^b(\SBim_n)$ is a complex of Soergel bimodules then
\[
\HH^k(X)(-2n)\cong \HH^{n-k}(X^\vee)^\vee \cong \HH_{n-k}(X)
\]
naturally.
\begin{definition}
For each $X\in \KC^b(\SBim_n)$, let $\HH^\bullet(X):=\bigoplus_{k=0}^n \HH^k(X)$ and $\HH_\bullet(X):=\bigoplus_{k=0}^n \HH_k(X)$.
\end{definition}
The homology of $\HH^\bullet(X)$ is often denoted $\HHH^\bullet(X)$, but we will not need this.
\begin{lemma}[Markov moves]\label{lemma:Markov2}
Let $X,Y\in \KC^b(\SBim_{n-1})$ be arbitrary. Then
\[
\HH^k(X\otimes \rouq_{n-1}\otimes Y)\simeq \HH^k(X\otimes Y)[-1](1)
\]
and
\[
\HH^k(X\otimes \rouq_{n-1}\inv\otimes Y)\simeq \HH^{k-1}(X\otimes Y)(-1).
\]
\end{lemma}
\begin{proof}
Standard; see \cite{Kh07} and Proposition 3.10 in \cite{Hog18-GT}.
\end{proof}
\begin{corollary}
\label{cor:negative stabilization}
In the notation of Lemma \ref{lemma:Markov2} we have
\[
\HH_0(X\otimes \rouq_{n-1} \otimes Y)\simeq 0,\qquad\qquad \HH^0(X\otimes \rouq_{n-1}\inv\otimes Y)\simeq 0.
\]
\end{corollary}
We also record an important connection of $\HH$ to the Jones-Ocneanu trace:
\begin{theorem}[\cite{Kh07}]
The Euler characteristic of $\HH$ equals the Jones-Ocneanu trace:
\[
\sum_i a^{i} \chi(\HH^i(X)(-2i))=\Tr([X])
\]
for all $X\in \KC^b(\SBim_n)$, where $\chi$ is a graded Euler characteristic and
$[X]$ is the class of $X$ in the Grothendieck group $K_0(\SBim_n)\cong\mathbb{H}_n$.
\end{theorem}
\begin{corollary}
We have $\chi(\HH^0(X))=\Tr^0([X])$ and $\chi(\HH^n(X))=\Tr^n([X])$.
\end{corollary}
\section{Decompositions of categories}
We first recall some background on categorical idempotents and semi-orthogonal decompositions.
\subsection{Adjoints to inclusions}
\label{subsec:adjointsgeneral}
Let $\AS$ be a category and $\BS$ a category with functor $\sigma:\BS\rightarrow \AS$. The following is classical; its proof is an exercise in category theory and Yoneda embedding. See also Theorem 1 in \S IV.3 of \cite{Maclane}.
\begin{lemma}\label{lemma:adjunctions yield idempotents}
Suppose $\sigma:\BS\rightarrow \AS$ has a left adjoint $\pi_L:\AS\rightarrow \BS$. Then $\sigma$ is fully faithful if and only if the counit of the adjunction is an isomorphism $\pi_L\circ \sigma\rightarrow \Id_{\BS}$.
Dually, if $\sigma$ has a right adjoint $\pi_R:\AS\rightarrow \BS$, then $\sigma$ is fully faithful if and only if the unit of the adjunction is an isomorphism $\Id_\BS\rightarrow \pi_R\circ \sigma$.\qed
\end{lemma}
If these equivalent conditions hold, then $\EB_L:=\sigma\circ \pi_L$ and $\EB_R:=\sigma\circ \pi_R$ are idempotent functors $\AS\rightarrow \AS$ with essential image equivalent to $\BS$. We discuss these next, after an example.
\begin{example}
Let $A$ be an algebra and $I\subset A$ a two-sided ideal. Let $\AS$ be the category of $A$-modules and $\BS\subset \AS$ the category of objects on which $I$ acts by zero. Then the inclusion $\BS\rightarrow \AS$ has a left adjoint sending an $A$-module $M$ to $M/IM$, and a right adjoint sending $M$ to the annihilator of $I$ in $M$. Both functors can be regarded as idempotent endofunctors of $\AS$ with image $\BS$.
\end{example}
\begin{definition}\label{def:localization}
A \emph{localization functor} on a category $\CS$ is a pair $(\EB,\eta)$ consisting of an endofunctor $\EB:\CS\rightarrow \CS$ and a natural transformation $\eta:\Id_{\CS}\rightarrow \EB$ such that $\EB\eta$ and $\eta \EB$ are isomorphisms $\EB\buildrel \cong \over \rightarrow \EB\EB$.
A \emph{colocalization functor} on $\CS$ is a pair $(\EB,\e)$ consisting of an endofunctor $\EB:\CS\rightarrow \CS$ and a natural transformation $\e:\EB\rightarrow \Id_{\CS}$ such that $\EB\e$ and $\e \EB$ are isomorphisms $\EB\EB \buildrel\cong\over \rightarrow \EB$.
\end{definition}
\begin{lemma}\label{lemma:idempotents yield adjunctions}
Let $\EB$ be an endofunctor of a category $\CS$ with essential image $\BS\subset \CS$. If $\eta:\Id_\CS\rightarrow \EB$
(resp. $\e:\EB\rightarrow \Id_{\CS}$)
gives $\EB$ the structure of a localication (resp.~colocalization) endofunctor, then $\EB$ defines a left (resp.~right) adjoint to the inclusion $\BS\rightarrow \CS$.
Conversely if $\BS\subset \CS$ is a full subcategory such that the inclusion $\sigma:\BS\rightarrow \CS$ admits a left (resp.~right) adjont $\pi:\CS\rightarrow \BS$, then the $\sigma \pi$ is a localization (resp.~colocalization) endofunctor.
\end{lemma}
\begin{proof}
Suppose that $\EB:\CS\rightarrow \CS$ is a localization functor. We must show that if $\EB(Y)\cong Y$ and $X\in \AS$ is arbitrary, then the unit map $\eta_X:X\rightarrow \EB(X)$ induces an isomorphism
\[
\Hom_{\AS}(X,Y)\cong \Hom_{\BS}(\EB(X),Y)=\Hom_{\AS}(\EB(X),Y).
\]
This is proven, for example in \cite{Krause10}, Proposition 2.4.1.
The statement about colocalization functors follows by taking opposite categories.
Conversely, if $\sigma:\BS\rightarrow \CS$ admits a left adjoint $\pi:\CS\rightarrow \BS$, then the counit of the adjunction $\e:\pi\sigma\rightarrow \Id_{\BS}$ is an isomorphism of functors. Let $\eta:\Id_\CS\rightarrow \sigma\pi$ be the unit of the adjunction. Then $\eta \sigma\pi, \sigma\pi\eta: \sigma\pi \rightarrow \sigma\pi \sigma\pi$ are isomorphisms with inverse $\sigma\e \pi$.
\end{proof}
\begin{remark}
In linear algebra there can be many idempotent endomorphisms of a vector space $V$ which project onto a given subspace $W\subset V$ (the embedding $W\rightarrow V$ has many left inverses). In the realm of category theory, idempotents are quite a bit more rigid. Indeed, the inclusion $\BS\rightarrow \AS$ of a full subcategory can be the image of at most one localization functor and at most one colocalization functor (because left and right adjoints to a given functor are unique when they exist).
\end{remark}
\newcommand{\ES}{\EuScript{E}}
\newcommand{\IB}{\mathbf{I}}
\subsection{Semi-orthogonal decompositions}
\label{subsec:semiortho}
In the setting of triangulated categories it is possible to discuss the notion of complementary idempotent functors. Let $\MS$ a triangulated category and let $\AS$ be a triangulated monoidal category which acts on $\MS$ by exact endofunctors (see \cite{Hog17b} for details).
\begin{example}
The only example we will need in this paper is the following. Let $\CS$ be an additive $\k$-linear category with $\End(\CS)$ the category of linear endofunctors. Then the homotopy category of complexes $\MS:=\KC^b(\CS)$ is an $\AS$-module category where $\AS:=\KC^b(\End(\CS))$. In other words, complexes of endofunctors of $\CS$ can be thought of as endofunctors of $\KC^b(\CS)$. For instance if $F:\CS\rightarrow \CS$, then $F$ can be thought of as a complex in $\KC^b(\End(\CS))$ of degree zero, and the action of $F$ on complexes is the usual:
\[
F(X) \ := \ \cdots \buildrel F(d) \over \longrightarrow F(X^k) \buildrel F(d) \over \longrightarrow F(X^{k+1}) \buildrel F(d) \over \longrightarrow\cdots
\]
\end{example}
\begin{definition}\label{def:complementary idempotents}
An \emph{idempotent triangle} in $\AS$ is a distinguished triangle of the form
\begin{equation}\label{eq:idemp triangle}
\PB\buildrel \e \over \rightarrow \IB\buildrel \eta \over \rightarrow \QB\buildrel\d\over \rightarrow \PB[1]
\end{equation}
such that $\PB\QB \simeq 0 \simeq \QB\PB$ in $\AS$. In such a triangle, we refer to $\PB$ and $\QB$ as \emph{complementary idempotents} in $\AS$.
A \emph{unital idempotent} in $\AS$ is an object $\QB\in \AS$ with a map $\eta:\IB\rightarrow \QB$ such that $\eta\QB$ and $\QB\eta$ are isomorphisms $\QB\rightarrow \QB\QB$. A \emph{counital idempotent} in $\AS$ is an object $\PB\in \AS$ with a map $\e:\PB\rightarrow \IB$ such that $\PB\e$ and $\e\PB$ are isomorphisms $\PB\PB\rightarrow \PB$.
\end{definition}
We will denote the identity of $\AS$ by $\IB$, and we will write the monoidal structure in $\AS$ simply by juxtaposition of functors. Moreover, if $f$ is a morphism in $\AS$ and $\XB$ is an object of $\AS$, then we write $f \XB$ and $\XB f$ for $f\otimes \Id_{\XB}$ and $\Id_{\XB} \otimes f$, respectively. This notation is compatible with the usual conventions for writing horizontal compositions of functors and natural transformations.
\begin{remark}
If $\EB\in \AS$ has the structure of a (co)unital idempotent, then its action on $\MS$ is a (co)localization functor.
\end{remark}
Observe that if \eqref{eq:idemp triangle} is an idempotent triangle in $\AS$ and $X\in \MS$ is an object of the $\AS$-module category $\MS$, then $X$ fits into a distinguished triangle
\[
\PB(X)\rightarrow X\rightarrow \QB(X)\rightarrow \PB(X)[1].
\]
If $\QB(X)\simeq 0$ then the first map is an isomorphism in $\MS$ $\PB(X)\rightarrow X$, while if $\PB(X)\simeq 0$ then the second map is an isomorphism $X\rightarrow \QB(X)$, by properties of distinguished triangles. Thus, we arrive at the following:
\begin{observation}
The essential image of $\PB$ acting on $\MS$ coincides with the kernel of $\QB$ acting on $\MS$, and vice versa.
\end{observation}
\begin{lemma}\label{lemma:idemp triangles}
In any idempotent triangle \eqref{eq:idemp triangle} the maps $\e$ and $\eta$ give $\PB$ and $\QB$ the structure of a counital and unital idempotent, respectively. Conversely:
\begin{enumerate}
\item if $(\QB,\eta)$ is a unital idempotent in $\AS$, then $\PB:=\Cone(\eta)[-1]$ has the structure of a counital idempotent.
\item If $(\PB,\e)$ is a counital idempotent in $\AS$ then $\QB:=\Cone(\e)$ has the structure of a unital idempotent in $\AS$.
\end{enumerate}
\end{lemma}
\begin{proof}
Straightforward.
\end{proof}
\begin{definition}\label{def:complement}
If $(\PB,\e)$ is a counital idempotent, then $\PB^c:=\Cone(\e)$ is called the \emph{(unital) complement} of $\PB$. If $(\QB,\eta)$ is a unital idempotent, then $\QB^c:=\Cone(\eta)[-1]$ is called the \emph{(counital) complement} of $\QB$.
\end{definition}
Note that $(\PB^c)^c\simeq \PB$ and $(\QB^c)^c\simeq \QB$.
Next we discuss the relation between categorical idempotents and semi-orthogonal decompositions. The following property is key.
\begin{lemma}\label{lemma:idempt semi ortho}
Let $(\PB,\e)$ be a counital idempotent in $\AS$ with complement $\QB:=\PB^c$. Then for each $X,Y\in \MS$ we have
\[
\Hom_{\MS}(\PB(X),\QB(Y))\cong 0.
\]
\end{lemma}
\begin{proof}
See Theorem 4.13 in \cite{Hog17b}.
\end{proof}
This semi-orthogonality property is closely tied with the discussion of adjoints to inclusions in the previous section. Let $\QB(\MS)$ denote the essential image of $\QB$, and let $Y\in \QB(\MS)$ be given, so that $\QB(Y)\simeq Y$. Let $X$ be arbitrary. Then Lemma \ref{lemma:idempt semi ortho} implies that $\Hom_\MS(\PB(X),Y)\cong 0$. Applying $\Hom_{\MS}(-,\QB(Y))$ to the distinguished triangle $\PB(X)\rightarrow X\rightarrow \QB(X)\rightarrow \PB(X)[1]$, we find that precomposing with $\eta_X:X\rightarrow \QB(X)$ is an isomorphism
\[
\eta_X^\ast : \Hom_{\QB(\MS)}(\QB(X),Y)\buildrel \cong \over \rightarrow \Hom_\MS(X,Y)
\]
This shows that $\QB$, when regarded as a functor $\MS\rightarrow \QB(\MS)$ is the left adjoint to the inclusion $\QB(\MS)\rightarrow \MS$.
Similarly, if $X\in \PB(\MS)$ and $Y\in \MS$ is arbitrary then applying $\Hom_{\MS}(X,-)$ to the distinguished triangle $\PB(Y)\rightarrow Y\rightarrow \QB(Y)\rightarrow \PB(Y)[1]$ and using Lemma \ref{lemma:idempt semi ortho} shows that post-composing with $\e_Y:\PB(Y)\rightarrow Y$ is an isomorphism
\[
(\e_Y)_\ast : \Hom_{\PB(\MS)}(X,\PB(Y))\buildrel\cong\over \rightarrow \Hom_{\MS}(X,Y).
\]
Thus, $\PB$ defines the right adjoint to the inclusion $\PB(\MS)\rightarrow \MS$.
\begin{definition}\label{def:semiorthog}
Let $\NS_1,\NS_2\subset \MS$ be full triangulated subcategories. We say that $(\NS_1,\NS_2)$ is a \emph{semi-orthogonal decomposition} of $\MS$, written $\MS\simeq (\NS_1\rightarrow \NS_2)$, if
\begin{enumerate}
\item each object $X\in \MS$ fits into a distinguished triangle
\begin{equation}\label{eq:semiortho triangle}
Y_2\buildrel \a\over \rightarrow X\buildrel \b\over \rightarrow Y_1\buildrel \gamma\over \rightarrow Y_2[1],\qquad\qquad Y_i\in \NS_i.
\end{equation}
\item $\Hom_{\AS}(Y_2,Y_1)\cong 0$ for all $Y_i\in \NS_i$.
\end{enumerate}
\end{definition}
The following is an immediate corollary of Lemma \ref{lemma:idempt semi ortho} and the fact that the kernel of a (co)unital idempotent in $\AS$ is the image of its complement.
\begin{corollary}\label{cor:semiortho from idempts}
If $\PB$ (resp.~$\QB$) is a counital (resp.~unital) idempotent in $\AS$ then each $\AS$-module category $\MS$ inherits a semi-orthogonal decomposition $\MS\simeq (\operatorname{ker} \PB \rightarrow \operatorname{im} \PB)$ (resp.~$\MS\simeq (\operatorname{im} \QB \rightarrow \operatorname{ker}\QB)$).\qed
\end{corollary}
\begin{definition}\label{def:perp}
If $\NS\subset \MS$ is a full triangulated subcategory then $\NS^\perp\subset \MS$ and ${}^\perp\NS\subset \MS$ denote the full subcategories of objects $X$ such that $\Hom_\MS(B,X)\cong 0$ (resp.~$\Hom_\MS(X,B)\cong 0$) for all $B\in \NS$.
\end{definition}
\begin{theorem}\label{thm:semiorthogonality}
If $\MS\simeq (\NS_1\rightarrow \NS_2)$ is a semi-orthogonal decomposition then
\begin{enumerate}
\item Each distinguished triangle \eqref{eq:semiortho triangle} is unique up to unique isomorphism (extending the identity of $X$).
\item The categories $\NS_1$ and $\NS_2$ determine each other: $\NS_1={}^\perp\NS_2$ and $\NS_2=\NS_1^\perp$.
\end{enumerate}
\end{theorem}
It is also possible to show that in any semi-orthogonal decomposition $\MS\simeq (\NS_1\rightarrow \NS_2)$ the ``projections onto $\NS_i$'' are well-defined idempotent endofunctors of $\MS$ (the projection onto $\NS_1$ is unital, while projection onto $\NS_2$ is counital). However, we will not need this.
\begin{proof}
Let $\MS\simeq (\NS_1\rightarrow \NS_2)$ be a semi-orthogonal decomposition. We will first show (1). Let
\[
Z_2\buildrel \a'\over \rightarrow X \buildrel \b' \over \rightarrow Z_1 \buildrel \gamma'\over \rightarrow Z_2[1]
\]
be a distinguished triangle with $Z_i\in \NS_i$. We must construct a map to this triangle from \eqref{eq:semiortho triangle} and show that this map is unique. First, since $\Hom(Y_2,Z_1)\cong 0$ it follows that:
\begin{itemize}
\item post-composing with the map $\a':Z_2\rightarrow X$ gives an isomorphism $\Hom(Y_2,Z_2)\rightarrow \Hom(Y_2,X)$.
\item pre-composing with the map $\b:X\rightarrow Y_1$ gives an isomorphism $\Hom(Y_1,Z_1)\rightarrow \Hom(X,Z_1)$.
\end{itemize}
One sees this by applying $\Hom(Y_2,-)$ to the distinguished triangle $Z_2\to X\to Z_1\to $ and by applying $\Hom(-,Z_1)$ to the distinguished triangle $Y_2\to X\to Y_1\to$. Thus there is a unique $f_2:Y_2\rightarrow Z_2$ such that $\a'\circ f_2 = \a$ and a uniqe $f_1:Y_1\rightarrow Z_1$ such that $f_1\circ \b = \b'$. Thus, there is a unique map of triangles as claimed. The uniqueness also implies that all such maps are isomorphisms of triangles.
Now we show (2). The containment $\NS_2\subset \NS_1^\perp$ holds by the assumption of semiorthogonality. To prove the opposite containment $\NS_1^\perp\subset \NS_2$, suppose $Z_2\in \NS_1^\perp$ is given. We have a distinguished triangle
\[
Y_2\rightarrow Z_2\rightarrow Y_1\rightarrow Y_2[1]
\]
with $Y_i\in \NS_i$. We want to show that $Y_2=Z_2$. To do this it suffices to show that $Y_1=0$. Apply the cohomological functor $\Hom(-,Y_1)$, and consider the resulting long exact sequence in cohomology:
\[
\Hom_\MS(Y_1,Y_1[k])\rightarrow \Hom_{\MS}(Z_2,Y_1[k])\rightarrow \Hom_{\MS}(Y_2,Y_1[k])\rightarrow \Hom_{\MS}(Y_1,Y_1[k+1]).
\]
The second term is zero for all $k$ by hypothesis that $Z_2\in \NS_1^\perp$, and the third term is zero for all $k$ by semi-orthogonality between $\NS_1$ and $\NS_2$. It follows that $\End_{\MS}(Y_1)\cong 0$, hence $Y_1=0$, hence $Z_2=Y_2\in \NS_2$, as claimed.
A similar argument shows that $\NS_1={}^\perp \NS_2$.
\end{proof}
The following is useful in determining the images and kernels of (co)unital idempotents.
\begin{lemma}\label{lemma:kernels and images}
Let $\{X_i\}$ and $\{Y_j\}$ be a collection of objects in $\MS$. Let $\EB$ be a unital or counital idempotent in $\AS$, with complement $\EB^c$. Assume that $\EB(X_i)\simeq 0$ for all $i$ and $\EB(Y_i)\simeq Y_j$ for all $j$. Then:
\begin{enumerate}
\item $\EB(X)\simeq 0$ for all $X\in \ip{X_i}_i$.
\item $\EB(Y)\simeq Y$ for all $Y\in \ip{Y_j}_j$.
\item If $\{X_i\}_i$ and $\{Y_j\}_j$ together span all of $\CS$, then $\{X_i\}_i$ spans the kernel of $\EB$ while $\{Y_j\}_j$ spans the image of $\EB$.
\end{enumerate}
\end{lemma}
\begin{proof}
Statements (1) follows by the five lemma, and (2) is equivalent to (1) for the complement $\EB^c$. Now, for (3), let us assume for the sake of concreteness that $\EB$ is a counital idempotent with counit $\e:\EB\rightarrow \IB$. Now, let $Z\in \MS$ be given. We may express $Z$ as an iterated cone involving various $X_i$ and $Y_j$ (with shifts). But $\EB$ annihilates each $X_i$, which implies that $\EB(Z)$ can be expressed as some iterated cone involving $\EB(Y_j)\simeq Y_j$, since $\EB$ is assumed to be exact. It follows that the essential image of $\EB$ is spanned by the $Y_j$. The same argument applied to $\EB^c$ shows that the kernel of $\EB$ is spanned by the $X_i$.
\end{proof}
\section{Semi-orthogonal decompositions of the Hecke category}
\label{sec:semiortho}
\subsection{Two adjoints}
\label{subsec:twoadjoints}
Let $\CS_\one\subset \SBim_n$ denote the full subcategory generated by the trivial bimodule $\one=R$ and its shifts. In this section we make the key observation that the inclusion $\iota:\CS_\one\rightarrow \SBim_n$ has both a left adjoint and a right adjoint, given by $\HH_0$ and $\HH^0$.
First, for each $B\in \SBim_n$ we will identify $\HH^0(B)$ with the sub-bimodule of $B$ consisting of elements $b\in B$ with $fb=bf$ for all $f\in R$. This is free as a graded $R$-module, hence is isomorphic to a finite direct sum of shifted copies of $R$. Thus, $\HH^0$ can be thought of as a functor from $\SBim_n\rightarrow \CS_\one$, or as an endofunctor of $\SBim_n$. The inclusion $\HH^0(B)\subset B$ defines a natural transformation $\e:\HH^0\rightarrow \Id_{\SBim_n}$.
Also, for each $B$ recall that $\HH_0(B)$ is the quotient of $B$ by the sub-bimodule spanned by elements of the form $fb - bf$ for all $b\in B$ and all $f\in R$. This, too, is free as a graded $R$-module, hence can be regarded as an object of $\CS_\one$. The projection $B\rightarrow \HH_0(B)$ defines a natural transformation $\eta:\Id_{\SBim_n}\rightarrow \HH_0$.
\begin{lemma}
We have the following:
\begin{enumerate}
\item the map $\e:\HH^0\rightarrow \Id_{\SBim_n}$ makes $\HH^0$ into a counital idempotent endofunctor of $\SBim_n$ with image $\CS_\one$.
\item the map $\eta:\Id_{\SBim_n}\rightarrow \HH_0$ makes $\HH_0$ into a unital idempotent endofunctor of $\SBim_n$ with image $\CS_\one$.
\end{enumerate}
Furthermore, when regarded as functors $\SBim_n\rightarrow \CS_\one$, $\HH^0$ and $\HH_0$ are right and left adjoints to the inclusion $\CS_\one\rightarrow \SBim_n$, respectively.
\end{lemma}
\begin{proof}
It is clear that if $B$ is a direct sum of shifted copies of $R$, then $\HH^0(B) = B$ and $\e_B:\HH^0(B)\rightarrow B$ is the identity map. On the other hand, $\HH^0(B)\in \CS_\one$ for all $B\in \SBim_n$, so the image of $\HH^0$ is $\CS_\one$. A moment's thought confirms that $\e_{\HH^0(B)}$ and $\HH^0(\e_{B})$ are natural isomorphisms $\HH^0(\HH^0(B))\buildrel\cong\over\rightarrow \HH^0(B)$. This proves (1). The proof of (2) is similar.
The fact that $\HH^0$ and $\HH_0$ define right and left adjoints to the inclusion of $\CS_\one\rightarrow \SBim_n$ now follows from general properties of (co)unital idempotent endofunctors.
\end{proof}
\begin{example}
In case $n=2$, $\HH^0$ sends $R\mapsto R$ and $B_s\mapsto R(1)$, while $\HH_0$ sends $R\mapsto R$ and $B_s\mapsto R(-1)$.
\end{example}
\subsection{Triangulated perspective on $\HH^0$ and $\HH_0$}
\label{subsec:absolute semiortho decomp}
We already know that the \emph{images} of the idempotent functors $\HH^0$, $\HH_0$ are both $\CS_\one$. What about the kernels? To answer this extend the functors $\HH^0$ and $\HH_0$ to complexes. For each $\XB\in \KC^b(\SBim_n)$, we have:
\begin{enumerate}
\item $\HH^0(X)\subset X$ is the subcomplex consisting of those homogeneous elements $b$ with $fb=bf$ for all $f\in R$.
\item $\HH_0(X)$ is the quotient of $X$ by the subcomplex spanned by elements $fb-bf$ for all homogeneous $b\in X$, $f\in R$.
\end{enumerate}
Following Definition \ref{def:complement}, we define endofunctors $Q^{\pm}:\KC^b(\SBim_n)$ which are complementary to $\HH_0,\HH^0$, as follows. For each $X\in \KC^b(\SBim_n)$ define
\[
Q^-(X):=\Cone(\HH^0(X)\rightarrow X),
\]
\[
Q^+(X):=\Cone(B\rightarrow \HH_0(X))[-1].
\]
In order to use the setup of \S \ref{subsec:semiortho}, one should set $\MS:=\KC^b(\SBim_n)$ and $\AS:=\KC^b(\End(\SBim_n))$. Then $\HH_0$ is a unital idempotent in $\End(\SBim_n)\subset \AS$, and $Q^+$ is its complement. Similarly $\HH^0$ is a counital idempotent in $\AS$, and $Q^-$ is its complement.
\begin{example}
In case $n=2$, recall that $\HH_0$ sends $R\mapsto R$ and $B_s\mapsto R(1)$. The unit of $\HH_0$ consists of the identity map $R\rightarrow R$ and the dot map $B_s\rightarrow R(1)$. Thus, the complementary idempotent $Q^+$ sends $R$ to the contractible complex $\Cone(\Id_R)$, and sends $B_s$ to $\rouq_s$, the Rouquier complex associated to $s$.
A similar computation shows that $Q^-$ sends $R$ to a contractible complex and $B_s$ to $\rouq_s\inv$.
\end{example}
Next we intend to use Lemma \ref{lemma:kernels and images} to describe the images and kernels of $\HH^0,\HH_0$.
\begin{lemma}
\label{lem: negative and pos tw}
For all $w\neq 1$ one has $\HH^0(\rouq_w\inv)\simeq 0$ and $\HH_0(\rouq_w)\simeq 0$.
\end{lemma}
\begin{proof}
If $w\neq 1$, we can find an integer $1\leq k\leq n-1$ and a minimal length representative of $w$ which contains a unique copy of $s_{k}$ and no $s_j$ for $j>k$, so that $\rouq_w^{\pm 1}=A\otimes \rouq_k^{\pm 1}\otimes B$ for $A, B\in \KC^b(\SBim_{k})$. Then Corollary \ref{cor:negative stabilization} gives the desired relations.
\end{proof}
\begin{definition}\label{def:TScats}
Let $\TS^{+}:=\ip{\rouq_w}_{w\neq 1}$ and $\TS^{-}:=\ip{\rouq_w\inv}_{w\neq 1}$ be the full triangulated subcategories of $\KC^b(\SBim)$ generated by $\rouq_w$ (resp.~$\rouq_w\inv$) and their shifts, where $w$ ranges over all non-identity elements of $S_n$.
\end{definition}
\begin{theorem}
\label{th: semiortho 1}
The kernels of $\HH^0$ and $\HH_0$, regarded as endofunctors $\KC^b(\SBim_n)\rightarrow \KC^b(\SBim_n)$ are $\TS^{-}$ and $\TS^+$ respectively. That is to say, we have two semi-orthogonal decompositions
\begin{eqnarray*}
\KC^b(\SBim_n) & \simeq & (\TS^-\rightarrow \ip{\one})\\
& \simeq & (\ip{\one}\rightarrow \TS^+) .
\end{eqnarray*}
in which the projections onto $\ip{\one}$ are $\HH^0$ and $\HH_0$, respectively.
\end{theorem}
\begin{proof}
The triangulated category $\KC^b(\SBim_n)$ is spanned by $\{\rouq_w\inv\}_{w\in S_n}$. Now, since $\HH^0$ annihilates $\rouq_w\inv$ for $w\neq 1$ (Lemma \ref{lem: negative and pos tw}) and fixes $\rouq_1=\one$, it follows from Lemma \ref{lemma:kernels and images} that $\ip{\one}$ and $\TS^-$ are the image and kernel of $\HH^0$, respectively.
A similar argument shows that $\ip{\one}$ and $\TS^+$ are the image and kernel of $\HH_0$. This yields the semi-orthogonal decompositions in the statement and completes the proof.
\end{proof}
\begin{remark}
Theorem \ref{th: semiortho 1} can be regarded as a categorification of Lemma \ref{lem: trace as coefficient at 1}.
Indeed, $\HH^0$ is a counital idempotent functor whose image is $\ip{\one}$ and whose kernel is $\ip{\rouq_w\inv}_{w\neq 1}$. This is a categorification of the fact that the coefficient of $a^0$ in the Jones-Ocneanu trace picks out the coefficient of 1 in the $\{\rouq_w\inv\}$-expansion of $x\in \mathbb{H}_n$.
On the other hand, $\HH_0$ is a unital dempotent endofunctor with image $\ip{\one}$ and kernel $\ip{\rouq_w}_{w\neq 1}$.
Since $\HH_0(X)\cong \HH^n(X)(-2n)$, this yields a categorification of the fact that the coefficient of $a^n$ in Jones-Ocneanu trace picks out the coefficient of 1 in the $\{\rouq_w\}$-expansion of $x\in \mathbb{H}_n$.
\end{remark}
\subsection{Partial trace functors}
\label{subsec:partial traces}
We introduce some intermediate categories between $\SBim_n$ and $\CS_\one$. For each parabolic subgroup $P:=S_{m_1}\times\cdots\times S_{m_r}\subset S_n$, let $\SBim_{m_1,\ldots,m_r} = \SBim(P)$ denote the full subcategory of $\SBim_n$ spanned by the bimodules $B_w$ with $w\in P$ and their shifts. Note that $\SBim_{1,\ldots,1} =\CS_{\one}$.
We regard $\SBim_{m_1,\ldots,m_r}$ as a full monoidal subcategory of $\SBim_n$, and so $\SBim_n$ can be thought of as a \emph{bimodule category} over $\SBim_{m_1,\ldots,m_r}$.
\begin{definition}\label{def:ptr}
For $M\in \SBim_n$ define $\pi^-(M)$ and $\pi^+(M)$ to be the kernel and cokernel of $x_n\otimes 1 - 1\otimes x_n$ acting on $M$, respectively.
\end{definition}
The following is obvious.
\begin{lemma}\label{lemma:tr is bilinear}
The functors $\pi^\pm$ are $\SBim_{n-1,1}$-bilinear, in the sense that
\[
\pi^{\pm}(X\otimes Y\otimes Z) \cong X\otimes \pi^{\pm}(Y)\otimes Z
\]
for all $X,Z\in \SBim_{n-1,1}$ and all $Y\in \SBim_n$; this isomorphism is natural in $X,Y,Z$.\qed
\end{lemma}
\begin{lemma}\label{lemma:image of ptr}
If $M\in \SBim_n$, then $\pi^{\pm}(M)$ is an object of $\SBim_{n-1,1}$, so $\pi^{\pm}$ are well-defined functors $\SBim_n\rightarrow \SBim_{n-1,1}$.
\end{lemma}
\begin{proof}
An easy exercise shows that $\pi^{\pm}(B_{n-1})\cong R(\pm 1)$ and $\pi^{\pm}(R)=R$, hence
\begin{equation}
\label{eq:partial trace of B 1}
\pi^\pm(X) = X
\end{equation}
and
\begin{equation}\label{eq:partial trace of B 2}
\pi^{\pm} (X\otimes B_{n-1} \otimes Y)\cong X\otimes Y(\pm 1)
\end{equation}
for all $X,Y\in \SBim_{n-1,1}$. Every indecomposable Soergel bimodule appears as a direct summand of some Bott-Samelson bimodule in which $B_{n-1}$ appears at most once, so it follows that $\pi^\pm(Z)\in \SBim_{n-1,1}$ for all $Z\in \SBim_n$.
\end{proof}
Since $\pi^+(M)$ is a sub-bimodule of $M$, the inclusion $\pi^+(M)\hookrightarrow M$ defines a natural transformation $\e:\pi^+R\rightarrow \Id$. Similarly, the projection $M\twoheadrightarrow \pi^-(M)$ defines a natural transformation $\eta:\Id\rightarrow \pi^-$.
\begin{proposition}\label{prop:partial trace as adjoints}
The functors $\pi^+$ and $\pi^-$ are the left and right adjoints to the inclusion $\SBim_{n-1,1}\hookrightarrow \SBim_n$.
\end{proposition}
\begin{proof}
It is clear that $\pi^+$ and $\pi^-$ are strict idempotent functors: $(\pi^{\pm})^2=\pi^\pm$. In fact the inclusion $\pi^-(M)\rightarrow M$ makes $\pi^-$ into a localization functor and the projection $M\rightarrow \pi^+(M)$ makes $\pi^+$ into a colocalization functor.
Lemma \ref{lemma:idempotents yield adjunctions} then tells us that $\pi^-$ is right adjoint to the inclusion of its essential image $\SBim_{n-1,1}$, and similarly $\pi^+$ is left adjoint to the inclusion of its essential image, which again is $\SBim_{n-1,1}$.
\end{proof}
Since $\pi^+,\pi^-$ are additive functors $\SBim_n\rightarrow \SBim_{n-1,1}$ they can be extended to complexes. Next we record how these functors interact with Rouquier complexes.
\begin{lemma}\label{lemma:ptrRouquier}
We have the following ``Markov moves'' for $\pi^\pm$:
\begin{subequations}
\begin{equation}\label{eq:ptrMarkovZero}
\pi^+(X\otimes \rouq_{n-1} \otimes Y)\ \simeq \ 0 \ \simeq \ \pi^-(X\otimes \rouq_{n-1}\inv \otimes Y)
\end{equation}
\begin{equation}\label{eq:ptrMarkovNonzero1}
\pi^+(X\otimes \rouq_{n-1}\inv \otimes Y) \cong \Cone\left(X\otimes Y(-1)\buildrel f\over\rightarrow X\otimes Y(1)\right),
\end{equation}
\begin{equation}\label{eq:ptrMarkovNonzero2}
\pi^-(X\otimes \rouq_{n-1} \otimes Y) \cong \Cone\left(X\otimes Y(-1)\buildrel f\over\rightarrow X\otimes Y(1)\right)[1]
\end{equation}
\end{subequations}
for every $X,Y\in \KC^b(\SBim_{n-1,1})$, where $f$ is ``middle multiplication'' by $x_{n-1}-x_n$.\qed
\end{lemma}
\begin{remark}
If we forget the action of $x_n$, then we obtain the more recognizable Markov moves $\pi^+(X\otimes \rouq_{n-1}\inv \otimes Y)\simeq X\otimes Y(-1)[1]$ and $\pi^-(X\otimes \rouq_{n-1}\otimes Y)\simeq X\otimes Y(1)[-1]$.
\end{remark}
\begin{remark}
We refer to $\pTr^\pm$ as the \emph{partial trace functors}. They can be regarded as functors $\SBim_{m,1,\ldots,1}\rightarrow \SBim_{m-1,1,1,\ldots,1}$ for all $1\leq m\leq n$. Composing them yields
\[
\HH^0 = (\pi^-)^n : \SBim_n\rightarrow \SBim_{1,\ldots,1} = \CS_{\one}
\]
and
\[
\HH_0 = (\pi^+)^n : \SBim_n\rightarrow \SBim_{1,\ldots,1} = \CS_{\one}
\]
\end{remark}
\begin{remark}
Note that the inclusion $\SBim_{n-1}\rightarrow \SBim_{n-1,1}$ is not full: hom spaces in $\SBim_{n-1,1}$ are obtained from those in $\SBim_{n-1}$ by applying $-\otimes_\k \k[x_n]$. On the other hand, \emph{there is} a fully faithful functor on the level of homotopy categories $\sigma:\KC^b(\SBim_{n-1})\rightarrow \KC^b(\SBim_{n-1,1})$, sending
\[
X\mapsto \Cone\left( (X\sqcup \one_1) \buildrel x_n\over \longrightarrow (X\sqcup \one_1)\right).
\]
The Grothendieck groups of $\SBim_{n-1}$ and $\SBim_{n-1,1}$ are both naturally identified with $\mathbb{H}_{n-1}$, and $\sigma$ categorifies multiplication by $(1-q)$.
There is a forgetful functor $F:\SBim_{n-1,1}\rightarrow R_{n-1}\text{-bimod}$, where $R_{n-1}:=\k[x_1,\ldots,x_{n-1}]$. Note that $F(B_w\sqcup \one_1)=B_w[x_n]$ for all $B_w\in \SBim_{n-1}$, hence $F$ could be regarded (loosely speaking) as a categorification of multiplication by $\frac{1}{1-q}$. In any case $F(\sigma(X))\simeq X$ for all $X\in \KC^b(\SBim_{n-1})$.
This explains how the apparently mysterious factor of $(1-q)$ in \eqref{eq:twoisos} is built-in to the categorical picture.
\end{remark}
\subsection{Relative semi-orthogonal decompositions}
\label{subsec:relative semiortho}
In the previous section we constructed idempotent endofunctors $\pi^{\pm}$ if $\SBim_n$ which project onto $\SBim_{n-1,1}$. Now we would like to understand the semi-orthogonal decompositions they determine. That is to say, we want to understand the kernels of these functors after extending to complexes $\pi^\pm:\KC^b(\SBim_n)\rightarrow \KC^b(\SBim_n)$. To study the kernels, we define the complementary idempotents by the usual formulas (Definition \ref{def:complement}).
\[
Q^-(X):=\Cone(\pi^-(X)\rightarrow X),\qquad\qquad Q^+(X):=\Cone(X\rightarrow \pi^+(X))[-1].
\]
\begin{definition}\label{def:relativeTpm}
Let $\US^\pm\subset \KC^b(\SBim_n)$ denote the full triangulated subcategory spanned by the Rouquier complexes $F_w^\pm$ with $w\in S_n \setminus (S_{n-1}\times S_1)$.
Equivalently, $\US^\pm$ is the span of complexes of the form $X\otimes \rouq_{n-1}^\pm\otimes Y$ with $X,Y\in \KC^b(\SBim_{n-1,1})$.
\end{definition}
\begin{theorem}\label{thm:relative semiortho}
The kernel of the idempotent endofunctor $\pi^{\pm}:\KC^b(\SBim_n)\rightarrow \KC^b(\SBim_n)$ is $\US^{\pm}$, so that we have semi-orthogonal decompositions
\begin{eqnarray*}
\KC^b(\SBim_n)
& \simeq & \left(\KC^b(\SBim_{n-1,1})\rightarrow \US^+\right)\\
& \simeq & \left(\US^-\rightarrow \KC^b(\SBim_{n-1,1}) \right).
\end{eqnarray*}
\end{theorem}
With respect to these semi-orthogonal decompositions $\pi^\pm$ can be described as the projection onto $\KC^b(\SBim_{n-1,1})$ with kernel $\US^{\pm}$.
\begin{proof}
The proof is similar to the proof of Theorem \ref{th: semiortho 1}. If $w\in S_n\setminus (S_{n-1}\times S_1)$, then $w$ can be presented as
\[
w = w's_{n-1} w''
\]
with $w',w''\in S_{n-1}\times S_1$. For such $w$ we have
\[
\pi^+(F_w) \cong F_{w'}\otimes \pi^+(F_{n-1})\otimes F_{w''}\simeq 0
\]
and
\[
\pi^-(F_w\inv) \cong F_{w'}\inv\otimes \pi^+(F_{n-1}\inv)\otimes F_{w''}\inv\simeq 0.
\]
On the other hand, if $w\in S_{n-1}\times S_1$, then we have $F_w\in \KC^b(\SBim_{n-1,1})$, and so $\pi^\pm(F_w)\cong F_w$. Thus, Lemma \ref{lemma:kernels and images} tells us that the image and kernel of $\pi^{\pm}:\KC^b(\SBim_n)\rightarrow \KC^b(\SBim_n)$ are $\KC^b(\SBim_{n-1,1})$ and $\US^\pm$, respectively. This gives the desired semi-orthogonal decompositions and completes the proof.
\end{proof}
\section{Serre duality}
\label{sec:serre}
Let $\HT_n =\rouq_{w_0}$ denote the Rouquier complex associated to the half twist, and let $\FT_n:=\HT_n^{\otimes 2}$ denote the full twist. When the index $n$ is understood, it will be omitted.
The purpose of this section is to prove the following.
\begin{theorem}
\label{thm:left right}
We have $\HH^0(\FT\inv\otimes X)\simeq \HH_0(X)$ and $\HH^0(X)\simeq \HH_0(\FT\otimes X)$ as complexes of $R$-modules, for all $X\in \KC^-(\SBim_n)$.
\end{theorem}
We will prove this theorem as a corollary of a certain ``relative version.''
\subsection{Jucys-Murphy braids and the splitting map}
\label{subsec:splitting map}
Define braids $\CL_n\in \Br_n$ inductively by $\CL_1=\one_1$ and
\[
\CL_n = \sigma_{n-1}(\CL_{n-1}\sqcup \one_1)\sigma_{n-1},\qquad \qquad n\geq 2.
\]
We will denote the braid $\CL_n$ and the Rouquier complex $F(\CL_n)$ by the same notation. Note that
\[
\FT_n = \CL_2\otimes \CL_3\otimes\cdots \otimes \CL_n.
\]
For any $i$ we define the chain map $\psi_i:F_i\to F^{-1}_i$ by the following diagram:
\[
\begin{tikzcd}
F_i \arrow{d}{\psi_i} & = & {[\underline{B_i}} \arrow{r} \arrow{dr} & {R(1)]}\\
F^{-1}_i & = & {[R (-1)} \arrow{r} & {B_i]}\\
\end{tikzcd}
\]
Clearly, $\Cone[\psi_i]=[R(1)\to R(-1)]$.
By combining these maps we get a ``splitting map'' $\Psi:\CL_n\rightarrow \one_n$. This map was studied in \cite[Section 4]{GH}
in a more complicated ``$y$-ified'' version.
\begin{lemma}\label{lemma:maps from Ln}
The space of chain maps $\Homc(\CL_n,\one_n)$ is homotopy equivalent to $R$, generated by the ``splitting map'' $\Psi:\CL_n\rightarrow \one_n$. This splitting map becomes a homotopy equivalence after applying $\pi^+$.
\end{lemma}
\begin{proof}
The first statement follows from the second. Indeed, if $\Psi$ becomes a homotopy equivalence after applying $\pi^+$ then we have
\[
\Homc(\CL_n,\one)\cong \Homc(\pi^+(\CL_n),\one) \simeq \Homc(\one,\one) = R,
\]
generated by $\Psi$, as claimed. The first isomorphism above holds since $\pi^+$ is the left adjoint to the inclusion $\SBim_{n-1,1}\rightarrow \SBim_{n}$ (Proposition \ref{prop:partial trace as adjoints}).
To prove the second statement it suffices to show that the cone of the splitting map $\Psi:\CL_n\rightarrow \one_n$ is mapped to a contractible complex by $\pi^+$. However, $\Cone(\Psi)$ is in the triangulated hull of the Rouquier complexes
\[
F(\sigma_{n-1}\cdots\sigma_{k+1}\sigma_{k}\sigma_{k+1}\cdots \sigma_{n-1}) \ \ \simeq \ \ F(\sigma_k\cdots\sigma_{n-2}\sigma_{n-1}\sigma_{n-2}\cdots\sigma_k),\qquad 1\leq k\leq n-1,
\]
Each of these is annihilated by $\pi^+$ by Lemma \ref{lemma:ptrRouquier}, and the lemma follows.
\end{proof}
\subsection{Relative Serre duality}
\label{subsec:relative serre}
\begin{lemma}\label{lemma:Ln action}
Tensoring on the left (or right) with $\CL_n$ restricts to an equivalence of categories $\US^-\rightarrow \US^+$ with inverse given by tensoring on the left (or right) with $\CL_n\inv$, where $\US^\pm$ is as in Definition \ref{def:relativeTpm}.
\end{lemma}
\begin{proof}
It is clear that tensoring on the left with $\CL_n$ restricts to a functor $\US^-\rightarrow \US^+$, since $\CL_n \otimes F_{n-1}\inv \simeq F_{n-1}\otimes \CL_{n-1}$ and $\CL_n$ tensor commutes with all Rouquier complexes $F_w$ with $w\in S_{n-1}\times S_1$. Similarly, tensoring on the left with $\CL_n\inv$ restricts to a functor $\US^+\rightarrow \US^-$. These functors are inverse equivalences. A similar argument takes care of tensoring on the right with $\CL_n^\pm$.
\end{proof}
\begin{theorem}\label{thm:relativeSerre}
We have $\pi^-(X)\simeq \pi^+(\CL_n\otimes X)\simeq \pi^+(X\otimes \CL_n)$ for all $X\in \KC^b(\SBim_n)$. These homotopy equivalences are natural isomorphisms of functors $\KC^b(\SBim_n)\rightarrow \KC^b(\SBim_{n-1,1})$.
\end{theorem}
\begin{proof}
We only consider the equivalence $\pi^-(X)\simeq \pi^+(\CL_n\otimes X)$; the equivalence $\pi^-(X)\simeq \pi^+(X\otimes \CL_n)$ is proven similarly. Let $X\in \KC^b(\SBim_n)$ be given. We may as well assume that $X$ is expressed as
\[
X\simeq (\QB^-(X) \buildrel[1]\over\rightarrow \pi^-(X))
\]
where $\QB^-(X)\in \US^-$. Here, the label $[1]$ above an arrow indicates a chain map of degree 1 so for instance $B\simeq (C \buildrel[1]\over\rightarrow A)$ means that $B\simeq \Cone(C[-1]\rightarrow A)$ or, equivalently $B$ fits into a distinguished triangle of the form $A\rightarrow B\rightarrow C\rightarrow A[1]$. Tensoring with $\CL_n$ yields
\[
\CL_n\otimes X \ \simeq \ ( \CL_n\otimes \QB^-(X) \buildrel[1]\over\rightarrow \CL_n\otimes \pi^-(X) ).
\]
Since $\CL_n\otimes \QB^-(X)\in \US^+$, it follows that $\pi^+(\CL_n\otimes \QB^-(X))\simeq 0$, hence
\[
\pi^+(\CL_n\otimes X)\simeq \pi^+(\CL_n\otimes \pi^-(X))\simeq \pi^-(X).
\]
In this last equivalence we used Lemma \ref{lemma:maps from Ln}.
Now we consider the naturality of this homotopy equivalence. Let $X\rightarrow Y$ be a chain map. Then in terms of the decompositions $X\simeq (\pi^-(X)\rightarrow \QB^-(X))$ and $Y\simeq (\pi^-(Y)\rightarrow \QB^-(Y))$, the chain map $f$ can be written as
\[
\begin{tikzpicture}[baseline=-2.8em]
\tikzstyle{every node}=[font=\small]
\node (a) at (-1,0) {$X$};
\node at (0,0) {$\simeq$};
\node at (1.2,0) {$\Big($};
\node at (5.7,0) {$\Big)$};
\node (b) at (2,0) {$\QB^-(X)$};
\node (c) at (5,0) {$\pi^-(X)$};
\node (d) at (-1,-2.5) {$Y$};
\node at (0,-2.5) {$\simeq$};
\node at (1.2,-2.5) {$\Big($};
\node at (5.7,-2.5) {$\Big)$};
\node (e) at (2,-2.5) {$\QB^-(Y)$};
\node (f) at (5,-2.5) {$\pi^-(Y)$};
\path[->,>=stealth',shorten >=1pt,auto,node distance=1.8cm,
thick]
(b) edge node {$[1]$} (c)
(e) edge node {$[1]$} (f)
(a) edge node {$f$} (d)
(b) edge node {} (e)
(b) edge node {} (f)
(c) edge node {$\pi^-(f)$} (f);
\end{tikzpicture}.
\]
Applying $\pi^+(\CL_n\otimes -)$ and contracting the contractible terms $\pi^+(\CL_n\otimes \QB^-(X))$ and $\pi^+(\CL_n\otimes \QB^-(Y))$, we obtain a diagram which commutes up to homotopy:
\[
\begin{tikzpicture}[baseline=-2.8em]
\tikzstyle{every node}=[font=\small]
\node (a) at (-3,0) {$\pi^+(\CL_n\otimes X)$};
\node at (-.7,0) {$\simeq$};
\node (b) at (2,0) {$\pi^+(\CL_n\otimes \pi^-(X))$};
\node (c) at (7,0) {$\pi^-(X)$};
\node (d) at (-3,-2.5) {$\pi^+(\CL_n\otimes Y)$};
\node at (-.7,-2.5) {$\simeq$};
\node (e) at (2,-2.5) {$\pi^+(\CL_n\otimes \pi^-(Y))$};
\node at (4.5,0) {$\simeq$};
\node at (4.5,-2.5) {$\simeq$};
\node (f) at (7,-2.5) {$\pi^-(Y)$};
\path[->,>=stealth',shorten >=1pt,auto,node distance=1.8cm,
thick]
(a) edge node {$\pi^+(\CL_n\otimes f)$} (d)
(b) edge node {$\pi^+(\CL_n\otimes \pi^-(f))$} (e)
(c) edge node {$\pi^-(f)$} (f);
\end{tikzpicture}.
\]
The second square commutes up to homotopy because it is induced by the splitting map $\CL_n\rightarrow \one$ (together with the observation that $\pi^+(\pi^-(X))=\pi^-(X)$ naturally). This proves the statement about naturality.
\end{proof}
\subsection{Top versus bottom}
\label{subsec:top v bottom}
For each $1\leq r\leq n$, let $\US_r^\pm\subset \KC^b(\SBim_n)$ denote the full triangulated subcategories spanned by $X \otimes \rouq_{r-1}^\pm \otimes Y$ for all $X,Y\in \KC^b(\SBim(S_r\times S_1^{n-r})$.
Iterating the semi-orthogonal decompositions from \S \ref{subsec:relative semiortho} yields more sophisticated semi-orthogonal decompositions of $\KC^b(\SBim_n)$ of the form:
\[
\KC^b(\SBim_n) \ \simeq \ \left(\SBim(S_r\times S_1^{n-r}) \rightarrow \US_r^+\rightarrow \US_{r+1}^+\rightarrow\cdots \rightarrow \US_n^+\right),
\]
\[
\KC^b(\SBim_n) \ \simeq \ \left(\US_n^-\rightarrow \cdots\rightarrow \US_{r+1}^-\rightarrow \US_r^-\rightarrow \SBim(S_r\times S_1^{n-r})\right),
\]
or more generally
\[
\KC^b(\SBim_n) \ \simeq \ \left(\US_{l_b}^-\rightarrow \cdots\rightarrow \US_{l_1}^-\rightarrow \SBim(S_r\times S_1^{n-r}) \rightarrow \US_{k_1}^+\rightarrow \cdots \rightarrow \US_{k_a}^+\right),
\]
for any decomposition $\{r+1,\ldots,n\} = \{k_1<\cdots < k_a\}\sqcup \{l_1<\cdots<l_b\}$.
The case $r=1$ yields the semi-orthogonal decompositions considered in \S \ref{subsec:absolute semiortho decomp}:
\[
\KC^b(\SBim_n) \ \simeq \ \left(\SBim(S_1^n)\rightarrow \TS^+\right) \ \simeq \left(\TS^-\rightarrow \SBim(S_1^n)\right),
\]
where $\TS^\pm\subset \KC^b(\SBim_n)$ is the full triangulated category spanned by the Rouquier complexes $\rouq_w^{\pm 1}$ with $w\neq 1$.
\begin{proof}[Proof of Theorem \ref{thm:left right}]
We must show that
\[
\HH^0(X)\simeq \HH_0(\FT_n\otimes X)
\]
for all $X\in \KC^b(\SBim_n)$, and that these homotopy equivalences yield a natural isomorphism of functors $\KC^b(\SBim_n)\rightarrow \KC^b(R-\text{gmod})$.
We will prove this by induction on $n$. The base case $n=1$ is trivial. Note that $\FT_n = \CL_n\otimes \FT_{n-1}$, $\HH^0(X) =(\pi^-)^n(X)$, and $\HH_0(X) =(\pi^+)^n(X)$. Thus,
\begin{eqnarray*}
\HH^0(X)
&=& (\pi^-)^n(X) \\
&\simeq & \HH_0(\FT_{n-1} \otimes\pi^-(X))\\
& \simeq & \HH_0(\FT_{n-1}\otimes \pi^+(\CL_n \otimes X))\\
& \cong & \HH_0(\pi^+(\FT_{n-1}\otimes\CL_n \otimes X))\\
& =& \HH_0(\FT_n\otimes X).
\end{eqnarray*}
In the second line we used the induction hypothesis, in the third line we used Theorem \ref{thm:relativeSerre}, in the fourth we used Lemma \ref{lemma:tr is bilinear}, and the last line is clear.
\end{proof}
\begin{remark}
In light of the isomorphism $\HH_0(Y)\cong \HH^n(Y)(-2n)$, we prefer to view the result of Theorem \ref{thm:left right} as saying
\[
\HH^n(X)(-2n)\simeq \HH^0(\FT\inv\otimes X).
\]
\end{remark}
\begin{theorem}
\label{thm:serre}
The Rouquier complex for the full twist braid is a Serre functor for $\KC^b(\SBim_n)$. In other words,
for all $A,B\in \KC^b(\SBim_n)$ we have
$$
\Homc(A,B)=\Homc(B\otimes \FT,A)^{\vee},
$$
where the dual on the right hand side is defined using the left $R$-action on the hom complex.
\end{theorem}
\begin{proof}
Recall that $\SBim_n$ has duals, hence $\Homc(A,B)\cong\Homc(A\otimes B^\vee,\one)$ as complexes of left $R$-modules.
By Theorem \ref{thm:left right}, we have
$$
\Homc(A,B)\cong \Homc(A\otimes B^\vee,\one) \cong \HH_0(A\otimes B^\vee)^\vee
$$
as complexes of left $R$-modules (the right action of $R$ on $\Hom(A,B)$ corresponds to ``middle multiplication'' on $A\otimes B^\vee$. By Theorem \ref{thm:left right}, this latter complex is homotopy equivalent to
\[
\HH^0(A\otimes B^\vee\otimes \FT\inv)^\vee =\Homc(\one,A\otimes B^\vee\otimes \FT\inv)^\vee\cong \Homc(\FT\otimes B, A)^\vee
\]
where in the last complex we use the left $R$-action on $ \Homc(\FT\otimes B, A)$ when forming the dual $R$-module $(-)^\vee$. These are homotopy equivalences of complexes of left $R$-modules. \end{proof}
\subsection{Soergel modules}
\label{sec:smod}
In this section we review the Serre duality for the category of {\em Soergel modules} $\SMod$, which is closely related to the Bernstein-Gelfand-Gelfand category $\mathcal{O}$ for the Lie algebra $\mathfrak{gl}_n$. Soergel modules are obtained from Soergel bimodules as quotients by the right $R$-action. Given $M\in \SBim_n$, we define
$$
\overline{M}=M\otimes_{R} R/(x_1,\ldots,x_n)=M\otimes_{R}\k.
$$
For example, $\overline{R}=\k$. Note that the $R\otimes_\k R$ action on the Soergel bimodule $M$ factors through the quotient $R\otimes_{R^{S_n}} R$, so the residual $R\otimes_\k \k$-action on $\overline{M}$ factors through $R\otimes_{R^{S_n}}\k$. This latter ring is called the \emph{coinvariant ring}, denoted $C$; it is the quotient of $R$ by the ideal generated by positive degree symmetric functions in the $x_i$. By definition, a morphism in $\SMod$ is a homogeneous $C$-linear map.
\begin{remark}
Let us explain the connection between $\SMod$ and the BGG categories $\OC$. Let $\OC_0=\OC_0(\mathfrak{gl}_n)$ denote the principal block of the category $\OC$ for $\gl_n$ (i.e.~the block containing the trivial 1-dimensional representation). This category has a special projective module $P=P_{w_0}$, the \emph{anti-dominant projective}, whose endomorphism ring is isomorphic to the coinvariant ring $C$. Soergel \cite{Soergel} proved that the functor $\OC_0\rightarrow C\text{-mod}$ sending $M\mapsto \Hom_{\OC_0}(P,M)$ is fully faithful on projectives, hence identifies $D^b(\OC_0)$ ($\simeq$ the homotopy category of projectives) with a full subcategory of $\KC^b(C\text{-mod})$. This full subcategory is precisely $\KC^b(\SMod)$. An important consequence (and the original motivation) for such a description is that it yields a $\Z$-graded lift of $\OC_0$.
\end{remark}
Soergel modules do not form a monoidal category, but they form a module category over $\SBim_n$: given
$A,B\in \SBim_n$, we have
\begin{equation}
\label{eq: bimodule act on module}
\overline{AB}=A\otimes_{R} \overline{B}.
\end{equation}
The functor $\overline{\cdot}$ can be extended to complexes, and defines a functor $\KC^b(\SBim_n)\to \KC^b(\SMod_n)$.
Equation \eqref{eq: bimodule act on module} holds for complexes as well. As a consequence, left tensor multiplication with Rouquier complexes defines a braid group action on $\KC^b(\SMod_n)$.
\begin{lemma}
For $A,B\in \KC^b(\SBim_n)$ one has
$$
\Hom(\overline{A},\overline{B})=\Hom(A,B)\otimes_{R} \k,
$$
where we consider $\Hom(A,B)$ as a right $R$-module.
\end{lemma}
\begin{proof}
It is sufficient to prove the lemma for $A,B\in \SBim_n$. Then $A$ and $B$ are free as right $R$-modules, and
$\Hom(A,B)$ is free as a right $R$-module by Lemma \ref{lemma:free homs}, so the result follows.
\end{proof}
\begin{theorem}
\label{thm: serre for smod}
For all $A,B\in \KC^b(\SMod)$ one has
$$
\Homc(A,B)=\Homc(B,\FT^{-1}\otimes A)^{*},
$$
where in the right hand side we take a linear dual over $\k$.
\end{theorem}
Again, here we regard $\FT^{-1}$ as a complex of Soergel bimodules acting (on the left) on the complex of Soergel modules $A$.
\begin{proof}
We can assume $A=\overline{M}$ and $B=\overline{N}$ for $M,N\in \SBim_n$. Then
$$
\Homc(M,N)=\Homc(N,\FT^{-1}\otimes M)^{\vee},
$$
where we regard both sides as complexes of (right) $R$-modules. Now.
$$
\Homc(A,B)=\Homc(\overline{M},\overline{N})=\Homc(M,N)\otimes_{R}\k,
$$
$$
\Homc(B,\FT^{-1}\otimes A)=\Homc(\overline{N},\FT^{-1}\otimes \overline{M})=\Homc(\overline{N},\overline{\FT^{-1}\otimes M})=\Homc(N,\FT^{-1}\otimes M)\otimes_{R}\k,
$$
and
$$
\Homc(B,\FT^{-1}\otimes A)^*=\Homc(N,\FT^{-1}\otimes M)^{\vee}\otimes_{R}\k.
$$
\end{proof}
Theorem \ref{thm: serre for smod} was proved earlier in \cite{Bez,MS} by different methods and using the relation between $\SMod$ and the category
$\mathcal{O}$.
\begin{appendix}
\section{Semiorthogonal decompositions for Coxeter groups}
\label{app:anytype}
Let $W$ be a Coxeter group with the set of reflections $S$. Let $\frakh$ be a realization of $W$.
Define $R=\k[\frakh]$,
and $B_s=R\otimes_{R^s}R$ for $s\in S$. The category $\SBim_W$ of Soergel bimodules is the smallest full subcategory of
the category of $R-R$ bimodules containing $R$ and all $B_s$ and closed under direct sums, direct summands, tensor products and
grading shifts. For $W=S_n$ and $\frakh=\k^n$ we recover the category $\SBim_n$ defined in Section \ref{subsec:SBim}.
Rouquier complexes can be defined in the homotopy category $\KC^b(\SBim_W)$ similarly to Section \ref{subsec:rouquier}:
$$
\rouq_s=[B_s\to R],\ \rouq_s^{-1}=[R\to B_s]
$$
In \cite{Rouquier} Rouquier proved that they satisfy the relations in the braid group associated to $W$. Therefore for any $w\in W$ one can consider a Rouquier complex $\rouq_w$ corresponding to the positive permutation braid associated to any reduced expression of $w$. It does not depend on the choice of a reduced expression up to homotopy equivalence.
We define triangulated subcategories $\US_{<w}$ and $\US_{\le w}$ of $\KC^b(\SBim_W)$ generated by the Rouquier complexes $\rouq_v$ with $v<w$
(and $v\le w$) in Bruhat order.
For any $s\in S$, there exists a chain map $\psi_s:\rouq_s\to \rouq_s^{-1}$ such that
\begin{equation}
\label{eq:general skein}
\Cone[\rouq_s\to \rouq_s^{-1}]=[R\to R].
\end{equation}
As an immediate corollary, we get the following:
\begin{proposition}
\label{prop:positive to negative}
For any $w\in W$ there is a chain map $\psi_w: \rouq_w\to \rouq_{w^{-1}}^{-1}$.
The cone of $\psi_w$ is filtered by $\rouq_{u^{-1}}^{-1}$ for $u<w$ in Bruhat order.
\end{proposition}
In \cite{LW} Libedinsky and Williamson proved a much stronger statement (conjectured by Rouquier in \cite{Rouquier2}, p. 215 before Remark 4.12).
\begin{theorem}[\cite{LW}] \label{th:LW}
\label{th:standard costandard}
If $w\neq v$ then $\Hom(\rouq_v,\rouq^{-1}_{w^{-1}})=0$.
If $w=v$ then $\Hom(\rouq_w,\rouq^{-1}_{w^{-1}})=R$ is generated by the map $\psi_w$.
\end{theorem}
\begin{corollary} \label{cor:LW}
\label{cor:Hom vanishing in all types}
We have $\Hom(\rouq_w,R)=0$ for $w\neq 1$.
\end{corollary}
In type A this corollary is an easy consequence of Corollary \ref{cor:negative stabilization}. In fact, Theorem \ref{th:LW} also can be deduced:
\begin{prop}
Theorem \ref{th:LW} follows from Corollary \ref{cor:LW}.
\end{prop}
\begin{proof}
Assume that $\Hom(F_w,R)=0$ for $w\neq 1$. Note that
\[
\Hom(\rouq_v,\rouq^{-1}_{w^{-1}}) \cong \Hom(\rouq_v \rouq_{w^{-1}}, R).
\]
We induct on the number $\min(l(v), l(w))$. The base case follows from the assumption. Without loss of generality, we may assume $l(v)\leq l(w)$. Let $v=v's$ for a simple reflection $s$ and $l(v')=l(v)-1$. If $l(ws)>l(w)$ then $w\neq v$ and $ws\neq v'$, so we have
\[
\Hom(\rouq_v \rouq_{w^{-1}}, R) = \Hom(\rouq_{v'} \rouq_{s w^{-1}}, R) = \Hom(\rouq_{v'} \rouq_{(ws)^{-1}}, R) \cong 0,
\]
where the last equality follows from the induction hypothesis. So we assume $l(ws)<l(w)$. Let $w=w' s$.
The map $\psi_s:F_s\to F_s^{-1}$ induces a map
\[
\Hom(\rouq_{v'}, \rouq_{{w'}^{-1}}^{-1}) = \Hom(\rouq_{v'} \rouq_s^{-1}, \rouq_{{w'}^{-1}}^{-1} \rouq_s^{-1}) \to \Hom(\rouq_{v'} \rouq_s, \rouq_{{w'}^{-1}}^{-1} \rouq_s^{-1}) = \Hom(\rouq_v, \rouq_{w^{-1}}^{-1}),
\]
whose cone is filtered by $\Hom(\rouq_{v'}, \rouq_{w^{-1}}^{-1})$, which vanishes by the induction
hypothesis since $l(v')<l(v)\leq l(w)$. So we are reduced to the statement for the pair $v', w'$.
\end{proof}
We use Theorem \ref{th:standard costandard} to deduce a very important fact about Rouquier complexes which does not appear to be explicitly stated in the literature.
\begin{theorem}
\label{th:triangular hom}
We have $\Hom(\rouq_w,\rouq_v)=0$ unless $w\leq v$ in Bruhat order.
\end{theorem}
\begin{proof}
By Proposition \ref{prop:positive to negative} $\rouq_v$ is homotopy equivalent to a complex filtered by $\rouq_{u^{-1}}^{-1}$
with $u\le v$. Therefore $\Hom(\rouq_w,\rouq_v)=0$ unless $\Hom(\rouq_w,\rouq_{u^{-1}}^{-1})\neq 0$ for some $u\le v$.
But by Theorem \ref{th:standard costandard} this is possible only if $u=w$, and hence $w\le v$.
\end{proof}
\begin{corollary}
\label{cor:adding w}
For all $w$ we have semiorthogonal decompositions $\US_{\le w}=\langle \US_{<w}, \rouq_w\rangle$ and
$\US_{\le w}=\langle \rouq_{w^{-1}}^{-1}, \US_{<w}\rangle$.
\end{corollary}
\begin{proof}
By Proposition \ref{prop:positive to negative} the category $\US_{\le w}$ is generated by $\US_{<w}$ and $\rouq_w$, or, equivalently,
by $\US_{<w}$ and $\rouq_{w^{-1}}^{-1}$ (since $\Cone[\rouq_w\to \rouq_{w^{-1}}^{-1}]\in \US_{<w}$). Now for all $u<w$ we have
$\Hom(\rouq_w,\rouq_u)=0$ by Theorem \ref{th:triangular hom} and $\Hom(\rouq_{u},\rouq_{w^{-1}}^{-1})=0$ by Theorem \ref{th:standard costandard}.
\end{proof}
\begin{corollary}
\label{cor: left right adjoints Bruhat}
If $W$ is a finite Coxeter group then for all $w\in W$ the inclusion $\US_{\leq w}\hookrightarrow \KC^b(\SBim_W)$
has both left and right adjoints.
\end{corollary}
\begin{proof}
Fix an arbitrary total order $\prec$ on W refining the Bruhat order, let $w_0$ be the longest element in $W$. Then for all $w$ we have a chain
$$
w=w^{(1)}\prec w^{(2)}\prec \ldots \prec w^{(k)}=w_0.
$$
Similarly to Corollary \ref{cor:adding w}, the inclusions $\US_{\leq w^{(i)}}\hookrightarrow \US_{\leq w^{(i+1)}}$ have both left and right adjoints, and by combining these we get adjoints to the inclusion
$$
\US_{\leq w^{(i)}}\hookrightarrow \US_{\leq w_0}=\KC^b(\SBim_W).
$$
\end{proof}
If $W'$ is a parabolic subgroup of $W$, we can consider the category of Soergel bimodules $\SBim_{W'}$ associated to the same realization $\frakh$.
\begin{corollary}
Let $W$ be a finite Coxeter group and $W'$ a parabolic subgroup. Then the inclusion $\KC^b(\SBim_{W'})\to \KC^b(\SBim_W)$
has both left and right adjoints.
\end{corollary}
\begin{proof}
We have $\KC^b(\SBim_{W'})=\US_{\leq w}$ where $w$ is the longest element of $W'$.
\end{proof}
Note that in type A this gives an alternative construction of adjoints to inclusions of $\SBim_{n,1,\ldots,1}$ in $\SBim_{m}$.
However, it seems that the direct construction of adjoints in Section \ref{sec:semiortho} is easier to work with than the induction on Bruhat graph as in Corollary \ref{cor: left right adjoints Bruhat}.
\end{appendix}
\bibliographystyle{alpha}
\bibliography{refs}
\end{document}
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\begin{document}
\maketitle
\begin{abstract}
We study local regularity properties of local minimizer of scalar integral functionals of the form
$$
\mathcal F[u]:=\int_\Omega F(\nabla u)-f u\,dx
$$
where the convex integrand $F$ satisfies controlled $(p,q)$-growth conditions. We establish Lipschitz continuity under sharp assumptions on the forcing term $f$ and improved assumptions on the growth conditions on $F$ with respect to the existing literature. Along the way, we establish an $L^\infty$-$L^2$-estimate for solutions of linear uniformly elliptic equations in divergence form which is optimal with respect to the ellipticity contrast of the coefficients.
\end{abstract}
\section{Introduction}
In this note, we revisit the question of Lipschitz-regularity for local minimizers of integral functionals of the form
\begin{equation}\label{eq:int}
w\mapsto \mathcal F(w;\Omega):=\int_\Omega F(\nabla w)-fw\,dx.
\end{equation}
We recall in the case $F(z)=\frac12|z|^2$ local minimizer of \eqref{eq:int} satisfy $-\Delta w=f$. A classic theorem due to Stein \cite{stein} implies
\begin{equation}\label{eq:Deltaln1}
-\Delta w=f\in L^{n,1}(\Omega)\qquad\Rightarrow\qquad \nabla w\in L^\infty_{\rm loc}(\Omega),
\end{equation}
where $L^{n,1}(\Omega)$ denotes the Lorentz space (see below for a definition). In view of \cite{Cianchi92} the condition $f\in L^{n,1}(\Omega)$ is optimal in the Lorentz-space scale for the conclusion in \eqref{eq:Deltaln1}. In the last decade the implication in \eqref{eq:Deltaln1} was greatly generalized by replacing the linear operator $\Delta$ by possibly degenerate/singular uniformly elliptic nonlinear operators, see \cite{B16,KM13,KM14,CianMaz11}. More recently, those results where extended to a wide range of non-uniformly elliptic variational problems by Beck and Mingione in \cite{BM20} (see also \cite{DM21,Filippis21} for related results for non-autonomous or non-convex vector valued problems).
In this paper, we are interested in a specific class of non-uniformly elliptic problems, namely functionals with so-called $(p,q)$-growth conditions which are described in the following
\begin{assumption}\label{ass}
Let $0<\nu\leq \Lambda<\infty$, $1<p\leq q<\infty$ and $\mu\in[0,1]$ be given. Suppose that $F:\R^n\to[0,\infty)$ is convex, locally $C^2$-regular in $\R^n\setminus\{0\}$ and satisfies
\begin{equation}\label{ass:Fpq}
\begin{cases}
\nu(\mu^2+|z|^2)^\frac{p}2\leq F(z)\leq \Lambda(\mu^2+|z|^2)^\frac{q}2+\Lambda (\mu^2+|z|^2)^\frac{p}2\\
|\partial^2 F(z)|\leq \Lambda(\mu^2+|z|^2)^\frac{q-2}2+\Lambda (\mu^2+|z|^2)^\frac{p-2}2,\\
\nu(\mu^2+|z|^2)^\frac{p-2}2|\xi|^2\leq \langle \partial^2 F(z)\xi,\xi\rangle,
\end{cases}
\end{equation}
for every choice of $z,\xi\in\R^n$ with $|z|>0$.
\end{assumption}
Regularity properties of local minimizers of \eqref{eq:int} in the case $p=q$ are classical, see, e.g.,\ \cite{Giu}. A systematic regularity theory in the case $p<q$ was initiated by Marcellini in \cite{Mar89,Mar91}, see \cite{MR21} for a recent overview.
Before we state our main result, we recall a standard notion of local minimality in the context of integral functionals with $(p,q)$-growth
\begin{definition}\label{def:localmin}
We call $u\in W_{\rm loc}^{1,1}(\Omega)$ a local minimizer of $\mathcal F$ given in \eqref{eq:int} with $f\in L_{\rm loc}^n(\Omega)$ if for every open set $\Omega'\Subset\Omega$ the following is true:
\begin{equation*}
\mathcal F(u,\Omega')<\infty
\end{equation*}
and
\begin{equation*}
\mathcal F(u,\Omega')\leq \mathcal F(u+\varphi,\Omega')
\end{equation*}
for any $\varphi\in W^{1,1}(\Omega)$ satisfying ${\rm supp}\;\varphi\Subset \Omega'$.
\end{definition}
The main result of the present paper is
\begin{theorem}\label{T:1}
Let $\Omega\subset\R^n$, $n\geq3$ be an open bounded domain and suppose Assumption~\ref{ass} is satisfied with $1<p<q<\infty$ such that
\begin{equation}\label{eq:pqrhs}
\frac{q}p<1+\min\biggl\{\frac{2}{n-1},\frac{4(p-1)}{p(n-3)}\biggr\}.
\end{equation}
Let $u\in W_{\rm loc}^{1,1}(\Omega)$ be a local minimizer of the functional $\mathcal F$ given in \eqref{eq:int} with $f\in L^{n,1}(\Omega)$. Then $\nabla u$ is locally bounded in $\Omega$. When $p\geq 2-\frac{4}{n+1}$ or when $f\equiv0$ condition \eqref{eq:pqrhs} can be replaced by
\begin{equation}\label{eq:pq}
\frac{q}p<1+\frac2{n-1}.
\end{equation}
\end{theorem}
\begin{remark}
Theorem~\ref{T:1} should be compared to the findings of the recent papers \cite{BM20} and \cite{BS19c}: In \cite{BM20}, Beck and Mingione proved (among many other things) the conclusion of Theorem~\ref{T:1} under the more restrictive relation
$$
\frac{q}p<1+\min\biggl\{\frac2n,\frac{4(p-1)}{p(n-2)}\biggr\}.
$$
Hence, we obtain here an improvement in the gap conditions on $\frac{q}p$. In \cite{BS19c}, we proved Theorem~\ref{T:1} in the specific case $f\equiv 0$, $p\geq2$ and $\mu=1$.
\end{remark}
The proof of Theorem~\ref{T:1} closely follows the strategy presented in \cite{BM20} and relies on careful estimates for certain \textit{uniformly elliptic} problems. To illustrate this, let us consider the case that $F$ satisfies Assumption~\ref{ass} with $p=2<q$ and $f\equiv0$: Let $u$ be a local minimizer of $\mathcal F(\cdot,B_1)$ and assume that $u$ is smooth. Standard arguments yield
\begin{equation}\label{intro:lineq}
\divv (A(x)\nabla \partial_i u)=0\qquad\mbox{where}\qquad A:=\partial^2F(\nabla u).
\end{equation}
Hence, $\partial_i u$ satisfies a linear elliptic equation where the ellipticity ratio $\mathcal R(x)$ of the coefficients, that is the quotient of the highest and lowest eigenvalue of $A(x)$, is determined by the size of $|\nabla u(x)|$. More precisely, \eqref{ass:Fpq} with $p=2<q$ implies $\mathcal R(x)\sim (1+|\nabla u(x)|^{q-2})$. By standard theory for uniformly elliptic equations applied to \eqref{intro:lineq}, there exists an exponent $m=m(n)>0$ such that
\begin{equation}\label{intro:linftyl2}
\|\partial_i u\|_{L^\infty(B_\frac12)}\lesssim \|\mathcal R\|_{L^\infty(B_1)}^m\|\partial_i u\|_{L^2(B_1)}\lesssim(1+ \|\nabla u\|_{L^\infty(B_1)}^{q-2})^{m}\|\partial_i u\|_{L^2(B_1)}.
\end{equation}
Appealing to some well-known iteration arguments, it is possible to absorb the $(1+ \|\nabla u\|_{L^\infty(B_1)}^{q-2})^{m}$-prefactor on the right-hand side in \eqref{intro:linftyl2} provided that $(q-2)m<1$, which yields the restriction $\frac{q}2<1+\frac{1}{2m}$. Once an a priori Lipschitz estimate for smooth minimizer is established, the proof of Theorem~\ref{T:1} follows by a careful regularization and approximation procedure (as in e.g.\ \cite{BM20}).
The main technical achievement of the present manuscript -- using a method introduced in \cite{BS19a} -- is an improvement of the exponent $m$ in \eqref{intro:linftyl2} compared to the previous results. This improvement is \textit{optimal} for $n\geq 4$ and essentially optimal for $n=3$. More precisely, we have
\begin{proposition}\label{P:2}
Let $B=B_R(x_0)\subset\R^n$, $n\geq3$ and $\kappa\in(0,\frac12)$. There exists $c=c(n,\kappa)<\infty$, where $c=c(n)$ provided $n\geq4$, such that the following is true. Let $0<\nu\leq \lambda<\infty$ and suppose $a\in L^\infty(B;\R^{n\times n})$ satisfies that $a(x)$ is symmetric for almost every $x\in B$ and uniformly elliptic in the sense
\begin{equation}\label{def:ellipt}
\nu|z|^2\leq a(x)z\cdot z\leq \Lambda|z|^2\quad\mbox{for every $z\in\R^n$ and almost every $x\in B$.}
\end{equation}
Let $v\in W^{1,2}(B)$ be a subsolution, that is it satisfies
\begin{equation}\label{def:subsolution}
\int_{B}a\nabla v\cdot\nabla \varphi\leq0\qquad\mbox{for all $\varphi\in C_c^1(B)$ with $\varphi\geq0$.}
\end{equation}
Then,
\begin{equation}\label{est:P1}
\sup_{\frac12 B} v\leq c\biggl(\frac{\Lambda}{\nu}\biggr)^m\biggl(\fint_{B}(v_+)^2\,dx\biggr)^\frac12,\qquad\mbox{where}\quad m:=\frac12\begin{cases}\frac{n-1}2&\mbox{for $n\geq4$}\\1+\kappa&\mbox{for $n=3$}\end{cases}.
\end{equation}
Moreover, for $n\geq4$ the exponent $m=\frac{n-1}4$ is optimal for the estimate in \eqref{est:P1} and for $n=3$ the exponent is essentially optimal in the sense that the estimate in \eqref{est:P1} is in general false for $m<\frac12$.
\end{proposition}
While estimate \eqref{est:P1} is in some sense optimal, the condition \eqref{eq:pqrhs} in Theorem~\ref{T:1} is in general not optimal. To see this, we recall a result of \cite{BF}: Suppose Assumption~\ref{ass} is satisfied with $\mu=1$ and $F(0)=0$, $\partial F(0)=0$, then \textit{bounded} local minimizer of \eqref{eq:int} with $f\equiv0$ are locally Lipschitz provided $q<p+2$. This can be combined with the recent local boundedness \cite{HS19}, where it is proven that under Assumption~\ref{ass} local minimizers of \eqref{eq:int} (with $f=0$) are locally bounded provided $\frac1p-\frac1q\leq \frac1{n-1}$ and this condition is sharp in view of \cite[Theorem 6.1]{Mar91}. Combining Theorem~\ref{T:1} with the above mentioned results of \cite{BF,HS19}, we deduce that Assumption~\ref{ass} (together with some mild technical extra assumptions) with $1<p\leq q$ and
\begin{equation}\label{eq:pqnew}
\frac{q}p<1+\max\biggl\{\frac{2}{n-1},\min\biggl\{\frac{2}{p},\frac{p}{n-1-p}\biggr\}\biggr\}
\end{equation}
implies that local minimizer of \eqref{eq:int} with $f\equiv 0$ are locally Lipschitz (see also \cite{AT21} for related discussion). While it is not clear whether \eqref{eq:pqnew} is optimal, it strictly improves condition \eqref{eq:pq} for example in the cases $p=3$ and $n\geq5$. Let us mention that condition \eqref{eq:pq} also appears in \cite{S21} in the context of higher integrability results for vectorial problems (where Lipschitz regularity fails even in the case $p=q$, see \cite{SY02}). While also in that case bounded minimizer enjoy higher gradient integrability under the condition $q<p+2$, see \cite{CKP11}, this result cannot be combined with an a priori local boundedness result which fails in the vectorial case already for $p=q$.
For non-autonomous functionals, i.e., $\int_\Omega f(x,Du)\,dx$, rather precise sufficient \& necessary conditions are established in \cite{ELM04}, where the conditions on $p,q$ and $n$ have to be balanced with the (H\"older)-regularity in space of the integrand. Currently, regularity theory for non-autonomous integrands with non-standard growth, e.g.\ $p(x)$-Laplacian or double phase functionals, are a very active field of research, see, e.g.,\ the recent papers \cite{BCM18,BR20,BO20,CMMP21,CFK20,CM15,DM19,DM21,DM22,EMM19,HHT17,HO21,Koch,MaPa21} and \cite{BDS20,BGS21} for related results about the Lavrentiev phenomena.
\section{Preliminaries}
\subsection{Preliminary lemmata}
A crucial technical ingredient in the proof of Theorem~\ref{T:1} is the following lemma which can be found in \cite[Lemma~3]{BS19c}.
\begin{lemma}[\cite{BS19c}]\label{L:optimcutoff}
Fix $n\geq2$. For given $0<\rho<\sigma<\infty$ and $v\in L^1(B_\sigma)$ consider
\begin{equation*}
J(\rho,\sigma,v):=\inf\left\{\int_{B_\sigma}|v||\nabla \eta|^2\,dx \;|\;\eta\in C_0^1(B_\sigma),\,\eta\geq0,\,\eta=1\mbox{ in $B_\rho$}\right\}.
\end{equation*}
Then for every $\delta\in(0,1]$
\begin{equation}\label{1dmin}
J(\rho,\sigma,v)\leq (\sigma-\rho)^{-(1+\frac1\delta)} \biggl(\int_{\rho}^\sigma \left(\int_{S_r} |v|\,d\mathcal H^{n-1}\right)^\delta\,dr\biggr)^\frac1\delta,
\end{equation}
where
$$
S_r:=\{x\,:\,|x|=r\}
$$
\end{lemma}
Moreover, we recall here the following classical iteration lemma
\begin{lemma}[Lemma~6.1, \cite{Giu}]\label{L:holefilling}
Let $Z(t)$ be a bounded non-negative function in the interval $[\rho,\sigma]$. Assume that for every $\rho\leq s<t\leq \sigma$ it holds
\begin{equation*}
Z(s)\leq \theta Z(t)+(t-s)^{-\alpha} A+B,
\end{equation*}
with $A,B\geq0$, $\alpha>0$ and $\theta\in[0,1)$. Then, there exists $c=c(\alpha,\theta)\in[1,\infty)$ such that
\begin{equation*}
Z(s)\leq c((t-s)^{-\alpha} A+B).
\end{equation*}
\end{lemma}
\subsection{Non-increasing rearrangement and Lorentz-spaces}
We recall the definition and useful properties of the non-increasing rearrangement $f^*$ of a measurable function $f$ and Lorentz spaces, see e.g.\ \cite[Section~22]{TatarBook}. For a measurable function $f:\R^n\to\R$, the non-increasing rearrangement is defined by
\begin{equation*}
f^*(t):=\inf\{\sigma\in(0,\infty)\,:\,|\{x\in\R^n\,:\,|f(x)|>\sigma\}|\leq t\}.
\end{equation*}
Let $f:\R^n\to\R$ be a measurable function with ${\rm supp}f\subset \Omega$, then it holds for all $p\in[1,\infty)$
\begin{equation}\label{eq:ff*}
\int_\Omega |f(x)|^p\,dx=\int_0^{|\Omega|}(f^*(t))^p\,dt.
\end{equation}
A simple consequence of \eqref{eq:ff*} and the fact $f\leq g$ implies $f^*\leq g^*$ is the following inequality
\begin{equation}\label{est:omegat}
\sup_{|A|\leq t\atop A\subset\Omega}\int_A|f(x)|^p\leq \int_0^t(f_\Omega^*(t))^p\,dt,
\end{equation}
where $f_\Omega^*$ denotes the non-increasing rearrangement of $f_\Omega:=f\chi_\Omega$ (inequality \eqref{est:omegat} is in fact an \textit{equality} but for our purpose the upper bound suffices).
The Lorentz space $L^{n,1}(\R^d)$ can be defined as the space of measurable functions $f:\R^n\to\R$ satisfying
\begin{equation*}
\|f\|_{L^{n,1}(\R^n)}:=\int_0^\infty t^\frac1n f^*(t)\,\frac{dt}t<\infty.
\end{equation*}
Moreover, for $\Omega\subset\R^n$ and a measurable function $f:\R^n\to\R$, we set
\begin{equation*}
\|f\|_{L^{n,1}(\Omega)}:=\int_0^{|\Omega|} t^\frac1n f_\Omega^*(t)\,\frac{dt}t<\infty,
\end{equation*}
where $f_\Omega$ is defined as above. Let us recall that $L^{n+\e}(\Omega)\subset L^{n,1}(\Omega)\subset L^n(\Omega)$ for every $\e>0$, where $L^{n,1}(\Omega)$ is the space of all measurable functions $f:\Omega\to\R$ satisfying $ \|f\|_{L^{n,1}(\Omega)}<\infty$ (here we identify $f$ with its extension by zero to $\R^n\setminus \Omega$).
\section{Nonlinear iteration lemma and proof of Proposition~\ref{P:2}}\label{Sec:nonlineariteration}
In this section, we provide a nonlinear iteration lemma which will eventually be the main workhorse in the proof of Theorem~\ref{T:1}:
\begin{lemma}\label{L:basiciteration}
Let $B=B_R(x_0)\subset\R^n$, $n\geq3$, $\kappa\in(0,\frac12)$ and let $v\in W^{1,2}(B)\cap C(B)$ be non-negative and $f\in L^2(\R^n)$. Suppose there exists $ M_1\geq1$, $M_2,c_m>0$ and $k_0\geq0$ such that for all $k\geq k_0$ and for all $\eta\in C_c^1(B)$ with $\eta\geq0$ it holds
\begin{align}\label{L:basic:caccio}
\int_{B}|\nabla(v-k)_+|^2\eta^2\,dx\leq c_m^2M_1^2\int_{B}(v-k)_+^2|\nabla \eta|^2\,dx+c_m^2M_2^2\int_{B\cap\{v>k\}}\eta^2|f|^2.
\end{align}
Then there exist $c=c(c_m,n)\in[1,\infty)$ for $n\geq4$ and $c=c(c_m,\kappa)\in[1,\infty)$ for $n=3$ such that
\begin{equation}\label{L:basic:claim}
\|(v)_+\|_{L^\infty(\frac14 B)}\leq k_0+cM_1^{1+\max\{\kappa,\frac{n-3}2\}}\biggl(\fint_B (v-k_0)_+^2\,dx\biggr)^\frac12+cM_1^{\max\{\kappa,\frac{n-3}2\}}M_2 \|f\|_{L^{n,1}(B)}.
\end{equation}
\end{lemma}
\begin{remark}\label{rem:Loptim}[Optimality]
The exponent in the factor $M_1^{1+\max\{\kappa,\frac{n-3}2\}}$ is optimal in dimensions $n\geq4$ and almost optimal for $n=3$. Indeed, consider $v(x):=x_n^2+1-\Lambda |x'|^2$, where $x=(x_1,\dots,x_{n-1},x_n)=:(x',x_n)$ which clearly satisfies
\begin{equation}\label{eq:counterexample}
-\nabla\cdot a\nabla v=0\qquad\mbox{where}\qquad a:={\rm diag}(1,\dots,1,(n-1)\Lambda).
\end{equation}
Hence, by classical computations (using the symmetry of $a$) $v$ satisfies the Caccioppoli inequality \eqref{L:basic:caccio} with
\begin{equation}\label{eq:remoptim}
c_m^2=4(n-1),\qquad M_1=\Lambda^\frac12,\qquad f\equiv0.
\end{equation}
Obviously, we have $\sup_{B_\frac14}\, v\geq v(0)\geq1$ and
\begin{align*}
\fint_{B_1}(v_+)^2\lesssim& \int_{-1}^1\int_0^{\Lambda^{-1/2}(1+x_n^2)^{1/2}}r^{n-2}(x_n^2+1)^2+\Lambda^2r^{n+2}\,dr\,dx_n\\
\lesssim&\Lambda^{-\frac{n-1}2}\int_{-1}^1(1+x_n^2)^{\frac{n+3}2}\lesssim \Lambda^{-\frac{n-1}2},
\end{align*}
where $\lesssim$ means $\leq$ up to a multiplicative constant depending only on $n$. In particular, we have
$$
\frac{\|v_+\|_{L^\infty(B_\frac14)}}{\|v_+\|_{L^2(B_1)}}\geq c(n)\Lambda^{\frac{n-1}{4}}\stackrel{\eqref{eq:remoptim}}=c(n)M_1^{\frac{n-1}{2}}
$$
which matches exactly the scaling in \eqref{L:basic:claim} for $n\geq4$ and almost matches in the case $n=3$. We mention that $-(v)_+$ appeared already in \cite{connor} in the context of optimal dependencies in Krylov-Safanov estimates.
\end{remark}
\begin{remark} Lemma~\ref{L:basiciteration} should be compared with \cite[Lemma~3.1]{BM20}, where starting from a similar Caccioppoli inequality as \eqref{L:basiciteration} a pointwise bound for $v$ in terms of the $L^2$-norm $v$ and the Riesz potential of $f$ is deduced. The main improvement of Lemma~\ref{L:basiciteration} compared to \cite[Lemma~3.1]{BM20} lies in the dependence of $M_1$ (from $M_1^{\max\{\kappa,\frac{n-2}2\}}$ in \cite{BM20} to $M_1^{\max\{\kappa,\frac{n-3}2\}}$) which in view of Remark~\ref{rem:Loptim} is essentially optimal. Let us remark that variants of \cite[Lemma~3.1]{BM20} also play an important role in \cite{DM21,DM22} where non-autonomus functionals are considered.
\end{remark}
Before, we prove Lemma~\ref{L:basiciteration} let us observe that Proposition~\ref{P:2} is implied by Lemma~\ref{L:basiciteration}.
\begin{proof}[Proof of Proposition~\ref{P:2}]
By density arguments, we are allowed to use $\varphi=\eta^2(v-k)_+$, $k\geq0$ and $\eta\in C_c^1(B)$ with $\eta\geq0$ in \eqref{def:subsolution} which implies
\begin{equation*}
\int_{B}a\nabla (u-k)_+\cdot \nabla (v-k)_+\eta^2\,dx\leq-2\int_Ba\nabla (v-k)_+\cdot (v-k)_+\eta\nabla \eta\,dx.
\end{equation*}
Assumption \eqref{def:ellipt} combined with Cauchy-Schwarz and Youngs inequality imply \eqref{L:basic:caccio} with $c_m^2=4$ and $M_1^2=\frac{\Lambda}\nu$. The claimed estimate \eqref{est:P1} now follows directly from Lemma~\ref{L:basiciteration} and the claimed optimality is a consequence of the discussion in Remark~\ref{rem:Loptim}.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{L:basiciteration}]
Without loss of generality, we only consider the case $B=B_2$.
Throughout the proof we set
\begin{equation}\label{def:2star}
2^*=2^*(n,\kappa):=\begin{cases}\frac{2(n-1)}{n-3}&\mbox{if $n\geq4$}\\2+\frac2\kappa&\mbox{if $n=3$}\end{cases}.
\end{equation}
\step 1 Optimization Argument. We claim that there exists $c_1=c_1(n,2^*)\in[1,\infty)$ such that for all $k>h\geq0$ and all $\frac12\leq \rho<\sigma\leq 1$
\begin{equation}\label{L:basiciteratio:S1claim}
\|\nabla (v-k)_+\|_{L^2(B_\rho)}\leq \frac{c_1c_m M_1}{(\sigma-\rho)^{\alpha}} \frac{\norm{(v-h)_+}_{W^{1,2}(B_\sigma\setminus B_\rho)}^{\frac{2^*}2}}{(k-h)^{\frac{2^*}2-1}}+c_m M_2\omega_f(|A_{k,\sigma}|),
\end{equation}
where $\alpha=\frac12+\frac{2^*}4>0$,
\begin{equation}\label{def:Alr}
A_{l,r}:=B_r\cap \{x\in\Omega\, :\,v(x)>l\}\qquad\mbox{for all $r>0$ and $l>0$,}
\end{equation}
and $\omega_f:[0,|B|]\to[0,\infty)$ is defined by
\begin{equation}\label{def:omegaf}
\omega_f(t):=\biggl(\int_0^t((f\chi_B)^*(s))^2\,ds\biggr)^\frac12.
\end{equation}
We optimize the right-hand side of \eqref{L:basic:caccio} with respect to $\eta$ satisfying $\eta\in C_0^1(B_\sigma)$ and $\eta=1$ in $B_\rho$: we use Lemma~\ref{L:optimcutoff} and $1\leq\frac{v(x)-h}{k-h}$ whenever $v(x)\geq k$ in the form
\begin{align}\label{est:opteta}
&\inf_{\eta\in\mathcal A_{\rho,\sigma}}\int_{B_1}|\nabla \eta|^2(v-k)_+^2\,dx\notag\\
\leq&\frac1{(\sigma-\rho)^{1+\frac1\delta}}\biggl(\int_\rho^\sigma \biggl(\int_{S_r\cap A_{k,\sigma}}(v-k)_+^2\,d\mathcal H^{n-1}\biggr)^\delta\,dr\biggr)^\frac1\delta\notag\\
\leq&\frac1{(\sigma-\rho)^{1+\frac1\delta}}\frac1{(k-h)^{2^*-2}}\biggl(\int_\rho^\sigma \biggl(\int_{S_r\cap A_{k,\sigma}}(v-h)_+^{2^*}\,d\mathcal H^{n-1}\biggr)^\delta\,dr\biggr)^\frac1\delta
\end{align}
for every $\delta>0$. Appealing to Sobolev inequality on spheres, we find $c=c(n,2^*)\in[1,\infty)$ such that for almost every $r\in[\rho,\sigma]$ it holds
\begin{equation}\label{est:sobsphere}
\biggl(\int_{S_r}(v-h)_+^{2^*}\,d\mathcal H^{n-1}\biggr)^\frac{1}{2^*}\leq c\biggl(\int_{S_r}(v-h)_+^2+|\nabla (v-h)_+|^2\,d\mathcal H^{n-1}\biggr)^\frac{1}{2}.
\end{equation}
Inserting \eqref{est:sobsphere} in \eqref{est:opteta} with $\delta=\frac{2}{2^*}$, we obtain
\begin{equation}\label{est:opteta1}
\inf_{\eta\in\mathcal A_{\rho,\sigma}}\int_{B_1}(v-k)_+^2|\nabla \eta|^2\,dx\leq \frac{c}{(\sigma-\rho)^{2\alpha}} \frac{\norm{(v-h)_+}_{W^{1,2}(B_\sigma\setminus B_\rho)}^{2^*-2}}{(k-h)^{2^*-2}}\norm{(v-h)_+}_{W^{1,2}(B_\sigma\setminus B_\rho)}^2
\end{equation}
The claim \eqref{L:basiciteratio:S1claim} follows from \eqref{L:basic:caccio} and \eqref{est:opteta1} combined with
\begin{equation*}
\int_{B_1\cap\{v>k\}}\eta^2|f|^2\,dx\leq \sup\biggl\{\int_A |f|^2\,dx\,:\,|A| \leq |A_{k,\sigma}|,\, A\subset B_1\biggr\}\stackrel{\eqref{est:omegat}}{\leq}\int_0^{|A_{k,\sigma}|}((f\chi_B)^*(s))^2\,ds
\end{equation*}
and the definition of $\omega_f$, see \eqref{def:omegaf}.
\step 2 One-Step improvement.
We claim that there exists $c_2=c_2(n)\in[1,\infty)$ such that
\begin{equation}\label{est:onestepL}
J(k,\rho)\leq \frac{c_1c_m M_1J(h,\sigma)^{\frac{2^*}2-1}}{(k-h)^{\frac{2^*}{2}-1}}\frac{J(h,\sigma)}{(\sigma-\rho)^\alpha}+c_mM_2\omega_{f}\biggl(\frac{c_2 J(h,\sigma)^{2_n^*}}{(k-h)^{2_n^*}}\biggr)+\frac{c_2J(h,\sigma)^{1+\frac{2_n^*}n}}{(k-h)^\frac{2_n^*}n}
\end{equation}
where $2_n^*:=\frac{2n}{n-2}$,
\begin{equation}
J(t,r):=\|(v-t)_+\|_{W^{1,2}(B_r)}\qquad \forall t\geq0,\, r\in(0,1]
\end{equation}
and $\omega_f$ is defined in \eqref{def:omegaf}. Note that $k-h<v-h$ on $A_{k,r}$ for every $r>0$ and thus with help of Sobolev inequality (this time on the $n$-dimensional ball $B_\sigma$)
\begin{equation}\label{est:Aksigma}
|A_{k,\sigma}|\leq \int_{A_{k,\sigma}}\biggl(\frac{v(x)-h}{k-h}\biggr)^{2_n^*}\,dx\leq \frac{\|(v-h)_+\|_{L^{2_n^*}(B_\sigma)}^{2_n^*}}{(k-h)^{2_n^*}}\leq c_2 \frac{J(h,\sigma)^{2_n^*}}{(k-h)^{2_n^*}},
\end{equation}
where $c_2=c_2(n)\in[1,\infty)$. Combining \eqref{est:Aksigma} with \eqref{L:basiciteratio:S1claim}, we obtain
\begin{equation}\label{est:onestep1}
\|\nabla (v-k)_+\|_{L^2(B_\rho)}\leq \frac{c_1c_m M_1}{(\sigma-\rho)^{\alpha}} \frac{J(h,\sigma)^{\frac{2^*}2}}{(k-h)^{\frac{2^*}2-1}}+c_m M_2\omega_{f}\biggl(\frac{c_2 J(h,\sigma)^{2_n^*}}{(k-h)^{2_n^*}}\biggr).
\end{equation}
It remains to estimate $\|(v-k)_+\|_{L^2(B_\rho)}$: A combination of H\"older inequality, Sobolev inequality and \eqref{est:Aksigma} yield
\begin{align}\label{est:onestep2}
\| (v-k)_+\|_{L^2(B_\rho)}\leq \|(v-h)_+\|_{L^{2_n^*}(B_\sigma)}|A_{k,\sigma}|^{\frac{1}{n}}\leq c(n) \frac{J(h,\sigma)^{1+\frac{2_n^*}n}}{(k-h)^\frac{2_n^*}n}.
\end{align}
Combining \eqref{est:onestep1} and \eqref{est:onestep2}, we obtain \eqref{est:onestepL}.
\step 3 Iteration
For $k_0\geq0$ and a sequence $(\Delta_\ell)_{\ell\in\mathbb N}\subset [0,\infty)$ specified below, we set
\begin{equation}\label{def:kell}
k_\ell:=k_0+\sum_{i=1}^\ell\Delta_i,\quad \sigma_\ell=\frac12+\frac1{2^{\ell+1}}.
\end{equation}
For every $\ell\in\mathbb N\cup\{0\}$, we set $J_\ell:=J(k_\ell,\sigma_\ell)$. From \eqref{est:onestepL}, we deduce for every $\ell\in\mathbb N$
\begin{align}\label{est:iteration}
J_\ell\leq c_1c_m M_12^{(\ell+1)\alpha}\biggl(\frac{J_{\ell-1}}{\Delta_\ell}\biggr)^{\frac{2^*}2-1}J_{\ell-1} +c_m M_2\omega_f\biggl(\frac{c_2 J_{\ell-1}^{2_n^*}}{(\Delta_\ell)^{2_n^*}}\biggr)+c_2\biggl(\frac{J_{\ell-1}}{\Delta_\ell}\biggr)^{\frac{2_n^*}n}J_{\ell-1},
\end{align}
where $c_1$ and $c_2$ are as in Step~2. Fix $\tau=\tau(n,\kappa)\in(0,\frac12)$ such that
\begin{equation}\label{def:tau}
(2\tau)^{\frac{2^*}2-1}=2^{-\alpha}.
\end{equation}
We claim that we can choose $\{\Delta_\ell\}_{\ell\in\mathbb N}$ satisfying
\begin{equation}\label{est:sumdeltaell}
\sum_{\ell\in\mathbb N}\Delta_\ell<\infty
\end{equation}
in such a way that
\begin{equation}\label{ass:iterationJell}
J_\ell\leq \tau^\ell J_0\qquad\mbox{for all $\ell\in\mathbb N\cup\{0\}$}.
\end{equation}
Obviously, \eqref{def:kell}, \eqref{ass:iterationJell} and $\tau\in(0,1)$ yield boundedness of $v$ in $B_\frac12$.
For $\ell\in\mathbb N$, we set $\Delta_\ell=\Delta_\ell^{(1)}+\Delta_\ell^{(2)}+\Delta_\ell^{(3)}$ with
\begin{equation}\label{def:deltaell}
\Delta_\ell^{(1)}=\biggl(2^\alpha 3c_1 c_m M_1 \tau^{-(\frac{2^*}2-1)-1}\biggr)^\frac1{\frac{2^*}2-1}J_02^{-\ell},\qquad \Delta_\ell^{(3)}=\biggl(\frac{3 c_2}\tau\biggr)^{\frac{n}{2_n^*}}\tau^{\ell-1}J_0
\end{equation}
and $\Delta_\ell^{(2)}$ being the smallest value such that
\begin{align}\label{def:deltaell1}
c_m M_2\omega_f\biggl(\frac{c_2 J_{\ell-1}^{2_n^*}}{(\Delta_\ell^{(2)})^{2_n^*}}\biggr)\leq \frac13\tau^\ell J_0
\end{align}
is valid. The choice of $\tau$ (see \eqref{def:tau}), $\Delta_\ell$ and estimate \eqref{est:iteration} combined with a straightforward induction argument yield \eqref{ass:iterationJell}. Indeed, assuming $J_{\ell-1}\leq \tau^{\ell-1}J_0$, we obtain
\begin{align*}
J_\ell\leq& \frac{c_1c_m M_12^{(\ell+1)\alpha}}{2^\alpha 3c_1 c_m M_1 \tau^{-(\frac{2^*}2-1)-1}}\biggl(\tau^{\ell-1}2^{\ell}\biggr)^{\frac{2^*}2-1}\tau^{\ell-1}J_0+\frac13\tau^\ell J_0+\frac{c_2\tau}{3c_3}\tau^{\ell-1}J_0\\
=&\frac13\tau 2^{\alpha \ell}\biggl((2\tau)^\ell\biggr)^{\frac{2^*}2-1}\tau^{\ell-1}J_0+\frac23\tau^\ell J_0\stackrel{\eqref{def:tau}}=\tau^\ell J_0.
\end{align*}
Using $\sum_{\ell\in\mathbb N}(2^{-\ell}+\tau^\ell)<\infty$, we deduce from \eqref{def:deltaell1} and \eqref{def:deltaell}
\begin{align}\label{sum:deltaell}
\sum_{\ell\in\mathbb N}\Delta_\ell \leq \sum_{\ell\in\mathbb N}\frac{c_2^{\frac1{2_n^*}}\tau^{\ell-1}J_0}{(\omega_f^{-1}(\frac{\tau^\ell J_0}{3c_m M_2}))^{\frac1{2_n^*}}}+c(1+M_1^\frac{1}{\frac{2^*}2-1}) J_0,
\end{align}
where $c=c(n,\kappa,c_m)\in[1,\infty)$. Next, we show that $f\in L^{n,1}(B_1)$ ensures that the first term on the right-hand side of \eqref{sum:deltaell} is bounded and thus \eqref{est:sumdeltaell} is valid. Indeed,
\begin{align}\label{sum:omega1}
\sum_{\ell\in\mathbb N}\frac{\tau^{\ell}J_0}{(\omega_f^{-1}(\frac{\tau^\ell J_0}{3c_m M_2}))^{\frac12-\frac1n}}\leq& \frac1\tau\int_1^\infty \frac{\tau^x J_0}{(\omega_f^{-1}(\frac{\tau^xJ_0}{3c_mM_2}))^{\frac1{2}-\frac1n}}\,dx\notag\\
=&\frac1{\tau|\log \tau|}\int_0^\tau\frac{tJ_0}{(\omega_f^{-1}(\frac{tJ_0}{3c_m M_2}))^{\frac12-\frac1n}}\,\frac{dt}t\notag\\
\leq&\frac{3c_mM_2}{\tau|\log\tau|}\int_0^{\omega_f^{-1}({\frac{\tau J_0}{3c_m M_2}})} \frac{\omega_f'(s)}{s^{\frac12-\frac1n}}\,ds.
\end{align}
Recall $\omega_f(t)=(\int_0^t((f\chi_{B_1})^*(s))^2\,ds)^\frac12$ and thus $\omega_f'(t)=\frac12 \frac1{\omega_f(t)}(f\chi_{B_1})^*(t))^2$. Since $(f\chi_{B_1})^*$ is non-increasing, we have $\omega_f(t)\geq t^\frac12(f\chi_{B_1})^*(t)$ and
\begin{align}\label{sum:omega2}
\int_0^{\omega^{-1}({\frac{\tau J_0}{3c_mM_2}})} \frac{\omega_f'(s)}{s^{\frac12-\frac1n}}\,ds
\leq&\frac12\int_0^{\infty}s^{\frac1n}(f\chi_{B_1})^*(s)\,\frac{ds}s=\frac12\|f\|_{L^{n,1}(B_1)}.
\end{align}
Notice that \eqref{ass:iterationJell} implies
\begin{equation*}
\|(v-k_0-\sum_{\ell\in\mathbb N}\Delta_\ell)_+\|_{L^2(B_\frac12)}=0\quad \Rightarrow\quad \sup_{B_\frac12}v\leq k_0+\sum_{\ell\in\mathbb N}\Delta_\ell.
\end{equation*}
Hence, appealing to \eqref{sum:deltaell}-\eqref{sum:omega2} and $M_1\geq1$, we find $c=c(c_m,n,\kappa)\in[1,\infty)$ such that
\begin{equation*}
\sup_{B_\frac12}v\leq k_0+ cM_1^\frac{1}{\frac{2^*}2-1}\|(u-k_0)_+\|_{W^{1,2}(B_1)}+cc_m M_2\|f\|_{L^{n,1}(B_1)}.
\end{equation*}
The claimed estimate \eqref{L:basic:claim} with $B=B_2$, follows by the definition of $2^*$, see \eqref{def:2star} and a further application of \eqref{L:basic:caccio} with $\eta\in C_c^1(B_2)$ with $\eta=1$ in $B_1$ with $|\nabla \eta|\lesssim1$.
\end{proof}
\section{A priori estimate for regularized integrand}
In this section, we derive a priori estimates for regular weak solutions $u\in W^{1,\infty}_{\rm loc}(B)$ of the equation
\begin{equation}\label{eq:aeps}
-\nabla \cdot \bfa_\e(\nabla u)=f\qquad \mbox{in $B\subset\R^n$, $f\in L^\infty(\R^n)$,}
\end{equation}
where $\bfa_\e:\R^n\to\R^n$ is (a possibly regularized version of) $\partial F$ and satisfies
\begin{assumption}\label{ass:reg}
Let $0<\nu\leq \Lambda<\infty$, $1<p\leq q<\infty$, $\mu\in[0,1]$ and $\e,T>0$ be given. Suppose that $\bfa_\e\in C^1_{\rm loc}(\R^n,\R^n)$ satisfies
\begin{equation}\label{ass:bfaeps}
\begin{cases}
\partial_i (\bfa_\e)_j(z)=\partial_j (\bfa_\e)_i(z)&\mbox{for all $z\in\R^n$, and $i,j\in\{1,\dots,n\}$}\\
g_{1}(|z|)|\xi|^2\leq \langle\partial \bfa_\e(z)\xi,\xi\rangle \leq g_{2,\e}(|z|)|\xi|^2&\mbox{for all $z,\xi\in\R^n$ with $|z|\geq T$}\\
\mbox{$\partial\bfa_\e(z)$ is strictly positive definite}&\mbox{for all $z\in\R^n$ with $|z|\leq 2T$}
\end{cases}
\end{equation}
where for all $s\geq0$
\begin{align*}
g_1(s):=\nu(\mu^2+s^2)^{\frac{p-2}2},\qquad g_{2,\e}(s):=\Lambda(\mu^2+s^2)^{\frac{p-2}2}+\Lambda (\mu^2+s^2)^{\frac{q-2}2}+\e\Lambda(1+s^2)^\frac{\min\{p-2,0\}}2
\end{align*}
\end{assumption}
Moreover, we introduce (following \cite{BM20}) the quantity
\begin{equation}
G_T(t):=\int_T^{\max\{t,T\}} g_1(s)s\,ds\qquad (T>0).
\end{equation}
\begin{lemma}[Caccioppoli inequality]\label{L:caccio}
Suppose Assumption~\ref{ass:reg} is satisfied for some $\e,T>0$. There exists $c=c(n)\in[1,\infty)$ such that the following is true: Let $u\in W^{1,\infty}_{\rm loc}(B)$ be a weak solution to \eqref{eq:aeps}. Then it holds for all $\eta\in C_c^1(B)$
\begin{align}\label{eq:caccioaprio}
\int_{B}|\nabla((G_T(|\nabla u|)-k)_+)|^2\eta^2\,dx\leq& c\int_{B}\frac{g_{2,\e}(|\nabla u|)}{g_1(|\nabla u|)}(G_T(|\nabla u|)-k)_+^2|\nabla \eta|^2\,dx\notag\\
&+c\int_{B\cap\{G_T(|\nabla u|)>k\}}\eta^2|\nabla u|^2 |f|^2\
\end{align}
\end{lemma}
The proof of Lemma~\ref{L:caccio} follows almost verbatim the proof of \cite[Lemma 4.5]{BM20} where \eqref{eq:caccioaprio} is proven for a specific choice of $\eta$.
\begin{proof}
The following computations are essentially a translation of the proof of \cite[Lemma 4.5]{BM20} to the present situation.
\step 0 Preliminaries.
Since we suppose that $u$ is Lipschitz continuous and $\bfa_\e\in C_{\rm loc}^1(\R^n,\R^n)$ satisfies the uniform estimate $\langle\partial \bfa_\e(z)\xi,\xi\rangle\geq\nu_\e|\xi|^2$ for some $\nu_\e>0$ for all $z,\xi\in\R^n$ (which follows from \eqref{ass:bfaeps}), we obtain from standard regularity theory that
\begin{equation*}
u\in W_{\rm loc}^{2,2}(B),\quad u\in C_{\rm loc}^{1,\alpha}(B)\,\mbox{for some $\alpha\in(0,1)$},\quad \bfa_\e(\nabla u)\in W_{\rm loc}^{1,2}(B,\R^n).
\end{equation*}
Hence, we can differentiate equation \eqref{eq:aeps} and obtain for all $s\in\{1,\dots,n\}$ and every $\varphi\in W_0^{1,2}(B)$ with compact support in $B$ that
\begin{equation}\label{L:aprio:eulerdiffesp}
\int_{B}\langle \partial \bfa_\e(\nabla u)\nabla \partial_s u,\nabla \varphi\rangle \,dx=-\int_B f\partial_s \varphi\,dx.
\end{equation}
We test \eqref{L:aprio:eulerdiffesp} with
$$
\varphi:=\varphi_s:=\eta^2 (G_T(|\nabla u|)-k)_+\partial_s u,\qquad k\geq0,\quad\eta\in C_c^1(B).
$$
On the set $\{G_T(|\nabla u|)>k\}$ holds
\begin{equation}\label{L:caccio:nablaphis}
\nabla \varphi_s=\eta^2 (G_T(|\nabla u|)-k)_+\nabla \partial_s u+\eta^2 \partial_s u\nabla (G_T(|\nabla u|)-k)_++2\eta (G_T(|\nabla u|)-k)_+\partial_s u\nabla \eta
\end{equation}
and
\begin{equation}\label{eq:GTg1}
\nabla (G_T(|\nabla u|-k)_+=\nabla (G_T(|\nabla u|))=g_1(|\nabla u|)\sum_{s=1}^n\partial_s u\nabla \partial_s u\quad\mbox{and}\quad g_1(|\nabla u|)>0.
\end{equation}
We compute,
\begin{align*}
&\sum_{s=1}^n \int_B\langle\partial \bfa_\e(\nabla u)\nabla \partial_s u,\nabla \partial_s u\rangle (G_T(|\nabla u|)-k)_+\eta^2\,dx\\
&+\int_B g_1(|\nabla u|)^{-1}\langle \partial \bfa_\e(\nabla u)\nabla (G_T(|\nabla u|)-k)_+,\nabla (G_T(|\nabla u|)-k)_+)\rangle \eta^2\,dx\\
\stackrel{\eqref{eq:GTg1}}{=}&\sum_{s=1}^n \int_B\langle\partial \bfa_\e(\nabla u)\nabla \partial_s u,(G_T(|\nabla u|)-k)_+\nabla \partial_s u\rangle \eta^2\,dx\\
&+\sum_{s=1}^n\int_B \langle \partial \bfa_\e(\nabla u)\nabla \partial_s u,(\partial_s u)\nabla (G_T(|\nabla u|)-k)_+)\rangle \eta^2\,dx\\
\stackrel{\eqref{L:aprio:eulerdiffesp}}=&-2\sum_{s=1}^n \int_B\langle\partial \bfa_\e(\nabla u)\nabla \partial_s u,\nabla \eta\rangle (\partial_s u)(G_T(|\nabla u|)-k)_+\eta\,dx-\sum_{s=1}^n\int_B f\partial_s\varphi_s\,dx\\
\stackrel{\eqref{eq:GTg1}}=&-2 \int_Bg_1(|\nabla u|)^{-1}\langle\partial\bfa_\e(\nabla u)\nabla((G_T(|\nabla u|)-k)_+),\nabla \eta\rangle (G_T(|\nabla u|)-k)_+\eta\,dx-\sum_{s=1}^n\int_B f\partial_s\varphi_s\,dx.
\end{align*}
The symmetry of $\partial \bfa_\e$ and Cauchy-Schwarz inequality yield
\begin{align*}
&-2 \int_Bg_1(|\nabla u|)^{-1}\langle\partial\bfa_\e(\nabla u)\nabla((G_T(|\nabla u|)-k)_+),\nabla \eta\rangle (G_T(|\nabla u|)-k)_+\eta\,dx\\
\leq&\frac12\int_Bg_1(|\nabla u|)^{-1}\langle\partial\bfa_\e(\nabla u)\nabla((G_T(|\nabla u|)-k)_+),\nabla((G_T(|\nabla u|)-k)_+)\rangle\eta^2\,dx\\
&+2\int_Bg_1(|\nabla u|)^{-1}\langle\partial\bfa_\e(\nabla u)\nabla\eta,\nabla\eta\rangle((G_T(|\nabla u|)-k)_+)^2\,dx.
\end{align*}
The previous two (in)equalities combined with \eqref{ass:bfaeps} imply the existence of $c=c(n)<\infty$ such that
\begin{align}\label{L:caccio:almostfinal1}
&\int_B\biggl(g_1(|\nabla u|)(G_T(|\nabla u|)-k)_+|\nabla^2 u|^2+|\nabla (G_T(|\nabla u|)-k)_+|^2\biggr)\eta^2\,dx\notag\\
\leq&c\int_B\biggl(\frac{g_{2,\e}(\nabla u)}{g_1(|\nabla u|)}\biggr)(G_T(|\nabla u|)-k)_+^2|\nabla \eta|^2\,dx+c\sum_{s=1}^n\int_B|f||\nabla \varphi_s|\,dx.
\end{align}
Appealing to Youngs inequality and \eqref{L:caccio:nablaphis}, we estimate the last term on the right-hand side by
\begin{align}\label{L:caccio:almostfinal2}
&\sum_{s=1}^n\int_B|f||\nabla \varphi_s|\,dx\notag\\
\leq&\frac12 \int_B \biggl(g_1(|\nabla u|)(G_T(|\nabla u|)-k)_+|\nabla^2 u|^2+|\nabla (G_T(|\nabla u|)-k)_+|^2\biggr)\eta^2\,dx\notag\\
&+c\int_{B}((G_T(|\nabla u|)-k)_+)^2|\nabla \eta|^2\,dx\notag\\
&+c\int_{B\cap \{G_T(|\nabla u|)>k\}}|f|^2\big(g_1(|\nabla u|)^{-1}(G_T(|\nabla u|)-k)_++|\nabla u|^2\big)\eta^2\,dx,
\end{align}
where $c=c(n)\in[1,\infty)$. The monotonicity of $s\mapsto g_1(s)s$ yields $G_T(t)\leq g_1(t)t(t-T)\leq g_1(t)t^2$ and thus
\begin{align}\label{L:caccio:almostfinal3}
\int_{B\cap \{G_T(|\nabla u|)>k\}}|f|^2(g_1(|\nabla u|)^{-1}(G_T(|\nabla u|)-k)_++|\nabla u|^2)\eta^2\,dx\leq 2\int_{B\cap \{G_T(|\nabla u|)>k\}}|f|^2|\nabla u|^2\eta^2\,dx
\end{align}
Combining \eqref{L:caccio:almostfinal1}-\eqref{L:caccio:almostfinal3}, we obtain
\begin{align*}
&\int_B\biggl(g_1(|\nabla u|)(G_T(|\nabla u|)-k)_+|\nabla^2 u|^2+|\nabla (G_T(|\nabla u|)-k)_+|^2\biggr)\eta^2\,dx\\
\leq&c\int_B\biggl(\frac{g_{2,\e}(\nabla u)}{g_1(|\nabla u|)}+1\biggr)(G_T(|\nabla u|)-k)_+^2|\nabla \eta|^2\,dx+c\int_{B\cap \{G_T(|\nabla u|)>k\}}|f|^2|\nabla u|^2\eta^2\,dx
\end{align*}
and the claim follows from $1\leq \frac{g_{2,\e}(t)}{g_1(t)}$ for all $t>0$.
\end{proof}
Combining the Caccioppoli inequality of Lemma~\ref{L:caccio} with Lemma~\ref{L:basiciteration}, we obtain the following local $L^\infty$-bound on $G_T(|\nabla u|)$
\begin{lemma}\label{L:apriorilipschitz}
Suppose Assumption~\ref{ass:reg} is satisfied for some $\e,T\in(0,1]$ and let $u\in W^{1,\infty}_{\rm loc}(B)$ be a weak solution to \eqref{eq:aeps}. Set
\begin{equation}\label{def:gammavartheta}
\gamma:=\frac12\frac{q-p}p\max\biggl\{\kappa,\frac{n-3}2\biggr\}+\frac{q}{2p},\qquad\tilde \gamma:=\frac12\frac{q-p}p\max\biggl\{\kappa,\frac{n-3}2\biggr\}+\frac1p
\end{equation}
and suppose that $\gamma,\gamma'\in(0,1)$. Then there exists $c=c(\gamma,\kappa,\Lambda,n,p,q)\in[1,\infty)$ such that
\begin{align}\label{claim:L:apriorilipschitz}
\|G_T(|\nabla u|)\|_{L^\infty(\frac12B)}\leq& c(1+\tfrac{\e}{(\mu^2+T^2)^\frac{p-2}2})^{\max\{\kappa,\frac{n-3}2\}+1}\fint_B G_T(|\nabla u|)+c\biggl(\fint_BG_T(|\nabla u|)\biggr)^{\frac12\frac1{1-\gamma}}\notag\\
&+c(1+\tfrac{\e}{(\mu^2+T^2)^\frac{p-2}2})^{\frac12\max\{\kappa,\frac{n-3}2\}}((T^2+\mu^2)^\frac12+(T^2+\mu^2)^{\frac12\tilde \gamma p})\|f\|_{L^{n,1}(B)}\notag\\
&+c\biggl((1+\tfrac{\e}{(\mu^2+T^2)^\frac{p-2}2})^{\frac12\max\{\kappa,\frac{n-3}2\}}\|f\|_{L^{n,1}(B)}\biggr)^\frac{p}{p-1}\notag\\
&+c\|f\|_{L^{n,1}(B)}^\frac1{1-\tilde \gamma}.
\end{align}
\end{lemma}
\begin{remark}\label{rem:gamma}
Note that $1<p\leq q$ and $\kappa\in(0,\frac12)$ imply that $\gamma$ and $\tilde \gamma$ defined in \eqref{def:gammavartheta} are positive. Moreover, for $n\geq4$ relation \eqref{eq:pqrhs} imply $\gamma,\tilde \gamma<1$. Indeed, we have
\begin{align*}
\frac{q}p<1+\frac2{n-1}\quad\Rightarrow&\quad \gamma=\frac12\frac{q-p}p\frac{n-3}2+\frac{q}{2p}<\frac12\frac{2}{n-1}\frac{n-3}2+\frac12+\frac1{n-1}=1\\
\frac{q}p<1+\frac{4(p-1)}{p(n-3)}\quad\Rightarrow&\quad \tilde \gamma=\frac12\frac{q-p}p\frac{n-3}2+\frac1p<\frac12\frac{4(p-1)}{p(n-3)}\frac{n-3}2+\frac1p=1.
\end{align*}
In dimension $n=3$ and $\kappa\in(0,\frac12)$, a straightforward computation yields that $\kappa<\frac{2p-q}{q-p}$ implies $\gamma<1$ and $\kappa<2\frac{p-1}{q-p}$ implies $\tilde \gamma<1$.
\end{remark}
\begin{proof}[Proof of Lemma~\ref{L:apriorilipschitz}] Throughout the proof we write $\lesssim$ if $\leq$ holds up to a multiplicative constant depending only on $\kappa,\Lambda,\nu,n,p$ and $q$.
\step 0 Technical estimates.
There exists $c_1=c_1(\nu,\Lambda,p,q)\in[1,\infty)$ such that for all $t\geq T>0$ holds
\begin{equation}\label{eq:relateg1gT}
\frac{g_{2,\e}(t)}{g_1(t)}\leq c_1 (G_T(t)^\frac{q-p}p+1+\frac{\e}{(\mu^2+T^2)^{\frac{p-2}2}})\quad\mbox{and}\quad t\leq c_1 G_T(t)^\frac1p+(\mu^2+T^2)^\frac12.
\end{equation}
To establish \eqref{eq:relateg1gT}, we first compute for all $t\geq T$
\begin{equation}\label{GTt}
G_T(t)=\nu\int_T^{t}(\mu^2+s^2)^\frac{p-2}2s\,ds=\frac\nu{p}(\mu^2+t^2)^\frac{p}2-\frac\nu{p}(\mu^2+T^2)^\frac{p}2
\end{equation}
and thus
\begin{align*}
\frac{g_{2,\e}(t)}{g_1(t)}=&\frac{\Lambda}\nu(\mu^2+t^2)^{\frac{q-p}2}+\frac{\Lambda}\nu+\e\frac{\Lambda}\nu\frac{(1+t^2)^\frac{\min\{p-2,0\}}2}{(\mu^2+t^2)^{\frac{p-2}2}}\\
\leq& \frac{\Lambda}\nu \biggl(\frac{p}\nu G_T(t)+(\mu^2+T^2)^\frac{p}2\biggr)^{\frac{q-p}p}+\frac\Lambda\nu+\e\frac{\Lambda}{\nu}(1+(\mu^2+T^2)^{-\frac{p-2}2}),
\end{align*}
which implies the first estimate of \eqref{eq:relateg1gT} (recall $\mu\in[0,1]$ and $\e,T\in(0,1]$). The second estimate of \eqref{eq:relateg1gT} follows from \eqref{GTt} in the form: For $t\geq T>0$ holds
$$
t^p\leq \frac{p}\nu G_T(t)+(\mu^2+T^2)^\frac{p}2
$$
and the second estimate of \eqref{eq:relateg1gT} follows by taking the $p$-th root.
\step 1 In this step, we suppose $B_1\Subset B$ and prove
\begin{align}\label{est:keyapprio}
&\|G_T(|\nabla u|)\|_{L^\infty(B_\frac14)}\notag\\
\lesssim&\|G_T(|\nabla u|)\|_{L^\infty(B_1)}^\gamma\|G_T(|\nabla u|)\|_{L^1(B_1)}^\frac12\notag\\
&+(1+\tfrac{\e}{(\mu^2+T^2)^\frac{p-2}2})^{\frac12\max\{\kappa,\frac{n-3}2\}+\frac12}\|G_T(|\nabla u|)\|_{L^\infty(B_1)}^\frac12\|G_T(|\nabla u|)\|_{L^1(B_1)}^{\frac12}\notag\\
&+(1+\tfrac{\e}{(\mu^2+T^2)^\frac{p-2}2})^{\frac12\max\{\kappa,\frac{n-3}2\}}(\|G_T(|\nabla u|)\|_{L^\infty(B_1)}^\frac1p+(\mu^2+T^2)^\frac12)\|f\|_{L^{n,1}(B_1)}\notag\\
&+(\|G_T(|\nabla u|)\|_{L^\infty(B_1)}^{\tilde \gamma}+(\mu^2+T^2)^{\frac12 \tilde \gamma p})\|f\|_{L^{n,1}(B_1)}
\end{align}
where $\gamma$ and $\tilde \gamma$ are defined in \eqref{def:gammavartheta}.
A direct consequence of the Caccioppoli inequality of Lemma~\ref{L:caccio} and the iteration Lemma~\ref{L:basiciteration} with the choice
$$
M_1^2=\biggl\|\frac{g_{2,\e}(|\nabla u|)}{g_1(|\nabla u|)}\biggr\|_{L^\infty(B_1\cap\{|\nabla u| \geq T\})}\qquad\mbox{and}\qquad M_2^2=\|\nabla u\|_{L^\infty(B_1)}^2
$$
is the following Lipschitz estimate
\begin{align}\label{est:keyapprio:prep1}
&\|G_T(|\nabla u|)\|_{L^\infty(B_\frac14)}\notag\\
\lesssim&\biggl(\biggl\|\frac{g_{2,\e}(|\nabla u|)}{g_1(|\nabla u|)}\biggr\|_{L^\infty(B_1\cap\{|\nabla u| \geq T\})}\biggr)^{\frac12+\frac12\max\{\kappa,\frac{n-3}2\}}\|G_T(|\nabla u|)\|_{L^{2}(B_1)}\notag\\%+\|\nabla u\|_{L^\infty(B_1 )}\|f\|_{L^{n,1}(B_1)}\notag\\
&+\biggl(\biggl\|\frac{g_{2,\e}(|\nabla u|)}{g_1(|\nabla u|)}\biggr\|_{L^\infty(B_1\cap\{|\nabla u| \geq T\})}\biggr)^{\frac12\max\{\kappa,\frac{n-3}2\}}\|\nabla u\|_{L^\infty(B_1 )}\|f\|_{L^{n,1}(B_1)}.
\end{align}
Estimate \eqref{est:keyapprio} follows from \eqref{est:keyapprio:prep1} in combination with \eqref{eq:relateg1gT} in the form
\begin{align}\label{est:keyapprio:prep2}
\biggl\|\frac{g_{2,\e}(|\nabla u|)}{g_1(|\nabla u|)}\biggr\|_{L^\infty(B_1\cap\{|\nabla u| \geq T\})}\lesssim \|G_T(|\nabla u|)\|_{L^\infty(B_1)}^\frac{q-p}p+1+\frac{\e}{(\mu^2+T^2)^{\frac{p-2}2}}
\end{align}
and
\begin{equation*}
\|\nabla u\|_{L^\infty(B_1)}\leq c_1 \|G_T(|\nabla u|)\|_{L^\infty(B_1)}^\frac1p+(\mu^2+T^2)^\frac12,
\end{equation*}
and the elementary interpolation inequality $\|\cdot\|_{L^2}\leq (\|\cdot\|_{L^\infty}\|\cdot\|_{L^1})^\frac12$.
\step 2 Conclusion
Appealing to standard scaling and covering arguments, we deduce from Step~1 the following: For every $x_0\in B$ and $0<\rho<\sigma$ satisfying $B_\sigma(x_0)\Subset B$ it holds
\begin{align}\label{est:keyapprio1}
&\|G_T(|\nabla u|)\|_{L^\infty(B_\rho(x_0))}\notag\\
\lesssim&(\sigma-\rho)^{-\frac{n}2}\|G_T(|\nabla u|)\|_{L^\infty(B_\sigma(x_0))}^\gamma\|G_T(|\nabla u|)\|_{L^1(B_\sigma(x_0))}^\frac12\notag\\
&+(\sigma-\rho)^{-\frac{n}2}(1+\tfrac{\e}{(\mu^2+T^2)^\frac{p-2}2})^{\frac12\max\{\kappa,\frac{n-3}2\}+\frac12}\|G_T(|\nabla u|)\|_{L^\infty(B_\sigma(x_0))}^{\frac12}\|G_T(|\nabla u|)\|_{L^1(B_\sigma(x_0))}^{\frac12}\notag\\
&+(1+\tfrac{\e}{(\mu^2+T^2)^\frac{p-2}2})^{\frac12\max\{\kappa,\frac{n-3}2\}}(\|G_T(|\nabla u|)\|_{L^\infty(B_\sigma(x_0))}^\frac1p+(\mu^2+T^2)^\frac12)\|f\|_{L^{n,1}(B_\sigma(x_0))}\notag\\
&+(\|G_T(|\nabla u|)\|_{L^\infty(B_\sigma(x_0))}^{\tilde \gamma}+(\mu^2+T^2)^{\frac12 \tilde \gamma p})\|f\|_{L^{n,1}(B_\sigma(x_0))}
\end{align}
From estimate \eqref{est:keyapprio1} in combination with Young inequality and assumption $\gamma,\tilde \gamma\in(0,1)$, we obtain the existence of $c=c(\kappa,\nu,\Lambda,n,p,q)\in[1,\infty)$ such that
\begin{align*}
\|G_T(|\nabla u|)\|_{L^\infty(B_\rho(x_0))}\leq& \frac12\|G_T(|\nabla u|)\|_{L^\infty(B_\sigma(x_0))}+c\frac{\|G_T(|\nabla u|)\|_{L^1(B_\sigma(x_0))}^{\frac12\frac1{1-\gamma}}}{(\sigma-\rho)^{\frac1{1-\gamma}\frac{n}2}}\\
&+ c\frac{(1+\tfrac{\e}{(\mu^2+T^2)^{\frac{p-2}2}})^{\max\{\kappa,\frac{n-3}2\}+1}\|G_T(|\nabla u|)\|_{L^1(B_\sigma(x_0))}}{(\sigma-\rho)^{n}}\\
&+(1+\tfrac{\e}{(\mu^2+T^2)^\frac{p-2}2})^{\frac12\max\{\kappa,\frac{n-3}2\}}((T^2+\mu^2)^\frac12+(T^2+\mu^2)^{\frac12\tilde \gamma p})\|f\|_{L^{n,1}(B_\sigma(x_0))}\\
&+c\biggl((1+\tfrac{\e}{(\mu^2+T^2)^\frac{p-2}2})^{\frac12\max\{\kappa,\frac{n-3}2\}}\|f\|_{L^{n,1}(B_\sigma(x_0))}\biggr)^\frac{p}{p-1}\\
&+c\|f\|_{L^{n,1}(B_\sigma(x_0))}^\frac{1}{1-\tilde \gamma}.
\end{align*}
The claimed inequality \eqref{claim:L:apriorilipschitz} (for $B=B_1$) now follows from Lemma~\ref{L:holefilling}.
\end{proof}
\section{Proof of Theorem~\ref{T:1}}
In this section, we prove Theorem~\ref{T:1} together with a suitable gradient estimate. More precisely, we show the following result which obviously contains the statement of Theorem~\ref{T:1}
\begin{theorem}\label{T:2}
Let $\Omega\subset\R^n$, $n\geq3$ be an open bounded domain and suppose Assumption~\ref{ass} is satisfied with $1<p<q<\infty$ such that \eqref{eq:pqrhs}. Let $u\in W_{\rm loc}^{1,1}(\Omega)$ be a local minimizer of the functional $\mathcal F$ given in \eqref{eq:int} with $f\in L^{n,1}(\Omega)$. Then $\nabla u$ is locally bounded in $\Omega$. Moreover, for every $\kappa\in(0,\min\{\frac12,\frac{2p-q}{q-p},2\frac{p-1}{q-p}\})$ there exists $c=c(\kappa,\Lambda,\nu,n,p,q)\in[1,\infty)$ such that for all $B\Subset \Omega$ it holds
\begin{align}\label{est:T1}
\|\nabla u\|_{L^\infty(\frac12 B)}\leq& c\biggl(\fint_B F(\nabla u)\,dx+\|f\|_{L^{n,1}(B)}^\frac{p}{p-1}\biggr)^\frac1p\notag\\
&+c\biggl(\fint_B F(\nabla u)\,dx+\|f\|_{L^{n,1}(B)}^\frac{p}{p-1}\biggr)^{\alpha_n}\notag\\
&+c\|f\|_{L^{n,1}(B)}^{\beta_n}
\end{align}
where
\begin{align}\label{def:alphabetan}
\alpha_n:=\begin{cases}\frac{2}{(n+1)p-(n-1)q}&\mbox{if $n\geq4$}\\ \frac1{2p-q-(q-p)\kappa}&\mbox{if $n=3$}\end{cases},\quad \beta_n:=\begin{cases}\frac{4}{4(p-1)-(q-p)(n-3)}&\mbox{if $n\geq4$}\\\frac2{2(p-1)-(q-p)\kappa}&\mbox{if $n=3$}\end{cases}
\end{align}
In the case $n\geq4$ the constant $c$ in \eqref{est:T1} is independent of $\kappa$. When $p\geq 2-\frac{4}{n+1}$ or when $f\equiv0$ condition \eqref{eq:pqrhs} can be replaced by \eqref{eq:pq}.
\end{theorem}
\begin{proof}[Proof of Theorem~\ref{T:2}]
Throughout the proof we write $\lesssim$ if $\leq$ holds up to a multiplicative constant that depends only on $\kappa,\Lambda,\nu,n,p$ and $q$. We assume that $B_2\Subset\Omega$ and show
\begin{align}\label{est:apriorilipschitzu0}
&\|\nabla u\|_{L^\infty(B_\frac12)}\notag\\%\|G_T(\nabla u_{m})\|_{L^\infty(B_\frac12)}\notag\\
\lesssim&\biggl(\fint_{B_{1}}F(\nabla u)\,dx+\|f\|_{L^{n,1}(B_1)}^\frac{ p}{ p-1}\biggr)^\frac1p\notag\\
&+\biggl(\fint_{B_{1}}F(\nabla u)\,dx+\|f\|_{L^n(B_1)}^\frac{ p}{ p-1}\biggr)^{\frac1{2p}\frac1{1-\gamma}}+\|f\|_{L^{n,1}(B_1)}^{\frac1p\frac1{1-\tilde \gamma}},
\end{align}
where $\gamma$ and $\tilde\gamma$ are given in \eqref{def:gammavartheta}. Clearly, the conclusion follows from a standard scaling, translation and covering arguments using $\alpha_n={\frac1{2p}\frac1{1-\gamma}}$ and $\beta_n={\frac1p\frac1{1-\tilde \gamma}}$.
\step 0 Preliminaries.
Following \cite{BM20}, we introduce various regularizations on the minimizer $u$, the integrand $F$ and the forcing term $f$: For this we choose a decreasing sequence $(\e_m)_{m\in\mathbb N}\subset(0,1)$ satisfying $\e_m\to0$ as $m\to\infty$. We set $\overline u_m:=u\ast \varphi_{\e_m}$ with $\varphi_\e:=\e^{-n}\varphi(\frac{\cdot}\e)$ and $\varphi$ being a non-negative, radially symmetric mollifier, i.e. it satisfies
$$
\varphi\geq0,\quad {\rm supp}\; \varphi\subset B_1,\quad \int_{\R^n}\varphi(x)\,dx=1,\quad \varphi(\cdot)=\widetilde \varphi(|\cdot|)\quad \mbox{for some $\widetilde\varphi\in C^\infty(\R)$}.
$$
Moreover, we denote by $f_m$ the truncated forcing $f_m(x)=\min\{\max\{f(x),-m\},m\}$ and consider the functional
\begin{equation}
\mathcal F_{m}(w,B):=\int_B[F_{\e_m}(\nabla w)-f_mw]\,dx,
\end{equation}
where for all $z\in\R^n$
$$F_\e(z):=\widetilde F_{\e}(z)+\e L_p(z)\quad\mbox{with $L_p(z):=\frac12|z|^2$ for $p\geq2$ and $L_p(z):=(1+|z|^2)^\frac{p}2-1$ for $p\in(1,2)$}$$
and $\widetilde F_\e$ satisfies for all $\e\in(0,\e_0]$ with $\e_0=\e_0(F,T)\in(0,1]$
\begin{equation}\label{ass:tildeFeps}
\widetilde F_\e\geq0,\quad \mbox{$\widetilde F_\e\in C^2_{\rm loc}(\R^n)$ is convex and $\widetilde F_\e=F$ on $\R^n\setminus B_\frac{T}2$}.
\end{equation}
(obviously $\widetilde F_\e$ and thus $F_\e$ depends also on $T>0$ which is suppressed in the notation) and it holds
\begin{equation}\label{ass:tildeFeps1}
\limsup_{\e\to0}\sup_{z\leq T}|\widetilde F_\e(z)-F(z)|=0.
\end{equation}
In the case $F\in C^2_{\rm loc}(\R^n)$, we simply set $\widetilde F_\e\equiv F$ and in the case that $F$ is singular at zero we give (a standard) smoothing and gluing construction in Step~3 below.
Clearly, the functionals $\mathcal F_{m}$ are strictly convex and we denote by $u_m\in W^{1,1}(B)$ the unique function satisfying
\begin{equation}
\mathcal F_m(u_m,B)\leq \mathcal F_m(v,B)\qquad\mbox{for all $v\in \overline u_m+W_0^{1,1}(B)$}
\end{equation}
Appealing to \cite[Theorem~4.10]{BM20} (based on \cite{BB16}), we have $u_m\in W_{\rm loc}^{1,\infty}(B)$. In particular it follows that $u_m$ satisfies the Euler-Lagrange equation
$$
-\divv (\partial F_{\e_m}(\nabla u_m))=f_m
$$
and since $\bfa_\e:=\partial F_{\e_m}$ satisfies Assumption~\ref{ass:reg} (with $\e=\e_m$) we can apply Lemma~\ref{L:apriorilipschitz}. Note that in view of Remark~\ref{rem:gamma}, the assumptions on $p,q$ and $\kappa$ ensure $\gamma,\tilde \gamma\in (0,1)$.
\step 1 We claim that
\begin{align}\label{est:apriorilipschitzum}
&\|\nabla u_m\|_{L^\infty(B_\frac12)}^p\notag\\%\|G_T(\nabla u_{m})\|_{L^\infty(B_\frac12)}\notag\\
\lesssim&(1+\tfrac{\e_m}{(\mu^2+T^2)^\frac{p-2}2})^{\max\{\kappa,\frac{n-3}2\}+1}\biggl(\int_{B_{1+\e_m}}\widetilde F_{\e_m}(\nabla u)\,dx+\e_m\int_{B_1}L_p(\nabla \overline u_m)\,dx+\|f\|_{L^n(B_1)}^\frac{ p}{ p-1}+T^{ p}+\mu^{ p}\biggr)\notag\\
&+\biggl(\int_{B_{1+\e_m}}\widetilde F_{\e_m}(\nabla u)\,dx+\e_m\int_{B_1}L_p(\nabla \overline u_m)\,dx+\|f\|_{L^n(B_1)}^\frac{ p}{ p-1}+T^{ p}+\mu^{ p}\biggr)^{\frac12\frac1{1-\gamma}}\notag\\
&+(1+\tfrac{\e_m}{(\mu^2+T^2)^\frac{p-2}2})^{\frac12\max\{\kappa,\frac{n-3}2\}}((T^2+\mu^2)^\frac12+(T^2+\mu^2)^{\frac12\tilde \gamma p})\|f\|_{L^{n,1}(B_1)}\notag\\
&+\biggl((1+\tfrac{\e_m}{(\mu^2+T^2)^\frac{p-2}2})^{\frac12\max\{\kappa,\frac{n-3}2\}}\|f\|_{L^{n,1}(B_1)}\biggr)^\frac{p}{p-1}\notag\\
&+\|f\|_{L^{n,1}(B_1)}^\frac1{1-\tilde \gamma}
\end{align}
and
\begin{equation}\label{est:eneum}
\int_{B_1}F_{\e_m}(\nabla u_{m})\,dx\lesssim \int_{B_{1+\e_m}}\widetilde F_{\e_m}(\nabla u)\,dx+\e_m\int_{B_1}L_p(\nabla \overline u_m)\,dx+\|f\|_{L^n(B_1)}^{\frac{ p}{ p-1}}+(\mu^2+T^2)^\frac{p}2.
\end{equation}
A combination of H\"older and Sobolev inequality with the elementary inequality
$$
\nu |z|^p\leq \nu(\mu^2+T^2)^\frac{p}2+F_{\e_m}(z)
$$
(which follows from the definition of $F_\e$, \eqref{ass:tildeFeps} and \eqref{ass:Fpq}) yields
\begin{align*}
\|f_m(u_{m}-\overline u_m)\|_{L^1(B_1)}\leq& \|f_m\|_{L^n(B_1)}\|u_m-\overline u_m\|_{L^{\frac{n}{n-1}}(B_1)}\\
\leq&c(n, p) \|f_m\|_{L^n(B_1)}\|\nabla (u_m-\overline u_m)\|_{L^{p}(B_1)}\\
\leq&c\|f_m\|_{L^n(B_1)}\biggl(\int_{B_1}F_{\e_m}(\nabla u_{m})+F_{\e_m}(\nabla \overline u_m)\,dx+(\mu^2+T^2)^\frac{p}2\biggr)^\frac1{ p}\\
\leq&\tfrac1{ p}\biggl(\int_{B_1}F_{\e_m}(\nabla u_{m})+F_{\e_m}(\nabla \overline u_m)\,dx+(\mu^2+T^2)^\frac{p}2\biggr)+(1-\tfrac1{ p})(c\|f_m\|_{L^n(B_1)})^\frac{ p}{ p -1}
\end{align*}
where $c=c(n,\nu,\Lambda,p,q)\in[1,\infty)$. Combining the above estimate with the minimality of $u_{m}$ and the convexity of $\widetilde F_\e$ in the form
\begin{align}\label{est:eneum2}
\int_{B_1}F_{\e_m}(\nabla u_{m})\,dx\leq& \int_{B_1}F_{\e_m}(\nabla \overline u_m)-f_m(\overline u_m-u_{m})\,dx\notag\\
\leq& \int_{B_{1+\e_m}}\widetilde F_{\e_m}(\nabla u)\,dx+\int_{B_1}\e_mL_p(\nabla \overline u_m)-f_m(\overline u_m-u_{m})\,dx
\end{align}
we obtain \eqref{est:eneum}. The claimed Lipschitz-estimate \eqref{est:apriorilipschitzum} follows from Lemma~\ref{L:apriorilipschitz}, estimates \eqref{eq:relateg1gT}, \eqref{est:eneum} and
\begin{align*}
0\leq G_T(|\nabla u_m|)\stackrel{\eqref{GTt}}\lesssim& (\mu^2+|\nabla u_m|^2)^\frac{p}2\stackrel{\eqref{ass:Fpq}}{\lesssim} F_{\e_m}(\nabla u_m)+(\mu^2+T^2)^\frac{p}2.
\end{align*}
\step 2 Passing to the limit.
\substep{2.1} We claim
\begin{equation}\label{claim:umem}
\lim_{m\to\infty}\e_m\int_{B_1}L_p(\nabla \overline u_m)\,dx=0
\end{equation}
and
\begin{equation}\label{claim:umem0}
\lim_{m\to\infty}\int_{B_{1+\e_m}}\widetilde F_{\e_m}(\nabla u)\,dx= \int_{B_1}F(\nabla u)\,dx.
\end{equation}
We first note that $F(\nabla u)\in L^1_{\rm loc}(B_2)$. Indeed, by Definition~\ref{def:localmin} combined with H\"older and Sobolev inequality, we have for every $\tilde B\Subset B_2$
\begin{align*}
\int_{\tilde B}F(\nabla u)\,dx=&\mathcal F(u,\tilde B)+\int_{\tilde B}fu\leq \mathcal F(u,\tilde B)+\|f\|_{L^n(\tilde B)}\|u\|_{L^{\frac{n}{n-1}}(\tilde B)}\\
\lesssim& \mathcal F(u,\tilde B)+\|f\|_{L^n(\tilde B)}\|u\|_{W^{1,1}(\tilde B)}<\infty
\end{align*}
For $p\geq2$, equation \eqref{claim:umem} follows from
$$
\int_{B_1}L_p(\nabla\overline u_m)\,dx=\frac12\|\nabla \overline u_m\|_{L^2(B_1)}^2\lesssim \|\nabla u\|_{L^2(B_\frac32)}^2\lesssim \biggl(\int_{B_\frac32}F(\nabla u)\,dx\biggr)^\frac{2}{ p}<\infty.
$$
In the case $p\in(1,2)$, equation \eqref{claim:umem} is a consequence of
$$
\|L_p(\nabla \overline u_m)\|_{L^1(B_1)}\lesssim 1+\|\nabla u\|_{L^p(B_\frac32)}^p\lesssim 1+\int_{B_\frac32}F(\nabla u)\,dx<\infty.
$$
The argument for \eqref{claim:umem0} follows from the identity
$$
\int_{B_{1+\e_m}}\widetilde F_{\e_m}(\nabla u)\,dx= \int_{B_{1+\e_m}}F(\nabla u)\,dx+\int_{B_{1+\e_m}\cap \{|\nabla u|\leq T\}}\widetilde F_{\e_m}(\nabla u)-F(\nabla u)\,dx.
$$
together with the uniform convergence \eqref{ass:tildeFeps1} and $F(\nabla u)\in L^1_{\rm loc}(B_2)$.
\substep{2.2} Proof of \eqref{est:apriorilipschitzu0}.
From \eqref{est:apriorilipschitzum}, \eqref{est:eneum} and \eqref{claim:umem}, we deduce the existence of a subsequence and $\overline u\in u+W^{1,1}_0(B)$ such that
\begin{align*}
u_m\rightharpoonup \overline u&\qquad\mbox{weakly in $W^{1, p}(B_1)$}\\
u_m\rightharpoonup \overline u&\qquad\mbox{weakly$^*$ in $W^{1,\infty}(B_\frac12)$}
\end{align*}
In view of \eqref{est:apriorilipschitzum}, \eqref{claim:umem}, \eqref{claim:umem0} and the weak$^*$ lower semicontinuity of norms $\overline u$ satisfies
\begin{align}\label{est:apriorilipschitzu}
\|\nabla \overline u\|_{L^\infty(B_\frac12)}
\lesssim&\biggl(\int_{B_{1}}F(\nabla u)\,dx+\|f\|_{L^n(B_1)}^\frac{p}{ p-1}+T^p+\mu^p\biggr)^\frac1p\notag\\
&+\biggl(\int_{B_{1}}F(\nabla u)\,dx+\|f\|_{L^n(B_1)}^\frac{ p}{ p-1}+T^{ p}+\mu^{ p}\biggr)^{\frac1{2p}\frac1{1-\gamma}}\notag\\
&+((T^2+\mu^2)^\frac1{2p}+(T^2+\mu^2)^{\frac12\tilde \gamma })\|f\|_{L^{n,1}(B_1)}^\frac1p
+\|f\|_{L^{n,1}(B_1)}^{\frac1p\frac1{1-\tilde \gamma}}\notag\\
\lesssim&\biggl(\int_{B_{1}}F(\nabla u)\,dx+\|f\|_{L^{n,1}(B_1)}^\frac{p}{ p-1}+T^p+\mu^p\biggr)^\frac1p\notag\\
&+\biggl(\int_{B_{1}}F(\nabla u)\,dx+\|f\|_{L^{n}(B_1)}^\frac{ p}{ p-1}+T^{ p}+\mu^{ p}\biggr)^{\frac1{2p}\frac1{1-\gamma}}
+\|f\|_{L^{n,1}(B_1)}^{\frac1p\frac1{1-\tilde \gamma}},
\end{align}
where we use in the last estimate $\|f\|_{L^n(B_1)}\lesssim\|f\|_{L^{n,1}(B_1)}$ and Youngs inequality in the form
$$
((T^2+\mu^2)^\frac1{2p}+(T^2+\mu^2)^{\frac12\tilde \gamma })\|f\|_{L^{n,1}(B_1)}^\frac1p\lesssim ((T^2+\mu^2)^\frac1{2}+\|f\|_{L^{n,1}(B_1)}^\frac1{p-1}+\|f\|_{L^{n,1}(B_1)}^{\frac1p\frac1{1-\tilde \gamma}}.
$$
By the definition of $F_\e$ we have
\begin{align*}
\int_{B_1}F_{\e_m}(\nabla u_m)\,dx\geq& \int_{B_1}\widetilde F_{\e_m}(\nabla u_m)\,dx\\
=&\int_{B_{1}}F(\nabla u_m)\,dx+\int_{B_{1}\cap \{|\nabla u|\leq T\}}\widetilde F_{\e_m}(\nabla u_m)-F(\nabla u_m)\,dx\\
\geq&\int_{B_{1}}F(\nabla u_m)\,dx-|B_1|\sup_{z\leq T}|\widetilde F_\e(z)-F(z)|.
\end{align*}
Hence, using convexity of $F$ and the uniform convergence \eqref{ass:tildeFeps1}, we can pass to the limit (along the above chosen subsequence) in \eqref{est:eneum2} and obtain with help of \eqref{claim:umem} and \eqref{claim:umem0}
\begin{align*}
\int_{B_1}F(\nabla \overline u)\,dx\leq \int_{B_{1}}F(\nabla u)-f(u-\overline u)\,dx
\end{align*}
and thus
$$
\mathcal F(\overline u,B_1)\leq \mathcal F(u,B_1).
$$
The above inequality combined with $\overline u\in u+W_0^{1,p}(B_1)$ and the strict convexity of $\mathcal F(\cdot,B_1)$ implies $\overline u=u$. The claimed estimate \eqref{est:apriorilipschitzu0} follows by sending $T$ to $0$ in \eqref{est:apriorilipschitzu} combined with \eqref{ass:Fpq} in the form $\nu\mu^p\leq \fint_{B}F(\nabla u)\,dx$.
\step 3 Construction of $\widetilde F_\e$.
Let $\rho\in C^\infty(\R,[0,1])$ be such that $\rho\equiv1$ on $(-\infty,\frac{T}4)$ and $\rho\equiv0$ on $(\frac{T}3,\infty)$.
We set $\hat F_\e:=F\ast \varphi_{\e}$ where $\varphi_\e$ is as in Step~1 and $\widetilde F_\e=\rho \hat F_\e+(1-\rho)F$. By general properties of the mollification, we have that $\hat F_\e$ is smooth, non-negative and convex, and thus $\widetilde F_\e$ is non-negative, locally $C^2$ and it holds $\widetilde F_\e\equiv F$ on $\R^n\setminus B_{T/2}$. Since $F$ is convex and locally bounded, we also have \eqref{ass:tildeFeps1}. It remains to show that $\widetilde F_\e$ is convex for $\e>0$ sufficiently small. For this we observe that
$$
\nabla^2 \widetilde F=\rho \nabla^2 \hat F_\e+(1-\rho)\nabla^2 F+(\hat F_\e-F_\e)\nabla^2\rho +\nabla(\hat F_\e-F_\e)\otimes\nabla \rho+\nabla \rho\otimes \nabla(\hat F_\e-F_\e)
$$
is strictly positive definite since $\rho \nabla^2 \hat F_\e+(1-\rho)\nabla^2 F$ is strictly positive definite and the remainder tends to zero as $\e\to0$.
\step 4 The case $f\equiv 0$. It is straightforward to check that the restriction $\tilde\gamma\in(0,1)$ is not needed in Lemma~\ref{L:apriorilipschitz} if $f\equiv0$ and since we checked in Remark~\ref{rem:gamma} that \eqref{eq:pq} suffices to ensure $\gamma\in(0,1)$ the claim follows.
\end{proof}
\section*{Acknowledgments}
PB was partially supported by the German Science Foundation DFG in context of the Emmy Noether Junior Research Group BE 5922/1-1.
| 97,719
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Syrian Army on the verge of taking control of Damascus-Golan Heights border: map
BE.
| 69,304
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TITLE: Minimal polynomial and field extension
QUESTION [3 upvotes]: Suppose the degree of a field extension $[\mathbb{Q}(\alpha):\mathbb{Q}]=n\gt 1$ and $\alpha$ is a root of a monic polynomial $f \in \mathbb{Q}[T]$ and the degree of $f$ is $n$.
Does the above imply that $f$ is irreducible and hence the minimal polynomial of the extension?
I'm wondering if that is the case.
REPLY [4 votes]: If $\alpha$ is a root of $f$, then the minimal polynomial $p_\alpha$ of $\alpha$ divides $f$. Recall that $p_\alpha$ has degree $n=[\mathbb Q(\alpha):\mathbb Q]$, since $\mathbb Q(\alpha)\simeq \mathbb Q[x]/(p_\alpha)$.
Since $f$ is monic and a multiple of $p_\alpha$ of the same degree, it follows that $f=p_\alpha$ so that $f$ is irreducible. However, notice that we've used that $f$ is the minimal polynomial to deduce irreducibility -- I'm not sure about proving irreducibility first, without using some variant of the fact that $f$ must be the minimal polynomial.
Edit: Alternately, we can note that if $f(x)=g(x)h(x)$ for $g$, $h$ nonconstant polynomials, then $\alpha$ is a root of either $g$ or $h$. This implies that the minimal polynomial of $\alpha$ over $\mathbb Q$ has degree less than $[\mathbb Q(\alpha):\mathbb Q]$, a contradiction.
| 31,009
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\begin{document}
\title{Continuous time random walks modeling of quantum measurement
and fractional equations of quantum stochastic filtering and control}
\author{Vassili N. Kolokoltsov\thanks{Department of Statistics, University of Warwick,
Coventry CV4 7AL UK, associate member of HSE, Moscow,
Email: v.kolokoltsov@warwick.ac.uk}}
\maketitle
\begin{abstract}
Initially developed in the framework of quantum stochastic calculus, the main equations of quantum
stochastic filtering were later on derived as the limits of Markov models of discrete measurements
under appropriate scaling. In many branches of modern physics it became popular to extend random
walk modeling to the continuous time random walk (CTRW) modeling, where the time between discrete
events is taken to be non-exponential. In the present paper we apply the CTRW modeling to the
continuous quantum measurements yielding the new fractional in time evolution equations of quantum
filtering and thus new fractional equations of quantum mechanics of open systems. The related quantum
control problems and games turn out to be described by the fractional Hamilton-Jacobi-Bellman (HJB)
equations on Riemannian manifolds. By-passing we provide a full derivation of the standard quantum
filtering equations, in a modified way as compared with existing texts, which (i) provides
explicit rates of convergence (that are not available via the tightness of martingales approach
developed previously) and (ii) allows for the direct applications of the basic results of CTRWs to
deduce the final fractional filtering equations.
\end{abstract}
{\bf Key words:} CTRW, quantum stochastic filtering, fractional quantum control, Belavkin equation, fractional quantum mechanics,
fractional quantum mean field games, fractional Hamilton-Jacobi-Bellman-Isaacs equation on manifolds.
{\bf MSC2010:} 35R11, 81Q93, 93E11, 93E20.
\section{Introduction}
Direct continuous observations are known to destroy quantum evolutions
(so-called quantum Zeno paradox), so that continuous quantum measurements have to be indirect,
and the results of the observation are assessed via quantum filtering. Initially developed in the framework
of quantum stochastic calculus by Belavkin in the 80s of the last century in \cite{Bel87},
\cite{Bel88}, \cite{Bel92}, see \cite{BoutHanJamQuantFilt} for a readable modern account,
the main equations of quantum stochastic filtering, often referred
to as the Belavkin equations, were later on derived
via more elementary approach, as the limit of standard discrete measurements under appropriate scaling,
see e.g. \cite{Be195}, \cite{BelKol}, \cite{Pellegrini}.
The scaling arises from the basic Markovian assumption that the times between measurement are either fixed or
exponentially distributed, like in a standard random walk. Since such Markovian assumption has no a priori
justification, in many branches of modern physics it became popular
to extend random walk modeling to the continuous time random walk (CTRW) modeling, where the time between
discrete events is taken to be non-exponential, usually from the domain of attraction of a stable law.
In the present paper we apply the CTRW modeling to the continuous quantum measurements yielding
the new fractional in time evolution equations of quantum filtering in the scaling limit. The related quantum
control problems turn out to be described by the fractional Hamilton-Jacobi-Bellman (HJB) equations on
Riemannian manifolds (complex projective spaces in the case of finite-dimensional quantum mechanics)
or the fractional Isaacs equation in the case of competitive control. By-passing
we provide a full derivation of the standard quantum filtering equations
(explaining from scratch all underlying quantum mechanical rules used)
in a slightly modified and simplified way
yielding also new explicit rates of convergence (which are not available via
the tightness of martingales approach developed previously) and tailored in a way that allows for the direct
applications of the basic results of CTRWs to deduce the final fractional filtering equations.
Several general comments on a wider context are in order.
(i) The fractional equations of quantum stochastic filtering derived here can be considered as an alternative
formulation of fractional quantum mechanics, which is different from the framework of fractional
Schr\"odinger equations suggested in \cite{Lask02} and extensively studied recently. This leads also
to a different class of quantum control problems, as those related to fractional Schr\"odinger formulation,
as discussed e. g. in \cite{Wang12}.
(ii) The fractional versions of the classical stochastic filtering (see \cite{BainCrisbook} for the basics)
has been actively studied recently, see e.g. \cite{Umarov14}.
(iii) The quantum mean-field games as developed by the author in \cite{KolquaMFG} can now be extended to the
theory of fractional quantum mean-field games. The classical versions of fractional mean-field games just
started to appear in the literature, see \cite{Camilli19}. On the other hand, the application
of classical stochastic filtering in the study of mean-field games has also started to appear, see \cite{CainesFilt}.
(iv) Fractional modeling and CTRW become very popular in almost all domains of physics, as well as economics
and finances, see e.g. \cite{Scalabook}, \cite{MetzlerBark14},
\cite{UchaiBook13}, \cite{WestComplexBook} for some representative references.
The content of the paper is as follows. In Section \ref{secnot} we recall the basic notions
and notations of finite-dimensional quantum mechanics, and in Section \ref{secMarchainindirodserv}
we introduce the Markov chain of sequential indirect quantum measurements, which is the standard starting
point for dealing with continuous measurements. In Sections \ref{seccountcase} and \ref{secdifcase}
we derive the main quantum filtering equations in the cases of so-called counting and diffusive
observations. As was already mentioned, though the derivation of the filtering equations from the approximating Markov chain
is well known by now (see e. g. \cite{Pellegrini10a}) our approach is new and yields
explicit rates of convergence. In Section \ref{secseveralchan} the limiting equation is derived in a general case
of mixed counting and diffusive observations via a multichannel measuring device.
This preparatory work allows us to derive our main results, fractional equations of quantum filtering and control,
in a more or less straightforward way, by applying the established techniques of CTRW to the setting
of the Markov chains of sequential quantum measurements, as developed in Sections \ref{seccountcase}
- \ref{secseveralchan}. This is done in Sections \ref{secfraceq} and \ref{secfraceqHJB}.
In Section \ref{secunbound} we briefly describe a slightly different Markov chain approximation to
continuous measurement that can be used to derive filtering equations in certain cases of unbounded operators
involved.
In Appendices A,B,C several (known) probabilistic techniques are presented
in a concise form tailored to our purposes. They are used in the main body of the paper.
Some basic notations to be used throughout the text are as follows.
For two Banach spaces $B$ and $D$ equipped with norms $\|.\|_B$ and $\|.\|_B$ respectively,
let us denote by $\LC(D,B)$ the Banach space of
bounded linear operators in $B$ equipped with the usual operator norm $\|.\|_{D\to B}$.
We shall also write $\LC(B)$ for $\LC(B,B)$.
The scalar product of operators in a Hilbert space is given by the trace:
$(R,S)={\tr} (RS)$.
For $K=\R^d$ or a convex closed subset of $\R^d$ we denote $C(K)$ the Banach space of continuous
bounded functions on $K$, equipped with the sup-norm and $C^k(K)$ the Banach space of $k$ times
continuously differentiable functions on $K$ (with the derivatives at the boundary understood as
the continuous extensions of the derivatives in the inner points), with the norm being the sum
of the sup-norms of the functions and all their partial derivatives of order not exceeding $k$.
\section{Notations for quantum states and tensor products}
\label{secnot}
Recall that a general isolated quantum system is described by a Hilbert space $\HC$ and a self-adjoint operator $H$ in it,
the Hamiltonian. The pure states of the system are unit vectors in $\HC$ and the general mixed states are density matrices,
that is, non-negative operators in $\HC$ with unit trace. Let us denote $S(H)$ the set of all such mixed states in $H$.
To a pure state there corresponds a density matrix according
to the rule $\psi \to \ga=\psi\otimes \bar \psi$, also denoted in Dirac's notation as $|\psi \rangle \langle \psi|$.
This density matrix is the on-dimensional orthogonal projector on the line generated by $\psi$.
Pure states evolve in time according to the rule $\psi \to e^{-itH} \psi$
and the mixed state according to the rule $\ga \to e^{-itH} \ga e^{itH}$.
If two systems living in spaces $\HC_0$ and $\HC_1$ are brought to interaction, the combined system
has the tensor product Hilbert space $\HC_0 \otimes \HC_1$ as the state space. Recall that,
in the coordinate description of tensor products, if $\HC_0$ and $\HC_1$
have orthonormal bases $\{e_j\}$ and $\{f_j\}$ respectively, the tensor product is the space
with an orthnormal basis $\{e_k\otimes f_j\}$. In particular, if $\HC_0$ and $\HC_1$ have finite dimensions $n$ and $k$,
the space $\HC_0 \otimes \HC_1$ has the dimension $nk$.
The operators $A$ in $\HC_0 \otimes \HC_1$ can be given by matrices $A^{i_1i_2}_{j_1j_2}$, so that
\[
A (e_{i_1}\otimes f_{i_2})=\sum_{j_1,j_2} A_{i_1i_2}^{j_1j_2} e_{j_1}\otimes f_{j_2}.
\]
Or equivalently, if $X\in \HC_0 \otimes \HC_1$ has coordinates $X^{kj}$ in the basis $\{e_k\otimes f_j\}$,
the vector $AX$ has the coordinates $\sum_{m,l} A^{kj}_{ml} X^{ml}$ in this basis.
A product $A\otimes B$ of two operators $A$ and $B$ acting in $\HC_0$ and $\HC_1$ respectively
is defined by its action on tensor products as
\[
(A\otimes B)(e\otimes f)=Ae\otimes Bf.
\]
In the coordinate description $A\otimes B$ has the matrix elements expressed as $A^{i_1}_{j_1}B^{i_2}_{j_2}$
in terms of the matrix elements of $A$ and $B$.
An operator $A$ in $\HC_0$ has the natural lifting $A\otimes I$ (where $I$ is the unit operator) to $\HC_0\otimes \HC_1$.
Similarly an operator $B$ in $\HC_1$ has the natural lifting $I\otimes B$ to $\HC_0\otimes \HC_1$.
The key notion of the theory of interacting systems is that of the {\it partial trace}. For an operator
$A$ in $\HC_0 \otimes \HC_1$ the partial trace with respect to the second system is the operator ${\tr}_{p1} A$
in $\HC_0$ given by the matrix
\begin{equation}
\label{eqdefparttr}
({\tr}_{p1} A)^i_j=\sum_k A^{ik}_{jk}.
\end{equation}
This partial trace is interpreted as the state of the first system given the state of the coupled one.
Therefore it can be looked at as the quantum analog of the notion of marginal distribution of classical
probability. Similarly, the partial trace with respect to the first system is the operator ${\tr}_{p0} A$
in $\HC_1$ given by the matrix
\[
({\tr}_{p0} A)^i_j=\sum_k A^{ki}_{kj}.
\]
Clearly,
\[
{\tr} ({\tr}_{p0} A)= {\tr} ({\tr}_{p1} A)= {\tr} (A).
\]
In a two-dimensional Hilbert spaces
$\C^2$ one usually chooses the standard basis $e_0=(1,0)$, $e_1=(0,1)$,
and represents the Hilbert product space $\HC_0\otimes \C^2$ by the natural
decomposition
\[
\HC_0 \otimes \C^2=\HC_{00}\oplus H_{01}=\HC_0 \otimes e_0 \oplus \HC_0 \otimes e_1.
\]
Every operator $A$ in this space has the block decomposition
\[
A=
\begin{pmatrix}
A_{0\to 0} & A_{1\to 0} \\
A_{0\to 1} & A_{1\to 1}
\end{pmatrix}
=
\begin{pmatrix}
(A_{j0}^{i0}) & (A_{j1}^{i0}) \\
(A_{j0}^{i1}) & (A_{j1}^{i1})
\end{pmatrix}
\]
where the operators $A_{i\to j}$ act from $\HC_{0i}$ to $\HC_{0j}$, $i,j=0,1$. The trace
\eqref{eqdefparttr} gets the expression
\begin{equation}
\label{eqdefparttrwithtwodim}
({\tr}_{p1} A)^i_j=A^{i0}_{j0}+A^{i1}_{j1}.
\end{equation}
In particular, we shall use the following block representations:
\begin{equation}
\label{eqblockmatrix}
A\otimes I=
\begin{pmatrix} A & 0 \\ 0 & A \end{pmatrix},
\quad A\otimes \Om=
\begin{pmatrix} A & 0 \\ 0 & 0 \end{pmatrix},
\quad
C\otimes
\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}
=\begin{pmatrix} 0 & 0 \\ C & 0 \end{pmatrix},
\quad
C\otimes
\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}
=\begin{pmatrix} 0 & C \\ 0 & 0 \end{pmatrix}
\end{equation}
More generally,
if $B=(B^i_j)$ is a matrix in $\C^2$, then the matrix of $I\times B$ in $\HC\otimes \C^2$
has the block decomposition
\begin{equation}
\label{eqblockmatrep}
\begin{pmatrix} B^0_0 I & B_1^0 I \\
B^1_0 I & B_1^1 I
\end{pmatrix}.
\end{equation}
To conclude this section let us write down the simple small time asymptotic
formula for the evolutions $e^{-itH}$
that we shall use repeatedly. Namely, up to the terms
of order higher than $t^2$ in small $t$, we have
\[
e^{-itH}\rho e^{itH}
=(1-it H -\frac12 t^2 H^2)\rho (1+it H-\frac12 t^2 H^2)
=\rho-it [H,\rho]-\frac12 t^2 H^2\rho -\frac12 t^2 \rho H^2+t^2 H\rho H
\]
\begin{equation}
\label{smalltimegroup}
=\rho-it [H,\rho]+t^2 (H\rho H-\frac12 \{H^2,\rho\}).
\end{equation}
\section{The starting point: Markov chains of sequential indirect observations}
\label{secMarchainindirodserv}
Here we describe the
Markov chains of sequential indirect observations (rather standard by now, at least after paper \cite{Attal})
in discrete and continuous time recalling first quickly the main notions related to quantum measurements.
{\it Physical observables} are given by self-adjoint operators $A$ in $\HC$. If $A$ has
a discrete spectrum (which is always the case in finite-dimensional $\HC$, that we shall mostly work with),
then $A$ has the spectral decomposition $A=\sum_j \la_j P_j$, where
$P_j$ are orthogonal projections on the eigenspaces of $A$ corresponding to
the eigenvalues $\la_j$. According to the {\it basic postulate of quantum measurement}
\index{basic postulate of quantum measurement}, measuring observable $A$ in a state $\ga$
(often referred to as the {\it Stern-Gerlach experiment}\index{ Stern-Gerlach experiment})
can yield each of the eigenvalue $\la_j$ with the probability
\begin{equation}
\label{eqantiunitdress1}
{\tr} \, (\ga P_j)={\tr} \, (P_j \ga P_j),
\end{equation}
and, if the value $\la_j$ was obtained, the state
of the system changes (instantaneously) to the reduced state
\[
P_j\ga P_j/ {\tr} \, (\ga P_j).
\]
In particular, if the state $\rho$ was pure, $\ga=|\psi\rangle \langle \psi|$, then the
probability to get $\la_j$ as the result of the measurement becomes $(\psi_,P_j\psi)$
and the reduced state also remains pure and is given by the vector $P_j\psi$.
If the interaction with the apparatus was preformed 'without reading the results',
the state $\rho$ is said to be subject to a {\it non-selective measurement}
\index{non-selective measurement} that changes $\ga$ to the state $\sum_j P_j\rho P_j$.
Indirect measurements of a chosen quantum system in the initial space $\HC_0$,
which we shall often referred to as an atom, are organised in the following way.
One couples the atom with another quantum system, a measuring devise, specified
by another Hilbert space $\HC$. Namely the combined system lives in the tensor product
Hilbert space $\HC_0\times \HC$ and its evolution is given by certain self-adjoint
operator $H$ in $\HC_0\times \HC$. In the measuring device some fixed vector $\varphi \in \HC$
is chosen, called the vacuum and interpreted as the stationary state of the devise when
no interaction is involved. The corresponding density matrix will be denoted $\Om=|\varphi \rangle \langle \varphi|$.
Indirect measurements of the states of the atom
are performed by measuring the coupled system via an observable of the second system
and then projecting the resulting state to the atom via the partial trace.
Namely it is described by an operator $R$ in $\HC$ with the spectral decomposition
$R=\sum_j \la_j P_j$ and is performed in two steps: given a state $\ga$ in $\HC_0\times \HC$
one performs a measurement of $R$ lifted as $I\otimes R$ to $\HC_0\times \HC$
yielding values $\la_j$ and new states
\[
(I\otimes P_j)\ga (I\otimes P_j)/ {\tr} \, (\ga (I\otimes P_j))
\]
with probabilities $p_j= {\tr} \, (\ga (I\otimes P_j))$, and then one projects these states to $\HC_0$
via the partial trace producing the states
\begin{equation}
\label{eqindirmeas}
{\tr}_{p1} [(I\otimes P_j)\ga (I\otimes P_j)/ {\tr} \, (\ga (I\otimes P_j))].
\end{equation}
The discrete time {\it Markov chain of successive indirect observations} (or measurements) evolves according
to the following procedure specified by a triple: a self-adjoint operator $H$ in $\HC_0\times \HC$,
a self-adjoint operator $R$ in $\HC$ and the vacuum vector $\Om$ in $\HC$.
(i) Starting with an initial state $\rho$ of $\HC_0$ one couples it with the device in its vacuum state $\Om$
producing the state $\ga=\rho\otimes \Om$ in $\HC_0\times \HC$,
(ii) During a fixed period of time $t$ one evolves the system according to the operator $H$ producing the state
$\ga_t=e^{-itH}\ga e^{itH}$ in $\HC_0\times \HC$,
(iii) One performs the indirect measurement with the state $\ga_t$ yielding the states
\begin{equation}
\label{eqMarkchain}
\rho_t^j={\tr}_{p1} \frac{(I\otimes P_j)\ga_t (I\otimes P_j)}{p_j(t)}
={\tr}_{p1} \frac{(I\otimes P_j)e^{-itH}(\rho\otimes \Om) e^{itH} (I\otimes P_j)}{p_j(t)}
\end{equation}
with the probabilities
\begin{equation}
\label{eqMarkchain1}
p_j(t)={\tr} \, (\ga_t (I\otimes P_j))
={\tr} \, (e^{-itH}(\rho\otimes \Om) e^{itH} (I\otimes P_j)).
\end{equation}
Then the same repeats starting with $\rho_t$ as the initial state. Let us denote $U_t$ the transition
operator of this Markov chain that acts on the set of continuous functions on $S(H)$ as
\begin{equation}
\label{eqMarkchain2}
U_t f(\rho)=\E f(\rho_t)=\sum_j p_j(t) f(\rho_t^j).
\end{equation}
Similarly one can define the continuous time {\it Markov chain of successive indirect observations} (or measurements)
$O^{\rho}_{t,\la}$ and the corresponding Markov semigroup $T_t^{\la}$ on $C(H(S))$
evolving according to the same rules, with only difference that the times $t$ between successive measurements
are not fixed, but represent exponential random variables $\tau$ with some fixed intensity $\la$: $\P(\tau>t)=e^{-\la t}$.
The generator $L^{\la}$ of this Markov process is bounded in $C(S(H))$ and acts as
\begin{equation}
\label{eqMarkchain3}
L^{\la}f(\rho)=\frac{(U_{\la}f-f)(\rho)}{\la}=\frac{1}{\la}\sum_j p_j(t) (f(\rho_t^j)-f(\rho)).
\end{equation}
All "quantum content" of the theory is now captured in the explicit formula \eqref{eqMarkchain}.
What follows will be
the pure classical probability analysis of these Markov chains, their scaling limits and control.
In this paper we shall work with the measuring devises of the simplest form living in two-dimensional Hilbert spaces
$\C^2$ or more generally the tensor products of these spaces. Choosing the standard basis $e_0=(1,0)$, $e_1=(0,1)$,
we shall use the decomposition
\[
\HC_0\otimes \C^2=\HC_{00}\oplus H_{01}=\HC_0 \otimes e_0 \oplus \HC_0 \otimes e_1,
\]
and we shall choose the vacuum vector $\varphi=e_0$, so that
\[
\Om=\begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix}
\]
\section{Belavkin equations for a counting observation}
\label{seccountcase}
For simplicity we shall work exclusively with finite-dimensional Hilbert spaces $\HC_0=\C^n$,
making occasionally some comments about more general case.
The set of states $S(\C^n)$ is a compact convex set in the Euclidean space $\R^{n^2}$,
the space of complex Hermitian $n\times n$ matrices.
Let us choose an arbitrary self-adjoint operator in $\HC_0\otimes \C^2$ given by its matrix representation
\[
H= \begin{pmatrix} A & 0 \\ 0 & B \end{pmatrix}
+ \begin{pmatrix} 0 & -iC^* \\ iC & 0 \end{pmatrix}.
\]
We are aiming at calculating the small time asymptotics of the Markov transition operators defined by
\eqref{eqMarkchain}.
The main idea for obtaining sensible asymptotic limits suggests enhancing the interaction part $C$ of $H$ by
replacing it with the scaled version $C/\sqrt t$. Thus we choose the Hamiltonian in the form
\[
H= \begin{pmatrix} A & 0 \\ 0 & B \end{pmatrix}
+ \frac{1}{\sqrt t} \begin{pmatrix} 0 & -iC^* \\ iC & 0 \end{pmatrix}.
\]
\begin{remark}
The idea of the scaling comes from the analysis of the so-called quantum Zeno paradox.
Its essence is a rather simple observation that if one performs repeated measurements with reduction
\eqref{eqantiunitdress1} and pass to the limit,
as time between measurements tends to zero, then the state effectively remains in the initial
state all the time irrespectively of the dynamics. This effect is also referred to as the watch dog effect.
Therefore the only way to get a sensible dynamics that takes into account both dynamics and observation is
to enhance the interaction part of the dynamics to make its effect comparable with that of the repeated reduction
\eqref{eqantiunitdress1}. Thus one can suggest scaling $C$ as $C/t^{\al}$ with some $\al>0$. As calculations show
(one can repeat the calculations below with an arbitrary $\al$)
only with $\al=1/2$ a sensible limit is obtained.
\end{remark}
By the second equation in \eqref{eqblockmatrix}, we get
\[
\rho \otimes \Om =\begin{pmatrix} \rho & 0 \\ 0 & 0 \end{pmatrix},
\quad
\left[H, \begin{pmatrix} \rho & 0 \\ 0 & 0 \end{pmatrix}\right]
=\begin{pmatrix} [A,\rho] & + i\rho C^*/\sqrt t \\ iC\rho/\sqrt t & 0 \end{pmatrix}
\]
\[
H\begin{pmatrix} \rho & 0 \\ 0 & 0 \end{pmatrix}H
=\begin{pmatrix} A\rho A & -i A\rho C^*/\sqrt t\\ iC\rho A/\sqrt t & C\rho C^*/t \end{pmatrix},
\quad
H^2 =\begin{pmatrix} A^2+C^*C/t & -i(AC^* +C^* B)/\sqrt t \\ i(CA+BC)/\sqrt t & B^2+CC^*/t \end{pmatrix},
\]
\[
\{H^2, \rho \otimes \Om\}=\begin{pmatrix} \{A^2+C^*C/t,\rho\} & -i\rho (AC^*+C^*B)/\sqrt t \\ i(CA+BC)\rho/\sqrt t & 0 \end{pmatrix}.
\]
where $\{C,D\}=CD+DC$ denotes the anti-commutator.
Using \eqref{smalltimegroup}, and keeping terms of order not exceeding $t$ we get the approximation
\begin{equation}
\label{eqdressedrho1}
e^{-itH} (\rho\otimes \Om) e^{itH}
=\begin{pmatrix} \rho -it[A, \rho] -\frac12 t\{C^*C,\rho\} & \sqrt t \rho C^* \\ \sqrt t C\rho & tC\rho C^* \end{pmatrix},
\end{equation}
which is the key formula for what follows.
As it turns out, the limiting processes are of two types, depending on whether
the projectors $P_0$ and $P_1$ of the spectral decomposition of $R$ are diagonal, that is
\begin{equation}
\label{eqdiagonproj}
P_0= \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \quad P_1=\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}
\end{equation}
or otherwise. Let us start with the case of projectors \eqref{eqdiagonproj}.
We have
\[
I\otimes P_0= \begin{pmatrix} I & 0 \\ 0 & 0 \end{pmatrix},
\quad I\otimes P_1 =\begin{pmatrix} 0 & 0 \\ 0 & I \end{pmatrix},
\]
and
\[
(I\otimes P_0) e^{-itH} \begin{pmatrix} \rho & 0 \\ 0 & 0 \end{pmatrix} e^{itH} (I\otimes P_0)
= \rho -it[A, \rho] -\frac12 t\{C^*C,\rho\},
\]
\[
(I\otimes P_1) e^{-itH} \begin{pmatrix} \rho & 0 \\ 0 & 0 \end{pmatrix} e^{itH} (I\otimes P_1)
= tC\rho C^*.
\]
Hence the non-normalized new states are
\[
\tilde \rho_1=\rho -it[A, \rho] -\frac12 t\{C^*C,\rho\},
\quad
\tilde \rho_2= tC\rho C^*,
\]
occurring with the probabilities
\[
p_1=1-t \, {\tr} (C^*C \rho), \quad p_2=t \, {\tr} (C^*C \rho).
\]
Aiming at using Proposition \ref{propconvsemigr} (ii) we are looking for
the limit of the operator $(U_h-1)/h$ for $h \to 0$.
Denoting $T= {\tr} (C^*C \rho)$ we can write up to terms of order $t$ that
\[
\frac{U_h-1}{h} f(\rho)=\frac{1}{h} (1-hT)\left[f(\frac{\tilde \rho_1}{1-hT})-f(\rho)\right]
+\frac{1}{h} h\, T \left[(f(\frac{\tilde \rho_2}{hT})-f(\rho))\right]
\]
\[
\approx \frac{1}{h} (1-hT)[f(\rho -it[A, \rho] -\frac12 t\{C^*C,\rho\}+h\rho T)-f(\rho)]
+T \left[f(\frac{C\rho C^*}{T})-f(\rho)\right]\approx L_{count}f,
\]
with
\begin{equation}
\label{eqjumpgener}
L_{count}f(\rho)=-(f'(\rho), i[A, \rho] +\frac12 \{C^*C,\rho\}-\rho T)
+T \left[f(\frac{C\rho C^*}{T})-f(\rho)\right].
\end{equation}
Summarising by looking carefully at the small terms ignored, we can conclude the following.
\begin{lemma}
\label{lemmaongencount}
Under the setting considered,
\begin{equation}
\label{eqjumpgener1}
\|\frac{U_h-1}{h} f -Lf\|\le \sqrt h \ka \|f\|_{C^2(S(\HC_0))}
\end{equation}
for $f\in C^2(S(\HC_0))$, with $L_{count}$ given by \eqref{eqjumpgener} and a constant $\ka$.
\end{lemma}
We can prove now our first result.
\begin{theorem}
\label{thBeleqcount}
Let $\HC_0=\C^n$ and $A$, $C$ be $n\times n$ square matrices with $A$ being Hermitian. Then:
(i) The operator \eqref{eqjumpgener} generates a Feller process $O_t^{\rho}$ in $S(\HC_0)$ and
the corresponding Feller semigroup $T_t$ in $C(S(\HC_0))$ having the spaces
$C^1(S(\HC_0))$ and $C^2(S(\HC_0))$ as invariant cores, and $T_s$ are bounded in these spaces
uniformly for $s\in [0,t]$ with any $t>0$.
(ii) The scaled discrete semigroups $(U_h)^{[s/h]}$ converge to the semigroup $T_s$, as $h\to 0$, so that the corresponding
processes converge in distribution, with the following rates of convergence:
\begin{equation}
\label{eq1thBeleqcount}
\|(U_h)^{[s/h]} -T_sf\| \le \sqrt h s \ka(t) \|f\|_{C^2(S(\HC_0))},
\end{equation}
where the constant $\ka(t)$ depends on the dimension $n$ and the norms of $A$ and $C$.
(iii) The scaled semigroups $T_s^{\la}$ converge to the semigroup $T_s$, as $\la\to 0$, so that the corresponding
processes converge in distribution, with the following rates of convergence:
\begin{equation}
\label{eq2thBeleqcount}
\|T_s^{\la}f -T_sf\| \le \sqrt \la s \ka(t) \|f\|_{C^2(S(\HC_0))}.
\end{equation}
\end{theorem}
\begin{proof}
(i) This is a consequence of Proposition \ref{propdetjumpdom}. To make this conclusion one needs to show
property \eqref{eqBrezis} with $K=S(\C^n)$ and
\[
b(\rho)= -i[A, \rho] -\frac12 \{C^*C,\rho\}+ {\tr} (C^*C\rho) \rho.
\]
It is straightforward to see that the solutions to the ODE $\dot \rho=b(\rho)$ preserve the affine set of
Hermitian matrices with unit trace. So the key point is the preservation of positivity. It turns out
that a stronger version of \eqref{eqBrezis} holds, namely that
$d(\rho +h b(\rho),K)=0$
for any $\rho$ from the boundary of $K$ and all sufficiently small $h$. By the compactness of a unit ball in $\C^n$,
this claim follows from the following one. If $\rho$ belongs to the boundary of $K$,
that is, there exists a nonempty set $V(\rho)$ of unit vectors such that $\rho v=0$ for $v\in V(\rho)$,
then $(v,(\rho +h b(\rho))v)\ge 0$ for any unit vector $v$ and all sufficiently small $h$. But this property
is obvious for $v\notin V(\rho)$. On the other hand $(v,b(\rho)v)=0$ for $v\in V(\rho)$ implying that
$(v,(\rho +h b(\rho))v)= 0$ for all $h>0$ and all $v\in V(\rho)$.
(ii) This is a consequence of (i), Proposition \ref{propconvsemigr} (ii) and the observation that \eqref{eq5propconvsemigr}
holds here with the triple of spaces $C^2(S(\HC_0))\subset C^1(S(\HC_0)) \subset C(S(\HC_0))$.
(iii) This is a consequence of (i), formula \eqref{eqMarkchain3} and Proposition \ref{propconvsemigr} (i),
with $B=C(S(\HC_0))$, $D=C^2(S(\HC_0))$.
\end{proof}
\begin{remark}
\label{remoninfindim}
This result extends almost automatically to the case of an arbitrary separable Hilbert space $\HC_0$
and arbitrary bounded operators $H,C$, with the derivatives understood in the Fr\'echet sense. The only point
where the finite-dimensional setting was used was in proving statement (i) using compactness of a unit ball in
$\C^n$ and the Brezis theorem. In infinite-dimensional case one can use the compactness of a unit ball in
a Hilbert space in the weak topology and the Banach-space version of the Brezis theorem, as presented in
\cite{Martin} and \cite{Lakshm}.
\end{remark}
As is seen directly via Ito's formula, the Feller process $O_t^{\rho}$ generated by \eqref{eqjumpgener} can be
described as solving the jump type SDE
\begin{equation}
\label{eqBeleqcount}
d\rho=(- i[A, \rho] -\frac12 \{C^*C,\rho\}+ {\tr} (C\rho C^*) \rho ) dt
+\left(\frac{C\rho C^*}{{\tr} (C\rho C^*)}-\rho\right) dN_t,
\end{equation}
with the counting process $N_t$ with the position dependent intensity ${\tr} (C^*C\rho)$,
so that the compensated process $N_t-\int_0^t {\tr} (C^*C\rho_s) \, ds$ is a martingale.
Equation \eqref{eqBeleqcount} is the {\it Belavkin quantum filtering SDE} corresponding
to the {\it counting type observation} (because the driving process $N_t$ is a counting process).
Representation via the generator is an equivalent way of specifying the process of
continuous quantum observation and filtering.
\begin{remark} Equation \eqref{eqBeleqcount} is slightly nonstandard as the driving noise $N_t$ is itself
position dependent. However there is a natural way to rewrite it in terms of an independent driving noise.
Namely, with a standard Poisson random measure process $N(dx \,dt)$ on $\R_+\times \R_+$ (with Lebesgue measure
as intensity) one can rewrite equation \eqref{eqBeleqcount} in the following equivalent form:
\begin{equation}
\label{eqBeleqcountindepnoise}
d\rho=(- i[A, \rho] -\frac12 \{C^*C,\rho\}+ {\tr} (C\rho C^*) \rho ) dt
+\left(\frac{C\rho C^*}{{\tr} (C\rho C^*)}-\rho\right) \1({\tr} (C^*C\rho)\le x) N(dx\, dt),
\end{equation}
see details of this construction in \cite{Pellegrini10a}. Alternatively, one can make sense
of \eqref{eqBeleqcount} in terms of the general theory of weak SDEs from \cite{Kol11}.
\end{remark}
\begin{remark} The meaning of the term 'counting observation' (as well as 'diffusive type' of the next section)
becomes more concrete in a more advanced
treatment of the process of quantum measurement, see e.g. \cite{BoutHanJamQuantFilt}.
\end{remark}
\section{Belavkin equations for a diffusive observation}
\label{secdifcase}
Let us turn to the second case of choosing orthogonal projectors $P_0,P_1$,
when they differ from the diagonal choice \eqref{eqdiagonproj}.
General couple of two orthogonal projectors in $\C^2$ is easily seen to be of the form
\[
P_0=\begin{pmatrix} \cos^2 \phi & \sin\phi \cos \phi e^{i\psi}\\ \sin\phi \cos \phi e^{i\psi} & \sin^2\phi \end{pmatrix},
\quad
P_1=\begin{pmatrix} \sin^2 \phi & -\sin\phi \cos \phi e^{i\psi}\\ -\sin\phi \cos \phi e^{i\psi} & \cos^2\phi \end{pmatrix}.
\]
The phase terms with $\psi$ does not make much difference, so we choose further $\psi=0$. Moreover, to avoid
diagonal case we assume $\phi \neq \pi k/2$, $k\in N$.
By \eqref{eqblockmatrep},
\[
I\times P_0
= \begin{pmatrix} \cos^2 \phi I & \sin\phi \cos \phi I \\ \sin\phi \cos \phi I & \sin^2\phi I\end{pmatrix},
\quad
I\times P_1
= \begin{pmatrix} \sin^2 \phi I & -\sin\phi \cos \phi I \\ -\sin\phi \cos \phi I & \cos^2\phi I\end{pmatrix}.
\]
Hence, for arbitrary matrices $a,b,c,d$, we have
\[
(I\times P_0) \begin{pmatrix} a & b \\ c & d \end{pmatrix}
= \begin{pmatrix} \cos^2 \phi \, a + \sin\phi \cos \phi \, c & \cos^2 \phi \, b + \sin\phi \cos \phi \, d \\
\sin\phi \cos \phi \, a + \sin ^2\phi \, c & \sin\phi \cos \phi \, b + \sin^2\phi \, d\end{pmatrix}
\]
and
\[
(I\times P_0) \begin{pmatrix} a & b \\ c & d \end{pmatrix} (I\times P_0)
=\begin{pmatrix} \cos^2 \phi \, \om_{\phi} & \sin\phi \cos \phi \, \om_{\phi} \\
\sin\phi \cos \phi \, \om_{\phi} & \sin^2\phi \, \om_{\phi} \end{pmatrix}
\]
with
\[
\om_{\phi}=\om_{\phi}(a,b,c,d)=
\cos^2 \phi \, a+ \sin\phi \cos \phi (b+c)+ \sin^2\phi \, d.
\]
Since $P_1$ is obtained from $P_0$ by changing $\phi$ to $\phi+\pi/2$, it follows that
\[
(I\times P_1) \begin{pmatrix} a & b \\ c & d \end{pmatrix} (I\times P_1)
=\begin{pmatrix} \sin^2 \phi \, \tilde \om_{\phi} & -\sin\phi \cos \phi \, \tilde \om_{\phi} \\
-\sin\phi \cos \phi \, \tilde \om_{\phi} & \cos^2\phi \, \tilde \om_{\phi} \end{pmatrix}
\]
with
\[
\tilde \om_{\phi}=\om_{\phi+\pi/2}
= \sin^2 \phi \, a -\sin\phi \cos \phi (b+c)+ \cos^2\phi \, d.
\]
By \eqref{eqdefparttrwithtwodim} we get
\[
{\tr}_{p1} [(I\times P_0) \begin{pmatrix} a & b \\ c & d \end{pmatrix} (I\times P_0)]
\]
\[
=\om_{\phi} = \cos^2 \phi \, a+ \sin\phi \cos \phi (b+c)+ \sin^2\phi \, d,
\]
\[
{\tr}_{p1} [(I\times P_1) \begin{pmatrix} a & b \\ c & d \end{pmatrix} (I\times P_1)]
\]
\[
=\tilde \om_{\phi} = \sin^2 \phi \, a- \sin\phi \cos \phi (b+c)+ \cos^2\phi \, d.
\]
To get new states we have to take $a,b,c,d$ from \eqref{eqdressedrho1}.
Hence for the non-normalized states we get the approximate formulas (up to terms of order $t$):
\[
\tilde \rho_1 = \cos^2 \phi (\rho -it[A, \rho] -\frac12 t\{C^*C,\rho\})
+ \sqrt t \sin\phi \cos \phi (\rho C^* + C\rho)+ t \sin^2\phi \,C\rho C^*,
\]
\[
\tilde \rho_2 = \sin^2 \phi (\rho -it[A, \rho] -\frac12 t\{C^*C,\rho\})
- \sqrt t \sin\phi \cos \phi (\rho C^* + C\rho)+ t\cos^2\phi \, C\rho C^*.
\]
These states occur
with the probabilities
\[
p_1=\cos^2\phi(1-tT)+\sqrt t \sin\phi \cos \phi \, {\tr} (\rho C^* + C\rho)+t T \sin^2\phi,
\]
\[
p_2=\sin^2\phi(1-tT)-\sqrt t \sin\phi \cos \phi \, {\tr} (\rho C^* + C\rho)+t T \cos^2\phi.
\]
For arbitrary numbers $a,b,c$, one can write up to terms of order $t$, that
\[
\frac{1}{a+b\sqrt t +ct}
=\frac{1}{a} \frac{1}{1+(b/a) \sqrt t+(c/a) t}
=\frac{1}{a}(1-(b/a) \sqrt t-(c/a) t+(b/a)^2 t).
\]
Consequently, with this order of approximation,
\[
\frac{1}{p_1}=\frac{1}{\cos^2 \phi}(1- \tan \phi \sqrt t \, {\tr} (\rho C^* + C\rho)-T(\tan^2\phi-1) t
+\tan^2 \phi \, [{\tr} (\rho C^* + C\rho)]^2 t),
\]
\[
\frac{1}{p_2}=\frac{1}{\sin^2 \phi}(1+ \cot \phi \sqrt t \, {\tr} (\rho C^* + C\rho) -T(\cot^2\phi-1) t
+\cot^2 \phi \, [{\tr} (\rho C^* + C\rho)]^2 t),
\]
and therefore the normalized states are given by the formulas
\[
\rho_1=\frac{\tilde \rho_1}{p_1}
= [\rho -it[A, \rho] -\frac12 t\{C^*C,\rho\}
+ \sqrt t \tan\phi (\rho C^* + C\rho)+ t \tan^2\phi \,C\rho C^*]
\]
\[
\times(1- \tan \phi \sqrt t \, {\tr} (\rho C^* + C\rho)-T(\tan^2\phi-1) t
+\tan^2 \phi \, [{\tr} (\rho C^* + C\rho)]^2 t)
\]
\[
=\rho + \sqrt t \tan\phi (\rho C^* + C\rho-\Om \rho)+tB_1
\]
with
\[
\Om= {\tr} (\rho C^* + C\rho)
\]
and
\[
B_1=
-i[A, \rho] -\frac12 \{C^*C,\rho\}+ T\rho
+\tan^2\phi (C\rho C^*- (\rho C^* + C\rho) \Om -T\rho+ \Om^2 \rho),
\]
and
\[
\rho_2=\frac{\tilde \rho_2}{p_2}
= [\rho -it[A, \rho] -\frac12 t\{C^*C,\rho\}
- \sqrt t \cot\phi (\rho C^* + C\rho)+ t \cot^2\phi \,C\rho C^*]
\]
\[
\times(1+ \cot \phi \sqrt t \, {\tr} (\rho C^* + C\rho)-T(\cot^2\phi-1) t
+\cot^2 \phi \, [{\tr} (\rho C^* + C\rho)]^2 t)
\]
\[
=\rho - \sqrt t \cot\phi (\rho C^* + C\rho-\Om\rho)+tB_2
\]
with
\[
B_2=
-i[A, \rho] -\frac12 \{C^*C,\rho\}+T\rho
+ \cot^2\phi (C\rho C^*- (\rho C^* + C\rho) \Om -T\rho+\Om^2 \rho).
\]
The terms of order $t$ in $p_j$ give contributions of lower order, so that
to the main order in small $h$ we have
\[
\frac{U_h-1}{h} f(\rho)=\frac{1}{h} p_1\left[f(\rho_1)-f(\rho)\right]
+ \frac{1}{h} p_1\left[f(\rho_2)-f(\rho)\right].
\]
\[
=\frac{1}{h}(\cos^2\phi+\sqrt h \sin\phi \cos \phi \Om)
\]
\[
\times \left[(f'(\rho),\sqrt h \tan\phi (\rho C^* + C\rho-\Om \rho)+tB_1)
+\frac12 \tan^2\phi [(\rho C^* + C\rho-\Om \rho)f''(\rho) (\rho C^* + C\rho-\Om \rho)]h\right]
\]
\[
+\frac{1}{h}(\sin^2\phi-\sqrt h \sin\phi \cos \phi \Om)
\]
\[
\times \left[ (f'(\rho),- \sqrt h \cot\phi (\rho C^* + C\rho-\Om\rho)+tB_2)
+\frac12 \cot^2\phi [(\rho C^* + C\rho-\Om \rho)f''(\rho) (\rho C^* + C\rho-\Om \rho)]h\right],
\]
where
\[
[Af''(\rho) A]=\sum_{ijkl} A_{ij} \frac{\pa^2 f}{\pa \rho_{ij}\pa \rho_{kl}} A_{kl}.
\]
The terms of order $h^{-1/2}$ cancel and we get in the main term
\[
\frac{U_h-1}{h} f(\rho)\approx \frac12 [(\rho C^* + C\rho-\Om \rho)f''(\rho) (\rho C^* + C\rho-\Om \rho)]
\]
\[
+(f'(\rho),\Om (\rho C^* + C\rho-\Om \rho)+\cos^2\phi B_1+\sin^2\phi B_2)\approx L_{dif}f(\rho)
\]
with
\begin{equation}
\label{eqdifgener}
L_{dif}f(\rho)=\frac12 [(\rho C^* + C\rho-\Om \rho)f''(\rho) (\rho C^* + C\rho-\Om \rho)]
+(f'(\rho),-i[A, \rho] -\frac12 \{C^*C,\rho\}+ C\rho C^* ),
\end{equation}
which is remarkably independent of $\phi$! Thus, taking into account the terms that were ignored
withinn the approximation, we obtained the following counterpart of Lemma \ref{lemmaongencount}:
\begin{lemma}
\label{lemmaongendif}
Under the setting considered, and for any $\phi\neq \pi k/2$, $k\in \Z$,
\begin{equation}
\label{eqdifgener1}
\|\frac{U_h-1}{h} f -Lf\|\le \sqrt h \ka \|f\|_{C^3(S(\HC_0))}
\end{equation}
for $f\in C^3(S(\HC_0))$, with $L_{dif}$ given by \eqref{eqdifgener}.
\end{lemma}
Unlike the jump-type limiting processes analysed in the previous section,
where a straightforward pure analytic proof of the well-posedness of the process generated by $L$ is available,
here an approach using SDEs is handy. Ito's formula shows that a process generated by \eqref{eqdifgener}
can arise from solving the following Ito's SDE:
\begin{equation}
\label{eqBeleqdif}
d\rho=(-i[A, \rho] -\frac12 \{C^*C,\rho\}+ C\rho C^* ) \, dt
+\left(\rho C^* + C\rho-{\tr}\, (\rho C^* + C\rho)\rho\right) \, dW_t,
\end{equation}
where $W_t$ is a standard one-dimensional Wiener process.
This SDE is the {\it Belavkin quantum filtering SDE} for normalized states corresponding
to the {\it diffusive type observation}.
\begin{theorem}
\label{thBeleqdif}
Let $\HC_0=\C^n$ and $A$, $C$ be $n\times n$ square matrices with $A$ being Hermitian. Then:
(i) The operator \eqref{eqdifgener} generates a Feller process $O_t^{\rho}$ in $S(\HC_0)$ and
the corresponding Feller semigroup $T_t$ in $C(S(\HC_0))$ having the spaces
$C^2(S(\HC_0))$ and $C^3(S(\HC_0))$ as invariant cores, and $T_s$ are bounded in these spaces
uniformly for $s\in [0,t]$ with any $t>0$. This process is given by the solutions to SDE
\eqref{eqBeleqdif}, which is well posed as a diffusion equation in $S(\HC_0)$.
(ii) The scaled discrete semigroups $(U_h)^{[s/h]}$ converge to the semigroup $T_s$, as $h\to 0$, so that the corresponding
processes converge in distribution, with the following rates of convergence:
\begin{equation}
\label{eq1thBeleqdif}
\|(U_h)^{[s/h]} -T_sf\| \le \sqrt h s \ka(t) \|f\|_{C^3(S(\HC_0))},
\end{equation}
where the constant $\ka(t)$ depends on the norms of $A$ and $C$.
(iii) The scaled semigroups $T_s^{\la}$ converge to the semigroup $T_s$, as $\la\to 0$, so that the corresponding
processes converge in distribution, with the following rates of convergence:
\begin{equation}
\label{eq2thBeleqdif}
\|T_s^{\la}f -T_sf\| \le \la \sqrt s \ka(t) \|f\|_{C^3(S(\HC_0))}.
\end{equation}
\end{theorem}
\begin{proof} Parts (ii) and (iii) are obtained by the same arguments as in the proof of Theorem \ref{thBeleqcount}.
One only has to mention that estimate \eqref{eq3propconvsemigr} needed to apply Proposition \ref{propconvsemigr}
follows from the standard fact of the theory of diffusion that $\E((X_t(x)-x)^2)\le Ct$ for any diffusion $X_t(x)$
with bounded smooth coefficients.
So we need only to prove (i). All claims follow if one can construct a diffusion in $S(\HC_0)$ solving
\eqref{eqBeleqdif}, because in $S(\HC_0)$ all coefficients are bounded, and then both the uniqueness of solution and the
required smoothness of solutions with respect to initial data follow automatically from the smoothness of
the coefficients by the standard tools of Ito's SDEs.
The main difficulty here lies in proving that solutions to \eqref{eqBeleqdif} preserve the set of positive matrices.
But the fact that SDE \eqref{eqBeleqdif} is well-posed in $S(\HC_0)$ is a well known fact, see e.g.
Section 3.4.1 in monograph \cite{BarchBook}. Thus one can complete a proof of Theorem \ref{thBeleqdif}
by referring to this result.
However, a proof of \cite{BarchBook} is indirect, and the fact is really crucial.
Therefore, for completeness we sketch below a different direct proof that the
solutions to \eqref{eqBeleqdif} preserve the set of positive matrices.
In this approach we shall consider the coefficients of the equation
\eqref{eqBeleqdif} to be given as they are only for nonnegative $\rho$ of unit trace and continued smoothly to all
Hermitian $\rho$ in such a way that these coefficients vanish outside some neighborhood of this set. The modified equations
\eqref{eqBeleqdif} have globally bounded smooth coefficients and hence have unique well defined global solutions. Thus
we really only need to show the preservation of positivity.
Our method is based on the Stratonovich integral. Recall that the Stratonovich differential $\circ dX$
is lined with Ito's differential by the formula $Z\circ dX =Z \, dX +(1/2)dZ \, dX$. Hence
denoting
\[
B(\rho)=\rho C^* + C\rho-{\tr}\, (\rho C^* + C\rho)\rho,
\]
equation \eqref{eqBeleqdif} rewrites in Stratonovich form as
\[
d\rho=(-i[A, \rho] -\frac12 \{C^*C,\rho\}+ C\rho C^* ) \, dt
+B(\rho) \circ dW_t-\frac12 dB(\rho) \,dW_t
\]
\[
=(-i[A, \rho] -\frac12 \{C^*C,\rho\}+ C\rho C^* ) \, dt
+B(\rho) \circ dW_t
\]
\[
-\frac12 [B(\rho) C^*+CB(\rho)-{\tr} \,(\rho C^*+C\rho) B(\rho)
- {\tr}\,(B(\rho)C^*+CB(\rho))\rho] dt.
\]
Using the fundamental result of the Stratonovich integral, stating that solutions to
Stratonovich SDEs can be obtained as the limits of the solutions to the ODEs obtained
by approximating the white noise with smooth functions, we can state that the
solutions to this Stratonovich equation preserve positivity of matrices, if
the equations
\[
\dot \rho
=-i[A, \rho] -\frac12 \{C^*C,\rho\}+ C\rho C^*
+B(\rho) \phi_t
\]
\begin{equation}
\label{eq2thBeleqdifStappr}
-\frac12 [B(\rho) C^*+CB(\rho)-{\tr} \,(\rho C^*+C\rho) B(\rho)
- {\tr}\,(B(\rho)C^*+CB(\rho))\rho]
\end{equation}
preserve the set of positive matrices for any continuous function $\phi_t$.
But this follows by the Brezis Theorem \ref{thBrezis}.
To see this we substitute the expression for $B(\rho)$ in the first three places of the
last square bracket yielding the equation
\[
\dot \rho
=-i[A, \rho] -\frac12 \{C^*C,\rho\}
+B(\rho) \phi_t
\]
\begin{equation}
\label{eq2thBeleqdifStappr1}
-\frac12 [\rho (C^*)^2+C^2\rho +({\tr} \,(\rho C^*+C\rho))^2 \rho
- {\tr}\,(B(\rho)C^*+CB(\rho))\rho]
\end{equation}
(the key point is that the 'nasty' term $C\rho C^*$ cancels).
It is seen that Theorem \ref{thBrezis} applies, because whenever $(v,\rho v)=0$, the r.h.s.
$\om_t(\rho)$ of equation \eqref{eq2thBeleqdifStappr1} satisfies $(v,\om_t(\rho) v)=0$ for
any function $\phi_t$. The details of the argument are the same as in the proof of Theorem \ref{thBeleqcount}.
\end{proof}
\begin{remark} The methods developed can be used to extend this result to infinite dimensional $\HC_0$.
However, unlike the situation with counting observations, explained in Remark \ref{remoninfindim},
there is some subtlety here in working with SDEs in the space of trace class operators, which are
not going to discuss in this paper.
\end{remark}
A remarkable property of the SDEs \eqref{eqBeleqcount} and \eqref{eqBeleqdif} is that they preserve the pure states.
Namely if the initial state $\rho$ was pure, $\rho=\psi\otimes \bar \psi$, then it remains pure for all times.
Namely, one can check by a direct application of Ito's formula that if $\phi$ satisfies the SDE
\begin{equation}
\label{eqqufiBnonlin}
d\phi=-[i(A-\langle Re \, C\rangle_{\phi} \, Im \, C)
+\frac12(LC-\langle Re \, C\rangle_{\phi})^*(C-\langle Re \, C\rangle_{\phi})]\phi \, dt
+ (C-\langle Re \, C\rangle_{\phi})\phi \, dW_t,
\end{equation}
then $\rho=\psi\otimes \bar \psi$ satisfies equation \eqref{eqBeleqdif}. Equation \eqref{eqqufiBnonlin}
is the Belavkin quantum filtering equation for pure states. It looks much simpler for the most important case
of self-adjoint $C$:
\begin{equation}
\label{eqqufiBnonlins}
d\phi=-[iA+\frac12(L-\langle C\rangle_{\phi})^2]\phi \, dt
+ (C-\langle C\rangle_{\phi})\phi \, dW_t.
\end{equation}
Another key observation is that there exists an equivalent linear version of \eqref{eqBeleqdif}.
Namely assume that $\xi$ solves the following {\it Belavkin quantum filtering SDE} for non-normalized states:
\begin{equation}
\label{eqBeleqdiflin}
d\xi=(-i[A, \xi] -\frac12 \{C^*C,\xi\}+ C\xi C^* ) \, dt
+(\xi C^* + C\xi) \, dY_t,
\end{equation}
where $Y_t$ is a Brownian motion under a certain measure. Applying Ito's formula to $\rho=\xi/{\tr} \, \xi$ one finds that
$\rho$ satisfies \eqref{eqBeleqdif} with the process $W$ satisfying the equation
\begin{equation}
\label{eqBeleqdiflinlink}
dW_t=dY_t-{\tr} \,(\xi C^* + C\xi) \, dt.
\end{equation}
It follows from the famous Girsanov formula that if $Y_t$ was a Wiener process, then $W_t$ would be also a Wiener
process under some different but equivalent measure with respect to one defining $Y_t$. Hence a solution $\xi_t$ to
the linear equation \eqref{eqBeleqdiflin} with some Brownian motion $Y_t$ yields
the solution $\rho=\xi/{\tr} \, \xi$ to \eqref{eqBeleqdif} with some other Brownian motion $W_t$.
\section{Observations via different channels}
\label{secseveralchan}
Let us now extend the theory to the case of several channels of observation. Namely,
we take
\begin{equation}
\label{interHam}
\HC=\HC_0\otimes \C^2 \otimes \cdots \otimes \C^2, \quad (K \, \text{multipliers}\, \C^2),
\end{equation}
and the atom (system with Hilbert space $\HC_0$) is supposed to interact with each of the $K$
measuring devices with the state space $\C^2$. Each of the devises is equipped with
the standard basis $(e_0^j,e_1^j)$ with $e_0^j$ chosen as a vacuum vector, that is as its stationary state,
with the corresponding density matrix being $\Om_j= |e_0^j\rangle \langle e_0^j|$.
The Hamiltonian is given by the sum $H=H_0+\sum_{k=1}^KH_k$, where
$H_0=A\otimes I^{\otimes k}$ describes the free dynamics of the atom, and $H_j$ connects the atom
with the $j$th device. The same scaling $1/\sqrt t$ applies to the interaction parts.
Thus $H$ is specified by $k+1$ operators $A,C_1, \cdots, C_K$ in $\HC_0$, so that $H_j$ are
give by the formulas:
\begin{equation}
\label{interHam1}
\begin{aligned}
& H_0 (h\otimes e_{i_1}^1 \otimes \cdots \otimes e_{i_K}^K)=A h \otimes e^1_{i_1} \otimes \cdots \otimes e_{i_K}^K, \\
& H_j (h\otimes e_{i_1}^1 \otimes \cdots \otimes e_{i_K}^K)|_{e_{i_j}^j=e_1^j}
=-\frac{i}{\sqrt t} C_j^* h\otimes e_{i_1}^1 \otimes \cdots \otimes e_{i_K}^K)|_{e^j_{i_j}=e_0^j}, \quad j>0, \\
& H_j (h\otimes e_{i_1}^1 \otimes \cdots \otimes e_{i_K}^K)|_{e_{i_j}^j=e_0^j}
= \frac{i}{\sqrt t} C_j (h\otimes e_{i_1}^1 \otimes \cdots \otimes e_{i_K}^K)|_{e_{i_j}^j=e_1^j}, \quad j>0.
\end{aligned}
\end{equation}
At a starting time of an interaction the devices are supposed to be set to their vacuum states,
so that a state $\rho$ on $\HC_0=\C^n$ lifts to $\HC$ as
\[
\rho_{\HC}=\rho \otimes \Om_1 \otimes \cdots \otimes \Om_K.
\]
The observation procedure can be specified by choosing two orthogonal projectors
$P_0^j$ and $P_1^j$ in the space $\C^2$ of each device (that is in each channel of observation)
arising from some observables with the spectral decompositions $\sum_l \la_l P_l^j$.
This choice yields the totality of $2^K$ orthogonal projectors in $\HC$,
\[
I\otimes P_{i_1}^1 \otimes \cdots \otimes P_{i_K}^K,
\]
so that the possible new non-normalized states after each step of interaction and measurement are
\begin{equation}
\label{newstatedouble}
\tilde \rho_t^{i_1 \cdots i_K}
={\tr}_{p1\cdots K} [(I\otimes P_{i_1}^1 \otimes \cdots \otimes P_{i_K}^K)
e^{-itH} \rho_{\HC} e^{itH} (I\otimes P_{i_1}^1 \otimes \cdots \otimes P_{i_K}^K)],
\end{equation}
where
\begin{equation}
\label{eqMarkchain1m}
\ga_t= e^{-itH} \rho_{\HC} e^{itH} = e^{-itH}(\rho \otimes \Om_1 \otimes \cdots \otimes \Om_K) e^{itH},
\end{equation}
and ${\tr}_{p1\cdots K}$ is the partial trace with respect to all spaces, but for $\HC_0$.
These states may occur with the probabilities
\begin{equation}
\label{eqMarkchain2m}
p_{i_1 \cdots i_K}(t)={\tr} \, [\ga_t (I\otimes P_{i_1} \otimes \cdots \otimes P_{i_K})]
={\tr} \tilde \rho_t^{i_1 \cdots i_K}.
\end{equation}
Therefore the multichannel extension of the discrete time {\it Markov chain of successive indirect observations} given by \eqref{eqMarkchain} and \eqref{eqMarkchain1} is given by $2^K$ possible transitions of $\rho$ to the states
\begin{equation}
\label{eqMarkchainmult}
\rho_t^{i_1 \cdots i_K}= \frac{1}{p_{i_1 \cdots i_K}} {\tr}_{p1\cdots K}
[(I\otimes P_{i_1} \otimes \cdots \otimes P_{i_K})
\ga_t (I\otimes P_{i_1} \otimes \cdots \otimes P_{i_K})],
\end{equation}
where $\ga_t$ and the probabilities $p_{i_1 \cdots i_K}$ are given by
\eqref{eqMarkchain1m} and \eqref{eqMarkchain2m}. The transition operator of this Markov chain writes down as
\begin{equation}
\label{eqMarkchain2mult}
U_t f(\rho)=\E f(\rho_t)=\sum_{i_1 \cdots i_K} p_{i_1 \cdots i_K}(t) f(\rho_t^{i_1 \cdots i_K}).
\end{equation}
The operators in $\HC$ are best described in terms of blocks. Namely, writing $\HC=\oplus \HC_{i_1 \cdots i_K}$,
with $\HC_{i_1 \cdots i_K}$ generated by $\HC_0\otimes e_{i_1} \otimes \cdots \otimes e_{i_K}$,
we can represent an operator $\LC$ in $\HC$ by $4^K$ operators
$L_{i_1 \cdots i_K}^{j_1 \cdots j_K}$ in $\HC$, so that
\[
\LC ( h^{i_1 \cdots i_K}\otimes e_{i_1} \otimes \cdots \otimes e_{i_K})
=\sum_{j_1 \cdots j_K} L_{i_1 \cdots i_K}^{j_1 \cdots j_K} h^{i_1 \cdots i_K}
\otimes e_{j_1} \otimes \cdots \otimes e_{j_K}.
\]
The composition and partial trace in this notations are expressed by the following formulas:
\begin{equation}
\label{eqblockcom}
(\LC_1 \LC_2)_{i_1 \cdots i_K}^{j_1 \cdots j_K}
=\sum_{m_1 \cdots m_K} (\LC_1)_{m_1 \cdots m_K}^{j_1 \cdots j_K}
(\LC_2)^{m_1 \cdots m_K}_{j_1 \cdots j_K},
\end{equation}
\begin{equation}
\label{eqblocktr}
{\tr}_{p1\cdots K} \LC
=\sum_{j_1 \cdots j_K} L_{j_1 \cdots j_K}^{j_1 \cdots j_K}.
\end{equation}
For simplicity let us perform detailed calculations for $K=2$ (they are quite similar in the general case).
Thus $\HC=\C^n\otimes \C^2 \otimes \C^2$ and $H=H_0+H_1+H_2$. Let us denote the bases of the two devices
$\{e_k\}$ and $\{f_k\}$ respectively. Formulas \eqref{interHam1} rewrite in a simpler way as
\[
H_0 (h\otimes e_k \otimes f_j)=A h \otimes e_k \otimes f_j,
\]
\[
H_1 (h\otimes e_1 \otimes f_j)=-iC_1^* h \otimes e_0 \otimes f_j/ \sqrt t,
\quad
H_1 (h\otimes e_0 \otimes f_j)=iC_1 h \otimes e_1 \otimes f_j \sqrt t,
\]
\[
H_2 (h\otimes e_j \otimes f_1)=-iC_2^* h \otimes e_j \otimes f_0 /\sqrt t,
\quad
H_2 (h\otimes e_j \otimes f_0)=iC_2 h \otimes e_j \otimes f_1 /\sqrt t,
\]
With the chosen vacuum vectors $e_0=(1,0)$ in the first device and $f_0=(1,0)$ in the second device,
a state $\rho$ on $\HC_0=\C^n$ lifts to $\HC$ as
\[
\rho_{\HC}=\rho \otimes |e_0\rangle \langle e_0| \otimes |f_0\rangle \langle f_0|.
\]
The operators $\LC$ in $\HC$ are described by 16 operators
$L_{jk}^{lm}$ in $\HC$.
To shorten the formulas, let us perform calculations without scaling $C_j$ (without the factor $1/\sqrt t$)
and will restore the scaling at the end.
In term of the blocks we can write:
\[
(\rho_{\HC})^{ml}_{jk}= \de^m_0 \de^l_0 \de^0_j \de^0_k \rho.
\]
\[
(H_1)^{ml}_{jk}=i\de^l_k \de^m_{\bar j}(C_1 \de^0_j- C_1^* \de^1_j),
\quad
(H_2)^{ml}_{jk}=i\de^m_j \de^l_{\bar k}(C_2 \de^0_k- C_2^* \de^1_k),
\]
where we have introduced the following notations: for $i$ being $0$ or $1$ we denote
$\bar i$ as being $1$ and $0$ respectively.
By \eqref{eqblockcom} it follows that
\[
[H_1, \rho_{\HC}]^{ml}_{jk}
=i\sum \de^l_q \de^m_{\bar p}(C_1 \de^0_p- C_1^* \de^1_p)\, \de^p_0 \de^q_0 \de^0_j \de^0_k \rho
-i\sum \de^m_0 \de^l_0 \de^0_p \de^0_q \rho \, \de^q_k \de^p_{\bar j}(C_1 \de^0_j- C_1^* \de^1_j)
\]
\[
=i \de^0_k \de^l_0 (\de^0_j \de^m_1 C_1\rho+\de^1_j \de^m_0 \rho C_1^*)
=i \de^0_k \de^l_0 \de^m_{\bar j} (\de^0_j C_1\rho+\de^1_j \rho C_1^*).
\]
Next
\[
(H_1^2)^{ml}_{jk}=\sum (H_1)^{ml}_{pq} (H_1)^{pq}_{jk}
=-\sum \de^l_q \de^m_{\bar p}(C_1 \de^0_p- C_1^* \de^1_p)
\de^q_k \de^p_{\bar j}(C_1 \de^0_j- C_1^* \de^1_j)
\]
\[
=-\de^l_k \de^m_j (C_1\de^1_j-C_1^*\de_j^0)(C_1\de^0_j-C_1^* \de^1_j)
=\de^l_k \de^m_j ( \de^1_j C_1C_1^*+ \de^0_jC_1^* C_1),
\]
\[
(H_2^2)^{ml}_{jk}=\sum (H_2)^{ml}_{pq} (H_2)^{pq}_{jk}
=-\de^m_p \de^l_{\bar q}(C_2 \de^0_q- C_2^* \de^1_q)
\de^p_j \de^q_{\bar k}(C_2 \de^0_k- C_2^* \de^1_k)
\]
\[
=-\de^m_j \de^l_k (C_2 \de^1_k- C_2^* \de_k^0)(C_2 \de^0_k- C_2^* \de^1_k)
=\de^m_j \de^l_k (\de^1_k C_2 C_2^*+ \de^0_kC_2^* C_2),
\]
\[
(H_1 H_2)^{ml}_{jk}=\sum (H_1)^{ml}_{pq} (H_2)^{pq}_{jk}
=-\sum \de^l_q \de^m_{\bar p}(C_1 \de^0_p- C_1^* \de^1_p)
\de^p_j \de^q_{\bar k}(C_2 \de^0_k- C_2^* \de^1_k)
\]
\[
=-\de^l_{\bar k} \de^m_{\bar j}(C_1 \de^0_j- C_1^* \de^1_j)(C_2 \de^0_k- C_2^* \de^1_k),
\]
\[
(H_2 H_1)^{ml}_{jk}=\sum (H_2)^{ml}_{pq} (H_1)^{pq}_{jk}
=-\de^m_p \de^l_{\bar q}(C_2 \de^0_q- C_2^* \de^1_q)
\de^q_k \de^p_{\bar j}(C_1 \de^0_j- C_1^* \de^1_j)
\]
\[
=-\de^l_{\bar k} \de^m_{\bar j}(C_2 \de^0_k- C_2^* \de^1_k)(C_1 \de^0_j- C_1^* \de^1_j),
\]
and
\[
(H_1\rho_{\HC}H_1)^{ml}_{jk}
=(H_1\rho_{\HC})^{ml}_{pq} (H_1)^{pq}_{jk}
=-\de^0_q \de^l_0 \de^m_{\bar p} \de^0_p C_1\rho \,
\de^q_k \de^p_{\bar j}(C_1 \de^0_j- C_1^* \de^1_j)
\]
\[
=\de^0_k \de^l_0 \de^1_j \de^m_1 C_1\rho C_1^*,
\]
\[
(H_2\rho_{\HC}H_2)^{ml}_{jk}
=(H_2\rho_{\HC})^{ml}_{pq} (H_2)^{pq}_{jk}
=-\de^m_0\de^0_p \de^l_{\bar q}\de^0_q C_2\rho \,
\de^p_j \de^q_{\bar k}(C_2 \de^0_k- C_2^* \de^1_k)
\]
\[
=\de^1_k \de^l_1 \de^0_j \de^m_0 C_2\rho C_2^*,
\]
\[
(H_1\rho_{\HC}H_2)^{ml}_{jk}
=(H_1\rho_{\HC})^{ml}_{pq} (H_2)^{pq}_{jk}
=-\de^0_q \de^l_0 \de^m_{\bar p} \de^0_p C_1\rho \,
\de^p_j \de^q_{\bar k}(C_2 \de^0_k- C_2^* \de^1_k)
\]
\[
= \de^1_0 \de^m_1 \de^0_j \de^1_k C_1\rho C_2^*,
\]
\[
(H_2\rho_{\HC}H_1)^{ml}_{jk}
=(H_2\rho_{\HC})^{ml}_{pq} (H_1)^{pq}_{jk}
=-\de^m_0\de^0_p \de^l_{\bar q}\de^0_q C_2\rho \,
\de^q_k \de^p_{\bar j}(C_1 \de^0_j- C_1^* \de^1_j)
\]
\[
= \de^1_1 \de^m_0 \de^1_j \de^0_k C_2\rho C_1^*.
\]
Therefore
\[
(H_1+H_2)\rho_{\HC}(H_1+H_2)^{ml}_{jk}=\de^0_k \de^l_0 \de^1_j \de^m_1 C_1\rho C_1^*
+\de^1_k \de^l_1 \de^0_j \de^m_0 C_2\rho C_2^*
\]
\[
+\de^1_0 \de^m_1 \de^0_j \de^1_k C_1\rho C_2^*
+ \de^1_1 \de^m_0 \de^1_j \de^0_k C_2\rho C_1^*.
\]
Next,
\[
\{H_1^2, \rho_{\HC}\}^{ml}_{jk}=(H_1^2)^{ml}_{pq}(\rho_{\HC})^{pq}_{jk}
+(\rho_{\HC})^{ml}_{pq}(H_1^2)^{pq}_{jk}
\]
\[
=\de^l_q \de^m_p (\de^1_p C_1C_1^* +\de^0_p C_1^*C_1) \, \de^p_0 \de^q_0 \de^0_j \de^0_k \rho
+\de^m_0 \de^l_0 \de^0_p \de^0_q \rho \, \de^q_k \de^p_j (\de^1_j C_1C_1^* +\de^0_j C_1^*C_1)
=\de^m_0 \de^l_0 \de^0_j \de^0_k\{C_1^* C_1, \rho\},
\]
\[
\{H_2^2, \rho_{\HC}\}^{ml}_{jk}=(H_2^2)^{ml}_{pq}(\rho_{\HC})^{pq}_{jk}
+(\rho_{\HC})^{ml}_{pq}(H_2^2)^{pq}_{jk}
\]
\[
=\de^m_p \de^l_q (\de^1_q C_2 C_2^*+ \de^0_qC_2^* C_2)\, \de^p_0 \de^q_0 \de^0_j \de^0_k \rho
+\de^m_0 \de^l_0 \de^0_p \de^0_q \rho \, \de^p_j \de^q_k (\de^1_k C_2 C_2^*+ \de^0_kC_2^* C_2)
=\de^m_0 \de^l_0 \de^0_j \de^0_k\{C_2^* C_2, \rho\},
\]
and
\[
\{H_1 H_2, \rho_{\HC}\}^{ml}_{jk}=(H_1H_2)^{ml}_{pq}(\rho_{\HC})^{pq}_{jk}
+(\rho_{\HC})^{ml}_{pq}(H_1H_2)^{pq}_{jk}
\]
\[
=-\de^l_{\bar q} \de^m_{\bar p}(C_1 \de^0_p- C_1^* \de^1_p)(C_2 \de^0_q- C_2^* \de^1_q)
\de^p_0 \de^q_0 \de^0_j \de^0_k \rho
-\de^m_0 \de^l_0 \de^0_p \de^0_q \rho
\de^q_{\bar k} \de^p_{\bar j}(C_1 \de^0_j- C_1^* \de^1_j)(C_2 \de^0_k- C_2^* \de^1_k)
\]
\[
= -\de^l_1 \de^m_1 \de^0_j \de^0_k C_1C_2\rho - \de^l_0 \de^m_0 \de^1_j \de^1_k \rho C_1^*C_2^*,
\]
\[
\{H_2 H_1, \rho_{\HC}\}^{ml}_{jk}=(H_2H_1)^{ml}_{pq}(\rho_{\HC})^{pq}_{jk}
+(\rho_{\HC})^{ml}_{pq}(H_2H_1)^{pq}_{jk}
\]
\[
-\de^l_{\bar q} \de^m_{\bar p}(C_2 \de^0_p- C_2^* \de^1_q)(C_1 \de^0_p- C_1^* \de^1_q)
\de^p_0 \de^q_0 \de^0_j \de^0_k \rho
-\de^m_0 \de^l_0 \de^0_p \de^0_q \rho
\de^q_{\bar k} \de^p_{\bar j}(C_2 \de^0_k- C_2^* \de^1_k)(C_1 \de^0_j- C_1^* \de^1_j)
\]
\[
=- \de^l_1 \de^m_1 \de^0_j \de^0_k C_2C_1 \rho -\de^l_0 \de^m_0 \de^1_j \de^1_k\rho C_2^*C_1^*
\]
Thus,
\[
\{(H_1+H_2)^2,\rho_{\HC}\}^{ml}_{jk}
=\{H_1^2+H_2^2+H_1H_2+H_2H_1, \rho_{\HC}\}^{ml}_{jk}
\]
\[
=\de^m_0 \de^l_0 \de^0_j \de^0_k\{C_1^*C_1 +C_2^*C_2, \rho\}
-\de^l_1 \de^m_1 \de^0_j \de^0_k \{C_1,C_2\}\rho
-\de^l_0 \de^m_0 \de^1_j \de^1_k\rho \{C_1^*, C_2^*\}.
\]
Thus all parts of \eqref{smalltimegroup} are collected.
Let us turn to \eqref{newstatedouble}. From the calculations with a single channel we know that one has to
distinguish diagonal and non-diagonal projectors $P^j_k$. Let us start with the case, when in both devises the projectors
are diagonal, that is
\[
P_0^1=P_0^2=\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix},
\quad
P_1^1=P_1^2=\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}.
\]
Let us calculate
\[
(I\otimes P_j^1 \otimes P_k^2) \LC (I\otimes P_j^1 \otimes P_k^2)
\]
for arbitrary $\LC$.
We have
\[
(I\otimes P_i^1 \otimes P_r^2) \sum h^{jk} \otimes e_j\otimes f_k=h^{ir},
\]
\[
(I\otimes P_i^1 \otimes P_r^2)^{ml}_{jk}=\de^i_j \de^r_k \de^m_i \de^1_r.
\]
So
\[
((I\otimes P_i^1 \otimes P_r^2) \LC)^{ml}_{jk}
=(I\otimes P_i^1 \otimes P_r^2)^{ml}_{pq} \LC^{pq}_{jk}
=\de^i_p \de^r_q \de^m_i \de^1_r \LC^{pq}_{jk}
= \de^m_i \de^1_r \LC^{ir}_{jk}
\]
and
\[
((I\otimes P_i^1 \otimes P_r^2) \LC (I\otimes P_i^1 \otimes P_r^2))^{ml}_{jk}
=((I\otimes P_i^1 \otimes P_r^2) \LC)^{ml}_{pq}
(I\otimes P_i^1 \otimes P_r^2)^{pq}_{jk}
\]
\[
= \de^m_i \de^1_r \LC^{ir}_{pq} \de^i_j \de^r_k \de^p_i \de^q_r
=\de^m_i \de^1_r \de^i_j \de^r_k \LC^{ir}_{ir}.
\]
Thus
\[
{\tr}_{p12} ((I\otimes P_i^1 \otimes P_r^2) \LC (I\otimes P_i^1 \otimes P_r^2))
=\LC^{ir}_{ir},
\]
and
\[
\tilde \rho_{ir}=(e^{-itH} \rho e^{itH})^{ir}_{ir},
\quad
p_{ir} ={\tr} (e^{-itH} \rho e^{itH})^{ir}_{ir}.
\]
Thus we have
\[
[H_1+H_2, \rho_{\HC}]^{jk}_{jk}=0,
\]
\[
(H_1+H_2)\rho_{\HC}(H_1+H_2)^{jk}_{jk}=\de^0_k \de^1_j C_1\rho C_1^*
+\de^1_k \de^0_j C_2\rho C_2^*,
\]
\[
\{H_1^2+H_2^2+H_1H_2+H_2H_1, \rho_{\HC}\}^{jk}_{jk}
= \de^0_j \de^0_k\{C_1^*C_1 +C_2^*C_2, \rho\}.
\]
Restoring scaling $C \to C/\sqrt t$ yields approximately
\[
(e^{-itH} \rho_{\HC} e^{itH})^{jk}_{jk}
=(\rho_{\HC}-it [H,\rho_{\HC}]+t^2 (H\rho_{\HC} H-\frac12 \{H^2,\rho_{\HC}\}))^{jk}_{jk}
\]
\[
= \de^0_j \de^0_k (\rho-it[A,\rho])+t [\de^0_k \de^1_j C_1\rho C_1^*
+\de^1_k \de^0_j C_2\rho C_2^*-\frac12 \de^0_j \de^0_k\{C_1^*C_1 +C_2^*C_2, \rho\}]
\]
and thus
\[
\tilde \rho_{jk}= \de^0_j \de^0_k (\rho-it[A,\rho])+t [\de^0_k \de^1_j C_1\rho C_1^*
+\de^1_k \de^0_j C_2\rho C_2^*-\frac12 \de^0_j \de^0_k\{C_1^*C_1 +C_2^*C_2, \rho\}],
\]
\[
p_{jk} =\de^0_j \de^0_k +t [\de^0_k \de^1_j {\tr} (C_1\rho C_1^*)
+\de^1_k \de^0_j {\tr} (C_2\rho C_2^*)-\de^0_j \de^0_k {\tr}((C_1^*C_1 +C_2^*C_2) \rho)].
\]
Thus $p_{11}=0$,
\[
\rho_{00}=\frac{\tilde \rho_{00}}{p_{00}}
=(\rho-it[A,\rho] -\frac12 t\{C_1^*C_1 +C_2^*C_2, \rho\})(1+ t \,{\tr}((C_1^*C_1 +C_2^*C_2) \rho)),
\]
\[
=\rho -it[A,\rho]-\frac12 t\{C_1^*C_1 +C_2^*C_2, \rho\}+t \, {\tr}((C_1^*C_1 +C_2^*C_2) \rho) \rho,
\]
\[
\rho_{10}=\frac{\tilde \rho_{10}}{p_{10}}=\frac {C_1\rho C_1^*} {{\tr} (C_1\rho C_1^*)},
\quad
\rho_{01}=\frac{\tilde \rho_{01}}{p_{01}}=\frac {C_2\rho C_2^*} {{\tr} (C_2\rho C_2^*)}.
\]
Thus we get, up to terms of order $h$ in small $h$, that
\[
\frac{U_h-1}{h} f(\rho)=\frac{1}{h}\sum_{jk} p_{jk} \left[f(\rho_{jk})-f(\rho)\right]
\]
\[
=\frac{1}{h} p_{00}[f(\rho-it[A,\rho]-\frac12 t\{C_1^*C_1 +C_2^*C_2, \rho\}+h \, {\tr}((C_1^*C_1 +C_2^*C_2) \rho) \rho)-f(\rho)]
\]
\[
+\frac{1}{h} p_{10} \left[ f\left(\frac {C_1\rho C_1^*} {{\tr} (C_1\rho C_1^*)}\right) -f(\rho)\right]
+\frac{1}{h} p_{01} \left[ f\left(\frac {C_2\rho C_2^*} {{\tr} (C_2\rho C_2^*)}\right) -f(\rho)\right]
\]
\[
= \left( f'(\rho), -\frac12 \{C_1^*C_1, \rho\}+ {\tr}(C_1 \rho C_1^*) \rho
-\frac12 \{C_2^*C_2, \rho\}+ {\tr}(C_2 \rho C_2^*) \rho\right)
\]
\[
+ {\tr} (C_1\rho C_1^*)\left[ f\left(\frac {C_1\rho C_1^*} {{\tr} (C_1\rho C_1^*)}\right) -f(\rho)\right]
+{\tr} (C_2\rho C_2^*) \left[ f\left(\frac {C_2\rho C_2^*} {{\tr} (C_2\rho C_2^*)}\right) -f(\rho)\right].
\]
Summarising and extending to arbitrary number of channels $k$ we can conclude that we proved the following
extension of Lemma \ref{lemmaongencount}.
\begin{lemma}
\label{lemmaongencountmultich}
Under the setting considered,
\begin{equation}
\label{eqjumpgener1m}
\|\frac{U_h-1}{h} f -Lf\|\le \sqrt h \ka \|f\|_{C^2(S(\HC_0))}
\end{equation}
for $f\in C^2(S(\HC_0))$, with $L$ given by
\[
L_{count}f(\rho)=-i[A, \rho] \, dt+\sum_{j=1}^K \left( f'(\rho), -\frac12 \{C_j^*C_j, \rho\}+ {\tr}(C_j \rho C_j^*) \rho\right)
\]
\begin{equation}
\label{eqjumpgener2m}
+ \sum_{j=1}^K \,{\tr}\, (C_j\rho C_j^*)\left[ f\left(\frac {C_j\rho C_j^*} {{\tr} (C_j\rho C_j^*)}\right) -f(\rho)\right].
\end{equation}
\end{lemma}
As a consequence we get the following direct extension of Theorem \ref{thBeleqcount}.
\begin{theorem}
\label{thBeleqcountm}
Let $\HC_0=\C^n$ and $A,C_1, \cdots, C_K$ be operators in $\HC_0$ with $A$ being Hermitian.
Let the projectors defining the measurements be chosen to be diagonal in each channel:
\begin{equation}
\label{eqthBeleqcountm}
P_0^j=\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix},
\quad
P_1^j=\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}
\end{equation}
for all $j=1, \cdots, K$.
Then all statements of Theorem \ref{thBeleqcount} hold for the operator
\eqref{eqjumpgener2m} and Markov semigroups described by the transition operator \eqref{eqMarkchain2mult}.
In particular, estimates \eqref{eq1thBeleqcount} and \eqref{eq2thBeleqcount} hold.
\end{theorem}
\begin{remark} As explained in Remark \ref{remoninfindim} this result extends automatically to the case
of arbitrary separable Hilbert space $\HC$ and bounded operators $A,C_1, \cdots, C_K$ in it.
\end{remark}
As in the case of a single channel, the process generated by \eqref{eqjumpgener2m}
can be described by the solutions to the SDE of jump type, which takes now the form
\begin{equation}
\label{eqBeleqcountm}
d\rho=- i[A, \rho]\, dt +\sum_j (-\frac12 \{C^*_jC_j,\rho\}+ {\tr} (C_j\rho C^*_j) \rho ) \, dt
+\sum_j\left(\frac{C_j\rho C^*_j}{{\tr} (C_j\rho C^*_j)}-\rho\right) dN^j_t,
\end{equation}
with the counting processes $N_t^j$ are independent and have the position dependent intensities ${\tr} (C^*_jC_j\rho)$.
Equation \eqref{eqBeleqcountm} is the {\it Belavkin quantum filtering SDE} corresponding
to the {\it counting type observation via several channels}.
As suggested by Theorem \ref{thBeleqdif}, exploiting non diagonal pairs of projectors $P_0^j, P_1^j$
should lead to the limiting generator of diffusive type. In fact, performing similar calculations
(which we omit) one arrives at the following general result.
\begin{theorem}
\label{thBeleqmixm}
Let $\HC_0=\C^n$ and $A,C_1, \cdots, C_K$ be operators in $\HC_0$ with $A$ being Hermitian.
Let the projectors defining the measurements are chosen to be diagonal, that is of type
\eqref{eqthBeleqcountm}, for a subset $I\subset \{1, \cdots,K\}$
of the set of channels. And for $j\notin I$ these channels are chosen as non-diagonal,
that is of the form
\begin{equation}
\label{eq1thBeleqmixtm}
P_0^j=\begin{pmatrix} \cos^2 \phi_j & \sin\phi_j \cos \phi_j \\ \sin\phi_j \cos \phi_j & \sin^2\phi \end{pmatrix},
\quad
P_1^j=\begin{pmatrix} \sin^2 \phi_j & -\sin\phi_j \cos \phi_j \\ -\sin\phi_j \cos \phi_j & \cos^2\phi_j \end{pmatrix},
\end{equation}
with $\phi_j\neq k\pi/2$, $k\in \N$.
Then the limiting generator for the semigroup with the transition operator \eqref{eqMarkchain2mult}
gets the expression
\[
L_{mix}f(\rho)=\sum_{j\in I} \left( f'(\rho), -\frac12 \{C_j^*C_j, \rho\}+ {\tr}(C_j \rho C_j^*) \rho\right)
+ \sum_{j\in I} \,{\tr}\, (C_j\rho C_j^*)\left[ f\left(\frac {C_j\rho C_j^*} {{\tr} (C_j\rho C_j^*)}\right) -f(\rho)\right]
\]
\[
+\frac12 \sum_{j\notin I}[(\rho C_j^* + C_j\rho- {\tr} (\rho C^*_j + C_j\rho) \rho)f''(\rho)
(\rho C_j^* + C_j\rho- {\tr} (\rho C^*_j + C_j\rho) \rho)]
\]
\begin{equation}
\label{eq2thBeleqmixtm}
+\sum_{j\notin I} \left(f'(\rho), -\frac12 \{C^*_jC_j,\rho\}+ C_j\rho C_j^* \right)
-(f'(\rho), i[A, \rho]).
\end{equation}
This operator generates
a Feller process $O_t^{\rho}$ in $S(\HC_0)$ and
the corresponding Feller semigroup $T_t$ in $C(S(\HC_0))$ such that claims
(ii) and (iii) of Theorem \ref{thBeleqdif} hold.
The Markov process generated by \eqref{eq2thBeleqmixtm} can be given by
the solutions of the following SDEs in $S(\HC_0)$:
\[
d\rho=-i[A, \rho] \, dt
+ \sum_{j\in I} (-\frac12 \{C^*_jC_j,\rho\}+ {\tr} (C_j\rho C^*_j) \rho ) \, dt
+\sum_{j\in I}\left(\frac{C_j\rho C^*_j}{{\tr} (C_j\rho C^*_j)}-\rho\right) dN^j_t
\]
\begin{equation}
\label{eqBeleqmmix}
+\sum_{j\notin I}(-\frac12 \{C_j^*C_j,\rho\}+ C_j\rho C_j^* ) \, dt
+\sum_{j\notin I}\left(\rho C_j^* + C_j\rho-{\tr}\, (\rho C_j^* + C_j\rho)\rho\right) \, dW^j_t,
\end{equation}
where $W_j$ are independent Wiener processes and $N_t^i$ independent jump process of intensity
${\tr} (C_j\rho C^*_j)$.
\end{theorem}
\begin{proof} In the pure diffusive case, that is with empty $I$, the proof is exactly the
same as in Theorem \ref{thBeleqdif}. For the general case one only has to show
that operator $L_{mix}$ generates a Feller process in $S(\HC_0)$ preserving the sets of smooth functions
(other arguments are again the same).
Two proofs for proving this fact can be suggested.
(i) One starts with generator $\tilde L_{mix}$ obtained from \eqref{eq2thBeleqmixtm}
by ignoring the jump part. This is a well-defined diffusion operator and by the same methods as
in Theorem \ref{thBeleqdif} one shows that it generates a Feller processes in $S(\HC_0)$.
But the jump part of \eqref{eq2thBeleqmixtm} is a bounded operator preserving positivity and smoothness.
Hence it can be dealt with straightforwardly via the perturbation theory.
(ii) Each of the two parts of \eqref{eq2thBeleqmixtm}, related to $I$ and its complement,
generates a well-defined Feller process in $S(\HC_0)$ preserving smoothness (of arbitrary order in fact).
Hence one can derive that the sum of these operators generates a well-defined Feller process in $S(\HC_0)$
via the Lie-Trotter formula, namely from Theorem 5.3.1 of \cite{Kolbook11}.
\end{proof}
\begin{remark}
The Markov chain of multichannel measurement that we are using is a bit different from the one used in
\cite{Pellegrini10a}, where measurement is based on a single operator $R$ in the device (no different channels),
and counting and diffusive parts of the generator arise from different projectors linked to different eigenspaces
of this operator. As was already mentioned the method of \cite{Pellegrini10a}
did not provide the rates of convergence.
\end{remark}
When $I$ is empty, $L_{mix}$ turns to $L_{dif}$ describing the multichannel observations of diffusive type.
\section{Fractional quantum stochastic filtering}
\label{secfraceq}
Now everything is ready for our main result: the derivation of the fractional equations of quantum stochastic filtering.
As was shown above the standard Belavkin equations of quantum filtering can be obtained as the scaled limits of
the sequences of discrete observations. The main assumption for each of the approximating processes was that the time
between successive measurement is either constant (discrete Markov chain approximation) or is exponentially distributed
(continuous time Markov chain approximation). Of course there is no a priori reasons for these assumptions.
And in fact in several domains of physics it turned out to be more appropriate to model times between successive
events by random variables from the domains of attraction of a stable law, that is via CTRW.
Our next result is a direct consequence of Theorem \ref{thBeleqmixm} and Proposition \ref{propCTRW}.
\begin{theorem}
\label{mainth}
Under the assumptions of Theorem \ref{thBeleqmixm} let the Markov chain \eqref{eqMarkchainmult}
is modified in such a way that the laws of transitions $\rho \to \rho_t^{i_1 \cdots i_K}$
remain unchanged, by the time between transitions is taken as scaled random variable from the domain
of attraction of a $\be$-stable law, that is as $T_i^h=h^{1/\be} T_i$ from Proposition \ref{propCTRW}.
Then the corresponding generalized CTRW processes \eqref{ctrwposdep} built from the transition operator
\eqref{eqMarkchain2mult} converge to the process $O^{\rho}_{\si_t}$ obtained from the process
$O^{\rho}_t$ of Theorem \ref{thBeleqmixm} via subordination by the inverse stable process
$\si_t=\max \{y: S_y \le t\}$. Moreover, the functions $f_t(x)=\E (T_{\si_t}f)(x)$
satisfy the fractional Caputo-Djerbashian equation
\eqref{eqfraceq} with the generator $L=L_{mix}$ given by \eqref{eq2thBeleqmixtm}.
\end{theorem}
As noted at the end of Appendix C the fractional derivative $D^{\be}_{0+\star}$ is a particular case of a class
of mixed fractional derivatives \eqref{eqfraceq}. Therefore, under appropriately organised scaled times
between the acts of measurements the limiting evolution will satisfy a more general fractional equation
\begin{equation}
\label{eqfraceqquBe}
D^{(\nu)}_{0+\star}f_t(x)=L_{mix}f_t(x), \quad f_0(x)=f(x),
\end{equation}
with $D^{\nu}$ given by \eqref{eqfraceq2}.
When only one type of observation channels is used, equation \eqref{eqfraceqquBe} simplifies to the case,
when either $L_{count}$ or $L_{dif}$ are places instead of $L_{mix}$.
Equations \eqref{eqfraceqquBe} (and their particular cases with fractional derivative $D^{\be}$ of order $\be$)
represent the fractional analogs of the process of quantum stochastic filtering. These equations can be
also considered as the new equations of fractional quantum mechanics. They are different from the fractional
Schr\"odinger equations suggested in \cite{Lask02} and extensively studied recently.
Equations \eqref{eqfraceqquBe} describe the process of continuous quantum control and filtering on the level
of the evolution of averages. On the 'micro-level' of SDEs \eqref{eqBeleqmmix} these equations correspond
to stopping the solutions of these SDEs at a random time $\si_t$ given by the inverse of a L\'evy subordinator.
\section{Fractional quantum control and games}
\label{secfraceqHJB}
The theory of quantum filtering reduces the analysis of quantum dynamic control and games
to the controlled version of evolutions \eqref{eqBeleqmmix}. The simplest situation concerns the case
when the homodyne device is fixed, that is the operators $C_j$ and the projectors $P_i^j$ are fixed,
and the players can control the individual Hamiltonian $H_0$ of the atom, say,
by applying appropriate electric or magnetic fields to the atom. Thus equations
\eqref{eqBeleqmmix} become modified by allowing $H_0$ to depend on one or several control parameters.
The so-called separation principle states
(see \cite{BoutHanQuantumSepar}) that the effective control of an observed quantum system
(that can be based in principle on the whole history of the interaction of the atom and optical devices)
can be reduced to the Markovian feedback control of the quantum filtering equation, with the feedback
at each moment depending only on the current (filtered) state of the atom.
In the present case of CTRW modeling of the process of measurements the problem of control becomes the
problem of control of scaled CTRW. The theory of such control was built in the series of papers
\cite{KolVer14} - \cite{KolVer17}. The main result is that in the scaling limit the cost functions
is a solution of the fractional Hamilton-Jacobi equation. In the present context and in game-theoretic setting
it implies the following.
Let us consider the controlled version of the process $O^{\rho}_{\si_t}$ from Theorem \ref{mainth},
where the individual Hamiltonian is now $\tilde H_0=H_0+uH_0^1+v H_0^2$ and it depends on control parameters
$u,v$ of two players from compact sets $U$ and $V$ respectively. Suppose that it is possible
to choose new $u,v$ directly after each act of measurement, and thus a control strategy is the sequence
$(u_1, v_1), (u_2,v_2), \cdots )$ of controls applied after each act of measurement, with each $(u_j,v_j)$
applied after $j$th act of measurement and depending on the history of the process until that time.
The case of a pure control (not a game) corresponds to the choice $V=0$ and is thus automatically included.
Assume that players $I$ and $II$ play a standard dynamic zero-sum game with a finite time horizon $T$
meaning that the objective of $I$ is to maximize the payoff
\begin{equation}
\label{eqcostfun}
P(t; u(.), v(.)) =\E [\int_t^T {\tr} \, (J \rho_s) \, ds +{\tr} \, (F \rho_T)],
\end{equation}
where $J$ and $F$ are some operators expressing the current and the terminal costs of the game
(they may depend on $u$ and $v$, but we exclude this case just for simplicity) and $W$ is the collection
of all noises involved in \eqref{eqBeleqmmix} (both diffusive and Poisson).
Then under the scaling limit of Theorem \ref{mainth} the optimal cost function
\begin{equation}
\label{eqcostfunopt}
S_t(\rho) =\max_{u(.)} \min_{v(.)}P(t; u(.), v(.))=\min_{v(.)} \max_{u(.)} P(t; u(.), v(.))
\end{equation}
will satisfy the following {\it fractional HJB-Isaacs equation of the CTRW modeling of quantum games}:
\begin{equation}
\label{eqHJB}
D^{\nu}_{0+\star}S_t(\rho)=\max_u (f'(\rho), i[\rho, uH_0^1]) +\min_v (f'(\rho), i[\rho, vH_0^2])
+{\tr} \, (J \rho_t)+ L_{mix}S_t(\rho).
\end{equation}
In \cite{KolVer14} this equation was derived heuristically, in the general framework of controlled
CTRW by the dynamic programming approach. As usual in optimal control theory, to justify the derivation
one has to show the well-posedness of the limiting HJB equation
and then to prove the verification theorem, a classical reference is \cite{FlemSon}. For some cases
of CTRWs this was performed in \cite{KolVer17}.
In the present fractional quantum case this problem will be considered elsewhere.
The additional complexity of this equation is related to the fact that the state space is a
rather nontrivial set of positive matrices with the unit trace. One can reduce the complexity
by looking at the dynamics of pure states only. But the set of pure states is not a Euclidean space,
but a manifold. In the finite-dimensional setting this manifold is the complex projective space $\C P^n$.
Let us mention that in the non-fractional case, that is with the usual derivative $\pa/\pa t$ instead of
$D^{\nu}_{0+\star}$ in \eqref{eqHJB}, the well-posedness of (the analogs of) equation \eqref{eqHJB} was proved
in \cite{Kol92}, for a special model of pumping a laser with a counting measurement, with some particular
solutions calculated explicitly,
and in \cite{KolDynQuGames}, for a special arrangements of diffusive measuring devises that ensured that the
diffusive part of operator $L_{dif}$ was nondegenerate and therefore the optimal control problem was reduced
to the drift control of the diffusions on a Riemannian manifold $\C P^n$.
\section{Other Markov approximations and unbounded generators}
\label{secunbound}
We commented above on the possible extension to infinite-dimensional Hilbert spaces. However, for all approximations
the assumption of boundedness of all operators involved seemed to be essential in the derivation given,
at least of the coupling operators $C_j$ (unboundedness of $A$ can be possibly treated via the interaction representation).
However, the quantum filtering equations are used also in the standard setting of quantum mechanics. The mostly studied
case is that of the standard Hamiltonian $H=-\De+V(x)$ in $L^2(\R^d)$ and the coupling operators being either position
(multiplication by $x$) or momentum operators. Different Markov chain approximations may be used to derive the
filtering equation in this case.
A powerful approach was suggested by Belavkin in \cite{Be195}: to use the von Neumann model
of unsharp measurement. In this model the effect of measurement for the product state $\phi(x)f(y)$ of an atom and
a measuring device, a pointer, is given by the shift
\[
U: \phi (x) f(y) \mapsto \phi(x) f(y-ax).
\]
Here both $\phi$ and $f$ are from $L^2(\R^d)$, and $f>0$ describes the stationary state of a pointer
(the analog of the vacuum state in our modeling above).
Projecting on the state of an atom this yields the transition
\begin{equation}
\label{ea0}
G(y) \colon \quad \phi(x) \mapsto \phi_y(x)=\phi(x) f(y-ax)/f(y),
\end{equation}
depending on the observed position $y$ of the pointer. Assuming the evolution of the atom
during time $t$ between the moments of measurements to be given by a Hamiltonian $A$,
the transition of a Markov chain of sequential measurements become
\begin{equation}
\label{ea1}
\phi \mapsto \phi_{t,y}(x)=(e^{-iAt}\phi)(x) f(y-ax)/f(y).
\end{equation}
After an appropriate scaling from this Markov chain one derives the diffusive filtering SDE
\eqref{eqqufiBnonlins} with $C=x$ (the multiplication operator), that is directly the filtering equation
for pure states, see detail in Appendix to \cite{BelKol}.
The model can be extended to more general situations, but seems to be linked with a specific von Neumann
instantaneous interaction. For the well-posedness of these kind of diffusive SDEs we can refer to
\cite{Holevo91}, \cite{FagMor} and references therein.
The derivation of the fractional version of this equation, as well as the fractional
control of Section \ref{secfraceqHJB} can be performed in this setting in the same way as above.
\section{Appendix A. Convergence of semigroups}
\label{secconbergsem}
Here we collect the results on the convergence of Markov semigroups and CTRW, which form the
the theoretical basis for our derivations of the filtering equations.
It is well known that the convergence of the generators on the core of the limiting generator
implies the convergence of semigroups. We shall use a version of this result with the rates,
namely the following result, given in Theorem 8.1.1 of \cite{Kolbook11}.
\begin{prop}
\label{propconvsemigr}
Let $F_t=e^{tL}$ be a strongly continuous semigroup in a Banach space $B$ with a norm $\|.\|_B$,
generate by an operator $L$,
having a core $D$, which is itself a Banach space with a norm $\|.\|_D\ge \|.\|_B$ so that $L\in \LC(D,B)$.
Let $F_t$ be also a bounded semigroup in $D$
such that $\|F_t\|_{D\to D} \le C_D(T)$ with a constant $C_D(T)$ uniformly for $t\in [0,T]$.
(i) Let $F_t^h$, $h>0$, be a family of strongly continuous contraction semigroups in a Banach space $B$
with bounded generators $L_h$ such that
\[
\|L_hf-Lf\|_B \le \ep_h \|f\|_D
\]
for all $f\in D$ and some $\ep_h$ such that $\ep_h\to 0$ as $h\to 0$.
Then the semigroups $F_t^h$ converge strongly to the semigroup $F_t$, as $h\to 0$, and
\begin{equation}
\label{eq1propconvsemigr}
\|F_t^hf -F_tf\|_B \le t \ep_h C_D(T)\|L\|_{D\to B}.
\end{equation}
(ii) Let $U_h$ be a family of contractions in $B$ such that
\begin{equation}
\label{eq2propconvsemigr}
\|\left(\frac{U_h-1}{h} -L\right)f\|_B \le \ep_h \|f\|_D,
\end{equation}
and
\begin{equation}
\label{eq3propconvsemigr}
\|\left(\frac{F_h-1}{h} -L\right)f\|_B \le \ka_h \|f\|_D,
\end{equation}
with $\ep_h \to 0$ and $\ka_h\to 0$, as $h\to 0$.
Then the scaled discrete semigroups $(U_h)^{[t/h]}$ converge to the semigroup $F_t$
and moreover
\begin{equation}
\label{eq4propconvsemigr}
\sup_{s\le t}\|(U_h)^{[s/h]} -F_sf\|_B \le (\ka_h+\ep_h)t \|f\|_B.
\end{equation}
\end{prop}
Additional condition \eqref{eq3propconvsemigr} makes working with discrete approximation a bit more subtle,
than with the continuous chain approximations. Effectively to get \eqref{eq3propconvsemigr}
one needs a deeper regularity. Namely one should have another core $\tilde D$ such that $D\subset \tilde D\subset B$
with $L\in \LC(D,\tilde D) \cap \LC(\tilde D,B)$. In this case it is easy to see that
\begin{equation}
\label{eq5propconvsemigr}
\|\left(\frac{F_h-1}{h} -L\right)f\|_B \le h \|L\|_{D,\tilde D} \|L\|_{\tilde D,B}\|f\|_D.
\end{equation}
\section{Appendix B. Deterministic motions with random jumps}
\label{secdetandjump}
Let us look at the Cauchy problem
\begin{equation}
\label{eqdetjump}
\frac{\pa f_t}{\pa t} =(\nabla f_t, b(x))+Lf_t(x), \quad f_0(x) \, \text{given},
\end{equation}
with the simplest jump-type operator
\[
L_f(x)=\sum_{j=1}^J f(Y_j(x)-x),
\]
where $x\in \R^d$, $\nabla f=\pa f/\pa x$ and $b,Y_j:\R^d\to \R^d$ are given bounded smooth functions.
It is more or less obvious that the resolving operators of the Cauchy problem \eqref{eqdetjump}
form a semigroup of contractions in the space $C(\R^d)$ preserving the spaces of
smooth functions. Let us make a precise statement. The simplest way to see it is via the 'interaction
representation'. Namely, let $X_t(x)$ denote the solution to the Cauchy problem $\dot X_t(x)=b(X_t(x))$,
$X_0(x)=x$, and let us change the unknown function $f$ in \eqref{eqdetjump} to $\phi$ via the equation
$f(x)=\phi(X_t(x))$. Direct substitution shows that $\phi$ solves the Cauchy problem
\begin{equation}
\label{eqdetjump1}
\frac{\pa \phi_t}{\pa t} =L_t\phi_t(x)=\sum_{j=1}^J \phi((X_t(Y_j(X_{-t}(x))))-x), \quad \phi_0=f_0.
\end{equation}
Since $L_t$ is a bounded operator, this Cauchy problem can be solved by the convergence series
over the powers of $L_t$. This leads to the following result.
\begin{prop}
\label{propdetjump}
Let $b, Y_j\in C^2(\R^d)$, $j=1, \cdots, J$. Then the resolving operators $R_t$ of the Cauchy problem
\eqref{eqdetjump} form a semigroups of contractions in $C(\R^d)$ such that the spaces $C^1(\R^d)$ and $C^2(\R^d)$
are invariant and $R_t$ form semigroups of operators in these spaces that are uniformly bounded for $\in [0,T]$ with any $T$.
\end{prop}
We need an extension of this result for the subsets of $\R^d$. The main tool is the following classical theorem of Brezis,
which we formulate in its simplest form referring to proofs, extensions and history to \cite{Redheffer}.
\begin{theorem}
\label{thBrezis}
Let $b(x):K\to \R^d $ be a Lipschitz continuous function, where $K$ is a convex closed subset of $\R^d$, such that
\begin{equation}
\label{eqBrezis}
\lim_{h\to 0_+} \frac{d(y+hb(x),K)}{h}=0
\end{equation}
for any $x\in K$, where $d(z,K)$ denotes the distance between a point $z$ and the set $K$.
Then $K$ is flow invariant. More precisely, for any $x\in K$ there exists a unique solution $X_t(x)$
of the equation $\dot X_t(x)=b(X_t(x))$ with the initial condition $x$ that belongs to $K$ for all $t$.
\end{theorem}
As a direct consequence we get the following extension of Proposition \ref{propdetjump}.
\begin{prop}
\label{propdetjumpdom}
Let $K$ be a convex compact subset of $\R^d$ and $b:K\to \R^d$, $Y_j:K\to K$ be twice continuously differentiable functions.
Let $b$ satisfy the assumptions of Theorem \ref{thBrezis}. Then the resolving operators $R_t$ of the Cauchy problem
\eqref{eqdetjump} form a semigroups of contractions in $C(K)$ such that the spaces $C^1(K)$ and $C^2(K)$
are invariant and $R_t$ are uniformly bounded operators in these spaces for $\in [0,T]$ with any $T$.
\end{prop}
\section{Appendix C. Position dependent CTRW}
\label{secCTRW}
Here we recall the basic result on the convergence of continuous time random walks (CTRW).
Suppose $T_1^h,T_2^h, \cdots $ is a sequence of i.i.d.
random variables in $\R_+$ such that the distribution of each $T_i^h$
is given by a probability measure $\mu_{time}^h (dt)$
on $\R_+$, that depend on a positive (scaling) parameter $h$. Let
\begin{equation}
\label{eqinversewalk}
N_t^h=\max \{ n: \sum_{i=1}^n T_i^h \le t\}.
\end{equation}
Suppose $X_1^h,X_2^h, \cdots $ is a sequence of i.i.d.
random variables in $\R^d$, such that the distribution of each $X_i^h$
is given by a probability measure $\mu_{space}^h (dt)$, that depends on $h$.
The standard (scaled) {\it continuous time random walk}\index{continuous time random walk (CTRW)} (CTRW)
is a random process given by the random sum
\[
\sum_{j=1}^{N_t^h} X_i^h.
\]
In position dependent CTRW the jumps $X_i^h$ are not independent, but each $X_i^h$
depends on the position of the process before this jump. The natural general formulation
can be given in terms of discrete Markov chains as follows.
Let $U_h$ be a transition operator of a discrete time Markov chain $O^h_n(x)$ in $\R^d$
depending on a positive parameter $h$, so that
\begin{equation}
\label{transitoperMar}
U_hf(x)=\E O^h_1(x)=\int f(y) \mu^h(x, dy),
\end{equation}
with some family of stochastic kernels $\mu^h(x, dy)$ such that $U_h$
is a bounded operator either in the space $C(K)$ with a compact convex subset $K$ of $\R^d$
or in the space $C_{\infty}(\R^d)$ of continuous functions vanishing at infinity.
For our purposes we need only
the operators of the type
\[
U_hf(x)=\E O^h_1(x)=\sum_{j=1}^J f(Y_j^h(x)) p_j(x)^h,
\]
with a family of continuous mappings $Y_j^h:\R^d\to \R^d$ and the probability laws $\{p_1^h, \cdots, p_J^h\}$.
Suppose $T_1^h,T_2^h, \cdots $ is a sequence of random variables introduced above, and independent of $O^h_n(x)$.
The process
\begin{equation}
\label{ctrwposdep}
O^h_{N_t^h}(x)
\end{equation}
is a generalized scaled (position dependent)
{\it continuous time random walk}\index{continuous time random walk (CTRW)} (CTRW) arising from
$U_h$ and $\mu^h_{time}$.
The CTRW were introduced in \cite{MW}. They found numerous applications in physics.
The scaling limits of these CTRW were analysed by many authors, see e.g. \cite{KKU}, \cite{MS},
\cite{MS2}. The scaling limit for the position dependent CTRW was developed in \cite{Kol7}.
Formally in \cite{Kol7} it was developed not in full generality, but for the case of the spacial
process $O^h_n(x)$ converging to a stable process. However, the arguments of \cite{Kol7} were completely
general and did not depend on this assumption. The only point used was that $O^h_n(x)$ converge in
the sense of Proposition \ref{propconvsemigr} (ii). For completeness let us formulate the result
\cite{Kol7} in a slightly modified version that we need in this paper and present a short proof
with essentially simplified arguments from \cite{Kol7} (see also Chapter 8 in \cite{Kolbook11}).
As an auxiliary result we need the standard functional limit theorem for the random-walk-approximation
of stable laws, see e.g. \cite{GnedKor} and \cite{MS2} and references therein for various proofs.
\begin{prop}
\label{limtheorstable}
Let a positive random variables $T$ belong to the domain of attraction of a $\be$-stable law,
$\be\in (0,1)$, in the sense that
\begin{equation}
\label{eq1propCTRW}
\P (T>m)\sim \frac{1}{\be m^{\be}}
\end{equation}
(the sign $\sim$ means here that the ratio tends to $1$, as $m\to \infty$).
Let $T_i$ be a sequence of i.i.d. random variables from
the domain of attraction of a $\be$-stable law and let
\[
\Phi_t^h=\sum_{i=1}^{[t/h]} h^{1/\al} T_i
\]
be a scaled random walk based on $T_i$, $h>0$, and $S_t$ a $\be$-stable L\'evy subordinator, that is
a L\'evy process in $\R_+$ generated by the stable generator
\[
L_{\be}(x)=\int \frac{f(x+y)-f(x)}{y^{1+\be}} dy
\]
(which up to a multiplier represents the fractional derivative $d^{\be}/d(-x)^{\be}$).
Then $\Phi_t^h \to S_t$ in distribution, as $h\to 0$.
\end{prop}
The next result is from \cite{Kol7}, though modified and simplified.
\begin{prop}
\label{propCTRW}
Let the random variables $T_i^h=h^{1/\be} T_i$, where i.i.d. random variables $T_i$ belong to the
domain of attraction of a $\be$-stable law, $S_t$ be a $\be$-stable L\'evy suboridinator
and
\[
\si_y=\max \{t: S_t \le y\}
\]
be its inverse process.
Let a family of contractions \eqref{transitoperMar}
satisfy \eqref{eq2propconvsemigr} with an operator $L$ generating a Feller process $F_t$.
Then
\[
\E U_h^s|_{s=[N_t^h/h]} \to \E F_{\si_t}, \quad h\to 0,
\]
strongly as contraction operators in $C(K)$ or $C_{\infty}(\R^d)$.
\end{prop}
\begin{remark} This Proposition directly implies the following statement about the processes:
the subordinated Markov chains \eqref{ctrwposdep}, that is the scaled CTRW, converge in distribution
to the process generated by $L$ and subordinated by the inverse of the L\'evy $\be$-subordinator.
\end{remark}
\begin{proof}
By the density arguments it is sufficient to show that
\[
\|\E U_h^{[s/h]}|_{s=N_t^h}f - \E F_{\si_t}f\| \to 0
\]
for functions $f$ from the domain of $L$.
We have
\[
\|\E U_h^{[s/h]}|_{s=N_t^h}f - \E F_{\si_t}f\| \le I+II,
\]
with
\[
I=\|\E U_h^{[s/h]}|_{s=N_t^h}f - \E F_{N^h_t}f\|, \quad II=\|\E F_{N^h_t}f - \E F_{\si_t}f\|.
\]
To estimate I we write
\[
I=\int_0^{\infty}(U_h^{[s/h]}f-F_sf)\mu^h_t(ds)=\int_0^K(U_h^{[s/h]}f-F_sf)\mu^h_y(ds)
+\int_K^{\infty}(U_h^{[s/h]}f-F_sf)\mu^h_t(ds),
\]
where $\mu^h_t$ is the distribution of $N_t^h$. Choosing $K$ large
enough we can make the second integral arbitrary small uniformly in $h$. And then by \eqref{eq4propconvsemigr}
we can make the first integral arbitrary small by choosing small enough $h$ (and uniformly in $t$ from
compact sets).
It remains II. Integrating by parts we get the following:
\[
II=\| \E e^{N_t^h L}f-\E e^{\si_t L} f\|
\]
\[
=\|\int_0^{\infty} \frac{\pa}{\pa s} (e^{sL}f) (\P(\si_t\le s) -\P(N_t^h\le s)) \, ds\|
\]
\[
= \|\int_0^{\infty} L e^{sL}f (\P(S_s> t) -\P(\Phi_s^h>t)) \, ds\|.
\]
By \eqref{limtheorstable}, $\P(S_s> t) \to \P(\Phi_s^h>t)$ as $h\to 0$. Therefore $II \to 0$ by the dominated
convergence, as $h\to 0$.
\end{proof}
\begin{remark}
From this proof it is seen how to get some explicit rates of convergence. We are not going to give details.
\end{remark}
It is well known, see e.g. \cite{SZ} and detailed presentations in monographs \cite{Meerbook}, \cite{Kolbook19},
that the subordinated limiting evolution described by the operators
$\E F_{\si_t}$ solves fractional in time differential equations. Namely,
under the conditions of Proposition \ref{propCTRW}, the function $f_t(x)=\E (F_{\si_t}f)(x)$
satisfies the equation
\begin{equation}
\label{eqfraceq}
D^{\be}_{0+\star}f_t(x)=Lf(x), \quad f_0(x)=f(x),
\end{equation}
where $D^{\be}_{0+\star}$ is the Caputo-Djerbashian derivative of order $\be$ acting on the variable $t$, and the
operator $L$ acts on the variable $x$.
Recall that a L\'evy subordinator is a process generated by the operator
\begin{equation}
\label{eqLevysubord}
L_{\nu}f(x)=\int_0^{\infty} f(x+y)-f(x)) \nu (dy),
\end{equation}
where $\nu$ is a one-sided L\'evy measure, that is , it satisfies the condition $\int \min(1,y) \nu(dy)<\infty$.
Proposition \ref{propCTRW} is based on the central limit for stable laws stating the convergence $\Phi_t^h \to S_t$
of random walks approximations to a stable L\'evy subordinator. If scaled random walks $\Phi_t^h$ are designed in such a way
that they approximate an arbitrary L\'evy subordinator, that is,
$\Phi_t^h \to S_t$ with $S_t$ generated by \eqref{eqLevysubord},
then similar arguments show that
\[
\E U_h^s|_{s=[N_t^h/h]} \to \E F_{\si_t}, \quad h\to 0,
\]
where
\[
\si_y=\max \{t: S_t \le y\}, \quad
\N_y^h=\max \{t: \Phi_t^h \le y\}.
\]
In this case the functions $f_t(x)=\E (F_{\si_t}f)(x)$
satisfy the equation
\begin{equation}
\label{eqfraceq1}
D^{(\nu)}_{0+\star}f_t(x)=L_tf(x), \quad f_0(x)=f(x),
\end{equation}
see e.g. \cite{Kol7}, \cite{Kol15},
where $D^{(\nu)}_{0+\star}$ is the generalised Caputo-type mixed fractional derivative
defined by the equation
\begin{equation}
\label{eqfraceq2}
D^{(\nu)}_{0+\star}f_t=\int_0^t (f_{t-s}-f_t)\nu(ds)+ (f_0-f_t)\int_t^{\infty} \nu(ds).
\end{equation}
The derivative $D^{\be}_{0+\star}$ in \eqref{eqfraceq} corresponds to $\nu(dy)=y^{-1-\be} dy$.
| 96,418
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77 Comments
- inactive, on 10/12/2007, -7/+44This happens a lot with the scientific unknown...
They said the atmosphere would be set on fire by above ground nuclear testing...
They said that planes would tear themselves apart attempting to break the unbreakable sound barrier (a few did, but we eventually got there)...
They said that we would starve ourselves to death with genetic modifications of crops...
The list goes on and on... but for the most party "they" are usually wrong.
- inactive, on 10/12/2007, -3/+38If they create a black hole in Australia will it rotate counter clockwise?
I should get a grant to figure this out.
- scott1, on 10/12/2007, -3/+29If we all do die we won't have the time to blame the scientist. We should all be sucked into infite point of singularity within a second.
And if we all do die, Pluto will be happy that Earth is no longer a planet.
- DjOverEZ, on 10/12/2007, -3/+27Am I the only one who now reads "Wii" everytime a headline says "Will?"
- NinjAlt, on 10/12/2007, -1/+20Maybe they should print "Dont Panic" in large friendly letters on the cover.
- MSTK, on 10/12/2007, -8/+27Actually most of time, "they" are usually right. It's just the times that they are wrong that we remember.
- syberghost, on 10/12/2007, -0/+14Well what the hell good is it then?
- Computer_Kid, on 10/12/2007, -0/+12famous last words?
- doublebackslash, on 10/12/2007, -1/+12Here is a little bit of science for everyone at the table:
Black holes emit Hawking radiation. The reason for this is that particle/anti-particle pairs are constantly being created & destroyed at all points in space (also photon pairs with opposite spin and shifted 90 degrees in frequency). Normally they cancel each other out, it is as simple as 1 + (-1) = 0; the event horizon of a black hole, however, is special. The outer edge of a black hole is space curved to an extreme. When a pair of particles are created at the horizon one tends to loose energy as it is accelerated away from its partner by the black hole and when they meet up to annihilate there is an imbalance of energy due to blue-shift. This imbalance results in energy (and therefore, via relativity, mass. E=Mc^2 and all that) being released from the black hole in the same way other quantum particles do: photons. This is similar to the way electrons drop states, or radioactive decay works, except not exactly. It is a lot of terribly ugly physics and some frankly beautiful math. The Heisenberg uncertainly principle, plank's constant, the speed of light, E=Mc^2, etc all get to play. The point of it all is that the steeper the curve in space around the event horizon the greater the imbalance in energy possible between the two particles. Large black holes have smoother curves around them than do smaller ones and therefore do not leach as much energy from the virtual particle pairs and therefore do not have to give up as much energy to maintain the law of conservation of energy (energy can neither be created nor destroyed).
In the end this mens that smaller black holes 'evaporate' much, much faster than larger ones. In fact the growth is cubic. Small back holes of even appreciable mass burn away in a fraction of a second.
The black holes that will be created in this particle accelerator will burn away quickly and brightly, spilling all manner of interesting particles into detectors for scientists to pour over. The creation of mini black holes my in fact allow physicists to study particles at energies greatly exceeding that normally produced by the accelerator, since black hole decay can be a faster release of energy than a mere collision.
This is truly a good day for science, but , I would like to note, also science fiction. The power source of choice for Romulan star ships in Star Trek was a small back hole; as I have explained here a small back hole is actually a very good source of energy, and it can be fed any sort of matter for fuel =)
Good stuff.
- longman2g, on 10/12/2007, -3/+13How many times have people said that the earth would be destoryed? Thousands, at a minimum. How many times has it been destroyed? 0. Saying they are usually wrong is an understatement.
- wurzelgummage, on 10/12/2007, -0/+9I have to warn you!! don't turn it on .. there will be a TIME BOUNCE.. I'm running out of t
- versionist, on 10/12/2007, -1/+10John Titor!
- inactive, on 10/12/2007, -0/+8Dr. Octopus is unavailable for comment.
- Web_Weasel, on 10/12/2007, -0/+7Had to digg that comment just because if I had a time machine I would go back in time and make a hoax about myself.
- TenebrousX, on 10/12/2007, -0/+7@ Poco: you're thinking of the wrong "they"
the original post was refering to people who do not understand things, and make outlandish predictions
- CKR600, on 10/12/2007, -0/+6Scientist 1: did it work? is that a black hole?
S2: I don't know, I can't see it.
S1: Well go check it out.
S2: ***** that, you go.. no wait...
S3: Hey sorry I'm late... did you guys start?
S1: No no not yet, hey can you go do some final diagnostics under the accelerator? Thanks.
- BowieX, on 10/12/2007, -0/+5I love the subsidiary article:
Top 10 Ways to Destroy Earth
- md81544, on 10/12/2007, -0/+5I can't help thinking that many of the arguments presented thus are directly analogous to:
"we've played Russian roulette loads of times before, and we've never blown our heads off, therefore we never will..."
- inactive, on 10/12/2007, -1/+6Yeah this suuuuuucks...
- MrViklund, on 10/12/2007, -4/+8You can't speak for them all. And fact is that most that we know today have come through testing/trail and error.
- Dakoman, on 10/12/2007, -0/+4"At its maximum, each particle beam the collider fires will pack as much energy as a 400-ton train traveling at 120 mph"
wow.
- 0v3rk1ll, on 10/12/2007, -0/+4Great. Everyone grab a shotgun and lots of shells, 'cause if Hell gets spawned here we're gonna have a serious problem on our hands... :-/
- escheppa, on 10/12/2007, -0/+4clock in at 9 and out at a singularity a typical day at the old black hole factory
- Mosatii, on 10/12/2007, -0/+4It's time to wake up Gordon. Time to wake up, and smell--the ashes.
- Bob042, on 10/12/2007, -0/+4I like the "Top 10 ways to destroy the Earth" list linked to on that page.
- dziban303, on 10/12/2007, -1/+4Actually, it was strangelets at the RHIC.
- Dufresne, on 10/12/2007, -2/+5This shows humanity's biggest weakness: we value knowledge and technology over our wellbeing; there is no use for a black hole, and yet we make one anyways. Why? to see what happens.
- chrono13, on 10/12/2007, -3/+6While my completely uneducated guess is that this will not kill us all, people who use past examples as proof of future unknowns is as bad as those who use fear and uncertainty as proof of the worst-possible-scenario.
It is an undeniable fact that "they" (the worst-case folks) only have to be right once for it to be a Very Bad Thing™.
If 100 coins were flipped for a year, once a second, and each coin landed on heads every time.
What are the odds then, after all that, that one more coin flipped would land on heads?
The same as any other coin flip: 50/50.
I'm all for discussion. Now lets discuss why we believe it won't destroy us, or why we believe it will. Lets leave superstition, and past short-sightedness out of it.
How about
"Does anyone know what happens if something unexpected happens - does it's safey rely on everything going perfectly?"
"What do the experts think?"
"Does anyone know if it is scalable or able to be modified to be a doomsday weapon?"
Discuss.
- Zreitan, on 10/12/2007, -0/+3I smell a High Budget Sci-fi Action Thriller!
- Atomic1fire, on 10/12/2007, -0/+3or general zod
- AnteChronos, on 10/12/2007, -2/+5"there is no use for a black hole, and yet we make one anyways. Why?"
Because a better understanding of physics results in more advanced technology which eventually makes our lives better. Saying that there is "no use" for the advancement of knowledge is very short sighted.
- Tawni, on 10/12/2007, -5/+8Why worry about a blackhole when you have a President that could get you blown up long before a blackhole would destroy us.
- Snakedal337, on 10/12/2007, -0/+2Dont worry, itll only destory the world on our line ;-)
- msoya, on 10/12/2007, -0/+2I'd rather they corrected their theories if they turned out to be wrong that clung to them, denying reality and refusing to change their minds. That is, after all, what science is all about.
At present, though, they seem to be doing pretty well with their "simpLeton's" approach.
- benjaminbr, on 10/12/2007, -1/+3Do it!
- CoolDude330, on 10/12/2007, -0/+2Hey how about a bigger version of that first pic?
That'd make a nice wallpaper.
- floppyparty, on 10/12/2007, -1/+3Wow, there are actually rumors about that? Hilarious. We all know the only thing we need to worry about is inadvertently opening a gateway into some hellish dimension.
- msoya, on 10/12/2007, -0/+2S1: Climb up and start the rotors.
S1: Very good. We’ll take it from here.
S2: Power to stage one emitters in three... two... one...
S2: I’m seeing predictable phase arrays.
S2: Stage two emitters activating... now.
S1: I have just been informed that the sample is ready, Gordon.
- Web_Weasel, on 10/12/2007, -0/+2That is good news. It should make the premiums on Destroying the Earth insurance much lower.
- budderfly, on 10/12/2007, -0/+2Boy, that was uplifting...!
- drog, on 10/12/2007, -1/+3How about some alternative theories. You accuse these people of arrogance yet propose nothing.
- scott1, on 10/12/2007, -0/+2I don't know but I don't think any of the other planets will be comming to our funeral. Everone will just be having a party with pluto.
I'll just say "we got pwn'd"
- MrViklund, on 10/12/2007, -1/+3Hehe, very interesting article :)
- subscribtion, on 10/12/2007, -0/+2Damn. We will just have to keep trying until we get it right!
- DiggidtyDog, on 10/12/2007, -0/+2The Lifeboat Foundation, a nonprofit organization devoted to safeguarding humanity from what it considers threats to our existence, has stated that artificial black holes could "threaten all life on Earth" and so it proposes to set up "self-sustaining colonies elsewhere."
And just where would these "colonies" be located?
- malkir, on 10/12/2007, -0/+2What I like about this is on the one hand if Stephen Hawkings is right, we won't destroy ourselves with a black hole generator. On the other hand, he thinks we need to settle other planets in the next 100 years or risk the extinction of humanity. Right or wrong.. we're *****.
- Knife720, on 10/12/2007, -0/+1Anyone else remember when they thought that the Atom bomb would ignite the atmosphere?
Yeah, I predict that this is a similar reaction.
- baalzebub, on 10/12/2007, -0/+1after some thought i have a feeling this will fail...
- bedlam17, on 10/12/2007, -0/+1Like singularity.com?
- inactive, on 10/12/2007, -1/+2It doesn't matter if they were right all those hundreds of times before, if we stuff it just once, then bye bye humanity.
- Show 51 - 77 of 77 discussions
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\begin{document}
\maketitle
\pagenumbering{arabic}
\begin{abstract}
Despite the superior empirical success of deep meta-learning, theoretical understanding of overparameterized meta-learning is still limited. This paper studies the generalization of a widely used meta-learning approach, Model-Agnostic Meta-Learning (MAML), which aims to find a good initialization for fast adaptation to new tasks.
Under a mixed linear regression model, we analyze the generalization properties of MAML trained with SGD in the overparameterized regime.
We provide both upper and lower bounds for the excess risk of MAML, which captures how SGD dynamics affect these generalization bounds. With such sharp characterizations, we further explore how various learning parameters impact the generalization capability of overparameterized MAML, including explicitly identifying typical data and task distributions that can achieve diminishing generalization error with overparameterization, and characterizing the impact of adaptation learning rate on both excess risk and the early stopping time. Our theoretical findings are further validated by experiments.
\end{abstract}
\section{Introduction}
Meta-learning~\cite{hospedales2020meta} is a learning paradigm which aims to design algorithms that are capable of gaining knowledge from many previous tasks and then using it to improve the performance on future tasks efficiently. It has exhibited great power in various machine learning applications spanning over few-shot image classification~\cite{ren2018meta,rusu2018meta}, reinforcement learning~\cite{gupta2018meta} and intelligent medicine~\cite{gu2018meta}.
One prominent type of meta-learning approaches is an optimization-based method, Model-Agnostic Meta-Learning (MAML)~\cite{finn2017model}, which achieves impressive results in different tasks~\cite{obamuyide2019model,bao2019few,antoniou2018train}. The idea of MAML is to learn a good initialization $\boldsymbol{\omega}^{*}$, such that for a new task we can adapt quickly to
a good task parameter starting from $\boldsymbol{\omega}^{*}$. MAML takes a bi-level implementation: the inner-level initializes at the meta parameter and takes task-specific updates using a few steps of gradient descent (GD), and the outer-level optimizes the meta parameter across all tasks.
With the superior empirical success, theoretical justifications have been provided for MAML and its variants over the past few years from both optimization~\cite{finn2019online,wang2020globala,fallah2020convergence,ji2022theoretical} and generalization perspectives~\cite{amit2018meta,denevi2019learning,fallah2021generalization,chen2021generalization}. However, most existing analyses
did not take overparameterization into consideration, which we deem as crucial to demystify the remarkable generalization ability of deep meta-learning~\cite{zhang2021understanding, hospedales2020meta}. More recently, \cite{wang2020globalb} studied the MAML with overparameterized deep neural nets and derived a complexity-based bound to quantify the difference between the empirical and population loss functions at their optimal solutions. However, complexity-based generalization bounds tend to be weak in the high dimensional, especially in the overparameterized regime. Recent works~\cite{bernacchia2021meta,zou2021unraveling} developed more precise bounds for overparameterized setting under a mixed linear regression model, and identified the effect of adaptation learning rate on the generalization. Yet, they considered only the simple isotropic covariance for data and tasks, and did not explicitly capture how the generalization performance of MAML depends on the data and task distributions.
Therefore, the following important problem still remains largely open:
\begin{center}
\emph{ Can \textbf{overparameterized} MAML generalize well to a new task, under general data and task distributions?}
\end{center}
In this work, we utilize the mixed linear regression, which is widely adopted in theoretical studies for meta-learning~\cite{kong2020meta,bernacchia2021meta,denevi2018learning,bai2021important}, as a proxy to address the above question. In particular, we assume that each task $\tau$ is a noisy linear regression and the associated weight vector is sampled from a common distribution. Under this model, we consider one-step MAML meta-trained with stochastic gradient descent (SGD), where
we minimize the loss evaluated at single GD step further ahead for each task.
Such settings correspond to real-world implementations of MAML~\cite{finn2017meta,li2017meta,hospedales2020meta} and are extensively considered in theoretical analysis~\cite{fallah2020convergence,chen2022bayesian,fallah2021generalization}.
The focus of this work is the overparameterized regime, i.e., the data dimension $d$ is far larger than the meta-training iterations $T$ ($d\gg T$).
\subsection{Our Contributions}
Our goal is to characterize the generalization behaviours of the MAML output in the overparameterized regime, and to explore how different problem parameters, such as data and task distributions, the adaptation learning rate $\btr$, affect the test error. The main contributions are highlighted below.
\begin{itemize}
\item Our first contribution is a sharp characterization (both upper and lower bounds) of the excess risk of MAML trained by SGD. The results are presented in a general manner, which depend on a new notion of effective meta weight, data spectrum, task covariance matrix, and other hyperparameters such as training and test learning rates.
In particular, the {\bf effective meta weight} captures an essential property of MAML, where the inner-loop gradient updates have distinctive effects on different dimensions of data eigenspace, i.e., the importance of "leading" space will be magnified whereas the "tail" space will be suppressed.
\item We investigate the influence of data and task distributions on the excess risk of MAML. For $\log$-decay data spectrum, our upper and lower bounds establish a sharp phase transition of the generalization. Namely, the excess risk vanishes for large $T$ (where benign fitting occurs) if the data spectrum decay rate is faster than the task diversity rate, and non-vanishing risk occurs otherwise. In contrast, for polynomial or exponential data spectrum decays, excess risk always vanishes for large $T$ irrespective of the task diversity spectrum.
\item We showcase the important role the adaptation learning rate $\btr$ plays in the excess risk and the early stopping time of MAML. We provably identify a novel tradeoff between the different impacts of $\btr$ on the "leading" and "tail" data spectrum spaces as the main reason behind the phenomena that the excess risk will first increase then decrease as $\btr$ changes from negative to positive values under general data settings. This complements the explanation based only on the "leading" data spectrum space given in~\cite{bernacchia2021meta} for the isotropic case.
We further theoretically illustrate that $\btr$ plays a similar role in determining the early stopping time, i.e., the iteration at which MAML achieves steady generalization error.
\end{itemize}
\textbf{Notations.}
We will use bold lowercase and capital letters for vectors and matrices respectively. $\mathcal{N}\left(0, \sigma^{2}\right)$ denotes the Gaussian distribution with mean $0$ and variance $\sigma^2$. We use $f(x) \lesssim g(x)$ to denote the case $f(x) \leq c g(x)$ for some constant $c>0$. We use the standard big-O notation and its variants: $\mathcal{O}(\cdot), \Omega(\cdot)$, where $T$ is the problem parameter that becomes large. Occasionally, we use the symbol $\widetilde{\mathcal{O}}(\cdot)$
to hide $\polylog(T)$ factors. $\mathbf{1}_{(\cdot)}$ denotes the indicator function. Let $x^{+}=\max\{x,0\}$.
\section{Related Work}
\label{sec-related}
\paragraph{Optimization theory for MAML-type approaches} Theoretical guarantee of MAML was initially provided in~\cite{finn2017meta} by proving a universal approximation property under certain conditions. One line of theoretical works have focused on the optimization perspective. \cite{fallah2020convergence} established the convergence guarantee of one-step MAML for general nonconvex functions, and \cite{ji2022theoretical} extended such results to the multi-step setting. \cite{finn2019online} analyzed the regret bound for online MAML. \cite{wang2020globala,wang2020globalb} studied the global optimality of MAML with sufficiently wide deep neural nets (DNN). Recently, \cite{collins2022maml} studied MAML from a representation point of view, and showed that MAML can provably recover the ground-truth subspace. h
\paragraph{Statistical theory for MAML-type approaches.}
One line of theoretical analyses lie in the statistical aspect. \cite{fallah2021generalization} studied the generalization of MAML
on recurring and unseen tasks. Information theory-type generalization bounds for MAML were developed in~\cite{jose2021information,chen2021generalization}.
\cite{chen2022bayesian} characterized the gap of generalization error between MAML and Bayes MAML. \cite{wang2020globalb} provided the statistical error bound for MAML with overparameterized DNN. Our work falls into this category,
where the overparameterization has been rarely considered in previous works. Note that \cite{wang2020globalb} only derived the generalization bound from the complexity-based perspective to study the difference between the empirical and population losses for the obtained optimization solutions. Such complexity bound is typically related to the data dimension~\cite{neyshabur2018towards} and may yield vacuous bound in the high dimensional regime. However, our work show that the generalization error of MAML can be small even the data dimension is sufficiently large.
\paragraph{Overparamterized meta-learning.}
\cite{du2020few,sun2021towards} studied overparameterized meta-learning from a representation learning perspective.
The most relevant papers to our work are~\cite{zou2021unraveling,bernacchia2021meta}, where they derived the population risk
in overparameterized settings to show the effect of the adaptation learning rate for MAML. Our analysis differs from these works from two essential perspectives: \romannum{1}).\ we analyze the excess risk of MAML based on the optimization trajectory of SGD in non-asymptotic regime, highlighting the dependence of iterations $T$, while they directly solved the MAML objective asymptotically; \romannum{2}). \cite{zou2021unraveling,bernacchia2021meta} mainly focused on the simple isotropic case for data and task covariance, while
we explicitly explore the role of data and task distributions under general settings.
\paragraph{Overparameterized linear model.} There has been several recent progress in theoretical understanding of overparameterized linear model under different scenarios, where the main goal is to provide non-asymptotic generalization guarantees, such as studies of linear regression ~\cite{bartlett2020benign}, ridge regression~\cite{tsigler2020benign}, constant-stepsize SGD~\cite{zou2021benign}, decaying-stepsize SGD~\cite{wu2021last}, GD~\cite{xu2022relaxing}, Gaussian Mixture models~\cite{wang2021benign}. This paper aims to derive the non-asymptotic excess risk bound for MAML under mixed linear model, which can be independent of data dimension $d$ and still converge as the iteration $T$ enlarges.
\section{Preliminary}\label{sec-form}
\subsection{Meta Learning Formulation}
In this work, we consider a standard meta-learning setting~\cite{fallah2021generalization}, where a number of tasks share some similarities, and the learner aims to find a good model prior by leveraging task similarities, so that the learner can quickly find a desirable model for a new task by adapting from such an initial prior.
{\bf Learning a proper initialization.}
Suppose we are given a collection of tasks $\textstyle\{\tau_t\}^{T}_{t=1}$ sampled from some distribution $\mathcal{T}$. For each task $\tau_t$, we observe $N$ samples $\textstyle\mathcal{D}_{t}\triangleq (\mathbf{X}_t,\mathbf{y}_{t})=\left\{\left(\mathbf{x}_{t, j}, y_{t, j}\right) \in \mathbb{R}^{d} \times \mathbb{R}\right\}_{j \in\left[N\right]}\stackrel{i.i.d.}{\sim} \mathbb{P}_{\phi_{t}}(y|\mathbf{x}) \mathbb{P}(\mathbf{x})$, where $\phi_t$ is the model parameter for the $t$-th task. The collection of $\{\mathcal{D}_{t}\}^{T}_{t=1}$ is denoted as $\mathcal{D}$. Suppose that $\mathcal{D}_{t}$ is randomly split into training and validation sets, denoted respectively as $\mathcal{D}^{\text{in}}_{t}\triangleq (\Xb^{\text{in}}_t,\yb_t^{\text{in}})$ and $\mathcal{D}^{\text{out}}_{t}\triangleq (\Xb^{\text{out}}_t,\yb_t^{\text{out}})$, correspondingly containing $n_{1}$ and $n_2$ samples (i.e., $N=n_1+n_2$).
We let $\boldsymbol{\omega}\in\mathbb{R}^{d}$ denote the initialization variable. Each task $\tau_t$ applies an inner algorithm $\mathcal{A}$ with such an initial and obtains an output $\mathcal{A}(\boldsymbol{\omega};\mathcal{D}^{\text{in}}_{t})$. Thus, the adaptation performance of $\boldsymbol{\omega}$ for task $\tau_t$ can be measured by the mean squared loss over the validation set given by $\textstyle\ell(\mathcal{A}(\boldsymbol{\omega};\mathcal{D}^{\text{in}}_{t});\mathcal{D}^{\text{out}}_{t}):= \frac{1}{2n_2}\sum^{n_2}_{j=1} \left(\left\langle \mathbf{x}^{\text{out}}_{t,j}, \mathcal{A}(\boldsymbol{\omega};\mathcal{D}^{\text{in}}_{t})\right\rangle-y^{\text{out}}_{t,j}\right)^{2}$. The goal of meta-learning is to find an optimal
initialization $\hat{\boldsymbol{\omega}}^{*}\in\mathbb{R}^{d}$ by minimizing the following empirical meta-training loss:
\begin{align}
\min_{\boldsymbol{\omega}\in\mathbb{R}^{d}} \widehat{\mathcal{L}}(\mathcal{A},\boldsymbol{\omega};\mathcal{D}) \quad \text{ where }\;
\widehat{\mathcal{L}}(\mathcal{A},\boldsymbol{\omega};\mathcal{D})&=\frac{1}{T}\sum^{T}_{t=1}\ell(\mathcal{A}(\boldsymbol{\omega};\mathcal{D}^{\text{in}}_{t});\mathcal{D}^{\text{out}}_{t})\label{emp_loss}.
\end{align}
In the testing process, suppose a new task $\tau$ sampled from $\mathcal{T}$ is given, which is associated with the dataset $\mathcal{Z}$ consisting of $m$ points with the task. We apply the learned initial $\hat{\boldsymbol{\omega}}^{*}$
as well as the inner algorithm
$\mathcal{A}$ on $\mathcal{Z}$ to produce a task predictor. Then the test performance can be evaluated via the following population loss:
\begin{align}
\mathcal{L}(\mathcal{A},\boldsymbol{\omega})=\mathbb{E}_{\tau \sim \mathcal{T}} \mathbb{E}_{\mathcal{Z},(\mathbf{x}, y)\sim \mathbb{P}_{\phi}(y \mid \mathbf{x}) \mathbb{P}(\mathbf{x})} \left[\ell\left(\mathcal{A}\left(\boldsymbol{\omega}; \mathcal{Z}\right);(\mathbf{x},y)\right)\right].
\label{obj}
\end{align}
\paragraph{Inner Loop with one-step GD.}
Our focus of this paper is the popular meta-learning algorithm MAML~\cite{finn2017model}, where inner stage takes a few steps of GD update initialized from $\boldsymbol{\omega}$. We consider one step for simplicity, which is commonly adopted in the previous studies~\cite{bernacchia2021meta,collins2022maml,gao2020modeling}. Formally, for any $\boldsymbol{\omega}\in\mathbb{R}^d$, and any dataset $(\mathbf{X},\mathbf{y})$ with $n$ samples, the inner loop algorithm for MAML with a learning rate $\beta$ is given by
\begin{align}
\mathcal{A}(\boldsymbol{\omega};(\mathbf{X},\mathbf{y})):= \boldsymbol{\omega}-\beta \nabla_{\boldsymbol{\omega}} \ell\left(\boldsymbol{\omega};(\mathbf{X},\mathbf{y})\right)=
(\mathbf{I}-\frac{\beta}{n}\mathbf{X}^{\top}\mathbf{X})\boldsymbol{\omega}+\frac{\beta}{n}\mathbf{X}^{\top}\mathbf{y}.
\end{align}
We allow the learning rate to differ at the meta-training and testing stages, denoted as $\beta^{\text{tr}}$ and $\beta^{\text{te}}$ respectively. Moreover, in subsequent analysis, we will include the dependence on the learning rate to the inner loop algorithm and loss functions as
$\mathcal{A}(\boldsymbol{\omega},\beta;(\mathbf{X},\mathbf{y}))$, $\widehat{\mathcal{L}}(\mathcal{A},\boldsymbol{\omega},\beta;\mathcal{D})$ and $\mathcal{L}(\mathcal{A},\boldsymbol{\omega},\beta)$.
\paragraph{Outer Loop with SGD.}
We adopt SGD to iteratively update the meta initialization variable $\boldsymbol{\omega}$ based on the empirical meta-training loss \cref{emp_loss}, which is how MAML is implemented in practice~\cite{finn2017meta}.
Specifically, we use the constant stepsize SGD with iterative averaging~\cite{fallah2021generalization, denevi2018learning,denevi2019learning},
and the algorithm is summarized in \Cref{alg-meta-sgd}. Note that at each iteration, we use one task for updating the meta parameter, which can be easily generalized to the case with a mini-batch tasks for each iteration.
\begin{algorithm}[ht]
\caption{MAML with SGD}\label{alg-meta-sgd}
\begin{algorithmic}
\REQUIRE Stepsize $\alpha>0$, meta learning rate $\beta^{\text{tr}}>0$
\ENSURE $\boldsymbol{\omega}_{0}$
\FOR{$t=1$ to $T$}
\STATE Receive task $\tau_t$ with data $\mathcal{D}_t$
\STATE Randomly divided into training and validation set: $\mathcal{D}^{in}_{t}=(\mathbf{X}^{in}_t, \mathbf{y}^{in}_t)$, $\mathcal{D}^{out}_{t}=(\mathbf{X}^{out}_t, \mathbf{y}^{out}_t)$
\STATE Update $\boldsymbol{\omega}_{t+1} =\boldsymbol{\omega}_{t}-\alpha \nabla \ell(\mathcal{A}(\boldsymbol{\omega},\beta^{\text{tr}};\mathcal{D}^{\text{in}}_{t});\mathcal{D}^{\text{out}}_{t})$
\ENDFOR
\RETURN $\overline{\boldsymbol{\omega}}_T=\frac{1}{T}\sum^{T-1}_{t=0} \boldsymbol{\omega}_{t}$
\end{algorithmic}
\end{algorithm}
\paragraph{Meta Excess Risk of SGD.} Let $\boldsymbol{\omega}^{*}$ denote the optimal solution to the population meta-test error
\cref{obj}.
We define the following excess risk
for the output $\overline{\boldsymbol{\omega}}_T$ of SGD:
\begin{align}
R(\overline{\boldsymbol{\omega}}_T,\beta^{\text{te}})\triangleq \mathbb{E}\left[\mathcal{L}(\mathcal{A},\overline{\boldsymbol{\omega}}_T,\beta^{\text{te}})\right]-\mathcal{L}(\mathcal{A},\boldsymbol{\omega}^{*},\beta^{\text{te}})\label{excess}
\end{align}
which identifies the difference between adapting from the SGD output $\overline{\boldsymbol{\omega}}_T$ and from the optimal initialization $\boldsymbol{\omega}^{*}$. Assuming that each task contains a fixed constant number of samples, the total number of samples over all tasks is $\mathcal{O}(T)$. Hence, the overparameterized regime can be identified as $d\gg T$, which is the focus of this paper, and is in contrast to the well studied underparameterized setting with finite dimension $d$ $(d\ll T)$.
The goal of this work is to characterize the impact of SGD dynamics, demonstrating how the iteration $T$ affects the excess risk, which has not been considered in the previous overparameterized MAML analysis~\cite{bernacchia2021meta,zou2021unraveling}.
\subsection{Task and Data Distributions}
To gain more explicit knowledge of MAML, we specify the task and data distributions in this section.
{\bf Mixed Linear Regression.}
We consider a canonical case in which the tasks are linear regressions. This setting has been commonly adopted recently in~\cite{bernacchia2021meta,bai2021important,kong2020meta}.
Given a task $\tau$, its model parameter $\phi$ is determined by
$\boldsymbol{\theta}\in\mathbb{R}^{d}$,
and the output response is generated as follows:
\begin{align}
y=\boldsymbol{\theta}^{\top} \mathbf{x}+z, \quad \xb\sim\mathcal{P}_{\xb},\quad z\sim \mathcal{P}_{z}
\end{align}
where $\xb$ is the input feature, which follows the same distribution $\mathcal{P}_{\xb}$ across different tasks, and $z$ is the i.i.d.\ Gaussian noise sampled from $\mathcal{N}(0,\sigma^2)$. The task signal $\boldsymbol{\theta}$ has the mean $\boldsymbol{\theta}^{*}$ and the covariance $\Sigma_{\boldsymbol{\theta}}\triangleq \mathbb{E}[\boldsymbol{\theta}\boldsymbol{\theta}^{\top}]$. Denote the distribution of $\boldsymbol{\theta}$ as $\mathcal{P}_{\boldsymbol{\theta}}$. We do not make any additional assumptions on $\mathcal{P}_{\boldsymbol{\theta}}$, whereas recent studies on MAML~\cite{bernacchia2021meta,zou2021unraveling} assume it to be Gaussian and isotropic.
{\bf Data distribution.} For the data distribution $\mathcal{P}_{\xb}$, we
first introduce some mild regularity conditions:
\begin{enumerate}
\item $\xb\in\mathbb{R}^d$ is mean zero with covariance operator $\bSigma=\mathbb{E}[\xb \xb^{\top}]$;
\item The spectral decomposition of $\bSigma$ is $\boldsymbol{V} \boldsymbol{\Lambda} \boldsymbol{V}^{\top}=\sum_{i>0} \lambda_{i} \boldsymbol{v}_{i} \boldsymbol{v}_{i}^{\top}$, with decreasing eigenvalues $\lambda_1\geq \cdots\geq\lambda_d>0$, and suppose $\sum_{i>0}\lambda_{i} <\infty $.
\item $\bSigma^{-\frac{1}{2}} \mathbf{x}$ is $\sigma_{\xb}$-subGaussian.
\end{enumerate}
To analyze the stochastic approximation method SGD
, we take the following standard fourth moment condition~\cite{zou2021benign, jain2017markov, berthier2020tight}.
\begin{assumption}[Fourth moment condition] There exist positive constants $c_1,b_1>0$, such that for any positive semidefinite (PSD)
matrix $\mathbf{A}$, it holds that
\begin{align*}
b_1 \operatorname{tr}(\bSigma \mathbf{A}) \Sigma+\Sigma \mathbf{A }\bSigma \preceq\mathbb{E}_{\mathbf{x} \sim \mathcal{P}_{\mathbf{x}}}\left[\mathbf{x x}^{\top} \mathbf{A} \mathbf{x} \mathbf{x}^{\top}\right] \preceq c_1 \operatorname{tr}(\bSigma \mathbf{A}) \Sigma
\end{align*}
For the Gaussian distribution, it suffices to take $c_1=3,b_1=2.$
\end{assumption}
\subsection{Connection to a Meta Least Square Problem.} After instantiating our study on the task and data distributions in the last section, note that $\textstyle\nabla\ell(\mathcal{A}(\boldsymbol{\omega},\beta^{\text{tr}};\mathcal{D}^{\text{in}}_{t});\mathcal{D}^{\text{out}}_{t})$ is linear
with respect to $\boldsymbol{\omega}$. Hence, we can reformulate the problem \cref{emp_loss} as a least square (LS) problem with transformed meta inputs and output responses.
\begin{proposition}[Meta LS Problem]\label{prop1} Under the mixed linear regression model,
the expectation of the meta-training loss \cref{emp_loss} taken over task and data distributions can be rewritten as:
\begin{align}\label{linear-loss}
\mathbb{E}\left[\widehat{\mathcal{L}}(\mathcal{A},\boldsymbol{\omega},\beta^{\text{tr}};\mathcal{D})\right]= \mathcal{L}(\mathcal{A},\boldsymbol{\omega},\beta^{\text{tr}})=
\mathbb{E}_{\Bb , \boldsymbol{\gamma}} \frac{1}{2}\left[\left\|\Bb\boldsymbol{\omega}-\boldsymbol{\gamma}\right\|^{2}\right].
\end{align}
The meta data are given by
\begin{align}\Bb =& \frac{1}{\sqrt{n_2}}\Xb^{out}\Big(\mathbf{I}-\frac{\beta^{\text{tr}}}{n_1} {\Xb^{\text{in}}}^{T} {\Xb^{\text{in}}}\Big)\nonumber\\ \boldsymbol{\gamma} =& \frac{1}{\sqrt{n_2}}\Big( \Xb^{\text{out}}\Big( \mathbf{I}-\frac{\beta^{\text{tr}}}{n_1} {\Xb^{\text{in}}}^{T} {\Xb^{\text{in}}}\Big)\boldsymbol{\theta}+\zb^{out}-\frac{\beta^{\text{tr}}}{n_1} \Xb^{\text{out}}{\Xb^{\text{in}}}^{\top}\zb^{\text{in}}\Big)\label{meta-data} \end{align}
where $\Xb^{\text{in}}\in \mathbb{R}^{n_1\times d}$,$\zb^{\text{in}}\in \mathbb{R}^{n_1}$,$\Xb^{\text{out}}\in \mathbb{R}^{n_2\times d}$ and $\zb^{\text{out}}\in \mathbb{R}^{n_2}$ denote the inputs and noise for training and validation.
Furthermore,
we have
\begin{align}
\boldsymbol{\gamma} = \Bb\boldsymbol{\theta}^{*}+\boldsymbol{\xi}\quad \text{ with meta noise } \mathbb{E}[\boldsymbol{\xi}\mid\Bb]=0. \label{linear}
\end{align}
\end{proposition}
Therefore, the meta-training objective
is equivalent to searching for a $\boldsymbol{\omega}$, which is close to the task mean $\boldsymbol{\theta}^{*}$.
Moreover,
with the specified data and task model, the optimal solution for meta-test loss \cref{obj} can be directly calculated~\cite{gao2020modeling}, and we obtain $\boldsymbol{\omega}^{*}=\mathbb{E}[\btheta] =\btheta^{*}$. Hence, the meta excess risk~\cref{excess} is identical to the standard excess risk~\cite{bartlett2020benign} for the linear model \cref{linear}, i.e., $ R(\overline{\boldsymbol{\omega}}_T,\beta^{\text{te}})= \mathbb{E}_{\Bb , \boldsymbol{\gamma}} \frac{1}{2}\left[\left\|\Bb\wl_{T}-\boldsymbol{\gamma}\right\|^{2}-\left\|\Bb\btheta^{*}-\boldsymbol{\gamma}\right\|^{2}\right]$, but with more complicated input and output data expressions. The following analysis will focus on this transformed linear model.
Furthermore, we can calculate the statistical properties of the reformed input $\Bb$, and obtain the meta-covariance: $$ E[\Bb^{\top}\Bb]=(\Ib-\beta^{\text{tr}}\bSigma)^2\bSigma+\frac{{\beta^{\text{tr}}}^2}{n_1}(F-\bSigma^3)$$ where $F=E[\xb\xb^{\top}\Sigma \xb\xb^{\top}]$. Let $\Xb\in\mathbb{R}^{n\times d}$ denote the collection of $n$ i.i.d.\ samples from $\mathcal{P}_{\xb}$, and denote
$$
\Hb_{n,\beta}=\mathbb{E}[(\mathbf{I}-\frac{\beta}{n}\Xb^{\top}\Xb)\Sigma(\mathbf{I}-\frac{\beta}{n}\Xb^{\top}\Xb)
]=(\Ib-\beta\bSigma)^2\bSigma+\frac{\beta^2}{n}(F-\bSigma^3).
$$
We can then write $E[\Bb^{\top}\Bb]=\Hb_{n_1,\beta^{\text{tr}}}$.
Regarding the form of $\Bb$ and $\Hb_{n_1,\beta^{\text{tr}}}$, we need some
further conditions on the higher order moments of the data distribution.
\begin{assumption}[Commutity
]\label{ass-comm}
$F=E[\xb\xb^{\top}\bSigma \xb\xb^{\top}]$ commutes with the data covariance $\bSigma$.
\end{assumption}
\Cref{ass-comm} holds for Gaussian data. Such commutity of $\bSigma$ has also been considered in ~\cite{zou2021benign}.
\begin{assumption}[Higher order moment condition]\label{ass:higherorder}
Given $|\beta|<\frac{1}{\lambda_1}$ and $\bSigma$, there exists a constant $C(\beta,\bSigma)>0$, for large $n>0$, s.t. for any unit vector $\vb\in\mathbb{R}^d$, we have:
\begin{align}\label{hoc}
\mathbb{E}[\|\vb^{\top}\Hb^{-\frac{1}{2}}_{n,\beta}(\mathbf{I}-\frac{\beta}{n}\Xb^{\top}\Xb)\bSigma (\mathbf{I}-\frac{\beta}{n}\Xb^{\top}\Xb)\Hb^{-\frac{1}{2}}_{n,\beta}\vb\|^2]< C(\beta,\bSigma).
\end{align}
\end{assumption}
In \Cref{ass:higherorder}, the analytical form of $C(\beta,\bSigma)$ can be derived if $\bSigma^{-\frac{1}{2}}\mathbf{x}$ is Gaussian.
Moreover, if $\beta=0$, then we obtain $C(\beta,\bSigma)=1$. Further technical discussions
are presented in Appendix.
\section{Main Results}\label{sec-main}
In this section, we present our analyses on generalization properties of MAML optimized by average SGD and derive insights on the effect of various parameters. Specifically, our results consist of three parts. First, we characterize the meta excess risk of MAML trained with SGD. Then, we establish the generalization error bound for various types of data and task distributions, to reveal which kind of overparameterization regarding data and task is essential for diminishing meta excess risk. Finally, we explore how the adaptation learning rate $\btr$ affects the excess risk and the training dynamics.
\subsection{Performance Bounds}
Before starting our results, we first introduce relevant notations and concepts. We define the following rates of interest (See \Cref{remark-f} for further discussions)
\begin{align*} c(\beta,\bSigma) &:= c_1(1+8|\beta|\lambda_1\sqrt{C(\beta,\bSigma)}\sigma_x^2+ 64\sqrt{C(\beta,\bSigma)}\sigma_x^4\beta^2\operatorname{tr}(\bSigma^2))\\
f(\beta,n,\sigma,\bSigma,\bSigma_{\boldsymbol{\theta}})&:=c(\beta,\bSigma)\operatorname{tr}({\bSigma_{\boldsymbol{\theta}}\bSigma})+4c_1\sigma^2\sigma_x^2\beta^2\sqrt{C(\beta,\bSigma)}\operatorname{tr}(\bSigma^2)+\sigma^2/n\\
g(\beta,n, \sigma,\bSigma, \bSigma_{\btheta}) & :={\sigma^2+b_1\operatorname{tr}(\bSigma_{\btheta}\Hb_{n,\beta})+\beta^2\mathbf{1}_{\beta\leq 0} b_1 \operatorname{tr}(\bSigma^2)/{n}}.
\end{align*}
Moreover, for a positive semi-definite matrix $\Hb$, s.t. $\Hb$ and $\bSigma$ can be diagonalized simultaneously, let $\mu_i(\Hb)$ denote its corresponding eigenvalues for $\vb_i$, i.e. $\Hb = \sum_{i}\mu_i(\Hb)\vb_i\vb_i^{\top}$ (Recall $\vb_i$ is the $i$-th eigenvector of $\bSigma$).
We next introduce the following new notion of the \emph{effective meta weight}, which will serve as an important quantity for capturing the generalization of MAML.
\begin{definition}[Effective Meta Weights]\label{meta-weight}
For $|\btr|,|\bte|<1/\lambda_1$, given step size $\alpha $ and iteration $T$, define
\begin{equation}
\Xi_i (\bSigma
,\alpha,T)=\begin{cases}
\mu_i(\Hb_{m,\beta^{\text{te}}})/\left(T \mu_i(\Hb_{n_1,\beta^{\text{tr}}})\right) & \mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})\geq \frac{1}{\alpha T}; \\
T\alpha^2 \mu_i(\Hb_{n_1,\btr})\mu_i(\Hb_{m,\beta^{\text{te}}})& \mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})< \frac{1}{\alpha T}.
\end{cases}
\end{equation}
We call $ \mu_i(\Hb_{m,\beta^{\text{te}}})/ \mu_i(\Hb_{n_1,\beta^{\text{tr}}})$ and $ \mu_i(\Hb_{m,\beta^{\text{te}}})\mu_i(\Hb_{n_1,\beta^{\text{tr}}})$ the \textbf{meta ratio} (See \Cref{remark-weight}).
\end{definition}
We omit the arguments of the effective meta weight $\Xi_i$ for simplicity in the following analysis.
Our first results characterize matching upper and lower bounds on the meta excess risk of MAML in terms of the effective meta weight.
\begin{theorem}[Upper Bound]\label{thm-upper} Let $\omega_i=\left\langle\boldsymbol{\omega}_{0}-\boldsymbol{\theta}^{*}, \mathbf{v}_{i}\right\rangle$. If $|\btr|,|\bte|<1/\lambda_1$, $n_1$ is large ensuring that $\mu_i(\Hb_{n_1,\beta^{\text{tr}}})>0$, $\forall i$ and
$\alpha<1/\left(c(\btr,\bSigma) \operatorname{tr}(\bSigma)\right)$, then the meta excess risk $R(\overline{\boldsymbol{\omega}}_T,\bte)$ is bounded above as follows
\[R(\wl_{T},\bte)\leq \text{Bias}+ \text{Var} \]
where
\begin{align*}
\text{Bias} & = \frac{2}{\alpha^2 T} \sum_{i}\Xi_i \frac{\omega_i^2}{\mu_i(\Hb_{n_1,\beta^{\text{tr}}})} \\
\text{Var} &= \frac{2}{(1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma))}\left(\sum_{i}\Xi_{i} \right)
\\
\quad \times & [\underbrace{f(\btr,n_2,\sigma,\bSigma_{\boldsymbol{\theta}},\bSigma)}_{V_1}+\underbrace{ 2c(\btr,\bSigma)
\sum_{i}\left( \frac{\mathbf{1}_{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})\geq \frac{1}{\alpha T}}}{T\alpha \mu_i(\Hb_{n_1,\beta^{\text{tr}}})}+\mathbf{1}_{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})< \frac{1}{\alpha T} } \right) \lambda_{i}\omega^2_i}_{V_2}]
\end{align*}
\end{theorem}
\begin{remark}\label{remark-1}
The primary error source of the upper bound are two folds. The bias term corresponds to the error if we directly implement GD updates towards the meta objective~\cref{linear-loss}. The variance error is composed of the disturbance of meta
noise $\boldsymbol{\xi}$ (the $V_1$ term), and the randomness of SGD itself (the $V_2$ term). Regardless of data or task distributions, for proper stepsize $\alpha$, we can easily derive that the bias term is $\mathcal{O}(\frac{1}{T})$, and the $V_2$ term is also $\mathcal{O}(\frac{1}{T})$, which is dominated by $V_1$ term ($\Omega(1)$). Hence, to achieve the vanishing risk, we need to understand the roles of $\Xi_i$ and $f(\cdot)$
\end{remark}
\begin{remark}[Effective Meta Weights]\label{remark-weight}
By \Cref{meta-weight}, we separate the data eigenspace into “\textbf{leading}” $(\geq \frac{1}{\alpha T})$ and “{\bf tail}” $(< \frac{1}{\alpha T})$ spectrum spaces with different meta weights.
The meta ratios
indicate the impact of one-step gradient update. For large $n$,
$\mu_i(\Hb_{n,\beta})\approx (1-\beta\lambda_i)^2\lambda_i$, and hence a larger $\btr$ in training will increase the weight for “leading” space and decrease the weight for “tail” space, while a larger $\bte$ always decreases the weight.
\end{remark}
\begin{remark}[Role of $f(\cdot)$]\label{remark-f} $f(\cdot)$ in variance term consists of various sources of meta noise $\boldsymbol{\xi}$, including inner gradient updates ($\beta$), task diversity ($\bSigma_{\btheta}$) and noise from regression tasks ($\sigma$). As mentioned in \Cref{remark-1}, understanding $f(\cdot)$ is critical in our analysis. Yet,
due to the multiple randomness origins, techniques for classic linear regression~\cite{zou2021benign,jain2017markov} cannot be directly applied here.
Our analysis overcomes such non-trivial challenges. $g(\cdot)$ in \Cref{thm-lower} plays a similar role to $f(\cdot)$.
\end{remark}
Therefore, \Cref{thm-upper} implies that overparameterization is crucial for diminishing risk under the following conditions:
\begin{itemize}[itemsep=2pt,topsep=0pt,parsep=0pt]
\item For $f(\cdot)$: $\operatorname{tr}(\bSigma\bSigma_{\btheta})$ and $\operatorname{tr}(\bSigma^2)$ is small compared to $T$;
\item For $\Xi_i$: the dimension of "leading" space is $o(T)$, and the summation of meta ratio over "tail" space is $o(\frac{1}{T})$.
\end{itemize}
We next provide a lower bound on the meta excess risk, which matches the upper bound in order.
\begin{theorem}[Lower Bound]~\label{thm-lower}
Following the similar notations in ~\Cref{thm-upper}, Then
\begin{align*}
R(\overline{\boldsymbol{\omega}}_T,\bte) \ge & \frac{1}{100\alpha^2 T} \sum_{i}\Xi_i \frac{\omega_i^2}{\mu_i(\Hb_{n_1,\beta^{\text{tr}}})} +\frac{1}{n_2}\cdot \frac{1}{(1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma))}\sum_{i}\Xi_{i} \\
\times & [\frac{1}{100} g(\btr,n_1, \bSigma, \bSigma_{\btheta})+\frac{b_1}{1000}
\sum_{i}\Big( \frac{\mathbf{1}_{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})\geq \frac{1}{\alpha T}}}{T\alpha \mu_i(\Hb_{n_1,\beta^{\text{tr}}})}+\mathbf{1}_{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})< \frac{1}{\alpha T} } \Big) \lambda_{i}\omega^2_i].
\end{align*}
\end{theorem}
Our lower bound can also be decomposed into bias and variance terms as the upper bound. The bias term well matches the upper bound up to absolute constants. The variance term differs from the upper bound only by $\frac{1}{n_2}$, where $n_2$ is the batch size of each task, and is treated as a constant (i.e., does not scale with $T$)~\cite{jain2018parallelizing,shalev2013accelerated} in practice. Hence, in the overparameterized regime where $d\gg T$ and $T$ tends to be sufficiently large, the variance term also matches that in the upper bound w.r.t.\ $T$.
\subsection{The Effects of Task Diversity}\label{sec-main-task}
From \Cref{thm-upper} and \Cref{thm-lower}, we observe that the task diversity $\bSigma_{\btheta}$ in $f(\cdot)$ and $g(\cdot)$ plays a crucial role in the performance guarantees for MAML. In this section,
we explore several types of data distributions to further characterize the effects of the task diversity.
We take the single task setting as a comparison with meta-learning, where the task diversity diminishes (tentatively say $\bSigma_{\btheta}\rightarrow\mathbf{0}$), i.e., each task parameter $\boldsymbol{\theta}=\boldsymbol{\theta}^{*}$. In such a case, it is unnecessary to do one-step gradient in the inner loop and we set $\btr=0$, which is equivalent to directly running SGD. Formally, the {\bf single task setting} can be described as outputting $\overline{\boldsymbol{\omega}}^{\text{sin}}_{T}$ with iterative SGD that minimizes $\widehat{\mathcal{L}}(\mathcal{A},\boldsymbol{\omega},0;\mathcal{D})$ with meta linear model as $\boldsymbol{\gamma} = \frac{1}{\sqrt{n_2}}(\Xb^{out}\boldsymbol{\theta}^{*}+\zb^{\text{out}})$.
\Cref{thm-upper} implies that the data spectrum should decay fast, which leads to a small dimension of "leading" space and small meta ratio summation over "tail" space. Let us first consider a relatively slow decaying case: $\lambda_k=k^{-1}\log^{-p}(k+1)$ for some $p>1$. Applying \Cref{thm-upper}, we immediately derive the theoretical guarantees for single task:
\begin{lemma}[Single Task]\label{lem-single}
If $|\beta^{\text{te}}|<\frac{1}{\lambda_1}$ and if the spectrum of $\bSigma$ satisfies $\lambda_k=k^{-1}\log^{-p}(k+1)$, then $R(\overline{\boldsymbol{\omega}}^{\text{sin}}_T,\beta^{\text{te}})=\mathcal{O}(\frac{1}{\log^{p}(T)})$
\end{lemma}
At the test stage, if we set $\beta^{\text{te}}=0$, then the meta excess risk for the single task setting, i.e., $R(\overline{\boldsymbol{\omega}}^{\text{sin}}_T,0)$, is exactly the excess risk in classical linear regression~\cite{zou2021benign}. \Cref{lem-single} can be regarded as a generalized version of Corollary 2.3 in \cite{zou2021benign}, where they provide the upper bound for $R(\overline{\boldsymbol{\omega}}^{\text{sin}}_T,0)$, while we allow a one-step fine-tuning for testing.
Lemma~\ref{lem-single} suggests that the $\log$-decay is
sufficient to assure that $R(\overline{\boldsymbol{\omega}}^{\text{sin}}_T,0)$ is diminishing when $d\gg T$.
However, in meta-learning with multi-tasks, the task diversity captured by the task spectral distribution can highly affect the meta excess risk. In the following, our \Cref{thm-upper} and \Cref{thm-lower} (i.e., upper and lower bounds) establish a sharp phase transition of the generalization for MAML for the same data spectrum considered in Lemma~\ref{lem-single}, which is in contrast to the single task setting (see \Cref{lem-single}), where $\log$-decay data spectrum always yields vanishing excess risk.
\begin{proposition}[MAML, $\log$-Decay Data Spectrum]\label{prop-hard}
Given $|\btr|, |\beta^{\text{te}}|<\frac{1}{\lambda_1}$, under the same data distribution as in \Cref{lem-single} with $d=\mathcal{O}(\operatorname{poly}(T))$ and the spectrum of $\bSigma_{\btheta}$, denoted as $\nu_i$, satisfies $\nu_k=\log^{r}(k+1)$ for some $r>0$, then
$$R(\overline{\boldsymbol{\omega}}_T,\beta^{\text{te}})=\begin{cases}
\Omega(\log^{r-2p+1}(T))&{r\geq 2p-1}\\ \mathcal{O}(\frac{1}{\log^{p-(r-p+1)^{+}}(T)}) &{r<2p-1}
\end{cases}$$
\end{proposition}
\Cref{prop-hard} implies that under $\log$-decay data spectrum parameterized by $p$, the meta excess risk of MAML experiences a phase transition determined by the spectrum parameter $r$ of task diversity. While slower task diversity rate $r < 2p-1$ guarantees vanishing excess risk, faster task diversity rate $r \ge 2p-1$ necessarily results in non-vanishing excess risk.
\Cref{prop-hard} and \Cref{lem-single} together indicate that while $\log$-decay data spectrum always yields benign fitting (vanishing risk) in the single task setting, it can yield non-vanishing risk in meta learning due to fast task diversity rate.
We further validate our theoretical results in \Cref{prop-hard} by experiments.
We consider the case $p=2$. As shown in \Cref{fig:subfig:1}, when $r<2p-1$, the test error quickly converges to the Bayes error. When $r>2p-1$, \Cref{fig:subfig:2} illustrates that MAML already
converges on the training samples, but the test error (which is further zoomed in \Cref{fig:subfig:3}) levels off and does not vanish, showing MAML generalizes poorly when $r>2p-1$.
\begin{figure}[ht]
\centering
\begin{subfigure}{.31\textwidth}
\centering
\includegraphics[width=\linewidth]{task1-te.png}
\caption{\small$\nu_i=0.25\log^{1.5}(i+1)$}
\label{fig:subfig:1}
\end{subfigure}
\begin{subfigure}{.31\textwidth}
\centering
\includegraphics[width=\linewidth]{task3.png}
\caption{\small$\nu_i=0.25\log^{8}(i+1)$}
\label{fig:subfig:2}
\end{subfigure}
\begin{subfigure}{.31\textwidth}
\centering
\includegraphics[width=\linewidth]{task3-te.png}
\caption{\small $\nu_i=0.25\log^{8}(i+1)$}
\label{fig:subfig:3}
\end{subfigure}
\caption{The effects of task diversity. $d=500$, $T=300$, $\lambda_i = \frac{1}{i\log(i+1)^2}$, $\btr=0.02$, $\bte=0.2$
}
\label{fig:twopicture}
\vspace{-0.3cm}
\end{figure}
Furthermore, we show that the above phase transition that occurs for $\log$-decay data distributions no longer exists for data distributions with faster decaying spectrum.
\begin{proposition}[MAML, Fast-Decay Data Spectrum]\label{prop-fast} Under the same task distribution as in \Cref{prop-hard}, i.e., the spectrum of $\bSigma_{\btheta}$, denoted as $\nu_i$, satisfies $\nu_k=\log^{r}(k+1)=\widetilde{O}(1)$ for some $r>0$, and the data distribution with $d=\mathcal{O}(\operatorname{poly}(T))$ satisfies:
\begin{enumerate}[itemsep=2pt,topsep=0pt,parsep=0pt]
\item $\lambda_k=k^{-q}$ for some $q>1$, $R(\overline{\boldsymbol{\omega}}^{\text{sin}}_T,\beta^{\text{te}})=\mathcal{O}\left(\frac{1}{T^{\frac{q-1}{q}}}\right)$ and $R(\overline{\boldsymbol{\omega}}_T,\beta^{\text{te}})=\widetilde{\mathcal{O}}\left(\frac{1}{T^{\frac{q-1}{q}}}\right)$;
\item $\lambda_k=e^{-k}$, $R(\overline{\boldsymbol{\omega}}^{\text{sin}}_T,\beta^{\text{te}})=\widetilde{\mathcal{O}}(\frac{1}{T})$ and $R(\overline{\boldsymbol{\omega}}_T,\beta^{\text{te}})=\widetilde{\mathcal{O}}(\frac{1}{T})$.
\end{enumerate}
\end{proposition}
\subsection{On the Role of Adaptation Learning Rate}\label{sec-main-stopping}
The analysis in \cite{bernacchia2021meta} suggests a surprising observation that a negative learning rate (i.e., when $\beta^{\text{tr}}$ takes a negative value) optimizes the generalization for MAML under mixed linear regression models. Their results indicate that the testing risk initially increases and then decreases as $\beta^{\text{tr}}$ varies from negative to positive values around zero for Gaussian isotropic input data and tasks. Our following proposition supports such a trend, but with a novel tradeoff in SGD dynamics as a new reason for the trend, under more general data distributions.
Denote $\overline{\boldsymbol{\omega}}^{\beta}_T$ as the average SGD solution of MAML after $T$ iterations that uses $\beta$ as the inner loop learning rate.
\begin{proposition}\label{prop-tradeoff}
Let $s=T\log^{-p}(T)$ and $d=T\log^{q}(T)$, where $p,q>0$.
If the spectrum of $\bSigma$ satisfies
$$\lambda_{k}= \begin{cases}1 / s, & k \leq s \\ 1 /(d-s), & s+1 \leq k \leq d. \end{cases}
$$
Suppose the spectral parameter $\nu_i$ of $\bSigma_{\btheta}$ is $O(1)$, and let the step size $\alpha=\frac{1}{2 c(\btr, \bSigma) \operatorname{tr}(\bSigma)}$. Then for large $n_1$, $|\beta^{\text{tr}}|, |\beta^{\text{te}}|<\frac{1}{\lambda_1}$, we have
\begin{align}\label{eq-tradeoff}
R(\overline{\boldsymbol{\omega}}^{\beta^{\text{tr}}}_T,\bte)\lesssim
& \mathcal{O}\Big(\frac{1}{\log^{p}(T)}\Big) \frac{1}{(1-\btr \lambda_{1})^{2}}+\mathcal{O}\Big(\frac{1}{\log^{q} (T)}\Big)\Big(1-\btr \lambda_{d}\Big)^{2}
+\widetilde{\mathcal{O}}(\frac{1}{T}).
\end{align}
\end{proposition}
The first two terms in the bound of \cref{eq-tradeoff} correspond to the impact of effective meta weights $\Xi_i$ on the "leading" and "tail" spaces, respectively, as we discuss in \Cref{remark-weight}. Clearly, the learning rate $\btr$ plays a tradeoff role in these two terms, particularly when $p$ is close to $q$. This explains the fact that the test error first increases and then decreases as $\btr$ varies from negative to positive values around zero. Such a tradeoff also serves as the reason for the first-increase-then-decrease trend of the test error under more general data distributions as we demonstrate in \Cref{fig:tradeoff}. This complements the reason suggested in \cite{bernacchia2021meta}, which captures only the quadratic form $\frac{1}{\left(1-\btr \lambda_{1}\right)^{2}}$ of $\btr$ for isotropic $\bSigma$, where there exists only the "leading" space without "tail" space.
\begin{figure}[ht]
\centering
\begin{subfigure}{.31\textwidth}
\centering
\includegraphics[width=\linewidth]{beta_1.pdf}
\caption{$\lambda_i=\frac{1}{i\log(i+1)^2}$}
\label{fig:tradeoff:1}
\end{subfigure}
\begin{subfigure}{.31\textwidth}
\centering
\includegraphics[width=\linewidth]{beta_3.pdf}
\caption{$\lambda_i=\frac{1}{i\log(i+1)^3}$}
\label{fig:tradeoff:2}
\end{subfigure}
\begin{subfigure}{.31\textwidth}
\centering
\includegraphics[width=\linewidth]{beta_2.pdf}
\caption{$\lambda_i=\frac{1}{i^2}$}
\label{fig:tradeoff:3}
\end{subfigure}
\caption{$R(\overline{\boldsymbol{\omega}}^{\beta^{\text{tr}}}_T,\bte)$ as a function of $\beta^{\text{tr}}$. $d=200$, $T=100$, $\bSigma_{\btheta}=\frac{0.8^2}{d}\mathbf{I}$, $\bte=0.2$
}
\label{fig:tradeoff}
\end{figure}
Based on the above results, incorporating with our dynamics analysis, we surprisingly find that $\btr$ not only affects the final risk, but also plays a pivot role towards the early iteration that the testing error tends to be steady. To formally study such a property, we define the stopping time as follows.
\begin{definition}[Stopping time]
Given $\btr,\bte$, for any $\epsilon>0$, the corresponding stopping time $t_{\epsilon}(\btr,\bte)$ is defined as:
\[
t_{\epsilon}(\btr,\bte) = \min t\quad
\text{s.t. }\; R(\wl^{\btr}_t;\bte)<\epsilon.
\]
\end{definition}
In the sequel, we may omit the arguments in $t_{\epsilon}$ for simplicity. We consider the similar data distribution in \Cref{prop-tradeoff} but parameterized by $K$, i.e., $s=K\log^{-p}(K)$ and $d=K\log^{q}(K)$, where $p,q> 0$. Then we can derive the following characterization for $t_{\epsilon}$.
\begin{corollary}~\label{col-stop}
If the assumptions in \Cref{prop-tradeoff} hold and $p=q$. Further, let $\bSigma_{\btheta}=\eta^2\mathbf{I}$, and $|\beta^{\text{tr}}|<\frac{1}{\lambda_1}$. Then for $t_{\epsilon}(\btr,\bte)\in (s, K]$, we have:
\begin{align}
\exp\Big(\epsilon^{-\frac{1}{p}} \Big[\frac{L_{l}}{(1-\btr\lambda_1)^2}+ L_{t} (1-\btr\lambda_d)^2\Big]^{\frac{1}{p}}\Big)
\leq t_{\epsilon}\leq \exp\Big(\epsilon^{-\frac{1}{p}}\Big[\frac{U_{l}}{(1-\btr\lambda_1)^2}+ U_{t} (1-\btr\lambda_d)^2\Big]^{\frac{1}{p}}\Big) \label{eq-stopping}
\end{align}
where $L_l$, $L_t$, $U_l$, $U_t>0$ are factors for "leading" and "tail" spaces that are independent of $K$\footnote{Such terms have been suppressed for clarity. Details are presented in the appendix.}.
\end{corollary}
\Cref{eq-stopping} suggests that the early stopping time $t_{\epsilon}$ is also controlled by the tradeoff role that $\btr$ plays in the "leading" ($U_l,L_l$) and "tail" spaces ($U_t,L_t$), which takes a similar form as the bound in \Cref{prop-tradeoff}. Therefore, the trend for $t_{\epsilon}$ in terms of $\btr$ will exhibit similar behaviours as the final excess risk, and hence the optimal $\btr$ for the final excess risk will lead to an earliest stopping time. We plot the training and test errors for different $\btr$ in Figure~\ref{fig:stopping}, under the same data distributions as \Cref{fig:tradeoff:1} to validate our theoretical findings. As shown in \Cref{fig:stopping:1}, $\btr$ does not make much difference in the training stage (the process converges for all $\btr$ when $T$ is larger than $100$). However, in \Cref{fig:stopping:2} at test stage, $\btr$ significantly affects the iteration when the test error starts to become relatively flat.
Such an early stopping time first increases then decreases as $\btr$ varies from $-0.5$ to $0.7$, which resembles the change of final excess risk in \Cref{fig:tradeoff:1}.
\begin{figure}[H]
\centering
\begin{subfigure}{.4\textwidth}
\centering
\includegraphics[width=\linewidth]{stopping-tr.pdf}
\caption{Training Risk}
\label{fig:stopping:1}
\end{subfigure}
\begin{subfigure}{.4\textwidth}
\centering
\includegraphics[width=\linewidth]{stopping-te.pdf}
\caption{Test Error}
\label{fig:stopping:2}
\end{subfigure}
\caption{Training and test curves for different $\beta^{\text{tr}}$. $d=500$, $\lambda_i=\frac{1}{i\log^2(i+1)}$,$\bSigma_{\btheta}=\frac{0.8^2}{d}\mathbf{I}$, $\bte=0.2$
}
\label{fig:stopping}
\end{figure}
\section{Conclusions }\label{sec-conclusion}
In this work, we give the theoretical treatment towards the generalization property of MAML based on their optimization trajectory in non-asymptotic and overparameterized regime.
We provide both upper and lower bounds on the excess risk of MAML trained with average SGD. Furthermore, we explore which type of data and task distributions are
crucial for diminishing error with overparameterization, and discover the influence of adaption learning rate both on the generalization error and the dynamics, which brings novel insights towards the distinct effects of MAML's one-step gradient updates on "leading" and "tail" parts of data eigenspace.
\bibliographystyle{plain}
\bibliography{main}
\appendix
\newpage
\renewcommand{\appendixpagename}{\centering \sffamily \LARGE Appendix}
\appendixpage
\vspace{5mm}
\section{Proof of Proposition~\ref{prop1}}
We first show how to connect the loss function associated with MAML to a Meta Least Square Problem.
\begin{proposition}[\Cref{prop1} Restated] Under the mixed linear regression model,
the expectation of the meta-training loss
taken over task and data distributions can be rewritten as:
\begin{align}
\mathbb{E}\left[\widehat{\mathcal{L}}(\mathcal{A},\boldsymbol{\omega},\beta^{\text{tr}};\mathcal{D})\right]= \mathcal{L}(\mathcal{A},\boldsymbol{\omega},\beta^{\text{tr}})=
\mathbb{E}_{\Bb , \boldsymbol{\gamma}} \frac{1}{2}\left[\left\|\Bb\boldsymbol{\omega}-\boldsymbol{\gamma}\right\|^{2}\right].
\end{align}
The meta data are given by
\begin{align}\Bb =& \frac{1}{\sqrt{n_2}}\Xb^{out}\Big(\mathbf{I}-\frac{\beta^{\text{tr}}}{n_1} {\Xb^{\text{in}}}^{T} {\Xb^{\text{in}}}\Big)\label{ap-eq-data}\\ \boldsymbol{\gamma} =& \frac{1}{\sqrt{n_2}}\Big( \Xb^{\text{out}}\Big( \mathbf{I}-\frac{\beta^{\text{tr}}}{n_1} {\Xb^{\text{in}}}^{T} {\Xb^{\text{in}}}\Big)\boldsymbol{\theta}+\zb^{out}-\frac{\beta^{\text{tr}}}{n_1} \Xb^{\text{out}}{\Xb^{\text{in}}}^{\top}\zb^{\text{in}}\Big) \end{align}
where $\Xb^{\text{in}}\in \mathbb{R}^{n_1\times d}$,$\zb^{\text{in}}\in \mathbb{R}^{n_1}$,$\Xb^{\text{out}}\in \mathbb{R}^{n_2\times d}$ and $\zb^{\text{out}}\in \mathbb{R}^{n_2}$ denote the inputs and noise for training and validation.
Furthermore,
we have
\begin{align}
\boldsymbol{\gamma} = \Bb\boldsymbol{\theta}^{*}+\boldsymbol{\xi}\quad \text{ with meta noise } \mathbb{E}[\boldsymbol{\xi}\mid\Bb]=0. \label{ap-linear}
\end{align}
\end{proposition}
\begin{proof}
We first rewrite $\mathcal{L}(\mathcal{A},\boldsymbol{\omega},\beta^{\text{tr}})$ as follows:
\begin{align*}
\mathcal{L}(\mathcal{A},\boldsymbol{\omega},\beta^{\text{tr}})
&=\mathbb{E}\left[ \ell(\mathcal{A}(\boldsymbol{\omega},\btr;\mathcal{D}^{\text{in}});\mathcal{D}^{\text{out}})\right]\\
&=\mathbb{E}\left[ \frac{1}{2n_2}\sum^{n_2}_{j=1} \left(\left\langle \mathbf{x}^{\text{out}}_{j}, \mathcal{A}(\boldsymbol{\omega},\btr;\mathcal{D}^{\text{in}})\right\rangle-y^{\text{out}}_{j}\right)^{2}\right]\\
&=\mathbb{E}\left[ \frac{1}{2n_2}\|\Xb^{\text{out}}\Big(\mathbf{I}-\frac{\beta^{\text{tr}}}{n_1} {\Xb^{\text{in}}}^{T} {\Xb^{\text{in}}}\Big)\bomega+\frac{\btr}{n_1}{\Xb^{\text{in}}}^{T}\yb^{\text{in}}-\yb^{\text{out}} \|^2\right].
\end{align*}
Using the mixed linear model:
\begin{align}
\yb^{\text{in}}= \mathbf{X}^{\text{in}}\boldsymbol{\theta}+\zb^{\text{in}},\quad \yb^{\text{out}}= \mathbf{X}^{\text{out}}\boldsymbol{\theta}+\zb^{\text{out}},
\end{align}
we have
\begin{align*}
\mathcal{L}&(\mathcal{A},\boldsymbol{\omega},\beta^{\text{tr}})\\
&=\mathbb{E}\left[ \frac{1}{2n_2}\|\Xb^{\text{out}}\Big(\mathbf{I}-\frac{\beta^{\text{tr}}}{n_1} {\Xb^{\text{in}}}^{T} {\Xb^{\text{in}}}\Big)\bomega\right.\\
&-\left. \Big( \Xb^{\text{out}}\Big( \mathbf{I}-\frac{\beta^{\text{tr}}}{n_1} {\Xb^{\text{in}}}^{T} {\Xb^{\text{in}}}\Big)\boldsymbol{\theta}+\zb^{out}-\frac{\beta^{\text{tr}}}{n_1} \Xb^{\text{out}}{\Xb^{\text{in}}}^{\top}\zb^{\text{in}}\Big)\|^2\right]\\
&= \mathbb{E}_{\Bb , \boldsymbol{\gamma}} \frac{1}{2}\left[\left\|\Bb\boldsymbol{\omega}-\boldsymbol{\gamma}\right\|^{2}\right].
\end{align*}
Moreover, note that $\btheta-\btheta^{*}$ has mean zero and is independent of data and noise, and define
\begin{align}
\boldsymbol{\xi}=\frac{1}{\sqrt{n_2}}\left( \Xb^{\text{out}}\left( \mathbf{I}-\frac{\btr}{n_1} {\Xb^{\text{in}}}^{T} {\Xb^{\text{in}}}\right)(\btheta-\btheta^{*})+\zb^{\text{out}}-\frac{\btr}{n_1} \Xb^{\text{out}}{\Xb^{\text{in}}}^{\top}\zb^{in}\right).\label{ap-eq-xi}
\end{align}
We call $\boldsymbol{\xi}$ as meta noise, and then we have
\begin{align*}
&\boldsymbol{\gamma} = \Bb\boldsymbol{\theta}^{*}+\boldsymbol{\xi}\quad \text{ and }\quad \mathbb{E}[\boldsymbol{\xi}\mid\Bb]=0.
\end{align*}
\end{proof}
\begin{lemma}[Meta Excess Risk]\label{ap-lemma-excess}
Under the mixed linear regression model, the meta excess risk can be rewritten as follows:
\begin{align*}
R(\wl_T, \bte)=\frac{1}{2}\mathbb{E}\|\wl_T-\btheta^{*}\|^2_{\Hte}
\end{align*}
where $\|\ab\|_{\Ab}^{2}=\ab^{T} \Ab \ab$. Moreover, the Bayes error is given by
\begin{align*}
\mathcal{L}(\mathcal{A},\boldsymbol{\omega}^{*},\beta^{\text{te}})=\frac{1}{2}\operatorname{tr}(\bSigma_{\btheta}\Hte)+\frac{\sigma^2{\bte} ^2}{2m}+\frac{\sigma^2}{2}.
\end{align*}
\end{lemma}
\begin{proof}
Recall that
\begin{align*}
R(\overline{\boldsymbol{\omega}}_T,\beta^{\text{te}})\triangleq \mathbb{E}\left[\mathcal{L}(\mathcal{A},\overline{\boldsymbol{\omega}}_T,\beta^{\text{te}})\right]-\mathcal{L}(\mathcal{A},\boldsymbol{\omega}^{*},\beta^{\text{te}})
\end{align*}
where $\boldsymbol{\omega}^{*}$ denotes the optimal solution to the population meta-test error. Under the mixed linear model, such a solution can be directly calculated~\cite{gao2020modeling}, and we obtain $\boldsymbol{\omega}^{*}=\mathbb{E}[\btheta] =\btheta^{*}$. Hence,
\[R(\wl_T, \bte)=\mathbb{E}_{\Bb , \boldsymbol{\gamma}} \frac{1}{2}\left[\left\|\Bb\wl_{T}-\boldsymbol{\gamma}\right\|^{2}-\left\|\Bb\btheta^{*}-\boldsymbol{\gamma}\right\|^{2}\right],\]
where \begin{align}\Bb =& {\xb^{\text{out}}}^{\top}\Big(\mathbf{I}-\frac{\beta^{\text{te}}}{m} {\Xb^{\text{in}}}^{T} {\Xb^{\text{in}}}\Big)\nonumber\\ \boldsymbol{\gamma} =& {\xb^{\text{out}}}^{\top}\Big( \mathbf{I}-\frac{\beta^{\text{te}}}{m} {\Xb^{\text{in}}}^{T} {\Xb^{\text{in}}}\Big)\boldsymbol{\theta}+\zb^{\text{out}}-\frac{\beta^{\text{te}}}{m} {\xb^{\text{out}}}^{\top}{\Xb^{\text{in}}}^{\top}\zb^{\text{in}}, \end{align}
and
$\xb^{\text{out}}\in\mathbb{R}^{d}$, $\zb^{\text{out}}\in\mathbb{R}^{d}$, $\Xb^{\text{in}}\in\mathbb{R}^{m\times d}$ and $\zb^{\text{in}}\in\mathbb{R}^{m}$. The forms of $\Bb$ and $\bgamma$ are slightly different since we allow a new adaptation rate $\bte$ and the inner loop has $m$ samples at test stage. Similarly
\begin{align}
\xi=\left(\underbrace{ {\xb^{\text{out}}}^{\top}\left( \mathbf{I}-\frac{\bte}{m} {\Xb^{\text{in}}}^{T} {\Xb^{\text{in}}}\right)(\btheta-\btheta^{*})}_{\xi_1}+\underbrace{\zb^{\text{out}}}_{\xi_2}\underbrace{-\frac{\btr}{m} {\xb^{\text{out}}}^{\top}{\Xb^{\text{in}}}^{\top}\zb^{in}}_{\xi_3}\right).
\end{align}
Then we have
\begin{align*}
R(\wl_T, \bte)&=\mathbb{E}_{\Bb , \boldsymbol{\gamma}} \frac{1}{2}\left[\left\|\Bb\wl_{T}-\boldsymbol{\gamma}\right\|^{2}-\left\|\Bb\btheta^{*}-\boldsymbol{\gamma}\right\|^{2}\right]\\
&=\mathbb{E}_{\Bb , \boldsymbol{\gamma}} \frac{1}{2}\left[\| \Bb(\wl_{T}-\btheta^{*})\|^2\right]\\
&=\frac{1}{2}\mathbb{E}\|\wl_T-\btheta^{*}\|^2_{\Hte}
\end{align*}
where the last equality follows because $\mathbb{E}\left[\Bb^{\top}\Bb\right]=\Hte$ at the test stage.
The Bayes error can be calculated as follows:
\begin{align*}
\mathcal{L}(\mathcal{A},\boldsymbol{\omega}^{*},\beta^{\text{te}})&= \mathbb{E}_{\Bb , \boldsymbol{\gamma}} \frac{1}{2}\left[\left\|\Bb\btheta^{*}-\boldsymbol{\gamma}\right\|^{2}\right]=\mathbb{E}_{\Bb , \boldsymbol{\gamma}} \frac{1}{2}\left[\xi^2\right]\\
&\overset{(a)}{=}\frac{1}{2}\left(\mathbb{E}\left[\xi_1^{2}\right]+\mathbb{E}\left[\xi_2^{2}\right]+\mathbb{E}\left[\xi_3^{2}\right]\right)\\
&=\frac{1}{2}(\operatorname{tr}(\bSigma_{\btheta}\Hte)+\frac{{\bte}^2\sigma^2}{m}+\sigma^2)
\end{align*}
where $(a)$ follows because $\xi_1,\xi_2,\xi_3$ are independent and have zero mean conditioned on $\Xb^{\text{in}}$ and $\xb^{\text{out}}$.
\end{proof}
\section{Analysis for Upper Bound (Theorem~\ref{thm-upper}) }
\subsection{Preliminaries}
We first introduce some additional notations.
\begin{definition}[Inner product of matrices]
For any two matrices $\Cb,\Db$, the inner product of them is defined as
$$
\langle \Cb,\Db \rangle = \operatorname{tr}(\Cb^{\top}\Db).
$$
\end{definition}
We will use the following property about the inner product of matrices throughout our proof.
\begin{property}
If $\mathbf{C} \succeq 0$ and $\mathbf{D} \succeq \mathbf{D}^{\prime}$, then we have $\langle\mathbf{C}, \mathbf{D}\rangle \geq\left\langle\mathbf{C}, \mathbf{D}^{\prime}\right\rangle$.
\end{property}
\begin{definition}[Linear operator]
Let $\otimes$ denote the tensor product. Define the following linear operators on symmetric matrices:
$$
\begin{gathered}
\mathcal{M}=\mathbb{E}\left[ \Bb^{\top}\otimes\Bb^{\top}\otimes \Bb\otimes\Bb\right]\quad
\widetilde{\mathcal{M}}:= \Htr\otimes \Htr
\quad
\mathcal{I}:= \mathbf{I} \otimes \mathbf{I}
\\ \mathcal{T}:=\Hb_{n_1,\btr} \otimes \mathbf{I}+\mathbf{I} \otimes \Hb_{n_1,\btr}-\alpha \mathcal{M}, \quad \widetilde{\mathcal{T}}=\Hb_{n_1,\btr} \otimes \mathbf{I}+\mathbf{I} \otimes \Hb_{n_1,\btr}-\alpha \Hb_{n_1,\btr} \otimes \Hb_{n_1,\btr}.
\end{gathered}
$$
\end{definition}
We next define the operation of the above linear operators on a symmetric matrix $\Ab$ as follows.
$$
\begin{gathered}
\mathcal{M} \circ \mathbf{A}=\mathbb{E}\left[\mathbf{B}^{\top}\Bb \mathbf{A} \Bb^{\top}\Bb\right], \quad \widetilde{\mathcal{M}} \circ \mathbf{A}=\Htr \mathbf{A} \Htr, \quad \mathcal{I} \circ \mathbf{A}=\mathbf{A},
\\
\mathcal{T}\circ \Ab = \Htr\Ab+\Ab\Htr -\alpha \mathbb{E}\left[\mathbf{B}^{\top}\Bb \mathbf{A} \Bb^{\top}\Bb\right]\\
\tilde{\mathcal{T}}\circ \Ab = \Htr\Ab+\Ab\Htr -\alpha\Htr\Ab\Htr.
\end{gathered}
$$
Based on the above definitions, we have the following equations hold.
$$
\begin{gathered}
(\mathcal{I}-\alpha \mathcal{T}) \circ \mathbf{A}=\mathbb{E}\left[\left(\mathbf{I}-\alpha \Bb^{\top}\Bb\right) \mathbf{A}\left(\mathbf{I}-\alpha \Bb^{\top}\Bb\right)\right]\\(\mathcal{I}-\alpha \tilde{\mathcal{T}}) \circ \mathbf{A}=(\mathbf{I}-\alpha \Htr) \mathbf{A}(\mathbf{I}-\alpha \Htr).
\end{gathered}
$$
For the linear operators, we have the following technical lemma.
\begin{lemma}\label{lemma-linearop} We call the linear operator $\mathcal{O}$ a PSD mapping, if for every symmetric PSD matrix $\mathbf{A}$, $\mathcal{O}\circ \Ab$ is also PSD matrix. Then we have:
\begin{enumerate}[label=\roman*]
\item[(i)] $ \mathcal{M}$, $\widetilde{\mathcal{M}}$ and
$(\mathcal{M}-\widetilde{\mathcal{M}}) $ are all PSD mappings.
\item[(ii)] $\tilde{\mathcal{T}}-\mathcal{T}$, $\mathcal{I}-\alpha \mathcal{T}$ and $\mathcal{I}-\alpha \tilde{\mathcal{T}}$ are all PSD mappings.
\item[(iii)] If $0<\alpha< \frac{1} { \max_{i}\{\mu_{i}(\Htr)\}}$, then $\tilde{\mathcal{T}}^{-1}$ exists, and is a PSD mapping.
\item[(iv)] If $0<\alpha<\frac{1} { \max_{i}\{\mu_{i}(\Htr)\}}$, $\tilde{\mathcal{T}}^{-1} \circ \Htr\preceq \mathbf{I}$.
\item[(v)] If $0<\alpha<\frac{1}{c(\btr,\bSigma) \operatorname{tr}(\bSigma
)}$, then $\mathcal{T}^{-1} \circ \mathbf{A}$ exists for PSD matrix $\mathbf{A}$, and $\mathcal{T}^{-1}$ is a PSD mapping.
\end{enumerate}
\end{lemma}
\begin{proof}
Items (i) and (iii) directly follow from the proofs in \cite{jain2017markov,zou2021benign}.
For $(\romannum{4})$, by the existence of $\tilde{\mathcal{T}}^{-1}$, we have
\begin{align*}
\tilde{\mathcal{T}}^{-1} \circ \Htr&=\sum_{t=0}^{\infty} \alpha (\mathcal{I}- \alpha\tilde{\mathcal{T}})^{t}\circ \Htr\\
&=\sum_{t=0}^{\infty} \alpha (\mathbf{I}-\alpha \Htr)^{t}\Htr(\mathbf{I}-\alpha \Htr)^{t}\\
&\preceq \sum_{t=0}^{\infty} \alpha (\mathbf{I}-\alpha \Htr)^{t}\Htr=\mathbf{I}.
\end{align*}
For $(\romannum{5})$, for any PSD matrix $\mathbf{A}$, consider
$$
\mathcal{T}^{-1} \circ \mathbf{A}=\alpha \sum_{k=0}^{\infty}(\mathcal{I}-\alpha \mathcal{T})^{k} \circ \mathbf{A}.
$$
We first show that $\sum_{k=0}^{\infty}(\mathcal{I}-\alpha \mathcal{T})^{k} \circ \mathbf{A}$ is finite, and then it suffices to show that the trace is finite, i.e.,
\begin{align}
\sum_{k=0}^{\infty} \operatorname{tr}\left((\mathcal{I}-\alpha \mathcal{T})^{k} \circ \mathbf{A}\right)<\infty. \label{b1-fin}
\end{align}
Let $\mathbf{A}_k=(\mathcal{I}-\gamma \mathcal{T})^{k} \circ \mathbf{A}$. Combining with the definition of $\mathcal{T}$, we obtain
$$
\begin{aligned}
\operatorname{tr}\left(\mathbf{A}_{k}\right) &=\operatorname{tr}\left(\mathbf{A}_{k-1}\right)-2\alpha \operatorname{tr}\left(\Htr \mathbf{A}_{k-1}\right)+\alpha^{2} \operatorname{tr}\left(\mathbf{A} \mathbb{E}\left[\Bb^{\top}\Bb \Bb^{\top}\Bb \right]\right).
\end{aligned}
$$
Letting $\Ab=\Ib$ in \Cref{prop-4}
, we have $\mathbb{E}\left[\Bb^{\top}\Bb \Bb^{\top}\Bb \right]\preceq c(\btr,\bSigma
) \operatorname{tr}(\bSigma) \Htr$. Hence
$$
\begin{aligned}
\operatorname{tr}\left(\mathbf{A}_{k}\right) & \leq \operatorname{tr}\left(\mathbf{A}_{k-1}\right)-\left(2 \alpha-\alpha^{2} c(\btr,\bSigma
) \operatorname{tr}(\bSigma)\right) \operatorname{tr}\left(\Htr \mathbf{A}_{k-1}\right) \\
& \leq \operatorname{tr}\left((\mathbf{I}-\alpha \Htr) \mathbf{A}_{k-1}\right)\quad \text{ by } \alpha< \frac{1}{c(\btr,\bSigma) \operatorname{tr}(\bSigma
)}\\
& \leq\left(1-\alpha \min_{i}\{\mu_{i}(\Htr)\}\right) \operatorname{tr}\left(\mathbf{A}_{k-1}\right).
\end{aligned}
$$
If $\alpha<\frac{1}{\min_{i}\{\mu_{i}(\Htr)\}}$, then we substitute it into \cref{b1-fin} and obtain
$$
\sum_{k=0}^{\infty} \operatorname{tr}\left((\mathcal{I}-\alpha \mathcal{T})^{k} \circ \mathbf{A}\right)=\sum_{k=0}^{\infty} \operatorname{tr}\left(\mathbf{A}_{k}\right) \leq \frac{\operatorname{tr}(\mathbf{A})}{\alpha\min_{i}\{\mu_{i}(\Htr)\}}<\infty
$$
which guarantees the existence of $\mathcal{T}^{-1}$. Moreover, $\Ab_k$ is a PSD matrix for every $k$ since $\mathcal{I}-\alpha \mathcal{T}$ is a PSD mapping. The $\mathcal{T}^{-1} \circ \mathbf{A}=\alpha \sum_{k=0}^{\infty} \Ab_k$ must be a PSD matrix, which implies that $\mathcal{T}^{-1}$ is PSD mapping.
\end{proof}
\begin{property}[Commutity]
Suppose Assumption $2$ holds, then for all $n>0$, $|\beta|<1/\lambda_1$, $\mathbf{H}_{n,\beta}$ with different $n$ and $\beta$ commute with each other.
\end{property}
\subsection{Fourth Moment Upper Bound for Meta Data}
In this section, we provide a technical result for the fourth moment of meta data $\Bb$, which is essential throughout the proof of our upper bound.
\begin{proposition}\label{prop-4}
Suppose Assumptions 1-3 hold. Given $|\beta|<\frac{1}{\lambda_1}$, for any PSD matrix $\Ab$, we have
\begin{align*}
\mathbb{E}\left[\Bb^{\top}\Bb \mathbf{A} \Bb^{\top}\Bb \right]\preceq c(\btr,\bSigma)\mathbb{E}\left[\operatorname{tr}(\mathbf{A} \bSigma) \right] \Htr
\end{align*}
where $c(\beta,\bSigma):= c_1\left(1+ 8|\beta|\lambda_1\sqrt{C(\beta,\bSigma)}\sigma_x^2+64\sqrt{C(\beta,\bSigma)}\sigma_x^4\beta^2\operatorname{tr}(\bSigma^2)\right)$.
\end{proposition}
\begin{proof}
Recall that $\Bb=\frac{1}{\sqrt{n_2}} \Xb^{\text{out}} (\mathbf{I}-\frac{\beta}{n_1}{\mathbf{X}^{\text{in}}}^{\top}\mathbf{X}^{\text{in}})$. With a slight abuse of notations, we write $\btr$ as $\beta$, $\mathbf{X}^{\text{in}}$ as $\mathbf{X}$ in this proof. First consider the case $\beta\geq 0$. By the definition of $\Bb$, we have
\begin{align*}
\mathbb{E}&\left[\Bb^{\top}\Bb \mathbf{A} \Bb^{\top}\Bb \right]\\ &= \mathbb{E}\left[(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\frac{1}{n_2}{\Xb^{\text{out}}}^{\top} \Xb^{\text{out}}(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\mathbf{A} (\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\frac{1}{n_2}{\Xb^{\text{out}}}^{\top} \Xb^{\text{out}}(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}\mathbf{X})\right] \\
&\preceq c_1 \mathbb{E}\left[\operatorname{tr}\left((\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\mathbf{A} (\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\bSigma\right) (\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\bSigma(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\right]
\\&\preceq c_1 \mathbb{E}\left[\operatorname{tr}\left(\mathbf{A} (\bSigma+ \frac{\beta^2}{n_1^2} \mathbf{X}^{\top}\mathbf{X}\bSigma \mathbf{X}^{\top}\mathbf{X})\right) (\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\bSigma(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\right]
\end{align*}
where the second inequality follows from Assumption 1. Let $\xb_i$ denote the $i$-th row of $\Xb$. Note that $\xb_i =\Sigma^{\frac{1}{2}}\zb_i$, where $\zb_i$ is independent $\sigma_x$-sub-gaussian vector.
For any $\xb_{i_1},\xb_{i_2},\xb_{i_3},\xb_{i_4}$, where $1\leq i_1,i_2,i_3,i_4\leq n_1 $, we have:
\begin{align*}
&\mathbb{E}\left[\operatorname{tr}(\Ab\xb_{i_1}\xb_{i_2}^{\top}\bSigma \xb_{i_3}\xb_{i_4}^{\top})(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\bSigma(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\right]\\
&= \mathbb{E}\left[\operatorname{tr}(\bSigma^{\frac{1}{2}}\Ab\bSigma^{\frac{1}{2}}\zb_{i_1}\zb_{i_2}^{\top}\bSigma^{2}\zb_{i_3}\zb_{i_4}^{\top})(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\bSigma(\mathbf{I}-\frac{\beta}{n _1}\mathbf{X}^{\top}\mathbf{X})\right]\\
& = \sum_{k,j}\mu_k\lambda^2_j \mathbb{E}\left[(\zb_{i_4}^{\top}\ub_k)(\zb_{i_1}^{\top}\ub_k)(\zb_{i_4}^{\top}\vb_j)(\zb_{i_1}^{\top}\vb_j)(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\Sigma(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\right]
\end{align*}
where the SVD of $\bSigma^{\frac{1}{2}}\Ab\bSigma^{\frac{1}{2}}$ is $\sum_{j} \mu_{j} \ub_{j}\ub^{\top}_{j}$, the SVD of $\bSigma$ is $\sum_{j} \lambda_{j} \vb_{j}\vb^{\top}_{j}$. For any unit vector $\wb\in\mathbb{R}^{d}$, we have:
\begin{align*}
\wb^{\top}&\mathbb{E}\left[\Hb^{-\frac{1}{2}}_{n_1,\beta}\operatorname{tr}(\Ab\xb_{i_1}\xb_{i_2}^{\top}\bSigma \xb_{i_3}\xb_{i_4}^{\top})(\mathbf{I}-\frac{\beta}{n}\mathbf{X}^{\top}\mathbf{X})\bSigma(\mathbf{I}-\frac{\beta}{n}\mathbf{X}^{\top}\mathbf{X})\Hb^{-\frac{1}{2}}_{n,\beta}\right]\wb\\
& \leq \sum_{k,j}\mu_k\lambda^2_j \sqrt{\mathbb{E}\left[\left((\zb_{i_4}^{\top}\ub_k)(\zb_{i_1}^{\top}\ub_k)(\zb_{i_4}^{\top}\vb_j)(\zb_{i_1}^{\top}\vb_j)^2\right)\right] } \\
&\quad\times \sqrt{\mathbb{E}\left[\|\wb^{\top}\Hb^{-\frac{1}{2}}_{n_1,\beta}(\mathbf{I}-\frac{\beta}{n_1}\Xb^{\top}\Xb)\Sigma (\mathbf{I}-\frac{\beta}{n_1}\Xb^{\top}\Xb)\Hb^{-\frac{1}{2}}_{n_1,\beta}\wb\|^2\right]}\\
&\leq 64\sqrt{C(\beta,\bSigma)}\sigma_x^4 \operatorname{tr}(A\bSigma)\operatorname{tr}(\bSigma^2)
\end{align*}
where the first inequality follows from the Cauchy Schwarz inequality; the last inequality is due to Assumption 3 and the property of sub-Gaussian distributions~\cite{vershynin2018high}.
Therefore,
\begin{align*}
\mathbb{E}&\left[\Hb^{-\frac{1}{2}}_{n_1,\beta}\operatorname{tr}(\Ab\xb_{i_1}\xb_{i_2}^{\top}\bSigma \xb_{i_3}\xb_{i_4}^{\top})(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\bSigma(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\Hb^{-\frac{1}{2}}_{n_1,\beta}\right]\\
&\preceq 64\sqrt{C(\beta,\bSigma)}\sigma_x^4 \operatorname{tr}(A\bSigma^2)\mathbf{I}
\end{align*}
which implies
$$\mathbb{E}\left[\operatorname{tr}(\Ab\xb_{i_1}\xb_{i_2}^{\top}\bSigma \xb_{i_3}\xb_{i_4}^{\top})(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\bSigma(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\right]\preceq 64\sqrt{C(\beta,\bSigma)}\sigma_x^4 \operatorname{tr}(A\bSigma^2) \Hb_{n_1,\beta}.$$
Hence,
\begin{align*}
& \mathbb{E}\left[\Bb^{\top}\Bb \mathbf{A} \Bb^{\top}\Bb \right]\\
&\preceq c_1\mathbb{E}\left[\operatorname{tr}\left(\mathbf{A} (\bSigma+ 64\sqrt{C}\sigma_x^4\beta^2 \bSigma\operatorname{tr}(\bSigma^2))\right) (\mathbf{I}-\frac{\beta}{n}\mathbf{X}^{\top}\mathbf{X})\bSigma(\mathbf{I}-\frac{\beta}{n}\mathbf{X}^{\top}\mathbf{X})\right]\\
&\preceq c_1(1+64\sqrt{C(\beta,\bSigma)}\sigma_x^4\beta^2\operatorname{tr}(\bSigma^2))\mathbb{E}\left[\operatorname{tr}(\mathbf{A} \bSigma) \right] \mathbf{H}_{n_1,\beta}.
\end{align*}
Now we turn to $\beta<0$, and derive
\begin{align*}
\mathbb{E}&\left[\Bb^{\top}\Bb \mathbf{A} \Bb^{\top}\Bb \right] \\
&\preceq c_1 \mathbb{E}\left[\operatorname{tr}\left((\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\mathbf{A} (\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\bSigma\right) (\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\bSigma(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\right]
\\&= c_1 \mathbb{E}\left[\operatorname{tr}\left(\mathbf{A} (\bSigma-\underbrace{\frac{\beta}{n_1}(\mathbf{X}^{\top}\mathbf{X}\bSigma +\bSigma \mathbf{X}^{\top}\mathbf{X})}_{\Jb_1}+ \frac{\beta^2}{n_1^2} \mathbf{X}^{\top}\mathbf{X}\bSigma \mathbf{X}^{\top}\mathbf{X})\right)\right.\\
&\quad\cdot \left. (\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\bSigma(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\right].
\end{align*}
We can bound the extra term $\Jb_1$ in the similar way as $\beta>0$.
For any $\xb_{i}$, $1\leq i\leq n_1$, we have
\begin{align*}
\mathbb{E}&\left[\operatorname{tr}\left(\Ab\xb_{i}\xb_{i}^{\top}\bSigma\right)(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\bSigma(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\right]\\
&= \mathbb{E}\left[\operatorname{tr}\left(\zb_{i}^{\top}\bSigma^{\frac{3}{2}}\Ab\bSigma^{\frac{1}{2}}\zb_{i}\right)(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\bSigma(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\right]\\
& = \sum_{k}\iota_k \mathbb{E}\left[(\zb_{i}^{\top}\boldsymbol{\kappa}_k)^2(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\bSigma(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\right]
\end{align*}
where the SVD of $\bSigma^{\frac{3}{2}}\Ab\bSigma^{\frac{1}{2}}$ is $\sum_{k} \iota_{k} \boldsymbol{\kappa}_{k}\boldsymbol{\kappa}^{\top}_{k}$. Similarly, for any unit vector $\wb\in\mathbb{R}^{d}$, we can obtain
\begin{align*}
\wb^{\top}& \mathbb{E}\left[\Hb^{-\frac{1}{2}}_{n_1,\beta}\operatorname{tr}\left(\Ab\xb_{i}\xb_{i}^{\top}\bSigma)(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X}\right)\bSigma(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\Hb^{-\frac{1}{2}}_{n_1,\beta}\right]\wb\\
&\leq \sum_{k}\iota_k \sqrt{\mathbb{E}[(\zb_{i}^{\top}\boldsymbol{\kappa}_k)^4]}\sqrt{\mathbb{E}[\|\wb^{\top}\Hb^{-\frac{1}{2}}_{n_1,\beta}(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\bSigma(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\Hb^{-\frac{1}{2}}_{n_1,\beta}\wb\|^2]}\\
&\leq 4 \sqrt{C(\beta,\bSigma)}\sigma^2_x\operatorname{tr}(A\bSigma^2)
\end{align*}
which implies:
\begin{align*}
\mathbb{E}\left[\operatorname{tr}\left(\Ab\xb_{i}\xb_{i}^{\top}\bSigma)(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X}\right)\bSigma(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\right]\preceq 4\sqrt{C(\beta,\bSigma)}\sigma^2_x\operatorname{tr}(\Ab\bSigma^2) \Hb_{n_1,\beta}.
\end{align*}
Hence,
\begin{align*}
& \mathbb{E}\left[\Bb^{\top}\Bb \mathbf{A} \Bb^{\top}\Bb \right]\\
&\preceq c_1\mathbb{E}\left[\operatorname{tr}\left(\mathbf{A} (\bSigma-8\beta\sqrt{C}\sigma_x^2\bSigma^2+ 64\sqrt{C}\sigma_x^4\beta^2 \bSigma\operatorname{tr}(\bSigma^2))\right) (\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\bSigma(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\right]\\
&\preceq c_1\left(1-8\beta\lambda_1\sqrt{C(\beta,\bSigma)}\sigma_x^2+ 64\sqrt{C(\beta,\bSigma)}\sigma_x^4\beta^2\operatorname{tr}(\bSigma^2)\right)\mathbb{E}\left[\operatorname{tr}(\mathbf{A} \bSigma) \right] \mathbf{H}_{n_1,\beta}.
\end{align*}
Together with the discussions for $\beta>0$, we have $$c(\beta,\bSigma
)=c_1(1+8|\beta|\lambda_1\sqrt{C(\beta,\bSigma)}\sigma_x^2+ 64\sqrt{C(\beta,\bSigma)}\sigma_x^4\beta^2\operatorname{tr}(\bSigma^2)),$$ which completes the proof.
\end{proof}
\subsection{Bias-Variance Decomposition}
We will use the bias-variance decomposition similar to theoretical studies of classic linear regression~\cite{jain2017markov,dieuleveut2017harder,zou2021benign}. Consider the error at each iteration: $\brho_t=\bomega_t-\btheta^{*}$, where $\bomega_t$ is the SGD output at each iteration $t$. Then the update rule can be written as:
$$
\brho_{t}:= (\Ib-\alpha\Bb^{\top}_t\Bb_t)\brho_{t-1}+\alpha \Bb^{\top}_{t}\boldsymbol{\xi}_{t}$$
where $\Bb_t,\bxi_t$ are the meta data and noise at iteration $t$ (see \cref{ap-eq-data,ap-eq-xi}). It is helpful to consider $\brho_{t}$ as the sum of the following two random processes:
\begin{itemize}
\item If there is no meta noise, the error comes from the bias:
$$
\brho^{\text{bias}}_{t}:= (\Ib-\alpha\Bb^{\top}_t\Bb_t)\brho^{\text{bias}}_{t-1}\quad \brho^{\text{bias}}_{t}=\brho_{0}.$$
\item If the SGD trajectory starts from $\btheta^{*}$, the error originates from the variance:
$$
\brho^{\text{var}}_{t}:= (\Ib-\alpha\Bb^{\top}_t\Bb_t)\brho^{\text{var}}_{t-1}+\alpha \Bb^{\top}_{t}\boldsymbol{\xi}_{t}\quad \brho^{\text{var}}=\mathbf{0}
$$
and $\mathbb{E}[\brho^{\text{var}}_{t}]=0$.
\end{itemize}
With slightly abused notations, we have:
$$
\brho_{t}= \brho^{\text{bias}}_{t}+\brho^{\text{var}}_{t}.
$$
Define the averaged output of $\brho^{\text{bias}}_{t}$, $\brho^{\text{var}}_{t}$ and $\brho_t$ after $T$ iterations as:
\begin{align}\label{eq-rho}
\rhob^{\text{bias}}_T=\frac{1}{T}\sum_{t=1}^{T} \brho^{\text{bias}}_{t},\quad
\rhob^{\text{var}}_T=\frac{1}{T}\sum_{t=1}^{T} \brho^{\text{var}}_{t},\quad \rhob_T=\frac{1}{T}\sum_{t=1}^{T} \brho_{t}.
\end{align}
Similarly, we have
$$
\rhob_{T}= \rhob^{\text{bias}}_{T}+\rhob^{\text{var}}_{T}.
$$
Now we are ready to introduce the bias-variance decomposition for the excess risk.
\begin{lemma}[Bias-variance decomposition]\label{lemma-bv}
Following the notations in \cref{eq-rho}, then the excess risk can be decomposed as
\begin{align*}
R(\wl_T, \bte)\leq 2\mathcal{E}_\text{bias}+2\mathcal{E}_\text{var}
\end{align*}
where
\begin{align}
\mathcal{E}_\text{bias}=\frac{1}{2} \langle\Hte, \mathbb{E}[\rhob^{\text{bias}}_T\otimes\rhob^{\text{bias}}_T] \rangle, \quad \mathcal{E}_\text{var} = \frac{1}{2} \langle\Hte, \mathbb{E}[\rhob^{\text{var}}_T\otimes\rhob^{\text{var}}_T] \rangle.
\end{align}
\end{lemma}
\begin{proof}
By \Cref{ap-lemma-excess}, we have
\begin{align*}
R(\wl_T, \bte)&=\frac{1}{2} \langle\Hte, \mathbb{E}[\rhob_T\otimes\rhob_T] \rangle \\
&=\frac{1}{2} \langle\Hte, \mathbb{E}[(\rhob^{\text{bias}}_T+\rhob^{\text{var}}_T)\otimes(\rhob^{\text{bias}}_T+\rhob^{\text{var}}_T)] \rangle\\
&\leq 2\left( \frac{1}{2} \langle\Hte, \mathbb{E}[\rhob^{\text{bias}}_T\otimes\rhob^{\text{bias}}_T] \rangle + \frac{1}{2} \langle\Hte, \mathbb{E}[\rhob^{\text{var}}_T\otimes\rhob^{\text{var}}_T] \rangle\right)
\end{align*}
where the last inequality follows because for vector-valued random variables $\ub$ and $\vb$, $\mathbb{E}\|\ub+\vb\|_{H}^{2} \leq\left(\sqrt{\mathbb{E}\|\ub\|_{H}^{2}}+\sqrt{\mathbb{E}\|\vb\|_{H}^{2}}\right)^{2}$ and from Cauchy-Schwarz inequality.
\end{proof}
For $t=0,1,\cdots,T-1$,
consider the following bias and variance iterates:
\begin{align}
\mathbf{D} _{t}=(\mathcal{I}-\alpha\mathcal{T}) \circ \Db _{t-1} \quad& \text { and } \quad \Db _{0}= (\boldsymbol{\omega}_t-\btheta^{*})(\boldsymbol{\omega}_t-\btheta^{*})^{\top}\nonumber\\ \mathbf{V} _{t}=(\mathcal{I}-\alpha\mathcal{T}) \circ \Vb _{t-1}+\alpha^{2} \Pi \quad &\text { and } \quad \Vb _{0}=\mathbf{0}\label{eq-bv}
\end{align}
where $\Pi=\mathbb{E}[\Bb^{\top}\boldsymbol{\xi}\boldsymbol{\xi}^{\top}\Bb]$. One can verify that
$$
\mathbf{D}_{t}=\mathbb{E}\left[\brho_{t}^{\text {bias }} \otimes \brho_{t}^{\text {bias }}\right], \quad \mathbf{V}_{t}=\mathbb{E}\left[\brho_{t}^{\text {var }} \otimes \brho_{t}^{\text {var }}\right].
$$
With such notations, we can further bound the bias and variance terms.
\begin{lemma}\label{lemma-further-bv}
Following the notations in \cref{eq-bv}, we have
\begin{align}
\mathcal{E}_\text { bias }& \leq \frac{1}{\alpha T^{2}} \left\langle \left(\Ib-(\mathbf{I}-\alpha\Htr)^{T}\right )\Htr^{-1}\Hte, \sum_{t=0}^{T-1}\Db _{t}\right\rangle,\\
\mathcal{E}_\text { var } &\leq \frac{1}{T^{2}} \sum_{t=0}^{T-1} \sum_{k=t}^{T-1}\left\langle(\mathbf{I}-\alpha \Htr)^{k-t} \Hte, \mathbf{V}_{t}\right\rangle.
\end{align}
\end{lemma}
\begin{proof}
Similar calculations have appeared in the prior works~\cite{jain2017markov,zou2021benign}. However, our meta linear model contains additional terms, and hence we provide a proof here for completeness. We first have
\begin{align*}
\mathbb{E}[\rhob^{\text{var}}_T\otimes\rhob^{\text{var}}_T]& =\frac{1}{T^{2}} \sum_{t=0}^{T-1} \sum_{k=0}^{T-1} \mathbb{E}[\brho^{\text{var}}_t\otimes\brho^{\text{var}}_k]\\
& \preceq \frac{1}{T^{2}} \sum_{t=0}^{T-1} \sum_{k=t}^{T-1} \mathbb{E}[\brho^{\text{var}}_t\otimes\brho^{\text{var}}_k]+\mathbb{E}[\brho^{\text{var}}_k\otimes\brho^{\text{var}}_t]
\end{align*}
where the last inequality follows because we double count the diagonal terms $t=k$.
For $t\leq k$, $\mathbb{E}[\brho^{\text{var}}_k|\brho^{\text{var}}_t]=(\mathbf{I}-\alpha\Htr)^{k-t} \brho^{\text{var}}_t$, since $\mathbb{E}[\Bb_t^{\top}\bxi_t|\brho_{t-1}]=\mathbf{0}$. From this, we have
\begin{align*}
\mathbb{E}[\rhob^{\text{var}}_T\otimes\rhob^{\text{var}}_T]
& \preceq \frac{1}{T^{2}} \sum_{t=0}^{T-1} \sum_{k=t}^{T-1} \Vb_t(\mathbf{I}-\alpha\Htr)^{k-t}+\Vb_t (\mathbf{I}-\alpha\Htr)^{k-t}.
\end{align*}
Substituting the above inequality into $\frac{1}{2} \langle\Hte, \mathbb{E}[\rhob^{\text{var}}_T\otimes\rhob^{\text{var}}_T] \rangle$, we obtain:
\begin{align*}
\mathcal{E}_\text{var} &=\frac{1}{2} \langle\Hte, \mathbb{E}[\rhob^{\text{var}}_T\otimes\rhob^{\text{var}}_T] \rangle\\
&\leq \frac{1}{2T^{2}} \sum_{t=0}^{T-1} \sum_{k=t}^{T-1} \langle \Hte, \Vb_t(\mathbf{I}-\alpha\Htr)^{k-t}\rangle + \langle \Hte,\Vb_t (\mathbf{I}-\alpha\Htr)^{k-t}\rangle\\
&=\frac{1}{T^{2}} \sum_{t=0}^{T-1} \sum_{k=t}^{T-1} \langle (\mathbf{I}-\alpha\Htr)^{k-t}\Hte, \Vb_t\rangle
\end{align*}
where the last inequality follows from \Cref{ass-comm} that $F$ and $\bSigma
$ commute, and hence $\Hte$ and $\mathbf{I}-\alpha\Htr$ commute.
For the bias term, similarly we have:
\begin{align}
\mathcal{E}_\text{bias}&\leq \frac{1}{T^{2}} \sum_{t=0}^{T-1} \sum_{k=t}^{T-1} \langle (\mathbf{I}-\alpha\Htr)^{k-t}\Hte, \Db_t\rangle\\
&=\frac{1}{\alpha T^2} \sum_{t=0}^{T-1} \langle \left(\Ib-(\mathbf{I}-\alpha\Htr)^{T-t}\right)\Htr^{-1} \Hte, \Db_t\rangle\\
&\leq \frac{1}{\alpha T^2} \langle \left(\Ib-(\mathbf{I}-\alpha\Htr)^{T}\right)\Htr^{-1} \Hte,\sum_{t=0}^{T-1} \Db_t\rangle
\end{align}
which completes the proof.
\end{proof}
\subsection{Bounding the Bias}
Now we start to bound the bias term. By \Cref{lemma-further-bv}, we focus on bounding the summation of $\Db_t$, i.e. $\sum_{t=0}^{T-1} \Db_t$. Consider $\mathbf{S}_{t}:=\sum_{k=0}^{t-1} \Db _{k}$, and the following lemma shows the properties of $\mathbf{S}_{t}$
\begin{lemma}
$\mathbf{S}_{t}$ satisfies the recursion form:
$$
\mathbf{S}_{t}=(\mathcal{I}-\alpha \mathcal{T}) \circ \mathbf{S}_{t-1}+\Db_{0}.
$$
Moreover, if $\alpha<\frac{1}{c(\btr,\bSigma)\operatorname{tr}(\bSigma
)}$, then we have:
$$
\Db_{0}=\mathbf{S}_{0} \preceq \mathbf{S}_{1} \preceq \cdots \preceq \mathbf{S}_{\infty}
$$
where $\mathbf{S}_{\infty}:=\sum_{k=0}^{\infty}(\mathcal{I}-\alpha\mathcal{T})^k \circ \Db_{0}=\alpha^{-1} \mathcal{T}^{-1} \circ \Db_{0}$.
\end{lemma}
\begin{proof}
By \cref{eq-bv}, we have
\begin{align*}
\mathbf{S}_{t}&=\sum_{k=0}^{t-1} \Db _{k}= \sum_{k=0}^{t-1}(\mathcal{I}-\alpha\mathcal{T})^{k} \circ \Db _{0}\\
&= \Db _{0}+(\mathcal{I}-\alpha\mathcal{T})\circ \left(\sum_{k=0}^{t-2}(\mathcal{I}-\alpha\mathcal{T})^{k} \circ \Db _{0}\right)\\
&= \Db _{0}+(\mathcal{I}-\alpha \mathcal{T}) \circ \mathbf{S}_{t-1}.
\end{align*}
By \Cref{lemma-linearop}, $(\mathcal{I}-\alpha \mathcal{T})$ is PSD mapping, and hence $\Db_t=(\mathcal{I}-\alpha \mathcal{T})\circ \Db_{t-1}$ is a PSD matirx for every $t$, which implies $\Sb_{t-1}\preceq \Sb_{t-1}+\Db_t=\Sb_{t}$.
The form of $\Sb_{\infty}$ can be directly obtained by \Cref{lemma-linearop}.
\end{proof}
Then we can decompose $\Sb_t$ as follows:
\begin{align}
\Sb_t& = \Db _{0}+(\mathcal{I}-\alpha \widetilde{\mathcal{T}}) \circ \mathbf{S}_{t-1}+ \alpha(\widetilde{\mathcal{T}}-\mathcal{T})\circ \mathbf{S}_{t-1}\nonumber\\
&=\Db _{0}+(\mathcal{I}-\alpha \widetilde{\mathcal{T}}) \circ \mathbf{S}_{t-1}+ \alpha^2(\mathcal{M}-\widetilde{\mathcal{M}})\circ \mathbf{S}_{t-1}\nonumber\\
&\preceq \Db _{0}+(\mathcal{I}-\alpha \widetilde{\mathcal{T}}) \circ \mathbf{S}_{t-1}+ \alpha^2\mathcal{M}\circ \mathbf{S}_{T}\nonumber\\
&=\sum^{t-1}_{k=0} (\mathcal{I}-\alpha \widetilde{\mathcal{T}})^{k} \circ (\Db_0+\alpha^2\mathcal{M}\circ \mathbf{S}_{T})\label{eq-st}
\end{align}
where the inequality follows because $\Sb_t\preceq \Sb_{T}$ for any $t\leq T$. Therefore, it is crucial to understand $\mathcal{M}\circ \mathbf{S}_{T}$.
\begin{lemma}\label{lemma-ms-pre}
For any symmetric matrix $\mathbf{A}$, if $\alpha<\frac{1}{c(\btr,\bSigma)\operatorname{tr}(\bSigma
)}$, it holds that
$$
\mathcal{M} \circ \mathcal{T}^{-1} \circ \mathbf{A} \preceq \frac{c(\btr,\bSigma)\operatorname{tr}\left(\bSigma \Htr^{-1} \mathbf{A}\right)}{1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma)} \cdot \Htr .
$$
\end{lemma}
\begin{proof}
Denote $\mathbf{C}=\mathcal{T}^{-1} \circ \mathbf{A} $. Recalling $\tilde{\mathcal{T}}=\mathcal{T}+\alpha \mathcal{M}-\alpha \widetilde{\mathcal{M}}$, we have
$$
\begin{aligned}
\widetilde{\mathcal{T}} \circ \mathbf{C}&=\mathcal{T} \circ \mathbf{C}+\alpha \mathcal{M} \circ \mathbf{C}-\alpha\widetilde{\mathcal{M}} \circ \mathbf{C}\\
& \preceq \mathbf{A}+\alpha \mathcal{M} \circ \mathbf{C}.
\end{aligned}
$$
Recalling that $\widetilde{\mathcal{T}}^{-1}$ exists and is a PSD mapping, we then have
\begin{align}
\mathcal{M} \circ \mathbf{C} &\preceq \alpha \mathcal{M} \circ \tilde{\mathcal{T}}^{-1} \circ \mathcal{M} \circ \mathbf{C}+\mathcal{M} \circ \tilde{\mathcal{T}}^{-1} \circ \mathbf{A}\nonumber\\
&\preceq \sum^{\infty}_{k=0} (\alpha \mathcal{M} \circ \tilde{\mathcal{T}}^{-1})^{k} \circ (\mathcal{M} \circ \tilde{\mathcal{T}}^{-1} \circ \mathbf{A}).\label{eq-mta}
\end{align}
By \Cref{prop-4}, we have $\mathcal{M} \circ \widetilde{\mathcal{T}}^{-1} \circ \mathbf{A}\preceq \underbrace{c(\btr,\bSigma)\operatorname{tr}( \bSigma\widetilde{\mathcal{T}}^{-1} \circ \mathbf{A})}_{J_2}\Htr$. Substituting back into \cref{eq-mta}, we obtain:
\begin{align*}
\sum^{\infty}_{k=0}& (\alpha \mathcal{M} \circ \tilde{\mathcal{T}}^{-1})^{k} \circ (\mathcal{M} \circ \tilde{\mathcal{T}}^{-1} \circ \mathbf{A})\preceq \sum^{\infty}_{k=0} (\alpha \mathcal{M} \circ \tilde{\mathcal{T}}^{-1})^{k} \circ (J_2\Htr)\\
&\preceq J_2\sum^{\infty}_{k=0} (\alpha c(\btr,\bSigma)\operatorname{tr}(\bSigma))^{k} \Htr\preceq \frac{J_2}{1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma)}\Htr
\end{align*}
where the second inequality follows since $\tilde{\mathcal{T}}^{-1}\circ \Htr \preceq \mathbf{I}$ (\Cref{lemma-linearop}) and $\mathcal{M} \circ \mathbf{I} \preceq c(\btr,\bSigma)\operatorname{tr}(\bSigma) \Htr$ (\Cref{prop-4}).
Finally, we bound $J_2$ as follows:
$$
\begin{aligned}
\operatorname{tr}\left(\bSigma \widetilde{\mathcal{T}}^{-1} \circ \mathbf{A}\right) &=\alpha\operatorname{tr}\left(\sum_{k=0}^{\infty} \bSigma(\mathbf{I}-\alpha \Htr)^{k} \mathbf{A}(\mathbf{I}-\alpha \Htr)^{k}\right) \\
&=\alpha \operatorname{tr}\left(\sum_{k=0}^{\infty} \bSigma(\mathbf{I}-\alpha \Htr)^{2 k} \mathbf{A}\right) \\
&=\operatorname{tr}\left(\bSigma\left(2 \Htr-\alpha \Htr^{2}\right)^{-1} \mathbf{A}\right)\\
&\leq \operatorname{tr}\left(\bSigma \Htr^{-1} \mathbf{A}\right)
\end{aligned}
$$
where the second equality follows because $\bSigma$ and $\Htr$ commute, and the last inequality holds since $\alpha<\frac{1} { \max_{i}\{\mu_{i}(\Htr)\}}$. Putting all these results together completes the proof.
\end{proof}
\begin{lemma}[Bounding $\mathcal{M}\circ \mathbf{S}_{T}$]\label{lemma-ms}
$$
\mathcal{M} \circ \mathbf{S}_{T} \preceq \frac{c(\btr,\bSigma) \cdot \operatorname{tr}\left(\bSigma \Htr^{-1}\left[\mathcal{I}-(\mathcal{I}-\alpha \tilde{\mathcal{T}})^{T}\right] \circ \Db _{0}\right)}{\alpha(1-c(\btr,\bSigma) \alpha \operatorname{tr}(\bSigma))} \cdot \Htr.
$$
\end{lemma}
\begin{proof}
$\mathbf{S}_{T}$ can be further derived as follows:
$$
\mathbf{S}_{T}=\sum_{k=0}^{T-1}(\mathcal{I}-\alpha \mathcal{T})^{k} \circ \Db _{0}=\alpha^{-1} \mathcal{T}^{-1} \circ\left[\mathcal{I}-(\mathcal{I}-\alpha \mathcal{T})^{T}\right]\circ \Db _{0}.
$$
Since $\tilde{\mathcal{T}}-\mathcal{T}$ is a PSD mapping by \Cref{lemma-linearop}, we have
$\mathcal{I}-\alpha \tilde{\mathcal{T}} \leq \mathcal{I}-\alpha \mathcal{T}$. Hence $\mathcal{I}-(\mathcal{I}-\alpha \mathcal{T})^{T} \preceq \mathcal{I}-(\mathcal{I}-\alpha \tilde{\mathcal{T}})^{T}$. Combining with the fact that $\mathcal{T}^{-1}$ is also a PSD mapping, we have:
$$
\mathbf{S}_{T} \preceq \alpha^{-1} \mathcal{T}^{-1} \circ\left[\mathcal{I}-(\mathcal{I}-\alpha \widetilde{\mathcal{T}})^{T}\right] \circ \Db _{0}.
$$
Letting $\Ab=\left[\mathcal{I}-(\mathcal{I}- \alpha\widetilde{\mathcal{T}})^{T}\right] \circ \Db _{0}$ in \Cref{lemma-ms-pre}, we obtain:
\begin{align*}
\mathcal{M} \circ \mathbf{S}_{T}& \preceq \alpha^{-1} \mathcal{M} \circ \mathcal{T}^{-1} \circ\left[\mathcal{I}-(\mathcal{I}-\alpha \tilde{\mathcal{T}})^{T}\right] \circ \Db _{0} \\
&\preceq \frac{c(\btr,\bSigma
) \cdot \operatorname{tr}\left(\bSigma \Htr^{-1}\left[\mathcal{I}-(\mathcal{I}-\alpha \tilde{\mathcal{T}})^{T}\right] \circ \Db _{0}\right)}{\alpha(1-c(\btr,\bSigma) \alpha \operatorname{tr}(\bSigma))} \cdot \Htr .
\end{align*}
\end{proof}
Now we are ready to derive the upper bound on the bias term.
\begin{lemma}[Bounding the bias]\label{lemma-bias}
If $\alpha<\frac{1}{c(\btr,\bSigma)\operatorname{tr}(\bSigma)}$, for sufficiently large $n_1$, s.t. $\mu_i(\Htr)>0$, $\forall i$, then we have
\begin{align*}
\mathcal{E}_\text{bias}&\leq \sum_{i}\left(\frac{1}{\alpha^{2} T^{2}}\mathbf{1}_{\mu_i(\Htr)\geq \frac{1}{\alpha T}}+ \mu_i^2(\Htr)\mathbf{1}_{\mu_i(\Htr)< \frac{1}{\alpha T} } \right)\frac{\omega_i^2\mu_i(\Hte)}{\mu_i(\Htr)^2}\\
&+ \frac{2 c(\btr,\bSigma)}{T \alpha\left(1-c(\btr,\bSigma)\alpha \operatorname{tr}(\bSigma)\right)}
\sum_{i}{\left(\frac{1}{\mu_i(\Htr)}\mathbf{1}_{\mu_i(\Htr)\geq \frac{1}{\alpha T}}+T\alpha \mathbf{1}_{\mu_i(\Htr)< \frac{1}{\alpha T} } \right) \cdot \lambda_i\omega_i^2}
\\&\times
\sum_{i}\left(\frac{1}{T}\mathbf{1}_{\mu_i(\Htr)\geq \frac{1}{\alpha T}}+T\alpha^2 \mu_i(\Htr)^2\mathbf{1}_{\mu_i(\Htr)< \frac{1}{\alpha T} } \right)\cdot\frac{\mu_i(\Htr)}{\mu_i(\Hte)}.
\end{align*}
\end{lemma}
\begin{proof} Applying \Cref{lemma-ms} to \cref{eq-st}, we can obtain:
$$
\begin{aligned}
\mathbf{S}_{t}
&\preceq\sum_{k=0}^{t-1}(\mathcal{I}-\alpha \widetilde{\mathcal{T}})^{k} \circ
\left(\frac{\alpha c(\btr,\bSigma)\cdot \operatorname{tr}\left(\bSigma \Htr^{-1}\left[\mathcal{I}-(\mathcal{I}-\alpha \tilde{\mathcal{T}})^{T}\right] \circ \Db _{0}\right)}{1-c(\beta,\Sigma) \alpha \operatorname{tr}(\Sigma)} \cdot \Htr + \Db_{0}\right) \\
&=\sum_{k=0}^{t-1}(\mathbf{I}-\alpha \Htr)^{k}\cdot \\
&\left(\underbrace{\frac{\alpha c(\btr,\bSigma) \cdot \operatorname{tr}\left(\bSigma\Htr^{-1}(\Db_0-(\mathbf{I}-\alpha \Htr)^{T}\Db _{0}(\mathbf{I}-\alpha \mathbf{H}_n)^{T})\right)}{1-c(\btr,\bSigma) \alpha \operatorname{tr}(\bSigma)} \cdot \Htr}_{\mathbf{G}_1} +\underbrace{\Db_{0}}_{\Gb_2}\right)\\
&\cdot (\mathbf{I}-\alpha \Htr)^{k}.
\end{aligned}
$$
Letting $t=T$, and substituting the upper bound of $\mathbf{S}_{T}$ into the bias term in \Cref{lemma-further-bv}, we obtain:
$$
\begin{aligned}
\mathcal{E}_\text { bias } & \leq \frac{1}{ \alpha T^{2}} \sum_{k=0}^{T-1}\left\langle ((\mathbf{I}- \alpha \Htr)^{2 k}-(\mathbf{I}- \alpha \mathbf{H}_{n,\beta})^{T+2 k})\mathbf{H}_{n,\beta}^{-1}\Hte,\mathbf{G}_1+\Gb_2\right\rangle\\
&\leq \frac{1}{\alpha T^{2}} \sum_{k=0}^{T-1}\left\langle ((\mathbf{I}- \alpha \Htr)^{ k}-(\mathbf{I}- \alpha \Htr)^{T+ k})\Htr^{-1}\Hte,\mathbf{G}_1+\Gb_2\right\rangle.
\end{aligned}
$$
We first consider
\begin{align*}
d_1= \frac{1}{\alpha T^{2}} \sum_{k=0}^{T-1}\left\langle ((\mathbf{I}- \alpha \Htr)^{ k}-(\mathbf{I}- \alpha \Htr)^{T+ k})\Htr^{-1}\Hte,\mathbf{G}_1\right\rangle.
\end{align*}
Since $\Htr$, $\Hte$ and $\mathbf{I}-\alpha \Htr$ commute,
we have
\begin{align*}
d_1=& \frac{c(\btr,\bSigma) \cdot \operatorname{tr}\left(\bSigma\Htr^{-1}(\Db_0-(\mathbf{I}-\alpha \Htr)^{T}\Db _{0}(\mathbf{I}-\alpha \Htr)^{T})\right)}{(1-c(\btr,\bSigma) \alpha \operatorname{tr}(\bSigma)) T^2}\\
&\times \sum_{k=0}^{T-1}\left\langle\left( (\mathbf{I}-\alpha \Htr)^{k}-(\mathbf{I}-\alpha \Htr)^{T+k} \right),\Hte\right\rangle.
\end{align*}
For the first term, since $\bSigma$, $\Htr$ and $\mathbf{I}-\alpha \Htr$ can be diagonalized simultaneously, considering the eigen-decompositions under the basis of $\bSigma$ and recalling $\bSigma
=\Vb\bLambda\Vb^{\top}$, we have:
$$
\begin{aligned}
&\operatorname{tr}\left(\bSigma
\Htr^{-1}[\Db _{0}-(\mathbf{I}-\alpha \Htr)^{T} \Db _{0}(\mathbf{I}-\alpha\Htr)^{T}]\right)\\
&=\sum_{i}{\left(1-\left(1-\alpha \mu_{i}(\Htr)\right)^{2 T}\right) \cdot\left(\left\langle\mathbf{w}_{0}-\mathbf{w}^{*}, \mathbf{v}_{i}\right\rangle\right)^{2}}\frac{\lambda_i}{\mu_i(\Htr)}\\
&\leq 2\sum_{i}\left(\mathbf{1}_{\lambda_{i}(\Hb_{n,\beta})\geq \frac{1}{\alpha T}}+T\alpha \mu_{i}(\Hb_{n,\beta})\mathbf{1}_{\mu_{i}(\Htr)< \frac{1}{\alpha T}} \right) \cdot \left(\left\langle\mathbf{w}_{0}-\mathbf{w}^{*}, \mathbf{v}_{i}\right\rangle\right)^{2}\frac{\lambda_i}{\mu_i(\Htr)}
\end{aligned}$$
where the last inequality holds since $1-(1-\alpha x)^{2T}\leq\min\{2, 2T\alpha x\}$.
For the second term, similarly, $\Hte$ and $\mathbf{I}-\alpha \Htr$ can be diagonalized simultaneously. We then have
\begin{align*}
\sum_{k=0}^{T-1}&\left\langle\left( (\mathbf{I}-\alpha \Htr)^{k}-(\mathbf{I}-\alpha \Htr)^{T+k} \right),\Hte\right\rangle\\
\leq & \sum^{T-1}_{k=0}\sum_{i} [(1-\alpha\mu_{i}(\Htr))^{k}-(1-\alpha\mu_{i}(\Htr))^{T+k}] \mu_{i}(\Hte)\\
=&\frac{1}{\alpha} \sum_{i} [1-(1-\alpha\mu_{i}(\Htr))^{T}]^2 \frac{\mu_{i}(\Hte)}{\mu_{i}(\Htr)}\\
\leq &\frac{1}{\alpha} \sum_i \left(\mathbf{1}_{\lambda_{i}(\Hb_{n,\beta})\geq \frac{1}{\alpha T}}+T^2\alpha^2 \lambda_{i}(\Hb_{n,\beta})\mathbf{1}_{\lambda_{i}(\Hb_{n,\beta})< \frac{1}{\alpha T}} \right)\frac{\mu_{i}(\Hte)}{\mu_{i}(\Htr)}.
\end{align*}
Now we turn to:
\begin{align*}
d_2&= \frac{1}{\alpha T^{2}} \sum_{k=0}^{T-1}\left\langle ((\mathbf{I}- \alpha \Htr)^{ k}-(\mathbf{I}- \alpha \Htr)^{T+ k})\Htr^{-1}\Hte,\mathbf{G}_2\right\rangle.
\end{align*}
Considering the orthogonal decompositions of $\Hte$ and $\Htr$ under $\Vb$, $\Htr=\Vb\bLambda_1\Vb^{\top}$, $\Hte=\Vb\bLambda_2\Vb^{\top}$, where the diagonal entries of $\bLambda_1$ are $\mu_i(\Htr)$ (and $\mu_i(\Hte)$ for $\bLambda_2$). Then we have:
\begin{align*}
d_2&=\frac{1}{\alpha T^{2}} \sum_{k=0}^{T-1}\left\langle\underbrace{\left( (\mathbf{I}- \alpha \bLambda_1)^{ k}-(\mathbf{I}- \alpha \bLambda_1)^{T+ k}\right)\bLambda_1^{-1}\bLambda_2}_{\Jb_3}, \Vb^{\top}\Db_0\Vb\right\rangle \\
&=\frac{1}{\alpha T^{2}} \sum_{k=0}^{T-1} \sum_{i}\left[\left(1-\alpha \mu_{i}(\Htr)\right)^{k}-\left(1-\alpha \mu_{i}(\Htr)\right)^{T+k}\right]
\frac{\omega_i^2\mu_{i}(\Hte)}{\mu_{i}(\Htr)}\\
&=\frac{1}{\alpha^{2} T^{2}} \sum_{i}\left[1-\left(1-\alpha \mu_{i}(\Htr)\right)^{T}\right]^{2}\frac{\omega_i^2\mu_{i}(\Hte)}{\mu_{i}^2(\Htr)}\\
&\leq \frac{1}{\alpha^{2} T^{2}}\sum_{i}\left(\mathbf{1}_{\mu_i(\Htr)\geq \frac{1}{\alpha T}}+ \alpha^{2} T^{2}\mu_i^2(\Htr)\mathbf{1}_{\mu_i(\Htr)< \frac{1}{\alpha T} } \right)\frac{\omega_i^2\mu_i(\Hte)}{\mu^2_i(\Htr)}
\end{align*}
where $\omega_i=\langle\bomega_0-\btheta^{*},\vb_i\rangle$ is the diagonal entry of $\Vb^{\top}\Db_0\Vb$ and the second equality holds since $\Jb_3$ is a diagonal matrix.
\end{proof}
\subsection{Bounding the Variance}
Note that the noisy part $\Pi=\mathbb{E}[\Bb^{\top}\boldsymbol{\xi}\boldsymbol{\xi}^{\top}\Bb]$ in \cref{eq-bv} is important in the variance iterates. In order to analyze the variance term, we first understand the role of $\Pi$ by the following lemma.
\begin{lemma}[Bounding the noise]\label{lemma-noise}
\begin{align*}
& \Pi=\mathbb{E}[\Bb^{\top}\boldsymbol{\xi}\boldsymbol{\xi}^{\top}\Bb]\preceq f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta}) \Htr
\end{align*}
where $f(\beta,n,\sigma,\bSigma,\bSigma_{\btheta})=[c(\beta,\bSigma)\operatorname{tr}({\bSigma_{\btheta}\bSigma})+4c_1\sigma^2\sigma_x^2\beta^2\sqrt{C(\beta,\bSigma)}\operatorname{tr}(\bSigma^2)+{\sigma^2}/n]$.
\end{lemma}
\begin{proof} With a slight abuse of notations, we write $\btr$ as $\beta$ in this proof. By definition of meta data and noise, we have
\begin{align*}
& \Pi=\mathbb{E}[\Bb^{\top}\boldsymbol{\xi}\boldsymbol{\xi}^{\top}\Bb]\\
&=\frac{\sigma^2}{n_2}\Hb_{n_1,\beta}+\mathbb{E}[\Bb^{\top}\Bb\Sigma_{\btheta}\Bb^{\top}\Bb]+\sigma^2\cdot \frac{\beta^{2}}{n_2 n_1^2}\mathbb{E}[\Bb^{\top}\Xb^{\text{out}}{\Xb^{\text{in}}}^{\top}\Xb^{\text{in}}{\Xb^{\text{out}}}^{\top}\Bb].
\end{align*}
The second term can be directly bounded by \Cref{prop-4}:
$$
\mathbb{E}[\Bb^{\top}\Bb\Sigma_{\btheta}\Bb^{\top}\Bb]\preceq c(\beta,\bSigma) \operatorname{tr}({\bSigma_{\btheta}\bSigma})\Hb_{n_1,\beta}.
$$
For the third term, we utilize the technique similar to \Cref{prop-4}, and by Assumption 1, we have:
\begin{align*}
& \sigma^2\cdot \frac{\beta^{2}}{n_2 n_1^2}\mathbb{E}\left[\Bb^{\top}\Xb^{\text{out}}{\Xb^{\text{in}}}^{\top}\Xb^{\text{in}}{\Xb^{\text{out}}}^{\top}\Bb\right]\\
&\preceq \sigma^2c_1\cdot \frac{\beta^{2}}{n_1^2}\mathbb{E}\left[\operatorname{tr}({\Xb^{\text{in}}}^{\top}\Xb^{\text{in}}\Sigma)(\mathbf{I}-\frac{\beta}{n_1}{\Xb^{\text{in}}}^{\top}\Xb^{\text{in}})\bSigma (\mathbf{I}-\frac{\beta}{n_1}{\Xb^{\text{in}}}^{\top}\Xb^{\text{in}})\right].
\end{align*}
Following the analysis for $\Jb_1$ in the proof of \Cref{prop-4}, and letting $\Ab=\mathbf{I}$, we obtain:
$$
\frac{1}{n_1^2}\mathbb{E}\left[\operatorname{tr}({\Xb^{\text{in}}}^{\top}\Xb^{\text{in}}\Sigma)(\mathbf{I}-\frac{\beta}{n_1}{\Xb^{\text{in}}}^{\top}\Xb^{\text{in}})\bSigma (\mathbf{I}-\frac{\beta}{n_1}{\Xb^{\text{in}}}^{\top}\Xb^{\text{in}})\right] \preceq 4\sqrt{C(\beta,\bSigma)}\sigma^2_x\operatorname{tr}(\bSigma^2) \Hb_{n_1,\beta}.
$$
Putting all these results together completes the proof.
\end{proof}
\begin{lemma}[Property of $\Vb_t$]\label{lemma-v-prop}
If the stepsize satisfies $\alpha<\frac{1} {c(\btr,\bSigma)\operatorname{tr}(\bSigma)}$, it holds that
$$
\mathbf{0}=\mathbf{V}_{0} \preceq \mathbf{V}_{1} \preceq \cdots \preceq \mathbf{V}_{\infty} \preceq \frac{\alpha f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta}) }{1-\alpha c(\btr,\bSigma)\operatorname{tr}(\bSigma)}\mathbf{I}.
$$
\end{lemma}
\begin{proof}
Similar calculations has appeared in prior works~\cite{jain2017markov,zou2021benign}. However, our analysis of the meta linear model needs to handle the complicated meta noise, and hence we provide a proof here for completeness.
We first show that $\mathbf{V}_{t-1}\preceq\Vb_t$. By recursion:
$$
\begin{aligned}
\mathbf{V}_{t} &=(\mathcal{I}-\alpha \mathcal{T}) \circ \mathbf{V}_{t-1}+\alpha^{2} \Pi \\
&\overset{(a)}{=}\alpha^{2} \sum_{k=0}^{t-1}(\mathcal{I}-\alpha \mathcal{T})^{k} \circ \Pi \\
&=\mathbf{V}_{t-1}+\alpha^{2}(\mathcal{I}-\alpha \mathcal{T})^{t-1} \circ \Pi \\
& \overset{(b)}{\succeq} \mathbf{V}_{t-1}
\end{aligned}
$$
where $(a)$ holds by solving the recursion and $(b)$ follows because $\mathcal{I}-\alpha \mathcal{T}$ is a PSD mapping.
The existence of $\mathbf{V}_{\infty}$ can be shown in the way similar to the proof of \Cref{lemma-linearop}. We first have
$$
\mathbf{V}_{t}=\alpha^{2} \sum_{k=0}^{t-1}(\mathcal{I}-\alpha \mathcal{T})^{k} \circ \Pi \preceq \alpha^{2} \sum_{k=0}^{\infty}\underbrace{(\mathcal{I}-\alpha \mathcal{T})^{k} \circ \Pi}_{\mathbf{A}_k}.
$$
By previous analysis in \Cref{lemma-linearop} , if $\alpha<\frac{1}{c(\btr,\bSigma)\operatorname{tr}(\bSigma)}$, we have
$$
\begin{aligned}
\operatorname{tr}\left(\mathbf{A}_{k}\right)
& \leq\left(1-\alpha\min_{i}\{\mu_{i}(\Htr)\}\right) \operatorname{tr}\left(\mathbf{A}_{t-1}\right).
\end{aligned}
$$
Therefore,
$$
\operatorname{tr}\left(\mathbf{V}_{t}\right) \leq \alpha^{2} \sum_{k=0}^{\infty} \operatorname{tr}\left(\mathbf{A}_{k}\right) \leq \frac{\alpha \operatorname{tr}(\Pi)}{\min_{i}\{\mu_{i}(\Htr)\}}<\infty.
$$
The trace of $\Vb_t$ is uniformly bounded from above, which indicates that $\Vb_{\infty}$ exists.
Finally, we bound $\mathbf{V}_{\infty}$. Note that $\Vb_{\infty}$ is the solution to:
$$
\mathbf{V}_{\infty} =(\mathcal{I}-\alpha \mathcal{T}) \circ \mathbf{V}_{\infty}+\alpha^{2} \Pi.
$$
Then we can write $\mathbf{V}_{\infty}$ as $\Vb_{\infty} = \mathcal{T}^{-1}\circ \alpha\Pi $. Following the analysis in the proof of \Cref{lemma-ms-pre}, we have:
\begin{align*}
\tilde{\mathcal{T}} \circ \mathbf{V}_{\infty}&=\tilde{\mathcal{T}}\circ \mathcal{T}^{-1}\circ \alpha\Pi\\
&\preceq \alpha \Pi+\alpha\mathcal{M} \circ \Vb_{\infty}\\
&\preceq \alpha f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta}) \Htr+\alpha\mathcal{M} \circ \mathbf{V}_{\infty}
\end{align*}
where the last inequality follows from \Cref{lemma-noise}.
Applying $\tilde{\mathcal{T}}^{-1}$, which exists and is a PSD mapping, to the both sides, we have
$$
\begin{aligned}
\mathbf{V}_{\infty} & \preceq \alpha f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})\cdot \tilde{\mathcal{T}}^{-1} \circ \Htr+\alpha \tilde{\mathcal{T}}^{-1} \circ \mathcal{M} \circ \mathbf{V}_{\infty} \\
& \overset{(a)}{\preceq} \alpha f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta}) \cdot \sum_{t=0}^{\infty}\left(\alpha \tilde{\mathcal{T}}^{-1} \circ \mathcal{M}\right)^{t} \circ \tilde{\mathcal{T}}^{-1} \circ \Htr\\
&\overset{(b)}{\preceq} \alpha f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})\sum^{\infty}_{t=0} (\alpha c(\btr,\bSigma)\operatorname{tr}(\bSigma))^{t}\mathbf{I}\\
&= \frac{\alpha f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})}{1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma)}\mathbf{I}
\end{aligned}
$$
where $(a)$ holds by directly solving the recursion; $(b)$ follows from the fact that $\tilde{\mathcal{T}}^{-1} \circ \Htr\preceq \mathbf{I}$ from \Cref{lemma-linearop} and $\mathcal{M} \circ\mathbf{I}\preceq c(\btr,\bSigma)\operatorname{tr}(\bSigma
)\Htr$ by letting $\Ab=\mathbf{I}$ in \Cref{prop-4}.
\end{proof}
Now we are ready to provide the upper bound on the variance term.
\begin{lemma}[Bounding the Variance]\label{lemma-var} If $\alpha<\frac{1}{c(\btr,\bSigma)\operatorname{tr}(\bSigma)}$, for sufficiently large $n_1$, s.t. $\mu_i(\Htr)>0$, $\forall i$, then we have
\begin{align*}
\mathcal{E}_\text{var}\leq& \frac{ f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})}{(1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma))}
\\
&\times\sum_{i}\left(\frac{1}{T}\mathbf{1}_{\mu_{i}(\Htr)\geq \frac{1}{\alpha T}}+T\alpha^2\mu_{i}^2(\Htr)\mathbf{1}_{\mu_{i}(\Htr)< \frac{1}{\alpha T} } \right)\frac{\mu_i(\Hte)}{\mu_i(\Htr)}.
\end{align*}
\end{lemma}
\begin{proof}
Recall
\begin{align}
\mathbf{V}_{t} &=(\mathcal{I}-\alpha \mathcal{T}) \circ \mathbf{V}_{t-1}+\alpha ^{2} \Pi \nonumber \\
&=(\mathcal{I}-\alpha \widetilde{\mathcal{T}}) \circ \mathbf{V}_{t-1}+\alpha ^{2}(\mathcal{M}-\widetilde{\mathcal{M}}) \circ \mathbf{V}_{t-1}+\alpha ^{2} \Pi \nonumber \\
& \preceq(\mathcal{I}-\alpha \widetilde{\mathcal{T}}) \circ \mathbf{V}_{t-1}+\alpha ^{2} \mathcal{M} \circ \mathbf{V}_{t-1}+\alpha ^{2} \Pi. \label{eq-v}
\end{align}
By the uniform bound on $\mathbf{V}_t$ and $\mathcal{M}$ is a PSD mapping, we have:
\begin{align*}
\mathcal{M} \circ \mathbf{V}_{t} &\preceq \mathcal{M} \circ \mathbf{V}_{\infty}\\
&\overset{(a)}{\preceq} \mathcal{M} \circ\frac{\alpha f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})}{1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma)}\mathbf{I}\\
&\overset{(b)}{\preceq} \frac{\alpha f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})c(\btr,\bSigma) \operatorname{tr}(\bSigma)}{1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma)}\cdot \Htr
\end{align*}
where $(a)$ directly follows from \Cref{lemma-v-prop}; $(b)$ holds because $\mathcal{M} \circ\mathbf{I}\preceq c(\btr,\bSigma)\operatorname{tr}(\bSigma
)\Htr$ (letting $\Ab=\mathbf{I}$ in \Cref{prop-4}). Substituting it back into \cref{eq-v}, we have:
\begin{align*}
\mathbf{V}_t&\preceq (\mathcal{I}-\alpha \tilde{\mathcal{T}}) \circ \mathbf{V}_{t-1}+\alpha^2 \frac{\alpha
f c(\btr,\bSigma) \operatorname{tr}(\bSigma)}{1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma)}\cdot \Htr+\alpha^2f \Htr\\
&=(\mathcal{I}-\alpha \tilde{\mathcal{T}}) \circ \mathbf{V}_{t-1}+\frac{\alpha^2 f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})
}{1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma)}\Htr\\
&\overset{(a)}{=} \frac{\alpha^2 f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})}{1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma)}\sum^{t-1}_{k=0} (\mathbf{I}-\alpha \tilde{\mathcal{T}})^{k}\circ \Htr\\
&\overset{(b)}{\preceq} \frac{\alpha f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})}{1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma)}(\mathbf{I}-(\mathbf{I}-\alpha \mathbf{H}_{n,\beta})^{t})
\end{align*}
where $(a)$ holds by solving the recursion and $(b)$ is due to the fact that
\begin{align*}
\sum^{t-1}_{k=0} (\mathbf{I}-\alpha \tilde{\mathcal{T}})^{k}\circ \Htr&= \sum^{t-1}_{k=0} (\mathbf{I}-\alpha\Htr)^{k}\Htr (\mathbf{I}-\alpha\Htr)^{k}\\
&\preceq \sum^{t-1}_{k=0} (\mathbf{I}-\alpha\Htr)^{k}\Htr\\
&=\frac{1}{\alpha}[\mathbf{I}-(\mathbf{I}-\alpha\Htr)^{t}].
\end{align*}
Substituting the bound for $\Vb_t$ back into the variance term in \Cref{lemma-further-bv}, we have
$$
\begin{aligned}
\mathcal{E}_\text { var } & \leq \frac{1}{T^{2}} \sum_{t=0}^{T-1} \sum_{k=t}^{T-1}\left\langle(\mathbf{I}-\alpha \Htr)^{k-t} \Hte, \mathbf{V}_{t}\right\rangle \\
&=\frac{1}{\alpha T^{2}} \sum_{t=0}^{T-1}\left\langle (\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{T-t})\Htr^{-1}\Hte, \mathbf{V}_{t}\right\rangle \\
& \leq \frac{ f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})}{(1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma))T^2}\sum_{t=0}^{T-1}\left\langle\mathbf{I}-(\mathbf{I}-\alpha \mathbf{H}_{n,\beta})^{T-t},\left(\mathbf{I}-(\mathbf{I}-\alpha \mathbf{H}_{n,\beta})^{t}\right)\mathbf{H}_{n,\beta}^{-1}\mathbf{H}_{m,\eta}\right\rangle .
\end{aligned}
$$
Simultaneously diagonalizing $\Htr$ and $\Hte$ as the analysis in \Cref{lemma-bias}, we have
$$
\begin{aligned}
\mathcal{E}_\text{var}\leq&\frac{ f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})}{(1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma))T^2}\\
&\cdot\sum_{i} \sum_{t=0}^{T-1}\left(1-\left(1-\alpha \mu_{i}(\Htr)\right)^{T-t}\right)\left(1-\left(1-\alpha \mu_{i}(\Htr)\right)^{t}\right)\frac{\mu_{i}(\Hte)}{\mu_{i}(\Htr)} \\
\leq &\frac{ f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})}{(1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma))T^2} \\
&\cdot\sum_{i} \sum_{t=0}^{T-1}\left(1-\left(1-\alpha \mu_{i}(\Htr)\right)^{T}\right)\left(1-\left(1-\alpha \mu_{i}(\Htr)\right)^{T}\right)\frac{\mu_{i}(\Hte)}{\mu_{i}(\Htr)} \\
=& \frac{ f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})}{(1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma))T} \sum_{i} \left(1-\left(1-\alpha \mu_{i}(\Htr)\right)^{T}\right)^{2}\frac{\mu_{i}(\Hte)}{\mu_{i}(\Htr)}\\
\leq& \frac{ f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})}{(1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma))T} \sum_{i} \left(\min \left\{1, \alpha T \mu_{i}(\Htr)\right\}\right)^2\frac{\mu_{i}(\Hte)}{\mu_{i}(\Htr)}\\
\leq &\frac{ f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})}{(1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma))}\\
&\cdot\sum_{i}\left(\frac{1}{T}\mathbf{1}_{\mu_{i}(\Htr)\geq \frac{1}{\alpha T}}+T\alpha^2\mu_{i}^2(\Htr)\mathbf{1}_{\mu{i}(\Hb_{n,\beta})< \frac{1}{\alpha T} } \right)\frac{\mu_i(\Hte)}{\mu_i(\Htr)},
\end{aligned}
$$
which completes the proof.
\end{proof}
\subsection{Proof of Theorem~\ref{thm-upper}}
\begin{theorem}[\Cref{thm-upper} Restated]\label{ap-thm-upper} Let $\omega_i=\left\langle\boldsymbol{\omega}_{0}-\boldsymbol{\theta}^{*}, \mathbf{v}_{i}\right\rangle$. If $|\btr|,|\bte|<1/\lambda_1$, $n_1$ is large ensuring that $\mu_i(\Hb_{n_1,\beta^{\text{tr}}})>0$, $\forall i$ and
$\alpha<1/\left(c(\btr,\bSigma) \operatorname{tr}(\bSigma)\right)$, then the meta excess risk $R(\overline{\boldsymbol{\omega}}_T,\bte)$ is bounded above as follows
\[R(\wl_{T},\bte)\leq \text{Bias}+ \text{Var} \]
where
\begin{align*}
\text{Bias} & = \frac{2}{\alpha^2 T} \sum_{i}\Xi_i \frac{\omega_i^2}{\mu_i(\Hb_{n_1,\beta^{\text{tr}}})} \\
\text{Var} &= \frac{2}{(1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma))}\left(\sum_{i}\Xi_{i} \right)
\\
\quad \times & [{f(\btr,n_2,\sigma,\bSigma_{\boldsymbol{\theta}},\bSigma)}+{\textstyle 2c(\btr,\bSigma)
\sum_{i}\left( \frac{\mathbf{1}_{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})\geq \frac{1}{\alpha T}}}{T\alpha \mu_i(\Hb_{n_1,\beta^{\text{tr}}})}+\mathbf{1}_{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})< \frac{1}{\alpha T} } \right) \lambda_{i}\omega^2_i}].
\end{align*}
\end{theorem}
\begin{proof}
By \Cref{lemma-bv}, we have
\begin{align*}
R(\wl_T, \bte)\leq 2\mathcal{E}_\text{bias}+2\mathcal{E}_\text{var}.
\end{align*}
Using \Cref{lemma-bias} to bound $\mathcal{E}_\text{bias}$, and \Cref{lemma-var} to bound $\mathcal{E}_\text{var}$, we have
\begin{align*}
& R(\wl_T, \bte)\\&\leq \frac{2 f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})}{(1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma))}\\
&\times \sum_{i}\left(\frac{1}{T}\mathbf{1}_{\mu_{i}(\Htr)\geq \frac{1}{\alpha T}}+T\alpha^2\mu_{i}^2(\Htr)\mathbf{1}_{\mu_{i}(\Htr)< \frac{1}{\alpha T} } \right)\frac{\mu_i(\Hte)}{\mu_i(\Htr)}\\
&+\frac{4 c(\btr,\bSigma)}{T \alpha(1-c(\btr,\bSigma)\alpha \operatorname{tr}(\bSigma))}
\sum_{i}\left(\frac{1}{T}\mathbf{1}_{\mu_i(\Htr)\geq \frac{1}{\alpha T}}+T\alpha^2 \mu_i(\Htr)^2\mathbf{1}_{\mu_i(\Htr)< \frac{1}{\alpha T} } \right)
\\&\times
\sum_{i}{\left(\frac{1}{\mu_i(\Htr)}\mathbf{1}_{\mu_i(\Htr)\geq \frac{1}{\alpha T}}+T\alpha \mathbf{1}_{\mu_i(\Htr)< \frac{1}{\alpha T} } \right) \cdot \lambda_i\left(\left\langle\bomega_{0}-\btheta^{*}, \mathbf{v}_{i}\right\rangle\right)^{2}}\\
&+ 2 \sum_{i}\left(\frac{1}{\alpha^{2} T^{2}}\mathbf{1}_{\mu_i(\Htr)\geq \frac{1}{\alpha T}}+ \mu_i^2(\Htr)\mathbf{1}_{\mu_i(\Htr)< \frac{1}{\alpha T} } \right)\frac{\omega_i^2\mu_i(\Hte)}{\mu_i(\Htr)^2}.
\end{align*}
Incorporating with the definition of effective meta weight
\begin{equation}
\Xi_i (\bSigma
,\alpha,T)=\begin{cases}
\mu_i(\Hb_{m,\beta^{\text{te}}})/\left(T \mu_i(\Hb_{n_1,\beta^{\text{tr}}})\right) & \mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})\geq \frac{1}{\alpha T}; \\
T\alpha^2 \mu_i(\Hb_{n_1,\btr})\mu_i(\Hb_{m,\beta^{\text{te}}})& \mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})< \frac{1}{\alpha T},
\end{cases}
\end{equation}
we obtain
$$
\left(\frac{1}{T}\mathbf{1}_{\mu_{i}(\Htr)\geq \frac{1}{\alpha T}}+T\alpha^2\mu_{i}^2(\Htr)\mathbf{1}_{\mu_{i}(\Htr)< \frac{1}{\alpha T} } \right)\frac{\mu_i(\Hte)}{\mu_i(\Htr)}=\Xi_i(\bSigma
,\alpha,T).
$$
Therefore,
\[ R(\wl_T, \bte)\leq \text{Bias}+\text{Var}\]
where
\begin{align*}
\text{Bias} & = \frac{2}{\alpha^2 T} \sum_{i}\Xi_i \frac{\omega_i^2}{\mu_i(\Htr)} \\
\text{Var} &= \frac{2}{(1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma))}\left(\sum_{i}\Xi_{i} \right)
\\
\quad \times & [{f(\btr,n_2,\sigma,\bSigma_{\boldsymbol{\theta}},\bSigma)}+\underbrace{ 2c(\btr,\bSigma)
\sum_{i}\left( \frac{\mathbf{1}_{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})\geq \frac{1}{\alpha T}}}{T\alpha \mu_i(\Hb_{n_1,\beta^{\text{tr}}})}+\mathbf{1}_{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})< \frac{1}{\alpha T} } \right) \lambda_{i}\omega^2_i}_{V_2}].
\end{align*}
Note that the term $V_2$ is obtained by our analysis for $\mathcal{E}_{\text{bias}}$. However, it originates from the stochasticity of SGD, and hence we treat this term as the variance in our final results.
\end{proof}
\section{Analysis for Lower Bound (Theorem~\ref{thm-lower}) }
\subsection{Fourth Moment Lower Bound for Meta Nosie}
Similarly to upper bound, we need some technical results for the fourth moment of meta data $\Bb$ and noise $\bxi$ to proceed the lower bound analysis.
\begin{lemma}\label{lemma-4l}
Suppose Assumption 1-3 hold. Given $|\btr|<\frac{1}{\lambda_1}$, for any PSD matrix $\Ab$, we have
\begin{align}\mathbb{E}[\Bb^{\top}\Bb\Ab\Bb^{\top}\Bb] &\succeq \Htr \mathbf{A }\Htr+\frac{b_1}{n_2} \operatorname{tr}(\Htr \mathbf{A}) \Htr\label{eq-lower-data}\\
\Pi&\succeq \frac{1}{n_2}g(\btr,n_1,\sigma,\bSigma_{\boldsymbol{\theta}},\bSigma)\Htr
\end{align}
where $g(\beta,n, \sigma,\bSigma, \bSigma_{\btheta}) :={\sigma^2+b_1\operatorname{tr}(\bSigma_{\btheta}\Hb_{n,\beta})+\beta^2 \mathbf{1}_{\beta\leq 0} b_1 \operatorname{tr}(\bSigma^2)/{n}}$.
\end{lemma}
\begin{proof}
With a slight abuse of notations, we write $\btr$ as $\beta$, $\mathbf{X}^{\text{in}}$ as $\mathbf{X}$ in this proof. Note that $\xb\in\mathbb{R}^{d}\sim\mathcal{P}_{\xb}$ is independent of $\Xb^{\text{in}}$. We first derive
\begin{align*}
\mathbb{E}&[\Bb^{\top}\Bb\Ab\Bb^{\top}\Bb]\\
&= \frac{1}{n_2} \mathbb{E}\left[(\mathbf{I}-\frac{\beta}{n_1}\Xb^{\top}\Xb)\mathbf{x x}^{\top} (\mathbf{I}-\frac{\beta}{n_1}\Xb^{\top}\Xb) \mathbf{A}(\mathbf{I}-\frac{\beta}{n_1}\Xb^{\top}\Xb) \mathbf{x} \mathbf{x}^{\top}(\mathbf{I}-\frac{\beta}{n_1}\Xb^{\top}\Xb)\right]\\
&+\frac{n_2-1}{n_2}\mathbb{E}\left[(\mathbf{I}-\frac{\beta}{n_2}\Xb^{\top}\Xb)\bSigma (\mathbf{I}-\frac{\beta}{n_1}\Xb^{\top}\Xb) \mathbf{A}(\mathbf{I}-\frac{\beta}{n_1}\Xb^{\top}\Xb)\bSigma(\mathbf{I}-\frac{\beta}{n_1}\Xb^{\top}\Xb)\right]\\
&\overset{(a)}{\succeq} \frac{b_1}{n_2} \mathbb{E}\left[\operatorname{tr}(\mathbf{A}(\mathbf{I}-\frac{\beta}{n}\Xb^{\top}\Xb) \Sigma(\mathbf{I}-\frac{\beta}{n}\Xb^{\top}\Xb))(\mathbf{I}-\frac{\beta}{n}\Xb^{\top}\Xb)\Sigma(\mathbf{I}-\frac{\beta}{n}\Xb^{\top}\Xb)\right]\\
&+\Hb_{n_1,\beta} \mathbf{A }\Hb_{n_1,\beta}\\
&{\succeq} \frac{b_1}{n_2} \operatorname{tr}(\Hb_{n_1,\beta} \mathbf{A}) \Hb_{n_1,\beta}+\Hb_{n_1,\beta} \mathbf{A }\Hb_{n_1,\beta}
\end{align*}
where $(a)$ is implied by Assumption 1.
Recall that $\Pi$ takes the following form:
\begin{align*}
\Pi&=\frac{\sigma^2}{n_2}\Hb_{n_1,\beta}+\mathbb{E}[\Bb^{\top}\Bb\bSigma_{\btheta}\Bb^{\top}\Bb]+\sigma^2\cdot \frac{\beta^{2}}{n_2 n_1^2}\mathbb{E}[\Bb^{\top}\Xb^{\text{out}}{\Xb}^{\top}\Xb{\Xb^{\text{out}}}^{\top}\Bb].
\end{align*}
The second term can be directly bounded by letting $\Ab=\bSigma_{\btheta}$ in \cref{eq-lower-data}, and we have:
$$\mathbb{E}[\Bb^{\top}\Bb\bSigma_{\btheta}\Bb^{\top}\Bb]\succeq \frac{b_1}{n_2} \operatorname{tr}(\Hb_{n_1,\beta} \bSigma_{\btheta}) \Hb_{n_1,\beta}.$$
For the third term:
\begin{align*}
& \frac{1}{n_2}\mathbb{E}[\Bb^{\top}\Xb^{\text{out}}{\Xb}^{\top}\Xb{\Xb^{\text{out}}}^{\top}\Bb]\\
&=\frac{1}{n_2}\mathbb{E}\left[(\mathbf{I}-\frac{\beta}{n_1}\Xb^{\top}\Xb)\mathbf{x x}^{\top}\Xb^{\top}\Xb \mathbf{x} \mathbf{x}^{\top}(\mathbf{I}-\frac{\beta}{n_1}\Xb^{\top}\Xb)\right]\\
&+\frac{n_2-1}{n_2}\mathbb{E}\left[(\mathbf{I}-\frac{\beta}{n_1}\Xb^{\top}\Xb)\bSigma\Xb^{\top}\Xb \bSigma(\mathbf{I}-\frac{\beta}{n_1}\Xb^{\top}\Xb)\right]\\
&\succeq \frac{n_1 b_1\operatorname{tr}(\bSigma^2)}{n_2}\Hb_{n_1,\beta}\mathbf{1}_{\beta\leq 0}
\end{align*}
Putting these results together completes the proof.
\end{proof}
\subsection{Bias-Variance Decomposition}
For the lower bound analysis, we also decompose the excess risk into bias and variance terms.
\begin{lemma}[Bias-variance decomposition, lower bound]\label{lemma-bv-lower}
Following the notations in \cref{eq-bv}, the excess risk can be decomposed as follows:
$$
\begin{aligned}
R(\wl_{T},\bte)\geq \underline{\mathcal{E}_{bias}}+\underline{\mathcal{E}_{var}}
\end{aligned}
$$
where
\begin{align*}
\underline{\mathcal{E}_{bias}}= & \frac{1}{2 T^{2}} \cdot \sum_{t=0}^{T-1} \sum_{k=t}^{T-1}\left\langle(\mathbf{I}-\alpha \Htr)^{k-t} \Hte, \mathbf{D}_{t}\right\rangle, \\
\underline{\mathcal{E}_{var}} =&\frac{1}{2 T^{2}} \cdot \sum_{t=0}^{T-1} \sum_{k=t}^{T-1}\left\langle(\mathbf{I}-\alpha \Htr)^{k-t} \Hte, \mathbf{V}_{t}\right\rangle.
\end{align*}
\end{lemma}
\begin{proof}
The proof is similar to that for \Cref{lemma-further-bv}, and the inequality sign is reversed since we only calculate the half of summation. In particular,
\begin{align*}
\mathbb{E}[\rhob^{\text{var}}_T\otimes\rhob^{\text{var}}_T]& =\frac{1}{T^{2}} \sum_{1\leq t<k\leq T-1} \mathbb{E}[\brho^{\text{var}}_t\otimes\brho^{\text{var}}_k]+\frac{1}{T^{2}} \sum_{1\leq k<t\leq T-1} \mathbb{E}[\brho^{\text{var}}_t\otimes\brho^{\text{var}}_k]\\
& \succeq\frac{1}{T^{2}} \sum_{1\leq t<k\leq T-1} \mathbb{E}[\brho^{\text{var}}_t\otimes\brho^{\text{var}}_k].
\end{align*}
For $t\leq k$, $\mathbb{E}[\brho^{\text{var}}_k|\brho^{\text{var}}_t]=(\mathbf{I}-\alpha\Htr)^{k-t} \brho^{\text{var}}_t$, since $\mathbb{E}[\Bb_t^{\top}\bxi_t|\brho_{t-1}]=\mathbf{0}$. From this
\begin{align*}
\mathbb{E}[\rhob^{\text{var}}_T\otimes\rhob^{\text{var}}_T]
& \succeq \frac{1}{T^{2}} \sum_{t=0}^{T-1} \sum_{k=t}^{T-1} \Vb_t(\mathbf{I}-\alpha\Htr)^{k-t}.
\end{align*}
Plugging this into $\frac{1}{2} \langle\Hte, \mathbb{E}[\rhob^{\text{var}}_T\otimes\rhob^{\text{var}}_T] \rangle$, we obtain:
\begin{align*}
&\frac{1}{2} \langle\Hte, \mathbb{E}[\rhob^{\text{var}}_T\otimes\rhob^{\text{var}}_T] \rangle\\
&\geq \frac{1}{2T^{2}} \sum_{t=0}^{T-1} \sum_{k=t+1}^{T-1} \langle \Hte, \Vb_t(\mathbf{I}-\alpha\Htr)^{k-t}\rangle \\
&=\frac{1}{2T^{2}} \sum_{t=0}^{T-1} \sum_{k=t}^{T-1} \langle (\mathbf{I}-\alpha\Htr)^{k-t}\Hte, \Vb_t\rangle
\\&=\underline{\mathcal{E}_\text{var}}.
\end{align*}
The proof is the same for the term $\underline{\mathcal{E}_\text{bias}}$.
\end{proof}
\subsection{Bounding the Bias}
We first bound the summation of $\Db_t$, i.e. $\Sb_k= \sum^{k-1}_{t=0}\Db_t$.
\begin{lemma}[Bounding $\mathbf{S}_t$]\label{lemma-sk}
If the stepsize satisfies $\alpha <1 / (2\max_{i}\{\mu_{i}(\Htr)\})$, then for any $k \geq 2$, it holds that
\begin{align*}
\mathbf{S}_{k} &\succeq \frac{b_1}{4n_2} \operatorname{tr}\left(\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{k / 2}\right) \mathbf{D}_{0}\right) \cdot\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{k / 2}\right)\\
&+\sum_{t=0}^{k-1}(\mathbf{I}-\alpha \Htr)^{t} \cdot \mathbf{D}_{0} \cdot(\mathbf{I}-\alpha \Htr)^{t}.
\end{align*}
\end{lemma}
\begin{proof} By \cref{eq-st}, since $\tilde{\mathcal{M}}-\mathcal{M}$ is a PSD mapping, we have
\begin{align}
\Sb_k &=\Db _{0}+(\mathcal{I}-\alpha \widetilde{\mathcal{T}}) \circ \mathbf{S}_{k-1}+ \alpha^2(\mathcal{M}-\widetilde{\mathcal{M}})\circ \mathbf{S}_{k-1}\label{eq-sk}\\
& \succeq \sum^{k-1}_{t=0} (\mathcal{I}-\alpha \widetilde{\mathcal{T}})^{t}
\circ \Db_0\nonumber\\
&=\sum_{t=0}^{k-1}(\mathbf{I}-\alpha \Htr)^{t} \cdot \mathbf{D}_{0} \cdot(\mathbf{I}-\alpha \Htr)^{t}.\nonumber
\end{align}
Note that for PSD $\Ab$, $$(\mathcal{M}-\widetilde{\mathcal{M}}) \circ\Ab= \mathbb{E}[\Bb^{\top}\Bb\Ab\Bb^{\top}\Bb] - \Htr \mathbf{A }\Htr
$$ By \Cref{lemma-4l}, we have
\begin{align}
(\mathcal{M}-\widetilde{\mathcal{M}}) \circ \mathbf{S}_{k} & \succeq \frac{b_1}{n_2} \operatorname{tr}\left(\Htr \mathbf{S}_{k}\right) \Htr\nonumber \\
& \succeq \frac{b_1}{n_2} \operatorname{tr}\left(\sum_{t=0}^{k-1}(\mathbf{I}-\alpha \Htr)^{2 t} \Htr \cdot \mathbf{D}_{0}\right) \Htr\nonumber \\
& \succeq \frac{b_1}{n_2} \operatorname{tr}\left(\sum_{t=0}^{k-1}(\mathbf{I}-2 \alpha\Htr)^{t} \Htr\cdot \mathbf{D}_{0}\right) \Htr\nonumber \\
& \succeq \frac{b_1}{2 n_2 \alpha} \operatorname{tr}\left(\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{k}\right) \mathbf{D}_{0}\right) \Htr.\label{eq-crude}
\end{align}
Substituting \cref{eq-crude} back into \cref{eq-sk}, and solving the recursion, we obtain
$$
\begin{aligned}
\mathbf{S}_{k} \succeq & \sum_{t=0}^{k-1}(\mathcal{I}-\alpha \tilde{\mathcal{T}})^{t} \circ\left\{\frac{b_1 \alpha}{2n_2} \operatorname{tr}\left(\left(\mathbf{I}-(\mathbf{I}-\alpha\Htr)^{k-1-t}\right) \mathbf{D}_{0}\right) \mathbf{H}+\mathbf{D}_{0}\right\} \\
=& \frac{b_1 \alpha}{2n_2} \underbrace{\sum_{t=0}^{k-1} \operatorname{tr}\left(\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{k-1-t}\right) \mathbf{D}_{0}\right) \cdot(\mathbf{I}-\alpha\Htr)^{2 t} \Htr}_{\Jb_4} \\
&+\sum_{t=0}^{k-1}(\mathbf{I}-\alpha\Htr)^{t} \cdot \mathbf{D}_{0} \cdot(\mathbf{I}-\alpha \Htr)^{t}.
\end{aligned}
$$
The term $\Jb_4$ can be further bounded by the following:
\begin{align*}
\Jb_4&\succeq \sum_{t=0}^{k-1} \operatorname{tr}\left(\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{k-1-t}\right) \mathbf{D}_{0}\right) \cdot(\mathbf{I}-2\alpha\Htr)^{t}\Htr\\
&\succeq \operatorname{tr}\left(\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{k/2}\right) \mathbf{D}_{0}\right) \cdot\sum_{t=0}^{k/2-1}(\mathbf{I}-2\alpha\Htr)^{t}\Htr\\
&\succeq \frac{1}{2\alpha}\operatorname{tr}\left(\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{k/2}\right) \mathbf{D}_{0}\right) \cdot\left(\mathbf{I}-(\mathbf{I}-\alpha\Htr)^{k/2}\right)
\end{align*}
which completes the proof.
\end{proof}
Then we can bound the bias term.
\begin{lemma}[Bounding the bias]\label{lemma-biasl}
Let $\omega_i=\left\langle\boldsymbol{\omega}_{0}-\boldsymbol{\theta}^{*}, \mathbf{v}_{i}\right\rangle$. If $\alpha<\frac{1}{c(\btr,\bSigma)\operatorname{tr}(\bSigma)}$, for sufficiently large $n_1$, s.t. $\mu_i(\Htr)>0$, $\forall i$, then we have
\begin{align*}
\underline{\mathcal{E}_{\text{bias}}}\ge& \frac{1}{100\alpha^2 T} \sum_{i}\Xi_i \frac{\omega_i^2}{\mu_i(\Hb_{n_1,\beta^{\text{tr}}})} + \frac{b_1}{1000n_2(1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma))}\sum_{i}\Xi_{i} \\
&\times
\sum_{i}\Big( \frac{\mathbf{1}_{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})\geq \frac{1}{\alpha T}}}{T\alpha \mu_i(\Hb_{n_1,\beta^{\text{tr}}})}+\mathbf{1}_{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})< \frac{1}{\alpha T} } \Big) \lambda_{i}\omega^2_i.
\end{align*}
\end{lemma}
\begin{proof}
From \Cref{lemma-bv-lower}, we have
\begin{align*}
\underline{\mathcal{E}_{bias}}&= \frac{1}{2 T^{2}} \cdot \sum_{t=0}^{T-1} \sum_{k=t}^{T-1}\left\langle(\mathbf{I}-\alpha \Htr)^{k-t} \Hte, \mathbf{D}_{t}\right\rangle \\
&=\frac{1}{2\alpha T^{2}} \cdot \sum_{t=0}^{T-1} \left\langle\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{T-t}\right)\Htr^{-1} \Hte, \mathbf{D}_{t}\right\rangle\\
&\ge \frac{1}{2 \alpha T^{2}} \left\langle\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{T/2}\right)\Htr^{-1} \Hte, \sum_{t=0}^{T/2}\mathbf{D}_{t}\right\rangle\\
&\ge \frac{1}{2\alpha T^{2}} \left\langle\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{T/2}\right)\Htr^{-1} \Hte, \mathbf{S}_{\frac{T}{2}}\right\rangle.
\end{align*}
Applying \Cref{lemma-sk} to $\mathbf{S}_{\frac{T}{2}}$, we obtain:
\begin{align*}
\underline{\mathcal{E}_{bias}}
\ge& \underline{d_1}+\underline{d_2}
\end{align*}
where
\begin{align*}
\underline{d_1}&=\frac{b_1}{8\alpha n_2T^2} \operatorname{tr}\left(\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{T / 4}\right) \mathbf{D}_{0}\right)\\ &\times\left\langle\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{T/2}\right)\Htr^{-1} \Hte,\right.
\left. \left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{T /4}\right)\right\rangle\\
\underline{d_2}&=\frac{1}{2\alpha T^2 }\left\langle\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{T/2}\right)\Htr^{-1} \Hte,\right.
\\&\left. \sum_{t=0}^{T/2-1}(\mathbf{I}-\alpha \Htr)^{t} \cdot \mathbf{D}_{0} \cdot(\mathbf{I}-\alpha \Htr)^{t}\right\rangle.
\end{align*}
Moreover,
\begin{align*}
\underline{d_2}&\ge\frac{1}{2\alpha T^2 }\left\langle\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{T/2}\right)\Htr^{-1} \Hte,
\sum_{t=0}^{T/2-1}(\mathbf{I}-2\alpha \Htr)^{t} \mathbf{D}_{0} \right\rangle;\\
&\ge\frac{1}{4\alpha^2 T^2 }\left\langle\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{T/2}\right)^2\Htr^{-2} \Hte,
\mathbf{D}_{0} \right\rangle.
\end{align*}
Using the diagonalizing technique similar to the proof for \Cref{lemma-bias}, we have
\begin{align}
\underline{d_1}&\ge \frac{b_1}{8 \alpha n_2 T^{2}}\left(\sum_{i}\left(1-\left(1-\alpha \mu_{i}(\Htr)\right)^{T / 4}\right) \omega_{i}^{2}\right)\\
&\times\left(\sum_{i}\left(1-\left(1-\alpha \mu_{i}(\Htr)\right)^{T / 4}\right)^{2} \frac{\mu_i(\Hte)}{\mu_i(\Htr)}\right)\label{b1},\\
\underline{d_2}&\geq \frac{1}{4 \alpha^{2} T^{2}} \sum_{i}\left(1-\left(1-\alpha \mu_{i}(\Htr)\right)^{T / 4}\right)^{2} \frac{\mu_i(\Hte)}{\mu_i^2(\Htr) } \omega_{i}^{2}.\label{b2}
\end{align}
We use the following fact to bound the polynomial term. For $h_1(x)=1-(1-x)^{\frac{T}{4}}$, we have
$$
h_1(x)\ge\begin{cases}
\frac{1}{5}& x\ge 1/T\\
\frac{T}{5} x& x< 1/T
\end{cases}
$$
i.e., $1-\left(1-\alpha \mu_{i}(\Htr)\right)^{T / 4}\geq \left(\frac{1}{5}\mathbf{1}_{\alpha \mu_{i}(\Htr)\ge \frac{1}{T}}+\frac{\alpha \mu_{i}(\Htr)}{5}\mathbf{1}_{\alpha \mu_{i}(\Htr)< \frac{1}{T}}\right)$. Substituting this back into \cref{b1,b2}, and using the definition of effective meta weight $\Xi_i$ complete the proof.
\end{proof}
\subsection{Bounding the Variance}
We first bound the term $\Vb_t$.
\begin{lemma}[Bounding $\mathbf{V}_{t}$]\label{lemma-vt}
If the stepsize satisfies $\alpha <1 / (\max_{i}\{\mu_{i}(\Htr)\})$, it holds that
$$
\mathbf{V}_{t} \succeq \frac{\alpha g(\btr,n_1, \bSigma, \bSigma_{\btheta})}{2} \cdot\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{2 t}\right).
$$
\end{lemma}
\begin{proof}
With a slight abuse of notations, we write $g(\btr,n_1, \bSigma, \bSigma_{\btheta})$ as $g$. By definition,
$$
\begin{aligned}
\mathbf{V}_{t} &=(\mathcal{I}-\alpha \mathcal{T}) \circ \mathbf{V}_{t-1}+\alpha ^{2} \Pi \\
&=(\mathcal{I}-\alpha \widetilde{\mathcal{T}}) \circ \mathbf{V}_{t-1}+(\mathcal{M}-\widetilde{\mathcal{M}}) \circ \mathbf{V}_{t-1}+\alpha ^{2} \Pi \\
&\overset{(a)}{\succeq}(\mathcal{I}-\alpha \widetilde{\mathcal{T}}) \circ \mathbf{V}_{t-1}+\alpha ^{2} g \Htr \\
&\overset{(b)}{=}\alpha ^{2} g \cdot \sum_{k=0}^{t-1}(\mathcal{I}-\alpha \widetilde{\mathcal{T}})^{k} \circ \Htr \\
&=\alpha ^{2}g \cdot \sum_{k=0}^{t-1}(\mathbf{I}-\alpha \Htr)^{k} \Htr(\mathbf{I}-\alpha \Htr)^{k} \quad \text { (by the definition of } \mathcal{I}-\alpha \widetilde{\mathcal{T}} ) \\
&=\alpha g \cdot\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{2 t}\right) \cdot\left(2 \mathbf{I}-\alpha \Htr\right)^{-1} \\
& \overset{(c)}{\succeq} \frac{\alpha g}{2} \cdot\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{2 t}\right)
\end{aligned}
$$
where $(a)$ follows from the \Cref{lemma-4l}, $(b)$ follows by solving the recursion and $(c)$ holds since we directly replace $\left(2 \mathbf{I}-\alpha \Htr\right)^{-1} $ by $(2\Ib)^{-1}$.
\end{proof}
\begin{lemma}[Bounding the variance]\label{lemma-varl}
Let $\omega_i=\left\langle\boldsymbol{\omega}_{0}-\boldsymbol{\theta}^{*}, \mathbf{v}_{i}\right\rangle$. If $\alpha<\frac{1}{c(\btr,\bSigma)\operatorname{tr}(\bSigma)}$, for sufficiently large $n_1$, s.t. $\mu_i(\Htr)>0$, $\forall i$, for $T>10$, then we have
\begin{align*}
\underline{\mathcal{E}_{\text{var}}}&\ge \frac{g(\btr,n_1, \bSigma, \bSigma_{\btheta})}{100n_2(1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma))}\sum_{i}\Xi_{i}.
\end{align*}
\end{lemma}
\begin{proof}
From \Cref{lemma-bv-lower}, we have
\begin{align*}
\underline{\mathcal{E}_{var}}&= \frac{1}{2 T^{2}} \cdot \sum_{t=0}^{T-1} \sum_{k=t}^{T-1}\left\langle(\mathbf{I}-\alpha \Htr)^{k-t} \Hte, \mathbf{V}_{t}\right\rangle \\
&=\frac{1}{2\alpha T^{2}} \cdot \sum_{t=0}^{T-1} \left\langle\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{T-t}\right)\Htr^{-1} \Hte, \mathbf{V}_{t}\right\rangle.
\end{align*}
Then applying \Cref{lemma-vt}, and writting $g(\btr,n_1, \bSigma, \bSigma_{\btheta})$ as $g$, we obtain
\begin{align}
\underline{\mathcal{E}_{var}}&\ge \frac{g}{4 T^{2}} \cdot \sum_{t=0}^{T-1} \left\langle\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{T-t}\right)\Htr^{-1} \Hte, \left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{2 t}\right)\right\rangle\nonumber\\
&=\frac{g}{4 T^{2}} \sum_{i}\frac{\mu_i(\Hte)}{\mu_i(\Htr)}\sum_{t=0}^{T-1}(1-(1-\alpha \mu_i(\Htr)^{T-t}))(1-(1-\alpha \mu_i(\Htr)^{2t}))\nonumber\\
&\ge \frac{g}{4 T^{2}} \sum_{i}\frac{\mu_i(\Hte)}{\mu_i(\Htr)}\sum_{t=0}^{T-1}(1-(1-\alpha \mu_i(\Htr)^{T-t-1}))(1-(1-\alpha \mu_i(\Htr)^{t})) \label{eq-var}
\end{align}
where the equality holds by applying the diagonalizing technique again. Following the trick similar to that in \cite{zou2021benign} to lower bound the function $h_2(x):=\sum_{t=0}^{T-1}\left(1-(1-x)^{T-t-1}\right)\left(1-(1-x)^{t}\right)$ defined on $x\in(0,1)$, for $T>10$, we have
$$
f(x) \geq \begin{cases}\frac{T}{20}, & \frac{1}{T} \leq x<1 \\ \frac{3 T^{3}}{50} x^{2}, & 0<x<\frac{1}{T}\end{cases}
$$
Substituting this back into \cref{eq-var}, and using the definition of effective meta weight $\Xi_i$ completes the proof.
\end{proof}
\subsection{Proof of Theorem~\ref{thm-lower}}
\begin{theorem}[\Cref{thm-lower} Restated]\label{ap-thm-lower}
Let $\omega_i=\left\langle\boldsymbol{\omega}_{0}-\boldsymbol{\theta}^{*}, \mathbf{v}_{i}\right\rangle$. If $|\btr|,|\bte|<1/\lambda_1$, $n_1$ is large ensuring that $\mu_i(\Hb_{n_1,\beta^{\text{tr}}})>0$, $\forall i$ and
$\alpha<1/\left(c(\btr,\bSigma) \operatorname{tr}(\bSigma)\right)$. For $T>10$, the meta excess risk $R(\overline{\boldsymbol{\omega}}_T,\bte)$ is bounded below as follows
\begin{align*}
R(\overline{\boldsymbol{\omega}}_T,\bte) \ge &\frac{1}{100\alpha^2 T} \sum_{i}\Xi_i \frac{\omega_i^2}{\mu_i(\Hb_{n_1,\beta^{\text{tr}}})} +\frac{1}{n_2}\cdot \frac{1}{(1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma))}\sum_{i}\Xi_{i} \\
\times & [\frac{1}{100} g(\btr,n_1, \bSigma, \bSigma_{\btheta})+\frac{b_1}{1000}
\sum_{i}\Big( \frac{\mathbf{1}_{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})\geq \frac{1}{\alpha T}}}{T\alpha \mu_i(\Hb_{n_1,\beta^{\text{tr}}})}+\mathbf{1}_{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})< \frac{1}{\alpha T} } \Big) \lambda_{i}\omega^2_i].
\end{align*}
\end{theorem}
\begin{proof}
The proof can be completed by combining \Cref{lemma-biasl,lemma-varl}.
\end{proof}
\section{Proofs for Section~\ref{sec-main-task}}
\subsection{Proof of Lemma~\ref{lem-single}}
\begin{proof}[Proof of \Cref{lem-single}]
For the single task setting, we first simplify our notations in \Cref{thm-upper} as follows.
\begin{align*} c(0,\bSigma) = c_1,\quad
f(0,n_2,\sigma,\bSigma,\mathbf{0})=\sigma^2/n_2,\quad \Htr = \bSigma.
\end{align*}
By \Cref{thm-upper}, we have
\begin{align*}
\text{Bias} & = \frac{2}{\alpha^2 T} \sum_{i}\left(\frac{1}{T}\mathbf{1}_{\lambda_i\geq \frac{1}{\alpha T}}+T\alpha^2\lambda^2_i\mathbf{1}_{\lambda_i< \frac{1}{\alpha T} } \right) \frac{\omega_i^2\mu_i(\Hte)}{\lambda^2_i}\\
&\leq \frac{2}{\alpha^2 T} \sum_{i}(\alpha \lambda_i\mathbf{1}_{\lambda_i\geq \frac{1}{\alpha T}}+\alpha\lambda_i\mathbf{1}_{\lambda_i< \frac{1}{\alpha T} } )\frac{\omega_i^2\mu_i(\Hte)}{\lambda^2_i}.
\end{align*}
For large $m$, we have $\mu_i(\Hte)=(1-\bte\lambda_i)^2\lambda_i+o(1)$. Therefore,
\begin{align*}
\text{Bias}
\leq \frac{2(1-\bte\lambda_d)^2}{\alpha^2 T} \sum_{i} {\omega_i^2}\leq \mathcal{O}(\frac{1}{T}).
\end{align*}
For the variance term,
\begin{align*}
\text{Var} &= \frac{2}{(1-\alpha c_1 \operatorname{tr}(\bSigma))}\underbrace{\sum_{i}\left(\frac{1}{T}\mathbf{1}_{\lambda_i\geq \frac{1}{\alpha T}}+T\alpha^2\lambda^2_i\mathbf{1}_{\lambda_i< \frac{1}{\alpha T} } \right) \frac{\mu_i(\Hte)}{\lambda_i}}_{J_5}
\\
\quad \times & [\frac{\sigma^2}{n_2}+ 2c_1
\sum_{i}\left( \frac{\mathbf{1}_{\lambda_i\geq \frac{1}{\alpha T}}}{T\alpha \lambda_i}+\mathbf{1}_{\lambda_i< \frac{1}{\alpha T} } \right) \lambda_{i}\omega^2_i].
\end{align*}
It is easy to check that
$$
\sum_{i}\left( \frac{\mathbf{1}_{\lambda_i\geq \frac{1}{\alpha T}}}{T\alpha \lambda_i }+\mathbf{1}_{\lambda_i< \frac{1}{\alpha T} } \right) \lambda_{i}\omega^2_i\leq \sum_{i}\left( \frac{\mathbf{1}_{\lambda_i\geq \frac{1}{\alpha T}}}{T\alpha }+\frac{1}{\alpha T}\mathbf{1}_{\lambda_i< \frac{1}{\alpha T} } \right) \omega^2_i \leq \mathcal{O}(1/T).
$$
Moreover,
$$
J_5\leq (1-\bte\lambda_d)^2 \sum_{i}\left(\frac{1}{T}\mathbf{1}_{\lambda_i\geq \frac{1}{\alpha T}}+T\alpha^2\lambda^2_i\mathbf{1}_{\lambda_i< \frac{1}{\alpha T} } \right).
$$
The term $\sum_{i}\left(\frac{1}{T}\mathbf{1}_{\lambda_i\geq \frac{1}{\alpha T}}+T\alpha^2\lambda^2_i\mathbf{1}_{\lambda_i< \frac{1}{\alpha T} } \right)$ has the form similar to Corollary 2.3 in \cite{zou2021benign}
and we directly have
$J_5=\mathcal{O}\left(\log^{-p}(T)\right)$, which implies
\begin{align*}
\text{Var} =\mathcal{O}\left(\log^{-p}(T)\right).
\end{align*}
Thus we complete the proof.
\end{proof}
\subsection{Proof of Proposition~\ref{prop-hard}}
\begin{proof}[Proof of \Cref{prop-hard}]\label{proof-prop2}
We first consider the bias term in \Cref{thm-upper,thm-lower} (up to absolute constants):
\begin{align*}
\text{Bias}&= \frac{2}{\alpha^{2} T} \sum_{i}\left(\frac{1}{ T}\mathbf{1}_{\mu_i(\Htr)\geq \frac{1}{\alpha T}}+\alpha^{2} T \mu_i^2(\Htr)\mathbf{1}_{\mu_i(\Htr)< \frac{1}{\alpha T} } \right)\frac{\omega_i^2\mu_i(\Hte)}{\mu_i(\Htr)^2}.
\end{align*}
If $\mu_i(\Htr)\geq \frac{1}{\alpha T}$, $\frac{1}{T}\leq \alpha\mu_i(\Htr) $; and if $\mu_i(\Htr)< \frac{1}{\alpha T}$, then $\alpha^{2} T \mu_i^2(\Htr)<\alpha \mu_i(\Htr) $. Hence
\begin{align*}
\text{Bias}\le \frac{1}{\alpha^{2} T} \sum_{i}\frac{\omega_i^2\mu_i(\Hte)}{\mu_i(\Htr)}\le \frac{2}{\alpha^{2} T}\cdot\max_{i}\frac{\mu_i(\Hte)}{\mu_i(\Htr)} \|\bomega_0-\btheta^{*}\|^2=\mathcal{O}(\frac{1}{T}).
\end{align*}
Moreover,
\begin{align*}
\sum_{i}&\left( \frac{\mathbf{1}_{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})\geq \frac{1}{\alpha T}}}{T\alpha \mu_i(\Hb_{n_1,\beta^{\text{tr}}})}+\mathbf{1}_{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})< \frac{1}{\alpha T} } \right) \lambda_{i}\omega^2_i
\\
&\overset{(a)}{\le} \frac{1}{\alpha T}\sum_{i} \frac{\lambda_{i}}{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})}\omega^2_i\\ &\le \frac{1}{\alpha T}\max_{i} \frac{\lambda_{i}}{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})}\|\bomega_0-\btheta^{*}\|^2=\mathcal{O}(\frac{1}{T})
\end{align*}
where $(a)$ holds since we directly upper bound $\mu_{i}(\Htr)$ by $\frac{1}{\alpha T}$ when $\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})< \frac{1}{\alpha T}$. Therefore, it is essential to analyze $ f(\btr,n_2,\sigma,\bSigma,\bSigma_{\boldsymbol{\theta}})\left(\sum_{i}\Xi_{i} \right)$ and $ g(\btr,n_1,\sigma,\bSigma,\bSigma_{\boldsymbol{\theta}})\left(\sum_{i}\Xi_{i} \right)$ from variance term in the upper and lower bounds respectively.
Then we calculate some rates of interesting in \Cref{thm-upper,thm-lower} under the specific data and task distributions in \Cref{prop-hard}.
If the spectrum of $\bSigma$ satisfies $\lambda_{k}=k^{-1} \log ^{-p}(k+1)$, then it is easily verified that $\operatorname{tr}(\bSigma^{s})=O(1)$ for $s=1,\cdots,4$. By discussions on \Cref{ass:higherorder} in \Cref{sec-diss}, we have $C(\beta,\bSigma
)=\Theta(1)$ for given $\beta$. Hence,
\begin{align*} c(\btr,\bSigma) &= \Theta(1)\\
f(\btr,n_2,\sigma,\bSigma,\bSigma_{\boldsymbol{\theta}})&=c(\btr,\bSigma)\operatorname{tr}({\bSigma_{\boldsymbol{\theta}}\bSigma})+ \Theta(1)\\
g(\btr,n_1, \sigma,\bSigma, \bSigma_{\btheta}) & =b_1\operatorname{tr}(\bSigma_{\btheta}\Htr)+\Theta(1).
\end{align*}
If $r\ge 2p-1$, then we have $g(\btr,n_1, \sigma,\bSigma, \bSigma_{\btheta})\ge \Omega\left(\log^{r-p+1}(d)\right)\ge \Omega\left(\log^{r-p+1}(T)\right)$.
Let $k^{\dagger}:= \operatorname{card}\{i: \mu_{i}(\Htr)\ge 1/\alpha T\}$. For large $n_1$, we have $\mu_i(\Htr)=(1-\btr\lambda_i)^2\lambda_i+o(1)$. If $k^{\dagger}=\mathcal{O}\left(T/\log^{p}(T+1)\right)$, then
$$\min_{1 \le i\le k^{\dagger}+1 }\mu_{i}(\Htr)=\omega\left(\frac{\log^{p}(T)}{T[\log(T)-p\log(\log(T))]^p}\right)=\omega\left(\frac{1}{T}\right)$$
which contradicts the definition of $k^{\dagger}$. Hence $k^{\dagger}=\Omega\left(T/\log^{p}(T+1)\right)$. Then
\begin{align*}
\sum_{i}\Xi_{i}\ge \Omega\left(k^{\dagger}\cdot \frac{1}{T} \right)=\Omega\left(\frac{1}{\log^{p}(T)}\right).
\end{align*}
Therefore, by \Cref{thm-lower}, $R(\wl,\bte)=\Omega\left(\log^{r-2p+1}(T)\right)$.
For $r< 2p-1$, if $d=T^l$, where $l$ can be sufficiently large ($d\gg T$) but still finite, then
\begin{itemize}
\item If $p-1<r< 2p-1$, $f(\btr,n_2,\sigma,\bSigma,\bSigma_{\boldsymbol{\theta}})\le \mathcal{O}(\log^{r-p+1} T)$;
\item If $r\leq p-1$, $f(\btr,n_2,\sigma,\bSigma,\bSigma_{\boldsymbol{\theta}})\le \mathcal{O}\Big(\log\left(\log(T)\right)\Big)$.
\end{itemize}
Following the analysis similar to that for Corollary 2.3 in \cite{zou2021benign}, we have $ \sum_{i}\Xi_{i}= \mathcal{O}(\frac{1}{\log^{p}(T)})$. Then by \Cref{thm-upper}
$$
R(\overline{\boldsymbol{\omega}}_T,\beta^{\text{te}})= \mathcal{O}\left(\frac{1}{\log^{p-(r-p+1)^{+}}(T)}\right).
$$
\end{proof}
\subsection{Proof of Proposition~\ref{prop-fast}}
\begin{proof}[Proof of \Cref{prop-fast}]
Following the analysis in \Cref{proof-prop2}, it is essential to analyze $ f(\btr,n_2,\sigma,\bSigma,\bSigma_{\boldsymbol{\theta}})\left(\sum_{i}\Xi_{i} \right)$.
If $d=T^l$, where $l$ can be sufficiently large but still finite, then $$f(\btr,n_2,\sigma,\bSigma,\bSigma_{\boldsymbol{\theta}})=\tilde{\Theta}(1)$$ for $\lambda_{k}=k^{q}$ $(q>1)$ or $\lambda_k=e^{-k}$.
Following the analysis similar to that for Corollary 2.3 in \cite{zou2021benign}, we have
\begin{itemize}
\item If $\lambda_{k}=k^{q}$ $(q>1)$, then $\sum_{i}\Xi_{i}=\mathcal{O}\left(\frac{1}{T^{\frac{q-1}{q}}}\right)$;
\item If $\lambda_k=e^{-k}$, then $\sum_{i}\Xi_{i}=\mathcal{O}\left(\frac{\log(T)}{T}\right)$.
\end{itemize}
Substituting these results back into \Cref{thm-upper}, we obtain
\begin{itemize}
\item If $\lambda_{k}=k^{q}$ $(q>1)$, then $ R(\overline{\boldsymbol{\omega}}_T,\bte)=\tilde{\mathcal{O}}\left(\frac{1}{T^{\frac{q-1}{q}}}\right)$;
\item If $\lambda_k=e^{-k}$, then $ R(\overline{\boldsymbol{\omega}}_T,\bte)=\tilde{\mathcal{O}}\left(\frac{1}{T}\right)$.
\end{itemize}
\end{proof}
\section{Proofs for Section~\ref{sec-main-stopping}}
\subsection{Proof of Proposition~\ref{prop-tradeoff}}\label{sec-prop4}
\begin{proof}[Proof of \Cref{prop-tradeoff}]
Following the analysis in \Cref{proof-prop2}, it is crucial to analyze $ f(\btr,n_2,\sigma,\bSigma,\bSigma_{\boldsymbol{\theta}})\left(\sum_{i}\Xi_{i} \right)$.
Then we calculate the rate of interest in \Cref{thm-upper,thm-lower} under some specific data and task distributions in Proposition 4. We have $\operatorname{tr}(\bSigma^2)=\frac{1}{s}+\frac{1}{d-s}=\Theta(\frac{\log^{p}(T)}{T})$. Moreover, by discussions on Assumption 3 in \Cref{sec-diss}, $C(\beta,\bSigma)=\Theta(1)$.
Hence
\begin{align*} c(\beta,\bSigma) &:= c_1+\tilde{\mathcal{O}}(\frac{1}{T});\\
f(\beta,n,\sigma,\bSigma,\bSigma_{\boldsymbol{\theta}})&:=2c_1\mathcal{O}(1)+\frac{\sigma^2}{n}+ \tilde{\mathcal{O}}\left(\frac{1}{T}\right).
\end{align*}
By the definition of $\Xi_i$, we have
\begin{align*}
\sum_{i}\Xi_i&=\mathcal{O}\left(s\cdot \frac{\mu_1(\Hte)}{T\mu_1(\Htr)}+\frac{1}{d-s}\cdot T \frac{\mu_d(\Htr)\mu_d(\Hte)}{\lambda^2_d}\right)\\
&= \mathcal{O}\left(\frac{1}{\log^{p}(T)} \right)\frac{\mu_1(\Hte)}{\mu_1(\Htr)}+\mathcal{O}\Big(\frac{1}{\log^{q}(T)} \Big)\frac{\mu_d(\Htr)\mu_d(\Hte)}{\lambda^2_d}\\
&=\mathcal{O}\left(\frac{1}{\log^{p}(T)} \right)\frac{(1-\bte\lambda_1)^2}{(1-\btr\lambda_1)^2}+\mathcal{O}\Big(\frac{1}{\log^{q}(T)} \Big)(1-\bte\lambda_d)^2(1-\btr\lambda_d)^2
\end{align*}
where the last equality follows from the fact that for large $n$, we have $\mu_i(\Hb_{n,\beta})=(1-\beta\lambda_i)^2\lambda_i+o(1)$. Combining with the bias term which is $\mathcal{O}(\frac{1}{T})$, and applying \Cref{thm-upper} completes the proof.
\end{proof}
\subsection{Proof of Corollary~\ref{col-stop}}
\begin{proof}[Proof of \Cref{col-stop}]
For $t\in (s, K]$,
by \Cref{thm-lower}, one can verify that $t=\tilde{\Theta}(K)$ for diminishing risk. Let $t=K\log^{-l}(K)$, where $p>l>0$. Following the analysis in \Cref{sec-prop4}, we have
\begin{align}
&R(\overline{\boldsymbol{\omega}}^{\beta^{\text{tr}}}_t,\bte)\lesssim \widetilde{\mathcal{O}}(\frac{1}{K})
+(2c_1\nu^2+\frac{\sigma^2}{n_2})\\
&\times \left[\mathcal{O}\Big(\frac{1}{\log^{p-l}(K)}\Big) \frac{(1-\bte\lambda_1)^2}{(1-\btr \lambda_{1})^{2}}+\mathcal{O}\Big(\frac{1}{\log^{p+l} (K)}\Big)\Big(1-\btr \lambda_{d}\Big)^{2}\Big(1-\bte \lambda_{d}\Big)^{2}\right].
\end{align}
To clearly illustrate the trade-off in the stopping time, we let $l=0$ for convenience. If $R(\overline{\boldsymbol{\omega}}^{\beta^{\text{tr}}}_t,\bte)<\epsilon$, we have
\begin{align*}
t_{\epsilon}\leq \exp\Big(\epsilon^{-\frac{1}{p}}\Big[\frac{U_{l}}{(1-\btr\lambda_1)^2}+ U_{t} (1-\btr\lambda_d)^2\Big]^{\frac{1}{p}}\Big)
\end{align*}
where
\begin{align*}
U_{l}=\mathcal{O}\Big( (2c_1\nu^2+\frac{\sigma^2}{n_2})(1-\bte\lambda_1)^2\Big) \quad { and }\quad U_{l} =\mathcal{O}\Big( (2c_1\nu^2+\frac{\sigma^2}{n_2})(1-\bte\lambda_d)^2\Big).
\end{align*}
The arguments are similar for the lower bound, and we can obtain:
\begin{align*}
L_{l}=\mathcal{O}\Big( (2\frac{b_1\nu^2}{n_2}+\frac{\sigma^2}{n_2})(1-\bte\lambda_1)^2\Big) \quad { and }\quad L_{l} =\mathcal{O}\Big( (2\frac{b_1\nu^2}{n_2}+\frac{\sigma^2}{n_2})(1-\bte\lambda_d)^2\Big).
\end{align*}
\end{proof}
\section{Discussions on Assumptions}\label{sec-diss}
\paragraph{Discussions on \Cref{ass-comm}}
If $\mathcal{P}_{\xb}$ is Gaussian distribution, then we have
$$
F=\mathbb{E}[\xb\xb^{\top}\bSigma\xb\xb^{\top} ]= 2\bSigma^3+\bSigma\operatorname{tr}(\bSigma^2).
$$
This implies that $F$ and $\bSigma$ commute because $\bSigma^3$ and $\bSigma$ commute. Moreover, in this case
$$
\frac{\beta^2}{n}(F-\bSigma^3)=\frac{\beta^2}{n}( \bSigma^3+\bSigma\operatorname{tr}(\bSigma^2)).
$$
Therefore, if $n\gg \lambda_1(\lambda^2_1+\operatorname{tr}(\bSigma^2))$, then the eigen-space of $\Hb_{n,\beta}$ will be dominated by $(\Ib-\beta\bSigma)^2\bSigma$.
\paragraph{Discussions on \Cref{ass:higherorder}} \Cref{ass:higherorder} is an eighth moment condition for $\xb:=\bSigma^{\frac{1}{2}}\zb$, where $\zb$ is a $\sigma_x$ sub-Gaussian vector. Given $\beta$, for sufficiently large $n$ s.t. $\mu_i(\Hb_{n,b})>0$, $\forall i$, and if $\operatorname{tr}(\bSigma
^{k})$ are all $O(1)$ for $k=1,\cdots,4$, then by the quadratic form and the sub-Gaussian property, which has finite higher order moments, we can conclude that $C(\beta,\Sigma)=\Theta(1)$.
The following lemma further shows that if $\mathcal{P}_{\xb}$ is a Gaussian distribution, we can derive the analytical form for $C(\beta,\bSigma
)$.
\begin{lemma}\label{lemma-C}
Given $|\beta|<\frac{1}{\lambda_1}$, for sufficiently large $n$ s.t. $\mu_i(\Hb_{n,b})>0$, $\forall i$, and if $\mathcal{P}_{\xb}$ is a Gaussian distribution, assuming $\bSigma
$ is diagonal, we have:
$$ C(\beta, \bSigma)=210( 1+\frac{\beta^4\operatorname{tr}(\bSigma^2)^2}{(1-\beta\lambda_1)^4}).$$
\end{lemma}
\begin{proof} Let $\eb_i\in\mathbb{R}^{d}$ denote the vector that the $i$-th coordinate is $1$, and all other coordinates equal $0$. For $\xb\sim\mathcal{P}_{\xb}$, denote $\xb\xb^{\top}=[x_{ij}]_{1\leq i,j\leq d}$. Then we have:
\begin{align*}
& \mathbb{E}[\|\eb_{i}^{\top}\Hb^{-\frac{1}{2}}_{n,\beta}(\mathbf{I}-\frac{\beta}{n}\Xb^{\top}\Xb)\bSigma (\mathbf{I}-\frac{\beta}{n}\Xb^{\top}\Xb)\Hb^{-\frac{1}{2}}_{n,\beta}\eb_{i}\|^2]\\
&\leq \mathbb{E}[\|\eb_{i}^{\top}\Hb^{-\frac{1}{2}}_{n,\beta}(\mathbf{I}-\beta\xb\xb^{\top})\bSigma (\mathbf{I}-\beta\xb\xb^{\top})\Hb^{-\frac{1}{2}}_{n,\beta}\eb_{i}\|^2]\\
&=\mathbb{E}\left[(\eb_{i}^{\top}\Hb^{-1}_{n,\beta}\eb_{i})^2\left(\sum_{j\neq i }\beta^2\lambda_jx^2_{ij}+ \lambda_i(1-\beta x_{ii})^2\right)^2\right]
\end{align*}
For Gaussian distributions, we have $$
\mathbb{E}[x^2_{ij}x^2_{ik}]=\begin{cases}9\lambda^2_{i}\lambda^2_{j} & j=k \text{ and } \neq i\\
105\lambda_{i}^4& i=j=k\\
3\lambda^{2}_i\lambda_j\lambda_k & i\neq j\neq k
\end{cases}$$
We can further obtain:
\begin{align*}
&\mathbb{E}[\|\eb_{i}^{\top}\Hb^{-\frac{1}{2}}_{n,\beta}(\mathbf{I}-\frac{\beta}{n}\Xb^{\top}\Xb)\bSigma (\mathbf{I}-\frac{\beta}{n}\Xb^{\top}\Xb)\Hb^{-\frac{1}{2}}_{n,\beta}\eb_{i}\|^2]\\
&\leq 105(\eb_{i}^{\top}\Hb^{-1}_{n,\beta}\eb_{i})^2\left(\sum_{j\neq i }\beta^2 \lambda^2_{j}+ (1-\beta \lambda_{i})^2 \right)^2\\
&\overset{(a)}{\leq} 210(\eb_{i}^{\top}\Hb^{-1}_{n,\beta}\eb_{i})^2[\beta^4\operatorname{tr}(\bSigma^2
)^2+ (1-\beta \lambda_{i})^4 ]\\
&\overset{(b)}{\leq} 210[(\eb_{i}^{\top}\Hb^{-1}_{n,\beta}\eb_{i})^2\beta^4\operatorname{tr}(\bSigma^2
)^2+ 1 ]
\end{align*}
where $(a)$ follows from the Cauchy-Schwarz inequality, and $(b)$ follows the fact that $(\eb_{i}^{\top}\Hb^{-1}_{n,\beta}\eb_{i})^2=\frac{1}{[(1-\beta\lambda_i)\lambda_i^2+\frac{\beta^2}{n}(\lambda_i^2+\operatorname{tr}(\bSigma
^2)\lambda_i)]^2}\leq 1/(1-\beta\lambda_i)^4$.
Therefore, for any unit $\vb\in\mathbb{R}^{d}$, we have
\begin{align*}
& \mathbb{E}[\|\vb^{\top}\Hb^{-\frac{1}{2}}_{n,\beta}(\mathbf{I}-\frac{\beta}{n}\Xb^{\top}\Xb)\bSigma (\mathbf{I}-\frac{\beta}{n}\Xb^{\top}\Xb)\Hb^{-\frac{1}{2}}_{n,\beta}\vb\|^2]\\
&\leq \max_{\vb} 210[(\vb^{\top}\Hb^{-1}_{n,\beta}\vb)^2\beta^4\operatorname{tr}(\bSigma^2
)^2+ 1 ]\leq 210\left( 1+\frac{\beta^4\operatorname{tr}(\bSigma^2)^2}{(1-\beta\lambda_1)^4}\right).
\end{align*}
\end{proof}
\end{document}
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San Diego Water - Signs You Need Water Treatment Posted on Apr 10, 2013 When to Consider San Diego Water Treatment In San Diego, water treatment is an important consideration. Water contaminants vary greatly based on your ... Posted By Anderson Plumbing, Heating & Air Read More San Diego Drain Repair Tips to Fix 3 Common Drain Clogs Posted on Apr 3, 2013 A clogged sink drain is not only a nuisance but also a health hazard. While the plumbing professionals at Anderson Plumbing, Heating & Air are ... Posted By Anderson Plumbing, Heating & Air Read More Page of 1
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Volunteer Calendar
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A design begins with a discussion with a client about how a room is used and what the needs are. It is also spurred by something the client sees and likes.
This is a sun room in a home I designed several years ago. The home is finished and furnished but sometimes the clients get tired toward the end of a project and just can’t make any more decisions so we have to come back to it perhaps when they see something that excites them and because the area that was left unfinished is not functioning. This project got re-ignited because the client saw some red wicker furniture on a porch that she loved. She was also tired of the messy unorganized look of the room.
The sun room with an entrance from the garage. It has become a catch all for dropping what ever is being carried in the house. The project started with a drawing of the furniture placement and the design of a piece of furniture to organize all that stuff. It has storage for boots and shoes under the bench for sitting. It has a closet and pegs for hanging and shelves for baskets to hold mittens, dog leashes, hats and games. It will be built of rustic alder, the same wood used in the interior doors and all the wood trim.
The color scheme is based on the rest of the home which is warm neutrals, nature stone colors and natural cherry and alder woods. The wall colors are molasses, some ceilings are natural cherry, trim and doors are alder. The kitchen cabinets are bisque with a caramel tone wash. The accent is a rusty berry red. We have selected a hopsack cotton print in tones of beige, red, taupe and green this will be upholstered on red wicker chairs. The multi color check will be upholstered on an ottoman, pillows and seat cushions. I have also a designed a replica table. It was an antique table that the client saw on the back porch of my studio many years ago. I found the table in a garage at a local antique store. It was used in a kitchen at some point in it’s life. I am having this table reproduced with a reclaimed pine plank top and finishing the legs, apron and drawer in a worn and chipped, off white color. I am very lucky to work with a furniture builder, Ron Sween from Northshore Wood Products that has always been able to build anything that I can design. He is building the back entry piece as well as the replica table.
Stay tuned for the finished project!
| 269,032
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\begin{document}
\maketitle
\begin{abstract}
We give yet another proof for Fa\`{a} di Bruno's formula for higher derivatives of composite functions. Our proof technique relies on reinterpreting the composition of two power series as the generating function for weighted integer compositions, for which a Fa\`{a} di Bruno-like formula is quite naturally established.
\end{abstract}
\section{Introduction}
According to Fa\`{a} di Bruno's formula, the $n$th derivative of a composite function $G\circ F$ is given by
\begin{align}\label{eq:faa}
\frac{d^n}{dx^n}G(F(x))=\sum\frac{n!}{b_1!\cdots b_n!}G^{(r)}(F(x))\prod_{i=1}^n\Bigl(\frac{F^{(i)}(x)}{i!}\Bigr)^{b_i},
\end{align}
where the sum ranges over all different solutions in nonnegative
integers $b_1,\ldots,b_n$ of $b_1+2b_2+\cdots+nb_n=n$ and where $r$ is
defined as $r=b_1+\cdots+b_n$. For example, for $n=3$, the three solutions for $(b_1,b_2,b_3)$ are $(0,0,1)$, $(1,1,0)$ and $(3,0,0)$, which correctly yields
\begin{align*}
G'(F(x))\cdot F'''(x) + 3G''(F(x))\cdot F'(x)F''(x)+G'''(F(x))\cdot (F'(x))^3
\end{align*}
as third derivative of $G\circ F$.
Many proofs of formula \eqref{eq:faa} have been given, both based on combinatorial arguments --- such as via Bell polynomials \cite{comtet} or set partitions --- as well as on analytical; the latter, for example, using Taylor's theorem \cite{johnson}.
Roman \cite{roman} gives a proof based on the umbral calculus. Johnson \cite{johnson} summarizes the historical discoveries and re-discoveries of the formula as well as a variety of different proof techniques.
Herein,
we give (yet) another proof of the formula, one that is based on the combinatorics of integer compositions and a particular interpretation of the composition of power series.
The essence of our derivation is as follows: First, we consider the number $C_{f,g}(n)$ of (doubly weighted) integer compositions of the positive integer $n$, for which we derive a closed-form formula; this requires some notation and introduction of terminology, but the derivation and combinatorial interpretation of the formula is quite intuitive. Then,
for two arbitrary power series $G(x)=\sum_{n\ge 0} g_nx^n$ and $F(x)=\sum_{n\ge 0}f_nx^n$, we argue that $G\circ F$ has a natural interpretation of
denoting
the \emph{generating function}
\begin{align*}
C(x)=\sum_{n\ge 0} C_{f,g}(n)x^n
\end{align*}
for $C_{f,g}(n)$. Hence, $\frac{1}{n!}\frac{d^n}{dx^n}C(0)=C_{f,g}(n)$. This yields formula \eqref{eq:faa} for $x=0$, but we argue that it is clear that the formula must indeed hold for any $x$.
Two remarks are in order: first, as indicated, our derivation does not apply to arbitrary functions $F$ and $G$, but only to power series.
While this may be considered a restriction, many interesting functions can indeed be represented as power series (those functions even have a name, real analytical functions).
We also remark that, throughout, we ignore matters of convergence and treat all series as \emph{formal} and assume that functions have sufficiently many derivatives.
Finally, while we think that many derivations of Fa\`{a} di Bruno's formula given in the literature are similar to the one we outline, we believe the particular approach that we suggest, based on integer compositions and a reinterpretation of the composition of power series, to be novel.\footnote{Technically, the approach most
similar to our own appears to be the one due to Flanders \cite{flanders}, which is, however, conceptually substantially different from our own.}
\section{Integer compositions and partitions}
An \emph{integer composition} of a positive integer $n$ is a tuple of positive integers $(\pi_1,\ldots,\pi_k)$, typically called \emph{parts}, whose sum is $n$. For example, the eight integer compositions of $n=4$ are
\begin{align*}
(4),(1,3),(3,1),(2,2),(1,1,2),(1,2,1),(2,1,1),(1,1,1,1).
\end{align*}
An \emph{integer partition} of $n$ is a tuple of positive integers $(\pi_1,\ldots,\pi_k)$ whose sum is $n$ and such that $\pi_1\ge \pi_2\ge \cdots\ge \pi_k$. For instance, there are (only) five integer partitions of $n=4$, namely
\begin{align*}
(4),(3,1),(2,2),(2,1,1),(1,1,1,1).
\end{align*}
Both integer compositions and partitions are well-studied objects in combinatorics \cite{andrews,heubach}. Instead of considering ordinary partitions and compositions as defined, we may consider \emph{weighted} compositions \cite{abramson,eger} and partitions, where each part value $\pi_i\in\nn=\set{1,2,3,\ldots}$ may have attributed with it a weight $f(\pi_i)\in\real$, where $\real$ denotes the set of real numbers. If weights are nonnegative integers, they may be interpreted as colors. For instance, when $f(3)=2$ and $f(1)=f(2)=f(4)=f(5)=\cdots=1$, then there are ten $f$-weighted compositions and six $f$-weighted partitions of $n=4$. These are
\begin{align*}
(4),(1,3),(1,3^*),(3,1),(3^*,1),(2,2),(1,1,2),(1,2,1),(2,1,1),(1,1,1,1)
\end{align*}
and
\begin{align*}
(4),(3,1),(3^*,1),(2,2),(2,1,1),(1,1,1,1),
\end{align*}
respectively, where we use a star ($*$) to differentiate between the two colors of part value $3$. When weights are nonintegral real numbers, they may simply be interpreted as ordinary `weights' --- possibly as probabilities if the range of $f$ is the unit interval $[0,1]$.
Let us note that integer partitions of an integer $n$ admit an alternative, equivalent representation. Instead of writing a partition of $n$ as a tuple $(\pi_1,\ldots,\pi_k)$ with $\pi_1\ge\cdots\ge \pi_k$, we may represent it as a tuple $(k_1,\ldots,k_n)$, with $0\le k_i\le n$, for all $i=1,\ldots,n$, whereby $k_i$ denotes the \emph{multiplicity} of (type) $i\in\set{1,2,\ldots,n}$ in the composition of $n$. For instance, the above five integer partitions of $n=4$ may be represented as
\begin{align*}
(0,0,0,1),(1,0,1,0),(0,2,0,0),(2,1,0,0),(4,0,0,0).
\end{align*}
Obviously, each such tuple $(k_1,\ldots,k_n)$ must satisfy $1\cdot k_1+2\cdot k_2+\cdots+n\cdot k_n=n$. Assuming that the weighting function $f$ takes on only integral values, for the moment, how many $f$-weighted integer partitions of $n$ are there? Apparently, this number is given by
\begin{align}\label{eq:partition1}
\sum_{k_1+2k_2+\cdots+nk_n=n}f(1)^{k_1}\dots f(n)^{k_n},
\end{align}
since the solutions, in positive numbers, of $k_1+2k_2+\cdots+nk_n=n$ are precisely the integer partitions of $n$ and the product $f(1)^{k_1}\cdots f(n)^{k_n}$ assigns the different colors to a given partition $(k_1,\ldots,k_n)$. How many $f$-weighted integer compositions of $n$ are there? Note that, in the representation $(k_1,\ldots,k_n)$ of a partition, $k_1$ denotes the number of `type' $1$, $k_2$ denotes the number of `type' $2$, ..., and $k_n$ denotes the number of `type' $n$ used in the partition of $n$. Since compositions are ordered partitions, for compositions, we need to distribute the $k_1$ types $1$, ..., $k_n$ types $n$ in a sequence of length $(k_1+\cdots+k_n)$. Therefore, the number of $f$-weighted integer compositions is simply:
\begin{align}\label{eq:composition1}
\sum_{k_1+2k_2+\cdots+nk_n=n}\binom{k_1+\cdots+k_n}{k_1,\ldots,k_n}f(1)^{k_1}\dots f(n)^{k_n},
\end{align}
where $\binom{r}{k_1,\ldots,k_n}=\frac{r!}{k_1!\cdots k_n!}$ (for $r=k_1+\cdots+k_n$) denote the \emph{multinomial coefficients}.
Finally, let us assume that integer partitions/compositions with a \emph{given, fixed number $k$ of parts} are (additionally) weighted (e.g., colored) by $g(k)$, for a weighting function $g:\nn\goesto\real$. For instance, we might double count the $f$-weighted partitions/compositions with exactly $k_1+\ldots+k_n=4$ parts (or assign them higher/lower probability). Then, the number of $f$-weighted integer compositions of $n$ where parts are $g$-weighted is simply given by
\begin{align}\label{eq:combinatorial}
C_{f,g}(n)=\sum_{k_1+2k_2+\cdots+nk_n=n}\binom{k_1+\cdots+k_n}{k_1,\ldots,k_n}g(k_1+\cdots+ k_n)\prod_{i=1}^n f(i)^{k_i}.
\end{align}
If $f$ and/or $g$ take on nonintegral values, \eqref{eq:partition1} and \eqref{eq:composition1} denote the total weight of all $f$-weighted integer partitions/compositions, and \eqref{eq:combinatorial} denotes the total weight of all $f$-weighted integer compositions of $n$ where parts are $g$-weighted. Henceforth, for brevity, we also call such compositions simply $(f,g)$-weighted.
\section{Derivation of Fa\`{a} di Bruno's formula}
We assume that $F(x)$ and $G(x)$ are the power series
\begin{align*}
F(x) &= f_0+f_1x^1+f_2x^2+\ldots = \sum_{n\ge 0} f_nx^n,\\
G(x) &= g_0+g_1x^1+g_2x^2+\ldots = \sum_{n\ge 0}g_nx^n,
\end{align*}
for some real coefficients $f_0,f_1,f_2,\ldots$ and $g_0,g_1,g_2,\ldots$.
In the remainder, for ease of interpretation, we speak of the $f_n$ and $g_n$ values as if they were nonnegative and integral, but keep in mind that they may be arbitrary real numbers.
We interpret $F$ and $G$ as follows. The function $F$ is the generating function for the number of $f$-weighted integer compositions of $n$ with exactly one part, whereby $f(n)=f_n$. In fact, the coefficient $f_n$ of $x^n$ of $F(x)$ gives the number of $f$-weighted integer compositions of $n$ with exactly one part. We assume that $f_0=0$ (that is, integer compositions admit only positive integers).
In the context $G\circ F$, we interpret the function $G$ as follows: $G\circ F$ represents, for $G(x)=x^k$, the generating function for the number of $f$-weighted integer compositions with exactly $k$ parts;
for $G(x)=a_kx^k$, it represents the generating function for the number of $f$-weighted integer compositions with exactly $k$ parts, where $f$-weighted compositions with $k$ parts are weighted by the factor $a_k$; and, finally, for $G(x)=x^j+x^k$, it represents the generating function for the number of $f$-weighted integer compositions with either $j$ or $k$ parts (union). This interpretation of $G$, in the context $G\circ F$, is a natural interpretation, since, for example, the coefficients of $x^n$ of $(F(x))^2$ have the form $\sum_{i=0}^nf_{n-i}f_i$, and all combinations of the number of $f$-weighted compositions of $n-i$ with one part and the number of $f$-weighted compositions of $i$ with one part yield the number of $f$-weighted compositions of $(n-i)+i=n$ with two parts.\footnote{This is in fact a critical point of our proof; if we interpreted $F(x)$ as the generating function for other combinatorial objects, such as integer partitions, then $G(x)=x^k$, in the context $G\circ F$, could not have the same interpretation as the one we have outlined.} Then, the interpretation of $(F(x))^k$ follows inductively. Similarly, if $(F(x))^k$ denotes the generating function for the number of $f$-weighted compositions of $n$ with exactly $k$ parts and $(F(x))^j$ denotes the analogous generating function for $j$ parts, then their sum obviously denotes the generating function for $k$ or $j$ parts.
Hence, to summarize, we interpret $G\circ F$ as the generating function for the number of $f$-weighted integer compositions with arbitrary number of parts (recall that the `$+$' mean union over number of parts) and where compositions with $k$ parts are weighted by $g(k)=g_k$. Then, by virtue of the definition of generating functions, we know that $\frac{1}{n!}\frac{d^n}{dx^n}(G\circ F)(0)$ gives the number of $(f,g)$-weighted integer compositions of $n$.
This is the number $C_{f,g}(n)$, whence by formula \eqref{eq:combinatorial}, we know that
\begin{align*}
\frac{1}{n!}\frac{d^n}{dx^n}(G\circ F)(0) = C_{f,g}(n)= \sum_{k_1+2k_2+\cdots+nk_n=n}\binom{k_1+\cdots+k_n}{k_1,\ldots,k_n}g(k_1+\cdots+ k_n)\prod_{i=1}^n f(i)^{k_i},
\end{align*}
or, equivalently,
\begin{align}\label{eq:interm}
\frac{d^n}{dx^n}(G\circ F)(0) = \sum_{k_1+2k_2+\cdots+nk_n=n}\frac{n!}{k_1!\cdots k_n!}r!g(r)\prod_{i=1}^n f(i)^{k_i},
\end{align}
where we write $r=k_1+\cdots+k_n$.
We note that
\begin{align*}
f(i) &= \frac{1}{i!}F^{(i)}(0), \quad\forall\:i=1,\ldots,n,\\
r!g(r) &= G^{(r)}(0) = G^{(r)}(F(0)),
\end{align*}
whence we can rewrite \eqref{eq:interm} as
\begin{align}\label{eq:final}
\frac{d^n}{dx^n}(G\circ F)(0) = \sum_{k_1+2k_2+\cdots+nk_n=n}\frac{n!}{k_1!\cdots k_n!}G^{(r)}(F(0))\prod_{i=1}^n \bigl(\frac{F^{(i)}(0)}{i!}\bigr)^{k_i},
\end{align}
which is Fa\`{a} di Bruno's formula \eqref{eq:faa} evaluated at $x=0$.
Now,
from $(G\circ F)'(x)=G'(F(x))\cdot F'(x)$, and then $(G\circ F)''(x)=G''(F(x))F'(x)+G'(F(x))F''(x)$, etc., it
is clear that
$\frac{d^n}{dx^n}(G\circ F)(x)$ is a sum of products of factors $G^{(j)}(F(x))$ and $F^{(m)}(x)$. It is also clear that, whatever the precise form of $\frac{d^n}{dx^n}(G\circ F)(x)$, evaluating it at $x=0$ will simply yield the same sum of products of factors $G^{(j)}(F(x))$ and $F^{(m)}(x)$, evaluated at $x=0$. Hence, \eqref{eq:final} must in fact hold for all $x$, not only for $x=0$.
\section{Discussion}
We argued that $G\circ F$ has, for arbitrary power series $G$ and $F$ with coefficients $g_n$ and $f_n$, respectively, a natural interpretation as denoting the generating function for $(f,g)$-weighted integer compositions, whereby $f(n)=f_n$ and $g(n)=g_n$, for whose coefficients Fa\`{a} di Bruno-like formulas quite effortlessly arise.
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TITLE: Differentiability of electric field due to bounded volume charge distribution
QUESTION [3 upvotes]: In books on electromagnetism, one often sees expressions of Maxwell's equations like $\nabla \cdot \mathbf{E}$ and $\nabla \times \mathbf{E}$. These expressions make sense if $\mathbf{E}$ (which is due to bounded volume charge distribution) is differentiable. I ask this question because in all the textbooks on electromagnetism which I have seen, expressions like $\nabla \cdot \mathbf{E}$ and $\nabla \times \mathbf{E}$ are used and nowhere do they prove the differentiability of $\mathbf{E}$. How can it be justified?
Is the differentiability of $\mathbf{E}$ such a trivial case? If yes, why is it so? If no, why do the books ignore discussing the differentiability of $\mathbf{E}$?
REPLY [0 votes]: Differentiability of the EM fields, like for many other quantities introduced in Physics, is not a property of the world but it is part of the mathematical model we find useful to describe the world.
As such, it is a property whose validity may be judged on an experimental basis. Until the model is in agreement with experiments, the property is valid. As soon we find a significant departure from the experiments we should be ready to change our model.
Actually, for EM fields we know that continuity properties may hold only on a coarse grained scale where the spatial average property of any real measurement allows to deal with smoothly varying quantities. However at the level of a microscopic QFT description a classical EM field emerges only when we can neglect the effect on the local values of the fluctuations of the underlying quantum fields.
Notice, that even much before QM or QFT entered into the toolbox of physicists, the smoothness of macroscopic fields was considered as a consequence of averaging over spatial regions large with respect to typical atomic length-scales but small with respect to the typical length of variation of the fields.
This is exactly the analog of a classical trajectory. We know that it does not exist at the QM level, still it is a very good approximation, with all possible smoothness, if we have to deal with planetary orbits.
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She’s so edgy! Miley Cyrus is overseas doing some performances, and lastnight before performing her current singe, 7 Things, Cyrus dedicated the song to “all the girls (or guys) out there who hate their ex boyfriends, or current boyfriends.”
She then ended the song and with her back to the crowd yelled out, “I still hate you.”
Dang, what did Nick Jonas do to deserve all the hate? I figure 15 is a little to young to be filled with relationship hate, but who are we kidding, this girl is 25 trapped in a 15 year old’s body anyways.
p.s. Thanks to whoever cut this video so we didn’t have to sit through the song.
Tags: miley cyrus hate, miley cyrus nick jonas drama, miley cyrus nick jonas news
You can leave a response, or trackback from your own site.
you go girl i hate dat faget 2 xD
[...] reportedly been trying to make it very clear that she supports diabetes awareness (even though she hates Nick Jonas, who battles with the condition) by frequently stating the phrase “drink water, not [...]
[...] ella apoya el movimiento relacionado al conocimiento y buen manejo de la diabetes (aunque odia a odia a Nick Jonas, su ex, quien sufre de la misma enfermedad), y numerosas veces ha dicho: “tomen agua y no [...]
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About Centiva Solid Vinyl Tile
Centiva’s history is an exciting story of an entrepreneurial company that started small, got bigger, and joined a larger company to become even better.
We are not satisfied with our progress and probably never will be, knowing
that remaining eager for change and incorporating new ideas is essential to be
the company we need to be for tomorrow’s customers.
We joined Tarkett in 2010, a flooring company that we had admired for years, and are now excited to be a genuine part of. Tarkett is a worldwide leader of innovative and sustainable flooring. Together we are further developing Centiva, expanding our product line, and implementing new ideas faster than ever.
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First Things.
(Excerpts from Akathist Hymn Glory to God for All Things)
Secondary Things
Were I a Roman Catholic, I think I’d very unhappy and uneasy right about now.
Hypotheticals are always tricky, but I thought “If I were a Catholic, I think I’d be a Benedict XVI Catholic.” I also used the think that if I weren’t Christian, I’d be Bahai. Go figure. When I was young and foolish, I was young and foolish.
So why “right about now”?
Well, four Cardinals apparently thought that Pope Francis had sown some uncertainty in the Church with his encyclical Amoris Laetitia. So they exercised a canonical option by posing four dubia to the Pope, seeking clarification. His Holiness declined to answer.
Meanwhile one of those four dubia authors, the combative traditionalist, Cardinal Raymond Burke, gave an interview suggesting that papal silence might require a “formal act of correction” from the cardinals — something without obvious precedent in Catholic history.
(Ross Douthat) Now
Archbishop.
(Rod Dreher) The standoff could be quite — ahem! — consequential.
Q: How can the doctrine of papal infallibility survive this?
A: Fans of logic will note that it can’t. If Pope Francis continues on the course he has chosen, he will prove, empirically, that this teaching was never true in the first place.
Q: What will that mean for the First Vatican Council?
A: That council, and every other council the Catholic Church has held since the great Schism with the Orthodox in 1054, will be called into question. The Orthodox theory, that it was Rome which went off the rails back then, will start looking pretty persuasive. Last time I checked, making the case for that was not the Roman pontiff’s job.
If you want more background, read Rod Dreher. I’m with him up to the point where he begins “You, reader, might be thinking: Rod, as a former Catholic turned Orthodox, must be pretty happy with this.” Rod isn’t happy and explains why.
His unhappiness is viscerally conservative, amounting, in my paraphrase, to “Rome is wrong on divorce and remarriage, but Pope Francis is flirting with reform in the right direction for the wrong reason of mollifying post-Christians who misunderstand sex and marriage. And goodness knows what collateral damage could be done by reform on this point, which may be more integral to the whole edifice than you think.”
But can that understandable gut reaction withstand reason, when Rod concedes that Francis is talking in ways that echo the correct Orthodox position? Can the bad consequences of abandoning the wrongful dogma of infallibility justify continuing to hold it (or should I say “abandon it much more slowly and deliberately”)?
I’m not at all sure it can. But I’ve thought for a long time that Vatican I’s dogma of Papal Infallibility painted the Roman Church into a corner from which I, a bystander, could see no escape without putting the whole structure at the gravest possible risk.
Tertiary Things
Pascal-Emmanuel Gobry has a bone to pick with “the middle ages” — not the era, but the term:
Today, we are told a very simple story about the grand sweep of European history. It goes something like this: There was once the Roman Empire, technologically advanced and sophisticated; after the Roman Empire fell, Europe fell into a millennium of darkness, poverty, and religious superstition; then came the Renaissance, when the West recovered the glories of Greco-Roman thought and science, and the wheel of progress started turning again, leading to the “Enlightenment” when philosophers threw off the fetters of irrational religion to advocate for free inquiry, human rights, and so on..
But I would like to make a modest proposal: Let’s retire the phrase “Middle Ages.” It’s not just misleading and ideologically biased, it’s also, on its own terms, entirely meaningless. We can do better.
I appreciate his point and I like his better alternative, for which you’ll need to read his piece.
As you may already know, every couple of years Congress considers whether to pass the Employment Non-Discrimination Act, a federal statute that would ban employment discrimination on the basis of sexual orientation, much as discrimination on the basis of race, sex, etc. are already banned. So far, the legislation has not yet been enacted, and that was about where things stood when I was in law school, although since then one attempt, H.R. 3685, passed the House in 2007 by a 235-to-184 vote, and another, S. 815, passed the Senate in 2013 by a 64-to-32 vote.
So I have been somewhat intrigued in more recent years as an argument has emerged quite prominently in some federal courts that employment discrimination on the basis of sexual orientation is already illegal. The argument is that Title VII’s ban on sex discrimination also entails a ban on discrimination on the basis of sexual orientation, and it has gotten enough momentum that it is coming to the en banc U.S. Court of Appeals for the 7th Circuit on Wednesday in Hively v. Ivy Tech Community College.
While it may seem surprising for a decades-old statute to suddenly be discovered to have an important new implication like this, the argument has something going for it ….
(Will Baude at Volokh Conspiracy)
Well, yes, it would seem “surprising for a decades-old statute to suddenly be discovered to have an important new implication like this.” Although “How convenient!” might be better than “it would seem surprising.”
The ability of courts (aided, of course, by the sophists of the ACLU or the like) to find progressive policy in the constitution or prior law after long failure to the legislatures to enact them, should strike anyone as very, very fishy — prima facie, albeit not ultima facie.
I suspect that such proclivities contributed to the election of President Disaster-Waiting-to-Happen.
Just when I start longing to go to France, I learn that this video was banned (as “not of general interest”) from French airwaves:
It seems that the French have a particular horror about Down Syndrome, aborting fully 96% of Down Syndrome children when the condition is discovered prenatally and using it as a way to mock and satirize. Yes, I mean Charlie Hebdo. Details here (headline hyperbole alert).
* * * * *
“In learning as in traveling and, of course, in lovemaking, all the charm lies in not coming too quickly to the point, but in meandering around for a while.” (Eva Brann)
Some succinct standing advice on recurring themes.
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YOGA: There’s a crispness in the air this week which suggests that autumn is not too far away and already information is dropping through my letterbox about courses beginning next month. One of them is a Yoga class to be held on Mondays from 10am.
WHAT’S ON: This Sunday String Theory will bring live music to the pub and on Friday August 28 there will be the usual monthly Folk Night, also at The Star. St Bartholomew’s.
| 3,016
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/cdn.vox-cdn.com/uploads/chorus_image/image/60809877/16099353553_19bbb8da7a_o.0.jpg)
USF confirmed after Saturday’s scrimmage that Ole Miss tight end transfer Jacob Mathis has been granted a waiver from the NCAA and will be eligible to play during the 2018 season. Our own Nathan Bond broke the news on Twitter during Fan Fest.
The news came just four days after he was added to the roster on Tuesday.
Got my eligibility to play this year. thanks everyone who came out to fan day #Home #Gobulls— Jacob (YG) Mathis (@Cuddie_J) August 12, 2018
A hometown kid out of Berkeley Prep in Tampa, Mathis was a consensus four-star prospect in the 2016 recruiting cycle, catching 46 passes for 691 yards and eight touchdowns during his senior season.
Rated as the No. 1 TE-H prospect in the nation by ESPN, the Under-Armour All-American chose Ole Miss over offers from a plethora of schools, including Michigan, Alabama, Florida, Georgia, Ohio State and of course, the hometown Bulls.
He would redshirt in Oxford as a true freshman in 2016 and according to our friends over at Red Cup Rebellion, he simply got lost on the depth chart with the emergence of fellow 2016 signees Dawson Knox and Octavious Cooley.
He decided to transfer after spring practice in April, following other former Rebels like quarterback Shea Patterson and wide receiver Van Jefferson, who made the decision to leave after the school was hit with sanctions by the NCAA in December.
Mathis’s eligibility adds immediate depth to a thin USF tight end core that includes veteran Mitchell Wilcox, redshirt freshman Frederick Lloyd Jr., and true freshman Chris Carter Jr.
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@barrock: Well all of the shows on Netflix are dubbed so you should be ok there. How much anime have you seen? Like could I recommend some anime ass anime to you or are you just starting out? I would say start off with Cowboy Bebop because that is a great place to start and the dub is great but it is not on Netflix or Hulu. If you don't mind some anime ass anime I would say watch Kill la Kill or Gurren Lagann because they are great but they are very crazy and over the top. Psycho Pass is also pretty great but is much more serious and dark. Madoka Magic is a good series and I would say to just watch it without knowing too much first. Death Note is pretty great and is also serous and dark. A lot of out mastering and out thinking people and such in that. Attack on Titan is a good beginner anime I think. It looks cool and has some good action but the plot isn't the best. And lastly, while I have not seen it myself Full Metal Alchemist Brotherhood is supposed to be pretty good. That is what I would start off with on Netflix at least. As for Hulu there is more there but I don't know what is dubbed and what is not.
I'm definitely down for some anime ass anime. I've not really watched much anime at all. Never watched any series, just a movie or two here and there. I own the first volume of Soul Eater, Hellsing,, Black Lagoon, and SamuraiChampo all on blu-ray. They were on sale and I had money burning a hole in my pocket I guess. ;)
Does FMA Brotherhood occur after the original series?
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\begin{document}
\title
[Commuting variety]
{On a variety related to the commuting variety of a reductive Lie algebra.}
\author[Jean-Yves Charbonnel]{Jean-Yves Charbonnel}
\address{Jean-Yves Charbonnel, Universit\'e Paris Diderot - CNRS \\
Institut de Math\'ematiques de Jussieu - Paris Rive Gauche\\
UMR 7586 \\ Groupes, repr\'esentations et g\'eom\'etrie \\
B\^atiment Sophie Germain \\ Case 7012 \\
75205 Paris Cedex 13, France}
\email{jean-yves.charbonnel@imj-prg.fr}
\subjclass
{14A10, 14L17, 22E20, 22E46 }
\keywords
{polynomial algebra, complex, commuting variety, desingularization, Gorenstein,
Cohen-Macaulay, rational singularities, cohomology}
\date\today
\begin{abstract}
For a reductive Lie algbera over an algbraically closed field of charasteristic zero,
we consider a Borel subgroup $B$ of its adjoint group, a Cartan subalgebra contained in
the Lie algebra of $B$ and the closure $X$ of its orbit under $B$ in the Grassmannian.
The variety $X$ plays an important role in the study of the commuting variety. In this
note, we prove that $X$ is Gorenstein with rational singularities.
\end{abstract}
\maketitle
\tableofcontents
\section{Introduction} \label{int}
In this note, the base field $\k$ is algebraically closed of characteristic $0$,
${\goth g}$ is a reductive Lie algebra of finite dimension, $\rg$ is its rank,
$\dim {\goth g}=\rg + 2n$ and $G$ is its adjoint group. As usual, ${\goth b}$ denotes a
Borel subalgebra of ${\goth g}$, ${\goth h}$ a Cartan subalgebra of ${\goth g}$,
contained in ${\goth b}$, and $B$ the normalizer of ${\goth b}$ in $G$.
\subsection{Main results.} \label{int1}
Let $X$ be the closure in $\ec {Gr}g{}{}{\rg}$ of the orbit of ${\goth h}$ under the
action of $B$. By a well known result, $G.X$ is the closure in $\ec {Gr}g{}{}{\rg}$
of the orbit of ${\goth h}$ under the action of $G$. By~\cite{Ric}, the commuting
variety of ${\goth g}$ is the image by the canonical projection of the restriction to
$G.X$ of the canonical vector bundle of rank $2\rg$ over $\ec {Gr}g{}{}{\rg}$. So
$X$ and $G.X$ play an important role in the study of the commuting variety. As it is
explained in \cite{CZ}, $X$ and $G.X$ play the same role for the so called generalized
commuting varieties and the so called generalized isospectral commuting varieties. The
main result of this note is the following theorem:
\begin{theo}\label{tint}
The variety $X$ is Gorenstein with rational singulatrities.
\end{theo}
An induction is used to prove this theorem. So we introduce the categories
${\cal C}'_{{\goth t}}$ and ${\cal C}_{{\goth t}}$ with ${\goth t}$ a commutative Lie
algebra of finite dimension. Their objects are nilpotent Lie algebras of finite
dimension, normalized by ${\goth t}$ with additional conditions analogous to those of the
action of ${\goth h}$ in ${\goth u}$. In particular the minimal dimension of the objects
in ${\cal C}_{{\goth t}}$ is the dimension of ${\goth t}$ and an object of dimension
$\dim {\goth t}$ is a commutative algebra. The category ${\cal C}_{{\goth t}}$ is a
full subcategory of ${\cal C}'_{{\goth t}}$. For ${\goth a}$ in ${\cal C}'_{{\goth t}}$,
we consider the solvable Lie algebra ${\goth r} := {\goth t}+{\goth a}$ and $R$ the
adjoint group of ${\goth r}$. Denoting by $X_{R}$ the closure in
$\ec {Gr}r{}{}{\dim {\goth t}}$ of the orbit of ${\goth t}$ under $R$, we prove by
induction on $\dim {\goth a}$ the following theorem:
\begin{theo}\label{t2int}
The variety $X_{R}$ is normal and Cohen-Macaulay.
\end{theo}
The result for the category ${\cal C}'_{{\goth t}}$ is easily deduced from the
result for the category ${\cal C}_{{\goth t}}$ by Corollary~\ref{csa1}.
One of the key argument in the proof is the consideration of the fixed points under the
action of $R$ in $X_{R}$. As a matter of fact, since the closure of all orbit under $R$
in $X_{R}$ contains a fixed point, $X_{R}$ is Cohen-Macaulay if so are the fixed points
by openness of the set of Cohen-Macaulay points. Then, by Serre's normality criterion, it
suffices to prove that $X_{R}$ is smooth in codimension $1$. For that purpose the
consideration of the restriction to $X_{R}$ of the tautological vector bundle of rank
$\dim {\goth t}$ over $\ec {Gr}r{}{}{\dim {\goth t}}$ is very useful.
For the study of the fixed points, we introduce Property $({\bf P})$ and Property
$({\bf P}_{1})$ for the objects of ${\cal C}'_{{\goth t}}$:
\begin{itemize}
\item Property $({\bf P})$ for ${\goth a}$ in ${\cal C}'_{{\goth t}}$ says that for $V$
in $X_{R}$, contained in the centralizer ${\goth r}^{s}$ of an element $s$ of
${\goth t}$, $V$ is in the closure of the orbit of ${\goth t}$ under the centralizer
$R^{s}$ of $s$ in $R$,
\item Property $({\bf P}_{1})$ for ${\goth a}$ in ${\cal C}'_{{\goth t}}$ says that
for $V$ in $X_{R}$ normalized by ${\goth t}$ and such that $V\cap {\goth t}$ is the
center of ${\goth r}$, then the non zero weights of ${\goth t}$ in $V$ are linearly
independent.
\end{itemize}
Property $({\bf P}_{1})$ for ${\goth a}$ results from Property $({\bf P})$ for
${\goth a}$ and Property $({\bf P})$ for ${\goth a}$ results from Property $({\bf P}_{1})$
for ${\goth a}$ and Property $({\bf P})$ for the objects of ${\cal C}'_{{\goth t}}$ of
dimension smaller than $\dim {\goth a}$. So, the main result for the objects of
${\cal C}'_{{\goth t}}$ is the following proposition:
\begin{prop}\label{pint}
The objects of ${\cal C}'_{{\goth t}}$ have Property $({\bf P})$.
\end{prop}
From this proposition, we deduce some structure property for the points of $X_{R}$.
The second part of Theorem~\ref{tint}, that is Gorensteinness property and Rational
singularities, is obtained by considering a subcategory ${\cal C}_{{\goth t},*}$ of
${\cal C}_{{\goth t}}$. This category is defined by an additional condition on
the objects. The main point for ${\goth a}$ in ${\cal C}_{{\goth t},*}$ is the following
result:
\begin{prop}\label{p2int}
Let $k\geq 2$ be an integer. Denote by ${\cal E}^{(k)}$ the $R$-equivariant vector
subbundle of $X_{R}\times {\goth r}^{k}$ whose fiber at ${\goth t}$ is ${\goth t}^{k}$.
Then there exists on the smooth locus of ${\cal E}^{(k)}$ a regular differential form of
top degree without zero.
\end{prop}
From Proposition~\ref{p2int} and Theorem~\ref{t2int}, we deduce that ${\cal E}^{(k)}$
and $X_{R}$ are Gorenstein with rational singularities.
This note is organized as follows. In Section~\ref{sa}, categories ${\cal C}'_{{\goth t}}$
and ${\cal C}_{{\goth t}}$ are introduced for some space ${\goth t}$. In particular,
${\goth u}$ is an object of ${\cal C}_{{\goth h}}$. In Subsection~\ref{sa3}, we define
Property $({\bf P})$ for the objects of ${\cal C}'_{{\goth t}}$ and we deduce some
result on the structure of points of $X_{R}$. In Subsection~\ref{sa4}, we define
Property $({\bf P}_{1})$ for the objects of ${\cal C}'_{{\goth t}}$ and we prove
that Property $({\bf P})_{1}$ is a consequence of Property $({\bf P})$. In
Subsection~\ref{sa5}, we give some geometric constructions to prove
Property $({\bf P})$ by induction on the dimension of ${\goth a}$. At last, in
Subsection~\ref{sa6}, we prove Proposition~\ref{pint}. In particular, the proof of
\cite[Lemma 4.4,(i)]{CZ} is completed. In Section~\ref{sav}, we are interested
in the singular locus of $X_{R}$. In Subsection~\ref{sav3}, regularity in codimension $1$
is proved with some additional properties analogous to those of~\cite[Section 3]{CZ}.
Moreover, the constructions of Subsection~\ref{sa5} are used to prove the results by
induction on the dimension of ${\goth a}$. In Section~\ref{ns}, Cohen-Macaulayness
property is proved by induction. In Section~\ref{rss}, the category
${\cal C}_{{\goth t},*}$ is introduced and Proposition~\ref{p2int} is proved. Then with
some results given in the appendix, we finish the proof of Theorem~\ref{tint}.
\subsection{Notations} \label{int2}
$\bullet$ An algebraic variety is a reduced scheme over $\k$ of finite type. For $X$ an
algebraic variety, its smooth locus is denoted by $X_{\loc}$.
$\bullet$ Set $\k^{*} := \k \setminus \{0\}$. For $V$ a vector space, its dual is
denoted by $V^{*}$.
$\bullet$
All topological terms refer to the Zariski topology. If $Y$ is a subset of a topological
space $X$, denote by $\overline{Y}$ the closure of $Y$ in $X$. For $Y$ an open subset
of the algebraic variety $X$, $Y$ is called {\it a big open subset} if the codimension
of $X\setminus Y$ in $X$ is at least $2$. For $Y$ a closed subset of an algebraic
variety $X$, its dimension is the biggest dimension of its irreducible components and its
codimension in $X$ is the smallest codimension in $X$ of its irreducible components. For
$X$ an algebraic variety, $\an X{}$ is its structural sheaf, $\k[X]$ is the algebra of
regular functions on $X$, $\k(X)$ is the field of rational functions on $X$ when $X$
is irreducible and $\Omega _{X}$ is the sheaf of regular differential forms of top
degree on $X$ when $X$ is smooth and irreducible.
$\bullet$
If $E$ is a subset of a vector space $V$, denote by span($E$) the vector subspace of
$V$ generated by $E$. The grassmannian of all $d$-dimensional subspaces of $V$ is denoted
by Gr$_d(V)$.
$\bullet$ For ${\goth a}$ a Lie algebra,$V$ a subspace of ${\goth a}$ and $x$ in
${\goth a}$, $V^{x}$ denotes the centralizer of $x$ in $V$. For $A$ a subgroup of
the group of automorphisms of ${\goth a}$, $A^{x}$ denotes the centralizer of $x$ in
$A$. An element $x$ of ${\goth g}$ is regular if ${\goth g}^{x}$
has dimension $\rg$ and the set of regular elements of ${\goth g}$ is denoted by
${\goth g}_{\r}$.
$\bullet$ The Lie algebra of an algberaic torus is also called a torus. In this note,
a torus denoted by a gothic letter means the Lie algebra of an algebraic torus.
$\bullet$ For ${\goth a}$ a Lie algebra, the Lie algebra of derivations of
${\goth a}$ is denoted by ${\mathrm {Der}}({\goth a})$. By definition
${\mathrm {Der}({\goth a})}$ is the Lie algebra of the group ${\mathrm {Aut}({\goth a})}$
of the automorphisms of ${\goth a}$.
$\bullet$
Let ${\goth b}$ be a Borel subalgebra of ${\goth g}$, ${\goth h}$ a Cartan
subalgebra of ${\goth g}$ contained in ${\goth b}$ and ${\goth u}$ the nilpotent
radical of ${\goth b}$.
\section{On solvable algebras} \label{sa}
Let ${\goth t}$ be a vector space of positive dimension $d$. Denote by
$\tilde{{\cal C}}_{{\goth t}}$ the subcategory of the category of finite dimensional
Lie algebras whose objects are finite dimensional nilpotent Lie algebras
${\goth a}$ such that there exists a morphism
$$\xymatrix{ {\goth t} \ar[rr]^{\varphi _{{\goth a}}} && {\mathrm {Der}({\goth a})}}$$
whose image is the Lie algebra of a subtorus of ${\mathrm {Aut}}({\goth a})$. For
${\goth a}$ and ${\goth a}'$ in $\tilde{{\cal C}}_{{\goth t}}$, a
morphism $\psi $ from ${\goth a}$ to ${\goth a}'$ is a morphism of Lie algebras such
that $\psi \rond \varphi _{{\goth a}}(t)= \varphi _{{\goth a}'}(t)\rond \psi $ for all
$t$ in ${\goth t}$. For $x$ in ${\goth t}$, $x$ is a semisimple derivation of
${\goth a}$. Denote by ${\cal R}_{{\goth t},{\goth a}}$ the set of weights of ${\goth t}$
in ${\goth a}$. Let ${\cal C}'_{{\goth t}}$ be the full subcategory of objects
${\goth a}$ of $\tilde{{\cal C}}_{{\goth t}}$ verifying the following conditions:
\begin{itemize}
\item [{\rm (1)}] $0$ is not in ${\cal R}_{{\goth t},{\goth a}}$,
\item [{\rm (2)}] for $\alpha $ in ${\cal R}_{{\goth t},{\goth a}}$, the weight space of
weight $\alpha $ has dimension $1$,
\item [{\rm (3)}] for $\alpha $ in ${\cal R}_{{\goth t},{\goth a}}$,
$\k \alpha \cap ({\cal R}_{{\goth t},{\goth a}}\setminus \{\alpha \})$ is empty.
\end{itemize}
For ${\goth a}$ in ${\cal C}'_{{\goth t}}$ and ${\goth a}'$ a subalgebra of
${\goth a}$, invariant under the adjoint action of ${\goth t}$, ${\goth a}'$ is in
${\cal C}'_{{\goth t}}$. Denote by ${\cal C}_{{\goth t}}$ the full subcategory of objects
${\goth a}$ of ${\cal C}'_{{\goth t}}$ such that $\varphi _{{\goth a}}$ is an embedding.
For example ${\goth u}$ is in ${\cal C}_{{\goth h}}$.
For ${\goth a}$ in $\tilde{{\cal C}}_{{\goth t}}$, denote by
${\goth r}_{{\goth t},{\goth a}}$ the solvable algebra ${\goth t}+{\goth a}$,
$\pi _{{\goth t},{\goth a}}$ the quotient morphism from ${\goth r}_{{\goth t},{\goth a}}$
to ${\goth t}$, $R_{{\goth t},{\goth a}}$ the adjoint group of
${\goth r}_{{\goth t},{\goth a}}$, $A_{{\goth t},{\goth a}}$ the connected
closed subgroup of $R_{{\goth t},{\goth a}}$ whose Lie algebra is $\ad {\goth a}$,
$X_{R_{{\goth t},{\goth a}}}$ the closure in $\ec {Gr}r{}{{\goth t},{\goth a}}{d}$ of the
orbit of ${\goth t}$ under $R_{{\goth t},{\goth a}}$ and ${\cal E}_{{\goth t},{\goth a}}$
the restriction to $X_{R_{{\goth t},{\goth a}}}$ of the tautological vector bundle over
$\ec {Gr}r{}{{\goth t},{\goth a}}d$. The variety $X_{R_{{\goth t},{\goth a}}}$ is called
{\it the main variety related to ${\goth r}_{{\goth t},{\goth a}}$}. For $\alpha $ in
${\cal R}_{{\goth t},{\goth a}}$, let ${\goth a}^{\alpha }$ be the weight space of weight
$\alpha $ under the action of ${\goth t}$ in ${\goth a}$.
In the following subsections, a vector space ${\goth t}$ of positive dimension $d$ and an
object ${\goth a}$ of ${\cal C}'_{{\goth t}}$ are fixed. We set:
$$ {\cal R} := {\cal R}_{{\goth t},{\goth a}}, \qquad
{\goth r} := {\goth r}_{{\goth t},{\goth a}} \qquad \pi := \pi _{{\goth t},{\goth a}},
\qquad R := R_{{\goth t},{\goth a}}, \qquad A := A_{{\goth t},{\goth a}}, \qquad
n := \dim {\goth a}.$$
Let ${\goth z}$ be the orthogonal complement of ${\cal R}$ in ${\goth t}$ and $d^{\#}$
its codimension in ${\goth t}$. Then $n\geq d^{\#}$.
\subsection{General remarks on ${\cal C}'_{{\goth t}}$} \label{sa1}
For $x$ in ${\goth r}$, we say that $x$ is semisimple if so is $\ad x$ and $x$ is
nilpotent if so is $\ad x$. For ${\goth s}$ a commutative subalgebra
of ${\goth r}$, we say that ${\goth s}$ is a torus if $\ad {\goth s}$ is the Lie algebra
of a subtorus of ${\mathrm {GL}}({\goth r})$.
\begin{lemma}\label{lsa1}
Let $x$ be in ${\goth r}$ and ${\goth s}$ a commutative subalgebra of ${\goth r}$.
{\rm (i)} The center of ${\goth r}$ is equal to ${\goth z}$.
{\rm (ii)} The element $x$ is semisimple if and only if $R.x\cap {\goth t}$ is not empty.
{\rm (iii)} The element $x$ is nilpotent if and only if $x$ is in ${\goth z}+{\goth a}$.
{\rm (iv)} The algebra ${\goth a}$ is in ${\cal C}_{{\goth t}}$ if and only if
${\goth z}=\{0\}$. In this case, $x$ has a unique decomposition $x=x_{\s}+x_{\n}$ with
$[x_{\s},x_{\n}]=0$, $x_{\s}$ semisimple and $x_{\n}$ nilpotent.
{\rm (v)} The algebra ${\goth s}$ is a torus if and only if
${\goth s}\cap {\goth a} = \{0\}$ and $\pi ({\goth s})$ is a subtorus of ${\goth t}$. In
this case, ${\goth s}$ and $\pi ({\goth s})$ are conjugate under $R$.
\end{lemma}
\begin{proof}
By definition $\ad {\goth r}_{{\goth t},{\goth a}}$ is an algebraic solvable
subalgebra of ${\goth {gl}}({\goth r}_{{\goth t},{\goth a}})$ and $\ad {\goth t}$ is
a maximal subtorus of $\ad {\goth r}_{{\goth t},{\goth a}}$.
(i) Let ${\goth z}'$ be the center of ${\goth r}$. As $[{\goth t},{\goth z}']=\{0\}$,
$$ {\goth z}' = {\goth z}'\cap {\goth t} \oplus
\bigoplus _{\alpha \in {\cal R}} {\goth z}'\cap {\goth a}^{\alpha }. $$
So, by Condition (1), ${\goth z}'$ is contained in ${\goth t}$. For $t$ in ${\goth t}$,
$t$ is in ${\goth z}'$ if and only if $\alpha (t)=0$ for all $\alpha $ in
${\cal R}_{{\goth t},{\goth a}}$, whence ${\goth z}'={\goth z}$.
(ii) As the elements of ${\goth t}$ are semisimple by defintion, the condition is
sufficient since the set of semisimple elements of ${\goth r}$ is invariant under the
adjoint action of $R$. Suppose that $x$ is semisimple. By ~\cite[Ch. VII]{Hu}, for some
$g$ in $R$, $\Ad g(x)$ is in $\ad {\goth t}$, whence $g(x)$ is in ${\goth t}$ by (i).
(iii) As $\ad {\goth a}$ is the set of nilpotent elements of
$\ad {\goth r}$, $x$ is in ${\goth z}+{\goth a}$ if and only if it is nilpotent by (i).
(iv) By definition, ${\goth z}$ is the kernel of $\varphi _{{\goth a}}$. Hence
${\goth z}=\{0\}$ if and only if ${\goth a}$ is in ${\cal C}_{{\goth t}}$. As
$\ad {\goth r}$ is an algebraic subalgebra of ${\mathrm {gl}}({\goth r})$, it contains
the components of the Jordan decomposition of $\ad x$. As a result, when ${\goth a}$ is
in ${\cal C}_{{\goth t}}$, $x$ has a unique decomposition $x=x_{\s}+x_{\n}$ with
$[x_{\s},x_{\n}]=0$, $x_{\s}$ semisimple and $x_{\n}$ nilpotent.
(v) Suppose that ${\goth s}$ is a torus. By (i), ${\goth s}\cap {\goth a}=\{0\}$ and
by~\cite[Ch. VII]{Hu}, for some $g$ in $R$, $\ad g({\goth s})$
is contained in $\ad {\goth t}$ since $\ad {\goth t}$ is a maximal torus of
$\ad {\goth r}$. Then, by (i), $g({\goth s})$ is a subtorus of ${\goth t}$. Moreover,
$g({\goth s})=\pi ({\goth s})$ since $g(y)-y$ is in ${\goth a}$ for all $y$ in
${\goth r}$. Conversely, if ${\goth s}\cap {\goth a} = \{0\}$ and $\pi ({\goth s})$ is a
subtorus of ${\goth t}$, $\ad {\goth s}$ is conjugate to the subtorus
$\ad \pi ({\goth s})$ of $\ad {\goth t}$ by~\cite[Ch. VII]{Hu} so that ${\goth s}$ and
$\pi ({\goth s})$ are conjugate under $R$.
\end{proof}
Denoting by ${\goth t}^{\#}$ a complement to ${\goth z}$ in ${\goth t}$, ${\goth a}$ is
an object of ${\cal C}_{{\goth t}^{\#}}$ since
$\varphi _{{\goth a}}({\goth t})=\varphi _{{\goth a}}({\goth t}^{\#})$ and the
restriction of $\varphi _{{\goth a}}$ to ${\goth t}^{\#}$ is injective. Set
${\goth r}^{\#} := {\goth t}^{\#}+{\goth a}$ and denote by $R^{\#}$ the adjoint group
of ${\goth r}^{\#}$. Let $X_{R^{\#}}$ be the closure in
$\ec {Gr}{}{{{\goth r}^{\#}}}{}{{d^{\#}}}$ of the orbit of ${\goth t}^{\#}$ under
$R^{\#}$.
\begin{coro}\label{csa1}
All element of $X_{R}$ is a commutative algebra containing ${\goth z}$. Moreover,
the map
$$ \xymatrix{ X_{R^{\#}} \ar[rr] && X_{R}}, \qquad V \longmapsto V \oplus {\goth z}$$
is an isomorphism.
\end{coro}
\begin{proof}
As the set of commutative subalgebras of dimension $d$ of ${\goth r}$ is a closed subset
of $\ec {Gr}r{}{}d$ containing ${\goth t}$ and invariant under $R$, all element of
$X_{R}$ is a commutative algebra. According to Lemma~\ref{lsa1}(i), all element of
$R.{\goth t}$ contains ${\goth z}$ and so does all element of $X_{R}$. For $g$ in
$R$, denote by $\overline{g}$ the image of $g$ in $R^{\#}$ by the restriction morphism.
Then
$$g({\goth t}) = \overline{g}({\goth t}^{\#})+{\goth z} \quad \text{and} \quad
\overline{g}({\goth t}^{\#}) = g({\goth t}) \cap {\goth r}^{\#} .$$
Hence the map
$$ \xymatrix{ X_{R^{\#}} \ar[rr] && X_{R}}, \qquad V \longmapsto V \oplus {\goth z}$$
is an isomorphism whose inverse is the map $V\mapsto V\cap {\goth r}^{\#}$.
\end{proof}
For ${\goth a}$ of dimension $d^{\#}$,
${\cal R} := \{\poi {\beta }1{,\ldots,}{{d^{\#}}}{}{}{}\}$, and for $I$ subset of
$\{1,\ldots,d^{\#}\}$, denote $X_{R,I}$ the image of $\k^{I}$ by the map
$$ \xymatrix{\k^{I} \ar[rr] && X_{R}}, \qquad
(z_{i},\; i\in I) \longmapsto {\goth z}\oplus
{\mathrm {span}}(\{t_{i}+z_{i}x_{i}, \; i\in I\}) \oplus
\bigoplus _{i \not \in I} {\goth a}^{\beta _{i}}$$
with $x_{i}$ in ${\goth a}^{\beta _{i}}$ for $i=1,\ldots,d^{\#}$ and
$\poi t1{,\ldots,}{d^{\#}}{}{}{}$ in ${\goth t}$ such that
$\beta _{i}(t_{j})=\delta _{i,j}$ for $1\leq i,j\leq d^{\#}$, with $\delta _{i,j}$ the
Kronecker symbol.
\begin{lemma}\label{l2sa1}
Suppose that ${\goth a}$ has dimension $d^{\#}$. Denote by
$\poi {\beta }1{,\ldots,}{{d^{\#}}}{}{}{}$ the elements of ${\cal R}$.
{\rm (i)} The algebra ${\goth a}$ is commutative.
{\rm (ii)} The set $X_{R}$ is the union of $X_{R,I}, \; I \subset \{1,\ldots,d^{\#}\}$.
\end{lemma}
\begin{proof}
(i) As ${\goth z}$ has codimension $d^{\#}$ in ${\goth t}$,
$\poi {\beta }1{,\ldots,}{{d^{\#}}}{}{}{}$ are linearly independent. Hence for $i\neq j$,
$\beta _{i}+\beta _{j}$ is not in ${\cal R}$. As a result, ${\goth a}$ is commutative.
(ii) According to Corollary~\ref{csa1}, we can suppose $d=d^{\#}$ so that
$\poi t1{,\ldots,}{d}{}{}{}$ is the dual basis of $\poi {\beta }1{,\ldots,}{d}{}{}{}$.
For $I$ subset of $\{1,\ldots,d\}$, denote by $I'$ the complement to $I$ in
$\{1,\ldots,d\}$ and ${\goth z}_{I'}$ the orthogonal complement to
$\beta _{i},\; i\in I'$ in ${\goth t}$ and set:
$$ V_{I} := {\goth z}_{I'} \oplus \bigoplus _{i\in I'} {\goth a}^{\beta _{i}} .$$
By (i), for $i$ in $I$,
$$ \exp(z_{1}\ad x_{1} + \cdots + z_{d}\ad x_{d})(t_{i}) = t_{i}-z_{i}x_{i} .$$
Hence $X_{R,I}$ is the orbit of $V_{I}$ under $A$ and its closure in $X_{R}$ is
the union of $X_{R,J}, \; J \subset I$. As a result, $X_{R}$ is the
union of $X_{R,I},\; I\subset \{1,\ldots,d\}$ since $X_{R,\{1,\ldots,d\}}$ is the orbit
of ${\goth t}$ under $A$.
\end{proof}
\subsection{On some subsets of ${\cal R}$} \label{sa2}
For $\alpha $ in ${\cal R}$, let $x_{\alpha }$ be in ${\goth a}^{\alpha }\setminus \{0\}$.
For $\Lambda $ subset of ${\cal R}$, denote by ${\goth t}_{\Lambda }$ the intersection
of the kernels of its elements and set:
$$ {\goth a}_{\Lambda } := \bigoplus _{\alpha \in \Lambda } {\goth a}^{\alpha }
\quad \text{and} \quad {\goth r}_{\Lambda } := {\goth t}\oplus {\goth a}_{\Lambda } .$$
When $\Lambda $ has only one element $\alpha $, set
${\goth t}_{\alpha } := {\goth t}_{\Lambda }$.
\begin{defi}\label{dsa2}
Let $\Lambda $ be a subset of ${\cal R}$. We say that $\Lambda $ is a complete subset of
${\cal R}$ if it contains all element of ${\cal R}$ whose kernel contains
${\goth t}_{\Lambda }$
\end{defi}
For $\Lambda $ complete subset of ${\cal R}$, ${\goth a}_{\Lambda }$ is a subalgebra
of ${\goth a}$ and ${\goth r}_{\Lambda }$ is a subalgebra of ${\goth r}$. In particular,
${\goth a}_{\Lambda }$ is in ${\cal C}'_{{\goth t}}$. In this case, denote by
$R_{\Lambda }$ the connected closed subgroup of $R$ whose Lie algebra is
$\ad {\goth r}_{\Lambda }$.
\begin{lemma}\label{lsa2}
Let $\Lambda $ be a complete subset of ${\cal R}$, strictly contained in ${\cal R}$.
Then ${\goth a}_{\Lambda }$ is contained in an ideal ${\goth a}'$ of ${\goth r}$ of
dimension $\dim {\goth a}-1$ and contained in ${\goth a}$.
\end{lemma}
\begin{proof}
As $\Lambda $ is complete and strictly contained in ${\cal R}$, ${\goth a}_{\Lambda }$ is
a subalgebra of ${\goth r}$, strictly contained in ${\goth a}$. Then, by Lie's Theorem,
there is a sequence
$$ {\goth a}_{\Lambda }=\poi {{\goth a}}0{\subset \cdots \subset}{m}{}{}{}={\goth a}$$
of subalgebras of ${\goth r}$ such that ${\goth a}_{i}$ is an ideal of codimension $1$ of
${\goth a}_{i+1}$ for $i=0,\ldots,m-1$, whence the lemma.
\end{proof}
For $s$ in ${\goth t}$, denote by $\Lambda _{s}$ the subset of elements of ${\cal R}$
whose kernel contains $s$.
\begin{lemma}\label{l2sa2}
Let $s$ be in ${\goth t}$.
{\rm (i)} The centralizer ${\goth r}^{s}$ of $s$ in ${\goth r}$ is the direct sum of
${\goth t}$ and ${\goth a}_{\Lambda _{s}}$.
{\rm (ii)} The center of ${\goth r}^{s}$ is equal to ${\goth t}_{\Lambda _{s}}$.
\end{lemma}
\begin{proof}
By definition, $\Lambda _{s}$ is a complete subset of ${\cal R}$. Let $x$ be in
${\goth r}$. Then $x$ has a unique decomposition
$$ x = x_{0} + \sum_{\alpha \in {\cal R}} c_{\alpha }x_{\alpha }$$
with $x_{0}$ in ${\goth t}$ and $c_{\alpha },\alpha \in {\cal R}$ in $\k$.
(i) Since $s$ is in ${\goth t}$, $x$ is in ${\goth r}^{s}$ if and
only if $c_{\alpha }=0$ for $\alpha \in {\cal R}\setminus \Lambda _{s}$, whence the
assertion.
(ii) The algebra ${\goth a}_{\Lambda _{s}}$ is in ${\cal C}'_{{\goth t}}$ and
${\goth t}_{\Lambda _{s}}$ is the orthogonal complement to $\Lambda _{s}$ in ${\goth t}$.
So, by (i) and Lemma~\ref{lsa1}(i), ${\goth t}_{\Lambda _{s}}$ is the center of
${\goth r}^{s}$.
\end{proof}
\subsection{Property $({\bf P})$ for objects of ${\cal C}_{{\goth t}}$.} \label{sa3}
Let ${\bf T}$ be the connected closed subgroup of $R$ whose Lie algebra is
$\ad {\goth t}$. For $s$ in ${\goth t}$, denote by $X_{R}^{s}$ the subset of elements of
$X_{R}$ contained in ${\goth r}^{s}$ and $\overline{R^{s}.{\goth t}}$ the closure
in $\ec {Gr}r{}{}d$ of the orbit of ${\goth t}$ under $R^{s}$. Then
$\overline{R^{s}.{\goth t}}$ is contained in $X_{R}^{s}$.
\begin{defi}\label{dsa3}
Say that ${\goth a}$ has Property $({\bf P})$ if $X_{R}^{s}$ is equal to
$\overline{R^{s}.{\goth t}}$ for all $s$ in ${\goth t}$.
\end{defi}
By Corollary~\ref{csa1}, ${\goth a}$ has Property $({\bf P})$ if and only if the object
${\goth a}$ of ${\cal C}_{{\goth t}^{\#}}$ has Property $({\bf P})$.
\begin{lemma}\label{lsa3}
If ${\goth a}$ has dimension $d^{\#}$, then ${\goth a}$ has Property $({\bf P})$.
\end{lemma}
\begin{proof}
According to Corollary~\ref{csa1}, we can suppose $d=d^{\#}$. Denote by
$\poi {\beta }1{,\ldots,}{d}{}{}{}$ the elements of ${\cal R}$. Then
$\poi {\beta }1{,\ldots,}{d}{}{}{}$ is a basis of ${\goth t}^{*}$. Let
$\poi t1{,\ldots,}{d}{}{}{}$ be the dual basis, $s$ in ${\goth t}$ and $V$ in
$X_{R}^{s}$. By Lemma~\ref{l2sa1}(ii), for some subset $I$ of $\{1,\ldots,d\}$, $V$ is
in $X_{R,I}$. Then for some $(z_{i},\; i\in I)$,
$$ V = {\mathrm {span}}(\{t_{i}+z_{i}x_{i} \; i\in I\}) \oplus \bigoplus _{i\in I'}
{\goth a}^{\beta _{i}}$$
with $I'$ the complement to $I$ in $\{1,\ldots,d\}$ and $x_{i}$ in
${\goth a}^{\beta _{i}}$ for $i=1,\ldots,d$. Setting
$$ I'' := I' \cup \{i \in I \; \vert \; z_{i} \neq 0\},$$
for $i$ in $\{1,\ldots,d\}$, $i$ is in $I''$ if and only if $\beta _{i}(s)=0$. So,
by Lemma~\ref{lsa2}(i),
$$ {\goth r}^{s} = {\goth t} \oplus \bigoplus _{i\in I''} {\goth a}^{\beta _{i}} .$$
Then by Lemma~\ref{l2sa1}(ii), $V$ is in $\overline{R^{s}.{\goth t}}$.
\end{proof}
By definition, an {\it algebraic subalgebra} ${\goth k}$ of ${\goth r}$ is the
semi-direct product of a torus ${\goth s}$ contained in ${\goth k}$ and
${\goth k}\cap {\goth a}$.
\begin{lemma}\label{l2sa3}
Suppose that ${\goth a}$ has Property $({\bf P})$. Let $V$ be in $X_{R}$, $x$ in
$V$ and $y$ in ${\goth r}$ such that $\ad y$ is the semisimple component of $\ad x$.
Then the center of ${\goth r}^{y}$ is contained in $V$.
\end{lemma}
\begin{proof}
By Corollary~\ref{csa1}, we can suppose ${\goth a}$ in ${\cal C}_{{\goth t}}$ so that $y$
is the semisimple component of $x$ by Lemma~\ref{lsa1}(iv). By Lemma~\ref{lsa1}(ii), for
some $g$ in $R$, $g(y)$ is in ${\goth t}$. Denote by ${\goth z}_{g(y)}$ the center of
${\goth r}^{g(y)}$. By Lemma~\ref{l2sa2}(ii), ${\goth z}_{g(y)}$ is contained in
${\goth t}$. As $V$ is a commutative algebra, $g(V)$ is in $X_{R}^{g(y)}$. So, by
Property $({\bf P})$, ${\goth z}_{g(y)}$ is contained in $g(V)$ since ${\goth z}_{g(y)}$
is in $k({\goth t})$ for all $k$ in $R^{g(y)}$, whence the lemma.
\end{proof}
\begin{coro}\label{csa3}
Suppose that ${\goth a}$ has Property $({\bf P})$. Let $V$ be in $X_{R}$. Then $V$ is a
commutative algebraic subalgebra of ${\goth r}$ and for some subset $\Lambda $ of
${\cal R}$, the biggest torus contained in $V$ is conjugate to ${\goth t}_{\Lambda }$
under $R$.
\end{coro}
\begin{proof}
According to Corollary~\ref{csa1}, $V$ is a commutative subalgebra of ${\goth r}$ and we
can suppose $d=d^{\#}$. Let ${\goth s}$ be the set of semisimple elements of $V$. Then
${\goth s}$ is a subspace of $V$. By Lemma~\ref{l2sa3}, $V$ contains the semisimple
components of its elements so that $V$ is the direct sum of ${\goth s}$ and
$V\cap {\goth a}$. Let $s$ be in ${\goth s}$ such that the center of ${\goth r}^{s}$ has
maximal dimension. After conjugation by an element of $R$, we can suppose that $s$ is in
${\goth t}$. By Lemma~\ref{l2sa2}(ii), ${\goth t}_{\Lambda _{s}}$ is the center of
${\goth r}^{s}$. Hence, by Lemma~\ref{l2sa3}, ${\goth t}_{\Lambda _{s}}$ is contained in
${\goth s}$. Suppose that the inclusion is strict. A contradiction is expected. Let $s'$
be in ${\goth s}\setminus {\goth t}_{\Lambda _{s}}$. Since $V$ is
contained in ${\goth r}^{s}$, for some $g$ in $R^{s}$, $g(s')$ is in ${\goth t}$.
Moreover, $g({\goth s})$ is the set of semisimple elements of $g(V)$ and
${\goth t}_{\Lambda _{s}}$ is contained in $g({\goth s})$. Denoting by $\Lambda '$ the
set of elements of $\Lambda _{s}$ whose kernel contains $g(s')$, for some $z$ in
$\k^{*}$, $\Lambda '$ is the set of elements of ${\cal R}$ such that
$\alpha (s+zg(s'))=0$. By Lemma~\ref{l2sa3}, ${\goth t}_{\Lambda '}$ is contained in
$g(V)$. So, by minimality of $\vert \Lambda _{s}\vert$, $\Lambda '=\Lambda _{s}$ and
$g(s')$ is in ${\goth t}_{\Lambda _{s}}$, whence the contradiction since $g(s')$ is in
$g({\goth s})\setminus {\goth t}_{\Lambda _{s}}$. As a result,
${\goth t}_{\Lambda _{s}}={\goth s}$ and $V={\goth t}_{\Lambda _{s}}+V\cap {\goth a}$,
whence the corollary.
\end{proof}
\subsection{Fixed points in $X_{R}$ under ${\bf T}$ and $R$} \label{sa4}
For $V$ subspace of dimension $d$ of ${\goth r}$, denote by ${\cal R}_{V}$ the
set of elements $\beta $ of ${\cal R}$ such that ${\goth a}^{\beta }$
is contained in $V$, $r_{V}$ the rank of ${\cal R}_{V}$ and ${\goth z}_{V}$ its
orthogonal complement in ${\goth t}$ so that $\dim {\goth z}_{V}=d-r_{V}$.
As $\ec {Gr}r{}{}d$ and $X_{R}$ are projective varieties, the actions of ${\bf T}$ and
$R$ in these varieties have fixed points since ${\bf T}$ and $R$ are connected and
solvable.
\begin{defi}\label{dsa4}
We say that ${\goth a}$ has Property $({\bf P}_{1})$ if for $V$ fixed point under
${\bf T}$ in $X_{R}$ such that $V\cap {\goth t}={\goth z}$,
$r_{V}= \vert {\cal R}_{V}\vert$.
\end{defi}
\begin{lemma}\label{lsa4}
Suppose that ${\goth a}$ has Property $({\bf P})$. Let $V$ be in $\ec {Gr}r{}{}d$.
{\rm (i)} The element $V$ is a fixed point under ${\bf T}$ in $X_{R}$ if
and only if $V$ is a commutative subalgebra of ${\goth r}$ and
$$ V = {\goth z}_{V} \oplus \bigoplus _{\beta \in {\cal R}_{V}} {\goth a}^{\beta } .$$
In this case, $r_{V} = \vert{\cal R}_{V} \vert$.
{\rm (ii)} The element $V$ is a fixed point under $R$ in $X_{R}$ if and only
if $V$ is a commutative ideal of ${\goth r}$ and ${\goth z}$ is the orthogonal complement
of ${\cal R}_{V}$ in ${\goth t}$. In this case, $r_{V}=\vert {\cal R}_{V}\vert = d^{\#}$.
\end{lemma}
\begin{proof}
If $V$ is a fixed point under ${\bf T}$,
$$ V = V\cap {\goth t} \oplus \bigoplus _{\beta \in {\cal R}_{V}} {\goth a}^{\beta } .$$
(i) Suppose that $V$ is a fixed point under ${\bf T}$ in $X_{R}\setminus \{{\goth t}\}$.
Then ${\cal R}_{V}$ is not empty. Let $s$ be an element of ${\goth z}_{V}$ such that
$\beta (s)\neq 0$ if $\beta $ is not a linear combination of elements of ${\cal R}_{V}$.
Then $V$ is contained in ${\goth r}^{s}$. So, by Property $({\bf P})$, $V$ is in
$\overline{R^{s}.{\goth t}}$. By Lemmma~\ref{l2sa2}(i), ${\goth z}_{V}$ is the center of
${\goth r}^{s}$. Hence ${\goth z}_{V}$ is contained in $V$ and
${\goth z}_{V}=V\cap {\goth t}$ since $V\cap {\goth t}$ is contained in ${\goth z}_{V}$.
As a result, ${\goth z}_{V}$ has dimension $d-\vert {\cal R}_{V} \vert$ and
$r_{V}=\vert {\cal R}_{V} \vert$.
Conversely, suppose that $V$ is a commutative algebra and
$$ V = {\goth z}_{V} \oplus \bigoplus _{\beta \in {\cal R}_{V}} {\goth a}^{\beta } .$$
Set:
$$ {\goth a}_{V} := \bigoplus _{\beta \in {\cal R}_{V}} {\goth a}^{\beta }, \qquad
{\goth r}_{V} := {\goth t} \oplus {\goth a}_{V} .$$
Then ${\goth a}_{V}$ is a commutative Lie algebra and ${\goth a}_{V}$ is in
${\cal C}'_{{\goth t}}$. Moreover, ${\goth z}_{V}$ is the center of ${\goth r}_{V}$
by Lemma~\ref{lsa1}(i). By Lemma~\ref{l2sa1}(ii), $V$ is in the closure
of the orbit of ${\goth t}$ under the action of the adjoint group of ${\goth r}_{V}$
in $\ec {Gr}r{}Vd$, whence the assertion.
(ii) The element $V$ of $\ec {Gr}r{}{}d$ is a fixed point under $R$ if and only if
$V$ is an ideal of ${\goth r}$. So, by (i), the condition is sufficient. Suppose that $V$
is a fixed point under the action of $R$ in $X_{R}$. By (i),
$$ V = {\goth z}_{V} \oplus \bigoplus _{\beta \in {\cal R}_{V}} {\goth a}^{\beta } .$$
As $V$ is an ideal of ${\goth r}$, ${\goth z}_{V}$ is contained in the kernel of all
elements of ${\cal R}$ so that ${\goth z}_{V}={\goth z}$. In particular,
$\vert {\cal R}_{V} \vert = d^{\#}$ and the elements of ${\cal R}_{V}$ are linearly
independent.
\end{proof}
\subsection{On some varieties related to $X_{R}$} \label{sa5}
Let ${\goth a}'$ be an ideal of ${\goth r}$ of dimension $\dim {\goth a}-1$ and contained
in ${\goth a}$. As a subalgebra of ${\goth a}$ normalized by ${\goth t}$, ${\goth a}'$
is in ${\cal C}'_{{\goth t}}$. Denote by ${\goth r}'$ the subalgebra
${\goth t}+{\goth a}'$ of ${\goth r}$, $A'$ and $R'$ the connected closed subgroups of
$R$ whose Lie algebras are $\ad {\goth a}'$ and $\ad {\goth r'}$ respectively. Let
$X_{R'}$ be the closure in $\ec {Gr}r{}{}d$ of the orbit of ${\goth t}$ under $R'$ and
$\alpha $ the element of ${\cal R}$ such that
$$ {\goth a} = {\goth a}' \oplus {\goth a}^{\alpha } .$$
For $\delta $ in ${\cal R}$ denote again by $\delta $ the character of ${\bf T}$ whose
differential at the identity is $\ad x \longmapsto \delta (x)$.
Setting:
$$ {\goth G}_{d-1,d,d,d+1} :=
\ec {Gr}r{}{}{d-1}\times \ec {Gr}r{}{}d\times \ec {Gr}r{}{}d\times \ec {Gr}r{}{}{d+1}
\quad \text{and} \quad
{\goth G}_{d-1,d,d+1} :=
\ec {Gr}r{}{}{d-1}\times \ec {Gr}r{}{}d\times \ec {Gr}r{}{}{d+1},$$
denote by $\thetaup _{\alpha }$ and $\thetaup '_{\alpha }$ the maps
$$ \xymatrix{ \k\times A' \ar[rr]^{\thetaup _{\alpha }} && {\goth G}_{d-1,d,d,d+1}},
\quad
(z,g) \longmapsto (g.{\goth t}_{\alpha },g.{\goth t},g\exp(z \ad x_{\alpha }).{\goth t}
,g.({\goth t}+{\goth a}^{\alpha })) ,$$
$$ \xymatrix{ A' \ar[rr]^{\thetaup '_{\alpha }} && {\goth G}_{d-1,d,d+1}},
\quad
g \longmapsto (g.{\goth t}_{\alpha },g.{\goth t},g.({\goth t}+{\goth a}^{\alpha })) .$$
Let $I_{\alpha }$ and $S_{\alpha }$ be the closures in $\ec {Gr}r{}{}{d-1}$ and
$\ec {Gr}r{}{}{d+1}$ of the orbits of ${\goth t}_{\alpha }$ and
${\goth t}+{\goth a}^{\alpha }$ under $A'$ respectively.
\begin{lemma}\label{lsa5}
Let $\Gamma $ and $\Gamma '$ be the closures in ${\goth G}_{d-1,d,d,d+1}$ and
${\goth G}_{d-1,d,d+1}$ of the images of $\thetaup _{\alpha }$ and $\thetaup '_{\alpha }$.
{\rm (i)} The varieties $\Gamma $ and $\Gamma '$ have dimension $n$ and $n-1$
respectively. Moreover, they are invariant under the diagonal actions of $R'$ in
${\goth G}_{d-1,d,d,d+1}$ and ${\goth G}_{d-1,d,d+1}$.
{\rm (ii)} The image of $\Gamma $ by the first, second, third and fourth projections are
equal to $I_{\alpha }$, $X_{R'}$, $X_{R}$, $S_{\alpha }$ respectively.
{\rm (iii)} The set $\Gamma '$ is the image of $\Gamma $ by the projection
$$ \xymatrix{{\goth G}_{d-1,d,d,d+1} \ar[rr]^{\varpi } && {\goth G}_{d-1,d,d+1}}, \qquad
(V_{1},V',V,W) \longmapsto (V_{1},V',W) .$$
{\rm (iv)} For all $(V_{1},V',V,W)$ in $\Gamma $, $V_{1}$ is contained in $V'\cap V$
and $V'+V$ is contained in $W$.
{\rm (v)} Let $(V_{1},V',V,W)$ be in $\Gamma $ such that $V'$ is contained in
${\goth t}_{\alpha }+{\goth a}'$. Then $W$ is contained in
${\goth t}_{\alpha }+{\goth a}$.
{\rm (vi)} Let $(V_{1},V',V,W)$ be in $\Gamma $ such that $V'$ is not contained in
${\goth t}_{\alpha }+{\goth a}$. Then $W$ is not commutative.
\end{lemma}
\begin{proof}
(i) The maps $\thetaup _{\alpha }$ and $\thetaup '_{\alpha }$ are injective since
${\goth t}$ is the normalizer of ${\goth t}$ in ${\goth r}$ by Condition (1), whence
$\dim \Gamma = n$ and $\dim \Gamma ' = n-1$. For $(z,g,k)$ in $\k\times A'\times A'$,
$\thetaup _{\alpha }(z,kg) = k.\thetaup _{\alpha }(z,g)$ and
$\thetaup '_{\alpha }(kg) = k.\thetaup '_{\alpha }(g)$. Hence $\Gamma $ and $\Gamma '$
are invariant under the diagonal action of $A'$ in ${\goth G}_{d-1,d,d,d+1}$ and
${\goth G}_{d-1,d,d+1}$. Let $k$ be in ${\bf T}$. For all $(z,g)$ in $\k \times A'$,
$$ kg.{\goth t}_{\alpha } = kgk^{-1}({\goth t}_{\alpha }), \qquad
kg.{\goth t} = kgk^{-1}({\goth t}), $$ $$
kg.({\goth t}+{\goth a}^{\alpha }) = kgk^{-1}.({\goth t}+{\goth a}^{\alpha }), \qquad
kg\exp (z\ad x_{\alpha }).{\goth t} =
kgk^{-1}\exp (z\alpha (k)\ad x_{\alpha }).{\goth t}$$
so that the images of $\thetaup _{\alpha }$ and $\thetaup '_{\alpha }$ are invariant
under ${\bf T}$, whence the assertion.
(ii) Since $\ec {Gr}r{}{}d$, $\ec {Gr}r{}{}{d-1}$, $\ec {Gr}r{}{}{d+1}$ are projective
varieties, the images of $\Gamma $ by the first, second, third and
fourth projections are closed subsets of their target varieties. Since the image of
$\thetaup _{\alpha }$ is contained in the closed subset
$I_{\alpha }\times X_{R'}\times X_{R}\times S_{\alpha }$ of ${\goth G}_{d-1,d,d,d+1}$,
they are contained in $I_{\alpha }$, $X_{R'}$, $X_{R}$ and $S_{\alpha }$ respectively. By
definition, $R'.{\goth t}_{\alpha }$, $R'.{\goth t}$ and
$R'.({\goth t}+{\goth a}^{\alpha })$ are contained in the images of $\Gamma $ by the
first, second and fourth projections and $R.{\goth t}$ is contained in the image of
$\Gamma $ by the third projection since $A$ is the image of $\k\times A'$ by the map
$(z,g) \mapsto g\exp(z\ad x_{\alpha })$, whence the assertion.
(iii) As $\ec {Gr}r{}{}d$ is a projective variety, $\varpi (\Gamma )$ is a closed
subset of ${\goth G}_{d-1,d,d+1}$ containing the image of $\thetaup '_{\alpha }$ since
$\varpi \rond \thetaup _{\alpha }(z,g)=\thetaup '_{\alpha }(g)$ for all $(z,g)$ in
$\k\times A'$. Moreover, $\Gamma $ is contained in $\varpi ^{-1}(\Gamma ')$, whence
$\Gamma '=\varpi (\Gamma )$.
(iv) The subset $\widetilde{\Gamma }$ of elements $(V_{1},V',V,W)$ of
${\goth G}_{d-1,d,d,d+1}$ such that $V_{1}$ is contained in $V'$ and $V$ and such that
$V'$ and $V$ are contained in $W$, is closed. For all $(z,g)$ in $\k\times A'$,
$$ g\exp (z\ad x_{\alpha }).({\goth t}+{\goth a}^{\alpha }) =
g.({\goth t}+{\goth a}^{\alpha }) .$$
Hence the image of $\thetaup _{\alpha }$ and $\Gamma $ are contained in
$\widetilde{\Gamma }$ so that $V_{1}$ and $V+V'$ are contained in $V'\cap V$ and $W$
respectively for all $(V_{1},V',V,W)$ in $\Gamma $.
(v) Denote by $\Gamma _{*}$ the closure in $\ec {Gr}r{}{}d\times \ec {Gr}r{}{}{d+1}$
of the image of the map
$$ \xymatrix{ A' \ar[rr]^{\thetaup _{\alpha ,*}} &&
\ec {Gr}r{}{}d \times \ec {Gr}r{}{}{d+1}},
\qquad g \longmapsto (g({\goth t}),g({\goth t}+{\goth a}^{\alpha })) .$$
For all $(T_{1},T',T,T_{2})$ in the image of $\thetaup _{\alpha }$, $(T',T_{2})$ is in
the image of $\thetaup _{\alpha ,*}$. Then $\Gamma _{*}$ is the image of $\Gamma $
by the projection
$$ \xymatrix{ {\goth G}_{d-1,d,d,d+1} \ar[rr] &&
\ec {Gr}r{}{}d \times \ec {Gr}r{}{}{d+1}}, \qquad
(T_{1},T',T,T_{2}) \longmapsto (T',T_{2}) .$$
Denote by $\tau $ the quotient morphism
$$ \xymatrix{ {\goth r} \ar[rr]^{\tau } && {\goth r}/{\goth a}' =
{\goth t}+{\goth a}^{\alpha }}.$$
For $g$ in $A'$ and $x$ in ${\goth r}$, $\tau \rond g(x) = \tau (x)$. Set:
$$ X := \{(g,t,z,z',v,w) \in A'\times {\goth t}_{\alpha }\times \k ^{2}\times
{\goth r}'\times {\goth r} ; \vert \; v = g(zs + t), \; w = g(zs+t+z'x_{\alpha }) \}$$
and denote by $Y$ the closure in ${\goth r}'\times {\goth r}$ of the image of $X$ by the
canonical projection
$$ \xymatrix{ A'\times {\goth t}_{\alpha }\times \k ^{2}\times {\goth r}'\times {\goth r}
\ar[rr] && {\goth r}'\times {\goth r} } .$$
As for all $(g,t,z,z',v,w)$ in $X$,
$$ \tau (v) = zs + t \quad \text{and} \quad \tau (w) = zs + t + z'x_{\alpha },$$
$$\alpha \rond \pi \rond \tau (v) = \alpha \rond \pi \rond \tau (w)$$
for all $(v,w)$ in $Y$. By definition, for all $(T,T')$ in $\Gamma _{*}$,
$T\times T'$ is contained in $Y$. By hypothesis, $V'$ is contained in the kernel of
$\alpha \rond \pi $ and $(V',W)$ is in $\Gamma _{*}$. Hence $W$ is contained in
the kernel of $\alpha \rond \pi $.
(vi) Denote by $\Gamma '_{*}$ the closure in
${\goth G}_{d-1,d,d,d+1}\times \ec {Gr}r{}{}1$ of the image of the map
$$ \xymatrix{ \k \times A' \ar[rr]^{\thetaup '_{\alpha ,*}} &&
{\goth G}_{d-1,d,d,d+1}\times \ec {Gr}r{}{}1},
\qquad (z,g) \longmapsto (\thetaup _{\alpha }(z,g),g({\goth a}^{\alpha })) $$
and $\Gamma '_{**}$ the closure in $\ec {Gr}{}{{\goth r}'}{}d\times \ec {Gr}r{}{}1$ of
the image of the map
$$ \xymatrix{ A' \ar[rr] && \ec {Gr}{}{{\goth r}'}{}d \times \ec {Gr}r{}{}1},
\qquad g \longmapsto (g({\goth t}),g({\goth a}^{\alpha }) .$$
For all $(T_{1},T',T,T_{2},T'_{2})$ in the image of $\thetaup '_{\alpha ,*}$, $T'+T'_{2}$
is contained in $T_{2}$. Then so is it for all $(T_{1},T',T,T_{2},T'_{2})$ in
$\Gamma '_{*}$. As ${\goth G}_{d-1,d,d,d+1}$ and $\ec {Gr}r{}{}1$ are projective
varieties, $\Gamma $ and $\Gamma '_{**}$ are the images of $\Gamma '_{*}$ by the
projections
$$ \xymatrix{ {\goth G}_{d-1,d,d,d+1}\times \ec {Gr}r{}{}1 \ar[rr] &&
{\goth G}_{d-1,d,d,d+1}}, \qquad
(T_{1},T',T,T_{2},T'_{2}) \longmapsto (T_{1},T',T,T_{2}) ,$$
$$ \xymatrix{ {\goth G}_{d-1,d,d,d+1}\times \ec {Gr}r{}{}1 \ar[rr] &&
\ec {Gr}{}{{\goth r}'}{}d\times \ec {Gr}r{}{}1}, \qquad
(T_{1},T',T,T_{2},T'_{2}) \longmapsto (T',T'_{2}) $$
respectively.
Set:
$$ X' := \{(g,t,z,v,w) \in
A'\times {\goth t}\times \k \times {\goth r}'\times {\goth r} \; \vert \;
v = g(t), \; w = g(z x_{\alpha }) \} $$
and denote by $Y'$ the closure in ${\goth r}'\times {\goth r}$ of the image of $X'$ by
the canonical projection
$$ \xymatrix{A'\times {\goth t}\times \k \times {\goth r}'\times {\goth r} \ar[rr] &&
{\goth r}'\times {\goth r}}.$$
As for all $(g,t,z,v,w)$ in $X'$,
$$ [v,w] = g([t,zx_{\alpha }]) = \alpha (t) g(zx_{\alpha }) =
\alpha \rond \pi (v) w ,$$
$[v,w] = \alpha \rond \pi (v) w$ for all $(v,w)$ in $Y'$. By definition, for all
$(T,T')$ in $\Gamma '_{**}$, $T\times T'$ is contained in $Y'$. For some $W'$ in
$\ec {Gr}r{}{}1$, $(V_{1},V',V,W,W')$ is in $\Gamma '_{*}$. By hypothesis, $V'$ is not
contained in the kernel of $\alpha \rond \pi $. Hence, for some $v$ in $V'$ and
$w$ in $W\setminus \{0\}$, $\alpha \rond \pi (v)\neq 0$ and
$[v,w]=\alpha \rond \pi (v)w$.
\end{proof}
\begin{coro}\label{csa5}
Suppose that ${\goth a}'$ has Property $({\bf P})$. Let $s$ be in ${\goth t}$ such that
${\goth r}^{s}$ is contained in ${\goth a}'$ and $(V_{1},V',V,W)$ be in $\Gamma $ such
that $V$ is contained in ${\goth r}^{s}$ and $[s,V']$ is contained in $V'$.
{\rm (i)} If $W$ is not commutative then $V'=V$ and $V$ is in
$\overline{R^{s}.{\goth t}}$.
{\rm (ii)} Suppose that for some $v$ in ${\goth a}$, $s+v$ is in $V$. Then
$V'=V$ and $V$ is in $\overline{R^{s}.{\goth t}}$.
\end{coro}
\begin{proof}
By Lemma~\ref{lsa5}(ii), $V$ and $V'$ are in $X_{R}$ and $X_{R'}$ respectively.
(i) If $V'=V$, $V$ is in $\overline{R^{s}.{\goth t}}$ by Property $({\bf P})$ for
${\goth a}'$. Suppose $V'\neq V$. A contradiction is expected. Then, by
Lemma~\ref{lsa5}(iv), for some $x$ and $y$ in $W$,
$$ V = V_{1}\oplus \k x, \quad V' = V_{1} \oplus \k y, \quad
W = V_{1} \oplus \k x \oplus \k y .$$
Moreover, as $V$ is contained in ${\goth r}^{s}$ and $[s,V']\subset V'$, $W$ is contained
in ${\goth r}'$ and we can choose $y$ so that $[s,y]\in \k y$. Since $V$ and $V'$ are
commutative subalgebras of ${\goth r}$, $[x,y]\neq 0$. We have two cases to consider:
\begin{itemize}
\item [{\rm (a,1)}] $V'$ is contained in ${\goth r}^{s}$,
\item [{\rm (a,2)}] $V'$ is not contained in ${\goth r}^{s}$.
\end{itemize}
(a,1) By Property $({\bf P})$ for ${\goth a}'$, $s$ is in $V'$. Hence $s=ty+v$ for some
in $(t,v)$ in $\k\times V_{1}$. As $V$ is a commutative subalgebra of ${\goth r}^{s}$,
containing $V_{1}$ and $x$,
$$ 0 = [x,s] = t[x,y] .$$
Then $s=v$ is in $V_{1}$, whence a contradiction since $\alpha (s) \neq 0$ and $V_{1}$ is
contained in ${\goth t}_{\alpha }+{\goth a}'$ by Lemma~\ref{lsa5}(ii).
(a,2) For some $a$ in $\k^{*}$, $[s,y]=ay$. Then $y$ is in ${\goth a}'$ and $V'$ is
contained in ${\goth t}_{\alpha }+{\goth a}'$ since so is $V_{1}$. As a result,
by Lemma~\ref{lsa5}(v), $V$ and $W$ are contained in ${\goth t}_{\alpha }+{\goth a}'$
since $V$ is contained in ${\goth a}'$. As $[s,[x,y]]=a[x,y]$, $[x,y]=by$ for some $b$ in
$\k^{*}$ since the eigenspace of eigenvalue $a$ of the restriction of $\ad s$ to $V'$ is
generated by $y$. Then $\ad x$ is not nilpotent. Let $x_{\s}$ be in ${\goth r}'$ such
that $\ad x_{\s}$ is the semisimple component of $\ad x$. Then $x_{\s}$ is in
${\goth t}_{\alpha }+{\goth a}'$, $[s,x_{s}]=0$ and $[x_{\s},V_{1}]=\{0\}$ since
$[s,x]=0$ and $[x,V_{1}]=\{0\}$. Moreover, $[ax_{\s}-bs,y]=0$. Then $ax_{\s}-bs$ is a
semisimple element of ${\goth r}'$ such that $[ax_{\s}-bs,V']=\{0\}$. As it is conjugate
under $R'$ to an element of ${\goth t}$ by Lemma~\ref{lsa1}(ii), by Property $({\bf P})$
for ${\goth a}'$, $ax_{\s}-bs$ is in $V'$, whence a contradiction since $V'$ is contained
in ${\goth t}_{\alpha }+{\goth a}'$ and $ax_{\s}-bs$ is not in
${\goth t}_{\alpha }+{\goth a}'$.
(ii) If $V=V'$, $V$ is in $\overline{R^{s}.{\goth t}}$ by Property $({\bf P})$ for
${\goth a}'$. Suppose $V\neq V'$. A contradiction is expected. As $V$ is contained in
${\goth r}^{s}$, $[s,v]=0$. Let $x$ and $y$ be as in (i). As $V_{1}$ is contained in
${\goth t}_{\alpha }+{\goth a}'$, $s+v$ is not in $V_{1}$ since $\alpha (s)\neq 0$. So we
can choose $s+v=x$. By (i), $W$ is commutative. Then $[s+v,y]=0$ and $[\ad s,\ad y]=0$
since $\ad s$ is the semisimple component of $\ad (s+v)$.
Hence, by Lemma~\ref{lsa1}(i), $[s,y]=0$ since $[s,y]$ is in ${\goth a}$. As a result,
$V'$ is contained in ${\goth r}^{s}$ since so is $V_{1}$. So, by Property $({\bf P})$ for
${\goth a}'$, $s$ is in $V'$ and $W$ is not commutative by Lemma~\ref{lsa5}(vi), whence
a contradiction.
\end{proof}
For $(T_{1},T',T_{2})$ in $\Gamma '$, denote by $\Gamma _{T_{1},T',T_{2}}$ the subset of
elements $(T_{1},T',T,T_{2})$ of ${\goth G}_{d-1,d,d,d+1}$ such that $T$ is contained in
$T_{2}$ and contains $T_{1}$. Then $\Gamma _{T_{1},T',T_{2}}$ is a closed subvariety of
${\goth G}_{d-1,d,d,d+1}$, isomorphic to ${\Bbb P}^{1}(\k)$. Let $(V_{1},V',V,W)$ be a
fixed point under ${\bf T}$ of $\Gamma $.
\begin{lemma}\label{l2sa5}
{\rm (i)} For some affine open neighborhood $\Omega $ of $(V_{1},V',W)$ in $\Gamma '$,
$\Omega $ is invariant under ${\bf T}$.
{\rm (ii)} For $i=0,\ldots,n-2$, there exist $Y_{i}$ and $O_{i}$ such that
\begin{itemize}
\item [{\rm (a)}] $Y_{i}$ is an irreducible closed subset of dimension $n-1-i$ of
$\Omega $, containing $(V_{1},V',W)$ and invariant under ${\bf T}$,
\item [{\rm (b)}] $O_{i}$ is a locally closed subvariety of dimension $n-1-i$ of $A'$,
invariant under ${\bf T}$ by conjugation,
\item [{\rm (c)}] $\thetaup '_{\alpha }(O_{i})$ is contained in $Y_{i}$ and
$(V_{1},V',V,W)$ is in the closure of $\thetaup _{\alpha }(\k\times O_{i})$ in $\Gamma $.
\end{itemize}
{\rm (iii)} There exist a smooth projective curve $C$, an action of ${\bf T}$ on
$C$, $\poi x1{,\ldots,}{m}{}{}{}$ in $C$ and two morphisms
$$ \xymatrix{ C\setminus \{\poi x1{,\ldots,}{m}{}{}{}\} \ar[rr]^{\mu } && A'},
\qquad \xymatrix{ C \ar[rr]^{\nu } && \Gamma '}$$
satisfying the following conditions:
\begin{itemize}
\item [{\rm (a)}] $\poi x1{,\ldots,}{m}{}{}{}$ are the fixed points under ${\bf T}$ in
$C$,
\item [{\rm (b)}] for $g$ in ${\bf T}$ and $x$ in
$C\setminus \{\poi x1{,\ldots,}{m}{}{}{}\}$, $\mu (g.x) = g\mu (x)g^{-1}$ and
$\nu (g.x)=g.\nu (x)$,
\item [{\rm (c)}] $\nu (x_{1})=(V_{1},V',W)$,
\item [{\rm (d)}] $(V_{1},V',V,W)$ is in the closure of the image of
$\k\times (C\setminus \{\poi x1{,\ldots,}{m}{}{}{}\})$ by the map
$(z,x) \longmapsto \thetaup _{\alpha }(z,\mu (x))$.
\end{itemize}
\end{lemma}
\begin{proof}
(i) As $\Gamma '$ is a projective variety with a ${\bf T}$ action and $(V_{1},V',W)$
is a fixed point under ${\bf T}$, there exists an affine open neighborhood $\Omega $
of $(V_{1},V',W)$ in $\Gamma '$, invariant under ${\bf T}$.
(ii) Prove the assertion by induction on $i$. For $i=0$, $Y_{i}=\Omega $ and
$O_{i}$ is the inverse image of $\Omega $ by $\thetaup '_{\alpha }$. Suppose that
$Y_{i}$ and $O_{i}$ are known. Let $Y'_{i}$ be the closure in $\Gamma $ of
$\thetaup _{\alpha }(\k\times O_{i})$. By Condition (c), $Y'_{i}$ is invariant under
${\bf T}$ and $\thetaup _{\alpha }(\k\times O_{i})$ is a ${\bf T}$-invariant dense
subset of $Y'_{i}$. So, it contains an ${\bf T}$-invariant dense open subset $O'_{i}$ of
$Y'_{i}$. As $\thetaup '_{\alpha }$ is an orbital injective morphism,
$\thetaup '_{\alpha }(O_{i})$ is a dense open subset of $Y_{i}$. Set:
$$Z':=Y'_{i}\setminus O'_{i}, \quad Z := Y_{i}\setminus \theta _{\alpha }(O'_{i}),
\quad Z_{*} := \Omega \cap (\varpi (Z)\cup Z') .$$
Then $Z_{*}$ is a ${\bf T}$-invariant closed subset of $Y_{i}$, containing
$(V_{1},V',W)$.
Denote by $Z_{**}$ the union of irreducible components of dimension
$\dim Y_{i}-1$ of $Z_{*}$ and $I$ the union of the ideals of definition in $\k[Y_{i}]$ of
the irreducible components of $Z_{**}$. Let $p$ be in $\k[Y_{i}]\setminus I$,
semiinvariant under ${\bf T}$ and such that $p((V_{1},V',W))=0$. Denote by $Y'_{i+1}$
an irreducible component of the nullvariety of $p$ in $Y'_{i}\cap \varpi ^{-1}(\Omega )$,
containing $(V_{1},V',V,W)$ and $Y_{i+1}$ the closure in $\Omega $ of
$\varpi (Y'_{i+1})$. Then $Y_{i+1}$ has dimension $n-i-1$ and its intersection with
$\thetaup '_{\alpha }(O_{i})$ is not empty so that
$O_{i+1}:={\thetaup '_{\alpha }}^{-1}(O_{i}\cap\thetaup '_{\alpha }(O_{i}))$
is a nonempty locally closed subset of dimension $n-i-1$ of $A'$. Moreover,
$Y_{i+1}$ and $O_{i+1}$ are invariant under ${\bf T}$ since $p$ is semiinvariant
under ${\bf T}$. As $\thetaup _{\alpha }(\k\times O_{i+1})$ is the intersection of
$Y'_{i+1}$ and $\thetaup _{\alpha }(\k\times O_{i})$, it is dense in $Y'_{i+1}$ so that
$(V_{1},V',V,W)$ is in the closure of $\thetaup _{\alpha }(\k\times O_{i+1})$ and
$(V_{1},V',W)$ is in $Y_{i+1}$.
(iii) Let $\overline{Y_{n-2}}$ be the closure of $Y_{n-2}$, $C$ its normalization and
$\nu $ the normalization morphism. Then $C$ is a smooth projective curve. As $Y_{n-2}$ is
invariant under ${\bf T}$, so is $\overline{Y_{n-2}}$ and there is an action of
${\bf T}$ on $C$ such that $\nu $ is an equivariant morphism. As the restriction of
$\thetaup '_{\alpha }$ to $O_{n-2}$ is an isomorphism onto a dense open subset of
$Y_{n-2}$, the actions of ${\bf T}$ on $\overline{Y_{n-2}}$ and $C$ are not trivial
since $\thetaup '_{\alpha }$ is equivariant under the actions of ${\bf T}$. As
a result, ${\bf T}$ has an open orbit $O_{*}$ in $\overline{Y_{n-2}}$ and
$\overline{Y_{n-2}}\setminus O_{*}$ is the set of fixed points under ${\bf T}$ of
$\overline{Y_{n-2}}$ since $\overline{Y_{n-2}}$ has dimension $1$. Hence the
restriction of $\nu $ to $\nu ^{-1}(O_{*})$ is an isomorphism,
$C\setminus \nu ^{-1}(O_{*})$ is finite, its elements are fixed under ${\bf T}$ and
there exists a ${\bf T}$-equivariant morphism $\mu $ from $\nu ^{-1}(O_{*})$ to
$A'$ such that $\thetaup '_{\alpha }\rond \mu = \nu $. As $(V_{1},V',W)$ is a fixed point
under ${\bf T}$, for some $x_{1}$ in $C\setminus \nu ^{-1}(O_{*})$,
$\nu (x_{1})=(V_{1},V',W)$ since $(V_{1},V',W)$ is in $\nu (C)$. Moreover,
$(V_{1},V',V,W)$ is in the closure of the map
$$ \xymatrix{ \k \times (C\setminus \nu ^{-1}(O_{*})) \ar[rr] && \Gamma },
\qquad (z,x) \longmapsto \thetaup _{\alpha }(z,\mu (x))$$
since it is in $\overline{\thetaup _{\alpha }(\k\times O_{n-2})}$.
\end{proof}
Denote by $\eta $ the morphism
$$ \xymatrix{ \k \times (C\setminus \{\poi x1{,\ldots,}{m}{}{}{}\}) \ar[rr]^{\eta } &&
\Gamma }, \qquad (z,x) \longmapsto \thetaup _{\alpha }(z,\mu (x)) $$
and $\Delta $ the closure of the graph of $\eta $ in
${\Bbb P}^{1}(\k)\times C\times \Gamma $. Let $\upsilon $ be the restriction to
$\Delta $ of the canonical projection
$$ \xymatrix{ {\Bbb P}^{1}(\k)\times C\times \Gamma \ar[rr] && {\Bbb P}^{1}(\k)\times C}$$
and for $(z,x)$ in ${\Bbb P}^{1}(\k)\times C$, let $F_{z,x}$ be the subset of $\Gamma $
such that $\{(z,x)\}\times F_{z,x}$ is the fiber of $\upsilon $ at $(z,x)$. We have
an action of ${\bf T}$ in ${\Bbb P}^{1}(\k)$ given by
$$ t.z := \left \{ \begin{array}{ccc} \alpha (t)z & \mbox{ if } & z \in \k^{*} \\
z & \mbox{ if } & z \in \{0,\infty \} \end{array} \right. .$$
\begin{lemma}\label{l3sa5}
Let $\Delta _{\nu }$ be the graph of $\nu $.
{\rm (i)} The set $\Delta _{\nu }$ is the image of $\Delta $ by the map
$(z,x,y)\mapsto (x,\varpi (y))$.
{\rm (ii)} For $t$ in ${\bf T}$ and $(z,x,y)$ in $\Delta $, $t.(z,x,y) := (t.z,t.x,t.y)$
is in $\Delta $.
{\rm (iii)} For $(z,x)$ in ${\Bbb P}^{1}(\k)\times C$, $\eta $ is regular at $(z,x)$
if and only if $F_{z,x}$ has dimension $0$. In this case, $\vert F_{z,x} \vert=1$.
{\rm (iv)} For $(z,x)$ in
${\Bbb P}^{1}(\k)\times C\setminus \{0,\infty \}\times \{\poi x1{,\ldots,}{m}{}{}{}\}$,
$\eta $ is regular at $(z,x)$.
{\rm (v)} For $i=1,\ldots,m$, there exists a regular map $\eta _{i}$ from
${\Bbb P}^{1}(\k)$ to $\Gamma $ such that $\eta _{i}(z)=\eta (z,x_{i})$ for
all $z$ in $\k^{*}$. Moreover, its image is contained in
$\varpi ^{-1}(\{\nu (x_{i})\})\cap \Gamma $.
\end{lemma}
\begin{proof}
(i) As ${\Bbb P}^{1}(\k)$ and $\ec {Gr}r{}{}d$ are projective varieties, the image of
$\Delta $ by the map $(z,x,y) \mapsto (x,\varpi (y))$ is a closed subset of
$C\times \Gamma '$ contained in $\Delta _{\nu }$ since
$\varpi \rond \eta (z,x)=\nu (x)$ for all $(z,x)$ in
$\k\times (C\setminus \{\poi x1{,\ldots,}{m}{}{}{}\})$, whence the assertion since
the inverse image of $\Delta _{\nu }$ by this map is a closed subset of
${\Bbb P}^{1}(\k)\times C\times \Gamma $ containing the graph of $\eta $.
(ii) From the equality
$$ t \exp (z\ad x_{\alpha }) t^{-1} = \exp (\alpha (t)z\ad x_{\alpha })$$
for all $(t,z)$ in ${\bf T}\times \k$, we deduce the equality
$$ t.\eta (z,x) = t.\thetaup _{\alpha }(z,\mu (x))=
\thetaup _{\alpha }(\alpha (t)z,\mu (t.x) = \eta (t.z,t.x)$$
for all $(t,z,x)$ in ${\bf T}\times \k \times (C\setminus \{\poi x1{,\ldots,}{m}{}{}{}\})$
since $\thetaup _{\alpha }$ and $\mu $ are ${\bf T}$-equivariant, whence the assertion.
(iii) As $\Gamma $ is a projective variety, $\upsilon $ is a projective morphism.
Moreover, it is birational since $\Delta $ is the closure of the graph of $\eta $. So, by
Zariski's Main Theorem \cite[Ch. III, Corollary 11.4]{Ha}, the fibers of $\upsilon $ are
connected of dimension $0$ or $1$ since ${\Bbb P}^{1}(\k)\times C$ is normal of
dimension $2$. Let $(z,x)$ be in ${\Bbb P}^{1}(\k)\times C$ such that $F_{z,x}$ dimension
$0$. There exists a neighborhood $O_{z,x}$ of $(z,x)$ in ${\Bbb P}^{1}(\k)\times C$ such
that $F_{y}$ has dimension $0$ for $y$ in $O_{z,x}$. In other words, the restriction of
$\upsilon $ to $\upsilon ^{-1}(O_{z,x})$ is a quasi finite morphism. Moreover, it is
birational and surjective. So, again by Zariski's Main Theorem~\cite[\S 9]{Mu}, it is an
isomorphism. Hence $\eta $ is regular at $(z,x)$. Conversely, if $\eta $ is regular at
$(z,x)$, $\eta (z,x)$ is an isolated point in $F_{z,x}$, whence $F_{z,x}=\{\eta (z,x)\}$
since $F_{z,x}$ is connected.
(iv) The variety $\k\times (C\setminus \{\poi x1{,\ldots,}{m}{}{}{}\})$ is an open
subset of the smooth variety ${\Bbb P}^{1}(\k)\times C$ and $\Gamma $
is a projective variety. Hence $\eta $ has a regular extension to a big open subset of
${\Bbb P}^{1}(\k)\times C$ by~\cite[Ch. 6, Theorem 6.1]{Sh}. By Condition (a) of
Lemma~\ref{l2sa5}(iii), $\{0,\infty \}\times \{\poi x1{,\ldots,}{m}{}{}{}\}$ is the set
of fixed points under ${\bf T}$ in ${\Bbb P}^{1}(\k)\times C$ and by (ii),
$t.\eta (z,x)=\eta (t.z,t.x)$ for all $(t,z,x)$ in
${\bf T}\times {\Bbb P}^{1}(\k)\times (C\setminus \{\poi x1{,\ldots,}{m}{}{}{}\})$. Hence
$\eta $ is regular on
$P^{1}(\k)\times C\setminus \{0,\infty \}\times \{\poi x1{,\ldots,}{m}{}{}{}\}$.
(v) The restriction of $\eta $ to $\k^{*}\times \{x_{i}\}$ is a regular
map from a dense open subset of the smooth variety ${\Bbb P}^{1}(\k)\times \{x_{i}\}$
to the projective variety $\Gamma $. So, again by~\cite[Ch. 6, Theorem 6.1]{Sh},
this map has regular extension to ${\Bbb P}^{1}(\k)\times \{x_{i}\}$, whence the
assertion by (i).
\end{proof}
Let $I$ be the set of indices such that $\nu (x_{i})=(V_{1},V',W)$. Denote by $S$ the
image of $\Delta $ by the canonical projection
$\xymatrix{{\Bbb P}^{1}(\k)\times C\times \Gamma \ar[r] & \Gamma }$,
$S_{\n}$ its normalization, $\sigma $ the normalization morphism, $S^{{\bf T}}$ and
$S_{\n}^{{\bf T}}$ the sets of fixed points under ${\bf T}$ in $S$ and $S_{\n}$
respectively. Set
$$ C_{*} := {\Bbb P}^{1}(\k)\times C\setminus
\{(0,\infty \}\times \{\poi x1{,\ldots,}{m}{}{}{}\} .$$
By Lemma~\ref{l2sa5}(iv), $\eta $ is a dominant morphism from $C_{*}$ to $S$, whence
a commutative diagram
$$ \xymatrix{ C_{*} \ar[rr]^{\eta _{\n}} \ar[rrd]_{\eta } && S_{\n} \ar[d]^{\sigma }\\
&& S}$$
since $C_{*}$ is smooth. Let $\Delta _{\n}$ be the closure in
${\Bbb P}^{1}(\k)\times C\times S_{\n}$ of the graph of $\eta _{\n}$ and $\upsilon _{2}$
the restriction to $\Delta _{\n}$ of the canonical projection
$$\xymatrix{ {\Bbb P}^{1}(\k)\times C\times S_{\n} \ar[rr] && S_{\n}}.$$
\begin{lemma}\label{l4sa5}
Suppose $V'\neq V$ and $V$ and $V'$ contained in ${\goth z}+{\goth a}$.
{\rm (i)} The variety $\Delta $ is the image of $\Delta _{\n}$ by the map
$(z,x,y)\mapsto (z,x,\sigma (y))$.
{\rm (ii)} The morphism $\upsilon _{2}$ is projective and birational.
{\rm (iii)} There exists a ${\bf T}$-equivariant morphism
$$ \xymatrix{ (S_{\n}\setminus S_{\n}^{{\bf T}}) \ar[rr]^{\varphi } && C_{*}}$$
such that $\eta \rond \varphi $ is the restriction of $\sigma $ to
$S_{\n}\setminus S_{\n}^{{\bf T}}$.
{\rm (iv)} For some $i$ in $I$, $\eta _{i}(1)$ is not invariant under ${\bf T}$.
\end{lemma}
\begin{proof}
(i) As $S$ is a projective variety, so are $S_{\n}$,
${\Bbb P}^{1}(\k)\times C\times S_{\n}$, $\Delta _{\n}$ and the image of
$\Delta _{\n}$ by the map $(z,x,y)\mapsto (z,x,\sigma (y))$, whence the assertion since
the image of the graph of $\eta _{\n}$ by this map is the graph of $\eta $.
(ii) As $\Delta _{\n}$ is projective so is $\upsilon _{2}$. Since $\thetaup _{\alpha }$
is injective, so is the restriction of $\eta $ to
$\k \times (C\setminus \{\poi x1{,\ldots,}{m}{}{}{}\})$. Hence $\upsilon _{2}$ is
birational.
(iii) By (ii) and Zariski's Main Theorem~\cite[Ch. III, Corollary 11.4]{Ha}, the fibers
of $\upsilon _{2}$ are connected. For $y$ in $S_{\n}\setminus S_{\n}^{{\bf T}}$ and
$(z,x)$ in ${\Bbb P}^{1}(\k)\times C$ such that $(z,x,y)$ is in $\Delta _{\n}$,
$\varpi \rond \sigma (y)=\nu (x)$ by (i). If $x$ is not in
$\{\poi x1{,\ldots,}{m}{}{}{}\}$, $\nu ^{-1}(\varpi \rond \sigma (y))=\{x\}$ by Condition
(b) of Lemma~\ref{l2sa5}(iii) and $z$ is the element of $\k$ such that
$\thetaup _{\alpha }(z,\mu (x))=\sigma (y)$. Suppose $x=x_{i}$ for some $i=1,\ldots,m$.
Let $z$ and $z'$ be in $\k^{*}$ such that $(z,x_{i},y)$ and $(z',x_{i},y)$ are in
$\Delta _{\n}$. Then $(z,x_{i},\sigma (y))$ and $(z',x_{i},\sigma (y))$ are in $\Delta $.
By Lemma~\ref{l3sa5},(iii) and (iv), $\sigma (y)=\eta (z,x_{i})=\eta (z',x_{i})$. For
some $t$ in ${\bf T}$, $z'=t.z$ so that $t.\sigma (y)=\sigma (y)$. As $y$ is not
invariant under ${\bf T}$ so is $\sigma (y)$ since the fibers of $\sigma $ are finite.
Hence the stabilizer of $\sigma (y)$ in ${\bf T}$ is finite and so is the fiber of
$\upsilon _{2}$ at $y$. As a result, the restriction of $\upsilon _{2}$ to
$\Delta _{\n}\setminus {\Bbb P}^{1}(\k)\times C\times S_{\n}^{{\bf T}}$ is an injective
morphism. So, again by Zariski's Main Theorem~\cite[\S 9]{Mu}, this map
is an isomorphism, whence a morphism
$$ \xymatrix{ (S_{\n}\setminus S_{\n}^{{\bf T}}) \ar[rr]^{\varphi } && C_{*}}.$$
Moreover, $\varphi $ is ${\bf T}$-equivariant since so is $\upsilon _{2}$. For
$y$ in $S_{\n}$ such that $\sigma (y)=\eta (z,x)$ for some $(z,x)$ in
$\k^{*}\times (C\setminus \{\poi x1{,\ldots,}{m}{}{}{}\})$, $(z,x,y)$ is the unique
element of $\Delta _{\n}$ above $y$. Hence $\eta \rond \varphi =\sigma $.
(iv) Suppose that for all $i$ in $I$, $\eta _{i}(1)$ is invariant under ${\bf T}$. A
contradiction is expected. As $V\neq V'$, $V_{1}=V\cap V'$ and $V+V'=W$ by
Lemma~\ref{lsa5}(iv). Moreover, since $V$ and $V'$ are contained in
${\goth z}+{\goth a}$, for some $\beta $ and $\gamma $ in ${\cal R}$,
$$ V = V_{1}+{\goth a}^{\beta } \quad \text{and} \quad V' = V_{1}+{\goth a}^{\gamma }.$$
Then $\Gamma _{V_{1},V',W}$ is invariant under ${\bf T}$. More precisely,
$\Gamma _{V_{1},V',W}$ is a union of one orbit of dimension $1$ and the set
$\{(V_{1},V',V',W),(V_{1},V',V,W)\}$ of fixed points. As a result,
$\Gamma _{V_{1},V',W}\cap S$ is equal to $\{(V_{1},V',V',W),(V_{1},V',V,W)\}$ or
$\Gamma _{V_{1},V',W}$ since $S$ is invariant under ${\bf T}$. By Lemma~\ref{l3sa5},(ii)
and (v), for $i$ in $I$, $\eta _{i}({\Bbb P}^{1}(\k))$ is equal to $(V_{1},V',V',W)$ or
$(V_{1},V',V,W)$ since $\nu (x_{i})=(V_{1},V',W)$.
Suppose $\Gamma _{V_{1},V',W}\cap S=\{(V_{1},V',V',W),(V_{1},V',V,W)\}$. By
Lemma~\ref{l3sa5},(v) and (iii), for all $i$ in $I$, $\eta $ is regular at $(0,x_{i})$
and $(\infty ,x_{i})$ since $\nu (x_{i})=(V_{1},V',W)$, whence
$$ \lim _{z\rightarrow 0} \eta _{i}(0) = (V_{1},V',V',W) \quad \text{and} \quad
\lim _{z\rightarrow \infty } \eta _{i}(\infty ) = (V_{1},V',V,W) ,$$
whence a contradiction.
Suppose $\Gamma _{V_{1},V',W}\cap S = \Gamma _{V_{1},V',W}$. Let $y$ be in $S_{\n}$
such that
$$\sigma (y) \in \Gamma _{V_{1},V',W}\setminus \{(V_{1},V',V',W),(V_{1},V',V,W)\}.$$
By (iii), for some $i$ in $I$ and some $z$ in $\k^{*}$, $\varphi (t.y)=(t.z,x_{i})$ and
$t.\sigma (y)=t.\eta (z,x_{i})=t.\eta _{i}(z)$ for all $t$ in ${\bf T}$, whence a
contradiction since $(V_{1},V',V',W)$ and $(V_{1},V',V,W)$ are in
$\overline{{\bf T}.\sigma (y)}$.
\end{proof}
\begin{coro}\label{c2sa5}
Let $(V_{1},V',V,W)$ be a fixed point under ${\bf T}$ of $\Gamma $ such that
$V\cap {\goth t}=V'\cap {\goth t}={\goth z}$. Then $V'=V$.
\end{coro}
\begin{proof}
Suppose $V'\neq V$. A contradiction is expected. By Lemma~\ref{lsa5}(iv),
$V_{1}=V\cap V'$ and $W=V+V'$. As $V\cap {\goth t}=V'\cap {\goth t}={\goth z}$, $V$ and
$V'$ are contained in ${\goth z}+{\goth a}$. So, for some $\beta $ in ${\cal R}$ and
$\gamma $ in
${\cal R}\setminus \{\alpha \}$,
$$ V = V_{1} \oplus {\goth a}^{\beta } \quad \text{and} \quad
V' = V_{1}\oplus {\goth a}^{\gamma }.$$
since $(V_{1},V',V,W)$ is invariant under ${\bf T}$. By Lemma~\ref{l4sa5}(iv), for
some $i$ in $I$, $\eta _{i}(1)$ is not fixed under ${\bf T}$. Then, by
Lemma~\ref{lsa5}(ii), $\eta _{i}({\Bbb P}^{1}(\k))=\Gamma _{V_{1},V',W}$. Denoting by
$\eta _{i}(z)_{3}$ the third component of $\eta _{i}(z)$, for all $z$ in
${\Bbb P}^{1}(\k)$, $V_{1}$ is contained in $\eta _{i}(z)_{3}$ and $\eta _{i}(z)_{3}$ is
contained in $W$. Hence for some $a$ in $\k^{*}$,
$$ \eta _{i}(1)_{3} = V_{1} \oplus \k (x_{\beta }+ax_{\gamma }) \quad \text{and} \quad
\eta _{i}(\alpha (t))_{3} = V_{1} \oplus \k (\beta (t)x_{\beta }+\gamma (t)ax_{\gamma })$$
for all $t$ in ${\bf T}$. For some $t_{1}$ and $t_{2}$ in ${\bf T}$, for all $\delta $
in ${\cal R}$, $\delta (t_{1})$ and $\delta (t_{2})$ are positive rational numbers and
$$ \alpha (t_{1}) > 1, \qquad \alpha (t_{2}) > 1, \qquad \beta (t_{1}) < \gamma (t_{1}),
\qquad \beta (t_{2}) > \gamma (t_{2}) .$$
Then
$$ \lim _{k \rightarrow \infty } V_{1} \oplus
\k (\beta (t_{1}^{k})x_{\beta }+\gamma (t_{1}^{k})ax_{\gamma }) =
V_{1} \oplus {\goth a}^{\gamma }, \quad
\lim _{k \rightarrow \infty } V_{1} \oplus
\k (\beta (t_{2}^{k})x_{\beta }+\gamma (t_{2}^{k})ax_{\gamma }) =
V_{1} \oplus {\goth a}^{\beta }, $$ $$
\lim _{k \rightarrow \infty } \eta _{i}(\alpha (t_{1}^{k}) =
\lim _{k \rightarrow \infty } \eta _{i}(\alpha (t_{2}^{k}) = \eta _{i}(\infty ),$$
whence $V=V'$ and the contradiction.
\end{proof}
\subsection{Property $({\bf P})$ and Property $({\bf P}_{1})$} \label{sa6}
In this subsection we suppose that all objects of ${\cal C}'_{{\goth t}}$ of dimension
smaller than $n$ has Property $({\bf P})$. For $V$ a fixed point of $X_{R}$ under
${\bf T}$, denote by $\Lambda _{V}$ the orthogonal complement to ${\goth z}_{V}$ in
${\cal R}$ and set:
$${\goth r}_{V} := {\goth r}_{\Lambda _{V}}, \qquad R_{V} := R_{\Lambda _{V}} .$$
\begin{lemma}\label{lsa6}
Let $V$ be a fixed point under ${\bf T}$ in $X_{R}$.
{\rm (i)} The action of $R_{V}$ in $\overline{R_{V}.V}$ has fixed points. For
$V_{\infty }$ such a point,
$$ V_{\infty } = V\cap {\goth t} \oplus
\bigoplus _{\beta \in {\cal R}_{V_{\infty }}} {\goth a}^{\beta }, \quad
\vert {\cal R}_{V} \vert = \vert {\cal R}_{V_{\infty }} \vert, \quad
\rk V \geq \rk {V_{\infty }}.$$
{\rm (ii)} The set ${\cal R}_{V}$ has rank at least $\vert {\cal R}_{V} \vert-1$.
{\rm (iii)} Suppose that ${\goth a}$ has Property $({\bf P}_{1})$. Then ${\cal R}_{V}$ has
rank $\vert {\cal R}_{V} \vert$.
{\rm (iv)} If ${\goth a}$ has Property $({\bf P}_{1})$, for $s$ in ${\goth t}$ such that
$V$ is contained in ${\goth r}^{s}$, $V$ is in $\overline{R^{s}.{\goth t}}$.
\end{lemma}
\begin{proof}
(i) As $\overline{R_{V}.V}$ is a projective variety and $R_{V}$ is connected and
solvable, $R_{V}$ has fixed points in $\overline{R_{V}.V}$. Denote by $V_{\infty }$
such a point. Since $V$ is fixed under ${\bf T}$,
$$ V = V\cap {\goth t} \oplus \bigoplus _{\beta \in {\cal R}_{V}} {\goth a}^{\beta } .$$
Moreover, $V\cap {\goth t}$ is contained in ${\goth z}_{V}$ since $V$ is commutative.
By Lemma~\ref{l2sa2}(ii), ${\goth z}_{V}$ is the center of ${\goth r}_{V}$. Hence
$V\cap {\goth t}$ is contained in all element of $R_{V}.V$. Moreover, all element of
$R_{V}.V$ is contained in $V\cap {\goth t}+{\goth a}_{\Lambda _{V}}$. Then
$$ V_{\infty } = V\cap {\goth t} \oplus
\bigoplus _{\beta \in {\cal R}_{V_{\infty }}} {\goth a}^{\beta } ,$$
whence $\vert {\cal R}_{V} \vert = \vert {\cal R}_{V_{\infty }} \vert$. Since
${\cal R}_{V_{\infty }}$ is contained in $\Lambda _{V}$ and
$\rk V = d-\dim {\goth z}_{V}$, $\rk V \geq \rk {V_{\infty }}$.
(ii) By (i), we can suppose that $V$ is invariant under $R_{V}$. By Lemma~\ref{lsa2},
${\goth a}_{\Lambda _{V}}$ is contained in an ideal ${\goth a}'$ of
${\goth r}$ of dimension $\dim {\goth a}-1$ and contained in ${\goth a}$. We then use
the notations of Lemma~\ref{lsa5}. Set $\Gamma _{V} := \varpi _{3}^{-1}(V)$. By
Lemma~\ref{lsa5}(i), $\Gamma _{V}$ is a projective variety invariant under $R_{V}$ since
so is $V$. Then $R_{V}$ has a fixed point in $\Gamma _{V}$. Let $(V_{1},V',V,W)$ be such
a point. As ${\goth a}'$ has Property $({\bf P})$, by Lemma~\ref{lsa4}(i),
$$ V' = {\goth z}_{V'} \oplus \bigoplus _{\beta \in {\cal R}_{V'}} {\goth a}^{\beta } .$$
and the elements of ${\cal R}_{V'}$ are linearly independent.
If $V'=V$ then ${\cal R}_{V'}={\cal R}_{V}$ so that
$\rk V = \rk {V'}=\vert {\cal R}_{V} \vert$. Suppose $V'\neq V$. Then, by
Lemma~\ref{lsa5}(iv),
$$ V_{1} = {\goth z}_{V'}\cap V\cap {\goth t} \oplus
\bigoplus _{\beta \in {\cal R}_{V}\cap {\cal R}_{V'}} {\goth a}^{\beta } .$$
As $V_{1}$ has codimension $1$ in $V$ and $V'$, ${\cal R}_{V'}={\cal R}_{V}$ or
${\goth z}_{V'}=V\cap {\goth t}$. In the first case, $\rk V=\vert {\cal R}_{V} \vert$
and in the second case,
$$\vert {\cal R}_{V} \cap {\cal R}_{V'} \vert = \vert {\cal R}_{V} \vert -1
= \vert {\cal R}_{V'} \vert -1,$$
whence $\rk V \geq \vert {\cal R}_{V} \vert-1$ since the elements of ${\cal R}_{V'}$
are linearly independent.
(iii) Prove the assertion by induction on $\dim {\goth z}_{V}$. If
${\goth z}_{V}={\goth z}$, then $r_{V}=\vert {\cal R}_{V} \vert$ by Property
$({\bf P}_{1})$. Suppose $\dim {\goth z}_{V}=\dim {\goth z}+1$ and
$V\cap {\goth t}={\goth z}$. Then $\vert {\cal R}_{V} \vert=d^{\#}$ and
$r_{V}=d^{\#}-1$. By Property $({\bf P}_{1})$, it is impossible. Hence
$V\cap {\goth t}={\goth z}_{V}$ since $V\cap {\goth t}$ is contained in ${\goth z}_{V}$.
As a result $r_{V}=\vert {\cal R}_{V} \vert$.
Suppose $\dim {\goth z}_{V}\geq 2+\dim {\goth z}$, the assertion true for the integers
smaller than $\dim {\goth z}_{V}$ and $r_{V}<\vert {\cal R}_{V} \vert$. A contradiction
is expected. By (ii), $V\cap {\goth t}$ has dimension at least $\dim {\goth z}_{V}-1$.
Then, for some $\alpha $ in ${\cal R}$, $V\cap {\goth t}_{\alpha }$ is strictly contained
in $V\cap {\goth t}$. Let $\Lambda $ be the orthogonal complement to
${\goth z}_{V}\cap {\goth t}_{\alpha }$ in ${\cal R}$. As $\overline{R_{\Lambda }.V}$ is
a projective variety and $R_{\Lambda }$ is connected, $R_{\Lambda }$ has a fixed point in
$\overline{R_{\Lambda }.V}$. Let $V_{\infty }$ be such a point. By
Lemma~\ref{l2sa2}(ii), ${\goth z}_{V}\cap {\goth t}_{\alpha }$ is the center of
${\goth r}_{\Lambda }$. Hence $V\cap {\goth t}_{\alpha }$ is contained in all element of
$R_{\Lambda }.V$. Moreover, all element of $R_{\Lambda }.V$ is contained in
$V\cap {\goth t}+{\goth a}_{\Lambda }$. As $V_{\infty }$ is an ideal of
${\goth r}_{\Lambda }$, $V\cap {\goth t}$ is not contained in $V_{\infty }$ since it is
not contained in the kernel of $\alpha $. Then
$$ V_{\infty } = V\cap {\goth t}_{\alpha } \oplus
\bigoplus _{\beta \in {\cal R}_{V_{\infty }}} {\goth a}^{\beta } .$$
By (ii), $r_{V_{\infty }}\geq \vert {\cal R}_{V_{\infty }} \vert -1$, whence
$$ \dim {\goth z}_{V_{\infty }} \leq \dim V\cap {\goth t}_{\alpha } + 1 =
\dim V\cap {\goth t} < \dim {\goth z}_{V} .$$
So, by induction hypothesis, $\vert {\cal R}_{V_{\infty }} \vert = \rk {V_{\infty }}$ and
${\goth z}_{V_{\infty }}=V\cap {\goth t}_{\alpha }$. Since
${\goth z}_{V}\cap {\goth t}_{\alpha }$ is the center of ${\goth r}_{\Lambda }$,
${\goth z}_{V}\cap {\goth t}_{\alpha }$ is contained in ${\goth z}_{V_{\infty }}$,
whence
$$ \dim {\goth z}_{V}-1 \leq \dim V\cap {\goth t}_{\alpha } = \dim V\cap {\goth t} - 1 .$$
As a result, ${\goth z}_{V}=V\cap {\goth t}$ since $V\cap {\goth t}$ is contained in
${\goth z}_{V}$, whence a contradiction.
(iv) Suppose that ${\goth a}$ has Property $({\bf P}_{1})$. By (iii),
$$ V = {\goth z}_{V} \oplus \bigoplus _{\beta \in {\cal R}_{V}} {\goth a}^{\beta } $$
and $\rk V = \vert {\cal R}_{V} \vert$. As a result, the centralizer of $V$ in
${\goth t}$ is equal to ${\goth z}_{V}$. Set
$$ {\goth a}'_{V} = \bigoplus _{\beta \in {\cal R}_{V}} {\goth a}^{\beta }, \qquad
{\goth r}'_{V} := {\goth t} + {\goth a}'_{V} .$$
Denote by $R'_{V}$ the connected closed subgroup of $R$ whose Lie algebra is
$\ad {\goth r}'_{V}$. The algebra ${\goth a}'_{V}$ is in
${\cal C}'_{{\goth t}}$ and has dimension $d-\dim {\goth z}_{V}$. Then, by
Lemma~\ref{l2sa1}(ii), $V$ is in $\overline{R'_{V}.{\goth t}}$, whence the assertion
since ${\goth r}'_{V}$ is contained in ${\goth r}^{s}$.
\end{proof}
\begin{coro}\label{csa6}
Suppose that ${\goth a}$ has Property $({\bf P}_{1})$. Then ${\goth a}$ has Property
$({\bf P})$.
\end{coro}
\begin{proof}
Let $V$ be in $X_{R}$ and $s$ in ${\goth t}\setminus {\goth z}$ such that $V$ is
contained in ${\goth r}^{s}$. As $\overline{{\bf T}.V}$ is a projective variety and
${\bf T}$ is a connected commutative group, ${\bf T}$ has a fixed point in
$\overline{{\bf T}.V}$. Let $V_{\infty }$ be such a point. Since all element of
${\bf T}.V$ is contained in ${\goth r}^{s}$, so is $V_{\infty }$. Then, by
Lemma~\ref{lsa6}(iv), $V_{\infty }$ is in $\overline{R^{s}.{\goth t}}$. In particular,
$s$ is in $V_{\infty }$. Let $E$ a complement to $V_{\infty }$ in ${\goth r}$, invariant
under ${\bf T}$. The map
$$\xymatrix{ {\mathrm {Hom}}_{\k}(V_{\infty },E) \ar[rr]^{\kappa } && \ec {Gr}r{}{}d},
\qquad \varphi \longmapsto
\kappa (\varphi ) := {\mathrm {span}}(\{v+\varphi (v) \; \vert \; v \in V_{\infty }\}) $$
is an isomorphism onto an open neighborhood $\Omega _{E}$ of $V_{\infty }$ in
$\ec {Gr}r{}{}d$. For $\varphi $ in ${\mathrm {Hom}}_{\k}(V_{\infty },E)$ such that
$\kappa (\varphi )$ is in ${\bf T}.V$, $\varphi (s)$ is in ${\goth a}^{s}$. Then, for
some $g$ in ${\bf T}$ and for some $v$ in ${\goth a}^{s}$, $s+v$ is in $g(V)$ and the
semisimple component of $\ad (s+v)$ is different from $0$ since $s$ is not in
${\goth z}$. Let $x$ be in ${\goth r}^{s}$ such that $\ad x$ is the semisimple component
of $\ad (s+v)$. By Lemma~\ref{lsa1}(ii), for some $k$ in $R^{s}$, $k(x)$ is in
${\goth t}$. Then, by Corollary~\ref{csa5}(ii), $kg(V)$ is in
$\overline{R^{k(x)}.{\goth t}}$. As $k(x)$ is not in ${\goth z}$, ${\goth a}^{k(x)}$
is an object of ${\cal C}'_{{\goth t}}$ of dimension smaller than $n$. By hypothesis,
${\goth a}^{k(x)}$ has Property $({\bf P})$. Moreover, $kg(V)$ is contained in
${\goth r}^{s}\cap {\goth r}^{k(x)}$. Hence, by Property $({\bf P})$ for
${\goth a}^{k(x)}$, $kg(V)$ is in $\overline{R^{s}.{\goth t}}$, whence $V$ is in
$\overline{R^{s}.{\goth t}}$ since $kg$ is in $R^{s}$.
\end{proof}
\begin{prop}\label{psa6}
The objects of ${\cal C}'_{{\goth t}}$ have Property $({\bf P})$.
\end{prop}
\begin{proof}
Prove by induction on $n$ that ${\goth a}$ has Property $({\bf P})$. By
Lemma~\ref{lsa3}, it is true for $n=d^{\#}$. Suppose that it is true for the integers
smaller than $n$. By Corollary~\ref{csa6}, it remains to prove that ${\goth a}$ has
Property $({\bf P}_{1})$.
Suppose that ${\goth a}$ has not Property $({\bf P}_{1})$. A contradiction is expected.
For some fixed point $V$ under ${\bf T}$ in $X_{R}$ such that
$V\cap {\goth t}={\goth z}$, $r_{V}\neq \vert {\cal R}_{V} \vert$. By
Lemma~\ref{lsa6}(ii), $r_{V}=\vert {\cal R}_{V} \vert -1$. Then the orthogonal complement
of ${\cal R}_{V}$ in ${\goth t}$ is generated by ${\goth z}$ and an element $s$ in
${\goth t}\setminus {\goth z}$. In particular, $V$ is contained in ${\goth r}^{s}$.
According to Lemma~\ref{lsa2}, for some ideal ${\goth a}'$ of codimension $1$ of
${\goth a}$, normalized by ${\goth t}$, ${\goth a}^{s}$ is contained in ${\goth a}'$.
Denote by $\alpha $ the element of ${\cal R}$ such that
$$ {\goth a} = {\goth a}'\oplus {\goth a}^{\alpha } $$
and consider $\thetaup _{\alpha }$ and $\Gamma $ as in Subsection~\ref{sa5}. Denote
by $\Gamma _{V}$ the set of elements of $\Gamma $ whose image by the projection
$$ \xymatrix{ \Gamma \ar[rr] && \ec {Gr}r{}{}d}, \qquad (T_{1},T',T,T_{2}) \longmapsto T$$
is equal to $V$. By Lemma~\ref{lsa5}(ii), $\Gamma _{V}$ is not empty and it is invariant
under ${\bf T}$ by Lemma~\ref{lsa5}(i). As it is a projective variety, it has a fixed
point under ${\bf T}$. Denote by $(V_{1},V',V,W)$ such a point. As ${\goth a}'$ has
Property $({\bf P})$, it has Property $({\bf P}_{1})$ by Lemma~\ref{lsa4}. Hence
$r_{V'}=\vert {\cal R}_{V'} \vert$ and $V'\neq V$ since $r_{V}\neq {\cal R}_{V}$. Then,
by Lemma~\ref{lsa5}(iv),
$$ V_{1} = V\cap V' \quad \text{and} \quad W = V'+V .$$
As a result, $V'\cap {\goth t}=V\cap {\goth t}={\goth z}$ since
${\cal R}_{V'}\neq {\cal R}_{V}$ and $V_{1}$ has codimension $1$ in $V$ and $V'$.
Then $V'=V$ by Corollary~\ref{c2sa5}, whence a contradiction.
\end{proof}
The following corollary results from Proposition~\ref{psa6}, Corollary~\ref{csa3} and
Lemma~\ref{lsa4}.
\begin{coro}\label{c2sa6}
Let $V$ be in $X_{R}$.
{\rm (i)} The space $V$ is a commutative algebraic subalgebra of ${\goth r}$ and for some
subset $\Lambda $ of ${\cal R}$, the biggest torus contained in $V$ is conjugate to
${\goth t}_{\Lambda }$ under $R$.
{\rm (ii)} If $V$ is a fixed point under $R$, then $V$ is an ideal of ${\goth r}$ and the
elements of ${\cal R}_{V}$ are linearly independent.
\end{coro}
\section{Solvable algebras and main varieties} \label{sav}
Let ${\goth t}$ be a vector space of positive dimension $d$ and ${\goth a}$ in
${\cal C}_{{\goth t}}$. Set:
$$ {\cal R} := {\cal R}_{{\goth t},{\goth a}}, \qquad
{\goth r} := {\goth r}_{{\goth t},{\goth a}} \qquad \pi := \pi _{{\goth t},{\goth a}},
\qquad R := R_{{\goth t},{\goth a}}, \qquad A := A_{{\goth t},{\goth a}}, \qquad
{\cal E} := {\cal E}_{{\goth t},{\goth a}}, \qquad n := \dim {\goth a} .$$
In this section, we give some results on the singular locus of $X_{R}$.
For ${\goth a}'$ in ${\cal C}_{{\goth t}}$, denote by $X_{R_{{\goth t},{\goth a}'},\n}$
the subset of elements of $X_{R_{{\goth t},{\goth a}'}}$ contained in ${\goth a}'$.
\subsection{Subvarieties of $X_{R}$} \label{sav1}
Denote by ${\cal P}_{c}({\cal R})$ the set of complete subsets of ${\cal R}$ and
for $\Lambda $ in ${\cal P}_{c}({\cal R})$ denote by $X_{R_{\Lambda }}$ the closure in
$\ec {Gr}r{}{}d$ of the orbit $R_{\Lambda }.{\goth t}$.
\begin{prop}\label{psav1}
Let $Z$ be an irreducible closed subset of $X_{R}$, invariant under $R$.
{\rm (i)} For a well defined complete subset $\Lambda $ of ${\cal R}$, all element of
a dense open subset of $Z$ is conjugate under $R$ to the sum of ${\goth t}_{\Lambda }$
and a subspace of ${\goth a}$.
{\rm (ii)} All element of $Z$ is contained in ${\goth t}_{\Lambda }\oplus {\goth a}$.
{\rm (iii)} For some irreducible closed subset $Z_{\Lambda }$ of $X_{R_{\Lambda }}$,
invariant under $R_{\Lambda }$, $R.Z_{\Lambda }$ is dense in $Z$.
\end{prop}
\begin{proof}
(i) For $\Lambda $ in ${\cal P}_{c}({\cal R})$, let $Y_{\Lambda }$ be the subset
of elements $V$ of $Z$ such that $\pi (V)={\goth t}_{\Lambda }$. Since $Z$ is invariant
under $R$, so is $Y_{\Lambda }$. Moreover, by Corollary~\ref{c2sa6}(i),
$$\overline{Y_{\Lambda }} \subset Y_{\Lambda } \cup
\bigcup _{\mycom {\Lambda '\in {\cal P}_{c}({\cal R})}
{\Lambda ' \supsetneq \Lambda }} Y_{\Lambda '} . $$
According to Corollary~\ref{c2sa6}(i), $Z$ is the union of
$Y_{\Lambda },\Lambda \in {\cal P}_{c}({\cal R})$. As a result, since
${\cal R}$ is finite and $Z$ is irreducible, for a well defined complete subset
$\Lambda $ of ${\cal R}$, $Y_{\Lambda }$ contains a dense open subset of $Z$. By
Lemma~\ref{lsa1}(v), all element of $Y_{\Lambda }$ is conjugate under
$R$ to the sum of ${\goth t}_{\Lambda }$ and a subspace of ${\goth a}$.
(ii) By (i), for all $V$ in a dense subset of $Z$, $V$ is contained in
${\goth t}_{\Lambda }\oplus {\goth a}$, whence the assertion.
(iii) Let $Z_{*}$ be the subset of elements of $Z$, containing ${\goth t}_{\Lambda }$.
Denote by $s$ an element of ${\goth t}_{\Lambda }$ such that $\alpha (s)\neq 0$ for all
$\alpha $ in ${\cal R}\setminus \Lambda $. By Lemma~\ref{l2sa2}(i),
$${\goth r}^{s}={\goth t}\oplus {\goth a}_{\Lambda }.$$
Hence $Z_{*}$ is contained in $X_{R_{\Lambda }}$ by Proposition~\ref{psa6}. Moreover,
$Z_{*}$ is invariant under $R_{\Lambda }$ since $Z$ is invariant under $R$. By (i),
$R.Z_{*}$ is dense in $Z$. So, for some irreducible component $Z_{\Lambda }$ of
$Z_{*}$, $R.Z_{\Lambda }$ is dense in $Z$. Moreover, $Z_{\Lambda }$ is
invariant under $R_{\Lambda }$ since so is $Z_{*}$.
\end{proof}
For $\Lambda $ in ${\cal P}_{c}({\cal R})$, denote by ${\goth t}_{\Lambda }^{\#}$ a
complement to ${\goth t}_{\Lambda }$ in ${\goth t}$ and set:
$$ {\goth r}^{\#}_{\Lambda } := {\goth t}^{\#}_{\Lambda } + {\goth a}_{\Lambda } .$$
Let $R^{\#}_{\Lambda }$ be the adjoint group of ${\goth r}^{\#}_{\Lambda }$
and $A^{\#}_{\Lambda }$ the connected closed subgroup of $R^{\#}_{\Lambda }$ whose Lie
algebra is $\ad {\goth a}_{\Lambda }$.
\begin{lemma}\label{lsav1}
Let $\Lambda $ be in ${\cal P}_{c}({\cal R})$, nonempty and strictly contained in
${\cal R}$.
{\rm (i)} The tori ${\goth t}_{\Lambda }$ and ${\goth t}_{\Lambda }^{\#}$ have positive
dimension and ${\goth a}_{\Lambda }$ is in ${\cal C}_{{\goth t}_{\Lambda }^{\#}}$.
Moreover,
$$\dim {\goth a}_{\Lambda } - \dim {\goth t}_{\Lambda }^{\#} \leq
\dim {\goth a} - \dim {\goth t} .$$
{\rm (ii)} The map $V\mapsto V \oplus {\goth t}_{\Lambda }$ is an isomorphism from
$X_{R^{\#}_{\Lambda }}$ onto $X_{R_{\Lambda }}$.
\end{lemma}
\begin{proof}
Since $\Lambda $ is a complete subset of ${\cal R}$ strictly contained in ${\cal R}$,
${\goth t}_{\Lambda }$ has positive dimension and since $\Lambda $ is not empty,
${\goth t}_{\Lambda }$ is strictly contained in ${\goth t}$. By definition, $\Lambda $
is the set of weights of ${\goth t}$ in ${\goth a}_{\Lambda }$ so that
${\goth a}_{\Lambda }$ is in ${\cal C}'_{{\goth t}}$. Then ${\goth a}_{\Lambda }$
is in ${\cal C}_{{\goth t}^{\#}_{\Lambda }}$ and Assertion (ii) results from
Corollary~\ref{csa1}.
By Lemma~\ref{lsa1},(i) and (iv), ${\cal R}$ generates ${\goth t}^{*}$. Hence
$$ \vert \Lambda \vert + \dim {\goth t}_{\Lambda } \leq \vert {\cal R} \vert.$$
By Condition (2) of Section~\ref{sa}, ${\goth a}$ has dimension $\vert {\cal R} \vert$
and ${\goth a}_{\Lambda }$ has dimension $\vert \Lambda \vert$. As a result,
$$ \dim {\goth a} - \dim {\goth t} = \vert {\cal R} \vert - \dim {\goth t}_{\Lambda }
- \dim {\goth t}_{\Lambda }^{\#} \geq \dim {\goth a}_{\Lambda } -
\dim {\goth t}_{\Lambda }^{\#} .$$
\end{proof}
\subsection{Smooth points of $X_{R}$ and commutators} \label{sav2}
Denote by ${\goth t}_{\r}$ the complement in ${\goth t}$ to the union of
${\goth t}_{\alpha }, \alpha \in {\cal R}$ and ${\goth r}_{\r}$ the set of elements $x$
of ${\goth r}$ such that ${\goth r}^{x}$ has minimal dimension.
\begin{lemma}\label{lsav2}
{\rm (i)} The set ${\goth t}_{\r}$ is a dense open subset of ${\goth t}$, contained in
${\goth r}_{\r}$. Moreover, $R.{\goth t}_{\r}$ is a dense open subset of ${\goth r}$.
{\rm (ii)} For all $x$ in ${\goth r}_{\r}$, ${\goth r}^{x}$ is in $X_{R}$.
{\rm (iii)} The set ${\goth r}_{\r}$ is a big open subset of ${\goth r}$.
\end{lemma}
\begin{proof}
(i) By definition, ${\goth t}_{\r}$ is a dense open subset of ${\goth t}$. According to
Lemma~\ref{l2sa2}(i), for $x$ in ${\goth t}_{\r}$, ${\goth r}^{x}={\goth t}$. Then
$R.x = A.x=x+{\goth a}$ since $A.x$ is a closed subset of $x+{\goth a}$ of
dimension $\dim {\goth a}$. As a result, $R.{\goth t}_{\r}={\goth t}_{\r}+{\goth a}$ is a
dense open subset of ${\goth r}$. Hence $R.{\goth t}_{\r}$ is contained in
${\goth r}_{\r}$ since ${\goth r}^{x}$ is conjugate to ${\goth t}$ for all $x$ in
$R.{\goth t}_{\r}$ and ${\goth r}_{\r}$ is a dense open subset of ${\goth r}$.
(ii) By (i), for all $x$ in ${\goth r}_{\r}$, ${\goth r}^{x}$ has dimension $d$, whence a
regular map
$$ \xymatrix{ {\goth r}_{\r} \ar[rr]^{\theta } && \ec {Gr}r{}{}d}, \qquad
x \longmapsto {\goth r}^{x} .$$
As a result, by (i), for all $x$ in ${\goth r}_{\r}$, ${\goth r}^{x}$ is in $X_{R}$.
(iii) Suppose that ${\goth r}_{\r}$ is not a big open subset of ${\goth r}$. A
contradiction is expected. Let $\Sigma $ be an irreducible component of codimension $1$
of ${\goth r}\setminus {\goth r}_{\r}$. Since $\Sigma \cap A.{\goth t}_{\r}$ is empty,
$\pi (\Sigma )$ is contained in ${\goth t}_{\alpha }$ for some $\alpha $ in ${\goth r}$.
Then $\Sigma ={\goth t}_{\alpha }+{\goth a}$ since $\Sigma $ has codimension $1$ in
${\goth r}$. By Condition (3) of Section~\ref{sa}, for some $s$ in ${\goth t}_{\alpha }$,
$\gamma (s)\neq 0$ for all $\gamma $ in ${\cal R}\setminus \{\alpha \}$. Then
${\goth r}^{s+x_{\alpha }}={\goth t}_{\alpha }+{\goth a}^{\alpha }$ so that
$s+x_{\alpha }$ is in ${\goth r}_{\r}$ by (i) and Condition (2) of Section~\ref{sa},
whence the contradiction.
\end{proof}
Denote by $X'_{R}$ the image of $\theta $.
\begin{prop}\label{psav2}
{\rm (i)} The complement to $R.{\goth t}$ in $X_{R}$ is equidimensional of dimension
$\dim {\goth a}-1$.
{\rm (ii)} The set $X'_{R}$ is a smooth open subset of $X_{R}$, containing $R.{\goth t}$.
\end{prop}
\begin{proof}
(i) As $R$ is solvable and $R.{\goth t}$ is dense in $X_{R}$, $R.{\goth t}$ is an affine
open subset of $X_{R}$. So, by \cite[Corollaire 21.12.7]{Gro1},
$X_{R}\setminus R.{\goth t}$ is equidimensional of dimension $\dim {\goth a}-1$ since
$X_{R}$ has dimension $\dim {\goth a}$.
(ii) By definition, ${\cal E}$ is the subvariety of elements $(V,x)$ of
$X_{R}\times {\goth r}$ such that $x$ is in $V$. Let $\Gamma $ be the image of the
graph of $\theta $ by the isomorphism
$$ \xymatrix{{\goth r}\times \ec {Gr}r{}{}d \ar[rr] && \ec {Gr}r{}{}d\times {\goth r}},
\qquad (x,V) \longmapsto (V,x) .$$
Then $\Gamma $ is the intersection of ${\cal E}$ and $X_{R}\times {\goth r}_{\r}$. Since
$\Gamma $ is isomorphic to ${\goth r}_{\r}$, $\Gamma $ is a smooth open subset of
${\cal E}$ whose image by the bundle projection is $X'_{R}$. As a result, $X'_{R}$ is a
smooth open subset of $X_{R}$ by~\cite[Ch. 8, Theorem 23.7]{Mat}.
\end{proof}
For $\alpha $ in ${\cal R}$, set
$V_{\alpha } := {\goth t}_{\alpha }\oplus {\goth a}^{\alpha }$ and denote by
$\theta _{\alpha }$ the map
$$ \xymatrix{\k \ar[rr]^{\theta _{\alpha }} && \ec {Gr}r{}{}d}, \qquad
z \longmapsto \exp( z\ad x_{\alpha })({\goth t}) ,$$
By Condition (2) of Section~\ref{sa}, $V_{\alpha }$ has dimension $d$.
\begin{lemma}\label{l2sav2}
Let $\alpha $ be in ${\cal R}$. Set $X_{R,\alpha } := \overline{A.V_{\alpha }}$.
{\rm (i)} The map $\theta _{\alpha }$ has a regular extension to ${\Bbb P}^{1}(\k)$ such
that $\theta _{\alpha }(\infty )=V_{\alpha }$.
{\rm (ii)} The variety $X_{R,\alpha }$ has dimension
$\dim {\goth a}-1$ and it is an irreducible component of $X_{R}\setminus R.{\goth t}$.
{\rm (iii)} The intersection $X_{R,\alpha }\cap X'_{R}$ is not empty.
\end{lemma}
\begin{proof}
(i) Let $h_{\alpha }$ be in ${\goth t}$ such that $\alpha (h_{\alpha })=1$.
Since $X_{R}$ is a projective variety, the map $\theta _{\alpha }$ has a regular
extension to ${\Bbb P}^{1}(\k)$ by~\cite[Ch. 6, Theorem 6.1]{Sh}. For $z$ in $\k$,
$$ \theta _{\alpha }(z) = {\goth t}_{\alpha } \oplus \k (h_{\alpha }-zx_{\alpha }) .$$
Hence $\theta _{\alpha }(\infty )=V_{\alpha }$.
(ii) By (i), $X_{R,\alpha }$ is contained in $X_{R}$ and its elements are contained in
${\goth t}_{\alpha }\oplus {\goth a}$ so that $X_{R,\alpha }$ is contained in
$X_{R}\setminus R.{\goth t}$. By Condition (3) of Section~\ref{sa}, for $\gamma $ in
${\cal R}\setminus \{\alpha \}$ and $v$ in ${\goth a}^{\gamma }$,
$[{\goth t}_{\alpha },v]=\k v$ so that no element of ${\goth a}^{\gamma }$ normalizes
$V_{\alpha }$. As a result, the normalizer of $V_{\alpha }$ in ${\goth r}$ is equal to
${\goth t}+{\goth a}^{\alpha }$ so that $X_{R,\alpha }$ has dimension
$\dim {\goth a} -1$. Hence $X_{R,\alpha }$ is an irreducible component
$X_{R}\setminus R.{\goth t}$.
(iii) According to Condition (3) of Section~\ref{sa}, for some $s$ in
${\goth t}_{\alpha }$, $\gamma (s)\neq 0$ for all $\gamma $ in
${\cal R}\setminus \{\alpha \}$. Then $V_{\alpha } = {\goth r}^{s+x_{\alpha }}$ so that
$s+x_{\alpha }$ is in ${\goth r}_{\r}$, whence the assertion.
\end{proof}
\subsection{On the singular locus of $X_{R}$} \label{sav3}
In this subsection we suppose $\dim {\goth a} > d$ and we fix an ideal ${\goth a}'$ of
codimension $1$ in ${\goth a}$, normalized by ${\goth t}$ and such that ${\goth a}'$ is
in ${\cal C}_{{\goth t}}$. For example, all ideal of ${\goth r}$ of dimension
$\dim {\goth a}-1$, contained in ${\goth a}$ and containing a fixed point under $R$ in
$X_{R}$ is in ${\cal C}_{{\goth t}}$ by Corollary~\ref{c2sa6}(ii).
Set:
$$ {\goth r}' := {\goth r}_{{\goth t},{\goth a}'} \qquad
\pi ':= \pi _{{\goth t},{\goth a}'},
\qquad R' := R_{{\goth t},{\goth a}'}, \qquad A' := A_{{\goth t},{\goth a}'}, \qquad
{\cal R}' := {\cal R}_{{\goth t},{\goth a}'}.$$
Let $\alpha $ be in ${\cal R}$ such that
$$ {\goth a} = {\goth a}' \oplus {\goth a}^{\alpha } $$
and $\Gamma $ as in Subsection~\ref{sa5}. Denote by $\varpi _{1}$, $\varpi _{2}$,
$\varpi _{3}$, $\varpi _{4}$ the restrictions to $\Gamma $ of the first, second, third,
fourth projections. Let $Z$ be an irreducible component of $X_{R,\n}$. According to
Lemma~\ref{lsa5}(ii), for some irreducible component $T$ of $\varpi _{3}^{-1}(Z)$,
$\varpi _{3}(T)=Z$. Denote by $Z'$ the image of $T$ by $\varpi _{2}$ and by $T_{1}$ the
image of $T$ by the projection $\varpi _{1}\mul \varpi _{4}$. Then $Z'$ and $T_{1}$ are
irreducible closed subsets of $\ec {Gr}r{}{}d$ and
$\ec {Gr}r{}{}{d-1}\times \ec {Gr}r{}{}{d+1}$ respectively. Let $T_{0}$ be the subset of
elements $(V_{1},V',V,W)$ of $T$ such that $V'=V$. Then $T_{0}$ is a closed subset of
$T$. If $T_{0}=T$, $Z'=Z$ and $Z$ is contained in $X_{R',\n}$.
Otherwise, $O:=T\setminus T_{0}$ is a dense open subset of $T$. According to
Lemma~\ref{lsa5}(iv), for all $(V_{1},V',V,W)$ in $O$, $V_{1}=V'\cap V$ and $V'+V=W$.
Denote by $O_{1}$ an open subset of $T_{1}$, contained and dense in
$\varpi _{1}\mul \varpi _{4}(O)$.
Let $(V_{1},W)$ be in $O_{1}$. Denote by $E$ a complement to $V_{1}$ in ${\goth r}$ and
by $E'$ a complement to $W$ in ${\goth r}$ contained in $E$. Let $\kappa $ be the map
\begin{eqnarray*}
\xymatrix{ {\mathrm {Hom}}_{\k}(V_{1},W\cap E)\times {\mathrm {Hom}}_{\k}(W,E')
\ar[rr]^{\kappa } && \ec {Gr}r{}{}{d-1}\times \ec {Gr}r{}{}{d+1} }, \\
(\varphi ,\psi ) \longmapsto
({\mathrm {span}}(\{v+\varphi (v)+\psi (v)+\psi \rond \varphi (v)
\; \vert \; v \in V_{1}\}), {\mathrm {span}}(\{v+\psi (v) \; \vert \; v \in W \})) .
\end{eqnarray*}
Then $\kappa $ is an isomorphism from its source to an open neighborhood of $(V_{1},W)$
in the subvariety of elements $(W_{1},W_{2})$ of
$\ec {Gr}r{}{}{d-1}\times \ec {Gr}r{}{}{d+1}$ such that $W_{1}$ is contained in $W_{2}$.
Denote by $\Omega $ the inverse image by $\kappa $ of the intersection of $T_{1}$ and the
image of $\kappa $. Let $(e_{1},e_{2})$ be a basis of $W\cap E$ and let $\kappa _{*}$ be
the map
$$ \xymatrix{ \Omega \times (\k^{2}\setminus \{(0,0)\}) \ar[rr]^{\kappa _{*}} &&
\ec {Gr}r{}{}d}, $$
$$ (\varphi ,\psi ,x_{1},x_{2}) \longmapsto
{\mathrm {span}}(\{v+\varphi (v)+\psi (v)+\psi \rond \varphi (v) \; \vert \; v \in V_{1}\}
\cup \{x_{1}(e_{1}+\psi (e_{1}))+x_{2}(e_{2}+\psi (e_{2}))\}) .$$
\begin{lemma}\label{lsav3}
Suppose that $O$ is not empty. Denote by $\widetilde{\Omega }$ the image of $\kappa _{*}$
and $\tilde{Z}$ the closure of $\widetilde{\Omega }$ in $\ec {Gr}r{}{}d$.
{\rm (i)} The intersections $\widetilde{\Omega }\cap Z'$ and
$\widetilde{\Omega }\cap Z$ are dense in $Z'$ and $Z$ respectively. In particular $Z'$
and $Z$ are contained in $\tilde{Z}$.
{\rm (ii)} For $V$ in $\widetilde{\Omega }$, there exists $(V',V'')$ in
$Z'\times Z$ such that
$$ V'\cap V'' \subset V, \qquad V \subset V'+V'', \qquad
(V'\cap V'',V'+V'') \in \kappa (\Omega ) .$$
{\rm (iii)} Let $F'$ be the fiber of $\kappa _{*}$ at some element $V$ of
$\kappa _{*}(\Omega )$. Denote by $F$ the subset of elements $(\varphi ,\psi )$ of
$\Omega $ such that $V$ contains the first component of $\kappa (\varphi ,\psi )$ and is
contained in the second component of $\kappa (\varphi ,\psi )$. Then
$F' = F \times \k^{*}(x_{1},x_{2})$ for some $(x_{1},x_{2})$ in
$\k^{2}\setminus \{(0,0)\}$.
{\rm (iv)} The varieties $\tilde{Z}$ and $Z$ have dimension at most $\dim Z' + 1$.
\end{lemma}
\begin{proof}
(i) Since $T$ is irreducible so are $T_{1}$ and $\Omega $. Hence $\tilde{Z}$ is
irreducible. For some $(V',V)$ in $Z'\times Z$, $V_{1}$ is contained in $V'$ and $V$ and
$V'$ and $V$ are contained in $W$. Since $\kappa (\Omega )$ is an open neighbourhood of
$(V_{1},W)$ in $T_{1}$,
$$\varpi _{2}(\varpi _{1}\mul \varpi _{4}^{-1}(\kappa (\Omega ))\cap T)
\quad \text{and} \quad
\varpi _{3}(\varpi _{1}\mul \varpi _{4}^{-1}(\kappa (\Omega ))\cap T) $$
are dense subsets of $Z'$ and $Z$ respectively. For all $(\varphi ,\psi )$ in $\Omega $,
all element of
$\varpi _{2}(\varpi _{1}\mul \varpi _{4}^{-1}(\kappa (\varphi ,\psi ))\cap T)$
contains the first component of $\kappa (\varphi ,\psi )$ and is contained in
the second component of $\kappa (\varphi ,\psi )$. Hence all element of
$\varpi _{2}(\varpi _{1}\mul \varpi _{4}^{-1}(\kappa (\Omega ))\cap T)$ is in the image
of $\kappa _{*}$. As a result, $\widetilde{\Omega }\cap Z'$ is dense in $Z'$ and $Z'$ is
contained in $\tilde{Z}$. In the same way, $\widetilde{\Omega }\cap Z$ is dense in $Z$
and $Z$ is contained in $\tilde{Z}$.
(ii) According to Lemma~\ref{lsa5}(iv), for all $(V'_{1},V',V,W')$ in $O$,
$V'_{1}=V'\cap V$ and $W'=V'+V$. By definition, $\kappa (\Omega )$ is contained in
$\varpi _{1}\mul \varpi _{4}(O)$ and for $V$ in $\widetilde{\Omega }$, $V'_{1}\subset V$
and $V\subset W'$ for some $(V'_{1},W')$ in $\kappa (\Omega )$, whence the assertion.
(iii) For $(\varphi ,\psi )$ in $F$ and for $(x_{1},x_{2})$ in
$\k^{2}\setminus \{(0,0)\}$ such that
$$ V = \kappa _{*}(\varphi ,\psi ,x_{1},x_{2}),$$
the subset of elements $(y_{1},y_{2})$ of $\k^{2}$ such that
$(\varphi ,\psi ,y_{1},y_{2})$ is in $F'$ is equal to $\k^{*}.(x_{1},x_{2})$. Moreover,
for all $(\varphi ,\psi ,y_{1},y_{2})$ in $F'$, $(\varphi ,\psi )$ is in $F$, whence the
assertion.
(iv) In (iii), we can choose $V$ such that $F'$ has minimal dimension so that
$$ \dim \tilde{Z} = \dim \Omega +2 - (\dim F+1) = \dim \Omega - \dim F +1 .$$
By (ii), for some $V'$ in $Z'$, for all $(\varphi ,\psi )$ in $F$, $V'$ contains the
first component of $\kappa (\varphi ,\psi )$ and is contained in the second component of
$\kappa (\varphi ,\psi )$. So, again by (iii) and (ii),
$$ \dim Z' \geq \dim \Omega - \dim F,$$
whence $\dim \tilde{Z} \leq \dim Z' +1$ and $\dim Z \leq \dim Z' +1$ since $Z$ is
contained in $\tilde{Z}$ by (i).
\end{proof}
\begin{prop}\label{psav3}
The variety $X_{R,\n}$ has dimension at most $n-d$.
\end{prop}
\begin{proof}
Prove this by induction on $n$. According to Lemma~\ref{l2sa1}(ii), it is true
for $n-d=0$. Suppose that $n-d$ is positive and that it is true for all integer smaller
than $n-d$. In particular, $X_{R',\n}$ has dimension at most $n-d-1$. Let $Z$ be an
irreducible component of $X_{R,\n}$. According to Lemma~\ref{lsa5}(ii), for some
irreducible component $T$ of $\varpi _{3}^{-1}(Z)$, $\varpi _{3}(T)=Z$. Denote by
$Z'$ the image of $T$ by $\varpi _{2}$. Let $T_{0}$ be the subset of elements
$(V_{1},V',V,W)$ of $T$ such that $V'=V$. Consider the following cases:
\begin{itemize}
\item [{\rm (a)}] $T_{0}=T$,
\item [{\rm (b)}] $T_{0}\neq T$ and $Z'$ is contained in $X_{R',\n}$,
\item [{\rm (c)}] $Z'$ is not contained in $X_{R',\n}$.
\end{itemize}
(a) In this case, $Z'=Z$ and $\dim Z \leq n-d-1$ by induction hypothesis.
(b) By induction hypothesis, $\dim Z' \leq n-d-1$ and by Lemma~\ref{lsav3}(iv),
$\dim Z \leq \dim Z'+1$, whence $\dim Z \leq n-d$.
(c) In this case, $T_{0}\neq T$, whence $\dim Z \leq \dim Z'+1$ by Lemma~\ref{lsav3}(iv).
Since $Z$ is an irreducible component of $X_{R,\n}$, $Z$ is invariant under $R$. By
Lemma~\ref{lsa5}(i), $\varpi _{2}$ and $\varpi _{3}$ are equivariant under the action
of $R'$ in $\Gamma $ so that $T$ and $Z'$ are invariant under $R'$. For all
$(V_{1},V',V,W)$ in $T\setminus T_{0}$, $V_{1}=V'\cap V$. Hence all element of
a dense open subset of $Z'$ contains a subspace of dimension $d-1$ of ${\goth a}'$.
Then, by Proposition~\ref{psav1}, for some complete subset $\Lambda $ of ${\cal R}'$ such
that ${\goth t}_{\Lambda }$ has dimension $1$ and for some closed subset $Z_{\Lambda }$
of $X_{R_{\Lambda }}$, $R'.Z_{\Lambda }$ is dense in $Z'$ so that
$$ \dim Z' \leq \dim Z_{\Lambda } + \dim {\goth a}'-\dim {\goth a}_{\Lambda }.$$
If $\dim {\goth a}_{\Lambda }-\dim {\goth t}+1 = n-d$, then $\Lambda ={\cal R}'$.
In this case, since ${\goth a}'$ is in ${\cal C}_{{\goth t}}$, $\Lambda $
generates ${\goth t}^{*}$. As ${\goth t}_{\Lambda }$ has dimension $1$, it is impossible.
As a result,
$$ \dim Z_{\Lambda } \leq \dim {\goth a}_{\Lambda }-\dim {\goth t}+1
\quad \text{and} \quad \dim Z' \leq n-d $$
by Lemma~\ref{lsav1} and induction hypothesis for ${\goth a}_{\Lambda }$. Then
$\dim Z \leq n-d+1$. According to Lemma~\ref{lsav3},(i) and (iv), $\tilde{Z}$ is an
irreducible variety of dimension at most $\dim Z'+1$, containing $Z'$ and $Z$. If
$\dim Z'=n-d$ and $\dim Z = n-d+1$, then $Z=\tilde{Z}$. In particular, $Z'$ is contained
in $Z$. It is impossible since all element of $Z$ is contained in ${\goth a}$. As a
result, $\dim Z \leq n-d$, whence the proposition.
\end{proof}
\begin{coro}\label{csav3}
{\rm (i)} The irreducible components of $X_{R}\setminus R.{\goth t}$ are the
$X_{R,\alpha },\alpha \in {\cal R}$.
{\rm (ii)} The set $X'_{R}$ is a smooth big open subset of $X_{R}$, containing
$R.{\goth t}$.
\end{coro}
\begin{proof}
According to Proposition~\ref{psav2}(ii) and Lemma~\ref{l2sav2}(iii), Assertion (ii)
results from Assertion (i). Prove Assertion (i) by induction on $n=\dim {\goth a}$.
For $n=1$, $d=1$ by Lemma~\ref{lsa1},(i) and (iv) so that $X_{R}$ is the union of
$R.{\goth t}$ and ${\goth a}^{\alpha }$, whence Assertion (i) in this case. Suppose
$n\geq 2$ and the assertion true for the integers smaller than $n$. By
Lemma~\ref{lsa1}(i), Condition (2) and Condition (3) of Section~\ref{sa}, $d\geq 2$.
According to Lemma~\ref{l2sav2}(ii), for all $\alpha $ in ${\cal R}$, $X_{R,\alpha }$ is
an irreducible component of $X_{R}\setminus R.{\goth t}$. Let $Z$ be an irreducible
component of $X_{R}\setminus R.{\goth t}$. By Proposition~\ref{psav2}(i), $Z$ has
dimension $n-1$. So, by Proposition~\ref{psav3}, $Z$ is not contained in $X_{R,\n}$.
Moreover, $Z$ is invariant under $R$. Then, by Proposition~\ref{psav1}, for some complete
subset $\Lambda $ of ${\cal R}$, strictly contained in ${\cal R}$ and for some
irreducible closed subset $Z_{\Lambda }$ of $X_{R_{\Lambda }}$, invariant under
$R_{\Lambda }$, $R.Z_{\Lambda }$ is dense in $Z$. By Lemma~\ref{lsav1},
${\goth a}_{\Lambda }$ is in ${\cal C}_{{\goth t}_{\Lambda }^{\#}}$
and $Z_{\Lambda }$ is the image of a closed subset $Z'_{\Lambda }$ of
$X_{R^{\#}_{\Lambda }}$, invariant by $R^{\#}_{\Lambda }$, by the map
$V\mapsto V\oplus {\goth t}_{\Lambda }$. Since $Z_{\Lambda }$ is contained in $Z$,
$Z'_{\Lambda }\cap R^{\#}_{\Lambda }.{\goth t}_{\Lambda }^{\#}$ is empty. As
$\Lambda $ is strictly contained in ${\cal R}$, $\dim {\goth a}_{\Lambda }$ is smaller
than $n$. So, by induction hypothesis, for some $\alpha $ in $\Lambda $,
$Z'_{\Lambda }$ is contained in $X_{R^{\#}_{\Lambda },\alpha }$. As a result,
$Z_{\Lambda }$ and $Z$ are contained in $X_{R,\alpha }$, whence $Z=X_{R,\alpha }$ since
$Z$ is an irreducible component of $X_{R}\setminus R.{\goth t}$.
\end{proof}
\section{Normality for solvable Lie algebras} \label{ns}
Let ${\goth t}$ be a vector space of positive dimension $d$ and ${\goth a}$ in
${\cal C}_{{\goth t}}$. Set:
$$ {\cal R} := {\cal R}_{{\goth t},{\goth a}}, \qquad
{\goth r} := {\goth r}_{{\goth t},{\goth a}} \qquad \pi := \pi _{{\goth t},{\goth a}},
\qquad R := R_{{\goth t},{\goth a}}, \qquad A := A_{{\goth t},{\goth a}}, \qquad
{\cal E} := {\cal E}_{{\goth t},{\goth a}}, \qquad n := \dim {\goth a} .$$
The goal of the section is to prove that $X_{R}$ is normal and Cohen-Macaulay.
\subsection{The case $\dim {\goth a}=\dim {\goth t}$} \label{ns1}
By Condition (2) of Section~\ref{sa}, ${\cal R}$ has $d$ elements
$\poi {\beta }1{,\ldots,}{d}{}{}{}$ linearly independent. Denote by
$\poi t1{,\ldots,}{d}{}{}{}$ the dual basis in ${\goth t}$. For $i=1,\ldots,d$, let
$v_{i}$ be a generator of ${\goth a}^{\beta _{i}}$.
\begin{lemma}\label{lns1}
If $\dim {\goth a}=\dim {\goth t}$ then $X_{R}$ is a smooth variety. Moreover, for all
$(\poi z1{,\ldots,}{d}{}{}{})$ in $\k^{d}$, the subspace generated by
$v_{1}+z_{1}t_{1},\ldots,v_{d}+z_{d}t_{d}$ is in $X_{R}$.
\end{lemma}
\begin{proof}
According to Lemma~\ref{l2sa1}, ${\goth a}$ is in in $X_{R}$ and the map
$$ \xymatrix{ \k^{d} \ar[rr] && X_{R}}, \qquad
(\poi z1{,\ldots,}{d}{}{}{}) \longmapsto
{\mathrm {span}}(\{v_{1}+z_{1}t_{1},\ldots,v_{d}+z_{d}x_{d}\})$$
is an isomorphism onto an open neighborhood of ${\goth a}$ in $X_{R}$. Hence
${\goth a}$ is a smooth point of $X_{R}$. By Corollary~\ref{c2sa6}, $R$ has only one
fixed point ${\goth a}$ in $X_{R}$. Since for all $V$ in $X_{R}$, $R$ has a fixed point
in $\overline{R.V}$ and ${X_{R}}_{\loc}$ is an open subset of $X_{R}$, invariant under
$R$, $X_{R}={X_{R}}_{\loc}$.
\end{proof}
\subsection{Cohen-Macaulayness property for some algebras} \label{ns2}
Let $A_{*}$ be an integral domain and a local commutative $\k$-algebra with maximal ideal
${\goth m}$ and $\poi u1{,\ldots,}{s}{}{}{}$ a regular sequence in $A_{*}$ of elements of
${\goth m}$. Let $\poi T1{,\ldots,}{s}{}{}{}$ be indeterminates. Set
$B_{s} := A_{*}[\poi T1{,\ldots,}{s}{}{}{}]$ and denote by $P_{s}$ and $P'_{s}$ the
ideals of $B_{s}$ generated by the sequences
$u_{j}T_{k}-u_{k}T_{j},1\leq j,k\leq s$ and $u_{j}T_{1}-u_{1}T_{j},1\leq j\leq s$
respectively.
\begin{lemma}\label{lns2}
The ideal $P_{s}$ is a prime ideal of $B_{s}$.
\end{lemma}
\begin{proof}
For $s=1$, $P_{s}=\{0\}$. Suppose $s\geq 2$. Let $\tilde{P}$ be the ideal of
$B_{s}[T_{1}^{-1}]$ generated by $P_{s}$. For $1\leq j,k\leq s$,
$$ T_{1}(u_{j}T_{k}-u_{k}T_{j}) = T_{k}(u_{j}T_{1}-u_{1}T_{j}) +
T_{j}(u_{1}T_{k}-u_{k}T_{1}).$$
Hence $\tilde{P}$ is the ideal of $B_{s}[T_{1}^{-1}]$ generated by
$P'_{s}$. Setting $S_{j} := T_{j}/T_{1}$ for $j=2,\ldots,s$,
denote by $C$ the polynomial algebra $A_{*}[\poi S2{,\ldots,}{s}{}{}{}]$ over $A_{*}$
so that $B_{s}[T_{1}^{-1}] = C[T_{1},T_{1}^{-1}]$ and $\tilde{P}$
is the ideal of $B_{s}[T_{1}^{-1}]$ generated by $u_{j}-u_{1}S_{j},j=2,\ldots,s$.
\begin{claim}\label{clns2}
Let $Q$ be the ideal of $C$ generated by $u_{j}-u_{1}S_{j},j=2,\ldots,s$. Then $Q$
is prime.
\end{claim}
\begin{proof}{[Proof of Claim~\ref{clns2}]}
Let $\tilde{Q}$ be the ideal of $C[u_{1}^{-1}]$ generated by $Q$. Then
$\tilde{Q}$ is prime since it is generated by $u_{j}u_{1}^{-1}-S_{j},j=2,\ldots,s$. As a
result, for $p$ and $q$ in $C$ such that $pq$ is in $Q$, for some nonnegative
integer $m$, $u_{1}^{m}p$ or $u_{1}^{m}q$ is in $Q$. So it remains to prove that
for $p$ in $C$, $p$ is in $Q$ if so is $u_{1}p$.
Let $p$ be in $C$ such that $u_{1}p$ is in $Q$. For some $\poi q2{,\ldots,}{s}{}{}{}$ in
$C$,
$$ u_{1}p = \sum_{j=2}^{s} q_{j}(u_{j}-u_{1}S_{j}) \quad \text{whence} \quad
\sum_{j=1}^{s} q_{j}u_{j} = 0 \quad \text{with} \quad
q_{1} := -(p + \sum_{j=2}^{s} q_{j}S_{j}) .$$
By hypothesis, the sequence $\poi u1{,\ldots,}{s}{}{}{}$ is regular in $C$. So for some
sequence $q_{j,k},1\leq j,k\leq s$ in $C$ such that $q_{j,k}=-q_{k,j}$,
$$ q_{j} = \sum_{k=1}^{s} q_{j,k}u_{k}$$
for $j=1,\ldots,s$. As a result,
\begin{eqnarray*}
u_{1}p = & \sum_{j=2}^{s} \sum_{k=1}^{s} q_{j,k}u_{k} (u_{j}-u_{1}S_{j}) \\
= & \sum_{j=2}^{s} q_{j,1}u_{j}u_{1} -
\sum_{j=2}^{s} \sum_{k=1}^{s} q_{j,k}u_{k}u_{1}S_{j} \\ = &
u_{1} (\sum_{j=2}^{s} q_{j,1}(u_{j}-u_{1}S_{j}) +
\sum_{2\leq j<k\leq s} q_{j,k}(u_{j}S_{k} - u_{k}S_{j})) .
\end{eqnarray*}
For $2\leq j,k\leq s$,
$$ u_{j}S_{k} - u_{k}S_{j} = (u_{j}-u_{1}S_{j})S_{k} - (u_{k}-u_{1}S_{k})S_{j}
\in Q,$$
whence the claim.
\end{proof}
By the claim, $\tilde{P}$ is a prime ideal of $B_{s}[T_{1}^{-1}]$ since it is
generated by $Q$. As a result for $p$ and $q$ in $B_{s}$ such that $pq$ is in $P_{s}$,
for some nonnegative integer $m$, $T_{1}^{m}p$ or $T_{1}^{m}q$ is in $P'_{s}$ since
$T_{1}P_{s}$ is contained in $P'_{s}$ by the equality
$$ T_{1}(u_{j}T_{k}-u_{k}T_{j}) = T_{k}(u_{j}T_{1}-u_{1}T_{j})
+T_{j}(u_{1}T_{k}-u_{k}T_{1})$$
for $1\leq i,j\leq s$. So it remains to prove that for $p$ in $B_{s}$, $p$ is in $P_{s}$
if $T_{1}p$ is in $P'_{s}$.
Let $p$ be in $B_{s}$ such that $T_{1}p$ is in $P'_{s}$. For some
$\poi r2{,\ldots,}{s}{}{}{}$ in $B_{s}$,
$$ T_{1}p = \sum_{j=2}^{s} r_{j}(u_{j}T_{1}-u_{1}T_{j}) .$$
For $j=2,\ldots,s$, $r_{j}$ has an expansion
$$ r_{j} = r_{j,0} + T_{1} r_{j,1}$$
with $r_{j,0}$ and $r_{j,1}$ in $B'_{s}:=A_{*}[\poi T2{,\ldots,}{s}{}{}{}]$ and $B_{s}$
respectively. Set:
$$ p' := p - \sum_{j=2}^{s} r_{j,1}(u_{j}T_{1}-u_{1}T_{j}) .$$
Then
$$T_{1}p'= \sum_{j=2}^{s} r_{j,0} (u_{j}T_{1}-u_{1}T_{j}) $$
so that the element
$$ \sum_{j=2}^{s} r_{j,0} u_{1}T_{j} \in B'_{s}$$
is divisible by $T_{1}$ in $B_{s}$, whence
$$ \sum_{j=2}^{s} r_{j,0} T_{j} = 0 .$$
As $\poi T2{,\ldots,}{s}{}{}{}$ is a regular sequence in $B_{s}$, for some sequence
$r_{j,k,0},2\leq j,k\leq s$ in $B_{s}$ such that $r_{j,k,0}=-r_{k,j,0}$ for all
$(j,k)$,
$$ r_{j,0} = \sum_{k=2}^{s} r_{j,k,0}T_{k}$$
for $j=2,\ldots,s$. Then
$$ T_{1}p' = \sum_{2\leq j,k\leq s} r_{j,k,0}T_{k}(u_{j}T_{1}-u_{1}T_{j})
= T_{1} \sum_{2\leq j<k\leq s} r_{j,k,0}(T_{k}u_{j}-T_{j}u_{k}) .$$
As a result $p'$ and $p$ are in $P_{s}$, whence the lemma.
\end{proof}
Denote by $P''_{s}$ the ideal of $B_{s}$ generated by $P_{s-1}$ and
$u_{s}T_{1}-u_{1}T_{s}$. Let ${\goth B}_{s}$ and ${\goth B}'_{s}$ be the quotients of
$B_{s}$ by $P_{s}$ and $P''_{s}$ respectively. The restrictions to $A_{*}$ of the
quotient morphisms $\xymatrix{B_{s} \ar[r] & {\goth B}'_{s}}$ and
$\xymatrix{ B_{s} \ar[r] & {\goth B}_{s}}$ are embeddings. For $j=1,\ldots,s$, denote
again by $T_{j}$ its images in ${\goth B}'_{s}$ and ${\goth B}_{s}$ by these morphisms.
\begin{lemma}\label{l2ns2}
Denote by $\overline{P_{s}}$ the image in ${\goth B}'_{s}$ of $P_{s}$ by the quotient
morphism.
{\rm (i)} The intersection of $\overline{P_{s}}$ and $T_{1}{\goth B}'_{s}$ is equal
to $\{0\}$.
{\rm (ii)} The ${\goth B}'_{s}$-modules $T_{1}{\goth B}'_{s}$ and ${\goth B}_{s}$ are
isomorphic.
\end{lemma}
\begin{proof}
Let $a$ be in $B_{s}$ such that $T_{1}a$ is in $P_{s}$. According to Lemma~\ref{lns2},
$P_{s}$ is a prime ideal of $B_{s}$. Hence $a$ is in $P_{s}$ since $T_{1}$ is not in
$P_{s}$. Moreover, for $j=1,\ldots,s$,
$$ T_{1}(u_{j}T_{s}-u_{s}T_{j}) = T_{s}(u_{j}T_{1}-u_{1}T_{j})
+T_{j}(u_{1}T_{s}-u_{s}T_{1}).$$
Hence $T_{1}P_{s}$ is contained in $P''_{s}$. As a result, $\overline{P_{s}}$ is the
kernel of the endomorphism $a\mapsto T_{1}a$ of ${\goth B}'_{s}$ and the intersection
of $\overline{P_{s}}$ and $T_{1}{\goth B}'_{s}$ is equal to $\{0\}$. As
${\goth B}_{s}$ is the quotient of ${\goth B}'_{s}$ by $\overline{P_{s}}$, the
endomorphism $a\mapsto T_{1}a$ defines through the quotient an isomorphism
$$\xymatrix{{\goth B}_{s} \ar[rr] && T_{1}{\goth B}'_{s}}$$
of ${\goth B}'_{s}$-modules.
\end{proof}
Let $Q_{s}$ be the ideal of the polynomial algebra $A_{*}[\poi T2{,\ldots,}{s}{}{}{}]$
generated by the sequence $u_{i}T_{k}-u_{k}T_{i},\; 2\leq i,k\leq s$ and denote by
${\goth B}_{s}^{\#}$ the quotient of $A_{*}[\poi T2{,\ldots,}{s}{}{}{}]$ by $Q_{s}$.
\begin{lemma}\label{l3ns2}
{\rm (i)} The quotient of the algebra ${\goth B}_{s}/T_{1}{\goth B}_{s}$ by the ideal
generated by $u_{1}$ is equal to the quotient of ${\goth B}_{s}^{\#}$ by the ideal
generated by $u_{1}$.
{\rm (ii)} The canonical map $\xymatrix{ A_{*} \ar[r] & {\goth B}_{s}/T_{1}{\goth B}_{s}}$
is an embedding.
{\rm (iii)} The ideal of ${\goth B}_{s}/T_{1}{\goth B}_{s}$ generated by $u_{1}$ is
isomorphic to $A_{*}$
\end{lemma}
\begin{proof}
Denote by $Q'_{s}$ the ideal of $B_{s}$ generated by $P_{s}$ and $T_{1}$.
(i) As the ideal of $B_{s}$ generated by $Q'_{s}$ and $u_{1}$ is equal to the
ideal generated by $u_{1}$, $T_{1}$ and $Q_{s}$,
${\goth B}_{s}^{\#}/u_{1}{\goth B}_{s}^{\#}$ is equal to the quotient of
${\goth B}_{s}/T_{1}{\goth B}_{s}$ by the ideal generated by $u_{1}$.
(ii) Since the intersection of $A_{*}$ and $Q'_{s}$ is equal to $\{0\}$, the canonical
map $\xymatrix{ A_{*} \ar[r] & {\goth B}_{s}/T_{1}{\goth B}_{s}}$ is an embedding.
(iii) For $k=2,\ldots,s$, $u_{1}T_{k}$ is in $Q'_{s}$. Hence $u_{1}B_{s}$ is contained
in the sum of $u_{1}A_{*}$ and $Q'_{s}$. As a result, $u_{1}A_{*}$ is equal to
$u_{1}{\goth B}_{s}/T_{1}{\goth B}_{s}$ by (ii), whence the assertion since $A_{*}$ is
an integral domain.
\end{proof}
\begin{prop}\label{pns2}
Suppose that $A_{*}$ is Cohen-Macaulay.
{\rm (i)} The algebra ${\goth B}_{s}$ is an integral domain and a Cohen-Macaulay algebra
of dimension $\dim A_{*}+1$.
{\rm (ii)} For $\poi a1{,\ldots,}{m}{}{}{}$ regular sequence in $A_{*}$ of elements of
${\goth m}$ an for ${\goth p}$ prime ideal of ${\goth B}_{s}$ containing it,
$\poi a1{,\ldots,}{m}{}{}{}$ is a regular sequence in the localization of
${\goth B}_{s}$ at ${\goth p}$.
\end{prop}
\begin{proof}
(i) Prove the assertion by induction on $s$. As ${\goth B}_{1}$ is the polynomial algebra
$A_{*}[T_{1}]$, the assertion is true for $s=1$ since $A_{*}$ an integral domain and a
Cohen-Macaulay algebra. Suppose the assertion true for $s-1$. By induction hypothesis,
${\goth B}_{s-1}[T_{s}]$ is an integral domain and a Cohen-Macaulay algebra as a
polynomial algebra over ${\goth B}_{s-1}$ and its dimension is equal to $\dim A_{*}+2$.
As a result, ${\goth B}'_{s}$ is Cohen-Macaulay of dimension $\dim A_{*}+1$ as the
quotient of the integral domain and a Cohen-Macaulay algebra ${\goth B}_{s-1}[T_{s}]$ by
the ideal generated by $T_{s}u_{1}-T_{1}u_{s}$. As ${\goth B}_{s}$ is the quotient of
${\goth B}'_{s}$ by $\overline{P_{s}}$, ${\goth B}_{s}$ has dimension at most
$\dim A_{*}+1$. By Lemma~\ref{lns2}, ${\goth B}_{s}$ is an integral domain so that
${\goth B}_{s}/T_{1}{\goth B}_{s}$ has dimension at most $\dim A_{*}$.
By induction hypothesis again, ${\goth B}_{s}^{\#}$ is an integral domain and a Cohen-
Macaulay algebra of dimension $\dim A_{*}+1$. Hence
${\goth B}_{s}^{\#}/u_{1}{\goth B}_{s}^{\#}$ is Cohen-Macaulay of dimension $\dim A_{*}$.
According to Lemma~\ref{l3ns2}, we have a short exact sequence
$$ \xymatrix{ 0 \ar[r] & A_{*} \ar[r] & {\goth B}_{s}/T_{1}{\goth B}_{s} \ar[r] &
{\goth B}_{s}^{\#}/u_{1}{\goth B}_{s}^{\#} \ar[r] & 0} .$$
Hence the algebra ${\goth B}_{s}/T_{1}{\goth B}_{s}$ is Cohen-Macaulay of dimension
$\dim A_{*}$ since $A_{*}$ and ${\goth B}_{s}^{\#}/u_{1}{\goth B}_{s}^{\#}$ are
Cohen-Macaulay of dimension $\dim A_{*}$ and ${\goth B}_{s}/T_{1}{\goth B}_{s}$ has
dimension at most $\dim A_{*}$. As a result, ${\goth B}_{s}$ has dimension
$\dim A_{*}+1$. As ${\goth B}_{s}$ is the quotient of
${\goth B}'_{s}$ by $\overline{P_{s}}$, we have a short exact sequence
$$ \xymatrix{ 0 \ar[r] & \overline{P_{s}}+T_{1}{\goth B}'_{s} \ar[r] &
{\goth B}'_{s} \ar[r] & {\goth B}_{s}/T_{1}{\goth B}_{s} \ar[r] & 0 }.$$
Then, setting $M := \overline{P_{s}}+T_{1}{\goth B}'_{s}$ and denoting by $M_{*}$ the
localization of $M$ at a maximal ideal of ${\goth B}'_{s}$, containing $T_{1}$,
$$ {\mathrm {Ext}}^{j}(\k,M_{*}) = 0$$
for $j\leq \dim A_{*}$ since ${\goth B}'_{s}$ and ${\goth B}_{s}/T_{1}{\goth B}_{s}$
have dimension $\dim A_{*}+1$ and $\dim A_{*}$. By Lemma~\ref{l2ns2}(i),
$M$ is the direct sum $\overline{P_{s}}$ and $T_{1}{\goth B}'_{s}$. So, denoting
by $(T_{1}{\goth B}'_{s})_{*}$ the localization of $T_{1}{\goth B}'_{s}$ at a maximal
ideal of ${\goth B}'_{s}$,
$$ {\mathrm {Ext}}^{j}(\k,(T_{1}{\goth B}'_{s})_{*}) = 0$$
for $j\leq \dim A_{*}$ since $(T_{1}{\goth B}'_{s})_{*}$ is the localization of
${\goth B}'_{s}$ at this maximal ideal when it does not contain $T_{1}$. As a result,
by Lemma~\ref{l2ns2}(ii), ${\goth B}_{s}$ is Cohen-Macaulay since it has dimension
$\dim A_{*}+1$.
(ii) Let ${\goth q}$ be a minimal prime ideal of ${\goth B}_{s}$, containing
$\poi a1{,\ldots,}{m}{}{}{}$. Since $A_{*}$ is embedded in ${\goth B}_{s}$,
${\goth q}\cap A_{*}$ is a prime ideal of $A_{*}$ containing
$\poi a1{,\ldots,}{m}{}{}{}$. As $A_{*}$ is Cohen-Macaulay and
$\poi a1{,\ldots,}{m}{}{}{}$ is a regular sequence in $A_{*}$, ${\goth q}\cap A_{*}$ has
height at least $m$ and $A_{*}/{\goth q}\cap A_{*}$ has dimension at most
$\dim A_{*}-m$ by \cite[Ch. 6, Theorem 17.4]{Mat}. Then ${\goth B}_{s}/{\goth q}$ has
dimension at most $\dim A_{*}+1-m$ since the fraction field of ${\goth B}_{s}/{\goth q}$
is generated by the fraction field of $A_{*}/{\goth q}\cap A_{*}$ and the image of $T_{1}$
by the quotient morphism $\xymatrix{ B_{s} \ar[r] & {\goth B}_{s}/{\goth q}}$.
As a result, by (i) and \cite[Ch. 6, Theorem 17.4]{Mat}, ${\goth q}$ has height at least
$m$. As a minimal prime ideal of ${\goth B}_{s}$ containing $m$ elements, ${\goth q}$ has
height at most $m$. Hence all minimal prime ideal of ${\goth B}_{s}$, containing
$\poi a1{,\ldots,}{m}{}{}{}$, has height $m$. So, by (i) and
\cite[Ch. 6, Theorem 17.4]{Mat}, $\poi a1{,\ldots,}{m}{}{}{}$ is a regular sequence in
the localization of ${\goth B}_{s}$ at ${\goth p}$.
\end{proof}
\subsection{Normality and Cohen-Macaulayness property for $X_{R}$} \label{ns3}
Let $V_{0}$ be a fixed point under the action of $R$ in $X_{R}$ and
$\poi {\beta }1{,\ldots,}{d}{}{}{}$ the elements of ${\cal R}_{V_{0}}$. By
Corollary~\ref{c2sa6}(ii), $\poi {\beta }1{,\ldots,}{d}{}{}{}$ is a basis of
${\goth t}^{*}$. Let $\poi t1{,\ldots,}{d}{}{}{}$ be the dual basis. Denote by $m$ the
codimension of $V_{0}$ in ${\goth a}$. According to Lie's Theorem, for $m>0$, the
elements $\poi {\gamma }1{,\ldots,}{m}{}{}{}$ of
${\cal R}\setminus \{\poi {\beta }1{,\ldots,}{d}{}{}{}\}$ can be ordered so that
$$ {\goth a}_{i} :=
V_{0} \oplus {\goth a}^{\gamma _{1}} \oplus \cdots \oplus {\goth a}^{\gamma _{i}}$$
is an algebra of codimension $m-i$ of ${\goth a}$ for $i=1,\ldots,m$. Set:
$$ {\cal R}' := {\cal R}\setminus \{\gamma _{m}\}, \qquad {\goth a}'={\goth a}_{m-1},
\qquad {\goth r}' := {\goth r}_{{\goth t},{\goth a}'}, \qquad
\pi ':= \pi _{{\goth t},{\goth a}'}, \qquad R' := R_{{\goth t},{\goth a}'}, \qquad
A' := A_{{\goth t},{\goth a}'},$$
$$ E := \bigoplus _{i=1}^{m} {\goth a}^{\gamma _{i}}, \qquad E' := E\cap {\goth a}' .$$
Denote by $\kappa $ the map
$$ \xymatrix{ {\mathrm {Hom}}_{\k}(V_{0},E\oplus {\goth t}) \ar[rr]^{\kappa } &&
\ec {Gr}r{}{}d}, \qquad \varphi \longmapsto
{\mathrm {span}}(\{v+\varphi (v) \; \vert \; v \in V_{0}\}) .$$
Then $\kappa $ is an isomorphism from ${\mathrm {Hom}}_{\k}(V_{0},E\oplus {\goth t})$ onto
an affine open neighbourhood of $V_{0}$ in $\ec {Gr}r{}{}d$. Moreover, there is a short
exact sequence
$$ \xymatrix{0 \ar[r] & {\mathrm {Hom}}_{\k}(V_{0},\k x_{\gamma _{m}}) \ar[r] &
{\mathrm {Hom}}_{\k}(V_{0},E\oplus {\goth t}) \ar[r]^{p} &
{\mathrm {Hom}}_{\k}(V_{0},E'\oplus {\goth t}) \ar[r] & 0}.$$
Let $\Omega $ and $\Omega '$ be the inverse images by $\kappa $ of the intersections of
the image of $\kappa $ with $X_{R}$ and $X_{R'}$ respectively. For $\varphi $ in
$\Omega $ and $i=1,\ldots,d$
$$ \varphi (v_{i}) = \sum_{i=1}^{d} z_{i,j}(\varphi ) t_{j} +
\sum_{j=1}^{m} a_{i,j}(\varphi ) x_{\gamma _{j}}$$
so that the $z_{i,j}$'s, $1\leq i,j\leq d$ and the $a_{i,j}$'s, $1\leq i\leq d$ and
$1\leq j\leq m$ are regular functions on $\Omega $.
Let $\psi $ be the map
$$ \xymatrix{\k \times \Omega ' \ar[rr]^{\psi } && X_{R} }, \qquad
(s,\varphi ) \longmapsto \exp(s\ad x_{\gamma _{m}}).\kappa (\varphi ) .$$
\begin{lemma}\label{lns3}
Let $O$ be the subset of elements $(s,\varphi )$ of $\k\times \Omega '$ such that
$\psi (s,\varphi )$ is in $\kappa (\Omega )$.
{\rm (i)} The subset $O$ of $\k\times \Omega '$ is open and contains
$\{0\}\times \Omega '$.
{\rm (ii)} The map
$$ \xymatrix{ O \ar[rr]^{\overline{\psi }} && \Omega },\qquad
(s,\varphi ) \longmapsto \kappa ^{-1}\rond \psi (s,\varphi )$$
is a birational morphism from $O$ to $\Omega $. In particular, the function
$(s,\varphi )\mapsto s$ is in $\k(\Omega )$.
\end{lemma}
\begin{proof}
(i) As $\kappa (\Omega )$ is an open neighborhood of $V_{0}$ in $X_{R}$, $O$ is
an open subset of $\k\times \Omega '$, containing $\{0\}\times \Omega '$ since
$\psi $ is a regular map such that $\psi (0,\varphi )=\kappa (\varphi )$ for all
$\varphi $ in $\Omega '$.
(ii) Let $\Omega ^{c}$ be the subset of elements $\varphi $ of $\Omega $ such that
$\kappa (\varphi )$ is in $A.{\goth t}$. Then $\Omega ^{c}$ is a dense open subset of
$\Omega $. Let $O'$ be the inverse image of $\Omega ^{c}$ by $\overline{\psi }$.
Let $(s,\varphi )$ and $(s',\varphi ')$ be in $O'$ such that
$\overline{\psi }(s,\varphi )=\overline{\psi }(s',\varphi ')$, that is
$$ \exp(s\ad x_{\gamma _{m}}).\kappa (\varphi ) =
\exp(s'\ad x_{\gamma _{m}}).\kappa (\varphi ') \quad \text{whence} \quad
\exp((s-s')\ad x_{\gamma _{m}}).\kappa (\varphi ) = \kappa (\varphi ') . $$
According to the above notations, for $i=1,\ldots,d$,
$$ \varphi (v_{i}) = \sum_{j=1}^{d} z_{i,j}(\varphi ) t_{j} +
\sum_{j=1}^{m-1} a_{i,j}(\varphi ) x_{\gamma _{j}}.$$
Since $\kappa (\varphi )$ is in $A.{\goth t}$,
$$ \det ([z_{i,j}(\varphi ),1\leq i,j\leq d]) \neq 0 .$$
For $i=1,\ldots,d$,
$$ \exp((s-s')\ad x_{\gamma _{m}})(\sum_{j=1}^{d} z_{i,j}(\varphi )t_{j}) =
\sum_{j=1}^{d} z_{i,j}(\varphi )t_{j} -(s-s')(\sum_{j=1}^{d} z_{i,j}(\varphi )
\gamma _{m}(t_{j}))x_{\gamma _{m}} .$$
For some $j$, $\gamma _{m}(t_{j}) \neq 0$, whence $s=s'$ since $\kappa (\varphi ')$ is
contained in ${\goth r}'$. As a result, the restriction of $\overline{\psi }$ to $O'$
is injective, whence the assertion since $\overline{\psi }$ is a dominant morphism.
\end{proof}
For $i=1,\ldots,d$ and $\gamma $ in ${\goth t}^{*}$, denote by $u_{i,\gamma }$ the
function on $\Omega $,
$$ u_{i,j} := z_{i,1}\gamma (t_{1}) + \cdots + z_{i,d} \gamma (t_{d}) .$$
Let ${\goth A}$ be the subalgebra of $\k[\Omega ]$ generated by the functions
$z_{i,j}$'s, $1\leq i,j\leq d$ and $a_{i,j}$'s, $1\leq i\leq d$ and
$1\leq j\leq m-1$.
\begin{lemma}\label{l2ns3}
Let $\iota $ be the restriction morphism from $\Omega $ to $\Omega '$.
{\rm (i)} The restriction of $\iota $ to ${\goth A}$ is an isomorphism onto
$\k[\Omega ']$.
{\rm (ii)} For $1\leq i,j\leq d$,
$u_{i,\gamma _{m}}a_{j,m}-u_{j,\gamma _{m}}a_{i,m}$ is equal to $0$.
{\rm (iii)} For $i=1,\ldots,d$ and $\gamma $ in ${\goth t}^{*}$, if
$\gamma (t_{i})\neq 0$ then $u_{i,\gamma }$ is different from $0$.
\end{lemma}
\begin{proof}
(i) For $1\leq i,j\leq d$, denote by
$z'_{i,j}$ the restriction of $z_{i,j}$ to $\Omega '$ and for $1\leq i\leq d$ and
$1\leq j\leq m-1$ denote by $a'_{i,j}$ the restriction of $a_{i,j}$ to $\Omega '$. Since
$\k[\Omega ']$ is generated by the functions
$$z'_{i,j}, \ 1\leq i,j\leq d \quad \text{and} \quad a'_{i,j}, \
1\leq i\leq d,1\leq j\leq m-1,$$
the restriction of $\iota $ to ${\goth A}$ is surjective. Let ${\goth p}$ be the kernel
of the restriction of $\iota $ to ${\goth A}$. It remains to prove ${\goth p}=\{0\}$.
For $1\leq i,j\leq d$ and $k=1,\ldots,m-1$, denote by $\overline{z}_{i,j}$ and
$\overline{a}_{i,k}$ the functions on $\k \times \Omega '$ such that
\begin{eqnarray*}
\exp(s\ad x_{\gamma _{m}})(v_{i}+\sum_{j=1}^{d} z'_{i,j}(\varphi ) t_{j} +
\sum_{k=1}^{m-1} a'_{i,k}(\varphi ) x_{\gamma _{k}}) - \\
(\sum_{j=1}^{d} \overline{z}_{i,j}(s,\varphi ) t_{j} -
\sum_{j=1}^{d} sz_{i,j}(\varphi )\gamma _{m}(t_{j})x_{\gamma _{m}} +
\sum_{k=1}^{m-1} \overline{a}_{i,k}(s,\varphi ) x_{\gamma _{k}}) \in V_{0} .
\end{eqnarray*}
Then $\overline{z}_{i,j}$ and $\overline{a}_{i,k}$ are regular functions on
$\k\times \Omega '$ as restrictions to $\k\times \Omega '$ of regular functions on
$\k\times {\mathrm {Hom}}(V_{0},E'\oplus {\goth t})$. Let $\overline{{\goth A}}$ be the
subalgebra of $\k[\Omega '][s]$ generated by the functions
$$\overline{z}_{i,j},i,j=1,\ldots,d \quad \text{and} \quad
\overline{a}_{i,k},i=1,\ldots,d,k=1,\ldots,m-1 .$$
Since $z'_{i,j}(\varphi )=\overline{z}_{i,j}(0,\varphi )$ and
$a'_{i,k}(\varphi )=\overline{a}_{i,k}(0,\varphi )$ for all $\varphi $ in $\Omega '$, the
restriction to $\overline{{\goth A}}$ of the quotient morphism
$\xymatrix{\k[\Omega '][s] \ar[r]& \k[\Omega ']}$ is surjective. As a result,
$\overline{{\goth A}}$ has dimension $n$ or $n-1$ since $\Omega '$ and
$\k[\Omega '][s]$ have dimension $n-1$ and $n$ respectively. As
$\exp(s\ad x_{\gamma _{m}})(v_{i})$ is not necessarily equal to $v_{i}$,
$$p\rond \psi \neq
(\overline{z}_{i,j},\overline{a}_{i,j}, \; 1\leq i\leq d,1\leq j \leq m-1) .$$
Moreover, $\Omega '$ is contained in $p(\Omega )$ by Lemma~\ref{lns3}(i) but the
inclusion may be strict.
\begin{claim}\label{clns3}
The algebra $\overline{{\goth A}}$ has dimension $n-1$.
\end{claim}
\begin{proof}{[Proof of Claim~\ref{clns3}]}
There are two cases to consider:
\begin{itemize}
\item [{\rm (1)}] for $i=1,\ldots,m-1$,
$[{\goth a}^{\gamma _{m}},{\goth a}^{\gamma _{i}}]$ is contained in $V_{0}$,
\item [{\rm (2)}] for some $i$ in $\{1,\ldots,m-1\}$,
$[{\goth a}^{\gamma _{m}},{\goth a}^{\gamma _{i}}]$ is not contained in $V_{0}$.
\end{itemize}
In the first case, $\overline{{\goth A}}=\k[\Omega ']$. Otherwise, denote by
$j$ the biggest integer such that $[{\goth a}^{\gamma _{m}},{\goth a}^{\gamma _{j}}]$ is
not contained in $V_{0}$ and $a'_{i,j}\neq 0$ for some $i=1,\ldots,d$. Then, for some
$j'$ smaller than $j$, $\gamma _{m}+\gamma _{j}=\gamma _{j'}$. Furthermore,
for $k<j$ such that $[{\goth a}^{\gamma _{m}},{\goth a}^{\gamma _{k}}]$ is not contained
in $V_{0}$, $\gamma _{m}+\gamma _{k}$ is in
${\cal R}\setminus \{\poi {\gamma }{j'}{,\ldots,}{m}{}{}{}\}$. Then for $k\geq j'$ and
$i=1,\ldots,d$, $a'_{i,k}=\overline{a}_{i,k}$ and for all $(s,\varphi )$ in
$\k\times \Omega '$,
$$ \overline{a}_{i,j'}(s,\varphi ) = a'_{i,j'}(\varphi ) + s a'_{i,j}(\varphi ) .$$
As a result, by induction on $m-k$, for $i=1,\ldots,d$,
$$ a'_{i,k} - \overline{a}_{i,k} \in s\overline{{\goth A}}[s] .$$
Hence $\k[\Omega '][s]=\overline{{\goth A}}[s]$ and there exists a surjective morphism
$\xymatrix{ \k[\Omega '] \ar[r]& \overline{{\goth A}}}$ so that $\overline{{\goth A}}$
has dimension $n-1$.
\end{proof}
According to Lemma~\ref{lns3}(ii), the comorphism of $\overline{\psi }$ is an embedding
of $\k[\Omega ]$ into $\k[O]$ and from this embedding results an isomorphism from
$\k(\Omega )$ onto $\k(\Omega ')(s)$. Moreover, $\overline{{\goth A}}$ is the image of
${\goth A}$ by this embedding so that ${\goth A}$ has dimension $n-1$. As a result,
${\goth p}=\{0\}$ since $\iota $ is surjective and $\Omega '$ has dimension $n-1$.
(ii) Let $\varphi $ be in $\Omega $. Since $\kappa (\varphi )$ is a commutative algebra,
for $1\leq i,j\leq d$,
$$ 0 = [v_{i}+\varphi (v_{i}),v_{j}+\varphi (v_{j})] = [v_{i},\varphi (v_{j})] +
[\varphi (v_{i}),v_{j}] + [\varphi (v_{i}),\varphi (v_{j})] .$$
The component on $x_{\gamma _{m}}$ of the right hand side is
$$ \sum_{k=1}^{d} (z_{i,k}a_{j,m}(\varphi )-z_{j,k}a_{i,m}(\varphi ))
[t_{k},x_{\gamma _{m}}] =
(u_{i,\gamma _{m}}a_{j,m}-u_{j,\gamma _{m}}a_{i,m})(\varphi )x_{\gamma _{m}},$$
whence the assertion.
(iii) Denote by $R_{0}$ the adjoint group of
${\goth r}_{0} := {\goth t}+V_{0}$ and $X_{R_{0}}$ the closure in $\ec {Gr}r{}0{d}$ of
$R_{0}.{\goth t}$. Let $\Omega _{0}$ be the inverse image of $X_{R_{0}}$ by $\kappa $.
According to Lemma~\ref{lns1}, for $i,j=1,\ldots,d$, the restriction to $\Omega _{0}$ of
$z_{i,j}$ is equal to $0$ if $j\neq i$, otherwise it is different from $0$. As a result,
for $i=1,\ldots,d$ and $\gamma $ in ${\goth t}^{*}$, the restriction of $u_{i,\gamma }$
to $\Omega _{0}$ is equal to $\overline{z_{i,i}}\gamma (t_{i})$ with $\overline{z_{i,i}}$
the restriction of $z_{i,i}$ to $\Omega _{0}$, whence the assertion.
\end{proof}
For $\gamma $ in ${\goth t}^{*}$, set:
$$ I_{\gamma } := \{j \in \{1,\ldots,d\} \ \vert \ \gamma (t_{j}) \neq 0 \}.$$
\begin{prop}\label{pns3}
Denote by $\k[\Omega ]_{0}$ the localization of $\k[\Omega ]$ at $0$.
{\rm (i)} The local algebra $\k[\Omega ]_{0}$ is Cohen-Macaulay.
{\rm (ii)} For $\gamma $ in ${\goth t}^{*}$, $u_{i,\gamma },i\in I_{\gamma }$ is a
regular sequence in $\k[\Omega ]_{0}$ of elements of its maximal ideal.
\end{prop}
\begin{proof}
Prove the proposition by induction on $m$. By Lemma~\ref{lns1}, for $m=0$, $\k[\Omega ]$
is a polynomial algebra of dimension $d$, generated by
$\poi z{1,1,0}{,\ldots,}{d,d,0}{}{}{}$. Moreover, for $i=1,\ldots,d$ and $\gamma $ in
${\goth t}^{*}$, $u_{i,\gamma }=z_{i,i}\gamma (t_{i})$, whence the proposition for $m=0$.
Suppose $m>0$ and the proposition true for $m-1$ and use the notations of
Lemma~\ref{l2ns3}.
According to Lemma~\ref{l2ns3}(i) and the induction hypothesis, the localization
${\goth A}_{*}$ of ${\goth A}$ at $0$ is Cohen-Macaulay and for $\gamma $ in
${\goth t}^{*}$, $u_{i,\gamma },i\in I_{\gamma }$ is a regular sequence in
${\goth A}_{*}$ of elements of its maximal ideal. Denote by ${\goth B}$ the polynomial
algebra ${\goth A}_{*}[T_{i},i\in I_{\gamma _{m}}]$ and by $P$ the ideal of ${\goth B}$
generated by the sequence
$u_{i,\gamma _{m}}T_{j}-u_{j,\gamma _{m}}T_{i},(i,j)\in I_{\gamma _{m}}^{2}$.
According to Condition (3) of Section~\ref{sa},
$s := \vert I_{\gamma _{m}} \vert \geq 2$. By Lemma~\ref{l2ns3}(ii),
$\k[\Omega ]_{0}$ is a quotient of the localization at $0$ of ${\goth B}/P$ and by
Lemma~\ref{lns2}, $P$ is a prime ideal of ${\goth B}$. By Proposition~\ref{pns2}(i),
${\goth B}/P$ is an integral domain and a Cohen-Macaulay algebra of dimension $n$ since
$\k[\Omega ']$ has dimension $n-1$. Hence $\k[\Omega ]_{0}$ is the localization of
${\goth B}/P$ at $0$ since $\k[\Omega ]_{0}$ is an integral domain of dimension $n$. As a
result, $\k[\Omega ]_{0}$ is Cohen-Macaulay and by Proposition~\ref{pns2}(ii), for
$\gamma $ in ${\goth t}^{*}$, the sequence $u_{i,\gamma },i\in I_{\gamma }$ is regular in
$\k[\Omega ]_{0}$.
\end{proof}
\begin{theo}\label{tns3}
The variety $X_{R}$ is normal and Cohen-Macaulay.
\end{theo}
\begin{proof}
By Corollary~\ref{csav3}, $X_{R}$ is smooth in codimension $1$. So, by Serre's normality
criterion~\cite[\S 1,no 10, Th\'eor\`eme 4]{Bou1}, it suffices to prove that $X_{R}$ is
Cohen-Macaulay. According to~\cite[Ch. 8, Theorem 24.5]{Mat}, the set of points $x$ of
$X_{R}$ such that $\an {X_{R}}x$ is Cohen-Macaulay, is open. For $x$ in $X_{R}$, the
closure in $X_{R}$ of $R.x$ contains a fixed point. So it suffices to prove that for $x$
a fixed point under the action of $R$ in $X_{R}$, $\an {X_{R}}x$ is Cohen-Macaulay. Let
$V_{0}$ and $\Omega $ be as in Lemma~\ref{lns3}. Then $\Omega $ is an
affine open neighborhood of $V_{0}$ in $X_{R}$. By Proposition~\ref{pns3}(i),
$\an {\Omega }0$ is Cohen-Macaulay, whence the theorem since $\kappa $ is an isomorphism
from $\Omega $ onto an open neighborhood of $V_{0}$ in $X_{R}$ and $\kappa (0)=V_{0}$.
\end{proof}
\subsection{Nipotent cone and regular sequence in ${\cal O}_{{\cal E}}$}\label{ns4}
Let $\poi {\beta }1{,\ldots,}{d}{}{}{}$ be a basis of ${\goth t}^{*}$. For
$i=1,\ldots,d$, denote again by $\beta _{i}$ the element of ${\goth r}^{*}$ extending
$\beta _{i}$ and equal to $0$ on ${\goth a}$. For $\Lambda $ a complete subset of
${\cal R}$, denote by ${\goth t}_{\Lambda }^{\#}$ a complement to
${\goth t}_{\Lambda }$ in ${\goth t}$ and set
$$ R'_{\Lambda } := R_{{\goth t}_{\Lambda }^{\#},{\goth a}_{\Lambda }}
\quad \text{and} \quad {\cal E}_{\Lambda } :=
{\cal E}_{{\goth t}_{\Lambda }^{\#},{\goth a}_{\Lambda }}.$$
For $Y$ closed subset of $X_{R'_{\Lambda }}$, denote by ${\cal E}_{\Lambda ,Y}$ the
restriction of ${\cal E}_{\Lambda }$ to $Y$. Let ${\cal N}'_{\Lambda }$ be the image
of the map
$$ \xymatrix{ {\cal E}_{\Lambda , X_{R'_{\Lambda },\n}} \ar[rr] && {\cal E}}, \qquad
(V,x) \longmapsto (V\oplus {\goth t}_{\Lambda },x) $$
and ${\cal N}_{\Lambda }$ the closure in ${\cal E}$ of $R.{\cal N}'_{\Lambda }$.
\begin{lemma}\label{lns4}
For $i=1,\ldots,d$, let $\tilde{\beta }_{i}$ be the function on ${\cal E}$ defined by
$\tilde{\beta }_{i}(V,x)=\beta _{i}(x)$. Denote by ${\cal N}$ the nullvariety of
$\poi {\tilde{\beta }}1{,\ldots,}{d}{}{}{}$ in ${\cal E}$.
{\rm (i)} For all complete subset $\Lambda $ of ${\cal R}$, ${\cal N}_{\Lambda }$ is
a subvariety of ${\cal N}$ of dimension at most $n$.
{\rm (ii)} The varieyt ${\cal N}$ is the union of
${\cal N}_{\Lambda }, \Lambda \in {\cal P}_{c}({\cal R})$.
{\rm (iii)} The variety ${\cal N}$ is equidimensional of dimension $n$.
\end{lemma}
\begin{proof}
(i) Since ${\goth a}$ is the nullvariety of $\poi {\beta }1{,\ldots,}{d}{}{}{}$ in
${\goth r}$, ${\cal N}$ is the intersection of ${\cal E}$ and
$X_{R}\times {\goth a}$. By definition ${\cal N}'_{\Lambda }$ is contained in
$X_{R}\times {\goth a}$. Hence ${\cal N}_{\Lambda }$ is contained in ${\cal N}$.
By Proposition~\ref{psav3},
$$ \dim {\cal N}'_{\Lambda } = \dim {\goth t}_{\Lambda }^{\#} + \dim X_{R'_{\Lambda },\n}
\leq \dim {\goth a}_{\Lambda }.$$
Since the image of $X_{R'_{\Lambda },\n}$ by the map
$V\mapsto V\oplus {\goth t}_{\Lambda }$ is invariant by $R_{\Lambda }$,
$$ \dim {\cal N}_{\Lambda } \leq \dim {\cal N}'_{\Lambda }+
\dim {\goth a}-\dim {\goth a}_{\Lambda } \leq \dim {\goth a} .$$
(ii) Let $\varpi _{1}$ be the bundle projection of the vector bundle ${\cal E}$ over
$X_{R}$ and $\tau _{1}$ the restriction to ${\cal E}$ of the projection
$\xymatrix{X_{R}\times {\goth r} \ar[r] & {\goth r}}$. Let $T$ be an irreducible
component of ${\cal N}$. For all $V$ in $\varpi _{1}(T)$,
$\tau _{1}(\varpi _{1}^{-1}(V)\cap T)$ is a closed cone of ${\goth a}$. Hence
$\varpi _{1}(T)\times \{0\}$ is the intersection of $T$ and $X_{R}\times \{0\}$ so that
$\varpi _{1}(T)$ is a closed subset of $X_{R}$. Since ${\cal N}$ is the intersection of
${\cal E}$ and $X_{R}\times {\goth a}$, ${\cal N}$ and its irreducible components
are invariant under $R$. As a result, $\varpi _{1}(T)$ is invariant under $R$ and by
Proposition~\ref{psav1}, for some complete subset $\Lambda $ of ${\cal R}$ and for some
closed subset of $Z_{\Lambda }$ of $X_{R_{\Lambda }}$,
$\varpi _{1}(T)=\overline{R.Z_{\Lambda }}$. Moreover, by Lemma~\ref{lsav1}, for some
closed subset $Z'_{\Lambda }$ of $X_{R'_{\Lambda },\n}$, $Z_{\Lambda }$ is the image of
$Z'_{\Lambda }$ by the map $V\mapsto V\oplus {\goth t}_{\Lambda }$. As a result,
$$ {\cal E}_{\Lambda ,Z'_{\Lambda }} \subset {\cal E}_{\Lambda ,X_{R'_{\Lambda },\n}}
\quad \text{and} \quad
\varpi _{1}^{-1}(Z_{\Lambda })\cap X_{R}\times {\goth a} \subset {\cal N}'_{\Lambda } .$$
Then $T$ is contained in ${\cal N}_{\Lambda }$, whence the assertion by (i).
(iii) By (i) and (ii), since ${\cal R}$ is finite, the irreducible components of
${\cal N}$ have dimension at most $n$. As the nullvariety of $d$ functions
on the irreducible variety ${\cal E}_{X_{R}}$, the irreducible components of ${\cal N}$
have dimension at least $n$, whence the assertion.
\end{proof}
For $x$ in ${\cal E}$, denote by $I_{x}$ the subset of elements $i$ of
$\{1,\ldots,d\}$ such that $\tilde{\beta }_{i}(x)=0$.
\begin{coro}\label{cns4}
For all $x$ in ${\cal E}$, the sequence $\tilde{\beta }_{i},i\in I_{x}$ is regular
in $\an {{\cal E}}x$.
\end{coro}
\begin{proof}
According to Lemma~\ref{lns4}, for all subset $I$ of $\{1,\ldots,d\}$, the nullvariety
of $\tilde{\beta }_{i},i\in I$ in ${\cal E}$ is equidimensional of dimension
$n+d-\vert I \vert$. By Theorem~\ref{tns3} and Lemma~\ref{lsi}(iii),
${\cal E}$ is Cohen-Macaulay as a vector bundle over a Cohen-Macaulay variety, whence
the corollary by \cite[Ch. 6, Theorem 17.4]{Mat}.
\end{proof}
\section{Rational singularities for solvable Lie algebras} \label{rss}
Let ${\goth t}$ be a vector space of positive dimension $d$. Denote by
${\cal C}_{{\goth t},*}$ the full subcategory of ${\cal C}_{{\goth t}}$ whose objects
${\goth a}$ satisfy the following condition:
\begin{itemize}
\item [{\rm (4)}] there exist regular maps $\poi {\varepsilon }1{,\ldots,}{d}{}{}{}$ from
${\goth r}_{{\goth t},{\goth a}}$ to ${\goth r}_{{\goth t},{\goth a}}$ such that
$\poi x{}{,\ldots,}{}{\varepsilon }{1}{d}$ is a basis of
${\goth r}_{{\goth t},{\goth a}}^{x}$ for all $x$ in
${{\goth r}_{{\goth t},{\goth a}}}_{\r}$.
\end{itemize}
According to~\cite[Theorem 9]{Ko}, ${\goth u}$ is in ${\cal C}_{{\goth h},*}$.
\begin{lemma}\label{lrss}
Let ${\goth a}$ be in ${\cal C}_{{\goth t},*}$ and ${\goth a}'$
an ideal of ${\goth t}+{\goth a}$, contained in ${\goth a}$ and containing a fixed point
under the action of $R_{{\goth t},{\goth a}}$ in $X_{R_{{\goth t},{\goth a}}}$. Then
${\goth a}'$ is in ${\cal C}_{{\goth t},*}$.
\end{lemma}
\begin{proof}
Set ${\goth r}:={\goth t}+{\goth a}$ and ${\goth r}':={\goth t}+{\goth a}'$. According
to Corollary~\ref{c2sa6}(ii), ${\goth a}'$ is in ${\cal C}_{{\goth t}}$ since it is in
${\cal C}'_{{\goth t}}$. Set ${\goth t}_{\r}:={\goth r}_{\r}\cap {\goth t}$. As
${\cal R}_{{\goth t},{\goth a}'}$ is contained in ${\cal R}_{{\goth t},{\goth a}}$,
${\goth t}_{\r}$ is contained in ${\goth r}'_{\r}$ by Lemma~\ref{lsav2}(i). Then
${\goth r}'_{\r}$ is contained in ${\goth r}_{\r}$ and for all $x$ in
$A_{{\goth t},{\goth a'}}.{\goth t}_{\r}$, ${\goth r}^{x}={{\goth r}'}^{x}$ since
$A_{{\goth t},{\goth a'}}.{\goth t}_{\r}$ is a dense open subset of ${\goth r}'$ by
Lemma~\ref{lsav2}(i). So, for all regular map $\varepsilon $ from ${\goth r}$ to
${\goth r}$ such that $[x,\varepsilon (x)]=0$ for all $x$ in ${\goth r}$,
$\varepsilon (x)$ is in ${\goth r}'$ for all $x$ in ${\goth r}'$, whence the lemma.
\end{proof}
Let ${\goth a}$ be in ${\cal C}_{{\goth t},*}$. Set:
$$ {\cal R} := {\cal R}_{{\goth t},{\goth a}}, \qquad
{\goth r} := {\goth r}_{{\goth t},{\goth a}} \qquad \pi := \pi _{{\goth t},{\goth a}},
\qquad R := R_{{\goth t},{\goth a}}, \qquad A := A_{{\goth t},{\goth a}}, \qquad
{\cal E} := {\cal E}_{{\goth t},{\goth a}}, \qquad n := \dim {\goth a}.$$
The goal of the section is to prove that $X_{R}$ is Gorenstein with rational
singularities.
For $k$ positive integer, set:
$$ {\cal E}^{(k)} := \{(u,\poi x1{,\ldots,}{k}{}{}{}) \in X_{R}\times {\goth r}^{k}
\ \vert \ \poi {u\ni x}1{,\ldots,}{k}{}{}{}\}$$
and denote by ${\goth X}_{R,k}$ the image of ${\cal E}^{(k)}$ by the projection
$$ (u,\poi x1{,\ldots,}{k}{}{}{}) \longmapsto (\poi x1{,\ldots,}{k}{}{}{}) .$$
Since $X_{R}$ is a projective variety, ${\goth X}_{R,k}$ is a closed subset of
${\goth r}^{k}$, invariant under the diagonal action of $R$ in ${\goth r}^{k}$.
\subsection{Differential forms on some smooth open subsets of ${\goth X}_{R,k}$}
\label{rss1}
For $j=1,\ldots,k$, let $V_{j}^{(k)}$ be the subset of elements of ${\goth X}_{R,k}$ whose
$j$-th component is in ${\goth r}_{\r}$.
\begin{lemma}\label{lrss1}
For $j=1,\ldots,k$, $V_{j}^{(k)}$ is a smooth open subset of ${\goth X}_{R,k}$. Moreover,
$\Omega _{V_{j}^{(k)}}$ has a global section without zero.
\end{lemma}
\begin{proof}
Denoting by $\sigma _{j}$ the automorphism of ${\goth r}_{k}$ which permutes the first
and the $j$-th component, ${\goth X}_{R,k}$ is invariant under $\sigma _{j}$ and
$\sigma _{j}(V_{1}^{(k)}) = V_{j}^{(k)}$ so that we can suppose $j=1$. Moreover, for
$k=1$, ${\goth X}_{R,k}={\goth r}$ so that we can suppose $k\geq 2$. By definition,
$V_{1}^{(k)}$ is the intersection of ${\goth r}_{\r}\times {\goth r}^{k-1}$ and
${\goth X}_{R,k}$. Hence $V_{1}^{(k)}$ is an open susbet of ${\goth X}_{R,k}$ since
${\goth r}_{\r}$ is an open subset of ${\goth r}$.
Let $\poi {\varepsilon }1{,\ldots,}{d}{}{}{}$ satisfying Condition (4) with respect to
${\goth r}$. Let $\theta $ be the map
$$ \xymatrix{ {\goth r}_{\r}\times {\mathrm {M}}_{k-1,d}(\k) \ar[rr]^{\theta } &&
{\goth r}^{k}}, \qquad (x,a_{i,j},2\leq i\leq k,1\leq j\leq d) \longmapsto
(x,\sum_{j=1}^{d} a_{i,j}\varepsilon _{j}(x)) .$$
Since for all $(x,\poi x2{,\ldots,}{k}{}{}{})$ in $V_{1}^{(k)}$,
$\poi x2{,\ldots,}{k}{}{}{}$ are in ${\goth r}^{x}$, $\theta $ is a bijective map
onto $V_{1}^{(k)}$. The open subset ${\goth r}_{\r}$ has a cover by open subsets
$V$ such that for some $\poi e1{,\ldots,}{n}{}{}{}$ in ${\goth r}$,
$\poi x{}{,\ldots,}{}{\varepsilon }{1}{d},\poi e1{,\ldots,}{n}{}{}{}$ is a basis of
${\goth r}$ for all $x$ in $V$. Then there exist regular functions
$\poi {\varphi }1{,\ldots,}{d}{}{}{}$ on $V\times {\goth r}$ such that
$$ v-\sum_{j=1}^{\rg} \varphi _{j}(x,v)\varepsilon _{j}(x) \in
{\mathrm {span}}(\{\poi e1{,\ldots,}{n}{}{}{}\})$$
for all $(x,v)$ in $V\times {\goth r}$, so that the restriction of $\theta $ to
$V\times {\mathrm {M}_{k-1,d}}(\k)$ is an isomorphism onto
${\goth X}_{R,k}\cap V\times {\goth r}^{k-1}$ whose inverse is
$$(\poi x1{,\ldots,}{k}{}{}{}) \longmapsto
(x_{1},((\poi {x_{1},x_{i}}{}{,\ldots,}{}{\varphi }{1}{d}),i=2,\ldots,k))$$
As a result, $\theta $ is an isomorphism and $V_{1}^{(k)}$ is a smooth variety.
Since ${\goth r}_{\r}$ is a smooth open subset of the vector space ${\goth r}$, there
exists a regular differential form $\omega $ of top degree on
${\goth r}_{\r}\times {\mathrm {M}_{k-1,\rg}}(\k)$, without zero. Then
$\theta _{*}(\omega )$ is a regular differential form of top degree on
$V_{1}^{(k)}$, without zero.
\end{proof}
For $k\geq 2$ set:
$$V^{(k)} := V_{1}^{(k)} \cup V_{2}^{(k)} \quad \text{and} \quad
V_{1,2}^{(k)} := V_{1}^{(k)}\cap V_{2}^{(k)} .$$
For $2\leq k'\leq k$, the projection
$$ \xymatrix{ {\goth r}^{k} \ar[rr] && {\goth r}^{k'}}, \qquad
(\poi x1{,\ldots,}{k}{}{}{}) \longmapsto (\poi x1{,\ldots,}{k'}{}{}{})$$
induces the projection
$$ \xymatrix{ {\goth X}_{R,k} \ar[rr] && {\goth X}_{R,k'}}, \qquad
\xymatrix{ V_{j}^{(k)} \ar[rr] && V_{j}^{(k')}} $$
for $j=1,\ldots,k'$.
\begin{lemma}\label{l2rss1}
Suppose $k\geq 2$. Let $\omega $ be a regular differential form of top degree on
$V_{1}^{(k)}$, without zero. Denote by $\omega '$ its restriction to $V_{1,2}^{(k)}$.
{\rm (i)} For $\varphi $ in $\k[V_{1}^{(k)}]$, if $\varphi $ has no zero then $\varphi $
is in $\k^{*}$.
{\rm (ii)} For some invertible element $\psi $ of $\k[V_{1,2}^{(2)}]$,
$\omega ' = \psi {\sigma _{2}}_{*}(\omega ')$.
{\rm (iii)} The function $\psi (\psi \rond \sigma _{2})$ on $V_{1,2}^{(k)}$ is equal
to $1$.
\end{lemma}
\begin{proof}
The existence of $\omega $ results from Lemma~\ref{lrss1}.
(i) According to Lemma~\ref{lrss1}, there is an isomorphism $\theta $ from
${\goth r}_{\r}\times {\mathrm {M}_{k-1,d}}(\k)$ onto $V_{1}^{(k)}$. Since
$\varphi $ is invertible, $\varphi \rond \theta $ is an invertible element of
$\k[{\goth r}_{\r}]$. According to Lemma~\ref{lsav2}(iii),
$\k[{\goth r}_{\r}]=\k[{\goth r}]$. Hence $\varphi $ is in $\k^{*}$.
(ii) The open subset $V_{1,2}^{(k)}$ is invariant under $\sigma _{2}$ so that
$\omega '$ and ${\sigma _{2}}_{*}(\omega ')$ are regular differential forms of top
degree on $V_{1,2}^{(k)}$, without zero. Then for some invertible element $\psi $
of $\k[V_{1,2}^{(k)}]$, $\omega ' = \psi {\sigma _{2}}_{*}(\omega ')$. Let
$O_{2}$ be the set of elements $(x,a_{i,j},1\leq i \leq k-1,1\leq j\leq d)$ of
${\goth r}_{\r}\times {\mathrm {M}_{k-1,d}}(\k)$ such that
$$ a_{1,1}\varepsilon _{1}(x) + \cdots + a_{1,\rg} \varepsilon _{\rg}(x) \in
{\goth r}_{\r}.$$
Then $O_{2}$ is the inverse image of $V_{1,2}^{(k)}$ by $\theta $. As a result,
$\k[V_{1,2}^{(k)}]$ is a polynomial algebra over $\k[V_{1,2}^{(2)}]$ since for
$k=2$, $O_{2}$ is the inverse image by $\theta $ of $V_{1,2}^{(2)}$. Hence
$\psi $ is in $\k[V_{1,2}^{(2)}]$ since $\psi $ is invertible.
(iii) Since the restriction of $\sigma _{2}$ to $V_{1,2}^{(k)}$ is an involution,
$$ {\sigma _{2}}_{*}(\omega ') = (\psi \rond \sigma _{2}) \omega ' =
(\psi \rond \sigma _{2})\psi {\sigma _{2}}_{*}(\omega '),$$
whence $(\psi \rond \sigma _{2})\psi = 1$.
\end{proof}
\begin{coro}\label{crss1}
The function $\psi $ is invariant under the action of $R$ in $V_{1,2}^{(k)}$ and for
some sequence $m_{\alpha },\alpha \in {\cal R}$ in ${\Bbb Z}$,
$$ \psi (\poi x1{,\ldots,}{k}{}{}{}) = \pm \prod_{\alpha \in {\cal R}}
(\alpha (x_{1})\alpha (x_{2})^{-1})^{m_{\alpha }},$$
for all $(\poi x1{,\ldots,}{k}{}{}{})$ in ${\goth t}_{\r}^{2}\times {\goth t}^{k-2}$.
\end{coro}
\begin{proof}
First of all, since $V_{1}^{(k)}$ and $V_{2}^{(k)}$ are invariant under the action of
$R$ in ${\goth X}_{R,k}$, so is $V_{1,2}^{(k)}$. Let $g$ be in $R$. As $\omega $ has
no zero, $g.\omega =p_{g}\omega $ for some invertible element $p_{g}$ of
$\k[V_{1}^{(k)}]$. By Lemma~\ref{l2rss1}(i), $p_{g}$ is in $\k^{*}$. Since
$\sigma _{2}$ is a $R$-equivariant isomorphism from $V_{1}^{(k)}$ onto $V_{2}^{(k)}$,
$$ g.{\sigma _{2}}_{*}(\omega )=p_{g}{\sigma _{2}}_{*}(\omega ) \quad \text{and} \quad
p_{g} \omega ' = g.\omega ' = (g.\psi ) g.{\sigma _{2}}_{*}(\omega ') =
p_{g} (g.\psi ) {\sigma _{2}}_{*}(\omega '),$$
whence $g.\psi =\psi $.
The open subset ${\goth t}_{\r}^{2}$ of ${\goth t}^{2}$ is the complement to the
nullvariety of the function
$$ (x,y) \longmapsto \prod_{\alpha \in {\cal R}} \alpha (x)\alpha (y).$$
Then, by Lemma~\ref{l2rss1}(ii), for some $a$ in $\k^{*}$ and for some
sequences $m_{\alpha },\alpha \in {\cal R}$ and
$n_{\alpha },\alpha \in {\cal R}$ in ${\Bbb Z}$,
$$ \psi (\poi x1{,\ldots,}{k}{}{}{}) = a \prod_{\alpha \in {\cal R}}
\alpha (x_{1})^{m_{\alpha }}\alpha (x_{2})^{n_{\alpha }},$$
for all $(\poi x1{,\ldots,}{k}{}{}{})$ in ${\goth t}_{\r}^{2}\times {\goth t}^{k-2}$.
By Lemma~\ref{l2rss1}(iii),
$$ a^{2} \prod_{\alpha \in {\cal R}} \alpha (x)^{m_{\alpha }+n_{\alpha }}
\alpha (y)^{m_{\alpha }+n_{\alpha }} = 1,$$
for all $(x,y)$ in ${\goth t}_{\r}^{2}$. Hence $a^{2}=1$ and $m_{\alpha }+n_{\alpha }=0$
for all $\alpha $ in ${\cal R}$.
\end{proof}
According to Lemma~\ref{l2sav2}(i), for $\alpha $ in ${\cal R}$, $\theta _{\alpha }$ is
a bijective regular map from ${\Bbb P}^{1}(\k)$ onto the closed subset $Z_{\alpha }$ of
$X_{R}$ such that $\theta _{\alpha }(\infty )=V_{\alpha }$. Recall that $x_{\alpha }$ is
a generator of ${\goth a}^{\alpha }$ and $h_{\alpha }$ is an element of ${\goth t}$ such
that $\alpha (h_{\alpha })=1$. Denote by ${\goth t}'_{\alpha }$ the subset of elements
$x$ of ${\goth t}_{\alpha }$ such that $\gamma (x) \neq 0$ for all $\gamma $ in
${\cal R}\setminus \{\alpha \}$. According to Condition (3) of Section~\ref{sa},
${\goth t}'_{\alpha }$ is a dense open subset of ${\goth t}_{\alpha }$. Let
$x_{-\alpha }$ be in ${\goth r}^{*}$ orthogonal to ${\goth t}+{\goth a}^{\gamma }$ for
all $\gamma $ in ${\cal R}\setminus \{\alpha \}$ and such that
$x_{-\alpha }(x_{\alpha })=1$.
\begin{lemma}\label{l3rss1}
Suppose $k\geq 2$. Let $\alpha $ be in ${\cal R}$, $x_{0}$ and $y_{0}$ in
${\goth t}'_{\alpha }$. Set:
$$ E := \k x_{0} \oplus \k h_{\alpha } \oplus {\goth a}^{\alpha }, \quad
E_{*} := x_{0} \oplus \k h_{\alpha } \oplus {\goth a}^{\alpha } , \quad
E_{*,1} := x_{0} \oplus \k h_{\alpha } \oplus ({\goth a}^{\alpha }\setminus \{0\}),
\quad
E_{*,2} = y_{0} \oplus \k h_{\alpha } \oplus ({\goth a}^{\alpha }\setminus \{0\}).$$
{\rm (i)} For $x$ in $E_{*}$, ${\goth r}^{x}$ is contained in ${\goth t}_{\alpha }+E$.
{\rm (ii)} For $V$ subspace of dimension $d$ of ${\goth t}_{\alpha }+E$, $V$ is in
$X_{R}$ if and only if it is in $Z_{\alpha }$.
{\rm (iii)} The intersection of $E_{*,1}\times E_{*,2}$ and ${\goth X}_{R,2}$ is the
nullvariety of the function
$$ (x,y) \longmapsto x_{-\alpha }(y) \alpha (x) - x_{-\alpha }(x) \alpha (y) $$
on $E_{*,1}\times E_{*,2}$.
\end{lemma}
\begin{proof}
(i) If $x$ is regular semisimple, its component on $h_{\alpha }$ is different
from $0$ so that ${\goth r}^{x}=\theta _{\alpha }(z)$ for some $z$ in $\k$. Suppose that
$x$ is not regular semisimple. Then $x$ is in $x_{0}+{\goth a}^{\alpha }$. Hence
${\goth r}^{x}$ is contained in ${\goth t}_{\alpha }+E$ since so is ${\goth r}^{x_{0}}$.
(ii) All element of $Z_{\alpha }$ is contained in ${\goth t}_{\alpha }+E$. Let $V$ be an
element of $X_{R}$, contained in ${\goth t}_{\alpha }+E$. According
to Corollary~\ref{c2sa6}(i), $V$ is an algebraic commutative
subalgebra of dimension $d$ of ${\goth r}$. By (i), $V=\theta _{\alpha }(z)$ for some $z$
in $\k$ if $V$ is in $A.{\goth t}$. Otherwise, $x_{\alpha }$ is in $V$. Then
$V=\theta _{\alpha }(\infty )$ since $\theta _{\alpha }(\infty )$ is the centralizer
of $x_{\alpha }$ in ${\goth t}_{\alpha }+E$.
(iii) Let $(x,y)$ be in $E_{*,1}\times E_{*,2}\cap {\goth X}_{R,2}$. By definition,
for some $V$ in $X_{R}$, $x$ and $y$ are in $V$. By (i) and (ii),
$V=\theta_{\alpha }(z)$ for some $z$ in ${\Bbb P}^{1}(\k)$. For $z$ in $\k$,
$$ x = x_{0}+s(h_{\alpha }-z x_{\alpha }) \quad \text{and} \quad
y = y_{0} + s'(h_{\alpha } - z x_{\alpha })$$
for some $s$, $s'$ in $\k$, whence the equality of the assertion. For
$z=\infty $, $$ x = x_{0}+sx_{\alpha } \quad \text{and} \quad
y = y_{0} + s'x_{\alpha } $$
for some $s$, $s'$ in $\k$ so that $\alpha (x)=\alpha (y) = 0$. Conversely,
let $(x,y)$ be in $E_{*,1}\times E_{*,2}$ such that
$$ x_{-\alpha }(y) \alpha (x) - x_{-\alpha }(x) \alpha (y) = 0.$$
If $\alpha (x)=0$ then $\alpha (y) = 0$ and $x$ and $y$ are in
$V_{\alpha }=\theta _{\alpha }(\infty )$. If $\alpha (x)\neq 0$, then
$\alpha (y) \neq 0$ and
$$x \in \theta _{\alpha } (-\frac{x_{-\alpha }(x)}{\alpha (x)})
\quad \text{and} \quad
y \in \theta _{\alpha } (-\frac{x_{-\alpha }(x)}{\alpha (x)}) ,$$
whence the assertion.
\end{proof}
Set $V^{(1)}:={\goth r}_{\r}$.
\begin{prop}\label{prss1}
For $k$ positive integer, there exists on $V^{(k)}$ a regular differential form of top
degree without zero.
\end{prop}
\begin{proof}
For $k=1$, it is true since ${\goth r}_{\r}$ is an open subset of the vector sapce
${\goth r}$. So we can suppose $k\geq 2$. According to Corollary~\ref{crss1}, it suffices
to prove $m_{\alpha }=0$ for all $\alpha $ in ${\cal R}$. Indeed, if so, by
Corollary~\ref{crss1}, $\psi =\pm 1$ on the open subset
$R.({\goth t}_{\r}^{2}\times {\goth t}^{k-2})$ of $V^{(k)}$ so that $\psi =\pm 1$ on
$V_{1,2}^{(k)}$. Then, by Lemma~\ref{l2rss1}(ii), $\omega $ and
$\pm {\sigma _{2}}_{*}(\omega )$ have the same restriction to $V_{1,2}^{(k)}$ so that
there exists a regular differential form of top degree $\tilde{\omega }$ on $V^{(k)}$
whose restrictions to $V_{1}^{(k)}$ and $V_{2}^{(k)}$ are $\omega $ and
$\pm {\sigma _{2}}_{*}(\omega )$ respectively. Moreover, $\tilde{\omega }$ has no zero
since so has $\omega $.
Since $\psi $ is in $\k[V_{1,2}^{(2)}]$ by Lemma~\ref{l2rss1}(ii), we can suppose $k=2$.
Let $\alpha $ be in ${\cal R}$, $E$, $E_{*}$,$E_{*,1}$, $E_{*,2}$ as in
Lemma~\ref{l2rss1}. Suppose $m_{\alpha }\neq 0$. A contradiction is expected. The
restriction of $\psi $ to $E_{*,1}\times E_{*,2}\cap V_{1,2}^{(2)}$ is given by
$$ \psi (x,y) = a x_{-\alpha }(x)^{m}x_{-\alpha }(y)^{n} ,$$
with $a$ in $\k^{*}$ and $(m,n)$ in ${\Bbb Z}^{2}$ since $\psi $ is an invertible element
of $\k[V_{1,2}^{(2)}]$. According to Lemma~\ref{l2rss1}(iii), $n=-m$ and $a=\pm 1$.
Interchanging the role of $x$ and $y$, we can suppose $m$ in ${\Bbb N}$. For $(x,y)$ in
$E_{*,1}\times E_{*,2}\cap V_{1,2}^{(2)}$ such that $\alpha (x)\neq 0$,
$\alpha (y)\neq 0$ and
$$ \psi (x,y) = \pm x_{-\alpha }(x)^{m}
(\frac{x_{-\alpha }(x) \alpha (y)}{\alpha (x)})^{-m} =
\pm \alpha (x)^{m} \alpha (y)^{-m} .$$
As a result, by Corollary~\ref{crss1}, for $x$ in $x_{0}+\k^{*} h_{\alpha }$ and
$y$ in $y_{0}+\k^{*}h_{\alpha }$,
\begin{eqnarray}\label{eqrss1}
\pm \alpha (x)^{m}\alpha (y)^{-m} = \pm \prod_{\gamma \in {\cal R}}
\gamma (x)^{m_{\gamma }}\gamma (y)^{-m_{\gamma }} .
\end{eqnarray}
For $\gamma $ in ${\cal R}$,
$$ \gamma (x) = \gamma (x_{0}) + \alpha (x)\gamma (h_{\alpha })
\quad \text{and} \quad
\gamma (y) = \gamma (y_{0}) + \alpha (y)\gamma (h_{\alpha }) .$$
Since $m$ is in ${\Bbb N}$,
\begin{eqnarray}\label{eq2rss1}
\pm \alpha (x)^{m} \prod_{\mycom{\gamma \in {\cal R}}{m_{\gamma }>0}}
(\gamma (y_{0}) + \alpha (y)\gamma (h_{\alpha }))^{m_{\gamma }}
\prod_{\mycom{\gamma \in {\cal R}}{m_{\gamma }< 0}}
(\gamma (x_{0}) + \alpha (x)\gamma (h_{\alpha }))^{-m_{\gamma }} = \\
\nonumber \pm \alpha (y)^{m} \prod_{\mycom{\gamma \in {\cal R}}{m_{\gamma }>0}}
(\gamma (x_{0}) + \alpha (x)\gamma (h_{\alpha }))^{m_{\gamma }}
\prod_{\mycom{\gamma \in {\cal R}}{m_{\gamma } < 0}}
(\gamma (y_{0}) + \alpha (y)\gamma (h_{\alpha }))^{-m_{\gamma }} .
\end{eqnarray}
For $m_{\alpha }$ positive, the terms of lowest degree in $(\alpha (x),\alpha (y))$ of
left and right sides are
$$ \pm \alpha (x)^{m} \alpha (y)^{m_{\alpha }}
\prod_{\mycom{\gamma \in {\cal R}\setminus \{\alpha \}}{m_{\gamma }>0}}
\gamma (y_{0})^{m_{\gamma }}
\prod_{\mycom{\gamma \in {\cal R}\setminus \{\alpha \}}{m_{\gamma }<0}}
\gamma (x_{0})^{-m_{\gamma }}
\quad \text{and} \quad
\pm \alpha (y)^{m} \alpha (x)^{m_{\alpha }}
\prod_{\mycom{\gamma \in {\cal R}\setminus \{\alpha \}}{m_{\gamma }>0}}
\gamma (x_{0})^{m_{\gamma }}
\prod_{\mycom{\gamma \in {\cal R}\setminus \{\alpha \}}{m_{\gamma }<0}}
\gamma (y_{0})^{-m_{\gamma }} $$
respectively and for $m_{\alpha }$ negative, the terms of lowest degree in
$(\alpha (x),\alpha (y))$ of left and right sides are
$$ \pm \alpha (x)^{m+m_{\alpha }}
\prod_{\mycom{\gamma \in {\cal R}\setminus \{\alpha \}}{m_{\gamma }>0}}
\gamma (y_{0})^{m_{\gamma }}
\prod_{\mycom{\gamma \in {\cal R}\setminus \{\alpha \}}{m_{\gamma }<0}}
\gamma (x_{0})^{-m_{\gamma }}
\quad \text{and} \quad
\pm \alpha (y)^{m+m_{\alpha }}
\prod_{\mycom{\gamma \in {\cal R}\setminus \{\alpha \}}{m_{\gamma }>0}}
\gamma (x_{0})^{m_{\gamma }}
\prod_{\mycom{\gamma \in {\cal R}\setminus \{\alpha \}}{m_{\gamma }<0}}
\gamma (y_{0})^{-m_{\gamma }} $$
respectively. From the equality of these terms, we deduce $m = \pm m_{\alpha }$ and
$$ \prod_{\mycom{\gamma \in {\cal R}\setminus \{\alpha \}}{m_{\gamma }>0}}
\gamma (y_{0})^{m_{\gamma }}
\prod_{\mycom{\gamma \in {\cal R}\setminus \{\alpha \}}{m_{\gamma }<0}}
\gamma (x_{0})^{-m_{\gamma }} =
\pm \prod_{\mycom{\gamma \in {\cal R}\setminus \{\alpha \}}{m_{\gamma }>0}}
\gamma (x_{0})^{m_{\gamma }}
\prod_{\mycom{\gamma \in {\cal R}\setminus \{\alpha \}}{m_{\gamma }<0}}
\gamma (y_{0})^{-m_{\gamma }} .$$
Since the last equality does not depend on the choice of $x_{0}$ and $y_{0}$ in
${\goth t}'_{\alpha }$, this equality remains true for all
$(x_{0},y_{0})$ in ${\goth t}_{\alpha }\times {\goth t}_{\alpha }$. As a result,
as the degrees in $\alpha (x)$ of the left and right sides of Equality (\ref{eq2rss1})
are the same,
\begin{eqnarray}\label{eq3rss1}
m - \sum_{\mycom{\gamma \in {\cal R}}{m_{\gamma }< 0\; \text{and} \;
\gamma (h_{\alpha })\neq 0}} m_{\gamma } =
\sum_{\mycom{\gamma \in {\cal R}}{m_{\gamma } > 0 \; \text{and} \;
\gamma (h_{\alpha })\neq 0}} m_{\gamma } .
\end{eqnarray}
Suppose $m=m_{\alpha }$. By Equality (\ref{eqrss1}),
$$ \prod_{\gamma \in {\cal R}\setminus \{\alpha \}}
\gamma (x)^{m_{\gamma }}\gamma (y)^{-m_{\gamma }} = \pm 1.$$
Since this equality does not depend on the choice of $x_{0}$ and $y_{0}$ in
${\goth t}'_{\alpha }$, it holds for all $(x,y)$ in
${\goth t}_{\r}\times {\goth t}_{\r}$. Hence $m_{\gamma }=0$ for all $\gamma $ in
${\cal R}\setminus \{\alpha \}$ and $m=0$ by Equality (\ref{eq3rss1}). It is
impossible since $m_{\alpha }\neq 0$. Hence $m=-m_{\alpha }$. Then, by
Equality (\ref{eqrss1})
$$ \prod_{\gamma \in {\cal R}\setminus \{\alpha \}}
\gamma (x)^{m_{\gamma }}\gamma (y)^{-m_{\gamma }} =
\pm \alpha (x)^{2m}\alpha (y)^{-2m} .$$
Since this equality does not depend on the choice of $x_{0}$ and $y_{0}$ in
${\goth t}'_{\alpha }$, it holds for all $(x,y)$ in
${\goth t}_{\r}\times {\goth t}_{\r}$. Then $m=0$, whence the contradiction.
\end{proof}
\subsection{Rational singularities and Gorensteinness of $X_{R}$} \label{rss2}
For $Y$ subvariety of $\ec {Gr}r{}{}d$, denote by ${\cal E}_{Y}$ the restriction to $Y$
of the tautological vector bundle of rank $d$ over $\ec {Gr}r{}{}d$. In particular, for
$Y$ contained in $X_{R}$, ${\cal E}_{Y}$ is a subvariety of ${\cal E}$. For $k$ positive
integer, denote by $\tau _{k}$ and $\varpi _{k}$ the restrictions to ${\cal E}^{(k)}$ of
the canonical projections
$$ \xymatrix{X_{R}\times {\goth r}^{k}\ar[rr]^{\tau _{k}} && {\goth r}^{k}}
\quad \text{and} \quad
\xymatrix{X_{R}\times {\goth r}^{k}\ar[rr]^{\varpi _{k}} && X_{R}} . $$
\begin{lemma}\label{lrss2}
{\rm (i)} The morphism $\tau _{k}$ is a projective and birational morphism onto
${\goth X}_{R,k}$.
{\rm (ii)} The sets $V^{(k)}$ and $\tau _{k}^{-1}(V^{(k)})$ are smooth open subsets
of ${\goth X}_{R,k}$ and ${\cal E}^{(k)}$. Moreover, for $k\geq 2$, they are big open subsets
of ${\goth X}_{R,k}$ and ${\cal E}^{(k)}$.
{\rm (iii)} The restriction of $\tau _{k}$ to $\tau _{k}^{-1}(V^{(k)})$ is an isomorphism
onto $V^{(k)}$.
\end{lemma}
\begin{proof}
Since $X_{R}$ is a projective variety, $\tau _{k}$ is projective and its image
is ${\goth X}_{R,k}$ by definition. For $(\poi x1{,\ldots,}{k}{}{}{})$ in $V^{(k)}$ and
$(u,\poi x1{,\ldots,}{k}{}{}{})$ in $\tau _{k}^{-1}((\poi x1{,\ldots,}{k}{}{}{}))$,
$u={\goth r}^{x_{1}}$ if $x_{1}$ is in ${\goth r}_{\r}$ and $u={\goth r}^{x_{2}}$ if
$x_{2}$ is in ${\goth r}_{\r}$. As a result, the restriction of $\tau _{k}$ to
$\tau _{k}^{-1}(V^{(k)})$ is a bijective morphism onto $V^{(k)}$. Hence $\tau _{k}$
is a birational morphism and by Zariski's Main Theorem~\cite[\S 9]{Mu}, this restriction
is an isomorphism since $V^{(k)}$ is a smooth variety by Lemma~\ref{lrss1}. So it
remains to prove that for $k\geq 2$, $\tau _{k}^{-1}(V^{(k)})$ is a big open subset of
${\cal E}^{(k)}$
Suppose that ${\cal E}^{(k)}\setminus \tau _{k}^{-1}(V^{(k)})$ has an irreducible
component $\Sigma $ of dimension $\dim {\cal E}^{(k)}-1$. A contradiction is expected.
Since ${\cal E}^{(k)}$ and $\tau _{k}^{-1}(V_{k})$ are invariant under the automorphisms
of $X_{R}\times {\goth r}^{k}$,
$$ (u,\poi x1{,\ldots,}{k}{}{}{}) \longmapsto (u,\poi {tx}1{,\ldots,}{k}{}{}{}),
\qquad (t\in \k^{*}),$$
so is $\Sigma $. Then $\Sigma \cap X_{R}\times \{0\}=\varpi _{k}(\Sigma )\times \{0\}$ so
that $\varpi _{k}(\Sigma )$ is a closed subset of $X_{R}$. Since
$\dim \Sigma = \dim {\cal E}^{(k)}-1$,
$\dim \varpi _{k}(\Sigma )\geq \dim X_{R}-1$. Suppose $\dim \Sigma =\dim X_{R}-1$. Then
for all $u$ in $\varpi _{k}(\Sigma )$, $\{u\}\times u^{k}$ is in $\Sigma $. It is
impossible since for all $u$ in a dense open subset of $\varpi _{k}(\Sigma )$,
$u={\goth r}^{x}$ for some $x$ in ${\goth r}_{\r}$ by Corollary~\ref{csav3}. Hence
$\varpi _{k}(\Sigma )=X_{R}$. Then for all $u$ in a dense open subset of $X'_{R}$,
$\{u\}\times u^{k}\cap \Sigma $ has codimension $1$ in $\{u\}\times u^{k}$. Since
the image of $\{u\}\times u^{k}\cap \Sigma $ by the projection
$$ (u,\poi x1{,\ldots,}{k}{}{}{}) \longmapsto x_{1}$$
is not dense in $u$, for all $x_{1}$ in a dense open subset of its image,
$\{u\}\times \{x_{1}\}\times u^{k-1}$ is contained in $\Sigma $, whence the contradiction
since $u\cap {\goth r}_{\r}$ is not empty.
\end{proof}
By definition, ${\cal E}^{(k)}$ is the inverse image of $X_{R}$ by the bundle projection
of the vector bundle
$$ \{u,\poi x1{,\ldots,}{k}{}{}{}) \in \ec {Gr}r{}{}d\times {\goth r}^{k} \; \vert \;
\poi {u\ni x}1{,\ldots,}{k}{}{}{}\} $$
over $\ec {Gr}r{}{}d$ so that ${\cal E}^{(k)}$ is vector bundle of rank $kd$ over $X_{R}$.
In particular, ${\cal E}^{(1)}={\cal E}$. According to~\cite{Hi}, there exists a
desingulartization $\Gamma $ of $X_{R}$ with morphism $\rho $ such that the restriction
of $\rho $ to $\rho ^{-1}({X_{R}}_{\loc})$ is an isomorphism onto ${X_{R}}_{\loc}$.
Let $\widetilde{{\cal E}^{(1)}}$ be the following fiber product
$$ \xymatrix{ \widetilde{{\cal E}^{(1)}} \ar[rr]^{\overline{\rho }} \ar[d] &&
{\cal E}^{(1)} \ar[d]^{\varpi _{1}}\\ \Gamma \ar[rr]_{\rho } && X_{R}}$$
with $\overline{\rho }$ the restriction map. Then $\widetilde{{\cal E}^{(1)}}$ is
a vector bundle of rank $d$ over $\Gamma $. In particular, it is a smooth variety
since $\Gamma $ is smooth.
Let $O$ be a trivialization open subset of the vector bundle ${\cal E}^{(1)}$ and
let $\Phi _{1}$ be a local trivialization over $O$ of
${\cal E}^{(1)}$, whence the following commutative diagram
$$ \xymatrix{ \varpi _{1}^{-1}(O) \ar[rr]^{\Phi _{1}} \ar[rrd]_{\varpi _{1}} &&
O \times \k^{d} \ar[d]^{\pr {1}} \\ && O} .$$
Then $O$ is a trivialization open subset of the vector bundle ${\cal E}^{(k)}$. The
variety ${\cal E}^{(1)}$ is a closed subbundle of ${\cal E}^{(k)}$ over
$X_{R}$ and for some local trivialization $\Phi $ over $O$ of
${\cal E}^{(k)}$, we have the following commutative diagram
$$ \xymatrix{ \varpi _{k} ^{-1}(O) \ar[rr]^{\Phi } \ar[rrd]_{\varpi _{k}} &&
O \times \k^{kd} \ar[d]^{\pr {1}} \\ && O} ,$$
$\Phi _{1}$ is the restriction of $\Phi $ to $\varpi _{1}^{-1}(O)$ and
$\Phi (\varpi _{1}^{-1}(O)) = O \times \k^{d}\times \{0\}$.
\begin{lemma}\label{l2rss2}
Suppose $k\geq 2$. Denote by $\mu $ a generator of $\Omega _{\k^{kd}}$ and by
$\tilde{\rho }$ the restriction of $\rho \mul {\mathrm {id}}_{\k^{kd}}$ to
$\rho ^{-1}(O)\times \k^{kd}$.
{\rm (i)} The sheaf $\Omega _{{{\cal E}^{(k)}}_{\loc}}$ has a global section
$\omega $ without zero.
{\rm (ii)} The sheaf $\Omega _{O_{{\mathrm {\loc}}}}$ has a global section $\omega _{O}$
without zero.
{\rm (iii)} For some $p$ in $\k[O\times \k^{kd}]\setminus \{0\}$,
$\tilde{\rho }^{*}(p(\omega _{O}\wedge \mu ))$ has a regular extension to
$\rho ^{-1}(O)\times \k^{kd}$.
\end{lemma}
\begin{proof}
(i) According to Proposition~\ref{prss1} and Lemma~\ref{lrss2}(iii),
$\Omega _{\tau _{k}^{-1}(V^{(k)})}$ has a global section without zero.
By Lemma~\ref{lrss2}(ii), $\tau _{k}^{-1}(V^{(k)})$ is a smooth big open subset
of ${\cal E}^{(k)}$. So, by Lemma~\ref{lars},
$\Omega _{{{\cal E}^{(k)}}_{\loc}}$ has a global section without zero.
(ii) Since $\mu $ is a generator of $\Omega _{\k^{kd}}$, there exists a unique $\nu $ in
$\tk {\k}{\k[\k^{kd}]}\Gamma (O_{\loc},\Omega _{O_{\loc}})$ such that
$$ \Phi _{*}(\omega \left \vert \right._{\varpi _{k}^{-1}(O_{\loc})}) =
\nu \wedge \mu .$$
Moreover, $\nu $ has no zero since so has $\omega $. Let $V$ be an affine open subset of
$O_{\loc}$ such that the restriction of $\Omega _{O_{\loc}}$ to $V$ is locally free,
generated by the local section $\omega _{V}$. Then for some $p_{V}$ in
$\k[V\times \k^{kd}]$,
\begin{eqnarray}\label{eqrss2}
\Phi _{*}(\omega \left \vert \right._{\varpi _{k}^{-1}(V)}) = p_{V}\omega _{V}\wedge \mu .
\end{eqnarray}
Then $p_{V}$ has no zero since so has $\nu \wedge \mu $. As a result, $p_{V}$ is in
$\k[V]$ and $p_{V}\omega _{V}$ is a local section of $\Omega _{O_{\loc}}$ without zero.
By the unicity of the decomposition (\ref{eqrss2}), for two different affine open subsets
$V$ and $V'$ as above, the differential forms $p_{V}\omega _{V}$ and $p_{V'}\omega _{V'}$
have the same restriction to $V\cap V'$. As a result, since such affine open subsets
cover $O_{\loc}$, for some global section $\omega _{O}$ of $\Omega _{O_{\loc}}$,
$$ \Phi _{*}(\omega \left \vert \right._{\varpi _{k}^{-1}(O_{\loc})}) =
\omega _{O} \wedge \mu .$$
Moreover, $\omega _{O}$ is unique and has no zero.
(iii) Let $\omega _{1}$ be a generator of $\Omega _{{\goth r}}$ and let $\mu _{1}$ be
a generator of $\Omega _{\k^{d}}$. By (i), $\omega _{O}\wedge \mu _{1}$ is a global
section of $\Omega _{O_{\loc}\times \k^{d}}$, without zero. So for some regular
function $p$ on $O_{\loc}\times \k^{d}$,
\begin{eqnarray}\label{eq2rs}
{\Phi _{1}}_{*}((\tau _{1})^{*}(\omega _{1})
\left \vert \right. _{\varpi _{1}^{-1}(O_{\loc})}) = p \omega _{O}\wedge \mu _{1}
. \end{eqnarray}
According to Theorem~\ref{tns3}, $X_{R}$ is normal. Then so is $O$ and $p$ has a regular
extension to $O\times \k^{d}$. Denote again by $p$ this extension. According to Equality
(\ref{eq2rs}), the differential form
$\tilde{\rho }^{*}(p\omega _{O}\wedge \mu _{1})$ on $\rho ^{-1}(O_{\loc})\times \k^{d}$
has a regular extension to $\rho ^{-1}(O)\times \k^{d}$. In fact, denoting by
$\overline{\Phi _{1}}$ the local trivialization over $\rho ^{-1}(O)$ of
$\widetilde{{\cal E}^{(1)}}$ such that the following diagram
$$ \xymatrix{ (\varpi _{1}\rond \overline{\rho }^{-1})(O)
\ar[rr]^{\overline{\Phi _{1}}} \ar[d]_{\overline{\rho }} &&
\rho ^{-1}(O)\times \k^{d} \ar[d]^{\tilde{\rho }} \\
\varpi _{1}^{-1}(O) \ar[rr]_{\Phi _{1}} && O\times \k^{d} }$$
is commutative, it is the restriction to $\rho ^{-1}(O_{\loc})\times \k^{d}$ of
$$\overline{\Phi _{1}}_{*}((\tau_{1}\rond \overline{\rho })^{*}(\omega _{1})
\left \vert \right. _{(\varpi _{1}\rond \overline{\rho }^{-1})^{-1}(O)} ).$$
For some generator $\mu '$ of $\Omega _{\k^{(k-1)d}}$, $\mu = \mu _{1}\wedge \mu '$
and $\k[O\times \k^{d}]$ is naturally embedded in $\k[O\times \k^{kd}]$.
As a result, $\tilde{\rho }^{*}(p\omega _{O}\wedge \mu )$
has a regular extension to $\rho ^{-1}(O)\times \k^{kd}$.
\end{proof}
\begin{prop}\label{prss2}
The variety $X_{R}$ is Gorenstein with rational singularities.
\end{prop}
\begin{proof}
According to Theorem~\ref{tns3}, $X_{R}$ is normal and Cohen-Macaulay. Then
by Lemma~\ref{l2rss2},(ii) and (iii) and Corollary~\ref{cars},
$O\times \k^{kd}$ is Gorenstein with rational singularities. Then so is $O$ by
Lemma~\ref{lsi},(i) and (ii). Since there is a cover of $X_{R}$ by open subsets as $O$,
$X_{R}$ is Gorenstein with rational singularities.
\end{proof}
As already mentioned, ${\goth u}$ is in ${\cal C}_{{\goth h},*}$, whence
Theorem~\ref{tint} by Proposition~\ref{prss2}.
\appendix
\section{Rational Singularities} \label{ars}
Let $X$ be an affine irreducible normal variety.
\begin{lemma}\label{lars}
Let $Y$ be a smooth big open subset of $X$.
{\rm (i)} All regular differential form of top degree on $Y$ has a unique regular
extension to $X_{\loc}$.
{\rm (ii)} Suppose that $\omega $ is a regular differential form of top degree on $Y$,
without zero. Then the regular extension of $\omega $ to $X_{\loc}$ has no zero.
\end{lemma}
\begin{proof}
(i) Since $\Omega _{X_{\loc}}$ is a locally free module of rank one, there is an affine
open cover $\poi O1{,\ldots,}{k}{}{}{}$ of $X_{\loc}$ such that the restriction of
$\Omega _{X_{\loc}}$ to $O_{i}$ is a free $\an {O_{i}}{}$-module generated by some
section $\omega _{i}$. For $i=1,\ldots,k$, set $O'_{i} := O_{i}\cap Y$. Let $\omega $ be
a regular differential form of top degree on $Y$. For $i=1,\ldots,k$, for some regular
function $a_{i}$ on $O'_{i}$, $a_{i}\omega _{i}$ is the restriction of $\omega $ to
$O'_{i}$. As $Y$ is a big open subset of $X$, $O'_{i}$ is a big open subset of $O_{i}$.
Hence $a_{i}$ has a regular extension to $O_{i}$ since $O_{i}$ is normal. Denoting again
by $a_{i}$ this extension, for $1\leq i,j\leq k$, $a_{i}\omega _{i}$ and
$a_{j}\omega _{j}$ have the same restriction to $O'_{i}\cap O'_{j}$ and $O_{i}\cap O_{j}$ since $\Omega _{X_{\loc}}$ is torsion free as a locally free module. Let $\omega '$ be
the global section of $\Omega _{X_{\loc}}$ extending the $a_{i}\omega _{i}$'s. Then
$\omega '$ is a regular extension of $\omega $ to $X_{\loc}$ and this extension is unique
since $Y$ is dense in $X_{\loc}$ and $\Omega _{X_{\loc}}$ is torsion free.
(ii) Suppose that $\omega $ has no zero. Let $\Sigma $ be the nullvariety of $\omega '$
in $X_{\loc}$. If it is not empty, $\Sigma $ has codimension $1$ in $X_{\loc}$. As $Y$ is
a big open subset of $X$, $\Sigma \cap X_{\loc}$ is not empty if so is $\Sigma $. As a
result, $\Sigma $ is empty.
\end{proof}
Denote by $\iota $ the canonical injection from $X_{\loc}$ into $X$.
\begin{lemma}\label{l2ars}
Suppose that $\Omega _{X_{\loc}}$ has a global section $\omega $ without zero. Then
the $\an X{}$-module $\iota _{*}(\Omega _{X_{\loc}})$ is free of rank $1$.
More precisely, the morphism $\theta $:
$$ \xymatrix{ \an X{} \ar[rr]^{\theta } && \iota _{*}(\Omega _{X_{\loc}})}, \qquad
\psi \longmapsto \psi \omega $$
is an isomorphism.
\end{lemma}
\begin{proof}
For $\varphi $ a local section of $\iota _{*}(\Omega _{X_{\loc}})$ above the open subset
$U$ of $X$, for some regular function $\psi $ on $U\cap X_{\loc}$,
$\psi \omega $ is the restriction of $\varphi $ to $U\cap X_{\loc}$. Since $X$ is normal,
so is $U$ and $U\cap X_{\loc}$ is a big open subset of $U$. Hence $\psi $ has a regular
extension to $U$. As a result, there exists a well defined morphism from
$\iota _{*}(\Omega _{X_{\loc}})$ to $\an X{}$ whose inverse is $\theta $.
\end{proof}
According to \cite{Hi}, $X$ has a desingularization $Z$ with morphism $\tau $ such that
the restriction of $\tau $ to $\tau ^{-1}(X_{\loc})$ is an isomorphism onto $X_{\loc}$.
For $U$ open subset of $X$, denote by $\tau _{U}$ the restriction of $\tau $ to
$\tau ^{-1}(U)$.
\begin{prop}\label{pars}
Suppose that $X$ is Cohen-Macaulay and that there exists a morphism
$\mu : \xymatrix{ \an Z{} \ar[r] & \Omega _{Z}}$
such that for some $p$ in $\k[X]$, $\tau _{*}\mu $ is an isomorphism onto
$p\tau _{*}(\Omega _{Z})$. Then $X$ has rational singularities.
\end{prop}
The following proof is the weak variation of the proof of~\cite[Lemma 2.3]{Hin}.
\begin{proof}
Since $Z$ and $X$ are varieties over $\k$, we have the commutative diagram
$$\xymatrix{ Z \ar[rr]^{\tau } \ar[rd]_{p} &&
X \ar[ld]^{q} \\ & {\mathrm {Spec}}(\k) & } .$$
According to ~\cite[V. \S 10.2]{Ha0}, $p^{!}(\k)$ and $q^{!}(\k)$ are dualizing complexes
over $Z$ and $X$ respectively. Furthermore, by ~\cite[VII, 3.4]{Ha0} or
\cite[4.3,(ii)]{Hin}, $p^{!}(\k)[-\dim Z]$ equals $\Omega _{Z}$. Set
$\mathpzc{D} := q^{!}(\k)[-\dim Z]$ so that
$\tau ^{!}(\mathpzc{D})=\Omega _{Z}$ by ~\cite[VII, 3.4]{Ha0} or \cite[4.3,(iv)]{Hin}.
In particular, $\mathpzc{D}$ is dualizing over $X$.
Since $\tau $ is a projective morphism, we have the isomorphism
\begin{eqnarray}\label{eqars}
{\mathrm {R}}\tau _{*}({\mathrm {R}}\hhom_{Z}(\Omega _{Z},\Omega _{Z}))
\longrightarrow {\mathrm {R}}\hhom_{X}
({\mathrm {R}}(\tau )_{*}(\Omega _{Z}),\mathpzc{D})
\end{eqnarray}
by ~\cite[VII, 3.4]{Ha0} or \cite[4.3,(iii)]{Hin}.
Since ${\mathrm {H}}^{i}({\mathrm {R}}\hhom_{Z}(\Omega _{Z},\Omega _{Z}))=\an Z{}$ for
$i=0$ and $0$ for $i>0$, the left hand side of (\ref{eqars}) can be identified with
${\mathrm {R}\tau _{*}}(\an Z{})$.
According to Grauert-Riemenschneider Theorem \cite{GR},
${\mathrm {R}}\tau _{*}(\Omega _{Z})$ has only cohomology in degree $0$. Since
$\tau $ is projective and birational and $Z$ is normal, $\tau _{*}(\an Z{})=\an X{}$.
So by assumption of the proposition,
$$ {\mathrm {R}}\tau _{*}(\Omega _{Z}) \approx \frac{1}{p}\an X{}{},$$
whence
$${\mathrm {R}}\hhom_{X}({\mathrm {R}}(\tau )_{*}(\Omega _{Z}),\mathpzc{D})
\approx \tk{\an X{}}{p\an X{}{}}\mathpzc{D}$$
and (\ref{eqars}) can be rewritten as
\begin{eqnarray}\label{eq2ars}
{\mathrm {R}}\tau _{*}(\an Z{}) \approx \tk{\an X{}}{p\an X{}{}}\mathpzc{D} .
\end{eqnarray}
Since $X$ is Cohen-Macaulay, $\mathpzc{D}$ has cohomology in only one degree. So,
by flatness of the $\an X{}$-module $p\an X{}$, $\tk{\an X{}}{p\an X{}{}}\mathpzc{D}$
has cohomology in only one degree. As a result, by (\ref{eq2ars}),
${\mathrm {R}}^{i}\tau _{*}(\an Z{})=0$ for $i>0$, that is $X$ has rational singularities.
\end{proof}
Denote by ${\cal M}$ the cohomology in degree $0$ of $\mathpzc{D}$.
\begin{lemma}\label{l3ars}
Suppose that $X$ has rational singularities. Then the $\an X{}$-modules
$\tau _{*}(\Omega _{Z})$ and ${\cal M}$ are isomorphic. In particular,
$\tau _{*}(\Omega _{Z})$ has finite injective dimension.
\end{lemma}
\begin{proof}
Since $X$ has rational singularities, ${\mathrm {R}\tau _{*}}(\an Z{})=\an X{}$ and
$\mathpzc{D}$ has only cohomology in degree $0$. Moreover, by Grauert-Riemenschneider
Theorem \cite{GR},
${\mathrm {R}}\tau _{*}(\Omega _{Z})$ has only cohomology in degree $0$, whence
$R\tau _{*}(\Omega _{Z}) = \tau _{*}(\Omega _{Z})$. Then, by (\ref{eqars}), we have
the isomorphism
$$ \xymatrix{ \an X{} \ar[rr] && \hhom_{X} ((\tau )_{*}(\Omega _{Z}),{\cal M})} .$$
As $\mathpzc{D}$ is dualizing, we have the isomorphism
$$ \xymatrix{ R\tau _{*}(\Omega _{Z}) \ar[rr] &&
{\mathrm {R}}\hhom_{X}({\mathrm {R}}\hhom_{X}(R\tau _{*}(\Omega _{Z}),\mathpzc{D}),
\mathpzc{D})}$$
whence the isomorphism $\xymatrix{ \tau _{*}(\Omega _{Z}) \ar[r] & {\cal M}}$ by
(\ref{eqars}). As a result, $\tau _{*}(\Omega _{Z})$ has finite injective dimension since
so has ${\cal M}$.
\end{proof}
\begin{coro}\label{cars}
Let $Y$ be a smooth big open subset of $X$. Suppose that the following conditions are
verified:
\begin{itemize}
\item [{\rm (1)}] $X$ is Cohen-Macaulay,
\item [{\rm (2)}] $\Omega _{Y}$ has a global section $\omega $ without zero,
\item [{\rm (3)}] for some global section $\omega _{Z}$ of $\Omega _{Z}$ and for some $p$
in $\k[X]\setminus \{0\}$, the restriction of $\omega _{Z}$ to $\tau ^{-1}(Y)$ is equal
to $p\tau _{Y}^{*}(\omega )$.
\end{itemize}
Then $X$ is Gorenstein with rational singularities. Moreover, its canonical module is free
of rank $1$.
\end{coro}
\begin{proof}
According to Lemma~\ref{lars}(ii), $\omega $ has a unique regular extension to
$X_{\loc}$ and this extension has no zero. Denote again by $\omega $ this extension.
Since $Z$ is irreducible, $\tau ^{-1}(Y)$ is dense in $\tau ^{-1}(X_{\loc})$ so that the
restriction of $\omega _{Z}$ to $\tau ^{-1}(X_{\loc})$ is equal to
$p\tau _{X_{\loc}}^{*}(\omega )$ since $\Omega _{Z}$ has no torsion. Denote by $\mu $ the
morphism
$$ \xymatrix{ \an Z{} \ar[rr]^{\mu } && \Omega _{Z}}, \qquad
\varphi \longmapsto \varphi \omega _{Z} .$$
Let $U$ be an open subset of $X$ and $\nu $ a local section of $\tau _{*}(\Omega _{Z})$
above $U$. Since $\omega $ has no zero and $\tau _{U_{\loc}}$ is an isomorphism onto
$U_{\loc}$,
$$ \nu \left \vert \right. _{\tau ^{-1}(U_{\loc})} = \tau _{U_{\loc}}^{*}
(\varphi \omega \left \vert \right. _{U_{\loc}}) $$
for some $\varphi $ in $\k[U_{\loc}]$, whence
$$ p \nu \left \vert \right. _{\tau ^{-1}(U_{\loc})} = \varphi \rond \tau _{U_{\loc}}
(\omega _{Z} \left \vert \right. _{\tau ^{-1}(U_{\loc})}) $$
by Condition (3). Since $X$ is normal, so is $U$ and $U_{\loc}$ is a big open subset of
$U$. Hence $\varphi $ has a regular extension to $U$. Denoting again by $\varphi $ this
extension,
$$p\nu = \varphi \rond \tau _{U}(\omega _{Z} \left \vert \right. _{\tau ^{-1}(U)})$$
since $Z$ is irreducible and $\Omega _{Z}$ has no torsion. As a result the morphism
$$ \tau _{*}\mu : \xymatrix{ \tau _{*}(\an Z{}) \ar[rr] && p\tau _{*}(\Omega _{Z})}$$
is an isomorphism since it is obviously injective. So, by Proposition~\ref{pars},
$X$ has rational singularities. In particular, by~\cite[p.50]{KK},
$\tau _{*}(\Omega _{X})=\iota _{*}(\Omega _{X})$. Then, by Lemma~\ref{l2ars},
the canonical module of $X$ is free of rank $1$ and by Lemma~\ref{l3ars}, $X$ is
Gorenstein.
\end{proof}
\section{About singularities} \label{si}
In this section we recall a well known result. Let $X$ be a variety and $Y$ a vector
bundle over $X$. Denote by $\tau $ the bundle projection.
\begin{lemma}\label{lsi}
{\rm (i)} If $Y$ is Gorenstein, then $X$ is Gorenstein.
{\rm (ii)} The variety $X$ has rational singularities if and only if so has $Y$.
{\rm (iii)} If $X$ is Cohen-Macaualay, then so is $Y$.
\end{lemma}
\begin{proof}
Let $y$ be in $Y$, $x:= \tau (y)$. Denote by $\widehat{\an Xx{}}$ and
$\widehat{\an Yy{}}$ the completions of the local rings $\an Xx$ and $\an Yy$
respectively.
(i) Since $Y$ is a vector bundle over $X$, $\widehat{\an Yy}$ is a ring of formal series
over $\widehat{\an Xx}$. By~\cite[Proposition 3.1.19,(c)]{Br}, $\widehat{\an Yy{}}$ is
Gorenstein. So, by~\cite[Proposition 3.1.19,(b)]{Br}, $\widehat{\an Xx}$ is Gorenstein.
Then by \cite[Proposition 3.1.19,(c)]{Br}, $\an Xx$ is Gorenstein, whence the assertion.
(ii) Since $Y$ is a vector bundle over $X$, then there exists a cover of $X$ by open
subsets $O$, such that $\tau ^{-1}(O)$ is isomorphic to $O\times \k^{m}$ with
$m=\dim Y - \dim X$. According to~\cite[p.50]{KK}, $O\times \k^{m}$ has rational
singularities if and only if so has $O$, whence the assertion since a variety has
rational singularities if and only it has a cover by open subsets having rational
singularities.
(iii) According to~\cite[Ch. 6, Theorem 17.7]{Mat}, a polynomial algebra over a
Cohen-Macaulay algebra is Cohen-Macaulay. Hence for $O$ open subset of $X$ as in (ii),
$\tau ^{-1}(O)$ is Cohen-Macaulay, whence the assertion since there is a cover of $Y$
by open subsets as $\tau ^{-1}(O)$.
\end{proof}
| 129,775
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The Bank of Japan (BoJ), the country’s central bank, has put its top economist in charge of the department responsible for research and development into central bank digital currencies (CBDCs).
According to a July 31 report by Reuters, Kazushige Kamiyama, the former Director General of the BoJ’s Research Statistics Development, has been moved to head the Payments and Settlements Systems Department. Since the beginning of the year, this department is heavily involved in a digital currency working group alongside 5 other central banks. It also manages the task force, set up earlier this month, which examines the possible implications of issuing a CBDC in Japan.
Kamiyama has served at the BoJ for more than 6 years, including 2 years at the central bank’s New York offices. During his term as the head of research department, Kamiyama advocated for the bank to use more “Big Data” in analyzing the country’s economy. This approach helped the bank to catch real-time changes affecting the economy amid the COVID-19 pandemic. The announcement also noted that Seisaku Kameda will take over Kamiyama’s position as the central bank’s top economist and head of its statistics department.
Though the BoJ claims it has no immediate plans to launch a digital Yen, research into the potentials of CBDCs is still underway. This latest move indicates how seriously the Japanese central bank is taking the research. Kamiyama’s recent appointment may mark a shift away from the BoJ’s previously cautious nature on digital currencies.
Japan has been cautious about moving quickly into issuing a digital yen, considering the social disruptions it may cause in a country with the most cash-loving population in the world. However, China’s steady progress towards issuing a digital currency has prompted Japan to look more closely into the idea of issuing its own CBDC.
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Rudolf Nureyev
Famous Russian Ballet Dancer. By Olivia.C
Rudolf Nureyev
Rudolf Nureyev was born March 17, 1938. After graduating from the Leningrad Ballet School, he became a soloist with the Kirov Ballet. In 1961, he made his London debut at Margot Fonteyn's yearly gala for the Royal Academy of Dancing. Rudolf nureyev died on January 6, 1993 in Paris.Ballet dancer and choreographer Rudolf Nureyev was born, the youngest child and only son to a peasant family of Tartar heritage, on March 17, 1938, in Irkutsk, Russia. At the age of 11, Nureyev started ballet classes under Anna Udeltsova. A year and a half later, he moved on to teacher Elena Vaitovich.Nureyev started dancing professionally as an extra at the local opera when he was 15. When Nureyev turned 17, he got into the Leningrad Ballet School, where Alexander Pushkin became his teacher. Concurrent with his success as a dancer, Nureyev took his first stab at choreography in 1964 with revised versions of Swan Lake.
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ELIOT, Maine — The Town of Eliot submitted a Good Neighbor Petition to the Environmental Protection Agency Tuesday calling for an investigation of dangerous air pollution from the Schiller Station coal-fired power plant in neighboring New Hampshire.
The filing, officially a Clean Air Act Section 126 Petition, comes after a town vote in June where residents overwhelmingly supported a measure to file the petition and require the EPA to investigate the problem of harmful pollution crossing state lines. If the EPA finds that pollution from the plant could cause unhealthy levels of air pollution in Maine, they could force the Public Service Company of New Hampshire (PSNH) the plant’s owner, to clean it up.
Tuesday’s filing is seen as a success for some local residents who have been working for decades to seek relief from what they claim is the plant’s harmful pollution responsible for coating homes with black soot, pumping dangerous sulfur dioxide into the air and making people in Eliot sick.
In response Kim Richards, leader of Citizens for Clean Air in Eliot, Maine made the following statement:
“We are proud of our Board of Selectmen for taking this important step to protect our health and our families from dangerous pollution. The petition is a result of decades of action by citizens to raise concerns about the harmful effects of living next to a dirty coal plant. Now we can finally look forward to getting answers.”
“Today’s (Tuesday’s) filing is a victory for people over polluters and we look forward to our Environmental Protection Agency’s timely assessment and action to clean up our air. This is proof that when people come together, we can do great things for our communities despite the efforts of large, polluting corporations to stop us. We all deserve clean air and now the people of Eliot are one step closer to holding PSNH accountable for their harmful air pollution.”
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Can You Trust Your Uber Ride?
Uber has become an incredibly popular transportation service, but now that the “newness” has worn off, many are starting to realize that it may not be as awesome as they hoped.
First and foremost, despite their attempts at security, Uber is a national organization with no local oversight. Additionally, reports of attacks, infrastructure turmoil, and rampant sexism are highlighting that the company’s
services may not be as safe or reliable as you hope.
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Whenever you need a transportation service you can trust, know that you have a local company who values your business and your safety. Rather than gambling with an unknown amateur, you gain the confidence of a provider who has a demonstrated reputation. To learn more or to request a pick-up, contact us today!
| 8,299
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BLOOMBERG/HASAN SHAABAN
Fitch Rating has downgraded Lebanon’s Bank Audi and Byblos Bank due to the impact of rising political tensions following nearly two weeks of nationwide protests, triggered in part by a currency crisis and struggling economy.
The rating agency downgraded Bank Audi and Byblos Bank’s long-term issuer default rating (IDR) to CCC- from CCC and both lenders were also placed on a negative rating watch, a day after Lebanon’s Prime Minister Saad Hariri stepped down—bowing down to demonstrators’ demands who said they would accept nothing less than the resignation of the government and key officials.
Fitch stated that the downgrade and the negative rating watch reflect heightened liquidity risks confronting Lebanese banks in view of increasing political tensions and social unrest in Lebanon.
The demonstrations have caused a closure of domestic banks for operations since 18 October 2019 and Fitch said that deposit stability is now at greater risk as depositor confidence has also suffered.
Influential Lebanese economists are calling for the government to impose formal restrictions on the movement of money to defend the country’s dollar peg and prevent a run on the banks when they open their doors after nationwide protests.
The negative rating watch reflects the probability of a further downgrade if potential funding stress materially affects the bank's liquidity profile, we expect to resolve the negative rating watch in the near term depending on the evolution of the bank's funding and liquidity position in the current stressful operating environment, said Fitch.
According to Fitch, around 88 per cent of Bank Audi’s deposits and 69 per cent of Byblos Bank deposits were in foreign currency at the end of H1 2019, which increases the vulnerability of the lenders to unexpected outflows.
The longer the banks remain shut, however, the more a backlog in dollar demand builds and speculation swirls about the measures the banks will need to take to avert financial collapse.
Lebanon has been gripped by widescale protests for the past two weeks demanding an end to nepotism and corruption that have led the country’s public debt to soar to $86 billion equivalent to 150 per cent of gross domestic product, one of the highest in the world.
Fitch said that access to foreign currency in the market is stretched and the foreign currency-liquidity management of the banks is largely dependent on Banque du Liban’s ability to meet foreign currency obligations.
The central bank Governor said that a political solution is needed within ‘days’ to avoid economic collapse and restore public confidence but violence and the resignation of Prime Minister Hariri has added to the uncertainty among investors.
The government’s proposals to almost wipe out Lebanon’s budget deficit next year and reduce wasteful spending was criticised by economists as unrealistic, with Moody’s cautioning that the reliance on BdL financing could undermine the country’s currency peg.
The International Monetary Fund said that it was assessing that emergency reform package, adding that reforms should be implemented urgently given the country’s high debt levels and fiscal deficits.
The Washington-based fund expects Lebanon's economic growth to decrease by 0.2 per cent in 2019.
MOST READ
| 271,351
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Gallery Holdings
Molded Tile of a Celestial Dancer on Lucite Stand
Item No. 3997
Yuan Dynasty (1280-1368), N/A, China
Earthenware
12.75" x 11.25" x 3.75"
( 32.385 x 28.575 x 9.525 cm)
(H x W x D)
The Literati in China built vast estates with sumptuous gardens, lakes and grottoes. The interest of the Chinese in creating these estates and devoting large areas to such pursuits can be traced back to the Han dynasty (206 B.C.-A.D. 220) when, as Robert Mowry writes, "Chinese connoisseurs began using large stones to decorate their gardens and courtyards. There are also references to the special qualities of garden rocks and individual stones in poems dating as far back as the Tang dynasty (618-907). Scholars' Rocks is the most common English name given to the small, individual stones that have been appreciated by educated and artistic Chinese at least since the Song dynasty (960-1270). They evolved from appreciation of the larger garden rocks, but their smaller size enabled the Chinese literati to carry them indoors where they could be admired and meditated over in their sparse studios.Scholar's Rocks (or Gongshi) began as stones that resembled or represented mythological and famous mountains, or even whole mountain ranges in China. Some are also mountains into their studios for meditation and contemplation while they wrote or painted. So smaller stones with the same qualities were found and initially received as gifts. They gained great favor among the literati and the Imperial court and have remained popular for over 1,000 years.""The earliest garden displays of rocks occurred during the Han dynasty and were most likely representative of the fanciful paradises known as Penglai, or the Eastern Isle of the Immortals. These paradises were actually perceived to be three or more mountains isolated in the Eastern Sea. The mountains were tall with "craggy, inaccessible peaks" and isolated - even from each other. Since the immortals could fly from one to another, they could easily carry on social commerce among themselves, however, to mere mortals these paradise isles were completely inaccessible. In Chinese paintings these gods are often depicted as cranes flying to or between the tall mountains."This desire to create models of Penglai were probbly, much later, combined with ideas from Chinese landscape design, garden design and the wish for the inclusion of lakes and grottoes to construct vast areas of Literati estates dedicated to the contemplation of nature in all its forms. It was seen as an aid in the creation of Literati works, as a sort of muse which aided the literati to write poetry, create paintings and more.Molded stoneware tiles such as these were used to decorate these settings. Many tiles were decorated with Buddhist themes of celestial dancers, musicians, and mythological beasts. This tile represents a celestial dancer. These dancers were portrayed as graceful, lively and spiritual beings in various versions, with most playing musical instruments and painted in lovely poses in paintings, decorating pottery, on the panels of temple interiors, and temple ceilings. The figure is set within a decorative graceful molded frame which adds to the lyrical quality of the piece. Sources:Robert Mowry, "A Brief History of Scholar's Rocks," on
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\begin{document}
\maketitle
\begin{abstract}
We construct examples of groups that are $FP_2(\Q)$ and $FP_2(\Z/p\Z)$ for all primes $p$ but not of type $FP_2(\Z)$.
\end{abstract}
\section{Introduction}
We begin with a definition:
\begin{definition}
A group $G$ is of {\em type $FP_n(R)$} if there is an exact sequence:
$$P_n\to\dots\to P_2\to P_1\to P_0\to R\to 0$$
of projective $R G$-modules such that $P_i$ is finitely generated and $R$ is the trivial $R G$-module.
\end{definition}
Using the chain complex of the universal cover of a presentation 2-complex we see that finitely presented groups are of type $FP_2(\Z)$.
Moreover, if a group is of type $FP_2(\Z)$, then it is of type $FP_2(R)$ for any ring $R$.
In \cite{BeBr}, the first examples of groups that are of type $FP_2(\Z)$ but not finitely presented are given.
More recently, there have been many new constructions of groups of type $FP_2$ with interesting properties, see \cite{BeBr, Lea, Lea2, KLS, Kropholler, LBrown}.
In particular, there are various constructions of uncountable families of groups of type $FP_2(\Z)$ \cite{Lea, Kropholler, LBrown}.
It is also possible to use the examples of \cite{BeBr, Lea} to give examples of groups that are of type $FP_2(R)$ but not $FP_2(\Z)$ for certain rings $R$.
The construction of \cite{BeBr}, takes in a connected flag complex $L$ and constructs a group $BB_L$ that is of type $FP_2(R)$ if and only if $H_1(L; R) = 0$.
Since these flag complexes are finite, it follows that if $H_1(L; \Z/p\Z) = 0$ for all primes $p$, then $H_1(L; \Z) = 0$.
Thus, if $BB_L$ is $FP_2(\Z/p\Z)$ for all primes $p$, then it is type $FP_2(\Z)$.
Similar results can be obtained for the groups constructed in \cite{Lea}.
In this paper we build on the work of \cite{Lea}. Leary built uncountably many groups of type $FP_2$ by taking branched covers of a cube complex $X$.
Leary's construction takes as input a flag complex $L$ and a set $S\subset \Z$. It outputs a cube complex $X_L^{(S)}$ with a height function $f^{(S)}$. These have the property that if a vertex has height in $S$, then the ascending and descending links at $v$ are $L$. If the height of a vertex is not in $S$, then the ascending and descending links are $\tilde{L}$, the universal cover of $L$.
We build on this construction by varying the covers that can be taken at each height. Our construction is the following:
\begin{restatable}{construction}{constrmain}\label{constrmain}
Let $L$ be a flag complex. Let $\sigma\colon \Z\to \CC$ be a function, where $\CC$ is the collection of normal covers of $L$.
Then there is a cube complex $X_L^\sigma$ and a height function $f_\sigma$ such that if $f_{\sigma}(v) = n$, then the ascending and descending links of $v$ are exactly $\sigma(n)$.
\end{restatable}
This cube complex arises as a branched cover and there is a group of deck tranformations $G_L(\sigma)$. Thus we can use the cube complex $X_L^{\sigma}$ to investigate the finiteness properties of $G_L(\sigma)$. We obtain the following theorem:
\begin{customthm}{\cref{thm:typefpn}}
Let $\sigma, L, \CC$ be as above. Suppose that $\pi_1(L)/\pi_1(\sigma(n))$ is of type $FP_k(R)$ for all $n$. Then $G_L(\sigma)$ is type $FP_k(R)$ if and only if $\tilde{H}_i(\sigma(n); R)$ vanishes for all but finitely pairs $(i, n)$ with $n\in \Z$ and $i < k$.
Similarly, suppose $\pi_1(L)/\pi_1(\sigma(n))$ is of type $FP(R)$ for all $n$. Then $G_L(\sigma)$ is type $FP(R)$ if and only if $\tilde{H}_i(\sigma(n), R)$ vanishes for all but finitely pairs $(i, n)$ with $n\in \Z$ and $i\in\N$.
\end{customthm}
We can use this to construct new examples of groups of type $FP_2(R)$ over various rings $R$. Here we detail two such constructions.
As pointed out previously, if $G$ is of type $FP_2(\Z)$, then $G$ is of type $FP_2(R)$ for all rings $R$.
One may hope that there is a collection of rings $\mathcal{R}$ such that if $G$ is of type $FP_2(R)$ for all $R\in \mathcal{R}$, then $G$ is of type $FP_2(\Z)$.
One possible candidate is the collection of fields.
We show that this is not the case, proving the following.
\begin{theorem}
There are groups that are of type $FP_2(F)$ for all fields $F$, but not of type $FP_2(\Z)$.
\end{theorem}
In fact, it is enough to study $\Q$ and $\Z/p\Z$ for all primes $p$. Since if $P$ is the prime subfield of $F$, then $FP_2(P)$ implies $FP_2(F)$.
Thus, we prove the following:
\begin{customthm}{\cref{thm:acyclicoverQandP}}
There exist groups that are of type $FP_2(\Q)$ and $FP_2(\Z/p\Z)$ for all primes $p$ that are not $FP_2(\Z)$.
\end{customthm}
Moreover, we are able to prove the above theorem for arbitrary sets of primes.
\begin{customthm}{\cref{thm:setofprimes}}
Let $S$ be a set of primes. Then there exists a group which is type $FP_2(\Z/p\Z)$ if and only if $p\notin S$.
\end{customthm}
We highlight one particularly novel corollary to this theorem:
\begin{corollary}
There exists a group $G$ that is type $FP_2(\Q)$ but not of type $FP_2(\Z/p\Z)$ for any prime $p$.
\end{corollary}
The second theorem of this paper concerns the constructions from \cite{Lea, Kropholler}.
In both papers, uncountably many groups of type $FP_2(\Z)$ are constructed by considering subpresentations of an initial group that is known to be of type $FP_2(\Z)$.
One may believe removing relators from an almost finitely presented group should result in an almost finitely presented group.
It is clear that one has to be careful when removing relations from a group.
Indeed, every $n$ generated group appears as a subpresentation of the trivial group presented as $\langle x_1, \dots, x_n\mid F(x_1, \dots, x_n)\rangle$.
However, in the examples from \cite{Lea, Kropholler} care is taken when considering subpresentations.
In both cases a generating set for the relation module of the initial group is retained and this is enough to ensure the resulting group is of type $FP_2(\Z)$.
For finitely presented groups, it is true that there is a finite set of relations, that if retained ensure that the subpresentation gives a finitely presented group.
It is then of interest to know whether if one retains an appropriate finite set of relations, say generators for the relation module, does one retain the property of being of type $FP_2(\Z)$? We show that this is not the case:
\begin{customthm}{\cref{thm:subpresnotfp2}}
There exists a presentation $G = \langle X\mid R\rangle$ of a group of type $FP_2(\Z)$ such that for any finite subset $T\subset R$, we can find $S$ with $T\subset S\subset R$ and $\langle X\mid S\rangle$ is not of type $FP_2(R)$ for any ring $R$.
\end{customthm}
{\bf Acknowledgements: } The author is thankful to Kevin Schreve for posing questions leading to this article. The author is also grateful to Peter Kropholler and Ian Leary for helpful conversations.
The author was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy EXC 2044--390685587, Mathematics M\"unster: Dynamics--Geometry--Structure.
\section{Preliminaries}
\subsection{Flag complexes and spherical doubles}
\begin{definition}
Let $L$ be flag complex.
The \emph{spherical double} of $L$, denoted $\S(L)$, is defined by replacing every simplex of $L$ by an appropriately triangulated sphere of the same dimension, in the following way.
Let $\{v_1,\dots,v_n\}$ be the vertices of $L$.
The vertex set of $\S(L)$ is a set $\{v_1^+,v_1^-,\dots,v_n^+,v_n^-\}$.
Thus, each $0$--simplex $\{v_i\}$ of $L$ corresponds to a $0$--sphere $\{v_i^+,v_i^-\}$ in $\S(L)$.
If $\tau\subset L$ is an $m$--dimensional simplex of $L$, it can be represented as the join $\tau=\{v_{i_0}\}*\dots*\{v_{i_m}\}$ of a collection of $m+1$ vertices of $L$.
Let $\S(\tau)$ be the join $\S(\tau)=\{v_{i_0}^+,v_{i_0}^-\}*\dots*\{v_{i_m}^+,v_{i_m}^-\}$.
We have that, $\S(\tau)$ is homeomorphic to an $m$--dimensional sphere.
We define $\S(L)=\bigcup_{\tau\subseteq L}\S(\tau)$, where $\tau$ ranges through all simplices of $L$.
\end{definition}
It can be shown that $\S(L)$ is a simplicial complex, which is flag if and only if $L$ is flag \cite[Lemma~5.8]{BeBr}.
Also the map $v_i^t\mapsto v_i$, gives a retraction $\S(L)\to L$.
The following two results are analogous to Proposition 7.1 and Corollary 7.2 of \cite{Lea}, we include the proofs for completeness.
\begin{proposition}\label{prop:covers}
Let $L$ be a flag complex.
Let $\bar{L}$ be a cover of $L$.
Then $\S(\bar{L})$ is a cover of $\S(L)$.
\end{proposition}
\begin{proof}
This follows since $\S(\bar{L})$ can be seen as the pullback in the square:
$$\begin{tikzcd}
\S(\bar{L})\arrow[r]\arrow[d]& \S(L)\arrow[d, "r"] \\
\bar{L}\arrow[r]& L
\end{tikzcd}$$
The lower map is a covering this, we have a pull back of a covering map which is also a covering map.
\end{proof}
\begin{corollary}\label{cor:subgroupscovers}
Let $r\colon \S(L)\to L$ be the retraction above.
Then $\pi_1(\S(\bar{L})) = r_*^{-1}(\pi_1(\bar{L}))$.
\end{corollary}
\begin{proof}
Since the diagram in the proof of \Cref{prop:covers} commutes we see that $\pi_1(\S(\bar{L})) \subset r_*^{-1}(\pi_1(\bar{L}))$.
Let $\gamma$ be a loop in $\S({L})$ such that $r\circ \gamma$ is an element of $\pi_1(\bar{L})$.
Then we can lift $r\circ\gamma$ to a loop $\gamma'$ in $\bar{L}$.
Then $(\gamma, \gamma')$ defines a loop in $\S(\bar{L})$ which maps to $\gamma$.
Thus $\pi_1(\S(\bar{L})) \supset r_*^{-1}(\pi_1(\bar{L}))$.
\end{proof}
\subsection{Morse theory}
For full details, we refer the reader to \cite{BeBr}.
A map $f\colon X\to\R$ defined on a cube complex $X$ is a \emph{Morse function} if
\begin{itemize}
\item for every cell $e$ of $X$, with characteristic map $\chi_e\colon [0,1]^m\to e$, the composition $f\circ\chi_e\colon [0,1]^m\to\R$ extends to an affine map $\R^m\to \R$ and $f\circ\chi_e$ is constant only when $\dim e=0$;
\item the image of the $0$--skeleton of $X$ is discrete in $\R$.
\end{itemize}
Suppose $X$ is a cube complex, $f\colon X\to \R$ a Morse function.
The {\em ascending link} of a vertex $v$, denoted $\Lk_{\uparrow}(v, X)$ is the subcomplex of $\Lk(v, X)$ corresponding to cubes $C$ such that $f|_C$ attains its minimum at $v$. The {\em descending link}, $\Lk_{\downarrow}(v, X)$, is defined similarly replacing minimum with maximum.
\subsection{Right-angled Artin and Bestvina-Brady groups}
\begin{definition}
Let $L$ be a flag complex.
The {\em right-angled Artin group}, or RAAG, associated to $L$ is given by the presentation:
\[
A_L=\langle v\in\Vertices(L) \mid [a_i,a_j]=1 \text{ if } \{a_i,a_j\} \text{ is an edge of }L\rangle.
\]
\end{definition}
Given a right-angled Artin group $A_L$, the {\em Salvetti complex}, $S_L$ associated to $A_L$ is a cube complex $S_L$ defined as follows.
For each $v\in\Vertices(L)$ let $S^1_{v}$ be a circle endowed with a structure of a CW complex having a single $0$--cell and a single $1$--cell.
Let $T=\prod_{v} S^1_{v}$ be an $n$--dimensional torus with the product CW structure.
For every simplex $K\subset L$, define a $k$--dimensional torus $T_K$ as a Cartesian product of CW complexes:
$T_K=\prod_{v\in K} S^1_{v}$
and observe that $T_K$ can be identified as a combinatorial subcomplex of $T$.
Then the {\em Salvetti complex} is
\[
S_L=\bigcup\big\{T_K\subset T\mid K \text{ is a simplex of }L\big\}.
\]
The link of the single vertex of $S_L$ is $\S(L)$.
This is a flag simplicial complex, and hence $S_L$ is a non-positively curved cube complex.
It follows that the universal cover $X_L = \tilde S_L$ is a CAT(0) cube complex.
We can define a homomorphism $\phi\colon A_L\to\Z$ by sending each generator to 1.
We can realise this topologically as a map $f\colon S_L\to S^1$ by restricting the map $\hat{f}\colon T\to S^1$ given by $(x_1, \dots, x_n)\mapsto x_1 + \dots + x_n$.
Let $BB_L\vcentcolon = \ker(\phi)$.
We can lift $f$ to a map which we also call $f\colon X_L/BB_L\to \R$.
This is a Morse function.
The ascending and decending links are copies of $L$ spanned by $v_i^+$ and $v_i^-$ respectively.
\section{New sequences of covers}
In \cite{Lea}, branched covers of $X_L/BB_L$ were taken to obtain uncountably many groups $G_L(S)$ depending on $S\subset\Z$, whose finiteness properties are controlled by the topology of $L$.
In this section, we generalise this machinery to construct various new groups.
Throughout, let $L$ be a flag complex.
Let $\CC$ be the set of normal covers of $L$.
Let $\CZ = \CC^\Z$ be the set of functions $\sigma\colon\Z\to \CC$.
Define a partial ordering on $\CZ$ by $\sigma \preceq \sigma'$ if $\sigma(n)$ is a cover of $\sigma'(n)$.
We will associate to each element $\sigma\in \CZ$ a group as follows.
{\bf Construction of $G_L(\sigma)$:}
For each integer $n$ there is a single vertex $v$ of $X_L/BB_L$ such that $f(v) = n$.
Let $\gamma_{i, n}$ be a sequence of loops in $\S(L)$ that normally generate $G_n = \pi_1(\S(\sigma(n)))\leq \pi_1(\S(L))$.
Let $V$ be the vertex set of $X_L/BB_L$ and let $V(\sigma) \vcentcolon = \{v\in X_L/BB_L\mid \sigma(f(v))\neq L\}\subset V$.
By identifying $\Lk(v, X)$ with the boundary of the $\frac{1}{4}$ neighbourhood of $v$ we can consider $\gamma_{i, n}$ as loops in $X_L/BB_L$.
Let $Y_L(\sigma)$ be the complex obtained from $(X_L/BB_L)\smallsetminus V(\sigma)$ by attaching disks to all the loops $\gamma_{i, n}$.
Let $G_L(\sigma) = \pi_1(Y_L(\sigma))$.
We can also view the group $G_L(\sigma)$ as a group of deck transformations of a branched cover of cube complexes.
\begin{theorem}
There is a CAT(0) cube complex $X_L^{\sigma}$ with a branched covering map $b\colon X_L^\sigma\to X_L/BB_L$ such that $G_L(\sigma)$ is the group of deck transformations of this branched cover.
\end{theorem}
\begin{proof}
We construct $X_L^\sigma$ as follows.
Let $\widetilde{Y_L(\sigma)}$ be the universal cover of $Y_L(\sigma)$.
Let $\mathcal{D}$ be the collection of open disks added to $(X_L/BB_L)\smallsetminus V(\sigma)$ to obtain $Y_L(\sigma)$ and let $\tilde{\mathcal{D}}$ be the collection of lifts of $\mathcal{D}$ to $\widetilde{Y_L(\sigma)}$.
Let $Z_L(\sigma)$ be $\widetilde{Y_L(\sigma)}\smallsetminus\tilde{\mathcal{D}}$.
Then the covering map $p\colon \widetilde{Y_L(\sigma)}\to Y_L(\sigma)$ restricts to a covering map $Z_L\to (X_L/BB_L)\smallsetminus V(\sigma)$.
We can now lift the metric and complete to obtain a branched cover $b\colon X_L^\sigma\to X_L/BB_L$.
The deck group is exactly the deck group of the covering $\widetilde{Y_L(\sigma)}\to Y_L(\sigma)$, this is $G_L(\sigma)$.
Given a vertex $v\in X_L^\sigma$ we obtain a covering map $b_v\colon \Lk(v, X_L^\sigma)\to \Lk(b(v), X_L)$.
Since the cover of a flag complex is a flag complex we see that $X_L^\sigma$ is non-positively curved.
Taking the completion adds in the missing vertices of $Z_L$.
The vertices added cone off their links.
As such the boundary of each disk in $\tilde{\mathcal{D}}$ is trivial in $X_L^\sigma$ and thus $\pi_1(\widetilde{Y_L(\sigma)})$ surjects $\pi_1(X_L^\sigma)$.
We conclude, $X_L^\sigma$ is simply connected.
Thus, $X_L^{\sigma}$ is non-positively curved and simply connected and hence CAT(0).
\end{proof}
There is a Morse function $f_\sigma\colon X_L^\sigma\to \R$ given by composition $f\circ b$.
Since $G_L(\sigma)$ is the covering group of $Z_L\to X_L\smallsetminus V(X_L)$, we see that it acts on $X_L^\sigma$ cellularly and freely away from the vertex set.
Moreover, $G_L(\sigma)$ acts properly, freely and cocompactly on $f_\sigma^{-1}(\frac{1}{2})$.
Thus by understanding this level set we can understand finiteness properties of $G_L(\sigma)$.
We will proceed by understanding the ascending and descending links of the Morse function $f_\sigma$.
\begin{lemma}
Let $v$ be a vertex of $X_L^\sigma$ such that $f_\sigma(v) = n$.
Then $\Lk(v, X_L^\sigma) = \S(\sigma(n))$ and $\Lk_\uparrow(v, X_L^\sigma) = \Lk_\downarrow(v, X_L^\sigma) = \sigma(n)$.
\end{lemma}
\begin{proof}
Let $\mathcal{D}_v$ be the collection of disks glued at $v$ in $\widetilde{Y_L(\sigma)}$.
There is a retraction $X_L^\sigma\smallsetminus \{v\}\to N(v, X_L^\sigma)\smallsetminus \{v\}$ where $N(v, X_L^\sigma)$ is a neighbourhood of $v$.
Let $\gamma$ be the attaching map of a disk in $\tilde{\mathcal{D}}\smallsetminus\mathcal{D}_v$, then this bounds a disk in $X_L^\sigma\smallsetminus \{v\}$.
Thus we can extend the retraction over elements of $\tilde{\mathcal{D}}\smallsetminus\mathcal{D}_v$.
This gives a retraction $\widetilde{Y_L(\sigma)}\to Y_L(\sigma, v)$, where $Y_L(\sigma, v) = (N(v, X_L^\sigma)\smallsetminus \{v\})\cup \mathcal{D}_v$
Since the former is simply connected, so is the latter.
We see that $Y_L(\sigma, v)$ is homotopy equivalent to a cover $\overline{\S({L})}$ of $\S(L)$ together with disks glued to each lift of $\gamma_{i, n}$.
Since this is simply connected we see that $\overline{\S({L})}$ is the cover corresponding $\langle \langle\gamma_{i, n}\rangle\rangle$.
Since $\{ \gamma_{i, n}\}$ normally generates $\pi_1(\S(\sigma(n)))$ we see that $\overline{\S({L})} = \S(\sigma(n))$ and the ascending link is the preimage of $L$ which is exactly $\sigma(n)$.
Similarly the descending link is $\sigma(n)$.
\end{proof}
This allows us to understand the finiteness properties of $G_L(\sigma)$.
Firstly, we recall a simplified version of Brown's criterion \cite{BrownCrit} (from \cite{Lea}) for a group to be of type $FP_k(R)$.
\begin{theorem}\label{thm:brown}
Suppose that $X$ is a finite-dimensional $R$-acyclic G-CW-complex, and that $G$ acts freely except possibly that some vertices have isotropy subgroups that are of type $FP(R)$ (resp. $FP_k(R)$). Suppose also that $X = \cup_{m\in \N} X(m)$ where $X(m)\subset X(m+1)\subset\dots\subset X$ is an ascending sequence of $G$-subcomplexes, each of which contains only finitely many orbits of cells. In this case $G$ is $FP(R)$ (resp. $FP_k(R)$) if and only if for all $i$ (resp. for all $i < k$) the sequence $\tilde{H}_i(X(m); R)$ of reduced homology groups is essentially trivial.
\end{theorem}
\begin{theorem}\label{thm:typefpn}
Let $\sigma, L, \CC$ be as above. Suppose that $\pi_1(L)/\pi_1(\sigma(n))$ is of type $FP_k(R)$ for all $n$. Then $G_L(\sigma)$ is type $FP_k(R)$ if and only if $\tilde{H}_i(\sigma(n), R)$ vanishes for all but finitely pairs $(i, n)$ with $n\in \Z$ and $i < k$.
Similarly, suppose $\pi_1(L)/\pi_1(\sigma(n))$ is of type $FP(R)$ for all $n$. Then $G_L(\sigma)$ is type $FP(R)$ if and only if $\tilde{H}_i(\sigma(n), R)$ vanishes for all but finitely pairs $(i, n)$ with $n\in \Z$ and $i\in\N$.
\end{theorem}
\begin{proof}
We focus on the proof for $FP_k(R)$, the proof for $FP(R)$ is similar.
The group $G_L(\sigma)$ acts on $X_L^\sigma$ freely away from vertices.
For vertices at height $n$, the stabiliser is the deck group of the covering $\sigma(n)\to L$.
This is exactly $\pi_1(L)/\pi_1(\sigma(n))$ which is of type $FP_k(R)$.
Let $X(m) = f_\sigma^{-1}([-m-\frac{1}{2}, m + \frac{1}{2}])$.
We are now in the situation of \cref{thm:brown}, thus $G_L(\sigma)$ is of type $FP_k(R)$ if and only if for all $i<k$ the sequence $\tilde{H}_i(X(m); R)$ of reduced homology groups is essentially trivial.
If $\tilde{H}_i(\sigma(n), R)$ vanishes for all but finitely pairs $(i, n)$ with $n\in \Z$ and $i < k$, then by \cite[Corollary 2.6]{BeBr} we can find an $l$ such that for all $m > l$ the inclusion $X(l) \to X(m)$ induces an isomorphism on all $H_i$ for $i< k$.
Since homology commutes with direct limits we see that $H_i(X(l)) = H_i(X_L^\sigma) = 0$ for all $i<k$.
Thus the system is essentially trivial.
Now conversely suppose that there are infinitely many $n$ such that $H_i(\sigma(n), R)\neq 0$ for some $i<k$.
Thus for each $l>0$ we can find $l' > l + 1$ and $v$ such that $H_i(\sigma(l'); R)$ or $H_i(\sigma(-l'); R)$ is non-trivial for some $i<k$.
We will assume that $\sigma(l')$ has the non-trivial homology group.
Let $v$ be a vertex at height $l'$.
There is a map from $X(l'-1)$ to $\Lk(v, X_L^\sigma)$.
We can further compose with the retraction $\Lk(v, X_L^\sigma)\to \sigma(l')$.
By extending geodesics from $v$ downwards, we can view $\sigma(l')$ as a subspace of $X(l'-1)$ and this composition will be a retraction.
However, the inclusion $\sigma(l')\to X(l')$ induces the trivial map on homology, thus we have an element in the kernel of the map $H_i(X(l'-1); R) \to H_i(X(l'); R)$.
Thus the system of homology groups is not essentially trivial and $G_L(\sigma)$ is not of type $FP_k(R)$.
\end{proof}
We can also prove similar results about finite presentability of $G_L(\sigma)$.
\begin{theorem}
Suppose that $\pi_1(L)/\pi_1(\sigma(n))$ is finitely presented for all $n$.
Then $G_L(\sigma)$ is finitely presented if and only if $\pi_1(\sigma(n))$ vanishes for all but finitely many $n$.
\end{theorem}
\begin{proof}
For one direction, suppose that $G_L(\sigma)$ is finitely presented.
Since it acts freely and cocompactly on $f_\sigma^{-1}(\frac{1}{2})$, there are finitely many orbits of loops which normally generate $\pi_1(f_\sigma^{-1}(\frac{1}{2}))$.
Since the limit of $X(m)$ is simply connected, we see that there is an $l$ such that each of these loops is trivial in $X(l)$.
Thus the inclusion $f_\sigma^{-1}(\frac{1}{2})\to X(l)$ is trivial on fundamental groups.
By \cite[Corollary 2.6]{BeBr}, we have that $f_\sigma^{-1}(\frac{1}{2})\to X(l)$ is also a surjection.
Thus, $X(l)$ is simply connected.
Now using the retraction from the proof of \cref{thm:typefpn} we obtain a $\pi_1$-surjective map $X(l)\to \sigma(m)$ for all $m$ such that $|m|>l$.
Thus, we see that for all $m$ such that $|m|>l$, we must have $\sigma(m)$ is simply connected.
For the other direction, suppose that there is an $l$ such that if $|m|>l$ we have that $\sigma(m)$ is simply connected.
Then, by \cite[Corollary 2.6]{BeBr}, $X(l)$ is simply connected and has a cocompact action by $G_L(\sigma)$.
We now have a cocompact action of $G_L(\sigma)$ on a simply connected CW complex where stabilisers of cells are finitely presented.
Thus $G_L(\sigma)$ is finitely presented.
\end{proof}
\section{Presentations for $G_L(\sigma)$}
This section closely follows Section 14 of \cite{Lea}.
We begin by describing presentations of the groups $G_L(\sigma)$ obtained in the previous section.
\begin{notation}
Let $L$ be a simplicial complex.
Let $E$ be the set of edges of $L$.
For a loop $c = (e_1, \dots, e_l)$ in $L$. Let $c^{[k]}$ denote the word $e_1^ke_2^k\dots e_l^k$ in $F(E)$.
\end{notation}
\begin{theorem}\label{thm:presentationforgroups}
Let $\gamma_{i, n}$ be a collection of loops that normally generate $\pi_1(\sigma(n))$.
Let $E$ be the edges of $L$.
Suppose that $\sigma(0) = L$.
Then $G_L(\sigma)$ has the following presentation:
$$\langle E\mid efg, gfe \text{ for each triangle $e, f, g$ in $L$ }, \gamma_{i, n}^{[n]} \text{ for $n\in \Z$ and all } i \rangle. $$
\end{theorem}
\begin{proof}
Recall that we have a Morse function $f\colon X_L/BB_L\to \R$.
Let $Y_0 = f^{-1}(0)$.
Since $\sigma(0) = L$ we have that $Y_0\subset Y_L(\sigma)$
By \cite[Corollary 10.4]{Lea}, we have that the inclusion $Y_0\to Y_L(\sigma)$ is a surjection on $\pi_1$.
From \cite[Theorem 14.1]{Lea}, we have a presentation for $\pi_1(Y_0)$ which is exactly,
$$\langle E\mid efg, gfe \text{ for each triangle $e, f, g$ in $L$ }\rangle. $$
Let $Z = X_L\smallsetminus\{v\in X_L^{(0)}\mid f(v)\neq 0\}$.
Then $\pi_1(Z) = \pi_1(Y_0)$.
Thus to obtain a presentation of $G_L(\sigma)$ we add in the relations coming from the disks in $Y_L(\sigma)$.
By \cite[Lemma 14.3]{Lea}, we see that if $D$ is the disk glued to the word $\gamma_{i, n}$, then this corresponds exactly to the relation $\gamma_{i, n}^{[n]}$.
Thus we arrive at the desired presentation.
\end{proof}
Later we will show that there are uncountably many such groups.
For this purpose it will be useful to know the following:
\begin{lemma}\label{lem:trivial}
Let $\gamma$ be an edge loop in $L$.
Then $\gamma^{[n]}$ is trivial in $G_L(\sigma)$ if and only if $\gamma$ lifts to a loop in $\sigma(n)$.
\end{lemma}
\begin{proof}
The loop $\gamma^{[n]}$ in $Y_L(\sigma)$ is homotopic to the loop $\gamma$ in the link of the vertex $v$ of $X_L$ at height $n$.
We will label this $\gamma_n$.
Note that $\gamma_n$ belongs to the ascending or descending link of $v$ (depending on the sign of $n$) and this subspace is exactly $L$.
If $\gamma$ lifts to a loop in $\sigma(n)$, then $\gamma$ defines an element of $\pi_1(\sigma(n))$ and is homotopic to a product of the loops $\gamma_{i, n}$.
Thus together with the triangle relations we see that $\gamma^{[n]}$ is trivial in $\pi_1(Y_L(\sigma))$.
Now suppose that $\gamma^{[n]}$ is trivial in $\pi_1(Y_L(\sigma))$, then the loop $\gamma^{[n]}$ lifts to a loop in $\widetilde{Y_L(\sigma)}$.
We can now lift the homotopy and see that $\gamma_n$ also lifts to a loop in $\widetilde{Y_L(\sigma)}$ which we will call $\lambda_n$.
This loop must be in the ascending or descending link of some vertex $w$ of $\widetilde{Y_L(\sigma)}$.
However, this link is exactly $\sigma(n)$, thus $\lambda_n$ is a loop in $\sigma(n)$ which is a lift of $\gamma$.
\end{proof}
\section{Maps between $G_L(\sigma)$}
Recall that we partially order the set $\CZ$ by $\sigma\preceq\sigma'$ if $\sigma(n)$ is a cover of $\sigma'(n)$.
Let $\gamma_{i, n}$ be the set of loops that generate $\pi_1(\sigma(n))$ and $\gamma_{i, n}'$ be the set of loops that normally generate $\pi_1(\sigma'(n))$.
In the case that $\sigma(n)$ is a cover of $\sigma'(n)$ we can assume that $\{\gamma_{i, n}\}\subset\{\gamma_{i, n}'\}$.
Thus, one can see from the presentations in \Cref{thm:presentationforgroups} that there are surjective maps $G_L(\sigma)\to G_L(\sigma')$ whenever, $\sigma\preceq\sigma'$.
This map can also be realised at the level of cube complexes.
\begin{theorem}\label{thm:covers}
Let $\sigma, \sigma'\in \CZ$.
Suppose that $\sigma(n)\preceq\sigma'(n)$.
Then there is a branched cover of cube complexes $X_L^\sigma\to X_L^{\sigma'}$ which preserves level sets.
\end{theorem}
\begin{proof}
To see this recall $Y_L(\sigma)$ is obtained from $X_L$ by removing all vertices and gluing disks in the link at level $n$ to generators for $\pi_1(\S(\sigma(n)))$.
Thus take as our generating set for $\pi_1(\S(\sigma(n)))$ a generating set for $\pi_1(\S(\sigma'(n)))$ along with extra generators.
This way we obtain an inclusion $Y_L(\sigma)\to Y_L(\sigma')$.
Let $\mathcal{D}$ be the set of disks added to obtain $Y_L(\sigma)$.
Let $\mathcal{D}'$ be the set of disks added to $Y_L(\sigma)$ to obtain $Y_L(\sigma')$.
Let $\widetilde{Y_L(\sigma')}$ be the universal cover of $Y_L(\sigma')$.
Let $Y_1$ be the space obtained from $\widetilde{Y_L(\sigma')}$ by removing the lifts of disks in $\mathcal{D}'$.
Then $Y_1$ is the cover of $Y_L(\sigma)$ corresponding to the kernel of the surjection $G_L(\sigma)\to G_L(\sigma')$.
Thus, the universal cover of $Y_1$ is the universal cover of $Y_L(\sigma)$.
When we remove the disks and complete we get a branched cover of cube complexes ${X_L^\sigma}\to {X_L^{\sigma'}}$.
\end{proof}
\begin{corollary}
Let $\sigma, \sigma'\in\CZ$ with $\sigma\prec\sigma'$.
Suppose that $\sigma(0) = \sigma'(0) = L$.
Then the Cayley graph of $G_L(\sigma)$ is a cover of the Cayley graph for $G_L(\sigma')$, where both groups have as generating sets the edges of $L$.
\end{corollary}
\begin{proof}
In the case that $\sigma(0) = L$ we have that $G_L(\sigma)$ acts freely on $f_{\sigma}^{-1}(0)$. Moreover, it acts transitively on vertices. Thus, the 1-skeleton of $f_{\sigma}^{-1}(0)$ is the Cayley graph for $G_L(\sigma)$.
Similar statements hold for $G_L(\sigma')$.
Now \cref{thm:covers}, we have a covering map $X_L^\sigma\to X_L(\sigma')$ which preserves level sets. Thus we get a covering of Cayley graphs.
\end{proof}
\begin{theorem}\label{thm:subpresnotfp2}
There exists a presentation $G = \langle X\mid R\rangle$ of a group of type $FP_2(\Z)$ such that for any finite subset $T\subset R$, we can find $S$ such that $T\subset S\subset R$ and $\langle X\mid S\rangle$ is not of type $FP_2(R)$ for any ring $R$.
\end{theorem}
\begin{proof}
Throughout the proof let $G = \Z^3\rtimes SL_3(\Z)$ and $N = \Z^3\triangleleft G$.
There are two properties of this pair we will use namely, that $G$ is perfect (see for instance \cite{condersl3}) and so $H_1(G; R) = 0$ for all rings $R$ and $H_1(N; R) = R^3$ for all rings $R$.
Let $L$ be a flag complex with no local cut points and fundamental group $G$.
Let $K$ be the cover corresponding to $N$.
Let $E$ be the set of edges of $L$.
Let $\alpha_i = (e_{1,i}, \dots, e_{n_i,i})$ be a sequence of loops in $L$ that generate $N$.
Let $\beta_i = (f_{1, i}, \dots, f_{m_i, i})$ be a sequence of loops in $L$ that generate $G$.
We can obtain the following presentation for $BB_L$ from \cite{Dicks}:
$$\langle E\mid efg, gfe \text{ for each triangle $e, f, g$ in $L$ }, \alpha_i^{[n]}, \beta_i^{[n]} \text{ for $n\in \Z$ and all } i \rangle.$$
Since $G$ is perfect, we obtain from \cite{BeBr} that $BB_L$ is of type $FP_2(\Z)$.
Let $T$ be a finite subset of the relations of $BB_L$.
Let $F = \{n\mid \exists i$ such that $\beta_i^{[n]}\in T\}$.
Let $S$ be the the union of the following sets of relations:
\begin{itemize}
\item $T$,
\item all triangle relations,
\item $\alpha_i^{[n]}$ for all $n\in \Z$,
\item $\beta_i^{[n]}$ for $n\in F$
\end{itemize}
Consider the subpresentation $$\langle E\mid S\rangle$$
This is a presentation of $G_L(\sigma)$, where
$$\sigma(n) =
\begin{cases}
K, &\text{ if } n\notin F, \\
L, &\text{ if } n\in F.
\end{cases}$$
Since $F$ is a finite set there are infinitely many vertices such that the ascending and descending link have non-trivial first homology with coefficients in $R$.
Thus $G_L(\sigma)$ is not of type $FP_2(R)$ by \cref{thm:typefpn}.
\end{proof}
\section{Groups that are $FP_2$ over fields}
We are now ready to prove the following theorem.
\begin{theorem}\label{thm:acyclicoverQandP}
There exists groups that are of type $FP_2(\Q)$ and $FP_2(\Z/p\Z)$ for all $p$ but not of type $FP_2(\Z)$.
\end{theorem}
\begin{proof}
To do this we find a finitely presented group $G$ with a sequence of subgroups $G_n$ such that
$$H_1(G_n ; \Z) = \begin{cases}
\Z_n^m, &\text{ if $n$ is prime,}\\
0 , &\text{ if $n$ is not prime.}
\end{cases}$$
Let $G = SL_3(\Z)$.
Let $G_p$ be the level $p$ congruence subgroup.
By \cite{Lee}, we have that $H_1(G_p ; \Z) = \Z/p\Z^8$.
Now let $L$ be a flag complex with no local cut points with fundamental group $G$.
Let $G_n$ be the level $n$ congruence subgroup if $n > 2$ is prime and the trivial subgroup otherwise.
Let $L_n$ be the cover corresponding to $G_n$.
Let $\sigma$ be the function assigning $n$ to $L_n$.
In this case $L_n$ is a trivial or finite cover of $L$.
In either case, the quotient is finitely presented and hence of type $FP_2$ over any ring.
Since all the homology groups considered are finite we see that $H_1(\sigma(n); \Q)$ vanishes for all $n$.
Also $H_1(\sigma(n); \Z/p\Z)$ is non-trivial if and only if $n = p$.
Thus we can apply \cref{thm:typefpn} to see that $G_L(\sigma)$ is of type $FP_2(\Q)$ and $FP_2(\Z/p\Z)$ for all $p$.
However, there are infinitely many $n$ with $H_1(\sigma(n); \Z)$ non-trivial. Thus, by \Cref{thm:typefpn} we see that $G_L(\sigma)$ is not of type $FP_2(\Z)$.
\end{proof}
One would imagine that it is possible to prove the corresponding result for $FP_k$ or even $FP$.
The above theorem gives a template for how to do this.
To prove the above theorem for type $FP_k$ one would need a flag complex $L$ and a sequence of normal covers $L_n$ satisfying the following conditions:
\begin{itemize}
\item For infinitely many $n$, there is an $i < k$ such that $H_i(L_n; \Z)$ does not vanish.
\item For all but finitely many pairs $(i, n)$ with $i<k$, we have that $H_i(L_n; \Q)$ vanishes.
\item For each prime $p$, we have that for all but finitely many pairs $(i, n)$, with $i<k$, we have that $H_i(L_n; \Z/p\Z)$ vanishes.
\item For all $n$ and all $p$ the quotient $\pi_1(L)/\pi_1(L_n)$ is of type $FP_k(\Q)$ and $FP_k(\Z/p\Z)$.
\end{itemize}
For the $FP$ result replace $FP_k$ by $FP$ and remove the $i<k$ assumption throughout.
In a similar way to \cref{thm:acyclicoverQandP}, we can prove the following theorem.
\begin{theorem}\label{thm:setofprimes}
Let $\mathcal{P}$ be the set of primes.
For each subset $S$ of $\mathcal{P}$ there is a group which is type $FP_2(\Z/p\Z)$ if and only if $p\notin S$.
Moreover, we can construct such a group that has a proper action on a 3-dimensional CAT(0) cube complex.
\end{theorem}
\begin{proof}
Let $L$ be a flag complex with fundamental group $GL_3(\Z)$.
For $p>2$, let $G_p$ denote the level $p$ congruence subgroup in $SL_3(\Z)$.
Note that $G_p$ is still finite index and normal in $GL_3(\Z)$.
Let $G_2 = GL_3(\Z)$.
Let $\bar{L}$ be the cover of $L$ corresponding to $SL_3(\Z)$.
Let $L_{p}$ be the cover of $L$ corresponding to $G_p$.
Thus $H_1(L_p; \Z)$ is a finite $p$-group.
Let $S = \{p_1, p_2, p_3, \dots\}$
Let $(a_n)_{n\in \N}$ be any sequence which contains $p_i$ infinitely many times for each $p_i\in S$.
For instance, we can take the sequence
$$p_1, p_1, p_2, p_1, p_2, p_3, p_1, p_2, p_3, p_4, p_1, \dots.$$
Define $\sigma\colon \Z\to \CC$ as follows
$$\sigma(n) =
\begin{cases}
\bar{L}, &\text{if $n<0$},\\
L_{a_n}, &\text{if $n\geq 0$}.
\end{cases}$$
Since each $p_i$ appears infinitely many times in $(a_n)$ we have by \cref{thm:typefpn}, that $G_L(\sigma)$ is $FP_2(\Z/p\Z)$ if and only if $p\notin S$.
Since all the covers taken were finite index we see that all the vertex stabilisers are finite. Thus the action of $G_L(\sigma)$ on $X_L^\sigma$ is proper.
\end{proof}
\section{Uncountably many quasi-isometry classes}
We can show that there are uncountably many quasi-isometry classes of groups as in \cref{thm:setofprimes}.
The proof given here closely follows that of \cite{KLS}.
The idea of the proof is to interleave the sequence of covers from \cref{thm:setofprimes}, with the sequences of universal covers as in \cite{KLS}.
Thus, we can use the sequence from covers from \cref{thm:setofprimes} to obtain the desired finiteness properties and we use the relations (or lack thereof) from the universal covers to obtain uncountably many quasi-isometry classes.
Let $\sigma, \sigma'\in \CZ$ with $\sigma\prec\sigma'$.
Let $M(\sigma, \sigma') = \min\{|n| \mid \sigma(n)\neq \sigma'(n)\}$.
\begin{lemma}\label{lem:minlength}
Suppose that $L$ is $d$-dimensional and $\sigma, \sigma'\in \CZ$ with $\sigma\prec\sigma'$ and $\sigma(0) = \sigma'(0) = L$, and take the standard generating set for $G_L(\sigma)$ and $G_L(\sigma')$. The word length of any non-identity element in the kernel of the map $G_L(\sigma)\to G_L(\sigma')$ is at least $M(\sigma, \sigma')\sqrt{2/d+1}.$
\end{lemma}
\begin{proof}
This follows from Lemmas 3.1 and 3.2 of \cite{KLS}.
\end{proof}
We will use the taut loop length spectrum of Bowditch \cite{bowditch}.
\begin{definition}
Let $\Gamma$ be a graph and $l\in \N$.
Let $\Gamma_l$ denote the 2-complex with 1-skeleton $\Gamma$ and a 2-cell attached to each loop of length $<l$.
An edge loop of length $l$ is {\em taut} if it is not null-homotopic in $\Gamma_l$.
Bowditch's {\em taut loop length spectrum}, $H(\Gamma)$ is the set of lengths of taut loops.
\end{definition}
\begin{definition}
Let $H, H'$ be two sets of natural numbers. We say that $H$ and $H'$ are {\em $k$-related} if for all $l\geq k^2 +2k +2 $, whenever $l\in H$ then there is some $l'\in H'$ such that $\frac{l}{k}\leq l'\leq lk$ and vice versa.
\end{definition}
The key element from \cite{bowditch} is the following relating the taut loop length spectrum to quasi-isometries.
\begin{lemma}
If (connected) graphs $\Gamma$ and $\Lambda$ are $k$-quasi-isometric, then $H(\Gamma)$ and $H(\Lambda)$ are $k$-related.
\end{lemma}
We are now ready to prove the main theorem of this section.
The proof is similar to that of Theorem 5.2 in \cite{KLS}.
Here is a brief outline.
For each $F\subset \N$ we will construct a function $\sigma_F$.
We will then give a rough computation of the taut loop length spectrum for the Cayley graph for $G_L(\sigma_F)$.
This will show that if $G_L(\sigma_F)$ is quasi-isometric to $G_L(\sigma_{F'})$, then $F\triangle F'$ is finite and conclude the desired result.
The key change from \cite{KLS} is that due to the sequence of covers we cannot take a single constant $C$ and must take a sequence of constants $C_i$ satisfying certain conditions relating to the function $\sigma_F$.
Comparing to the proof of Theorem 5.2 in \cite{KLS} we have replaced the constant $\beta$ by the sequence of constants $r_p$ and chosen $C_i$ to ensure the arguments contained there still work.
\begin{theorem}\label{thm:uncountable}
Let $S$ be a set of primes.
Then there are uncountably many quasi-isometry classes of groups that are of type $FP_2(\Z/p\Z)$ if and only if $p\in S$.
\end{theorem}
\begin{proof}
Let $L$ be a flag complex with fundamental group $GL_3(\Z)$.
For each prime $p$, let $L_p$ be the cover of $L$ corresponding to the level-$p$ congruence subgroup in $SL_3(\Z)$.
Let $L_1$ be the cover corresponding to $SL_3(\Z)$.
Let $L_0$ be the universal cover $\tilde{L}$ of $L$.
Given a finite set $\Omega$ of loops that normally generate $\pi_1(L_p)$, define $r_p(\Omega)$ as the maximum length of a loop in $\Omega$.
Define $r_p$ to be the minimum of $r_p(\Omega)$ as $\Omega$ runs over all possible normal generating sets for $\pi_1(L_p)$.
Let $(b_n)$ be the following sequence:
$$b_{n} =
\begin{cases}
a_m, & \text{if $n = 2m+1$}, \\
1, & \text{if $n = 2m$},\\
\end{cases}$$
Let $\alpha = \sqrt{2/(d+1)}$, where $d$ is the dimension of $L$.
Let $C_n$ be a sequence of integers satisfying the following conditions:
\begin{itemize}
\item $C_1\alpha > 3$,
\item $C_n\alpha > r_{b_{n-1}}$,
\item $C_n\alpha > r_{b_n}$
\item $C_n >C_{n-1}$.
\end{itemize}
From this we can deduce that $\alpha C_n^{2^n} > r_{b_{n-1}}C_{n-1}^{2^{n-1}}$.
Let $F\subset \N$.
Let $\overline{F} = \{2n\mid n\in F\}\cup\{2n+1\mid n\in \N\}$.
Define $\sigma_F$ as follows:
$$\sigma_F(n) = \begin{cases}
L_{b_i}, &\text{if $n = {C_i^{2^{i}}}$ and $i\in \overline{F}$,}\\
L, &\text{if $n = 0$,}\\
{L_0}, &\text{otherwise.}
\end{cases}$$
Let $\Gamma_F$ be the Cayley graph of $G_L(\sigma_F)$ with generating set the edges of $L$.
We will prove the following two propositions about $H(\Gamma_F)$:
\begin{itemize}
\item If $k\in \overline{F}$, then $H(\Gamma_F)\cap [C_k^{2^k}\alpha, C_k^{2^k}r_{b_k}] \neq \emptyset$.
\item If $k>3$ and $k \notin \cup_{l\in \overline{F}}[C_l^{2^l}\alpha, C_l^{2^l}r_{b_l}]$, then $k\notin H(\Gamma_F)$.
\end{itemize}
Suppose $k\in \overline{F}$, let $F' = F\smallsetminus\{k\}$.
There is a surjection $G_L(\sigma_F)\to G_L(\sigma_{F'})$.
Let $K$ be the kernel of this surjection.
By \cref{lem:minlength}, any non-identity element of $K$ has length at least $\alpha C_k^{2^k}$.
We also know that there is an element of length $r_kC_k^{2^k}$ in $K$.
The length of the shortest element of $K$ defines a member of $H(\Gamma_F)$.
Thus we obtain the first statement.
For the second statement, let $k\notin F$.
Choose $n\in \N$ maximal such that $r_{b_n}C_n^{2^n} < k$ or $-1$ otherwise.
Let $F' = F\cap [0, n]$.
Let $K$ be the kernel of the map $G_L(\sigma_{F'})\to G_L(\sigma_F)$.
Consider the covering map $\Gamma_{F'}\to \Gamma_F$ coming from \cref{thm:covers}.
Every relator in $G_L(\sigma_{F'})$ has length $\leq r_{b_n}C_n^{2^n}$.
Now suppose that $\gamma$ is a loop of length $k$ in $\Gamma_F$.
We can lift $\gamma$ to $\Gamma_{F'}$.
If $\gamma$ lifts to a loop in $\Gamma_{F'}$, then it must be a consequence of loops of length $\leq r_{b_n}C_n^{2^n}$.
Thus, $\gamma$ cannot be taut.
Now suppose that $\gamma$ lifts to a non-closed path.
In this case $\gamma$ defines an element of $K$.
The shortest such element has length $\geq \alpha C_m^{2^m}$ where $m = M(\sigma_F, \sigma_{F'})$.
By choice of $n$ we have that $k\leq r_mC_m^{2^m}$.
Thus, we obtain that $k\in [\alpha C_m^{2^m}, r_{b_m}C_m^{2^m}]$.
However, this contradicts the choice of $k$ and thus $\gamma$ is not taut.
To complete the proof, suppose that $l\in [\alpha C_m^{2^m}, r_{b_m}C_m^{2^m}]$ and $l'\in [\alpha C_{m+n}^{2^{m+n}}, r_{b_{m+n}}C_{m+n}^{2^{m+n}}]$ for some $n > 0$.
Then $$\frac{l'}{l} \geq \frac{\alpha C_{m+n}^{2^{m+n}}}{r_{b_m}C_m^{2^m}} > \frac{\alpha C_m^{2^{m+n}}}{r_{b_m}C_m^{2^m}} \geq \frac{\alpha C_m^{2^{m} - 1}} {r_{b_m}} > C_m^{2^{m}-2}.$$
Suppose that $H(\Gamma_F)$ and $H(\Gamma_{F'})$ are $k$-related. If $n\in F\triangle F'$, then $C_n^{2^{2n}-2}\leq k$.
Thus if $F\triangle F'$ is infinite, then $H(\Gamma_F)$ and $H(\Gamma_{F'})$ are not $k$-related.
Hence, $G_L(\sigma_F)$ is not quasi-isometric to $G_L(\sigma_{F'})$.
\end{proof}
While we have used the framework of Bowditch's taut loop length spectrum it is also possible to use the work of \cite{MOW}.
Let $\mathcal{G}$ be the space of marked groups.
Combining \cref{lem:trivial} and \cref{lem:minlength} we see that $\CZ\to \mathcal{G}$ given by $\sigma\mapsto G_L(\sigma)$ is a continuous injection with perfect image.
Thus, by \cite[Theorem 1.1]{MOW} we obtain uncountably many quasi-isometry classes of groups of the form $G_L(\sigma)$.
By carefully choosing subsets of $\CZ$ to ensure that the image is still perfect one can proves analogues of \cref{thm:uncountable} for other properties satisfied by the $G_L(\sigma)$.
\bibliographystyle{plain}
\bibliography{bib}
\end{document}
| 156,018
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Yard. 2015 © NewBay Media, LLC. 28 East 28th Street, 12th floor, New York, NY 10016 T (212) 378-0400 F (917) 281-4704
| 119,971
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You are in
FileFormat.Info » Info » Unicode » Characters » U+A093
This is a list of fonts that support Unicode Character 'YI SYLLABLE HMI' (U+A093).
If you are a font author and would like your font listed here, please let me know.
The default image is using the Code2000 font
View All
| 54,264
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There are often outdoor functions that need some kind of protection for the event. It could occur for an event at a local school event or for a corporate event, or even a smaller family gathering that needs the buffet area to be covered.
In these instances, you'll be required to recognize the differences between gazebos that are heavy-duty as well as instant marquees.
The movable marquee tent is often used to create outdoor displays. There is a wide range of individuals who could make use of these gazebos.
Image Source: Google
These gazebos that are heavy-duty are ideal for real estate agents who are promoting their properties during fairs and are ideal to host a family-friendly event in which seating arrangements and buffet arrangements are required.
It can also be that is used at fairs for commercial purposes where the entire event is conducted outdoors. In such instances, heavy-duty structures are beneficial as they offer ample space. They're also durable and are able to withstand elements, such as powerful wind gusts or sudden rainstorms.
There is also the option of having sides walls that are attached to them to provide more protection. This can also help keep the temperature in the gazebo by using heaters or fans.
Instant marquees don't require anyone else to put them up. When you lease them, they will come with a set of directions that are simple to follow. You will get them set up in just a few minutes. They are a great option for school parties, event planners wedding planners, firms of different sizes, marketing firms and churches, clubs, and DJs for outdoor events.
| 3,658
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If you purchase an independently reviewed product or service through a link on our website, we may receive an affiliate commission.
$130 is a price that is beyond fair for one the best-selling Alexa ready thermostats on Amazon. Unlike the Nest and similar options, the Sensi Smart Thermostat is so simple and familiar that anyone can figure out how to use it in no time. But the real fun begins thanks to the Alexa integration. Should you really have to get up off your couch in order to adjust your heating or air conditioning? No, no you shouldn’t. And right now, Amazon is slashing its already fair price on the Sensi’s Smart Thermostat all the way down to $99. Don’t let this one slip by.
Here are some highlights from the Amazon
- Works with Amazon Alexa for voice control (Alexa device sold separately) and Wink app (no Wink hub.
| 67,707
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TITLE: Could the concept of "finite free groups" be possible?
QUESTION [12 upvotes]: Is it possible to define "finite free groups" ? could that make it easier to deal with group presentations ?
REPLY [1 votes]: Groups have a first-order axiomatization, and one could repeat for groups the definition of pseudo-finite fields, which are the infinite fields satisfying the same sentences as (all) finite fields in the first-order theory of commutative fields.
[Edit. Search shows that the theory exists as expected, but is much more complicated than the theory of pseudofinite fields.]
The free $n$-generator pseudo-finite group would be the group containing $n$ distinct elements with no other properties except the ones that follow from the first-order theory of pseudofinite groups and the axiom "there exist $n$ distinct non-identity elements".
| 91,238
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PALMER (AP) -- An 8-year-old third-grader is king of the cabbages this year at the Alaska State Fair.
Seth Dinkel's entry in the Giant Cabbage Weigh-Off was measured at 92.5 pounds, short of a record but half again Seth's own 60-pound weight.
Second place was won by Robert Thom's 71.4-pounder; third place went to Mary Evans, who entered a 70.4-pound cabbage.
Twenty-nine adults and children entered the open and junior levels of competition, which took place Friday. Seth was by far the youngest in the Open division. He won $2,000, which works out to about $21.62 a pound.
His entry wasn't even as big as the one he brought last year, a 94.4-pounder that took second place. Both times he entered in the Open division, rather than joining the other kids in the Max Sherrod Junior Cabbage Growers Competition.
''It wouldn't be fair. I'd win,'' said Seth, a third-generation monster cabbage grower.
His cabbage wasn't the prettiest one in the patch. It looked rumpled, and some of the leaves at the base were turning yellow. But looks aren't everything. Last year's winner, grown by Barbara Everingham, set a fair record at 105.6 pounds.
It ''looked like something Captain Kirk would empty his phaser into,'' recalled Mike Campbell, the official weigher.
Joe Dubler wasn't real pleased with his 48.4-pounder; it would have weighed a lot more if the slugs hadn't been at it. The cabbage was so riddled with holes it looked strafed. That's because Dubler fished a lot this summer.
''I had fun fishing,'' he said, ''but I've got an ugly cabbage now.''
The prettiest entry was the fourth-place winner, a 64.2-pounder grown by Scott Robb. It had a perfectly round, unblemished head and crisp and graceful leaves without a touch of wilt.
''It's not a fluke. You have to work at it,'' said Robb, who last year grew a world-record rutabaga that almost made it onto ''The David Letterman Show.'' This year, he has two other unofficial world record-holders at the fair: a 42.4-pound kale and a 43.7-pound kohlrabi.
''It's all genetic,'' Robb said. ''You've got to have the right seed.''
Seth isn't saying what kind of seed he uses. It's a Dinkel family secret. This year he had several huge cabbages, but ''I wasn't sure (I) could break 90 pounds.''
| 139,941
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My Thanksgiving edition Courant arrived weighing more than most people's turkeys. Despite the wet snow, it was remarkably dry, which I hope was not the case for most people's turkeys.
I have no idea how our carrier managed to handle such shifty, oversized parcels, let alone bend the laws of physics to make them fit inside teensy bags, but when I opened my door I was pleasantly surprised to find my paper and not some wayward relative.
For all the talk of Thanksgiving work schedules, there was little mention of the people who bring us the news daily and are often forgotten about due to nightly hours. This holiday season I will remember to tip my carrier for exceptional service on even the darkest and dreariest nights.
Sara Walker, Manchester
| 143,172
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TITLE: Mathematical Induction (product of $n$ consecutive numbers)
QUESTION [2 upvotes]: Assumption:
$$(n+1)(n+2) \cdots (2n) = (2^n)\cdot 1 \cdot 3 \cdot 5 \cdots (2n-1)$$
Prove for $n+1$:
$$(n+2)(n+3) \cdots (2(n+1)) = (2^{n+1}) \cdot 1 \cdot 3 \cdot 5 \cdots (2(n+1)-1)$$
Using the assumption, I divide both sides by $(n+1)$ and substitute RHS into my $n+1$ equation, however it does not equate.
REPLY [2 votes]: HINT $\ $ Dividing the second equation by the first yields the identity
$$\rm\frac{(2\:n+1)\ (2\:n+2)}{n+1}\ =\ \ 2\ (2\:n+1) $$
Thus the second equation is simply $\rm\ 2\ (2\:n+1)\ $ times the first equation.
Alternatively one can easily reduce the induction to a trivial induction that a product of 1's equals 1, see my prior posts on (multiplicative) telescopy.
| 119,623
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TITLE: Uncountable unions of nested sets in a sigma field
QUESTION [6 upvotes]: I know that in a $\sigma$-algebra, uncountable unions may not exist. However suppose I have a directed system $\{A_i,i\in I\}$ where for each $i\in I$, $A_i\in\mathcal A$ (the $\sigma$-algebra) with the property that for any $i,j\in I$, we have $A_i\subseteq A_j$ or $A_j\subseteq A_i$. Then I believe $\cup_{i\in I}A_i\in\mathcal A$ but I am having difficulty in writing a proof. Notice here that $I$ may be uncountable.
REPLY [5 votes]: Here is a very general counterexample. Suppose $\mathcal A$ is any proper subcollection of the power set $\mathcal P(E)$ and suppose that all finite subsets of $E$ belong to $\mathcal A.$ Choose an element $X\in\mathcal P(E)\setminus\mathcal A$ of minimum cardinality. By well-ordering the infinite set $X$ we can write it as the union of a chain of subsets of smaller cardinality. Thus $X$ is the union of a chain of elements of $\mathcal A,$ but $X\notin\mathcal A.$
| 119,775
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Psittacosis
Psittacosis is an infection caused by Chlamydophila psittaci, a type of bacteria found in the droppings of birds. Birds spread the infection to humans.
Causes of Psittacosis
Psittacosis is a rare disease. Only 100 to. are usually not prescribed for children until after all their permanent teeth have started to grow in, because they can permanently discolor teeth that are still forming. These medicines are also not prescribed to pregnant women. Other antibiotics are used in these situations.
Prognosis (Outlook)
A full recovery is expected if you do not have any other conditions that affect your health.
Avoid exposure to birds that may carry these bacteria, such as imported parakeets. Medical problems that lead to a weak immune system increase your risk for this disease and should be treated appropriately.
References
Limper AH. Overview of pneumonia. In: Goldman L, Schafer AI, eds. Goldman's Cecil Medicine. 24th ed. Philadelphia, PA: Elsevier Saunders; 2011:chap 97.
Torres A. Pyogenic bacterial pneumonia and lung abscess. In: Mason RJ, Broaddus CV, Martin TR, et al. Murray & Nadel's Textbook of Respiratory Medicine. 5th ed. Philadelphia, PA: Elsevier Saunders; 2010:chap 32.
| 318,780
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Patch Over Network
Contents
Project Goal
An alternative approach to "hacking" the appletv without opening the case. This method would use the build in Apple software updater to install custom hacks and patches.
Even when the USB method is successful (and that would still be awesome), this method may be better because it would allow server based updates on an on going basis without creating a whole new updating system.
Making this a reality with DNS/HOSTS, IIS/Apache
Furthermore, deployment and scalability can be incorporated and images can be deployed very simply.
We can trick the Apple TV to download our patched image by using the HOSTS file or a DNS server which redirects mesu.apple.com to an IP on our LAN or an update server. This would be done by creating an A record.
The advantage of using DNS over HOSTS is that the Apple TV unit does not need a modified hosts file, which requires SSH or hard drive access. Therefore, it is easier for the end user.
Therefore, we can install the Windows DNS server on a box, and add IIS. We can create a new website, setting our host header value to (not required, but if you're running multiple sites on one IP).
Next we will need to add to our website, a version.xml file and a modified update in the form of a DMG. To gain an insight into the structure of these two files, see the known info which is listed below.
Once this is done, the user can change the DNS Settings IP Addresss to that of the nameserver, which is probably hosting the update on a webserver too.
We're a couple of points off though:
- We need to gain a more comprehensive form of the version.xml structure. - We need to be able to create a custom DMG file that has a patch.
We also need to consider how realistic this is. It is realistic if we have a hosted DNS server downloading the DMG off of a server, but not if the home-user is expected to manage this on their lan.
--Sam 17:30, 18 June 2009 (CEST)
Known Info
When "Update Software" is selected from the settings menu, this file is requested: .
From 2007-04-03 until 2007-06-19 there were no software updates and the file contained the following...
<?xml version="1.0" encoding="UTF-8"?> <!DOCTYPE plist PUBLIC "-//Apple Computer//DTD PLIST 1.0//EN" ""> <plist version="1.0"> <dict> </dict> </plist>
By adding mesu.apple.com to the hosts file to point to a local web server, we are able to modify this file and make the AppleTV unit download the modified file.
When downloading this file, the following is recorded by syslog...
Apr 3 20:38:47 appletv.local AppleTV FrontRow[113]: T:[0x193fa00] UPD: update check starting Apr 3 20:38:47 appletv.local AppleTV FrontRow[113]: UPD: checking version info at. Apr 3 20:38:48 appletv.local AppleTV FrontRow[113]: T:[0x193fa00] downloading file Apr 3 20:38:48 appletv.local AppleTV FrontRow[113]: finished downloading file Apr 3 20:38:48 appletv.local AppleTV FrontRow[113]: VERS: comparing OS 10.4.7 with (null) Apr 3 20:38:49 appletv.local AppleTV FrontRow[113]: VERS: comparing OS build 8N5107 with (null) Apr 3 20:38:49 appletv.local AppleTV FrontRow[113]: UPD: versions available: OS:(null)/(null) EFI:(null) IR:(null) SI:(null)/(null) valid:1 Apr 3 20:38:49 appletv.local AppleTV FrontRow[113]: T:[0x193fa00] UPD: updating check complete
It seems to be looking for OS, build, EFI, IR, and SI version numbers in the xml file. If it can be determined the format of this xml file, it would be possible to have the AppleTV unit download and install patched versions of the OS automatically.
On 2007-06-20 Apple rolled out their first software update to add the YouTube functionality. The file at now contains:
<?xml version="1.0" encoding="UTF-8"?> <!DOCTYPE plist PUBLIC "-//Apple Computer//DTD PLIST 1.0//EN" ""> <plist version="1.0"> <dict> <key>OS</key> <dict> <key>BuildVersion</key> <string>8N5239</string> <key>UpdateURL</key> <string></string> <key>Version</key> <string>10.4.7</string> </dict> </dict> </plist>
We can see that the update is keyed with "OS", presumably because it is a general Operating System update. It also details the BuildVersion, Version and most importantly the UpdateURL which points to a disk image .dmg file. The file in question appears to be 179MB and presumably is the only file the AppleTV needs to update itself with the new functionality. I have made no attempt analyse the disk image.
Attempting to point a web browser to results in a Service Unavailable message. However, the disk image may be downloaded to be analyzed using CURL: "curl > atv.dmg". (Update: IE6 on Windows XP is able to download the image too. Update 2: the Safari Beta 3.0.2 on Mac OSX 10.4.10 can download the image by just clicking the link too) At first glance, this image seems to contain a slimmed down OS install to boot and patch the Apple TV.
From the syslog above we see:
Apr 3 20:38:48 appletv.local AppleTV FrontRow[113]: VERS: comparing OS 10.4.7 with (null) Apr 3 20:38:49 appletv.local AppleTV FrontRow[113]: VERS: comparing OS build 8N5107 with (null)
The first line is comparing "10.4.7" with "null". We can see in the new version.xml file that the key "Version" has the string "10.4.7".
The second line is comparing "8N5107" with "null". Again, from the version.xml we see that the key "BuildVersion" has the string "8N5239".
It appears that in the syslog we are seeing the output of a check to ensure the update applied is a later version than the current system.
Prior to downloading the disk image, AppleTV downloads a signature file ( for the 20 June 2007 update). Next, the disk image is downloaded and presumably it's signature is compared with the downloaded one.
The next step is to record the syslog when someone applies this software update to their AppleTV.
Software Update in MacOS X
"Real" Mac OS X connects to swscan.apple.com
wget
(It seems that modifying user-agent is not necessary)
It is possible that AppleTV software update expects a similar xml catalogue.
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(* Title: JinjaThreads/Framework/FWLTS.thy
Author: Andreas Lochbihler
*)
section \<open>The multithreaded semantics as a labelled transition system\<close>
theory FWLTS
imports
FWProgressAux
FWLifting
LTS
begin
sublocale multithreaded_base < trsys "r t" for t .
sublocale multithreaded_base < mthr: trsys redT .
\<comment> \<open>Move to FWSemantics?\<close>
definition redT_upd_\<epsilon> :: "('l,'t,'x,'m,'w) state \<Rightarrow> 't \<Rightarrow> 'x \<Rightarrow> 'm \<Rightarrow> ('l,'t,'x,'m,'w) state"
where [simp]: "redT_upd_\<epsilon> s t x' m' = (locks s, ((thr s)(t \<mapsto> (x', snd (the (thr s t)))), m'), wset s, interrupts s)"
lemma redT_upd_\<epsilon>_redT_upd:
"redT_upd s t \<epsilon> x' m' (redT_upd_\<epsilon> s t x' m')"
by(auto simp add: redT_updLns_def redT_updWs_def)
context multithreaded begin
sublocale trsys "r t" for t .
sublocale mthr: trsys redT .
end
subsection \<open>The multithreaded semantics with internal actions\<close>
type_synonym
('l,'t,'x,'m,'w,'o) \<tau>moves =
"'x \<times> 'm \<Rightarrow> ('l,'t,'x,'m,'w,'o) thread_action \<Rightarrow> 'x \<times> 'm \<Rightarrow> bool"
text \<open>pretty printing for \<open>\<tau>moves\<close>\<close>
print_translation \<open>
let
fun tr' [(Const (@{type_syntax "prod"}, _) $ x1 $ m1),
(Const (@{type_syntax "fun"}, _) $
(Const (@{type_syntax "prod"}, _) $
(Const (@{type_syntax "finfun"}, _) $ l $
(Const (@{type_syntax "list"}, _) $ Const (@{type_syntax "lock_action"}, _))) $
(Const (@{type_syntax "prod"},_) $
(Const (@{type_syntax "list"}, _) $ (Const (@{type_syntax "new_thread_action"}, _) $ t1 $ x2 $ m2)) $
(Const (@{type_syntax "prod"}, _) $
(Const (@{type_syntax "list"}, _) $ (Const (@{type_syntax "conditional_action"}, _) $ t2)) $
(Const (@{type_syntax "prod"}, _) $
(Const (@{type_syntax "list"}, _) $ (Const (@{type_syntax "wait_set_action"}, _) $ t3 $ w)) $
(Const (@{type_syntax prod}, _) $
(Const (@{type_syntax list}, _) $ (Const (@{type_syntax "interrupt_action"}, _) $ t4)) $
(Const (@{type_syntax "list"}, _) $ o1)))))) $
(Const (@{type_syntax "fun"}, _) $
(Const (@{type_syntax "prod"}, _) $ x3 $ m3) $
(Const (@{type_syntax "bool"}, _))))] =
if x1 = x2 andalso x1 = x3 andalso m1 = m2 andalso m1 = m3 andalso t1 = t2 andalso t2 = t3 andalso t3 = t4
then Syntax.const (@{type_syntax "\<tau>moves"}) $ l $ t1 $ x1 $ m1 $ w $ o1
else raise Match;
in [(@{type_syntax "fun"}, K tr')]
end
\<close>
typ "('l,'t,'x,'m,'w,'o) \<tau>moves"
locale \<tau>multithreaded = multithreaded_base +
constrains final :: "'x \<Rightarrow> bool"
and r :: "('l,'t,'x,'m,'w,'o) semantics"
and convert_RA :: "'l released_locks \<Rightarrow> 'o list"
fixes \<tau>move :: "('l,'t,'x,'m,'w,'o) \<tau>moves"
sublocale \<tau>multithreaded < \<tau>trsys "r t" \<tau>move for t .
context \<tau>multithreaded begin
inductive m\<tau>move :: "(('l,'t,'x,'m,'w) state, 't \<times> ('l,'t,'x,'m,'w,'o) thread_action) trsys"
where
"\<lbrakk> thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>; thr s' t = \<lfloor>(x', ln')\<rfloor>; \<tau>move (x, shr s) ta (x', shr s') \<rbrakk>
\<Longrightarrow> m\<tau>move s (t, ta) s'"
end
sublocale \<tau>multithreaded < mthr: \<tau>trsys redT m\<tau>move .
context \<tau>multithreaded begin
abbreviation \<tau>mredT :: "('l,'t,'x,'m,'w) state \<Rightarrow> ('l,'t,'x,'m,'w) state \<Rightarrow> bool"
where "\<tau>mredT == mthr.silent_move"
end
lemma (in multithreaded_base) \<tau>rtrancl3p_redT_thread_not_disappear:
assumes "\<tau>trsys.\<tau>rtrancl3p redT \<tau>move s ttas s'" "thr s t \<noteq> None"
shows "thr s' t \<noteq> None"
proof -
interpret T: \<tau>trsys redT \<tau>move .
show ?thesis
proof
assume "thr s' t = None"
with \<open>\<tau>trsys.\<tau>rtrancl3p redT \<tau>move s ttas s'\<close> have "thr s t = None"
by(induct rule: T.\<tau>rtrancl3p.induct)(auto simp add: split_paired_all dest: redT_thread_not_disappear)
with \<open>thr s t \<noteq> None\<close> show False by contradiction
qed
qed
lemma m\<tau>move_False: "\<tau>multithreaded.m\<tau>move (\<lambda>s ta s'. False) = (\<lambda>s ta s'. False)"
by(auto intro!: ext elim: \<tau>multithreaded.m\<tau>move.cases)
declare split_paired_Ex [simp del]
locale \<tau>multithreaded_wf =
\<tau>multithreaded _ _ _ \<tau>move +
multithreaded final r convert_RA
for \<tau>move :: "('l,'t,'x,'m,'w,'o) \<tau>moves" +
assumes \<tau>move_heap: "\<lbrakk> t \<turnstile> (x, m) -ta\<rightarrow> (x', m'); \<tau>move (x, m) ta (x', m') \<rbrakk> \<Longrightarrow> m = m'"
assumes silent_tl: "\<tau>move s ta s' \<Longrightarrow> ta = \<epsilon>"
begin
lemma m\<tau>move_silentD: "m\<tau>move s (t, ta) s' \<Longrightarrow> ta = (K$ [], [], [], [], [], [])"
by(auto elim!: m\<tau>move.cases dest: silent_tl)
lemma m\<tau>move_heap:
assumes redT: "redT s (t, ta) s'"
and m\<tau>move: "m\<tau>move s (t, ta) s'"
shows "shr s' = shr s"
using m\<tau>move redT
by cases(auto dest: \<tau>move_heap elim!: redT.cases)
lemma \<tau>mredT_thread_preserved:
"\<tau>mredT s s' \<Longrightarrow> thr s t = None \<longleftrightarrow> thr s' t = None"
by(auto simp add: mthr.silent_move_iff elim!: redT.cases dest!: m\<tau>move_silentD split: if_split_asm)
lemma \<tau>mRedT_thread_preserved:
"\<tau>mredT^** s s' \<Longrightarrow> thr s t = None \<longleftrightarrow> thr s' t = None"
by(induct rule: rtranclp.induct)(auto dest: \<tau>mredT_thread_preserved[where t=t])
lemma \<tau>mtRedT_thread_preserved:
"\<tau>mredT^++ s s' \<Longrightarrow> thr s t = None \<longleftrightarrow> thr s' t = None"
by(induct rule: tranclp.induct)(auto dest: \<tau>mredT_thread_preserved[where t=t])
lemma \<tau>mredT_add_thread_inv:
assumes \<tau>red: "\<tau>mredT s s'" and tst: "thr s t = None"
shows "\<tau>mredT (locks s, ((thr s)(t \<mapsto> xln), shr s), wset s, interrupts s) (locks s', ((thr s')(t \<mapsto> xln), shr s'), wset s', interrupts s')"
proof -
obtain ls ts m ws "is" where s: "s = (ls, (ts, m), ws, is)" by(cases s) fastforce
obtain ls' ts' m' ws' is' where s': "s' = (ls', (ts', m'), ws', is')" by(cases s') fastforce
from \<tau>red s s' obtain t' where red: "(ls, (ts, m), ws, is) -t'\<triangleright>\<epsilon>\<rightarrow> (ls', (ts', m'), ws', is')"
and \<tau>: "m\<tau>move (ls, (ts, m), ws, is) (t', \<epsilon>) (ls', (ts', m'), ws', is')"
by(auto simp add: mthr.silent_move_iff dest: m\<tau>move_silentD)
from red have "(ls, (ts(t \<mapsto> xln), m), ws, is) -t'\<triangleright>\<epsilon>\<rightarrow> (ls', (ts'(t \<mapsto> xln), m'), ws', is')"
proof(cases rule: redT_elims)
case (normal x x' m') with tst s show ?thesis
by-(rule redT_normal, auto simp add: fun_upd_twist elim!: rtrancl3p_cases)
next
case (acquire x ln n)
with tst s show ?thesis
unfolding \<open>\<epsilon> = (K$ [], [], [], [], [], convert_RA ln)\<close>
by -(rule redT_acquire, auto intro: fun_upd_twist)
qed
moreover from red tst s have tt': "t \<noteq> t'" by(cases) auto
have "(\<lambda>t''. (ts(t \<mapsto> xln)) t'' \<noteq> None \<and> (ts(t \<mapsto> xln)) t'' \<noteq> (ts'(t \<mapsto> xln)) t'') =
(\<lambda>t''. ts t'' \<noteq> None \<and> ts t'' \<noteq> ts' t'')" using tst s by(auto simp add: fun_eq_iff)
with \<tau> tst tt' have "m\<tau>move (ls, (ts(t \<mapsto> xln), m), ws, is) (t', \<epsilon>) (ls', (ts'(t \<mapsto> xln), m'), ws', is')"
by cases(rule m\<tau>move.intros, auto)
ultimately show ?thesis unfolding s s' by auto
qed
lemma \<tau>mRedT_add_thread_inv:
"\<lbrakk> \<tau>mredT^** s s'; thr s t = None \<rbrakk>
\<Longrightarrow> \<tau>mredT^** (locks s, ((thr s)(t \<mapsto> xln), shr s), wset s, interrupts s) (locks s', ((thr s')(t \<mapsto> xln), shr s'), wset s', interrupts s')"
apply(induct rule: rtranclp_induct)
apply(blast dest: \<tau>mRedT_thread_preserved[where t=t] \<tau>mredT_add_thread_inv[where xln=xln] intro: rtranclp.rtrancl_into_rtrancl)+
done
lemma \<tau>mtRed_add_thread_inv:
"\<lbrakk> \<tau>mredT^++ s s'; thr s t = None \<rbrakk>
\<Longrightarrow> \<tau>mredT^++ (locks s, ((thr s)(t \<mapsto> xln), shr s), wset s, interrupts s) (locks s', ((thr s')(t \<mapsto> xln), shr s'), wset s', interrupts s')"
apply(induct rule: tranclp_induct)
apply(blast dest: \<tau>mtRedT_thread_preserved[where t=t] \<tau>mredT_add_thread_inv[where xln=xln] intro: tranclp.trancl_into_trancl)+
done
lemma silent_move_into_RedT_\<tau>_inv:
assumes move: "silent_move t (x, shr s) (x', m')"
and state: "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>" "wset s t = None"
shows "\<tau>mredT s (redT_upd_\<epsilon> s t x' m')"
proof -
from move obtain red: "t \<turnstile> (x, shr s) -\<epsilon>\<rightarrow> (x', m')" and \<tau>: "\<tau>move (x, shr s) \<epsilon> (x', m')"
by(auto simp add: silent_move_iff dest: silent_tl)
from red state have "s -t\<triangleright>\<epsilon>\<rightarrow> redT_upd_\<epsilon> s t x' m'"
by -(rule redT_normal, auto simp add: redT_updLns_def o_def finfun_Diag_const2 redT_updWs_def)
moreover from \<tau> red state have "m\<tau>move s (t, \<epsilon>) (redT_upd_\<epsilon> s t x' m')"
by -(rule m\<tau>move.intros, auto dest: \<tau>move_heap simp add: redT_updLns_def)
ultimately show ?thesis by auto
qed
lemma silent_moves_into_RedT_\<tau>_inv:
assumes major: "silent_moves t (x, shr s) (x', m')"
and state: "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>" "wset s t = None"
shows "\<tau>mredT^** s (redT_upd_\<epsilon> s t x' m')"
using major
proof(induct rule: rtranclp_induct2)
case refl with state show ?case by(cases s)(auto simp add: fun_upd_idem)
next
case (step x' m' x'' m'')
from \<open>silent_move t (x', m') (x'', m'')\<close> state
have "\<tau>mredT (redT_upd_\<epsilon> s t x' m') (redT_upd_\<epsilon> (redT_upd_\<epsilon> s t x' m') t x'' m'')"
by -(rule silent_move_into_RedT_\<tau>_inv, auto)
hence "\<tau>mredT (redT_upd_\<epsilon> s t x' m') (redT_upd_\<epsilon> s t x'' m'')" by(simp)
with \<open>\<tau>mredT^** s (redT_upd_\<epsilon> s t x' m')\<close> show ?case ..
qed
lemma red_rtrancl_\<tau>_heapD_inv:
"\<lbrakk> silent_moves t s s'; wfs t s \<rbrakk> \<Longrightarrow> snd s' = snd s"
proof(induct rule: rtranclp_induct)
case base show ?case ..
next
case (step s' s'')
thus ?case by(cases s, cases s', cases s'')(auto dest: \<tau>move_heap)
qed
lemma red_trancl_\<tau>_heapD_inv:
"\<lbrakk> silent_movet t s s'; wfs t s \<rbrakk> \<Longrightarrow> snd s' = snd s"
proof(induct rule: tranclp_induct)
case (base s') thus ?case by(cases s')(cases s, auto simp add: silent_move_iff dest: \<tau>move_heap)
next
case (step s' s'')
thus ?case by(cases s, cases s', cases s'')(auto simp add: silent_move_iff dest: \<tau>move_heap)
qed
lemma red_trancl_\<tau>_into_RedT_\<tau>_inv:
assumes major: "silent_movet t (x, shr s) (x', m')"
and state: "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>" "wset s t = None"
shows "\<tau>mredT^++ s (redT_upd_\<epsilon> s t x' m')"
using major
proof(induct rule: tranclp_induct2)
case (base x' m')
thus ?case using state
by -(rule tranclp.r_into_trancl, rule silent_move_into_RedT_\<tau>_inv, auto)
next
case (step x' m' x'' m'')
hence "\<tau>mredT^++ s (redT_upd_\<epsilon> s t x' m')" by blast
moreover from \<open>silent_move t (x', m') (x'', m'')\<close> state
have "\<tau>mredT (redT_upd_\<epsilon> s t x' m') (redT_upd_\<epsilon> (redT_upd_\<epsilon> s t x' m') t x'' m'')"
by -(rule silent_move_into_RedT_\<tau>_inv, auto simp add: redT_updLns_def)
hence "\<tau>mredT (redT_upd_\<epsilon> s t x' m') (redT_upd_\<epsilon> s t x'' m'')"
by(simp add: redT_updLns_def)
ultimately show ?case ..
qed
lemma \<tau>diverge_into_\<tau>mredT:
assumes "\<tau>diverge t (x, shr s)"
and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>" "wset s t = None"
shows "mthr.\<tau>diverge s"
using assms
proof(coinduction arbitrary: s x)
case (\<tau>diverge s x)
note tst = \<open>thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>\<close>
from \<open>\<tau>diverge t (x, shr s)\<close> obtain x' m' where "silent_move t (x, shr s) (x', m')"
and "\<tau>diverge t (x', m')" by cases auto
from \<open>silent_move t (x, shr s) (x', m')\<close> tst \<open>wset s t = None\<close>
have "\<tau>mredT s (redT_upd_\<epsilon> s t x' m')" by(rule silent_move_into_RedT_\<tau>_inv)
moreover have "thr (redT_upd_\<epsilon> s t x' m') t = \<lfloor>(x', no_wait_locks)\<rfloor>"
using tst by(auto simp add: redT_updLns_def)
moreover have "wset (redT_upd_\<epsilon> s t x' m') t = None" using \<open>wset s t = None\<close> by simp
moreover from \<open>\<tau>diverge t (x', m')\<close> have "\<tau>diverge t (x', shr (redT_upd_\<epsilon> s t x' m'))" by simp
ultimately show ?case using \<open>\<tau>diverge t (x', m')\<close> by blast
qed
lemma \<tau>diverge_\<tau>mredTD:
assumes div: "mthr.\<tau>diverge s"
and fin: "finite (dom (thr s))"
shows "\<exists>t x. thr s t = \<lfloor>(x, no_wait_locks)\<rfloor> \<and> wset s t = None \<and> \<tau>diverge t (x, shr s)"
using fin div
proof(induct A\<equiv>"dom (thr s)" arbitrary: s rule: finite_induct)
case empty
from \<open>mthr.\<tau>diverge s\<close> obtain s' where "\<tau>mredT s s'" by cases auto
with \<open>{} = dom (thr s)\<close>[symmetric] have False by(auto elim!: mthr.silent_move.cases redT.cases)
thus ?case ..
next
case (insert t A)
note IH = \<open>\<And>s. \<lbrakk> A = dom (thr s); mthr.\<tau>diverge s \<rbrakk>
\<Longrightarrow> \<exists>t x. thr s t = \<lfloor>(x, no_wait_locks)\<rfloor> \<and> wset s t = None \<and> \<tau>diverge t (x, shr s)\<close>
from \<open>insert t A = dom (thr s)\<close>
obtain x ln where tst: "thr s t = \<lfloor>(x, ln)\<rfloor>" by(fastforce simp add: dom_def)
define s' where "s' = (locks s, ((thr s)(t := None), shr s), wset s, interrupts s)"
show ?case
proof(cases "ln = no_wait_locks \<and> \<tau>diverge t (x, shr s) \<and> wset s t = None")
case True
with tst show ?thesis by blast
next
case False
define xm where "xm = (x, shr s)"
define xm' where "xm' = (x, shr s)"
have "A = dom (thr s')" using \<open>t \<notin> A\<close> \<open>insert t A = dom (thr s)\<close>
unfolding s'_def by auto
moreover {
from xm'_def tst \<open>mthr.\<tau>diverge s\<close> False
have "\<exists>s x. thr s t = \<lfloor>(x, ln)\<rfloor> \<and> (ln \<noteq> no_wait_locks \<or> wset s t \<noteq> None \<or> \<not> \<tau>diverge t xm') \<and>
s' = (locks s, ((thr s)(t := None), shr s), wset s, interrupts s) \<and> xm = (x, shr s) \<and>
mthr.\<tau>diverge s \<and> silent_moves t xm' xm"
unfolding s'_def xm_def by blast
moreover
from False have "wfP (if \<tau>diverge t xm' then (\<lambda>s s'. False) else flip (silent_move_from t xm'))"
using \<tau>diverge_neq_wfP_silent_move_from[of t "(x, shr s)"] unfolding xm'_def by(auto)
ultimately have "mthr.\<tau>diverge s'"
proof(coinduct s' xm rule: mthr.\<tau>diverge_trancl_measure_coinduct)
case (\<tau>diverge s' xm)
then obtain s x where tst: "thr s t = \<lfloor>(x, ln)\<rfloor>"
and blocked: "ln \<noteq> no_wait_locks \<or> wset s t \<noteq> None \<or> \<not> \<tau>diverge t xm'"
and s'_def: "s' = (locks s, ((thr s)(t := None), shr s), wset s, interrupts s)"
and xm_def: "xm = (x, shr s)"
and xmxm': "silent_moves t xm' (x, shr s)"
and "mthr.\<tau>diverge s" by blast
from \<open>mthr.\<tau>diverge s\<close> obtain s'' where "\<tau>mredT s s''" "mthr.\<tau>diverge s''" by cases auto
from \<open>\<tau>mredT s s''\<close> obtain t' ta where "s -t'\<triangleright>ta\<rightarrow> s''" and "m\<tau>move s (t', ta) s''" by auto
then obtain x' x'' m'' where red: "t' \<turnstile> \<langle>x', shr s\<rangle> -ta\<rightarrow> \<langle>x'', m''\<rangle>"
and tst': "thr s t' = \<lfloor>(x', no_wait_locks)\<rfloor>"
and aoe: "actions_ok s t' ta"
and s'': "redT_upd s t' ta x'' m'' s''"
by cases(fastforce elim: m\<tau>move.cases)+
from \<open>m\<tau>move s (t', ta) s''\<close> have [simp]: "ta = \<epsilon>"
by(auto elim!: m\<tau>move.cases dest!: silent_tl)
hence wst': "wset s t' = None" using aoe by auto
from \<open>m\<tau>move s (t', ta) s''\<close> tst' s''
have "\<tau>move (x', shr s) \<epsilon> (x'', m'')" by(auto elim: m\<tau>move.cases)
show ?case
proof(cases "t' = t")
case False
with tst' wst' have "thr s' t' = \<lfloor>(x', no_wait_locks)\<rfloor>"
"wset s' t' = None" "shr s' = shr s" unfolding s'_def by auto
with red have "s' -t'\<triangleright>\<epsilon>\<rightarrow> redT_upd_\<epsilon> s' t' x'' m''"
by -(rule redT_normal, auto simp add: redT_updLns_def o_def finfun_Diag_const2 redT_updWs_def)
moreover from \<open>\<tau>move (x', shr s) \<epsilon> (x'', m'')\<close> \<open>thr s' t' = \<lfloor>(x', no_wait_locks)\<rfloor>\<close> \<open>shr s' = shr s\<close>
have "m\<tau>move s' (t', ta) (redT_upd_\<epsilon> s' t' x'' m'')"
by -(rule m\<tau>move.intros, auto)
ultimately have "\<tau>mredT s' (redT_upd_\<epsilon> s' t' x'' m'')"
unfolding \<open>ta = \<epsilon>\<close> by(rule mthr.silent_move.intros)
hence "\<tau>mredT^++ s' (redT_upd_\<epsilon> s' t' x'' m'')" ..
moreover have "thr s'' t = \<lfloor>(x, ln)\<rfloor>"
using tst \<open>t' \<noteq> t\<close> s'' by auto
moreover from \<open>\<tau>move (x', shr s) \<epsilon> (x'', m'')\<close> red
have [simp]: "m'' = shr s" by(auto dest: \<tau>move_heap)
hence "shr s = shr s''" using s'' by(auto)
have "ln \<noteq> no_wait_locks \<or> wset s'' t \<noteq> None \<or> \<not> \<tau>diverge t xm'"
using blocked s'' by(auto simp add: redT_updWs_def elim!: rtrancl3p_cases)
moreover have "redT_upd_\<epsilon> s' t' x'' m'' = (locks s'', ((thr s'')(t := None), shr s''), wset s'', interrupts s'')"
unfolding s'_def using tst s'' \<open>t' \<noteq> t\<close>
by(auto intro: ext elim!: rtrancl3p_cases simp add: redT_updLns_def redT_updWs_def)
ultimately show ?thesis using \<open>mthr.\<tau>diverge s''\<close> xmxm'
unfolding \<open>shr s = shr s''\<close> by blast
next
case True
with tst tst' wst' blocked have "\<not> \<tau>diverge t xm'"
and [simp]: "x' = x" by auto
moreover from red \<open>\<tau>move (x', shr s) \<epsilon> (x'', m'')\<close> True
have "silent_move t (x, shr s) (x'', m'')" by auto
with xmxm' have "silent_move_from t xm' (x, shr s) (x'', m'')"
by(rule silent_move_fromI)
ultimately have "(if \<tau>diverge t xm' then \<lambda>s s'. False else flip (silent_move_from t xm')) (x'', m'') xm"
by(auto simp add: flip_conv xm_def)
moreover have "thr s'' t = \<lfloor>(x'', ln)\<rfloor>" using tst True s''
by(auto simp add: redT_updLns_def)
moreover from \<open>\<tau>move (x', shr s) \<epsilon> (x'', m'')\<close> red
have [simp]: "m'' = shr s" by(auto dest: \<tau>move_heap)
hence "shr s = shr s''" using s'' by auto
have "s' = (locks s'', ((thr s'')(t := None), shr s''), wset s'', interrupts s'')"
using True s'' unfolding s'_def
by(auto intro: ext elim!: rtrancl3p_cases simp add: redT_updLns_def redT_updWs_def)
moreover have "(x'', m'') = (x'', shr s'')" using s'' by auto
moreover from xmxm' \<open>silent_move t (x, shr s) (x'', m'')\<close>
have "silent_moves t xm' (x'', shr s'')"
unfolding \<open>m'' = shr s\<close> \<open>shr s = shr s''\<close> by auto
ultimately show ?thesis using \<open>\<not> \<tau>diverge t xm'\<close> \<open>mthr.\<tau>diverge s''\<close> by blast
qed
qed }
ultimately have "\<exists>t x. thr s' t = \<lfloor>(x, no_wait_locks)\<rfloor> \<and> wset s' t = None \<and> \<tau>diverge t (x, shr s')" by(rule IH)
then obtain t' x' where "thr s' t' = \<lfloor>(x', no_wait_locks)\<rfloor>"
and "wset s' t' = None" and "\<tau>diverge t' (x', shr s')" by blast
moreover with False have "t' \<noteq> t" by(auto simp add: s'_def)
ultimately have "thr s t' = \<lfloor>(x', no_wait_locks)\<rfloor>" "wset s t' = None" "\<tau>diverge t' (x', shr s)"
unfolding s'_def by auto
thus ?thesis by blast
qed
qed
lemma \<tau>mredT_preserves_final_thread:
"\<lbrakk> \<tau>mredT s s'; final_thread s t \<rbrakk> \<Longrightarrow> final_thread s' t"
by(auto elim: mthr.silent_move.cases intro: redT_preserves_final_thread)
lemma \<tau>mRedT_preserves_final_thread:
"\<lbrakk> \<tau>mredT^** s s'; final_thread s t \<rbrakk> \<Longrightarrow> final_thread s' t"
by(induct rule: rtranclp.induct)(blast intro: \<tau>mredT_preserves_final_thread)+
lemma silent_moves2_silentD:
assumes "rtrancl3p mthr.silent_move2 s ttas s'"
and "(t, ta) \<in> set ttas"
shows "ta = \<epsilon>"
using assms
by(induct)(auto simp add: mthr.silent_move2_def dest: m\<tau>move_silentD)
lemma inf_step_silentD:
assumes step: "trsys.inf_step mthr.silent_move2 s ttas"
and lset: "(t, ta) \<in> lset ttas"
shows "ta = \<epsilon>"
using lset step
by(induct arbitrary: s rule: lset_induct)(fastforce elim: trsys.inf_step.cases simp add: mthr.silent_move2_def dest: m\<tau>move_silentD)+
end
subsection \<open>The multithreaded semantics with a well-founded relation on states\<close>
locale multithreaded_base_measure = multithreaded_base +
constrains final :: "'x \<Rightarrow> bool"
and r :: "('l,'t,'x,'m,'w,'o) semantics"
and convert_RA :: "'l released_locks \<Rightarrow> 'o list"
fixes \<mu> :: "('x \<times> 'm) \<Rightarrow> ('x \<times> 'm) \<Rightarrow> bool"
begin
inductive m\<mu>t :: "'m \<Rightarrow> ('l,'t,'x) thread_info \<Rightarrow> ('l,'t,'x) thread_info \<Rightarrow> bool"
for m and ts and ts'
where
m\<mu>tI:
"\<And>ln. \<lbrakk> finite (dom ts); ts t = \<lfloor>(x, ln)\<rfloor>; ts' t = \<lfloor>(x', ln')\<rfloor>; \<mu> (x, m) (x', m); \<And>t'. t' \<noteq> t \<Longrightarrow> ts t' = ts' t' \<rbrakk>
\<Longrightarrow> m\<mu>t m ts ts'"
definition m\<mu> :: "('l,'t,'x,'m,'w) state \<Rightarrow> ('l,'t,'x,'m,'w) state \<Rightarrow> bool"
where "m\<mu> s s' \<longleftrightarrow> shr s = shr s' \<and> m\<mu>t (shr s) (thr s) (thr s')"
lemma m\<mu>t_thr_dom_eq: "m\<mu>t m ts ts' \<Longrightarrow> dom ts = dom ts'"
apply(erule m\<mu>t.cases)
apply(rule equalityI)
apply(rule subsetI)
apply(case_tac "xa = t")
apply(auto)[2]
apply(rule subsetI)
apply(case_tac "xa = t")
apply auto
done
lemma m\<mu>_finite_thrD:
assumes "m\<mu>t m ts ts'"
shows "finite (dom ts)" "finite (dom ts')"
using assms
by(simp_all add: m\<mu>t_thr_dom_eq[symmetric])(auto elim: m\<mu>t.cases)
end
locale multithreaded_base_measure_wf = multithreaded_base_measure +
constrains final :: "'x \<Rightarrow> bool"
and r :: "('l,'t,'x,'m,'w,'o) semantics"
and convert_RA :: "'l released_locks \<Rightarrow> 'o list"
and \<mu> :: "('x \<times> 'm) \<Rightarrow> ('x \<times> 'm) \<Rightarrow> bool"
assumes wf_\<mu>: "wfP \<mu>"
begin
lemma wf_m\<mu>t: "wfP (m\<mu>t m)"
unfolding wfP_eq_minimal
proof(intro strip)
fix Q :: "('l,'t,'x) thread_info set" and ts
assume "ts \<in> Q"
show "\<exists>z\<in>Q. \<forall>y. m\<mu>t m y z \<longrightarrow> y \<notin> Q"
proof(cases "finite (dom ts)")
case False
hence "\<forall>y. m\<mu>t m y ts \<longrightarrow> y \<notin> Q" by(auto dest: m\<mu>_finite_thrD)
thus ?thesis using \<open>ts \<in> Q\<close> by blast
next
case True
thus ?thesis using \<open>ts \<in> Q\<close>
proof(induct A\<equiv>"dom ts" arbitrary: ts Q rule: finite_induct)
case empty
hence "dom ts = {}" by simp
with \<open>ts \<in> Q\<close> show ?case by(auto elim: m\<mu>t.cases)
next
case (insert t A)
note IH = \<open>\<And>ts Q. \<lbrakk>A = dom ts; ts \<in> Q\<rbrakk> \<Longrightarrow> \<exists>z\<in>Q. \<forall>y. m\<mu>t m y z \<longrightarrow> y \<notin> Q\<close>
define Q' where "Q' = {ts. ts t = None \<and> (\<exists>xln. ts(t \<mapsto> xln) \<in> Q)}"
let ?ts' = "ts(t := None)"
from \<open>insert t A = dom ts\<close> \<open>t \<notin> A\<close> have "A = dom ?ts'" by auto
moreover from \<open>insert t A = dom ts\<close> obtain xln where "ts t = \<lfloor>xln\<rfloor>" by(cases "ts t") auto
hence "ts(t \<mapsto> xln) = ts" by(auto simp add: fun_eq_iff)
with \<open>ts \<in> Q\<close> have "ts(t \<mapsto> xln) \<in> Q" by(auto)
hence "?ts' \<in> Q'" unfolding Q'_def by(auto simp del: split_paired_Ex)
ultimately have "\<exists>z\<in>Q'. \<forall>y. m\<mu>t m y z \<longrightarrow> y \<notin> Q'" by(rule IH)
then obtain ts' where "ts' \<in> Q'"
and min: "\<And>ts''. m\<mu>t m ts'' ts' \<Longrightarrow> ts'' \<notin> Q'" by blast
from \<open>ts' \<in> Q'\<close> obtain x' ln' where "ts' t = None" "ts'(t \<mapsto> (x', ln')) \<in> Q"
unfolding Q'_def by auto
define Q'' where "Q'' = {(x, m)|x. \<exists>ln. ts'(t \<mapsto> (x, ln)) \<in> Q}"
from \<open>ts'(t \<mapsto> (x', ln')) \<in> Q\<close> have "(x', m) \<in> Q''" unfolding Q''_def by blast
hence "\<exists>xm''\<in>Q''. \<forall>xm'''. \<mu> xm''' xm'' \<longrightarrow> xm''' \<notin> Q''" by(rule wf_\<mu>[unfolded wfP_eq_minimal, rule_format])
then obtain xm'' where "xm'' \<in> Q''" and min': "\<And>xm'''. \<mu> xm''' xm'' \<Longrightarrow> xm''' \<notin> Q''" by blast
from \<open>xm'' \<in> Q''\<close> obtain x'' ln'' where "xm'' = (x'', m)" "ts'(t \<mapsto> (x'', ln'')) \<in> Q" unfolding Q''_def by blast
moreover {
fix ts''
assume "m\<mu>t m ts'' (ts'(t \<mapsto> (x'', ln'')))"
then obtain T X'' LN'' X' LN'
where "finite (dom ts'')" "ts'' T = \<lfloor>(X'', LN'')\<rfloor>"
and "(ts'(t \<mapsto> (x'', ln''))) T = \<lfloor>(X', LN')\<rfloor>" "\<mu> (X'', m) (X', m)"
and eq: "\<And>t'. t' \<noteq> T \<Longrightarrow> ts'' t' = (ts'(t \<mapsto> (x'', ln''))) t'" by(cases) blast
have "ts'' \<notin> Q"
proof(cases "T = t")
case True
from True \<open>(ts'(t \<mapsto> (x'', ln''))) T = \<lfloor>(X', LN')\<rfloor>\<close> have "X' = x''" by simp
with \<open>\<mu> (X'', m) (X', m)\<close> have "(X'', m) \<notin> Q''" by(auto dest: min'[unfolded \<open>xm'' = (x'', m)\<close>])
hence "\<forall>ln. ts'(t \<mapsto> (X'', ln)) \<notin> Q" by(simp add: Q''_def)
moreover from \<open>ts' t = None\<close> eq True
have "ts''(t := None) = ts'" by(auto simp add: fun_eq_iff)
with \<open>ts'' T = \<lfloor>(X'', LN'')\<rfloor>\<close> True
have ts'': "ts'' = ts'(t \<mapsto> (X'', LN''))" by(auto intro!: ext)
ultimately show ?thesis by blast
next
case False
from \<open>finite (dom ts'')\<close> have "finite (dom (ts''(t := None)))" by simp
moreover from \<open>ts'' T = \<lfloor>(X'', LN'')\<rfloor>\<close> False
have "(ts''(t := None)) T = \<lfloor>(X'', LN'')\<rfloor>" by simp
moreover from \<open>(ts'(t \<mapsto> (x'', ln''))) T = \<lfloor>(X', LN')\<rfloor>\<close> False
have "ts' T = \<lfloor>(X', LN')\<rfloor>" by simp
ultimately have "m\<mu>t m (ts''(t := None)) ts'" using \<open>\<mu> (X'', m) (X', m)\<close>
proof(rule m\<mu>tI)
fix t'
assume "t' \<noteq> T"
with eq[OF False[symmetric]] eq[OF this] \<open>ts' t = None\<close>
show "(ts''(t := None)) t' = ts' t'" by auto
qed
hence "ts''(t := None) \<notin> Q'" by(rule min)
thus ?thesis
proof(rule contrapos_nn)
assume "ts'' \<in> Q"
from eq[OF False[symmetric]] have "ts'' t = \<lfloor>(x'', ln'')\<rfloor>" by simp
hence ts'': "ts''(t \<mapsto> (x'', ln'')) = ts''" by(auto simp add: fun_eq_iff)
from \<open>ts'' \<in> Q\<close> have "ts''(t \<mapsto> (x'', ln'')) \<in> Q" unfolding ts'' .
thus "ts''(t := None) \<in> Q'" unfolding Q'_def by auto
qed
qed
}
ultimately show ?case by blast
qed
qed
qed
lemma wf_m\<mu>: "wfP m\<mu>"
proof -
have "wf (inv_image (same_fst (\<lambda>m. True) (\<lambda>m. {(ts, ts'). m\<mu>t m ts ts'})) (\<lambda>s. (shr s, thr s)))"
by(rule wf_inv_image)(rule wf_same_fst, rule wf_m\<mu>t[unfolded wfP_def])
also have "inv_image (same_fst (\<lambda>m. True) (\<lambda>m. {(ts, ts'). m\<mu>t m ts ts'})) (\<lambda>s. (shr s, thr s)) = {(s, s'). m\<mu> s s'}"
by(auto simp add: m\<mu>_def same_fst_def)
finally show ?thesis by(simp add: wfP_def)
qed
end
end
| 85,600
|
Atlanta Speakers
Opening Keynote
Jennifer BonnettJennifer Bonnett is a technology entrepreneur with over 20 years experience in
Guest Speakers
Vincent BaskervilleVincent Jordan Is the co-founder of two companies: TripLingo, Glancely, and founded 8bitFeed He is a 10+ year industry specialist & adrenaline junkie whom has traveled to 12 countries, and is known for creating highly intuitive-engaging digital products. Vincent has consulted and managed teams for many Fortune 500 clients like Disney, BB&T and Lockheed Martin regarding products for their online and broadcast mediums as well as working with the Army National Guard. In addition t
Maurice CherryMaur. Since 2005, the Black Weblog Awards has conferred over 200 awards in over 40 categories, and has an active user base spanning over 90 countries. In addition to his duties with the Black Weblog Awards, Maurice is also Creative P
Charles RossMr. Ross is a director with the National Association of Seed and Venture Funds, an organization focused on creating and growing innovation capital through out the U.S. He is also a director of The Center for Working Families, an employment and asset-building initiative to improve the economic well-being of Atlanta's inner city residents. Currently, Mr. Ross is also the General Manager of the Advanced Technology Development Center and Director of Entrepreneurial Services for Georgia Tech's Enterprise Innovation Institute. He is responsible for leading and providing the overall strategy for f
Tanika Gray p
Aerial M EllisEmerging among the brightest in the industry, Aerial has shaped Urbane Imagery to become a lead agency in public relations for emerging companies and entrepreneurs. With a growing commitment to startups in the technology space, she has led social brand awareness and online media relations strategies for web, mobile, entertainment technology startups to include Beat Kangz Electronics and LL Cool J's BoomDizzle. Clients have trusted her distinct ability to align partnerships with giants such as Burger King, NBA and BET and enhance their public image through media coverage with outle
Howard FranklinHoward Franklin is one of the South's most sought-after political strategists. He has distinguished himself by running successful campaigns for first-time candidates, including State Sen. Jason Carter's (grandson to President Jimmy Carter) 2010 special election and a 2009 race in the Cayman Islands, winning nine seats in Parliament to form a new government. Franklin has also played a role in major policy decisions, from helping Atlanta City Council win approval of a pension reform compromise to persuading Georgia’s largest counties to include foreclosures in property assessments. He founded Fi
Quentin B “Que” LynchQue, an
| 167,307
|
Gold Medal, 2019 San Francisco Chronicle Competition
Hints of nutmeg and clove tingle your senses while berry, cherry pie, and rhubarb characters shape the flavor profile. This wine has a gentle entry, focused yet balanced middle followed by a clean and lasting finish.
Appellation: Sonoma Valley
Fruit Source: Estate Grown
Varietal: 95% Zinfandel, 3% Syrah, 2% Petite Sirah
Alcohol: 15.4%
Residual Sugar: 0.09
pH: 3.71
TA: 0.59
Production: 198 cases
| 408,570
|
PU Epoxy Grouting Machine
Price And Quantity
- Unit/Units
- 3-4
- Unit/Units
- 85000INR
Product Specifications
- Smooth
- Eco Friendly
- Stailness Steel
- Electric
- Silver and Blue
Product Description
We have become the leading company in manufacturing supreme quality PU Epoxy Grouting Machine. Our Grouting Machines are made of premium grade component and are in-line with the latest technology. The team of experienced technocrats make sure to work in the most creative and scientific way to bring the best range of products that stand tall in terms of high-pressure working condition, high impact strength, abrasion resistance, controlled discharge flow and power usage. Most importantly, our machines are high demand and widely used for grouting of dam and reservoir for projects related to water conservation and to combat water leakage problems.
PU Epoxy Grouting Machine Details:
PU Epoxy Grouting Machine Details:
- Brand/Make : Graco
- Model/Type : 190
- Discharge : 0.9 lpm
- Gross Weight : 14 kg
- Pipe : 5 m
- Maximum Tip Size : 0.021
- Maximum gpm : 0.47 (1.8)
- Cycles : 680 Cycles Per Gallon
- Gross weight : 9.00 Kg
- Start-up : Less than 7500 PSI
- Application : Waterproofing, water sealing
- Power Requirements : 100V/120V AC,50/60 Hz,11A,1 phase
- Maximum Working Pressure : 3300 psi
- Generator Minimum Wattage : 3000 minimum
- Motar hp : 5/8 DC
Trade Information
- Cash Advance (CA), Cash in Advance (CID), Cheque
- 50 Per Week
- 10-15 Days
- No
- Carton Box
- Africa, Middle East, Australia
- Maharashtra
218, Sagar, Prabhat Co-operative Industrial Estate, Dahisar Check Naka, Dahisar (East) Mumbai Maharashtra India +918037301900
Phone :+918037301900
| 258,652
|
Bitcoin wallet desktop reddit
The ultimate guide to the Best Cryptocurrency Wallet, the Best Bitcoin Wallet and the Best Ethereum Wallet.
What is the Bitcoin wallet? - QuoraTo ensure the safety of the Bitcoin ecosystem, Bitcoin Gold has implemented full replay protection and unique wallet addresses, essential features that protect users.
Get BitPay – Secure Bitcoin Wallet - Microsoft Store
Reddit ditches Bitcoin payment option in wake of Coinbase
Which Bitcoin Wallet Is Best? Desktop Choices - Bitsonline
This service is added to the Ethereum and all ERC20 tokens already available.
With Bitcoin wallet you can send and receive money via mobile phone, computer, tablet or other devices.
I was looking for a desktop wallet that supports many. //steemit.com/bitcoin/@mrilevi/the-top-5-best-altcoin-and...
4 Bitcoin Wallet Software For Windows 10 - I Love Free
Find out how this wallet works today in our Electrum Bitcoin Wallet review.Top 10 Best Desktop Cryptocurrency Wallets July 7,. and is a free and decentralized open-source bitcoin wallet option that is very fast thanks to using servers.
Best Bitcoin Hardware Wallets - BitPremier
How To Choose The Best Bitcoin Wallet For You: Top 4
Coin Wallets - Coin Clarity
What Is A Bitcoin Wallet? - The Best Explanation EVERAccording to complaints lodged by users on Reddit, they are unable to access their bitcoin cash wallets, cannot view their balances or make transactions.
Litecoin.com - Open source P2P digital currency
Bitcoin Private is a hard fork of Bitcoin combined with the privacy of zk-snarks. Full Node Desktop Wallet, Feb 20.
Electrum Wallet Review - are you looking trusted Bitcoin wallets for your Bitcoin holding the check out electrum Bitcoins wallet review with detail features.Bitcoin wallet startup Blockchain is expanding its service to support ether, the cryptocurrency of the ethereum network.
ArcBit ArcBit is designed to be simple and easy to use, while giving users full control over their money.
Which wallets support Bitcoin Cash? - Bitcoin Stack Exchange
Desktop Bitcoin Wallet now available on Eidoo - blog.eidoo.ioThe Bitcoin Green wallet allows you to send and receive BITG currency with anyone, anywhere in the world.Hardware wallets are by far the most secure option for keeping your valuable Bitcoin secure.
Bitcoin's Biggest Software Wallet Blockchain - CoinDesk
If you read this post you probably have already purchased bitcoins from a bitcoin ATM and instead of using your own previously created wallet address you let the.
2018’s best cryptocurrency wallets | 70+ compared | finder.com
How to use a printed paper wallet from a bitcoin ATM
| 24,366
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December 3, 2020
Federal health officials predict the coming months could be the most difficult in the public health history of the U.S.
SHOW TRANSCRIPT
New record numbers as the U.S. continues to battle the pandemic. The country reported 3,157 deaths Wednesday. That's a single-day record. And this morning, 100,000 Americans are fighting COVID from a hospital bed. The head of the CDC is warning the months ahead will be difficult.
Dr. Robert Redfield, Jr.: "The reality is December, January and February are going to be rough times. I actually believe they're going to be the most difficult in the public health history of this nation, largely because of the stress its going to put on our health care system."
In 30 states, over 1,000 people are in the hospital due to COVID.
| 315,275
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