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1,116,691,500,815 | arxiv | \section{Introduction}
Chest X-Ray (CXR) image report generation aims to automatically generate detailed findings and diagnoses for given images, which has attracted growing attention in recent years~\cite{wang2018tienet,jing2017automatic,li2018hybrid}.
This technique can greatly reduce the workload of radiologists for interpreting CXR images and writing corresponding reports.
In spite of the progress made in this area, it is still challenging for computers to accurately write reports.
Besides the difficulties in detecting lesions from images, the complex structure of textual reports can prevent the success of automatic report generation.
As shown in Figure~\ref{fig:example}, the report for a CXR image usually comprises two major sections: \emph{Findings} and \emph{Impression}.
\emph{Findings} section records detailed descriptions about normal and abnormal findings, such as lesions (e.g. increased lung marking).
\emph{Impression} section concludes diseases (e.g. pneumonia) from \emph{Findings} and forms a diagnostic conclusion, consisting of abnormal and normal conclusions.
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth]{example.pdf}
\caption{An example of chest X-ray image along with its report. In the report, the \emph{Findings} section records detailed descriptions for normal and abnormal findings; the \emph{Impression} section provides a diagnostic conclusion. The underlined sentence is an abnormal finding.}
\label{fig:example}
\end{figure}
Existing methods~\cite{wang2018tienet,jing2017automatic,li2018hybrid} ignored the relationship between \emph{Findings} and \emph{Impression}, as well as the different distributions between normal and abnormal findings/conclusions.
In addressing this problem, we present a novel framework for automatic report generation by exploiting the structure of the reports.
Firstly, considering the fact that \emph{Impression} is a summarization of \emph{Findings}, we propose a two-stage modeling strategy given in Figure~\ref{fig:show}, where we borrow strength from image captioning task and text summarization task for generating \emph{Impression}.
Secondly, we decompose the generation process of both \emph{Findings} and \emph{Impression} into the following recurrent sub-tasks:
1) examine an area in the image (or a sentence in \emph{Findings}) and decide if an abnormality appears;
2) write detailed (normal or abnormal) descriptions for the examined area.
In order to model the above generation process, we propose a novel Co-operative Multi-Agent System (CMAS), which consists of three agents: Planner (PL), Abnormality Writer (AW) and Normality Writer (NW).
Given an image, the system will run several loops until PL decides to stop the process.
Within each loop, the agents co-operate with each other in the following fashion:
1) PL examines an area of the input image (or a sentence of \emph{Findings}), and decides whether the examined area contains lesions.
2) Either AW or NW will generate a sentence for the area based on the order given by PL.
To train the system, REINFORCE algorithm~\cite{williams1992simple} is applied to optimize the reward (e.g. BLEU-4 ~\cite{papineni2002bleu}).
To the best of our knowledge, our work is the first effort to investigate the structure of CXR reports.
The major contributions of our work are summarized as follows.
First, we propose a two-stage framework by exploiting the structure of the reports.
Second, We propose a novel Co-operative Multi-Agent System (CMAS) for modeling the sentence generation process of each section.
Third, we perform extensive quantitative experiments to evaluate the overall quality of the generated reports, as well as the model's ability for detecting medical abnormality terms.
Finally, we perform substantial qualitative experiments to further understand the quality and properties of the generated reports.
\section{Related Work}\label{related_work}
\paragraph{Visual Captioning}
The goal of visual captioning is to generate a textual description for a given image or video.
For one-sentence caption generation, almost all deep learning methods~\cite{mao2014deep, vinyals2015show, donahue2015long,karpathy2015deep} were based on Convolutional Neural Network (CNN) - Recurrent Neural Network (RNN) architecture.
Inspired by the attention mechanism in human brains, attention-based models, such as visual attention~\cite{xu2015show} and semantic attention~\cite{you2016image}, were proposed for improving the performances.
Some other efforts have been made for building variants of the hierarchical Long-Short-Term-Memory (LSTM) network~\cite{hochreiter1997long} to generate paragraphs~\cite{krause2017hierarchical, yu2016video, liang2017recurrent}.
Recently, deep reinforcement learning has attracted growing attention in the field of visual captioning~\cite{ren2017deep,rennie2017self,liu2017improved,wang2018video}.
Additionally, other tasks related to visual captioning, (e.g., dense captioning~\cite{johnson2016densecap}, multi-task learning~\cite{pasunuru2017multi}) also attracted a lot of research attention.
\paragraph{Chest X-ray Image Report Generation}
\newcite{shin2016learning} first proposed a variant of CNN-RNN framework to predict tags (location and severity) of chest X-ray images.
\newcite{wang2018tienet} proposed a joint framework for generating reference reports and performing disease classification at the same time.
However, this method was based on a single-sentence generation model~\cite{xu2015show}, and obtained low BLEU scores.
\newcite{jing2017automatic} proposed a hierarchical language model equipped with co-attention to better model the paragraphs, but it tended to produce normal findings.
Despite \newcite{li2018hybrid} enhanced language diversity and model's ability in detecting abnormalities through a hybrid of template retrieval module and text generation module, manually designing templates is costly and they ignored the template's change over time.
\paragraph{Multi-Agent Reinforcement Learning}
The target of multi-agent reinforcement learning is to solve complex problems by integrating multiple agents that focus on different sub-tasks.
In general, there are two types of multi-agent systems: independent and cooperative systems~\cite{tan1993multi}.
Powered by the development of deep learning, deep multi-agent reinforcement learning has gained increasing popularity.
\newcite{tampuu2017multiagent} extended Deep Q-Network (DQN)~\cite{mnih2013playing} into a multi-agent DQN for Pong game; \newcite{foerster2016learning, sukhbaatar2016learning} explored communication protocol among agents; \newcite{zhang2018fully} further studied fully decentralized multi-agent system.
Despite these many attempts, the multi-agent system for long paragraph generation still remains unexplored.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{overview.png}
\caption{Overview of the proposed Cooperative Multi-Agent System (CMAS).}
\label{fig:overview}
\end{figure*}
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{show.png}
\caption{Show, Describe and Conclude.}
\label{fig:show}
\end{figure}
\section{Overall Framework}\label{framework}
As shown in Figure~\ref{fig:show}, the proposed framework is comprised of two modules: Findings and Impression.
Given a CXR image, the Findings module will examine different areas of the image and generate descriptions for them.
When findings are generated, the Impression module will give a conclusion based on findings and the input CXR image.
The proposed two-stage framework explicitly models the fact that \emph{Impression} is a conclusive summarization of \emph{Findings}.
Within each module, we propose a Co-operative Multi-Agent System (CMAS) (see Section ~\ref{approach}) to model the text generation process for each section.
\section{Co-operative Multi-Agent System}\label{approach}
\subsection{Overview}
The proposed Co-operative Multi-Agent System (CMAS) consists of three agents: Planner (PL), Normality Writer (NW) and Abnormality Writer (AW).
These agents work cooperatively to generate findings or impressions for given chest X-ray images.
PL is responsible for determining whether an examined area contains abnormality, while NW and AW are responsible for describing normality or abnormality in detail (Figure~\ref{fig:overview}).
The generation process consists of several loops, and each loop contains a sequence of actions taken by the agents.
In the $n$-th loop, the writers first share their local states $LS_{n-1,T} = \{w_{n-1, t}\}_{t=1}^{T}$ (actions taken in the previous loop) to form a shared global state $GS_{n}=(I, \{s_i\}_{i=1}^{n-1})$, where $I$ is the input image, $s_i$ is the $i$-th generated sentence, and $w_{i,t}$ is the $t$-th word in the $i$-th sentence of length $T$.
Based on the global state $GS_{n}$, PL decides whether to stop the generation process or to choose a writer (NW or AW) to produce the next sentence $s_n$ .
If a writer is selected, then it will refresh its memory by $GS_{n}$ and generate a sequence of words $\{w_{n,t}\}_{t=1}^T$ based on the sequence of local state $LS_{n, t}=\{w_{n, 1},\cdots, w_{n, t-1}$\}.
Once the generation process is terminated, the reward module will compute a reward by comparing the generated report with the ground-truth report.
Given the reward, the whole system is trained via REINFORCE algorithm~\cite{williams1992simple}.
\subsection{Policy Network}
\subsubsection{Global State Encoder}\label{GSE}
During the generation process, each agent will make decisions based on the global state $GS_{n}$.
Since $GS_{n}$ contains a list of sentences $\{s_i\}_{i=1}^{n-1}$, a common practice is to build a hierarchical LSTM as Global State Encoder (GSE) for encoding it.
Equipping such an encoder with an excessive number of parameters for each agent in CMAS would be computation-consuming.
We address this problem in two steps.
First, we tie weights of GSE across the three agents.
Second, instead of encoding previous sentences from scratch, GSE dynamically encodes $GS_{n}$ based on $GS_{n-1}$.
Specifically, we propose a single layer LSTM with soft-attention~\cite{xu2015show} as GSE.
It takes a multi-modal context vector $\mathbf{ctx}_n\in\mathbb{R}^H$ as input, which is obtained by jointly embedding sentence $s_{n-1}$ and image $I$ to a hidden space of dimension $H$, and then generates the global hidden state vector $\mathbf{gs}_n\in\mathbb{R}^H$ for the $n$-th loop by:
\begin{equation}
\mathbf{gs}_n = \text{LSTM}(\mathbf{gs}_{n-1}, \mathbf{ctx}_n)
\end{equation}
We adopt a visual attention module for producing context vector $\mathbf{ctx}_n$, given its capability of capturing the correlation between languages and images~\cite{lu2017knowing,xu2015show}.
The inputs to the attention module are visual feature vectors $\{\mathbf{v}_p\}_{p=1}^{P}\in\mathbb{R}^C$ and local state vector $\mathbf{ls}_{n-1}$ of sentence $s_{n-1}$.
Here, $\{\mathbf{v}_p\}_{p=1}^{P}$ are extracted from an intermediate layer of a CNN, $C$ and $p$ are the number of channels and the position index of $\mathbf{v}_p$.
$\mathbf{ls}_{n-1}$ is the final hidden state of a writer (defined in section~\ref{writer}).
Formally, the context vector $\mathbf{ctx}_n$ is computed by the following equations:
\begin{equation}\label{va1}
\mathbf{h}_{p} = \tanh(\mathbf{W}_h[\mathbf{ls}_{n-1}; \mathbf{gs}_{n-1}])
\end{equation}
\begin{equation}\label{va2}
\alpha_{p} = \frac{\exp(\mathbf{W}_{att}\mathbf{h}_{p})}{\sum_{q=1}^{P}\exp(\mathbf{W}_{att}\mathbf{h}_{q})}
\end{equation}
\begin{equation}\label{va3}
\mathbf{v}_{att} = \sum_{p=1}^P\alpha_{p}\mathbf{v}_p
\end{equation}
\begin{equation}\label{va4}
\mathbf{ctx}_{n} = \tanh(\mathbf{W}_{ctx}[\mathbf{v}_{att}; \mathbf{ls}_{n-1}])
\end{equation}
where $\mathbf{W}_{h}$, $\mathbf{W}_{att}$ and $\mathbf{W}_{ctx}$ are parameter matrices; $\{\alpha_p\}_{p=1}^P$ are weights for visual features; and $[;]$ denotes concatenation operation.
At the beginning of the generation process, the global state is $GS_1=(I)$.
Let $\mathbf{\bar{v}}=\frac{1}{P}\sum_{i=1}^{P}\mathbf{v}_i$, the initial global state $\mathbf{gs}_0$ and cell state $\mathbf{c}_{0}$ are computed by two single-layer neural networks:
\begin{align}
\mathbf{gs}_0 &= \tanh(\mathbf{W}_{gs}\mathbf{\bar{v}})\\
\mathbf{c}_{0} &= \tanh(\mathbf{W}_{c}\mathbf{\bar{v}})
\end{align}
where $\mathbf{W}_{gs}$ and $\mathbf{W}_{c}$ are parameter matrices.
\subsubsection{Planner}
After examining an area, Planner (PL) determines: 1) whether to terminate the generation process; 2) which writer should generate the next sentence.
Specifically, besides the shared Global State Encoder (GSE), the rest part of PL is modeled by a two-layer feed-forward network:
\begin{align}
\mathbf{h}_n &= \tanh(\mathbf{W}_2\tanh(\mathbf{W}_1\mathbf{gs}_n))\\
idx_n &= \arg \text{max}(\text{softmax}(\mathbf{W}_3\mathbf{h}_n))
\end{align}
where $W_1$, $W_2$, and $W_3$ are parameter matrices; $idx_n\in\{0, 1, 2\}$ denotes the indicator, where $0$ is for STOP, $1$ for NW and $2$ for AW. Namely, if $idx_n=0$, the system will be terminated; else, NW ($idx_n=1$) or AW ($idx_n=2$) will generate the next sentence $s_n$.
\subsubsection{Writers}\label{writer}
The number of normal sentences is usually 4-12 times to the number of abnormal sentences for each report.
With this highly unbalanced distribution, using only one decoder to model all of the sentences would make the generation of normal sentences dominant.
To solve this problem, we design two writers, i.e., Normality Writer (NW) and Abnormality Writer (AW), to model normal and abnormal sentences.
Practically, the architectures of NW and AW can be different. In our practice, we adopt a single-layer LSTM for both NW and AW given the principle of parsimony.
Given a global state vector $\mathbf{gs}_n$, CMAS first chooses a writer for generating a sentence based on $idx_n$.
The chosen writer will re-initialize its memory by taking $\mathbf{gs}_n$ and a special token BOS (Begin of Sentence) as its first two inputs.
The procedure for generating words is:
\begin{align}
\mathbf{h}_{t} &= \text{LSTM}(\mathbf{h}_{t-1}, \mathbf{W_e}\mathbf{y}_{w_{t-1}})\\
\mathbf{p}_t &= \text{softmax}(\mathbf{W}_{out}\mathbf{h}_t)\\
w_{t} &=\arg\text{max} (\mathbf{p}_t)
\end{align}
where $\mathbf{y}_{w_{t-1}}$ is the one-hot encoding vector of word $w_{t-1}$; $\mathbf{h}_{t-1}, \mathbf{h}_t\in\mathbb{R}^H$ are hidden states of LSTM; $\mathbf{W_e}$ is the word embedding matrix and $\mathbf{W}_{out}$ is a parameter matrix. $\mathbf{p}_t$ gives the output probability score over the vocabulary.
Upon the completion of the procedure (either token EOS (End of Sentence) is produced or the maximum time step $T$ is reached), the last hidden state of LSTM will be used as local state vector $\mathbf{ls}_n$, which will be fed into GSE for generating next global state vector $GS_{n+1}$.
\subsection{Reward Module}
We use BLEU-4~\cite{papineni2002bleu} to design rewards for all agents in CMAS.
A generated paragraph is a collection $(\mathbf{s}^{ab}, \mathbf{s}^{nr})$ of normal sentences $\mathbf{s}^{nr}= \{s^{nr}_1, \dots, s^{nr}_{N_{nr}}\}$ and abnormal sentences $\mathbf{s}^{ab} = \{s^{ab}_1, \dots, s^{ab}_{N_{ab}}\}$, where $N_{ab}$ and $N_{nr}$ are the number of abnormal sentences and the number of normal sentences, respectively.
Similarly, the ground truth paragraph corresponding to the generated paragraph $(\mathbf{s}^{ab}, \mathbf{s}^{nr})$ is $(\mathbf{s}^{\ast ab}, \mathbf{s}^{\ast nr})$.
We compute BLEU-4 scores separately for abnormal and normal sentences.
For the first $n$ generated abnormal and normal sentences, we have:
\begin{align}
f(s^{ab}_n) &= \text{BLEU}(\{s^{ab}_1,\cdots, s^{ab}_n\}, \mathbf{s}^{\ast ab})\\
f(s^{nr}_n) &= \text{BLEU}(\{s^{nr}_1,\cdots, s^{nr}_n\}, \mathbf{s}^{\ast nr})
\end{align}
Then, the immediate reward for $s_n$ ($s^{ab}_n$ or $s^{nr}_n$) is $r(s_n) = f(s_n) - f(s_{n-1})$.
Finally, the discounted reward for $s_n$ is defined as:
\begin{equation}\label{eq_reward}
R(s_n) = \sum_{i=0}^{\infty}\gamma^ir(s_{n+i})
\end{equation}
where $\gamma\in[0, 1]$ denotes discounted factor, and $r(s_1) = \text{BLEU}(\{s_1\}, \mathbf{s}^{\ast})$.
\subsection{Learning} \label{Learning}
\subsubsection{Reinforcement Learning}
Given an input image $I$, three agents (PL, NW and AW) in CMAS work simultaneously to generate a paragraph $\mathbf{s}$ = $\{s_1, s_2, \dots, s_N\}$ with the joint goal of maximizing the discounted reward $R(s_n)$ (Equation~\ref{eq_reward}) for each sentence $s_n$.
The loss of a paragraph $\mathbf{s}$ is negative expected reward:
\begin{equation}\label{eq_loss_all}
L(\theta) = -\mathbb{E}_{n,s_n\sim\pi_\theta}[R(s_n)]
\end{equation}
where $\pi_\theta$ denotes the entire policy network of CMAS.
Following the standard REINFORCE algorithm~\cite{williams1992simple}, the gradient for the expectation $\mathbb{E}_{n, s_n\sim\pi_\theta}[R(s_n)]$ in Equation~\ref{eq_loss_all} can be written as:
\begin{equation}\label{eq_grad}
\nabla_\theta L(\theta) = \mathbb{E}_{n, s_n\sim\pi_\theta}[R(s_n)\nabla_\theta -\log \pi_\theta(s_n, idx_n)]
\end{equation}
where $-\log\pi_{\theta}(s_n, idx_n)$ is joint negative log-likelihood of sentence $s_n$ and its indicator $idx_n$, and it can be decomposed as:
\begin{equation}
\begin{split}
&-\log\pi_\theta(s_n, idx_n) \\
=& \mathds{1}_{\{idx_n=AW\}}L_{AW} + \mathds{1}_{\{idx_n=NW\}}L_{NW} + L_{PL} \\
=& - \mathds{1}_{\{idx_n=AW\}}\sum_{t=1}^{T}\log p_{AW}(w_{n,t}) \\
&- \mathds{1}_{\{idx_n=NW\}}\sum_{t=1}^{T}\log p_{NW}(w_{n,t}) \\
&-\log p_{PL}(idx_n)
\end{split}
\end{equation}
where $L_{AW}$, $L_{NW}$ and $L_{PL}$ are negative log-likelihoods; $p_{AW}$, $p_{NW}$ and $p_{PL}$ are probabilities of taking an action; $\mathds{1}$ denotes indicator function.
Therefore, Equation~\ref{eq_grad} can be re-written as:
\begin{equation}\label{loss_rl}
\begin{split}
\nabla_\theta L(\theta) &=
\mathbb{E}_{n, s_n\sim\pi_\theta}[R(s_n)(\mathds{1}_{\{idx_n=AW\}}\nabla L_{AW}\\
&+ \mathds{1}_{\{idx_n=NW\}}\nabla L_{NW} + \nabla L_{PL})]
\end{split}
\end{equation}
\subsubsection{Imitation Learning}
It is very hard to train agents using reinforcement learning from scratch, therefore a good initialization for policy network is usually required~\cite{bahdanau2016actor, silver2016mastering, wang2018video}.
We apply imitation learning with cross-entropy loss to pre-train the policy network.
Formally, the cross-entropy loss is defined as:
\begin{equation}\label{loss_ce}
\begin{split}
&L_{CE}(\theta) = -\lambda_{PL}\sum_{n=1}^{N}\{\log p_{PL}(idx_n^\ast)\}\\
-&\lambda_{NW}\sum_{n=1}^{N}\{\mathds{1}_{\{idx_n^\ast=NW\}}\sum_{t=1}^{T}\log p_{NW}(w_{n,t}^\ast)\} \\
-& \lambda_{AW}\sum_{n=1}^{N}\{\mathds{1}_{\{idx_n^\ast=AW\}}\sum_{t=1}^{T}\log p_{AW}(w_{n,t}^\ast)\} \\
\end{split}
\end{equation}
where $w^\ast$ and $idx^\ast$ denote ground-truth word and indicator respectively; $\lambda_{PL}$, $\lambda_{NW}$ and $\lambda_{AW}$ are balancing coefficients among agents; $N$ and $T$ are the number of sentences and the number of words within a sentence, respectively.
\subsection{CMAS for Impression}
Different from the Findings module, the inputs of the Impression module not only contain images $I$ but also the generated findings $\mathbf{f}=\{f_1,f_2,\dots,f_{N_{f}}\}$, where $N_f$ is the total number of sentences.
Thus, for the Impression module, the $n$-th global state becomes $GS_n=(I, \mathbf{f}, \{s_i\}_{i=1}^{n-1})$.
The rest part of CMAS for the Impression module is exactly the same as CMAS for the Findings module.
To encode $\mathbf{f}$, we extend the definition of multi-modal context vector $\mathbf{ctx}_n$ (Equation~\ref{va4}) to:
\begin{equation}\label{va24}
\mathbf{ctx}_{n} = \tanh(\mathbf{W}_{ctx}[\mathbf{v}_{att}; \mathbf{f}_{att}; \mathbf{ls}_{n-1}])
\end{equation}
where $\mathbf{f}_{att}$ is the soft attention~\cite{bahdanau2014neural,xu2015show} vector, which is obtained similar as $\mathbf{v}_{att}$ (Equation~\ref{va2} and~\ref{va3}).
\section{Experiments}\label{experiments}
\subsection{Datasets}
\paragraph{IU-Xray} Indiana University Chest X-Ray Collection~\cite{demner2015preparing} is a public dataset containing 3,955 fully de-identified radiology reports collected from the Indiana Network for Patient Care, each of which is associated with a frontal and/or lateral chest X-ray images, and there are 7,470 chest X-ray images in total.
Each report is comprised of several sections: \emph{Impression}, \emph{Findings} and \emph{Indication} etc. We preprocess the reports by tokenizing, converting tokens into lower-cases and removing non-alpha tokens.
\paragraph{CX-CHR} CX-CHR~\cite{li2018hybrid} is a proprietary internal dataset, which is a Chinese chest X-ray report dataset collected from a professional medical examination institution.
This dataset contains examination records for 35,500 unique patients, each of which consists of one or multiple chest X-ray images as well as a textual report written by professional radiologists.
Each textual report has sections such as \emph{Complain}, \emph{Findings} and \emph{Impression}.
The textual reports are preprocessed through tokenizing by ``jieba''\footnote{https://github.com/fxsjy/jieba.}, a Chinese text segmentation tool, and filtering rare tokens.
For both datasets, we used the same data splits as \citeauthor{li2018hybrid}.
\subsection{Experimental Setup}
\paragraph{Abnormality Term Extraction}
Human experts helped manually design patterns for most frequent medical abnormality terms in the datasets.
These patterns are used for labeling abnormality and normality of sentences, and also for evaluating models' ability to detect abnormality terms.
The abnormality terms in \emph{Findings} and \emph{Impression} are different to some degree.
This is because many abnormality terms in \emph{Findings} are descriptions rather than specific disease names.
For examples, ``low lung volumes'' and ``thoracic degenerative'' usually appear in \emph{Findings} but not in \emph{Impression}.
\paragraph{Evaluation Metrics}
We evaluate our proposed method and baseline methods on: BLEU~\cite{papineni2002bleu}, ROUGE~\cite{lin2004rouge} and CIDEr~\cite{vedantam2015cider}.
The results based on these metrics are obtained by the standard image captioning evaluation tool\footnote{https://github.com/tylin/coco-caption}.
We also calculate precision and average False Positive Rate (FPR) for abnormality detection in generated textual reports on both datasets.
\paragraph{Implementation Details}
The dimensions of all hidden states in Abnormality Writer, Normality Writer, Planner and shared Global State Encoder are set to 512.
The dimension of word embedding is also set as 512.
We adopt ResNet-50~\cite{he2016deep} as image encoder, and visual features are extracted from its last convolutional layer, which yields a $7\times7\times2048$ feature map.
The image encoder is pretrained on ImageNet~\cite{deng2009imagenet}).
For the IU-Xray dataset, the image encoder is fine-tuned on ChestX-ray14 dataset~\cite{wang2017chestx}, since the IU-Xray dataset is too small.
For the CX-CHR dataset, the image encoder is fine-tuned on its training set.
The weights of the image encoder are then fixed for the rest of the training process.
During the imitation learning stage, the cross-entropy loss (Equation~\ref{loss_ce}) is adopted for all of the agents, where $\lambda_{PL}$, $\lambda_{AW}$ and $\lambda_{NW}$ are set as 1.0.
We use Adam optimizer~\cite{kingma2014adam} with a learning rate of $5\times10^{-4}$ for both datasets.
During the reinforcement learning stage, the gradients of weights are calculated based on Equation~\ref{loss_rl}.
We also adopt Adam optimizer for both datasets and the learning rate is fixed as $10^{-6}$.
\paragraph{Comparison Methods}
For \emph{Findings} section, we compare our proposed method with state-of-the-art methods for CXR imaging report generation:
CoAtt~\cite{jing2017automatic} and HGRG-Agent~\cite{li2018hybrid}, as well as several state-of-the-art image captioning models: CNN-RNN~\cite{vinyals2015show}, LRCN~\cite{donahue2015long}, AdaAtt~\cite{lu2017knowing}, Att2in~\cite{rennie2017self}.
In addition, we implement several ablated versions of the proposed CMAS to evaluate different components in it:
$\text{CMAS}_\text{W}$ is a single agent system containing only one writer, but it is trained on both normal and abnormal findings.
$\text{CMAS}_\text{NW,AW}$ is a simple concatenation of two single agent systems $\text{CMAS}_\text{NW}$ and $\text{CMAS}_\text{AW}$, which are respectively trained on only normal findings and only abnormal findings.
Finally, we show CMAS's performances with imitation learning (CMAS-IL) and reinforcement learning (CMAS-RL).
For \emph{Impression} section, we compare our method with \newcite{xu2015show}: SoftAtt$_\text{vision}$ and SoftAtt$_\text{text}$, which are trained with visual input only (no findings) and textual input only (no images).
We also report CMAS trained only on visual and textual input: $\text{CMAS}_\text{text}$ and $\text{CMAS}_\text{vision}$.
Finally, we also compare CMAS-IL with CMAS-RL.
\begin{table*}[t!]
\centering
\small
\begin{tabular}{c|l|c c c c c c}
\hline
Dataset & Methods & BLEU-1 & BLEU-2 & BLEU-3 & BLEU-4 & ROUGE & CIDEr\\
\hline
\multirow{10}{*}{CX-CHR}
& CNN-RNN~\cite{vinyals2015show} & 0.590 & 0.506 & 0.450 & 0.411 & 0.577 & 1.580\\
& LRCN~\cite{donahue2015long} & 0.593 & 0.508 & 0.452 & 0.413 & 0.577 & 1.588\\
& AdaAtt~\cite{lu2017knowing} & 0.588 & 0.503 & 0.446 & 0.409 & 0.575 & 1.568\\
& Att2in~\cite{rennie2017self} & 0.587 & 0.503 & 0.446 & 0.408 & 0.576 & 1.566\\
& CoAtt~\cite{jing2017automatic} & 0.651 & 0.568 & 0.521 & 0.469 & 0.602 & 2.532\\
& HGRG-Agent~\cite{li2018hybrid} & 0.673 & 0.587 & 0.530 & 0.486 & 0.612 & 2.895\\
\cline{2-8}
& CMAS$_\text{W}$ & 0.659 & 0.585 & 0.534 & 0.497 & 0.627 & 2.564\\
& CMAS$_\text{NW,AW}$ & 0.657 & 0.579 & 0.522 & 0.479 & 0.585 & 1.532\\
& CMAS-IL & 0.663 & 0.592 & 0.543 & 0.507 & 0.628 & 2.475\\
& CMAS-RL & \textbf{0.693} & \textbf{0.626} & \textbf{0.580} & \textbf{0.545} & \textbf{0.661} & \textbf{2.900}\\
\hline
\hline
\multirow{10}{*}{IU-Xray}
& CNN-RNN \cite{vinyals2015show} & 0.216 & 0.124 & 0.087 & 0.066 & 0.306 & 0.294\\
& LRCN \cite{donahue2015long} & 0.223 & 0.128 & 0.089 & 0.067 & 0.305 & 0.284\\
& AdaAtt~\cite{lu2017knowing} & 0.220 & 0.127 & 0.089 & 0.068 & 0.308 & 0.295\\
& Att2in \cite{rennie2017self} & 0.224 & 0.129 & 0.089 & 0.068 & 0.308 & 0.297\\
& CoAtt~\cite{jing2017automatic} & 0.455 & 0.288 & 0.205 & 0.154 & 0.369 & 0.277\\
& HGRG-Agent~\cite{li2018hybrid} & 0.438 & 0.298 & 0.208 & 0.151 & 0.322 & \textbf{0.343}\\
\cline{2-8}
& CMAS$_\text{W}$ & 0.440 & 0.292 & 0.204 & 0.147 & 0.365 & 0.252\\
& CMAS$_\text{NW,AW}$ & 0.451 & 0.286 & 0.199 & 0.146 & 0.366 & 0.269\\
& CMAS-IL & 0.454 & 0.283 & 0.195 & 0.143 & 0.353 & 0.266\\
& CMAS-RL & \textbf{0.464} & \textbf{0.301} & \textbf{0.210} & \textbf{0.154} & \textbf{0.362} & 0.275\\
\hline
\end{tabular}
\caption{Main results for findings generation on the CX-CHR (upper) and IU-Xray (lower) datasets. BLEU-n denotes the BLEU score that uses up to n-grams. }
\label{tab:main_results}
\end{table*}
\begin{table*}[t!]
\centering
\small
\begin{tabular}{c|l|c c c c c c}
\hline
Dataset & Methods & BLEU-1 & BLEU-2 & BLEU-3 & BLEU-4 & ROUGE & CIDEr\\
\hline
\multirow{6}{*}{CX-CHR}
& SoftAtt$_\text{text}$~\cite{xu2015show} & 0.112 & 0.044 & 0.016 & 0.005 & 0.142 & 0.038\\
& SoftAtt$_\text{vision}$~\cite{xu2015show} & 0.408 & 0.300 & 0.247 & 0.208 & 0.466 & 0.932\\
\cline{2-8}
& CMAS$_\text{text}$ & 0.182 & 0.141 & 0.127 & 0.119 & 0.356 & 2.162\\
& CMAS$_\text{vision}$ & 0.415 & 0.357 & \textbf{0.323} & \textbf{0.296} & \textbf{0.511} & \textbf{3.124}\\
& CMAS-IL & 0.426 & 0.360 & 0.322 & 0.290 & 0.504 & 3.080\\
& CMAS-RL & \textbf{0.428} & \textbf{0.361} & \textbf{0.323} & 0.290 & 0.504 & 2.968\\
\hline
\hline
\multirow{6}{*}{IU-Xray}
& SoftAtt$_\text{text}$~\cite{xu2015show} & 0.179 & 0.047 & 0.006 & 0.000 & 0.161 & 0.032\\
& SoftAtt$_\text{vision}$~\cite{xu2015show} & 0.224 & 0.103 & 0.045 & 0.022 & 0.210 & 0.046\\
\cline{2-8}
& CMAS$_\text{text}$ & 0.316 & 0.235 & 0.187 & 0.148 & \textbf{0.537} & \textbf{1.562}\\
& CMAS$_\text{vision}$ & 0.379 & 0.270 & 0.203 & 0.151 & 0.513 & 1.401\\
& CMAS-IL & 0.399 & 0.285 & 0.214 & 0.158 & 0.517 & 1.407\\
& CMAS-RL & \textbf{0.401} & \textbf{0.290} & \textbf{0.220} & \textbf{0.166} & 0.521 & 1.457\\
\hline
\end{tabular}
\caption{Main results for impression generation on the CX-CHR (upper) and IU-Xray (lower) datasets. BLEU-n denotes the BLEU score that uses up to n-grams.}
\label{tab:main_results2}
\end{table*}
\subsection{Main Results}\label{main_results}
\paragraph{Comparison to State-of-the-art}
Table~\ref{tab:main_results} shows results on the automatic metrics for the Findings module.
On both datasets, CMAS outperforms all baseline methods on almost all metrics, which indicates its overall efficacy for generating reports that resemble those written by human experts.
The methods can be divided into two different groups: single sentence models (CNN-RNN, LRCN, AdaAtt, Att2in) and hierarchical models (CoAtt, HGRG-Agent, CMAS).
Hierarchical models consistently outperform single sentence models on both datasets, suggesting that the hierarchical models are better for modeling paragraphs.
The leading performances of CMAS-IL and CMAS-RL over the rest of hierarchical models demonstrate the validity of our practice in exploiting the structure information within sections.
\begin{table*}[h!]
\centering
\scriptsize
\begin{tabular}{c|c|c|c|c|c|c|c|c}
\hline
Dataset & \multicolumn{4}{c}{CX-CHR} & \multicolumn{4}{|c}{IU-Xray}\\
\hline
Methods & \newcite{li2018hybrid} & CMAS$_\text{NW,AW}$ & CMAS-IL & CMAS-RL & \newcite{li2018hybrid} & CMAS$_\text{NW,AW}$ & CMAS-IL & CMAS-RL \\
\hline
Precision & 0.292 & 0.173 & 0.272 & \textbf{0.309} & 0.121 & 0.070 & 0.094 & \textbf{0.128}\\
\hline
FPR & 0.059 & 0.076 & 0.063 & \textbf{0.051} & 0.043 & 0.044 & 0.012 & \textbf{0.007}\\
\hline
\end{tabular}
\caption{Average precision and average False Positive Rate (FPR) for abnormality detection. (\emph{Findings})}
\label{tab:acc}
\end{table*}
\begin{table*}[h!]
\centering
\scriptsize
\begin{tabular}{c|c|c|c|c|c|c|c|c}
\hline
Dataset & \multicolumn{4}{c}{CX-CHR} & \multicolumn{4}{|c}{IU-Xray}\\
\hline
Methods & CMAS$_\text{text}$ & CMAS$_\text{vision}$ & CMAS-IL & CMAS-RL & CMAS$_\text{text}$ & CMAS$_\text{vision}$ & CMAS-IL & CMAS-RL \\
\hline
Precision & 0.067 & 0.171 & 0.184 & \textbf{0.187} & 0.054 & 0.160 & 0.162 & \textbf{0.165}\\
\hline
FPR & \textbf{0.067} & 0.142 & 0.170 & 0.168 & \textbf{0.023} & 0.024 & 0.024 & 0.024\\
\hline
\end{tabular}
\caption{Average precision and average False Positive Rate (FPR) for abnormality detection. (\emph{Impression})}
\label{tab:acc2}
\end{table*}
\paragraph{Ablation Study}
$\text{CMAS}_\text{W}$ has only one writer, which is trained on both normal and abnormal findings.
Table~\ref{tab:main_results} shows that $\text{CMAS}_\text{W}$ can achieve competitive performances to the state-of-the-art methods.
$\text{CMAS}_\text{NW, AW}$ is a simple concatenation of two single agent models $\text{CMAS}_\text{NW}$ and $\text{CMAS}_\text{AW}$, where $\text{CMAS}_\text{NW}$ is trained only on normal findings and $\text{CMAS}_\text{AW}$ is trained only on abnormal findings.
At test time, the final paragraph of $\text{CMAS}_\text{NW, AW}$ is simply a concatenation of normal and abnormal findings generated by $\text{CMAS}_\text{NW}$ and $\text{CMAS}_\text{AW}$ respectively.
Surprisingly, $\text{CMAS}_\text{NW, AW}$ performs worse than $\text{CMAS}_\text{W}$ on the CX-CHR dataset.
We believe the main reason is the missing communication protocol between the two agents, which could cause conflicts when they take actions independently.
For example, for an image, NW might think ``the heart size is normal'', while AW believes ``the heart is enlarged''.
Such conflict would negatively affect their joint performances.
As evidently shown in Table~\ref{tab:main_results}, CMAS-IL achieves higher scores than $\text{CMAS}_\text{NW, AW}$, directly proving the importance of communication between agents and thus the importance of PL.
Finally, it can be observed from Table~\ref{tab:main_results} that CMAS-RL consistently outperforms CMAS-IL on all metrics, which demonstrates the effectiveness of reinforcement learning.
\paragraph{Impression Module}
As shown in Table~\ref{tab:main_results2}, CMAS$_\text{vision}$ and CMAS$_\text{text}$ have higher scores than SoftAtt$_\text{vision}$ and SoftAtt$_\text{text}$, indicating the effectiveness of CMAS.
It can also be observed from Table~\ref{tab:main_results2} that images provide better information than text, since CMAS$_\text{vision}$ and SoftAtt$_\text{vision}$ exceed the scores of CMAS$_\text{text}$ and SoftAtt$_\text{text}$ to a large margin on most of the metrics.
However, further comparison among CMAS-IL, CMAS$_\text{text}$ and CMAS$_\text{vision}$ shows that text information can help improve the model's performance to some degree.
\begin{figure*}[h!]
\centering
\includegraphics[width=1.\textwidth]{samples_p.png}
\caption{Examples of findings generated by CMAS-RL and CMAS$_\text{W}$ on IU-Xray dataset, along with their corresponding CXR images and ground-truth reports.
Highlighted sentences are abnormal findings.}
\label{fig:samples}
\end{figure*}
\subsection{Abnormality Detection}
The automatic evaluation metrics (e.g. BLEU) are based on n-gram similarity between the generated sentences and the ground-truth sentences.
A model can easily obtain high scores on these automatic evaluation metrics by generating normal findings~\cite{jing2017automatic}.
To better understand CMAS's ability in detecting abnormalities, we report its precision and average False Positive Rate (FPR) for abnormality term detection in Table~\ref{tab:acc} and Table~\ref{tab:acc2}.
Table~\ref{tab:acc} shows that CMAS-RL obtains the highest precision and the lowest average FPR on both datasets, indicating the advantage of CMAS-RL for detecting abnormalities.
Table~\ref{tab:acc2} shows that CMAS-RL achieves the highest precision scores, but not the lowest FPR.
However, FPR can be lowered by simply generating normal sentences, which is exactly the behavior of CMAS$_\text{text}$.
\subsection{Qualitative Analysis}
In this section, we evaluate the overall quality of generated reports through several examples.
Figure~\ref{fig:samples} presents 5 reports generated by CMAS-RL and CMAS$_\text{W}$, where the top 4 images contain abnormalities and the bottom image is a normal case.
It can be observed from the top 4 examples that the reports generated by CMAS-RL successfully detect the major abnormalities, such as ``cardiomegaly'', ``low lung volumes'' and ``calcified granulomas''.
However, CMAS-RL might miss secondary abnormalities sometimes.
For instance, in the third example, the ``right lower lobe'' is wrongly-written as ``right upper lobe'' by CMAS-RL.
We find that both CMAS-RL and $\text{CMAS}_\text{W}$ are capable of producing accurate normal findings since the generated reports highly resemble those written by radiologists (as shown in the last example in Figure~\ref{fig:samples}).
Additionally, $\text{CMAS}_\text{W}$ tends to produce normal findings, which results from the overwhelming normal findings in the dataset.
\subsection{Template Learning}
Radiologists tend to use reference templates when writing reports, especially for normal findings.
Manually designing a template database can be costly and time-consuming.
By comparing the most frequently generated sentences by CMAS with the most used template sentences in the ground-truth reports, we show that the Normality Writer (NW) in the proposed CMAS is capable of learning these templates automatically.
Several most frequently used template sentences~\cite{li2018hybrid} in the IU-Xray dataset are shown in Table~\ref{tab:gt_template}.
The top 10 template sentences generated by NW are presented in Table~\ref{tab:learned_template}.
In general, the templates sentences generated by NW are similar to those top templates in ground-truth reports.
\begin{table}[h!]
\centering
\scriptsize
\begin{tabular}{l}
\hline
The lungs are clear. \\
Lungs are clear. \\
The lung are clear bilaterally.\\
\hline
No pneumothorax or pleural effusion. \\
No pleural effusion or pneumothorax. \\
There is no pleural effusion or pneumothorax.\\
\hline
No evidence of focal consolidation, pneumothorax, or pleural effusion.\\
No focal consolidation, pneumothorax or large pleural effusion. \\
No focal consolidation, pleural effusion, or pneumothorax identified..\\
\hline
\end{tabular}
\caption{Most commonly used templates in IU-Xray.
Template sentences are clustered by their topics.}
\label{tab:gt_template}
\end{table}
\begin{table}[h!]
\centering
\scriptsize
\begin{tabular}{l}
\hline
The lungs are clear. \\
\hline
The heart is normal in size. \\
Heart size is normal.\\
\hline
There is no acute bony abnormality. \\
\hline
There is no pleural effusion or pneumothorax. \\
There is no pneumothorax.\\
No pleural effusion or pneumothorax.\\
\hline
There is no focal air space effusion to suggest a areas.\\
No focal consolidation.\\
Trachea no evidence of focal consolidation pneumothorax or pneumothorax.\\
\hline
\end{tabular}
\caption{Top 10 sentences generated by CMAS.
The sentences are clustered by their topics.}
\label{tab:learned_template}
\end{table}
\section{Conclusion}\label{conclusion}
In this paper, we proposed a novel framework for accurately generating chest X-ray imaging reports by exploiting the structure information in the reports.
We explicitly modeled the between-section structure by a two-stage framework, and implicitly captured the within-section structure with a novel Co-operative Multi-Agent System (CMAS) comprising three agents: Planner (PL), Abnormality Writer (AW) and Normality Writer (NW).
The entire system was trained with REINFORCE algorithm.
Extensive quantitative and qualitative experiments demonstrated that the proposed CMAS not only could generate meaningful and fluent reports, but also could accurately describe the detected abnormalities.
|
1,116,691,500,816 | arxiv | \section{The spectral densities of $Z^+_{cs}$}
In order to calculate the spectral density of the operator product expansion (OPE) side, the light quark ($q=u$, $d$ or $s$) and heavy-quark ($Q=c$ or $b$) full propagators $S^q_{i j}(x)$ and $S^Q_{i j}(p)$ are employed, say
\begin{eqnarray}
S^q_{j k}(x) \! \! & = & \! \! \frac{i \delta_{j k} x\!\!\!\slash}{2 \pi^2
x^4} - \frac{\delta_{jk} m_q}{4 \pi^2 x^2} - \frac{i t^a_{j k} G^a_{\alpha\beta}
}{32 \pi^2 x^2}(\sigma^{\alpha \beta} x\!\!\!\slash
+ x\!\!\!\slash \sigma^{\alpha \beta}) - \frac{\delta_{jk}}{12} \langle\bar{q} q \rangle + \frac{i\delta_{j k}
x\!\!\!\slash}{48} m_q \langle \bar{q}q \rangle - \frac{\delta_{j k} x^2}{192} \langle g_s \bar{q} \sigma \cdot G q \rangle \nonumber \\ &+& \frac{i \delta_{jk} x^2 x\!\!\!\slash}{1152} m_q \langle g_s \bar{q} \sigma \cdot G q \rangle - \frac{t^a_{j k} \sigma_{\alpha \beta}}{192}
\langle g_s \bar{q} \sigma \cdot G q \rangle
+ \frac{i t^a_{jk}}{768} (\sigma_{\alpha \beta} x \!\!\!\slash + x\!\!\!\slash \sigma_{\alpha \beta}) m_q \langle
g_s \bar{q} \sigma \cdot G q \rangle \;,
\end{eqnarray}
\begin{eqnarray}
S^Q_{j k}(p) \! \! & = & \! \! \frac{i \delta_{j k}(p\!\!\!\slash + m_Q)}{p^2 - m_Q^2} - \frac{i}{4} \frac{t^a_{j k} G^a_{\alpha\beta} }{(p^2 - m_Q^2)^2} [\sigma^{\alpha \beta}
(p\!\!\!\slash + m_Q)
+ (p\!\!\!\slash + m_Q) \sigma^{\alpha \beta}] \nonumber \\ &+& \frac{i\delta_{jk}m_Q \langle g_s^2 G^2\rangle}{12(p^2 - m_Q^2)^3}\bigg[ 1 + \frac{m_Q (p\!\!\!\slash + m_Q)}{p^2 - m_Q^2} \bigg] \nonumber \\ &+& \frac{i \delta_{j k}}{48} \bigg\{ \frac{(p\!\!\!\slash +
m_Q) [p\!\!\!\slash (p^2 - 3 m_Q^2) + 2 m_Q (2 p^2 - m_Q^2)] }{(p^2 - m_Q^2)^6}
\times (p\!\!\!\slash + m_Q)\bigg\} \langle g_s^3 G^3 \rangle \; .
\end{eqnarray}
Here, the vacuum condensates are explicitly shown. For more explanation on above propagators, readers may refer to Refs.~\cite{Albuquerque:2013ija}.
\begin{eqnarray}
\rho^{OPE}(s) & = & \rho^{pert}(s) + \rho^{\langle \bar{q} q
\rangle}(s) + \rho^{\langle \bar{s} s
\rangle}(s) + \rho^{\langle G^2 \rangle}(s) + \rho^{\langle \bar{q} G q \rangle}(s) \nonumber \\
&+& \rho^{\langle \bar{s} G s \rangle}(s)
+ \rho^{\langle \bar{q} q \rangle \langle \bar{s} s \rangle}(s) + \rho^{\langle G^3 \rangle}(s)+ \rho^{\langle \bar{q} q\rangle \langle G^2 \rangle}(s) + \rho^{\langle \bar{s} s\rangle \langle G^2 \rangle}(s) \nonumber\\
&+& \rho^{\langle \bar{s} s \rangle \langle \bar{q} G q \rangle}(s)+ \rho^{\langle \bar{q} q \rangle \langle \bar{s} G s \rangle}(s)\; , \label{rho-OPE} \\
\Pi^{sum}(q^2) &=& \Pi^{\langle \bar{q} q \rangle \langle \bar{s} s \rangle}(q^2)+ \Pi^{\langle G^3 \rangle}(q^2) +\Pi^{\langle \bar{q} q\rangle \langle G^2 \rangle}(q^2)+\Pi^{\langle \bar{s} s\rangle \langle G^2 \rangle}(q^2)\nonumber\\
&+& \Pi^{\langle \bar{q} q \rangle \langle \bar{s} G s \rangle}(q^2) + \Pi^{\langle \bar{s} s \rangle \langle \bar{q} G q \rangle}(q^2) \; . \label{rho-OPE-Pi}
\end{eqnarray}
\subsection{The spectral densities for molecular state}
The spectral density $\rho^{OPE}(s)$ is calculated up to dimension eight.
\begin{eqnarray}
\rho^{pert}(s) &=& \frac{1}{3 \times 2^{13} \pi^6}\int^{\alpha_{max}}_{\alpha_{min}} \frac{d \alpha}{\alpha^3} \int^{1 - \alpha}_{\beta_{min}} \frac{d \beta}{\beta^3} {\cal F}^3_{\alpha \beta} (\alpha + \beta - 1) \bigg{\{} 18 {\cal A}^2 \bigg[{\cal F}_{\alpha \beta}(1+\alpha+\beta)- 4m_c [2 \alpha m_q\nonumber\\
&+&\beta m_s (1+\alpha+\beta)]\bigg] + 18 {\cal B}^2 \bigg[{\cal F}_{\alpha \beta}(1+\alpha+\beta) -4m_c [2 \beta m_s +\alpha m_q (1+\alpha+\beta)]\bigg] \nonumber\\
&+& 6 \bigg[ 3({\cal C}^2+{\cal D}^2){\cal F}_{\alpha \beta}(1+\alpha+\beta) +4m_c^2(\alpha + \beta - 1)[3{\cal C}^2+{\cal D}^2(2+\alpha+\beta)] \bigg]\nonumber\\
&+&{\cal AC}\bigg[3{\cal F}_{\alpha \beta}(1+\alpha+\beta)+12 m_c\big[m_c(\alpha + \beta - 1)-(m_q+m_s)(\beta^2+\beta+\alpha\beta+2\alpha)\big]\bigg]\nonumber\\
&+&{\cal BC}\bigg[3{\cal F}_{\alpha \beta}(1+\alpha+\beta)+12 m_c\big[m_c(\alpha + \beta - 1)-(m_q+m_s)(\alpha^2+\alpha+\alpha\beta+2\beta)\big]\bigg]\nonumber\\
&+&{\cal AD}\bigg[3{\cal F}_{\alpha \beta}(1+\alpha+\beta)+4 m_c\big[m_c(\alpha + \beta - 1)(\alpha + \beta + 2)\nonumber\\
&-&3(\alpha+\beta)[2m_q+m_s(1+\alpha+\beta)]\big]\bigg]+{\cal BD}\bigg[3{\cal F}_{\alpha \beta}(1+\alpha+\beta)\nonumber\\
&+&4 m_c\big[m_c(\alpha + \beta - 1)(\alpha + \beta + 2)-3(\alpha+\beta)[2m_s+m_q(1+\alpha+\beta)]\big]\bigg]\bigg{\}}\; ,\\
\rho^{\langle \bar{q} q \rangle}(s) &=& \frac{\langle \bar{q} q \rangle}{2^9\pi^4} \int^{\alpha_{max}}_{\alpha_{min}} d \alpha \bigg{\{} \int^{1 - \alpha}_{\beta_{min}} d \beta \bigg{\{} \frac{6{\cal A}^2{\cal F}_{\alpha \beta}(2m_c {\cal F}_{\alpha \beta}+\beta m_q {\cal F}_{\alpha \beta}-4\beta m_c^2 m_s )}{\alpha\beta^2}\nonumber\\
&+&\frac{6{\cal B}^2{\cal F}_{\alpha \beta}(2m_c (\alpha+\beta) {\cal F}_{\alpha \beta}+\beta m_q {\cal F}_{\alpha \beta}-4\beta m_c^2 m_s )}{\alpha\beta^2}+\frac{6{\cal D}^2{\cal F}_{\alpha \beta}({\cal F}_{\alpha \beta}+2m_c^2)(m_q-2m_s)}{\alpha\beta}\nonumber\\
&+&\frac{6{\cal C}^2{\cal F}_{\alpha \beta}(m_q {\cal F}_{\alpha \beta}+2 m_q m_c^2(\alpha+\beta)-4 m_c^2 m_s )}{\alpha\beta}\nonumber\\
&+&\frac{{\cal AC}{\cal F}_{\alpha \beta}[{\cal F}_{\alpha \beta}(\alpha\beta m_q +2m_c(\alpha+\beta))+2\alpha\beta m_c^2(m_q(\alpha+\beta)-2m_s)]}{\alpha^2\beta^2}\nonumber\\
&+&\frac{{\cal AD}{\cal F}_{\alpha \beta}[\alpha\beta({\cal F}_{\alpha \beta} + 2m_c^2)(m_q-2m_s) +2m_c{\cal F}_{\alpha \beta}(\alpha+\alpha\beta+\beta^2)]}{\alpha^2\beta^2}\nonumber\\
&+&\frac{{\cal BC}{\cal F}_{\alpha \beta}[{\cal F}_{\alpha \beta}(\alpha\beta m_q +2m_c(\alpha+\beta)^2)+2\alpha\beta m_c^2(m_q(\alpha+\beta)-2m_s)]}{\alpha^2\beta^2}\nonumber\\
&+&\frac{{\cal BD}{\cal F}_{\alpha \beta}[\alpha\beta({\cal F}_{\alpha \beta} + 2m_c^2)(m_q-2m_s) +2m_c{\cal F}_{\alpha \beta}(\alpha^2+\alpha\beta+\beta)]}{\alpha^2\beta^2}\bigg{\}}\nonumber\\
&+&\frac{{\cal H}_\alpha^2}{\alpha(\alpha-1)}\bigg[6({\cal A}^2+{\cal B}^2+{\cal C}^2+{\cal D}^2)m_q-12{\cal C}^2m_s\nonumber\\
&+&({\cal AC+AD+BC+BD})(m_q-2m_s)\bigg]\bigg{\}}\;,\\
\rho^{\langle \bar{s} s \rangle}(s) &=& \frac{\langle \bar{s} s \rangle}{2^9\pi^4} \int^{\alpha_{max}}_{\alpha_{min}} d \alpha \bigg{\{} \int^{1 - \alpha}_{\beta_{min}} d \beta \bigg{\{} \frac{ -24 {\cal A}^2 m_c^2 m_q{\cal F}_{\alpha \beta}}{\alpha\beta}-\frac{6{\cal D}^2{\cal F}_{\alpha\beta}({\cal F}_{\alpha\beta}+2 m_c^2)(2m_q-m_s)}{\alpha\beta}\nonumber\\
&+&\frac{6{\cal C}^2{\cal F}_{\alpha\beta}(m_s {\cal F}_{\alpha\beta}-4m_c^2m_q+2m_c^2m_s(\alpha+\beta))}{\alpha\beta}\nonumber\\
&+&\frac{6{\cal B}^2{\cal F}_{\alpha\beta}(2m_c {\cal F}_{\alpha\beta}-4\alpha m_c^2m_q+\alpha m_s {\cal F}_{\alpha\beta})}{\alpha^2\beta}\nonumber\\
&+&\frac{{\cal AC}{\cal F}_{\alpha \beta}[{\cal F}_{\alpha \beta}(\alpha\beta m_s +2m_c(\alpha+\beta)^2)+2\alpha\beta m_c^2(m_s(\alpha+\beta)-2m_q)]}{\alpha^2\beta^2}\nonumber\\
&+&\frac{{\cal AD}{\cal F}_{\alpha \beta}[\alpha\beta({\cal F}_{\alpha \beta} + 2m_c^2)(m_s-2m_q) +2m_c{\cal F}_{\alpha \beta}(\alpha+\alpha\beta+\beta^2)]}{\alpha^2\beta^2}\nonumber\\
&+&\frac{{\cal BC}{\cal F}_{\alpha \beta}[{\cal F}_{\alpha \beta}(\alpha\beta m_s +2m_c(\alpha+\beta))+2\alpha\beta m_c^2(m_s(\alpha+\beta)-2m_q)]}{\alpha^2\beta^2}\nonumber\\
&+&\frac{{\cal BD}{\cal F}_{\alpha \beta}[\alpha\beta({\cal F}_{\alpha \beta} + 2m_c^2)(m_s-2m_q) +2m_c{\cal F}_{\alpha \beta}(\alpha^2+\alpha\beta+\beta)]}{\alpha^2\beta^2}\bigg{\}}\nonumber\\
&+&\frac{{\cal H}_\alpha^2}{\alpha(\alpha-1)}\bigg[6({\cal B}^2+{\cal C}^2-{\cal D}^2)m_s-12({\cal C}^2+{\cal D}^2)m_q\nonumber\\
&+&({\cal AC+AD+BC+BD})(m_s-2m_q)\bigg]\bigg{\}}\;,\\
\rho^{\langle G^2 \rangle}(s) &=& \frac{\langle g_s^2 G^2\rangle}{3^2\times2^{14}\pi^6} \int^{\alpha_{max}}_{\alpha_{min}} d \alpha \bigg{\{} \int^{1 - \alpha}_{\beta_{min}} \frac{d \beta}{\alpha^3\beta^3} \bigg{\{} 36{\cal A}^2\bigg[\alpha\beta{\cal F}_{\alpha\beta}\big[2\alpha\beta m_c m_s (\alpha+\beta-2)\nonumber\\
&-&{\cal F}_{\alpha\beta}(\alpha^2-2\alpha+4\alpha\beta+3\beta^2)\big]-m_c(\alpha+\beta-1)(\alpha+\beta+1)\big[ 3\beta^3m_s{\cal F}_{\alpha\beta}\nonumber\\
&-&m_c({\cal F}_{\alpha\beta}+\beta m_c m_s)(\alpha^3+\beta^3) \big]\bigg]+ 36{\cal B}^2\bigg[\alpha\beta{\cal F}_{\alpha\beta}[2\beta{\cal F}_{\alpha\beta}+6\alpha\beta m_c m_s\nonumber\\
&-&(\alpha+\beta)(3\alpha+\beta){\cal F}_{\alpha\beta}]+(\alpha+\beta-1)m_c[-6\beta^3 m_s {\cal F}_{\alpha\beta}+(\alpha+\beta)(1+\alpha+\beta)\nonumber\\
&\times&(\alpha^2-\alpha\beta+\beta^2)m_c{\cal F}_{\alpha\beta}-2 \beta m_c^2 m_s (\alpha^3+\beta^3) ]\bigg]+{\cal C}^2 \bigg[ 12m_c^4 (\alpha+\beta-1)^2\nonumber\\
&\times&(\alpha+\beta)(\alpha+\beta+2)(\alpha^2-\alpha\beta+\beta^2)+18\alpha\beta{\cal F}^2_{\alpha\beta}(3\alpha^2+3\beta^2+8\alpha\beta-3)\nonumber\\
&+&36m_c^2{\cal F}_{\alpha\beta}[2\alpha^5+7\alpha^4\beta-4\alpha^3+13\alpha^3\beta^2+2\beta^2-4\beta^3+2\beta^5\nonumber\\
&+&\alpha^2 \beta ((\beta - 1) (13\beta + 9) + 2) + \alpha\beta (7\beta^3 - 9\beta + 4)] \bigg] +{\cal D}^2\bigg[18\alpha\beta{\cal F}_{\alpha\beta}({\cal F}_{\alpha\beta} (8 \alpha \beta\nonumber\\
&+&(\alpha -4) \alpha +\beta ^2-4 \beta +3)+12 \alpha \beta m_c^2)+ 36 m_c^2 (\alpha +\beta -1) [{\cal F}_{\alpha\beta} (\alpha ^4\nonumber\\
&+&\alpha ^3 (\beta +4)+3 \alpha ^2 (\beta -1)+\alpha \beta ^2 (\beta +3)+\beta ^2 (\beta (\beta +4)-3))\nonumber\\
&+&m_c^2 (\alpha +\beta -1) (\alpha +\beta )(\alpha ^2-\alpha \beta +\beta ^2)] \bigg]-{\cal AC}\bigg[6m_c(\alpha+\beta-1)[3m_s{\cal F}_{\alpha\beta}(2 \alpha ^3\nonumber\\
&+&(\alpha +1) \beta ^3+\beta ^4)-m_c^3(\alpha^3+\beta^3)(m_c(\alpha+\beta-1)-m_s(\beta^2+\beta+2\alpha\nonumber\\
&+&\alpha\beta))-{\cal F}_{\alpha\beta}m_c(\alpha ^4+\alpha ^3 (\beta +4)
+3 \alpha ^2 (\beta -1)+\alpha \beta ^2 (\beta +3)\nonumber\\
&+&\beta ^2 (\beta (\beta +4)-3))]+3\alpha\beta{\cal F}_{\alpha\beta}[{\cal F}_{\alpha\beta}(5 \alpha ^2+\alpha (8 \beta -4)+\beta (5 \beta -4)+3)\nonumber\\
&+&2m_c(2m_c(2 (\alpha -1) \beta ^2+2 (\alpha -1) \alpha \beta +\alpha ((\alpha -2) \alpha +3)+\beta ^3+3 \beta -2)\nonumber\\
&+&m_s(\alpha+\beta)(\alpha ^2+2 \alpha (\beta +2)+\beta (\beta +4)-3))]\bigg]+{\cal AD}\bigg[-3\alpha\beta{\cal F}^2_{\alpha\beta}[3 \alpha ^2\nonumber\\
&+&\alpha (8 \beta -4)+\beta (11 \beta +4)-3]+6 m_c^3 (\alpha +\beta -1) (\alpha +\beta ) (\alpha ^2\nonumber\\
&-&\alpha \beta +\beta ^2) [m_c (\alpha +\beta -1)-m_s (\alpha (\beta +2)
+\beta ^2+\beta)]+6m_c {\cal F}^2_{\alpha\beta}[m_c (\alpha ^5\nonumber\\
&+&\alpha ^4 (6 \beta +3)+\alpha ^3 (\beta +1) (9 \beta -7)+\alpha ^2 (\beta +1) (\beta (5 \beta -9)+3)\nonumber\\
&+&2 \alpha \beta ^2 ((\beta -3) \beta +3)+(\beta -1) \beta ^2 (\beta (\beta +4)-3))-m_s (\alpha +\beta ) (2 \alpha ^3 (\beta +3)\nonumber\\
&-&3 \alpha ^2 \left(\beta ^2+2\right)
-2 \alpha (\beta -1) \beta (\beta +3)+3 \beta ^2 (\beta ^2-1))]\bigg]+{\cal BC}\bigg[2 m_c^3 (\alpha +\beta\nonumber\\
&-&1) (\alpha +\beta ) (\alpha ^2-\alpha \beta +\beta ^2) [m_c (\alpha+\beta -1) (\alpha +\beta +2)-6 m_s (\alpha +\beta )]\nonumber\\
&-&3\alpha\beta{\cal F}^2_{\alpha\beta}[5 \alpha ^2+\alpha (8 \beta -4)+\beta (5 \beta -4)+3]
+6m_c{\cal F}_{\alpha\beta}[m_c (2 \alpha ^2 (\alpha ^3-2 \alpha +1)\nonumber\\
&+&3 \alpha \beta ^4+(\alpha (\alpha +4)-4) \beta ^3+(\alpha (\alpha (\alpha +4)-9)+2) \beta ^2\nonumber\\
&+&\alpha (\alpha (\alpha (3 \alpha +4)-9)+4) \beta +2 \beta ^5)-6 m_s (\alpha +\beta ) ((\alpha -1) \alpha ^2+(\beta -1) \beta ^2)]\bigg]\nonumber\\
&+&{\cal BD}\bigg[-3 \alpha\beta{\cal F}_{\alpha\beta}[11 \alpha ^2+\alpha (8 \beta +4)+\beta (3 \beta -4)-3]\nonumber\\
&+&6m_c{\cal F}_{\alpha\beta}[m_c (2 \alpha ^2 ((\alpha -3) \alpha +3) \beta
+(\alpha -1) \alpha ^2 (\alpha (\alpha +4)-3)\nonumber\\
&+&3 (2 \alpha +1) \beta ^4+(\alpha +1) (9 \alpha -7) \beta ^3+(\alpha +1) (\alpha (5 \alpha -9)+3) \beta ^2\nonumber\\
&+&\beta ^5)-m_s (\alpha +\beta ) (-3 (\alpha ^2+2) \beta ^2+3 \alpha ^2 (\alpha ^2-1)+2 (\alpha +3) \beta ^3\nonumber\\
&-&2 (\alpha -1) \alpha (\alpha +3) \beta )]
+6m_c^3(\alpha+\beta-1)(\alpha+\beta)(\alpha ^2-\alpha \beta +\beta ^2)[m_c (\alpha\nonumber\\
&+&\beta -1)-m_s (\alpha (\alpha +\beta +1)+2 \beta )]\bigg]\bigg{\}}\nonumber\\
&+&\frac{6{\cal H}^2_\alpha(-{\cal C} ({\cal A} + {\cal B} + 6 {\cal C}) + 3 ({\cal A} + {\cal B}) {\cal D} + 18 {\cal D}^2)}{(\alpha -1) \alpha}\bigg{\}}\;,\\
\rho^{\langle \bar{q} G q \rangle}(s) &=& \frac{\langle g_s \bar{q} \sigma \cdot G q \rangle}{3\times2^9\pi^4} \int_{\alpha_{min}}^{\alpha_{max}} \bigg{\{} \int_{\beta_{min}}^{1 - \alpha} d \beta \bigg{\{}\frac{18 {\cal B}^2 m_c (2\beta m_c m_s -{\cal F}_{\alpha\beta}(2\alpha+3\beta))}{\beta^2} \nonumber\\
&-& 6{\cal D}^2 m_c^2 m_q +{\cal AC}\frac{2 {\cal F}_{\alpha\beta}(\alpha+\beta)(m_c(\alpha+2\beta-1)-\alpha m_s)-\alpha^2\beta m_c^2 m_q}{\alpha^2\beta}\nonumber\\
&+&{\cal AD}\frac{-{\cal F}_{\alpha\beta} m_c (\alpha -5 \beta +1)+{\cal F}_{\alpha\beta} m_s (\alpha +3 \beta )-2 \beta m_c^3 (\alpha +\beta )+6 \beta m_c^2 m_s}{\alpha\beta}\nonumber\\
&-&{\cal BC}\frac{m_c \left({\cal F}_{\alpha\beta} \left(6 \alpha ^2+11 \alpha \beta +2 \beta ^2\right)+2 \beta m_c (\alpha +\beta ) (\beta m_c-3 m_s)\right)}{\alpha\beta^2}\nonumber\\
&+&\frac{{\cal BD}}{\alpha^2\beta^2}[{\cal F}_{\alpha\beta} \left(\alpha \beta m_s (3 \alpha +\beta )-m_c \left(6 \alpha ^3+2 (1-3 \alpha ) \beta ^2+(4 \alpha +1) \alpha \beta -4 \beta ^3\right)\right)\nonumber\\
&-&2 \alpha \beta m_c^2 (\beta m_c (\alpha +\beta )-3 \alpha m_s)]\bigg{\}}+6{\cal C}^2[2m_c^2(m_q-3m_s)+{\cal H}_\alpha(9m_s-2m_q)]\nonumber\\
&-&12{\cal D}^2({\cal H}_\alpha-m_c^2)(m_q-3m_s)+6({\cal A}^2+{\cal B}^2)[\frac{3m_c {\cal H}_\alpha}{1-\alpha}- 2 m_q {\cal H}_\alpha+m_c^2(m_q-3m_s)]\nonumber\\
&+&{\cal AC}[2 m_c^2 (m_q-3m_s)+{\cal H}_\alpha(9 m_s-2 m_q)-\frac{{\cal H}_\alpha(2m_c-m_s)}{\alpha(\alpha-1)}]\nonumber\\
&+&{\cal AD}[2 m_c^2 (m_q-3m_s)+\frac{{\cal H}_\alpha(3m_s+2m_c(2+\alpha)+2\alpha(m_s+m_q(\alpha-1)-3\alpha m_s))}{\alpha(\alpha-1)}]\nonumber\\
&+&{\cal BC}\frac{-2 \alpha {\cal H}_\alpha (m_c+(\alpha -1) m_q)+(6 (\alpha -1) \alpha -3) {\cal H}_\alpha m_s+2 (\alpha -1) \alpha m_c^2 (m_q-3 m_s)}{\alpha(\alpha-1)}\nonumber\\
&+&\frac{{\cal BD}}{\alpha(\alpha-1)}[{\cal H}_\alpha (2 (\alpha -3) m_c+2 \alpha (-\alpha m_q+m_q+(3 \alpha -5) m_s)+m_s)\nonumber\\
&+&2 (\alpha -1) \alpha m_c^2 (m_q-3 m_s)]\bigg{\}}\;,\\
\rho^{\langle \bar{s} G s \rangle}(s) &=& \frac{\langle g_s \bar{s} \sigma \cdot G s \rangle}{3^2\times2^9\pi^4} \int_{\alpha_{min}}^{\alpha_{max}} \bigg{\{} \int_{\beta_{min}}^{1 - \alpha} d \beta \bigg{\{} -\frac{48{\cal F}_{\alpha\beta}m_c(3\alpha+2\beta)}{\alpha^2}\nonumber\\
&-&3{\cal AC}\frac{(\alpha+\beta)(3m_c^2m_s\alpha (\alpha+\beta) + {\cal F}_{\alpha\beta}(8\alpha m_c+ 3\alpha m_s+ 6\beta m_c))}{\alpha^2\beta}\nonumber\\
&+& \frac{3{\cal AD}}{\alpha^2\beta^2} [-\alpha \beta m_c^2 \left(2 \beta m_c (\alpha +\beta )+m_s \left(-2 \alpha ^2-2 \alpha \beta +\alpha +3 \beta \right)\right)\nonumber\\
&+&{\cal F}_{\alpha\beta}(m_c \left(4 \alpha ^3+\alpha ^2 (3 \beta -2)-\alpha \beta (\beta +1)-6 \beta ^3\right)-\alpha \beta m_s (\alpha +3 \beta ))]\nonumber\\
&+&\frac{3{\cal BC}}{\alpha\beta^2} [{\cal F}_{\alpha\beta}(m_c \left(4 \alpha ^2+\alpha (6 \beta -2)+\beta (5 \beta -2)\right)-\beta m_s (\alpha +\beta ))\nonumber\\
&-&m_c^2\beta(2 \beta m_c (\alpha +\beta )+m_s \left((\alpha -2) \beta +(\alpha -2) \alpha +\beta ^2\right))]\nonumber\\
&+&\frac{3{\cal BD}}{\alpha\beta}[{\cal F}_{\alpha\beta}(5 \alpha m_c-\beta (m_c+m_s)-\text{mc}-3 \alpha m_s)\nonumber\\
&-&m_c^2(2 \alpha m_c (\alpha +\beta )+m_s \left(-2 \alpha \beta +3 \alpha -2 \beta ^2+\beta \right))] \bigg{\}}\nonumber\\
&+&48({\cal A}^2+{\cal B}^2) \frac{m_c({\cal H}_\alpha-\alpha m_c m_q)}{\alpha} + 48{\cal C}^2m_q(3{\cal H}_\alpha-2m_c^2)\nonumber\\
&+&96{\cal D}^2 m_q({\cal H}_\alpha-m_c^2) +{\cal AC}[-2 m_c^2 (9 m_q+m_s) +3{\cal H}_\alpha(9 m_q\nonumber\\
&+&2m_s +\frac{3 m_s}{\alpha(\alpha-1)}) ]+3{\cal AD}[-2 m_c^2 (3 m_q-4m_s) +{\cal H}_\alpha(6m_q)\nonumber\\
&+&\frac{m_s(3+4\alpha-8\alpha^2)-2m_c(\alpha+2)}{\alpha(\alpha-1)}] -3{\cal BC}[6m_c^2m_q\nonumber\\
&+&\frac{{\cal H}_\alpha[m_s+2(m_c+3m_q+m_s)\alpha+2\alpha^2(3m_q+m_s)]}{\alpha(\alpha-1)}]+3{\cal BD}[-2m_c^2(3m_q-4m_s)\nonumber\\
&+&\frac{{\cal H}_\alpha(2 (\alpha -3)m_c+6 (\alpha -1) \alpha m_q+(4 (3-2 \alpha ) \alpha -1) m_s)}{\alpha(\alpha-1)}]\bigg{\}}\;,\\
\rho^{\langle \bar{q} q \rangle \langle \bar{s} s \rangle}(s) &=& \frac{\langle \bar{q} q \rangle \langle \bar{s} s \rangle}{3\times2^6\pi^2} \int_{\alpha_{min}}^{\alpha_{max}} d \alpha \bigg{\{}6{\cal A}^2 m_c(-2m_c+m_q-\alpha m_q+2\alpha m_s)\nonumber\\
&+&6{\cal B}^2 m_c(-2m_c+2m_q-2\alpha m_q+\alpha m_s)+{\cal C}^2(36{\cal H}_\alpha-24m_c^2)+{\cal D}^2(24{\cal H}_\alpha-24m_c^2)\nonumber\\
&+&{\cal AC}(6 {\cal H}_\alpha+m_c (-4 m_c+m_q+2m_s))+{\cal AD}(4 {\cal H}_\alpha+m_c (-4 m_c+m_q(1+\alpha)\nonumber\\
&+&m_s(1+\alpha)))+{\cal BC}(6 {\cal H}_\alpha+m_c (-4 m_c+2m_q+m_s))\nonumber\\
&+&{\cal BD}(4 {\cal H}_\alpha+m_c (-4 m_c+m_q(2-\alpha)+m_s(2-\alpha)))\bigg{\}}\;,\\
\rho^{\langle G^3 \rangle}(s) &=& \frac{\langle g_s^3 G^3 \rangle}{3\times2^{15}\pi^6} \int_{\alpha_{min}}^{\alpha_{max}} \frac{d\alpha}{\alpha^3} \int_{\beta_{min}}^{1 - \alpha} \frac{d \beta}{\beta^3} \bigg{\{} 6{\cal A}^2 (\alpha +\beta -1) (\alpha +\beta +1)\nonumber\\
&\times& [{\cal F}_{\alpha\beta} \left(\alpha ^3+\beta ^3\right)+m_c \left(2 m_c \left(\alpha ^4+\beta ^4\right)-\beta m_s \left(\alpha ^3+6 \beta ^3\right)\right)]\nonumber\\
&+& 6{\cal B}^2 (\alpha +\beta -1) [{\cal F}_{\alpha\beta} (\alpha +\beta ) (\alpha +\beta +1) \left(\alpha ^2-\alpha \beta +\beta ^2\right)\nonumber\\
&+&2 m_c \left(m_c (\alpha +\beta +1) \left(\alpha ^4+\beta ^4\right)-\beta m_s \left(\alpha ^3+6 \beta ^3\right)\right)]\nonumber\\
&+&6{\cal C}^2 (\alpha +\beta -1)[{\cal F}_{\alpha\beta} (\alpha +\beta ) (\alpha +\beta +1) \left(\alpha ^2-\alpha \beta +\beta ^2\right)\nonumber\\
&+&2 m_c^2 (2 \alpha ^5+\alpha ^4 (3 \beta +2)+\alpha ^3 \left(\beta ^2+\beta -2\right)+\alpha ^2 \beta ^3\nonumber\\
&+&\alpha \beta ^3 (3 \beta +1)+2 \beta ^3 (\beta ^2+\beta -1))]+6{\cal D}^2 (\alpha +\beta -1)[{\cal F}_{\alpha\beta} (\alpha +\beta ) (\alpha +\beta +1)\nonumber\\
&\times& \left(\alpha ^2-\alpha \beta +\beta ^2\right)+2 m_c^2 (\alpha ^5+\alpha ^4 (\beta +4)+3 \alpha ^3 (\beta -1)+\alpha \beta ^3 (\beta +3)\nonumber\\
&+&\beta ^3 (\beta (\beta +4)-3))]+{\cal AC}(\alpha +\beta -1)[{\cal F}_{\alpha\beta} (\alpha +\beta ) (\alpha +\beta +1) \left(\alpha ^2-\alpha \beta +\beta ^2\right)\nonumber\\
&+&m_c (2 m_c (\alpha ^3 \beta ^2+\alpha ^3 (3 \alpha +1) \beta +\left(\alpha ^2+\alpha -2\right) \beta ^3+2 \alpha ^3 \left(\alpha ^2+\alpha -1\right)\nonumber\\
&+&(3 \alpha +2) \beta ^4+2 \beta ^5)-m_s (\alpha +\beta +1) \left(2 \alpha ^2+3 \alpha \beta +2 \beta ^2\right) \left(3 \alpha ^2-4 \alpha \beta +3 \beta ^2\right))]\nonumber\\
&+&{\cal AD}(\alpha +\beta -1)[{\cal F}_{\alpha\beta} (\alpha +\beta ) (\alpha +\beta +1) \left(\alpha ^2-\alpha \beta +\beta ^2\right)\nonumber\\
&+&m_c (2m_c \left(\alpha ^5+\alpha ^4 (\beta +4)+3 \alpha ^3 (\beta -1)+\alpha \beta ^3 (\beta +3)+\beta ^3 (\beta (\beta +4)-3)\right)\nonumber\\
&-&m_s \left(\alpha ^4 (\beta +12)+\alpha ^3 \beta (\beta +1)+2 \alpha \beta ^3 (3 \beta +1)+6 \beta ^4 (\beta +1)\right))]\nonumber\\
&+&{\cal BC}(\alpha +\beta -1)[{\cal F}_{\alpha\beta} (\alpha +\beta ) (\alpha +\beta +1) \left(\alpha ^2-\alpha \beta +\beta ^2\right)\nonumber\\
&+&2 m_c (m_c (\alpha ^3 \beta ^2+\alpha ^3 (3 \alpha +1) \beta +\left(\alpha ^2+\alpha -2\right) \beta ^3+2 \alpha ^3 \left(\alpha ^2+\alpha -1\right)\nonumber\\
&+&(3 \alpha +2) \beta ^4+2 \beta ^5)-m_s \left(6 \alpha ^4+\alpha ^3 \beta +\alpha \beta ^3+6 \beta ^4\right))]\nonumber\\
&+&{\cal BD}(\alpha +\beta -1)[{\cal F}_{\alpha\beta} (\alpha +\beta ) (\alpha +\beta +1) \left(\alpha ^2-\alpha \beta +\beta ^2\right)\nonumber\\
&+&m_c (2 m_c \left(\alpha ^5+\alpha ^4 (\beta +4)+3 \alpha ^3 (\beta -1)+\alpha \beta ^3 (\beta +3)+\beta ^3 (\beta (\beta +4)-3)\right)\nonumber\\
&-&m_s \left(6 (\alpha +1) \alpha ^4+2 (3 \alpha +1) \alpha ^3 \beta +(\alpha +12) \beta ^4+(\alpha +1) \alpha \beta ^3\right))]\bigg{\}}\;,\\
\rho^{\langle \bar{q} q\rangle \langle G^2 \rangle}(s) &=& \frac{\langle \bar{q} q\rangle \langle G^2 \rangle}{3^2\times2^{10}\pi^4} \int_{\alpha_{min}}^{\alpha_{max}} d\alpha \bigg{\{}\int_{\beta_{min}}^{1 - \alpha} d \beta \bigg{\{}-18{\cal A}^2\frac{\alpha m_c}{\beta^2} +6{\cal B}^2\frac{m_c(3\alpha^2+3\alpha\beta-\beta^2)}{\beta^2}\nonumber\\
&-&6{\cal D}^2m_s+3{\cal AC}\frac{m_c(\alpha+\beta)(\alpha^2+\beta^2)}{\alpha^2\beta^2}+{\cal AD}[\frac{m_c(\alpha+\beta)(3\alpha^2+3\beta^2-2\alpha\beta(1+\beta))}{\alpha^2\beta^2}\nonumber\\
&-&m_s]+{\cal BC}\frac{m_c(\alpha+\beta)^2(3\alpha^2-\alpha\beta+3\beta^2)}{\alpha^2\beta^2}+{\cal BD}[-m_s\nonumber\\
&+&\frac{m_c(\alpha+\beta)(3\alpha^2+3\beta^2-2\alpha\beta(1+\alpha))}{\alpha^2\beta^2}] \bigg{\}} +(18{\cal A}^2-6{\cal B}^2) m_c +(18{\cal C}^2-6{\cal D}^2) m_s\nonumber\\
&+&3{\cal AC}[m_s+\frac{m_c}{\alpha(1-\alpha)}]+{\cal AD}[-m_s+\frac{m_c(3-4\alpha)}{\alpha(1-\alpha)}]+{\cal BC}[3m_s+\frac{-m_c}{\alpha(1-\alpha)}]\nonumber\\
&+&{\cal BD}[-m_s+\frac{m_c(4\alpha-1)}{\alpha(1-\alpha)}]\bigg{\}}\;,\\
\rho^{\langle \bar{s} s\rangle \langle G^2 \rangle}(s) &=& \frac{\langle \bar{s} s\rangle \langle G^2 \rangle}{3^2\times2^{11}\pi^4} \int_{\alpha_{min}}^{\alpha_{max}} d\alpha \bigg{\{}\int_{\beta_{min}}^{1 - \alpha} d \beta \bigg{\{} 12{\cal A}^2 \frac{m_c(3\beta^2+3\alpha\beta-\alpha^2)}{\alpha^2} + 36{\cal B}^2\frac{m_c \beta}{\alpha^2}\nonumber\\
&+&18{\cal C}^2m_s+6{\cal D}^2m_s +{\cal AC}[3ms + \frac{2m_c(\alpha+\beta)^2(3\alpha^2-\alpha\beta+3\beta^2)}{\alpha^2\beta^2}] \nonumber\\
&+&{\cal AD}[ms+\frac{2m_c(\alpha+\beta)(3\alpha^2+3\beta^2-2\alpha\beta(1+\beta))}{\alpha^2\beta^2}]\nonumber\\
&+&{\cal BC}[3ms + \frac{6m_c(\alpha+\beta)(\alpha^2+3\beta)}{\alpha^2\beta^2}]+{\cal BD}[ms\nonumber\\
&+&\frac{2m_c(\alpha+\beta)(3\alpha^2+3\beta^2-2\alpha\beta(1+\alpha))}{\alpha^2\beta^2}] \bigg{\}} +{\cal A}^2(18\alpha m_s -12m_c) \nonumber\\
&-&18{\cal C}^2 m_s+6{\cal D}^2 m_s +2{\cal AC}\frac{m_c}{\alpha(\alpha-1)}+2{\cal AD}[(3\alpha-1)m_s\nonumber\\
&+&\frac{m_c(4\alpha-3)}{\alpha(\alpha-1)}]-6{\cal BC}[m_s+\frac{m_c}{\alpha(1-\alpha)}]\nonumber\\
&+&2{\cal BD}\frac{m_c(1-4\alpha)+\alpha m_s(-2+5\alpha-3\alpha^2)}{\alpha(\alpha-1)}\bigg{\}}\;,\\
\rho^{\langle \bar{q} q \rangle \langle \bar{s} G s \rangle}(s) &=& \frac{\langle \bar{q} q \rangle \langle \bar{s} G s \rangle}{2^7\pi^2} \int_{\alpha_{min}}^{\alpha_{max}} d \alpha \bigg{\{}\alpha(\alpha-1)(2{\cal AC}+2{\cal BC}+{\cal AD}+{\cal BD}+8{\cal D}^2)\bigg{\}}\;,\\
\rho^{\langle \bar{s} s \rangle \langle \bar{q} G q \rangle}(s) &=& \frac{\langle \bar{s} s \rangle \langle \bar{q} G q \rangle}{2^7\pi^2} \int_{\alpha_{min}}^{\alpha_{max}} d \alpha \bigg{\{}\alpha(\alpha-1)(24{\cal C}^2+8{\cal D}^2)\bigg{\}}\;,\\
\Pi^{\langle \bar{q} q \rangle \langle \bar{s} s \rangle} (M_B^2) &=& \frac{m_c^3\langle \bar{q} q \rangle \langle \bar{s} s \rangle}{3\times 2^6\pi^2} \int_{0}^1 d \alpha \; e^{- \frac{m_c^2}{M_B^2 (1 - \alpha) \alpha}}\bigg{\{} -({\cal AC+AD+BC+BD})(m_q+m_s) \nonumber\\
&+&6({\cal A}^2+{\cal B}^2)[(\alpha-1)m_q-\alpha m_s] \bigg{\}}\;,\\
\Pi^{\langle G^3 \rangle} (M_B^2) &=& \frac{m_c^3\langle g_s^3 G^3 \rangle}{3^2\times 2^{14}\pi^6} \int_{0}^{1} \frac{d \alpha}{\alpha^4} \int_{0}^{1 - \alpha} \frac{d \beta}{\beta^4} \; e^{- \frac{m_c^2 (\alpha + \beta)}{M_B^2 \alpha \beta}}(\alpha+\beta-1)\bigg{\{} 18{\cal A}^2 \beta m_s [\alpha ^5+\alpha ^4 (\beta -1)\nonumber\\
&+&\alpha \beta ^3 (\beta +2)+\beta ^4 (\beta +1)] +36{\cal B}^2(\alpha^4+\beta^4)+{\cal C}^2(-6 \alpha ^6 m_c\nonumber\\
&-&12 \alpha ^5 \beta m_c-6 \alpha ^5 m_c-6 \alpha ^4 \beta ^2 m_c-6 \alpha ^4 \beta m_c+12 \alpha ^4 m_c-6 \alpha ^2 \beta ^4 m_c-12 \alpha \beta ^5 m_c\nonumber\\
&-&6 \alpha \beta ^4 m_c-6 \beta ^6 m_c-6 \beta ^5 m_c+12 \beta ^4 m_c)+{\cal D}^2(-18 \alpha ^5 m_c-18 \alpha ^4 \beta m_c\nonumber\\
&+&18 \alpha ^4 m_c-18 \alpha \beta ^4 m_c-18 \beta ^5 m_c+18 \beta ^4 m_c)+{\cal AC}[3 m_s (\alpha +\beta ) (\alpha +\beta +1) (\alpha ^4+\beta ^4 )\nonumber\\
&-&m_c (\alpha +\beta -1) (\alpha ^4 \beta +(\alpha +2) \alpha ^4+3 \beta ^4 )]+{\cal AD}[3 m_s (\alpha ^4+\beta ^4 ) (\alpha (\beta +2)+\beta ^2+\beta )\nonumber\\
&-&m_c (\alpha +\beta -1) (3 \alpha ^4+3 \beta ^4 )]+{\cal BC}[6 m_s (\alpha +\beta ) (\alpha ^4+\beta ^4 )\nonumber\\
&-&m_c (\alpha +\beta -1) (\alpha ^4 \beta +(\alpha +2) \alpha ^4+3 \beta ^4 )]+{\cal BD}[3 m_s (\alpha ^4+\beta ^4 ) (\alpha (\alpha +\beta +1)+2 \beta )\nonumber\\
&-&m_c (\alpha +\beta -1) (3 \alpha ^4+3 \beta ^4 )] \bigg{\}}\;,\\
\Pi^{\langle \bar{q} q\rangle \langle G^2 \rangle} (M_B^2) &=& \frac{m_c^2 \langle \bar{q} q\rangle \langle G^2 \rangle}{3^2\times 2^{11}\pi^4} \int_{0}^{1} d \alpha \bigg{\{} \int_{0}^{1 - \alpha} \frac{d \beta}{\alpha^3\beta^3M_B^2} \; e^{- \frac{m_c^2 (\alpha + \beta)}{M_B^2 \alpha \beta}}\bigg{\{} 12{\cal A}^2[-\alpha M_B^2 (m_c (\alpha ^3+\beta ^3 )\nonumber\\
&-&3 \beta m_s (\alpha ^2+\beta ^2 ) )-m_c^2 m_s (\alpha ^3+\beta ^3 )] +12{\cal B}^2[-\alpha M_B^2 (m_c (\alpha ^4+\alpha ^3 \beta +\alpha \beta ^3+\beta ^4 )\nonumber\\
&-&3 \beta m_s (\alpha ^2+\beta ^2 ) )-m_c^2 m_s (\alpha ^3+\beta ^3 )]+12{\cal C}^2[3 \alpha \beta m_s (\alpha +\beta )^2 M_B^2\nonumber\\
&-&m_c^2 m_s (\alpha ^3+\beta ^3 )]+12{\cal D}^2m_s[\alpha \beta ((\alpha +3) \alpha ^2+\beta ^2 (\beta +3) ) M_B^2\nonumber\\
&-&m_c^2 (\alpha ^3+\beta ^3 )]+2{\cal AC}[-(\alpha +\beta )^2 M_B^2 (m_c (\alpha ^2-\alpha \beta +\beta ^2 )-3 \alpha \beta m_s )\nonumber\\
&-&m_c^2 m_s (\alpha ^3+\beta ^3 )]+2{\cal AD}[M_B^2 (- (m_c (\alpha +\beta ) (\alpha ^2-\alpha \beta +\beta ^2 ) (\alpha \beta +\alpha +\beta ^2 )\nonumber\\
&-&\alpha \beta m_s ((\alpha +3) \alpha ^2+\beta ^2 (\beta +3) ) ) ) -m_c^2 m_s (\alpha ^3+\beta ^3 )] \nonumber\\
&+& 2{\cal BC}[-(\alpha +\beta )^2 M_B^2 (m_c (\alpha ^3+\beta ^3 )-3 \alpha \beta m_s )-m_c^2 m_s (\alpha ^3+\beta ^3 )]\nonumber\\
&+&2{\cal BD}[M_B^2 (- (m_c (\alpha +\beta ) (\alpha ^2-\alpha \beta +\beta ^2 ) (\alpha (\alpha +\beta )+\beta )-\alpha \beta m_s ((\alpha +3) \alpha ^2\nonumber\\
&+&\beta ^2 (\beta +3) ) ) )-m_c^2 m_s (\alpha ^3+\beta ^3 )]\bigg{\}}+ \frac{e^{- \frac{m_c^2}{M_B^2 (1 - \alpha) \alpha}}}{\alpha^2(\alpha-1)^2} \bigg{\{} ({\cal C}^2+{\cal D}^2) m_s (-12\nonumber\\
&+&36\alpha-36\alpha^2)+2({\cal AC+AD+BC+BD})[1+3\alpha(\alpha-1)] \bigg{\}}\bigg{\}}\;,\\
\Pi^{\langle \bar{s} s\rangle \langle G^2 \rangle} (M_B^2) &=& \frac{m_c^2 \langle \bar{s} s\rangle \langle G^2 \rangle}{3^2\times 2^{11}\pi^4} \int_{0}^{1} d \alpha \bigg{\{} \int_{0}^{1 - \alpha} \frac{d \beta}{\alpha^3\beta^3M_B^2} \; e^{- \frac{m_c^2 (\alpha + \beta)}{M_B^2 \alpha \beta}}\bigg{\{} 6\beta{\cal A}^2M_B^2\left(\alpha ^3+\beta ^3\right)\nonumber\\
&\times& (2 m_c (\alpha +\beta )+\alpha \text{ms}) +6\beta{\cal B}^2M_B^2\left(\alpha ^3+\beta ^3\right) (2 m_c+\alpha m_s) \nonumber\\
&+& 6{\cal C}^2m_s \left(m_c^2 \left(\alpha ^4+\alpha ^3 \beta +\alpha \beta ^3+\beta ^4\right)-\alpha \beta \left(4 \alpha ^3+3 \alpha ^2 \beta +3 \alpha \beta (\beta +2)+4 \beta ^3\right) M_B^2\right)\nonumber\\
&+&6{\cal D}^2m_s (m_c^2 (\alpha ^3+\beta ^3 )-\alpha \beta ((\alpha +3) \alpha ^2+\beta ^2 (\beta +3) ) M_B^2 )\nonumber\\
&+&{\cal AC}(\alpha +\beta ) (m_c^2 m_s (\alpha ^3+\beta ^3 )-2 M_B^2 (m_c (\alpha ^4+\alpha ^3 \beta +\alpha \beta ^3+\beta ^4 )\nonumber\\
&+&2 \alpha \beta m_s (\alpha ^2+\alpha \beta +\beta ^2 ) ) ) +{\cal AD}[m_c^2 m_s (\alpha ^3+\beta ^3 )\nonumber\\
&-&M_B^2 (2 m_c (\alpha +\beta ) (\alpha \beta +\alpha +\beta ^2 ) (\alpha ^2-\alpha \beta +\beta ^2 )\nonumber\\
&+&\alpha \beta m_s (\alpha ^3+\alpha ^2 (2 \beta +3)+\beta ^2 (\beta +3) ) )]+{\cal BC}(\alpha +\beta ) (m_c^2 m_s (\alpha ^3+\beta ^3 )\nonumber\\
&-&2 M_B^2 (m_c (\alpha ^3+\beta ^3 )+\alpha \beta m_s (2 \alpha ^2+\alpha \beta +2 \beta ^2 ) ) )+{\cal BD}[m_c^2 m_s (\alpha ^3+\beta ^3 )\nonumber\\
&-&M_B^2 (2 m_c (\alpha +\beta ) (\alpha ^2-\alpha \beta +\beta ^2 ) (\alpha (\alpha +\beta )+\beta )\nonumber\\
&+&\alpha \beta m_s (\alpha ^2 (\alpha +3)+(2 \alpha +3) \beta ^2+\beta ^3 ) )]\bigg{\}}\nonumber\\
& +& \frac{m_s e^{- \frac{m_c^2}{M_B^2 (1 - \alpha) \alpha}}}{\alpha^2(\alpha-1)^2} \bigg{\{} {\cal A}^2(-6 \alpha ^3+24 \alpha ^2-18 \alpha +6) +{\cal B}^2(-6 \alpha ^3+24 \alpha ^2-18 \alpha +6)\nonumber\\
&+&({\cal C}^2+{\cal D}^2)(18 \alpha ^2-18 \alpha +6) +{\cal AC}((\alpha -1) \alpha +1)+{\cal AD}((\alpha -1) \alpha (8 \alpha -9)-1)\nonumber\\
&+&{\cal BC}(9 (\alpha -1) \alpha +1)+{\cal BD}((\alpha -1) \alpha (8 \alpha +1)+1)\bigg{\}} \bigg{\}}\;,\\
\Pi^{\langle \bar{q} q \rangle \langle \bar{s} G s \rangle} (M_B^2) &=& \frac{m_c\langle \bar{q} q \rangle \langle \bar{s} G s \rangle}{3^2\times 2^8\pi^2} \int_{0}^1 \frac{d \alpha}{\alpha^2(\alpha-1)^2M_B^4} \; e^{- \frac{m_c^2}{M_B^2 (1 - \alpha) \alpha}}\bigg{\{}18{\cal A}^2 m_c[2 (\alpha -2) (\alpha -1) \alpha M_B^4\nonumber\\
&+&\alpha m_c M_B^2 (-2 m_c-\alpha m_q+m_q)+m_c^3 m_q]+18{\cal B}^2[2 \alpha M_B^2 (m_c\nonumber\\
&+&(\alpha -1) m_q) ((\alpha -1) \alpha M_B^2-m_c^2 )+m_c^4 m_q]+36{\cal C}^2m_c[19 (\alpha -1) \alpha M_B^2+5 m_c^2]\nonumber\\
&+&36{\cal D}^2m_c[18 (\alpha -1) \alpha M_B^2+5 m_c^2]-{\cal AC}[(\alpha -1) \alpha M_B^2 (m_c^2 (12 m_c\nonumber\\
&-&3 m_q-4 m_s)-2 (\alpha -1) \alpha M_B^2 (15 m_c-2 m_s) )+m_c^4 (3 m_q+2 m_s)]\nonumber\\
&-&{\cal AD}[(\alpha -1) \alpha M_B^2 (2 (\alpha -1) \alpha M_B^2 (-12 m_c+3 \alpha m_q+2 \alpha m_s)\nonumber\\
&+&m_c^2 (12 m_c-(\alpha +1) (3 m_q+2 m_s)) )+m_c^4 (3 m_q+2 m_s)]\nonumber\\
&+&{\cal BC}[2 (\alpha -1) \alpha M_B^2 (3 (\alpha -1) \alpha M_B^2 (5 m_c-m_q)+m_c^2 (-6 m_c+3 m_q+m_s) )\nonumber\\
&-&m_c^4 (3 m_q+2 m_s)]+{\cal BD}[(\alpha -1) \alpha M_B^2 (2 (\alpha -1) \alpha M_B^2 (12 m_c+(\alpha -1) (3 m_q+2 m_s))\nonumber\\
&+&m_c^2 (-12 m_c-(\alpha -2) (3 m_q+2 m_s)) )-m_c^4 (3 m_q+2 m_s)]\bigg{\}}\;,\\
\Pi^{\langle \bar{s} s \rangle \langle \bar{q} G q \rangle} (M_B^2) &=& \frac{m_c\langle \bar{s} s \rangle \langle \bar{q} G q \rangle}{ 2^7\pi^2} \int_{0}^1 \frac{d \alpha}{\alpha^2(\alpha-1)^2M_B^4} \; e^{- \frac{m_c^2}{M_B^2 (1 - \alpha) \alpha}}\bigg{\{}-{\cal A}^2[(\alpha -1) \alpha M_B^2 \nonumber\\
&\times&(2 (\alpha -1) \alpha M_B^2 (\alpha m_s-m_c)+m_c^2 (2 m_c+(\alpha -1) m_q-2 \alpha m_s) )\nonumber\\
&+&m_c^4 (-\alpha m_q+m_q+\alpha m_s)]+{\cal B}^2[\alpha m_c^2 M_B^2 (-2 (\alpha -1) m_c-2 (\alpha -1)^2 m_q\nonumber\\
&+&\alpha (\alpha +1) m_s )+(\alpha -1) \alpha ^2 M_B^4 (2 (\alpha +1) m_c+(\alpha -1) (2 (\alpha -1) m_q-\alpha m_s))\nonumber\\
&+&m_c^4 ((\alpha -1) m_q-\alpha m_s)]+2{\cal C}^2 m_c[5m_c^2+19\alpha(\alpha-1)M_B^2]\nonumber\\
&+&2{\cal D}^2 m_c[5m_c^2+18\alpha(\alpha-1)M_B^2]\bigg{\}}\;,
\end{eqnarray}
where $M_B$ is the Borel parameter introduced by the Borel
transformation, $q = u$ or $d$. Here, we also have the following definitions:
\begin{eqnarray}
{\cal F}_{\alpha \beta} &=& (\alpha + \beta) m_c^2 - \alpha \beta s \; , {\cal H}_\alpha = m_c^2 - \alpha (1 - \alpha) s \; , \\
\alpha_{min} &=& \left(1 - \sqrt{1 - 4 m_c^2/s} \right) / 2, \; , \alpha_{max} = \left(1 + \sqrt{1 - 4 m_c^2 / s} \right) / 2 \; , \\
\beta_{min} &=& \alpha m_c^2 /(s \alpha - m_c^2).
\end{eqnarray}
\subsection{The spectral densities for tetraquark state}
\begin{eqnarray}
\rho^{pert}(s) &=& \frac{1}{3 \times 2^{10} \pi^6}\int^{\alpha_{max}}_{\alpha_{min}} \frac{d \alpha}{\alpha^3} \int^{1 - \alpha}_{\beta_{min}} \frac{d \beta}{\beta^3} {\cal F}^2_{\alpha \beta} (\alpha + \beta - 1) \bigg{\{} 3 {\cal A}^2 \bigg[{\cal F}_{\alpha \beta}^2(1+\alpha+\beta)- 4m_c {\cal F}_{\alpha \beta} [2 \beta m_s\nonumber\\
&+&\alpha m_q (1+\alpha+\beta)] + 24 \alpha \beta m_q m_s m_c^2\bigg] + 3 {\cal B}^2 \bigg[{\cal F}_{\alpha \beta}^2(1+\alpha+\beta) -4m_c {\cal F}_{\alpha \beta} [2 \alpha m_s \nonumber\\
&+&\beta m_q (1+\alpha+\beta)] +24 \alpha \beta m_q m_s m_c^2 \bigg] + 3 \bigg[ ({\cal C}^2+{\cal D}^2)({\cal F}_{\alpha \beta}^2(1+\alpha+\beta)+24 \alpha \beta m_q m_s m_c^2)\nonumber\\
&+&4 m_c {\cal F}_{\alpha \beta}[\alpha m_q (2{\cal D}^2+{\cal C}^2(1+\alpha+\beta))+\beta m_s (2{\cal C}^2+{\cal D}^2(1+\alpha+\beta))] \bigg]\nonumber\\
&+&4({\cal AD+BC}){\cal F}_{\alpha \beta}\bigg[6\alpha\beta m_q m_s - (\alpha+\beta-1)^2m_c^2-3 m_c(\alpha+\beta-1)(\alpha m_s-\beta m_q) \bigg]\bigg{\}}\; ,\\
\rho^{\langle \bar{q} q \rangle}(s) &=& \frac{\langle \bar{q} q \rangle}{2^6\pi^4} \int^{\alpha_{max}}_{\alpha_{min}} d \alpha \bigg{\{} \int^{1 - \alpha}_{\beta_{min}} d \beta \frac{{\cal F}_{\alpha \beta}}{\alpha^2\beta^2} \bigg{\{} {\cal A}^2\bigg[ -2 \alpha^2 m_c {\cal F}_{\alpha \beta} -2 \alpha\beta m_c {\cal F}_{\alpha \beta} -\alpha \beta m_q {\cal F}_{\alpha \beta}\nonumber\\
&+&4\alpha\beta m_s m_c^2 \bigg] +{\cal B}^2\bigg[ -2 \alpha m_c {\cal F}_{\alpha \beta} +2 \alpha\beta^2 m_c m_s m_q -\alpha \beta m_q {\cal F}_{\alpha \beta} +4\alpha\beta m_s m_c^2 \bigg]\nonumber\\
&+&{\cal C}^2\bigg[ 2 \alpha^2 m_c {\cal F}_{\alpha \beta} +2 \alpha\beta^2 m_c {\cal F}_{\alpha \beta} -\alpha \beta m_q {\cal F}_{\alpha \beta} +4\alpha\beta m_s m_c^2 \bigg]\nonumber\\
&+& {\cal D}^2\bigg[ 2 \alpha m_c {\cal F}_{\alpha \beta} -2 \alpha\beta^2 m_c m_s m_q -\alpha \beta m_q {\cal F}_{\alpha \beta} +4\alpha\beta m_s m_c^2 \bigg]\nonumber\\
&+&{\cal AD}\bigg[ (2\alpha\beta m_q m_c+2\beta {\cal F}_{\alpha \beta})(\alpha m_s +(\alpha+\beta-1)m_c) \bigg]\nonumber\\
&+&{\cal BC}\bigg[ (2\alpha\beta m_q m_c-2\beta {\cal F}_{\alpha \beta})(-\alpha m_s +(\alpha+\beta-1)m_c) \bigg]\bigg{\}}\;, \\
&-&\frac{m_q {\cal H}_\alpha}{\alpha(\alpha-1)}\bigg[({\cal A}^2+{\cal B}^2+{\cal C}^2+{\cal D}^2){\cal H}_\alpha+2(\alpha-1)m_s m_c({\cal A}^2+{\cal B}^2-{\cal C}^2-{\cal D}^2)\bigg]\bigg{\}}\;,\\
\rho^{\langle \bar{s} s \rangle}(s) &=& \frac{\langle \bar{s} s \rangle}{2^6\pi^4} \int^{\alpha_{max}}_{\alpha_{min}} d \alpha \bigg{\{} \int^{1 - \alpha}_{\beta_{min}} d \beta \frac{{\cal F}_{\alpha \beta}}{\alpha^2\beta^2} \bigg{\{}- {\cal A}^2\bigg[ 2 \beta m_c {\cal F}_{\alpha \beta} -2 \alpha^2\beta m_c m_q m_s +\alpha \beta m_s {\cal F}_{\alpha \beta} \nonumber\\
& -&4\alpha\beta m_s m_c^2 \bigg]-{\cal B}^2\bigg[ 2 \alpha\beta m_c {\cal F}_{\alpha \beta} +2 \beta^2 {\cal F}_{\alpha \beta} +\alpha \beta m_s {\cal F}_{\alpha \beta} -4\alpha\beta m_q m_c^2 \bigg]\nonumber\\
&-&{\cal C}^2\bigg[ -2 \beta m_c {\cal F}_{\alpha \beta} +2 \alpha^2\beta m_c m_q m_s +\alpha \beta m_s {\cal F}_{\alpha \beta} -4\alpha\beta m_q m_c^2 \bigg]\nonumber\\
&-& {\cal D}^2\bigg[ - 2 \alpha\beta m_c {\cal F}_{\alpha \beta} -2 \beta^2 {\cal F}_{\alpha \beta}+\alpha \beta m_s {\cal F}_{\alpha \beta} -4\alpha\beta m_q m_c^2 \bigg]\nonumber\\
&+&{\cal AD}\bigg[ (2\alpha\beta m_s m_c-2\alpha {\cal F}_{\alpha \beta})(-\beta m_q +(\alpha+\beta-1)m_c) \bigg]\nonumber\\
&+&{\cal BC}\bigg[ (2\alpha\beta m_s m_c+2\alpha {\cal F}_{\alpha \beta})(\beta m_q +(\alpha+\beta-1)m_c) \bigg]\bigg{\}}\;, \\
&-&\frac{m_q {\cal H}_\alpha}{\alpha(\alpha-1)}\bigg[({\cal A}^2+{\cal B}^2+{\cal C}^2+{\cal D}^2){\cal H}_\alpha-2\alpha m_q m_c({\cal A}^2+{\cal B}^2-{\cal C}^2-{\cal D}^2)\bigg]\bigg{\}}\;,\\
\rho^{\langle G^2 \rangle}(s) &=& \frac{\langle g_s^2 G^2\rangle}{3^2\times2^{12}\pi^6} \int^{\alpha_{max}}_{\alpha_{min}} d \alpha \bigg{\{} \int^{1 - \alpha}_{\beta_{min}} \frac{d \beta}{\alpha^3\beta^3} \bigg{\{} 6{\cal A}^2\bigg[\alpha\beta {\cal F}_{\alpha\beta}^2 ((\alpha + \beta) (3\alpha + \beta) - 2\beta) \nonumber\\
&-& 2 {\cal F}_{\alpha\beta} m_c ( m_c (\alpha + \beta - 1) (\alpha + \beta) (\alpha + \beta + 1) (\alpha^2 - \alpha\beta + \beta^2) + \alpha^2 m_q (-6\alpha^2\beta \nonumber\\
&-& 3\alpha (\alpha^2 - 1)- 2 (\alpha + 1)\beta^2 + \beta^3) + 3\beta^2 m_s (\alpha^2 - 2\alpha\beta - 2 (\beta - 1)\beta )) + 2 m_c^2 (\alpha + \beta \nonumber\\
&-& 1) ( m_c (\alpha^3 + \beta^3) (\alpha m_q (\alpha + \beta + 1) + 2\beta m_s) - 6\alpha\beta m_q m_s (\alpha^2 + \beta^2))\bigg]\nonumber\\
&+&6{\cal B}^2\bigg[\alpha\beta {\cal F}_{\alpha\beta}^2 (4\alpha\beta + (\alpha - 2)\alpha + 3\beta^2) - 2 {\cal F} _ {\alpha\beta} m_c (3\alpha^2 m_q (-2\alpha\beta - 2 (\alpha - 1)\alpha \nonumber\\
&+& \beta^2) + m_c (\alpha + \beta - 1) (\alpha + \beta) (\alpha + \beta + 1) (\alpha^2 - \alpha\beta + \beta^2) + \beta^2 m_s (\alpha^3 \nonumber\\
&-& 2\alpha^2 (\beta + 1) - 6\alpha\beta^2 - 3\beta^3 + 3\beta )) + 2 m_c^2 (\alpha + \beta - 1)(2\alpha m_q ( m_c (\alpha^3 + \beta^3) \nonumber\\
&-& 3\beta m_s (\alpha^2 + \beta^2)) + \beta m_c m_s (\alpha +\beta) (\alpha + \beta + 1) (\alpha^2 - \alpha\beta + \beta^2)) \bigg]\nonumber\\
&+&6{\cal C}^2\bigg[\alpha\beta {\cal F} _ {\alpha\beta}^2 ((\alpha + \beta) (3\alpha + \beta) - 2\beta) - 2 {\cal F} _ {\alpha\beta} m_c (\alpha^2 m_q (6\alpha^2\beta + 3\alpha (\alpha^2 - 1) \nonumber\\
&+& 2 (\alpha + 1)\beta^2 - \beta^3) +m_c (\alpha + \beta - 1) (\alpha + \beta) (\alpha + \beta + 1) (\alpha^2 - \alpha\beta + \beta^2) \nonumber\\
&+& 3\beta^2 m_s (-\alpha^2 + 2\alpha\beta + 2 (\beta - 1)\beta )) - 2 m_c^2 (\alpha + \beta - 1) (\alpha m_q ( m_c (\alpha + \beta) (\alpha \nonumber\\
&+& \beta +1) (\alpha^2- \alpha\beta + \beta^2) + 6\beta m_s (\alpha^2 + \beta^2)) + 2\beta m_Q m_s (\alpha^3 + \beta^3))\bigg]\nonumber\\
&+&6{\cal D}^2\bigg[ \alpha\beta {\cal F}_{\alpha\beta}^2 (4\alpha\beta + (\alpha - 2)\alpha + 3\beta^2) - {\cal F}_{\alpha\beta} m_c (3\alpha^2 m_q (2\alpha\beta + 2 (\alpha - 1)\alpha - \beta^2)\nonumber\\
& +& m_c (\alpha + \beta - 1) (\alpha + \beta) (\alpha + \beta + 1) (\alpha^2 - \alpha\beta + \beta^2) \nonumber\\
&+& \beta^2 m_s (-\alpha^3 + 2\alpha^2 (\beta + 1) +6\alpha\beta^2 + 3\beta (\beta^2 - 1))) - m_c^2 (\alpha + \beta - 1) (2\alpha m_q ( m_c (\alpha^3 + \beta^3) \nonumber\\
&+& 3\beta m_s (\alpha^2 + \beta^2)) + \beta m_c m_s (\alpha + \beta) (\alpha + \beta + 1) (\alpha^2 - \alpha\beta + \beta^2)) \bigg] \nonumber\\
&+& {\cal AB}\alpha\beta\bigg[ {\cal F} _ {\alpha\beta} (\beta^3 m_c (4 m_c - 3 m_q) + 3\beta^2 (3 m_c - 2 m_q) (2\alpha m_c + m_c - \alpha m_s) \nonumber\\
&-& 3\beta ( m_q - 2 m_c) ((\alpha (3\alpha + 4) - 3) m_c - 2\alpha (\alpha + 1)m_s)+ (\alpha - 1) m_c ((\alpha (4\alpha + 13) - 5) m_q\nonumber\\
& -& 3\alpha (\alpha + 3) m_s))\bigg]+{\cal AD }\bigg[ m_c (\alpha + \beta - 1) ({\cal F} _ {\alpha\beta} ( m_c (\alpha + \beta - 1) (12 (\alpha - 1)\alpha^2 \nonumber\\
&-& (5\alpha + 12)\beta^2 - 5 (\alpha - 1)\alpha\beta + 12\beta^3) + 36 ((\alpha - 1)\alpha^3 m_s - (\beta - 1)\beta^3 m_q)) \nonumber\\
&+& 4 m_c (\alpha^3 + \beta^3) ( m_c^2 (\alpha + \beta - 1)^2 + 3 m_c (\alpha + \beta - 1) (\alpha m_s - \beta m_q) - 6\alpha\beta m_q m_s))\bigg]\nonumber\\
&+&{\cal BC }\bigg[m_c (\alpha + \beta - 1) ({\cal F} _ {\alpha\beta} ( m_c (\alpha + \beta - 1) (12 (\alpha - 1)\alpha^2 \nonumber\\
&-& (5\alpha + 12)\beta^2 - 5 (\alpha - 1)\alpha\beta + 12\beta^3) + 36 (-(\alpha - 1)\alpha^3 m_s + (\beta - 1)\beta^3 m_q)) \nonumber\\
&+& 4 m_c (\alpha^3 + \beta^3) ( m_c^2 (\alpha + \beta - 1)^2 - 3 m_c (\alpha + \beta - 1) (\alpha m_s - \beta m_q) - 6\alpha\beta m_q m_s))\bigg]\nonumber\\
&+& {\cal CD}\alpha\beta\bigg[ {\cal F} _ {\alpha\beta} (\beta^3 m_c (4 m_c + 3 m_q) + 3\beta^2 (3 m_c+ 2 m_q) (2\alpha m_c + m_c + \alpha m_s) \nonumber\\
&+& 3\beta ( m_q + 2 m_c) ((\alpha (3\alpha + 4) - 3) m_c + 2\alpha (\alpha + 1)m_s)+ (\alpha - 1) m_c ((\alpha (4\alpha + 13) - 5) m_c\nonumber\\
& +& 3\alpha (\alpha + 3) m_s))\bigg]\bigg{\}}\bigg{\}}\;,\\
\rho^{\langle \bar{q} G q \rangle}(s) &=& \frac{\langle g_s \bar{q} \sigma \cdot G q \rangle}{3\times2^8\pi^4} \int_{\alpha_{min}}^{\alpha_{max}} \bigg{\{} \int_{\beta_{min}}^{1 - \alpha} d \beta\frac{1}{\alpha^2\beta^2} \bigg{\{} 12{\cal A}^2 \alpha^2 m_c [(\alpha+2\beta){\cal F}_{\alpha\beta}-\beta m_c m_s]\nonumber\\
&-&12{\cal C}^2 \alpha^2 m_c [(\alpha+2\beta){\cal F}_{\alpha\beta}+\beta m_c m_s] + {\cal AB} \beta^2 {\cal F}_{\alpha\beta}[(2\alpha +\beta-1)m_c-\alpha m_s]\nonumber\\
&-&{\cal CD} \beta^2 {\cal F}_{\alpha\beta}[(2\alpha +\beta+1)m_c+\alpha m_s]+{\cal AD}\beta[6\alpha{\cal F}_{\alpha\beta}(m_c-\alpha m_c-\alpha m_s)\nonumber\\
&-&m_c({\cal F}_{\alpha\beta}(1+19\alpha)+4\alpha^2 m_q m_c)-\beta^2 m_c {\cal F}_{\alpha\beta}]+{\cal BC}\beta[-6\alpha{\cal F}_{\alpha\beta}(m_c-\alpha m_c+\alpha m_s)\nonumber\\
&+&m_c({\cal F}_{\alpha\beta}(1+19\alpha)-4\alpha^2 m_q m_c)+\beta^2 m_c {\cal F}_{\alpha\beta}]\bigg{\}}+\frac{4}{\alpha-1}\bigg{\{}{\cal A}^2[{\cal H}_\alpha(3m_c+2m_q(\alpha-1))\nonumber\\
&+&m_c(\alpha-1)(-m_c m_q+3m_c m_s+m_s m_q(\alpha-1))]+{\cal B}^2[{\cal H}_\alpha(3m_c+2m_q(\alpha-1))\nonumber\\
&+&m_c(\alpha-1)(-m_c m_q+3m_c m_s+2m_s m_q(\alpha-1))]+{\cal C}^2[{\cal H}_\alpha(-3m_c+2m_q(\alpha-1))\nonumber\\
&-&m_c(\alpha-1)(m_c m_q-3m_c m_s+2m_s m_q(\alpha-1))]+{\cal D}^2[{\cal H}_\alpha(-3m_c+2m_q(\alpha-1))\nonumber\\
&-&m_c(\alpha-1)(m_c m_q-3m_c m_s+m_s m_q(\alpha-1))]+{\cal AD}m_s(\alpha-1)(3{\cal H}_\alpha+\alpha m_c m_q)\nonumber\\
& +&{\cal BC}m_s(\alpha-1)(3{\cal H}_\alpha-\alpha m_c m_q)+2{\cal H}_\alpha[{\cal AB}(-m_c+ms)+{\cal CD}(m_c+m_s)]\bigg{\}}\bigg{\}}\;,\\
\rho^{\langle \bar{s} G s \rangle}(s) &=& \frac{\langle g_s \bar{s} \sigma \cdot G s \rangle}{3\times2^8\pi^4} \int_{\alpha_{min}}^{\alpha_{max}} \bigg{\{} \int_{\beta_{min}}^{1 - \alpha} d \beta\frac{1}{\alpha^2\beta^2} \bigg{\{} 12{\cal B}^2 \beta^2 m_c [(2\alpha+\beta){\cal F}_{\alpha\beta}-\alpha m_c m_s]\nonumber\\
&-&12{\cal D}^2 \beta^2 m_c [(2\alpha+\beta){\cal F}_{\alpha\beta}+\alpha m_c m_s] + {\cal AB} \alpha^2 {\cal F}_{\alpha\beta}[(\alpha +2\beta+1)m_c-\beta m_s]\nonumber\\
&-&{\cal CD} \alpha^2 {\cal F}_{\alpha\beta}[(\alpha +2\beta+1)m_c+\alpha m_s]+{\cal AD}\alpha[-6\beta^2{\cal F}_{\alpha\beta}m_q -4 \alpha\beta^2 m_c^2 m_s\nonumber\\
&-&m_c{\cal F}_{\alpha\beta}(\alpha+\alpha^2+19\alpha\beta+6\beta^2-6\beta)]-{\cal BC}\alpha[6\beta^2{\cal F}_{\alpha\beta}m_q +4 \alpha\beta^2 m_c^2 m_s\nonumber\\
&+&m_c{\cal F}_{\alpha\beta}(\alpha+\alpha^2+19\alpha\beta+6\beta^2-6\beta)]\bigg{\}}+\frac{4}{\alpha}\bigg{\{}{\cal A}^2[{\cal H}_\alpha(-3m_c+2m_q\alpha)\nonumber\\
&+&m_c\alpha(-m_c m_s+3m_c m_q-m_s m_q\alpha)]+{\cal B}^2[{\cal H}_\alpha(-3m_c+2m_q\alpha)\nonumber\\
&+&m_c\alpha(-m_c m_s+3m_c m_q-2m_s m_q\alpha)]+{\cal C}^2[{\cal H}_\alpha(3m_c+2m_q\alpha)\nonumber\\
&+&m_c\alpha(-m_c m_s+3m_c m_s+m_s m_q\alpha)]+{\cal D}^2[{\cal H}_\alpha(3m_c+2m_q\alpha)\nonumber\\
&+&m_c\alpha(-m_c m_s+3m_c m_s+2m_s m_q\alpha)]+{\cal AD}m_q\alpha(3{\cal H}_\alpha+(\alpha-1) m_c m_q)\nonumber\\
& +&{\cal BC}m_q\alpha(3{\cal H}_\alpha-(\alpha-1) m_c m_s)+2{\cal H}_\alpha[{\cal AB}(-m_c+ms)+{\cal CD}(m_c+m_s)]\bigg{\}}\bigg{\}}\;,\\
\rho^{\langle \bar{q} q \rangle \langle \bar{s} s \rangle}(s) &=& \frac{\langle \bar{q} q \rangle \langle \bar{s} s \rangle}{3\times2^4\pi^2} \int_{\alpha_{min}}^{\alpha_{max}} d \alpha \bigg{\{}{\cal A}^2[2m_c(2m_c-\alpha m_s)+(\alpha-1)m_q(4m_c-3\alpha m_s)]\nonumber\\
&+&{\cal B}^2[4m_c(m_c-\alpha m_s)+(\alpha-1)m_q(2m_c-3\alpha m_s)]\nonumber\\
&+&{\cal C}^2[2m_c(2m_c+\alpha m_s)-(\alpha-1)m_q(4m_c+3\alpha m_s)]\nonumber\\
&+&{\cal D}^2[4m_c(m_c+\alpha m_s)-(\alpha-1)m_q(2m_c+3\alpha m_s)]\nonumber\\
&+&{\cal AD}[4{\cal H}_\alpha +2 m_c m_s(\alpha-1)+2m_c m_q \alpha]\nonumber\\
&+&{\cal BC}[4{\cal H}_\alpha - 2m_c m_s(\alpha-1)-2m_c m_q \alpha]\bigg{\}}\;,\\
\rho^{\langle G^3 \rangle}(s) &=& \frac{\langle g_s^3 G^3 \rangle}{3\times2^{12}\pi^6} \int_{\alpha_{min}}^{\alpha_{max}} d\alpha \int_{\beta_{min}}^{1 - \alpha} \frac{d \beta}{\beta^3} (\alpha+\beta-1)\bigg{\{} {\cal A}^2\bigg[2{\cal F}_{\alpha\beta}(\alpha+\beta+1)\nonumber\\
&+&m_c[4\alpha m_c(\alpha+\beta+1)-2m_s(6\alpha+\beta)-m_q(6\alpha^2+6\alpha+7\alpha\beta+\beta+\beta^2)]\bigg]\nonumber\\
&+&{\cal B}^2\bigg[2{\cal F}_{\alpha\beta}(\alpha+\beta+1)+m_c[4\alpha m_c(\alpha+\beta+1)-2m_q(6\alpha+\beta)\nonumber\\
&-&m_s(6\alpha^2+6\alpha+7\alpha\beta+\beta+\beta^2)]\bigg]+{\cal C}^2\bigg[2{\cal F}_{\alpha\beta}(\alpha+\beta+1)\nonumber\\
&+&m_c[4\alpha m_c(\alpha+\beta+1)+2m_s(6\alpha+\beta)+m_q(6\alpha^2+6\alpha+7\alpha\beta+\beta+\beta^2)]\bigg]\nonumber\\
&+&{\cal D}^2\bigg[2{\cal F}_{\alpha\beta}(\alpha+\beta+1)+m_c[4\alpha m_c(\alpha+\beta+1)+2m_q(6\alpha+\beta)\nonumber\\
&+&m_s(6\alpha^2+6\alpha+7\alpha\beta+\beta+\beta^2)]\bigg] +\bigg[ 4({\cal AD}+{\cal BC})[\alpha\beta m_q m_s-m_c^2(\alpha+\beta-1)^2]\nonumber\\
&+&({\cal AD}-{\cal BC})m_c(m_q-m_s)(6\alpha^2-6\alpha+7\alpha\beta+\beta^2-\beta) \bigg]\bigg{\}}\;,\\
\rho^{\langle \bar{q} q \rangle \langle \bar{s} G s \rangle}(s) &=& \frac{\langle \bar{q} q \rangle \langle \bar{s} G s \rangle}{3\times2^6\pi^2} \int_{\alpha_{min}}^{\alpha_{max}} d \alpha \bigg{\{}\alpha({\cal AB}+{\cal CD})-2({\cal AD}+{\cal BC})(6\alpha^2-7\alpha-1)\bigg{\}}\;,\\
\rho^{\langle \bar{s} s \rangle \langle \bar{q} G q \rangle}(s) &=& \frac{\langle \bar{s} s \rangle \langle \bar{q} G q \rangle}{3\times2^6\pi^2} \int_{\alpha_{min}}^{\alpha_{max}} d \alpha \bigg{\{}(1-\alpha)({\cal AB}+{\cal CD})-2({\cal AD}+{\cal BC})(6\alpha^2-5\alpha)\bigg{\}}\;,\\
\Pi^{\langle G^2 \rangle} (M_B^2) &=& \frac{m_c^4 m_q m_s\langle g_s^2 G^2 \rangle}{3\times 2^{9}\pi^6} \int_{0}^{1} \frac{d \alpha}{\alpha^3} \int_{0}^{1 - \alpha} \frac{d \beta}{\beta^3} \; e^{- \frac{m_c^2 (\alpha + \beta)}{M_B^2 \alpha \beta}}({\cal A}^2+{\cal B}^2+{\cal C}^2+{\cal D}^2)\nonumber\\
&\times&[\alpha^4+\alpha^3(\beta-1)+\alpha\beta^3+\beta^3(\beta-1)]\;,\\
\Pi^{\langle \bar{q} G q \rangle}(M_B^2) &=& \frac{m_c^3 m_q m_s\langle g_s \bar{q} \sigma \cdot G q \rangle}{3\times2^6\pi^4} \int_{0}^{1} \bigg{\{} -\frac{e^{ \frac{m_c^2}{M_B^2 \alpha(\alpha-1)}}}{\alpha}({\cal A}^2+{\cal B}^2-{\cal C}^2-{\cal D}^2) \bigg{\}}\;,\\
\Pi^{\langle \bar{s} G s \rangle}(M_B^2) &=& \frac{m_c^3 m_q m_s\langle g_s \bar{s} \sigma \cdot G s \rangle}{3\times2^6\pi^4} \int_{0}^{1} \bigg{\{} \frac{e^{ \frac{m_c^2}{M_B^2 \alpha(\alpha-1)}}}{\alpha-1}({\cal A}^2+{\cal B}^2-{\cal C}^2-{\cal D}^2) \bigg{\}}\;,\\
\Pi^{\langle \bar{q} q\rangle\langle\bar{s} s \rangle}(M_B^2) &=& \frac{m_c^2 \langle \bar{q} q\rangle\langle\bar{s} s \rangle}{3\times2^4\pi^2} \int_{0}^{1}\frac{e^{ \frac{m_c^2}{M_B^2 \alpha(\alpha-1)}}}{\alpha(\alpha-1)} \bigg{\{} ({\cal A}^2+{\cal B}^2)[2\alpha m_c m_s +m_q (\alpha-1)(-2m_c + 3\alpha m_s)]\nonumber\\
&+&({\cal C}^2+{\cal D}^2)[-2\alpha m_c m_s +m_q (\alpha-1)(2m_c + 3\alpha m_s)]\nonumber\\
&+&{\cal AD+BC}m_q m_s \alpha(\alpha-1) \bigg{\}}\;,\\
\Pi^{\langle G^3 \rangle} (M_B^2) &=& \frac{m_c^2\langle g_s^3 G^3 \rangle}{3^2\times 2^{11}\pi^6} \int_{0}^{1} d \alpha \int_{0}^{1 - \alpha} \frac{d \beta}{M_B^2\beta^4} \; e^{- \frac{m_c^2 (\alpha + \beta)}{M_B^2 \alpha \beta}}(\alpha+\beta-1)\bigg{\{}{\cal A}^2[6(\alpha+\beta)m_c m_s M_B^2\nonumber\\
&+&3m_q(4m_c^2 m_s-12\beta m_s M_B^2+m_c M_B^2(\alpha+\beta+1)(\alpha+\beta))]+{\cal B}^2[6(\alpha+\beta)\nonumber\\
&\times&m_c m_q M_B^2+3m_s(4m_c^2 m_q-12\beta m_q M_B^2+m_c M_B^2(\alpha+\beta+1)(\alpha+\beta))]\nonumber\\
&-&{\cal C}^2[6(\alpha+\beta)m_c m_s M_B^2+3m_q(-4m_c^2 m_s+12\beta m_s M_B^2+m_c M_B^2(\alpha+\beta\nonumber\\
&+&1)(\alpha+\beta))] -{\cal D}^2[6(\alpha+\beta)m_c m_q M_B^2+3m_s(-4m_c^2 m_q+12\beta m_q M_B^2\nonumber\\
&+&m_c M_B^2(\alpha+\beta+1)(\alpha+\beta))] +({\cal AD+BC})M_B^2[-12\alpha\beta m_q m_s +m_c^2(\alpha+\beta-1)^2]\nonumber\\
&+&3({\cal AD-BC})M_B^2 m_c(m_q-m_s)(\alpha+\beta)(\alpha+\beta-1)\bigg{\}}\;,\\
\Pi^{\langle \bar{q} q \rangle \langle \bar{s} G s \rangle} (M_B^2) &=& \frac{m_c\langle \bar{q} q \rangle \langle \bar{s} G s \rangle}{3^2\times 2^7\pi^2} \int_{0}^1 \frac{d \alpha}{\alpha^2(\alpha-1)^3M_B^6} \; e^{- \frac{m_c^2}{M_B^2 (1 - \alpha) \alpha}}\bigg{\{}-4{\cal A}^2(\alpha-1)[m_c m_s\nonumber\\
&+&3(\alpha-1)M_B^2] \bigg[ 2\alpha m_c M_B^2 (-m_c^2+\alpha (\alpha-1)M_B^2) +m_q(m_c^4 -2\alpha(\alpha-1) m_c^2 M_B^2\nonumber\\
&+&2\alpha^2(\alpha-1)^2 M_B^4) \bigg]+4{\cal B}^2(\alpha-1)(-m_c m_q+2\alpha M_B^2)\bigg[m_c^4 m_s+m_c^2 M_B^2 (\alpha\nonumber\\
&-&1)(3m_c-2\alpha m_s) +(\alpha-1)^2 M_B^2 (-3 m_c (\alpha-1) +2 \alpha^2 m_s) \bigg] \nonumber\\
&+&4{\cal C}^2(\alpha-1)[-m_c m_s +3(\alpha-1)M_B^2] \bigg[ 2\alpha m_c M_B^2 (m_c^2-\alpha (\alpha-1)M_B^2) \nonumber\\
&+&m_q(m_c^4 -2\alpha(\alpha-1) m_c^2 M_B^2+2\alpha^2(\alpha-1)^2 M_B^4) \bigg] \nonumber\\
&-&4{\cal D}^2(\alpha-1)(m_c m_q+2\alpha M_B^2)\bigg[m_c^4 m_s-m_c^2 M_B^2 (\alpha\nonumber\\
&-&1)(3m_c+2\alpha m_s) +(\alpha-1)^2 M_B^2 (3 m_c (\alpha-1) +2 \alpha^2 m_s) \bigg] \nonumber\\
&+&{\cal AB}\alpha^2(\alpha-1)M_B^4[-2m_q m_c^2+M_B^2(\alpha-1)(-4m_c+\alpha m_q)]\nonumber\\
&+&{\cal CD}\alpha^2(\alpha-1)M_B^4[2m_q m_c^2-M_B^2(\alpha-1)(4m_c+\alpha m_q)]\nonumber\\
&+&{\cal AD} \alpha (\alpha-1)M_B^2\bigg[ 4(\alpha-1)m_c^3 m_q m_s +2M_B^2 m_c( 4(\alpha-1)^2 m_c m_s \nonumber\\
&-&2\alpha(\alpha-1)^2m_q m_s +\alpha(6\alpha-7)m_q m_c) -\alpha(\alpha-1)M_B^4((\alpha (24\alpha - 31)\nonumber\\
&+& 6) m_q + 8 (\alpha - 1) (2 (\alpha - 1) m_s - 3 m_c)) \bigg]\nonumber\\
&+&{\cal BC} \alpha (\alpha-1)M_B^2\bigg[ 4(\alpha-1)m_c^3 m_q m_s -2M_B^2 m_c( 4(\alpha-1)^2 m_c m_s \nonumber\\
&+&2\alpha(\alpha-1)^2m_q m_s +\alpha(6\alpha-7)m_q m_c) +\alpha(\alpha-1)M_B^4((\alpha (24\alpha - 31)\nonumber\\
&+& 6) m_q + 8 (\alpha - 1) (2 (\alpha - 1) m_s + 3 m_c)) \bigg] \bigg{\}}\;,\\
\Pi^{\langle \bar{s} s \rangle \langle \bar{q} G q \rangle} (M_B^2) &=& \frac{m_c\langle \bar{s} s \rangle \langle \bar{q} G q \rangle}{3^2\times 2^7\pi^2} \int_{0}^1 \frac{d \alpha}{\alpha^2(\alpha-1)^3M_B^6} \; e^{- \frac{m_c^2}{M_B^2 (1 - \alpha) \alpha}}\bigg{\{}-4{\cal A}^2(\alpha-1)[m_c m_s\nonumber\\
&+&2(\alpha-1)M_B^2] \bigg[ 3\alpha m_c M_B^2 (-m_c^2+\alpha^2 M_B^2) +m_q(m_c^4 -2\alpha(\alpha-1) m_c^2 M_B^2\nonumber\\
&+&2\alpha^2(\alpha-1)^2 M_B^4) \bigg]+4{\cal B}^2(\alpha-1)(-m_c m_q+3\alpha M_B^2)\bigg[m_c^4 m_s+m_c^2 M_B^2 (\alpha\nonumber\\
&-&1)(m_c-\alpha m_s) +2\alpha(\alpha-1)^2 M_B^2 (- m_c + \alpha m_s) \bigg] \nonumber\\
&+&4{\cal C}^2(\alpha-1)[-m_c m_s +2(\alpha-1)M_B^2] \bigg[ 3\alpha m_c M_B^2 (m_c^2-\alpha^2 M_B^2) \nonumber\\
&+&m_q(m_c^4 -2\alpha(\alpha-1) m_c^2 M_B^2+2\alpha^2(\alpha-1)^2 M_B^4) \bigg] \nonumber\\
&-&4{\cal D}^2(\alpha-1)(m_c m_q+3\alpha M_B^2)\bigg[m_c^4 m_s-2m_c^2 M_B^2 (\alpha\nonumber\\
&-&1)3m_c+\alpha m_s) +2\alpha(\alpha-1)^2 M_B^2 ( m_c \alpha + \alpha m_s) \bigg] \nonumber\\
&+&{\cal AB}(\alpha-1)^3 M_B^4[-2m_q m_c^2+M_B^2\alpha(4m_c+(\alpha-1) m_s)]\nonumber\\
&+&{\cal CD}(\alpha-1)^3 M_B^4[2m_q m_c^2+M_B^2\alpha(4m_c+-(\alpha-1) m_s)]\nonumber\\
&+&{\cal AD} (\alpha-1)^2M_B^2\bigg[ 4\alpha m_c^3 m_q m_s +2M_B^2 m_c( 4\alpha^2 m_c m_s \nonumber\\
&-&2\alpha^2(\alpha-1) m_q m_s +(\alpha-1)(6\alpha+1)m_s m_c) -\alpha(\alpha-1)M_B^4((\alpha (24\alpha - 17)\nonumber\\
&-& 1) m_s + 8 \alpha (2 \alpha m_s - 3 m_c)) \bigg]\nonumber\\
&+&{\cal BC} (\alpha-1)^2M_B^2\bigg[ 4\alpha m_c^3 m_q m_s -2M_B^2 m_c( 4\alpha^2 m_c m_s \nonumber\\
&+&2\alpha^2(\alpha-1) m_q m_s +(\alpha-1)(6\alpha+1)m_s m_c) +\alpha(\alpha-1)M_B^4((\alpha (24\alpha - 17)\nonumber\\
&-& 1) m_s + 8 \alpha (2 \alpha m_s + 3 m_c)) \bigg] \bigg{\}}\;,
\end{eqnarray}
\section{The decay spectral densities of $Z^+_{cs}$}
\subsection{The decay spectral densities of $Z^+_{cs}$ for molecular state}
To calculate the decay spectral densities of $Z^+_{cs}$ in molecular structure, we isolate the $g_{\mu\nu}$ structure of both side of
Eqs.(\ref{three-point-p}) and (\ref{three-point-o}).
On the OPE side of QCD sum rules for the current (\ref{current-mixing1}), the three-point function of $Z^+_{cs} \to \bar{D}^\ast D_s^+$ after Borel transformation may write:
\begin{eqnarray}
\Pi^{pert}(s_0,v_0,M_{1B}^2,M_{2B}^2) &=& \frac{9 {\cal A}}{8\pi^2} \int_{m_c^2}^{s_0} d s \int_{m_c^2}^{v_0} d v \int_0^\Lambda d \alpha \int_0^\Lambda d \beta\nonumber\\
&\times& e^{-\frac{s}{M_{1B}^2}-\frac{v}{M_{2B}^2}}[m_c^2-(1-\alpha) s][m_c^2-(1-\beta)v]\;,\\
\Pi^{\langle\bar{q}q\rangle}(s_0,v_0,M_{1B}^2,M_{2B}^2) &=&\frac{3 m_c \langle\bar{q}q\rangle {\cal A} e^{\frac{-m_c^2}{M_{2B}^2}} }{\pi^2} \int_{m_c^2}^{s_0} d s \int_0^\Lambda d \alpha \; e^{-\frac{s}{M_{1B}^2}}[m_c^2-(1-\alpha) s]\;,\\
\Pi^{\langle\bar{s}s\rangle}(s_0,v_0,M_{1B}^2,M_{2B}^2) &=&\frac{3 m_c \langle\bar{s}s\rangle {\cal A} e^{\frac{-m_c^2}{M_{1B}^2}} }{2 \pi^2} \int_{m_c^2}^{v_0} d v \int_0^\Lambda d \beta \; e^{-\frac{v}{M_{2B}^2}} [m_c^2-(1-\beta) v]\;,\\
\Pi^{\langle G^2 \rangle}(s_0,v_0,M_{1B}^2,M_{2B}^2) &=&-\frac{m_c^2 \langle G^2 \rangle {\cal A}}{64\pi^4}\bigg{\{} \int_{m_c^2}^{s_0} d s \int_0^\Lambda d \alpha \int_0^1 d \beta\ \;e^{-\frac{s}{M_{1B}^2}-\frac{m_c^2}{(1-\beta)M_{2B}^2}}\nonumber\\
&\times&\frac{(m_c^2-(1-\alpha)s)}{{(1-\beta)^2M_{2B}^2}}+\int_{m_c^2}^{v_0} d v \int_0^1 d \alpha \int_0^\Lambda d \beta\ \;e^{-\frac{v}{M_{2B}^2}-\frac{m_c^2}{(1-\alpha)M_{1B}^2}}\nonumber\\
&\times&\frac{(m_c^2-(1-\beta)v)}{{(1-\alpha)^2M_{1B}^2}}\bigg{\}}\;,\\
\Pi^{\langle \bar{q} G q \rangle}(s_0,v_0,M_{1B}^2,M_{2B}^2)&=&\frac{-3m_c^3 \langle \bar{q} G q \rangle {\cal A}\;e^{\frac{-m_c^2}{M_{2B}^2}}}{4\pi^2 M_{2B}^4}\int_{m_c^2}^{s_0} d s \int_0^\Lambda d \alpha e^{-\frac{s}{M_{1B}^2}} [m_c^2-(1-\alpha) s]\;,\\
\Pi^{\langle \bar{s} G s \rangle}(s_0,v_0,M_{1B}^2,M_{2B}^2)&=&\frac{-3m_c^3 \langle \bar{q} G q \rangle {\cal A}\;e^{\frac{-m_c^2}{M_{1B}^2}}}{8\pi^2 M_{2B}^4}\int_{m_c^2}^{v_0} d v \int_0^\Lambda d \beta \; e^{-\frac{v}{M_{1B}^2}} [m_c^2-(1-\beta) v]\;,\\
\Pi^{\langle\bar{q}q\rangle\langle\bar{s}s\rangle}(s_0,v_0,M_{1B}^2,M_{2B}^2) &=&4 m_c^2 {\cal A} \langle\bar{q}q\rangle\langle\bar{s}s\rangle e^{-\frac{m_c^2}{M_{1B}^2}-\frac{m_c^2}{M_{1B}^2}}\;,
\end{eqnarray}
where $s_0$ and $v_0$ are the continuum threshold of $D_s$ and $\bar{D}^\ast$, and $M_{1B}^2$ and $M_{2B}^2$ are their Borel parameters, respectively, and $\Lambda=(1-m_c^2/s)$. Since $m_{D_s}\approx m_{\bar{D}^\ast}$, we can set $s_0=v_0$ and $M_{1B}^2=M_{2B}^2\equiv M_B^2$.
After Borel transformation on the Eq.(\ref{three-point-p}), the phenomenological side of three-point is then obtained as
\begin{eqnarray}\label{three-point-p-B}
\Pi^{phen}(s_0,M_B^2)&=& \frac{3\lambda_{Z^+_{cs}} m_{D^\ast} f_{D^\ast} f_{D_s} m_{D_s}^2 g_{Z^+_{cs} \bar{D}^\ast D_s} }{4(m_c+m_s)(m_{Z^+_{cs}}^2/4-m_{D^\ast}^2)} \; \nonumber\\
&\times&(e^{-m^2_{D^\ast}/M_B^2}-e^{-m^2_{Z^+_{cs}}/(4 M_B^2)})e^{-m^2_{D_s}/M_B^2}\; .
\end{eqnarray}
In our calculation, $\sqrt{s_0} = 2.1\; \rm{GeV}$ and $1.5\; \rm{GeV}^2 \le M_B^2 \le 2.5\; \rm{GeV}^2$ is the proper Borel window for decay process.
The OPE side of the three-point function of the decay process $Z^+_{cs} \to \bar{D} D_s^\ast$ is just the transformation of three-point function of the decay process $Z^+_{cs} \to \bar{D}^\ast D_s^+$ with $\langle\bar{q}q\rangle \leftrightarrow \langle\bar{s}s\rangle$, $\langle \bar{q} G q \rangle \leftrightarrow \langle \bar{s} G s \rangle$, and $\cal{A} \to \cal{B}$, and the phenomenological side of the three-point function of $Z^+_{cs} \to \bar{D} D_s^\ast$ is just the transformation of Eq.(\ref{three-point-p-B}) with $D_s^+ \to \bar{D}$ and $\bar{D}^\ast \to D_s^\ast$. In numerical analysis, $\sqrt{s_0} = 2.5\; \rm{GeV}$ and $1.7\; \rm{GeV}^2 \le M_B^2 \le 2.3\; \rm{GeV}^2$ is the proper Borel window for decay process.
The OPE side of the three-point function of the decay process $Z^+_{cs} \to J/\psi K^+$ reads as
\begin{eqnarray}
\Pi^{pert}(s_0,v_0,M_{1B}^2,M_{2B}^2) &=&\frac{9 \cal{C}}{16 \pi^4} \int_{4m_c^2}^{s_0} d s \int_{m_s^2}^{v_0} d v \int_{\alpha_{min}}^{\alpha_{max}} d \alpha\; e^{-\frac{s}{M_{1B}^2}-\frac{v}{M_{2B}^2}} (m_c^2-{\cal H}_\alpha) v \;,\\
\Pi^{\langle\bar{q}q\rangle}(s_0,v_0,M_{1B}^2,M_{2B}^2) &=&\frac{9 m_s \langle\bar{q}q\rangle {\cal C}}{2 \pi^2} \int_{4m_c^2}^{s_0} d s \int_{\alpha_{min}}^{\alpha_{max}} d \alpha\; e^{-\frac{s}{M_{1B}^2}}({\cal H}_\alpha - m_c^2)\;,\\
\Pi^{\langle\bar{s}s\rangle}(s_0,v_0,M_{1B}^2,M_{2B}^2) &=&\frac{9 m_s \langle\bar{s}s\rangle {\cal C}}{4 \pi^2} \int_{4m_c^2}^{s_0} d s \int_{\alpha_{min}}^{\alpha_{max}} d \alpha\; e^{-\frac{s}{M_{1B}^2}}( m_c^2-{\cal H}_\alpha)\;,\\
\Pi^{\langle G^2 \rangle}(s_0,v_0,M_{1B}^2,M_{2B}^2) &=&\frac{\langle G^2 \rangle {\cal C}}{128 \pi^4} \bigg{\{}\int_{m_s^2}^{v_0} d v \int_0^1 d \alpha \; e^{-\frac{m_c^2}{\alpha(1-\alpha)M_{1B}^2}-\frac{v}{M_{2B}^2}} \bigg( \frac{3 v (m_c^2-\alpha(\alpha-1)M_{1B}^2)}{128\alpha(\alpha-1)M_{1B}^2}\nonumber\\
&+&\frac{3v m_c^2[m_c^2(1-3\alpha+3\alpha^2)+2\alpha M_{1B}^2 (2\alpha^3-4\alpha^2+3\alpha-1)]}{128\alpha^3(\alpha-1)^3M_{1B}^4} \bigg)\nonumber\\
&+& \int_{4m_c^2}^{s_0} d s \int_{\alpha_{min}}^{\alpha_{max}} d \alpha 18 e^{\frac{s}{M_{1B}^2}}(m_c^2-{\cal H}_\alpha) \bigg{\}}\;,\\
\Pi^{\langle G^3 \rangle}(s_0,v_0,M_{1B}^2,M_{2B}^2) &=&\frac{\langle G^3 \rangle{\cal C}}{512\pi^4}\int_{m_s^2}^{v_0} d v \int_0^1 d \alpha \frac{v m_c^2\; e^{-\frac{m_c^2}{\alpha(1-\alpha)M_{1B}^2}-\frac{v}{M_{2B}^2}}}{\alpha^4(\alpha-1)^4M_{1B}^6} [6m_c^2(2 \alpha ^4\nonumber\\
&-&4 \alpha ^3+6 \alpha ^2-4 \alpha +1) + \alpha M_{1B}^2 (4 \alpha ^5-12 \alpha ^4\nonumber\\
&+&59 \alpha ^3-98 \alpha ^2+62 \alpha -15) ]\;,
\end{eqnarray}
where $s_0$ and $v_0$ are the continuum threshold of $J/\psi$ and $K^+$, and $M_{1B}^2$ and $M_{2B}^2$ are their Borel parameters, respectively. After employing Borel transformation to Eq.(\ref{three-point-p}) with $\bar{D}^\ast \to J/\psi$ and $D_s \to K^+$, the phenomenological side of three-point function of $Z^+_{cs} \to J/\psi K^+$ will be obtained.
\begin{eqnarray}
\Pi^{phen}(s_0,v_0,M_{1B}^2,M_{2B}^2)&=&\frac{3 \lambda_{Z_{cs}^+}m_{J/\psi}f_{J/\psi}f_{K^+}m_{K^+}^2g_{Z_{cs}^+J/\psi K^+}}{m_s(m_{Z_{cs}^+}^2-m_{J/\psi}^2)}\nonumber\\
&\times&(e^{-m_{Z_{cs}}^2/M_{1B}^2}-e^{-m_{J/\psi}^2/M_{1B}^2})e^{-m_{K^+}^2/M_{2B}^2}\;.\label{three-point-p-BJ}
\end{eqnarray}
In numerical analysis, $\sqrt{s_0} = 3.2\; \rm{GeV}$, $\sqrt{v_0} = 0.6\; \rm{GeV}$, $2.0\; \rm{GeV}^2 \le M_{1B}^2 \le 3.0\; \rm{GeV}^2$, and $2.0\; \rm{GeV}^2 \le M_{2B}^2 \le 3.0\; \rm{GeV}^2$ is the proper Borel window for decay process. Here we set $v_0=(m_K+m_{\pi})^2$.
The OPE side of the three-point function of the decay process $Z^+_{cs} \to\eta_c K^{+\ast}$ reads:
\begin{eqnarray}
\Pi^{pert}(s_0,v_0,M_{1B}^2,M_{2B}^2) &=&\frac{3 \cal{D}}{16 \pi^4} \int_{4m_c^2}^{s_0} d s \int_{m_s^2}^{v_0} d v \int_{\alpha_{min}}^{\alpha_{max}} d \alpha \;e^{-\frac{s}{M_{1B}^2}-\frac{v}{M_{2B}^2}} (2m_c^2-3{\cal H}_\alpha) v \;,\\
\Pi^{\langle\bar{q}q\rangle}(s_0,v_0,M_{1B}^2,M_{2B}^2) &=&\frac{3 m_s \langle\bar{q}q\rangle {\cal D}}{2 \pi^2} \int_{4m_c^2}^{s_0} d s \int_{\alpha_{min}}^{\alpha_{max}} d \alpha \;e^{-\frac{s}{M_{1B}^2}}(3{\cal H}_\alpha - 2m_c^2)\;,\\
\Pi^{\langle\bar{s}s\rangle}(s_0,v_0,M_{1B}^2,M_{2B}^2) &=&\frac{3 m_s \langle\bar{s}s\rangle {\cal D}}{4 \pi^2} \int_{4m_c^2}^{s_0} d s \int_{\alpha_{min}}^{\alpha_{max}} d \alpha\; e^{-\frac{s}{M_{1B}^2}}( 2m_c^2-3{\cal H}_\alpha)\;,\\
\Pi^{\langle G^2 \rangle}(s_0,v_0,M_{1B}^2,M_{2B}^2) &=&\frac{\langle G^2 \rangle {\cal D}}{128 \pi^4} \bigg{\{}\int_{m_s^2}^{v_0} d v \int_0^1 d \alpha \; e^{-\frac{m_c^2}{\alpha(1-\alpha)M_{1B}^2}-\frac{v}{M_{2B}^2}} \bigg( \frac{3 v (m_c^2-\alpha(\alpha-1)M_{1B}^2)}{128\alpha(\alpha-1)M_{1B}^2}\nonumber\\
&+&\frac{v m_c^2[m_c^2(1-6\alpha+6\alpha^2)+3\alpha M_{1B}^2 (2\alpha^3-4\alpha^2+3\alpha-1)]}{128\alpha^3(\alpha-1)^3M_{1B}^4} \bigg)\nonumber\\
&+& \int_{4m_c^2}^{s_0} d s \int_{\alpha_{min}}^{\alpha_{max}} d \alpha 6 e^{\frac{s}{M_{1B}^2}}(2m_c^2-3{\cal H}_\alpha) \bigg{\}}\;,\\
\Pi^{\langle G^3 \rangle}(s_0,v_0,M_{1B}^2,M_{2B}^2) &=&\frac{\langle G^3 \rangle{\cal D}}{512\pi^4}\int_{m_s^2}^{v_0} d v \int_0^1 d \alpha \frac{v m_c^2 \;e^{-\frac{m_c^2}{\alpha(1-\alpha)M_{1B}^2}-\frac{v}{M_{2B}^2}}}{\alpha^4(\alpha-1)^4M_{1B}^6} [4m_c^2(2 \alpha ^4\nonumber\\
&-&4 \alpha ^3+6 \alpha ^2-4 \alpha +1) + \alpha M_{1B}^2 (4 \alpha ^5-12 \alpha ^4\nonumber\\
&+&41 \alpha ^3-62 \alpha ^2+38 \alpha -9) ]\;,
\end{eqnarray}
where $s_0$ and $v_0$ are the continuum threshold of $\eta_c$ and $K^{+\ast}$, and $M_{1B}^2$ and $M_{2B}^2$ are their Borel parameters, respectively. Taking the transformation of Eq.(\ref{three-point-p-BJ}) with $J/\psi \to K^{+\ast}$ and $K^+ \to \eta_c$, the phenomenological side of decay process $Z^+_{cs} \to\eta_c K^{+\ast}$ will be obtained. In numerical analysis, $\sqrt{s_0} = 3.1\; \rm{GeV}$, $\sqrt{v_0} = 1.0\; \rm{GeV}$, $2.0\; \rm{GeV}^2 \le M_{1B}^2 \le 3.0\; \rm{GeV}^2$, and $2.0\; \rm{GeV}^2 \le M_{2B}^2 \le 3.0\; \rm{GeV}^2$ is the proper Borel window for decay process $Z^+_{cs} \to\eta_c K^{+\ast}$.
\subsection{The decay spectral densities of $Z^+_{cs}$ for tetraquark state}
The Ref. \cite{Dias:2013qga} discussed that only the color-connected diagrams which give the nontrivial color-structure should be considered for tetraquark states, which suggests that only the condensates of $ \langle \bar{q} G q \rangle$ and $\langle \bar{s} G s \rangle$ of OPE side need to be considered. For the color-connected diagrams, we need to isolate the $q_1^\mu q_2^\nu$ structure of both sides of Eqs.(\ref{three-point-p}) and (\ref{three-point-o}).
On the OPE side of QCD sum rules for the current Eq.(\ref{current-mixing2}), the three-point function of $Z^+_{cs} \to \bar{D}^\ast D_s^+$ after Borel transformation is
\begin{eqnarray}
\Pi^{\langle \bar{q} G q \rangle+\langle \bar{s} G s \rangle}(M_B^2)&=&\frac{m_c (\langle \bar{q} G q \rangle+\langle \bar{s} G s \rangle)}{64\pi^2}({\cal A-B+C-D}) \int_0^1\; d \alpha\; e^\frac{(2-\alpha)m_c^2}{(\alpha-1)M_B^2}\; \frac{1+\alpha}{1-\alpha}\nonumber\\
&+&\frac{m_c \langle \bar{s} G s \rangle}{32\pi^2}({\cal A-B}) \int_0^1\; d \alpha\; e^\frac{(2-\alpha)m_c^2}{(\alpha-1)M_B^2}\; \frac{1-3\alpha}{1-\alpha}
\end{eqnarray}
After Borel transformation on the Eq.(\ref{three-point-p}), the phenomenological side of three-point reads:
\begin{eqnarray}\label{three-point-p-C}
\Pi^{phen}(M_B^2)&=& \frac{3\lambda_{Z^+_{cs}} m_{D^\ast} f_{D^\ast} f_{D_s} m_{D_s}^2 g_{Z^+_{cs} \bar{D}^\ast D_s} }{4m_{Z^+_{cs}}^2(m_c+m_s)(m_{Z^+_{cs}}^2/4-m_{D^\ast}^2)} \; \\ \nonumber
&\times&(e^{-m^2_{D^\ast}/M_B^2}-e^{-m^2_{Z^+_{cs}}/(4 M_B^2)})e^{-m^2_{D_s}/M_B^2}\; .
\end{eqnarray}
In our calculation $2\; \rm{GeV}^2 \le M_B^2 \le 3\; \rm{GeV}^2$ is the proper Borel window for decay process.
The OPE side of the three-point function of the decay process $Z^+_{cs} \to \bar{D} D_s^\ast$ is just the transformation of three-point function of the decay process $Z^+_{cs} \to \bar{D}^\ast D_s^+$ with $\langle \bar{q} G q \rangle \leftrightarrow \langle \bar{s} G s \rangle$, and the phenomenological side of the three-point function of $Z^+_{cs} \to \bar{D} D_s^\ast$ is just the transformation of Eq.(\ref{three-point-p-C})
with $D_s^+ \to \bar{D}$ and $\bar{D}^\ast \to D_s^\ast$. In numerical analysis $2\; \rm{GeV}^2 \le M_B^2 \le 3\; \rm{GeV}^2$ is the proper Borel window for decay process.
The OPE side of the three-point function of the decay process $Z^+_{cs} \to J/\psi K^+$ is
\begin{eqnarray}
\Pi^{\langle \bar{q} G q \rangle+\langle \bar{s} G s \rangle}(M_B^2)&=&\frac{m_c (\langle \bar{q} G q \rangle+\langle \bar{s} G s \rangle)}{32\pi^2}({\cal A-B+C-D}) \int_0^1\; d \alpha\; e^\frac{m_c^2}{\alpha(\alpha-1)M_B^2}\; \frac{1}{\alpha}\;.
\end{eqnarray}
After employing Borel transformation to Eq.(\ref{three-point-p}) with $\bar{D}^\ast \to J/\psi$ and $D_s \to K^+$, the phenomenological side of three-point function of $Z^+_{cs} \to J/\psi K^+$ will be obtained.
\begin{eqnarray}\label{three-point-p-CJ}
\Pi^{phen}(M_{B}^2)&=&\frac{3 \lambda_{Z_{cs}^+}m_{J/\psi}f_{J/\psi}f_{K^+}m_{K^+}^2g_{Z_{cs}^+J/\psi K^+}}{m_s m_{Z_{cs}^+}^2 (m_{Z_{cs}^+}^2-m_{J/\psi}^2)}\nonumber\\
&\times&(e^{-m_{Z_{cs}}^2/M_{1B}^2}-e^{-m_{J/\psi}^2/M_{1B}^2})\;.
\end{eqnarray}
While the $m_{K^+}$ is very small, the contribution of the $e^{-m_{K^+}^2/M_{B}^2}$ is very close to $1$ which is neglecting in the
Eq.(\ref{three-point-p-CJ}). In our calculation $2\; \rm{GeV}^2 \le M_B^2 \le 3\; \rm{GeV}^2$ is the proper Borel window for decay process.
The OPE side of the three-point function of the decay process $Z^+_{cs} \to\eta_c K^{+\ast}$ writes as
\begin{eqnarray}
\Pi^{\langle \bar{q} G q \rangle+\langle \bar{s} G s \rangle}(M_B^2)&=&\frac{m_c (\langle \bar{q} G q \rangle+\langle \bar{s} G s \rangle)}{32\pi^2}({\cal A-B-C+D}) \int_0^1\; d \alpha\; e^\frac{m_c^2}{\alpha(\alpha-1)M_B^2}\; \frac{1}{1-\alpha}\;.
\end{eqnarray}
Taking the transformation of Eq.(\ref{three-point-p-CJ}) with $J/\psi \to K^{+\ast}$ and $K^+ \to \eta_c$, the phenomenological side of decay process $Z^+_{cs} \to\eta_c K^{+\ast}$ will be obtained. In our calculation $2\; \rm{GeV}^2 \le M_B^2 \le 3\; \rm{GeV}^2$ is the proper Borel window for decay process.
\end{widetext}
\end{document}
|
1,116,691,500,817 | arxiv | \section{Introduction}
The fractional Laplace operator $(-\triangle)^{\frac{\alpha}{2}}$ for $\alpha\in (0,2)$
in $\mathbb{R}^n $ is defined on the Schwartz class through the Fourier transform as
$$\widehat{((-\triangle)^{\frac{\alpha}{2}} f)} (\xi)=(2\pi |\xi|)^{\alpha}\widehat{f}(\xi),$$
where $\widehat{f}(\xi)=\int_{\mathbb{R}^n}e^{-2\pi ix\cdot \xi } f(x) dx$ is the Fourier transform of $f,$
or via the Riesz potential as
$$
(-\triangle)^{\frac{\alpha}{2}} f(x)=\frac{\alpha2^{\alpha}\Gamma\left({(n+\alpha)}/{2}\right)}{2\Gamma(1-\alpha/2)\pi^{{n}/{2}}}\hbox{P.V.}\int_{\mathbb{R}^n}\frac{f(x)-f(y)}{|x-y|^{n+\alpha}}dy.$$
Here $\Gamma(\cdot) $ is the usual Gamma function and $\hbox{P.V.}$ denotes the Cauchy principal value.
The fractional Laplacian has been widely applied in probability, finance, physical systems, and engineering problems.
The fractional Laplacian is a nonlocal operator because the value of $(-\triangle)^{\frac{\alpha}{2}} f$ at $x$ depends on the value of $f$ at infinity. This nonlocal property may cause some issues.
Caffarelli and Silvestre in \cite{Caffarelli} localized the nonlocal operator $(-\triangle)^{\frac{\alpha}{2}} $ by adding another variable. They provided the characterization for the
fractional Laplacian $(-\triangle)^{\frac{\alpha}{2}}$ by solving the harmonic extension problem to the upper half-space as the weighted operator that maps the Dirichlet boundary condition to the Neumann condition.
Let $f$ be a regular function in $\mathbb{R}^n.$ We say that $u(x,t)=P_{\alpha} f(x,t)$ is the Caffarelli-Silvestre extension of $f$ to the upper half-space $\mathbb{R}^{n+1}_{+}:=\mathbb{R}^n\times (0,\infty),$ if $u$ is a solution to the problem
\begin{equation}\label{1}
\left\{
\begin{aligned}
\hbox{div}(t^{1-\alpha}\nabla u)&= 0,\quad \hbox{in}\quad \mathbb{R}^{n+1}_{+};\\
u &= f, \quad \hbox{on} \quad \mathbb{R}^n\times\{t=0\}.
\end{aligned}
\right.
\end{equation}
The Caffarelli-Silvestre extension is well defined for smooth functions through the Poisson kernel $$p^{\alpha}_t(x)=\frac{c(n,\alpha)t^\alpha}{(|x|^2+t^2)^{{(n+\alpha)}/{2}}}$$ as
$$P_{\alpha} f(x,t)=p^{\alpha}_t\ast f(x,t)
=c(n,\alpha)\int_{\mathbb{R}^n}\frac{f(y)t^{\alpha}}{(|x-y|^2+t^2)^{{(n+\alpha)}/{2}}}dy.$$
Here $f\ast g$ means the convolution of $f$ and $g,$ and $c(n,\alpha)=\frac{\Gamma({(n+\alpha)}/{2})}{\pi^{n/2}\Gamma({\alpha}/{2})}$ is the normalized constant such that $\int_{\mathbb{R}^n}p^{\alpha}_t(x)dx=1.$
Caffarelli and Silvestre \cite{Caffarelli} proved that
\begin{equation}\label{2}
(-\triangle)^{\frac{\alpha}{2}} f(x)=-c_{\alpha}\lim_{t\rightarrow{0^+}}t^{1-\alpha}\partial_t u(x,t),\ c_{\alpha}=\frac{\Gamma(\alpha/2)}{2^{1-\alpha}\Gamma(1-\alpha/2)}.
\end{equation}
This characterization has dramatically popularized the application of the fractional Laplacian.
The identity (\ref{2}) can be viewed as the consequence of the equality of the energy functionals
$
\int_{\mathbb{R}^n}|2\pi\xi|^{\alpha}|\widehat{f}(\xi)|^2d\xi =D(n,\alpha)\int_{\mathbb{R}^{n+1}_{+}}|\nabla P_{\alpha} f(x,t)|^2t^{1-\alpha}dxdt
$$
which is equivalent to
\begin{equation}\label{3}
\|f\|_{\dot{W}^{\alpha/2,2}(\mathbb{R}^n)} =\|P_{\alpha} f(x,t)\|_{\dot{\mathbb{W}}^{1,2}_{\alpha}(\mathbb{R}^{n+1}_{+})}
\end{equation}
up to a multiplication constant. Here, $\dot{\mathbb{W}}^{1,2}_{\alpha}(\mathbb{R}^{n+1}_{+})$ is the weighted Sobolev space defined as
$$\dot{\mathbb{W}}^{1,2}_{\alpha}(\mathbb{R}^{n+1}_{+})=\left\{u(x,t)\in W^{1,1}_{loc}(\mathbb{R}^{n+1}_{+}): \int_{\mathbb{R}^{n+1}_{+}}|\nabla u(x,t)|^2t^{1-\alpha}dxdt<\infty\right\}.$$
Let $C^{\infty}_{0}(\mathbb R^{n})$ stand for all infinitely smooth functions with compact support in $\mathbb R^{n}$. The homogeneous Sobolev space $\dot{W}^{\beta,p}(\mathbb{R}^{n})$ is the completion of $C_0^\infty(\mathbb{R}^n)$ with respect to the norm
\begin{equation*}
\|f\|_{\dot{W}^{\beta,p}(\mathbb{R}^{n})}=\left\{
\begin{aligned}
&\|(-\triangle)^{\beta/2}f\|_{L^p(\mathbb{R}^n)}, \quad p\in (1,n/\beta);\\
&\left(\int_{\mathbb{R}^n}\frac{\|\triangle^k_{h}f\|^p_{L^p(\mathbb{R}^n)}}{|h|^{n+p\beta}}dh\right)^{1/p}, \quad p=1 \text{ or } p=n/\beta,\ \beta\in (0,n),
\end{aligned}
\right.
\end{equation*}
where $k=1+[\beta], \beta=[\beta]+\{\beta\}$ with $[\beta]\in \mathbb{Z}_+,$ $\{\beta\}\in(0,1)$ and
\begin{equation*}
\triangle^k_{h}f(x)=
\left\{\begin{aligned}
&\triangle^1_h\triangle^{k-1}_h f(x),\quad k>1;\\
&f(x+h)-f(x), \quad k=1.
\end{aligned}\right.
\end{equation*}
Equality (\ref{3}) allows us to identify fractional (logarithmic) Sobolev inequalities as fractional (logarithmic) Sobolev trace inequalities, see \cite{Brandle, Nguyen}. It also provides us a way to view the fractional perimeters of a Borel set $E\subset\mathbb{R}^n$ as the norm of the Caffarelli-Silvestre extension of the $1_E$ (the characteristic function of $E$). See \cite{CaffarelliRoquejoffreSavin, Fusco} and the references therein for more details on the fractional perimeter.
Motivated by (\ref{3}), in this paper, we characterize the following two embedding relations via the Caffarelli-Silvestre extension:
{\bf Embedding I: } Given $\alpha\in (0,2)$ and a nonnegative Radon measure $\mu$ on $\mathbb{R}^{n+1}_{+}$,
\begin{equation}\label{5}
\|P_{\alpha} f(\cdot,\cdot)\|_{L^{q}(\mathbb{R}^{n+1}_+,\mu)}\lesssim \|f\|_{L^p(\mathbb R^n)}.
\end{equation}
For $0<p,q<\infty$ and a nonnegative Radon measure $\mu$ on $\mathbb{R}^{n+1}_+,$
$L^{q,p}(\mathbb{R}^{n+1}_+,\mu)$ and $L^q(\mathbb{R}^{n+1}_+,\mu)$ denote the Lorentz space and the Lebesgue space of all functions on $\mathbb{R}^{n+1}_+$, respectively, for which
$$\|g\|_{L^{q,p}(\mathbb{R}^{n+1}_+,\mu)}=\left\{\int_0^\infty\left(\mu\left(\{x\in \mathbb{R}^{n+1}_+:\ |g(x)|>s \}\right)\right)^{p/q}ds^p\right\}^{1/p}<\infty$$
and
$$\|g|_{L^q(\mathbb{R}^{n+1}_+,\mu)}=\left(\int_{\mathbb{R}^{n+1}_+}|g(x)|^qd\mu\right)^{1/q}<\infty,$$
respectively. Moreover, we denote by $L^{q,\infty}(\mathbb{R}^{n+1}_+,\mu)$ the set of all $\mu-$measurable functions $g$ on $\mathbb{R}^{n+1}_+$ with
$$\|g\|_{L^{q,\infty}(\mathbb{R}^{n+1}_+,\mu)}=\sup_{s>0}s\left(\mu\left(\left\{x\in \mathbb{R}^{n+1}_+: |g(x)|>s \right\}\right)\right)^{1/q}<\infty.$$
The embedding (\ref{5}) will be characterized by conditions in terms of capacities and Hedberg-Wolff potentials of $\mu.$
Firstly, we introduce the $L^p-$capacity associated with the Caffarelli-Silvestre extension.
\begin{definition}
Let $1\leq p<\infty$. For a subset $E$ of $\mathbb R^{n+1}_{+}$, let
$$C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E):=\inf\left\{\|f\|^{p}_{L^{p}(\mathbb R^{n})}:\ f\geq 0\ \&\ P_{\alpha}f(x,t)\geq 1\text{ for all }(x,t)\in E\right\}.$$
When the set $\Big\{f\in {L^{p}(\mathbb R^{n})}:\ f\geq 0\ \&\ P_{\alpha}f(x,t)\geq 1\text{ for all }(x,t)\in E\Big\}$ is empty, we set $C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E)=0$.
\end{definition}
Then, we establish the mixed norm estimate of $P_{\alpha}f(x,t),$ the dual form of $C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(\cdot),$ some basic properties of $C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(\cdot)$ and a capacitary strong type inequality.
{\bf Embedding II: } Given $\alpha\in (0,2)$ and a nonnegative Radon measure $\mu$ on $\mathbb{R}^{n+1}_{+}$,
\begin{equation}\label{4}
\|P_{\alpha} f(\cdot,\cdot)\|_{L^{q}(\mathbb{R}^{n+1}_+,\mu)}\lesssim \|f\|_{\dot{W}^{\beta,p}(\mathbb{R}^{n})}
\end{equation}
for $0<\beta<n, 1\leq p\leq {n}/{\beta}$ and $1<q<\infty.$
We will show that the embedding (\ref{4}) when $p=1$ can be characterized by conditions in terms of the fractional perimeter of open sets for the endpoint cases and fractional capacities for general cases. The fractional perimeter and the fractional capacity are defined, respectively, as follows.
\begin{definition}
Let $s\in(0,1)$. The fractional perimeter is defined as
$$Per_s(E)=\int_E\int_{\mathbb{R}^n\backslash{E}}\frac{1}{|x-y|^{n+s}}dxdy$$
for a given measurable set $E\subseteq \mathbb{R}^n.$
\end{definition}
It follows from the definition of $\|f\|_{\dot{W}^{s,1}(\mathbb{R}^n)}$ that
$Per_{s}(E)=\frac{1}{2}\|1_E\|_{\dot{W}^{s,1}(\mathbb{R}^n)}.$ On the other hand, there holds
$$Per_s(E)=\frac{\Gamma((n+s)/2)}{2\pi^{n/2}\Gamma(s/2)}\int_{\mathbb{R}^{n+1}_+}|\nabla u_E(x,t)|^{2}t^{1-s}dxdt,$$
where $ u_E(\cdot,\cdot)$ is the solution to equation (\ref{1}) with $f=1_E.$
For the fractional perimeter, Ambrosio-DePhilippis-Martinazzi in \cite{Ambrosio Philippis Martinazzi} proved the generalized coarea formula:
\begin{equation} \label{7}
\|f\|_{\dot{W}^{s,1}(\mathbb{R}^n)}=2\int_0^\infty Per_s\Big(\Big\{x:\ f(x)>t\Big\}\Big)dt
\end{equation}
for every nonnegative $f\in \dot{W}^{s,1}(\mathbb{R}^n).$
Denote by $T(O)$ the tent based on an open subset $O$ of $\mathbb{R}^n:$
$$T(O)=\left\{(x,r)\in \mathbb{R}^{n+1}_+:\ B(x,r)\subseteq O\right\}
$$
with $B(x,r)$ the open ball centered at $x\in\mathbb{R}^n$ with radius $r>0$.
\begin{definition}
Let $\beta \in (0,n)$ and $p\in[1,n/\beta].$
\item{(i)} The fractional capacity of an arbitrary set $S\subset \mathbb{R}^n,$
denoted by $Cap_{\mathbb{R}^n}^{\beta,p}(S),$ is defined as
$$Cap_{\mathbb{R}^n}^{\beta,p}(S):=\inf\left\{\|f\|^p_{\dot{W}^{\beta,p}(\mathbb{R}^n)}: \ f\in C_0^\infty(\mathbb{R}^{n}),\ f\geq 0\text{ on }\mathbb{R}^n\ \&\ f\geq 1_S\right\}.$$
\item{(ii)} For $t\in (0,\infty),$ the $(p,\beta)-$fractional capacity minimizing function associated with both $\dot{W}^{\beta,p}(\mathbb{R}^{n})$ and a nonnegative measure $\mu$ on $\mathbb{R}^{n+1}_+,$ denoted by $c^\beta_p(\mu,t),$ is set as
$$c^\beta_p(\mu,t):=\inf\left\{Cap_{\mathbb{R}^n}^{\beta,p}(O):\ \text{ bounded open } O\subseteq \mathbb{R}^n,\ \mu(T(O))>t\right\}.$$
\end{definition}
This article is mainly motivated by the work on embeddings like (\ref{5}) and (\ref{4}) via classical/fractional heat equations. Xiao in \cite{Xiao} studied the embeddings of the homogeneous Sobolev space $\dot{W}^{1,p}(\mathbb{R}^{n})$ into the Lebesgue space $L^{q} (\mathbb{R}^{n+1}_{+}
,\mu),$ under
$(p, q) \in (1,\infty) \times \mathbb{R}_{+},$ via the Gauss-Weierstrass heat
kernel. For fractional diffusion equations, motivated by Xiao \cite{Xiao}, Zhai in \cite{Zhai} explored the embeddings of the homogeneous Sobolev space $\dot{W}^{\beta,p}(\mathbb{R}^{n})$ into the Lebesgue space $L^{q} (\mathbb{R}^{n+1}_{+}
,\mu).$ By using the $L^p-$capacities associated with the fractional heat kernel, Chang-Xiao in \cite{Chang Xiao} and Shi-Xiao in \cite{Shi Xiao} established embeddings similar to (\ref{5}). For applications of Poisson like kernel in function spaces, see \cite{Bui,Lenzmann} and the references therein.
This article will be organized as follows. In Section 2.1, we investigate the dual form and basic properties of the $L^p-$capacity $C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(\cdot)$, and Section \ref{sec-2.2} is devoted to a capacitary strong type inequality corresponding to $C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(\cdot)$. Section \ref{sec-2.3} describes several technical lemmas on the fractional capacity $Cap_{\mathbb{R}^n}^{\beta,p}(\cdot)$.
In Section \ref{sec-3}, we deduce the embedding (\ref{5}) for two cases $p\leq q$ and $p>q,$ respectively. Section \ref{sec-4} studies the embedding (\ref{4}) for two cases $p\leq q$ and $p>q$.
{\it Some notations}: Let us agree to more conventions. ${\mathsf U}\approx{\mathsf V}$ represents that
there is a constant $c>0$ such that $c^{-1}{\mathsf V}\le{\mathsf
U}\le c{\mathsf V}$ whose right inequality is also written as
${\mathsf U}\lesssim{\mathsf V}$. Similarly, one writes ${\mathsf V}\gtrsim{\mathsf U}$ for
${\mathsf V}\ge c{\mathsf U}$.
\section{Preliminaries on capacities}\label{sec-2}
\subsection{$L^p-$ capacities associated with the Caffarelli-Silvestre extension}\label{sec-2.1}
In this section, $\alpha\in (0,2).$ We will first establish the dual form of the $L^p-$capacity associated with the Caffarelli-Silvestre extension. Then, we will prove some basic properties of the $L^p-$capacity and the capacitary strong type inequality for $C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(\cdot)$.
To establish the adjoint formulation of the foregoing definition, we need to find out the adjoint operator of $P_{\alpha}$. Note that
$$\int_{\mathbb R^{n+1}_{+}}P_{\alpha}f(x,t)g(x,t)dtdx=\int_{\mathbb R^{n}}f(x)\left(\int_{\mathbb R^{n+1}_{+}}p^{\alpha}_{t}(x-z)g(z,t)dzdt\right)dx$$
holds for all $f\in C^{\infty}_{0}(\mathbb R^{n})$ and $g\in C^{\infty}_{0}(\mathbb R^{n+1}_{+})$.
The adjoint operator denoted by $P^{\ast}_{\alpha}$ can be defined as
$$(P^{\ast}_{\alpha}g)(x):=\int_{\mathbb R^{n+1}_{+}}p^{\alpha}_{t}(x-z)g(z,t)dzdt,\quad g\in C^{\infty}_{0}(\mathbb R^{n+1}_{+}).$$
For a Borel measure $\mu$ with compact support in $\mathbb R^{n+1}_{+}$, we define
$$P^{\ast}_{\alpha}\mu(x):=\int_{\mathbb R^{n+1}_{+}}p^{\alpha}_{t}(x-z)d\mu(z,t).$$
\begin{proposition}\label{prop-4.1}
Given $p\in (1,\infty)$ and a compact subset $K$ of $\mathbb R^{n+1}_{+}$, let $p'=p/(p-1)$ and $\mathcal M_{+}(K)$ be the class of nonnegative Radon measures supported on $K$. Then
\item{\rm (i)}
$$C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K)=\sup\left\{\|\mu\|^{p}_{1}:\ \mu\in \mathcal M_{+}(K)\ \&\ \|P^{\ast}_{\alpha}\mu\|_{L^{p'}(\mathbb R^{n})}\leq 1\right\}.$$
\item{\rm (ii)} There exists a measure $\mu_{K}\in\mathcal M_{+}(K)$ such that
$$
\mu_{K}(K)=\int_{\mathbb R^{n}}(P^{\ast}_{\alpha}\mu_{K}(x))^{p'}dx=\int_{\mathbb R^{n+1}_{+}}P_{\alpha}\left(P^{\ast}_{\alpha}\mu_{K}(x)\right)^{p'-1}d\mu_{K}=C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K).
$$
\end{proposition}
\begin{proof}
(i) We set
$$\widetilde{C}^{\alpha,p}_{\mathbb{R}^{n+1}_+}:=\sup\left\{\|\mu\|^{p}_{1}:\ \mu\in \mathcal M_{+}(K)\ \&\ \|P^{\ast}_{\alpha}\mu\|_{L^{p'}(\mathbb R^{n})}\leq 1\right\}.$$
Let $\mu\in \mathcal M_{+}(K)$ satisfy $\|P^{\ast}_{\alpha}\mu\|_{L^{p'}(\mathbb R^{n})}\leq 1$.
Since $\|\mu\|_{1}=\mu(K)$, for any $f\geq 0$ and $P_{\alpha}f\geq 1_{K}$,
\begin{eqnarray*}
\|\mu\|_{1}&\leq&\int_{\mathbb R^{n+1}_{+}}P_{\alpha}fd\mu=\int_{\mathbb R^{n}}f(x)P^{\ast}_{\alpha}\mu(x)dx
\leq\|f\|_{L^{p}(\mathbb R^{n})}\|P^{\ast}_{\alpha}\mu\|_{L^{p'}(\mathbb R^{n})}\leq\|f\|_{L^{p}(\mathbb R^{n})},
\end{eqnarray*}
which implies $\widetilde{C}^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K)\leq C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K)$.
Conversely, define
\begin{equation*}
\left\{\begin{array}{ll}
&\mathcal{X}:=\left\{\mu:\ \mu\in\mathcal M_{+}(K)\quad \&\quad \mu(K)=1\right\};\\
&\mathcal{Y}:=\left\{f:\ 0< f\in L^{p}(\mathbb R^{n})\quad \&\quad \|f\|_{L^{p}(\mathbb R^{n})}\leq 1\right\};\\
&\mathcal Z:=\left\{f:\ 0\leq f\in L^{p}(\mathbb R^{n})\quad \&\quad P_{\alpha}f\geq 1_{K}\right\};\\
&E(\mu, f):=\int_{\mathbb R^{n}}(P^{\ast}_{\alpha}\mu)(x)f(x)dx=\int_{\mathbb R^{n+1}_{+}} P_{\alpha}f(x,t)d\mu(x,t).
\end{array}
\right.
\end{equation*}
By \cite[Thorem 2.4.1]{AH}, $\min_{\mu\in\mathcal X}\sup_{f\in\mathcal Y}E(\mu, f)
=\sup_{f\in\mathcal Y}\min_{\mu\in\mathcal X}E(\mu, f)$. We can get
\begin{eqnarray*}
\min_{\mu\in\mathcal M_{+}(K)}\frac{\|P^{\ast}_{\alpha}\mu\|_{L^{p'}(\mathbb R^{n})}}{\mu(K)}
&=&\min_{\mu\in\mathcal M_{+}(K)}\frac{\sup_{0\leq f\in L^{p}(\mathbb R^{n}) \& \|f\|_{L^{p}(\mathbb R^{n})}\leq 1}\int_{\mathbb{R}^{n}}f(x) P^{\ast}_{\alpha}\mu (x) dx}{\mu(K)}\\
&=&\min_{\mu\in\mathcal M_{+}(K)}\frac{\sup_{ f\in L^{p}(\mathbb R^{n}) \& \|f\|_{L^{p}(\mathbb R^{n})}=1}\int_{\mathbb{R}^{n}}f(x) P^{\ast}_{\alpha}\mu (x) dx}{\mu(K)}\\
&\leq &\min_{\mu\in\mathcal M_{+}(K)}\sup_{f\in\mathcal Y}\frac{\int_{\mathbb{R}^{n}}f(x) P^{\ast}_{\alpha}\mu (x) dx}{\|f\|_{L^{p}(\mathbb R^{n})}\mu(K)}\\
&=&\sup_{f\in\mathcal Y}\min_{\mu\in\mathcal X}\frac{1}{\|f\|_{L^{p}(\mathbb R^{n})}}\left\{\int_{K}P_{\alpha}f(x,t)d\mu(x,t)\right\}\\
&=&\sup_{f\in\mathcal Y}\frac{1}{\|f\|_{L^{p}(\mathbb R^{n})}}\left(\min_{(x,t)\in K}P_{\alpha}f(x,t)\right)\min_{\mu\in\mathcal X}\mu(K)\\
&\leq&\sup_{0< f\in L^{p}(\mathbb R^{n}) }\frac{\left(\min_{(x,t)\in K}P_{\alpha}f(x,t)\right)}{\|f\|_{L^{p}(\mathbb R^{n})}}\\
&\leq & \left(\inf_{0\leq f \in L^{p}(\mathbb R^{n}) \ \&\ \min P_\alpha f=1}\|f\|_{L^{p}(\mathbb R^{n})}\right)^{-1}\\
&=&\left(C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K)\right)^{-1/p}.
\end{eqnarray*}
For any $\mu\in\mathcal M_{+}(K)$, take $\mu_{1}:=\|P^{\ast}_{\alpha}\mu\|^{-1}_{L^{p'}(\mathbb R^{n})}\mu.$
It is obvious that $\|P^{\ast}_{\alpha}\mu_{1}\|_{L^{p'}(\mathbb R^{n})}=1$, and consequently,
$$\Big(\widetilde{C}^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K)\Big)^{1/p}\geq\sup\left\{\frac{\|\mu\|_{1}}{\|P^{\ast}_{\alpha}\mu\|_{L^{p'}(\mathbb R^{n})}}:\ \mu\in\mathcal M_{+}(K)\right\}=\sup\Big\{\left\|\mu_{1}\right\|_{1},\ \mu_{1}\in\mathcal M_{+}(K)\Big\},$$
which implies
$$\min_{\mu\in\mathcal M_{+}(K)}\left\{\frac{\|P^{\ast}_{\alpha}\mu\|_{L^{p'}(\mathbb R^{n})}}{\mu(K)}\right\}=\min_{\mu\in\mathcal M_{+}(K)}\left\{\frac{\|P^{\ast}_{\alpha}\mu\|_{L^{p'}(\mathbb R^{n})}}{\|\mu\|_{1}}\right\}\geq \left(\widetilde{C}^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K)\right)^{-1/p}.$$
This gives
$\left({C}^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K)\right)^{-1/p}\geq\left(\widetilde{C}^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K)\right)^{-1/p}$. The proof of (i) is completed.
Next let us verify (ii). According to (i),
we select a sequence $\{\mu_{j}\}\subset \mathcal M_{+}(K)$ such that
$$\begin{cases}
&\lim\limits_{j\rightarrow\infty}(\mu_{j}(K))^{p}=C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K);\\
&\|P^{\ast}_{\alpha}\mu_{j}\|_{L^{p'}(\mathbb R^{n})}\leq 1.
\end{cases}$$
A direct computation implies that
$$C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K)=\sup\left\{\|\mu\|_{1}^{p}:\ \mu\in\mathcal M_{+}(K)\ \&\ \|P^{\ast}_{\alpha}\mu\|_{L^{p'}(\mathbb R^{n})}=1\right\}.$$
Then, using the fact $\|P^{\ast}_{\alpha}\mu_{j}\|_{L^{p'}(\mathbb R^{n})}=1$, we get
\begin{eqnarray*}
\left|\int_{\mathbb R^{n+1}_{+}}P_{\alpha}f(x,t)d\mu_{j}(x,t)\right|=\left|\int_{\mathbb R^{n}}f(x)P^{\ast}_{\alpha}\mu_{j}(x)dx\right|\leq \|P^{\ast}_{\alpha}\mu_{j}\|_{L^{p'}(\mathbb R^{n})}\|f\|_{L^{p}(\mathbb R^{n})}
\leq\|f\|_{L^{p}(\mathbb R^{n})}.
\end{eqnarray*}
There exists $\mu\in\mathcal M_{+}(K)$ such that $\mu_{j}$ weak $\ast$ convergence to $\mu$. Hence $\mu^{p}(K)=C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K)$. Since the Lebesgue space $L^{p}(\mathbb R^{n})$ is uniformly convex for $1<p<\infty$, the set
$$\Big\{f\in C^{\infty}_{0}(\mathbb R^{n}):\ f\geq0\text{ on }\mathbb R^{n}\text{ and } f\geq1\Big\}$$
is a convex set. Following the procedure of \cite[ Theorem 2.3.10]{AH}, we can prove that there exists a unique function denoted by $f_{K}$ such that
\begin{equation}\label{eq-2.1}
\left\{\begin{aligned}
&f_{K}\in L^{p}(\mathbb R^{n});\\
&C_{\mathbb R^{n+1}_{+}}^{\alpha,p}(\{(x,t)\in K:\ P_{\alpha}f_{K}< 1\})=0; \\
&\|f_{K}\|^{p}_{L^{p}(\mathbb R^{n})}=C_{\mathbb R^{n+1}_{+}}^{\alpha,p}(K).
\end{aligned}\right.
\end{equation}
Then we have
\begin{eqnarray}\label{eq-2.2}
(C_{\mathbb R^{n+1}_{+}}^{\alpha,p}(K))^{1/p}=\mu(K)
&\leq&\int_{\mathbb R^{n+1}_{+}}P_{\alpha}f(x,t)d\mu(x,t)\\
&\leq&\int_{\mathbb R^{n}}f_{K}(x)P^{\ast}_{\alpha}\mu(x)dx\nonumber\\
&\leq&\|f_{K}\|_{L^{p}(\mathbb R^{n})}\|P^{\ast}_{\alpha}\mu\|_{L^{p'}(\mathbb R^{n})}\nonumber\\
&=&(C_{\mathbb R^{n+1}_{+}}^{\alpha,p}(K))^{1/p}\|P^{\ast}_{\alpha}\mu\|_{L^{p'}(\mathbb R^{n})},\nonumber
\end{eqnarray}
which implies that $\|P^{\ast}_{\alpha}\mu\|_{L^{p'}(\mathbb R^{n})}\geq 1$. On the other hand, following the procedure of \cite[Proposition 2.3.2, (b)]{AH}, we can prove that $\mu\longmapsto P^{\ast}_{\alpha}\mu(x)$ is lower semi-continuous on $ M_{+}(\mathbb R^{n+1}_{+})$ in the weak* topology, i.e.,
$$P^{\ast}_{\alpha}\mu(x)\leq \liminf_{j\rightarrow\infty}P^{\ast}_{\alpha}\mu_{j}(x),$$
which, together with $\|P^{\ast}_{\alpha}\mu_{j}\|_{L^{p'}(\mathbb R^{n})}\leq 1$, gives $\|P^{\ast}_{\alpha}\mu\|_{L^{p'}(\mathbb R^{n})}\leq 1$. Hence
$\|P^{\ast}_{\alpha}\mu\|_{L^{p'}(\mathbb R^{n})}=1$. Taking $\mu_{K}=(C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K))^{1/p'}\mu$ yields
\begin{eqnarray*}
\mu_{K}(K)&=&\int_{K}(C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K))^{1/p'}d\mu
=(C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K))^{1/p'}\mu(K)\\
&=&(C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K))^{1/p'}(C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K))^{1/p}=C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K).
\end{eqnarray*}
On the other hand,
\begin{eqnarray*}
\int_{\mathbb R^{n}}(P^{\ast}_{\alpha}\mu_{K}(x))^{p'}dx=\|P^{\ast}_{\alpha}\mu_{K}\|^{p'}_{L^{p'}(\mathbb R^{n})}=(C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K))\|P^{\ast}_{\alpha}\mu\|^{p'}_{L^{p'}(\mathbb R^{n})}=C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K).
\end{eqnarray*}
This indicates that
\begin{equation}\label{eq-4.1}
\mu_{K}(K)=\int_{\mathbb R^{n}}(P^{\ast}_{\alpha}\mu_{K}(x))^{p'}dx=C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K).
\end{equation}
Let $f_{K}$ be the function mentioned above. Then
$\mu_{K}\left(\left\{(x,t)\in K:\ P_{\alpha}f_{K}(x,t)\leq 1\right\}\right)=0$.
By H\"older's inequality, we can get
\begin{eqnarray}\label{eq-4.2}
C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K)&=&\mu_{K}(K)\leq\int_{K}P_{\alpha}f_{K}d\mu_{K}\\
&\leq&\int_{\mathbb R^{n}}f_{K}(x)P^{\ast}_{\alpha}\mu_{K}(x)dx\nonumber\\
&\leq&\|f_{K}\|_{L^{p}(\mathbb R^{n})}\|P^{\ast}_{\alpha}\mu_{K}\|_{L^{p'}(\mathbb
R^{n})}\nonumber\\
&=&(C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K))^{1/p}(C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K))^{1/p'}=C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K).\nonumber
\end{eqnarray}
It follows from (\ref{eq-4.2}) that
$$C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K)=\int_{\mathbb R^{n}}f_{K}P^{\ast}_{\alpha}\mu_{K}dx.$$
Hence
$$\int_{\mathbb R^{n}}f_{K}P^{\ast}_{\alpha}\mu_{K}dx=\int_{\mathbb R^{n}}(P^{\ast}_{\alpha}\mu_{K})^{p'}dx=C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K).$$
The above identity implies that
$$C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K)=\int_{\mathbb R^{n+1}_{+}}P_{\alpha}f_{K}d\mu_{K}=\int_{\mathbb R^{n+1}_{+}}P_{\alpha}(P^{\ast}_{\alpha}\mu_{K})^{p'-1}d\mu_{K},$$
which completes the proof of Proposition \ref{prop-4.1}.
\end{proof}
\begin{remark}
The inequality (\ref{eq-4.2}) implies that
$$\int_{\mathbb R^{n}}f_{K}(x)P^{\ast}_{\alpha}\mu_{K}(x)dx=\|f_{K}\|_{L^{p}(\mathbb R^{n})}\|P^{\ast}_{\alpha}\mu_{K}\|_{L^{p'}(\mathbb R^{n})},$$
i.e., we have equality in H\"older inequality. This means that, due to the normalization chosen, $(f_{K})^{p}=(P^{\ast}_{\alpha}\mu_{K})^{p'}$.
\end{remark}
Below we investigate some basic properties of $C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(\cdot)$.
\begin{proposition}\label{prop-4.2}
\item{\rm (i)} $C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(\varnothing)=0$;
\item{\rm (ii)} If $E_{1}\subseteq E_{2}\subset\mathbb R^{n+1}_{+}$, then $C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E_{1})\leq C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E_{2})$;
\item{\rm (iii)} For any sequence $\{E_{j}\}^{\infty}_{j=1}$ of subsets of $\mathbb R^{n+1}_{+}$,
$$C^{\alpha,p}_{\mathbb{R}^{n+1}_+}\left(\bigcup^{\infty}_{j=1}E_{j}\right)\leq \sum^{\infty}_{j=1}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E_{j});$$
\item{\rm (iv)} For any $E\subset \mathbb R^{n+1}_{+}$ and any $x_{0}\in \mathbb R^{n}$, $C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E+(x_{0},0))=C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E)$.
\end{proposition}
\begin{proof}
The statements (i)\ \&\ (ii) can be deduced from the definition of $C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(\cdot)$ immediately.
For (iii), let $\epsilon>0$. Take $f_{j}\geq0$ such that $P_{\alpha}f_{j}\geq 1$ on $E_{j}$ and $\int_{\mathbb R^{n}}|f_{j}(x)|^{p}dx\leq C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E_{j})+\epsilon/2^{j}$. Let $f=\sup\limits_{j\in\mathbb N_{+}}f_{j}$. For any $(x,t)\in \cup^{\infty}_{j=1}E_{j}$, there exists a $j_{0}$ such that $(x,t)\in E_{j_{0}}$ and $P_{\alpha}f_{j_{0}}(x,t)\geq 1$. Hence $P_{\alpha}f(x,t)\geq 1$ on $\cup^{\infty}_{j=1}E_{j}$. On the other hand,
$$\|f\|^{p}_{L^{p}(\mathbb R^{n})}=\int_{\mathbb R^{n}}|f(x)|^{p}dx\leq \sum^{\infty}_{j=1}\int_{\mathbb R^{n}}|f_{j}(x)|^{p}dx=\sum^{\infty}_{j=1}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E_{j})+\epsilon,$$
which indicates $C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(\cup^{\infty}_{j=1}E_{j})\leq \sum\limits^{\infty}_{j=1}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E_{j}).$
Now we verify (iv). Define $f_{x_{0}}(x)=f(x+x_{0})$. Then $\|f\|_{L^{p}(\mathbb R^{n})}=\|f_{x_{0}}\|_{L^{p}(\mathbb R^{n})}$. If $(x,t)\in E+(x_{0}, 0)$, then $(x-x_{0}, t)\in E$ and vice versa. Take $f\geq0$ such that $P_{\alpha}f\geq 1_{E}$. Changing of variables reaches
\begin{eqnarray*}
P_{\alpha}f_{x_{0}}(x,t)&=&\int_{\mathbb R^{n}}p^{\alpha}_{t}(x-y)f_{x_{0}}(y)dy=P_{\alpha}f(x-x_{0},t),
\end{eqnarray*}
which implies that
$P_{\alpha}f_{x_{0}}(x,t)\geq 1_{E+(x_{0},0)}$ is equivalent to $P_{\alpha}f(x-x_{0},t)\geq 1_{E}.$
This gives $C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E+(x_{0},0))=C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E)$ and the proof of Proposition \ref{prop-4.2} is completed.
\end{proof}
We then prove that the capacity $C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(\cdot)$ is an outer capacity, i.e.,
\begin{proposition}\label{prop-2.1}
For any $E\subset\mathbb R^{n+1}_{+}$,
$$C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E)=\inf\Big\{C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(O):\ O\supset E, O\text{ open}\Big\}.$$
\end{proposition}
\begin{proof}
Without loss of generality, we assume that $C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E)<\infty$. By (ii) of Proposition \ref{prop-4.2},
$$C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E)\leq\inf\Big\{C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(O):\ O\supset E, O\text{ open}\Big\}.$$
For $\epsilon\in(0,1)$, there exists a measurable, nonnegative function $f$ such that $P_{\alpha}f\geq 1$ on $E$ and
$$\int_{\mathbb R^{n}}|f(x)|^{p}dx\leq C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E)+\epsilon.$$
Since $P_{\alpha}f$ is lower semi-continuous, then the set $O_{\epsilon}:=\{(x,t)\in\mathbb R^{n+1}_{+}:\ P_{\alpha}f(x,t)>1-\epsilon\}$ is an open set.
On the other hand, $E\subset O_{\epsilon}$. This implies that
$$C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(O_{\epsilon})\leq \frac{1}{(1-\epsilon)^{p}}\int_{\mathbb R^{n}}|f(x)|^{p}dx<\frac{1}{(1-\epsilon)^{p}}(C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E)+\epsilon).$$
The arbitrariness of $\epsilon$ indicates that
$$C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E)\geq\inf\Big\{C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(O):\ O\supset E, O\text{ open}\Big\}.$$
\end{proof}
An immediate corollary of Proposition \ref{prop-2.1} is the follow result.
\begin{corollary}
If $\{K_{j}\}^{\infty}_{j=1}$ is a decreasing sequence of compact sets, then
$$C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(\cap^{\infty}_{j=1}K_{j})=\lim_{j\rightarrow\infty} C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K_{j}).$$
\end{corollary}
\begin{proof}
Let $O$ be any open set satisfying $\cap^{\infty}_{j=1}K_{j}\subset O$. Then for some $j$, $K_{j}\subset O$, which, together with Proposition \ref{prop-2.1}, gives
$$C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(\cap^{\infty}_{j=1}K_{j})\leq \lim_{j\rightarrow\infty} C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K_{j})\leq
\inf_{\cap^{\infty}_{j=1}K_{j}\subset O}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(O)=C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(\cap^{\infty}_{j=1}K_{j}).$$
\end{proof}
\begin{proposition}\label{prop-2.2}
Let $1<p<\infty$. If $\{E_{j}\}^{\infty}_{j=1}$ is an increasing sequence of arbitrary subsets of $\mathbb R^{n}$, then
$$C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(\cup^{\infty}_{j=1}E_{j})=\lim_{j\rightarrow\infty}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E_{j}).$$
\end{proposition}
\begin{proof}
Since $\{E_{j}\}^{\infty}_{j=1}$ is increasing, then
$$C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(\cup^{\infty}_{j=1}E_{j})\geq\lim_{j\rightarrow\infty}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E_{j}).$$
Conversely, without loss generality, we assume that $\lim_{j\rightarrow\infty}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E_{j})$ is finite. For each $j$, let $f_{E_{j}}$ be the unique function such that $f_{E_{j}}\geq1$ on $E_{j}$ and $\|f_{E_{j}}\|^{p}_{L^{p}}=C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E_{j})$. Then for $i<j$, it holds that $P_{\alpha}f_{E_{j}}\geq 1$ on $E_{i}$ and further, $P_{\alpha}((f_{E_{i}}+f_{E_{j}})/2)\geq 1$ on $E_{i}$, which means that
$$\int_{\mathbb R^{n}}((f_{E_{i}}+f_{E_{j}})/2)^{p}dx\geq C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E_{i}).$$
By \cite[Corollary 1.3.3]{AH}, the sequence $\{f_{E_{j}}\}^{\infty}_{j=1}$ converges strongly to a function $f$ satisfying
$$\|f\|_{L^{p}(\mathbb{R}^n)}^{p}=\lim_{j\rightarrow\infty}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E_{j}).$$
Similar to \cite[Proposition 2.3.12]{AH}, we can prove that $P_{\alpha}f\geq 1$ on $\cup^{\infty}_{j=1}E_{j}$, except possibly on a countable union of sets with $C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(\cdot)$ zero. Hence
$$\lim_{j\rightarrow\infty}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E_{j})\geq\int_{\mathbb R^{n}}|f(x)|^{p}dx\geq C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(\cup^{\infty}_{j=1}E_{j}).$$
\end{proof}
As a corollary of Proposition \ref{prop-2.2}, we can get
\begin{corollary}
Let $O$ be an open subset of $\mathbb R^{n+1}_{+}$. Then
$$C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(O)=\sup\left\{C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K):\ \text{ compact }K\subset O\right\}.$$
\end{corollary}
\begin{proof}
It is obvious that
$$C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(O)\geq\sup\left\{C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K):\ \text{ compact }K\subset O\right\}.$$
Conversely, any open set $O$ in $\mathbb R^{n}$ is the union of an increasing sequence of compact sets denoted by $\{K_{j}\}_{j=1}^{\infty}$. Then it follows from Proposition \ref{prop-2.2} that
\begin{eqnarray*}
C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(O)&=&C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(\cup^{\infty}_{j=1}K_{j})=\lim_{j\rightarrow\infty}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K_{j})\\
&\leq&\sup\left\{C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K):\ \text{ compact }K\subset O\right\}.
\end{eqnarray*}
\end{proof}
For $t_{0}\geq 0$, $x_{0}\in \mathbb R^{n}$ and $r_{0}>0$, define the ball in $\mathbb R^{n+1}_{+}$ as
$$B_{r_{0}}(x_{0},t_{0})=\left\{(x,t)\in \mathbb R^{n+1}_{+}:\ |x-x_{0}|<r_{0}/2,\ r_{0}<t-t_{0}<2r_{0} \right\}.$$
Let $t=r_{0}s$, $x=r_{0}y$ and $f_{r_{0}}(x)=f(r_{0}x)$. We can get $(x,t)\in B_{r_{0}}(0,0)$ is equivalent to $(y,s)\in B_{1}(0,0)$. A direct computation, together with a change of variable $z=r_{0}u$, gives
\begin{eqnarray*}
P_{\alpha}f(x,t)&=&\int_{\mathbb R^{n}}\frac{t^{\alpha}}{(t^{2}+|x-z|^{2})^{(n+\alpha)/2}}f(z)dz\\
&=&\int_{\mathbb R^{n}}\frac{s^{\alpha}}{(|y-u|^{2}+s^{2})^{(n+\alpha)/2}}f_{r_{0}}(u)du= P_{\alpha}f_{r_{0}}(y,r).
\end{eqnarray*}
Then
$$P_{\alpha}f(x,t)\geq 1 \quad \forall (x,t)\in B_{r_{0}}(0,0)\Longleftrightarrow P_{\alpha}f_{r_{0}}(s,y)\geq 1 \quad \forall (y,s)\in B_{1}(0,0).$$
This means that
$$C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(B_{r_{0}}(0,0))=r^{n}_{0}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(B_{1}(0,0)).$$
Now we investigate the $L^{p}$-capacity of general balls $B_{r_{0}}(x_{0}, t_{0})$. We first give a space-time estimate for $P_{\alpha} f$.
\begin{lemma}\label{le-4.1}
Let $1\leq r\leq p\leq\infty.$ Then
$\|P_{\alpha}f(\cdot,t)\|_{L^{p}(\mathbb{R}^n)}\leq t^{n(1/p-1/r)}\|f\|_{L^{r}(\mathbb{R}^n)}.$
\end{lemma}
\begin{proof}
Let $q$ obey $1/p+1=1/r+1/q$.
It is easy to see that
\begin{eqnarray*}
\|p^{\alpha}_{t}(\cdot)\|_{L^{q}(\mathbb{R}^n)}=\left(\int_{\mathbb R^{n}}\left|\frac{t^{\alpha}}{(t^{2}+|x|^{2})^{(n+\alpha)/2}}\right|^{q}dx\right)^{1/q} \lesssim t^{n(1/q-1)}.
\end{eqnarray*}
It follows from Young's inequality that
\begin{eqnarray*}
\|P_{\alpha}f(\cdot,t)\|_{L^{p}(\mathbb{R}^n)}\leq\|p^{\alpha}_{t}\ast f\|_{L^{p}(\mathbb{R}^n)}\leq\|p^{\alpha}_{t}(\cdot)\|_{L^{q}(\mathbb{R}^n)}\|f\|_{L^r(\mathbb{R}^n)} \leq t^{n(1/p-1/r)}\|f\|_{L^{r}(\mathbb{R}^n)}.
\end{eqnarray*}
\end{proof}
\begin{theorem}
Let $(q,p,r)$ be a triple satisfying $1/q=n(1/r-1/p)$, where
$$1<r\leq p<
\begin{cases}
nr/(n-1), n>2r;\\
\infty, n\leq 2r.
\end{cases}$$
Given $f\in L^{r}(\mathbb R^{n})$. Then for $I=(0,T)$ with $0<T\leq \infty$,
$P_{\alpha}f(\cdot,\cdot)\in L^{q}(I; L^{p}(\mathbb{R}^n))\cap
C_{b}(I; L^{r}(\mathbb{R}^n))$ with the estimate
$$\|P_{\alpha} f(\cdot,\cdot)\|_{L^{q}(I; L^{p}(\mathbb{R}^n))}\lesssim \|f\|_{L^{r}(\mathbb{R}^n)}.$$
Here $C_{b}(I; L^{r}(\mathbb{R}^n))$ denotes the space of bounded continuous functions from $I$ to $L^{r}(\mathbb{R}^n).$
\end{theorem}
\begin{proof}
{\it Case 1: $p=r\ \&\ q=\infty$}. By Lemma \ref{le-4.1}, we can get
\begin{eqnarray*}
\|P_{\alpha}f\|_{L^{\infty}(I; L^{r}(\mathbb{R}^n))}&=&\sup_{t>0}\|p^{\alpha}_{t}\ast f\|_{L^{r}(\mathbb{R}^n)}
\leq\sup_{t>0}t^{-n(1/r-1/r)}\|f\|_{L^{r}(\mathbb{R}^n)}
\leq \|f\|_{L^{r}(\mathbb{R}^n)}.
\end{eqnarray*}
{\it Case 2: $p\neq r$}. Denote by $F_{t}(f)=\|p^{\alpha}_{t}\ast f\|_{L^{p}(\mathbb{R}^n)}$. Applying Lemma \ref{le-4.1} again, we also obtain
\begin{eqnarray*}
F_{t}(f)&=&\|p^{\alpha}_{t}\ast f\|_{L^{p}(\mathbb{R}^n)}\leq t^{-n(1/r-1/p)}\|f\|_{L^{r}(\mathbb{R}^n)}\leq t^{-1/q}\|f\|_{L^{r}(\mathbb{R}^n)}.
\end{eqnarray*}
On the other hand,
\begin{eqnarray*}
\left|\left\{t:\ |F_{t}(f)|>\tau\right\}\right|&\leq& \left|\left\{t:\ t^{-1/q}\|f\|_{L^{r}(\mathbb{R}^n)}>\tau\right\}\right|
\leq\left|\left\{t:\ \|f\|^{q}_{L^{r}(\mathbb{R}^n)}/\tau^{q}>t\right\}\right|
\leq \|f\|^{q}_{L^{r}(\mathbb{R}^n)}/\tau^{q}.
\end{eqnarray*}
The above estimate implies that $F_{t}$ is a weak $(r,q)$ type operator. Noticing that
$$|p^{\alpha}_{t}\ast f(x)|\leq\int_{\mathbb R^{n}}\frac{t^{\alpha}}{(t^{2}+|x-y|^{2})^{(n+\alpha)/2}}|f(y)|dy,$$
we have
$$|F_{t}(f)|=\|P_{\alpha}f(\cdot,t)\|_{L^{p}(\mathbb{R}^n)}\leq t^{n(1/p-1/p)}\|f\|_{L^{p}(\mathbb{R}^n)},$$
i.e., $F_{t}$ is also a $(p,\infty)$ type operator. For any triplet $(q,p,r)$, we can choose another triplet $(q_{1}, p_{1}, r_{1})$ such that $q_{1}<q<\infty$, $r_{1}<r<p$ and
$$\begin{cases}
1/q=\theta/q_{1}+(1-\theta)/\infty,\\
1/r=\theta/r_{1}+(1-\theta)/p.
\end{cases}$$
The Marcinkiewicz interpolation theorem implies that $F_{t}$ is a strong $(r,q)$ type operator and satisfies
\begin{equation}\label{eq-4.3}
\|P_{\alpha}f(\cdot,\cdot)\|_{L^{q}(I; L^{p}(\mathbb{R}^n))}\leq \|f\|_{L^{r}(\mathbb{R}^n)}.
\end{equation}
Below we prove that $P_{\alpha}f(\cdot,\cdot)\in C_{b}(I; L^{r}(\mathbb{R}^n))$. Since $u(\cdot,\cdot):=P_{\alpha}f(\cdot,\cdot)$ satisfies the equations (\ref{1}). Then by a direct computation, we can verify that $u(\cdot,\cdot)$ also satisfies:
\begin{equation}\label{eq-2.4}
\begin{cases}
\Delta_{x}u+\frac{1-\alpha}{t}u_{t}+u_{tt}=0,&\ \ (x,t)\in\ \mathbb{R}^{n}\times(0,\infty);\\
u(x,0)=f(x),&\ \ x\in\ \mathbb{R}^{n},
\end{cases}
\end{equation}
see \cite{Caffarelli}. Then by the Poisson type formula obtained by P. Stinga and J. Torrea (cf. \cite[(1.9)]{sti}),
\begin{eqnarray*}
u(x,t)\equiv P_{\alpha}f(x) = C_{\alpha}\int^{\infty}_{0}e^{-s}e^{-t^{2}(-\Delta)/4s}f(x)s^{\alpha/2-1}ds,\quad x\in\mathbb{R}^{n}\ \&\ t>0.
\end{eqnarray*}
By Minkowski's inequality, we get
\begin{eqnarray*}
\Big|\|u(\cdot,t)\|_{L^{r}(\mathbb{R}^n)}-\|u(\cdot,t_{0})\|_{L^{r}(\mathbb{R}^n)}\Big|&\leq&\|u(\cdot,t)-u(\cdot,t_{0})\|_{L^{r}(\mathbb{R}^n)}\\
&\leq&C_{\alpha}\int^{\infty}_{0}e^{-s}\|e^{-t^{2}(-\Delta)/4s}f(\cdot)-e^{-t_{0}^{2}(-\Delta)/4s}f(\cdot)\|_{L^{r}(\mathbb{R}^n)}s^{\alpha/2-1}ds.
\end{eqnarray*}
Notice that
$$\|e^{-t^{2}(-\Delta)/4s}f(\cdot)-e^{-t_{0}^{2}(-\Delta)/4s}f(\cdot)\|_{L^{r}(\mathbb{R}^n)}\leq C\|f\|_{L^{r}(\mathbb{R}^n)}.$$
Since $\{e^{-t(-\Delta)}\}_{t>0}$ is a strongly continuous semigroups in $L^{r}(\mathbb R^{n})$, then by the dominate convergence theorem, we can deduce that
$$\lim_{t\rightarrow t_{0}}\|u(\cdot,t)\|_{L^{r}(\mathbb{R}^n)}=\|u(\cdot,t_{0})\|_{L^{r}(\mathbb{R}^n)}.$$
\end{proof}
For $p\in (1,\infty)$, choose $(\tilde{p}, \tilde{q})$ such that
$$\begin{cases}
1\leq p\leq \tilde{p}<{np}/{(n-1)};\\
1/\tilde{q}=n(1/p-1/\tilde{p}).
\end{cases}$$
\begin{theorem}\label{th-4.1}
If $ 1\leq p<\infty,$ then for $(x_0,r_0)\in\mathbb R^{n+1}_+$,
$$ r_0^{n}\lesssim C^{\alpha,p}_{\mathbb{R}^{n+1}_+}\left(B_{r_0}(x_0, t_0)\right)\lesssim
(t_0+r_0)^{pn}r_{0}^{n(1-p)}.$$
Particularly, if $t_0 \lesssim r_0$, then
$$r_0^{n}\lesssim C^{\alpha,p}_{\mathbb{R}^{n+1}_+}\left(B_{r_0}(x_0,t_0)\right)\lesssim r_0^{n}.$$
\end{theorem}
\begin{proof}
Let $B_{r_{0}}(x_{0},t_{0})$ be a ball in $\mathbb R^{n+1}_{+}$. For any $f\geq 0$ satisfying $P_{\alpha}(f)\geq 1_{B_{r_{0}}(x_{0}, t_{0})}$,
by (\ref{eq-4.3}), we can get
\begin{eqnarray*}
r_{0}^{n/\tilde{p}+1/\tilde{q}}
&\lesssim&\left(\int_{r_{0}<t-t_{0}<2r_{0}}\left(\int_{|x-x_{0}|<r_{0}/2}|P_{\alpha}f(x,t)|^{\tilde{p}}dx\right)^{\tilde{q}/\tilde{p}}dt\right)^{1/\tilde{q}}
\lesssim\|f\|_{L^{p}(\mathbb{R}^n)},
\end{eqnarray*}
which gives $r_{0}^{(n/\tilde{p}+1/\tilde{q})p}\leq \|f\|_{L^p(\mathbb{R}^n)}^{p}$ and $r_{0}^{n}\leq C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(B_{r_{0}}(x_{0}, t_{0}))$.
For the converse, choose $f=1_{\{x\in\mathbb R^{n}:\ |x-x_{0}|<r_{0}/2\}}$. If $(x,t)\in B_{r_{0}}(x_{0}, t_{0})$, then $|x-x_{0}|<r_{0}/2$ and $r_{0}<t-t_{0}<2r_{0}$, i.e., $r_{0}+t_{0}\leq t<t_{0}+2r_{0}$. We can get there exists a constant $c_{\alpha}$ such that
$$\frac{t^{\alpha}}{(t^{2}+|x-y|^{2})^{(n+\alpha)/2}}\geq \frac{(t_{0}+r_{0})^{\alpha}}{(t_{0}+2r_{0}+3r_{0}/2)^{(n+\alpha)/2}}\geq \frac{c_{\alpha}}{(t_{0}+r_{0})^{n}},$$
which gives
\begin{eqnarray*}
P_{\alpha}f(x,t)&=&\int_{|y-x_{0}|<r_{0}/2}\frac{t^{\alpha}}{(t^{2}+|x-y|^{2})^{(n+\alpha)/2}}dy\geq\frac{c_{\alpha}r^{n}_{0}}{(t_{0}+r_{0})^{n}},
\end{eqnarray*}
equivalently,
$P_{\alpha}\left(\left(1+{t_{0}}/{r_{0}}\right)^{n}c^{-1}_{\alpha}f\right)\geq 1.$
By the definition of $C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(\cdot)$, we obtain
\begin{eqnarray*}
C^{\alpha,p}_{\mathbb{R}^{n+1}_+}\left(B_{r_{0}}(x_{0}, t_{0})\right)&\leq&\Big\|\frac{(t_{0}+r_{0})^{n}}{r^{n}_{0}}f\Big\|^{p}_{L^{p}(\mathbb{R}^n)}\\
&=&\frac{(t_{0}+r_{0})^{pn}}{r^{pn}_{0}}\int_{\mathbb R^{n}}\left(1_{\{x\in\mathbb R^{n}:\ |x-x_{0}|<r_{0}/2\}}(y)\right)^{p}dy\\
&=&(t_{0}+r_{0})^{pn}r_{0}^{(1-p)n},
\end{eqnarray*}
which completes the proof of Theorem \ref{th-4.1}.
\end{proof}
\subsection{Capacitary strong type inequality}\label{sec-2.2}
In order to establish the embeddings of $L^{p}(\mathbb R^{n})$ to $L^{q}(\mathbb R^{n+1}_{+})$ via $P_{\alpha}$, we need to prove the capacitary strong and weak inequalities. Let $L^{p}_{+}(\mathbb R^{n})$ denote the class of all nonnegative functions in $L^{p}(\mathbb R^{n})$.
\begin{lemma}\label{le-4.2}
Let $p\in (1,\infty)$. Then
\begin{equation}\label{eq-2.3}
\int^{\infty}_{0}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}\left(\left\{(x,t)\in\mathbb R^{n+1}_{+}:\ P_{\alpha}f(x,t)\geq\lambda\right\}\right)d\lambda^{p}\lesssim \|f\|_{L^{p}(\mathbb{R}^n)}^{p}\quad \forall\ f\in L^{p}_{+}(\mathbb R^{n}).
\end{equation}
\end{lemma}
\begin{proof}
Firstly, we verify the inequality for any nonnegative function in $C^{\infty}_{0}(\mathbb R^{n})$. Let $f\in C^{\infty}_{0}(\mathbb R^{n})$ and for $r>0$, set
$$E_{j}:=\left\{(x,t)\in \overline{B_{r}(0,0)}:\ P_{\alpha}f(x,t)\geq 2^{j}\right\}.$$
Note that $E_j$ depends on $r.$ Since $f\in C^{\infty}_{0}(\mathbb R^{n}),$ $P_{\alpha}f(x,t)$ is continuous. Thus, $E_j$ is the intersection of $\overline{B_r(0,0)}$ and a close set and so $E_j$ is compact.
Let $\mu_{j}$ stand for the measure corresponding to $E_{j}$ such that
$$\mu_{j}(E_{j})=\int_{\mathbb R^{n}}(P_{\alpha}^{\ast}\mu_{j}(x))^{p'}dx=\int_{\mathbb R^{n+1}_{+}}P_{\alpha}(P^{\ast}_{\alpha}\mu_{j})^{p'-1}d\mu_{j}=C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E_{j}).$$
Let $S:=\sum\limits^{\infty}_{j=-\infty}2^{jp}\mu_{j}(E_{j})$ and $T:=\big\|\sum\limits^{\infty}_{j=-\infty}2^{j(p-1)}(P^{\ast}_{\alpha}\mu_{j})\big\|^{p'}_{L^{p'}}$.
Because $f\in C^{\infty}_{0}(\mathbb R^{n})$, there exists a positive integer $j_{0}$ such that $E_{j}$ are empty sets for $j>j_{0}$, i.e., $\mu_{j}(E_{j})=C^{\alpha,p}_{\mathbb R^{n+1}_{+}}(E_{j})=0, j>j_{0}$. Hence $\sum\limits^{\infty}_{j=1}2^{jp}\mu_{j}(E_{j})<\infty$. On the other hand,
\begin{eqnarray*}
\sum^{-1}_{j=\infty}2^{jp}\mu_{j}(E_{j})&=&\sum^{-1}_{j=\infty}2^{jp}C^{\alpha,p}_{\mathbb R^{n+1}_{+}}(E_{j})\\
&\leq&\sum^{-1}_{j=\infty}2^{jp}C^{\alpha,p}_{\mathbb R^{n+1}_{+}}(B_{r}(0,0))\\
&\lesssim&r^{n}\sum^{-1}_{j=\infty}2^{jp},
\end{eqnarray*}
which means that $S<\infty$. Then by H\"older's inequality,
\begin{equation}\label{eq-2.5}
S\leq\sum^{\infty}_{j=-\infty}2^{j(p-1)}\int_{\mathbb R^{n+1}_{+}}P_{\alpha}fd\mu_{j}
\leq\int_{\mathbb R^{n}}f\left(\sum^{\infty}_{j=-\infty}2^{j(p-1)}(P^{\ast}_{\alpha}\mu_{j})\right)dx
\leq T^{1/p'}\|f\|_{L^{p}(\mathbb{R}^n)}.
\end{equation}
Below we claim that
\begin{equation}\label{eq-2.6}
T\lesssim S.
\end{equation}
We divide the proof of this claim into two cases.
{\it Case 1: $2\leq p<\infty$.} For $k=0, \pm1, \pm2,\ldots,$ let
$$\begin{cases}
\sigma_{k}(x)=\sum\limits^{\infty}_{j=k}2^{j(p-1)}P^{\ast}_{\alpha}\mu_{j}(x);\\
\sigma(x)=\sum\limits^{\infty}_{j=-\infty}2^{j(p-1)}P^{\ast}_{\alpha}\mu_{j}(x).
\end{cases}$$
Since $f\in C^{\infty}_{0}(\mathbb R^{n})$, the sets $E_{j}$ are empty for sufficiently large $j$, i.e., there exists a positive integer $j_{0}$ such that
$C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E_{j})=0,\ j> j_{0}$. Then by (ii) of Proposition \ref{prop-4.1},
\begin{eqnarray*}
\|\sigma_{k}\|_{L^{p'}(\mathbb{R}^n)}&=&\Big\|\sum\limits^{\infty}_{j=k}2^{j(p-1)}P^{\ast}_{\alpha}\mu_{j}\Big\|_{L^{p'}(\mathbb{R}^n)}\\
&\leq&\sum\limits^{\infty}_{j=k}2^{j(p-1)}\left\|P^{\ast}_{\alpha}\mu_{j}\right\|_{L^{p'}(\mathbb{R}^n)}\\
&\simeq&\sum\limits^{j_{0}}_{j=k}2^{j(p-1)}(C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E_{j}))^{1/p'}<\infty,
\end{eqnarray*}
which implies that for any $k$, $\sigma_{k}\in L^{p'}(\mathbb R^{n})$. For $x\in\mathbb R^{n}$, the sequence $\{\sigma_{k}(x)\}$ is increasing as $k\rightarrow-\infty$ since $P^{\ast}_{\alpha}\mu_{j}(x)\geq0$. If there exists an upper bound for $\{\sigma_{k}(x)\}$, we know that $\sigma_{k}(x)$ convergences to the series $\sum\limits^{\infty}_{j=-\infty}2^{j(p-1)}P^{\ast}_{\alpha}\mu_{j}(x)$. If the sequence $\{\sigma_{k}(x)\}$ is unbounded, then $\sigma_{k}(x)$ tends to $\infty$ as $k\rightarrow-\infty$. Without loss of generality, formally, we write $\lim\limits_{k\rightarrow -\infty}\sigma_{k}(x)=\sigma(x)$. We can get
\begin{eqnarray*}
\sum^{\infty}_{k=n}\Big(\sigma_{k}(x)^{p'}-\sigma_{k+1}(x)^{p'}\Big)&=&\sum^{\infty}_{k=n}\sigma_{k}(x)^{p'}-\sum^{\infty}_{k=n+1}\sigma_{k}(x)^{p'}
=\sigma_{n}(x)^{p'}.
\end{eqnarray*}
By the mean value theorem, noticing that $\sigma_{k}(x)\geq\sigma_{k+1}(x)$, we have
\begin{eqnarray}\label{eq-2.7}
\sigma(x)^{p'}&=&\left(\sum^{\infty}_{j=-\infty}2^{j(p-1)}P^{\ast}_{\alpha}\mu_{j}(x)\right)^{p'}=p'\sum^{\infty}_{k=-\infty}(\sigma_{k}(x))^{p'-1}\\
&=&\lim_{n\rightarrow-\infty}\sigma_{n}(x)^{p'}\nonumber\\
&=&\lim_{n\rightarrow-\infty}\sum^{\infty}_{k=n}\Big(\sigma_{k}(x)^{p'}-\sigma_{k+1}(x)^{p'}\Big)\nonumber\\
&\leq&p'(\sigma_{k}(x))^{p'-1}\Big(\sigma_{k}(x)-\sigma_{k+1}(x)\Big)\nonumber\\
&=&p'\sum^{\infty}_{k=-\infty}(\sigma_{k}(x))^{p'-1}
2^{k(p-1)}P^{\ast}_{\alpha}\mu_{k}(x).\nonumber
\end{eqnarray}
Using Cauchy-Schwartz's inequality, we obtain
\begin{eqnarray*}
T&=&p'\int_{\mathbb R^{n}}\sum^{\infty}_{k=-\infty}\sigma_{k}^{p'-1}(x)
2^{k(p-1)}P^{\ast}_{\alpha}\mu_{k}(x)dx\\
&\lesssim&p'\int_{\mathbb R^{n}}\left(\sum^{\infty}_{k=-\infty}(\sigma_{k}(x))2^{k\frac{1}{(p-1)}\frac{1}{p'-1}}(P^{\ast}_{\alpha}\mu_{k})^{p'-1}\right)^{p'-1}\\
&&\times\left(\sum^{\infty}_{k=-\infty}2^{k(p-1)\frac{p(p-2)}{(p-1)^{2}}\frac{1}{2-p'}}(P^{\ast}_{\alpha}\mu_{k}(x))^{p'}\right)^{2-p'}dx\\
&=&p'\int_{\mathbb R^{n}}\left(\sum^{\infty}_{k=-\infty}\sigma_{k}(x)2^{k}(P^{\ast}_{\alpha}\mu_{k})^{p'-1}\right)^{p'-1}
\left(\sum^{\infty}_{k=-\infty}2^{kp}(P^{\ast}_{\alpha}\mu_{k}(x))^{p'}\right)^{2-p'}dx,
\end{eqnarray*}
which, together with H\"older's inequality, indicates that $T\lesssim p' T_{1}^{2-p'}T^{p'-1}_{2}$, where
\begin{equation}
\left\{ \begin{aligned}
&T_{1}:=\int_{\mathbb R^{n}}\left(\sum^{\infty}_{k=-\infty}2^{kp}(P^{\ast}_{\alpha}\mu_{k}(x))^{p'}\right)dx;\nonumber\\
&T_{2}:=\int_{\mathbb R^{n}}\left(\sum^{\infty}_{k=-\infty}\sigma_{k}(x)2^{k}(P^{\ast}_{\alpha}\mu_{k})^{p'-1}\right)dx.\nonumber
\end{aligned} \right.
\end{equation}
For $T_{1}$, we have
\begin{eqnarray*}
T_{1}&=&\sum^{\infty}_{k=-\infty}2^{kp}\int_{\mathbb R^{n}}\left(P^{\ast}_{\alpha}\mu_{k}(x)\right)^{p'}dx
=\sum^{\infty}_{k=-\infty}2^{kp}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E_{k})\\
&\lesssim&\sum^{\infty}_{k=-\infty}\int^{2^{k}}_{2^{k-1}}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}\left(\left\{(x,t)\in \overline{B_{r}(0,0)}
:\ P_{\alpha}f(x,t)\geq 2^{k}\right\}\right)d\lambda^{p}\\
&\lesssim&\sum^{\infty}_{k=-\infty}\int^{2^{k}}_{2^{k-1}}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}\left(\left\{(x,t)\in \overline{B_{r}(0,0)}:\ P_{\alpha}f(x,t)\geq \lambda\right\}\right)d\lambda^{p}\\
&\lesssim&\int^{\infty}_{0}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}\left(\left\{(x,t)\in \overline{B_{r}(0,0)}:\ P_{\alpha}f(x,t)\geq \lambda\right\}\right)d\lambda^{p}
\lesssim S.
\end{eqnarray*}
For $T_{2}$, we can get
\begin{eqnarray*}
T_{2}&=&\sum^{\infty}_{k=-\infty}2^{k}\int_{\mathbb R^{n}}\left(\sum^{\infty}_{j\geq k}2^{j(p-1)}P^{\ast}_{\alpha}\mu_{j}(x)\right)(P^{\ast}_{\alpha}\mu_{k})^{p'-1}dx\\
&=&\sum^{\infty}_{k=-\infty}\sum^{\infty}_{j\geq k}2^{k+j(p-1)}\int_{\mathbb R^{n}}P^{\ast}_{\alpha}\mu_{j}(x)(P^{\ast}_{\alpha}\mu_{k})^{p'-1}dx\\
&=&\sum^{\infty}_{k=-\infty}\sum^{\infty}_{j\geq k}2^{k+j(p-1)}\int_{\mathbb R^{n+1}_{+}}P_{\alpha}(P^{\ast}_{\alpha}\mu_{k})^{p'-1}d\mu_{j}(x).
\end{eqnarray*}
Note that $E_{k}$ is compact subsets. Let $f_{E_{k}}$ be the function satisfying (\ref{eq-2.1}). Suppose that $P_{\alpha}f_{E_{k}}(x_{0},t_{0})>1$, by the lower-semicontinuity, $P_{\alpha}f_{E_{j}}\geq 1+\delta>1$ on some neighborhood $U$ of $(x_{0},t_{0})$. On the other hand, denote by $\Omega_{E_{k}}$ the set $\{(x,t)\in E_{k}:\ P_{\alpha}f_{E_{k}}(x,t)< 1\}$ and let $F$ be any compact subset of $\Omega_{E_{k}}$. For any $0\leq f\in L^{p}(\mathbb R^{n})$ such that $P_{\alpha}f\geq 1$ on $F$
\begin{eqnarray*}
\mu_{k}(F)&\leq&\int_{\mathbb R^{n+1}_{+}}P_{\alpha}fd\mu_{k}\\
&=&\int_{\mathbb R^{n}}f P^{\ast}_{\alpha}\mu_{k}dx\\
&\leq&\|P^{\ast}_{\alpha}\mu_{k}\|_{L^{p'}(\mathbb{R}^n)}\|f\|_{L^{p}(\mathbb{R}^n)},
\end{eqnarray*}
which, together with (\ref{eq-2.1}), gives
$$\mu_{k}(F)\leq\|P^{\ast}_{\alpha}\mu_{k}\|_{L^{p'}(\mathbb{R}^n)}C^{\alpha,p}_{\mathbb R^{n+1}_{+}}(F)\leq \|P^{\ast}_{\alpha}\mu_{k}\|_{L^{p'}(\mathbb{R}^n)}C^{\alpha,p}_{\mathbb R^{n+1}_{+}}(\Omega_{E_{k}})=0.$$
Hence, $\mu_{k}(\Omega_{E_{k}})=0$, i.e., $P_{\alpha}f_{E_{k}}\geq 1$ on $E_{k}$ for $\mu_{k}$ a.e. We can deduce that
\begin{eqnarray*}
C^{\alpha,p}_{\mathbb R^{n+1}_{+}}({E_{k}})&=&\int_{\mathbb R^{n+1}_{+}}P_{\alpha}(P^{\ast}_{\alpha}\mu_{k})^{p'-1}d\mu_{k}\\
&=&\int_{\mathbb R^{n+1}_{+}}P_{\alpha}f_{E_{k}}d\mu_{k}\\
&\geq&(1+\delta)\mu_{k}(U)+\mu_{k}(E_{k}\setminus U)\\
&\geq&\delta\mu_{k}(U)+\mu_{k}(E_{k}),
\end{eqnarray*}
which gives $\mu_{k}(U)=0$, i.e., $(x_{0},t_{0})\notin \text{ supp }\mu_{k}$. Equivalently, for all $(x,t)\in \text{ supp }\mu_{k}$, $$P_{\alpha}(P^{\ast}_{\alpha}\mu_{k})^{p'-1}=P_{\alpha}f_{E_{k}}\leq 1.$$
Since $E_{j}\subset E_{k}$ for $j\geq k$, we obtain
\begin{eqnarray*}
T_{2}&\lesssim&\sum^{\infty}_{k=-\infty}\sum^{\infty}_{j\geq k}2^{k+j(p-1)}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E_{j})\\
&=&\sum^{\infty}_{j=-\infty}2^{j(p-1)}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E_{j})\sum^{j}_{k=-\infty}2^{k}\\
&\lesssim&\sum^{\infty}_{j=-\infty}2^{jp}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E_{j})\lesssim S.
\end{eqnarray*}
The estimates for $T_{1}$ and $T_{2}$ imply (\ref{eq-2.6}). It can be deduced from (\ref{eq-2.5})\ \&\ (\ref{eq-2.6}) that
$$\Big\|\sum\limits^{\infty}_{j=-\infty}2^{j(p-1)}(P^{\ast}_{\alpha}\mu_{j})\Big\|^{p'}_{L^{p'}(\mathbb{R}^n)}\lesssim \|f\|_{L^{p}(\mathbb{R}^n)}^{p},$$
i.e., $\sigma\in L^{p'}(\mathbb R^{n})$ and the series $\sum\limits^{\infty}_{j=-\infty}2^{j(p-1)}P^{\ast}_{\alpha}\mu_{j}(x)<\infty \text{ a.e. }x\in\mathbb R^{n}$.
{\it Case 2: $1<p<2$.} For $k=0,\pm1,\pm2,\ldots,$ let
$$\begin{cases}
\sigma_{k}(x)=\sum\limits^{k}_{j=-\infty}2^{j(p-1)}P^{\ast}_{\alpha}\mu_{j}(x);\\
\sigma(x)=\sum\limits^{\infty}_{j=-\infty}2^{j(p-1)}P^{\ast}_{\alpha}\mu_{j}(x).
\end{cases}$$
Similarly, because $\sigma_{k}(x)\geq 0$ for $x\in \mathbb R^{n}$, we can also write $\lim\limits_{k\rightarrow\infty}\sigma(x)=\sigma(x)$ formally.
Below, similar to {\it Case 1}, we will prove (\ref{eq-2.6}) and $\sigma\in L^{p'}(\mathbb R^{n})$. Following the procedure of (\ref{eq-2.7}), we can deduce that $$(\sigma(x))^{p'}=\left(\sum^{\infty}_{j=-\infty}2^{j(p-1)}P^{\ast}_{\alpha}\mu_{j}(x)\right)^{p'}\leq p'\sum^{\infty}_{k=-\infty}(\sigma_{k}(x))^{p'-1}
2^{k(p-1)}P^{\ast}_{\alpha}\mu_{k}(x).$$
We get
\begin{eqnarray*}
T
&=&p'\sum^{\infty}_{k=-\infty}2^{k(p-1)}\int_{\mathbb R^{n}}P^{\ast}_{\alpha}\mu_{k}(x)\left(\sum\limits^{k}_{j=-\infty}2^{j(p-1)}P^{\ast}_{\alpha}\mu_{j}(x)\right)^{p'-1}dx\\
&=&p'\sum^{\infty}_{k=-\infty}2^{k(p-1)}\int_{\mathbb R^{n}}\left(\sum\limits^{k}_{j=-\infty}2^{j(p-1)}\left(P^{\ast}_{\alpha}\mu_{k}(x)\right)^{1/(p'-1)}P^{\ast}_{\alpha}\mu_{j}(x)\right)^{p'-1}dx\\
&=&p'\sum^{\infty}_{k=-\infty}2^{k(p-1)}\left\|\sum\limits^{k}_{j=-\infty}2^{j(p-1)}\left(P^{\ast}_{\alpha}\mu_{k}(x)\right)^{1/(p'-1)}
P^{\ast}_{\alpha}\mu_{j}(x)\right\|_{L^{p'-1}(\mathbb{R}^n)}^{p'-1}\\
&\lesssim&p'\sum^{\infty}_{k=-\infty}2^{k(p-1)}\left\{\sum\limits^{k}_{j=-\infty}2^{j(p-1)}\left\|\left(P^{\ast}_{\alpha}\mu_{k}(x)\right)^{1/(p'-1)}
P^{\ast}_{\alpha}\mu_{j}(x)\right\|_{L^{p'-1}(\mathbb{R}^n)}\right\}^{p'-1}.
\end{eqnarray*}
Also, for $j\leq k$, if $x\in E_{k}$ then $x\in E_{j}$. Similarly, we can deduce that
$$P_{\alpha}\left(P^{\ast}_{\alpha}\mu_{j}(x)\right)^{p'-1}(x)\leq1,\ x\in E_{k}.$$
This indicates that
\begin{eqnarray*}
\int_{\mathbb R^{n}}\left(P^{\ast}_{\alpha}\mu_{k}(x)\right)\left(P^{\ast}_{\alpha}\mu_{j}(x)\right)^{p'-1}dx
&=& \int_{\mathbb R^{n+1}_{+}}P_{\alpha}\left(P^{\ast}_{\alpha}\mu_{j}(x)\right)^{p'-1}d\mu_{k}\\
&\leq&\mu_{k}(E_{k})=C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E_{k}).
\end{eqnarray*}
We obtain
\begin{eqnarray*}
T&\lesssim& \sum^{\infty}_{k=-\infty}2^{k(p-1)}\left\{\sum\limits^{k}_{j=-\infty}2^{j(p-1)}\left(\int_{\mathbb R^{n}}P^{\ast}_{\alpha}\mu_{k}(x)
\left(P^{\ast}_{\alpha}\mu_{j}(x)\right)^{p'-1}dx\right)^{1/(p'-1)}\right\}^{p'-1}\\
&\lesssim& \sum^{\infty}_{k=-\infty}2^{k(p-1)}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E_{k})\left(\sum^{k}_{j=-\infty}2^{j(p-1)}\right)^{p'-1}\\
&\lesssim&\sum^{\infty}_{k=-\infty}2^{kp}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E_{k})\lesssim S,
\end{eqnarray*}
which gives (\ref{eq-2.6}) for $1<p<2$.
It follows from (\ref{eq-2.5})\ \&\ (\ref{eq-2.6}) that
$$S\lesssim \|f\|_{L^{p}(\mathbb{R}^n)}T^{1/p'}\lesssim \|f\|_{L^{p}(\mathbb{R}^n)}S^{1/p'}.$$
We can get $S\lesssim \|f\|_{L^{p}(\mathbb{R}^n)}^{p}$ and consequently, $T<\infty$. Then $\sigma\in L^{p'}(\mathbb R^{n}),$ for $1<p<2$.
By the fact that $f\in C^{\infty}_{0}(\mathbb R^{n})$, there exists an integer $j_{0}$ such that
\begin{eqnarray*}
\Big\|\sum\limits^{\infty}_{j=0}2^{j(p-1)}P^{\ast}_{\alpha}\mu_{j}\Big\|_{L^{p'}(\mathbb{R}^n)}
&\leq&\sum\limits^{\infty}_{j=0}2^{j(p-1)}\left\|P^{\ast}_{\alpha}\mu_{j}\right\|_{L^{p'}(\mathbb{R}^n)}\\
&\simeq&\sum\limits^{j_{0}}_{j=0}2^{j(p-1)}(C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E_{j}))^{1/p'}<\infty.
\end{eqnarray*}
This indicates that $\sum\limits^{-1}_{j=-\infty}2^{j(p-1)}P^{\ast}_{\alpha}\mu_{j}\in L^{p'}(\mathbb R^{n})$. For $\sigma_{k}$, if $k\leq -1$, because $P^{\ast}_{\alpha}\mu_{j}(x)\geq 0$, then
$$\|\sigma_{k}\|_{L^{p'}(\mathbb{R}^n)}\leq \Big\|\sum\limits^{-1}_{j=-\infty}2^{j(p-1)}P^{\ast}_{\alpha}\mu_{j}\Big\|_{L^{p'}(\mathbb{R}^n)}$$
and $\sigma_{k}\in L^{p'}(\mathbb R^{n})$. If $k\geq 0$, noticing that
$$\sigma_{k}=\sum\limits^{-1}_{j=-\infty}2^{j(p-1)}P^{\ast}_{\alpha}\mu_{j}+\sum\limits^{j_{0}}_{j=0}2^{j(p-1)}P^{\ast}_{\alpha}\mu_{j},$$
we can also get $\sigma_{k}\in L^{p}(\mathbb R^{n})$.
Further, we obtain that
\begin{eqnarray*}
&&\int^{\infty}_{0}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}\left(\left\{(x,t)\in \overline{B_{r}(0,0)}:\ P_{\alpha}f(x,t)\geq \lambda\right\}\right)d\lambda^{p}\\
&&\quad =\sum^{\infty}_{j=-\infty}\int^{2^{j+1}}_{2^{j}}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}\left(\left\{(x,t)\in \overline{B_{r}(0,0)}:\ P_{\alpha}f(x,t)\geq \lambda\right\}\right)d\lambda^{p}\\
&&\quad\lesssim \sum^{\infty}_{j=-\infty}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E_{j})\int^{2^{j+1}}_{2^{j}}d\lambda^{p}\\
&&\quad\lesssim \sum^{\infty}_{j=-\infty}2^{pj}\mu_{j}(E_{j})\lesssim \|f\|^{p}_{L^{p}(\mathbb{R}^n)}.
\end{eqnarray*}
Hence we can deduce from Proposition \ref{prop-2.2} and Fauto's lemma that
\begin{eqnarray*}
&&\int^{\infty}_{0}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}\left(\left\{(x,t)\in \mathbb R^{n+1}_{+}:\ P_{\alpha}f(x,t)>\lambda\right\}\right)d\lambda^{p}\\
&&\quad=\int^{\infty}_{0}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}\left(\cup^{\infty}_{r=1}\left\{(x,t)\in \overline{B_{r}(0,0)}:\ P_{\alpha}f(x,t)>\lambda\right\}\right)d\lambda^{p}\\
&&\quad=\int^{\infty}_{0}\lim_{r\rightarrow\infty}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}\left(\left\{(x,t)\in \overline{ B_{r}(0,0)}:\ P_{\alpha}f(x,t)>\lambda\right\}\right)d\lambda^{p}\\
&&\quad\leq \underline{\lim}_{r\rightarrow\infty}\int^{\infty}_{0}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}\left(\left\{(x,t)\in \overline{ B_{r}(0,0)}:\ P_{\alpha}f(x,t)>\lambda\right\}\right)d\lambda^{p}\\
&&\quad\leq \underline{\lim}_{r\rightarrow\infty}\int^{\infty}_{0}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}\left(\left\{(x,t)\in \overline{B_{r}(0,0)}
:\ P_{\alpha}f(x,t)\geq \lambda\right\}\right)d\lambda^{p}\\
&&\quad\lesssim\|f\|^{p}_{L^{p}(\mathbb{R}^n)},
\end{eqnarray*}
which proves (\ref{eq-2.3}) for the functions in $C^{\infty}_{0}(\mathbb R^{n})$.
At last, we prove that (\ref{eq-2.3}) holds for the functions in $L^{p}_{+}(\mathbb R^{n})$. Let $f\in L^{p}_{+}(\mathbb R^{n})$. Since $C^{\infty}_{0}(\mathbb R^{n})$ is dense in $L^{p}_{+}(\mathbb R^{n})$, there exists a sequence of $\{f_{j}\}_{j=1}^{\infty}$ such that $f_{n}\rightarrow f$ in $L^{p}(\mathbb R^{n})$ as $n\rightarrow \infty$. Then it is obvious that $\|f_{j}\|_{L^{p}(\mathbb{R}^n)}\rightarrow\|f\|_{L^{p}(\mathbb{R}^n)}$ as $n\rightarrow \infty$. We can select a subsequence denoted by $\{f_{i_{j}}\}$ such that $\|f_{i_{j}}-f\|_{L^{p}(\mathbb{R}^n)}<4^{-j}$. For any $\lambda>0$, there exists an $m_{1}\in\mathbb N$ such that for $j\geq m_{1}$,
\begin{eqnarray*}
C^{\alpha,p}_{\mathbb R^{n+1}_{+}}\Big(\Big\{(x,t):\ P_{\alpha}(|f_{i_{j}}-f|)(x,t)>\lambda/2\Big\}\Big)
&&\leq C^{\alpha,p}_{\mathbb R^{n+1}_{+}}\Big(\Big\{(x,t):\ P_{\alpha}(|f_{i_{j}}-f|)(x,t)>2^{-j}\Big\}\Big)\\
&&\lesssim 2^{jp}\|f_{i_{j}}-f\|^{p}_{L^{p}(\mathbb{R}^n)}\\
&&\lesssim 2^{-jp}.
\end{eqnarray*}
On the other hand, there exists an $m_{2}\in\mathbb N$ such that
$$2^{-jp}\leq\frac{1}{2} C^{\alpha,p}_{\mathbb R^{n+1}_{+}}\Big(\Big\{(x,t):\ P_{\alpha}(|f_{i_{j}}-f|)(x,t)>\lambda\Big\}\Big).$$
Take $m:=\max\{m_{1}, m_{2}\}$. Then
\begin{eqnarray*}
C^{\alpha,p}_{\mathbb R^{n+1}_{+}}\Big(\Big\{(x,t):\ P_{\alpha}(f)(x,t)>\lambda\Big\}\Big)&\leq&C^{\alpha,p}_{\mathbb R^{n+1}_{+}}\Big(\Big\{(x,t):\ P_{\alpha}(f_{i_j})(x,t)>\lambda/2\Big\}\Big)\\
&&+C^{\alpha,p}_{\mathbb R^{n+1}_{+}}\Big(\Big\{(x,t):\ P_{\alpha}(|f_{i_{j}}-f|)(x,t)>\lambda/2\Big\}\Big)\\
&\leq&C^{\alpha,p}_{\mathbb R^{n+1}_{+}}\Big(\Big\{(x,t):\ P_{\alpha}(f_{i_j})(x,t)>\lambda/2\Big\}\Big)\\
&&+\frac{1}{2} C^{\alpha,p}_{\mathbb R^{n+1}_{+}}\Big(\Big\{(x,t):\ P_{\alpha}(|f_{i_{j}}-f|)(x,t)>\lambda\Big\}\Big),
\end{eqnarray*}
which means that $\forall j\geq m$,
$$C^{\alpha,p}_{\mathbb R^{n+1}_{+}}\Big(\Big\{(x,t):\ |P_{\alpha}(f)|(x,t)>\lambda\Big\}\Big)\leq 2C^{\alpha,p}_{\mathbb R^{n+1}_{+}}\Big(\Big\{(x,t):\ |P_{\alpha}(f_{i_j})|(x,t)>\lambda/2\Big\}\Big).$$
Then
$$C^{\alpha,p}_{\mathbb R^{n+1}_{+}}\Big(\Big\{(x,t):\ |P_{\alpha}(f)|(x,t)>\lambda\Big\}\Big)\leq 2\liminf_{j\rightarrow\infty}C^{\alpha,p}_{\mathbb R^{n+1}_{+}}\Big(\Big\{(x,t):\ |P_{\alpha}(f_{i_j})|(x,t)>\lambda/2\Big\}\Big).$$
By Fauto's lemma, we can get
\begin{eqnarray*}
&&\int^{\infty}_{0}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}\left(\left\{(x,t)\in\mathbb R^{n+1}_{+}:\ P_{\alpha}f(x,t)\geq\lambda\right\}\right)d\lambda^{p}\\
&&\lesssim \int^{\infty}_{0}\liminf_{j\rightarrow\infty}C^{\alpha,p}_{\mathbb R^{n+1}_{+}}\Big(\Big\{(x,t):\ |P_{\alpha}(f_{j})|(x,t)>\lambda/2\Big\}\Big)d\lambda^{p}\\
&&\lesssim \liminf_{j\rightarrow\infty}\int^{\infty}_{0}C^{\alpha,p}_{\mathbb R^{n+1}_{+}}\Big(\Big\{(x,t):\ P_{\alpha}(f_{i_j})(x,t)>\lambda/2\Big\}\Big)d\lambda^{p}\\
&&\lesssim \liminf_{j\rightarrow\infty}\|f_{i_{j}}\|^{p}_{L^{p}(\mathbb{R}^n)}\\
&&\lesssim \|f\|_{L^{p}(\mathbb{R}^n)}^{p}.
\end{eqnarray*}
\end{proof}
\subsection{Preliminary lemmas on the fractional capacity}\label{sec-2.3}
Let $\mathcal M_{+}(\mathbb R^{n+1}_{+})$ represent the class of all nonnegative Radon measures on $\mathbb R^{n+1}_{+}$.
\begin{lemma}\label{lemma 2}
Let $\alpha \in(0,2) $ and $\beta\in (0,n).$ Given $f\in \dot{W}^{\beta,p}(\mathbb{R}^n), s>0,$ and $\mu\in\mathcal M_{+}(\mathbb R^{n+1}_{+}),$ let
$$L^{\alpha,\beta}_{s}(f)=\left\{(x,t)\in \mathbb{R}^{n+1}_+:\ |P_{\alpha} f(x,t)|>s\right\}$$
and
$$R^{\alpha,\beta}_{s}(f)=\Big\{y\in \mathbb{R}^{n}:\ \sup_{|y-x|<t}|P_{\alpha} f(x,t)|>s\Big\}.$$
Then the following four statements are true.
\item{\rm(i)} For any natural number $k$
$$\mu\left(L^{\alpha,\beta}_{s}(f)\cap T(B(0,k))\right)\leq \mu\left(T\left(R^{\alpha,\beta}_{s}(f)\cap B(0,k)\right)\right).$$
\item{\rm (ii)} For any natural number $k,$
$$Cap_{\mathbb{R}^n}^{\beta,p}\left(R^{\alpha,\beta}_{s}(f)\cap B(0,k)\right)\geq c^\beta_p(\mu, \mu\left(T\left(R^{\alpha,\beta}_{s}(f)\cap B(0,k)\right)\right).$$
\item{\rm (iii)} There exists a constant $\theta_\alpha>0$ such that
$$\sup_{|y-x|<t}|P_{\alpha} f (y,t)|\leq \theta_\alpha \mathcal{M} f(x),\quad x\in \mathbb{R}^n,$$
where $\mathcal{M}$ denotes the Hardy-Littlewood maximal operator:
$$\mathcal{M}f(x)=\sup_{r>0}r^{-n}\int_{B(x,r)}|f(y)|dy,\ x\in \mathbb{R}^n.$$
\item{\rm (iv)} There exists a constant $\eta_{n,\alpha}>0$ such that
$$(x,t)\in T(O)\Rightarrow (p^{\alpha}_t\ast
|f|)(x,t) \geq
\eta_{n,\alpha},$$
where $O$ is a bounded open set contained in $\hbox{Int}(\{x\in \mathbb{R}^n: f(x)\geq 1\}).$
\end{lemma}
\begin{proof}
(i) Since $\sup\limits_{|y-x|<t}|P_{\alpha} f(x,t)|$ is lower semicontinuous on $\mathbb{R}^n$, we can see that $R^{\alpha,\beta}_s(f)$ is an open subset of $\mathbb{R}^n$ and
$$\left\{\begin{aligned}
&L^{\alpha,\beta}_s(f)\subseteq T(R^{\alpha,\beta}_s(f));\\
&\mu(L^{\alpha,\beta}_s(f))\leq \mu(T(R^{\alpha,\beta}_s(f))).
\end{aligned}\right.$$
Then
\begin{eqnarray*}
\mu(L^{\alpha,\beta}_s(f)\cap T(B(0,k)))&\leq& \mu(T(R^{\alpha,\beta}_s(f)\cap T(B(0,k))))=\mu(T(R^{\alpha,\beta}_s(f)\cap B(0,k))).
\end{eqnarray*}
(ii) It follows from the definition of $c^\beta_p(\mu;t).$
(iii) Since
$p^{\alpha}_{t}(x)=\frac{1}{t^{n}}\psi(\frac{x}{t})$, where $\psi(x):=(1+|x|^{2})^{-(n+\alpha)/2}$ is radial bounded and integrable on $\mathbb{R}^n,$ it follows from \cite[p.57, Proposition]{Stein} that $$\sup_{|y-x|<t}|P_{\alpha}f(y,t)|\leq \theta_\alpha\mathcal{M}f(x).$$
(iv) For any $(x,t)\in T(O),$ we have
$$B(x,t)\subseteq O\subseteq\hbox{Int }(\{x: f(x)>1\}).$$
There exist $\sigma$ and $C$ which depend only on $n$ and $\alpha$ such that $
\inf\{p^\alpha_t(x): |x|<\sigma t\}\geq Ct^{-n}.$
Then $$p^{\alpha}_t\ast |f|(x,t)\geq Ct^{-n}\int_{B(x,\sigma t)\cap \text{ Int }(\{x: f(x)\geq 1\})} |f|(y)dy.
$$
If $\sigma>1,$ then
$$B(x,\sigma t)\cap \text{ Int }(\{x: f(x)\geq 1\})\supseteq B(x, t)\cap \text{ Int }(\{x: f(x)\geq 1\})=B(x,t).$$
If $\sigma\leq 1,$ then
$$B(x,\sigma t)\cap \text{ Int }(\{x: f(x)\geq 1\})=B(x,\sigma t).$$
Thus $p^\alpha_t\ast|f|(x,t)\geq \eta_{n,\alpha}$ for a constant $\eta_{n,\alpha}>0.$
\end{proof}
The following result provides the capacitary strong estimates for $Cap_{\mathbb{R}^n}^{\beta,p}(\cdot)$. For the proof, we refer the reader to
\cite[Lemma 2.1]{Zhai} and the references therein.
\begin{lemma}\label{lemma 4}
Let $\beta\in (0,n)$ and $p\in [1, {n}/{\beta}]$.
\item{\rm (i)} For $f\in C_0^\infty(\mathbb{R}^n)$,
$$\int_0^\infty Cap_{\mathbb{R}^n}^{\beta,p}\left(\left\{x\in \mathbb{R}^n: |f(x)|\geq s \right\}\right)ds^p\lesssim \|f\|^p_{\dot{W}^{\beta,p}(\mathbb{R}^n)}.$$
\item{\rm (ii)} For $f\in C_0^\infty(\mathbb{R}^n)$,
$$\int_0^\infty Cap_{\mathbb{R}^n}^{\beta,p}\left(\left\{x\in \mathbb{R}^n: |\mathcal{M}f(x)|\geq s \right\}\right)ds^p\lesssim \|f\|^p_{\dot{W}^{\beta,p}(\mathbb{R}^n)}.$$
\end{lemma}
For handling the endpoint case $p=n/\beta$ when $p>q,$ we need the following Riesz potentials on $\mathbb{R}^{2n}$, see Adam-Xiao \cite{Adams Xiao} and Adam \cite{Adams} . For $\gamma\in (0,2n),$
$$I^{(2n)}_\gamma\ast f(z)=\int_{\mathbb{R}^{2n}}|z-y|^{\gamma-2n}f(y)dy, \quad z\in \mathbb{R}^{2n}.$$
For $\gamma\in (0,2n),$ $\dot{\mathcal{L}}^p_{\gamma}(\mathbb{R}^{2n})$ is defined as the completion of $I^{2n}_\gamma\ast C^\infty_0(\mathbb{R}^{2n})$ with $\|I^{2n}_\gamma\ast f\|_{\dot{\mathcal{L}}^p_{\gamma}(\mathbb{R}^{2n})}=\|f\|_{L^{p}(\mathbb{R}^{2n})}.$
The following result is a special case of \cite[Theorem 5.2]{Adams} or \cite[Theorem A]{Adams Xiao}.
\begin{lemma}\label{lemma 3}
Let $\beta\in (0,n).$ Then there are a linear extension operator
$$\mathcal{E}:\ \dot{W}^{\beta,n/\beta}(\mathbb{R}^n)\longrightarrow\dot{\mathcal{L}}^{n/\beta}_{2\beta}(\mathbb{R}^{2n}),$$
and a linear restriction operator
$$\mathcal{R}:\ \dot{\mathcal{L}}^{n/\beta}_{2\beta}(\mathbb{R}^{2n})\longrightarrow\dot{W}^{\beta,n/\beta}(\mathbb{R}^n)$$
such that $\mathcal{R}\mathcal{E}$ is the identity. Moreover,
\item{\rm (i)} For $f\in \dot{W}^{\beta,n/\beta}(\mathbb{R}^n)$,
$ \|\mathcal{E} f\|_{\dot{\mathcal{L}}^{n\beta}_{2\beta}(\mathbb{R}^{2n})}\lesssim\|f\|_{\dot{W}^{\beta,n/\beta}(\mathbb{R}^n)}$.
\item{\rm (ii)} For $g\in \dot{\mathcal{L}}^{n/\beta}_{2\beta}(\mathbb{R}^{2n})$,
$ \|\mathcal{R} g\|_{\dot{W}^{\beta,n/\beta}(\mathbb{R}^n)}\lesssim\|g\|_{\dot{\mathcal{L}}^{n/\beta}_{2\beta}(\mathbb{R}^{2n})}.$
\end{lemma}
\begin{lemma}
\label{lemma 1}
If $\alpha\in(0,2), \beta\in(0,n)$ and $(x,t)\in \mathbb{R}^{n+1}_+,$ then
$$\int_{\mathbb{R}^n}p^{\alpha}_t(y) |y-x|^{\beta-n}dy\lesssim (t^2+|x|^2)^{{(\beta-n)}/{2}}.$$
\end{lemma}
\begin{proof}
Define $$J(x,t)=\int_{\mathbb{R}^n}p^{\alpha}_t(y) |y-x|^{\beta-n}dy
=c(n,\alpha)\int_{\mathbb{R}^n}\frac{t^\alpha|y-x|^{\beta-n}}{(|y|^2+t^2)^{{(n+\alpha)}/{2}}} dy.
$$
Via the change of variables: $x\longrightarrow tx\ \&\ y\longrightarrow ty,$ it is sufficient to show that
$$J(x,1)\lesssim (1+|x|^2)^{{(\beta-n)}/{2}}.$$
Since $J(0,1)\lesssim 1,$ we may assume that $|x|>0.$ Write $J(x,1)\lesssim I_1(x)+I_2(x)$, where
$$\left\{\begin{aligned}
&I_{1}(x):= \int_{B(x,|x|/2)}\frac{|y-x|^{\beta-n}}{(|y|^2+1)^{{(n+\alpha)}/{2}}}dy;\\
&I_{2}(x):=\int_{\mathbb{R}^n\backslash B(x,|x|/2)}\frac{|y-x|^{\beta-n}}{(|y|^2+1)^{{(n+\alpha)}/{2}}}dy.
\end{aligned}\right.$$
Since $|x-y|\leq |x|/2$ implies that $|y|\approx |x|,$ we have
\begin{eqnarray*}
I_1(x)
&\lesssim& (1+|x|^2)^{-{(n+\alpha)}/{2}}\int_{B(x,|x|/2)}|y-x|^{\beta-n}dy\\
&\lesssim& (1+|x|^2)^{{(\beta-n)}/{2}}.
\end{eqnarray*}
If $|x-y|>|x|/2,$ then $|y-x|^{\beta-n}\lesssim |x|^{\beta-n},$ thus
\begin{eqnarray*}
I_2(x)&\lesssim&|x|^{\beta-n}\int_{\mathbb{R}^n\backslash B(x,|x|/2)}\frac{1}{(|y|^2+1)^{{(n+\alpha)}/{2}}}dy\ \lesssim\ |x|^{\beta-n}.
\end{eqnarray*}
For $|x-y|>|x|/2$, it holds $|y|<3|x-y|$ and
\begin{eqnarray*}
I_2(x)=\int_{\mathbb{R}^n\backslash B(x,|x|/2)}\frac{1}{(|y|^2+1)^{{(n+\alpha)}/{2}}|y|^{n-\beta}}dy\lesssim1.
\end{eqnarray*}
So, $I_2(x)\lesssim (1+|x|^2)^{{(\beta-n)}/{2}}$ and $J(x,1)\lesssim (1+|x|^2)^{{(\beta-n)}/{2}}.$
\end{proof}
\section{Embeddings of $L^p(\mathbb{R}^{n})$ to $L^q(\mathbb{R}^{n+1}_+,\mu)$}\label{sec-3}
In this section, we focus on the embedding \eqref{5}:
$$\|P_{\alpha} f(\cdot,\cdot)\|_{L^{q}(\mathbb{R}^{n+1}_+,\mu)}\lesssim \|f\|_{L^p(\mathbb R^n)}
.$$
\subsection{Embeddings of $L^p(\mathbb{R}^{n})$ to $L^q(\mathbb{R}^{n+1}_+,\mu)$ when $p\leq q$}
Given $1<p\leq q<\infty$. For $\lambda>0$, define
$$c_{\alpha,p}(\mu; \lambda):=\inf\left\{C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K):\ \text{ compact }K\subset \mathbb R^{n+1}_{+},\quad \mu(K)\geq \lambda\right\}.$$
\begin{theorem}\label{thm-3.1}
Let $1<p\leq q<\infty$ and $\mu\in \mathcal{M}_{+}(\mathbb R^{n+1}_{+})$.
\item{\rm (i)} The embedding \eqref{5} holds if and only if $$\sup_{\lambda\in(0,\infty)}\lambda^{p/q}/c_{\alpha,p}(\mu;\lambda)<\infty.$$
\item{\rm (ii)} If $1<p<q<\infty$, then $\sup_{\lambda\in(0,\infty)}\lambda^{p/q}/c_{\alpha,p}(\mu;\lambda)<\infty$ can be replaced by
$$\sup_{ (r, x_{0}, t_{0})\in(0,\infty)\times\mathbb R^{n+1}_+, t_{0}\lesssim r}\frac{\mu(B_{r}(x_{0}, t_{0})) }{r^{qn/p}}<\infty.$$
\end{theorem}
\begin{proof}
(i) Suppose that \eqref{5} is true. Let $K$ be a compact set in $\mathbb R^{n+1}_{+}$. Denote by $\mu\mid_{K}$ the restriction of $\mu$ to the set $K$.
\begin{eqnarray*}
\int_{\mathbb R^{n}}f(x)P^{\ast}_{\alpha}\mu\mid_{K}(x)dx &=&\int_{\mathbb R^{n+1}_{+}}P_{\alpha}f(x,t)d\mu\mid_{K}(x,t)\\
&\lesssim& \|P_{\alpha}f\|_{L^{q}(\mathbb R^{n+1}_{+},\mu)}(\mu(K))^{1/q'}
\lesssim \|f\|_{L^{p}(\mathbb R^{n})}(\mu(K))^{1/q'},
\end{eqnarray*}
which means that
$\|P^{\ast}_{\alpha}\mu\mid_{K}\|_{L^{p'}(\mathbb R^{n})}\lesssim (\mu(K))^{1/q'}$. Define
$$E_{\lambda}(f)=\left\{(x,t)\in\mathbb R^{n+1}_{+}:\ |P_{\alpha}f(x,t)|\geq\lambda\right\}.$$
We can get
\begin{eqnarray*}
\lambda\mu(E_{\lambda}(f))&\leq&\int_{E_{\lambda}(f)}|P_{\alpha}f(x,t)|d\mu\mid_{E_{\lambda}(f)}(x,t)\\
&\leq&\|f\|_{L^{p}(\mathbb R^{n})}\|P^{\ast}_{\alpha}\mu\mid_{E_{\lambda}(f)}\|_{L^{p'}}\\
&\leq&\|f\|_{L^{p}(\mathbb R^{n})}(\mu(E_{\lambda}(f)))^{1/q'}.
\end{eqnarray*}
This implies $$\sup_{\lambda\in(0,\infty)}\lambda^{q}(\mu(E_{\lambda}(f)))\lesssim \|f\|_{L^{p}(\mathbb R^{n})}^{q}.$$
Choose a function $f\in L^{p}(\mathbb R^{n})$ such that $P_{\alpha}f\geq1$ on a given compact set $K\subset\mathbb R^{n+1}_{+}$, i.e., $K\subset E_{1}(f)$. We have
\begin{equation}\label{eq-4.4}
(\mu(K))^{1/q}\leq (\mu(E_{1}(f)))^{1/q}\lesssim \|f\|_{L^{p}(\mathbb R^{n})}.
\end{equation}
Take the infimum on both sides of (\ref{eq-4.4}), we obtain $(\mu(K))^{1/q}\lesssim (C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K))^{1/p}$. If $K$ is compact and $\mu(K)\geq \lambda$, then $\lambda^{p/q}\lesssim C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K)$ and hence, $\lambda^{p/q}\lesssim c(\mu;\lambda)$ due to
$$\lambda^{p/q}\lesssim\inf\left\{C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K),\ K \text{ is compact and }\mu(K)\geq\lambda \right\}.$$
Conversely, if
$\sup\limits_{\lambda\in(0,\infty)}\lambda^{p/q}/c_{\alpha,p}(\mu;\lambda)<\infty,$
then for any $\lambda>0$, $\lambda^{p/q}\lesssim C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K)$.
For any $\tau$, according to Lemma \ref{le-4.2}, one has
\begin{eqnarray*}
\tau^{p}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E_{\tau})&=&\int^{\tau}_{0}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}\left(\left\{(x,t)\in\mathbb R^{n+1}_{+}:\ P_{\alpha}f(x,t)\geq\tau\right\}\right)d\lambda^{p}\\
&\lesssim& \int^{\tau}_{0}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}\left(\left\{(x,t)\in\mathbb R^{n+1}_{+}:\ P_{\alpha}f(x,t)\geq\lambda\right\}\right)d\lambda^{p}\\
&\lesssim& \int^{\infty}_{0}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}\left(\left\{(x,t)\in\mathbb R^{n+1}_{+}:\ P_{\alpha}f(x,t)\geq\lambda\right\}\right)d\lambda^{p}\lesssim \|f\|_{L^{p}(\mathbb R^{n})}^{p}.
\end{eqnarray*}
By the layer cake representation, this implies
\begin{eqnarray*}
\int_{\mathbb R^{n+1}_{+}}|P_{\alpha}f(x,t)|^{q}d\mu(x,t)&=&\int^{\infty}_{0}\mu(E_{\lambda})d\lambda^{q}\\
&\lesssim&\int^{\infty}_{0}\left(C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E_{\lambda})\right)^{q/p}d\lambda^{q}\\
&\lesssim&\int^{\infty}_{0}\left(\lambda^{-p}\|f\|_{L^{p}(\mathbb R^{n})}^{p}\right)^{q/p-1}\left(C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E_{\lambda})\right)\lambda^{q-1}d\lambda\\
&\lesssim&\|f\|^{q-p}_{L^{p}(\mathbb R^{n})}\int^{\infty}_{0}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(E_{\lambda})d\lambda^{p}\\
&\lesssim&\|f\|^{q}_{L^{p}(\mathbb R^{n})}.
\end{eqnarray*}
(ii) If $\lambda^{p/q}\lesssim c(\mu;\lambda)$, then $\mu^{1/q}(K)\lesssim (C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K))^{1/p}$ for any compact set $K\subset \mathbb R^{n+1}_{+}$. Let $K=B_{r}(x_{0}, t_{0})$. By Theorem \ref{th-4.1}, for $t_{0}\lesssim r$,
$$(\mu(B_{r}(x_{0}, t_{0})))^{1/q}\lesssim (C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(B_{r}(x_{0}, t_{0})))^{1/p}\lesssim r^{n/p}.$$
For the reverse, take a compact set $K$ such that $\mu(K) \geq \lambda$. If $(x,t)\in B_{r}(x_{0}, t_{0})$, $|x-x_{0}|<r/2$ and $r+t_{0}\leq t\leq t_{0}+2r$. If $|x-x_{0}|<r<t$, we can get $|x-x_{0}|/t<1$ and
\begin{eqnarray*}
p^{\alpha}_{t}(x-x_{0})&=&\frac{t^{\alpha}}{(t^{2}+|x-x_{0}|^{2})^{(n+\alpha)/2}}\\
&=&\frac{1}{t^{n}}\frac{1}{(1+|x-x_{0}|^{2}/t^{2})^{(n+\alpha)/2}}\\
&\geq&\frac{1}{2^{(n+\alpha)/2}}\frac{1}{t^{n}}\\
&\gtrsim&\frac{1}{3^{n}r^{n}}\frac{1}{2^{(n+\alpha)/2}},
\end{eqnarray*}
equivalently, there exists a constant $c$ such that $r\geq c(p_{t}^{\alpha}(x-x_{0}))^{-1/n}$. Hence if $(x,t)\in B_{r}(t_{0}, x_{0})$, then
$r\in (c(p_{t}^{\alpha}(x-x_{0}))^{-1/n}, \infty)$. For this case, the characteristic functions
$$1_{B_{r}(x_{0},t_{0})}(x,t)=1_{(c(p_{t}^{\alpha}(x-x_{0}))^{-1/n}, \infty)}(r)=1.$$
By Fubini's theorem, we can get
\begin{eqnarray*}
\int^{\infty}_{0}\mu\mid_{K}(B_{r}(x_{0},t_{0}))\frac{dr}{r^{n+1}}
&=&\int^{\infty}_{0}\left\{\int_{\mathbb R^{n+1}_{+}}1_{B_{r}(x_{0},t_{0})}(x,t)d\mu\mid_{K}(x,t)\right\}\frac{dr}{r^{n+1}}\\
&=&\int_{\mathbb R^{n+1}_{+}}\left\{\int^{\infty}_{0}1_{(c(p_{t}^{\alpha}(x-x_{0}))^{-1/n}, \infty)}(r)\frac{dr}{r^{n+1}}\right\}d\mu\mid_{K}(x,t)\\
&=&\int_{\mathbb R^{n+1}_{+}}\left\{\int^{\infty}_{(c(p_{t}^{\alpha}(x-x_{0}))^{-1/n}}\frac{dr}{r^{n+1}}\right\}d\mu\mid_{K}(x,t)\\
&\gtrsim&\int_{\mathbb R^{n+1}_{+}}p^{\alpha}_{t}(x_{0}-x)d\mu\mid_{K}(x,t)\\
&=&P^{\ast}_{\alpha}\mu\mid_{K}(x_{0}).
\end{eqnarray*}
By Minkowski's inequality, we have
$$\|P^{\ast}_{\alpha}\mu\mid_{K}\|_{L^{p'}(\mathbb R^{n})}\lesssim \int^{\infty}_{0}\|\mu\mid_{K}(B_{r_{0}}(\cdot,t_{0}))\|_{L^{p'}(\mathbb R^{n})}\frac{dr}{r^{n+1}}.$$
Take $\delta=(\mu(K))^{p/nq}$. On the one hand,
\begin{eqnarray*}
\|\mu\mid_{K}(B_{r}(\cdot,t_{0}))\|_{L^{p'}(\mathbb R^{n})}^{p'}
&=&\int_{\mathbb R^{n}}\left(\mu(K\cap B_{r}(x_{0}, t_{0}))\right)^{p'}dx_{0}\\
&\lesssim&(\mu(K))^{p'-1}\int_{\mathbb R^{n}}\mu(K\cap B_{r}(x_{0}, t_{0}))dx_{0}\\
&=&(\mu(K))^{p'-1}\int_{\mathbb R^{n}}\left(\int_{K\cap B_{r}(x_{0}, t_{0})}1d\mu(t,y)\right)dx_{0}\\
&\lesssim&(\mu(K))^{p'-1}\int_{K}\left(\int_{|y-x_{0}|<r}dx_{0}\right)d\mu(t,y)\\
&\lesssim&(\mu(K))^{p'}r^{n}.
\end{eqnarray*}
The above estimate gives
\begin{eqnarray*}
\int^{\infty}_{\delta}\|\mu\mid_{K}(B_{r}(\cdot,t_{0}))\|_{L^{p'}(\mathbb R^{n})}\frac{dr}{r^{n+1}}&\lesssim&\int^{\infty}_{\delta}\mu(K)r^{n/p'-n-1}dr
\lesssim \mu(K)\delta^{-n/p}.
\end{eqnarray*}
Meanwhile, on the other hand, since $\mu(B_{r}(x_{0}, t_{0}))\lesssim r^{nq/p}$, we can obtain
\begin{eqnarray*}
\|\mu\mid_{K}(B_{r}(\cdot,t_{0}))\|_{L^{p'}(\mathbb R^{n})}^{p'} &\lesssim& r^{nq(p'-1)/p}\int_{\mathbb R^{n}}\mu\mid_{K}(B_{r}(t_{0},x_{0}))dx_{0}\\
&=&r^{nq(p'-1)/p}\int_{\mathbb R^{n}}\left(\int_{K\cap B_{r}(x_{0}, t_{0})}1d\mu(t,y)\right)dx_{0}\\
&\lesssim&r^{nq(p'-1)/p}\int_{K}\left(\int_{|y-x_{0}|<r}dx_{0}\right)d\mu(t,y)\\
&\lesssim& \mu(K)r^{n+nq/(p^{2}-p)}.
\end{eqnarray*}
Hence, noting that $q>p$, we obtain
$$\int^{\delta}_{0}\|\mu\mid_{K}(B_{r}(\cdot,t_{0}))\|_{L^{p'}(\mathbb R^{n})}\frac{dr}{r^{1+n}}\lesssim (\mu(K))^{1/p'}\delta^{n(q-p)/p^{2}}.$$
Finally, we get
\begin{eqnarray}\label{eq-3.3}
\|P^{\ast}_{\alpha}\mu\mid_{K}\|_{L^{p'}(\mathbb R^{n})}&\lesssim&\int^{\infty}_{0}\|\mu\mid_{K}(B_{r}(\cdot,t_{0}))\|_{L^{p'}(\mathbb R^{n})}\frac{dr}{r^{n+1}}\\
&=&\left(\int^{\delta}_{0}+\int_{\delta}^{\infty}\right)\|\mu\mid_{K}(B_{r}(\cdot,t_{0}))\|_{L^{p'}(\mathbb R^{n})}\frac{dr}{r^{n+1}}\nonumber\\
&\lesssim&(\mu(K))^{1/p'}\delta^{n(q-p)/p^{2}}+(\mu(K))\delta^{-n/p}\nonumber\\
&\lesssim&(\mu(K))^{1/q'}.\nonumber
\end{eqnarray}
Below we prove that
$(\mu(K))^{1/q}\leq (C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K))^{1/p}$ for any compact set $K\subset \mathbb R^{n+1}_{+}$.
In fact, let $f\in C^{\infty}_{0}(\mathbb R^{n})$ and $E_{\lambda,K}=\{(x,t)\in K:\ |P_{\alpha}\ast f(x,t)|\geq \lambda\}$. Then the set $E_{\lambda,K}$ is compact.
It follows from (\ref{eq-3.3}) that
\begin{eqnarray}\label{eq-3.4}
\lambda\mu(E_{\lambda,K})&\leq&\int_{\mathbb R^{n}}|P_{\alpha}f(x,t)|d\mu\mid_{E_{\lambda,K}}\\
&\leq&\|f\|_{L^{p}(\mathbb{R}^n)}\|P^{\ast}_{\alpha}\mu\mid_{E_{\lambda,K}}\|_{L^{p'}}\nonumber\\
&\lesssim&\|f\|_{L^{p}(\mathbb{R}^n)}(\mu(E_{\lambda,K}))^{1/q'}.\nonumber
\end{eqnarray}
For an arbitrary $f\in L^{p}(\mathbb R^{n})$, via approximating $f$ by a sequence from $C^{\infty}_{0}(\mathbb R^{n})$ in the $L^{p}$-norm, we can prove that (\ref{eq-3.4}) holds for $f$. For $f\in L^{p}(\mathbb R^{n})$ such that $P_{\alpha}f\geq 1$ on $K,$ we have $E_{\lambda,K}=K.$ Then (\ref{eq-3.4}) implies $(\mu(K))^{1/q}\lesssim \|f\|_{L^{p}(\mathbb{R}^n)}$, which gives $(\mu(K))^{1/q}\leq (C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K))^{1/p}.$ Recall that $\mu(K)\geq \lambda$. Then taking the infimum over the compact sets $K$ such that $\mu(K) \geq \lambda$, we get
$\lambda^{p/q}\lesssim C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K)$, i.e.,
$$\lambda^{p/q}\lesssim \inf\left\{C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K):\ \text{ compact }K\subset \mathbb R^{n+1}_{+},\quad \mu(K)\geq \lambda\right\}=c_{\alpha,p}(\mu; \lambda).$$
\end{proof}
\subsection{Embeddings of $L^p(\mathbb{R}^{n})$ to $L^q(\mathbb{R}^{n+1}_+,\mu)$ when $p>q$ }
Let $M\mu(x)=\sup_{r>0}r^{-n}\mu\left({B_{r}(x, r)}\right)$ be the parabolic maximal function of a nonnegative Radon measure $\mu$ on $\mathbb R^{n+1}_+$. We show the embedding for $p>q$ inspired by some ideas from \cite{Shi Xiao}, which needs the following $L^p$-boundedness of $M\mu$.
\begin{lemma}\label{l21} Let $1<p<\infty$. Then
$\|M\mu\|_{L^{p}(\mathbb{R}^{n})}\approx \|P^\ast_\alpha\mu\|_{L^p(\mathbb R^n)}.$
\end{lemma}
\begin{proof}
It is easy to check that for any $r>0$,
$$
P^\ast_\alpha\mu(x)\gtrsim \int_{B_{r}(x, r)}\frac{t^{\alpha}}{(t^{2}+|x-y|^{2})^{{(n+\alpha)}/{2}}}d\mu(y,t)\gtrsim r^{-n}\mu(B_{r}(x, r)),
$$
and so that
$ \|M\mu\|_{L^{p}(\mathbb{R}^{n})}\lesssim \|P^\ast_{\alpha}\mu\|_{L^{p}(\mathbb{R}^{n})}.$
Next, we are about to prove the converse inequality by a slight modification of \cite[Theorem 3.6.1]{AH}.
Denote by $E_{\mu}(T,\lambda,r)=\{x\in \mathbb{R}^{n}:T\mu(x)>\lambda r\}$ for an operator $T$ and $(\lambda,r) \in (\mathbb{R}, \mathbb{R})$. Then
by the famous good $\lambda$ inequality, there exist two constants $c_{1}>1$ and $c_{2}>0$ such that
$$
\left |E_{\mu}(P^\ast_{\alpha},c_{1},\rho)\right|\leq c_{2}\theta^{{(n+\alpha)}/{n}}\left|E_{\mu}(P^\ast_{\alpha},1,\rho)\right|
+\left|E_{\mu}(M,\theta,\rho)\right|\,\,\,\,\hbox{for any}\,\, \rho>0\,\,\hbox{and}\,\, 0<\theta\leq 1.
$$
Hence, for any $\tau>0$, we get
$$
\int_{0}^{\tau}\left|E_{\mu}(P^\ast_{\alpha},c_{1},\rho)\right|\rho^{p-1}d\rho \leq c_{2}\theta^{{(n+\alpha)}/{n}}\int_{0}^{\tau}\left|E_{\mu}(P^\ast_{\alpha},1,\rho)\right|\rho^{p-1}d\rho+\int_{0}^{\tau}\left|E_{\mu}(M,\theta,\rho)\right|\rho^{p-1}d\rho.
$$
Namely,
$$
c_{1}^{-p}\int_{0}^{c_{1}\tau}\left|E_{\mu}(P^\ast_{\alpha},1,\rho)\right|\rho^{p-1}d\rho\leq c_{2}\theta^{{(n+\alpha)}/{n}}\int_{0}^{\tau}\left|E_{\mu}(P^\ast_{\alpha},1,\rho)\right|\rho^{p-1}d\rho+\theta^{-p}\int_{0}^{\theta \tau}\left|E_{\mu}(M,1,\rho)\right|\rho^{p-1}d\rho.
$$
Accordingly,
$$
c_{1}^{-p}\int_{\mathbb{R}^{n}}(P^\ast_{\alpha}\mu(x))^{p}dx\leq 2\theta^{-p}\int_{\mathbb{R}^{n}}(M\mu(x))^{p}dx
$$
by letting $\theta$ be so small such that $c_{2}\theta^{{(n+\alpha)}/{n}}\leq \frac{1}{2}c_{1}^{-p}$ and $\tau\to\infty$.
The foregoing analysis yields
$$
\|M\mu\|_{L^{p}(\mathbb{R}^{n})}\gtrsim \|P^\ast_{\alpha}\mu\|_{L^{p}(\mathbb{R}^{n})},
$$
which is the desired one.
\end{proof}
\begin{lemma}\label{l22}
Denote by
$$ H_{p}\mu(x,t):=\int_{0}^{\infty}\left(r^{-n}\mu(B_{r}(x, t))\right)^{p'-1}r^{-1}dr$$
the Hedberg-Wolff potential of $\mu.$ Let $1<p<\infty$ and $\mu\in \mathcal M_{+}(\mathbb R^{n+1}_{+}).$ Then one has
$$\|P_{\alpha}^{*}\mu\|_{L^{p'}(\mathbb{R}^{n})}^{p'}\approx \int_{\mathbb{R}_{+}^{n+1}}H_{p}\mu\, d\mu.$$
\end{lemma}
\begin{proof}
We conclude from the fact
$$
\frac{\mu(B_{r}(x, r))}{r^{n}}\lesssim \left(\int_{r}^{2r}\left(\frac{\mu(B_{s}(x, s))}{s^{n}}\right)^{p'}\frac{ds}{s}\right)^{{1}/{p'}}
$$
that
$$
M\mu(x)\lesssim \left(\int_{0}^{\infty}\left(\frac{\mu(B_{s}(x, s))}{s^{n}}\right)^{p'}\frac{ds}{s}\right)^{{1}/{p'}}.
$$
Since
$$
\int_{\mathbb{R}^{n}}\int_{0}^{\infty}\left(\frac{\mu(B_{r}(x, r))}{r^{n}}\right)^{p'}\frac{drdx}{r}=\int_{0}^{\infty}\int_{\mathbb{R}^{n}}\frac{\mu(B_{r}(x, r))^{p'}}{r^{np'+1}}dxdr,$$
by the Fubini theorem,
$$
\int_{\mathbb{R}^{n}}\mu(B_{r}(x, r))^{p'}dx \lesssim\int_{B_{r}(y, r)}\int_{\mathbb{R}^{n}}\mu(B_{r}(x, r))^{p'-1}dxd\mu\lesssim r^{n}\int_{B_{\frac{3r}{2}}(y, \frac{r}{2})}\mu(B_{r}(y, r))^{p'-1}d\mu.
$$
Accordingly,
$$
\int_{0}^{\infty}\int_{\mathbb{R}^{n}}\frac{\mu(B_{r}(x, r))^{p'}}{r^{np'+1}}dxdr
\lesssim \int_{\mathbb{R}_{+}^{1+n}}\left(\int_{0}^{\infty}\left(\frac{\mu(B_{r}(x, r))}{r^{n}}\right)^{p'-1}\frac{dr}{r}\right)d\mu,
$$
which, together with Lemma \ref{l21}, gives
$$\|P^\ast_{\alpha}\mu\|_{L^{p'}(\mathbb{R}^{n})}^{p'}\lesssim \int_{\mathbb{R}_{+}^{n+1}}H_{p}\mu\,d\mu.$$
Write
$$B(y,2^{-m})=\left\{x\in\mathbb R^n:\ |x-y|<2^{-m}\ \&\ 2^{-m}<t<2^{1-m}\right\}\quad \forall\,\, m\in\mathbb Z\equiv\left\{0,\pm 1,\pm 2,\ldots\right\}.$$
The converse inequality is a by-product of the following estimate
\begin{align*}
\int_{\mathbb{R}^{n}}(P^\ast_{\alpha}\mu(x))^{p'-1}p_{t}^{\alpha}(x-y)dx
&\quad\approx\int_{\mathbb{R}^{n}}\frac{t^{\alpha}}{(t^{2}+|x-y|^{2})^{{(n+\alpha)}/ {2}}}
\left(\int_{\mathbb{R}_{+}^{n+1}}\frac{s^{\alpha}}{(s^{2}+|x-z|^{2})^{{(n+\alpha)}/{2}}}d\mu\right)^{p'-1}dx\\
&\quad\gtrsim \sum_{m\in \mathbb{Z}}\int_{B(y,2^{-m})}t^{-n}
\left(\int_{B_{2^{-m}}(y, t)}s^{-n}d\mu\right)^{p'-1}dx\\
&\quad\gtrsim \sum_{m\in \mathbb{Z}}\int_{B(y,2^{-m})}{2^{mn}}
\left(\frac{\mu(B_{2^{-m}}(y, t))}{2^{-mn}}\right)^{p'-1}dx\\
&\quad\gtrsim \int_{0}^{\infty}
\left(\frac{\mu(B_{r}(y, t))}{r^{n}}\right)^{p'-1}\frac{dr}{r}
\end{align*}
since
$$
\|P^\ast_{\alpha}\mu\|_{L^{p'}(\mathbb{R}^{n})}^{p'}=\int_{\mathbb{R}^{n}}\left(P^\ast_{\alpha}\mu(x)\right)^{p'-1}(P^\ast_{\alpha}\mu(x))dx=\int_{\mathbb{R}_{+}^{n+1}}\int_{\mathbb{R}^{n}}(P^\ast_{\alpha}\mu(x))^{p'-1}p_{t}^{\alpha}(x-y)dx\,d\mu(y,t).
$$
Therefore,
$$\|P^\ast_{\alpha}\mu\|_{L^{p'}(\mathbb{R}^{n})}^{p'}\gtrsim \int_{\mathbb{R}_{+}^{n+1}}H_{p}\mu\,d\mu
$$
as desired.
\end{proof}
Now, we are ready to characterize the embedding \eqref{5} for $1<q<p<\infty$ as follows.
\begin{theorem}\label{Them 3.4}
Let $1<q<p<\infty$ and $\mu\in\mathcal M_{+}(\mathbb R^{n+1}_{+}).$ The following statements are true.
\item{\rm (i)} The embedding \eqref{5} holds if and only if
$$\int^{\infty}_{0}\left(\frac{\lambda^{p/q}}{c(\mu; \lambda)}\right)^{q/(p-q)}\frac{d\lambda}{\lambda}<\infty.$$
\item{\rm (ii)} The embedding \eqref{5} holds if and only if
$$\int_{\mathbb R^{n+1}_+}\left\{\int_0^\infty \left(\frac{\mu(B_{r}(x_{0}, t_{0}))}{C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(B_{r}(x_{0}, t_{0}))}\right)^{1/(p-1)}\,\frac{dr}{r}\right\}^{{q(p-1)}/{(p-q)}}\,d\mu(x_0,t_0)<\infty.
$$
\item{\rm (iii)} The embedding \eqref{5} holds if and only if $H_{p}\mu \in L^{q(p-1)/(p-q)}(\mathbb{R}_{+}^{n+1}, \mu).$
\end{theorem}
\begin{proof}
It is easy to see that the statement (ii) is a consequence of the statement (iii). We only need to show statements (i)\ $\&$\ (iii). The rest of the proof is divided into two parts.
{\it Part I: Proof of statement (i)}.
Let $P_{\alpha}: L^{p}(\mathbb R^{n})\rightarrow L^{q}(\mathbb R^{n+1}_{+},\mu)$ be bounded. Then
$$\left(\int_{\mathbb R^{n+1}_{+}}|P_{\alpha}f(x,t)|^{q}d\mu\right)^{1/q}\lesssim \|f\|_{L^{p}(\mathbb R^{n})}.$$
By Markov's inequality, we can get
$$\sup_{\lambda>0}\lambda\left(\mu(E_{\lambda}(f))\right)^{1/q}\lesssim \|f\|_{L^{p}(\mathbb R^{n})},$$
where $E_\lambda(f)=\{(x,t)\in
\mathbb{R}^{n+1}_+: |P_{\alpha}f(x,t)|\geq \lambda\}.$
By the layer cake representation, we get
$$\left(\int^{\lambda}_{0}\mu(E_{\tau}(f))\tau^{q-1}d\tau\right)^{1/q}\lesssim\|f\|_{L^{p}(\mathbb R^{n})}\quad \forall\ \lambda>0.$$
Then we have
$$(\mu(E_{\lambda}(f)))^{1/q}\left(\int^{\lambda}_{0}\tau^{q-1}d\tau\right)^{1/q}\lesssim\|f\|_{L^{p}(\mathbb R^{n})}.$$
For each integer $j$, there is a compact set $K_{j}\subset \mathbb R^{n+1}_{+}$ and a nonnegative function $f_{j}\in L^{p}(\mathbb R^{n})$ such that
$$\begin{cases}
C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K_{j})\leq 2c_{\alpha,p}(\mu; 2^{j});\\
\mu(K_{j})>2^{j};\\
P_{\alpha}f_{j}\geq 1_{K_{j}};\\
\|f_{j}\|^{p}_{L^{p}(\mathbb R^{n})}\leq 2C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K_{j}).
\end{cases}$$
For the integers $i,k$ with $i<k$, let
$f_{i,k}:=\sup\limits_{i\leq j\leq k}\left(\frac{2^{j}}{c_{\alpha,p}(\mu; 2^{j})}\right)^{1/(p-q)}f_{j}.$
Then
\begin{eqnarray}\label{eq-3.5}
\|f_{i,k}\|^{p}_{L^{p}(\mathbb R^{n})}&\leq&\sum^{k}_{j=i}\left(\frac{2^{j}}{c_{\alpha,p}(\mu; 2^{j})}\right)^{{p}/{(p-q)}}\|f_{j}\|^{p}_{L^{p}(\mathbb R^{n})}\\
&\lesssim&\sum^{k}_{j=i}\left(\frac{2^{j}}{c_{\alpha,p}(\mu; 2^{j})}\right)^{{p}/{(p-q)}}C^{\alpha,p}_{\mathbb{R}^{n+1}_+}(K_{j})\nonumber\\
&\lesssim&\sum^{k}_{j=i}\left(\frac{2^{j}}{c_{\alpha,p}(\mu; 2^{j})}\right)^{{p}/{(p-q)}}c_{\alpha,p}(\mu; 2^{j}).\nonumber
\end{eqnarray}
Note that if $(x,t)\in K_{j}$, then
\begin{eqnarray}\label{eq-3.2}
P_{\alpha}f_{i,k}(x,t)&=&P_{\alpha}\left(\sup_{i\leq l\leq k}\left(\frac{2^{j}}{c_{\alpha,p}(\mu; 2^{l})}\right)^{1/(p-q)}f_{l}(x,t)\right)\\
&\geq&\left(\frac{2^{j}}{c_{\alpha,p}(\mu; 2^{j})}\right)^{1/(p-q)}P_{\alpha}f_{j}(x,t)\nonumber\\
&\geq&\left(\frac{2^{j}}{c_{\alpha,p}(\mu; 2^{j})}\right)^{1/(p-q)}.\nonumber
\end{eqnarray}
It can be seen from (\ref{eq-3.2}) that
$$K_{j}\subset \left\{(x,t)\in\mathbb R^{n+1}_{+}:\ P_{\alpha}f_{i,k}(x,t)\geq\left(\frac{2^{j}}{c_{\alpha,p}(\mu; 2^{j})}\right)^{1/(p-q)}\right\}.$$
This means that
\begin{equation}\label{eq-3.6}
2^{j}<\mu(K_{j})\leq\mu\left(E_{({2^{j}}/{c_{\alpha,p}(\mu; 2^{j})})^{1/(p-q)}}(f_{i,k})\right).
\end{equation}
In view of the embedding (\ref{5}), we can use the (nonsymmetric) decreasing rearrangement (\cite[Theorem 1.8]{Bennett}), and then a simple computation to obtain
\begin{eqnarray*}
\|f_{i,k}\|_{L^{p}(\mathbb R^{n})}^{q}&\gtrsim&\int_{\mathbb R^{n+1}_{+}}|P_{\alpha}(f_{i,k})(x,t)|^{q}d\mu(x,t)\\
&\approx&\int^{\infty}_{0}\left(\inf\left\{\lambda:\ \mu(E_{\lambda}(f_{i,k}))\leq s\right\}\right)^{q}ds.
\end{eqnarray*}
Notice that (\ref{eq-3.6}) indicates that
$$\inf \Big\{ \lambda:\ \mu(E_\lambda(f_{i,k}) \leq 2^j \Big\} > \Big(\frac{2^j}{c_{\alpha,p}(\mu;2^j)}\Big)^{1/(p-q)}.$$
Then we split the integral to get
\begin{eqnarray*}
\|f_{i,k}\|_{L^{p}(\mathbb R^{n})}^{q}&\geq&\sum^{k}_{j=i}2^{j}\left(\inf\left\{\lambda:\ \mu(E_{\lambda}(f_{i,k}))\leq 2^{j}\right\}\right)^{q}\geq\sum^{k}_{j=i}2^{j}\left(\frac{2^{j}}{c_{\alpha,p}(\mu; 2^{j})}\right)^{q/(p-q)}.
\end{eqnarray*}
Finally, applying (\ref{eq-3.5}) and a direct computation, we have
\begin{eqnarray*}
\|f_{i,k}\|_{L^{p}(\mathbb R^{n})}^{q}&\geq&\left(\frac{\sum\limits_{j=i}^k\left({2^j}/{c_{\alpha,p}(\mu;2^j)}\right)^{{q}/{(p-q)}}2^j}{\left(\sum\limits_{j=i}^k
\left({2^j}/{c_{\alpha,p}(\mu;2^j)}\right)
^{{p}/{(p-q)}}c_{\alpha,p}(\mu;2^j)\right)^{{q}/{p}}}\right)\|f_{i,k}\|_{L^p(\mathbb{R}^{n})}^q\\
&\approx& \left(\sum_{j=i}^k\frac{2^{{jp}/{(p-q)}}}{\left(c_{\alpha,p}(\mu;2^j)\right)^{{q}/{(p-q)}}}\right)^{{(p-q)}/{p}}
\|f_{i,k}\|_{L^p(\mathbb{R}^{n})}^q,
\end{eqnarray*}
which implies
$$\sum_{j=i}^k\frac{2^{{jp}/{(p-q)}}}{\left(c_{\alpha,p}(\mu;2^j)\right)^{{q}/{(p-q)}}}\lesssim 1,$$
and hence
$$
\int_0^\infty\left(\lambda^{p/q}/c_{\alpha,p}(\mu;\lambda)\right)^{{q}/{(p-q)}}\lambda^{-1}\,d\lambda\lesssim
\sum_{j=-\infty}^\infty\frac{2^{{jp}/{(p-q)}}}{\left(c_{\alpha,p}(\mu;2^j)\right)^{{q}/{(p-q)}}}\lesssim
1.
$$
Conversely, let
$$
I_{p,q}(\mu)=\int_0^\infty\left(\frac{\lambda^{p/q}}{c_{\alpha,p}(\mu;\lambda)}\right)^{{q}/{(p-q)}}\frac{d\lambda}{\lambda}<\infty.
$$
Now for $f\in C^{\infty}_0(\mathbb{R}^{n})$, let
$$
S_{p,q}(\mu;f)=\sum_{j=-\infty}^\infty
\frac{\left(\mu\left(E_{2^j}(f)\right)-\mu\left(E_{2^{j+1}}(f)\right)\right)^{{p}/{(p-q)}}}{\left(C^{\alpha,p}_{\mathbb R^{n+1}_{+}}\left(E_{2^j}(f)\right)\right)^{{q}/{(p-q)}}}.
$$
Using layer-cake representation, H\"older's inequality and Lemma \ref{le-4.2}, we obtain
\begin{align*}
&\int_{ \mathbb{R}^{n+1}_+}|P_{\alpha}f(x,t)|^q\, d\mu(x,t)\\
&\ \ =q\int_{0}^{\infty}\lambda^{q-1}\mu(E_\lambda(f))\, d\lambda\\
&\ \ \lesssim\sum_{j=-\infty}^\infty\left(\mu\left(E_{2^j}(f)\right)-\mu(E_{2^{j+1}}(f)\right)2^{jq}\\
&\ \ \lesssim(S_{p,q}(\mu;f))^{{(p-q)}/{p}}
\left(\sum_{j=-\infty}^\infty 2^{jp}C^{\alpha,p}_{\mathbb R^{n+1}_{+}}\left(E_{2^j}(f)\right)\right)^{q/p}\\
&\ \ \lesssim(S_{p,q}(\mu;f))^{{(p-q)}/{p}}\left(\int_0^\infty C^{\alpha,p}_{\mathbb R^{n+1}_{+}}\left(\left\{(x,t)\in \mathbb{R}^{n+1}_+:\ |P_{\alpha}f(x,t)|>\lambda\right\}\right)\,d\lambda^p\right)^{{q}/{p}}\\
&\ \ \lesssim(S_{p,q}(\mu;f))^{{(p-q)}/{p}}\|f\|_{L^p(\mathbb{R}^{n})}^q.
\end{align*}
Note also that
\begin{align*}
\left(S_{p,q}(\mu;f)\right)^{{(p-q)}/{p}}
&\ \ =\left\{\sum_{j=-\infty}^\infty\frac{\left(\mu(E_{2^j}(f))-\mu(E_{2^{j+1}}(f))\right)^{{p}/{(p-q)}}}
{(C^{\alpha,p}_{\mathbb R^{n+1}_{+}}\left(E_{2^j}(f)\right))^{{q}/{(p-q)}}}\right\}^{{(p-q)}/{p}}\\
&\ \ \lesssim\left\{\sum_{j=-\infty}^\infty\frac{\left(\mu(E_{2^j}(f))-\mu(E_{2^{j+1}}(f))\right)^{{p}/{(p-q)}}}
{\left(c_{\alpha,p}(\mu;\mu\left(E_{2^j}(f)\right))\right)^{{q}/{(p-q)}}}\right\}^{{(p-q)}/{p}}\\
&\ \ \lesssim\left\{\sum_{j=-\infty}^\infty\frac{\left(\mu(E_{2^j}(f))\right)^{{p}/{(p-q)}}-\left(\mu(E_{2^{j+1}}(f))\right)^{{p}/{(p-q)}}}
{\left(c_{\alpha,p}(\mu;\mu\left(E_{2^j}(f)\right))\right)^{{q}/{(p-q)}}}\right\}^{{(p-q)}/{p}}\\
&\ \ \lesssim\left(\int_0^\infty\frac{d s^{{p}/{(p-q)}}}{\left(c_{\alpha,p}(\mu; s)\right)^{{q}/{(p-q)}}}\right)^{{(p-q)}/{p}}\\
&\ \ \simeq \left(I_{p,q}(\mu)\right)^{{(p-q)}/{p}}.
\end{align*}
Therefore,
$$
\left(\int_{ \mathbb{R}^{n+1}_+}|P_{\alpha}f(x,t)|^q\,d\mu(x,t)\right)^{1/q}\lesssim \left(I_{p,q}(\mu)\right)^{{(p-q)}/{pq}}\|f\|_{L^p(\mathbb{R}^{n})}.
$$
{\it Part II: Proof of statements (iii)}.
Similar to \cite[Theorem 2.1]{COV},
this part consists of two steps.
{\it Step 1} - proving that
$
\label{right} \eqref{5}\Rightarrow H_{p}\mu \in L_\mu^{q(p-1)/(p-q)}(\mathbb{R}_{+}^{n+1}).
$
For $m_{0}\in \mathbb Z_{+}={0,1,2,...},$ and $m_{k}\in \mathbb Z$, $k=1,2,...$, denote by $\Theta_{l}$ the following dyadic cube with side length $l\equiv l(\Theta_{l})$:
$$\Theta_{l}\equiv \left(m_{1}l, (m_{1}+1)l\right)\times\cdots\times\left(m_{n}l, (m_{n}+1)l\right)\times \left(m_{0}l, (m_{0}+1)l\right).$$
Let $\mathbf{\Theta}=\{\Theta_{l}\}$ be the family of all the above-defined-dyadic cubes in $\mathbb R^{n+1}_+$. Setting
$$H_{p}^{d}\mu:=H_{p}^{d}\mu(x,t)=\sum\limits_{\Theta_{l}\in\mathbf{\Theta}}\left({\mu(\Theta_{l})}/{l^{n}}\right)^{p'-1}\textbf{1}_{\Theta_{l}}(x,t),$$
we first show that
\begin{equation}\label{4.23}
\eqref{5}\Rightarrow\int_{\mathbb{R}_{+}^{n+1}}(H_{p}^{d}\mu)^{{q(p-1)}/{(p-q)}}d\mu<\infty.
\end{equation}
In fact,
$(\ref{5})$ is equivalent to the following inequality by the duality
$$
\|P_{\alpha}^{*}(fd\mu)\|_{L^{p'}(\mathbb{R}^{n})}^{p'}\lesssim \|f\|_{L^{q'}(\mathbb{R}_{+}^{n+1}, \mu)}^{p'} \quad\forall f\in L^{q'}(\mathbb{R}_{+}^{n+1}, \mu).
$$
Applying Lemma \ref{l22} to $H_{p}^{d}\mu$ and $f d\mu$, and using Jensen's inequality, we thus get
$$
\|P_{\alpha}^{*}(f d\mu)\|_{L^{p'}(\mathbb{R}^{n})}^{p'}\gtrsim \int_{\mathbb{R}_{+}^{n+1}}H_{p}^{d}(fd\mu)(x,t)f(x,t)d\mu
\gtrsim \sum_{\Theta_{l}}\left\{{\int_{\Theta_{l}}f(x,t)d\mu}\right\}^{p'}l^{n(1-p')},
$$
and hence
$$
\sum_{\Theta_{l}}\left\{{\int_{\Theta_{l}}f(x,t)d\mu}\right\}^{p'}l^{n(1-p')}\lesssim \|f\|_{L_\mu^{q'}(\mathbb{R}_{+}^{n+1})}^{p'},
$$
which is equivalent to
\begin{equation}\label{4.25''}
\sum_{\Theta_{l}}g_{\Theta_{l}}\left({\int_{\Theta_{l}}f(x,t)\,d\mu(x,t)}\right)^{p'}\left(\mu(\Theta_{l})\right)^{-p'}\lesssim \|f\|_{L_\mu^{q'}(\mathbb{R}_{+}^{n+1})}^{p'} \quad\hbox{with}\quad
g_{\Theta_{l}}=\left(\mu(\Theta_{l})\right)^{p'}l^{n(1-p')}.
\end{equation}
For $0\le \widetilde{f}\in L_\mu^{{q'}/{p'}}(\mathbb{R}_{+}^{n+1})$, set
$M(x,t)=(M_{\mu}^{d}\widetilde{f})^{{1}/{p'}}(x,t).$
Here $M_{\mu}^{d}$ is the dyadic Hardy-Littlewood maximal function defined as
$$
M_{\mu}^{d}f(x,t)=\sup_{(x,t)\in \Theta_{l}\in \mathbf{\Theta}}\frac{1}{\mu(\Theta_{l})}\int_{\Theta_{l}}|f(y,s)|d\mu.
$$
Applying $\eqref{4.25''}$ to $M$ in place of $f$, and using $
\|M\|_{L^{q'}(\mathbb{R}_{+}^{n+1}, \mu)}^{p'}\lesssim \|\widetilde{f}\|_{L^{{q'}/{p'}}(\mathbb{R}_{+}^{n+1},\mu)},
$ we find
$$
\sum_{\Theta_{l}}g_{\Theta_{l}} \left(\frac{1}{\mu(\Theta_{l})}\int_{\Theta_{l}}f(x,t)d\mu(x,y)\right)^{p'}\lesssim \|M\|_{L^{q'}(\mathbb{R}_{+}^{n+1}, \mu)}^{p'}\lesssim \|\widetilde{f}\|_{L^{{q'}/{p'}}(\mathbb{R}_{+}^{n+1},\mu)}.
$$
Then using
$$
\left(\frac{1}{\mu(\Theta_{l})}\int_{\Theta_{l}}M(x,t)d\mu\right)^{p'}\gtrsim \frac{1}{\mu(\Theta_{l})}\int_{\Theta_{l}}\widetilde{f}(x,t)d\mu,
$$
one has
$$
\sum_{\Theta_{l}}\frac{g_{\Theta_{l}}}{\mu(\Theta_{l})}\int_{\Theta_{l}}\widetilde{f}(x,t)d\mu\lesssim \|\widetilde{f}\|_{L^{{q'}/{p'}}(\mathbb{R}_{+}^{n+1},\mu)}.
$$
Thereby getting by the duality that
$$
\int_{\mathbb{R}_{+}^{n+1}}\sum_{\Theta_{l}}\frac{g_{\Theta_{l}}}{\mu(\Theta_{l})}\textbf{1}_{\Theta_{l}}\widetilde{f}(x,t)d\mu=\sum_{\Theta_{l}}\frac{g_{\Theta_{l}}}{\mu(\Theta_{l})}\textbf{1}_{\Theta_{l}}\in L^{{q'}/{(q'-p')}}(\mathbb{R}_{+}^{n+1},\mu)$$
i.e.$$
\sum_{\Theta_{l}}\left(\frac{\mu(\Theta_{l})}{l^{n}}\right)^{p'-1}\textbf{1}_{\Theta_{l}}\in L^{{q(p-1)}/{(p-q)}}(\mathbb{R}_{+}^{n+1},\mu),
$$
which shows \eqref{4.23} as desired.
We proceed the proof by letting
$$\begin{cases}
H_{p}^{d,\lambda}\mu(x,t)=\sum_{\Theta_{l}\in\mathbf{\Theta}_{\lambda}}\left(\frac{\mu(\Theta_{l})}{l^{n}}\right)^{p'-1}\textbf{1}_{\Theta_{l}}(x,t);\\
\mathbf{\Theta}_{\lambda}=\mathbf{\Theta}+\lambda=\{\Theta_{l}+\lambda\}_{\Theta_{l}\in \mathbf{\Theta}};\\
\Theta_{l}+\lambda=\{(x,t)+\lambda:\ (x,t)\in \Theta_{l}\}.
\end{cases}$$
Then
\begin{equation}\label{4.25}
\sup_{\lambda\in \mathbb{R}_{+}^{n+1}}\int_{\mathbb{R}_{+}^{n+1}}\left(H_{p}^{d,\lambda}\mu(x,t)\right)^{{q(p-1)}/{(p-q)}}d\mu<\infty
\end{equation}
by the arguments leading to \eqref{4.23}(apply just the same for $\Theta_{l}$ instead of $\Theta$). Hence Step 1 is completed by showing that
\begin{equation}\label{4.25'}
H_{p}\mu \in L_\mu^{q(p-1)/(p-q)}(\mathbb{R}_{+}^{n+1}),
\end{equation}
which will be considered in two cases.
{\it Case 1. $\mu$ is a doubling measure.} $(\ref{4.25'})$ can be obtained by $(\ref{4.23})$ and the following estimate
$$
H_{p}\mu(x,t)\lesssim \sum_{\Theta_{l}}\left(\frac{\mu(\Theta_{l}^{*})}{l^{n}}\right)^{p'-1}\textbf{1}_{\Theta_{l}}(x,t),
$$
where $\Theta^{*}_{l}$ is the cube with the same center as $\Theta_{l}$ and side length two times as $\Theta_{l}$.
{\it Case 2. $\mu$ is a possibly non-doubling measure.} In this case, we first claim that for any $\delta>0$,
\begin{equation}\label{4.26}
H_{p,\delta}\mu(x,t):=\int_{0}^{\delta}\left(\frac{\mu(B_{r}(x,t))}{r^{n}}\right)^{p'-1}\frac{dr}{r}\lesssim \delta^{-(n+1)}\int_{|\lambda|\lesssim\delta}H_{p}^{d,\lambda}\mu(x,t)d\lambda.
\end{equation}
Indeed, for fixed $x\in \mathbb{R}^{n}$ and $\delta>0$ with $2^{i-1}\xi\leq \delta< 2^{i}\xi$, where $i\in\mathbb{Z}$ and $\xi>0$ will be determined later, we have
$$
H_{p,\delta}\mu(x,t)\lesssim \sum_{j=-\infty}^{i}\left(\frac{\mu(B_{2^{j}\xi}(x,t))}{(2^{j}\xi)^{n}}\right)^{p'-1}.
$$
Assume that $\Theta_{h,j}$ is a cube centered at $x$, of length $h$, with $2^{j-1}<h\leq2^{j}$ for $j\leq i$, then $B_{2^{j}\xi}(x,t)\subseteq \Theta_{h,j}$ for sufficiently small $\xi$ and $\xi$ does not depend on $x$. Suppose that $F=\{\lambda: \lambda\in \mathbb{R}_{+}^{n+1}, \,\,|\lambda|\lesssim \delta\}$ and there exists $\Theta_{l}^{\lambda}\in \Theta_{l}$ satisfying $l=2^{j+1}$ and $\Theta_{l,j}\subseteq \Theta^{\lambda}_{l}$. Then it is immediate that there is a constant $c(n)>0$ such that $|F|\ge c(n)\delta^{n+1}$ by a geometric consideration. Therefore, one has
\begin{align*}
\mu(B_{2^{j}\xi}(x,t))^{p'-1}&\lesssim |F|^{-1}\int_{F}\sum_{\Theta^{\lambda}_{l}\in \Theta^{\lambda}}\mu(\Theta^{\lambda}_{l})^{p'-1}\textbf{1}_{\Theta^{\lambda}_{l}}(x,t)d\lambda\\
&\lesssim \delta^{-(n+1)}\int_{|\lambda|\lesssim \delta}\sum_{\Theta^{\lambda}_{l}\in \Theta^{\lambda}}\mu(\Theta^{\lambda}_{l})^{p'-1}\textbf{1}_{\Theta^{\lambda}_{l}}(x,t)d\lambda,
\end{align*}
with $l=2^{j+1}$ and $B_{2^{j}\xi}(x,t)\subset \Theta^{\lambda}_{l}$, which clearly implies
\begin{align*}
H_{p,\delta}\mu(x,t)&\lesssim \delta^{-(n+1)}\int_{|\lambda|\lesssim\delta} \sum_{j=-\infty}^{i}\sum_{l=2^{j+1}}\left(\frac{\mu(\Theta^{\lambda}_{l})}{(2^{j}\xi)^{n}}\right)^{p'-1}\textbf{1}_{\Theta^{\lambda}_{l}}(x,t)ds\lesssim \delta^{-(n+1)}\int_{|\lambda|\lesssim\delta}H_{p}^{d,\lambda}\mu(x,t)d\lambda
\end{align*}
as the desired inequality $(\ref{4.26})$. This, along with H\"{o}lder's inequality and the Fubini theorem, implies
\begin{align*}
&\int_{\mathbb{R}_{+}^{n+1}}\left(H_{p,\delta}\mu(x,t)\right)^{{q(p-1)}/{(p-q)}}d\mu(x,t)\\
& \lesssim \int_{\mathbb{R}_{+}^{n+1}} \left\{\frac{1}{\delta^{(n+1)}}\left(\int_{|\lambda|\lesssim C\delta} \left(H_{p}^{d,\lambda}\mu(x,t)\right)^{{q(p-1)}/{(p-q)}}d\lambda\right)^{{(p-q)}/{[q(p-1)]}}
\left(\int_{|\lambda|\lesssim\delta}d\lambda\right)^{{p(q-1)}/{[q(p-1)]}}\right\}^{{q(p-1)}/{(p-q)}}d\mu\\
& \lesssim\delta^{-(n+1)}\int_{|\lambda|\lesssim\delta}\left( \int_{\mathbb{R}_{+}^{n+1}}\left(H_{p}^{d,\lambda}\mu(x,t)\right)^{{q(p-1)}/{(p-q)}}d\mu\right) d\lambda\\
& \le C(n).
\end{align*}
In the last step, we used $(\ref{4.25}).$
The constant $C(n)$ is independent of $\delta$.
$(\ref{4.25'})$ follows readily from the monotone convergence theorem and the above inequality by letting $\delta\rightarrow \infty$.
{\it Step 2} - showing that
$
H_{p}\mu \in L_\mu^{q(p-1)/(p-q)}(\mathbb{R}_{+}^{n+1})\Rightarrow\eqref{5}.
$
Utilizing Lemma \ref{l22}, it is sufficient to show that $$H_{p}\mu \in L_\mu^{q(p-1)/(p-q)}(\mathbb{R}_{+}^{n+1})
\Rightarrow\int_{\mathbb{R}_{+}^{n+1}}H_{p}(fd\mu)(x,t)f(x,t)d\mu(x,t)\lesssim \|f\|_{L^{q'}_{\mu}(\mathbb{R}_{+}^{n+1})}^{p'} \quad\forall f\in L_\mu^{q'}(\mathbb{R}_{+}^{n+1}),
$$
since \eqref{5} is equivalent to
$$
\|P_{\alpha}^{*}(fd\mu)\|_{L^{p'}(\mathbb{R}^{n})}\lesssim \|f\|_{L_\mu^{q'}(\mathbb{R}_{+}^{n+1})}.
$$
Without loss of generality, we assume $f\geq 0$ in the following analysis. Let
$$
M_{\mu}f(x,t)=\sup_{r>0}\frac{1}{\mu(B_{r}(x,t))}\int_{B_{r}(x,t)}f(y,s)d\mu(y,s)
$$
denote the centered Hardy-Littlewood maximal function of $f$ with respect to $\mu$. Then the H\"{o}lder inequality shows
\begin{align*}
&\int_{\mathbb{R}_{+}^{n+1}}H_{p}(fd\mu)(x,t)f(x,t)d\mu(x,t)\\
&\ \ \lesssim \int_{\mathbb{R}_{+}^{n+1}}\left(M_{\mu}f(x,t)\right)^{p'-1}H_{p}\mu(x,t)f(x,t)d\mu(x,t)\\
&\ \ \lesssim \left(\int_{\mathbb{R}_{+}^{n+1}}\left(M_{\mu}f(x,t)\right)^{q'}d\mu(x,t)\right)^{{(p'-1)}/{q'}} \left(\int_{\mathbb{R}_{+}^{n+1}}\left(f(x,t)H_{p}\mu(x,t)\right)^{{q'}/{(q'-p'+1)}}d\mu(x,t)\right)^{{(q'-p'+1)}/{q'}}
\end{align*}
via the following observation
\begin{align*}
H_{p}(fd\mu)(x,t)
&\approx\int_{0}^{\infty} \left(\frac{\mu(B_{r}(x,t))}{r^{n}}\right)^{p'-1}\left(\frac{1}{\mu(B_{r}(x,t))}\int_{B_{r}(x,t)}f(x,t)d\mu(x,t)\right)^{p'-1}\frac{dr}{r}\\
&\lesssim \left(M_{\mu}f(x,t)\right)^{p'-1}H_{p}\mu(x,t).
\end{align*}
We conclude from the $L_\mu^{q'}(\mathbb{R}_{+}^{n+1})$-boundedness of $M_{\mu}$ (cf.\cite{F}) and a further use of the H\"{o}lder inequality that
$$
\int_{\mathbb{R}_{+}^{n+1}}H_{p}(fd\mu)(x,t)f(x,t)d\mu(x,t)\lesssim \|f\|_{L_\mu^{q'}(\mathbb{R}_{+}^{n+1})}^{p'}\left(\int_{\mathbb{R}_{+}^{n+1}}\left(H_{p}\mu\right)^{{q(p-1)}/{(p-q)}}d\mu(x,t)\right)^{{(p-q)}/{[q(p-1)]}}
$$
as desired.
\end{proof}
\section{Embeddings of $\dot{W}^{\beta,p}(\mathbb{R}^n)$ to $L^q(\mathbb{R}^{n+1}_+,\mu)$}\label{sec-4}
In this section, we will characterize the embedding (\ref{4})
$$ \|P_{\alpha} f(\cdot,\cdot)\|_{L^{q}(\mathbb{R}^{n+1}_+,\mu)}\lesssim \|f\|_{\dot{W}^{\beta,p}(\mathbb{R}^{n})}
$$
in terms of the capacity and the fractional perimeter of open balls.
\subsection{Embeddings of $\dot{W}^{\beta,p}(\mathbb{R}^n)$ to $L^q(\mathbb{R}^{n+1}_+,\mu)$ when $1\leq p\leq \min\{n/\beta,q\}$}
\begin{proposition}\label{proposition 1}
Let $\beta\in (0,n), 1\leq p\leq \min\{n/\beta,q\},$ and $\mu\in\mathcal M_{+}(\mathbb R^{n+1}_{+}).$ Then the following statements are equivalent:
\item{\rm (i)} $\|P_{\alpha} f(\cdot,\cdot)\|_{L^{q,p}(\mathbb{R}^{n+1}_+,\mu)}\lesssim \|f\|_{\dot{W}^{\beta,p}(\mathbb{R}^{n})} \quad \forall\ f\in C_0^{\infty}(\mathbb{R}^{n});$
\item{\rm (ii)} $\|P_{\alpha} f(\cdot,\cdot)\|_{L^{q}(\mathbb{R}^{n+1}_+,\mu)}\lesssim \|f\|_{\dot{W}^{\beta,p}(\mathbb{R}^{n})} \quad \forall\ f\in C_0^{\infty}(\mathbb{R}^{n});$
\item{\rm (iii)} $\|P_{\alpha} f(\cdot,\cdot)\|_{L^{q,\infty}(\mathbb{R}^{n+1}_+,\mu)}\lesssim \|f\|_{\dot{W}^{\beta,p}(\mathbb{R}^{n})} \quad \forall\ f\in C_0^{\infty}(\mathbb{R}^{n});$
\item{\rm (iv)} $\sup\limits_{t>0}\frac{t^{p/q}}{c^\beta_p(\mu;t)}<\infty;$
\item{\rm (v)} $(\mu(T(O)))^{p/q}\lesssim Cap_{\mathbb{R}^n}^{\beta,p}(O)$ holds for any bounded open set $ O\subseteq \mathbb{R}^n.$
\end{proposition}
\begin{proof}
The implications (i) $\Longrightarrow $ (ii) $ \Longrightarrow$ (iii) can be deduced from
$$\left(s^q\mu(L^{\alpha,\beta}_{s}(f))\right)^{p/q}
\leq \left(q\int_0^\infty \mu(L^{\alpha,\beta}_{s}(f))s^{q-1}
ds \right)^{p/q}\leq \int_0^\infty \left(\mu(L^{\alpha,\beta}_{s}(f))\right)^{p/q}ds^p$$
since
$$q\mu(L^{\alpha,\beta}_{s}(f))s^{q-1}
\leq
\frac{d}{ds}
\left(\int_0^s (\mu(L^{\alpha,\beta}_{t}(f)))^{p/q}dt^p
\right)^{q/p}.$$
Now, we prove (iii) $\Longrightarrow$ (v) $\Longrightarrow$ (i). If (iii) is true,
$$K_{p,q}(\mu)=\sup_{f\in C^\infty_0(\mathbb{R}^n)\ \& \ \|f\|_{\dot{W}^{\beta, p}(\mathbb{R}^n)}>0 }\frac{\sup_{s>0}s\left(\mu\left(\left\{(x,t)\in \mathbb{R}^{n+1}_+:\ |P_{\alpha} f(x,t)|>s\right\}\right)\right)^{1/q}}{\|f\|_{\dot{W}^{\beta, p}(\mathbb{R}^n)}}<\infty.$$
Then, for any $f \in C^\infty_0(\mathbb{R}^n)$ and any open set $O\subseteq \hbox{ Int }(\{x\in \mathbb{R}^n: f(x)\geq 1\}),$ (iv) of Lemma \ref{lemma 2} implies
$T(O)\subset L^{\alpha,\beta}_{\eta_{n,\alpha}}(f), $ thus
\begin{equation}\label{51}
\mu(T(O))\leq \mu(L^{\alpha,\beta}_{\eta_{n,\alpha}}(f))\lesssim (K_{p,q}(\mu)\|f\|_{\dot{W}^{\beta, p}(\mathbb{R}^n)})^q.\end{equation}
Thus (v) holds.
For (v) $\Longrightarrow$ (i), denote
$$Q_{p,q}(\mu):=\sup\left\{\frac{(\mu(T(O)))^{p/q}}{Cap_{\mathbb{R}^n}^{\beta,p}(O)}:\ \text{ bounded open } O\subseteq \mathbb{R}^n\right\}<\infty.$$ Lemmas \ref{lemma 2}\ \&\ \ref{lemma 4} imply
\begin{eqnarray*}
&&\int_0^\infty(\mu(L^{\alpha,\beta}_s)(f)\cap T(B(0,k)))^{p/q}ds^p\\
&&\quad\leq \int_0^\infty(\mu(T(R^{\alpha,\beta}_s(f)\cap B(0,k)))^{p/q}ds^p \\
&&\quad\leq \int_0^\infty(\mu(T(\{x\in \mathbb{R}^n: \theta_\alpha\mathcal{M}(f)(x)>s\}\cap B(0,k)))^{p/q}ds^p \\
&&\quad\leq Q_{p,q}(\mu)\int_0^\infty(Cap_{\mathbb{R}^n}^{\beta,p}(\{x\in \mathbb{R}^n: \theta_\alpha\mathcal{M}(f)(x)>s\}))ds^p \\
&&\quad\leq Q_{p,q}(\mu)\|f\|^p_{\dot{W}^{\beta,p}(\mathbb{R}^n)}
\end{eqnarray*}
for any $f\in C^\infty_0(\mathbb{R}^n).$ Letting $k\longrightarrow \infty$ reaches (i).
Now, we will prove (iii) $\Longrightarrow$ (iv) $\Longrightarrow$ (i).
If (iii) is true, then (\ref{51}) implies
$t^{p/q}\lesssim (K_{p,q}(\mu))^pCap_{\mathbb{R}^n}^{\beta,p}(O)$ whenever $t\in (0,\mu(T(O))).$ So, $t^{p/q}\lesssim (K_{p,q}(\mu))^pc^\beta_p(\mu;t).$
Thus, (iv) is true.
Assume (iv) is true.
By Lemma \ref{lemma 2} and Lemma \ref{lemma 4}, for any $f\in C^\infty_0(\mathbb{R}^n)$,
\begin{eqnarray*}
&&\int_0^\infty(\mu(L^{\alpha,\beta}_s)(f)\cap T(B(0,k)))^{p/q}ds^p\\
&&\quad \leq \int_0^\infty\frac{(\mu(L^{\alpha,\beta}_s(f)\cap B(0,k)))^{p/q}}{c^\beta_p(\mu;\mu(L^{\alpha,\beta}_s(f)\cap B(0,k)) )} Cap_{\mathbb{R}^n}^{\beta,p}\left(R^{\alpha,\beta}_s(f)\cap B(0,k)\right)ds^p \\
&&\quad\lesssim \sup_{t>0}\frac{t^{p/q}}{c^\beta_p(\mu;t)} \int_0^\infty Cap_{\mathbb{R}^n}^{\beta,p}\left(\left\{x\in \mathbb{R}^n: \theta_\alpha\mathcal{M}(f)(x)>s\right\}\cap B(0,k)\right)ds^p \\
&&\quad\lesssim \sup_{t>0}\frac{t^{p/q}}{c^\beta_p(\mu;t)} \|f\|^p_{\dot{W}^{\beta,p}(\mathbb{R}^n)},
\end{eqnarray*}
which gives (i) via letting $k\longrightarrow \infty$.
\end{proof}
\begin{remark}\label{4.2}
Given $\beta\in (0,n),$ $1=p\leq \min\{n/\beta,q\}<\infty$ or $1<p<\min\{n/\beta,q\}.$ Following the idea of Xiao in \cite[Theorem 4.2]{Xiao 1} or that of Zhai in \cite[Theorem 1.4]{Zhai}, we can deduce from Lemma \ref{lemma 1} that the condition (v) in the equivalence of (i)$\equiv$(v) of Proposition
\ref{proposition 1} can be replaced by \begin{equation}\label{4.111}
\sup_{x\in\mathbb{R}^n,r>0}\frac{(\mu(T(B(x,r))))^{1/q}}{Cap_{\mathbb{R}^n}^{\beta,p}(B(x,r))}<\infty.
\end{equation}
Specially, when $1=p\leq\min\{n/\beta, q\}<\infty$ and $\beta\in (0,n),$ the equivalence $Cap_{\dot{W}^{\beta,1}}\approx H ^{n-\beta}_\infty$ implies that $Cap_{\mathbb{R}^n}^{\beta,p}(B(x,r))$ in (\ref{4.111}) can be replaced by $H ^{n-\beta}_\infty(B(x,r)).$ Here $H ^{d}_\infty(\cdot)$ denotes the $d-$dimensional Hausdorff capacity.
\end{remark}
For the endpoint cases $p=1$ and $\beta\in(0,1),$ based on \cite[Theorem 2]{Xiao 2}, we can replace condition (v) of Proposition \ref{proposition 1} by a condition in terms of the fractional perimeter of bounded open sets.
\begin{theorem}\cite[Theorem 2]{Xiao 2}\label{Them 2 Jie Xiao}
If $K$ is a compact subset of $\mathbb{R}^n,$ then
$$Cap_{\mathbb{R}^n}^{\beta,1}(K)=2\inf_{O\in O^\infty(K)}Per_\beta(O),$$ where $O^\infty(K)$ is the class of all open sets with $C^\infty$ boundary that contain $K.$
\end{theorem}
\begin{theorem}\label{them 4}
Let $\beta\in (0,1), 1=p\leq q< \infty$ and $\mu\in\mathcal M_{+}(\mathbb R^{n+1}_{+}).$ Then
(\ref{4}) is equivalent to
\begin{equation}\label{periOpen}
\mu(T(O))^{1/q}\lesssim Per
_{\beta}(O)
\end{equation}
for all bounded open sets $ O\subseteq \mathbb{R}^n.$
\end{theorem}
\begin{proof}
It follows from Proposition \ref{proposition 1} that (\ref{4}) is equivalent to
\begin{equation}\label{8}
(\mu(T(O)))^{1/q}\lesssim Cap_{\mathbb{R}^n}^{\beta,1}(O)
\end{equation}
for all bounded open sets $O\subseteq \mathbb{R}^n.$ Thus it suffices to show that $(\ref{8})\Longrightarrow (\ref{periOpen})\Longrightarrow (\ref{4}).$
Firstly, we show that $(\ref{8})\Longrightarrow (\ref{periOpen}).$ Ponce-Spector in \cite{Ponce Spector} proved that
$$H^{n-\beta}_\infty(O)\lesssim Per_\beta(O)$$
holds for every bounded open sets $O\subseteq \mathbb{R}^n.$
According to \cite{Adams 1989, Adams 1998}, we have
$Cap_{\mathbb{R}^n}^{\beta,1}(\cdot)\approx H^{n-\beta}_\infty(\cdot).$
Then (\ref{periOpen}) follows from
$$(\mu(T(O)))^{1//q}\lesssim Cap_{\mathbb{R}^n}^{\beta,1}(O)\lesssim H^{n-\beta}_\infty(O)\lesssim Per_\beta(O).$$
For $(\ref{periOpen})\Longrightarrow(\ref{4}),$
denote
$$Q_{q}(\mu)\coloneqq\sup\frac{(\mu(T(O)))^{1/q}}{Per_\beta(O)}<\infty,$$
where the supermum is taken over all bounded open sets $O\subseteq \mathbb{R}^n.$
For any $f\in C^\infty_0(\mathbb{R}^n)$ and any integer $k,$ denote $$V=\{x\in \mathbb{R}^n: \theta_\alpha\mathcal{M}(f)(x)>s\}\cap B(0,k).$$
It follows from Theorem \ref{Them 2 Jie Xiao} that $$Cap_{\mathbb{R}^n}^{\beta,1}(\overline{V})=2\inf_{O\in O^\infty(K)}Per_\beta(\overline{V}),$$ where $O^\infty(K)$ is the class of all open sets with $C^\infty$ boundary that contain $\overline{V}.$
Thus, we have \begin{eqnarray*}
\mu(T(V))^{1/q}&\leq&\mu(T(\overline{V}))^{1/q}\\
&\leq&
\inf_{O\in O^\infty(\overline{V})}\mu(T(O))^{1/q}\\
&\leq& Q_{q}(\mu)\inf_{O\in O^\infty(\overline{V})}Per_\beta(O) \\ &=&\frac{Q_{q}(\mu)}{2}Cap_{\mathbb{R}^n}^{\beta,1}(\overline{V})\\
&\lesssim& Cap_{\mathbb{R}^n}^{\beta,1}(\{x\in \mathbb{R}^n: \theta_\alpha\mathcal{M}(f)(x)\geq s\}).
\end{eqnarray*}
It follows from Lemma \ref{lemma 2} and Lemma \ref{lemma 4} that, for any $f\in C^\infty_0(\mathbb{R}^n)$,
\begin{eqnarray*}
\int_0^\infty\left(\mu(L^{\alpha,\beta}_s)(f)\cap T(B(0,k))\right)^{1/q}ds
& \leq&\int_0^\infty(\mu(T(R^{\alpha,\beta}_s(f)\cap B(0,k)))^{1/q}ds\\
& \leq & \int_0^\infty(\mu(T(\{x\in \mathbb{R}^n: \theta_\alpha\mathcal{M}(f)(x)>s\}\cap B(0,k)))^{1/q}ds \\
& \lesssim& \int_0^\infty Cap_{\mathbb{R}^n}^{\beta,1}(\{x\in \mathbb{R}^n: \theta_\alpha\mathcal{M}(f)(x)\geq s\})ds \\
& \lesssim& \|f\|_{\dot{W}^{\beta,1}(\mathbb{R}^n)},
\end{eqnarray*}
which reaches (\ref{4}) by letting $k\longrightarrow \infty.$
\end{proof}
\begin{remark}
It follows from Remark \ref{4.2} and $Cap_{\mathbb{R}^n}^{\beta,p}(B(x,r))= r^{n-\beta}Per_\beta(B(0,1)),$
see Xiao \cite[Themrem 4]{Xiao 2}, that $Per_\beta(O)$ in (\ref{periOpen}) of Theorem \ref{them 4}
can be also replaced by $r^{n-\beta}Per_\beta(B(0,1))$ when $\beta\in (0,1).$
\end{remark}
\subsection{Embeddings of $\dot{W}^{\beta,p}(\mathbb{R}^n)$ to $L^q(\mathbb{R}^{n+1}_+,\mu)$ when $\max\{q,1\}<p\leq n/ \beta$}
\begin{theorem}\label{them 1}
Let $\beta\in (0,n), 0<q<p,1<p\leq n/\beta$ and $\mu\in\mathcal M_{+}(\mathbb R^{n+1}_{+}). $ Then (\ref{4}) is equivalent to
\begin{equation}\label{integralCondition}
\int_0^\infty\left(\frac{t^{p/q}}{c^\beta_{p}(\mu;t)}\right)^{q/(p-q)}\frac{dt}{t}<\infty.
\end{equation}
\end{theorem}
\begin{proof}
$(\ref{integralCondition})\Rightarrow (\ref{4}).$ If
$$I_{p,q}(\mu)\coloneqq\int_0^\infty\left(\frac{t^{p/q}}{c^\beta_{p}(\mu;t)}\right)^{q/(p-q)}\frac{dt}{t}<\infty,$$
then for each $f\in C^\infty_0(\mathbb{R}^n),$ each $j=0,\pm 1, \pm 2,\ldots$, and each natural number $k,$ Lemma \ref{lemma 2} (iii) implies
$$Cap_{\mathbb{R}^n}^{\beta,p}\left(R^{\alpha,\beta}_{2^j}(f)\cap(B(0,k))\right)\leq Cap_{\mathbb{R}^n}^{\beta,p}
\left(\left\{x\in \mathbb{R}^n: \theta_\alpha\mathcal{M}f(x)>2^j \right\}\cap B(0,k)\right).$$
Define
$$\left\{\begin{aligned}
&\mu_{j,k}(f)=\mu\left(T(R^{\alpha,\beta}_{2^j}(f)\cap(B(0,k)))\right),\\
&S_{p,q,k}(\mu;f)=\sum_{j=-\infty}^\infty\frac{(\mu_{j,k}(f)-\mu_{j+1,k}(f))^{{p}/{(p-q)}}}
{(Cap_{\mathbb{R}^n}^{\beta,p}(R^{\alpha,\beta}_{2^j}(f)\cap(B(0,k))))^{{q}/{(p-q)}}}.
\end{aligned}\right.$$
It follows from (ii) of Lemma \ref{lemma 2} that
\begin{eqnarray*}
\left(S_{p,q,k}(\mu;f)\right)^{{(p-q)}/{p}}
&\lesssim &
\left(\sum_{j=-\infty}^\infty\frac{\mu^{{q}/{(p-q)}}_{j,k}(f)-\mu^{{p}/{(p-q)}}_{j+1,k}(f)}{\left(c^\beta_{p}(\mu;\mu_{j,k}(f))
\right)^{{q}/{(p-q)}}}
\right)^{{(p-q)}/{p}}\\
&\lesssim &
\left(\int_0^\infty\frac{1}{\left(c^\beta_{p}(\mu;s)\right)^{{q}/{(p-q)}}}ds^{p/(p-q)}\right)^{{(p-q)}/{p}}\\
&\simeq& (I_{p,q}(\mu))^{{(p-q)}/{p}}.
\end{eqnarray*}
On the other hand, it follows from the H\"{o}lder inequality, (ii) of Lemma \ref{lemma 4} and (ii)-(iii) of Lemma \ref{lemma 2} that
\begin{eqnarray*}
&&\int_{T(B(0,k))}|P_{\alpha} f(x,t)|^{q}d\mu(x,t)\\
&&\quad= \int_0^\infty
\mu(L^{\alpha,\beta}_s(f)\cap T(B(0,k)))ds^q\\
&&\quad\lesssim \sum_{j=-\infty}^\infty (\mu_{j,k}(f)-\mu_{j+1,k}(f))2^{jq}\\
&&\quad\lesssim\left(S_{p,q,k}(\mu;f)\right)^{{(p-q)}/{p}}\left(\sum_{j=-\infty}^{\infty} 2^{jp}Cap_{\mathbb{R}^n}^{\beta,p}(R^{\alpha,\beta}_{2j}(f)\cap(B(0,k))\right)^{q/p} \\
&&\quad\lesssim\left(S_{p,q,k}(\mu;f)\right)^{{(p-q)}/{p}}\left(\int_0^\infty Cap_{\mathbb{R}^n}^{\beta,p}\left(\left\{x\in \mathbb{R}^N: \theta_\alpha\mathcal{M}f(x)>s\right\}\cap(B(0,k)\right)ds^p\right)^{q/p} \\
&&\quad\lesssim\left(S_{p,q,k}(\mu;f)\right)^{{(p-q)}/{p}}\|f\|^q_{\dot{W}^{\beta,p}(\mathbb{R}^n)}.
\end{eqnarray*}
So, we get
$$\left(\int_{T(B(0,k))}|P_{\alpha} f(x,t)|^{q}d\mu(x,t)\right)^{1/q}\lesssim (I_{p,q}(\mu))^{{(p-q)}/{pq}}\|f\|_{\dot{W}^{\beta,p}(\mathbb{R}^n)}.$$
Letting $k\longrightarrow \infty$ derives (\ref{4}).
$(\ref{4})\Rightarrow(\ref{integralCondition}). $ If (\ref{4}) holds, then
$$C_{p,q}(\mu)\coloneqq\sup_{f\in C_0^\infty(\mathbb{R}^n),\ \|f\|_{\dot{W}^{\beta,p}(\mathbb{R}^n)}>0}\frac{1}{\|f\|_{\dot{W}^{\beta,p}(\mathbb{R}^n)}}\left(\int_{\mathbb{R}^{n+1}_+}|P_{\alpha} f(x,t)|^{q} d\mu(x,t)\right)^{1/q}<\infty.$$
Then for each $f\in C_0^\infty(\mathbb{R}^n)$ with $ \|f\|_{\dot{W}^{\beta,p}(\mathbb{R}^n)}>0,$ there holds
$$\left(\int_{\mathbb{R}^{n+1}_+}|P_{\alpha} f(x,t)|^{q}d\mu(x,t)\right)^{1/q}\leq C_{p,q}(\mu)\|f\|_{\dot{W}^{\beta,p}(\mathbb{R}^n)},$$
which implies by Markov's inequality
$$\sup_{s>0}s\left(\mu(L^{\alpha,\beta}_s(f))\right)^{1/q}\lesssim C_{p,q}(\mu)\|f\|_{\dot{W}^{\beta,p}(\mathbb{R}^n)}.$$
This, together with (iv) of Lemma \ref{lemma 2}, implies that for fixed $f\in C_0^\infty(\mathbb{R}^n),$
$$\mu(T(O))\lesssim C^q_{p,q}(\mu)\|f\|^q_{\dot{W}^{\beta,p}(\mathbb{R}^n)}$$
for any bounded open set $O\subseteq \hbox{Int}(\{x\in \mathbb{R}^n: f(x)\geq 1\}).$
The definition of $c^\beta_p(\mu;t)$ implies that $ c^\beta_p(\mu;t)>0$, and for $t\in (0,\infty)$ and every $j$ there exists a bounded open set $O_j\subseteq \mathbb{R}^n$ such that
$Cap_{\mathbb{R}^n}^{\beta,p}(O_j)\leq 2 c^\beta_p(\mu;2^j)$
and $\mu(T(O_j))>2^j.$
When $p\in (1,n/\beta)$, according to \cite[Theorem 2, P. 347]{Mazya}, $f\in \dot{W}^{\beta,p}(\mathbb{R}^n)$ when $p\in (1,n/\beta)$ if and only if $f=c I_{\beta}\ast g(x)$ for some $f\in L^{p}(\mathbb{R}^n).$ Here $I_{\beta}\ast g(x)=\int_{\mathbb{R}^n}\frac{f(y)}{|x-y|^{n-\beta}}dy.$ Thus, $\|f\|_{\dot{W}^{\beta,p}(\mathbb{R}^n)}=\|(-\triangle)^{\beta/2}f\|_{L^p(\mathbb{R}^n)}\approx\|(-\triangle)^{\beta/2}(I_{\beta}\ast g)\|_{L^p(\mathbb{R}^n)}\approx\| g\|_{L^p(\mathbb{R}^n)}.$ So, we have $$Cap_{\mathbb{R}^n}^{\beta,p}(S)\approx \inf\left\{\|g\|^p_{L^p(\mathbb{R}^n)}: g\in C^\infty_0(\mathbb{R}^n), g\geq 0, S\subseteq \hbox{Int}(\{x\in \mathbb{R}^n: I_{\beta}\ast g(x)\geq 1\})\right\}.$$
Then, there exists $g_j\in C^\infty_0(\mathbb{R}^n)$ such that $ I_{\beta}\ast g_j(x)\geq 1$ on $ O_j,$ and $$\|g_{j}\|^p_{L^p(\mathbb{R}^n)}\leq 2Cap_{\mathbb{R}^n}^{\beta,p}(O_j)\leq 4 c^\beta_p(\mu;2^j).$$
For the integers $ i,k$ with $i<k,$ define
$$g_{i,k}=\sup_{i\leq j\leq k}\left(\frac{2^j}{c^\beta_p(\mu;2^j)}\right)^{\frac{1}{p-q}} g_j.$$
Then $g_{i,k}\in L^p(\mathbb{R}^n)\cap L^1_{loc}(\mathbb{R}^n)$
with
\begin{equation}\label{normofgik}
\|g_{i,k}\|^p_{L^p(\mathbb{R}^n)}\lesssim \sum_{j=i}^k\left(\frac{2^j}{c^\beta_p(\mu;2^j)}\right)^{\frac{p}{p-q}} c^\beta_p(\mu;2^j).\end{equation}
On the other hand, $I_{\beta}\ast g_{i,k}\in \dot{W}^{\beta,p}(\mathbb{R}^n)$
since $I_\beta: \dot{F}_{p,2}^0(\mathbb{R}^n)=L^p(\mathbb{R}^n) \to \dot{F}_{p,2}^{\beta}=\dot{W}^{\beta,p}(\mathbb{R}^n), $ see \cite[Theorem 1, page 242]{Hans}
Since for $i\leq j\leq k,$
$$x\in O_j\Rightarrow I_{\beta}\ast g_{i,k}(x)\geq \left(\frac{2^j}{c^\beta_p(\mu;2^j)}\right)^{\frac{1}{p-q}},$$
(iv) of Lemma \ref{lemma 2} implies
that there exists a constant $\eta_{n,\alpha}$ such that
$$(x,t)\in T(O_j)\Rightarrow |u_{i,k}(x,t)|\geq \left(\frac{2^j}{c^\beta_p(\mu;2^j)}\right)^{\frac{1}{p-q}}\eta_{n,\alpha}.$$
Here $u_{i,k}$ denotes the Caffarelli-Silvestre extension of $I_{\beta}\ast g_{i,k}.$
Thus, with $s=\left(\frac{2^j}{c^\beta_p(\mu;2^j)}\right)^{\frac{1}{p-q}}\frac{\eta_{n,\alpha}}{2},$
$$2^j\leq \mu(T(O_j))\leq \mu (L^{\alpha,\beta}_{s}(I_\beta\ast g_{i,k})).$$
Similar to that in the the proof of (i) in Theorem \ref{Them 3.4}, we can get
\begin{eqnarray*}
(C_{p,q}(\mu)\|g_{i,k}\|_{L^p(\mathbb{R}^n)})^q
&\gtrsim&
\int_{\mathbb{R}^{n+1}_+}|u_{i,k}(x,t)|d\mu\\
&\approx& \int_0^\infty\left(\inf\left\{s: \mu(L^{\alpha,\beta}_s)(I_\beta\ast g_{i,k})\leq s\right\}\right)^qds\\
&\gtrsim&\sum_{j=i}^k\left(\inf\left\{s: \mu(L^{\alpha,\beta}_s)(I_\beta\ast g_{i,k})\leq2^j \right\}\right)^q2^j\\
&\gtrsim&
\sum_{j=i}^k\left(\frac{2^j}{c^\beta_p(\mu;2^j)}\right)^{\frac{q}{p-q}}2^j\\
&\gtrsim&
\left(\sum_{j=i}^k\frac{2^{{jp}/{(p-q)}}}{c^\beta_p(\mu;2^j)^{{q}/{(p-q)}}}\right)^{\frac{p-q}{p}}\|g_{i,k}\|^q_{L^p(\mathbb{R}^n)},
\end{eqnarray*}
where in the last step, we used (\ref{normofgik}).
Thus, $$\sum_{j=i}^k\frac{2^{{jp}/{(p-q)}}}{c^\beta_p(\mu;2^j)^{{q}/{(p-q)}}}\lesssim(C_{p,q}(\mu))^{{pq}/{(p-q)}}.$$
When $p={n}/{\beta},$ the definition of $Cap_{\mathbb{R}^n}^{\beta,p}(\cdot)$ implies that there exists a positive $f_j\in C^\infty_0(\mathbb{R}^n)$ such that $f_j\geq 1$ on $O_j$ and $$\|f_j\|^p_{\dot{W}^{\beta,p}(\mathbb{R}^n)}\leq 2 Cap_{\mathbb{R}^n}^{\beta,p}(O_j)\leq 4c^\beta_p(\mu;2^j).$$
Define $g_j(x,h)\coloneqq|h|^{-2\beta}\triangle^k_hf_j(x).$ Then $g_j\in L^p(\mathbb{R}^{2n})\cap L^1_{loc}(\mathbb{R}^{2n}). $ On the other hand, \cite[Theorem 5.1]{Adams} implies $f_j(x)=CI_{2\beta}^{(2n)}\ast g_j(x,0),$ for a.e. $x\in \mathbb{R}^n.$
For the integers $ i,k$ with $i<k,$ define
$$g_{i,k}(x,h)=\sup_{i\leq j\leq k}\left(\frac{2^j}{c^\beta_p(\mu;2^j)}\right)^{\frac{1}{p-q}} g_j(x,h).$$
Then $g_{i,k}\in L^{p}(\mathbb{R}^{2n})\cap L^1_{loc}(\mathbb{R}^{2n}),$
and Lemma \ref{lemma 3} implies
\begin{eqnarray*}
\|\mathcal{R}I^{(2n)}_{2\beta}\ast g_{i,k}\|^p_{\dot{W}^{\beta,p}(\mathbb{R}^n)}
&\lesssim& \sum_{j=i}^k\left(\frac{2^j}{c^\beta_p(\mu;2^j)}\right)^{\frac{p}{p-q}} \|I^{(2n)}_{2\beta}\ast g_{i,k}\|^p_{\dot{\mathcal{L}}^p(\mathbb{R}^{2n})}\\
&\lesssim& \sum_{j=i}^k\left(\frac{2^j}{c^\beta_p(\mu;2^j)}\right)^{\frac{p}{p-q}} \|f_j\|^p_{\dot{W}^{\beta,p}(\mathbb{R}^n)}\\
&\lesssim& \sum_{j=i}^k\left(\frac{2^j}{c^\beta_p(\mu;2^j)}\right)^{\frac{p}{p-q}} c^\beta_p(\mu;2^j).
\end{eqnarray*}
Let $u_{i,k}(x,t)$ be the Caffarelli-Silvestre extension of $\mathcal{R}(I^{(2n)}_{2\beta}g_{i,k})(x)$. Then, similar to the previous case,
for $i\leq j\leq k,$
$$x\in O_j\Rightarrow \mathcal{R}(I^{(2n)}_{2\beta}g_{i,k})(x)\geq \left(\frac{2^j}{c^\beta_p(\mu;2^j)}\right)^{\frac{1}{p-q}}.$$
(iv) of Lemma \ref{lemma 2} implies
that there exists a constant $\eta_{n,\alpha}$ such that
$$(x,t)\in T(O_j)\Rightarrow |u_{i,k}(x,t)|\geq \left(\frac{2^j}{c^\beta_p(\mu;2^j)}\right)^{\frac{1}{p-q}}\eta_{n,\alpha}$$
and with $s=\left(\frac{2^j}{c^\beta_p(\mu;2^j)}\right)^{\frac{1}{p-q}}\frac{\eta_{n,\alpha}}{2},$
$$2^j\leq \mu(T(O_j))\leq \mu (L^{\alpha,\beta}_{s}(\mathcal{R}(I^{(2n)}_{2\beta}g_{i,k}))).$$
We can get
\begin{eqnarray*}
(C_{p,q}(\mu)\|\mathcal{R}(I^{(2n)}_{2\beta}g_{i,k})\|_{\dot{W}^{\beta,p}(\mathbb{R}^n)})^q
&\gtrsim&
\int_{\mathbb{R}^{n+1}_+}|u_{i,k}(x,t)|d\mu\\
&\approx& \int_0^\infty\left(\inf\left\{s: \mu(L^{\alpha,\beta}_s(\mathcal{R}(I^{(2n)}_{2\beta}g_{i,k})))\leq s\right\}\right)^qds\\
&\gtrsim&\sum_{j=i}^k\left(\inf\left\{s: \mu(L^{\alpha,\beta}_s(\mathcal{R}(I^{(2n)}_{2\beta}g_{i,k})))\leq2^j \right\}\right)^q2^j\\
&\gtrsim&
\sum_{j=i}^k\left(\frac{2^j}{c^\beta_p(\mu;2^j)}\right)^{\frac{q}{p-q}}2^j\\
&\gtrsim&
\left(\sum_{j=i}^k\frac{2^{{jp}/{(p-q)}}}{c^\beta_p(\mu;2^j)^{{q}/{(p-q)}}}\right)^{\frac{p-q}{p}}\|\mathcal{R}(I^{(2n)}_{2\beta}g_{i,k}\|^q_{\dot{W}^{\beta,p}(\mathbb{R}^n)}.
\end{eqnarray*}
Thus, we have
$$\sum_{j=i}^k\frac{2^{\frac{jp}{(p-q)}}}{c^\beta_p(\mu;2^j)^{\frac{q}{p-q}}}\lesssim\left(C_{p,q}(\mu)\right)^{\frac{pq}{p-q}}.$$
Letting $i,k\longrightarrow \infty,$ we reach
$$\int_0^\infty\left(\frac{t^{p/q}}{c^\beta_p(\mu;t)}\right)^{\frac{q}{p-q}}\lesssim \sum_{-\infty}^\infty\frac{2^{\frac{jp}{p-q}}}{(c^\beta_p(\mu;2^j))^{\frac{q}{(p-q)}}}\lesssim (C_{p,q}(\mu))^{\frac{pq}{p-q}},$$ which implies (\ref{integralCondition}).
\end{proof}
When $q<p=1,$ we can establish the following necessary conditions for the embedding (\ref{4}).
\begin{proposition}\label{them 5}
Let $\beta\in (0,n), 0<q<p=1$ and $\mu\in\mathcal M_{+}(\mathbb R^{n+1}_{+}). $ Then
(\ref{4}) $\Rightarrow$ (i) $\Rightarrow$ (ii) $\Rightarrow$ (iii):
\item{\rm (i)} $\|P_{\alpha} f(\cdot,\cdot)\|_{L^{q,\infty}(\mathbb{R}^{n+1}_+,\mu)}\lesssim \|f\|_{\dot{W}^{\beta,1}(\mathbb{R}^{n})} \quad \forall f\in C_0^{\infty}(\mathbb{R}^{n});$
\item{\rm (ii)}
$\sup\left\{\frac{(\mu(T(O)))^{1/q}}{Cap_{\mathbb{R}^n}^{\beta,1}(O)}: \text{ bounded open set} \, O\subseteq \mathbb{R}^n\right\}<\infty;$
\item{\rm (iii)} $\|P_{\alpha} f(\cdot,\cdot)\|_{L^{q,1}(\mathbb{R}^{n+1}_+,\mu)}\lesssim \|f\|_{\dot{W}^{\beta,1}(\mathbb{R}^{n})} \quad \forall f\in C_0^{\infty}(\mathbb{R}^{n}).$
\end{proposition}
\begin{proof}
The proofs of (\ref{4}) $\Longrightarrow$ (i) $\Longrightarrow$ (ii) are similar to those of (ii) $\Longrightarrow$ (iii) $\Longrightarrow$ (v) in the proof of Proposition \ref{proposition 1}. For the implication (ii) $\Longrightarrow$ (iii), firstly, there holds
\begin{eqnarray*}
\mu(L^{\alpha,\beta}_s(f))&\leq& \mu(T(R^{\alpha,\beta}_s(f)))\\
&\leq& \mu\left(T\left(\left\{x\in \mathbb{R}^n:\ \theta_\alpha\mathcal{M}f(x)>s\right\}\right)\right)\\
&\lesssim & \left(Cap_{\mathbb{R}^n}^{\beta,1}\left(\left\{x\in \mathbb{R}^n: \theta_\alpha\mathcal{M}f(x)>s\right\}\right)\right)^q,
\end{eqnarray*}
due to Lemma \ref{lemma 2}.
Then, applying (ii) of Lemma \ref{lemma 4} we can show (ii)$\Longrightarrow$(iii) in a similar way as the proof of (v)$\Longrightarrow$(i) in Proposition \ref{proposition 1}.
\end{proof}
\section*{Acknowledgement}
The authors are very grateful to the anonymous reviewers, especially one of whom gave numerous useful suggestions that lead to a substantial improvement of this manuscript.
|
1,116,691,500,818 | arxiv | \section{Introduction}
A real analytic Levi-flat set $M$ in $\mathbb C^{N}$ is a real analytic set such that its regular part is a Levi-flat CR manifold of hypersurface type.
An important special case (closely related to the theory of holomorphic foliations) arises when $M$ is a hypersurface. The local geometry of a Levi-flat hypersurface near its singular locus has been studied by several authors \cite{Bru1, Bru2, CerNeto, CerSad, Fe, Le, PSS, SS}. One of the main questions here concerns an extension of the Levi foliation of the regular part of $M$ as a (singular) holomorphic foliation (or, more generally, a singular holomorphic web) to a full neighbourhood of a singular point. The existence of such an extension allows one to use the holomorphic resolution of singularities results for the study of local geometry of singular Levi-flat hypersurfaces.
The present paper is concerned with local properties of real analytic Levi-flat sets near their singularities. These sets arise in the study of Levi-flat hypersurfaces when lifted to the projectivization of the cotangent bundle of the ambient space. Our main result gives a complete characterization of dicritical singularities of such sets in terms of their Segre varieties. Our method is a straightforward
generalization of arguments in~\cite{PSS,SS} where the case of Levi-flat hypersurfaces is considered.
\section{ Levi-flat subsets, Segre varieties}
In this section we provide relevant background material on real analytic Levi-flat sets (of higher codimension) and their Segre varieties. To the best of our knowledge this topic has not been considered in detail before; for
convenience of the reader we provide some details.
\subsection{Real and complex analytic sets.}
Let $\Omega$ be a domain in $\mathbb C^{N}$. We denote by $z = (z_1,...,z_{N})$ the standard complex coordinates. A closed subset $M \subset \Omega$ is called {\it a real (resp. complex) analytic subset} in
$\Omega$ if it is locally defined by a finite collection of real analytic (resp. holomorphic) functions.
For a real analytic $M$ this means that for every point $q \in \Omega$ there exist a neighbourhood $U$ of $q$ and real analytic vector function
$\rho = (\rho_1,...,\rho_k): U \to \mathbb R^k$ such that
\begin{eqnarray}
\label{DefEq}
M \cap U = \rho^{-1}(0) = \{z \in U: \rho_j(z,\overline{z}) = 0,\ j = 1,...,k \} .
\end{eqnarray}
In fact, one can reduce the situation to the case $k=1$ by considering the defining function $\rho_1^2 + ...+ \rho_k^2$. Without loss of generality assume $q = 0$ and choose a neighbourhood $U$ in (\ref{DefEq}) in the form of a polydisc
$\Delta(\varepsilon) = \{ z \in \mathbb C^{N}: \vert z_j \vert < \varepsilon \}$
of radius $ \varepsilon > 0$. Then, for $\varepsilon$ small enough, the (vector-valued) function $\rho$ admits the Taylor expansion
convergent in $U$:
\begin{eqnarray}\label{exp}
\rho(z,\overline z) = \sum_{IJ} c_{IJ}z^I \overline{z}^J, \ c_{IJ}\in{\mathbb{C}}, \ \ I,J \in \mathbb N^N.
\end{eqnarray}
Here and below we use the multi-index notation $I = (i_1,....,i_N)$ and $\vert I \vert = i_1 +...+i_N$. The ($\mathbb C^k$-valued) coefficients $c_{IJ}$ satisfy the condition
\begin{eqnarray}
\label{coef}
\overline c_{IJ} = c_{JI},
\end{eqnarray}
since $\rho$ is a real ($\mathbb R^k$-valued) function.
An analytic subset $M$ is called {\it irreducible} if it cannot be represented as a union $M = M_1 \cup M_2$ where $M_j$ are analytic subsets of $\Omega$ different from $M$. Similarly, an analytic subset is irreducible as a germ at a point $p \in M$ if its germ cannot be represented as a union of germs of two real analytic sets. All considerations of the present paper are local and we
always assume irreducibility of germs even if it is not specified explicitly.
A set $M$ can be decomposed into a disjoint union $M = M_{reg} \cup M_{sing}$, the regular and the singular part respectively. The regular part $M_{reg}$ is a nonempty and open subset of $M$. In the real analytic case we adopt the following convention: $M$ is a real analytic submanifold of maximal dimension in a neighbourhood of every point of $M_{reg}$. This dimension is called the dimension of $M$ and is denoted by $\dim M$. The set $M_{sing}$ is a real semianalytic subset of
$\Omega$ of dimension $< \dim M$. Unlike complex analytic sets, for a real analytic~$M$, the set $M_{sing}$ may contain manifolds of smaller dimension which are not in the closure of $M_{reg}$, as seen in the classical example of the Whitney umbrella. Therefore, in general $M_{reg}$ is not dense in $M$.
Recall that the dimension of a complex analytic set $A$ at a point $a \in A$ is defined as
$$
\dim_a A : = \underset{{A_{reg} \ni z \to a}}{\overline{\lim}} \dim_z A ,
$$
and that the the function $z \mapsto \dim_z A$ is upper semicontinuous. Suppose that $A$ is an irreducible complex analytic subset of a domain $\Omega$ and let $F: A \to X$ be a holomorphic mapping into some complex manifold $X$. The local dimension of $F$ at a point $z\in A$
is defined as $\dim_{z} F = \dim A - \dim_{z} F^{-1}( F(z))$ and the dimension of $F$ is set to be $\dim F = \max_{z \in A}\dim_{z} F$. Note that the equality $\dim_{z} F = \dim F$ holds on a Zariski open subset of
$A$, and that $\dim F$ coincides with the rank of the map~$F$, see \cite{Ch}.
\subsection{Complexification and Segre varieties.}
Let $M$ be the germ at the origin of an irreducible real analytic subset of $\mathbb C^{N}$ defined by~\eqref{exp}.
We are interested in the geometry of $M$ in an arbitrarily small neighbourhood of $0$. We may consider a sufficiently
small open neighbourhood $U$ of the origin and a representative of the germ which is also irreducible, see \cite{N} for
details. In what follows we will not distinguish between the germ of $M$ and its particular representative in a suitable
neighbourhood of the origin.
Denote by $J$ the standard complex structure of $\mathbb C^{N}$ and consider the opposite structure $-J$.
Consider the space $\mathbb C^{2N}_\bullet := (\mathbb C^{N}_z,J) \times (\mathbb C^{N}_w,-J)$ and the diagonal
$$
\Delta = \left\{ (z,w) \in \mathbb C^{2N}_\bullet : z=w \right\} .
$$
The set $M$ can be lifted to $\mathbb C^{2N}_\bullet$ as the real analytic subset
$$
\hat{M} := \left\{ (z,z) \in \mathbb C^{2N}_\bullet : z \in M \right\} .
$$
There exists a unique irreducible complex analytic subset $M^{\mathbb C}$ in $\mathbb C^{2N}_\bullet$ of complex dimension equal
to the real dimension of $M$ such that
$\hat{M} = M^{\mathbb C} \cap \Delta$ (see \cite{N}). The set $M^{\mathbb C}$ is called the {\it complexification} of $M$. The
antiholomorphic involution
$$
\tau: \mathbb C^{2N}_\bullet \to \mathbb C^{2N}_\bullet, \,\,\, \tau: (z,w) \mapsto (w,z)
$$
leaves $M^\mathbb C$ invariant and $\hat{M}$ is the set of fixed points of $\tau\vert_{M^{\mathbb C}}$.
The complexification $M^\mathbb C$ is equipped with two canonical holomorphic projections $\pi_z:(z,w) \mapsto z$ and $\pi_w:(z,w) \mapsto w$. We always suppose by convention that the domain of these projections is $M^\mathbb C$. The triple $(M^\mathbb C,\pi_z,\pi_w)$ is represented by the following diagram
\begin{center}
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=3em,
column sep=2em, text height=2ex, text depth=0.25ex]
{ & M^{\mathbb C} & \\
(\mathbb C^{N},J) & & (\mathbb C^{N},-J) \\
};
\path[->] (m-1-2) edge node[auto,swap] {$ \pi_z $} (m-2-1);
\path[->] (m-1-2) edge node[auto] {$ \pi_w $} (m-2-3);
\end{tikzpicture}
\end{center}
which leads to the central notion of the present paper. The {\it Segre variety} of a point $w \in \mathbb C^{N}$
is defined as
$$
Q_w := (\pi_z \circ \pi^{-1}_w)(w) = \left\{ z \in \mathbb C^{N}: (z,w) \in M^\mathbb C\right\} .
$$
When $M$ is a hypersurface defined by (\ref{DefEq}) (with $k = 1$) this definition coincides with the usual definition $$Q_w = \left\{ z: \rho(z,\overline{w}) = 0 \right\}.$$
Of course, here we suppose that $\rho$ is a minimal function, that is, it generates the ideal of real analytic functions vanishing on $M$.
The following properties of Segre varieties are well-known for hypersurfaces.
\begin{prop}\label{SegreProp}
Let $M$ be the germ of a real analytic subset in $\mathbb C^N$. Then
\begin{itemize}
\item[(a)] $z \in Q_z \Longleftrightarrow z \in M$.
\item[(b)] $z \in Q_w \Longleftrightarrow w \in Q_z$,
\item[(c)] (invariance property) Let $M_1$ be a real analytic CR manifold, and $M_2$ be a real analytic germ
in $\mathbb C^N$ and $\mathbb C^K$ respectively. Let $p \in M_1$, $q\in M_2$, and $U_1 \ni p$, $U_2 \ni q$ be small neighbourhoods.
Let also
$f:U_1 \to U_2$ be a holomorphic map such that $f(M_1 \cap U_1) \subset M_2 \cap U_2$. Then
$$f(Q_w^1) \subset Q^2_{f(w)}$$
for all $w$ close to $p$. If, in addition, $M_2$ is nonsingular and $f:U_1 \to U_2$ is biholomorphic, then $f(Q_w^1) = Q_{f(w)}^2$.
Here $Q_w^1$ and $Q_{f(w)}^2$ are Segre varieties associated with $M_1$ and $M_2$ respectively.
\end{itemize}
\end{prop}
\begin{proof} (a) Note that $z \in Q_z$ if and only if $(z,z) \in M^\mathbb C$, which is equivalent to $(z,z) \in \hat M$.
(b) The relation $(z,w) \in M^\mathbb C$ holds if and only if $\tau(z,w) = (w,z) \in M^\mathbb C$.
(c) Suppose that $M_1$ is defined near $p$ by the equations $\rho_1 = ...=\rho_k = 0$ and
$d\rho_1 \wedge ... \wedge d\rho_k \neq 0$. Similarly, suppose that $M_2$ is defined by the equations
$\phi_1 = ... = \phi_l = 0$.
Then the Segre varieties are respectively given by
$Q_w^1 = \{ z: \rho_j(z,\overline{w}) = 0,\ j = 1,...,k\}$ and $Q_w^2 = \{ z: \phi_s (z,\overline{w}) = 0, \ s=1,...,l \}$. By assumption we have
$\phi_s(f(z),\overline{f(z)}) = 0$ when $z \in M_1$. This implies that there exist real analytic functions $\lambda_j$, $j=1,...,k$, such that
$$
\phi_s(f(z),\overline{f(z)}) = \sum_1^k \lambda_{sj}(z,\overline{z})\rho_j(z,\overline{z}) .
$$
Consider first the case where $M_1$ is a generic manifold, that is, $\partial \rho_1 \wedge ... \wedge \partial \rho_k\neq 0$.
Let $f^*$ be a holomorphic function such that $f^*(\overline{w}) = \overline{f(w)}$. Since $M_1$ is generic, it is the
uniqueness set for holomorphic functions. Therefore, by analyticity we have
$$
\phi_s(f(z),f^*(\overline{w})) = \sum_1^k \lambda_{sj}(z,\overline{w})\rho_j(z,\overline{w}),
$$
for all $z$ and $w$. This concludes the proof for the case when $M_1$ is generic.
Let now $M_1$ be a CR manifold which is not generic. Then, by real analyticity, $M_1$ can be represented as the graph of (the restriction of) a holomorphic (vector) function over a real analytic generic manifold $\tilde M_1$ of real codimension $l$ in $\mathbb C^d$, for some $l$ and $d$. More precisely, set $z = (z',z'')$, $z' = (z_1,...,z_d)$, $z'' = (z_{d+1},...,z_N)$.
Then $\tilde M_1 = \{ z': \psi_j(z',\overline{ z'}) =0,\ j=1,...,l \}$ and $M_1 = \{ (z',z''): z' \in N_1, z'' = g(z') \}$, where $\psi_j$ are real analytic functions and $g$ is a holomorphic (vector) function. Then every Segre variety $Q_w^1$ of $M_1$ is the graph of $g$ over the Segre variety of $\tilde M_1$. Indeed,
$Q_w^1 = \{ (z',z''): \phi_j(z',\overline{w'}) = 0, j= 1,...,l, z'' = g(z') \}$. The holomorphic map $\tilde f(z') = f(z',g(z'))$ transforms the generating manifold $\tilde M_1$ to $M_2$. Since we already proved the result for generic submanifolds in the
source, we conclude that the map $\tilde f$ transforms Segre varieties of the manifold $\tilde M_1$ to Segre varieties of the manifold
$M_2$. This implies the required statement.
\end{proof}
\subsection{Levi-flat sets}
We say that an irreducible real analytic set $M\subset \mathbb C^{n+m}$ is {\it Levi-flat} if $\dim M = 2n-1$ and $M_{reg}$ is locally foliated by complex manifolds of complex dimension $n-1$. In particular, $M_{reg}$ is a CR manifold of hypersurface type.
The most known case arises when $m=0$, i.e., when $M$ is a Levi-flat hypersurface in $\mathbb C^n$.
We use the notation $z'' = (z_{n+1},...,z_{n+m})$, and similarly for the $w$ variable.
It follows from the Frobenius theorem and the implicit function theorem that for every
point $q \in M_{reg}$ there exist an open neighbourhood $U$ and a local biholomorphic
change of coordinates $F: (U,q) \to (F(U),0)$ such that $F(M)$ has the form
\begin{eqnarray}
\label{NormForm1}
\{ z \in F(U): z_n + \overline{z}_n = 0, \ z'' = 0 \} .
\end{eqnarray}
The subspace $F(M)$ is foliated by complex affine subspaces $L_c = \{z_n= ic,\ z'' = 0, \ c \in \mathbb R\}$,
which gives a foliation of $ M_{reg} \cap U$ by complex submanifolds $F^{-1}(L_c)$. This defines a
foliation on $M_{reg}$ which is called {\it the Levi foliation} and denoted by~$\mathcal L$. Every leaf of $\mathcal L$ is
tangent to the complex tangent space of $M_{reg}$. The complex affine subspaces
\begin{eqnarray}
\label{SegrePlane}
\left\{ z_n = c, z'' = 0 \right\}, \,\,\ c \in \mathbb C,
\end{eqnarray}
in local coordinates given by
(\ref{NormForm1}) are precisely the Segre varieties of $M$ for every complex $c$. Thus, the Levi foliation is closely related to
Segre varieties. The complexification $M^\mathbb C$ is given by
\begin{eqnarray}\label{compl}
M^\mathbb C = \{ (z,w): z_n + \overline{w}_n = 0, z'' =0, w'' = 0 \}.
\end{eqnarray}
For $M$ defined by (\ref{NormForm1}) its Segre varieties (\ref{SegrePlane})
fill the complex subspace $z'' = 0$ of $\mathbb C^{n+m}$. In particular, if $w$ is not in this subspace, then
$Q_w$ is empty.
In arbitrary coordinates, in a neighbourhood $U \subset \mathbb C^{n+m}(z)$ of a regular point $z^0 \in M$ the Levi flat set is the transverse intersection of a real analytic hypersurface with a complex n-dimensional manifold, that is
\begin{eqnarray}
\label{compl1}
M = \{ z \in U: h_j(z) = 0, \ j=1,...,m , \,\, r(z,\overline z) = 0 \} .
\end{eqnarray}
Here $h_j$ are functions holomorphic on $U$ and $r:U \longrightarrow \mathbb R$ is a real analytic function.
Furthermore, $\partial r \wedge dh_1 \wedge...\wedge dh_m \neq 0$. Then
\begin{eqnarray}
\label{compl2}
M^\mathbb C = \{ (z,w) \in U \times U: r(z,\overline w ) = 0, h_j(z) = 0, h_j(\overline w) = 0, j=1,...,m \}
\end{eqnarray}
in a neighbourhood $U \times U$ of $(z^0,{\overline z}^0)$.
We need to study some general properties of projections $\pi_z$ and $\pi_w$. Let $\pi$ be one of the projections $\pi_z$ or $\pi_w$. Introduce
the dimension of $\pi$ by setting $\dim \pi = \max_{(z,w) \in M^\mathbb C}\dim_{(z,w)} \pi$.
If $M$ is irreducible as a germ, then so is $M^\mathbb C$ (see~\cite[p.92]{N}). Hence, $(M^\mathbb C)_{reg}$ is a connected complex manifold of dimension $2n-1$. Then the equality $\dim_{(z,w)} \pi = \dim \pi$ holds on a Zariski open set
\begin{equation}\label{e.mstarc}
M^\mathbb C_*: = (M^\mathbb C \setminus X) \subset (M^\mathbb C)_{reg},
\end{equation}
where $X$ is a complex analytic subset of dimension $< 2n-1$. Here $\dim \pi$ coincides with the rank of $\pi \vert_{M^\mathbb C_*}$.
Furthermore, $\dim (\pi \vert (M^\mathbb C)_{sing}) \le \dim \pi$.
\begin{lemma}
\label{dim1}
Let $\pi$ be one of the projections $\pi_z$ or $\pi_w$.
\begin{itemize}
\item[(a)] We have $\dim \pi = n$.
\item[(b)] The image $\pi(M^\mathbb C)$ is contained in the (at most) countable union of complex analytic sets of dimension $\le n$.
\end{itemize}
\end{lemma}
\begin{proof}
(a) Consider the case where $\pi = \pi_w$. In view of (\ref{compl2})
the image of an open neighbourhood a regular point
$(z^0,{\overline z}^0)$ in $M^\mathbb C$ coincides with the complex $n$-dimensional manifold
$\{w: h_j(\overline w) = 0, j=1,...,m \}$. This implies (a).
(b) This is a consequence of (a), see \cite{Ch}.
\end{proof}
It follows from the lemma above that generically, i.e., for $w \in \pi_w(M^\mathbb C_*)$, the complex analytic set $Q_w$ has
dimension $n-1$, and that $Q_w$ can have dimension $n$ if $(z,w)\notin M^\mathbb C_*$.
Of course, $Q_w$ is empty if $w$ does not belong to $ \pi_w(M^\mathbb C)$.
A singular point $q \in M$ is called {\it Segre degenerate} if $\dim Q_q = n$.
Note that the set of Segre degenerate points is contained in a complex analytic subset of dimension $n-2$. The proof,
which we omit, is quite similar to that in \cite{SS}, where this claim is established for hypersurfaces.
Let $q \in M_{reg}$. Denote by ${\mathcal L}_q$ the leaf of the Levi foliation through $q$. Note that
by definition this is a connected complex submanifold of complex dimension $n-1$ that is closed in $M_{reg}$.
Denote by $M_* \subset M_{reg}$ the image of $\hat M \cap M^\mathbb C_*$ under the projection $\pi$,
where $M^\mathbb C_*$ is defined as in~\eqref{e.mstarc}. This set coincides with $M_{reg} \setminus A$ for some proper
complex analytic subset $A$.
As a simple consequence of Proposition \ref{SegreProp} we have quite similarly
to~\cite{SS} the following.
\begin{cor}
\label{Segre+Levi}
Let $a \in M_*$. Then the following holds:
\begin{itemize}
\item[(a)] The leaf ${\mathcal L}_a$ is contained in a unique irreducible component $S_a$ of
$Q_a$ of dimension $n-1$. In particular, $Q_a$ is a nonempty complex analytic set of pure dimension $n-1$. In a small neighbourhood $U$ of $a$ the intersection $S_a \cap U$ is also a unique complex submanifold of complex dimension $n-1$ through $a$ which is contained in $M$.
\item[(b)] For every $a \in M_*$ the complex variety $S_a$ is contained in $M$;
\item[(c)] For every $a, b \in M_*$ one has $b \in S_a \Longleftrightarrow S_a = S_b$.
\item[(d)] Suppose that $a \in M_*$ and ${\mathcal L}_a$ touches a point $q \in M$ (the point $q$ may be singular). Then $Q_q$ contains $S_a$. If, additionally, $\dim_\mathbb C Q_q = n-1$, then $S_a$ is an irreducible component of~$Q_q$.
\end{itemize}
\end{cor}
\begin{proof}
(a) We first make an elementary but important observation. Suppose that $M$ is a representative in a domain $U$ of the germ
of a real analytic set $\{\rho = 0\}$. Let $a\in M$ and $V$ be a neighbourhood of $a$, $V\subset U$. Suppose that we used a different function $\tilde \rho$ to define $M\cap V$, i.e., $M\cap V = \{\tilde\rho = 0\}$. Applying Proposition~\ref{SegreProp}
to the inclusion map $V \hookrightarrow U$ we conclude that the Segre varieties of $M\cap V$ defined by complexifying $\tilde\rho$
are contained in the Segre varieties of $M$ defined by complexifying the function $\rho$. More precisely, the Segre varieties with
respect to $\tilde\rho$ are coincide with the intersection with $V$ of some components of Segre varieties with respect to $\rho$.
In view of the invariance of the Levi form under biholomorphic maps, the Levi foliation is an intrinsic notion, i.e., it is
independent of the choice of (local) holomorphic coordinates. Similarly, in view of the biholomorphic invariance of Segre
varieties described in Proposition~\ref{SegreProp}(c), these are also intrinsic objects. There exist a small neighbourhood
$U$ of $a$ and a holomorphic map which takes $a$ to the origin and is one-to-one between $U$ and a neighbourhood $U'$
of the origin, such that the image of $M$ has the form (\ref{NormForm1}). Hence, without loss of generality we may assume
that $a = 0$ and may view (\ref{NormForm1}) as a representation of $M \cap U$ in the above local coordinates. Then
$Q_0 \cap U = \{ z_n = 0, z'' = 0 \}$. Hence, going back to the original coordinates, we obtain, by the invariance of Segre
varieties, that the intersection $Q_a \cap U$ is a complex submanifold of dimension $n-1$ in $M \cap U$ which coincides
with ${\mathcal L}_a \cap U$. In particular, it belongs to a unique irreducible component of $Q_a$ of dimension $n-1$. It follows also from (\ref{NormForm1}) that it is a unique complex submanifold of dimension $n-1$ through $a$ contained in a
neighbourhood of $a$ in $M$.
(b) Recall that we consider $M$ defined by (\ref{DefEq}). Since $S_a$ is contained in $M$ near $a$, it follows by analyticity of $\rho$ and the uniqueness that
$\rho \vert_{S_a} \equiv 0$, i.e., $S_a$ is contained in $M$.
(c) By part (b), the complex submanifold $S_a$ is contained in $M$. Therefore, in a small neighbourhood of $b$ we have $S_a = S_b$
by part (a). Then also globally $S_a = S_b$ by the uniqueness theorem for irreducible complex analytic sets.
(d) Since $q \in Q_a$, we have $a \in Q_q$. The same holds for every point $a'\in{\mathcal L}_a$ in a neighbourhood of $a$.
Hence, $S_a$ is contained in $Q_q$ by the uniqueness theorem for complex analytic sets. Suppose now that
$\dim_{\mathbb C} Q_q = n-1$. Since $a$ is a regular point of $M$, the leaf ${\mathcal L}_a$ is not contained in the set of singular points of $M$; hence, regular points of
$M$ form an open dense subset in this leaf. Consider a sequence of points
$q^m \in {\mathcal L}_a \cap M_{reg}$ converging to $q$. It follows by (c) that the complex
$n-1$-submanifold $S_a = S_{q^m}$ is independent of $m$
and by (a) $S_{q^m}$ is an irreducible component of $Q_{q^m}$. Passing to the limit we obtain that $S_a$ is contained in $Q_q$ as an
irreducible component.
\end{proof}
\section{Dicritical singularities of Levi-flat subsets}
Let $M$ be a real analytic Levi-flat subset of dimension $2n-1$ in $ \mathbb C^{n+m}$. A singular point $q\in M$ is called
{\it dicritical} if $q$ belongs to infinitely many geometrically different leaves ${\mathcal L}_a$.
Singular points which are not dicritical are called {\it nondicritical}.
Our main result is the following.
\begin{theorem}\label{DicritTheo}
Let $M$ be a real analytic Levi-flat subset of dimension $2n-1$ in $\mathbb C^{n+m}$, irreducible as a germ at $0 \in \overline{M_{reg}}$ . Then $0$ is a dicritical point if and only if $\dim_\mathbb C Q_0 = n$.
\end{theorem}
For hypersurfaces this result is obtained in \cite{PSS}.
A dicritical point is Segre degenerate; this follows by Corollary ~\ref{Segre+Levi}(d).
From now on we assume that $0$ is a Segre degenerate point; our goal is to prove that $0$ is a dicritical point.
For every point $w \in \pi_w(M^\mathbb C)$ denote by $Q_w^c$ the union of irreducible components of $Q_w$ containing the origin; we call it {\it the canonical Segre variety}. Note that by Proposition~\ref{SegreProp}(b), for every $w$ from a neighbourhood of the origin in $Q_0$ its canonical Segre variety $Q_w^c$ is a nonempty complex analytic subset. Consider the set
$$\Sigma = \{ (z,w) \in M^\mathbb C_*: z \notin Q_{w}^c \} .$$
If $\Sigma$ is empty, then for every point $w$ from a neighbourhood of the origin in $\pi_w(M^\mathbb C_*)$ the Segre variety $Q_{w}$ coincides
with the canonical Segre variety $Q_w^c$, i.e., all components of $Q_w$ contain the origin. But for a regular point $w$
of $M$, its Levi leaf is a component of $Q_{w}$. Therefore, every Levi leaf contains the origin which is then necessarily a dicritical point. Thus, in order to prove the theorem, it suffices to establish the following
\begin{prop}
\label{empty}
$\Sigma$ is the empty set.
\end{prop}
Arguing by contradiction assume that $\Sigma$ is not empty. The proof consists in two main steps. First, we prove that the boundary of $\Sigma$ is ``small enough", and so is a removable singularity for $\Sigma$. Second, we prove that the complement of $\Sigma$ is not empty. This will lead to a contradiction.
To begin, we need some technical preliminaries. Consider the complex $2n+m$ dimensional analytic set
$$
Z = \mathbb C^{n+m} \times Q_0 = \{ (z,w): w \in Q_0 \} .
$$
Here we view a copy of $Q_0$ in $\mathbb C^{n+m}(w)$ that is defined by $\pi_w \circ \pi_z^{-1}(0)$.
\begin{lemma}
\label{Q0}
One has $M^\mathbb C \subset Z$. As a consequence, $0 \in Q_w$ for every $(z,w) \in M^\mathbb C$.
\end{lemma}
Essentially this result was proved by Brunella \cite{Bru1}. For the convenience of readers we include the proof.
\begin{proof} Denote by $X$ the proper complex analytic subset of $M^\mathbb C$ where the dimension of fibres of $\pi_w$ is $\ge n$.
Thus for every $(z,w) \in M^\mathbb C \setminus X$ the dimension of the fibres $\pi_w^{-1}(w)$ is equal to $n-1$.
Note that the lift $\tilde Q_0 = \{ (z,w): z= 0, w \in Q_0 \}$ is contained in $M^\mathbb C$. First we claim that the intersection $\tilde Q_0 \cap (M^\mathbb C \setminus X)$ is not empty. Arguing by contradiction, assume that $\tilde Q_0$ is contained in $X$. Then the dimension of the fibre of $\pi_w$ at every point of $\tilde Q_0$ is $\ge n$. Since $\dim \tilde Q_0 = n$, the dimension of $M^\mathbb C$ must be $\ge 2n$, which is a contradiction.
Let $(0,w^0) \in M^\mathbb C \setminus X$. Slightly perturbing $w^0$ one can assume that $w^0$ is a regular point of $Q_0$. Let $U$ be a sufficiently small open neighbourhood of $w^0$
in $\mathbb C^{n+m}$. The fibres of $\pi_w$ over $Q_0 \cap U$ have the dimension $n-1$ so the preimage $\pi_w^{-1}(Q_0 \cap U)$ contains an open piece (of dimension $2n-1$) of $M^\mathbb C$. Since $M^\mathbb C$ is irreducible, we conclude by the uniqueness theorem that $M^\mathbb C \subset Z$ globally.
\end{proof}
The first step of proof consists of the following.
\begin{lemma}
\label{boundary}
We have
\begin{itemize}
\item[(a)] $\Sigma$ is an open subset of $M^\mathbb C$.
\item[(b)] The boundary of $\Sigma$ is contained in a complex hypersurface in $M^\mathbb C$.
\end{itemize}
\end{lemma}
\begin{proof} (a) The fact that the set $\Sigma$ is open in $ M^\mathbb C_*$ follows immediately because the defining functions of a complex variety $Q_{w}$ depend continuously on the parameter $w$.
(b) We are going to describe the boundary of $\Sigma$. Let $(z^k,w^k)$ , $k=1,2,...,$ be a sequence of points from $\Sigma$ converging to some $(z^0,w^0) \in M^\mathbb C_*$. Every Segre variety $Q_{{w}}$ (for $w = w^0$ or $w = w^k$) is a complex analytic subset of dimension $n-1$ containing the origin. Assume that $(z^0,w^0)$ does not belong to $\Sigma$, that is, $(z^0,w^0)$ is a boundary point of $\Sigma$.
The point $(z^0,w^0)$ is a regular point for $M^\mathbb C$, and the point $z^0$ is a regular point of the Segre variety $Q_{{w^0}}$;
we may assume that the same holds for every $(z^k,w^k)$.
For $w = w^0$ or $w = w^k$ denote by $K(w)$ the unique irreducible component of $Q_{w}$ containing $z^0$ or $z^k$ respectively. It follows from the definition of $\Sigma$ that $K(w^0)$ contains the origin, while $K(w^k)$ does not contain the origin, $k=1,2,...$.
The limit as $k\to\infty$ (with respect to the Hausdorff distance) of the sequence $\{K(w^k)\}$ of complex analytic subsets is an $(n-1)$ dimensional complex analytic subset containing $K(w^0)$ as an irreducible component. Indeed, this is true in a neighbourhood of the point $z^0$ and then holds globally by the uniqueness theorem for irreducible complex analytic subsets.
We use the notation $z = (z',z_n,z'') = (z_1,...,z_{n-1},z_n,z_{n+1},...,z_{n+m})$. Performing a complex linear change of coordinates in $\mathbb C^{n+m}(z)$ if necessary,
we can assume that the intersection of $Q_{{w^0}}$ with
the complex linear subspace $ \{ z: z' = 0' \} $ is a discrete set. Denote by $\mathbb D(z_{i_1},...,z_{i_l})$ the unit polydisc $\{ \vert z_{i_j} \vert < 1, j=1,...,l \}$ in the space
$\mathbb C(z_{i_1},...,z_{i_l})$. Choose $\delta > 0$ small enough such that
\begin{equation}
\label{e.fibre}
\{ z: (0',z''): z'' \in \delta \mathbb D(z_n,z'') \} \cap Q_{w^0} = \{ 0 \} .
\end{equation}
Using the notation $w = ('w,''w)$, where $'w = (w_{i_1},...,w_{i_n})$, choose a suitable complex affine subspace
$ \{ ''w = {''w}^0 \}$ of $\mathbb C^{n+m}$ of dimension $n$ such that the canonical projection of $Q_0 \subset \mathbb C^{n+m}(w)$ on $\mathbb C^n('w)$ is proper. Recall that $\dim Q_0=n$. Shrinking $\delta$, one can assume that
$$
Q_0 \cap \{ w: {'w} = {'w}^0, {''w} \in {''w}^0 + \delta\mathbb D({''w}) \} = \{ w^0 \}.
$$
Denote by $\pi$ the projection
$$\pi: (z,w) \mapsto (z', {'w}).$$
Then the intersection $\pi^{-1}(0',{'w}^0) \cap M^\mathbb C$ is discrete (we use here Lemma \ref{Q0}).
Thus, there exist small enough neighbourhoods $U'$ of $0'$ in $\mathbb C^{n-1}(z')$, $'V$ of
$'w^0$ in $\mathbb C^{n}('w)$, and $\delta > 0$ such that the restriction
$$\pi: M^\mathbb C \cap (U' \times \delta\mathbb D(z_n,z'') \times 'V \times (''w^0+\delta \mathbb D(''w))) \longrightarrow U' \times 'V$$
is a proper map. This means that we have the following defining equations for $M^\mathbb C \cap (U' \times \delta\mathbb D(z_n,z'') \times 'V \times (''w^0+\delta \mathbb D(''w)))$:
\begin{eqnarray}\label{gamma1}
\left\{ (z,w) : \Phi_{I}(z','{\overline w})(z_n,z'', {''w}):= \sum_{ \vert J \vert \le d} \phi_{IJ} (z',
'\overline {w})(z_n, z'', ({''w}-{''w}^0))^J = 0, \, \, \, \vert I \vert = d \right\},
\end{eqnarray}
where the coefficients $ \phi_{IJ}(z','{\overline w})$ are holomorphic in $(U' \times 'V)$. The Segre varieties are obtained by fixing $'w$ in the above equations:
\begin{eqnarray}\label{Q1}
Q_{w}= \left\{ z : \Phi_{I}(z','{\overline w})(z_n,z'',{''w}):= \sum_{ \vert J \vert \le d} \phi_{IJ}(z','{\overline w}) (z_n,z'',
({''w} - {''w}^0))^J = 0, \, \, \, \vert I \vert = d \right\} .
\end{eqnarray}
Note that
$ \phi_{I0} (0','{\overline w}) = 0$ for all $I$ and
all $'w$ because every Segre variety contains the origin.
Denote by $\pi_j$ the projection
$$\pi_j: (z,w) \mapsto (z',z_j,w), \,\,\, j=n,n+1,...,n+m .$$
Then the restrictions
$$\pi_j: Q_{w} \cap (U' \times \delta\mathbb D(z_n,z'') ) \longrightarrow U' \times \delta \mathbb D(z_j)$$
are proper for every $w \in 'V \times (''w^0 + \delta \mathbb D(''w))$. The image $\pi_j(Q_w)$ is a complex hypersurface in $U' \times \delta \mathbb D(z_j)$ with a proper projection on $U'$. Hence
\begin{eqnarray}\label{Q2}
\pi_j(Q_{w})= \left\{ (z',z_j): P_j(z', '\overline {w})(z_j):=z_j^{d_j} + a_{j d_j-1}(z', '{\overline w})z_j^{d_j-1}+ ...+ a_{j 0}(z', '{\overline w}) = 0 \right\} ,
\end{eqnarray}
where the coefficients $a_{js}$ are holomorphic in $(z', '{\overline w}) \in U' \times V$. Indeed, the equations (\ref{Q2}) are obtained from the equations (\ref{Q1}) by the standard elimination construction using the resultants of pseudopolynomials $\Phi_{I}$ from (\ref{Q1}), see \cite{Ch}. This assures the holomorphic dependence of the coefficients with respect to the parameter $'{\overline w}$. Note that the first coefficient of each $P_j$ can be made equal to 1 since the projections are proper.
Recall that $K(w^k)$ does not contain the origin in $\mathbb C^{n+m}$ for $k = 1,2,...$. Therefore, for each $k$ there exists some
$j\in\{n, n+1, \dots, n+m\}$ such that the $z_j$ coordinate of some point of the fibre $\pi^{-1}(0') \cap K(w^k)$ does not vanish.
After passing to a subsequence and relabeling the coordinates one can assume that for every $k = 1,2,...,$ the $z_n$-coordinate of some point of the fibre $\pi^{-1}(0') \cap K(w^k)$ does not vanish. The $z_n$-coordinate of every point of the fibre $\pi^{-1}(0') \cap K(w^k)$ satisfies the equation $(\ref{Q2})$ with $j= n$, $z' = 0'$ and $w = w^k$.
Hence, for every $k$ this equation admits a nonzero solution. Passing again to a subsequence one can also assume that there exists $s$ such that for every $k = 1,2,...,$ one has $a_{n s}(0','{\overline w}^k) \neq 0$.
Set $z' = 0$ in (\ref{Q2}). Consider $(\zeta,'w) \in \mathbb C \times 'V$ satisfying the equation
\begin{eqnarray}
\label{algebr}
\zeta^{d_n} + a_{n d_n-1}(0', '{\overline w})\zeta^{d_n-1}+ ...+ a_{n 0}(0', '{\overline w}) = 0 .
\end{eqnarray}
This equation defines a $d_n$-valued algebroid function $'w \mapsto \zeta('w)$. Given $'w \in 'V$ the algebroid function $\zeta$
associates to it the set $\zeta('w) = \{ \zeta_1('w),...,\zeta_s('w))\}$, $s = s('w) \le d_n$, of distinct roots of the equation (\ref{algebr}). Let $j$ be the smallest index such that the coefficient $a_{nj}(0', '{\overline w})$ does not vanish identically. Dividing (\ref{algebr}) by $\zeta^j$ we obtain
\begin{eqnarray}\label{Q3}
\zeta^{d_n-j} + a_{n d_n-1}(0', '{\overline w})\zeta^{d_n-j-1}+ ...+ a_{n j}(0', '{\overline w}) = 0 .
\end{eqnarray}
Every non-zero value of $ \zeta$ satisfies this equation.
For each $w = w^k$, $k=1,2,...$ or $w = w^0$ the fibre $ \{ (z',z_n): z' = 0 \} \cap \pi_n(K(w))$ is a finite set $ \{ p^1(w),...,p^l(w) \}$, $l = l(k)$ in $\mathbb C^n(z',z_n)$. Each non-vanishing $n$-th coordinate $p^\mu_n(w^k)$, $k=1,2,...$ is a value of the algebroid function $ \zeta$ at $'w^k$. By our assumption we have $p^\nu_n(w^k) \neq 0$ for some $\nu$ and every $k=1,2,...$; one can assume that $\nu$ is the same for all $k$. These $p^\nu_n(w^k)$ satisfy the equation (\ref{Q3}) with $'w = 'w^k$. On the other hand, it follows
by~\eqref{e.fibre} that the fibre $ \{ (z',z_n): z' = 0 \} \cap \pi_n(K(w^0))$ is the singleton $\{ 0 \}$ in $\mathbb C^n(z',z_n)$; hence $p^\mu_n(w^0) = 0 $ for all $\mu$. The sequence $(p_n^\nu(w^k))$, $k=1,2,....$ tends to some $p_n^\nu(w^0)$ as $w^k \longrightarrow w^0$. Therefore, every such $p^\nu_n(w^0)$ also satisfies ( \ref{Q3}) with $'w = 'w^0$. But $p^\mu_n(w^0) = 0 $ for all $\mu$ and, in particular, $p_n^\nu(w^0) = 0$. This means that $a_{nj}(0', '{\overline w}^0) = 0$.
Thus the boundary of $ \Sigma$ in $M^\mathbb C_*$ is contained in the complex analytic hypersurface
$$A_1 = (z,w) \in M^\mathbb C_*: a_{nj}(0', '{\overline w}) = 0 \} = 0.$$
The union $A_2 = A_1 \cup M^\mathbb C_{sing} \cup (M^\mathbb C \setminus M^\mathbb C_*)$ is a complex hypersurface in $M^\mathbb C$ with the following property: if $(z,w) \in M^\mathbb C$ is close enough to $(z^0,w^0)$ and belongs to the closure of $\Sigma$ but
does not belong to $\Sigma$, then $(z,w) \in A_2$. \end{proof}
The second step of the proof is given by the following
\begin{lemma}\label{l.compl}
The complement of $\overline\Sigma$ has a nonempty interior in $M^\mathbb C$.
\end{lemma}
\begin{proof} We begin with the choice of a suitable point $w^*$. First fix any point
$(z^*, w^*) \in M^\mathbb C_*$. We can assume that the rank of the projection $ \pi_w$ is maximal and is equal to $n$ in a neighbourhood $O$ of $(z^*,w^*)$ in $M^\mathbb C_*$; denote by $S$ the image $\pi_w(O)$. As above, we use the notation $w = ('w,''w)$ and suppose that the projection $ \sigma: S \ni ('w,''w) \longrightarrow {'w}$ is one-to-one on a neighbourhood $'W$ of $'w^*$ in $ \mathbb C^n$.
Let $l$ be the maximal number of components of $Q_{w}$ for $w$ with $'w \in {'W}$, and let $w^*$ be such that $Q_{ {w^*}}$ has exactly $l$ geometrically distinct components. One can assume that a neighbourhood ${'W}$ of $'w^*$ is chosen
such that $Q_w$ has exactly $l$ components for all $w \in \sigma^{-1}({'W}) \subset S$. Let $K_1(w), \dots, K_l(w)$ be the irreducible components
of $Q_{w}$. Note that the components $K_j(w)$ depend continuously on $w$.
Consider the sets $F_j = \{ 'w \in {'W} : 0 \in K_j( \sigma^{-1}('w)) \}$. Every set $F_j$ is closed in $'W$.
Since $0\in Q_w$ for every $w \in S$ (by Lemma~\ref{Q0}), we have
$\cup_j F_j = {'W}$. Therefore, one of this sets, say, $F_1$, has a nonempty interior in $'W$. This means that there
exists a small ball $B$ in $ \mathbb C^n('w)$
centred at some $'\tilde w$ such that $K_1(w)$ contains $0$ for all $'w \in B \cap {'W}$.
Choose a regular point $\tilde z$ in $K_1( \tilde w)$ where $ \tilde w =\sigma^{-1}('\tilde w))$. Then for every
$(z,w) \in M^\mathbb C$ near $(\tilde z, \tilde w)$, we have $z \in K_1(w)$, i.e., $(z,w)\notin \Sigma$. Hence,
the complement of $\overline\Sigma$ has a nonempty interior. \end{proof}
Now by Lemma~\ref{boundary}(b) and the Remmert-Stein theorem the closure
$\overline\Sigma$ of $\Sigma$ coincides with an irreducible component of $M^\mathbb C$.
Since the complexification $M^\mathbb C$ is irreducible, the closure $\overline\Sigma$ of $\Sigma$ coincides with the whole $M^\mathbb C$. This contradiction with Lemma~\ref{l.compl} concludes the proof of Proposition \ref{empty} and proves Theorem \ref{DicritTheo}.
\
|
1,116,691,500,819 | arxiv | \section{Introduction}
Neutral kaon and neutrino oscillations are discussed in
theoretical physics for more than half a century already. The
standard theoretical approach to describing these phenomena is
based on considering the states with definite strangeness or
lepton number ($K^0, \bar K^0$ and the neutrino flavor states)
that are treated as superpositions of the corresponding mass
eigenstates. It is assumed that it is the former states that are
produced in the strong and weak interactions. Considering their
time evolution in accordance with the Schr\"odinger equation, one
gets the oscillating probabilities of processes. This was first
demonstrated for neutral kaons in paper \cite{Pais:1955sm} and
later for neutrinos in papers
\cite{Pontecorvo:1957cp,Gribov:1968kq}. The present day status of
neutrino oscillations is discussed in detail in textbook
\cite{Bilenky:2010zza} and in review article \cite{Petcov}.
However, although this approach to describing neutral kaon and
neutrino oscillations is physically transparent and simple, it is
not rigorous and in many papers it was argued that it should be
considered as a phenomenological one (see, e.g.
\cite{Giunti:1993se,Grimus:1996av,Lobanov:2015esa,Naumov:2010um,Slad:2016axy}).
Its most noticeable flaw is the violation of the energy-momentum
conservation in the processes, in which $K^0, \bar K^0$ or a
neutrino flavor state are produced, because in local quantum field
theory, where the four-momentum is conserved in any interaction
vertex, different mass-eigenstate components of these states must
have different momenta as well as different energies. Thus, if the
incoming particles have definite four-momenta, then the produced
particles must also have definite four-momenta, which implies
that neutral kaons and neutrinos can be produced only in the mass
eigenstates. Apparently, the spread of the momenta of the
incoming particles (the use of wave packets) does not solve the
problem, if the interaction remains local.
A simple physical idea for solving the problem with the violation
of energy-momentum conservation is to go off the mass shell.
However, a specific realization of this idea encountered serious
difficulties. Its first implementation in QFT was given in paper
\cite{Giunti:1993se}, where it was suggested that the produced
neutrino mass eigenstates are virtual and their motion to the
detection point should be described by the Feynman propagators.
The authors managed to get through rather complicated calculations
and reproduced the results of the standard quantum mechanical
approach. Later this idea was developed in paper
\cite{Grimus:1996av}. The authors of this paper proved an
important theorem that, for large macroscopic distances between
the production and the detection points, the virtual neutrinos are
almost on the mass shell, and thus reduced the description in
terms of the virtual neutrinos to the standard one. Recently, the
approach with virtual neutrinos was also discussed in paper
\cite{Naumov:2010um}, where the authors used a formalism of
relativistic wave packets to facilitate the calculations.
Nevertheless, the calculations in all these papers are too bulky
and the final results are not quite convincing. This is due to the
standard S-matrix formalism of QFT used in these papers, which is
not appropriate for describing processes at finite distances and
finite time intervals.
In the present paper we will modify the standard perturbative
S-matrix formalism so as to be able to calculate the amplitudes of
the processes at finite distances and finite time intervals and
apply it to describing neutrino and neutral kaon oscillations. We
start with discussing neutrino oscillation processes, which is
simpler because in these processes the massive neutrinos can be
treated as stable particles \cite{Petcov:1976ff}.
\section{Neutrino oscillations}
We consider the minimal extension of the Standard Model (SM) by
the right neutrino singlets. After the diagonalization of the
terms sesquilinear in the neutrino fields, the charged current
interaction Lagrangian of leptons takes the form
\begin{equation}\label{L_cc}
L_{cc} = - \frac{g }{2\sqrt{2}}\left(\sum_{i,k = 1}^3 \bar l_i
\gamma^\mu (1 - \gamma^5)U_{ik}\nu_k W^{-}_\mu + h.c.\right),
\end{equation}
where $l_i$ denotes the field of the charged lepton of the i-th
generation, $\nu_i$ denotes the field of the neutrino mass
eigenstate most strongly coupled to $l_i$ and $U_{ik}$ stands for
the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix.
We consider the case, where the neutrinos are produced and
detected in the charged current interaction with electrons. Due
to the structure of the interaction Lagrangian any process
involving the production of a neutrino at one point and its
detection at another point, when treated perturbatively, includes
a subprocess described by the following diagram \vspace*{0.5cm}
\begin{center}
\begin{picture}(193,81)(0,0)
\ArrowLine(40.5,88.0)(40.5,64.5) \Text(33.5,65.5)[r]{$x$}
\Photon(13.5,41.0)(40.5,64.5){2}{3.0} \Text(12.0,41.0)[r]{$W^+$}
\Vertex (40.5,64.5){2} \ArrowLine(40.5,64.5)(167.5,64.5) \Vertex
(167.5,64.5){2} \Text(104.8,70.5)[b]{$\nu_i$}
\ArrowLine(167.5,64.5)(194.5,88.0) \Text(197.0,88.0)[l]{$e^-$}
\Text(175.0,64.5)[l]{$y$} \Photon(167.5,64.5)(194.5,41.0){2}{3.0}
\Text(197.0,41.0)[l]{$W^+$}
\Text(330.0,60.5)[b]{\addtocounter{equation}{1}(\arabic{equation})}
\label{diag}
\end{picture}
\end{center}
\vspace*{-1cm} for all the three neutrino mass eigenstates.
Depending on the neutrino production process, the external fermion
line at the point $x$ can either correspond to an incoming
electron or an outgoing positron. An important point is that there
is no need to add the diagram with interchanged W-bosons, because
they are associated with different spatially separated processes
and therefore should not be considered as identical particles.
Without loss of generality we can assume that the incoming
particles have definite momenta. Therefore all the three virtual
neutrino eigenstates and the outgoing particles also have definite
momenta.
The amplitude in the coordinate representation corresponding to
diagram (\ref{diag}) can be easily written out using the Feynman
rules formulated in \S 23 of textbook \cite{BOSH}. In the present
paper we will not calculate the complete amplitude, we will rather
calculate only its oscillating part that depends on the distance
between the points $x$ and $y$. This means that we can drop the
fermion wave functions that depend only on the particle momenta,
the polarization vectors of the W-bosons, as well as the coupling
constants and the $\gamma$-matrix structures in the vertices.
What remains of the amplitude looks as follows:
\begin{equation}\label{amp_x}
|U_{1i}|^2 e^{-ipx} S^c_i(y - x) e^{iqy},
\end{equation}
where $S^c_i(y - x)$ is the Feynman propagator of the neutrino
mass eigenstate $\nu_i$, $p$ is the sum of the momenta of the
external lines at $x$ and $q$ is the sum of the momenta of the
external lines at $y$.
According to the prescriptions of the standard perturbative
S-matrix theory (\cite{BOSH}, \S 24), next we would have to
integrate with respect to $x$ and $y$ over the Minkowski space,
which would give us the part of the scattering amplitude
corresponding to the inner fermion line of diagram (\ref{diag}).
However, in this case we would get the amplitude of the process
lasting an infinite amount of time and loose the information about
the distance between the production and detection points. In
order to retain this information, we have to integrate with
respect to $x$ and $y$ in such a way that the distance between
these points along the direction of the neutrino propagation
remains fixed. Of course, this is at variance with the standard
S-matrix formalism. However, we recall that the diagram technique
in the coordinate representation was developed by R.~Feynman
\cite{Feynman:1949zx} without reference to S-matrix theory. Thus,
the Feynman diagrams in the coordinate representation make sense
beyond this theory, and for this reason we may integrate with
respect to $x$ and $y$ in any way depending on the physical
problem at hand. In particular, in the case under consideration we
have to integrate in such a way that the distance between the
points $x$ and $y$ along the direction of the neutrino propagation
equals to $L$. This can be achieved by introducing the delta
function $\delta(\vec p(\vec y -\vec x) - |\vec p|L)$ into the
integral, which gives the amplitude
\begin{equation}\label{amp_L}
|U_{1i}|^2 \int dx dy\, e^{-ipx} S^c_i(y - x)\, e^{iqy}\,
\delta(\vec p(\vec y -\vec x) - |\vec p|L).
\end{equation}
It is convenient to make the change of variables
\begin{equation}\label{chevar}
x = u - \frac{z}{2}, \quad y = u + \frac{z}{2}, \quad
\frac{D(x,y)}{D(u,z)} = 1
\end{equation}
and to integrate with respect to $u$ in the integral in formula
(\ref{amp_L}). This transformation shows that the amplitude is
proportional to the delta function of energy-momentum conservation
$(2\pi)^4~\delta(q - p)$, which is unimportant for our further
considerations and will be dropped, and the distance-dependent
propagator of the neutrino mass eigenstate $\nu_i$ in the momentum
representation, which will be denoted by $S^c_i(p,L)$ and is
defined by the remaining integral:
\begin{equation}\label{prop_L_mom}
S^c_i(p,L) = \int dz\, e^{ipz} S^c_i(z)\, \delta(\vec p\vec z -
|\vec p|L).
\end{equation}
Obviously, the integration of this distance-dependent fermion
propagator with respect to $|\vec p|L$ from minus infinity to
infinity gives the standard Feynman fermion propagator in the
momentum representation. Thus, the distance-dependent fermion
propagator can be considered as the distribution of the Feynman
propagator with respect to $L$ along the direction of momentum
$\vec p$.
The integral in formula (\ref{prop_L_mom}) is evaluated exactly
in Appendix 1, and for $\vec p^{\,2} > m_i^2 - p^2$ the result is
given by
\begin{equation}\label{prop_L_mom_a}
S^c_i(p,L) = i\,\frac{\hat p + \vec \gamma \vec
p\left(1 - \sqrt{1 + \frac{p^2 - m_i^2}{\vec p^{\,2}}}\,\right) + m_i }{2|\vec
p|\sqrt{\vec p^{\,2} + p^2 - m_i^2}}\, e^{-i\left(|\vec p| -
\sqrt{\vec p^{\,2} + p^2 - m_i^2}\,\right) L} \,.
\end{equation}
We emphasize that this distance-dependent fermion propagator makes
sense only for macroscopic distances. As we have already mentioned
in the Introduction, the results of paper \cite{Grimus:1996av}
imply that the virtual particles propagating at macroscopic
distances are almost on the mass shell. This means that $|p^2 -
m_i^2|/ \vec p^{\,2} \ll 1$ and we can expand the square roots to
the first order in $(p^2 - m_i^2)/ \vec p^{\,2}$. It is clear that
this term can be dropped everywhere, except in the exponential,
where it is multiplied by a large macroscopic distance $L$, which
results in
\begin{equation}\label{prop_L_mom_bm}
S^c_i(p,L) = i\, \frac{\hat p + m_i }{ 2 \vec
p^{\,2}}\,e^{-i\frac{m_i^2 - p^2}{2|\vec p|} L}\,.
\end{equation}
It is worth noting that this distance-dependent fermion propagator
taken on the mass shell has no pole and does not depend on the
distance, which is also true for the exact propagator in formula
(\ref{prop_L_mom_a}). The integration of this propagator with
respect to $|\vec p|L$ from zero to infinity gives one half of the
Feynman fermion propagator in the momentum representation (the
other half can be obtained by the integration of $S^c_i(p,-L)$
defined in accordance with formula (\ref{prop_L_mom})).
Since the neutrinos produced in the standard processes with
electrons or positrons are ultrarelativistic, the terms $m_i/|\vec
p|$ are negligibly small and we can also drop them. In this
approximation distance-dependent propagator (\ref{prop_L_mom_bm})
takes the simple form
\begin{equation}\label{prop_L_mom_b}
S^c_i(p,L) = i\, \frac{\hat p }{ 2 \vec p^{\,2}}\,e^{-i\frac{m_i^2
- p^2}{2|\vec p|} L}\,.
\end{equation}
Accordingly the amplitude of any process, where the neutrinos are
produced and detected in the charged current interaction with
electrons, is the sum of the amplitudes with all the three
virtual neutrino mass eigenstates and contains, due to
(\ref{amp_L}), (\ref{prop_L_mom}), (\ref{prop_L_mom_b}), the
scalar factor
\begin{equation}\label{amp_L_s}
\sum_{i = 1}^3|U_{1i}|^2\, e^{-i\frac{m_i^2 - p^2}{2|\vec p|}
L}\,.
\end{equation}
Therefore the probability of the process is proportional to
\begin{equation}\label{prob_L_s}
\left|\sum_{i = 1}^3|U_{1i}|^2\, e^{-i\frac{m_i^2 - p^2}{2|\vec
p|} L}\,\right|^2 = 1 - 4 \sum_{i,j = 1, i < j}^3|U_{1i}|^2
|U_{1j}|^2 \sin^2\left(\frac{m_j^2 - m_i^2}{4|\vec p|}\,L\right).
\end{equation}
In this way we have obtained the survival probability of the
electron, which coincides with the survival probability of the
electron neutrino, and reproduced the result of the standard
quantum mechanical approach based on the notion of the neutrino
flavor states \cite{Giunti:2007ry}, although these states have not
been used in the derivation of formula (\ref{prob_L_s}).
Similarly, we can find the transition probability from the charged
lepton of the k-th generation to the charged lepton of the l-th
generation. The corresponding diagram looks like diagram
(\ref{diag}) with other incoming and outgoing fermions. If the
momentum of the neutrino line satisfies the conditions $m_i/|\vec
p| \ll 1$, the amplitude of this process is proportional to
\begin{equation}\label{amp_L_s_kl}
\sum_{i = 1}^3\bar U_{ki}U_{li}\, e^{-i\frac{m_i^2 - p^2}{2|\vec
p|} L}\,,
\end{equation}
and one can easily obtain the general formulas for the oscillating
survival probabilities of charged leptons and transition
probabilities from any charged lepton to another charged lepton,
which again coincide with the corresponding standard formulas for
neutrino flavor states \cite{Giunti:2007ry}; we will not write
them out here.
Several remarks are in order. First, in the SM only the sum of all
three diagrams (\ref{diag}) with all the neutrino mass eigenstates
is gauge invariant, and therefore it is impossible to derive by
this method the formula for the oscillation of only two neutrino
flavors as it is usually done in the standard approach. Second, if
the neutrino production and detection processes are specified, one
can calculate the amplitude of the complete process by drawing the
corresponding Feynman diagram, constructing the amplitude in the
momentum space in accordance with the standard Feynman rules and
replacing the standard fermion propagator in the latter by the
distance-dependent propagator in formula (\ref{prop_L_mom_bm}). By
this method, one can also calculate the loop corrections to this
amplitude that will determine the actual accuracy of formula
(\ref{prob_L_s}). Third, to calculate the survival and transition
probabilities of charged leptons in realistic experiments one has
to generalize the approach by considering the amplitudes of
processes, where the distance between the production and detection
points lies between $L_1$ and $L_2$. Such an amplitude can be
obtained by integrating the amplitude in formula (\ref{amp_L})
with respect to $|\vec p|L$ from $|\vec p|L_1$ to $|\vec p|L_2$.
However, in the case $L_2 - L_1 \ll L_1$, which is specific for
experiments with neutrinos, the result is trivial: the amplitude
corresponding to the distance $L_1$ is just multiplied by $|\vec
p|(L_2 - L_1)$. We note once again that the integration of the
amplitude in formula (\ref{amp_L}) with respect to $|\vec p|L$
from minus infinity to infinity gives the standard amplitude with
the Feynman propagators. Thus, the amplitude integrated from
$|\vec p|L_1$ to $|\vec p|L_2$ can be considered as a portion of
the standard Feynman amplitude. Fourth, formula (\ref{prob_L_s})
is formally valid for any value of the macroscopic distance $L$.
The momentum spread of the produced neutrinos gives rise to the
standard coherence length, which limits its range of
applicability. In the framework of the present approach there may
exist one more coherence length arising due to a different reason.
As we have already noted, the virtual particles propagating at
macroscopic distances are almost on the mass shell. However, the
theorem proved in paper \cite{Grimus:1996av} does not define
explicitly the admissible deviation of the virtual particle
momentum squared from the mass shell depending on the distance
traveled. If such an admissible deviation exists, one can imagine
the situation, where, for certain large distances, the admissible
deviation from the mass shell is less than the minimal absolute
value of the differences of the neutrino masses squared. In this
case one should expect that either only two of the three diagrams
contribute to the amplitude of the process or only one of the
three diagrams contributes to the amplitude of the process, which
means that there would be either a change of the interference
pattern at a certain large distance or no interference at all
between the contributions of different neutrino mass eigenstates
and, therefore, no oscillation of the process probability
depending on the distance. For this reason it would be very
interesting to improve the theorem of paper \cite{Grimus:1996av}
so as to estimate the admissible deviation of virtual particles
from the mass shell depending on the distance.
\section{Neutral kaon oscillations}
Now we turn to discussing neutral kaon oscillations. The
production of neutral kaons by 24~$GeV$ protons hitting a platinum
target and their subsequent decay to $\pi^+ \pi^-$ pairs were
studied in paper \cite{Geweniger:1973di}. It was shown in this
paper that the probability of detecting a $\pi^+ \pi^-$ pair
includes an oscillating component.
The standard theoretical explanation of this phenomenon is based
on the assumption that the state with strangeness -1 denoted $K^0$
is predominantly produced in the original interaction of protons
with platinum nuclei. This state has no definite mass and can be
treated as a superposition of states with definite masses. The
time evolution of this superposition of the mass eigenstates gives
rise to the oscillating decay probability to $\pi^+ \pi^-$ pairs
\cite{Pais:1955sm,Belusevic:1998pw}.
However, as we have already mentioned in the previous section, in
local QFT the production of a state without definite mass violates
the energy-momentum conservation. Below we will show, how this
problem can be solved by considering distance-dependent and
time-dependent amplitudes of processes mediated by the particles
$K^0_S$ and $K^0_L$. The situation here is more complicated than
in the case of neutrinos for two reason: neutral kaons are states
of quark-antiquark pairs bound by the strong interaction and they
are unstable.
Let us denote the mass and the width of $K^0_S$ by $m_S, \Gamma_S$
and those of $K^0_L$ by $m_L, \Gamma_L$. Since these particles are
produced in the strong interaction processes, we cannot isolate
the subprocesses of $K^0_S, K^0_L$ production as definitely, as in
the case of only weakly interacting neutrinos. However, we can
assume that either a virtual $K^0_S$ or a virtual $K^0_L$ is
produced at a point $x$ and decays to $\pi^+ \pi^-$ pair at a
point $y$, which can be represented by the following diagram:
\vspace*{0.5cm}
\begin{center}
\begin{picture}(193,81)(0,0)
\Text(33.5,65.5)[r]{$x$}\Vertex (40.5,64.5){5}
\DashLine(40.5,64.5)(167.5,64.5){5} \Vertex (167.5,64.5){2}
\Text(104.8,70.5)[b]{$K^0_S (K^0_L)$}
\DashLine(167.5,64.5)(194.5,88.0){3} \Text(197.0,88.0)[l]{$\pi^+$}
\Text(175.0,64.5)[l]{$y$} \DashLine(167.5,64.5)(194.5,41.0){3}
\Text(197.0,41.0)[l]{$\pi^-$}
\Text(330.0,60.5)[b]{\addtocounter{equation}{1}(\arabic{equation})}
\label{diag_K}
\end{picture}
\end{center}
\vspace*{-1cm} The amplitudes of the $\pi^+ \pi^-$ pair
production processes mediated by $K^0_S$ and $K^0_L$ differ in the
production amplitudes, the propagators and the decay amplitudes to
$\pi^+ \pi^-$. Denoting by $P_S$ the production amplitude of
$K^0_S$ and by $A_{+-,S} $ its decay amplitude to $\pi^+ \pi^-$,
as well as by $P_L$ the production amplitude of $K^0_L$ and by
$A_{+-,L} $ its decay amplitude to $\pi^+ \pi^-$, we can write
down the amplitudes corresponding to diagram (10) as follows:
\begin{equation}\label{amp_x_K}
P_S\, e^{-ipx} D^c_S(x - y) e^{iqy} A_{+-,S} ,\quad P_L \,
e^{-ipx} D^c_L(x - y) e^{iqy} A_{+-,L},
\end{equation}
where $ D^c_S(x - y)$ and $ D^c_L(x - y)$ stand for the
propagators of $K^0_S$ and $K^0_L$, $p$ is the total momentum
entering the vertex at the point $x$ and $q$ is the sum of the
momenta of $\pi^+ \pi^-$ pair.
To find the distance-dependent amplitude of the process under
consideration, we again have to integrate these amplitudes with
respect to $x$ and $y$ in such a way that the distance between
the points $x$ and $y$ along the direction of the neutral kaon
propagation equals to $L$. The amplitudes due to virtual $K^0_S$
and $K^0_L$ having absolutely the same form, we will do it in
detail only for the amplitude due to $K^0_S$ that looks like
\begin{equation}\label{amp_L_K}
P_S A_{+-,S}\,\int dx dy\, e^{-ipx}D^c_S(x - y) e^{iqy} \,
\delta(\vec p(\vec y -\vec x) - |\vec p|L)\,.
\end{equation}
Again making the change of variables in accordance with
(\ref{chevar}), we obtain that the amplitude is proportional to
the delta function of energy-momentum conservation $(2\pi)^4
\delta(q -p)$, which is dropped, and the distance-dependent
propagator of $K^0_S$ in the momentum representation that will be
denoted by $D^c_S(p,L)$ and is defined by the integral
\begin{equation}\label{prop_L_K}
D^c_S(p,L) = \int dz\, e^{ipz}D^c_S(z) \, \delta(\vec p\vec z -
|\vec p|L)\,.
\end{equation}
This integral is evaluated exactly in Appendix 2, the result for
$\vec p^{\,2} > m_S^2 - p^2 $ being given by
\begin{equation}\label{prop_L_mom_Ka}
D^c_S(p,L) = \frac{i}{2|\vec p|\sqrt{\vec p^{\,2} + p^2 - m_S^2 +
i m_S \Gamma_S}}\, e^{-i\left(|\vec p| - \sqrt{\vec p^{\,2} + p^2
- m_S^2 + i m_S \Gamma_S}\,\right) L} \,.
\end{equation}
We recall once more that the virtual particles propagating at
macroscopic distances are almost on the mass shell
\cite{Grimus:1996av}, and, therefore, $|p^2 - m_S^2|/ \vec p^{\,2}
\ll 1$. The ratio $m_S\Gamma_S/ \vec p^{\,2}$ being also very
small, we actually have $|p^2 - m_S^2 + im_S\Gamma_S|/ \vec
p^{\,2} \ll 1$ and can drop the term $p^2 - m_S^2 + i m_S
\Gamma_S$ in the square root in the denominator and expand the
square root in the exponential to the first order in this term.
In this approximation formula (\ref{prop_L_mom_Ka}) for the
distance-dependent propagator of $K^0_S$ can be written as
\begin{equation}\label{prop_L_mom_Kf}
D^c_S(p,L) = \frac{i}{2\vec p^{\,2}}\, e^{-i\frac{m_S^2 - p^2 -
i m_S \Gamma_S }{2|\vec p|} L} \,.
\end{equation}
The same calculations can be easily repeated for the
distance-dependent propagator of $K^0_L$, and from
(\ref{amp_x_K}), (\ref{amp_L_K}),(\ref{prop_L_mom_Kf}) one gets
that the total amplitude is proportional to
\begin{equation}\label{amp_L_mom_K_tot}
e^{-\frac{ m_S \Gamma_S}{2|\vec p|} L} \, e^{-i\frac{ m_S^2 - p^2}
{2|\vec p|} L} + \frac{P_L }{P_S} \eta_{+-} e^{-\frac{ m_L
\Gamma_L}{2|\vec p|} L} \, e^{-i\frac{m_L^2 - p^2}{2|\vec p|}
L}\,,
\end{equation}
where $\eta_{+-} = A_{+-,L}/ A_{+-,S}$\, is the standard parameter
for describing the decay of neutral kaons to $\pi^+ \pi^-$ pairs
\cite{Belusevic:1998pw}. It is convenient to isolate the phases of
$\eta_{+-}$ and ${P_L }/{P_S}$ in accordance with the
parameterizations
\begin{equation}\label{phases}
\eta_{+-} = |\eta_{+-}|e^{i\phi_{+-}}, \quad \frac{P_L }{P_S} =
\left|\frac{P_L }{P_S}\right| e^{i\phi_{P}}.
\end{equation}
The specificity of the experiments with neutral
kaons suggests that their results should be represented not in
terms of the distance $L$, but rather in terms of the time $T = L
p^0/|\vec p|$ \cite{Geweniger:1973di}. Substituting $L$ in terms
of $T$ into formula (\ref{amp_L_mom_K_tot}), we get
\begin{equation}\label{amp_LT_mom_K_tot}
e^{-\frac{ m_S \Gamma_S}{2p^0} \,T} \, e^{-i\frac{m_S^2 -
p^2}{2p^0} \,T} + \frac{P_L }{P_S} \eta_{+-} e^{-\frac{ m_L
\Gamma_L}{2p^0} \,T} \, e^{-i\frac{m_L^2 - p^2}{2p^0} \,T} \,.
\end{equation}
However, it is clear that time-dependent amplitudes can be
obtained directly from amplitudes (\ref{amp_x_K}) by multiplying
them by the delta function $\delta(\beta(y^0 - x^0 - T))$ and then
integrating with respect to $x$ and $y$. Here the parameter
$\beta$ has dimension of mass and will be defined later. Similar
to the case of the distance-dependent amplitude, the integral
entering the expression for the time-dependent amplitude due to
virtual $K^0_S$ can be represented as the product of the delta
function of energy-momentum conservation and the time-dependent
propagator of $K^0_S$ in the momentum representation $D^c_S(p,T)$
defined by the formula
\begin{equation}\label{prop_T_K}
D^c_S(p,T) = \int dz \, e^{ipz}D^c_S(z) \, \delta(\beta (z^0 -
T))\,.
\end{equation}
The integral in this formula is evaluated exactly in Appendix 3,
and the resulting time-dependent propagator looks like
\begin{equation}\label{prop_T_mom_4_K_S}
D^c_S(p,T) = \frac{i}{2\beta\sqrt{(p^0)^2 + m_S^2 - p^2 - i m_S
\Gamma_S}}\, e^{i\left(p^0 -\sqrt{(p^0)^2 + m_S^2 - p^2 - i m_S
\Gamma_S}\,\right) T} \,.
\end{equation}
Again we have $| m_S^2 - p^2 - im_S\Gamma_S|/ (p^0)^2 \ll 1$ and
for this reason we can drop the term $m_S^2 - p^2 - im_S\Gamma_S$
in the square root in the denominator and expand the square root
in the exponential to the first order in this term. In this
approximation formula (\ref{prop_T_mom_4_K_S}) takes the form
\begin{equation}\label{prop_T_mom_Kf}
D^c_S(p,T) = \frac{i}{2\beta p^{0}}\, e^{-i\frac{m_S^2 - p^2 -
im_S \Gamma_S}{2p^0} \,T}\,,
\end{equation}
which coincides with (\ref{prop_L_mom_Kf}), if we choose $\beta =
\vec p^{\,2}/p^0$ and substitute $T = L p^0/ |\vec p|$. Thus, if
one uses the matched delta functions in the propagator
definitions, the distance-dependent propagator and the
time-dependent propagator at macroscopic distances and time
intervals are, in fact, the same and differ only in the
parametrization. It is worth noting that this is also true for the
case of neutrinos, where formula (\ref{prop_L_mom_b}) can be
obtained by substituting $\beta = \vec p^{\,2}/p^0, \, T = L p^0/
|\vec p|$\,\, into the corresponding time-dependent propagator.
Moreover, in the case of ultrarelativistic neutrinos one can put
$\beta = p^0$ and get the time-dependent propagator with the same
accuracy.
Summing the time-dependent amplitudes due to $K^0_S$ and $K^0_L$,
one again gets that the total amplitude is proportional to the
expression in formula (\ref{amp_LT_mom_K_tot}). It is customary to
write this formula in terms of the proper time, which, in the case
under consideration, should be defined as follows:
\begin{equation}\label{p_time}
t_p = \frac{2 p^0 \,T}{m_S + m_L}.
\end{equation}
Formula (\ref{amp_LT_mom_K_tot}) rewritten in terms of the proper
time looks like
\begin{equation}\label{amp_t_p_mom_K_tot}
e^{-\frac{ m_S \Gamma_S}{m_S +\, m_L} \,t_p} \, e^{-i\frac{m_S^2 -
p^2 }{m_S +\, m_L} \,t_p} + {\left|\frac{P_L }{P_S}\right|
|\eta_{+-}|}\, e^{i(\phi_P + \phi_{+-})} e^{-\frac{ m_L
\Gamma_L}{m_S +\, m_L} \,t_p} \, e^{-i\frac{ m_L^2 - p^2}{m_S +\,
m_L} \,t_p} \,,
\end{equation}
where the phases $\phi_P$ and $\phi_{+-}$ have been defined in
(\ref{phases}). Therefore the probability of the $\pi^+ \pi^-$
production process is proportional to
\begin{eqnarray}\nonumber
&&e^{-\frac{2 m_S \Gamma_S}{m_S +\, m_L} \,t_p} + {\left|\frac{P_L
}{P_S}\right|^2 |\eta_{+-}|^2 }\, e^{-\frac{2 m_L \Gamma_L}{m_S
+\, m_L} \,t_p} + \\ \label{prob_t_K} &+& 2 \left|\frac{P_L
}{P_S}\right| |\eta_{+-}| e^{-\frac{ m_S \Gamma_S + \, m_L
\Gamma_L}{m_S +\, m_L} \,t_p} \cos\left( \Delta m_{L,S}\, t_p -
\phi_P - \phi_{+-} \right),
\end{eqnarray}
where $\Delta m_{L,S} = m_L - m_S $. For $P_L = P_S$ and $P_L =
-P_S$ this expression coincides exactly with those presented in
\cite{Belusevic:1998pw}, \S 4.3, if one neglects the difference
between $m_L$ and $m_S$ in all the exponentials. This formula can
also be rewritten in the form
\begin{eqnarray}\nonumber
&& \left(e^{-\frac{ m_S \Gamma_S}{m_S +\, m_L} \,t_p} +
{\left|\frac{P_L }{P_S}\right| |\eta_{+-}| }\, e^{-\frac{ m_L
\Gamma_L}{m_S +\, m_L} \,t_p}\right)^2 - \\ \label{prob_t_K1} &-&
4 \left|\frac{P_L }{P_S}\right| |\eta_{+-}| e^{-\frac{ m_S
\Gamma_S + \, m_L \Gamma_L}{m_S +\, m_L} \,t_p} \sin^2\left(
\frac{\Delta m_{L,S}}{2}\, t_p - \frac{\phi_P + \phi_{+-}}{2}
\right),
\end{eqnarray}
which is most similar to formula (\ref{prob_L_s}) for neutrino
oscillations. Thus, we have again reproduced the results of the
standard approach without reference to the states with definite
strangeness $K^0, \bar K^0.$
\section{Conclusion}
In the present paper we have shown that it is possible to give a
consistent quantum field-theoretical description of neutrino
oscillations in the SM minimally extended by the right neutrino
singlets. To this end we had just to adapt the standard
perturbative S-matrix formalism for calculating the amplitudes of
the processes passing at finite distances and finite time
intervals. The developed approach is physically transparent and,
unlike the standard one, has the advantage of not violating the
energy-momentum conservation. In its framework, the calculations
of amplitudes are much simpler than in papers
\cite{Giunti:1993se,Naumov:2010um}, where a similar approach based
on the standard S-matrix description of the propagation of virtual
neutrinos is used.
The application of this modified formalism to describing the
neutrino and neutral kaon oscillation processes showed that the
standard results can be easily and consistently obtained using
only the mass eigenstates of these particles. Therefore, the
neutrino flavor states and the neutral kaon states with definite
strangeness are redundant in the theory and should be amputated by
Occam's razor.
Although the results obtained within the developed approach
coincide with the standard results of the neutrino and neutral
kaon oscillation theory, the physical picture of the phenomena
changes radically. If there are no neutrino flavor states, there
is no neutrino oscillation, and the term can be retained only as a
historical one. What remains of this theory is the observable
oscillating survival probabilities of charged leptons and the
observable oscillating transition probabilities from one charged
lepton to another charged lepton arising due to the interference
of the amplitudes of the processes mediated by different neutrino
mass eigenstates. Thus, it is the mass eigenstates $\nu_i, \, i =
1,2,3,$ that are the only observable neutrino states in the SM.
These states can be consistently treated as particles, and it is
natural to call them after the charged leptons, to which they are
most strongly coupled, the electron neutrino, the muon neutrino
and the tau neutrino respectively, which restores the quark-lepton
symmetry in the SM.
In the present paper we have considered only the case of Dirac
neutrinos. However, all the results can be transferred, \emph{
mutatis mutandis}, to the case of Majorana neutrinos, because
their propagators have essentially the same structure.
A similar situation takes place in the case of neutral kaons,
where only the states $K^0_S$ and $K^0_L$ can be considered to be
particles. The states $K^0$ and $\bar K^0$ are artificial abstract
entities that are unnecessary in QFT and cannot exist or even be
produced in Nature, where, unlike in the theory, the weak
interaction cannot be switched off. Therefore, we have to admit
that there is no neutral kaon oscillations, but there is an
interference of the amplitudes with virtual $K^0_S$ and $K^0_L$
instead.
Finally, we note that one can construct the amplitudes and find
the cross sections of processes with any particles, whose
production and detection sites are separated by a macroscopic
distance, by drawing the corresponding Feynman diagrams,
constructing the amplitudes in the momentum space in accordance
with the standard Feynman rules and replacing the standard
propagators in the latter by the corresponding distance-dependent
or time-dependent propagators introduced in the present paper.
Such calculations may be useful for analyzing events in the
experiments, where detectors are situated at macroscopic distances
from the interaction points.
\bigskip
{\large \bf Acknowledgments}
\medskip \\
\noindent The author is grateful to E. Boos, A. Lobanov and M.
Smolyakov for reading the manuscript and making important
comments. The work was supported by grant NSh-7989.2016.2 of the
President of Russian Federation.
\setcounter{equation}{0}
\renewcommand{\theequation}{A\arabic{equation}}
\section*{Appendix 1}
Substituting the standard integral representations for the
fermion propagator and the delta function into the integral
defining the distance-dependent propagator in formula
(\ref{prop_L_mom}), we get
\begin{equation}\label{prop_L_mom_1}
S^c_i(p,L) = \frac{1}{(2\pi)^5} \int dz dk dt\, e^{ipz} e^{-ikz}
e^{i(\vec p\vec z - |\vec p|L)t} \frac{\hat k + m_i }{m^2_i - k^2
- i\epsilon}\,,
\end{equation}
where $m_i$ denotes the mass of the neutrino mass eigenstate
$\nu_i$.
Next we make the change of variable $t = \omega/|\vec p|$ and
integrate with respect to $z$, which results in
\begin{equation}\label{prop_L_mom_2}
S^c_i(p,L) = \frac{ 1}{2\pi |\vec p|} \int d\omega dk \,
\delta(p^0 - k^0) \delta(\vec k - \vec p + \vec n \omega )
e^{-i\omega L} \frac{\hat k + m_i }{m^2_i - k^2 - i\epsilon}\,,
\end{equation}
where $\vec n = \vec p / |\vec p|$.
The integration with respect
to $k$ is trivial due to the $\delta$-functions:
\begin{equation}\label{prop_L_mom_3}
S^c_i(p,L) = \frac{ 1}{2\pi |\vec p|} \int d\omega \,
e^{-i\omega L} \frac{\hat p + \vec \gamma \vec n \omega + m_i
}{m^2_i - p^2 - 2 |\vec p| \omega + \omega^2 - i\epsilon}\,.
\end{equation}
The last integral can be evaluated for $ \vec p^{\,2} > m_i^2 -
p^2$ by closing the integration contour in the lower complex
half-plane and calculating the residue at the simple pole, which
gives:
\begin{equation}\label{prop_L_mom_4}
S^c_i(p,L) = i\, \frac{\hat p + \vec \gamma \vec
p\left(1 - \sqrt{1 + \frac{p^2 - m_i^2}{\vec p^{\,2}}}\right) + m_i }{2|\vec
p|\sqrt{\vec p^{\,2} + p^2 - m_i^2}}\, e^{-i\left(|\vec p| -
\sqrt{\vec p^{\,2} + p^2 - m_i^2}\,\right) L} \,.
\end{equation}
\section*{Appendix 2}
Again substituting the standard integral representations for the
scalar field propagator and the delta function into the integral
in formula (\ref{prop_L_K}), we get
\begin{equation}\label{prop_L_mom_K}
D^c_S(p,L) = \frac{1}{(2\pi)^5}\, \int dz dk dt\, e^{ipz} e^{ikz}
e^{i(\vec p\vec z - |\vec p|L)t} \frac{1 }{m^2_S - k^2 - im_S
\Gamma_S}\,.
\end{equation}
Then we make the change of variable $t = \omega/|\vec p|$ and
integrate with respect to $z$ and $k$ similar to
(\ref{prop_L_mom_2}), (\ref{prop_L_mom_3}), which gives
\begin{equation}\label{prop_L_mom_3_K}
D^c_S(p,L) = \frac{1}{2\pi |\vec p|} \int d\omega \, e^{-i\omega
L} \frac{1}{m^2_S - p^2 - 2 |\vec p| \omega + \omega^2 - im_S
\Gamma_S}\,.
\end{equation}
For $ \vec p^{\,2} > m_S^2 - p^2$, the residue integration method
gives:
\begin{equation}\label{prop_L_mom_4_K}
D^c_S(p,L) = \frac{i}{2|\vec p|\sqrt{\vec p^{\,2} + p^2 - m_S^2 +
i m_S \Gamma_S}}\, e^{-i\left(|\vec p| -\sqrt{\vec p^{\,2} + p^2 -
m_S^2 + i m_S \Gamma_S}\,\right) L} \,.
\end{equation}
\section*{Appendix 3}
Once more substituting the standard integral representations for
the scalar field propagator and the delta function into the
integral in formula (\ref{prop_T_K}), we get
\begin{equation}\label{prop_T_mom_K}
D^c_S(p,T) = \frac{1}{(2\pi)^5}\, \int dz dk dt\, e^{ipz}
e^{ikz} e^{i\beta ( z^0 - T)t}\, \frac{1 }{m^2_S - k^2 - im_S
\Gamma_S}\,.
\end{equation}
To evaluate this integral we make the change of variable $t =
\omega/\beta $, then integrate with respect to $z$ and $k$
similar to (\ref{prop_L_mom_2}), (\ref{prop_L_mom_3}) and arrive
at the result
\begin{equation}\label{prop_T_mom_3_K}
D^c_S(p,T) = \frac{1}{2\pi \beta} \int d\omega \, e^{-i\omega T}
\frac{1}{m^2_S - p^2 - 2 p^0 \omega - \omega^2 - im_S
\Gamma_S}\,.
\end{equation}
The residue integration method gives:
\begin{equation}\label{prop_T_mom_4_K}
D^c_S(p,T) = \frac{i}{2\beta\sqrt{(p^0)^2 + m_S^2 - p^2 - i m_S
\Gamma_S}}\, e^{i\left(p^0 -\sqrt{(p^0)^2 + m_S^2 - p^2 - i m_S
\Gamma_S}\,\right) T} \,.
\end{equation}
|
1,116,691,500,820 | arxiv | \section{SUPPLEMENTARY MATERIAL:\\Orthogonality catastrophe as a consequence of the quantum speed limit}
\section{Survival probability behavior as a function of $\lambda$}
As commented in the main text, the behavior of the survival probability shows markedly different characteristics depending on which phase the LMG bath is initialized in. In Fig.~\ref{FigSuppmat}(a) we consider a large environment, $N\!=\!1000$, and sweep through a range of values for $\lambda$. The qualitative and quantitative difference between the two phases is evident. For $\lambda\!<\!1$ we see no evidence of the orthogonality catastrophe, while conversely, as soon as the system is in the critical phase the seemingly small perturbation introduced by quenching on the interaction with the impurity is sufficient to drive the total state to orthogonality.
\section{Energy spectrum of the LMG model}
\label{LMGspectrum}
In Fig.~\ref{FigSuppmat}(b) we show the energy spectrum for the isotropic LMG model, which is characterized by a cascade of energy level crossings in the critical phase, $\lambda\!>\!1$. We see that in the aligned phase the ground state is a single energy level, and since we are writing eigenstates of the LMG model in terms of the collective spin operator $S_z$, this eigenstate then corresponds to the $\ket{-\tfrac{N}{2}}$ state. When we enter the critical phase we see the energy level crossings. The value of $\lambda$ at which these crossings occur is dependent on the size of the system as evident by comparing the spectra for $N\!=\!10$ and $100$. This cascade has a significant effect on the resulting variance of the energy as discussed in the main text and shown in the inset.
\section{Quantum speed limit and work}
\label{sec:fogartyQSL}
Here we provide some additional details for deriving Eq.~\eqref{fogartyQSL}. Consider again the Bures angle $\mc{L}$ for which we have that
\begin{equation}
\partial_t\mathcal{L}(\ket{\psi_0},\ket{\psi_\tau}) \leq \vert \partial_t \mathcal{L}(\ket{\psi_0},\ket{\psi_\tau}) \vert = \left\vert \frac{ \partial_t \bra{\psi_0}\psi_t\rangle }{\sqrt{1-\bra{\psi_0}\psi_t\rangle^2 }} \right\vert
\end{equation}
and therefore
\begin{equation}
\dot{\mathcal{L}} \sin\mathcal{L} \leq \vert \partial_t \bra{\psi_0}\psi_t\rangle \vert.
\end{equation}
Taking the spectral decomposition of the initial (final) total Hamiltonian, $H_{i(f)}=\sum_j \tfrac{E^{i(f)}_j}{\hbar} \ket{\phi^{i(f)}_j}\!\bra{\phi^{i(f)}_j}$. For a sudden quench the evolved state is simply given by
\begin{equation}
\ket{\psi_t} = \sum_j e^{-i E^{f}_j t/\hbar} \ket{\phi^{f}_j}\!\bra{\phi^{f}_j} \psi_0 \Big>.
\end{equation}
Assuming the system is initially in the ground state, the survival probability is essentially the absolute value squared of the characteristic function of the work distribution~\cite{Silva2008,Talkner2016}
\begin{equation}
\chi(t) = \sum_j e^{i (E^{f}_j - E^i_0)t/\hbar}\, \Big\vert \bra{\phi^{f}_j} {\psi_0}\Big>\,\Big\vert^2,
\end{equation}
and we obtain
\begin{equation}
\dot{\mathcal{L}} \sin\mathcal{L} \leq \vert \partial_t \chi(t) \vert = \left\vert \sum_j \frac{(E^{f}_j-E^{i}_0)}{\hbar} e^{i (E^{f}_j - E^i_0)t/\hbar} \left\vert \bra{\phi^{f}_j}\psi_0\Big> \right\vert^2 \right\vert.
\end{equation}
Integrating and simplifying we find
\begin{equation}
\label{fogartyQSL2}
\tau_\mrm{W} = \frac{\hbar\left(1 - \vert\chi(t)\vert \right)}{\vert \langle W \rangle \vert}.
\end{equation}
where $\langle W \rangle=\partial_t \chi(t) \vert_{t=0}$ is the average work done in performing the quench. Indeed for the simple case of the trap quench with the non-interacting Fermi gas, the average work is given as $\langle W \rangle=\frac{N}{4}\left[\eta^2-1\right]$ therefore exhibiting similar scaling to $\tau_\mrm{QSL}$, see Fig.\ref{FigSuppmat}(c).
\begin{figure}[b]
{\bf (a)} \hskip0.5\columnwidth {\bf (b)}
\includegraphics[width=0.45\columnwidth]{figSupp_a.jpg}~~\includegraphics[width=0.6\columnwidth]{figSupp_b.pdf}\\
{\bf (c)} \\
\includegraphics[width=0.5\columnwidth]{figSupp_c.png}
\caption{(a) $\mathcal{F}$ as a function of time, $t$, and environmental coupling, $\lambda$, for a large environment, $N\!\!=\!\!1000$. (b) Energy spectra for the isotropic LMG model for $N\!=\!10$ (thick, blue) and $N\!=\!100$ (thin, red). {\it Inset:} Behavior of $\Delta H$, Eq.~\eqref{varF} for $N\!=\!10$ (red), $100$ (blue), and 1000 (green). (c) QSL time for the fermionic trap quench of $\omega_2\!=\!4$ with the Mandelstam-Tamm (blue triangles), Margolus-Levitin (green circles), and the average work bound (yellow diamonds).}
\label{FigSuppmat}
\end{figure}
\end{document} |
1,116,691,500,821 | arxiv | \section{Introduction}
We describe two recent additions to SFIRE, a fire spread model coupled with
WRF \cite{Mandel-2009-DAW,Mandel-2011-CAF}. This model builds on the earlier
CAWFE code \cite{Clark-1996-CAF-x,Clark-2004-DCA,Coen-2005-SBE}; see
\cite{Mandel-2011-CAF} and
\url{http://www.openwfm.org/wiki/WRF-Fire_development_notes} for further
historical details and acknowledgements. The coupled model is available from
\url{OpenWFM.org}. An earlier version of the code is included in WRF release
as WRF-Fire.
\section{Fuel moisture model}
\label{sec:moisture}
Fire spread rate depends strongly on the moisture contents of the fuel. In
fact, the spread rate drops to zero when the moisture reaches the so-called
extinction value \cite{Pyne-1996-IWF}. For this reason, we have coupled the
fire spread model with a simple fuel moisture model integrated in SFIRE and
run independently at points of the mesh. See
\cite{Nelson-2000-PDC,Weise-2005-CTM} for other, much more sophisticated
models. \cite{Coen-2005-SBE} used an assumed diurnal dependence of fuel moisture
on time. Empirical models
\cite{Fosberg-1971-DHT,VanWagner-1985-EFP} attempt to predict fuel moisture
from meteorological conditions measured daily. We use a simple timelag
differential equation at every point of the domain. This equations has
solutions which approach an equilibrium exponentially, if the equilibrium does
not change. In general, the solutions track a changing equilibrium with a delay.
\label{sec:equilibrium}First, relative humidity is computed from the
atmospheric temperature $T$ (K), waver vapor contents $Q$ (kg$/$kg), and
pressure $P$ (Pa). The saturated water vapor pressure $P_{WS}$ (Pa) is
approximated, following \cite[Eq. (10)]{Murphy-2005-RVP}, as $P_{WS}%
=\exp(54.842763-6763.22/T-4.210\log T+0.000367T+\tanh\left\{ 0.0415\left(
T-218.8\right) \right\} (53.878-1331.22/T-9.44523\log
T+0.014025T)),123<T<332\mathrm{K}$. The water vapor pressure is $P_{W}=PQ/\left(
\varepsilon+\left( \ 1-\varepsilon\right) Q \right) $, where $\varepsilon
=0.622$ is the ratio of the molecular weight of water ($18.02$ g/mol) to
molecular weight of dry air ($28.97$ g/mol). We then obtain the relative
humidity $H=100P_{W}/P_{WS}.$The temperature and the relative humidity of the
air are then used to estimate the drying and wetting fuel equilibrium moisture
contents \cite[eq. (4), (5)]{VanWagner-1985-EFP} \cite[eq. (7), (8)]%
{Viney-1991-RFF},
\begin{align*}
E_{d} & =0.924H^{0.679}+0.000499e^{0.1H}+0.18(21.1+273.15-T)(1-e^{-0.115H}%
),\\
E_{w} & =0.618H^{0.753}+0.000454e^{0.1H}+0.18(21.1+273.15-T)(1-e^{-0.115H}).
\end{align*}
The fuel is considered as a combination of time-lag classes, and the fuel
moisture contents $m_{k}\left( t\right) $ in each class $k=1,\ldots,N$ is
then modeled by the standard time-lag equation%
\begin{equation}
\frac{dm_{k}}{dt}=\left\{
\begin{array}
[c]{c}%
\frac{E_{d}-m_{k}}{T_{k}}\text{ if }m_{k}>E_{d}\\
0\text{ if }E_{d}<m_{k}<E_{w}\\
\frac{E_{w}-m_{k}}{T_{k}}\text{ if }m_{k}<E_{w}%
\end{array}
\right. \label{eq:moisture-ode}%
\end{equation}
where $T_{k}$ is the lag time. We use the standard model with the fuel
consisting of components with $T_{k}=1$, $10$, and $100$ hour lag time, with
the proportions $w_{k}\geq0,$ $\sum_{k=1}^{N}w_{k}=1$, given by the fuel
category description \cite{Scott-2005-SFB}. The overall fuel moisture then is
the weighted average $m=\sum_{k=1}^{N}w_{k}m_{k}$.
\begin{figure}[ptb]
\begin{center}%
\begin{tabular}
[c]{cc}%
\includegraphics[width=3in]{figures/finefuels.png} &
\includegraphics[width=3in]{figures/timelag14.png}\\
(a) & (b)
\end{tabular}
\end{center}
\caption{Response of fine fuels to rain over 24 hours (a) following
\cite{VanWagner-1985-EFP} (b) from the time-lag model (\ref{eq:rain-ode}) by a
calibration of coefficients.}%
\label{fig:finerain}%
\end{figure}
During rain, the equilibrium moisture $E_{d}$ or $E_{w}$ is replaced by the
saturation moisture contents $S$, and equation (\ref{eq:moisture-ode}) is
modified to achieve the rain-wetting lag time $T_{rk}$ for heavy rain only
asymptotically, when the rain intensity $r$ (mm/h) is large:%
\begin{equation}
\frac{dm_{k}}{dt}=\frac{S-m_{k}}{T_{rk}}\left( 1-\exp\left( -\frac{r-r_{0}%
}{r_{k}}\right) \right) ,\text{ if }r>r_{0}, \label{eq:rain-ode}%
\end{equation}
where $r_{0}$ is the threshold rain intensity below which no perceptible
wetting occurs, and $r_{k}$ is the saturation rain intensity for fuel
component $k$. At the saturation rain intensity, $1-1/e\approx63\%$ of the
maximal rain-wetting rate is achieved. The coefficients can be calibrated to
achieve a similar behavior as accepted empirical models
\cite{Fosberg-1971-DHT,VanWagner-1985-EFP}. This model estimates the fuel
moisture as a function of the initial moisture contents and rain accumulation
over 24 hours. Assuming steady rain over the 24 hours, we have obtained a
reasonable match with the Canadian fire danger rating system
\cite{VanWagner-1985-EFP} using $S=250\%,T_{rk}=14$ h, $r_{0}=0.05$ mm$/$h and
$r_{k}=8$ mm$/$h, cf., Fig.~\ref{fig:finerain}. Of course, since fuel may dry
up between episodes of rain, in the case of intermittent rain, the result of
the model depends also on the temporal rain distribution, not only the total
accumulation -- just like fuel moisture in reality.
\label{sec:numerical}Because we want the model to support an arbitrarily long
time step, an adaptive exponential method was implemented. The method is exact
for long time step when the atmospheric variables and the rain intensity are
constant in time. Equations (\ref{eq:moisture-ode}) and (\ref{eq:rain-ode}%
)\ have the common form%
\begin{equation}
\frac{dm}{dt}=\frac{E-m}{T}. \label{eq:common}%
\end{equation}
For longer time steps, use the exact solution of (\ref{eq:common}) over the
time step interfal $\left[ t_{n},t_{n+1}\right] $, with constant
coefficients taken as their values at $t_{n+1/2}=t_{n}+\Delta t/2$, which
gives $M_{n+1}=M_{n}+\left( E_{n+1/2}-M_{n}\right) \left( 1-e^{-\Delta
t/T_{n+1/2}}\right) $, where $E_{n+1/2}$ and $T_{n+1/2}$ are computed at
$t_{n+1/2}$. This method is of second order and it is particularly useful when
the time step is comparable to or even larger than the time lag and the
coefficients $E$ and $T$ vary slowly, such as in the case of fast drying just
after the rain ends. For very short time steps, however, the rounding error in
the subtraction of the almost equal quantities in $1-e^{-\Delta t/T_{n+1/2}}$
will pollute the solution. Thus, for $\Delta t/T_{n+1/2}<\varepsilon=0.01$, we
replace the exponential by a truncated Taylor expansion, and use the second
order method $M_{n+1}=M_{n}+\left( E_{n+1/2}-M_{n}\right) \frac{\Delta
t}{T_{n+1/2}}\left( 1-\frac{1}{2}\frac{\Delta t}{T_{n+1/2}}\right) .$
\begin{figure}[t]
{\centering
\begin{tabular}
[c]{cc}%
\hspace*{-0.5in} \includegraphics[height=3in]{figures/fmc200hours.png}
\hspace*{-0.5in} &
\ \includegraphics[height=2.25in,trim=0 -100 0 0, clip]{figures/Sfire_smoke.png}\\
(a) & (b)
\end{tabular}
} \caption{(a) Simulation of fuel moisture contents at a single point for fine
fuels. The drying and wetting equilibria are computed from the WRF state. The
fuel moisture contents does not change when it is between the two equilibria.
The red vertical lines correspond to periods of rain, where the equilibrium is
2.5 (above the range shown). The fuel moisture contents increases during rain
and exhibits diurnal variability. The flat part between 100 and 200 hours is due
to one day of missing data. (b) Advection of smoke by tracers injected
in WRF-Chem, in an ideal example.}%
\label{fig:2}%
\end{figure}
Because the time scale of the moisture changes (hours) is very different from
the time scales of the atmoshere (minutes(\ and fire (seconds), the moisture
model runs at a multiple of the WRF time step. WRF variables \texttt{P}
(pressure, Pa), \texttt{T2} (temperature at 2m, K), and \texttt{Q2} (water
vapor mixing ratio, kg/kg) at the beginning and the end of the moisture model
time step are averaged to obtain the values of $P$, $T$, and $Q$ used in the
timestep of the moisture model,%
\[
P_{n+1/2}=\frac{\mathtt{P}\left( t_{n+1}\right) +\mathtt{P}\left(
t_{n}\right) }{2},\quad T_{n+1/2}=\frac{\mathtt{T2}\left( t_{n+1}\right)
+\mathtt{T2}\left( t_{n}\right) }{2},\quad Q_{nn+1/2}=\frac{\mathtt{Q2}%
\left( t_{n+1}\right) +\mathtt{Q2}\left( t_{n}\right) }{2},
\]
This is done for 2nd order accuracy as well as for compatibility with the
computation of the rain intensity from the difference of the accumulated rain,%
\[
r_{n+1/2}=\frac{\mathtt{RAINC}(t_{n+1})+\mathtt{RAINNC}\left( t_{n+1}\right)
-\mathtt{RAINC}(t_{n})-\mathtt{RAINNC}\left( t_{n}\right) }{\Delta t}.
\]
These values are then used to compute the equilibrium moisture contents and
the time lag, and one time step is performed, as described above. See
Fig.~\ref{fig:2}i(a) for a typical simulation result.
The fire model runs on a finer grid than the atmospheric model, typically
refined by a factor of $10$ or more. The moisture model consists of two steps.
In the first step, we compute the moisture content $m_{k}$ of the fuel
components on the nodes of the atmospheric grid on the Earth surface for
several reasons. (1) The WRF variables are known at the nodes of the
atmospheric grid and no interpolation is needed. (2) The atmospheric grid is
relatively coarse, so the added costs of storage of several surface moisture
fields $m_{k}$ and of the computation are not significant. (3) The computation
is done for each fuel component indivudually over the whole domain and it does
not depend on the actual fuel map. (4) The fire model does not need to run for
the computation of $m_{k}$.
In the second step, we interpolate the values of $m_{k}$ to the fire grid and
compute the weighted averages on the fire grid following using the actual fuel
map. There are several options how to run the fuel moisture model: (1) Turn on
both steps of the moisture model: compute the moisture fields $m_{k}$ and
interpolate on the fire grid, as the fire model runs. This option is intended
for the actual fire simulation. (2) Turn on the first step, computation of the
moisture fields $m_{k}$ only, for an extended run (many days) to evolve the
$m_{k}$ in response to the simulated weather, into a dynamic equilibrium. The
moisture fields are stored in WRF state in the output and the restart files.
(3) Run the first step, computation of the moisture fields $m_{k}$, as a
standalone executable from stored output files from standard WRF runs, adding
the moisture fields $m_{k}$ to the files. (4) Start the coupled
atmosphere-fire-moisture simulation from the moisture fields evolved over time
as above. (5) Run the fire model as a standalone exectutable, without feedback
on the atmosphere, with fuel moisture computed using the second step only from
the fields $m_{k}$ in WRF output files produced in advance.
\section{Coupling with WRF-Chem}
Coupling with WRF-Chem is implemented by inserting the smoke intensity in the
WRF-Chem arrays \texttt{emis\_ant} and \texttt{tracer} at the ground layer and
with the species index \texttt{p\_smoke}. WRF needs to be configured with the
\texttt{chem} option and built with \texttt{em\_fire} or \texttt{em\_real},
and the appropriate options \texttt{trace\_opt=1} and \texttt{chem\_opt=14}
set in the namelist. A sample visualization of an ideal run with smoke
transport is in Fig.~\ref{fig:2}(b). See the WRF SFIRE Users' Guide at
\url{http://www.openwfm.org/wiki/Users_guide} for further details and more
information as the code develops further.
Currently, the smoke inserted into WRF-Chem is simply proportional to the fire
heat flux. However, a more sophisticated scheme following the FEPS (Fire
Emission Production Simulator) is being implemented \cite{Anderson-2004-FEP}.
In this new method, the fire emission will be treated as the fluxes of carbon
dioxide (CO2), carbon monoxide (CO), methane (CH4) and particulate matter (PM
2.5) released into the atmosphere at the location of the fire. The total
emission for each of these species will be computed based on the fuel burn
rate (amount of fuel burnt per unit time), fuel load, and the smoldering
correction computed based from the wind speed and relative humidity. The PM
2.5 will be portioned into the the accumulation and nuclei mode and released
into the atmosphere as PM25J and PM25I respectively.
Aside from the obvious application to smoke dispersal, which is important in
practice, taking into account the composition of the fire emitted smoke will
allow for treating it as chemically and physically active. The species
released into the atmosphere by the fire will undergo chemical reactions
allowing for capturing the secondary aerosol effects. The particular mater
will interact with atmospheric radiation, and have a potential to serve as
condensation nuclei when suitable cloud microphysics is available. This may be
of great importance in the simulation of visible plumes and pyrocumulus
clouds, under development in \cite{Peace-2011-CSP}.
{\small
\bibliographystyle{ametsoc}
|
1,116,691,500,822 | arxiv | \section{Introduction}
\label{sec:intro}
Wireless Acoustic Sensor Network (WASN) based animal monitoring is of great importance for biologists to monitor real-time wildlife behavior for long periods and under variable weather/climate conditions. The acquired animal voice can provide valuable information for researchers, such as the density and diversity of different species of animals~\cite{hao2013monitoring,luque2016evaluation,akyildiz2005underwater}. For example, Hu et al. proposed a WASN application to census the populations of native frogs and the invasive introduced species (Cane Toad) in Australia~\cite{hu2009design}.
There are also several important commercial applications of acoustic animal detection. For instance, America imports billions of dollars of timber from Aisa every year. However, the inadvertent introduction of the Asian Longhorn Beetle has cost USA government millions of dollars to eradicate the Beetle population~\cite{nowak2001potential}. Therefore, a wireless monitoring system is imperative to detect the distribution of these insects.
There are a large volume of audio monitoring systems in the literature~\cite{hu2009design,anderson1996template,kogan1998automated,fagerlund2007bird,guo2003content,huang2009frog,acevedo2009automated,banerjee2014partial,dutta2013energy,diaz2012use}. In the early stage, biologists have traditionally deployed audio recording systems over the natural environment where their research projects were developed~\cite{anderson1996template,kogan1998automated}.
However this procedure requires human presence in the area of interest at certain moments. In recent years, with the development of WSN, some researchers have proposed remotely accessible systems in order to minimize the impact of the presence of human beings in the habitat of interest~\cite{hu2009design,banerjee2014partial,dutta2013energy,diaz2012use}.
Despite much effort in this area, previous studies suffer from several disadvantages. First, traditional methods usually first extract a number of appropriate features and then employ classic machine learning methods such as Support Vector Machine (SVM) or K-Nearest Neighbours (KNN) to detect the species of the animals. Features, such as statistical features through statistical anlaysis (e.g., variance, mean, median), Fast Fourier Transmission (FFT) spectrum, spectrograms, Wigner-Ville distribution (WVD), Mel-frequecy cepstrum coefficient (MFCC) and wavelets have been broadly used. However, extracting robust features to recognizing noisy field recordings is non-trivial. While these features may work well for one , it is not clear whether they generalize to other species. The specific features for one application do not necessarily generalize to others. Moreover, a significant number of calibrations are required for both manually feature extraction and the classification algorithms. This is because the performance of the traditional classifiers such as SVM and KNN~\cite{fagerlund2007bird,huang2009frog,acevedo2009automated} highly depends on the quality of the extracted features. However, handcrafting features relies on a significant amount of domain and engineering knowledge to translate insights into algorithmic methods. Additionally, manual selection of good features is slow and costly in effort. Therefore, these approaches lack scalability for new applications. Deep learning technologies can solve these problems by using deep architectures to learn feature hierarchies. The features that are higher up in deep hierarchies are formed by the composition of features on lower levels. These multi-level representations allow a deep architecture to learn the complex functions that map the input (such as digital audio) to output (e.g. classes), without the need of dependence on manual handcrafted features.
Secondly, these approaches suffer from accuracy degradation in real-world applications because of the impact of environmental noise. The voice recorded from field usually contains much noise which poses a big challenge to real deployment of such system. To address this problem, Wei et al.~\cite{wei2013real} proposed an \textit{in-situ} animal classification system by applying sparse representation-based classification (SRC). SRC uses $\ell_1$-optimization to make animal voice recognition robust to environmental noise. However, it is known that $\ell_1$-optimization is computationally expensive~\cite{shen2014face,xu2016sensor}, which limits the application of their system in resource-limited sensor nodes. Additionally, in order to make SRC achieve high accuracy, a large amount of training data is required. This means a wireless sensor node can only store a limited number of training classes because of the limited storage.
Recently, deep learning has emerged as a powerful tool to solve various recognition tasks such as face recognition~\cite{sun2014deep}, human speech recognition~\cite{hinton2012deep,graves2013speech} and natural language processing~\cite{collobert2008unified}. The application of deep learning in audio signal is not new; however, most previous studies focus on human speech analysis to obtain context information~\cite{hinton2012deep,graves2013speech,wang2016attention}. Limited efforts have been devoted to applying deep learning in WASN to classify different species of animals. To bridge this gap, we aim to design and implement a acoustic classification framework for WASN by employing deep learning techniques. Convolutional Neural Network (CNN), as a typical deep learning algorithm, has been widely used in high-level representative feature learning. In detail, CNN is enabled to capture the local spatial coherence from the input data. In our case, the spatial information refers to the spectral amplitude of the audio signal. However, one drawback of the standard CNN structure is that the filter length of the convolution operation is fixed. As a result, the convolutional filter can only discover the spatial features with the fixed filter range. For example, CNN may explore the short-term feature but fail to capture the middle- and long-term features. In this paper, we propose a multi-view CNN framework which contains three convolution operation with three different filter length in parallel in order to extract the short-, middle-, and long-term information at the same time. We conduct extensive experiments to evaluate the system on real datasets. More importantly, we implement the proposed framework on a testbed and conduct a case study to analyse the system performance in real environment. To the best of our knowledge, this is the first work that designs and implements a deep learning based acoustic classification system for WASN.
The main contributions of this paper are threefold:
\begin{itemize}
\item We design a deep learning-based acoustic classification framework for WASN, which adopts a multi-view convolution neural network in order to automatically learn the latent and high-level features from short-, middle- and long-term audio signals in parallel.
\item We conduct extensive evaluation on two real dataset (Forg dataset and Cricket dataset) to demonstrate the classification accuracy and robustness of the proposed framework to environmental noise. Evaluation results show that the proposed system can achieve high recognition accuracy and outperform traditional methods significantly especially in low SNR scenarios.
\item We implement the proposed system on a testbed and conduct a case study to evaluate the performance in real world environments. The case study demonstrate that the proposed framework can achieve high accuracy in real applications.
\end{itemize}
The rest of this paper is organized as follows. Section~\ref{sec:relatedwork} introduces related work. Then, we describe system architecture in Section~\ref{sec:system} and evaluate the system performance in Section~\ref{sec:evaluation}. We implement the system on a testbed and conduct user study to evaluate the system in Section~\ref{sec:testbed}. Finally, Section~\ref{sec:conclusion} concludes the paper.
\begin{figure*}[!ht]
\centering
\includegraphics[width=6.5in]{system_new.png}
\caption{System Overview.}
\label{fig:systemoverview}
\end{figure*}
\section{Related Work}
\label{sec:relatedwork}
Animal voice classification has been extensively studied in the literature. At the highest level, most work extract sets of features from the data, and use these features as inputs for standard classification algorithms such as SVM, KNN, decision tree, or Bayesian classifier. Previous studies have involved a wide range of species which include farm animals~\cite{jahns1997sound}, bats~\cite{preatoni2005identifying}, birds~\cite{fagerlund2007bird,acevedo2009automated,hodon2015monitoring}, pests~\cite{eliopoulos2016estimation}, insects~\cite{ntalampiras2019automatic} and anurans~\cite{vaca2010using}. The works of Anderson et al.~\cite{anderson1996template} and Kogan et al.~\cite{kogan1998automated} were among the first attempts to recognize bird species automatically by their sounds. They applied dynamic time warping and hidden Markov models for automatic song recognition of Zebra Finche and Indigo Punting. In~\cite{luque2016evaluation}, the authors focus on classifying two anuran species: Alytes obstetricans and Epidalea calamita using generic descriptors based on an MPEG-7 standard. Their evaluation demonstrate that MPEG-7 descriptors are suitable to be used in the recognition of different patterns, allowing a high scalability. In~\cite{hao2013monitoring}, the authors propose to classify animal sounds in the visual space, by treating the texture of animal sonograms as an acoustic fingerprint. Their method can obviate the complex feature selection process. They also show that by searching for the most representative acoustic fingerprint, they can significantly outperform other techniques in terms of speed and accuracy.
The WSN has been massively applied in sensing the environment and transferring collected samples to the server. However, it is challenging to realize in-network classification system because of the limited computational ability of wireless sensor node. Recently, several research works regarding in-network classification have been proposed. Sun et al.~\cite{sun2008dynamic} dynamically select the feature space in order to accelerate the classification process. A hybrid sensor networks is designed by Hu et al.~\cite{hu2009design} for in-network and energy-efficient classification in order to monitor amphibian population. Wei et al.~\cite{wei2013real} proposed a sparse representation classification method for acoustic classification on WSN. A dictionary reduction method was designed in order to improve the classification efficiency. The sparse representation classification method was also used by face recognition on resource-constrained smart phones to improve the classification performance~\cite{shen2014face,xu2016sensor}.
Deep learning has achieved great success over the past several years for the excellent ability on high-level feature learning and representative information discovering. Specifically, deep learning has been widely used in a number of areas, such as computer version \cite{wen2016discriminative}, activity recognition \cite{chen2018interpretable,luo2019brush}, sensory signal classification \cite{zhang2018multi,lan2019entrans,xu2019energy}, and brain computer interface \cite{zhang2018mindid}. Wen et al. \cite{wen2016discriminative} propose a new supervision signal, called center loss, for face recognition task. The proposed center loss function is demonstrated to enhance the discriminative power of the deeply learned features. Chen et al. \cite{chen2018interpretable} propose an interpretable parallel recurrent neural network with convolutional attentions to improve the activity recognition performance based on Inertial
Measurement Unit signals. Zhang et al. \cite{zhang2018multi} combine deep learning and reinforcement learning to deal with multi-modal sensory data (e.g., RFID, acceleration) and extract the latent information for better classification. Recently, deep learning involves in the brain signal mining in brain computer interface (BCI). Zhang et al. \cite{zhang2018mindid} propose an attention-based Encoder-Decoder RNNs (Recurrent Neural Networks) structure in order to improve the robustness and adaptability of the brainwave based identification system.
There are also several works that apply deep learning techniques in embedded devices. Lane et al.~\cite{lane2015can} propose low-power Deep Neural Network (DNN) model for mobile sensing. CPU and DSP in one mobile device are exploited for activity recognition. Lane et al.~\cite{lane2015can} also design a DNN model for audio sensing in mobile phone by using dataset from 168 places for the training purpose. A framework DeepX is further proposed for software accelerating on mobile devices~\cite{lane2016deepx}.
In terms of animal voice classification, Zhang et al.~\cite{zhang2018automatic}, Oikarinen et al.~\cite{oikarinen2018deep} study animal voice classification using deep learning techniques. Our method is different from these two works. Our studies focus on voice classification in noisy environment while the voice data in~\cite{zhang2018automatic} are collected from controlled room without environmental noise. Instead of classifying different animals, \cite{oikarinen2018deep} analyses different call types of marmoset monkeys such as Trill, Twitter, Phee and Chatter. Moreover, we implement the proposed system on a testbed and evaluate its performance in real world environment. In another work~\cite{ntalampiras2018bird}, Stavros Ntalampiras used transfer learning to improve the accuracy of bird classification by exploiting music genres. Their result show that the transfer learning scheme can improve classification accuracy by $11.2\%$. Although the goal of this work and our study is to improve the recognition accuracy with deep learning technology, the methodologies are different. Our approach analyses the inherent features of audio signal and propose a multi-view CNN model to improve the accuracy. Instead of looking at the bird audio signals alone, Stavros Ntalampiras proposed to statistically analyse audio signals using their similarities with music genres. Their method, however, is only effective for a limited number of bird species because they need to perform feature transformation again when a new bird species comes in. In comparison, our approach is applicable for a large number of bird species. A number of studies also apply deep learning technologies in bird voice classification~\cite{potamitis2016deep,koops2014deep,goeau2016lifeclef}, however, they only use conventional deep learning approaches such as CNN and do not make any novel improvement. In this paper, we propose a multi-view CNN model and evaluation results show that the proposed model outperforms the conventional CNN.
\section{System Design}
\label{sec:system}
\subsection{System Overview}
As shown in Figure~\ref{fig:systemoverview}, our proposed framework consists of two parts: WASN and server. In the WASN, the wireless nodes will detect and record animal voices and then perform local processing which include silence removal, segmentation and FFT. We process signal \emph{in-situ} before uploading because of the high sampling frequency of audio signal and energy inefficiency of wireless communication~\cite{barr2006energy,wei2013real}. The spectrum signal obtained from FFT can save half spaces since FFT is symmetric.
On the server side, the spectrum signal will be fed into a deep neural network to obtain the species of the animal. The classification results can be used by biologists to analyze the density, distribution and behavior of animals.
Wireless sensors are usually resource-poor relative to server, and not able to run computationally expensive algorithms such as deep learning models. Therefore, we assume all the wireless sensors can connect to a server via wireless communication technologies, such as ZigBee, Wi-Fi, and LoRa~\cite{xu2019measurement}. However, there may be network failure, server failure, power failure, or other disruption makes offload impossible. While such failures will hopefully be rare, they cannot be ignored in a cloud-based system. In this case, the node can transmit the data to the gateway or a nearby server which are usually resource-rich and capable of running deep learning models. Alternatively, the classification can be performed in the node to recognize only a few species, pre-defined by the user. When offloading becomes possible again, the system can revert to recognizing its full range of species.
In the following parts, we will describe the design details of each component.
\subsection{Local Processing}
\textbf{Silence Removal.} The collected audio signal usually contains a large amount of silent signal when this is no animal present. Therefore, we apply a simple silence removal method on the raw signal to delete not-of-interest area. The procedure is explained in Algorithm~\ref{al:silence}. We first calculate the root mean square (RMS) of each window which contains 1s samples and then compare it with a pre-defined threshold learned from the environment.
The windows of samples whose RMS above the threshold will be kept. The threshold is determined by exhaustive search. To be specific, we increase the threshold from 0 to 0.5 with an increment of 0.01, then choose the one that can achieve the best performance (0.03 in this paper).
\begin{algorithm}[ht]
\caption{Silence Removal}
\label{al:silence}
\begin{algorithmic}[1]
\State{\textbf{Input}: Audio Segment $S_{i = 1:N} \in \mathbb{R} > 1$}, where $N$ is the total number of segments and $\rho$ is the threshold
\For {$i = 1:N$}
\If{RMS $(S_i) < \rho$}
\State Remove $(S_i)$\;
\EndIf
\EndFor
\end{algorithmic}
\end{algorithm}
Figure~\ref{fig:silenceremoval} shows an example of silence removal on an animal voice recording. We can see that it can effectively remove the silent periods and detect the present of animals.
\begin{figure}[!ht]
\centering
\includegraphics[width=3in]{silenceremoval.pdf}
\caption{Silence removal.}
\label{fig:silenceremoval}
\end{figure}
\textbf{Segmentation and FFT.} After silence removal, we obtain audio signals containing animal vocalization only.
The audio signal is segmented into consecutive sliding windows with $50\%$ overlap. Hamming window is used in this paper to avoid spectral leakage. Each window contains $2^{14}$ samples which is chosen to balance the trade-off between classification accuracy and latency as discussed in Section~\ref{sec:evaluation}. The overlap in sliding window is used to capture changes or transitions around the window limits. Then we perform FFT on each segment to calculate spectrum energy (i.e. the magnitude of the FFT coefficients). As an example, Figure~\ref{fig:fft} shows the sound in time and frequency domain of two different frog species: Cultripes and Litoria Caerulea. It is conspicuous that they have different spectrum distributions. The graphs are plotted by audio signal analysis software Audacity.
\subsection{Multi-view Convolutional Neural Networks}
\begin{figure*}[!ht]
\centering
\subfigure[Cultripes]{
\includegraphics[width=1.6in,height=1.28in]{Cultripes.jpg}
\label{fig:1}
}
\subfigure[Cultripes: time domain]{
\includegraphics[width=1.6in,height=1.28in]{1.png}
\label{fig:1}
}
\subfigure[Cultripes: spectrum]{
\includegraphics[width=1.6in]{2.png}
\label{fig:2}
}
\subfigure[Litoria Caerulea]{
\includegraphics[width=1.6in,height=1.28in]{LitoriaCaerulea.jpg}
\label{fig:3}
}
\subfigure[Litoria Caerulea: time domain]{
\includegraphics[width=1.6in,height=1.28in]{3.png}
\label{fig:3}
}
\subfigure[Litoria Caerulea: spectrum]{
\includegraphics[width=1.6in]{4.png}
\label{fig:4}
}
\caption{Audio signal of two species of frog.}
\label{fig:fft}
\end{figure*}
We propose a deep learning framework in order to automatically learn the latent and high-level features from the processed audio signals for better classification performance. Among deep learning algorithms, CNN is widely used to discover the latent spatial information in applications such as image recognition \cite{ciresan2011convolutional}, ubiquitous \cite{ning2018deepmag}, and object searching \cite{ren2017faster}, due to their salient features such as regularized structure, good spatial locality and translation invariance. CNN applies a convolution operation to the input, passing the result to the next layer. Specifically, CNN captures the distinctive dependencies among the patterns associated to different audio categories. However, one drawback of the standard CNN structure is that the filter length of the convolution operation is fixed. As a result, the convolutional filter can only discover the spatial features with the fixed filter range. For example, CNN may explore the short-term feature but fail to capture the middle- and long-term features.
\begin{figure*}[!ht]
\centering
\includegraphics[width=5in]{CNN_flowwork_horizontal.png}
\caption{Multi-view CNN workflow.The audio signal is feed into three views for short-term, middle-term, and long-term spatial dependencies learning. $Conv_h^k$ denotes the $k$-th convolutional operation in the $h$-th view. The learned features from the multi-view structure are processed by the max pooling layer for dimension reduction, which are followed by the fully-connected layer, softmax layer, and at last predict the animal species as output.}
\label{fig:CNN_workflow}
\end{figure*}
To address the mentioned challenge, we propose a multi-view CNN framework which applies three different filter length to extract the short-, middle-, and long-term features in parallel. As shown in Figure~\ref{fig:CNN_workflow}, the proposed framework regards the processed audio signals as input and feed into three views at the same time. Each view contains three convolutional layers. $Conv_h^k$ denotes the $k$-th convolutional operation in the $h$-th view. The convolutional layer contains a set of filters to convolve the audio data followed by the nonlinear transformation to extract the geographical features.
The filter length keeps invariant in the same view while varies in different views. The extracted features from the multi-view pipe are stacked together and then through the max pooling operation for dimension reduction. Afterward, a fully-connected layer, a softmax layer and the output layer work as a classifier to predict the audio label. The proposed multi-view CNN has several key differences from the inception module~\cite{szegedy2015going} although the ideas are similar. First, ~\cite{szegedy2015going} has a $1 \times 1$ convolutional filter in the module in order to prevent the information corruption brought by inter-channel convolutions. The proposed multi-view CNN does not has this component. This is because in our case the input data are naturally formed as a vector which represents the spectral information of the acoustic signals. Moreover, ~\cite{szegedy2015going} adds an alternative parallel pooling path in the middle layer to acquire additional beneficial effect. However, we believe this may cause information loss and only perform the pooling operation after the concentration of the results of various views.
\begin{table*}[h]
\caption{Species used in the experiments.}
\label{tab:species}
\centering
\small
\resizebox{6.5in}{!}{
\begin{tabular}{|c|c|c|c|}
\hline
\multicolumn{2}{|c|}{Frog dataset} & \multicolumn{2}{c|}{Cricket dataset (1 belongs to Gryllidae, 2 belongs to Tettigoniidae)} \\ \hline
Cyclorana Cryptotis & Cyclorana Cultripes & Acheta$^{1}$ & Aglaothorax$^{2}$ \\ \hline
Limnodynastes Convexiusculus & Litoria Caerulea &Allonemobius$^{1}$ & Amblycorypha$^{2}$ \\ \hline
Litoria Inermis & Litoria Nasuta & Anaxipha$^{1}$ & Anaulacomera$^{2}$ \\ \hline
Litoria Pallida & Litoria Rubella & Anurogryllus$^{1}$ & Arethaea$^{2}$ \\ \hline
Litoria Tornieri & Notaden Melanoscaphus & Cyrtoxipha$^{1}$ & Atlanticus$^{2}$ \\ \hline
Ranidella Bilingua & Ranidella Deserticola & Eunemobius$^{1}$ & Belocephalus$^{2}$ \\ \hline
Uperoleia Lithomoda & Bufo Marinus & Gryllus$^{1}$ & Borinquenula$^{2}$ \\ \hline
& &Hapithus$^{1}$ & Bucrates$^{2}$ \\ \hline
& &Capnobotes$^{2}$ & Caribophyllum$^{2}$ \\ \hline
& & Ceraia$^{2}$ & Conocephalus$^{2}$ \\ \hline
\end{tabular}
}
\end{table*}
Suppose the input audio data $\bm{E}$ has shape $[M, L]$ with depth as $1$. The chosen three convolutional filters with size in short-, middle-, and long-term views are $[M, 10]$, $[M, 15]$, $[M, 20]$, respectively. The stride sizes keep $[1, 1]$ for all the convolutional layers. The stride denotes the x-movements and y-movements distance of the filters. Since the audio signals are arranged as 1-dimension data, we set $M = 1$. Same shape zero padding is used, which keeps the sample shape constant during the
the convolution calculation. In the convolutional operation, the feature maps from the input layer are convolved with the learnable filters and fed to the activation function to generate the output feature map. For a specific convolutional area (also called perceptive area) $\bm{x}$ which has the same shape as the filter, the convolutional operation can be described as
$$\bm{x}' = tanh(\sum_{i}\sum_{j}\bm{f}_{ij}*\bm{x}_{ij})$$
where $\bm{x}'$ denotes the filtered results while $\bm{f}_{ij}$ denotes the $i$-th row and the $j$-th column element in the trainable filter. We adopt the widely used \textit{tanh} activation function for nonlinearity.
The depth of input sample transfers to $D$ through the convolutional layer and the sample shape is changed to $[M, L, D]$. In particular, the corresponding depth $D_h= 2, 4, 8$ for three convolutional layers. The features learned from the filters are concatenated and flattened to $[1, M*L*\sum_{h=1}^{3}D_h]$. The max pooling has $[1, 3]$ as both pooling length and strides. Therefore, the features with shape $[1, M*L*\sum_{h=1}^{3}D_h/3]$ after the pooling operation, which are forwarded to the fully-connected layer. The operation between the fully-connected layer and the output layer can be represented by
$$\bm{y} = softmax(\bar{\bm{w}}\bm{E^{FC}} + \bar{\bm{b}})$$
where $FC$ denotes the fully-connected layer while the $\bar{\bm{w}}$ and $\bar{\bm{b}}$ denote the corresponding weights matrix and biases.
The softmax function is used for activation. For each sample, the corresponding label information is presented by one-hot label $\bm{y} \in \mathbb{R}^H$ where $H$ denotes the category number of acoustic signals. The error between the predicted results and the ground truth is evaluated by cross-entropy
$$loss = - \sum_{h=1}^{H}\bm{y}_h log(p_h)$$
where $p_h$ denotes the predicted probability of observation of an object belonging to category $h$.
The calculated error is optimized by the AdamOptimizer algorithm~\cite{kingma2014adam}. To minimize the possibility of overfitting, we adopt the dropout strategy and set the drop rate to 80\%.
\begin{figure*}[!ht]
\centering
\subfigure[Impact of window size.]{
\includegraphics[width=2in]{impact_windowsize.pdf}
\label{fig:impactofwindow_Frog}
}
\subfigure[Impact of iterations.]{
\includegraphics[width=2in]{impact_iterations.pdf}
\label{fig:impactofiteration_Frog}
}
\subfigure[Impact of dropout rate.]{
\includegraphics[width=2in,height=1.71in]{impact_dropoutrate.pdf}
\label{fig:impactofdropout_Frog}
}
\subfigure[Impact of learning rate.]{
\includegraphics[width=2in,height=1.71in]{impact_learningrate.pdf}
\label{fig:impactoflearningrate_Frog}
}
\subfigure[Impact of training dataset.]{
\includegraphics[width=2in,height=1.71in]{impact_trainingsize.pdf}
\label{fig:impactoftrainingsize_Frog}
}
\subfigure[Comparison with other methods.]{
\includegraphics[width=2in,height=1.71in]{comparisonwithothers.pdf}
\label{fig:comparisonwithothers_Frog}
}
\caption{Evaluation results of frog dataset.}
\label{fig:frogdataset}
\end{figure*}
\section{Evaluation}
\label{sec:evaluation}
\subsection{Goals, Metrics, and Methodology}
\label{subsec:goals}
In this section, we evaluate the performance of the proposed system based on two real datasets. The goals of the evaluation are twofold: 1) evaluate the performance of the proposed system under different settings; 2) compare the proposed system with previous animal vocalization system.
\begin{figure}[!ht]
\centering
\includegraphics[width=3.2in]{confusionmatrix.png}
\caption{Confusion matrix of frog dataset.}
\label{fig:confusionmatrix_frog}
\end{figure}
We use two datasets collected from real-world for evaluation. The first dataset contains audio signals recorded from fourteen different species of frogs. The sampling frequency for this dataset is 24Khz. More details about this dataset can be found in~\cite{wei2013real}. The second dataset~\footnote{http://alumni.cs.ucr.edu/~yhao/animalsoundfingerprint.html} contains audio signals recorded from different species of crickets. The data consists of twenty species of crickets, eight of which are Gryllidae and twelve of which are Tettigoniidae. The sampling frequency is also 24Khz. More details about this dataset can be found in~\cite{hao2013monitoring}. For completeness, Table.~\ref{tab:species} lists all the species we used in the experiments.
\begin{table*}[h]
\caption{Performance of different methods on frog dataset (SNR=-6dB).}
\label{tab:comparison_frog}
\centering
\small
\begin{tabular}{cccccccc}
\toprule
& Our method &CNN & SRC & SVM-MFCC & SVM-Spectrum & KNN-MFCC & KNN-Spectrum \\ \hline
Accuracy & \textbf{94.7\%} &82.7\% & 53.4\% & 24.4\% & 40.5\% & 20.1\% & 26.4\% \\ \hline
Precision & \textbf{93.1\%} &81.6\% & 54.2\% & 25.9\% & 43.5\% & 19.9\% & 25.1\% \\ \hline
Recall & \textbf{94.3\%} &82.4\% & 53.7\% & 24.7\% & 41.2\% & 21.5\% & 27.1\% \\ \hline
F1-score & \textbf{92.9\%} &81.2 & 52.1\% & 25.1\% & 39.6\% & 20.7\% & 25.7\% \\ \bottomrule
\end{tabular}
\end{table*}
In this paper, we use SVM and KNN to benchmark ASN classification because they have been widely used in WASN classification systems~\cite{fagerlund2007bird,huang2009frog,acevedo2009automated}. We evaluate the performance of SVM and KNN by using frequency domain and Mel-frequency cepstral coefficients (MFCCs), respectively. The parameters in SVM and KNN are well tuned to give highest accuracy. In addition, we compare the accuracy of our system with a recent work which is based on SRC~\cite{wei2013real} and conventional CNN. In total, we compare our method with six classifiers: CNN, SRC, SVM-MFCC, SVM-spectrum, KNN-MFCC and KNN-spectrum. For each classifier, we perform 10-fold cross-validation on the collected dataset. In the original dataset, the data only contain little environment noise. Therefore, to demonstrate the robustness of the proposed framework, we add different scales of environmental noise to create different SNRs. This is used to simulate the real environment because the recorded animal voices are usually deteriorated by environmental noise in real WASN. In the evaluation, it is done by adding different scales of random Gaussian noise to the original audio data.
\begin{figure*}[!ht]
\centering
\subfigure[Impact of window size.]{
\includegraphics[width=2in]{impact_windowsize_cricket.pdf}
\label{fig:impactofwindow_cricket}
}
\subfigure[Impact of iterations.]{
\includegraphics[width=2in]{impact_iterations_cricket.pdf}
\label{fig:impactofiteration_cricket}
}
\subfigure[Impact of dropout rate.]{
\includegraphics[width=2in,height=1.71in]{impact_dropoutrate_cricket.pdf}
\label{fig:impactofdropout_cricket}
}
\subfigure[Impact of learning rate.]{
\includegraphics[width=2in,height=1.71in]{impact_learningrate_cricket.pdf}
\label{fig:impactoflearningrate_cricket}
}
\subfigure[Impact of training dataset.]{
\includegraphics[width=2in,height=1.71in]{impact_trainingsize_cricket.pdf}
\label{fig:impactoftrainingsize_cricket}
}
\caption{Evaluation results of cricket dataset.}
\label{fig:cricketdataset}
\end{figure*}
In this paper, we focus on the following four metrics: \textit{accuracy}, \textit{precision}, \textit{recall} and \textit{F1-score}. We plot the results of the average values and stand deviation obtained from 10 folds cross-validation.
\subsection{Performance of Frog Dataset}
\subsubsection{Impact of parameters}
We first evaluate the impact of important parameters in our system. On the node's side, the important parameters include window size of segment. On the server's side, the important parameters include the number of iterations in training, the dropout rate and learning rate in CNN, the size of training dataset. Dropout is a technique where randomly selected neurons are ignored during training. For example, the dropout rate of $80\%$ means that we randomly select $20\%$ of the neurons and drop them (force the values of them as 0). The dropout strategy is widely used to enhance the generalization of a machine learning model and prevent overfitting. The learning rate is a hyper-parameter that controls how much we are adjusting the weights of the neuron network with respect to the loss gradient.
To evaluate the impact of window size, we vary the window size from $2^{11}$ to $2^{15}$ samples and calculate the accuracy of our scheme. From the results in Figure~\ref{fig:impactofwindow_Frog}, we can see that there is a performance gain when we increase the window size and the improvement reduces after $2^{14}$ samples. Although we can achieve higher accuracy with more samples, the resource consumption of FFT operation which runs on the wireless sensor node also increases. Therefore, we choose to use $2^{14}$ window size to balance the trade-off between accuracy and resource consumption.
Figure~\ref{fig:impactofiteration_Frog} shows the accuracy along with different training iterations. We can see that the proposed method converges to its highest accuracy in less than 200 iterations. The results show that the proposed framework can finish training quickly. Figure~\ref{fig:impactofdropout_Frog} plots the accuracy of various dropout rates. We can observe that the accuracy fluctuates first and then becomes stable after the dropout rate is greater than 0.8. Therefore, we set the default dropout rate to be 0.8. Moreover, we can infer from Figure~\ref{fig:impactofdropout_Frog} that our model is not very sensitive to the dropout rate. This is because the Frog dataset matches well with the proposed multi-view CNN, as a result, the convergence suffers less from overfitting which can be demonstrated by the good convergence property as shown in Figure~\ref{fig:impactofiteration_Frog}. Figure~\ref{fig:impactoflearningrate_Frog} shows the accuracy under different learning rates. We can see that it achieves the highest accuracy when the learning rate is $0.5\times 10^{-3}$ and $1 \times 10^{-3}$. Correspondingly, we choose 0.001 to reduce the training time because the smaller the learning rate is, the slower the training process is. From Figure~\ref{fig:impactoflearningrate_Frog}, we can observe that the performance varies dramatically with the increasing of learning rate. One possible reason for this is that the gradient surface of our loss function is not smooth and very sensitive to the learning rate. The optimiser is easy to step over the local optima while the learning rate is larger than a threshold.
Next, we evaluate the accuracy of the proposed system under different sizes of training dataset. In this experiment, we use different proportions of the whole dataset for training, and use the left dataset for testing. The proportion increases from $10\%$ to $90\%$ with an increment of $10\%$. For example, the proportion of $10\%$ means we use $10\%$ of the dataset for training, and use the left dataset for testing. For comparison purpose, we also calculate the accuracy of CNN, SRC, SVM and KNN. From the results in Figure~\ref{fig:impactoftrainingsize_Frog}, we can see that our method continuously achieves the highest accuracy, and the accuracy becomes relatively stable after $60\%$ of the dataset is used for training. We also notice that the improvement of our method from $10\%$ to $90\%$ is remarkable. More specifically, when the proportion of the training dataset increases from $10\%$ to $90\%$, the accuracy improvement of our method is $40.1\%$ while the improvement of CNN, SRC, SVM and KNN are $34.7\%$, $29.3\%$, $26.7\%$ and $22.4\%$, respectively. In this experiment, we do not test SVM-MFCC and KNN-MFCC because their accuracy is poor as will be shown later.
\begin{table*}[!h]
\caption{Performance of different methods on cricket dataset (SNR=-6dB).}
\label{tab:comparison_cricket}
\centering
\small
\begin{tabular}{cccccccc}
\toprule
& Our method &CNN & SRC & SVM-MFCC & SVM-Spectrum & KNN-MFCC & KNN-Spectrum \\ \hline
Accuracy & \textbf{86.4\%} &76.6\% & 42.4\% & 22.1\% & 36.8\% & 19.4\% & 28.5\% \\ \hline
Precision & \textbf{86.9\%} &76.2\% & 42.5\% & 23.6\% & 36.3\% & 18.2\% & 28.6\% \\ \hline
Recall & \textbf{85.1\%} &75.3\% & 41.2\% & 22.7\% & 37.3\% & 19.9\% & 29.8\% \\ \hline
F1-score & \textbf{86.1\%} &74.6\% & 41.7\% & 21.8\% & 38.1\% & 19.5\% & 29.5\% \\ \bottomrule
\end{tabular}
\end{table*}
\subsubsection{Comparison With Other Methods}
We now compare the performance of proposed scheme with previous approaches. As mentioned above, we compare the accuracy of the proposed system with conventional CNN, SRC, SVM-MFCC, SVM-spectrum, KNN-MFCC and KNN-spectrum. The MFCC of each window is calculated by transforming the power spectrum of each window into the logarithmic mel-frequency spectrum. We calculate the accuracy of different methods under different SNRs by adding different scales of environmental noise.
As we can see from Figure~\ref{fig:comparisonwithothers_Frog}, SVM-MFCC and KNN-MFCC performs the worst which suggests that different frog species are not distinguishable in MFCC feature space. The results also explains why MFCC-based methods usually requires other carefully selected features~\cite{vaca2010using}. We find that when the animal voice is overwhelmed by environmental noise (low SNR), the accuracy of our system is significantly higher than the other methods. For example, when $SNR=-6dB$, the accuracy of our method is $12\%$ higher than CNN, $41\%$ higher than SRC, $70\%$ higher than SVM-MFCC, $53.9\%$ higher than SVM-spectrum, $74.3\%$ higher than KNN-MFCC, and $68\%$ higher than KNN-spectrum. The robustness to noise makes the proposed system suitable for real deployment in noisy environments. Moreover, the results also indicate that our system needs less sensors to cover a certain area because our system can classify low SNR signals which are usually collected from longer distance.
To take a closer look at the result, we summarize the results of different methods in Table~\ref{tab:comparison_frog} and plot confusion matrix in Figure~\ref{fig:confusionmatrix_frog} when SNR is -6dB. We can see that each class can achieve high accuracy and the overall average accuracy is $94.7\%$.
\begin{figure}[!ht]
\centering
\subfigure[Two-class classification.]{
\includegraphics[width=2.2in]{comparisonwithothers_cricket_binary.pdf}
\label{fig:comparison_cricket2}
}
\subfigure[Twenty-class classification.]{
\includegraphics[width=2.2in]{comparisonwithothers_cricket.pdf}
\label{fig:comparison_cricket20}
}
\caption{2 class classification vs 20 class classification.}
\label{fig:cricketdataset}
\end{figure}
\subsection{Performance of Cricket Dataset}
\begin{figure*}[!ht]
\centering
\subfigure[Network topology]{
\includegraphics[scale = 0.3]{testbed.jpg}
\label{fig:local}
}
\subfigure[Bird species]{
\includegraphics[width=3in]{bird.png}
\label{fig:bird}
}
\caption{Testbed.}
\label{fig:casestudy}
\end{figure*}
Similar to Frog dataset, we also evaluate the impact of window size, the number of iterations, the dropout rate and learning rate in CNN, and the size of training dataset using Cricket dataset. The procedures are the same as above and the results are shown in Figure~\ref{fig:cricketdataset}. We can see that it shows similar patterns as Frog dataset which suggests that the proposed framework is robust to different species. In terms of the dropout rate and learning rate, the optimal values for dropout rate and learning rate are 0.7 and 0.0005 which is slightly different from that of Frog dataset.
As mentioned in~\cite{hao2013monitoring}, the cricket dataset consists of twenty species of insects, eight of which are Gryllidae and twelve of which are Tettigoniidae. Thus, we can treat the problem as either a two-class genus level problem, or twenty-class species level problem. We first treat the classification as a two-class level problem and calcualte the accuracy of different methods under different SNRs. From the results in Figure~\ref{fig:comparison_cricket2}, we can see that our method, SRC, SVM-spectrum and KNN-spectrum can achieve high accuracy. However, our method still outperforms all the other classifiers. Thereafter, we treat the classification as a twenty-class species level problem and plot the accuracy of different methods in Figure~\ref{fig:comparison_cricket2}. We can see that the proposed method significantly outperforms the other methods when SNR is low. Table~\ref{tab:comparison_cricket} summarizes the results of each method in detail. The results above demonstrate the advantage of our method in classifying more species in noisy environment.
\section{Case Study on Testbed}
\label{sec:testbed}
To validate the feasibility of the proposed framework in real environment, we implement the system on an outdoor ASN testbed which is located in Brisbane, Australia. As shown in Figure~\ref{fig:local}, the testbed is composed of five nodes which are configured as \emph{Ad-hoc} mode with a star network topology. Its task is to evaluate the system's capability of recognizing bird vocalization in real world environment.
\begin{table}[htb]
\centering
\caption{Power Consumption.}
\label{tab:powerconsumption}
\begin{tabular}{|l| c| }
\hline
\textbf{Module} & \textbf{Consumption (W)} \\
\hline
CPU & 2.05\\
\hline
CPU + microphone & 2.1\\
\hline
CPU + Wifi (idle) & 2.45 \\
\hline
CPU + Wifi (Rx) & 2.67 \\
\hline
CPU + Wifi (Tx) & 2.78 \\
\hline
\end{tabular}
\label{tab:energy}
\end{table}
The hardware platform used in the testbed is based on a Pandaboard ES with an 1.2Ghz OMAP 4460, 1GB Ram and 4GB SD-card. Additionally, Pandaboard includes an 802.11 interface for wireless connection. Microphones are connected to the Pandaboard via USB port to record bird voice with 24Khz sampling rate. All the nodes are connected via the local Wi-Fi network. The data collected from Node 2, 3, 4 and 5 will be first transferred to Node 1. Then, all the data will be uploaded from Node 1 to the local server. The acoustic date from different nodes are classified separately in the system.
In the testbed, each node is powered by a rechargeable battery (12V, 7.2Ah), and an optional solar panel (5W, 12V). The power consumption of each module is given in Table~\ref{tab:powerconsumption}. Compared to SolarStore testbed~\cite{solarstore} which consumes 10W (low load) and 15W (high load) energy, our testbed is approximately 3.5 to 5.4 times more energy efficient. Without solar panel, a node in our ASN testbed will run continuously for more than 31 hours, which is significantly longer than the previous platforms such as ENSBox \cite{Girod2006:AENSBox}. We find that if a solar panel is exposed to direct sunlight for 8 hours per day, the node can maintain a 50\% duty cycle at 85\% solar charge efficiency.
The nodes use Network Time Protocol (NTP) for time synchronization. We use one node as the NTP server, and the other nodes as the NTP clients. The NTP clients send request for time synchronization every 10 seconds. The accuracy of time synchronization is about 25 ms, which is good enough for our distributed real-time system because the length of each testing signal segment is 400ms.
During deployment, we found that the recorded voice is deteriorated by wind. To solve this problem, we take two measures. First, we install foam and fur windscreen around each microphone. Second, we apply a Butterworth high pass filter with 200Hz cut-off frequency to filter out unwanted noise. This is because most of the wind audio energy lies in the frequency band below 200Hz, while most of the vocalization energy of the birds is in the frequency band higher than 200Hz.
After implementing the proposed framework on the testbed, we calculate the computation time on the node's side and classification accuracy on the server's side. On the node's side, we find that the node in our testbed can process all the captured acoustic data in real time. From Table~\ref{tab:performancetestbed}, we can see the silence removal and FFT take 20.38 ms and 15.33 ms, respectively.
\begin{table}[]
\centering
\small
\caption{Computation time of local processing.}
\label{tab:performancetestbed}
\begin{tabular}{ccc}
\toprule
& Silence Removal & FFT \\ \hline
Time (ms) & 20.38 $\pm$ 2.04 & 15.33 $\pm$ 0.63 \\ \bottomrule
\end{tabular}
\end{table}
\begin{table*}[!h]
\centering
\small
\caption{Performance on testbed.}
\label{tab:performancetestbed}
\begin{tabular}{cccccc}
\toprule
& Our system &CNN & SRC &SVM-Spectrum & KNN-Spectrum\\ \hline
Accuracy & \textbf{90.3\%} &84.4\% &72.3\% & 65.7\% & 68.8\% \\ \hline
Precision & \textbf{91.2\%} &82.1\% & 72.6\% & 66.4\% & 69.2\% \\ \hline
Recall & \textbf{89.4\%} &84.6\% & 70.9\% & 65.6\% & 67.1\% \\ \hline
F1-score & \textbf{91.1\%} &83.7\% & 71.8\% & 66.4\% & 70.5\% \\ \bottomrule
\end{tabular}
\end{table*}
In this study, we choose two common bird species in the area of interest: Anseriformes and Galliformes (Figure~\ref{fig:bird}).
Our goal is to classify the voice into three classes: Anseriformes, Galliformes and others. The testbed runs for 30 days and the data is labeled manually.
Table~\ref{tab:performancetestbed} lists the results of different methods for classification in the server. We find that the proposed system achieve $90.3\%$ classification accuracy which outperforms other methods significantly. The results in turn suggest that the proposed framework is robust to environmental noise and can achieve high classification accuracy in real-world WASN. We also notice that the results of the case study is slightly lower than the simulation results in Section~\ref{sec:evaluation}. This is because the public dataset are collected in a controlled manner and the signals are well trimmed and processed. However, the data we used in our case study are collected in a totally automatic manner.
\section{Conclusion}
\label{sec:conclusion}
In this paper, we design and implement a CNN-based acoustic classification system for WASN. To improve the accuracy in noisy environment, we propose a multi-view CNN framework which contains three convolution operation with three different filter length in parallel in order to extract the short-, middle-, and long-term information at the same time. Extensive evaluations on two real datasets show that the proposed system significantly outperforms previous methods. To demonstrate the performance of the proposed system in real world environment, we conduct a case study by implementing our system in a public testbed. The results show that our system works well and can achieve high accuracy in real deployments. In our future work, we will deploy the proposed framework in wider area and evaluate its performance in different environments.
\section*{Acknowledgement}
The work described in this paper was fully supported by a grant from City University of Hong Kong (Project No.7200642)
\balance
|
1,116,691,500,823 | arxiv | \section{Introduction} \label{sec:intro}
Let $F$ be a field, and $f\in F[x]$ be an irreducible polynomial. Results from classical field theory tells us that we can always find some field extension $F\subseteq L$ such that $f$ contains a root in $L$. The construction of this field extension is quite simple, we just take the ring of polynomials over $F$ and quotient out be the ideal generated by $f$, that is $L=F[x]/\langle f\rangle$ is a field extension of $F$ that contains a root to $f$. Thus, given some field $F$, and a root $\root$ of some polynomial $f\in F[x]$, we can make sense of the field $F(\root)$ as the smallest field extension of $F$ containing the root $\root$. Furthermore, this extension is unique up to isomorphism. This minimal extension plays an important role in Galois theory.
\bigskip
In this paper, we set out to answer an analogous question over hyperfields. That is, given a polynomial $k$ over some hyperfield $K$, is there a hyperfield extension of $K$ that contains some root to $k$. Furthermore, if such extensions exist are the minimal extensions of $K$ containing a root unique up to isomorphism. That is for a hyperfield $K$ and a root $\xi$ to some polynomial $k$ can we join the root $\xi$ to $K$ and make sense of the hyperfield $K(\xi)$. As addition over hyperfields is a multivalued operation, there are two definitions one can take for a hyperfield extension namely we can require that the sum of two elements in the ground hyperfield is equal to sum in the extension. Alternatively, we could require set inclusion. When equality is held we shall say that we have a \textit{strong hyperfield extension}, and when inclusion is held we shall say it is a \textit{weak hyperfield extension}.
\subsection{Statement of Results} In this paper, we will develop a method for constructing strong hyperfield extensions containing roots for quotient hyperfields. Furthermore, we shall exhibit an example of a hyperfield and a polynomial that contains two non-isomorphic minimal extensions. That is, for a hyperfield $K$ and a root $\xi$ to a polynomial $k$, adjoining the root to form $K(\xi)$ is not a well-defined operation. Without the uniqueness of minimal extensions, trying to develop a Galois theory over hyperfields becomes a challenge.
\subsection{Content Overview} In section \ref{sec:prelim} of this paper, we give the definitions of hyperfields, polynomials over hyperfields, and other related concepts. Furthermore, we shall review the construction of quotient hyperfields developed by Krasner in \cite{krasner1983class}. This construction will play a key role in our construction of extensions. In section \ref{sec:ext}, we construct strong hyperfield extensions for quotient hyperfields and prove that our construction is valid.
\begin{comment}
Then we preform a calculation that will suggest that strong extensions of hyperfields might not exist for non-quotient hyperfields.
\end{comment}
In section \ref{sec:minext}, we use the weak hyperfield $\mathbb{W}$ and a polynomial with no roots over $\mathbb{W}$ to construct two extensions containing roots. Then we show that one of the two extensions is minimal, but not contained in the other. This implies that there must be two non-isomorphic minimal extensions of $\mathbb{W}$ containing a root.
\section{Preliminaries}\label{sec:prelim}
\subsection{Hyperfields}
Our main object of interest are hyperfields which were introduced by M. Krassner in 1956 \cite{krasner1956approximation}. Hyperfields are a generalization of fields with the difference being that addition in hyperfields is a multivalued operation. Thus, we can think of addition as a binary operator from $K\times K$ going to the nonempty subsets of $K$; formally, we can say that $\boxplus:K\times K\rightarrow {\mathcal P}^\times(K)$. As addition is multivalued, we must make sense of associativity. We will first give the formal definition of a hyperfield and then give a discussion on associativity.
\begin{defn}
A \textit{hyperfield} is a $5$-tuple $(K,\boxplus,\odot,0,1)$ that satisfies the following conditions:
\begin{enumerate}
\item $(K^\times=K\backslash\{0\},\odot, 1)$ forms an abelian group.
\item For all $x\in K$ we have that $0\odot x=x\odot 0 = 0$
\item The following distributive laws hold for all $x,y,z\in K$:
\begin{enumerate}
\item $x\odot(y\boxplus z)=(x\odot y)\boxplus (x\odot z)$
\item $(x\boxplus y)\odot z=(x\odot z)\boxplus (y\odot z)$
\end{enumerate}
\item $(K,\boxplus, 0)$ forms a commutative hypergroup that is:
\begin{enumerate}
\item $\boxplus$ is an associative that is for all $x,y,z\in K$, we have that $(x\boxplus y)\boxplus z= x\boxplus(y\boxplus z)$
\item $\boxplus$ is commutative that is for all $x,y\in K$, we have that $x\boxplus y= y\boxplus x$
\item $0$ is an additive identity that is for all $x\in K$ we have that $0\boxplus x=x\boxplus 0=\{x\}$
\item For all $x\in K$ there is a unique element $-x\in K$ such that $0\in x\boxplus -x=-x\boxplus x$
\item $x\in y\boxplus z$ if and only if $-y \in -x \boxplus z$
\end{enumerate}
\end{enumerate}
\end{defn}
{
Since $a\boxplus b$ is a set, we will need to define what it means to preform hyperaddition on a set $A$ and an element $c$, so we define:
\[
A\boxplus c=\bigcup_{x\in A} x\boxplus c
\]
We can further generalize this to say what happens when we preform hyperaddition on two sets $A$ and $B$:
\[
A\boxplus B=\bigcup_{x\in A, y\in B} x\boxplus y
\]
That is we can think of the hypersum of two sets as the union of all hypersums of pairs of elements from the sets.
These definitions allow us to recursively define arbitrarily long hypersums. That is we can recursively define $x_1\boxplus x_2\boxplus...\boxplus x_n$ as:
\[
x_1\boxplus x_2\boxplus...\boxplus x_n=\bigcup_{y\in x_1\boxplus x_2\boxplus...\boxplus x_{n-1}}y\boxplus x_n
\]
}
{
Now that we have defined hyperfield, let us give some examples.
\begin{example}
(Fields) A field $F$ forms a hyperfield with the hyperaddition as the original addition in the field that is $x\boxplus y=\{x+y\}$. The multiplication is just that of the field that is $x\odot y=x\cdot y$
\end{example}
\begin{example}
(Krasner Hyperfield) The Krasner hyperfield $\KK$ is a hyperfield over the set $\{0,1\}$ where $0$ acts as the additive identity and $1$ acts as the multiplicative identity. Thus, we have that $0\boxplus 1=1\boxplus 0=\{1\}$, and we have $1\boxplus 1=\{0,1\}$.
\end{example}
\begin{example}
(Hyperfield of Signs) The hyperfield of signs $\SS$ is a hyperfield over the set $\{0,1,-1\}$. The addition and multiplication are given by the following Cayley tables:
\begin{table}[h]
\parbox{.45\linewidth}{
\centering
\begin{tabular}{|l|l|l|l|}
\hline
$\boxplus$ & 0 & 1 & -1 \\ \hline
0 & \{0\} & \{1\} & \{-1\} \\ \hline
1 & \{1\} & \{1\} & \{0,1,-1\} \\ \hline
-1 & \{-1\} & \{0,1,-1\} & \{-1\} \\ \hline
\end{tabular}
\caption{Cayley table for addition for $\SS$}
}
\hfill
\parbox{.45\linewidth}{
\centering
\begin{tabular}{|l|l|l|}
\hline
$\odot$ & 1 & -1 \\ \hline
1 & 1 & -1 \\ \hline
-1 & -1 & 1 \\ \hline
\end{tabular}
\caption{Cayley table for multiplication for $\SS$}
}
\end{table}
The way to think of the hyperfield of signs is to interpret the element $1$ as any positive number, think of the element $-1$ as any negative number, and think of $0$ as $0$.
So multiplication is defined in the obvious manner as $1\odot 1=1$ is interpreted as the product of two positive numbers is positive. The product $1\odot -1=-1$ is interpreted as the product of a positive and a negative is negative, and $-1\odot -1=1$ is interpreted as the product of two negative numbers is positive.
We interpret addition as follows $1\boxplus 1=\{1\}$ we view this as the sum of two positive numbers is positive. We interpret $-1\boxplus -1=\{-1\}$ as the sum of two negative numbers is negative. The interesting case is the sum $1 \boxplus -1=\{0,1,-1\}$ which we interpret as the sum of a positive and a negative number can be either positive, negative, or zero. We will see later that this interpretation is quite natural due to the quotient hyperfield structure $\SS=\mathbb{R}/\mathbb{R}_{>0}$.
\end{example}
\begin{example}
(Weak Hyperfield of Signs) The weak hyperfield of signs or simply the weak hyperfield, $\mathbb{W}$, is a hyperfield over the set $\{0,1,-1\}$ where multiplication is defined in the same manner as the hyperfield of signs, but addition is defined by $1\boxplus 1=\{1,-1\}$,\\ $-1\boxplus -1=\{1,-1\}$, and $1\boxplus -1=\{0,1,-1\}$.
\end{example}
}
\subsection{Quotient Hyperfields}
Each of the examples we gave above are a special class of hyperfields called a quotient hyperfields. A quotient hyperfield is a hyperfield that is constructed from a field by quotienting out by a multiplicative subgroup. For a long time, it was an open question on whether or not all hyperfields admit a quotient structure; however, there exist non-quotient hyperfields see \cite{massouros1985methods}. As the construction of quotient hyperfields is important to understanding our construction of strong hyperfield extensions, we will go over the construction of quotient hyperfields; however, we will omit the proof that the quotient is a hyperfield see \cite{krasner1983class}.
Given a field $F$ and $G\leq F^\times$ a multiplicative subgroup of $F$, we construct the following equivalence relation on the elements of $F$. We say that $x\sim y$ if and only if there is some element $g\in G$ such that $x=gy$. We note that for $x\in F^\times$, $[x]$ is the coset of $F^\times/G$, so we can think of the equivalence classes as the multiplicative cosets where $[0]=\{0\}$.
We now form the quotient hyperfield $K=F/G$ by modding out by the equivalence $\sim$. The elements of $K$ are the equivalence classes under $\sim$, and we have the following induced operations:
\begin{enumerate}
\item $[x]\odot [y]=[x\cdot y]$
\item $[z]\in [x]\boxplus[y]$ if and only if there is some $z'\in[z]$ such that $z'=x'+y'$ for some $x'\in[x]$ and $y'\in[y]$
\end{enumerate}
The hyperaddition formulation might seem complicated; however, another way to think of the hypersum $[x]\boxplus [y]$ would be to think of $[x]$ and $[y]$ as cosets, then define \\$[x]+[y]=\{x'+y': x\in[x], y\in[y]\}$. That is $[x]+[y]$ is the set sum, so the hypersum $[x]\boxplus [y]=\{[z]:z\in [x]+[y]\}$, so after taking the set sum, we take all coset representatives.
\begin{remark}\label{rem:rem1}
We note that if $z\in [x_1]+[x_2]+...+[x_n]$, then $[z]\in [x_1]\boxplus[x_2]\boxplus...\boxplus[x_n]$.
\end{remark}
We know that these operations are well-defined and that this quotient forms a hyperfield by \cite{krasner1983class}. When we are working with a quotient hyperfield $K=F/G$, we make the convention to denote the elements of $K$ using the equivalence class notation $[x]\in K$, so we can distinguish between the element $[x]\in K$ and the element $x\in F$.
\begin{example}
(Fields) Fields form a quotient hyperfield with the quotient structure \\$F=F/\{1\}$. We see this as $x\sim y$ if and only if $x=y\cdot 1=y$. Thus, we have that $[x]=\{x\}$ for all $x\in F$.
\end{example}
\begin{example}
(Krasner Hyperfield) The Krassner hyperfield can be obtained as a quotient of any field $F$ by quotienting out by the entire multiplicative subgroup, that is $\KK= F/F^\times$. We can see this as our two equivalence classes will be $[0],[1]$ where $[0]$ corresponds to the additive identity in $F$, and $[1]$ corresponds to all of $F^\times$ that is $[1]$ corresponds to any non-zero element.
\end{example}
\begin{example}
(Hyperfield of Signs) The hyperfield of signs can be obtained as a quotient of $\mathbb{R}$ by $\mathbb{R}_{>0}$ (or in general any ordered field $(F,<)$ as $F/F_{>0}$). When we look at the equivalence classes of $\SS=\mathbb{R}/\mathbb{R}_{>0}$, we have $[1]=\{x:x>0\}$, $[-1]=\{x:x<0\}$, and $[0]=\{0\}$ we have the three elements we saw corresponding to the positive numbers, the negative numbers, and zero. Thus, we now see why it made sense to interpret $[1]$ as any positive number, $[-1]$ as any negative number, and $[0]$ as zero.
\end{example}
\begin{example}\label{ex:ex1}
(Weak Hyperfield of Signs) The weak hyperfield of signs admits a quotient structure $\mathbb{W}=\mathbb{F}_p/(\mathbb{F}_p^\times)^2$ where $p$ is a prime such that $p\geq 7$ and $p\equiv 3\pmod 4$. The fact that this quotient structure works for different primes will be key to showing that we can find two non-isomorphic minimal extensions of a hyperfield containing a root to some polynomial. We will see that taking different quotient structures can lead to different minimal extensions.
\end{example}
\begin{example}
(Phase Hyperfield) The phase hyperfield is a hyperfield with quotient structure $\mathbb{P}=\mathbb{C}/\mathbb{R}_{>0}$. Since we are modding out by the reals, we can think of the phase hyperfield as $S^1\cup \{0\}$ where $S^1$ is the unit circle in the complex plane. We should think of the operations geometrically. Thus, multiplication is the usual multiplication over $\mathbb{C}$ that is given two points $[x],[y]\in S^1$, $[x]\odot [y]$ is the point on $S^1$ given by adding the angles of $[x]$ and $[y]$. Then addition is given by $[x]\boxplus [0] = \{[x]\}$. For inverses we have $[x]\boxplus[-x]=\{[0],[x],[-x]\}$, we should think of this as since the line through $[x]$ and $[-x]$ goes through the origin the only possible phases you can get are $[x]$ and $[-x]$, and we include $[0]$ as the line goes through the origin. Then for $[x],[y]\neq [0]$ and $[y]\neq[-x]$ we have that $[x]\boxplus[y]$ is the shorter of the two arcs between the points $[x]$ and $[y]$ on $S^1$.
\end{example}
\subsection{Strong and Weak Hyperfield Extensions}
Since addition is multivalued, there are two natural notions of hyperfield extension that one can take. We shall call them \textit{weak hyperfield extensions} and \textit{strong hyperfield extensions}. The two definitions come from requiring either inclusion or equality. We shall define the hyperfield extensions in terms of homomorphisms of hyperfields. Let $K$ and $L$ be hyperfields and denote $\boxplus_K$ the hyperaddition in $K$ and $\boxplus_L$ the hyperaddition in $L$. Similarly, $\odot_K$ and $\odot_L$ are the multiplications in $K$ and $L$.
\begin{defn}
A \textit{weak hyperfield homomorphism} is a function $\phi:K\rightarrow L$ that satisfies the following:
$\phi(0)=0$
$\phi(1)=1$
$\phi(x\boxplus_K y)\subseteq \phi(x)\boxplus_L\phi(y)$
$\phi(x\odot_K y)=\phi(x)\odot_L \phi(y)$
\end{defn}
\begin{defn}
We say that $\phi: K\rightarrow L$ is a \textit{strong hyperfield homomorphsim} if in addition to being a weak hyperfield homomorphism we have that $\phi(x\boxplus_K y)= \phi(x)\boxplus_L\phi(y)$
\end{defn}
\begin{remark}
We remark that unlike classical field theory, homomorphisms need not be injective. For example, it is not to hard to show that for any hyperfield $K$ that $\phi:K\rightarrow \KK$, given by $\phi(0)=[0]$ and $\phi(x)=[1]$ for $x\neq 0$ is always a weak hyperfield homomorphism.
\end{remark}
Now we shall define hyperfield extensions in terms of homomorpisms of hyperfields; however, we shall require that these homomorphisms be injective. Since we have two notions of homomorphism, we will define two types of extensions.
\begin{defn}
For hyperfields $K$ and $L$, we say that $L$ is a \textit{weak hyperfield extension} of $K$ if there is an injective weak homomorphism of hyperfields $\phi: K\hookrightarrow L$. In this case, we say that $K$ is \textit{weak subhyperfield} of $L$.
If $\phi$ is a strong homomorphism of hyperfields, we sat that $L$ is a \textit{strong hyperfield extension} of $K$ and we say that $K$ is a \textit{strong subhyperfield} of $L$
\end{defn}
The terminology of weak and strong is natural as if $\phi:K\rightarrow L$ is a strong hyperfield homomorphism, then it is also a weak hyperfield homomorphism. Similarly, strong hyperfield extensions are also weak hyperfield extensions. Ideally, we would like to find strong hyperfield extensions that will contain roots to polynomials. We shall do this for quotient hyperfields. Now let us give some examples of hyperfield extensions.
\begin{example}
The weak hyperfield of signs $\mathbb{W}$ is a weak extension of the hyperfield of signs $\SS$. We see this via the homomorphism $\phi:\SS\hookrightarrow\mathbb{W}$ given by $\phi([x])=[x]$. It is a simple exercise to see that this is a weak extension of the hyperfield of signs, but is not a strong extension.
\end{example}
We note, as in the above example, that for weak hyperfield extensions, we can have the size of the hyperfields be the same, but changing the way addition is defined can create extensions. We now give an example of a strong extension.
\begin{example}
The phase hyperfield $\mathbb{P}=\mathbb{C}/\mathbb{R}_{>0}$ is a strong hyperfield extension of the hyperfield of signs $\SS=\mathbb{R}/\mathbb{R}_{>0}$. We see this via the homomorphism $\phi:\SS\hookrightarrow\mathbb{P}$ given by $\phi([x])=[x]$. This is indeed the case, since the equivalence classes $[0]=\{0\}$, \\$[1]=\{x\in\mathbb{R}: x>0\}$, $[-1]=\{x\in\mathbb{R}:x<0\}$ are the same in both $\SS$ and $\mathbb{P}$. Since the equivalence classes are the same, the operation of $\boxplus_\mathbb{P}$ and $\odot_\mathbb{P}$ is the same as in $\SS$. Thus, we have a strong extension.
\end{example}
We note that $\mathbb{P}$ and $\SS$ have a similar structure as quotient hyperfields. Namely, $\mathbb{P}$ and $\SS$ have the form $F/G$ and $L/G$ for $L$ a field extension of $F$. We can generalize this into the following lemma.
\begin{lemma}\label{lem:lem1}
For field extension $L/F$, for $G\leq F^\times$, we have that the quotient hyperfield $L/G$ is a strong hyperfield extension of $F/G$.
\end{lemma}
\begin{proof}
We shall construct a strong homomorphism of hyperfields. Let us define \\$\phi: F/G\rightarrow L/G$ given by $\phi([x])=[x]$. We shall first show that $[x]_{F/G}=[x]_{L/G}$ where $[x]_{F/G}$ is the equivalence class of $x$ in $F/G$, and $[x]_{L/G}$ is the equivalence class of $x$ in $L/G$. To see this, we see that $[x]_{F/G}=\{xg: g\in G\}$ and $[x]_{L/G}=\{xg: g\in G\}$, since the multiplicative subgroup $G$ is the same in both cases, we see that $[x]_{F/G}=[x]_{L/G}$. We shall now drop the subscript of $[x]$. Since the equivalence classes are the same in $F/G$ and $L/G$ we have that the map $\phi$ is well-defined and injective. Furthermore, we have that the strong homomorphism property holds since
\[
\phi([x]\odot_{F/G}[y])=\phi([xy])=[xy]=[x]\odot_{L/G}[y]=\phi(x)\odot_{L/G}\phi(y)
\]
and
\[
\phi([x]\boxplus_{F/G}[y])=\phi(\{[z]:z\in [x]+[y]\})=\{[z]:z\in [x]+[y]\}=[x]\boxplus_{L/G}[y]=\phi([x])\boxplus_{L/G}\phi([y])
\]
Thus, we have that $L/G$ is a strong hyperfield extension of $F/G$.
\end{proof}
\subsection{Polynomials over Hyperfields}
Now there are several issues that we encounter when moving to polynomials over hyperfields. In field theory, polynomials over fields form rings; however, we will see that polynomials over hyperfields do not form hyperrings. We will see that the multiplication of polynomials over hyperfields is actually a multivalued operation. Thus, as we are not working over a hyperring, so we can't simply mimic the construction of field extensions as we do not have ideals generated by polynomials over hyperfields.
\begin{defn}
A \textit{polynomial over a hyperfield} $K$ in variable $T$ is a formal sum
\[
k(T)=\op_{i=0}^n a_iT^i
\]
where each $a_i\in K$. We denote the set of all polynomials over a hyperfield $K$ in variable $T$ by $K[T]$.
\end{defn}
Thus, we see that polynomials look exactly the same over hyperfield as fields with the main difference being that we are taking a hypersum; thus, evaluating a polynomial at an element of $K$ will result in a set rather than a single element. We note that there have been several studies focused more on the algebraic structure of polynomials over hyperfield for example see \cite{article} or \cite{doi:10.1080/09720529.2003.10697978}.
We will make the convention that we will use $x$ as our variable for polynomials over fields, and $T$ will be our variable when we work over hyperfields.
Now the next issue that we encounter encounter is defining roots to polynomials, as when we plug in an element into our polynomial rather than getting back an element, we get a set. So we simply require that $0$ is in the set $k(\xi)$ for $\xi$ to be a root of $k$. We note that a study on the multiplicities of roots to polynomials over hyperfields was developed in \cite{baker2018descartes}.
\begin{defn}
For a polynomial $k\in K[T]$, we say that $\xi$ is a \textit{root} of $k$ if $0\in k(\xi)$.
\end{defn}
We will show that for a quotient hyperfield $K=F/G$, if $k\in K[T]$ does not have any roots, then we can find a strong hyperfield extension $L$ of $K$ such that $k$ contains a root in $L$. One might try the classical construction of modding out by the ideal generated by the polynomial; however, the next example shows that multiplication of polynomials over hyperfields is a multivalued operation. Thus, trying to construct an extension by modding out by an ideal would result in an algebraic structure with a multi-valued multiplication.
\begin{example}
(Hyperfield of Signs) Consider the polynomials $p(T)=1\boxplus T$ and\\ $q(T)=-1\boxplus T$ over the hyperfield of signs. If we were to multiply these polynomials in the usual way we see that:
\[
(1\boxplus T)(-1\boxplus T)=-1 \boxplus ((-1)T\boxplus T)\boxplus T^2 = -1\boxplus (1\boxplus -1) T\boxplus T^2
\]
That is $p(T)q(T)=\{-1\boxplus T^2, -1\boxplus -T\boxplus T^2, -1 \boxplus T\boxplus T^2\}$, so multiplication of polynomials over hyperfield is not a single valued operation.
\end{example}
Since quotient hyperfields arise from fields, for a quotient hyperfield $K=F/G$ there is a natural way to get a polynomial $\overline{f}\in K[T]$ from a polynomial $f\in F[x]$.
\begin{defn}
For a quotient hyperfield $K=F/G$, for a polynomial $f\in F[x]$, we define the \textit{induced polynomial} $\overline{f}\in K[T]$ as follows:
\[
f(x)=\sum_{i=0}^na_ix^i\leadsto \overline{f}=\op_{i=0}^n[a_i]T^i
\]
\end{defn}
\section{Constructing Hyperfield Extensions}\label{sec:ext}
\subsection{Strong Extensions for Quotient Hyperfields}
For this entire section, let $F$ be a field, $G\leq F^\times$, and let $K=F/G$ be the corresponding quotient hyperfield.
Let $k(T)\in K[T]$ be a polynomial containing no roots in $K$. In this section, we will construct a strong hyperfield extension to $K$ that contains some root $[\root]$ to $k$.
We begin with a key lemma that will be used in our construction.
\begin{lemma}\label{lem:lem2}
If $f\in F[x]$ with $f(x)=\sum_{i=0}^na_ix^i$, then for $\gamma\in F$, if $f(\gamma)=\alpha$, then for the induced polynomial $\overline{f}$, we have that $[\alpha]\in \overline{f}([\gamma])$.
\end{lemma}
\begin{proof}
We want to show that $[\alpha]\in \overline{f}([\gamma])$, now
\[
\overline{f}([\gamma])=\op_{i=0}^n[a_i][\gamma]^i=\op_{i=0}^n[a_i\gamma^i]
\]
Since $\alpha=\sum_{i=0}^n a_i\gamma^i$, we have that $\alpha\in [a_0\gamma^0]+[a_1\gamma^1]+...+[a_n\gamma^n]$, so by Remark \ref{rem:rem1}, we have that $[\alpha]\in\op_{i=0}^n[a_i\gamma^i]$ that is $[\alpha]\in\overline{f}([\gamma])$.
\end{proof}
\begin{comment}
Now $[\alpha]\in \overline{f}([\gamma])$ if and only if
\[
[\alpha]\cap ([a_0\gamma^0]+[a_1\gamma]+...+[a_n\gamma^n])\neq \emptyset
\]
Since $a_i\gamma^i\in[a_i\gamma^i]$, we have that $\alpha=\sum_{i=0}^n a_i\gamma^i\in [a_0\gamma^0]+[a_1\gamma^1]+...+[a_n\gamma^n]$. Thus, as $\alpha\in[\alpha]$, we have that and we $[\alpha]\cap ([a_0\gamma^0]+[a_1\gamma]+...+[a_n\gamma^n])\neq \emptyset$, so $[\alpha]\in\overline{f}([\gamma])$
\end{comment}
\begin{corollary}\label{cor:cor1}
If $f\in F[x]$ has a root, then $\overline{f}\in K[T]$ has a root.
\end{corollary}
\begin{proof}
Since $f$ has a root, we have that there is some $\root$ such that $f(\root)=0$, so by Lemma \ref{lem:lem2} above $[0]\in \overline{f}([\root])$.
\end{proof}
\begin{lemma}\label{lem:lem3}
If $k\in K[T]$ contains no roots, then there exists a polynomial $f\in F[x]$ with no roots in $F$ such that $\overline{f}=k$
\end{lemma}
\begin{proof}
Let $k(T)=\boxplus_{i=0}^n[a_i]T^i$ be a polynomial over $K$, then consider the polynomial $f(x)=\sum_{i=0}^na_ix^i$. Clearly, $\overline{f}=k$, now assume for sake of contradiction that $f$ has some root $\root$ in $F$, but by the above corollary, this would imply that $\overline{f}=k$ would have a root, but this contradicts the assumption that $k$ does not have a root.
\end{proof}
The idea behind our construction will be to find a polynomial $f\in F[x]$ with no roots such that $\overline{f}=k$. Then as $f$ will have is over the field $F$, we can apply classical field theory to find an extension $F'/F$ that contains a root to $f$, then we will form the quotient $F'/G$ and show that this is a strong hyperfield extension of $K$.
\begin{theorem}\label{thm:thm1}
If $k\in K[T]$ contains, no root, then there is a strong hyperfield extensions $K\subseteq L$.
\end{theorem}
\begin{proof}
Let us construct the hyperfield extension. Since $k(T)$ contains no roots, by Lemma \ref{lem:lem3} there is a polynomial $f\in F[x]$ that contains no roots over $F$ such that $\overline{f}=k$. Since $f\in F[x]$ contains no roots over $F[x]$, there is a field extension $F\subseteq F'$ such that $f$ contains a root $\root\in F'$. Now let $L=F'/G$ be the quotient hyperfield taken by modding out by the same multiplicative subgroup by Lemma \ref{lem:lem1} we have that $L$ is a strong hyperfield extension of $K$. Now since $f$ has a root in $F'$ by Corollary \ref{cor:cor1} we have that $\overline{f}=k$ has a root in $L$. Thus, $L$ is a strong hyperfield extension of $K$ that contains a root to $k$.
\end{proof}
\begin{example}
The polynomial $1\boxplus T^2$ contains no roots over $\SS$. Since $\SS$ has the quotient structure of $\mathbb{R}/\mathbb{R}_{>0}$, we lift $1\boxplus T^2$ to $1+x^2\in \mathbb{R}[x]$. Now we take the field extension containing a root to $1+x^2$, namely $\mathbb{C}$. Thus, our extension is $\mathbb{C}/\mathbb{R}_{>0}=\mathbb{P}$.
\end{example}
The construction that we gave relies on the fact that we could lift our polynomial over a hyperfield to a polynomial with no roots over a field. Thus, we do not have a construction for extensions for non-quotient hyperfields.
\begin{question}
Given a polynomial over a hyperfield is there a hyperfield extensions (weak or strong) that contains a root to the polynomial?
\end{question}
\begin{comment}
\subsection{Extensions for the Massouros Hyperfield} In this section, we will give some computations involving a certain class of non-quotient hyperfields which might give evidence towards non-quotient hyperfields that contain no strong extensions containing a root.
In \cite{massouros1985methods}, Massouros constructs a non-quotient hyperfield in the .following manner. Let $G$ be an abelian group under $\odot$, then $H=G\cup \{0\}$ will is the underlying set of the hyperfield. Where multiplication is defined using $\odot$, and $0$ is an absorbing element that is $0\odot x=x\odot 0=0$ for all $x\in H$. Now addition is defined using the following rules:
\begin{enumerate}
\item $x\boxplus y=\{x,y\}$ if $x,y\neq 0$ and $x\neq y$
\item $x\boxplus x=H\backslash \{x\}$ if $x\neq 0$
\item $x\boxplus 0=0\boxplus x=x$ for all $x\in H$
\end{enumerate}
\begin{lemma}
$(H,\boxplus,\odot)$ is a hyperfield that is not obtained from a quotient of a field.
\end{lemma}
For a proof of this lemma, see \cite{massouros1985methods}. We turn our attention to the specific example when we take $G=\mathbb{Z}/2\mathbb{Z}$, we will note that since we consider $M=\mathbb{Z}/2\mathbb{Z}\cup \{0\}$ we will denote the elements as $0,1,2$, where $1$ represents $[1]$ and $2$ represents $[2]=[0]$. We note that the element $0$ that we add to $\mathbb{Z}/2\mathbb{Z}$ is distinct from $[0]\in\mathbb{Z}/2\mathbb{Z}$. We write the Cayley tables:
\begin{table}[h]
\parbox{.45\linewidth}{
\centering
\begin{tabular}{|l|l|l|l|}
\hline
$\boxplus$ & 0 & 1 & 2 \\ \hline
0 & \{0\} & \{1\} & \{-1\} \\ \hline
1 & \{1\} & \{0,2\} & \{1,2\} \\ \hline
2 & \{-1\} & \{1,2\} & \{0,1\} \\ \hline
\end{tabular}
\caption{Cayley table for addition in $M$}
}
\hfill
\parbox{.45\linewidth}{
\centering
\begin{tabular}{|l|l|l|}
\hline
$\odot$ & 1 & 2 \\ \hline
1 & 1 & 2 \\ \hline
2 & 2 & 1 \\ \hline
\end{tabular}
\caption{Cayley table for multiplication in $M$}
}
\end{table}
\todo{Write section}
\end{comment}
\section{An Example of Two Non-Isomorphic Minimal Extensions}\label{sec:minext}
In this section we will give an explicit example of a polynomial over a hyperfield such that there are two non-isomorphic minimal extensions containing a root to this polynomial. To do this, we will show that a certain extension is minimal, and show that the other extension does not contain the minimal extension as a subhyperfield (in both the weak and strong sense).
Now for our construction, we will consider the weak hyperfield $\mathbb{W}$. As we saw in Example \ref{ex:ex1}, we know that if $p$ is a prime such that $p\geq 7$, and $p\equiv 3\pmod 4$, then $\mathbb{W}$ has the quotient structure $\mathbb{F}_p/(\mathbb{F}^\times_p)^2$ where $(\mathbb{F}^\times_p)^2$ denotes the non-zero squares.
The polynomial $k(T)=1\boxplus T^2$ contains no roots over $\mathbb{W}$. Thus, by taking the two different quotient structures of $\mathbb{W}=\mathbb{F}_7/(\mathbb{F}_7^\times)^2$ and $\mathbb{W}=\mathbb{F}_{11}/(\mathbb{F}_{11}^\times)^2$, we get strong extensions containing roots using the construction from Theorem \ref{thm:thm1}. The extensions have the quotient structures $\mathbb{F}_{49}/(\mathbb{F}_7^\times)^2$ and $\mathbb{F}_{121}/(\mathbb{F}_{11}^\times)^2$ respectively.
We will first show that $\mathbb{F}_{49}/(\mathbb{F}_7^\times)^2$ is a minimal extension, then we shall show that $\mathbb{F}_{121}/(\mathbb{F}_{11}^\times)^2$ does not contain $\mathbb{F}_{49}/(\mathbb{F}_7^\times)^2$ as a subhyperfield, so any the minimal subhyperfield of $\mathbb{F}_{121}/(\mathbb{F}_{11}^\times)^2$ containing a root to $k(T)$ is not isomorphic to $\mathbb{F}_{49}/(\mathbb{F}_7^\times)^2$; thus, there are two non-isomorphic minimal extensions.
\begin{theorem}\label{lem:lem4}
The extension $\mathbb{W}\subseteq \mathbb{F}_{49}/(\mathbb{F}_7^\times)^2$ is a minimal extension of $\mathbb{W}$ containing a root to $k(T)$.
\end{theorem}
\begin{proof}
Let us denote $L=\mathbb{F}_{49}/(\mathbb{F}_7^\times)^2$. We first note that by construction $L$ contains a root to $k(T)$, so we simply need to show that this extension is indeed minimal. It will be helpful to think of $\mathbb{F}_{49}$ as $\mathbb{F}_7[i]$ where $i^2=-1$. Furthermore, we note that $(\mathbb{F}_{7}^\times)^2=\{1,2,-3\}$. Now to see that this extension is minimal, let us consider the equivalence classes:
{
$[0]=\{0\}$
$[1]=\{1,2,-3\}$
$[-1]=\{-1,-2,3\}$
$[i]=\{i,2i,-3i\}$
$[-i]=\{-i,-2i,3i\}$
$[i+1]=\{i+1,2i+2,-3i-3\}$
$[-i+1]=\{-i+1,-2i+2,3i-3\}$
$[i+2]=\{i+2,2i-3,-3i+1\}$
$[-i+2]=\{-i+2,-2i-3,3i+1\}$
$[i+3]=\{i+3,2i-1,-3i-2\}$
$[-i+3]=\{-i+3,-2i-1,3i-2\}$
$[i-1]=\{i-1,2i-2,-3i+3\}$
$[-i-1]=\{-i-1,-2i-2,3i+3\}$
$[i-2]=\{i-2,2i+3,-3i-1\}$
$[-i-2]=\{-i-2,-2i+3,3i-1\}$
$[i-3]=\{i-3,2i+1,-3i+2\}$
$[-i-3]=\{-i-3,-2i+1,3i+2\}$
}
We note that the elements of $L^\times$ form a cyclic group of order $16$. We know this as $((\mathbb{F}_{49})^\times, \cdot)$ is a cyclic group, and $L^\times$ is a quotient group. Hence, it is cyclic.
Now let $K$ be a weak subhyperfield of $L$ such that $\mathbb{W}\subseteq K\subseteq \mathbb{F}_{49}/(\mathbb{F}_7^\times)^2$, and such that $K$ contains a root to $k(T)$. Our goal will be to show that $K^\times=L^\times$, that is, $K$ contains all nonzero elements of $L$. This will prove that $K=L$, and $L$ is a minimal extension containing a root to $k(T)$. We note that as $K\subseteq L$, we have that $K^\times \leq L^\times$; thus, by Lagrange's theorem $\vert K^\times \vert \mid 16$.
Since $\mathbb{W}\subseteq K$, we know that $[0],[1],[-1]\in K$. Furthermore, since $K$ contains a root to k(T) and $K\subseteq L$ we know that either $[i]$ or $[-i]$ is an element of $K$ as these are the roots of $k(T)$ in $L$. However, we have that both of these elements are in $K$ as $[-i]=[-1]\odot [i]$ and as $K$ is a hyperfield, it is closed under $\odot$.
Now since $L$ is a weak hyperfield extension of $K$ and $[1],[i]\in K$, we have that:
\[
[1]\boxplus_K [i]\subseteq [1]\boxplus_L [i]=\{[i+1],[i+2],[i-3]\}
\]
Thus, $K$ contains some element $y\in\{[i+1],[i+2],[i-3]\}$, so $[1],[-1],[i],[-i],y\in K$. That is $K$ contains at least $5$ nonzero elements. Since $\vert K^\times \vert\mid 16$, $\vert K\vert =8$ or $\vert K\vert = 16$.
Assume for sake of contradiction that $K\neq L$, then $\vert K^\times \vert=8$, so $K^\times$ will contain all elements of order at most $8$. Thus, we have would have that $K^\times=\{[1],[-1],[i],[-i],[1+i],$\\$ [i-1],[-i+1],[-i-1]\}$ as these are the only elements of order at most $8$ in $L^\times$.
However, since $L$ is a weak hyperfield extension of $K$, we have that:
\[
[i]\boxplus_K[i+1]\subseteq [i]\boxplus_L[i+1]=\{[i-3],[-i-3],[-i+2]\}
\]
However, none of these elements are in $K$; thus, $K$ cannot be a hyperfield a contradiction. Hence, we have that $K=L$. Namely, $L$ is a minimal extension of $\mathbb{W}$ containing a root to $k(T)$.
\end{proof}
Now that we have shown that $\mathbb{F}_{49}/(\mathbb{F}_7^\times)^2$ is a minimal extension. We shall show that $\mathbb{F}_{49}/(\mathbb{F}_7^\times)^2$ is not contained as a subhyperfield of $\mathbb{F}_{121}/(\mathbb{F}_{11}^\times)^2$.
\begin{theorem}
$\mathbb{F}_{49}/(\mathbb{F}_7^\times)^2$ is not contained as a subhyperfield of $\mathbb{F}_{121}/(\mathbb{F}_{11}^\times)^2$. Thus, minimal extensions containing roots are not unique for polynomials over hyperfields.
\end{theorem}
\begin{proof}
Assume for sake of contradiction that $ \mathbb{F}_{121}/(\mathbb{F}_{11}^\times)^2$ is a hyperfield extension of $\mathbb{F}_{49}/(\mathbb{F}_7^\times)^2$. Then $(\mathbb{F}_{49}/(\mathbb{F}_7^\times)^2)^\times \leq (\mathbb{F}_{121}/(\mathbb{F}_{11}^\times)^2)^\times$, so $16\mid \vert(\mathbb{F}_{121}/(\mathbb{F}_{11}^\times)^2)^\times\vert=24$ a contradiction. Thus, $\mathbb{F}_{49}/(\mathbb{F}_7^\times)^2$ is not a subhyperfield of $\mathbb{F}_{121}/(\mathbb{F}_{11}^\times)^2$.
Namely, the minimal subhyperfield of $\mathbb{F}_{121}/(\mathbb{F}_{11}^\times)^2$ containing a root to $k(T)$ is not isomorphic to the minimal extension $\mathbb{F}_{49}/(\mathbb{F}_7^\times)^2$, that is, minimal extensions containing roots need not be unique for polynomials over hyperfields.
\end{proof}
\bibliographystyle{alpha}
|
1,116,691,500,824 | arxiv | \section{\label{introduction}Introduction}
Quark flavor mixing in the Standard Model arises from unitary matrices
which diagonalize the corresponding hermitian mass matrices. The CKM
quark-flavor mixing matrix $V$ in the physical basis~\cite{2a,2b,2c}, is
given by $V= V_{\rm u}V^\dagger_{\rm d}$, where the unitary matrices
$V_{\rm u}$ and $V_{\rm d}$ diagonalize the up-quark and down-quark
Dirac mass matrices $M_{\rm u}$ and $M_{\rm d}$, respectively. Given
this circumstance, complete knowledge of the mass matrices fully
determines the corresponding mixing matrices. In practice, however, the
mass matrices are guessed at, while experiment can only determine the
moduli of the CKM matrix elements.
Recently, evolved quark masses have been given at various energy
scales~\cite{xing07}. The quark masses at the
$M_W\,(=80.403\,{\rm GeV})$, $M_Z\,(=91.1876\,{\rm GeV})$, and
$m_{\rm t}\,(=172.5\,{\rm GeV})$ scales are quite similar. For
our analysis we use as input the quark masses at the
$M_Z$~energy scale, which lies between the $M_W$ and the $m_t$
scales. So, in this paper we shall use as experimental
restrictions the reported values of the CKM matrix
elements $|V_{\alpha\, j}|$~\cite{pdg06}, the Jarlskog invariant
$J(V)$~\cite{pdg06}, and the quark masses at the $M_Z$ energy
scale~\cite{xing07}, to obtain constraints on the elements of
the quark mass matrices in five specific cases for three
generations. As a check of the stability of our type of
analysis, under the evolution of quark masses, we have repeated
it with quark masses at 2~GeV scale~\cite{xing07}.
Section~\ref{notation} gives the notation and the basic formulas or
expressions needed and the general procedure adopted for the analysis.
In Sec.~\ref{fritzsch}, we consider the Fritzsch-type of mass
matrices~\cite{fritzsch70,fritzsch78,fritzsch86} in the physical basis.
As pointed out there $M_{\rm u}$ can be chosen to be real with three
parameters while $M_{\rm d}$ has five, two of which are phase angles. In
Sec.~\ref{stech}, Stech-type matrices~\cite{stech83} are considered in
the basis in which $M_{\rm u}$ is diagonal, while $M_{\rm d}=p\,M_{\rm
u}+i\,S$. Here $p$ is a real number and $S$ is a non-diagonal matrix,
satisfying $S^{\rm T}=-S$, with three real parameters. In
Secs.~\ref{diagonal} and \ref{nondiagonal} we consider mass matrices
which are a simple generalization of the Fritzsch-type matrix in that
$M_{\rm u}$ and $M_{\rm d}$ have an additional parameter. These were
considered recently, in the $M_{\rm u}$ ($M_{\rm d}$) diagonal
basis~\cite{chaturvedi07}. In Sec.~\ref{diagonal} we consider these
cases again and fit them to the data. In Sec.~\ref{nondiagonal} we
consider the case of these type of mass matrices in the physical basis.
Results obtained in the above cases are compared and discussed in the
final section~\ref{conclusions}. Here also the results of the two
energy scales ($M_Z$ and 2~GeV) are compared and discussed.
\section{\label{notation}Notation and basic formulas}
The $3\times 3$ hermitian quark mass matrix $M_{\rm q}$ is diagonalized
by $V_{\rm q}$ so that $M_{\rm q}=V_{\rm q}^{\dagger}\hat{M_{\rm
q}}V_{\rm q}\,$, q=u,d. The eigenvalues are denoted by ($\lambda_{\rm
u}$,$\lambda_{\rm c}$,$\lambda_{\rm t}$) and ($\lambda_{\rm
d}$,$\lambda_{\rm s}$,$\lambda_{\rm b}$) for the up and down quark mass
matrices, respectively. Note that the eigenvalues are real but not
necessarily positive. Each mass matrix can be expressed in terms of its
projectors. Thus,
\begin{equation}
M_{\rm u}=\sum_{\alpha={\rm u, c, t}}\lambda_{\alpha}N_{\alpha}
\qquad
{\rm and}
\qquad
M_{\rm d}=\sum_{j={\rm d, s, b}}\lambda_{j}N_{j}.
\label{projectors}
\end{equation}
\noindent
Since $V= V_{\rm u}V^\dagger_{\rm d}$, it follows that~\cite{18}
\begin{equation}
|V_{\alpha\, j}|^2={\rm Tr}[N_\alpha N_j],
\label{5}
\end{equation}
\noindent
where
\begin{equation}
N_\alpha= \frac{(\lambda_\beta-M_{\rm u})(\lambda_\gamma
-M_{\rm u})}{(\lambda_\beta-\lambda_\alpha) (\lambda_\gamma
-\lambda_\alpha)}
\label{4}
\end{equation}
\noindent
and
\begin{equation}
N_j=
\frac{(\lambda_k-M_{\rm d})(\lambda_l -M_{\rm
d})}{(\lambda_k-\lambda_j) (\lambda_l -\lambda_j)},
\label{4p}
\end{equation}
\noindent
with ($\alpha$,$\beta$,$\gamma$) and ($j$,$k$,$l$) any permutation of
(u,c,t) and (d,s,b), respectively.
The Jarlskog invariant $J(V)$, which is a measure of CP-violation can be
directly expressed in terms of $M_{\rm u}$ and $M_{\rm d}$ and their
eigenvalues~\cite{4}, thus
\begin{equation}
{\rm Det}([M_{\rm u},M_{\rm d}])=2i\,D(\lambda_\alpha)D(\lambda_j) J(V),
\label{12}
\end{equation}
\noindent
where
\begin{equation}
D(\lambda_\alpha)=
(\lambda_{\rm c}-\lambda_{\rm u})(\lambda_{\rm t}-\lambda_{\rm u})
(\lambda_{\rm t}-\lambda_{\rm c})
\label{8}
\end{equation}
\noindent
and
\begin{equation}
D(\lambda_j)=
(\lambda_{\rm s}-\lambda_{\rm d})(\lambda_{\rm b}-\lambda_{\rm d})
(\lambda_{\rm b}-\lambda_{\rm s}).
\label{8p}
\end{equation}
The bases when $M_{\rm u}$ or $M_{\rm d}$ is diagonal are of special
interest for the mass matrices considered in Secs.~\ref{stech} and
\ref{diagonal}. For the case, $M_{\rm u}$ diagonal and $M_{\rm d}=M$,
Eq.~(\ref{12}) reduces to~\cite{chaturvedi07}
\begin{equation}
J(V)=
\frac{{\rm Im}(M_{12}\,M_{23}\,M^{*}_{13})}{D(\lambda_j)}.
\label{jdiag}
\end{equation}
\noindent
There is a similar formula for the case $M_{\rm d}$ diagonal.
This result shows that to obtain CP-violation, the mass matrix for
up-quark (down-quark) must have ${\rm
Im}(M_{12}\,M_{23}\,M^{*}_{13})$ non-zero in a basis in which the
down-quark (up-quark) mass matrix is diagonal. Consequently, the
Fritzsch type of mass matrices can only be used in the physical basis.
In general, our procedure in each case is to first determine the
elements of the quark mass matrices in terms of the
eigenvalues and then to form a $\chi^2$-function which contains
eleven summands. The first five compare the theoretical
expressions as functions of the elements of the quark mass
matrices of the four best measured moduli, namely, $|V_{\rm
ud}|$, $|V_{\rm us}|$, $|V_{\rm cd}|$, and $|V_{\rm cs}|$, and of
the Jarlskog invariant $J(V)$, with their experimental
values~\cite{pdg06}. The last six summands constrain the quark
mass matrices eigenvalues to the experimentally deduced quark
masses values at the $M_Z$ energy scale~\cite{xing07}.
\section{\label{fritzsch}Fritzsch type mass matrices}
We consider first the well-known case of the Fritzsch mass
matrices~\cite{fritzsch70,fritzsch78,fritzsch86}, given by the hermitian
matrices
\begin{equation}
M_{\rm u}=
\begin{pmatrix}
0 & A & 0 \\
A^* & 0 & B \\
0 & B^* & C
\end{pmatrix}
,
\quad\quad
M_{\rm d}=
\begin{pmatrix}
0 & A' & 0 \\
A'^* & 0 & B' \\
0 & B'^* & C'
\end{pmatrix}
.
\label{13}
\end{equation}
\noindent
Without lack of generality we may take $C$ and $C'$ to be positive. Since
we may rotate $M_{\rm u}$ and $M_{\rm d}$ with the same unitary matrix $X$
without changing the physics~\cite{Jarlskog87,Jarlskog87b}, we can make
$M_{\rm u}$ real with positive elements by choosing
\begin{equation}
X=
\begin{pmatrix}
e^{-i\phi_A} & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & e^{i\phi_B}
\end{pmatrix}
,
\label{13p}
\end{equation}
\noindent
with $\phi_A$ and $\phi_B$ the phases of $A$ and $B$, respectively. Then,
$M_{\rm u}$ and $M_{\rm d}$ have eight parameters, $A$, $B$, $C$, $|A'|$,
$|B'|$, $C'$, and the phases $\phi_{A'}$ and $\phi_{B'}$.
From the characteristic equation of $M_{\rm u}$, we have
\[
C=\lambda_{\rm u}+\lambda_{\rm c}+\lambda_{\rm t},
\]
\begin{equation}
-A^2-B^2=\lambda_{\rm u}\lambda_{\rm c} +\lambda_{\rm u}\lambda_{\rm
t}+\lambda_{\rm c}\lambda_{\rm t},
\label{15}
\end{equation}
\[
-A^2C=\lambda_{\rm u}\lambda_{\rm c}\lambda_{\rm t}.
\]
\noindent
Solving for $A$ and $B$ yields
\[
A=\left[
-\frac{\lambda_{\rm u}\lambda_{\rm c}\lambda_{\rm t}}
{\lambda_{\rm u}+\lambda_{\rm c}+\lambda_{\rm t}
}
\right]^{1/2}
,
\]
\begin{equation}
B=\left[-
\frac{
(\lambda_{\rm t}+\lambda_{\rm c})
(\lambda_{\rm t}+\lambda_{\rm u})
(\lambda_{\rm c}+\lambda_{\rm u})}
{\lambda_{\rm u}+\lambda_{\rm c}+\lambda_{\rm t}
}
\right]^{1/2}
.
\label{15b}
\end{equation}
\noindent Similarly for $M_{\rm d}$, the parameters $|A'|$, $|B'|$, and
$C'$ are obtained from~(\ref{15}) and (\ref{15b}) by replacing
($A$,$B$,$C$) by ($|A'|$,$|B'|$,$C'$) and ($\lambda_{\rm
u}$,$\lambda_{\rm c}$,$\lambda_{\rm t}$) by ($\lambda_{\rm
d}$,$\lambda_{\rm s}$,$\lambda_{\rm b}$).
According to~(\ref{5}), the magnitudes of the unitary quark
mixing matrix elements are given in the Fritzsch case by
\begin{eqnarray}
|V_{\alpha\,j}|^2&=&
\big{[}
(\lambda_{\alpha} - \lambda_{\beta})
(\lambda_{\alpha} - \lambda_{\gamma})
(\lambda_j - \lambda_k)
(\lambda_j - \lambda_l)
\big{]}^{-1}\times
\nonumber
\\
&\ &
\bigg\{
\big{(}\lambda_{\beta}\lambda_{\gamma} + A^2 + B^2\big{)}
\big{(}\lambda_k\lambda_l + {|A'|}^2 + {|B'|}^2\big{)}
+
\big{(}\lambda_{\beta}\lambda_{\gamma} + A^2\big{)}
\big{(}\lambda_k\lambda_l + {|A'|}^2\big{)}
\nonumber
\\
&\ &+\,
\Big{[}
\left(\lambda_{\alpha} + \lambda_{\beta}\right)
\left(\lambda_{\alpha} + \lambda_{\gamma}\right)
+ B^2
\Big{]}
\left[
\left(\lambda_j + \lambda_k\right)
\left(\lambda_j + \lambda_l\right)
+ {|B'|}^2
\right]
\nonumber
\\
&\ &+\,
2
\left(\lambda_{\beta} + \lambda_{\gamma}\right)
\left(\lambda_k + \lambda_l)\,
A{|A'|}\cos(\phi_{A'}\right)
\nonumber
\\
&\ &+\,
2\lambda_{\alpha}\lambda_j\,B{|B'|}\cos(\phi_{B'}) +
2B\,A{|B'|}{|A'|}\cos(\phi_{A'} + \phi_{B'})
\bigg\}\,,
\label{vaj}
\end{eqnarray}
\noindent
where ($\alpha$,$\beta$,$\gamma$) is any permutation of (u,c,t) and
($j$,$k$,$l$) any permutation of (d,s,b). By unitarity only four of
the nine $|V_{\alpha\,j}|^2$ are independent. As mentioned in
Sec.~\ref{introduction}, we shall use the four best experimentally
measured magnitudes $|V_{\rm ud}|$, $|V_{\rm us}|$, $|V_{\rm cd}|$, and
$|V_{\rm cs}|$\,.
Finally, the Jarslkog invariant $J(V)$ given by Eq.~(\ref{12}) translates
for the Fritzsch case into,
\begin{eqnarray}
J(V)&=&
-\big{[}
(\lambda_{\rm t} - \lambda_{\rm c}) (\lambda_{\rm t} -
\lambda_{\rm u}) (\lambda_{\rm c} - \lambda_{\rm u})(\lambda_{\rm b} -
\lambda_{\rm s}) (\lambda_{\rm b} - \lambda_{\rm d})(\lambda_{\rm s} -
\lambda_{\rm d})
\big{]}^{-1} \times
\nonumber
\\ &\ &
\bigg\{
\Big[
B\,{|B'|}\sin(\phi_{B'}) - A\,{|A'|}\sin(\phi_{A'})
\Big]
\nonumber
\\ &\ & \quad \times
\Big[
A^2{|B'|}^2+B^2 {|A'|}^2-2\,A\,B\,{|A'|}\,{|B'|}\cos(\phi_{A'}+\phi_{B'})
\Big]
\nonumber
\\ &\ & \quad + \,
A\,{|A'|}\sin(\phi_{A'})
\Big[
C^2 {|B'|}^2 + B^2{C'}^2 - 2\,C\,B\,{C'}\,{|B'|}\cos(\phi_{B'})
\Big]
\bigg\}\,.
\label{jv}
\end{eqnarray}
The mass matrices $M_{\rm u}$ and $M_{\rm d}$ in Eq.~(\ref{13}) do not
have positive definite eigenvalues. However, a viable mass matrix does
not need to have positive definite eigenvalues~\cite{fritzsch78}. These
eigenvalues are real but not necessarily positive. Thus, $\lambda^2_{\rm
u}=m^2_{\rm u}$, $\lambda^2_{\rm d}=m^2_{\rm d}$, etc., where $m_{\rm u}$
is the (positive) mass of the up quark, etc.
In this case it is possible to fix the relative phases between the
eigenvalues and the quark masses. For the up quark sector (and
analogously for the down one) we need a solution with the mass hierarchy
$|\lambda_{\rm u}|<<|\lambda_{\rm c}|<<|\lambda_{\rm t}|$. From the first
relation~(\ref{15}) we see that $\lambda_{\rm t}=m_{\rm t}$, $C$ being
positive. This coupled with the second relation in~(\ref{15}) require
$\lambda_{\rm u}=m_{\rm u}$ and $\lambda_{\rm c}=-m_{\rm c}<0$. Then, for
the Fritzsch case the relative phases between $\lambda$'s and $m$'s are
\begin{equation}
(\lambda_{\rm u}\,,\lambda_{\rm c}\,,\lambda_{\rm t}) =
(m_{\rm u}\,,-m_{\rm c}\,,m_{\rm t})
\quad
{\rm and}
\quad
(\lambda_{\rm d}\,,\lambda_{\rm s}\,,\lambda_{\rm b}) =
(m_{\rm d}\,,-m_{\rm s}\,,m_{\rm b})\,.
\label{17p}
\end{equation}
From the above formulation, we are now in position to apply the procedure
described at the end of Sec.~\ref{introduction} to determine the
parameters of the quark mass matrices in the Fritzsch case. The
parameters to be estimated are the six eigenvalues ($\lambda_{\rm
u}$,$\lambda_{\rm c}$,$\lambda_{\rm t}$) and ($\lambda_{\rm
d}$,$\lambda_{\rm s}$,$\lambda_{\rm b}$), and the two phases $\phi_{A'}$
and $\phi_{B'}$.
Using Eqs.~(\ref{15b}) for the up and down quark sectors the
$|V_{\alpha\, j}|$ and $J(V)$ can be expressed as functions of the six
eigenvalues and the phases $\phi_{A'}$ and $\phi_{B'}$. We now fit these
theoretical expressions to the experimental values of the four moduli
$|V_{\rm ud}|$, $|V_{\rm us}|$, $|V_{\rm cd}|$, and $|V_{\rm cs}|$, and
$J(V)$~\cite{pdg06}. In doing so we constrain the eigenvalues to the experimentally
determined values of the quark masses at the $M_Z$
energy scale~\cite{xing07} displayed in Column~2 of
Table~\ref{table1}, with the relative phases as given
by~(\ref{17p}). The fitted values obtained for the eigenvalues
are given in Column~3 of Table~\ref{table1}. Column~3 also gives
the values predicted for $\phi_{A'}$, $\phi_{B'}$, the four
moduli and $J(V)$. The corresponding $\Delta\chi^2$ for the
eigenvalues, $|V_{\alpha\, j}|$, and $J(V)$ are given in the
last column. Note that $\phi_{A'}$ and $\phi_{B'}$ are unknown
to begin with. The total $\chi^2/({\rm dof}) = 4.23/3=1.41$.
Also, from the relations in Eq.~(\ref{15b}) for the up and down quark
sectors and the entries of Table~\ref{table1} we can determine for the
derived parameters $A$, $B$, $C$, $|A'|$, $|B'|$, and $C'$, their \lq\lq
experimental" values (using the experimental constraints on the quark
masses) and their predicted values (using the fitted values of the
eigenvalues), along with their corresponding $\Delta\chi^2$
contributions. These numbers are shown in Table~\ref{table1b}.
We observe from Table~\ref{table1} that the fitted values obtained for
the phases $\phi_{A'}$ and $\phi_{B'}$ are compatible with
$-\pi/2$ and $0$, respectively. In Tables~\ref{table1c} and
\ref{table1d} we display the corresponding results obtained for this
particular choice. In this six parameters fit (the six eigenvalues) the
total $\chi^2/({\rm dof}) = 4.84/5=0.97$.
\section{\label{stech}Stech type mass matrix}
The second case we consider is the model of Stech~\cite{stech83},
\begin{equation}
M_{\rm u}=
\begin{pmatrix}
\lambda_{\rm u} & 0 & 0 \\
0 & \lambda_{\rm c} & 0 \\
0 & 0 & \lambda_{\rm t}
\end{pmatrix}
,
\quad\quad
M_{\rm d}=p\,M_{\rm u}+i\,
\begin{pmatrix}
0 & a & d \\
-a & 0 & b \\
-d & -b & 0
\end{pmatrix}
.
\label{13st}
\end{equation}
\noindent
The mass matrices are hermitian, the $\lambda$'s are the
eigenvalues of $M_{\rm u}$, and $p$ is constant. $a$, $b$, and $d$ are
real and $a$ and $b$ can be made positive like $A$ and $B$ in
Eq.~(\ref{13}). The Stech model has seven parameters.
Using the characteristic equation of $M_{\rm d}$, we may express $p$, $a$
and $b$ in terms of the eigenvalues of $M_{\rm u}$ and $M_{\rm d}$ and
the parameter $d$,
\begin{equation}
p=\frac{\lambda_{\rm d} + \lambda_{\rm s} + \lambda_{\rm b}}
{\lambda_{\rm u} + \lambda_{\rm c} + \lambda_{\rm t}},
\label{17st}
\end{equation}
\begin{equation}
a^2=\frac{-\lambda_{\rm u}\,E_1 + E_2 - (\lambda_{\rm c}-\lambda_{\rm
u})\,d^2}{\lambda_{\rm t}-\lambda_{\rm u}},
\label{14st}
\end{equation}
\begin{equation}
b^2=\frac{\lambda_{\rm t}\,E_1 - E_2 - (\lambda_{\rm t}-\lambda_{\rm
c})\,d^2}{\lambda_{\rm t}-\lambda_{\rm u}},
\label{15st}
\end{equation}
\noindent
where,
\begin{equation}
E_1=\frac{1}{2}[\lambda^2_{\rm d} + \lambda^2_{\rm s} + \lambda^2_{\rm b}
-p^2(\lambda^2_{\rm u} + \lambda^2_{\rm c} + \lambda^2_{\rm t})],
\quad
E_2=\frac{1}{p}(p^3\lambda_{\rm u}\,\lambda_{\rm c}\,\lambda_{\rm
t} - \lambda_{\rm d}\,\lambda_{\rm s}\,\lambda_{\rm b}).
\label{16st}
\end{equation}
From~(\ref{5}), the moduli of the unitary quark mixing matrix elements in
this case are,
\begin{eqnarray}
|V_{\alpha\,j}|^2&=&
\big{[}
(\lambda_k - \lambda_j)
(\lambda_l - \lambda_j)\big{]}^{-1}\times
\nonumber
\\
&\ &
\big{[}
(\lambda_k - p\,\lambda_{\alpha})(\lambda_l - p\,\lambda_{\alpha}) +
(a^2 + d^2)\,\delta_{\alpha,{\rm u}}
+
(a^2 + b^2)\,\delta_{\alpha,{\rm c}}
+
(b^2 + d^2)\,\delta_{\alpha,{\rm t}}
\big{]}
\,,
\label{vajst}
\end{eqnarray}
\noindent
again ($j$,$k$,$l$) is any permutation of (d,s,b) and we shall use
only the four independent expressions corresponding to the best
experimentally measured magnitudes $|V_{\rm ud}|$, $|V_{\rm us}|$,
$|V_{\rm cd}|$, and $|V_{\rm cs}|$\,.
For the Stech case the Jarslkog invariant $J(V)$ given by (\ref{12}) is
simply~\cite{chaturvedi06}
\begin{equation}
J(V) =
\frac{a\,b\,d}{(\lambda_{\rm b} - \lambda_{\rm s}) (\lambda_{\rm b} -
\lambda_{\rm d})(\lambda_{\rm s} - \lambda_{\rm d})}.
\label{jvst}
\end{equation}
In order to determine the parameters of the mass matrices in the Stech
case, notice first that using Eqs.~(\ref{17st}) - (\ref{16st}) the
original set of seven parameters ($\lambda_{\rm u}$, $\lambda_{\rm c}$,
$\lambda_{\rm t}$, $p$, $a$, $b$, $d$) can be replaced by the set
($\lambda_{\rm u}$, $\lambda_{\rm c}$, $\lambda_{\rm t}$, $\lambda_{\rm
d}$, $\lambda_{\rm s}$, $\lambda_{\rm b}$, $d$).
As mentioned before, $\lambda^2_{\rm u}=m^2_{\rm u}$, $\lambda^2_{\rm
d}=m^2_{\rm d}$, etc., and we need a solution with the mass hierarchies
$|\lambda_{\rm u}|<<|\lambda_{\rm c}|<<|\lambda_{\rm t}|$ for the up
quark sector and $|\lambda_{\rm d}|<<|\lambda_{\rm s}|<<|\lambda_{\rm
b}|$ for the down one. In this case the relative phases between the
$\lambda$'s and the quark masses can not be fixed {\it a priori} and all
the two possible signs in front of each of the quark masses must be
explored.
As in Sec.~\ref{fritzsch}, in the $\chi^2$ function we shall compare the
theoretical expressions of $|V_{\rm ud}|$\,, $|V_{\rm us}|$\,, $|V_{\rm
cd}|$\,, $|V_{\rm cs}|$ (from~(\ref{vajst})), and $J(V)$
(from~(\ref{jvst})) as functions of the six eigenvalues and $d$ (using
(\ref{17st}) - (\ref{16st})) with their experimental
counterparts~\cite{pdg06}. The $|\lambda|$'s are constrained to the
experimentally determined quark masses~\cite{xing07}.
Our best fit for the Stech case is obtained for the
identification~(\ref{17p}) of the relative phases between $\lambda$'s
and $m$'s, that is,
\begin{equation}
(\lambda_{\rm u}\,,\lambda_{\rm c}\,,\lambda_{\rm t}) =
(m_{\rm u}\,,-m_{\rm c}\,,m_{\rm t})\,,
\quad
(\lambda_{\rm d}\,,\lambda_{\rm s}\,,\lambda_{\rm b}) =
(m_{\rm d}\,,-m_{\rm s}\,,m_{\rm b})\,.
\label{17pst}
\end{equation}
\noindent
The corresponding numerical results are displayed in
Table~\ref{table2}. The total $\chi^2/({\rm dof}) = 9.10/4=2.28$.
Using the experimental constraints and the values of the fitted
parameters of Table~\ref{table2}, in Table~\ref{table2b} we show
the experimental, predicted, and $\Delta\chi^2$ values for the
derived parameter $p$ of Eq.~(\ref{17st}), and the predicted
values for $a$ and $b$ of Eqs.~(\ref{14st}) and (\ref{15st}),
respectively.
\section{\label{diagonal}new type os mass matrices}
Recently~\cite{chaturvedi07} mass matrices which are a simple
generalization of the Fritzsch mass matrix with one extra parameter in
$M_{\rm u}$ and $M_{\rm d}$ were considered. For want of a short name we
call it the CGS type mass matrix. The extra parameter comes from choosing
the 13 (and 31) matrix element to be non-zero and complex. This choice
gives CP-violation (see Eq.~(\ref{jdiag})), unlike the Fritzsch case,
even if one works in either up-quark or down-quark diagonal basis. In
subsections~\ref{down} and \ref{up} we consider this type of matrix in
the down-quark and up-quark diagonal basis, respectively. In
Sec.~\ref{nondiagonal} we do present fits with the CGS-type matrices in
the physical basis.
We now consider a basis in which the up-quark (down-quark) mass matrix
$M_{\rm u}$ ($M_{\rm d}$) is diagonal and the remaining mass matrix
$M_{\rm d}$ ($M_{\rm u}$) is hermitian of the CGS-type, namely,
\begin{equation}
M=
\begin{pmatrix}
0 & a & d \\
a^* & 0 & b \\
d^* & b^* & c
\end{pmatrix}
.
\label{di13}
\end{equation}
\noindent
For $d=0$ this reduces to the Fritzsch-type mass matrix and will give
$J(U)=0$, where $U$ is the corresponding diagonalizing unitary matrix.
Let $\lambda_{1,2,3}$ the eigenvalues of $M$, from the characteristic
equation we have
\[
c=\lambda_1+\lambda_2+\lambda_3\,,
\]
\begin{equation}
-(|a|^2+|b|^2+|d|^2)=\lambda_1\lambda_2+
\lambda_1\lambda_3+\lambda_2\lambda_3\,,
\label{di15}
\end{equation}
\[
-c|a|^2+2{\rm Re}(abd^*)=\lambda_1\lambda_2\lambda_3\,.
\]
\noindent As before, for the quark sector we need the mass hierarchy
$|\lambda_1|<<|\lambda_2|<<|\lambda_3|$ and from Eqs.~(\ref{di15}) it is
required that $\lambda_1,\lambda_3>0$ and $\lambda_2<0$, assuming $c>0$,
for both up and down quarks. For simplicity we take $a$ and $b$ to be
real and positive and $d$ as pure imaginary. Solving for $a$, $b$, and
$c$ yields
\begin{equation}
a=\left[
-
\frac{\lambda_1\lambda_2\lambda_3}
{\lambda_1+\lambda_2+\lambda_3}
\right]^{1/2}
\,,
\qquad
c=\lambda_1+\lambda_2+\lambda_3
\,,
\label{di15b}
\end{equation}
\begin{equation}
b=\left[-
\frac{
(\lambda_{\rm 3}+\lambda_{\rm 2})
(\lambda_{\rm 3}+\lambda_{\rm 1})
(\lambda_{\rm 2}+\lambda_{\rm 1})}
{\lambda_{\rm 1}+\lambda_{\rm 2}+\lambda_{\rm 3}
} - |d|^2
\right]^{1/2}
\,.
\label{di15c}
\end{equation}
\noindent
The CGS-type mass matrix in the up or down quark diagonal basis has in
principle, only 4 parameters, the eigenvalues $\lambda_1$,
$\lambda_2$, and $\lambda_3$ of $M$, and the magnitude of $d$.
Nevertheless, the total number of parameters is 7 because this
number comes from the non-zero elements of the two mass matrices. These
are 3 parameters from the diagonal mass matrix plus the 4 in the
other non-diagonal mass matrix. The number of constraints is 11 as
before. We now investigate the viability of $M$ in both the up-quark
and down-quark diagonal basis.
\subsection{\label{down}CGS-type mass matrix in down quark diagonal
basis}
In this case $M=M_{\rm u}$ is the up-quark mass matrix which is
diagonalized by $V_{\rm u}$. So the CKM-matrix $V=V_{\rm u}$
since $V_{\rm d}=I$. According to~(\ref{5}), the magnitudes of the
unitary quark mixing matrix elements are given by
\begin{eqnarray}
|V_{\alpha\,j}|^2&=&
\big{[}
(\lambda_{\beta} - \lambda_{\alpha})(\lambda_{\gamma} - \lambda_{\alpha})
\big{]}^{-1}\times
\nonumber
\\
&\ &
\big{\{}
(a^2 + |d|^2 + \lambda_{\beta}\lambda_{\gamma})\,\delta_{j,{\rm d}}
+
(a^2 + b^2 + \lambda_{\beta}\lambda_{\gamma})\,\delta_{j,{\rm s}}
\label{vajdown}
\\
&\ & +
\big{[}b^2 + |d|^2 + (\lambda_{\beta} + \lambda_{\alpha})(\lambda_{\gamma} +
\lambda_{\alpha})\big{]}\,\delta_{j,{\rm b}}
\big{\}}
\,,
\nonumber
\end{eqnarray}
\noindent
where ($\alpha$,$\beta$,$\gamma$) is any permutation of (u,c,t). As
mentioned in Sec.~\ref{introduction}, we shall use the four best
experimentally measured magnitudes $|V_{\rm ud}|$, $|V_{\rm us}|$,
$|V_{\rm cd}|$, and $|V_{\rm cs}|$\,.
Note that $\lambda_{\rm c}<0$ would imply $J(V)=J(V_{\rm u})<0$, so we
choose $d=-i|d|$ in this case. The Jarslkog invariant $J(V)$ given by
(\ref{12}), is then~\cite{chaturvedi06}
\begin{equation}
J(V) = -
\frac{a\,b\,|d|}{(\lambda_{\rm t} - \lambda_{\rm c}) (\lambda_{\rm t} -
\lambda_{\rm u})(\lambda_{\rm c} - \lambda_{\rm u})}.
\label{jvdown}
\end{equation}
The relative phases between the eigenvalues and the up quark masses are
\begin{equation}
(\lambda_{\rm u}\,,\lambda_{\rm c}\,,\lambda_{\rm t}) =
(m_{\rm u}\,,-m_{\rm c}\,,m_{\rm t})\,.
\label{17pdown}
\end{equation}
\noindent
The parameters to be estimated in this case are the three
eigenvalues ($\lambda_{\rm u}$,$\lambda_{\rm c}$,$\lambda_{\rm t}$) and
$|d|$.
In the $\chi^2$ function we shall compare the theoretical expressions of
$|V_{\rm ud}|$\,, $|V_{\rm us}|$\,, $|V_{\rm cd}|$\,, $|V_{\rm cs}|$
(from (\ref{vajdown})), and $J(V)$ (from (\ref{jvdown})) as functions of
the above parameters (using relations (\ref{di15b}) and (\ref{di15c})),
with their experimental counterparts~\cite{pdg06}. We shall also
constrain the $\lambda$'s to the experimentally determined quark
masses~\cite{xing07} with the relative phases as given
by~(\ref{17pdown}).
The results of our best fit with the four parameters and the eight
constraints used are displayed in Table~\ref{table3}. The total
$\chi^2/({\rm dof}) = 5.92/4=1.48$. Using the experimental
constraints and the values of the fitted parameters of
Table~\ref{table3}, in Table~\ref{table3b} we show the
experimental, predicted, and $\Delta\chi^2$ values for the
derived parameters $a$ and $c$ of Eqs.~(\ref{di15b}) and the
predicted value for $b$ from Eq.~(\ref{di15c}).
\subsection{\label{up}CGS-type mass matrix in up quark diagonal basis}
In this case $M=M_{\rm d}$ is the down-quark mass matrix which is
diagonalized by $V_{\rm d}$. So the CKM-matrix $V=V^{\dag}_{\rm d}$
since $V_{\rm u}=I$. According to~(\ref{5}), the magnitudes of the
unitary quark mixing matrix elements are given by
\begin{eqnarray}
|V_{\alpha\,j}|^2&=&
\big{[}
(\lambda_{k} - \lambda_{j})(\lambda_{l} - \lambda_{j})
\big{]}^{-1}\times
\nonumber
\\
&\ &
\big{\{}
(a^2 + |d|^2 + \lambda_{k}\lambda_{l})\,\delta_{\alpha,{\rm u}}
+
(a^2 + b^2 + \lambda_{k}\lambda_{l})\,\delta_{\alpha,{\rm c}}
\label{vajup}
\\
&\ & +
\big{[}b^2 + |d|^2 + (\lambda_{k} + \lambda_{j})(\lambda_{l} +
\lambda_{j})\big{]}\,\delta_{\alpha,{\rm t}}
\big{\}}
\,,
\nonumber
\end{eqnarray}
\noindent
where ($j$,$k$,$l$) is any permutation of (d,s,b). As
mentioned in Sec.~\ref{introduction}, we shall use the four best
experimentally measured magnitudes $|V_{\rm ud}|$, $|V_{\rm us}|$,
$|V_{\rm cd}|$, and $|V_{\rm cs}|$\,.
Note that $\lambda_{\rm s}<0$ imply $J(V)=J(V^{\dagger}_{\rm d})>0$, so
we choose $d=i|d|$ in this case. The Jarslkog invariant $J(V)$ given by
(\ref{12}), is then~\cite{chaturvedi06}
\begin{equation}
J(V) = -
\frac{
a\,b\,|d|
}{
(\lambda_{\rm b} - \lambda_{\rm s}) (\lambda_{\rm b} -
\lambda_{\rm d})(\lambda_{\rm s} - \lambda_{\rm d})
}.
\label{jvup}
\end{equation}
The relative phases between the eigenvalues and the down quark masses are
\begin{equation}
(\lambda_{\rm d}\,,\lambda_{\rm s}\,,\lambda_{\rm b}) =
(m_{\rm d}\,,-m_{\rm s}\,,m_{\rm b})\,.
\label{17pup}
\end{equation}
\noindent
The parameters to be estimated in this case are the three
eigenvalues ($\lambda_{\rm d}$,$\lambda_{\rm s}$,$\lambda_{\rm b}$) and
$|d|$.
In the $\chi^2$ function we shall compare the theoretical expressions of
$|V_{\rm ud}|$\,, $|V_{\rm us}|$\,, $|V_{\rm cd}|$\,, $|V_{\rm cs}|$
(from (\ref{vajup})), and $J(V)$ (from (\ref{jvup})) as functions of the
above parameters (using (\ref{di15b}) and (\ref{di15c})), with
their experimental counterparts~\cite{pdg06}. We shall also constrain
the $\lambda$'s to the experimentally determined quark
masses~\cite{xing07} with the relative phases as given
by~(\ref{17pup}).
The results of our best fit with the four parameters and the eight
constraints used are displayed in Table~\ref{table4}. The total
$\chi^2/({\rm dof}) = 15.50/4=3.88$. Using the experimental
constraints and the values of the fitted parameters of
Table~\ref{table4}, in Table~\ref{table4b} we show the
experimental, predicted, and $\Delta\chi^2$ values for the
derived parameters $a$ and $c$ of Eqs.~(\ref{di15b}) and the
predicted value for $b$ from Eq.~(\ref{di15c}).
\section{\label{nondiagonal}CGS-type mass matrix in physical basis}
We conclude by considering a small variation of the Fritzsch case, given
by the hermitian matrices
\begin{equation}
M_{\rm u}=
\begin{pmatrix}
0 & a & 0 \\
a & 0 & b \\
0 & b & c
\end{pmatrix}
,
\quad\quad
M_{\rm d}=
\begin{pmatrix}
0 & a' & i|d'| \\
a' & 0 & b' \\
-i|d'| & b' & c'
\end{pmatrix}
.
\label{13non}
\end{equation}
\noindent
The parameters $a$, $b$, $c$, $a'$, $b'$, and $c'$ are considered real
and positive. Here $M_{\rm d}$ is CGS-type and $M_{\rm u}$ is
Fritzsch-type so that we have 7 real parameters.
From the characteristic equations of $M_{\rm u}$ and $M_{\rm d}$ we can
solve for the matrix elements in terms of the corresponding eigenvalues
and $|d'|$,
\[
a=\left[
-\frac{\lambda_{\rm u}\lambda_{\rm c}\lambda_{\rm t}}
{\lambda_{\rm u}+\lambda_{\rm c}+\lambda_{\rm t}
}
\right]^{1/2}
,
\qquad
c=\lambda_{\rm u}+\lambda_{\rm c}+\lambda_{\rm t}\,,
\]
\begin{equation}
b=\left[-
\frac{
(\lambda_{\rm t}+\lambda_{\rm c})
(\lambda_{\rm t}+\lambda_{\rm u})
(\lambda_{\rm c}+\lambda_{\rm u})}
{\lambda_{\rm u}+\lambda_{\rm c}+\lambda_{\rm t}
}
\right]^{1/2}
,
\label{15bnon}
\end{equation}
\noindent
and
\[
a'=\left[
-\frac{\lambda_{\rm d}\lambda_{\rm s}\lambda_{\rm b}}
{\lambda_{\rm d}+\lambda_{\rm s}+\lambda_{\rm b}
}
\right]^{1/2}
,
\qquad
c'=\lambda_{\rm d}+\lambda_{\rm s}+\lambda_{\rm b}\,,
\]
\begin{equation}
b'=\left[-
\frac{
(\lambda_{\rm b}+\lambda_{\rm s})
(\lambda_{\rm b}+\lambda_{\rm d})
(\lambda_{\rm s}+\lambda_{\rm d})}
{\lambda_{\rm d}+\lambda_{\rm s}+\lambda_{\rm b}
} - |d'|^2
\right]^{1/2}
.
\label{15bpnon}
\end{equation}
From~(\ref{5}), the magnitudes of the unitary quark
mixing matrix elements in this case are
\begin{eqnarray}
|V_{\alpha\,j}|^2&=&
\big{[}(\lambda_{\beta} - \lambda_{\alpha})
(\lambda_{\gamma} - \lambda_{\alpha})
(\lambda_k - \lambda_j)
(\lambda_l - \lambda_j)\big{]}^{-1}\times
\nonumber
\\
&\ &
\big{[}
(a\,b'+b\,a')^2
+ 2(a^2a'^2+b^2b'^2)+(a^2+b^2)|d'|^2
\nonumber
\\
&\ &+\,
2\,a\,a'(\lambda_{\beta} + \lambda_{\gamma})
(\lambda_k + \lambda_l)
+\lambda_{\beta}\lambda_{\gamma}
\lambda_{k} \lambda_l
\nonumber
\\
&\ &+\,
\lambda_{k} \lambda_l
(
a^2+b^2+\lambda^2_{\alpha}
)+
\lambda_{\beta}\lambda_{\gamma}
(
a'^2+b'^2+|d'|^2+\lambda^2_j
)
\nonumber
\\
&\ &+\,
c\,\lambda_{\alpha}(b'^2+|d'|^2)
+c'\lambda_{j}\,b^2
+\lambda_{\alpha}\lambda_{j}(c\,c'+2b\,b')
\big{]}
\,,
\label{vajnon}
\end{eqnarray}
\noindent
where ($\alpha$,$\beta$,$\gamma$) is any permutation of (u,c,t)
and ($j$,$k$,$l$) any permutation of (d,s,b). We shall use the four best
experimentally measured magnitudes $|V_{\rm ud}|$, $|V_{\rm us}|$,
$|V_{\rm cd}|$, and $|V_{\rm cs}|$\,.
The Jarslkog invariant $J(V)$ given by Eq.~(\ref{12}) translates
into,
\begin{equation}
J(V) = -
\frac{
b\,|d'|\,\big{[}(a'b - a\,b')\,(b\,c' - b'c) +
a\,c\,|d'|^2\big{]}
}{
(\lambda_{\rm b} - \lambda_{\rm s}) (\lambda_{\rm b} -
\lambda_{\rm d})(\lambda_{\rm s} - \lambda_{\rm d})
(\lambda_{\rm t} - \lambda_{\rm c}) (\lambda_{\rm t} -
\lambda_{\rm u})(\lambda_{\rm c} - \lambda_{\rm u})
}.
\label{jvnon}
\end{equation}
As in the Fritzsch case it is possible to fix the relative phases between
the eigenvalues and the quark masses using Eqs.~(\ref{15bnon}) and
(\ref{15bpnon}),
\begin{equation}
(\lambda_{\rm u}\,,\lambda_{\rm c}\,,\lambda_{\rm t}) =
(m_{\rm u}\,,-m_{\rm c}\,,m_{\rm t})
\quad
{\rm and}
\quad
(\lambda_{\rm d}\,,\lambda_{\rm s}\,,\lambda_{\rm b}) =
(m_{\rm d}\,,-m_{\rm s}\,,m_{\rm b})\,.
\label{17pnon}
\end{equation}
\noindent
The parameters to be estimated are the six eigenvalues ($\lambda_{\rm
u}$,$\lambda_{\rm c}$,$\lambda_{\rm t}$) and ($\lambda_{\rm
d}$,$\lambda_{\rm s}$,$\lambda_{\rm b}$), and $|d'|$.
In the $\chi^2$ function we shall compare the theoretical expressions of
$|V_{\rm ud}|$\,, $|V_{\rm us}|$\,, $|V_{\rm cd}|$\,, $|V_{\rm cs}|$
(from (\ref{vajnon})), and $J(V)$ (from (\ref{jvnon})) as functions of
the above parameters (using relations (\ref{15bnon}) and
(\ref{15bpnon})), with their experimental counterparts~\cite{pdg06}.
We shall also constrain the $\lambda$'s to the experimentally
determined quark masses~\cite{xing07} with the relative phases as
given by~(\ref{17pnon}).
The results of our best fit are displayed in Table~\ref{table5}.
Using the experimental constraints and the values of the fitted
parameters of Table~\ref{table5}, in Table~\ref{table5b} we show
the experimental, predicted, and $\Delta\chi^2$ values for the
derived parameters $a$, $b$, $c$, $a'$, and $c'$ of
Eqs.~(\ref{15bnon}) and (\ref{15bpnon}), and the predicted value
for $b'$ of Eq.~(\ref{15bpnon}).
For this case, the total $\chi^2/({\rm dof}) = 1.89/4=0.47$ which
is much better than the Fritzsch case or the Stech case. The fit
for the mixed case, namely $M_{\rm u}$ CGS-type and $M_{\rm d}$
Fritzsch-type (with 7 real parameters) gives the same
$\chi^2/({\rm dof})$ as the fit given above. The reason is that
there exists an unitary matrix $Y$ such that it can rotate the
mass matrices in Eq.~(\ref{13non}) to mass matrices $N_{\rm
u}=Y^{\dagger}M_{\rm u}Y$ and $N_{\rm d}=Y^{\dagger}M_{\rm d}Y$
which are CGS-type and Fritzsch-type, respectively. Similarly,
we can rotate $M_{\rm u}$ and $M_{\rm d}$ in Eq.~(\ref{13non}),
using an unitary matrix so that both the up and down quark mass
matrices are of the CGS-type. Consequently, the case when both
mass matrices are of the CGS-type with 8 parameters does not
give any improvement in the quality of the fit.
\section{\label{conclusions}Concluding Remarks and discussion of results}
Working with different specific textures of quark mass matrices, in this
work we have determined bounds on the mass matrices elements for each
case using as experimental restrictions the values of the four
independent best measured moduli of the quark mixing matrix
elements, the Jarlskog invariant $J(V)$, and the quark masses at
the $M_Z$ energy scale.
The results of the fits are presented in
Tables~\ref{table1}-\ref{table5b}. The $\chi^2/({\rm dof})$ for the
various cases are tabulated in Table~\ref{table6} for
easy comparison. As can be seen, the fits for the Fritzsch and
Stech type are comparable even though the former has 8
parameters compared to the 7 of the latter. The CGS-type
matrix in the $M_{\rm u}$ diagonal basis is poorer despite having
7 parameters, but in the $M_{\rm d}$ diagonal basis it gives a
fit comparable to the Fritzsch and Stech type matrices. However,
the choice of different type of mass matrices in
Sec.~\ref{nondiagonal} (last row of Table~\ref{table6}) for up
and down quarks gives a very good fit despite having only 7
parameters. This is very encouraging phenomenologically.
An important question is whether the bounds obtained on the
structure of the quark mass matrices are affected by the
evolution of the quark masses. To check the stability of our type
of analysis we repeated it using as restrictions the values
of the quark masses at the 2~GeV energy scale~\cite{xing07}. The
results for $\chi^2/({\rm dof})$ for the various cases, at 2~GeV
scale, are summarized in Table~\ref{table6-2gev}. Results in
Table~\ref{table6-2gev} are very similar to the results in
Table~\ref{table6} implying that our results are not affected by
the evolution of the quark masses from 2 to
91~GeV~\cite{footnote}.
A simply way of understanding this behaviour is to note that if
all quark masses are scaled by a common factor, then the
algebraic expressions for the dimensionless numbers $J(V)$ and
the four moduli $|V_{\alpha\, j}|$ ($\alpha={\rm u}, {\rm c}$;
$j={\rm d}, {\rm s}$) will be unaffected. Actually, the ratio of
the quark masses at 2~GeV and $M_Z$ energy scales are $m_q(2~{\rm
GeV})/m_q(M_Z) = 1.71 - 1.74$ ($q = {\rm u}, {\rm d},
{\rm s}, {\rm c}, {\rm b}$) and 1.85 for $q = {\rm
t}$~\cite{xing07}.
Phenomenologically, our results are encouraging. However, the
basic problem remains, namely to find a fundamental theoretical
mechanism to generate quark mass matrices.
\begin{acknowledgments}
One of us (G.~S-C) is grateful to the Faculty of Mathematics,
Autonomous University of Yucat\'an, M\'exico, for hospitality where part
of this work was done. VG and G.~S-C would like to thank CONACyT
(M\'exico) for partial support.
\end{acknowledgments}
|
1,116,691,500,825 | arxiv | \section{Introduction}
The tiny value of the cosmological constant (dark energy), which is responsible
for the accelerated expansion of the Universe and constitutes $70\%-73\%$
of its energy density, is a major puzzle for modern particle physics.
A fit to the recent data shows that its value is
$\Lambda \sim 10^{-123}M_{Pl}^4 \sim 10^{-55} M_Z^4$ \cite{1}.
At the same time much larger contributions should come
from electroweak symmetry breaking ($\sim 10^{-67}M_{Pl}^4$)
and QCD condensates ($\sim 10^{-79}M_{Pl}^4$). Moreover the
contribution of zero--modes is expected to push the
vacuum energy density even higher up to $\sim M_{Pl}^4$, i.e.
\begin{equation}
\rho_{\Lambda}\simeq \sum_{bosons}\frac{\omega_b}{2}- \sum_{fermions}\frac{\omega_f}{2}=
\int_{0}^{\Omega}\biggl[\sum_b\sqrt{|\vec{k}|^2+m_b^2}
-\sum_f\sqrt{|\vec{k}|^2+m_f^2}\biggr]\frac{d^3\vec{k}}{2(2\pi)^3}\sim -\Omega^4\,,
\label{01}
\end{equation}
where the $m_b$ and $m_f$ are the masses of bosons and fermions while
$\Omega\sim M_{Pl}$. Because of the enormous cancellation needed between
the contributions of different condensates to $\Lambda$, the smallness of
the cosmological constant should be regarded as a fine--tuning problem.
The smallness of the cosmological constant could be related to an almost
exact symmetry. An exact global supersymmetry (SUSY) ensures zero value
for the energy density at the minimum of the potential of the scalar fields.
However the breakdown of supersymmetry induces a huge and positive
contribution to the total vacuum energy density of order $M_{S}^4$, where
$M_{S}$ is the SUSY breaking scale. Because superpartners of quarks and
leptons have not yet been observed, $M_{S}$ is expected to be higher
than $100\,\mbox{GeV}$.
Since we are interested in the value of the cosmological constant, we must
include gravity and thus local supersymmetry in our consideration.
In supergravity (SUGRA) an enormous fine--tuning is also required to keep
the cosmological constant around its observed value \cite{2}. Our basic
scenario for evaluating the tiny value of the cosmological constant implies
that the SUGRA scalar potential of the hidden sector has a supersymmetric
Minkowski minimum (second vacuum), in addition to the physical vacuum in
which we live. In this second vacuum the supersymmetry is expected to be
broken dynamically in the observable sector when the supersymmetric QCD
interaction becomes non-perturbative. The physical and the second vacua
are assumed to be degenerate, so that the estimated vacuum energy density
in the second vacuum is then transferred to our vacuum.
The assumed degeneracy of the vacua is supposed to be justified by the
so-called Multiple Point Principle (MPP). According to the MPP, Nature
chooses values of coupling constants such that many phases of the
underlying theory should coexist \cite{10}. On the phase diagram of the
theory it corresponds to a special point -- the multiple point.
The vacuum energy densities of these different phases are degenerate
at the multiple point. In this paper we shall apply MPP to $(N=1)$
supergravity to give degeneracy between the two vacua mentioned above.
This would normally require an extra fine-tuning associated with the
presence of the second vacuum \cite{Froggatt:2003jm}. However, the breakdown
of global symmetries, which are chosen to protect a zero value for the
cosmological constant in SUGRA models, may lead to a set of degenerate
vacua with broken and unbroken supersymmetry (SUSY) whose vacuum energy
densities vanish in the leading approximation
\cite{Froggatt:2004gc}-\cite{Froggatt:2005nb}, resulting in a natural
realization of the MPP conditions \cite{10}. In our previous articles
the MPP assumption was used to estimate the value of the cosmological
constant in such SUGRA models inspired by degenerate vacua
\cite{Froggatt:2003jm}-\cite{talks}. It is the main purpose of the
present article to use the same idea for estimating the cosmological
constant, but now under the assumption of {\em a Split SUSY picture}.
Instead of just postulating the MPP, a large set of approximately degenerate
vacua with broken and unbroken supersymmetry can also arise from the vast
landscape of string theory vacua, if the vacua with large cosmological
constants are not allowed. Recently it was realized that the presence of an
enormous number of long-lived metastable vacua favors high-scale breaking
of supersymmetry, which motivated the introduction of the Split SUSY
scenario. In this paper we attempt to estimate the value of the dark energy
in the Split SUSY model, assuming the degeneracy of vacua with
broken and unbroken supersymmetry (i.e. MPP). We argue that the observed value
of the cosmological constant can be reproduced in the considered case,
if the SUSY breaking scale is $M_S\sim 10^{9}-10^{10}\,\mbox{GeV}$. This
Split SUSY scenario predicts extremely long-lived gluinos.
The paper is organized as follows. In the next section we present $(N=1)$
SUGRA models in which the MPP conditions are fulfilled. In section 3 we discuss
the Split SUSY scenario, and in section 4 we present some numerical estimates
of the vacuum energy density and consider possible phenomenological
implications. Our results are summarized in section 5.
\section{SUGRA models inspired by degenerate vacua}
\label{noscale}
The full $(N=1)$ SUGRA Lagrangian \cite{4}-\cite{3} is specified
in terms of an analytic gauge kinetic function $f_a(\phi_{M})$ and
a real gauge-invariant K$\Ddot{a}$hler function
$G(\phi_{M},\phi_{M}^{*})$, which depend on the chiral superfields
$\phi_M$. The function $f_{a}(\phi_M)$ determines the gauge
coupling constants $Re f_a(\phi_M)=1/g_a^2$, where the index $a$
designates different gauge groups. The K$\Ddot{a}$hler function is
a combination of two functions
\begin{equation}
G(\phi_{M},\phi_{M}^{*})=K(\phi_{M},\phi_{M}^{*})+\ln|W(\phi_M)|^2\,,
\label{1}
\end{equation}
where $K(\phi_{M},\phi_{M}^{*})$ is the K$\Ddot{a}$hler potential
while $W(\phi_M)$ is the superpotential of the considered SUSY model.
Here we use standard supergravity mass units: $\displaystyle\frac{M_{Pl}}{\sqrt{8\pi}}=1$.
The SUGRA scalar potential is given by
\begin{equation}
\begin{array}{c}
V(\phi_M,\phi^{*}_M)=\sum_{M,\,\bar{N}} e^{G}\left(G_{M}G^{M\bar{N}}
G_{\bar{N}}-3\right)+\displaystyle\frac{1}{2}\sum_{a}(D^{a})^2,\,\,\,
D^{a}=g_{a}\sum_{i,\,j}\left(G_i T^a_{ij}\phi_j\right)\,,\\[2mm]
G_M \equiv \displaystyle\frac{\partial G}{\partial \phi_M}\,,\quad
G_{\bar{M}}\equiv \displaystyle\frac{\partial G}{\partial \phi^{*}_M}\,,\quad
G_{\bar{N}M}\equiv \displaystyle\frac{\partial^2 G}{\partial \phi^{*}_N \partial \phi_M}\,,\quad
G^{M\bar{N}}=G_{\bar{N}M}^{-1}\,,
\end{array}
\label{2}
\end{equation}
where $g_a$ is the gauge coupling constant associated with the
generator $T^a$ of the gauge transformations. In order to break
supersymmetry in $(N=1)$ SUGRA models, a hidden sector is
introduced. It is assumed that the superfields of the hidden
sector $(z_i)$ interact with the observable ones only by means of
gravity. At the minimum of the scalar potential (\ref{2}), hidden
sector fields acquire vacuum expectation values (VEVs) so that at least
one of their auxiliary fields
\begin{equation}
F^{M}=e^{G/2}G^{M\bar{P}}G_{\bar{P}}
\label{201}
\end{equation}
gets a non-vanishing VEV, giving rise to the breakdown of local SUSY and
generating a non--zero gravitino mass $m_{3/2}=<e^{G/2}>$.
As mentioned in section 1, the successful implementation of the MPP in
$(N=1)$ supergravity requires us to assume the existence of a vacuum in
which the low--energy limit of the considered theory is described by
a pure supersymmetric model in flat Minkowski space. According to the
MPP this vacuum and the physical one must be degenerate. Such a second
vacuum is realised only if the SUGRA scalar potential has a minimum
where $m_{3/2}=0$. The corresponding minimum is achieved when
the superpotential $W$ for the hidden sector and its derivatives
vanish, i.e.
\begin{equation}
W(z_m^{(2)})=0\,,
\label{202}
\end{equation}
\begin{equation}
\frac{\partial W(z_i)}{\partial z_m}\Biggl|_{z_m=z_m^{(2)}}=0
\label{203}
\end{equation}
where $z_m^{(2)}$ denote vacuum expectation values of the hidden sector
fields in the second vacuum. In general Eq.~(\ref{202}) represents
the extra fine-tuning associated with the presence of the
supersymmetric Minkowski vacuum.
The simplest K$\Ddot{a}$hler potential and superpotential
that satisfies conditions (\ref{202}) and (\ref{203})
can be written as
\begin{equation}
K(z,\,z^{*})=|z|^2\,,\qquad\qquad W(z)=m_0(z+\beta)^2\,.
\label{204}
\end{equation}
The hidden sector of this SUGRA model contains only one singlet
superfield $z$. If the parameter $\beta=\beta_0=-\sqrt{3}+2\sqrt{2}$,
the SUGRA scalar potential of the considered model possesses
two degenerate minima with zero energy density at the classical
level. One of them is a supersymmetric Minkowski minimum that
corresponds to $z^{(2)}=-\beta$. In the other minimum of the
SUGRA scalar potential ($z^{(1)}=\sqrt{3}-\sqrt{2}$)
local supersymmetry is broken; so it can be associated
with the physical vacuum. Varying the parameter $\beta$ around
$\beta_0$ one can obtain a positive or a negative contribution
from the hidden sector to the total energy density of the physical
vacuum. Thus $\beta$ can be fine--tuned so that the
physical and second vacua are degenerate.
The obvious drawback of the model discussed above is related
with the degree of the fine-tuning which is required, in order to
arrange for the presence of a supersymmetric Minkowski vacuum
as well as for the degeneracy of the physical and second vacua.
Nevertheless in some SUGRA models this fine-tuning can be
alleviated. Let us consider the no--scale inspired SUGRA model
with two hidden sector fields ($T$ and $z$) and a set of
chiral supermultiplets $\varphi_{\sigma}$ in the observable sector.
These superfields transform differently under the imaginary translations
($T\to T+i\beta,\, z\to z,\, \varphi_{\sigma}\to \varphi_{\sigma}$)
and dilatations
($T\to\alpha^2 T,\, z\to\alpha\,z,\,\varphi_{\sigma}\to\alpha\,\varphi_{\sigma}$),
which are subgroups of the $SU(1,1)$ group \cite{15}--\cite{12}
\footnote{The Lagrangian for a no--scale SUGRA model is invariant
under imaginary translations and dilatations. As a consequence the
vacuum energy density goes to zero near global minima of the
tree--level scalar potential which vanishes identically along
some directions \cite{4}, \cite{12}--\cite{11}. Thus imaginary
translations and dilatations protect a zero value for the
cosmological constant in supergravity. However these symmetries
also preserve supersymmetry in all vacua, which has to be broken
in any phenomenologically acceptable theory.}.
In order to ensure the appropriate breakdown of local supersymmetry,
we assume that there is a weak breaking of the dilatation invariance
of the hidden sector superpotential characterised by an hierarchically
small parameter $\varkappa$. The full superpotential of the model
is given by \cite{Froggatt:2005nb}:
\begin{equation}
\begin{array}{c}
W(z,\,\varphi_{\alpha})=W_{hid} + W_{obs}\,,\\[2mm]
W_{hid}=\varkappa\biggl(z^3+\mu_0 z^2+\sum_{n=4}^{\infty}c_n z^n\biggr),\qquad
W_{obs}=\displaystyle\sum_{\sigma,\beta,\gamma}\displaystyle\frac{1}{6}
Y_{\sigma\beta\gamma}\varphi_{\sigma}\varphi_{\beta}\varphi_{\gamma}\,.
\end{array}
\label{191}
\end{equation}
The superpotential (\ref{191}) contains a bilinear mass term for the
superfield $z$ and higher order terms $c_n z^n$ that spoil dilatation
invariance. A term proportional to $z$ is not included. It can be
forbidden by a gauge symmetry of the hidden sector, if $z$ transforms
non--trivially under the corresponding gauge transformations. Here we
do not allow the breakdown of dilatation invariance in the superpotential
of the observable sector, in order to avoid the appearance of potentially
dangerous terms which lead, for instance, to the so--called $\mu$--problem.
We also assume that the dilatation invariance is broken in the K$\ddot{a}$hler
potential of the observable sector, so that the full K$\ddot{a}$hler
potential takes the form \cite{Froggatt:2005nb}:
\begin{equation}
\begin{array}{rcl}
K(\phi_{M},\phi_{M}^{*})&=&\displaystyle-3\ln\biggl[T+\overline{T}
-|z|^2-\sum_{\alpha}\zeta_{\alpha}|\varphi_{\alpha}|^2\biggr]
+\\[2mm]
&+&\sum_{\alpha, \beta}\biggl(\displaystyle\frac{\eta_{\alpha\beta}}
{2}\,\varphi_{\alpha}\,\varphi_{\beta}+h.c.\biggr)+
\sum_{\beta}\xi_{\beta}|\varphi_{\beta}|^2\,,
\end{array}
\label{192}
\end{equation}
where $\zeta_{\alpha}$, $\eta_{\alpha\beta}$, $\xi_{\beta}$ are some
constants. In the limit when $\eta_{\alpha\beta}$, $\xi_{\beta}$
and $\varkappa$ go to zero, the dilatation invariance is restored,
protecting supersymmetry and a zero value of the cosmological constant.
It is worth noticing that we only allow the breakdown of the dilatation
invariance in the K$\ddot{a}$hler potential of the observable sector,
since any variations in the K$\ddot{a}$hler potential of the hidden
sector may spoil the vanishing of the vacuum energy density in global
minima. We restrict our consideration to the simplest set of terms
that break dilatation invariance in the K$\ddot{a}$hler potential.
Additional terms which are proportional to $|\varphi_{\alpha}|^2$
normally appear in minimal SUGRA models \cite{minimal-sugra}. The
other terms $\eta_{\alpha\beta}\varphi_{\alpha} \varphi_{\beta}$
introduced in the K$\Ddot{a}$hler potential (\ref{192}) give rise
to effective $\mu$ terms after the spontaneous breakdown of local
supersymmetry, solving the $\mu$ problem \cite{30}.
In the considered SUGRA model the scalar potential is positive definite
\begin{equation}
V=V_{hid}+V_{obs},
\label{1921}
\end{equation}
\begin{equation}
V_{hid}=\frac{1}{3(T+\overline{T}-|z|^2)^2}
\biggl|\frac{\partial W_{hid}(z)}{\partial z}\biggr|^2,
\label{1922}
\end{equation}
\begin{equation}
V_{obs}=\sum_{\alpha}\biggl|\displaystyle\frac{\partial
W_{eff}(y_{\beta})}{\partial y_{\alpha}}
+m_{\alpha}y^{*}_{\alpha}\biggr|^2+\displaystyle\frac{1}{2}\sum_{a}(D^{a})^2\,,
\label{193}
\end{equation}
so that the vacuum energy density vanishes near its global minima.
In Eq.~(\ref{193}) $y_{\alpha}$ are canonically normalized scalar fields
\begin{equation}
\begin{array}{c}
y_{\alpha}=\tilde{C}_{\alpha}\varphi_{\alpha}\,,\qquad
\tilde{C}_{\alpha}=\xi_{\alpha}\biggl(1+\displaystyle\frac{1}
{x_{\alpha}}\biggr)\,,\qquad x_{\alpha}
=\frac{\xi_{\alpha}<(T+\overline{T}-|z|^2)>}{3\zeta_{\alpha}}\,.
\end{array}
\label{196}
\end{equation}
The mass parameters $m_{\alpha}$ are given by \cite{Froggatt:2005nb}
\begin{equation}
m_{\alpha} = \frac{m_{3/2} x_{\alpha}}{1+x_{\alpha}},
\end{equation}
where $m_{3/2}$ denotes the gravitino mass.
The effective superpotential, which describes the interactions of
observable superfields at low energies, is given by
\begin{equation}
\begin{array}{c}
W_{eff}=\sum_{\alpha,\,\beta}\displaystyle\frac{\mu_{\alpha\beta}}{2}\,
y_{\alpha}\,y_{\beta}+\sum_{\alpha,\,\beta,\,\gamma}
\displaystyle\frac{h_{\alpha\beta\gamma}}{6}\,y_{\alpha}\,y_{\beta}\,y_{\gamma}\,,\\[3mm]
\mu_{\alpha\beta}=m_{3/2}\eta_{\alpha\beta}(\tilde{C}_{\alpha}
\tilde{C}_{\beta})^{-1}\,,\qquad\qquad
h_{\alpha\beta\gamma}=\displaystyle\frac{Y_{\alpha\beta\gamma}
(\tilde{C}_{\alpha}\tilde{C}_{\beta}\tilde{C}_{\gamma})^{-1}}
{<(T+\overline{T}-|z|^2)^{3/2}>}\,.
\end{array}
\label{197}
\end{equation}
The form of the superpotential (\ref{191}) guarantees that there is
always a supersymmetric Minkowski minimum at $z=0$, because the
conditions (\ref{202}) and (\ref{203}) are fulfilled for $z=0$
without any extra fine-tuning. In the simplest case when $c_n=0$,
$V_{hid}$ has two minima, at $z=0$ and $z=-\frac{2\mu_0}{3}$.
At these points the scalar potential (\ref{1922}) achieves its
absolute minimal value i.e.~zero. In the first vacuum, where $z=-\frac{2\mu_0}{3}$,
local supersymmetry is broken so that the gravitino becomes massive
\begin{equation}
m_{3/2}=\biggl<\frac{W(z)}{(T+\overline{T}-|z|^2)^{3/2}}\biggr>
=\frac{4\kappa\mu_0^3}{27\biggl<\biggl(T+\overline{T}
-\frac{4\mu_0^2}{9}\biggr)^{3/2}\biggr>}\,.
\label{7}
\end{equation}
and all scalar particles get non--zero masses
$m_{\sigma}\sim \frac{m_{3/2} \xi_{\sigma} }{\zeta_{\sigma}}$. Assuming
that $\xi_{\alpha}$, $\zeta_{\alpha}$, $\mu_0$ and $<T>$ are all of
order unity, a supersymmetry breaking scale $M_S\sim 1\,\mbox{TeV}$
can only be obtained for extremely small values of $\varkappa\simeq 10^{-15}$.
In the second minimum, with $z=0$, the superpotential of the hidden sector
vanishes and local SUSY remains intact, so that the low--energy limit of
this theory is described by a pure SUSY model in flat Minkowski space.
If the high order terms $c_n z^n$ are present in Eqs.~(\ref{191}),
the scalar potential of the hidden sector may have many degenerate vacua,
with broken and unbroken supersymmetry, in which the vacuum energy
density vanishes.
In the first vacuum, where the gravitino gains a non-zero mass $m_{3/2}$,
the breakdown of electroweak (EW) symmetry can also be achieved, if the
Higgs sector involves two Higgs doublets ($H_u$ and $H_d$) and a
singlet field $S$. The most general superpotential that describes
the interactions of the Higgs superfields at low energies can be
written as
\begin{equation}
W_{H}=\lambda S (H_u H_d) - \dfrac{\sigma}{3} S^3 - \mu (H_u H_d)
- \dfrac{\mu'}{2} S^2.
\label{8}
\end{equation}
We assume that the Lagrangian of the considered model is invariant under
permutation symmetry $H_u\leftrightarrow H_d$ so that the soft scalar
masses $m_{H_u}^2$ and $m_{H_d}^2$ are equal, i.e. $m_{H_u}=m_{H_d}=m_H$.
Then the scalar potential (\ref{193}) has a global minimum with zero
vacuum energy density at which the EW symmetry is broken by
the vacuum expectation values (VEVs) of the Higgs fields
\begin{equation}
<S>=s=\dfrac{\mu-m_H}{\lambda}\,,\qquad\quad
<H_u>=<H_d>=\pm\sqrt{\dfrac{\sigma s^2 + \mu' s - m_S s}{\lambda}}\,,
\label{9}
\end{equation}
where $m_S$ is the soft scalar mass of the singlet field $S$. The scalar
potential also has another minimum with the same energy density, where
the Higgs field VEVs all vanish and in which the EW symmetry remains intact.
In the supersymmetric vacuum $\mu$, $\mu'$, $m_H$ and $m_S$ vanish and
the Higgs fields do not acquire VEVs, so that EW symmetry does not get broken.
Thus we find that there are three degenerate vacua: the supersymmetric
Minkowski vacuum and two with SUSY broken in the hidden sector, one of
which has unbroken EW symmetry and the other has broken EW symmetry as
in the physical vacuum. As a consequence the MPP conditions are fulfilled
automatically without any extra fine-tuning at the tree--level.
It is worth emphasizing here that, even after the EW symmetry breaking,
the total vacuum energy density remains zero at the minimum where SUSY is
broken so that vacua with broken and unbroken supersymmetry are degenerate.
Thus, in the considered case, the breakdown of dilatation invariance provides
a mechanism that leads to the cancellation of the contribution to the
total vacuum energy density arising from the breakdown of the EW symmetry.
At the same time this breakdown of dilatation invariance also ensures the
cancellation of the vacuum energy density which originates from the breakdown
of local supersymmetry.
Of course, this remarkable cancellation takes place at the tree level only.
The inclusion of perturbative and non--perturbative corrections to the considered
SUGRA Lagrangian, which should depend on the structure of the underlying theory,
are expected to spoil the degeneracy of vacua inducing a huge energy density
in the vacuum where SUSY is broken. Moreover the considered SUGRA model has
other serious shortcomings as well. In particular, the mechanism for the
stabilization of the vacuum expectation value of the hidden sector field $T$
remains unclear. As a result the gravitino mass (see Eq.~(\ref{7})) and
the supersymmetry breaking scale are not fixed in the vacuum where SUSY is broken.
Therefore the SUGRA model discussed above should be considered as a toy example.
This model demonstrates that, in $(N=1)$ supergravity, there might be a mechanism
which ensures the cancellation of different contributions to the total vacuum
energy density in the physical vacuum. This mechanism may also lead to a set
of degenerate vacua with broken and unbroken supersymmetry,
resulting in a natural realization of the multiple point principle.
\section{Split SUSY and the Landscape}
\label{landscape}
In the present section we want to discuss the Split SUSY scenario,
which we shall use in section 4 for our estimate of the cosmological
constant.
For this estimation it is crucial that we assume that the
gauge couplings $1/g_a^2= Re f_a(T,\, z)$ are the same in the phase
in which we live and in the supersymmetric phase. This assumption is of course
only justified if the kinetic functions $f_a(T,\, z)$ are approximately
constant. Due to the
mild dependence of $f_a(T,\, z)$ on $T$ and $z$ the derivatives of the gauge
kinetic function tend to be small. As a result the gauginos are typically
substantially lighter than the scalar particles in the considered SUSY
models, i.e. $M_a\ll m_{\alpha}$. Such a hierarchical structure of the
particle spectrum naturally appears in models with Split Supersymmetry
\footnote{It is worth noting that it seems to be rather problematic to
generate a large mass hierarchy between the gauginos and sfermions within
the toy SUGRA models discussed in section 2. Indeed, even in the limit,
when $f_a(T,\, z)=const$, the non--zero gaugino masses are expected to get
induced \cite{AMSB}. Therefore, in our case an additional fine-tuning is
probably required to ensure that $\frac{M_a}{m_{\alpha}}\ll 10^{-2}$.}.
In Split SUSY one gives up on the naturalness criterion in dealing with
the electroweak scale \cite{ArkaniHamed:2004fb}-\cite{Giudice:2004tc}.
The corresponding idea has its origin in the observation that all known
models of electroweak symmetry breaking, including supersymmetric ones,
require an incredible amount of fine-tuning of the vacuum energy, such that
the resulting cosmological constant is as small as observed. This amount of
fine-tuning is much more severe than what is required to make the Higgs
boson mass free of quadratic divergence. This suggests the possibility that
our notions of naturalness might be misleading, and that other fine-tuning
mechanisms may be at work in nature.
In the Split SUSY scenario supersymmetry is not used to stabilize the weak
scale. This stabilization is supposed to be provided by some other
fine-tuning mechanism, which anyway is needed to explain the value of
the cosmological constant. Therefore, in the
considered models, the SUSY breaking scale $M_S$ is taken to be much above
10 TeV. All scalar particles acquire masses at this high scale $M_S$, except
for a single neutral Higgs boson, whose mass is either finely-tuned or is
preserved by some other mechanism. The gauginos and Higgsinos of this theory
are chosen to lie near the TeV scale, so as to ensure gauge coupling unification at
$M_{GUT}\sim 10^{16}\,\mbox{GeV}$. Indeed, if all sfermions are heavy, the
pattern of grand unification is unchanged as compared with the MSSM, since
heavy sfermions form complete $SU(5)$ representations \cite{Giudice:2004tc}.
Such a modification of the MSSM spectrum does not necessarily affect the mass
parameters of gauginos and Higgsinos, which can be protected by a combination
of an R-symmetry and Peccei Quinn symmetry, and thus they can have weak-scale masses.
Hence, a TeV-scale lightest neutralino can be an appropriate cold dark matter
candidate \cite{Giudice:2004tc}-\cite{dm-split-susy}. A Split SUSY spectrum
naturally arises in frameworks where the mechanism of SUSY breaking preserves
an R-symmetry and/or forbids Gaugino and Higgsino mass terms \cite{ArkaniHamed:2004fb},
\cite{ArkaniHamed:2004yi}, \cite{split-susy-spectrum}.
Thus the Split SUSY scenario retains the successes of the MSSM, aside from the
tuning of the light Higgs mass. At the same time some flaws inherent to the MSSM
elegantly disappear, when the scalar superpartners decouple. The ultra-heavy scalars,
whose masses are a priori undetermined and can in principle range from hundreds
of TeV up to $10^{13}\,\mbox{GeV}$ \cite{Giudice:2004tc}, guarantee the absence
of large supersymmetric flavor changing interactions and CP violation. The
generic constraints from flavour and electric dipole moment data, which set
$M_S>100-1000\,\mbox{TeV}$, are easily satisfied within this class of models.
The dimension-five operators which mediate proton decay are also suppressed,
delaying proton decay which now occurs via dimension-six operators.
The increase in the predicted proton lifetime is also in better
agreement with data \cite{Murayama:2001ur}.
Nevertheless, since the sfermions are ultra-heavy, the large quadratic corrections to
the mass of the Higgs are not cancelled in the manner present in weak-scale
supersymmetry. As a consequence, in Split SUSY the Higgs sector remains
extremely fine-tuned, often with the understanding that this fine-tuning
could be resolved by some anthropic-like selection effects \cite{Weinberg:1987dv}.
In other words the solution to both the hierarchy and cosmological constant
problems might not involve natural cancellations, but follow from a
completely different reasoning, such as the idea that galaxy and
star formation, chemistry and biology, are simply impossible without these
scales having the values found in our Universe
\cite{Weinberg:1987dv}-\cite{anthropic-principle}. In this case SUSY may have
nothing to do with the naturalness problem, although it may be a necessary
ingredient in a fundamental theory of nature such as String Theory.
It has been argued that String Theory can have an enormous number of long-lived metastable
vacua \cite{landscape}-\cite{Susskind:2004uv}. The space of such string theory vacua
is called the ``landscape". The number of discrete vacua in String Theory is measured
not in the millions or billions but in googles ($\sim 10^{100}$)
\cite{Bousso:2000xa}-\cite{Susskind:2004uv}. Recent developments in String Theory
applied a statistical approach to the large multitude of universes, corresponding to
the ``landscape" of vacua present in the theory
\cite{large-number-of-vacua}-\cite{high-energy-susy-breaking}.
These investigations indicate that among the vast number of metastable vacua, there
can be a small subset exhibiting low scale SUSY breaking. However the fine-tuning
required to achieve a small cosmological constant implies the need for a much larger
number of such vacua. Remarkably, the total number of vacua in String Theory can
be large enough to fine-tune both the cosmological constant and the Higgs mass,
favoring a high-scale breaking of supersymmetry
\cite{high-energy-susy-breaking}-\cite{Susskind:2004uv}. It is thus statistically
feasible in String Theory for us to live in a universe fine-tuned in the way we
find it, thereby having both a small cosmological constant and the electroweak scale
stabilized in the 100 GeV range. In the vast landscape of possible string theory vacua,
we may find ourselves in the observed ground state simply because of a cosmic
selection rule, i.e. the anthropic principle \cite{Weinberg:1987dv}.
The idea of the multiple point principle and the landscape paradigm have at least two
things in common. Both approaches imply the existence of a large number of vacua with
broken and unbroken supersymmetry. The landscape paradigm and MPP also imply that the
parameters of the theory, which leads to the Standard Model (SM) at low energies, can be extremely
fine-tuned so as to ensure a tiny vacuum energy density and a large hierarchy between $M_{Pl}$
and the electroweak scale. Moreover the MPP assumption may originate from the landscape of
string theory vacua, if all vacua with a large cosmological constant are forbidden
for some reason so that all the allowed string vacua, with broken and unbroken supersymmetry,
are approximately degenerate. If this is the case, then the breaking of supersymmetry at
high scales is probably still favored. Therefore our attempt to estimate the value of
the cosmological constant within the Split SUSY scenario assuming the degeneracy of
vacua with broken and unbroken supersymmetry might not be so inconsistent with the
string landscape picture.
If one assumes that for some reason numerically large cosmological constants get
forbidden, the landscape model leads to what we might call a landscape and forbiddenness
based MPP. In fact only a narrow band of values around zero cosmological constant would
then be allowed and the surviving vacua would obey MPP to the accuracy of the width
$w$ of this remaining band. However, this landscape and forbiddenness based MPP
would never be sufficiently accurate to become relevant for the main point of
the present article, according to which MPP ``transfers'' the cosmological
constant $\Lambda_{2}$ of the supersymmetric ``second vacuum'' to the physical
vacuum.
\section{Cosmological constant in Split SUSY scenario}
\label{cosmological}
In section 2 we argued that the supersymmetric Minkowski and physical vacua
can be degenerate in SUGRA models. We now try to
estimate the value of the cosmological constant in such models.
Thus we assume that the MPP is realized in Nature, leading to the formation
of a set of degenerate vacua with broken and unbroken local supersymmetry.
In the SUGRA models considered the low energy limit
of the theory in the vacuum with unbroken local supersymmetry is
described by a SUSY model in flat Minkowski space. So the
vacuum energy density of the corresponding supersymmetric states vanishes.
Because all vacua in the MPP inspired SUGRA models are
degenerate, the cosmological constant problem is thereby solved to first
approximation by assumption.
However the value of the cosmological constant may differ from zero in the
considered models. This occurs if non--perturbative effects in the ``observable
sector'' of the supersymmetric phase give rise to the breakdown of SUSY
at low energies. Then the MPP assumption implies that the physical
phase, in which local supersymmetry is broken in the hidden sector, has
the same energy density as a second phase, where supersymmetry
breakdown takes place in the observable sector.
The non-perturbative effects in the simplest SUSY models, like the minimal
supersymmetric standard model (MSSM), are extremely weak. Our strategy is to
estimate these effects in the vacuum with unbroken local supersymmetry
(the second vacuum) and thereby estimate the energy density in
this phase. This value of the cosmological constant
can then be interpreted as the physical value in our phase, by virtue of
the MPP.
If supersymmetry breaking takes place in the second vacuum, it is caused by the strong interactions. We assume the gauge couplings at high energies are
identical in both vacua. Consequently the renormalization group running of
the gauge couplings down to the scale
$M_S$, corresponding to the SUSY breaking scale in the physical vacuum,
are also the same in both vacua \footnote{ The gauge couplings
obey the renormalization group equations
$\displaystyle\frac{d\log{\alpha_i(Q)}}{d\log{Q^2}}=\frac{b_i\alpha_i(Q)}{4\pi}$,
where $\alpha_i(Q)=g_i^2(Q)/(4\pi)$.}. Below the scale $M_S$ all
squarks and sleptons in the physical vacuum decouple and the corresponding beta functions
change. At the TeV scale, the beta functions in the physical vacuum change once again because of
the gradual decoupling of the gluino, neutralino and chargino. Using the value of
$\alpha^{(1)}_3(M_Z)\approx 0.1184\pm 0.0007$ and the matching condition
$\alpha^{(2)}_3(M_S)=\alpha^{(1)}_3(M_S)$, one finds the value of the strong coupling
in the second vacuum:
\begin{equation}
\displaystyle\frac{1}{\alpha^{(2)}_3(M_S)}=\displaystyle\frac{1}{\alpha^{(1)}_3(M_Z)}-
\frac{\tilde{b}_3}{4\pi}\ln\frac{M^2_{g}}{M_Z^2}-\frac{b'_3}{4\pi}\ln\frac{M^2_{S}}{M_g^2}\, .
\label{22}
\end{equation}
In Eq.(\ref{22}) $\alpha^{(1)}_3$ and $\alpha^{(2)}_3$ are the values of the strong gauge
couplings in the physical and second vacua, $M_g$ is the mass of the gluino, while $\tilde{b}_3=-7$
and $b'_3=-5$ are the one--loop beta functions of the strong gauge coupling in the SM and
Split SUSY scenario respectively.
The particles of the MSSM are, of course, all massless in the second supersymmetric phase,
where the EW symmetry is unbroken. So,
in the second vacuum, the renormalization group (RG) flow of the strong gauge coupling is determined
by the corresponding MSSM beta function, which exhibits asymptotically free behaviour ($b_3=-3$).
As a consequence $\alpha^{(2)}_3(Q)$ increases in the infrared region. Moreover the top quark
is massless in the second vacuum and the top quark Yukawa coupling grows in the infrared
with the increasing of $\alpha^{(2)}_3(Q)$. At the scale
\begin{equation}
\Lambda_{SQCD}=M_{S}\exp\left[{\frac{2\pi}{b_3\alpha_3^{(2)}(M_{S})}}\right]\,,
\label{23}
\end{equation}
where the supersymmetric QCD interactions become strong in the second vacuum, the
top quark Yukawa coupling is of the same order of magnitude as the strong gauge coupling.
The large Yukawa coupling of the top quark may result in the formation of a quark
condensate\footnote{Two of us have speculated \cite{nbs} that such a new phase might even
exist in the SM with a condensate of a 6 top and 6 antitop quark bound state.}
that breaks supersymmetry, inducing a non--zero positive value for the cosmological constant
\begin{equation}
\Lambda \simeq \Lambda_{SQCD}^4\, .
\label{24}
\end{equation}
In Fig.~1 the dependence of $\Lambda_{SQCD}$ for the second vacuum on the SUSY breaking
scale $M_S$ in the physical vacuum is examined.
In our numerical analysis we set $M_g=500\,\mbox{GeV}$. Because $b_3 > b'_3$ the
QCD gauge coupling below $M_S$ is larger in the physical vacuum than in the second vacuum.
Therefore the value of $\Lambda_{SQCD}$ is much lower than the QCD scale in the Standard
Model and diminishes with increasing $M_S$. When $M_S$ is of the order of 1 TeV, we obtain
$\Lambda_{SQCD}=10^{-26}M_{Pl} \simeq 100\,\mbox{eV}$. This leads to a vacuum energy density
($\Lambda\simeq 10^{-104} M_{Pl}^4$) which is much smaller than the electroweak scale
contribution in our vacuum $v^4 \simeq 10^{-62}M_{Pl}$. From Fig.~1 one can see that the
measured value of the cosmological constant is reproduced when $M_S\sim 10^{10}\,\mbox{GeV}$.
The value of the SUSY breaking scale, which leads to a reasonable agreement with the
observed vacuum energy density, depends on $\alpha_3(M_Z)$ and the gluino mass. As one can
see from Fig.~1, the dependence of $M_S$ on $\alpha_3(M_Z)$ is rather weak. With increasing
gluino mass the value of the SUSY breaking scale, which results in an appropriate value of the
cosmological constant, decreases. The results of our numerical analysis indicate that,
for $\alpha_3(M_Z)=0.116-0.121$ and $M_g=500-2500\,\mbox{GeV}$, the value of $M_S$ varies from
$2\cdot 10^9\,\mbox{GeV}$ up to $3\cdot 10^{10}\,\mbox{GeV}$.
The obtained prediction for the supersymmetry breaking scale can be tested. A striking
feature of the Split SUSY model is the extremely long lifetime of the gluino. In the
considered case, the gluino decays through a virtual squark to a quark antiquark pair
and a neutralino $\tilde{g} \rightarrow q\bar{q}+\chi_1^0$.
The large mass of the squarks then implies a long lifetime for the gluino. This lifetime
is given approximately by \cite{Dawson:1983fw}-\cite{Hewett:2004nw}
\begin{equation}
\tau \sim 8\biggl(\displaystyle\frac{M_S}{10^9\,\mbox{GeV}}\biggr)^4
\biggl(\displaystyle\frac{1\,\mbox{TeV}}{M_g}\biggr)^5\,s.
\label{241}
\end{equation}
When $M_S\gtrsim 10^{13}\,\mbox{GeV}$ the gluino lifetime becomes larger than the age of
the Universe. So long-lived gluinos would have been copiously produced during the very
early epochs of the Big Bang. They would have survived annihilation and would subsequently have
been confined in nuclear isotopes, which should be present in terrestrial matter \cite{43}.
Because the presence of such stable relics are ruled out by different experiments
(for $M_g\lesssim 10\,\mbox{TeV}$) \cite{42}, the supersymmetry breaking scale in the Split
SUSY scenario should not exceed $10^{13}\,\mbox{GeV}$ \cite{Giudice:2004tc}.
If, as is predicted, the SUSY breaking scale varies from $2\cdot 10^9\,\mbox{GeV}$
($M_g=2500\,\mbox{GeV}$) to $3\cdot 10^{10}\,\mbox{GeV}$ ($M_g=500\,\mbox{GeV}$),
then the gluino lifetime changes from $1\,\mbox{sec.}$ to $2\cdot 10^8\,\mbox{sec.}$
($1000\,\mbox{years}$). Thus the measurement of the gluino lifetime will allow an estimate
to be made of the value of $M_S$ in the Split SUSY model. The experimental signatures of long-lived
gluinos at colliders and in cosmic rays within the Split Supersymmetry scenario were explored
in \cite{Hewett:2004nw},\cite{split-susy-gluino}.
The observed value of the cosmological constant can also be reproduced for a much lower
supersymmetry breaking scale, if the MSSM particle content is supplemented by an
additional pair of $5+\bar{5}$ supermultiplets. The extra bosons and fermions would not affect
gauge coupling unification, because they form complete representations of $SU(5)$
(see for example \cite{29}). In the physical vacuum new bosonic states gain masses
around the supersymmetry breaking scale, while their fermionic partners survive to low
energies. In our numerical studies we assume that the masses of any new quarks are of the order
of the gluino mass, i.e. we set $M_q\simeq M_g\simeq 500\,\mbox{GeV}$. In the supersymmetric
phase the new bosons and fermions remain massless. The extra fermionic states do not much affect
the RG flow of gauge couplings in the Split SUSY scenario. For example, the one--loop beta
function that determines the running of the strong gauge coupling from the SUSY breaking
scale down to the TeV scale changes from $-5$ to $-13/3$. At the same time, the extra $5+\bar{5}$
supermultiplets give a considerable contribution to $b_3$ in the supersymmetric phase.
The corresponding one--loop beta function becomes $b_3=-2$. As a result $\alpha_3(Q)$ and
$\Lambda_{SQCD}$ decrease. In this case the observed value of the cosmological constant
can be reproduced even for $M_S\simeq 1\,\mbox{TeV}$ (see Fig.~1). However the Split
SUSY scenario has the advantage of avoiding the need for any new particles beyond those of
the MSSM, provided that $M_S\simeq 10^{10}\,\mbox{GeV}$.
\begin{figure}
\hspace*{-0cm}{\Large $\log[\Lambda_{SQCD}/M_{Pl}]$}\\[1mm]
\includegraphics[height=95mm,keepaspectratio=true]{lambda-split-susy-2.eps}
\hspace*{7.5cm}{\Large $\log[M_S]$}\\[3mm]
\caption{The value of $\log\left[\Lambda_{SQCD}/M_{Pl}\right]$ versus $\log M_S$
for $M_q=M_g=500\,\mbox{GeV}$. The thin and thick solid lines correspond to the
Split SUSY scenarios with the pure MSSM particle content and the MSSM particle
content supplemented by an additional pair of $5+\bar{5}$ multiplets respectively.
The dashed and dash--dotted lines represent the uncertainty in $\alpha_3(M_Z)$.
The thin and thick solid lines are obtained for $\alpha_3(M_Z)=0.1184$,
the upper (lower) dashed and dash-dotted lines correspond to
$\alpha_3(M_Z)=0.116$ ($\alpha_3(M_Z)=0.121$). The horizontal line represents the
observed value of $\Lambda^{1/4}$. The SUSY breaking scale $M_S$ is measured in GeV.}
\label{essmfig1}
\end{figure}
\section{Conclusions}
\label{conclusion}
In the present article we have tried to estimate the value of the
cosmological constant in the Split Supersymmetry model, assuming
the existence of a set of degenerate vacua with broken and unbroken SUSY.
Using a sufficiently accurate principle of these vacua being degenerate
(MPP), a small cosmological constant is calculated in one of the vacua
and transferred to the other ones, especially to the physical one in
which we live. The idea is that we can calculate the cosmological
constant in the supersymmetric Minkowski (second) vacuum, having unbroken
local supersymmetry up to non-perturbative corrections coming from the
supersymmetric QCD scale in that vacuum. With reasonable Split SUSY
parameters - a SUSY breaking scale of $M_S \sim 10^{10}$ GeV - we obtain
the observed value of the cosmological constant.
We have argued that a set of such degenerate vacua can appear in $N=1$ supergravity
models. In general the presence of degenerate vacua in these models requires
an extra fine-tuning. Moreover an extra fine--tuning is normally needed to
ensure the existence of the vacuum in which the low--energy limit of the
considered theory is described by a pure supersymmetric model in flat
Minkowski space. The fine--tuning mentioned above can be alleviated within
the no--scale inspired SUGRA models in which the global symmetries,
that protect a zero value for the cosmological constant in all vacua and
preserve local supersymmetry, can be broken so that the MPP conditions are
fulfilled automatically without any extra fine-tuning at the tree--level.
In these models there can be a vacuum with softly broken supersymmetry
and spontaneously broken EW symmetry in the observable sector just like
in the physical vacuum in which we live. We demonstrated that the contributions
to the vacuum energy density, which originate from the breakdown of supersymmetry
and EW symmetry breaking in the physical vacuum, might get cancelled at the tree
level in these models.
However the inclusion of perturbative and non-perturbative corrections would
spoil the degeneracy of vacua. As a consequence an enormous fine tuning is
required, in general, to keep the vacuum energy densities in the vacua with
broken and unbroken supersymmetry to be the same. Nevertheless for our estimation
of the dark energy density we adopt such a fine-tuning assumption.
In other words we assume that there might be a mechanism (approximate global
symmetry, MPP or something else) in Nature that guarantees the
degeneracy of the vacua mentioned above. Moreover we restrict our consideration
to the simplest SUGRA models in which $f_a(T,\, z)\approx const$. In this limit
the gauginos are much lighter than squarks and sleptons, which is a
characteristic feature of the Split SUSY type spectrum.
The landscape of the metastable string theory vacua favors high-scale
SUSY breaking which could lead to Split SUSY. The enormous number of vacua
in String Theory can also lead to a large set of approximately degenerate
vacua, with broken and unbroken supersymmetry and tiny values of the vacuum
energy densities, if the vacua with large cosmological constants are forbidden.
Thus the Split SUSY scenario, landscape paradigm and the multiple point principle
might be different parts of the same picture. This landscape based MPP is
however not sufficiently accurate for the main calculation in this paper.
The main calculation in the present article concerns two of the several
vacua degenerate according to the MPP assumption. The first one is the
physical vacuum in which we live and in which we assume the physics is
described by the Split SUSY scenario. The second vacuum is one with unbroken
supersymmetry to first approximation. At high energy the physics is the same
in both these vacua, so that especially the running gauge couplings are equal
above the SUSY breaking scale $M_S$ in the physical vacuum. Also, in the second
vacuum the QCD ($SU(3)_C$) gauge coupling and the top quark Yukawa coupling
become large towards the infrared. At the scale $\Lambda_{SQCD}$ a top quark
condensate may get formed, giving rise to the breakdown of SUSY in
the supersymmetric phase and resulting in a non-zero and positive vacuum
energy density. The corresponding energy density is determined by the value
of the $SU(3)_C$ gauge coupling in the physical vacuum at high energies.
In order to calculate this energy density from the experimentally known
QCD coupling, the only unknown parameter is the SUSY breaking scale $M_S$
in the physical vacuum. For any value of
$M_S$ the calculated vacuum energy density in the supersymmetric phase
is rather small ($\lesssim 10^{-100}\,M_{Pl}^4$). The measured value of the
cosmological constant can be reproduced when $M_S\sim 10^{9}-10^{10}\,\mbox{GeV}$,
which leads to extremely long-lived gluinos. The gluino lifetime varies
from $1\,\mbox{sec.}$ to $2\cdot 10^8\,\mbox{sec.}$ This prediction can be
tested if such long-lived gluinos are observed in future experiments.
Previously \cite{Froggatt:2005nb} we showed that a reasonable value for
the dark energy density can be obtained even for $M_S\sim 1\,\mbox{TeV}$
if the MSSM particle content is supplemented by an extra pair of $5+\bar{5}$
supermultiplets. The scenario with extra $5+\bar{5}$ multiplets of matter and
the supersymmetry breaking scale in the $\mbox{TeV}$ range can also be tested
at the LHC in the near future. But with the Split SUSY scenario we avoid these
ad hoc $5+\bar{5}$ particles.
\section*{Acknowledgements}
R.N. would like to thank M.~Drees, K.~R.~Dienes, D.~Gorbunov, D.~R.~T.~Jones,
S.~F.~King, J.~P.~Kumar, S.~Pakvasa, D.~G.~Sutherland, X.~R.~Tata, B.~Thomas
and M.~I.~Vysotsky for fruitful discussions. The work of R.N. was supported
by the U.S. Department of Energy under Contract DE-FG02-04ER41291.
Also C.D.F. would like to acknowledge support from STFC in UK.
\newpage
|
1,116,691,500,826 | arxiv | \section{Introduction}
The non-strictly thermal spectrum by Parikh and Wilczek \cite{key-1,key-2}
of Hawking radiation \cite{key-3} implies that emissions of subsequent
Hawking quanta are countable \cite{key-4,key-5,key-17,key-20,key-21,key-22,key-23,key-24},
and, in turn, generates a natural correspondence between Hawking radiation
and BH QNMs \cite{key-4,key-5,key-17}, permitting to naturally interpret
QNMs as quantum levels \cite{key-4,key-5,key-17}. In fact, Parikh
and Wilczek \cite{key-1,key-2} obtained their important result in
the tunnelling framework, which has been an elegant and largely used
approach to obtain Hawking radiation in recent years, see for example
\cite{key-1,key-2,key-19,key-28,key-29,key-30,key-31} and refs. within.
In the tunnelling framework, Hawking's mechanism of particles creation
by BHs \cite{key-3} can be described as tunnelling arising from vacuum
fluctuations near the BH horizon \cite{key-1,key-2,key-19,key-28,key-29,key-30,key-31}.
If a virtual particle pair is created just inside the horizon, the
virtual particle with positive energy can tunnel out. Then, it materializes
outside the BH as a real particle. Analogously, if a virtual particle
pair is created just outside the horizon, the particle with negative
energy can tunnel inwards. In both of the situations, the BH absorbs
the particle with negative energy. The absorptions of such particles
with negative energy are exactly the perturbations generating the
QNMs. The result will be that the BH mass decreases and the particle
with positive energy propagates towards infinity. Thus, subsequent
emissions of quanta appear as Hawking radiation. QNM can be naturally
considered in terms of quantum levels if one interprets the absolute
value of a QNM frequency as the total energy emitted at that level
\cite{key-4,key-5,key-17}. In other words, QNMs frequencies are the
eigenvalues of the system. The Hawking quanta are then interpreted
as the ``jumps'' among the levels. This key point agrees with the
idea that, in an underlying quantum gravity theory, BHs result in
highly excited states.
We recently used this important issue to re-analyse the spectrum of
BH QNMs through the introduction of a BH \emph{effective temperature
}\cite{key-4,key-5}\emph{.} In our analysis the formula of the horizon's
area quantization and the number of quanta of area resulted to be
functions of the quantum QNMs ``overtone'' number $n$ \cite{key-4,key-5},
i.e. of the BH quantum level. Consequently, Bekenstein-Hawking entropy,
its sub-leading corrections and the number of micro-states resulted
functions of $n\;$ too \cite{key-4,key-5}.
Here the analysis is improved, removing some approximations that have
been implicitly used in previous works \cite{key-4,key-5} and obtaining
the corrected expressions for the cited formulas like functions of
$n.$ An approximation concerning the maximum value of $n\;$ is also
corrected. As our previous results \cite{key-4,key-5} were strictly
corrected only for scalar and gravitational perturbations, in this
work we show that the analysis holds also for vector perturbations.
We shortly discuss potential important implications also for the BH
information paradox \cite{key-18}.
The analysis in this paper is totally consistent with the general
conviction that BHs result in highly excited states representing both
the ``hydrogen atom'' and the ``quasi-thermal emission'' in quantum
gravity. Previous results in the literature \cite{key-6,key-14,key-15},
obtained in strictly thermal approximation, are consistent with the
results in this paper.
At the present time, we do not yet have a full theory of quantum gravity.
Thus, we have to be content with the semi-classical approximation.
In fact, as for large $n\:$ \emph{Bohr's correspondence principle}
\cite{key-25,key-26,key-27} holds, such a semi-classical description
is adequate. In this framework, our BH model is somewhat similar to
the semi-classical Bohr's model of the structure of a hydrogen atom
\cite{key-26,key-27}. In our BH model, during a quantum jump a discrete
amount of energy is radiated and for large values of the principal
quantum number $n$ the analysis becomes independent from the other
quantum numbers. In a certain sense, QNMs represent the \textquotedbl{}electron\textquotedbl{}
which jumps from a level to another one and the absolute values of
the QNMs frequencies represent the energy \textquotedbl{}shells\textquotedbl{}.
In Bohr's model \cite{key-26,key-27}, electrons can only gain and
lose energy by jumping from one allowed energy shell to another, absorbing
or emitting radiation with an energy difference of the levels according
to the Planck relation $E=hf$, where $h$ is the Planck constant
and $f$ the transition frequency. In our BH model, QNMs can only
gain and lose energy by jumping from one allowed energy shell to another,
absorbing or emitting radiation (emitted radiation is given by Hawking
quanta) with an energy difference of the levels. On the other hand,
Bohr's model is an approximated model of the hydrogen atom with respect
to the valence shell atom model of full quantum mechanics. In the
same way, our model should be an approximated model of the emitting
BH with respect to the definitive, but at the present time unknown,
model of full quantum gravity theory.
\section{Quasi-normal modes in non-strictly thermal approximation}
\noindent Working with $G=c=k_{B}=\hbar=\frac{1}{4\pi\epsilon_{0}}=1$
(Planck units), in strictly thermal approximation the probability
of emission of Hawking quanta is \cite{key-1,key-2,key-3}
\begin{equation}
\Gamma\sim\exp(-\frac{\omega}{T_{H}}),\label{eq: hawking probability}
\end{equation}
\noindent where $T_{H}\equiv\frac{1}{8\pi M}$ is the Hawking temperature
and $\omega$ the energy-frequency of the emitted radiation.
\noindent The important correction, due to the BH varying geometry
yields \cite{key-1,key-2}
\noindent
\begin{equation}
\Gamma\sim\exp[-\frac{\omega}{T_{H}}(1-\frac{\omega}{2M})].\label{eq: Parikh Correction}
\end{equation}
\noindent This result takes into account the BH back reaction and
adds the term $\frac{\omega}{2M}$ like correction \cite{key-1,key-2}.
In a recent paper \cite{key-19} we have improved the tunnelling picture
in \cite{key-1,key-2}. In fact, we have shown that the probability
of emission (\ref{eq: Parikh Correction}) is indeed associated to
the two distributions \cite{key-19}
\begin{equation}
<n>_{boson}=\frac{1}{\exp\left[-4\pi n\left(M-\omega\right)\omega\right]-1},\;\;<n>_{fermion}=\frac{1}{\exp\left[-4\pi n\left(M-\omega\right)\omega\right]+1},\label{eq: final distributions}
\end{equation}
for bosons and fermions respectively, which are \emph{non} strictly
thermal.
\noindent As in various fields of physics and astrophysics the deviation
of the spectrum of an emitting body from the strict thermality is
taken into account by introducing an \emph{effective temperature}
(i.e. the temperature of a black body emitting the same total amount
of radiation) we introduced the effective temperature in BH physics
too \cite{key-4,key-5}
\noindent
\begin{equation}
T_{E}(\omega)\equiv\frac{2M}{2M-\omega}T_{H}=\frac{1}{4\pi(2M-\omega)}.\label{eq: Corda Temperature}
\end{equation}
\noindent One sees that $T_{E}$ depends on the frequency of the emitted
radiation. Therefore, one can rewrite eq. (\ref{eq: Parikh Correction})
in Boltzmann-like form as \cite{key-4,key-5}
\noindent
\begin{equation}
\Gamma\sim\exp[-\beta_{E}(\omega)\omega]=\exp(-\frac{\omega}{T_{E}(\omega)}),\label{eq: Corda Probability}
\end{equation}
\noindent where $\beta_{E}(\omega)\equiv\frac{1}{T_{E}(\omega)}$
and $\exp[-\beta_{E}(\omega)\omega]$ is the \emph{effective Boltzmann
factor} appropriate for a BH with inverse effective temperature $T_{E}(\omega)$
\cite{key-4,key-5}. The deviation of the BH radiation spectrum from
the strict thermality is given by the ratio $\frac{T_{E}(\omega)}{T_{H}}=\frac{2M}{2M-\omega}$
\cite{key-4,key-5}. The introduction of $T_{E}(\omega)$ permits
the introduction of the \emph{effective mass }and of the \emph{effective
horizon} \cite{key-4,key-5}
\begin{equation}
M_{E}\equiv M-\frac{\omega}{2},\mbox{ }r_{E}\equiv2M_{E}\label{eq: effective quantities}
\end{equation}
\noindent of the BH \emph{during} the emission of the particle, i.e.
\emph{during} the BH contraction phase \cite{key-4,key-5}. They are
average values of the mass and the horizon \emph{before} and \emph{after}
the emission \cite{key-4,key-5}.
\noindent We have shown that the correction to the thermal spectrum
is also very important for the physical interpretation of BH QNMs,
which, in turn, is very important to realize the underlying quantum
gravity theory as BHs represent theoretical laboratories for developing
quantum gravity and BH QNMs are the best candidates like quantum levels
\cite{key-4,key-5,key-6}.
\noindent QNMs are radial spin$-j$ perturbations ($j=0,1,2$ for
scalar, vector and gravitational perturbation respectively) of the
Schwarzschild background usually labelled as $\omega_{nl},$ being
$l$ the angular momentum quantum number \cite{key-4,key-5,key-6,key-7}.
For each $l$$\geq2$ for BH perturbations, there is a countable sequence
of QNMs, labelled by the ``overtone'' number $n$ ($n=1,2,...$)
\cite{key-4,key-5,key-7}. For large $n$ the QNMs of the Schwarzschild
BH become independent of $l$. In strictly thermal approximation and
for scalar and gravitational perturbations their frequencies are given
by \cite{key-4,key-5,key-6,key-7}
\noindent
\begin{equation}
\begin{array}{c}
\omega_{n}=\ln3\times T_{H}+2\pi i(n+\frac{1}{2})\times T_{H}+\mathcal{O}(n^{-\frac{1}{2}})=\\
\\
=\frac{\ln3}{8\pi M}+\frac{2\pi i}{8\pi M}(n+\frac{1}{2})+\mathcal{O}(n^{-\frac{1}{2}}).
\end{array}\label{eq: quasinormal modes}
\end{equation}
\noindent The introduction of the effective temperature $T_{E}(\omega)$
is useful also to analyse BH QNMs \cite{key-4,key-5}. An important
issue is that eq. (\ref{eq: quasinormal modes}) is an approximation
as it assumes that the BH radiation spectrum is strictly thermal.
If one wants to take into account the deviation from the thermal spectrum
the Hawking temperature $T_{H}$ must be replaced by the effective
temperature $T_{E}$ in eq. (\ref{eq: quasinormal modes}) \cite{key-4,key-5}.
Thus, for scalar and gravitational perturbations the correct expression
for the Schwarzschild BH QNMs, which takes into account the non-strict
thermality of the spectrum is \cite{key-4,key-5}
\noindent
\begin{equation}
\begin{array}{c}
\omega_{n}=\ln3\times T_{E}(\omega_{n})+2\pi i(n+\frac{1}{2})\times T_{E}(\omega_{n})+\mathcal{O}(n^{-\frac{1}{2}})=\\
\\
=\frac{\ln3}{4\pi\left[2M-(\omega_{0})_{n}\right]}+\frac{2\pi i}{4\pi\left[2M-(\omega_{0})_{n}\right]}(n+\frac{1}{2})+\mathcal{O}(n^{-\frac{1}{2}})\simeq\frac{2\pi in}{4\pi\left[2M-(\omega_{0})_{n}\right]},
\end{array}\label{eq: quasinormal modes corrected}
\end{equation}
\noindent where $(\omega_{0})_{n}\equiv|\omega_{n}|.$ We derived
eq. (\ref{eq: quasinormal modes corrected}) in \cite{key-4,key-5}.
A more rigorous derivation of it can be found in detail in the Appendix
of this paper. In that Appendix we also show that the behavior
\noindent
\begin{equation}
\omega_{n}\simeq\frac{2\pi in}{4\pi\left[2M-(\omega_{0})_{n}\right]}\label{eq: andamento asintotico}
\end{equation}
also holds for $j=1$ (vector perturbations) and this is in full agreement
with Bohr's correspondence principle \cite{key-25,key-26,key-27}
which states that \emph{``transition frequencies at large quantum
numbers should equal classical oscillation frequencies}'' \cite{key-10}.
\noindent The physical solution for $(\omega_{0})_{n}$ in eq. (\ref{eq: quasinormal modes corrected})
is \cite{key-4,key-5}
\begin{equation}
(\omega_{0})_{n}=M-\sqrt{M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(n+\frac{1}{2})^{2}}}\simeq M-\sqrt{M^{2}-\frac{1}{2}(n+\frac{1}{2})},\label{eq: radice fisica}
\end{equation}
where the term $M-\sqrt{M^{2}-\frac{1}{2}(n+\frac{1}{2})}$ represent
the solution of eq. (\ref{eq: andamento asintotico}), which in turn
hols for $j=0,1,2.$
\section{A note on black hole's remnants}
\noindent As $(\omega_{0})_{n}$ is interpreted like the total energy
emitted at level $n$ \cite{key-4,key-5}, one needs also
\begin{equation}
M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(n+\frac{1}{2})^{2}}\simeq M^{2}-\frac{1}{2}(n+\frac{1}{2})\geq0\label{eq: need}
\end{equation}
in eq. (\ref{eq: radice fisica}). In fact, BHs cannot emit more energy
than their total mass. The expression (\ref{eq: need}) is solved
giving a maximum value for the overtone number $n$ \cite{key-4}
\begin{equation}
n\leq n_{max}=2\pi^{2}\left(\sqrt{16M^{4}-(\frac{\ln3}{\pi})^{2}}-1\right)\simeq2\pi^{2}\pi^{2}(4M^{2}-1),\label{eq: n max}
\end{equation}
which corresponds to $(\omega_{0})_{n_{max}}=M.$ Thus, the countable
sequence of QNMs for emitted energies cannot be infinity although
$n$ can be extremely large \cite{key-4}. On the other hand, we recall
that, by using the Generalized Uncertainty Principle, Adler, Chen
and Santiago \cite{key-8} have shown that the total BH evaporation
is prevented in exactly the same way that the Uncertainty Principle
prevents the hydrogen atom from total collapse. In fact, the collapse
is prevented, not by symmetry, but by dynamics, as the \emph{Planck
distance} and the \emph{Planck mass} are approached \cite{key-8}.
That important result implies that eq. (\ref{eq: need}) has to be
slightly modified, becoming (the \emph{Planck mass} is equal to $1$
in Planck units)
\begin{equation}
M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(n+\frac{1}{2})^{2}}\simeq M^{2}-\frac{1}{2}(n+\frac{1}{2})\geq1.\label{eq: need 1}
\end{equation}
By solving eq. (\ref{eq: need 1}) one gets a different value of the
maximum value for the overtone number $n$
\begin{equation}
n\leq n_{max}=2\pi^{2}\left(\sqrt{16(M^{2}-1)^{2}-(\frac{\ln3}{\pi})^{2}}-1\right)\simeq2\pi^{2}(4M^{2}-5).\label{eq: n max 1}
\end{equation}
The result in eq. (\ref{eq: n max 1}) improves the one of eq. (\ref{eq: n max})
that we originally derived in \cite{key-4}.
\section{Quasi-normal modes like natural quantum levels}
\noindent Bekenstein \cite{key-9} has shown that the Schwarzschild
BH area quantum is $\triangle A=8\pi$ (the \emph{Planck length} $l_{p}=1.616\times10^{-33}\mbox{ }cm$
is equal to one in Planck units). By analysing Schwarzschild BH QNs,
Hod found a different numerical coefficient \cite{key-10}. Hod's
work was refined by Maggiore \cite{key-6}, who re-obtained the original
result by Bekenstein. In \cite{key-4,key-5} we further improved the
result by Maggiore taking into account the deviation from the strictly
thermal feature. In fact, in \cite{key-4,key-5} we used eq. (\ref{eq: quasinormal modes corrected})
instead of eq. (\ref{eq: quasinormal modes}) \cite{key-4,key-5}.
From eq. (\ref{eq: radice fisica}) one gets that an emission involving
$n$ and $n-1$ gives a variation of energy \cite{key-4,key-5}
\begin{equation}
\triangle M_{n}=(\omega_{0})_{n-1}-(\omega_{0})_{n}=-f_{n}(M,n)\label{eq: variazione}
\end{equation}
\noindent where one defines \cite{key-4,key-5}
\noindent
\begin{equation}
\begin{array}{c}
f_{n}(M,n)\equiv\\
\\
\sqrt{M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(n-\frac{1}{2})^{2}}}-\sqrt{M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(n+\frac{1}{2})^{2}}}\simeq\\
\\
\simeq\sqrt{M^{2}-\frac{1}{2}(n-\frac{1}{2})}-\sqrt{M^{2}-\frac{1}{2}(n+\frac{1}{2})}
\end{array}\label{eq: f(M,n)}
\end{equation}
\noindent Together with collaborators, we have recently shown that
the results (\ref{eq: variazione}) (\ref{eq: f(M,n)}) also hold
for Kerr BHs in the case $M^{2}\gg J,$ where $J$ is the angular
momentum of the BH \cite{key-17}.
\noindent Recalling that, as for the Schwarzschild BH the \emph{horizon
area} $A$ is related to the mass through the relation $A=16\pi M^{2},$
a variation $\triangle M\,$ in the mass generates a variation
\noindent
\begin{equation}
\triangle A=32\pi M\triangle M\label{eq: variazione area}
\end{equation}
\noindent in the area. Combining eqs. (\ref{eq: variazione area})
and (\ref{eq: variazione}) one gets \cite{key-4,key-5}
\noindent
\begin{equation}
\triangle A=32\pi M\triangle M_{n}=-32\pi M\times f_{n}(M,n).\label{eq: area quantum}
\end{equation}
\noindent As we consider large $n,\:$ eq. (\ref{eq: f(M,n)}) is
well approximated by $f_{n}(M,n)\approx\frac{1}{4M}$ \cite{key-4,key-5}
and eq. (\ref{eq: area quantum}) becomes $\triangle A\approx-8\pi$
which is the original result by Bekenstein for the area quantization
(a part a sign because we consider an emission instead of an absorption).
Then, only when $n$ is enough large the levels are approximately
equally spaced \cite{key-4,key-5}. Instead, for smaller $n\,$ there
are deviations.
\noindent One assumes that, for large $n$, the horizon area is quantized
\cite{key-4,key-5,key-6} with a quantum $|\triangle A|.$ The total
horizon area must be $A=N|\triangle A|$ where the integer $N$ is
the number of quanta of area. One gets \cite{key-4,key-5}
\noindent
\begin{equation}
N=\frac{A}{|\triangle A|}=\frac{16\pi M^{2}}{32\pi M\cdot f_{n}(M,n)}=\frac{M}{2f_{n}(M,n)}.\label{eq: N}
\end{equation}
This permits to write the famous formula of Bekenstein-Hawking entropy
\cite{key-3,key-9} like a function of the quantum overtone number
$n$ \cite{key-4,key-5}
\noindent
\begin{equation}
S_{BH}=\frac{A}{4}=8\pi NM\cdot f_{n}(M,n).\label{eq: Bekenstein-Hawking}
\end{equation}
\noindent As we consider large $n,$ the approximation $f_{n}(M,n)\approx\frac{1}{4M}\:$
permits to re-obtain the standard result \cite{key-6,key-11,key-12,key-13}
\noindent
\begin{equation}
S_{BH}\rightarrow2\pi N,\label{eq: entropia circa}
\end{equation}
\noindent like good approximation. Recent results show that that BH
entropy contains three parts which are important to realize the underlying
quantum gravity theory. They are the Bekenstein-Hawking entropy and
two sub-leading corrections: the logarithmic term and the inverse
area term \cite{key-14,key-15}
\noindent
\begin{equation}
S_{total}=S_{BH}-\ln S_{BH}+\frac{3}{2A}.\label{eq: entropia totale}
\end{equation}
\noindent Thus, eq. (\ref{eq: Bekenstein-Hawking}) permits to re-write
eq. (\ref{eq: entropia totale}) like \cite{key-4,key-5}
\noindent
\begin{equation}
S_{total}=8\pi NM\cdot f_{n}(M,n)-\ln\left[8\pi NM\cdot f_{n}(M,n)\right]+\frac{3}{64\pi NM\cdot f_{n}(M,n)}\label{eq: entropia totale 2}
\end{equation}
\noindent that, as one considers large $n,\:$ is well approximated
by \cite{key-4,key-5}
\noindent
\begin{equation}
S_{total}\simeq2\pi N-\ln2\pi N+\frac{3}{16\pi N}.\label{eq: entropia totale approssimata}
\end{equation}
\noindent Thus, at a level $n-1,$ the BH has a number of micro-states
\cite{key-4,key-5}
\noindent
\begin{equation}
g(N)\propto\exp\left\{ 8\pi NM\cdot f_{n}(M,n)-\ln\left[8\pi NM\cdot f_{n}(M,n)\right]+\frac{3}{64\pi NM\cdot f_{n}(M,n)}\right\} ,\label{eq: microstati}
\end{equation}
\noindent that, for large $n,\:$ is well approximated by \cite{key-4,key-5}
\noindent
\begin{equation}
g(N)\propto\exp\left[2\pi N-\ln\left(2\pi N\right)+\frac{3}{16\pi N}\right].\label{eq: microstati circa}
\end{equation}
Eqs. (\ref{eq: entropia circa}), (\ref{eq: entropia totale approssimata})
and (\ref{eq: microstati circa}) are in agreement with previous literature
\cite{key-6,key-14,key-15}, in which the strictly thermal approximation
were used.
Actually, in previous discussion, and hence also in \cite{key-4,key-5},
we used an implicit simplification. Now, we improve the analysis by
removing such a simplification and by giving the correct results.
In fact, we note that, after an high number of emissions (and potential
absorptions as the BH can capture neighboring particles), the BH mass
changes from $M\;$ to
\begin{equation}
M_{n-1}\equiv M-(\omega_{0})_{n-1},\label{eq: me-1}
\end{equation}
where $(\omega_{0})_{n-1}$ is the total energy emitted by the BH
at that time, and the BH is excited at a level $n-1$. In the transition
from the state with $n-1$ to the state with $n\;$ the BH mass changes
again from $M_{n-1}$ to
\begin{equation}
M_{n}\equiv M-(\omega_{0})_{n-1}+\triangle M_{n},\label{eq: me}
\end{equation}
which, by using eq. (\ref{eq: variazione}), becomes
\begin{equation}
\begin{array}{c}
M_{n}=M-(\omega_{0})_{n-1}-f_{n}(M,n)=\\
\\
=M-(\omega_{0})_{n-1}+(\omega_{0})_{n-1}-(\omega_{0})_{n}=M-(\omega_{0})_{n}.
\end{array}\label{eq: me 2}
\end{equation}
Now, the BH is excited at a level $n$. By considering eq. (\ref{eq: radice fisica}),
eqs. (\ref{eq: me-1}) and (\ref{eq: me 2}) read
\begin{equation}
M_{n-1}=\sqrt{M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(n-\frac{1}{2})^{2}}}\simeq\sqrt{M^{2}-\frac{1}{2}(n-\frac{1}{2})}\label{eq: me 3}
\end{equation}
and
\begin{equation}
M_{n}=\sqrt{M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(n+\frac{1}{2})^{2}}}\simeq\sqrt{M^{2}-\frac{1}{2}(n+\frac{1}{2})}.\label{eq: me 4}
\end{equation}
For extremely large $n\quad$ the condition $M_{n}\simeq M$, that
we implicitly used in previous discussion, does not hold because the
BH has emitted a large amount of mass. This implies that, if one uses
eqs. (\ref{eq: radice fisica}) and eq. (\ref{eq: me 3}), eq. (\ref{eq: area quantum})
has to be correctly rewritten as
\begin{equation}
\triangle A_{n-1}\equiv32\pi M_{n-1}\triangle M_{n}=-32\pi M_{n-1}\times f_{n}(M,n)\label{eq: area quantum e}
\end{equation}
This equation should give the area quantum of an excited BH for an
emission from the level $n-1$ to the level $n$ in function of the
quantum number $n$ and of the initial BH mass. Actually, there is
a problem in eq. (\ref{eq: area quantum e}). In fact, an absorption
from the level $n$ to the level $n-1$ is now possible, with an absorbed
energy \cite{key-5}
\begin{equation}
(\omega_{0})_{n}-(\omega_{0})_{n-1}=f_{n}(M,n)=-\triangle M_{n}.\label{eq: absorbed}
\end{equation}
In that case, the quantum of area should be
\begin{equation}
\triangle A_{n}\equiv-32\pi M_{n}\triangle M_{n}=32\pi M_{n}\times f_{n}(M,n),\label{eq: area quantum a}
\end{equation}
and the absolute value of the area quantum for an absorption from
the level $n$ to the level $n-1$ is different from the absolute
value of the area quantum for an emission from the level $n-1$ to
the level $n\;$ because $M_{n-1}\neq M_{n}.$ Clearly, one indeed
expects the area spectrum to be the same for absorption and emission.
This inconsistency is solved if, once again, one considers the \emph{effective
mass} which correspond to the transitions between the two levels $n\;$
and $n-1$, which is the same for emission and absorption
\begin{equation}
\begin{array}{c}
M_{E(n,\; n-1)}\equiv\frac{1}{2}\left(M_{n-1}+M_{n}\right)=\\
\\
=\frac{1}{2}\left(\sqrt{M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(n-\frac{1}{2})^{2}}}+\sqrt{M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(n+\frac{1}{2})^{2}}}\right)\simeq\\
\\
\simeq\frac{1}{2}\left(\sqrt{M^{2}-\frac{1}{2}(n-\frac{1}{2})}+\sqrt{M^{2}+\frac{1}{2}(n-\frac{1}{2})}\right)
\end{array}.\label{eq: massa effettiva n}
\end{equation}
Replacing $M_{n-1}$ with $M_{E(n,\; n-1)}$ in eq. (\ref{eq: area quantum e})
and $M_{n}$ with $M_{E(n,\; n-1)}$ in eq. (\ref{eq: area quantum a})
we obtain
\begin{equation}
\begin{array}{c}
\triangle A_{n-1}\equiv32\pi M_{E(n,\; n-1)}\triangle M_{n}=-32\pi M_{E(n,\; n-1)}\times f_{n}(M,n)\qquad emission\\
\\
\triangle A_{n}\equiv-32\pi M_{E(n,\; n-1)}\triangle M_{n}=32\pi M_{E(n,\; n-1)}\times f_{n}(M,n)\qquad absorption
\end{array}\label{eq: expects}
\end{equation}
and now one gets $\alpha=|\triangle A_{n}|=|\triangle A_{n-1}|.$
By using eqs. (\ref{eq: f(M,n)}) and (\ref{eq: massa effettiva n})
one finds
\begin{equation}
\begin{array}{c}
\alpha=|\triangle A_{n}|=|\triangle A_{n-1}|\\
\\
=4\left(\sqrt{(\ln3)^{2}+4\pi^{2}(n+\frac{1}{2})^{2}}-\sqrt{(\ln3)^{2}+4\pi^{2}(n-\frac{1}{2})^{2}}\right)\simeq8\pi.
\end{array}\label{eq: 8 pi planck}
\end{equation}
Hence, the introduction of the effective temperature and of the effective
mass does not degrade the importance of the Hawking temperature. Indeed,
the effective temperature and the effective mass are introduced because
the values of the Hawking temperature and of the mass change with
discrete behavior in time. Thus, it is not clear which value of the
Hawking temperature and of the mass have to be associated to the emission
(or to the absorption) of the particle. Has one to consider the values
of the Hawking temperature and of the mass \emph{before} the emission
(absorption) or the values \emph{after} the emission (absorption)?
The answer is that one must consider \emph{intermediate} values, the
effective temperature and the effective mass. In a certain sense,
they represent the values of the BH temperature and BH mass\emph{
during} the emission.
We note that, as we consider large $n$, the result (\ref{eq: 8 pi planck})
is $\simeq8\pi,$ which also holds for vector perturbations. Thus,
one can take the result (\ref{eq: 8 pi planck}) as the quantization
of the area of the horizon of a Schwarzschild BH. Again, the original
famous result by Bekenstein \cite{key-9} is a good approximation
and a confirmation of the correctness of the current analysis. This
is also in full agreement with Bohr's correspondence principle \cite{key-25,key-26,key-27}.
In previous analysis and in \cite{key-4,key-5}, the simplification
$(\omega_{0})_{n-1}\ll M$ has been implicitly used, i.e. the energy
associated to the QNM is much less than the original mass-energy of
the BH. Clearly, in that case the correction given by eq. (\ref{eq: area quantum e})
results non-essential, as one can neglect the difference between the
initial BH mass $M$ and the mass of the excited BH $M_{n-1}$, but
it becomes very important when $M_{n}\simeq M\;$ does not hold, i.e.
for very highly excited BHs. In that case, for example in the latest
stages of the BH evaporation (but before arriving at the Planck scale,
where our semi-classical approximation breaks down and a full quantum
gravity theory is needed), it could be $(\omega_{0})_{n}\lesssim M$,
and further corrections on previous formulas are needed. Putting $A_{n-1}\equiv16\pi M_{n-1}^{2}$
and $A_{n}\equiv16\pi M_{n}^{2}$, the formulas of the number of quanta
of area and of the Bekenstein-Hawking entropy become
\noindent
\begin{equation}
N_{n-1}\equiv\frac{A_{n-1}}{|\triangle A_{n-1}|}=\frac{16\pi M_{n-1}^{2}}{32\pi M_{E(n,\; n-1)}\cdot f_{n}(M,n)}=\frac{M_{n-1}^{2}}{2M_{E(n,\; n-1)}\cdot f_{n}(M,n)}\label{eq: N n-1}
\end{equation}
\noindent before the emission, and
\begin{equation}
N_{n}\equiv\frac{A_{n}}{|\triangle A_{n}|}=\frac{16\pi M_{n}^{2}}{M_{E(n,\; n-1)}}=\frac{M_{n}^{2}}{2M_{E(n,\; n-1)}\cdot f_{n}(M,n)}\label{eq: N n}
\end{equation}
after the emission respectively. One can easily check that
\noindent
\begin{equation}
N_{n}-N_{n-1}=\frac{M_{n}^{2}-M_{n-1}^{2}}{2M_{E(n,\; n-1)}\cdot f_{n}(M,n)}=\frac{f_{n}(M,n)\left(M_{n-1}+M_{n}\right)}{2M_{E(n,\; n-1)}\cdot f_{n}(M,n)}=1\label{eq: check}
\end{equation}
as one expects. Hence, the formulas of the Bekenstein-Hawking entropy
read
\begin{equation}
\begin{array}{c}
\left(S_{BH}\right)_{n-1}\equiv\frac{A_{n-1}}{4}=8\pi N_{n-1}M_{n-1}|\triangle M_{n}|=8\pi N_{n-1}M_{n-1}\cdot f_{n}(M,n)\\
\\
=4\pi\left(M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(n-\frac{1}{2})^{2}}\right),
\end{array}\label{eq: Bekenstein-Hawking n-1}
\end{equation}
before the emission and
\begin{equation}
\begin{array}{c}
\left(S_{BH}\right)_{n}\equiv\frac{A_{n}}{4}=8\pi N_{n}M_{n}|\triangle M_{n}|=8\pi N_{n}M_{n}\cdot f_{n}(M,n)\\
\\
=4\pi\left(M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(n+\frac{1}{2})^{2}}\right)
\end{array}\label{eq: Bekenstein-Hawking n}
\end{equation}
after the emission respectively. Notice that, as $n\gg1$, one obtains
\begin{equation}
\left(S_{BH}\right)_{n}\simeq\left(S_{BH}\right)_{n-1}\simeq4\pi\left(M^{2}-\frac{n}{2}\right).\label{eq: sb circa}
\end{equation}
\noindent Again, formula (\ref{eq: sb circa}) works for all $j=0,1,2.$
The formulas of the total entropy that takes into account the sub-leading
corrections to Bekenstein-Hawking entropy become
\noindent
\begin{equation}
\begin{array}{c}
\left(S_{total}\right)_{n-1}=8\pi N_{n-1}M_{n-1}\cdot f_{n}(M,n)\\
\\
-\ln\left[8\pi N_{n-1}M_{n-1}\cdot f_{n}(M,n)\right]+\frac{3}{64\pi N_{n-1}M_{n-1}\cdot f_{n}(M,n)}
\end{array}\label{eq: entropia n-1}
\end{equation}
\noindent before the emission, and
\begin{equation}
\begin{array}{c}
\left(S_{total}\right)_{n}=8\pi N_{n}M_{n}\cdot f_{n}(M,n)\\
\\
-\ln\left[8\pi N_{n}M_{n}\cdot f_{n}(M,n)\right]+\frac{3}{64\pi N_{n}M_{n}\cdot f_{n}(M,n)}
\end{array}\label{eq: entropia n}
\end{equation}
after the emission, respectively.
Hence, at level $n-1$ the BH has a number of micro-states
\begin{equation}
\begin{array}{c}
g(N_{n-1})\propto\exp\{8\pi N_{n-1}M_{n-1}\cdot f_{n}(M,n)+\\
\\
-\ln\left[8\pi N_{n-1}M_{n-1}\cdot f_{n}(M,n)\right]+\frac{3}{64\pi N_{n-1}M_{n-1}\cdot f_{n}(M,n)}
\end{array}\label{eq: microstati n-1}
\end{equation}
and, at level $n$, after the emission, the number of micro-states
is
\begin{equation}
\begin{array}{c}
g(N_{n})\propto\exp\{8\pi N_{n}M_{n}\cdot f_{n}(M,n)+\\
\\
-\ln\left[8\pi N_{n}M_{n}\cdot f_{n}(M,n)\right]+\frac{3}{64\pi N_{n}M_{n}\cdot f_{n}(M,n)}
\end{array}\label{eq: microstati n}
\end{equation}
All these corrections, which represent the correct formulas of an
excited BH for a transition between the levels $n-1$ and $n$ in
function of the quantum number $n$, result very important for very
highly excited BHs, when $n$ becomes extremely large and $M_{n}\simeq M\;$
does not hold (in particular in the last stages of the BH evaporation,
before arriving at the Planck scale, when $(\omega_{0})_{n}\lesssim M$).
Instead, when $(\omega_{0})_{n}\ll M$, formulas (\ref{eq: area quantum}),
(\ref{eq: N}), (\ref{eq: entropia totale 2}) and (\ref{eq: microstati}),
are a good approximation. Such formulas are a better approximation
with respect to formulas (\ref{eq: entropia circa}), (\ref{eq: entropia totale approssimata})
and (\ref{eq: microstati circa}) which were used in previous results
in the literature \cite{key-6,key-14,key-15}. In those works the
strictly thermal approximation was used. We also see that, for $(\omega_{0})_{n-1}\ll M$,
$M_{n}\simeq M_{n-1}\simeq M$, and eqs. (\ref{eq: area quantum}),
(\ref{eq: N}), (\ref{eq: entropia totale 2}) and (\ref{eq: microstati}),
are easily recovered.
\section{Final discussion and conclusion remarks}
We explain the way in which our BH model works. Let us consider a
BH original mass $M.$ After an high number of emissions (and potential
absorptions as the BH can capture neighboring particles), the BH will
be at an excited level $n-1$ and its mass will be $M_{n-1}\equiv M-(\omega_{0})_{n-1}$
where $(\omega_{0})_{n-1},$ is the absolute value of the frequency
of the QNM associated to the excited level $n-1.\;$ $(\omega_{0})_{n-1}\;$
is also the total energy emitted at that time. The BH can further
emit an energy to jump to the subsequent level: $\triangle M_{n}=(\omega_{0})_{n-1}-(\omega_{0})_{n}.$
Now, the BHat an excited level $n$ and the BH mass will be
\begin{equation}
\begin{array}{c}
M_{n}\equiv M-(\omega_{0})_{n-1}+\triangle M_{n-1}=\\
\\
=M-(\omega_{0})_{n-1}+(\omega_{0})_{n-1}-(\omega_{0})_{n}=M-(\omega_{0})_{n}.
\end{array}\label{eq: masse}
\end{equation}
The BH can, in principle, return to the level $n-1$ by absorbing
an energy $-\triangle M_{n}=(\omega_{0})_{n}-(\omega_{0})_{n-1}$.
We have also shown that the quantum of area is \emph{the same} for
both absorption and emission, given by eq. (\ref{eq: 8 pi planck}),
as one expects.
There are three different physical situations for excited BHs ($n\gg1$):
\begin{enumerate}
\item $n\;$ is large, but not enough large. It is also $(\omega_{0})_{n}\ll M_{n}\simeq M$
and one can use eqs. (\ref{eq: area quantum}), (\ref{eq: Bekenstein-Hawking}),
(\ref{eq: entropia totale 2}) and (\ref{eq: microstati}) which results
a better approximation than eqs. (\ref{eq: entropia circa}), (\ref{eq: entropia totale approssimata})
and (\ref{eq: microstati circa}) which were used in previous literature
in strictly thermal approximation \cite{key-6,key-14,key-15}
\item $n\;$ is very much larger than in point 1, but before arriving at
the Planck scale. In that case, it can be $(\omega_{0})_{n}\lesssim M,$
while $M_{n}\simeq M$ does not hold and one must use the eqs. (37),
and from (41) to (47).
\item At the Planck scale $n\;$ is larger also than in point 2, we need
a full theory of quantum gravity.
\end{enumerate}
In summary, in this paper we analyzed BH QNMs in terms of quantum
levels following the idea that, in an underlying quantum gravity theory,
BHs result in highly excited states. By using the concept of effective
temperature, we took into account the important issue that QNMs spectrum
is not strictly thermal.
The obtained results improve our previous analysis in \cite{key-4,key-5}
because here we removed some approximations that we implicitly used
in \cite{key-4,key-5}. The results look particularly intriguing as
important modifies on BH quantum physics have been realized. In fact,
we found the correct formulas of the horizon's area quantization and
of the number of quanta of area like functions of the quantum ``overtone''
number $n$. Consequently, we also found the correct expressions of
Bekenstein-Hawking entropy, its sub-leading corrections and the number
of micro-states, i.e. quantities which are fundamental to realize
the underlying quantum gravity theory, like functions of the quantum
``overtone'' number $n$. In other words, the cited important quantities
result to depend on the excited BH quantum state. An approximation
concerning the maximum value of $n\;$ has been also corrected. As
our previous results in \cite{key-4,key-5} were strictly corrected
only for scalar and gravitational perturbations, the results of this
work have shown that the analysis holds also for vector perturbations.
We stress that the analysis is totally consistent with the general
conviction that BHs result in highly excited states representing both
the ``hydrogen atom'' and the ``quasi-thermal emission'' in quantum
gravity and that previous results in the literature, where the thermal
approximation has been used \cite{key-6,key-14,key-15}, are consistent
with the results in this paper. As at the present time we do not yet
have a full theory of quantum gravity. Then, we must be content with
the semi-classical approximation. As for large $n$ Bohr's correspondence
principle holds \cite{key-25,key-26,key-27}, such a semi-classical
description is adequate. Thus, we can consider an intriguing analogy
in which our BH model is somewhat similar to the semi-classical Bohr's
model of the structure of a hydrogen atom \cite{key-26,key-27}. In
our BH model, during a quantum jump a discrete amount of energy is
indeed radiated and for large values of the principal quantum number
$n$ the analysis becomes independent from the other quantum numbers.
In a certain sense, QNMs represent the \textquotedbl{}electron\textquotedbl{}
which jumps from a level to another one and the absolute values of
the QNMs frequencies represent the energy \textquotedbl{}shells\textquotedbl{}.
In Bohr's model \cite{key-26,key-27}, electrons can only gain and
lose energy by jumping from one allowed energy shell to another, absorbing
or emitting radiation with an energy difference of the levels according
to the Planck relation $E=hf$, where $\: h\:$ is the Planck constant
and $f\:$ the transition frequency. In our BH model, QNMs can only
gain and lose energy by jumping from one allowed energy shell to another,
absorbing or emitting radiation (emitted radiation is given by Hawking
quanta) with an energy difference of the levels according to eqs.
(\ref{eq: variazione}) and (\ref{eq: f(M,n)}). On the other hand,
Bohr's model is an approximated model of the hydrogen atom with respect
to the valence shell atom model of full quantum mechanics. In the
same way, our BH model should be an approximated model of the emitting
BH with respect to the definitive, but at the present time unknown,
model of full quantum gravity theory.
It is also important to emphasize that the results in this paper could
have important implications for the BH information paradox. The more
important arguments that information can be lost during BH evaporation
rely indeed on the assumption of strict thermal behavior of the spectrum
\cite{key-18}. On the other hand, the results in this paper show
that BHs seem to be well defined quantum mechanical systems, which
have ordered, discrete quantum spectra. This important point is surely
consistent with the unitarity of the underlying quantum gravity theory
and endorses the idea that information should come out in BH evaporation
\cite{key-20,key-21,key-22,key-23,key-24,key-32,key-33}.
\section{Acknowledgements}
It is a pleasure to thank my colleague and collaborator Hossein Hendi
and my students Reza Katebi and Nathan Schmidt for useful discussions
on BH physics.
\section*{Appendix. Derivation of the quasi-normal modes equation in non-strictly
thermal approximation}
\noindent Being frequencies of the radial spin$-j=0,1,2$ perturbations
$\phi$ of the Schwarzschild space-time, QNMs are governed by the
Schrodinger-like equation \cite{key-4,key-5,key-16}
\begin{equation}
\left(-\frac{\partial^{2}}{\partial x^{2}}+V(x)-\omega^{2}\right)\phi.\label{eq: diff.}
\end{equation}
In strictly thermal approximation one introduces the Regge-Wheeler
potential
\noindent
\begin{equation}
V(x)=V\left[x(r)\right]=\left(1-\frac{2M}{r}\right)\left(\frac{l(l+1)}{r^{2}}+2\frac{(1-j^{2})M}{r^{3}}\right).\label{eq: Regge-Wheeler-1}
\end{equation}
We recall that $j=0,1,2$ for scalar, vector and gravitational perturbation
respectively.
\noindent The relation between the Regge-Wheeler ``tortoise'' coordinate
$x$ and the radial coordinate $r\;$ is \cite{key-4,key-5,key-16}
\noindent
\begin{equation}
\begin{array}{c}
x=r+2M\ln\left(\frac{r}{2M}-1\right)\\
\\
\frac{\partial}{\partial x}=\left(1-\frac{2M}{r}\right)\frac{\partial}{\partial r}.
\end{array}\label{eq:original tortoise}
\end{equation}
\noindent These states are analogous to quasi-stationary states in
quantum mechanics \cite{key-16}. Thus, their frequency is allowed
to be complex \cite{key-16}. They must have purely outgoing boundary
conditions both at the horizon ($r=2M$) and in the asymptotic region
($r\rightarrow\infty$) \cite{key-16}
\noindent
\begin{equation}
\phi(x)\sim c_{\pm}\exp\left(\mp i\omega x\right)\quad for\quad x=\pm\infty.\label{eq: condizioni contorno}
\end{equation}
\noindent Considering the non-strictly thermal behavior of BHs, one
substitutes the original BHs $M$ in eqs. (\ref{eq: diff.}) and (\ref{eq: Regge-Wheeler-1})
with the effective mass of the contracting BH defined in eq. (\ref{eq: effective quantities})
\cite{key-4,key-5}. Hence, eqs. (\ref{eq: Regge-Wheeler-1}) and
(\ref{eq:original tortoise}) are replaced by the \emph{effective
equations }\cite{key-4,key-5}
\begin{equation}
V(x)=V\left[x(r)\right]=\left(1-\frac{2M_{E}}{r}\right)\left(\frac{l(l+1)}{r^{2}}+2\frac{(1-j^{2})M}{r^{3}}\right)\label{eq: effettiva 1}
\end{equation}
\noindent and
\begin{equation}
\begin{array}{c}
x=r+2M_{E}\ln\left(\frac{r}{2M_{E}}-1\right)\\
\\
\frac{\partial}{\partial x}=\left(1-\frac{2M_{E}}{r}\right)\frac{\partial}{\partial r}.
\end{array}\label{eq: effettiva 2}
\end{equation}
\noindent In order to streamline the formulas, here we also set
\begin{equation}
2M_{E}=r_{E}\equiv1\;\;\;\; and\;\;\;\; m\equiv n+1.\label{eq: set}
\end{equation}
\noindent As the \emph{Planck mass} $m_{p}\:$ is equal to $1$ in
Planck units one rewrites (\ref{eq: quasinormal modes corrected})
as
\begin{equation}
\frac{\omega_{m}}{m_{p}^{2}}=\frac{\ln3}{4\pi}+\frac{i}{2}(m-\frac{1}{2})+\mathcal{O}(m^{-\frac{1}{2}}),\quad for\: m\gg1,\label{eq: intuizione}
\end{equation}
where now $m_{p}\neq1.$ Putting
\noindent
\begin{equation}
\tilde{\omega}_{m}\equiv\frac{\omega_{m}}{m_{p}^{2}},\label{eq: important definition}
\end{equation}
\noindent eqs. (\ref{eq: quasinormal modes corrected}), (\ref{eq: diff.}),
(\ref{eq: effettiva 1}) and (\ref{eq: effettiva 2}) become
\begin{equation}
\tilde{\omega}_{m}=\frac{\ln3}{4\pi}+\frac{i}{2}(m-\frac{1}{2})+\mathcal{O}(m^{-\frac{1}{2}}),\quad for\: m\gg1,\label{eq: rottura}
\end{equation}
\noindent
\begin{equation}
\left(-\frac{\partial^{2}}{\partial x^{2}}+V(x)-\tilde{\omega}^{2}\right)\phi,\label{eq: rottura 0}
\end{equation}
\noindent
\begin{equation}
V(x)=V\left[x(r)\right]=\left(1-\frac{1}{r}\right)\left(\frac{l(l+1)}{r^{2}}-\frac{3(1-j^{2})}{r^{3}}\right)\label{eq: rottura 2}
\end{equation}
\noindent and
\begin{equation}
\begin{array}{c}
x=r+\ln\left(r-1\right)\\
\\
\frac{\partial}{\partial x}=\left(1-\frac{1}{r}\right)\frac{\partial}{\partial r}
\end{array}\label{eq: rottura 3}
\end{equation}
respectively. Although $M_{E}$ and $r_{E}$ (and consequently the
tortoise coordinate and the Regge-Wheeler potential) are frequency
dependent, eq. (\ref{eq: set}) translates such a frequency dependence
into a continually rescaled mass unit in the following discussion.
We will show at the end of this Appendix that such a rescaling is
extremely slow and always included within a factor $2.$ Thus, it
does not influence the following analysis.
\noindent We emphasize that here after we closely follow ref. \cite{key-16}.
The solution of eq. (\ref{eq: rottura 0}) can be expanded as \cite{key-16}
\begin{equation}
\phi(r)=\left(\frac{r-1}{r^{2}}\right)^{i\tilde{\omega}}\exp\left[\text{-}i\text{\ensuremath{\tilde{\omega}}}(r\text{-}1)\right]\sum_{m}a_{m}\left(\frac{r-1}{r}\right)^{m}.\label{eq: fi di r}
\end{equation}
The pre-factor has to satisfy the boundary conditions (\ref{eq: condizioni contorno})
both at the effective horizon ($r=1$) and in the asymptotic region
($r=\infty$) \cite{key-16}:
\begin{enumerate}
\item One needs $\exp\left[-i\text{\ensuremath{\tilde{\omega}}}(r-1)\right]$
for the correct leading evolution at $r\rightarrow\infty$
\item $\left(r-1\right)^{i\tilde{\omega}}$ fixes the evolution at $r\rightarrow1^{+}$
\item $\left(\frac{1}{r^{2}}\right)^{i\tilde{\omega}}$ arranges the sub-leading
evolution at $r\rightarrow\infty$ which arises from the logarithmic
term in eq. (\ref{eq: rottura 3}).
\end{enumerate}
The power series (\ref{eq: fi di r}) converges for $\frac{1}{2}<r\leq\infty$
assuming that the boundary conditions at $r=\infty$ are preserved
\cite{key-16}. On the other hand, eq. (\ref{eq: diff.}) is equivalent
to the recursion relation \cite{key-16}
\begin{equation}
c_{0}(m,\tilde{\omega})a_{m}+c_{1}(m,\text{\ensuremath{\tilde{\omega}}})a_{m}-1+c_{2}(m,\text{\ensuremath{\tilde{\omega}}})a_{m}-2=0.\label{eq: recursione}
\end{equation}
One can extract the coefficients $\; c_{k}(m,\tilde{\omega})$ from
eq. (\ref{eq: recursione}) and rewrite them in a more convenient
way \cite{key-16}:
\begin{equation}
c_{0}(m,\text{\ensuremath{\tilde{\omega}}})=m(m+2i\text{\ensuremath{\tilde{\omega}}})\label{eq: c0}
\end{equation}
\begin{equation}
c_{1}(m,\tilde{\omega})=-2(m+2i\text{\ensuremath{\tilde{\omega}}}-\frac{1}{2})^{2}+j^{2}-\frac{1}{2}\label{eq:c1}
\end{equation}
\begin{equation}
c_{2}(m,\tilde{\omega})=(m+2i\text{\ensuremath{\tilde{\omega}}}-1)^{2}-j^{2}.\label{eq: c3}
\end{equation}
We see that, except for $c_{0},$ the coefficients $c_{k}$ depend
on $m,\tilde{\omega}\:$ only through their combination $m+2i\text{\ensuremath{\tilde{\omega}}}.$
The preservation of the boundary conditions at $r=1\;$ is guaranteed
by the initial conditions for the recursion relation \cite{key-16}.
Such conditions are $a_{0}=1$ (in general any non-zero constant that
we set equal to the unity for the sake of simplicity), and $a_{-1}=0$
(that, together with eq. (\ref{eq: recursione}) implies $a_{\lyxmathsym{\textminus}m}=0$
for all positive $m$) \cite{key-15}. One also defines \cite{key-16}
\begin{equation}
R_{m}=-\frac{a_{m}}{a_{m-1}},\label{eq: Rm}
\end{equation}
where we choose the minus sign in agreement with \cite{key-16}. By
using eq. (\ref{eq: recursione}), one gets \cite{key-16}
\begin{equation}
c_{1}(m,\tilde{\omega})-c_{0}(m,\text{\ensuremath{\tilde{\omega}}})R_{m}=\frac{c_{2}(m,\tilde{\omega})}{R_{m-1}}\label{eq: 1-0}
\end{equation}
which can be rewritten as
\begin{equation}
R_{m-1}=\frac{c_{2}(m,\tilde{\omega})}{c_{1}(m,\tilde{\omega})-c_{0}(m,\text{\ensuremath{\tilde{\omega}}})R_{m}}.\label{eq: 1-0 bis}
\end{equation}
Therefore, one can write $R_{m}\:$ in terms of a continued fraction.
The condition $a_{-1}=0\:$ becomes \cite{key-16}
\begin{equation}
R_{0}=\infty\;\;\;\rightarrow\;\;\; c_{1}(1,\tilde{\omega})-c_{0}(1,\text{\ensuremath{\tilde{\omega}}})R_{1}=0.\label{eq: Therefore}
\end{equation}
As eq. (\ref{eq: fi di r}) converges at $r=\infty,$ there is a particular
asymptotic form of $R_{m}\:$ for large $m$ (with $|R_{m}|<1$ for
very large $m$) \cite{key-16}. Thus, the boundary conditions require
eq. (\ref{eq: Therefore}). One can write down eq. (\ref{eq: Therefore})
in terms of continued fractions \cite{key-16}
\begin{equation}
0=c_{1}(1,\tilde{\omega})-c_{0}(1,\text{\ensuremath{\tilde{\omega}}})\frac{c_{2}(2,\tilde{\omega})}{c_{1}(2,\tilde{\omega})-c_{0}(2,\text{\ensuremath{\tilde{\omega}}})\frac{....}{....}}....,\label{eq: continued fractions}
\end{equation}
which is a condition for the existence of the quasi-normal modes.
As $R_{m}\rightarrow-1\:$ for large $m\;$ and the changes of $R_{m}$
slow down for large $|\tilde{\omega}|$, assuming that $R_{m}$ changes
adiabatically is an excellent approximation and one gets \cite{key-16}
\begin{equation}
\frac{R_{m}}{R_{m-1}}\simeq1+\mathcal{O}(\tilde{\omega}^{-\frac{1}{2}}).\label{eq: ordine}
\end{equation}
This approximation works for both $Re\left(m+2i\text{\ensuremath{\tilde{\omega}}}\right)>0,$
when one computes $R_{m}$ recursively from $R_{\infty}=\text{\textminus}1$,
and for $Re\left(m+2i\text{\ensuremath{\tilde{\omega}}}\right)<0,$
when one starts to compute from $R_{0}=\text{\ensuremath{\infty} }$
\cite{key-16}. If one inserts $R_{m-1}=R_{m}\;$ in eq. (\ref{eq: 1-0}),
one gets a quadratic equation, having the solutions (at the leading
terms for large $m$) \cite{key-16}
\begin{equation}
R_{m}^{\pm}=\frac{-(m+2i\text{\ensuremath{\tilde{\omega}}})\pm\sqrt{2i\text{\ensuremath{\tilde{\omega}}}(m+2i\text{\ensuremath{\tilde{\omega}}})}}{m}+\mathcal{O}(m^{-\frac{1}{2}}).\label{eq: ordinata}
\end{equation}
The approximated solution (\ref{eq: ordinata}) can be carefully checked
by using Mathematica \cite{key-16}. The issue that $R_{m}$ must
satisfy eq. (\ref{eq: ordinata}) for one of the signs is a necessary
condition \cite{key-16}. One needs a more deep discussion in order
to see if that condition is also sufficient \cite{key-16}. When $Re\left(m+2i\text{\ensuremath{\tilde{\omega}}}\right)<0,$
the sign arises from the condition $R_{1}$ is small. Two terms in
eq. (\ref{eq: ordinata}) are approximately deleted. The sign for
$Re\left(m+2i\text{\ensuremath{\tilde{\omega}}}\right)>0$ arises
from the condition $|R_{m}|<1$ for very large $m$ \cite{key-16}.
When $|\tilde{\omega}|$ is very large (but minor than the total mass
of the black hole), one chooses an integer $N\:$ such that \cite{key-16}
\begin{equation}
1\ll N\ll|\tilde{\omega}|.\label{eq: molto minore}
\end{equation}
For the values of $N\:$ in eq. (\ref{eq: molto minore}), eq. (\ref{eq: ordinata})
can be used to determine $R_{\left[-2i\tilde{\omega}\right]\pm N}\;$
\cite{key-16} and only the second term in the RHS of eq. (\ref{eq: ordinata})
results to be relevant (the symbol for the integer part $\left[-2i\tilde{\omega}\right]$
represents an arbitrary integer differing from $-2i\tilde{\omega}\;$
by a number much smaller than $N\;$ which is assumed to be even)
\cite{key-16}. Such a relevant term in eq. (\ref{eq: ordinata})
implies \cite{key-16}
\begin{equation}
R_{\left[-2i\tilde{\omega}\right]+x}\propto\pm\frac{i\sqrt{x}}{\sqrt{-2i\tilde{\omega}}}\;\; for\;\;1\ll x\ll|\tilde{\omega}|,\label{eq: for}
\end{equation}
while the first term in eq. (\ref{eq: ordinata}) is $\propto\frac{x}{\tilde{\omega}}\;$
and results subleading \cite{key-16}. Neglecting such a first term
is equivalent to neglecting $c_{1}(m,\tilde{\omega})$ in the original
equation (\ref{eq: 1-0}). In fact, this term is irrelevant for all
the $m\:$ for large $|\tilde{\omega}|$, with the possible exclusion
of some purely imaginary frequencies where $c_{0}(m)$ or $c_{2}(m)$
vanish \cite{key-16}. The ratio $R_{\left[-2i\tilde{\omega}\right]\pm N}\;$
is computed from eq. (\ref{eq: ordinata}) like the ratio of $\sqrt{x}\;$
for $x=\pm N$ \cite{key-16}
\begin{equation}
\frac{R_{\left[-2i\tilde{\omega}\right]+N}}{R_{\left[-2i\tilde{\omega}\right]-N}}=\pm i+\mathcal{O}(\tilde{\omega}^{-\frac{1}{2}}).\label{eq: ratio}
\end{equation}
The assumptions that permitted to obtain eq. (\ref{eq: ordinata})
break down when $|m+2i\text{\ensuremath{\tilde{\omega}}}|\sim1$\cite{key-16}.
In that case, the coefficients $c_{0}(m),\; c_{1}(m),\; c_{2}(m)$
contain terms of order $1$ that cannot be neglected \cite{key-16}.
They also strongly depend on $m$. The adiabatic approximation breaks
down in this region and the quantities $R_{\left[-2i\tilde{\omega}\right]\pm N}\;$
have to be related through the original continued fraction. Below,
we will calculate the continued fraction exactly in the limit of very
large $|\tilde{\omega}|$. The continued fraction will give the same
result of eq. (\ref{eq: ratio}) like the adiabatic argument. In fact,
the two solutions will eventually \textquotedblleft{}connect\textquotedblright{}
\cite{key-16}. Such a connection will release a non-trivial constraint
on $\tilde{\omega}$.
As $R_{\left[-2i\tilde{\omega}\right]+x}\propto\frac{1}{\sqrt{\tilde{\omega}}}$,
$\; c_{1}(m)$ in the denominator of eq. (\ref{eq: 1-0 bis}) is negligible
when compared to the other term (which results $\propto\sqrt{\tilde{\omega}}$
) \cite{key-16}. By fixing $N$, one understands that the effect
of $c_{1}(m)$ in eq. (\ref{eq: 1-0 bis}) vanishes for large $|\tilde{\omega}|$\cite{key-16}.
An exception should appear when $c_{0}(m)$ and/or $c_{2}(m)$ $\rightarrow0$
for some $m,$ but we will show that this exception cannot occur when
$Re(\tilde{\omega})\neq0$.
We note that the orbital angular momentum $l\:$ is irrelevant because
$c_{1}(m)$ does not affect the asymptotic frequencies \cite{key-16}.
This is not surprising as it is in agreement with Bohr correspondence
principle \cite{key-10,key-25,key-26,key-27}. One can replace the
factor $m$ in eq. (\ref{eq: c0}) with $\left[-2i\tilde{\omega}\right].$
In fact, $|\tilde{\omega}|$ becomes extremely large when one studies
only a relatively small neighborhood of $m\sim\left[-2i\tilde{\omega}\right]$
\cite{key-16}. We also see that the continued fraction is simplified
into an ordinary fraction. If one inserts eq. (\ref{eq: 1-0 bis})
recursively into itself $2N$ times, one gets \cite{key-16}
\begin{equation}
R_{\left[-2i\tilde{\omega}\right]-N}=\prod_{k=1}^{N}\left(\frac{c_{2}(\left[-2i\tilde{\omega}\right]-N+2k-1)c_{0}(\left[-2i\tilde{\omega}\right]-N+2k)}{c_{0}(\left[-2i\tilde{\omega}\right]-N+2k-1)c_{2}(\left[-2i\tilde{\omega}\right]-N+2k)}\right)R_{\left[-2i\tilde{\omega}\right]+N}.\label{eq: Produttore}
\end{equation}
The dependence of the $\; c_{k}(m)$ on the frequency is suppressed.
Thus, one can combine eqs. (\ref{eq: ratio}) and (\ref{eq: Produttore})
to eliminate the other $R_{m}$. We require that the generic solution
(\ref{eq: ordinata}), which holds almost everywhere, is \textquotedblleft{}patched\textquotedblright{}
with the solution (\ref{eq: Produttore}), which holds for $m\sim\left[-2i\tilde{\omega}\right]$
and for $|\tilde{\omega}|$ extremely large \cite{key-16}.
One can express the products of $c_{0}(m)$ and $c_{2}(m)$ in terms
of the gamma function $\Gamma,$ which is an extension of the factorial
function to real and complex numbers \cite{key-16}. As $c_{2}(m)$
is bilinear, the four factors of eq. (\ref{eq: Produttore}) lead
to a product of $6$ factors, i.e. $(2+1+1+2).$ Each of those equals
a ratio of two $\Gamma$ functions. In other words, one gets a ratio
of twelve $\Gamma$ functions \cite{key-16}.
One can write down the resulting condition in terms of a shifted frequency
defined by $-2if\equiv-2i\tilde{\omega}-\left[-2i\tilde{\omega}\right]$
\cite{key-16}:
\begin{equation}
\pm i=\begin{array}{c}
\frac{\Gamma\left(\frac{N+2if+j}{2}\right)\Gamma\left(\frac{N+2if-j}{2}\right)\Gamma\left(\frac{-N+2if+1}{2}\right)}{\Gamma\left(\frac{-N+2if+j}{2}\right)\Gamma\left(\frac{-N+2if-j}{2}\right)\Gamma\left(\frac{+N+2if+1}{2}\right)}\\
\\
\frac{\Gamma\left(\frac{-N+2if+j+1}{2}\right)\Gamma\left(\frac{-N+2if-j+1}{2}\right)\Gamma\left(\frac{+N+2if+2}{2}\right)}{\Gamma\left(\frac{N+2if+j+1}{2}\right)\Gamma\left(\frac{N+2if-j+1}{2}\right)\Gamma\left(\frac{-N+2if+2}{2}\right)}
\end{array}\label{eq: gamma-casini}
\end{equation}
Considering eq. (\ref{eq: c3}), the factors $m$ result cancelled.
Six of the $\Gamma$ functions in eq. (\ref{eq: gamma-casini}) show
an argument with a huge negative real part. Hence, they can be converted
into $\Gamma$ of positive numbers by using the formula \cite{key-16}
\begin{equation}
\Gamma(x)=\frac{\pi}{\sin\left(\pi x\right)\Gamma(1-x)}.\label{eq: gamma formula}
\end{equation}
Thus, the $\pi$ factors cancel like the Stirling approximations for
the $\;\Gamma$ functions which have a huge positive argument \cite{key-16}
\begin{equation}
\Gamma(m+1)\approx\sqrt{2\text{\ensuremath{\pi}}n}\left(\frac{m}{e}\right)^{m},\label{eq: Stirling}
\end{equation}
while the factors with $\sin\left(x\right)$ survive. The necessary
condition for regular frequencies for large $m\;$ (the frequencies
for which the analysis is valid. We will return on this point later)
reads \cite{key-16}
\begin{equation}
\pm i=\frac{\sin\left[\pi(if+1)\right]\sin\left[\pi(if+\frac{j}{2})\right]\sin\left[\pi(if-\frac{j}{2})\right]}{\sin\left[\pi(if+\frac{1}{2})\right]\sin\left[\pi(if+\frac{1+j}{2})\right]\sin\left[\pi(if+\frac{1-j}{2})\right]}.\label{eq: i}
\end{equation}
Choosing $N\in4\mathbb{Z},$ one erases $N\;$ from the arguments
of the trigonometric functions. One also replaces $f\:$ by $\tilde{\omega}$
again as the functions in eq. (\ref{eq: i}) are periodic with the
right periodicity and the number $\left[-2i\tilde{\omega}\right]$
can be chosen even. We can use the terms $\frac{\pi}{2}\;$ in the
denominator in order to convert the $\sin$ functions into $\cos$.
Multiplying eq. (\ref{eq: i}) by the denominator of the RHS and expanding
the $\sin$ functions in terms of the exponentials (one has to be
careful about the signs) the result is ($\epsilon(y)$ is the sign
function) \cite{key-16}
\begin{equation}
\exp\left[\epsilon Re\left(\omega\right)\cdot4\pi\tilde{\omega}\right]=-1-2\cos(\pi j).\label{eq: implica}
\end{equation}
Hence, for scalar or gravitational perturbations the allowed frequencies
are \cite{key-16}
\begin{equation}
\tilde{\omega}_{m}=\frac{i}{2}(m-\frac{1}{2})\pm\frac{\ln3}{4\pi}+\mathcal{O}(m^{-\frac{1}{2}}),\label{eq: piuomeno rottura}
\end{equation}
while for vector perturbations one gets
\begin{equation}
\tilde{\omega}_{m}=\frac{im}{2}+\mathcal{O}(m^{-\frac{1}{2}}).\label{eq: vettoriali}
\end{equation}
We note that, as we are in the large $m$ regime, one gets $\tilde{\omega}_{m}\simeq\frac{im}{2}\:$
independent of $j.$ This implies the correctness of eq. (\ref{eq: andamento asintotico})
and the important issue that the quantum of area (\ref{eq: 8 pi planck})
is an intrinsic property of Schwarzschild BHs. Again, this is in full
agreement with Bohr correspondence principle \cite{key-10,key-25,key-26,key-27}.
In order to finalize the analysis one has to resolve the question
marks concerning the special frequencies where the used approximation,
which neglects $c_{1}(m)$, breaks down \cite{key-16}.
The first step is to argue that the \textquotedblleft{}regular\textquotedblright{}
solutions must exist \cite{key-16}. In fact, if one can relate the
remainders $R_{\left[-2i\tilde{\omega}\right]\pm N}$ by using the
continued fraction, where $c_{1}(m)$ can be neglected, one can also
extrapolate them to eq. (\ref{eq: ordinata}) \cite{key-16}.
From the boundary conditions $R_{0}=\infty$, $R_{\infty}=\lyxmathsym{\textminus}1^{+},$
one sees that a specific sign of the square root in eq. (\ref{eq: ordinata})
has to be separately chosen for $\; m<\left[-2i\tilde{\omega}\right]$
and for $\; m>\left[-2i\tilde{\omega}\right]$ \cite{key-16}. But
one finds that the signs agree with the signs of $\pm i\;$ that automatically
lead to the solutions \cite{key-16}.
The condition for $\tilde{\omega}$ is both \emph{necessary and sufficient
}\cite{key-16}. This kind of solutions are only the $\ln(3$) solutions
from eq. (\ref{eq: rottura}). The existence of those solutions is
guaranteed \cite{key-16}.
As one needs to find all irregular solutions, let us recall two useful
points \cite{key-16}.
\begin{enumerate}
\item The continued fractions of eq. (\ref{eq: continued fractions}) depend
on the coefficients $c_{0}(m)$ and $c_{2}(m+1)$ \emph{only} through
their product $c_{0}(m)c_{2}(m+1)$ \cite{key-16}.
\item If one finds zeroes in eq. (\ref{eq: Produttore}) exclusively in
the numerator or in the denominator, the ratio $\frac{R_{\left[-2i\tilde{\omega}\right]-N}}{R_{\left[-2i\tilde{\omega}\right]+N}}$
can be only either zero, or infinite. When one takes into account
$c_{1}(m)$, \textquotedblleft{}zero\textquotedblright{} or \textquotedblleft{}infinity\textquotedblright{}
results to be replaced by a negative or positive power of $|\tilde{\omega}|$,
respectively \cite{key-16}.
\end{enumerate}
Point 2. means that one can obtain irregular solutions only by finding
$\omega$ such that eq. (\ref{eq: Produttore}) becomes an indeterminate
form $\frac{0}{0}$ \cite{key-16}.
As $c_{0}(m)$ can be null at most for one value of $m$, one finds
that there is at least one value of $m$ where $c_{2}(m)$ vanishes
\cite{key-16}. Thus, eq. (\ref{eq: c3}) implies that one between
$2\left(i\tilde{\omega}+1\right)$ and $\;2\left(i\tilde{\omega}-1\right)$
(maybe both) must be integer \cite{key-16}. As the two conditions
are equivalent, both numbers $2\left(i\tilde{\omega}\pm1\right)$
must be integers to give a chance to exist to the quasi-normal frequency
\cite{key-16}. Thus, the two numbers differ by an even number. Then,
both the vanishing factors of $\; c_{2}(m)$ must appear in the numerator
of eq. (\ref{eq: Produttore}), or, alternatively, both must appear
in the denominator of such an equation \cite{key-16}. One can assume,
for example, that they appear in the denominator without loss of generality
\cite{key-16}. Thus, one finds the indeterminate form only if the
vanishing $c_{0}(m)$ appears in the numerator \cite{key-16}. Clearly,
$2\left(i\tilde{\omega}\pm1\right)$ and $2i\tilde{\omega}$ are different
modulo two. Thus, the effect of $c_{1}(m)$ gives the desired result,
confirming that the regular states of eq. (\ref{eq: piuomeno rottura})
are the \emph{only} solutions \cite{key-16}. By using eqs. (\ref{eq: important definition})
and (\ref{eq: set}) one easily returns to Planck units and obtains
eq. (\ref{eq: quasinormal modes corrected}).
Now, we show that the continually rescaled mass unit in the above
discussion, which is due to the frequency dependence of $M_{E}$ and
$r_{E}$, did not influence the analysis. We note that, although $\tilde{\omega}$
in the analysis can be very large because of definition (\ref{eq: important definition}),
$\omega$ must instead be always minor than the BH initial mass as
BHs cannot emit more energy than their total mass. Inserting this
constrain in eq. (\ref{eq: effective quantities}) we obtain the range
of permitted values of $M_{E}(|\omega_{n}|)$ as
\begin{equation}
\frac{M}{2}\leq M_{E}(|\omega_{n}|)\leq M.\label{eq: range 1}
\end{equation}
Thus, setting $2M_{E}(|\omega_{n}|)=r_{E}(|\omega_{n}|)\equiv1(|\omega_{n}|)$
one sees that the range of permitted values of the continually rescaled
mass unit is always included within a factor $2.$ On the other hand,
we recall that the countable sequence of QNMs is very large, see Section
3 and \cite{key-4}. Thus, the mass unit's rescaling is extremely
slow. Hence, one can easily check, by reviewing the above discussion
step by step, that the continually rescaled mass unit did not influence
the analysis.
Another argument which remarks the correctness of the analysis in
this Appendix is the following. One can choose to consider $M_{E}$
as being constant within the range (\ref{eq: range 1}). In that case,
it is easy to show that such an approximation is indeed very good.
In fact, eq. (\ref{eq: range 1}) implies that the range of permitted
values of $T_{E}(|\omega_{n}|)$ is
\begin{equation}
T_{H}=T_{E}(0)\leq T_{E}(|\omega_{n}|)\leq2T_{H}=T_{E}(|\omega_{n_{max}}|),\label{eq: range 2}
\end{equation}
where $T_{H}$ is the initial BH Hawking temperature. Therefore, if
one fixes $M_{E}=\frac{M}{2}$ in the analysis, the approximate result
is
\begin{equation}
\omega_{n}\simeq2\pi in\times2T_{H}.\label{eq: approssima 1}
\end{equation}
On the other hand, if one fixes $M_{E}=M\:$ (thermal approximation),
the approximate result is
\begin{equation}
\omega_{n}\simeq2\pi in\times T_{H}.\label{eq: approssima 2}
\end{equation}
As both the approximate results in correspondence of the extreme values
in the range (\ref{eq: range 1}) have the same order of magnitude,
fixing $2M_{E}=r_{E}\equiv1$ does not change the order of magnitude
of the final (approximated) result with respect to the exact result.
In particular, if we set $T_{E}=\frac{3}{2}T_{H}$ the uncertainty
in the final result is $0.33$, while in the result of the thermal
approximation (\ref{eq: approssima 2}) the uncertainty is $2.$ Thus,
even considering $M_{E}$ as constant, our result is more precise
than the thermal approximation of previous literature and the order
of magnitude of the total emitted energies (\ref{eq: radice fisica})
is correct.
|
1,116,691,500,827 | arxiv | \section{Introduction}\label{section_intro}
Recall that a \textit{combinatorial $d$-manifold} is a simplicial complex in which the link of every vertex is PL homeomorphic to the $(d-1)$-dimensional sphere.
In 1987 Brehm and K\"uhnel~\cite{BrKu87} proved that any combinatorial $d$-manifold with less than $3d/2+3$ vertices is PL homeomorphic to the sphere. Moreover, they proved that combinatorial manifolds~$K$ that have exactly $3d/2+3$ vertices and are not homeomorphic to~$S^d$ may exist only in dimensions $d\in\{2,4,8,16\}$. Such combinatorial manifolds must satisfy the following conditions:
\begin{itemize}
\item[(a)] If $d=2$, then $K$ is isomorphic to the $6$-vertex triangulation~$\mathbb{RP}^2_6$ of the real projective plane obtained by taking the quotient of the boundary of the regular icosahedron by the antipodal involution. If $d=4$, then $K$ is isomorphic to the K\"uhnel $9$-vertex triangulation~$\mathbb{CP}^2_9$ of the complex projective plane, see~\cite{KuBa83}, \cite{KuLa83}. If $d=8$ or $d = 16$, then $K$ is a \textit{manifold like a projective plane} in the sense of Eells and Kuiper~\cite{EeKu62}, which means that $K$ admits a PL Morse function with exactly three critical points. In particular, this implies that $K$ admits a CW decomposition into three cells of dimensions~$0$, $d/2$, and~$d$ and therefore
$$
H^*(K;R)\cong R[x]/\bigl(x^3\bigr),\qquad \deg x = \frac{d}{2}\,,
$$
for any coefficients ring~$R$.
\item[(b)] $K$ is \textit{$s$-neighborly} with $s=d/2+1$, which means that every $s$-element set of vertices spans a simplex.
\item[(c)] $K$ satisfies the following \textit{complementarity} (or \textit{duality}) \textit{condition}: For each subset~$W$ of the set of vertices~$V$ of~$K$ exactly one of the sets~$W$ or~$V\setminus W$ spans a simplex of~$K$.
\item[(d)] The numbers~$f_k$ of $k$-simplices of~$K$ are as shown in Table~\ref{table_f}.
\begin{table}[t]
\caption{The face numbers of combinatorial $d$-manifolds that have $3d/2+3$ vertices and are not homeomorphic to the sphere}\label{table_f}
\begin{align*}
\boldsymbol{d}&\boldsymbol{=2\!:}& \boldsymbol{d}&\boldsymbol{=4\!:}
&\boldsymbol{d}&\boldsymbol{=8\!:} & \boldsymbol{d}&\boldsymbol{=16\!:}\\
f_0&= 6 & f_0 &=9 & f_0&=15 & f_0&=27& f_9 &=8335899\\
f_1&=15 & f_1&=36 & f_1&=105 & f_1&=351& f_{10} &=12184614\\
f_2&=10 & f_2&=84 & f_2&=455 & f_2&=2925& f_{11} &=14074164\\
&& f_3&=90 & f_3 &=1365 & f_3&=17550& f_{12} &=12301200\\
&& f_4&= 36 & f_4&= 3003 & f_4&= 80730& f_{13} &=7757100\\
&&&& f_5 &= 4515 & f_5&=296010& f_{14} &=3309696\\
&&&& f_6 &= 4230 & f_6&= 888030& f_{15} &=853281\\
&&&& f_7&= 2205 & f_7&=2220075& f_{16} &=100386\\
&&&& f_8&=490 & f_8&=4686825 &&
\end{align*}
\end{table}
\item[(e)] The canonical embedding $K\subset \mathbb{R}^{3d/2+2}$ obtained by regarding $K$ as the subcomplex of the standard $(3d/2+2)$-simplex is \textit{tight}. This means that, for each open half-space $h\subset \mathbb{R}^{3d/2+2}$ and any field of coefficients~$F$, the inclusion $K\cap h \subset K$ induces an injective map in homology $H_*(K\cap h;F)\to H_*(K;F)$; for more information on tight embeddings, see~\cite{Kuh95}.
\end{itemize}
The uniqueness of the $6$-vertex combinatorial $2$-manifold that is not a sphere is checked easily. The uniqueness of the $9$-vertex combinatorial $4$-manifold that is not a sphere was first proved in~\cite{KuLa83}; for simpler proofs, see~\cite[\S 19]{ArMa91},~\cite{BaDa01}. Condition~(a) for $d=8$ or $d=16$ is proved in~\cite{BrKu87}, the complementarity condition~(c) is proved in~\cite[\S 17,\,\S 20]{ArMa91}, the neighborliness condition~(b) follows immediately from~(c), since there are no simplices of dimensions greater than~$d$, and the tightness condition~(e) is deduced from~(c) in~\cite[Proposition~3]{BrKu92}. The face numbers~$f_i(K)$ can be computed using the complementarity condition, the Dehn--Sommerville equations, and the known value of the Euler characteristic $\chi(K)=3$ (for $d>2$). This computation can be found in~\cite{BrKu92} for $d=8$ and in~\cite[Sect.~4C]{Kuh95} for $d=16$, cf.~\cite{ChMa13}.
In the case~$d=8$, Brehm and K\"uhnel~\cite{BrKu92} constructed three $15$-vertex combinatorial manifolds like the quaternionic projective plane that satisfy conditions~(a)--(e). Nevertheless, they could not decide whether these combinatorial manifolds are homeomorphic to the quaternionic projective plane~$\mathbb{HP}^2$. This problem remained open for a long time until Gorodkov proved that the Brehm--K\"uhnel complexes are indeed PL homeomorphic to~$\mathbb{HP}^2$, see~\cite{Gor16},~\cite{Gor19}.
For a simplicial complex~$K$, we denote by $\mathop{\mathrm{Sym}}\nolimits(K)$ the \textit{symmetry group} of~$K$, that is, the group consisting of all permutations of vertices of~$K$ that take simplices to simplices and non-simplices to non-simplices. A simplicial complex is called \textit{vertex-transitive} if its symmetry group acts transitively on the vertices. Vertex-transitive triangulations of manifolds are especially interesting, see~\cite{KoLu05}.
Note that~$\mathbb{RP}^2_6$ and $\mathbb{CP}^2_9$ are vertex-transitive. In the case $d=8$, one of the three Brehm--K\"uhnel triangulations is vertex-transitive, and the other two are not. We denote the vertex-transitive triangulation by~$\mathbb{HP}^2_{15}$. Moreover, Brehm proved that $\mathbb{HP}^2_{15}$ is the only vertex-transitive $15$-vertex triangulation of a $8$-manifold like the quaternionic projective plane, cf.~\cite[Corollary~11]{KoLu05}.
Until now, the case $d=16$ has remained completely open. Namely, there were no candidates for a $27$-vertex combinatorial triangulation of the octonionic projective plane~$\mathbb{OP}^2$ or a manifold like the octonionic projective plane. The problem of whether there exists a $27$-vertex combinatorial $16$-manifold like the octonionic projective plane and (if exists) whether it is homeomorphic to~$\mathbb{OP}^2$ was posed in~\cite{BrKu87} (cf.~\cite{ArMa91}, \cite{BrKu92}, \cite{Kuh95}, \cite{Lut05}, \cite{ChMa13}). In the present paper we solve the first part of this problem. Namely, we construct $27$-vertex combinatorial $16$-manifolds like the octonionic projective plane. However, we are in the same situation as Brehm and K\"uhnel were in 1992 in the quaternionic case---we cannot decide yet whether our simplicial complexes are indeed homeomorphic to~$\mathbb{OP}^2$.
Kramer~\cite{Kra03} classified PL manifolds like a projective plane in dimensions~$8$ and~$16$, see Remark~\ref{remark_classification} for more detail. He showed that in dimension~$8$ such manifolds are distinguished by their Pontryagin numbers, and in dimension~$16$ by their Pontryagin numbers and certain exotic PL characteristic number. Gorodkov's result in the quaternionic case (see~\cite{Gor16},~\cite{Gor19}) was based on the computation of the first Pontryagin class of the Brehm--K\"uhnel combinatorial manifolds using an explicit combinatorial formula due to the author of the present paper, cf.~\cite{Gai04}, \cite{Gai08}, \cite{Gai10}, \cite{GaGo19}. However, to decide whether a $16$-manifold like a projective plane is indeed homeomorphic to~$\mathbb{OP}^2$ one needs to compute the second Pontryagin class and the first exotic PL characteristic class of it. There is currently no known efficient way to do this.
\begin{theorem}\label{theorem_main}
There are at least
$$
\frac1{351}\cdot\left(2^{351}+13\cdot 2^{118}+81\cdot 2^{29}\right)+2 \approx 1.3\cdot 10^{103}
$$
combinatorially distinct $27$-vertex combinatorial triangulations of $16$-manifolds like the octonionic projective plane. The symmetry groups of these triangulations are $C_3^3\rtimes C_{13}$, $C_3^3$, $C_{13}$, $C_3^2$, $C_3$, and the trivial group~$1$; the numbers of triangulations with each symmetry group are given in Table~\ref{table_num_triang}. The $634$ triangulations with symmetry groups~$C_3^3\rtimes C_{13}$ and $C_3^3$ are vertex-transitive, and all other triangulations are not. In addition, the four triangulations~$K_1$, $K_2$, $K_3$, and~$K_4$ with the largest symmetry group~$C_3^3\rtimes C_{13}$ have the following properties:
\begin{enumerate}
\item $\mathop{\mathrm{Sym}}\nolimits(K_i)$ acts transitively and freely on the $351$ undirected edges of~$K_i$,
\item $\mathop{\mathrm{Sym}}\nolimits(K_i)$ acts freely on the set of $16$-simplices of~$K_i$, so $K_i$ contains exactly $286$ orbits of $16$-dimensional simplices, each consisting of $351$ simplex.
\end{enumerate}
Besides, $K_1$, $K_2$, $K_3$, and~$K_4$ are the only (up to isomorphism) $27$-vertex combinatorial triangulations of $16$-manifolds like a projective plane whose symmetry groups contain a subgroup isomorphic to $C_3^3\rtimes C_{13}$.
Two of the four combinatorial manifolds $K_1$, $K_2$, $K_3$, and~$K_4$ (namely, $K_2$ and $K_3$ in our notation), and all mentioned above combinatorial manifolds with smaller symmetry groups are PL homeomorphic to each other.
\end{theorem}
\begin{table}
\caption{Numbers of $27$-vertex triangulations of $16$-manifolds like the octonionic projective plane with the given symmetry groups}\label{table_num_triang}
\begin{tabular}{|c|l|c|}
\hline
Symmetry group & Number of triangulations \\
\hline
$\vphantom{\strut_3^3}C_3^3\rtimes C_{13}$ & $4$\\
\hline
$\vphantom{\strut_3^3}C_3^3$ & $\ge 630$ \\
\hline
$\vphantom{\strut_3^3}C_{13}$ & $\ge 2^{27}-2$\\
\hline
$\vphantom{\strut_3^3}C_{3}^2$ & $\ge (2^{39}-2^{13})/3$\\
\hline
$\vphantom{\strut_3^3}C_{3}$ & $\ge (2^{117}-2^{41}+3\cdot 2^{13})/9$\\
\hline
$\vphantom{\strut_3^3}1$ & $\ge \bigl(2^{351}-13\cdot 2^{117}+39\cdot 2^{39}- 27\cdot (2^{27}+2^{13}-2)\bigr)/351$ \\
\hline
\end{tabular}
\end{table}
\begin{remark}
For the two other combinatorial manifolds~$K_1$ and~$K_4$, we do not know whether they are PL homeomorphic to~$K_2$ and~$K_3$ or to each other.
\end{remark}
As we have mentioned in the beginning of this section, conditions~(a)--(e) hold whenever a combinatorial $d$-manifold not homeomorphic to the sphere has the critical number of vertices $3d/2+3$. Hence, all the triangulations in Theorem~\ref{theorem_main} satisfy these conditions.
Note that there is a unique (up to isomorphism) semi-direct product $C_3^3\rtimes C_{13}$ that is not direct, so we may not specify the action of~$C_{13}$ on~$C_3^3$. An explicit description of the four simplicial complexes~$K_1$, $K_2$, $K_3$, and~$K_4$ and their symmetry group will be given in Section~\ref{section_result}, see Theorem~\ref{theorem_main_detail}. All the triangulations in Theorem~\ref{theorem_main} with smaller symmetry groups are obtained from~$K_2$ and~$K_3$ by \textit{triple flips} i.\,e. the replacements of a subcomplex of the form
\begin{equation}\label{eq_sc}
(\Delta_1*\partial\Delta_2)\cup(\Delta_2*\partial\Delta_3)\cup(\Delta_3*\partial\Delta_1),
\end{equation}
where $\Delta_1$, $\Delta_2$, and~$\Delta_3$ are $8$-simplices, with the subcomplex
$$
(\partial\Delta_1*\Delta_2)\cup(\partial\Delta_2*\Delta_3)\cup(\partial\Delta_3*\Delta_1),
$$
see Section~\ref{section_moves}. The same method does not work for~$K_1$ and~$K_4$, since they contain no subcomplexes of the form~\eqref{eq_sc}. Note that the same triple flips (with $\dim\Delta_i=4$) were used by Brehm and K\"uhnel to construct two $15$-vertex triangulations of manifolds like the quaternionic projective plane with smaller symmetry groups, starting from the most symmetric triangulation with symmetry group~$A_5$.
\begin{remark}\label{remark_classification}
Recall the classification of PL manifolds like projective planes up to PL homeomorphism. In dimensions~$2$ any such manifold is obviously PL homeomorphic to~$\mathbb{RP}^2$. In dimension~$4$, it follows from Freedman's classification of topological simply-connected $4$-manifolds that any PL manifold like the complex projective plane is homeomorphic to~$\mathbb{CP}^2$. However, the problem of classification of PL (equivalently, smooth) structures on~$\mathbb{CP}^2$ is still open. In dimensions $8$ and~$16$ there exist countably many pairwise non-homeomorphic PL manifolds like projective planes. First examples of such manifolds are due to Milnor~\cite{Mil56} in dimension~$8$ and Shimada~\cite{Shi57} in dimension~$16$. Namely, if $d=2m$, where $m$ is either $4$ or~$8$, then one can consider $m$-dimensional real vector bundles~$\xi$ over the sphere~$S^m$. If the Euler number $\langle e(\xi),[S^m]\rangle$ is $\pm 1$ (one can make it to be~$1$ by reversing the orientation of~$\xi$), then the Thom space~$Y=T(\xi)$ has a natural structure of a $d$-dimensional PL manifold and is a manifold like a projective plane. Oriented $\mathbb{R}^m$-vector bundles over $S^m$ with $e(\xi)=1$ are classified by their Pontryagin numbers $\langle p_1(\xi),[S^4]\rangle$ and $\langle p_2(\xi),[S^8]\rangle$, respectively, which can take and any integral value ${}\equiv 2\pmod 4$ in the case $m=4$ and any integral value ${}\equiv 6\pmod{12}$ in the case $m=8$. This construction yields a countable series of PL $d$-manifolds~$Y^d_h=Y^d_{1-h}$ like a projective plane indexed by $h\in\mathbb{Z}$ with characteristic numbers
\begin{align}
p_1^2\left[Y^8_h\right] &= \left\langle p_1^2\left(Y^8_h\right),\left[Y^8_h\right] \right\rangle = 4(2h-1)^2,\label{eq_pont8}\\
p_2^2\left[Y^{16}_h\right]&=\left\langle p_2^2\left(Y^{16}_h\right),\left[Y^{16}_h\right] \right\rangle = 36(2h-1)^2.\label{eq_pont16}
\end{align}
(These values of the characteristic numbers correspond to the orientation of~$Y^d_h$ chosen so that the signature of~$Y^d_h$ is $1$.) The Hirzebruch signature theorem implies that the $L$-genus of any manifold like a projective plane is equal to~$1$, so all Pontryagin numbers of~$Y^d_h$ are uniquely determined by~\eqref{eq_pont8},~\eqref{eq_pont16}. The standard projective planes~$\mathbb{HP}^2$ and~$\mathbb{OP}^2$ correspond to $h=1$. Eells and Kuiper~\cite{EeKu62} explored the manifolds~$Y^d_h$ in more detail. In particular, they proved that there are exactly $6$ and $60$ homotopy classes of such manifolds in the cases $d=8$ and $d=16$, respectively. A complete classification of PL manifolds like projective planes in dimensions~$8$ and~$16$ was obtained by Kramer~\cite{Kra03}:
\begin{itemize}
\item In dimension~$8$, he showed that any such manifold is PL homeomorphic to one of Milnor's manifolds~$Y^8_h$.
\item On the contrary, in dimension~$16$, he showed that Shimada's manifolds~$Y^{16}_h$ do not exhaust all PL manifolds like the octonionic projective plane. Namely, in the construction described above, one can replace the vector bundle~$\xi$ with an arbitrary PL $\mathbb{R}^8$-bundle~$\xi$ over~$S^8$. For such bundles, the second Pontryagin class~$p_2(\xi)$ should no longer be integral. However, the class $\frac76p_2(\xi)$ is always integral. Besides, in dimension~$8$, there is the first exotic PL characteristic class $\kappa\in H^8(\mathrm{BPL};\mathbb{Z}/4\mathbb{Z})$, which is not a reduction of any polynomial in Pontryagin classes. Kramer constructed a series of PL manifolds~$Y_{h,\theta}^{16}=Y_{1-h,\theta}^{16}$ like the octonionic projective plane indexed by $h\in\frac17\mathbb{Z}$ and $\theta\in\mathbb{Z}/4\mathbb{Z}$ with characteristic numbers
\begin{gather*}
p_2^2\left[Y^{16}_{h,\theta}\right] = 36(2h-1)^2,\qquad
\left(\left(\frac76p_2\right)\kappa\right) \left[Y^{16}_{h,\theta}\right] = \theta,
\end{gather*}
and proved that any PL manifold like the octonionic projective plane is PL homeomorphic to one of these manifolds. (The manifolds $Y_{h,0}^{16}$ with $h\in\mathbb{Z}$ are exactly Shimada's manifolds.)
\end{itemize}
Thus, our combinatorial manifolds from Theorem~\ref{theorem_main} are PL homeomorphic to some of Kramer's manifolds~$Y_{h,\theta}^{16}$.
\end{remark}
\begin{conj}
The combinatorial manifolds~$K_1$, $K_2$, $K_3$, and~$K_4$ are PL homeomorphic to~$\mathbb{OP}^2$.
\end{conj}
\begin{remark}
There is a nice and intriguing parallel between $(3d/2+3)$-vertex triangulations of manifolds not homeomorphic to the sphere and theory of Severi varieties in complex projective geometry. This parallel was noticed and explored by Chapoton and Manivel~\cite{ChMa13}. Nevertheless, there are many still open questions and numerical coincidences that are yet to be explained. Let $X\subset \mathbb{CP}^{n-1}$ be a smooth irreducible complex projective variety of dimension~$d$ that is not contained in a hyperplane. The \textit{secant variety} $\mathop{\mathrm{Sec}}\nolimits(X)$ is the Zariski closure of the union of the lines joining any two points of~$X$. The following result is due to Zak~\cite{Zak81},~\cite{Zak85}:
\begin{enumerate}
\item If $n < 3d/2+3$, then $\mathop{\mathrm{Sec}}\nolimits(X)=\mathbb{CP}^{n-1}$.
\item If $n= 3d/2+3$, then either $\mathop{\mathrm{Sec}}\nolimits(X)=\mathbb{CP}^{n-1}$ or $d\in\{2,4,8,16\}$. Moreover, there are exactly four examples, one for every $d=2,4,8,16$, of varieties with $\mathop{\mathrm{Sec}}\nolimits(X)\ne \mathbb{CP}^{n-1}$:
\begin{itemize}
\item the Veronese surface $\mathbb{CP}^2\hookrightarrow\mathbb{CP}^5$,
\item the Segre embedding $\mathbb{CP}^2\times\mathbb{CP}^2\hookrightarrow\mathbb{CP}^8$,
\item the Pl\"ucker embedding $G_{\mathbb{C}}(2,6)\hookrightarrow\mathbb{CP}^{14}$
\item the $16$-dimensional variety $E\subset \mathbb{CP}^{26}$ discovered by Lazarsfeld~\cite{Laz81}.
\end{itemize}
\end{enumerate}
Note that the four varieties~$\mathbb{CP}^2$, $\mathbb{CP}^2\times\mathbb{CP}^2$, $G_{\mathbb{C}}(2,6)$, and~$E$ can be naturally regarded as the complexifications of the projective planes~$\mathbb{RP}^2$, $\mathbb{CP}^2$, $\mathbb{HP}^2$, and~$\mathbb{OP}^2$, respectively (see, e.\,g.,~\cite{LaMa01}).
\end{remark}
\begin{remark}
It is not so surprising that the group $C_3^3\rtimes C_{13}$ arises as the symmetry group of combinatorial manifolds that are most likely homeomorphic to~$\mathbb{OP}^2$. The matter is that this group can be naturally realized as a subgroup in the isometry group~$\mathop{\mathrm{Isom}}\nolimits^+(\mathbb{OP}^2)$ of~$\mathbb{OP}^2$ endowed with the standard Fubini--Study metric. By a result of Borel~\cite{Bor50} (cf.~\cite{Bae01} and references therein), the group~$\mathop{\mathrm{Isom}}\nolimits^+(\mathbb{OP}^2)$ is isomorphic to the $52$-dimensional exceptional simply connected compact Lie group~$F_4$. Alekseevskii~\cite{Ale74} studied Jordan subgroups of complex Lie groups. We are not going to give a precise definition of a Jordan subgroup here. Let us only mention that any Jordan subgroup~$A$ of a Lie group~$G$ is a commutative finite subgroup with finite normalizer~$N_G(A)$. In particular, this means that $A$ is not contained in any torus in~$G$. In the case of~$F_4$, Alekseevskii proved that the only (up to conjugation) Jordan subgroup~$A$ is isomorphic to~$C_3^3$ and the normalizer~$N_{F_4}(A)$ is isomorphic to~$\mathrm{SL}_3(\mathbb{F}_3)$ and acts in the standard way on~$A\cong C_3^3\cong\mathbb{F}_3^3$. This group~$A$ was later rediscovered by several authors, see~\cite{Gri91} and references therein. Hence the group $C_3^3\rtimes \mathrm{SL}_3(\mathbb{F}_3)$ becomes realized as a finite isometry group of~$\mathbb{OP}^2$. Choosing a subgroup of~$\mathrm{SL}_3(\mathbb{F}_3)$ isomorphic to~$C_{13}$, we arrive to an isometry group of~$\mathbb{OP}^2$ isomorphic to~$C_3^3\rtimes C_{13}$.
\end{remark}
Recall that, for~$\mathbb{RP}^2_6$, $\mathbb{CP}^2_9$, and the three Brehm--K\"uhnel triangulations~$\mathbb{HP}^2_{15}$, $\widetilde{\HP}\vphantom{\HP}^2_{15}$ and~$\widetilde{\widetilde{\HP}}\vphantom{\HP}^2_{15}$, we have
\begin{gather*}
\mathop{\mathrm{Sym}}\nolimits (\mathbb{RP}^2_6)\cong A_5, \qquad
\mathop{\mathrm{Sym}}\nolimits (\mathbb{CP}^2_9)\cong C_3^2\rtimes C_6\cong\mathrm{He}_3\rtimes C_2, \\
\mathop{\mathrm{Sym}}\nolimits (\mathbb{HP}^2_{15})\cong A_5, \qquad
\mathop{\mathrm{Sym}}\nolimits \left(\widetilde{\HP}\vphantom{\HP}^2_{15}\right)\cong A_4, \qquad
\mathop{\mathrm{Sym}}\nolimits \left(\widetilde{\widetilde{\HP}}\vphantom{\HP}^2_{15}\right)\cong S_3,
\end{gather*}
where $ \mathrm{He}_3$ is the Heisenberg group of $3\times 3$ upper unitriangular matrices with entries in~$\mathbb{F}_3$, cf.~\cite{KuBa83},~\cite{BrKu92}. In the semi-direct product $C_3^2\rtimes C_6$ the action of the generator of~$C_6$ on the standard generators~$a$ and~$b$ of~$C_3^2$ is given by $a\mapsto a^{-1}b^{-1}$, $b\mapsto b^{-1}$. In the semi-direct product $\mathrm{He}_3\rtimes C_2$ the action of the generator of~$C_2$ on~$\mathrm{He}_3$ is given by
$$
\begin{pmatrix}
1 & a & c\\
0 & 1 & b\\
0 & 0 & 1
\end{pmatrix}
\mapsto
\begin{pmatrix*}[r]
1 & -a & -c\\
0 & 1 & b\\
0 & 0 & 1
\end{pmatrix*}\,.
$$
We see that there is a strange alternation. The symmetry groups of~$\mathbb{RP}^2_6$ and~$\mathbb{HP}^2_{15}$ are simple, while the symmetry groups of~$\mathbb{CP}^2_9$ and each of our triangulations~$K_i$ are solvable. We do not know any explanation for this phenomenon.
\begin{problem}
Enumerate all $15$-vertex $8$-dimensional and $27$-vertex $16$-dimensional combinatorial manifolds like a projective plane. Which groups can occur as the symmetry groups of such combinatorial manifolds?
\end{problem}
In the case $d=16$, this problem seems to be hard, since by Theorem~\ref{theorem_main} the number of combinatorially distinct $27$-vertex combinatorial $16$-manifolds is huge. The problem that seems to be accessible is as follows.
\begin{problem}
Enumerate all vertex-transitive $27$-vertex combinatorial $16$-manifolds like the octonionic projective plane. Which groups can occur as the symmetry groups of such combinatorial manifolds?
\end{problem}
Note that the transitivity of the action of~$\mathop{\mathrm{Sym}}\nolimits(K_i)$ on the set of undirected edges of~$K_i$ is a completely new phenomenon. Neither $\mathbb{CP}^2_9$ nor any of the three Brehm--K\"uhnel triangulations of~$\mathbb{HP}^2$ possess this property. (The case of~$\mathbb{RP}^2_6$ is very special; the group $\mathop{\mathrm{Sym}}\nolimits(\mathbb{RP}^2_6)$ acts transitively even on the set of directed edges and, moreover, on the set of all complete flags of simplices $\sigma^0\subset\sigma^1\subset\sigma^2$.)
Also, a new phenomenon is that $\mathop{\mathrm{Sym}}\nolimits(K_i)$ acts freely on the set of maximal simplices. None of the previously known triangulations of manifolds like a projective plane possesses this property. Namely,
\begin{itemize}
\item the $60$-element group $\mathop{\mathrm{Sym}}\nolimits(\mathbb{RP}^2_6)$ acts transitively on the $10$-element set of $2$-simplices,
\item the $54$-element group $\mathop{\mathrm{Sym}}\nolimits(\mathbb{CP}^2_9)$ acts on the $36$-element set of $4$-simplices with two orbits consisting of $27$ and~$9$ simplices, respectively,
\item the $60$-element group $\mathop{\mathrm{Sym}}\nolimits(\mathbb{HP}^2_{15})$ acts on the $490$-element set of $8$-simplices with $5$ orbits of $60$ elements, $4$ orbits of $30$ elements, $1$ orbit of $20$ elements, $2$ orbits of $15$ elements, and $2$ orbits of~$10$ elements,
\item the $12$-element group $\mathop{\mathrm{Sym}}\nolimits\left(\widetilde{\HP}\vphantom{\HP}^2_{15}\right)$ acts on the $490$-element set of $8$-simplices with $36$ orbits of $12$~elements, $6$~orbits of $6$~elements, $4$~orbits of $4$~elements, and $2$~orbits of $3$~elements.
\item the $6$-element group $\mathop{\mathrm{Sym}}\nolimits\left(\widetilde{\widetilde{\HP}}\vphantom{\HP}^2_{15}\right)$ acts on the $490$-element set of $8$-simplices with $75$ orbits of $6$~elements, $12$~orbits of $3$~elements, $1$~orbit of $2$~elements, and $2$~orbits of $1$~element.
\end{itemize}
\begin{remark}
A \textit{minimal triangulation} of a manifold is a triangulation of this manifold with the smallest possible number of vertices. There are not so many classes of manifolds for which minimal triangulations are known, see~\cite{Lut05}. Certainly, $\mathbb{RP}^2_6$, $\mathbb{CP}^2_9$, the three Brehm--K\"uhnel triangulations of~$\mathbb{HP}^2$, and all our triangulations from Theorem~\ref{theorem_main} are minimal, since all triangulations with smaller numbers of vertices are spheres. Alongside with the problem of finding minimal triangulations, there is an interesting problem of finding triangulations that are minimal in the class of triangulations admitting a regular coloring of vertices in $d+1$ colors, where $d$ is the dimension. In~\cite{Gai09}, the author constructed triangulations~$\mathbb{RP}^2_9$ and~$\mathbb{CP}^2_{15}$ that are minimal in this class; they have symmetry groups $S_3\times C_2$ and $S_4\times S_3$, respectively. It would be interesting to study the corresponding problem for~$\mathbb{HP}^2$ and~$\mathbb{OP}^2$ or manifolds like projective planes. In particular, it would be nice to understand if the equality of the numbers of vertices of the triangulations~$\mathbb{RP}^2_9$ and~$\mathbb{CP}^2_{15}$ with the numbers of vertices of the minimal triangulation~$\mathbb{CP}^2_9$ and~$\mathbb{HP}^2_{15}$, respectively, is a coincidence or has a reason.
\end{remark}
The present paper is organized as follows. In Section~\ref{section_result} we give a more detailed formulation of the part of Theorem~\ref{theorem_main} concerning the four triangulations~$K_1$, $K_2$, $K_3$, and~$K_4$ with the largest symmetry group $C_3^3\rtimes C_{13}$ (Theorem~\ref{theorem_main_detail}). In Section~\ref{section_scheme} we give the scheme of the proof of Theorem~\ref{theorem_main_detail}. Sections~\ref{section_find}--\ref{section_no_add_sym} contain a detailed proof of Theorem~\ref{theorem_main_detail} following this scheme. In Section~\ref{section_moves} we construct triangulations with smaller symmetry groups by applying triple flips to~$K_2$ and~$K_3$, and prove that all the obtained triangulations are PL homeomorphic to each other and to~$K_2$ and~$K_3$. Theorems~\ref{theorem_main_detail} and~\ref{theorem_smaller_groups} together constitute Theorem~\ref{theorem_main}.
Most of our proofs are computer assisted. In the paper we describe the required algorithms and present the results of their implementations. Explicit implementations of all the algorithms used in the present paper can be found in~\cite{Gai-prog}.
This work arose from multiple discussions with Denis Gorodkov, Alexander Kuznetsov, and Dmitri Orlov of the results of~\cite{ChMa13}. I am highly indebted to them for these really fruitful discussions. I am also grateful to Semyon Abramyan, Victor Buchstaber, Vasilii Rozhdestvenskii, Constantin Shramov, Igor Spiridonov, and Alexander Veselov for useful comments.
\section{Triangulations with symmetry group $C_3^3\rtimes C_{13}$}\label{section_result}
Recall terminology concerning abstract simplicial complexes. An (\textit{abstract}) \textit{simplicial complex} on the vertex set~$V$ is a set~$K$ of subsets of~$V$ such that
\begin{itemize}
\item $\varnothing\in K$,
\item if $\sigma\in K$ and~$\tau\subset\sigma$, then $\tau\in K$.
\end{itemize}
We will assume that all simplicial complexes under consideration contain no \textit{ghost vertices}, that is, all $1$-element subsets of~$V$ belong to~$K$.
The \textit{dimension} of a simplex $\sigma\in K$ is, by definition, the cardinality of~$\sigma$ minus~$1$. The \textit{link} of a simplex $\sigma\in K$ is the simplicial complex
$$
\mathop{\mathrm{link}}\nolimits(\sigma,K)=\bigl\{\tau\subseteq V\setminus\sigma \colon \sigma\cup\tau\in K \bigr\}.
$$
If $K_1$ and~$K_2$ are simplicial complexes on the vertex sets~$V_1$ and~$V_2$, then an \textit{isomorphism} of~$K_1$ and~$K_2$ is a bijection $f\colon V_1\to V_2$ that takes simplices of~$K_1$ to simplices of~$K_2$ and non-simplices of~$K_1$ to non-simplices of~$K_2$. The automorphisms of a simplicial complex~$K$ will be called \textit{symmetries} of~$K$; we denote the symmetry group of~$K$ by~$\mathop{\mathrm{Sym}}\nolimits(K)$.
A simplicial complex~$K$ is said to be \textit{pure of dimension~$d$} if any maximal (with respect to the inclusion) simplex of~$K$ is $d$-dimensional. A simplicial complex~$K$ is called a \textit{weak $d$-pseudomanifold} if $K$ is pure of dimension~$d$ and any $(d-1)$-simplex of~$K$ is contained in exactly two $d$-simplices of~$K$. A simplicial complex~$K$ is called a \textit{combinatorial $d$-sphere} (respectively, a \textit{combinatorial $d$-ball}) if $K$ (more precisely, the geometric realization of~$K$) is PL homeomorphic to the standard sphere~$S^d$ (respectively, to the standard disk~$D^d$). A simplicial complex~$K$ is called a \textit{combinatorial $d$-manifold} if the link of every vertex of~$K$ is a combinatorial $(d-1)$-sphere; then the link of any $k$-simplex of~$K$ is a combinatorial $(d-k-1)$-sphere, see~\cite[Corollary~1 of Lemma~9]{Zee63}. For further background material on simplicial complexes, see~\cite{RoSa72} or~\cite{Zee63}.
We denote by~$C_n$ the cyclic group of order~$n$, by~$S_n$ the symmetric group of degree~$n$, and by~$A_n$ the alternating group of degree~$n$.
Consider a field $\mathbb{F}_{27}$ of $27$ elements. The additive group $\mathbb{F}_{27}\cong C_3^3$ and the multiplicative group~$\mathbb{F}_{27}^{\times}\cong C_{13}\times C_2$ act on~$\mathbb{F}_{27}$ by additive shifts and multiplicative shifts, respectively. These two actions together yield the action on~$\mathbb{F}_{27}$ of the $702$-element general affine group $\mathrm{GA}_1(\mathbb{F}_{27})=\mathbb{F}_{27}\rtimes\mathbb{F}_{27}^{\times}$. We will be interested in a twice less group, namely, the $351$-element group
$$
G_{351}=\mathbb{F}_{27}\rtimes(\mathbb{F}_{27}^{\times})^2\cong C_3^3\rtimes C_{13},
$$
where $(\mathbb{F}_{27}^{\times})^2\subset \mathbb{F}_{27}^{\times}$ is the index~$2$ subgroup consisting of squares.
Our aim is to construct $G_{351}$-invariant combinatorial manifolds on the vertex set $\mathbb{F}_{27}$ that will be triangulations of manifolds like the octonionic projective plane. To work with simplicial complexes on the set $\mathbb{F}_{27}$ we need to number the elements of this set, that is, to choose a bijection $\varphi\colon \mathbb{F}_{27}\to [27]$, where we use notation $[n]=\{1,\ldots,n\}$. We will always conveniently identify $\mathbb{F}_{27}=\mathbb{F}_3(\alpha)$, where $\alpha^3-\alpha-1 = 0$. Then $\alpha^{13} = 1$ and hence $\alpha^k$, where $k=0,\ldots,12$, are all nonzero squares in~$\mathbb{F}_{27}$, and $-\alpha^k$, where $k=0,\ldots,12$, are all non-squares in~$\mathbb{F}_{27}$. We choose the numbering~$\varphi$ as follows:
\begin{align*}
\varphi(\alpha^k)&=k+1,&k&=0,\ldots,12,\\
\varphi(-\alpha^k)&=k+14,&k&=0,\ldots,12,\\
\varphi(0)&=27.&&
\end{align*}
It is easy to see that the group $G_{351}$ is generated by the multiplicative shift by~$\alpha$ and the additive shift by~$1$. In the chosen numbering, these two elements of~$G_{351}$ correspond to the permutations
\begin{gather*}
A = (1\ 2\ 3\ 4\ 5\ 6\ 7\ 8\ 9\ 10\ 11\ 12\ 13)(14\ 15\ 16\ 17\ 18\ 19\ 20\ 21\ 22\ 23\ 24\ 25\ 26),\\
B = (1\ 14\ 27)(2\ 4\ 10)(3\ 22\ 13)(5\ 6\ 21)(7\ 25\ 11)(8\ 19\ 18)(9\ 16\ 26)(12\ 20\ 24)(15\ 23\ 17),
\end{gather*}
respectively. So $G_{351}$ can be regarded as the subgroup of~$S_{27}$ generated by~$A$ and~$B$.
Now, consider the normalizer~$N(G_{351})$ of~$G_{351}$ in~$S_{27}$. It is easy to point out two permutations of~$\mathbb{F}_{27}$ that normalize~$G_{351}$, the sign reversal $x\mapsto -x$ and the Frobenius automorphism $x\mapsto x^3$. In the numbering we use they correspond to the permutations
\begin{gather*}
S =(1\ 14)(2\ 15)(3\ 16)(4\ 17)(5\ 18)(6\ 19)(7\ 20)(8\ 21)(9\ 22)(10\ 23)(11\ 24)(12\ 25)(13\ 26),\\
F=(2\ 4\ 10)(3\ 7\ 6)(5\ 13\ 11)(8\ 9\ 12)(15\ 17\ 23)(16\ 20\ 19)(18\ 26\ 24)(21\ 22\ 25),
\end{gather*}
respectively.
\begin{propos}\label{propos_normalizer}
The normalizer~$N(G_{351})$ is the $2106$-element group $G_{351}\rtimes (C_2\times C_3)$, where the multiples~$C_2$ and~$C_3$ are generated by~$S$ and~$F$, respectively. Therefore, $$N(G_{351})/G_{351}\cong C_2\times C_3.$$
\end{propos}
The proof of this proposition will be given in Section~\ref{section_no_add_sym}.
Consider the set $\mathcal{S}$ of all simplicial complexes on the vertex set~$[27]$ that are invariant with respect to the subgroup~$G_{351}\subset S_{27}$.
The normalizer~$N(G_{351})$ acts on~$\mathcal{S}$ and the action of the group~$G_{351}$ on~$\mathcal{S}$ is trivial. So we obtain the action of the $6$-element quotient group $N(G_{351})/G_{351}\cong C_2\times C_3$ on~$\mathcal{S}$.
\begin{theorem}\label{theorem_main_detail}
There are exactly $24$ simplicial complexes on the vertex set~$[27]$ that are
$G_{351}$-invariant combinatorial $16$-manifolds not homeomorphic to the sphere. The group $N(G_{351})/G_{351}\cong C_2\times C_3$ acts freely on the set of these $24$ simplicial complexes; hence, they are divided into $4$ orbits of $6$ complexes so that the complexes in each orbit are isomorphic to each other. The combinatorial manifolds~$K_1$, $K_2$, $K_3$, and~$K_4$ whose $G_{351}$-orbits of $16$-simplices are listed in Tables~\mbox{\ref{table_1234}--\ref{table_4}} are representatives of these four orbits. They are vertex-transitive, satisfy properties~(1) and~(2) in Theorem~\ref{theorem_main}, and are pairwise non-isomorphic. Besides, for each~$i$, the symmetry group~$\mathop{\mathrm{Sym}}\nolimits(K_i)$ is exactly~$G_{351}$.
\end{theorem}
\begin{remark}
Tables~\mbox{\ref{table_1234}--\ref{table_4}} are organized as follows. Every simplex is encoded by a row of $27$ binary digits. The $i$\textsuperscript{th} from the left digit is $1$ whenever $i$ is a vertex of the simplex, and is~$0$ otherwise.
Tables~\mbox{\ref{table_1234}--\ref{table_4}} contain not all $16$-simplices of the simplicial complexes~$K_1$, $K_2$, $K_3$, and~$K_4$ but exactly one representative of every $G_{351}$-orbit of such simplices. We conveniently interpret every row of $27$ binary digits in Tables~\mbox{\ref{table_1234}--\ref{table_4}} as a reversed binary notation for a number. For instance, the first row $$111111111110111011100000000$$ in Table~\ref{table_1234} corresponds to the number
$$1110111011111111111_2=489472.$$
With this interpretation, each of Tables~\mbox{\ref{table_1234}--\ref{table_4}} contains the smallest representatives of the orbits; in addition, each table uses ordering from smallest to largest.
\end{remark}
\begin{remark}
Attached to this paper are four text files \texttt{K1.dat}, \texttt{K2.dat}, \texttt{K3.dat}, and \texttt{K4.dat} containing the lists of (the smallest) representatives for the $G_{351}$-orbits of $16$-simplices of $K_1$, $K_2$, $K_3$, and~$K_4$, respectively, in the same format as in Tables~\mbox{\ref{table_1234}--\ref{table_4}}. The first row of each of the files contains the number $286$ of the orbits of $16$-simplices. These files fit the format for simplicial complexes used in our programs, see~\cite{Gai-prog}.
\end{remark}
\begin{remark}
Note that the numbering of the four combinatorial manifolds~$K_1$, $K_2$, $K_3$, and~$K_4$ is chosen so that to stress the following interesting property: If a $16$-simplex~$\sigma$ belongs to~$K_i$ and~$K_j$ for $i<j$, then $\sigma$ also belongs to all intermediate complexes~$K_k$ with $i<k<j$. We see that
\begin{itemize}
\item $K_2$ is obtained from~$K_1$ by replacing certain $62$ \ $G_{351}$-orbits of $16$-simplices with other $62$ \ $G_{351}$-orbits of $16$-simplices,
\item $K_3$ is obtained from~$K_2$ by replacing certain $27$ \ $G_{351}$-orbits of $16$-simplices with other $27$ \ $G_{351}$-orbits of $16$-simplices,
\item $K_4$ is obtained from~$K_3$ by replacing certain $115$ \ $G_{351}$-orbits of $16$-simplices with other $115$ \ $G_{351}$-orbits of $16$-simplices.
\end{itemize}
The relationship between~$K_2$ and~$K_3$ is clear, see Section~\ref{section_moves}. Namely, $K_3$ is obtained from~$K_2$ by replacing $351$ subcomplexes of the form
$$
(\Delta_1*\partial\Delta_2)\cup(\Delta_2*\partial\Delta_3)\cup(\Delta_3*\partial\Delta_1),
$$
where $\dim\Delta_1=\dim\Delta_2=\dim\Delta_3=8$, with the corresponding subcomplexes
$$
(\partial\Delta_1*\Delta_2)\cup(\partial\Delta_2*\Delta_3)\cup(\partial\Delta_3*\Delta_1),
$$
In particular, this implies that $K_2$ and~$K_3$ are PL homeomorphic.
The relationships between~$K_1$ and~$K_2$ and between~$K_3$ and~$K_4$ are less clear, and seem to be interesting to study further. At the moment, the author does not know whether it is true that $K_1$ or~$K_4$ is PL homeomorphic to~$K_2$ and~$K_3$.
If we consider all the $24$ combinatorial manifolds $qK_i$ from Theorem~\ref{theorem_main_detail}, where $q$ runs over the elements of the group~$C_2\times C_3$ generated by~$S$ and~$F$, then it is interesting to note that the four combinatorial manifolds~$K_1$, $K_2$, $K_3$, and~$K_4$ form a cluster in the sense that there pairwise intersections are large enough, namely consists of at least~$112$ \ $G_{351}$-orbits of $16$-simplices. On the contrary, each intersection $q_1K_i\cap q_2 K_j$ with $q_1\ne q_2$ contains sufficiently small number of $16$-simplices, namely, at most~$12$ \ $G_{351}$-orbits of $16$-simplices. A precise numbers of $G_{351}$-orbits of common $16$-simplices are given in Table~\ref{table_intersections}. (Certainly, the intersections~$q_1K_i\cap q_2 K_j$ and~$qq_1K_i\cap qq_2 K_j$ are isomorphic for each $q\in C_2\times C_3$, so the information in this table is enough to restore the numbers of orbits of $16$-simplices in all intersections~$q_1K_i\cap q_2 K_j$.)
\begin{table}
\caption{Numbers of common orbits of $16$-simplices}\label{table_intersections}
\begin{tabular}{|c|cccccccc|}
\hline
$\phantom{\strut^2}$ & $K_1$ & $K_2$ & $K_3$ & $K_4$ &
$SK_1$ & $SK_2$ & $SK_3$ & $SK_4$ \\
\hline
$K_1$ & 286 & 224 & 201 & 112 & 12 & 8 & 8 & 4 \\
$K_2$ & 224 & 286 & 259 & 149 & 8 & 5 & 5 & 3 \\
$K_3$ & 201 & 259 & 286 & 171 & 8 & 5 & 5 & 3 \\
$K_4$ & 112 & 149 & 171 & 286 & 4 & 3 & 3 & 3 \\
\hline
\hline
$\phantom{\strut^2}$ & $FK_1$ & $FK_2$ & $FK_3$ & $FK_4$ &
$SFK_1$ & $SFK_2$ & $SFK_3$ & $SFK_4$ \\
\hline
$K_1$ & 9 & 6 & 6 & 5 & 2 & 0 & 0 & 0 \\
$K_2$ & 7 & 6 & 6 & 5 & 5 & 2 & 1 & 1 \\
$K_3$ & 7 & 6 & 6 & 4 & 5 & 4 & 3 & 3 \\
$K_4$ & 9 & 7 & 7 & 7 & 6 & 6 & 5 & 6 \\
\hline
\end{tabular}
\end{table}
\end{remark}
\section{Scheme of proof of Theorem~\ref{theorem_main_detail}}\label{section_scheme}
The proof will consist of 5 steps.
\textsl{Step 1.} We will give an algorithm that, being given
\begin{itemize}
\item a number $m$ and a finite subgroup $G\subset S_m$,
\item a dimension~$d$,
\item a positive integer~$N$,
\end{itemize}
produces the list of all simplicial complexes~$K$ on the vertex set~$[m]$ satisfying the following properties:
\begin{itemize}
\item $K$ is a $G$-invariant weak $d$-pseudomanifold,
\item for any two simplices $\sigma,\tau\in K$, $\sigma\cup\tau$ is not the whole set~$[m]$,
\item the number of $d$-simplices of~$K$ is at least $N$.
\end{itemize}
We are interested in the case
\begin{equation}\label{eq_main_case_init}
m=27,\qquad d =16,\qquad G=G_{351},\qquad N=100386.
\end{equation}
So our aim is an algorithm that, when implemented on a computer, will complete in reasonable amount of time in this case. We will describe such an algorithm in Section~\ref{section_find}. In the case~\eqref{eq_main_case_init}, a computer implementation of this algorithm yields the following result.
\begin{propos}\label{propos_24_pseudo}
There are exactly $24$ simplicial complexes~$K$ on the vertex set~$[27]$ with the following properties:
\begin{itemize}
\item $K$ is a $G_{351}$-invariant weak $16$-pseudomanifold,
\item for any two simplices $\sigma,\tau\in K$, $\sigma\cup\tau$ is not the whole set~$[m]$,
\item $K$ has at least $100386$ top-dimensional simplices.
\end{itemize}
Four of these $24$ complexes are the complexes~$K_1$, $K_2$, $K_3$, and~$K_4$ whose orbits of simplices are listed in Tables~\mbox{\ref{table_1234}--\ref{table_4}}; the other $20$ complexes are obtained from those four by the action of the group~$C_2\times C_3$ generated by~$S$ and~$F$.
\end{propos}
At the moment, we do not know yet that the simplicial complexes~$K_1$, $K_2$, $K_3$, and~$K_4$ provided by our algorithm are combinatorial manifolds that are not homeomorphic to the sphere. Indeed, they could be not combinatorial manifolds or they could be homeomorphic to~$S^{16}$. However, recall that every $G_{351}$-invariant combinatorial $16$-manifold not homeomorphic to~$S^{16}$ (if exists) must satisfy the complementarity condition and have exactly $100386$ top-dimensional simplices. So all such combinatorial manifolds are in the list of the $24$ pseudomanifolds from Proposition~\ref{propos_24_pseudo}. Therefore, up to isomorphism they are in the list $K_1$, $K_2$, $K_3$, $K_4$. So these four simplicial complexes constitute a list of candidates that we need to study further.
For each $i$, a direct computation shows that the face numbers~$f_k(K_i)$ are exactly as in Table~\ref{table_f} and hence $\chi(K_i)=3$. Therefore, $K_i$ is not homeomorphic to~$S^{16}$.
\textsl{Step 2.} Next, we need to prove the following proposition.
\begin{propos}\label{propos_combman}
The weak pseudomanifolds~$K_1$, $K_2$, $K_3$, and~$K_4$ are combinatorial manifolds.
\end{propos}
To prove that $K_i$ is a combinatorial manifold, we need to check that the links of its vertices are PL homeomorphic to~$S^{15}$. There are several standard methods for doing this, mostly implemented in the system \texttt{GAP}, such as the program \texttt{BISTELLAR} by Bj\"orner and Lutz, see~\cite{BjLu00}, and several programs based on discrete Morse functions, see~\cite{BeLu14} and references therein. Nevertheless, most of these methods are useless in our situation. The matter is that the link of a vertex of each~$K_i$ has a very large size, namely, it has dimension~$15$, $26$ vertices, $63206$ top-dimensional simplices, and $29193216$ (non-empty) simplices of all dimensions. In addition, this link has few symmetries, namely, its symmetry group is only~$C_{13}$. In such a situation, it is hopeless to try to simplify the complex by bistellar moves or any other combinatorial moves. Also, standard methods for constructing discrete Morse functions are too slow in our situation, since they require working with simplices of all dimensions whose number is really huge. Instead, we use an approach based on the notion of nonevasiveness of simplicial complexes introduced by Kahn, Saks, and Sturtevant~\cite{KSS84}. Its advantage is that we can check the nonevasiveness of a simplicial complex by working only with its vertices and maximal simplices, without referring directly to simplices of intermediate dimensions. Conceptually, our approach is close to that of Engstr\"om~\cite{Eng09}, though we prefer to prove the nonevasiveness of complexes directly from the definition rather than to construct any kind of discrete Morse functions.
An explicit algorithm that, when implemented on a computer, checks that $K_1$, $K_2$, $K_3$, and~$K_4$ are combinatorial manifolds is described in Section~\ref{section_check_manifold}. Since $K_1$, $K_2$, $K_3$, and~$K_4$ are not homeomorphic to~$S^{16}$, it now follows from a result of Brehm and K\"uhnel~\cite{BrKu87} that they are manifolds like the octonionic projective plane and satisfy conditions~(a)--(e) in the introduction.
\textsl{Step 3.} Now, let us show that the simplicial complexes $K_1$, $K_2$, $K_3$, and~$K_4$ are pairwise non-isomorphic. There are several ways to do this. Probably, the easiest one is as follows. Each simplicial complex~$K_i$ has $f_{14}=3309696$ simplices of dimension~$14$. For each $14$-simplex $\rho\in K_i$, we can compute the number~$s(\rho)$ of $16$-simplices $\sigma\in K_i$ such that $\rho\subset \sigma$. It turns out that the number $s(\rho)$ varies from $3$ to~$9$ for each of the complexes~$K_i$. Now, we can study the distribution of the function~$s(\rho)$, that is, compute the number of simplices $\rho$ with each given value of~$s(\rho)$. The result of the computation is shown in Table~\ref{table_distribution}. We see that the distributions in the four columns are pairwise different and hence the simplicial complexes~$K_1$, $K_2$, $K_3$, and~$K_4$ are pairwise non-isomorphic.
\begin{table}
\caption{Numbers of $14$-simplices~$\rho$ with the given number~$s(\rho)$ of adjacent $16$-simplices}\label{table_distribution}
\begin{tabular}{|c|c|c|c|c|}
\hline
$s(\rho)$ & $K_1$ & $K_2$ & $K_3$ & $K_4$\\
\hline
3 & 849771 & 953316 & 940446 & 868140\\
4 & 1509651 & 1364688 & 1363986 & 1447524\\
5 & 697788 & 690066 & 724815 & 764829\\
6 & 201942 & 237978 & 220662 & 180297\\
7 & 40716 & 55107 & 51597 & 40716 \\
8 & 9477 & 7020 & 7722 & 6669\\
9 & 351 & 1521 & 468 & 1521\\
\hline
\end{tabular}
\end{table}
\textsl{Step 4.} Since each $K_i$ is $G_{351}$-invariant and $G_{351}$ acts transitively on~$\mathbb{F}_{27}$, we see that $K_i$ is vertex-transitive. Let us prove that $K_i$ satisfies conditions~(1) and~(2) from Theorem~\ref{theorem_main}.
Recall that the group~$G_{351}$ coincides with the group of all affine transformations of~$\mathbb{F}_{27}$ of the form $x\mapsto ax+b$, where $a\in(\mathbb{F}_{27}^{\times})^2$ and $b\in \mathbb{F}_{27}$. We need to prove that $G_{351}$ acts transitively on the set of undirected edges, that is, on the set of two-element subsets of~$\mathbb{F}_{27}$. Consider an arbitrary subset $\{u,v\}\subset \mathbb{F}_{27}$, where $u\ne v$. Swapping $u$ and~$v$, we may achieve that $v-u$ is a square. Then the transformation $x\mapsto (v-u)x+u$ belongs to~$G_{351}$ and takes $\{0,1\}$ to~$\{u,v\}$, hence the required transitivity.
To show that $G_{351}$ acts freely on the set of $16$-simplices of every~$K_i$, we prove a more general assertion.
\begin{propos}\label{propos_free}
The group~$G_{351}$ acts freely on the set of all $17$-element subsets of~$\mathbb{F}_{27}$.
\end{propos}
\begin{proof}
Any nontrivial transformation $g\colon x\mapsto ax+b$ in~$G_{351}$ can be written in the form
$$
x\mapsto
\left\{
\begin{aligned}
&b\left(b^{-1}x+1\right)&&\text{if}\ a=1,\,b\in (\mathbb{F}_{27}^{\times})^2 \\
&-b\left(-b^{-1}x-1\right)&&\text{if}\ a=1,\,b\notin (\mathbb{F}_{27}^{\times})^2,\\
&a(x+c)-c&&\text{if}\ a\ne 1,
\end{aligned}
\right.
$$
where in the third case $c=(a-1)^{-1}b$. Since $a$ is a square, in the third case we have $a=\alpha^k$ with $k\in\{1,\ldots,12\}$. Hence, $g$ is conjugate in~$G_{351}$ to one of the elements~$B^{\pm 1}$ and~$A^k$, where $k=1,\ldots 12$. Therefore, the permutation~$g\in S_{27}$ has the same cycle type as either~$A$ or~$B$. Now, a permutation of either of these cycle types cannot stabilize a $17$-element subset, since no union of its cycles has the sum of lengths~$17$.
\end{proof}
\textsl{Step 5.} Finally, we need to prove the following proposition.
\begin{propos}\label{propos_SymK}
For every $i=1,2,3,4$, $\mathop{\mathrm{Sym}}\nolimits(K_i)=G_{351}$.
\end{propos}
By construction, $K_i$ is invariant under the action of~$G_{351}$. So to prove Proposition~\ref{propos_SymK}, we need to check that $K_i$ has no additional symmetries. This will be done in Section~\ref{section_no_add_sym}.
\smallskip
Thus, we have proved Theorem~\ref{theorem_main_detail} modulo Propositions~\ref{propos_24_pseudo}, \ref{propos_combman}, and \ref{propos_SymK}. These three propositions will be proved in Sections~\ref{section_find}, \ref{section_check_manifold} and~\ref{section_no_add_sym}, respectively.
\section{Algorithm for finding triangulations}\label{section_find}
In this section we describe an algorithm that solves the problem from Step~$1$ in the scheme of the proof of Theorem~\ref{theorem_main_detail}. Recall that the algorithm should produce the list of all weak pseudomanifolds~$K$ of the given dimension~$d$ on the given set of vertices~$[m]$ such that $K$ is invariant with respect to the given subgroup~$G\subset S_m$, contains at least the given number~$N$ of $d$-simplices, and satisfies the following half of the complementarity condition:
\begin{itemize}
\item for any two simplices $\sigma,\tau\in K$, $\sigma\cup\tau$ is not the whole set~$[m]$.
\end{itemize}
We are mostly interested in the case
\begin{equation}\label{eq_main_case}
m=27,\qquad d =16,\qquad G=G_{351},\qquad N=100386,
\end{equation}
and our aim is an algorithm that, when implemented on a computer, will complete in reasonable amount of time in this case.
We conveniently reformulate our problem as follows. Let $\mathcal{O}$ be the set of all $G$-orbits of $(d+1)$-element subsets of~$[m]$. Each desired simplicial complex~$K$ is uniquely characterized by the set of its $d$-simplices. Moreover, this set must be $G$-invariant, so each orbit $\mathfrak{o}\in\mathcal{O}$ is either completely included in~$K$, or none of subsets in~$\mathfrak{o}$ is included in~$K$. So our goal is equivalently reformulated in the following way:
\smallskip
\textit{Produce the list of all subsets $\mathfrak{X}\subset \mathcal{O}$ satisfying the following properties:
\begin{itemize}
\item[(i)] For each $d$-element subset $\rho\subset [m]$, the number of $(d+1)$-element subsets $\sigma$ that contain~$\rho$ and belong to one of the orbits in~$\mathfrak{X}$ is either $0$ or~$2$.
\item[(ii)] For any two subsets $\sigma$ and~$\tau$ each of which belongs to one of the orbits in~$\mathfrak{X}$, $\sigma\cup\tau\ne [m]$.
\item[(iii)] The total number of subsets in all orbits belonging to~$\mathfrak{X}$ is at least~$N$.
\end{itemize}
}
The cardinality~$|\mathcal{O}|$ is approximately $\frac{1}{|G|}{m\choose d+1}$. In the most interesting for us case~\eqref{eq_main_case}, the action of~$G_{351}$ on the set of $17$-element subsets of~$[27]$ is free by Proposition~\ref{propos_free}. So in this case, we have an exact equality
\begin{gather*}
|\mathcal{O}|=\frac{1}{|G|}{m\choose d+1}=\frac{1}{351}\cdot{27\choose 17}=24035.
\end{gather*}
\begin{defin}
We will say that an orbit $\mathfrak{o}\in\mathcal{O}$ is \textit{admissible} if it satisfies neither of the following two conditions:
\begin{itemize}
\item there exist subsets $\sigma,\tau\in\mathfrak{o}$ such that $\sigma\cup\tau=[m]$,
\item there exists a $d$-element subset $\rho\subset [m]$ that is contained in at least $3$ subsets in~$\mathfrak{o}$.
\end{itemize}
\end{defin}
We denote by $\mathcal{O}_{\mathrm{adm}}$ the subset of~$\mathcal{O}$ consisting of all admissible orbits.
Only admissible orbits can enter a set~$\mathfrak{X}$ satisfying properties~(i) and~(ii). So we start with listing all the admissible orbits. This operation is very fast, since it requires only a simple enumeration of orbits in~$\mathcal{O}$. In the case~\eqref{eq_main_case}, we have $18546$ admissible orbits. Technically, each subset is encoded by a binary number as in Tables~\ref{table_1234}--\ref{table_4} and listing the admissible orbits means listing of the smallest representatives of them.
Further algorithm consists of two stages.
\textsl{Stage 1. Initial prohibition of some pairs of orbits and listing of all adjacency groups.} We are interested in finding pairs~$\{\mathfrak{a},\mathfrak{b}\}$ of (different) admissible orbits that are \textit{prohibited} in the sense that they cannot enter~$\mathfrak{X}$ simultaneously. There are two reasons for such prohibition.
The first one is as follows. If there are subsets $\sigma\in\mathfrak{a}$ and~$\tau\in\mathfrak{b}$ such that $\sigma\cup\tau=[m]$, then the pair~$\{\mathfrak{a},\mathfrak{b}\}$ is prohibited because its presence in~$\mathfrak{X}$ would violate condition~(ii). All prohibited pairs of orbits of this kind are found by a direct enumeration over all pairs of admissible orbits.
To describe the second reason for prohibition of a pair of orbits, we need to introduce some terminology. Suppose that $\mathfrak{o}\in \mathcal{O}_{\mathrm{adm}}$ and $\rho\subset [m]$ is a $d$-element subset. Since $\mathfrak{o}$ is admissible, the number of subsets $\sigma \in\mathfrak{o}$ that contain~$\rho$ does not exceed~$2$. We will say that an orbit $\mathfrak{o}$ is \textit{once adjacent} to~$\rho$ if $\mathfrak{o}$ contains exactly one subset~$\sigma$ such that $\sigma\supset\rho$ and is \textit{twice adjacent} to~$\rho$ if $\mathfrak{o}$ contains two subsets~$\sigma$ such that $\sigma\supset\rho$.
Now, if $\mathfrak{a}$ is twice adjacent to some $d$-element subset~$\rho$ and $\mathfrak{b}$ is either once or twice adjacent to the same~$\rho$, then the pair~$\{\mathfrak{a},\mathfrak{b}\}$ is prohibited because its presence in~$\mathfrak{X}$ would violate condition~(i). All prohibited pairs of orbits of this kind are found by a direct enumeration over~$\rho$. The adjacency of an orbit in~$\mathcal{O}_{\mathrm{adm}}$ to~$\rho$ will not change if replace $\rho$ with another $d$-element subset $\rho'$ in the $G$-orbit of~$\rho$. So, in fact, the enumeration is not over all $d$-element subsets but only over all their $G$-orbits. Moreover, we can restrict ourselves to only those~$\rho$ that are contained in at least one of the smallest representatives of admissible orbits.
In addition, at the same stage, we find all \textit{adjacency groups}~$\mathcal{A}_{\rho}$, where $\rho$ runs over $G$-orbits of $d$-element subsets. By definition, $\mathcal{A}_{\rho}$ is the subset of~$\mathcal{O}_{\mathrm{adm}}$ consisting of all orbits that are once adjacent to~$\rho$. The role of these groups in the further algorithm is as follows. By condition~(i), the set $\mathfrak{X}$ must contain either no or exactly two orbits in each adjacency groups. Linking adjacency groups to subsets~$\rho$ is no longer important to us, we need only the list of all non-empty adjacency groups $\mathcal{A}_1,\ldots,\mathcal{A}_q$. In the case~\eqref{eq_main_case}, we obtain $36059$ non-empty adjacency groups.
\begin{remark}
In fact, in out implementation of the algorithm we never check that the generated adjacency groups are pairwise different. It may happen that different orbits of subsets~$\rho$ lead to identical adjacency groups. We prefer to work with a list of adjacency groups that can contain repetitions, rather than eliminating repetitions.
\end{remark}
From now on, we may forget about the nature of the elements of~$\mathcal{O}_{\mathrm{adm}}$. Indeed, our goal is reduced to the following combinatorial problem:
\smallskip
\textit{We have a finite set~$\mathcal{O}_{\mathrm{adm}}$. To each $\mathfrak{o}\in\mathcal{O}_{\mathrm{adm}}$ is assigned a positive integer~$|\mathfrak{o}|$ called its size. Some pairs of elements of~$\mathcal{O}_{\mathrm{adm}}$ are prohibited. Besides, we have a list $\mathcal{A}_1,\ldots,\mathcal{A}_q$ of subsets of~$\mathcal{O}_{\mathrm{adm}}$ called adjacency groups. Our aim is to list all subsets $\mathfrak{X}\subset \mathcal{O}_{\mathrm{adm}}$ such that
\begin{itemize}
\item each intersection $\mathfrak{X}\cap\mathcal{A}_i$ is either empty or consists of exactly two elements,
\item $\mathfrak{X}$ is not allowed to contain simultaneously both elements of any prohibited pair,
\item $\sum_{\mathfrak{o}\in\mathfrak{X}}|\mathfrak{o}|\ge N$.
\end{itemize}
}
\textsl{Stage 2. Selection of orbits.} At this stage we solve the above combinatorial problem. We will not use the nature of elements of~$\mathcal{O}_{\mathrm{adm}}$ any more. However, for convenience we will continue to call them orbits.
Suppose that, at some moment, we have the following situation:
\begin{itemize}
\item Some orbits in~$\mathcal{O}_{\mathrm{adm}}$ are \textit{taken}, which means that we have decided that they are included in~$\mathfrak{X}$. Some orbits in~$\mathcal{O}_{\mathrm{adm}}$ are \textit{removed}, which means that we have decided that they are not included in~$\mathfrak{X}$. All other orbits are called \textit{indeterminate}.
\item Some unordered pairs of indeterminate orbits are \textit{prohibited}, which means that they are not allowed to enter~$\mathfrak{X}$ simultaneously.
\item Some ordered pairs of indeterminate orbits form \textit{requirements}. A requirement from $\mathfrak{a}$ to~$\mathfrak{b}$ (denoted~$\mathfrak{a}\to\mathfrak{b}$) means that we must include~$\mathfrak{b}$ to~$\mathfrak{X}$ whenever we include~$\mathfrak{a}$.
\end{itemize}
These data should satisfy natural logical conditions, namely (a) if $\mathfrak{a}\to\mathfrak{b}$, then $\{\mathfrak{a},\mathfrak{b}\}$ is not prohibited, (b) if $\mathfrak{a}\to\mathfrak{b}$ and $\mathfrak{b}\to\mathfrak{c}$, then $\mathfrak{a}\to\mathfrak{c}$, and (c) if $\mathfrak{a}\to\mathfrak{b}$ and $\{\mathfrak{b},\mathfrak{c}\}$ is prohibited, then $\{\mathfrak{a},\mathfrak{c}\}$ is also prohibited.
Note that the initial situation is of this type. Initially all orbits are indeterminate, we have several prohibited pars and no requirements.
Having a situation of the describe type, we want to improve our knowledge by (1) increasing the numbers of taken and removed orbits or (2) increasing the numbers of prohibited pairs and requirements without changing taken and removed orbits. To do this we study the adjacency groups.
For an adjacency group~$\mathcal{A}$, we may have several situations:
\smallskip
\textsl{Case 1. More than two orbits in~$\mathcal{A}$ are already taken.} This means that our task is inconsistent and we should stop seeking for~$\mathfrak{X}$.
\smallskip
\textsl{Case 2. Exactly two orbits in~$ \mathcal{A}$ are already taken.} This means that all other orbits in~$ \mathcal{A}$ cannot enter~$\mathfrak{X}$, so we should remove them. Note that removing an orbit~$\mathfrak{a}$, we always simultaneously remove all orbits~$\mathfrak{b}$ (not necessarily belonging to~$ \mathcal{A}$) such that $\mathfrak{b}\to\mathfrak{a}$. After doing this, we may forget about the adjacency group~$ \mathcal{A}$. It will not give us any more information in the future.
\smallskip
\textsl{Case 3. Exactly one orbit in~$ \mathcal{A}$ is already taken.} Let $\mathfrak{a}$ be this taken orbit, and let $\mathfrak{b}_1,\ldots,\mathfrak{b}_k$ be all indeterminate orbits in~$ \mathcal{A}$. First, if $k=0$, then our task is inconsistent again, and we stop. Second, if $k=1$, then the unique indeterminate orbit $\mathfrak{b}$ should be taken. Note that taking an orbit~$\mathfrak{b}$, we always simultaneously take all orbits~$\mathfrak{c}$ such that $\mathfrak{b}\to\mathfrak{c}$ and remove all orbits~$\mathfrak{d}$ such that the pair~$\{\mathfrak{b},\mathfrak{d}\}$ was prohibited. Third, suppose that $k\ge 2$. In this case we cannot take or remove any of the orbits $\mathfrak{b}_1,\ldots,\mathfrak{b}_k$. However, we can guarantee that any two of these orbits cannot enter~$\mathfrak{X}$ simultaneously. Hence, we should prohibit all pairs $\{\mathfrak{b}_i,\mathfrak{b}_j\}$ with $i\ne j$. Note that prohibiting a pair $\{\mathfrak{b},\mathfrak{b}'\}$, we always prohibit simultaneously all pairs $\{\mathfrak{c},\mathfrak{c}'\}$ such that $\mathfrak{c}\to\mathfrak{b}$ (or $\mathfrak{c}=\mathfrak{b}$) and $\mathfrak{c}'\to\mathfrak{b}'$ (or $\mathfrak{c}'=\mathfrak{b}'$). Moreover, if on this way we are faced with the need to prohibit a pair $\{\mathfrak{c},\mathfrak{c}\}$ (say, if $\mathfrak{c}\to\mathfrak{b}$ and simultaneously $\mathfrak{c}\to \mathfrak{b}'$) , then we should remove~$\mathfrak{c}$.
\smallskip
\textsl{Case 4. No orbit in~$ \mathcal{A}$ is taken.} Consider 3 subcases:
\smallskip
\textsl{Subcase 4a. There are three distinct indeterminate orbits $\mathfrak{a}$, $\mathfrak{b}$, and~$\mathfrak{c}$ in~$ \mathcal{A}$ such that $\mathfrak{a}\to \mathfrak{b}$ and $\mathfrak{a}\to\mathfrak{c}$.} If $\mathfrak{a}$ were in~$\mathfrak{X}$, then both orbits~$\mathfrak{b}$ and~$\mathfrak{c}$ should also be in~$\mathfrak{X}$, and the intersection $\mathfrak{X}\cap \mathcal{A}$ would contain more than two orbits, which is impossible. Therefore, we should remove the orbit~$\mathfrak{a}$ (and all orbits that require it).
\smallskip
\textsl{Subcase 4b. We are not in Subcase~4a but there are two distinct indeterminate orbits $\mathfrak{a}$ and~$\mathfrak{b}$ in~$ \mathcal{A}$ such that $\mathfrak{a}\to \mathfrak{b}$.} If $\mathfrak{a}$ were in~$\mathfrak{X}$, then~$\mathfrak{b}$ would also be in~$\mathfrak{X}$ and hence no other orbit in~$ \mathcal{A}$ could be in~$\mathfrak{X}$. Hence, we should prohibit all pairs $\{\mathfrak{a},\mathfrak{c}\}$, where $\mathfrak{c}$ is an indeterminate orbit in~$ \mathcal{A}$ different from~$\mathfrak{a}$ and~$\mathfrak{b}$. (Again, together with any such pair of orbits we prohibit all dependent pairs as in Case~3.)
\smallskip
\textsl{Subcase 4c. There is an indeterminate orbit~$\mathfrak{a}$ in~$ \mathcal{A}$ such that the pairs $\{\mathfrak{a},\mathfrak{b}\}$ are prohibited for all other indeterminate orbits~$\mathfrak{b}$ in~$ \mathcal{A}$.} (In particular, this subcase occurs if there is only one indeterminate orbits in~$ \mathcal{A}$.) Then we should remove~$\mathfrak{a}$ and all orbits that require it.
\smallskip
\textsl{Subcase 4d. There is an indeterminate orbit~$\mathfrak{a}$ in~$ \mathcal{A}$ such that the pairs $\{\mathfrak{a},\mathfrak{b}\}$ are prohibited for all but one other indeterminate orbits~$\mathfrak{b}$ in~$ \mathcal{A}$.} (In particular, this subcase occurs if there are exactly two indeterminate orbits in~$ \mathcal{A}$.) Let $\mathfrak{c}$ be the only indeterminate orbit in~$ \mathcal{A}$ such that $\mathfrak{c}\ne\mathfrak{a}$ and the pair $\{\mathfrak{a},\mathfrak{c}\}$ is not prohibited. Then $\mathfrak{c}$ will belong to~$\mathfrak{X}$ whenever $\mathfrak{a}$ belongs to~$\mathfrak{X}$. So we should add a requirement $\mathfrak{a}\to \mathfrak{c}$. Note that adding such a requirement, we simultaneously add all requirements $\mathfrak{a}'\to\mathfrak{c}'$, where $\mathfrak{a}'\to\mathfrak{a}$ (or $\mathfrak{a}'=\mathfrak{a}$) and $\mathfrak{c}\to\mathfrak{c}'$ (or $\mathfrak{c}=\mathfrak{c}'$). Besides, we simultaneously prohibit all pairs~$\{\mathfrak{a}',\mathfrak{d}\}$, where $\mathfrak{a}'\to\mathfrak{a}$ (or $\mathfrak{a}'=\mathfrak{a}$) and the pair $\{\mathfrak{c},\mathfrak{d}\}$ was prohibited. Also, as in Case~3, if on this way we are faced with the need to prohibit a pair $\{\mathfrak{d},\mathfrak{d}\}$, then we remove~$\mathfrak{d}$.
\smallskip
When we have examined an adjacency group~$\mathcal{A}$ as described above in Cases~1--4, it makes no sense for us to return to the same adjacency group~$\mathcal{A}$ until a new indeterminate orbit in~$\mathcal{A}$ is taken or removed or a new pair of orbits in~$\mathcal{A}$ is prohibited or a new requirement between orbits in~$\mathcal{A}$ is added.
So the algorithm is organized as follows. At every moment we have a waiting list of adjacency groups that are to be examined. Initially all adjacency groups are in the list. Adjacency groups in the list are examined one by one as described above. After an adjacency group is examined, we expel it from the list. However, every time a new indeterminate orbit~$\mathfrak{a}$ becomes either taken or removed, we put back to the waiting list all adjacency groups containing~$\mathfrak{a}$. Similarly, every time a new pair of orbits~$\{\mathfrak{a},\mathfrak{b}\}$ is prohibited or a new requirement $\mathfrak{a}\to\mathfrak{b}$ is created, we put back to the waiting list all adjacency groups containing both~$\mathfrak{a}$ and~$\mathfrak{b}$. This process stops when the waiting list becomes empty. If at this moment all orbits are either taken or removed, then a set~$\mathfrak{X}$ with the required properties is found. If not, then our algorithm branches. We choose an indeterminate orbit~$\mathfrak{o}$ and try to first take it and then to remove it. In either case, we start the same process again.
When the algorithm branches, we have the freedom to choose for~$\mathfrak{o}$ any of the indeterminate orbits. It turns out that this choice significantly affects the running time of the algorithm. For an indeterminate orbit~$\mathfrak{a}$, we denote by~$p(\mathfrak{a})$ the number of prohibited pairs~$\{\mathfrak{a},\mathfrak{b}\}$ and by~$r(\mathfrak{a})$ the number of requirements $\mathfrak{a}\to\mathfrak{b}$, where in both cases $\mathfrak{b}$ runs over all indeterminate orbits. It seems advantageous to take for~$\mathfrak{o}$ an orbit with large~$p(\mathfrak{o})$ and~$r(\mathfrak{o})$, since then taking of~$\mathfrak{o}$ will immediately lead to removing and taking of many orbits. Moreover, the intuition is that taken orbits are more important than removed ones, since they give more potential to analyze adjacency groups. So we use the following heuristic. We choose for~$\mathfrak{o}$ the orbit (any if several) with the largest value of $p(\mathfrak{o})+10r(\mathfrak{o})$. Practical calculations show that such a heuristic significantly improves the speed of the algorithm.
Until now, we have never used the restriction
\begin{equation}\label{eq_restr}
\sum_{\mathfrak{o}\in\mathfrak{X}}|\mathfrak{o}|\ge N.
\end{equation}
Nevertheless, it plays an important role in bounding the running time of the algorithm and the size of the output. When we implemented the described algorithm in the most interesting for us case $m=27$, $d=16$, $G=G_{351}$ without any restriction of the form~\eqref{eq_restr}, it began to give out an enormous list of weak pseudo-manifolds satisfying conditions~(i) and~(ii) with the numbers of $16$-simplices mostly in the range from $15000$ to~$30000$, and did not stop in a reasonable time.
So we need to make the following improvements to the algorithm:
\begin{enumerate}
\item Throughout the whole algorithm we look after the sum~$S$ of sizes $|\mathfrak{o}|$ of all orbits that are either taken or indeterminate, and stop considering any branch that leads us to a situation with this sum less than~$N$.
\item When the waiting list of adjacency groups becomes empty, before starting branching the algorithm, we may try to resolve the situation in the following way. For each indeterminate orbit~$\mathfrak{a}$, we compute the sum~$P_{\mathfrak{a}}$ of the sizes of all indeterminate orbits~$\mathfrak{b}$ such that the pair $\{\mathfrak{a},\mathfrak{b}\}$ is prohibited. Then we remove all indeterminate orbits~$\mathfrak{a}$ with $S-P_{\mathfrak{a}}<N$. (The reason for doing this is that, if $\mathfrak{a}$ entered~$\mathfrak{X}$, the maximal possible sum of sizes of orbits would become less than~$N$.)
\end{enumerate}
Procedure~(2) is not always helpful. The matter is that the computation of all the sum~$P_{\mathfrak{a}}$ requires enumeration over pairs of indeterminate orbits. When the number~$M$ of indeterminate orbits is large, this is a rather time-consuming procedure. Besides, when $M$ is large enough, then the sum~$S$ is much larger than~$N$ and it is unlikely that we will find at least one orbit~$\mathfrak{a}$ with $S-P_{\mathfrak{a}}<N$. So, most probably, the procedure will not be effective in this case. On the other hand, the procedure can be very useful in the case of a sufficiently small number of indeterminate orbits, that is, when the sum~$S$ is close to~$N$. (Typically, there are very few taken orbits, see Remark~\ref{remark_bound_level} below, so their contribution to~$S$ is negligible.)
We use the following heuristic. If at the moment when the waiting list of adjacency groups becomes empty, the inequality
\begin{equation}\label{eq_heuristic_bound}
M\le \frac{5N}{|G|}\,
\end{equation}
(which is approximately equivalent to $S\le 5N$) is satisfied, then we apply Procedure~(2) and start branching the algorithm only if Procedure~(2) is useless, that is, $S-P_{\mathfrak{a}}\ge N$ for all indeterminate orbits~$\mathfrak{a}$. However, if inequality~\eqref{eq_heuristic_bound} is not satisfied, then we do not apply Procedure~(2) and start branching the algorithm immediately.
Now, the description of the algorithm is completely finished. Being implemented in the case~\eqref{eq_main_case}, this algorithm yields Proposition~\ref{propos_24_pseudo}.
\begin{remark}\label{remark_bound_level}
At first glance, it may seem that our algorithm will branch in an uncontrolled way. However, this does not happen. The matter is that if the group~$G$ is large enough, then we get a lot of prohibited pairs from the very beginning. Therefore, every time an orbit is taken, a lot of other orbits become removed immediately. This leads to the fact that branches of the algorithm in which we choose to take an orbit several times end very quickly. We conveniently say that our algorithm is on \textit{level} $k$ if $k-1$ choices in favor of taking an orbit have been made so far. (The number of choices in favor of removing an orbit is unimportant.) In the case~\eqref{eq_main_case} the algorithm never goes beyond level~$5$, and spends most of its time at levels~$1$ and~$2$.
\end{remark}
\begin{remark}\label{remark_implement}
We have implemented the algorithm as a C++ program on a usual PC without any attempt to make parallel computing. (Note, however, that the algorithm can be very easily parallelized if necessary.) The computation was performed on one processor core of clock frequency 1.8 GHz. Under such conditions the running time of the program in the case~\eqref{eq_main_case} was about 7.5 hours.
\end{remark}
\section{Checking that $K_1$, $K_2$, $K_3$, and~$K_4$ are combinatorial manifolds}\label{section_check_manifold}
Our approach will be based on the properties of \textit{collapsibility} and \textit{nonevasiveness} for simplicial complexes. Recall the definitions of these properties (see~\cite{KSS84}).
First, introduce some notation. Suppose that $K$ is a simplicial complex on the vertex set~$V$. For a simplex~$\sigma\in K$, its \textit{link} and \textit{contrastar} are the simplicial complexes
\begin{align*}
\mathop{\mathrm{link}}\nolimits(\sigma,K)&=\bigl\{\tau\subseteq V\setminus\sigma \colon \sigma\cup\tau\in K \bigr\},\\
\mathop{\mathrm{cost}}\nolimits(\sigma,K)&=\bigl\{\tau\subseteq V\setminus\sigma \colon \tau\in K \bigr\},
\end{align*}
respectively. For a subset~$U\subset V$, the \textit{full subcomplex} of~$K$ spanned by~$U$ is the complex
$$
K_U=\bigl\{\tau\subseteq U \colon \tau\in K \bigr\}.
$$
Then $\mathop{\mathrm{cost}}\nolimits(\sigma,K)=K_{V\setminus\sigma}$.
\begin{defin}[Collapsibility]
Suppose that $\sigma$ is a maximal (w.\,r.\,t. inclusion) simplex of a simplicial complex~$K$. A codimension $1$ face $\tau\subset\sigma$ is called a \textit{free facet} if it is contained in no other simplex of~$K$. An \textit{elementary collapse} is the process of removing from~$K$ of some pair $(\sigma,\tau)$, where $\sigma$ is a maximal simplex of~$K$ and $\tau$ is a free facet of~$\sigma$. We say that a simplicial complex~$K$ \textit{collapses} to a subcomplex~$L$ and write $K\searrow L$ if $L$ can be obtained from~$K$ by a finite sequence of elementary collapses. A simplicial complex is called \textit{collapsible} if it collapses to a point.
\end{defin}
\begin{defin}[Nonevasiveness]
The properties of \textit{evasiveness} and \textit{nonevasiveness} of finite simplicial complexes can be defined recursively on the number of vertices in the following way. The empty simplicial complex~$\varnothing$ is, by definition, \textit{evasive}. A point, i.\,e., a $0$-simplex is, by definition, \textit{nonevasive}. Assume that $n\ge 2$ and the property of being evasive or nonevasive has already been defined for all simplicial complexes with less than $n$ vertices. Then a simplicial complex~$K$ with $n$ vertices is said to be \textit{nonevasive} if and only if there is a vertex $v$ of~$K$ such that both $\mathop{\mathrm{link}}\nolimits(v,K)$ and~$\mathop{\mathrm{cost}}\nolimits(v,K)$ are nonevasive. Otherwise, $K$ is said to be \textit{evasive}.
\end{defin}
We need the following two results from PL topology.
\begin{propos}[\cite{BrKu87}, Corollary~5]\label{propos_2collapse}
Let $K$ be a combinatorial manifold on the vertex set~$V$. Suppose that there exists a subset $U\subset V$ such that both complexes~$K_U$ and~$K_{V\setminus U}$ are collapsible. Then $K$ is a combinatorial sphere.
\end{propos}
\begin{cor}\label{cor_2collapse}
Suppose that a combinatorial manifold~$K$ contains a simplex~$\sigma$ such that $\mathop{\mathrm{cost}}\nolimits\sigma$ is collapsible. Then $K$ is a combinatorial sphere.
\end{cor}
\begin{remark}
The proof of Proposition~\ref{propos_2collapse} relies on theory of regular neighborhoods. Recall that the \textit{first derived neighborhood} of a full subcomplex~$L$ of a combinatorial manifold~$K$ is the subcomplex~$N(L,K')$ of the first barycentric subdivision~$K'$ of~$K$ consisting of all simplices that meet~$L$ and all their faces. By~\cite[Theorem~5]{Zee63}, $N(L,K')$ is a combinatorial ball whenever $L$ is collapsible. So under the conditions of Proposition~\ref{propos_2collapse}, the complex~$K'$ is the union of the two combinatorial balls~$N(K_U,K')$ and~$N(K_{V\setminus U},K')$ along their common boundary. Therefore, $K'$ and hence~$K$ are combinatorial spheres.
\end{remark}
\begin{propos}[\cite{KSS84}, Proposition 1]\label{propos_nonev}
A nonevasive simplicial complex is collapsible.
\end{propos}
Proposition~\ref{propos_nonev} is important for us, since checking nonevasiveness is significantly faster than checking collapsibility. Indeed, the nonevasiveness can be checked directly by definition, and this procedure requires working with vertices and maximal simplices only without referring directly to simplices of intermediate dimensions.
Suppose that $K$ is a pure $d$-dimensional simplicial complex and we would like to check that $K$ is a combinatorial manifold. To a pair of simplices $\rho\subset\sigma$ of~$K$ such that $\dim\rho<\dim\sigma=d$, we assign the simplicial complex
$$
L_{\rho,\sigma} =\mathop{\mathrm{cost}}\nolimits\bigl(\sigma\setminus\rho,\mathop{\mathrm{link}}\nolimits(\rho,K)\bigr)=\{\eta\subseteq V\setminus\sigma \colon \rho\cup\eta\in K\}.
$$
The following proposition gives a sufficient condition for~$K$ to be a combinatorial manifold.
\begin{propos}\label{propos_combman_suff}
Suppose that, for each nonempty simplex $\rho\in K$ with $\dim\rho<d$, there exists a $d$-simplex $\sigma\in K$ such that $\sigma\supset\rho$ and $L_{\rho,\sigma}$ is collapsible. Then $K$ is a combinatorial manifold.
\end{propos}
\begin{proof}
Let us prove by the reverse induction on~$k=d-1,\ldots,0$ that the link of every $k$-simplex $\rho\in K$ is a combinatorial $(d-k-1)$-sphere.
\textsl{Base of induction.} Suppose that $\dim\rho=d-1$. We know that there is a $d$-simplex $\sigma\supset\rho$ such that $L_{\rho,\sigma}$ is collapsible. Since $\dim L_{\rho,\sigma}=0$, it follows that $L_{\rho,\sigma}$ is a point. Hence, $\mathop{\mathrm{link}}\nolimits(\rho,K)$ is a pair of points, i.\,e., a $0$-dimensional sphere.
\textsl{Inductive step.} Suppose that $k<d-1$ and assume that we have already proved that links of all simplices of $K$ of dimensions greater than $k$ are combinatorial spheres. Consider a $k$-simplex $\rho\in K$. For each vertex $v$ of $\mathop{\mathrm{link}}\nolimits(\rho,K)$, the complex
$$
\mathop{\mathrm{link}}\nolimits\bigl( v, \mathop{\mathrm{link}}\nolimits(\rho,K)\bigr)=\mathop{\mathrm{link}}\nolimits(\{v\}\cup\rho,K)
$$
is a combinatorial $(d-k-2)$-sphere. Therefore, $\mathop{\mathrm{link}}\nolimits(\rho,K)$ is a combinatorial $(d-k-1)$-manifold. We know that there is a $d$-simplex $\sigma\supset\rho$ such that the complex $$L_{\rho,\sigma}=\mathop{\mathrm{cost}}\nolimits\bigl(\sigma\setminus\rho,\mathop{\mathrm{link}}\nolimits(\rho,K)\bigr)$$ is collapsible. So, by Corollary~\ref{cor_2collapse}, $\mathop{\mathrm{link}}\nolimits(\rho,K)$ is a combinatorial $(d-k-1)$-sphere.
For $k=0$, we obtain that the links of all vertices of~$K$ are combinatorial $(d-1)$-spheres. Thus, $K$ is a combinatorial $d$-manifold.
\end{proof}
The following proposition is checked by a computer enumeration.
\begin{propos}\label{propos_check_nonev}
Let $K$ be one of the four simplicial complexes $K_1$, $K_2$, $K_3$, and~$K_4$ described in Tables~\ref{table_1234}--\ref{table_4}. Then, for each nonempty simplex $\rho\in K$ with $\dim\rho<16$, there exists a $16$-dimensional simplex $\sigma\in K$ such that $\sigma\supset\rho$ and $L_{\rho,\sigma}$ is nonevasive and hence collapsible.
\end{propos}
It follows from Propositions~\ref{propos_combman_suff} and~\ref{propos_check_nonev} that $K_1$, $K_2$, $K_3$, and~$K_4$ are $16$-dimensional combinatorial manifolds, which gives Proposition~\ref{propos_combman}.
\begin{remark}
Let us give a more detailed description of the algorithm whose implementation yields Proposition~\ref{propos_check_nonev}. Recall that each simplicial complex $K=K_i$ is given by the list $\sigma_1,\ldots,\sigma_{286}$ of representatives of the $G_{351}$-orbits of its $16$-simplices. Since $L_{g(\rho),g(\sigma)}\cong L_{\rho,\sigma}$ for all $g\in G_{351}$, it is sufficient to examine the nonevasiveness of the complexes $L_{\rho,\sigma_s}$ only, where $\rho$ runs over nonempty proper faces of~$\sigma_s$. Moreover, once we have checked that certain complex $L_{\rho,\sigma_s}$ is nonevasive, we do not need any more to examine the nonevasiveness of other complexes of the form $L_{\rho',\sigma_t}$ such that $\rho'$ lies in the $G_{351}$-orbit of~$\rho$. However, it turns out that generating complexes~$L_{\rho,\sigma_s}$ is a more time-consuming operation than checking that these complexes are nonevasive. Indeed, the simplices of~$L_{\rho,\sigma_s}$ are exactly the simplices $\tau\setminus\sigma_s$, where $\tau\in K$ and $\tau\supset\rho$. So to find all maximal simplices of~$L_{\rho,\sigma_s}$, we need to enumerate the $100386$ maximal simplices $\tau\in K$. If we tried to perform this enumeration separately for each pair $\rho\subset\sigma_s$, this would be rather time-consuming. So we proceed as follows. For each $\sigma_s$, we perform the enumeration over all the $100386$ maximal simplices $\tau\in K$ only once, and for each such~$\tau$, we add the simplex $\tau\setminus \sigma_s$ to all simplicial complexes $L_{\rho,\sigma_s}$ such that $\rho\subseteq \sigma_s\cap\tau$. Using this procedure, we generate all the simplicial complexes $L_{\rho,\sigma_s}$ with given~$s$ simultaneously. Then we start checking whether the generated complexes~$L_{\rho,\sigma_s}$ are nonevasive, and once a complex $L_{\rho,\sigma_s}$ is checked to be nonevasive, we remove all complexes $L_{\rho',\sigma_t}$ with $\rho'$ in the $G_{351}$-orbit of~$\rho$ from the list of complexes to be examined. Being implemented as a C++ program under the same conditions as in Remark~\ref{remark_implement}, the running time of this algorithm is about $40$ minutes for each complex~$K_i$.
\end{remark}
\section{The normalizer of~$G_{351}$ and absence of additional symmetries}\label{section_no_add_sym}
In this section we prove Propositions~\ref{propos_normalizer} and~\ref{propos_SymK}.
Let $\Gamma$ be the oriented graph on the vertex set~$\mathbb{F}_{27}$ such that an oriented edge goes from~$a$ to~$b$ if and only if $b-a$ is a nonzero square. We denote the oriented edge from~$a$ to~$b$ by~$(a,b)$. For any nonzero element $c\in\mathbb{F}_{27}$, exactly on of the two elements~$\pm c$ is a square; hence any two distinct elements $a,b\in\mathbb{F}_{27}$ are connected by an oriented edge in exactly one of the two directions. For a vertex~$a\in\Gamma$, we denote by~$O_a$ (respectively, by~$I_a$) the set consisting of all vertices $b\ne a$ such that the graph~$\Gamma$ contains the outgoing edge~$(a,b)$ (respectively, the incoming edge~$(b,a)$). Then $|O_a|=|I_a|=13$. We denote by~$\mathop{\mathrm{Sym}}\nolimits(\Gamma)$ the symmetry group of the oriented graph~$\Gamma$.
\begin{propos}\label{propos_orgraph_symmetries}
We have $\mathop{\mathrm{Sym}}\nolimits(\Gamma)=G_{351}\rtimes C_3$, where the multiple~$C_3$ is generated by the Frobenius automorphism $F\colon x\mapsto x^3$.
\end{propos}
The proof of this proposition will be decomposed into several lemmas.
\begin{lem}\label{lem_orgraph_symmetries_inclusion}
We have, $\mathop{\mathrm{Sym}}\nolimits(\Gamma)\supseteq G_{351}\rtimes C_3$. Besides, the subgroup~$G_{351}\subset \mathop{\mathrm{Sym}}\nolimits(\Gamma)$ acts transitively on the edges of~$\Gamma$.
\end{lem}
\begin{proof}
It follows immediately from the definition of~$\Gamma$ that it is symmetric with respect to additive shifts $x\mapsto x+c$, $c\in F_{27}$, multiplicative shifts by squares $x\mapsto cx$, $c\in(\mathbb{F}_{27}^{\times})^2$, and all automorphisms of the field~$\mathbb{F}_{27}$.
Besides, if $(a,b)$ is an edge of~$\Gamma$, then $b-a$ is a square and hence the transformation $x\mapsto (b-a)x+a$ belongs to~$G_{351}$ and takes the edge~$(0,1)$ to~$(a,b)$.
\end{proof}
\begin{lem}\label{lem_sym_stab_mnogo}
Suppose that a symmetry $g\in\mathop{\mathrm{Sym}}\nolimits(\Gamma)$ stabilizes every of the vertices~$0$, $1$, $\alpha$, $\alpha^3$, and~$\alpha^9$. Then $g=\mathrm{id}$.
\end{lem}
\begin{proof}
Whenever $g$ stabilizes a vertex~$x$, it necessarily stabilizes each of the sets~$O_x$ and~$I_x$. So $g$ stabilizes every of the sets~$O_0$, $O_1$, $O_{\alpha}$, $O_{\alpha^3}$, and~$O_{\alpha^9}$, and the corresponding sets~$I_x$. Since $O_0=(\mathbb{F}_{27}^{\times})^2$, we obtain that $g$ takes squares to squares and non-squares to non-squares.
First, let us prove that $g$ stabilizes every square~$\alpha^k$. To do this, for every square~$\alpha^k$ different from~$1$, $\alpha$, $\alpha^3$, and~$\alpha^9$, find out which of the sets $O_1$, $O_{\alpha}$, $O_{\alpha^3}$, and~$O_{\alpha^9}$ contain~$\alpha^k$, that is, which of the elements $\alpha^k-1$, $\alpha^k-\alpha$, $\alpha^k-\alpha^3$, and $\alpha^k-\alpha^9$ are squares. The result of an easy calculation is shown in Table~\ref{table_alpha_member}. In this table $+$ (respectively, $-$) indicates that the element belongs (respectively, does not belong) to the set. We see that the columns in the table are pairwise different, so the squares $\alpha^2$, $\alpha^4$, $\alpha^5$, $\alpha^6$, $\alpha^7$, $\alpha^8$, $\alpha^{10}$, $\alpha^{11}$, and~$\alpha^{12}$ are pairwise distinguished by membership in the sets $O_1$, $O_{\alpha}$, $O_{\alpha^3}$, and~$O_{\alpha^9}$. Thus, $g$ stabilizes every square~$\alpha^k$.
\begin{table}
\caption{Membership of elements in sets}\label{table_alpha_member}
\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|}
\hline
$\phantom{\strut^2}$ & $\alpha^2$ & $\alpha^4$ & $\alpha^5$ & $\alpha^6$ & $\alpha^7$ & $\alpha^8$ & $\alpha^{10}$ & $\alpha^{11}$ & $\alpha^{12}$\\
\hline
$O_1$ & $+$ & $-$ & $+$ & $+$ & $-$ & $-$ & $-$ & $-$ & $-$\\
\hline
$O_{\alpha}$ & $+$ & $+$ & $-$ & $+$ & $+$ & $-$ & $+$ & $-$ & $-$\\
\hline
$O_{\alpha^3}$ & $-$ & $+$ & $+$ & $+$ & $-$ & $+$ & $-$ & $-$ & $+$\\
\hline
$O_{\alpha^9}$ & $+$ & $-$ & $+$ & $-$ & $-$ & $-$ & $+$ & $+$ & $+$\\
\hline
\end{tabular}
\end{table}
Now, consider all non-squares $-\alpha^k$, where $k=0,\ldots,12$. The set $O_{-1}\cap O_0$ consists of all elements $x\in\mathbb{F}_{27}$ such that both $x$ and~$x+1$ are nonzero squares. It is easy to see that $O_{-1}\cap O_0=\{\alpha,\alpha^3,\alpha^4,\alpha^9,\alpha^{10},\alpha^{12}\}$. The sets $O_{-\alpha^k}\cap O_0$ are obtained from $O_{-1}\cap O_0$ by multiplication by~$\alpha^k$ and hence are pairwise different. On the other hand, each set~$O_{-\alpha^k}\cap O_0$ consists of squares and therefore is stabilized by~$g$. So for all non-squares~$x$, we have
$$
O_x\cap O_0=g( O_x\cap O_0 ) = O_{g(x)}\cap O_0
$$
and hence $g(x)=x$.
\end{proof}
\begin{lem}\label{lem_sym_stab_malo}
A symmetry $g\in\mathop{\mathrm{Sym}}\nolimits(\Gamma)$ stabilizes each of the vertices~$0$ and~$1$ if and only if $g$ is a power of the Frobenius automorphism~$F$.
\end{lem}
\begin{proof}
Obviously, $F(0)=0$ and $F(1)=1$. Now, we assume that $g(0)=0$ and $g(1)=1$ and prove that $g$ is a power of~$F$. Since $g(0)=0$ and $g(1)=1$, we see that $g$ stabilizes the set~$O_0\cap O_1$. This set consists of all elements $x\in\mathbb{F}_{27}$ such that both $x$ and~$x-1$ are nonzero squares. We have $O_0\cap O_1=\{\alpha,\alpha^2,\alpha^3,\alpha^5,\alpha^6,\alpha^9\}$. If $g(x)=y$, then $g$ maps the set $O_0\cap O_1\cap O_x$ bijectively onto the set $O_0\cap O_1\cap O_y$. However, it is easy to see that each of the sets $O_0\cap O_1\cap O_{\alpha^k}$ with $k=1,3,9$ consists of three elements, while each of the sets $O_0\cap O_1\cap O_{\alpha^k}$ with $k=2,5,6$ consists of two elements. Indeed, $O_0\cap O_1\cap O_{\alpha}=\{\alpha^2,\alpha^3,\alpha^6\}$, $O_0\cap O_1\cap O_{\alpha^2}=\{\alpha^3,\alpha^5\}$, and the other four sets are obtained from those two by the action of the Frobenius automorphism, which permutes cyclically $\alpha$, $\alpha^3$, and $\alpha^9$ and permutes cyclically $\alpha^2$, $\alpha^6$, and $\alpha^5$. Thus, $g$ stabilizes each of the two sets $\{\alpha,\alpha^3,\alpha^9\}$ and $\{\alpha^2,\alpha^5,\alpha^6\}$. Hence there is a number~$m$ such that $F^mg(\alpha)=\alpha$. Then $F^mg(\alpha^3)$ is either $\alpha^3$ or~$\alpha^9$. Since the set $O_0\cap O_1\cap O_{\alpha}=\{\alpha^2,\alpha^3,\alpha^6\}$ is stabilized by~$g$, it follows that $F^mg(\alpha^3)=\alpha^3$ and hence $F^mg(\alpha^9)=\alpha^9$. By Lemma~\ref{lem_sym_stab_mnogo}, $F^mg=\mathrm{id}$. Therefore, $g=F^{-m}$.
\end{proof}
\begin{lem}\label{lem_size_Sym}
$|\mathop{\mathrm{Sym}}\nolimits(\Gamma)|= 3\cdot 351=1053$.
\end{lem}
\begin{proof}
The equality follows, since $\mathop{\mathrm{Sym}}\nolimits(\Gamma)$ acts transitively on the set of ${27\choose 2}=351$ oriented edges of~$\Gamma$ and, by Lemma~\ref{lem_sym_stab_malo}, the stabilizer of the edge $(0,1)$ has order $3$.
\end{proof}
Proposition~\ref{propos_orgraph_symmetries} follows from Lemmas~\ref{lem_orgraph_symmetries_inclusion} and~\ref{lem_size_Sym}.
\begin{proof}[Proof of Proposition~\ref{propos_normalizer}]
Let $\mathop{\mathrm{Sym}}\nolimits^{\pm}(\Gamma)$ be the group consisting of all maps of~$\mathbb{F}_{27}$ onto itself that either preserve the directions of all edges of~$\Gamma$ or reverse the directions of all edges of~$\Gamma$ simultaneously. Since $-1$ is not a square in~$\mathbb{F}_{27}$, the sign reversal $S\colon x\mapsto -x$ reverses the directions of all edges of~$\Gamma$, so $\mathop{\mathrm{Sym}}\nolimits(\Gamma)$ is an index~$2$ subgroup of $\mathop{\mathrm{Sym}}\nolimits^{\pm}(\Gamma)$ and $\mathop{\mathrm{Sym}}\nolimits^{\pm}(\Gamma)$ is generated by $\mathop{\mathrm{Sym}}\nolimits(\Gamma)$ and~$S$. It follows from Proposition~\ref{propos_orgraph_symmetries} that $\mathop{\mathrm{Sym}}\nolimits^{\pm}(\Gamma)$ is exactly the $2106$-element group $G_{351}\rtimes (C_2\times C_3)$ from Proposition~\ref{propos_normalizer}. So we need to prove that $N(G_{351})=\mathop{\mathrm{Sym}}\nolimits^{\pm}(\Gamma)$. We have $\mathop{\mathrm{Sym}}\nolimits^{\pm}(\Gamma)\subseteq N(G_{351})$, since the permutations~$F$ and~$S$ normalize the subgroup~$G_{351}$. Hence we need only to prove the inverse inclusion.
Suppose that a permutation $h$ of the set~$\mathbb{F}_{27}$ does not belong to~$\mathop{\mathrm{Sym}}\nolimits^{\pm}(\Gamma)$ and prove that then it does not belong to~$N(G_{351})$. Since $h\notin\mathop{\mathrm{Sym}}\nolimits^{\pm}(\Gamma)$, there exist edges~$e_1$ and~$e_2$ of~$\Gamma$ such that $h(e_1)$ is an edge of~$\Gamma$ and~$h(e_2)$ is not an edge of~$\Gamma$. By Lemma~\ref{lem_orgraph_symmetries_inclusion}, the group~$G_{351}$ acts transitively on edges of~$\Gamma$; hence, there is an element $g\in G_{351}$ satisfying $g(e_1)=e_2$. Then the permutation $hgh^{-1}$ takes the edge~$h(e_1)$ to the non-edge~$h(e_2)$. Therefore, $hgh^{-1}\notin \mathop{\mathrm{Sym}}\nolimits(\Gamma)$ and hence $hgh^{-1}\notin G_{351}$. Thus, $h\notin N(G_{351})$.
\end{proof}
To prove Proposition~~\ref{propos_SymK}, we need the following lemma.
\begin{lem}\label{lem_not_transitive}
For every $i=1,2,3,4$, the group~$\mathop{\mathrm{Sym}}\nolimits(K_i)$ does not act transitively on the set of directed edges of~$K_i$.
\end{lem}
\begin{proof}
Since we have already proved in Section~\ref{section_result} that the group~$\mathop{\mathrm{Sym}}\nolimits(K_i)$ acts transitively on the set of undirected edges of~$K_i$, we need only to show that $\mathop{\mathrm{Sym}}\nolimits(K_i)$ does not contain an element that swaps the endpoints of an edge of~$K_i$. Moreover, we suffice to show this only for one edge of~$K_i$, say for the edge $\{1,2\}$ in the numbering introduced in Section~\ref{section_result}. To do this, we improve our method in Step~3 of the scheme of proof of Theorem~\ref{theorem_main_detail}. Namely, for each $15$-simplex $\tau\in K_i$ that contains both~$1$ and~$2$, we can consider the two $14$-dimensional faces of it, $\rho_1=\tau\setminus\{1\}$ and~$\rho_2=\tau\setminus\{2\}$. For each of them, we can compute the number~$s(\rho_k)$ of $16$-simplices $\sigma\in K_i$ such that $\sigma\supset\rho_k$. Thus, the ordered pair of numbers $s(\tau)=\bigl(s(\rho_1),s(\rho_2)\bigr)$ corresponds to~$\tau$. Now, for each ordered pair of positive integers~$(p,q)$, we can compute the number~$N_{pq}$ of $15$-simplices~$\tau$ such that $\tau\supset\{1,2\}$ and $s(\tau)=(p,q)$. The matrices $(N_{pq})$ for the triangulations~$K_1$, $K_2$, $K_3$, and~$K_4$ are given in Tables~\ref{table_N_matrix_1}--\ref{table_N_matrix_4}. If there existed a symmetry of~$K_i$ that swaps~$1$ and~$2$, then the corresponding matrix~$(N_{pq})$ would be symmetric. The lemma follows, since none of the matrices in Tables~\ref{table_N_matrix_1}--\ref{table_N_matrix_4} is symmetric.
\end{proof}
\begin{table}
\caption{The matrix $(N_{pq})$ for~$K_1$}\label{table_N_matrix_1}
\begin{tabular}{|c|ccccccc|}
\hline
& 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline
3 & 10860 & 22509 & 13809 & 5375 & 1374 & 360 & 16\\
4 & 22504 & 62261 & 31272 & 10187 & 2247 & 624 & 21\\
5 & 14207 & 31196 & 20867 & 6602 & 1481 & 372 & 13\\
6 & 5267 & 10221 & 6553 & 2905 & 737 & 183 & 13\\
7 & 1400 & 2166 & 1488 & 656 & 242 & 59 & 6\\
8 & 388 & 567 & 359 & 171 & 76 & 40 & 1\\
9 & 16 & 24 & 14 & 5 & 6 & 0 & 0\\
\hline
\end{tabular}
\bigskip
\bigskip
\caption{The matrix $(N_{pq})$ for~$K_2$}\label{table_N_matrix_2}
\begin{tabular}{|c|ccccccc|}
\hline
& 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline
3&14532 & 21847 & 15078 & 6996 & 2090 & 322 & 89\\
4 &21975 & 52378 & 28416 & 10895 & 2745 & 389 & 79\\
5 &15408 & 28271 & 20266 & 7554 & 1862 & 268 & 63 \\
6 &6956 & 10614 & 7825 & 3847 & 1088 & 138 & 48 \\
7 &2003 & 2796 & 1886 & 1035 & 400 & 69 & 20 \\
8 &320 & 391 & 237 & 146 & 73 & 19 & 5 \\
9 & 72 & 106 & 50 & 31 & 18 & 4 & 0 \\
\hline
\end{tabular}
\bigskip
\bigskip
\caption{The matrix $(N_{pq})$ for~$K_3$}\label{table_N_matrix_3}
\begin{tabular}{|c|ccccccc|}
\hline
& 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline
3 & 14304 & 21634 & 15634 & 6324 & 1901 & 368 & 28\\
4 & 21663 & 51651 & 30266 & 10093 & 2596 & 408 & 28\\
5 & 15911 & 30125 & 21767 & 7505 & 1886 & 300 & 15\\
6 & 6320 & 9940 & 7467 & 3358 & 928 & 163 & 13\\
7 & 1806 & 2673 & 1904 & 936 & 328 & 71 & 7\\
8 & 353 & 397 & 313 & 168 & 60 & 19 & 0\\
9 & 20 & 35 & 15 & 7 & 11 & 1 & 0\\
\hline
\end{tabular}
\bigskip
\bigskip
\caption{The matrix $(N_{pq})$ for~$K_4$}\label{table_N_matrix_4}
\begin{tabular}{|c|ccccccc|}
\hline
& 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline
3 & 12288 & 21553 & 15249 & 4858 & 1361 & 276 & 75\\
4 & 21376 & 56670 & 34129 & 8632 & 2339 & 406 & 99\\
5 & 15438 & 34057 & 23961 & 6323 & 1526 & 307 & 71\\
6 & 4953 & 8724 & 6454 & 2408 & 586 & 84 & 36\\
7 & 1259 & 2286 & 1586 & 625 & 247 & 53 & 8\\
8 & 251 & 409 & 284 & 112 & 48 & 17 & 9\\
9 & 75 & 90 & 79 & 27 & 9 & 7 & 0\\
\hline
\end{tabular}
\end{table}
\begin{cor}\label{cor_sym_cont}
For every $i=1,2,3,4$, the group~$\mathop{\mathrm{Sym}}\nolimits(K_i)$ is contained in~$\mathop{\mathrm{Sym}}\nolimits(\Gamma)$.
\end{cor}
\begin{proof}
We know that~$\mathop{\mathrm{Sym}}\nolimits(K_i)$ contains the subgroup~$G_{351}$, which acts transitively on the edges of~$\Gamma$. So if~$\mathop{\mathrm{Sym}}\nolimits(K_i)$ contained an element not belonging to~$\mathop{\mathrm{Sym}}\nolimits(\Gamma)$, then $\mathop{\mathrm{Sym}}\nolimits(K_i)$ would act transitively on directed edges of~$K_i$, which is not true by Lemma~\ref{lem_not_transitive}.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{propos_SymK}]
By Proposition~\ref{propos_orgraph_symmetries} and Corollary~\ref{cor_sym_cont}, we have
$$
G_{351}\subseteq \mathop{\mathrm{Sym}}\nolimits(K_i)\subseteq \mathop{\mathrm{Sym}}\nolimits(\Gamma)=G_{351}\rtimes C_3,
$$
where $C_3$ is generated by the Frobenius automorphism~$F$. Hence $\mathop{\mathrm{Sym}}\nolimits(K_i)$ is either $G_{351}$ or $G_{351}\rtimes C_3$. However, as we have already mentioned in Section~\ref{section_result}, a direct computation shows that the group~$C_3$ generated by~$F$ acts freely on the set of the $24$ simplicial complexes in Theorem~\ref{theorem_main_detail}. In particular, this means that $F\notin\mathop{\mathrm{Sym}}\nolimits(K_i)$. Therefore, $\mathop{\mathrm{Sym}}\nolimits(K_i)=G_{351}$.
\end{proof}
\section{Triangulations with smaller symmetry groups}
\label{section_moves}
Let $K$ be a $(3d/2+3)$-vertex triangulation of a $d$-manifold like a projective plane. Each of the previously known examples $K=\mathbb{RP}^2_6$, $\mathbb{CP}^2_9$, $\mathbb{HP}^2_{15}$, $\widetilde{\HP}\vphantom{\HP}^2_{15}$, and~$\widetilde{\widetilde{\HP}}\vphantom{\HP}^2_{15}$ satisfies the following condition, see~\cite[Section~3]{BrKu92}:
\begin{itemize}
\item[$(*)$] $K$ contains three $(d/2)$-simplices~$\Delta_1$, $\Delta_2$, and~$\Delta_3$ such that
\begin{equation}\label{eq_triple}
\mathop{\mathrm{link}}\nolimits(\Delta_1,K)=\partial\Delta_2,\qquad
\mathop{\mathrm{link}}\nolimits(\Delta_2,K)=\partial\Delta_3,\qquad
\mathop{\mathrm{link}}\nolimits(\Delta_3,K)=\partial\Delta_1.
\end{equation}
\end{itemize}
The author is indebted to Denis Gorodkov for pointing out the importance of this condition.
We shall conveniently use the following terminology. A triple~$(\Delta_1,\Delta_2,\Delta_3)$ of $(d/2)$-simplices satisfying~\eqref{eq_triple} will be called a \textit{distinguished triple}. If $(\Delta_1,\Delta_2,\Delta_3)$ is a distinguished triple, then $K$ contains the subcomplex
\begin{equation}\label{eq_distinguished_subcomp}
J=(\Delta_1*\partial\Delta_2)\cup(\Delta_2*\partial\Delta_3)\cup(\Delta_3*\partial\Delta_1),\end{equation}
where $*$ denotes the join of simplicial complexes.
This subcomplex will be referred to as a \textit{distinguished subcomplex}.
Brehm and K\"uhnel made the following observation.
\begin{propos}[\cite{BrKu92}, Lemma 1]\label{propos_move}
Suppose that a combinatorial $d$-manifold~$K$ contains a distinguished subcomplex of the form~\eqref{eq_distinguished_subcomp}. Then a replacement of this subcomplex by
\begin{equation}\label{eq_distinguished_replace}
\widetilde{J}=(\partial\Delta_1*\Delta_2)\cup(\partial\Delta_2*\Delta_3)\cup(\partial\Delta_3*\Delta_1)
\end{equation}
yields a combinatorial manifold~$\widetilde{K}$ that is PL homeomorphic to~$K$.
\end{propos}
\begin{remark}
A replacement of a subcomplex of the form $\Delta_1*\partial\Delta_2$ by~$\partial\Delta_1*\Delta_2$ is called a \textit{bistellar move} (or \textit{flip}). Since both $\Delta_1*\partial\Delta_2$ and~$\partial\Delta_1*\Delta_2$ are combinatorial balls with the same boundary $\partial\Delta_1*\partial\Delta_2$, this operation yields a PL manifold homeomorphic to the initial one. Nevertheless, there is an important difference between a single bistellar move and the move in Proposition~\ref{propos_move} consisting of three simultaneous bistellar moves. Namely, a single bistellar move $\Delta_1*\partial\Delta_2\rightsquigarrow\partial\Delta_1*\Delta_2$ is well defined only when $\Delta_1*\partial\Delta_2$ is a full subcomplex of the initial manifold~$K$, i.\,e., $\Delta_2\notin K$. On the contrary, the operation in Proposition~\ref{propos_move} is always defined. To show this, one only needs to check that any simplex~$\sigma$ that belongs to~$K$ and to the new subcomplex~$\widetilde{J}$ belongs to the old subcomplex~$J$ as well. Indeed, up to a cyclic permutation of~$\Delta_1$, $\Delta_2$, and~$\Delta_3$, any simplex~$\sigma\in\widetilde{J}$ has the form $\sigma=\sigma_1*\sigma_2$, where $\sigma_1\subsetneq\Delta_1$ and $\sigma_2\subseteq\Delta_2$. If $\sigma_1=\varnothing$ or $\sigma_2\ne \Delta_2$, then $\sigma\in J$. If $\sigma_1\ne\varnothing$ and $\sigma_2=\Delta_2$, then $\sigma\notin K$, since $\sigma_1\notin\mathop{\mathrm{link}}\nolimits (\Delta_2,K)=\partial\Delta_3$.
\end{remark}
The replacement of a subcomplex~$J$ of the form~\eqref{eq_distinguished_subcomp} by the subcomplex~$\widetilde{J}$ of the form~\eqref{eq_distinguished_replace} will be called a \textit{triple flip} corresponding to the distinguished triple~$(\Delta_1,\Delta_2,\Delta_3)$.
Before studying the octonionic case, we recall how the distinguished subcomplexes look like in the real, complex, and quaternionic cases.
\smallskip
\textsl{1. Suppose that $d=2$ and~$K=\mathbb{RP}^2_6$.} Then any one-dimensional simplex of~$\mathbb{RP}^2_6$ enters a distinguished triple and all the $15$ one-dimensional simplices of~$\mathbb{RP}^2_6$ are decomposed into $5$ distinguished triples. The triple flip corresponding to any of these distinguished triples transforms~$\mathbb{RP}^2_6$ to a combinatorial manifold that is isomorphic to~$\mathbb{RP}^2_6$ again.
\smallskip
\textsl{2. Suppose that $d=4$ and~$K=\mathbb{CP}^2_9$.} Then $\mathbb{CP}^2_9$ contains a unique distinguished triple, cf.~\cite[\S 3]{MoYo91}. Recall that the $54$-element group $\mathop{\mathrm{Sym}}\nolimits(\mathbb{CP}^2_9)$ act on the $36$ four-dimensional simplices with two orbits consisting of $27$ and~$9$ simplices, respectively. The $9$ four-dimensional simplices in the second orbit and their faces form the distinguished subcomplex~$J$. So $J$ is invariant under the action of $\mathop{\mathrm{Sym}}\nolimits(\mathbb{CP}^2_9)$. The corresponding triple flip transforms~$\mathbb{CP}^2_9$ to a combinatorial manifold that is isomorphic to~$\mathbb{CP}^2_9$ again.
\smallskip
\textsl{2. Suppose that $d=8$ and~$K=\mathbb{HP}^2_{15}$ is the most symmetric of the three Brehm--K\"uhnel triangulations.} Then $\mathbb{HP}^2_{15}$ contains exactly five distinguished triples and the group $\mathop{\mathrm{Sym}}\nolimits(\mathbb{HP}^2_{15})\cong A_5$ acts transitively on them, see~\cite[Section~3]{BrKu92}. The corresponding five distinguished subcomplexes~$J_1,\ldots,J_5$ have pairwise disjoint interiors. So we may perform the corresponding five triple flips independently of each other. If one performs simultaneously all the five triple flips corresponding to~$J_1,\ldots,J_5$, then the obtained combinatorial manifold will be isomorphic to~$\mathbb{HP}^2_{15}$ again. Nevertheless, performing only some of the five triple flips, one can obtain new triangulations. This is exactly the way how Brehm and K\"uhnel constructed the two other triangulations. Namely, performing any one (or any four) of the five triple flips leads to a combinatorial manifold isomorphic to~$\widetilde{\HP}\vphantom{\HP}^2_{15}$, and performing any two (or any three) of the five triple flips leads to a combinatorial manifold isomorphic to~$\widetilde{\widetilde{\HP}}\vphantom{\HP}^2_{15}$. For each $k$, the result is independent of which $k$ of the $5$ flips are performed, since the group $\mathop{\mathrm{Sym}}\nolimits(\mathbb{HP}^2_{15})\cong A_5$ acts transitively on the $k$-element subsets of $\{1,2,3,4,5\}$. Note that the symmetry groups $\mathop{\mathrm{Sym}}\nolimits\left(\widetilde{\HP}\vphantom{\HP}^2_{15}\right)\cong A_4$ and $\mathop{\mathrm{Sym}}\nolimits\left(\widetilde{\widetilde{\HP}}\vphantom{\HP}^2_{15}\right)\cong S_3$ are exactly the subgroups of~$A_5$ that stabilize a distinguished subcomplex and the union of two distinguished subcomplexes, respectively.
\smallskip
Now, we are ready to consider the case $d=16$, that is, the four combinatorial manifolds $K_1$, $K_2$, $K_3$, and~$K_4$ from Theorem~\ref{theorem_main_detail}.
\begin{theorem}\label{theorem_smaller_groups}
\textnormal{(1)} The combinatorial manifolds~$K_1$ and~$K_4$ contain no distinguished subcomplexes.
\textnormal{(2)} The combinatorial manifold~$K_2$ contains exactly $351$ distinguished subcomplexes. One of them is the subcomplex
$$
J=(\Delta_1*\partial\Delta_2)\cup(\Delta_2*\partial\Delta_3)\cup(\Delta_3*\partial\Delta_1),
$$
where
\begin{align*}
\Delta_1&=\{1,2,5,6,9,10,11,16,25\},\\
\Delta_2&=\{3,4,7,8,12,13,14,15,17 \},\\
\Delta_3&=\{18,19,20,21,22,23,24,26,27\}.
\end{align*}
The $351$ subcomplexes $g(J)$, where $g\in \mathop{\mathrm{Sym}}\nolimits(K_2)=G_{351}$, have pairwise disjoint interiors and are exactly all the distinguished subcomplexes of~$K_2$.
\textnormal{(3)} For a subset $S\subseteq G_{351}$, let $K_S$ be the simplicial complex obtained from~$K_2$ by replacing all subcomplexes $g(J)$ such that $g\in S$ by the corresponding complexes $g\bigl(\widetilde{J}\,\bigr)$, where
$$
\widetilde{J}=(\partial\Delta_1*\Delta_2)\cup(\partial\Delta_2*\Delta_3)\cup(\partial\Delta_3*\Delta_1).
$$
Then $K_S$ is a $27$-vertex combinatorial $16$-manifold like the octonionic projective plane and is PL homeomorphic to~$K_2=K_{\varnothing}$. The symmetry group~$\mathop{\mathrm{Sym}}\nolimits(K_S)$ is exactly the subgroup of~$G_{351}$ consisting of all $g$ satisfying $gS=S$. Two combinatorial manifolds~$K_{S_1}$ and $K_{S_2}$ are isomorphic if and only if there exists $g\in G_{351}$ such that $gS_1=S_2$. Besides, $g(J)$, where $g\in G_{351}\setminus S$, and $g\bigl(\widetilde{J}\,\bigr)$, where $g\in S$, are exactly all distinguished subcomplexes of~$K_S$.
\textnormal{(4)} $K_{G_{351}}=K_3$.
\textnormal{(5)} The number of combinatorially distinct combinatorial manifolds of the form~$K_S$ is exactly
$$
\frac1{351}\cdot\left(2^{351}+13\cdot 2^{118}+81\cdot 2^{29}\right).
$$
So together with the two exceptional combinatorial manifolds~$K_1$ and~$K_4$, we obtain
$$
\frac1{351}\cdot\left(2^{351}+13\cdot 2^{118}+81\cdot 2^{29}\right)+2
$$
$27$-vertex combinatorial $16$-manifolds like the octonionic projective plane. Moreover, the distribution of these combinatorial manifolds with respect to symmetry groups is exactly as in Table~\ref{table_num_triang}.
\end{theorem}
We will prove this theorem in the next section. Now, we would like to stress that four new phenomena occur in dimension~$16$:
\begin{itemize}
\item $K_1$ and~$K_4$ are the first examples of $(3d/2+3)$-vertex combinatorial $d$-manifolds like a projective plane without distinguished subcomplexes.
\item If we perform in~$K_2$ all the possible triple flips simultaneously, we obtain a combinatorial manifold~$K_3$ not isomorphic to~$K_2$. In all previously known examples in dimensions~$2$, $4$, and~$8$ performing all the possible triple flips simultaneously leads to the combinatorial manifold isomorphic to the initial one.
\item The simplices $\Delta_1,\Delta_2,\Delta_3\in K_2$ that constitute the distinguished triple from Claim~(2) of Theorem~\ref{theorem_smaller_groups} lie in pairwise different $\mathop{\mathrm{Sym}}\nolimits(K_2)$-orbits. (This follows immediately from the fact that the subcomplexes~$g(J)$ with $g\in \mathop{\mathrm{Sym}}\nolimits(K_2)$ are pairwise different.) On the contrary, in all previously known examples of distinguished triples~$(\Delta_1,\Delta_2,\Delta_3)$ in dimensions~$2$, $4$, and~$8$ there existed an element of the symmetry group permuting $\Delta_1$, $\Delta_2$, and~$\Delta_3$ cyclically.
\item Also a new phenomenon is the existence of (a huge amount of) $27$-vertex combinatorial $16$-manifolds like the octonionic projective plane with trivial symmetry groups. All known examples of $(3d/2+3)$-vertex combinatorial $d$-manifolds like a projective plane in dimensions $d=2,4,8$ have nontrivial symmetry groups.
\end{itemize}
\section{Proof of Theorem~\ref{theorem_smaller_groups}}
Suppose that $X$ is a pure $d$-dimensional simplicial complex on a finite vertex set~$V$. Let $\sigma$ be a $d$-simplex of~$X$ and let $v$ be a vertex of~$X$ not belonging to~$\sigma$. We denote by $\nu_X(\sigma,v)$ the number of $d$-simplices $\tau\in X$ such that $v\in\tau$ and $\dim(\sigma\cap\tau)=d-1$. The numbers $\nu_X(\sigma,v)$ for various $v\notin\sigma$ will be called \textit{$\nu$-parameters} for~$\sigma$.
If $X$ is a weak $d$-pseudomanifold, then it contains exactly $d+1$ simplices $\tau$ such that $\dim\tau=d$ and $\dim(\sigma\cap\tau)=d-1$. Hence,
\begin{equation}
\sum_{v\in V\setminus\sigma} \nu_X(\sigma,v)=d+1.
\end{equation}
\begin{propos}\label{propos_nu_param}
Suppose that $K$ is a combinatorial $d$-manifold like a projective plane with $3d/2+3$ vertices. Then
\begin{enumerate}
\item $\nu_K(\sigma,v)\le d/2$ for all~$\sigma$ and~$v$,
\item a $d$-simplex $\sigma\in K$ belongs to a distinguished subcomplex if and only if there is a vertex $v\notin\sigma$ satisfying $\nu_K(\sigma,v)=d/2$,
\item moreover, a $d$-simplex $\sigma\in K$ belongs to exactly $q$ different distinguished subcomplexes, where $q$ is the number of vertices $v\notin\sigma$ satisfying $\nu_K(\sigma,v)=d/2$.
\end{enumerate}
\end{propos}
\begin{proof}
Recall that $K$ satisfies the neighborliness condition~(b) and the complementarity condition~(c) from Section~\ref{section_intro}.
Let $\sigma$ be a $d$-simplex of~$K$. Suppose that $\nu_K(\sigma,v)=k$. Then there are exactly $k$ vertices $u\in \sigma$ satisfying $$(\sigma\setminus\{u\})\cup\{v\}\in K.$$ We denote by~$\rho$ the set consisting of all such vertices~$u$, and put
$$\Delta_1=\sigma\setminus\rho,
\qquad
\Delta_2=\rho\cup\{v\},
\qquad
\Delta_3=V\setminus(\sigma\cup\{v\}).
$$
Then any set obtained from $\Delta_1\cup\Delta_2$ by removing a vertex from~$\Delta_2$ belongs to~$K$, so $K$ contains the subcomplex $\Delta_1*\partial\Delta_2$. By the complementarity condition, we obtain that $K$ contains no simplex of the form $\Delta_3\cup\{u\}$ with $u\in \Delta_2$. On the other hand, the set~$\Delta_3$ consists of $d/2+1$ elements and hence, by the neighborliness condition, is a $(d/2)$-simplex of~$K$. It follows that $\mathop{\mathrm{link}}\nolimits(\Delta_3,K)$ is contained in~$\Delta_1$, so it has at most $d+1-k$ vertices. However, since $K$ is a combinatorial $d$-manifold, we obtain that $\mathop{\mathrm{link}}\nolimits(\Delta_3,K)$ is a combinatorial $(d/2-1)$-sphere and hence has at least $d/2+1$ vertices. Therefore $k\le d/2$.
Moreover, if $k=d/2$, then $\mathop{\mathrm{link}}\nolimits(\Delta_3,K)=\partial\Delta_1$. So $K$ contains the subcomplex $\Delta_3*\partial\Delta_1$. Repeating the same argument, we similarly obtain that $K$ contains the subcomplex $\Delta_2*\partial\Delta_3$, too. Then $K$ contains the distinguished subcomplex
$$
(\Delta_1*\partial\Delta_2)\cup(\Delta_2*\partial\Delta_3)\cup(\Delta_3*\partial\Delta_1).
$$
Thus, to each vertex $v\notin \sigma$ with $\nu_K(\sigma,v)=d/2$, we have assigned a distinguished subcomplex containing~$\sigma$. Vice versa, suppose that
$$
J=(\Delta_1*\partial\Delta_2)\cup(\Delta_2*\partial\Delta_3)\cup(\Delta_3*\partial\Delta_1)
$$
is a distinguished subcomplex containing~$\sigma$. Permuting cyclically~$\Delta_1$, $\Delta_2$, and~$\Delta_3$, we may achieve that $\sigma$ is contained in~$\Delta_1*\partial\Delta_2$. We assign to~$J$ a unique vertex~$v\in\Delta_2$ that does not belong to~$\sigma$. Then $\nu_K(\sigma,v)=d/2$.
It is easy to see that the constructed correspondence between vertices $v\notin\sigma$ satisfying $\nu_K(\sigma,v)=d/2$ and distinguished subcomplexes containing~$\sigma$ is bijective, which completes the proof of the proposition.
\end{proof}
For each of the four combinatorial $16$-manifolds~$K_1$, $K_2$, $K_3$, and~$K_4$, all $\nu$-parameters can be easily calculated using a computer. The result of this calculation is as follows.
\begin{propos}\label{propos_nu_param_calc}
\textnormal{(1)} All $\nu$-parameters $\nu_{K_1}(\sigma,v)$ and $\nu_{K_4}(\sigma,v)$ do not exceed~$7$.
\textnormal{(2)} For each $16$-simplex $\sigma\in K_i$, where $i$ is either~$2$ or~$3$, at most one of the corresponding $\nu$-parameters $\nu_{K_i}(\sigma,v)$ can be equal to~$8$.
\textnormal{(3)} There are exactly $27$ \ $G_{351}$-orbits of $16$-simplices $\sigma \in K_2$ for which one of the $\nu$-parameters $\nu_{K_2}(\sigma,v)$ is equal to~$8$. These are exactly all $16$-simplices $\sigma \in K_2$ that do not belong to~$K_3$. Moreover, these $27\cdot 351=9477$ simplices~$\sigma$ are exactly all $16$-simplices that belong to the union $\bigcup_{g\in G_{351}}g(J)$, where $J$ is the complex from Claim~(2) of Theorem~\ref{theorem_smaller_groups}.
\textnormal{(4)} The same remains true if we swap~$K_2$ and~$K_3$, and replace~$J$ with the complex~$\widetilde{J}$ from Claim~(3) of Theorem~\ref{theorem_smaller_groups}.
\end{propos}
Claims~(1), (2), and~(4) of Theorem~\ref{theorem_smaller_groups} immediately follow from Propositions~\ref{propos_nu_param} and~\ref{propos_nu_param_calc}.
Since $K_2$ and~$K_3$ are subcomplexes of the same $26$-simplex with the vertex set~$[27]$, we can consider the simplicial complexes $K_2\cup K_3$ and $K_2\cap K_3$. The former of them is pure but the latter is not, so instead of the intersection~$K_2\cap K_3$ we conveniently consider the following smaller complex. Namely, let $K_{2,3}$ be the simplicial complex consisting of all $16$-simplices belonging to both $K_2$ and~$K_3$, and all their faces. Then $K_{2,3}$ is pure. To prove Claim~(3) of Theorem~\ref{theorem_smaller_groups}, we need several auxiliary propositions.
\begin{propos}\label{propos_Sym23}
$\mathop{\mathrm{Sym}}\nolimits(K_{2,3})=G_{351}$.
\end{propos}
\begin{proof}
The proof of this proposition repeats the proof of Proposition~\ref{propos_SymK} (see Section~\ref{section_no_add_sym}) almost literally. Namely, first, $\mathop{\mathrm{Sym}}\nolimits(K_{2,3})\supseteq G_{351}$, since $\mathop{\mathrm{Sym}}\nolimits(K_2)=\mathop{\mathrm{Sym}}\nolimits(K_3)=G_{351}$. Second, as in the proof of Lemma~\ref{lem_not_transitive}, we can compute the matrix $(N_{pq})$ for~$K_{2,3}$, the result of computation is shown in Table~\ref{table_N_matrix_23}.
\begin{table}
\caption{The matrix $(N_{pq})$ for~$K_{2,3}$}\label{table_N_matrix_23}
\begin{tabular}{|c|ccccccccc|}
\hline
& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline
1 & 17 & 318 & 241 & 128 & 37 & 4 & 2 & 0 & 0 \\
2 & 353 & 3975 & 4336 & 3787 & 1703 & 462 & 133 & 12 & 2 \\
3 & 312 & 4671 & 18186 & 23937 & 13733 & 4676 & 1401 & 237 & 28 \\
4 & 146 & 3929 & 24376 & 48766 & 23576 & 6948 & 1821 & 259 & 28 \\
5 & 41 & 1655 & 14156 & 23770 & 14830 & 4470 & 1095 & 147 & 15 \\
6 & 4 & 397 & 4678 & 6910 & 4622 & 1776 & 521 & 78 & 12 \\
7 & 0 & 112 & 1356 & 1887 & 1134 & 492 & 190 & 42 & 6 \\
8 & 0 & 14 & 228 & 258 & 156 & 74 & 36 & 9 & 0 \\
9 & 0 & 1 & 22 & 38 & 15 & 5 & 7 & 1 & 0 \\
\hline
\end{tabular}
\end{table}
Since this matrix is not symmetric, we obtain that $\mathop{\mathrm{Sym}}\nolimits(K_{2,3})$ does not act transitively on the set of directed edges and hence
$$
G_{351}\subseteq \mathop{\mathrm{Sym}}\nolimits(K_{2,3})\subseteq \mathop{\mathrm{Sym}}\nolimits(\Gamma)=G_{351}\rtimes C_3.
$$
So we only need to show that the Frobenius automorphism~$F$ does not belong to~$\mathop{\mathrm{Sym}}\nolimits(K_{2,3})$. To this end, note that $K_{2,3}$ contains $259$~\ $G_{351}$-orbits of $16$-simplices, and the intersection~$K_2\cap F(K_2)$ contains only $6$~\ $G_{351}$-orbits of $16$-simplices, see Table~\ref{table_intersections}. Therefore, $K_{2,3}$ is not contained in $K_2\cap F(K_2)$ and hence $F\notin\mathop{\mathrm{Sym}}\nolimits(K_{2,3})$. Thus, $\mathop{\mathrm{Sym}}\nolimits(K_{2,3})=G_{351}$.
\end{proof}
It follows immediately from the construction that $K_S$ contains at least $351$ distinguished subcomplexes, namely, the subcomplexes~$g(J)$, where $g\in G_{351}\setminus S$, and~$g(\widetilde{J}\,)$, where $g\in S$. To prove Claim~(3) of Theorem~\ref{theorem_smaller_groups} we, in particular, need to check that $K_S$ contains no other distinguished subcomplexes. Certainly, we cannot check this directly by computing the $\nu$-parameters for all~$K_S$, since the number of complexes~$K_S$ is huge. Nevertheless, we can compute the $\nu$-parameters for the pure simplicial complex~$K_2\cup K_3$, which contains all combinatorial manifolds~$K_S$. The result of computation is as follows.
\begin{propos}\label{propos_nu_param_calc2+3}
\textnormal{(1)} For each $16$-simplex $\sigma\in K_2\cap K_3$, we have
$\nu_{K_2\cup K_3}(\sigma,v)\le 7$ for all $v\notin \sigma$.
\textnormal{(2)} For each $16$-simplex $\sigma\in K_2\cup K_3$ that does not belong to~$K_2\cap K_3$, there exists a unique vertex $v_{\sigma}\notin\sigma$ satisfying $\nu_{K_2\cup K_3}(\sigma,v_{\sigma})= 17$, and $\nu_{K_2\cup K_3}(\sigma,v)\le 7$ for all other vertices $v\notin \sigma$.
\end{propos}
Since $K_S\subset K_2\cup K_3$, we have $\nu_{K_S}(\sigma,v)\le\nu_{K_2\cup K_3}(\sigma,v)$ whenever $\sigma\in K_S$ and $v\notin\sigma$. Besides, $\nu_{K_S}(\sigma,v)\le 8$ by Proposition~\ref{propos_nu_param}(1). Also we know that if $\sigma\notin K_2\cap K_3$, i.\,e., $\sigma$ lies in one of the subcomplexes $g(J)$ and~$g(\widetilde{J}\,)$, then $\sigma$ lies in a distinguished subcomplex of~$K_S$ and hence there is a vertex $v\notin\sigma$ satisfying $\nu_{K_S}(\sigma,v)=8$. This yields the following proposition.
\begin{propos}\label{propos_nu_param_calcS}
\textnormal{(1)} For each $16$-simplex $\sigma\in K_2\cap K_3$, we have
$\nu_{K_S}(\sigma,v)\le 7$ for all $v\notin \sigma$.
\textnormal{(2)} For each $16$-simplex $\sigma\in K_S$ that does not belong to~$K_2\cap K_3$, there exists a unique vertex $v_{\sigma}\notin\sigma$ satisfying $\nu_{K_S}(\sigma,v_{\sigma})= 8$ and $\nu_{K_S}(\sigma,v)\le 7$ for all other vertices $v\notin \sigma$.
\end{propos}
\begin{cor}\label{cor_distinguished}
For each subset $S\subseteq G_{351}$, the combinatorial manifold~$K_S$ contains exactly $351$ distinguished subcomplexes, namely, the subcomplexes~$g(J)$, where $g\in G_{351}\setminus S$, and~$g(\widetilde{J}\,)$, where $g\in S$.
\end{cor}
\begin{proof}[Proof of Claim~(3) of Theorem~\ref{theorem_smaller_groups}]
It follows from Claim~(2) of Theorem~\ref{theorem_smaller_groups} and Proposition~\ref{propos_move} that each~$K_S$ is a well-defined $27$-vertex combinatorial $16$-manifold like the octonionic projective plane and is PL homeomorphic to~$K_2$.
Suppose that $f\colon K_{S_1}\to K_{S_2}$ is an isomorphism. We conveniently identify~$f$ with the corresponding permutation of vertices. Since any isomorphism of combinatorial manifold preserves the $\nu$-parameters, it follows from Proposition~\ref{propos_nu_param_calcS} that $f$ takes $16$-simplexes belonging to~$K_2\cap K_3$ to $16$-simplexes belonging to~$K_2\cap K_3$ and $16$-simplexes not belonging to~$K_2\cap K_3$ to $16$-simplexes not belonging to~$K_2\cap K_3$. Hence, $f$ belongs to the subgroup $\mathop{\mathrm{Sym}}\nolimits(K_{2,3})\subset S_{27}$, which by Proposition~\ref{propos_Sym23} coincides with~$G_{351}$. Now, for each $g\in[27]\setminus S_1$, the permutation~$f$ takes the subcomplex $g(J)\subset K_{S_1}$ to the complex $fg (J)$, which therefore must be contained in~$K_{S_2}$. Hence, $f([27]\setminus S_1)\subseteq [27]\setminus S_2$. Similarly, considering the subcomplexes $g(\widetilde{J}\,)\subset K_{S_1}$ for $g\in S_1$, we obtain that $fS_1\subseteq S_2$. Thus, $fS_1=S_2$. In particular, taking $S_1=S_2=S$, we see that $\mathop{\mathrm{Sym}}\nolimits(K_S)$ is exactly the subgroup of~$G_{351}$ consisting of all~$f$ satisfying~$fS=S$. These assertions together with Corollary~\ref{cor_distinguished} constitute Claim~(3) of Theorem~\ref{theorem_smaller_groups}.
\end{proof}
\begin{proof}[Proof of Claim~(5) of Theorem~\ref{theorem_smaller_groups}]
Set $G=G_{351}$. The group~$G$ acts by left shifts on itself and hence on subsets $S\subseteq G$. We denote the stabilizer of~$S$ in~$G$ by~$G_S$. By Claim~(3) of Theorem~\ref{theorem_smaller_groups}, $\mathop{\mathrm{Sym}}\nolimits(K_S)=G_S$ and $K_{S_1}$ is isomorphic to~$K_{S_2}$ if and only if $gS_1=S_2$ for some $g\in G$. Hence, for a subgroup $H\subseteq G$, the number~$m_H$ of isomorphism classes of triangulations~$K_S$ with $\mathop{\mathrm{Sym}}\nolimits(K_S)$ conjugate to~$H$ is equal to the number of $G$-orbits of subsets $S$ with stabilizers conjugate to~$H$. Therefore,
\begin{equation}\label{eq_mH}
m_H=\frac{r_Hn_H}{[G:H]}=\frac{n_H}{[N(H):H]}\,,
\end{equation}
where $n_H$ is the number of subsets $S\subseteq G$ such that $G_S=H$, $N(H)$ is the normalizer of~$H$ in~$G$, and $r_H=[G:N(H)]$ is the number of conjugates of~$H$.
Now,
\begin{equation}\label{eq_nH}
n_H=n_{\ge H}-\sum_{H\subsetneq Q\subseteq G} n_{Q},
\end{equation}
where $n_{\ge H}$ is the number of subsets $S\subseteq G$ such that $G_S\supseteq H$. Obviously, $G_S\supseteq H$ if and only if $S$ is a union of right cosets of~$H$ in~$G$. Hence,
\begin{equation}\label{eq_n>=H}
n_{\ge H}=2^{[G:H]}.
\end{equation}
Formulae~\eqref{eq_mH}--\eqref{eq_n>=H} allow us to compute the required numbers~$m_H$ for all conjugacy classes of subgroups of~$G$.
It is easy to check that the group $G=C_3^3\rtimes C_{13}$ contains exactly $6$ conjugacy classes of subgroups:
\begin{enumerate}
\item The whole group $G$. We have $m_G=n_G=n_{\ge G}=2$.
\item The normal subgroup~$C_3^3$. We have
\begin{align*}
n_{C_3^3}&=n_{\ge C_3^3}-n_G=2^{13}-2,\\
m_{C_3^3}&=\frac{n_{C_3^3}}{13}=630.
\end{align*}
\item The conjugacy class consisting of $27$ subgroups isomorphic to~$C_{13}$. We have $N(C_{13})=G$ and
$$
m_{C_{13}}=n_{C_{13}}=n_{\ge C_{13}}-n_G=2^{27}-2.
$$
\item The conjugacy class consisting of $13$ subgroups isomorphic to~$C_{3}^2$. We have $N(C_{3}^2)=C_3^3$ and
\begin{align*}
n_{C_3^2}&=n_{\ge C_3^2}-n_{C_3^3}-n_G=2^{39}-2^{13},\\
m_{C_3^2}&=\frac{n_{C_3^2}}{3}=\frac{1}{3}(2^{39}-2^{13}).
\end{align*}
\item The conjugacy class consisting of $13$ subgroups isomorphic to~$C_{3}$. We have $N(C_{3})=C_3^3$. Besides, each subgroup~$C_3$ is contained in exactly $4$ subgroups isomorphic to~$C_3^2$. Therefore,
\begin{align*}
n_{C_3}&=n_{\ge C_3}-4n_{C_3^2}-n_{C_3^3}-n_G=2^{117}-2^{41}+3\cdot 2^{13},\\
m_{C_3^2}&=\frac{n_{C_3}}{9}=\frac19(2^{117}-2^{41}+3\cdot 2^{13}).
\end{align*}
\item The trivial subgroup~$1$. We have
\begin{align*}
n_{1}&=n_{\ge 1}-13n_{C_3}-13n_{C_3^2}-27n_{C_{13}}-n_{C_3^3}-n_G\\
{}&=2^{351}-13\cdot 2^{117}+39\cdot 2^{39}- 27\cdot (2^{27}+2^{13}-2),\\
m_{1}&=\frac{n_{1}}{351}=\frac1{351}\bigl(2^{351}-13\cdot 2^{117}+39\cdot 2^{39}- 27\cdot (2^{27}+2^{13}-2)\bigr).
\end{align*}
\end{enumerate}
This completes the proof of Claim~(5) of Theorem~\ref{theorem_smaller_groups}.
\end{proof}
\begin{remark}
It follows from Theorem~\ref{theorem_main_detail} that $K_1$, $K_2$, $K_3$, and~$K_4$ are the only up to isomorphism $27$-vertex combinatorial $16$-manifolds like the octonionic projective plane with the symmetry group $G_{351}\subset S_{27}$. However, Table~\ref{table_num_triang} contains a stronger assertion. Namely, we claim that $K_1$, $K_2$, $K_3$, and~$K_4$ are the only $27$-vertex combinatorial $16$-manifolds like the octonionic projective plane with the symmetry groups isomorphic to~$G_{351}$. To prove this we need to check that the group $G_{351}$ has a unique up to isomorphism effective action on $27$ points. Indeed, all transitive actions of~$G_{351}$ on~$[27]$ are isomorphic, since all index~$27$ subgroups $C_{13}\subset G_{351}$ are conjugate to each other. Besides, if an action of~$G_{351}$ on~$[27]$ is not transitive, then all orbits of this action have cardinalities~$1$ or~$13$ and therefore the normal subgroup $C_3^3\subset G_{351}$ lies in the kernel of the action. So the action is not effective.
\end{remark}
\clearpage
|
1,116,691,500,828 | arxiv | \section{Introduction} \label{Introduction}
Fiber-tracking is probably one of the most fascinating applications in diffusion MRI (dMRI), gathering a lot of attention since its introduction because of its ability to reconstruct the main neuronal bundles of the brain from the acquired data.
In fact, the random movement of the molecules in the white matter can be exploited for mapping brain connectivity, and structures otherwise invisible with other imaging modalities can be highlighted.
The study of this structural connectivity is of major importance in a fundamental neuroscience perspective, for developing our understanding of the brain, but also in a clinical perspective, with particular applications for the study of a wide range of neurological disorders.
The most powerful acquisition modality is diffusion spectrum imaging (DSI) \citep{Wedeen:2005aa}. It relies on cartesian signal sampling and is known to provide good imaging quality, but it is too time-consuming to be of real interest in a clinical perspective. Diffusion tensor imaging (DTI) \citep{Basser:1994aa} is always preferred instead. DTI is a very fast model-based technique providing valuable diagnostic information but, on the contrary, it is unable to model multiple fiber populations in a voxel. In a global connectivity analysis perspective, this constitutes a key limiting factor. Accelerated acquisitions relying on a smaller number of samples while providing accurate estimations of the intra-voxel fiber configuration thus represent an important challenge.
Recently, an increasing number of high angular resolution diffusion imaging (HARDI) approaches have been proposed for tackling this problem. In particular, spherical deconvolution (SD) based methods formed a very active area in this field \citep{Tournier:2004aa, Alexander:2005aa, Tournier:2007aa, Dellacqua:2007aa}. These methods rely on the assumption that the signal attenuation acquired with diffusion MRI can be expressed as the convolution of a given response function with the fiber orientation distribution (FOD). The FOD is a real-valued function on the unit sphere $\left(\mathrm{S}^2\right)$ giving the \textit{orientation} and the \textit{volume fraction} of the fiber populations present in a voxel. The response function, or kernel, describes the dMRI signal attenuation generated by an isolated single fiber population; it can be estimated from the data or represented by means of parametric functions.
SD approaches represented a big step in reducing the acquisition time of diffusion MRI, but are known to suffer heavily from noise and intrinsic instabilities in solving the deconvolution problem. For this reason, a regularization scheme is normally employed. A variety of approaches have been proposed, which are generally based on the two assumptions that the FOD is (i) a non-negative function and (ii) sparse, i.e. with only a few nonzero values, either explicitly or implicitly.
In fact, at the imaging resolution available nowadays, diffusion MRI is sensitive only to the major fiber bundles and it is commonly accepted that it can reliably \textit{disentangle up to 2-3 different fiber populations} inside a voxel \citep{Jeurissen:2010aa, Schultz:2012aa}. Hence, the FOD can reasonably be considered sparse in nature.
\MOD{In particular, the state-of-the-art Constrained Spherical Deconvolution (CSD) approach of \citet{Tournier:2007aa} resorts to Tikhonov regularization, based on an $\ell_2$-norm prior. While its primary purpose is to ensure the positivity of the FOD, it actually also implicitly promotes sparsity, but only a weak version of it.}
The recent advent of \emph{compressed sensing} (CS) theory \citep{Donoho:2006aa,Candes:2006aa,Baraniuk:2007aa} provided a mathematical framework for the reconstruction of sparse signals from under-sampled measurements mainly in the context of convex optimization.
CS has inspired new advanced approaches in the last few years for solving the reconstruction problem in diffusion MRI and allowed a further dramatic reduction in the number of samples needed to accurately infer the fiber structure in each voxel\MOD{, by promoting sparsity explicitly}.
For instance, \citet{Tristan-Vega:2011aa} and \citet{Michailovich:2011aa} recovered the orientation distribution function (ODF) by using different representations for the response function, while \citet{Merlet:2011aa} and \citet{Rathi:2011aa} focused on the full ensemble average propagator (EAP) of the diffusion process. In this work, however, we focus on spherical deconvolution based-methods and the quantity of interest is the FOD.
In general these methods are based on $\ell_1$ minimization, where the $\ell_1$ norm is defined as $|| \vect{x} ||_1 = \sum_i^n |\vect{x}_i|$ for any vector $\vect{x} \in \mathbb{R}^n$, and the common goal is to recover the FOD with fewest non-zeros that is compatible with the acquired dMRI data \citep{Ramirez-Manzanares:2007aa, Pu:2011aa, Landman:2012aa, Mani:2012aa}.
However, a minimum $\ell_1$-norm prior is inconsistent with the physical constraint that the sum of the volume fractions of the compartments inside a voxel is intrinsically equal to unity.
In this paper, we propose to exploit the versatility of compressed sensing and convex optimization to solve what we understand as \textit{the $\ell_1$ inconsistency}\MOD{, while simultaneously exploiting sparsity more optimally than the approaches based on the $\ell_2$ prior,} and improve the quality of FOD reconstruction in the white matter.
Our approach is as follows.
Strictly speaking, the FOD sparsity is the number of fiber populations, thus identified by the $\ell_0$ norm of the FOD.
$\ell_0$-norm problems are generally intractable as they are non convex, which explains the usual convex $\ell_1$-norm relaxation in the framework of compressed sensing.
To this end, some greedy algorithms have been proposed to approximate the $\ell_0$ norm through a sequence of incremental approximations of the solution, such as Matching Pursuit~\citep{Mallat:1993aa} and Orthogonal Matching Pursuit~\citep{Pati:1993aa}.
However, the greedy and local nature of these algorithms, i.e. in the sense that compartments are identified sequentially, makes them suboptimal as compared to more robust approaches based on convex optimization, which are global in nature. In particular, a reweighted $\ell_1$ minimization scheme was developed by \citet{Candes:2008aa} in order to approach $\ell_0$ minimization by a sequence of convex weighted-$\ell_1$ problems.
We thus solve the $\ell_0$ minimization problem by making use of a reweighting scheme \MOD{and evaluate the effectiveness of the proposed formulation} in comparison with \MOD{state-of-the-art aprroaches based on either $\ell_2$ or $\ell_1$ priors}. We report results on both synthetic and real data.
\section{Materials and methods} \label{MaterialsAndMethods}
\subsection{Intra-voxel structure recovery via spherical deconvolution} \label{Background}
As shown by \citet{Jian:2007aa}, spherical deconvolution methods can be cast into the following computational framework:
\begin{equation} \label{eqn:DeconvUnifiedFramework}
S(b,\vect{\hat{q}})/S_0 = \int R_{\vect{\hat{q}}}(\vect{\hat{p}}) \; f(\vect{\hat{p}}) \; d\Omega(\vect{\hat{p}}) ,
\end{equation}
where $f$ is the FOD to be estimated, $R_{\vect{\hat{q}}}$ the response function rotated in direction $\vect{\hat{q}} \in \mathrm{S}^2$ and the integration is performed over the unit sphere with $\vect{\hat{p}} = (\phi,\theta) \in \mathrm{S}^2$ and $d\Omega = \sin{\phi} \, d\phi \, d\theta$.
$S(b,\vect{\hat{q}})$ represents the dMRI signal measured on the q-space shell acquired with b-value $b$ in direction $\vect{\hat{q}} \in \mathrm{S}^2$, while $S_0$ is the signal acquired without diffusion weighting.
\MOD{The FOD $f$ is normally expressed as a linear combination of basis functions, e.g. spherical harmonics, as $f(\vect{\hat{p}}) = \sum_j w_j f_j(\vect{\hat{p}})$.}
The measurement process can thus be expressed in terms of the general formulation:
\begin{equation} \label{eqn:Recon-Problem}
\vect{y} = \Phi \vect{x} + \eta ,
\end{equation}
where $\vect{x} \in \mathbb{R}_+^n$ are the coefficients of the FOD, $\vect{y} \in \mathbb{R}^m$ is the vector with the dMRI signal measured in the voxel with $y_i = S(b,\vect{\hat{q}}_i)/S_0$ for $i \in \{ 1, \ldots, m \}$, $\eta$ represents the acquisition noise and \MOD{$\Phi = \{ \phi_{ij} \}\in \mathbb{R}^{m \times n}$ is the observation matrix modeling explicitly the convolution operator with the response function $R$, with $\phi_{ij} = \int R_{\vect{\hat{q}_i}}(\vect{\hat{p}}) \, f_j(\vect{\hat{p}}) \, d\Omega(\vect{\hat{p}})$.
Several choices for the convolution kernels and basis functions exist in the literature; more details will be provided on the specific $\Phi$ used with each algorithm considered in this work.}
\subsection{\MOD{\texorpdfstring{$\ell_2$}{Lg} prior}}
In the original formulation of \citet{Tournier:2004aa}, the FOD $\vect{x}$ and the measurements $\vect{y}$ were expressed by means of spherical harmonics (SH), and the deconvolution problem was solved by a simple matrix inversion. To reduce noise artifacts, a low-pass filter was applied for attenuating the high harmonic frequencies.
The method was improved in \citet{Tournier:2007aa} by reformulating the problem as an iterative procedure where, at each iteration, the current solution $\vect{x}^{(t)}$ is used to drive to zero the negative amplitudes of the FOD at the next iteration with a Tikhonov regularization \citep{Tikhonov:1977aa}:
\begin{equation} \label{eqn:SD-Problem}
\vect{x}^{(t+1)} = \underset{\vect{x}}{\operatorname{argmin}} \; || \Phi \vect{x} - \vect{y} ||_2^2 + \lambda^2 || L^{(t)} \vect{x}||_2^2 ,
\end{equation}
where $||\cdot||_p$ are the usual $\ell_p$ norms in $\mathbb{R}^n$, the free parameter $\lambda$ controls the degree of regularization and $L^{(t)}$ can be understood as a simple binary mask preserving only the directions of negative or small values of $x^{(t)}$. The $\ell_2$-norm regularization term therefore tends to send these values to zero, as probably spurious, hence \MOD{favoring large positive} values. Interestingly,
\MOD{beyond the claimed purpose of enforcing positivity}
, the operator $L$ thus \MOD{also} implicitly promotes a weak version of sparsity. However, this \MOD{$\ell_2$ prior} does not \MOD{explicitly} guarantee either positivity or sparsity in the recovered FOD.
In \citet{Alexander:2005aa} a maximum-entropy regularization was proposed to recover the FOD as the function that exhibits the minimum information content. The method showed higher robustness to noise than previous approaches, but was limited by the very high computational cost and did not promote sparsity.
Other regularization schemes have been proposed in the literature, but FOD sparsity has never been addressed with a rigorous mathematical formulation.
\subsection{\MOD{\texorpdfstring{$\ell_1$}{Lg} prior}}
Compressed sensing provides a powerful mathematical framework for the reconstruction of sparse signals from a low number of data \citep{Donoho:2006aa,Candes:2006aa}, mainly in the context of convex optimization.
According to this theory, it is possible to recover a signal from fewer samples than the number required by the Nyquist sampling theorem, provided that the signal is sparse in some \emph{sparsity basis} $\Psi$.
Let $\vect{x} \in \mathbb{R}^n$ be the signal to be recovered from the $m \ll n$ linear measurements $\vect{y} = \Phi \vect{x} \in \mathbb{R}^m$ and $\vect{\alpha} \in \mathbb{R}^n$ a sparse representation of $\vect{x}$ through $\Psi \in \mathbb{R}^{n \times n}$. If the observations $\vect{y}$ are corrupted by noise and $\Phi$ obeys some randomness and incoherence conditions, then the signal $\vect{x} = \Psi \vect{\alpha}$ can be recovered by solving the convex $\ell_1$ optimization problem:
\begin{equation} \label{eqn:CSProblem}
\underset{\vect{\alpha}}{\operatorname{argmin}} \; || \vect{\alpha} ||_1 \;\; \textrm{subject to} \;\; || \Phi \, \Psi \, \vect{\alpha} - \vect{y} ||_2 \le \epsilon ,
\end{equation}
where $\epsilon$ is a bound on the noise level.
\MOD{Assuming Gaussian noise, the square $\ell_2$ norm of the residual represents the log-likelihood of the data and follows a $\chi^2$ distribution. For a sufficiently large number of measurements, this distribution is extremely concentrated around its mean value. This fact is related to the well-known phenomenon of concentration of measure in statistics. Consequently, $\epsilon$ can be precisely defined by the mean of the $\chi^2$.}
In the context of FOD reconstructions, the sparsity basis $\Psi$ boils down to the identity matrix, thus $\vect{x} = \vect{\alpha}$.
In \citet{Ramirez-Manzanares:2007aa} and \citet{Jian:2007aa} the sensing basis $\Phi$, also called \emph{dictionary}, is generated by applying a set of rotations to a given Gaussian kernel (i.e. diffusion tensor) and the sparsest coefficients $\vect{x}$ of this linear combination best matching the measurements $\vect{y}$ are recovered by solving the following constrained minimization problem:
\begin{equation} \label{eqn:BPDNProblem}
\underset{\vect{x} \ge 0}{\operatorname{argmin}} \; || \vect{x} ||_1 \;\; \textrm{subject to} \;\; || \Phi \, \vect{x} - \vect{y} ||_2 \le \epsilon ,
\end{equation}
where the positivity constraint on the FOD values was directly embedded in the formulation of the convex problem.
\MOD{For $\mathrm{SNR}>2$ the noise in the magnitude dMRI images can be assumed Gaussian-distributed\footnote{Since dMRI data is commonly normalized by the baseline $S_0$, the $S_0$ image must be accurately estimated in order to keep the same noise statistics of the non-normalized signal. This is normally the case, though, as multiple $S_0$ volumes are commonly acquired in practice.} \citep{Gudbjartsson:1995aa}.
However, a statistical estimation of $\epsilon$ is not reliable, precisely because the number of measurements is very small and the $\chi^2$ is not really concentrated around its mean value. Thus, $\epsilon$ becomes an arbitrary parameter of the algorithm.
At very low SNR, one can also extrapolate the choice of the $\ell_2$ norm as a simple penalization term independent of statistical considerations.}
\MOD{The reconstruction problem can also be re-formulated as} a regularized (as opposed to constrained) $\ell_1$ minimization as in \citet{Landman:2012aa} and \citet{Pu:2011aa}:
\begin{equation} \label{eqn:1-normProblemRegularized}
\underset{\vect{x} \ge 0}{\operatorname{argmin}} \; || \Phi \vect{x} - \vect{y} ||_2^2 \; + \; \beta \; || \vect{x} ||_1 ,
\end{equation}
where the free parameter $\beta$ controls the trade-off between the data and the sparsity constraints.
In general, $\beta$ depends on the acquisition scheme and the noise level and it must be empirically optimized.
\MOD{Following the general CS approach, problems \eqref{eqn:BPDNProblem} and \eqref{eqn:1-normProblemRegularized} consider an $\ell_1$-norm prior on the FOD $\vect{x}$. However, in the dMRI context, a minimum $\ell_1$-norm prior is inconsistent with the physical constraint that the sum of the volume fractions of the compartments inside a voxel is intrinsically equal to unity, i.e. $|| \vect{x} ||_1 \equiv \sum_i \vect{x}_i = 1$.}
For this reason, we reckon that \MOD{also} these $\ell_1$-based formulations are intrinsically suboptimal.
\MOD{Fig.~\ref{fig:SumX} illustrates this inconsistency by reporting the $\ell_1$ norm of reconstructed FODs as a function of the amplitude of measurement noise.}
\MOD{Our main goal in this work is to demonstrate the suboptimalities of the approaches based on $\ell_2$ and $\ell_1$ priors and to suggest a new formulation, based on an $\ell_0$ prior, adequately characterizing the actual sparsity lying in the FOD.}
\subsection{\MOD{\texorpdfstring{$\ell_0$}{Lg} prior}} \label{ReweightedL1Minimization}
In the aim of adequately characterizing the FOD sparsity, we re-formulate the reconstruction problem as a constrained $\ell_0$ minimization problem:
\begin{equation} \label{eqn:0-normProblem}
\underset{\vect{x} \ge 0}{\operatorname{argmin}} \; || \Phi \vect{x} - \vect{y} ||_2^2 \;\; \textrm{subject to} \;\; || \vect{x} ||_0 \leq k,
\end{equation}
where $||\cdot||_0$ explicitly counts the number of nonzero coefficients and $k$ represents \MOD{an upper bound on} the expected number of fiber populations in a voxel.
As already stated, the $\ell_0$ problems as such are intractable. The reweighting scheme proposed by \citet{Candes:2008aa} proceeds by sequentially solving weighted $\ell_1$ problems of the form \eqref{eqn:0-normProblem}, where the $\ell_0$ norm is substituted by a weighted $\ell_1$ norm defined as $||\vect{w} \vect{\alpha}||_1 = \sum_i \; \vect{w}_i \, |\vect{\alpha}_i|$, for positive weights $\vect{w}_i$ and where $i$ indexes vector components.
At each iteration, the weights are set as the inverse of the values of the solution of the previous problem, i.e. $\vect{w}_i^{(t)} \approx 1 / \vect{x}_i^{(t-1)}$.
At convergence, this set of weights makes the weighted $\ell_1$ norm independent of the precise value of the nonzero components, thus mimicking the $\ell_0$ norm while preserving the tractability of the problem with convex optimization tools.
Of course, it is not possible to have infinite weights for null coefficients; so a stability parameter $\tau$ must be added to the coefficients in the selection of the weights.
\begin{algorithm}[htbp]
\caption{Reweighted $\ell_1$ minimization for intra-voxel structure recovery}
\label{alg:ReweightedL1}
\small
\begin{algorithmic}
\Require
\hspace{0.5cm}Diffusion MRI signal $\vect{y} \in \mathbb{R}^m$ and sensing basis $\Phi \in \mathbb{R}^{m \times n}$
\Ensure
\hspace{0.2cm}FOD $\vect{x} \in \mathbb{R}^n$
\State Set the initial status: \\
\hspace{0.5cm} $t \gets 0$ \hspace{0.1cm} and \hspace{0.1cm} $\vect{w}_i^{(0)} \gets 1, \;\; i = 1, \ldots, n$ \hspace{1cm} (the symbol $\gets$ denotes assignment)
\Repeat
\State Solve the problem: \\
\hspace{1.0cm}$\vect{x}^{(t)} \gets \underset{\vect{x} \ge 0}{\operatorname{argmin}} \; || \Phi \vect{x} - \vect{y} ||_2^2 \;\; \textrm{subject to} \;\; || \vect{w}^{(t)} \vect{x} ||_1 \leq k$
\State Update the weights: \\
\hspace{1.5cm} $\vect{w}_i^{(t+1)} = \frac{1}{|\vect{x}_i^{(t)}| + \tau}$
\State $t \gets t + 1$
\Until{stopping criterion is satisfied}
\State $\vect{x} \gets \vect{x}^{(t-1)}$
\end{algorithmic}
\end{algorithm}
The \textit{main steps of the reweighted scheme} are reported in the algorithm \ref{alg:ReweightedL1}; in the remaining of the manuscript we will refer to it as \texttt{L2L0}\xspace, as it is based on a $\ell_0$ prior.
We empirically set $\tau = 10^{-\MOD{3}}$ and the procedure was stopped if $\frac{||\vect{x}^{(t)} - \vect{x}^{(t-1)}||_1}{||\vect{x}^{(t-1)}||_1} < 10^{-3}$ between two successive iterations or after 20 iterations.
At the first iteration the weighted $\ell_1$ norm is the standard $\ell_1$ norm given $\vect{w}=1$, and therefore the constraint $|| \vect{w}^{(0)} \vect{x} ||_1 \leq k$ is a weak bound on the sum of the fiber compartments and does not constitute a limitation in the procedure.
\MOD{The proposed $\ell_0$ approach thus strongly promotes sparsity (by opposition with the $\ell_2$ approach) and circumvents the $\ell_1$ inconsistency. It is noteworthy that our formulation at least partially addresses the problem of arbitrary parameters such as $\epsilon$ in \eqref{eqn:BPDNProblem} and $\beta$ in \eqref{eqn:1-normProblemRegularized}, or $\lambda$ in \eqref{eqn:SD-Problem}. Our parameter $k$ indeed explicitly identifies an upper bound on the number of fibers.}
As discussed before and largely assumed in the literature, we can expect to have at maximum 2-3 fiber compartments in each voxel. The algorithm was found to be quite robust to the choice of $k$, and differences were not observed for values up to $k = 5$.
Finally, an explicit constraint $\sum_i \vect{x}_i = 1$ might have been added, as it represents the physical property that the volume fractions must sum up to unity.
For the sake of simplicity, in this work this constraint was not included (as it is always the case), assuming it is carried over by the data and well-designed bases as pointed out by \citet{Ramirez-Manzanares:2007aa}. \MOD{In sections \ref{VolumeFractionsAndL1} and \ref{RealDataVolumeFractionsAndL1}} we will provide evidence that actually this physical constraint is not met when using \MOD{$\ell_2$ or $\ell_1$ priors}, whereas it is correctly satisfied with our \MOD{proposed} $\ell_0$ formulation. This might have severe consequences on the reconstruction quality.
\subsection{Comparison framework}
\MOD{We compared our $\ell_0$ approach based on problem \eqref{eqn:0-normProblem} against state-of-the-art $\ell_2$ and $\ell_1$ approaches respectively based on problems \eqref{eqn:SD-Problem} and \eqref{eqn:1-normProblemRegularized}, and referred to as \texttt{L2L2}\xspace and \texttt{L2L1}\xspace.}
To run \texttt{L2L2}\xspace reconstructions we made use of the original \texttt{mrtrix} implementation of \cite{Tournier:2012aa}, setting the optimal parameters as suggested by the software itself. To solve the \texttt{L2L1}\xspace and \texttt{L2L0}\xspace problems we used the SPArse Modeling Software (SPAMS)\footnote{http://spams-devel.gforge.inria.fr}, an open-source toolbox written in C++ for solving various sparse recovery problems. SPAMS contains a very fast implementation of the LARS algorithm \citep{Efron:2004aa} for solving the LASSO problem and its variants as \MOD{the \texttt{L2L1}\xspace problem in equation \eqref{eqn:1-normProblemRegularized} and the weighted $\ell_1$ minimizations required for our \texttt{L2L0}\xspace approach in equation \eqref{eqn:0-normProblem}}.
Numerical simulations on synthetic data were performed to quantitatively assess the performance of \texttt{L2L2}\xspace, \texttt{L2L1}\xspace and \texttt{L2L0}\xspace under controlled conditions. \MOD{The effectiveness of the three priors was also assessed in case of real human brain data}.
\subsection{Numerical simulations} \label{SyntheticData}
Independent voxels with two fiber populations crossing at specific angles ($30^\circ - 90^\circ$ range) and with equal volume fractions were synthetically generated. The signal $S$ corresponding to each voxel configuration was simulated by using the exact expression given in \citet{Soderman:1995aa} for the dMRI signal attenuation from particles diffusing in a restricted cylindrical geometry of radius $\rho$ and length $L$ with free diffusion coefficient $D_0$. The following parameters were used \citep{Ozarslan:2006aa,Jian:2007aa}: $L = 5 \; \mathrm{mm}$, $\rho = 5 \; \mathrm{\mu m}$, $D_0 = 2.02 \; \times \; 10^{-3} \; \mathrm{mm^2/s}$, $\Delta = 20.8 \; \mathrm{ms}$, $\delta = 2.4 \; \mathrm{ms}$.
The signal $S$ was contaminated with \emph{Rician noise} \citep{Gudbjartsson:1995aa} as follows:
\begin{equation}
S_{\mathrm{noisy}} = \sqrt{(S+\xi_1)^2 + (\xi_2)^2} ,
\end{equation}
where $\xi_1,\xi_2 \sim \mathcal{N}(0,\sigma^2)$ and $\sigma = S_0 / \mathrm{SNR}$ corresponds to a given signal-to-noise ratio on the $S_0$ image. We assumed $S_0 = 1$ without loss of generality.
Because of this assumption, we have implicitly considered a constant echo-time for acquisitions with different b-values, thus ignoring the fact that higher b-values normally require longer echo-times and therefore the images have a lower signal-to-noise ratio.
The study of the impact of the echo-time on different regularization priors is beyond the scope of our investigation.
For each voxel configuration, the signal was simulated at different b-values, $b \in \left\{ 500, 1000, \ldots, 4000 \right\} \mathrm{s/mm^2}$, and seven q-space sampling schemes were tested, respectively with 6, 10, 15, 20, 25, 30 and 50 samples equally distributed on half the unit sphere using electrostatic repulsion \citep{Jones:1999aa} assuming antipodal symmetry in diffusion signal.
Six different noise levels were considered, $\mathrm{SNR} = {5, 10, \ldots, 30}$. For every SNR, 100 repetitions of the same voxel were generated using different realizations of the noise. In our experiments, the actual signal-to-noise ratio in the simulated signal was always in a range where the Gaussian assumption on the noise holds. In the extreme setting with a $\textrm{SNR} = 5$ on the $S_0$ and $b = 4000 \; \mathrm{s/mm^2}$ the actual signal-to-noise ratio in the diffusion weighted signal was about 1.4.
\subsection{Evaluation criteria} \label{EvaluationCriteria}
As one of the aims of this work is to improve SD reconstructions, we adopted standard metrics widely used in the literature \citep{Ramirez-Manzanares:2008aa,Landman:2012aa,Michailovich:2011aa} to assess the quality of the reconstructions with respect to number and orientation of the fiber populations:
\begin{itemize}
\item \emph{Probability of false fiber detection}. This metric quantifies the correct assessment of the real number $M$ of populations inside a voxel:
\begin{equation}
\mathrm{P_d} = \frac{ | M - \tilde{M} | }{ M } \cdot 100 \%,
\end{equation}
where $\tilde{M}$ is the estimated number of compartments.
As $\mathrm{P_d}$ does not distinguish between missed fibers and extra compartments found by the reconstruction, we also make use of the following two quantities where needed, $\mathrm{n^-}$ and $\mathrm{n^+}$, explicitly counting the number of \emph{under-} and \emph{over-estimated} compartments, respectively.
\item \emph{Angular error}. This metric quantifies the angular accuracy in the estimation of the directions of the fiber populations in a voxel:
\begin{equation}
\epsilon_{\theta} = \frac{180}{\pi} \, \arccos( \, | \vect{d} \, \cdot \, \tilde{\vect{d}} \, | ) ,
\end{equation}
where $\vect{d}$ is a true direction and $\tilde{\vect{d}}$ is its closest estimate.
The final value is an average over all fiber compartments \MOD{by first matching the estimated directions to the ground-truth without using twice the same direction}.
\end{itemize}
Peaks detection was performed using a local maxima search algorithm on the recovered FOD, considering a neighborhood of orientations within a cone of $15^{\circ}$ around every direction.
For this reason, evaluation metrics are not sensitive for small crossing angles and results are reported in a conservative range $30^\circ\textrm{--}90^\circ$.
To filter out spurious peaks, values smaller than $10\%$ of the largest peak were discarded; in the case of \texttt{L2L2}\xspace we had to increase this threshold to $20\%$, as suggested in \citet{Tournier:2007aa}, in order to compare with the other methods.
\subsection{Real data} \label{RealData}
The human brain data have been acquired from 3 young healthy volunteers on a 3T Magnetom Trio system (Siemens, Germany) equipped with a 32-channel head coil using standard protocols routinely used in clinical practice. Each dataset corresponds to a distinct subject.
Two DTI scans (referred in the following as $\mathtt{dti}_\mathtt{30}$\xspace and $\mathtt{dti}_\mathtt{20}$\xspace) were acquired at $b = 1000 \; \mathrm{s}/\mathrm{mm}^2$ using 30 and 20 diffusion gradient directions, respectively, uniformly distributed on half the unit sphere using electrostatic repulsion \citep{Jones:1999aa}.
Other acquisition parameters were as follows: $\mathrm{TR}/\mathrm{TE} = 7000/82 \; \mathrm{ms}$ and spatial resolution = $2.5 \times 2.5 \times 2.5 \; \mathrm{mm}$ for dataset $\mathtt{dti}_\mathtt{30}$\xspace, while $\mathrm{TR}/\mathrm{TE} = 6000/99 \; \mathrm{ms}$ and spatial resolution = $2.2 \times 2.2 \times 3 \; \mathrm{mm}$ for dataset $\mathtt{dti}_\mathtt{20}$\xspace.
One HARDI dataset (referred as $\mathtt{hardi}_\mathtt{256}$\xspace) was acquired at $b = 3000 \; \mathrm{s}/\mathrm{mm}^2$ using 256 directions uniformly distributed on half the unit sphere \citep{Jones:1999aa}, $\mathrm{TR}/\mathrm{TE} = 7000/108 \; \mathrm{ms}$ and spatial resolution = $2.5 \times 2.5 \times 2.5 \; \mathrm{mm}$.
To study the robustness of the three algorithms to different under-sampling rates, the $\mathtt{hardi}_\mathtt{256}$\xspace dataset has been retrospectively under-sampled and two additional datasets ($\mathtt{hardi}_\mathtt{50}$\xspace and $\mathtt{hardi}_\mathtt{20}$\xspace) have been created, consisting of only 50 and 20 diffusion directions, respectively. These subsets of directions were randomly selected in order to be as much equally distributed on half the unit sphere as possible.
The actual SNR in the $b = 0$ images, computed as the ratio of the mean value in a region-of-interest placed in the white matter and the standard deviation of the noise estimated in the background, was about 60 in $\mathtt{dti}_\mathtt{30}$\xspace, 30 in $\mathtt{dti}_\mathtt{20}$\xspace and 30 in $\mathtt{hardi}_\mathtt{256}$\xspace.
\subsection{Implementation details} \label{ImplementationDetails}
In all our experiments, the \textit{response function was estimated from the data} following the procedure described in \citet{Tournier:2007aa}.
A different response function was estimated for every combination of experimental conditions (number of samples, b-value, SNR), which was then used consistently in the three reconstruction methods.
Specifically, the 300 voxels with the highest fractional anisotropy were selected as expected to contain only one fiber population, and a tensor was fitted from the dMRI signal in each.
In the case of numerical simulations, an additional set of data containing 300 voxels with a single fiber compartment was generated for this scope.
The estimated coefficients were then averaged to provide a robust estimation of the signal profile for the response function.
As we used the tool \texttt{estimate\_response} of \texttt{mrtrix} for these operations, the estimated kernel was already suitable to be fed into the \texttt{L2L2}\xspace algorithm.
\MOD{Note that the fiber directions rely on a maxima identification from the SH coefficients, which can take any continuous position on the sphere.}
Conversely, in the case of both \texttt{L2L1}\xspace and \texttt{L2L0}\xspace, \MOD{the estimated kernel} was used to create the dictionary $\Phi$ by rotating it along 200 orientations uniformly distributed on half the unit sphere.
\MOD{Because of this discretization, the resulting grid resolution is about $10^\circ$ and thus the intrinsic average error when measuring the angular accuracy is about $5^\circ$.
In other words, the precision of both \texttt{L2L1}\xspace and \texttt{L2L0}\xspace is limited by the resolution of the grid used to construct the dictionary.
For this reason differences between methods below this threshold will be considered not significant.
Note that, to improve the precision it would be sufficient to increase the number of directions of the discretization which, however, would have serious consequences on the efficiency and stability of the minimization algorithm. Interestingly, recent works of \citet{Tang:2012aa} and \citet{Candes:2012aa} explored a novel theory of CS with continuous dictionaries, in the context of which FOD peaks could be thought to be located with infinite precision. This topic will be the subject of future research.}
Finally, in order to model adequately any partial/full contamination with cerebrospinal fluid (CSF) that may occur in real data, an \textit{additional isotropic compartment} has been considered by adding a column to $\Phi$.
This compartment was estimated by fitting an isotropic tensor in voxels within the lateral ventricles.
\MOD{The} free parameter controlling the degree of regularization had to be estimated for both \texttt{L2L2}\xspace and \texttt{L2L1}\xspace algorithms.
For the former we used the default values suggested in the original implementation available in the \texttt{mrtrix} software.
For the latter, the regularization parameter $\beta$ was empirically estimated following the guidelines of \citet{Landman:2012aa}, in order to place the method in its best conditions. In numerical simulations, we created an additional training dataset for every combination of experimental conditions (number of samples, b-value, SNR) and 50 reconstructions were performed varying the parameter $\beta$ from $10^{-4} \; \beta^*$ to $\beta^*$, with $\beta^* = || 2 \Phi^T \vect{y}||_\infty$ computed independently in each voxel. The value providing the best reconstructions (according to the above metrics) was then used to run \texttt{L2L1}\xspace on the actual data used for the final comparison. We did not observe any improvement in the reconstructions outside this range.
\MOD{In real data, we tested different values for $\beta$ but, as the ground-truth is unknown, the optimal value was chosen on the basis of a qualitative inspection of the reconstructions considering their shape, spatial coherence and adherence to anatomy. Nonetheless, we found that the algorithm was quite robust to the choice of $\beta$ and the value providing visually the best results was always very close to $\beta = 0.1 \cdot \beta^*$, as suggested in the same work. Therefore this value was used in all real data experiments. This stability might be probably due to the adaptive strategy of estimating $\beta^*$ in each voxel from the signal $\vect{y}$. }
\MOD{As already emphasized}, \texttt{L2L0}\xspace \textit{does not require any free parameter} to be tuned. In fact, in numerical simulations $k$ can be fixed \MOD{in all iterations} to 3 while we can safely assume $k=5$ in real data, hence larger than the 2-3 fibers normally assumed.
\section{Results and discussion} \label{Results}
\subsection{Numerical simulations}
We quantitatively compared the three approaches on synthetic data with the aim of assessing the impact on the reconstructions of each regularization scheme (i.e. $\ell_2$, $\ell_1$ and $\ell_0$ \MOD{priors}) under controlled conditions.
In particular, the quality of the reconstructions was evaluated using the metrics introduced above and selectively varying (i) the number of samples and (ii) the b-value of the acquisition scheme, (iii) the noise level and (iv) the crossing angle between the fiber compartments.
Results are reported independently for each experimental condition.
\subsubsection{Volume fractions and \texorpdfstring{$\ell_1$ norm}{Lg}} \label{VolumeFractionsAndL1}
As previously stated, the physical constraint that the volume fractions sum to unity is normally omitted in every problem formulation, as it is expected to be carried over by the data and properly designed bases \citep{Ramirez-Manzanares:2007aa}.
In Fig.~\ref{fig:SumX} we explicitly tested whether this property is actually satisfied by the algorithms considered in this work.
A more detailed analysis of the performance of each prior is performed in the following sections.
\begin{figure}[ht]
\centering
\includegraphics[width=11.5cm]{SumX}
\caption{\textit{Sum of volume fractions and impact on the reconstructions}. Top plots report the $||\vect{x}||_1$ of the FODs reconstructed by \texttt{L2L2}\xspace (blue boxes), \texttt{L2L1}\xspace (red diamonds) and \texttt{L2L0}\xspace (green circles), while the NMSE of the recovered signal is shown at the bottom.
The reference value $||\vect{x}||_1 = 1$ is plotted in magenta.
Results are reported as a function of the SNR in 2 experimental settings with 30 samples: $b = 1000 \; \mathrm{s/mm^2}$ (left) and $b = 3000 \; \mathrm{s/mm^2}$ (right).}
\label{fig:SumX}
\end{figure}
The figure reports the average value for the sum of the volume fractions of the reconstructed FODs (i.e. $|| \vect{x} ||_1$), as a function of the noise level, for two acquisition schemes with 30 samples at $b = 1000 \; \mathrm{s/mm^2}$ and $b = 3000 \; \mathrm{s/mm^2}$, respectively.
The impact on the reconstructions is shown by means of the normalized mean-squared error \MOD{$\mathrm{NMSE = || \vect{y} - \tilde{\vect{y}} ||_2^2 / || \vect{y} ||_2^2}$ \citep{Michailovich:2011aa} between the measured signal $\vect{y}$ and its estimate $\tilde{\vect{y}}$}.
The image clearly demonstrates that both \texttt{L2L2}\xspace and \texttt{L2L1}\xspace reconstructions do not fulfill the $\sum_i \vect{x}_i = 1$ physical constraint, as the sum of the recovered volume fractions always tends to be over-estimated by \texttt{L2L2}\xspace and under-estimated by \texttt{L2L1}\xspace.
\MOD{This is a clear effect of the weakness of the sparsity constraint in the \texttt{L2L2}\xspace approach and of the inconsistency of the $\ell_1$ prior in \texttt{L2L1}\xspace.}
On the contrary our \texttt{L2L0}\xspace approach appears to correctly satisfy the constraint, with deviations from unity only with very high noise levels (SNR $\approx$ 5).
With high quality data this over/under-estimation behavior is fairly mild (at SNR = 30, $||\vect{x}||_1 \approx 0.7$ for \texttt{L2L1}\xspace and $||\vect{x}||_1 \approx 1.2$ for \texttt{L2L2}\xspace), but it progressively intensifies as the noise level increases.
The trend is even amplified with high b-value data, in which case the $||\vect{x}||_1$ can be as high as $\approx 2.1$ for \texttt{L2L2}\xspace and as low as $\approx 0.25$ for \texttt{L2L1}\xspace.
Despite showing quite different behaviors with respect to the $||\vect{x}||_1$, \texttt{L2L2}\xspace and \texttt{L2L0}\xspace exhibit very similar NMSE values.
On the contrary, \texttt{L2L1}\xspace shows significantly higher reconstruction errors than both \texttt{L2L2}\xspace and \texttt{L2L0}\xspace, pointing to the aforementioned \textit{$\ell_1$ inconsistency}.
Debiasing methods~\citep{Zou:2006aa} have been proposed with the aim to correct the magnitude of the recovered coefficients and mitigate this effect. Nonetheless, a critical step for applying these techniques consists in the proper identification of the support of the solution, otherwise this procedure can lead to really bad results. As we will show in the next sections, this is the case in this work, as the three methods differ significantly in their ability to estimate the number of fiber populations.
As the very same data and reconstruction basis have been used for all the methods, we can conclude that any deviation from the unit sum has to be attributed to the different regularization employed in each algorithm.
In the following \textit{we will investigate the consequences on the reconstructions} of using different regularization schemes.
\subsubsection{Comparison as a function of the number of samples}
Fig.~\ref{fig:performances-vs-nS} reports the performance of the three reconstruction methods as the number of samples changes. We considered seven acquisition schemes from 6 to 50 samples and results are reported for a standard scenario, specifically a shell at $b = 2000 \; \mathrm{s/mm^2}$ with a $\mathrm{SNR} = 25$.
The dependence on the b-value and the robustness to noise will be investigated in detail in the following sections.
The quality metrics are reported here as the average value computed over all simulated crossing angles ($30^\circ\textrm{--}90^\circ$).
\begin{figure}[ht!]
\centering
\includegraphics[width=11.6cm]{performances-vs-nS}
\caption{\textit{Quantitative comparison as a function of the number of samples.} The values of the four quality metrics are reported for \texttt{L2L2}\xspace (blue boxes), \texttt{L2L1}\xspace (red diamonds) and \texttt{L2L0}\xspace (green circles) as the number of samples changes. Values shown here correspond to an experimental setting with $b = 2000 \; \mathrm{s/mm^2}$ and $\mathrm{SNR} = 25$.}
\label{fig:performances-vs-nS}
\end{figure}
Looking at the plots the benefits of using an $\ell_0$ prior are clear: \texttt{L2L0}\xspace always outperforms both \texttt{L2L2}\xspace and \texttt{L2L1}\xspace in identifying the correct number of fiber populations ($P_d$) and results are consistent for all number of samples considered.
The main benefit of \texttt{L2L0}\xspace seems to be the drastically decreased number of missed fibers (smaller $n^-$), even though also the number of over-estimated compartments ($n^+$) is significantly reduced.
Concerning the angular accuracy of the recovered fiber populations ($\epsilon_\theta$), reconstructions with \texttt{L2L0}\xspace always resulted in smaller errors as compared to both \texttt{L2L2}\xspace and \texttt{L2L1}\xspace.
Although the difference with respect to \texttt{L2L1}\xspace is \MOD{not significant as always within the intrinsic grid precision}, both methods showed a substantial improvement over \texttt{L2L2}\xspace, which appeared to suffer from a sudden and significant deterioration of the reconstructions ($\approx 10^\circ\textrm{--}15^\circ$) for less than 30 samples.
This can be explained with the SH representation used internally by \texttt{L2L2}\xspace. In fact, even though the FOD is a function on the sphere containing high-resolution features by definition, a maximum SH order $l_{max}=4$ (or less) can be used for acquisitions with less than 30 samples, hence drastically reducing the intrinsic angular resolution of the recovered FOD. At least 30 to 60 samples are normally advised for using \texttt{L2L2}\xspace, so in our experiments we have actually tested \texttt{L2L2}\xspace beyond its applicability range. On the contrary, \texttt{L2L1}\xspace and \texttt{L2L0}\xspace do not make use of SH and the reconstruction quality degrades more smoothly with the under-sampling rate of the dMRI data.
In the following we will focus on two acquisition schemes to further analyze the performance of three methods: (i) in a normal setting with 30 samples and (ii) in a regime of high under-sampling with only 15 samples.
\subsubsection{Comparison as a function of the crossing angle}
In Fig.~\ref{fig:performances-vs-CA} the performances of \texttt{L2L2}\xspace, \texttt{L2L1}\xspace and \texttt{L2L0}\xspace are plotted in detail as a function of the crossing angle between the fiber populations.
Results are shown for two acquisitions with 30 and 15 samples, both simulated at $b = 2000 \; \mathrm{s/mm^2}$ and $\mathrm{SNR} = 25$.
\begin{figure}[pt!]
\centering
\includegraphics[width=\textwidth]{performances-vs-CA}
\caption{\textit{Quantitative comparison as a function of the crossing angle.} The performances of the three reconstruction methods are detailed separately for each crossing angle used in the simulations. Results are reported for 30 and 15 samples, using the same experimental configuration of Fig.~\ref{fig:performances-vs-nS}, i.e. $b = 2000 \; \mathrm{s/mm^2}$ and $\mathrm{SNR} = 25$.}
\label{fig:performances-vs-CA}
\end{figure}
With 30 samples, the major source of errors for both \texttt{L2L2}\xspace and \texttt{L2L1}\xspace is represented by \textit{under-estimation} ($n^-$), although spurious orientations are not negligible ($n^+ \approx 0.2$).
In particular, both methods start to severely miss fibers for crossing angles below $60^\circ$, where they tend to recover a single peak lying between the two real fiber directions. In these situations, the maximum angular error for the sole estimated peak is generally upper bounded by half the angle separating the two fibers; for this reason the overall $\epsilon_\theta$ performances of \texttt{L2L2}\xspace and \texttt{L2L1}\xspace do not differ significantly from \texttt{L2L0}\xspace despite the drastic improvement in terms of $P_d$, $n^-$ and $n^+$.
On the other hand, in an under-sampling scenario with 15 samples \texttt{L2L2}\xspace and \texttt{L2L1}\xspace exhibit much higher $P_d$ values and a stronger tendency to \textit{over-estimate} compartments, usually in completely arbitrary orientations not even close to the true fiber directions. The overall improvement in the angular accuracy of \texttt{L2L0}\xspace is more evident, with an average enhancement up to $5^\circ$ with respect to \texttt{L2L1}\xspace, whereas \texttt{L2L2}\xspace exhibits a severe drop of the performance mainly due to modeling limitations, as previously pointed out.
These differences can have dramatic consequences for fiber-tracking applications.
In fact, tractography algorithms are particularly prone to these estimation inaccuracies, i.e. number and orientation of fiber populations, because the propagation of these (perhaps locally small) errors can lead to completely wrong final trajectories. For instance, a missed compartment might stop prematurely a trajectory, while a spurious peak might lead to create an anatomically incorrect fiber tract.
Hence, the ability to accurately recover the intra-voxel fiber geometry is of utmost importance.
\subsubsection{Comparison as a function of the b-value}
So far \texttt{L2L2}\xspace, \texttt{L2L1}\xspace and \texttt{L2L0}\xspace have been compared for given acquisition schemes at a fixed $b = 2000 \; \mathrm{s/mm^2}$.
Fig.~\ref{fig:performances-vs-b} reports the quality of the reconstructions with the three approaches as a function of the b-value. The results are shown for 30 and 15 samples with a $\mathrm{SNR} = 25$.
\begin{figure}[ht!]
\centering
\includegraphics[width=11.2cm]{performances-vs-b}
\caption{\textit{Quantitative comparison as a function of the b-value.} The dependence of the reconstruction quality on the b-values used in the acquisition is reported here for 30 and 15 samples with a $\mathrm{SNR} = 25$.}
\label{fig:performances-vs-b}
\end{figure}
\texttt{L2L2}\xspace tends to miss compartments for low b-values and over-estimate them at higher $b$ ($n^+$ and $n^-$ are not shown here for brevity).
This is even more apparent when decreasing the number of samples in the acquisition to 15, where \texttt{L2L2}\xspace estimates a lot of spurious peaks at high b-values (high $n^+$) and thus the angular accuracy of the estimated fiber directions drops considerably.
Interestingly, \texttt{L2L1}\xspace shows the opposite behavior, under-estimating at high b and over-estimating at low b, although at a smaller rate thus preventing the performance to degrade significantly.
Again, in comparison, \texttt{L2L0}\xspace shows a very stable estimation of the number of fibers.
Concerning the angular accuracy, all methods showed a minimum for $\epsilon_\theta$ corresponding to $b \approx 1500-2500 \; \mathrm{s/mm^2}$, representing a sort of trade-off between the loss in angular resolution happening at small b-values and the stronger noise influence at higher $b$. In fact, as in this work we report the noise level as the SNR of the $S_0$ dataset, images at high b-values will have lower actual signal-to-noise ratio, and thus the noise effects will be inherently stronger.
Overall, \texttt{L2L0}\xspace always results in smaller angular errors than the other two methods.
The improvement with respect to \texttt{L2L1}\xspace is not \MOD{significant}, while the difference with \texttt{L2L2}\xspace is \MOD{much} more pronounced (up to $20^\circ$) especially as the b-value increases.
\subsubsection{Comparison as a function of the SNR}
\begin{figure}[htb]
\centering
\includegraphics[width=\textwidth]{performances-vs-SNR}
\caption{\textit{Quantitative comparison as a function of the SNR.} The robustness to noise of the three reconstruction methods in shown for 30 and 15 samples at $b = 2000 \; \mathrm{s/mm^2}$. Reported values for the SNR correspond to the signal-to-noise ratio of the $S_0$ dataset.}
\label{fig:performances-vs-SNR}
\end{figure}
Finally, Fig.~\ref{fig:performances-vs-SNR} compares the robustness to noise of the three methods. Six noise levels have been considered, with the SNR of the $S_0$ dataset varying from 5 to 30. The comparison is reported for 30 and 15 samples at $b = 2000 \; \mathrm{s/mm^2}$.
The results show that \texttt{L2L0}\xspace clearly outclasses the other two methods concerning the estimation of the number of compartments ($P_d$) and results are consistent as the SNR changes, both with 30 and 15 samples.
In terms of angular accuracy, \texttt{L2L0}\xspace and \texttt{L2L1}\xspace have very similar $\epsilon_\theta$ performances, almost indistinguishable from one another.
On the contrary, \texttt{L2L2}\xspace systematically obtains significantly higher $\epsilon_\theta$ values at all considered SNRs (up to $6^\circ$ with 30 samples).
In a high under-sampling regime (right plots), the angular accuracy drastically degrades in the case of \texttt{L2L2}\xspace and it appears almost independent of the noise level.
This is again consistent with the limitations of the SH representation for acquisitions with very few samples.
\subsection{Real data}
\begin{figure}[p!]
\centering
\includegraphics[height=17cm]{RealData}
\caption{\textit{Qualitative comparison on DTI human brain data.} Reconstructions of the FODs in the corona radiata region are shown for: \texttt{L2L2}\xspace (\textsf{A} and \textsf{D}), \texttt{L2L1}\xspace (\textsf{B} and \textsf{E}) and \texttt{L2L0}\xspace (\textsf{C} and \textsf{F}). FODs in subplots \textsf{A--C} correspond to dMRI images acquired using 30 samples, superimposed on the ADC map, while \textsf{D--F} are relative to the acquisition with 20 samples, superimposed on the FA map. All images have been acquired at $b = 1000 \; \mathrm{s/mm^2}$.}
\label{fig:RealData}
\end{figure}
\begin{figure}[p!]
\centering
\includegraphics[width=\textwidth]{RealDataHighBValue}
\caption{\textit{Qualitative comparison on HARDI human brain data.} Reconstructions of the FODs in the corona radiata region are shown for: \texttt{L2L2}\xspace (\textsf{A}, \textsf{D} and \textsf{G}), \texttt{L2L1}\xspace (\textsf{B}, \textsf{E} and \textsf{H}) and \texttt{L2L0}\xspace (\textsf{C}, \textsf{F} and \textsf{I}). Subplots \textsf{A--C} correspond to the fully-sampled dataset $\mathtt{hardi}_\mathtt{256}$\xspace (256 samples), \textsf{D--F} to the dataset $\mathtt{hardi}_\mathtt{50}$\xspace (50 samples) while \textsf{G--I} are relative to $\mathtt{hardi}_\mathtt{20}$\xspace (20 samples). Images have been acquired at $b = 3000 \; \mathrm{s/mm^2}$.}
\label{fig:RealDataWithHighBValue}
\end{figure}
\subsubsection{\MOD{Qualitative evaluation on DTI data}}
\MOD{Fig.~\ref{fig:RealData} compares the reconstructions\footnote{\MOD{The images have been created using the tool \texttt{mrview} of \texttt{mrtrix}. As a consequence, the FODs from \texttt{L2L1}\xspace and \texttt{L2L0}\xspace had to be converted to SH, and this operation caused some blur in the sparse reconstructions of these two methods.}} obtained with the three regularization schemes in the case of real data acquired with a typical DTI protocol.}
Subplots \textsf{A}, \textsf{B} and \textsf{C} correspond to the $\mathtt{dti}_\mathtt{30}$\xspace dataset acquired using 30 samples.
Even though the acquisition scheme used for this dataset is not the setting where our numerical simulations highlighted the most substantial differences between the three methods, important conclusions can be drawn in favor of \texttt{L2L0}\xspace.
Looking at the regions in the white circles, the ability of both \texttt{L2L1}\xspace and \texttt{L2L0}\xspace to properly model the isotropic compartment in voxels with full or partial contamination with CSF is clearly visible.
On the contrary, as \texttt{L2L2}\xspace does not explicitly model any CSF compartment, it appears unable to adequately characterize the signal in these cases, but it rather approximates any isotropic contribution with a set of random and incoherent fiber compartments.
Besides, comparing \textsf{B} and \textsf{C} we can observe that \texttt{L2L0}\xspace successfully differentiates gray matter (light gray regions) from CSF voxels with pure isotropic and fast diffusion (very bright areas), whereas \texttt{L2L1}\xspace appears unable to distinguish them.
The yellow frames highlight the corona radiata, a well-known region in the white matter containing crossing fibers.
As expected from our simulations at this still relatively high number of samples, differences are not obvious between the three methods.
However, we observe that \texttt{L2L0}\xspace clearly results in sharper and more defined profiles than \texttt{L2L1}\xspace, whereas the improvements with respect to \texttt{L2L2}\xspace are confined only to few voxels. The not so good performance of \texttt{L2L1}\xspace might be related to the value chosen for $\beta$. In contrast, no free parameter has to be empirically optimized in our approach.
When decreasing the acquisition samples to 20 (subplots \textsf{D}, \textsf{E} and \textsf{F} corresponding to $\mathtt{dti}_\mathtt{20}$\xspace dataset), \MOD{fiber directions are definitely much better resolved with \texttt{L2L0}\xspace than with both \texttt{L2L2}\xspace and \texttt{L2L1}\xspace}.
In fact \texttt{L2L2}\xspace clearly breaks, missing many fiber compartments probably due to the aforementioned limitations of the SH representation. The same happens to \texttt{L2L1}\xspace, whose reconstructions appear very blurred and noisy.
\subsubsection{\MOD{Qualitative evaluation on HARDI data}}
The comparison with high b-value data is reported in Fig.~\ref{fig:RealDataWithHighBValue}. The figure shows also the robustness to different under-sampling rates of each scheme.
Subplots \textsf{A}, \textsf{B} and \textsf{C} correspond to the fully-sampled dataset $\mathtt{hardi}_\mathtt{256}$\xspace.
In this situation, no evident differences between the three approaches can be observed as they perform essentially the same.
With moderate under-sampled data (subplots \textsf{D}, \textsf{E} and \textsf{F} corresponding to $\mathtt{hardi}_\mathtt{50}$\xspace) both \texttt{L2L2}\xspace and \texttt{L2L0}\xspace do not show any significant difference in the quality of the reconstructions, so far exposing neat and sharp profiles.
On the other hand, the FODs reconstructed by \texttt{L2L1}\xspace show some signs of progressive degradation, appearing a little more blurred as compared to those reconstructed from fully-sampled data (compare subplots \textsf{E} and \textsf{B}).
The situation changes drastically with highly under-sampled data, as easily noticeable by comparing the subplots \textsf{G}, \textsf{H} and \textsf{I}, which correspond to the reconstructions performed with only $8\%$ of the original data.
In fact, while \texttt{L2L0}\xspace does not show yet any significant degradation of the FODs, both \texttt{L2L2}\xspace and \texttt{L2L1}\xspace clearly do not provide as sharp and accurate reconstructions as in the case of fully-sampled data (compare \textsf{G} to \textsf{A} and \textsf{H} to \textsf{B}).
In addition, in the case of \texttt{L2L2}\xspace we can observe a higher incidence of negative peaks (identified in the plots by small yellow spikes), a clear sign of augmented modeling errors.
\subsubsection{\MOD{Quantitative comparison: volume fractions and \texorpdfstring{$\ell_1$}{Lg} norm}} \label{RealDataVolumeFractionsAndL1}
\MOD{In Fig.~\ref{fig:RealDataSumx} we tested whether the physical constraint of unit sum is satisfied also in case of real data.
The images confirm the observations previously made with synthetic data (cf. Fig~\ref{fig:SumX}).
In fact, the sum of the recovered volume fractions tends to be over-estimated by \texttt{L2L2}\xspace (subplots \textsf{A} and \textsf{B}) and under-estimated by \texttt{L2L1}\xspace (subplots \textsf{C} and \textsf{D}), whereas \texttt{L2L0}\xspace reconstructions (subplots \textsf{E} and \textsf{F}) appear to meet the property of unit sum as expected.
All methods coherently show a mild over-estimation in the corpus callosum, compatible with the highly-packed axonal structure in this region.
Finally, \texttt{L2L2}\xspace seems to suffer from over-estimation more with fully- than with under-sampled data, which might be related to the SH order employed for different number of samples.}
\begin{figure}[htb]
\centering
\includegraphics[width=10cm]{RealDataSumX}
\caption{\MOD{\textit{Sum of volume fractions in real data}. The sum of the volume fractions of the FODs reconstructed with \texttt{L2L2}\xspace (\textsf{A} and \textsf{B}), \texttt{L2L1}\xspace (\textsf{C} and \textsf{D}) and \texttt{L2L0}\xspace (\textsf{E} and \textsf{F}) is reported in a representative slice of the high b-value data. The top row corresponds to the fully-sampled dataset ($\mathtt{hardi}_\mathtt{256}$\xspace) while the bottom to the under-sampled one with 20 samples ($\mathtt{hardi}_\mathtt{20}$\xspace).}}
\label{fig:RealDataSumx}
\end{figure}
\subsubsection{\MOD{Quantitative comparison: fully- vs under-sampled data}} \label{RealDataQuantitative}
\MOD{We compared the reconstructions obtained from under-sampled data (i.e. $\mathtt{hardi}_\mathtt{50}$\xspace and $\mathtt{hardi}_\mathtt{20}$\xspace) to those with fully-sampled data (i.e. $\mathtt{hardi}_\mathtt{256}$\xspace), considering this latter as ground-truth, as done by \citet{Yeh:2013aa}.
In agreement with the results from numerical simulations, no significant difference was found between the three approaches in terms of angular accuracy.
The average error using 50 samples was $10.9^\circ \pm 9.9^\circ$ (mean $\pm$ standard deviation) for \texttt{L2L2}\xspace, $8.8^\circ \pm 8.1^\circ$ for \texttt{L2L1}\xspace and $10.0^\circ \pm 11.3^\circ$ for \texttt{L2L0}\xspace.
The reconstructions using 20 samples clearly showed higher angular errors. The differences between \texttt{L2L1}\xspace and \texttt{L2L0}\xspace were below the resolution of the sphere discretization used in this study: $11.6^\circ \pm 9.1^\circ$ and $12.6^\circ \pm 12.4^\circ$ respectively.
\texttt{L2L2}\xspace revealed significantly higher $\epsilon_\theta$ values: $17.2^\circ \pm 12.8^\circ$.}
\MOD{On the other hand, results definitely confirmed the superior performance of \texttt{L2L0}\xspace in terms of $P_d$ that was previously observed in synthetic experiments.
With 50 samples \texttt{L2L0}\xspace had an average $P_d = 4.0\% \pm 13.9\%$ as opposed to sensibly higher values for \texttt{L2L2}\xspace and \texttt{L2L1}\xspace, respectively $17.8\% \pm 32.6\%$ and $17.3\% \pm 24.3\%$.
For 20 samples, the performance of \texttt{L2L2}\xspace and \texttt{L2L1}\xspace visibly deteriorated, $42.1\% \pm 43.6\%$ for the former and $21.3\% \pm 27.7\%$ for the latter. \texttt{L2L0}\xspace reconstructions appeared very stable with an average $P_d = 5.5\% \pm 15.8\%$. These enlightening results are illustrated in Fig.~\ref{fig:RealDataQuantitative}.}
\begin{figure}[htb]
\centering
\includegraphics[width=10cm]{RealDataQuantitative}
\caption{\MOD{\textit{Comparison between fully- and under-sampled real data}. The performance of \texttt{L2L2}\xspace (\textsf{A} and \textsf{B}), \texttt{L2L1}\xspace (\textsf{C} and \textsf{D}) and \texttt{L2L0}\xspace (\textsf{E} and \textsf{F}) was quantified by considering $\mathtt{hardi}_\mathtt{256}$\xspace as ground-truth and computing $P_d$ for the reconstructions on under-sampled data. The top row corresponds to the dataset with 50 samples ($\mathtt{hardi}_\mathtt{50}$\xspace) and the bottom to 20 ($\mathtt{hardi}_\mathtt{20}$\xspace).}}
\label{fig:RealDataQuantitative}
\end{figure}
\subsection{Limitations and future work}
Our proposed \MOD{formulation} represents an extension of classical spherical deconvolution and sparse reconstruction methods \citep{Tournier:2007aa,Landman:2012aa} and, as such, it also inherits all the intrinsic limitations and shortcomings of this class of techniques.
Like all its predecessors, in fact, our method is based on the assumption that the response function can be estimated from the data and especially that it is adequate for characterizing the diffusion process in all the voxels of the brain.
Moreover, the validity of these approaches has yet to be properly assessed with more critical intra-voxel configurations \citep{Sotiropoulos:2012aa} or pathological brain conditions.
Yet, as these methods are currently widely used in this field, we have shown in this work that by \textit{expressing adequately the regularization prior used for promoting sparsity} the quality of the reconstructions can significantly be improved, with no additional cost.
Some of the aforementioned limitations might be addressed by enhancing the estimation of the dictionary accounting for more complex configurations, such as using different response functions for different brain regions and/or pathological tissues and including specific kernels which explicitly model fiber fanning/bending.
In addition, even though we focused here on single voxel experiments, future work will be devoted to study the applicability and the effectiveness of our approach in more sophisticated frameworks exploiting the spatial coherence of the data.
Finally, future research will investigate the use of the recently proposed continuous CS theory \citep{Tang:2012aa, Candes:2012aa} with the aim of further improving the accuracy of the reconstructions and reducing the acquisition time.
\section{Conclusion}\label{Conclusions}
In this paper we focused on \MOD{spherical deconvolution} methods currently used in diffusion MRI for recovering the FOD and estimating the intra-voxel configuration in white matter.
In particular, we investigated the effectiveness of state-of-the-art regularization schemes based on $\ell_2$ and $\ell_1$ priors and provided evidence that these formulations are intrinsically suboptimal: \MOD{the former because it does not explicitly promote sparsity in the FOD, the latter because it is inconsistent with the fact that the fiber compartments must sum up to unity}.
\MOD{We proposed a formulation that rather places a strict bound on the number of expected fibers in the voxel through a bound on the $\ell_0$ norm of the FOD, relying on a reweighted $\ell_1$ scheme}.
We compared our \MOD{\texttt{L2L0}\xspace} approach with \MOD{the} state-of-the-art \MOD{\texttt{L2L2}\xspace and \texttt{L2L1}\xspace methods}, both on synthetic and real human brain data.
Results showed that our proposed formulation significantly improves \MOD{single-voxel FOD} reconstructions, with no additional overheads.
This evolution is most remarkable in a high q-space under-sampling regime, thus driving the acquisition cost of HARDI closer to DTI.
\section*{Acknowledgments}
This work was supported by the Center for Biomedical Imaging (CIBM) of the Geneva and Lausanne Universities, EPFL, the Leenaards and Louis-Jeantet foundations, the EPFL-Merck Serono Alliance award and the Swiss National Science Foundation (SNSF) under grant PP00P2-123438.
\section*{References}
\bibliographystyle{model2-names}
|
1,116,691,500,829 | arxiv | \subsection{Derivation of Divergence-Preserving Coarse-Graining}\label{sec:DP_derivation}
With the definitions given in \cref{sec:defs}, we may derive a condition on the averaging weights for $\EXP{\cdot}^\beta$ to be a divergence-preserving coarse-graining. Combining \cref{eq:div_def} with \cref{eq:div_condition} gives us:
\algn{
\label{eq:filt_div}
\sum_{v\in\mathcal{V}} \alpha(\bar{v},v)\frac{1}{2\abs{A_{v}}}\sum_{c\in\mathcal{C}(v)}\B{v}_{c}\cdot\Delta \B{x}_{v,c} & = \frac{1}{2\abs{\bar{A}_{\bar{v}}}}\sum_{\bar{c}\in\bar{\mathcal{C}}(\bar{v})} \sum_{c\in\mathcal{C}}\beta(\bar{c},c)\B{v}_{c}\cdot\Delta \bar{\B{x}}_{\bar{v},\bar{c}}
}
and \cref{eq:filt_div} must hold for all vector fields $\B{v}$. We therefore re-arrange the sums in order to equate coefficients of $\B{v}_{c}$. On the left-hand side of \cref{eq:filt_div} we sum firstly over all fine-grid vertices $v\in \mathcal{V}$; then for each such vertex $v$ we sum over all cells attached to that vertex. This is equivalent to summing over all fine-grid cells $c$ then over all vertices in $\mathcal{V}$ that border the cell $c$. On the right-hand side the two sums are independent and therefore we can simply switch them. Thus:
\algn{
\begin{split}
\label{eq:filt_div_re-arrange}
\sum_{c\in \mathcal{C}}\sum_{\lrc{v\in \mathcal{V}: c\in\mathcal{C}(v)}} \frac{\abs{\bar{A}_{\bar{v}}}}{\abs{A_{v}}} \alpha(\bar{v},v)\B{v}_{c}\cdot\Delta \B{x}_{v,c}
= \sum_{c\in \mathcal{C} }\sum_{\bar{c}\in \bar{\mathcal{C}}(\bar{v})} \beta(\bar{c},c)\B{v}_{c}\cdot \Delta \bar{\B{x}}_{\bar{v},\bar{c}}
\end{split}
}
Equating coefficients of $\B{v}_{c}$ in \cref{eq:filt_div_re-arrange}, we find that $\EXP{\cdot}^{\beta}$ is divergence-preserving at vertex $\bar{v} \in\bar{\mathcal{V}}$ if there exists a coarse-graining $\lrc{\cdot}^{\alpha}$ such that for all $c\in\mathcal{C}$:
\algn{
\sum_{\lrc{v\in \mathcal{V}: c\in\mathcal{C}(v)}} \frac{\abs{\bar{A}_{\bar{v}}}}{\abs{A_{v}}} \alpha(\bar{v},v)\Delta \B{x}_{v,c} & = \sum_{\bar{c}\in \bar{\mathcal{C}}(\bar{v})} \beta(\bar{c},c)^{T}\Delta \bar{\B{x}}_{\bar{v},\bar{c}} \label{eq:DP_general}}
\Cref{eq:DP_general} is quite general and even holds in the case of irregular meshes, the cells of which are of any convex shape (in that case $\Delta \B{x}_{v,c}$ is orthogonal to the vector connecting the two vertices of cell $c$ that share an edge with $v$).\\
\par
Here, however, we only consider regular triangular meshes. Let us suppose that the vertex $\bar{v}$ is situated at some point in the domain which is far from the boundary so that we may assume a translational symmetry in the grid. Therefore, $\beta(\bar{c},c)$ should depend only on the relative position of the centres of cells $\bar{c}$ and $c$, not on the absolute position within the grid. Similarly, $\alpha\lrb{\bar{v},v}$ should not depend on the position of $\bar{v}$, only the relative position of the vertices $\bar{v}$ and $v$. \\ \par
It is convenient to define a metric $d:\mathcal{G}\times\mathcal{G}\rightarrow\mathbb{N}\cup \lrc{0}$ on vertices such that $d(v,v')$ is a integer equal to the minimum number of fine-grid edges we must traverse in order to arrive at vertex $v'$ starting from vertex $v$. Because of the translational symmetry we suppose $\alpha(\bar{v},v)$ depends only on $d(\bar{v},v)$ and therefore we write $\alpha(\bar{v},v) =: \alpha_{d(\bar{v},v)}$. Further, we assume that $\alpha_{n} = 0$ for $n\geq N$; that is, we set $\mathcal{V}(\bar{v}) = \lrc{v\in\mathcal{V} : d(\bar{v},v)<N}$. Then there are $N$ variables $\alpha_{0},\alpha_1,...,\alpha_{N-1}$ to determine. We also specify $\mathcal{C}(\bar{c})$ by setting $\beta(\bar{c},c)=0$ unless $c$ is contained in $\bar{c}$. With this assumption the sum on the right-hand side of \cref{eq:DP_general} has only one term.\\
\par
Since the mesh is regular and triangular there are (up to a minus sign) only three possible values for $\Delta \B{x}_{v,c}$. We denote these vectors $\Delta \B{x}_{i}$ and note that $\sum_{i=1}^3 \Delta \B{x}_i = 0$. On the coarse-grid the corresponding vectors are simply rescaled by the ratio of resolutions: $\Delta \bar{\B{x}}_i = N\Delta \B{x}_i$. \\
\par
Suppose $c$ is an upward-pointing fine-grid cell contained in an upward-pointing coarse-grid cell $\bar{c}$, as shown in \cref{fig:DP_N}. And let us consider the coarse-grid vertices $\bar{v}_{i}$ in turn. Then \cref{eq:DP_general} gives, for $i=1,2,3$:
\algn{
\begin{split} \label{eq:beta_eq_gen}
\beta(\bar{c}, c)^T \Delta \B{x}_{i} = &N\alpha\lrb{\bar{v}_{i},v_{1}}\Delta \B{x}_{1} + N\alpha\lrb{\bar{v}_{i},v_{2}}\Delta \B{x}_{2} + N\alpha\lrb{\bar{v}_{i},v_{3}}\Delta \B{x}_{3}
\end{split}
}
where we have used the fact that on a regular grid with $\bar{v}, v$ away from the boundary the ratio of areas is $\abs{\bar{A}_{\bar{v}}}/\abs{A_{v}} = N^{2}$. Further, we notice that since $c$ is an upward-pointing triangle, we have $d(\bar{v}_{i},v_{i\pm1}) = d(\bar{v}_{i},v_{i}) + 1$. We may therefore use the fact that $\sum_{i=1}^{3}\Delta \B{x}_i = 0 $ to write:
\algn{
\beta(\bar{c}, c)^T \Delta \B{x}_{i} & = N\lrb{\alpha_{d(\bar{v}_{i},v_{i})} - \alpha_{d(\bar{v}_{i},v_{i})+1}}\Delta \B{x}_{i} \label{eq:be_i}
}
We obtain an identical result when $c$ is a downward-pointing triangle. Summing \cref{eq:be_i} over $i=1,2,3$ allows us to deduce that:
\algn{
\alpha_{d(\bar{v}_{i\pm1},v_{i\pm1})} - \alpha_{d(\bar{v}_{i\pm1},v_{i\pm1})+1} = \alpha_{d(\bar{v}_{i},v_{i})} - \alpha_{d(\bar{v}_{i},v_{i})+1} \label{eq:all_c}
}
\Cref{eq:all_c} holds for $v_{i},v_{i\pm1}$ being the vertices of any cell $c$ contained in the cell $\bar{c}$ with vertices $\bar{v}_{i}, \bar{v}_{i\pm 1}$. A direct consequence of \cref{eq:all_c} when combined with \cref{eq:be_i} is that $\beta(\bar{c},c)$ must all be equal and proportional to the identity. But $\mathcal{C}(\bar{c})$ contains $N^2$ elements therefore by \cref{eq:be_conditions} we find:
\algn{
\beta(\bar{c},c) = \frac{\chi_{\mathcal{C}(\bar{c})}(c)}{N^2}\mathbb{I} \label{eq:solution_be_DP}
}
where $\chi_{\mathcal{C}(\bar{c})}(c)$ is the indicator function, equal to $1$ if $c\in\mathcal{C}(\bar{c})$ and $0$ otherwise. It is readily verified that the solution given by \cref{eq:solution_be_DP} satisfies \cref{eq:be_conditionsi}. Furthermore, we deduce from \cref{eq:be_i,eq:solution_be_DP} that for all $n\leq N$ we have:
\algn{
\alpha_{n} - \alpha_{n+1} = \frac{1}{N^{3}}
}
Then we can use the fact that $\alpha_{N} = 0$ to solve inductively, giving the result, for $n<N$:
\algn{
\alpha_{n} &= \frac{1}{N^2}\lrb{1-\frac{n}{N}} \label{eq:al_solution_DP}
}
For $n>0$ there are $6n$ vertices at distance $n$ from a coarse-grid vertex $\bar{v}$, so we can verify \cref{eq:al_conditions} since $\sum_{v\in\mathcal{V}(\bar{v})}\alpha(\bar{v},v) = \frac{1}{N^2}\lrb{1 + 6\sum_{n=1}^{N-1}n\lrb{1-\frac{n}{N}} }= 1$. \\
\par
Moreover, since we are considering a regular grid, the areas obey $\abs{A_v}/\abs{\bar{A}_{\bar{v}}} = \abs{A_c}/\abs{\bar{A}_{\bar{c}}} = 1/N^2$ for cells and vertices located away from the boundaries. Therefore, \cref{eq:al_area} also holds for vertices $v$ away from the boundary, as does \cref{eq:al_sum1_new} with $\kappa(\bar{v},v) = \lrb{1 - \frac{d(\bar{v},v)}{N}}\chi_{\mathcal{V}(\bar{v})}(v)$ which satisfies $\sum_{\bar{v}\in\mathcal{V}}\kappa(\bar{v},v)=1$. $\beta$ similarly obeys \cref{eq:be_area,eq:be_sum1_new} for $c$ away from the boundary, with $\lambda(\bar{c},c) = \chi_{\mathcal{C}(\bar{c})}(c)\mathbb{I}$. Therefore, we find that at points away from the boundaries, the solutions for $\alpha,\beta$ given in \cref{eq:al_solution_DP,eq:solution_be_DP} define a divergence-preserving coarse-graining; these coarse-grainings also give a good approximation to the original fields, as they obey \cref{eq:al_conditionsi,eq:be_conditionsi}; and they preserve integrals, since \cref{eq:al_area,eq:be_area} also hold.
\\
\par
We may also extend our solution to the boundaries provided the boundaries of the fine and coarse grids coincide. If $\bar{v}$ is a boundary vertex in the coarse grid then its corresponding vertex area $\abs{\bar{A}_{\bar{v}}}$ will not be the same as the vertex area of a vertex located off the boundary. Therefore we use the following generalisation of \cref{eq:al_solution_DP}:
\algn{
\alpha(\bar{v},v) = \lrb{1 - \frac{d(\bar{v},v)}{N}}\frac{\abs{A_v}}{\abs{\bar{A}_{\bar{v}}}} \chi_{\mathcal{V}(\bar{v})}(v)\label{eq:al_solution_gen}
}
It can be shown using similar arguments to before that the choice given in \cref{eq:al_solution_gen} obeys \cref{eq:DP_general} with the same $\beta(\bar{c},c)$ as before. Moreover, \cref{eq:al_solution_gen} can be shown to satisfy \cref{eq:al_area,eq:al_sum1_new} with $\kappa(\bar{v},v) = \lrb{1 - \frac{d(\bar{v},v)}{N}}\chi_{\mathcal{V}(\bar{v})}(v)$; \cref{eq:al_conditions} is also satisfied on a regular grid. The averaging weights given by \cref{eq:al_solution_gen} do not however, satisfy \cref{eq:al_conditionsi}. \\
\par
A difficulty arises where the fine-grid boundary contains cells which lie outside the boundary of the coarse-grid. Such is the case for the model we use for our numerical results, in which the north-east and south-west corners of the domain are resolved differently in the fine and coarse grids; the south-western corner is shown in \cref{fig:corner} for $N=2$. The $\alpha,\beta$ derived above will not satisfy \cref{eq:DP_general} and \cref{eq:be_conditions} at the corners because, for example, the $\alpha$-averaging onto vertex $\bar{v}_1$ includes vertices which lie outside the boundary of the coarse grid, while the averaging onto coarse-grid cells $\bar{c}_0, \bar{c}_{\pm1}$ does not include the cells which lie outside the coarse-grid boundary. It is unclear at present how to define $\mathcal{C}(\bar{c})$ and $\mathcal{V}(\bar{v})$ at these points such that it is possible to find $\alpha(\B{v},\cdot)$ and $\beta(\bar{c},\cdot)$ obeying \cref{eq:DP_general}. Further investigation is needed into the divergence-preserving problem at the boundary and this is a potential subject for future study.\\
\par
It is possible to define an integral-preserving coarse-graining on vector fields at such boundaries by, for example, choosing $\mathcal{C}(\bar{c}_0)$ to include the cells contained in $\bar{c}_0$, along with the adjacent fine-grid cells which lie outside the coarse-grid boundary. \Cref{eq:be_area} then gives $\beta(\bar{c}_0,c)=\frac{1}{N^2}\mathbb{I}$ on these cells. With this choice \cref{eq:be_sum1_new} holds with $\lambda(\bar{c}_0,c) = \mathbb{I}\chi_{\mathcal{C}(\bar{c}_0)}(c)$ and so this defines an integral-preserving coarse graining.
\subsection{Effect of Curvature}
\label{sec:curvature}
A further potential to extend the ideas explored herein would be to take into account irregular meshes, or those with curvature. As mentioned previously, the formula for computing divergences in FESOM2 is the same as that given in \cref{eq:div_def}, but now $\abs{A_c}$, $\abs{A_v}$ and $\Delta \B{x}_{v,c}$ vary with latitude. Concretely, we write $\abs{A_c} = \abs{A}\cos y_c$, where $\abs{A}$ is a constant area and $y_c$ denotes the latitude of the centre of cell $c$. We also have a distortion of the normal vectors: $\Delta \B{x}_{v,c} = D(c)\Delta \tl{\B{x}}_{v,c}$, where $D(c) = \twobytwomatrix{1 }{0}{0}{\cos y_{c}}$ and $\Delta \tl{\B{x}}_{v,c}$ are the corresponding flat- grid vectors, as used previously. It is convenient to define re-scaled $\alpha$ and $\beta$, so we let $\tl{\alpha}(\bar{v},v):= \frac{\abs{\bar{A}_{\bar{v}}}}{N^2\abs{A_v}}\alpha(\ol{v},v)$ and $\tl{\beta}(\bar{c},c) := D(\bar{c})\beta(\bar{c},c)D(c)^{-1}$. By doing this we recover \cref{eq:beta_eq_gen} but with $\alpha\rightarrow\tl{\alpha}$, $\beta\rightarrow\tl{\beta}$, and $\Delta \B{x}_{v,c}\rightarrow\Delta \tl{\B{x}}_{v,c}$. Therefore the divergence-preserving solution is given by $\tl{\alpha}(\bar{v},v) = \frac{1}{N^2}\lrb{1-\frac{d(\bar{v},v)}{N}}\chi_{\mathcal{V}(\bar{v})}(v)$ and $\tl{\beta}(\bar{c},c) = \frac{\chi_{\mathcal{C}(\bar{c})}(c)}{N^2}\mathbb{I}$ away from the boundaries; equivalently $\alpha(\bar{v},v) = \frac{\abs{A_v}}{\abs{\bar{A}_{\bar{v}}}}\lrb{1-\frac{d(\bar{v},v)}{N}}\chi_{\mathcal{V}(\bar{v})}(v)$ and $\beta(\bar{c},c) = \frac{\chi_{\mathcal{C}(\bar{c})}(c)}{N^2}D(\bar{c})^{-1}D(c)$. This choice of $\alpha$ obeys \cref{eq:al_area,eq:al_sum1_new} away from the corners, with $\kappa(\bar{v},v) = \lrb{1-\frac{d(\bar{v},v)}{N}}\chi_{\mathcal{V}(\bar{v}}(v)$. Thus, the coarse-graining defined by $\alpha$ is integral-preserving away from the corners. The coarse-graining defined by $\beta$ obeys neither \cref{eq:be_conditions} nor \cref{eq:be_sum1_new}. We can, however, show that $\sum_{c\in\mathcal{C}(\bar{c})}\cos y_c = N^2 \cos\bar{y}_{c}\lrb{1 + \mathcal{O}(N^2 \Delta x^2)}$ and therefore $\sum_{c\in\mathcal{C}(\bar{c})}\beta(\bar{c},c) = \mathbb{I} + \twobytwomatrix{1}{0}{0}{\mathcal{O}(N^2\Delta x^2)}$, so that \cref{eq:be_conditions} is satisfied in the $x$-direction, and satisfied up to $\mathcal{O}(N^2\Delta x^2)$ in the $y$-direction.
\iffalse
\\
\par
The difference from the flat-space case is that now we obtain a different equation for the sum of the re-scaled averaging weights. Since we have the re-scaled cell areas, we no longer expect the total area of the fine-grid to be the same as the total area in the fine grid. We therefore use \cref{eq:al_sum1_new,eq:be_sum1_new} and find, in terms of $\tl{\alpha},\tl{\beta}$:
\eqsyslab{
\sum_{v\in\mathcal{V}}\tl{\alpha}(\bar{v},v)\abs{A_{v}}
& = \frac{1}{N^2}\sum_{v\in\mathcal{V}}\kappa(\bar{v},v)\abs{A_v}\label{eq:newsums_al}\\
\sum_{c\in\mathcal{C}}\tl{\beta}(\bar{c},c)D(c)
& = \frac{1}{\abs{\bar{A}_{\bar{c}}}}\sum_{c\in\mathcal{C}}D(\bar{c})\lambda(\bar{c},c)\abs{A_c} \label{eq:newsums_be}
}{newsums}
For the choice of $\tl{\alpha}$ required for $\EXP{\cdot}^{\beta}$ to be divergence-preserving, we have $\tl{\alpha}(\bar{v},v) = \frac{1}{N^2}\lrb{1-\frac{d(\bar{v},v)}{N}}$ so $\kappa(\bar{v},v) = \lrb{1-\frac{d(\bar{v},v)}{N}}\chi_{\mathcal{V}(\bar{v}}(v)$, which, as we have shown before, satisfies $\sum_{\bar{v}\in\mathcal{V}}\kappa(\bar{v},v)=1$. Thus the $\alpha$ coarse-graining is integral-preserving. However, for preservation of divergence we need $\tl{\beta}(\bar{c},c) = \frac{1}{N^2}\mathbb{I}$, or equivalently $\beta(\bar{c},c) = $. However, this solution obeys neither \cref{eq:be_conditions} nor \cref{eq:newsums_be}. We can, however, show that $\sum_{c\in\mathcal{C}(\bar{c})}\cos y_c = N^2 \cos\bar{y}_{c}\lrb{1 + \mathcal{O}(N^2 \Delta x^2)}$ and therefore \cref{eq:be_conditions} is satisfied up to this approximation.
\fi \\
\par
For $\EXP{\cdot}^\beta$ to be integral-preserving we would need $\beta(\bar{c},c) =\frac{\abs{A_c}}{\abs{\bar{A}_{\bar{c}}}}\chi_{\mathcal{C}(\bar{c})}(c)\mathbb{I}$ so that \cref{eq:be_area} is satisfied as long as for each $c\in\mathcal{C}$ there is exactly one $\bar{c}\in\mathcal{C}(\bar{c})$ with $c\in\mathcal{C}(\bar{c})$. In this case \cref{eq:be_sum1_new} also holds with $\lambda(\bar{c},c) = \chi_{\mathcal{C}(\bar{c})}(c)\mathbb{I}$. At the corners we can use the same prescription as before, averaging the cells outside the coarse boundary onto the adjacent cell inside the boundary. This solution, however, will not be divergence-preserving. Thus when coarse-graining vector fields on grids with curvature we can choose a divergence-preserving coarse-graining which approximately obeys \cref{eq:be_conditions} and which does not preserve integrals; or we can choose an integral-preserving coarse-graining which does not preserve divergences.
\subsection{ALE coordinates}
One of our motivations for defining a divergence-preserving coarse graining was in order to give the correct vertical velocity when computed on the coarse grid. However, FESOM2 may be configured to use Arbitrary Lagrangian-Eulerian (ALE) coordinates, which allows the thicknesses of the vertical layers to vary in time. Details on how ALE coordinates are implemented can be found in \citep{danilov_sidorenko_wang_jung_2016}, but here we repeat some of the relevant aspects. In particular, the vertical velocities, since they are now defined at moving vertical coordinates, are given by:
\algn{\label{eq:ALE}
w_{k} = w_{k+1} - \partial_{t}h_{k} - \lrsq{\nabla\cdot\lrb{h\B{u}}}_k
}
where $h_k$ is the thickness of layer $k$ (defined at vertices), and $k$ increases with depth so that $h_0$ is the thickness of the surface layer, etc. The evolution of the layer thicknesses, $\partial_t h_k$ is specified as one of several options. The fixed-layer case $\partial_t h_k = 0$ we have already considered. Another popular option is to use so-called $z^*$ coordinates in which the change in the total layer thickness $\hat{h} := \sum_k h_k - H$ is distributed over all layers, so that we impose $\partial_t h_k = \frac{h_k^0}{H}\partial_{t}\hat{h}$, where $h_k^0$ is the initial thickness of layer $k$. The evolution of the total layer thickness is given by $\partial_{t}\hat{h} = -\nabla\cdot\int_{-H}^0 \B{u} dz = -\nabla\cdot\sum_k (h \B{u})_k $ and so in this case the vertical velocity is given by:
\algn{\label{eq:zstar}
w_{k} = w_{k+1} + \frac{h_k^0}{H}\lrsq{\nabla\cdot \sum_k (h \B{u})_k } - \nabla\cdot\lrb{h\B{u}}_k
}
In the expression $h\B{u}$, $h$ must be defined at cell centres, and in FESOM2 this is calculated this via an average over surrounding vertices: $h_c := \frac{1}{3}\sum_{v\in\mathcal{V}(c)} h_v$. There are two divergences to be calculated in \cref{eq:zstar}, and we note that these may be preserved if we coarse grain the velocities according to $\bar{\B{u}}_{\bar{c}} := {\EXP{h\B{u}}}_{\bar{c}}/{\bar{h}_{\bar{c}}}$, where $\EXP{\cdot}$ is a divergence-preserving coarse-graining and $\bar{h}_{\bar{c}}$ is the layer thickness at cell $\bar{c}$ in the coarse grid. Thus by making a small modification we are able to adapt the coarse-graining method to obtain the correct vertical velocity when $z^*$ coordinates are used. \\
\par
Other versions of ALE coordinates are available, and the method for defining the coarse-grained velocity is likely to be different in those cases; indeed it may not be possible for all methods to define an appropriate coarse-graining that preserves the vertical velocity. However, we do not consider every possible choice in the present work, but merely include this as an example of how the method may be adapted.
\subsection{Numerical results} \label{sec:Num_results}
Here we take the special case $N=2$ and verify our results numerically using output from FESOM2 \citep{danilov_sidorenko_wang_jung_2016}. The definition of divergence used in the model is slightly different from that given in \cref{eq:div_def} because FESOM2 takes into account curvature and so there are additional factors of $\cos y$ (where $y$ is the latitude) in the cell and vertex cell areas, and in the normals. Therefore, we do not expect our coarse-graining to preserve the divergence exactly. For our model set-up we use a domain $[0,40^{\circ}]\times[30^{\circ},60^{\circ}]\times[0,-1600m]$ in the longitude, latitude and vertical directions respectively. For the fine-grid we use vertices spaced at intervals of $1/4^{\circ}$ in the latitude and longitude directions. For the coarse-grid we use a spacing of $1/2^{\circ}$. In both grids the vertical resolution contains 23 layers, spaced closer together towards the surface of the domain. We do not use ALE coordinates, so layer thicknesses are fixed. The flow is driven by a wind-forcing at the surface and temperature relaxation at the northern and southern boundaries. \\ \par
We run the fine-grid model for one year (after 20 years of spin-up) and save the output horizontal velocities every 5 days. We then consider a coarse-graining consisting of the four-point average shown in \cref{fig:Cc}, i.e. in order to obtain the velocity on a coarse-grid cell, we average over all fine-grid cells contained in that cell. According to the results of \cref{sec:DP_derivation}, the divergence-preserving coarse-graining will have $\beta_{0}=\beta_{1}=\frac{1}{4}$. Here we use a range of values: $\beta_{1} = 0,1/6,1/4,1/3$ so that $\beta_{0}=1,1/2,1/4,0$ (since $\beta_{0}=1-3\beta_{1}$). We also consider a further coarse-graining, called the `vertex-cell averaging' (henceforth VC averaging) , which consist of averaging from cells (weighted by cell area) onto vertices, then averaging back onto cells (this is described in Appendix C of \cite{juricke_danilov_kutsenko_oliver_2019}). Furthermore, we include a divergence-preserving solution modified to take into account the curvature, i.e. $\beta(\bar{c},c) = \frac{1}{N^2}D(\bar{c})^{-1}D(c)\chi_{\mathcal{C}(\bar{c})}(c)$ with $\mathcal{C}(c)$ consisting of exactly the fine-grid cells contained in $\bar{c}$. Finally, we consider an integral-preserving coarse-graining $\beta(\bar{c},c) = \frac{\abs{A_c}}{\abs{\bar{A}_{\bar{c}}}}\chi_{\mathcal{C}(\bar{c})}(c)\mathbb{I}$ where $\mathcal{C}(\bar{c})$ consists of those fine grid cells contained in $\bar{c}$, but if $\bar{c}$ lies on a boundary such that there are fine grid cells outside the boundary of the fine grid adjacent to $\bar{c}$, then these cells are also included in $\mathcal{C}(\bar{c})$. \\
\par
After computing the coarse-grained solutions we run FESOM2 on the coarse grid and load in the coarse-grained velocities, so that the corresponding vertical velocities are calculated online. The resulting $w$ fields in the surface layer are shown in \cref{fig:w_filt}. We see that taking $\beta_{1}=\beta_{0}=1/4$, as we derived in \cref{sec:DP_derivation}, gives a noticeable improvement over the other values and over the VC averaging. In these latter cases we see grid effects appearing in the vertical velocities calculated on the coarse grid. However, when we average with $\beta_{1}=1/4$ these effects largely disappear, leaving a field strongly resembling the reference solution. A significant discrepancy is the presence of large vertical velocities in the north-east and south-west corners of the domain; this is a result of the fact that the boundary of the fine grid differs slightly from the boundary of the coarse grid at these points. There also seem to be some artificial streaks in the domain, with some of the large-scale structures distorted. The vertical velocity field in the top layer which results from the integral-preserving choice, $\beta(\bar{c},c) = \beta(\bar{c},c) = \frac{\abs{A_c}}{\abs{\bar{A}_{\bar{c}}}}\chi_{\mathcal{C}(\bar{c})}(c)\mathbb{I}$ is almost identical to the vertical velocity in the case $\beta=1/4$, though the error in the north-east corner seems to have grown; the large-scale streaks are also present in this case. For the coarse-graining defined by $\beta(\bar{c},c) = \frac{1}{N^2}D(\bar{c})^{-1}D(c)\chi_{\mathcal{C}(\bar{c})}(c)$ the large-scale streaks seem to have been eliminated, indicating that this source of error was indeed caused by not taking the curvature into account. Some errors remain in the corners of the domain. \\
\par
The relative $L^{1}$ errors between the coarse-grid vertical velocities in the top layer and those of the reference solution are shown in \cref{fig:w_L1}. There we see that there is a significantly smaller error in the case $\beta_{1}=1/4$ than in the cases with other choices of $\beta_1$ and a much smaller error than the VC averaging. However, a slightly smaller error is achieved with the integral-preserving coarse-graining, while a significant reduction in error occurs in the coarse-graining taking curvature into account. \Cref{fig:w_L1_depth} shows that, apart from in the top layer, the difference between $\beta_{1}=1/4$, $\beta(\bar{c},c) = \frac{1}{N^2}D(\bar{c})^{-1}D(c)\chi_{\mathcal{C}(\bar{c})}(c)$ and $\beta(\bar{c},c) = \beta(\bar{c},c) = \frac{\abs{A_c}}{\abs{\bar{A}_{\bar{c}}}}\chi_{\mathcal{C}(\bar{c})}(c)\mathbb{I}$ is minimal. Towards the top layers these have a lower error than all other coarse-grainings. In the lower depths, however, the case $\beta_{1}=1/3$ gives a slight improvement over these methods. The reason for this could be that the divergence-preserving solution does not obey \cref{eq:al_conditions} exactly and the other two are not exactly divergence-preserving. Moreover, at lower depths there are more small-scale spatial features, which means that relatively large point-wise errors may arise; thus since $\beta_1=1/3$ gives a smoother field the overall error is reduced.
\subsection{Derivation of Gradient-Preserving Coarse-Graining}\label{sec:GP_derivation}
We also wish to compute a coarse-graining on scalar quantities such that the gradient is preserved, and we follow a similar approach to that used in \Cref{sec:DP}. Applying the definitions of coarse-graining on scalar and vector quantities, we find a condition analogous to \cref{eq:DP_general}. That is, $\lrc{\cdot}^{\alpha'}$ is a gradient-preserving coarse-graining at $\bar{c}\in\mathcal{C}(\bar{c})$ if there exists a coarse-graining $\EXP{\cdot}^{\beta'}$ such that for all fine-grid vertices $v\in\mathcal{V}$:
\algn{
\sum_{ \lrc{c:v\in\mathcal{V}(c)}} \frac{\abs{\bar{A}_{\bar{c}}}}{\abs{A_{c}}} \beta'(\bar{c},c) \Delta \B{x}_{v,c} & = \sum_{\bar{v}\in\bar{\mathcal{V}}(\bar{c})} \alpha'(\bar{v},v) \Delta \bar{\B{x}}_{\bar{v},\bar{c}} \label{eq:grad_condition_general}
}
\Cref{eq:grad_condition_general} is similar to \cref{eq:DP_general}, but there is a difference in the cells and vertices over which the sums range; therefore, the choices of $\mathcal{V}(\bar{v})$ and $\mathcal{C}(\bar{c})$ may also be different. \\
\par
Let us consider the simplest possible choice for $\mathcal{V}(\bar{v})$, which is the set containing only the fine grid vertex which has the same location as the coarse-grid vertex $\bar{v}$; this means the scalar field is projected directly onto the coarse grid with no averaging. We can do this since the coarse-grid vertices are all contained in the fine grid. Explicitly, we have $\lrc{\phi}_{\bar{v}}^{\alpha'} = \phi_{\bar{v}}$, or equivalently $\alpha'(\bar{v},v) = \delta_{\bar{v},v}$. With this choice there is, for each $v\in\mathcal{V}$, at most one term on the right hand side of \cref{eq:grad_condition_general}. For the cell averaging let $\mathcal{C}(\bar{c})$ contain precisely the fine-grid cells contained in $\bar{c}$.\\
\par
If we first consider the case $N=2$ so that, for example, if $\bar{c}$ is the cell shown in \cref{fig:Mesh_gradient}, then we have $\mathcal{C}(\bar{c})=\lrc{c_{0},c_{1}, c_{2},c_{3}}$ and $\mathcal{V}(\bar{v}_{i}) = \lrc{\bar{v}_{i}}$ for $i=1,2,3$. Then \cref{eq:grad_condition_general} becomes, for $i=1,2,3$:
\eqsyslab{
\beta'_{i}\Delta \B{x}_{i} & = \frac{1}{2} \Delta \B{x}_{i} \\
\beta'_{0}\Delta \B{x}_{i} & = \beta'_{(i+1)}\Delta \B{x}_{i-1} + \beta'_{(i-1)}\Delta \B{x}_{i+1} \label{eq:direct_proj_be0}
}{direct_proj}
where $\beta_j := \beta(\bar{c},c_j)$ for $j=0,1,2,3$. Moreover, since the matrices $\beta'_{i}$ are symmetric and positive semi-definite, their eigenvectors must be perpendicular; in particular, $\beta'_{i}$ has one eigenvector $\Delta \B{x}_{i}$ so the other must be in the perpendicular direction, which we shall denote $\Delta \B{x}_{i}^{\perp}$. Writing the three vectors explicitly we have $\Delta \B{x}_1 := (1,1)\Delta x$, $\Delta \B{x}_1^\perp := (-1,1)\Delta x$, $\Delta \B{x}_2 := (-1,0)\Delta x$, $\Delta \B{x}_2^\perp := (0,-1)\Delta x$, $\Delta \B{x}_3 := (0,-1)\Delta x$, and $\Delta \B{x}_3^\perp := (-1,0)\Delta x$. Let the eigenvalue corresponding to $\Delta \B{x}_i^\perp$ be denoted $\mu_{i}\geq 0$. Then \cref{eq:direct_proj_be0} becomes:
\eqsys{
0 & = \lrb{\beta'_{0} + \mu_{2}}\Delta \B{x}_{3} + \lrb{\beta'_{0} + \mu_{3}}\Delta \B{x}_{2} \\
\lrb{\beta'_{0} + \mu_{1} + \mu_{3} - \frac{1}{2}}\Delta \B{x}_{1}^{\perp} & = \lrb{\beta'_{0} + \mu_{3}}\Delta \B{x}_{1}\\
\lrb{\beta'_{0} + \mu_{1} + \mu_{2} - \frac{1}{2}}\Delta \B{x}_{1}^{\perp} & = - \lrb{\beta'_{0} + \mu_{2}}\Delta \B{x}_{1}
}
This has a solution:
\algn{
\begin{split}\label{eq:solution_GP_beta}
\beta'_{0} & = \frac{1}{4}\twobytwomatrix{1-4\mu_{3}-2\mu_{1} }{2\mu_{1}-1}{2\mu_{1}-1}{1-4\mu_{2}-2\mu_{1}} \\
\beta'_{1} &= \frac{1}{4}\twobytwomatrix{1+2\mu_{1}}{1-2\mu_{1}}{1-2\mu_{1}}{1+2\mu_{1}} \\
\beta'_{2} &= \frac{1}{2}\twobytwomatrix{1}{0}{0}{2\mu_{2}} \quad \quad \beta'_{3} = \frac{1}{2}\twobytwomatrix{2\mu_{3}}{0}{0}{1}
\end{split}
}
In order that $\beta'_{0}$ be positive semi-definite, we have the following restrictions on the values of $\mu_{i}$:
\eqsys{\label{eq:conditions_GP_mu}
\mu_{1}+\mu_{2}+\mu_{3}& \leq \frac{1}{2} \\
\lrb{\mu_{2} + \mu_{3}}\lrb{1-2\mu_{1}} &\leq 4\mu_{2}\mu_{3}
}
Some re-arranging of \cref{eq:conditions_GP_mu} shows that the only non-negative values of $\mu_{i}$ that satisfy these conditions have $\mu_{2}=\mu_{3}=0$ and $ 0\leq \mu_{1} \leq \tfrac{1}{2}$.
However, the solution \cref{eq:solution_GP_beta} satisfies \cref{eq:be_conditionsi} if and only if $\mu_2=\mu_3=1/2$. Thus the direct projection cannot satisfy both \cref{eq:be_conditions_posdef} and \cref{eq:be_conditionsi} simultaneously. However, the fact that \cref{eq:be_conditionsi} is satisfied for the latter solution gives us that $\bar{\nabla}\lrc{\phi}^{\alpha'} =\EXP{\nabla \phi}^{\beta'} \approx \nabla \phi$, up to order $N^2\Delta x^2$, where $\lrc{\cdot}^{\alpha'}$ is the direct projection onto coarse-grid vertices. In both choices for $\EXP{\cdot}^{\beta'}$ the parameter $\mu_1$ is undetermined, and so there is a family of coarse-grainings that will all give the same result when applied to a scalar gradient. We therefore have several different options for defining the coarse-graining $\EXP{\cdot}^{\beta'}$, though none of these is entirely satisfactory. We shall therefore verify numerically in the next section that the coarse-graining by direct projection on scalar quantities does give a smaller error in the gradient than other methods of coarse-graining.
\\
\par
We may generalise the results above to general $N$. Let us again suppose that $\alpha'(\bar{v},v) = \delta_{\bar{v},v}$ and that $\beta'\lrb{{\bar{c},c}}=0$ unless $c$ is contained in $\bar{c}$. Consider first an upward-pointing coarse-grid cell $\bar{c}$ with the fine-grid cells contained therein labelled as shown in \cref{fig:GP_general}, so that $c_{ij}^+$ is the upward pointing-cell with its right-angled vertex at position $(i, j)$ relative to the right-angled vertex of $\bar{c}$. We write $c_{ij}^-$ to denote the downward-pointing cell that shares a hypotenuse with $c_{ij}^+$. Considering condition \cref{eq:grad_condition_general} at each of the vertices of $\bar{c}$, we find:
\eqsys{
&\beta'\lrb{\bar{c},c_{0,0}^{+}} \Delta x_1 = \frac{1}{N}\Delta x_1 \\ &\beta'\lrb{\bar{c},c_{N-1,0}^{+}} \Delta x_2 = \frac{1}{N}\Delta x_2 \\ &\beta'\lrb{\bar{c},c_{0,N-1}^{+}} \Delta x_3 = \frac{1}{N}\Delta x_3
}
At the fine-grid vertices on the boundary of the cell we find, for $i, j = 1,...,N-1$:
\eqsys{
& \beta'\lrb{\bar{c},c_{i-1,0}^{-}} \Delta x_3 = \beta'\lrb{\bar{c},c_{i-1,0}^{+}}\Delta x_2 + \beta'\lrb{\bar{c},c_{i,0}^{+}}\Delta x_1 \\
& \beta'\lrb{\bar{c},c_{0,j-1}^{-}} \Delta x_2 = \beta'\lrb{\bar{c},c_{0,j-1}^{+}}\Delta x_3 + \beta'\lrb{\bar{c},c_{0,j}^{+}}\Delta x_1\\
& \beta'\lrb{\bar{c},c_{i-1,N-i-1}^{-}} \Delta x_1 = \beta'\lrb{\bar{c},c_{i-1,N-i}^{+}}\Delta x_2 + \beta'\lrb{\bar{c},c_{i,N-i-1}^{+}}\Delta x_3
}
And considering all other vertices, we have, for $i,j=1,...,N-2$ and $i+j\leq N-1$:
\begin{subequations}
\algn{
\begin{split}
0 = &\lrb{\beta'\lrb{\bar{c},c_{i,j}^{+}} - \beta'\lrb{\bar{c},c_{i-1,j-1}^{-}}}\Delta x_1 \\
&+\lrb{\beta'\lrb{\bar{c},c_{i-1,j}^{+}} - \beta'\lrb{\bar{c},c_{i,j-1}^{-}}}\Delta x_2 \\
&+\lrb{\beta'\lrb{\bar{c},c_{i,j-1}^{+}} - \beta'\lrb{\bar{c},c_{i-1,j}^{-}}}\Delta x_3
\end{split}
}
\end{subequations}
It may be verified that one possible solution to conditions is given by taking:
\eqsyslab{
\beta'\lrb{\bar{c},c_{0,0}^+ } &= \twobytwomatrix{1/N}{0}{0}{1/N} \\
\beta'\lrb{\bar{c},c^+_{i,0}} &= \twobytwomatrix{1/N}{0}{0}{0} \quad \text{for}\quad i>0 \\
\beta'\lrb{\bar{c},c^+_{0,j}} &= \twobytwomatrix{0}{0}{0}{1/N} \quad \text{for}\quad j>0
}{be_sol_N}
and $\beta'\lrb{\bar{c},c^+_{i,j}}=0$ for $ij>0$, $\beta'\lrb{\bar{c},c^-_{i,j}}=0$ for all $i,j$. What we have therefore shown is that if we compute the gradient after a direct projection, this gives the same result as averaging the meridional (zonal) fine-grid gradients calculated on cells adjacent to the meridional (zonal) boundaries of the coarse-grid cell. \Cref{eq:be_sol_N} in the case $N=2$ corresponds to setting $\mu_1 = \frac{1}{2}$, $\mu_2=\mu_3=0$ in \cref{eq:solution_GP_beta}. We see therefore that this method of averaging is by no means unique, and there may be others which give the same average gradient at the centre of $\bar{c}$. However, as we saw in the $N=2$ case this solution does not satisfy \cref{eq:be_conditionsi}. An alternative is to set $\beta'(\bar{c},c^{\pm}_{i,j}) = \pm\frac{1}{N}\mathbb{I}$. This satisfies \cref{eq:be_conditionsi}, though it violates \cref{eq:be_conditions_posdef}. It is clear that all of the proposed choices for $\tl{\alpha},\tl{\beta}$ here are not integral-preserving and so are not suitable for comparing domain-averaged quantities. It remains unclear whether or not a more satisfactory solution exists, but we proceed numerically and observe in the next section that the direct projection does give a good approximation to the fine-grid gradient.
\subsection{Numerical Results}
Here we use the same model set-up as in \cref{sec:DP} to verify our results. The pressure is calculated according to the formula given in \citep{danilov_sidorenko_wang_jung_2016}, but with a linear equation of state $b(T)=\kappa(T-T_0)$ so that $p(\B{x},z) = g\lrb{\eta - \kappa \int_{z}^{0}(T- T_0) dz'}$, where $\eta$ is the sea surface height, $T$ is the temperature, and $\kappa,T_0$ are constants. We do the coarse-graining on $\eta$ and $T$, and since pressure depends linearly on these variables, this is equivalent to coarse-graining $p$ directly. If the equation of state were not linear then this would not be possible and the method for coarse-graining may vary depending on the application. This is because it is not necessarily the case that $b(\lrc{T}^{\alpha'}) =\lrc{b(T)}^{\alpha'}$. Thus if we want to preserve the buoyancy gradient we may need to search for coarse-grainings that obey $\EXP{\nabla b(T)}^{\beta'} = \bar{\nabla} b\lrb{\lrc{T}^{\alpha'}}$. However, this will depend on what the equation of state is and whether such a coarse-graining will exist is unclear, and beyond the scope of the present work. \\
\par
We consider several options for the coarse graining, with the averaging done according to that shown in \cref{fig:Vv}. Explicitly, for each vertex we perform a seven-point average with weights $\alpha'_{0}$ on the central vertex and $\alpha'_{1}$ on the remaining vertices. Note that $\alpha'_{0}=1-6\alpha'_{1}$, so $0\leq \alpha'_{1} \leq 1/6$; therefore, we consider the set of values: $\alpha'_{1}=0,1/12,1/10,1/8,1/6$. The case $\alpha'_{1}=0$ corresponds to direct projection and should preserve the gradient, according to the results derived in \cref{sec:GP_derivation}. At the boundaries we use $\alpha_0'\rightarrow \frac{\alpha_0'}{\alpha_0' +n\alpha_1'}$ and $\alpha_1'\rightarrow \frac{\alpha_1'}{\alpha_0' +n\alpha_1'}$ where $n$ is the number of vertices connected by a fine-grid edge to the boundary vertex around which we average. However, in our calculations of error we exclude the boundaries so this choice will not affect our conclusions.\\
\par
If we plot the coarse-grid gradients of the coarse-grained pressures, then we find that there is almost no visible difference between these and the fine-grid pressure gradient. We therefore do not include a plot of this. However, we plot the $L^{1}$ error of the surface pressure gradient in \cref{fig:grad_p_L1}. We see here that there is a small difference in error between the different coarse graining methods, but generally the direct projection $\alpha'_{1}=0$ gives the lowest error, as expected. We also see in \cref{fig:dp_L1_depth} that at other depths the direct projection continues to be the best at reducing the error and in fact the improvement becomes more noticeable at these depths.
\subsection{Problem Statement} \label{sec:defs}
The discretisation of the mesh used in the present study is based on that of FESOM2 \citep{danilov_sidorenko_wang_jung_2016}. Therefore we consider a triangular mesh with a placement of variables according to an Arakawa-B \citep{ARAKAWA_1977} grid. This means that scalar quantities (temperature, sea surface height, vertical velocity, etc) are defined at vertices, while vector quantities (horizontal velocity, pressure gradient, etc) are defined at cell centres, as shown in \cref{fig:meshdefs}. Unless stated otherwise, we shall assume for the purposes of this work that the grid has a regular discretisation in the horizontal directions so that vertices are spaced equally. On the fine grid, which we shall call $\mathcal{G}$, the spacing shall be taken to be $\Delta x$, while on the coarse grid $\bar{\mathcal{G}}$ we suppose the separation is $N\Delta x$ for some integer $N\geq 2$. The same vertical discretisation is used for both grids. The fine grid consists of a set of vertices $\mathcal{V}$ connected by edges, and a set of cells $\mathcal{C}$. The corresponding sets on the coarse grid will be denoted with an over-bar, so that the sets of vertices and cells in the coarse grid are $\bar{\mathcal{V}}$ and $\bar{\mathcal{C}}$ respectively. It is assumed that every vertex in the coarse grid is also a vertex in the fine grid, and that every cell in the coarse grid is the union of $N^2$ cells in the fine grid. The $N=2$ case is shown in \cref{fig:mesh2}, in which each coarse-grid cell contains $N^2 = 4$ fine-grid cells.\\
\par
A \emph{scalar field} $\phi$ on $\mathcal{G}$ is a map $\phi: \mathcal{V}\rightarrow\mathbb{R}$. We shall denote the field $\phi$ evaluated at vertex $v$ as $\phi_v$. Let $\mathcal{F}$ denote the set of all scalar fields on $\mathcal{G}$ and let $\bar{\mathcal{F}}$ be the set of scalar fields on $\mathcal{G}$. We define a \emph{coarse-graining} to be a map $\lrc{\cdot}^{\alpha} : \mathcal{F}\rightarrow\bar{\mathcal{F}}$ which sends $\phi\mapsto\lrc{\phi}^\alpha$ such that $\lrc{\phi}^\alpha$ is a scalar field on the coarse grid which is specified by means of an average over cells on the fine grid:
\eq{
\lrc{\phi}^\alpha_{\bar{v}} := \sum_{v\in\mathcal{V}} \alpha(\bar{v},v)\phi_{v} \label{eq:al_def}
}
where $\bar{v}$ is a coarse-grid vertex and $v\in\mathcal{V}$ are fine-grid vertices. $\alpha(\ol{v},v)$ are averaging weights and for this to be a `good' coarse-graining, $\alpha$ must satisfy certain properties. In particular, $\lrc{\phi}^\alpha$ should be a good approximation to $\phi$. If we suppose that $\phi$ is a discrete approximation of some smooth scalar field then we can do a Taylor expansion of $\phi_v$ around vertex $\bar{v}$ to find:
\algn{
\lrc{\phi}^\alpha_{\bar{v}} = \sum_{v\in\mathcal{V}} \alpha(\bar{v},v)\lrsq{ \phi_{\bar{v}} + \lrb{\B{x}_v - \B{x}_{\bar{v}}}\cdot\lrb{\nabla \phi}_{\bar{v}} + \mathcal{O}\lrb{ \abs{\B{x}_v - \B{x}_{\bar{v}}}^2 } } \label{eq:al_approx}
}
where $\B{x}_v$, $\B{x}_{\bar{v}}$ are the coordinates of fine-grid vertex $v$ and coarse-grid vertex $\bar{v}$ respectively. So, in order to have an approximation up to second order, $\alpha$ should satisfy the following conditions:
\begin{subequations}
\algn{
\sum_{v\in\mathcal{V}} \alpha(\bar{v},v) &= 1 \label{eq:al_conditions} \\
\sum_{v\in\mathcal{V}} \alpha(\bar{v},v)\B{x}_v & = \B{x}_{\bar{v}} \label{eq:al_conditionsi}
}
The averaging weights should also decay as $v$ gets further from $\bar{v}$. Thus we define $\mathcal{V}(\bar{v}) := \lrc{v\in\mathcal{V}: \alpha(\bar{v},v)\neq 0}$, which will be some set of vertices neighbouring $\bar{v}$ in the fine grid. Moreover, in order for $\lrc{\cdot}^\alpha$ to be an average the averaging weights should all be positive:
\algn{
\alpha(\bar{v},v) &\geq 0 \label{eq:al_conditions_posdef}
}
Another property of coarse-graining that may be advantageous is to preserve integrals over the domain. This is the method used in \citep{mana2014toward} and imposes the following condition on the averaging weights:
\algn{
\sum_{\bar{v}\in\bar{\mathcal{V}}} \abs{\bar{A}_{\bar{v}}}\alpha(\bar{v},v) & = \abs{A_v} \label{eq:al_area}
}
for all fine-grid vertices $v$, where $A_{v}$ the vertex cell as defined in \citep{danilov_sidorenko_wang_jung_2016}. i.e. it is the region enclosed by the boundary formed by connecting cell centres to the midpoints of the adjacent edges, see \cref{fig:div_def}. The magnitude of this area is given by $\abs{A_v} = \frac{1}{3}\sum_{c\in\mathcal{C}(v)}\abs{A_c}$, where $\mathcal{C}(v)$ is the set of cells surrounding vertex $v$ and $\abs{A_c}$ is the area of cell $c$. We may observe that \cref{eq:al_area}, when combined with \cref{eq:al_conditions} implies that $\sum_{\bar{v}\in\bar{\mathcal{V}}} \abs{\bar{A}_{\bar{v}}} = \sum_{v\in\mathcal{V}}\abs{A_v}$, i.e. the total volumes of the two domains are equal. However, this is not always the case; for example, if the boundaries of the fine and coarse grids do not exactly coincide, or if the mesh is on a curved space then the sums of all vertex areas may not be equal in the two grids. We may therefore consider replacing \cref{eq:al_conditions} with the following:
\algn{
\sum_{v\in\mathcal{V}}\alpha(\bar{v},v) &= \frac{1}{\abs{\bar{A}_{\bar{v}}}}\sum_{v\in\mathcal{V}} \kappa(\bar{v},v) \abs{A_v} \tag{\ref*{eq:al_conditions}$'$} \label{eq:al_sum1_new}
}
\end{subequations}
where $\kappa$ satisfies $\sum_{\bar{v}\in\bar{\mathcal{V}}}\kappa(\bar{v},v) = 1$. \Cref{eq:al_sum1_new} is a necessary, but not sufficient condition for $\lrc{\cdot}^{\alpha}$ to be an integral-preserving coarse-graining. Further, it reduces to \cref{eq:al_conditions} if $\kappa$ also obeys $\sum_{v\in\mathcal{V}}\kappa(\bar{v},v)\abs{A_v} = \abs{\bar{A}_{\bar{v}}} $.
Thus the modified condition can be thought of as an alternative to \cref{eq:al_conditions} that allows preservation of the total integral. However, using \cref{eq:al_sum1_new} will not in general keep the coefficient of $\phi_{\bar{v}}$ equal to $1$ in \cref{eq:al_approx}. The choice of which condition to use will therefore depend on the required application. \\
\par
Similarly, we may define a \emph{vector field} $\B{v}$ on $\mathcal{G}$ to be a map $\B{v}:\mathcal{C} \rightarrow \mathbb{R}^2$. The vector field evaluated at cell $c$ shall be denoted $\B{v}_c$ and we write $ \mathbb{V}$ to be the set of all vector fields on $\mathcal{G}$. The corresponding sets for $\bar{\mathcal{G}}$ shall be denoted with an over-bar. Thus, we define a coarse-graining on vector quantities to be a map $\EXP{\cdot}^\beta: \mathbb{V}\rightarrow\bar{\mathbb{V}}$ such that $\EXP{\B{v}}^\beta$ is a vector field on the coarse grid given by:
\eq{
\EXP{\B{v}}_{\bar{c}}^\beta = \sum_{c\in\mathcal{C}} \beta(\bar{c},c)\B{v}_{c} \label{eq:be_def}
}
In this case the averaging weights $\beta(\bar{c},c)$ are $2\times 2$ matrices and the conditions analogous to \cref{eq:al_conditions,eq:al_conditionsi,eq:al_conditions_posdef} are:
\begin{subequations}
\algn{
\sum_{c\in\mathcal{C}} \beta(\bar{c},c) &= \mathbb{I} \label{eq:be_conditions}\\
\sum_{c\in\mathcal{C}} \beta_{ij}(\bar{c},c) \B{x}_{c} & = \B{x}_{\bar{c}}\delta_{ij} \label{eq:be_conditionsi}\\
\beta(\bar{c},c) &\geq 0 \label{eq:be_conditions_posdef}
}
\Cref{eq:be_conditions_posdef} is to be interpreted as saying that $\beta(\bar{c},c)$ is symmetric and positive semi-definite. $\B{x}_c$ is the coordinate of the centre of the fine grid cell $c$, $\B{x}_{\bar{c}}$ is the coordinate of the centre of the coarse grid cell $\bar{c}$ and $\delta_{ij}$ is the Kronecker delta. We denote the set of cells on which $\beta(\bar{c},\cdot)$ is non-zero as $\mathcal{C}(\bar{c}):=\lrc{c\in\mathcal{C}: \beta(\bar{c},c)\neq 0}$. In order for $\EXP{\cdot}^\beta$ to be an integral-preserving coarse-graining the weights would need to satisfy:
\algn{
\sum_{\bar{c}\in\bar{\mathcal{C}}} \abs{\bar{A}_{\bar{c}}}\beta(\bar{c},c) &= \abs{A_c} \mathbb{I} \label{eq:be_area}
}
In order for this to hold everywhere we also need to impose the following condition, analogous to \cref{eq:al_sum1_new}:
\algn{
\sum_{c\in\mathcal{C}}\beta(\bar{c},c) &= \frac{1}{\abs{\bar{A}_{\bar{c}}}}\sum_{c\in\mathcal{C}} \lambda(\bar{c},c)\abs{A_c} \tag{\ref*{eq:be_conditions}$'$} \label{eq:be_sum1_new}
}
where $\lambda(\bar{c},c)$ are $2\times2$ matrices which satisfy $\sum_{\bar{c}\in\bar{\mathcal{C}}}\lambda(\bar{c},c) = \mathbb{I}$. If, additionally, $\lambda$ obeys $\sum_{c\in\mathcal{C}}\lambda(\bar{c},c)\abs{A_c} = \abs{\bar{A}_{\bar{c}}}\mathbb{I} $ then \cref{eq:be_sum1_new} reduces to \cref{eq:be_conditions}. Again, which of these we use will depend on the desired application.
\end{subequations}
\\
\par
In order to define the divergence, we follow the method used for FESOM2, as described in \cite{danilov_sidorenko_wang_jung_2016}. In particular, since this is a finite-volume \citep{barth2018finite,DANILOV201214} model we have a constant velocity $\B{v}_{c}$ on each cell, and it is assumed that the divergence at vertex $v$ multiplied by the area of the corresponding vertex cell $A_{v}$ is approximately equal to the integral of the divergence over this cell. The divergence theorem applied to the integral over this region, gives the divergence as an integral over the boundary of the vertex cell, as shown in \cref{fig:div_def}. Therefore the divergence is calculated as:
\algn{\label{eq:div_def}
\begin{split}
\lrsq{\nabla\cdot\B{v}}_{v} & \approx \frac{1}{\abs{A_{v}}} \int_{A_{v}} \nabla\cdot\B{v} \, dA \\
& = \frac{1}{\abs{A_{v}}} \oint_{\partial A_{v}} \B{v}\cdot\B{n} \, dl \\
& = \frac{1}{2\abs{A_{v}}}\sum_{c\in\mathcal{C}(v)}\B{v}_{c}\cdot\Delta \B{x}_{v,c}
\end{split}
}
where $\mathcal{C}(v)$ is the set containing the six cells surrounding vertex $v$ and $\Delta \B{x}_{v,c}$ is a vector which is outward-pointing, orthogonal to the edge opposite vertex $v$ in cell $c$, and of magnitude equal to the length of that edge. The coarse-grid divergence $\bar{\nabla}\cdot $ is defined with a formula analogous to that in \cref{eq:div_def}. We remark here that in FESOM2 the areas $\abs{A_c}$ and the vectors $\Delta \B{x}_{v,c}$ vary with latitude in order to account for the curvature of the earth. We comment on the case with curvature in \cref{sec:curvature}; however, for the remainder of this section we shall consider the case in which they are constant over the grid, unless stated otherwise. \\
\par
With these definitions in mind we say that a coarse graining $\EXP{\cdot}^\beta$ on vector quantities is \emph{divergence-preserving} at $\bar{v}\in\bar{\mathcal{V}}$ if there exists a coarse-graining $\lrc{\cdot}^\alpha$ on scalar quantities such that for any vector field $\B{v}$ defined at cell centres on the fine grid:
\algn{
\lrsq{\bar{\nabla}\cdot \EXP{\B{v}}^\beta}_{\bar{v}} = \lrc{\nabla \cdot \B{v}}^\alpha_{\bar{v}} \label{eq:div_condition}
}
In \cref{sec:DP_derivation} we shall find averaging weights $\alpha$,$\beta$ such that the corresponding coarse-grainings obey \cref{eq:div_condition} for all vector fields $\B{v}$.\\
\par
Here we remark that the definition of the vertical component of the curl in FESOM2, $\PD{u_2}{x} - \PD{u_1}{y}$, is almost identical to the definition of divergence \cref{eq:div_def}, but instead of $\Delta \B{x}_{v,c}$ there is another vector of the same magnitude, pointing along the direction of integration rather than normal to it. The explicit value of these vectors will not be important, only the fact that they point in one of three directions (up to a minus sign) and sum to zero. Therefore, any coarse-graining that preserves divergence will also preserve the $z$-component of the curl. \\
\par
We also use the definition of gradient from FESOM2; this is calculated at cell $c$ by integrating over the cell, and then applying Stokes' Theorem to get an integral along edges, as shown in \cref{fig:grad_def}. The value of the scalar field at each edge is obtained by averaging the field from the vertices attached to the edge:
\begin{align}
\begin{split} \label{eq:gradient_definition}
\lrsq{\nabla \phi}_{c} &\approx \frac{1}{\abs{A_{c}}}\int_{A_{c}} \nabla \phi \, dA \\
& = \frac{1}{\abs{A_{c}}}\oint_{\partial A_{c}} \phi \B{n} \, dl \\
& = \frac{1}{2\abs{A_{c}}}\sum_{v\in \mathcal{V}(c)} \phi_{v} \Delta \B{x}_{v,c}
\end{split}
\end{align}
where $A_{c}$ is the area of cell $c$, $\B{n}$ is the outward-pointing normal, and $\mathcal{V}(c)$ is the set of vertices attached to cell $c$. The gradient $\bar{\nabla}$ on the coarse grid is defined similarly. As with the divergence, the gradient in FESOM2 also accounts for the curvature, though the present work only considers the flat space case. \\
\par
We say that a coarse-graining $\lrc{\cdot}^{\alpha'}$ on scalar quantities is \emph{gradient-preserving} at $\bar{c}\in\bar{\mathcal{C}}$ if there exists a coarse-graining $\EXP{\cdot}^{\beta'}$ on vector quantities such that:
\algn{
\lrsq{\bar{\nabla}\lrc{\phi}^{\alpha'}}_{\bar{c}} = \EXP{\nabla \phi}^{\beta'}_{\bar{c}} \label{eq:grad_condition}
}
In \cref{sec:GP_derivation} we shall find averaging weights $\alpha', \beta'$ such that the corresponding coarse-grainings obey \cref{eq:grad_condition} for all scalar fields $\phi$ defined on the fine grid. We then verify numerically that our proposed coarse-graining is successful in reducing the error.
\section{Introduction}
\label{sec:intro}
\input{Introduction}
\section{Divergence-Preserving Coarse-Graining}
\label{sec:DP}
\input{DP}
\section{Gradient-Preserving Coarse-Graining}
\label{sec:GP}
\input{GP}
\section{Summary and Discussion}
\label{sec:conclusion}
\input{conclusion}
\section*{Acknowledgements}
The author was funded by the UK Engineering and Physical Sciences Research Council (EPSRC) Centre for Doctoral Training in Mathematics of Planet Earth (MPE-CDT). He would like to thank the three anonymous reviewers for their constructive comments, as well as Pavel Berloff and Stephan Juricke for feedback on the manuscript. Also acknowledged are Darryl Holm and all the members of the Geometric Mechanics research group at Imperial College for the stimulating discussions at our Friday group meetings, as well as Colin Cotter, Peter Korn, Ruiao Hu and Pavel Perezhogin for useful discussions.
\section{Figures}
\input{figures}
\newpage
\bibliographystyle{elsarticle-num}
|
1,116,691,500,830 | arxiv | \section{ Introduction } \label{s1}
Recent LEP measurements \cite{LEP} fix the number of light neutrinos
to three, therefore a simple further repetition of fermion families
is excluded.
Extensions of the minimal Standard Model by heavy fermions are,
however, possible.
Examples are: a fourth fermion family with heavy neutrino (for a
recent reference see \cite{MARCI}), or a duplication of the three
light families by heavy mirror families \cite{MIRFAM,ROMAPR}.
Some limitations on the number of heavy fermions follow from
studies of 1-loop radiative corrections \cite{PESTAK,ALTBAR}
because of the non-decoupling of heavy fermions.
The question of non-decoupling in higher loop orders is, however,
open.
In fact, one of the goals of lattice studies is to investigate this
in the nonperturbative regime of couplings.
The lattice formulation of the electroweak Standard Model is
difficult because of the fermion doublers \cite{NIENIN}.
In fact, at present no completely satisfactory formulation is known
\cite{TSUKPR}: if one insists on explicit chiral gauge invariance,
then mirror fermion fields have to be introduced \cite{CHFER},
otherwise one has to fix the gauge as in the ``Rome-approach''
\cite{ROMA2}.
(In a recently proposed method \cite{KAPLAN} a fifth extra dimension
has to be introduced.)
There is, however, an interesting limit of the Standard Model
which can be numerically simulated by present techniques.
Namely, if the $\rm SU(3)_{colour} \otimes U(1)_{hypercharge}$
gauge couplings are neglected, then, as a consequence of the
pseudo-reality of SU(2) representations, mirror fermions can be
transformed to normal fermions by charge conjugation, and Yukawa models
with an even number ($N_f$) of degenerate fermion doublets can be
simulated \cite{TSUKPR}.
(For instance, $N_f=4$ corresponds to a heavy degenerate fermion
family.)
This can be done at nonzero Yukawa-couplings for both fermion
($G_\psi$) and mirror fermion ($G_\chi$), or by keeping $G_\chi$ and
the fermion-mirror-fermion mixing mass $\mu_{\psi\chi}$
at zero, and thereby decoupling the mirror fermion exactly from
the physical spectrum \cite{ROMA1}.
In the present paper we choose this second way, where the exact
decoupling in the continuum limit is assured by the Golterman-Petcher
fermion shift symmetry \cite{GOLPET}.
This symmetry is exact at $G_\chi=\mu_{\psi\chi}=0$, and implies
a set of identities, which makes the parameter tuning easier.
Another interpretation of the decoupling limit also deserves attention.
Namely, interchanging the r\^oles of fermion and mirror fermion by
considering $\chi$ to be the fermion and $\psi$ the mirror fermion, the
decoupling scenario turns out to be a rather good approximation of the
situation in phenomenological models with mirror fermions
\cite{MIRFAM}.
This is due to the fact that all known physical fermions have very
small Yukawa-couplings.
The only fermion states with strong Yukawa-couplings would be the
members of the mirror families, if they would exist.
In fact, the smallness of the known fermion masses on the electroweak
scale could then be explained by the approximate validity of the
Golterman-Petcher shift symmetry.
An important set of questions for the numerical simulations is
concentrated around the ``allowed range of renormalized couplings'',
which is cut-off dependent and shrinks to zero for infinite cut-off
if the continuum limit is trivial.
It is expected on the basis of 1-loop perturbation theory that,
as a function of the renormalized Yukawa-coupling, the allowed
region for the renormalized quartic coupling is limited by an
upper bound obtained at infinite bare quartic coupling
($\lambda=\infty$), and by a lower bound called ``vacuum stability
bound'' reached at zero bare quartic coupling ($\lambda=0$)
(for discussions see \cite{FKLMMM,LMMW,TSUKPR}).
In the present work the numerical simulations are restricted to
$\lambda=\infty$, and first results on the behaviour of the upper bound
are obtained.
The study of the $\lambda \to 0$ limit is postponed to future work.
\section{ Lattice action, decoupling, physical quantities } \label{s2}
Our numerical simulations were performed in the chiral
$\rm SU(2)_L \otimes SU(2)_R$ symmetric Yukawa model with
$N_f = 2$ mirror pairs of fermion doublet fields in the decoupling
limit.
The conventions in the lattice action and the definition of different
renormalized physical quantities closely follow our previous papers
on $\rm U(1)_L \otimes U(1)_R$ \cite{FKLMMM,LMMW} and
$\rm SU(2)_L \otimes SU(2)_R$ \cite{LINWIT} symmetric models.
Therefore we only repeat here the most essential formulae, and
include the definitions specific to the present investigation.
The lattice action is a sum of the O(4)
($\cong \rm SU(2)_L \otimes SU(2)_R$) symmetric pure scalar part
$S_\varphi$ and fermionic part $S_\Psi$:
\begin{equation} \label{eq01}
S = S_\varphi + S_\Psi \ .
\end{equation}
$\varphi_x$ is the $2 \otimes 2$ matrix scalar field, and
$\Psi_x \equiv (\psi_x, \chi_x)$ stands for the mirror pair of fermion
doublet fields (usually $\psi$ is the fermion doublet and $\chi$ the
mirror fermion doublet).
In the usual normalization conventions for numerical simulations we
have
$$
S_\varphi = \sum_x \left\{ \frac{1}{2} {\rm Tr\,}(\varphi_x^+\varphi_x) +
\lambda \left[ \frac{1}{2}{\rm Tr\,}(\varphi_x^+\varphi_x) - 1\right]^2
- \kappa\sum_{\mu=1}^4
{\rm Tr\,}(\varphi^+_{x+\hat{\mu}}\varphi_x)
\right\} \ ,
$$
$$
S_\Psi = \sum_x \left\{ \mu_{\psi\chi} \left[
(\overline{\chi}_x\psi_x) + (\overline{\psi}_x\chi_x) \right]
\right.
$$
$$
- K \sum_{\mu=\pm 1}^{\pm 4} \left[
(\overline{\psi}_{x+\hat{\mu}} \gamma_\mu \psi_x) +
(\overline{\chi}_{x+\hat{\mu}} \gamma_\mu \chi_x)
+ r \left( (\overline{\chi}_{x+\hat{\mu}}\psi_x)
- (\overline{\chi}_x\psi_x)
+ (\overline{\psi}_{x+\hat{\mu}}\chi_x)
- (\overline{\psi}_x\chi_x) \right) \right]
$$
\begin{equation} \label{eq02}
\left.
+ G_\psi \left[ (\overline{\psi}_{Rx}\varphi^+_x\psi_{Lx}) +
(\overline{\psi}_{Lx}\varphi_x\psi_{Rx}) \right]
+ G_\chi \left[ (\overline{\chi}_{Rx}\varphi_x\chi_{Lx}) +
(\overline{\chi}_{Lx}\varphi^+_x\chi_{Rx}) \right]
\right\} \ .
\end{equation}
Here $K$ is the fermion hopping parameter, $r$ the Wilson-parameter,
which will be fixed to $r=1$ in the numerical simulations, and the
indices $L,R$ denote, as usual, the chiral components of fermion
fields.
In this normalization the fermion-mirror-fermion mixing mass is
$\mu_{\psi\chi}=1-8rK$.
In the limit $\lambda \to \infty$ the length of the scalar field is
frozen to unity, therefore in $S_\varphi$ only the term proportional
to $\kappa$ is relevant.
The consequence of the Golterman-Petcher identities is that
at $G_\chi=0$ all higher vertex functions containing the $\chi$-field
vanish identically, and the $\chi$-$\chi$ and $\chi$-$\psi$
components of the inverse fermion propagator $\tilde{\Gamma}_\Psi(p)$
are equal to the corresponding components of the free inverse
propagator \cite{GOLREV,LINWIT}.
In the broken phase the small momentum ($p \to 0$) behaviour of
$\tilde{\Gamma}_\Psi$ defines the renormalized $\psi$-mass
$\mu_{R\psi}$ and wave function renormalization factor $Z_\psi$,
therefore
$$
\tilde{\Gamma}_\Psi(p) \equiv M + i \gamma \cdot p\, N + O(p^2)
$$
\begin{equation} \label{eq03}
= \left(
\begin{array}{cc}
(\mu_{R\psi} + i\gamma \cdot \bar{p} + O(p^2))Z_\psi^{-1} &
\mu_0 + \frac{r}{2}\hat{p}^2 \\[1em]
\mu_0 + \frac{r}{2}\hat{p}^2 & i\gamma \cdot \bar{p}
\end{array} \right) \ .
\end{equation}
Here $\mu_0 \equiv \mu_{\psi\chi}/(2K) = (1-8rK)/(2K)$ and, as usual,
$\bar{p}_\mu \equiv \sin p_\mu$ and
$\hat{p}_\mu \equiv 2\sin \frac{1}{2} p_\mu$.
The propagator is the inverse of $\tilde{\Gamma}_\Psi$.
With the notation
\begin{equation} \label{eq04}
\mu_p \equiv \mu_0 + \frac{r}{2} \hat{p}^2
\end{equation}
we have
$$
\tilde{\Delta}_\Psi(p) = \tilde{\Gamma}_\Psi(p)^{-1}
\equiv A - i \gamma \cdot p\, B + O(p^2)
= \left[ (\bar{p}^2 + Z_\psi\mu_p^2)^2 + \mu_{R\psi}^2 \bar{p}^2
\right]^{-1}
$$
\begin{equation} \label{eq05}
\cdot \left(
\begin{array}{cc}
Z_\psi [ \mu_{R\psi} \bar{p}^2 - i\gamma \cdot \bar{p}
(\bar{p}^2 + Z_\psi\mu_p^2) ] &
Z_\psi\mu_p (\bar{p}^2 + Z_\psi\mu_p^2 + i\gamma \cdot \bar{p}\,
\mu_{R\psi}) \\[1em]
Z_\psi\mu_p (\bar{p}^2 + Z_\psi\mu_p^2 + i\gamma \cdot \bar{p}\,
\mu_{R\psi}) &
-Z_\psi\mu_p^2\mu_{R\psi} - i\gamma \cdot \bar{p}
(\bar{p}^2 + \mu_{R\psi}^2 + Z_\psi\mu_p^2)
\end{array} \right) + O(p^2) \ .
\end{equation}
This shows that in the broken phase near $p=0$ the elements of
$\tilde{\Delta}_\Psi(p)$ are rapidly changing, unless $\mu_p$ is very
small.
Consider, for instance, $\tilde{\Delta}_{\psi\psi}$:
\begin{equation} \label{eq06}
\tilde{\Delta}_{\psi\psi} = Z_\psi \frac{ \mu_{R\psi}
\left( 1 + Z_\psi \frac{\mu_p^2}{\bar{p}^2} \right)^{-1}
- i\gamma \cdot \bar{p} }
{ \bar{p}^2 + Z_\psi\mu_p^2 + \mu_{R\psi}^2
\left( 1 + Z_\psi \frac{\mu_p^2}{\bar{p}^2} \right)^{-1} } + O(p^2) \ .
\end{equation}
One sees that in the phase with broken symmetry ($\mu_{R\psi} \ne 0$)
for $\mu_0$ and $p$ both being small there is a qualitative change
of the behaviour as a function of $p$ around $p^2 \simeq \mu_0$.
The limits $\mu_0 \to 0$ and $p \to 0$ cannot be interchanged.
The correct order is to take first $\mu_0 \to 0$ and then $p \to 0$.
Smooth behaviour near $\mu_0=0$ is reached only if
\begin{equation} \label{eq07}
\mu_0 = O(p^2) \ .
\end{equation}
In this case $\mu_p^2/\bar{p}^2 = O(p^2)$, and the components of the
propagator are
$$
\tilde{\Delta}_{\psi\psi}(p) = Z_\psi \frac{ \mu_{R\psi} -
i\gamma \cdot \bar{p} } { \mu_{R\psi}^2 + \bar{p}^2 } + O(p^2) \ ,
$$
\begin{equation} \label{eq08}
\tilde{\Delta}_{\psi\chi}(p) = \tilde{\Delta}_{\chi\psi}(p)
= i\gamma\cdot\bar{p} \frac{\mu_p Z_\psi \mu_{R\psi}}
{\bar{p}^2 (\mu_{R\psi}^2 + \bar{p}^2)} = O(p) \ ,
\hspace{3em}
\tilde{\Delta}_{\chi\chi}(p) = \frac{-i\gamma \cdot \bar{p}}
{\bar{p}^2} + O(p^2) \ .
\end{equation}
This limit is best approximated if for a given momentum $p$ we choose
\begin{equation} \label{eq09}
\mu_p = \mu_0 + \frac{r}{2} \hat{p}^2 = 0 \ .
\end{equation}
In fact, since our fermionic renormalized quantities are defined at
the smallest timelike momentum $p_{min}$ (and zero spacelike momenta),
we took
\begin{equation} \label{eq10}
\mu_{p_{min}}=0 \ .
\end{equation}
Note that this also implies
$\tilde{\Delta}_{\psi\chi}(p_{min}) = 0$, which ensures that no
mixing between $\psi$ and $\chi$ occurs.
On a lattice with time extension $T$ the smallest fermion momentum
in the timelike direction with our antiperiodic boundary conditions
is $p_{min}=\pi/T$, therefore this condition gives a $T$-dependent
hopping parameter $K > K_{cr} \equiv 1/(8r)$.
For $T \to \infty$ one has, of course, $K \to K_{cr}$.
In summary: since in the broken phase the decoupling situation is
reached only for very small $\mu_0$, in terms of the hopping parameter
one has to be so close to $K=K_{cr} \equiv 1/(8r)$ that trying to
perform simulations at $K < K_{cr}$ and to extrapolate to $K=K_{cr}$
does not pay in the parameter region we studied (see in next section).
Therefore in our simulations we always took
$\mu_0 \simeq \mu_{p_{min}}=0$ ($K \simeq K_{cr}=0.125$).
In numerical simulations one can determine the fermion propagator
$\tilde{\Delta}_\Psi$.
For the renormalized quantities one needs the inverse propagator
$\tilde{\Gamma}_\Psi$.
This could, in principle, be obtained for a given momentum by the
numerical inversion of the $8 \otimes 8$ matrix in
spinor-$\psi$-$\chi$ space, but this would introduce large
fluctuations in the results.
In \cite{FKLMMM,LMMW} we used an analytic inversion up to $O(p^2)$.
Here it is better to take
$$
M = (A + \bar{p}^2 B A^{-1} B)^{-1} \ ,
$$
\begin{equation} \label{eq11}
N = A^{-1} B M = A^{-1} B (A + \bar{p}^2 B A^{-1} B )^{-1} \ ,
\end{equation}
which follows from the spinor structure
$\tilde{\Delta}_\Psi = A - i\gamma \cdot \bar{p}B$, and has at most
$O(p^4)$ corrections.
In addition, for the exactly known elements of $\tilde{\Gamma}_\Psi$
we took the values in (\ref{eq03}), and (\ref{eq11}) was used only
for the $\psi$-$\psi$ component of $\tilde{\Gamma}_\Psi$.
The definition of the renormalized physical quantities can be
taken over in most cases with trivial modifications from the
$\rm U(1)_L \otimes U(1)_R$ model \cite{FKLMMM,LMMW}.
In the broken phase, where most of our runs were performed, we also
use the ``constraint correlations'' obtained after an SU(2)
rotation of the average $\varphi$-field into a fixed (``$\sigma$'')
direction.
(The three Goldstone boson components of the $\varphi$-field
perpendicular to $\sigma$ are denoted by $\pi_a\; (a=1,2,3)$.)
In order to avoid infrared singularities, external $\pi$-legs are
usually set to the smallest nonzero momentum on our $L^3 \cdot T$
lattices.
The renormalized Yukawa-couplings can be defined in different ways.
One definition is given by the ratio of the mass to the renormalized
vacuum expectation value, as for instance
$G_{R\psi}=\mu_{R\psi}/v_R$.
It is interesting to compare this to
the renormalized Yukawa couplings obtained through
Goldstone-fermion-antifermion vertex functions.
These renormalized Yukawa couplings, denoted as $G^{(3)}_{aR\psi}$ and
$G^{(3)}_{aR\chi}$, where $a=1,2,3$, are defined by
\begin{equation} \label{eq12}
\left(
\begin{array}{cc}
i\, \gamma_5\tau_a\, G^{(3)}_{aR\psi} & 0\\
0 &-i\,\gamma_5\tau_a\, G^{(3)}_{aR\chi}
\end{array}
\right) \delta_{k,-p+q}
= {{\hat k}^2_4\over\sqrt{Z_T}}\,\tilde{\Gamma}_R(p_4)\, Z_\Psi^{-1/2}
\,G^{(c)}_{a}\, (Z_\Psi^{-1/2})^T\, \tilde{\Gamma}_R(q_4)\, \ ,
\end{equation}
where no summation over $a$ is applied, and $k_4$, $p_4$, $q_4$ are
the $4$th components of the momenta of Goldstone boson, fermion and
anti-fermion, respectively.
We have set the spatial components of all momenta to zero.
The appearance of the Kronecker-delta above is due to
energy-momentum conservation.
The renormalized 2-point fermion vertex function $\tilde{\Gamma}_R$
at small $p=(0,0,0,p_4)$ is given as
$$
\tilde{\Gamma}_R(p_4) \simeq i\,\gamma_4\,{\bar p}_4\,+\, M_R\, ,
\hspace{3em}
M_R = \left(
\begin{array}{cc}
G_{R\psi}v_R& \mu_R\\
\mu_R & G_{R\chi}v_R
\end{array}
\right)
\,\, ,
$$
and
$$
G^{(c)}_{a} = \frac{1}{L^3T}\,
\sum_{x,y,z}\,e^{-ik_4x_4}\,e^{-ip_4y_4}\,
e^{iq_4z_4}\Bigl<\pi_a(x)\Psi(y)\bar\Psi(z)\Bigr >_c
$$
is the connected part of the $\pi_a$-$\Psi$-$\bar\Psi$ 3-point Green's
function and $\pi_a(x)$ are the Goldstone fields
($\Psi(y) \equiv \Psi_y$ is the fermion field).
Using the fact that the renormalized couplings are the same for all
three Goldstone bosons, and
$$
{\rm Tr}_{Dirac}\,(\gamma_5^2) = 4\, , \hspace{2em}
{\rm Tr}_{SU(2)}\,(\tau_a\tau_b)=2\delta_{ab}\,\, ,
$$
we obtain
$$
\left(
\begin{array}{cc}
G^{(3)}_{R\psi} & 0 \\
0 & -G^{(3)}_{R\chi}
\end{array}
\right) \delta_{k,-p+q}
={-i{\hat k}^2_4\over {24\sqrt{Z_T}}}
$$
\begin{equation} \label{eq13}
\cdot \sum_{a=1}^3\,{\rm Tr}_{SU(2)}\,\Bigl\{\tau_a\,
{\rm Tr}_{Dirac}\,\bigl [\gamma_5\,
\tilde{\Gamma}_R(p_4)\, Z_\Psi^{-1/2}\, G^{(c)}_{a}\,(Z_\Psi^{-1/2})^T\,
\tilde{\Gamma}_R(q_4)\bigr ]\Bigr\}
\,\, .
\end{equation}
Because of the existence of massless Goldstone bosons in the broken
phase, renormalized quantities cannot be defined at zero momentum.
For instance, the connected 3-point $\pi$-$\Psi$-$\bar{\Psi}$
Green's function has an infrared singularity on the external
$\pi$-leg.
Therefore in our simulations on $L^3 \cdot T$ lattices we choose
$$
k_4 = {2\pi\over T}\,\, ,\,\,
p_4 = -{\pi\over T}\,\, ,\,\, q_4={\pi\over T}
\,\, .
$$
After carrying out all the matrix multiplications in (\ref{eq13}), we
get $G^{(3)}_{R\psi}$ and $G^{(3)}_{R\chi}$.
The expressions are too voluminous to be displayed here.
\section{ Numerical simulation } \label{s3}
We used the Hybrid Monte Carlo algorithm \cite{DKPR}.
This requires the flavour duplication of the fermion spectrum.
If the fermion matrix in the action (\ref{eq02}) is denoted by $Q$,
then the replica flavours have $Q^\dagger$, therefore for them the
r\^oles of $\psi$ and $\chi$ are interchanged: $\chi$ is the
``fermion'' and $\psi$ the ``mirror fermion''.
Since in the model under consideration the fermions are equivalent to
mirror fermions, in the decoupling limit $G_\chi=\mu_{\psi\chi}=0$
the model describes two degenerate fermion doublets
(corresponding to the $\psi$-fields) and two massless ``sterile''
doublets (belonging to the $\chi$'s), which have no interactions with
the physical sector.
The commonly used algorithm for the inversion of the fermion matrix
is the conjugate gradient algorithm (CGA).
Motivated by the paper of Gupta et al. \cite{GUPTA} we were also
testing matrix inversion by minimal residual algorithm
(MRA) with odd-even ($o$-$e$) decomposition in our SU(2) symmetric
Higgs-Yukawa model with mirror pairs of fermion fields.
One difference between our case and that of \cite{GUPTA} is that we
have a nontrivial $Q_{ee}$ and $Q_{oo}$ instead of their matrix
$M{\bf 1}$.
But the requirements for a successful implementation of the MRA are
satisfied in our case too.
Namely we can invert $Q_{ee}$ and $Q_{oo}$ explicitly with an amount
of time that is negligible in comparison with the algorithmic
inversion of the full matrix $Q$.
To compare the MRA with the CGA we solve the equation $Q^+ Qp=v$ for
some scalar field configuration $\phi$ at different values of
Yukawa-couplings, hopping parameter, lattice size and different
convergence parameter.
This last quantity is defined by:
\begin{equation} \label{eq14}
\delta = \frac{|Q^+Qp-v|^2}{|v|^2} \ .
\end{equation}
Using the CGA the solution $p$ is accepted as soon as $\delta$ is
smaller than some prescribed $\delta_0$.
The solution by MRA is done in two steps, first solving $Q^+\bar{p}=v$
and then $Qp=\bar{p}$, both with a bound for $\delta$ that is a
factor of 100 smaller than $\delta_0$.
We were also testing the ``polynomial preconditioning'' for the MRA,
described in \cite{GUPTA}.
Preconditioning of order $n$, denoted by MRA($n$), is characterized by
the fact that the matrix, which is to be inverted, contains the
fermion hopping parameter $K$ to the power $2n$.
The tests were performed at $\lambda = 1.0$ and $\lambda = 10^{-6}$.
No important differences were observed for different $\lambda$.
As a few test runs showed, at $\lambda=\infty$ the algorithm behaves
similarly to $\lambda = 1.0$.
\begin{table}[tb]
\caption{ \label{tb1}
Comparison of matrix inversion algorithms for various $\delta_0$
and $G_{\chi}$.\newline
$8^3 \cdot 16$ lattice, $G_{\psi}=0.1$, $\kappa=0.15$, $K=0.1$,
$\lambda=1.0$ . }
\begin{center}
\begin{tabular}{|l||c|r|r|r|r|}
\hline
$\delta_0$ & $G_{\chi}$ & CGA & MRA(1) & MRA(2) & MRA(3) \\
\hline\hline
$10^{-8}$ & 0.0 & 6.5 & 3.6 & 4.5 & 5.5 \\
$10^{-15}$ & 0.0 & 12.6 & 6.1 & 7.0 & 8.5 \\
$10^{-20}$ & 0.0 & 16.8 & 7.9 & 8.9 & 10.7 \\
$10^{-25}$ & 0.0 & 21.1 & 9.7 & 10.7 & 12.6 \\
$10^{-30}$ & 0.0 & 25.2 & 11.8 & 12.5 & 14.9 \\
$10^{-8}$ & -0.1 & 5.8 & 3.5 & 4.1 & 5.1 \\
$10^{-15}$ & -0.1 & 10.5 & 5.5 & 6.5 & 7.7 \\
$10^{-30}$ & -0.1 & 20.5 & 10.4 & 11.1 & 13.0 \\
$10^{-8}$ & 0.2 & 11.4 & 5.0 & 6.1 & 7.8 \\
$10^{-15}$ & 0.2 & 24.4 & 8.8 & 10.2 & 12.2 \\
$10^{-30}$ & 0.2 & 52.3 & 17.6 & 19.0 & 22.0 \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[tb]
\caption{ \label{tb2}
Comparison of matrix inversion algorithms for various $K$.\newline
$4^3 \cdot 8$ lattice,
$\delta_0=10^{-8}$, $G_{\psi}=0.3$, $G_{\chi}=0.0$, $\kappa=0.09$,
$\lambda=10^{-6}$ . }
\begin{center}
\begin{tabular}{|l||c|c|c|c|}
\hline
K & CGA & MRA(1) & MRA(2) & MRA(3) \\
\hline\hline
0.10 & 0.59 & 0.24 & 0.30 & 0.37 \\
0.11 & 0.87 & 0.35 & 0.41 & 0.59 \\
0.12 & 1.51 & 0.91 & 1.14 & 1.41 \\
0.123 & 1.70 & 2.41 & 2.87 & 2.80 \\
0.124 & 1.74 & 4.37 & 4.45 & 3.90 \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[tb]
\caption{ \label{tb3}
Comparison of different preconditionings at $K=0.125$.\newline
$4^3\cdot 8$ lattice,
$\delta_0=10^{-8}$, $G_{\psi}=0.3$, $G_{\chi}=0.0$, $\kappa=0.09$,
$\lambda=10^{-6}$ . }
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
CGA & \multicolumn{7}{c|}{MRA(n)} \\
\cline{2-8}
& n=1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline\hline
1.87 & 11.30 & 8.58 & 6.11 & 3.95 & 2.29 & 12.90 &
15.00 \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[tb]
\caption{ \label{tb4}
Comparison of matrix inversion algorithms on different lattice
sizes\newline
$\delta_0=10^{-15}$, $G_{\psi}= 0.1$, $G_{\chi}=0.0$, $\kappa=0.15$,
$K=0.1$, $\lambda=1.0$ . }
\begin{center}
\begin{tabular}{|c||r|c|c|}
\hline
$L^3 \cdot T$ & CGA & MRA(1) & MRA(2) \\
\hline\hline
$4^3 \cdot 8 $ & 0.67 & 0.41 & 0.46 \\
$6^3 \cdot 12$ & 3.84 & 1.97 & 2.24 \\
$8^3 \cdot 16$ & 12.60 & 6.10 & 7.00 \\
\hline
\end{tabular}
\end{center}
\end{table}
Our results are summarized in tables \ref{tb1} to \ref{tb4},
where the CPU time necessary for solving $Q^+Qp=v$ is given in seconds.
In these tests we could not find any gain using overrelaxation.
Table \ref{tb1} shows for $K=0.1$ that the smaller $\delta_0$ becomes,
the better is the MRA compared to CGA.
Comparing different preconditionings MRA(1) is the best.
This picture changes, if one looks at higher $K$ as can be seen from
tables \ref{tb2} and \ref{tb3}.
Near the critical value of $K$ simple preconditioning is no longer the
best choice and even with an optimal preconditioning of MRA the CGA
performs better.
Table \ref{tb4} shows a comparison between MRA and CGA at $K=0.1$ for
different lattice sizes.
One observes that the gain of using MRA increases with the lattice
size.
Our conclusion is that in the case of small $K$ ($K \leq 0.1$) the MRA
has to be preferred.
For some choices of parameters, e.g.\ in the symmetric phase, the gain
by using MRA can be so large that it would be advantageous to do
calculations at different small values of $K$ (e.g. $0.1,0.11,0.12$)
and then extrapolate to the value under investigation (e.g. $0.125$).
However for $K$ deep in the broken phase the CGA seems to be still the
best choice.
For the investigation of the decoupling limit ($K \simeq K_{cr}=0.125$)
in the broken phase it was necessary to be very close to $K_{cr}$, and
we always used CGA.
The technical advantage of using the decoupling method is that the
number of tuned parameters is less, because the fermion hopping
parameter is fixed at $K=K_{cr}$.
At $\lambda=\infty$ and at a fixed value of the bare Yukawa-coupling
$G_\psi$ one has to tune only the scalar hopping parameter $\kappa$.
The difficulty is that the presence of massless $\chi$-fermions
slows down the convergence of the fermion matrix inversion.
We also tried to improve on this by using the free fermion propagator
for preconditioning in momentum space.
In this way the number of iterations in CGA can be reduced by a factor
not larger than two.
On the other hand, the computer time required for performing the
necessary Fourier transformations is so large that the gain is
completely counteracted, except for very small Yukawa-couplings.
This is presumably due to the ``roughness'' of typical scalar field
configurations.
Another attempt to improve the matrix inversion was to supply the CGA
with an ``educated guess'' for the start vector in the iterations.
This was done by means of a hopping parameter expansion of $Q^{-1}$ up
to some order $n$:
$$
Q^{-1} \approx
Q^{-1}_{n} = D^{-1} \left( 1 - \sum_{k=1}^n (M D^{-1})^k \right) \ ,
$$
where $D$ and $M$ are the diagonal and off-diagonal parts of $Q$
with respect to site indices, and $n$ was optimized for given
parameters $K$ and $G_{\psi}$.
For $K=0.1$ and $G_{\psi} < 2.4$ the speed of the algorithm can be
increased up to a factor of three in this way.
On the other hand, for $K$ above its critical value this improvement is
not applicable.
\section{ Results } \label{s4}
The first step in the numerical simulations was to check the
phase structure at $\lambda=\infty$ and $K = K_{cr}$.
On the basis of experience with several other lattice Yukawa models
\cite{GOLREV,SHIREV}, and our own previous work \cite{LIMOWI}, this is
expected to possess several phase transitions between the
{\it ``ferromagnetic''} (FM), {\it ``antiferromagnetic''} (AFM),
{\it ``paramagnetic''} (PM) and {\it ``ferrimagnetic''} (FI) phases.
The resulting picture in the ($G_\psi,\kappa$)-plane is shown in
fig.~1.
Due to CPU-time limitations we did not try to disentangle the details
of the structure near the meeting point of the four phases, neither did
we follow the shape of the FI-phase for very strong bare
Yukawa-coupling beyond $G_\psi=1.5$.
The physical phase is FM with spontaneously broken chiral symmetry.
Therefore we fixed $G_\psi=0.3,\; 0.6,\; 1.0$ and performed a series
of runs in the $\kappa$-ranges shown in fig.~1 by the dashed lines.
Most of the time $4^3 \cdot 8$ and $6^3 \cdot 12$ lattices were
taken.
In a few particularly important points, for instance at
($\kappa=0.27,\, G_\psi=0.3$) and ($\kappa=0.15,\, G_\psi=0.6$),
in addition to $4^3 \cdot 8$ and $6^3 \cdot 12$ also an $8^3 \cdot 16$
run was performed.
The typical run consisted of about 1000-2000 equilibrating and
4000-10000 measured HMC trajectories.
The length of trajectories was randomly changed by the number of
classical dynamics steps between 3 and 10.
The step length was chosen such that the average acceptance rate per
trajectory stayed near 0.75.
On our lattices the expectation value of the average of the scalar
field in the $\sigma$-direction
$v \equiv \langle \sigma_x \rangle = \langle \phi_{Lx} \rangle$
(in short ``magnetization'') has a smooth behaviour across the
physically important PM-FM phase transition (part of our data is
shown in fig.~2).
Furthermore, the magnetization always decreased with increasing
lattice size, in the same way as in case of the pure O(4)-symmetric
$\phi^4$ model.
This agrees with the expected second order phase transition, which
is well suited for the definition of a continuum limit.
In all data points the mass of the mirror fermion, $\mu_{R\chi}$
and the renormalized mixing mass, $\mu_R$ were consistent with zero
within errors, in agreement with the consequences of the
Golterman-Petcher relations.
The behaviour of the fermion mass $\mu_{R\psi}$ and Higgs-boson mass
in lattice units is shown in figs.~3 and 4, respectively.
The fermion mass ($\mu_{R\psi}$) is decreasing monotonically to zero,
as one approaches the phase transition from the FM-side (decreasing
$\kappa$).
This agrees with the expectation, since in the PM phase, due to
$K \simeq K_{cr}$, the fermion mass is nearly zero (at $K=K_{cr}$ on an
infinite lattice it would be exactly zero).
At the same time the fermion mass is increasing with $G_\psi$, in
such a way that at $G_\psi=1.0$ within our limited computer time
we were unable to find points with really small masses.
This could presumably be cured by investigating more points on
larger lattices.
In fact, the fermion mass shows in this point strong finite size
effects, implying smaller masses on larger lattices.
The masses of the doubler fermions for both $\psi$ and $\chi$ at
nonzero corners of the Brillouin-zone were also determined and turned
out to be always above 1.5, with a slight decreasing tendency for
increasing $G_\psi$.
The values of the Higgs-boson mass ($m_{R\sigma} \equiv m_L$)
were determined by fitting the constraint correlation in the
$\sigma$-channel by a form $\cosh() + const.$ in the range
$1 \le t \le T/2$.
The dependence of $m_{R\sigma}$ on $\kappa$ in fig.~4 shows a gradual
decrease and then a sharp increase for decreasing $\kappa$.
On larger lattices the values are smaller, but there is a substantial
increase with increasing $G_\psi$ if the lattice size is kept fixed.
This is depicted in fig.~5, where the averages of a few points with
lowest Higgs-boson mass are shown.
This figure also displays the strong finite size effects present
on these lattices: in the limit of infinitely large volumes the
minimum of $m_{R\sigma}$ is expected to be zero at the second order
phase transition between the FM-PM phases.
A plausible interpretation of fig.~5 is that the finite size effects
become stronger for larger $G_\psi$, because the renormalized
couplings become stronger.
In any case, the large values at the minima represent a difficulty
for the numerical simulations in the critical region, because large
lattices are needed.
Our experience shows (see also table~\ref{tb5}), that reasonably small
masses $m_{R\sigma} \simeq 0.5-0.7$ can be achieved at $G_\psi=0.3$ on
$6^3 \cdot 12$, at $G_\psi=0.6$ on $8^3 \cdot 16$ lattices.
Presumably at $G_\psi=1.0$ lattices with spatial extension of
at least $16^3$ are necessary.
In general, for a physical interpretation of the results in the broken
phase on a lattice of given size one should stay with $\kappa$ above
the value where $m_{R\sigma}$ takes its minimum.
This expectation is strengthened by the comparison of $6^3 \cdot 12$
and $8^3 \cdot 16$ results at ($\kappa=0.27,\, G_\psi=0.3$), which
are both in the broken phase, and within statistical errors show
no finite size effects of the renormalized couplings (see below).
The behaviour of the $\sigma$- (Higgs-boson) and $\pi$-
(Goldstone-boson) inverse propagators as a function of momentum is
shown at $G_\psi=0.6,\kappa=0.15$ in fig.~6.
The method of measurement is the same as in Ref.~\cite{BDFJT}.
The Goldstone- ($\pi$-) and $\sigma$-propagators in momentum space
are defined by
\begin{eqnarray}
\tilde{G}_{\pi}(p) &=& \left\langle \frac{1}{3L^3} \sum_{x,y}
\sum_{a=1}^{3} \pi_{ax} \pi_{ay}
\exp\{ i p \cdot (x-y) \} \right\rangle \; ,
\label{eq15} \\
\tilde{G}_{\sigma}(p) &=& \left\langle \frac{1}{L^3} \sum_{x,y}
\sigma_x \sigma_y
\exp\{ i p \cdot (x-y) \} \right\rangle \; .
\label{eq16}
\end{eqnarray}
In order to limit computer time and storage we actually measure
$\tilde{G}(p)$ for only one of multiple 4-momenta $p$ giving degenerate
$\hat{p}^2$ according to the assumption that, at least for small
momenta, $\tilde{G}(p)$ is just a function of $\hat{p}^2$.
The values are blocked during the MC runs with a block length
of typically 100 configurations.
The error bars are estimated with the jackknife method.
As one can see, in this point the inverse $\pi$-propagator extrapolates
reasonably well to zero for zero momentum.
The extrapolation of the inverse $\sigma$-propagator to zero gives
a Higgs-boson mass $m_{R\sigma}=0.98 \pm 0.05$, in good agreement with
the value obtained from a fit of the time-dependence of timeslices.
The curvature of the inverse propagators at this $(\kappa,G_\psi)$
value is not strong.
This allows a reasonably accurate determination of the renormalized
quantities by the formulae in \cite{LMMW}, assuming a linear
dependence between zero and the lowest nonzero momentum.
For larger values of the momentum the propagators are quite smooth,
therefore the effect of heavy $\psi$-fermion doublers is not strong.
Closer to the phase transition the curvature of the inverse propagators
near zero momentum becomes stronger, and the $\pi$-propagator starts
to show an increasingly nonzero mass.
We interpret the latter as a finite size effect.
Taking the Goldstone-boson field renormalization factor
$Z_\pi \equiv Z_T$ from the $\pi$-propagator, one can determine the
renormalized scalar vacuum expectation value
$v_R=\langle \sigma_x \rangle /\sqrt{Z_\pi}$, which in turn gives the
renormalized quartic- ($g_R$) and Yukawa-couplings ($G_{R\psi}$) by
\begin{equation} \label{eq17}
g_R \equiv \frac{3m_{R\sigma}^2}{v_R^2} \ , \hspace{4em}
G_{R\psi} \equiv \frac{\mu_{R\psi}}{v_R} \ .
\end{equation}
These are shown in figs.~7 and 8.
Although the errors are quite large, one can observe a strong
increase of $g_R$ for decreasing $\kappa$.
In the $\kappa$ region above the minimum of $m_{R\sigma}$ there is
much less variation.
The values on these ``plateaus'' show a moderate increase for
increasing $G_\psi$.
The renormalized Yukawa-coupling $G_{R\psi}$ is rather flat as a
function of $\kappa$, but increases definitely with $G_\psi$.
Considering only the points above the minimum of $m_{R\sigma}$ on the
given lattice size as being in the broken phase, one can make a
tentative first estimate of the upper bound on the renormalized
quartic coupling (or Higgs-boson mass), as a function of the
renormalized Yukawa-coupling (or fermion mass).
Such an estimate is shown by fig.~9 together with the perturbative
estimates based on the 1-loop $\beta$-functions.
The agreement with the perturbative results at $G_\psi=0.3$ and
0.6 is good, although the renormalized couplings are quite strong,
i.~e. close to the tree unitarity bound.
The $8^3 \cdot 16$ points have, unfortunately, larger statistical
errors.
At strong coupling the good agreement could partly be due to a fixed
point in the ratio $g_R/G_{R\psi}^2$, which implies that this ratio is
insensitive to the cut-off.
A few measured physical quantities in selected typical points are
collected in table \ref{tb5}.
Comparing the results at $G_\psi=0.6,\kappa=0.15$ on $4^3 \cdot 8$,
$6^3 \cdot 12$ and $8^3 \cdot 16$ lattices with label b, c and d,
respectively, one can see the evolution of the finite size effects.
Between c and d there is much less change than between b and c, but
point d on the $8^3 \cdot 16$ lattice still does somewhat differ
from the infinite volume limit.
This may explain why the corresponding point in fig.~9 is higher
than the $6^3 \cdot 12$ points at larger $\kappa$.
The situation is better if one compares points C and D, where the
deviation of the renormalized couplings is within statistical errors.
\begin{table}[tb]
\caption{ \label{tb5}
The main renormalized quantities and the bare magnetization
$\langle \sigma \rangle \equiv \langle|\varphi|\rangle$ for several
bare couplings $G_\psi$, and $\kappa$-values near the minimum scalar
mass attainable for the given lattice size. Points labelled by capital
letters are at $G_\psi=0.3$, whereas lower case and greek letters
denote data obtained for $G_\psi=0.6$ and $G_\psi=1.0$, respectively. }
\begin{center}
\begin{tabular}
{||c|r@{$\cdot$}l|r@{.}l|r@{.}l|r@{.}l|r@{.}l|r@{.}l|r@{(}l|r@{.}l||}
\hline
\hline
&$L^3$&$\,T$ & \multicolumn{2}{c|}{$\kappa$}
&\multicolumn{2}{c|}{$\langle\sigma\rangle$} & \multicolumn{2}{c|}{$v_R$}
&\multicolumn{2}{c|}{$m_{R\sigma}$} &\multicolumn{2}{c|}{$\mu_{R\psi}$}
&\multicolumn{2}{c|}{$g_R$} & \multicolumn{2}{c||}{$G_{R\psi}$} \\
\hline
A & $4^3$&$\, 8$ & 0&24 & 0&2807(15) & 0&307(4) & 1&23(1)
& 0&34(2) & 48&3) & 1&09(7) \\
B & $6^3$&$\,12$ & 0&24 & 0&146(6) & 0&18(1) & 0&73(5)
& 0&21(4) & 53&16) & 1&2(4) \\
C & $6^3$&$\,12$ & 0&27 & 0&303(3) & 0&309(13)& 0&80(5)
& 0&39(2) & 20&4) & 1&25(5) \\
D & $8^3$&$\,16$ & 0&27 & 0&270(2) & 0&25(1) & 0&77(3)
& 0&342(2)& 31&4) & 1&35(6) \\
E & $6^3$&$\,12$ & 0&30 & 0&4391(14) & 0&400(13)& 1&17(7)
& 0&55(2) & 26&7) & 1&36(6) \\
\hline
a & $8^3$&$\,16$ & 0&12 & 0&118(5) & 0&136(15)& 0&63(8)
& 0&61(2) & 80&50) & 4&5(3) \\
b & $4^3$&$\, 8$ & 0&15 & 0&3358(16) & 0&361(8) & 1&60(6)
& 1&9(3) & 59&7) & 4&9(5) \\
c & $6^3$&$\,12$ & 0&15 & 0&248(2) & 0&25(1) & 1&14(5)
& 0&67(6) & 63&10) & 2&7(3) \\
d & $8^3$&$\,16$ & 0&15 & 0&218(3) & 0&217(17)& 0&86(6)
& 0&54(4) & 52&9) & 2&5(3) \\
e & $6^3$&$\,12$ & 0&18 & 0&3524(18) & 0&36(2) & 1&23(8)
& 0&86(8) & 36&6) & 2&4(3) \\
f & $6^3$&$\,12$ & 0&21 & 0&4390(17) & 0&41(2) & 1&34(8)
& 1&11(3) & 32&5) & 2&71(13)\\
\hline
$\alpha$
& $6^3$&$\,12$ & $-0$&12 & 0&189(2) & 0&243(14)& 1&79(15)
& 0&95(9) & 180&40) & 3&9(4) \\
\hline
\hline
\end{tabular}
\end{center}
\end{table}
A qualitative relation of the masses at the strongest Yukawa-coupling
($G_\psi = 1.0$) is that the mass of the Higgs-boson $m_{R\sigma}$
is roughly twice as large as the fermion mass $\mu_{R\psi}$.
Because of finite size effects and limited statistics it cannot be
decided at present whether the $\sigma$-particle is a two-fermion bound
state or a resonance near threshold.
In the latter case, due to the fast decay into a fermion pair, the
physical Higgs-boson could become a very broad resonance.
An interesting question is the behaviour of the renormalized
Yukawa-coupling $G^{(3)}_{R\psi}$ defined by the 3-point vertex
function in (\ref{eq13}).
$G^{(3)}_{R\psi}$ is smaller than $G_{R\psi}$ in all points.
The measured values on $6^3 \cdot 12$ lattice are, for instance,
$G^{(3)}_{R\psi}=1.0 \pm 0.6$ at point c and
$G^{(3)}_{R\psi}=1.8 \pm 0.5$ at point $\alpha$.
Therefore the ratio
\begin{equation} \label{eq18}
S_3 \equiv \frac{G^{(3)}_{R\psi}}{G_{R\psi}}
\end{equation}
is smaller than 1.
On the $8^3 \cdot 16$ lattice, within our statistics
$G^{(3)}_{R\psi}$ turned out to be difficult to measure.
An exception is point D where we obtained
$G^{(3)}_{R\psi}=0.74 \pm 0.05$.
This is also smaller by about a factor of 2 than the corresponding
value of $G_{R\psi}$.
The deviation could partly be due to the nonzero momentum value
where $G^{(3)}_{R\psi}$ was extracted.
The measured values of $G_{R\chi}$ were always consistent with zero
within small errors, in agreement with the consequences of the
Golterman-Petcher relations.
\section{ Conclusions } \label{s5}
The important trends seen in our numerical data on $4^3 \cdot 8$,
$6^3 \cdot 12$ and $8^3 \cdot 16$ lattices are the following:
\begin{itemize}
\item
The phase structure at $(\lambda=\infty,\, K=K_{cr})$ is qualitatively
the same as in other lattice Yukawa-models with FM, PM, AFM and
FI phases (fig.~1).
\item
The FM-PM phase transition at $\lambda=\infty$ is smooth, probably of
second order (fig.~2).
\item
On most of our lattices there are strong finite size effects.
In particular, large lattices are needed in order to bring the minimum
of the Higgs-boson mass on a given lattice size down to interesting
values below 1.
As our simulations show, at moderate values of the bare
Yukawa-coupling $8^3 \cdot 16$ might be enough, but for large values
near $G_\psi \ge 1.0$ one will need at least presumably something like
$16^3 \cdot 32$.
\item
Considering only the $\kappa$-values above the minimum of the
Higgs-boson mass on a given lattice size, where in one point we also
have evidence that finite size effects are not very strong, we
obtained a first tentative estimate of the upper bound on the
renormalized quartic coupling as a function of the renormalized
Yukawa-coupling (fig.~9).
Up to renormalized Yukawa-couplings at the tree unitarity limit,
which is reached near $G_\psi=0.6$, this agrees well with 1-loop
perturbation theory, but further investigations are necessary in order
to check finite size effects and extend the results towards larger
$G_\psi$.
\item
At the strongest Yukawa-coupling $G_\psi=1.0$ the mass of the physical
Higgs-boson is roughly equal to twice the heavy fermion mass.
This could imply that the Higgs-boson is a very broad resonance,
which decays very fast into a heavy fermion pair.
\item
For a given lattice size the renormalized Yukawa-coupling $G_{R\psi}$
defined by the fermion mass increases more or less linearly with
$G_\psi$ up to $G_\psi=1.0$, where it becomes almost twice the tree
unitarity bound $\simeq 2.5$.
$G^{(3)}_{R\psi}$ defined by the 3-point vertex function is smaller
than $G_{R\psi}$ on the $6^3 \cdot 12$ and $8^3 \cdot 16$ lattices.
As discussed above, the finite size effects are particularly strong
for $G_\psi > 0.6$, therefore large lattices are needed for
confirmation of the values of $G_{R\psi}$.
\end{itemize}
The question of the possible influence of heavy fermions in the
Standard Model is important and very interesting.
By numerical simulations at $\lambda=\infty$ one can obtain
information on the upper limit on the Higgs-boson mass.
The extension to smaller values of the bare quartic coupling, in
particular to $\lambda \simeq 0$ gives a lower bound related to
vacuum stability.
{\em Note added:} In writing this paper we received a recent preprint
of Bock, Smit and Vink, where the same continuum ``target'' theory
as ours has been numerically investigated in a staggered fermion
formulation \cite{BOSMVI}.
\vspace{1cm}
\large\bf Acknowledgements \normalsize\rm \newline
\vspace{3pt}
\noindent
We thank Jiri Jers\'ak for active help and useful discussions.
The Monte Carlo calculations for this work have been performed
on the CRAY Y-MP/832 of HLRZ J\"ulich and the S-600 of RWTH Aachen.
\newpage
|
1,116,691,500,831 | arxiv | \section{Introduction\label{sec:1}}
Nonparametric Bayesian methods like the Dirichlet process~\citep{ferguson} and the Gaussian process~\citep{rasmussen} allow practitioners to specify flexible priors over infinite-dimensional objects like functions and probability densities.
These have seen wide success in applied disciplines like biostatistics~\citep{dunson}, document modeling~\citep{teh} and image modeling~\citep{sudderth}, with an accompanying rich literature on theoretical properties and computational strategies.
The flexibility of these models
however often comes at the cost of interpretibility, and they can be quite challenging to elicit from applied scientists.
Even more problematic is that such flexible priors imply statistical properties of the objects of interest that are incompatible with expert knowledge. This expert knowledge is often quantified by probability distributions over functionals of the infinite-dimensional objects of interest.
The challenge is then to specify nonparametric Bayesian priors subject to constraints on the distribution of such functionals.
This problem can be quite general~\citep{kessler}, and in this work, we focus on a specific setting.
Consider observations lying in a $d$-dimensional Euclidean space $\Re^d$, with the observations drawn i.i.d.\ from an unknown probability density.
We model this density, an infinite-dimensional object, with a nonparametric Bayesian prior, but now wish to constrain the marginal distribution of a subset of the coordinates of the observations.
{\Cref{fig:marginal constraint graphic interpretation} presents a graphical illustration of this problem. The two-dimensional contour plot represents a complex probability density $p(X_1,X_2)$, drawn from a nonparametric prior. The one-dimensional densities to the top and right, plotted with broken lines, show the corresponding marginal densities of $X_1$ and $X_2$ respectively.
If the first component $X_1$ is known to follow some known density (e.g. a Gaussian, shown with the continuous curve), then it is important to modify the prior to satisfy this, while still remaining flexible about the rest of the density.}
\begin{figure}
\centerline{
\includegraphics[width=0.8\textwidth]{Untitle21copy102_1.pdf}
}
\caption{Contour plot of a two-dimensional density $p(X_1, X_2)$ drawn from a nonparametric prior. Right is the corresponding marginal density of $X_2$, while top is the corresponding marginal density of $X_1$, along with a target constraint $p(X_1)$. \label{fig:marginal constraint graphic interpretation}}
\end{figure}
As a motivating example, following~\citet{schifeling}, we consider the 2012 American Community Survey Public Use Microdata Sample of North Carolina survey data (ACS PUM).
This dataset, obtained from the United States Census Bureau's website (\url{https://data.census.gov/mdat/#/}), comprises features like gender, age, and educational attainment.
The marginal distribution of age in the population may already be known empirically from external census data sources and, due to sampling effects, may differ from the empirical distribution of age in the survey data.
Modeling this dataset with an off-the-shelf nonparametric model will imply a marginal distribution over age that also differs from the prior knowledge.
Incorporating the marginal distribution of age into the off-the-shelf prior can result in more accurate inferences, as predictive datasets from the model will satisfy
the marginal distribution of age group in the population while {preserving} the dependence structure in the original multivariate data.
More broadly, such an approach is also useful in a variety of modern tasks in statistics and machine learning.
A topic of increasing interest in machine learning is the problem of distribution shift, when the target distribution does not align with the underlying distribution of the training data.
\citet{dai2022marginal} proposed an approach for such a setting, but for discrete set modeling
(see~\cref{sec:related}).
Similarly, concerns about fairness and privacy might make it preferable to introduce simplifying constraints on certain variables, rather than modeling the observed data as accurately as possible.
\section{Related work} \label{sec:related}
An important step towards incorporating marginal constraints into nonparametric models, motivated by the ACS PUM survey dataset mentioned earlier, is the work of~\citet{schifeling}.
Here, the authors used a Dirichlet process mixture of products of multinomials to model a dataset of discrete values.
To enforce marginal constraints on a subset of the coordinates, the authors proposed a {\em hypothetical records augmentation method}, which essentially amounts simulating values of these coordinates from the specified marginal distribution.
These simulated values serve as an auxiliary dataset, which when combined with the original dataset, guide results towards respecting the marginal constraint.
This data-augmentation approach, while conceptually simple, has a number of limitations.
First, it only approximately enforces the marginal constraint, with the constraint enforced exactly only in the limit as the number of augmenting datapoints tends to infinity.
The authors do not provide any clear guidelines about choosing the size of the augmenting dataset.
Since the unconstrained coordinates are missing on the augmenting dataset, posterior inference involves additional complexity, and the scheme in~\citet{schifeling} is tailored only to categorial data.
Finally, to simulate the augmenting dataset, the marginal density must be known exactly.
In many settings, one only wishes to constrain the parametric family the marginal distribution belongs to, with its parameters themselves learned from the data.
For instance, prior knowledge might suggest that interarrival times in a queue follows an exponential distribution, with the actual arrival rate unknown.
In such a setting,
one cannot directly apply the methodology from~\citet{schifeling}.
Our proposed approach links the problem of enforcing marginal constraints to the problem of nonparametric conditional density modeling~\citep{dunson2008kernel,chung2009nonparametric,pati2013posterior,ghosh2010bayesian,tokdar2011dimension}.
Rather than indirectly induce a marginal constraint on a subset of variables using an auxiliary dataset, we propose to directly model that subset using the specified marginal distribution, placing a prior on any unknown parameters.
We then place a nonparametric prior on the conditional density of the remaining variables, with the resulting joint distribution 1) satisfying the marginal constraint {\em by construction}, and 2) inheriting the large support and flexibility of the nonparamtric prior.
Our contribution in this paper can thus also be viewed as a novel model for nonparametric conditional density modeling, with an associated novel MCMC sampling algorithm.
As we outline below, there already exist a number of approaches to nonparametric conditional density modeling in the literature.
We note that when the marginal constraint is known exactly, our problem reduces to a conditional density modeling problem, and any of these existing methods can be used.
When only the parametric form of the marginal constraint is known, then our model affords a little more flexibility by allowing the conditional distribution to vary with the marginal.
As an additional benefit, by building on the work of~\citet{adams}, our model has an associated {\em exact} MCMC sampling with no asymptotic bias, something that is lacking for most existing conditional density models.
There is a rich and growing literature on nonparametric conditional density modeling, with a large number of approaches based on {predictor-dependent} stick-breaking process priors. {To estimate the conditional density of a response variable $Y$, these use the mixture specification
$p(Y|X)=\int g(Y|X, \psi) dG_X(\psi)$, with known parametric densities $g(Y|X,\psi)$.
The random probability measure $G_X$ is a predictor-dependent mixture
distribution taking the form
$$G_X=\sum_{h=1}^\infty \pi_h(X)\delta_{\psi_h},\; \psi_h\sim G_0,$$ where $G_0$ is a basis measure and $\pi_h (X)$ is a predictor-dependent probability weight constructed from a predictor-dependent stick-breaking process$$\pi_h(X)=V_h(X)\prod_{s<h}\{1-V_s(X)\}$$ with $V_h$
confined to the unit interval for all $X$. Different construction choices of $V_h$ have been proposed, among others works, in~\citet{dunson2008kernel} and \citet{chung2009nonparametric}.}
While well understood theoretically\citep{pati2013posterior}, computation with these models typically requires truncating the number of mixture components to some finite number, or implementing involved slice-sampling algorithms.
An alternative approach based on logistic Gaussian process is introduced in~\citet{ghosh2010bayesian,tokdar2011dimension}.
The authors extend logistic Gaussian process priors originally introduced and
studied by~\citet{lenk1988logistic,lenk1991towards,leonard1978density} to model a nonparametric conditional density. Again however, exact posterior computation with logistic Gaussian process priors is a difficult problem as indicated in~\citet{ghosh2010bayesian, tokdar2007towards}. To avoid this issue, we extend the sigmoid Gaussian process prior introduced in~\citet{adams} for modeling a nonparametric conditional density, and show how one can carry out exact posterior inference when this class of models is extended to modeling conditional distributions.
In~\citet{kessler}, the authors considered a version of the broader problem stated at the start of this paper: given a prior $\Pi_0(\theta)$ on some joint space $\Theta$, and a marginal distribution $p_1$ on some functional $f$ of $\theta$, what is the probability measure `closest' to $\Pi_0$ that satisfies the marginal constraint.
Using the Kullback-Leibler divergence as the measure of closeness, and writing
$\mathcal{H}$ for the set of probability measures on $\Theta$ with $f-$marginal distribution $p_1$, the authors arrived at the following solution:
\begin{align}
\Pi_1(\theta) &:= \mathop{\rm argmin}_{\Pi \in \mathcal{H}}\text{KL}(\Pi(\theta)|\Pi_0(\theta)) =\Pi_0(\theta|f)p_1(f). \label{eq:kessler}
\end{align}
The form of this solution allowed the authors to carry out posterior sampling via a small modification to an MCMC algorithm for the unconstrained prior $\Pi_0(\theta)$.
While conceptually simple, this correction involves computing an intractable marginal, and in practice, this algorithm must be run as an approximate one, requiring a kernel density estimate of this marginal probability in order to calculate the acceptance probability.
We can attempt to cast our problem as a special instance of this general scheme, where $\Theta$ is the space of densities on $\Re^d$ and $f$ projects $\theta$ onto a density on a subset of these coordinates.
Note though that with a known marginal constraint, the projected variable is distributed as a Dirac delta function.
As a consequence of this hard constraint,
the earlier MCMC scheme will make proposals attempting to hit a submanifold of measure 0, resulting in an acceptance probability of $0$.
Our proposed solution nevertheless builds on the form given in~\cref{eq:kessler}, and links the problem of marginal constraints to that of nonparametric conditional density modeling.
{As mentioned at the end of~\cref{sec:related}, \citet{dai2022marginal} consider a similar problem is the machine learning setting of marginal distribution shift. Their approach is restricted to discrete sets, and different from our problem, the authors try to learn a generative model to approximate the distribution of a random discrete {\em set},
given information about element marginals, i.e., the occurrence frequency of particular elements.
Their goal is to efficiently adapt a previously learned generative model without constraints to respect the element marginals,
without having to retrain the entire generative model from scratch.}
\section{Problem statement and proposed approach}
In the following, we denote random variables with uppercase letters and their values with lowercase letters.
We seek to model a dataset $\mathbf{X} = (X^1,\dotsc,X^n)$ comprising $n$ observations, each of dimension $d$, with the $i$th observation written as $X^i = (X^i_1,\dotsc, X^i_d)$.
As this notation indicates, we use superscripts to index individual observations in the dataset, and subscripts to index covariates.
For an increasing, ordered subset $A \subset \{1,\dotsc,d\}$, we will use $X_A$ to refer the subvector $(X_j)_{j \in A}$ of $X$, and write $X_{A^c}=(X_j)_{j \notin A}$ for the complement of the components $X_A$.
We model the observations as independent and identical draws from some unknown probability density $p$, subject to the following constraint: the subset $X_A$ follows a known marginal distribution $p_A$.
This marginal constraint on $X_A$, obtained from some external source, can take the form of a completely specified distribution (e.g.\ $p_A$ might be the standard normal distribution $N(0,1)$).
More generally, domain knowledge about the distribution of $X_A$ might take the form of some parametric distribution with the parameter value unknown, for example, $X_A$ might be marginally distributed as a Gaussian with unknown mean and variance.
We refer to the former as a {\em distribution constraint}, and the latter as a {\em family constraint}.
The latter arises in fields like queuing theory or genetics, where structural knowledge like memorylessness or independence can result in distributions like the Poisson, Gaussian or exponential.
In the first real example presented in the~\cref{sec:3}, the modeler knows that concentrations of pollutants follow a lognormal distribution, with the mean and variance unknown.
Note that the hypothetical records approach of~\citet{schifeling} no longer works, since know we no longer have a specific distribution to generate auxiliary samples.
Taking a nonparametric Bayesian approach, the problem now is to place a nonparametric Bayesian prior on the unknown density $p$, while ensuring that the induced marginal distribution of $X_A$ is consistent with this side knowledge.
For the family constraint, one must also place a prior on the unknown parameters.
\subsection{ Proposed nonparametric Bayesian Model}
\label{section:newmodel}
Our modeling approach, which can also be viewed as a contribution to the literature on nonparametric conditional density modeling, proceeds as follows.
Inspired by~\cref{eq:kessler}, we factor the joint probability as $p(X_A,X_{A^c}) = p_A(X_A)p(X_{A^c}|X_A)$.
We then directly model the subset $X_A$ of variables according to their specified marginal distribution,
and then place a flexible nonparametric prior on the conditional density of the remaining variables given a realization of $X_A$.
Specifically, with $p_A(\;\cdot\;|\;\phi)$ the known marginal constraint (parametrized by $\phi$), we model $X_A$ as
\begin{align}
X_A \sim p_A(\;\cdot\;|\;\phi), \qquad \phi \sim p_\phi(\cdot).
\end{align}
When the prior $p_\phi$ on $\phi$ is a Dirac delta (that is, $\phi$ is known), we are in the distributional constraint setting.
For the family constraint setting, the prior $p_\phi$ on $\phi$ might either reflect domain knowledge about $\phi$, or can be a weakly informative or uninformative distribution.
We now have to specify a prior on the conditional distribution of $X_{A^c}$ given $X_A$.
As outlined earlier, there exist a number of approaches in the literature to do this, though posterior inference for these typically involve discretization or truncation approximations.
After specifying our prior below, we will show in the next section how it is possible to carry out {\em exact} MCMC inference for this model.
We next introduce some more notation. Let $\pi_0(X_{A^c}|X_A, \theta, \phi)$ be a simple conditional distribution that is strictly positive, continuous, and parameterized by $\theta$ and $\phi$.
This will serve as a {\em centering distribution}.
We will model the true conditional distribution of $X_{A^c}$ given $X_A$ as a perturbation of this centering distribution, with a transformed Gaussian process (GP)~\citep{rasmussen} prior on the perturbation function.
Specifically, let $\sigma(x) = 1/(1+\exp(-x))$ be the sigmoid function, and let $\lambda(\cdot)$ be a realization of a Gaussian process with mean function $\mu(\cdot)$ and covariance kernel $ k(\cdot,\cdot)$.
Then, we set the conditional distribution $p(X_{A^c}|X_A,\theta,\phi) \propto \pi_0(X_{A^c}|X_A, \theta,\phi)\sigma(\lambda(X_A,X_{A^c})) $.
Recall that $\lambda$ being a Gaussian process implies that for every finite subset of random vectors $z_1, \dots, z_m \in \Re^d$, the corresponding vector $\lambda(z_1),\dotsc,\lambda(z_m)$ follows a Gaussian distribution as below, see~\citet{rasmussen} for more details:
\begin{align*}
\begin{pmatrix}
\lambda(z_1) \\
\vdots\\
\lambda(z_m) \\
\end{pmatrix}\sim \mathcal{N}\left(\begin{pmatrix}
\mu(z_1) \\
\vdots\\
\mu(z_m) \\
\end{pmatrix},\begin{pmatrix}
k(z_1, z_1) & \dots & k(z_1, z_m) \\
&\ddots&\\
k(z_m, z_1) &\dots & k(z_m, z_m) \\
\end{pmatrix}\right).
\end{align*}
The Gaussian property above shows that sample paths of a GP can take negative values, and the sigmoid transformation serves as a {\em link function} to keep $p(X_{A^c}|X_A,\theta,\phi)$ positive.
There are a number of other choices for the link function in the literature, the most popular being the exponential function in~{\citet{tokdar2007towards}}.
However, we show that along these lines of~\citet{adams}, the use of the sigmoid function allows us to sample exactly from this model.
Following~\citet{rao}, we can carry out exact MCMC inference without any approximation error.
First, we write down the overall model below:
\begin{align}
\theta &\sim p_\theta(\cdot), \quad \phi \sim p_\phi(\cdot), \quad \lambda(\cdot) \sim \mathcal{GP}(\mu(\cdot), k(\cdot, \cdot)) \label{eq:eqn9}\\
X_A &\sim p_A(\;\cdot\;|\;\phi)\label{eq:eqn10}\\
X_{A^c}|X_A & \sim \frac{\pi_0(X_{A^c}|X_A,\theta, \phi)\sigma(\lambda(X_A, X_{A^c}))}{\int \pi_0(X_{A^c}|X_A,\theta,\phi)\sigma(\lambda(X_A,X_{A^c})) dX_{A^c}}.
\label{eq:eqn11}
\end{align}
The Gaussian process is a well studied nonparametric prior on functions, and can be shown to possess desirable large support properties~\citep{choudhuri2007nonparametric}.
By modeling the modulation function with a GP prior, one can expect the resulting conditional density also to inherit similar large support properties. We show this below.
\subsection{Large support and consistency properties}
Assume that $\mathcal{X}_A\times \mathcal{X}_{A^c}$ is a compact subset of {$\Re^d$}.
For simplicity, we assume that the parameters $\theta$ and $\phi$ are fixed and known, and drop them from all notation, writing the centering distribution and the marginal constraining distribution simply as $\pi_0(\cdot|\cdot)$ and $p_A(\cdot)$.
Write $\mathcal{F}$ for the space of all joint densities on $\mathcal{X}_A\times \mathcal{X}_{A^c}$ with {their corresponding} marginal densities on $\mathcal{X}_A$ equal to $p_A(\cdot)$, and conditional densities jointly continuous on $\mathcal{X}_A\times \mathcal{X}_{A^c}$.
The Gaussian process prior on $\lambda$ induces a prior $\Pi$ on $\mathcal{F}$ through the map {from $\lambda$ to $f_\lambda$}
\begin{align}
&\lambda\rightarrow f_\lambda(X_A, X_{A^c})=p_A(X_A)\cdot\cfrac{\pi_0(X_{A^c}| X_A)\sigma(\lambda(X_A,X_{A^c}))}{\int_{\mathcal{X}_{A^c}} \pi_0(X_{A^c}| X_A)\sigma(\lambda(X_A,X_{A^c}))dX_{A^c}}.\label{eq:map}
\end{align}
In~\cref{thm:largesupport}, we prove a `large-support' property of the conditional density in~\cref{eq:eqn11}, showing that any density function in $\mathcal{F}$ is in the KL support of the induced prior $\Pi$.
First, we state a key intermediate result from~\citet{ghosal2006posterior} that we will use to prove the theorem:
\begin{restatable}{lemma}{xyz}{[Theorem 4 in \citet{ghosal2006posterior}]}
Assume that $\lambda(X_A,X_{A^c})$ on the compact index set $\mathcal{X}_A \times \mathcal{X}_{A^c}$ is a Gaussian process (GP) with continuous sample paths.
Assume that the GP mean function and a function $\lambda_0(X_A, X_{A^c})$ on $\mathcal{X}_A \times \mathcal{X}_{A^c}$ belong to the RKHS of the covariance kernel of the Gaussian process. Then
\begin{align*}
P(\lambda:\sup_{\{X_A\in \mathcal{X}_A,\, X_{A^c} \in \mathcal{X}_{A^c}\}}|\lambda(X_A, X_{A^c})-\lambda_0(X_A,X_{A^c})|<\delta)>0\;\; \forall \delta>0.
\end{align*}\label{lm:gaussian process property1}
\end{restatable}
We can then state our result, which we prove in Appendix A~\ref{appendix A}.
\begin{restatable}{theorem}{abc}
Suppose the Gaussian process
on the compact space $\mathcal{X}_A\times\mathcal{X}_{A^c}$ satisfies the assumptions in~\cref{lm:gaussian process property1}.
Assume that its mean function is continuous and that the RKHS associated with its covariance kernel equals the set of all continuous functions on $\mathcal{X}_A\times\mathcal{X}_{A^c}$.
Also assume $\pi_0(X_{A^c}|X_A)$ and $p_A(X_A)$ in the map of equation~\ref{eq:map} are strictly positive on $\mathcal{X}_A\times\mathcal{X}_{A^c}$ and $\mathcal{X}_A$, and additionally $\pi_0(X_{A^c}|X_A)$ is continuous on $\mathcal{X}_A\times\mathcal{X}_{A^c}$. Then any density function belonging to $\mathcal{F}$ is in the KL support of $\Pi$.\label{thm:largesupport}
\end{restatable}
We remark that for many covariance kernels, the associated RKHS equals the set of all continuous functions, see theorems 4.3-4.5 of \citet{tokdar2007posterior}.
If a kernel on a one-dimensional space can be written as $k(s,t)=\psi(s-t)$ for some nonzero, continuous density function $\psi$, then \citet{tokdar2007posterior} showed its RKHS is the set of continuous functions.
For spaces with dimension larger than 1, if the covariance kernel is the Kronecker product of one-dimensional kernels, with each one-dimensional kernel having the set of all continuous functions as its RKHS, then
\citet{tokdar2007posterior} showed that the RKHS of the covariance kernel also equals the set of all continuous functions.
We finish this section with a well-known result from~\citet{schwartz1965bayes}, showing that the large support property of the prior distribution from~\cref{thm:largesupport} translates to asymptotic consistency of the posterior distribution.
Specifically, assume the product of $p_A(X_A)$ and the true conditional density $q(X_{A^c}|X_A)$ lies in $\mathcal{F}$, then the posterior $\Pi(f_\lambda|(X_A^1,X_{A^c}^1),\dots,(X_A^n,X_{A^c}^n))$ will be weakly consistent at the joint density $p_A(X_A)\cdot q(X_{A^c}|X_A)$ as $n\rightarrow \infty$.
\begin{restatable}{theorem}{ef}
(\citet{schwartz1965bayes})
If the true density $f_0$ is in the Kullback-Leibler support of $\Pi$, then the posterior is weakly consistent at $f_0$. \label{thm:weak_cons}
\end{restatable}
\subsection{Exact prior simulation}
We next explain how to draw samples from this model. This step, useful in itself, is also key to our MCMC algorithm for posterior simulation.
Along the lines of~\citet{adams}, we exploit the fact that the logistic function $\sigma(x)$ satisfies $\sigma(x) \le 1$, so that
$\pi_0(X_{A^c}|X_A; \theta,\phi) \sigma(\lambda(X_A, X_{A^c})) \le \pi_0(X_{A^c}|X_A; \theta,\phi)$. We note that this bounding property does not hold for most typical link functions used in the literature.
The bounding property allows us to generate samples from the nonparametric conditional through a simple rejection sampling scheme: first propose from $ \pi_0(X_{A^c}|X_A; \theta,\phi)$ and then accept or reject with probability $\sigma(\lambda(X_A,X_{A^c}))$.
The accepted samples form a realization from the probability density proportional to $\pi_0(X_{A^c}|X_A; \theta,\phi)\sigma(\lambda(X_A, X_{A^c}))$.
Crucially, generating a dataset of $n$ observations in such a fashion only requires evaluating the Gaussian process at a finite set of points: the locations of the data points as well as the rejected proposals that were produced along the way.
Since the values of a GP on a finite set of {points} follow a multivariate {normal} distribution, these can easily be sequentially sampled from the corresponding conditional distributions.
Importantly, this does not require integrating the transformed GP as in the denominator of~\cref{eq:eqn11}.
We describe this in detail in~\cref{algo:4}.
\begin{algorithm}
\KwIn{the marginal density family $p_A(\cdot)$, a tractable conditional density family $\pi_0(\cdot|\cdot)$, priors $p_\phi(\cdot)$ and $p_\theta(\cdot)$ on $\phi$ and $\theta$, the Gaussian process mean and covariance functions $\mu(\cdot)$ and $k(\cdot,\cdot)$, the sample size $n$.}
\KwOut{Multivariate samples $(X_{A}^1, X_{A^c}^1), \dots, (X_{A}^n, X_{A^c}^n)$ with $X^i_A\sim p_A(\cdot)$.}
Set $\lambda_{pre}$ as null and $S=\{1,2,\dots,n\}$.
Sample
$\phi\sim p_\phi(\cdot)$ and $\theta\sim p_\theta(\cdot)$.
Sample $X_{A}^1,\dots, X_{A}^n \stackrel{i.i.d}{\sim} p_A(\cdot|\phi)$.
\SetAlgoLined\SetArgSty{}
\Repeat{$S$ is null}
{
Sample $y^{i} \sim \pi_0(\cdot|X_{A}^i; \theta,\phi)$ $\forall i\in S$, and denote $T=\{(X_{A}^i, y^{i}),i \in S\}$;
Sample $\lambda_T|\lambda_{pre}$ conditionally from the Gaussian process, and add $\lambda_T$ into $\lambda_{pre}$;
Set $\alpha^i=1$ with probability $\sigma(\lambda(X_{A}^i, y^{i}))$ $\forall i \in S$;
Set $X_{A^c}^i \gets y^{i}$ and delete index $i$ from $S$ if $\alpha^i=1 \,\forall i \in S$.
}
\caption{Algorithm to generate prior samples from the proposed model (Equations~\eqref{eq:eqn9} -~\eqref{eq:eqn11})}
\label{algo:4}
\end{algorithm}
\subsubsection{Placing a prior distribution on { $\theta$ and $\phi$:} the parameters of the centering distribution {$\pi_0(\;\cdot\;|\;\cdot\;;\theta, \phi)$}}\label{section:prior on conditional}
As the results in~\cref{thm:largesupport} and~\cref{thm:weak_cons} show, it is sufficient for the centering density $\pi_0(\;\cdot\;|\;\cdot\;;\theta, \phi)$ to be a strictly positive, {continuous} conditional distribution for our model to possess desirable asymptotic properties.
Even so, a poor choice of $\pi_0(\;\cdot\;|\;\cdot\;;\theta, \phi)$ can impact the efficiency of~\cref{algo:3}.
Specifically, if there exists a region of space that has high probability under the true density, but low probability under the centering distribution, then it will take a large number of rejected proposals before a sample is finally accepted.
While ultimately avoiding such issues requires a careful choice of the centering distribution, in our experiments, we found that the additional flexibility gained from allowing a location and/or scale parameter of the centering distribution to vary significantly improves performance.
Accordingly, we place {priors $p_\theta(\cdot)$ and $p_\phi(\cdot)$} on the parameters of $\pi_0(\;\cdot\;|\;\cdot\;;\theta, \phi)$.
\section{Posterior inference}\label{section:posterior computation}
Having specified the model, we now move to the problem of posterior computation: given a dataset $(X^1,\dotsc,X^n)$ of observations from an unknown density, if we assume the data-generating density lies in $\mathcal{F}$ and model it using our marginally constrained nonparametric prior, how do we characterize resulting the posterior distribution?
We take a Markov chain Monte Carlo (MCMC) approach, devising a Markov chain with this posterior distribution as its stationary distribution.
The unknown latent variables of interest in the model specified above are the GP-distributed function $\lambda$, and any unknown parameters $\theta$ and $\phi$ of the centering distribution and the marginal constraining distribution respectively.
For simplicity, we ignore any unknown hyperparameters of the kernel of the Gaussian process; these can easily be simulated given realizations of $\lambda$.
The posterior distribution for $(\lambda,\phi,\theta)$ is
\begin{align}
p(\lambda,\phi, \theta|X^1,\dots,X^n)
&\propto p_\phi(\phi)\cdot p_\theta(\theta)\cdot \mathcal{GP}(\lambda)\cdot \prod_{i=1}^n p(X^i|\lambda,\phi,\theta)\nonumber\\
&\propto p_\phi(\phi) \cdot p_\theta(\theta)\cdot \mathcal{GP}(\lambda)\cdot\prod_{i=1}^n p_A(X_A^i|\phi)\cdot\prod_{i=1}^n \frac{\pi_0(X_{A^c}^i|X_A^i,\theta, \phi)\sigma(\lambda(X_{A}^i, X_{A^c}^i))}{Z(\lambda, \theta, \phi, X_{A}^i)}
\label{eq:eqn121}
\end{align}
where $\mathcal{GP}(\lambda)$ denotes the Gaussian process prior and $Z(\lambda, \theta, \phi, X_{A}^i)=\int \pi_0(X_{A^c}^i|X_{A}^i,\theta, \phi)\sigma(\lambda(X_{A}^i, X_{A^c}^i)) dX_{A^c}^i$.
Note that
the dependence of the intractable denominator $Z(\lambda, \theta, \phi, X_{A}^i)$ on $\theta$, $\phi$ and $\lambda$ makes their posterior distribution an example of a {\em doubly intractable probability distribution}~\citep{murray2012mcmc}, so that standard MCMC methods cannot directly be used.
Specifically, for a Metropolis-Hastings algorithm that proposes new parameters $\theta^*$, $\phi^*$ and a new function $\lambda^*$, the acceptance probability involves the ratios $Z(\lambda, \theta, \phi, X_A^i)/Z(\lambda^*, \theta^*, \phi^*, X_A^i)$, something that is clearly impossible to calculate.
To solve this, we follow a data augmentation scheme based on an approach proposed in~\citet{rao}.
At a high level, our approach is to also instantiate the rejected proposals from $\pi_0$. Write $\mathcal{Y}^i=\{y^{i1}, \dots, y^{i\left|\mathcal{Y}^i\right|}\}$ as the set of rejected samples preceding the $i$th observation $(X^i_A,X^i_{A^c})$.
Instead of computing $p(\lambda,\phi, \theta|(X_{A}^1, X_{A^c}^1), \dots, (X_{A}^n, X_{A^c}^n))$, we simulate from $p(\lambda,\phi,\theta,\mathcal{Y}^1, \dots, \mathcal{Y}^n| (X_{A}^1, X_{A^c}^1), \dots, (X_{A}^n, X_{A^c}^n))$.
Observe that the latter has the former as its marginal distribution, so that having produced samples from the latter, we can just discard the rejected samples $(\mathcal{Y}^1,\dotsc,\mathcal{Y}^n)$.
Importantly, we will see that given the rejected samples $(\mathcal{Y}^1,\dotsc,\mathcal{Y}^n)$, the variables $\lambda$, $\phi$ and $\theta$ can be updated using standard MCMC techniques.
Our overall approach to produce samples from $p(\lambda,\phi,\theta,\mathcal{Y}^1, \dots, \mathcal{Y}^n| (X_{A}^1, X_{A^c}^1), \dots, (X_{A}^n, X_{A^c}^n))$ is a Gibbs sampling algorithm described in ~\cref{algo:3}, involving three main steps:
\begin{description}
\item[Sample $\mathcal{Y}^1, \dots, \mathcal{Y}^n \,|\, \lambda,\phi, \theta, (X_{A}^1, X_{A^c}^1) \dots, (X_{A}^n, X_{A^c}^n)$:]
Note that under our model, the number of rejected samples preceding each observation is a random quantity following a geometric distribution, whose success probability equals the acceptance probability.
Noting that the acceptance probability equals $\int \pi_0(y|X_A^i;\theta,\phi)\sigma(\lambda(y,X_A^i))dy$ and is intractable, we avoid evaluating this by directly simulating the number and values of the rejected samples preceding an observation.
To do so, we just simulate a new observation following~\cref{algo:4}, discard the accepted sample and keep the rejected samples.
The validity of this scheme was proved in~\citet{rao} who showed that crucially, these quantities are independent of the value of the accepted sample.
\item[Sample $\lambda\, |\,\phi,\theta, \mathcal{Y}^1, \dots, \mathcal{Y}^n, (X_{A}^1, X_{A^c}^1) \dots, (X_{A}^n, X_{A^c}^n)$:]
This conditional distribution no longer involves any intractable normalizers:
\begin{align} p(\lambda|\,\phi,\theta, \mathcal{Y}^1, \dots, \mathcal{Y}^n, (X_{A}^1, X_{A^c}^1), \dots, (X_{A}^n, X_{A^c}^n)) \propto p(\lambda) \cdot \prod_{i=1}^n \left\{\sigma(\lambda(X_{A}^i, X_{A^c}^i))\prod_{j=1}^{|\mathcal{Y}^i|}\left[1-\sigma(\lambda(X_{A}^i, y^{ij}))\right]\right\}.\label{eq:lambda}
\end{align}
Effectively, $\sigma(\lambda)$ serves as a classification function to separate accepted and rejected proposals. We can generate a new random function $\lambda^*$ by a standard MCMC methods for GPs, for example, elliptical slice sampling~\citep{murray2010elliptical} or Hamiltonian Monte Carlo~\citep{neal2011mcmc}.
\item[Sample $\theta,\phi|\,\lambda, \mathcal{Y}^1, \dots, \mathcal{Y}^n, (X_{A}^1, X_{A^c}^1) \dots, (X_{A}^n, X_{A^c}^n)$:] Given all other variables, $\theta$ and $\phi$ are dependent on each other, and follow distributions
\begin{align}
p(\phi\,|\,\theta,\mathcal{Y}^1, \dots, \mathcal{Y}^n, (X_{A}^1, X_{A^c}^1), \dots, (X_{A}^n, X_{A^c}^n)) &\propto p_\phi(\phi)\cdot \prod_{i=1}^n p(X_A^i|\phi) \cdot \prod_{i=1}^n \left\{\pi_0(X_{A^c}^i|X_A^i,\theta,\phi)\prod_{j=1}^{|\mathcal{Y}^i|}\pi_0(y^{ij}|X_A^i,\theta,\phi)\right\}, \label{eq:phi} \\
p(\theta\,|\,\phi, \mathcal{Y}^1, \dots, \mathcal{Y}^n, (X_{A}^1, X_{A^c}^1), \dots, (X_{A}^n, X_{A^c}^n)) &\propto p_\theta(\theta) \cdot \prod_{i=1}^n \left\{\pi_0(X_{A^c}^i|X_A^i,\theta, \phi)\prod_{j=1}^{|\mathcal{Y}^i|}\pi_0(y^{ij}|X_A^i,\theta,\phi)\right\}.\label{eq:theta}
\end{align}
These can be simulated using standard MCMC techniques such as Metropolis-Hastings, Hamiltonian Monte Carlo or slice sampling.
A simplifying assumption is to make the centering distribution $\pi_0$ independent of the parameter $\phi$.
While we do not make this assumption, it does hold in the setting of distribution constraints where $\phi$ is fixed.
In such instances, it might be possible to choose a prior $p_\phi(\cdot)$ that is conjugate to the constraint family and a prior $p_\theta(\cdot)$ conjugate to the centering distribution.
Then, given the rejected samples, the posterior belongs to the same family as the prior and is typically easy to sample from.
\end{description}
\begin{algorithm}[H]
\KwIn{The observations $\mathcal{X}=\left\{(X_{A}^1, X_{A^c}^1), \dots, (X_{A}^n, X_{A^c}^n)\right\}$, set of rejected samples $\Tilde{\mathcal{Y}}=\{(X_{A}^1, \Tilde{y}^{11}),\dots,(X_{A}^1, \Tilde{y}^{1|\Tilde{\mathcal{Y}}^1|}),\dots, (X_{A}^n,\Tilde{y}^{n1}),\dots,(X_{A}^n,\Tilde{y}^{n|\Tilde{\mathcal{Y}}^n|}) \}$, where $\Tilde{\mathcal{Y}}^1=\{\Tilde{y}^{11},\dots, \Tilde{y}^{1|\Tilde{\mathcal{Y}}^1|}\},\dots, \Tilde{\mathcal{Y}}^n=\{\Tilde{y}^{n1},\dots, \Tilde{y}^{n|\Tilde{\mathcal{Y}}^n|}\}$, the current random function values $\lambda_{\mathcal{X}\cup \Tilde{\mathcal{Y}}}$, the current parameters $\Tilde{\phi}$ and $\Tilde{\theta}$. }
\KwOut{ A new set of rejected samples $\mathcal{Y}=\{(X_{A}^1, y^{11}),\dots,(X_{A}^1, y^{1|\mathcal{Y}^1|}), \dots,(X_A^n,y^{n1}), \dots,(X_{A}^n,y^{n|\mathcal{Y}^n|}) \}$, where $\mathcal{Y}^1=\{y^{11},\dots, y^{1|\mathcal{Y}^1|}\}, \dots, \mathcal{Y}^n=\{y^{n1},\dots, y^{n|\mathcal{Y}^n|}\}$, and a new instantiation $\lambda_{\mathcal{X}\cup\mathcal{Y}}$ of the GP on $\mathcal{X}\cup\mathcal{Y}$, new parameters $\phi$ and $\theta$.}
Set $S\gets\{1,\dots, n\}$, $\lambda_{pre}=\lambda_{\mathcal{X}\cup\mathcal{\Tilde{Y}}}$, and $\mathcal{Y}\gets\{\}$.
\SetAlgoLined\SetArgSty{}
\Repeat{$S$ is null}
{
Sample $y^{i}\sim \pi_0(\cdot|X_{A}^i; \Tilde{\theta}, \Tilde{\phi})$ $\forall i\in S$, and denote $T=\{(X_{A}^i,y^i), i\in S \}$;
Sample $\lambda_T|\lambda_{pre}$ conditionally from the Gaussian process, and add $\lambda_T$ into $\lambda_{pre}$;
Set $\alpha^i=1$ with probability $\sigma(\lambda(X_{A}^i, y^{i}))$ $\forall i \in S$;
Set $\mathcal{Y}\gets\mathcal{Y}\cup(X_{A}^i,y^i)$ if $\alpha^i=0$; otherwise, delete index $i$ from $S$ $\forall i \in S$.
}
Restrict $\lambda_{pre}$ to $\lambda_{\mathcal{X}\cup\mathcal{Y}}$.
Update $\lambda_{\mathcal{X}\cup\mathcal{Y}}$ with a Markov kernel having a stationary distribution as~\cref{eq:lambda}.
Update $\Tilde{\phi}$ to $\phi$ with a Markov kernel having a stationary distribution as~\cref{eq:phi}.
Update $\Tilde{\theta}$ to $\theta$ with a Markov kernel having a stationary distribution as~\cref{eq:theta}.
\caption{An iteration of the Markov chain for posterior inference for $p(\lambda,\phi, \theta|X_{A}^1, X_{A^c}^1, \dots, X_{A}^n, X_{A^c}^n)$}
\label{algo:3}
\end{algorithm}
\section{Experiments\label{sec:3}}
In this section, we present two synthetic examples and two real examples to demonstrate the usefulness of our proposed methodology. For the first synthetic example, we incorporate a specific marginal {\em distribution} constraint into the model and assume the empirical marginal distribution of observations agree with the marginal distribution constraint. The second example demonstrates the additional flexibility of our approach over standard conditional density modeling, by including an additional coupling between the variances of the constrained and unconstrained coordinates.
For both real examples, we incorporate marginal {\em family} constraints into the models.
We implemented our algorithm in Python and ran it on Purdue Community Cluster Workbench, an interactive compute environment for non-batch big data analysis and simulation, consisting of Dell compute nodes with 24-core AMD EPYC 7401P processors (24 cores per node), and 512 GB of memory.
%
For all examples, we ran a total of 5000 MCMC iterations, and treated the first 1000 iterations as burn-in.
We compared our proposed marginally constrained model with a fully nonparametric model without any marginal constraints, specifically, the model of~\citet{adams} that ours is based upon.
For both models, we used Gaussian processes with a squared exponential kernel: $k_{SE}(x,x')=\sigma^2\exp\left({-\cfrac{\|x-x'\|^2}{2l^2}}\right)$.
In our experiments, we set the parameter $\sigma^2$ to $1$ and updated lengthscale parameter $l$ via Hamiltonian Monte Carlo (HMC) under a weakly informative prior.
{In the family constraint setting, we also compared our proposed marginally constrained model with a simple parametric model that satisfies the marginal family constraint.}
We carried out both quantitative and qualitative evaluations of the models; for the former, we used the likelihood of a held-out test dataset.
\subsection{Synthetic Example 1}
\begin{figure}
\centerline{
\includegraphics[width=0.33\textwidth]{grountruth2.pdf}
\includegraphics[width=0.33\textwidth]{normalinvgamma_hmc.pdf}
\includegraphics[width=0.33\textwidth]{fullynonparametric_syn1.pdf}
}
\centerline{
\includegraphics[width=0.33\textwidth]{grountruth2_20_2.pdf}
\includegraphics[width=0.33\textwidth]{normalinvgamma_hmc_20.pdf}
\includegraphics[width=0.33\textwidth]{fullynonparametric_syn1_20.pdf}
}
\caption{(top left): the groundtruth density $0.5\,\mathcal{N}\left(\begin{pmatrix}
13 \\
-20 \\
\end{pmatrix},\begin{pmatrix}
1 &
\frac{3\sqrt{5}}{5}\\
\frac{3\sqrt{5}}{5} &20 \\
\end{pmatrix}\right)+0.5\,\mathcal{N}\left(\begin{pmatrix}
13 \\
20 \\
\end{pmatrix},\begin{pmatrix}
1 & \frac{3\sqrt{5}}{5}\\
\frac{3\sqrt{5}}{5} &20 \\
\end{pmatrix}\right)$; (top middle): the posterior mean density based on 100 observations drawn from the true density using our proposed model; (top right): the posterior mean density based on 100 observations drawn from the true density with posterior samples of lengthscale parameter via HMC using the fully nonparametric model. The bottom panels are similar, now based on 20 observations drawn from the true density.}\label{fig:normalinvgamma}
\end{figure}
\begin{comment}
\begin{figure}
\centerline{
\includegraphics[width=0.6\textwidth]{mu_y_vec1_syn1_l5.pdf}
\includegraphics[width=0.6\textwidth]{rho_vec1_syn1_l5.pdf}
}
\centerline{
\includegraphics[width=0.6\textwidth]{sigmasq_y_vec1_syn1_l5.pdf}
\includegraphics[width=0.6\textwidth]{traceplot_syn1_l5.pdf}
}
\caption{The two rows are traceplots with fixed lengthscale parameter of 5 for posterior samples of $\theta$ and the latent GP at the midpoint among selected grid points using our proposed model.}\label{fig:traceplotssyn1}
\end{figure}
\end{comment}
\begin{figure}
\centerline{
\includegraphics[width=0.4\textwidth]{Untitle21copy113_2.pdf}
\includegraphics[width=0.4\textwidth]{Untitle21copy113_4.pdf}
\includegraphics[width=0.4\textwidth]{Untitle21copy113_5.pdf}
}
\caption{MCMC traceplots for posterior samples of the latent GP at the midpoint along with parameters $\rho$ and $\mu_2$ for the first synthetic example.}\label{fig:traceplotssyn2}
\end{figure}
Here, we generated 2-dimensional datasets of size 100 and 20 from the following mixture of two Gaussians: $$ (X_1, X_2)\sim 0.5\,\mathcal{N}\left(\begin{pmatrix}
13 \\
-20 \\
\end{pmatrix},\begin{pmatrix}
1 &
\frac{3\sqrt{5}}{5}\\
\frac{3\sqrt{5}}{5} &20 \\
\end{pmatrix}\right)+0.5\,\mathcal{N}\left(\begin{pmatrix}
13 \\
20 \\
\end{pmatrix},\begin{pmatrix}
1 & \frac{3\sqrt{5}}{5}\\
\frac{3\sqrt{5}}{5} &20 \\
\end{pmatrix}\right).$$
The leftmost panel of~\cref{fig:normalinvgamma} shows the contours of this probability density with different sets of observations.
Observe that the first coordinate of these observations follows a Gaussian distribution
$p_1(X_1)=\mathcal{N}(X_1\,|\,\mu_1,\sigma_1^2),$
where $\mu_1=13$ and $\sigma_1^2=1$.
We assume this marginal constraint is known, and model
{each} dataset with our marginally constrained nonparametric prior.
We choose the centering distribution $\pi_0(X_2|X_1;\theta, \phi)$ as a conditional normal distribution, namely $$\pi_0(X_2|X_1=x_1;\theta, \phi)=\mathcal{N}\left(X_2 \,;\mu_2+\rho\sigma_2 \sigma_1^{-1}(x_1-\mu_1), \left(1-\rho^2\right)\sigma_2^2\right)$$ where $\theta=(\rho, \mu_2, \sigma_2^2)$, and $\phi=(\mu_1, \sigma^2_1)$, which are known values here.
We place a Normal-Inverse-Gamma-Uniform prior on $\theta=(\rho, \mu_2, \sigma_2^2)$, with $\mu_0=0$, $k=0.001$, $\alpha_0=0.001$ and $\beta_0=0.001$:
$$p(\rho, \mu_2, \sigma_2^2)\propto \mathcal{N}\left(\mu_2\,;\mu_0,\cfrac{\sigma_2^2}{k_0}\right)\cdot\text{ Inv-Gamma} (\sigma_2^2\,;\alpha_0, \beta_0)\cdot\mathbbm{1}_{[-1,1]}(\rho).$$
Running our MCMC sampler from~\cref{algo:3} (including a Hamiltonian Monte Carlo update for the lengthscale parameter in the kernel covariance matrix), we produce 4000 posterior samples for $\theta$ and $\lambda$ for each of the two datasets and then use those posterior samples to compute the mean of data densities, which is presented in the middle column of~\cref{fig:normalinvgamma}. {We include the traceplots for the posterior samples corresponding to the dataset of size 100 in figure~\ref{fig:traceplotssyn2}. For comparison, we also compute the mean of data densities for the fully nonparametric model, which is displayed at the rightmost column of~\cref{fig:normalinvgamma}. We can observe that the fully nonparametric model does not satisfy the marginal distribution constraint exactly, which is further supported in the quantitative comparison below.}
To quantitatively compare posterior results of our proposed marginally constrained model and the fully nonparametric model, we generate a test dataset of size 60 and 5 training datasets of size 20, 60 and 100 respectively from the mixture of normal distribution.
For each model and each training dataset,
we produce 4000 posterior samples and then use those posterior samples to compute `marginal' and `joint' loglikelihoods of the test dataset. {Here, the joint loglikehood refers to the standard logarithmic probability of the test dataset, while the marginal loglikelihood describes the logarithmic probability of the first component of the test datapoints (viz.\ the constrained component). }
{Finally, for each model
and each training sample size, the median of
average loglikelihoods over posterior samples across the 5 training-test splits is reported in ~\cref{table:marginalsyn} and~\cref{table:jointsyn}.}
\begin{table*}
\footnotesize
\begin{center}
\caption{Average marginal loglikelihood in the first synthetic example}
\label{table:marginalsyn}
\begin{threeparttable}
\begin{tabular}
{
@{\kern-.5\arrayrulewidth}
|p{\dimexpr3.5cm-4\tabcolsep-.5\arrayrulewidth}
|p{\dimexpr4.5cm-5\tabcolsep-.5\arrayrulewidth}
|p{\dimexpr4.5cm-5\tabcolsep-.5\arrayrulewidth}
|@{\kern-.5\arrayrulewidth}
}
\hline
Training dataset size & Truth/Our proposed model & Fully nonparametric model
\\ [0.5ex]
\hline
\textbf{100} &\textbf{-82.79}& \textbf{-83.57}\\
\hline
\textbf{60} &\textbf{-82.79}& \textbf{-84.02}\\% (-86.38, -82.74)}\\
\hline
\textbf{20} &\textbf{-82.79}& \textbf{-86.88 }\\%(-94.38, -82.94)}\\
\hline
\end{tabular}
\end{threeparttable}
\end{center}
\end{table*}
\begin{table*}
\footnotesize
\begin{center}
\caption{ Average joint loglikelihood in the first synthetic example}
\label{table:jointsyn}
\begin{threeparttable}
\begin{tabular}
{
@{\kern-.5\arrayrulewidth}
|p{\dimexpr3.5cm-4\tabcolsep-.5\arrayrulewidth}
|p{\dimexpr1.5cm-2\tabcolsep-.5\arrayrulewidth}
|p{\dimexpr3.5cm-4\tabcolsep-.5\arrayrulewidth}
|p{\dimexpr4.5cm-5\tabcolsep-.5\arrayrulewidth}
|@{\kern-.5\arrayrulewidth}
}
\hline
Training dataset size & Truth & Our proposed model & Fully nonparametric model\\ [0.5ex]
\hline
\textbf{100} &\textbf{-299.63} &\textbf{-304.69} &\textbf{-305.67}\\
\hline
\textbf{60} &\textbf{-299.63}& \textbf{-306.70} &\textbf{-308.64}\\
\hline
\textbf{20} &\textbf{-299.63} &\textbf{-318.41} &\textbf{-322.40} \\
\hline
\end{tabular}
\end{threeparttable}
\end{center}
\end{table*}
As reported in the two tables, we conclude that, compared with the fully nonparametric model, both joint and marginal loglikelihoods for our proposed marginally constrained model are always closer to the truth. The difference of either joint or marginal loglikelihoods between the two models increases as the size of traning datasets diminishes.
\subsection{Synthetic example 2}
For this example, we consider a setting that requires a bit more structure than our original model.
Specifically, we assume that two random variables $X_1$ and $X_2$, where $X_1$ is known to follow a normal distribution with unknown parameters $\mu_1$ and $\sigma_1^2$, i.e., $p_1(X_1)=\mathcal{N}(X_1\,;\mu_1,\sigma_1^2).$
While the conditional distribution of $X_2$ is unknown, it is known to have a variance of the same order as $X_1$; this is a reasonable assumption in many settings. We now seek to model a dataset of observations of $(X_1, X_2)$, while incorporating both pieces of information into the joint model.
Our original model already allows the marginal to be incorporated, and a simple modification to incorporate the variance constraint is by setting
the centering distribution $\pi_0(X_2|X_1; \theta,\phi)$ to a conditional normal distribution as below:
$$\pi_0(X_2|X_1=x_1;\theta,\phi)=\mathcal{N}(X_2 \,;\mu_2+\rho(x_1-\mu_1), (1-\rho^2)\sigma_1^2).$$
Observe that we use the same variance $\sigma^2_1$ as the marginal constraint.
We place the normal-inverse-gamma prior on $\theta=(\rho, \mu_2, \sigma_1^2)$, specifically, with $\mu_0=0$, $k_0=0.001$, $\alpha_0=0.001$ and $\beta_0=0.001$, we set
$$P(\rho, \mu_2, \sigma_1^2) \propto N(\mu_2;\mu_0,\frac{\sigma_1^2}{k_0}) \text{Inv-Gamma}(\sigma_1^2;\alpha_0, \beta_0)\mathbbm{1}_{[-1,1] }(\rho).$$
Simultaneously, with $\mu_{x_0}=-10$ and $k_{x_0}=0.01$, we place the following normal prior on $\phi=\mu_1$:
\begin{align*}
\mu_1|\sigma_1^2&\sim N(\mu_{x_0},\frac{\sigma_1^2}{k_{x_0}}).
\end{align*}
Next, we draw 15 observations from the normal distribution shown at topleft of~\cref{fig:variance equivalence2}, namely,$$(X_1, X_2)\sim
\mathcal{N}\left(\begin{pmatrix}
13 \\
-5 \\
\end{pmatrix},\begin{pmatrix}
20 & 6\\
6 &20 \\
\end{pmatrix}\right) $$
Observe that this distributions has same marginal variances for each component. We apply the modified model described above, the fully nonparametric model and our original model to the observations and compute the mean of densities estimated from 4000 posterior samples for $(\rho, \mu_2, \sigma_1^2, \mu_1, \lambda)$ as shown in~\cref{fig:variance equivalence2}.
{Our MCMC sampler here involves a straightforward modification of~\cref{eq:theta} in ~\cref{algo:3}, now having an additional term since the parameter $\sigma_1^2$ now depends on both the marginal distribution of $X_1$ and the prior distribution of $\mu_1$.
}
The posterior mean densities for all models show that the performance of the modified model surpasses the other two models with respect to the similarity with the true density. In absolute terms, the recovered density does a good job approximating the truth, despite there being 15 observations.
Apart from the straight qualitative results, we use the same metric in the first synthetic example, namely the joint loglikelihood, to compare the three different models. We generate 10 pairs of training and test datasets of size 15 from the true normal distribution. The lengthscale parameter is updated via HMC. For each model and each pair of datasets, we produce 4000 posterior samples and compute the joint loglikelihoods of the test dataset.~\Cref{table:jointsyn3} reports quantiles of joint loglikelihoods among the 10 pairs of datasets and shows that the modified model outpeforms the other two models.
\begin{comment}
\begin{figure}
\centerline{
\includegraphics[width=0.4\textwidth]{Untitle21-Copy120_2.pdf}
\includegraphics[width=0.4\textwidth]{Untitle21-Copy120_3.pdf}
}
\centerline{
\includegraphics[width=0.4\textwidth]{Untitle21-copy121_1.pdf}
\includegraphics[width=0.4\textwidth]{Untitle12-copy2.pdf}
\includegraphics[width=0.4\textwidth]{Untitle21-Copy122.pdf}}
\caption{Topleft is the true density of $0.5\,\mathcal{N}\left(\begin{pmatrix}
13 \\
-5 \\
\end{pmatrix},\begin{pmatrix}
20 &
\frac{3\sqrt{5}}{5}\\
\frac{3\sqrt{5}}{5} &1 \\
\end{pmatrix}\right)+0.5\,\mathcal{N}\left(\begin{pmatrix}
13 \\
5 \\
\end{pmatrix},\begin{pmatrix}
20 & \frac{3\sqrt{5}}{5}\\
\frac{3\sqrt{5}}{5} &1 \\
\end{pmatrix}\right) $; Topright is the posterior mean density based on 100 observations drawn from the true density with lengthscale parameter of 1; Bottomleft is the posterior mean density based on 100 observations drawn from the true density with lengthscale parameter of 5; Bottomiddle is the posterior mean density based on 100 observations drawn from the true density with lengthscale parameter of 5 based on fully nonparametric model; Bottomright is the posterior mean density based on 100 observations drawn from the true density with lengthscale parameter of 5 based on our proposed model.} \label{fig:variance equivalence}
\end{figure}
\end{comment}
\begin{figure}
\centerline{
\includegraphics[width=0.65\textwidth]{Untitle21-Copy124-2.pdf}
\includegraphics[width=0.65\textwidth]{Untitle21-Copy124-1.pdf}
}
\centerline{
\includegraphics[width=0.65\textwidth]{Untitle12-Copy5.pdf}
\includegraphics[width=0.65\textwidth]{Untitle21-Copy122.pdf}}
\caption{(top left): the true density $\mathcal{N}\left(\begin{pmatrix}
13 \\
-5 \\
\end{pmatrix},\begin{pmatrix}
20 & 6\\
6 &20 \\
\end{pmatrix}\right) $; (top right): the posterior mean density given 15 observations drawn from the true density using our modified model; (bottom left): the posterior mean density given 15 observations using the fully nonparametric model; (bottom right): the posterior mean density using 15 observations using our original proposed model.} \label{fig:variance equivalence2}
\end{figure}
\begin{table*}
\footnotesize
\begin{center}
\caption{Average joint loglikelihood in the second synthetic example. The three columns present joint loglikelihoods in the format 0.50 quantile (0.25 quantile, 0.75 quantile) using 10 pairs of training/test datasets.}
\label{table:jointsyn3}
\begin{threeparttable}
\begin{tabular}
{
@{\kern-.5\arrayrulewidth}
|p{\dimexpr4cm-4.5\tabcolsep-.5\arrayrulewidth}
|p{\dimexpr4.5cm-5\tabcolsep-.5\arrayrulewidth}
|p{\dimexpr7cm-7.5\tabcolsep-.5\arrayrulewidth}
|@{\kern-.5\arrayrulewidth}
}
\hline
The modified model& Fully nonparametric model& Our proposed marginally constrained model\\ [0.5ex]
\hline
\textbf{-88.45\,(-90.07,\,-85.68)}&\textbf{-90.25\,(-95.52, \,-87.71)}&\textbf{-89.33\,(-92.03,\, -84.78)}\\
\hline
\end{tabular}
\end{threeparttable}
\end{center}
\end{table*}
\subsection{Real Example 1} \label{sec:pm}
{ Particulate matter 2.5 (PM 2.5) refers to a category of particles in the air that are 2.5 micrometers or less in size~\citep{ott1990physical}. Their ability to penetrate deeply into the lung makes them dangerous to human health.}
It is common to model concentration levels of PM 2.5 with a lognormal distribution and this concentration level is known to be highly correlated with outdoor temperature; see for example~\citep{ott1990physical}.
We obtain measurements of the daily average concentration level of PM 2.5 and outdoor temperature in Clinton Drive in Houston, TX(CAM 55) for the year 2020\footnote{from the website https://www.tceq.texas.gov/cgi-bin/compliance/monops/yearly\_summary.pl}.
The original dataset consists of 366 daily observations, which we filtered
to eliminate outliers and missing data points to obtain the final dataset of 356 observations. This is plotted in~\cref{fig:PM2.5dataset}.
\begin{figure}
\centerline{
\includegraphics[width=0.6\textwidth]{scatterafteroutlier.pdf}
\includegraphics[width=0.6\textwidth]{Untitle21copy117_6withsamples.pdf}
}\caption{(left): the PM2.5 dataset showing PM2.5 levels versus temperature; (right): the posterior mean density based on 4000 MCMC samples.
}\label{fig:PM2.5dataset}
\end{figure}
Let $X_1$ denote the daily average concentration levels of PM 2.5 and $X_2$ denote the daily average outdoor temperature.
We applied our model to this dataset of $(X_1,X_2)$ pairs, imposing a lognormal family constraint on the PM 2.5 concentration levels.
{In this example, a minor modification to the centering distribution is required and an additional prior is placed on the parameters of the family constraint.
Denoting the parameters of the lognormal family as $\phi=(\mu_x, \sigma_x^2)$, we placed a conjugate normal-inverse-chi-squared prior on these:
\begin{align*}
\sigma_x^2&\sim \cfrac{v_{x0}\,\sigma^2_{x0}}{\chi^2_{v_{x0}}}\,, \, \quad
\mu_x\,|\,\sigma_x^2\sim \mathcal{N}(\mu_{x0}\, ,\, \cfrac{\sigma_x^2}{k_{x0}}).
\end{align*}
We set $\mu_{x0}=-10, k_{x0}=0.01,v_{x0}=0.001$ and $\sigma^2_{x0}=5$.
For our centering distribution, we use
\begin{align}
\pi_0(X_2|X_1=x_1, \theta, \phi)=\mathcal{N}\left(X_2 \,;\mu_2+\rho\sigma_2 s_x^{-1} \left(x_1- m_x \right), \left(1-\rho^2\right)\sigma_2^2\right), \label{eq:centering distribution}
\end{align}
where $\theta=(\rho, \mu_2, \sigma_2^2)$, $s_x^2=\mathrm{Var}\left[X_1\right]=\exp\left\{(\sigma_x^2-1)(2\mu_x+\sigma_x^2)\right\}$, and $m_x={\mathbb{E}}\left[X_1\right]=\exp\left\{\mu_x+\frac{\sigma_x^2}{2}\right\}$.
We place a normal-inverse-gamma prior on $\theta=(\rho, \mu_2, \sigma_2^2)$: $$P(\rho, \mu_2, \sigma_2^2)\propto \mathcal{N}\left(\mu_2\,;\mu_0,\cfrac{\sigma_2^2}{k_0}\right)\cdot\text{ Inv-Gamma} (\sigma_2^2\,;\alpha_0, \beta_0)\cdot\mathbbm{1}_{[-1,1]}(\rho)$$
where $\mu_0=0$, $k=0.001$, $\alpha_0=0.001$ and $\beta_0=0.001$.
}
Using our MCMC sampler with this model, we draw samples from the posterior distribution given the PM 2.5 dataset, plotting the posterior mean density in~\cref{fig:PM2.5dataset}. {We see that the posterior mean density captures the characteristics of the dataset reasonably well.}
The model does struggle to capture some of the outliers along the $X_1$-component, though this is more a reflection of the marginal lognormal constraint, rather than the nonparametric component.
Modeling both components together allows practitioners to assess this limitation for different values of the temperature variable, and the figure suggests that failures of the parametric assumption occur at large values of the temperature variable.
Analogous to the quantitative comparison in the first synthetic example, we also perform a comparison among our proposed marginally constrained model, a fully nonparametric model and a parametric model.
For the fully nonparametric model and our proposed model, the lengthscale parameter is
updated via HMC.
We also fit a bivariate normal parametric model to fit temperature and the log-transformed PM 2.5 variable; note that the parametric model satisfies the lognormal family constraint on PM 2.5. We repeat splitting the dataset into a training dataset of size 296 and a test dataset of size 60 5 times and obtain 5 pairs of training and test datasets. For each model
and each pair of training and test datasets, we produce 4000 posterior samples according to the matching training dataset and then use those posterior samples to compute the joint and marginal loglikelihoods of the corresponding test dataset. Finally, for each mode
, the median of the average loglikelihoods over posterior samples across the 5 pairs of training and testing datasets are reported in~\cref{table:jointreal} and~\cref{table:marginalreal}.
Both tables illustrate that our proposed marginally constrained model always behaves the best, {demonstrating the importance of flexibility in preserving the dependence structure between the two variables as well as incorporating prior information through marginal constraints in data-poor settings.}
\begin{table*}
\footnotesize
\begin{center}
\caption{Average joint loglikelihood in the PM 2.5 data from 5 training-test splits}
\label{table:jointreal}
\begin{threeparttable}
\begin{tabular}
{
@{\kern-.5\arrayrulewidth}
|p{\dimexpr3.8cm-4.3\tabcolsep-.5\arrayrulewidth}
|p{\dimexpr4.5cm-5\tabcolsep-.5\arrayrulewidth}
|p{\dimexpr7cm-7.5\tabcolsep-.5\arrayrulewidth}
|p{\dimexpr3cm-3.5\tabcolsep-.5\arrayrulewidth}
|@{\kern-.5\arrayrulewidth}
}
\hline
Lengthscale parameter& Fully nonparametric model& Our proposed marginally constrained model & Parametric model\\ [0.5ex]
\hline
\textbf{HMC}&\textbf{-385.57} &\textbf{-382.24} &\textbf{-384.52} \\
\hline
\end{tabular}
\end{threeparttable}
\end{center}
\end{table*}
\begin{table*}
\footnotesize
\begin{center}
\caption{ Average marginal loglikelihood for the PM 2.5 data for 5 training-test splits}
\label{table:marginalreal}
\begin{threeparttable}
\begin{tabular}
{
@{\kern-.5\arrayrulewidth}
|p{\dimexpr3.8cm-4.3\tabcolsep-.5\arrayrulewidth}
|p{\dimexpr4.5cm-5\tabcolsep-.5\arrayrulewidth}
|p{\dimexpr7cm-7.5\tabcolsep-.5\arrayrulewidth}
|p{\dimexpr3cm-3.5\tabcolsep-.5\arrayrulewidth}
|@{\kern-.5\arrayrulewidth}
}
\hline
Lengthscale parameter& Fully nonparametric model& Our proposed marginally constrained model &Parametric model\\ [0.5ex]
\hline
\textbf{HMC}&\textbf{-169.33} &\textbf{-164.11} &\textbf{-164.14} \\
\hline
\end{tabular}
\end{threeparttable}
\end{center}
\end{table*}
\subsection{Real Example 2}
{In our final experiment, we consider modeling
earthquake data.
Following~\citep{dehghani2020probabilistic}, we are interested in modeling the bivariate distribution of earthquake recurrence time and magnitude, while simultaneously ensuring that the recurrence time follows an exponential distribution~\citep{ferraes2003conditional}.}
We obtain a dataset of 45 observations from~\citet{ferraes2003conditional} (table 1) and run our proposed marginally constrained model with a family constraint.
Denoting the rate parameter of the exponential family constraint on the recurrence time as $r$, we place a weakly informative gamma prior with both shape and rate parameters to be $0.1$.
To apply our maginally constrained model, we choose a slightly different centering distribution with a same normal-inverse-gamma prior placed on its parameters as that in the first real example described in section~\ref{sec:pm}.
We use the same centering distribution as described in equation~\ref{eq:centering distribution}, where $\theta=(\rho, \mu_2, \sigma_2^2)$, $\phi=r$, $m_x=\frac{1}{r}$ and $s_x^2=\frac{1}{r^2}$.
The posterior mean density is presented in~\cref{fig:earthquake}.
We see that other than three outliers, the model succeeds in capturing the underlying observation pattern, and that the failure to model the observations arises from the parametric exponential constraint, which effectively robustifies the model against these outliers.
We do not report quantitative performance measures here, essentially, depending on whether or not the outliers are part of the test set, either the fully nonparametric model or our model performs best.
\begin{figure}
\centerline{
\includegraphics[width=0.6\textwidth]{posterior2_earthquake_HMC.pdf}
}
\centerline{
\includegraphics[width=0.6\textwidth]{rho_vec1_earthquake.pdf}
\includegraphics[width=0.6\textwidth]{traceplot_earthquake.pdf}
}
\centerline{
\includegraphics[width=0.6\textwidth]{rate_vec1_earthquake.pdf}
\includegraphics[width=0.6\textwidth]{mu_y_vec1_earthquake.pdf}
}
\caption{In the top row is the posterior mean density with posterior samples of the lengthscale parameter.
The rest are the corresponding traceplots for posterior samples of $\theta$, $\phi$ and the latent GP at the midpoint of the selected gridpoints.}\label{fig:earthquake}
\end{figure}
To assess MCMC mixing, we evaluate the latent GP on a grid of 3540 points and run MCMC for 5000 iterations with a burn-in period of 1000 iterations. In~\cref{table:ess}, we report the minimum ESS, maximum ESS and ESS at the midpoint among all the 3540 grid points. In~\cref{fig:earthquake}, we present the traceplots of posterior samples for $\phi$, $\theta$ and the latent GP at the midpoint.
\begin{table*}
\footnotesize
\begin{center}
\caption{Effective sample size per 4000 iterations for the second real example}
\label{table:ess}
\begin{tabular}
{
@{\kern-.5\arrayrulewidth}
|p{\dimexpr5.5cm-6\tabcolsep-.5\arrayrulewidth}
|p{\dimexpr5.5cm-6\tabcolsep-.5\arrayrulewidth}
|p{\dimexpr5.5cm-6\tabcolsep-.5\arrayrulewidth}
|@{\kern-.5\arrayrulewidth}
}
\hline
Minimum ESS & Maximum ESS& ESS at midpoint \\ [0.5ex]
\hline\hline
16.76&
242.32&
107.09\\
\hline
\end{tabular}
\end{center}
\end{table*}
\section{Conclusions\label{sec:4}}
In this work, we propose a nonparametric Bayesian approach for density modeling while enforcing constraints on the marginal distribution of a subset of components. Our approach, closely tied to conditional density modeling introduces a novel constrained Bayesian model based on a transformed Gaussian process, satisfying the marginal constraining distribution exactly and inducing large support prior. For posterior sampling, we devise an exact MCMC algorithm without any approximation/discretization errors, which is additional attraction of our approach over existing conditional density modeling approaches.
In the present paper, we are only focused on placing one marginal density constraint on a subset of variables.
In some settings, partial prior beliefs are available about different subsets of variables, which requires simultaneously imposing multiple marginal constraints on those subsets of variables.
Since the dependence between these subsets is unavailable, our approach doesn't extend to it in a straightforward manner. As mentioned at the start of this paper, a more general problem is to constrain some functional of the data distribution. For example, we might have prior information about the mean of the distribution, either in the form of fixed values or prior distribution. In future work, it is of interest to extend our framework to solve these problems.
There are some other open issues to be considered. First, in this paper we have not discussed sufficient conditions for strong consistency and rates of the convergence of the posterior distribution. \citet{ghosal2000convergence} presents general results on the rates of convergence of the posterior distribution, which can be adapted to our specific transformed Gaussian process prior. Second, we can think about using our proposed transformed Gaussian process prior to solve nonparametric conditional density modeling problems like density regressions. \citet{tokdar2010bayesian} develops a framework for modeling conditional densities and offering dimension reduction of predictors by combining the logistic Gaussian process and subspace projection. A similar future work worth consideration is to connect our proposed transformed Gaussian process prior to subspace projection. More generally, it is of interest to leverage the vast literature on scalability of Gaussian processes to improve the scalability of our proposed model.
\section*{Acknowledgments}
We thank the Editor, Associate Editor and referees, as well as our financial sponsors.
\section*{Appendix}
\section{Appendix A: Weak Posterior Consistency}\label{appendix A}
\subsection{Basics of consistency}
Let $D_1,\dots, D_n$ be i.i.d. with true density $f_0$ belonging to a space of densities $\mathcal{F}$ with weak topology. Let $\Pi$ be a prior on $\mathcal{F}$. For a density $f$, let $P_{f}$ stand for the probability measure corresponding to $f$. Then for any measurable subset $A$ of $\mathcal{F}$, the posterior probability
of $A$ given $D_1, \dots, D_n$ is
$$\Pi(A|D_1, \dots, D_n)=\frac{\int_A \prod_{i=1}^n f(D_i)\Pi(df)}{\int_\mathcal{F}\prod_{i=1}^n f(D_i)\Pi(df)}$$
\begin{definition}[weak neighborhood]
A weak neighborhood of $f_0$ is a set
$$V_\epsilon(f_0) =\left\{f\in \mathcal{F}: \left|\int\phi f-\int \phi f_0 \right|<\epsilon , \text{for any bounded and continuous function}\, \phi\right\}$$.
\end{definition}
\begin{definition}[weak posterior consistency]
A prior $\Pi$ is said to achieve weak posterior consistency at $f_0$, if $$\Pi(U|X_1,\dots,X_n)\to 1\, \text{almost surely under}\, P_{f_0}$$ for all weak neighborhoods $U$ of $f_0$.
\end{definition}
\begin{definition}[KL support]
$f_0$ is said to be in the KL support of $\Pi$ if $\forall \epsilon>0,\,\Pi\left(K_\epsilon(f_0)\right)>0$ , where $K_\epsilon(f_0) = \{f: \int f_0\log(f_0/f)<\epsilon\}$ is a KL neighborhood of $f_0$ .
\end{definition}
\subsection{A formal proof of weak posterior convergence for our proposed model}
\begin{lemma}
\label{lm:corresponding theorem 4.1}
For any two functions $\lambda_1$ and $\lambda_2$ on the index set $\mathcal{X}_A\times\mathcal{X}_{A^c}$ and for any $\delta > 0$, $$\|\lambda_1-\lambda_2\|_\infty<\delta\Rightarrow \left\| \log\frac{f_{\lambda_1}}{f_{\lambda_2}}\right\|_\infty \le 2\delta$$
\end{lemma}
\begin{proof}[\textbf{\upshape Proof:}]
It follows from $\|\lambda_1-\lambda_2\|_\infty<\delta$ that $\lambda_2\left(X_A,X_{A^c}\right)-\delta < \lambda_1\left(X_A, X_{A^c}\right) <\lambda_2\left(X_A,X_{A^c}\right)+\delta$
for any $X_A, X_{A^c} \in\mathcal{X}_A\times\mathcal{X}_{A^c}$.
Then, from the monotonicity of the sigmoid function $\sigma$, we have that
$\sigma\left(\lambda_2\left(X_A,X_{A^c}\right)-\delta\right)<\sigma\left(\lambda_1\left(X_A, X_{A^c}\right)\right)<\sigma\left(\lambda_2\left(X_A,X_{A^c}\right)+\delta\right)$.
Next, observe that the sigmoid function satisfies $\sigma(x-\delta) > \exp(-\delta)\sigma(x)$ for any $\delta > 0$.
Combining this with the previous result, we obtain \begin{align*}\exp\left(-\delta\right)\sigma\left(\lambda_2\left(X_A,X_{A^c}\right)\right)<\sigma\left(\lambda_1\left(X_A,X_{A^c}\right)\right)<\exp\left(\delta\right)\sigma\left(\lambda_2\left(X_A,X_{A^c}\right)\right).\end{align*}
It follows that for any density $\pi_0(X_{A^c}|X_A)$, we have
\begin{align*}
\exp\left(-\delta\right)\int_{\mathcal{X}_{A^c}}\pi_0\left(X_{A^c}|X_A\right)\sigma\left(\lambda_2\left(X_A,X_{A^c}\right)\right)dX_{A^c} & <\int_{\mathcal{X}_{A^c}}\pi_0\left(X_{A^c}|X_A\right)\sigma\left(\lambda_1\left(X_A,X_{A^c}\right)\right)dX_{A^c} \\
& <\exp\left(\delta\right)\int_{\mathcal{X}_{A^c}}\pi_0\left(X_{A^c}|X_A\right)\sigma\left(\lambda_2\left(X_A,X_{A^c}\right)\right)dX_{A^c}
\end{align*}
From the above two inequalities, it follows that
\begin{align*}
\exp\left(-2\delta\right)\cfrac{\sigma\left(\lambda_2\left(X_A,X_{A^c}\right)\right)}{\int_{\mathcal{X}_{A^c}}\pi_0\left(X_{A^c}|X_A\right)\sigma\left(\lambda_2\left(X_A,X_{A^c}\right)\right)dX_{A^c}}
& < \cfrac{\sigma\left(\lambda_1\left(X_A,X_{A^c}\right)\right)}{\int_{\mathcal{X}_{A^c}}\pi_0\left(X_{A^c}|X_A\right)\sigma\left(\lambda_1\left(X_A,X_{A^c}\right)\right)dX_{A^c}} \\
& <\exp\left(2\delta\right)\cfrac{\sigma\left(\lambda_2\left(X_A,X_{A^c}\right)\right)}{\int_{\mathcal{X}_{A^c}}\pi_0\left(X_{A^c}|X_A\right)\sigma\left(\lambda_2\left(X_A,X_{A^c}\right)\right)dX_{A^c}},
\end{align*}
so that for all $(X_A,X_{A^c})$,
\begin{align*}
\exp\left(-2\delta\right) < \frac{f_{\lambda_1}\left(X_A,X_{A^c}\right)}{f_{\lambda_2}\left(X_A,X_{A^c}\right)} < \exp\left(2\delta\right)
\end{align*}
The result then follows.
\end{proof}
\abc*
\begin{proof}[\textbf{\upshape Proof:}]
By definition, any density that belongs to $\mathcal{F}$ takes the form $p_A(X_A)\cdot q(X_{A^c}|X_A)$.
For any $\epsilon>0$, set $\delta=\cfrac{\epsilon}{4}$, and choose $\xi>0$ such that $\log(1+\xi)<\frac{\epsilon}{2}$.
Define a strictly positive density $q_0$ as
$$
q_0\left(X_{A^c}|X_A\right)=\cfrac{q\left(X_{A^c}|X_A\right)+\xi}{1+\xi} \qquad\forall X_A\in\mathcal{X}_A, X_{A^c}\in \mathcal{X}_{A^c}.
$$
Consider the ratio $\cfrac{q_0\left(X_{A^c}|X_A\right)}{\pi_0\left(X_{A^c}|X_A\right)}$.
As both $q_0$ and $\pi_0$ are continuous functions on $\mathcal{X}_A\times\mathcal{X}_{A^c}$, and as $\pi_0$ does not vanish, this is also a continuous function on $\mathcal{X}_A\times\mathcal{X}_{A^c}$.
Due to the compactness of $\mathcal{X}_A\times\mathcal{X}_{A^c}$, we additionally have that $M:=\sup_{\left\{X_A\in \mathcal{X}_A, X_{A^c} \in \mathcal{X}_{A^c}\right\}} \cfrac{q_0\left(X_{A^c}| X_A\right)}{\pi_0\left(X_{A^c}|X_A\right)}<\infty$.
Define $\lambda_0(X_A,X_{A^c})=\sigma^{-1}(\frac{q_0(X_{A^c}| X_A)}{M\pi_0(X_{A^c}|X_A)})$.
The definition of $q_0$ and $M$, and the fact that $\pi_0$ does not vanish ensures that $\frac{q_0(X_{A^c}| X_A)}{M\pi_0(X_{A^c}|X_A)} \in (0,1)$, and thus lies in the domain of $\sigma^{-1}(\cdot)$.
Recalling the mapping $f_\lambda$ is defined in~\cref{eq:map}, it is easy to see that the function $\lambda_0$ satisfies
$f_{\lambda_0}(X_A,X_{A^c}) = p_A(X_A) q_0(X_{A'}|X_A)$.
From the assumptions of the theorem, it follows from~\cref{lm:gaussian process property1} that
$$P(\lambda:\sup_{\left\{X_A\in \mathcal{X}_A, X_{A^c} \in \mathcal{X}_{A^c}\right\}}|\lambda(X_A, X_{A^c})-\lambda_0(X_A,X_{A^c})|<\delta)>0.$$
From~\cref{lm:corresponding theorem 4.1}, we then obtain
\begin{align*}
&\Pi\left(f_\lambda:\left\|\log\cfrac{f_{\lambda_0}(X_A, X_{A^c})}{f_\lambda(X_A, X_{A^c})}\right\|_\infty<\cfrac{\epsilon}{2}\right)\geq P(\lambda:\sup_{\left\{X_A \in \mathcal{X}_A,X_{A^c}\in \mathcal{X}_{A^c}\right\}}|\lambda\left(X_A, X_{A^c}\right)-\lambda_0\left(X_A,X_{A^c}\right)|<\delta)>0.
\end{align*}
Now, recognizing that $\log\cfrac{q\left(X_{A^c}|X_A\right)}{q\left(X_{A^c}|X_A\right)+\xi} < 0$, we have:
\begin{align*}
\text{KL}\,&\left(p_A\left(X_A\right)\cdot q\left(X_{A^c}|X_A\right),f_\lambda\left(X_A, X_{A^c}\right)\right)=\int_{\mathcal{X}_A}\int_{\mathcal{X}_{A^c}}p_A\left(X_A\right)q\left(X_{A^c}|X_A\right)\log\cfrac{p_A\left(X_A\right)q\left(X_{A^c}|X_A\right)}{f_\lambda\left(X_A, X_{A^c}\right)}dX_{A^c}dX_A\\
&=\int_{\mathcal{X}_A}\int_{\mathcal{X}_{A^c}}P_A\left(X_A\right)q\left(X_{A^c}|X_A\right)\log\cfrac{q\left(X_{A^c}|X_A\right)}{q_0\left(X_{A^c}|X_A\right)}dX_{A^c}dX_A+\int_{\mathcal{X}_A}\int_{\mathcal{X}_{A^c}}p_A\left(X_A\right)q\left(X_{A^c}|X_A\right)\log\cfrac{p_A\left(X_A\right)q_0\left(X_{A^c}|X_A\right)}{f_\lambda\left(X_A, X_{A^c}\right)}dX_{A^c}dX_A\\
&=\int_{\mathcal{X}_A}\int_{\mathcal{X}_{A^c}}p_A\left(X_A\right)q\left(X_{A^c}|X_A\right)\log\cfrac{q\left(X_{A^c}|X_A\right)}{q\left(X_{A^c}|X_A\right)+\xi}dX_{A^c}dX_A+\log\left(1+\xi\right)\\
&\quad +\int_{\mathcal{X}_A}\int_{\mathcal{X}_{A^c}}p_A\left(X_A\right)q\left(X_{A^c}|X_A\right)\log\cfrac{p_A\left(X_A\right)q_0\left(X_{A^c}|X_A\right)}{f_\lambda\left(X_A, X_{A^c}\right)}dX_{A^c}dX_A\\
&< \frac{\epsilon}{2}+\left\|\log\cfrac{f_{\lambda_0}\left(X_A, X_{A^c}\right)}{f_\lambda\left(X_A, X_{A^c}\right)}\right\|_\infty.
\end{align*}
It follows that
$\Pi(f_\lambda:\text{KL}(p_A\cdot q,f_\lambda)<\epsilon)\geq \Pi\left(f_\lambda:\left\|\log\cfrac{f_{\lambda_0}(X_1, X_2)}{f_\lambda(X_1, X_2)}\right\|_\infty<\cfrac{\epsilon}{2}\right)>0$, completing the proof.
\end{proof}
\bibliographystyle{unsrtnat}
|
1,116,691,500,832 | arxiv | \section{Introduction}
Program proof is the application of deductive verification techniques to
programs. Industrial acceptability of such tools relies on the high degree of
automation provided by modern automatic provers, in particular SMT solvers,
when the source code is restricted to a suitable language
subset. SPARK~\footnote{\url{https://www.adacore.com/sparkpro}} is an example
of such a FLOSS industrial tool for Ada programs. It is available freely online
as part of GNAT Community
Edition~\footnote{\url{https://www.adacore.com/download}}.
While striving to offer the most automation to our users, we have also
recognized early on the need for interactions when the analyzer cannot complete
the verification automatically. In those cases, the task for the analyzer is to
display to the user the right information that will allow her to provide in
exchange the pieces of information that are required for the analyzer to
complete the verification.
Over the years, we have come up with a variety of solutions to address this
challenge, trying to adapt these solutions to the degree of expertise of the
user. All these solutions came from discussions (usually by email) with
industrial users of SPARK, as part of the support activity that they subscribed
to: a user asks about a puzzling message or a problem with proving a property;
we explain the possible problem and show how this can be investigated; through
discussion we together come up with a way to include part of this explanation
in the messages of the analyzer. We have lots of anecdotal evidence that this
helps users, in the form of positive feedback to our message improvements. We
are presenting these solutions in this article, in the hope that they can be
useful to others, and serve as a basis for better future solutions, as the
challenges presented here are common to most similar analyzers.
\section{The Nurse: Providing First Aid}
At a minimum, the analyzer should help the user understand what is the problem,
using in particular clear messages with precise locations. This is not always
enough, due to the complexity of the detailed program semantics at play which
may elude the user. This difficulty is compounded by the lack of expertise for
novice users of the programming language. The solution we have adopted in SPARK
is to augment the basic message with additional information explaining the
immediate cause of the problem.
Let's consider a procedure \verb|Erase| which replaces every character in a
string with the blank character:
\begin{lstlisting}
procedure Erase (S : out String) is
begin
for J in 1 .. S'Length loop
S (J) := ' ';
end loop;
end Erase;
\end{lstlisting}
\noindent
When running GNATprove (the name of the analysis tool for SPARK) on that code,
it issues the following message:
\begin{verbatim}
strings.adb:6:13: medium: array index check might fail
6 | S (J) := ' ';
| ^ here
e.g. when J = 1
and S'First = 2
reason for check: value must be a valid index into the array
\end{verbatim}
The main message line contains the precise location of the message, its
severity, and the message text indicating which property might be
violated. Here, this is an array index check, which corresponds to the run-time
check when accessing an array in SPARK. The next two lines are only present in
command-line usage, and help visually locate the issue outside of an IDE. So
far, this is what every tool diagnostic should contain.
The purpose of the remaining lines is to help the user understand the reason
for the message to be issued. The counterexample introduced by ``e.g.'' gives
concrete values of variables which lead to a property violation. Here, when
\verb|J| is one and the string \verb|S| starts at index two (denoted in SPARK
by \verb|S'First|), the assignment attempts to write outside of the bounds of
the array. This is reinforced by the last line, which gives the reason for a
check at this location: assigning into an array in SPARK requires the index to
be within bounds.
These messages are particularly useful for beginners, who may not know that
arrays (and among them, strings) in SPARK may start at other indexes than one,
or who may not realize that assigning into an array involves implicitly a
run-time check that this assignment is within bounds.
The ``reason for the check'' provides additional context which can be useful
also to more advanced users, in cases where the details of the language
semantics requiring a check are more complex. It also facilitates understanding
exactly in which part of a longer expression a check originates, as there might
be many similar checks on the same line.
Only part of the counterexample is displayed above as part of the message. A
counterexample is really a trace inside the subprogram consisting in multiple
program points with values of variables at each program point. In the GNAT
Studio IDE, the user can choose to display the trace with a simple click on a
magnify icon next to the message.
Still in this category of first-aid help, GNATprove strives to present the user
with the smallest sub-property that cannot be proved, in cases where the
property to prove is a conjunction of smaller sub-properties or a universally
quantified property. This is particularly useful when attempting to prove the
precondition of a call or a loop invariant, which typically are a conjunction
of sub-properties. This requires GNATprove to split unproved properties further
until a leaf sub-property is not proved.
\section{The Investigator: Looking for Probable Cause}
\label{sec:the-investigator}
Once the user understands the immediate cause of the problem, the next step is
to understand its root cause. Similar to the use of backtrace when debugging a
program, the analyzer should provide information on the context of the problem
that helps identify the missing link in the chain of deductions that the user
does in her head. In particular, programmers rely on operational semantics to
understand program executions, while the analyzer relies on axiomatic semantics
which may abstract away crucial details of the operational semantics. The
solution we have adopted in SPARK is to output additional information related
to the problem when there is a chance that it might have been overlooked by the
user.
Let's fix our implementation of \verb|Erase| by iterating over the range of
string S:
\begin{lstlisting}
procedure Erase (S : out String) is
begin
for J in S'Range loop
S (J) := ' ';
end loop;
end Erase;
\end{lstlisting}
\noindent
and let's add a contract to \verb|Erase| specifying in a postcondition that all
characters in \verb|S| should be blank on return:
\begin{lstlisting}
procedure Erase (S : out String)
with Post => All_Blanks (S);
\end{lstlisting}
\noindent
which is defined recursively over the range of \verb|S|:
\begin{lstlisting}
function All_Blanks (S : String) return Boolean is
(if S = "" then True
else S (S'First) = ' '
and then All_Blanks (S (S'First + 1 .. S'Last)));
\end{lstlisting}
\noindent
When running GNATprove on that code, it issues the following messages:
\begin{verbatim}
strings.ads:9:19: medium: postcondition might fail
9 | with Post => All_Blanks (S);
| ^~~~~~~~~~~~~
possible fix: loop at strings.adb:5 should mention S in a loop invariant
5 | for J in S'Range loop
| ^ here
\end{verbatim}
The three last lines are new compared to the messages in the previous
section. GNATprove points here to a possible cause for the failure to prove the
postcondition, which is that the loop in \verb|Erase| has no loop invariant. It
comes to this conclusion by looking at the variables which are mentioned
(explicitly or implicitly) in the property to prove, here only \verb|S|, and
traverses the code in reverse execution order from the program point where the
property should hold. During this traversal, it correctly identifies here that
the loop modifies \verb|S| without specifying how those changes impact the
value of \verb|S| inside a loop invariant, which is a likely cause for not
being able to prove the postcondition.
A common pitfall of program proof is the frame problem. For automatic provers
to be able to reason about formulas that represent the program semantics, these
formulas necessarily must encode small parts of the whole program
semantics. Thus GNATprove defines a frame for each property to check, that only
presents a subset of the information available in the program, abstracting in
particular subprogram calls as the corresponding subprogram contract and loop
iterations as the loop invariant for that loop. When the user did not write a
contract for a subprogram, or did not write a loop invariant for a loop,
GNATprove may still be able to analyze the corresponding call/loop precisely by
inlining the call or unrolling the loop. But this is not always the case, which
raises the question of how these internal tool decisions are communicated to
the user. Because this kind of information was a source of confusion for
beginners, GNATprove only outputs it when instructed to do so with the switch
\verb|--info|, in which case it issues here the following messages:
\begin{verbatim}
strings.adb:5:24: info: cannot unroll loop (too many loop iterations)
strings.ads:6:18: info: expression function body not available for proof
("All_Blanks" might not return)
\end{verbatim}
The first line informs the user that the loop in \verb|Erase| could not be
unrolled (hence it requires a loop invariant) because doing so would require
too many loop iterations. Indeed, a string in SPARK is an array over the range
of positive (32-bits) integers, a range much too large to unroll the loop.
The second line informs the user of another problem here: although function
\verb|All_Blanks| was defined as an expression function (a purely functional
expression for a function, which can readily be translated into an axiom for
proof), its defining expression cannot be used for interpreting the
postcondition here. Indeed, \verb|All_Blanks| is defined recursively, which
makes it possible that it does not return on some inputs. In such cases, it
would be unsound for GNATprove to treat its defining expression as an axiom in
proof, which prevents using it here. There are multiple ways to solve this
problem, either by providing a subprogram variant in order to prove
termination, or by expressing \verb|All_Blanks| differently without recursion.
Still in this category of probable cause, GNATprove can attempt to detect
inconsistencies in specifications or code, by trying to prove that the logical
context for a given branch in the specification or the program entails the
False proposition. As this involves additional calls to automatic provers,
hence has an impact on running time, this is only done when the user chooses to
do so with the switch \verb|--proof-warnings|.
\section{The Magician: Suggesting a Possible Fix}
The ultimate goal of interactivity is to suggest a possible fix to the user, in
those (alas, few!) cases where it is possible, either because some information
is clearly missing, or because a faulty pattern can be recognized. Our
experience with SPARK has shown a few such cases where the analyzer just stops
short of fixing the code itself.
Let's add a loop invariant to the loop in \verb|Erase|:
\begin{lstlisting}
procedure Erase (S : out String) is
begin
for J in S'Range loop
S (J) := ' ';
pragma Loop_Invariant (for all K in S'First .. J => S (K) = ' ');
end loop;
end Erase;
\end{lstlisting}
\noindent
and reimplement \verb|All_Blanks| without recursion as follows:
\begin{lstlisting}
function All_Blanks (S : String) return Boolean is
begin
for J in S'Range loop
if S (J) /= ' ' then
return False;
end if;
end loop;
return True;
end All_Blanks;
\end{lstlisting}
\noindent
When running GNATprove on that code, it issues the following messages:
\begin{verbatim}
strings.ads:7:19: medium: postcondition might fail, cannot prove All_Blanks (S)
7 | with Post => All_Blanks (S);
| ^~~~~~~~~~~~~
possible fix: you should consider adding a postcondition to function All_Blanks
or turning it into an expression function
\end{verbatim}
The poscondition of \verb|Erase| still cannot be proved. GNATprove this time
has a more precise suggestion for the solution, which is to add a postcondition
to \verb|All_Blanks| or to turn it into an expression function. Indeed,
GNATprove handles differently regular functions, which may themselves contain
imperative constructs like loops, and so-called expression functions, which can
be readily interpreted in logical terms. Another option would be to add a
postcondition to \verb|All_Blanks|, which the message also mentions.
A similar case where GNATprove can suggest a precise fix to the user relates to
the choice made in GNATprove to prove the absence of run-time errors inside
preconditions independently from calling contexts. So if a subprogram has the
expression (A and B) as a precondition, neither the evaluation of A nor the
evaluation of B should lead to an error. In many cases though, evaluating B
might require that A evaluates to True, and in such cases the precondition
should be expressed using the shorthand connective ``and then'' as (A and then
B). GNATprove detects cases where the user could have used ``and then'' instead
of ``and'' in preconditions to protect against errors, and suggests this
possible fix.
Another such case is the well-known misuse of a conditional inside an
existential quantification, which beginners are almost certain to be bitten by
at some point. When it finds such a syntactic construct (for some X $=>$ (if P
then Q)) which will evaluate to True whenever P is False, GNATprove issues a
warning suggesting the likely fixes:
\begin{verbatim}
file:line:column: warning: suspicious expression
did you mean (for all X => (if P then Q))
or (for some X => P and then Q) instead?
\end{verbatim}
While this degree of feedback to the user is highly desirable, it is hard to
produce in general, outside of the specific common cases described above.
\section{The Surgeon: Looking at the Innards}
This exploration would not be complete if we did not present the way for users
to look at the innards of a Verification Condition, in cases where the analyzer
did not present the information needed to understand the problem. Note however
that the preferred means to investigate such unproved properties in SPARK is
through so-called auto-active verification, where the user states intermediate
properties through ghost code (assertions and lemmas).
Let's define \verb|All_Blanks| as an expression function whose body is the same
universally quantified property that we wrote in the loop invariant:
\begin{lstlisting}
function All_Blanks (S : String) return Boolean is
(for all J in S'Range => S (J) = ' ');
\end{lstlisting}
When running GNATprove on that code, it proves the postcondition of
\verb|Erase|, but issues messages related to possible reads of uninitialized
data (which were in fact issued on previous versions of the example), of the
form:
\begin{verbatim}
file:line:column: "S" might not be initialized
\end{verbatim}
The reason is that, by default, GNATprove checks correct data initialization by
data flow analysis instead of proof, which is not sufficient here to prove that
\verb|S| is progressively initialized in the loop, which ends with \verb|S|
being completely initialized, and that only the initialized part of \verb|S| is
read in the loop invariant. The solution here is to indicate to GNATprove that
we want it to treat \verb|S| as partially initialized, and to use proof to
demonstrate correct initialization before use:
\begin{lstlisting}
procedure Erase (S : out String)
with Post => All_Blanks (S),
Relaxed_Initialization => S;
\end{lstlisting}
The effect is not immediately visible, as GNATprove keeps issuing messages
about possible reads of uninitialized data. Ignoring for a moment that the
User's Guide explains how to deal with such cases, we can try to understand by
ourselves the underlying model used in proof to deal with
initialization. Through a contextual menu, we can start manual proof on one of
the unproved check, which opens multiple panels in the
IDE~\cite{dailler:hal-01936302}: a panel showing the proof tree (consisting in
the tree of transformations and sub-goals), a panel displaying the current goal
with names translated to reflect source code variable names (with hypotheses
and conclusion), and a panel to enter commands to interact with the tool. After
introducing quantified variables and hypotheses with the command
\verb|split_vc|, the goal looks like this:
\begin{verbatim}
goal def'vc : __attr__init (get2 S _f) = True
\end{verbatim}
\noindent
We can display the definition of \verb|get2|:
\begin{verbatim}
> print get2
function get2 (f:'a -> 'b) (x:'a) : 'b = f \@ x
\end{verbatim}
This is the application of a map representing the string to an index in order
to get the corresponding element. Thus, the Verification Condition here looks
at some attribute \verb|__attr__init| representing the initialization status of
this value, which should be the boolean True to denote that the value has been
initialized. We can search for occurrences of \verb|__attr__init| in the
background theory encoding the program semantics and in the hypotheses encoding
the subprogram execution, using the command \verb|search|:
\begin{verbatim}
> search __attr__init
type character__init_wrapper =
| character__init_wrapper'mk (rec__value:character) (__attr__init:bool)
function character__init_wrapper___attr__init__projection (a1:
character__init_wrapper) : bool = __attr__init a1
\end{verbatim}
Here it returns elements of the background theory which allow to attach an
initialization value to a character using type constructor
\verb|character__init_wrapper'mk| and to retrieve the corresponding value from
the pair using function
\verb|character__init_wrapper___attr__init__projection|. And indeed the
constructor is used to define what it means to initialize a character in
function \verb|to_wrapper|:
\begin{verbatim}
function to_wrapper (x:character) : character__init_wrapper =
character__init_wrapper'mk x True
\end{verbatim}
\noindent
which is used in one of the hypotheses to indicate that \verb|S(J)| is
initialized after the assignment inside the loop:
\begin{verbatim}
H1 : S = set2 S1 J (to_wrapper o)
\end{verbatim}
So we only get that the element at the current index \verb|J| of the string
\verb|S| is initialized, which is not sufficient here. What we need is to
specify in a loop invariant that elements up to the current index have been
initialized, using the attribute \verb|‘Initialized| in SPARK:
\begin{lstlisting}
pragma Loop_Invariant (for all K in S'First .. J => S (K)'Initialized);
\end{lstlisting}
With that additional loop invariant, everything is proved about \verb|Erase|,
including the additional postcondition that \verb|S| is fully initialized on
return:
\begin{lstlisting}
procedure Erase (S : out String)
with Post => All_Blanks (S) and then S'Initialized,
Relaxed_Initialization => S;
\end{lstlisting}
As visible from this example, looking at the innards of proof requires some
expertise which can only be acquired with time, to understand the mapping from
source language constructs to logical encodings. The use of source variable
names like\verb|S| and \verb|J| in the Verification Condition presented to the
user is a first step towards more systematic roundtrip translation into
constructs at the source code level, to facilitate this understanding.
\section{Related and Future Works}
Counterexamples are the main feature discussed in the context of interacting
with program proof tools~\cite{Cok2010,goues11sefm}. Ideally, a counterexample
captures in an understandable executable trace why a property cannot be proved,
by exhibiting a consistent example where the property does not hold. In
reality, after multiple person-year efforts to develop and improve
counterexamples in the context of SPARK~\cite{dailler:hal-01802488}, there is
still much to be desired here.
On the one hand, counterexamples are very valuable to beginners, to point at
implicit assumptions they might have about the language or program semantics as
well as misunderstandings about the way program proof in general or the
specific SPARK proof tools work. This is particularly valuable for proving
implicit properties of programs like absence of run-time errors, as programmers
are not used to thinking about non-happy paths: How can a signed integer
division overflow? How can a floating-point multiplication between positive
values give zero as a result? On the other hand, counterexamples can be
confusing when they are either spurious because they only reflect a limitation
of the underlying provers (e.g. regarding non-linear arithmetic) or they do not
represent a possible execution but only a limitation of the approach
(e.g. related to the frame problem).
To reduce the possibility of confusion, we decided recently in SPARK to only
enable counterexamples by default at higher levels of proof effort (when
multiple provers are invoked for more than a few seconds per Verification
Condition), so that counterexamples are generated in fewer cases and only in
cases deemed difficult to prove. In parallel, we are working on using symbolic
execution to verify the execution trace represented by a counterexample, in
order to filter out spurious counterexamples and to better label the underlying
error as a property violation or as a frame problem. Previous work on symbolic
or concrete execution have shown benefits for better exploiting
counterexamples~\cite{muller11fm,jacobs11nfm,Hentschel2016,christakis16tacas,petiot16tap}.
Our experience with SPARK is that engineers have difficulties understanding the
axiomatic semantics on which program proof is based. Their main mode of
reasoning about programs is through whole-program operational semantics, which
is supported by tools such as debuggers, fuzzers, profilers, etc. It is thus
critical to provide tool feedback highlighting gaps between the two semantics
which may explain why a property cannot be proved, such as the messages that we
presented in the section~\ref{sec:the-investigator}. This is an area where we
will continue to look for ideas and improvements in the coming years.
\section{Conclusion}
Program proof is intrinsically an interactive effort between a human and a
machine, as complete automation is not achievable. Thus, we are doomed to hit
the so-called Left-Over Principle of automation~\cite{leftover,leftover2},
which is that tasks that are not automated are precisely tasks where humans may
not fare well either, because they are very infrequent or complex. More
generally, the general understanding of the cooperation between the user and
the tool in program proof could be improved, which may require the help of
cognitive science~\cite{tenchallenges}. And notations used to convey
information to the tool could also benefit from the point of view of cognitive
science~\cite{cognitive}. Just looking at the specific issue of tool messages,
the lessons learned from research on compiler error messages have direct
implications for the design and implementation of program proof
tools~\cite{comperr}. In this article, we presented the current state of such
machine-to-human interactions in the SPARK technology, in the hope that it can
trigger interesting human-to-human interactions in the community around proof
tools.
\nocite{*}
\bibliographystyle{eptcs}
|
1,116,691,500,833 | arxiv | \section{Introduction}\label{introduction}
This paper considers finite-level M/G/1-type Markov chains, which belong to a special class of upper block-Hessenberg Markov chains. Finite-level M/G/1-type Markov chains appear in the analysis of finite semi-Markovian queues (see, e.g.,~\cite{Baio94a,Baio93b,Cuyt03,Herr01}). Except for a few special cases~\cite{Akar98, KimBara16}, the stationary distribution in finite-level M/G/1-type Markov chains does not have any simple and analytical expression. Hence, several researchers have derived approximate and/or asymptotic formulae for the stationary distribution in finite-level M/G/1-type Markov chains and related ones, such as finite-level GI/M/1-type Markov chains and finite-level quasi-birth-and-death processes (QBDs).
Miyazawa~et~al.~\cite{Miya07} present an asymptotic formula for the stationary probability of a finite-level QBD being in the maximum level. The asymptotic formula is used to investigate the asymptotic behavior of the loss probability of a MAP/MSP/$c$/$K+c$ queue. J.~Kim and B.~Kim~\cite{KimJeon07} extend the asymptotic formula in \cite{Miya07} to the finite-level GI/M/1-type
Markov chain. Ishizaki and Takine~\cite{Ishi99a} consider a special finite-level M/G/1-type Markov chain with level-decreasing jumps governed by a block matrix of rank one, and they show a direct relation of such a chain to its infinite-level limit. Baiocchi~\cite{Baio94a} derives a geometric asymptotic
formula for the loss probability of a MAP/G/1/$K$ queue, through the asymptotic analysis of a finite-level M/G/1-type Markov chain with light-tailed level increments.
Liu and Zhao \cite{LiuYuan13} present power-law asymptotic
formulas for the loss probability of an M/G/1/$N$ queue with vacations, where the embedded queue length process is a special finite-level M/G/1-type Markov chain with a single background state.
The main contribution of this paper is to present a subgeometric convergence formula for the stationary distribution of the finite-level M/G/1-type Markov chain as its maximum level goes to infinity. The infinite-level limit of a finite-level M/G/1-type Markov chain is an infinite-level (and thus ordinary) M/G/1-type Markov chain. For simplicity, we refer to finite- and infinite-level M/G/1-type Markov chains as {\it finite- and infinite-level chains}, respectively.
The key to our analysis is a probabilistically interpretable solution of the Poisson equation of the {\it deviation matrix} $\vc{D}$ (see, i.e., \cite{Cool02}). We refer to the probabilistically interpretable solution as the {\it fundamental deviation matrix} $\vc{H}$. The fundamental deviation matrix $\vc{H}$ yields a difference formula for the respective stationary distributions of the finite-level chain and its infinite-level limit. Moreover, the combination of the difference formula with the block decomposition of $\vc{H}$ yields a subgeometric convergence formula for the stationary distribution of the finite-level chain as its maximum level goes to infinity, provided that the equilibrium distribution of level increments is subexponential. Finally, the subgeometric convergence formula leads to an asymptotic formula for the loss probability of a MAP/G/1/$N+1$ queue with the subexponential equilibrium service-time distribution.
The rest of this paper consists of five sections. Section~\ref{sec-MG1} describes finite- and infinite-level M/G/1-type Markov chains. Section~\ref{sec-second-order-moment} provides basic results under the second-order moment condition for level increments of the infinite-level chain. Section~\ref{sec-diff} discusses the difference between the respective stationary distributions of the finite-level chain and its infinite-level limit, through the fundamental deviation matrix $\vc{H}$. Section~\ref{sec-convergence} presents the main results of this paper, which are concerned with the convergence of the stationary distribution of the finite-level chain as its maximum level goes to infinity. Section~\ref{sec-application} considers the application of the main results to the asymptotic analysis of the loss probability of the MAP/G/1/$N+1$ queue.
\section{Model description and basic results}\label{sec-MG1}
We introduce mathematical symbols and notation.
Let
\begin{align*}
\bbZ &=\{0, \pm1, \pm2,\ldots\}, &
\bbZ_+ &= \{0,1,2,\dots\}, &
\bbN &= \{1,2,3,\dots\},
\\
\bbZ_{\ge k} &= \{\ell \in \bbZ; \ell \ge k\}, & k &\in \bbZ,
\\
\bbZ_{[k,\ell]} &= \{k,k+1,\dots,\ell\}, & k,\ell &\in \bbZ,~ k \le \ell.
\end{align*}
We then define
\[
\bbM_0 = \bbZ_{[1,M_0]} = \{1, 2, \ldots,M_0\} \subset \bbN,
\qquad
\bbM_1 = \bbZ_{[1,M_1]} = \{1, 2, \ldots, M_1\} \subset \bbN.
\]
We also define $x \vmin y = \min(x,y)$ for $x,y \in (-\infty,\infty)$.
In addition, for matrices (including vectors), the absolute value operator $|\,\cdot\,|$ works on them elementwise.
Let $\vc{O}$ and $\vc{I}$ denote the zero matrix and identity matrix, respectively, with appropriate sizes (i.e., with appropriate numbers of rows and columns).
Finally, let $\one(\,\cdot\,)$ denote an indicator function that takes value of one if the statement in the parentheses is true; otherwise takes the value of zero.
\subsection{The infinite-level M/G/1-type Markov chain}
Let $\{(X_{\nu}, J_{\nu});\nu \in \bbZ_+\}$ denote
a discrete-time Markov chain on state space
$\bbS: = \bigcup_{k=0}^{\infty} \bbL_k$,
where $\bbL_k = \{k\} \times \bbM_{k \vmin 1}$ for $k \in \bbZ_+$. The subset $\bbL_k$ of state space $\bbS$ is referred to as {\it level $k$}. For later use, we define $\bbL_{\ge k}$ and $\bbL_{\le k}$, $k \in \bbZ_+$, as
\begin{align*}
&&&&
\bbL_{\ge k} &= \bigcup_{\ell=k}^{\infty} \bbL_{\ell},
&
\bbL_{\le k} &= \bigcup_{\ell=0}^k \bbL_{\ell},
&
k \in \bbZ_+.
&&&&
\end{align*}
Let $\vc{P}$ denote the transition probability matrix of the Markov chain $\{(X_{\nu}, J_{\nu})\}$, which is in the form:
\begin{equation}
\vc{P}
=\bordermatrix{
& \bbL_0
& \bbL_1
& \bbL_2
& \bbL_3
& \cdots
\cr
\bbL_0
& \bm{B}(0)
& \bm{B}(1)
& \bm{B}(2)
& \bm{B}(3)
& \cdots
\cr
\bbL_1
& \bm{B}(-1)
& \bm{A}(0)
& \bm{A}(1)
& \bm{A}(2)
& \cdots
\cr
\bbL_2
& \bm{O}
& \bm{A}(-1)
& \bm{A}(0)
& \bm{A}(1)
& \cdots
\cr
\bbL_3
& \bm{O}
& \bm{O}
& \bm{A}(-1)
& \bm{A}(0)
& \cdots
\cr
~\vdots
& \vdots
& \vdots
& \vdots
& \vdots
& \ddots
}.
\label{defn-P}
\end{equation}
By definition,
\begin{align}
\sum_{k=-1}^{\infty}\vc{A}(k)\vc{e} &= \vc{e},
\qquad
\sum_{k=0}^{\infty}\vc{B}(k)\vc{e} = \vc{e},
\label{cond-finite-MG1-01}
\\
\vc{B}(-1)\vc{e} &= \vc{A}(-1)\vc{e},
\label{cond-finite-MG1-03}
\end{align}
where $\vc{e}$ denotes the column vector of ones with an appropriate dimension.
The Markov chain $\{(X_{\nu}, J_{\nu})\}$ is referred to as an {\it M/G/1-type Markov chain} (see \cite{Neut89}). To distinguish this Markov chain from its finite-level version (described later), we call $\{(X_{\nu}, J_{\nu})\}$ the {\it infinite-level M/G/1-type Markov chain} or {\it infinite-level chain} for short.
For later use, we define some symbols.
Let
\begin{align}
&&&&
\ol{\vc{A}}(k)
&= \sum_{\ell=k+1}^{\infty} \vc{A}(\ell),
&&&&
k \in \bbZ_{\ge -2},
\label{defn-ol{A}(k)}
\\
&&&&
\ol{\vc{B}}(k)
&= \sum_{\ell=k+1}^{\infty} \vc{B}(\ell),
&&&&
k \in \bbZ_+.
&&&&
\label{defn-ol{B}(k)}
\end{align}
It then follows from (\ref{cond-finite-MG1-01}) that
\begin{eqnarray}
\lim_{k\to\infty}\ol{\vc{A}}(k) &=& \vc{O},
\qquad
\lim_{k\to\infty}\ol{\vc{B}}(k) = \vc{O}.
\label{lim-ol{A}(k)=O}
\end{eqnarray}
Furthermore, let
\begin{eqnarray}
\vc{A} &=& \sum_{\ell=-1}^{\infty}\vc{A}(\ell),
\qquad
\vc{\beta}_{\! A} = \sum_{k=-1}^{\infty} k \vc{A}(k)\vc{e},
\label{defn-beta_A}
\end{eqnarray}
where $\vc{A}$ is a stochastic matrix. We then make the following assumption, throughout this paper.
\begin{assumpt} \label{assumpt-ergodic-MG1}
(i) The transition probability matrix $\vc{P}$ is irreducible;
(ii) $\vc{A}$ is irreducible;
(iii) $\sum_{k=1}^{\infty} k\vc{B}(k)\vc{e}$ is finite; and
(iv) $\sigma :=\vc{\varpi}\vc{\beta}_{\! A} < 0$,
where $\vc{\varpi}$ denotes the unique stationary distribution vector of $\vc{A}$.\end{assumpt}
Assumption~\ref{assumpt-ergodic-MG1} ensures (see, e.g., \cite[Chapter XI, Proposition~3.1]{Asmu03}) that $\{(X_{\nu}, J_{\nu})\}$ is irreducible and positive recurrent, and thus it has the unique stationary distribution, denoted by $\vc{\pi} = (\pi(k,i))_{(k,i)\in\bbS}$.
To describe the stationary distribution $\vc{\pi}$, we introduce the $G$- and $R$-matrices of the infinite-level M/G/1-type Markov chain. Let $\vc{G}:=(G_{i,j})_{i,j\in\bbM_1}$ denote an $M_1 \times M_1$ matrix such that
\[
G_{i,j}
=
\PP (J_{T_n} = j \mid (X_0, J_0) = (n+1, i) \in \bbL_{\ge 2}),
\]
where $T_n = \inf\{\nu \in \bbN; X_{\nu} = n\}$ for $n \in \bbZ_+$.
Assumption~\ref{assumpt-ergodic-MG1} (ii) and (iv) ensure that $\vc{G}$ is a stochastic matrix that is the minimal nonnegative
solution of the matrix equation (see \cite[Eq.~(2.3.3) and Theorem~2.3.1]{Neut89}):
\begin{equation*}
\mathscr{G}
=
\sum_{m=-1}^{\infty}
\vc{A}(m)\mathscr{G}^{m+1}.
\end{equation*}
Furthermore, $\vc{G}$ has the unique closed communicating class
\cite[Proposition 2.1]{Kimu10} and thus the unique stationary distribution vector, denoted by $\vc{g}$.
For $k \in \bbN$, let $\vc{R}(k):=(R_{i,j}(k))_{(i,j)\in\bbM_1\times\bbM_1}$ and $\vc{R}_0(k):=(R_{0,i,j}(k))_{(i,j)\in\bbM_0\times\bbM_1}$ denote
\begin{alignat}{2}
\vc{R}(k)
&=
\sum_{m=k}^{\infty}
\vc{A}(m)\vc{G}^{m-k} (\vc{I}-\vc{\Phi}(0))^{-1},& \qquad k & \in \bbN,
\label{def-R(k)}
\\
\vc{R}_0(k)
&=
\sum_{m=k}^{\infty}
\vc{B}(m)\vc{G}^{m-k} (\vc{I}-\vc{\Phi}(0))^{-1},& \qquad k & \in \bbN,
\label{def-R_0(k)}
\end{alignat}
respectively, where
\begin{equation}
\vc{\Phi}(0)
= \sum_{m=0}^{\infty} \vc{A}(m)\vc{G}^{m}.
\label{defn-Phi(0)}
\end{equation}
The matrices $\vc{R}(k)$ and $\vc{R}_0(k)$ have the
following probabilistic interpretations:
\begin{align*}
R_{i,j}(k)
&=
\EE_{(n,i)} \!\! \left[\,\, \sum_{\nu=1}^{T_{\le n+k-1}}
\one((X_{\nu}, J_{\nu}) = (n+k, j))\right] \quad \mbox{with $(n,i) \in \bbL_{\ge 1}$},
\\
R_{0,i,j}(k)
&=
\EE_{(0,i)} \!\! \left[\,\, \sum_{\nu=1}^{T_{\le k-1}}
\one((X_{\nu}, J_{\nu}) = (k,j))\right],
\end{align*}
where $T_{\le k} = \inf\{\nu \in \bbN; X_{\nu} \le k\}$ for $k \in \bbZ_+$, and where
\begin{align*}
\EE_{(k,i)}[\,\cdot\,]
&= \EE[\,\cdot\mid (X_0, J_0)=(k,i)], \qquad (k,i)\in\bbS.
\end{align*}
Let $\vc{\pi}(k) = (\pi(k,1),\pi(k,2),\dots,\pi(k,M_{k \vmin 1}))$ for $k \in \bbZ_+$. We then have
\begin{equation}
\vc{\pi}(k) = \vc{\pi}(0)\vc{R}_0(k)
+ \sum_{\ell=1}^{k-1}\vc{\pi}(\ell)\vc{R}(k-\ell),
\qquad k \in \bbN,
\label{eqn-pi(k)}
\end{equation}
which is referred to as {\it Ramaswami's recursion} \cite{Rama88}.
\subsection{The finite-level M/G/1-type Markov chain}
For $N\in\bbN$,
let $\vc{P}^{(N)}$ denote a stochastic matrix such that
\begin{equation}
\vc{P}^{(N)} =
\left(
\begin{array}{@{}c@{~}c@{~}c@{~}c@{~}c@{~}c@{~}c@{}}
\vc{B}(0)
& \vc{B}(1)
& \vc{B}(2)
& \cdots
& \vc{B}(N-2)
& \vc{B}(N-1)
& \vc{B}^{(N)}(N)
\\
\vc{B}(-1)
& \vc{A}(0)
& \vc{A}(1)
& \cdots
& \vc{A}(N-3)
& \vc{A}(N-2)
& \vc{A}^{(N)}(N-1)
\\
\vc{O}
& \vc{A}(-1)
& \vc{A}(0)
& \cdots
& \vc{A}(N-4)
& \vc{A}(N-3)
& \vc{A}^{(N)}(N-2)
\\
\vdots
& \vdots
& \vdots
& \ddots
& \vdots
& \vdots
& \vdots
\\
\vc{O}
& \vc{O}
& \vc{O}
& \cdots
& \vc{A}(0)
& \vc{A}(1)
& \vc{A}^{(N)}(2)
\\
\vc{O}
& \vc{O}
& \vc{O}
& \cdots
& \vc{A}(-1)
& \vc{A}(0)
& \vc{A}^{(N)}(1)
\\
\vc{O}
& \vc{O}
& \vc{O}
& \cdots
& \vc{O}
& \vc{A}(-1)
& \vc{A}^{(N)}(0)
\end{array}
\right),
\label{defn-P^{(N)}}
\end{equation}
where $\vc{B}^{(N)}(N)$ and $\vc{A}^{(N)}(k)$, $k\in\bbZ_{[0,N-1]}$, are $M_0 \times M_1$ and $M_1 \times M_1$ substochastic matrices such that
\begin{alignat}{2}
\vc{B}^{(N)}(N)\vc{e}
&= \ol{\vc{B}}(N-1)\vc{e},
\label{eqn-wt{B}(k)e}
\\
\vc{A}^{(N)}(k)\vc{e}
&= \ol{\vc{A}}(k-1)\vc{e},
&\qquad
k &\in \bbZ_{[0,N-1]}.
\label{eqn-wt{A}(k)e}
\end{alignat}
It then follows from (\ref{lim-ol{A}(k)=O}) and (\ref{eqn-wt{B}(k)e}) that
\begin{equation}
\lim_{k\to\infty}\vc{B}^{(N)}(N) = \vc{O}.
\label{limit-wt{A}(k)-wt{B}(k)}
\end{equation}
A Markov chain with driven by $\vc{P}^{(N)}$ is referred to as a {\it finite-level M/G/1-type Markov chain}. The finite-level M/G/1-type Markov chain always has at least one stationary distribution. We define $\vc{\pi}^{(N)}:=(x^{(N)}(k,i))_{(k,i)\in\bbL_{\le N}}$ as an arbitrary stationary distribution (vector) of $\vc{P}^{(N)}$, i.e.,
\begin{equation}
\vc{\pi}^{(N)}\vc{P}^{(N)} = \vc{\pi}^{(N)},
\qquad \vc{\pi}^{(N)}\vc{e}=\vc{e}.
\label{defn-pi^{(N)}}
\end{equation}
As mentioned in the introduction, this paper discusses the difference between $\vc{\pi}^{(N)}$ and $\vc{\pi}$. For this purpose, if necessary, we append zeros to $\vc{\pi}^{(N)}$ and $\vc{P}^{(N)}$ (keeping their original elements in the original positions) so that they have the same sizes as those of $\vc{\pi}$ and $\vc{P}$, respectively. According to this rule, for example, $\vc{\pi}^{(N)} - \vc{\pi}$ and $\vc{P}^{(N)} - \vc{P}$ are well-defined. Furthermore, (\ref{defn-P}) and (\ref{defn-P^{(N)}}) lead to
\begin{equation}
\lim_{N\to\infty} \vc{P}^{(N)} = \vc{P},
\label{lim-P^{(N)}}
\end{equation}
where this limit holds elementwise.
\begin{rem}
If $\vc{A}^{(N)}(k) = \ol{\vc{A}}(k-1)$ for $k \in \bbZ_{[0,N-1]}$ and $\vc{B}^{(N)}(N) = \ol{\vc{B}}(N-1)$, then $\vc{P}^{(N)}$, $N \in \bbN$, is the {\it last-column-block-augmented truncations} of $\vc{P}$ (see \cite{Masu15-AAP,Masu16-SIMAX,Masu17-LAA,Masu17-JORSJ}).
\end{rem}
\section{The second-order moment condition for level increments}\label{sec-second-order-moment}
This section provides some preliminary results on finite- and infinite-level M/G/1-type Markov chains under the {\it second-order moment condition for level increments}.
\begin{assumpt}[Second-order moment condition for level increments] \label{assum-2nd-moment}
\hfill\\
$\sum_{k=1}^{\infty} k^2 \vc{A}(k)$ and $\sum_{k=1}^{\infty} k^2 \vc{B}(k)$ are finite.
\end{assumpt}
Assumption~\ref{assum-2nd-moment} (together with Assumption~\ref{assumpt-ergodic-MG1}) enables us to establish a Foster-Lyapunov drift condition on $\vc{P}$, which is used to prove that $\sum_{k=1}^{\infty}k\vc{\pi}(k)\vc{e}$ is finite.
The Foster-Lyapunov drift condition ensures that the infinite-level chain reaches level zero in a finite expected time if it starts with the stationary distribution $\vc{\pi}$. These results contribute to the discussion on the convergence of $\{\vc{\pi}^{(N)};N\in\bbN\}$ to $\vc{\pi}$ in the subsequent sections.
\subsection{Foster-Lyapunov drift condition}
We begin with the following proposition.
\begin{prop}\label{prop-Poisson}
If Assumption~\ref{assumpt-ergodic-MG1} (ii) holds, then, for any $c \in (-\infty,\infty)$, the vector $\vc{a} := (a(i))_{i \in \bbM_1}$, given by
\begin{equation}
\vc{a}= (\vc{I} - \vc{A} + \vc{e}\vc{\varpi})^{-1}\vc{\beta}_{\! A}
+ c \vc{e},
\label{defn-alpha}
\end{equation}
is a solution of the following Poisson equation:
\begin{equation}
(\vc{I}- \vc{A})\vc{x}
= -\sigma\vc{e} + \vc{\beta}_{\! A}.
\label{Poisson-EQ-A}
\end{equation}
\end{prop}
\begin{proof}
The inverse $(\vc{I} - \vc{A} + \vc{e}\vc{\varpi})^{-1}$ exists and $\vc{\varpi}(\vc{I} - \vc{A} + \vc{e}\vc{\varpi})^{-1}=\vc{\varpi}$ (see, e.g., \cite[Section 6.1, Theorem~6.2]{Brem99}). Using this equation and $\sigma=\vc{\varpi}\vc{\beta}_{\! A}$, we can confirm that $\vc{x}=\vc{a}$ satisfies (\ref{Poisson-EQ-A}). \qed
\end{proof}
To proceed further, we assume that $c > 0$ is sufficiently large so that $\vc{a} > \vc{0}$. We then define $\vc{v}:=(v(k,i))_{(k,i)\in\bbS}$ and $\vc{f}:=(f(k,i))_{(k,i)\in\bbS}$ as column vectors such that
\begin{alignat}{2}
\vc{v}(k):= (v(k,i))_{i\in\bbM_{k\vmin1}}
&=
\left\{
\begin{array}{ll}
\vc{0}, & k =0,
\\
\dm{1 \over -\sigma} \left(k^2\vc{e} + 2k\vc{a} \right), & k \in\bbN,
\end{array}
\right.
\label{defn-v(k)}
\\
\vc{f}(k):= (f(k,i))_{i\in\bbM_{k\vmin1}}
&=\left\{
\begin{array}{ll}
\vc{e}, & k=0,
\\
(k+1)\vc{e}, & k \in\bbZ_+,
\end{array}
\right.
\label{defn-f(k)}
\end{alignat}
respectively, where the size of $\vc{v}(0)$ and $\vc{f}(0)$ is different from that of $\vc{v}(k)$ and $\vc{f}(k)$, $k \in \bbN$. We also define $\vc{1}_{\bbC}:=(1_{\bbC}(k,i))_{(k,i)\in\bbS}$, $\bbC \subseteq \bbS$, as a column vector such that
\[
1_{\bbC}(k,i) =
\left\{
\begin{array}{ll}
1, & (k,i) \in \bbC,
\\
0, & (k,i) \not\in \bbC.
\end{array}
\right.
\]
\begin{lem}\label{lem-drift-cond}
If Assumptions~\ref{assumpt-ergodic-MG1} and \ref{assum-2nd-moment} hold, then there exist some $b \in (0,\infty)$ and $K \in \bbN$ such that
\begin{align}
\vc{P}\vc{v}
\le \vc{v} - \vc{f} + b \vc{1}_{\bbL_{\le K}}.
\label{drift-cond-Pv}
\end{align}
\end{lem}
\begin{proof}
Let $\vc{P}(k;\ell)$ $k,\ell \in \bbZ_+$, denote a submatrix of $\vc{P}$ that contains the transition probabilities from level $k$ to level $\ell$.
It then follows from (\ref{defn-P}), (\ref{defn-beta_A}), and (\ref{defn-v(k)}) that, for all $k \in \bbZ_{\ge 2}$,
\begin{align}
&
\sum_{\ell=0}^{\infty}\vc{P}(k;\ell)\vc{v}(\ell)
\nonumber
\\
&\quad =
\sum_{\ell=-1}^{\infty} \vc{A}(\ell)\vc{v}(k+\ell)
\nonumber
\\
&\quad =
{1 \over -\sigma}
\left[
\sum_{\ell=-1}^{\infty} (k+\ell)^2 \vc{A}(\ell)\vc{e}
+ 2\sum_{\ell=-1}^{\infty} (k+\ell) \vc{A}(\ell)\vc{a}
\right]
\nonumber
\\
&\quad =
{1 \over -\sigma}
\left[ k^2 \vc{e}
+ 2 k \left(
\vc{\beta}_{\! A} + \vc{A}\vc{a}
\right)
\right]
+ {1 \over -\sigma}
\left[
\sum_{\ell=-1}^{\infty} \ell^2 \vc{A}(\ell)\vc{e}
+ 2\sum_{\ell=-1}^{\infty} \ell \vc{A}(\ell)\vc{a}
\right]
\nonumber
\\
&\quad =
{1 \over -\sigma}
\left[
k^2 \vc{e} + 2k ( \vc{a} + \sigma\vc{e} )
\right]
+
{1 \over -\sigma}
\left[
\sum_{\ell=-1}^{\infty} \ell^2 \vc{A}(\ell)\vc{e}
+ 2\sum_{\ell=-1}^{\infty} \ell \vc{A}(\ell)\vc{a}
\right],
\label{eqn-180628-01}
\end{align}
where the last equality holds because $\vc{\beta}_{\! A} + \vc{A}\vc{a}
= \vc{a} + \sigma\vc{e}$, which follows from Proposition~\ref{prop-Poisson}.
It also follows from (\ref{eqn-180628-01}) and Assumption~\ref{assum-2nd-moment} that $\sum_{\ell=0}^{\infty}\vc{P}(k;\ell)\vc{v}(\ell)$ is finite for all $k \in \bbZ_{\ge 2}$. Similarly, we can confirm that $\sum_{\ell=0}^{\infty}\vc{P}(0;\ell)\vc{v}(\ell)$ and $\sum_{\ell=0}^{\infty}\vc{P}(1;\ell)\vc{v}(\ell)$ are finite.
It remains to show that there exists some $k \in \bbN$ such that
\begin{eqnarray}
\sum_{\ell=0}^{\infty}\vc{P}(k;\ell)\vc{v}(\ell)
&\le& \vc{v}(k) - \vc{f}(k)\quad \mbox{for all $k \in \bbZ_{\ge K+1}$.}
\label{ineqn-sum-P(k;l)v(l)}
\end{eqnarray}
Using (\ref{defn-v(k)}) and (\ref{defn-f(k)}), we rewrite (\ref{eqn-180628-01}) as
\begin{eqnarray}
\lefteqn{
\sum_{\ell=0}^{\infty}\vc{P}(k;\ell)\vc{v}(\ell)
}
\quad &&
\nonumber
\\
&=& \vc{v}(k) - 2k\vc{e}
+ {1 \over -\sigma}
\left[
\sum_{\ell=-1}^{\infty} \ell^2 \vc{A}(\ell)\vc{e}
+ 2\sum_{\ell=-1}^{\infty} \ell \vc{A}(\ell)\vc{a}
\right]
\nonumber
\\
&=& \vc{v}(k) - \vc{f}(k) - (k-1)\vc{e}
+
{1 \over -\sigma}
\left[
\sum_{\ell=-1}^{\infty} \ell^2 \vc{A}(\ell)\vc{e}
+ 2\sum_{\ell=-1}^{\infty} \ell \vc{A}(\ell)\vc{a}
\right],~~ k \in \bbZ_{\ge 2}.\quad~~
\label{eqn-180628-02}
\end{eqnarray}
Clearly, there exists some $K \in \bbN$ such that, for all $k \in \bbZ_{\ge K+1}$,
\[
- (k-1)\vc{e}
+
{1 \over -\sigma}
\left[
\sum_{\ell=-1}^{\infty} \ell^2 \vc{A}(\ell)\vc{e}
+ 2\sum_{\ell=-1}^{\infty} \ell \vc{A}(\ell)\vc{a}
\right]
< \vc{0}.
\]
Combining this and (\ref{eqn-180628-02}) results in (\ref{ineqn-sum-P(k;l)v(l)}). Thus, we have proved that (\ref{drift-cond-Pv}) holds. \qed
\end{proof}
We now make the following assumption.
\begin{assumpt}\label{assumpt-P^{(N)}}
\begin{equation}
\lim_{N\to\infty} N \sup_{k \in \bbZ_{[0,N-1]}}
\left\| \vc{A}^{(N)}(k) - \ol{\vc{A}}(k-1) \right\|_{\infty} = 0,
\label{lim-A^{(N)}(k)}
\end{equation}
where $\|\,\cdot\,\|_{\infty}$ denotes the infinity-norm for matrices.
\end{assumpt}
\begin{lem}\label{lem-drift-cond-02}
Suppose that Assumptions~\ref{assumpt-ergodic-MG1} and \ref{assum-2nd-moment} are satisfied. If Assumption~\ref{assumpt-P^{(N)}} holds, then, for any $\varep > 0$, there exists some $N_{\varep} \in \bbN$ such that
\begin{alignat}{2}
\vc{P}^{(N)}\vc{v}
&\le \vc{v} -(1 - \varep) \vc{f} + b \vc{1}_{\bbL_{\le K}}
&& \quad \mbox{for all $N \in \bbZ_{\ge N_{\varep}}$},
\label{drift-cond-P^{(N)}v}
\end{alignat}
where $b \in (0,\infty)$ and $K \in \bbN$ are the parameters satisfying (\ref{drift-cond-Pv}).
\end{lem}
\begin{proof}
Let $\vc{P}^{(N)}(k;\ell)$ $k,\ell \in \bbZ_{[0,N]}$, denote a submatrix of $\vc{P}^{(N)}$ that contains the transition probabilities from level $k$ to level $\ell$. It then follows from (\ref{defn-P}) and (\ref{defn-P^{(N)}}) that, for $N \in \bbN$,
\begin{align}
&
\sum_{\ell=0}^N \vc{P}^{(N)}(k;\ell)\vc{v}(\ell)
\nonumber
\\
&~~ =
\sum_{\ell=0}^{\infty} \vc{P}(k;\ell)\vc{v}(\ell)
+ \vc{P}^{(N)}(k;N)\vc{v}(N)
- \sum_{\ell=N}^{\infty} \vc{P}(k;\ell)\vc{v}(\ell)
\nonumber
\\
&~~ =
\sum_{\ell=0}^{\infty} \vc{P}(k;\ell)\vc{v}(\ell)
+
\left\{
\begin{array}{@{}ll}
\vc{B}^{(N)}(N-k)\vc{v}(N)
- \dm\sum_{\ell=N}^{\infty} \vc{B}(\ell-k)\vc{v}(\ell), & k=0,
\\
\vc{A}^{(N)}(N-k)\vc{v}(N)
- \dm\sum_{\ell=N}^{\infty} \vc{A}(\ell-k)\vc{v}(\ell), & k \in \bbZ_{[1,N]}.
\end{array}
\right.
\label{eqn-190715-01}
\end{align}
We estimate the second and third terms on the right hand side of (\ref{eqn-190715-01}). Using (\ref{defn-ol{A}(k)}), (\ref{eqn-wt{A}(k)e}), and (\ref{defn-v(k)}), we obtain, for $N \in \bbN$ and $k \in \bbZ_{[1,N]}$,
\begin{align}
&
\vc{A}^{(N)}(N-k)\vc{v}(N)
- \sum_{\ell=N}^{\infty} \vc{A}(\ell-k)\vc{v}(\ell)
\nonumber
\\
&\quad =
{1 \over -\sigma}
\vc{A}^{(N)}(N-k) (N^2\vc{e} + 2N\vc{a})
- {1 \over -\sigma}
\sum_{\ell=N}^{\infty} \vc{A}(\ell-k) ( \ell^2\vc{e} + 2\ell\vc{a} )
\nonumber
\\
&\quad \le
{1 \over -\sigma}
\vc{A}^{(N)}(N-k) (N^2\vc{e} + 2N\vc{a})
- {1 \over -\sigma}
\ol{\vc{A}}(N-k-1) ( N^2\vc{e} + 2N\vc{a} )
\nonumber
\\
&\quad =
{2 \over -\sigma}
N
\left[ \vc{A}^{(N)}(N-k) - \ol{\vc{A}}(N-k-1) \right] \vc{a}
\to \vc{0}~~~\mbox{as $N\to \infty$},
\label{eqn-190717-01}
\end{align}
where the limit as $N\to \infty$ converges uniformly over $k\in \bbZ_{[1,N]}$ due to
Assumption~\ref{assumpt-P^{(N)}}.
Similarly, we have
\begin{align}
&
\vc{B}^{(N)}(N)\vc{v}(N)
- \sum_{\ell=N}^{\infty} \vc{B}(\ell)\vc{v}(\ell)
\nonumber
\\
&\quad
\le
{2 \over -\sigma}N
\left[ \vc{B}^{(N)}(N) - \ol{\vc{B}}(N-1) \right] \vc{a}
\to \vc{0}~~~\mbox{as $N\to \infty$},
\label{eqn-190717-02}
\end{align}
where the limit as $N\to \infty$ of the right hand side follows from (\ref{eqn-wt{B}(k)e}) and Assumption~\ref{assum-2nd-moment}.
The following holds due to (\ref{eqn-190715-01})--(\ref{eqn-190717-02}): For any $\varep > 0$, there exists some $N_{\varep} \in \bbN$ such that
\begin{eqnarray}
\vc{P}^{(N)}\vc{v} \le \vc{P}\vc{v} + \varep\vc{e}
\quad\mbox{for all $N \in \bbZ_{\ge N_{\varep}}$}.
\label{ineqn-P^{(N)}}
\end{eqnarray}
Applying (\ref{drift-cond-Pv}) to (\ref{ineqn-P^{(N)}}), and using $\vc{f} \ge \vc{e}$, we obtain
\begin{eqnarray*}
\vc{P}^{(N)}\vc{v}
&\le& \vc{v} -\vc{f} + \varep\vc{e} + b \vc{1}_{\bbZ_{[0,K]}}
\nonumber
\\
&=& \vc{v} -(1-\varep)\vc{f} - \varep(\vc{f} - \vc{e}) + b \vc{1}_{\bbZ_{[0,K]}}
\nonumber
\\
&\le& \vc{v} -(1-\varep)\vc{f} + b \vc{1}_{\bbZ_{[0,K]}}
\quad\mbox{for all $N \in \bbZ_{\ge N_{\varep}}$},
\end{eqnarray*}
which shows that (\ref{drift-cond-P^{(N)}v}) holds. \qed
\end{proof}
\begin{thm}\label{thm-mean}
Under Assumption~\ref{assumpt-ergodic-MG1}, the following hold:
\begin{enumerate}
\item Assumption~\ref{assum-2nd-moment}
holds if and only if
\begin{eqnarray}
\sum_{k=1}^{\infty} k \vc{\pi}(k) \vc{e} &<& \infty.
\label{sum-k*pi(k)e<b}
\end{eqnarray}
\item If Assumptions~\ref{assum-2nd-moment} and \ref{assumpt-P^{(N)}} hold, then
\begin{eqnarray}
\sup_{N \in \bbN}\sum_{k=1}^N k \vc{\pi}^{(N)}(k) \vc{e} &<& \infty.
\label{sum-k*pi^{(N)}(k)e<b}
\end{eqnarray}
\end{enumerate}
\end{thm}
\begin{proof}
Suppose that Assumption~\ref{assumpt-ergodic-MG1} is satisfied.
If Assumptions~\ref{assum-2nd-moment} and \ref{assumpt-P^{(N)}} are satisfied, then Lemma~\ref{lem-drift-cond} together with (\ref{defn-pi^{(N)}}) implies that, for all sufficiently large $N \in \bbN$,
\begin{align*}
\vc{\pi}^{(N)}\vc{v}
&= \vc{\pi}^{(N)} \vc{P}^{(N)}\vc{v}
\le \vc{\pi}^{(N)} \vc{v} - (1 - \varep) \vc{\pi}^{(N)}\vc{f} + b,
\end{align*}
and thus $\vc{\pi}^{(N)}\vc{f} \le b / (1 - \varep)$. Combining this with (\ref{defn-f(k)}) results in (\ref{sum-k*pi^{(N)}(k)e<b}). The statement (ii) has been proved. It thus remains to prove that the statement (i) holds. The ``only if" part of the statement (i) can be proved in a similar way to the proof of the statement (ii). Therefore, we prove the ``if" part of the statement (i); that is, (\ref{sum-k*pi(k)e<b}) implies Assumption \ref{assum-2nd-moment}, under Assumption~\ref{assumpt-ergodic-MG1}.
It follows from (\ref{eqn-pi(k)}) that
\begin{align*}
\sum_{k=1}^{\infty} k\vc{\pi}(k)\vc{e}
&=\vc{\pi}(0) \sum_{k=1}^{\infty} k\vc{R}_0(k)\vc{e}
+
\sum_{\ell=1}^{\infty} \vc{\pi}(\ell)
\sum_{k=\ell+1}^{\infty} k \vc{R}(k-\ell)\vc{e}
\nonumber
\\
&=\vc{\pi}(0) \sum_{k=1}^{\infty} k\vc{R}_0(k)\vc{e}
+
\sum_{\ell=1}^{\infty} \vc{\pi}(\ell)
\sum_{k=\ell+1}^{\infty} (k-\ell) \vc{R}(k-\ell)\vc{e}
\nonumber
\\
&{} \quad +
\sum_{\ell=1}^{\infty} \ell \vc{\pi}(\ell)
\sum_{k=\ell+1}^{\infty} \vc{R}(k-\ell)\vc{e},
\end{align*}
which is finite due to (\ref{sum-k*pi(k)e<b}). Thus, $\sum_{k=1}^{\infty} k\vc{R}(k)\vc{e} $ and $\sum_{k=1}^{\infty} k\vc{R}_0(k)\vc{e}$ are finite. It also follows from (\ref{def-R(k)}), $\vc{G}\vc{e}=\vc{e}$ and $(\vc{I} - \vc{\Phi}(0))^{-1}\vc{e} \ge \vc{e}$ that
\begin{eqnarray*}
\sum_{k=1}^{\infty} k\vc{R}(k)\vc{e}
&=&
\sum_{k=1}^{\infty} k
\sum_{m=k}^{\infty}
\vc{A}(m)\vc{G}^{m-k} (\vc{I}-\vc{\Phi}(0))^{-1}\vc{e}
\nonumber
\\
&\ge&
\sum_{k=1}^{\infty} k
\sum_{m=k}^{\infty}
\vc{A}(m)\vc{G}^{m-k}\vc{e}
= \sum_{k=1}^{\infty} k
\sum_{m=k}^{\infty}
\vc{A}(m)\vc{e}
\nonumber
\\
&=&
{1 \over 2}\sum_{m=1}^{\infty} m(m+1)
\vc{A}(m)\vc{e}.
\end{eqnarray*}
Therefore, $\sum_{m=1}^{\infty} m^2\vc{A}(m)\vc{e}$ is finite.
Similarly, using (\ref{def-R_0(k)}) and the finiteness of $\sum_{k=1}^{\infty} k\vc{R}_0(k)\vc{e}$, we can prove that $\sum_{m=1}^{\infty} m^2\vc{B}(m)\vc{e}$ is finite. As a result, Assumption~\ref{assum-2nd-moment} holds. \qed
\end{proof}
\subsection{The mean first passage time to level zero}
This subsection provides basic results on the mean first passage time to level zero. Some of them may be found in a paper or book, but we cannot find such an appropriate reference. We thus provides them below, for the reader's convenience.
\begin{lem}\label{lem-u(k)}
Suppose that Assumption~\ref{assumpt-ergodic-MG1} holds, and let
$\vc{u}(k):=(u(k,i))_{i\in\bbM_{k\vmin1}}$, $k\in\bbZ_+$, denote a column vector such that
\begin{equation}
u(k,i) = \EE_{(k,i)}[T_0] \ge 1,\qquad (k,i) \in \bbS.
\label{defn-u(k,i)}
\end{equation}
We then have
\begin{eqnarray}
\vc{u}(0) &=& \vc{e} + \sum_{m=1}^{\infty} \vc{B}(m)
(\vc{I} - \vc{G}^m)
(\vc{I} - \vc{A} - \vc{\beta}_{\! A}\vc{g} )^{-1}\vc{e}
+ \sum_{m=1}^{\infty} {m\vc{B}(m) \over -\sigma}\vc{e},
\label{eqn-u(0)}
\\
\vc{u}(k) &=& (\vc{I} - \vc{G}^k)(\vc{I} - \vc{A} - \vc{\beta}_{\! A}\vc{g} )^{-1}\vc{e}
+ {k \over -\sigma}\vc{e},
\qquad k\in\bbN.
\label{eqn-u(k)}
\end{eqnarray}
\end{lem}
\begin{proof}
Let $\wh{\vc{G}}(z):=(\wh{G}_{i,j}(z))_{(i,j)\in\bbM_1 \times \bbM_1}$,
$\wh{\vc{G}}_0(z):=(\wh{G}_{0,i,j}(z))_{(i,j)\in\bbM_1 \times \bbM_0}$
and $\wh{\vc{K}}(z):= \break (\wh{K}_{i,j}(z))_{(i,j)\in\bbM_0 \times \bbM_0}$ denote $M_1 \times M_1$, $M_1 \times M_0$ and $M_0 \times M_0$ matrices, respectively, such that
\begin{alignat*}{2}
\wh{G}_{i,j}(z)
&= \EE_{(n+1,i)} \!\left [z^{T_n}\dd{1}(X_{T_n} = j) \right]
\quad \mbox{with $n\in\bbN$},
\\
\wh{G}_{0,i,j}(z)
&= \EE_{(1,i)} \!\left [z^{T_0}\dd{1}(X_{T_0} = j) \right],
\\
\wh{K}_{i,j}(z)
&= \EE_{(0,i)} \!\left [z^{T_0}\dd{1}(X_{T_0} = j) \right],
\end{alignat*}
It then follows from (\ref{defn-u(k,i)}) that
\begin{eqnarray}
\vc{u}(0)
&=& \left. {\rd \over \rd z} \wh{\vc{K}}(z) \right|_{z=1} \vc{e},
\label{eqn-u(0)-02}
\\
\vc{u}(k)
&=& \left. {\rd \over \rd z} [ \wh{\vc{G}}(z) ]^{k-1}\wh{\vc{G}}_0(z) \right|_{z=1} \vc{e},\qquad k \in \bbN.
\label{eqn-u(k)-02}
\end{eqnarray}
Note here (see \cite[Eqs.~(2.2.9), (2.4.3), and (2.4.8)]{Neut89}) that
\begin{eqnarray}
\wh{\vc{G}}(z)
&=&
\left[
\vc{I} - z\sum_{m=0}^{\infty}\vc{A}(m) \{ \wh{\vc{G}}(z) \}^m
\right]^{-1} z\vc{A}(-1),
\label{eqn-wh{G}(z)}
\\
\wh{\vc{G}}_0(z)
&=&
\left[
\vc{I} - z\sum_{m=0}^{\infty}\vc{A}(m) \{ \wh{\vc{G}}(z) \}^m
\right]^{-1} z\vc{B}(-1),
\label{eqn-wh{G}_0(z)}
\\
\wh{\vc{K}}(z)
&=& z\vc{B}(0)
+ z\sum_{m=1}^{\infty}\vc{B}(m) \{ \wh{\vc{G}}(z) \}^{m-1} \wh{\vc{G}}_0(z).
\label{eqn-wh{K}(z)}
\end{eqnarray}
From (\ref{eqn-wh{G}(z)}), (\ref{eqn-wh{G}_0(z)}), and (\ref{cond-finite-MG1-03}), we have
\begin{equation}
\wh{\vc{G}}_0(z)\vc{e}
= \wh{\vc{G}}(z)\vc{e}.
\label{eqn-wh{G}_0(z)e}
\end{equation}
Substituting (\ref{eqn-wh{G}_0(z)e}) into (\ref{eqn-u(k)-02}) leads to
\begin{eqnarray}
\vc{u}(k)
&=& \left. {\rd \over \rd z} [ \wh{\vc{G}}(z) ]^k \right|_{z=1} \vc{e},\qquad k \in \bbN.
\label{eqn-u(k)-03}
\end{eqnarray}
Moreover, post-multiplying both sides of (\ref{eqn-wh{K}(z)}) by $\vc{e}$, and using (\ref{eqn-wh{G}_0(z)e}), we obtain
\begin{eqnarray*}
\wh{\vc{K}}(z)\vc{e}
&=& z\vc{B}(0)\vc{e}
+ z\sum_{m=1}^{\infty}\vc{B}(m) \{ \wh{\vc{G}}(z) \}^m \vc{e}.
\end{eqnarray*}
Combining this and (\ref{eqn-u(0)-02}) yields
\begin{align}
\vc{u}(0)
&= \vc{B}(0) \vc{e}
+ \sum_{m=1}^{\infty}\vc{B}(m)\vc{e}
+ \sum_{m=1}^{\infty}\vc{B}(m)
\left. {\rd \over \rd z} [ \wh{\vc{G}}(z) ]^m \right|_{z=1} \vc{e}
\nonumber
\\
&= \vc{e} + \sum_{m=1}^{\infty}\vc{B}(m) \vc{u}(m),
\label{eqn-u(0)-00}
\end{align}
where the second equality holds due to (\ref{eqn-u(k)-03}) and $\sum_{m=0}^{\infty}\vc{B}(m)\vc{e} = \vc{e}$.
We note that (\ref{eqn-u(0)}) is obtained by substituting (\ref{eqn-u(k)}) into (\ref{eqn-u(0)-00}). It thus suffices to prove that (\ref{eqn-u(k)}) holds. From (\ref{eqn-u(k)-03}) and $\wh{\vc{G}}(1)\vc{e}=\vc{G}\vc{e}=\vc{e}$, we have
\begin{equation*}
\vc{u}(k)
= \sum_{n=0}^{k-1} \vc{G}^n \left. {\rd \over \rd z} \wh{\vc{G}}(z) \right|_{z=1} \vc{e} ,
\qquad k \in \bbN.
\label{eqn-u(k)-04}
\end{equation*}
We also have (see \cite[Eqs.~(3.1.3), (3.1.12), and (3.1.14)]{Neut89})
\begin{eqnarray}
\left. {\rd \over \rd z} \wh{\vc{G}}(z) \right|_{z=1} \vc{e}
&=&
(\vc{I} - \vc{G} + \vc{e}\vc{g})
(\vc{I} - \vc{A} - \vc{\beta}_{\! A}\vc{g} )^{-1}\vc{e}
\nonumber
\\
&=&
(\vc{I} - \vc{G} )
(\vc{I} - \vc{A} - \vc{\beta}_{\! A}\vc{g} )^{-1}\vc{e}
+ {1 \over -\sigma}\vc{e}.
\label{eqn-u(1)}
\end{eqnarray}
Combining these two equations yields
\begin{eqnarray*}
\vc{u}(k)
&=& \sum_{n=0}^{k-1} \vc{G}^n(\vc{I} - \vc{G})
(\vc{I} - \vc{A} - \vc{\beta}_{\! A}\vc{g} )^{-1}\vc{e}
+ {k \over -\sigma}\vc{e}
\nonumber
\\
&=& (\vc{I} - \vc{G}^k)(\vc{I} - \vc{A} - \vc{\beta}_{\! A}\vc{g} )^{-1}\vc{e}
+ {k \over -\sigma}\vc{e}
\qquad k \in \bbN,
\end{eqnarray*}
which shows that (\ref{eqn-u(k)}) holds. \qed
\end{proof}
\begin{thm}\label{thm-u(k)}
Suppose that Assumption~\ref{assumpt-ergodic-MG1} is satisfied.
Assumption~\ref{assum-2nd-moment}
holds if and only if
\begin{equation}
\sum_{k=0}^{\infty} \vc{\pi}(k) \vc{u}(k) < \infty.
\label{sum-k*u(k)e-finite}
\end{equation}
\end{thm}
\begin{proof}
Equation (\ref{eqn-u(k)}) leads to
\begin{equation}
\lim_{k\to\infty}{\vc{u}(k) \over k}
= {1 \over -\sigma}\vc{e}.
\label{lim-u(k)}
\end{equation}
Moreover, Assumption~\ref{assumpt-ergodic-MG1} and (\ref{defn-beta_A}) yield
\begin{align*}
0 < -\sigma \le \vc{\varpi} \vc{A}(-1)\vc{e}
\le \vc{\varpi}\vc{e} = 1.
\end{align*}
These results imply that (\ref{sum-k*u(k)e-finite}) is equivalent to (\ref{sum-k*pi(k)e<b}). Consequently, Theorem~\ref{thm-u(k)} follows from Theorem~\ref{thm-mean}~(i). \qed
\end{proof}
\section{The difference between the stationary distributions of the finite-level chain and its infinite-level limit}\label{sec-diff}
This section presents a difference formula for $\vc{\pi}^{(N)}$ and $\vc{\pi}$ through the {\it fundamental deviation matrix} $\vc{H}$ (introduced in this section) of the transition probability matrix $\vc{P}$. The fundamental deviation matrix is a solution of the Poisson equation satisfied by the deviation matrix $\vc{D}$ (see, e.g., \cite{Cool02}). The M/G/1-type structure of $\vc{P}$ enables us to obtain a block decomposition of $\vc{H}$, which is used, together with the difference formula for $\vc{\pi}^{(N)}$ and $\vc{\pi}$, on order to derive an asymptotic formula for $\vc{\pi}^{(N)} - \vc{\pi}$ in the next section.
\subsection{The difference formula via the fundamental deviation matrix}
The following is the basic difference formula for $\vc{\pi}^{(N)}$ and $\vc{\pi}$
\begin{prop}[Basic difference formula]\label{prop-diff}
Let $\mathscr{H}$ denote an arbitrary solution of the Poisson equation (see, e.g., \cite{Mako02}):
\begin{equation}
(\vc{I} - \vc{P}) \mathscr{H} = \vc{I} - \vc{e}\vc{\pi}.
\label{Poisson-EQ}
\end{equation}
We then have
\begin{equation}
\vc{\pi}^{(N)}-\vc{\pi} = \vc{\pi}^{(N)} ( \vc{P}^{(N)} - \vc{P} ) \mathscr{H}.
\label{diff-formula}
\end{equation}
\end{prop}
\begin{proof}
It is implied in \cite{LiuYuan18-AAP} and \cite{LLM2018} that this proposition holds. However, for completeness, we provide the proof: Using (\ref{Poisson-EQ}) and $\vc{\pi}^{(N)}=\vc{P}^{(N)}\vc{\pi}^{(N)}$, we have
\begin{eqnarray*}
\vc{\pi}^{(N)} ( \vc{P}^{(N)} - \vc{P} ) \mathscr{H}
&=& \vc{\pi}^{(N)} ( \vc{I} - \vc{P} ) \mathscr{H}
= \vc{\pi}^{(N)} (\vc{I} - \vc{e}\vc{\pi})
= \vc{\pi}^{(N)} - \vc{\pi},
\end{eqnarray*}
which shows that (\ref{diff-formula}) holds. \qed
\end{proof}
Fix $(k',i') \in \bbS$ arbitrarily, and let $\vc{H}:=(H(k,i;\ell,j))_{(k,i;\ell,j)\in\bbS^2}$ denote a matrix such that
\begin{equation}
H(k,i;\ell,j)
=\EE_{(k,i)}\!\!
\left[
\sum_{\nu=0}^{T_{(k',i')}-1} \dd{1}((X_{\nu},J_{\nu})=(\ell,j))
\right]
- \pi(\ell,j)\EE_{(k,i)}\!
\left[ T_{(k',i')} \right],
\label{defn-H}
\end{equation}
where $T_{(k',i')} = \inf\{\nu \in \bbN; (X_{\nu},J_{\nu}) = (k',i')\}$. Under Assumption~\ref{assumpt-ergodic-MG1}, $\vc{H}$ is well-defined, and its subvector $\vc{h}(\ell,j):=(H(k,i;\ell,j))_{(k,i)\in\bbS}$ satisfies the following (see, e.g., \cite[Lemma~2.1]{Dend13}):
\[
(\vc{I} - \vc{P}) \vc{h}(\ell,j) = \vc{1}_{\{(\ell,j)\}} - \pi(\ell,j)\vc{e},
\qquad (\ell,j) \in \bbS,
\]
which shows that $\vc{H}$ is a solution of Poisson equation (\ref{Poisson-EQ}). Proposition~\ref{prop-diff} thus yields a concrete difference formula:
\begin{equation}
\vc{\pi}^{(N)}-\vc{\pi} = \vc{\pi}^{(N)} ( \vc{P}^{(N)} - \vc{P} ) \vc{H}.
\label{diff-formula-H}
\end{equation}
The matrix $\vc{H}$ is closely related to the deviation matrix $\vc{D}$, which is defined as
\begin{equation*}
\vc{D} = \sum_{n=0}^{\infty} ( \vc{P}^n - \vc{e}\vc{\pi} ).
\end{equation*}
The deviation matrix $\vc{D}$ (if it exists) is the unique solution of Poisson equation (\ref{Poisson-EQ}) with the constraints (see, e.g., \cite[Lemma~2.7]{Dend13}):
\begin{eqnarray}
&&\mbox{$\vc{\pi}\mathscr{H} = \vc{0}$},
\label{ineqn-const-01}
\\
&&\mbox{$\vc{\pi}|\mathscr{H}|$ is finite}.
\label{ineqn-const-02}
\end{eqnarray}
In summary, $\vc{D}$ (if exists) and $\vc{H}$ are the solutions of the same Poisson equation (\ref{Poisson-EQ}).
Proposition~\ref{prop-D-H} below shows that the existence condition of $\vc{D}$ is stronger than that of $\vc{H}$. More specifically, $\vc{D}$ requires the aperiodicity of $\vc{P}$ and Assumption~\ref{assum-2nd-moment} whereas $\vc{H}$ does not necessarily them. Furthermore, $\vc{D}$ can be expressed with $\vc{H}$ (see Remark~\ref{rem-D-H} below). These facts imply that $\vc{H}$ is more {\it fundamental} than the deviation matrix $\vc{D}$. This is why we focus on $\vc{H}$ instead of $\vc{D}$. For convenience, we refer to $\vc{H}$ as the {\it fundamental deviation matrix}.
\begin{prop}\label{prop-D-H}
Suppose that Assumption~\ref{assumpt-ergodic-MG1} is satisfied and $\vc{P}$ is aperiodic. Assumption~\ref{assum-2nd-moment} holds if and only if $\vc{D}$ exists.
\end{prop}
\begin{proof}
Assumption~\ref{assumpt-ergodic-MG1}, together with the aperiodicity of $\vc{P}$, ensures that the Markov chain $\{(X_{\nu},J_{\nu})\}$ is ergodic. It thus follows from \cite[Theorem~4.1]{Cool02} that the deviation matrix $\vc{D}$ exists if and only if
\begin{equation}
\sum_{(k,i)\in\bbS} \pi(k,i)\EE_{(k,i)}[T_{(0,j)}] < \infty.
\label{sum-pi(k,i)*E[T_{(0,j)}]-finite}
\end{equation}
Moreover, for all $j \in \bbM_0$ and $(k,i) \in \bbS$,
\begin{eqnarray*}
\EE_{(k,i)}[T_0]
\le
\EE_{(k,i)}[T_{(0,j)}]
&=& \EE_{(k,i)}[T_0 \dd{1}(J_{T_0} = j)]
\nonumber
\\
&& {} + \EE_{(k,i)}
\left[\dd{1}(J_{T_0} \neq j)
\{ T_0 + \EE[ T_{(0,j)} - T_0 \mid J_{T_0} \neq j] \}
\right]
\nonumber
\\
&\le& \EE_{(k,i)}[T_0] + \EE[ T_{(0,j)} \mid X_0 = 0, J_0 \neq j],
\end{eqnarray*}
where $\EE[ T_{(0,j)} \mid X_0 = 0, J_0 \neq j] < \infty$ due to
the ergodicity of $\{(X_{\nu},J_{\nu})\}$. Thus, (\ref{sum-pi(k,i)*E[T_{(0,j)}]-finite}) holds if and only if
\begin{equation*}
\sum_{(k,i)\in\bbS} \pi(k,i)\EE_{(k,i)}[T_0]
= \sum_{k=0}^{\infty} \vc{\pi}(k) \vc{u}(k) < \infty,
\end{equation*}
where the equality is due to (\ref{defn-u(k,i)}). As a result, $\sum_{k=0}^{\infty} \vc{\pi}(k) \vc{u}(k) < \infty$ and thus Assumption~\ref{assum-2nd-moment} holds (see Theorem~\ref{thm-u(k)}) if and only if the deviation matrix $\vc{D}$ exists. \qed
\end{proof}
\subsection{Block decomposition of the fundamental deviation matrix}
\label{sec-deviation}
We partition $\vc{P}$ in (\ref{defn-P}) as
\begin{equation}
\vc{P} =
\bordermatrix{
& \bbL_0& \bbL_{\ge1} \cr
\bbL_0& \vc{B}(0) & \vc{P}_{0+} \cr
\bbL_{\ge1} & \vc{P}_{+0} & \vc{P}_{+}
},
\label{partition-P}
\end{equation}
where
\begin{align}
\vc{P}_{0+} &= \begin{pmatrix}
\vc{B}(1) & \vc{B}(2) & \vc{B}(3) & \cdots
\end{pmatrix},
\label{defn-P_{0+}}
\\
\vc{P}_{+0} &= \begin{pmatrix}
\vc{B}(-1) \\
\vc{O} \\
\vc{O} \\
\vdots
\end{pmatrix},
\qquad
\vc{P}_{+} =
\begin{pmatrix}
\vc{A}(0) &
\vc{A}(1) &
\vc{A}(2) &
\vc{A}(3) &
\cdots
\\
\vc{A}(-1) &
\vc{A}(0) &
\vc{A}(1) &
\vc{A}(2) &
\cdots
\\
\vc{O} &
\vc{A}(-1) &
\vc{A}(0) &
\vc{A}(1) &
\cdots
\\
\vc{O} &
\vc{O} &
\vc{A}(-1) &
\vc{A}(0) &
\cdots
\\
\vdots &
\vdots &
\vdots &
\vdots &
\ddots
\end{pmatrix}.
\nonumber
\end{align}
We then define $\vc{F}_+:=(F_+(k,i;\ell,j))_{(k,i;\ell,j)\in(\bbL_{\ge1})^2}$ as the fundamental matrix of $\vc{P}_{+}$, i.e.,
\begin{equation}
\vc{F}_{+} =
\sum_{m=0}^{\infty} \vc{P}_{+}^m = (\vc{I} - \vc{P}_{+})^{-1}.
\label{defn-F_+}
\end{equation}
Note that
\begin{equation}
F_+(k,i;\ell,j)
= \EE_{(k,i)}\!\!
\left[
\sum_{\nu=0}^{T_0-1} \one((X_{\nu},J_{\nu}) = (\ell,j))
\right],\qquad (k,i;\ell,j) \in (\bbL_{\ge 1})^2,
\label{eqn-F_+}
\end{equation}
where $\{(X_{\nu}, J_{\nu})\}$ is the infinite-level chain with the transition probability matrix $\vc{P}$.
From (\ref{partition-P}), (\ref{defn-F_+}) and $\vc{\pi}\vc{P}=\vc{\pi}$, we have
\[
(\vc{\pi}(1),\vc{\pi}(2),\dots)
= \vc{\pi}(0)\vc{P}_{0+} \vc{F}_{+}.
\]
This equation and (\ref{defn-P_{0+}}) yield
\begin{equation}
\vc{\pi}(\ell) = \vc{\pi}(0)\sum_{m=1}^{\infty}\vc{B}(m)\vc{F}_{+}(m;\ell),
\qquad \ell \in \bbN,
\label{eqn-pi(l)}
\end{equation}
where $\vc{F}_{+}(k;\ell) =(F_+(k,i;\ell,j))_{(i,j) \in (\bbM_1)^2}$ for $k,\ell \in \bbN$. The block matrices $\vc{F}_{+}(k;\ell)$, $k,\ell \in \bbN$, of $\vc{F}_{+}$ satisfy the following recursive formula.
\begin{prop}[{}{\cite[Theorem 9]{Zhao03}}]\label{prop-F_+}
If Assumption~\ref{assumpt-ergodic-MG1} holds, then, for $k,\ell \in \bbN$,
\begin{eqnarray}
\vc{F}_{+}(k;\ell)
&=&
\begin{cases}
\vc{G}\vc{F}_{+}(k-1;\ell), & 1 \le \ell \le k-1,
\\
\vc{F}_{+}(1;1) + \vc{G}\vc{F}_{+}(k-1;k),
& \ell=k,
\\
\displaystyle \sum_{n=1}^{\ell-1} \vc{F}_{+}(k;n) \vc{R}(\ell-n),
& \ell \ge k+1,
\end{cases}
\label{eqn-WH{H}(k;l)}
\end{eqnarray}
where $\vc{F}_{+}(0;m) = \vc{O}$ for $m\in\bbN$.
\end{prop}
We note that $\vc{\Phi}(0)$ in (\ref{defn-Phi(0)}) is a substochastic matrix that contains the transition probabilities of an absorbing Markov chain obtained by observing the infinite-level Markov chain $\{(X_{\nu},J_{\nu})\}$ only when it is in $\bbL_1$ and until it reaches $\bbL_0$. It thus follows from (\ref{eqn-F_+}) that
\begin{equation*}
\vc{F}_{+}(1;1) = \left(\vc{I}-\vc{\Phi}(0)\right)^{-1}.
\end{equation*}
From this equation and (\ref{eqn-WH{H}(k;l)}), we have
\begin{equation}
\vc{F}_{+}(k;1) = \vc{G}^{k-1}(\vc{I} - \vc{\Phi}(0))^{-1},
\qquad k \in \bbN.
\label{eqn-F_{+}(k;1)}
\end{equation}
To describe the block decomposition of $\vc{H}$, we introduce a stochastic matrix associated with $\vc{F}_{+}$.
Let
\begin{equation*}
\wt{\vc{P}}_0 = \vc{B}(0) + \vc{P}_{0+}\vc{F}_{+}\vc{P}_{+0}.
\end{equation*}
The matrix $\wt{\vc{P}}_0$ can be interpreted as the transition probability matrix of a censored Markov chain obtained by observing the Markov chain $\{(X_{\nu},J_{\nu})\}$ only when it is in $\bbL_0$ (see \cite[Theorem~2]{Zhao00}). Therefore, the probability vector
\begin{equation}
\wt{\vc{\pi}}_0 := {\vc{\pi}(0) \over \vc{\pi}(0)\vc{e}}
\label{defn-pi_0^ddagg}
\end{equation}
is the unique stationary distribution of $\wt{\vc{P}}_0$, and $\wt{\vc{\pi}}_0\wt{\vc{P}}_0 = \wt{\vc{\pi}}_0$.
Using Proposition~\ref{prop-F_+}, we obtain the following theorem.
\begin{thm} \label{thm-dev-MG1}
Suppose that Assumption~\ref{assumpt-ergodic-MG1} holds. Let $\vc{H}(k;\ell)$, $k,\ell\in\bbZ_+$, denote the block matrix $(H(k,i;\ell,j))_{(i,j) \in \bbM_{k \vmin 1} \times \bbM_{\ell \vmin 1}}$ of the fundamental deviation matrix $\vc{H}$. We then have
\begin{align}
(\vc{I}-\wt{\vc{P}}_0)\vc{H}(0;0)
&= \vc{I} - \vc{u}(0) \vc{\pi}(0),
\label{eqn-H(0;0)-02}
\\
(\vc{I}-\wt{\vc{P}}_0)\vc{H}(0;\ell)
&=
\sum_{m=1}^{\infty} \vc{B}(m)\vc{F}_+(m;\ell) - \vc{u}(0) \vc{\pi}(\ell),
\qquad \ell \in \bbN,
\label{eqn-H(0;l)-02}
\end{align}
and, for $k \in \bbN$,
\begin{eqnarray}
\vc{H}(k;\ell)
&=&
(1 - \delta_{0,\ell})
\vc{F}_{+}(k;\ell)
\nonumber
\\
&& {} + \vc{G}^{k-1} (\vc{I}-\vc{\Phi}(0))^{-1} \vc{B}(-1)\vc{H}(0;\ell)
-\vc{u}(k) \vc{\pi}(\ell),\qquad \ell \in \bbZ_+,
\label{eqn-H(k;l)-02}
\end{eqnarray}
where $\delta_{k,\ell}$ denotes the Kronecker delta, and
where the $\vc{u}(m)$, $m\in\bbZ_+$, are given in (\ref{eqn-u(0)}) and (\ref{eqn-u(k)}).
\end{thm}
\begin{proof}
Let $\wt{\vc{H}}(k;\ell):=(\wt{H}(k,i;\ell,j))_{(i,j)\in\bbM_{k\vmin1}\times\bbM_{\ell\vmin1}}$, $k,\ell\in\bbZ_+$, denote a matrix such that
\begin{equation}
\wt{H}(k,i;\ell,j)
=\EE_{(k,i)}\!\!
\left[
\sum_{\nu=0}^{T_0-1} \one((X_{\nu},J_{\nu}) = (\ell,j))
\right].
\label{defn-wt{H}}
\end{equation}
It follows from (\ref{eqn-F_+}) and (\ref{defn-wt{H}}) that
\begin{equation}
\wt{\vc{H}}(k;\ell)
= \left\{
\begin{array}{lll}
\delta_{k,0}\vc{I}, & k \in \bbZ_+, & \ell=0,
\\
\dm\sum_{m=1}^{\infty} \vc{B}(m)\vc{F}_{+}(m;\ell),
& k=0, & \ell \in \bbN,
\\
\vc{F}_{+}(k;\ell),
& k\in \bbN, & \ell \in \bbN.
\end{array}
\right.
\label{eqn-wt{H}}
\end{equation}
It also follows from \cite[Theorem~2.5]{Dend13} that
\begin{align}
\vc{H}(0;\ell)
&= \wt{\vc{H}}(0;\ell) - \vc{u}(0)\vc{\pi}(\ell) + \wt{\vc{P}}_0
\vc{H}(0;\ell),
& \ell &\in \bbZ_+,
\label{eqn-H(0;l)}
\\
\vc{H}(k;\ell) &= \wt{\vc{H}}(k;\ell) - \vc{u}(k)\vc{\pi}(\ell)
+ \vc{F}_+(k;1)\vc{B}(-1)\vc{H}(0;\ell),
&
k \in \bbN,~\ell &\in \bbZ_+.
\label{eqn-H(k;l)}
\end{align}
Combining (\ref{eqn-wt{H}}) and (\ref{eqn-H(0;l)}) leads to (\ref{eqn-H(0;0)-02}) and (\ref{eqn-H(0;l)-02}). Furthermore, substituting (\ref{eqn-F_{+}(k;1)}) and (\ref{eqn-wt{H}}) into (\ref{eqn-H(k;l)}) results in (\ref{eqn-H(k;l)-02}). \qed
\end{proof}
Theorem~\ref{thm-dev-MG1} yields the following corollary.
\begin{coro}\label{coro-H-01}
Suppose that Assumption~\ref{assumpt-ergodic-MG1} is satisfied. Let
\begin{eqnarray}
\vc{E}(k;\ell)
&=& (1-\delta_{\ell,0})\vc{G}^{k-\ell}\vc{F}_+(\ell;\ell)
+ \vc{G}^{k-1} (\vc{I}-\vc{\Phi}(0))^{-1} \vc{B}(-1)\vc{H}(0;\ell)
\nonumber
\\
&&{}
- (\vc{I} - \vc{G}^k)
(\vc{I} - \vc{A} - \vc{\beta}_{\! A}\vc{g} )^{-1}\vc{e}\vc{\pi}(\ell),
\quad k \in \bbZ_{\ge \ell} \cap \bbN.\quad
\label{defn-E(k;l)}
\end{eqnarray}
Under these conditions, the following hold: (i) For any $\ell \in \bbZ_+$,
\begin{eqnarray}
\vc{H}(k;\ell)
&=& -{k \over -\sigma}\vc{e}\vc{\pi}(\ell) + \vc{E}(k;\ell),
\qquad k \in \bbZ_{\ge \ell} \cap \bbN,
\label{eqn-decomposition-H(k;l)}
\end{eqnarray}
and (ii) there exists some constant $C_E(\ell) > 0$ such that
\begin{equation}
|\vc{E}(k;\ell)|\le C_E(\ell) \vc{e}\vc{e}^{\top}
\quad\mbox{for all $k \in \bbZ_{\ge \ell} \cap \bbN$}.
\label{lim-E(k;l)}
\end{equation}
(iii) In addition, Assumption~\ref{assum-2nd-moment}
holds if and only if
\begin{eqnarray}
\sum_{(k,i)\in\bbS} \pi(k,i) |H(k,i;\ell,j)| < \infty,\qquad (\ell,j)\in\bbS.
\label{sum-pi*|H|}
\end{eqnarray}
\end{coro}
\begin{proof}
It follows from (\ref{eqn-WH{H}(k;l)}) that
\begin{equation}
\vc{F}_{+}(k;\ell)
= \vc{G}^{k-\ell}\vc{F}_{+}(\ell;\ell),
\qquad \ell \in\bbN,~k \in \bbZ_{\ge \ell}.
\label{eq-H(m;k)}
\end{equation}
Equations (\ref{eq-H(m;k)}), (\ref{eqn-u(k)}), and (\ref{eqn-H(k;l)-02}) yield, for $\ell \in\bbZ_+$,
\begin{eqnarray*}
\vc{H}(k;\ell)
&=& -{k \over -\sigma}\vc{e}\vc{\pi}(\ell)
\nonumber
\\
&&{}
+ (1-\delta_{\ell,0})\vc{G}^{k-\ell}\vc{F}_+(\ell;\ell)
+ \vc{G}^{k-1} (\vc{I}-\vc{\Phi}(0))^{-1} \vc{B}(-1)\vc{H}(0;\ell)
\nonumber
\\
&&{}
- (\vc{I} - \vc{G}^k)
(\vc{I} - \vc{A} - \vc{\beta}_{\! A}\vc{g} )^{-1}\vc{e}\vc{\pi}(\ell),
\qquad k \in \bbZ_{\ge \ell} \cap \bbN,\quad
\label{eqn-H(N;k)}
\end{eqnarray*}
which shows that the statement (i) holds. The statement (ii) follows from (\ref{defn-E(k;l)}). Furthermore, the statements (i) and (ii) imply that, for any fixed $\ell \in \bbZ_+$,
\[
\lim_{k\to\infty}
{|\vc{H}(k;\ell)| \over k}
= {1 \over -\sigma}\vc{e}\vc{\pi}(\ell) > \vc{O}.
\]
Consequently, (\ref{sum-pi*|H|}) holds if and only if $\sum_{k=1}^{\infty} k\vc{\pi}(k)\vc{e}$ is finite. This equivalence and Theorem~\ref{thm-mean} imply the statement (iii). \qed
\end{proof}
\begin{rem}\label{rem-D-H}
It follows from (\ref{sum-pi*|H|}) and \cite[Proposition 1.1]{Glyn96} that there exists some row vector $\vc{\eta}:=(\eta(k,i))_{(k,i) \in \bbS}$ such that
\begin{equation}
\vc{D}
= \vc{H} + \vc{e} \vc{\eta}.
\label{eqn-D-H}
\end{equation}
Since $\vc{D}$ satisfies $\vc{\pi}\vc{D}=\vc{0}$ due to the constraint (\ref{ineqn-const-01}), we have $\vc{\eta} = - \vc{\pi}\vc{H}$.
Substituting this into (\ref{eqn-D-H}) yields
\begin{equation}
\vc{D} = (\vc{I} - \vc{e}\vc{\pi})\vc{H}.
\label{eqn-D-H-02}
\end{equation}
Using (\ref{eqn-D-H-02}), we can obtain a block decomposition of $\vc{D}$, which is not necessary for this paper. Thus, we omit its details. There are some studies on the block decomposition of the deviation matrices of block structured Markov chains \cite{Dend13,LiuYuan14,LiuYuan18-STM}.
\end{rem}
\section{Subgeometric convergence in the infinite-level limit}\label{sec-convergence}
This section discusses the subgeometric convergence in the infinite-level limit of the finite-level chain. We first prove $\lim_{N\to\infty}\vc{\pi}^{(N)} = \vc{\pi}$ under the condition $\sum_{k=1}^{\infty}k\vc{\pi}(k)\vc{e} < \infty$; that is, Assumption~\ref{assum-2nd-moment} (see Theorem~\ref{thm-mean}). We then derive a subgeometric convergence formula for $\vc{\pi}^{(N)} - \vc{\pi}$ under Assumption~\ref{assumpt-tail} (introduced below) in addition to Assumption~\ref{assum-2nd-moment}.
\subsection{Basic results on convergence}
Let $\vc{v}':=(v'(k,i))_{(k,i) \in \bbS}$ denote a column vector such that
\begin{equation}
\vc{v}'(k):= (v'(k,i))_{i\in\bbM_{k\vmin1}}
=
\left\{
\begin{array}{ll}
\vc{0}, & k=0,
\\
\dm{1 \over |\sigma|}(k\vc{e} + \vc{a}), & k \in\bbN,
\end{array}
\right.
\label{defn-v'(k)}
\end{equation}
where $\vc{a}$ is given in (\ref{defn-alpha}).
\begin{lem}\label{lem-Pv'}
If Assumptions~\ref{assumpt-ergodic-MG1} and \ref{assumpt-P^{(N)}} are satisfied, then the following hold: For any $\varep > 0$, there exist some $b \in (0,\infty)$, $K' \in \bbN$ and $N_{\varep} \in \bbN$ such that
\begin{alignat}{2}
\vc{P}\vc{v}' &\le \vc{v}' - \vc{e} + b' \vc{1}_{\bbL_{\le K'}}
&\quad
&\mbox{for all $N \in \bbN$},
\label{drift-cond-02a}
\\
\vc{P}^{(N)}\vc{v}' &\le (1-\varep)\vc{v}' - \vc{e} + b' \vc{1}_{\bbL_{\le K'}}
&\quad
&\mbox{for all $N \in \bbZ_{\ge N_{\varep}}$}.
\label{drift-cond-02b}
\end{alignat}
\end{lem}
\begin{proof}
It follows from (\ref{defn-P}), (\ref{defn-v'(k)}), and $\vc{A}\vc{a}+\vc{\beta}_{\! A}=\vc{a}+\sigma\vc{e}$ that, for $k \in \bbZ_{\ge 2}$,
\begin{eqnarray*}
\sum_{\ell=0}^{\infty}\vc{P}(k;\ell)\vc{v}'(\ell)
&=& {1 \over |\sigma|}
\left(
\sum_{\ell=-1}^{\infty} (\ell+k) \vc{A}(\ell)\vc{e}
+ \sum_{\ell=-1}^{\infty} \vc{A}(\ell)\vc{a}
\right)
\nonumber
\\
&=& {1 \over |\sigma|}
\left[
k\vc{e}
+ \left(
\vc{A}\vc{a} + \vc{\beta}_{\! A}
\right)
\right]
\nonumber
\\
&=& {1 \over |\sigma|} (k\vc{e} + \vc{a})
+ {1 \over |\sigma|}
\left(
\vc{A}\vc{a} + \vc{\beta}_{\! A} - \vc{a}
\right)
\nonumber
\\
&=& {1 \over |\sigma|} (k\vc{e} + \vc{a}) + {\sigma \over |\sigma|}\vc{e}
\nonumber
\\
&=& \vc{v}'(k) - \vc{e},
\end{eqnarray*}
which implies that, for some $b' \in (0,\infty)$ and $K' \in \bbN$,
\[
\vc{P}\vc{v}'
\le \vc{v}' - \vc{e} + b' \vc{1}_{\bbL_{\le K'}}.
\]
Therefore, we have (\ref{drift-cond-02a}). Using (\ref{drift-cond-02a}) and following the proof of Lemma~\ref{lem-Pv'}, we can readily prove (\ref{drift-cond-02b}). We thus omit its details. \qed
\end{proof}
Lemma~\ref{lem-Pv'} yields the following lemma.
\begin{lem}\label{lem-asymp-pi^(N)(k)}
If Assumptions~\ref{assumpt-ergodic-MG1} and \ref{assum-2nd-moment}
hold, then there exists some function $E: \bbN \to \bbR_+$ such that $\lim_{N\to\infty} E(N) = 0$ and
\begin{equation}
| \vc{\pi}^{(N)}(k) - \vc{\pi}(k) |
\le E(N) \vc{\pi}(k) \quad \mbox{for all $N \in \bbN$ and $k \in \bbZ_{[0,N-1]}$}.
\label{asymp-pi^{(N)}(k)}
\end{equation}
\end{lem}
\begin{proof}
According to (\ref{defn-H}), the fundamental deviation matrix $\vc{H}$ depends on $T_{(k',i')}$. To prove this lemma, we fix $(k',i')=(\ell,j) \in \bbS$. It then follows from (\ref{defn-H}) that
\begin{align*}
H(k,i;\ell,j)
&=\EE_{(k,i)}\!\!
\left[
\sum_{\nu=0}^{T_{(\ell,j)}-1} \dd{1}((X_{\nu},J_{\nu})=(\ell,j))
\right]
- \pi(\ell,j)\EE_{(k,i)}\!
\left[ T_{(\ell,j)} \right]
\nonumber
\\
&= 1_{(\ell,j)}(k,i)
- \pi(\ell,j)\EE_{(k,i)}\!
\left[ T_{(\ell,j)} \right], \qquad (k,i;\ell,j) \in \bbS^2.
\end{align*}
Using this and (\ref{diff-formula-H}), we obtain
\begin{align}
&\pi^{(N)}(\ell,j) - \pi(\ell,j)
\nonumber
\\
&\quad = \vc{\pi}^{(N)} ( \vc{P}^{(N)} - \vc{P} ) \vc{H}\vc{1}_{\{(\ell,j)\}}
\nonumber
\\
&\quad = \vc{\pi}^{(N)} ( \vc{P}^{(N)} - \vc{P} )
\left[
\vc{1} _{\{(\ell,j)\}} - \pi(\ell,j) \vc{w}(\ell,j)
\right],\qquad (\ell,j) \in \bbS,
\label{eqn-pi^{(N)}(k',i')-02}
\end{align}
where $\vc{w}(\ell,j):=(w(k,i;\ell,j))_{(k,i) \in \bbS}$ is given by
\begin{equation}
w(k,i;\ell,j) = \EE_{(k,i)}[ T_{(\ell,j)}].
\label{defn-w}
\end{equation}
Moreover, from (\ref{defn-P}) and (\ref{defn-P^{(N)}}), we have
\begin{eqnarray}
\lefteqn{
\vc{P}^{(N)} - \vc{P}
}
&&
\nonumber
\\
&=&
\!
\bordermatrix{
& \bbL_{\le N-1}
& \bbL_N
& \bbL_{N+1}
& \bbL_{N+2}
& \cdots
\cr
\bbL_0 &
\vc{O} &
\vc{B}^{(N)}(N) -\vc{B}(N) &
\!\!\!\!\!\! -\vc{B}(N+1) &
\!\!\! -\vc{B}(N+2) &
\cdots
\cr
\bbL_1 &
\vc{O} &
\!\!\!\!\!\! \vc{A}^{(N)}(N-1) -\vc{A}(N-1) &
-\vc{A}(N) &
\!\!\! -\vc{A}(N+1) &
\cdots
\cr
\vdots &
\vdots &
\vdots &
\vdots &
\vdots
\cr
\bbL_{N-1}\! &
\vc{O} &
\vc{A}^{(N)}(1) -\vc{A}(1) &
-\vc{A}(2) &
-\vc{A}(3) &
\cdots
\cr
\bbL_N &
\vc{O} &
\vc{A}^{(N)}(0) -\vc{A}(0) &
-\vc{A}(1) &
-\vc{A}(2) &
\cdots
\cr
\bbL_{N+1}\! &
\vc{O} &
-\vc{A}(-1) &
-\vc{A}(0) &
-\vc{A}(1) &
\cdots
\cr
\bbL_{N+2}\! &
\vc{O} &
\vc{O} &
-\vc{A}(-1) &
-\vc{A}(0) &
\cr
\bbL_{N+3}\! &
\vc{O} &
\vc{O} &
\vc{O} &
-\vc{A}(-1) &
\cdots
\cr
~~~\vdots &
\vdots &
\vdots &
\vdots &
\vdots &
\ddots
},
\label{hm-eqn-01}
\qquad
\end{eqnarray}
and thus
\begin{equation*}
\vc{\pi}^{(N)} ( \vc{P}^{(N)} - \vc{P} ) \vc{1} _{\{(\ell,j)\}}
= 0,\qquad (\ell,j) \in \bbL_{\le N-1}.
\end{equation*}
Substituting this into (\ref{eqn-pi^{(N)}(k',i')-02}) yields
\begin{align}
\pi^{(N)}(\ell,j) - \pi(\ell,j)
= -\pi(\ell,j)\vc{\pi}^{(N)} ( \vc{P}^{(N)} - \vc{P} ) \vc{w}(\ell,j),
\quad (\ell,j) \in \bbL_{\le N-1}.
\label{eqn-pi^{(N)}(k',i')-04}
\end{align}
Furthermore, it follows from (\ref{defn-w}), Lemma~\ref{lem-Pv'} and \cite[Theorem~14.2.3]{Meyn09} that there exists some $C > 0$ such that
\begin{equation*}
\vc{w}(\ell,j) \le \vc{v}' + C\vc{e}\quad \mbox{for all $(\ell,j) \in \bbS$.}
\end{equation*}
This inequality and (\ref{eqn-pi^{(N)}(k',i')-04}) imply that
\begin{align}
|\pi^{(N)}(\ell,j) - \pi(\ell,j) |
\le \pi(\ell,j)\vc{\pi}^{(N)} | \vc{P}^{(N)} - \vc{P} |
(\vc{v}' + C\vc{e}), ~~~~ (\ell,j) \in \bbL_{\le N-1}.
\label{eqn-pi^{(N)}(k',i')-05}
\end{align}
As a result, to prove (\ref{asymp-pi^{(N)}(k)}), it suffices to show that
\begin{eqnarray}
\lim_{N \to \infty}\vc{\pi}^{(N)}\,|\vc{P}^{(N)} - \vc{P} |\, \vc{v}' &=& 0.
\label{eqn-180710-03}
\end{eqnarray}
Lemma~\ref{lem-Pv'} implies that
\[
|\vc{P}^{(N)} - \vc{P} |\, \vc{v}' \le \vc{v}' + b'\vc{e},
\qquad N \in \bbN.
\]
From this inequality, (\ref{sum-k*pi^{(N)}(k)e<b}), and (\ref{defn-v'(k)}), we obtain
\begin{eqnarray*}
\sup_{N\in\bbN}
\vc{\pi}^{(N)}\,|\vc{P}^{(N)} - \vc{P} |\, \vc{v}'
&\le& \sup_{N\in\bbN}\vc{\pi}^{(N)} (\vc{v}' + b'\vc{e})
\nonumber
\\
&=&
\sup_{N\in\bbN}\sum_{k=1}^{\infty} k \vc{\pi}^{(N)}(k) \vc{e}
+ \sup_{N\in\bbN}\sum_{k=0}^{\infty} \vc{\pi}^{(N)}(k) \vc{a}
+ b' + 1 < \infty.
\end{eqnarray*}
Therefore, using $\lim_{N\to\infty}|\vc{P}^{(N)} - \vc{P}| = \vc{O}$ (due to (\ref{lim-P^{(N)}})) and the dominated convergence theorem, we have (\ref{eqn-180710-03}). \qed
\end{proof}
\subsection{A subgeometric convergence formula}\label{sec-subgeometric}
We introduce two classes of discrete long-tailed distributions.
\begin{defn}\label{defn-long-tailed}
A discrete distribution $F$ on $\bbZ_+$ is said to be {\it long-tailed} if
$\ol{F}(k) :=1-F(k)>0$ for
all $k\in\bbZ_+$ and
\begin{equation*}
\lim_{k \to \infty} {\ol{F}(k+\ell) \over \ol{F}(k)} = 1
\quad \mbox{for any fixed $\ell \in \bbN$}.
\end{equation*}
The set of long-tailed distributions is denoted by $\calL$.
\end{defn}
\begin{defn}\label{defn-subexp}
A discrete distribution $F$ on $\bbZ_+$ is said to be {\it subexponential} if
$\ol{F}(k)>0$ for all $k\in\bbZ_+$ and
\begin{equation*}
\lim_{k\to\infty} { 1 - F^{\ast 2}(k) \over 1 - F(k) } = 2,
\end{equation*}
where $F^{\ast 2}$ is the two-fold convolution of $F$; that is,
\begin{equation*}
F^{\ast 2}(k)
=
\sum_{\ell=0}^{k} F(k-\ell) F(\ell),
\qquad k\in\bbZ_+.
\end{equation*}
The set of subexponential distributions is denoted by $\calS$.
\end{defn}
\begin{rem}
Definitions~\ref{defn-long-tailed} and \ref{defn-subexp} are the discrete versions of the definitions of long-tailed and subexponential distributions in $\bbR_+$ (see, e.g., \cite[Definitions~2.21 and 3.1]{Foss11}). Indeed, if the distribution of a random variable $Y$ in $\bbR_+$ is long tailed (resp.\ subexponential), then those of $\lceil Y \rceil$ and $\lfloor Y \rfloor$ are in $\calL$ specified by Definition~\ref{defn-long-tailed} (resp.\ $\calS$ specified by Definition~\ref{defn-subexp}), because the long-tailed and subexponential classes are closed with respect to tail equivalence (see, e.g., \cite[Lemma~2.23 and Corollary~3.13]{Foss11}).
\end{rem}
To proceed further, we assume the following.
\begin{assumpt} \label{assumpt-tail}
There exists some $F \in \calS$ such that
\begin{eqnarray} \label{eq-assum-tail}
\lim_{N\to\infty} {\ool{A}(N)\vc{e} \over \ol{F}(N)} &=& \vc{c}_A,
\label{asymp-ool{A}(k)}
\\
\lim_{N\to\infty} {\ool{B}(N)\vc{e} \over \ol{F}(N)} &=& \vc{c}_B,
\label{asymp-ool{B}(k)}
\end{eqnarray}
where either $\vc{c}_A\neq\vc{0}$ or $\vc{c}_B\neq\vc{0}$; and
\begin{equation}
\ool{A}(n) = \sum_{m=n+1}^{\infty} \ol{\vc{A}}(m), \qquad
\ool{B}(n) = \sum_{m=n+1}^{\infty} \ol{\vc{B}}(m), \qquad
n \in \bbZ_{\ge -1}.
\label{eq-def-ool}
\end{equation}
\end{assumpt}
Under Assumption~\ref{assumpt-tail}, we have a subexponential asymptotic formula for the stationary distribution $\vc{\pi}$ of the infinite-level M/G/1-type Markov chain.
\begin{prop}[{}{\cite[Theorem 3.1]{Masu16-ANOR}}] \label{prop-masu}
If Assumptions~\ref{assumpt-ergodic-MG1} and \ref{assumpt-tail} hold, then
\begin{equation*}
\lim_{N\to\infty} {\ol{\vc{\pi}}(N) \over \ol{F}(N)}
= {
\vc{\pi}(0) \vc{c}_B + \ol{\vc{\pi}}(0) \vc{c}_A
\over
-\sigma
} \vc{\varpi},
\end{equation*}
where $\ol{\vc{\pi}}(k) =\sum_{\ell=k+1}^{\infty}\vc{\pi}(\ell)$ for all $k\in\bbZ_+$.
\end{prop}
We also assume the following.
\begin{assumpt}\label{assumpt-P^{(N)}-02}
For $F \in \calS$ in Assumption~\ref{assumpt-tail},
\begin{align}
\sum_{k=0}^{\infty} \ol{F}(k) &< \infty,
\label{sum-ol{F}(N)}
\\
\lim_{N\to\infty} \sup_{ k \in \bbZ_{[0,N-1]}}
{ \left\| \vc{A}^{(N)}(k) - \ol{\vc{A}}(k-1) \right\|_{\infty}
\over
\ol{F}(N)} &= 0.
\label{lim-A^{(N)}(k)-subexp}
\end{align}
\end{assumpt}
\begin{rem}
It follows from (\ref{asymp-ool{A}(k)}), (\ref{asymp-ool{B}(k)}), and (\ref{sum-ol{F}(N)}) that $\sum_{k=0}^{\infty}\ool{A}(k)\vc{e}$ and $\sum_{k=0}^{\infty}\ool{B}(k)\vc{e}$ are finite, which implies that Assumption~\ref{assum-2nd-moment} holds. Moreover, (\ref{sum-ol{F}(N)}) implies that
\[
\lim_{N\to\infty} {\ol{F}(N) \over N^{-1}} = 0.
\]
Therefore, Assumption~\ref{assumpt-P^{(N)}-02} implies Assumption~\ref{assumpt-P^{(N)}}.
\end{rem}
\begin{lem}\label{lem-asymp-pi^{(N)}(N)}
If Assumptions~\ref{assumpt-ergodic-MG1}, \ref{assumpt-tail}, and \ref{assumpt-P^{(N)}-02} hold, then
\begin{equation}
\lim_{N\to\infty}
{\vc{\pi}^{(N)}(N) \over \ol{F}(N)}
= \vc{0}.
\label{asymp-pi^{(N)}(N)}
\end{equation}
\end{lem}
\begin{proof}
By definition, $\vc{\pi}^{(N)}(\ell) = \vc{0}$ for all $\ell \ge N+1$.
It thus follows from (\ref{eqn-pi^{(N)}(k',i')-02}) and (\ref{hm-eqn-01}) that
\begin{align}
\pi^{(N)}(N,j) - \pi(N,j)
&= \vc{\pi}^{(N)} ( \vc{P}^{(N)} - \vc{P} )
\left[
\vc{1} _{\{(N,j)\}} - \pi(N,j) \vc{w}(N,j)
\right]
\nonumber
\\
&= \vc{\pi}^{(N)}(0)(\vc{B}^{(N)}(N) - \vc{B}(N))\vc{e}_j
\nonumber
\\
&\quad +
\sum_{\ell=1}^{N-1} \vc{\pi}^{(N)}(\ell)
(\vc{A}^{(N)}(N-\ell) - \vc{A}(N-\ell))\vc{e}_j
\nonumber
\\
&\quad + \vc{\pi}^{(N)}(N)
(\vc{A}^{(N)}(0) - \vc{A}(0))\vc{e}_j
\nonumber
\\
&\quad +
\vc{\pi}^{(N)} ( \vc{P}^{(N)} - \vc{P} )\vc{w}(N,j)\cdot \pi(N,j),
\label{eqn-190903-01}
\end{align}
where $\vc{e}_j$ denote the $j$-th unit column vector.
Recall here that $\vc{P}$ is irreducible. Therefore, the spectral radius of the substochastic matrix $\ol{\vc{A}}(-1) = \sum_{\ell=0}^{\infty}\vc{A}(\ell)$ is less than one, which implies that $[\vc{I} - \ol{\vc{A}}(-1)]^{-1}$ exists. Note also that $\vc{A}^{(N)}(0) - \vc{A}(0)$ is a substochastic matrix. These facts, together with (\ref{eqn-wt{A}(k)e}), imply that $[\vc{I} - (\vc{A}^{(N)}(0) - \vc{A}(0))]^{-1}$ exists. Thus, (\ref{eqn-190903-01}) leads to
\begin{align}
\vc{\pi}^{(N)}(N)
&=
\Bigg[
\vc{\pi}(N) + \vc{\pi}^{(N)}(0)(\vc{B}^{(N)}(N) - \vc{B}(N))
\nonumber
\\
&\qquad +
\sum_{\ell=1}^{N-1}
\vc{\pi}^{(N)}(\ell)
( \vc{A}^{(N)}(N-\ell) - \vc{A}(N-\ell) )
\nonumber
\\
&\qquad +
\vc{\pi}^{(N)} ( \vc{P}^{(N)} - \vc{P} )\vc{Z}(N)
\Bigg] [\vc{I} - (\vc{A}^{(N)}(0) - \vc{A}(0))]^{-1},
\label{eqn-190903-02}
\end{align}
where $\vc{Z}(N)$ denotes a matrix such that its $j$-th column is equal to $\vc{w}(N,j)\pi(N,j)$; that is,
\begin{equation}
\vc{Z}(N)
=
\left(
\pi(N,1)\vc{w}(N,1)~~
\pi(N,2)\vc{w}(N,2)~~
\cdots~~
\pi(N,M_1)\vc{w}(N,M_1)
\right).
\label{defn-Z(N)}
\end{equation}
According to the proof of Lemma~\ref{lem-asymp-pi^(N)(k)}, we have
\begin{equation}
\lim_{N\to\infty}
\vc{\pi}^{(N)} ( \vc{P}^{(N)} - \vc{P} ) \vc{w}(N,j)
=0,\qquad j \in \bbM_1.
\end{equation}
Proposition~\ref{prop-masu} yields
\begin{equation}
\lim_{N\to\infty} {\vc{\pi}(N) \over \ol{F}(N)}
=
\lim_{N\to\infty} {\ol{\vc{\pi}}(N-1) - \ol{\vc{\pi}}(N) \over \ol{F}(N)}
= \vc{0}.
\label{asymp-pi(N)}
\end{equation}
Combining (\ref{defn-Z(N)})--(\ref{asymp-pi(N)}), we obtain
\begin{equation}
\lim_{N\to\infty}
{
\vc{\pi}^{(N)} ( \vc{P}^{(N)} - \vc{P} )\vc{Z}(N)
\over \ol{F}(N)
}
=\vc{0}.
\label{eqn-190904-01}
\end{equation}
Moreover, it follows from (\ref{eqn-wt{B}(k)e}), (\ref{asymp-pi^{(N)}(k)}), and (\ref{asymp-ool{B}(k)}) that
\begin{align*}
\limsup_{N\to\infty}
{\vc{\pi}^{(N)}(0) \left| \vc{B}^{(N)}(N) - \vc{B}(N) \right|\vc{e}
\over
\ol{F}(N)
}
& \le
\limsup_{N\to\infty}
{\vc{\pi}(0) \ol{\vc{B}}(N-1)\vc{e}
\over
\ol{F}(N)
}
\nonumber
\\
& =
\lim_{N\to\infty}
{\vc{\pi}(0) ( \ool{\vc{B}}(N-2) - \ool{\vc{B}}(N-1) ) \vc{e}
\over
\ol{F}(N)
}
= 0,
\end{align*}
which leads to
\begin{equation}
\limsup_{N\to\infty}
{\vc{\pi}^{(N)}(0) \left( \vc{B}^{(N)}(N) - \vc{B}(N) \right)
\over
\ol{F}(N)
}
= \vc{0}.
\label{eqn-190904-02}
\end{equation}
It also follows from (\ref{lim-A^{(N)}(k)-subexp}) that
\begin{align}
\lim_{N\to\infty}
[\vc{I} - (\vc{A}^{(N)}(0) - \vc{A}(0))]^{-1}
&= [\vc{I} - \ol{\vc{A}}(-1) + \vc{A}(0))]^{-1}
\nonumber
\\
&= ( \vc{I} - \ol{\vc{A}}(0) )^{-1}.
\label{eqn-190904-03}
\end{align}
Applying (\ref{asymp-pi(N)})--(\ref{eqn-190904-03}) to (\ref{eqn-190903-02}) yields
\begin{align}
\lim_{N\to\infty}
{ \vc{\pi}^{(N)}(N) \over \ol{F}(N)}
&=
\lim_{N\to\infty}
\sum_{\ell=1}^{N-1}
{
\vc{\pi}^{(N)}(\ell)
( \vc{A}^{(N)}(N-\ell) - \vc{A}(N-\ell) )
\over
\ol{F}(N)}
( \vc{I} - \ol{\vc{A}}(0) )^{-1}.
\label{eqn-190904-04}
\end{align}
Therefore, to prove (\ref{asymp-pi^{(N)}(N)}), it suffices to show that
\begin{equation}
\lim_{N\to\infty}
\sum_{\ell=1}^{N-1}
{
\vc{\pi}^{(N)}(\ell)
\left| \vc{A}^{(N)}(N-\ell) - \vc{A}(N-\ell) \right|
\over
\ol{F}(N)}
= \vc{0}.
\label{eqn-190904-05}
\end{equation}
Fix $\varep > 0$ arbitrarily. It then follows from Lemma~\ref{lem-asymp-pi^(N)(k)} and (\ref{lim-A^{(N)}(k)-subexp}) that there exists some $N^*:=N_{\varep}^* \in \bbN$ such that, for all $N \in \bbZ_{\ge N^*}$ and $k \in \bbZ_{[0,N-1]}$,
\begin{align}
(1-\varep)\vc{\pi}(k)
\le
\vc{\pi}^{(N)}(k)
&\le (1+\varep)\vc{\pi}(k),
\label{ineqn-pi^{(N)}(k)}
\\
\left| \vc{A}^{(N)}(k) - \ol{\vc{A}}(k-1) \right|
&\le \varep \ol{F}(N)\vc{e}\vc{e}^{\top}.
\label{ineqn-A^{(N)}(k)}
\end{align}
Using (\ref{ineqn-pi^{(N)}(k)}) and (\ref{ineqn-A^{(N)}(k)}), we obtain
\begin{align}
&
\sum_{\ell=1}^{N-1}
{
\vc{\pi}^{(N)}(\ell)
\left| \vc{A}^{(N)}(N-\ell) - \vc{A}(N-\ell) \right|
\over
\ol{F}(N)
}
\nonumber
\\
&\quad \le (1+\varep)
\sum_{\ell=1}^{N-1}
{
\vc{\pi}(\ell)
\left( \ol{\vc{A}}(N-\ell-1) - \vc{A}(N-\ell) + \varep \ol{F}(N)\vc{e}\vc{e}^{\top}
\right)
\over
\ol{F}(N)
}
\nonumber
\\
&\quad
\le (1+\varep)
\sum_{\ell=1}^{N-1}
{
\vc{\pi}(\ell) \ol{\vc{A}}(N-\ell)
\over
\ol{F}(N)
}
+
\varep(1+\varep)
\sum_{\ell=1}^{N-1}
\vc{\pi}(\ell) \vc{e}\vc{e}^{\top}
\nonumber
\\
&\quad
\le (1+\varep)
\sum_{\ell=1}^{N-1}
{
\vc{\pi}(\ell) \ol{\vc{A}}(N-\ell)
\over
\ol{F}(N)
}
+
\varep(1+\varep) \vc{e}^{\top},
\label{eqn-190904-06}
\end{align}
for all $N \in \bbZ_{\ge N^*}$ and $k \in \bbZ_{[0,N-1]}$.
Let
\[
\gamma(k)
=
\sum_{\ell=1}^{k-1}
\vc{\pi}(\ell) \ol{\vc{A}}(k-\ell)\vc{e},
\qquad k \ge 2.
\]
It then follows from (\ref{lim-A^{(N)}(k)-subexp}), Proposition~\ref{prop-masu}, and \cite[Lemma~6]{Jele98} that
\[
\lim_{N\to\infty}
{ \sum_{k=N}^{\infty}\gamma(k) \over \ol{F}(N)}
= \ol{\vc{\pi}}(0)\vc{c}_A
+ {\vc{\pi}(0)\vc{c}_B + \ol{\vc{\pi}}(0)\vc{c}_A \over -\sigma}
\vc{\varpi} \ool{\vc{A}}(0)\vc{e},
\]
and thus
\[
\lim_{N\to\infty}
{ \gamma(N) \over \ol{F}(N)}
= 0,
\]
which leads to
\begin{equation*}
\lim_{N\to\infty}
\sum_{\ell=1}^{N-1}
{
\vc{\pi}(\ell) \ol{\vc{A}}(N-\ell)
\over
\ol{F}(N)
}
=\vc{0}.
\end{equation*}
Applying this to (\ref{eqn-190904-06}) and letting $\varep \to 0$, we obtain (\ref{eqn-190904-05}). The proof has been completed. \qed
\end{proof}
Using Proposition~\ref{prop-masu}, Lemmas~\ref{lem-asymp-pi^(N)(k)} and \ref{lem-asymp-pi^{(N)}(N)} (together with the technical lemmas presented in Appendix~\ref{sec-proof-main-thm}), we prove the following theorem.
\begin{thm} \label{thm-asym}
If Assumptions~\ref{assumpt-ergodic-MG1}, \ref{assumpt-tail}, and \ref{assumpt-P^{(N)}-02} hold, then
\begin{equation}
\lim_{N\to\infty}
{\vc{\pi}^{(N)}(k) -\vc{\pi}(k) \over \ol{\vc{\pi}}(N)\vc{e}}
= \vc{\pi}(k)
\quad
\mbox{for any fixed $k \in \bbZ_+$}.
\label{eq-thm-lim-xN-x/P}
\end{equation}
\end{thm}
\begin{proof}
See Appendix~\ref{sec-proof-main-thm}. \qed
\end{proof}
\section{Application to the loss probability of the MAP/G/1 queue}\label{sec-application}
We first describe the Markovian arrival process (MAP) \cite{Luca90}.
Let $\{J(t);t \ge 0\}$ denote the background
Markov chain of the MAP, which is defined on the state space $\bbM:=\{1,2,\dots,M\}$. Let $\{U(t);t \ge 0\}$ denote the counting process of the MAP; that is, $U(t)$ is the total number of
arrivals in time interval $(0,t]$, where $U(0) = 0$ is assumed. The bivariate stochastic process $\{(U(t),J(t));t\ge0\}$ is a continuous-time Markov chain on state space $\bbZ_+ \times \bbM$, such that
\begin{equation}
\left(
\EE[z^{U(t)} \one(J(t)=j) \mid J(0)=i]
\right)_{i,j\in\bbM}
= \exp\{(\vc{\varLambda}_0 + z\vc{\varLambda}_1 ) t\},
\label{defn-Q}
\end{equation}
\if0
\begin{equation}
\left(
\begin{array}{ccccc}
\vc{\varLambda}_0 & \vc{\varLambda}_1 & \vc{O} & \vc{O} & \cdots
\\
\vc{O} & \vc{\varLambda}_0 & \vc{\varLambda}_1 & \vc{O} & \cdots
\\
\vc{O} & \vc{O} & \vc{\varLambda}_0 & \vc{\varLambda}_1 & \cdots
\\
\vc{O} & \vc{O} & \vc{O} & \vc{\varLambda}_0 & \ddots
\\
\vdots & \vdots &\vdots & \ddots & \ddots
\end{array}
\right),
\label{defn-Q}
\end{equation}
\fi
where $\vc{\varLambda}_1$ is an $M \times M$ nonnegative
matrix, and where $\vc{\varLambda}_0$ is an $M \times M$ matrix with negative
diagonal elements and nonnegative off-diagonal ones. We denote by MAP
$(\vc{\varLambda}_0, \vc{\varLambda}_1)$ the MAP characterized in
(\ref{defn-Q}).
By definition, $\vc{\varLambda}:=\vc{\varLambda}_0 + \vc{\varLambda}_1$ is the infinitesimal generator of the background Markov chain $\{J(t)\}$.
For analytical convenience, we assume that $\vc{\varLambda}$ is
irreducible, and we then define $\vc{\varpi} > \vc{0}$ as the unique
stationary probability vector of $\vc{\varLambda}$. We also define
$\lambda$ as the mean arrival rate, i.e., $\lambda = \vc{\varpi} \vc{\varLambda}_1 \vc{e}$.
To exclude trivial cases, we assume $\lambda > 0$ and thus $\vc{\varLambda}_1 \neq \vc{O}$.
Next we describe the MAP/G/1/$N+1$ queue. The system has a single server and a buffer of capacity $N$, and thus the system capacity is equal to $N+1$. Customers arrive at the system according to MAP $(\vc{\varLambda}_0, \vc{\varLambda}_1)$.
Arriving customers are allowed to join the system until the queue length reaches the system capacity $N+1$. Accepted customers are served on a first-come-first-served basis, and their service times are
independent of MAP $(\vc{\varLambda}_0, \vc{\varLambda}_1)$ and are independent
and identically distributed (i.i.d.) according to a general
distribution function $\beta$ on $\bbR_+$ with mean $\beta_1 \in (0,\infty)$.
Thus, $\rho:= \lambda \beta_1$, which is the traffic intensity.
In what follows,
we discuss the loss probability, denoted by $P_{\rm loss}^{(N)}$, of the MAP/G/\break 1/$N+1$ queue, described above.
Let $X^{(N)}(t)$, $t \in \bbR_+$, denote the queue length at time $t$.
Let $p^{(N)}(0)$ denote
\[
p^{(N)}(0) = \lim_{t\to\infty}{1 \over t}\int_0^t \dd{1}(X^{(N)}(u) = 0) \rd u.
\]
It then follows that
\begin{equation}
P_{\rm loss}^{(N)} = 1 - {1 - p^{(N)}(0) \over \rho}.
\label{defn-P_{loss}(N)}
\end{equation}
To express $P_{\rm loss}^{(N)}$ more specifically, we introduce the
embedded queue length process in the MAP/G/1/$N+1$ queue.
Let $X_{\nu}^{(N)} = X^{(N)}(t_{\nu}+)$ and $J_{\nu} = J(t_{\nu}+)$ for $\nu \in \bbZ_+$, where $t_{\nu}$ is the $\nu$-th service completion time. It is known that $\{(X_{\nu}^{(N)}, J_{\nu});\nu \in \bbZ_+\}$ is a Markov chain with the following transition probability matrix:
\begin{equation}
\vc{P}^{(N)} =
\begin{pmatrix}
\vc{B}(0)
& \vc{B}(1)
& \vc{B}(2)
& \cdots
& \vc{B}(N-2)
& \vc{B}(N-1)
& \ol{\vc{B}}(N-1)
\\
\vc{A}(-1)
& \vc{A}(0)
& \vc{A}(1)
& \cdots
& \vc{A}(N-3)
& \vc{A}(N-2)
& \ol{\vc{A}}(N-2)
\\
\vc{O}
& \vc{A}(-1)
& \vc{A}(0)
& \cdots
& \vc{A}(N-4)
& \vc{A}(N-3)
& \ol{\vc{A}}(N-3)
\\
\vdots
& \vdots
& \vdots
& \ddots
& \vdots
& \vdots
& \vdots
\\
\vc{O}
& \vc{O}
& \vc{O}
& \cdots
& \vc{A}(0)
& \vc{A}(1)
& \ol{\vc{A}}(1)
\\
\vc{O}
& \vc{O}
& \vc{O}
& \cdots
& \vc{A}(-1)
& \vc{A}(0)
& \ol{\vc{A}}(0)
\\
\vc{O}
& \vc{O}
& \vc{O}
& \cdots
& \vc{O}
& \vc{A}(-1)
& \ol{\vc{A}}(-1)
\end{pmatrix},
\label{defn-P^{(N)}-MAP-GI-1}
\end{equation}
where
\begin{eqnarray}
\vc{B}(k) = (-\vc{\varLambda}_0)^{-1}\vc{A}(k+1),\qquad k \in \bbZ_+,
\label{eqn-B(k)-MAP-G-1}
\end{eqnarray}
and where $\vc{A}(k)$, $k \in \bbZ_{\ge -1}$, denotes an $M \times M$ matrix
such that
\begin{equation}
\widehat{\vc{A}}(z) :=
\sum_{k=-1}^{\infty}z^k \vc{A}(k)
= z^{-1}
\int_0^{\infty}
\exp\{(\vc{\varLambda}_0 + z\vc{\varLambda}_1) x\}\rd \beta(x).
\label{defn-A(k)}
\end{equation}
Since $\vc{\varLambda}= \vc{\varLambda}_0 + \vc{\varLambda}_1$ is irreducible and
\begin{eqnarray}
\ol{\vc{A}}(1)
&=& \sum_{k=2}^{\infty}\vc{A}(k)
\nonumber
\\
&\ge&
\int_0^{\infty} \rd \beta(x)
\int_0^x \rd x_2 \int_0^{x_2} \rd x_1
e^{\vc{\varLambda}x_1} \vc{\varLambda}_1
e^{\vc{\varLambda}(x_2-x_1)} \vc{\varLambda}_1
e^{\vc{\varLambda}(x-x_2)} > \vc{O}.
\end{eqnarray}
Furthermore, all the diagonal elements of $\vc{A}(-1) = \int_0^{\infty} \rd \beta(x) \exp\{\vc{\varLambda}_0 x\}$ are positive. As a result, the stochastic matrix $\vc{P}^{(N)}$ in (\ref{defn-P^{(N)}-MAP-GI-1}) is irreducible and thus has the unique stationary distribution, denoted by $\vc{\pi}^{(N)}:=(\pi^{(N)}(k,i))_{(k,i)\in\bbZ_{[0,N]}\times \bbM}$. It is known (see \cite[Page 873]{Baio94a}) that
\begin{equation}
p^{(N)}(0)
= {\vc{\pi}^{(N)}(0)(-\vc{\varLambda}_0)^{-1}\vc{e}
\over
\vc{\pi}^{(N)}(0)(-\vc{\varLambda}_0)^{-1}\vc{e} + \beta_1},
\label{eqn-p^{(N)}(0)}
\end{equation}
where $\vc{\pi}^{(N)}(k) = (\pi^{(N)}(k,i))_{i\in\bbM}$ for $k \in \bbZ_{[0,N]}$.
Substituting (\ref{eqn-p^{(N)}(0)}) into (\ref{defn-P_{loss}(N)}) yields
\begin{equation}
P_{\rm loss}^{(N)}
= 1 - {1 \over \rho}
{ \beta_1 \over \vc{\pi}^{(N)}(0)(-\vc{\varLambda}_0)^{-1}\vc{e} + \beta_1}.
\label{eqn-P_{loss}(N)}
\end{equation}
We now assume the following.
\begin{cond}\label{cond-MAP-GI-1-arrival}
\begin{equation}
\sigma := \rho - 1
= \vc{\varpi}\sum_{k=-1}^{\infty} k \vc{A}(k) \vc{e}
< 0.
\label{eqn-rho}
\end{equation}
\end{cond}
It follows from (\ref{eqn-B(k)-MAP-G-1}) and (\ref{eqn-rho}) that $\sum_{k=1}^{\infty}k \vc{B}(k)\vc{e}$ is finite. Therefore, Condition~\ref{cond-MAP-GI-1-arrival} ensures that the stochastic matrix
\begin{equation}
\vc{P} :=
\lim_{N\to\infty}\vc{P}^{(N)}
=
\left(
\begin{array}{ccccc}
\vc{B}(0)
& \vc{B}(1)
& \vc{B}(2)
& \vc{B}(3)
& \cdots
\\
\vc{A}(-1)
& \vc{A}(0)
& \vc{A}(1)
& \vc{A}(2)
& \cdots
\\
\vc{O}
& \vc{A}(-1)
& \vc{A}(0)
& \vc{A}(1)
& \cdots
\\
\vc{O}
& \vc{O}
& \vc{A}(-1)
& \vc{A}(0)
& \cdots
\\
\vdots
& \vdots
& \vdots
& \vdots
& \ddots
\end{array}
\right)
\label{defn-P-MAP-GI-1}
\end{equation}
satisfies Assumption~\ref{assumpt-ergodic-MG1} except aperiodicity, and $\vc{P}$ in (\ref{defn-P-MAP-GI-1}) thus
has the unique stationary distribution (see, e.g., \cite[Chapter XI, Proposition~3.1]{Asmu03}), denoted by $\vc{\pi}:=(\pi(k,i))_{(k,i) \in \bbZ_+\times \bbM}$.
Let $\vc{\pi}(k) = (\pi(k,i))_{i\in\bbM}$ and $\ol{\vc{\pi}}(k)=\sum_{\ell=k+1}^{\infty}\vc{\pi}(\ell)$ for $k \in \bbZ_+$. It then follows from \cite[Theorem 4.2]{Masu16-ANOR} that
\begin{equation}
\lim_{k\to\infty}{ \ol{\vc{\pi}}(k) \over \PP(S_{\re} > k/\lambda)}
= {\rho \over 1 - \rho}\vc{\varpi},
\label{asymp-ol{pi}}
\end{equation}
under Condition \ref{cond-MAP-GI-1-service} below together with Condition~\ref{cond-MAP-GI-1-arrival}.
\begin{cond}\label{cond-MAP-GI-1-service}
Let $S$ denote a generic random variable for i.i.d.\ service times according to distribution $\beta$. Let $S_{\re}$ denote the equilibrium random variable of $S$, which is distributed as follows:
\[
\PP(S_{\re} \le x) = {1 \over \beta_1}\int_0^x (1 - \beta(y)) \rd y,
\qquad x \in \bbR_+.
\]
In addition, (i) $S_{\re}$ is subexponential; and (ii) $\sqrt{S_{\re}}$ is long-tailed.
\end{cond}
\medskip
From (\ref{asymp-ol{pi}}) and Theorem~\ref{thm-asym}, we obtain an asymptotic formula for $ P_{\rm loss}^{(N)} $.
\begin{thm}
Consider the MAP/G/1/$N+1$ queue, and suppose that Conditions~\ref{cond-MAP-GI-1-arrival} and \ref{cond-MAP-GI-1-service} hold. If $\EE[S^2] < \infty$ or equivalently, $\EE[S_{\re}] < \infty$, then
\begin{equation}
P_{\rm loss}^{(N)}
\simhm{N}
{\rho\PP(S_{\re} > N/\lambda) \over 1 + \rho\PP(S_{\re} > N/\lambda)},
\label{asymp-P_{loss}(N)}
\end{equation}
where $\xi_1(x) \simhm{x} \xi_2(x)$ represents $\lim_{x\to\infty}\xi_1(x)/ \xi_2(x) = 1$.
\end{thm}
\begin{proof}
It follows from (\ref{asymp-ol{pi}}) and $\EE[S_{\re}] < \infty$ that
\[
\sum_{k=0}^{\infty}\ol{\vc{\pi}}(k)\vc{e}
=\sum_{k=1}^{\infty} k\vc{\pi}(k)\vc{e} < \infty.
\]
Theorem~\ref{thm-mean} thus implies that Assumption~\ref{assum-2nd-moment} is satisfied. Applying Theorem~\ref{thm-asym} to (\ref{eqn-P_{loss}(N)}) yields
\begin{equation}
P_{\rm loss}^{(N)}
\simhm{N} 1 - {1 \over \rho}
{ \beta_1 \over \{1 + \ol{\vc{\pi}}(N)\vc{e}\}\vc{\pi}(0)(-\vc{\varLambda}_0)^{-1}\vc{e} + \beta_1}.
\label{asymp-P_{loss}(N)-02}
\end{equation}
Furthermore, applying the Markov renewal theory (see, e.g., \cite[Chapter~10, Theorem~4.3]{Cinl75}, and see also \cite{Luca90}) to the queue length process in the MAP/G/1 queue (with infinite capacity), we have
\begin{eqnarray}
{1 \over \lambda}
&=& \vc{\pi}(0) \{ (-\vc{\varLambda}_0)^{-1}\vc{e} + \beta_1 \}
+ \sum_{k=1}^{\infty} \vc{\pi}(k)\vc{e} \cdot \beta_1
\nonumber
\\
&=& \vc{\pi}(0)(-\vc{\varLambda}_0)^{-1}\vc{e} + \beta_1.
\label{eqn-1/lambda}
\end{eqnarray}
Combining (\ref{asymp-P_{loss}(N)-02}) and (\ref{eqn-1/lambda}), and using $\rho = \lambda \beta_1$ and (\ref{asymp-ol{pi}}), we obtain
\begin{eqnarray*}
P_{\rm loss}^{(N)}
&\simhm{N}&
1 - {1 \over \rho}
{
\beta_1
\over
\{1 + \ol{\vc{\pi}}(N)\vc{e}\} (\lambda^{-1} - \beta_1) + \beta_1}
\nonumber
\\
&=& 1 -
{
1
\over
\{1 + \ol{\vc{\pi}}(N)\vc{e}\} (1 - \rho) + \rho}
=
{
(1 - \rho) \ol{\vc{\pi}}(N)\vc{e}
\over
1 + (1 - \rho) \ol{\vc{\pi}}(N)\vc{e}}
\nonumber
\\
&\simhm{N}&
{\rho\PP(S_{\re} > N/\lambda) \over 1 + \rho\PP(S_{\re} > N/\lambda)},
\end{eqnarray*}
which shows that (\ref{asymp-P_{loss}(N)}) holds. \qed
\end{proof}
|
1,116,691,500,834 | arxiv | \section{Introduction}
\label{sec:intro}
It is now well established that the high redshift
intergalactic medium (IGM) is enriched with heavy metals
to metallicites of $10^{-3}$ to $10^{-2}$~${\rm Z}_{\odot}$
\citep[e.g.,][]{cowie95, schaye03, simcoe04, aguirre08}.
While metals only constitute a fraction of the total baryon
budget, they play an integral role in our understanding
of galaxy formation and evolution by providing a fossil record
of star formation, and by impacting upon cooling-rates which can
alter structure on many scales \citep[e.g.,][]{haas13a}.
Metals are synthesized in and released
from stars located in very overdense environments,
therefore they need to travel large distances to reach the
diffuse IGM, and this transport is likely driven by
feedback from star formation and active galactic nuclei (AGN).
Simulations have shown that metal pollution by galactic winds yields
reasonable enrichment statistics, without destroying the filamentary pattern
that gives rise to the \hone\ \lya\ forest \citep{theuns02}.
Although the need for inclusion of these processes
is clear, the mechanisms responsible
are not resolved, even in state-of-the-art cosmological simulations,
making their implementation uncertain.
By comparing observed and theoretical
metal-line absorption in the IGM, we may be able to constrain
enrichment mechanisms such as outflows.
Models and simulations of the IGM have been used
to make predictions about sources of metal pollution.
\citet{booth12} established that the observations of \citet{schaye03} of
\cfour\ associated with weak \hone\ at $z\approx3$
can only be explained if the low-density IGM was enriched primarily by low-mass
galaxies ($M_{\text{halo}} \leq 10^{10}$~\msol) that drive
outflows to distances of $\sim10^2$~proper~kiloparsecs (pkpc),
and calculated that
$>10$\% of the simulated volume and $>50$\% of the baryonic mass
in their successful model
was polluted by metals.
The simulations studied by \citet{wiersma10} indicate that
at least half of the metals found in the $z=2$ IGM were ejected from
galaxies at $z\geq3$, and that the haloes hosting these galaxies had masses less than
$M_{\text{halo}} = 10^{11}$~\msol.
This picture is consistent with that inferred from observations by
\citet{simcoe04}, who estimate that half of all baryons are enriched
to metallities $>10^{-3.5}$~Z$_{\odot}$ by $z\sim2.5$.
Studies of the IGM using the direct detection of individual metal lines
can typically probe only relatively
overdense gas, which constitutes a very small volume fraction of the
Universe. In this work, we employ an approach known as the pixel
optical depth method \citep{cowie98, ellison00, schaye00a, aguirre02, schaye03, turner14},
and provide a public version of the code at
\url{https://github.com/turnerm/podpy}.
This technique is a valuable tool for studying the IGM, as it enables
us to detect metals statistically even in low-density gas.
At the redshifts studied in this work, direct detection of metal-line absorption
in regions of the spectrum contaminated by \hone\ is nearly
impossible due to the density of the forest of absorption features. By using
the pixel optical depth method, we can correct for contamination and derive
statistical properties of the absorption by metals in this region.
In this work we take advantage of the fact that this technique is
fast and objective, and can be applied uniformly to both observations
and simulations.
Our observational sample consists of new spectra of eight $3.62\leq z \leq 3.92$ QSOs
with uniform coverage and high signal-to-noise (S/N). We compare the results to the
Evolution and Assembly of Galaxies and their Environments (EAGLE)
cosmological hydrodynamical simulations \citep{schaye15, crain15}.
The EAGLE simulations are ideal for studying metal-line absorption in the IGM, as they
have been run at relatively high resolution in a cosmologically
representative volume ($2\times1504^3$ particles in a 100~cMpc box).
The fiducial EAGLE model is able to reproduce the present-day galaxy stellar mass function, galaxy
sizes and the Tully Fisher relation \citep{schaye15}, and has been found to match observations
of galaxy colours \citep{trayford15} and the evolution of galaxy stellar masses \citep{furlong15}
and sizes \citep{furlong16}.
Furthermore, the simulations are in good agreement with a number of relevant observables,
including the properties of \hone\ absorption at $z\sim2$--$3$ \citep{rahmati15},
as well as the \osix\ and \cfour\ \citep{schaye15}
and the \hone\ \citep{crain16} column density distribution functions (CDDFs) at $z\sim0$ (although
for the latter we note that a higher than fiducial resolution is needed to
achieve agreement).
\citet{rahmati16} compared observed metal-line CDDFs
for various ions and redshifts with the predictions from EAGLE, finding generally
good agreement. Their study also included CIV, which was compared to the observations of
\citet[$z=1.5\text{--}5.5$]{songaila05},
\citet[$z=1.5\text{--}4.0$]{dodorico10}, and
\citet[$z=1.6\text{--}4.4$]{boksenberg15}.
Although the redshift ranges of these surveys overlap with ours, they are
much wider and hence likely affected by evolutionary trends. These observational
studies each used different, subjective methods to decompose the absorption
into Voigt profiles, which limits their utility for testing models. Indeed,
the different studies do not agree with each other, making it relatively easy
for a single model to agree with the data. Here we will instead employ a like-for-like
comparison of new data with virtual EAGLE observations tailored to mimic the
characteristics of our own quasar spectra and analysed using the same automated
method. We will consider relations between the pixel optical depths of
\hone, \cthree, \cfour, \sithree, \sifour\ and \osix.
This paper is structured as follows.
In \S~\ref{sec:method}, we describe the observations and simulations.
We also summarize the pixel optical depth method,
and how it is applied.
The results are presented in \S~\ref{sec:results},
and we give a discussion and conclusions in \S~\ref{sec:discussion}
and \ref{sec:conclusion}, respectively.
Throughout this work, we denote proper and comoving
distances as pMpc and cMpc, respectively.
Both simulations and observations use cosmological parameters
determined from the \textit{Planck} mission \citep{planck13}, i.e.
$H_{\rm 0}=67.8$~\kmps~Mpc$^{-1}$, $\Omega_{\rm m} = 0.307$,
and $\Omega_{\Lambda} = 0.693$.
\section{Method}
\label{sec:method}
\subsection{Observations}
We analyze a sample of eight QSOs with $3.62\leq z_{\text{QSO}} \leq 3.922$.
They were selected based on their redshift and the existence
of substantial, high S/N data taken with VLT/UVES. Initially,
there were already 76.0 hours of UVES data, excluding overheads,
of the QSOs.
Follow-up observations to fill in the gaps and improve S/N were
completed in 62.7 hours of on-source time in
programmes 091.A-0833, 092.A-0011 and 093.A-0575 (P.I. Schaye).
We note that for Q1422$+$23, the gaps in the UVES data were filled
using archival observations with Keck/HIRES of comparable S/N and resolution
(which is $\approx8.5$~\kmps).
The properties of the QSOs and the S/N of the spectra
are summarized in Table~\ref{tab:qso}.
The reduction of the UVES data was performed using the \texttt{UVES\_headsort} and
\texttt{UVES\_popler} software by
Michael T. Murphy, and binned to have a uniform velocity dispersion
of 1.3~\kmps.
The HIRES data was reduced using T. Barlow's \texttt{MAKEE} package, and binned on to 2.8~\kmps\ pixels.
The continuum fits for the spectra were performed by hand by M.\ Turner.
Any DLAs or Lyman break regions (i.e., due to
strong absorbers in \hone) were masked out, with the exception of DLAs in the \lya\ forest,
which were unmasked when recovering the \hone\ to be used for
subtraction of contaminating absorption by higher-order Lyman
series lines from \osix\ and \cthree\ optical depths.
To homogenize the continuum fitting errors, we implemented the automated
sigma-clipping procedure of \citet{schaye03} at wavelengths greater than
that of the QSO's \lya\ emission, which was applied to both the observed and simulated spectra.
The spectrum is divided into rest-frame bins of $\Delta \lambda = 20$~\AA,
which have central wavelength $\lambda_i$ and median flux $\bar{f_i}$.
A B-spline of order 3 is then interpolated through all $\bar{f_i}$ values, and any pixels with flux
$N_{\sigma}^{\rm cf}\times\sigma$ below the interpolated values are discarded, where $\sigma$ is
the normalized noise array.
We then recalculate $\bar{f_i}$ without the discarded pixels,
and repeat the procedure until convergence is reached.
We use $N_{\sigma}^{\rm cf}=2$, which has been shown to be optimal in the \cfour\ region
for spectra with a quality similar to ours,
as it induces errors that are smaller than the noise by at least an order of magnitude \citep{schaye03}.
\begin{table*}
\caption{Properties of the QSOs used in this work, and the median S/N in the \hone\ \lya\ and \cfour\
recovery regions. The columns list, from left to right,
name, right ascension, declination, redshift, magnitude and magnitude band (either $V$-band (V), $R$-band (R),
or photographic (P)) from \citet{veron10},
median signal-to-noise in the \lya\ forest
region and \cfour\ region, respectively (see \S~\ref{sec:redshift}
for the definition of these regions),
the median optical depth of \hone\ in the \lya\ forest, the factor $C_{\rm UVB}$ used to scale the UVB in the simulations,
and the redshift midpoint of eq.~\ref{eq:zlim}
that defines the \lya\ forest region.}
\begin{center}
\begin{tabular}{l l l l l l r r r r r}
\hline
Name & R.A. & Dec & $z_\text{QSO}$ & Mag & Band & S/N$_{\lyam}$ & S/N$_{\cfourm}$ & $\log_{10}\tau^{\rm med}_{\honem}$ & $C_{\rm UVB}$ & $z^{\rm mid}_{\lyam}$\\
\hline \hline
Q1422$+$23 & \tt{ 14:24:38 } & \tt{ +22:56:01 } & 3.620 & 15.84 & V & 87 & 82 & $-0.605$ & 0.896 & 3.24\\
Q0055$-$269 & \tt{ 00:57:58 } & \tt{ -26:43:14 } & 3.655 & 17.47 & P & 60 & 79 & $-0.583$ & 1.123 & 3.27\\
Q1317$-$0507 & \tt{ 13:20:30 } & \tt{ -18:36:25 } & 3.700 & 18.10 & P & 59 & 90 & $-0.549$ & 1.009 & 3.31\\
Q1621$-$0042 & \tt{ 16:21:17 } & \tt{ -23:17:10 } & 3.709 & 17.97 & V & 78 & 92 & $-0.546$ & 0.807 & 3.32\\
QB2000$-$330 & \tt{ 20:03:24 } & \tt{ -32:51:44 } & 3.773 & 17.30 & R & 105 & 83 & $-0.501$ & 0.809 & 3.38\\
PKS1937$-$101 & \tt{ 19:39:57 } & \tt{ -13:57:19 } & 3.787 & 17.00 & R & 96 & 64 & $-0.494$ & 1.293 & 3.39\\
J0124$+$0044 & \tt{ 01:24:03 } & \tt{ +00:44:32 } & 3.834 & 18.71 & V & 48 & 59 & $-0.445$ & 0.898 & 3.43\\
BRI1108$-$07 & \tt{ 11:11:13 } & \tt{ -15:55:58 } & 3.922 & 18.10 & R & 29 & 29 & $-0.390$ & 0.808 & 3.51\\
\hline
\end{tabular}
\end{center}
\label{tab:qso}
\end{table*}
\subsection{Simulations}
We compare the observations to predictions from the EAGLE cosmological
hydrodynamical simulations. EAGLE was run with a substantially modified version of the $N$-body TreePM
smoothed particle hydrodynamics (SPH) code \texttt{GADGET}~3
(last described in \citealt{springel05a}). EAGLE uses the package of hydrodynamics updates
``Anarchy'' (Dalla Vecchia, in prep.; see Appendix~A1 of \citealt{schaye15}) which invokes the pressure-entropy formulation
of SPH from \citet{hopkins13}, the time-step limiter from \citet{durier12},
the artificial viscosity switch from \citet{cullen10}, an artificial conduction
switch close to that of \citet{price08}, and the $C^2$ \citet{wendland95} kernel.
The influence of the updates is explored by \citet{schaller15}.
The fiducial EAGLE model
is run in a 100~cMpc periodic box with $1504^3$ of both dark matter and baryonic particles,
and is denoted Ref-L1001504. To test convergence with resolution and box size,
runs varying the number of particles and box size were also conducted, and are
listed in Table~\ref{tab:sims}.
The stellar feedback in EAGLE is implemented as described by \citet{dallavecchia12},
where thermal energy is injected
stochastically. While the temperature of heated particles is always increased
by $10^{7.5}$~K, the probability of heating varies with the local metallicity and density \citep{schaye15, crain15}.
The simulations include thermal AGN feedback \citep{booth09},
also implemented stochastically \citep{schaye15}.
Both stellar and AGN feedback have been calibrated such that the simulations
match the observed $z\sim0$ stellar mass function and galaxy--black hole mass relation,
and give sensible disc galaxy sizes. We note that of the two highest-resolution runs,
Ref-L025N072 has been realized with the same subgrid parameters used in the fiducial model,
while for the Recal-L025N072 the subgrid parameters were re-calibrated to better match
the observed galaxy stellar mass function.
EAGLE also includes a subgrid model for photo-heating
and radiative cooling via eleven elements: hydrogen, helium, carbon, nitrogen, oxygen,
neon, magnesium, silicon, sulphur, calcium and iron \citep{wiersma09a},
assuming a \citet{haardt01} UV and X-ray background.
Star formation is implemented with a gas metallicity-dependent density threshold \citep{schaye04}
as described by \citet{schaye08}, followed by stellar evolution
and enrichment from \citet{wiersma09b}.
Finally, details of the subgrid model for black hole seeding and growth
can be found in \citet{springel05b, rosas13} and \citet{schaye15}.
For each of our eight observed QSOs in Table~\ref{tab:qso},
we synthesize 100 corresponding mock spectra using the
\texttt{SPECWIZARD} package by Schaye, Booth, and Theuns (implemented
as in \citealt{schaye03}, see also Appendix~A4 of \citealt{theuns98}).
To create mock spectra that resemble the observed QSOs and whose absorption features
span a large redshift range, we follow \citet{schaye03} and stitch together
the physical state of the gas intersecting uncorrelated sightlines from
snapshots with different redshifts. The ionization balance of each gas particle
is estimated using interpolation tables generated from \texttt{Cloudy} \citep[][version 13.03]{ferland13}
assuming uniform illumination by an ultra-violet background (UVB).
We take the QSO+galaxy \citet{haardt01} UVB (denoted as ``HM01'')
to be our fiducial model,\footnote{ Our choice of the HM01 UVB is primarily
to maintain consistency with the EAGLE simulations, in which the \citet{haardt01} UVB is
also used to calculate radiative cooling rates.
While more recent
UVBs are available \citep[e.g.,][]{fauchergiguere09, haardt12}, they
do not necessarily provide a better match to observations \citep[e.g.,][]{kollmeier14}.
}
and have plotted the intensity
as a function of energy at $z=3.5$ in Fig.~\ref{fig:uvb}.
We also consider
the \citet{haardt01} background using quasars only (``Q-only''), which
is much harder than the fiducial model above $\approx4$~Ryd.
Furthermore, to explore the possible effects of a
delayed \hetwo\ reionization, we consider a UVB that is
significantly softer above 4~Ryd. To implement this,
we use the QSO+galaxy model and reduce the intensity above 4~Ryd
by a factor of 100, which we denote as ``4Ryd-100''.
Self-shielding for \hone\ was included by modifying the ionization fraction
using the fitting functions of \citet{rahmati13a}.
The normalization of the UVB is set such that the median recovered \hone\ \lya\ optical
depth of the simulated QSOs agrees with that of the observations at the same redshift.
A unique value of $C_{\rm UVB}$ is determined for each
observed QSO and corresponding set of 100 mock spectra,
and the values are presented in Table~\ref{tab:qso}.
In the EAGLE simulations, the dense particles
that represent the multiphase interstellar
medium (ISM, $n_{\rm H} > 0.1$~cm$^{-3}$) are not allowed to cool
below an effective equation of state (EoS),
and their temperature can be interpreted as a proxy for the pressure at which the warm
and cool ISM phases equilibriate.
We set their
temperatures to $10^4$~K when generating
the mock spectra, although we note that due to the small cross-section
of such dense absorbers the effect of including them is negligible.
\begin{figure}
\includegraphics[width=0.5\textwidth]{figures/uvb.pdf}
\caption{The intensity as a function of energy for the different UVB models.
The different models are: HM01 QSO+galaxy \citep{haardt01}, which is our
fiducial model; Q-only, which is also by \citet{haardt01} but
only considers an ionizing contribution from QSOs; and
4Ryd-100, the same as the fiducial model except that
the intensity is reduced by a factor of 100 above 4~Ryd.
The vertical light grey lines indicate the ionization energies
of ions of interest, where increasing line thickness denotes silicon,
carbon and oxygen ions, respectively.
All of the UVBs have been normalized
to have the same intensity as HM01 at 1~Ryd.
}
\label{fig:uvb}
\end{figure}
Each set of 100 mock spectra is synthesized to have redshifts
identical to that of their corresponding observed QSO, and we consider absorption
ranging from $1.5 < z < z_{\text{QSO}}$ in every case.
We include contributions from 31 \hone\ Lyman series transitions beginning with \lya,
and metal-line absorption from
\ctwo, \cthree, \cfour, \fetwo, \nfive, \osix, \sitwo, \sithree, and
\sifour\ (see Appendix~\ref{app:trans} for the rest wavelengths and oscillator strengths
of these transitions).
To match the spectral properties of the observations, the simulated spectra are convolved with a Gaussian
with a FWHM of 6.6~\kmps, and resampled on to pixels of 1.3~\kmps. For each observed
QSO, we have measured the RMS noise in bins of 150~\AA\ in wavelength and 0.2 in normalized flux.
We then use these measurements to add random Gaussian noise with the same variance to the simulations.
\begin{table*}
\caption{Characteristics of the EAGLE simulations. From left to right, the columns list the simulation
name, box size, number of particles,
initial baryonic particle mass, dark matter particle mass, comoving (Plummer-equivalent) gravitational softening, and
maximum physical softening.}
\begin{center}
\begin{tabular}{rrrccccl}
\hline
Simulation & $L$ & $N$ & $m_{\rm b}$ & $m_{\rm dm}$ & $\epsilon_{\rm com}$ & $\epsilon_{\rm prop}$ \\
& [cMpc] & & [\msol] & [\msol] & [ckpc] & [pkpc] \\
\hline
\hline
{Ref-L100N1504} & 100 & $2\times1504^3$ & $1.81 \times 10^6$ & $9.70 \times 10^6$ & 2.66 & 0.70 \\
{Ref-L050N0752} & 50 & $2\times752^3$ & $1.81 \times 10^6$ & $ 9.70 \times 10^6$ & 2.66 & 0.70 \\
{Ref-L025N0376} & 25 & $2\times376^3$ & $1.81 \times 10^6$ & $ 9.70 \times 10^6$ & 2.66 & 0.70 \\
{Ref-L025N0752} & 25 & $2\times752^3$ & $2.26 \times 10^5$ & $ 1.21 \times 10^6$ & 1.33 & 0.35 \\
{Recal-L025N0752} & 25 & $2\times752^3$ & $2.26 \times 10^5$ & $ 1.21 \times 10^6$ & 1.33 & 0.35 \\
\hline
\end{tabular}
\end{center}
\label{tab:sims}
\end{table*}
\subsection{Analyzed redshift range}
\label{sec:redshift}
The first step for the pixel optical depth
recovery involves choosing optimal
redshift limits. The fiducial redshift
range is selected to lie in the \lya\ forest, defined to be:
\begin{equation}
(1 + \zqsom) \dfrac{\lambda_{\lybm} }{ \lambda_{\lyam}}- 1 \leq z \leq \zqsom - (1+\zqsom) \dfrac{3000\,\kmpsm}{c}
\label{eq:zlim}
\end{equation}
where $\lambda_{\lyam}=1215.7$~\AA\ and $\lambda_{\lybm}=1025.7$~\AA\
are the \hone\ \lya\ and \lyb\ rest wavelengths, respectively.
The lower limit was chosen to avoid the \lyb\ forest and corresponds
to the \lyb\ transition at the
redshift of the QSO, while the upper limit is 3000~\kmps\ bluewards of the QSO redshift
to avoid any proximity effects.
For \hone, \cfour\ ($\lrestm = [1548.2, 1550.8]$~\AA) and
\cthree\ ($\lrestm = 977.0$~\AA) we use the above
redshift limits. For the remaining ions, we make slight modifications,
listed below,
in order to homogenize the contamination.
We use the notation $\lambda_{Z,k}$ to denote the rest wavelength of
multiplet component $k$ of the ion $Z$.
\begin{enumerate}
\item \osix\ ($\lrestm=[1031.9, 1037.6]$~\AA):
We limit the recovery to where \osix\ overlaps with the \lyb\ forest and
place a cut-off at the \lya\ forest region, which leads to
$z_{\rm max} = (1 + \zqsom) \lambda_{\honem,\lybm} / \lambda_{\osixm,2} - 1$
\item \sithree\ ($\lrestm=1206.6$~\AA):
We constrain the recovered optical depth region to not extend
outside of the \lya\ forest. For \sithree, which extends
slightly bluewards into the \lyb\ forest, we set
$z_{\rm min} =(1 + \zqsom) \lambda_{\lybm} / \lambda_{\sithreem}$.
\item \sifour\ ($\lrestm = [1393.8, 1402.8]$~\AA):
To avoid contamination by the \lya\ forest, we limit the blue end of the \sifour\ recovery
by setting
$z_{\rm min} = (1+\zqsom)\lambda_{\lyam}/\lambda_{\sifourm,1}-1$.
\end{enumerate}
\subsection{Pixel optical depth method}
\label{sec:pod}
We employ the pixel optical depth method, which we
use to study absorption on an individual pixel basis rather than
by fitting Voigt profiles to individual lines.
The goal is to obtain statistics on absorption by \hone\ and
various metal ions in the IGM, and on how their absorption relates to one another.
Our implementation is close to that of \citet{aguirre02}, but with
the improvements of \citet{turner14}, and the effects of varying the
chosen parameters can be found in both works.
The exact methodology
is described in full detail in Appendix~A of \citet{turner14},
and is summarized below.
A working version of the code can be found at
\url{https://github.com/turnerm/podpy}.
After restricting the redshift range, the next step is to convert the flux of every
pixel of ion $Z$ and multiplet component $k$ to an optical depth
$\tau_{Z,k}(z) = -\ln(F)$, where $F(\lambda)$ is the normalised flux at
$\lambda = \lambda_{k}(1+z)$. Then, depending on the ion, corrections are made
for saturation or contamination, as described below.
\begin{enumerate}
\item For \hone\ \lya, while there is very little contamination in the \lya\ forest, the absorption in
many of the pixels will be saturated, and we use the higher order Lyman series
transitions to correct for this. Specifically, if we consider a \lya\ pixel to be saturated,
we look to $N=16$ higher-order Lyman lines (beginning with \hone\ \lyb),
and take the minimum optical depth, scaled to that of \lya, of all unsaturated pixels at the same redshift (if any).
If we are unable to correct the pixel due to saturation of the higher-order transitions,
we set it to $10^4$ (so that these pixels can still be used for computing the median). Finally, we search for and discard
any contaminated pixels, by checking that higher-order transitions
do not have optical depth values significantly below what would be expected from the scaled \hone\ \lya\ optical depth.
\item For \osix\ and \cthree, we can use the corrected \hone\ \lya\ optical depths
to estimate and subtract contamination by \hone.
We do so beginning with \hone\ \lyb\ ($N=2$) and use higher-order Lyman series orders up to $N=5$.
For saturated \osix\ and \cthree\ pixels, the optical depth is not well defined
and therefore the above subtraction is not performed. Instead, we leave the pixel
uncorrected, unless the saturation can be attributed to \hone, in which case
the pixel is discarded.
\item \sifour\ and \osix\ are both closely-spaced doublets, and we can
use this fact to correct for contamination. To do so, we scale the optical depth
of the weak component to match that of the strong component, and take the minimum of the
two components modulo noise.
We only take the scaled optical depth of the weaker component if it is significantly
lower (when taking into account the noise array) than the stronger component.
\item For \cfour, which is a strong transition redward of the \lya\ forest,
the largest source of contamination is by its own doublet.
To correct for this, we perform an iterative self-contamination correction.
We first discard any pixels determined to be contaminated by other ions, by checking
whether the optical depth of a pixel is too high
to be explained by half of the associated stronger component combined with twice
the associated weaker component. We then subtract the estimated contribution of the
weaker component from each pixel, iterating until convergence is reached.
\end{enumerate}
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth]{figures/c4_vs_h1_q03_leg.png}
\end{center}
\caption{Example analysis of the \cfour(\hone) relation for Q1317$-$507.
The measured optical depths from the observations are shown by the points, and those recovered from the mock
spectra are indicated by the curves,
with the different colours and line styles representing variations in the model. $1\sigma$ errors are indicated by the bars (shaded region)
for the observations (fiducial simulation), and were determined by bootstrap resampling spectral chunks (mock spectra).
We indicate \taumin\ by the horizontal dashed and dotted lines for the observations and
simulations, respectively.
In Appendix~\ref{sec:single} we provide individual results for all eight QSOs,
and for all relations examined in this work.
}
\label{fig:example}
\end{figure}
\subsection{Analysis}
\label{sec:pod_as}
For the analysis, we would like to see how the absorption from one ion
correlates with that from another. The procedures used here are also described in
\S~3.4 and 4.2 of \citet{aguirre04}.
As an illustrative example, we will consider the ions
\cfour\ and \hone. For a single observed QSO, we use the recovered pixel
optical depths to construct a set of pixel pairs where each pair shares the same redshift.
We then divide the ions into bins of $\log\tau_{\honem}$, and take the median
$\tau_{\honem}$ and $\tau_{\cfourm}$ in each bin , to obtain
$\tau_{\cfourm}^{\text{med}}(\tau_{\honem})$,
which from this point forward we will denote as \cfour(\hone).
The result of this procedure applied to
one of our QSOs is shown in Fig.~\ref{fig:example},
and we briefly describe the characteristics here.
We note that the results from individual QSOs for all
relations examined here are given in Appendix~\ref{sec:single}.
We make note of two different regimes within the \cfour(\hone) relation.
The first is on the right-hand side of Fig.~\ref{fig:example},
where $\tau_{\honem}\gtrsim1$. Here, the median \cfour\ optical
depth increases with \hone, which indicates that the pixels are probing gas enriched by \cfour.
The value of $\tau_{\cfourm}^{\text{med}}$ constrains the
number density ratio of \cfour\ to \hone.
Next, we turn to the region with $\tau_{\honem}\lesssim1$, where
$\tau_{\cfourm}^{\text{med}}$ is approximately constant.
This behaviour arises because the median \cfour\ optical depths reach values
below the flat level \taumin, which is essentially a detection limit
set by noise, contamination, and/or continuum fitting errors.
An important caveat to keep in mind throughout this work is that
the median recovered metal-line optical depth is not necessarily representative
of \textit{typical} intrinsic pixel optical depths for a given \hone\ bin.
In particular, as the metal-line optical
depths approach the flat level, it is likely that many individual pixels in a given
\hone\ bin have intrinsic metal optical depths at or below the flat level itself. In this case, the median
recovered metal optical depth will be determined by the fraction of pixels
that have optical depths above the flat level.
To construct the \cfour(\hone) relation for the
observed spectra, \hone\ bins containing fewer than 25 pixels in total
are discarded.
Furthermore, we divide each spectrum into chunks of 5~\AA\ (chosen to be much greater than the line widths),
and discard any bins containing fewer than 5 unique chunks. This is done to ensure that the
median optical depths are obtained from more than just a few pixels in a very small spectral region.
To measure errors on $\tau_{\cfourm}^{\text{med}}$,
we create new spectra by bootstrap
resampling the chunks 1000 times with replacement. We then measure
\cfour(\hone)
for each bootstrap realization of the spectrum and take the error in each $\tau_{\honem}$ bin
to be the $1\sigma$ confidence interval of all
realizations.
For the simulated spectra, we measure
\cfour(\hone)
for each mock spectrum, and require that each $\tau_{\honem}$ bin
have at least 5 pixels in total.
Next, we combine the results for all 100 mock spectra
associated with a single observed QSO by measuring
the median \cfour\ optical depth in each $\tau_{\honem}$ bin
for all spectra, and we discard any bin containing contributions
from fewer than 5 spectra.
Errors are calculated by bootstrap resampling the spectra 1000 times.
Next, we compute the flat levels \taumin\ by taking
the median of all pixels that have
$\tau< \tau_c$, where we choose $\tau_c$ to be an optical depth below
which we do not find any correlations. As in \citet{aguirre08}, we take $\tau_c=0.1$ when binning in \hone,
and $0.01$ when binning in \cfour\ and \sifour. To estimate the error
on \taumin, for the observations we again divide the spectrum
into 5~\AA\ chunks, measure \taumin\ for 1000 bootstrap realizations,
and take the $1\sigma$ confidence interval. For the simulations, we calculate
\taumin\ for each spectrum, and take the final value to be the median
value from all 100 spectra.
Finally, below we outline the steps for combining the results from the different QSOs,
which is applied to both the observed relations as well as their respective counterparts from the mocks.
Because our sample is uniform in terms of S/N,
we simply combine the binned data points directly without subtracting \taumin.
However, because the implementation of the noise, continuum fitting errors and contamination
in simulations is not completely accurate, the flat levels differ from the observations.
To account for this offset, we linearly add the difference between flat levels
($\tauminm^{\text{obs}} - \tauminm^{\text{sims}}$) to the median
optical depths in the simulations. We have verified that performing this step
before the QSOs are combined does not modify the results.
Next, to measure the combined median values,
we perform $\chi^2$ fitting of a single value of $\tau_{\cfourm}^{\text{med}}$ to
all points in the bin, which is plotted against the central value of each \hone\ bin
(in contrast to the results from individual QSOs, which are plotted against
the median of all \hone\ pixel optical depths in each bin).
We discard any data points that have contributions from fewer than four QSOs,
and the $1\sigma$ errors are estimated by bootstrap resampling the QSOs.
The combined results for \cfour(\hone) can be seen in the left-hand panel of Fig.~\ref{fig:hone}.
\begin{figure*}
\includegraphics[width=0.33\textwidth]{{figures/c4_vs_h1_allqso_p50_force_tau_min_comb_leg}.pdf}
\includegraphics[width=0.33\textwidth]{{figures/si4_vs_h1_allqso_p50_force_tau_min_comb}.pdf}
\includegraphics[width=0.33\textwidth]{{figures/o6_vs_h1_allqso_p50_force_tau_min_comb}.pdf}
\caption{Median recovered pixel optical depths binned by \hone\ for \cfour\ (left),
\sifour\ (centre) and \osix\ (right).
Note that the dynamic range shown along the $y$-axis decreases strongly
from left to right.
The data from eight QSOs have been
combined, and the $1\sigma$ error bars are measured by bootstrap resampling the QSOs.
The black circles show the data, while the curves denote the results from
simulations, where different colours and line styles represent variations in the model and
we show the $1\sigma$ error region around the fiducial HM01 model. The flat level \taumin,
which is the same for the observations and simulations by construction,
is indicated by the dashed horizontal line. The data from the observations
is provided in Table~\ref{tab:data_0}.
The
simulation run with the fiducial UVB systematically underpredicts the median
\cfour\ and \sifour\ optical depths. The discrepancy is
lessened by invoking a softer UVB (4Ryd-100), higher metallicity (Zrel-10),
or unresolved turbulent broadening (bturb-100),
although the metal-line optical depths associated with the strongest \hone\ absorption
are still underestimated by $\approx0.5$~dex. In contrast, the predicted
\osix(\hone) relation (right-hand panel) is insensitive
to the UVB models, and in good agreement with the observations.
}
\label{fig:hone}
\end{figure*}
\begin{table}
\caption{Observational data shown in Fig.~\ref{fig:hone}. The left column
indicates the central $\log_{10}\tau_{\honem}$ value of each bin, and the next three columns list the $\log_{10}$ median
recovered optical depths for the \cfour(\hone), \sifour(\hone), and \osix(\hone) relations,
respectively, along with the $1\sigma$ errors. We note that for the bin where
$\log_{10}\tau_{\honem}=0.70$, the metal optical depths are
not defined because there were contributions from fewer that four of our eight quasars.}
\begin{center}
\input{tables/data_0.tab}
\end{center}
\label{tab:data_0}
\end{table}
\section{Results}
\label{sec:results}
\subsection{$\tau_{Z}$ as a function of $\tau_{\rm HI}$}
We begin by examining median metal-line pixel optical
depths as a function of \hone\ pixel optical depth in Fig.~\ref{fig:hone}, where we have plotted
\cfour(\hone), \sifour(\hone) and \osix(\hone) from left to right.
The grey points with error bars represent the observations,
while the curves show results from simulations, with different
colours indicating variations in the model. The data from
the observations is presented in Table~\ref{tab:data_0}.
The relations displayed in Fig.~\ref{fig:hone} depend on the following quantities
in the simulations, using \cfour\ as an illustrative example:
\begin{equation}
\log_{10} \dfrac{\tau_{\cfourm}}{\tau_{\honem}} =
\log_{10} \dfrac{(f\lambda)_{\cfourm}}{(f\lambda)_{\honem}}
\dfrac{n_{\cfourm}}{n_{\rm C}}
\dfrac{n_{\rm H}}{n_{\honem}}
+ [\text{C}/\text{H}]
+ (\text{C}/\text{H})_{\odot} ,
\label{eq:metallicity}
\end{equation}
where $f$ is the oscillator strength, $\lambda$ is the rest wavelength, and
$n$ is the number density.
While the oscillator strength and rest wavelength are fixed
empirical quantities, the element abundances are predicted, and
the ionization fractions (i.e., the ratio of ionized to total number density)
are determined using the particle temperatures and densities in the same manner
as was done to compute the cooling rates used in the simulation.
In the following analysis, we will consider the results in
two different regimes, separated by $\tau_{\honem}\approx10$. The reasons for this are:
(1) For $\tau_{\honem}\gtrsim10$ \hone\ pixel optical depths
will be highly saturated, and even though this is corrected for in our recovery procedure, the final values
still suffer from large uncertainties compared to their unsaturated counterparts.
(2) If the gas being probed is mainly in photoionization equilibrium,
which is a reasonable assumption for \cfour\ and \sifour\ \citep{schaye03, aguirre04}, then
\hone\ is considered a good tracer of the density \citep{schaye01}, even on an individual
pixel basis \citep{aguirre02}.
This means that higher \hone\ optical
depths probe dense regions closer to galaxies,
rather than the diffuse IGM.
To touch on this point more quantitatively, we have calculated
the optical depth-weighted overdensities for
our full sample of mock spectra. Fig.~\ref{fig:delta} shows the median optical depth-weighted
overdensity as a function of \hone\ for
\hone, \cfour, \sifour, and \osix, and we find a strong correlation
in every case. We have performed ordinary least-squares
fits to the curves using a power-law function:
\begin{equation}
\log_{10} \delta =A \times \log_{10} \tau_{\honem} + B,
\label{eq:delta_vs_hone}
\end{equation}
and give the resulting parameters in Table~\ref{tab:delta}. We note that our aforementioned division of
the \hone\ optical depths into two regimes at $\tau_{\honem}\approx10$ corresponds to an overdensity
of $\approx8$ for the gas responsible for the \hone\ absorption, but that the overdensity of the metal
ions at the same redshifts as the selected \hone\ pixels is typically a factor of a few lower.
We also caution that the quantities presented here are pixel-weighted,
and may therefore be biased to high temperatures since higher temperature lines will be more broadened
and therefore cover more pixels. This may also bias the densities to lower values, as intergalactic gas with $T\gtrsim10^5$~K
tends to correspond to lower densities than that found at lower temperatures.
\citep[e.g., Fig.~1 from][]{vandevoort11a}.
The fitted values for \hone\ can be compared to eq.~8 of \citet{rakic12}, who used
the relations of \citet{schaye01} to obtain an expression for overdensity as a
function of \hone\ \lya\ line centre optical depth $\tau_{0,\lyam}$, under the
assumption that the absorbers have
sizes of the order of the local Jeans length. Modifying their eq.~8
using the parameters from this work, we obtain:
\begin{equation}
\begin{split}
\delta \approx& \, 0.74\, \tau_{0,\lyam}^{2/3}\
\left(\dfrac{\Gamma_{12}}{0.85\times10^{12}\text{ s}^{-1}} \right)^{2/3}
\left( \dfrac{1+z}{4.36}\right)^{-3} \\
\times& \left(\dfrac{T}{2\times10^4\text{ K}} \right)^{0.17}
\left( \dfrac{f_{\rm g}}{0.15} \right)^{-1/3}
\left( \dfrac{b}{20 \text{ \kmps}} \right)^{2/3}.
\end{split}
\label{eq:rakic_eq8}
\end{equation}
To determine the redshift $z$, for each QSO we considered the
redshift at the centre of the \lya\ forest as defined by eq.~\ref{eq:zlim},
and took the mean of all of the values. To obtain the photoionization rate
$\Gamma_{12}$, we multiplied the photoionization rate from \citet{haardt01} at
each of the above redshifts
by the scale factor that was used to bring the median recovered \hone\ optical depth of
the mock spectra into agreement with that of the corresponding observed spectrum, and took the mean for
all set of mocks. As in \citet{rakic12}, we chose a temperature typical of
a moderately-dense region in IGM \citep[e.g.,][]{schaye00b, lidz10, becker11, rudie12b},
and assumed that the gas fraction $f_{\rm g}$ corresponds to the universal value of $\Omega_{\rm b}/\Omega_{\rm m}$.
Finally, we have taken $b$ to be $20$~\kmps, which is similar to
values measured by \citet{rudie12} for $z\approx2.5$.
Putting eq.~\ref{eq:rakic_eq8} into the form of eq.~\ref{eq:delta_vs_hone}, we obtain
$A=0.67$ and $B=-0.13$, compared with $A=0.66\pm0.01$ and $B=0.16\pm0.01$ measured from our simulations.
Hence, the slope $A$ is in excellent agreement with the theoretical scaling relation implied by the model
of \citet{schaye01}. The normalization agrees to within a factor of two, which we consider
good agreement given the uncertainties in the fiducial parameter values.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth]{{figures/delta_leg}.pdf}
\end{center}
\caption{Median optical depth-weighted overdensity as a function of \hone\ optical depth
for the simulated spectra. The different curves indicate the ion used for the
weighting. We have indicated the one $1\sigma$ error region around the \hone-weighted overdensity.
The parameters from an ordinary least-squares fits of the form
$\log_{10} \delta =A \times \log_{10} \tau_{\honem} +B$ to each of the relations
are given in Table~\ref{tab:delta}. The metal-line absorption typically arises in lower-density
gas than the \hone\ absorption with the same redshift.
}
\label{fig:delta}
\end{figure}
\begin{table}
\caption{Results from fitting a power-law $\log_{10} \delta =A \times \log_{10} \tau_{\honem} +B$ to
the relations in Fig.~\ref{fig:delta}, using ordinary least-squares.}
\begin{center}
\begin{tabular}{lrr}
\hline
Ion & $A$ & $B$ \\
\hline \hline
\hone & $ 0.66\pm 0.01$ & $ 0.16\pm 0.01$ \\
\cfour & $ 0.37\pm 0.02$ & $-0.02\pm 0.01$ \\
\sifour & $ 0.52\pm 0.01$ & $-0.24\pm 0.01$ \\
\osix & $ 0.36\pm 0.02$ & $-0.05\pm 0.01$ \\
\hline
\end{tabular}
\end{center}
\label{tab:delta}
\end{table}
With the above in mind, we can interpret the results of Fig.~\ref{fig:hone}.
Focusing first on the left-hand panel, we find that at fixed \hone, the observed
median \cfour\ optical depths are significantly higher than in the fiducial HM01 model.
The discrepancy increases from $\approx0.1$~dex at $\tau_{\honem}=1$ to $\approx0.5$~dex
at $\tau_{\honem}=10$ and $\approx1$~dex at $\tau_{\honem}=10^2$.
This suggests that at a given
gas overdensity, there is less \cfour\ in the simulations by $\approx0.5$~dex ($\approx1.0$~dex)
for $\tau_{\honem}\lesssim10$ ($\tau_{\honem}\gtrsim10$).
Turning to the different UVB models, while the harder Q-only background
provides a poorer match to the observations,
4Ryd-100 fares much better. Although this model still falls short of the
observed $\tau_{\cfourm}^{\text{med}}$ by about $\approx0.5$~dex
in the highest $\tau_{\honem}$ bin, the softest background is nearly
fully consistent with the observations for $\tau_{\honem}\lesssim10$.
From eq.~\ref{eq:metallicity},
it is apparent that an increase in [C/H] will lead to higher \cfour\ optical depths
at fixed \hone. Therefore, we have run \texttt{SPECWIZARD} with the
elemental abundances scaled linearly by a factor of ten, denoted as ``Zrel-10''.
Such a modification can be partially motivated by the fact that we expect some uncertainty
in the nucleosynthetic yields, of about a factor of two. We have chosen a factor much larger than this
since we find that increasing the metallicities does not scale the median recovered optical depth by the same factor
(contrary to what one might expect from eq.~\ref{eq:metallicity}).
This is because many pixels contributing to the median optical depth are noise dominated
(particularly at low \hone), and some fraction of these pixels have a true optical depth of zero,
so no matter how much the metallicity is increased, the optical depth will never change and exceed the noise.
Although Zrel-10 provides much better agreement between the model and data, even
with such an extreme choice of multiplicative factor it cannot fully account
for the discrepancy between the simulations and data.
This suggests that in the simulation too many \hone\ clouds have negligible metallicity.
To check this, we have calculated the mass
and volume filling factors of the metals using eq.~1 from
\citet{booth12}\footnote{We compute the mass filling factors
using SPH smoothed metallicities, but to avoid smoothing twice,
we compute the volume filling fractions using the particle
metallicities (see \citealt{wiersma09b} for a discussion on the use of
SPH-smoothed versus particle metallicities).}
The authors determined that $>10$\% of the volume and $>50$\% of
the mass need to be enriched to metallicities $Z> 10^{-3}$~Z$_\odot$ to
achieve agreement with observations of \cfour\ in the low-density IGM
at $z\approx3$. For the fiducial L100N1504 box at $z=3.5$ and $Z>0$ we
find volume and mass filling factors of 42\% and 68\%, but for
$Z>10^{-3}$~Z$_\odot$ these are reduced to 10\% and 19\%, respectively.
This suggests that while the fractions of the volume and mass with
non-zero metallicity may be sufficiently high, the metallicities
in the photo-ionised IGM are typically far too low.
Next, we describe another modification to our fiducial model denoted as
``bturb-100'', for which we have considered an unresolved
turbulent broadening term in addition to the
usual thermal broadening. Specifically, we add
$b_{\rm turb} = 100$~\kmps\ in quadrature
to the already included thermal broadening
$b_{\rm therm} (T, m)$ which is calculated for every ion
at each spectral pixel, and
depends on the local temperature $T$ and inversely on atomic mass $m$.
Because of this inverse dependence on atomic mass,
metal ions will be much more strongly affected by the
inclusion of turbulent broadening than \hone.\footnote{
We initially added the thermal broadening term to both \hone\ and metal ions. However,
this led to unphysical values for the flat level of metal ions recovered from
regions bluewards of the \lya\ forest, due to the extreme abundance of \hone\ lines.
Thus, for the model presented in this work we have added turbulent broadening only to
the metals and not to \hone, which
allows us to examine the effect of broadening on \osix, \cthree\ and \sithree\ more clearly.
This is likely also more physical, as the metal-bearing gas may well
be more turbulent than the gas that dominates the neutral hydrogen
absorption \citep[e.g.,][]{theuns02}.
We have confirmed that both methods produce equivalent results
for the \cfour(\hone) and \sifour(\hone) relations, likely because
the metal-line optical depth signal is mostly associated with the strongest and most clustered
hydrogen systems that are not significantly affected by an increased $b$-parameter.
}
Indeed, we find that bturb-100 provides a somewhat better
match to the observed \cfour(\hone) relation.
However, we note that our chosen broadening value $100$~\kmps\ should be considered
a very conservative upper limit, as individual
\cfour\ and \sifour\ components are not usually detected with such large $b$-parameters.
In the centre panel of Fig.~\ref{fig:hone}, we show \sifour(\hone), and find
results that are similar to those for \cfour(\hone).
For $\tau_{\honem}\gtrsim10$ the \sifour\ optical
depths are underestimated by the fiducial HM01 model by a factor
ranging from $\approx0.2$~dex at $\tau_{\honem}=10$
up to $\approx0.8$~dex at $\tau_{\honem}=10^2$.
Invoking 4Ryd-100 leads to near agreement
for all but the highest \hone\ optical depth, while Zrel-10 and bturb-100
also fare markedly better than the fiducial model.
We now consider the \osix(\hone) relation in the right-hand panel of Fig.~\ref{fig:hone}.
While \cfour\ and \sifour\ are expected to mainly probe
cool photoionized gas ($T\sim10^4$~K), \osix\ reaches its peak ionization fraction of 0.2 at
$T=3\times10^5$~K, which is close to the temperatures expected of shocks
associated with accretion events or winds.
Simulations predict that \osix\ around galaxies
is primarily collisionally ionized \citep[e.g.][]{tepper-garcia11, stinson12, ford13, shen13}.
Applying ionization modelling to observations also provides
evidence that \osix\ near moderate to strong \hone\ preferentially probes this
hot gas phase \citep[e.g,][]{simcoe04, aguirre08, danforth08, savage14, turner15}.
Indeed, the results from the right-hand panel of Fig.~\ref{fig:hone} differ considerably
from the previous two relations. Firstly, the simulation realized with the
fiducial model is almost fully consistent with the observations, with any discrepant
points offset by a maximum of 0.2~dex (note the smaller dynamic range
of the y-axis compared to the previous panels).
While
the alternate UVBs (Q-only and 4Ryd-100) have slightly lower $\tau_{\osixm}$ than the fiducial case,
overall we do not find significant differences between these models.
This suggests that in EAGLE the \osix(\hone) relation may be probing a
primarily collisionally ionized gas phase, for which variations in the ionization background
do not have a significant impact on the results.
Furthermore, the fact that the median \osix\ optical depth
appears to be largely insensitive to the addition of turbulent broadening
could indicate that \osix\ is already significantly thermally broadened.
We note that if the pixel optical
depths do not originate predominantly from photoionized gas, then $\tau_{\honem}$ can no
longer be used as a measure of the density.
\begin{figure*}
\includegraphics[width=0.45\textwidth]{{figures/c3_vs_c4_allqso_p50_force_tau_min_comb_leg}.pdf}
\includegraphics[width=0.45\textwidth]{{figures/si3_vs_si4_allqso_p50_force_tau_min_comb}.pdf}\\
\caption{The same as Fig.~\ref{fig:hone}, but showing \cthree(\cfour)
and \sithree(\sifour), and with the data from observations presented in
Table~\ref{tab:data_1}. Unlike for relations binned by \hone,
different ionization states of the same element are not sensitive to
the metallicity of the gas. We find that for \cthree(\cfour), the simulations and data
are in good agreement for the fiducial ionizing background, and the observations
particularly disfavour 4Ryd-100 and Zrel-10. The \sithree(\sifour) relation is somewhat
less constraining, and while the median \sithree\ optical depths from HM01 model
are slightly above the observed values, the discrepancy is no more than 0.1~dex and only
seen in the highest \sifour\ bins.
This indicates that
the temperature and density of the gas probed by pixels with detected \cfour\ and and \sifour\ is
well captured by the simulations, without needing to invoke modifications to the model.}
\label{fig:temp}
\end{figure*}
\begin{table*}
\caption{Observational data from Figs.~\ref{fig:temp} and \ref{fig:rel}.
The format is the same as Table~\ref{tab:data_0}, but here we present
relations binned by either \cfour\ or \sifour\ optical depths. The left column
indicates the central value of the \cfour\ or \sifour\ bin, and the
subsequent columns list the median recovered optical depths for the relation
denoted in the top row. }
\input{tables/data_1.tab}
\label{tab:data_1}
\end{table*}
\subsection{$\tau_Z$ as a function of $\tau_{\rm CIV}$ and $\tau_{\rm SiIV}$}
While metal ions as a function of $\tau_{\honem}$ can probe the metallicity-density relation,
examining different ionization states of a single element can provide insight into
the physical properties of the gas, because the ionization fractions that
set the relative optical depths will only depend on the temperature, the density,
and the UV radiation field (and not on the metallicity, but see below). These optical depth ratios
have previously been used to establish that the gas probed by \cfour\ and \sifour\ is
consistent with being in photoionization equilibrium \citep{schaye03, aguirre04}.
Fig.~\ref{fig:temp} examines \cthree(\cfour) and \sithree(\sifour), and the
observational data is provided in Table~\ref{tab:data_1}.
Looking first at \cthree(\cfour), we find that HM01 is consistent
with all of the \cfour\ bins. Both the bturb-100 and Q-only models
also agree with the data, which is notable in particular for Q-only as it is the most disfavoured by the
\cfour(\hone) relation.
Finally, we find that the 4Ryd-100 and Zrel-10 models fare
particularly poorly in this relation, and produce median \cthree\ optical depths lower than
the observations by up to 0.4~dex. In particular, for Zrel-10 such a discrepancy
may seem surprising, since changing the carbon abundance should not affect
the amount of one ionization state against another. However, the reason this occurs
is because \cfour\ increases more than \cthree, which is
due to the fact that many more \cthree\ pixels are contaminated and hence do not change.
Next, we find the \sithree(\sifour) relation
to be somewhat less constraining. While the fiducial HM01 model demonstrates one of the largest
discrepancies with the data,
the difference is not more than $\approx0.1$~dex when the errors are considered,
and is only seen in the highest \sifour\ bins.
Thus, we find good agreement between the data and the fiducial model for
both relations, which suggests that the temperature and density of the gas probed by
\cfour\ and \sifour\ pixels is consistent between the observations and simulations.
We note that the above result is not in tension with the results shown in Fig.~\ref{fig:hone}. Namely,
while Fig.~\ref{fig:hone} indicates that there is a lack of \cfour\ and \sifour\ in the simulations,
Fig.~\ref{fig:temp} demonstrates that the \sifour\ and \cfour\ that we do find in the mock spectra,
regardless of the amount,
likely resides in gas with similar temperature and density as in the observations.
\begin{figure*}
\includegraphics[width=0.33\textwidth]{{figures/si4_vs_c4_allqso_p50_force_tau_min_comb}.pdf}
\includegraphics[width=0.33\textwidth]{{figures/o6_vs_c4_allqso_p50_force_tau_min_comb_leg}.pdf}
\includegraphics[width=0.33\textwidth]{{figures/o6_vs_si4_allqso_p50_force_tau_min_comb}.pdf}
\caption{The same as Fig.~\ref{fig:hone}, except showing \sifour(\cfour),
\osix(\cfour) and \osix(\sifour) from left to right, which probe relative abundances.
The data from the observations is given in Table~\ref{tab:data_1}.
In the left-hand panel, we find that $\tau_{\sifourm}^{\text{med}}$
is underestimated by all models except bturb-100, and is insensitive to the
choice of UVB.
Next, for \osix(\cfour) and \osix(\sifour) we observe a stronger
sensitivity to different ionizing background models and turbulent broadening
compared to \osix(\hone). For these relations, we
observe a better match between the fiducial and hardest UVB models (HM01 and Q-only),
in tension with the results from \cfour(\hone) and \sifour(\hone) relations,
where we find a strong preference for the softer ionization backgrounds (see Fig.~\ref{fig:hone}).
}
\label{fig:rel}
\end{figure*}
In Fig.~\ref{fig:rel} we examine
relations between different metal ions, which trace relative abundances
and physical conditions. The data for this figure is provided in Table~\ref{tab:data_1}.
For example, Si/C, which can be estimated using the \sifour(\cfour) relation,
has been found to be greater than solar by a
factor of a few in the IGM \citep[e.g.,][]{songaila01, boksenberg03, aguirre04}.
In the left-hand panel of Fig.~\ref{fig:rel}, we plot the median
\sifour\ optical depth against \cfour. While the results are not very
sensitive to the choice of ionizing background, all models except bturb-100
present a paucity of \sifour\ with respect
to the observations. In particular, the median \sifour\ optical depth from the Zrel-10 model
shows an offset of $\leq-1$~dex from the observations at fixed
\cfour. Again, this is due to the fact that for \sifour,
many more pixels that contribute to the median optical depth are noise dominated compared to \cfour,
and therefore do not change when the metallicity is increased.
For the remaining models that use the metallicities directly from the simulations,
this may indicate that at $z\sim3.5$ the simulations have lower [Si/C] than observed.
Additionally, as evidenced by the bturb-100 model, turbulent
broadening, which has the strongest influence on the heavy silicon atoms,
could perhaps be invoked to alleviate this discrepancy.
We briefly draw attention to the bin centred at
$\log_{10}\tau_{\cfourm}=-0.3$, where the observed median \sifour\ optical
depth deviates starkly from the rest of the points. The same behaviour
is also seen in the central panel of Fig.~\ref{fig:rel}, in which
we examine \osix(\cfour). To find the origin of this inconsistency,
we turn to the relations of individual QSOs, in Figs.~\ref{fig:app_si4_vs_c4} and
\ref{fig:app_o6_vs_c4}. In the case of Q1317$-$507 (the upper right-hand panel
of both figures), the median \sifour\ and \osix\ optical depths are
unusually low in this \cfour\ bin, while having relatively
small error bars (the median optical depths of different QSOs are
combined in linear space). We conclude that these points from Q1317$-$507,
likely the result of small number statistics, are responsible for the
anomaly in the \sifour(\hone) and \cfour(\hone) relations.
The centre panel of Fig.~\ref{fig:rel}
shows $\tau_{\osixm}^{\text{med}}$ binned by $\tau_{\cfourm}$.
In contrast to the \osix(\honem) relation (Fig.~\ref{fig:hone}, right-hand panel),
it is apparent that the median \osix(\cfour) optical depth
depends strongly on the choice of UVB, and is sensitive to both an increase in the elemental
abundances and the addition
of turbulent broadening. This is consistent
with the picture that \cfour\ primarily traces photoionized gas,
which will depend on the choice ionizing background, and will not be
as thermally broadened as hot, collisionally ionized gas.
We find that the fiducial HM01 model is in broad agreement
with the data for this relation, even when including the bin centred
at $\log_{10}\tau_{\cfourm}=-0.3$,
while softer UVB models predict too weak \osix\ at high $\tau_{\cfourm}$.
If the \osix\ that is coincident with strong \cfour\ were photoionized,
then this would be a useful constraint. However, unlike \cfour, \osix\ may
well be collisionally ionized.
Finally, in the right-hand panel of Fig.~\ref{fig:rel} we show
\osix(\sifour). Except for $\log_{10}\tau_{\sifourm}\gtrsim-0.5$, we observe a much weaker
dependence on the models than for \osix(\cfour), but we still find that
HM01, in addition to Q-only and Zrel-10, provides the best match to the data.
\subsection{Physical conditions of the gas}
\begin{figure*}
\begin{center}
\includegraphics[width=0.45\textwidth]{{figures/odhist_temp_lin_h1min1.0max10.0_taucutnoise_leg}.pdf}
\includegraphics[width=0.45\textwidth]{{figures/odhist_delta_lin_h1min1.0max10.0_taucutnoise_leg}.pdf}
\end{center}
\caption{PDFs of the optical depth-weighted temperature (left) and overdensity (right), where the line
colour and style indicates the ion used for weighting. The narrower lines represent the full
sample of pixels that have $\tau_{\honem}>10$, and the thicker line PDFs were
calculated using this same sample but with the additional constraint that the metal-line optical
depth had to be $3\sigma$ above the flat level, \taumin. This second cut is used to isolate pixels that
have a significant optical depth detection. The PDF offset to the left of the main distribution denotes
pixels that have zero temperature and density. By definition these pixels have zero metallicity,
but a detected optical depth due to noise or contamination. We find that \cfour\ and \sifour\ probe
a bimodal range of temperatures, while pixels with higher \osix\ optical depths
primarily arise in hot gas.
}
\label{fig:pdf}
\end{figure*}
In this section, we examine the temperatures and densities of the
gas probed by our optical depth relations. In Fig.~\ref{fig:pdf}, we have plotted
the probability density functions (PDFs) of the optical depth-weighted temperatures (left) and
densities (right) from our simulations.
In particular, we are interested
in the physical properties of the regions from which we detect signal in Fig.~\ref{fig:hone}.
We therefore only consider pixels with
$\tau_{\honem}>10$, to focus on areas where
we find the largest discrepancy between
observations and mock spectra in this figure.\footnote{While we have used the
recovered optical depths for the \hone\ and
metal-line optical depth cuts made in Fig.~\ref{fig:pdf}, we note that the results
are unchanged when we use the true optical depths instead.}
The resulting PDFs are shown as the solid lines in Fig.~\ref{fig:pdf}, and demonstrate
high temperatures, which may be surprising for \sifour\ and \cfour, which we
expected to be at least somewhat photoionized.
However, these PDFs are biased to high temperatures because higher temperature gas is
more broadened. For a single absorption line at high temperature, the optical depths will hence
be spread over more pixels, and the individual pixel optical depth values will be lower
than for a lower temperature region.
This effect is further amplified by the fact that the pixel optical depth-weighted temperatures are averages
over the linear (rather than log) temperatures of different gas elements. This means
that gas elements with relatively low ion fractions but high temperatures can affect the
weighted means. Furthermore, at low density the ionization fraction peaks with temperature
become much less prominent if photoionization is included (see e.g. Fig.~1 of \citealt{rahmati16}).
In an effort to combat this bias,
we then make an additional cut, where we only take pixels that have a metal-line optical
depths $3\sigma$ above the corresponding flat level \taumin.
The result of making this additional cut is shown as the dashed line in Fig.~\ref{fig:pdf}.
First, we find that the temperature and density PDF for \osix\ is unchanged,
which indicates that most of the signal for \osix\ truly comes from gas with
higher temperatures (although the distribution is still quite broad, consistent
with \citealt{oppenheimer16}). On the other hand, for \cfour- and \sifour-weighted
quantities this cut reveals a bi-modal temperature distribution, where many of the pixels probe
cooler, $\approx10^{4.5}$~K gas.
Likewise, the \osix-weighted overdensities are not affected by the $3\sigma$ cut,
while for \cfour\ and \cfour\ the overdensity PDFs are still unimodal but have shifted to higher values.
Overall, this figure indicates that while most of the signal for \osix(\hone) comes from
hot ($T\gtrsim10^5$) gas with $\delta\sim1$--$10$, a substantial portion of the pixels that lead to the
\cfour(\hone) and \sifour(\hone) relations arise from cooler, likely photoionized
gas at $\sim10^{4.5}$~K with overdensities $\gtrsim10$.
\section{Discussion}
\label{sec:discussion}
In the previous section, we compared observations of pixel optical depth
relations to the EAGLE simulations. We considered a fiducial QSO+galaxy HM01 UVB \citep{haardt01},
as well a harder QSO-only model, and a softer UVB with
reduced intensity above 4~Ryd by a factor of 100.
For \osix(\hone), we found an insensitivity to the ionizing background model, and
saw good agreement between the simulations and the data. However,
the observed median optical depths from the \cfour(\hone) and \sifour(\hone) relations
were measured to be systematically higher than those derived from the simulations
using the fiducial UVB. The discrepancy is smaller than $\approx0.5$~dex below $\tau_{\honem}=10$
but can reach up to 1~dex for \hone\ bins above this threshold. For \sifour(\hone), invoking
4Ryd-100 fully alleviates the tension, while for \cfour(\hone)
we find this model still falls short of the data, but only for $\tau_{\honem}\gtrsim10$.
We also find that increasing the metallicity by a factor of ten (Zrel-10)
and manually broadening the absorption lines to take unresolved turbulence into consideration (bturb-100)
do not fully resolve the discrepancy.
In this section, we discuss in more detail possible reasons for the observed
mismatch.
Can the discrepancies between the observations and simulations
be attributed to differences in the UVB?
We have indeed found better agreement with the observed \cfour(\hone) and \sifour(\hone)
relations using our softest UVB intended to explore the effects of delayed
\hetwo\ reionization, 4Ryd-100. The reduced
intensity above 4~Ryd disfavours ionization to higher states,
increasing the abundances of \sifour\ and \cfour.
While \citet{haardt01} models take \hetwo\ reionization into account
and predict that the \hethree\ fraction already reaches 50\% at $z\approx6$,
recent studies suggest that the reionization process is patchy,
with \hetwo\ optical depths still high above $z\gtrsim3$
\citep[e.g.,][]{shull10, worseck11}. Thus, the work presented here probes
the epoch where the observed gas may be subject to a strongly fluctuating UVB
above 4~Ryd.
The much better match of the 4Ryd-100 UVB suggests that
\hetwo\ reionization could be complete too early in the simulations.
Turning to other optical depth relations, we find that \cthree(\cfour)
and \sithree(\sifour) do not strongly rule out the 4Ryd-100
model. While these soft UVBs are inconsistent with \osix(\cfour),
the problem occurs only for $\log_{10}\tau_{\cfourm}\gtrsim-0.7$, which
is higher than relevant for Fig.~\ref{fig:hone}.
An alternative effect could be the presence of ionization due
to stellar light from nearby galaxies, which is thought to be important for absorbers as rare as Lyman
limit systems \citep{schaye06, rahmati13b}. The strength of the ionizing radiation
emitted by galaxies drops sharply above 4~Ryd, but could strongly ionize \hone,
lowering the typical optical depths. If \hone\ optical depths are lower,
than at a fixed \hone\ the metal-line optical depths will be higher.
This could explain the larger discrepancy seen at
$\tau_{\honem}\gtrsim10$, where the pixel optical depths are probing denser gas at small
galactocentric distances compared to lower \hone\ optical
depths. However, since it is difficult to estimate the shape
and normalization of this ionizing radiation (and it likely should
not be applied uniformly), we leave testing of this explanation
to a future work.
We have also considered the effects of turbulent broadening.
It is certainly true that our fiducial model misses the unresolved
turbulence from the dense particles
on the imposed EoS (where by fixing the temperatures
to $10^4$~K we neglect a possibly
significant fraction of the energy), and possible
that turbulence in other regions is also underestimated.
Indeed, we find that by artificially broadening metal ion absorption lines, we are able to
bring the \cfour(\hone) and \sifour(\hone)
relations into slightly better agreement with observations. Furthermore, the inclusion of $b_{\rm turb}$
may help alleviate the tension between the observed and simulated \sifour(\cfour). However, we stress that our implementation
provides a very conservative upper limit on this effect, because (a) we use a very high
$b$-parameter (100~\kmps) and (b) we apply the turbulent term to all metal-line absorption pixels,
not just those in very high density regions or with contributions from particles on the EoS.
Another possibility for the observed discrepancy
is that the metallicity of the intergalactic gas in the simulations
is too low. Using our Zrel-10 model, we have examined the effect of increasing
the elemental abundances linearly by a factor of ten. While such a change in metallicity is
larger than the expected metal yield uncertainties,
it is still unable to increase the simulated median \cfour\ and \sifour\ at fixed
\hone\ enough to agree with observations. Furthermore, increasing the
metallicities of both carbon and silicon by equal amounts leads to disagreement
between the simulated and observed \sifour(\cfour) relation, since the optical depths
do not scale directly with metallicity in the same way due to differences in
contamination and noise.
A related issue may be that the that the metals are not transported far enough
into the IGM. An insufficient volume filling fraction of
enriched gas could arise if the simulations
do not resolve the low-mass galaxies thought to be important for metal
pollution \citep[e.g.,][]{wiersma10, booth12}. In Appendix~\ref{sec:restest},
we examined results from simulations with higher resolution than our fiducial model
(the Ref- and Recal-L025N0752 runs). These simulations can resolve galaxies (containing
at least 100 star particles) with $M_{\star}=2.3\times10^7$~\msol,
almost an order of magnitude below that of our fiducial model, where a
100 star particle galaxy would have stellar mass of $1.8\times10^8$~\msol.
Indeed, we find that relations involving \cfour\ are not fully converged
at our fiducial resolution, and invoking the highest-resolution model
for \cfour(\hone) results in an increase in $\tau_{\cfourm}^{\text{med}}$
of up to $\approx0.3$~dex in the highest \hone\ bins.
To investigate the reason for the resolution dependence, we have imposed the same metallicity-density
relation on both Ref-L025N0752 and Ref-L025N0376, finding no significant differences in the optical depth relations.
This implies that the better agreement with observations with increasing resolution is not caused by changes in the
density-temperature structure of the gas. We have also calculated the average metallicites for the various resolutions
in the 25~cMpc volume, and find that the differences are small (varying at most by a factor of 1.2).
Furthermore, we find that the mean gas-particle metallicity is lowest in Recal-L025N0752, and highest in Ref-L025N0188.
Therefore, the increase in \cfour(\hone) with resolution is not due to an increase in the total amount of metals,
but rather to an increase of the metallicity of the IGM. This suggests that winds ejected from galaxies with stellar masses below
$\approx1.8\times10^8$~\msol\ are likely important for IGM pollution.
While a simulation with higher resolution may bring the observations
and simulations closer to agreement, the effect does not appear to be strong enough to
fully explain the differences seen in the \cfour(\hone) relation, and furthermore,
the \sifour(\hone) relation shows almost no change when the resolution is
increased. Therefore, we believe that additional factors may be at play.
An important piece of information to consider is the much better
agreement between the observed and simulated \osix(\hone) relations.
The insensitivity of $\tau_{\osixm}^{\text{med}}$ (when binned by \hone) to the different UVB models
suggests that the gas is primarily collisionally ionized, and hence that \osix(\hone)
is probing a hotter ($T\gtrsim10^5$~K) gas phase than \cfour(\hone) and \sifour(\hone),
as also found for $\tau_{\honem}>10$ by \citet{aguirre08}.
From this, we can conclude that for \textit{hot} gas, the physical properties probed by the
pixel optical depth relations are consistent with observations of
the IGM at $z\sim3.5$. The lack of \cfour\ and \sifour, on the other hand,
may not be due to a too low metallicity or volume filling fraction,
but rather to an incorrect gas phase. If too much of the enriched gas
is excessively hot, then too much carbon and silicon will be ionized to states above
\cfour\ and \sifour, reducing the number of pixels with detectable
\cfour\ and \sifour\ absorption.
\citet{aguirre05} found an even more severe underestimation
of simulated median \cfour\ optical depths, with the tension also
being alleviated by invoking a softer UVB. In contrast to EAGLE,
the simulations in \citet{aguirre05} did not include metal-line cooling,
and that study found that most of the metals resided in an unrealistically-hot gas
phase ($10^5\lesssim T \lesssim 10^7$~K). The authors speculated that the simulations
could be brought into agreement with the observations by implementing
metal-line cooling, but here we have shown that this is not the case.
However, the inclusion of metal-line cooling may have aided in resolving
other issues. While we find good agreement between our observations and simulations
of the \cthree(\cfour) relation, \citet{aguirre05} measured a far too
low $\tau_{\cthreem}^{\text{med}}$, indicating a much stronger mismatch
in the temperature and/or density of the gas in their simulations.
It may be that the temperature of the metal-enriched gas in our
simulations is sensitive to the
details of the stellar feedback.
It is implemented thermally, using a
stochastic prescription in which the temperature of the directly heated
gas is guaranteed to initially exceed $10^{7.5}$~K \citep{dallavecchia12}. The probability
of heating events was calibrated to observations
of galaxy stellar masses and disc sizes at $z\approx0$,
but observations of the CGM were not considered.
In Fig.~\ref{fig:pdf} we established that much of the signal for the
\cfour(\hone) and \sifour(\hone)
relations comes from pixels with temperatures $T~\sim10^{4.5}$~K and
overdensities $\delta\gtrsim10$. This suggests that at $z\sim3.5$, the outflows driven
by stellar feedback may not entrain enough cool ($T\sim10^4$~K) gas.
While \citet{furlong15} found that galaxy star
formation rate densities and stellar masses are in good agreement with observations
for $z\sim3$--$4$, the work presented here suggests that other indicators may be needed to
test fully the feedback implementation at these redshifts.
\section{Conclusion}
\label{sec:conclusion}
In this work we used pixel optical depth relations to study the $z\sim3.5$ IGM,
using new, very high-quality data for a sample of eight $\langle z_{\text{QSO}} \rangle=3.75$ QSOs,
and compared our results with the EAGLE hydrodynamical simulations of galaxy formation.
The QSOs were observed with VLT/UVES, and their spectra all have similar
S/N and coverage. We employed the pixel optical depth technique to obtain
\hone\ and metal-line absorption partially corrected for the effects of
noise, contamination, and saturation. A public version of the code used
can be found at
\url{https://github.com/turnerm/podpy}.
The resulting pixel optical depth
relations were compared to those derived from mock spectra
generated from the EAGLE simulations. The mock spectra were synthesized to have a
resolution, pixel size, S/N and wavelength coverage closely matched to the observations.
We have considered a fiducial QSO+galaxy UVB \citep{haardt01},
as well as a harder QSO-only model and a model for which the intensity above 4~Ryd was reduced by
a factor of 100.
The fiducial EAGLE model was run in a cosmologically
representative box size (100~cMpc) at a relatively high resolution ($2\times1504^3$ particles), and
the feedback from star formation and AGN was calibrated to reproduce the
$z\approx0$ galaxy stellar mass function, galaxy-black hole mass relation,
and galaxy disc sizes.
Our conclusions are listed below.
\begin{itemize}
\item We detect strong correlations for the observed median
\cfour(\hone), \sifour(\hone), \osix(\hone) pixel optical depth relations (Fig.~\ref{fig:hone}), as well as for
\cthree(\cfour) and \sithree(\sifour) (Fig.~\ref{fig:temp}), and for
\sifour(\cfour), \osix(\cfour) and \osix(\sifour) (Fig.~\ref{fig:rel}).
\item We find that for the \cfour(\hone) and
\sifour(\hone) relations, the observed metal-line optical
depths are higher than in the simulations run with the fiducial HM01 UVB.
For \cfour(\hone), we find a discrepancy of up to $\approx0.1$~dex at $\tau_{\honem}=1$,
$\approx0.5$~dex at $\tau_{\honem}=10$, and $\approx1$~dex at $\tau_{\honem}=10^2$,
where we believe we are probing gas at high densities and small galactocentric
distances. For \sifour(\hone), while the agreement is slightly better,
the behaviour is qualitatively similar to that of \cfour(\hone),
and we find that the observed optical depths are higher than seen in the simulations
by up to $\approx0.2$~dex at $\tau_{\honem}=10$
and by up to $\approx0.8$~dex at $\tau_{\honem}=10^2$.
In contrast, \osix(\hone), which probes a hotter gas phase, exhibits
much better agreement (i.e. differences smaller than 0.2~dex) with the data for all \hone\ bins (Fig.~\ref{fig:hone}).
\item We consider UVBs that differ from the fiducial HM01 model, including a harder quasar-only background
(Q-only) and softer backgrounds with 100 times reduced intensity above 4~Ryd (4Ryd-100).
The softer models, which may be more realistic
than our fiducial background if \hetwo\ is still partially ionized at $z\sim3.5$,
are a better match to the \cfour(\hone) and \sifour(\hone) relations,
and can nearly reproduce the observations for $\tau_{\honem}\lesssim10$.
The results of the \osix(\hone) relation are however insensitive to the change
in UVB, which suggests that \osix\ is tracing predominantly collisionally ionized gas (Fig.~\ref{fig:hone}).
\item We also test a model where the elemental abundances are increased
linearly by a factor of 10 (Zrel-10), and a model where the absorption is broadened by
100~\kmps\ (bturb-100). These variations are meant to explore the effects of uncertainties
in the metal yields and of unresolved turbulence, respectively. In both
cases we find that the simulated \cfour(\hone) and \sifour(\hone) relations are in better (but not full)
agreement with the observations, and stress that in we have chosen very aggressive values
for our models such that we would expect the actual effects from uncertainties to be smaller (Fig.~\ref{fig:hone}).
\item Examining relations that investigate different ionization states of the same element, \cthree(\cfour) and
\sithree(\sifour), we find good agreement between the observations and simulations
for the fiducial HM01 model as well as Q-only and bturb-100. However, the observations disfavour
the 4Ryd-100 and Zrel-10 models (Fig.~\ref{fig:temp}).
\item Most models demonstrate a mild paucity of $\tau_{\sifourm}$ in the \sifour(\cfour) relation,
which suggests the simulations may have a slightly too low [Si/C]. The two exceptions are bturb-100, which is in good
agreement with the observations, and Zrel-100, which demonstrates substantially too little \sifour\ at
fixed \cfour\ (left-hand panel of Fig.~\ref{fig:rel}).
\item Unlike \osix(\hone), the \osix(\cfour) and \osix(\sifour) relations
exhibit sensitivity to the UVB for $\tau_{\cfourm}\gg10^{-1}$ and $\tau_{\sifourm}\gtrsim1$,
and we find that \osix(\cfour) is best described by the hardest models
(the fiducial HM01 and Q-only). The dependence on the ionizing background
suggests that strong
\cfour\ and \sifour\ typically probe a cooler ($T\sim10^4$~K), photoionized gas phase compared to the gas traced
by \osix(\hone) (centre and right-hand panels of Fig.~\ref{fig:rel}).
\item We use the simulations to examine the PDFs of the optical depth-weighted temperatures and densities of the pixels
responsible for the high optical depth values in Fig.~\ref{fig:hone}. We find that while
\cfour, \sifour\ and \osix\ all have a component probing hot $T\gtrsim10^5$ gas,
the \cfour\ and \sifour\ optical depths mainly arise from a phase of
cooler ($\sim10^{4.5}$~K) gas with $\delta\gtrsim10$ (Fig.~\ref{fig:pdf}).
\item We discuss possible reasons why \cfour\ and \sifour\ optical depths with
associated \hone\ are underestimated by the fiducial simulations, and we consider that perhaps
a combination of a number of explanations are responsible:
\begin{enumerate}
\item Ionization by local sources, which the simulations do not account for,
may play an important role.
Since the strength of the radiation emitted by stars typically falls sharply above 4~Ryd,
this would ionize \hone\ while having a much weaker effect on the metals, which
would increase the median metal-line absorption for a fixed \hone\ optical
depth (see e.g.\ \S~4.2 and Fig.~9 in \citealt{turner15}). This explanation
is particularly viable for $\tau_{\honem}\gtrsim10$, where we may
be probing small galactocentric distances.
\item The completion of \hetwo\ reionization in the HM01 simulations may
occur too early, or it may be too uniform, since the observations indicate that
it could be quite patchy around $z\sim3.5$ \citep[e.g.,][]{shull10, worseck11}.
This explanation is supported by the better agreement between the
4Ryd-100 model and the \cfour(\hone) and \sifour(\hone) observations.
However, even the 4Ryd-100 model cannot fully explain the \cfour(\hone) observations
for $\tau_{\honem}\gtrsim10$.
\item The magnitude of line-broadening, particularly in dense star-forming regions,
could be underestimated due to unresolved
turbulence in the simulations. While artificially adding a large turbulent broadening
term slightly increases the median \cfour\ and \sifour\ optical depths when binned by \hone,
the effect is not large enough to explain the observed discrepancy.
\item The metallicities may not be high enough in the simulations,
due to uncertainties in the yields. However, even scaling the metallicities
by a factor of 10 is not enough to achieve agreement in the case of \cfour(\hone). Furthermore,
this scaling creates tension in the \sifour(\cfour) relation, due to the fact that the median
recovered optical
\item The simulations may not resolve the low-mass galaxies
required to pollute the diffuse IGM. We find that the highest-resolution
simulations, Ref- and Recal-L025N0752, exhibit superior agreement with the observed \cfour(\hone) relation
by $\approx0.3$~dex at $\tau_{\honem}\approx10^2$. While resolution likely plays a role,
the magnitude of the effect does not appear large enough to explain fully the discrepancy,
particularly for the \sifour(\hone) relation, which we find to be almost insensitive to the
resolution increase.
\item The stellar feedback in the simulations may be driving outflows that
contain insufficient cool gas ($T\sim10^4$~K). The relatively good
agreement between the observed and simulated
\osix(\hone) relation, which probably traces collisionally ionized gas, indicates that
the simulations correctly capture
this hotter gas phase ($T\gtrsim10^5$~K), and that it contains enough metals.
However, if too much of the enriched gas is hot with respect to the observations, then more
\cfour\ and \sifour\ will be ionized to higher energy levels, leading to a paucity of pixels with
detected $\tau_{\cfourm}$ and $\tau_{\sifourm}$.
\end{enumerate}
\end{itemize}
Overall, while the EAGLE simulations qualitatively reproduce all of the pixel optical depth
correlations seen in our sample of QSOs, the mock spectra are found
to have less \cfour\ and \sifour\ at a given density than in the observations.
This suggests that the simulations are still missing one or more important components,
which we have tested in this work:
a more rigorous treatment of \hetwo\ reionization to create a softer UVB,
the resolution required to model turbulence that contribute to line broadening,
and/or higher metallicities and volume filling factors.
However, we do not find that any
of the above models are individually able to match the observations.
While it is possible that the addition of enhanced photoioniziation
of \hone\ by sources close to the absorbers may play an important role, the fact that the simulations agree with
the observed \osix(\hone) relation indicates that the fiducial model is at least able
to capture the hot gas phase correctly.
Therefore, we believe that it is likely that the outflows created by
energetic stellar feedback in the simulations entrain insufficient cool gas.
\section*{Acknowledgements}
This work used the DiRAC Data Centric system at Durham University,
operated by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC
Facility (www.dirac.ac.uk). This equipment was funded by BIS National E-infrastructure
capital grant ST/K00042X/1, STFC capital grants ST/H008519/1 and ST/K00087X/1, STFC DiRAC
Operations grant ST/K003267/1 and Durham University. DiRAC is part of the National
E-Infrastructure. We also gratefully acknowledge PRACE for
awarding us access to the resource Curie based in France at Tr\`{e}s
Grand Centre de Calcul. This work was sponsored by the Dutch
National Computing Facilities Foundation (NCF) for the use of supercomputer
facilities, with financial support from the Netherlands
Organization for Scientific Research (NWO). The research was supported in part by the
European Research Council under the European Union's Seventh
Framework Programme (FP7/2007-2013)/ERC grant agreement
278594-GasAroundGalaxies and the Interuniversity Attraction
Poles Programme of the Belgian Science Policy Office [AP
P7/08 CHARM]. RAC is a Royal Society URF.
\bibliographystyle{mnras}
|
1,116,691,500,835 | arxiv | \section{Introduction and Contributions}
In mobile cellular systems of the fourth (4G)- and fifth generation (5G), the number of \acp{BS} per unit area is expected to grow substantially~\citep{6736747}. One of the main performance limiting factors in such dense networks is \emph{aggregate interference}. Hence, its accurate statistical characterization becomes imperative for network design and analysis. Although abstraction models such as the Wyner model and the hexagonal grid have been reported two \citep{340450}- or even five decades ago \citep{BellLabsIdeaFactory}, mathematically tractable interference statistics are still the exception rather than the rule. Moreover, the emerging network topologies fundamentally challenge various time-honored aspects of traditional network modeling~\citep{ghosh2012heterogeneous}.
In current literature, \ac{BS} deployment models mainly follow the trend away from being fully deterministic towards complete spatial randomness \citep{6171992,6779696}.
However, even the new approaches only yield known expression for the \ac{PDF} of the aggregate interference, if particular combinations of spatial node distributions, path loss models and user locations are given~\citep{elsawy2013stochastic}.
For example, a finite, typically small number of interferers together with certain fading distributions, such as Rayleigh, lognormal or Gamma allows to exploit literature on the sum of \acp{RV}~ \citep{966578,1556824,6292935,C:Sagias_C2_2006,1096243,1673666,1583918,747812,729390,4524273,TorrieriV12,moschopoulos,1962,3717,693785,Coelho199886,abu1994outage,beaulieu1995estimating,hu2005accurate,mehta2007approximating}.
Otherwise, tractable interference statistics have mainly been reported in the field of stochastic geometry. This powerful mathematical framework recently gained momentum as the only available tool that provides
a rigorous approach to modeling, analysis and design of networks with a substantial amount of nodes per unit area~\citep{Baccelli96stochasticgeometry,895048,StochasticGeomAndArchi,Haenggi:2009,4802198,BaccelliVolI,BaccelliVolII,5621983,6747823,guo2013spatial,6516171,5226957,haenggi2012stochastic,NET-032,BaiBlockage}. However, when closed-form expressions are desired, it imposes its own particular limitations, typically including spatial stationarity and isotropy of the scenario~\citep{elsawy2013stochastic,Haenggi:2009,5226957}. Hence, the potential to consider an asymmetric interference impact is very limited and notions such as \emph{cell-center} and \emph{cell-edge} are, in general, not accessible.
The contributions of this paper outline as follows:
\begin{itemize}
\item A new \emph{circular interference model} is introduced. The key idea is to map arbitrary out-of-cell interferer deployments onto circles of uniformly spaced nodes such that the original aggregate interference statistics can accurately be reproduced. The model greatly reduces complexity as the number of participating interferers is significantly reduced.
\item A \emph{mapping scheme} that specifies a procedure for determining the power profiles of arbitrary interferer deployments is proposed. Its performance is evaluated by means of Kolmogorov-Smirnov statistics. The test scenarios are modeled by \acp{PPP} so as to confront the regular circular structure with complete spatial randomness. It is shown that the individual spatial realizations exhibit largely diverging power profiles.
\item A new finite sum representation for the \ac{PDF} of the \emph{sum of Gamma \acp{RV} with integer-valued shape parameter} is introduced to further enhance and validate interference analysis with the circular model. Its restriction to integer-valued shape parameters is driven by relevant use cases for wireless communication engineering and the availability of \emph{exact} solutions.
The key strength of the proposed approach lies in the ability to decompose the interference distribution into the contributions of the individual interferers.
\item Statistics of aggregate interference with \emph{asymmetric interference impact} are investigated. The asymmetry is induced by eccentrically placing a user in a generic, isotropic scenario. This setup is achieved by applying the introduced circular model with uniform power profiles. On top of that, the model enables to employ the proposed finite sum representation. It is shown that the partition of the interference distribution is particularly useful to identify candidate \acp{BS} for user-centric \ac{BS} collaboration schemes. Moreover, the framework allows to predict the corresponding \ac{SIR}- and rate statistics.
\end{itemize}
This paper is organized as follows. \cref{Sec:SGN_CircleModel,Sec:SGN_SumOfGammaTheorem} introduce the circular interference model and the new finite sum representation for the sum of Gamma \acp{RV} with integer-valued shape parameter, respectively.
\cref{Sec:SGN_SamplingArbitraryNetworks} presents a mapping scheme and validates the applicability of the circular model. \cref{Sec:SGN_GenericScenario} investigates aggregate interference statistics and the performance of \ac{BS} collaboration schemes at eccentric user locations. \cref{Sec:SGN_Summary} concludes the work. The main focus of this paper is on downlink transmission in cellular networks. A comparable framework for the uplink is found in \citep{6378490}.
\section{Circular Interference Model}
\label{Sec:SGN_CircleModel}
\begin{figure}
\centering
\includegraphics[width=.6\columnwidth]{./SGN_JRingScenario.pdf}
\caption{Circular interference model with $C$ circles of radius $R_c$ and phase $\phi_c$, $c\in\{1,\ldots,C\}$, and user at $(r,0)$. $\mathcal{T}_{c,n}$ denotes the nodes of the model.}
\label{Fig:JRingScenario}
\end{figure}
Consider the serving \ac{BS} to be located at the origin.
The proposed circular interference model is composed of $C$ concentric circles of interferers, as shown in \cref{Fig:JRingScenario}. On circle $c\in\{1,\ldots,C\}$ of radius $R_c$, $N_c$ interfering nodes are spread out equidistantly. The interferer locations are expressed in terms of polar coordinates as $(R_c,\Psi_{c,n})$, where $\Psi_{c,n} = 2\pi n/N_c-\phi_c$, with $n\in\{1,\ldots,N_c\} $ and $\phi_c \in[0,2\pi)$. Each node is unambiguously assigned to a tuple $(c,n)$ and labeled as $\mathcal{T}_{c,n}$. The central \ac{BS} is denoted as $\mathcal{T}_{0,0}$. Some of the interferers on the circles may also become serving nodes when \ac{BS} collaboration schemes are applied, as will be shown later in \cref{Sec:SGN_TransmitterCollabSchemes}.
\begin{figure}
\centering
\includegraphics[width=.75\textwidth]{./SGN_CircularModelExampleSimple02.pdf}
\caption{Circular interference model with two circles, i.e., $C=2$. Characteristics of an arbitrary heterogeneous interferer deployment are condensed to circles of equidistantly spaced nodes $\mathcal{T}_{c,n}$ such that the original interference statistics can accurately be reproduced. A mapping scheme is presented in \cref{Sec:SGN_SamplingArbitraryNetworks}. The original \acp{BS} are distributed within an annulus of inner radius $R_{\rm in}$ and outer radius $R_{\rm out}$.}
\label{Fig:CircularModelExample}
\end{figure}
The interferers on the circles do not necessarily represent real physical sources. As illustrated in \cref{Fig:CircularModelExample}, they rather correspond to the $N_c$ mapping points of an angle-dependent \emph{power profile} $p_c[n]$, with $\sum_{n=1}^{N_c} p_c[n] = 1$. Exemplary profiles of a single circle are shown in \cref{Fig:PowerProfiles}. Intuitively, $p_c[n]$ condenses the interferer characteristics of an annulus with inner radius $R_{\rm in}$ and possibly infinite outer radius $R_{\rm out}$ such that the circular model equivalently reproduces the original \ac{BS} deployment in terms of interference statistics. This technique enables to represent substantially large networks by a \emph{finite- and well-defined} constellation of nodes. By reducing the number of relevant interferers, it greatly reduces complexity and thus allows to apply finite sum-representations as those introduced in \cref{Sec:SGN_SumOfGammaTheorem}.
\begin{table}
\centering
\caption{Parameters of the circular interference model.}
\begin{tabular}{ r | l }
\hline
\textbf{Symbol} & \textbf{Annotation} \\
\hline
$R_{\rm in}$ & Inner radius of mapping region, $R_{\rm in} \geq 0$ \\
$R_{\rm out}$ & Outer radius of mapping region, $R_{\rm out} > R_{\rm in} $ \\
\hline
$C$ & Number of interferer circles, $C\in\mathbb{N}^+$\\
$R_c$ & Radius of circle $c$, $c \in \{1,\ldots,C\}$, $R_c > 0 $\\
$\phi_c$ & Phase of circle $c$, $c\in\{1,\ldots,C\}$ $\phi_c\in[-\frac{\pi}{N_c}, \frac{\pi}{N_c}]$\\
$N_c$ & Number of mapping points, $c\in\{1,\ldots,C\}$, $N_c\in\mathbb{N}^+$\\
\hline
$P_c$ & Total transmit power of circle $c$, $c\in\{1,\ldots,C\}$, $P_c >0$\\
$p_c[n]$ &Power profile of circle $c$, $c\in\{1,\ldots,C\}$, $n\in\{1,\ldots,N_c\}$, $p_c[n]\in[0,1]$\\
\hline
\end{tabular}
\label{Tab:SGN_Notation}
\end{table}
\cref{Tab:SGN_Notation} summarizes the parameters of the model. Typically, the size of the mapping region, as specified by $R_{\rm in}$ and $R_{\rm out}$, is predetermined by the scenario. The freely selectable variables are the amount of circles $C$ and, for each circle, the phase $\phi_c$, the radius $R_c$ and the number of mapping points $N_c$, respectively. \cref{Sec:SGN_SamplingArbitraryNetworks} presents systematic experiments to provide a reference for the parameter setting and proposes a mapping scheme to determine power profiles $p_c[n]$ and transmit powers $P_c$, respectively.
\newcommand{.4\columnwidth}{.4\columnwidth}
\begin{figure}
\centering
\subfloat[10 interferers]{
\includegraphics[width = .4\columnwidth]{./SGN_PP_10.pdf}
\label{subfig:SGN_PP_10}
}
\subfloat[100 interferers]{
\includegraphics[width = .4\columnwidth]{./SGN_PP_100.pdf}
\label{subfig:SGN_PP_100}
}\\
\subfloat[1\,000 interferers]{
\includegraphics[width = .4\columnwidth]{./SGN_PP_1000.pdf}
\label{subfig:SGN_PP_1000}
}
\subfloat[Hexagonal grid]{
\includegraphics[width = .4\columnwidth]{./SGN_PP_Hex.pdf}
\label{subfig:SGN_PP_Hex}
}
\caption{Power profiles of circular models with one circle, i.e, $C=1$, for three stochastic interference scenarios ((a)-(c)) with $N_1 = 20$ mapping points, and for a hexagonal grid with $N_1 = 12$ mapping points, respectively. The stochastic \ac{BS} distributions are modeled by a \ac{PPP} of intensity $\lambda = 0.01/{\rm Unit\ Area}$. The expected number of interferers as denoted by the figure labels, is varied by altering the scenario size.}
\label{Fig:PowerProfiles}
\end{figure}
A signal from node $\mathcal{T}_{c,n}$, located at $(R_c, \Psi_{c,n})$, to a user at $(r,0)$ experiences \emph{path loss} $\ell(d_{c,n}(r))$, where $d_{c,n}(r)=\sqrt{R_c^2 + r^2 - 2R_c r \cos(\Psi_{c,n})}$ (conf. \cref{Fig:JRingScenario}) and $\ell(\cdot)$ is an arbitrary distance-dependent path loss law, as well as \emph{fading}, which is modeled by statistically independent \acp{RV} $G_{c,n}$. The received power from node $\mathcal{T}_{c,n}$ at position $(r,0)$ is determined as
\begin{equation}
P_{{\rm Rx},c,n}(r) = P_c\, p_c[n]\, \ell(d_{c,n}(r)) \, G_{c,n},
\label{Eq:pRxSingle}
\end{equation}
where $P_c$ denotes the total transmit power of circle $c$. It is important to note that the term $P_{{\rm Rx},c,n}(r)$ can be interpreted as a \ac{RV} $G_{c,n}$, which is scaled by a factor of $P_c\, p_c[n]\, \ell(d_{c,n}(r))$.
The nodes employ omnidirectional antennas with unit antenna gain. Characteristics of antenna directivity are incorporated into the power profile. In general, the central cell will have an irregular shape that can be determined by Voronoi tesselation \citep{6515339}. For simplicity, the \emph{small ball} approximation from \citep{6515339} is applied. A user is considered as \emph{cell-edge user}, if it is located at the edge of the central Voronoi cell's inscribing ball. This approximation misses some poorly covered areas at the actual cell-edge with marginal loss of accuracy \citep{6515339}.
Let $\mathcal{S}$ and $\mathcal{I}$ denote the sets of nodes $\mathcal{T}_{c,n}$ corresponding to desired signal and interference, respectively. Then, the aggregate signal- and interference powers are calculated as
\begin{align}
S(r)= \sum_{\{(c,n)|\mathcal{T}_{c,n}\in\mathcal{S}\}}P_{{\rm Rx},c,n} (r) \label{Eq:SignalRxPower},\\
I(r) = \sum_{\{(c,n)|\mathcal{T}_{c,n}\in\mathcal{I}\}}P_{{\rm Rx},c,n} (r) \label{Eq:InterfRxPower},
\end{align}
with $P_{{\rm Rx},c,n}(r)$ from \cref{Eq:pRxSingle}. The set $\mathcal{S}$ may include the central node $\mathcal{T}_{0,0}$ as well as nodes on the circles, if collaboration among the \acp{BS} is employed. The incoherence assumption is exploited for a more realistic assessment of the co-channel interference \citep{289422}. Following the interpretation of \cref{Eq:pRxSingle}, \cref{Eq:SignalRxPower,Eq:InterfRxPower} can be viewed as sums of scaled \acp{RV}, which are supported by a vast amount of literature for certain fading distributions such as Rayleigh, log-normal and Nakagami-m ~\citep{966578,1556824,6292935,C:Sagias_C2_2006,1096243,1673666,1583918,747812,729390,4524273,TorrieriV12,moschopoulos,1962,3717,693785,Coelho199886,abu1994outage,beaulieu1995estimating,hu2005accurate,mehta2007approximating}.
The present work places particular focus upon the Gamma distribution due to its wide range of useful features for wireless communication engineering. The next section provides preliminary information and introduces a new theorem on the sum of Gamma \acp{RV}. The theorem is introduced before validating the accuracy of the circular model as it is later exploited for this purpose.
\section{Distribution of the Sum of Gamma \aclp{RV}}
\label{Sec:SGN_SumOfGammaTheorem}
\subsection{Preliminaries}
\label{Sec:SGN_Preliminaries}
The \ac{PDF} of a Gamma distributed \ac{RV} $X$ with \emph{shape parameter} $k$ and \emph{scale parameter} $\theta$, i.e., $X\sim\Gamma[k,\theta]$, is defined as
\begin{equation}
f_X(x) = \frac{1}{\theta^k \Gamma(k)} x^{k-1}e^{-x/\theta},
\label{Eq:GammaDistribution}
\end{equation}
with $k>0$ and $\theta>0$, respectively. The Gamma distribution exhibits the \emph{scaling property}, i.e., if $X\sim\Gamma[k,\theta]$, then $aX\sim[k,a\theta]$, $\forall a>0$, as well as the \emph{summation property}, i.e., if $X_i\sim\Gamma[k_i,\theta]$ with $i=1,2,\ldots,N$, then $\sum_{i=1}^N X_i\sim\Gamma[\sum_{i=1}^N k_i, \theta]$.
While the latter is convenient to apply, it is the \emph{sum of Gamma \acp{RV}} with \emph{distinct} scale parameters that has attracted a lot of attention in describing wireless communications though.
Most commonly, it emerged in the performance analysis of diversity combining receivers and the study of \emph{aggregate co-channel interference under Rayleigh fading} \citep{966578,1556824,6292935,C:Sagias_C2_2006,1096243,1673666,1583918,747812,729390,4524273,TorrieriV12}. Therefore, communication engineers have considerably pushed the search for closed form statistics.
Representatively, Moschopoulos' much-cited series expansion in \citep{moschopoulos} was extended for correlated Gamma \acp{RV} in~\citep{966578}. Other approaches based on the inverse Mellin transform (e.g., \citep{mabrouk,provost}) paved the way for representations with a single integral as shown, e.g., in~\citep{6292935} or a Lauricella hypergeometric series as employed, e.g., in~\citep{1556824,1583918}.
All the aforementioned contributions focus on the sum of Gamma \acp{RV} with \emph{real-valued} shape parameter.
The resulting integrals and \emph{infinite} series, despite being composed of elementary functions, typically yield a slow rate of convergence. Therefore, an accurate approximation by a truncated series requires to keep a high amount of terms and complicates further analysis.
The sum of Gamma \acp{RV} with \emph{integer} shape parameter has mainly been reported in statistical literature. Initial approaches focused on the moment generating function and results were obtained in the form of series expansions \citep{1962}. Based on the work of \citep{3717}, \citep{693785} was among the first to formulate a convenient closed form solution. Soon after, the \ac{GIG} distribution was published in \citep{Coelho199886}. This approach was also adopted in wireless communication engineering \citep{C:Sagias_C2_2006,1673666}. In comparison to \acp{RV} with \emph{real-valued} shape parameter, the \ac{PDF} of the sum of \acp{RV} with \emph{integer} shape parameter allows an \emph{exact} representation by a \emph{finite} series.
\subsection{Proposed Finite Sum Representation}
In the analysis of aggregate interference statistics, it is particularly desirable to identify the main distribution-shaping factors, i.e., the interfering sources with the highest impact. However, the expressions in \citep{C:Sagias_C2_2006} and \citep{1673666} are not suitable for this task due to multiple nested sums and recursions. The proposed finite-sum representation in this work avoids recursive functions and enables to \emph{straightforwardly trace the main determinants} of the distribution characteristics.
\addtocounter{footnote}{1}
\begin{shaded}
\begin{theorem}\label{Thm:Theorem1}
Let $G_l\sim\Gamma[k_l,\theta_l]$ be $L$ independent Gamma \acp{RV} with $k_l\in\mathbb{N}^+$ and all $\theta_l$ different\footnotemark[\value{footnote}]. Then, the \ac{PDF} of $Y=G_1+\cdots+G_L$ can be expressed as
\begin{equation}
f_Y(y) = \sum_{l=1}^{L} \frac{\Lambda_l}{\theta_l^{k_l}} h_{k_l-1,l}(0) e^{-y/\theta_l}
\label{Eq:Theorem1}
\end{equation}
with
\begin{align}
\Lambda_l = \frac{(-1)^{k_l+1}}{(k_l-1)!}\prod_{i=1, i\neq l}^{L}\left(1-\frac{\theta_i}{\theta_l}\right)^{-k_i} \label{Eq:LambdaEll}, &\qquad l=1,\ldots,L \\
h_{\delta+1,l}(\zeta) = h_{1,l}(\zeta) h_{\delta,l}(\zeta) + \frac{d}{d\zeta} h_{\delta,l}(\zeta) \label{eq:htau}, &\qquad \delta=0,\ldots,k_l-1
\end{align}
and
\begin{align}
h_{1,l}(0) = -y + \sum_{i=1, i\neq l}^{L} k_i \left(\frac{1}{\theta_i}-\frac{1}{\theta_l}\right)^{-1}\label{eq:h1t}, & \qquad l=1,\ldots,L\\
h_{1,l}^{(m)}(0) = m! \sum_{i=1, i\neq l}^{L} k_i \left(\frac{1}{\theta_i}-\frac{1}{\theta_l}\right)^{-m - 1}\label{eq:h1taut}, &\qquad m=1,\ldots, k_{l}-1
\end{align}
\end{theorem}
\end{shaded}
\footnotetext[\value{footnote}]{The uniqueness of $\theta_l$ can be assumed without loss of generality. In case of some $\theta_l$ being equal, the corresponding \acp{RV} are added up by virtue of the \emph{summation property of Gamma \acp{RV}} (conf. \cref{Sec:SGN_Preliminaries}).}
\begin{proof}
The proof is provided in \cref{App:ProofOfTheorem}.
\end{proof}
Superscript $(m)$ of $h_{1,l}^{(m)}(\zeta)$ denotes the $m$-th derivative of $h_{1,l}(\zeta)$. The recursive determination of $h_{\delta,l}(\zeta)$ in \cref{eq:htau} seemingly interrupts the straightforward calculation of $f_Y(y)$. However, $h_{\delta,l}(\zeta)$ is a function of only $h_{1,l}(\zeta)$ and its higher order derivatives. Therefore, the function series in \cref{eq:htau} can be evaluated \emph{in advance} up to the highest required degree $\delta_{\max}= \max_l k_l-1$.
Thus, the proposed approach enables the \emph{exact} calculation of $f_Y(y)$ in a \emph{component-wise} manner
\footnote{A \emph{Mathematica\textregistered} implementation is provided at \emph{https://www.nt.tuwien.ac.at/downloads/?key=g1Y9anw3Dcletqb57RhoiH7ZleE1YlbG}.
The code is conveniently separated into the pre-calculation, storing and reloading of the auxiliary functions in \cref{eq:htau}, and the computation of the actual distribution function.}. In the next step, it is shown how to apply \Cref{Thm:Theorem1} in the proposed circular model.
\subsection{Application in Circular Interference Model}
Assume that $G_{c,n}\sim\Gamma[k_{c,n},\theta_{c,n}]$ in \cref{Eq:pRxSingle}, with $k_{c,n}\in\mathbb{N}^+$ and $\theta_{c,n}>0$. Then, \cref{Eq:SignalRxPower,Eq:InterfRxPower} represent sums of scaled Gamma \acp{RV} $P_{{\rm Rx},c,n}(r)\sim\Gamma[k_{c,n}, \theta_{c,n}'(r)]$, where $\theta_{c,n}'(r) = P_c\, p_c[n]\, \ell(d_{c,n}(r))\,\theta_{c,n}$.
Therefore, their \acp{PDF} can be determined by applying \Cref{Thm:Theorem1}.
The theorem requires all scale parameters to be different. Thus, let $\boldsymbol{\theta}_{\mathcal{I}}(r)$ denote the vector of unique scale parameters $\theta_{c,n}'(r)$ with $(c,n)$ from the set $\{(c,n)|\mathcal{T}_{c,n} \in \mathcal{I}\}$. A second vector $\mathbf{k}_\mathcal{I}$ contains the corresponding shape parameters. By virtue of the \emph{summation property}, if $\theta_{c,n}'(r)$ occurs multiple times in the set, the respective shape parameter in $\mathbf{k}_\mathcal{I}$ is calculated as the sum of shape parameters $k_{c,n}$ of the according entries. The vectors $\boldsymbol{\theta}_\mathcal{S}(r)$ and $\mathbf{k}_\mathcal{S}$ are obtained equivalently.
Then, the \acp{PDF} of $S(r)$ and $I(r)$ are expressed as
\begin{align}
f_S(\gamma; r) &= \sum_{l=1}^{L_\mathcal{S}} \frac{\Lambda_{l}}{\theta_{l}(r)^{k_{l}}} h_{k_{l}-1,l}(0)e^{-\gamma/\theta_{l}(r)}\label{Eq:pdfS}, \\
f_I(\gamma; r) &= \sum_{l=1}^{L_\mathcal{I}} \frac{\Lambda_{l}}{\theta_{l}(r)^{k_{l}}} h_{k_{l}-1,l}(0)e^{-\gamma/\theta_{l}(r)}\label{Eq:pdfI},
\end{align}
with $\Lambda_{l}$ and $h_{\delta,l}(\cdot)$ as defined in \cref{Eq:LambdaEll,eq:htau}. Subscript $l$ indicates the $l$-th components of the vectors $\mathbf{k}_\mathcal{S}$ ($\boldsymbol{\theta}_\mathcal{S}(r)$) and $\mathbf{k}_\mathcal{I}$ ($\boldsymbol{\theta}_\mathcal{I}(r)$) and $L_\mathcal{S}$ and $L_\mathcal{I}$ are their corresponding lengths, respectively.
Hence, employing \Cref{Thm:Theorem1} allows to evaluate the \emph{exact} distributions of the aggregate signal- and interference from the circular model by \emph{finite} sums. In the following section, this fact is exploited to verify the accuracy of the model.
\section{Mapping Scheme for Stochastic Network Deployments}
\label{Sec:SGN_SamplingArbitraryNetworks}
This section presents a procedure to determine the power profiles $p_c[n]$ and the corresponding powers $P_c$ of the circular model for completely random interferer distributions.
Then, systematic experiments are carried out to provide a reference for selecting the free variables $C$ and $N_c$, respectively. The parameters $R_c$ and $\phi_c$ are also specified by the procedure. The accuracy of the approximation is measured by means of the Kolmogorov-Smirnov distance. It is defined as
\begin{equation}
D(r) = {\rm sup}_{x} \left| F_{I,{\rm original}}(x;r) - F_{I,{\rm circular}}(x;r)\right|,
\end{equation}
where $r$ refers to the user's eccentricity and $F_{I,{\rm original}}(x;r)$ and $F_{I,{\rm circular}}(x;r)$ denote the aggregate-interference \acp{CDF}\footnote{The \ac{CDF} of a \ac{RV} $X$ with \ac{PDF} $X$ is determined as $F_X(x) = \int_{-\infty}^x f_X(x')dx'$.} of the original deployment and the circular model, respectively. The corresponding \acp{PDF} are obtained by \Cref{Thm:Theorem1}.
\subsection{Mapping Procedure}
\label{Sec:SamplingScheme}
\begin{algorithm}
\KwData{number of circles $C$; nodes per circle $N_c$\;
\hspace{28pt} original base station deployment $\mathcal{N}$\;
\hspace{28pt} inner- and outer radius of mapping region $\mathcal{A}$: $R_{\rm in}$ and $R_{\rm out}$\;}
\KwResult{$P_c$, $p_c[n]$, $R_c$ and $\phi_c$ for all $c \leq C$\;}
\For {$c = 1\ {\rm to}\ C$}{
determine $R_c$ and $\phi_c$ based on the strongest interferer that has not yet been mapped\;
}
\For {$c = 1\ {\rm to}\ C$}{
specify mapping region $\mathcal{A}_c$ with inner radius $R_c$ and outer radius $R_{c+1}$\;
\textbf{if} $c=1$ \textbf{then} set inner radius of $\mathcal{A}_c$ to $R_{\rm in}$; \textbf{end} \\
\textbf{if} $c=C$ \textbf{then} set outer radius of $\mathcal{A}_c$ to $R_{\rm out}$; \textbf{end} \\
compute $P_c$ and $p_c[n]$ for $\mathcal{A}_c$\;
}
\caption{Mapping procedure for circular model.}
\label{Alg:MappingProcedure}
\end{algorithm}
Let $\mathcal{N}$ denote a (possibly heterogeneous) set of \acp{BS}\footnote{A deployment is denoted as \emph{heterogeneous}, if the network encompasses different types of \acp{BS}. The part of a network that is associated to a certain type of \ac{BS} is denoted as \emph{tier}.}
that are arbitrarily distributed within an annulus $\mathcal{A}$ of inner radius $R_{\rm in}$ and outer radius $R_{\rm out}$, as shown in \cref{Fig:CircularModelExample}. Radius $R_{\rm out}$ as well as the number of nodes in $\mathcal{N}$ could be substantially large. Given a circular model with $C$ circles and $N_c$ nodes per circle, the parameters $P_c$, $R_c$ and $\phi_c$ as well as the power profile $p_c[n]$ can be determined by \cref{Alg:MappingProcedure}.
The presented procedure employs the origin as a reference point and therefore does \emph{not} depend on the user location.
The computation of $P_c$ and $p_c[n]$ outlines as follows. Let $\mathcal{T}_{c,n}$ denote node $n$ on circle $c$. Assume that its associated mapping area $\mathcal{A}_{c,n}$ is bounded by the circles of radius $R_c$ and $R_{c+1}$ (in the case of $c=1$, the inner radius is set to $R_{\rm in}$; for $c=C$ the outer radius is set to $R_{\rm out}$) as well as the perpendicular bisectors of the two line segments $\overline{\mathcal{T}_{c,n}\mathcal{T}_{c,n-1}}$, and $\overline{\mathcal{T}_{c,n}\mathcal{T}_{c,n+1}}$, as illustrated in \cref{Fig:CircularModelExample}. This yields an even division of circle $c$'s mapping area $\mathcal{A}_c$, which can be formulated as $\mathcal{A}_c = \bigcup_{n\in\{1,\ldots,N_c\}} \mathcal{A}_{c,n}$. The average received power at the origin from all considered \acp{BS} in $\mathcal{A}_c$ is calculated as
\begin{equation}
P_{{\rm Rx},\mathcal{A}_c} = \sum_{i\, \in \, \mathcal{N}\cap\mathcal{A}_c}P_{{\rm Tx},i}\, \ell(d_i) \mathbb{E}[G_{i}],
\label{Eq:AvgRxPowerccp}
\end{equation}
where $P_{{\rm Tx},i}$, $d_i$ and $G_i$ correspond to transmit power, distance and experienced fading of interferer $i$, respectively. Then, the total transmit power $P_c$ is obtained by mapping $P_{{\rm Rx},\mathcal{A}_c}$ back on the circle, which formulates as $P_c = P_{{\rm Rx},\mathcal{A}_c}\, \ell(R_c)^{-1}$. Hence, in this scheme the average received powers from the original deployment and the circular model are equivalent at the origin. The segmentation of $\mathcal{A}_c$ into areas $\mathcal{A}_{c,n}$ yields the corresponding power profile
\begin{equation}
p_c[n] =\frac{1}{ P_{{\rm Rx},\mathcal{A}_c}} \left(\sum_{i\, \in\, \mathcal{N}\cap\mathcal{A}_{c,n}} P_{{\rm Tx},i}\, \ell(d_i)\right),
\label{Eq:PowerProfileExSample}
\end{equation}
with $P_{{\rm Rx},\mathcal{A}_c}$ from \cref{Eq:AvgRxPowerccp}.
In the presented procedure, the parameters $R_c$ and $\phi_c$ are set such that the $c$-th dominant interferer coincides with a node on circle $c$, as illustrated in \cref{Fig:CircularModelExample}. This ensures that $R_1\geq R_{\rm in}$ (in a heterogeneous network, as investigated in \cref{Sec:MappingPerformanceHetNet}, non-dominant interferers between $R_{\rm in}$ and $R_1$ are mapped "back" on circle $1$ by the receive-power dependent weighting in \cref{Eq:PowerProfileExSample}) and $R_C\leq R_{\rm out}$, and is especially suitable for completely random interferer distributions, as demonstrated in the next section. In fully regular scenarios, on the other hand, a circle comprises multiple, equally dominant nodes, making it expedient to specify $R_c$ and $\phi_c$ according to the structure of the grid. For example, the circular model allows to \emph{perfectly} represent a hexagonal grid setup, when the number of mapping points is set as a multiple of six. Then, the nodes on the circle coincide with the actual interferer locations. An exemplary power profile for $N_1 = 12$ is shown in \cref{subfig:SGN_PP_Hex}.
\cref{Alg:MappingProcedure} is one of many possible mapping approaches. It is a heuristic, based on the authors' experience and observations and is thus \emph{not claimed} to be optimal and its refinement yields an interesting topic for further work. The next two sections perform systematic experiments in \emph{completely random} scenarios to provide a reference for setting $C$ and $N_c$. For reasons of clarity, \cref{Sec:SamplingPerformanceEval} is limited to \emph{homogeneous} \ac{BS} deployments. Heterogeneous setups are then evaluated in \cref{Sec:MappingPerformanceHetNet}.
It is refrained from stochastic scenarios with a certain degree of regularity, since measuring spatial inhomogeneity is an ongoing topic of research~\citep{5621983}. Completely random- and fully regular scenarios are considered as limiting cases, encompassing every conceivable practical deployment.
\subsection{Performance Evaluation for Homogeneous Base Station Deployments}
\label{Sec:SamplingPerformanceEval}
\begin{table}
\centering
\caption{System setup for evaluation.}
\begin{tabular}{ r | l }
\hline
\textbf{Parameter} & \textbf{Value} \\
\hline
Transmit power & $P_{T1} = 1$ ($P_{T2} = 0.01$) \\
Node density & $\lambda = \{0.05, 0.1\}/{\rm Unit\ Area}$ ($\lambda_2 = 1/{\rm Unit\ Area}$) \\
Expected number of interferers & $N_{\rm I}=\{100,1000\}$ \\
\hline
Antenna configuration & $N_{\rm Tx}\times N_{\rm Rx}=2\times 1$; omnidirectional\\
Path loss & $\ell(x)=\max(c_{\rm B}, c_{\rm PL}\, x^{-4})$, $c_{\rm B}=1$, $c_{\rm PL} = 1,x> 0$\\
Fading & $G_{c,n}\sim\Gamma[2,1]$\\
\hline
\end{tabular}
\label{Tab:ParameterSetup}
\end{table}
The \emph{original} interferer deployment $\mathcal{N}$ is modeled by a \ac{PPP} of intensity $\lambda$. Such process is considered \emph{most challenging} for the regularly structured circular model, as it represents complete spatial randomness. Signal attenuation is modeled by a log-distance dependent path loss law $\ell(x) = \max(c_{\rm B}, c_{\rm PL}\, x^{-4})$, and Gamma fading with $k=2$ and $\theta=1$, referring to a $2\times1$ \ac{MISO} setup and maximum ratio transmission. Without loss of generality, normalized distance values $x$ are used. The dimension of the network is incorporated in the intercept $c_{\rm B}$ and the constant $c_{\rm PL}$, respectively. In this work, $c_{\rm B}=1$ and $c_{\rm PL}=1$ for simplicity\footnote{Consider the examples as presented in normalized setups, i.e., relative to multiples of the wavelength.}. The \acp{BS} transmit with unit power $P_{T1}=1$ and are distributed within an annular regions of inner radius $R_{\rm in} = 2$ and $R_{\rm out}=\sqrt{N_{\rm I}/(\pi \lambda) +R_{\rm in}^2}$. Radius $R_{\rm in}$ ensures a unit central cell size, assuming that the central \ac{BS} also transmits with $P_{T1}$. The outer radii $R_{\rm out}$ are chosen such that, on average, $N_{\rm I}$ \acp{BS} locations are generated within the corresponding annulus\footnote{Consider a \ac{PPP} of intensity $\lambda$ within an annulus of inner radius $R_{\rm in}$ and outer radius $R_{\rm out}$. The expected number of generated nodes is calculated as $N_{\rm I} = \lambda(R_{\rm out}^2 - R_{\rm in}^2)\pi$.}. In order to cover a wide range of scenarios, $N_{\rm I} = \{100,1000\}$ and $\lambda=\{0.1, 0.05\}/{\rm Unit\ Area}$ are studied. The parameter settings are summarized in \cref{Tab:ParameterSetup}.
\begin{figure}
\centering
\subfloat[$N_c = 10$, $\lambda = 0.1/{\rm Unit\ Area}$]{\includegraphics[width = .45\columnwidth]{./SGN_KS_distance_10_01.pdf}
\label{subfig:SGN_KS_distance_10_01}}
\subfloat[$N_c = 20$, $\lambda = 0.1/{\rm Unit\ Area}$]{\includegraphics[width = .45\columnwidth]{./SGN_KS_distance_20_01.pdf}
\label{subfig:SGN_KS_distance_20_01}}\\
\subfloat[$N_c = 20$, $\lambda = 0.05/{\rm Unit\ Area}$]{\includegraphics[width = .45\columnwidth]{./SGN_KS_distance_20_005.pdf}
\label{subfig:SGN_KS_distance_20_005}}
\subfloat[$N_c = 40$, $\lambda = 0.05/{\rm Unit\ Area}$]{\includegraphics[width = .45\columnwidth]{./SGN_KS_distance_40_005.pdf}
\label{subfig:SGN_KS_distance_40_005}}\\
\caption{Kolmogorov-Smirnov distance over user eccentricity $r$. Plot markers $\{\bigtriangledown,\Box\}$ refer to various scenario sizes $N_{\rm I}=\{100,1\,000\}$ expected interferers, respectively. Different line styles denote circular models with $C = \{1,2,3,4,5\}$. Figure labels refer to the corresponding number of nodes per circle, $N_c$, and the density $\lambda$ of the original interferer deployment. Black bars depict $95\%$ confidence intervals.}
\label{Fig:KS_distances_over_r}
\end{figure}
For each scenario snapshot, ten circular models with $C = \{1,2,3,4,5\}$ and two distinct values of $N_c$ are set up according to \cref{Sec:SamplingScheme}. In the case of $\lambda=0.1/{\rm Unit\ Area}$, $N_c=\{10,20\}$ and, for $\lambda = 0.05/{\rm Unit\ Area}$, $N_c=\{20,40\}$, respectively. Then, the \emph{aggregate interference distributions} are determined. The distributions for the original interferer deployment are only obtained via simulations (by averaging over $1\,000$ spatial realizations and $10\,000$ fading realizations), since the vast amount of nodes hampers the application of \Cref{Thm:Theorem1} due to complexity issues. On the other hand, the circular models comprise at most $44$ \emph{active} nodes and therefore enable to utilize the theorem. This number is obtained for $C=5$ and $N_c=40$, and stems from the fact that in a homogeneous \ac{BS} deployment, the dominant interferers are also the closest ones. Therefore, the presented scheme only maps a single \ac{BS} on each circle $c<C$, i.e., except for $c=C$ there is only one active node per circle.
\cref{Fig:KS_distances_over_r} depicts Kolmogorov-Smirnov distances over the user eccentricity $r$. The first important observation is that the accuracy considerably improves with an increasing number of circles $C$. This mainly results from accurately capturing the first few dominant \acp{BS} that have the largest impact on the aggregate interference distribution, as later shown in \cref{Sec:SGN_GenericScenario}. A second remarkable observation is that doubling the amount of nodes per circle from $N_c = 10$ to $N_c=20$ for $\lambda = 0.1/{\rm Unit\ Area}$ (conf. \cref{subfig:SGN_KS_distance_10_01,subfig:SGN_KS_distance_20_01}), and from $N_c = 20$ to $N_c=40$ for $\lambda = 0.05/{\rm Unit\ Area}$ (conf. \cref{subfig:SGN_KS_distance_20_005,subfig:SGN_KS_distance_40_005}) does not achieve smaller Kolmogorov-Smirnov distances, respectively. This result indicates that it is rather the number of circles $C$ than the number of nodes per circle $N_c$ that impacts the accuracy. As shown in the examples, good operating points are $N_c = \lfloor 1/\lambda\rfloor$ and $C={\rm arg\, min}_c |\ell(d_c)/\ell(d_1) - 2\,\lambda|$, where $d_c$ denotes the average distance of the $c$-th dominant interferer to the origin~\citep{moltchanov2012distance}. Lastly, it should be noted that the circular model allows to represent $1\,000$ and more interferers by some $10$ nodes with Kolmogorov-Smirnov distances at the cell-edge not exceeding $0.05$.
\subsection{Performance Evaluation of Heterogeneous Base Station Deployments}
\label{Sec:MappingPerformanceHetNet}
\begin{figure}
\centering
\subfloat[$N_c = 10$]{\includegraphics[width = .45\columnwidth]{./SGN_KS_distance_10_1.pdf}
\label{subfig:SGN_KS_distance_10_1}}
\subfloat[$N_c = 20$]{\includegraphics[width = .45\columnwidth]{./SGN_KS_distance_20_1.pdf}
\label{subfig:SGN_KS_distance_20_1}}
\caption{Kolmogorov-Smirnov distance over user eccentricity $r$ for \emph{heterogeneous} \ac{PPP} scenarios with $\lambda =0.1/{\rm Unit Area}$ ($P_{T1}=1$) and $\lambda_2 = 1/{\rm Unit\ Area}$ ($P_{T2}=0.01$). Plot markers $\{\bigtriangledown,\Box\}$ refer to various scenario sizes with $\{1\,100,11\,000\}$ expected interferers, respectively. Different line styles denote circular models with $C = \{1,2,3,4,5\}$. Figure labels refer to the corresponding number of nodes per circle $N_c$. Black bars depict $95\%$ confidence intervals. }
\label{Fig:KS_distances_over_r_het}
\end{figure}
In this section, a second independent \ac{PPP} of intensity $\lambda_2 = 1/{\rm Unit\ Area}$ is added on top of the \ac{PPP} scenarios with $\lambda=0.1$ in \cref{Sec:SamplingPerformanceEval}. The corresponding nodes transmit with normalized power $P_{T2} = 0.01$, thus representing a dense overlay of low power \acp{BS}. For simplicity, they are distributed within annuli of inner radius $R_{\rm in}$ and outer radii $R_{\rm out}$ as specified above\footnote{To ensure a unit central cell size, an inner radius of $1+(P_{T1}/P_{T2})^{-1/\alpha}$ would be sufficient.}. Then, the total number of expected interferers calculates as $\{1\,100,11\,000\}$, respectively. For each snapshot, \cref{Alg:MappingProcedure} is applied with $C=\{1,2,3,4,5\}$ and $N_c = \{10,20\}$. The performance evaluation is carried out along the lines of \cref{Sec:SamplingPerformanceEval} and the parameters are summarized in \cref{Tab:ParameterSetup}.
\cref{Fig:KS_distances_over_r_het} depicts the results in terms of Kolmogorov Smirnov distances. It is observed that employing the same parameters $C$ and $N_c$ as for the homogeneous scenarios only slightly decreases the performance (conf. \cref{subfig:SGN_KS_distance_10_01,subfig:SGN_KS_distance_20_01}), although mapping $11$ times as many interferers.
Hence, applying the recommendations in \cref{Sec:SamplingPerformanceEval} with respect to the \ac{PPP} that models the \acp{BS} with the highest power, yields a good initial operating point.
\subsection{Power Profiles of \ac{PPP} Snapshots}
\label{Sec:InterferenceCharacteristicsPPP}
As indicated in \cref{Fig:PowerProfiles}, power profiles of homogeneous \ac{PPP} scenarios are characterized by one or a few large amplitudes. To quantify this claim, \cref{Fig:SGN_PAPR_ECDF} shows the empirical distributions of the power-profile peak-to-average ratios as obtained from the \acp{PPP} in \cref{Sec:SamplingPerformanceEval} with $\lambda=0.1/{\rm Unit\ Area}$. The corresponding circular models encompass a single circle (i.e., $C=1$) with $N_1 = 20$ mapping points. It is observed that the peak-to-average ratios range from $3$ to $19$ with the medians being located around $9.5$. The presence of dominant interferers results in a large asymmetry of the interference impact. However, in modeling approaches that are based on stochastic geometry, the differences between scenarios at both ends of the scale are concealed by spatial averaging. What is more, such approaches commonly require user-centric isotropy of the setup in order to obtain exact solutions (e.g., circularly symmetric exclusion regions~\citep{6515339}). Hence, the differences between interference characteristics in the \emph{center of the cell} and at \emph{cell-edge} are generally not accessible. The next section applies the circular model to generate a generic, circularly symmetric scenario and, by employing \Cref{Thm:Theorem1}, analyzes the impact of user eccentricity
\begin{figure}
\centering
\includegraphics[width=.55\columnwidth]{./SGN_PAPR_ECDF.pdf}
\caption{Peak-to-average ratio of power profiles of \ac{PPP} scenarios with intensity $\lambda = 0.1/{\rm Unit\ Area}$ and $N_{\rm I} = \{100,1000\}$ expected interferers. The corresponding circular models are obtained by \cref{Alg:MappingProcedure} with $C=1$ and $N_1 = 20$. Bold dots denote the mean ratios.
\label{Fig:SGN_PAPR_ECDF}
\end{figure}
\section{Interference and Rate at Eccentric User Locations}
\label{Sec:SGN_GenericScenario}
This section investigates user-centric \ac{BS} collaboration schemes in scenarios with \emph{asymmetric interferer impact}. The asymmetry can either arise from \emph{non-uniform} power profiles or user locations outside the center of an otherwise isotropic scenario. The particular emphasis of this section is on the latter, since it is found less frequently in literature. In order to generate a generic, circularly symmetric scenario\footnote{In fact, the circular model generates a \emph{rotationally symmetric} scenario due to the finite number of nodes. However, by setting $N_c$ sufficiently large, the scenario can be considered as \emph{quasi-circularly symmetric}.}, the introduced circular model is applied, which enables to employ \Cref{Thm:Theorem1} for the analysis of the interference statistics.
\subsection{Generic Circularly Symmetric Scenario}
\label{Sec:GenericCircularModel}
The network is composed of a central \ac{BS} and two circles of interferers with $R_1=2$ and $R_2 = 4$, as depicted in \cref{Fig:AppScenario}. Each circle employs $10$ interferers, a uniform power profile, i.e., $p_c[n]=1/10$, and unit total transmit power, i.e., $P_c= 1$. The interferer locations are assumed to be rotated by $\phi_1 = -\pi/10$ and $\phi_2=0$, respectively. \ac{BS} $\mathcal{T}_{0,0}$ is located at the origin and $P_0=0.1$. The normalized system parameters are employed to emulate a unit central cell size and to facilitate reproducibility.
The parameters of the circular model are summarized in \cref{Tab:ParamsOfCirc} and the modeling of the signal propagation is referred from \cref{Tab:ParameterSetup}, respectively. The first goal is to identify the nodes, which dominate the interference statistics at eccentric user locations. Then, these insights are applied for user-centric \ac{BS} coordination and -cooperation.
\begin{figure}
\centering
\includegraphics[width=.55\textwidth]{SGN_AppScenario}
\caption{Circular model with two two circles of radius $R_1 = 2$ and $R_2=4$, respectively. Each circle employs $10$ transmitters. The transmitter positions are rotated by $\phi_1 = -\pi/10$ and $\phi_2=0$. Users at $r=0.5$ and $r=1$ are denoted as bold dots and refer to \emph{middle of cell} and \emph{cell-edge}, respectively.}
\label{Fig:AppScenario}
\end{figure}
\subsection{Components of Asymmetric Interference}
\label{Sec:InterferenceComponents}
\begin{table}
\centering
\caption{Parameters of circular model for numerical evaluation.}
\begin{tabular}{c | l l l l l l}
\hline
\textbf{Circle} & \multicolumn{5}{|l}{\textbf{Values}} \\
\hline
1 & $R_1 = 2$ & $N_1 = 10$ & $P_1 = 1$ & $\phi_1 = -\frac{\pi}{10}$ & $p_1[n] = \frac{1}{10}$ & $n\in\{1,\ldots,10\}$\\
2 & $R_2 = 4$ & $N_2 = 10$ & $P_2 = 1$ & $\phi_2 = 0$ & $p_2[n] = \frac{1}{10}$ & $n\in\{1,\ldots,10\}$ \\
\hline
\end{tabular}
\label{Tab:ParamsOfCirc}
\end{table}
In the first step, only the inner circle of interferers is assumed to be present, i.e., the set $\mathcal{I}$ comprises the $10$ nodes $\mathcal{T}_{1,n},\, n=1,\ldots,10$, of circle $1$. The target is to determine the impact of the closest nodes on the aggregate interference statistics. For this purpose, two representative user locations at $r=R_1/4$ and $r=R_1/2 $ are investigated, referring to \emph{middle of cell} and \emph{cell-edge}, respectively.
The \ac{PDF} of the aggregate interference is obtained by \Cref{Thm:Theorem1}. Its evaluation is simplified by the scenario's symmetry about the $x$-axis: (i) equal node-to-user distances from upper- and lower semicircle, i.e., $d_{1,n} = d_{1,10-n+1}$, (ii) uniform power profile $p_1(n) = 1/10$, and (iii) equal scale parameters $\theta_{1,n} = 1$. Thus, $\theta'_{1,n}(r)=\theta'_{1,10-n+1}(r)$, with $\theta'_{1n}(r) = P_1/10\, \ell(d_{1,n}(r))$. The vectors $\boldsymbol{\theta}_\mathcal{I}(r)$ and $\mathbf{k}_\mathcal{I}$
are of length $L^\mathcal{I}=5$, with $\left[\boldsymbol{\theta}_\mathcal{I}(r)\right]_l = \theta'_{1,l}(r)$ and $\left[\mathbf{k}_\mathcal{I}\right]_l = 4$, respectively. Hence, the distribution of aggregate interference at distance $r$ from the center formulates as
\begin{equation}
f_I(x;r) = \sum_{l=1}^5 \frac{\Lambda_l}{\theta'_{1,l}(r)^4} h_{3,l}(0) e^{-x/\theta'_{1,l}(r)},
\label{Eq:interferenceDistroOneRing}
\end{equation}
where
\begin{align}
\Lambda_l &= -\frac{1}{6}\prod_{i=1, i\neq l}^{5}\left(1-\frac{\theta_i}{\theta_l}\right)^{-4} \label{Eq:LambdaEllEx}, \qquad l=1,\ldots,5, \\
h_{3,l} &= \left(h_{1,l}(0)\right)^3 + 3h_{1,l}(0) h_{1,l}^{(1)}(0) + h_{1,l}^{(2)}(0),
\end{align}
with
\begin{align}
h_{1,l}(0) &= -y + 4 \sum_{i=1, i\neq l}^{5} \left(\frac{1}{\theta_i}-\frac{1}{\theta_l}\right)^{-1}\label{eq:h1tEx},\\
h_{1,l}^{(1)}(0) &= 4 \sum_{i=1, i\neq l}^{5} \left(\frac{1}{\theta_i}-\frac{1}{\theta_l}\right)^{-2}, \\
h_{1,l}^{(2)}(0) &= 8 \sum_{i=1, i\neq l}^{5} \left(\frac{1}{\theta_i}-\frac{1}{\theta_l}\right)^{-3}.
\end{align}
\cref{Fig:InterferenceComponents} shows $f_I(x;r)$ for $r=0.5$ (narrow solid curve) and $r=1$ (wide solid curve), referring to middle of cell and cell-edge, respectively. The dots denote results as obtained with the approach in \citep{6292935}, which requires numerical evaluation of a line-integral and confirms the accuracy of the proposed finite-sum representation.
\begin{figure}
\centering
\includegraphics[width=.6\columnwidth]{SGN_InterferenceComponents}
\caption{Distribution of aggregate interference at user distances $r=0.5$ and $r=1$, respectively. Dots refer to results as obtained with the approach in \citep{6292935}. Dashed curves show contribution from dominant interferers.}
\label{Fig:InterferenceComponents}
\end{figure}
In \cref{Eq:interferenceDistroOneRing}, each sum term refers to a pair of transmitters $\{\mathcal{T}_{1,l},\, \mathcal{T}_{1,10-l+1}\}$. The contribution of each pair to the final \ac{PDF} is rendered visible by truncating the sum in \cref{Eq:interferenceDistroOneRing} at $L'$ with $L'\in\{1,\ldots,5\}$, i.e., only the first $L'$ sum terms are taken into account. Dashed curves in \cref{Fig:InterferenceComponents} depict results for $L'=1$ and $L'=2$.
It is observed that (i) in the \emph{middle of the cell}, body and tail of the \ac{PDF} are mainly shaped by the \emph{four closest} interferers while (ii) at \emph{cell-edge} the distribution is largely dominated by the \emph{two closest} interferers, and (iii) interference at $r=1$ yields a larger variance than at $r=0.5$ due to higher diversity of the transmitter-to-user distances. The results verify link-level simulations in \citep{PDL06:SCVT}. They emphasize the strong impact of interference asymmetry due to an eccentric user location, which is commonly overlooked in stochastic geometry analysis. The next section exploits the above findings for \ac{BS} coordination and -cooperation and investigates the resulting \ac{SIR}- and rate statistics.
\subsection{Transmitter Collaboration Schemes}
\label{Sec:SGN_TransmitterCollabSchemes}
This subsection studies \ac{SIR}- and rate statistics in the full two-circle scenario, as shown in \cref{Fig:AppScenario}. Motivated by the observations in \cref{Sec:InterferenceComponents}, three schemes of \emph{\ac{BS} collaboration} are discussed:
\begin{enumerate}
\item \textit{No collaboration among nodes}: This scenario represents the \emph{baseline}, where $\mathcal{S} = \{\mathcal{T}_{0,0}\}$ and $\mathcal{I}$ comprises all nodes on the circle, i.e., $\mathcal{I}=\{\mathcal{T}_{c,n}\}$ with $c=1,2$ and $n\in\{1,\ldots,10\}$.
\item \textit{Interference coordination}\footnote{Conf., e.g., \ac{eICIC} in the \acs{3GPP} \acs{LTE} standard \citep{3gpp.36.839}.}: The nodes coordinate such that co-channel interference from the two strongest interferers of the inner circle, $\mathcal{T}_{1,1}$ and $\mathcal{T}_{1,10}$, is eliminated. This could be achieved, e.g., by joint scheduling. Then, $\mathcal{S}=\{\mathcal{T}_{0,0}\}$ and $\mathcal{I}$ is composed of $\{\mathcal{T}_{1,n}\}$ with $n\in\{2,\ldots,9\}$ and $\{\mathcal{T}_{2,n}\}$ with $n\in\{1,\ldots,10\}$.
\item \textit{Transmitter cooperation}\footnote{Conf., e.g., \ac{CoMP} in the \acs{3GPP} \acs{LTE} standard \citep{3gpp.36.819}.}: The signals from the two closest nodes of the inner circle, $\mathcal{T}_{1,1}$ and $\mathcal{T}_{1,10}$, can be exploited as useful signals and are incoherently combined with the signal from $\mathcal{T}_{0,0}$. Then, $\mathcal{S}=\{\mathcal{T}_{0,0}, \mathcal{T}_{1,1}, \mathcal{T}_{1,10}\}$ and, as above, $\mathcal{I}$ comprises $\{\mathcal{T}_{1,n}\}$ with $n\in\{2,\ldots,9\}$ and $\{\mathcal{T}_{2,n}\}$ with $n\in\{1,\ldots,10\}$.
\end{enumerate}
For each \emph{collaboration} scheme, the \acp{PDF} of aggregate signal and -interference, $f_S(x;r)$ and $f_I(x;r)$, are calculated using \Cref{Thm:Theorem1}.
The \emph{\ac{SIR}} at user location $(r,0)$ is defined as $\gamma(r) = {S(r)}/{I(r)}$. According to \citep{1941}, the \ac{PDF} of $\gamma(r)$ is calculated as
\begin{equation}
f_\gamma(\gamma; r) = \int_0^{\infty} z f_S(z\, \gamma; r) f_I(z;r) dz,
\label{Eq:QuotientDistribution}
\end{equation}
where $z$ is an auxiliary variable, $f_S(\cdot;r)$ and $f_I(\cdot; r)$ refer to \cref{Eq:pdfS,Eq:pdfI}, and the integration bounds are obtained by exploiting the fact that $f_S(\gamma; r) = 0$ and $f_I(\gamma;r) = 0$ for $x< 0$.
Evaluating \cref{Eq:pdfS,Eq:pdfI} yields sums of elementary functions of the form $a \gamma^b e^{-c\gamma}$, with the generic parameters $a\in\mathbb{R}$, $b\in\mathbb{N}^+$ and $c>0$. Therefore, $f_S(\gamma;r)$ and $f_I(\gamma;r)$ can generically be written as
\begin{align}
f_S(\gamma;r) &= \sum_{s}a_s \gamma^{b_s}e^{-c_s \gamma},\\
f_I(\gamma;r) &= \sum_{i}a_i \gamma^{b_i}e^{-c_i \gamma},
\end{align}
and allow to straightforwardly evaluate \cref{Eq:QuotientDistribution} as
\begin{align}
f_{\gamma}(\gamma;r) &= \sum_s\sum_i \int_0^\infty z\, a_s (z\gamma)^{b_s} e^{-c_s(\gamma z)}\, a_i z^{b_i}e^{-c_i z} dz \nonumber\\
&=\sum_s\sum_i a_s a_i \gamma^{b_s}(c_i+c_s \gamma)^{-i-b_s-b_i}\Gamma(i+b_s+b_i).
\label{Eq:GenericSIR}
\end{align}
The \emph{normalized rate} $\tau$ as a function of the \ac{SIR} $\gamma(r)$ is calculated by the well known Shannon formula $\tau(\gamma(r)) = \log_2 \left(1+\gamma(r)\right)$.
Note that $\tau(\cdot)$ is a function of the \ac{RV} $\gamma(r)$. Hence, its distribution is calculated by the transformation
\begin{align}
f_{\tau}(\tau;r) = \log_{e}(2) 2^\tau f_\gamma(2^\tau-1;r),
\label{Eq:RatePDF}
\end{align}
with $f_{\gamma}(\cdot;\cdot)$ from \cref{Eq:GenericSIR}.
The distributions $f_{\gamma}(\gamma;r)$ and $f_{\tau}(\tau;r)$ are analyzed at $r=0.5$ and $r=1$ referring to \emph{middle of the cell}, and \emph{cell-edge}, respectively. For reasons of clarity, \ac{CDF} curves are presented. In order to verify the analysis, Monte Carlo simulations are carried out, employing the system model from \cref{Sec:GenericCircularModel} and the signal propagation model from \cref{Tab:ParameterSetup}. The results are computed by averaging over $10^7$ channel realizations for each \ac{BS} collaboration scheme and each user location, and are denoted as bold dots in \cref{Fig:SIR_Distributions,Fig:Rate_Distributions}, respectively.
\begin{figure}
\centering
\includegraphics[width=.65\textwidth]{./SGN_SINR_Distributions.pdf}
\caption{\ac{SIR} \ac{CDF} curves for user locations in the \emph{middle of the cell} ($r=0.5$) and at \emph{cell-edge} ($r=1$), respectively. Three cases are depicted: (i) No collaboration among \acp{BS} (solid), (ii) interference coordination (dashed), (iii) cooperation among \acp{BS} (dotted).}
\label{Fig:SIR_Distributions}
\end{figure}
\cref{Fig:SIR_Distributions} shows the obtained \ac{SIR} distributions. It is observed that
\begin{itemize}
\item In the case of \emph{no collaboration} (solid lines in \cref{Fig:SIR_Distributions}), the curves have almost equal shape in the middle of the cell and at cell-edge. The distribution in the middle of the cell is slightly steeper due to the lower variance of the interferer impact. Their medians, hereafter used to represent the distributions' position, differ by 15.5\,dB.
\item When the central node $\mathcal{T}_{0,0}$ \textit{coordinates} its channel access with the user's two dominant interferers, $\mathcal{T}_{1,1}$ and $\mathcal{T}_{1,10}$, the \ac{SIR} improves by 2.4\,dB in the middle of the cell and 5.9\,dB at cell-edge (dashed curves in \cref{Fig:SIR_Distributions}), compared to \textit{no collaboration}.
\item \textit{\ac{BS} cooperation} enhances the \ac{SIR} by 10.2\,dB at cell-edge in comparison to \textit{no collaboration} (left dotted curve in \cref{Fig:SIR_Distributions}). Note that the \ac{CDF} curve also has a steeper slope than without coordination, indicating lower variance of the \ac{SIR}.
\item In the middle of the cell, \emph{cooperation} achieves hardly any additional improvement, as recognized from the overlapping rightmost curves in \cref{Fig:SIR_Distributions}. This remarkable result states that \emph{interference coordination} already performs close to optimal at this user location. Note that in realistic networks \emph{coordination} is typically far less complex than \emph{cooperation}.
\end{itemize}
The curves reflect findings from \citep{6482234}, stating that even in the best case, gains of transmitter cooperation are much smaller than largely envisioned. \cref{Fig:Rate_Distributions} depicts the corresponding rate distributions. The results show that
\begin{figure}
\centering
\includegraphics[width=.7\textwidth]{./SGN_Cap_Distributions.pdf}
\caption{Normalized-rate \ac{CDF} curves for user locations in the \emph{middle of the cell} ($r=0.5$) and at \emph{cell-edge} ($r=1$), respectively. Three cases are depicted: (i) No collaboration among \acp{BS} (solid), (ii) interference coordination (dashed), (iii) cooperation among \acp{BS} (dotted).}
\label{Fig:Rate_Distributions}
\end{figure}
\begin{itemize}
\item Notably, the rate statistics of all three collaboration schemes indicate lower variance at cell-edge than in the middle of the cell.
\item In terms of median value, \emph{\ac{BS} coordination} shows rate improvements by $18.7\, \%$ in the middle of the cell and by $167\, \%$ at cell-edge.
\item \emph{Cooperation} between the central node $\mathcal{T}_{0,0}$ and the user's two closest interferers, $\mathcal{T}_{1,1}$ and $\mathcal{T}_{1,10}$, achieves a rate enhancement of 19.8\,\% in the middle of the cell and 355.7\,\% at cell-edge. Similar to the \ac{SIR}, it is observed that in the middle of the cell, interference coordination already performs close to optimal.
\end{itemize}
In summary, collaboration among the \acp{BS} that were identified as main contributors to the shape of the interference distribution by \Cref{Thm:Theorem1}, achieved large performance enhancements in terms of \ac{SIR} and rate.
It was further shown that the efficiency of such schemes considerably depends on the user eccentricity, or equivalently, the asymmetry of the interference impact.
\section{Summary and Conclusions}
\label{Sec:SGN_Summary}
This paper introduced a new interference model that enables to represent substantially large interferer deployments by a well-defined circular structure in terms of interference statistics. The model applies angle-dependent power profiles, which require the specification of a mapping procedure. The presented scheme, despite not claimed to be optimal, achieved to accurately capture heterogeneous interferer deployments with hundreds or even thousands of base stations by a circular model with only several tens of nodes, reducing complexity substantially. Motivated by the desire to decompose the aggregate interference distribution into the contributions from the individual sources, a new representation for the sum of Gamma random variables with integer shape parameter was proposed. The approach enabled to identify candidate base stations for user-centric base station collaboration schemes and to predict the corresponding \ac{SIR}- and rate statistics at eccentric user locations. It was shown that the performance largely depends on the asymmetry of the interference impact. At the same time, power profiles of \ac{PPP} scenarios indicated the frequent presence of one or a few dominant interferers. Interference modeling based on stochastic geometry tends to conceal this diversity by spatial averaging and isotropic conditions. The authors are therefore confident that the introduced framework offers a convenient tool for investigating fifth generation mobile cellular networks, where the number of interferers and the diversity of scenarios is expected to grow substantially. Further work is mainly directed towards enhanced mapping schemes.
|
1,116,691,500,836 | arxiv | \section{Introduction}
The celebrated theory of linear response of systems close to
equilibrium has been developed in the first half of the XX century,
and has proven highly successful in describing a very wide range of macroscopic
phenomena, particularly thanks to the usefulness of the Fluctuation-Dissipation Theorems (FDT)
\cite{CTXX}. In the second half of the XX century, numerous generalizations have been proposed
to extend the classical theory beyond the linear regime, by
considering the microscopic evolution as represented by deterministic
dynamical systems, by dynamical systems perturbed by some appropriate noise, or by purely
stochastic processes, which reach a nonequilibrium steady state. In particular, relations known as Fluctuation Relations
have been introduced together with different nonequilibrium generalizations of the FDT.
One then investigates the variations of these dynamics under certain perturbations. Remarkably, it turns out that rather different
approaches lead to similar results, cf.\ Ref.\cite{BPRVetc} for recent
reviews of the historical developments and of the recent results. The recent developments,
however, do not constitute a consistent and comprehensive theory yet, therefore a variety
of different approaches is worthy investigating. For instance, Ruelle has shown \cite{CS07} that, for uniformly
hyperbolic dynamical systems, the linear response away from equilibrium is very similar to the linear response close to equilibrium: the Kramers-Kronig dispersion relations hold, and the FDT survives in a modified form, which accounts for the oscillations around
the relevant attractors. If the chaotic hypothesis does not hold, Ruelle concludes that
two new phenomena may arise. The first is a violation of linear response, in the sense that the nonequilibrium steady state does not depend differentiably on parameters. The second phenomenon is a violation of the dispersion relations: the susceptibility has singularities
in the upper half complex plane. These `acausal' singularities are actually due to `energy
nonconservation': for a small periodic perturbation of the system, the amplitude of the
linear response is arbitrarily large.
This means that the steady state of the dynamical system
under study is not `inert' but can give energy to the outside world, something rather
different from the behavior of an equilibrium state.
This approach is based on the smoothness property of SRB measures \cite{Young} along the unstable fibres of the attractors, something hardly realizable in the phase space probability distribution of physically relevant
nonequilibrium systems. Therefore, this approach is limited to the properties
of systems whose departure from the ideal uniform hyperbolicity does not seriously affect
the observed behavior \cite{CS07}.
From a different perspective, Vulpiani \textit{et al.} \cite{BLMV} have studied the response of dynamical systems
to finite amplitude perturbation, assuming that their phase space probability distributions
are not singular with respect to the Lebesgue measure, hence have an integrable density. These authors found that a generalized fluctuation-response relation holds, and it links the average
relaxation towards the steady state to the invariant measure of the system. Furthermore, this approach
points out the relevance of the amplitude of the initial perturbation, something which does not
pertain to the approach mentioned above. The use of regular distributions is justified by
considering that any physical phenomenon is affected by some noise, capable of smoothing out the
singularities produced by the dissipative deterministic dynamics.
However, one may also like to consider the noise as emerging from a projection procedure \cite{Zwanzig}, or may investigate the possibility that the distribution (not necessarily invariant) be expressed in terms of appropriate expansions around a reference one. In this work we pursue the latter approach and propose a method which resembles the Chapman-Enskog technique \cite{Chap} for the Boltzmann Equation (BE), which is suitable for the generalized Liouville Equation (LE) of deterministically thermostatted particle systems. The method takes inspiration from the Bogolyubov hypothesis \cite{Bog} of time scales separation and postulates that the dynamics of the single-particle distribution function is driven by the dynamics of a set of selected fields \cite{Matteo}. Analogously, in the context of many-particle systems, one can conjecture that the dynamics of the phase space probability density is triggered by the dynamics of some averaged phase space quantities.
In particular, in the theory of the BE, the notion of slow variables was corroborated by the assumption of local equilibrium.
At the level of the LE, on the other hand, a similar notion can be introduced under the assumption of a weakly dissipative dynamics, in order to let the resulting probability density be not too far from the Gibbsian density, which is required for the application of the method.
\section{The Chapman-Enskog method of solving the Boltzmann Equation}
\label{sec:sec2}
In this Section we review some of the main features of the Chapman-Enskog method, which we intend, next, to employ, in Sec.\ref{sec:sec3}, in the context of thermostatted particle systems, to study the evolution of the probability density in the full phase space. The approach, introduced by Enskog and popularized in a modified version by Chapman and Cowling \cite{Chap}, consists of a mathematical procedure meant to approximate the solution of the BE. The method deserves a special mention in kinetic theory of gases since the Boltzmann pioneering works, as it allows to derive the Navier-Stokes-Fourier equations of hydrodynamics from the BE, by elucidating and making use of concepts, such as the scaling (hydrodynamic) limit of a kinetic equation, the notion of local equilibrium and the time scales separation, which led to the systematic construction of projection operator methods in statistical mechanics as well as to the later development of special large deviation methods in the theory of stochastic processes \cite{Jona}.
Let us start by writing down the BE, which (in absence of external forces) reads:
\be
\partial_{t}f=-\textbf{v} \cdot \nabla f + C(f) \label{BE}
\ee
where $f=f(\textbf{r},\textbf{v},t)$ is the one particle distribution function, depending on position $\textbf{r}$, particle velocity $\textbf{v}$ and time $t$ and $C(f)$ is a nonlinear integral collision operator. By rescaling the BE with characteristic time and length scales, the Knudsen number $\epsilon=\frac{\lambda}{L}$ arises naturally from (\ref{BE}), which, in dimensionless form, becomes:
\be
\partial_{t}f=-\textbf{v} \cdot \nabla f + \frac{1}{\epsilon}C(f) \label{BECE}
\ee
where $L$ denotes a macroscopic length scale (e.g. the sice of the system) and $\lambda$ is the \textit{mean free path}, i.e. the average distance covered by a moving particle between successive collisions with other moving particles.
In the limit $\epsilon\rightarrow0$, which is commonly referred to as the \textit{hydrodynamic limit}, the fluid becomes dense enough that the dynamics, described by (\ref{BECE}), is dominated by the effect of the collisions.
By introducing the notion of \textit{local equilibrium}, (i.e. the existence of macroscopic regions each of which is in an equilibrium state which may be different from the equilibrium states found in other regions) it is possible to define, locally, a set of selected variables $\textbf{x}$, known as \textit{hydrodynamic fields} (which correspond to the collision invariants: number of particles density $n(\textbf{r},t)$, momentum density $n(\textbf{r},t)\textbf{u}(\textbf{r},t)$, and kinetic energy density $e(\textbf{r},t)=\frac{3}{2}n(\textbf{r},t) k_B T(\textbf{r},t)$), which enter the definition of Maxwell-Boltzmann statistics:
\be
f^{LM}(\textbf{r},\textbf{v},t)=n(\textbf{r},t) (\frac{m}{2 \pi
k_{B}T(\textbf{r},t)})^{\frac{3}{2}}e^{-\frac{m(\textbf{v}-\textbf{u}(\textbf{r},t))^{2}}{2
k_{B}T(\textbf{r},t)}} \label{LM} \ee
where $m$ is the mass of the particle. The local Maxwellians (\ref{LM}) are not, in general, solutions of (\ref{BE}). Nevertheless, in the hydrodynamic limit, where the length scale of the spacial inhomogeneities, $L\sim \epsilon^{-1}$, tends to diverge, they offer a good approximation to the exact solution. In fact, it is evident from Eq. (\ref{BECE}) that for $\epsilon\rightarrow0$ the last term on the right becomes singular. Then, the only way to avoid the singularity is that the collision term itself vanishes, which is guaranteed by the form of the local Maxwellians.
It is worth to notice that Eq. (\ref{BECE}) can also be considered, in the Fourier-Laplace space, as an eigenvalue problem for the operator $\Lambda=C(f)-i \epsilon\textbf{k} \cdot\textbf{v}$. As discussed in Ref. \cite{Blatt}, some technical problems arise when one attempts to solve the eigenvalue problem away from the strict hydrodynamic limit $\epsilon\rightarrow0$, and alternative techniques, based on projection operators, have been recently made available \cite{Matteo2}.
However, in those circumstances where $\epsilon \ll 1$, it makes sense to attempt a perturbative method to solve Eq. (\ref{BECE}), by introducing the following expansion:
\be
f=\sum_{l=0}^{\infty}\epsilon^{l} f^{(l)}\label{fexp}
\ee
Thus, by inserting (\ref{fexp}) into (\ref{BECE}), one may define a \textit{microscopic} time derivative $\partial_t^{micro}f$ as:
\be
\partial_t^{micro}f =-\textbf{v} \cdot \nabla[\sum_{l=0}^{\infty} f^{(l)}\epsilon^{l}]+\frac{1}{\epsilon}C(\sum_{l=0}^{\infty} f^{(l)}\epsilon^{l}) \label{microBE}
\ee
Following \cite{Matteo2}, in the theory of the BE it is also possible to introduce a \textit{macroscopic} time derivative, $\partial_t^{macro}f$, provided that the assumption of local equilibrium holds. The idea is to employ the notion of \textit{normal solutions}, which allows to express the spatial and temporal dependence of the terms $f^{(l)}$ through the hydrodynamic fields, $f^{(l)}(\textbf{r},\textbf{v},t)=f^{(l)}(\textbf{x}(\textbf{r},t),\textbf{v})$.
By applying a sort of chain rule, we can, then, write:
\be
\partial_t^{macro}f =\frac{\partial (\sum_{j=0}^{\infty}\epsilon^j f^{(j)})}{\partial \textbf{x}}\partial_{t}\textbf{x} \label{macroBE}
\ee
The time derivative of the hydrodynamics fields, in (\ref{macroBE}), is found to be given by:
\be
\partial_{t}\textbf{x}=\sum_{k=0}^{\infty}\epsilon^{k}\partial_{t}^{(k)}\textbf{x} \label{dtx}
\ee
where $\partial_{t}^{(k)}\textbf{x}=-\int \textbf{w}(\textbf{v})\textbf{v}\cdot \nabla f^{(k)} d^{3}\textbf{v}$ and where the $\textbf{w}$'s are lower order Sonine polynomials of $\textbf{v}$.
The method, then, requires to equalize the two time derivatives $\partial_t^{micro}f$ and $\partial_t^{macro}f$ (supplemented by (\ref{dtx})). The equality, in fact, provides a cascade of equations for the various terms of the expansion $f^{(l)}(\textbf{x}(\textbf{r},t),\textbf{v})$, whose leading order is represented by the family of Maxwellians (\ref{LM}), and whose first correction, $f^{(1)}$, can be obtained \cite{Chap} from:
\be
\frac{\partial f^{(0)}}{\partial \textbf{x}}\int \textbf{w}(\textbf{v}) \textbf{v} \cdot \nabla f^{(0)}(v) d^{3}\textbf{v}= \textbf{v} \cdot \nabla f^{(0)}- C(f^{(1)}) \label{1storder}
\ee
The lowest order of the equations of hydrodynamics (\ref{dtx}) is known as \textit{Euler} hydrodynamics and features vanishing transport coefficients, whereas, by adding the first correction $f^{(1)}$, one obtains the dissipative Navier-Stokes-Fourier equations, which are endowed with constitutive expressions for the stress tensor and the heat flux \cite{Matteo2}.
However, higher order corrections of the Chapman-Enskog method,
resulting in hydrodynamic equations with higher derivatives (Burnett
and super-Burnett hydrodynamic equations) face severe difficulties
both from theoretical, as well as from the practical point of view
and various regularization methods have been suggested \cite{Matteo,Bobyl}.
\section{Dissipative Liouville Equation with external forcing and nonequilibrium response}
\label{sec:sec3}
The method discussed in the previous Section and applied in the context of kinetic theory can
be extended to Hamiltonian particle systems, as it is formally possible to
construct perturbative techniques to solve the LE. Such perturbation theories allow
to compute the higher order corrections to the phase space counterpart of the \textit{local equilibrium} in the BE and lead to expressions of transport coefficients in terms of the detailed microscopic dynamics.
The standard method which succeeds in this derivation was originally proposed by Green and Kubo \cite{GK}. Any perturbation theory based on an expansion of the probability density in powers of a small parameter $\epsilon$ needs to extend and justify the assumption of \textit{local equilibrium} in phase space and to prove that higher order corrections are suitable to refine the lower order approximations. This is a delicate issue, as, formally, there is no indication of the convergence of such an expansion in phase space and, mostly, there is no analog of the Caflish Theorem \cite{Caf}, formally proving that a proper truncation of the series (\ref{fexp}) approximates the solution to the BE. Furthermore, as pointed out by the Authors in \cite{Marra}, in order to obtain finite transport coefficients, one needs to postulate a strong decay property of the time autocorrelations of mass, momentum and energy currents.
We intend to show, here, that, in the context of externally driven thermostatted particles systems, the analogy with kinetic theory can be, to some extent, retained and most of the concepts formerly employed in the derivation of hydrodynamics from the BE, can be used to obtain nonequilibrium response formulae from the LE. In particular, the dimensionless parameter $\epsilon$ which arises after a proper rescaling of the LE, does not constitute just a mathematical book-keeping device which is eventually set to unity, but it can be endowed with a physical content, such as it was in the case of the BE.
We consider a particle system, held, for time $t\in (-\infty,0]$, in equilibrium with an external bath at temperature $T$.
At time $t=0$ an external force $\textbf{F}^{ext}$, starts acting and, hence, some energy is ''pumped'' inside the system. At the same time a deterministic thermostat is switched on in order to remove part (or the whole) of the energy provided by the external force, and to achieve, on a large time scale, a nonequilibrium steady state. The dynamics we have in mind are the deterministic thermostatted dynamics of nonequilibrium molecular dynamics \cite{EM}.
In particular, we refer to those systems for which the dimensionless LE takes the form:
\be
\dot{\rho}= -\dot{\Gamma} \cdot\nabla \rho + \epsilon \kappa \rho
\label{Liouv}
\ee
with:
\be
\dot{\Gamma}=\dot{\Gamma}_{0}+\epsilon R(\Gamma)=S \cdot \nabla H_{0}+\epsilon R(\Gamma) \label{eom}
\ee
where $S$ is the symplectic matrix, $H_{0}$ is the Hamiltonian of the many-particle system, $\dot{\Gamma}_{0}$ denotes the corresponding Hamiltonian contribution to the dynamics and $R(\Gamma)$ is the term which spoils the conservativity of the equations of motion. Furthermore, in (\ref{Liouv}), we introduced a shorthand notation for the phase space contraction rate: $\kappa=- \nabla \cdot \dot{\Gamma}$ and $\epsilon$ is a dimensionless parameter which is proportional to the intensity of the external driving.
A paradigmatic example, where the parameter $\epsilon$ attains a sensible structure, is provided by the Gaussian thermostatted Lorentz gas of hard spheres, with Hamiltonian $H_{0}(\Gamma)=\frac{\textbf{p}^{2}}{2 m}$. By adding an external force, here represented by $\mathbf{F}^{ext}=q \mathbf{E}$, with $q$ the electric charge and $\textbf{E}$ the external electric field, and the thermostat, both entering the definition of $R$, in (\ref{eom}), one finds: $R =q \textbf{E}-\alpha(\textbf{p})\textbf{p}$, where $m$ the mass of the particle and $\alpha=\frac{q \textbf{E}\cdot \textbf{p}}{p^{2}}=\kappa(\Gamma)$. For simplicity, we consider just a single particle and we also do not take the presence of scatterers explicitly into account, as it does not contribute to the phase space contraction rate, which, as discussed below, is the only observable we are going to be concerned with, in our model.
By rescaling all dimensional quantities with proper characteristic scales, the resulting dimensionless LE takes precisely the structure indicated in (\ref{Liouv}), and, in analogy with the definition of the Knudsen number introduced in Sec. \ref{sec:sec2}, the parameter $\epsilon$ is, then, given by the ratio of the energy absorbed by the thermostat, $\Delta\mathcal{E}_{diss}$, to a characteristic energy of the system $K$:
\be
\epsilon=\frac{E L}{v^{2}}\sim\frac{\Delta\mathcal{E}_{diss}}{K} \label{eps}
\ee
where denotes $E$ a typical intensity of the applied electric field, $L$ a reference length scale, $v$ a characteristic velocity of the dynamics and $K$ may be identified by the kinetic energy of the system or, in systems enjoying local thermodynamic equilibrium, may be also related to the temperature $T$ of the system.
We observe that in the Zubarev's seminal papers \cite{Zub}, an infinitesimal source term, corresponding to our $\epsilon$, was introduced \textit{ad hoc} in the construction of the nonequilibrium statistical operator from the LE, whereas, here, in the context of thermostatted particle systems, a small parameter arises naturally after the adimensionalization.
The parameter $\epsilon$ can be small or large, depending on the details of the coupling with the thermostat. It is well known that, in strongly dissipative systems ($\epsilon\gg 1$), the support of the invariant measure is an attractor endowed with a very thin fractal structure, which appears strongly at variance with a regular distribution. On the other hand, in the limit of weak dissipation (i.e.: weak coupling with the thermostat), $\epsilon\ll1$, it makes sense to attempt a perturbative technique to solve Eq. (\ref{Liouv}), qhich is performed by expanding $\rho$ in powers of $\epsilon$:
\be
\rho=\sum_{l=0}^{\infty}\epsilon^{l} \rho^{(l)}\label{expan}
\ee
where $\rho^{(0)}$ obeys a purely conservative equilibrium dynamics (unaffected by the external field and by the thermostat):
\be
\rho^{(0)}(t)=e^{-\Lambda t}\rho^{(0)}(0)
\ee
with $\Lambda=\dot{\Gamma_{0}}\cdot \nabla$ the Liouvillian operator. Let us point out, here, the analogy with Sec. \ref{sec:sec2}. In the context of the BE, we have already remarked that, in the limit $\epsilon \rightarrow \infty$, it holds $f \rightarrow f^{LM}$, i.e., the local equilibria (\ref{LM}) are a good approximation of the exact statistics of the system.
Similarly, we find, here, that for vanishing $\epsilon$, $\rho\rightarrow\rho^{(0)}$, i.e. the density reduces to the purely Hamiltonian contribution.\\
With (\ref{eom}) and (\ref{expan}), Eq. (\ref{Liouv}) transforms into:
\be
\dot{\rho}= -\dot{\Gamma_{0}} \cdot \nabla (\rho^{(0)}+\epsilon \rho^{(1)}+...)-\epsilon(R\cdot\nabla -\kappa)(\rho^{(0)}+\epsilon \rho^{(1)}+...) \label{micro}
\ee
Moreover, if the assumption of time scales separation holds, one may wish to decompose the dynamics of the probability density into a contribution (expressed via a projection operator $P$, to be defined below in (\ref{projector})) which is solely dependent on some relevant \textit{slow} variables, plus a contribution given in terms of an orthogonal projector $Q=1-P$ (with $1$ the identity operator), which features the dynamics of the \textit{fast} variables.
The decomposition of the dynamics of the density as induced by a specific choice of the projector $P$ was provided by Robertson \cite{Rob}:
\be
\dot{\rho}=P\dot{\rho}+\int_{0}^{t}K(t,t') \rho(t') dt' \label{Rober}
\ee
where the term $K(t,t')\rho(t')$ includes the contribution of fluctuations of the overwhelming majority of the \textit{fast} degrees of freedom and makes Eq. (\ref{Rober}) essentially nonlocal in time.
One may be tempted to investigate under which general conditions the memory kernel in (\ref{Rober}) becomes negligible.
The answer is that the full dynamics of the probability density $\dot{\rho}$, in (\ref{Liouv}), can be properly approximated by the projected part $P \dot{\rho}$ in (\ref{Rober}), for those systems characterized by a rapid decay of fluctuations of the fast degrees of freedom. This approximation, which is referred to in the literature as the \textit{adiabatic approximation} \cite{Just}, paves the way to determine a manifold of slow variables in the phase space. As already shown in previous works addressing the BE \cite{Matteo2}, the adiabatic approximation, which postulates a decomposition between fast and solw variables, has no immediate connection with the order of magnitude of the parameter $\epsilon$. In fact, the standard Chapman-Enskog technique, which relies on the adiabatic approximation in order to compare (\ref{microBE}) with (\ref{macroBE}), works properly just for $\epsilon \ll 1$; on the other hand, the method discussed in \cite{Matteo,Matteo2}, which also disregards the contribution of the dynamics of the fast variables in (\ref{Rober}), was shown to correspond to an exact summation of the Chapman-Enskog expansion and does not impose any constraint on the magnitude of $\epsilon$.
Now, in the spirit of the standard Chapman-Enskog technique, we may assume that the probability density $\rho$ is a sufficiently smooth function whose time dependence is parameterized through a set of $N$ phase space functions $\langle A_{\mu}\rangle(t)$, with $\mu=1,...,N$.
We apply, then, the chain rule (summation over repeated indices is assumed) and write:
\be
P\dot{\rho}
=\sum_{j=0}^{\infty}\epsilon^{j}\frac{\delta \rho^{(j)}}{\delta \langle A_{\mu}\rangle} \partial_{t}\langle A_{\mu}\rangle \label{projector}
\ee
The dynamics of $\langle A_{\mu}\rangle$ (provided that the boundary terms, in the integration by parts, are negligible) is found to be given by:
\be
\partial_{t}\langle A_{\mu}\rangle=\sum_{k=0}^{\infty}\epsilon^{k}\partial_{t}^{(k)}\langle A_{\mu}\rangle
\ee
where:
\be
\partial_{t}^{(k)}\langle A_{\mu}\rangle=\int \nabla A_{\mu}\cdot \dot{\Gamma} \rho^{(k)} d\Gamma= \langle \nabla A_{\mu}\cdot \dot{\Gamma}\rangle^{(k)}\label{defin}
\ee
with $\dot{\Gamma}$ given by (\ref{eom}).
Therefore, we obtain:
\be
P \dot{\rho}=\sum_{j,k=0}^{\infty}\epsilon^{(j+k)}\partial_{t}^{(k)}\rho^{(j)} \label{macro}
\ee
with:
\be
\partial_{t}^{(k)}\rho^{(j)}=\frac{\delta \rho^{(j)}}{\delta \langle A_{\mu}\rangle}\partial_{t}^{(k)}\langle A_{\mu}\rangle
\ee
It is now time to specify the various observables $\langle A_{\mu}\rangle$. A crucial aspect of many nonequilibrium theories is, in fact, concerned with the definition of a proper set of coarse-grained variables triggering the dynamics of certain phenomena of interest. In the context of thermostatted many-particle systems, we suggest to consider the average phase space contraction rate $\langle \kappa \rangle$.
The latter appear to be a promising candidate, as it is a dynamical quantity which, in the steady state, attains a constant value related, in the thermodynamic limit, to the steady entropy production of Irreversible Thermodynamics \cite{RonCoh}.
Then, the protocol we follow is to reconstruct the dynamics of the probability density by matching, at any order of $\epsilon$, the original \textit{microscopic} dynamics given by (\ref{micro}) with the \textit{projected} dynamics (\ref{macro}). Thus, we obtain the following equation:
\be
\sum_{j,k=0}^{\infty}\epsilon^{(j+k)}\partial_{t}^{(k)}\rho^{(j)} =-\sum_{l=0}^{\infty}\epsilon^{l} \dot{\Gamma_{0}} \cdot \nabla \rho^{(l)} -\sum_{m=0}^{\infty}\epsilon^{(m+1)}(R\cdot\nabla -\kappa)\rho^{(m)} \label {main}
\ee
Since $\rho^{(0)}$ evolves according to purely Hamiltonian dynamics, at the first order, $\epsilon^{1}$, Eq. (\ref{main}) gives:
\be
-\dot{\Gamma_{0}} \cdot \nabla\rho^{(1)}-(R\cdot\nabla - \kappa) \rho^{(0)}=\frac{\delta \rho^{(1)}}{\delta \langle \kappa\rangle}\langle \nabla \kappa \cdot \dot{\Gamma_{0}} \rangle^{(0)} \label{good}
\ee
Equation (\ref{good}) addresses and defines the first order deviation from the reference Hamiltonian dynamics, by taking into account the effect of the external field and the phase space contraction induced by the thermostat. A remark is in order when discussing the derivative of the probability density $\rho^{(k)}$ with respect to a generic variable $\langle A_{\mu} \rangle$. The evaluation of the average is, in principle, performed through the knowledge of the full probability density. This is avoided, in the Chapman-Enskog theory, by requiring that the collision invariants are defined only through the local Maxwellians (\ref{LM}). In the present context, instead, the assumption of weak coupling limit ($\epsilon\ll 1$) is used to justify the equivalence: $\langle A_{\mu} \rangle\simeq \sum_{i=0}^{k}\epsilon^{i}\langle A_{\mu} \rangle^{(i)}$.
It is worth to mention that a historically relevant approximate solution of (\ref{good}), is given by the \textit{quasi-equilibrium} (or maximum-entropy) approximation. The technique employs a maximization of the Gibbs entropy, $S(t)=\int \rho\log \rho d\Gamma$, under some given constraints, and works at full equilibrium, i.e. for $\epsilon=0$ and in the limit $t\rightarrow \infty$ (this is, in fact, a method to derive the canonical ensemble in equilibrium statistical mechanics, cf. Ref. \cite{Zwanzig}). Nevertheless, it is not obvious that the technique also applies for $\epsilon > 0$, although some authors \cite{Zub2, Rob} consider it a valid principle in general. Furthermore, a major drawback of this maximization method, outside equilibrium, is that the set of variables and the corresponding constraints are usually not known (typical candidates are the invariants of motion and quantities prescribed by the boundary conditions). We can avoid employing this approximation, as, due to the specific choice of variables we made, we can attempt to tackle analytically Eq. (\ref{good}).
In the limit $t\rightarrow \infty$ and for $\epsilon \ll1$, we seek a solution of (\ref{good}) taking the modified Gibbs form:
\be
\rho\simeq\rho^{(0)}+\epsilon \rho^{(1)}=e^{-[\beta H_{0}(\Gamma)-\epsilon \psi_{\kappa} \kappa(\Gamma)]} \label{rho}
\ee
where the $\psi_{\kappa}$'s are Lagrange multipliers depending on the value of the chosen parameter $\langle \kappa\rangle$.
Let us observe that the functional form of $\rho$ in (\ref{rho}) resembles the expression proposed, through a different method, also in \cite{MLJ}.
For the projected time derivative in (\ref{good}), one finds:
\be
\frac{\delta\rho^{(1)}}{\delta \langle \kappa\rangle}\langle \nabla \kappa \cdot \dot{\Gamma_{0}} \rangle^{(0)}\simeq \rho \kappa(\Gamma) \langle \nabla \kappa \cdot \dot{\Gamma_{0}} \rangle^{(0)} \frac{\partial \psi_{\kappa}}{\partial \langle \kappa\rangle^{(1)}} \label {1}
\ee
On the other hand, the computation of the microscopic time derivative, corresponding to the left hand side of (\ref{good}), gives:
\be
\dot{\Gamma_{0}} \cdot \nabla \rho^{(1)}= (\dot{\Gamma}_{0} \cdot\nabla \kappa )\rho \psi_{\kappa}\label{2}
\ee
Then, by inserting (\ref{1}) and (\ref{2}) into (\ref{good}), and by integrating over the whole phase space, we obtain a differential equation for the $\psi_{\kappa}$'s:
\be
\frac{\partial \psi_{\kappa}}{\partial \langle \kappa\rangle^{(1)}}\langle \kappa\rangle^{(1)}=-\psi_{\kappa}[1+ O(\epsilon)]
\ee
Therefore, up to the first order in $\epsilon$, the density in (\ref{rho}) attains the form:
\be
\rho\simeq e^{-[\beta H_{0}(\Gamma)-\epsilon \frac{\kappa(\Gamma)}{\langle \kappa \rangle^{(1)}}]} \label{soluz}
\ee
For small external fields, the expression (\ref{soluz}) can be further linearized in the nonequilibrium parameter $\epsilon$, which enables us to deduce the nonequilibrium steady state ensemble average of a generic (smooth enough) observable $A$ for a weakly dissipative thermostatted particle system:
\be
\langle A\rangle \simeq \langle A \rangle^{eq}+ \epsilon \frac{\langle A \kappa \rangle^{eq}}{\langle \kappa \rangle^{(1)}} \label{NESS}
\ee
Eq. (\ref{NESS}) is our main result, showing that the steady state average of an observable $A$ comprises a first order nonequilibrium correction proportional to the equilibrium correlation function between phase space contraction rate and the observable itself. The response formula above can be interpreted as the adiabatic limit of the Green-Kubo formula \cite{CTXX}, such that the fast degrees of freedom have been projected out and the time integral of the equilibrium time-correlation function is replaced by an \textit{effective} equilibrium correlation function.
Eq. (\ref{NESS}) corresponds to a first order projection, on a manifold of slow variables, of the FDT for deterministic dissipative particle systems.
\begin{figure}
\begin{center}
\includegraphics[width=6.8cm, height=4.7cm]{Lorenz.eps}
\caption{Electrical conductivity $\sigma$ vs. electric field $E_x$.\\ \textit{Black dots}: numerical results of Lloyd \textit{et al.} \cite{Lloyd}.\\ \textit{Blue line}: adiabatic approximation of the Liouville Equation}\label{Lorenz}
\end{center}
\hspace{0.7cm}
\begin{center}
\includegraphics[width=6.8cm, height=4.7cm]{Lorenz2.eps}
\caption{Magnification of Fig. (\ref{Lorenz}) in the range where the theory \\ best fits the numerical data.}\label{Lorenz2}
\end{center}
\end{figure}
By setting $A=\kappa$ and by recalling that $\textbf{E}=\sigma \textbf{j}$, with $\textbf{j}=q\langle \textbf{p}\rangle$, we used Eq. (\ref{NESS}) to compute, for instance, the electrical conductivity $\sigma(E_{x})$ of a Gaussian thermostatted periodic Lorentz gas driven by an external field $E_{x}$ parallel to the $x-$axis. The numerical analysis performed by Lloyd \textit{et al.} in Ref.\cite{Lloyd} shows that the $\sigma(E_{x})$ decreases nonlinearly with the field strength $E_{x}$, which could not be justified in terms of standard linear response theory and was understood as a typical nonlinear (higher order) effect. In turn, Eq. (\ref{NESS}) predicts, at the first order, the power law relation $\sigma \sim \frac{1}{\sqrt{E_{x}}}$ see Fig. (\ref{Lorenz}). Let us point out that our method applies, by construction, to dissipative systems, hence the external field must not vanish. Consequently, the agreement with experimental or numerical data is expected to be sensible in a range where the external field is finite and not exceedingly strong (as our result is pertinent to a first order perturbation theory). These expectations are, in fact, corroborated by the results shown in Fig. (\ref{Lorenz2}). This tends to suggest, \textit{a posteriori}, that, in nonequilibrium response theory for thermostatted particle systems, the adiabatic limit stands as a plausible physical approximation.
\section{Conclusions}
We remark that the novelty of the proposed approach stems from a suitable combination of a perturbation theory with a projection operator technique in the context of deterministically thermostatted particle systems with nonvanishing phase space contraction rate.
Our work aimed to show that the hypothesis of a time scales separation in a thermostatted particle system enables to express the expectation value of an observable solely in terms of few selected relevant variables (in our case the average phase space contraction rate) which drive the dynamics of the probability density.
The issue of determining response formulae in terms of few relevant observables is crucial to establish a bridge between a macroscopic hydrodynamic-like description and the underlying Liouville Equation.
Alternative derivations which lead to response formulae deduced from the exact Liouville propagator, without resorting to any projection technique, are known \cite{Evans}. We believe it is convenient to also look at approximate formulae which might offer a complementary point of view and might be, in some cases, of more immediate application, as we showed above in computing the conductivity of the Lorentz gas.
It is also worth to notice that, regardless of the intensity of the external field, for any finite field the steady state attractor has a lower dimension than the embedding space. Thus, since the support of the steady state measure lies on the attractor, the measure is not smooth and one has to resort to SRB measures \cite{Young}. On the other hand, the procedure of projecting out the fast degrees of freedom, is believed to ensure regularity to the resulting density, as also discussed in \cite{Evans2}.
Therefore, the suggested technique seems, also, to corroborate the applicability of the method proposed by Vulpiani \textit{et al.} in \cite{BPRVetc}, without invoking unstable manifolds of Anosov systems or introducing any noise occurring in not isolated physical systems
\ack
The Author acknowledges many inspiring discussions with Lamberto Rondoni and Wolfram Just and wishes, also, to thank Angelo Vulpiani and Rainer Klages for helpful comments.
This work was supported by the Swiss National Science Foundation (SNSF).
\section*{References}
|
1,116,691,500,837 | arxiv | \section{Introduction}
Neutron-antineutron ($n-\bar n$) oscillations involve a change of the baryon number ($B$)
by two units ($|\Delta B| = 2$). An experimental observation would allow a glimpse on
physics beyond the standard model, see e.g.~\cite{Mohapatra:2009}.
Since in such oscillations $B$ is violated the process satisfies one of the
Sakharov conditions~\cite{Sakharov:1967} that have been formulated in
order to explain the observation that there is more matter than anti-matter in
the universe.
Given such important and far-reaching consequences it is not surprising that there
is a vast amount of literature on this topic \cite{Phillips:2014}.
Indeed, even within the past two years a wealth of papers have been published
that deal with various and quite different aspects of $n-\bar n$ oscillations
\cite{Bijnens:2017,Gardner:2018,Grojean:2018,Oosterhof:2019,Rinaldi:2019,Gardner:2019,Berezhiani:2019,Kerbikov:2019}.
The key quanitity in this subject is the free $n-\bar n$ oscillation time, $\tau_{n-\bar n}$.
The presently best experimental limit on it is $\tau_{n-\bar n} > 0.86 \times 10^8 \ {\rm s}
\approx 2.7$ yr (with 90\,\% C.L.) \cite{BaldoCeolin:1994}.
Additional information can be gained by studying $n-\bar n$ oscillations in a nuclear
environment. Corresponding experiments have been performed, e.g., for
$^{56}$Fe \cite{Chung:2002}, $^{16}$O \cite{Abe:2015}, and for the deuteron ($^{2}$H)
\cite{Aharmim:2017}, while others are planned \cite{Barrow:2019}. In such a case the
oscillation process is suppressed as compared to the free situation. The pertinent
lifetime $\tau_{\rm nuc}$ is commonly expressed in terms of the one in free space as
\cite{Phillips:2014}
\begin{equation}
\tau_{\rm nuc}=R\, \tau^2_{n-\bar n} \ ,
\end{equation}
where $R$ is an intranuclear suppression factor, also called reduced lifetime, that
depends on the specific nucleus. It can be calculated from nuclear theory and then
can be used to relate the measured lifetimes of
those nuclei with the free $n-\bar n$ oscillation time \cite{Phillips:2014}, see, e.g.,
Refs.~\cite{Sandars:1980,Dover:1983,Alberico:1991,Hufner:1998,Friedman:2008}.
For a long time the suppression factors published in Ref.~\cite{Dover:1983} have been
used as standard by experimentalists in the interpretation of their measurements
\cite{Chung:2002,Aharmim:2017}. For example, in case of the deuteron the corresponding
value is $R \sim (2.40-2.56) \times 10^{22}$ s$^{-1}$, a prediction based on the
phenomenological antinucleon-nucleon ($\bar NN$) potentials by Dover and
Richard~\cite{Dover:1980,Dover:1982}.
Recently, however, those values have been called into question in a work by Oosterhof
et al.~\cite{Oosterhof:2019}. In that study an effective field theory for the
$|\Delta B| = 2$ interaction is constructed and the quantity $R$ is evaluated within the
power counting scheme proposed by Kaplan, Savage, and Wise (KSW) \cite{Kaplan:1998,Kaplan:1999}
for the nucleon-nucleon ($NN$) and $\bar NN$ interactions. The value of $R$ for the deuteron
obtained in that approach is with $(1.1\pm 0.3) \times 10^{22}$ s$^{-1}$ about a factor $2$
smaller than the one by Dover et al.~\cite{Dover:1983}.
In the present paper we report on a calculation of the deuteron annihilation
lifetime, considering neutron-antineutron oscillations. The main motivation comes,
of course, from the aforementioned discrepancy and the work aims at providing if
possible a plausible explanation for the difference.
In our study we follow closely the benchmark calculations of
Sandars~\cite{Sandars:1980} and Dover et al.~\cite{Dover:1983}.
The essential new aspect is that we employ modern interactions for the involved
$NN$ and the $\bar NN$ systems. Modern means that these interactions have
been derived in a consistent and systematic framework, namely chiral effective field
theory (EFT) \cite{Epelbaum:2009}. And it means that these potentials are in line with
present-day empirical information. This concerns specifically the $\bar NN$ system
where a wealth of data on $\bar pp$ scattering and the charge-exchange reaction
$\bar pp \to \bar n n$ has accumulated in the years after the publication of Ref.~\cite{Dover:1983},
notably due to measurements at the LEAR facility at CERN \cite{Klempt:2002}.
That fact is accounted for by utilizing $\bar NN$ potentials which have been fitted to
up-to-date phase shifts and inelasticities provided by a recently
published phase-shift analysis of available $\bar pp$ scattering data \cite{Zhou:2012}.
The $\bar NN$ interactions used in the present study are taken from two works~\cite{Kang:2013,Dai:2017}.
In the first reference, $\bar NN$ potentials up to next-to-next-to-leading order (N$^2$LO)
were constructed, based on a modified Weinberg power counting, in close analogy to
pertinent studies of the $NN$ interaction \cite{Epelbaum:2005}.
In the second, the study was extended to next-to-next-to-next-to-leading order
(N$^3$LO) and, in addition, a new regularization scheme was implemented that
had been introduced in the $NN$ study of Ref.~\cite{Epelbaum:2015}.
In the actual calculations we rely mostly on the recent more refined potential
of higher order \cite{Dai:2017} that describes $\bar NN$ phase shifts and scattering
observables up to laboratory energies of $T_{lab}\sim 250$~MeV. However, additional
calculations with the other potential are performed in order to estimate possible
uncertainties that arise, e.g., from the employed regularization scheme.
The paper is structured in the following way: In Sect.~\ref{Formalism} a
basic description of the employed formalism is provided. Our results
are presented and discussed in Sect.~\ref{Results}. We compare also with
the works of \cite{Dover:1983} and \cite{Oosterhof:2019}.
The paper closes with a brief summary.
\section{Formalism}
\label{Formalism}
In our treatment of $n-\bar n$ oscillations in the deuteron we follow very closely the
formalism outlined in Refs.~\cite{Sandars:1980,Dover:1983}.
However, contrary to those works our calculation is performed in momentum
space and, therefore, we provide details about the main steps below. The
starting point is the eigenvalue (Schr\"odinger) equation \cite{Sandars:1980}
\begin{equation}
\begin{pmatrix}
H_0 + V_{np} & V_{n-\bar n}\\
V_{n-\bar n} & H_0 + V_{\bar np} \\
\end{pmatrix}
\begin{pmatrix}
|\psi_{np} \rangle \\
|\psi_{\bar np} \rangle \\
\end{pmatrix}
= (E - i \Gamma /2 )
\begin{pmatrix}
|\psi_{np} \rangle \\
|\psi_{\bar np} \rangle \\
\end{pmatrix} \ .
\label{eq:EW}
\end{equation}
Here, $V_{np}$ and $V_{\bar np}$ are the potentials in the $np$ and $\bar n p$
systems and $|\psi_{np} \rangle$ and $|\psi_{\bar np} \rangle$ are the corresponding
wave functions. The systems are coupled via $V_{n-\bar n}$ which is given by
the ${n-\bar n}$ transition matrix element $\delta m_{n-\bar n}$ where the latter
is proportional to the inverse of the $n-\bar n$ oscillation time,
i.e. $V_{n-\bar n} = \delta m_{n-\bar n} = \hbar / \tau_{n-\bar n}$ \cite{Phillips:2014}.
To leading order the decay width of the deuteron, $\Gamma_d$, is \cite{Sandars:1980}
\begin{equation}
\Gamma_d = -2\, V_{n-\bar n}\, {\rm Im} \langle \psi_{d} |\psi_{\bar np} \rangle \ ,
\label{eq:Gamma}
\end{equation}
where $|\psi_{d} \rangle$ is the deuteron wave function. The wave
function $|\psi_{\bar np} \rangle$ obeys the equation
\begin{equation}
(H_0 + V_{\bar n p} - E) |\psi_{\bar np} \rangle = -V_{n-\bar n} |\psi_{d} \rangle \ .
\label{eq:LS}
\end{equation}
We solve Eq.~(\ref{eq:LS}) in momentum space. Performing a partial wave
decomposition and taking into account the coupling of the $^3S_1$ and
$^3D_1$ channels, the above integral equation reads
\begin{eqnarray}
(2 E_p - 2 E_\kappa)\,\psi^{L}_{\bar np} (p)
&&+ \sum_{L'}\int \frac{dq\,q^2}{(2\pi)^3}
V^{L,L'}_{\bar n p}(p,q)\, \psi^{L'}_{\bar np} (q) \nonumber \\
&& = -V_{n-\bar n}\, \psi^L_{d}(p) \ ,
\label{eq:LSP}
\end{eqnarray}
with $L,\,L' = 0,\,2$.
Here, $2 E_\kappa$ is the total energy corresponding to the deuteron, i.e.
$2 E_\kappa - 2 m_N = 2 \sqrt{ m_N^2 - \kappa^2} - 2 m_N =-B_d$
where $B_d$ is the standard binding energy of $2.225$ MeV and $\kappa = \sqrt{m_N B_d} \simeq 45.7$~MeV
is the binding momentum. The deuteron wave function is normalized by
\begin{equation}
\int dp p^2\, \left[(\psi^0_{d}(p))^2 + \psi^2_{d}(p))^2\right] = 1 \ ,
\label{eq:DWF}
\end{equation}
and the width is provided by
\begin{equation}
\Gamma_d = -2\, V_{n-\bar n}\, {\rm Im} \sum_L \int dp p^2\, \psi^L_{d}(p) \, \psi^L_{\bar np}(p) \ .
\label{eq:Width}
\end{equation}
The deuteron lifetime $\tau_d$ is given by $\tau_d=\hbar /\Gamma_d$.
The interesting quantity is the so-called reduced lifetime $R$
\cite{Sandars:1980,Dover:1983,Friedman:2008} which relates the free $n-\bar n$
oscillation lifetime with that of the deuteron,
\begin{equation}
\tau_d = R\, \tau^2_{n-\bar n} \quad {\rm i.e.} \quad
R = \frac{\hbar} {\Gamma_d \tau^2_{n-\bar n} } \ .
\label{eq:TR}
\end{equation}
\section{Results and discussion}
\label{Results}
As already stated in the Introduction, we use the $\bar NN$ interactions
from Refs.~\cite{Kang:2013,Dai:2017} derived
within chiral EFT and the deuteron wave functions from the corresponding
$NN$ potentials \cite{Epelbaum:2005,Epelbaum:2015}, derived in the same
framework.
We consider the two $\bar NN$ interaction because they are based on
rather different regularization schemes. In the earlier potential
\cite{Kang:2013} a non-local exponential exponential regulator
was employed for the whole potential while in the recent work \cite{Dai:2017}
a local regulator was adopted for the evaluation of the one- and two-pion
contributions, see Refs.~\cite{Dai:2017,Epelbaum:2015} for details.
Comparing the pertinent results will allow us to shed light on the question
in how far the choice of the regulator influences the predictions.
For exploring further the sensitivity of the results to the deuteron wave
function we employ also those of two meson-exchange potentials
\cite{Haidenbauer:1993,Machleidt:2001}.
Our results are summarized in Table~\ref{tab:T1}. They are based on
our N$^3$LO interaction with cutoff $R_0 = 0.9$~fm from Ref.~\cite{Dai:2017}
and the N$^2$LO interaction with cutoff $\{\Lambda,\tilde\Lambda\} =
\{450,500\}$~MeV from Ref.~\cite{Kang:2013}. For details on those interactions
we refer the reader to the corresponding publications.
Besides the predictions for $R$ based on the chiral $\bar NN$ interactions
we list also the values given in Ref.~\cite{Dover:1983} where the $\bar NN$
potentials DR$_1$ and DR$_2$ by Dover-Richard~\cite{Dover:1980,Dover:1982}
have been utilized. Furthermore we include results from the calculation of
Oosterhof et al. performed directly within EFT on the basis of the KSW
approach. In this case $R$ can be represented in a compact analytical
form which reads up to NLO~\cite{Oosterhof:2019}
\begin{equation}
R = -\frac{\kappa}{m_N}\, \frac{1}{{\rm Im}\,a_{\bar np}}\,
\frac{1} {1+ 0.4+2 \kappa\, {\rm Re}\,a_{\bar np}-0.13\pm 0.4} \ .
\label{eq:Oost}
\end{equation}
Obviously, the only parameter here is the $\bar n p$ $^3S_1$ scattering length.
All other quantities that enter are well established $NN$ observables, cf.
Ref.~\cite{Oosterhof:2019} for details. Note that in that paper, the scattering length
${\rm Re}\,a_{\bar np}$ was taken from Ref.~\cite{Dai:2017}.
\begin{table*}[ht]
\caption{Reduced lifetime ${R}$ calculated for the $\chi$EFT $\bar NN$
potentials from Refs.~\cite{Kang:2013,Dai:2017}, together with information
on the pertinent $\bar n p$ $^3S_1$ scattering length.
Results for the Dover-Richard potentials DR$_1$ and DR$_2$
are taken from Ref.~\cite{Dover:1983}. The corresponding scattering lengths
are from Ref.~\cite{Carbonell:1992}.
Predictions based on Eq.~(\ref{eq:Oost}), i.e. on the KSW approach
applied in Ref.~\cite{Oosterhof:2019}, are indicated too.
}
\label{tab:T1}
\vskip 0.1cm
\renewcommand{\arraystretch}{1.4}
\begin{center}
\begin{tabular}{|c|cc|cc|}
\hline
\hline
& $\chi$EFT N$^2$LO \cite{Kang:2013} & $\chi$EFT N$^3$LO \cite{Dai:2017}
& DR$_1$ \cite{Dover:1983} & DR$_2$ \cite{Dover:1983} \\
\hline
\hline
${R}$ [s$^{-1}$] & $2.49\times 10^{22}$ & $2.56\times 10^{22}$
& $2.56\times 10^{22}$ & $2.40\times 10^{22}$\\
(Eq.~(\ref{eq:Oost})) & $(1.1\pm 0.3)\times 10^{22}$ & $(1.2\pm 0.3)\times 10^{22}$
& $(1.4\pm 0.4)\times 10^{22}$ & $(1.3\pm 0.3)\times 10^{22}$ \\
\hline
\hline
${a_{^3S_1}}$ [fm] & $0.44-{\rm i}\,0.91$ & $0.44-{\rm i}\,0.96$
& $0.87-{\rm i}\,0.66$ & $0.89-{\rm i}\,0.71$ \\
\hline
\hline
\end{tabular}
\end{center}
\renewcommand{\arraystretch}{1.0}
\end{table*}
As can be seen from Table~\ref{tab:T1} the values for $R$ predicted by
the chiral $\bar NN$ interactions are fairly similar to those obtained
for the DR potentials in the past. The values based on the framework
utilized by Oosterhof et al.~\cite{Oosterhof:2019}, on the other
hand, are rather different as already pointed out by the authors of that work.
The large discrepancy is definitely not due to differences in Im~$a_{\bar n p}$
as suggested in Ref.~\cite{Oosterhof:2019} where the scattering length
from our N$^3$LO chiral $\bar NN$ interaction \cite{Dai:2017} was used,
but clearly a consequence of the different approaches employed.
For illustration we present in Fig.~\ref{fig:3S1} the wave functions
of the $^3S_1$ component of the deuteron and of the corresponding
$\bar n p$ state with arbitrary normalizations. In case of the latter
the imaginary part is shown which is relevant for the determination of
the width, cf. Eq.~(\ref{eq:Width}). The impact of the different
regularization schemes used in Refs.~\cite{Epelbaum:2005,Kang:2013} and
\cite{Epelbaum:2015,Dai:2017}, respectively, can be clearly seen from the
different behavior of the wave functions for large momenta. Note, however,
that the bulk contribution to the integral in Eq.~(\ref{eq:Width}) comes
from momenta $p\le 300$~MeV/c. Contributions from larger momenta
to $\Gamma_d$ are only in the order of $5$\,\% as revealed by test
calculations.
\begin{figure}[h]
\begin{center}
\includegraphics[height=65mm,angle=0]{wave.pdf}
\caption{Wave functions of the deuteron ($\psi_d$; upper curves) and of
the $\bar n p$ state in the $^3S_1$ partial wave (imaginary part
of $\psi_{\bar n p}$; lower curves), multiplied by the momentum $p$
and with arbitrary normalization.
Solid and dashed (red) curves are for the N$^3$LO $\bar NN$ interaction \cite{Dai:2017}
with cutoff $R_0=0.9$~fm and
dash-dotted and dotted (black) curves for the N$^2$LO $\bar NN$ interacton \cite{Kang:2013}
with cutoff $\{\Lambda,\tilde\Lambda\}=\{450,500\}$~MeV.
}
\label{fig:3S1}
\end{center}
\end{figure}
In order to investigate the sensitivity of our results to the utilized
input we performed various exploratory calculations. First of all, we
employed the NLO and N$^2$LO variants of the considered $\bar NN$ (and $NN$)
interactions. The corresponding predictions for $R$ were found to lie
within a range of $(2.48 - 2.65)\times 10^{22}$~s$^{-1}$.
Taking this variation as measure for the uncertainty due to the nuclear structure,
i.e. the $NN$ and $\bar NN$ interactions (wave functions), leads to a value of
roughly $R = (2.6\pm 0.1)\times 10^{22}$~s$^{-1}$. Applying the method
proposed in Ref.~\cite{Epelbaum:2015} for estimating the uncertainty to the
calculation based on the $\bar NN$ interaction from 2017 \cite{Dai:2017},
say, would actually lead to a slightly smaller uncertainty.
We have also varied the deuteron wave functions alone. As an extreme case
we even took wave functions from phenomenological $NN$ potentials derived
in an entirely different framework, namely in the meson-exchange
picture \cite{Haidenbauer:1993,Machleidt:2001}. Also here the obtained
values for $R$ stayed within the range given above.
Finally, omitting the $D$-wave component of the deuteron wave function, which
is kept in our calculation and in the one by Dover et al.~\cite{Dover:1983}
causes a $5$~\% variation. But it leads to an increase of the value of $R$
and, thus, does not bring it closer to the values presented by Oosterhof
et al.
Overall, we confirm the observation by Dover et al. that the predictions for
$R$ are fairly insensitive to the details of the employed $\bar NN$
potentials \cite{Dover:1983}, provided that these potentials describe
the $\bar pp$ data at low energies.
Since in Ref.~\cite{Oosterhof:2019} the $\bar NN$ interaction is represented
by the leading-order term, i.e. the scattering length alone, we have
evaluated the effective range parameter for our $\bar NN$ interactions and
used it to extrapolate the $\bar NN$ amplitude to the deuteron pole.
The found variations are in the order of $10$~\% and can
certainly not explain the large difference documented in Table~\ref{tab:T1}.
We do not have a clear explanation for the difference of our results
(and those of Ref.~\cite{Dover:1983}) to the ones of Oosterhof et
al.~\cite{Oosterhof:2019}. However, we believe that it is due to the
fact that in the latter work the width $\Gamma_d$ is evaluated following the
perturbative scheme developed by Kapan, Savage, and Wise \cite{Kaplan:1999}.
In that scheme there is no proper deuteron wave function. Rather
one works with an effectively constructed wave function that is represented
in terms of an irreducible two-point function~\cite{Oosterhof:2019,Kaplan:1999}.
This seems to work well for some electromagnetic form factors of
the deuteron, at least at low momentum transfer~\cite{Kaplan:1999,Walzl:2001}.
On the other hand, the quadrupole moment of the deuteron is
overestimated by $40$\,\% \cite{Kaplan:1999}, which hints that the
properties of the wave function at large distances (small momenta) are
not that well represented in this scheme. Clearly, this should have an
impact on the quantity studied in the present work as well.
Note that a comparable agreement (mismatch) with regard to the KSW scheme
has been also observed in studies of the electric dipole moment (magnetic
quadrupole moment) of the deuteron~\cite{deVries:2011,Liu:2012,Bsaisou:2015}.
In any case, one should not forget that there is convergence problem of the
KSW approach for $NN$ partial waves where the tensor force from pion exchange
is present~\cite{Fleming:2000}. It affects specifically the $^3S_1$-$^3D_1$
channel where difficulties appear already for momenta around $100$~MeV/c,
see~\cite{Fleming:2000} and also the discussions in
Refs.~\cite{Epelbaum:2009,Bedaque:2002}.
Interestingly enough, this is the momentum region where the dominant
contributions in our calculation come from, see Fig.~\ref{fig:3S1}.
In contrast to \cite{Oosterhof:2019} in our study the $\bar NN$ ($NN$)
and $|\Delta B| =2$ interactions are not treated at the same order.
Guided by the success of the work by Epelbaum et al.
\cite{Epelbaum:2005,Epelbaum:2015} based on the Weinberg counting
the non-perturbative effects due to the $NN$ and $\bar NN$
interactions are fully take into account from the very beginning.
It is worth mentioning that in this approach the quadrupole moment
of the deuteron is very close to its empirical value already for
interactions at NLO \cite{Epelbaum:2005,Epelbaum:2015}.
Finally, note that in the present work we have only considered the contribution
from actual $n-\bar n$ oscillations to the deuteron decay rate. In principle, there
can be also contributions from a direct annihilation of the $NN$ system.
Moreover, there can be a $NN\to \bar NN$ transition involving both nucleons
and the $\bar NN$ state. The latter gives rise to a
$|\Delta B| = 2$ four-baryon contact term involving an unknown complex
low-energy constant, which has been discussed and considered in the work by
Oosterhof et al.~\cite{Oosterhof:2019}. We refrain from introducing such
a contribution in the present study which, anyway, is of higher
order in the employed counting scheme. However, we would like to mention
that according to Ref.~\cite{Oosterhof:2019} the deuteron decay rate
is indeed dominated by free $n-\bar n$ oscillations. Nonetheless, an
uncertainty in the order of $\pm 0.3 \times 10^{22}$~s$^{-1}$ for the
reduced lifetime $R$ due to additional $|\Delta B|=2$ mechanisms, as
suggested in that work, has to be certainly expected.
Note that additional decay mechanisms should always increase the decay width of
the deuteron and, thus, lead to smaller values of the reduced lifetime $R$.
\section{Summary}
In the present work we have considered neutron-antineutron
oscillations in the deuteron. In particular, we have calculated
the deuteron lifetime in terms of the free-space $n-\bar n$ oscillation time
based on $\bar NN$ \cite{Kang:2013,Dai:2017} and
$NN$ \cite{Epelbaum:2005,Epelbaum:2015} interactions derived within
chiral effective field theory. The value obtained for the
so-called reduced lifetime $R$ which relates the free-space $n-\bar n$
oscillation time $\tau_{n-\bar n}$ with the deuteron lifetime is found to
be $R=(2.6\pm 0.1) \times 10^{22}$~s$^{-1}$, where the quoted uncertainty is due
to the $NN$ and $\bar NN$ interactions (wave functions).
Our result for $R$ is close to the value obtained by Dover and collaborators
more than three decades ago \cite{Dover:1983} but disagrees with recent EFT
calculations, performed within the perturbative scheme proposed by Kaplan, Savage,
and Wise \cite{Oosterhof:2019}, by about a factor of~$2$.
We believe that the difference is due to the fact that in the perturbative
KSW scheme there is no proper deuteron wave function. Rather this ingredient
is represented effectively in terms of an irreducible two-point function.
It is known from past studies that the KSW approach fails to describe quantities
that depend more sensitively on the wave function like, for example, the quadrupole
moment of the deuteron \cite{Kaplan:1999}.
\section*{Acknowledgements}
We acknowledge stimulating discussions with Jordy de Vries, Andreas Nogga and Tom Luu.
This work is supported in part by the DFG and the NSFC through
funds provided to the Sino-German CRC 110 ``Symmetries and
the Emergence of Structure in QCD'' (DFG grant. no. TRR~110)
and the VolkswagenStiftung (grant no. 93562).
The work of UGM was supported in part by The Chinese Academy
of Sciences (CAS) President's International Fellowship Initiative (PIFI) (grant no.~2018DM0034).
|
1,116,691,500,838 | arxiv | \section{\uppercase{Introduction}}
\label{sec:introduction}
Discriminative CNNs, such as those constructed by \cite{kamnitsas2017efficient,havaei2017brain,ronneberger2015u}, have consistently ranked on the top of the leaderboard in many brain lesion segmentation challenges \cite{maier2017isles,winzeck2018isles,bakas2018identifying}. Commonly, they are formulated by assuming a voxel-wise Multinoulli distribution, conditional on the MRI intensities of neighboring voxels where the parameters are obtained and estimated with a CNN. For the case of $2$ labels, if $x_j$ denotes the 3D MRI intensities used in the prediction of voxel $j$ over a discrete domain $\Omega \subset \mathbb{R}^3$, $Y_j$ represents the random label at voxel $j$, and $\Theta$ are the parameters of a CNN architecture, then the model is formulated as
\begin{equation}
Y_j|x_j \sim Bernoulli(\pi_j = CNN(\Theta,x_j)). \label{bern}
\end{equation}
\noindent At prediction time, the translational invariance property of the CNN allows for a fast way to estimate all the voxel-wise conditional distributions, without having to separately feed their respective covariates. When a model is fully trained, the parameters in \eqref{bern} are replaced with an estimate: $\hat{\Theta}$. In this way, discriminative CNNs are capable of sampling segmentations by sampling the associated label at each voxel, conditionally independent of each other. Despite this source of variability, a decision rule of selecting the label with the highest probability is often applied rendering the segmentation deterministic.
Measuring the performance variation of trained discriminative CNNs in brain lesion segmentation on a patient by patient basis can help clinical practitioners instill a degree of confidence in the use of automated segmentation methods. One metric of model performance is the Dice coefficient \cite{dice1945measures,sorensen1948method}, which measures the similarity between two 3D voxel-wise labeled images: $S_1$ and $S_2$. The Dice coefficient is a real number in $[0,1]$ given by
\begin{equation}
Dice(S_1,S_2) = \frac{2TP}{2TP + FN + FP},
\label{dice}
\end{equation}
\noindent where a value of $0$ indicates no overlap and a value of $1$ indicates perfect similarity. TP, FP and FN refer to the number of true positive, false positive, and false negative voxels in the medical image respectively. Formulating this incentive mathematically, if $D$ denotes the Dice coefficient random variable which incorporates randomness across patients, and across model segmentations, and $x$ collects all the covariates in \eqref{bern} across all voxels in the image for the patient to from a high dimensional vector, then the interest lies in measuring the model variance
\begin{equation}
Var(D|x,\hat{\Theta}).
\label{condvar}
\end{equation}
\noindent Here the Dice random variable is theoretically measured on the true segmentation, which is never known, and the random segmentations produced by a trained discriminative CNN model on the particular patient. Intuitively, the observations of $D$ can be generated by: observing a new random patient whose radiological features are drawn independently from other patients, randomly generating a segmentation from the trained discriminative CNN given their radiological features, and calculating the Dice coefficient against the ground truth. In \eqref{condvar}, the first source of variation is eliminated by conditioning on the patient. Estimating \eqref{condvar} can also help suggest the complete abandonment of an algorithm in favor of another, as some CNNs may be better tuned to deal with specific brain lesion characteristics like small lesion volume.
Despite these advantages, some brain lesion segmentation challenges compute variability in the Dice metric across patient predictions \cite{maier2017isles}. Moreover, for a particular patient, only a single deterministic segmentation is produced by some decision rule. That is, $Var(D|\hat{\Theta})$ becomes the measure to be estimated, but this estimate does not solely account for model variability. It is important to note this measure is unconditional on the patient's radiological features, and thus inherently incorporates additional variability. The merit of this metric is in its applicability, as not all segmentation algorithm are capable of sampling segmentations. However in the case of discriminative CNNs, by its very nature, this metric can solely be used to quantify variation in the Dice coefficient across patients and could completely discard a model that may in fact perform well on certain kinds of patients. Another disadvantage is that this metric is obtained from the training set, and then must be generalized to arbitrary cases, which may present significant differences. Analysis of model performance variance on a single patient \eqref{condvar}, conditional on their radiological features, eliminates these extra sources of variation, and can be more useful in clinical practice by providing specialized patient care.
One step towards this direction was taken in the BraTS 2019 challenge, which is an extension of \cite{bakas2018identifying}. In particular, voxel-wise measures of segmentation uncertainty ranging from $0$ to $100$ were calculated from 3D MRIs for all patients individually. Then, at specified thresholds, uncertain voxels were filtered out in the calculation of the Dice coefficient. Depending on the structure of uncertainties, this method can reward or penalize the Dice score, but this can never be confirmed in practice since we never know the ground truth segmentation.
Segmentation sampling and consequent analysis of the inter-sample Dice distribution has been undertaken for generative models, for instance, by L{\^e} et al. \cite{le2016sampling}. In their work, they use a Gaussian process to produce segmentation samples based on a single expert manual segmentation of grade $4$ gliomas. The mean of the inter-sample Dice distribution and variability of the segmentations can be controlled by a single model parameter. The samples can then be used in radiotherapy planning by delivering radiation to certain voxels and avoiding dose to uncertain voxels, where perhaps there are more sensitive tissues.
The contribution mentions the applicability of their method in evaluating uncertainty in the performance of segmentation algorithms, by repeatedly sampling segmentations off the ground truth and calculating the variability against a deterministic predicted segmentation. Though this method can be applied to arbitrary segmentation algorithms, the source of variation is not produced by the model predictions, as in the case of discriminative CNNs. As a consequence, the method is effective in assessing the effectiveness of a particular segmentation produced from a model, but not the model itself. Another related work by Roy et al. \cite{roy2018inherent}, shows that the inter-sample Dice coefficient correlates with Dice performance using a discriminative CNN for the segmentation of brain scans from children with psychiatric disorders (CANDI-13 dataset), and suggests it as a measure for quantifying uncertainty.
In this proposed work, the inter-sample Dice observations are shown to be independent and identically distributed samples from a distribution with finite variance and mean, under certain conditions. Segmentations are sampled directly from a discriminative CNN. The mean with a confidence interval of the inter-sample Dice distribution is then estimated by the central limit theorem and used in place of \eqref{condvar} to decide on whether to reject or accept the CNN model on patients with ischemic stroke. This chapter is organized as follows: Section 2 describes the methods, Section 3 presents the results, and a discussion follows in Section 4.
\section{\uppercase{Methods}}
\subsection{Architecture}
The CNN architecture considered is Wider2DSeg, which was originally constructed by Kamnitsas et al. \cite{kamnitsas2017efficient} and is a two dimensional variant of their 3D deepMedic architecture. Raina et al. \cite{bioimaging20} tuned this architecture in conjunction with additional symmetry covariates obtained from a reflective registration step to yield a $0.62$ Dice coefficient over a 7-fold cross validation on the ISLES 2015 SISS dataset. All the implementation details in this work are exactly as in the Wider2DSeg implementation of Raina et al. \cite{bioimaging20} with nonlinear symmetry covariates.
\subsection{Segmentation sampling}
Referencing equation \eqref{bern}, suppose there are $V$ voxels in an image. The attention is restricted to the case of $2$ voxel labels, though this analysis can extend to an arbitrary finite number of labels. Let $S \in
\{0,1\}^V$ represent the segmentation random vector of a trained discriminative CNN model, where each element $S_k = Y_j$, for unique $j \in \Omega$. The segmentation vector is unconditional and incorporates randomness across patients. Although in what follows it does not matter which permutation in the assignment of voxels to array elements is used, the assignment must be chosen and fixed. Then, denote $x$ as the patient's radiological features as formulated in equation \eqref{condvar}, and $\hat{\Theta}$ as the estimated parameters of the CNN. Segmentations can be sampled conditional on a patient's radiological features if conditional independence is assumed across voxels. In this case, the distribution is completely known by
\begin{equation}
S|x \sim p(s;x,\hat{\Theta}) = \prod_{j=1}^{V}\pi_j(x_j,\hat{\Theta})^{s_j}(1-\pi_j(x_j,\hat{\Theta}))^{1-s_j}.
\label{conddist}
\end{equation}
\subsection{Inter-sample Dice distribution}
Let $S^*_1$ and $S^*_2$ represent two iid model segmentation random vectors for a given patient, each distributed according to \eqref{conddist}. By independence, the joint distribution of these samples is given by
\begin{equation}
(S^*_1,S^*_2) \sim p(s^*_1;x,\hat{\Theta})p(s^*_2;x,\hat{\Theta}).
\label{jointdist}
\end{equation}
\noindent In this way, the inter-sample Dice random variable is just a function of the random vectors. In particular, if $S^*_2$ is seen as the ground truth, then the Dice coefficient of $S^*_1$ can be computed against it, and vice-versa. Define $\Gamma$ to be the inter-sample Dice random variable. Then
\begin{equation}
\Gamma = Dice(S^*_1,S^*_2).
\label{isd}
\end{equation}
\noindent In fact, the complete distribution of $\Gamma$ can be obtained, but can be quite computationally expensive as the number of voxels are often large. That is, if $\mathcal{D}_{(\cdot)}$ represents the support of a random variable, then $|\mathcal{D}_{S^*_1}| \leq 2^V$, where the upper bound is obtained by having $0 < \pi_j < 1$ for all voxels in the image. By independence of $S^*_1$ and $S^*_2$, this further implies that $|\mathcal{D}_{(S^*_1,S^*_2)}| \leq 2^{2V}$. Subsequently, after applying the Dice transformation \eqref{isd} which may at most associate a unique Dice value for each element in $\mathcal{D}_{(S^*_1,S^*_2)}$, the inequality $|\mathcal{D}_{\Gamma}| \leq 2^{2V}$ is obtained. Hence, computing the exact Dice distribution may require the calculation of at most $2^{2V}$ probabilities, where $V$ tends to be in the millions. Though, since $\mathcal{D}_{\Gamma}$ is bounded by $2^{2V}$, $\Gamma$ has a discrete distribution with a finite and countable support.
In order to conduct statistical inference on $E[\Gamma]$ it must also be finite, however, this is not the case. Notice that if $(S^*_1,S^*_2) = (0,0)$, $\Gamma$ is undefined, and this outcome always has a non-zero probability. In fact, this is the only way for $\Gamma$ to be undefined. To correct this, two necessary modifications are introduced: 1) $\exists$ $\pi_j \neq 0$, for some $j$ and 2) The inter-sample Dice random variable is redefined as
\begin{equation}
\Gamma^* = \Gamma|(S^*_1,S^*_2) \neq (0,0).
\end{equation}
\noindent The first condition ensures that $(0,0)$ can never be the only sample generated, thereby permitting its removal and still retaining a distribution through the redefinition in the second condition. The second condition redefines the inter-sample Dice random variable to be conditional on all samples drawn from \eqref{jointdist} except $(0,0)$. Now, $\forall \gamma \in \mathcal{D}_{\Gamma^*}, 0 \leq \gamma \leq 1$, where $\mathcal{D}_{\Gamma^*} \neq \emptyset$. Then
\begin{equation}
E[(\Gamma^*)^2] < \infty.
\end{equation}
As a consequence, sampling observations from this distribution, and estimating its mean from a sample can be used. Independent and identically distributed pairs of segmentations can be sampled from \eqref{jointdist}, and can generate an iid sample of inter-sample Dice observations. To adjust for the second modification, any $(0,0)$ sample, or equivalently undefined Dice score, is removed in the sampling phase. Moreover, since the inter-sample Dice distribution has a finite mean and variance, the central limit theorem can be applied, and associated confidence intervals can be constructed.
Computationally, this method can be undertaken by first producing the estimated probability tensors. At each voxel in the probability tensor are the estimated label probabilities obtained from the features and CNN parameter estimates. Once this tensor is calculated, segmentations are sampled by sampling from the associated Multinoulli distribution and iterating over all voxels. Naturally, these will produce segmentation samples that can be appropriately viewed as such since the format of the probability tensors align with that of the actual image.
\subsection{Decision rule}
Let $\gamma_1,..,\gamma_n$ be an iid sample of realized inter-sample Dice coefficients for a given patient, and construct a $(1-\alpha) \%$ approximate confidence interval, based on the central limit theorem. For a specified threshold, reject the use of a discriminative CNN model on a patient if the confidence interval is entirely below the threshold. In this manner, the clinician can justify with $(1-\alpha) \%$ confidence that the true mean of the inter-sample Dice distribution is below the given threshold, and appropriate actions to reject the model in favor of another can be undertaken.
\section{Experiments and Results}
\subsection{Dataset}
The discriminative CNN model is trained and evaluated on the ISLES2015 (SISS) training data, which consists of $28$ patients with sub-acute ischemic stroke. The radiological features $x$ for a patient are 4 MR sequences: FLAIR, DWI, T1 and T1-contrast. Each image has a total of $230 \times 230 \times 154$ voxels. At each voxel, there are only two possible labels classes: lesion or non-lesion.
\subsection{Efficacy of decision rule}
The proposed segmentation sampling based decision rule is applied to the results of Raina et al. \cite{bioimaging20}, which yielded an average Dice coefficient (predicted against ground truth) of $0.62$ over a 7-fold cross-validation from single deterministic segmentations obtained by selecting the label with the highest probability. For each fold in the cross-validation and for each brain in the validation fold, $30$ inter-sample Dice observations are sampled, and
\begin{table}[!h]
\centering
\renewcommand{\arraystretch}{1.2}
\caption{Case by case hypothesis testing for Wider2dSeg on ISLES2015 SISS. The inter-sample Dice confidence interval is computed using the central limit theorem with $30$ samples. The bolded rows indicate rejecting the use of the CNN on the patient as per the decision rule.}\label{tab1}
\begin{tabular}{|c|c|c|}
\hline
\bfseries Patient No. & \bfseries ISD confidence & \bfseries Dice\\
\hline
1 & 0.940339 $\pm$ 0.000116 & 0.866798\\
\hline
2 & 0.955828 $\pm$ 0.000243 & 0.815299
\\
\hline
3 & 0.897922 $\pm$ 0.001002 & 0.736231
\\
\hline
4 & 0.946641 $\pm$ 0.000122 & 0.797607
\\
\hline
5 & 0.944956 $\pm$ 0.000133 & 0.857821
\\
\hline
6 & 0.965364 $\pm$ 0.000146 & 0.905630
\\
\hline
7 & 0.943856 $\pm$ 0.000178 & 0.822416
\\
\hline
8 & 0.909659 $\pm$ 0.000349 & 0.702697
\\
\hline
9 & 0.969197 $\pm$ 0.000076 & 0.854602
\\
\hline
10 & 0.875187 $\pm$ 0.000340 & 0.592438
\\
\hline
11 & 0.933384 $\pm$ 0.000399 & 0.775896
\\
\hline
12 & 0.888211 $\pm$ 0.000810 & 0.517170
\\
\hline
13 & 0.850324 $\pm$ 0.001449 & 0.277828
\\
\hline
14 & 0.980536 $\pm$ 0.000095 & 0.815283
\\
\hline
15 & 0.974488 $\pm$ 0.000179 & 0.887611
\\
\hline
\bfseries 16 & \bfseries 0.580227 $\pm$ 0.004708 & \bfseries 0.009584
\\
\hline
\bfseries 17 & \bfseries 0.508344 $\pm$ 0.005390 & \bfseries 0.164190
\\
\hline
18 & 0.949571 $\pm$ 0.000435 & 0.730066
\\
\hline
\bfseries 19 & \bfseries 0.730039 $\pm$ 0.003967 & \bfseries 0.487031
\\
\hline
20 & 0.950651 $\pm$ 0.000441 & 0.795393
\\
\hline
\bfseries 21 & \bfseries 0.687765 $\pm$ 0.003033 & \bfseries 0.434674
\\
\hline
22 & 0.863127 $\pm$ 0.000823 & 0.685439
\\
\hline
\bfseries 23 & \bfseries 0.754482 $\pm$ 0.002522 & \bfseries 0.634320
\\
\hline
24 & 0.861986 $\pm$ 0.001278 & 0.507946
\\
\hline
25 & 0.916227 $\pm$ 0.001130 & 0.761670
\\
\hline
\bfseries 26 & \bfseries 0.598502 $\pm$ 0.006355 & \bfseries 0.191637
\\
\hline
\bfseries 27 & \bfseries 0.824157 $\pm$ 0.000633 & \bfseries 0.000156
\\
\hline
28 & 0.887825 $\pm$ 0.001183 & 0.732581
\\
\hline
\end{tabular}
\end{table}
\noindent the central limit theorem confidence interval is computed. Table \ref{tab1} displays the patient by patient results. The Pearson correlation between mean ISD and Dice over the $28$ patients was computed to be $r = 0.81$. Figure \ref{fig:isdplot} plots the mean ISD against Dice score for each patient, and depicts the fitted regression line. The threshold was set to $0.85$ with reference to the regression line in Figure \ref{fig:isdplot}, and corresponds to a Dice score of $0.60$. In addition, the confidence level is $95 \%$. By removing the rejected predictions, the average Dice coefficient increased from $0.62$ to $0.74$.
\begin{figure}[!h]
\centering
\includegraphics[scale=0.41]{isdplot.png}
\caption{Plot of mean ISD against Dice score with regression line: $y = 1.66x - 0.82$, where $y$ is Dice score and $x$ is mean ISD. The F-test yielded a p-value of $1.4 \times 10^{-7}.$
\label{fig:isdplot}}
\end{figure}
\section{\uppercase{Discussion}}
The reason as to why ISD is highly correlated to Dice performance for discriminative CNNs, and can be used to detect weak segmentations is not entirely clear. As a counter-example, consider a model that always predicts $P(lesion) = 0$ for all but one voxel, which instead has $P(lesion) = 1$. Then, the ISD would always be $1$, since the model produces exactly the same segmentation with probability $1$, regardless of the input. However, it would be unexpected to see a high Dice score for this model. The coupling of CNN outputs and ISD for detecting uncertain segmentations requires further investigation for a deeper understanding of its performance.
One important remark is that the Dice metric can be substituted by other metrics such as the sensitivity, specificity, mean squared error, or precision and the preceding analysis would also follow for these distributions, thereby permitting hypothesis testing. Moreover, the computations considered in this work were over the entire brain, but could also be calculated on specific regions of interest (ROIs). Deciding on which metrics to use, and applying them to more detailed brain sub-regions could improve the decision-making potential, and is a possible area of development.
Another point to remark is that the proposed method can be used to rigorously test competing discriminative models based on their respective inter-sample mean Dice confidence intervals, and select the most robust one on an individualized patient basis. Applying this unifying technique for all competing CNNs in a brain lesion challenge may exhibit the best possible performance, without any consideration to the ground truth. Segmentation challenges have recently begun to incorporate uncertainty analysis, but further work is required to apply these techniques on various types of brain lesion structures.
\bibliographystyle{apalike}
{\small
|
1,116,691,500,839 | arxiv | \section{Introduction}\label{introduction}
The stochastic evolution of systems composed by elements
which interact among each other has always been of great interest in several areas of application,
e.g. in medicine a tumor growth is the evolution of a system of interacting cells~\cite{YakTso},
in socio-economics and life sciences a collective phenomenon reflects the result of the interactions among the individuals~\cite{NalParTos},
in physics the concentration of certain molecules within cells varies over time due to interactions between different cells~\cite{Pri}.
In the last decade several models have been proposed in which the elements of the system are represented by urns containing balls of different colors,
in which the urn proportions reflect the status of the elements,
and the evolution of the system is established by studying the dynamics at discrete times of
this collection of dependent urn processes.
The main reason of this popularity is concerned with the urn dynamics, which is
(i) suitable to describe random phenomena in different scientific fields (see e.g.~\cite{JohKot}),
(ii) flexible to cover a wide range of possible asymptotic behaviors,
(iii) intuitive and easy to be implemented in several fields of application.
The dynamics of a single urn typically consists in a sequential repetition of a \emph{sampling phase}, when a ball is sampled from the urn,
and a \emph{replacement phase}, when a certain quantity of balls is replaced in the urn.
The basic model is the P\'{o}lya's urn proposed in~\cite{EffPol}:
from an urn containing balls of two colors, balls are sequentially sampled and then replaced in the urn with a new ball of the same color.
This updating scheme is then iterated generating a sequence of urn proportions whose almost sure limit is random and Beta distributed.
Starting from this simple model, several interesting variations have been suggested by considering different distributions in
the sampling phase, e.g.~\cite{HilLanSud1,HilLanSud2}, or in the replacement phase, e.g.~\cite{Ath,Fri,Pem}.
In a general $K$-colors urn model,
the color sampled at time $n$ is usually represented by a vector $\mathbf{X}_n$
such that $X_{i,n}=1$ when the sampled color is $i\in\{1,{\ldots},K\}$, $X_{i,n}=0$ otherwise;
the quantities of balls replaced in the urn at time $n$ are typically
defined by a matrix $D_n$ such that $D_{ki,n}$ indicates the number of balls of color $k$ replaced in the urn when the color $i$ is sampled.
Considering $\{D_n;n\geq1\}$ as an i.i.d. sequence,
a crucial element to characterize the asymptotic behavior of the urn
is the mean replacement matrix $H:=\mathbf{E}[D_n]$, typically called \emph{generating matrix}.
The class of urn models considered in this paper is commonly denoted by
\emph{Generalized Friedman's Urn} (GFU).
The GFU model was introduced in~\cite{Fri} and its extensions and their asymptotic behavior have been studied in
several works, see e.g.~\cite{AthKar2,BaiHu,BaiHu2,Smy}.
The GFU considered in this paper is characterized by a non-negative irreducible generating matrix $H$
with average constant balance, i.e. the columns of $H$ sum up at the same constant, $\sum_{k=1}^KH_{ki}=c>0$ for any $i\in\{1,{\ldots},K\}$,
which implies that its maximum eigenvalues $\lambda_{\max}(H)=c$ has multiplicity one.
The irreducibility of $H$ distinguishes the GFU from the \emph{Randomly Reinforced Urn} (RRU) model,
which includes the classical P\`{o}lya's Urn, whose replacement matrix is diagonal: when the color $i$ is sampled,
the GFU replaces in the urn more colors
following the distribution of the $i^{th}$ column of $D_n$ while the RRU only adds balls of colors $i$;
hence, the probability to sample color $i$ at next step is reinforced in the RRU,
while it may increase or decrease according to the current urn composition in the GFU.
The asymptotic behavior is in general very different:
in a GFU the urn proportion converges to a deterministic equilibrium identified by $H$ (see e.g.~\cite{AthKar2,BaiHu,BaiHu2,Smy}),
while in a RRU the limit is random and its distribution depends on the initial composition (see e.g.~\cite{AleMaySec1,AleMaySec2,DurFloLi}).
The model proposed in this paper is a collection of $N\geq1$ GFUs that interact among each other during the sampling phase:
the probability to sample a color $i$ from an urn $j$ is a convex combination of the urn proportions of the entire system.
Hence, a crucial role to describe the system dynamics is played by the \emph{interacting matrix} $W$ made by the weights of those combinations.
Since the asymptotic properties of the single GFUs are typically determined by the corresponding generating matrices $\{H^j;1\leq j\leq N\}$
and since the interaction among them is ruled by $W$,
the system dynamics has been studied by defining a new object $\mathbf{Q}$ that merges the information contained in $\{H^j;1\leq j\leq N\}$ and $W$.
From the analysis of the eigen-structure of $\mathbf{Q}$,
we are able to establish the convergence and the second-order asymptotic behavior of the urn proportions in the entire system.
Hence, this paper extends the theory on GFU models in the sense that,
in the special case of no interaction, i.e. $W=I$,
the results presented for the system reduce to the well-known results for a single GFU.
Several interacting urn models have been proposed in the last decade, especially for RRU models.
An early work is represented by~\cite{PagSec} that considered a collection of two-colors RRU in which
the replacements depend on the colors sampled in the rest of the system and hence
the sequence $\{D_n;n\geq1\}$ is not i.i.d.
Consequently, the interaction in~\cite{PagSec}
is modeled through the definition of $D_n$, instead of $\mathbf{X}_n$ as in our model.
A completly different updating rule has been used in the two-color urn model proposed in~\cite{MarVal},
in which sampling color 1 in the urn $j$ increases the composition of color 1 in the urn $j$,
while sampling color 2 increases the composition of color 2 in the neighbor urns $i\neq j$ and the urn $j$ comes back to the initial composition.
Asymptotic properties for this system have been obtained in~\cite{MarVal}
where there is no convergence of the urn proportions.
Other models in which the interaction enters in the replacement matrices are for instance~\cite{BenBenCheLim,CheLuc,CirGalHus}.
Recently there have been more works concerning urn systems in which the interaction is modeled
through the sampling probabilities as in our model.
They differ from this paper since all of them consider RRUs and the interaction is only modeled as mean-field interaction tuned by a parameter $\alpha\in(0,1)$,
i.e. the urns interact among each other only through the average composition in the entire system.
As a consequence, their asymptotic results lead to the \textit{synchronization} property in which all the urn proportions of the system
converge to the same random limit.
In particular, in~\cite{Lau,LauLim} the asymptotic behavior of the urn system has been studied
for a model that defines the sampling probabilities through the exponential of the urn compositions.
In~\cite{CriDaiPraMin,DaiPraLouisMin} the sampling probabilities are defined directly using the urn compositions
and, in addition, the synchronization property has been proved;
moreover, different convergence rates and second-order asymptotic distributions for the urn proportion have been established
for different values of the tuning parameter $\alpha$.
Since we consider GFU models the asymptotic results established in this paper are totally different
from those proved in~\cite{CriDaiPraMin,DaiPraLouisMin}, e.g.
our limiting proportions are not random and they do not depend on the initial compositions.
It is also significant to highlight that this work allows a general structure for the urn interaction,
which reduces to the mean-field interaction only for a particular choice of the interacting matrix $W$.
Moreover, from the analysis of the structure of $W$ we are able to individuate subsystems of urns evolving with different behaviors
(see Subsection~\ref{subsection_W interaction}):
(i) the \textit{leading systems}, whose dynamics are independent of the rest of the system
and (ii) the \textit{following systems}, whose dynamics ``follow'' the evolution of other urns of the system;
in the special case of irreducible interacting matrix, which includes the mean-field
interaction considered in~\cite{CriDaiPraMin,DaiPraLouisMin},
there is a unique leading system and no following systems.
These two classes of systems have been studied separately
(leaders in Section~\ref{section_leading} and followers in Section~\ref{section_follower}),
in order to provide an exhaustive description of the asymptotic behavior in any part of the system.
In fact, since different systems may converge at different rates,
a unique central limit theorem would not be able
to characterize the convergence of any urn proportion.
Hence, through a careful analysis on the eigen-structure of $\mathbf{Q}$ realized in Subsection~\ref{subection_in_out_follower},
we individuate the components of the urn processes in the system that actually ``lead'' or influence the following systems,
so that we can establish the right convergence rate and a non-degenerate asymptotic distribution for any subsystem.
A pivotal technique in the proofs consists in revisiting the dynamics of the urn proportions of the system in the stochastic approximation (SA) framework,
as suggested for the composition of a single GFU in~\cite{LarPag}.
To this end, the dynamics of the urn compositions of the same subsystems
have been reformulated into a recursive stochastic algorithm (see Section~\ref{section_stochastic_approximation}).
Then, the dynamics of the urn proportions have been properly modified to embed the processes
of the urn proportions into the whole suitable space $\mathbb{R}^K$
(see Subsection~\ref{subection_redefining_simplex_leading} and~\ref{subection_redefining_simplex_follower}).\\
The main results of the paper starts at Section~\ref{section_leading}.
The first part of the paper is a necessary formulation of the problem in its general form,
together with all the assumptions and notations that may appear tough at a first reading.
We provide a guiding Example~\ref{example_1}, that is recovered in the Example~\ref{example_2} and Example~\ref{example_3},
to help the reader to appreciate the main results, although not in all their depth.
More precisely, the structure of the paper is the following.
In Section~\ref{section_model_assumptions} we present model and main assumptions concerning the interacting GFU system.
Specifically, in Subsection~\ref{subection_model} we describe how the composition of the colors in each urn of the system evolves at any time $n\geq1$.
Then, in Subsection~\ref{subection_main_assumptions} the main assumptions required to establish the results of the paper are presented.
Subsection~\ref{subection_preliminary} contains a preliminary result.
Subsection~\ref{subsection_W interaction} is dedicated to analyze the structure of the interacting matrix and
hence to define the leading and the following subsystems that compose the entire system.
Section~\ref{section_stochastic_approximation} is concerned with the dynamics of the interacting GFU system
expressed in the stochastic approximation framework.
In particular, in Subsection~\ref{subection_notation_SA} we introduce the notation
that combines the composition of the urns in the same subsystem.
Then, in Subsection~\ref{subection_dynamics_SA} the dynamics of the urn proportions
in any subsystem is reformulated into a recursive stochastic algorithm.
Section~\ref{section_leading} and Section~\ref{section_follower} contain the main results of the paper.
In particular, Section~\ref{section_leading} is concerned with the asymptotic behavior of the leading systems:
the convergence of the urn proportions is established in Subsection~\ref{subection_first_order_leading}
and the corresponding CLT is presented in Subsection~\ref{subection_second_order_leading}.
Then, Section~\ref{section_follower} is focused on the asymptotic behavior of the following systems:
in Subsection~\ref{subection_first_order_follower} we present the result on the convergence of the urn proportions,
while in Subsection~\ref{subection_second_order_follower} we establish the relative CLT.
Section~\ref{section_further_extensions} contains a brief discussion on further possible extensions of the interacting GFU model.
The proofs of all the results presented in the paper are contained in Section~\ref{section_proofs}.
Finally, in Appendix we report basic results of stochastic approximation that have been used in the main proofs.
\section{Model Setting and main Assumptions}\label{section_model_assumptions}
Consider a collection of $N\geq1$ urns containing balls of $K\geq1$ different colors.
At any time $n\geq0$ and for any urn $j\in\{1,{\ldots},N\}$,
let $Y^j_{k,n}>0$ be the real number denoting the amount balls of color $k\in\{1,{\ldots},K\}$,
let $T^j_n:=\sum_{k=1}^KY^j_{k,n}$ be the total number of balls
and let $Z^j_{k,n}:=Y^j_{k,n}/T^j_n$ be the proportion of color $k$.
\subsection{Model}\label{subection_model}
We now describe precisely how the system evolves at any time $n\geq1$.
Denote by $\mathcal{F}_{n-1}$ the $\sigma$-algebra generated by the urn compositions of the entire system up to time $(n-1)$, i.e.
\begin{equation*}
\mathcal{F}_{n-1}\ :=\ \sigma\left(\ X^j_{k,t}, Y^j_{k,t},\ 1\leq j\leq N,\ 1\leq k\leq K,\ 1\leq t\leq n-1\ \right).
\end{equation*}
The dynamics of the system is described by two main phases: \emph{sampling} and \emph{replacement}.\\
\emph{Sampling phase}:
for each urn $j\in\{1,{\ldots},N\}$, a ball is virtually sampled and its color is represented as follows:
$X^{j}_{i,n}=1$ indicates that the sampled ball is of color $i$, $X^{j}_{i,n}=0$ otherwise.
We denote by $\tilde{Z}^{j}_{i,n-1}$ the probability to sample a ball of color $i$ in the urn $j$ at time $n$, i.e.
\begin{equation*}
\tilde{Z}^{j}_{i,n-1} :=\mathbf{E}\left[\ X^{j}_{i,n}\ |\ \mathcal{F}_{n-1}\ \right].
\end{equation*}
Given the sampling probabilities $\{\tilde{Z}^{j}_{i,n-1},1\leq j\leq N,1\leq i\leq K\}$,
the colors are sampled independently in all the urns of the system and hence,
for any $i\in\{1,{\ldots},K\}$, $X^{1}_{i,n},{\ldots},X^{N}_{i,n}$ are independent conditionally on $\mathcal{F}_{n-1}$.
We define the sampling probabilities as convex combinations of the urn proportions of the system.
Formally, for any urn $j\in\{1,{\ldots},N\}$ we introduce the weights $\{w_{jh};1\leq h\leq N\}$ such that $0\leq w_{jh}\leq1$ and
$\sum_{h=1}^Nw_{jh}=1$.
Thus, the probability to sample the color $i$ in the urn $j$ is defined as follows
\begin{equation}\label{def:Z_tilde}
\tilde{Z}^{j}_{i,n-1}\ :=\ \sum_{h=1}^Nw_{jh}Z^{h}_{i,n-1}.
\end{equation}\\
\emph{Replacement phase}:
after that a ball of color $i$ has been sampled from the urn $j$,
we replace $D^j_{ki,n}$ balls of color $k\in\{1,{\ldots},K\}$ in the urn $j$.
For any urn $j$ we assume that $\{D_{n}^j;n\geq1\}$ is a sequence of i.i.d. non-negative random matrices, where $D_{n}^j:=[D^j_{ki,n}]_{ki}$.
We will refer to $D^j_n$ as \emph{replacement matrix} and to $H^j:=\mathbf{E}[D_{n}^j]$ as \emph{generating matrix}.
Notice that $H^j$ are time-independent since $\{D_{n}^j;n\geq1\}$ are identically distributed
(see Subsection~\ref{section_further_extensions} for possible extensions).
Moreover, we assume that at any time $n$ the replacement matrix for the urn $j$, i.e. $D_{n}^j$,
is independent of the sampled colors, i.e. $\{X_{i,n}^j;1\leq j\leq N, 1\leq i\leq K\}$, and
independent of the replacement matrices of the other urns of the system, i.e. $D_{n}^{j_0}$ with $j_0\neq j$.\\
In conclusion, the composition of the color $k\in\{1,{\ldots},K\}$ in the urn $j\in\{1,{\ldots},N\}$ evolves at time $n\geq1$ as follows:
\begin{equation}\label{eq:dynamics_Y}
Y^j_{k,n}\ =\ Y^j_{k,n-1}\ +\ \sum_{i=1}^K D^j_{ki,n} X^j_{i,n}.
\end{equation}
\subsection{Main Assumptions}\label{subection_main_assumptions}
We now present the main conditions required to establish the results of the paper.
The first assumption is concerned with bounds for the moments of the replacement distributions.
Specifically, we require the following condition:
\begin{itemize}
\item[(\secondmomentpideltaD)] there exists $\delta>0$ and a constant $0<C_{\delta}<\infty$ such that, for any $j\in\{1,{\ldots},N\}$ and any $k,i\in\{1,{\ldots},K\}$,
$\mathbf{E}[(D^j_{ki,n})^{2+\delta}]<C_{\delta}$.
\end{itemize}
Note that $C_{\delta}$ does not depend on $n$ since $\{D_{n}^j;n\geq1\}$ are identically distributed.
The second assumption is the average constant balance of the urns in the system and
it is imposed by the following condition on the generating matrices $H^1,{\ldots},H^N$:
\begin{itemize}
\item[(\constantbalanceH)] for any $j\in\{1,{\ldots},N\}$ and $i\in\{1,{\ldots},K\}$, there exists a constant $0<c^j<\infty$ such that
$\sum_{k=1}^K H^j_{ki}\ =\ c^j$.
\end{itemize}
Note that (\constantbalanceH) guarantees that the average number of balls replaced in any urn is constant,
regardless its composition.
Assumption (\constantbalanceH) is essential to obtain the asymptotic configuration of the system, i.e. the limiting urn proportions.
The second-order asymptotic properties of the interacting urn system, namely the rate of convergence and the limiting distributions,
are obtained by assuming a stricter assumption than (\constantbalanceH).
This condition is expressed as follows:
\begin{itemize}
\item[(\constantbalanceD)] for any $j\in\{1,{\ldots},N\}$, $i\in\{1,{\ldots},K\}$, $\mathbf{P}\left(\ \sum_{k=1}^K D^j_{ki,n}\ =\ c^j\ \right)=1$,
i.e. each urn is updated with a constant total amount of balls.
\end{itemize}
Naturally, (\constantbalanceD) implies (\secondmomentpideltaD) with $C_{\delta}=(\max_{j}\{c^j\})^{2+\delta}$.
Notice that, by defining $\widehat{Y}^j_{k,n}=(c^j)^{-1}Y^j_{k,n}$ and $\widehat{D}^j_{ki,n}=(c^j)^{-1}D^j_{ki,n}$ for all $n\geq1$, the urn dynamics in~\eqref{eq:dynamics_Y} can be expressed in the following equivalent form:
$$\widehat{Y}^j_{k,n}\ =\ \widehat{Y}^j_{k,n-1}+\sum_{i=1}^K\widehat{D}^j_{ki,n} \cdot X^j_{i,n},\ \ \ \ \ \ \
\widehat{Z}^j_{k,n-1}\ =\ \frac{\widehat{Y}^j_{k,n-1}}{\sum_{k=1}^K\widehat{Y}^j_{k,n-1}}\ =\
\frac{Y^j_{k,n-1}}{\sum_{k=1}^KY^j_{k,n-1}}\ =\ Z^j_{k,n-1}.$$
Therefore, from now on we will denote by $Y^j_{k,n}$ and $D^j_{ki,n}$ the normalized quantities $\widehat{Y}^j_{k,n}$ and $\widehat{D}^j_{ki,n}$
and hence (\constantbalanceH) and (\constantbalanceD) are replaced by the following conditions:
\begin{itemize}
\item[(\constantbalanceH)] for any $j\in\{1,{\ldots},N\}$ and $i\in\{1,{\ldots},K\}$, $\sum_{k=1}^K H^j_{ki}\ =\ 1$.
\end{itemize}
\begin{itemize}
\item[(\constantbalanceD)] for any $j\in\{1,{\ldots},N\}$ and $i\in\{1,{\ldots},K\}$, $\mathbf{P}\left(\ \sum_{k=1}^K D^j_{ki,n}\ =\ 1\ \right)=1$.
\end{itemize}
In this case, (\constantbalanceD) implies (\secondmomentpideltaD) with $C_{\delta}=1$.
Finally, we consider \textit{Generalized Friedman's Urns} (GFUs) with irreducible generating matrices,
as expressed in the following condition:
\begin{itemize}
\item[(\irreducibleH)] for any $j\in\{1,{\ldots},N\}$, $H^j$ is irreducible.
\end{itemize}
This assumption will guarantee deterministic asymptotic configurations for the urn proportions in the system.
\begin{rem}
Extensions to non-homogeneous generating matrices $\{H_n;n\geq0\}$
are possible, as discussed in Section~\ref{section_further_extensions}.
In that case, assumption (\constantbalanceH) should be referred to the
limiting matrix $H^j:=a.s.-\lim_{n\rightarrow\infty}H^j_n$.
\end{rem}
\subsection{A preliminary result}\label{subection_preliminary}
The assumptions (\constantbalanceH) and (\constantbalanceD) on the constant balance are essential to obtain the following result on
the total number of balls in the urns of the system:
\begin{theo}\label{thm:total_number}
Let $T^j_n=\sum_{k=1}^KY^j_{k,n}$ be the total number of balls contained in the urn $j$ at time $n$.
Then, under assumptions (\secondmomentpideltaD) and (\constantbalanceH), $\{T^j_{n}-n;n\geq1\}$ is an $L^2$ martingale and,
for any $\alpha<1/2$,
\begin{equation}\label{eq:total_number_convergence}
n^{\alpha}\left(\frac{T^j_{n}}{n}-1\right)\ \stackrel{a.s./L^2}{\longrightarrow}\ 0.
\end{equation}
Moreover, under assumption (\constantbalanceD), $T^j_{n}=T^j_{0}+n$ a.s. and hence~\eqref{eq:total_number_convergence} holds for any $\alpha<1$.
\end{theo}
\subsection{The interacting matrix}\label{subsection_W interaction}
The interaction among the urns of the system is modeled through the sampling probabilities $\tilde{Z}^j_{i,n-1}$,
that are defined in~\eqref{def:Z_tilde} as convex combinations of the urn proportions of the system.
Formally, we denote by $W$ the $N\times N$ matrix composed by the weights $\{w_{jh},1\leq j,h\leq N\}$ of such linear combinations
and we refer to it as \emph{interacting matrix}.
We now consider a particular decomposition of $W$ that individuates subsystems of urns evolving with different behaviors.
The same decomposition is typically applied to the transition matrices in the context of discrete time-homogeneous Markov chains (see~\cite{Nor})
to characterize the state space.
For this reason, we first present the decomposition of $W$ in this
framework,
and then we identify the subsystems of urns as the communicating classes of the state space.
Consider a discrete time-homogeneous Markov chain with state space $\{1,{\ldots},N\}$ and transition matrix $W$, i.e.
the element $w_{jh}$ now represents the probability of a Markov chain to move from state $j$ to state $h$ in one step.
It is well-known (see~\cite{Nor}) that the \emph{communication} relationship ($i\sim j$ if there exist $m,n\geq 0$ such that $[W^m]_{ij}>0$ and
$[W^n]_{ji}>0$) induces a partition of the state space into communicating classes (some of them are necessarily closed and recurrent,
with possibly some transient classes).
The maximum eigenvalue is $\lambda=1$ and its multiplicity reflects the number of recurrent classes.
Accordingly, let us denote by $\mathcal{L}$ the set of labels that identify the communicating classes, $n_L\geq1$ the multiplicity of $\lambda_{\max}(W)=1$,
and define the integers $n_F\geq 0$ and $1\leq r^{L_1}<\ldots<r^{L_{n_L}}<r^{F_1}<\ldots<r^{F_{n_F}}= N$
such that $W$ can be decomposed as follows (see \cite[Example 1.2.2]{Nor} for the analogous upper triangular case):
\begin{equation}\label{def:W}
\begin{gathered}
W:=\begin{pmatrix}
W^{L}&0\\
W^{FL}&W^{F}
\end{pmatrix},
\\
\begin{matrix}
W^L :=
\left(
\begin{smallmatrix}
W^{L_1}&0&\ldots&0\\
0&W^{L_2}&\ldots&\ldots\\
\ldots&\ldots&\ldots&0\\
0&0&\ldots&W^{L_{n_L}}
\end{smallmatrix} \right)
\\
W^{FL}:=
\left(
\begin{smallmatrix}
W^{F_1L_1}&\ldots&W^{F_1L_{n_L}}\\
\ldots&\ldots&\ldots\\
W^{F_{n_F}L_1}&\ldots&W^{F_{n_F}L_{n_L}}
\end{smallmatrix} \right)
,&
W^F := \left(
\begin{smallmatrix}
W^{F_1}&0&\ldots&0\\
W^{F_2F_1}&W^{F_2}&\ldots&0\\
\ldots&\ldots&\ldots&\ldots\\
W^{F_{n_F}F_1}&W^{F_{n_F}F_2}&\ldots&W^{F_{n_F}}
\end{smallmatrix} \right).
\end{matrix}
\end{gathered}
\end{equation}
where:
\begin{itemize}
\item[(1)] for any $l\in\mathcal{L}$, $W^{l}$ is an $s^{l}\times s^{l}$ irreducible matrix,
where we let $s^{l}:=r^{l}-r^{l^{-}}$ and $l^{-}$ indicates the element in $\mathcal{L}$ that precedes $l$
(by convention $L_1^{-}\equiv \emptyset$ and $F_1^{-}\equiv L_{n_L}$);
\item[(2)] $\mathcal{L}:=\mathcal{L}_L\cup\mathcal{L}_F$, $\mathcal{L}_L:=\{L^1,{\ldots},L^{n_L}\}$ and $\mathcal{L}_F:=\{F^1,{\ldots},F^{n_F}\}$ are sets of labels that identify, respectively, recurrent and transient communicating classes in the state space ($\mathcal{L}_F=\emptyset$ when $n_F=0$);
\item[(3)] for any $l_1\in\mathcal{L}_F$, there is at least an $l_2\in\mathcal{L}$, $l_1\neq l_2$, such that $W^{l_1l_2}\neq0$;
hence, $\lambda_{\max}(W^{l})=1$ if $l\in\mathcal{L}_L$ and $\lambda_{\max}(W^{l})<1$ if $l\in\mathcal{L}_F$.
\end{itemize}
Naturally, when $n_F=0$ the elements in $W^{FL}$ and $W^F$ do not exist and we consider $r^{L_{n_L}}=N$.
This occurs
when all the classes are closed and recurrent and hence the state space can be partitioned into
irreducible and disjoint subspaces.
In the case of $W$ irreducible, there is only one closed and recurrent class and hence $n_L=1$ and $r^{1}=N$.
In the framework of urn systems, $W$ indicates the interacting matrix and hence
the element $w_{jh}$ represents how the color sampled from the urn $j$ is influenced by the composition of the urn $h$.
Hence, the probability of the Markov process to move from $j$ to $h$ in the state space can be interpreted as
the influence that $h$ has on $j$ in the urn system.
As a consequence, recurrent classes may be seen as subsystems of urns which are not influenced by the rest of the system;
analogously, transient classes may represent subsystems of urns which are influenced by other urns of the system.
Hence, from an interacting matrix $W$ expressed as in~\eqref{def:W},
we can decompose the urn system in:
\begin{itemize}
\item[(i)] leading systems $\{S^l,l\in\mathcal{L}_L\}$, $S^{l}:=\{r^{l^{-}}+1< j\leq r^{l}\}$,
that evolve independently with respect to the rest of the system;
\item[(ii)] if $n_F\geq0$, following systems $\{S^l,l\in\mathcal{L}_F\}$, $S^{l}:=\{r^{l^{-}}+1< j\leq r^{l}\}$, that evolve depending on the proportions of the urns in the leaders $S^{L_1},{\ldots},S^{L_{n_L}}$ and their upper followers $S^{F_1},{\ldots},S^{l^-}$.
\end{itemize}
As we will see in the following sections, the asymptotic behaviors of the leading systems
and the following systems are quite different.
For completeness of the paper, we will present the results for both the types of systems, assuming that $n_F\geq 1$.
\begin{rem}
Extensions to random and time-dependent interacting matrices $\{W_n;n\geq0\}$
are possible, as discussed in Section~\ref{section_further_extensions}.
In that case, the structure presented in~\eqref{def:W} is concerned with the limiting matrix $W:=a.s.-\lim_{n\rightarrow\infty}W_n$.
\end{rem}
\section{The interacting urn system in the stochastic approximation framework}\label{section_stochastic_approximation}
A crucial technique to characterize the asymptotic behavior of the interacting urn system consists in
revisiting its dynamics into the stochastic approximation (SA) framework.
A similar approach has been adopted in~\cite{LarPag} to establish the asymptotic behavior of a single urn.
However, since here we deal with systems of urns,
we need to extend the dynamics~\eqref{eq:dynamics_Y} to jointly study the urns that interact among each other.
To this end, we first introduce in Subsection~\ref{subection_notation_SA} a compact notation
that combines
the composition of the urns in the same subsystem $S^l$, $l\in\mathcal{L}$.
Then, in Subsection~\ref{subection_dynamics_SA} we
embed each
subsystem dynamics into the classical SA form:
given a filtered probability space $(\Omega,{\mathcal A},(\mathcal{F}_n)_{n\geq0},\mathbf{P})$, we consider the following recursive procedure
\begin{equation}\label{SAP}
\forall\, n\geq1,\quad \theta_{n}=\theta_{n-1}-\frac{1}{n}f(\theta_{n-1})+\frac{1}{n}\left(\Delta M_{n}+R_{n}\right),
\end{equation}
where $f:\mathbb{R}^d\rightarrow \mathbb{R}^d$ is a locally Lipschitz continuous function, $\theta_{n}$ an $\mathcal{F}_{n}$-measurable finite random vector and, for every $n\ge 1$, $\Delta M_{n}$ is an $\mathcal{F}_{n-1}$-martingale increment and $R_{n}$ is an $\mathcal{F}_n$-adapted remainder term.
In our framework, the process $\theta_{n}$ satisfying~\eqref{SAP} will represent the proportions of the colors of the urns in the same subsystem.
In the next sections we
apply the ``ODE'' and the ``SDE'' methods for SA
reported in Theorem~\ref{ThmODE} and in Theorem~\ref{ThmCLT} (see Appendix),
that establish first and second-order asymptotic results for $\theta_{n}$.
Specifically, Theorem~\ref{ThmODE} states that, under suitable hypotheses on $\Delta M_{n}$ and $R_{n}$,
the set $\Theta^{\infty}$ of the limiting values of $\theta_n$ as $n\rightarrow+\infty$ is $a.s.$ a compact connected set, stable by the flow of
$ODE_f\equiv\dot{\theta}=-f(\theta)$; moreover, if $\theta^*\in\Theta^{\infty}$ is a uniformly stable equilibrium on $\Theta^{\infty}$ of $ODE_f$, then
$\theta_n\stackrel{a.s.}{\longrightarrow}\theta^*$.
In addition, under further assumptions on $\Delta M_{n}$ and $R_{n}$,
Theorem~\ref{ThmCLT} establishes the CLT for $\theta_{n}$ in which
the convergence rate and the asymptotic distribution depend on the eigen-structure of the
Jacobian matrix of $f(\theta)$ evaluated at the equilibrium point $\theta^*$.
\subsection{Notation}\label{subection_notation_SA}
The quantities related to the urn $j\in\{1,{\ldots},N\}$ at time $n$ are random variables denoted by:
\begin{itemize}
\item[(1)] $Y_{n}^{j}=(Y_{1,n}^{j},{\ldots},Y_{K,n}^{j})'\in\mathbb{R}_{+}^{K}$,
\item[(2)] $Z_{n}^{j}=(Z_{1,n}^{j},{\ldots},Z_{K,n}^{j})'\in\mathcal{S}^K$, where
$\mathcal{S}^{K}$ indicates the $K$-simplex,
\item[(3)] $\tilde{Z}_{n}^{j}=(\tilde{Z}_{1,n}^{j},{\ldots},\tilde{Z}_{K,n}^{j})'\in\mathcal{S}^K$,
\item[(4)] $X_{n}^{j}=(X_{1,n}^{j},{\ldots},X_{K,n}^{j})'\in\mathcal{S}^K\cap\{0,1\}^{K}$,
\end{itemize}
while the corresponding terms of the system $S^l$, $l\in\mathcal{L}$, given by the $s^l$ urns labeled by $\{r^{l^{-}}+1,{\ldots},r^{l}\}$,
are denoted by:
\begin{itemize}
\item[(1)] $\mathbf{Y}^{l}_n:=(Y_{n}^{r^{l^{-}}+1},{\ldots},Y_{n}^{r^{l}})'\in\mathbb{R}_{+}^{s^lK}$,
\item[(2)] $\mathbf{Z}^{l}_n:=(Z_{n}^{r^{l^{-}}+1},{\ldots},Z_{n}^{r^{l}})'\in\mathcal{S}^{s^lK}$, where
$\mathcal{S}^{s^lK}$ indicates the Cartesian product of $s^l$ $K$-simpleces where $Z_{n}^{r^{l^{-}}+1},{\ldots},Z_{n}^{r^{l}}$ are defined,
\item[(3)] $\tilde{\mathbf{Z}}^{l}_n:=(\tilde{Z}_{n}^{r^{l^{-}}+1},{\ldots},\tilde{Z}_{n}^{r^{l}})'\in\mathcal{S}^{s^lK}$,
\item[(4)] $\mathbf{X}^{l}_n:=(X_{n}^{r^{l^{-}}+1},{\ldots},X_{n}^{r^{l}})'\in\mathcal{S}^{s^lK}\cap\{0,1\}^{s^lK}$,
\item[(5)] $\mathbf{T}^{l}_n:=(T_{n}^{r^{l^{-}}+1}\mathbf{1}_K,{\ldots},T_{n}^{r^{l}}\mathbf{1}_K)'\in\mathbb{R}_{+}^{s^lK}$,
where $\mathbf{1}_K$ indicates the $K$-vector of all ones.
\end{itemize}
The replacement matrix for the system $S^l$ is defined by a non-negative
block diagonal matrix $\mathbf{D}^l_n$ of dimensions ${s^lK \times s^lK}$,
where the $s^l$ blocks are the replacement matrices of the urns $\{r^{l^{-}}+1,{\ldots},r^{l}\}$ in $S^l$, i.e. $D_{n}^{r^{l^{-}}+1},{\ldots},D_{n}^{r^{l}}$.
Analogously, the generating matrix for $S^l$ is defined by a
block diagonal matrix $\mathbf{H}^l$ of the same dimensions,
where the $s^l$ blocks are $H^{r^{l^{-}}+1},{\ldots},H_{n}^{r^{l}}$.
The interaction within the system $S^l$ is modeled by the $s^lK\times s^lK$ matrix $\mathbf{W}^l $ with values in $[0,1]$ defined as follows:
starting from $W^l$ in~\eqref{def:W}, each weight $w_{jh}$ is replaced by the corresponding diagonal matrix $w_{jh}I_K$,
where here $I_K$ indicates the $K\times K$-identity matrix.
Analogously, the interaction between a following system $S^{l_1}$, $l_1\in\mathcal{L}_F$, and another system $S^{l_2}$, $l_2\in\{L_1,{\ldots},l_1^-\}$,
is modeled by the matrix $\mathbf{W}^{l_1l_2}$,
obtained by replacing each weight $w_{jh}$ of $W^{l_1l_2}$ in~\eqref{def:W} with the corresponding diagonal matrix $w_{jh}I_K$.
Finally, we will denote by $\mathbf{I}$ the identity matrix composed by more matrices $I_K$.
\begin{Example}\label{example_1}
Consider a system of $N=2$ urns containing balls of $K=2$ colors.
Let the generating matrices $H^1$, $H^2$ and the interacting matrix $W$ be as follows:
\begin{equation}\label{def:example_H12_W}
H^1 :=
\begin{pmatrix}
3/4&1/2\\
1/4&1/2
\end{pmatrix},
\qquad
H^2 :=
\begin{pmatrix}
7/8&7/8\\
1/8&1/8
\end{pmatrix},
\qquad
W :=
\begin{pmatrix}
\alpha&1-\alpha\\
1-\beta&\beta
\end{pmatrix},
\end{equation}
where $\alpha$ and $\beta$ are given constants in $[0,1]$.
In the case of no interaction $\alpha=\beta=1$, from the classical theory on single GFUs (see~\cite{AthKar2,BaiHu,BaiHu2,Smy}), we have that
\begin{itemize}
\item[(1)] $Z^1_{n}=(Z^1_{1,n},Z^1_{2,n})'$ converges a.s. to $(2/3,1/3)'$, i.e. the right eigenvector of $H^1$ associated to $\lambda=1$; moreover
the convergence rate is $\sqrt{n}$, since the second eigenvalue of $H^1$ is $0.25$.
\item[(2)] $Z^2_{n}=(Z^2_{1,n},Z^2_{2,n})'$ converges a.s. to $(1/2,1/2)'$, i.e. the right eigenvector of $H^2$ associated to $\lambda=1$; moreover
the convergence rate is $n^{0.25}$, since the second eigenvalue of $H^2$ is $0.75$.
\end{itemize}
When both $\alpha<1$ and $\beta<1$, $W$ is irreducible.
Using the notation introduced in
Subsection~\ref{subsection_W interaction} and Subsection~\ref{subection_notation_SA},
in this case the two urns belong to the same leading subsystem $S^{L_1}=\{1,2\}$. We have
$s^{L_1}=2$,
$W=W^{L}= W^{L_1}$, and
the joint quantities read as follows:
$\mathbf{Z}^{1}_n:=(Z_{1,n}^{1},Z_{2,n}^{1},Z_{1,n}^{2},Z_{2,n}^{2})'\in\mathcal{S}^{2,2}$,
\begin{equation*}
\mathbf{H} :=
\begin{pmatrix}
\frac{3}{4} & \frac{1}{2} & 0 & 0\\ \frac{1}{4} & \frac{1}{2} & 0 & 0\\ 0 & 0 & \frac{7}{8} & \frac{1}{8}\\ 0 & 0 & \frac{1}{8} & \frac{7}{8}
\end{pmatrix}
,\qquad
\mathbf{W} :=
\begin{pmatrix}
{\alpha} & 0 & 1 - {\alpha} & 0\\ 0 & {\alpha} & 0 & 1 - {\alpha}\\ 1 - {\beta} & 0 & {\beta} & 0\\ 0 & 1 - {\beta} & 0 & {\beta}
\end{pmatrix}.
\end{equation*}
We will discuss the asymptotic properties of this system in Example~\ref{example_2}.
When $\alpha=1$ and $\beta<1$, the first urn forms a leading system, while the second one
exhibits the behavior of a following system (see Example~\ref{example_3}).
\end{Example}
\subsection{The system dynamics in the SA form}\label{subection_dynamics_SA}
For any system $S^l$, $l\in\mathcal{L}$,
the dynamics in~\eqref{eq:dynamics_Y} can be written, using the notation of Subsection~\ref{subection_notation_SA}, as follows:
\begin{equation}\label{eq:dynamics_Y_SA}
\mathbf{Y}^l_{n}\ =\ \mathbf{Y}^l_{n-1}\ +\ \mathbf{D}^l_{n}\mathbf{X}^l_{n}.
\end{equation}
We now express~\eqref{eq:dynamics_Y_SA} in the SA form~\eqref{SAP}, where the process $\{\theta_n;n\geq1\}$ is represented by the urn proportions of the system
$S^l$, i.e. $\{\mathbf{Z}^l_{n};n\geq1\}$.
Since $\mathbf{Y}^l_{n}=diag(\mathbf{T}^l_{n})\mathbf{Z}^l_{n}$ for any $n\geq1$, from~\eqref{eq:dynamics_Y_SA} we have
$$diag(\mathbf{T}^l_{n})\mathbf{Z}^l_{n}\ =\ diag(\mathbf{T}^l_{n-1})\mathbf{Z}^l_{n-1}\ +\ \mathbf{D}^l_{n}\mathbf{X}^l_{n},$$
that is equivalent to
\begin{equation}\label{eq:dynamics_Y_SA_2}
diag(\mathbf{T}^l_{n})(\mathbf{Z}^l_{n}-\mathbf{Z}^l_{n-1})\ =\ -diag(\mathbf{T}^l_{n}-\mathbf{T}^l_{n-1})\mathbf{Z}^l_{n-1}\ +\ \mathbf{D}^l_{n}\mathbf{X}^l_{n}.
\end{equation}
Now, notice that, for any $n\geq1$,
\begin{itemize}
\item[(1)] $\mathbf{E}[diag(\mathbf{T}^l_{n}-\mathbf{T}^l_{n-1})|\mathcal{F}_{n-1}]\ =\ \mathbf{I}$ by Theorem~\ref{thm:total_number};
\item[(2)] $\mathbf{E}[\mathbf{D}^l_{n}\mathbf{X}^l_{n}|\mathcal{F}_{n-1}]\ =\ \mathbf{E}[\mathbf{D}^l_{n}|\mathcal{F}_{n-1}]\mathbf{E}[\mathbf{X}^l_{n}|\mathcal{F}_{n-1}]\ =\ \mathbf{H}^l\tilde{\mathbf{Z}}^l_{n-1}$, since $\mathbf{D}^l_{n}$ and $\mathbf{X}^l_{n}$ are independent conditionally on $\mathcal{F}_{n-1}$.
\end{itemize}
Hence, defining the martingale increment
\begin{equation}\label{eq:martingale_increment_SA}
\Delta \mathbf{M}^l_n\ :=\ \mathbf{D}^l_{n}\mathbf{X}^l_{n} - \mathbf{H}^l\tilde{\mathbf{Z}}^l_{n-1}\ -\
(diag(\mathbf{T}^l_{n}-\mathbf{T}^l_{n-1})-\mathbf{I})\mathbf{Z}^l_{n-1},
\end{equation}
we can express~\eqref{eq:dynamics_Y_SA_2} as follows:
\begin{equation}\label{eq:dynamics_Y_SA_3}
diag(\mathbf{T}^l_{n})(\mathbf{Z}^l_{n}-\mathbf{Z}^l_{n-1})\ =\ -\mathbf{Z}^l_{n-1}\ +\ \mathbf{H}^l\tilde{\mathbf{Z}}^l_{n-1}\ +\ \Delta \mathbf{M}^l_n.
\end{equation}
Now, multiplying by $diag(\mathbf{T}^l_{n})^{-1}$ and defining the remainder term
\begin{equation}\label{eq:remainder_term_SA}
\mathbf{R}^l_n\ :=\ \left(n \cdot diag(\mathbf{T}^l_{n})^{-1}-\mathbf{I}\right)
\left(-\mathbf{Z}^l_{n-1}\ +\ \mathbf{H}^l\tilde{\mathbf{Z}}^l_{n-1}\ +\ \Delta \mathbf{M}^l_n\right),
\end{equation}
we can write~\eqref{eq:dynamics_Y_SA_3} as follows:
\begin{equation}\label{eq:dynamics_Y_SA_4}
\mathbf{Z}^l_{n}-\mathbf{Z}^l_{n-1}\ =\ -\frac{1}{n}(\mathbf{Z}^l_{n-1}\ -\ \mathbf{H}^l\tilde{\mathbf{Z}}^l_{n-1})\ +\
\frac{1}{n}\left(\Delta \mathbf{M}^l_n\ +\ \mathbf{R}^l_n\right).
\end{equation}
The term $(\mathbf{Z}^l_{n-1}-\mathbf{H}^l\tilde{\mathbf{Z}}^l_{n-1})$ in~\eqref{eq:dynamics_Y_SA_4}
should represent the function $f$ in~\eqref{SAP} in the SA form.
However, although in a leader $S^l$, $l\in\mathcal{L}_L$, we have that $\tilde{\mathbf{Z}}^l_{n-1}$ only depends on $\mathbf{Z}^l_{n-1}$,
in a follower $S^l$, $l\in\mathcal{L}_F$,
the term $\tilde{\mathbf{Z}}^l_{n-1}$ is in general a function of the composition of all the urns of the system, i.e.
$\mathbf{Z}^{L_1}_{n-1},{\ldots},\mathbf{Z}^{l}_{n-1}$.
Hence, the dynamics of a leading system can be expressed as in~\eqref{eq:dynamics_Y_SA_4},
while the dynamics of a following system needs to be incorporated with other systems to be fully described.
For this reason, the asymptotic behavior of these two types of systems are studied separately:
the leading systems in Section~\ref{section_leading} and the following systems in Section~\ref{section_follower}.
\section{Leading Systems}\label{section_leading}
In this section we present the main asymptotic results concerning the leading systems $S^l$, $l\in\mathcal{L}_L$.
We recall that these systems are characterized by irreducible interacting matrices $W^l$ such that $\lambda_{max}(W^l)=1$ (see~\eqref{def:W} in Subsection~\ref{subsection_W interaction}).
For this reason, their dynamics is independent of the rest of the system and hence,
by using $\tilde{\mathbf{Z}}^l_{n-1}=\mathbf{W}^l\mathbf{Z}^l_{n-1}$ in~\eqref{eq:dynamics_Y_SA_4}, we have
\begin{equation}\label{eq:dynamics_Y_SA_leading}
\begin{aligned}
\mathbf{Z}^l_{n}-\mathbf{Z}^l_{n-1}\ &&=&\ -\frac{1}{n}h^l(\mathbf{Z}^l_{n-1})\ +\ \frac{1}{n}\left(\Delta \mathbf{M}^l_n\ +\ \mathbf{R}^l_n\right),\\
h^l(\mathbf{x})\ &&:=&\ (\mathbf{I}-\mathbf{Q}^{l})\mathbf{x},\ \ \ \mathbf{Q}^{l}\ :=\ \mathbf{H}^l\mathbf{W}^l
\end{aligned}
\end{equation}
\subsection{Extension of the urn dynamics to $\mathbb{R}^{s^l K}$}\label{subection_redefining_simplex_leading}
Since $h^l$ is defined on $\mathbb{R}^{s^l K}$,
while the process $\{\mathbf{Z}^l_{n};n\geq0\}$ takes values in the subset $\mathcal{S}^{s^l K}$,
then applying theorems based on the SA directly to~\eqref{eq:dynamics_Y_SA_leading}
may lead to improper results for the process $\mathbf{Z}^l_{n}$.
To address this issue, we appropriately modify the dynamics~\eqref{eq:dynamics_Y_SA_leading}
by replacing $h^l$ with a suitable function $f_m^l:=h^l+mg^l$,
where $m>0$ is an arbitrary constant and $g^l$ is a function defined in $\mathbb{R}^{s^l K}$ that satisfies the following properties:
\begin{itemize}
\item[(i)] the derivative $\mathcal{D}g^l$ is positive semi-definite and its kernel is $Span\{(x-y):x,y\in\mathcal{S}^{s^l K}\}$:
hence, $g^l$ does not modify the eigen-structure of $\mathcal{D}h^l(\mathbf{x})$ on the subspace $\mathcal{S}^{s^l K}$,
where the process $\mathbf{Z}^l_{n}$ is defined, while it changes
the eigen-structure outside $\mathcal{S}^{s^l K}$,
where it can be arbitrary redefined;
\item[(ii)] $g^l(\mathbf{z})=0$ for any $\mathbf{z}\in\mathcal{S}^{s^l K}$:
hence, since $f_m^l(\mathbf{z})=h^l(\mathbf{z})$ for any $\mathbf{z}\in\mathcal{S}^{s^l K}$,
the modified dynamics restricted to the subset $\mathcal{S}^{s^l K}$
represents the same dynamics as in~\eqref{eq:dynamics_Y_SA_leading}.
\end{itemize}
Let us now provide an analytic expression of $g^l$.
First note that, since by definition of convex combination we always have $W^l\mathbf{1}_{s^l}=\mathbf{1}_{s^l}$,
the left eigenvectors of $W^l$ (possibly generalized) are such that $U_1'\mathbf{1}_{s^l}=1$ and $U_i'\mathbf{1}_{s^l}=0$ for all $i\neq 1$.
Denote by $Sp(A)$ the set of the eigenvalues of a matrix $A$ and note that,
since by $(\constantbalanceH)$ we always have $\mathbf{1}_K'H^j=\mathbf{1}_K'$,
then $Sp(W^l)\subset Sp(\mathbf{Q}^{l})$ and the $s^l$ left eigenvectors of $\mathbf{Q}^{l}$ associated to any $\lambda_i\in Sp(W^l)\subset Sp(\mathbf{Q}^{l})$,
$i\in\{1,{\ldots},s^l\}$, present the following structure: $\mathbf{U}_i:=(U_{i1}\mathbf{1}_K,{\ldots},U_{is^l}\mathbf{1}_K)'$.
As a consequence, for any $\mathbf{z}\in\mathcal{S}^{s^l K}$,
we have $\mathbf{U}_1'\mathbf{z}=U_1'\mathbf{1}_{s^l}=1$ and $\mathbf{U}_i'\mathbf{z}=U_i'\mathbf{1}_{s^l}=0$ for all $i\in\{2,{\ldots},s^l\}$.
Hence, denoting by $\mathbb{V}_2$ and $\mathbb{U}_2$ the matrices whose columns are $\mathbf{V}_2,{\ldots},\mathbf{V}_{s^l}$ and
$\mathbf{U}_2,{\ldots},\mathbf{U}_{s^l}$, respectively, we define the function $g^l$ as follows:
\begin{equation}\label{def:function_g}
g^l(\mathbf{x})\ :=\ \mathbf{V}_1\left(\mathbf{U}_1'\mathbf{x}-1\right)\ +\ \mathbb{V}_2\mathbb{U}_2'\mathbf{x},
\end{equation}
and the dynamics of the process $\mathbf{Z}^l_{n}$ in~\eqref{eq:dynamics_Y_SA_leading} can be replaced by the following:
\begin{equation}\label{eq:dynamics_Y_SA_leading_h}
\begin{aligned}
\mathbf{Z}^l_{n}-\mathbf{Z}^l_{n-1}\ &&=&\ -\frac{1}{n}f_m^l(\mathbf{Z}^l_{n-1})\ +\ \frac{1}{n}\left(\Delta \mathbf{M}^l_n\ +\ \mathbf{R}^l_n\right),\\
f_m^l(\mathbf{x})\ &&:=&\ (\mathbf{I}-\mathbf{Q}^{l})\mathbf{x}\ +\ m\mathbf{V}_1\left(\mathbf{U}_1'\mathbf{x}-1\right)\ +\ m\mathbb{V}_2\mathbb{U}_2'\mathbf{x}.
\end{aligned}
\end{equation}
\subsection{First-order asymptotic results}\label{subection_first_order_leading}
We now present the main convergence result concerning the limiting proportion of the urns in the leading systems.
\begin{theo}\label{thm:first_order_leading}
Assume $(\secondmomentpideltaD)$, $(\constantbalanceH)$ and $(\irreducibleH)$.
Thus, for any leading system $S^l$, $l\in\mathcal{L}_L$, we have that
\begin{equation}\label{eq:first_order_leading}
\mathbf{Z}^l_n\ \stackrel{a.s.}{\longrightarrow}\ \mathbf{Z}^l_{\infty}\ :=\ \mathbf{V}_1,
\end{equation}
where $\mathbf{V}_1$ indicates the right eigenvector associated to the simple eigenvalue $\lambda=1$ of the matrix $\mathbf{Q}^l$,
with $\sum_i V_{1i}=1$.
\end{theo}
\begin{rem}\label{rem:W_identity_Friedman_as}
Note that when the interacting matrix is the identity matrix, i.e. $W=I$, $n_L=N$ and $n_F=0$,
each urn represents a leading system and it evolves independently of the rest of the system.
In this case,~\eqref{eq:first_order_leading} expresses the usual result for a single GFU,
where the urn proportion converges to the right eigenvector associated to the maximum eigenvalue of the generating matrix, see e.g.~\cite{AthKar2,BaiHu,BaiHu2,Smy}.
\end{rem}
\begin{rem}\label{rem:simple_eigenvalue}
In Theorem~\ref{thm:first_order_leading}, condition $(\irreducibleH)$ implies that the maximum eigenvalue $\lambda=1$ of $\mathbf{Q}^l$ has multiplicity one,
which guarantees $\mathbf{V}_1$ to be the unique global attractor for the system $S^l$.
Without assumption $(\irreducibleH)$, there could be multiple attractors and hence the limiting proportions of the system would be a random variable,
as in~\cite{CriDaiPraMin,DaiPraLouisMin} where the RRU model is considered.
\end{rem}
\subsection{Second-order asymptotic results}\label{subection_second_order_leading}
We now establish the rate of convergence and the asymptotic distribution of the urn proportions in the leading systems $S^l$, $l\in\mathcal{L}_L$.
Since to obtain these results we need to apply the Central Limit Theorem of the SA (see Theorem~\ref{ThmCLT} in Appendix)
to the dynamics~\eqref{eq:dynamics_Y_SA_leading_h},
a crucial role is played by the spectrum of the $Ks^l\times Ks^l$-matrix of the first-order derivative of $f_m^l$ defined as follows:
for any $\mathbf{x}\in\mathbb{R}^{Ks^l}$
\begin{equation}\label{def:F_derivative}
\mathbf{F}^l_m\ :=\ \mathcal{D}f_m^l(\mathbf{x})\ =\ (\mathbf{I}-\mathbf{Q}^l)\ +\ m\mathbf{V}_1\mathbf{U}_1'\ +\ m\mathbb{V}_2\mathbb{U}_2'.
\end{equation}
Moreover, since the asymptotic variance depends on the second moments of the replacement matrices,
we denote by $C^j(i)$ the covariance matrix of the $i^{th}$ column of $D^j_n$, i.e.
$C^j(i):=\mathbf{C}ov[D^j_{\cdot i,n}]$, where $D^j_{\cdot i,n}:=(D^j_{1 i,n},{\ldots},D^j_{K i,n})'$;
note that $(\constantbalanceD)$ ensures the existence of $C^j(i)$.
Hence, denoting by $H^j(i):=E[H^j_{\cdot i}(H^j_{\cdot i})']$ where $H^j_{\cdot i}:=(H^j_{1 i},{\ldots},H^j_{K i})'$, we let
\begin{equation}\label{def:G}
G^j\ :=\ \sum_{i=1}^K\left(C^j(i)+H^j(i)\right)\tilde{Z}_{i,\infty}^j\ -\ Z_{\infty}^j(Z_{\infty}^j)^{'},
\end{equation}
where $\tilde{Z}_{i,\infty}^j=\sum_{h=1}^N w_{jh}Z_{i,\infty}^h$.
Then, for any leading system $S^l$, $l\in\mathcal{L}_L$,
we denote by $\mathbf{G}^l$ the block diagonal matrix made by the $s^l$ blocks $G^{r^{l^-}+1},{\ldots},G^{r^l}$.
The following theorem shows the rate of convergence and the limiting distribution of the urn proportions in the leading systems.
\begin{theo}\label{thm:second_order_leading}
Assume $(\constantbalanceD)$ and $(\irreducibleH)$.
For any leading system $S^l$, $l\in\mathcal{L}_L$, let $\lambda^{*l}$ be the eigenvalue of $Sp(\mathbf{Q}^l)\setminus Sp(W^l)$ with highest real part.
Thus, we have that $\Re e(\lambda^{*l})\equiv1-\Re e(Sp(\mathbf{F}^l_m))$ and
\begin{itemize}
\item[(a)] if $\Re e(\lambda^{*l})<1/2$, then
\begin{equation*}\label{eq:second_order_Z_less_leading}
\sqrt{n}(\mathbf{Z}^l_n-\mathbf{Z}^l_{\infty})\ \stackrel{d}{\longrightarrow}\ \mathcal{N}\left(0,\Sigma^l\right),\ \ \ \
\Sigma^l\ :=\ \lim_{m\rightarrow \infty}\int_0^{\infty}e^{u(\frac{\mathbf{I}}{2}-\mathbf{F}^l_m)}\mathbf{G}^le^{u(\frac{\mathbf{I}}{2}-\mathbf{F}^l_m)'}du.
\end{equation*}
\item[(b)] if $\Re e(\lambda^{*l})=1/2$, then
\begin{equation*}\label{eq:second_order_Z_equal_leading}
\sqrt{\frac{n}{\log(n)}}(\mathbf{Z}^l_n-\mathbf{Z}^l_{\infty})\ \stackrel{d}{\longrightarrow}\ \mathcal{N}\left(0,\Sigma^l\right).
\end{equation*}
\item[(c)] if $\Re e(\lambda^{*l})>1/2$, then there exists a finite random variable $\psi^l$ such that
\begin{equation*}\label{eq:second_order_Z_more_leading}
n^{1-\Re e(\lambda^{*l})}(\mathbf{Z}^l_n-\mathbf{Z}^l_{\infty})\ \stackrel{a.s.}{\longrightarrow}\ \psi^l.
\end{equation*}
\end{itemize}
\end{theo}
\begin{rem}\label{rem:W_identity_Friedman_clt}
When the interacting matrix $W$ is the identity matrix,
each urn represents a leading system and hence $W^l=1$ and $\mathbf{Q}^l\equiv H^l$.
In that case, $\lambda^*$ is the eigenvalue of $H^l$ with second highest real part
and hence Theorem~\ref{thm:second_order_leading} expresses the usual Central Limit Theorem for a single GFU, see e.g.~\cite{AthKar2,BaiHu,BaiHu2,Smy}.
\end{rem}
\begin{rem}\label{rem:role_W_in_Q}
The role of $\mathbf{Q}^{l}$ in Theorem~\ref{thm:second_order_leading} shows that
the convergence rate of the urns in $S^l$ does not depend only on their generating matrices $\{H^j,r^{l^-}+1\leq j\leq r^l\}$
but also on their interaction expressed in $W^l$.
For instance, consider two single GFUs whose generating matrices $H^1$ and $H^2$ are such that
the convergence rates of the urn proportions $Z_n^1$ and $Z^2_n$ without interactions are different.
Then, an interaction between these urns with an irreducible $W^l$ would make
$Z_n^1$ and $Z^2_n$ converge at the same rate,
which would depend on the choice of $W^l$.
\end{rem}
\begin{Example}[Continuation of Example~\ref{example_1}]\label{example_2}
When we introduce an interaction with an irreducible $W$,
the limit of the urn proportions changes as established in Theorem~\ref{thm:first_order_leading}.
For instance, if we consider $W$ as in~\eqref{def:example_H12_W} with $\alpha=\beta=0.8$ we have that $\mathbf{Z}_n=(Z^1_{1,n},Z^1_{2,n},Z^2_{1,n},Z^2_{2,n})'$
converges a.s. to $(0.66,0.34,0.56,0.44)'$, which is the right eigenvector of
\[
\mathbf{Q}=\mathbf{H}\mathbf{W}
=
\begin{pmatrix}
\alpha H^1 & (1-\alpha) H^1 \\
(1-\beta)H^2 & \beta H^2
\end{pmatrix}
=
\left(
\begin{smallmatrix}
\frac{3\, {\alpha}}{4} & \frac{{\alpha}}{2} & \frac{3}{4}(1 - {\alpha}) & \frac{1}{2} ( 1- {\alpha})\\
\frac{{\alpha}}{4} & \frac{{\alpha}}{2} & \frac{1}{4}(1 - {\alpha}) & \frac{1}{2} ( 1- {\alpha})\\
\frac{7}{8} ( 1 - {\beta}) & \frac{1}{8} ( 1 - {\beta}) & \frac{7\, {\beta}}{8} & \frac{{\beta}}{8}\\
\frac{1}{8} ( 1 - {\beta}) & \frac{7}{8} ( 1 - {\beta}) & \frac{{\beta}}{8} & \frac{7\, {\beta}}{8}
\end{smallmatrix}
\right),
\]
associated to $\lambda=1$.
Moreover, as explained in Remark~\ref{rem:role_W_in_Q}, the interaction makes the two urns
converge at the same rate, which depends on the interacting matrix, as established in Theorem~\ref{thm:second_order_leading}.
In this case $\alpha=\beta=0.8$, since $Sp(\mathbf{Q})=\{1, 0.62, 0.6, 0.18\}$ and $Sp(W)=\{1, 0.6\}$,
we have $\lambda^{*}=0.62$ and hence the convergence rate is $n^{0.38}$.
In addition, to underline the role of the interaction in the convergence rate of the system,
we note that
\begin{itemize}
\item[(i)] if $\alpha=(1-\beta)=0.8$, since $Sp(\mathbf{Q})=\{1,0.35,0,0\}$ and $Sp(W)=\{1,0\}$
we have $\lambda^{*}=0.35$ and hence the convergence rate is $\sqrt{n}$;
\item[(ii)] if $\alpha=\beta=0.5$, since $Sp(\mathbf{Q})=\{1,0.5,0,0\}$ and $Sp(W)=\{1,0\}$
we have $\lambda^{*}=0.5$ and hence the convergence rate is $\sqrt{n/\log(n)}$;
\item[(iii)] if $\alpha=(1-\beta)=0.2$, since $Sp(\mathbf{Q})=\{1,0.65,0,0\}$ and $Sp(W)=\{1,0\}$
we have $\lambda^{*}=0.65$ and hence the convergence rate is $n^{0.35}$.
\end{itemize}
\end{Example}
\section{Following Systems}\label{section_follower}
In this section we establish asymptotic properties concerning the following systems $S^l$, $l\in\mathcal{L}_F$.
As we have already underlined, the dynamics of these systems can be properly expressed in the SA form~\eqref{SAP}
only through a joint model with the urns in the systems $\{S^{L_1},{\ldots},S^l\}$.
Thus, we need a further notation to study collections of more systems.
In particular, we will replace the label $l$ with $(l)$ whenever an object is referred to the joint system
$S^{(l)}:=\{S^{L_1},{\ldots},S^l\}$ instead of the single system $S^l$.
For instance, the vector $\mathbf{Y}^{(l)}_n\in\mathbb{R}^{Kr^l}$ indicates $(\mathbf{Y}^{L_1}_n,{\ldots},\mathbf{Y}^{l}_n)'$,
and $\mathbf{D}^{(l)}_n$ indicates the block diagonal $(Kr^l\times Kr^l)$-matrix, whose blocks are made by $\mathbf{D}^{L_1}_n,{\ldots},\mathbf{D}^{l}_n$.
Then, from~\eqref{def:W} we can express the sampling probabilities in the follower $S^l$ as follows:
$$\tilde{\mathbf{Z}}^{l}_{n-1}=\sum_{i\in \{L_1,{\ldots}l^-\}} \mathbf{W}^{li}\mathbf{Z}^i_{n-1}+\mathbf{W}^{l}\mathbf{Z}^{l}_{n-1}.$$
Hence, from~\eqref{eq:dynamics_Y_SA_4} we obtain
\begin{equation}\label{eq:dynamics_Y_SA_following}
\begin{aligned}
\mathbf{Z}^{l}_{n}-\mathbf{Z}^{l}_{n-1}\ &&=&\ -\frac{1}{n}h^{l}(\mathbf{Z}^{(l^-)}_{n-1},\mathbf{Z}^{l}_{n-1})\ +
\ \frac{1}{n}\left(\Delta \mathbf{M}^{l}_n\ +\ \mathbf{R}^{l}_n\right),\\
h^{l}(\mathbf{x}_1,\mathbf{x}_2)\ &&:=&\ -\mathbf{Q}^{l(l^{-})}\mathbf{x}_1+(\mathbf{I}-\mathbf{Q}^{l})\mathbf{x}_2,\\
\mathbf{Q}^{l(l^{-})}\ &&:=&\ \left[\mathbf{H}^l\mathbf{W}^{lL_1}\ \ldots\ \mathbf{H}^l\mathbf{W}^{l\,l^-}\right],\ \
\mathbf{Q}^{l}\ :=\ \mathbf{H}^l\mathbf{W}^l
\end{aligned}
\end{equation}
Since $h^{l}$ is not only a function of $\mathbf{Z}^{l}_{n-1}$, the dynamics in~\eqref{eq:dynamics_Y_SA_following}
is not already expressed in the SA form~\eqref{SAP}.
To address this issue, we need to consider a joint model for the global system $S^{(l)}=S^{(l^-)}\cup S^l=S^{L_1}\cup{\ldots}\cup S^l$ as follows:
\begin{equation}\label{eq:dynamics_Y_SA_following_entire}\begin{aligned}
\mathbf{Z}^{(l)}_{n}-\mathbf{Z}^{(l)}_{n-1}\ &&=&\ -\frac{1}{n}h^{(l)}(\mathbf{Z}^{(l)}_{n-1})\ +\
\frac{1}{n}\left(\Delta \mathbf{M}^{(l)}_n\ +\ \mathbf{R}^{(l)}_n\right),\\
h^{(l)}(\mathbf{x})\ &&:=&\ \left(\mathbf{I}-\mathbf{Q}^{(l)}\right)\mathbf{x},
\end{aligned}\end{equation}
where $\mathbf{Q}^{(l)}$ can be recursively defined as follows:
\begin{equation}\label{def:Q_complete}
\mathbf{Q}^{(l)} :=
\begin{pmatrix}
\mathbf{Q}^{(l^{-})}&0\\
\mathbf{Q}^{l(l^{-})}&\mathbf{Q}^{l}
\end{pmatrix},
\qquad
\mathbf{Q}^{(L_{n_L})} :=
\begin{pmatrix}
\mathbf{Q}^{L_1}&{\ldots}&0\\
\ldots&\ldots&\ldots\\
0&{\ldots}&\mathbf{Q}^{L_{n_L}},
\end{pmatrix},
\end{equation}
where by convention $F_1^-=L_{n_L}$.
\subsection{Extension of the urn dynamics to $\mathbb{R}^{r^l K}$}\label{subection_redefining_simplex_follower}
We now apply to following systems similar considerations made for SA of the leading systems in Section~\ref{subection_redefining_simplex_leading}.
Note again that
$h^{(l)}$ in~\eqref{eq:dynamics_Y_SA_following_entire} is defined in $\mathbb{R}^{r^l K}$,
while the process $\{\mathbf{Z}^{(l)}_{n};n\geq0\}$ lies in the subspace $\mathcal{S}^{r^l K}$.
The application of the theorems based on the SA needs an extension of $h^{(l)}$, which takes into account the SA structure.
For this reason, we replace $h^{(l)}$ in~\eqref{eq:dynamics_Y_SA_following} with a suitable function $f_m^{(l)}:=h^{(l)}+mg^{(l)}$ such that
$m>0$ is an arbitrary constant and $g^{(l)}$ is a function defined as in~\eqref{def:function_g},
where in this case
$\{\mathbf{U}_i;1\leq i\leq r^l\}$ and $\{\mathbf{V}_i;1\leq i\leq r^l\}$,
indicate, respectively, the left and right eigenvectors of $\mathbf{Q}^{(l)}$ (possibly generalized).
Hence, the dynamics of the process $\mathbf{Z}^{(l)}_{n}$~\eqref{eq:dynamics_Y_SA_following} is replaced by the following:
\begin{equation}\label{eq:dynamics_Y_SA_following_h}\begin{aligned}
\mathbf{Z}^{(l)}_{n}-\mathbf{Z}^{(l)}_{n-1}\ &&=&\ -\frac{1}{n}f_m^{(l)}(\mathbf{Z}^{(l)}_{n-1})\ +\
\frac{1}{n}\left(\Delta \mathbf{M}^{(l)}_n\ +\ \mathbf{R}^{(l)}_n\right),\\
f_m^{(l)}(\mathbf{x})\ &&:=&\ \left(\mathbf{I}-\mathbf{Q}^{(l)}\right)\mathbf{x}\ +\ m\mathbf{V}_1\left(\mathbf{U}_1'\mathbf{x}-1\right)\ +\ m\mathbb{V}_2\mathbb{U}_2'\mathbf{x}.
\end{aligned}\end{equation}
Note that in the joint system $S^{(l)}$ the eigenvalue $\lambda=1$ of $\mathbf{Q}^{(l)}$ may not have multiplicity one;
in that case, $\mathbf{V}_1$ is univocally identified as the right eigenvector of $\mathbf{Q}^{(l)}$ associated to $\lambda=1$
such that, letting $\mathbf{U}_i:=(U_{i1}\mathbf{1}_K,{\ldots},U_{ir^l}\mathbf{1}_K)'$ and $U_i'W^{(l)}=\lambda_iU_i'$ for any $i\in\{1,{\ldots},r^l\}$,
we have $\mathbf{U}_1'\mathbf{V}_1=U_1'\mathbf{1}_{r^l}=1$ and $\mathbf{U}_i'\mathbf{V}_1=U_i'\mathbf{1}_{r^l}=0$ when $i\neq 1$.
\subsection{Removal of unnecessary components}\label{subection_in_out_follower}
The following system $S^l$ may not depend on all the components of $S^{(l^-)}$ and hence
the convergence in $S^l$ may be faster than the rate in $S^{(l^-)}$.
When this occurs, the asymptotic distribution obtained for the urn proportions in $S^{(l)}$ restricted to the urns in $S^l$ is degenerate.
To address this issue and characterize the asymptotic behavior in the following system $S^l$,
we need to reduce the dimensionality of $\mathbf{Z}^{(l)}_n$ by deleting those components which do not influence the dynamics of $\mathbf{Z}^l_n$.
Since the interaction between $S^l$ and the systems in $S^{(l^-)}$ is expressed by $\mathbf{Q^{l(l^-)}}$,
we exclude the components of $\mathbf{Z}^{(l^-)}_n$ defined on the null space of $\mathbf{Q}^{l(l^-)}$.
Formally, consider the following decomposition:
$$Sp(\mathbf{Q^{(l)}})\ =\ Sp(\mathbf{Q^{l}})\ \cup\ Sp(\mathbf{Q^{(l^-)}})\ =\ \mathcal{A}_{IN}\ \cup\ \mathcal{A}_{OUT},$$
where
\[\begin{aligned}
\mathcal{A}_{OUT}\ &&:=&\ \left\{\ \lambda\in Sp(\mathbf{Q^{(l^-)}})\ :\ \exists v \{\mathbf{Q^{(l^-)}}v=\lambda v\}\cap\{\mathbf{Q^{l(l^-)}}v=0\}\ \right\}\\
\mathcal{A}_{IN}\ &&:=&\ Sp(\mathbf{Q^{l}})\ \cup\ \left(Sp(\mathbf{Q^{(l^-)}})\setminus \mathcal{A}_{OUT}\right).
\end{aligned}\]
Then, the eigenspace of $\mathbf{Q^{(l)}}$ associated to $\lambda\in\mathcal{A}_{OUT}$ will be removed from the dynamics in~\eqref{eq:dynamics_Y_SA_following_h}.
To do this, let us denote by:
\begin{itemize}
\item[(1)] $\mathbf{U}_{IN}$ and $\mathbf{V}_{IN}$ the matrices whose columns are the left and right eigenvectors of $\mathbf{Q^{(l)}}$, respectively,
associated to eigenvalues in $\mathcal{A}_{IN}$;
\item[(2)] $\mathbf{U}_{OUT}$ and $\mathbf{V}_{OUT}$ the matrices whose columns are the left and right eigenvectors of $\mathbf{Q^{(l)}}$, respectively,
associated to eigenvalues in $\mathcal{A}_{OUT}$;
\end{itemize}
Since we do not want to modify the process $\mathbf{Z}^{(l)}_n$ on $S^l$, i.e. $\mathbf{Z}^{l}_n$,
we now construct two conjugate basis in $\mathcal{I}m(\mathbf{U}_{IN})$ and $\mathcal{I}m(\mathbf{V}_{IN})$ that are invariant on $S^l$.
Note that, since $Sp(\mathbf{Q^{l}})\subset\mathcal{A}_{IN}$,
there exists a non-singular matrix $\mathbf{P}$ such that the following decompositions hold:
\begin{equation*}
\mathbf{B}:=\mathbf{V}_{IN}\mathbf{P}=
\begin{pmatrix}
\mathbf{\hat{B}}&0\\
0&\mathbf{I}
\end{pmatrix},
\qquad
\mathbf{C}:=\mathbf{P}^{-1}\mathbf{U}_{IN}'=
\begin{pmatrix}
\mathbf{\hat{C}}&0\\
0&\mathbf{I}
\end{pmatrix}.
\end{equation*}
Since $\mathbf{\hat{C}}'\mathbf{\hat{B}}=\mathbf{I}$ and $\mathbf{\hat{B}}\mathbf{\hat{C}}'=\mathbf{V_{IN}}\mathbf{U_{IN}}'$,
$\mathbf{\hat{C}}$ and $\mathbf{\hat{B}}$ represent conjugate basis in $\mathcal{I}m(\mathbf{U}_{IN})$ and $\mathcal{I}m(\mathbf{V}_{IN})$, respectively.
Thus, for any $\mathbf{x}=(\mathbf{x}^{(l^-)},\mathbf{x}^l)'\in\mathbb{R}^{Kr^l}$, we have the following decomposition:
\begin{equation}\label{eq:decomposition_x}
\mathbf{x}\ =\ \mathbf{V_{IN}}\mathbf{U_{IN}}'\mathbf{x}\ +\ \mathbf{V_{OUT}}\mathbf{U_{OUT}}'\mathbf{x}\ =\
\mathbf{\hat{B}}\mathbf{\hat{x}}\ +\ \mathbf{V_{OUT}}\mathbf{x_{OUT}},\ \ \
\end{equation}
where
$$\mathbf{\hat{x}}:=\mathbf{\hat{C}}'\mathbf{x}=
\begin{pmatrix}
\mathbf{C}'\mathbf{x}^{(l^-)}\\
\mathbf{x}^{l}
\end{pmatrix},
\qquad \mathbf{x_{OUT}}:=\mathbf{U_{OUT}}'\mathbf{x}.$$
In particular, we consider the process $\{\mathbf{\hat{Z}}^{(l)}_{n},n\geq1\}$ defined as follows:
\begin{equation}\label{eq:decomposition_Z_hat}
\mathbf{\hat{Z}}^{(l)}_{n}:=\mathbf{\hat{C}}'\mathbf{Z}^{(l)}_{n}=
\begin{pmatrix}
\mathbf{C}'\mathbf{Z}_{n}^{(l^-)}\\
\mathbf{Z}_{n}^{l}
\end{pmatrix};
\end{equation}
now, multiplying by $\mathbf{\hat{C}}'$ to~\eqref{eq:dynamics_Y_SA_following_h}
and applying the decomposition~\eqref{eq:decomposition_x} in~\eqref{eq:dynamics_Y_SA_following_h},
since $\mathbf{\hat{C}}'\mathbb{V}_2\mathbb{U}_2'\mathbf{V_{OUT}}=0$, $\mathbf{U}_1'\mathbf{V_{OUT}}=0$ and
$\mathbf{\hat{C}}'\mathbf{V_{OUT}}=0$, we have that
\begin{equation}\label{eq:dynamics_Y_SA_following_hat}\begin{aligned}
\mathbf{\hat{Z}}^{(l)}_{n}-\mathbf{\hat{Z}}^{(l)}_{n-1}\ &&=&\ -\frac{1}{n}\hat{f}_m^{(l)}(\mathbf{\hat{Z}}^{(l)}_{n-1})\ +\
\frac{1}{n}\mathbf{\hat{C}}'\left(\Delta \mathbf{M}^{(l)}_n\ +\ \mathbf{R}^{(l)}_n\right),\\
\hat{f}_m^{(l)}(\mathbf{\hat{x}})\ &&:=&\ \left(\mathbf{I}-\mathbf{\hat{C}}'\mathbf{Q}^{(l)}\mathbf{\hat{B}}\right)\mathbf{\hat{x}}\ +\ m\mathbf{\hat{V}}_1\left(\mathbf{\hat{U}}_1'\mathbf{\hat{x}}-1\right)\ +\
m\mathbb{\hat{V}}_2\mathbb{\hat{U}}_2'\mathbf{\hat{x}},
\end{aligned}\end{equation}
where $\mathbf{\hat{U}}_1':=\mathbf{U}_1'\mathbf{\hat{B}}$, $\mathbb{\hat{U}}_2:=\mathbb{U}_2'\mathbf{\hat{B}}$, $\mathbf{\hat{V}}_1:=\mathbf{\hat{C}}'\mathbf{V}_1$ and $\mathbb{\hat{V}}_2:=\mathbf{\hat{C}}'\mathbb{V}_2$
represent the left and right eigenvectors of $\mathbf{\hat{C}}'\mathbf{Q}^{(l)}\mathbf{\hat{B}}$ associated to
$\lambda\in Sp(W^{(l)})\setminus \mathcal{A}_{OUT}$.
Since $\hat{f}_m^{(l)}$ is a function of $\mathbf{\hat{Z}}^{(l)}_{n}$,
the dynamics in~\eqref{eq:dynamics_Y_SA_following_hat} is now expressed in the SA form~\eqref{SAP}.
\begin{rem}
The interacting matrix $W$ lonely is not enough to individuate
the components of the system that actually influence a following system,
but it is necessary to study the eigen-structure of $\mathbf{Q}^{(l)}$, that joins the information of $W$ and of the generating matrices
$\{H^j,1\leq j\leq r^l\}$ of the urns in $S^{(l)}$.
This may be surprising since $W$ is the only element that defines the interaction among the urns in the system.
Nevertheless, when $H^j$ is singular, different values of $\tilde{Z}^j_{n}$ may give the same average replacements, $H^j\tilde{Z}^j_{n}$,
which is equivalent as having singularities in $W$, where different values of $\{Z^i_{n};1\leq i\leq r^l\}$
may give the same $\tilde{Z}^j_{k,n}$, and hence same $H^j\tilde{Z}^j_{n}$.
For instance, if all the columns of $H^j$ were equal to a given vector $v^j$,
the urn $j$ would be updated on average by $v^j$ regardless the value of $\tilde{Z}^j_{n-1}$
and hence the urns in $S^{(l^-)}$ would not play any role in the dynamics of the urn $j$
for any choice of $W$.
\end{rem}
\subsection{First-order asymptotic results}\label{subection_first_order_follower}
We now present the convergence result concerning the limiting proportion of the urns in the following systems.
The asymptotic behavior of $\mathbf{Z}_{n}^{(l)}$ is obtained recursively from
$\mathbf{Z}_{\infty}^{(l^-)}:=a.s.-\lim_{n\rightarrow\infty}\mathbf{Z}_{n}^{(l^-)}$.
\begin{theo}\label{thm:first_order_follower}
Assume $(\secondmomentpideltaD)$, $(\constantbalanceH)$ and $(\irreducibleH)$.
Thus, for any $l\in\mathcal{L}_F$, we have that
\begin{equation*}\label{eq:first_order_follower_joint}
\mathbf{\hat{Z}}^{(l)}_n\ \stackrel{a.s.}{\longrightarrow}\ \mathbf{\hat{Z}}^{(l)}_{\infty}\ :=\ \mathbf{\hat{V}}_1;
\end{equation*}
hence, from~\eqref{eq:decomposition_Z_hat}, in the following system $S^l$ we have that
\begin{equation*}\label{eq:first_order_follower}
\mathbf{Z}^l_n\ \stackrel{a.s.}{\longrightarrow}\ \mathbf{Z}^l_{\infty}\ :=\ \left(\mathbf{I}-\mathbf{Q}^{l}\right)^{-1}\mathbf{Q}^{l(l^-)}\mathbf{Z}^{(l^-)}_{\infty}.
\end{equation*}
\end{theo}
\subsection{Second-order asymptotic results}\label{subection_second_order_follower}
We now present the results concerning the rate of convergence and the asymptotic distribution of the urn proportions in the following systems.
To this end, let us introduce the $Ks^l\times Ks^l$-matrix of the first-order derivative of $\hat{f}_m^l$:
\begin{equation}\label{def:F_derivative_hat}
\begin{aligned}
\hat{\mathbf{F}}_m^{(l)}\ &&:=&\ \mathbf{\hat{C}}'\mathbf{F}^{(l)}_m\mathbf{\hat{B}}\\
&&=&\ (\mathbf{I}-\mathbf{\hat{C}}'\mathbf{Q}^{(l)}\mathbf{\hat{B}})\ +\ m\mathbf{\hat{V}}_1\mathbf{\hat{U}}_1'\ +\ m\mathbb{\hat{V}}_2\mathbb{\hat{U}}_2'.
\end{aligned}
\end{equation}
Moreover, the asymptotic variance will be based on the quantity $\mathbf{\hat{G}}^{(l)}:=\mathbf{\hat{C}}'\mathbf{G}^{(l)}\mathbf{\hat{B}}$,
where $\mathbf{G}^{(l)}$ is the block diagonal matrix made by $G^{1},{\ldots},G^{r^l}$ (see~\eqref{def:G}).
The following theorem shows the rate of convergence and the limiting distribution of the urn proportions in the following systems.
\begin{theo}\label{thm:second_order_follower}
Assume $(\constantbalanceD)$ and $(\irreducibleH)$.
For any following system $S^l$, $l\in\mathcal{L}_F$, let $\lambda^{*l}$ be the eigenvalue of
$Sp(\mathbf{Q}^{(l)})\setminus (Sp(W^{(l)})\cup \mathcal{A}_{OUT})$ with highest real part.
Thus, we have that $\Re e(\lambda^{*l})\equiv1-\Re e(Sp(\hat{\mathbf{F}}_m^{(l)}))$ and
\begin{itemize}
\item[(a)] if $\Re e(\lambda^{*l})<1/2$, then
\begin{equation*}\label{eq:second_order_Z_less_follower}
\sqrt{n}(\mathbf{\hat{Z}}^{(l)}_n-\mathbf{\hat{Z}}^{(l)}_{\infty})\ \stackrel{d}{\longrightarrow}\ \mathcal{N}\left(0,\hat{\Sigma}^{(l)}\right),\ \ \ \
\hat{\Sigma}^{(l)}\ :=\ \lim_{m\rightarrow \infty}\int_0^{\infty}e^{u(\frac{\mathbf{I}}{2}-\hat{\mathbf{F}}_m^{(l)})}\mathbf{\hat{G}}^{(l)}e^{u(\frac{\mathbf{I}}{2}-\hat{\mathbf{F}}_m^{(l)})'}du.
\end{equation*}
\item[(b)] if $\Re e(\lambda^{*l})=1/2$, then
\begin{equation*}\label{eq:second_order_Z_equal_follower}
\sqrt{\frac{n}{\log(n)}}(\mathbf{\hat{Z}}^{(l)}_n-\mathbf{\hat{Z}}^{(l)}_{\infty})\ \stackrel{d}{\longrightarrow}\ \mathcal{N}\left(0,\Sigma^{(l)}\right).
\end{equation*}
\item[(c)] if $\Re e(\lambda^{*l})>1/2$, then there exists a finite random variable $\psi^{(l)}$ such that
\begin{equation*}\label{eq:second_order_Z_more_follower}
n^{1-\Re e(\lambda^{*l})}(\mathbf{\hat{Z}}^{(l)}_n-\mathbf{\hat{Z}}^{(l)}_{\infty})\ \stackrel{a.s.}{\longrightarrow}\ \psi^{(l)}.
\end{equation*}
\end{itemize}
\end{theo}
\begin{rem}\label{rem:CLT_following_joint}
Note that, since from~\eqref{eq:decomposition_Z_hat} $\mathbf{\hat{Z}}^{(l)}_n=(\mathbf{C}'\mathbf{Z}^{(l^-)}_n,\mathbf{Z}^{l}_n)'$,
Theorem~\ref{thm:second_order_follower} explicitly states the limiting distribution and the asymptotic covariance structure
of the urn proportions in any following system $\mathbf{Z}^{l}_n$, $l\in\mathcal{L}_F$.
In addition, Theorem~\ref{thm:second_order_follower} also determines the correlations between $\mathbf{Z}^{l}_n$ and the components
of the urn proportions in the other systems $S^l$, $l\in\{L_1,{\ldots},l^-\}$, that actually influence the dynamics of $\mathbf{Z}^{l}_n$.
\end{rem}
\begin{rem}\label{rem:drop_irreducibility}
We highlight that condition $(\irreducibleH)$, i.e. irreducibility of the generating matrices $H^j$,
may be relaxed in Theorem~\ref{thm:first_order_follower} and Theorem~\ref{thm:second_order_follower},
by requiring $(\irreducibleH)$ only for the urns in the leading systems.
In fact, we can note from the proof that $(\irreducibleH)$
is not needed for the urns that belong to the following systems.
\end{rem}
\begin{Example}[Continuation of Example~\ref{example_1} and Example~~\ref{example_2}]\label{example_3}
Set $W$ as in~\eqref{def:example_H12_W} with $\alpha=1$ and $\beta<1$, and hence
\[
\mathbf{Q}=\mathbf{H}\mathbf{W}
=
\begin{pmatrix}
H^1 & 0 \\
(1-\beta)H^2 & \beta H^2
\end{pmatrix}.
\]
Urn 1 forms a leading system and urn 2 is a following system.
As a consequence, the asymptotic behavior of urn 1 does not depend on urn 2.
We have that
$\mathbf{Z}^{1}_{n}=(Z^1_{1,n},Z^1_{2,n})'$ converges a.s. to $\mathbf{Z}^{1}_{\infty}=(2/3,1/3)'$, and the convergence rate is $\sqrt{n}$,
see Example~\ref{example_1}.
Concerning urn 2, its limiting proportion depends also on urn 1 as established in Theorem~\ref{thm:first_order_follower},
where in this case:
\begin{equation}\label{def:example_Q_following}
\mathbf{Q}^1=H^1,\qquad\mathbf{Q}^{12}=(1-\beta)H^2,\qquad\mathbf{Q}^2=\beta H^2.
\end{equation}
For instance, if $\beta=0.5$ we have that $\mathbf{Z}^{2}_{n}=(Z^2_{1,n},Z^2_{2,n})'$
converges a.s.\ to $(\mathbf{I}-\mathbf{Q}^{2})^{-1}\mathbf{Q}^{12}\mathbf{Z}^{1}_{\infty} = (0.6,0.4)' $.
Moreover, the convergence rate of urn 2 is determined by the interaction as established in Theorem~\ref{thm:second_order_follower}.
With $\beta = 0.5$, since $Sp(\mathbf{Q})=\{1, 0.5, 0.375, 0.25\}$, $Sp(W)=\{1,0.5\}$ and $\mathcal{A}_{OUT}=\emptyset$,
we have $\lambda^{*}=0.375$ and hence the convergence rate is $\sqrt{n}$.
In addition, to underline the role of the interaction in the convergence rate of the following system,
we note that
\begin{itemize}
\item[(i)] if $\beta=0.2$, since $Sp(\mathbf{Q})=\{1,0.25,0.2,0.15\}$, $Sp(W)=\{1,0.2\}$ and $\mathcal{A}_{OUT}=\emptyset$,
we have $\lambda^{*}=0.25$ and hence the convergence rate is $\sqrt{n}$;
\item[(ii)] if $\beta=0.8$, since $Sp(\mathbf{Q})=\{1,0.8,0.6,0.25\}$ and $Sp(W)=\{1,0.8\}$ and $\mathcal{A}_{OUT}=\emptyset$,
we have $\lambda^{*}=0.6$ and hence the convergence rate is $n^{0.4}$.
\end{itemize}
If we compare these results with
the convergence rate of urn 2 without interaction ($n^{0.25}$, see Example~\ref{example_1}),
we can observe that, in this example, the interaction makes the following system converge faster.
\end{Example}
\section{Further extensions}\label{section_further_extensions}
In this section, we discuss some possible extensions of the interacting urn model presented in this paper.
\subsection{Random and time-dependent interacting matrix}
Although we consider a constant interacting matrix $W$,
the results of this paper may be extended to a system characterized by
a random sequence of interacting matrices $\{W_n;n\geq0\}$,
i.e. $W_n=[w_{jh,n}]\in\mathcal{F}_n$ and $\tilde{Z}^j_{i,n}=\sum_{h=1}^Nw_{jh,n}Z^h_{i,n}$ for any $i\in\{1,{\ldots},K\}$.
In that case, it is essential to assume the existence of a deterministic matrix $W$ such that $W_n\stackrel{a.s.}{\longrightarrow}W$,
which individuates the leading and the following systems, as in Subsection~\ref{subsection_W interaction}.
The dynamics with random and time-dependent interacting matrices could be also expressed in the SA form~\eqref{SAP},
by including the difference $(W_n-W)$ in the remainder term~\eqref{eq:remainder_term_SA}.
Naturally, the asymptotic behavior of the urn proportions would depend on the limiting interacting matrix $W$
and on the rate of convergence of the sequence $\{W_n;n\geq0\}$.
Specifically, the convergence of the urn proportions
could be obtained with the only assumption $W_n\stackrel{a.s.}{\longrightarrow}W$,
while extensions for the second-order results presented in this paper
would require $n\mathbf{E}[\|W_n-W\|^2]\rightarrow0 $ (cfr. \cite[Assumption (A5)]{LarPag}).
\subsection{Non-homogeneous generating matrices}
The independence and identically distribution of the replacement matrices is an assumption that could be relaxed
by assuming that the sequence of generating matrices $\{H_n^j;n\geq0\}$, $H^j_{n-1}:=\mathbf{E}[D^j_{n}|\mathcal{F}_{n-1}]$,
converges to some deterministic matrix $H^j$.
Thus, the urn dynamics could be expressed in the SA form~\eqref{SAP},
by including the difference $(H^j_n-H^j)$ in the remainder term~\eqref{eq:remainder_term_SA},
and the asymptotic behavior would depend on $H^j$ and on the rate of convergence of $H_n^j$.
Specifically, the second-order results would require an additional assumption as $n\mathbf{E}[\|H^j_n-H^j\|^2]\rightarrow0$
(cfr. \cite[Assumption (A5)]{LarPag}).
\section{Proofs}\label{section_proofs}
This section contains the proofs of all the results presented in the paper.
Initially, in Subsection~\ref{subsection_proofs_total_number} we prove Theorem~\ref{thm:total_number}
concerning the behavior of the total number of balls in the urns of the system.
Then, in Subsection~\ref{subsection_proofs_leading} we present the proofs of the results on the leading systems
described in Section~\ref{section_leading}.
Finally, Subsection~\ref{subsection_proofs_follower} contains the proofs of the results of Section~\ref{section_follower}
concerning the following systems.
The proofs of Subsection~\ref{subsection_proofs_leading} and~\ref{subsection_proofs_follower}
on the asymptotic behavior of the subsystems of urns are based on basic results of stochastic approximation,
which have been reported in Appendix as Theorem~\ref{ThmODE} and Theorem~\ref{ThmCLT}.
\subsection{Proof of Theorem~\ref{thm:total_number}}\label{subsection_proofs_total_number}
The proof of Theorem~\ref{thm:total_number} requires the following auxiliary result on the martingale convergence:
\begin{lem}\label{lem:preliminary}
Let $\{S_n;n\geq1\}$, $S_n:=\sum_{i=1}^n\Delta S_i$, be a zero-mean martingale with respect to a filtration $\{\mathcal{F}_n;n\geq1\}$ and
let $\{b_n;n\geq1\}$ be a non-decreasing sequence of positive numbers such that
\begin{equation}\label{eq:kronacker_assumption}
\sum_{i=1}^{\infty}b_i^{-2}\mathbf{E}[(\Delta S_i)^2|\mathcal{F}_{i-1}]\ <\ \infty,\ \ \ \ \ a.s.
\end{equation}
Then, $b_n^{-1}S_n\stackrel{a.s.}{\longrightarrow}0$.
\end{lem}
\proof
Let us define the zero-mean martingale $\tilde{S}_n:=\sum_{i=1}^n\Delta \tilde{S}_i$, with $\Delta \tilde{S}_i:=b_i^{-1}\Delta S_i$.
Equation~\eqref{eq:kronacker_assumption} states that $\sum_{i=1}^n\mathbf{E}[(\Delta \tilde{S}_i)^2|\mathcal{F}_{i-1}]<\infty$
and hence $\tilde{S}_n$ converges a.s. since its bracket $\langle\tilde{S}\rangle_{\infty}<\infty$ a.s. (see \cite[Theorem 12.13]{Will}).
Thus, the result follows by using Kronecker's Lemma (see \cite[Lemma IV.3.2]{Shi}).
\endproof
\proof[Proof of Theorem~\ref{thm:total_number}]
By using Lemma~\ref{lem:preliminary} with $b_n:=n^{1-\alpha}$ and $S_n:=T^j_{n}-n$,
the proof follows by showing that $T^j_{n}-n$ is a martingale whose increments have bounded second moments.
Now, since
$$T^j_{n}-T^j_{n-1}\ =\ \sum_{k=1}^K(Y^j_{k,n}-Y^j_{k,n-1})\ =\ \sum_{k=1}^K\sum_{i=1}^K(D^j_{ki,n} X^j_{i,n}),$$
the result follows by establishing that
\begin{itemize}
\item[(a)]\ $\sup_{n\geq1}\mathbf{E}\left[\left(\sum_{k=1}^K\sum_{i=1}^K D^j_{ki,n} X^j_{i,n}\right)^2\ \big|\mathcal{F}_{n-1}\right]<\infty$;
\item[(b)]\ $\sum_{k=1}^K\sum_{i=1}^K\mathbf{E}\left[D^j_{ki,n} X^j_{i,n}|\mathcal{F}_{n-1}\right]=1$.
\end{itemize}
For part (a), by using $|X^j_{i,n}|\leq1$ and (\secondmomentpideltaD), we have that
$$\sup_{n\geq1}\mathbf{E}\left[\left(\sum_{k=1}^K\sum_{i=1}^K (D^j_{ki,n} X^j_{i,n})\right)^2\ \big|\mathcal{F}_{n-1}\right]\ \leq\
K^2\sup_{n\geq1}\max_{j\in\{1,{\ldots},N\}}\max_{i,k\in\{1,{\ldots},K\}}\mathbf{E}\left[(D^j_{ki,n})^2\right]\ <\ \infty,$$
where the last passage follows by noticing that by Jensen's inequality and (\secondmomentpideltaD)
\begin{equation}\label{eq:relation_2_2+delta}
\mathbf{E}\left[(D^j_{ki,n})^2\right]^{\frac{1}{2}}\ \leq\ \mathbf{E}\left[(D^j_{ki,n})^{2+\delta}\right]^{\frac{1}{2+\delta}}\ <\ C_{\delta}^{\frac{1}{2+\delta}}.
\end{equation}
For part (b), since $\sum_{k=1}^KH^j_{ki}=1$ by (\constantbalanceH) and
since $D^j_{ki,n}$ and $X^j_{i,n}$ are independent conditionally on $\mathcal{F}_{n-1}$,
we obtain
$$\sum_{k=1}^K\sum_{i=1}^K\mathbf{E}\left[ D^j_{ki,n} X^j_{i,n}\ |\mathcal{F}_{n-1}\right]
=\ \sum_{k=1}^K\sum_{i=1}^K H^j_{ki} \tilde{Z}^j_{i,n-1}\
=\ \sum_{i=1}^K \tilde{Z}^j_{i,n-1} \sum_{k=1}^KH^j_{ki}\
=\ \sum_{i=1}^K \tilde{Z}^j_{i,n-1}.$$
Finally, by the definition of $\tilde{Z}^j_{i,n-1}$ in~\eqref{def:Z_tilde}, we have
$$\sum_{i=1}^K \tilde{Z}^j_{i,n-1}\ =\ \sum_{i=1}^K\sum_{h=1}^Nw_{jh}Z^{h}_{i,n-1}\ =\
\sum_{h=1}^Nw_{jh}\sum_{i=1}^KZ^{h}_{i,n-1}\ =\ \sum_{h=1}^Nw_{jh}\ =\ 1,$$
which concludes the proof of~\eqref{eq:total_number_convergence} for $\alpha<1/2$ under assumption (\constantbalanceH).
Concerning the proof of~\eqref{eq:total_number_convergence}
under assumption (\constantbalanceD), note that
$$T^j_{n}-T^j_{n-1}\ =\ \sum_{k=1}^K\sum_{i=1}^K(D^j_{ki,n} X^j_{i,n})\ =\ \sum_{i=1}^K X^j_{i,n}\ =\ 1;$$
hence, $T^j_{n}=T^j_{0}+n$ a.s. and, for any $\alpha<1$,
$$
n^{\alpha}\left(\frac{T^j_{n}}{n}-1\right)\ =\ \frac{T^j_{0}}{n^{1-\alpha}}\ \stackrel{a.s./L^2}{\longrightarrow}\ 0.\qedhere
$$
\endproof
\subsection{Proofs on the leading systems}\label{subsection_proofs_leading}
\proof[Proof of Theorem~\ref{thm:first_order_leading}]
Fix $l\in\mathcal{L}_L$ and consider the leading system $S^l=\{r^{l^-}+1\leq j\leq r^{l}\}$ with interacting matrix $W^l$.
Since the dynamic of the urn proportions $\mathbf{Z}^l_n$ in $S^l$ has been expressed in~\eqref{eq:dynamics_Y_SA_leading_h} in the SA form~\eqref{SAP},
we can establish the convergence result stated in Theorem~\ref{thm:first_order_leading} by applying Theorem~\ref{ThmODE} in Appendix.
To this end, we will show that the assumptions of Theorem~\ref{ThmODE} are satisfied by the process $\{\mathbf{Z}^l_n;n\geq1\}$ of the system $S^l$:
\begin{itemize}
\item[(1)] the function $f_m^l$ defined in~\eqref{eq:dynamics_Y_SA_leading_h} is a linear transformation and hence locally Lipschitz.
\item[(2)] from~\eqref{eq:martingale_increment_SA}, we have that
$\sup_{n\geq 1}\mathbf{E}\left[\left\|\Delta \mathbf{M}^l_n\right\|^2|\mathcal{F}_{n-1}\right]<\infty$ is satisfied by establishing
\begin{itemize}
\item[(2a)] $\sup_{n\geq 1}\mathbf{E}\left[\left\|\mathbf{D}^l_{n}\mathbf{X}^l_{n}\right\|^2|\mathcal{F}_{n-1}\right]<\infty$;
\item[(2b)] $\sup_{n\geq 1}\mathbf{E}\left[\left\|diag(\mathbf{T}^l_{n}-\mathbf{T}^l_{n-1})\mathbf{Z}^l_{n-1}\right\|^2
|\mathcal{F}_{n-1}\right]<\infty$.
\end{itemize}
Concerning (2a), since $X^j_{i,n}\in\{0,1\}$ a.s., we have that
$$\left\|\mathbf{D}^l_{n}\mathbf{X}^l_{n}\right\|^2\ \leq\ \sum_{j\in S^l}\sum_{k=1}^K\sum_{i=1}^K\left(D^j_{ki,n}\right)^2,\ \ a.s.$$
Thus, (2a) follows by assumption (\secondmomentpideltaD), since
$$\sup_{n\geq 1}\mathbf{E}\left[\left\|\mathbf{D}^l_{n}\mathbf{X}^l_{n}\right\|^2|\mathcal{F}_{n-1}\right]\ \leq\
\sum_{j\in S^l}\sum_{k=1}^K\sum_{i=1}^K\sup_{n\geq 1}\mathbf{E}\left[\left(D^j_{ki,n}\right)^2\right]\ \leq\ s^lK^2C_{\delta}^{\frac{2}{2+\delta}},$$
where $\mathbf{E}[(D^j_{ki,n})^2]\leq C_{\delta}^{\frac{2}{2+\delta}}$ follows by~\eqref{eq:relation_2_2+delta}.
Concerning (2b), since $\sum_{i=1}^K(Z^j_{i,n})^2\leq1$, we have
\begin{equation}\label{eq:step1_2b}
\left\|diag(\mathbf{T}^l_{n}-\mathbf{T}^l_{n-1})\mathbf{Z}^l_{n-1}\right\|^2\ \leq\ \sum_{j\in S^l}(T^j_n-T^j_{n-1})^2,\ \ a.s.
\end{equation}
where we recall that
\begin{equation}\label{eq:step2_2b}
T^j_{n}-T^j_{n-1}\ =\ \sum_{k=1}^K(Y^j_{k,n}-Y^j_{k,n-1})\ =\ \sum_{k=1}^K\sum_{i=1}^K(D^j_{ki,n} X^j_{i,n}).
\end{equation}
Hence, combining~\eqref{eq:step1_2b} and~\eqref{eq:step2_2b}, since $X^j_{i,n}\in\{0,1\}$ and $\sum_{i=1}^KX^j_{i,n}=1$ a.s., we obtain that
\begin{equation*}\begin{aligned}
\left\|diag(\mathbf{T}^l_{n}-\mathbf{T}^l_{n-1})\mathbf{Z}^l_{n-1}\right\|^2\
&&\leq&\ \sum_{j\in S^l}\left(\sum_{k=1}^K\sum_{i=1}^K (D^j_{ki,n}X^j_{i,n})\right)^2\\
&&\leq&\ \sum_{j\in S^l}\sum_{i=1}^K\left(\sum_{k=1}^K D^j_{ki,n}\right)^2,\ \ a.s.
\end{aligned}\end{equation*}
Finally, using the relation $(\sum_{k=1}^K a_k^2)\leq K^2(\sum_{k=1}^Ka_k^2)$, (2b) follows by assumption (\secondmomentpideltaD), since
\begin{multline*}
\sup_{n\geq 1}\mathbf{E}\left[\left\|diag(\mathbf{T}^l_{n}-\mathbf{T}^l_{n-1})\mathbf{Z}^l_{n-1}\right\|^2|\mathcal{F}_{n-1}\right]\\
\leq
\sup_{n\geq 1}\sum_{j\in S^l}\sum_{i=1}^KK^2\sum_{k=1}^K\mathbf{E}\left[\left(D^j_{ki,n}\right)^2\right]\ \leq\ s^lK^4 C_{\delta}^{\frac{2}{2+\delta}},
\end{multline*}
where $\mathbf{E}[(D^j_{ki,n})^2]\leq C_{\delta}^{\frac{2}{2+\delta}}$ follows by~\eqref{eq:relation_2_2+delta}.
\item[(3)] from~\eqref{eq:remainder_term_SA}, we show $\|\mathbf{R}^l_n\|\stackrel{a.s.}{\longrightarrow}0$ by establishing that,
for any $(2+\delta)^{-1}<\alpha<2^{-1}$,
\begin{itemize}
\item[(3a)] $n^{\alpha}\left\|n\cdot diag(\mathbf{T}^l_{n})^{-1}-\mathbf{I}\right\|\stackrel{a.s.}{\longrightarrow}0$,
\item[(3b)] $n^{-\alpha}\left\|\mathbf{Z}^l_{n-1}-\mathbf{H}^l\tilde{\mathbf{Z}}^l_{n-1}\right\|\stackrel{a.s.}{\longrightarrow}0$,
\item[(3c)] $n^{-\alpha}\left\|\Delta \mathbf{M}^l_n\right\|\stackrel{a.s.}{\longrightarrow}0$,
\end{itemize}
where we recall that $\delta>0$ is defined in Assumption (\secondmomentpideltaD) (see Subsection~\ref{subection_main_assumptions}).
Since (3a) follows straightforwardly by Theorem~\ref{thm:total_number}, consider (3b).
For any $\epsilon>0$, using Markov's inequality we obtain
\begin{equation*}
\mathbf{P}\left(\left\|\mathbf{Z}^l_{n-1}-\mathbf{H}^l\tilde{\mathbf{Z}}^l_{n-1}\right\|>\epsilon n^{\alpha}\right)\ \leq\
(\epsilon n^{\alpha})^{-(2+\delta)}\mathbf{E}\left[\left\|\mathbf{Z}^l_{n-1}-\mathbf{H}^l\tilde{\mathbf{Z}}^l_{n-1}\right\|^{(2+\delta)}\right].
\end{equation*}
Hence, (3b) follows by Borel-Cantelli Lemma since $\alpha\cdot(2+\delta)>1$ and
$$\sup_{n\geq0}\mathbf{E}\left[\left\|\mathbf{Z}^l_{n-1}-\mathbf{H}^l\tilde{\mathbf{Z}}^l_{n-1}\right\|^{(2+\delta)}\right]\
\leq\ \sum_{j\in S^l}2^{(2+\delta)}\ <\ \infty.$$
Concerning (3c), we can apply again Markov's inequality and the same arguments of part (3b) since by assumption (\secondmomentpideltaD) we have that
$$\sup_{n\geq0}\mathbf{E}\left[\left\|\mathbf{D}^l_{n}\mathbf{X}^l_{n} - \mathbf{H}^l\tilde{\mathbf{Z}}^l_{n-1}\right\|^{(2+\delta)}\right]\ \leq\
\sup_{n\geq0}\sum_{j\in S^l}\sum_{k=1}^K\sum_{i=1}^K\mathbf{E}\left[(D^l_{ki,n})^{(2+\delta)}\right] \ <\ \infty.$$
\end{itemize}
Thus, by applying Theorem~\ref{ThmODE} to the dynamics in~\eqref{eq:dynamics_Y_SA_leading_h},
we have that the limiting values of $\mathbf{Z}^l_n$ are included in the set
$$\left\{\ \mathbf{x}\in\mathbb{R}^{Ks_l}\ :\ f_m^l(\mathbf{x})=0\ \right\}.$$
Now, denote by $\mathbb{V}_3$ and $\mathbb{U}_3$ the matrices whose columns are, respectively, the right and left eigenvectors of $\mathbf{Q}^l$
(possibly generalized) associated to the eigenvalues $\lambda\in Sp(\mathbf{Q}^l)\setminus Sp(W^l)$.
Hence, we have the following decomposition
\begin{equation}\label{eq:decomposition_Q}
\mathbf{Q}^l\ =\ \mathbf{V}_1\mathbf{U}_1'\ +\ \mathbb{V}_2\mathbf{J}_2\mathbb{U}_2'\ +\ \mathbb{V}_3\mathbf{J}_3\mathbb{U}_3',
\end{equation}
where $\mathbf{J}_2$ and $\mathbf{J}_3$ represent the corresponding jordan blocks.
Since the eigenvectors of $\mathbf{Q}^l$ represent a basis of $\mathbb{R}^{Ks_l}$,
for any $\mathbf{x}\in\mathbb{R}^{Ks_l}$ there exists $a\in \mathbb{R}$, $b\in \mathbb{R}^{s^l-1}$ and $c\in \mathbb{R}^{s^l(K-1)}$ such that
\begin{equation}\label{eq:decomposition_x_initial}
\mathbf{x}\ =\ \mathbf{V}_1a\ +\ \mathbb{V}_2b\ +\ \mathbb{V}_3c.
\end{equation}
Hence, by using~\eqref{eq:decomposition_Q} and~\eqref{eq:decomposition_x_initial}, we obtain
\[\begin{aligned}
h^l(\mathbf{x})\ &&=&\ \mathbb{V}_2(\mathbf{I}-\mathbf{J}_2)b\ +\ \mathbb{V}_3(\mathbf{I}-\mathbf{J}_3)c,\\
g^l(\mathbf{x})\ &&=&\ \mathbf{V}_1(a-1)\ +\ \mathbb{V}_2b,
\end{aligned}\]
and then, since $f_m^l(\mathbf{x})=h^l(\mathbf{x})+mg^l(\mathbf{x})$, it gives us
\begin{equation}\label{eq:h_transformed_x}
f_m^l(\mathbf{x})\ =\ m\mathbf{V}_1(a-1)\ +\ \mathbb{V}_2((1+m)\mathbf{I}-\mathbf{J}_2)b\ +\ \mathbb{V}_3(\mathbf{I}-\mathbf{J}_3)c.
\end{equation}
From the irreducibility of $H^j$ assumed in (\irreducibleH), for all $\lambda\in Sp(\mathbf{Q}^l)\setminus Sp(W^l)$ we have $\lambda<1$
and hence $(\mathbf{I}-\mathbf{J}_3)$ is positive definite.
Therefore, since $m>0$, from~\eqref{eq:h_transformed_x} we have that $f_m^l(\mathbf{x})=0$ if and only if $a=1$ and $b=c=0$, i.e. $\mathbf{x}=\mathbf{V}_1$.
It
remains to prove that $\mathbf{V}_1$ is a global attractor in $\mathbb{R}^{Ks_l}$.
To this end, we will show that the Jacobian matrix
$\mathcal{D}f_m^l(\mathbf{x})$ is positive definite for any $\mathbf{x}\in\mathbb{R}^{Ks_l}$.
We recall that, from~\eqref{def:F_derivative} we have
\begin{equation}\label{eq:F_derivative_jordan}
\mathbf{F}^l_m\ =\ \mathcal{D}f_m^l(\mathbf{x})\ =\
m\mathbf{V}_1\mathbf{U}_1'\ +\ \mathbb{V}_2((1+m)\mathbf{I}-\mathbf{J}_2)\mathbb{U}_2'\ +\ \mathbb{V}_3(\mathbf{I}-\mathbf{J}_3)\mathbb{U}_3'.
\end{equation}
Hence, since $m>0$ and $(\mathbf{I}-\mathbf{J}_3)$ is positive definite by assumption (\irreducibleH),
we have that $\mathbf{F}^l_m$ is positive definite for any $m>0$.
This concludes the proof.
\endproof
\proof[Proof of Theorem~\ref{thm:second_order_leading}]
The proof consists in showing that the assumptions of the $CLT$ for processes in the $SA$ form (Theorem~\ref{ThmCLT} in Appendix) are satisfied by
the dynamics in~\eqref{eq:dynamics_Y_SA_following_h} of the urn proportions $\mathbf{Z}^l_{n}$ in the leading system $S^l$.\\
First, we show that condition $\{\Re e(Sp(Df(\theta^*)))>1/2\}$ in Theorem~\ref{ThmCLT} is equivalent to $\{\Re e(\lambda^{*l})<1/2\}$.
Note that the function $f$ of the SA form~\eqref{SAP} is represented in our case by $f_m^l$ defined in~\eqref{eq:dynamics_Y_SA_leading_h}.
Similarly, the term $\theta^*$ in Appendix indicates the deterministic limiting proportion $\mathbf{Z}^l_{\infty}$,
while $Dh(\theta^*)$ is represented by $\mathbf{F}^l_m$ defined in~\eqref{def:F_derivative}.
Now, consider the eigen-structure of $\mathbf{Q}^l$ and note that
$\mathbf{F}^l_m$ has been expressed in~\eqref{eq:F_derivative_jordan} as follows:
$$\mathbf{F}^l_m\ =\ m\mathbf{V}_1\mathbf{U}_1'\ +\ \mathbb{V}_2((1+m)\mathbf{I}-\mathbf{J}_2)\mathbb{U}_2'\ +\ \mathbb{V}_3(\mathbf{I}-\mathbf{J}_3)\mathbb{U}_3',$$
Hence, it is easy to see that the eigenvectors of $\mathbf{F}^l_m$ and $\mathbf{Q}^l$ are the same, since
\begin{itemize}
\item[(1)] $\mathbf{F}^l_m\mathbf{V}_1=m\mathbf{V}_1$,
\item[(2)] $\mathbf{F}^l_m\mathbb{V}_2=\mathbb{V}_2((1+m)\mathbf{I}-\mathbf{J}_2)$,
\item[(3)] $\mathbf{F}^l_m\mathbb{V}_3=\mathbb{V}_3(\mathbf{I}-\mathbf{J}_3)$.
\end{itemize}
Thus
$$Sp(\mathbf{F}^l_m)=\{m\}\cup \left\{(1+m)-\lambda,\lambda\in Sp(W^l)\setminus\{1\}\right\}\cup \left\{1-\lambda,\lambda\in Sp(\mathbf{Q}^l)\setminus Sp(W^l)\right\}.$$
By setting $m>0$ arbitrary large,
we obtain that
\[
\{\Re e(Sp(\mathcal{D}f(\theta^*)))>1/2\}\equiv\{\Re e(\lambda^{*l})<1/2\}.
\]
Condition~\eqref{HypDM1} of Theorem~\ref{ThmCLT} follows from
analogous arguments of point (2) in the proof of Theorem~\ref{thm:first_order_leading}.
In fact, since
$$\sup_{n\geq1}\mathbf{E}[\|\Delta\mathbf{M}^l_{n}\|^{2+\delta}|\mathcal{F}_{n-1}]\ \leq\
K^{2+\delta}\sum_{j=1}^N\sum_{i=1}^K\sum_{k=1}^K\sup_{n\geq1}\mathbf{E}[(D^j_{ki,n})^{2+\delta}]
\ \leq\ NK^{4+\delta}.$$
For what concerns condition~\eqref{HypDM2}, we will show in a moment that for any $l\in\mathcal{L}_L$
$$\mathbf{E}[\Delta\mathbf{M}^l_{n}(\Delta\mathbf{M}^l_{n})^{'}|\mathcal{F}_{n-1}]\stackrel{a.s.}{\longrightarrow}\mathbf{G}^l,\ \quad\
\mathbf{E}[\Delta\mathbf{M}^{l_1}_{n}(\Delta\mathbf{M}^{l_2}_{n})^{'}]=0\ \ \ \forall l_1\neq l_2.$$
To this end, we first show that, for any urn $j\in S^l$,
$\mathbf{E}[\Delta M_{n}^j(\Delta M_{n}^j)'|\mathcal{F}_{n-1}] \stackrel{a.s.}{\longrightarrow} G^j$.
Note that
$$
\mathbf{E}[\Delta M_{n}^j(\Delta M_{n}^j)^{'}|\mathcal{F}_{n-1}]\ =\ \mathbf{E}[(D_n^j X_{n}^j)(D_n^jX_{n}^j)^{'}|\mathcal{F}_{n-1}]\ -\
(H^j\tilde{Z}_{n-1}^j)(H^j\tilde{Z}_{n-1}^j)^{'},
$$
and the first term of the right-hand side can be written as
\[\begin{aligned}\mathbf{E}[(D_n^j X_{n}^j)(D_n^j X_{n}^j)^{'}|\mathcal{F}_{n-1}]\
&&=&\ \sum_{i=1}^K\mathbf{E}[D_{\cdot i,n}^j(D_{\cdot i,n}^j)^{'}|\mathcal{F}_{n-1}]\mathbf{P}(X_{i,n}^j=1|\mathcal{F}_{n-1})\\
&&=&\ \sum_{i=1}^K(C^j(i)+H^j(i))\tilde{Z}_{i,n}^j.
\end{aligned}\]
When
$n$ increases to infinity, from~\eqref{def:G} we obtain
$$\mathbf{E}[\Delta M_{n}^j(\Delta M_{n}^j)^{'}|\mathcal{F}_{n-1}]\ \stackrel{a.s.}{\longrightarrow}\
\sum_{i=1}^K(C^j(i)+H^j(i))\tilde{Z}_{i,\infty}^j\ -\
Z_{\infty}^j(Z_{\infty}^j)^{'}\ =\ G^j.$$
We recall that for any $j_1\neq j_2$, $D_n^{j_1} X_{n}^{j_1}$ and $D_n^{j_2} X_{n}^{j_2}$ are independent conditionally on $\mathcal{F}_{n-1}$.
As a consequence,
$\mathbf{E}[\Delta M_{n}^{j_1}(\Delta M_{n}^{j_2})^{'}|\mathcal{F}_{n-1}]=0$ and hence
$\mathbf{E}[\Delta\mathbf{M}^{l_1}_{n}(\Delta\mathbf{M}^{l_2}_{n})^{'}]=0$ for any $l_1\neq l_2$.\\
It remains to check that the remainder sequence $\{\mathbf{R}^l_n;n\geq1\}$ satisfies~(\ref{HypReste}) for any $\epsilon>0$, i.e.
\begin{equation}\label{eq:check_remainder_CLT}
\mathbf{E}\left[n\|\mathbf{R}^l_{n}\|^2\mathds{1}_{\{\left\|\mathbf{Z}^l_n-\mathbf{Z}^l_{\infty}\right\|\leq \epsilon\}}\right]\longrightarrow 0.
\end{equation}
Equation \eqref{eq:check_remainder_CLT} can be obtained by
combining~\eqref{eq:remainder_term_SA} and part (3b) in the proof of Theorem~\ref{thm:first_order_leading},
once we have observed that assumption (\constantbalanceD) in Theorem~\ref{thm:total_number} implies that
\begin{equation*}
\mathbf{E}\left[n\left\|n\cdot diag(\mathbf{T}^l_{n})^{-1}-\mathbf{I}\right\|^2\right]\longrightarrow0 .
\end{equation*}
Since the assumptions are all satisfied, we can apply Theorem~\ref{ThmCLT} to any leading system $S^l$, $l\in\mathcal{L}_L$,
so obtaining the CLT of Theorem~\ref{thm:second_order_leading}, with asymptotic variance
$$\Sigma_m^l\ :=\
\int_0^{\infty}e^{u(\frac{\mathbf{I}}{2}-\mathbf{F}^l_m)}\mathbf{G}^le^{u(\frac{\mathbf{I}}{2}-\mathbf{F}^l_m)'}du.$$
Finally, we need to fix $m>0$ to obtain the correct asymptotic variance $\Sigma^l$ for
$(\mathbf{Z}^l_n-\mathbf{Z}^l_{\infty})$ in $Span\{(x-y):x,y\in\mathcal{S}^{s^l K}\}$.
Since by construction the kernel of $\mathbf{U}'_i$, $i\in\{r^{l^-}+1,r^l\}$, is exactly $Span\{(x-y):x,y\in\mathcal{S}^{s^l K}\}$,
we impose that $\mathbf{U}'_i\Sigma_m^l\mathbf{U}_i=0$ so obtaining that
$\Sigma^l=\lim_{m\rightarrow\infty}\Sigma_m^l$.
This concludes the proof.
\endproof
\subsection{Proofs on the following systems}\label{subsection_proofs_follower}
\proof[Proof of Theorem~\ref{thm:first_order_follower}]
Consider the joint system $S^{(l)}=\cup_{i\in\{L_1,{\ldots}l\}}S^i$, for $l\in\mathcal{L}_F$,
composed by the leading systems $S^{L_1},{\ldots}S^{L_{n_L}}$ and the following systems $S^{F_1},{\ldots}S^{l}$, where we recall $S^{l}:=\{r^{l^-}+1\leq j\leq r^l\}$.
As explained in Section~\ref{section_follower}, we focus on the reduced process $\mathbf{\hat{Z}}_{n}^{(l)}:=\mathbf{\hat{C}}'\mathbf{Z}_{n}^{(l)}$,
whose dynamics is expressed in~\eqref{eq:dynamics_Y_SA_following_hat} as follows:
\begin{equation}\label{eq:dynamics_Y_SA_following_hat_proof_first}\begin{aligned}
\mathbf{\hat{Z}}^{(l)}_{n}-\mathbf{\hat{Z}}^{(l)}_{n-1}\ &&=&\ -\frac{1}{n}\hat{f}_m^{(l)}(\mathbf{\hat{Z}}^{(l)}_{n-1})\ +\
\frac{1}{n}\mathbf{\hat{C}}'\left(\Delta \mathbf{M}^{(l)}_n\ +\ \mathbf{R}^{(l)}_n\right),\\
\hat{f}_m^{(l)}(\mathbf{\hat{x}})\ &&:=&\ \left(\mathbf{I}-\mathbf{\hat{C}}'\mathbf{Q}^{(l)}\mathbf{\hat{B}}\right)\mathbf{\hat{x}}\ +\ m\mathbf{\hat{V}}_1\left(\mathbf{\hat{U}}_1'\mathbf{\hat{x}}-1\right)\ +\
m\mathbb{\hat{V}}_2\mathbb{\hat{U}}_2'\mathbf{\hat{x}},
\end{aligned}\end{equation}
where the function $f$ in the SA form~\eqref{SAP} is here represented by $\hat{f}_m^{(l)}$ that takes values in $Span\{\mathbf{V}_{IN}\}$.
Analogously to the proof of Theorem~\ref{thm:first_order_leading} for the leading systems, one can show that
all the assumptions of Theorem~\ref{ThmODE} are satisfied by the dynamics in~\eqref{eq:dynamics_Y_SA_following_hat_proof_first} and hence
the limiting values of $\mathbf{\hat{Z}}^{(l)}_{n}$ are represented by those $\mathbf{x}\in Span\{\mathbf{V}_{IN}\}$ such that $\hat{f}_m^{(l)}(\mathbf{x})=0$.
We use analogous decompositions of those in~\eqref{eq:decomposition_Q}
and in~\eqref{eq:decomposition_x_initial} for $\mathbf{\hat{C}}'\mathbf{Q}^{(l)}\mathbf{\hat{B}}$
and $\mathbf{x}\in Span\{\mathbf{V}_{IN}\}$ respectively, obtaining
\begin{equation}\label{eq:h_transformed_x_follower}
\hat{f}_m^l(\mathbf{x})\ =\ m\mathbf{\hat{V}}_1(a-1)\ +\ \mathbb{\hat{V}}_2((1+m)\mathbf{I}-\mathbf{\hat{J}}_2)b\ +\ \mathbb{\hat{V}}_3(\mathbf{I}-\mathbf{\hat{J}}_3)c,
\end{equation}
where $\mathbf{\hat{J}}_2:=\mathbf{\hat{C}}'\mathbf{J}_2\mathbf{\hat{B}}$ and $\mathbf{\hat{J}}_3:=\mathbf{\hat{C}}'\mathbf{J}_3\mathbf{\hat{B}}$.
By assumption (\irreducibleH), $H^j$ are irreducible.
Thus, $\lambda<1$ for all $\lambda\in \mathcal{A}_{IN}\setminus Sp(W^{(l)})$
and hence $(\mathbf{I}-\mathbf{\hat{J}}_3)$ is positive definite.
Therefore, since $m>0$, from~\eqref{eq:h_transformed_x_follower} we have that $\hat{f}_m^l(\mathbf{x})=0$ if and only if $a=1$ and $b=c=0$, i.e. $\mathbf{x}=\mathbf{\hat{V}}_1$.
By definition of $\mathbf{Q}^{(l)}$ (see \eqref{def:Q_complete}), we have that
$$
\mathbf{\hat{C}}'\mathbf{Q}^{(l)}\mathbf{\hat{B}} =
\begin{pmatrix}
\mathbf{C}'\mathbf{Q}^{(l^{-})}\mathbf{B}&0\\
\mathbf{Q}^{l(l^{-})}\mathbf{B}&\mathbf{Q}^{l}
\end{pmatrix},
$$
and hence we can express $\mathbf{\hat{V}}_1=(\mathbf{\hat{V}}^{(l^-)}_1,\mathbf{\hat{V}}^{l}_1)'$ as follows:
$$
\mathbf{\hat{V}}_1 =
\begin{pmatrix}
\mathbf{\hat{V}}^{(l^-)}_1\\
(\mathbf{I}-\mathbf{Q}^{l})^{-1}\mathbf{Q}^{l(l^-)}\mathbf{B}\mathbf{\hat{V}}^{(l^-)}_1
\end{pmatrix}
=
\begin{pmatrix}
\mathbf{C}'\mathbf{V}^{(l^-)}_1\\
(\mathbf{I}-\mathbf{Q}^{l})^{-1}\mathbf{Q}^{l(l^-)}\mathbf{B}\mathbf{C}'\mathbf{V}^{(l^-)}_1
\end{pmatrix}.
$$
Now, since $\mathbf{V}^{(l^-)}_1\in \mathcal{I}m(\mathbf{V}_{IN})$, we have
\[
\mathbf{B}\mathbf{C}'\mathbf{V}^{(l^-)}_1=\mathbf{V}_{IN}\mathbf{U}_{IN}'\mathbf{V}^{(l^-)}_1=\mathbf{V}^{(l^-)}_1.
\]
Finally, since from~\eqref{def:F_derivative_hat} $\hat{\mathbf{F}}_m^{(l)}=\mathbf{\hat{C}}'\mathbf{F}^{(l)}_m\mathbf{\hat{B}}$,
we have that $Sp(\hat{\mathbf{F}}^{(l)}_m)\subset Sp(\mathbf{F}^{(l)}_m)$
and hence $\hat{\mathbf{F}}_m^{(l)}$ is positive definite for any $m>0$.
As a consequence, $\mathbf{\hat{V}}_1$ is a global attractor in $Span\{\mathbf{V}_{IN}\}$ and
this concludes the proof.
\endproof
\begin{rem}
We highlight that, when (\irreducibleH) does not hold, the matrix $(\mathbf{I}-\mathbf{J}_3)$ in~\eqref{eq:h_transformed_x_follower}
may not be positive definite and hence the solution $\mathbf{\hat{V}}_1$ would not be unique.
However, since in the following systems $S^l$, $l\in\mathcal{L}_F$, we have $\lambda_{\max}(W^{l})<1$ and this implies $\lambda_{\max}(\mathbf{Q}^{l})<1$,
the irreducibility assumption of $H^j$ required in (\irreducibleH) is not necessary for the following systems,
but it is only essential in the leading systems in which $\lambda_{\max}(W^{l})=1$.
\end{rem}
\proof[Proof of Theorem~\ref{thm:second_order_follower}]
Consider the joint system $S^{(l)}=\cup_{i\in\{L_1,{\ldots}l\}}S^i$, for $l\in\mathcal{L}_F$,
composed by the leading systems $S^{L_1},{\ldots}S^{L_{n_L}}$ and the following systems $S^{F_1},{\ldots}S^{l}$, where we recall $S^{l}:=\{r^{l^-}+1\leq j\leq r^l\}$.
As explained in Section~\ref{section_follower}, we focus on the reduced process $\mathbf{\hat{Z}}_{n}^{(l)}:=\mathbf{\hat{C}}'\mathbf{Z}_{n}^{(l)}$,
whose dynamics is expressed in~\eqref{eq:dynamics_Y_SA_following_hat} as follows:
\[\begin{aligned}
\mathbf{\hat{Z}}^{(l)}_{n}-\mathbf{\hat{Z}}^{(l)}_{n-1}\ &&=&\ -\frac{1}{n}\hat{f}_m^{(l)}(\mathbf{\hat{Z}}^{(l)}_{n-1})\ +\
\frac{1}{n}\mathbf{\hat{C}}'\left(\Delta \mathbf{M}^{(l)}_n\ +\ \mathbf{R}^{(l)}_n\right),\\
\hat{f}_m^{(l)}(\mathbf{\hat{x}})\ &&:=&\ \left(\mathbf{I}-\mathbf{\hat{C}}'\mathbf{Q}^{(l)}\mathbf{\hat{B}}\right)\mathbf{\hat{x}}\ +\ m\mathbf{\hat{V}}_1\left(\mathbf{\hat{U}}_1'\mathbf{\hat{x}}-1\right)\ +\
m\mathbb{\hat{V}}_2\mathbb{\hat{U}}_2'\mathbf{\hat{x}},
\end{aligned}\]
where the function $f$ in the SA form in~\eqref{SAP} is here represented by $\hat{f}_m^{(l)}$.
The proof will be realized by showing that the assumptions of the Theorem~\ref{ThmCLT} in Appendix are satisfied by
the process $\mathbf{\hat{Z}}_{n}^{(l)}$, with $\theta^*$ replaced by the deterministic limiting proportion $\mathbf{\hat{Z}}^{(l)}_{\infty}$,
and $\mathcal{D}f(\theta^*)$ represented by $\hat{\mathbf{F}}_m^{(l)}$ defined in~\eqref{def:F_derivative_hat}.
To do this, we first show that condition $\{\Re e(Sp(Df(\theta^*)))>1/2\}$ in Theorem~\ref{ThmCLT} is equivalent to $\{\Re e(\lambda^{*l})<1/2\}$.
To this end, analogously to the proof of Theorem~\ref{thm:second_order_leading} for the leading systems,
note that
\begin{itemize}
\item[(1)] $\hat{\mathbf{F}}_m^{(l)}\mathbf{\hat{V}}_1=m\mathbf{\hat{V}}_1$,
\item[(2)] $\hat{\mathbf{F}}_m^{(l)}\mathbb{\hat{V}}_2=\mathbb{\hat{V}}_2((1+m)\mathbf{I}-\mathbf{J}_2)$,
\item[(3)] $\hat{\mathbf{F}}_m^{(l)}\mathbb{\hat{V}}_3=\mathbb{\hat{V}}_3(\mathbf{I}-\mathbf{J}_3)$.
\end{itemize}
Hence, the eigenvectors of $\hat{\mathbf{F}}_m^{(l)}$ and $\mathbf{\hat{C}}'\mathbf{Q}^{(l)}\mathbf{\hat{B}}$ are the same and then
\[\begin{aligned}
Sp(\hat{\mathbf{F}}_m^{(l)})\ &&=&\ \{m\}\ \cup\ \left\{(1+m)-\lambda,\lambda\in Sp(W^{(l)})\setminus(\{1\}\cup\mathcal{A}_{OUT})\right\}\\
&&\cup&\ \left\{1-\lambda,\lambda\in Sp(\mathbf{Q}^{(l)})\setminus (Sp(W^{(l)})\cup\mathcal{A}_{OUT})\right\},
\end{aligned}\]
which implies $\{\Re e(Sp(\mathcal{D}f(\theta^*)))>1/2\}\equiv\{\Re e(\lambda^{*l})<1/2\}$.
Then, by using analogous arguments of the proof of Theorem~\ref{thm:second_order_leading} for the leading systems,
it can be easily shown that
$$\mathbf{E}[\mathbf{\hat{C}}'\Delta\mathbf{M}^{(l)}_{n}(\Delta\mathbf{M}^{(l)}_{n})^{'}\mathbf{\hat{C}}|\mathcal{F}_{n-1}]\stackrel{a.s.}{\longrightarrow}
\mathbf{\hat{G}}^{(l)},\ \quad\
\mathbf{E}[\mathbf{\hat{C}}'\Delta\mathbf{M}^{(l_1)}_{n}(\Delta\mathbf{M}^{(l_2)}_{n})^{'}\mathbf{\hat{C}}]=0\ \ \ \forall l_1\neq l_2,$$
and for any $\epsilon>0$
\begin{equation*}
\mathbf{E}\left[n\|\mathbf{\hat{C}}'\mathbf{R}^{(l)}_{n}\|^2\mathds{1}_{\{\left\|\mathbf{\hat{Z}}^{(l)}_n-\mathbf{\hat{Z}}^{(l)}_{\infty}\right\|\leq \epsilon\}}\right]\longrightarrow 0.
\end{equation*}
We can then apply Theorem~\ref{ThmCLT} to obtain the CLT with asymptotic variance
$$\hat{\Sigma}^{(l)}\ :=\ \lim_{m\rightarrow\infty}\int_0^{\infty}e^{u(\frac{\mathbf{I}}{2}-\hat{\mathbf{F}}_m^{(l)})}\mathbf{\hat{G}}^{(l)}
e^{u(\frac{\mathbf{I}}{2}-\hat{\mathbf{F}}_m^{(l)})'}du.$$
This concludes the proof.
\endproof
\section*{Acknowledgement}
We thank the anonymous referees for their careful reading of our paper and their
insightful comments and suggestions.
|
1,116,691,500,840 | arxiv | \section{Introduction}
The integration of nonreciprocal photonic devices on Si or CMOS platforms has been challenging in the past decades.
The known optical nonreciprocity can be divided into two classes: forward-backward nonrecipricity (FBNR) and left-right nonrecipricity (LRNR). The first class of nonreciprocal component is well studied \cite{RPP67p717} and can be realized by various approaches using the magneto-optic effects \cite{fujita2000waveguide,haldane2008possible,zheng2009observation, OE20p18440,PRL107p173902,NaturePhotonOnchip}, nonlinearity \cite{Science335p447}, a modulation media \cite{NPhoton3p91,OptoAcousticIsolator1,OptoAcousticIsolator2}, optomechanics \cite{PRL102p213903,Optom2}, or magnetized plasmonic metal \cite{hadad2010magnetized,khanikaev2010one}. It can be used for optical isolators.
To date, the LRNR, indicating the nonreciprocal light flow between left and right ports of photonic circuits, has been discussed in an array of coupled waveguides by only a few research groups \cite{NPhys6p192,PRL103p093902,PRA82p043803}. If the complex optical potential causes the parity-time (PT) symmetry breaking, two coupled waveguides can show a LRNR light transfer in the transverse direction \cite{NPhys6p192,PRL103p093902}. The realization relies on the precise control of the active medium. Although not for optical isolators, the photonic circuit with the left-right nonrecipricity may switch or route the incident light beams.
The need for integration of optical nonreciprocal elements on a Si/CMOS chip platform is a long-standing problem. The realization of the nonreciprocal light propagation in a completely linear optical medium can strongly impact on both fundamental physics, and also vast applications for integrated optics because of the compatibility with the Si material and CMOS chips.
Again based on the PT symmetry breaking induced by a periodic modulation of complex optical potential, Feng et al. stated that they, for the first time, observed the nonreciprocal light propagation in a linear passive optical material \cite{Science333p729}. Unfortunately, they admitted their mistake \cite{Science333p38c} after Fan et al. commented on their work \cite{Science333p38b}. Another experimental realization of on-chip optical diodes using all-dielectric, passive, and linear silicon photonic crystal structures is reported by Wang et al. \cite{OE19p26948,SRep2p674}. They confidentially explained why it is possible to make optical diodes using a spatial symmetry breaking geometry in a passive and linear optical medium \cite{SRep2p674}.
In the LRNR, the input light is always localized in one waveguide \cite{NPhys6p192,PRL103p093902,PRA82p043803}. This is similar to the coherent population trapping in a three-level $\Lambda-$type atom \cite{CPT}. While the bending waveguide array can simulate well the quantum dynamics of atoms. The classical optical analogs of coherent population transfer \cite{ClassicCPTTheo1,ClassicCPTExp,LPR3p243} and population trapping in the continuum\cite{ClassicCPT} and Rabi oscillation \cite{ClassicRO} as well has been proved by Longhi's group. Although the trapping of equal light in two waveguides has been discussed \cite{ClassicCPTTheo1}, the results did not show a valid LRNR of light flow. A theory work indicated that the critical large nonlinearity is necessary to induce nonrecipricity \cite{PRA82p043803} in two evanescently coupled waveguides. However the LRNR in the transverse light flow has been recently observed \cite{NPhys6p192,PRL103p093902}. Moreover, the three/many-body systems behavior essentially different from the simple two-body system studied in \cite{PRA82p043803}. At least, the optical trapping in coupled three waveguides analogous to the atomic CPT can not be achieved in an optical system composing of two waveguides. It is interesting if one can realize the nonreciprocal wave propagation in evanescently coupled linear and passive waveguides. We expect to achieve the LRNR in a waveguide array. This is the motivation of our work.
Here we propose simple methods to generate the second class of optical nonreciprocity in an array of three coupled waveguides only making from linear, passive optical materials. We focus on the nonreciprocal transverse energy flow between left and right optical waveguides. This left-right nonreciprocity does not violate the Lorentz reciprocity theorem. Thanks to a small dispersion of a linear waveguide, our system can behave in a nonreciprocal manner in an ultrabroad band. The results by numerical simulation of beam propagating method (BPM) and solving the coupled mode equation (CME) demonstrate the breaking of symmetry of transverse energy flow.
\section{Setup and model}
Our system, shown in Fig.~\ref{fig:system}, is composed of three coupled waveguides embedded in a wafer of width $W$ and length $L$. The middle waveguide D$_3$ couples to the waveguides D$_1$ and D$_2$. We assume that the coupling between the waveguides D$_1$ and D$_2$ is negligible. We also assume that the two side waveguides are lossless but some loss can be included in the middle one. In our photonic system, eigenmodes in the individual waveguide exchange energy via their evanescent fields when two waveguides are close. The couplings are denoted as $\kappa_{13}$ and $\kappa_{23}$, and decrease as the distance $d_{13,23}$ between two waveguides increases. The coupling between D$_1$ and D$_2$ is assumed vanishing because these two waveguides are far enough from each other. The field in D$_3$ decays exponentially with a constant $\gamma$ that can be controlled \cite{TuneLoss1,TuneLoss2,NPhys6p192,Science333p729,PRL103p093902}.
\begin{figure}
\centering
\includegraphics[width=0.6\linewidth]{fig1.pdf}\\
\caption{Light trapping in a photonic circuit consisting of three waveguides embedded in a $W$ wide, $L$ long substrate. The coupling $\kappa_{13} \ll \kappa_{23}$. $d_{13,23}$ is the distance between two waveguides. The dielectric constants of substrate is $\varepsilon_s$, while $\varepsilon_{i}$ ($i\in \{1,2,3\}$) is the constant profile of corresponding individual waveguides D$_i$ without others. The widths of waveguides are $t_i$, respectively.}\label{fig:system}
\end{figure}
The evolution of power of field in photonic circuits can be studied either by numerical solving the Helmholtz equation or with the derived coupled mode theory. However the later presents a clearer physic understanding. We first present the approach based on Helmholz equation.
\subsection{Beam propagating method}
We consider our system to be two-dimensional (2D). This is reasonable if the size of waveguide in the $y$ direction is much larger than that in the $x$ direction. The propagation of the field $E$ in the photonic circuits in a 2D space can be described by the Helmholtz equation, which takes the form
\begin{equation} \label{eq:HelmholtzE}
\frac{\partial^2 E}{\partial z^2}+ \frac{\partial^2 E}{\partial x^2}+ k_0^2 \varepsilon (x,z)E=0\,,
\end{equation}
where $z$ is the propagating direction, $x$ is the transverse direction, and $k_0=\frac{2\pi}{\lambda}$ is the wave vector of field with wavelength $\lambda$ in the free space. The dielectric constant $\varepsilon(x,z)$ plays the role of the optical potential. In Eq.~\ref{eq:HelmholtzE}, we assume that $\frac{\partial E}{\partial t}\sim 0$ and it is reasonable for our interest in the steady-state behavior of the system. We can resolve the field $E$ into its slowly varying amplitude $\psi$ and a fast oscillating factor, $E=\psi e^{\pm j\beta z}$, where the propagation constant $\beta=k_0 n_{eff}$. $n_{eff}$ is the effective index of waveguides. The sign before $\beta$ indicates the propagating direction: minus (plus) for the propagation along the positive (negative) $z$ direction. In the paraxial approximation $\left|2j\beta \frac{\partial \psi}{\partial z} \right| \gg \left|\frac{\partial^2\psi}{\partial z^2} \right|$, the amplitude of the field in the photonic circuit evolves according to the equation \cite{CMT1}
\begin{equation} \label{eq:AE}
\mp 2j\beta\frac{\partial \psi}{\partial z} \approx \frac{\partial^2\psi}{\partial x^2}
+k_0^2 \left[\varepsilon (x,z)-n^2_{eff}\right]\psi \,.
\end{equation}
We numerically solve Eq.~(\ref{eq:AE}) with BPM to simulate the propagation of field with a spatial resolution $\delta z=1$~\micro \meter~ and $\delta x=0.1$~\micro \meter. A finer spatial grid gives the same results.
Throughout simulation, we use the zero-order eigenmode profile $\psi(x,x_0,z=0,L)$ of TE mode by solving the eigen equation \cite{CMT1}, where $x_0=x_0^{(i)}$ ($i\in\{1,2\}$) is the center position of input ports $P_{L1,L2,R1}$. This mode profile is very close to a Gaussian function $\exp\left(-\left(x-x_0 \right)^2/2 w_{\text{p}}^2\right)$ with a half width of waist $w_{\text{p}}=1$~\micro \meter~.
The profiles $\psi(x,x_0^{(1)}, 0)$ and $\psi(x,x_0^{(1)},L)$ corresponds to the input field launching into the waveguide D$_1$ from the port $P_{L1}$ at $z=0$ and $P_{R1}$ at $z=L$, respectively. While the profile $\psi(x,x_0^{(2)}, 0)$ means an input to the port $P_{L2}$ at $z=0$. The intensity of the field at the peak is unity. This profile is very close to the fundamental eigenmode of D$_1$ or D$_2$.
Our simulation focuses on a light with wavelength $\lambda_0=1.55$~\micro\meter~, which is of interest in optical communications. The photonic circuit can be integrated in a wafer with a substrate dielectric constant $\varepsilon_s=10.56\varepsilon_0$, where $\varepsilon_0$ is the permittivity of free space. We consider weakly guiding waveguides with $\varepsilon_{\text{core}}=10.76\varepsilon_0$ in order to ensure the valid of our BPM and CME method. The imaginary part of the dielectric constant in the middle waveguide D$_3$ is $\Im[\varepsilon_3]=-0.01\varepsilon_0$. This induces a loss of $\gamma=11.6$~\milli \meter$^{-1}$ according to our numerical simulation. The propagation constant is calculated by solving the eigenvalue equation \cite{CMT1} of TE mode.
In our numerical method, the Neumann boundary condition (NBC) is used to greatly suppress the reflection field from the transverse boundary. A small reflection, which can be a practical noise from the boundary of device in experiments, is responsible for the background noise of our numerical results.
\subsection{Coupled mode equation method}
Before discussing the results, we present the equivalent, but physically transparent, coupled mode equation approach to explain our system.
The light field $E$ in photonic circuits can be expressed as a supermode of eigenmodes $E_i$ of individual waveguide D$_i$, i.e.,
\begin{equation}
E = \sum_{i=1,2,3}A_iE_i= \sum_{i=1,2,3} A_i \psi_i e^{\pm j\beta_i z}\,,
\end{equation}
where the amplitude of eigenmode in the $i$th waveguide is denoted by $A_i$ ($i\in \{1, 2, 3\}$). Here $\psi_i$ is the slowly varying envelope of eigenmode $E_i$ and $\beta_i$ is the corresponding propagation constant, which can be controlled by designing the dielectric constant $\varepsilon_i$ of individual waveguide D$_i$ and its width $t_i$. This parameter also depends on the wavelength of light. According to the coupled mode theory \cite{CMT1,CMT2,CMT3} derived from the Helmholtz equation, the dynamics of the modal amplitudes is described by
\begin{subequations}\label{eq:CME}
\begin{align}
\frac{\partial A_1}{\partial z} &= j\Delta_{13} A_1+ j \kappa_{13}(z) A_3\\
\frac{\partial A_2}{\partial z} &= j\Delta_{23} A_1+j \kappa_{23}(z) A_3\\
\frac{\partial A_3}{\partial z} &= j \kappa_{13}(z) A_1 + j \kappa_{23}(z) A_2 -\gamma A_3\,,
\end{align}
\end{subequations}
where $\Delta_{13}=\beta_1-\beta_3$ and $\Delta_{23}=\beta_2-\beta_3$ are the phase mismatch between waveguides D$_1$ (D$_2)$ and D$_3$. The power in the $i$th waveguide is evaluated by $P_i=A_i A_i^*$. The total power in the system is $P=\sum_i P_i$. The phase mismatching and coupling in the CME Eq.~(\ref{eq:CME}) can be derived from Helmholtz equation Eq.~(\ref{eq:HelmholtzE}).
These parameters are dependent on $k_0$ and the optical potential $\varepsilon(x,z)$. The coupling is given by \cite{CMT1,CMT2,CMT3}
\begin{equation}\label{eq:coupling}
\kappa_{mn}=\frac{k_0^2\int_s \delta\varepsilon_n E_m^*E_n dS}{\beta_m\int_s E_m^*E_m dS} \,,
\end{equation}
where $s$ denotes the cross section of space and $\delta \varepsilon_n=\varepsilon(x,z)-\varepsilon_n$ at the cut position $z$.
For simplicity's sake, we have assumed weakly guiding waveguides and the relation $\kappa_{mn}=\kappa_{nm}$. We also neglect the second-order spatial derivatives of the amplitude $A_i$ and the small self phase shifts due to the perturbation of neighbor waveguides. A full study of the relation of parameter to the Helmholtz equation has been presented by Hardy et al. \cite{CMT2,CMT3}. Note that the coupled mode theory presents a general model. In contrast, the numerical results depend on the structure of system, and only provides one of many implementations. The different structures can lead to the same set of parameters in the CME. On the other hand, the numerical simulation presents a full picture of light in photonic circuits.
Next we turn to our idea about how to create optical nonreciprocity in the transversal energy flow in three coupled waveguides by giving a connection of our classic photonic circuit to a quantum system.
Due to the equivalence between the Helmholtz equation in photonic circuits and the Schr\"{o}dinger equation in quantum mechanics, the behavior of light propagating in a photonic circuit is similar to the dynamics of the internal atomic states of a quantum system \cite{ClassicCPTTheo1,ClassicCPTExp,LPR3p243}. For example, the normalized light power trapped in optical waveguides plays the role of atomic population. Optical nonreciprocities in an array of coupled waveguides can then be considered as the trapping of input light on demand. As is well known, in a $\Lambda-$type three-level atomic systems, we can adiabatically create a target state independent of the initial state of system via the so-called coherent population trapping (CPT) \cite{CPT}. Our system is analogous to such a $\Lambda$-type three-level system. Just as in CPT in the atomic system, we expect to trap most light energy in a selected optical waveguide by suitably controlling the coupling between the waveguides. This is the basis of the optical nonreciprocity studied in this paper.
Note that the LRNR we propose here is substantially different from the FBNR used for optical isolator on the basis of the breaking of Lorentz reciprocity theorem \cite{Haus, RPP67p717}. In the former case, both two sources input into ports in the left hand side but their responses come out from the right hand side, while the source and the response must exchange in the later. As a result, $\int {\bf J}_s^{(1)} \cdot {\bf E}_{R}^{(2)} d{\bf S} = \int {\bf J}_s^{(2)} \cdot {\bf E}_{R}^{(1)} d{\bf S} =0$ for our 2D case, where the response ${\bf E}_{R}^{(i)}$ at port $P_{R1}$ is the electric field created by a source ${\bf J}_s^{(j)}$ at port $P_{L1,L2}$ with $i\neq j$ and $i,j\in \{1,2\}$, because the source and response are separated in space, i.e. ${\bf J}_s^{(1)} \cdot {\bf E}_{R}^{(2)} = {\bf J}_s^{(2)} \cdot {\bf E}_{R}^{(1)} =0$. Thus the implementation of the LRNR in a linear, passive medium does not violate the Lorentz reciprocity theorem \cite{Haus, RPP67p717}, which requires the exchange of the place of source and response.
\subsection{Connection between two methods}
The structure of photonic circuit to create the nonreciprocal transverse energy flow of light is shown in Fig.~\ref{fig:coupling} (a). Through our system, we assume no loss in waveguide $D_1$ and $D_2$. The dielectric constant is $\varepsilon_{\text{s}}=10.56 \varepsilon_0$ in the substrate, while it is $\varepsilon_{\text{core}}=10.76 \varepsilon_0$ in core of $D_1$ and $D_2$. In waveguide $D_3$, the dielectric constant in core is $\varepsilon_3=(10.76-0.01 i)\varepsilon_0$.
To fit the numerical results, the coupling $\kappa_{13}$ and $\kappa_{23}$ are assumed to vary corresponding to the central positions $w_{1}(z),w_{2}(z)$ and $w_{3}(z)$ of waveguides. The other parameters for CME are given by:
\begin{align}
\gamma= &11.6~ \milli \meter^{-1} \,, \\
\Delta_{13}= &\Delta_{23}=-23 ~\milli \meter^{-1} \,.
\end{align}
The mismatch of propagating constant $\Delta_{13}= \Delta_{23}=-23$~\milli \meter$^{-1}$ is obtained by solving the eigenvalue of zero-order TE mode.
To simulate the varying gaps between waveguides, we assume two gradient changing coupling strength $\kappa_{13}$ and $\kappa_{23}$
in the propagating direction for Eq.~\ref{eq:CME} as shown in Fig.~\ref{fig:coupling} (b).
Two waveguides in the same chip always couples to each other even if the coupling strength is very small. To consider this coupling, we assume small values as the distance between two waveguides are large. The intensity in waveguides given by the CMEs change very slightly if we neglect this small coupling.
We note that, in the absence of loss in the waveguide $w_3$, the system is reciprocal (not shown here). However if we include loss in the middle waveguide, we create left-right nonreciprocities.
Let us assume that a light with unit amplitude is incident on the port $P_{L1}$ or $P_{L2}$ at $z=0$.
If the photonic circuit is reciprocal, the transmission from port $P_{R1}$ or $P_{R2}$ exchanges as well if the incident exchange.
However, in the case of left-right nonrecipricity, the light launched into port $P_{L1}$ and $P_{L2}$ always effectively transfer to the waveguide D$_1$ and comes out from port $P_{R1}$.
If we use constant couplings $\kappa_{13}$ and $\kappa_{23}$, the LRNR is obtained but the transmissions are small. The energy trapped in waveguide D$_1$ also decays because part of the energy couples to the middle waveguide from which the energy is lost into the environment at a rate $\gamma$. To avoid a strong coupling of energy between D$_1$ and D$_3$, we gradually change the distance between the two side waveguides and the middle one to guarantee an adiabatic process. In addition, large phase mismatchings $\Delta_{13}$ and $\Delta_{23}$ are used to suppress the energy coupling to waveguide D$_3$. In the output side, we decouple the waveguides D$_1$ and D$_3$ by introducing a large distance to keep the light energy in D$_1$ almost constant. The profile of the mode is also kept stable after $z/\lambda_0=1500$.
\begin{figure}
\begin{center}
\includegraphics[width=0.44\linewidth]{fig2-a.pdf}
\includegraphics[width=0.53\linewidth]{fig2-b.pdf}
\caption{(a) The waveguide structure for left-right nonrecipricity.
The straight waveguide D$_3$ is $4$~\micro\meter~ wide around its center $w_3(0)=26.7$~\micro\meter. The waveguides D$_1$ and D$_2$ with width $t_1=t_2=2$~\micro\meter~ are curves along their varying central positions $w_1(z)$ and $w_2(z)$ defined as $w_1[z][\micro\meter]=31$ for $z<1.1$~\milli\meter; $w_1[z][\micro\meter]=34$ for $z>1.6~\milli\meter$ and $w_1[z][\micro\meter]=31+3(1+\sin(2\pi(z-1350)/1000))/2$ for $1.1~ \milli\meter \leq z \leq 1.6~ \milli\meter$. $w_2[z]$ is constant $22~\micro\meter$ for $z < 0.1~\milli\meter$ and $23.4~\micro\meter$ for $z > 1.1~\milli\meter$. During the transition region, $w_2[z][\micro\meter]=22 + 1.4(1+\sin(2\pi(z-600)/2000))/2$ for $0.1~\milli\meter \leq z \leq 1.1 \milli\meter$.
(b) The coupling as a function of propagating distance $z$. Blue lines (i) for $\kappa_{13}$, red lines (ii) for $\kappa_{23}$. Solid lines for coupling rates are evaluated by Eq.~\ref{eq:coupling}, while dashed lines indicates coupling rates for fitting the numerical results. Detailedly, the coupling rates for fitting are
$\kappa_{13}(z)$[\milli\meter$^{-1}$]=$4$ for $z<1.1$~\milli\meter; $\kappa_{13}(z)$[\milli\meter$^{-1}$]=$0.03$ for $z>1.44$~\milli\meter~ and $\kappa_{13}(z)$[\milli\meter$^{-1}$]=$0.03+3.97(1-\sin(2\pi(z-1270)/680))/2.0$ for $1.1$~\milli\meter $\leq z \leq 1.44$~\milli\meter. While $\kappa_{23}(z)$ is $0.6$~\milli\meter$^{-1}$ for $z<0.24$~\milli\meter~ and $8.5$~\milli\meter$^{-1}$ for $z>1.1$~\milli\meter, and $0.6+ 7.9(1+\sin(2\pi(z-670)/1720))/2.0$ for $0.24$~\milli\meter $\leq z \leq 1.1$~\milli\meter. Here $\lambda_0=1.55~\micro\meter$. }\label{fig:coupling}
\end{center}
\end{figure}
The couplings used in the CME for fitting the following numerical results are dashed lines shown in Fig.~\ref{fig:coupling} (b). The solid lines are numerically evaluated by Eq.~\ref{eq:coupling}. These coupling are strongly dependent on the distance $d_{13,23}$. In spite the exact numerical solution of coupling rates $\kappa_{13}$ and $\kappa_{23}$ from Eq.~\ref{eq:coupling} is different from the numbers we use to fit the distribution of fields below, it provide us a good guide for the fitting function.
\section{Results}
Now we study the left-right nonreciprocity where we have nonreciprocal light transfer in the transverse direction \cite{NPhys6p192,PRL100p103904}. Similar to coherent population trapping in quantum optics, we can trap most light energy in the selected waveguide D$_1$ by designing a weak coupling $\kappa_{13}$ in comparison with $\kappa_{23}$. Our numerical results shown in Figs.~\ref{fig:NLRNon} (a) and (c) demonstrate a left-right nonreciprocal transverse energy flow. Whatever port $P_{L1}$ or $P_{L2}$ we choose to lauch the light into, most of the light is trapped in the waveguide D$_1$, and comes out from the same port $P_{R1}$ (blue lines). The transmission for light input into port $P_{L2}$ is about $25\%$ (Figs.~\ref{fig:NLRNon}(b)) but it increases to $40\%$ if the light is incident into port $P_{L1}$ (Figs.~\ref{fig:NLRNon}(d)). The light in the waveguide D$_2$ leaks to D$_3$ and subsequently is absorbed as it propagates. The contrast ratios of light intensities in waveguides D$_1$ and D$_2$ are higher than $29$~dB in both cases.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\linewidth]{fig3-a.pdf}
\includegraphics[width=0.45\linewidth]{fig3-b.pdf}\\
\includegraphics[width=0.45\linewidth]{fig3-c.pdf}
\includegraphics[width=0.45\linewidth]{fig3-d.pdf}\\
\caption{\label{fig:NLRNon} Left-right nonreciprocity corresponding to Fig.~\ref{fig:coupling} (a). The field propagates from left to right in photonic circuits. (a) Light incident into the waveguide D$_2$; (c) light enters the waveguide D$_1$; (b) and (d) Intensities of field at the middle of waveguides D$_1$ [blue lines (i)], D$_2$ [red lines (ii)] and D$_3$ [green lines (iii)]. Dashed thin lines are the corresponding plots by solving Eq.~(\ref{eq:CME}).}
\end{center}
\end{figure}
So far we presented results based on the numerical solution of the Helmholtz equation. Next we compare these results with the solution of the CME Eq.~(\ref{eq:CME}), see dashed lines in Figs.~ \ref{fig:NLRNon}(b) and~ \ref{fig:NLRNon}(d). Clearly, the coupled mode theory agrees well with the numerical simulation in detail. The intensity of light in waveguide D$_1$ in Figs. \ref{fig:NLRNon}(b) are slightly higher than the numerical results of BPM method. A small discrepancy is that the light in waveguide D$_2$ decays slower than the numerical results mainly because the loss in waveguide D$_2$ in the bending region is not included in the coupled mode theory Eq.~(\ref{eq:CME}). A full coupled mode theory \cite{CMT2,CMT3} involving higher modes and many parameters corresponding to the structure of system may present a better fitting of numerical results by the BPM. However, to provide a clearer physical understanding of the LRNR behavior we use the simple model, Eq. ~(\ref{eq:CME}). In spite of small discrepancy, the coupled mode theory still fits numerical results detailedly.
It is interesting to check whether the device displays the FBNR, a counterpart of LRNR, because the former is the basis for optical isolators.
According to the Lorentz reciprocity theorem \cite{Haus} and Fan et al. \cite{Science333p38b}, the FBNR is impossible in a linear, time-independent medium. Our numerical simulation agrees with it and demonstrates the forward-backward reciprocity, as shown in Fig.~\ref{fig:FBR}.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\linewidth]{fig4-a.pdf}
\includegraphics[width=0.45\linewidth]{fig4-b.pdf}\\
\caption{Numerical simulation of propagation of light in Fig.~\ref{fig:coupling}(a). The field incident into port P$_{R1}$ propagates from right to left. (a) Distribution of field (intensity), (b) Intensities of field at the middle of waveguides D$_1$ [blue lines (i)], D$_2$ [red lines (ii)],D$_3$ [green lines (iii)]. Dashed lines are the fitting plots evaluated by Eq.~(\ref{eq:CME}) using the same parameters as in Fig.~\ref{fig:NLRNon}.}\label{fig:FBR}
\end{center}
\end{figure}
To check the forward-backward reciprocity, we interchange the source and the response and solve Eqs. (\ref{eq:AE}) and (\ref{eq:CME}). The light is incident into the port P$_{R1}$ and the output from ports P$_{L1}$ and P$_{L2}$ are monitored. Again, the results by solving Eq. (\ref{eq:CME}) using the same parameter and coupling rates fit the numerical results by BPM well. As predicted by the Lorentz Reciprocity theorem, the transmission from P$_{R1}$ to P$_{L1}$ (P$_{L2}$) in the numerical simulations equals to those from P$_{L1}$ (P$_{L2}$) to P$_{R1}$. So the Lorentz Reciprocity theorem still rules the dynamics of our system.
The bandwidth in which the propagating light show nonreciprocal behavior is an important feature. We study the frequency dependence of the transmission spectra in Fig.~\ref{fig:NonrepFreq} by numerical simulations. We numerically calculate the transmissions using the same Gaussian profile for input in BPM method but solve the eigen equation \cite{CMT1} for the propagation constant for different wavelength. As the widths and refractive indices of the two side waveguides D$_1$ and D$_2$ are the same, the propagation constants $\beta_1$ and $\beta_2$ are equal. Thus the phase mismatch $\Delta_{13}$ is equal to $\Delta_{23}$ ideally. However each propagation constant itself and the coupling rates are dependent on the wavelength of input field. As a result, the transmission to P$_{R1}$ from P$_{L1}$ (P$_{L2}$) decreases (increases) gradually as the wavelength of incident light increases. Our system traps more than $24\%$ in waveguide D$_1$ from $1.56$~\micro \meter~ to $1.64$~\micro \meter. So it has an ultrabroadband nonreciprocal window over $80$~\nano\meter. In the nonreciprocal windows, the light in waveguide D$_2$ is always vanishing because it couples to the lossy channel D$_3$. Next we concentrate our discussion in the nonreciprocal window of interest. It can be clearly seen in Fig.~\ref{fig:NonrepFreq}, whatever waveguide the light is incident to, more than $24\%$ energy is trapped in D$_1$ and comes out of port $P_{R1}$.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\linewidth]{fig5.pdf}\\
\caption{\label{fig:NonrepFreq} Nonreciprocal transmission as a function of frequency of input light using structure as in Fig.~\ref{fig:coupling}(a). Thin blue lines (i) and (iii) show the light trapped in waveguide D$_1$, thick red lines (ii) and (iv) show the light energy in D$_2$. Solid lines for light launching to D$_2$, dashed lines for light input into D$_1$.}
\end{center}
\end{figure}
The frequency-dependence of transmission comes from the change of eigenmode profiles, propagation constants and their couplings $\kappa_{13}$ and $\kappa_{23}$, which are also dependent on the profiles of eigenmodes and wavelength [ref. to Eq.~(\ref{eq:coupling})]. The deviation in fabrication may result in a small difference between $\Delta_{13}$ and $\Delta_{23}$. However the transmission change slightly if $\Delta_{13} \approx \Delta_{23}$. A longer bending waveguide can tune the coupling between waveguides slower but is not necessary to provide a wider nonrecipricity window because it also changes the effective coupling length and the propagation constants are dependent on the wavelength as well.
For a practical application, the performance of device need be robust against small deviation in structure. Figure~\ref{fig:geom} shows how robust the nonreciprocal transfer of light is when the length, width of and gaps between waveguides change.
\begin{figure}
\begin{center}
\includegraphics[width=0.3\linewidth]{fig6-a.pdf}
\includegraphics[width=0.3\linewidth]{fig6-b.pdf}
\includegraphics[width=0.3\linewidth]{fig6-c.pdf}
\caption{\label{fig:geom} Nonreciprocal transmission in the structure as in Fig.~\ref{fig:coupling}(a) changes in the length (a), width (b) of, gap (c) between waveguide. (a) scale the device by $\eta_z$ in the z-direction; (b) scale the width of waveguides D$_1$ and D$_2$; (c) shift the center of waveguides by $g$ as $w_1[z] + g$ and $w_2[z] - g$. Thin blue lines (i) and (iii) show the light trapped in waveguide D$_1$, thick red lines (ii) and (iv) show the light energy in D$_2$. Solid lines for light launching to D$_2$, dashed lines for light input into D$_1$.}
\end{center}
\end{figure}
It can be seen from Fig.~\ref{fig:geom}(a) that the nonreciprocal performance varies slowly as the total length $\eta_z L$ of device changes. There is more than $24\%$ of input light is trapped in D$_1$ when the device scales in the z-direction from $\eta_z=1.0$ to $\eta_z=1.3$. As shown in Fig.~\ref{fig:geom}(b), the light trapped in D$_1$ oscillates as a function of the width $\eta_t t_1 =\eta_t t_2$ but it is stable for $1.0 \leqslant \eta_t \leqslant 1.02$, which means that the width of waveguides can vary $40$~\nano\meter. In contrast, the LRNR of our design is more sensitive to the distance between waveguides. The transmission is larger than $21\%$ if the shift/offset of waveguides $g$ is negative. It means a smaller distance between waveguides is preferable. While the nonreciprocity deteriorates rapidly as the distance increases.
The deviation of dielectric constant in fabrication changes the mistmatching of propagation constants and the coupling rate as well. First, we check the performance when the global dielectric constant $\varepsilon_{core}$ changes in all waveguides, as shown in Fig.~\ref{fig:DEC}(a). The dielectric constant $\varepsilon_{core}$ need be accurately engineered to pursue for a good LRNR behavior. Only the region $10.73 \leqslant\varepsilon_{core} / \varepsilon_0 \leqslant 10.77$ is useful to trap light in waveguide D$_1$. When $\varepsilon_{core} / \varepsilon_0$ changes from $10.68$ to $10.9$ corresponding to $\Delta n/n_{core} \sim 1\%$, the output from P$_{R1}$ is switched from ``on'' (``off'') to `` off'' (``on'') for input to P$_{L2}$ (P$_{L1}$). Then the output is investigated as the dielectric constant $\varepsilon_3$ of waveguide D$_3$ changes only. For $\Re[\varepsilon_3]/ \varepsilon_0<10.7$, no LRNR displays in our system. When $\Re[\varepsilon_3]$ is larger, the LRNR occurs and the light trapped in D$_1$ fluctuates as the refractive index increases. However, more than $25\%$ of light can be trapped in D$_1$ over the region of $10.76\leqslant \Re[\varepsilon_3]/ \varepsilon_0 \leqslant 10.78$. In contrast, the LRNR is very robust against the loss of waveguide D$_3$. The light in D$_2$ decreases rapidly as the loss increases. The light trapped in D$_1$ is stable for an input to P$_{L1}$ and decays exponentially for an input to P$_{L2}$. When $\Im[\varepsilon_3]/ \varepsilon_0 < -0.005$ corresponding to $\gamma \geqslant 60$~\centi\meter$^{-1}$, no light in D$_2$ and there is only light in D$_1$. In the range of $-0.01\leqslant \Im[\varepsilon_3] / \varepsilon_0\leqslant -0.005$, the system can trap more than $25\%$ of light in D$_1$.
Our simulations show a relative flexible parameters to the LRNR. The existing modern technology can fabricate the device in a much more accuracy. Therefore a linear, passive medium can display highly optical nonrecipricity in our design.
\begin{figure}
\begin{center}
\includegraphics[width=0.3\linewidth]{fig7-a.pdf}
\includegraphics[width=0.3\linewidth]{fig7-b.pdf}
\includegraphics[width=0.3\linewidth]{fig7-c.pdf}
\caption{\label{fig:DEC} Nonreciprocal transmission as the structure as in Fig.~\ref{fig:coupling}(a) changes of the global refractive index $\Re[\varepsilon_{core}]$ (a), refractive index $\Re[\varepsilon_3]$ (b) and loss $\Im[\varepsilon_{core}]$ of waveguide D$_3$. Thin blue lines (i) and (iii) show the light trapped in waveguide D$_1$, thick red lines (ii) and (iv) show the light energy in D$_2$. Solid lines for light launching to D$_2$, dashed lines for light input into D$_1$.}
\end{center}
\end{figure}
According to Eqs.~(\ref{eq:HelmholtzE}) and ~(\ref{eq:AE}), the structure of system is scalable in size to shift the frequency window of nonrecipricity, e.g. the LRNR around $\lambda_0=800$~\nano\meter. It can be seen from the coupled mode theory Eq.~(\ref{eq:CME}) that the LRNR occurs if all parameters are scaled in a similar structure according to the propagation constant $\beta$ for a different wavelength. In Fig.~\ref{fig:800nm}(a), we first scale the photonic circuit in the x-direction and then adjust the structure parameters and the width of input light for keeping the parameters mismatching and propagation constants close to those in Fig.~\ref{fig:system}(b). As a result, the distributions of field for the inputs into ports $P_{L1}$ and $P_{L2}$ are similar to Fig.~\ref{fig:NLRNon} (see Fig.~\ref{fig:800LRNR}). There is about $25\%$ of light trapped in the waveguide D$_1$, while the light from P$_{L2}$ is vanishing small. Then the intensities outcoming from ports $P_{R1}$ and $P_{R2}$ are scanned in wavelength between $700\sim900$~\nano\meter. It can been clearly seen from Fig.~\ref{fig:800nm}(b) that the second structure allow a high performance of LRNR over $40$~\nano\meter~ from $780$~\nano\meter to $820$~\nano\meter, allowing to control a ultrashort laser pulse with duration $1/\Delta \omega \sim 30$~\femto\second. Thus our scheme promises an ultrabroadband LRNR at difference wavelengths. A shorter wavelength means a larger loss in the bending region due to the stronger dipole radiation. To reduce this unwanted loss, we use a finer grid $\delta x=50$~\nano\meter~ and $\delta z=0.5$~\micro\meter, and adjust $w_p=375$~\nano\meter~ according to the eigen mode profile in our simulation.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\linewidth]{fig8-a.pdf}
\includegraphics[width=0.45\linewidth]{fig8-b.pdf}\\
\caption{\label{fig:800nm} (a) The waveguide structure for left-right nonreciprocity at $\lambda=800$~\nano\meter.
Similar to Fig.~\ref{fig:system}(a),
the straight waveguide D$_3$ is $1$~\micro\meter~ wide around its center $w_3(0)=26.7$~\micro\meter. The waveguides D$_1$ and D$_2$ with width $t_1=t_2=0.6$~\micro\meter~ are curves along their varying central positions $w_1(z)$ and $w_2(z)$ defined as $w_1[z][\micro\meter]=28.4$ for $z<1.2$~\milli\meter; $w_1[z][\micro\meter]=30.4$ for $z>3.1~\milli\meter$ and $w_1[z][\micro\meter]=28.4+2(1+\sin(2\pi(z-2150)/3800))/2$ for $1.2~ \milli\meter \leq z \leq 3.1~ \milli\meter$. $w_2[z]$ is constant $24~\micro\meter$ for $z < 0.2~\milli\meter$ and $23.4~\micro\meter$ for $z > 1.95~\milli\meter$. During the transition region, $w_2[z][\micro\meter]=24 + 1.4(1+\sin(2\pi(z-1075)/3500))/2$ for $0.2~\milli\meter \leq z \leq 1.95 \milli\meter$.
(b) Nonreciprocal transmission as a function of frequency of input light in (a). Thin blue lines (i) and (iii) show the light trapped in waveguide D$_1$, thick red lines (ii) and (iv) show the light energy in D$_2$. Solid lines for light launching to D$_2$, dashed lines for light input into D$_1$. The Gaussian profile of input light is adjusted to be $w_p=0.375$~\nano\meter wide.}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.45\linewidth]{fig9-a.pdf}
\includegraphics[width=0.45\linewidth]{fig9-b.pdf}\\
\includegraphics[width=0.45\linewidth]{fig9-c.pdf}
\includegraphics[width=0.45\linewidth]{fig9-d.pdf}\\
\caption{\label{fig:800LRNR} Left-right nonreciprocity corresponding to Fig.~\ref{fig:800nm}(a). The field propagates from left to right in photonic circuits. (a) Light incident into the waveguide D$_2$; (c) light enters the waveguide D$_1$; (b) and (d) Intensities of field at the middle of waveguides D$_1$ [blue lines (i)], D$_2$ [red lines (ii)] and D$_3$ [green lines (iii)].}
\end{center}
\end{figure}
\section{Discussion on two types of nonreciprocity}
Two different types of nonrecipricity, FBNR and LRNR, have been discovered in optical systems\cite{fujita2000waveguide,haldane2008possible,zheng2009observation, OE20p18440,PRL107p173902,NaturePhotonOnchip,Science335p447,NPhoton3p91,OptoAcousticIsolator1,OptoAcousticIsolator2,PRL102p213903,Optom2,hadad2010magnetized,khanikaev2010one,NPhys6p192,PRL103p093902,PRA82p043803}. The FBNR requires the breaking of the Lorentz reciprocity theorem and means that the forward and backward transmissions are not equal when the sources and responses interchange \cite{Science333p38b}. It is the basis of optical isolators. If the forward transmission $T_f$ is much larger than the backward transmission $T_{b}$, then the device with the FBNR can allow the forward propagating light to go through but block the back scattering light. Whereas the LRNR in our photonic circuit means that the light launching into different waveguides in the left hand side comes out from the same port from right. It does not violate the Lorentz reciprocity theorem \cite{Haus,RPP67p717} and as a result can not be used to isolate a light scattered backward.
In spite of the absence of ability for optical isolator, our scheme can dynamically route the light into difference paths.
Our device also provide a novel method to switch on/off the light via dynamically tuning the loss of waveguide, shown in Fig.~\ref{fig:DEC}(c). For $\Im[\varepsilon_3]/\varepsilon_0 <-0.05$ ($\gamma \geqslant 450$ \centi\meter$^{-1}$), the light input to P$_{L2}$ can be switched off, while it can effectively transfer to the output port P$_{R1}$ for $-0.005<\Im[\varepsilon_3]/\varepsilon_0 <-0.01$. More importantly, the loss of waveguide can be tune faster ($<1$~\pico\second) and more efficiently \cite{TuneLoss1,TuneLoss2} than the refractive index modulation \cite{Nature431p1081}. Moreover, the intensity of light outcoming from port P$_{R2}$ can be increased essentially using a shorter output length.
An alternative method to route light is to dynamically tune the refractive index of waveguides.
Although our nonreciprocal photonic circuit is discussed in a linear optical medium, we also can realize the setup in a nonlinear medium like fused silica \cite{router3D} or silicon \cite{Nature431p1081} using ultrashort laser pulses. Dynamically tuning the refractive index of waveguides, see Figs.~\ref{fig:DEC}(a) and (b), and subsequently the coupling strength and phase mismatch using ultrashort laser pulses \cite{router3D}, one can switching on/off the light outcoming from port P$_{R1}$ (Fig.~\ref{fig:DEC}(a)) or route the light incident to P$_{L2}$ into port P$_{R1}$ for $10.74 \leqslant\Re[\varepsilon_3]/\varepsilon_0 \leqslant 10.78$ or port P$_{R2}$ for $\Re[\varepsilon_3]/\varepsilon_0 \leqslant 10.6$ (Fig.~\ref{fig:DEC}(b)). The dynamical tuning range is $\Delta n <1\%$, which can be obtained using the existing technology \cite{router3D,Nature431p1081,PhysRep463p1}. Therefore, our nonreciprocal design can be a ultrafast, broadband optical router.
\section{Conclusion}
In conclusion, using a technique analogous to the coherent population trapping in quantum optics, we broke the symmetry of transverse light propagation in the photonic circuits of three coupled waveguides. Our proposed system is made only from linear, passive materials. Our simulations indicate the possibility of asymmetric transverse energy flow in an ultrabroadband window spanning over $80$~nm in frequency. Although our proposed system has a relatively large insertion loss, it opens a door to the possibility of highly efficient optical nonreciprocity in a linear, passive medium.
\section*{Acknowledgments}
This research is supported by a grant from the King Abdulaziz City for Science and Technology (KACST). One of us (MSZ) is grateful for the NPRP grant 5-102-1-071 from the Qatar National Research Fund (QNRF). KX also gratefully acknowledge the hospitality at ARC Center for Engineered Quantum Systems and Department of Physics and Astronomy, Macquarie University.
\end{document}
|
1,116,691,500,841 | arxiv | \section{Introduction}
Object detection, as a key component in a wide spectrum of vision-based intelligent applications~\cite{dalal2005histograms,felzenszwalb2008discriminatively}, aims to simultaneously locate and classify objects in images. Thanks to the strong ability of Convolutional Neural Networks (CNN)~\cite{krizhevsky2012imagenet}, numerous CNN-based object detection models have been recently proposed, which can be roughly divided into two categories, say one-stage and two-stage object detectors, according to their detection processes. Specifically, two-stage detectors first select possible object regions (proposals), and then classify these regions. The series of Region-based CNN (R-CNN)~\cite{girshick2014rich,girshick2015fast,ren2015faster} is the pioneer of two-stage object detectors with a variety of follow-ups~\cite{he2016deep,lin2017feature,dai2016r,cai2018cascade,he2017mask,liu2018path}, remarkably boosting the accuracy of detection. Given region-level features, these detectors for still images can be easily transferred to more complicated tasks such as segmentation and video object detection. However, due to the two-stage nature, the efficiency is a bottleneck for practical use, while for one-stage object detectors, the location and classification are jointly and directly produced by the dense prediction from feature maps. The YOLO family~\cite{redmon2016you,redmon2017yolo9000,bochkovskiy2020yolov4} and SSD~\cite{liu2016ssd} are representatives in this group. Without directly involving region proposals, compared to the aforementioned two-stage approaches, the speed of one-stage detectors is superior and suitable for scenarios with real-time requirements. Though the accuracy of one-stage detectors are typically inferior at the beginning, following designs~\cite{lin2017focal,ge2021yolox,ge2021ota,tian2019fcos} largely alleviate the accuracy gap.
Video object detection can be viewed as an advanced version of still image object detection. Intuitively, one can process video sequences via feeding frames one-by-one into still image object detectors. But, by this means, the temporal information across frames will be wasted, which could be the key to eliminating/reducing the ambiguity happened in a single image. As shown in Fig.~\ref{fig:cat_case}, degradation such as motion blur, camera defocus, and occlusion often appears in video frames, significantly increasing the difficulty of detection. For instance, via solely looking at the last frame in Fig.~\ref{fig:cat_case}, it is hard or even impossible for human beings to tell where and what the object is. On the other hand, video sequences can provide richer information than single still images. In other words, other frames in the same sequence can possibly support the prediction for a certain frame. Hence, \emph{how to effectively aggregate temporal messages from different frames is crucial to the accuracy}. As can be seen from Fig.~\ref{fig:cat_case}, our proposed method gives correct answers.
In the literature, there are two main types of frame aggregation, \emph{i.e.}, box-level and feature-level. These two technical routes can boost detection accuracy from different perspectives. Regarding box-level methods, they connect predictions by still object detectors through linking bounding boxes to form tubelet, and then refine the results in the same tubelet. The box-level methods can be viewed as post-processing, which are flexible to be applied to both one-stage and two-stage detectors. While for feature-level schemes, the features of a keyframe are enhanced by finding and aggregating similar features from other frames (\emph{a.k.a.}, reference frames). The two-stage manner endows proposals with explicit representation from the backbone feature map extracted by Region Proposal Network (RPN)~\cite{ren2015faster}. Benefiting from this nature, two-stage detectors can be easily migrated to the video object detection problem. Hence, most of video object detectors are built on two-stage detectors. However, because of the introduction of seeking the relationship between proposals, these two-stage video object detectors are further decelerated, and thus hardly meet the need of real-time scenarios.
Different from the two-stage bases, proposals are implicitly represented by elements of feature maps from a one-stage detector. Though, without explicit representations for objects, these elements of feature maps can still benefit from aggregating temporal information for the VID task. Again, as previously mentioned, the one-stage strategies usually execute faster than those two-stage ones. Driven by these considerations, a natural question arises: \emph{Can we make such region-level designs available to one-stage detectors that merely contain pixel-level features for building a practical (accurate and fast) video object detector}?
\textbf{Contribution.} This paper answers the above question via designing a simple yet effective strategy to aggregate features generated by one-stage detectors\footnote{Please note that our design is general and suitable to many detectors with different backbones.} (\emph{e.g.}, YOLOX~\cite{ge2021yolox} in this work to verify the primary claims). To connect the features of reference frames with those of the keyframe, we propose a feature similarity measurement module to construct an affinity matrix, which is then employed to guide the aggregation. To further mitigate the limitation of cosine similarity, an average pooling operator on reference features is customized. These two operations cost limited computational resources with significant gains in accuracy. Equipped with the proposed strategies, our model, termed as YOLOV, can achieve a promising accuracy 85.5$\%$ AP50 on the ImageNet VID dataset with 40+ FPS on a single 2080Ti GPU (please see Fig.~\ref{fig:performance} for details) without bells and whistles, which is attractive for practical scenarios. By further introducing post-processing, its accuracy reaches higher up to 87.5$\%$ AP50 at over 30 FPS.
\begin{figure*}[t]
\centering
\includegraphics[width=1\linewidth]{new_frame.pdf}
\caption{Framework of our design. Taking the YOLOX as an example base detector, the corresponding model is termed as YOLOV. We randomly sample a number of frames from a video and feed them into the base detector to extract features. According to the prediction of YOLOX, the Feature Selection Module (FSM) picks out top $k$ confident proposals and applies the NMS on the selected proposals for further refinement. All the features from the FSM are input into our Feature Aggregation Module (FAM) to produce information for final classification. The proposed strategy can be easily applied to other bases.}
\label{fig:yolov}
\end{figure*}
\section{Related Work}
This section will briefly review representative approaches in object detection closely related to this work.
\textbf{Object Detection in Still Images.} Thanks to the development of hardware, large-scale datasets~\cite{lin2014microsoft,krizhevsky2012imagenet} and sophisticated network structures~\cite{simonyan2014very,he2016deep,xie2017aggregated,wang2020cspnet}, the performance of object detection has continuously improved. Existing object detectors can be mainly divided into two-stage and one-stage schemes. Representative two-stage detectors such as RCNN~\cite{girshick2014rich}, Faster RCNN~\cite{ren2015faster}, R-FCN~\cite{dai2016r}, and Mask RCNN~\cite{he2017mask}. The methods in this group first select candidate regions through RPN and then extract features for the candidates through some feature extraction modules like RoIPooling~\cite{ren2015faster} and RoIAlign~\cite{he2017mask}. Finally, they complete the bounding box regression and classification through the detection head. There are also many successful one-stage detectors such as the YOLO series~\cite{redmon2016you,redmon2017yolo9000,bochkovskiy2020yolov4,ge2021yolox}, SSD~\cite{liu2016ssd}, RetinaNet~\cite{lin2017focal}, FCOS~\cite{tian2019fcos}. Different from the two-stage ones, the one-stage detectors perform dense prediction on feature maps and directly give the position and class probability without the step of region proposal.
One-stage detectors are usually faster than two-stage ones, owing to the end-to-end manner. However, they lack explicit region-level semantic features that are widely used for feature aggregation in video object detection. Our work attempts to explore the feasibility of aggregation over pixel-level features for one-stage detectors.
\textbf{Object Detection in Videos.} Compared to still image object detection, degradation may frequently appear in some video frames. When the keyframe is polluted, temporal information could be used for better detection. One branch of existing video object detectors concentrate on video-level post-processing \cite{han2016seq,belhassen2019improving,sabater2020robust}. The methods in this group try to refine the prediction results from the still image detector in consecutive frames by forming the object tubelets. The final classification will be uniformly adjusted to each tubelet in hand-crafted fashions. Another branch aims to enhance the features of keyframes, expecting to alleviate degradation via utilizing the features from (selected) reference frames. These approaches can be roughly classified as optical flow-based~\cite{zhu2017flow,zhu2018towards}, attention-based~\cite{wu2019sequence,deng2019relation,chen2020memory,gong2021temporal,sun2021mamba} and tracking-based~\cite{feichtenhofer2017detect,zhang2018integrated} methods. Deep feature flow~\cite{zhu2017deep} first introduces optimal flow for image-level feature alignment, and FGFA~\cite{zhu2017flow} adopts the optical flow to aggregate features along motion paths. Considering the computation cost of image-level feature aggregation, several attention-based methods have been developed. As a representative, SESLA~\cite{wu2019sequence} proposes a long-range feature aggregation scheme according to the semantic similarity between region-level features. Inspired by the relation module from Relation Networks~\cite{hu2018relation} for still image detection, RDN~\cite{deng2019relation} captures the relationship between objects in both spatial and temporal contexts. Furthermore, MEGA~\cite{chen2020memory} designs a memory enhanced global-local aggregation module for better modeling objects relationship. Alternatively, TROIA~\cite{gong2021temporal} executes the ROI alignment operation for fine-grained feature aggregation, while HVR-Net~\cite{han2020mining} integrates intra-video and inter-video proposal relations for further improvement. Moreover, MBMBA~\cite{sun2021mamba} enlarges the reference feature set by introducing memory bank. QueryProp~\cite{he2022queryprop} notices the high computational cost of video detectors and tries to speed up the process through a lightweight module. Besides the attention based methods, D\&T~\cite{feichtenhofer2017detect} tries solving video object detection in a tracking manner by constructing correlation maps of different frame features. Although these mentioned approaches boost the precision of detection, they are mostly based on two-stage detectors and hence suffer from the relatively slow inference speed.
\section{Methodology}
Considering the characteristics of videos (various degradation \emph{vs.} rich temporal information), instead of individually processing frames, how to seek supportive information from other frames for a target one (keyframe) plays a key role in boosting the accuracy of video detection. Recent attempts~\cite{deng2019relation,chen2020memory,wu2019sequence,he2022queryprop} with noticeable improvement in accuracy corroborate the importance of temporal aggregation to the problem. However, most of the existing methods are two-stage based techniques.
As previously discussed, their main drawback is relatively slow inference speed, compared to one-stage bases. To mitigate this limitation, we put the region/feature selection after the prediction head of one-stage detectors. In this section, we choose the YOLOX as base to present our main claims. Our proposed framework is schematically depicted in Fig.~\ref{fig:yolov}. Let us recall the traditional two-stage pipeline: 1) massive candidate regions are first ``selected" as proposals; and 2) determine if each proposal is an object or not and which class it belongs to. The computational bottleneck mainly comes from dealing with substantial low-confidence region candidates. As can be seen from Fig.~\ref{fig:yolov}, our pipeline also contains two stages. Differently, \emph{its first stage is prediction (having a large number of regions with low confidences discarded), while the second stage can be viewed as region-level refinement (taking advantage of other frames by aggregation).} By this principle, our design can simultaneously benefit from the efficiency of one-stage detectors and the accuracy gained from temporal aggregation. It is worth to emphasize that \emph{such a small difference in design leads to a huge difference in performance.} The proposed strategy can be generalized to many base detectors such as YOLOX~\cite{ge2021yolox}, FCOS~\cite{tian2019fcos} and PPYOLOE~\cite{xu2022pp}.
\subsection{Our Design}
From a possible perspective of human beings, the recognition procedure would first link \emph{related} instances in temporal and identify which class they belong to until sufficiently \emph{confident} messages are collected. Then, the determination can be broadcast to less confident cases. The mechanism of multi-head attention, as a key part of Transformers~\cite{vaswani2017attention}, seems to fit the situation well, which lifts the capability of long-range modeling. Given a sequence $Z$, the query, key, and value matrices are packed as $Q, K$ and $V$, respectively. The self-attention can be calculated via:
\begin{equation}
\operatorname{SA}(Z) = \operatorname{softmax}\left(A\right)V\quad \text{with} \quad A =QK^{T}/{\sqrt{d}},
\end{equation}
where $d$ is the dimension of each feature in Q (also K).
Executing $m$ self-attentions in parallel yields the multi-head attention by simply concatenating them together as follows:
\begin{equation}
\operatorname{MSA}(Z)=\operatorname{concat}\big(\operatorname{SA}_1(Z),\operatorname{SA}_2(Z),...,\operatorname{SA}_m(Z)\big).
\end{equation}
Modern two-stage based video object detectors typically obtain candidate regions for feature aggregation by, for example, RPN~\cite{ren2015faster}. As a representative, RelationNet~\cite{hu2018relation} first introduces the above multi-head attention to the still object detection task by viewing a sequence of proposals as input. The ROI pooling and/or alignment are applied to these proposals to extract region-level features. However, one-stage detectors make dense prediction directly from feature maps. Simply transferring the region-level feature aggregation to the whole feature maps of one-stage detectors will result in intensive computational cost. To address this issue, we propose an effective strategy for selecting features that suit the multi-head attention.
\subsubsection{FSM: Feature Selection Module}
As most of the predictions are of low confidence, the detection head of one-stage detectors is a natural and rational option to select (high-quality) candidates from the feature maps. Following the process of RPN, we first pick out top $k$ (\emph{e.g.}, 750) predictions according to the confidence scores. Then, a fixed quantity $a$ of predictions (\emph{e.g.}, $a=30$) are chosen by the Non-Maximum Suppression (NMS) for reducing the redundancy. To obtain features for video object classification, the accuracy of the base detector should be largely guaranteed. In practice, we found that directly aggregating the selected features in the classification branch and backpropagating the classification loss of the aggregated features will result in unstable training. To address the above concerns, we insert two $3 \times 3$ convolutional (Conv) layers into the model neck as a new branch, called video object classification branch, which generates features for aggregation. Then, we feed the features from both the video classification and regression branches with respect to the locations into our feature aggregation module.
\subsubsection{FAM: Feature Aggregation Module}
Now we come to the step of connecting related RoIs.
For a certain (key) frame, let $\mathcal{F} = \left\{{C}_{1}, {C}_{2}, ..., {C}_{f};{R}_{1}, {R}_{2}, ..., {R}_{f}\right\}$ denote the feature set selected by FSM. In addition, $C_i\in\mathbb{R}^{d_q\times a} = \left[\mathbf{c}_{1}^{i}, \mathbf{c}_{2}^{i}, ..., \mathbf{c}_{a}^{i}\right]$ and $R_i\in\mathbb{R}^{d_q\times a} = \left[\mathbf{r}_{1}^{i}, \mathbf{r}_{2}^{i}, ..., \mathbf{r}_{a}^{i}\right]$ designate the features of the $i$-th frame in $\mathcal{F}$ from the video classification and regression branches, respectively. The generalized cosine similarity is arguably the most widely used metric to compute the similarity between features or the attention weight~\cite{wu2019sequence,shvets2019leveraging,deng2019relation}. Merely referring to the cosine similarity will find features most similar to the target. However, when the keyframe suffers from some degradation, the selected proposals corresponding to those similar features very likely have the same problem. We name this phenomenon \emph{the homogeneity issue}.
To overcome this issue, we further take prediction confidences from the base into consideration, denoted as $\mathcal{P} = \left\{{P}_{1}, {P}_{2}, ..., {P}_{f}\right\}$ with $P_i = \left[\mathbf{p}_{1}^{i}, \mathbf{p}_{2}^{i}, ..., \mathbf{p}_{a}^{i}\right]$. In this work, each column of $P_i$ only contains 2 scores, \emph{i.e.}, the classification score and IoU score respectively from the classification and regression heads. Then, similarly to~\cite{vaswani2017attention}, the query, key, and value matrices are constructed and fed into the multi-head attention.
For instance, $Q_c$ and $Q_r$ are respectively formed by combining the features from the classification branch and the regression branch along the temporal dimension (\emph{i.e.}, $Q_c\in\mathbb{R}^{fa\times d}=\operatorname{LP}([{C}_{1}, {C}_{2}, ..., {C}_{f}]^T)$ and $Q_r\in\mathbb{R}^{fa\times d}=\operatorname{LP}([{R}_{1}, {R}_{2}, ..., {R}_{f}]^T)$, where $\operatorname{LP}(\cdot)$ is the linear projection operator), while the others are done analogously.
By the scaled dot-product in attention, we obtain the corresponding $A_c = Q_c K_c^{T} /\sqrt{d} $ and $A_r = Q_r K_r^{T} /\sqrt{d}$.
Gathering all the scores in $\mathcal{P}$ gives a matrix $[P_1,P_2,...,P_f]$ of size $2\times fa$. To make these scores fit into the attention weights, we build two matrices, say $S_r\in\mathbb{R}^{fa\times fa}$ and $S_c\in\mathbb{R}^{fa\times fa}$, through respectively repeating each of two rows $fa$ times. As a consequence, the self-attentions for the classification and regression branches turn out to be:
\begin{equation}
\begin{array}{c}
\operatorname{SA_c}(\mathcal{C})=\operatorname{softmax}\left({S_c \circ A_c }\right) V_c,\\
\operatorname{SA_r}(\mathcal{R})=\operatorname{softmax}\left({S_r \circ A_r }\right) V_c,
\label{eq:msa}
\end{array}
\end{equation}
where $\circ$ represents the Hadamard product. To be clearer, let $s^{ij}$ stand for the $(i,j)$-th element in $S$, while $Q^{i}$ and $K^{j}$ represent the $i$-th row of $Q$ and the $j$-th row of $K$, respectively. Each coefficient in the pre-softmax weight is $s^{ij}\cdot Q^i(K^{j})^T=Q^i(s^{ij}\cdot K^{j})^T$. This is to say, the self-attention considers not only the similarity between the query and key items, but also the quality of the key. Note that, since the main purpose is to refine the classification, $\operatorname{SA_c}(\mathcal{C})$ and $\operatorname{SA_r}(\mathcal{R})$ share the same value matrix $V_c$.
Our experiments demonstrate that replacing the original QK manner with Eq. \eqref{eq:msa} (called affinity manner) in multi-head attention can significantly boost the performance of video object detector. Besides, we concatenate $V_c$ with the outputs of Eq. \eqref{eq:msa} for better preserving initial representations via:
\begin{equation}
\begin{array}{c}
\operatorname{SA(\mathcal{F})}= \operatorname{concat}((\operatorname{SA_c}(\mathcal{C})+\operatorname{SA_r}(\mathcal{R}),V_c).
\end{array}
\end{equation}
Different from the traditional ViT, the positional information is not embedded, because the locations in a long temporal range would not be helpful as claimed in~\cite{chen2020memory}.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{new_ap.pdf}
\caption{Our average pooling over reference features.}
\label{fig:ave pooling fig}
\end{figure}
Moreover, considering the characteristic of softmax, may a small part of reference features hold a large portion of weights. In other words, it often overlooks the features with low weights, which limits the diversity of reference features for possible follow-up use. To avoid such risks, we introduce an \textbf{average pooling over reference features (A.P.)}. Concretely, we select all of the references with similarity scores above a threshold $\tau$ and apply the average pooling to these survivals. Note that the similarity in this work is computed via $\mathcal{N}(V_c) \mathcal{N}(V_c)^{T}$. The operator $\mathcal{N}(\cdot)$ means the layer normalization, which guarantees the values to be in a certain range and thus eliminates the influence of scale difference.
By doing so, we can maintain more information from related features. The average-pooled features and the key features are then transported into one linear projection layer for final classification. The procedure is illustrated in Fig.~\ref{fig:ave pooling fig}. One may ask that if $\mathcal{N}(Q_c) \mathcal{N}(K_c)^{T}$ or $\mathcal{N}(Q_r) \mathcal{N}(K_r)^{T}$ can perform as the similarity. Indeed, it is another option. But, in practice, it is not as stable as our choice during training due to the discrepancy between $Q$ and $K$.
\section{Experimental Validation}
\subsection{Implementation Details}
Similar to previous works, we also initialize our base detector from COCO pre-trained weights provided by YOLOX. Following~\cite{zhu2017flow,wu2019sequence,gong2021temporal}, we combine the videos in the ImageNet VID and those in the ImageNet DET with the same classes as our training data. Specifically, the ImageNet VID ~\cite{russakovsky2015imagenet} contains 3,862 videos for training and 555 videos for validation. There are 30 categories in the VID dataset, \emph{i.e.}, a subset of the 200 basic-level categories of ImageNet DET~\cite{russakovsky2015imagenet}. Considering the redundancy of video frames, we randomly sample 1/10 frames in the VID training set instead of using all of them. The base detectors are trained for 7 epochs by SGD with batch size of 16 on 2 GPUs. As for the learning rate, we adopt the cosine learning rate schedule used in YOLOX with one epoch for warming up and disable the strong data augmentation for the last 2 epochs. When integrating the feature aggregation module into the base detectors, we fine-tune them for 150K iterations with batch size of 16 on a single 2080Ti GPU. In addition, we use warm up with the first 15K iterations and cosine learning rate schedule for the rest iterations. Only the linear projection layers in YOLOX prediction head, the newly added video object classification branch and the multi-head attention are fine-tuned for the sake of largely excluding the influence from other factors. It is positive that further improvements could be obtained when more layers in the model participate in the BP procedure. For training the feature aggregation module, the number of frames $f$ is set to 16, and the threshold of NMS is set to $0.75$ for rough feature selection. While for producing final detection boxes, we alternatively set the threshold of NMS to $0.5$ for catching more confident candidates. In the training phase, the images are randomly resized from $352\times 352$ to $672\times 672$ with 32 strides. In the testing phase, the images are uniformly resized to $576\times576$. The AP50 and inference speed are two metrics to reflect the performance in terms of accuracy and efficiency, respectively. Regarding the inference speed, we test all of the models with FP16-precision on a 2080Ti GPU unless otherwise stated.
\subsection{Ablation Study}
\subsubsection{On the reference frame sampling strategy.} Investigating frame sampling strategies to balance accuracy and efficiency is crucial for video object detectors. Several global and local sampling schemes have been discussed in previous two-stage based methods~\cite{wu2019sequence,gong2021temporal,chen2020memory}. As for the global sampling schemes, $f_g$ frames are randomly selected from the whole video. With respect to the local sampling ones, $f_l$ consecutive frames are employed. To see the effect of different sampling strategies, we vary the numbers of reference frames in both global and local modes. The numerical results are reported in Table~\ref{table:effectiveness of local and global number}. The performance of using only 3 global reference frames has already outperformed that of using 39 local reference frames, which corroborates the evidence given in ~\cite{wu2019sequence,gong2021temporal}. As a trade-off, we adopt the global sampling strategy with $f_g= 31$ by default for the rest experiments according to Table~\ref{table:effectiveness of local and global number}.
\begin{figure*}[t]
\centering
\includegraphics[width=1\linewidth]{conf_compare.pdf}
\caption{Visual comparison between reference proposals selected by three different methods for given key proposals. We display 4 reference proposals that contribute most in aggregation.}
\label{fig:conf compare}
\end{figure*}
\setlength{\tabcolsep}{4pt}
\begin{table}[t]
\begin{center}
\caption{Effect of the numbers of global $f_g$ and local $f_l$ reference frames.}
\label{table:effectiveness of local and global number}
\begin{tabular}{c|cccccc}
\hline\noalign{\smallskip}
$f_g$ & 3 & 7 & 15 & 23 & \textbf{31} & 39\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
AP50 ($\%$) & 74.6 & 76.1 & 77.0 & 77.3 & \textbf{77.3} & 77.3 \\
\hline
\hline\noalign{\smallskip}
$f_l$ & 3 & 7 & 15 & 23 & 31 & 39\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
AP50 ($\%$) & 70.4 & 71.2 & 72.2 & 72.8 & 73.2 & 73.6 \\
\hline
\end{tabular}
\end{center}
\end{table}
\setlength{\tabcolsep}{4pt}
\begin{table}
\begin{center}
\caption{Effect of the number of proposals $a$ in FSM.}
\label{table:effectiveness of a in fsm}
\begin{tabular}{c|cccccc}
\hline\noalign{\smallskip}
$a$ & 10 & 20 & \textbf{30} & 50 & 75 & 100\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
AP50 ($\%$) & 76.1 & 77.0 & \textbf{77.3} & 77.3 & 77.4 & 77.4 \\
\hline
\noalign{\smallskip}
Time (ms) & 10.4 & 10.7 & \textbf{11.3} & 13.8 & 19.5 & 29.3 \\
\hline
\end{tabular}
\end{center}
\end{table}
\subsubsection{On the number of proposals in a single frame.}
In this experiment, we adjust the quantity of most confident proposals remained for each frame $a$ from 10 to 100 in the FSM, to see its influence on the performance. As shown in Table~\ref{table:effectiveness of a in fsm}, as $a$ increases, the accuracy continuously goes higher and levels off until reaching 75. As the complexity of self-attention is $O\left(n^{2}\right)$ with respect to the amount of input proposals, dragging too many proposals per frame will dramatically raise the time cost. By considering both the speed and accuracy, we adopt $a=30$ that is much smaller than the optimal setting with 75 proposals used in the two-stage based method RDN~\cite{deng2019relation}.
\subsubsection{On the threshold in average pooing of reference features.}
Here, we test the effect of different thresholds for average pooling over reference features. Table~\ref{table:effectiveness of thresh in average} lists the numerical results. As can be viewed, when all the features participate in the average pooling, \emph{i.e.} $\tau=0$, the AP50 is mere $73.1\%$. Lifting the selection standard results in better performance. When $\tau$ falls in $[0.75, 0.85]$, the accuracy keeps steady high to $77.3\%$. But, when $\tau=1$, the average pooling is equivalent to only duplicating $\operatorname{SA(\mathcal{F})}$, the accuracy of which drops to $76.9\%$. Dynamically determining the threshold for different cases is desired and left as our future work. For the rest experiments, we adopt $\tau=0.75$ as the default threshold.
\setlength{\tabcolsep}{4pt}
\begin{table}[t]
\begin{center}
\caption{Influence of the threshold $\tau$ in average pooling over reference features.}
\label{table:effectiveness of thresh in average}
\begin{tabular}{c|ccccccc}
\hline\noalign{\smallskip}
$\tau$ & 0 & 0.2 & 0.5 & 0.65 & \textbf{0.75} & 0.85 & 1\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
AP50 ($\%$) & 73.1 & 73.9 & 76.1 & 77.1 & \textbf{77.3} & 77.3 & 76.9 \\
\hline
\end{tabular}
\end{center}
\end{table}
\setlength{\tabcolsep}{4pt}
\begin{table}[t]
\begin{center}
\caption{Effectiveness of our affinity manner (A.M.) and average pooling over reference features (A.P.). }
\label{table:effectiveness of ours}
\begin{tabular}{c|cccc}
\hline\noalign{\smallskip}
Methods&A.M. &A.P.&Time (ms)&AP50 ($\%$)\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
YOLOX-S & - & - & 9.38 &$69.5$\\
YOLOV-S & \checkmark & - & 10.95 & $76.9_{\uparrow 7.4 }$ \\
{YOLOV-S} & \checkmark& \checkmark & {11.30} & {$77.3_{\uparrow 7.8 }$}\\
\hline
\end{tabular}
\end{center}
\end{table}
\setlength{\tabcolsep}{1.4pt}
\setlength{\tabcolsep}{4pt}
\begin{table}[t]
\begin{center}
\caption{Effectiveness of our strategy compared to bases. $\dagger$ indicates using strong augmentation.}
\label{table:effectiveness of diff model}
\begin{tabular}{c|cccc}
\hline\noalign{\smallskip}
Model&Params&GFLOPs&Time (ms)&AP50 ($\%$)\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
YOLOX-S & 8.95M & 21.63 & 9.4 &$69.5$\\
YOLOV-S & 10.28M & 26.18 & 11.3 & $77.3_{\uparrow 7.8 }$\\
YOLOX-L & 54.17M & 125.90 & 14.8 & $76.1$\\
YOLOV-L & 59.45M & 143.10 & 16.4 & $83.6_{\uparrow 7.5}$ \\
YOLOX-X & 99.02M & 228.15 & 20.4 & $77.8$\\
YOLOV-X & 107.26M & 254.72 & 22.7 & $85.0_{\uparrow 7.2 }$\\
YOLOV-X$^\dagger$ & 107.26M & 254.72 & 22.7 & $85.5_{\uparrow 7.7 }$\\
\hline
\end{tabular}
\end{center}
\end{table}
\setlength{\tabcolsep}{1.4pt}
\setlength{\tabcolsep}{4pt}
\begin{table}[t]
\begin{center}
\caption{Performance comparison in accuracy and efficiency. $\dagger$ indicates with strong augmentation, $T$ means the inference time is tested on a TITAN RTX GPU as reported in corresponding papers. MCSP stands for the Modified CSP v5 backbone adopted in YOLOX. The lower part involves post-processing while the upper does not.}
\label{table:compare w post}
\begin{tabular}{c|ccccc}
\hline\noalign{\smallskip}
Method & Backbone & AP50 ($\%$) & Time (ms) \\
\hline
\noalign{\smallskip}
FGFA & R101 & 76.3 & 104.2 \\
SELSA & X101 & 83.1 & 153.8 \\
RDN & R101 & 81.8 & 162.6 \\
RDN & X101 & 83.2 & - \\
MEGA & R101 & 82.9 & 230.4 \\
MEGA & X101 & 84.1 & - \\
TROIA & X101 & 84.3 & 285.7 \\
MAMBA & R101 & 84.6 & 110.3$(T)$ \\
HVR & X101 & 84.8 & - \\
TransVOD & R101 & 81.9 & - \\
QueryProp & R50 & 80.3 & 21.9$(T)$ \\
QueryProp & R101 & 82.3 & 30.8$(T)$ \\
\hline
\noalign{\smallskip}
\textbf{YOLOV-S} & MCSP & \textbf{77.3} & \textbf{11.3} \\
\textbf{YOLOV-L} & MCSP & \textbf{83.6} & \textbf{16.3} \\
\textbf{YOLOV-X} & MCSP & \textbf{85.0} & \textbf{22.7} \\
\textbf{YOLOV-X$^\dagger$} & MCSP & \textbf{85.5} & \textbf{22.7} \\
\hline
\hline
\noalign{\smallskip}
FGFA & R101 & 78.4 & - \\
SELSA & X101 & 83.7 & - \\
RDN & X101 & 84.7 & - \\
MEGA & X101 & 85.4 & - \\
HVR & X101 & 85.5 & -\\
\hline
\noalign{\smallskip}
\textbf{YOLOV-S} & MCSP & \textbf{80.1} & \textbf{11.3 + 6.9} \\
\textbf{YOLOV-L}& MCSP & \textbf{86.2} & \textbf{16.3 + 6.9}\\
\textbf{YOLOV-X}& MCSP & \textbf{87.2} & \textbf{22.7 + 6.1}\\
\textbf{YOLOV-X$^\dagger$} & MCSP & \textbf{87.5} & \textbf{22.7 + 6.1}\\
\hline
\end{tabular}
\end{center}
\vspace{-10pt}
\end{table}
\subsubsection{On the effectiveness of FAM.} To validate the effectiveness of the affinity manner (A.M.) and the average pooling over reference features (A.P.), we evaluate the performance with and without these modules. The results in Table~\ref{table:effectiveness of ours} reveal that these designs both can help the feature aggregation to catch better semantic representations from the one-stage detector. Compared to the YOLOX-S ($69.5\%$ AP50), the YOLOV-S only armed with A.M. gains $7.4\%$ in accuracy. While equipping YOLOV-S with both A.M. and A.P. (complete YOLOV-S), the performance reaches $77.3\%$ in AP50 with only about 2ms time cost in comparison with YOLOV-S. To be more convincing, we also plug our design into the YOLOX-L and YOLOX-X. Tabel~\ref{table:effectiveness of diff model} shows the detailed comparison between YOLOXs and YOLOVs. The $\dagger$ indicates using strong augmentation (like MixUp~\cite{zhang2018mixup} and Mosaic~\cite{bochkovskiy2020yolov4}) when fine-tuning our version. Our YOLOVs consistently outperform their respective bases by over $7\%$ in AP50.
Moreover, we provide two cases to intuitively exhibit the advance of our FAM. They are the lion case with rare pose and the fox case with motion blur as shown in Fig~\ref{fig:conf compare}. Without loss of generality, the top $4$ reference proposals are listed for different feature selection modes, including the cosine similarity, QK manner in multi-head attention, and our affinity manner. As previously analyzed, the cosine manner selects proposals most similar to the key proposal but suffering the same degradation problem as the key proposal. Though the QK manner alleviates this problem, it is obviously inferior to the affinity manner. By introducing the confidence scores as guidance, our method selects better proposals and further boosts the detection accuracy.
\subsection{Comparison with State-of-the-art Methods}
Table~\ref{table:compare w post} summarizes the detailed information of the competitors including FGFA~\cite{zhu2017flow}, SELSA~\cite{wu2019sequence}, RDN~\cite{deng2019relation}, MEGA~\cite{chen2020memory}, TROIA~\cite{gong2021temporal}, MAMBA~\cite{sun2021mamba}, HVR~\cite{han2020mining}, TransVOD~\cite{he2021end} and QueryProp~\cite{he2022queryprop}. As can be observed, our method can achieve $85.5\%$ AP50 with 21.1 ms per frame. With the help of REPP~\cite{sabater2020robust}, it reaches $87.5\%$ AP50 spending extra 6 ms. In terms of inference efficiency, our method is much faster than the other methods.
To be specific, in the upper part of Table~\ref{table:compare w post}, we report the performance of the involved competing models without adopting any post-processing. Thanks to the property of one-stage detectors and the effectiveness of our strategy, the YOLOVs can remarkably leverage the detection accuracy and inference efficiency. For fair comparison, all the models listed in Table~\ref{table:compare w post} are tested with the same hardware environment except for MAMBA and QueryProp\footnote{The codes and models of the methods with marks `T' (the reported results are from their papers) and '-' are not available when this work is prepared.}. The lower part of Table~\ref{table:compare w post} reports the results of our YOLOV and the other SOTA models with post-processing. The time cost of post-processing is tested on an i7-8700K CPU.
\subsection{Additional Concerns}
\subsubsection{Performance on different motion speeds.}
We separately evaluate the detection performance on the ImageNet VID categories of slow, medium and fast moving objects. The categories are defined by their average IoU scores between objects across nearby frames (Slow: IoU$>0.9$, Medium: $0.9\ge$IoU$\ge0.7$, Fast: $0.7>$IoU). Two other methods FGFA~\cite{zhu2017flow}, SELSA~\cite{wu2019sequence} and our base detectors~\cite{ge2021yolox} with the same post processing strategy~\cite{sabater2020robust} are involved in the comparison. As shown in Table~\ref{table:acc of speed}, the effectiveness of our models is clearly verified on each category. The improvement enlarges as the moving speed increases.
\subsubsection{Inference with different batches.}
In the main paper, we test our model with batch size = 1 and the reference frame features extracted for backbone are saved temporarily. Considering the offline situation, we can feed multiple frames to the model and give the result of these frames simultaneously, \emph{i.e.}, the batch inference manner. In this way, the inference speed can be further improved. In this setting, we select 32 frames randomly from a video to form a batch and test the speed of our models. Benefiting from the batch inference, the inference times of our small, large, and xlarge models are respectively deduced to 3.70 ms, 7.13 ms, and 12.10 ms, as shown in Table~\ref{table:model in offline mode}.
\setlength{\tabcolsep}{4pt}
\begin{table}[t]
\begin{center}
\caption{Accuracy of detecting objects with different speeds.}
\label{table:acc of speed}
\begin{tabular}{c|c|ccc}
\hline\noalign{\smallskip}
Models& Backbone &Slow &Medium &Fast \\
\noalign{\smallskip}
\hline
\noalign{\smallskip
FGFA & R101 &87.4 & 79.1 & 61.4\\
SELSA & R101 &88.7 & 83.3 & 71.1\\
YOLOX-S & MCSP & 80.1 & 71.4 & 55.3 \\
YOLOX-L & MCSP & 85.3 & 80.0 & 65.6 \\
YOLOX-X & MCSP &87.9 & 80.8 & 68.6\\
\hline
YOLOV-S & MCSP &84.6 & 78.6 & 63.7 \\
YOLOV-L & MCSP &89.3 & 85.8 & 72.6 \\
YOLOV-X & MCSP & \textbf{90.6} & \textbf{86.8} & \textbf{74.8}\\
\hline
\end{tabular}
\vspace{-10pt}
\end{center}
\end{table}
\setlength{\tabcolsep}{1.4pt}
\setlength{\tabcolsep}{4pt}
\begin{table}[t]
\begin{center}
\caption{Time cost in the offline mode with the batch inference. The post-processing is tested on an i7-8700K CPU.}
\label{table:model in offline mode}
\begin{tabular}{c|ccc}
\hline\noalign{\smallskip}
Models&B1 (ms) &B32 (ms)&AP50 ($\%$)\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
YOLOV-S & 11.3 & 3.70 & 77.3\\
YOLOV-L & 16.3 & 7.13 & 83.6\\
YOLOV-X & 22.7 & 12.10 & 85.5\\
YOLOV-X + Post & 22.7 + 6.10 & 12.10 + 6.10 & 87.5\\
\hline
\end{tabular}
\end{center}
\vspace{-10pt}
\end{table}
\setlength{\tabcolsep}{1.4pt}
\setlength{\tabcolsep}{4pt}
\begin{table}[t]
\begin{center}
\caption{Effectiveness of our strategy on other bases.}
\label{table:effectiveness of diff detector}
\begin{tabular}{c|ccc}
\hline\noalign{\smallskip}
Model&Params&GFLOPs&AP50 ($\%$)\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
PPYOLOE-S & 6.71M & 12.40 &$69.5$\\
PPYOLOE-S+Ours & 8.04M & 16.73 & $74.9_{\uparrow 5.6}$\\
PPYOLOE-L & 50.35M & 89.19 & $76.9$\\
PPYOLOE-L+Ours & 55.63M & 105.49 & $82.0_{\uparrow 5.1}$ \\
FCOS & 31.00M & 102.84& $67.0$\\
FCOS+Ours & 36.28M & 120.04 & $73.1_{\uparrow 6.1 }$\\
\hline
\end{tabular}
\vspace{-10pt}
\end{center}
\end{table}
\setlength{\tabcolsep}{1.4pt}
\subsubsection{Application to other base detectors.}
In order to validate the generalization ability of the proposed strategy, we also try it on widely-used one-stage detectors including PPYOLOE~\cite{xu2022pp} and FCOS~\cite{tian2019fcos}. Specifically for PPYOLOE, it has different channel numbers at different FPN levels. To achieve the multi-scale feature aggregation, we straightforwardly modify the channel numbers of the detection head at different scales to be the same. While for FCOS, the backbone is ResNet-50~\cite{he2016deep}. There exist 5 FPN levels in the original architecture for dealing with images with large image sizes (\emph{e.g.}, $1333\times 800$). To match the situation of the ImageNet VID, we maintain 3 FPN levels with the largest downsampling rate of 32. For the training procedure and other hyper-parameters settings, we simply keep them the same as those in the YOLOX. Table~\ref{table:effectiveness of diff detector} shows that our strategy can consistently improve different base detectors by over $5\%$ in terms of AP50. It is worth noting that searching more suitable hyper-parameters for different base detectors could obtain better performance.\\
\subsubsection{Evaluation on the OVIS dataset.}
Besides the experiments on the ImageNet VID, we also verify our video object detector on the Occluded Video Instance Segmentation (OVIS)~\cite{qi2022occluded} dataset. There are 607 videos for training and 140 for validation involving 25 classes in this dataset. With 4.72 objects per frame on average and a large portion of objects suffering severe occlusions, the OVIS dataset brings more challenging scenarios. For the training, the COCO pre-trained weights are finetuned for 10 epochs with batch size of 8 on 4 GPUs. The video object detectors are trained for 7 epochs. In addition, the proposal number per frame is set to 75 to match the object density. The images are randomly resized from $480 \times 720$ to $800 \times 1200$ with 32 strides (shorter side) for the multi-scale training. In the testing phase, the images are uniformly resized to $640 \times 960$. We keep the other settings the same as the VID implementation. Table~\ref{table:validation on ovis} shows the performance of ours and the base detectors evaluated at the server \href{https://codalab.lisn.upsaclay.fr/competitions/6567?secret_key=fe618589-31d6-4816-9a47-087ca89ce6c9}{OVIS}.
\setlength{\tabcolsep}{4pt}
\begin{table}
\begin{center}
\caption{Evaluation on the OVIS validation set.}
\label{table:validation on ovis}
\begin{tabular}{c|ccc}
\hline\noalign{\smallskip}
Model&AP&AP50&AP75\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
YOLOX-S & $37.6$ & $58.6$ & $38.7$\\
YOLOV-S & $40.9_{\uparrow 3.3}$ & $66.5_{\uparrow 7.9}$ & $42.0_{\uparrow 3.3}$\\
YOLOX-X & $51.3$ & $68.8$ & $53.7$\\
YOLOV-X & $54.7_{\uparrow 3.4}$ & $75.0_{\uparrow 6.2}$ & $57.2_{\uparrow 3.5}$\\
\hline
\end{tabular}
\vspace{-10pt}
\end{center}
\end{table}
\setlength{\tabcolsep}{1.4pt}
\begin{figure*}[!p]
\centering
\includegraphics[width=1\linewidth]{compare2_topdf.pdf}
\caption{Visual comparisons between YOLOX-X (base detector), our YOLOV-X, and MEGA-ResNet101. Three cases suffer from different types of degradation: (a) motion blur, (b) rare pose, and (c) occlusion. Our method exhibits its robustness against these challenging cases.}
\label{fig:degradation}
\end{figure*}
\begin{figure*}[!p]
\centering
\includegraphics[width=1\linewidth]{ovis_topdf.pdf}
\caption{More visual comparisons between the base detector (upper row) and ours (lower row) on the OVIS dataset.}
\label{fig: ovis comparsion}
\end{figure*}
\subsubsection{Visual comparisons.}
We offer visual results of detection on several samples from the VID dataset by our model, the base of our model-YOLOX, and a SOTA two-stage video object detector-MEGA in Fig.~\ref{fig:degradation}, for more intuitive comparisons. To reveal the robustness of our model against different types of degradation, we select three challenging cases with (a) motion blur, (b) rare poses, and (c) occlusion. As can be seen from the pictures, without temporal information, the base detector, even human beings, can hardly tell what the objects exactly are, \emph{e.g.}, the last frame in (a), the first frame in (b), and the third in (c). By equipping with our feature aggregation module, the prediction is significantly boosted precise. We would like to emphasize again that our model is about 10 times faster than MEGA.
Moreover, we also visually compare our model with the base detector on samples from the OVIS dataset (Fig.~\ref{fig: ovis comparsion}). As we can see, the severe occlusion in OVIS limits the performance of the base detector, while our method remarkably alleviates the problem.
\section{Conclusion}
In this paper, we developed a practical video object detector that jointly considers the detection accuracy and inference efficiency. To improve the detection accuracy, a feature aggregation module was designed to effectively aggregate temporal information. For saving computational resources, different from existing two-stage detectors, we proposed to put the region selection after the (rough) prediction. This subtle change makes our detectors significantly more efficient.
Experiments and ablation studies have been conducted to verify the effectiveness of our strategy, and its advance over previous arts. The core idea is simple and general, which can potentially inspire further research works and broaden the applicable scenarios related to video object detection.
|
1,116,691,500,842 | arxiv | \section{Critical properties of QCD}
The study of the QCD phase diagram in the baryonic chemical
potential-temperature plane is a subject of rapidly increasing
interest in the last decade. Recent investigations \cite{SRS}
suggest that in the real world where the $u$ and $d$ quarks have a
small current mass ($O(10 MeV)$) and the strange quark is much
heavier ($O(100 MeV)$) there is a second order critical point as
endpoint of a first order transition line. This critical endpoint is
located at low baryonic density (compared to the baryonic density in
the nuclear matter) and high temperature ($O(100 MeV)$) values. The
order parameter characterizing the critical behaviour has isoscalar
quantum numbers and the underlying symmetry which breaks
spontaneously at the critical point is the $Z(2)$ symmetry
classifying the QCD critical point in the $3-D$ Ising
universality class \cite{Ising}. However this symmetry does not
represent an obvious symmetry of the original QCD Langrangian
\cite{KLS} but it is rather an invariance of the effective thermal QCD
action.
The fluctuations of the condensate formed at the
critical point correspond to isoscalar particles which are
distributed in phase space producing a characteristic self-similar
pattern with fractal geometry determined by the isothermal critical
exponent of the $3-D$ Ising universality class \cite{Ant1}. The
properties of the isoscalar condensate $\sigma(\vec{x})$ are
strongly affected by the baryonic environment:
\begin{equation}
\sigma_{\rho} \approx \lambda \left( \frac{\rho - \rho_c}{\rho_c} \right)
\sigma_o
\label{eq:eq1}
\end{equation}
where $\rho$ is the baryonic density in the critical region, $\rho_c$ is the
critical baryonic density and $\lambda$ is a dimensionless parameter of
order one. Eq.(\ref{eq:eq1}) relates the isoscalar condensate at zero
baryonic density ($\sigma_o$) with its value at baryonic density $\rho$.
The form of eq.(\ref{eq:eq1}) suggests that the difference $\rho-\rho_c$ can be
considered as an alternative order parameter (besides the isoscalar
condensate $\sigma$) characterizing the QCD critical point. Projecting the
baryonic density onto the rapidity space and using the scaling properties of
the critical baryonic fluid formed in a $A+A$-collision process,
one obtains the relation \cite{Ant2}:
\begin{equation}
A_{\perp}^{-2/3} n_b = \Psi(z_c,\frac{\mu}{\mu_c},\rho_c)
\label{eq:eq2}
\end{equation}
where $A_{\perp}$ is the total number of nucleons of the $A+A$ system in the plane
transverse to the beam, $n_b$ is the net baryon density at midrapidity and $\Psi$
is a scaling function. The variable $z_c$ is defined as: $z_c = A_{\perp}^{-2/3}
A_t L^{-1}$ with $A_t$ the total number of participating nucleons
in the $A+A$ collision and $L$ the size of the system in rapidity space.
The scaling function $\Psi$ depends also on the ratio of the chemical potentials
$\frac{\mu}{\mu_c}$ and for $\mu=\mu_c$ simplifies to:
$$\Psi(z_c,1,\rho_c) =\left\{ \begin{array}{c} \rho_c +
\frac{2}{\pi} (z_c - \rho_c) + C (z_c -\rho_c)^4~~~;~~~z_c > \rho_c \\
z_c~~~;~~~z_c \leq \rho_c \end{array} \right.$$
In fact the scaling relation (\ref{eq:eq2}) represents an alternative description
of the QCD phase diagram in terms of measurable quantities \cite{Ant2}.
In Fig.~1
we present a plot of eq.(\ref{eq:eq2}) in the $(z_c,\xi)$ plane (we use the notation
$\xi=A_{\perp}^{-2/3} n_b$). In the same plot we show also the coordinate pairs
$(z_{c,i},\xi_i)$ for a variety of $A+A$ processes in running (NA49, RHIC),
passed (NA35) and future experiments (LHC). We also mark the QCD critical point in
this graph. One can easily see that the $C+C$ system at the SPS energies
($158~GeV/n$) is very close to the critical point.
\begin{figure}[ht]
\centerline{\epsfxsize=5.9in\epsfbox{procs-fig1.eps}}
\vspace{-12cm}
\caption{The QCD phase diagram in experimentally accessible variables
according eq.~(2). The various processes in recent and future heavy-ion
collision experiments are also displayed.
\label{fig1}}
\end{figure}
Notice that the remaining
$A+A$ processes at the SPS ($Si+Si$, $Pb+Pb$) are not so close to the critical
point although they still lie in the scaling region. It is therefore expected that
nonstatistical fluctuations will be present in all these processes and become stronger
as we approach the critical point. How to reveal these fluctuations, it will be discussed
in the following sections.
\section{Statistical description of the isoscalar condensate}
The isoscalar condensate formed at the critical point can be described as
a critical (Feynman-Wilson) fluid in local thermal equilibrium \cite{Ant3}.
Universality class arguments determine the effective action for the dynamics
of the condensate $\sigma(\vec{x})$ at energies $\approx T_c$ in $3-D$ as:
\begin{equation}
\Gamma_c[\sigma]=T_c^{-1} \int d^3
\vec{x}\left[\frac{1}{2}(\nabla \sigma)^2
+ GT_c^4 (T_c^{-1} \sigma)^{\delta +1}\right]
\label{eq:eq3}
\end{equation}
Eq.(\ref{eq:eq3}) leads to the correct equation of state:
$${\displaystyle{\frac{\delta \Gamma}{\delta \sigma}}}
\sim G \sigma^{\delta}$$
where $\delta =5$ is the isothermal critical exponent of the $3-D$
Ising model.
The coupling $G$ has been calculated in \cite{MT} for the $3-D$ Ising
model on the lattice leading to $G \approx 2$. The field $\sigma(\vec{x})$
in eq.(\ref{eq:eq3}) is in fact macroscopic, i.e. the quantum fluctuations
are integrated out to get the effective action (\ref{eq:eq3}) and therefore
it possesses classical properties. Following ref.(5) we recall here that
the experimentally accessible quantity is not the field
$\langle \sigma \rangle$ itself but
the quantity $\langle \sigma^2 \rangle$ which represents
density fluctuations of the $\sigma$-particles created at $T=T_c$.
Based on the effective action (\ref{eq:eq3}) we can now proceed to determine
the partition function of the condensates as a functional integral:
\begin{equation}
Z=\int {\mathcal{D}} [\sigma] e^{-\Gamma_c[\sigma]}
\label{eq:eq4}
\end{equation}
The path summation in the above equation is dominated by the saddle points
of the corresponding action which have an instanton-like form \cite{Ant1}.
Within this approximation we can determine the density-density correlation
of the critical system both in configuration as well as in momentum space.
Then using the calculated density-density correlation function we find the
distribution of the corresponding $\sigma$-particles in phase space. We end up
with a pattern formed through the overlapp of several self-similar clusters with
fractal mass dimension determined by the isothermal critical exponent $\delta$
\cite{Ant1}.
Using the Fourier transform of the spatial density-density correlation function
we obtain the corresponding quantity in momentum space. A similar pattern occurs
also in momentum space. The number of clusters as well as the multiplicity
within each cluster are the same in both spaces while the local fractal dimension
differs. Another property determining the geometrical features of the critical
system is the shape of its evolution. For a cylindrical evolution the number of
clusters is in general greater than one while in the case of spherical evolution
the system consists a single cluster. Also the corresponding fractal dimensions
are influenced by the geometrical shape of the evoluting system \cite{Ant3}.
A less influenced property is the fractal dimension of the transverse momentum
space which turns out to be $\approx 0.7$ for cylindrical systems and
$\approx 1$ for spherical systems.
\begin{center}
\noindent{\bf{The Critical Monte Carlo (CMC) event generator}}
\end{center}
Using the results of the saddle point approximation to the partition function
of the critical system one can develop a Monte-Carlo algorithm to simulate the
production of the critical $\sigma$-particles in an $A+A$ collision.
We restrict our interest to the
the distribution of the sigmas in momentum space since the coordinates in this
space are experimentally accessible. The momentum coordinates of the centers of
the $\sigma$-clusters are treated as random variables distributed according to
an exponential decay law with range determined by the critical temperature.
Within each cluster
the particles are strongly correlated and possess a fractal geometry. The
corresponding fractal dimension is given in terms of the exponent $\delta$ while
the multiplicity within each cluster is determined through the transverse radius
of the entire system, its size in rapidity, the critical coupling $G$ and the
critical temperature $T_c$. The momenta of the sigma-particles within each cluster
are generated using the tool of L\'{e}vy walks \cite{AZ}.
Exactly at the critical temperature $T_c$ the mass of the sigma particles is zero
(for an infinite system). As the system freezes out the sigma-mass increases and when it overcomes the
two pion threshold the sigmas decay into pions which constitute the experimentally
observable sector of the critical system. Unfortunately there is no theoretical
description of this process based on first principles. A possible treatment of the
$\sigma$-decay into pions is to introduce the probability density $P(m)$ for a sigma
to have mass $m$ and then using pure kinematics to determine the momenta of the
produced pions. The mass $m$ is assigned to the decaying sigmas randomly. A more
detailed description of the whole algorithm can be found in \cite{Ant3}.
\section{SPS data analysis (preliminary)}
If the mass of the decaying sigma is well above the two-pion threshold the
momenta of the produced pions are very distorted with respect to the momentum
of the initial sigma and the fractal geometry of the critical condensate is
not transfered to the pionic sector. Therefore in order to reveal the critical
fluctuations in an analysis of the final pions one has to isolate the part of
phase space for which the mass of the decaying sigmas is very close to the
two-pion threshold. In this case the fractal properties of the sigma-momenta
are transfered to final pions. Our proposal is to perform an event by event
analysis in an $A+A$-dataset forming for each event all the pairs of pions
with opposite charge and filtering out those pairs with invariant mass within
a narrow window just above the value $2 m_{\pi}$ \cite{Ant3}:
\begin{equation}
2 m_{\pi} \leq \sqrt{m^2_{\pi^+ \pi^-}} \leq 2 m_{\pi} + \epsilon~~~;~~~~~
m^2_{\pi^+ \pi^-}=(p_{\pi^+} + p_{\pi^-})^2
\label{eq:eq5}
\end{equation}
In (\ref{eq:eq5}) $p_{\pi^{\pm}}$ are the four-momenta of the positive
(negative) charged pions respectively. The parameter $\epsilon$ is assumed
to be very small compared to the pion mass. We apply first our analysis to
a large set of CMC generated events (100000). The CMC input parameters have
been chosen to meet the properties of the $C+C$ system at the SPS:
\begin{itemize}
\item{The size in rapidity $\Delta=6$, corresponding to $\sqrt{s}=158~GeV/n$}
\item{The transverse radius $R_{\perp}=15~fm$. With this choice we can fix
the mean multiplicity of charged pions to be $\approx 50$ close to the
corresponding value in the $C+C$-system at the SPS}
\item{The critical temperature $T_c \approx 140-170~MeV$}
\item{The self-couplig $G=2$ and the isothermal critical exponent $\delta=5$
detrmined by the universality class of the transition}
\end{itemize}
The essential parameters in our approach is the transverse radius $R_{\perp}$
which controls the mean pion multiplicity in the simulation of an
$A+A$-process, the size in rapidity $\Delta$ which controls the total energy
of the system and the isothermal exponent $\delta$ determining the fractal
geometry of the isoscalar fluctuations.
Having produced the $10^{5}$ CMC events we calculate the factorial moments in
transverse momentum space of the final produced chraged pions. For the decay
of the $\sigma$-s into pions we use a Gaussian $\sigma$-mass probability
distribution with a large mean value ($300~MeV$) and large standard deviation
($100~MeV$). With such a deviation in mass we expect that the critical
fluctuations present in the sigma-sector will be strongly suppressed in the
charged pion-sector. This choice of $P(m_{\sigma})$ may be quite conservative
but it consists a good test for the efficience of our data-analysis algorithm.
Then using the charged pion momenta of each event one can form $\pi^+ - \pi^-$
pairs with invariant mass very close to the two-pion threshold. In our actual
calculations we have used as a window $\epsilon=4~MeV$ to filter out pion
pairs with invariant mass in the range
$2 m_{\pi} \leq \sqrt{m_{\pi^+ \pi^-}^2} \leq 2 m_{\pi} + \epsilon$.
Then we consider each charged pion pair as a sigma-particle.
Performing the factorial moment analysis in the new momenta (of the
reconstructed sigmas) we expect a partial restoration of the critical
fluctuations. Indeed this characteristic behaviour is clearly shown in
Fig.~2a where we present the results of the calculation of the second
factorial moment in transverse momentum space for both the negative pions
as well as the sigmas. The effect of the restoration of the critical
fluctuations in the reconstructed sigma sector combined with the
large suppression of the fluctuations in the negative pions, as predicted, is
impressive. The theoretical expectation, for an infinite critical system, for
the corresponding intermittency index is $s_{2,cr}^{(2D)}\approx 0.67$ while our
analysis leads to $s_2^{(2D)}\approx 0.54$. We have applied the same analysis
to data sets obtained from the NA49 experiment at the SPS. We have analysed
13731 events of the $C+C$ system at $158~GeV/n$, 76065 events of the
$Si+Si$ system at the same energy, 13420 events of the $Pb+Pb$ system at
$40~GeV/n$, 384 events of the same system at $80~GeV/n$ and finally 5584
events of the $Pb+Pb$ system at $158~GeV/n$. It must be noted that all the
data sets used in our analysis are only preliminary and there is a need for
further investigations with improved data. In Fig.~2b we show the results for
the second moment in transverse momentum space in the $C+C$ system. There is
an impressive agreement between simulated and real data. In fact the slope
$s_2^{(2D)}$ of the $C+C$ system turns out to be $\approx 0.58$.
\begin{figure}[ht]
\centerline{\epsfxsize=4.9in\epsfbox{procs-fig2.eps}}
\vspace{-6cm}
\caption{The reconstruction of $\sigma$-momenta in (a) $10^{5}$ CMC
events and (b) 13731 preliminary $C+C$ data.
\label{fig2}}
\end{figure}
In Fig.~3 we show the second factorial moment for all the available NA49
experimental data sets both for negative pions as well as sigmas.
A gradual increament of the slope $s_{2}^{(2D)}$ as we approach the
$C+C$ system - and according to Fig.~1 the critical point - is observed close
to our theoretical expectations. For all the systems the effect of the
reconstruction of the critical fluctuations in the $\sigma$-sector is clearly
seen.
\begin{figure}[ht]
\centerline{\epsfxsize=3.9in\epsfbox{procs-fig3.eps}}
\caption{The second factorial moment in transverse momentum space for
all the analysed SPS processes. Represented are only the results
obtained after the reconstruction of the isoscalar sector.
\label{fig3}}
\end{figure}
The analysis described so far concerns a finite kinematic window above
the two pion threshold. It is interesting to extrapolate the properties
of the various systems exactly at the two pion threshold. In this case
no distortion due to the $\sigma$-decay into pions will be present and
we expect to reproduce the theoretically expected results for the
critical system. Therefore we have to take the limit $\epsilon \to 0$.
In order to extract this information one has to calculate $s_2^{(2D)}$ for
various values of the kinematical window $\epsilon$ and use an interpolating
function to extrapolate to $\epsilon=0$. The obtained value
$s_{2,o}^{(2D)}$ can be directly compared with the theoretical expected
value for $s_{2,cr}^{(2D)}$. To be able to perform this analysis one has to study
a system with very large charged pion multiplicity per event and/or to use
a very large dataset. For this reason we have applied our approach to two
systems: (i) the 5584 $Pb+Pb$ events at $158~GeV/n$ and (ii) the
$10^{5}$ CMC generated critical events (simulating the $C+C$ system
at $158~GeV/n$). The
results of our calculations are presented in Fig~4. The solid circles
are the values of $s_2^{(2D)}$ for the $Pb+Pb$ system while the open
triangles describe the CMC results for various values of $\epsilon$.
The dashed lines present a corresponding exponential fit.
\begin{figure}[ht]
\centerline{\epsfxsize=3.9in\epsfbox{procs-fig4.eps}}
\vspace{-2cm}
\caption{The slope $s_2^{(2D)}$ for different values of the kinematic
window $\epsilon$ both for the $10^{5}$ CMC events as well as for 5584
$Pb+Pb$ events at $158~GeV/n$ using preliminary SPS-NA49 data.
\label{fig4}}
\end{figure}
For the CMC events
we find $s_{2,o}^{(2D)}=0.69 \pm 0.03$ a value which is
very close to
the expected $s_{2,cr}=0.67$, while for
the $Pb+Pb$ at $158~GeV/n$ system we get $s_{2,o}^{(2D)}=0.34$.
The last value corresponds to a strong effect, owing to the fact
that the $Pb+Pb$ system lies
in the scaling region around the critical point. However it is clearly
smaller
than the theoretical value at the endpoint, in accordance with the fact that this
system freezes out in a distance from the critical point in terms of
the variables in Fig.~1.
In summary we have introduced an algorithm to detect critical fluctuations
related to the formation of an isoscalar condensate in $A+A$-collisions.
First analysis, using preliminary SPS-NA49 data, indicates
the proximity to the critical point of the freeze-out area in the
collisions with nuclei of medium size ($C+C$ or $Si+Si$).
\section*{Acknowledgments}
The authors thank the NA49 Collaboration for supplying the preliminary
experimental data from the SPS.
\vspace*{-9pt}
|
1,116,691,500,843 | arxiv | \subsection{The Distance Problem at mmWave-THz Bands}
\hl{Comment the main aspects relative to the distance problem, wireless comms oriented.}
\cite{akyildiz2018combating}\textcolor{red}{Odysseas}
\\
\\
\\
\hl{Describe the existence of prototypes for RIS, wireless comms oriented.}\textcolor{red}{Odysseas and Sergi}
\\
\\
\\
\hl{Briefly describe how those can help to address the distance problem.}\textcolor{red}{Odysseas}
\\
\\
\\
\hl{Mention that there is not much work specific for mmWave-THz frequencies. We are different and aim to bridge this gap.}\textcolor{red}{Odysseas}
\cite{tsilipakostoward, abadal2020programmable}
\cite{pitchappa2020frequency}
\\
\\
\\
\hl{Mention that graphene-based metasurfaces are the excellent candidate at mmWave-THz bands (especially THz) }\textcolor{red}{Ana}
\cite{Correas2017}
\cite{han2020complete}
\cite{cheng2019recent}
\subsection{Multi-band Coding Formulation}
The realization of the promising paradigm of widely tunable MS integrating graphene-based unit cells needs careful consideration of multiple aspects related to, among others, graphene technology and metasurface design. We discuss them next.
\subsection*{Technological implementation}
Although there are a rising number of methods for making various forms of graphene, the volume production for some of those methods remains limited. The focus will have to be on material quality and manufacturing cost. The main challenge lies in maintaining the quality of graphene, i.e. large $\tau$, in large areas. Defects on the graphene layer will significantly affect the electrical conductivity.
Therefore, choosing the most accurate method for producing graphene and good control over the graphene layer deposition is very important.
Various fabrication approaches have been utilized to develop graphene. The more efficient among them, Chemical Vapor Deposition (CVD), has been shown to provide high quality, single layer, uniform graphene in large scale samples~\cite{Choi2017116X}.
The contact of graphene with other materials, for example, dielectric substrates or metallic electrodes, modifies the doping level and chemical potential of graphene. In this case, encapsulating graphene in insulating materials such as hexagonal Boron Nitride (hBN) helps minimizing the influence of contacting materials and, hence, maintaining a high $\tau$ \cite{wang2019graphene}.
For the implementation of graphene-based MSs, it is usual to deploy a patterned scheme applied in a uniform sheet, as the patches discussed here. For the patterning, post-process techniques in pre-synthesized graphene are available, i.e., electron-beam, helium ion beam, lithography, nanostructure-assisted, or nanoimprint technology~\cite{Wei20201655}.
\subsection*{Graphene tuning mechanisms}
As mentioned before, the properties of graphene depend strongly on aspects such as relaxation time and chemical potential, the latter of which can be tuned by imposing an external stimulus. Thus one can modify graphene's conductivity by altering the chemical doping, by thermal tuning, external electrostatic or magnetic field, and optical pumping or self-action. Many of the works investigating the physical aspects of graphene's modulation capabilities focus on the global modification usually involving uniform graphene sheets. The most prevalent technique for locally manipulating the optical properties of graphene is controlling the biasing voltage, often enhanced with the use of an electrolytic medium. In this case, care must be taken in controlling the biasing voltage at each unit cell individually as required by the MS coding. This non-uniform modification can be achieved by properly placed transparent electrodes or by alternative techniques, for example, uneven ground planes or homogeneous dielectric spacers~\cite{DeAbajo2014133}, which in any case are non-trivial at the granularity of the proposed unit cells. In all cases, we note that the tuning speed of graphene (in the order of picoseconds) is enough to support not only slow-paced indoor applications, but also the most demanding outdoor applications, even vehicular ones, whose reconfiguration speeds may be in the the order of hundreds of microseconds \cite{akyildiz2018combating}.
\subsection*{Multi-band metasurface design}
Assuming that the MS operation in all three bands is concurrent, e.g., without any time-multiplexing scheme to decouple them, means that setting the graphene chemical potentials $(\mu_1,\mu_2)$ to optimize the four states for one frequency band will unavoidably simultaneously affect the graphene response in the other bands. In line with the vision outlined in this paper, an ideal unit cell would provide a large number of states with high amplitude and all possible combinations of equally spaced phases in the three target frequencies, i.e. $\varphi=\varphi_{0}+\{0,\tfrac{\pi}{2},\pi,\tfrac{3\pi}{2}\}$ at $f_{1},f_{2},f_{3}$. This would allow controlling the response at the three bands simultaneously and independently. However, such a unit cell would probably require heavy optimization to strike a delicate balance among multiple resonances.
In less ideal unit cells, the complex reflection coefficients in the other bands will in principle form a sub-optimal set of states deteriorating the performance. It can be intuitively understood that optimization of the unit-cell geometry can potentially lead to adequately broadband behavior; alternatively, the four states can be assigned to four $(\mu_1,\mu_2)$ sets that lead to a tolerable performance, \textit{on average} across the operation bands. In Fig.~\ref{fig:coupled_response}, we present two-beam splitting of a normally incident plane wave by a circular-aperture MS (diameter 1.4~mm) of 35~$\upmu$m-wide cells. Each panel in this figure corresponds to an operating frequency band ($\{a,b,c\}=\{0.65,0.85,1.05\}$~THz) for which the $(\mu_1,\mu_2)$ for the four states have been optimally selected. Evidently, the scattering patterns present two well-defined lobes at the desired directions, $(\theta,\varphi)=(30^\circ,0)$ and $(45^\circ,90^\circ)$. In each case, the response at the bands for which the MS has not been tuned, which is not shown for space constraints, is degraded. Specifically, the scattering directivity in the desired lobes was decreased by 1-2~dB (worst-case) whereas parasitic side lobes were raised by 6-7~dB.
\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{figures/Fig_CoupledResponse.pdf}
\caption{Scattering patterns for two-beam splitting of a normally incident plane wave by a circular-aperture MS ($\{a,b,c\}=\{0.65,0.85,1.05\}$~THz).}
\label{fig:coupled_response}
\end{figure}
\subsection*{Near-field and link-budget considerations}
The Huygens-Fresnel principle introduced previously can be used to accurately estimate the far-field as a superposition of elementary scattering patterns from each cell.
Even though this Huygens-Fresnel framework is strictly valid in the far-field, i.e., in a context similar to geometric-optics/ray-tracing, its applicability for performance evaluation extends well into the `transition' (or Fresnel) region, bounded by the near-field and the Fraunhofer ($r_{FF}$) regions. This means that a RIS can be envisioned as a part of the antenna system of a transceiver base station, providing digital beamforming that enhances path loss to virtual-line-of-sight mobile stations. Moreover, in real-world applications relevant to 6G THz communications, such as pico/femto-cells or short-range indoor channels, all incoming sources will practically be in the far-field of the MS, even for cm-sized apertures for which the Fraunhofer region is a few meters.
Finally, note that this multi-band RIS is capable of operating at frequency bands of almost an octave distance. In this sense, the path loss will exhibit a 6~dB deterioration switching from the lowest to the highest frequency, which could be considered for a short-range link budget. Nevertheless, this difference can be partially compensated by an increased directivity of the scattering pattern assuming that: (i) the RIS aperture is fixed, (ii) the channel is appreciably unobstructed so that a low exponent $\lambda^n$, $n\rightarrow2$, can be assumed, and (iii) the MS reflection coefficients are approximately similar across the frequency bands; the latter can be attained by engineering the unit-cell and careful selection of its states, and/or using materials with sufficiently broadband response, such as graphene.
\section*{Introduction}
\label{sec:intro}
\input{intro.tex}
\section*{Graphene-based Multi-Wideband Metasurface}
\label{sec:unit}
\input{unitCell.tex}
\section*{Implementation Considerations}
\label{sec:challenges}
\input{challenges.tex}
\section*{Conclusions}
\label{sec:conc}
\input{conclusions.tex}
\bibliographystyle{IEEEtran}
\subsection*{Capacity Analysis}
\section{Discussion}
\hl{Metasurface: how frequency modulation affects directivity and other metrics.}\textcolor{red}{Hamidreza and Alex}
\subsection*{Graphene Modeling}
\subsection*{Unit Cell Design}
Fig.~\ref{fig:unit cells} shows the proposed unit cell whose lateral size is $c=35~\upmu$m (around $\lambda/10$), small enough to allow for fine resolution of the reflection phase profile, characteristic of MSs over standard reflectarrays. From top to bottom, the unit cell consists of a multi-layer structure with four parasitic graphene patches ($a=15~\upmu$m, chemical potential $\mu_2$) on top of an alumina layer with $0.1~\upmu$m thickness. Synthesis of graphene/alumina composite materials enhances strength, toughness, and wear-resistance with a low-cost process \cite{Kim2014}. This composite is stacked on high-density polyethylene (HDPE) substrate, due to its particularly low losses in the terahertz band \cite{polym12092094}. HDPE permittivity is $\epsilon_r=2.37$ and its thickness is selected $h_1=15~\upmu$m. The combination of graphene and HDPE leads to high electrical conductivity with good thermal and mechanical properties \cite{polym12092094}. Another layer of graphene/alumina composite ($b=20~\upmu$m, the chemical potential $\mu_1$) is sandwiched between HDPE and silicon ($h_2=10~\upmu$m) to allow for supporting multiple resonances and, thereby, being able to operate in three disjoint bands.
\begin{figure}[!t]
\vspace{-0.4cm}
\centering
\includegraphics[width=0.5\textwidth]{figures/unitcell3.png}
\caption{Schematic representation of the unit cell layout. This unit cell is composed of two main dielectric layers (i.e., HDPE and silicon), four parasitic graphene patches on top and one graphene sheet sandwiched between the dielectrics. The unit cell is back plated with a thin layer of gold to be fully reflective.}
\label{fig:unit cells}
\vspace{-0.1cm}
\end{figure}
Due to the native oxidation of silicon, a thin layer of silicon dioxide is grown in the outer layer in a short time. To analyze a realistic design we have considered a layer of silicon dioxide with $0.3~\upmu$m thickness on top of the silicon layer and the layout is back plated with gold. Regarding the fabrication feasibility of the proposed low-profile structure, there are advanced silicon substrate thinning techniques that can be used to achieve an ultra-thinning down to 4~$\upmu$m without damage occurred due to thinning processes.
As mentioned above, the frequency and reflection phase of a graphene-based unit cell can be controlled via changes in its biasing voltage. This is modeled through the frequency-dependent surface conductivity of graphene $\sigma(f)$, which in the THz band is given by
\begin{equation}
\sigma\left(f\right)=\frac{e^{2}\mu}{\pi\hbar^2}\frac{i}{2\pi f+i\tau^{-1}},
\label{eq:sigma_graphene}
\end{equation}
where $e$ and $\hbar$ are constants, while $\mu$ and $\tau$ are variables that correspond to the chemical potential and the relaxation time of the graphene layer, respectively \cite{wang2019graphene}. When a graphene sheet forms one plate of a capacitor, any applied static voltage will alter the carrier density at its surface thus modulating its EM material properties, i.e., its complex-valued electric conductivity; the chemical potential, expressed in eV units, is typically used as the controllable variable, linked to the externally applied tuning voltage as $\mu\propto\sqrt{V_\mathrm{bias}}$. On the one hand, the amplitude response depends on the losses within graphene, $\mathrm{Re}\{\sigma\}$, which in turn depend on the quality of the graphene sheets as modeled by the relaxation time value $\tau$. For the purpose of this work, the relaxation time of graphene is assumed to be $\tau=1$~ps, which has been considered in multiple works and is realizable with state-of-the-art fabrication and encapsulation techniques \cite{Banszerus2015}. On the other hand, to modify the resonance frequency and the phase of the response, the key tuning variable is the chemical potential value $\mu$ or, in our case, two distinct values $\mu_{1}$ and $\mu_{2}$ on top and bottom patches as shown next.
All full-wave EM simulations at the unit-cell level were conducted in CST Microwave Studio (frequency domain solver).
\subsection*{Unit Cell States} \label{cell_states}
We explore the chemical potential design space by sweeping the values of $\mu_{1}$ and $\mu_{2}$ in the three desired operational frequencies. The inset figures in Fig.~\ref{fig:apf} show a trade-off between reflection amplitude and reflection phase versus chemical potential values at 0.65 THz as an example. Hence, by controlling the bias voltage we can tune the reflection characteristics.
\begin{figure*}[!ht]
\centering
\vspace{-0.5cm}
\includegraphics[width=0.65\textwidth]{figures/APF.png}
\caption{Reflection characteristics of the selected states at respective operation frequency. The insets represent the design space of different combinations of chemical potentials at 0.65 THz. The horizontal axis represents the chemical potential of sandwiched graphene patch (see Fig. 3) and the vertical axis represents the chemical potential of the graphene patches on top. Four states corresponding to the highest possible amplitude and spaced with $\pi/2$ phase are selected (black circles in the inset and blue circles in the outset figure). Reflection amplitude and phase at 0.85 and 1.05 THz versus chemical potentials have similar design spaces. Selected states are plotted with red stars (0.85 THz) and yellow diamonds (1.05 THz) in the outset figure.}
\label{fig:apf}
\end{figure*}
The results show the response of the MS for a continuous range of chemical potentials. However, to design a digital reconfigurable MS, we need to discretize the design space and select a set of chemical potentials to obtain a finite set of addressable states. Firstly, the number of unit cell states (per frequency band) has to be selected. Here, the number of states will determine the phase difference between consecutive unit cell states. As shown in our previous work \cite{9109701}, four states (with $\pi/2$ phase difference between them) can offer high-quality beam-steering performance, given that the unit cell is fairly subwavelength as in our case here. In the insets of Fig.~\ref{fig:apf}, black circles indicate the four selected states at 0.65 THz. These states are selected manually with two goals i) $\pi/2$ phase shift to each other and maximum reflection amplitude. The blue symbols in Fig.~\ref{fig:apf} represent the selected states at 0.65 THz, whereas the red and yellow symbols represent the other two frequencies.
We note that the selected states may need to change when the incidence angle changes. Assuming that a completely different set of states is needed to cover the neighborhoods of 0, 30, and 60 degrees \cite{9109701}, the minimum \emph{total} number of required states is 4 states per frequency band $\times$ 3 frequency bands $\times$ 3 incidence angle neighborhoods $=36$ states. Moreover, if the metasurface is brought in the near-field of transmitting/receiving antennas, the reflection phase/amplitude may vary and depend on the distance \cite{Danufane2021}. This behavior should be taken into account with additional states being foreseen to cover operation in such short distances.
\subsection*{Metasurface coding for beam steering} \label{sec:MScoding}
Having selected the four sets of chemical potentials $(\mu_1,\mu_2)$ that define the cell states, we retrieve the corresponding complex reflection coefficients (phase and amplitude) for each band, from charts similar to those in the insets of Fig.~\ref{fig:apf}, as calculated from full-wave EM simulations. In this work, we assume that the operation between the three bands is decoupled, i.e., the four $(\mu_1,\mu_2)$ sets that define the four states are different for each band. In this case, achieving independent control over the three bands is only possible via space multiplexing (large RIS panels divided into subpanels devoted to a particular frequency) or time multiplexing (slots reserved to each particular frequency) leveraging the fact that the response of graphene to the electrical tuning of its chemical potential is practically instantaneous. As discussed above, the phase-stepping is very close to $\pi/2$ along with the states of all three bands (horizontal axis of Fig.~\ref{fig:apf}), with a worst-case reflection coefficient of $-4$~dB; these metrics ensure qualitative agreement with theory and can be further improved with cell redesign.
The Huygens-Fresnel principle (HFP) \cite{9109701} allows us to evaluate the scattering pattern of the MS for a given configuration as the aggregated response of its unit cells to a given incidence (frequency, polarization, wavefront shape, and direction). For linearly polarized incidence waves, the scattered field can be expressed as
\begin{equation}
\begin{split}
E(\theta, \varphi) = \sum_{m=1}^{M} \sum_{n=1}^{N} A_{mn} p_{mn}(\theta_{mn}, \varphi_{mn})\\ R_{mn} p_{mn}(\theta, \varphi)
e^{jk_0\zeta_{mn}(\theta, \varphi)}
\end{split}
\label{eq:Huygens}
\end{equation}
where $\varphi$ and $\theta$ are the azimuth and elevation angles, $A_{mn}$ corresponds to the complex amplitude of the wave incident for the $mn$-th unit cell, $p_{mn}$ denotes the scattering pattern of the $mn$-th unit cell, and $R_{mn}$ is the complex reflection coefficient for the $mn$-th unit cell.
Finally, $\zeta_{mn}(\theta, \varphi)$ is the relative phase shift of the unit cells with respect to the radiation pattern coordinates and their geometric arrangement on the MS. Tuning of the two voltages applied at each $mn$-cell alters graphene's local $\mu_{1,2}$ and consequently the cell's response, $R_{mn}$; the collective effect of the tuning of all $R_{mn}$ is a tuning of the scattering pattern, $E(\theta, \varphi)$. The underlying approximation of the HFP is that unit cells are uncoupled, so that their $R_{mn}$ can be tuned independently, which is valid for the cell sizes considered here. Finally, we note that HFP is an analytical method implemented on MATLAB using parameters calculated from 3D geometry and/or extracted from full-wave unit-cell level EM simulations.
The ideal MS coding or configuration, that is, the unit cell amplitudes and phases required to provide a given functionality, can also be analytically calculated in the same framework \cite{9109701}. This procedure can be summarized as forward-propagating the source `rays' while back-propagating the desired radiation pattern (prescribing a functionality) onto the MS, and then taking the complex-valued ratio of the propagated wavefronts on each cell; these ratios denote the required reflection coefficient $R_{mn}$ profile to realize the functionality.
Then, the unit cell states closest to the ideal ones are selected. As an example, normally incident plane waves can be steered away from the specular (normal) reflection direction by configuring the MS cells in stripes of progressively increasing reflection phase, according to diffraction grating theory. More elaborate beamforming patterns can be engineered by more complicated configuration patterns; for instance, beam-splitting in two different directions can be accomplished by superimposing the \emph{stripes} that correspond to single-beam steering thus generating a pattern of rectangular \emph{supercells}.
In Fig.~\ref{fig:scatpats}, we present a set of scattering patterns that demonstrate the potential of the proposed multi-wideband multi-functional RIS, assuming $40\times40$ 35~$\upmu$m-wide cells. This figure contains a selection of configurations and functionalities interesting in 6G contexts, demonstrated in three different frequencies, namely:
\begin{itemize}
\item \textbf{Three-beam splitting} at 0.85 THz, Fig.~\ref{fig:scatpats}(a), shows a case where a normally incident plane wave is split in three beams, each one steered to a different direction that can be controlled independently of others: $(\theta,\varphi)=(15^\circ,315^\circ)$, $(30^\circ,90^\circ)$, and $(45^\circ,135^\circ)$. Fig.~\ref{fig:scatpats}(a) depicts the scattering pattern both in spherical (3D) and in cylindrical (2.5D) coordinates, in the main view and top-left inset, respectively, together with the unit cell states with color-coded squares in the left-hand insets. This functionality could be useful for multicasting in multi-band 6G networks.
\item \textbf{Random scattering} of normally incident radiation at 0.65~THz, Fig.~\ref{fig:scatpats}(b). This is accomplished by totally random (uniform distribution) phase states across the MS; note that to eliminate back-scattering (or monostatic radar cross-section) the phase-states must be clustered in supercells of $\lambda/2$ size, as smaller randomly-coded cells emulate a metallic reflector and produce a specular reflection. This functionality is useful for physical-layer security methods, blocking certain users.
\item \textbf{Collimation of a spherical wavefront towards two far-field directions} $(\theta,\varphi)=(30^\circ,0)$ and $(45^\circ,90^\circ)$ at 1.05~THz, Fig.~\ref{fig:scatpats}(c). In this case, the illuminating point-source is outside the near-field border, at 3~mm above the MS axis, so that emitted `rays' impinge almost perpendicularly on the MS. This functionality is useful for the building of antennas where an integrated RIS acts as an intelligent reflector.
\end{itemize}
\begin{figure}[!t]
\centering
\vspace{-.1cm}
\includegraphics[width=\columnwidth]{figures/Fig_ScatPats.pdf}
\caption{Scattering patterns for normal illumination of the multi-band multi-functional RIS; scattering heatmaps are normalized to theoretical maximum directivity. (a) 3D scattering pattern of a normally incident 0.85~THz plane-wave, split/steered in 3 predetermined directions: $(\theta,\varphi)=(15^\circ,315^\circ)$, $(30^\circ,90^\circ)$, and $(45^\circ,135^\circ)$; left-hand insets depict the cylindrical-coordinate representation of the same pattern, the orthogonal color-coded view of the MS cells configuration, and the color-mapping of the phase-states. (b) MS configuration and corresponding cylindrical scattering pattern for diffused scattering, and (c) MS configuration and corresponding cylindrical scattering pattern for collimation/steering of a spherical wavefront towards two distinct directions.}
\label{fig:scatpats}
\vspace{-.5cm}
\end{figure} |
1,116,691,500,844 | arxiv |
\section{Introduction}
\IEEEPARstart{E}{lectrical} machines have numerous real-world applications, ranging from home appliances to the automotive industry. Permanent magnet synchronous machines (PMSM) gained popularity in recent years due to their high efficiency, large power density, and high torque-current ratio, see for example \autoref{fig:data_vis}. However, the use of expensive materials such as rare earth magnets (neodymium, dysprosium, or terbium) is a significant cost operator in the manufacturing process. Therefore simulation based optimization, e.g. minimizing cost and maximizing performance, is an established technique in academia and industry \cite{Di-Barba_2010aa}. The finite element method (FEM) is at the heart of such numerical optimization workflows. In the automotive industry, FEM-based optimization has been used for decades to reduce the number of physical prototypes and to optimize the active parts of the electrical machine (stator, rotor, magnets, winding). However, it is a computationally expensive process, so there is a great need for complexity reduction.
\begin{figure}
\centering
\input{fig2/data_vis2}
\vspace{-0.5em}
\caption{Exemplary double-V PMSM geometry.}
\label{fig:data_vis}
\vspace{-1em}
\end{figure}
To this end, surrogate or meta modelling has become a research field on its own in the last decade \cite{Koziel_2013aa}. More recently, meta-modelling by machine learning has gained attention. Nowadays, deep neural networks (DNNs) are employed at various design stages because of their advantages such as scalability for high dimensional data, handling of big data, easy parallelization by graphical processing units (GPUs) and automatic feature extraction. For example, Yan et al.~\cite{8921886} describes the deep-learning-based approximation of the electromagnetic torque for PMSM. In another article, the DNN is used as a meta-model for predicting objectives obtained by FE output in the design phase for the optimization of flux switching machine to lower the computational burden \cite{8785344}. Another work \cite{8961095} demonstrates how a deep learning-assisted approach is used to predict efficiency maps of electric drives. The DNN-based two-step optimization method is investigated in \cite{9366983} to speed up the design process by inferring torque-related performance. The application of CNN to approximate performance characteristics of electric machines for multidimensional outputs is investigated in \cite{act10020028}. This paper builds on the findings in \cite{9333549}. It demonstrates how a deep learning based meta-models predict cross-domain key performance indicators (KPIs) effectively using different PMSM representations in a large input design space. The meta-model trained with parameter-based data has higher prediction accuracy than an image-based representation. The generation of (high-resolution) images and the training of the deep convolutional neural network requires more time and effort than the parameter-based approach. However, the image-based meta-model remains invariant to re-parametrization that does not hold for the parameter-based model. Recently, \cite{parekh2022variational} demonstrated how KPIs of PMSM topologies with different parameterizations can be predicted concurrently by mapping high dimensional scalar parameters into a lower-dimensional regularized latent space using a variational autoencoder.
In this paper, we present a novel physics and data-driven approach for characterizing the electromagnetic behavior of an electrical machine. A multi-branch DNN functions as a meta-model. It accepts high dimensional varying scalar parameter input, e.g. geometry, electrical, and material. The results of the FE simulations are stored in a database and the DNN is trained by supervised learning. In contrast to previous approaches, we do not predict KPIs by the DNN but only intermediate measures. The result, along with the system parameters, is fed into a physics-based analytical model to calculate characteristic maps and cross-domain KPIs, e.g., maximum torque, power, cost, etc. Finally, we quantitatively compare this hybrid model to existing approaches.
The article is organized as follows: \autoref{sec:DG} explains the problem formulation and data set specifics. \autoref{sec:MNAT} presents the methodology, network architecture, and training settings. \autoref{sec:res} provides quantitative analysis of results which is followed by the conclusion in \autoref{sec:conclusion}.
\begin{table}
\caption{Inputs (selection)}
\label{tab:input}
\begin{tabularx}{\linewidth}{|c|X|r|r|c|}
\hline
& \textbf{Parameter} & \textbf{Min} & \textbf{Max} & \textbf{Unit} \\ \hline
$p_{1}$ & Air gap & 0.50 & 2.00 & mm \\ \hline
$p_{2}$ & Height of inner magnet & 2.23 & 7.00 & mm \\ \hline
$p_{3}$ & Width of inner magnet & 8.00 & 25.00 & mm \\ \hline
$p_{4}$ & Height of tooth head & 12.00 & 20.00 & mm \\ \hline
$p_{5}$ & Rotor outer diameter & 159.00 & 165.00 & mm \\ \hline
$p_{6}$ & Input phase current $I$ & 0.00 & 1336.40 & A \\ \hline
$p_{7}$ & Control angle $\alpha$ & 0 & 360 & degree \\ \hline
\hline
$p_{8}$ & Input phase voltage & \multicolumn{2}{c|}{640} & V \\ \hline
$p_{9}$ & Slots per pole per phase & \multicolumn{2}{c|}{2} & - \\ \hline
$p_{10}$ & Pole pairs & \multicolumn{2}{c|}{4} & - \\ \hline
$p_{11}$ & Stator type & \multicolumn{2}{c|}{symmetric} & - \\ \hline
$p_{12}$ & Rotor type & \multicolumn{2}{c|}{double-V} & - \\ \hline
\end{tabularx}%
\end{table}
\section{Problem formulation and data set}\label{sec:DG} %
Each FE simulation requires a high-dimensional parameter-based input $\mathbf{p}$ to compute the intermediate measures $\mathbf{y}$ which then leads to the KPIs $\mathbf{z}$. Let us consider an electric machine simulation dataset
\begin{equation}
\begin{aligned}
\mathcal{D} := \Big\{ (\mathbf{p}^{(1)},\mathbf{y}^{(1)},\mathbf{z}^{(1)}),\ldots,(\mathbf{p}^{(N)},\mathbf{y}^{(N)},\mathbf{z}^{(N)}) \Big\}
\end{aligned}
\end{equation}
of $N_{\mathrm{D}}$ samples, where each input is generated by the realization of a $d$-dimensional random input vector $\mathbf{p}^{(i)}$ and leads to the $m$-dimensional vector $\mathbf{y}^{(i)}=\mathbf{F}(\mathbf{p}^{(i)})$ using the FE model $\mathbf{F}$ and finally to the KPIs $\mathbf{z}^{(i)}=\mathbf{K}(\mathbf{p}^{(i)},\mathbf{y}^{(i)})$ using an analytical post-processing, summarized in $\mathbf{K}$. The goal is now to learn a function $\hat{\mathbf{F}}$ that is cheap to evaluate and approximates the computationally heavy FE simulation. It will be used afterwards as a meta-model to predict the KPIs for new unseen machine designs.
\subsection{Dataset specifics}\label{sec:dataset}
In this study, the simulation dataset is generated for a double-V PMSM. The procedure for calculating KPIs is detailed in Section~2 of \cite{9333549}. A total of $N_{\mathrm{EM}}=44877$ machine designs have been simulated for 35 input parameters $p_i$, e.g. geometry and material. \autoref{fig:data_vis} depicts a representative geometry from the dataset. Each machine design was simulated using FE simulations for $N_{\mathrm{OP}}=37$ operating points, considering the magnetic state symmetry to reduce computational overhead. The operating points are interpreted as a variable electrical excitation input for the electrical machine, i.e., it is given by an input phase current $I$ and its control angle $\alpha$.
As a result, $N_{\mathcal{D}} = N_{\mathrm{OP}} \times N_{\mathrm{EM}} = 1660449$ samples are available in the dataset on which the meta-model will be trained and tested on. A few selected input parameters are listed in \autoref{tab:input}.
The simulated quantities, i.e., intermediate measures $y_i$, from the FE analysis are non-linear iron losses $P_{\mathrm{fe}}$, torque $T$ and fluxes $\Psi_{1}, \Psi_{2}, \Psi_{3}$ linked with the three coils for one electrical period, see \autoref{tab:intermediate}. Note, that the losses $P_\mathrm{fe}$ include two contributions from hysteresis and eddy currents, \cite{6689759,jordan1924ferromagnetischen}. A plot of two operating point calculations for one electrical period of a test sample from the dataset $\mathcal{D}$ is shown in \autoref{fig:evalopptsd}. Finally, examples of a few KPIs are given in \autoref{tab:kpis} and their distribution can be seen in \autoref{fig:DS}.
\begin{table}
\caption{Outputs (intermediate measures)}
\label{tab:intermediate}
\begin{tabularx}{\linewidth}{|c|X|r|r|c|}
\hline
& \textbf{Measure} & \textbf{Unit} \\ \hline
$y_{1}$ & Nonlinear iron losses $P_{\mathrm{fe}}$ & J \\ \hline
$y_{2}$ & Electromagnetic torque $T$ & Nm \\ \hline
$y_{3}$ & Flux linkage $\Psi_{1}$ coil 1 & Wb \\ \hline
$y_{4}$ & Flux linkage $\Psi_{2}$ coil 2 & Wb \\ \hline
$y_{5}$ & Flux linkage $\Psi_{3}$ coil 3 & Wb \\ \hline
\end{tabularx}%
\end{table}
\begin{table}
\caption{Key performance indicators}
\label{tab:kpis}
\begin{tabularx}{\linewidth}{|c|X|l|}
\hline
& \textbf{KPIs} & \textbf{Unit} \\ \hline
$z_{1}$ & Maximum torque on limit curve & Nm \\ \hline
$z_{2}$ & Max. shaft power & W \\ \hline
$z_{3}$ & Max. shaft power at max. speed & W \\ \hline
$z_{4}$ & Max. torque ripple on limit curve & Nm \\ \hline
$z_{5}$ & Material cost & Euro \\ \hline
$z_{6}$ & Mass of active parts & Kg \\ \hline
$z_{7}$ & Torque ripple deviation & Nm \\ \hline
\end{tabularx}%
\end{table}
\begin{figure*}[b]
\centering
\subfloat[Operating point: zero current $I$ and $\alpha = 0\degree$]{\input{fig2/OP_PT_0A_0Deg}%
\label{fig:op1d}}
\hfil
\subfloat[Operating point: maximal current $I$ and $\alpha = 0\degree$]{\input{fig2/OP_PT_1335.8654A_0Deg}%
\label{fig:op2d}}
\caption{Operating points for the test design.}
\label{fig:evalopptsd}
\end{figure*}
\begin{figure}
\centering
\input{fig2/data_distribution}
\vspace{-0.25em}
\caption{Visualization of parameter and KPIs distributions.
}
\label{fig:DS}
\end{figure}%
\section{Procedure, Network architecture and training}\label{sec:MNAT}
This section discusses the procedure for calculating KPIs using a multi-branch DNN, followed by specifics on its architecture and training.
\subsection{Procedure}\label{subsec:MTH}
\begin{figure}
\centering
\scalebox{0.85}{
\input{fig2/Methodolog1}
}
\caption{Block diagrams of different approaches for calculation of KPIs.
}
\label{fig:MTH}
\end{figure}
\autoref{fig:MTH} shows block diagrams of three different approaches for the calculation of KPIs. The classical approach is a conventional way of computing KPIs and is widely used in industry. In this method, as shown in the diagram, the input parameters $p_i$ are fed to the FE simulation, and then the result $y_i$, along with the inputs $p_i$, are given to a physics-based in-house post-processing tool to estimate final KPIs $z_i$. The true functions for computing FE outcomes and KPIs can be written abstractly as,
\begin{equation}
\mathbf{z}=\mathbf{K}\bigl(\mathbf{p},{\mathbf{F}}(\mathbf{p})\bigr).
\label{eq:true_KF}
\end{equation}
Until now classical data-driven approaches, e.g. \cite{9333549}, proposed to learn the function composition $\textbf{p}\to\mathbf{z}$, i.e.
\begin{equation}
\mathbf{z}\approx\hat{\mathbf{K}}_\varphi\bigl(\mathbf{p}\bigr)
\label{eq:approx_K}
\end{equation}
where $\varphi$ represents DNN model parameters. Here, the approximation $\hat{\mathbf{K}}_{\varphi}$ works as a meta model to directly predict the KPIs. This model is trained by optimizing the model parameters $\varphi$, i.e., minimizing the difference between the ground truth and the prediction. The training loss function using the $\ell_{1}$-norm can be written as
\begin{equation}
\min_\varphi {L}(\varphi) :=
\sum_i \|\mathbf{z}^{(i)} - \hat{\mathbf{K}}_{\varphi}(\mathbf{p}^{(i)}) \|_1
\label{eq:train_K}
\end{equation}
where $\mathbf{z}^{(i)}$ are true results for the KPIs from $\mathcal{D}$. On the other hand, in our hybrid approach, we use the meta-model only to approximate the computational expensive function $\mathbf{F}:\mathbf{p}\to\mathbf{y}$ for the FE calculation. Similar as before this approach can be written as
\begin{equation}
\mathbf{z}\approx\mathbf{K}\bigl(\mathbf{p},\hat{\mathbf{F}}_\theta(\mathbf{p})\bigr)
\label{eq:approx_F}
\end{equation}
where now $\theta$ represents the model parameters. The training loss function is now
\begin{equation}
\min_\theta {L}(\theta) :=
\sum_i \|\mathbf{y}^{(i)} - \hat{\mathbf{F}}_{\theta}(\mathbf{p}^{(i)}) \|_1
\label{eq:train_F}
\end{equation}
where $\mathbf{y}^{(i)}$ are true results for the intermediate measures from~$\mathcal{D}$.
\subsection{Network architecture and training}
\begin{figure*}
\centering
\input{fig2/network_architecture}
\vspace{-0.5em}
\caption{Proposed multi-branch DNN network.}
\label{fig:MDNN}
\end{figure*}
The performance of deep learning models depends strongly on the choice of hyperparameters. They are fixed before the training, whereas the model parameters, such as biases, weights of DNN are optimized during training. For better understanding, we divide the hyperparameters into two categories: model hyperparameters and learning hyperparameters. The model hyperparameters are specified as the width and height of different network layers, filter size, stride, type of layer (dense, convolutional, max-pooling, etc.), and so on. The learning hyperparameters include, for example, learning rate, activation functions, optimizer, loss function, batch size, epochs, etc. We propose the multi-branch DNN as depicted in \autoref{fig:MDNN}. Initially, this structure was obtained by trial and error, i.e., evaluating more than ten different configurations. The hyperparameters were then numerically optimized. The parameter space is given in~\autoref{tab:HPO}. The hyperparameter optimization (HPO) was performed using an in-house optimization tool that includes implementation of the Asynchronous Successive Halving Algorithm algorithm \cite{MLSYS2020_f4b9ec30} for the multi-objective case. This tool has evaluated 800 different configurations within search space defined in \autoref{tab:HPO}. The entire HPO process took approximately two days with parallelization across four GPUs. The final hyperparameter configuration is given in the last column of \autoref{tab:HPO}. As shown in \autoref{fig:MDNN}, the input layer has a size of $37\times 1$ as there are 37 varying input parameters. There are four common layers (CL: $1590 \rightarrow 1410 \rightarrow 810 \rightarrow 210$) to exploit correlation among all the output measures. There are total five distinct branches with different number of branch layers (BL). There are two identical branches for torque $\hat{T}$ and iron loss $\hat{P}_{\mathrm{fe}}$ prediction, and three equal size branches for flux $\hat{\Psi}$ prediction. The network layers for iron loss and torque are $1530 \rightarrow 1210 \rightarrow 900 \rightarrow 880 \rightarrow 750 \rightarrow 660 \rightarrow 610 \rightarrow 580 \rightarrow 550 \rightarrow 530$. The branch configuration for other three branches are $322 \rightarrow 278 \rightarrow 240$.
As stated earlier in the section \autoref{sec:dataset}, each branch predicts intermediate measures for every operating point. In this study, we predict 15 time steps by the flux and torque branches, so the size of output layers remains $15 \times 1$ for both output measures. These figures, however, can be modified as needed. The output layer for iron losses is $4 \times 1$ in size. The total number of trainable network parameters is noted to be around 2.3 million. The actual number of machine designs ${N_{\mathrm{EM}}}= 44877$ is divided into training, validation and testing. The training sample consists $N_{\mathrm{train}}= 40390$ ($\sim 90\%$), whereas validation ${N_{\mathrm{val}}} = 2243$ and test samples ${N_{\mathrm{test}}} = 2244$ comprises $\sim 5\%$ each. Since each machine design is evaluated for all operating points, the total number of training samples is obtained after multiplication by~$N_{\mathrm{OP}}=37$.
The model training for the final configuration was carried out on a NVIDIA Quadro M2000M GPU using standard back-propagation \cite{rumelhart1986learning}. Tensorflow2 \cite{tensorflow2015-whitepaper}, a deep learning framework, was used to implement the training pipeline. It took around $\sim2$ hours to finish the training with early stopping criteria (10 epochs over validation error) for a maximum of 300 epochs. The training and validation curve is illustrated in \autoref{fig:TRN}. To allow a fair comparison with the direct DL approach, we used the same network architecture and model hyperparameters from \cite{9333549}. The model learning hyperparameters are the same for both approaches.
\begin{table*}
\caption{Details of Hyperparameters}
\label{tab:HPO}
\begin{tabularx}{\linewidth}{|X|r|r|r|}
\hline
\textbf{Hyperparameter} & \textbf{Min} & \textbf{Max} & \textbf{Optimized values} \\ \hline
Learning rate & \hspace{2cm}$10^{-6}$ & $10^{-3}$ & $2.6\cdot10^{-4}$ \\ \hline
Average number of neurons per hidden layer & $50\phantom{^{-6}}$ & $1000\phantom{^{-3}}$ & $991$ \\ \hline
Number of common layers & $1\phantom{^{-6}}$ & $6\phantom{^{-3}}$ & $4$ \\ \hline
Number of branch layers & $3\phantom{^{-6}}$ & $12\phantom{^{-3}}$ & $10$ \\ \hline
Batchsize & $80\phantom{^{-6}}$ & $150\phantom{^{-3}}$ & $132$ \\ \hline
Activation functions & \multicolumn{2}{c|}{relu, elu, tanh, softplus} & elu \\ \hline
Optimizers & \multicolumn{2}{c|}{Adam, adamax, adagrad, nadam \cite{8769211}} & Adam \\ \hline
Loss functions & \multicolumn{2}{c|}{Mean absolute error, mean squared error, huber loss} & Mean absolute error \\ \hline
\end{tabularx}%
\end{table*}
\begin{figure}
\centering
\input{fig2/training_plot}
\vspace{-0.6em}
\caption{Training curve.}
\label{fig:TRN}
\end{figure}
\section{Results and analysis}\label{sec:res}
We first showcase the evaluation of the intermediate measures using classical machine learning and multi-branch DNN. Following that, we explain the quantitative study with empirical results of the hybrid and direct DL approach. The characterization of the performance of a PMSM is a non-linear multi-output regression problem, and thus we chose to evaluate the performance of meta-models using several evaluation metrics, i.e., the mean relative error (MRE), mean absolute error (MAE) and the Pearson correlation coefficient (PCC).
\subsection{Discussion of the intermediate measures prediction}
\begin{figure}[!t]
\centering
\includegraphics[width = 0.9\columnwidth]{fig2/Iron_loss.png}
\vspace{-0.6em}
\caption{Prediction plot of losses over test samples.}\label{fig:Iron_loss}
\end{figure}
\begin{table}
\caption{Intermediate measures over test samples
}
\label{tab:IMFT}
\begin{tabularx}{\linewidth}{|X|r|r|r|r|}
\hline
\multirow{2}{*}{\textbf{Measure}} & \multicolumn{2}{c|}{\textbf{Multi-branch DNN}}
& \multicolumn{2}{c|}{\textbf{Random forest}}
\\ \cline{2-5}
& \multicolumn{1}{c|}{$\varepsilon_{\textrm{MRE}}$}
& \multicolumn{1}{c|}{$\varepsilon_{\textrm{PCC}}$}
& \multicolumn{1}{c|}{$\varepsilon_{\textrm{MRE}}$}
& \multicolumn{1}{c|}{$\varepsilon_{\textrm{PCC}}$}
\\ \hline
eddy current loss (rotor) & $1.5\cdot10^{0\phantom{-}}$ & $0.98$ & $9.02\cdot10^{0\phantom{-}}$ & $0.97$ \\ \hline
eddy current loss (stator) & $5.9\cdot10^{-1}$ & $0.99$ & $7.32\cdot10^{0\phantom{-}}$ & $0.98$ \\ \hline
hysteresis loss (rotor) & $1.6\cdot10^{0\phantom{-}}$ & $0.98$ & $9.48\cdot10^{0\phantom{-}}$ & $0.97$ \\ \hline
hysteresis loss (stator) & $4.8\cdot10^{-1}$ & $0.99$ & $7.09\cdot10^{0\phantom{-}}$ & $0.98$ \\ \hline
\hline
& \multicolumn{1}{c|}{$\varepsilon_{\textrm{MAE}}$} & \multicolumn{1}{c|}{$\varepsilon_{\textrm{PCC}}$} & \multicolumn{1}{c|}{$\varepsilon_{\textrm{MAE}}$} & \multicolumn{1}{c|}{$\varepsilon_{\textrm{PCC}}$} \\ \hline
\hline
Electromagnetic\;torque\;$T$ & $6.5\cdot10^{-1}$ & $0.99$ & $8.54\cdot10^{0\phantom{-}}$ & $0.97$ \\ \hline
Flux linkage $\Psi_{1}$ coil 1 & $1.0\cdot10^{-6}$ & $0.99$ & $2.40\cdot10^{-3}$ & $0.98$ \\ \hline
Flux linkage $\Psi_{2}$ coil 2 & $1.0\cdot10^{-5}$ & $0.99$ & $2.30\cdot10^{-3}$ & $0.98$ \\ \hline
Flux linkage $\Psi_{3}$ coil 3 & $1.0\cdot10^{-6}$ & $0.99$ & $2.80\cdot10^{-3}$ & $0.98$ \\ \hline
\end{tabularx}%
\end{table}
The problem at hand requires multi-output regression and there are several algorithms and implementations available. We considered several state-of-the-art approaches using the sci-kit-learn library \cite{JMLR:v12:pedregosa11a,buitinck:hal-00856511} with their default settings. The library contains an implementation of K-Nearest Neighbour (with KNeighborsRegressor) \cite{Omohundro89fiveballtree,10.1145/361002.361007}, Support Vector Regression \cite{10.1145/1961189.1961199}, and Gaussian Process Regression \cite{C.E.Rasmussen}, which were unable to process the data due to memory constraints. These algorithms require higher computational power (ca. five times higher in our experiments) as compared to our existing resources (NVIDIA Quadro M2000M GPU) to process such a large volume of high dimensional data. Random forest \cite{breiman2001random} is an ensemble method, that combines multiple decision trees to train a meta-model. The final model makes predictions by taking the average of the results of the different trees (bootstrap aggregation). The algorithm succeeded to train the meta-model using the default settings (number of estimators is $100$, the evaluation criterion is squared error, the minimum number of samples for each split is $2$, the minimum samples leaf is $1$ and the bootstrap aggregation enabled). However, with low accuracy and high computational costs: the training time is almost $\sim 10$ higher than for the multi-branch DNN, and the final test accuracy is much lower compared to multi-branch DNN. The comparison between multi-branch DNN and random forest over test samples for intermediate measures is given in \autoref{tab:IMFT}. From the results, it is evident that the multi-branch DNN outperforms the prediction accuracy of random forest. We use the MRE to evaluate iron losses because its range varies largely, while torque and flux with MAE. The reason is that the torque $T$ and the fluxes $\Psi_{1}, \Psi_{2}, \Psi_{3}$ are more sensitive. For the testing, the MAE is evaluated over the mean of the predicted time steps for each operating point. \autoref{fig:Iron_loss} depicts the prediction curve over each operating point for all the test samples. It is observed that the multi-branch DNN predicts intermediate measures with high accuracy close to the ground truth for all the test samples. The evaluation time for new machine designs is $\sim100$ ms/sample, which is much lower in comparison to the actual FE simulation ($\sim3-5$ hours/sample). \autoref{fig:evaloppts} illustrates two operating points for different input electrical excitation conditions for one test machine design. Figure~\ref{fig:op1p} unveils that the multi-branch DNN has poor prediction accuracy for the operating point at zero input phase current and open circuit (no load) condition.
\subsection{Quantitative analysis}
\begin{figure*}[ht]
\centering
\subfloat[Operating point: zero current $I$ and $\alpha = 0\degree$]{\input{fig2/OP_PT_0A_0Deg_p}%
\label{fig:op1p}}
\hfil
\subfloat[Operating point: maximal current $I$ and $\alpha = 0\degree$]{\input{fig2/OP_PT_1335.8654A_0Deg_p}%
\label{fig:op2p}}
\caption{Operating points for the test design.}
\label{fig:evaloppts}
\end{figure*}
\begin{figure}
\centering
\input{fig2/Performance_curve}
\caption{Different performance curves for one test design.
}
\label{fig:PC}
\end{figure}
\begin{figure*}[!ht]
\centering
\subfloat[Classical approach]{\includegraphics[width=0.33\textwidth]{fig2/EM_org.pdf}%
\label{fig:emorg}}
\hfil
\subfloat[Hybrid approach]{\includegraphics[width=0.33\textwidth]{fig2/EM_pred.pdf}%
\label{fig:empred}}
\hfil
\subfloat[Absolute calculation difference]{\includegraphics[width=0.33\textwidth]{fig2/EM_dev.pdf}%
\label{fig:emdiff}}
\caption{Illustration of efficiency map calculation.
}
\label{fig:EM_map}
\end{figure*}
\begin{figure}[ht]
\centering
\includegraphics[width=0.95\linewidth]{fig2/final_resultp.pdf}
\caption{KPIs prediction plot over test samples.}
\label{fig:pred}
\end{figure}
\begin{figure}[!t]
\centering
\input{fig2/final_result}
\caption{KPIs evaluation over varying training set size.}
\label{fig:cmp}
\end{figure}
\begin{table}
\caption{Hybrid and direct DL approach over test samples}
\label{tab:cmphydl}
\begin{tabularx}{\linewidth}{|X|r|r|r|r|}
\hline
\multirow{2}{*}{\textbf{KPIs}} & \multicolumn{2}{c|}{\textbf{Multi-branch DNN}}
& \multicolumn{2}{c|}{\textbf{Direct DL approach}}
\\ \cline{2-5}
& \multicolumn{1}{c|}{$\varepsilon_{\textrm{MRE}}$}
& \multicolumn{1}{c|}{$\varepsilon_{\textrm{PCC}}$}
& \multicolumn{1}{c|}{$\varepsilon_{\textrm{MRE}}$}
& \multicolumn{1}{c|}{$\varepsilon_{\textrm{PCC}}$}
\\ \hline
$z_{1}$ & \hspace{1.5em}$0.35$ & \hspace{1.5em}$1.00$ &\hspace{1.5em} $0.39$ & \hspace{1.5em}$1.00$ \\ \hline
$z_{2}$ & $0.34$ & $1.00$ & $0.40$ & $1.00$ \\ \hline
$z_{3}$ & $0.60$ & $0.99$ & $0.55$ & $1.00$ \\ \hline
$z_{4}$ & $1.62$ & $0.99$ & $3.42$ & $0.98$ \\ \hline
$z_{5}$ & $0.14$ & $1.00$ & $0.26$ & $1.00$ \\ \hline
$z_{6}$ & $0.13$ & $1.00$ & $0.16$ & $1.00$ \\ \hline
$z_{7}$ & $2.93$ & $0.99$ & $6.06$ & $0.98$ \\ \hline
\end{tabularx}%
\end{table}
Let us compare the proposed hybrid approach \autoref{fig:MTH} with the parameter-based direct DL approach described in \cite{9333549}. \autoref{fig:cmp} shows evaluation with MRE over an increasing training size from $5\%$ to the total training size. The hybrid approach consistently performs better for KPIs $z_{1}, z_{2}, z_{4}, z_{5},z_{6}, z_{7}$, while the direct DL approach is slightly more accurate for $z_{3}$. Both meta-models predict KPIs for unseen machine designs. \autoref{fig:pred} illustrates the prediction plot over the test samples for meta-models trained on the full training set. As explained in \autoref{fig:MTH}, the training of multi-branch DNN is independent of the system parameters and solely relies on varying geometry, electrical excitation, and material parameters, whereas in the direct DL approach, model training with output KPIs implicitly involve fixed value for the system parameters. This makes the hybrid approach more flexible than the direct DL approach. The post-processing of FE output only takes very little time ($\sim 3-5$ min.) and is performed using a physics-based in-house post-processing tool, and hence the hybrid approach opens up new possibilities for further analysis. Demonstrative examples for one test sample are shown in \autoref{fig:PC} and \autoref{fig:EM_map}. \autoref{fig:PC} displays different performance curves, e.g., maximum torque curve, open circuit, and short circuit voltage characteristic, and maximum shaft power at different rotor speeds. \autoref{fig:EM_map} shows the efficiency map for the given test design, see \cite{7310051} for a detailed interpretation. The Figure~\ref{fig:emorg} shows the efficiency map for the FE simulation, Figure~\ref{fig:empred} is calculated from the intermediate measures predicted with multi-branch DNN, while Figure~\ref{fig:emdiff} gives details on the deviation between the two. The difference is close to zero with a maximum near the low torque region ($\sim20\%$). A possible explanation is that multi-branch DNN does not predict well for the open circuit operation mode at zero input current, which is also reflected in Figure~\ref{fig:op1p} for the same associated sample. In the open circuit mode, the associated torque range for the electrical machine is low ($\sim 10^{-1}$ to $10^{-3}$ Nm). If the multi-branch DNN predicts values in the range of $\sim 10^{-2}$ to $10^{-4}$ Nm this results in a relatively large error. However, the relatively poor prediction in this region does not have significant impact on the calculation of the overall efficiency of the electrical machine at other operating points (especially high-efficiency operating point regions as shown in Figure~\ref{fig:emdiff}).
There are also a few disadvantages of the proposed hybrid approach. The training time of the multi-branch DNN is roughly about $\sim2$ hours, which is about $6\times$ higher than the training time ($\sim20$ minutes) of the DNN defined for direct KPIs prediction \cite{9333549}. A possible explanation is that the hybrid approach must deal with a high number of samples for model training ($37\times$ compared to the direct DNN in this study), and also the number of model parameters is around $\sim2.3$ million. Therefore, the hybrid approach requires higher computational power compared to DNN for the direct KPIs prediction. Also, the time to estimate the KPIs in the hybrid approach is increased from milliseconds to seconds due to the post-processing tool. However, the computational time remains much lower than the time for a FE simulation, which takes around $\sim 3-5$ hours/sample on a single-core CPU.
\section{Conclusion}\label{sec:conclusion}
In this paper, we presented a physics and data-driven hybrid method for the performance analysis of the electrical machine. The dataset used for the research is generated from a real-world industry design workflow. The proposed multi-branch DNN is trained to predict the intermediate measures using supervised learning. This hybrid approach is better than the existing parameter-based direct DL approach in terms of KPIs estimation and flexibility. This gain can be explained by two facts. Firstly, learning a few time-steps of intermediate measures is expected to be simpler than multiple (possibly independent) cross-domain KPIs and secondly, the post-processing tool exploits the laws of physics and thus ensuring that the KPIs fulfill the right constraints. Also on the application side, this approach makes the calculation of KPIs independent from the system parameters by predicting intermediate measures during the optimization and it enables the analysis of electrical machines with the calculation of more complex performance measures, e.g. efficiency map and characteristic curves. We have demonstrated that the trained multi-branch DNN meta-model evaluates new designs at much lower computational effort than the FE simulation. In future work, the proposed hybrid approach can be applied to many query scenarios, e.g. uncertainty quantification, sensitivity analysis and multi-objective optimization.
\bibliographystyle{IEEEtran}
|
1,116,691,500,845 | arxiv | \section*{Introduction}
In the mathematical field, equation solving has been a key action to progress in the evolution of the human species. For most of the human history, this task was performed by hand, involving long paper-written equations and solutions to be computed. First digital electronics and then computers have turn equation solving task into a simpler paradigm, exploiting digital computation and algorithms to achieve the searched results. Beside the close-form solutions algorithm, the main algorithm to solve equations still relies on brute-force search, where a solution is searched by exhausting the entire solution space.
\\
However, it is straightforward that this approach brings many limitation, such as energy consumption as well as the huge amount of time needed, if the solution space is wide. Nowadays, a quantum annealer like D-wave or Coherent Ising Machine can solve NP-hard problems in polynomial time\cite{hamerly2019experimental,inagaki2016coherent} by relying on other mechanism, inherited by the physics of the devices themselves, although the size of both two solvers is bulky in reference to an integrated chip. Similar to many unsolved problems in Computing Science that may be solved by the P vs NP problem and Rieman Hypothesis, here we would like to introduce an analog solver to solve a specific class of Partial Differential Equations (PDEs) in small size and a time-of-flight of photon's speed.\\ \\
First of all, PDEs are a widely known way to mathematically describe various amounts of problems, physical laws, and even economic behaviours\cite{renardy2006introduction}. There is a long history of human beings solving PDEs since their definition was introduced in the 18th century by Euler, d’Alembert, Lagrange and Laplace\cite{widder2015laplace}. There are several examples where PDEs are used in models to precisely describe behaviors and laws, and their rapid solution is therefore of significant practical importance. In a few instances, models that use PDEs could have precise closed-form solutions easily computed using mathematical formulas (e.g. propagation of an electromagnetic plane wave in a uniform medium). In general for most problems, however, close solutions cannot be found, leaving numerical solutions as the only way to achieve approximate results (e.g. for many models in fluid dynamics models that rely on Navier–Stokes equations\cite{temam2001navier}, as shown in figure \ref{fig:fig1}A). In fact, mathematicians, scientists and engineers have spent innumerable time and effort optimizing the solving mechanism. \\
Currently, automatic digital computation has brought a powerful tool to solve numerically PDEs, allowing recursive interaction to find the approximate results\cite{braun1995numerical}. The rise of CMOS as a way to fabricate computing units (such as CPU), where billions of transistors can be set to perform millions of mathematical operations per second, has paved the way for solving PDEs models making use of commercial software\cite{yentis1996vlsi}. The use of digital solvers has several advantages, in particular accessibility to interactive recursive approximation algorithms capable of iteratively improving solution accuracy. However, this approach comes with a major drawback, namely the excessive time required when approaching complex PDEs. As a consequence, large data centers have been used to process complex models, for example in multi-physics problems. To overcome such limitations, analog solvers have been studied\cite{liebmann1950solution}, which mimic complex physical models and obtain solutions in a fraction of time required by digital solvers. \\ \\
Here, we present a tunable-weights Application-Specific Photonic Integrated Circuit (ASPIC) which offers a state-of-art option for solving PDEs. By using photonics we can take advantage of the extremely large bandwidth as well as the extremely low latency and power consumption to mimic the PDE models and obtain approximate results, with accuracy that reaches up to 93\% as benchmarked by commercial digital solvers. Moreover, our proposed solution consumes a negligible amount of energy compared to state-of-the-art electronic CPUs or GPUs (Figure \ref{fig:fig1}B). \\ \\
In this paper, we first introduce the PDE model we address, and the related ASPIC as a PDE solver using photonic integrated circuits, showing how the ASPIC can map the actual physical problem, as in figure \ref{fig:fig1}C. We follow with characterization of our photonic solution using the Kirchhoff's Photonic Nodes (KPNs), and a photonic network implemented with KPNs. The last part will show passive and active ASPIC networks that mimic the diffusion of heat into homogeneous (passive network) and non-homogeneous (active network) media, highlighting the high accuracy attainable by our solution. The last part is dedicated to discussion of future improvements.
\section{Result}
\paragraph{ASPIC as a PDE Solver}
$ $ \\
We aim to solve the heat distribution over a homogeneous medium for our demonstration by mapping this problem to our ASPIC. This PDE-based model can be configured with different parameters and boundary conditions, such as different thermal conductivity of the materials or the heating flux at the boundary of our model.\\
While electronics, in particular resistors networks, can be a good analog solver for PDEs, given the intrinsic behavior of electromagnetic waves on lumped elements, photonics finds itself lacking these possibilities, as the wave propagation does not affect the previous elements in the first place (i.e. a modulator after one branch of a 1:2 splitter does not affect the other branch). To break this limitation, we introduced a photonic matrix network whose crossing nodes are formed by a four-port Kirchhoff’s Photonic Node (KPN) \cite{sun2021induced}\cite{shen2021application}, shown in figure \ref{fig:fig2}B. This four-port component has the property to split the light from one port to the other in a quasi-even fashion. This node element mimics the Kirchoff’s current law in optics, while inheriting all the advantages that optics brings, such as high speed, low latency, and robustness to electromagnetic interference. \\
\paragraph{KPN using Silicon Photonics}
$ $ \\
A series of KPNs and a full 5x5 matrix were realized on a Silicon Photonics passive platform\cite{bogaerts2018silicon}. We chose 500 nm waveguide width, using periodic grating couplers\cite{wang2013universal} to couple the light from the light source to the chip, as well as from the chip to the photodetectors, as shown in figure \ref{fig:fig2}B. Similar to the electrical counterpart, we need to extract the total power out from all the nodes present in the network. To do so, while reducing the losses that a direct detection would introduce, we place a 90\%:10\% Directional Coupler (DC) at the output of all the node's ports, directing the smaller amount of light to a grating coupler. By doing so, we can compute the power in the node by summing the power coming from all the ports, while the larger part of the light signal will continue toward the next node. The design of the DC was performed following Lu et al. method\cite{lu2015broadband}. We fabricate several different KPNs having different DC configurations that reflect the ones present in the main matrix. For example, a KPN placed on the corner of the matrix will have just 2 DC, as the ports that are not facing another node do not require a splitter before the grating coupler. Figures \ref{fig:fig2}C show the spectra of two KPNs, where light is coupled from the north port. The effect of the DCs are clearly visible in the second spectrum, where East and South ports have a 10 dB reduction compared to the West one. To notice that, as we consider the DC presence or absence among a series of 9 different configurations of KPN, we statistically see a small difference, as shown in figure \ref{fig:fig2}D. Moreover, the errors due to fabrication variability are reasonably contained.\\
The general spectra of the KPN shown in the same figure \ref{fig:fig2}C show a response resembling that of a ring resonator coming from the structure of the used KPN. In particular, KPN responses have a Free Spectrum Range of 8 nm (998 GHz) at 1550 nm, an average bandwidth of 1.26 nm, resulting in a Q-factor of 1000. This component can reach a high Extinction Ratio of more than 15 dB. \\
\paragraph{Silicon Photonics ASPIC}
$ $ \\
Using the defined KPN, we design and fabricate a 5x5 passive silicon photonic matrix, as shown in figure \ref{fig:fig2}A. We placed each node to be 500 $\mu$m apart, linking them with a straight silicon waveguide, having a 2.1 dB/cm estimated propagation loss. The input of our matrix is placed in node 1-B (using a coordinate system from the figure). In our heating model, this will be the place where the heating source is set.
The matrix we implement is a larger version of the previous one (from \cite{sun2021induced}), allowing a better resolution of the model we solve. A larger matrix is still possible. However, propagation losses, as well as optical power dynamic range, must be taken into account. For example, considering using the commercial laser with a power output of 10 dBm and photodetectors with sensitivity of -70 dBm, we can compute that the largest matrix the approach could afford will be a 6x6 one considering node splitting, as well as propagation and coupling losses. Some strategies can be used to address this limitation, for example, by using Silicon Nitride (SiN) waveguides that permit lower propagation losses, as well as other strategies like CLIPP\cite{grillanda2014non} to detect the output power from a single node that does not require a DC. \\
\paragraph{Heat Model Benchmark}
$ $ \\
In this problem, we are addressing a subset of PDEs describing heat distribution within a uniform medium. In particular, we apply these heating equations to a 2D surface, having a heat source located at the input of our 5x5 matrix and heat flux at the boundary of the surface.
We obtain our benchmark model result from COMSOL multiphysics, a known commercial and solid numerical iterative solver. By using the COMSOL solution, we can benchmark the accuracy of the solutions of a Laplace homogeneous PDE obtained by our ASPIC. The COMSOL model is formed by a 25 m$^2$ square film made of a fixed material (whose heat conductivity is set). A fixed temperature point is applied at the left top of the geometry, applying a constant temperature (393 K) as a heat source. Simultaneously all the remaining edges are set to have a heat flux toward the outside, which is set at 293 K to represent the heat sink of this model. We set the computation domain to be divided into 25 smaller squares, represented by our 5x5 model and computed the average temperatures of all these squares. This model can simulate the actual thermal transfer profile across the domain under the Dirichlet boundary conditions (BC).
Besides our experimental ASPIC, we built a one-on-one 5x5 optical mesh on the Lumerical Interconnect software to simulate the ASPIC and imitate the heat transfer mesh structure. At the crossing nodes, we used the scattering matrix parameter (S-parameter) of the KPN, computed to reproduce its optical characteristics. Same as our Silicon Photonic ASPIC, we extract the power from each node by placing a DC in all the ports of the KPN. Moreover, all the nodes are connected by a waveguide model, whose propagation loss is set to the experimental value and tuned to match the heat conductivity of our model. \\
\paragraph{Passive ASPIC Results}
$ $ \\
To measure our passive ASPIC, we inject a 10 $mW$ tunable continuous-wave laser, centered at 1550 $nm$, through the grating coupler at the input node. We then collect all the power output from all the ports of all the nodes to compute the actual power for our model. A spectrum of the four outputs from one KPN is shown in figure \ref{fig:fig3}A. The spectra are similar to the one presented in the previous section, with the same FSR. However, in this case, we can notice that the notches do not have the same shape or bandwidth as those of a single KPN. Since this node is inside a matrix, its spectrum is the result of all the incoming light beams from adjacent nodes that have slightly de-tuned microring notches, that results in a more complex line shape. Moreover, the non-optimized grating couplers produce a Fabry-Perot ripple that we eliminate by averaging over one period, similar to having a large bandwidth laser at the input. \\
After collecting the data from the measurement and from the simulations, considering that those quantities are incoherent in terms of measured units, we opted for a normalization scheme (0-100). The data are shown in figure \ref{fig:fig3}B, as a heat map of COMSOL, Interconnect simulation, and experimental data from the passive ASPIC.\\ \\
From these results, we can define the accuracy as:
\begin{equation}
A=1-\sum_{i=1}^n \frac{N_{i,c}-N_{i,a}}{_{Ni,c}}, \quad
\end{equation}
where $A$ is the total accuracy of the solution, $n$ is the number of the nodes, $N_{i,c}$ and $N_{i,a}$ are respectively the node values of COMSOL solution and solver solution (simulated or measured), normalized to a 0-100\% range. With our model, we obtain a 89.5\% accuracy for the Interconnect solution, and a 86.4\% accuracy for the experimental ASPIC (Figure \ref{fig:fig3}C). \\
The obtained accuracy is slightly below the minimum standard required for a solver (90\% or above), and it is mainly due to the mismatch of optical properties over the matrix, as well as the passive structure that does not allow tuning the losses between nodes to match the physical heat properties of the material in the model. However, the ASPIC solver can compute the model with a latency of hundreds of ns, compared to the COMSOL model, which requires several ms, making the optical solution several orders of magnitude faster.
\paragraph{Active ASPIC as PDEs solver using PCM}
$ $ \\
To further improve the accuracy of our ASPIC, a tunable element has to be placed between each node pairs to increase the losses and by so matching any material properties and boundary conditions. Among all the components we can use for this scope, such as Mach-Zendher interferometers, Variable-Optical Attenuator, and so on, we chose a non-volatile tunable element based on an integrated Phase Change Material (PCM), whose optical properties can be modulated with its crystalline structure. This choice enables ultra-compact integration, reduced system complexity, and high energy efficiency taking advantage of the non-volatile nature of PCM. Among the many PCM materials, we select GSSe ($Ge_2Sb_2Se_5$) since it provides an excellent Extinction Ratio (ER) between its amorphous and crystalline states, while having an almost negligible Insertion Loss at the C-band.
\paragraph{Phase Change Materials for Tunable Component }
$ $ \\
The integration of the GSSe has been done by depositing a thin strip over the waveguide and implementing a heater scheme to provide the Joule heating energy to switch the material between its two states\cite{meng2020multi}. A graphical representation of the implementation is shown in figure \ref{Fig:fig4}A. We place this component between each KPN node, as shown in figure \ref{Fig:fig4}B, permitting full reconfigurability of the ASPIC. \\
For the fabrication, we deposited a GSSe strip with 40 nm thickness and 100 $\mu$m length on top of the silicon waveguide, covered with a 10 nm $Al_{2}O_{3}$ layer to prevent oxidation of GSSe. Then, a 350 nm thick Tungsten-Titanium (W/Ti) alloy heater with identical length is paralleled, laying 1 $\mu$m away from the waveguide, routing to two 100 $\mu$m $\times$ 100 $\mu$m contact pads. On top of the contact pads and route, we deposit a 200 nm thick Aluminum layer to minimize the resistance. After the fabrication process, we measured each heater, resulting in an 11 $k\Omega$ resistance. \\
A heuristic approach is taken to study the sufficient power required for switching the GSSe's state. In accordance with the measurement results, we observed a hybrid pulse comprised of a 12 $V$ high voltage for 500 $\mu$ s and a 9.6 V low voltage for 2 $ms$ could switch the GSSe strip from the amorphous state to the crystalline state (Figure \ref{Fig:fig4}C). The spectrum plotted indicates that the absorption is 8 dB, which implies a 0.08 dB/$\mu$m absorption coefficient of the GSSe on-chip. Significantly, the imaginary part of the effective refractive index in the amorphous state is $2.18 \times 10^{-5}$, leading to an exceedingly small unit of passive insertion loss \cite{meng2022electrical}.\\
\paragraph{Active ASPIC as PDE solver}
$ $ \\
The tunable PCM elements implemented in our circuit impart full programmability to the ASPIC. Following a heuristic approach of switching different states in single or multiple GSSe strips within the matrix, we monitor the accuracy variation and seek the highest accuracy value. Over the course of time, by selectively switching the GSSe strips from the amorphous state to the crystalline state, we successfully achieved the solution corresponding to the same problem, solved by the Interconnect solver and passive ASPIC solver, with an accuracy of 93.2\% (Figure \ref{Fig:fig4}D), increased by 7.8\% from the full passive ASPIC. \\ \\
To demonstrate the flexibility of our active ASPIC, we propose two more approximate models of heat shields present in all space vehicles\cite{rasky2012perspective}. In both cases, this shield has to protect the whole vehicles against high-temperature environments in different parts of the vehicle body, particularly during the re-entry phase into the atmosphere, when the vehicle gets exposed to temperatures of more than 1000°C\cite{gordon2014space}. In the first case, we emulated a thermal shield with low heat conductivity in dual high temperature sources situation, that has the main goal to shield the bottom right square from the higher temperature source in figure \ref{Fig:fig4}E. After obtaining the COMSOL result as a benchmark, we iteratively switched various PCM strips between different nodes in the photonic circuit following the same heuristic approach. Thus, we can program the ASPIC to adapt the new model with 88.9\% accuracy, as show in the latter heatmaps of figure \ref{Fig:fig4}E. Similarly, the second COMSOL model presents now a high temperature source, and a thermal shield. The first heatmap in figure \ref{Fig:fig4}F shows the actual temperature distribution and the strong thermal insulation provided by the heat shield, and the accuracy of this model is 93.2\%. We can also observe from the power map comparison that the highest error occurs at the first layer in the heat shield. The explanation is that our PCM “modulators” only have a fixed tunable extinction ratio of 8 dB, and so, we can not arbitrarily tune the extinction ratio precisely in various states. However, by means of applying more PCM strips with different lengths deposited on the waveguides, we will be allowed to address this problem and increase the accuracy.
\section{Discussion}
As we showed, it is possible to solve the PDE model using photonic integrated circuits, achieving high accuracy. However, the circuit shows some limitations that must be taken into account.\\
First of all, the separation between two adjacent nodes sets the minimum amount of loss (as propagation loss) that the ASPIC has. This reflects a boundary in the model we want to solve, as the heating material cannot have a heat conductivity larger than the equivalent value obtained from the photonic circuit. This power budget problem can be addressed in several ways, from using a low loss waveguide (large-core or ridge silicon waveguides, or silicon nitride (SiN) waveguides) for reducing the distance between nodes. \\
In fact, this latter one comes with the need to assess the optical power out from every single node. As shown in figure \ref{Fig:fig4}B, most of the space is occupied by the grating couples and directional couplers from each node's output ports. This design has several limitations, such as footprint occupation and power budget, since it requires 90:10 couplers and low cross-talking cross-nodes. Using integrated photodetectors could improve the overall KPN footprint. Solutions, such as CLIPP\cite{morichetti2014non, carminati2017design}, could further enhance this design, as they have the feature of being transparent, and they can be directly integrated on the waveguide to be monitored. \\
A third element limiting the current design comes from the PCM as a tunable attenuator. In our case, the GSSe material has an absorption coefficient varying from 0.07 dB/$\mu$m to ~ 0.2 dB/$\mu$m by applying variable voltage pulses. However, having a single strip does not allow sufficient variable attenuation, limiting our capabilities to match the materials' proprieties in the heat distribution model. A solution that implements four strips of GSSe with different lengths between two KPNs can achieve much larger dynamic range in terms of loss modulation and further improve our system's accuracy. Integrating other materials could open the circuit to more configuration options, for example, implementing a proper "heat insulation" or "heat reflecting shield" by increasing the ER or producing a variation of the effective index that will return a backpropagating wave.\\ \\
The last point focuses on the boundary conditions. In our circuit every port towards the outside of the matrix are directly connected to a grating coupler for coupling all the light out. This can be seen in the heat model as a heat flux toward the outside. However, to fully generalize our ASPIC to all the possible model, we have to permit different configurations at each external port, for example switching between a grating coupler, that extracts all the optical signal, and a Sagnac Loop\cite{fernandez2019universal}, whose function is to reflect completely the light back into the matrix. This later solution could be used as heat insulation, providing an ASPIC that can solve more complex PDEs models.
In summary, we design and test an Application-specific Photonic Integrated Circuit using KPN capable of solving the analog partial differential equation for a heat distribution problem, with over 90\% accuracy. The Silicon Photonics integration permits to obtain a chip with a size smaller than a fingernail, over 50 nm optical bandwidth, and most importantly, obtain the results in a nanosecond scale. We show a passive and active version of the ASPIC, showing the higher accuracy we can achieve once we can tune the Phase Change Material between two nodes to match the properties of the model materials. With the progressive Through Silicon Via (TSV) and multi-layer technology\cite{zhang2020scalable}, we will be able to design the optical and electrical routes through different layers, achieving more complicated functionalities. Also, by add the ring modulators to the matrix, it is possible to reconfigure each node to achieve different splitting ratios, so that different types of circuit can be configure, from optical filter, multiplexer, de-multiplexer, and more complicated version for artificial intelligence and neuromorphic computing\cite{shastri2021photonics}.
\section{Method}
\paragraph{Device fabrication}
$ $ \\
The passive silicon photonic circuits are fabricated using an electron-beam lithography (EBL) process on a silicon on insulator (SOI) wafer by Applied Nanotools Inc. The complete circuit was designed using Klayout, a design framework for integrated circuitry. The GSSe deposition process is done by single-source thermal evaporation following previously disclosed protocols.\cite{zhang2019broadband} The microheater fabrication process is accomplished in two steps in the nanofabrication and imaging center at George Washington University. A tungsten titanium layer with $350 nm$ thickness represents the heater, route, and contact pads are evaporated using a DC sputter process. Then a $200 nm$ thick Aluminum layer is deposited on top of the tungsten titanium route and pads. (Figure 4)
\paragraph{Measurement setup}
$ $ \\
For the optical measurement, we used the Single-Die Probe Station as well as the measuring system from Maple Leaf Photonics.
For optical power transferred outside the ASPIC, we used two 8-parallel-port fiber arrays with 127 $\mu m$ pitch to inject the light in and capture the light out through grating couplers. The laser source and detector devices we use are Keysight N7778C and N7745A, respectively, both functionally programmed in the Maple Leaf Photonics measuring system.
In the active measurement part, we use the National Instrument PXIe-5413 waveform generator module to generate the pulse for heating the PCM strips.
\paragraph{Heat transfer simulation}
$ $ \\
The heat transfer simulation mapping with the partial differential equation is performed using a two-dimensional finite element model in COMSOL Multiphysics. We use the stationary “heat transfer in solid” module, which elucidates the thermal boundary conditions. The simulated area is 5m x 5m, equally divided into 25 (5x5) squares using line segments. The equations applied are:
\begin{equation}
d_{z}\rho C_{p}u\cdot \nabla T + \nabla\cdot q = d_{z}Q+q_{0}+d_{z}Q_{ted}
\end{equation}
\begin{equation}
q = -d_{z}k\nabla T
\end{equation}
Where $\rho$ represents the density of the material ($kg/m^{3}$), $Q$ is the heat source, $u$ is the velocity, $q$ is the heat flux and $k$ is the thermal conductivity ($W(m\cdot K)$). In our stationary case, $d_{z}$ is out-of-plane thickness, $C_{p}$ represents the specific heat capacity($J/(kg\cdot K)$), and $Q_{ted}$ is the thermoelastic damping. \\
The initial value of the whole domain is 273 K, and the heat source temperature is 393 K, while all the remaining edges are set to have a heat flux toward the outside set at 293 K to represent the heat sink of this model.
\paragraph{Optical simulation}
$ $ \\
We use Lumerical Interconnect simulation software to emulate the photonic circuit on an integrated chip. S-parameter elements with user-defined files are employed to represent the Kirchhoff Photonic Node. The continuous-wave laser is placed to generate the light at a 1550 nm wavelength. Different S-parameter elements with simple parameter numbers are exploited to mimic the optical absorption change tuning function of the PCM. We also set the splitting ratio of optical waveguide coupler elements to characterize directional couplers. Oscilloscope elements are used to read and record the power output.
\section{Acknowledgement}
This work was performed in part at the George Washington University Nanofabrication and Imaging Center (GWNIC), as well as at the Harvard Center for Nanoscale Systems (CNS).
C.P. and J.H acknowledge funding support provided by NSF under award number 2132929.
\section{Contribution}
V.J.S. and T.E. conceived the original ideas, acquired the funds, and supervised the project. C.S designed and fabricated the photonic integrated circuit. C.P prepared the phase change material. C.S and N.P developed the relevant theories and analyses for the project, measured the PIC, and discussed the result. C.S processed the data and plotted the figures. J.M and X.M initiate the measurement environment. All authors discussed the results, contributed to writing the manuscript and provided they feedback.
\bibliographystyle{Science}
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1,116,691,500,846 | arxiv |
\section*{Acknowledgment}
The authors would like to thank Xiang Yi, Donggyu Kim, David Bono, and Cheng Peng at MIT for technical discussions and their assistance during prototyping and testing.
\section*{Appendix I: Estimation of the Contribution of the Microwave Phase Noise on the Sensor Sensitivity}\label{sec_appendix_2}
The microwave signal that drives the NV centers can be represented as:\begin{equation}
\label{eqn_VRF}
\begin{split}
V_{RF} = &V_0 \cos{(\omega_0t+\phi_n(t))},
\end{split}
\end{equation}
where $V_0$, and $\omega_0$ are the microwave signal amplitude and frequency, respectively.
$\phi_n(t)$ is the phase noise of the microwave frequency. The instantaneous frequency can be represented as:
\begin{equation}
\label{eqn_Omegat}
\begin{split}
\omega(t)= \omega_0+\frac{d\phi_n(t)}{dt},
\end{split}
\end{equation}
The frequency fluctuations around the operating frequency ($\omega_0$) is related to the phase noise by:
\begin{equation}
\label{eqn_deltaf}
\begin{split}
\delta\omega = \frac{d\phi_n(t)}{dt},
\end{split}
\end{equation}
In our case, a lock-in amplifier is used to observe the noise around the modulation frequency ($\omega_m$). Therefore, $\phi_n(t)$=$\phi_n\cos{2\pi f_m t}$ can be assumed to have a sinusoidal form, where $\phi_n$ is the phase noise of the microwave source at the modulation frequency ($f_m$) in dBc/Hz. This means the peak frequency fluctuation is $\approx 2\pi f_m \phi_n$. Therefore the voltage spectral density due to the PM to AM noise conversion is represented as: \begin{equation}
\label{eqn_Vn}
\begin{split}
\sqrt{\frac{\bar{V_n^2}}{\delta f}} \approx 2\pi f_m \phi_n m,
\end{split}
\end{equation}
where $m$ is the slope of the FM curve in V/Hz.
\section*{Appendix II: Estimation of Sensitivity for Future Hybrid CMOS-NV Magnetometers}\label{sec_appendix}
The sensor sensitivity is inversely proportional to the $SNR$ of the sensor output
\begin{equation}\label{eqn_sensitivity_App}
S \propto \frac{1}{SNR} = \frac{\sigma_i\cdot\Delta\nu}{\gamma_e\cdot I_r},
\end{equation}
where $\sigma_i$ is the total noise current density (unit: A/Hz$^{1/2}$), $\Delta\nu$ is the linewidth of the ODMR curve, $I_r$ is the photo-current due to the red fluorescence signal, and $\gamma_e$ is 28~GHz/T. From our experiments, the linewidth of the ODMR is 6~MHz. The noise is limited by the shot noise of either the green excitation (for low optical filtering ratio) or the red fluorescence (for high optical filtering ratio):
\begin{equation}\label{eqn_noise_App}
\sigma_i = \sqrt{2\cdot q \cdot (I_g + I_r)},
\end{equation}
where $q$ is the electronic charge, and $I_g$ and $I_r$ are the photo-currents due to the unfiltered green light and red fluorescence, respectively. The value of $I_g$ is estimated as:
\begin{equation}\label{eqn_Igreen_App}
I_g = P_g\cdot R_{PD}\cdot\eta_g,
\end{equation}
where $P_g$ is the input optical power, $R_{PD}$ is the photodiode responsivity, and $\eta_g$ is the green rejection of the optical filter. The value of $I_r$ is proportional to the total number $n_{nv}$ of NV centers in the diamond, which is proportional to the sensing area $A_{nv}$ and NV density $\rho_{nv}$. Lastly, the performance using the spin-controlled pulse sequence is estimated based on the fact that the equivalent transition linewidth (hence the zero-crossing slope $m$) of the configuration is reduced by $\sim$100$\times$ for AC magnetic field measurements \cite{Taylor2008High-SensitivityResolution}, \cite{barry2019sensitivity}.
\section{A Scalable CMOS-NV Magnetometer for Enhanced Sensitivity}\label{sec_arch}
In this section, the design details of a new-generation CMOS-NV magnetometer\cite{Ibrahim2019ACapability} are provided. The highly scalable architecture of our CMOS-NV magnetometer allows for the microwave driving of NVs over a large area. The photodiode noise is further reduced with the adoption of a new on-chip photonic filter. These structures are co-designed with the on-chip electronics.
\subsection{Systematic Architecture of the Chip}
The overall schematic of the CMOS-NV magnetometer is given in Fig.~\ref{fig_architecture}. Shown on the left of Fig.~\ref{fig_architecture} is an on-chip phase-locked loop (PLL) which generates the 2.87-GHz microwave signal (see Section~\ref{sec_PLL}). To drive the NVs with the on-chip generated microwave field, an array of current-driven linear wires are implemented using the M8 of the chip, of which the driving currents are toggled by the PLL output. Such a design addresses a major challenge regarding the uniformity of the microwave magnetic field over a large area. In Section~\ref{sec_launcher}, detailed explanations of the microwave launcher are provided. In this work, a diamond area of $\sim$500$~\times$~500~$\upmu$m$^2$ is excited by the microwave. The sensing area with uniform microwave excitation is limted to 300~$\times$~80~$\upmu$m$^2$ as discussed in Section~\ref{sec_co_design}. Shown in Section~\ref{sec_filter} and \ref{sec_co_design}, the current-driven wire array, along with additional two layers of metal gratings, also form a photonic filter in order to suppress the green light transmitted through the diamond placed on top of the chip. Finally, the spin-dependent red fluorescence of the NV centers is measured using a n-well/p-substrate photodiode.
\section{Conclusions}\label{sec_conclusion}
\begin{figure}[!b]
\centering
\includegraphics[width=3in]{figures/fig_sensitivity_comp.pdf}
\caption{Estimation of sensitivity for hybrid CMOS-NV magnetometers with different configurations.}\label{fig_sens_comp}
\end{figure}
In this paper, we present a vector-field magnetometer, which is based on a CMOS chip-scale platform with a hybrid integration of color centers (NV centers) in diamond. The chip consists of the essential components for manipulation and detection of the spin states of the NV centers in diamond. We demonstrate room-temperature magnetic field sensing with a sensitivity of 245~nT/Hz$^{1/2}$, which is 130$\times$ better than our previous prototype \cite{Kim2019a}. A microwave launcher with uniform microwave generation is introduced that can be scaled up for large diamond areas (i.e. large $SNR$). An enhanced multi-layer nanophotonic filter for lower background noise is also presented.
The scalable architecture and component designs presented in this paper provide a clear pathway to further push the sensitivity of the proposed sensor to sub-nT/Hz$^{1/2}$. This is shown in Fig.~\ref{fig_sens_comp}, which plots the estimated sensitivity with varying optical filtering performance, sensing areas and NV-centers densities (see Appendix II for more details). As Shown in Fig.~\ref{fig_grating_patterns}, current locations for lower optical grating layers are still not optimal due to the fixed BEOL stack configuration. In more advanced CMOS technology nodes, where a denser stack of thinner metal layers are available, the optical filtering performance can be further improved. For the microwave launcher, a better optimized configuration of the boundary wire arrays (i.e. Group \emph{D} in Fig.~\ref{fig_finite_wire_array_w_boundary}) can lead to high magnetic homogeneity over a larger proportion of the total launcher area. For example, we find that if each Group~\emph{D} consists of $m$=4 wires ($d$=4~$\upmu$m) with $I_D$=6.35~mA, and a gap of 12~$\upmu$m exists between Group~\emph{D} and Group~\emph{A}/\emph{C}, 95\% magnetic homogeneity is achieved for the entire -60 to 60-$\upmu$m space above Group \emph{A-B-C}. Homogeneity over large space also promises a significant sensitivity enhancement via spin-controlled pulse sequence \cite{barry2019sensitivity}. Lastly, the ability to manipulate the NV centers over a larger area also enables the integration of larger numbers of detectors for gradient magnetometry, multiplexed analytical nuclear-magnetic resonance (NMR) spectroscopy\cite{glenn2018high}, atomic gyroscopes\cite{Jaskula_2019}, and other quantum-sensing applications \cite{degen2017quantum}
\section{Introduction}\label{sec_introduction}
Solid-state quantum sensors are attracting broad interests thanks to a combination of excellent sensitivity and long-term stability. In particular, ensembles of nitrogen-vacancy (NV) centers in diamond have emerged as an outstanding \emph{room-temperature} sensor platform \cite{Gruber1997ScanningCenters,Balasubramanian2008,Taylor2008High-SensitivityResolution,LeSage2013OpticalCells,Steinert2013MagneticResolution}. Ensemble NV-based magnetometry has achieved sensitivities at or below the picotesla level\cite{PhysRevX.5.041001,Clevenson2014}, with applications ranging from bacteria magnetic imaging\cite{LeSage2013OpticalCells}, NMR spectroscopy\cite{bucher2018hyperpolarizationenhanced} to wide-field microscopy of superconducting materials \cite{PhysRevApplied.10.034032}. However, an impediment to fieldable devices lies in the co-integration of different subsystems needed for optically detected magnetic resonance (ODMR), including microwave generation/delivery to diamond, optical excitation, filtering and fluorescence-based spin detection (see Section~\ref{sec_physics}).
We recently addressed this integration problem by performing ODMR with a custom-designed CMOS circuit coupled to a diamond NV ensemble\cite{Ibrahim2018}, \cite{Kim2019a}. However, insufficient pump light rejection and limited area of homogeneous microwave driving fields for the ODMR measurements posed a major limitation on achievable magnetic field sensitivity. Here, we present a new CMOS prototype that addresses these problems to achieve a $>$100$\times$ sensitivity improvement, down to 245 nT/Hz$^{1/2}$\cite{Ibrahim2019ACapability}.
The manuscript is organized as follows. Section~\ref{sec_physics} summarizes the basics quantum magnetometry and reviews the first NV-CMOS prototype\cite{Ibrahim2018,Kim2019a}. Section~\ref{sec_arch} describes the improved chip architecture that incorporates high-homogeneity microwave delivery and diffraction- and plasmonic optical filtering, realized in a 65-nm CMOS process. Section~\ref{sec_measurement} presents performance measurements. Section~\ref{sec_conclusion} concludes with an outlook on further improvements to magnetometry and the incorporation of additional quantum sensing functions.
\section{Chip Prototype and Experimental Results}\label{sec_measurement}
The chip prototype is fabricated using a TSMC 65-nm CMOS technology. The chip micrograph is shown in Fig.~\ref{fig_chip_photo}. The chip area has an area of 1$\times$1.5~mm$^2$ and consumes 40~mW of power. A diamond slab with NV centers is placed and attached onto the top of the CMOS chip as shown in Fig.~\ref{fig_architecture}.
This diamond is a single crystalline CVD-grown diamond from Element 6. It is electronically irradiated with a dosage of $10^{18} \ \text{e}^- /\text{cm}^{2}$ at 1~MeV, and then annealed for 2~hours at 850$^{\circ}$C. Immersion oil is used to adhere the diamond slab to the chip. By bridging the difference of the refractive index, the oil also minimizes the fluorescence loss. The measured filter rejection for green light (532~nm) is $\sim$25~dB, which well match the simulation results in Section~\ref{sec_filter}. A 45$^{\circ}$ cut is introduced in the diamond’s corner to direct the vertical incident green laser horizontally to further enhance the overall green rejection ratio to $\sim$33~dB. The measured responsivity of the on-chip photodiode is 0.19~A/W.
\begin{figure}[h]
\centering
\includegraphics[width=3in]{figures/fig_chip_photo.pdf}
\caption{The micrograph of the CMOS quantum magnetometry chip.}\label{fig_chip_photo}
\end{figure}
\begin{figure} [!b]
\centering
\subfloat[]{\includegraphics[width=3.3in]{figures/fig_test_setup.pdf}\label{fig_test_setup}}\\
\subfloat[]{\includegraphics[width=2.6in]{figures/fig_wms.pdf}\label{fig_wms}}
\caption{(a) The test setup for the hybrid NV-CMOS magnetometer. (b) The frequency-modulation scheme used in the setup.}
\end{figure}
\subsection{Optical Detected Magnetic Resonance Measurements}
The measurement setup to generate the ODMR spectral line is shown in Fig.~\ref{fig_test_setup}. A 532-nm laser light is used to excite the NV centers in the diamond. The red fluorescence from the on-chip photodiode is recorded while sweeping the reference signal ($\sim$100~MHz) of the on-chip PLL. Due to the limited green-red suppression ratio of the on-chip filter, the red fluorescence signal is still superposed on a large green-light background at the output of the photodiode. Meanwhile, we notice the green-light excitation exhibits large power fluctuation, so to remove the fluctuating background signal, a frequency-modulation (FM) scheme is used, where the chip-generated microwave probing frequency $f_p(t)$ is periodically varied with a deviation of 6~MHz and repetition frequency $f_{m}$ of 3~kHz. That is done by modulating the 100-MHz reference signal of the on-chip PLL. As shown in Fig.~\ref{fig_wms}, the corresponding photodiode output changes at $f_{m}$ (with harmonics at 2$f_m$, 3$f_m$, etc.). Such a time-varying component mainly results from the transition line and the associated red-fluorescence change, which is then readily measured by a lock-in amplifier (SR865) with the reference at $f_m$. The lock-in amplifier also provides narrow-band filtering around $f_m$. Note that the FM technique is common in spectroscopy\cite{Wang2018} and atomic/molecular clocks\cite{Wang2018b}. Lastly, to further de-embed the excitation noise within the lock-in amplifier bandwidth, a differential detection scheme is adopted, where a split beam of the same laser is measured by an off-chip photodiode; such a duplicated noise signal is then taken by the lock-in amplifier and is subtracted from the chip output. Two identical resistive transimpedance amplifiers (TIA) with gain of 10K are used to convert the photodiode currents into voltage. This gain is chosen to prevent the saturation of the amplifier due to the green current background (40~$\upmu$A)
First, without an externally applied magnetic field, an ODMR plot is obtained as shown in Fig.~\ref{fig_ODMR_NoField}. A strong spin resonance, which results from all four NV centers in the diamond lattice, is detected at 2.87~GHz. The FM technique introduced previously can be interpreted as taking the first-order derivative of the regular Gaussian-shape transition line; as a result, the measured curve has the zigzag shape shown in Fig.~\ref{fig_ODMR_NoField}. The zero-crossing point corresponds to the actual transition frequency. Next, a 5.4-mT DC magnetic field is applied to the sensor from a nearby permanent magnet. This results in the Zeeman splitting for the spin states of the NV centers. As shown in the measured ODMR plot in Fig.~\ref{fig_ODMR_Field}, four pairs of resonances are observed. They correspond to the four NV-center orientations of the single crystalline diamond. The difference of the transition frequencies (i.e. zero-crossing points) in each resonance pair is proportional to the external static magnetic field along the associated NV-center orientation. The proportionality constant, as described in Section~\ref{sec_physics}, is $\gamma_e$=28~GHz/T. Accordingly, the magnetic fields along the four NV-center orientations are 0.77~mT, 2.27~mT, 3.25~mT, and 4.77~mT, respectively. Lastly, as the comparison between Fig.~\ref{fig_ODMR_NoField} and \ref{fig_ODMR_Field} shows, the chip output signal with external magnetic field is reduced due to the breaking of the degeneracy of the resonances from 2.87~GHz. Fig.~\ref{fig_ODMR_Field} clearly indicates that, by mapping the projections of the external magnetic field along the four NV-center orientations and certain frame of reference, one can construct the full vector magnetic field.
\begin{figure}
\centering
\subfloat[]{\hspace{-4mm}\includegraphics[width=2.8in]{figures/fig_ODMR_NoField.pdf}\label{fig_ODMR_NoField}}\\
\subfloat[]{\includegraphics[width=2.9in]{figures/fig_ODMR_Field.pdf}\label{fig_ODMR_Field}}
\caption{Measured optically-detected magnetic resonance (ODMR) plot from the CMOS chip, when (a) no external magnetic filed is applied, and (b) a 5.4-mT external magnetic field is applied from a certain angle.}
\end{figure}
One practical way to use the sensor for vector-field sensing, is to first ``bias" the sensor with a static magnetic field (e.g. a permanent magnet), which allows for the four resonances to be completely split (such as shown in Fig.~\ref{fig_ODMR_Field}). Then, we record the additional change of the sensor output voltage $\Delta v_{out,i}$ ($i$=$\pm$1, $\pm$2, $\pm$3, $\pm$4) around each zero-crossing point of the FM-based ODMR curve. Note that $\Delta v_{out,i}$ is caused by the shift of each resonance frequency $\Delta f_i$ due to the projection of the added field on the associated NV-center axis $\Delta B_{zi}$ (to be measured) on top of the bias static field, its expression is (for positive $i$):
\begin{align}\label{eqn_delta_vout}
\Delta v_{out,i}=m\Delta f_i=m\gamma_e\Delta B_{zi},
\end{align}
where $m$ is the slope of the ODMR curve at each transition zero-crossing point ($\sim$15~$\upmu$V/MHz in our case, see Fig.~\ref{fig_ODMR_Field}). The value of $\Delta B_{zi}$ can be derived from $\Delta B_{zi}=\Delta v_{out,i}$/$(m\gamma_e)$. Note that the transition frequencies of both $\ket{m_s=\pm 1}$ sub-levels are also temperature dependent (with a coefficient of -74~kHz/K\cite{Acosta2010TemperatureDiamond}). To cancel such temperature-induced drifts, it is better to measure the differential change of each pair of $\ket{m_s=\pm 1}$ transitions, and use the following equation for $\Delta B_{zi}$\cite{Kim2019a}:
\begin{align}
\Delta B_{zi}=\frac{\Delta v_{out,i}-\Delta v_{out,-i}}{2m\gamma_e}~~~(i=1, 2, 3, 4).
\end{align}
\subsection{Magnetic Sensitivity Estimation}\label{sec_noise_analysis}
In order to calculate the magnetic sensitivity of the sensor, the noise floor, $\sigma$ (unit: V/Hz$^{1/2}$), is measured. The sensor noise is measured by monitoring the read-out of noise in the lock-in amplifier while sweeping the modulation frequency $f_m$. The results are shown in Fig.~\ref{fig_noise_floor}, where the sensor shot noise floor is $\sim$0.1~$\upmu$V/Hz$^{1/2}$. The increased noise below 3~kHz is caused by the interference from the unshielded testing environment. This allows us to calculate the magnetometer sensitivity $S$ (unit: T/Hz$^{1/2}$), which is the minimum detectable magnetic field with 1-s integration time, using the following equation:
\begin{equation}\label{eqn_sensitivity}
S=\frac{\sigma}{\gamma_em}.
\end{equation}
The measured magnetometry sensitivity of this sensor at $f_m$=3~kHz is 245~nT/Hz$^{1/2}$.
The sensor performance is limited by the green light noise. The estimated theoretical magnetic sensitivity ($S_g$) of the sensor can be estimated using the following:
\begin{equation}\label{total_noise}
S_g = \frac{\sigma_g}{\gamma_e m} = \frac{\sqrt{2(2qi_gR_f^2 + 4KTR_f)}}{\gamma_e m}
\end{equation}
where $\sigma_g$ is the theoretical noise limit from the green light. The first term of $\sigma_g$ represents the shot noise of the green light detected by the photodiode. $q$ is the electronic charge, $i_g$ is the current measured at the photodiode due to the green light ($\sim$40~$\upmu$A), and $R_f$ is the feedback resistor of the TIA (10~k$\Omega$). The second term represents the noise of the TIA (assuming it is mainly limited by $R_f$). $K$ is the Boltzmann's constant, $T$ is the temperature. The factor of 2 is due to the addition of the noise power due to the two detection branches. The calculated theoretical value is ~$\sim$150~nT/Hz$^{1/2}$, which agrees to good extent with the measured value, and the difference is due to the residual un-canceled green light fluctuation. Even though this differential measurements technique increase the total shot noise limit by a factor of $\sqrt{2}$, it helps in canceling almost all the external laser fluctuation, which is almost one order of magnitude larger than the shot noise limit in our case.
\begin{figure}[t]
\centering
\includegraphics[width=3in]{figures/fig_noise_floor_New}
\caption{The measured output noise floor of the sensor.}\label{fig_noise_floor}
\end{figure}
To confirm that the green light noise is the dominant source of noise. The contributions of the other noise sources in the system are estimated as follows: \begin{enumerate}
\item The magnetic sensitivity due to the NV red fluorescence shot noise ($S_r$) is $\sim 10~\text{nT} / \sqrt{\text{Hz}}$ at $f_m =3 \text{ kHz}$ as given in:
\begin{equation}\label{red_noise}
S_r = \frac{\sigma_r}{\gamma_e m} = \frac{\sqrt{(2q i_r)R_f^2}}{\gamma_e m} \approx \frac{\sqrt{(2q\frac{V_{max}}{CR_f})R_f^2}}{\gamma_e m}.
\end{equation}
In (\ref{red_noise}), $V_{max}$ is the maximum voltage of the ODMR curve in Fig.~\ref{fig_ODMR_Field}, which is $\sim$45~$\upmu$V, and $C$ is the ODMR contrast of $\sim 0.01$.
\item The magnetic sensitivity ($S_m$) due to the amplitude noise converted from the microwave generator spectral purity is determined by (see Appendix I):
\begin{equation}
\label{eqn_Sm}
\begin{split}
S_m = \frac{\sigma_m}{\gamma_e m} \approx \frac{2\pi f_m \phi_n m}{\gamma_e m} \approx \frac{2\pi f_m \phi_n}{\gamma_e},
\end{split}
\end{equation}
where $m$ is the slope of the FM-ODMR curve. Based on a measured PLL phase noise ($\phi_n$) of -88~dBc/Hz at 3~kHz (FM modulation frequency $f_m$). The above magnetic sensitivity is $\sim 30~\text{pT}/\sqrt{\text{Hz}}$.
\end{enumerate}
\subsection{Generation of High-Homogeneity Magnetic Field}\label{sec_launcher}
Our NV-CMOS sensor interrogates an ensemble of NV centers to perform magnetometry. The microwave field strength determines both the ODMR resonance amplitude (contrast) and the resonance linewidth. Optimizing the sensitivity requires maximizing the contrast while minimizing the resonance linewidth \cite{Dreau2011AvoidingSensitivity}. For ensembles, the delivery of a homogeneous microwave magnetic field is critical in order to simultaneously perform this optimization across the entire area \cite{Eisenach_Loop}. Such microwave homogeneity is also critical to pulse-based coherent quantum control protocols, such as Ramsey-type sequence, which can significantly increase the sensitivity to time-varying external magnetic field \cite{Taylor2008High-SensitivityResolution}. Homogeneous microwave \emph{synchronously} rotate the spin states of a large number of NV centers on the Bloch sphere\cite{Bayat2014,Zhang2016}. Since the microwave-field strength determines the Rabi nutation frequency of each NV electron spin, spatial variation of the microwave field causes dephasing of the overall quantum ensemble.
Traditional microwave-launching structures include single straight wires\cite{Dreau2011AvoidingSensitivity,LeSage2013OpticalCells}, metal loops\cite{Stanwix2010CoherenceDiamond,Clevenson2014,Steinert2013MagneticResolution} and split-ring resonators\cite{Bayat2014,Zhang2016}. They can only keep the field homogeneity in an area that is much smaller than the launcher size. That is undesired for compact chip implementation when excitation of a large-size diamond is pursued. Meanwhile, the above structures also have poor power-delivery efficiency, hence watt-level microwave input power is common\cite{Clevenson2014,Bayat2014,Zhang2016}. We note the above solutions share one commonality in that they all rely on \emph{one piece of passive structure driven by a single electrical port}. That, unfortunately, makes it extremely difficult, if not impossible, to synthesize a certain desired current distribution\footnote{For electrically-small structures (size$\ll$$\lambda_{\text{2.87GHz}}$), the distributions of the current on the structure and the generated near-field magnetic wave follow a one-to-one mapping. Note that although displacement current also generates magnetic field, our following discussions are constrained to structures mainly with conduction current.}, because the only design variable is the structural geometry.
One distinct advantage of the implementation using integrated circuits is \emph{the highly flexible and tight integration between passive and active components.} A large number of biased transistors, when forming current sources, can be used to mandate the complex current values at various particular locations on the passive structure (Fig.~\ref{fig_field_distribution}). Compared to the aforementioned passive-only structures, this new design methodology provides abundant additional degrees of freedom through the positions and currents of the embedded transistors. A finer control of near-field wave distribution is therefore much easier. Note that similar concepts were already proposed and applied to millimeter-wave and terahertz integrated circuit designs\cite{Hajimiri2009,Sengupta2012ARadiators,Hu2018}.
\begin{figure}
\centering
\includegraphics[width=3.3in]{figures/fig_field_distribution.pdf}
\caption{Comparison of design methodologies for the microwave launcher in a quantum sensor. A distributed co-design of passive and active components is equivalent to adding more boundary conditions to an electromagnetic-solving problem, hence providing better control of the near-field pattern.}\label{fig_field_distribution}
\end{figure}
Following the above methodology, our microwave launcher design in Fig.~\ref{fig_architecture} evolves from an ideal, infinite sheet of uniformly-distributed surface current density $J_x$ (see Fig.~\ref{fig_infinite_current_sheet}), which generates a homogeneous field $B_y$ with a transverse ($y$-) direction and strength of $\mu_0J_x/2$ ($\mu_0$: permeability of vacuum). However, such current uniformity is hard to imitate in a single metal structure, due to the self-redistribution of current like skin effect. To prevent such redistribution, the current sheet is first transformed into an infinite array of wires, each driven by an identical current $I_0$ (Fig.~\ref{fig_infinite_wire_array}). With a tight wire center-to-center pitch $d$ (or equivalent current density of $I_0/d$), a uniform transverse magnetic field can still be obtained, with a field strength of $\mu_0I_0/2d$. The wire array also ensures high scalability in the longitudinal ($x$-) direction for large-area diamond.
\begin{figure}
\centering
\subfloat[]{\includegraphics[width=3in]{figures/fig_infinite_current_sheet.pdf}\label{fig_infinite_current_sheet}}\\
\subfloat[]{\includegraphics[width=3in]{figures/fig_infinite_wire_array.pdf}\label{fig_infinite_wire_array}}
\caption{Approaches to generate homogeneous magnetic field: (a) an infinite sheet of uniformly-distributed current, (b) an infinite array of wires with uniform driving current.}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=3in]{figures/fig_sim_finite_array_uniform.pdf}
\caption{Simulated and calculated field distribution ($f$=2.87~GHz) of a wire array with uniform driving current ($I_0$=1~mA).}\label{fig_sim_finite_array_uniform}
\end{figure}
\begin{figure}
\centering
\subfloat[]{\includegraphics[width=3.5in]{figures/fig_finite_wire_array.pdf}\label{fig_finite_wire_array}}\\
\subfloat[]{\includegraphics[width=3.5in]{figures/fig_finite_wire_array_w_boundary.pdf}\label{fig_finite_wire_array_w_boundary}}
\caption{(a) A finite array of wires with uniform driving current. (b) An additional pair of boundary wire arrays (Group~\emph{D}) for nulling the vertical magnetic field between $-L$ and $+L$.}
\end{figure}
In reality, only a finite wire array can be implemented. The introduced boundaries, however, break the above homogeneity. Fig.~\ref{fig_sim_finite_array_uniform} shows the simulated field distribution of a 48-wire array with $d$=4~$\upmu$m. Although the transverse component of the field $B_y$ maintains homogeneity, the vertical component $B_z$ becomes non-zero and exhibits large gradient over $y$-axis. To understand that, we focus at a reference point (Point~\emph{R}) located above an array of $N$ wires and closer to the left side (i.e. $y_{R}<0$ in Fig.~\ref{fig_finite_wire_array}). The wires are then divided to three groups: the \emph{n} wires located between $y_R$ and the left boundary $-L$ (Group~\emph{A}), another \emph{n} wires symmetrically located at the right side of Point~\emph{R} (between $y_R$ and 2$y_R+L$, Group~\emph{B}), and the rest (Group~\emph{C}). Next, note that at Point~\emph{R}, the vertical magnetic fields generated by Group~\emph{A} and \emph{B} cancel, and the $B_z$ generated by Group~$C$, by Ampere's Law, is:
\begin{equation}\label{eqn_array_field}
B_{z,C}(y_R)=-\frac{1}{2\pi}\int_{2y_R+L}^{+L}\frac{\mu_0J_xdy}{y-y_R}=\frac{\mu_0J_x}{2\pi}\ln\frac{L+y_R}{L-y_R},
\end{equation}
where the discreteness of the array is approximated as a uniform current sheet with a current density of $J_x=I_0/d$. Note that (\ref{eqn_array_field}) also applies for $y_R>0$. Shown in Fig.~\ref{fig_sim_finite_array_uniform}, the result calculated from (\ref{eqn_array_field}) well matches the simulation by HFSS\cite{ANSYSHFSS}. Intuitively, $B_z$ increases towards the edges because Group~\emph{C} has a larger total current there and gets closer to Point~\emph{R}.
\begin{figure}[t]
\centering
\includegraphics[width=3in]{figures/fig_cal_finite_boundary.pdf}
\caption{Calculated profiles of vertical field generated by Group~\emph{C} and \emph{D} (and the residue after the cancellation) of the array shown in Fig.~\ref{fig_finite_wire_array_w_boundary}. Note that the $B_z$ of Group~\emph{D} is plotted with its polarity reversed, to facilitate a straightforward comparison with $B_z$ from Group~\emph{C}.}
\label{fig_cal_finite_boundary}
\end{figure}
To create an opposite magnetic-field gradient for nulling the above $B_z$, a pair of additional $m$-wire arrays (Group~\emph{D} in Fig.~\ref{fig_finite_wire_array_w_boundary}), which are driven by $I_D$ per wire, are symmetrically placed at $y=-L-D$ to $-L$ and $y=L$ to $L+D$. Their combined impact on the vertical magnetic field at $y=y_R$ is derived as:
\begin{align}\label{eqn_array_boundary}
B_{z,D}(y_R) =&\int^{-L}_{-L-D}\frac{\mu_0 mI_Ddy}{2\pi D(y_R-y)}-\int^{L+D}_{L}\frac{\mu_0 mI_Ddy}{2\pi D(y-y_R)}\nonumber\\
=&\frac{m\mu_0I_D}{2\pi D}\ln\frac{(y_R+L+D)(L-y_R)}{(L+D-y_R)(y_R+L)},
\end{align}
of which some example plots (to be discussed more) is shown in Fig.~\ref{fig_cal_finite_boundary}, indicating the similarity of its curve shape to that from Group~\emph{C}. To further determine the value of $I_D$, we compare the derivatives of (\ref{eqn_array_field}) and (\ref{eqn_array_boundary}) around the center of the launcher:
\begin{align}
\frac{dB_z}{dy}|_{y=0}= \begin{cases}
\frac{\mu_0J_x}{\pi L} &\hspace{1cm}\textrm{for Group~\emph{C}}\\
-\frac{\mu_0 mI_D}{\pi L(L+D)} &\hspace{1cm}\textrm{for Group~\emph{D}}
\end{cases}.
\end{align}
Therefore, for zero $B_z$ around the array center, the value of $I_D$ should be:
\begin{equation}\label{eqn_array_condition}
I_{D,0}=\frac{L+D}{m}J_x=\frac{L+D}{md}I_0=(1+\frac{N}{2m})I_0.
\end{equation}
The last step of (\ref{eqn_array_condition}) assumes that the boundary arrays adopt the same wire pitch $d$ as the $N$-wire uniform array in the center
For a 32-wire uniform array ($N$=32) with $I_0$=1~mA and $d$=4~$\upmu$m, the vertical field $B_{z,D}$ generated by the boundary array pair with varying wire numbers $m$, as well as the residual of $B_z$ after cancellation, are plotted in Fig.~\ref{fig_cal_finite_boundary}. We see that a larger $m$ improves the area of uniformity at a modest expense of higher current consumption. Interestingly, if the $I_D$ value is slightly lower than the one calculated by (\ref{eqn_array_condition}) (e.g. for $m$=8, the calculated value for $I_D$ is 3~mA), the $B_z$ field around $y$=0 is not cancelled perfectly but the overall 95\% uniformity area is extended to $y$$\approx$$\pm$40~$\upmu$m. Lastly, Fig.~\ref{fig_sim_finite_array_final} shows the HFSS-simulated field distribution of the entire launcher ($N$=32, $m$=8, $d$=4~$\upmu$m), which is implemented at the Metal 8 (M8) interconnect layer of our chip prototype. The calculated profile in Fig.~\ref{fig_sim_finite_array_final} is achieved with $I_D$=2.8$I_0$. Note that the HFSS simulation is done with a slightly larger $I_D$ ($I_D$=3$I_0$) to account for the uncalculated effect of current in the return path. Compared to Fig.~\ref{fig_sim_finite_array_uniform}, the overall homogeneity of the magnetic vector field is significantly improved.
\begin{figure}
\centering
\includegraphics[width=3in]{figures/fig_sim_finite_array_final.pdf}
\caption{Simulated (solid line) and calculated (dashed line) field distribution of the entire launcher wire array. Here, $f$=2.87~GHz, $N$=32, $m$=8, $I_0$=1~mA, $I_D$$\approx$3~mA.}
\label{fig_sim_finite_array_final}
\end{figure}
\subsection{Co-Design of the Circuits, Microwave Launcher and Photonic Filter}\label{sec_co_design}
The full schematic of the microwave launcher with the the switches and current sources banks is shown in Fig.~\ref{fig_array_circuits}. NMOS current mirrors are used to control the current in each individual wire of the array. PMOS switches are used to convert the output voltage of the PLL into microwave currents in the array. The current sources are design to enable current control from $I_0$ = 0.1~mA to $I_0$ = 1~mA. At low current values our simulation in Fig.~\ref{fig_switch_comp} indicates that when the PMOS switches are off, the current of wire returns to zero at a slow speed. This is due to the large parasitic capacitance of the wire array and the limited discharging current. To increase the pull down speed, a row of NMOS switches are also used (see Fig.~\ref{fig_array_circuits}) provide a fast discharging (see the comparison in Fig.~\ref{fig_switch_comp}). The transistors dimensions ($W_1$/$L_1$, $W_2$/$L_2$, and $W_3$/$L_3$) are scaled up by a factor of 3 at the boundaries. As described in Section~\ref{sec_launcher}, the microwave launcher is implemented on M8 with wire width and spacing of 2~$\upmu$m. Therefore, all the transistor layouts are designed in a multiplier fashion so that they can fit in the tight wire pitch. A pair of wide return ground paths are used to close the current loop.
Part of the photonic filter is also implemented on M8 with unit width and spacing of 400~nm (shown in Fig.~\ref{fig_three_layer_filter}). The smallest dimension limited by the design rule of the technology is selected to enhance the coupling to the surface plasmon polariton mode. To cater the needs of both components, shown in Fig.~\ref{fig_array_circuits}, each of the microwave launcher wires other than the boundary wires is divided into three sub-wires with their two ends electrically connected. The width and the spacing of these sub-wires are 400~nm. In addition, each of the 2~$\upmu$m spacing in the microwave launcher is filled with another two floating dummy wires with 400-nm width and spacing (Fig.~\ref{fig_array_circuits}). These dummy wires do not affect the microwave generation/delivery operation, but along with the other sub-wires, they form the desired phontonic filter geometry, which is 350$\times$130~$\upmu$m$^2$ of total area. In this prototype, a conservative size was adopted for the sensing area, since the edge effects like leakage of light was uncertain. The size of the photodiode placed under the filter is 300$\times$80~$\upmu$m$^2$. In future designs, the photodiode area may increase for better $SNR$.
This array generates $>$95$\%$ field homogeneity over $>50\%$ of its area compared to 25$\%$ in \cite{Ibrahim2018}. Uniform array spacing is adopted in this design, however, in Section~\ref{sec_conclusion}, we show that a certain gap between group~\emph{D} and group~\emph{A}/\emph{C} can achieve homogeneity across a larger area. The design is only enabled by CMOS technologies, where we can integrate a very tight wire array with efficient switching and current control.
\begin{figure}[h]
\centering
\includegraphics[width=3.2in]{figures/fig_switch_comp.png}
\caption{Comparison the switching performance with and without extra NMOS switch. The first plot shows the input microwave signal ($V_{in}$) and the second one shows the voltage at the drain of the NMOS current source ($V_{MD}$).}\label{fig_switch_comp}
\end{figure}
\subsection{Nano-Photonic Filter on CMOS}\label{sec_filter}
Plasmonic filters based on a multi-stacked-layer gratings in the BEOL of CMOS technologies have been demonstrated previously in \cite{hong2017fully}. In our first CMOS quantum sensor\cite{Ibrahim2018}, a single-layer filter based on the same principle of wavelength-dependent plasmonic loss is adopted, which exhibits a green-to-red rejection ratio of $\sim$10~dB in our experiment. Based on the discussion in Section~\ref{sec_physics}, that ratio should be improved to enhance the magnetometry sensitivity. Next, we show how a wavelength-dependent diffraction pattern under the plasmonic filter layer is utilized to achieve the goal. To facilitate that discussions, we first revisit some details related to the plasmonic grating layer.
\begin{figure}[h]
\centering
\includegraphics[width=3in]{figures/fig_single_layer_filter_mod.pdf}
\caption{A single-layer plasmonic grating filter implemented on Metal~8\cite{Ibrahim2018}\label{fig_single_layer_filter}.}
\end{figure}
\begin{figure}[t]
\centering
\subfloat[]{\includegraphics[width=2.4in]{figures/fig_dispersion_mod.pdf}\label{fig_dispersion}}\hfill
\subfloat[]{\includegraphics[width=1in]{figures/fig_mode_coupling.pdf}\label{fig_mode_coupling}}
\caption{TEM and SPP modes inside a grating slit (modeled as a parallel-metal-plate waveguide): (a) dispersion relationships and (b) short-distance coupling.}
\end{figure}
\begin{figure}[!t]
\centering
\subfloat[]{\includegraphics[width=3.1in]{figures/fig_M8_grating_patterns.pdf}\label{fig_M8_grating_patterns}}\\
\subfloat[]{\includegraphics[width=3.1in]{figures/fig_M8_M6_grating_patterns.pdf}\label{fig_M8_M6_grating_patterns}}\\
\subfloat[]{\includegraphics[width=3.1in]{figures/fig_M8_M6_M3_grating_patterns.pdf}\label{fig_M8_M6_M3_grating_patterns}}\\
\caption{The simulated FDTD Poynting vector profile $|P|$=$|E\times H|$ for (a) single-layer grating in M8, (b) double gratings in M8 and M6, and (c) triple-layer gratings in M8, M6 and M3. {\hspace{3mm} Note:} the scale bar for green (right column) is 10$\times$ lower than that for red (left column), in order to count for the $\sim$10-dB plasmonic loss in the M8 grating and to better show M6 and M3 gratings further suppress the transmission at 535~nm.}\label{fig_grating_patterns}
\end{figure}
Figure~\ref{fig_single_layer_filter} shows the structure of the filter used in \cite{Ibrahim2018,Kim2019a}. Each slit is considered as a parallel plate waveguide transmitting light inside a dielectric (refractive index $\sqrt{\epsilon_d}$$\approx$1.5) in the $z$- direction. The incoming light is modeled as a plane wave with a transverse electrical field $E_x$ and a propagation constant $k_0=\omega\sqrt{\epsilon_d}/c$ ($c$ is the speed of light in vacuum). This light is coupled to the TEM mode of the parallel-plate waveguides, which has identical propagation constant $k_0$ (see Fig.~\ref{fig_dispersion}). As the propagating wave interacts with the metal, the surface plasmon polariton (SPP) mode at the metal-dielectric interface is excited. Note that the dispersion relation of SPP mode is:
\begin{align}\label{eqn_spp_dispersion}
k_{z,SPP} &=\beta_{SPP}+j\alpha_{SPP}=k_0\sqrt{\frac{\epsilon_m}{\epsilon_m+\epsilon_d}}.
\end{align}
In (\ref{eqn_spp_dispersion}), the permittivity ($\epsilon_m$=$\epsilon_{mr}+j\epsilon_{mi}$) of the metal (copper in our case) has a negative real part, which leads to $\beta_{SPP}>k_0$. Although coupling from the TEM waveguide mode and the SPP mode generally does not occur\footnote{Normally, dedicated material/geometrical configurations, such as the Otto and Kretschmann-Raether configurations, are needed to enable the excitation of the SPP mode.}, it still does in this case. That is because, by examining the value of $\beta_{SPP}$ in (\ref{eqn_spp_dispersion}):
\begin{align}\label{eqn_spp_beta}
\beta_{SPP}=\Re(k_0\sqrt{\frac{\epsilon_m}{\epsilon_m+\epsilon_d}})\approx k_0(1-\frac{\epsilon_d}{2\epsilon_{mr}}),
\end{align}
we see that since $\abs{\epsilon_{mr}}$ remains large, the relative mismatch between the $\beta_{SPP}$ and $\beta_{TEM}$=$k_0$, namely $\epsilon_d/2\epsilon_{mr}$ in (\ref{eqn_spp_beta}), are $\sim$0.07 for red (700~nm) and $\sim$0.22 for green (530~nm) (see Fig.~\ref{fig_dispersion}). These calculations are based on the Drude-Brendel-Bormann model \cite{Rakic1998OpticalDevices,Hagemann1975OpticalAl2O3}. Along the vertical propagation distance of the grating (i.e. the M8 grating thickness $d$=900~nm, or 2$\lambda_{red}$=2.5$\lambda_{green}$ in dielectric), the accumulated phase mismatch is:
\begin{align}\label{eqn_phase_mismatch}
\Delta\phi=2\pi\frac{\epsilon_d}{2\epsilon_{mr}}\frac{d}{\lambda},
\end{align}
or $\sim$50$^o$ (red) and $\sim$180$^o$ (green), respectively. Therefore, for red light, constructive coupling from the TEM mode to SPP mode still occurs within the single-layer grating. For green light, although phase mismatch appears to be large, the coupling is still effective in the M8 slit due to the large SPP loss; and when the TEM wave reaches the bottom half of the slit, its power is already heavily depleted. To quantify the loss of the SPP mode, we derive the attenuation factor $\alpha_{SPP}$ in (\ref{eqn_spp_dispersion}):
\begin{align}\label{eqn_spp_alpha}
\alpha_{SPP}=\Im(k_0\sqrt{\frac{\epsilon_m}{\epsilon_m+\epsilon_d}})\approx k_0\frac{\epsilon_d\epsilon_{mi}}{2\epsilon_{mr}^2},
\end{align}
$\alpha_{SPP}$ exhibits large difference between red ($\approx$0.006$k_0$) and green ($\approx$0.18$k_0$) as calculated based also on the Drude-Brendel-Bormann model.
\begin{figure}[!b]
\centering
\includegraphics[width=3.2in]{figures/fig_three_layer_filter.pdf}
\caption{Simulated transmission values through the filter. The total power is derived by surface integrals of Poynting vectors at varying depths of the chip.}\label{fig_three_layer_filter}
\end{figure}
\begin{figure}[!hb]
\centering
\includegraphics[width=3.2in]{figures/fig_filter_freq_response_new.pdf}
\caption{The transmission through the filter at different wavelength.}\label{fig_filter_freq_response}
\end{figure}
\begin{figure*}[!b]
\centering
\includegraphics[width=6in]{figures/fig_array_detailed.pdf}
\caption{Schematic of the microwave launcher with the switches and the current sources.}\label{fig_array_circuits}
\end{figure*}
Using a finite-difference time-domain (FDTD) solver, Lumerical, we simulate the green-to-red suppression ratio due to the slit in M8 layer ($d$=900~nm), which is $\sim$9~dB (Fig.~\ref{fig_M8_grating_patterns}). Increasing the slit thickness by stacking more grating layers (in M7, M6...) is expected to further increase the suppression of the green light\cite{hong2017fully}. However, that causes degradation of red-light transmission due to the scattering at the side walls formed by the sparse inter-layer via pillars. Moreover, as Section~\ref{sec_co_design} describes, the grating structure in M8 is also used for the microwave wire array (Section~\ref{sec_launcher}). Connecting it to the lower metal layers would significantly decrease the density and the uniformity of the RF current inside the wires. In our chip, an alternative approach based on the wavelength-dependent diffraction pattern of the grating is adopted.
As the simulation in Fig.~\ref{fig_M8_grating_patterns} shows, the diffraction of the light coming out of the grating in M8 causes repeated self-imaging patterns in the chip dielectric. That optical phenomenon is called Talbot effect\cite{Talbot1836FactsIV} and the vertical ($z$-) period of the self-images (i.e. Talbot length) is:
\begin{equation}\label{eqn_talbot_length}
z_T=\frac{\lambda}{1-\sqrt{1-\frac{\lambda^2}{t^2}}},
\end{equation}
where $\lambda$ is the wavelength in the dielectric and $t$ is the grating period (800~nm in our case). For the red (700~nm) and green (535~nm) light, $z_T$ is $\sim$2.5~$\upmu$m and $\sim$3.4~$\upmu$m (see Fig.~\ref{fig_M8_grating_patterns}). That means we may strategically place additional metal structures at positions which are dark in red diffraction pattern (so that the light is not affected) but bright for green (so that the light is blocked). In the 65~nm CMOS process used for this work, the distance between M8 and M6 layers is close to the half Talbot length for red; therefore, a M6 grating with 400~nm strip width and pitch is placed. With the positions of the metal strips right under the slits of M8, the M6 grating has little interference with the red-light transmission, while blocking a significant portion of the green light (Fig.~\ref{fig_M8_M6_grating_patterns}). It is noteworthy that the similar multi-grating concept was previously adopted for lensless 3D CMOS cameras\cite{Wang2012}, where the angle sensitivity of the structure is utilized. To the authors' best knowledge, this is the first time in CMOS that the idea of diffraction selectivity is applied into optical spectral filtering.
Following the same principle, a third grating layer in M3, of which the distance to M8 is about one Talbot length at 700~nm, is added to further enhance the green-light rejection (Fig.~\ref{fig_M8_M6_M3_grating_patterns}). The same grating strip width and pitch of 400~nm are used. Fig.~\ref{fig_three_layer_filter} provides the normalized transmitted power (Poynting vector integrated over a $x$-$y$ cross-sectional area) at varying chip depths. For green light, the plasmonic behavior attenuates the power by 9.7~dB, and the M6 and M3 gratings pose an additional 12.1~dB loss, leading to a total attenuation of 21.8~dB. Meanwhile, the total insertion loss for red light is only 2.1~dB. Note that the reflection loss at the top of M8, which is $\sim$3~dB due to the 50\% metal fill factor, is not included in both cases.
Lastly, the simulated transmission response (excluding the 50\% surface reflection) of the filter from 500~nm to 750~nm is plotted in Fig.~\ref{fig_filter_freq_response}. The spectrum of the NV-center red fluorescence is also shown. It can be seen that the majority of the fluorescence power between 650 to 725~nm is transmitted efficiently to the photodiode beneath the filter.
\section{Background and Prior Research}\label{sec_physics}
\begin{figure}
\centering
\subfloat[]{\includegraphics[width=3.4in]{figures/fig_nv_centers.pdf}\label{fig_nv_centers}}\\
\subfloat[]{\includegraphics[width=2.8in]{figures/fig_nv_energy.pdf}\label{fig_nv_energy}}
\caption{(a) Negatively-charged nitrogen-vacancy centers in a diamond lattice. The projections
of an external magnetic field
along the four nitrogen-vacancy axes are also shown. (b) The energy-level diagram of a nitrogen-vacancy center.}
\end{figure}
A negatively-charged nitrogen-vacancy (NV) center in diamond consists of a nitrogen atom adjacent to a vacancy in the carbon lattice (see Fig.~\ref{fig_nv_centers}). The hybridization of the two unpaired electrons leads to a quantum system with the energy-level diagram shown in Fig.~\ref{fig_nv_energy}. The spin magnetic triplet is formed at the ground state ($^3$A), consisting of a sub-level $\ket{m_s=0}$ at its lowest energy and another two zero-field degenerate sub-levels $\ket{m_s=\pm 1}$ raised by $\sim$2.87~GHz.
When an external magnetic field $\vv{B_{ext}}$ with a component $\vv{B_z}$ along the N-V axis (see Fig.~\ref{fig_nv_centers}) is applied, the $\ket{m_s=\pm 1}$ sub-levels are split apart (i.e. Zeeman effect).
The photon frequency $\Delta f$ associated with such an energy gap is proportional to $|\vv{B_z}|$:
\begin{equation}
\Delta f=f_+-f_-=2\gamma_e|\vv{B_z}|,
\end{equation}
where $\gamma_e$ is the gyromagnetic ratio and equals to 28~GHz/T, and $f_+$ and $f_-$ are the frequencies for the transitions from $\ket{m_s=0}$ to $\ket{m_s=+1}$ and $\ket{m_s=-1}$, respectively. We use $\Delta f$ to derive $\vv{B_z}$.
NV magnetometry is performed by determining $f_+$ and $f_-$ via optically-detected magnetic resonance (ODMR)\cite{Gruber1997ScanningCenters}. In an ODMR experiment, the NV center spins are stimulated to their excitation states ($^3$E in Fig.~\ref{fig_nv_energy}) with green light ($\lambda\approx$~532~nm), and then relax back to the ground state ($^3$A). The relaxation of the $\ket{m_s=0}$ state is accompanied by bright red fluorescence ($\lambda\approx~$600$\sim$800~nm). In contrast, when the $\ket{m_s=\pm 1}$ states are excited and relax back, they can undergo an intersystem crossing into a metastable spin-singlet state ($^1$A in Fig.~\ref{fig_nv_energy}), and then into the $\ket{m_s=0}$ ground level\cite{Kim2019a} reducing the red fluorescence intensity. Thus, static or slowly-varying magnetic fields $\vv{B_{z}}$, can be determined by sweeping a microwave frequency $f_{RF}$ around 2.87~GHz and monitoring the average intensity.
The observed resonances of Fig.~\ref{fig_zeeman} are $f_+$ and $f_-$, which give $\vv{B_z}$.
NV centers have four known orientations (Fig.~\ref{fig_nv_centers}) which separately lay along the tetrahedral axes of the host diamond. Accordingly, an external magnetic field $\vv{B_{ext}}$ has four projections -- $\vv{B_{z1}}$, $\vv{B_{z2}}$, $\vv{B_{z3}}$, $\vv{B_{z4}}$ -- along the NV orientations. This leads to four pairs of splitting in a single ODMR measurement (Fig.~\ref{fig_zeeman}). The magnetometer based on this principle, therefore, has a vector-field measurement capability by monitoring the different magnetic field projects and reconstructing $\vv{B_{ext}}$. That is advantageous over conventional Hall and fluxgate-based sensors, where three devices in \emph{x- y- z-} axes are needed for vector detection.
\begin{figure} [!t]
\centering
\includegraphics[width=2.6in]{figures/fig_zeeman.pdf}
\caption{The red-fluorescence intensity of the diamond at varying microwave frequency. An external magnetic-field bias with projections along the four N-V axes is assumed.}\label{fig_zeeman}
\end{figure}
\begin{figure*}[!t]
\centering
\includegraphics[width=6.5in]{figures/fig_architecture.pdf}
\caption{The overall schematic of the CMOS quantum magnetometer with high scalability.}\label{fig_architecture}
\end{figure*}
To explore the feasibility of a chip-scale, low-cost quantum magnetometer, a custom-designed CMOS prototype was realized and reported, for the first time, in \cite{Ibrahim2018}. This chip, using TSMC 65-nm CMOS technology, integrates most of the critical components (except the green light source) for the ODMR operation. Using this hybrid CMOS-NV-center integration platform, the ODMR spectrum of a nanodiamond layer attached on top of the chip is demonstrated. The estimated sensitivity of the system is 74~$\upmu$T/Hz$^{1/2}$. Since the lattice orientations in the nanodiamond particles are random, the amount of frequency splitting in each NV is also random. As a result, this prototype cannot be used for vector-field sensing. Recently, we attached a film of bulk diamond (with uniform and well-defined lattice structure) to the same CMOS chip and demonstrated vector-sensing capability with 32~$\upmu$T/Hz$^{1/2}$ sensitivity\cite{Kim2019a}.
Our first CMOS prototype demonstrates the basic concept of chip-scale miniaturization of NV-center quantum sensors. The achieved sensitivity is still limited by two main factors. (i) In our experiments we observed that the dominant noise source was the shot noise due to the green light despite the presence of the grating filter \cite{Kim2019a}. Ideally, this sensor would be limited to the red fluorescence shot noise and therefore, a higher green-to-red suppression ratio is required for the integrated photonic filter. We estimated that, for the diamonds used in \cite{Ibrahim2018,Kim2019a}, the intensity of red fluorescence is about 40$\sim$50~dB lower than that of the incident green light. Ultimately, an additional 30~dB out-of-band rejection is required for the photonic filter, so that the majority of photodiode noise is no longer generated by the green background. (ii) The sensing area used is only 50~$\upmu$m$~\times~$50~$\upmu$m. This limits the number of NV centers, $N$. Increasing $N$ improves the sensitivity due to larger red-fluorescence intensity (hence the \emph{SNR} of the ODMR spectrum). This can be achieved by using a larger sensing area and higher NV-center density in diamond. Note that the \emph{SNR} has the following dependency with $N$:
\begin{equation}
\text{\it SNR}~\text{is} \begin{cases}
\propto N &\text{when noise is green-light limited}\\
\propto \sqrt{N} &\text{when noise is red-light limited}
\end{cases}.
\end{equation}
\subsection{On-Chip Synthesis of the Microwave Frequency}\label{sec_PLL}
A PLL loop is used to stabilize the output of the voltage-controlled oscillator (VCO) that drives the NV centers. The full schematic of the PLL is shown in Fig.~\ref{fig_pll_schematic}. To ensure a uniform phase of the magnetic field across the wire array, the VCO is based on a tightly-coupled ring-oscillator array. Four sub-oscillators are placed with a spacing of $\sim$50~$\upmu$m (see Fig.~\ref{fig_architecture}). The two oscillators in the middle drive 32 wires, while the other two oscillators on the array boundaries drive 8 wires each. Compared to a centralized signal-generation scheme, the coupled-oscillator eliminates the phase variation caused by different lengths of the microwave-distributing traces. Here, ring oscillators, instead of LC oscillators, are chosen, in order to eliminate the need for extra inductors, as well as any magnetic coupling and degradation of field homogeneity in the diamond-sensing area. A ring oscillator also offers wider tuning range and layout compactness. Its inferior phase noise is improved by 6~dB due to the oscillator coupling, and in fact our analysis indicates that our sensor sensitivity is currently limited by the green and red light noise, as explained in Section~\ref{sec_noise_analysis}. The VCO frequency is tuned using varactors ($C_v$=9~to~42~fF) with a coefficient of 1.2~GHz/V. The charge pump current of the PLL is 5~mA and the loop filter parameters are $C_1$=1~nF,~$R_6$=220~$\Omega$,~and~$C_2$=20~pF.
\begin{figure}[h]
\centering
\includegraphics[width=3.5in]{figures/fig_pll_schematic.pdf}
\caption{Schematic of the 2.87-GHz PLL.}\label{fig_pll_schematic}
\end{figure}
|
1,116,691,500,847 | arxiv | |
1,116,691,500,848 | arxiv | \section{Introduction}
Multiple-input multiple-output (MIMO) detection is a fundamental problem in modern
digital communications~\cite{multiuser_detection,fifty_years_of_MIMO}.
The MIMO channel can be modeled as
\begin{equation}\label{eq:mimo}
\vr = \mH \vx^*+\vv,
\end{equation}
where $\vr\in \C^m$ is the vector of received signals,
$\mH\in \C^{m\times n}$ is a complex channel matrix,
$\vx^*$ is the vector of transmitted symbols, and
$\vv$ is the vector of additive Gaussian noises. Moreover,
each entry of $\vx^*$ is drawn from a discrete symbol set $\sS$
determined by the modulation scheme.
The MIMO detection problem is to recover the transmitted symbol vector $\vx^*$ from the noisy channel output $\vr$,
with the information of the symbol set $\sS$ and the channel matrix $\mH$.
Under the assumption that each entry of $\vx^*$ is drawn uniformly and independently from the
symbol set $\sS$,
it is known that the maximum likelihood detector can achieve the optimal detection error rate performance.
Mathematically, it can be formulated as a discrete least-squares problem:
\begin{equation}\label{eq:ML}
\begin{aligned}
\min_{\vx\in \C^n}\;\;\; &\|\mH\vx-\vr\|_2^2 \\
\mathrm{s.t.}\,\;\;\; & x_i\in \mathcal{S},\;i=1,2,\ldots,n,\\
\end{aligned}
\end{equation}
where $x_i$ denotes the $i$-th entry of the vector $\vx$ and $\|\cdot\|_2$ denotes the Euclidean norm.
In this paper, unless otherwise specified, we will focus on the $M$-ary phase shift keying~($M$-PSK) modulation, whose symbol set is given by
\begin{equation}\label{eq:MPSK symbol set}
\mathcal{S}_M:=\bigl\{z\in \C:\; |z|=1,\;\mathrm{arg}(z)\in \{2j\pi/M,\;j=0,1,\ldots,M-1\}\bigr\},
\end{equation}
where $|z|$ and $\mathrm{arg}(z)$ denote the modulus and argument of a complex number, respectively.
As in most practical digital communication systems, throughout the paper we require $M=2^b$ where $b\geq 1$ is an integer\footnote{Our
results in \cref{sec:tightness}
also hold for the more general case where $M$ is a multiple of four.}.
Many detection algorithms have been proposed to solve problem \cref{eq:ML} either exactly or approximately.
However, for
general $\mH$ and $\vr$, problem \cref{eq:ML}
has been proved to be NP-hard~\cite{computational_complexity}.
Hence, no polynomial-time algorithms can find the exact solution (unless $\text{P}=\text{NP}$).
Sphere decoding~\cite{sphere_decoder}, a classical combinatorial algorithm based on the branch-and-bound paradigm,
offers an efficient way to solve problem~\cref{eq:ML} exactly
when the problem size is small,
but its expected complexity is still exponential~\cite{complexity_of_SD}.
On the other hand, some suboptimal algorithms such as linear detectors~\cite{linear_ZF,layered_space_time} and
decision-feedback detectors~\cite{decision_feedback,decision_feedback_cdma} enjoy low complexity but
at the expense of substantial performance loss: see~\cite{fifty_years_of_MIMO} for an excellent review.
Over the past two decades, semidefinite relaxation (SDR) has gained increasing attention in non-convex optimization~\cite{sdr_maxcut,SDR_quadratic,phase_recovery_complex_SDP}.
It is a celebrated technique to tackle quadratic optimization problems arising from
various signal processing and wireless communication applications, such as beamforming design~\cite{transmit_beamforming,max_min_linear_transceiver},
sensor network localization~\cite{sensor_network_localization,sdp_sensor_network_localization,theory_sdp_snl}, and angular synchronization
\cite{angular_synchronization,tightness_sdp_angular_synchronization,near_optimal_phase_sync}.
Such SDR-based approaches can usually offer superior performance in both theory and practice while
maintaining polynomial-time worst-case complexity.
For MIMO detection problem \cref{eq:ML}, the first SDR detector~\cite{SDR_BPSK,SDR_BPSK2} was designed for the real MIMO channel and the binary symbol set $\mathcal{S}=\{+1,-1\}$.
Notably,
it is proved that
this detector can achieve the maximal possible diversity order~\cite{diversity_SDR},
meaning that it achieves an asymptotically optimal detection error rate
when the signal-to-noise ratio (SNR) is high.
It was later extended to the more general setting with a complex channel and an $M$-PSK symbol set in~\cite{SDR_QPSK,SDR_MPSK},
which we refer to as the conventional SDR or \cref{eq:CSDR}.
However, this conventional approach fails to fully utilize the structure in the symbol set $\mathcal{S}$.
To overcome this issue, researchers have developed various improved SDRs and we consider the two most popular classes
below. The first class proposed in~\cite{near_ml_decoding} is
based on an equivalent
zero-one integer programming formulation of problem \cref{eq:ML}.
Four SDR models were introduced and
two of them will be discussed in details later (see \cref{eq:Model II} and \cref{eq:Model III} further ahead).
The second class proposed in~\cite{tightness_enhanced} further enhances \cref{eq:CSDR}
by adding valid cuts, resulting in a complex SDR and a real SDR (see \cref{eq:CSDP2} and \cref{eq:ERSDP} later on).
In this paper, we focus on two key problems in SDR-based MIMO detection: the tightness of SDRs and
the relationship between different SDR models. Firstly,
note that SDR detectors are suboptimal algorithms as they replace
the original discrete optimization problem \cref{eq:ML} with tractable semidefinite programs (SDPs). Hence,
after solving an SDP, we need some rounding procedure to make final symbol decisions.
However, under some favorable conditions on $\mH$ and
$\vv$, an SDR can be \emph{tight}, i.e., it has an optimal rank-one solution corresponding to
the true vector of transmitted symbols.
Such tightness conditions are of great interest since they give theoretical guarantees
on the optimality of SDR detectors.
While it has been well studied for the simple case~\cite{detection_mimo,optimal_condition,performance_analysis_sdp,probabilistic_analysis_SDR} where $\mH\in \R^{m\times n}$, $\vv\in \R^{m}$, and $\mathcal{S}=\{+1,-1\}$, for the more general case where $\mH\in \C^{m\times n}$, $\vv\in \C^m$, and $\mathcal{S}=\mathcal{S}_M$ ($M\geq 4$),
tightness conditions for SDR detectors have remained unknown until very recently. The authors in~\cite{tightness_enhanced} showed
that \cref{eq:CSDR} is not tight with probability one under some mild conditions. On the other hand,
their proposed enhanced SDRs are tight\footnote{The definition of tightness in~\cite{tightness_enhanced} is slightly different from ours since
they also require the optimal solution of the SDR to be unique. } if the following condition is satisfied:
\begin{equation}\label{eq:CSDP2_suff}
\lambda_{\mathrm{min}}(\mH^\dagger \mH)\sin\left(\frac{\pi}{M}\right)
>\|\mH^\dagger \vv\|_{\infty},
\end{equation}
where $\lambda_{\mathrm{min}}(\cdot)$ denotes the smallest eigenvalue of a matrix,
$(\cdot)^\dagger$ denotes the conjugate transpose, and
$\|\cdot\|_{\infty}$ denotes the $L_{\infty}$-norm. To the best of our knowledge, this
is the best condition that guarantees a certain SDR to be tight for problem \cref{eq:ML} in the $M$-PSK settings.
Secondly, researchers have noticed some rather unexpected equivalence between
different SDR models independently developed in the
literature. The earliest one of such results is reported in~\cite{QAM_equivalence}, where three different SDRs for the high-order
quadrature amplitude modulation (QAM) symbol sets are proved to be equivalent.
Very recently, the authors in~\cite{liu2019equivalence} showed that
the enhanced real SDR proposed in~\cite{tightness_enhanced} is equivalent to
one SDR model in~\cite{near_ml_decoding}. It is worth noting that while these
two papers are of the same nature, the proof techniques are quite different and
it is unclear how to generalize their results at present.
In this paper, we make contributions to both problems.
For the tightness of SDRs, we
sharpen the analysis in~\cite{tightness_enhanced} to give the necessary and sufficient condition
for the complex enhanced SDR to be tight, and a necessary condition
for the real enhanced SDR to be tight. Specifically, for the case where $M\geq 4$, we show that the enhanced complex
SDR \cref{eq:CSDP2} is tight if and only if
\begin{equation}\label{eq:CSDP2_necc_and_suff_intro}
\mH^\dagger \mH + \Diag\big(\mathrm{Re}(\Diag(\vx^*)^{-1}\mH^\dagger \vv)\big)
-\cot\left(\frac{\pi}{M}\right)\Diag(|\mathrm{Im}(\Diag(\vx^*)^{-1}\mH^\dagger \vv)|\big) \succeq 0,
\end{equation}
while the enhanced real SDR \cref{eq:ERSDP} is tight only if
\begin{equation}\label{eq:ERSDP_necc_intro}
\mH^\dagger \mH + \Diag\big(\mathrm{Re}(\Diag(\vx^*)^{-1}\mH^\dagger \vv)\big)
-\cot\left(\frac{2\pi}{M}\right)\Diag(|\mathrm{Im}(\Diag(\vx^*)^{-1}\mH^\dagger \vv)|\big) \succeq 0,
\end{equation}
where $\mA \succeq 0$ means that the matrix $\mA$ is positive semidefinite (PSD),
$\Diag(\vx)$ denotes a diagonal matrix whose diagonals are the vector $\vx$,
and $\Real(\cdot)$, $\Imag(\cdot)$, and $|\cdot|$ denote the
entrywise real part, imaginary part, and absolute value of a number/vector/matrix, respectively.
Moreover, we prove that
one of the SDR models proposed in~\cite{near_ml_decoding} is generally not tight: under some
mild assumptions, its probability of being tight decays exponentially with respect to the number
of transmitted symbols $n$.
For the relationship between different SDR models, we propose a
general theorem showing the equivalence between two subsets of PSD cones.
Specifically, we prove the correspondence between
a subset of a high-dimensional PSD cone with a special ``separable'' structure
and the one in a lower dimension.
Our theorem covers both equivalence results in~\cite{QAM_equivalence} and~\cite{liu2019equivalence}
as special cases, and has the potential to find other
applications due to its generality.
The paper is organized as follows.
We introduce the existing SDRs for \cref{eq:ML} in \cref{sec:review_of_SDR} and analyze their tightness in \cref{sec:tightness}.
In \cref{sec:equiv},
we propose a general theorem that establishes the
equivalence between two subsets of PSD cones in different dimensions,
and discuss how
our theorem implies previous results.
\Cref{sec:experiments} provides some numerical results to validate our analysis.
Finally, \cref{sec:conclusions} concludes the paper.
We summarize some standard notations used in this paper.
We use
$x_i$ to denote the $i$-th entry of a vector $\vx$ and
$X_{i,j}$ to denote the $(i,j)$-th entry of
a matrix $\mX$.
We use $|\cdot|$, $\|\cdot\|_2$, and $\|\cdot\|_{\infty}$ to denote
the entrywise absolute value,
the Euclidean norm, and the $L_{\infty}$ norm
of a vector, respectively.
For a given number/vector/matrix, we use
$(\cdot)^\dagger$ to denote the conjugate transpose,
$(\cdot)^{\T}$ to denote the transpose, and
$\Real(\cdot)$/$\Imag(\cdot)$ to denote the entrywise
real/imaginary part.
We use $\Diag(\vx)$ to denote
the diagonal matrix whose diagonals are the vector $\vx$,
and $\diag(\mX)$ to denote the vector whose
entries are the diagonals of the matrix $\mX$.
Given an $m\times n$ matrix $\mA$ and the index sets
$\alpha \subset \{1,2,\ldots,m\}$ and $\beta \subset \{1,2,\ldots,n\}$,
we use $\mA[\alpha,\beta]$ to denote the submatrix with entires in the rows of $\mA$
indexed by $\alpha$ and the columns indexed by $\beta$.
Moreover, we denote the principal submatrix $\mA[\alpha,\alpha]$ by
$\mA[\alpha]$ in short.
For two matrices $\mA$ and $\mB$ of appropriate size,
$\langle \mA,\mB\rangle:=\Real(\mathrm{Tr}(\mA^\dagger \mB))$ denotes
the inner product, $\mA \otimes \mB$ denotes the Kronecker product,
and $\mA\succeq \mB$ means $\mA-\mB$ is PSD.
For a set $\mathcal{A}$ in a vector space,
we use $\mathrm{conv}(\mathcal{A})$ to denote its convex hull.
For a random variable $X$ and measurable sets
$\mathcal{B}$ and $\mathcal{C}$, $\Prob(X\in \mathcal{B})$ denotes the
probability of the event $\{X\in \mathcal{B}\}$,
$\Prob(X\in \mathcal{B}\givenp \mathcal{C})$ denotes
the conditional probability given $\mathcal{C}$,
and $\E[X]$ denotes the expectation
of $X$.
Finally,
the symbols $\imag$, $\1_n$, $\mI_n$, and $\semiS^n$ represent the
imaginary unit, the $n\times1$ all-one vector,
the $n\times n$ identity matrix, and the
$n$-dimensional PSD cone, respectively.
\input{sec2_review.tex}
\input{sec3_tight.tex}
\input{sec4_equiv.tex}
\input{sec5_exp.tex}
\section{Conclusions}
\label{sec:conclusions}
In this paper, we studied the tightness and
equivalence of various existing SDR models for the MIMO detection
problem \cref{eq:ML}.
For the two SDRs \cref{eq:CSDP2} and \cref{eq:ERSDP} proposed in~\cite{tightness_enhanced},
we improved their sufficient tightness condition and
showed that the former is tight if and only if \cref{eq:CSDP2_necc_and_suff_intro} holds
while the latter is tight only if \cref{eq:ERSDP_necc_intro} holds.
On the other hand, for the SDR \cref{eq:Model II} proposed in~\cite{near_ml_decoding},
we proved that its tightness probability decays to zero exponentially
fast with an increasing problem size under some mild assumptions.
Together with known results, our analysis provides a more complete understanding
of the tightness conditions for existing SDRs.
Moreover, we proposed a general theorem that
unifies previous results on the equivalence of SDRs~\cite{QAM_equivalence,liu2019equivalence}.
For a subset of PSD matrices with a special ``separable'' structure,
we showed its equivalence to another subset of PSD matrices in
a potentially much smaller dimension.
Our numerical results demonstrated
that we
could significantly improve the computational efficiency by using such equivalence.
Due to its generality,
we believe that our equivalence theorem can be applied to SDPs in other domains beyond MIMO detection
and we would like to put this as a future work.
Additionally, we noticed that
the SDRs for problem \cref{eq:ML} combined with some simple rounding
procedure can detect the transmitted symbols successfully even when
the optimal solution has rank more than one. Similar observations have also been made in~\cite{diversity_SDR}.
It would be interesting to extend our analysis to take the postprocessing procedure into account.
\section{Review of semidefinite relaxations}
\label{sec:review_of_SDR}
In this paper, we focus on the $M$\nobreakdash-PSK setting with the symbol set $\mathcal{S}_M$ given in \cref{eq:MPSK symbol set}.
To simplify the notations, we let $\vs\in \C^M$ be the vector of all symbols, where
\begin{equation*}
s_j = e^{\imag\theta_j}\;\text{and}\;\theta_j=\frac{(j-1)2\pi}{M},\;
j=1,2,\ldots,M,
\end{equation*}
and further we let $\vs_R=\Real(\vs)$ and $\vs_I=\Imag(\vs)$.
The objective in \cref{eq:ML} can be written as
\begin{equation*}
\|\mH\vx-\vr\|_2^2=\vx^\dagger \mQ \vx+2\mathrm{Re}(\vc^\dagger \vx)+\vr^\dagger \vr
= \langle \mQ, \vx\vx^\dagger \rangle+2\mathrm{Re}(\vc^\dagger \vx)+\vr^\dagger \vr,
\end{equation*}
where we define
\begin{equation}\label{eq:def of Q and c}
\mQ=\mH^\dagger \mH \; \text{and} \; \vc=-\mH^\dagger \vr.
\end{equation}
By introducing $\mX=\vx\vx^\dagger$ and discarding the constant $\vr^\dagger \vr$,
we can reformulate \cref{eq:ML} as
\begin{equation}\label{eq:reformulated ML}
\begin{aligned}
\min_{\vx,\mX}\;\;\; &\langle \mQ,\mX \rangle+2 \mathrm{Re}(\vc^\dagger \vx) \\
\mathrm{s.t.}\;\;\; & X_{i,i}=1,\;i=1,2,\ldots,n,\\
& x_i \in \mathcal{S}_M,\;i=1,2,\ldots,n,\\
&\mX= \vx\vx^\dagger,
\end{aligned}
\end{equation}
where the constraint $X_{i,i}=1$ comes from $X_{i,i}=|x_i|^2=1$.
The conventional SDR (CSDR) in~\cite{SDR_QPSK,SDR_MPSK} simply drops the discrete symbol constraints $x_i \in \mathcal{S}_M$
and relaxes the rank-one
constraint to $\mX\succeq \vx\vx^\dagger$, resulting in the following
relaxation:
\begin{equation}\label{eq:CSDR}\tag{CSDR}
\begin{aligned}
\min_{\vx,\mX}\;\;\; &\langle \mQ,\mX \rangle+2 \mathrm{Re}(\vc^\dagger \vx) \\
\mathrm{s.t.}\;\;\; & X_{i,i}=1,\;i=1,2,\ldots,n,\\
&\mX\succeq \vx\vx^\dagger,
\end{aligned}
\end{equation}
where $\vx\in \C^n$ and $\mX\in \C^{n\times n}$.
Since $\mX\succeq \vx\vx^\dagger$ is equivalent to
\begin{equation*}
\begin{bmatrix}
1 & \vx^\dagger \\
\vx & \mX
\end{bmatrix}
\succeq 0,
\end{equation*}
the above \cref{eq:CSDR} is an SDP on the complex domain.
Moreover, for the simple case where $\mH\in \R^{m\times n}$, $\vv\in \R^m$, and $M=2$, a real SDR similar to \cref{eq:CSDR} has the form:
\begin{equation}\label{eq:BSDR}
\begin{aligned}
\min_{\vx,\mX}\;\;\; &\langle \mQ,\mX \rangle+2 \vc^\T \vx \\
\mathrm{s.t.}\;\;\; & X_{i,i}=1,\;i=1,2,\ldots,n,\\
&\mX\succeq \vx\vx^{\T},
\end{aligned}
\end{equation}
where $\vx\in\R^n$, $\mX\in \R^{n\times n}$, and we redefine $\mQ=\mH^{\T}\mH$ and $\vc=-\mH^{\T}\vr$ (cf. \cref{eq:def of Q and c}).
The problem \cref{eq:BSDR} has also been extensively studied in the literature~\cite{SDR_BPSK,SDR_BPSK2,detection_mimo,optimal_condition,performance_analysis_sdp,probabilistic_analysis_SDR}.
It is proved in~\cite{detection_mimo,optimal_condition} that
\cref{eq:BSDR} is tight if and only if
\begin{equation}\label{eq:real_channel_binary}
\mH^{\T}\mH+[\Diag(\vx^*)]^{-1}\Diag(\mH^{\T}\vv)\succeq 0,
\end{equation}
while \cref{eq:CSDR} is not tight for $M\geq 4$ with probability one under some mild conditions~\cite{tightness_enhanced}.
Recently, a class of enhanced SDRs was proposed in~\cite{tightness_enhanced}.
Instead of simply dropping the constraints $x_i\in \sS_M$ as in \cref{eq:CSDR},
the authors replaced the discrete symbol set $\sS_M$ by its convex hull to get a
continuous relaxation:
\begin{equation}\label{eq:CSDP2}\tag{ESDR-$\mX$}
\begin{aligned}
\min_{\vt,\vx,\mX}\;\;\; &\langle \mQ,\mX \rangle+2 \mathrm{Re}(\vc^\dagger \vx) \\
\mathrm{s.t.}\,\;\;\; & X_{i,i}=1,\;i=1,2,\ldots,n,\\
& x_i = \sum_{j=1}^M t_{i,j}s_j,\;\sum_{j=1}^{M}t_{i,j}=1,\;i=1,2,\ldots,n,\\
& t_{i,j}\geq 0,\;j=1,2,\ldots,M,\;i=1,2,\ldots,n,\\
&\mX\succeq \vx\vx^\dagger,
\end{aligned}
\end{equation}
where $\vx\in\C^n$, $\mX\in\C^{n\times n}$, and
$\vt\in \R^{Mn}$ is the concatenation of $M$-dimensional vectors $\vt_1,\vt_2,\ldots,\vt_n$
with $\vt_i=[t_{i,1},t_{i,2},\ldots,t_{i,M}]^{\T}$.
The authors in~\cite{tightness_enhanced} further proved that \cref{eq:CSDP2} is tight if condition \cref{eq:CSDP2_suff} holds.
We term the above SDP as ``ESDR\nobreakdash-$\mX$'', where ``E'' stands for ``enhanced'' and
``$\mX$'' refers to the matrix variable. The same naming convention is adopted for
all the SDRs below.
We can also formulate \cref{eq:reformulated ML}
in the real domain and then use the same technique to get a real counterpart
of \cref{eq:CSDP2}. Let
\begin{equation}\label{eq:def of hat Q and hat c}
{\vy} =
\begin{bmatrix}
\Real(\vx) \\
\Imag(\vx)
\end{bmatrix},\;
\hat{\mQ}=
\begin{bmatrix}
\Real(\mQ) & -\Imag(\mQ) \\
\Imag(\mQ) & \Real(\mQ)
\end{bmatrix},\;\text{and}\;
\hat{\vc} =
\begin{bmatrix}
\Real(\vc) \\
\Imag(\vc)
\end{bmatrix},
\end{equation}
then the real enhanced SDR \cref{eq:ERSDP} is given by
\begin{equation}\label{eq:ERSDP}\tag{ESDR-$\mY$}
\begin{aligned}
\min_{\vt,\vy,\mY}\;\;\; &\langle \hat{\mQ},\mY \rangle+2 \hat{\vc}^{\T} \vy \\
\mathrm{s.t.}\;\;\;\; & \mathcal{Y}(i) = \sum_{j=1}^{M}t_{i,j}\mK_j,
\;\sum_{j=1}^{M}t_{i,j}=1, \;i=1,2,\ldots,n,\\
& t_{i,j}\geq 0,\;j=1,2,\ldots,M,\;i=1,2,\ldots,n,\\
& \mY\succeq \vy\vy^{\T},
\end{aligned}
\end{equation}
where $\vt\in \R^{Mn},\;\vy\in \R^{2n},\;\mY\in \R^{2n\times 2n}$, and we define
\begin{equation*}
\mathcal{Y}(i):=
\begin{bmatrix}
1 & y_i & y_{n+i} \\
y_i & Y_{i,i} & Y_{i,n+i} \\
y_{n+i} & Y_{n+i,i} & Y_{n+i,n+i}
\end{bmatrix},\;
i=1,2,\ldots,n.
\end{equation*}
In \cref{eq:ERSDP}, these $3\times 3$ matrices are constrained in a convex hull whose extreme points are
\begin{equation}\label{eq:def of K}
\mK_j =
\begin{bmatrix}
1 \\ s_{R,j} \\ s_{I,j}
\end{bmatrix}
\begin{bmatrix}
1 \\ s_{R,j} \\ s_{I,j}
\end{bmatrix}^{\T},\;
j=1,2,\ldots,M,
\end{equation}
where $s_{R,j}=\Real(s_j)$ and $s_{I,j}=\Imag(s_j)$.
It has been shown that \cref{eq:ERSDP} is tighter than \cref{eq:CSDP2}~\cite[Theorem~4.1]{tightness_enhanced}, and hence
\cref{eq:ERSDP} is tight whenever \cref{eq:CSDP2} is tight.
Now we turn to another class of SDRs developed from a different perspective in~\cite{near_ml_decoding},
which is applicable to a general symbol set.
The idea is to introduce
binary variables to express $x_i\in \mathcal{S}_M$ by
\begin{equation}\label{eq:express discrete by 01}
x_i = \vt_i^{\T}\vs,\;i=1,2,\ldots,n,
\end{equation}
where $\vt_i=[t_{i,1},t_{i,2},\ldots,t_{i,M}]^{\T}$, $\sum_{j=1}^M t_{i,j}=1$, and $t_{i,j}\in \{0,1\}$.
The above constraints~\cref{eq:express discrete by 01} can be rewritten in a compact form as
$\vx = {\mS} \vt$,
where $\mS=\mI_n \otimes \vs^{\T}$ and we concatenate all vectors $\vt_i$
to get $\vt=[\vt^{\T}_1,\ldots,\vt^{\T}_n]^{\T}\in \R^{Mn}$.
Similarly, we can also formulate \cref{eq:express discrete by 01} in the real domain as
$\vy = \hat{\mS}\vt$,
where
\begin{equation}\label{eq:def_of_S_hat}
\vy =
\begin{bmatrix}
\Real(\vx) \\
\Imag(\vx)
\end{bmatrix}\;\text{and}\;
\hat{\mS} =
\begin{bmatrix}
\Real(\mS) \\
\Imag(\mS)
\end{bmatrix}
=
\begin{bmatrix}
\mI_n \otimes \vs_R^{\T} \\
\mI_n \otimes \vs_I^{\T}
\end{bmatrix}.
\end{equation}
By introducing $\mT=\vt\vt^{\T}\in \R^{Mn\times Mn}$, the problem \cref{eq:ML} is equivalent to
\begin{equation}\label{eq:reformulated ML T}
\begin{aligned}
\min_{\vt,\mT}\;\;\; &\langle \bar{\mQ},\mT \rangle+2\bar{\vc}^{\T} \vt \\
\mathrm{s.t.}\;\;\; & \sum_{j=1}^M t_{i,j}=1,\;i=1,2,\ldots,n,\\
& t_{i,j}\in\{0,1\},\;\;j=1,2,\ldots,M,\;i=1,2,\ldots,n,\\
&\mT= \vt\vt^{\T},
\end{aligned}
\end{equation}
where
\begin{equation}\label{eq:def_of_bar_Q_bar_c}
\bar{\mQ} = \hat{\mS}^{\T}\hat{\mQ}\hat{\mS}\;\text{and}\;
\bar{\vc} = \hat{\mS}^{\T}\hat{\vc}.
\end{equation}
To derive an SDR for \cref{eq:reformulated ML T}, we first allow
$t_{i,j}$ to take any value between 0 and 1.
For the rank-one constraint $\mT=\vt\vt^{\T}$, the authors in~\cite{near_ml_decoding}
proposed four ways of relaxation and we will introduce two of them in the following\footnote{
Our formulations are slightly different from the original ones in~\cite{near_ml_decoding}
since they used the equality constraints
to eliminate one variable for each $t_i$ before relaxing the PSD constraint.
However, in numerical tests we found that this variation only causes
a negligible difference in the optimal solutions of the SDRs.
}.
We first partition $\mT$ as an $n\times n$ block matrix
\begin{equation*}
\mT=
\begin{bmatrix}
\mT_{1,1} & \mT_{1,2} & \ldots & \mT_{1,n} \\
\mT_{2,1} & \mT_{2,2} & \ldots & \mT_{2,n} \\
\vdots & \vdots & \ddots & \vdots \\
\mT_{n,1} & \mT_{n,2} & \ldots & \mT_{n,n}
\end{bmatrix},
\end{equation*}
where $\mT_{i,j}\in \R^{M\times M}$ for $i=1,2,\ldots,n$ and $j=1,2,\ldots,n$.
In the first model, we relax $\mT=\vt\vt^{\T}$ to $\mT\succeq \vt\vt^{\T}$ and
impose constraints on the diagonal elements:
\begin{equation}\label{eq:Model II}\tag{ESDR1-$\mT$}
\begin{aligned}
\min_{\vt,\mT}\;\;\; & \langle\bar{\mQ},\mT\rangle+2\bar{\vc}^{\T}\vt\\
\mathrm{s.t.}\;\;\; & t_{i,j}\geq 0,\;\sum_{j=1}^{M}t_{i,j}=1,\;j=1,2,\ldots,M,\;i=1,2,\ldots,n,\\
& \diag({\mT}_{i,i})={\vt}_{i},\;i=1,2,\ldots,n,\\
& \mT\succeq\vt\vt^{\T},
\end{aligned}
\end{equation}
where $\vt\in \R^{Mn}$ and $\mT\in \R^{Mn\times Mn}$.
The second model further requires $\mT_{i,i}$ to be a diagonal matrix, leading
to the following SDR:
\begin{equation}\label{eq:Model III}\tag{ESDR2-$\mT$}
\begin{aligned}
\min_{\vt,\mT}\;\;\; & \langle\bar{\mQ},\mT\rangle+2\bar{\vc}^{\T}\vt\\
\mathrm{s.t.}\;\;\; & t_{i,j}\geq 0,\;\sum_{j=1}^{M}t_{i,j}=1,\;j=1,2,\ldots,M,\;i=1,2,\ldots,n,\\
& {{\mT}_{i,i}=\Diag({\vt}_{i})},\;i=1,2,\ldots,n,\\
& \mT\succeq\vt\vt^{\T}.
\end{aligned}
\end{equation}
Since \cref{eq:Model III} puts more constraints on the variables $\vt$ and $\mT$,
\cref{eq:Model III} is tighter than \cref{eq:Model II}. Notably,
it is shown in~\cite{liu2019equivalence} that \cref{eq:Model III} is equivalent to \cref{eq:ERSDP},
and hence \cref{eq:CSDP2_suff} is also a sufficient condition for \cref{eq:Model III} to be tight.
\cref{tab:summary of sdrs} summarizes all SDR models discussed in this paper, where we highlight our
contributions on the tightness of different SDRs in bold.
\begin{table}[htbp]\label{tab:summary of sdrs}
{\footnotesize
\caption{Summary of SDR models in this paper.}
\begin{center}
\begin{tabular}{|c|C{19mm}|c|C{19mm}|p{43mm}|}
\hline
SDR model & Origin & Domain & Dimension of PSD cone & Comments \\ \hline
CSDR & Ma \textit{et al}.~\cite{SDR_MPSK} & $\C$ & $n+1$ & tight with probability 0~\cite{tightness_enhanced} \\ \hline
ESDR-$\mX$ & CSDP2 in Lu \textit{et al}.~\cite{tightness_enhanced} & $\C$ & $n+1$ & \textbf{tight if and only if \cref{eq:CSDP2_necc_and_suff_intro} holds} \\ \hline
ESDR-$\mY$ & ERSDP in Lu \textit{et al}.~\cite{tightness_enhanced} & $\R$ & $2n+1$ & \textbf{tight only if \cref{eq:ERSDP_necc_intro} holds} \\ \hline
ESDR1-$\mT$ & Model II in Mobasher \textit{et al}.~\cite{near_ml_decoding} & $\R$ & $Mn+1$ & \textbf{tight with probability no greater than~$(2/M)^n$} \\ \hline
ESDR2-$\mT$ & Model III in Mobasher \textit{et al}.~\cite{near_ml_decoding} & $\R$ & $Mn+1$ & equivalent to ESDR-$\mY$~\cite{liu2019equivalence} \\ \hline
\end{tabular}
\end{center}
}
\end{table}
\section{Tightness of semidefinite relaxations}
\label{sec:tightness}
\subsection{Tightness of \texorpdfstring{{\cref{eq:CSDP2}}}{(ESDR-X)}}
Let $\mX^*=\vx^*(\vx^*)^\dagger$, and the key
idea of showing the tightness of \cref{eq:CSDP2}
is to certify $(\vx^*,\mX^*)$ as the optimal solution by
considering the
Karush-Kuhn-Tucker (KKT) conditions of \cref{eq:CSDP2}.
Our derivation is based on~\cite[Theorem~4.2]{tightness_enhanced} and
we provide a simplified version for completeness.
\begin{theorem}[{\cite[Theorem~4.2]{tightness_enhanced}}]\label{thm:main thm for CSDP2}
Suppose that $M\geq 4$. Then
$(\vx^*,\mX^*)$ is the optimal solution of \cref{eq:CSDP2}
if and only if there exist
\begin{equation*}
\lambda_i\in \R,\;\mu_{i,-1}\geq 0,\;\text{and}\;\mu_{i,1}\geq 0,\;i=1,2,\ldots,n,
\end{equation*}
such that $\mH$ and $\vv$ in \cref{eq:mimo} satisfy
\begin{equation*}
(x_i^*)^{-1}(\mH^\dagger \vv)_i =
\lambda_i + \frac{\mu_{i,-1}}{2}e^{-\imag\frac{\pi}{M}} + \frac{\mu_{i,1}}{2}e^{\imag\frac{\pi}{M}},\;
i=1,2,\ldots,n,
\end{equation*}
and
$
\mQ + \Diag(\vlambda) \succeq 0
$.
\end{theorem}
The authors in~\cite{tightness_enhanced} further derived the sufficient condition
\cref{eq:CSDP2_suff}, under which the conditions in \cref{thm:main thm for CSDP2} are met by
choosing $\lambda_i=-\lambda_{\mathrm{min}}(\mQ)$ for $i=1,2,\ldots,n$.
To strengthen their analysis,
we view the conditions in \cref{thm:main thm for CSDP2} as
a semidefinite feasibility problem. To be specific, if we define
\begin{equation}\label{eq:def of z}
z_i = (x_i^*)^{-1}(\mH^\dagger \vv)_i,\;i=1,2,\ldots,n,
\end{equation}
and
\begin{equation*}
\mathcal{C}(\lambda)=\left\{
z\in \C :\; \exists\; \mu_{-1},\;\mu_{1}\geq 0\;\text{s.t.}\;
z=\lambda+
\frac{\mu_{-1}}{2}e^{-\imag\frac{\pi}{M}} + \frac{\mu_{1}}{2}e^{\imag\frac{\pi}{M}}
\right\},
\end{equation*}
then \cref{thm:main thm for CSDP2} states that
\cref{eq:CSDP2} is tight if and only if the following problem is feasible:
\begin{equation}\label{eq:feasibility CSDP2}
\begin{aligned}
\mathrm{find}\;\;\; & \vlambda\in \R^n\\
\mathrm{s.t.}\;\;\; &\mQ + \Diag(\vlambda) \succeq 0,\\
& z_i\in \mathcal{C}(\lambda_i),\;i=1,2,\ldots,n.
\end{aligned}
\end{equation}
Each constraint $z_i\in \mathcal{C}(\lambda_i)$ turns
out to be a simple inequality on $\lambda_i$.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.7\linewidth]{figs/visual_C3.pdf}
\caption{Illustration of $\mathcal{C}(\lambda_i)$ in the complex plane.}
\label{fig:visual}
\end{figure}
To see this, we plot $\mathcal{C}(\lambda_i)$ as
the shaded area in \cref{fig:visual}.
It is clear from the figure that
\begin{equation*}
z_i\in \mathcal{C}(\lambda_i)
\;\Leftrightarrow\;
|\Imag(z_i)|\leq \big(-\lambda_i+\Real(z_i)\big)\tan\left(\frac{\pi}{M}\right),
\end{equation*}
which leads to
\begin{equation*}
z_i\in \mathcal{C}(\lambda_i)
\;\Leftrightarrow\;
\lambda_i\leq \Real(z_i)-|\Imag(z_i)|\cot\left(\frac{\pi}{M}\right).
\end{equation*}
This, together with \cref{eq:feasibility CSDP2}, gives the necessary and
sufficient condition for \cref{eq:CSDP2} to be tight and
we formally state it in \cref{thm:CSDP2_necess_suff}.
\begin{theorem}\label{thm:CSDP2_necess_suff}
Suppose that $M\geq 4$. Then \cref{eq:CSDP2} is tight if and only if
\begin{equation}\label{eq:CSDP2_necc_and_suff}
\mQ + \Diag\big(\mathrm{Re}(\vz)\big)
-\cot\left(\frac{\pi}{M}\right)\Diag(|\mathrm{Im}(\vz)|\big) \succeq 0,
\end{equation}
where $\vz=[z_1,z_2,\ldots,z_n]^{\T}\in \C^n$.
\end{theorem}
Note that \cref{eq:CSDP2_necc_and_suff} is
exactly the same as \cref{eq:CSDP2_necc_and_suff_intro} if we recall the
definitions of $\mQ$ in \cref{eq:def of Q and c} and $z_i$ in \cref{eq:def of z}.
Furthermore, if we set $M=2$ and $\mH,\;\vv$ to be real in \cref{eq:CSDP2_necc_and_suff_intro},
it becomes the same as the previous result \cref{eq:real_channel_binary}.
Hence, our result extends \cref{eq:real_channel_binary} to the more general
case where $M\geq 4$ and $\mH,\;\vv$ are complex.
Finally,
the sufficient
condition \cref{eq:CSDP2_suff} in~\cite{tightness_enhanced} can be derived from
our result.
Since
\begin{align*}
\mathrm{Re}(z_i)-|\Imag(z_i)|\cot\left(\frac{\pi}{M}\right) &= \frac{1}{\sin\left(\frac{\pi}{M}\right)}
\left( \mathrm{Re}(z_i)\sin\left(\frac{\pi}{M}\right)-|\Imag(z_i)|\cos\left(\frac{\pi}{M}\right)\right) \\
& \geq -\frac{1}{\sin\left(\frac{\pi}{M}\right)}
|z_i|
\geq -\frac{1}{\sin\left(\frac{\pi}{M}\right)} \|\mH^\dagger \vv\|_{\infty},
\end{align*}
we have
\begin{equation*}
\Diag\big(\mathrm{Re}(\vz)\big)
-\cot\left(\frac{\pi}{M}\right)\Diag(|\mathrm{Im}(\vz)|\big) \succeq
-\frac{1}{\sin\left(\frac{\pi}{M}\right)} \|\mH^\dagger \vv\|_{\infty} \mI_n.
\end{equation*}
Combining this with
$\mQ\succeq \lambda_{\mathrm{min}}(\mQ)\mI_n$, we can see that
\cref{eq:CSDP2_suff} is a stronger condition on
$\mH$ and $\vv$ than \cref{eq:CSDP2_necc_and_suff_intro}.
\subsection{Tightness of \texorpdfstring{{\cref{eq:ERSDP}}}{(ESDR-Y)}}
Similar to \cref{thm:main thm for CSDP2}, we have the following
characterization for \cref{eq:ERSDP} to be tight.
Since the proof technique is essentially the same as that in~\cite{tightness_enhanced},
we put the proof in a separate technical report~\cite{companion_report}.
\begin{theorem}\label{thm:main thm for ERSDP}
Suppose that $M\geq 4$. Let the transmitted symbol vector $\vx^*$ be
\begin{equation*}
x_i^* =s_{u_i},\;u_i\in\{1,2,\ldots,M\},\;i=1,2,\ldots,n,
\end{equation*}
and define
\begin{align
\hat{\vv} &=
\begin{bmatrix}
\mathrm{Re}(\vv) \\
\mathrm{Im}(\vv)
\end{bmatrix}\in \R^{2n},\;
\hat{\mH} =
\begin{bmatrix}
\mathrm{Re}(\mH) & -\mathrm{Im}(\mH) \\
\mathrm{Im}(\mH) & \mathrm{Re}(\mH)
\end{bmatrix}\in \R^{2n\times 2n}, \label{eq:def hat v and hat H}\\
\vy^* &=
\begin{bmatrix}
\mathrm{Re}(\vx^*) \\
\mathrm{Im}(\vx^*)
\end{bmatrix}\in \R^{2n},\;
\mY^* = \vy^*(\vy^*)^\T
\in \R^{2n\times 2n}. \nonumbe
\end{align}
Then $(\vy^*,\mY^*)$ is the optimal solution of \cref{eq:ERSDP}
if and only if there exist
${\vlambda} \in \R^{2n}$, ${\vmu}\in \R^{n}$, and
${\vg} \in \R^{2n}$
that satisfy
\begin{gather}
\hat{\mH}^{\T} \hat{\vv} = {\vg}+({\mLambda}+{\mM})\vy^*, \label{eq:ERSDP equality}\\
\langle {\mGamma}_i,\mK_{u_i}\rangle
\geq \langle {\mGamma}_i,\mK_j\rangle,\;
j=1,2,\ldots,M,\;i=1,2,\ldots,n, \label{eq:ERSDP inequality}
\end{gather}
and
\begin{equation*}
\hat{\mQ}+{\mLambda}+{\mM} \succeq 0,
\end{equation*}
where $\mK_j$ is defined in \cref{eq:def of K}, $\hat{\mQ}$ is defined in \cref{eq:def of hat Q and hat c}, and
\begin{equation}\label{eq:def_of_Lambda_M_Gamma}
{\mLambda}=\Diag({\vlambda}),\;
{\mM}=
\begin{bmatrix}
\bm{0} & \Diag({\vmu}) \\
\Diag({\vmu}) & \bm{0}
\end{bmatrix},\;
{\mGamma}_i =
\begin{bmatrix}
0 & {g}_i & {g}_{n+i} \\
{g}_i & {\lambda}_i & {\mu}_i \\
{g}_{n+i} & {\mu}_i & {\lambda}_{n+i}
\end{bmatrix}.
\end{equation}
\end{theorem}
Furthermore, \cref{eq:ERSDP equality,eq:ERSDP inequality} in
\cref{thm:main thm for ERSDP} can be simplified to
the following inequalities on $(\vlambda,\vmu)$ (see \cref{appen:simplification}):
\begin{equation}\label{eq:simplified inequality}
\begin{aligned}
\sin^2\biggl(&\theta_{u_i}+\frac{\Delta\theta_j}{2}\biggr)\lambda_i+
\cos^2\left(\theta_{u_i}+\frac{\Delta\theta_j}{2}\right)\lambda_{n+i}
-
\sin\left(2\theta_{u_i}+\Delta\theta_j\right)\mu_j \\
&\leq \Real(z_i)-\cot\left(\frac{\Delta\theta_j}{2}\right)\Imag(z_i),\;
j\in\{1,2,\ldots,M\}\backslash\{u_i\},\;i=1,2,\ldots,n.
\end{aligned}
\end{equation}
Here $\Delta \theta_j=\theta_j-\theta_{u_i}$,
$\theta_{u_i}$ is the phase of the $i$-th transmitted symbol $x_i^*$,
and $z_i$ is defined in \cref{eq:def of z}.
Similar to \cref{eq:feasibility CSDP2}, we formulate the conditions
in \cref{thm:main thm for ERSDP} as a semidefinite feasibility problem as follows:
\begin{equation}\label{eq:ERSDP feasibility}
\begin{aligned}
\mathrm{find}\;\;\; & \vlambda\in \R^{2n}\;\text{and}\;\vmu\in \R^n \\
\mathrm{s.t.}\;\;\; & \hat{\mQ}+\mLambda+\mM\succeq 0,\\
& \text{\cref{eq:simplified inequality} is satisfied,}
\end{aligned}
\end{equation}
where $\mLambda$ and $\mM$ are defined in \cref{eq:def_of_Lambda_M_Gamma}.
However, unlike problem \cref{eq:feasibility CSDP2} where every inequality
only involves one dual variable, problem \cref{eq:ERSDP feasibility} has
inequalities with three variables coupled together and it is unclear how to choose the
``optimal'' $\vlambda$ and $\vmu$. In the following, we give
a simple necessary condition for \cref{eq:ERSDP} being tight
based on \cref{eq:ERSDP feasibility}.
\begin{theorem}\label{thm:necess of ERSDP}
Suppose that $M\geq 4$.
If \cref{eq:ERSDP} is tight, then
\begin{equation}\label{eq:ERSDP necessary}
\mQ + \Diag\big(\mathrm{Re}(\vz)\big)
-\cot\left(\frac{2\pi}{M}\right)\Diag(|\mathrm{Im}(\vz)|\big) \succeq 0,
\end{equation}
where $\mQ=\mH^\dagger\mH$ and $\vz$ is defined in \cref{eq:def of z}.
\end{theorem}
Before proving \cref{thm:necess of ERSDP}, we first introduce the following lemma.
\begin{lemma}\label{lem:real semidefinite to complex semidefinite}
Suppose that $\mV$ is a PSD matrix in $\R^{2n}$ and is partitioned as
\begin{equation*}
\mV =
\begin{bmatrix}
\mA & \mB \\
\mB^{\T} & \mC
\end{bmatrix},
\end{equation*}
where $\mA=\mA^{\T}$, $\mC=\mC^{\T}$, and $\mA,\mB,\mC\in \R^{n\times n}$. Then
\begin{equation*}
\mU = \frac{1}{2}(\mA+\mC)+\frac{\imag}{2}(\mB^{\T}-\mB)
\end{equation*}
is a PSD matrix in $\C^n$.
\end{lemma}
\begin{proof}
We observe that
\begin{equation*}
\mU = \frac{1}{2}\begin{bmatrix}
\mI_n & \imag \mI_n
\end{bmatrix}
\begin{bmatrix}
\mA & \mB \\
\mB^{\T} & \mC
\end{bmatrix}
\begin{bmatrix}
\mI_n \\ -\imag \mI_n
\end{bmatrix}.
\end{equation*}
The result follows immediately.
\end{proof}
\begin{proof}[Proof of \cref{thm:necess of ERSDP}]
If \cref{eq:ERSDP} is tight, we can find
${\vlambda}\in \R^{2n}$ and ${\vmu}\in \R^n$ that satisfy the
constraints in \cref{eq:ERSDP feasibility}.
By \cref{lem:real semidefinite to complex semidefinite},
the constraint $\hat{\mQ}+\mLambda+\mM\succeq 0$ implies
\begin{equation}\label{eq:semidefinite blambda}
\mQ+\Diag(\bvlambda) \succeq 0,
\end{equation}
where $\bvlambda\in \R^n$ is given by
\begin{equation*}
\blambda_i = \frac{1}{2}(\lambda_i+\lambda_{n+i}),\;i=1,2,\ldots,n.
\end{equation*}
Fix $i\in\{1,2,\ldots,n\}$ and let $\hat{z}_i$ and $\hat{z}_{n+i}$
denote $\Real(z_i)$ and $\Imag(z_i)$, respectively.
If $\hat{z}_{n+i}\geq 0$, we
set $\Delta \theta_j=\frac{2\pi}{M}$
in \cref{eq:simplified inequality} to get
\begin{equation*}
\sin^2\Bigl(\theta_{u_i}+\frac{\pi}{M}\Bigr)\lambda_i+
\cos^2\Bigl(\theta_{u_i}+\frac{\pi}{M}\Bigr)\lambda_{n+i}-
\sin\Bigl(2\theta_{u_i}+\frac{2\pi}{M}\Bigr)\mu_j
\leq \hat{z}_i-\cot\Bigl(\frac{\pi}{M}\Bigr)|\hat{z}_{n+i}|.
\end{equation*}
Since $M\geq 4$, we can also set
$\Delta \theta_j=\frac{2\pi}{M}+\pi$ to get
\begin{equation*}
\cos^2\Bigl(\theta_{u_i}+\frac{\pi}{M}\Bigr)\lambda_i+
\sin^2\Bigl(\theta_{u_i}+\frac{\pi}{M}\Bigr)\lambda_{n+i}+
\sin\Bigl(2\theta_{u_i}+\frac{2\pi}{M}\Bigr)\mu_j
\leq \hat{z}_i+\tan\Bigl(\frac{\pi}{M}\Bigr)|\hat{z}_{n+i}|.
\end{equation*}
Adding the above two inequalities and dividing both sides by two,
we have
\begin{equation}\label{eq:inequality on blambda}
\blambda_i=\frac{1}{2}(\lambda_i+\lambda_{n+i})\leq \hat{z}_i-\cot\left(\frac{2\pi}{M}\right)|\hat{z}_{n+i}|.
\end{equation}
If $\hat{z}_{n+i}<0$, we can also arrive at \cref{eq:inequality on blambda}
by setting
$\Delta\theta_j$ to be $-\frac{2\pi}{M}$ and $-\frac{2\pi}{M}+\pi$, respectively.
Finally, \cref{thm:necess of ERSDP} follows from \cref{eq:semidefinite blambda}
and \cref{eq:inequality on blambda}.
\end{proof}
Note that \cref{eq:ERSDP necessary} is the same as \cref{eq:ERSDP_necc_intro} if we
recall the definitions of $\mQ$ in \cref{eq:def of Q and c} and $z_i$ in \cref{eq:def of z}.
Moreover, since \cref{eq:ERSDP} is tighter than \cref{eq:CSDP2},
\cref{eq:ERSDP} will also be tight if \cref{eq:CSDP2_necc_and_suff_intro} holds. Therefore, we
have both a necessary condition \cref{eq:ERSDP_necc_intro}
and a sufficient condition \cref{eq:CSDP2_necc_and_suff_intro} for \cref{eq:ERSDP} to be tight.
\subsection{Tightness of \texorpdfstring{{\cref{eq:Model II}}}{(ESDR1-T)}}
In the same spirit,
we first give a necessary and sufficient condition for \cref{eq:Model II} to be tight.
Since the technique is essentially the same as that used in \cref{thm:main thm for ERSDP}, we omit the
proof details due to the space limitation.
\begin{theorem} \label{thm:main thm for Model II}
Suppose that $M\geq 4$. Let the transmitted symbol vector $\vx^*$ be
\begin{equation*}
x_i^* =s_{u_i},\;u_i\in\{1,2,\ldots,M\},\;i=1,2,\ldots,n,
\end{equation*}
and define
\begin{gather*}
\vt^*_{i,u_i} = 1,\; \vt^*_{i,j} = 0,\;j\neq u_i,\;i=1,2,\ldots,n, \\
\mT^* = \vt^* (\vt^*)^{\T}. \label{eq: rank 1}
\end{gather*}
Then $(\vt^{*},\mT^{*})$ is the optimal solution of \eqref{eq:Model II}
if and only if there exist ${\valpha}\in \mathbb{R}^n$ and ${{\vgamma}}\in\mathbb{R}^{Mn}$ such that
\begin{equation}\label{eq:Model II equality}
\Diag(1-2\vt^*){\vgamma}=-2{\hat{\mS}}^{\T}\hat{\mH}^{\T}\hat{\vv}+{\valpha}\otimes \1_M,
\end{equation}
and
\begin{equation}\label{eq:Model II semidefiniteness}
\bar{\mQ}+\Diag({{{\vgamma}}})\succeq 0.
\end{equation}
where $\hat{\mS}$ is defined in
\cref{eq:def_of_S_hat},
$\hat{\mH},\;\hat{\vv}$ are defined in \cref{eq:def hat v and hat H},
and $\bar{\mQ}$ is defined in \cref{eq:def_of_bar_Q_bar_c}.
\end{theorem}
Now we provide a corollary that will serve as our basis for further derivation.
\begin{corollary} \label{coro:Model II necessary condition}
If \eqref{eq:Model II} is tight, then there exist
\begin{equation*}
{\valpha}\in \R^n\;\text{and}\;{\vgamma}_1,{\vgamma}_2,\ldots,{\vgamma}_n\in \R^{M}
\end{equation*}
that satisfy
\begin{equation}\label{eq:Model II equality explicit}
{\gamma}_{i,j} =
\begin{cases}
-2 \mathrm{Re}[s_j^\dagger (\mH^\dagger \vv)_i]+{\alpha}_i,\;\;\; &\text{if}\;j\neq u_i, \\
2 \mathrm{Re}[s_j^\dagger (\mH^\dagger \vv)_i]-{\alpha}_i,\;\;\; &\text{if}\;j=u_i,
\end{cases}
\;j=1,2,\ldots,M,\;i=1,2,\ldots,n,
\end{equation}
and
\begin{equation*}
\vw^{\T}\Diag({\vgamma_i})\vw \geq 0,\;\;i=1,2,\ldots,n,
\end{equation*}
for any $\vw\in \R^M$ such that
$
\vw^{\T}\vs_{R}=\vw^{\T}\vs_{I}=0
$.
\end{corollary}
\begin{proof}
By \cref{thm:main thm for Model II}, if \cref{eq:Model II} is tight, we
can find ${\valpha}\in \R^n$ and ${\vgamma}\in \R^{Mn}$ that satisfy
\eqref{eq:Model II equality} and \eqref{eq:Model II semidefiniteness}. Let
$\vgamma$ be partitioned as
${\vgamma}=[{\vgamma}_1^{\T},{\vgamma}_2^{\T},\ldots,{\vgamma}_n^{\T}]^{\T}$
where ${\vgamma}_j\in \R^M$
is the $j$-th block of ${\vgamma}$.
It is straightforward to verify that \eqref{eq:Model II equality} is equivalent to
\eqref{eq:Model II equality explicit}.
Moreover, for any $i\in \{1,2,\ldots,n\}$ and
any $\vw\in \R^M$ that satisfies $\vw^{\T}\vs_{R}=\vw^{\T}\vs_{I}=0$, we set
$\bar{\vw}=[\bar{\vw}_1^{\T},\bar{\vw}_2^{\T},\ldots,\bar{\vw}_n^{\T}]^{\T}\in \R^{Mn}$ to be
\begin{equation*}
\bar{\vw}_j =
\begin{cases}
\0, & \text{if}\; j\neq i,\\
\vw, & \text{if}\; j=i.
\end{cases}
\end{equation*}
It is simple to check that $\hat{{\mS}}\bar{\vw}=\0$. Therefore, recalling that
$\bar{\mQ}=\hat{{\mS}}^{\T}\mhQ\hat{{\mS}}$, by \eqref{eq:Model II semidefiniteness}
we have
\begin{equation*}
\bar{\vw}^{\T} (\bar{\mQ}+\Diag({{{\vgamma}}}))\bar{\vw}
=\bar{\vw}^{\T}\Diag({{{\vgamma}}})\bar{\vw}
=\vw^{\T}\Diag({\vgamma_i})\vw \geq 0.
\end{equation*}
The proof is complete.
\end{proof}
In practice,
the symbol set $\sS$, such as the one in \cref{eq:MPSK symbol set} considered in this paper,
is symmetric with respect to the origin.
Therefore, we can find
$u_i'\in\{1,2,\ldots,M\}$ that satisfies $s_{u_i'}=-s_{u_i}$. Now let $\vw\in \R^M$ be
\begin{equation*}
w_j =
\begin{cases}
0, & \text{if}\; j\notin \{u_i,u_i'\},\\
1, & \text{if}\; j\in \{u_i,u_i'\},
\end{cases}\; j=1,2,\ldots,M.
\end{equation*}
We have $\vw^{\T}\vs_R=s_{R,u_i}+s_{R,u_i'}=0$ and $\vw^{\T}\vs_I=s_{I,u_i}+s_{I,u_i'}=0$.
Hence, when \cref{eq:Model II} is tight, \cref{coro:Model II necessary condition} implies that
\begin{equation*}
\vw^{\T}\Diag({\vgamma_i})\vw = {\gamma}_{u_i}+{\gamma}_{u_i'}=
4\mathrm{Re}[s_{u_i}^\dagger (\mH^\dagger \vv)_i] =
4\mathrm{Re}[(x_i^*)^\dagger (\mH^\dagger \vv)_i]
\geq 0.
\end{equation*}
This immediately leads to the following upper bound on the tightness probability
of \cref{eq:Model II}.
\begin{corollary}\label{coro:general bound}
Suppose that
the symbol set $\sS$ is symmetric with respect to the origin and $0\notin \sS$.
We further assume that
\begin{enumerate}[label=(\alph*)]
\item The entries of $\vx^*$ are drawn from $\sS$ uniformly and independently;
\item $\vx^*$, $\mH$, and $\vv$ are mutually independent; and
\item the distribution of $\mH$ and $\vv$ are continuous.
\end{enumerate}
Then we have
\begin{equation*}
\Prob\bigl(\text{\cref{eq:Model II} is tight}\bigr) \leq \left(\frac{1}{2}\right)^n.
\end{equation*}
\end{corollary}
\begin{proof}
Let $z_i=(x_i^*)^\dagger(\mH^\dagger \vv)_i$, $i=1,2,\ldots,n$.
Since $\mH$ and $\vv$ are independent continuous random variables,
the event
\begin{equation*}
\bigcup_{i=1}^n \bigcup_{s\in \sS} \{\mathrm{Re}(s^\dagger (\mH^\dagger \vv)_i)=0\}
\end{equation*}
happens with probability zero.
Hence, because of the symmetry of $\sS$,
with probability one exactly half of the symbols $s\in \sS$ satisfy
$
\mathrm{Re}(s^\dagger (\mH^\dagger \vv)_i)>0
$
for each $i\in\{1,2,\ldots,n\}$ when $\mH$ and $\vv$ are given. By the assumption that $x^*_i$ is uniformly distributed
over $\sS$, we obtain
\begin{equation*}
\Prob\bigl(\mathrm{Re}(z_i)\geq 0 \givenp \mH,\vv \bigr) = \frac{1}{2}\;\text{almost surely}.
\end{equation*}
Moreover, $\{z_i\}_{i=1}^n$ are mutually independent conditioned on $\mH$ and $\vv$.
This leads to
\begin{align*}
\Prob\bigl(\mathrm{Re}(z_i)\geq 0,\;i=1,2,\ldots,n \bigr)&=\E_{\mH,\vv}\bigl[\Prob\bigl(\mathrm{Re}(z_i)\geq 0,\;i=1,2,\ldots,n \givenp \mH,\vv\bigr)\bigr] \\
& = \E_{\mH,\vv}\bigl[\prod_{i=1}^n
\Prob\bigl(\mathrm{Re}(z_i)\geq 0 \givenp \mH,\vv\bigr)\bigr] \\
& = \left(\frac{1}{2}\right)^n.
\end{align*}
Finally, \cref{coro:general bound} follows from the fact that the tightness of \cref{eq:Model II} implies $\Real(z_i)\geq 0$, $i=1,2,\ldots,n$.
\end{proof}
It is worth noting that all the assumptions in \cref{coro:general bound} are mild: they are satisfied if
we use the $M$-PSK or QAM modulation scheme and the entries of $\mH$ and $\vv$ follow the complex Gaussian distribution.
Intuitively, we will expect that \cref{eq:Model II} is less likely
to recover the transmitted symbols with an increasing symbol set size $M$.
In the following, we present a more refined upper bound on the tightness
probability specific to the $M$-PSK setting and the proof can be found in~\cite{companion_report}.
\begin{theorem}\label{thm:MPSK bound}
Suppose that $M$-PSK is used with $M\geq 4$
and the same assumptions in
\cref{coro:general bound} hold.
Then we have
\begin{equation*
\Prob\bigl(\text{\cref{eq:Model II} is tight}\bigr) \leq \left(\frac{2}{M}\right)^n.
\end{equation*}
\end{theorem}
From \cref{coro:general bound,thm:MPSK bound}, we can see that the tightness probability
of \cref{eq:Model II} is bounded away from one regardless of the noise level, and
it tends to zero exponentially fast when the number of transmitted symbols $n$ increases.
This is in sharp contrast to \cref{eq:CSDP2,eq:ERSDP}, whose tightness probabilities will
approach one if the noise level is sufficiently small and the number of received signals $m$ is sufficiently large compared to $n$
\cite[Theorem~4.5]{tightness_enhanced}.
\section{Equivalence between different SDRs}
\label{sec:equiv}
In this section, we focus on the relationship between different SDR models of \cref{eq:ML}
Related to the SDRs discussed so far, a recent paper~\cite{liu2019equivalence} proved
that \cref{eq:Model III} is equivalent to \cref{eq:ERSDP} for the MIMO detection problem with
a general symbol set. An earlier paper~\cite{QAM_equivalence} compared
three different SDRs in the QAM setting and showed their equivalence.
Compared with those in \cref{sec:review_of_SDR},
the SDRs considered in~\cite{QAM_equivalence} differ greatly in their motivations and
structures, and the two equivalence results are
proved using different techniques.
In this section, we provide a more general equivalence theorem from which
both results follow as special cases.
This not only reveals the underlying connection between these two works, but also may potentially lead to
new equivalence between SDRs.
\subsection{Review of previous results}
In~\cite{liu2019equivalence}, the authors established the following correspondence
between a pair of feasible points of \cref{eq:Model III} and \cref{eq:ERSDP}:
\begin{equation}\label{eq:modelIII-ERSDP equiv}
\mY = \hat{\mS}\mT\hat{\mS}^{\T}\;\text{and}\;\vy=\hat{\mS}\vt,
\end{equation}
where $\hat{\mS}\in \R^{2n\times Mn}$ is defined in \cref{eq:def_of_S_hat}.
In~\cite{QAM_equivalence}, the authors considered the feasible set
of a virtually-antipodal SDR (VA-SDR):
\begin{equation}\label{eq:VA-SDR}\tag{VA-SDR}
\begin{aligned}
& \begin{bmatrix}
1 & \vb^{\T} \\
\vb & \mB
\end{bmatrix}\in \semiS^{qn+1} \\
\mathrm{s.t.}\;\;\; &
B_{i,i}=1,\;i=1,2,\ldots,qn,
\end{aligned}
\end{equation}
and that of a bounded-constrained SDR (BC-SDR):
\begin{equation}\label{eq:BC-SDR}\tag{BC-SDR}
\begin{aligned}
& \begin{bmatrix}
1 & \vx^{\T} \\
\vx & \mX
\end{bmatrix}\in \semiS^{n+1} \\
\mathrm{s.t.}\;\;\; &
1\leq X_{i,i}\leq (2^q-1)^2,\;i=1,2,\ldots,n,
\end{aligned}
\end{equation}
where $q\geq 1$ is an integer. We refer interested readers to~\cite{QAM_equivalence} and
references therein for their derivations.
The authors proved the equivalence between \cref{eq:VA-SDR} and \cref{eq:BC-SDR} by showing the following correspondence:
\begin{equation}\label{eq:VA-BC equiv}
\mX = {\mW}\mB{\mW}^{\T}\;\text{and}\;\vx={\mW}\vb,
\end{equation}
where
\begin{equation*}
\mW =
\begin{bmatrix}
\mI_n & 2\mI_n & 4\mI_n & \ldots & 2^{q-1}\mI_n
\end{bmatrix}
\in \R^{n\times qn}.
\end{equation*}
Note that both equivalence results in \eqref{eq:modelIII-ERSDP equiv}
and \eqref{eq:VA-BC equiv} fall into the following form:
\begin{equation*}
\left\{
\begin{bmatrix}
1 & \vy^{\T} \\
\vy & \mY
\end{bmatrix}\in \mathcal{F}_1
\right\}
=
\left\{
\begin{bmatrix}
1 & \vy^{\T} \\
\vy & \mY
\end{bmatrix}
=
\begin{bmatrix}
1 & \0 \\
\0 & \mP
\end{bmatrix}
\begin{bmatrix}
1 & \vt^{\T} \\
\vt & \mT
\end{bmatrix}
\begin{bmatrix}
1 & \0 \\
\0 & \mP^{\T}
\end{bmatrix}
:\;
\begin{bmatrix}
1 & \vt^{\T} \\
\vt & \mT
\end{bmatrix}
\in \mathcal{F}_2
\right\},
\end{equation*}
where $\mathcal{F}_1$ is a subset of $\semiS^{k+1}$,
$\mathcal{F}_2$ is a subset of $\semiS^{d+1}$, and we call $\mP\in \R^{k\times d}$
as the transformation matrix. Moreover, both the
transformation matrices $\hat{\mS}$ in \eqref{eq:modelIII-ERSDP equiv}
and $\mW$ in \eqref{eq:VA-BC equiv} have a special ``separable'' property that
we now define for ease of presentation.
\begin{definition}
A matrix $\mP\in\R^{k\times d}$ is called separable if there
exist a partition of rows $\alpha_1,\alpha_2,\ldots,\alpha_l$ and
a partition of columns $\beta_1,\beta_2,\ldots,\beta_l$ such that
\begin{equation*}
\mP[\alpha_i,\beta_j]=\mathbf{0},\qquad \forall\;i\neq j.
\end{equation*}
\end{definition}
In other words, a matrix is separable if, after possibly rearranging rows and
columns, it has a block diagonal structure. In particular, for
the transformation matrix $\hat{\mS}$ in \cref{eq:modelIII-ERSDP equiv},
the corresponding row and column partitions are given by
\begin{equation}\label{eq:parititon in ModelIII}
\alpha_i = \{i,n+i\},\;\beta_i=\{(i-1)M+1,(i-1)M+2,\ldots,iM\},\;i=1,2,\ldots,n;
\end{equation}
for the transformation matrix $\mW$ in \cref{eq:VA-BC equiv},
they are given by
\begin{equation}\label{eq:parititon in VA}
\alpha_i = \{i\},\;\beta_i=\{i,i+n,i+2n,\ldots,i+(q-1)n\},\;i=1,2,\ldots,n.
\end{equation}
\subsection{A general equivalence theorem} Now we are ready to present our main equivalence
result.
\begin{theorem}\label{thm:main equiv thm}
Suppose that the matrix $\mP\in \R^{k\times d}$ is separable with
row partition $\alpha_1,\alpha_2,\ldots, \alpha_l$
and column partition $\beta_1,\beta_2,\ldots, \beta_l$
Moreover, define
\begin{equation*}
k_i:=|\alpha_i|,\;d_i:=|\beta_i|,\;\text{and}\;\mP_i:=\mP[\alpha_i,\beta_i]\in \R^{k_i\times d_i},\;
i=1,2,\ldots,l,
\end{equation*}
where we use $|\cdot|$ to denote the cardinality of a set.
Then given arbitrary constraint sets ${\mathcal{A}_i\subset \R^{d_i\times d_i}}$
for $i=1,2,\ldots,l$,
the following set
\begin{equation}\label{eq:original}
\begin{aligned}
& \begin{bmatrix}
1 & \vy^{\T} \\
\vy & \mY
\end{bmatrix}\in \semiS^{k+1} \\
\mathrm{s.t.}\;\;\; &
\mY = \mP \mT \mP^{\T},\;\vy=\mP\vt,\\
& \begin{bmatrix}
1 & \vt^{\T} \\
\vt & \mT
\end{bmatrix}\in \semiS^{d+1},\\
& \begin{bmatrix}
1 & \vt[\beta_i]^{\T} \\
\vt[\beta_i] & \mT[\beta_i]
\end{bmatrix}\in \mathcal{A}_i,\;i=1,2,\ldots,l,
\end{aligned}
\end{equation}
where the variables are $\vy\in \R^k$, $\mY\in \R^{k\times k}$, $\vt\in \R^d$, and $\mT\in \R^{d\times d}$,
is the same as
\begin{equation}\label{eq:decomposed}
\begin{aligned}
& \begin{bmatrix}
1 & \vy^{\T} \\
\vy & \mY
\end{bmatrix}\in \semiS^{k+1} \\
\mathrm{s.t.}\;\;\; &
\mY[\alpha_i] = \mP_i \mT^{(i)} \mP_i^{\T},\;\vy[\alpha_i]=\mP_i\vt^{(i)},\\
& \begin{bmatrix}
1 & (\vt^{(i)})^{\T} \\
\vt^{(i)} & \mT^{(i)}
\end{bmatrix}\in \semiS^{d_i+1},\\
& \begin{bmatrix}
1 & (\vt^{(i)})^{\T} \\
\vt^{(i)} & \mT^{(i)}
\end{bmatrix} \in \mathcal{A}_i,\;i=1,2,\ldots,l,
\end{aligned}
\end{equation}
where the variables are $\vy\in \R^k$, $\mY\in \R^{k\times k}$, $\vt^{(i)}\in \R^{d_i}$, and $\mT^{(i)}\in \R^{d_i\times d_i}$
with $i=1,2,\ldots,l$.
\end{theorem}
The following lemma will be
useful in our proof.
\begin{lemma}[{\cite[Theorem~7.3.11]{matrix_analysis}}]\label{lem: two factorizations}
Let $\mA\in \R^{p\times n}$ and $\mB\in \R^{q\times n}$ where $p\leq q$. Then
$\mA^{\T}\mA=\mB^{\T}\mB$ if and only if there exists a matrix $\mU\in \R^{q\times p}$ with
$\mU^{\T}\mU=\mI_p$ such that $\mB=\mU\mA$.
\end{lemma}
\begin{proof}[Proof of \cref{thm:main equiv thm}]
Without loss of generality, we assume the transformation matrix $\mP\in \R^{k\times d}$
is in the form
\begin{equation*}
\mP =
\begin{bmatrix}
\mP_1 & & & \\
& \mP_2 & & \\
& & \ddots & \\
& & & \mP_l
\end{bmatrix},
\end{equation*}
where $\mP_i\in \R^{k_i\times d_i}$, $\sum_{i=1}^l k_i=k$, and $\sum_{i=1}^l d_i=d$.
For one direction, suppose that $(\vy, \mY, \vt,\mT)$ satisfies the constraints
in \cref{eq:original}. Then it is straightforward to see that $(\vy,\mY)$ also
satisfies the constraints in \cref{eq:decomposed} together with
\begin{equation*}
\vt^{(i)} = \vt[\beta_i],\;\mT^{(i)}=\mT[\beta_i],\;i=1,2,\ldots,l.
\end{equation*}
The other direction of the proof is more involved. Given $(\vy,\mY)$
and the variables $\{\vt^{(i)},\mT^{(i)}\}_{i=1}^l$ in \cref{eq:decomposed}, our goal is to construct
$(\vt,\mT)$ satisfying the conditions in \cref{eq:original}. To simplify the notations, we define
\begin{equation}\label{eq:augmented matrices}
\tilde{\mY} :=
\begin{bmatrix}
1 & \vy^{\T} \\
\vy & \mY
\end{bmatrix}
,\;
\tilde{\mT}^{(i)} :=
\begin{bmatrix}
1 & (\vt^{(i)})^{\T} \\
\vt^{(i)} & \mT^{(i)}
\end{bmatrix}
,\;
\text{and}\;
\tilde{\mP}_i :=
\begin{bmatrix}
1 & \0 \\
\0 & \mP_i
\end{bmatrix}.
\end{equation}
Let $r=\max\{k,d\}$.
Since $\tilde{\mY}\in \semiS^{k+1}$, it can be factorized as
\begin{equation}\label{eq:Y factorization}
\tilde{\mY} = \tilde{\mV}^{\T}\tilde{\mV},
\end{equation}
where $\tilde{\mV}\in \R^{(r+1)\times (k+1)}$. The above factorization can be done because $r\geq k$.
Further, we partition $\tilde{\mV}$ as
\begin{equation*
\tilde{\mV} =
\begin{bmatrix}
\vv & \mV_1 & \mV_2 & \ldots & \mV_l
\end{bmatrix},
\end{equation*}
where $\vv\in \R^{r+1}$ and $\mV_i\in \R^{(r+1)\times k_i}$ contains the columns of $\tilde{\mV}$ indexed by $\alpha_i$
for $i=1,2,\ldots,l$.
Moreover, we have $\vv^{\T}\vv=\tilde{\mY}_{1,1}=1$.
Similarly, $\tilde{\mT}^{(i)}$ can be factorized as
\begin{equation}\label{eq:T factorization}
\tilde{\mT}^{(i)} = (\tilde{\mZ}^{(i)})^{\T}\tilde{\mZ}^{(i)},\;i=1,2,\ldots,l,
\end{equation}
where $\tilde{\mZ}^{(i)}\in \R^{(d_i+1)\times (d_i+1)}$ and is partitioned as
\begin{equation}\label{eq:paritioned Z}
\tilde{\mZ}^{(i)} =
\begin{bmatrix}
\vz^{(i)} & \mZ^{(i)}
\end{bmatrix}.
\end{equation}
Combining \cref{eq:T factorization} with
the equality constraints in \cref{eq:decomposed}, we get
\begin{equation*}
\begin{bmatrix}
1 & \vy[\alpha_i]^{\T} \\
\vy[\alpha_i] & \mY[\alpha_i]
\end{bmatrix}
=\tilde{\mP}_i \tilde{\mT}^{(i)}\tilde{\mP}^{\T}_i
= \bigl(\tilde{\mZ}^{(i)}\tilde{\mP}^{\T}_i\bigr)^{\T}
\bigl(\tilde{\mZ}^{(i)}\tilde{\mP}^{\T}_i\bigr),\;i=1,2,\ldots,l,
\end{equation*}
where $\tilde{\mZ}^{(i)}\tilde{\mP}^{\T}_i\in \R^{(d_i+1)\times (k_i+1) }$.
On the other hand, the factorization in \cref{eq:Y factorization} implies
\begin{equation*}
\begin{bmatrix}
1 & \vy[\alpha_i]^{\T} \\
\vy[\alpha_i] & \mY[\alpha_i]
\end{bmatrix}
=
\begin{bmatrix}
\vv & \mV_i
\end{bmatrix}^{\T}
\begin{bmatrix}
\vv & \mV_i
\end{bmatrix},
\end{equation*}
where $ [
\vv \quad \mV_i
]\in \R^{(r+1)\times (k_i+1)}$. By
\cref{lem: two factorizations}, we can find $\mU_i\in \R^{(r+1)\times (d_i+1)}$
with $\mU_i^{\T}\mU_i=\mI_{d_i+1}$ such that
\begin{equation}\label{eq: equality of factors}
\begin{bmatrix}
\vv & \mV_i
\end{bmatrix}
=\mU_i \tilde{\mZ}^{(i)}\tilde{\mP}^{\T}_i.
\end{equation}
Substituting \cref{eq:augmented matrices} and \cref{eq:paritioned Z} into \cref{eq: equality of factors},
we get
\begin{equation}\label{eq: equality between V and Z}
\vv=\mU_i\vz^{(i)}\;\text{and}\;
\mV_i=\mU_i{\mZ}^{(i)}\mP_i^\T.
\end{equation}
Finally, we define
\begin{equation*}
\mR = \begin{bmatrix}
\vv & \mU_1{\mZ}^{(1)} & \mU_2{\mZ}^{(2)} & \ldots & \mU_l{\mZ}^{(l)}
\end{bmatrix}\in \R^{(r+1)\times (d+1)},
\end{equation*}
whose columns indexed by $\beta_i$ are given by $\mU_i{\mZ}^{(i)}$,
and construct $(\vt,\mT)$ by
\begin{equation} \label{eq:constructed t and T}
\begin{bmatrix}
1 & \vt^{\T} \\
\vt & \mT
\end{bmatrix}
=
\mR^{\T}\mR.
\end{equation}
Next we verify that $(\vt,\mT)$ in \cref{eq:constructed t and T} indeed satisfies
all the constraints in \cref{eq:original}.
The positive semidefiniteness is evident by our construction.
For the equality constraints, by using \cref{eq: equality between V and Z} we have
\begin{align*}
\mR
\begin{bmatrix}
1 & \0 \\
\0 & \mP^{\T}
\end{bmatrix}
&=
\begin{bmatrix}
\vv & \mU_1{\mZ}^{(1)}\mP_1^{\T} & \mU_1{\mZ}^{(2)}\mP_2^{\T} & \ldots & \mU_l{\mZ}^{(l)}\mP_l^{\T}
\end{bmatrix}\\
&=
\begin{bmatrix}
\vv & \mV_1 & \mV_2 & \ldots & \mV_l
\end{bmatrix} \\
&= \tilde{\mV}.
\end{align*}
Hence, we get
\begin{align*}
\begin{bmatrix}
1 & \0 \\
\0 & \mP
\end{bmatrix}
\begin{bmatrix}
1 & \vt^{\T} \\
\vt & \mT
\end{bmatrix}
\begin{bmatrix}
1 & \0 \\
\0 & \mP^{\T}
\end{bmatrix} & =
\begin{bmatrix}
1 & \0 \\
\0 & \mP
\end{bmatrix}
\mR^{\T}\mR
\begin{bmatrix}
1 & \0 \\
\0 & \mP^{\T}
\end{bmatrix} \\
& = \tilde{\mV}^{\T}\tilde{\mV}\\
& = \begin{bmatrix}
1 & \vy^{\T} \\
\vy & \mY
\end{bmatrix},
\end{align*}
which is equivalent to $\mY=\mP\mT\mP^{\T}$ and $\vy=\mP\vt$.
Lastly, note that
\begin{align}
\begin{bmatrix}
1 & \vt[\beta_i]^{\T} \\
\vt[\beta_i] & \mT[\beta_i]
\end{bmatrix}
&=
\begin{bmatrix}
\vv & \mU_i\mZ^{(i)}
\end{bmatrix}^{\T}
\begin{bmatrix}
\vv & \mU_i\mZ^{(i)}
\end{bmatrix} \label{eq:submatrix_of_T}\\
&= \begin{bmatrix}
\mU_i\vz^{(i)} & \mU_i\mZ^{(i)}
\end{bmatrix}^{\T}
\begin{bmatrix}
\mU_i\vz^{(i)} & \mU_i\mZ^{(i)}
\end{bmatrix} \label{eq:substitute v_1}\\
&= (\tilde{\mZ}^{(i)})^{\T}\mU_i^{\T}\mU_i\tilde{\mZ}^{(i)} \nonumber\\
& = (\tilde{\mZ}^{(i)})^{\T}\tilde{\mZ}^{(i)} \label{eq:orthogonal of U}\\
& = \tilde{\mT}^{(i)} =
\begin{bmatrix}
1 & (\vt^{(i)})^{\T} \\
\vt^{(i)} & \mT^{(i)}
\end{bmatrix},\nonumber
\end{align}
where we used \cref{eq:constructed t and T} in \cref{eq:submatrix_of_T}, $\vv=\mU_i\vz^{(i)}$ (cf. \cref{eq: equality between V and Z}) in \cref{eq:substitute v_1}, and
$\mU_i^{\T}\mU_i=\mI_{d_i+1}$ in \cref{eq:orthogonal of U}. Hence, the remaining
constraints in \cref{eq:original} are also satisfied because of the conditions on
$\tilde{\mT}^{(i)}$ in \cref{eq:decomposed}.
The proof of \cref{thm:main equiv thm} is complete.
\end{proof}
Two remarks are in order. Firstly, the variables in \cref{eq:original} are in
a high-dimensional PSD cone $\semiS^{d+1}$, while those in \cref{eq:decomposed} are in the
Cartesian product of smaller PSD cones
$
\semiS^{k+1}\times \semiS^{d_1+1}\times \semiS^{d_2+1}\times \cdots \times \semiS^{d_l+1}
$.
When $k,\;d_1,\; d_2,\ldots,d_l$ are much smaller than $d$, using \cref{eq:decomposed} instead of
\cref{eq:original} can achieve dimension reduction without any
additional cost. This can bring substantially higher computational efficiency for solving
the corresponding SDP in practice (see \cref{sec:experiments}).
Secondly, \cref{thm:main equiv thm} is very general and thus could be applicable to a potentially wide range
of problems. It is worth highlighting that we require no assumptions on the sets $\mathcal{A}_i$ that constrain the submatrices, as well
as the row and column partitions of the separable matrix $\mP$.
This enables us to accommodate both the equivalence results
\cref{eq:modelIII-ERSDP equiv} and \cref{eq:VA-BC equiv}, as we will
show next.
\subsubsection{Equivalence between \texorpdfstring{\cref{eq:Model III}}{(ESDR2-T)} and
\texorpdfstring{\cref{eq:ERSDP}}{(ESDR-Y)}}
As we noted before, the transformation matrix $\hat{\mS}$ in \cref{eq:modelIII-ERSDP equiv} is separable with the row
and column partitions given in \cref{eq:parititon in ModelIII}
and we have
\begin{equation*}
\hat{\mS}[\alpha_i,\beta_i]=
\begin{bmatrix}
\vs^{\T}_{R} \\
\vs^{\T}_{I}
\end{bmatrix}\in \R^{2\times M},\;i=1,2,\ldots,n.
\end{equation*}
Moreover, we can see that the feasible set of \cref{eq:Model III}
is in the form of \cref{eq:original} with the set $\mathcal{A}_i$ given by
\begin{align*}
\mathcal{A}_i &=
\biggl\{
\begin{bmatrix}
1 & \vt^{\T} \\
\vt & \Diag(\vt)
\end{bmatrix}:\;\vt\in\R^M,\;
\sum_{j=1}^M t_j=1,\;t_j\geq 0,\;j=1,2,\ldots,M
\biggr\} \\
&= \biggl\{
\sum_{j=1}^M t_j\mE_j:\;\sum_{j=1}^M t_j=1,\;t_j\geq 0,\;j=1,2,\ldots,M
\biggr\} \\
&= \mathrm{conv}\{\mE_1,\mE_2,\ldots,\mE_M\},
\end{align*}
where
\begin{equation*}
\mE_j =
\begin{bmatrix}
1 \\
\ve_j
\end{bmatrix}
\begin{bmatrix}
1 \\
\ve_j
\end{bmatrix}^{\T},\;j=1,2,\ldots,M,
\end{equation*}
and $\ve_j\in\R^M$ is the $j$-th unit vector.
Applying \cref{thm:main equiv thm} to
\cref{eq:Model III} gives the following equivalent formulation:
\begin{equation}\label{eq:equivalent of Model III}
\begin{aligned}
& \begin{bmatrix}
1 & \vy^{\T} \\
\vy & \mY
\end{bmatrix}\in \semiS^{2n+1} \\
\mathrm{s.t.}\;\;\; &
\begin{bmatrix}
1 & y_i & y_{n+i} \\
y_i & Y_{i,i} & Y_{i,n+i} \\
y_{n+i} & Y_{n+i,i} & Y_{n+i,n+i}
\end{bmatrix}
=\begin{bmatrix}
1 & \0 \\
\0 & \vs^{\T}_{R} \\
\0 & \vs^{\T}_{I}
\end{bmatrix}
\begin{bmatrix}
1 & (\vt^{(i)})^{\T} \\
\vt^{(i)} & \mT^{(i)}
\end{bmatrix}
\begin{bmatrix}
1 & \0 & \0 \\
\0 & \vs_{R} & \vs_{I}
\end{bmatrix}, \\
& \begin{bmatrix}
1 & (\vt^{(i)})^{\T} \\
\vt^{(i)} & \mT^{(i)}
\end{bmatrix}\in \semiS^{M+1},\\
& \begin{bmatrix}
1 & (\vt^{(i)})^{\T} \\
\vt^{(i)} & \mT^{(i)}
\end{bmatrix}\in \mathrm{conv}\{\mE_1,\mE_2,\ldots,\mE_M\},\;
i=1,2,\ldots,n.
\end{aligned}
\end{equation}
Since each matrix
$\mE_j$ is PSD, their convex hull is a subset of
$\semiS^{M+1}$ and hence the PSD constraints in \cref{eq:equivalent of Model III} are
redundant. Furthermore,
note that
\begin{align*}
\begin{bmatrix}
1 & \0 \\
\0 & \vs^{\T}_{R} \\
\0 & \vs^{\T}_{I}
\end{bmatrix}\mE_j
\begin{bmatrix}
1 & \0 & \0 \\
\0 & \vs_{R} & \vs_{I}
\end{bmatrix}
&=\begin{bmatrix}
1 & \0 \\
\0 & \vs^{\T}_{R} \\
\0 & \vs^{\T}_{I}
\end{bmatrix}
\begin{bmatrix}
1 \\
\ve_j
\end{bmatrix}
\begin{bmatrix}
1 \\
\ve_j
\end{bmatrix}^{\T}
\begin{bmatrix}
1 & \0 & \0 \\
\0 & \vs_{R} & \vs_{I}
\end{bmatrix}\\
&=
\begin{bmatrix}
1 \\ s_{R,j} \\ s_{I,j}
\end{bmatrix}
\begin{bmatrix}
1 & s_{R,j} & s_{I,j}
\end{bmatrix},
\end{align*}
which is exactly the matrix $\mK_j$ defined in \cref{eq:def of K}. Therefore,
we can conclude that \cref{eq:equivalent of Model III} is the same as
\cref{eq:ERSDP}, and hence
\cref{eq:Model III} and \cref{eq:ERSDP} are equivalent.
\subsubsection{Equivalence between \texorpdfstring{\cref{eq:VA-SDR}}{(VA-SDR)} and
\texorpdfstring{\cref{eq:BC-SDR}}{(BC-SDR)}}
Similarly, we observe that the transformation matrix $\mW$ in \cref{eq:VA-BC equiv} is separable with row and column
partitions given in \cref{eq:parititon in VA}, and let
\begin{equation}\label{eq:def of w}
\vw^{\T} := \mW[\alpha_i,\beta_i]=
\begin{bmatrix}
1 & 2 & 4 & \ldots & 2^{q-1}
\end{bmatrix}.
\end{equation}
The feasible set in \cref{eq:VA-SDR} conforms to \cref{eq:original} with the
set $\mathcal{A}_i$ given by
\begin{equation*}
\mathcal{A}_i =
\biggl\{
\begin{bmatrix}
1 & \vb^{\T} \\
\vb & \mB
\end{bmatrix}:\;
\vb\in \R^q,\;\mB\in\R^{q\times q},\;
\diag(\mB) = \1_q
\biggr\}.
\end{equation*}
Hence, by applying \cref{thm:main equiv thm} to \cref{eq:VA-SDR}, we get
the following equivalent formulation:
\begin{equation}\label{eq:equivalent of VA-SDR}
\begin{aligned}
& \begin{bmatrix}
1 & \vx^{\T} \\
\vx & \mX
\end{bmatrix}\in \semiS^{n+1} \\
\mathrm{s.t.}\;\;\; &
X_{i,i} = \vw^{\T} \mB^{(i)}\vw,\; x_i=\vw^{\T}\vb^{(i)},\\
& \begin{bmatrix}
1 & (\vb^{(i)})^{\T} \\
\vb^{(i)} & \mB^{(i)}
\end{bmatrix}\in \semiS^{q+1},\\
&
\diag(\mB^{(i)}) = \1_q,\;
i=1,2,\ldots,n.
\end{aligned}
\end{equation}
Next we argue that all the constraints $x_i=\vw^{\T}\vb^{(i)}$ are redundant, i.e.,
the set in \cref{eq:equivalent of VA-SDR} is equivalent to
\begin{equation}\label{eq:equivalent of VA-SDR 2}
\biggl\{
\begin{bmatrix}
1 & \vx^{\T} \\
\vx & \mX
\end{bmatrix}\in\semiS^{n+1}:\;
X_{i,i} = \vw^{\T}\mB^{(i)}\vw,\;
\mB^{(i)}\in \semiS^q,\;
\diag(\mB^{(i)}) = \1_q,\;
i=1,2,\ldots,n\biggr\}.
\end{equation}
To show this, we need to prove that, for any $\vx,\mX,\mB^{(1)},\ldots,\mB^{(n)}$
satisfying the constraints in \cref{eq:equivalent of VA-SDR 2}, there must exist
$\vb^{(i)}\in \R^q$ such that
\begin{equation}\label{eq:conditions on b}
x_i=\vw^{\T}\vb^{(i)}\;\text{and}\;\begin{bmatrix}
1 & (\vb^{(i)})^{\T} \\
\vb^{(i)} & \mB^{(i)}
\end{bmatrix}\succeq 0,\;i=1,2,\ldots,n.
\end{equation}
Fix $i\in\{1,2,\ldots,n\}$. Note that the PSD constraints in \cref{eq:equivalent of VA-SDR 2} implies
\begin{equation}\label{eq: inequality on x_i and X_ii}
\begin{bmatrix}
1 & x_i \\
x_i & X_{i,i}
\end{bmatrix}\succeq 0 \Leftrightarrow
x_i^2\leq X_{i,i}.
\end{equation}
When $X_{i,i}=0$, we must have $x_i=0$ and we can achieve \cref{eq:conditions on b} by
simply letting $\vb^{(i)}=\0$. Otherwise, we have $X_{i,i}>0$ and hence we can let
\begin{equation}\label{eq:equality of b}
\vb^{(i)} = \frac{x_i}{X_{i,i}}\mB^{(i)}\vw.
\end{equation}
Since $X_{i,i}=\vw^{\T}\mB^{(i)}\vw$, we can see that $\vw^{\T}\vb^{(i)}=(\vw^{\T}\mB^{(i)}\vw)x_i/X_{i,i}=x_i$.
To verify the PSD constraint in \cref{eq:conditions on b}, it suffices to show that
$
\mB^{(i)}\succeq \vb^{(i)}(\vb^{(i)})^{\T}
$.
Note that
\begin{equation*}
\begin{bmatrix}
X_{i,i} & \vw^{\T}\mB^{(i)} \\
\mB^{(i)}\vw & \mB^{(i)}
\end{bmatrix}=
\begin{bmatrix}
\vw^{\T}\mB^{(i)}\vw & \vw^{\T}\mB^{(i)} \\
\mB^{(i)}\vw & \mB^{(i)}
\end{bmatrix}
=\begin{bmatrix}
\vw^{\T} \\
\mI_{q}
\end{bmatrix}\mB^{(i)}
\begin{bmatrix}
\vw & \mI_{q}
\end{bmatrix}\succeq 0,
\end{equation*}
which implies the Schur complement is also PSD, i.e.,
\begin{equation*
\mB^{(i)}-\frac{1}{X_{i,i}}\mB^{(i)}\vw\vw^{\T}\mB^{(i)}\succeq 0.
\end{equation*}
This, together with \cref{eq: inequality on x_i and X_ii,eq:equality of b}, shows
\begin{equation*}
\vb^{(i)}(\vb^{(i)})^{\T} =\frac{x_i^2}{X_{i,i}^2}\mB^{(i)}\vw\vw^{\T}\mB^{(i)}
\preceq \frac{X_{i,i}}{X_{i,i}^2}\mB^{(i)}\vw\vw^{\T}\mB^{(i)}
\preceq \mB^{(i)}.
\end{equation*}
Hence both conditions in \cref{eq:conditions on b} are satisfied.
Finally, to show that \cref{eq:equivalent of VA-SDR 2} is the same as
\cref{eq:BC-SDR}, we need the following lemma.
\begin{lemma}\label{lem:bound on Xii}
Let $\vw\in\R^q$ be the vector defined in \cref{eq:def of w}. It holds that
\begin{equation}\label{eq:set_equiv}
\bigl\{x:\;\exists\;\mB\in\semiS^{q}\;\mathrm{s.t.}\; x=\vw^{\T}\mB\vw,\; \diag(\mB) = \1_q
\bigr\}
=\{1\leq x\leq (2^q-1)^2\}.
\end{equation}
\end{lemma}
\begin{proof}
See \cref{appen:proof of interval}.
\end{proof}
Putting all pieces together, we have proved that \cref{eq:VA-SDR} is equivalent to
\cref{eq:BC-SDR} by showing the correspondence \cref{eq:VA-BC equiv}.
\section{Numerical results}\label{sec:experiments}
In this section, we present some numerical results.
Following standard assumptions in the wireless communication literature (see, e.g.,~\cite[Chapter 7]{tse2005fundamentals}), we
assume that all entries of the channel matrix $\mH$ are
independent and identically distributed (i.i.d.)
following a complex circular Gaussian distribution with zero mean and
unit variance, and all entries of the additive noise $\vv$ are
i.i.d. following a complex circular Gaussian distribution with zero mean and
variance $\sigma^2$. Further, we choose the transmitted symbols $x^*_1,x^*_2,\ldots,x^*_n$
from
the symbol set $\sS_M$ in \cref{eq:MPSK symbol set} independently and uniformly.
We define the SNR as the received SNR per symbol:
\begin{equation*}
\mathrm{SNR}:= \frac{\E[\|\mH\vx^*\|_2^2]}{n\E[\|\vv\|_2^2]} = \frac{mn}{n\cdot m\sigma^2}=\frac{1}{\sigma^2}.
\end{equation*}
We first consider a MIMO system where $(m,n)=(16,10)$ and $M=8$.
To evaluate the empirical probabilities of SDRs not being tight,
we compute the optimal solutions
of \cref{eq:CSDP2}, \cref{eq:ERSDP}, and \cref{eq:Model II} by the general-purpose SDP solver SeDuMi~\cite{sedumi} with the desired accuracy set to $10^{-6}$.
The SDR is decided to be tight if the output $\hat{\vx}$ returned by the SDP solver\footnote{The output $\hat{\vx}$ is directly given by the optimal solution in \cref{eq:CSDP2},
while it is obtained from the relation \cref{eq:def of hat Q and hat c} between $\vx$ and $\vy$ in \cref{eq:ERSDP} and
the relation \cref{eq:express discrete by 01} between $\vx$ and $\vt$ in \cref{eq:Model II}.} satisfies $\|\hat{\vx}-\vx^*\|_\infty\leq 10^{-4} $.
We also evaluate the empirical probabilities of conditions \cref{eq:CSDP2_suff,eq:CSDP2_necc_and_suff_intro,eq:ERSDP_necc_intro}
not being satisfied.
We run the simulations at 8 SNR values in total ranging from 3 dB to 24 dB.
For each SNR value, 10,000 random instances are generated
and the averaged results are plotted in \cref{fig:tightness probability versus SNR}.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.9\linewidth]{figs/8PSK_m16_n10_10000_v2.pdf}
\caption{Error probabilities versus the SNR in a $16\times 10$ MIMO system with 8-PSK. }
\label{fig:tightness probability versus SNR}
\end{figure}
We can see from \cref{fig:tightness probability versus SNR} that our results
\cref{eq:CSDP2_necc_and_suff_intro,eq:ERSDP_necc_intro} provide better
characterizations than the previous tightness condition \cref{eq:CSDP2_suff} in~\cite{liu2019equivalence}.
The empirical probability of \cref{eq:CSDP2} not being tight
matches perfectly with our analysis given by
the necessary and sufficient condition \cref{eq:CSDP2_necc_and_suff_intro}.
The probability of \cref{eq:ERSDP_necc_intro} not being satisfied is also
a good approximation to the probability of \cref{eq:ERSDP} not being tight,
underestimating the latter roughly by a factor of 9.
Moreover, the numerical results also validate our analysis that
\cref{eq:Model II} is not tight with high probability.
In fact,
\cref{eq:Model II} fails to recover the vector of transmitted symbols
in all 80,000 instances.
Next, we compare the optimal values as well as the CPU time of solving
\cref{eq:ERSDP} and \cref{eq:Model III}. \cref{tab:relative diff} shows the relative
difference between the optimal values of \cref{eq:ERSDP} (denoted as $\mathrm{opt}_{\text{ESDR-}\mY}$) and \cref{eq:Model III} (denoted as $\mathrm{opt}_{\text{ESDR2-}\mT}$) averaged over 300 simulations, which is defined as
$|\mathrm{opt}_{\text{ESDR-}\mY}-\mathrm{opt}_{\text{ESDR2-}\mT}|/|\mathrm{opt}_{\text{ESDR2-}\mT}|$.
We can see from \cref{tab:relative diff} that the difference is consistently in the order 1e$-$7 in various settings, which verifies the equivalence between
\cref{eq:ERSDP} and \cref{eq:Model III}.
In \cref{fig:time complexity}, we plot the average CPU time consumed by solving \cref{eq:ERSDP} and \cref{eq:Model III} in an
8-PSK system with increasing problem size $n$. For fair comparison, both SDRs are implemented and solved by SeDuMi and we repeat
the simulations for 300 times. With the same error performance,
we can see that \cref{eq:ERSDP} indeed solves the MIMO detection problem \cref{eq:ML} more efficiently and saves roughly 90\% of the computational time in
our experiment.
\begin{table}[tbhp]\label{tab:relative diff}
{\footnotesize
\caption{Average relative difference between \cref{eq:ERSDP} and \cref{eq:Model III} in optimal objective values.}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
\multirow{2}{*}{SNR} & \multicolumn{4}{c|}{Relative diff. in optimal objective values.} \\ \cline{2-5}
& $(m,n)= (4,4)$ & $(m,n)=(6,4)$ & $(m,n)=(10,10)$ & $(m,n)=(15,10) $ \\ \hline
5dB & 4.62e$-$7 & 5.10e$-$7 & 6.26e$-$7 & 7.73e$-$7 \\ \hline
10dB & 5.67e$-$7 & 4.06e$-$7 & 6.50e$-$7 & 7.16e$-$7 \\ \hline
{15dB} & 5.94e$-$7 & 3.83e$-$7 & 7.80e$-$7 & 5.58e$-$7 \\ \hline
\end{tabular}
\end{center}
}
\end{table}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.9\linewidth]{figs/time_complexity_10_18_300_new.pdf}
\caption{Average CPU time of solving \cref{eq:ERSDP} and \cref{eq:Model III} when $M=8$.}
\label{fig:time complexity}
\end{figure}
|
1,116,691,500,849 | arxiv | \section{Introduction}
\label{sec:intro}
There have been continuous efforts on synthesizing high-fidelity speech by computer. These so-called text-to-speech (TTS) systems have various applications in human-computer interaction, assistive technology, media and entertainment.
In particular, deep learning-based TTS systems have evolved at a rapid pace, starting from having multiple finely engineered neural networks for different components~(e.g., phoneme duration, fundamental frequency predictions)~\citep{ze2013statistical, dv1, dv2}, to text-to-spectrogram models connected with a separate vocoder~\citep{tacotron, dv3, char2wav}.
ClariNet~\citep{Ping2018} is the first text-to-wave neural architecture for speech synthesis, which converts text to raw waveform using a single neural network trained from scratch.
However, the original ClariNet is a single speaker system and not aimed for generating various voices from different speakers.
In this paper, we augment ClariNet by adding the multi-speaker functionality to the system. Applying a similar technique as Deep Voice 3~\citep{dv3}, we add speaker embedding as a bias to each part of the network.
We demonstrate that this modified version of ClariNet can generate unique voices of more than one hundred speakers in higher quality than the state-of-the-art methods.
\section{Related work}
Deep learning-based TTS systems~\citep{ling2015deep} can be categorized according to the structure of the systems.
Earlier systems including Deep Voice 1~\citep{dv1} and Deep Voice 2~\citep{dv2} retain the traditional parametric TTS pipeline, which has separate grapheme-to-phoneme, segmentation, phoneme duration, frequency, and waveform synthesis models. Building and tuning such system require laborious feature engineering. In addition, each component within the system is optimized separately and their errors can accumulate through the pipeline, which may lead to sub-optimal performance, e.g., unnatural prosody.
Deep Voice 3~\citep{dv3}, Tacotron~\citep{tacotron}, Tacotron 2~\citep{tacotron2}, Char2Wav~\citep{char2wav}, and ParaNet~\citep{paranet} are attention-based seq2seq models, which convert text to acoustic features~(e.g., mel spectrogram) and have much more compact architectures.
To synthesize high-fidelity speech, one still needs to feed the predicted spectrogram to a separately trained neural vocoder ~\citep{dv3, tacotron2}, such as WaveNet~\citep{wavenet}.
As demonstrated in \citep{Ping2018}, it may still result in sub-optimal performance as text-to-spectrogram model and WaveNet are optimized separately.
In contrast, ClariNet~\citep{Ping2018} uses a variant of WaveNet conditioned on the hidden representation within the model, which is jointly trained with the whole model in an end-to-end manner.
Building multi-speaker TTS systems is a long-standing task in speech synthesis research.
In traditional parametric TTS system~\citep{yamagishi2009robust}, an average voice model is trained using all speakers' data, which is then adapted to different speakers.
There have been many recent works on neural network-based multi-speaker TTS models in addition to Deep Voice 2 and Deep Voice 3.
Several deep learning-based systems~\citep{fan2015multi,yang2016training} also build average voice model, with i-vectors as inputs and speaker-dependent output layers for different speakers.
Wu et al. \citep{wu2015study} empirically studies and compares DNN-based multi-speaker modeling methods.
VoiceLoop~\citep{voiceloop} focuses on training the model with in-the-wild dataset and shows that a simple buffer structure could produce high quality voices.
All of these systems are built on the pipelines with separately trained components.
Our model is the first multi-speaker text-to-wave system that is trained end-to-end from scratch and can produce high quality synthesized speech for a hundred speakers.
\section{Multi-speaker ClariNet}
\begin{figure*}[tb] \centering
\hspace{-.3cm}
\includegraphics[clip, width=0.7\textwidth]{multiClariNet.png}
\caption{Overall architecture of multi-speaker ClariNet.}
\label{fig:overall_arc}
\end{figure*}
\vspace{0.1em}
\begin{figure*}[tb] \centering
\hspace{-.3cm}
\includegraphics[clip, width=0.78\textwidth]{wavenet.png}
\caption{Overall architecture of the Gaussian autoregressive WaveNet. Shadowed arrow after the dilated convolution layers depicts the skip connections, and GAU stands for gated activation unit.}
\label{fig:wavenet}
\end{figure*}
\begin{figure*}[tb] \centering
\includegraphics[clip, width=0.37\textwidth]{bridgenet.png}
\hspace{10pt}
\includegraphics[clip, width=0.28\textwidth]{convolution_block.png}\\
\hspace{25pt} (a) \hspace{135pt} (b) \\
\includegraphics[clip, width=0.3\textwidth]{transconvolution_block.png}
\hspace{40pt}
\includegraphics[clip, width=0.28\textwidth]{dilatedconvolution_block.png}\\
\hspace{25pt} (c) \hspace{135pt} (d) \vspace{0pt}
\caption{Architectures of (a) Bridge-net, (b) Convolution block, (c) Transposed convolution block, and (d) Dilated convolution layer.}
\label{fig:convblocks}
\end{figure*}
\subsection{Architecture}
We follow a similar architecture as the original ClariNet, which trains the text-to-wave model in an end-to-end manner. Figure~\ref{fig:overall_arc} depicts the overall architecture of the multi-speaker ClariNet. Instead of connecting two separately trained models, one for generating spectral features~(e.g., mel spectrogram) and the other for synthesizing raw waveform, the model feeds the hidden state from the first component to the vocoder through the \emph{bridge-net} and trains the whole model jointly.
In addition, trainable speaker embeddings serve as conditional input throughout the whole network and are stored in an embedding lookup table. The projected embeddings after a softsign function are added as a bias to the convolution blocks within the encoder and decoder networks as in \citep{dv3}. They are also added to the bridge-net and the vocoder in a similar manner. Since ClariNet is an end-to-end network where everything is jointly learned, adding speaker embeddings to each component of the model allows them to learn better representations of the speaker characteristics, which in turn improves the quality of the generated audios.
Single-speaker ClariNet consists of four components including encoder, decoder, bridge-net, and vocoder. Since the same encoder-decoder structure as in Deep Voice 3~\citep{dv3} and ClariNet~\citep{Ping2018} is used for multi-speaker ClariNet, we skip the detailed description of the encoder and decoder due to the space constraint and focus on the architecture of the \textit{speaker-dependent} bridge-net and WaveNet.
\subsubsection{Bridge-net}
Bridge-net is a non-causal convolutional processing block that connects the two main parts of the system. It consists of a series of convolution blocks followed by a series of transposed convolution blocks. The speaker embedding is added as a bias to each block after being projected by a fully connected layer and softsign activated. The overall architecture of bridge-net is shown in Figure \ref{fig:convblocks} (a).
The non-causality of the convolution blocks allows utilizing future information, which gathers more knowledge for prediction. Also, transposed convolution is used to upsample the hidden representation obtained from the convolution blocks from frame-level to sample-level. We can see the diagrams for the convolution block and the transposed convolution block in Figures \ref{fig:convblocks} (b) and (c), respectively.
\subsubsection{Vocoder}
As in single-speaker ClariNet, a Gaussian autoregressive WaveNet is used as the vocoder (shown in Figure \ref{fig:wavenet}). WaveNet consists of multiple dilated convolution layers and the speaker embedding is fed to each layer as a bias. Figure \ref{fig:convblocks} (d) shows the detailed diagram of a single dilated convolution layer. The speaker embedding is also added to the input of each fully-connected layer at the end of the architecture after a series of dilated convolution layers. Note that the transposed convolution blocks in bridge-net upsample the frame-level hidden representations to sample-level and feed these information as the local conditioner input to the vocoder. In this paper, we only discuss the autoregressive WaveNet without knowledge distillation.
\subsubsection{Objective function}
We optimize a linear combination of the losses from decoder, bridge-net and vocoder. In our experiments, we set the linear coefficients to 1 for simplicity.
Instead of predicting the end of the utterance with the \emph{done} loss as in Deep Voice 3, we simply keep track of the most recent value of a monotonic attention layer and stop when it reaches the last character, which is the period (``.'').
\section{Experiments}
In this section, we present several experiments to evaluate the proposed multi-speaker ClariNet.~\footnote{Demo website: \url{https://multi-speaker-clarinet-demo.github.io/}}
\subsection{Data and preprocessing}
We train our multi-speaker ClariNet model on the VCTK dataset \citep{vctk2009}, which consists of audio recordings from 108 speakers, with approximately 400 utterances for each speaker and a total duration of $\sim$44 hours. The audios are downsampled from 48 KHz to 24 KHz. Also, we clip the leading silence of audios but keep the trailing silence. Keeping the trailing silence helps preventing the clicking sound or sharp noise at the end of the synthesized audios. We use log-mel spectrogram with 80 bands as decoder input.
·
\subsection{Multi-speaker ClariNet}
Similar to \citep{dv1}, the WaveNet component of our multi-speaker ClariNet model employs a stack of dilated convolution blocks, where each block has 10 layers and the dilation is doubled at each layer, $\{1,2,4,\ldots,512\}.$ We add the output hidden states of 128 channels from each layer through skip connections. We use a single Gaussian as the output distribution for the WaveNet vocoder and $-7$ as the lower bound for natural logarithm of the predicted Gaussian standard deviation \citep{Ping2018}.
We use 8 GPUs to train three multi-speaker ClariNet models with 20, 30, and 40 dilated convolution layers in its WaveNet component, respectively.
All three models are trained
for 1.5M steps
using Adam optimizer \citep{Kingma2015} with batch size $16$. The learning rate is initially set to 0.001, and it gets annealed by half at every 200K steps after the first 500K steps. Additional hyperparameter settings are listed in Table \ref{tab:params}.
\begin{table}[tb]
\caption{Shared hyperparameters used for the three multi-speaker ClariNet models in our experiment.}
\vspace{0.1em}
\centering
\begin{tabular}{l c}
\hline
Parameter & Value \\\hline
FFT Size & 2048\\
FFT Window Size / Shift & 1200 / 300\\
Reduction Factor & 4\\
Speaker Embedding Dim. & 32\\
Character Embedding Dim. & 256\\
Encoder Layers / Conv. Width / Channels & 7 / 5 / 128\\
Decoder Pre-net Affine Channels & 128, 256\\
Decoder Layers / Conv. Width & 6 / 5\\
Attention Channels & 256\\
Positional Encoding Weight / Initial Rate & 0.1 / 7.6\\
Bridge-net Conv Layers / Width / Channels & 6 / 5 / 256\\
Dropout Keep Probability & 0.95\\
Max. Gradient Norm & 100.0\\
Gradient Clipping Max. Value & 5.0\\\hline
\end{tabular}
\label{tab:params}
\end{table}
\begin{figure*}[tb]
\centering
\includegraphics[width=0.4\textwidth]{pca_gender_clarinet40_vctk_embed.png} \hspace{35pt}
\includegraphics[width=0.4\textwidth]{pca_region_clarinet40_vctk_embed.png} \vspace{0pt}
\caption{Principal components of the learned speaker embeddings of multi-speaker ClariNet with 40-layer WaveNet, shown with genders and regions of the speakers.}
\label{fig:pca}
\end{figure*}
\begin{table*}[t]
\caption{Mean Opinion Score (MOS) ratings with 95\% confidence intervals, speaker classification accuracy, and speaker verification EERs (using 1 and 5 enrollment audios, denoted as EER-1 and EER-5, respectively) of multi-speaker ClariNet, Deep Voice 2, and Deep Voice 3 trained on the VCTK dataset.
We train the speaker classification and verification models using the VCTK and LibriSpeech datasets, respectively.
The EER-1 and EER-5 on 50 held-out speakers from LibriSpeech are 6.30\% and 2.72\%, respectively.
}
\centering
\begin{tabular}{|c c|c|c|c|c|} \hline
\multicolumn{2}{|c|}{\textbf{Multi-speaker Model}} & \textbf{MOS} & \textbf{Accuracy} & \textbf{EER-1} & \textbf{EER-5} \\\hline
\multicolumn{2}{|c|}{VCTK Ground Truth} & ${\bf 4.26 \pm 0.38}$ & 100\% & 12.8\% & 5.4\% \\\hline
\multirow{3}{*}{Multi-speaker ClariNet} & 20-layer WaveNet & $3.75 \pm 0.42$ & 99.3\% & 12.2\% & 3.8\% \\
& 30-layer WaveNet & ${\bf 3.90 \pm 0.36}$ & 99.3\% & 12.3\% & 4.1\% \\
& 40-layer WaveNet & ${\bf 3.89 \pm 0.28}$ & 99.5\% & 12.1\% & 3.7\% \\\hline
\multirow{4}{*}{Deep Voice 2} & 20-layer WaveNet & $3.19 \pm 0.44$ & 98.8\% & 13.3\% & 4.2\% \\
& 40-layer WaveNet & $3.36 \pm 0.37$ & 99.9\% & 12.6\% & 3.7\% \\
& 60-layer WaveNet & $3.50 \pm 0.23$ & 99.5\% & 13.0\% & 3.9\% \\
& 80-layer WaveNet & $3.63 \pm 0.31$ & 99.9\% & 12.8\% & 3.7\% \\\hline
Deep Voice 3 & 30-layer WaveNet & $3.74 \pm 0.33$ & 99.9\% & 12.8\% & 4.0\% \\\hline
\end{tabular}
\label{tab:clarinet_result}
\end{table*}
We compare the proposed multi-speaker ClariNet to Deep~Voice~2~(DV2) and Deep~Voice~3~(DV3).
Four DV2 models are evaluated with 20, 40, 60 and 80 layers WaveNet, respectively.\footnote{Independently trained WaveNet vocoder with softmax output.\label{Wavenet}}
We also evaluate DV3 with 30-layer WaveNet vocoder~\textsuperscript{\ref{Wavenet}}.
\textbf{Naturalness:}
We randomly choose 10 speakers from the VCTK dataset and use a 10-sentence test set to synthesize a total of 100 audios for each model.
To evaluate the quality of these synthesized audio, we run 5-scale mean opinion score (MOS) evaluations using the crowdMOS toolkit~\citep{ribeiro2011crowdmos} on Amazon Mechanical Turk.
The batches of samples from all these models were presented to raters, which naturally induces a comparison between the models.
We purposefully include ground truth samples in the set being evaluated, and the ground truth MOS provides a reference for calibrating the subjective evaluation.
The results are shown in Table \ref{tab:clarinet_result}.
The results indicate that the quality of synthesized speech from multi-speaker ClariNet is higher than that of both DV2 and DV3 with WaveNet vocoder. The multi-speaker ClariNet with 20 layers (the fewest number of layers) outperforms DV2 with 80 layers (the largest number of layers), and the multi-speaker ClariNet with 30 layers outperforms DV3 with the same number of layers.
From the results of DV2 and multi-speaker ClariNet, we also observe that the quality generally improves as we add more layers of the dilated convolution in WaveNet.
\textbf{Speaker identity:}
A multi-speaker TTS system that generates high fidelity and indistinguishable voices would result in high MOS, but fail to meet the desired objective of reproducing the various voices accurately. To show that our models can achieve high audio
quality synthesis while preserving the speaker identities, we synthesize a total of 2484 audios for each model using a 23-sentence test set on all 108 speakers from the VCTK dataset and use well-trained speaker classification and verification models for evaluations as suggested by \citep{dv2, arik2018voiceclone}.
We follow the same setting to train a speaker classifier as in~\citep{dv2} on VCTK dataset to classify which of the 108 speakers a speech sample belongs to.
The classification results in Table \ref{tab:clarinet_result} demonstrate that samples generated from multi-speaker ClariNet models are as distinguishable as the ground truth audios as well as those generated from the DV2 and DV3 counterparts.
Speaker verification is the task of authenticating the claimed identity of a speaker, based on a test audio and enrolled audios from the speaker. In particular, it performs binary classification to identify whether the test audio and enrolled audios are from the same speaker~\citep{Snyder2016}. We train the same speaker verification model as in~\citep{arik2018voiceclone} using the LibriSpeech dataset, which consists of 820 hours of audio data from 2484 speakers. As a quantitative performance metric, the equal error-rate (EER) can be used to measure how close the audios generated from multi-speaker TTS models are to the ground truth audios. EERs are estimated by randomly pairing up utterances from the same or different speakers (50\% for each case) from the test audios generated by multi-speaker TTS models and the ground-truth samples from the VCTK dataset. We perform $40960$ trials to obtain the mean EERs using $1$ and $5$ enrollment audios, respectively. The results are listed in Table~\ref{tab:clarinet_result}, where the VCTK ground truth results represent EERs estimated from random pairing of utterances from the VCTK dataset. The EERs of multi-speaker ClariNet models are comparable to those of the VCTK ground truth, DV2, and DV3, which indicates that audios synthesized by multi-speaker ClariNet models are highly distinguishable. Note that, the EERs of the VCTK ground truth may be higher than those of the multi-speaker models, and this is caused by the discrepancy between the training LibriSpeech dataset and the testing VCTK dataset~\citep{arik2018voiceclone}.
\textbf{Speaker embedding:} In addition to evaluating the quality of the generated audio samples, we explore the latent space of the learned speaker embeddings by visualizing them in a lower dimensional space. The first two principal components of 32 dimensional speaker embeddings of multi-speaker ClariNet are plotted in Figure~\ref{fig:pca}. Each speaker is plotted as circle of a specific color depending on the gender or region of the speaker. Although the embeddings are randomly initialized and learned without any gender or region information of the speakers, the figure suggests that they encode discriminative information for our generative models. Gender of the speakers has such an apparent discriminative pattern that a linear classifier fit on the two dimensional space can classify the gender with high accuracy. We also observe apparent discriminative patterns for region of the speakers. In particular, Great Britain (red circle) and North America (green circle) regions can be perfectly separated by a linear classifier. Such apparent distinction among the gender and region groups also validates that our method can generate not only high quality but also distinguishable voices.
\section{Conclusion}
In this paper, we present multi-speaker ClariNet that can produce high-fidelity voices of various speakers.
The shared speaker embeddings can learn discriminative features of each speaker since they serve as a bias to each component of the model.
Also, by directly conditioning on the hidden state from the bridge-net to the vocoder, the model can generate raw waveform from the text input without a separately trained vocoder.
The results show that this end-to-end model with speaker embedding performs better than state-of-the-art multi-speaker models, even with fewer layers in the vocoder.
Despite the good performance of the model, we still think there exists a room for improvement. For instance, applying the parallel wave generation method described in \citep{Ping2018} to the current autoregressive vocoder could reduce huge amount of running time at inference.
In addition, multi-speaker ClariNet can be used for voice cloning. One simple approach described in \citep{arik2018voiceclone} is training a speaker encoder model to predict the speaker embedding of an unseen speaker, which can then be plugged into the multi-speaker model. However, training a speaker encoder requires a larger dataset with more speakers than the VCTK dataset, since it is difficult to generalize speaker characteristics with only a hundred speakers. In the future, we would like to train our multi-speaker ClariNet model on the LibriTTS dataset~\citep{libritts}, which consists of 585 hours of audio recordings from 2456 speakers.
\renewcommand{\bibsection}{\section*{References}} {
\bibliographystyle{abbrvnat}
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1,116,691,500,850 | arxiv |
\subsection{Our Contributions}
We show that relaxing the optimality requirement on the bound allows to recover doubly-exponential running times: an approximately optimal bound can be determined using the alternating color technique, which was introduced by Kupferman et al.\ to solve the decision problems for $\prompt$. To this end, we present an approximation algorithm with doubly-exponential running time with an approximation ratio of two. The algorithm has to solve at most doubly-exponentially many $\ltl$ realizability problems, each solvable in doubly-exponential time. We present the algorithm for $\prompt$, but it is applicable to stronger parameterized extensions of $\ltl$ like parametric $\ltl$~\cite{AlurEtessamiLaTorrePeled01} and parametric $\text{LDL}$~\cite{FaymonvilleZimmermann14}.
In many situations, approximating the optimal bound is sufficient, since the exact optimum depends on the granularity of the realizability problem at hand. This is even more true if the optimization problem indeed turns out to be harder than the decision variant, e.g., if it is ${\textsc{3Exptime}}$-hard. Then, the loss in quality is made up for by significant savings in running time. On the other hand, if the optimal bound is at most exponential in the size of the formula, then it can be exactly determined in doubly-exponential time~\cite{Zimmermann13}: the bound can be hardwired into a non-deterministic automaton capturing the specification, which has to be determinized to solve the realizability problem. This involves an exponential blow-up, which implies that this approach only yields a doubly-exponential time algorithm, if the bound is at most exponential.
Furthermore, we report on a proof-of-concept implementation of our algorithm. To handle the solution of the $\ltl$ realizability problems, we rely on the framework of bounded synthesis~\cite{FinkbeinerSchewe13}, which searches for a minimal-size finite-state winning strategy for a given specification. The evaluation of this implementation shows that, while it suffers from a significant increase in running time compared to $\ltl$ realizability, synthesis of prompt arbiters for a small number of clients is feasible.
In our experiments, a tradeoff between size and quality (measured in the bound on the prompt eventually operators) of winning strategies becomes apparent: one can trade size of the strategy for quality and vice versa. We conclude by studying this tradeoff in depth. First, we show that fixing the size of the strategy to $n$ (as it is done during bounded synthesis) implies an exponential upper bound (in $n$) on the sufficient bound~$k$ on the prompt eventually operators. This upper bound reduces the search space of our algorithm. The upper bound is then matched by a tight lower bound. Secondly, we present a family of formulas exhibiting a continuous tradeoff between size and quality with exponential extremal values, i.e., the specifications are realizable with exponential size and a linear bound or with constant size and an exponential bound and the tradeoff between these two points is continuous. Thirdly, by giving up the continuity, one can show even stronger tradeoffs: there is a family of specifications that is realizable with doubly-exponential size and bound zero or with size one and an exponential bound.
\section{Introduction}
\label{section_intro}
\input{intro}
\section{Definitions}
\label{section_defs}
\input{defs}
\subsection{Prompt-LTL}
\input{prompt}
\subsection{Prompt-LTL Realizability}
\label{subsection_realizability}
\input{realizability}
\subsection{The Alternating Color Technique}
\input{altcolor}
\section{Approximating Optimal Bounds in Doubly-Exponential Time}
\label{section_approx}
\input{approx}
\section{Empirical Evaluation}
\label{section_experiments}
\input{experiments}
\section{Trading Memory for Quality and Vice Versa}
\label{section_tradeoffs}
\input{tradeoffs}
\section{Conclusion}
\label{section_conclusion}
\input{conc}
\bibliographystyle{eptcs}
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1,116,691,500,851 | arxiv | \section{Introduction}
Due to its impact on revenues, costs, and more importantly, on service levels, logistics management has become increasingly critical in the apparel industry (Kiesmuller and Minner 2009\nocite{kiesmuller2009inventory}). As consumers demand greater product variety and higher levels of responsiveness at lower prices, effective management of logistics activities arises as a key competitive advantage for the retailers in this industry. The main challenges faced by these retailers are short selling seasons and unpredictable demands. Since forecasts are mostly inaccurate, firms usually have either excess inventories that are sold at markdown prices or stock-outs that lead to lost sales. The problem is exacerbated with short selling seasons which prevent firms to replenish their stocks. Therefore, an effective logistics strategy is key to avoid both of these undesirable outcomes.\\
Logistics decisions of apparel retailers include initial ordering before the season begins, allocation to the branches at the beginning of the season, and eventually phasing-out of the products at the end of the selling season. Increasingly, however, retailers are also practicing what is called ``transshipment" or ``transfer" policies, which involve the reallocation of products among retail branches in mid-season (Li et al. 2013). These policies help retailers to reduce stock-outs as well as excess inventories. This is the issue that is addressed in this paper. \\
The problem that we consider here is inspired by the logistics operations at the largest apparel retailer in Turkey, LC Waikiki, which has positioned itself as a ``fast fashion" retailer. The term fast fashion used to refer to inexpensive designs that appeared on catwalks and were quickly moved to store shelves. Fast fashion items based on the most recent trends have shaped mass-merchandized clothing collections. Therefore, mass-merchendize retailers compete to introduce latest fashion trends in their collections. Although the term was first used in the US in the 1980s, the expression did not receive worldwide adoption until popularized by the Spanish-based apparel giant Zara. The crucial issue in fast fashion is providing inexpensive collections that also respond to fast changing consumer tastes and trends. Therefore, the entire fast fashion supply chain must be sufficiently agile to operate with products for which life cycles are measured not in months but rather in weeks.\\
On the plus side, the speed at which fast fashion moves tends to help retailers avoid markdowns. Typically, these retailers do not place very large orders months before the actual selling season, but rather work with smaller initial orders and renew collections more frequently. On the negative side, however, the fast-paced environment calls for higher turnover and more frequent introduction of new designs, a setting that necessitates shorter design and production lead times. As a result, companies need to rely on more expensive local sources and accommodate large design teams. This fast-paced environment also creates new logistics challenges for retailers: When will these products be replaced? Should they be completely removed from the stores or kept at display at select stores? What will happen to the leftover items; reintroduced elsewhere, sold at discount, or simply written-off? Fast fashion companies need to deal with these issues much more frequently than traditional retailers. \\
Facing such challenges, leading fast fashion companies such as Zara and another Spanish company, Mango, the Japanese World Co., and Swedish H\&M have built supply chains aiming at quickly responding to consumers' changing demands while decreasing the excess inventories at branches and hence, lowering costs (Caro and Gallien 2007)\nocite{caro2007dynamic}. For instance, Zara developed a decision support system featuring demand updating and a dynamic optimization module for initial shipment decisions to avoid stock-outs as well as excess inventories (Gallien et al. 2015\nocite{gallien2015initial}). In addition to correct initial shipment decisions, the transfer or transshipment decisions among retail locations are also instrumental to reduce stock-outs and excess inventories.\\
The main benefit of transfer actions is better matching inventory and demand at different locations. It uses up-to-date sales information and inventory data and redistribute available inventory among the retail locations. Due to socio-economic and geographical differences among retailer locations, it is possible that a product sells very well in some stores while less so in others. Transfer actions can be adopted as a tool to increase the inventory levels at receiver stores while providing extra shelf space at sender stores. This action can be adopted by bypassing the central depot to facilitate the quick movement of merchandise. As a result, the revenues are increased while costs are reduced as compared to a system where no transshipment is utilized (Tagaras 1989). There are a number works in the literature that describe how retailers take advantage of transfers to improve their performances. For example, Archibald et al. (2009)\nocite{archibald2009index} and Archibald et al. (2010)\nocite{archibald2010use} address transshipment issues at a tire retailer that has a network of 50 stores. In another work, Hu and Yu (2014) \nocite{hu2014optimization} present a proactive transshipment problem for a famous fashion brand in China that has network for 43 retailers in Shanghai. The problem that we introduce here is motivated by the largest apparel retailer in Turkey. \\
In the next section, we provide a detailed background of our problem that includes the transfer practices at the company that motivated this work and a detailed description of the problem setting. In Section \ref{sec_LR} we give a brief literature review. Section \ref{PD} presents the progressive development of the mathematical model. Section \ref{SA} presents our solution methods that include a Lagrangian relaxation based upper bounding method and simulated annealing based metaheuristic to find good feasible solutions. Section \ref{CR} reports on our numerical experiments followed by a few concluding remarks in Section \ref{Concl}. \\
\section{Background}\label{BG}
Textile is one of the key sectors in the Turkish economy in terms of GDP, domestic employment, and exports. Textile accounts for 10\% of the Turkish GDP and 20\% of employment in the manufacturing sector\footnote{blog.tcp.gov.tr}. In 2016 Turkey exported around 15 Billion USD, mainly to the European Union countries and was ranked as the 6th biggest textile exporting country (see Figure \ref{growth_store})\footnote{www.wikipedia.com}. LC Waikiki, which has provided motivation to this work, is the largest textile retailer in Turkey with significant international presence. \\
LC Waikiki was founded in 1988 in France by a French designer and his friend. The LC Waikiki brand name is created by adding the word Waikiki, a famous beach in Hawaii, to LC, the abbreviation of the French word ``Les Copains" meaning ``friends". TEMA, a Turkey based group which was then a major supplier of the company, bought the LC Waikiki brand in 1997 undertook a major restructuring that included focusing on domestic market. In the same year, the group entered the Turkish fashion retail market with 21 stores. In 2009, it opened its first international store in Romania since the TEMA group had purchased the brand. Over the years, the group has followed an aggressive expansion strategy both domestically and internationally. Today, LC Waikiki has more than 370 stores in 34 countries in Asia, Africa, and Europe, in addition to over 480 stores in Turkey. In 2011, LC Waikiki became the leader of the ``Ready-to-Wear" market in Turkey and remains as the largest apparel retailer in terms of sales as well as the number of stores. Figure \ref{growth_store} depicts LC Waikiki's phenomenal growth in terms of the total number of stores over the years. \\
\begin{figure}[!b].
\hspace*{-3em}
\begin{minipage}{0.5\linewidth}
\includegraphics[width = 1.05\textwidth]{figure1a.jpg}
\end{minipage}
\hfill
\hspace*{-3em}
\begin{minipage}{0.505\linewidth}
\includegraphics[width = 1.05\textwidth]{figure1b.jpg}
\end{minipage}
\caption{Left: Global export market share, Right: Growth in the total number of stores}
\label{growth_store}
\end{figure}
LC Waikiki has a highly centralized order planning and logistics system in which all initial orders and subsequent distribution decisions are made by the headquarters. New merchandise is received at a single central warehouse located in Istanbul, which then distributes essentially the entire amount to the retail branches (there are varying practices for international stores which are excluded from the consideration in this study). The retail practice at LC Waikiki can be considered as fast fashion in that it aims to keep items in stores only for about six to eight weeks. During this period, if the sales realize below expectations, they may reduce prices or if the sales display disparities across the stores, they may utilize transfers among the stores. Finally, at the end of their shelf-life, products are returned to the central warehouse and later sent to outlet stores (about 40 of the 480 stores are designated as outlet stores) or simply given away to charities. Stock-outs and excess inventories are critical issues at LC Waikiki as in any fast fashion company due to forecast errors. Since LC Waikiki initially distributes all of the items to stores, transfer remains essentially as the only tool to deal with these issues by rebalancing inventories across the retail network. It is these transfer decisions that is the subject of this paper.\\
Currently, a group at the headquarters manages transfer decisions. This group utilizes a mathematical model accompanied with some pre- and post-processing activities. However, we cannot disclose the precise nature of the model and the activities due to proprietary nature of these information. After transfer solutions are obtained, orders are automatically generated and transmitted to the stores. Store employees collect the products that have been chosen for transfer from the shelves and move them to a storage room. In the storage room, products are put in the boxes, each destined to a specific store without any re-assortments. Since the storage room capacities are limited, stores cannot to transfer more than what they can hold at their storage room. Once boxing is finished, the logistics company picks up the boxes and delivers them to their destinations. The boxes are ideally delivered before the weekend so that the transferred items can be put on shelves for the weekend sales.\\
Although our work is motivated by LC Waikiki's logistics operations, we believe many of the features of our model would resonate with issues fast fashion retailers need to consider. In our model, we maximize a measure of the total profit which is the total revenue less the total logistics cost that includes transportation, handling, and inventory holding costs. We also include a number of operational constraints that represent the real practice of the company. For example, we consider a centrally managed system where stores have no control over the decisions, i.e., they may not refuse the transfer decisions. This is valid particularly for firms that own their stores and manage them centrally. We also restrict the total number of items and the total number of stores to which each store can make shipments. Both of these constraints are justified by the limited number of employees in the stores and sizes of the storage rooms. Furthermore, in our model once a product is decided to be transferred from one store to the other, the entire stock (all the available sizes) is sent to the same store. LC Waikiki justifies this practice by the simplicity of the picking operations, which otherwise would be too labor intensive. Here, we will also investigate the effects of these restrictions on system performance. \\
There are a number of issues relevant to the fashion logistics decisions that we leave out of the scope of this work: i) Initial allocation decisions, ii) Uncertainty in demand, and iii) Dynamic nature of the decision making process. At LC Waikiki the initial shipment decisions are made after a pilot sales experiment in which they obtain sales information from about 30 stores. They then make initial allocations in which they essentially distribute the entire stock to the stores. Certainly, the option of transfers might impact the initial allocation. However, we believe that the impact is small due to two aspects in this logistics system. First, the company has a policy to allocate almost all of the available inventory to the stores keeping none at the central depot. Therefore, the firm cannot use central depot for reallocation of products. Second, since the company has flat transportation cost rate independent of origin-destination pair, regional risk pooling effect becomes irrelevant. Therefore, the impact of subsequent transfer practice on initial allocation decisions has lessened. Demand uncertainty is always a concern particularly in the fashion industry, and in fact, it is the demand uncertainty that makes the transfer problem relevant. However, at the time a product is considered for transfer, there is demand information for at least for a couple of weekends. Therefore, the company is able to make much more accurate demand forecasts after this initial sales information, as compared to the time the initial allocation decisions are made. Finally, the transfer problem ideally should consider the fact that transfer decisions are made every week and hence, there are subsequent recourse opportunities. However, considering a multi-stage decision environment under demand uncertainty is simply beyond analysis for the sizes that we envision, particularly with complicating operational constraints. Instead, we envision a setting where the firm makes demand forecast until a product is planned to stay on shelves and the transfer problem is solved on a rolling-horizon basis. This setting, we believe, is a reasonable compromise given the other complexities of the system. Similarly, initial replenishment decisions are also important and they would be impacted by subsequent transfer options. However, considering transshipment and replenishment decisions jointly would also be extremely difficult considering the scale of our problem and particular operational constraints. As we will see, in the literature too, there are very few papers that consider these decisions jointly (see Paterson et al. 2011). \\
We have also assumed that each product's shelf life is known. This assumption is justified by the practice of the company where they keep merchandize for about six to eight weeks. Decision to continue displaying products on the shelves or removing them involves a number of other factors to consider. It requires information on new product designs as well as space considerations at the stores for different merchandize groups. These issues are also rather involved and therefore, kept out of the current study, but certainly worthwhile to consider in the future.
\section{Related literature}\label{sec_LR}
There is a vast literature as far back as the 1950's on lateral transshipment or, as we call here, transfer issues. Although both terms commonly describe the decisions considered here and we use them interchangeably, the term transshipment has a wider meaning and usage. Time and again, various studies have shown that transfer option between retailers improves supply chain performance in terms of costs, revenues, and service levels. For example, Tagaras (1989\nocite{tagaras1989effects}) shows that utilizing transfer in a system with two retail locations leads to significant cost reductions. Although transfers considerably increase transportation cost, systems with these options are superior to systems without them (Banerjee et al. 2003). \nocite{banerjee2003simulation} Furthermore, transfers enhance customer service levels without the burden of carrying extra safety stock at retail locations (Burton and Banerjee 2005)\nocite{burton2005cost}. \\
There are essentially two types of transfers: emergency or reactive transfers and preventive or proactive transfers, which are differentiated mainly with respect to their timing (Lee et al. 2007\nocite{lee2007effective}, Paterson et al. 2011\nocite{paterson2011inventory}, Seidscher and Minner 2013\nocite{seidscher2013semi}, and Ahmadi et al. 2016\nocite{ahmadi2016bi}). Reactive transfer refers to responding to realized stock-outs at a retail location by using available inventory at another location whereas proactive transfer refers to redistribution of inventories among locations before the actual demand is realized. The literature can be classified primarily along this dimension, although there are also works that consider them jointly. \\
Perhaps the earliest work that considers reactive transfers is by Krishnan and Rao (1965) \nocite{krishnan1965inventory} who study a centralized one-echelon inventory system with the objective of minimizing the total cost through transfers. One of the main motivations for reactive transfer models comes from spare parts distribution systems for repairable items, as exemplified by one of the more notable earlier works by Lee (1987) \nocite{lee1987multi} who studies a single-echelon model, which is then extended by Axs{\"a}ter (1990\nocite{axsater1990modelling}) to a two-echelon system. More recent works on spare parts systems can be attributed to van Wijk et al. (2019\nocite{van2019optimal}) who consider a two-location system with lateral transshipment as well as an outside emergency option and Boucherie et al. (2018\nocite{boucherie2018two}) who consider a complex two-echelon inventory system with multiple local warehouses.\\
Models with reactive transfer policies have also been studied for non-repairable items. A notable contribution is due to Robinson (1990), \nocite{robinson1990optimal} who provides structural results for a two-retailer system and develops a heuristic for the initial ordering decisions considering the subsequent transshipments. Herer et al. (2006\nocite{herer2006multilocation}) extend this work by considering more general cost structures and {\"O}zdemir et al. (2013) \nocite{ozdemir2013multi} extend it considering capacity constraints on the transportation network. More recently, transshipment policies in systems with perishable items have also attracted research (see for example, Nakandala et al. 2017 \nocite{nakandala2017lateral} and Dehghani and Abbasi 2018\nocite{dehghani2018age} for such recent works).\\
Proactive transfer is based on the concept of inventory rebalancing and is mostly utilized in periodic review inventory control framework. Allen (1958\nocite{allen1958redistribution}) provides perhaps the earliest model that considers proactive transfers in a single-period setting, which is then generalized by Das (1975) \nocite{das1975supply} who also considers the initial replenishment decision. There are also models that include the timing of the transshipment as decision in a dynamic setting (Agrawal et al. 2004\nocite{agrawal2004dynamic} and Tiacci and Saetta 2011\nocite{tiacci2011heuristic}) as well as in a static setting (Kiesmuller and Minner 2009\nocite{kiesmuller2009inventory}).\\
Although the type of transfers may be dictated by operational conditions of the setting, proactive transfer policies are found to be superior to purely reactive policies both in terms of costs and stock-out levels (see for example, Banerjee \nocite{banerjee2003simulation} et al. 2003 and Burton and Banerjee 2005\nocite{burton2005cost}). In some settings, however, companies may also have opportunities to implement these policies jointly (see for example, Lee et al. 2007\nocite{lee2007effective} for such a model). Finally, although all of the works mentioned above and majority of research on the transshipment issues, assume that the systems are centrally operated decentralized systems where retailers might refuse transshipment requests have also attracted research recently (see for example, \c{C}\"{o}mez et al. 2012 \nocite{ccomez2012season} and Li et al. 2013\nocite{li2013coordinating}).\\
As we have noted earlier, the literature on transshipment issues is vast with considerable growth in the last two decades. Since reviewing this voluminous literature is not possible here, we have only offered a very selective review. Aside from the types of the transfer (i.e., reactive vs. proactive), the literature on transshipment is also divided along two other important dimensions: Whether the models consider only transshipment decisions or jointly with replenishment decisions and whether they consider multiple locations or just two locations. We have classified aforementioned works and few others with respect to these characteristics as shown in Table \ref{literature}. We choose to put some classic and some more recent ones, but it is still far from portraying a complete picture. We refer the reader to a somewhat older, but an excellent review by Paterson et al. (2011) \nocite{paterson2011inventory} who also provide a more thorough classification and a comprehensive review up to its publication date. \nocite{karmarkar1977one} \nocite{diks1996controlling} \nocite{herer1999lateral} \nocite{tagaras2002effectiveness} \nocite{nonaas2007optimal} \nocite{bendoly2004integrated}\\
\begin{table}[!b]
\caption{Main characteristics of reviewed transfer-related papers}
\setlength{\tabcolsep}{.05em}
\scriptsize{
\begin{tabular}{ l l|m{3cm}|m{3.5cm}|m{2.8cm}|m{4.1cm}| }
\cline{3-6}
& & \multicolumn{2}{c }{\textbf{Single Period}} & \multicolumn{2}{ |c| }{\textbf{Multiple Period}}\\
\cline{3-6}
& \textbf{Replenishment}&\multicolumn{1}{c|}{\textbf{2 Retailers}} & \multicolumn{1}{c|}{\textbf{Multiple Retailers}} & \multicolumn{1}{c|}{\textbf{2 Retailers}} & \multicolumn{1}{c|}{\textbf{Multiple Retailers}} \\ \hline
\multicolumn{1}{|c}{\multirow{2}{*}{\textbf{Proactive}}}& \multicolumn{1}{|c|}{\textbf{Yes}} & Das 1975\newline Tagaras and Vlachos 2002 & Karmarkar and Patel 1977 \newline Hoadley and Heyman 1977\nocite{hoadley1977two} & Tiacci and Saetta 2011 \newline Abouee-Mehrizi et al. 2015\nocite{abouee2015optimal} & Diks and Kok 1996 \newline Ahmadi et al. 2016 \newline Feng et al. 2017\nocite{feng2017preventive} \\ \cline{2-6}
\multicolumn{1}{|c}{} & \multicolumn{1}{|c|}{\textbf{No}} & Kiesmuller and Minner 2009 \newline Li et al. 2013 & Allen 1958 \newline Agrawal et al. 2004 & Dan et al. 2016\nocite{dan2016ordering} & Bertrand and Bookbinder 1998\nocite{bertrand1998stock} \newline Banerjee et al 2003 \newline Burton and Banerjee 2005 \newline Acimovic and Graves 2014\nocite{acimovic2014making}\newline Peres et al. 2017 \\ \cline{1-6}
\multicolumn{1}{|c}{\multirow{2}{*}{\textbf{Reactive}}}& \multicolumn{1}{|c|}{\textbf{Yes}} & Herer and Rashit 1999 \newline Minner and Silver 2005\nocite{minner2005evaluation} \newline Liao et al. 2014\nocite{liao2014optimal}\newline Olsson 2015\nocite{olsson2015emergency} \newline Dehghani and Abbasi 2018 & Lee 1987 \newline Axs{\"a}ter 1990 \newline Herer et al. 2006 \newline Johansson and Olsson 2018\nocite{johansson2018age} \newline Boucherie et al. 2018 & Archibald et al. 1997\nocite{archibald1997optimal} \newline Herer and Tzur 2001\nocite{herer2001dynamic} \newline van Wijk et al. 2019\nocite{van2019optimal} & Archibald et al. 2009 \newline van Wijk et al. 2012\nocite{van2012approximate} \newline {\"O}zdemir et al. 2013\\ \cline{2-6}
\multicolumn{1}{|c}{} & \multicolumn{1}{|c|}{\textbf{No}} & Herer and Rashit 1995\nocite{herer1995lateral} \newline Shao et al. 2011\nocite{shao2011incentives} \newline Liao et al. 2014\nocite{liao2014optimal} & Non{\aa}s and J{\"o}rnsten 2007 \newline Hu and Yu 2014 \newline Patriarca et al. 2016\nocite{patriarca2016inventory} \newline Bhatnagar and Lin 2019\nocite{bhatnagar2019joint}&Tagaras 1989 \newline Comez et al. 2012 \newline Shao 2018\nocite{shao2018production} & Robinson 1990 \newline Banerjee et al. 2003 \newline Burton and Banerjee 2005 \newline Dijkstra et al. 2017\nocite{dijkstra2017transshipments}\\ \cline{1-6}
\end{tabular}}
\label{literature}
\end{table}
Despite many simplification attempts, solving transfer problem to optimality remains a challenge. Even in the presence of many simplifying assumptions such as single product, single-period, limited number of retail locations, static transfer timing and so on, past works can only provide approximate solutions. The problem that we present here considers proactive transfers, but since it is motivated by the actual logistics operations at a large fashion retailer, it has many complexities that would be quite challenging to resolve under demand uncertainty or in a dynamic fashion. Therefore, we need to make restrictive assumptions along these dimensions. Certainly we are not alone in this respect; there are numerous other works that consider deterministic demand for transshipment models. Not surprisingly, these works also contain other complicating factors. For example, Herer and Tzur (2001\nocite{herer2001dynamic} and 2003\nocite{herer2003optimal}) in their multi-period model consider fixed ordering costs in transshipments; Lim et al. (2005\nocite{lim2005transshipment}) and Ma et al. (2011)\nocite{ma2011crossdocking} study transshipment decisions via cross-docking locations under time windows; Qi (2006\nocite{qi2006logistics}) considers transshipment and production scheduling decisions jointly; Lee (2015\nocite{lee2015modeling}) considers concave production and transportation costs; Coelho et al. \nocite{coelho2012inventory} (2012), Mirzapour Al-e-hashem and Rekik (2014)\nocite{mirzapour2014multi}, and Peres et al. (2017\nocite{peres2017optimization}) consider routing issues alongside transshipments; Rahmouni et al. (2015\nocite{rahmouni2015multi}) and Feng et al. (2017) develop EOQ-based delivery scheduling models with transshipment while considering multiple products and resource constraints. Our setting too has a few operational practices that force us to model a static and deterministic problem.
\section{Problem description and model formulation}\label{PD}
We consider a retail logistics system that consists of a number of retail stores, each carrying a set of products of different sizes (SKUs). The firm has the precise stock information; that is, how many of each SKU the stores have and the projected demands of each SKU at each location during the remainder of the sales period. The problem is how to reallocate (some of) the products to maximize a profit measure that is total revenue less transfer, handling, and inventory holding costs. \\
The firm has a single price policy in that the same price is applied to a product at all locations, which is indeed the practice of many retail chains and particularly of LC Waikiki. Each product also has a fixed transfer cost regardless of the origin-destination pair. This assumption is also motivated by the practice at LC Waikiki which has outsourced the transportation operations to a logistics company. The transfers are made by standard sized boxes for which LC Waikiki pays a fixed amount regardless of its contents and the locations of the sender and receiver stores. Since the number of products that fit in a box depends on the volume of the product, transportation cost differs for each product, but not on origin-destination pairs. The transfer cost can be estimated by adding the handling cost to the transportation cost for each product. However, none of these assumptions are really essential either for modeling or for our solution method and they can easily be relaxed.\\
We also assume that transfer time has no effect on the sales. The main purpose is to finalize the delivery of transfer items before the weekend where the bulk of the sales materialize. Hence, delivering a day earlier or later presumably does not make much difference, as long as the products arrive for the weekend. Furthermore, the geography of Turkey does not allow wide variations in transfer times, but we also recognize that considering transfer time effects would be valuable in some settings. Finally, we assume that there are no replenishment opportunities from the central warehouse at the time of the transfer decisions. Since the company has a policy to allocate the entire inventory of a product to the stores at the beginning rather than keeping some at the warehouse, this assumption is well justified. As another operational practice, they do not consider a second replenishment option. This is a common practice among the fast-fashion retailers whose business practices involve speedy turnover of designs as exemplified by Zara's practice (see for example, Gallien et al. 2015). \\
In addition to these requirements, we assume a single-period setting and deterministic demand. At LC Waikiki, most of the sales occur at weekends and therefore, the inventory levels of each product are updated at the beginning of each week. Likewise, demand forecasts are also revised after observing weekend sales. As a result, LC Waikiki, solves the transshipment problem once a week which allows us to consider single-period assumption to decrease the complexity of the problem. Deterministic demand is assumed since the forecast from the company is fairly accurate. After two or three weekend sales, the company can have a fairly good idea about the demand in the rest of the products' shelf lives. It is stated that their forecast error is below 15\%. Many papers related to fast-fashion also state that forecast errors are considerably smaller towards the end of shelf lives of products (see for example, Caro and Gallien, 2010\nocite{caro2010inventory}) . Finally, as mentioned earlier, the company has a few operational practices that we include in our model: If a product is transferred from a store, its entire available inventory (all SKUs) is shipped to a single store. Also, there are limits on the total number of SKUs that can be transferred from a store and the number of different destinations to which a store can make transfers. All these assumptions could be relaxed or generalized, but we choose to stay with the company practices as much as possible.\\
As we will see shortly, without the aforementioned operational constraints, the problem can simply be formulated as a profit-maximizing transportation problem, which can easily be solved as a linear program. We are also ensured integer solutions if the demand and inventory values are integers. When we add the restriction on the total number of SKUs that can be transferred from a store, the problem can still be solved as a linear program. When we further add the restriction on the number of stores that a store can ship to, however, we need to introduce binary variables to keep track of whether a shipment is made from one store to another. Finally, when we include the single-destination constraint (i.e., when a product is shipped from one store to another, all the SKUs of the product must be shipped) the problem becomes much more difficult because we now also need to define a much larger set of binary decision variables to keep track of shipments between stores.\\
We now give the preliminary definitions, followed by the formulation of the model. We start with the base model without considering the operational requirements of the company and progressively extend the model by adding each of these constraints. We call two stores as ``connected" if at least one product is transferred from one store to the other.
\subsection*{Sets and indices:}
\begin{itemize}
\item[] $i,j \in I:$ Set of stores,
\item[] $p \in P:$ Set of products,
\item[] $k \in K_{p}:$ Set of sizes for each product $p \in P$.
\end{itemize}
\subsection*{Parameters:}
\begin{itemize}
\item[] $s_{ipk}:$ Stock level of size $k$ of product $p$ at store $i$,
\item[] $d_{ipk}:$ Demand of size $k$ of product $p$ at store $i$,
\item[] $r_p$: Unit net revenue of product $p$,
\item[] $c_p$: Unit transfer cost of product $p$,
\item[] $h_{p}:$ Holding cost of product $p$.
\end{itemize}
\subsection*{Decision variables:}
\begin{itemize}
\item[] $x_{ijpk} : \mbox{ Amount of size } k \mbox{ of product } p \mbox{ transfered from store } i \mbox{ to store } j,$
\item[] $z_{ipk} : \mbox{ Sales of size } k \mbox{ of product } p \mbox{ at store } i,$
\item[] $w_{ipk} : \mbox{ Amount of size } k \mbox{ of product } p \mbox{ store } i \mbox{ has after the transfers}.$
\end{itemize}
\subsection*{Relaxed model:}
\begin{subequations}
\begin{eqnarray}
\label{relaxed_objective}\max \Pi & = & \sum_{j \in I}\sum_{k \in k_{p}} \sum_{p \in P}r_{p} z_{jpk} - \sum_{i \in I} \sum_{\substack{j \in I \\ j\neq i}} \sum_{p \in P}\sum_{k \in K_p} c_p x_{ijpk} - \sum_{i \in I } \sum_{p \in P } \sum_{k \in K_p }h_{p}(w_{ipk}-z_{ipk}) \\
\label{relaxed_recieved} \mbox{s.t.} & & w_{ipk} = \sum_{j \in I}x_{jipk}, \mbox{ for all } i \in I , p \in P \mbox{ and } k \in K_{p},\\
\label{relaxed_sales_supply} & & z_{ipk} \leq w_{ipk} , \mbox{ for all } i \in I, p \in P \mbox{ and } k \in K_p, \\
\label{relaxed_sales_demand} & & z_{ipk} \leq d_{ipk}, \mbox{ for all } i \in I, p \in P \mbox{ and } k \in K_p, \\
\label{relaxed_shipment_number2} & & x_{ijpk} \leq s_{ipk} , \mbox{ for all } i,j \in I, p \in P \mbox{ and } k \in K_p, \\
\label{relaxed_binary_x} & & x_{ijpk} \geq 0, \mbox{ for all } i,j \in I, p \in P\mbox{ and } k \in K_p ,\\
\label{relaxed_binary_z} & & z_{ipk} \geq 0, \mbox{ for all } i \in I, p \in P \mbox{ and } k \in K_p.
\end{eqnarray}
\end{subequations}
The objective function (\ref{relaxed_objective}) maximizes the total profit where the first term represents the total revenue obtained from sales, the second term is the total transfer cost, and the last term is the total holding cost. Constraints (\ref{relaxed_recieved}) define the stock level of each SKU after the transfers are completed. Constraints (\ref{relaxed_sales_supply}) and (\ref{relaxed_sales_demand}) ensure that sales are less than or equal to demand or the available stock of SKUs after the transfers are made. Constraints (\ref{relaxed_shipment_number2}) guarantee that a store may not transfer more than its inventory. Constraints (\ref{relaxed_binary_x}) and (\ref{relaxed_binary_z}) define the decision variables.\\
As mentioned earlier, above problem is simply a profit maximizing transportation problem and can be easily solved by commercial optimizers. Now we extend the problem (\ref{relaxed_objective})-(\ref{relaxed_binary_z}) by adding one of the capacity constraints:
\begin{subequations}
\begin{eqnarray}
\label{relaxed_objective_ex1}\max &(\ref{relaxed_objective})& \\
\label{relaxed_knapsack_ex1} \mbox{s.t.} & & \sum_{\substack{j \in I \\ j\neq i}}\sum_{p \in P}\sum_{k \in K_p} x_{ijpk} \leq A_i, \mbox{ for all } i \in I, \\
\label{all_const_ex1} & & (\ref{relaxed_recieved})-(\ref{relaxed_binary_z}).
\end{eqnarray}
\end{subequations}
where Constraints (\ref{relaxed_knapsack_ex1}) ensure that a store does not transfer more SKUs than it is allowed. This constraint does not pose a challenge as the problem is still a linear program. \\
Next, we add the second capacity constraint to the current model. To do so, however, we need to introduce a binary decision variable $y_{ij}$ that represents if stores $i$ and $j$ are connected. The extended model is formulated as follows:
\begin{subequations}
\begin{eqnarray}
\label{relaxed_objective_ex2}\max &(\ref{relaxed_objective})& \\
\label{relaxed_knapsack_ex2} \mbox{s.t.} & & \sum_{\substack{j \in I \\ j\neq i}}\sum_{p \in P}\sum_{k \in K_p} x_{ijpk} \leq A_i, \mbox{ for all } i \in I, \\
\label{relaxed_shipment_number2_ex2} & & \sum_{\substack{j \in I \\ j\neq i}} y_{ij} \leq B_i , \mbox{ for all } i \in I, \\
\label{basic_shipment_number1_ex2} & & x_{ijpk} \leq s_{ipk} y_{ij} , \mbox{ for all } i,j \in I, j\neq i,p \in P \mbox{ and } k \in K_p , \\
\label{all_const_ex2} & & (\ref{relaxed_recieved})-(\ref{relaxed_sales_demand}),\; (\ref{relaxed_binary_x}), \mbox{ and } (\ref{relaxed_binary_z}).
\end{eqnarray}
\end{subequations}
where Constraints (\ref{relaxed_shipment_number2_ex2}) do not allow a particular store to transfer to more than a given number of stores. Constraints (\ref{basic_shipment_number1_ex2}) allow transfer between two stores only if they are connected; these constraints essentially replace Constraints (\ref{relaxed_shipment_number2}).\\
Finally, single-destination constraint is added to the model. This constraint requires a change in one decision variable set that represents the SKU flow. We now define a binary decision variable $x_{ijp}$ that represents if product $p$ is transfered from store $i$ to store $j$, or not. Then, $x_{ijpk} = s_{ipk} x_{ijp}$, which allows us to drop the original flow variables from the formulation. The final model is given below.
\subsection*{The final model:}
\begin{subequations}
\small
\begin{eqnarray}
\label{basic_objective_nl}\max \Pi & = & \sum_{i \in I}\sum_{k \in k_{p}} \sum_{p \in P}r_{p} z_{ipk} - \sum_{i \in I} \sum_{\substack{j \in I\\ j\neq i}} \sum_{p \in P} \sum_{k \in K_p} c_p s_{ipk} x_{ijp} - \sum_{i \in I } \sum_{p \in P } \sum_{k \in K_p }h_{p}(w_{ipk}-z_{ipk}) \\
\label{basic_sales_supply} \mbox{s.t.} & & z_{ipk} \leq \sum_{j \in I} s_{jpk} x_{jip}, \mbox{ for all } i \in I, p \in P \mbox{ and } k \in K_p, \\
\label{basic_sales_demand} & & z_{ipk} \leq d_{ipk}, \mbox{ for all } i \in I, p \in P \mbox{ and } k \in K_p, \\
\label{PM_pw_sales} & & w_{ipk} = \sum_{j \in I}x_{jip}s_{jpk}, \mbox{ for all } i \in I , p \in P \mbox{ and } k \in K_{p},\\
\label{basic_assignment1} & & \sum_{j \in J} x_{ijp} = 1, \mbox{ for all } i \in I, p \in P, \\
\label{basic_knapsack} & & \sum_{\substack{j \in I \\ j \neq i}} \sum_{p \in P} \sum_{k \in K_p} s_{ipk} x_{ijp} \leq A_i, \mbox{ for all } i \in I, \\
\label{basic_shipment_number2} & & \sum_{\substack{j \in I \\ j \neq i}} y_{ij} \leq B_i , \mbox{ for all } i \in I, \\
\label{basic_shipment_number1} & & x_{ijp} \leq y_{ij} , \mbox{ for all } i,j \in I, j\neq i \mbox{ and } p \in P, \\
\label{basic_binary_x} & & x_{ijp} \in \{0,1\}, \mbox{ for all } i,j \in I \mbox{ and } p \in P,\\
\label{basic_binary_y} & & y_{ij} \in \{0,1\}, \mbox{ for all } i,j \in I,\\
\label{basic_nonzero_z} & & z_{ipk} \geq 0, \mbox{ for all } i \in I, p \in P \mbox{ and } k \in K_p.
\end{eqnarray}
\end{subequations}
where Constraints (\ref{basic_assignment1}) ensure that if a product is transferred from a store, its entire inventory is moved to exactly one store. As a result, assignment to multiple stores is not allowed and similarly, a store may not also keep a portion of the inventory. As we have elaborated before, this ``single-destination'' practice is rather peculiar, but nonetheless it is the case at LC Waikiki. The company justify this practice on the grounds that without this they would have to devote too much of their sales personnels' times for collection, which they are not willing to do. Clearly, this assumption may have a substantial impact on the profit, as it may severely restrict options to better match demand with the supply. Indeed, in our numerical experiments we try to give a sense of the implications of this assumption. As we have also noted, this assumption also complicates the problem substantially, without which the problem can be solved much more effectively. \\
Before we move to the analysis of the problem, we like to point out that the final model is indeed quite difficult. The following proposition shows that the problem is NP-hard.\\
\newtheorem{prop}{Proposition}
\begin{prop}
Problem (\ref{basic_objective_nl})-(\ref{basic_nonzero_z}) is NP-hard.
\end{prop}
\emph{Proof:} We will prove the proposition by reduction. Assume that there is only one product ($P=\{1\}$), no holding cost, ($h=0$) and the product has only one size ($K_{1}=\{1\})$. Furthermore, assume that the unit net revenue of the product is zero, ($r=0$), and there is no limitation on the number of stores to which each store can be connected (unlimited $B_{i}$). Since $r=0$, Constraints (\ref{basic_sales_supply}) and (\ref{basic_sales_demand}) become redundant. Moreover, if $B_{i}$ is unlimited, Constraints (\ref{basic_shipment_number2}) become redundant. Consequently, since any $y_{ij}$ can be one, Constraints (\ref{basic_shipment_number1}) are also redundant. Now the problem reduces to:
\begin{proof}
\begin{subequations}
\begin{eqnarray}
\label{gen_assg_obj} \min \Phi & = & \sum_{\substack{i,j \in I\\j\neq i}} c x_{ij} \\
\label{gen_assg_cons1} \mbox{s.t.} & & \sum_{j \in I} x_{ij}=1, \mbox{ for all } i \in I, \\
\label{gen_assg_cons2} & & \sum_{\substack{j \in I\\j\neq i}}s_{i}x_{ij} \leq A_{i}, \mbox{ for all } i \in I ,\\
\label{gen_assg_cons3} & & x_{ij} \in \{0,1\}, \mbox{ for all } i,j \in I.
\end{eqnarray}
\end{subequations}
Problem (\ref{gen_assg_obj})-(\ref{gen_assg_cons3}) is the well-known generalized assignment problem which belongs to class of NP-hard problems (Savelsberg 1997\nocite{savelsbergh1997branch}).
\end{proof}
As the proposition shows, our problem (\ref{basic_objective_nl})-(\ref{basic_nonzero_z}) is a very difficult mixed integer linear problem. As we will present later, our experiments with a commercial optimizer demonstrated that this problem could not be solved effectively. At LC Waikiki, the number of products that are considered for transfer is about 2,000, on average. On the other hand, the number of stores is approximately 450 nationwide. Therefore, the proposed mixed integer linear program can be huge and a heuristic approach seems to be a reasonable way to proceed. The next section describes such a heuristic method.\\
\section{Solution approach}\label{SA}
We have developed a Lagrangian Relaxation (LR) based approach to obtain good upper bounds in reasonable time. LR has shown exceptional success in solving large scale combinatorial optimization problems (Fisher 1981\nocite{fisher1981lagrangian}). LR is also used in the context of transshipment and it is shown that it can provide acceptable bounds to the optimal solution (Wong et al. 2005\nocite{wong2005simple} and Wong et al. 2006\nocite{wong2006multi}). A solution of the Lagrangian dual provides an upper bound on the optimal solution of the problem (\ref{basic_objective_nl})-(\ref{basic_nonzero_z}). To obtain a lower bound (i.e., a feasible solution), we have developed a two-stage heuristic that consists of a construction heuristic and simulated annealing based metaheuristic to improve the solution. Different heuristic and metaheuristic methods are applied to transshipment problems. For example, Patriarca et al. (2016) and Peres et al. (2017) develop metaheuristics to solve transshipment in inventory-routing problems. The latter applied a variable neighborhood search based algorithm, while the former developed a genetic algorithm. Moreover, local search based methods are utilized in transshipment problems. For instance, Wong et al. (2005) and Wong et al. (2006) developed a simulated annealing based metaheuristic to find promising feasible solutions. Therefore, we have also opted for such metaheuristic. In the rest of this section, we describe these methods in detail.\\
\subsection{Obtaining upper bounds}
Note that in the formulation, Constraints (\ref{basic_sales_demand}), (\ref{basic_shipment_number2}), and (\ref{basic_knapsack}) are similar to knapsack constraints and Constraints (\ref{basic_assignment1}) are basic assignment constraints, all of which are well-known in the literature. On the other hand, Constraints (\ref{basic_sales_supply}) and (\ref{basic_shipment_number1}) complicate the problem because they connect ``$z$" variables to ``$x$" variables and ``$x$" variables to ``$y$" variables, respectively. Thus, problem (\ref{basic_objective_nl})-(\ref{basic_nonzero_z}) can be decomposed in well-known problems by relaxing these complicating constraints. \\
Let $\overline{\boldsymbol{\alpha}} = \{\alpha_{ipk} \in \mathbb{R}^{+}:i\in I, p \in P, k \in K_{p} \}$ and $\overline{\boldsymbol{\beta}}=\{\beta_{ijp}\in \mathbb{R}^{+}: i,j \in I,p \in P \}$ represent vectors of Lagrangian multipliers associated with Constraints (\ref{basic_sales_supply}) and (\ref{basic_shipment_number1}), respectively. Then the relaxed problem can be written as \\
\begin{subequations}
\small{
\begin{eqnarray}
\label{basic_objective_LR2}\max \Pi_{LR}(\overline{\boldsymbol{\alpha}},\overline{\boldsymbol{\beta}}) & = & \sum_{i \in I} \sum_{p \in P} \sum_{k \in K_p} r_p z_{ipk} - \sum_{i \in I} \sum_{\substack{j \in I\\ j\neq i}} \sum_{p \in P}\sum_{k \in K_p} c_p s_{ipk} x_{ijp} - \sum_{i \in I } \sum_{p \in P } \sum_{k \in K_p }h_{p}(w_{ipk}-z_{ipk}) \notag \\
& & -\sum_{i \in I}\sum_{p \in P}\sum_{k \in K_{p}} \alpha_{ipk}(z_{ipk}-\sum_{j \in I}s_{jpk}x_{jip}) - \sum_{i \in I}\sum_{\substack{j \in I \\ j\neq I}}\sum_{p \in P}\beta_{ijp}(x_{ijp}-y_{ij}) \\
\label{Constraints} \mbox{s.t.} & & (\ref{basic_sales_demand})-(\ref{basic_shipment_number2}) \mbox{ and } (\ref{basic_binary_x})-(\ref{basic_nonzero_z}) .
\end{eqnarray}}
\end{subequations}
\indent This problem can be decomposed into three subproblems, which are given as follows:
\begin{subequations}
\begin{eqnarray}
\hspace*{-1.7cm} \label{basic_objective_LR2_z} \mbox{\emph{Subproblem 1}: } \max \Pi_{LR}^{z}(\overline{\boldsymbol{\alpha}}) & = & \sum_{i \in I} \sum_{p \in P} \sum_{k \in K_p} z_{ipk}(r_{p}-\alpha_{ipk}+h_p) \\
\label{basic_sales_demand_LR2_z} \mbox{s.t.} & & z_{ipk} \leq d_{ipk}, \mbox{ for all } i \in I, p \in P \mbox{ and } k \in K_p, \\
\label{basic_nonzero_z_LR2_z} & & z_{ipk} \geq 0, \mbox{ for all } i \in I, p \in P \mbox{ and } k \in K_p.
\end{eqnarray}
\end{subequations}
\begin{subequations}
\begin{eqnarray}
\label{basic_objective_LR2_x} \mbox{\emph{Subproblem 2}: } \max \Pi_{LR}^{x}(\overline{\boldsymbol{\alpha}},\overline{\boldsymbol{\beta}}) & = & \sum_{i \in I}\sum_{\substack{j \in I\\ j \neq i}} \sum_{p \in P}x_{ijp}(\sum_{k \in K_{p}}((-c_{p}+\alpha_{jpk}-h_p)s_{ipk})-\beta_{ijp}) \notag \\
& & +\sum_{i \in I}\sum_{p \in P}\sum_{k \in K_{p}}(\alpha_{iok}-h_p)s_{iok}x_{iip} + \sum_{i \in I}\sum_{j \in I} \sum_{p \in P} h_{p}w_{ipk}\\
\label{basic_assignment1_LR2_x} \mbox{s.t.} & & \sum_{j \in J} x_{ijp} = 1, \mbox{ for all } i \in I, p \in P, \\
\label{basic_knapsack_LR2_x} & & \sum_{\substack{j \in I \\ j \neq i}} \sum_{p \in P} \sum_{k \in K_p} s_{ipk} x_{ijp} \leq A_i, \mbox{ for all } i \in I, \\
\label{PM_pw_sales_LR2_x} & & w_{ipk} = \sum_{j \in I}x_{jip}s_{jpk}, \mbox{ for all } i \in I , p \in P \mbox{ and } k \in K_{p},\\
\label{basic_binary_x_LR2_x} & & x_{ijp} \in \{0,1\}, \mbox{ for all } i,j \in I \mbox{ and } p \in P.
\end{eqnarray}
\end{subequations}
\begin{subequations}
\begin{eqnarray}
\hspace*{-4.4cm} \label{basic_objective_LR2_y} \mbox{\emph{Subproblem 3}: } \max \Pi_{LR}^{y}(\overline{\boldsymbol{\beta}}) & = & \sum_{i \in I}\sum_{\substack{j \in I\\j\neq I}}\sum_{p \in P}\beta_{ijp}y_{ij} \\
\label{basic_shipment_number2_LR2_y}\mbox{s.t.} & & \sum_{\substack{j \in I\\ j \neq i}} y_{ij} \leq B_i , \mbox{ for all } i \in I, \\
\label{basic_binary_y_LR2_y} & & y_{ij} \in \{0,1\} \mbox{ for all } i,j \in I.
\end{eqnarray}
\end{subequations}
\indent Among these problems, Problem (\ref{basic_objective_LR2_z})-(\ref{basic_nonzero_z_LR2_z}) is solvable by inspection. Problem (\ref{basic_objective_LR2_y})-(\ref{basic_binary_y_LR2_y}) can be decomposed into knapsack problems for each store. Problem (\ref{basic_objective_LR2_x})-(\ref{basic_binary_x_LR2_x}) seems to be computationally the most challenging of the three since this problem is similar to the generalized assignment problem. However, it is also separable for each store, which allows us to efficiently solve it. The subproblems for each $i \in I$ can be written as \\
\begin{subequations}
\begin{eqnarray}
\label{basic_objective_LR3_x}\max \Pi_{LR}^{x^{i}}(\overline{\boldsymbol{\alpha}},\overline{\boldsymbol{\beta}}) & = & \sum_{\substack{j \in I \\ j \neq i}} \sum_{p \in P}x_{ijp}((\sum_{k \in K_{p}}(-c_{p}+\alpha_{jpk}-h_p)s_{ipk})-\beta_{ijp}) \notag \\
& & +\sum_{p \in P}\sum_{k \in K_{p}}(\alpha_{ipk}-h_p)s_{ipk}x_{iip} + \sum_{j \in I} \sum_{p \in P} h_{p}w_{ipk}\\
\label{basic_assignment1_LR3_x} \mbox{s.t.} & & \sum_{j \in J} x_{ijp} = 1, \mbox{ for all } p \in P, \\
\label{basic_knapsack_LR3_x} & & \sum_{\substack{j \in I\\ j \neq i}} \sum_{p \in P} s_{ipk} x_{ijp} \leq A_i, \\
\label{PM_pw_sales_LR3_x} & & w_{ipk} = \sum_{j \in I}x_{jip}s_{jpk}, \mbox{ for all } p \in P \mbox{ and } k \in K_{p},\\
\label{basic_binary_x_LR3_x} & & x_{ijp} \in \{0,1\}, \mbox{ for all } j \in I \mbox{ and } p \in P.
\end{eqnarray}
\end{subequations}
Suppose that Lagrangian multipliers $\alpha_{ipk}$ and $\beta_{ijp}$ are set to some values. Then, let us define $\widehat{\text{z}}=\{\widehat{z}_{ipk}: i \in I, p \in P, k\in K_{p} \}$, $\widehat{\text{x}}=\{\widehat{x}_{ijp}: i,j \in I, p \in P \}$, and $\widehat{\text{y}}=\{\widehat{y}_{ij}: i,j\in I \}$ as the corresponding optimal solutions to the subproblems (\ref{basic_objective_LR2_z})-(\ref{basic_nonzero_z_LR2_z}), (\ref{basic_objective_LR2_x})-(\ref{basic_binary_x_LR2_x}) and (\ref{basic_objective_LR2_y})-(\ref{basic_binary_y_LR2_y}), respectively. We can then improve the Lagrangian bounds for a given solution by revising these Lagrangian multipliers. We achieve this by solving the Lagrangian dual while retaining the primal solutions. Interested reader can refer to Litvinchev (2007\nocite{litvinchev2007refinement}) for a detailed account of this approach. The Lagrangian dual can be formulated as follows: \\
\begin{subequations}
\begin{eqnarray}
\label{multipliers_objective}
\min_{\overline{\alpha},\overline{\beta}} \max \Delta(\widehat{\text{z}},\widehat{\text{x}},\widehat{\text{y}}) & = & \sum_{i \in I} \sum_{p \in P} \sum_{k \in K_p} \widehat{z}_{ipk}(r_{p}-\alpha_{ipk}+h_p)+ \sum_{i \in I}\sum_{\substack{j \in I\\j\neq I}}\sum_{p \in P}\beta_{ijp}\widehat{y}_{ij} +\sum_{i \in I}\sum_{p \in P}\sum_{k \in K_{p}}(\alpha_{ipk}-h_p)s_{ipk}\widehat{x}_{iip} \notag \\
& & +\sum_{i \in I}\sum_{\substack{j \in I\\ j \neq i}} \sum_{p \in P}\widehat{x}_{ijp}((\sum_{k \in K_{p}}(-c_{p}+\alpha_{jpk}-h_p)s_{ipk})-\beta_{ijp}) \\
\label{multipliers_cons1}\mbox{s.t.} & & \sum_{k \in K_{p}}(\alpha_{jpk}-c_{p}-h_p)s_{ipk}-\beta_{ijp} \leq \sum_{k \in K_{p}} (\alpha_{ipk}-h_p)s_{ipk}, , \notag \\
& & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \mbox{ for all } i\neq j \in I, p \in P, k \in K_{p} \mbox{ if } \widehat{x}_{ijp}=1, i= j, \\
\label{multipliers_cons2} & & \sum_{k \in K_{p}}(\alpha_{jpk}-c_{p}-h_p)s_{ipk}-\beta_{ijp} \leq \sum_{k \in K_{p}}(\alpha_{j^{*}pk}-c_{o}-h_p)s_{ipk}-\beta_{ij^{*}p} , \notag \\
& & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \mbox{for all } i,j \in I, p \in P, k \in K_{p}\mbox{ if } \widehat{x}_{ijp}=1, i\neq j, j^{*},\quad \quad \\
\label{multipliers_cons3} & & \sum_{k \in K_{p}} (\alpha_{ipk}-h_p)s_{ipk} \leq \sum_{k \in K_{p}}(\alpha_{j^{*}pk}-c_{p}-h_p)s_{ipk}-\beta_{ij^{*}p},\notag \\
& & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \mbox{ for all } i,j \in I, p \in P, k \in K_{p}\mbox{ if } \widehat{x}_{ijp}=1, i\neq j \neq j^{*}, \\
\label{multipliers_profit} & & \alpha_{ipk} \leq r_{p}+h_p, \mbox{ for all } i \in I, p \in P \mbox{ and } k \in K_{p}, \\
\label{multipliers_cost} & & \beta_{ijp} \leq \sum_{k \in K_{p}}((-c_{p}+r_p-h_p)s_{ipk}), \mbox{ for all } i,j \in I \mbox{ and } p \in P,\\
\label{multipliers_nonnegative_alpha} & & \alpha_{ipk} \geq 0, \mbox{ for all } i \in I, p \in P \mbox{ and } k \in K_{p}, \\
\label{multipliers_nonnegative_beta} & & \beta_{ijp} \geq 0, \mbox{ for all } i \in I, j \in I,\mbox{ and } p \in P.
\end{eqnarray}
\end{subequations}
\indent The objective function (\ref{multipliers_objective}) is the objective function of the dual problem. Since the solutions $\widehat{\text{x}}$, $\widehat{\text{y}}$, and $\widehat{\text{z}}$ are known, Constraints (\ref{multipliers_cons1})-(\ref{multipliers_nonnegative_beta}) are added to modify the Lagrangian multipliers while retaining the primal solutions. Constraints (\ref{multipliers_cons1}) ensure that if a product $p$ is sent from store $i$ to any other store $j$, then the coefficient of $\widehat{x}_{ijp}$ in the objective function must be less than the coefficient of $\widehat{x}_{iip}$. Similarly, the coefficient of $\widehat{x}_{ijp}$ must be less than the coefficient of any other $\widehat{x}_{ij^{*}p}$, which is guaranteed by Constraints (\ref{multipliers_cons2}). On the other hand, if $\widehat{x}_{iip}=1$, that is, product $p$ remains at its original location, then $\widehat{x}_{ijp}=0$, which is ensured by Constraints (\ref{multipliers_cons3}). Constraints (\ref{multipliers_profit}) guarantee that multipliers $\alpha_{ipk}$ are not greater than corresponding unit revenues plus holding cost to prevent $\widehat{z}_{ipk}$ to be zero. Likewise, Constraints (\ref{multipliers_cost}) set upper bounds on $\beta_{ijp}$. Finally, multipliers $\alpha_{ipk}$ and $\beta_{ijp}$ must be non-negative which are ensured by Constraints (\ref{multipliers_nonnegative_alpha}) and (\ref{multipliers_nonnegative_beta}), respectively.\\
The optimal solution to this problem is a tighter upper bound as compared to the solution obtained from the relaxed problem. Naturally, this solution provides an upper bound to the optimal solution of the original problem as well.
\subsection{Obtaining lower bounds}
As mentioned earlier, we obtain lower bounds, i.e., feasible solutions, via a construction heuristic followed by an improvement metaheuristic. The construction heuristic consists of two steps, in the first of which we iteratively connect stores until there is no improvement. We start by dividing all store-product combinations into two groups as sender and receiver based on their stock and demand levels without considering the sizes. For each product, if the stock level in a store is more than its demand, the store is classified as sender; otherwise, it is classified as a receiver. Note that, a store can be either in the sender group or in the receiver group for a product (or, in none of the groups in case the stock and demand levels are equal). We then sequentially connect senders to receivers by selecting products randomly. For each store in the sender group, we find a candidate store from the receiver group that creates the highest profit, i.e., revenue less implied costs. A transfer decision is made if Constraints (\ref{basic_knapsack}) and (\ref{basic_shipment_number2}) remain feasible. After all products are selected, we update the sender and receiver groups considering the current transfers. That is, a store that was initially in the sender group and sends its entire inventory to another store may be included in the receiver group in the next iteration. Moreover, a store in the receiver group can continue to stay in the same group, if it still has needs. Otherwise, it will not be considered as a sender or a receiver. This procedure is repeated until there is no improvement in the solution. In the second step, we further investigate profitable transfers that were not made in the previous step due to Constraints (\ref{basic_shipment_number2}). Now, we search for beneficial transfers by choosing among the destinations that a store is already connected, so that the constraint remains feasible, while the solution is improved. \\
At the improvement stage, we have developed a simulated annealing based metaheuristic. The proposed metaheuristic essentially destroys the current feasible solution by removing a transfer and then repairing it by inserting another transfer. It removes transfers according to three rules that are applied randomly. In the first rule, the transfer to be removed is also selected randomly. The other two rules use the ``residual demand" information for each store-product pair, i.e., the difference between the demand and the transfer it receives in the current solution. That is, those with the negative residual demand are the ones that receive more than their demand. The second rule randomly chooses a store-product pair among those that have negative residual demands. And finally, for the third rule we first list all store-product pairs that have negative residual demand. Then we select the product that appears the most in the list and then choose from the stores that also appears the most in the list and paired with this product. After selecting a store-product pair, that transfer is removed and another transfer is inserted while maintaining the feasibility of Constraints (\ref{basic_assignment1}). The destination store is chosen randomly among the ones that have \emph{positive} residual demand. The purpose of these rules is to enable moving to worse as well as better solutions than the current one.\\
The algorithm allows non-improving moves to include diversity as in the simulated annealing (SA) approach. It is adopted as follows: If the profit of the new transfer is greater than or equal to the profit of the removed one, the transfer is accepted. Otherwise, we accept it with probability $ e^{\frac{-(currentProfit - newProfit)}{temperature}}$, where $currentProfit$ and $newProfit$ denote the profits of the removed transfer and the newly added one, respectively and $temperature$ is the current temperature, which is a parameter of SA. Initially, $temperature$ is equal to the total profit of current solution so that the $probability$ becomes high and the chance of accepting a worse solution is high. The $temperature$ is decreased at each iteration using the formula $temperature = temperature \times \tau$, where $0 < \tau < 1$ is the $cooling$ $rate$. A $counter$ keeps the number of worse solutions accepted. The $cooling$ $ rate$ is calculated by $cooling$ $ rate = 1/ counter$. The best solution is kept and updated whenever a better solution is found. To avoid being trapped in local optima, the algorithm continues to search from either the best solution or second best solution if there is no improvement in a predefined number of iterations. The algorithm stops if either the total number of iterations reaches to its upper bound or the time limit hast been reached.
\section{Computational results}\label{CR}
In this section, we report on our computational experiments that consist of two main parts. In the first part, our purpose is to compare the effectiveness of our solution method to that of a commercial solver. Towards this purpose, the generated instances that are first solved by Gurobi 8.1 and then by our algorithm, which is implemented in Python 3.6 and the subproblems are also solved with Gurobi. All experiments are conducted on a High-Performance Computing (HPC) cluster with Linux system, 44 GB RAM, and 2.40 Ghz processors with two cores. In the second part, our purpose is to develop insights into the effects of the operational constraints on the system performance. Towards this end, we develop another set of instances, all of which are solved by Gurobi.\\
LC Waikiki has about 450 stores (excluding the outlet ones) and about 2,000 items considered for transfer at any week. Unfortunately, problems of this scale cannot be directly handled by Gurobi. Therefore, we have targeted 50 and 100 as the number of stores and 100, 200, 500, and 1000 as the number of products. The number of sizes varies according to the product. However, most of the products have five to 10 different sizes (e.g., S, M, L, XL, and XXL or 28, 30, 32, 34, 36, 38, 40, 42, 44, and 46 for two different products). Thus, in the test problems, the number of sizes is set to either five or 10. Detailed information on the combination of sizes of the instances are given in the first columns of Tables \ref{low}, \ref{medium}, and \ref{high}. In total, we have solved instances of 14 different size combinations. \\
We randomly generated demands ($d_{ipk}$) and initial stock levels ($s_{ipk}$) from a discrete uniform distribution that is defined between 0 and 10. The prices of the products are set between 20 and 50 Turkish Lira (TL) while the transfer costs for these products are set between 0.4 to 1.5 TL. We generated the selling prices and transfer costs randomly from uniform distributions with the bounds given above. To set the holding cost rate we should have also drawn unit costs but in the interest of simplicity we used the unit revenues. The holding cost per week is taken as the 0.5\% of the unit revenue, which corresponds to about 30\% or less, annually. Although we have not used any real data from LC Waikiki to develop these instances, we have decided on these values upon our conversations with the group that deals with the transfers; hence, we believe our instances are quite realistic. \\
In order to find the parameters of two of the operational constraints, first we solved each instance by ignoring all three restrictions. We then found the number of items each store sends and the number of stores it is connected. These are essentially, the maximum values when there are no operational constraints. Based on these numbers, we then set three levels of $A_i$ and $B_i$ for each store as low (1/3 of the maximum), medium (1/2 of the maximum), and high (2/3 of the maximum).\\
For each size and ($A_i$, $B_i$) combinations, we have randomly generated 10 instances. We have reached at this number through a small numerical experiment. We took three instance sizes, generated 100 random instances of each, solved them with our algorithm, and computed the average gaps progressively. It turned out that after 10th replication the progressive average of gaps becomes nearly constant for all three sets of instances as illustrated in Figure \ref{replication}. Therefore, we concluded that 10 instances would be enough to have a reliable performance metric in terms of average gaps. As a result, we have generated and tested a total of 14x3x10 = 420 instances.\\
\begin{figure}[!b]
\centering
\hspace*{-1em}
\includegraphics[width=0.78\textwidth]{figure2.jpg}
\caption{Sensitivity of progressive optimality gap to the number of replications}
\label{replication}
\end{figure}
The results are illustrated in Tables \ref{low}, \ref{medium}, and \ref{high}. The first column depicts the size of each instance with respect to the number of stores, products, and sizes. The Gurobi column reports the average of best feasible solutions, the minimum, maximum, and average optimality gaps of 10 replications that Gurobi achieved and the time limit that we set. We have given a one-hour time limit for smaller sized problems, two-hour time limit for medium sized ones, and six-hour time limit to the larger sized instances, in addition to the problem loading times to Gurobi. The metaheuristic column also reports the average of best feasible solutions, the minimum, maximum, and average optimality gaps of 10 replications and the time our algorithms spent to find the upper and lower bounds. We calculated the optimality gaps as $\frac{UB-LB}{LB}$ where $UB$ and $LB$ represent the upper and lower bounds found by Gurobi and our method.\\
\begin{table}[t]
\centering
\scriptsize
\caption{Results for Low level of $A_i$ and $B_i$}
\hspace*{-3em}
\begin{tabular}{ccccccccccc}
\hline
& \multicolumn{5}{c}{Gurobi} & \multicolumn{5}{c}{Metaheuristic} \bigstrut[b]\\
\cline{2-11} Instance & Average & \multicolumn{3}{c}{Gap} & Average & Average & \multicolumn{3}{c}{Gap} & Average \bigstrut[t]\\
$I$-$P$-$K$ & Lower Bound & Min & Average & Max & Runtime & Lower Bound & Min & Average & Max & Runtime \bigstrut[b]\\
\hline
50-100-5 & 848,013 & 0.71 & 0.79 & 0.89 & 3600 & 841,322 & 1.04 & 2.07 & 3.67 & 1285 \bigstrut[t]\\
50-100-10 & 307,259 & 0.00 & 0.00 & 0.01 & 7.04 & 302,956 & 0.50 & 2.41 & 5.33 & 1017\\
50-200-5 & 1,519,767 & 0.78 & 0.87 & 1.01 & 3600& 1,500,891 & 1.06 & 3.00 & 5.38 & 1424 \\
50-200-10 & 935,098 & 0.79 & 1.09 & 1.37 & 3600 & 924,970 & 1.77 & 3.39 & 8.32 & 1318 \\
50-500-5 & 2,317,320 & 282.10 & 288.27 & 296.74 & 3600 & 6,244,111 & 4.22 & 4.48 & 4.76 & 2001 \\
50-500-10 & 2,417,042 & 0.34 & 0.40 & 0.45 & 3600 & 2,398,677 & 0.98 & 2.40 & 3.65 & 1960 \\
50-1000-5 & 4,663,613 & 262.45 & 284.38 & 302.65 & 3600 & 11,074,571 & 2.50 & 3.76 & 4.29 & 1966 \\
50-1000-10 & 3,507,723 & 0.09 & 0.14 & 0.22 & 3600 & 3,487,356 & 0.76 & 1.04 & 1.44 & 2082 \\
100-100-5 & 950,596 & 283.04 & 287.71 & 293.21 & 7200& 2,695,145 & 3.47 & 5.30 & 6.20 & 1492 \\
100-100-10 & 1,890,957 & 0.22 & 0.31 & 0.41 & 7200 & 1,882,898 & 0.57 & 1.58 & 3.29 & 3707 \\
100-500-5 & 4,666,991 & 284.15 & 287.76 & 294.74 & 7200 & 12,141,278 & 6.10 & 6.57 & 6.96 & 4313 \\
100-500-10 & 7,809,377 & 0.71 & 0.91 & 1.11 & 7200 & 7,714,316 & 0.85 & 3.52 & 5.16 & 4414 \\
100-1000-5 & 9,303,094 & 284.71 & 288.34 & 291.90 & 21600 & 18,948,867 & 0.49 & 0.94 & 1.24 & 9245 \\
100-1000-10 & 12,997,254 & 0.39 & 0.99 & 1.28 & 21600 & 12,906,313 & 1.33 & 2.45 & 3.22 & 6475 \\
\hline
\end{tabular}%
\label{low}%
\end{table}%
\begin{table}[!b]
\centering
\scriptsize
\caption{Results for Medium level of $A_i$ and $B_i$}
\hspace*{-3em}
\begin{tabular}{ccccccccccc}
\hline
& \multicolumn{5}{c}{Gurobi} & \multicolumn{5}{c}{Metaheuristic} \bigstrut[b]\\
\cline{2-11} Instance & Average & \multicolumn{3}{c}{Gap} & Average & Average & \multicolumn{3}{c}{Gap} & Average \bigstrut[t]\\
$I$-$P$-$K$ & Lower Bound & Min & Average & Max & Runtime & Lower Bound & Min & Average & Max & Runtime \bigstrut[b]\\
\hline
50-100-5 & 1,023,543 & 0.42 & 0.49 & 0.56 & 3600 & 1,015,396 & 0.82 & 2.57 & 4.95 & 1274 \bigstrut[t]\\
50-100-10 & 547,922 & 0.01 & 0.01 & 0.01 & 169 & 542,032 & 0.63 & 1.74 & 3.05 & 1181 \\
50-200-5 & 1,794,245 & 0.72 & 0.83 & 0.95 & 3600 & 1,760,781 & 1.60 & 3.58 & 5.37 & 1404 \\
50-200-10 & 1,249,620 & 0.47 & 0.56 & 0.66 & 3600 & 1,224,849 & 1.09 & 3.36 & 6.11 & 1366 \\
50-500-5 & 5,085,082 & 6.65 & 90.41 & 287.53 & 3600 & 6,501,742 & 3.70 & 3.96 & 4.23 & 2000 \\
50-500-10 & 3,192,027 & 0.08 & 0.10 & 0.13 & 3600 & 3,149,863 & 0.73 & 3.53 & 4.67 & 1982 \\
50-1000-5 & 4,663,613 & 283.04 & 287.71 & 293.21 & 3600 & 12,553,494 & 1.12 & 1.80 & 2.93 & 1974 \\
50-1000-10 & 6,070,388 & 0.10 & 0.14 & 0.19 & 3600 & 5,984,386 & 1.08 & 1.75 & 2.56 & 2075 \\
100-100-5 & 3,008,779 & 0.52 & 0.82 & 1.18 & 7200 & 2,997,312 & 2.57 & 4.49 & 5.21 & 1475 \\
100-100-10 & 1,890,938 & 0.07 & 0.08 & 0.10 & 7200 & 1,879,030 & 0.34 & 2.79 & 4.13 & 3774 \\
100-500-5 & 4,636,738 & 284.96 & 289.79 & 294.74 & 7200 & 12,713,377 & 4.81 & 5.30 & 5.76 & 4754 \\
100-500-10 & 9,447,082 & 0.94 & 1.06 & 1.17 & 7200 & 9,363,384 & 2.09 & 2.74 & 3.43 & 4514 \\
100-1000-5 & 9,325,113 & 284.71 & 287.41 & 289.97 & 21600 & 20,854,343 & 1.83 & 2.50 & 2.92 & 9028 \\
100-1000-10 & 15,871,661 & 0.01 & 0.42 & 1.22 & 21600 & 15,780,238 & 1.11 & 1.55 & 2.52 & 6109 \\ \hline
\end{tabular}%
\label{medium}%
\end{table}%
\begin{table}[t]
\centering
\scriptsize
\caption{Results for High level of $A_i$ and $B_i$}
\hspace*{-3em}
\begin{tabular}{ccccccccccc}
\hline
& \multicolumn{5}{c}{Gurobi} & \multicolumn{5}{c}{Metaheuristic} \bigstrut[b]\\
\cline{2-11} Instance & Average & \multicolumn{3}{c}{Gap} & Average & Average & \multicolumn{3}{c}{Gap} & Average \bigstrut[t]\\
$I$-$P$-$K$ & Lower Bound & Min & Average & Max & Runtime & Lower Bound & Min & Average & Max & Runtime \bigstrut[b]\\
\hline
50-100-5 & 1,171,178 & 0.33 & 0.39 & 0.48 & 15 & 1,154,752 & 1.79 & 2.96 & 5.17 & 1290 \bigstrut[t]\\
50-100-10 & 819,047 & 0.01 & 0.02 & 0.06 & 2845 & 805,510 & 0.37 & 2.13 & 3.10 & 1333 \\
50-200-5 & 2,042,125 & 0.58 & 0.79 & 1.37 & 3600 & 2,017,624 & 1.52 & 3.36 & 5.10 & 1448 \\
50-200-10 & 1,507,772 & 0.31 & 0.37 & 0.46 & 3600 & 1,496,739 & 1.31 & 2.63 & 3.88 & 1423 \\
50-500-5 & 7,151,269 & 0.12 & 0.28 & 0.41 & 3600 & 7,120,120 & 0.72 & 1.06 & 1.97 & 1778 \\
50-500-10 & 4,456,310 & 0.01 & 0.01 & 0.01 & 154 & 4,296,653 & 3.37 & 4.28 & 4.77 & 2131 \\
50-1000-5 & 12,891,943 & 0.07 & 0.12 & 0.16 & 3600 & 12,839,782 & 0.46 & 0.75 & 1.13 & 1971 \\
50-1000-10 & 8,840,304 & 0.01 & 0.01 & 0.02 & 1963 & 8,462,620 & 1.46 & 2.12 & 2.98 & 2064 \\
100-100-5 & 3,021,758 & 0.20 & 0.30 & 0.46 & 7200& 3,010,490 & 3.76 & 4.55 & 5.64 & 1464 \\
100-100-10 & 1,891,297 & 0.53 & 0.56 & 0.63 & 7200 & 1,872,998 & 4.41 & 5.23 & 5.96 & 3647 \\
100-500-5 & 14,830,650 & 0.49 & 1.27 & 2.16 & 7200 & 14,685,243 & 2.66 & 3.87 & 5.58 & 4201 \\
100-500-10 & 9,492,130 & 0.02 & 0.14 & 0.29 & 7200 & 9,373,087 & 1.10 & 2.19 & 3.02 & 4512 \\
100-1000-5 & 29,909,041 & 0.56 & 0.82 & 1.56 & 21600 & 29,313,762 & 1.55 & 3.39 & 5.72 & 9014 \\
100-1000-10 & 19,301,509 & 0.00 & 0.01 & 0.01 & 11479 & 19,217,184 & 0.65 & 1.09 & 1.43 & 6004 \\ \hline
\end{tabular}%
\label{high}%
\end{table}%
The results show that our algorithm is comparable to, and in some cases much more effective than, Gurobi. First of all, our algorithm spends about one-third to one-half of the time that we give to Gurobi excluding the time it takes to load the problem to Gurobi, which could be rather substantial in larger instances. In terms of the optimality gaps, the results are somewhat mixed. There are many instance sets, for which Gurobi found better solutions (lower bounds) than our method did. However, Gurobi's solutions deteriorate faster than our method with increasing problem size. Although our approach also suffers, it can solve most medium-sized problems with around 1\% optimality gaps and large-sized problems with a maximum gap of about 7\%.\\
The most important problem with Gurobi, however, is that it is rather unreliable. In some cases the solutions it found were just terrible, with optimality gaps hovering around 300\%. Upon close inspection, we noticed that these poor results belong to the instances where there are five different sizes, whereas the instances with the same number of stores and products but 10 different sizes for each products, the behavior was quite the opposite. This was rather puzzling; after all, the latter instances are of larger size, but with closer inspection we were able to conclude that it was the combination of several factors that led to this unexpected results. First of all, since the transportation costs are quite low as compared to the revenues, as long as it is revenue-improving the optimal solution tends to have large number of transfers. Secondly, when there are 10 sizes for each product, there are fewer profitable opportunities for transfers as compared to the same number of stores and products with five sizes for each product. This might seem unclear at first, but single-destination constraint is mainly responsible for these results. For example, if there is only one size, then there would be many profitable opportunities for transfer. When there are two sizes, the opportunities would diminish because there would be more sales opportunities at the original sources and there would be fewer alternative stores that would have demand for both sizes. This would be even more prominent with increasing the number of sizes. This is indeed what we have observed in a simple experiment that we have conducted with 20 stores, 100 products and no operational constraints. We then set the number of sizes as 2, 5, 8, 10, and 15 and have randomly drawn 10 instances for each size. Table \ref{siz} reports the results, which confirm our intuition. Hence, since there are far fewer transfers in the optimal solution as the number of sizes increases most stores do not perform any transfers but keep at the source. As a result, Gurobi can eliminate a substantial number of potential transfers across the stores and finds solutions easier in instances where there are 10 sizes as compared to five. To conclude, while our approach does not produce better solutions than Gurobi all the time, due to its robustness to problem characteristics, it is a much better alternative of the two.\\
\begin{table}[!t]
\centering
\caption{Effect of the number of sizes on the number of transfers}
\hspace*{-1em}
\scalebox{0.75}{
\begin{tabular}{ccccccccccc}
\hline
$K$ & Instance 1 & Instance 2 & Instance 3 & Instance 4 & Instance 5 & Instance 6 & Instance 7 & Instance 8 & Instance 9 & Instance 10 \bigstrut\\
\hline
2 & 1,038 & 1,074 & 1,009 & 1,013 & 1,031 & 1,030 & 1,069 & 1,011 & 998 & 1,038 \bigstrut[t]\\
5 & 616 & 626 & 624 & 650 & 630 & 625 & 634 & 653 & 612 & 621 \\
8 & 327 & 371 & 334 & 366 & 352 & 353 & 315 & 335 & 315 & 354 \\
10 & 208 & 226 & 220 & 220 & 206 & 231 & 204 & 244 & 207 & 237 \\
15 & 42 & 55 & 65 & 57 & 48 & 53 & 59 & 57 & 78 & 50 \bigstrut[b]\\
\hline
\end{tabular}}
\label{siz}%
\end{table}%
In the second part of our experiments, our purpose is to shed some light into the effects of particular operational restrictions used by LC Waikiki. The restrictions include the two capacity constraints and the single-destination policy. Towards this end, we considered six problem sizes as illustrated in Figure \ref{managerial}. We randomly generated 10 instances for each of these problem sizes. As we have done in the first part, we have created further instances based on how tight the capacity constraints are. Similarly, we first solved each instance by ignoring all three restrictions, found the maximum values $A_i$ and $B_i$ can take for each store, and then created four combinations of ($A_i$, $B_i$) by setting them to either ``low" (1/3 of the maximum) or ``high" (2/3 of the maximum). Therefore, altogether we have solved a total of 240 instances. \\
\begin{figure}[!h]
\hspace*{-6em}
\begin{minipage}{0.5\linewidth}
\includegraphics[width = 1.3\textwidth]{figure3a.jpg}
\end{minipage}
\hfill
\hspace*{-6em}
\begin{minipage}{0.505\linewidth}
\includegraphics[width = 1.3\textwidth]{figure3b.jpg}
\end{minipage}
\hspace*{-6em}
\begin{minipage}{0.5\linewidth}
\includegraphics[width = 1.3\textwidth]{figure3c.jpg}
\end{minipage}
\hfill
\hspace*{-6em}
\begin{minipage}{0.505\linewidth}
\includegraphics[width = 1.3\textwidth]{figure3d.jpg}
\end{minipage}
\caption{Effect of adding capacity and single destination constraints on the objective function value (OFV). Top-left: low $A_i$ and low $B_i$, top-right: low $A_i$ and high $B_i$, bottom-left: high $A_i$ and low $B_i$, and bottom-right: high $A_i$ and high $B_i$.}
\label{managerial}
\end{figure}
The unconstrained version of the problem, i.e., \eqref{relaxed_objective}-\eqref{relaxed_binary_z}, is solved first, and then the problem with the transfer capacity constraint, i.e., \eqref{relaxed_objective_ex1}-\eqref{all_const_ex1}, which is followed by the problem with both capacity constraints, i.e., \eqref{relaxed_objective_ex2}-\eqref{all_const_ex2}, and finally, the full problem \eqref{basic_objective_nl}-\eqref{basic_nonzero_z} is solved. All these problems are solved with Gurobi. Although not all problems are solved to optimality, the gaps are rather small, so the results are quite reliable. \\
Figure \ref{managerial} depicts the summary results. Each graph in the figure contains results based on the combinations of ($A_i$, $B_i$) pairs. In the figure, optimum solution of each unconstrained instance is normalized to 100 and the objective functions of each instance's constrained versions are found as percentage of the optimal value of the unconstrained version. The graphs report the averages of these percentages over 10 instances. The effect of the constraint on the number of SKUs that can be transferred is quite clear. The optimal values reduces to roughly 40-60\% and 65-85\% of the maximum possible for low and high $A_i$'s, respectively. The addition of the second constraint that restricts the number of stores has almost no further deteriorating effect. Hence, it appears that the first constraint is already restrictive enough. The only exception to these results is when $A_i$ is high \emph{and} there are 10 sizes for each product. As the bottom-left graph shows, although adding the second constraint has a very slight effect when there are five sizes, its effect is substantial when it is 10. This result is actually quite intuitive because when there are 10 sizes there are simply more opportunities to match demand and supply as there are more sizes (because there is no single-destination constraint yet). Therefore, restricting the number of stores greatly eliminates those opportunities. \\
We can observe that the single destination constraint has a substantial deteriorating effect on the profit. Depending on the cases, it has roughly an additional 5-25\% negative effect on the maximum profit. The effect is somewhat less when $A_i$ is low, presumably the first constraint already has a great effect on reducing the number of destinations. However, the effect is quite substantial when $A_i$ is high regardless of the value of $B_i$.\\
To summarize the managerial implications, we can conclude that in general, i) it is the restriction on the total number of SKUs that can be transferred rather than the restrictions on the number of destinations that has a more negative effect, ii) single destination constraint has a more detrimental effect when the capacities are less restrictive, and finally, iii) all the negative effects are usually more pronounced when there are larger number of sizes per product. Therefore, if a firm wishes to relax the single-destination constraint, it should start from products with larger number of sizes and accompany it relaxing restrictions on the transfer capacities.
\section{Concluding remarks}\label{Concl}
In this paper we introduce a novel proactive transshipment problem motivated by the practice at the largest fast fashion retailer in Turkey, LC Waikiki. The company, after allocating the initial inventory to over 450 stores and observing sales for a few weeks, engages in lateral transshipments among the stores. When a product has different sales performances across the stores, lateral transshipments can improve the overall system performance. Not only such a practice helps the company better match supply with demand but also eliminates additional handling and transportation operations at its central depot. \\
Its large scale and particular operational restrictions necessitate the development of a novel model. We formulate the transfer problem of LC Waikiki as a mixed integer linear programming problem. With around 450 stores, 2,000 products, and a variety of operational constraints, this problem becomes a very large mixed integer program and solving it optimally becomes a challenge. Therefore, we have developed a simulated annealing based metaheuristic to solve the problem. We also applied Lagrangian relaxation with a primal-dual approach to obtain sharp upper bounds on the optimal solution of the original problem. We generated 420 problem instances of varying sizes to evaluate the performance of the proposed algorithm against the commercial optimizer Gurobi. Each instance is solved by the proposed algorithm and Gurobi. The results show that although the solutions prescribed by Gurobi are better than ours in small-size instances and those with loose capacities, the proposed algorithm outperforms Gurobi in instances that are characterized by having a large number of potentially beneficial transfers and tighter capacity constraints. These instances are the ones where the combinatorial nature of the problem becomes the most challenging. Gurobi fails spectacularly in these instances, while our algorithm performs without a significant loss in its performance. Hence, our algorithm is quite robust to changing program characteristics. Finally, our algorithm is also quicker in finding solutions, spending only about one-third to one-half of the time spent by Gurobi. This feature makes our approach particularly attractive when companies need speedy solutions to these problems. \\
We have also conducted a carefully designed numerical experiment to uncover the effect of the particular operational constraints of the company. First, we have solved the instances without any of those constraints and found the maximum potential gross revenue (i.e., the base gross revenue) that can be obtained with transshipments. We have then added those constraints one by one to observe their effects. We observe that constraints on the total number of products a store can send has a significant impact and depending on how tight those restrictions are, may reduce the gross revenue to around 40-85\% of its base level. The second capacity restriction, i.e., the number stores that a store can make shipments, usually has very little negative impact after the first capacity restriction is already imposed. Finally, we have also measured the effect of the single-destination practice and have found that it may reduce the revenues by another 5-25\% of the base revenue depending on how tight the capacity constraints are. When the capacity constraints are already tight, the negative impact of this practice is quite small, but where the capacities are loose the negative impact of this practice is quite significant. We have also observed that the number of sizes also plays a significant role in these results. In our sample instances we have used five and 10 as the number of sizes. Naturally, when there are 10 sizes of products, single-destination practice renders much fewer number of transfers as potentially beneficial and therefore, this practice becomes more detrimental to the base revenue when the number of sizes increases.\\
We had to make a number of simplifying assumptions to effectively deal with this very large problem that has complicating operational constraints. Therefore, there are a number of avenues for future research. While one may have accurate demand forecasts as in this case, there are always forecast errors, and therefore, considering demand uncertainty is naturally an important extension to this study. In a similar vein, initial shipment decisions under demand uncertainty may also be considered jointly with the transfer decisions. Another potentially important avenue is to develop integrated models that include transfer decisions as well as markdown decisions, an avenue that we are currently pursuing. Finally, the frequency at which the collections are renewed can also be made jointly with transfer as well as other logistical decisions. \\
\section*{Acknowledgments}
The authors first thank to their industrial partner, LC Waikiki and their key collaborator, Allocation Strategy and Process Development team- in particular, Murat Aksu (the team's previous director), Serkan Ulukaya, Mustafa Tilkat, Yavuz Selim Dola\c{s}, and Muhammed Can Konur. They are also particularly grateful to other employees at LC Waikiki, including Elif Hilal Akar, Aytu\u{g} Alten, and Sinan Erg\"{u}n for their support.
\section*{References}
|
1,116,691,500,852 | arxiv | \section{Introduction}
Distinguishing knots and links up to ambient isotopy is the central problem in knot theory. The main technique that a knot theorist uses is to compute some knot invariants and see if one of them can be of help. Over the last 35 years tremendous progress has been made in the development of several new knot invariants, starting with the Jones polynomial and the HOMFLY-PT polynomial\cite{HOMFLY, PT, de1986jones}. Recently even more sophisticated invariants such as Heegard-Floer homology groups \cite{manolescu2014introduction} and Khovanov homology groups \cite{bar2002khovanov} have been added to the toolkit. In the 1980s William Thurston's seminal result \cite[Corollary 2.5]{thurston1982three} that most knot complements have the structure of a hyperbolic manifold, combined with Mostow's rigidity theorem \cite[Theorem 3.1]{thurston1982three} giving uniqueness of such structures, establishes a strong connection between hyperbolic geometry and knot theory, since knots are determined by their complements. Indeed, any geometric invariant of a knot complement, such as the hyperbolic volume, becomes a topological invariant of the knot. Thus, investigating if data derived from the new knot invariants is related to natural differential geometric invariants becomes another natural problem. In this direction `volume conjecture' is one of the most challenging open problem. This conjecture has been tested for torus knots but for hyperbolic knots it has been verified only for a handful of knots. Weaving knots $W(p,n)$ of type $(p,n)$ for a pair of co-prime integers $p$ and $n$ are doubly infinite family of alternating, hyperbolic knots and share the same projection with torus knots. They can be thought of a prototype of hyperbolic knots. Thus an extensive study of this family of knots will provide an insight to `volume conjecture.' One of us in an earlier work\cite{mishra2017jones} have attempted recursive method of relating the HOMFLY-PT of $W(3,n)$. In a parallel paper\cite{RRV}, the closed form of HOMFLY-PT for $W(3,n)$ with explicit proof is provided. In this paper, we study the {\it hybrid family of weaving knots} denoted by $(\hat{W}_{3}(m,n))$ . Here, we use the approach of Reshitikhin and Turaev to evaluate the colored polynomials for knots and obtained the closed form expression for HOMFLY-PT polynomial for hybrid weaving knots. Further, we have computed the $[r]$-colored HOMFLY-PT polynomial for $W(3,n)$ which agrees when $[r]=[1]$ with the results in \cite{RRV}. Further we study the reformulated invariants in the context of topological string dualities and validate Oogur-Vafa conjecture\cite{OV, GV1, LMV}. Interestingly, we show that certain BPS integers of weaving knot $W(3,n)$ can be written in the Chebyshev coefficients of first kind.
The paper is organized as follows:
In section~\ref{s.int}, we will review Reshitikhin and Turaev (RT) method of constructing knot and link invariants which involves $\mathcal{R}$-matrices. This is followed by the subsection~\ref{s12.int} where we present the $\hat{\mathcal{R}}$-matrices in a block structure form for a three strands braid. In section~\ref{s3.int}, we used the properties of quantum $\hat{\mathcal R}_{2 \times 2}$ matrices, we succeeded in writing a closed form expression of HOMFLY-PT polynomial for $(\hat{W}_{3}(m,n))$. As a consequence , we showed the relation to the infinite set of Laurent polynomials called $\mathcal{V}_{n,t}[q]$ whose absolute coefficients are related to Fibonacci numbers . Section~\ref{s4.int} deals with $[r]$-colored HOMFLY-PT for weaving knots $W(3,n)$. Particularly, we could express the trace of product of 2 dimensional matrices as a Laurent polynomial. We explicitly calculate colored polynomials for weave knot up to representation $[r]= [3]$. In section~\ref{s5.int}, we verify that the reformulated invariants from these weave knot invariants indeed respect Ooguri-Vafa conjecture. The concluding section~\ref{s6.int} contains summary and related challenging open problems. There are two appendices with explicit data on colored HOMFLY-PT~\ref{app1} and reformulated invariants~\ref{app2} for $W(3,n)$.
\section{Knot invariants from quantum groups}
\label{s.int}
Recall Alexander theorem which states that any knot or link can be viewed as closure of $m$-strand braid. Hence the knot invariants can be constructed from the braid group $\mathcal B_m$ representations. The representations of the generators $\sigma_i$'s of $\mathcal B_m$: $$\mathcal R_1, \mathcal R_2, \ldots \mathcal R_i, \ldots \mathcal R_{m-1}$$ are derivable from the well-known universal $\check{\mathcal{R}}$-matrix of $U_q(\mathfrak{sl}_N)$ defined as
\begin{equation}
\check{\mathcal{R}} = q^{\sum\limits_{i,j}C_{ij}^{-1}H_i\otimes H_j}
\prod_{\textrm{positive root }\alpha} \exp_q[( 1-q^{-1}) E_\alpha\otimes F_\alpha]\,, \label {univR}
\end{equation}
where $q$ is complex number, $( C_{ij})$ is the Cartan matrix and and $\{H_i,E_i,F_i\}$ are generators of $U_q(sl_N)$. Braid group generators ${\mathcal R}_i$'s, depicted in \ref{RM}, in terms of (\ref{univR}) is
\begin{equation}
\label{Rmat}
\mathcal{R}_i = 1_{V_1}\otimes1_{V_2}\otimes\ldots\otimes P \check{\mathcal{R}}_{i,i+1} \otimes\ldots\otimes1_{V_m} \ \in \text{End}(V_1\otimes\ldots,\otimes V_m)~,
\end{equation}
where $P$ denotes the permutation operation: $P(x\otimes y) = y\otimes x$. Notice that the subscript $i, i+1$ on the universal quantum $\check{{\mathcal{R}}}$ in the above equation implies $\check{{\mathcal{R}}}$ acts only on the modules $V_i$ and $V_{i+1}$ of the $U_q(sl_N)$. The quantum $\mathcal{R}_i$ matrices discussed in \cite{KirResh}, \cite{Rosso:1993vn, lin2010hecke, Liu:2007kv} provides a braid group $\mathcal B_m$ representation. That is.,
\begin{eqnarray}
\begin{array}{rcl}
\pi: \mathcal B_m &\rightarrow& \text{End}(V_1\otimes\ldots,\otimes V_m) ~,\\
\pi(\sigma_i) &=& \mathcal{R}_i.
\end{array}
\end{eqnarray}
Graphically the braid group generator $\mathcal{R}_i$ as follows:
\begin{equation}\label{RM}
\begin{picture}(850,140)(-250,-90)
\put(-70,20){\line(0,-1){90}}
\put(-40,20){\line(0,-1){90}}
\put(0,20){\line(0,-1){30}}
\put(30,20){\line(0,-1){30}}
\put(0,-40){\line(0,-1){30}}
\put(30,-40){\line(0,-1){30}}
\put(70,20){\line(0,-1){90}}
\put(0,-10){\line(1,-1){30}}
\multiput(30,-10)(-17,-17){2}
{\line(-1,-1){13}}
\put(-105,-25){\mbox{$\mathcal{R}_i$ \ = \ }}
\put(45,-25){\mbox{$\ldots$}}
\put(-25,-25){\mbox{$\ldots$}}
\put(-75,25){\mbox{$V_1$}}
\put(-45,25){\mbox{$V_2$}}
\put(-5,25){\mbox{$V_i$}}
\put(20,25){\mbox{$V_{i+1}$}}
\put(60,25){\mbox{$V_m$}}
\end{picture}
\end{equation}
Algebriacally these generators in terms of (\ref{univR}). These operators $\mathcal{R}_i$ obeys the following relations :
\begin{eqnarray}
{\mathcal{R}}_i{{\mathcal{R}}}_j&=&{{\mathcal{R}}}_j{{\mathcal{R}}}_i~~ \text{for}~ |i- j|>1~,
\end{eqnarray}
\begin{eqnarray}\label{braidp}
{{\mathcal{R}}}_i{{\mathcal{R}}}_{i+1}{{\mathcal{R}}}_{i} &=&{ {\mathcal{R}}}_{i+1}{{\mathcal{R}}}_i{{\mathcal{R}}}_{i+1},~\text{for}~\ i=1,\ldots,m-2.
\end{eqnarray}
Graphically, the equation (b) is equivalent to the third Reidemeister move. According to Reshetikhin-Turaev approach \cite{RT1, RT2} the quantum group invariant, known as $[r]$-colored HOMFLY polynomial of the knot $\mathcal{K}$ denoted by $H_{[r]}^{\mathcal{K}}$ is defined as follows:
\begin{equation}
\label{HMF1}
H_{[
r]}^{\mathcal{K}} =
{}_q\text{tr}_{V_1\otimes\dots\otimes V_m}\left( \, \pi(\alpha_{\mathcal{K}}) \, \right),
\end{equation}
where $_q\text{tr}$ is the quantum trace ( \cite{Klimyk}) defined as follows:
\begin{equation}
\label{qtr}
_q\text{tr}_V(z) = \text{tr}_V(zK_{2\rho})~~~\forall z \in \text{End}(V),
\end{equation}
where $\vec{\rho}$ is the Weyl vector that can expressed in terms of simple roots $\vec{\alpha}_i$ is $2 \vec{\rho}=\sum_i a_i \vec{\alpha}_i$ and the $K_{2 \rho}$ is defined as
$$K_{2\rho} = K_1^{a_1} \, K_2^{a_2} \ldots K_{N-1}^{a_{N-1}}$$
where $K_p = q^{\vec{\alpha}_p . {\mathbf H}}$ having Cartan generators $H_1, H_2, \ldots
H_{N-1}$.
Note that the universal $\check R$ matrix is not diagonal and makes the computations of knot invariants very cumbersome. There is a modified RT-approach\cite{ModernRT1,Mironov:2011ym, Anokhina:2013wka} where the braiding generators can be written in a block structure form. This methodology gives a better control and simplify the computation of knot invariants. We will present the details of this modified RT method in the following section.
\subsection{ $\hat{\mathcal{R}}$-matrices with Block structure}\label{s11.int}
The modified RT approach fixes the block structure form for $\hat{\mathcal R}_i$'s from the study of the irreducible representation in the tensor product of symmetric representations $\underbrace{[r]\otimes [r]\otimes \ldots \otimes [r]}_m$:
\begin{eqnarray}\label{irrdec}
{[r]^{\bigotimes}}^{m} &=& \bigoplus_{\alpha,~\Xi_{\alpha} \vdash m |r|} ({\rm dim} {\mathcal{M}}^{1,2 \ldots m}_{\Xi_{\alpha}}) ~ \Xi_{\alpha}~,
\end{eqnarray}
where $\Xi_{\alpha}$ denote the irreducible representations labeled by index $\alpha$. The repetition in the irreducible representation called multiplicity( an irreducible representation occurs more than once) denoted by $\mathcal{M}_{\Xi_{\alpha}}^{1,2, \ldots m}$ that keep track of the subspace of the highest weight vectors\footnote{
Note that the Young diagram $\Xi_{\alpha}$ represented as $[\xi_1^\alpha,\xi_1^\alpha,\ldots,\xi_l^\alpha]$ partitioned by $\{ \xi_1^\alpha \geq \xi_2^\alpha \geq\ldots,\xi_{l-1}^\alpha \geq\xi_l^\alpha\geq 0\}$ , then the highest weights $\vec{\omega}_{\Xi_{\alpha}}$ of the corresponding representation are $\omega^{\alpha}_i= \xi_i^{\alpha}-\xi_{i+1}^{\alpha} \ \forall \, i=1,\dots,l$, and vice versa ${\xi_i^{\alpha}} = \sum_{k=i}^l \, \omega^{\alpha}_k$.} sharing same highest weights corresponding to Young diagram $\Xi_{\alpha}\vdash m |r|$\footnote{$\Xi_{\alpha} \vdash m |r|$ means a sum over all Young diagrams $\Xi_{\alpha}$ of the size equal to $m |r|$. Here,$ |r|$ is total number of boxes in the Young diagram $[r]={\tiny \underbrace{\yng(3)\ldots \yng(2)}_{r}}$}, which we indicate as $\Xi_{\alpha,\mu}$ with the index $\mu$, takes values $1,2,\ldots {\rm dim} \mathcal{M}_{\Xi_{\alpha}}^{1,2,\ldots m}$, keep track of the different highest weight vectors sharing the same highest weight $\vec {\omega}_{\Xi_{\alpha}}$.
To evaluate quantum trace(\ref{qtr}), we need to write the states in weight space incorporating the multiplicity as well. There are several paths leading to the state corresponding to the irreducible representations $\Xi_{\alpha}$. Pictorially depicted one such state in the weight space (see in (\ref{pic2})).
\begin{equation}
\hskip6cm
\begin{picture}(400,80)(-40,-32)\label{pic2}
\put(0,0){\line(-1,1){30}}
\put(-24,24){\vector(1,-1){0}}
\put(0,0){\line(1,1){30}}
\put(24,24){\vector(-1,-1){1}}
\put(-10,10){\line(1,1){20}}
\put(04,24){\vector(-1,-1){1}}
\put(-20,20){\line(1,1){10}}
\put(-10,30){\vector(-1,-1){6}}
\put(-37,35){\mbox{$[r]$}}
\put(-13,35){\mbox{$[r]$}}
\put(7,35){\mbox{$[r]$}}
\put(27,35){\mbox{$[r]$}}
\put(-27,9){\mbox{$\Lambda_{\alpha}$}}
\put(-22,-1){\mbox{$\Xi_{\alpha_1}$}}
\multiput(2,-2)(3,-3){3}{\circle*{2}}
\put(-9,-15){\mbox{$\Xi_{\alpha_2}$}}
\put(10,-10){\line(1,1){40}}
\put(44,24){\vector(-1,-1){0}}
\put(47,35){\mbox{$[r]$}}
\put(10,-10){\line(1,-1){10}}
\put(10,-10){\vector(1,-1){8}}
\put(17,-30){\mbox{$\Xi_{\alpha}$}}
\end{picture}
\end{equation}
and algebraically it can written as
\begin{equation}
\label{decomp}
\vert \left(\ldots \left(( [r] \otimes [r])_{\Lambda_{\alpha}} \otimes [r]\right)_{{\Xi}_{\alpha_1}} \ldots [r]\right)_{\Xi_{\alpha}}\rangle^{(\mu)} \equiv | \Xi_{\alpha}; \Xi_{\alpha,\mu}, \Lambda_{\alpha}\rangle
\cong~| \Xi_{\alpha,\mu},\Lambda_{\alpha}
\rangle \otimes |\Xi_{\alpha}\rangle~,
\end{equation}
where $[r]\otimes [r]=\oplus_{\alpha=0}^r \Lambda_{\alpha}\equiv [2 r-\alpha,\alpha]$. For clarity, in this paper, we denote the $\hat{\mathcal{R}}_{i}$'s- matrices corresponding to $\Xi_{\alpha}$ as $\hat{\mathcal{R}}_i^{\Xi_{\alpha}}$. Incidentally, the choice of state (\ref{decomp}) is an eigenstate of quantum
$\hat{\mathcal{ R}}^{\Xi_{\alpha}}_1$ matrix:
\begin{eqnarray}
\hat{\mathcal{R}}^{\Xi_{\alpha}}_1 \vert \Xi_{\alpha}; \Xi_{\alpha,\mu}, \Lambda_{\alpha} \rangle =
\lambda_{\Lambda_{\alpha},\mu}([r],[r])\vert \Xi_{\alpha}; \Xi_{\alpha,\mu}, \Lambda_{\alpha} \rangle \rangle\nonumber.
\end{eqnarray}
Hence we will denote the $\hat{\mathcal R}^{\Xi_{\alpha}}_1$ matrix which is diagonal in the above basis and the elements denoted by $\lambda_{\Lambda_{\alpha},\mu}([r],[r])$.
These elements are the braiding eigenvalues
whose explicit form is \cite{Klimyk,GZ}
\begin{equation}
\label{evR1}
\lambda_{\Lambda_{\alpha},\mu} ([r],[r])=
\epsilon_{\Lambda_{\alpha},\mu} q^{\varkappa(\Lambda_{\alpha})-
4\varkappa([r])-r N}~,
\end{equation}
where $\varkappa(\Lambda_{\alpha})=\tfrac{1}{2} \sum_j \alpha_j (\alpha_j+1-2j)$\footnote{The representation $\Lambda_{\alpha}$ whose Young diagram is denoted by $\alpha_1\geq \alpha_2\ldots,\geq\alpha_{N-1}$} is cut-and-join-operator eigenvalue of Young tableaux representation $\Lambda_{\alpha}$ that does not depend on the braid representation of the knot $\mathcal{K}$\cite{MMN,MMN1} and $\epsilon_{\Lambda_{\alpha},\mu}$ will be $\pm 1$.\footnote { The multiplicity subspace state $\Xi_{\alpha,\mu}$ is connected by $\Lambda_{\alpha}$ and zero otherwise.} From the eqn.(\ref{decomp},\ref{HMF1}), incorporating all the facts of $\hat{\mathcal{R}}$-matrix and the decomposition of states (\ref{decomp})\footnote{In facts, the action of $\hat{\mathcal{R}}$-matrix acts an identity operator on $|\Xi_{\alpha} \rangle$ and non-trivially on the subspace $\mathcal{M}_{\Xi_{\alpha}}^{1,2,\ldots m}$ and similarly on other way, the element $K_{2\rho}$ acts diagonally on $|\Xi_{\alpha}\rangle$ but as identity operator on subspace $\mathcal{M}_{\Xi_{\alpha}}^{1,2 \ldots m}$ as this space represent all possible highest weight vectors $\Xi_{\alpha,\mu}$ with the same weight $\vec {\omega}_{\Xi_{\alpha}}$.}, the unreduced [r]-colored HOMFLY-PT will become
\begin{eqnarray}
H_{[r]}^{\star \mathcal{K}} \left(q,\,A=q^N \right)
&=&
\text{tr}_{V_1\otimes\dots\otimes V_m}\left( \, \pi(\alpha_{\mathcal{K}}) \, K_{2\rho} \, \right) \ = \ \sum_{\alpha} \text{tr}_{\mathcal{M}_{\Xi_{\alpha}}^{1,2 \ldots m}}\left( \, \pi(\alpha_{\mathcal{K}}) \, \right) \cdot \text{tr}_{\Xi_{\alpha}}\left( \, K_{2\rho} \, \right) \nonumber\\
&=& \sum_{\alpha} \text{tr}_{\mathcal{M}_{\Xi_{\alpha}}^{1,2 \ldots m}}\left( \, \pi(\alpha_{\mathcal{K}}) \, \right) \cdot S^*_{\Xi_{\alpha}}~=\sum\limits_{\alpha, \Xi_{\alpha}\vdash m |r|} S^{*}_{\Xi_{\alpha}} C_{\Xi_{\alpha}}^{\mathcal{K}},
\label{HOMFLY}
\end{eqnarray}
where $\Xi_{\alpha}$ represent the irreducible representations in the product $[r]^{\otimes m}$, $m$ stands for number of braid strands, $[r]$ denotes the representation on each strand, $C_{\Xi_{\alpha}}$ having the trace of product of all $\hat{\mathcal{R}}$-matrices, and $S^*_{\Xi_{\alpha}}$ is the quantum dimension of the representation $\Xi_{\alpha}$ whose explicit form is given in terms of Schur polynomials \cite{Mironov:2011aa, Dhara:2018wqe}. Note that the notation $H_{[r]}^{\star \mathcal{K}}$ denote unreduced HOMFLY-PT of knot $\mathcal{K}$. The reduced [r]- colored HOMFLY-PT($H_{[r]}^{\mathcal{K}}$) is obtained by dividing the $[r]$-colored unknot invariant ($H_{[r]}^{unknot}$) i.e
\begin{eqnarray}\label{NHOM}
H_{[r]}^{\mathcal{K}}&=&\frac{H_{[r]}^{\star \mathcal{K}}}{H_{[r]}^{unknot}}=\frac{H_{[r]}^{\star \mathcal{K}}}{S^{*}_{[r]}}.
\end{eqnarray}
For clarity, we present the invariants of knots obtained from the simplest two-strand braids using this method in the following subsection.
\subsubsection{$[r]$-colored HOMFLY-PT polynomial for closure of two strand braids}
We will illustrate the $[r]$-colored HOMFLY-PT for knot $\mathcal{K}$ carrying symmetric representation $[r]$ obtained from the braid word $\sigma_1^{n}$ (\ref{2st}) where $n$ is odd integer. These knots known as torus knot $T_{(2,n)}$ and we have drawn as a example in Fig.\ref{TK}.The irreducible representation $\Lambda_{\alpha}$ in the tensor product of $[r]\bigotimes[ r]=\bigoplus_{\beta=0}^{r} \Lambda_{\alpha}=\bigoplus_{\beta=0}^{r} [2 r-\alpha,\alpha]$ has no multiplicity. Note that each irreducible representation occurs only once. So the $\hat{\mathcal R}$ are only eigenvalues and not matrices.
\begin{equation}
\hskip0cm
\begin{tikzpicture}{\label{2st}}
\braid[number of strands=2,rotate=90] (braid) a_1 a_1 a_1 a_1 a_1;
\end{tikzpicture}
\ldots n
\end{equation}
\begin{figure}[h]
\centering
\includegraphics[scale=.6]{91.jpg}
\caption{ Torus knot $T_{(2,9)}={\bf 9_{1}}$ knot}
\label{TK}
\end{figure}
Hence, using eqns.(\ref{HOMFLY}, \ref{evR1}), we can obtain colored HOMFLY-PT $H_{[r]}^{\mathcal{K}}(q,A=q^N)$ which involves the single diagonal $\hat{\mathcal{R}}$-matrix i.e $\hat{\mathcal{R}}_{1}^{n}$ whose explicit entries depicted from eqn.(\ref{evR1}).
\begin{eqnarray}{\label{egn}}
\lambda_{\Lambda_{\alpha}}([r],[r]) &=& (-1)^{\alpha}q^{(2 r^2- r (2 \alpha +1)+\alpha(\alpha-1)-rN)}.
\end{eqnarray}
The HOMFLY-PT for torus knot $T_{(2,n)}$
\begin{eqnarray}\label{HOMR}
H_{[r]}^{\mathcal{K}}(q,A)&=&\frac{\sum_{\alpha}\text{tr}_{\Lambda_{\alpha}}\hat{\mathcal{R}}^{n}S^*_{\Lambda_{\alpha}}}{ S^*_{[r]}}=\frac{1}{ S^*_{[r]}}\sum_{\alpha} \lambda_{\Lambda_{\alpha}}([r],[r])^{n} ~ S^*_{ \Lambda_{\alpha}},\nonumber\\
&=&\frac{A^{(-r n))}}{ S^*_{[r]}}\sum_{\alpha=0}^{r} (-1)^{n\alpha} q^{(2 r^2- r (2 \alpha +1)+\alpha(\alpha-1)) n}
~S^*_{ \Lambda_{\alpha}}.
\end{eqnarray}
The explicit form of quantum dimension $S^*_{ \Lambda_{\alpha}}$
\begin{equation*}
S^*_{\Lambda_{\alpha}} = \frac{[N+\alpha-2]_q!\,[N+2r-\alpha-1]_q!\,[2r-2\alpha+1]_q}{[\alpha]_q!\,[2r-\alpha+1]_q!\,[N-1]_q!\,[N-2]_q!} ,
\end{equation*}
where the factorial is defined as $[n]_{q}! = \prod_{i=1}^n [i]_q$ with $[0]_{q}!=1$ and the $q$-numbers for our computation will be given as,
\begin{equation}
[n]_{q} = \frac{q^{n} - q^{-n}}{q^{1} - q^{-1}}.
\label{q-number}
\end{equation}
The explicit polynomial form for colors $[r]=[1]$ and $[r]=[2]$ for this knot $T(2,9)={\bf 9_1}$ are
\begin{eqnarray*}
H^{{\bf 9_1}}_{[1]}(q,A)&=&A^8+\frac{A^8}{q^8}-\frac{A^{10}}{q^6}+\frac{A^8}{q^4}-\frac{A^{10}}{q^2}-A^{10} q^2+A^8 q^4-A^{10} q^6+A^8 q^8,\\
H^{{\bf 9_1}}_{[2]}(q,A)&=&(A^{16}-A^{18}+A^{20}+\frac{A^{16}}{q^{16}}-\frac{A^{18}}{q^{12}}+\frac{A^{16}}{q^{10}}-\frac{A^{18}}{q^{10}}+\frac{A^{16}}{q^8}-\frac{A^{18}}{q^6}+\frac{A^{20}}{q^6}+\frac{A^{16}}{q^4}\\&&-\frac{2
A^{18}}{q^4}+\frac{A^{16}}{q^2}-\frac{A^{18}}{q^2}+A^{16} q^2-2 A^{18} q^2+A^{20} q^2+A^{16} q^4-2 A^{18} q^4+A^{16} q^6\\&&-2 A^{18} q^6+A^{20} q^6+2
A^{16} q^8-2 A^{18} q^8+A^{20} q^8+A^{16} q^{10}-2 A^{18} q^{10}+A^{20} q^{10}\\&&+A^{16} q^{12}-3 A^{18} q^{12}+A^{20} q^{12}+2 A^{16} q^{14}-3 A^{18}
q^{14}+A^{20} q^{14}+2 A^{16} q^{16}\\&&-2 A^{18} q^{16}+A^{20} q^{16}+A^{16} q^{18}-3 A^{18} q^{18}+2 A^{20} q^{18}+2 A^{16} q^{20}-3 A^{18} q^{20}+\\&&A^{20}
q^{20}+A^{16} q^{22}-2 A^{18} q^{22}+A^{20} q^{22}+A^{16} q^{24}-2 A^{18} q^{24}+A^{20} q^{24}+A^{16} q^{26}\\&&-2 A^{18} q^{26}+A^{20} q^{26}+A^{16}
q^{28}-A^{18} q^{28}-A^{18} q^{30}+A^{20} q^{30}+A^{16} q^{32}-A^{18} q^{32}).
\end{eqnarray*}
If we go beyond two-strand braids, we need to deal with quantum $\hat{\mathcal R}_i's$ which could be matrices depending on the multiplicity sub-spaces. As our focus is on weaving knots $W(3,n)$ and their hybrid generalization, we will elaborate the steps of the modified RT method for three strands braid in the following section. Notice that, the braiding property eqn.(\ref{braidp}) means that both $\hat{\mathcal R}_1$ and $\hat{\mathcal R}_2$ cannot be simultaneously diagonal but related by a unitary matrix which can be identified with the $U_q(sl_N)$ Racah matrices.
\subsection{ $\hat{\mathcal{R}}$- matrices with Block structure for three strand braids}\label{s12.int}
For three strand braids and each strands carrying the symmetric representation,
the tensor product of representations $\underbrace{[r]\otimes [r]\otimes \ldots \otimes [r]}_m$ into the direct sum of irreducible representations($\Xi_{\alpha}$) is shown:
\begin{eqnarray*}
\label{irrdec}
\bigotimes^3 [1] &=& [3,0,0]\bigoplus [1,1,1]\bigoplus 2 [2,1,0],\\
\bigotimes^3 [2]& =& [6,0,0]\bigoplus [3,3,0]\bigoplus [4,1,1]\bigoplus 2 [5,1,0]\bigoplus 2 [3,2,1]\bigoplus 3 [4,2,0],\\
\bigotimes^3 [3]& =& [9,0,0]\bigoplus [7,1,1]\bigoplus [5,2,2]\bigoplus [4,4,1]\bigoplus [3,3,3]\bigoplus 3 [8,1,0]\bigoplus \\&&
2[4,3,2]\bigoplus 2 [6,2,1]\bigoplus 2 [5,4,0]\bigoplus 3 [7,2,0]\bigoplus 2 [5,3,1]\bigoplus 4 [6,3,0],\\
\ldots\\
\bigotimes^3 [r] &=&\sum_{\alpha}({\rm dim} {\mathcal{M}}^{1,2, 3}_{\Xi_{\alpha}}) \Xi_{\alpha},
\end{eqnarray*}
where $\Xi_{\alpha}\equiv[{\xi_1}^{\alpha},{\xi_2}^{\alpha},{\xi_3}^{\alpha}]$ is such that ${\xi_1}^{\alpha}+{\xi_2}^{\alpha}+{\xi_3}^{\alpha}=3 r$ and ${\xi_1}^{\alpha}\geq {\xi_2}^{\alpha}\geq{\xi_3}^{\alpha} \geq 0 $.
Let us discuss the path and block structure of $\hat{\mathcal{R}}$-matrix for irreducible representation [4,2,0]. Note that the multiplicity of the representation $[4,2,0]$ is equal to three which means there are three possible paths:
\begin{equation}
\begin{array}{llclclcl}
(i) & [2] & \rightarrow & [2] & \rightarrow & [4] & \rightarrow & [4,2,0] ~,\\
(ii) & [2] & \rightarrow & [2] & \rightarrow & [3,1] & \rightarrow & [4,2,0]~, \\
(iii) & [2] & \rightarrow & [2] & \rightarrow & [2,2] & \rightarrow & [4,2,0]. \\
\end{array}
\end{equation}
Let us choose $\hat{\mathcal{R}}^{[4,2,0]}_1$ to be diagonal whose entries defined by (\ref{evR1})
\begin{equation}
\label{R1}
\lambda_{[2,2]} ([2],[2])= A^{-2},~ \lambda_{[3,1]} ([2],[2])= -A^{-2} q^2,~\lambda_{[4]} ([2],[2])=A^{-2} q^6.
\end{equation}
The explicit form of $\hat{\mathcal{R}}^{[4,2,0]}_1$
\begin{equation}
\hat{\mathcal{R}}^{[4,2,0]}_1=A^{-2} \left(\begin{array}{ccc} 1& 0&0 \\ \\ 0 & -q^2&0\\ \\ 0 & 0&q^6\end{array}\right).
\end{equation}
$\hat{\mathcal{R}}_2$ is defined as
\begin{equation*}
\hat{\mathcal{R}}_2^{\Xi_{\alpha}}=\mathcal{U}^{\Xi_{\alpha}}\hat{\mathcal{R}}_1^{\Xi_{\alpha}}(\mathcal{U}^{ \Xi_{\alpha})^{\dagger}}.
\end{equation*}
Note that $\mathcal{U}^{\Xi_{\alpha}\dagger}$ denotes the conjugate-transpose of $\mathcal{U}^{\Xi_{\alpha}}$. This unitary matrix relate two equivalent basis states for irreducible representation $[4,2,0]$ as shown below in:
\begin{equation}\label{path}
\begin{picture}(200,75)(0,-35)
\put(0,0){\line(-1,1){30}}
\put(0,0){\line(1,1){30}}
\put(-15,15){\line(1,1){15}}
\put(-33,32){\mbox{$[2]$}}
\put(-3,32){\mbox{$[2]$}}
\put(27,32){\mbox{$[2]$}}
\put(-30,7){\mbox{$\Lambda_{\alpha}$}}
\put(-0,0){\line(1,-1){15}}
\put(0,-20){\mbox{$\Xi_{\alpha}=\{3[4,2,0] \}$}}
\put(58,0){\mbox{$\ \ \ \overset{\mathcal{U}^{[4,2,0]}}{\longrightarrow}$}}
\put(150,0){
\put(0,0){\line(-1,1){30}}
\put(0,0){\line(1,1){30}}
\put(15,15){\line(-1,1){15}}
\put(-33,32){\mbox{$[2]$}}
\put(-3,32){\mbox{$[2]$}}
\put(27,32){\mbox{$[2]$}}
\put(15,5){\mbox{$\Lambda_{\alpha'}$}}
\put(0,0){\line(1,-1){15}}
\put(8,-23){\mbox{$\Xi_{\alpha}=\{3[4,2,0] \}$}}
}
\end{picture}
\end{equation}
where, $\Lambda_\alpha ~\&~ \Lambda_\alpha' \in\{[4],[3,1],[2,2]\}$ and algebraically the transformation state for $\Xi_{\alpha}$ are:
\begin{equation*}
\label{asis}
|\left(\left( [r]\otimes [r]\right)_{\Lambda_{\alpha}}\otimes [r]\right)_{\Xi_{\alpha}} \rangle \xrightarrow[\text{}]{\mathcal{U}^{\Xi_{\alpha}}} \vert \left([r] \otimes \left( [r]\otimes [r] \right)_{\Lambda_{\alpha'}} \right)_{\Xi_{\alpha}}\rangle~,
\end{equation*}
where the elements of the transformation matrix $\mathcal{U}^{\Xi_{\alpha}}$ related to quantum Racah coefficients discuss in details \cite{Itoyama:2012re, Dhara:2017ukv, Dhara:2018wqe}. For completeness, Racah matrix involving $\Xi_{\alpha}\equiv [{\xi_1}^{\alpha},{\xi_2}^{\alpha},{\xi_3}^{\alpha}]$ (whose Young diagram has three rows) can be identified as $U_q(sl_2)$ Racah matrix:
\begin{eqnarray}
\label{symrac}
\mathcal{U}^{\Xi_{\alpha}\equiv [{\xi_1}^{\alpha},{\xi_2}^{\alpha},{\xi_3}^{\alpha}]}
&=&
U_{U_q(sl_2)}
\begin{bmatrix}
(r-{\xi_3}^{\alpha})/2 & (r-{\xi_3}^{\alpha})/2 \\
~&~\\
(r-{\xi_3}^{\alpha})/2 & ({\xi_1}^{\alpha}-{\xi_2}^{\alpha})/2.
\end{bmatrix}
\end{eqnarray}
The closed form expression of $U_q(sl_2)$ Racah coefficients \cite{KirResh}:
\begin{eqnarray*}
U^{U_q(sl_2)}_{j,l}
\begin{bmatrix}
j_1 &j_2\\
~&~\\
(j_3 & j_4 \end{bmatrix}&=&\sqrt{[2 j+1]_q [2 l+1]_q} (-1)^{j_1+j_2+j_3+j_4+j+l+1} \Delta(j_1, j_2, j)
\Delta(j_3,~j_4,j)\\&& \Delta(~j_4,~j_1, l) \Delta(~j_2,~j_3,l)F[j_1,j_2,j_3,j_4]~,\\
\end{eqnarray*}
where,
\begin{eqnarray*}
F[j_1,j_2,j_3,j_4]&=&\sum _{m\geq0}(-1)^m {[m+1]_q}!\{{[m-(j+~j_1+~j_2)]_q}! {[m-(j+~j_3+~j_4)]_q}! {[(m-(~j_1+~j_4+l))]_q}!\\&&{[m-(~j_2+~j_3+l)]_q}! {[(j+~j_1+~j_3+l)-m]_q}! ~{[(j+~j_2+~j_4+l)-m]_q}!\\&&~{[(~j_1+~j_2+~j_3+~j_4)-m]_q}!\}^{-1}\\
\Delta(a, b, c)&=&\sqrt{\frac{{[a-b+c]_q}!{[b-a+c]_q}!{[a+b-c]_q}!}{ {[a+b+c+1]_q}!}}~.
\end{eqnarray*}
Hence, from eqn.(\ref{symrac}) the explicit form of unitary matrix $\mathcal{U}^{[4,2,0]}$ defined as
\begin{equation}
\mathcal{U}^{[4,2,0]}=\left(
\begin{array}{ccc}
-\frac{1}{1+\frac{1}{q^2}+q^2} & -\frac{q}{\sqrt{1+q^2+q^4}} & -\frac{\sqrt{1+q^2+q^4+q^6+q^8}}{1+q^2+q^4} \\
-\frac{q}{\sqrt{1+q^2+q^4}} & -1+\frac{q^2}{1+q^4} & \frac{q^4 \sqrt{\left(1+\frac{1}{q^2}+q^2\right) \left(1+\frac{1}{q^4}+\frac{1}{q^2}+q^2+q^4\right)}}{\left(1+q^4\right)
\left(1+q^2+q^4\right)} \\
-\frac{\sqrt{1+q^2+q^4+q^6+q^8}}{1+q^2+q^4} & \frac{q^4 \sqrt{\left(1+\frac{1}{q^2}+q^2\right) \left(1+\frac{1}{q^4}+\frac{1}{q^2}+q^2+q^4\right)}}{\left(1+q^4\right)
\left(1+q^2+q^4\right)} & -\frac{q^4}{1+q^2+2 q^4+q^6+q^8} \\
\end{array}
\right).
\end{equation}
Hence, the $\hat{\mathcal{R}}_{2}$ matrix for $[4,2,0]$ is
$$\hat{\mathcal{R}}^{[4,2,0]}_2=\mathcal{U}^{[4,2,0]} \hat{\mathcal{R}}^{[4,2,0]}_1\mathcal{U}^{[4,2,0]\dagger}.$$
Explicit form of quantum $\hat{\mathcal{R}}_i$'s and $\mathcal{U}$ can be similarly worked out for other irreducible representations to compute $[r]$-colored HOMFLY-PT for hybrid weaving knots.
\section{Hybrid weaving knot $\hat{W}_3(m, n)$}\label{s3.int}
In this section, we discuss the hybrid weaving knot obtained from closure of three-strand braid whose braid word is
$$(\sigma_1^{m} \sigma_2^{-m})^n$$ which is pictorially seen in (\ref{GWK}). Note that the subscript 3 in $\hat{W}_3(m, n)$ indicates three-strand braid.
\begin{equation}
\begin{picture}(350,100)(-20,-50)\label{GWK}
\put(40,30){\vector(1,0){35}}
\put(40,0){\vector(1,0){55}}
\put(40,-30){\vector(1,0){35}}
\put(75,-30){\vector(1,0){80}}
\put(75,30){\vector(1,0){40}}
\put(95,0){\vector(1,0){20}}
\qbezier(115,30)(120,30)(120,25)
\qbezier(115,0)(120,0)(120,5)
\put(115,25){\line(1,0){20}}
\put(115,5){\line(0,1){20}}
\put(120,12){\mbox{$m$}}
\put(115,5){\line(1,0){20}}
\put(135,5){\line(0,1){20}}
\qbezier(130,25)(130,30)(135,30)
\qbezier(130,5)(130,0)(135,0)
\put(135,30){\vector(1,0){20}}
\put(135,0){\vector(1,0){20}}
\qbezier(155,-30)(160,-30)(160,-25)
\qbezier(155,0)(160,0)(160,-5)
\put(155,-25){\line(1,0){20}}
\put(155,-5){\line(0,-1){20}}
\put(157,-17){\mbox{$-m$}}
\put(155,-5){\line(1,0){20}}
\put(175,-5){\line(0,-1){20}}
\qbezier(170,-25)(170,-30)(175,-30)
\qbezier(170,-5)(170,0)(175,0)
\put(155,30){\line(1,0){40}}
\put(175,-30){\vector(1,0){60}}
\put(175,0){\vector(1,0){20}}
\qbezier(195,30)(200,30)(200,25)
\qbezier(195,0)(200,0)(200,5)
\put(195,25){\line(1,0){20}}
\put(195,5){\line(0,1){20}}
\put(200,12){\mbox{$m$}}
\put(195,5){\line(1,0){20}}
\put(215,5){\line(0,1){20}}
\qbezier(210,25)(210,30)(215,30)
\qbezier(210,5)(210,0)(215,0)
\put(215,30){\vector(1,0){60}}
\put(215,0){\vector(1,0){20}}
\qbezier(235,-30)(240,-30)(240,-25)
\qbezier(235,0)(240,0)(240,-5)
\put(235,-25){\line(1,0){20}}
\put(235,-5){\line(0,-1){20}}
\put(236,-17){\mbox{$-m$}}
\put(235,-5){\line(1,0){20}}
\put(255,-5){\line(0,-1){20}}
\qbezier(250,-25)(250,-30)(265,-30)
\qbezier(250,-5)(250,0)(255,0)
\put(255,-30){\vector(1,0){50}}
\put(255,0
){\vector(1,0){20}}
\qbezier(275,30)(280,30)(280,25)
\qbezier(275,0)(280,0)(280,5)
\put(275,25){\line(1,0){20}}
\put(275,5){\line(0,1){20}}
\put(280,12){\mbox{$m$}}
\put(330,0){\mbox{$\ldots n$}}
\put(275,5){\line(1,0){20}}
\put(295,5){\line(0,1){20}}
\qbezier(290,25)(290,30)(295,30)
\qbezier(290,5)(290,0)(295,0)
\put(295,-30){\vector(1,0){30}}
\put(295,0
){\vector(1,0){20}}
\put(295,30
){\vector(1,0){20}}
\end{picture}
\end{equation}
The classification of knots belongs to the hybrid weaving knot $\hat{W}_3(m,n)$ are tabulated below for some values of $m$ and $n$:
\begin{table}
\begin{center}
\begin{tabular} { | p {5 cm} | p {5 cm}| }
\hline
Notation & Knot \\
\hline
$\hat{W}_3(1,n)$& weaving knot of type $W(3,n)$ \\
$\hat{W}_3(m,1)$& $T_{(2,m)}\#T^*_{(2,m)}$ \\
$\hat{W}_3(3,2)$ & $12a1288$ Knot \\
\hline
\end{tabular}
\caption{ The classification of hybrid weaving knot $\hat{W}_3(m,n)$ }
\label{table1}
\end{center}
\end{table}
where $m$ is odd and $m\neq n>1$. When $m=1$,$\hat{W}_3(1, n)$ reduces to the weaving knot $W(3,n)$ discussed in \cite{mishra2017jones, RRV}. Well known examples of weaving knots(see in Fig.\ref{HKD}) are $$ W(3,2)={\bf 4_1}, W(3,4)={\bf 8_{18}} ~\text{ and}~ W(3,5)={\bf 10_{123}}.$$ For $m>3$ and $n\geq 2$, the crossing number exceeds 20 whose data are not available in the knot theory literature to validate.
\begin{figure}[h]
\centering
\includegraphics[scale=.6]{41.jpg}
\includegraphics[scale=.6]{818.jpg}
\includegraphics[scale=.6]{10123.jpg}
\includegraphics[scale=.6]{12a1288.jpg}
\caption{Snappy diagram representation for hybrid knots\cite{katlas, indiana}: (a) $\hat{W}_3(2,1)={\bf 4_{1}}$, (b) $\hat{W}_3(4,1)={\bf 8_{18}}$ knot, (c) $\hat{W}_3(5,1)={\bf 10_{123}}$ knot, and (d) $\hat{W}_3(3,2)={\bf 12a1288}$ knot}
\label{HKD}
\end{figure}
Now we will elaborate the modified RT method for hybrid weaving knots and achieve a closed form expression for their HOMFLY-PT polynomial.
\subsection{HOMFLY-PT for hybrid weaving knot $\hat{W}_3(m,n)$}
In this case, tensor product of fundamental representation of three strand braid: $$[1]^{\bigotimes 3}=[3]\bigoplus[1,1,1]\bigoplus 2 [2,1,0]$$ shows that representation $[2,1,0]$ has multiplicity two. Incorporating $2\times 2$ matrix form ${\hat{\mathcal{R}}}_1$ and ${\hat{\mathcal{R}}}_2$ for representation $[2,1,0]$ in eqn.(\ref{HOMFLY}), the HOMFLY-PT for $\hat{W}_3(m,n)$ is
\begin{eqnarray}
\mathcal{H}_{[1]}^{\hat{W}_3(m,n)} &=& \frac{1}{S^*_{[1]}}\sum_{\Xi_{\alpha}=\{[3],[1,1,1],[2,1,0]\}}S^*_{\Xi_{\alpha}}{\rm Tr}_{\Xi_{\alpha}} (\hat{\mathcal{R}}^{\Xi_{\alpha}}_1)^{m}(\hat{\mathcal{R}}^{\Xi_{\alpha}}_2)^{-m}\ldots (\hat{\mathcal{R}}^{\Xi_{\alpha}}_1)^{m}(\hat{\mathcal{R}}^{\Xi_{\alpha}}_2)^{-m}\nonumber\\
&=& \frac{1}{S^*_{[1]}} (S^*_{[3]}+ S^*_{[1,1,1]}+S^*_{[2,1]} {\rm Tr}_{[2,1,0]} (\hat{\mathcal{R}}^{[2,1,0]}_1)^{m}(\hat{\mathcal{R}}^{[2,1,0]}_2)^{-m})^n{\label{HW}}~.
\end{eqnarray}
Here $S^*_{[1]}=[N]_q$, $S^*_{[3]}=\frac{[N]_{q}[N+1]_{q}[N+2]_{q}}{[2]_{q}[3]_{q}}$, $S^*_{[111]}=\frac{[N]_{q}[N-1]_{q}[N-2]_{q}}{[2]_{q}[3]_{q}}$, and $S^*_{[21]}=\frac{[N]_{q}[N+1]_{q}[N-1]_{q}}{[3]_{q}}$.\\
In order to apply the formula (\ref{HW}) to evaluate the HOMFLYPT polynomial for $\hat{W}_3(m,n)$ we need to compute the trace of the matrix $\Psi^{[2,1,0]}[m,n]=((\hat{\mathcal{R}}^{[2,1,0]}_1)^{m}(\hat{\mathcal{R}}^{[2,1,0]}_2)^{-m}))^n$.
Using eqn(\ref{egn}) and eqn.(\ref{symrac}), we have,
\begin{eqnarray*}
{\hat{\mathcal{R}}}_1&=& A^{-1}\left(\begin{array}{cc} q & 0 \\ \\ 0 & -\frac{1}{q} \end{array}\right)~\text{and}~~ \hat{\mathcal{R}}_{2}
=A^{-1}\left(\begin{array}{cc} \frac{q^2 - [3]_q}{q {[2]_q}^2} &- \frac{\sqrt{[3]_q}}{[2]_{q}}\\ \\ - \frac{\sqrt{[3]_q}}{[2]_{q}}& \frac{1-q^2 [3]_q}{q {[2]_q}^2} \end{array}\right).\end{eqnarray*}
Thus
\begin{eqnarray*}
\hat{\mathcal{R}}^m_{1}{\hat{\mathcal{R}}_2}^{-m}
=\left(\begin{array}{cc} \frac{1-q^{2m}[3]_q}{([2]_q)^2} & -\frac{(1+q^{2 m})\sqrt{[3]_q}}{([2]_q)^2}\\ \\ \frac{(1+q^{2m})\sqrt{[3]_q}}{q^{2m}([2]_q)^2} & \frac{1-q^{-2m}[3]_q}{( [2]_q)^2} \end{array}\right)= \left(\begin{array}{cc} x_1 & -x_2\\ \\ \frac{x_2}{q^{2m}} & x_3 \end{array}\right),\end{eqnarray*}~
where $x_1=\frac{1-q^{2m}[3]_q}{([2]_q)^2}$, $x_2=\frac{(1+q^{2m})\sqrt{[3]_q}}{([2]_q)^2}$, and $x_3=\frac{1-q^{-2m}[3]_q}{( [2]_q)^2}.$
Interestingly, we have succeed in the writing of diagonal entries of the $nth$ power of the above matrix $((\hat{\mathcal{R}}^{[2,1,0]}_1)^{m}(\hat{\mathcal{R}}^{[2,1,0]}_2)^{-m})^n$ in a compact form $\Psi^{[2,1,0]}_{1}[m,n]$ and $\Psi^{[2,1,0]}_{2}[m,n]$ i.e
\begin{eqnarray}{\label{traceterm}}
\Psi^{[2,1,0]}_{1}[m,n]&=&x_1^{n}+\sum_{i=1
}^{\floor{\frac{n}{2}}}\sum_{k=1}^{n-i}(-1)^i \binom{k+i-2}{i-1} \binom{n-(k+i-1)}{i}x_1^{n-(2i+k-1)} x_3^{k-1}(\frac{x_2}{q^m})^{2i}~,\nonumber\\
\Psi^{[2,1,0]}_{2}[m,n]&=&x_3^{n}+\sum_{i=1
}^{\floor{\frac{n}{2}}}\sum_{k=1}^{n-i}(-1)^i \binom{k+i-2}{i-1} \binom{n-(k+i-1)}{i}x_3^{n-(2i+k-1)} x_1^{k-1}(\frac{x_2}{q^m})^{2i}\nonumber\\.
\end{eqnarray}
Hence the trace of the matrix $\Psi^{[2,1,0]}[m,n]$
\begin{eqnarray}{\label{tt}}
\Psi^{[2,1,0]}[m,n]=\Psi^{[2,1,0]}_{1}[m,n]+\Psi^{[2,1,0]}_{2}[m,n].
\end{eqnarray}
Using these binomial series for the trace, the closed form expression for HOMFLY-PT for hybrid weaving knot turns out to be
\begin{eqnarray}{\label{HWK}}
\mathcal{H}_{[1]}^{\hat{W}_3 (m,n)}&=& \frac{1}{S^*_{[1]}}(S^*_{[3]} + S^*_{[111]} + S^*_{[21]}\cdot( x_1^{n}+x_3^{n}+\sum_{i=1
}^{\floor{\frac{n}{2}}}\sum_{k=1}^{n-i}(-1)^i \binom{k+i-2}{i-1} \binom{n-(k+i-1)}{i}\nonumber\\&&(x_1^{(n+1-2i-k)} x_3^{(k-1)}+x_1^{(k-1)} x_3^{(n-2i-k+1)}))(\frac{x_2}{q^m})^{2i})).
\end{eqnarray}
The closed form expression is an important result providing a useful starting point to investigate $[r]$-colored HOMFLY-PT, knot-quiver correspondence for hybrid weaving knots which we will pursue in future.
Incidentally for $m=1$, $\Psi^{[2,1,0]}[m,n]$ in eqn(\ref{tt}) is a Laurent polynomial \cite{RRV} giving closed
form HOMFLY-PT for weaving knots $W(3,n)$. We propose such a Laurent polynomial structure will be seen for all the multiplicity two irreducible representation $\Xi_{\alpha}\in [r]^{\otimes 3}$ for symmetric colors $[r]>1$ as well.
\noindent
\textbf{Proposition 1.} Given a representation $\Xi_{\alpha}\equiv[{\xi_1}^{\alpha},{\xi_2}^{\alpha},{\xi_3}^{\alpha}]$ having multiplicity 2 with $\hat{\mathcal{R}}^{\Xi_{\alpha}}_1=\pm q^{m_1}A^{-r} \left(\begin{array}{cc} q^t & 0 \\ \\ 0 & -\frac{1}{q^t} \end{array}\right)$, ~and~ $\mathcal{U}^{\Xi_{\alpha}}=\left(\begin{array}{cc}
\frac{1}{[2]_{q^t}} & \frac{\sqrt{[3]_{q^t}}}{ [2]_{q^t}} \\
\frac{\sqrt{[3]_{q^t}}}{[2]_{q^t}} & -\frac{1}{[2]_{q^t}}
\end{array}\right)\ \ $, the Laurent polynomial $\mathcal{V}_{n,t}[q]$ is defined as
\begin{eqnarray}{\label{conj}}
\mathcal{V}_{n,t}[q]&=&{\rm Tr}(\hat{\mathcal{R}}_1^{\Xi_{\alpha}} \mathcal{U}^{\Xi_{\alpha}}(\hat{\mathcal{R}}_1^{\Xi_{\alpha}})^{-1}(\mathcal{U}^{\Xi_{\alpha}})^{\dagger})^n=\sum_{g=-n}^{n} (-1)^{g} \mathcal{S}_{n,n-\abs{g}}q^{2 g t}.
\end{eqnarray}
Here $t$ and $m_1$ are also an integer dependent on $\Xi_{\alpha}$ and the coefficients $\mathcal{S}_{n,j}$ are:
\begin{eqnarray*}
\mathcal{S}_{n,j}&=&\sum_{i=0}^{\floor{\frac{j}{2}}}\frac{n}{n-i}\binom{n-i}{n-j+ i} \binom{j-i-1}{i}~,
\end{eqnarray*}
where the parameters $n~\& j $ are positive integers and $|x|$ denote the absolute value of $x$ and $\floor{x}$ indicate the greatest integer $x$. For $m=1$ and fundamental representation $[r]=[1]$, the trace in eqn.(\ref{tt}) is
\begin{eqnarray*}
\Psi^{[2,1,0]}[1,n]=\mathcal{V}_{n,1}[q],
\end{eqnarray*}
exactly matching with the parallel work\cite{RRV}.
Further, we conjecture the sum of the absolute coefficient $\mathcal{S}_{n,n-|\delta|}$ given by $ \mathcal{O}_n$, satisfy the beautiful relation.
\vskip.1cm
\noindent
\textbf{Conjecture 1}:
\begin{eqnarray}{\label{febona}}
\mathcal{O}_n&=& \sum^{n}_{\delta=-n}\mathcal{S}_{n,n-\delta}=5\mathcal{F}_n^2+2 (-1)^{n}~,
\end{eqnarray}
\noindent
where $\mathcal{F}_n$ denotes Fibonacci numbers. The explicit form of $\mathcal{F}_{n}$ is given by \cite{pelitifibonacci}
\begin{eqnarray*}
\mathcal{F}_{n}&=&\frac{1}{\sqrt{5}}(\phi^{n}-\cos{(n \pi)}(\phi)^{-n})~.\\
\end{eqnarray*}
Here $\phi \approx 1.618$ is the golden ratio.
\noindent
We have checked this conjecture for large values of $n$. For values of $n\leq 8$, we have
presented the values of $\mathcal O_n, \mathcal F_n$ in Table.\ref{table2}. For completeness, we will briefly discuss the Fibonacci numbers and its properties.
The Fibonacci ($\mathcal{F}_n$) numbers are sequences satisfying the Fibonacci
recursion relation
\begin{eqnarray*}
\mathcal{F}_{n+1}=\mathcal{F}_n+\mathcal{F}_{n-1},
\end{eqnarray*}
with following initial conditions : $\mathcal{F}_0=0, \mathcal{F}_1=1$. Here $n$ is integer and it satisfy yhe following relation
\begin{eqnarray*}
\mathcal{F}_{-n}&=&(-1)^{n+1}\mathcal{F}_n.
\end{eqnarray*}
\begin{table}
\centering
\small{
\begin{tabular} { | p {1 cm}| p {1 cm}| p {1 cm}| p {1 cm}| p {1 cm}| p {1 cm}| p {1 cm}| p {1 cm}| p {1 cm}|}
\hline
$n$ &1& 2& 3& 4& 5& 6& 7& 8\\
\hline
$\mathcal{F}_n$ &1&1& 2& 3& 5& 8& 13& 21\\
\hline
$\mathcal{O}_n$ &3&7& 18& 47& 123& 322& 843& 2207\\
\hline
\end{tabular}}
\caption{ $\mathcal O_n$ and $\mathcal F_n$ for $ n\leq 8$ }
\label{table2}
\end{table}
\subsection{Examples}
For the hybrid weaving knots in Table.\ref{table1}, HOMFLY-PT are obtained using our closed form expression for $\hat W_3(m,n)$.\\
$\bullet$ (a) For $m=1$, the HOMFLY-PT polynomial is for weaving knots $W(3,n)$:
\begin{eqnarray}{\label{WEAVING}}
\mathcal{H}_{[1]}^{W(3,n)} (A,q)= \frac{1}{S^*_{[1]}}(S^*_{[3]} + S^*_{[111]} + S^*_{[21]} \mathcal{V}_{n,1}[q]).
\end{eqnarray}
Substituting $A=q^2$, we get the Jones polynomial:
\begin{eqnarray*}
\mathcal{J}^{W(3,n)}(q)=q^{-2}+q^2+\mathcal{V}_{n,1}[q]~.
\end{eqnarray*}
These results agree with the results in the parallel paper on weaving knots\cite{RRV}.\\
$\bullet$ (b) composite knot $T_{(2, m)}\#T^*_{(2, m)}$\\
For odd $m\geq 2$ and $n=1$, the knot belongs to composite knot of type $T_{(2, m)}\#T^*_{(2, m)}$\footnote{$T^*_{(2, m)}$ is the mirror of torus knot $T_{(2, m)}$ } . Hence, the HOMFLY-PT will be
\begin{eqnarray*}
H_{[1]}^{\hat{W}_3(m, 1)}&=&H_{[1]}^{T_{(2, m)}}(q,A) H_{[1]}^{T^*_{(2,m)}}(q,A))\\&&=
\frac{q^{2 - 2 m} (-1 + A^2 q^2 - A^2 q^{2 m} + q^{2 + 2 m}) (-A^2 + q^2 -
q^{2 m} + A^2 q^{2 + 2 m})}{A^2 (-1 + q)^2 (1 + q)^2 (1 + q^2)^2}.
\end{eqnarray*}
\begin{table}[h]
\begin{center}
\begin{tabular} { | p {2 cm} | p {2cm}| p {8cm}|}
\hline
m &KNOT& $H_{[1]}^{\hat{W}_3(m, 1)}(q,A)$ \\
\hline
3 & $3_1\# 3^*_1$ & $(A^{-2} q^{-4})(1 - A^2 q^2 + q^4) (A^2 - q^2 + A^2 q^4)$ \\
\hline
9 &$9_1\# 9^*_1$&$ A^{-2} q^{-16}(1 - A^2 q^2 + q^4 - A^2 q^6 + q^8 - A^2 q^{10} + q^{12} - A^2 q^{14} +
q^{16})(A^2 - q^2 + A^2 q^4 - q^6 + A^2 q^8 - q^{10} + A^2 q^{12} -
q^{14} + A^2 q^{16})$\\
\hline
\end{tabular}
\caption{HOMFLY-PT for $\hat{W}_3(m, 1)=T_{(2,m)}\# T^*_{(2,m)}$}\label{table3}
\end{center}
\end{table}
$\bullet$ (c) The $m=3$ and $n=2$ refers to a $12$ crossing knot $"12a1288"$ in the Rolfsen table whose HOMFLY-PT polynomial is\\
\begin{eqnarray*}
H_{[1]}^{W_3(3, 2)}&=&11+\frac{7}{A^2}+7 A^2-\frac{1}{q^{10}}+\frac{1}{q^8}+\frac{1}{A^2 q^8}+\frac{A^2}{q^8}-\frac{5}{q^6}-\frac{1}{A^2 q^6}-\frac{A^2}{q^6}+\frac{6}{q^4}\\&&+\frac{4}{A^2
q^4}+\frac{4 A^2}{q^4}-\frac{11}{q^2}-\frac{5}{A^2 q^2}-\frac{5 A^2}{q^2}-11 q^2-\frac{5 q^2}{A^2}-5 A^2 q^2+6 q^4+\frac{4 q^4}{A^2}\\&&+4 A^2 q^4-5
q^6-\frac{q^6}{A^2}-A^2 q^6+
q^8+\frac{q^8}{A^2}+A^2 q^8-q^{10}.
\end{eqnarray*}
In the following section, we will present $[r]$-colored HOMFLYPT for $W(3,n)$ for $[r]=2,3$
and verify our proposition 1.
\section{Colored HOMFLY-PT for weaving knot type $W(3,n)$}\label{s4.int}
We will use the data on $U^{\Xi_{\alpha}}$ matrices in section~\ref{s12.int} for three-strand braid where $\Xi_{\alpha} \in [2]^3$ and $\Xi_{\alpha} \in [3]^3 $ to compute colored HOMFLY-PT for the weaving knots.
\subsection{Representation $[r]=[2]$\label{se.symrac1}}
In this case, $\bigotimes^3 [2] = [6,0,0]\bigoplus [3,3,0]\bigoplus [4,1,1]\bigoplus 2 [5,1,0]\bigoplus 2 [3,2,1]\bigoplus 3 [4,2,0]$.\\
From the multiplicity, we can see that there one $3\times 3$ matrix, two $2\times 2$ matrices, three $1\times 1$ matrices as shown in the Table.{\ref{table4}}.
{\small \begin{table}[h]
\begin{center}
\begin{tabular}{ |p{7cm}|p{2cm}|p{2cm}| }
\hline
\hspace{3cm}$\Xi_{\alpha} \in [2]^3$ & Matrix size & $\# $ of matrices \\
\hline
[6,0,0], [4,1,1], [3,3]& 1 &3 \\
\hline
[5,1,0] ,[3,2,1]&2 & 2 \\
\hline
[4,2,0]& 1 & 3 \\
\hline
\end{tabular}
\caption{The multiplicity table for $\Xi_{\alpha} \in [2]^3$ }\label{table4}
\end{center}
\end{table}
}
Also the path and the block structure of $\Xi \in [2]^{\otimes 3}$ is shown(\ref{2stand})
\begin{equation}{\label{2stand}}
\definecolor{ffqqtt}{rgb}{1.,0.,0.2}
\definecolor{ffqqqq}{rgb}{1.,0.,0.}
\definecolor{sqsqsq}{rgb}{0.12549019607843137,0.12549019607843137,0.12549019607843137}
\definecolor{qqzzff}{rgb}{0.,0.6,1.}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=.8
0cm,y=1cm,scale=.23]
\draw (9.335336945031727,14.0
49996666857155) node[anchor=north west] {$\mathbf{[4,0]}$};
\draw (23.551228417899944,14.496702093213744) node[anchor=north west] {$\mathbf{[3,1]}$};
\draw (38.50723616983793,14.496702093213744) node[anchor=north west] {$\mathbf{[2,2]}$};
\draw [color=sqsqsq](23.35180981324878,20.072864883911418) node[anchor=north west] {\textbf{[2]$\otimes$[2]}};
\draw [->,line width=1.pt,color=sqsqsq] (25.37790697674419,18.147653467420902) -- (10.562015503875973,14.150560444165093);
\draw [->,line width=1.pt,color=sqsqsq] (25.37790697674419,18.147653467420902) -- (25.377906976744196,14.257149591451917);
\draw (39.12700361169839,8.054647829647852) node[anchor=north west] {$\mathbf{[4,2]}$};
\draw (24.684174154334052,8.061236976934673) node[anchor=north west] {$\mathbf{[5,1]}$};
\draw (9.068864076814673,8.4600771860655603) node[anchor=north west] {$\mathbf{[6,0]}$};
\draw [color=sqsqsq](22.938050123326303,8.08640077709867) node[anchor=north west] {\textbf{2}};
\draw [color=sqsqsq](37.27429043340382,8.042050930423045) node[anchor=north west] {\textbf{3}};
\draw [->,line width=1.pt,color=sqsqsq] (25.37790697674419,18.09435889377752) -- (39.980620155038764,14.363738738738757);
\draw [->,line width=1.pt,color=sqsqsq] (25.377906976744185,12.76490152943643) -- (25.324612403100776,8.128273622459684);
\draw [->,line width=1.pt,color=sqsqsq] (11.041666666666666,12.445134087575966) -- (25.324612403100776,8.128273622459684);
\draw [->,line width=1.pt,color=sqsqsq] (10.455426356589147,12.338544940289143) -- (10.508720930232558,8.607924785250383);
\draw [->,line width=1.pt,color=sqsqsq] (11.319909418973037,12.36103833034904) -- (40.30038759689923,7.968389901529451);
\draw [->,line width=1.pt,color=sqsqsq] (26.017441860465116,12.71160695579302) -- (40.30038759689923,7.968389901529451);
\draw [->,line width=1.pt,color=sqsqsq] (40.35368217054264,12.65831238214961) -- (40.30038759689923,7.968389901529451);
\end{tikzpicture}
\end{equation}
The eigenvalues and $U^{\Xi_{\alpha}}$ matrices in this case are
\begin{equation}\label{u0}
\hat{\mathcal{R}}^{[5,1,0]}_1=A^{-2}q^4\left(
\begin{array}{cc}
q^2 & 0 \\
0 & -q^{-2} \\
\end{array}
\right)\\,
\hat{\mathcal{R}}^{[3,2,1]}_1=A^{-2}q\left(
\begin{array}{cc}
q^{-1} & 0 \\
0 & -q \\
\end{array}
\right)\\,~~
\hat{\mathcal{R}}^{[4,2,0]}_1=A^{-2}\left(
\begin{array}{ccc}
1 & 0 & 0 \\
0 & -q^2 & 0 \\
0 & 0 & q^6 \\
\end{array}
\right)
\end{equation}
\begin{equation}\label{u1}
\mathcal{U}^{[5,1,0]}=\left(\begin{array}{cc}
\frac{1}{[2]_{q^2}} & \frac{\sqrt{[3]_{q^2}}}{[2]_{q^2}} \\
\frac{\sqrt{[3]_{q^2}}}{[2]_{q^2}} & -\frac{1}{[2]_{q^2}}
\end{array}\right),\ \
~\mathcal{U}^{[3,2,1]}=\left(\begin{array}{cc}
\frac{1}{[2]_{q}} & \frac{\sqrt{[3]_{q}}}{[2]_{q}} \\
\frac{\sqrt{[3]_{q}}}{[2]_{q}} & -\frac{1}{[2]_{q}}
\end{array}\right),\ \
\end{equation}
\begin{equation}{\label{u2}}
\mathcal{U}^{[4,2,0]}=\left(
\begin{array}{ccc}
-\frac{1}{1+\frac{1}{q^2}+q^2} & -\frac{q}{\sqrt{1+q^2+q^4}} & -\frac{\sqrt{1+q^2+q^4+q^6+q^8}}{1+q^2+q^4} \\
-\frac{q}{\sqrt{1+q^2+q^4}} & -1+\frac{q^2}{1+q^4} & \frac{q^4 \sqrt{\left(1+\frac{1}{q^2}+q^2\right) \left(1+\frac{1}{q^4}+\frac{1}{q^2}+q^2+q^4\right)}}{\left(1+q^4\right)
\left(1+q^2+q^4\right)} \\
-\frac{\sqrt{1+q^2+q^4+q^6+q^8}}{1+q^2+q^4} & \frac{q^4 \sqrt{\left(1+\frac{1}{q^2}+q^2\right) \left(1+\frac{1}{q^4}+\frac{1}{q^2}+q^2+q^4\right)}}{\left(1+q^4\right)
\left(1+q^2+q^4\right)} & -\frac{q^4}{1+q^2+2 q^4+q^6+q^8} \\
\end{array}
\right)
\end{equation}
From eqn.(\ref{HOMFLY}), $[2]$-HOMFLY-PT for $W(3,n)$:
\begin{eqnarray}
\mathcal{H}_{[2]}^{W(3,n)} &=& \frac{1}{S^*_{[2]}}\sum_{\alpha} S^*_{\Xi_{\alpha}} {\rm Tr}_{\Xi_{\alpha}} (\hat{\mathcal{R}}^{\Xi_{\alpha}}_1)(\hat{\mathcal{R}}^{\Xi_{\alpha}}_2)^{-1}\ldots (\hat{\mathcal{R}}^{\Xi_{\alpha}}_1)(\hat{\mathcal{R}}^{\Xi_{\alpha}}_2)^{-1}\nonumber\\
&=& \frac{1}{S^*_{[2]}}\sum_{\alpha}S^*_{\Xi_{\alpha}} {\rm Tr}_{\Xi_{\alpha}} (\hat{\mathcal{R}}^{\Xi_{\alpha}}_1)(\hat{\mathcal{R}}^{\Xi_{\alpha}}_2)^{-1})^n\nonumber\\&=& \frac{1}{S^*_{[2]}} (S^*_{[6]}+ S^*_{[3,3]}+S^*_{[4,1,1]} +S^*_{[5,1,0]}{\rm Tr}_{[5,1,0]} (\hat{\mathcal{R}}^{[5,1,0]})(\hat{\mathcal{R}}^{[5,1,0]}_2)^{-1})^n+S^*_{[3,2,1]}*\nonumber\\&&{\rm Tr}_{[3,2,1]} (\hat{\mathcal{R}}^{[3,2,1]})(\hat{\mathcal{R}}^{[3,2,1]}_2)^{-1})^n+S^*_{[4,2,0]}{\rm Tr}_{[4,2,0]} (\hat{\mathcal{R}}^{[4,2,0]})(\hat{\mathcal{R}}^{[4,2,0]}_2)^{-1})^n{\label{HW22}}.
\end{eqnarray}
Using eqns.$(\ref{u0})$ to $(\ref{u2})$, and $(\ref{conj})$, we can rewrite the equation~(\ref{HW22}) into neat formula
\begin{eqnarray}
\mathcal{H}_{[2]}^{W{(3,n)}} &=& \frac{1}{S^*_{[2]}} (S^*_{[6]}+ S^*_{[3,3]}+S^*_{[4,1,1]} +S^*_{[5,1,0]} \mathcal{V}_{n,2}[q]+S^*_{[3,2,1]}\mathcal{V}_{n,1}[q]+S^*_{[4,2,0]}{\rm Tr} (X^{[4,2,0]})^n\nonumber{\label{HW2}}~,
\end{eqnarray}
where,
\begin{equation}
X^{[4,2,0]}=\left(
\begin{array}{ccc}
\frac{1}{q^6+q^8+q^{10}} & -\frac{1}{q^5 \sqrt{1+q^2+q^4}} & \frac{\sqrt{1+q^2+q^4+q^6+q^8}}{q^2+q^4+q^6} \\
\frac{1}{q^3 \sqrt{1+q^2+q^4}} & \frac{-1+q^2-q^4}{q^2+q^6} & -\frac{q^3 \sqrt{\frac{1+q^2+q^4+q^6+q^8}{1+q^2+q^4}}}{1+q^4} \\
\frac{q^4 \sqrt{1+q^2+q^4+q^6+q^8}}{1+q^2+q^4} & \frac{q^7 \sqrt{\frac{1+q^2+q^4+q^6+q^8}{1+q^2+q^4}}}{1+q^4} & \frac{q^{14}}{1+q^2+2 q^4+q^6+q^8}
\\
\end{array}
\right).
\end{equation}
Using eqn.(\ref{conj}), the [2]-colored reduced HOMFLY-PT polynomials for $W(3,n)$.
We would like to emphasize that the polynomial form of this algebraic expression for arbitrary $n$ is easily computable. We have listed $[2]$ colored HOMFLY-PT in Appendix~\ref{app1} for some weaving knots.
\subsection{Representation [3]\label{t.symrac1}}
In this case, $\bigotimes^3 [3] = [9,0,0]\bigoplus [7,1,1]\bigoplus [5,2,2]\bigoplus [4,4,1]\bigoplus [3,3,3]\bigoplus 3 [8,1,0]\bigoplus \\ 2[4,3,2]\bigoplus 2 [6,2,1]\bigoplus 2 [5,4,0]\bigoplus 3 [7,2,0]\bigoplus 2 [5,3,1]\bigoplus 4 [6,3,0],\\$
Thus, there are two $3\times 3$ matrices, four $2\times 2$ matrices, five $1\times 1$ matrices and one $4\times 4$ matrix tabulated below.
{\small
\begin{table}[h]
\begin{center}
\begin{tabular}{ |p{7cm}|p{2cm}|p{2cm}| }
\hline
\hspace{3cm}$\Xi_{\alpha} \in [3]^3 $ & Matrix size & $\# $ of matrices \\
\hline
[9,0,0], [7,1,1], [5,2,2],[4,4,1],[3,3,3]& 1 &5 \\
\hline
[4,3,2],[6,2,1], [5,4,0],[5,3,1]& 2 & 4 \\
\hline
[8,1,0], [7,2,0]& 3 & 2 \\
\hline
[6,3,0]&4 & 1 \\
\hline
\end{tabular}
\caption{The multiplicity table for $\Xi_{\alpha} \in [3]^3$ }
\end{center}
\end{table}
}
The braiding and $U^{\Xi_{\alpha}}$ matrices in this case are
\begin{equation}{\label{col3}}
~{\hat{\mathcal{R}}^{[6,3,0]}=A^{-3}\left(
\begin{array}{cccc}
-q^3 & 0 & 0 & 0 \\
0 & q^5 & 0 & 0 \\
0 & 0 & -q^9 & 0 \\
0 & 0 & 0 & q^{15} \\
\end{array}
\right)}\\,~~{\hat{\mathcal{R}}^{[5,3,1]}=A^{-3}\left(
\begin{array}{ccc}
-q^3 & 0 & 0 \\
0 & q^5 & 0 \\
0 & 0 & -q^9 \\
\end{array}
\right)}\\,~~{\hat{\mathcal{R}}^{[7,2,0]}=A^{-3}\left(
\begin{array}{ccc}
q^5 & 0 & 0 \\
0 & -q^9 & 0 \\
0 & 0 & q^{15} \\
\end{array}
\right)}~,
\end{equation}
\begin{equation}{\label{col31}}
\hat{\mathcal{R}}^{[5,4,0]}=\hat{\mathcal{R}}^{[6,2,1]}=A^{-3}q^7\left(
\begin{array}{cc}
q^{-2} & 0 \\
0 & -q^2 \\
\end{array}
\right)\\,~\hat{\mathcal{R}}^{[4,3,2]}=A^{-3}q^4\left(
\begin{array}{cc}
-q^{-1} & 0 \\
0 & q \\
\end{array}
\right) \\,~
~{\hat{\mathcal{R}}^{[8,1,0]}=A^{-3}q^{12}\left(
\begin{array}{cc}
-q^{-3} & 0 \\
0 & q^{3} \\
\end{array}
\right)},
\end{equation}~
\begin{equation}{\label{col32}}
\mathcal{U}^{[5,3,1]}=\left(
\begin{array}{ccc}
-\frac{1}{1+\frac{1}{q^2}+q^2} & -\frac{q}{\sqrt{1+q^2+q^4}} & -\frac{\sqrt{1+q^2+q^4+q^6+q^8}}{1+q^2+q^4} \\
-\frac{q}{\sqrt{1+q^2+q^4}} & -1+\frac{q^2}{1+q^4} & \frac{q^4 \sqrt{\left(1+\frac{1}{q^2}+q^2\right) \left(1+\frac{1}{q^4}+\frac{1}{q^2}+q^2+q^4\right)}}{\left(1+q^4\right)
\left(1+q^2+q^4\right)} \\
-\frac{\sqrt{1+q^2+q^4+q^6+q^8}}{1+q^2+q^4} & \frac{q^4 \sqrt{\left(1+\frac{1}{q^2}+q^2\right) \left(1+\frac{1}{q^4}+\frac{1}{q^2}+q^2+q^4\right)}}{\left(1+q^4\right)
\left(1+q^2+q^4\right)} & -\frac{q^4}{1+q^2+2 q^4+q^6+q^8} \\
\end{array}
\right)~,
\end{equation}
\begin{equation}{\label{col33}}
\mathcal{U}^{[5,4,0]}=\mathcal{U}^{[3,6,2,1]}=\left(\begin{array}{cc}
\frac{1}{[2]_{q^2}} & \frac{\sqrt{[3]_{q^2}}}{[2]_{q^2}} \\
\frac{\sqrt{[3]_{q^2}}}{[2]_{q^2}} & -\frac{1}{[2]_{q^2}}
\end{array}\right),\ \
\end{equation}
\begin{equation}{\label{col34}}
\mathcal{U}^{[8,1,0]}=\left(\begin{array}{cc}
\frac{1}{[2]_{q^3}} & \frac{\sqrt{[3]_{q^3}}}{[2]_{q^3}} \\
\frac{\sqrt{[3]_{q^3}}}{[2]_{q^3}} & -\frac{1}{[2]_{q^3}}
\end{array}\right),\ \
\mathcal{U}^{[4,3,2]}=\left(\begin{array}{cc}
\frac{1}{[2]_{q}} & \frac{\sqrt{[3]_{q}}}{[2]_{q}} \\
\frac{\sqrt{[3]_{q}}}{[2]_{q}} & -\frac{1}{[2]_{q}}
\end{array}\right),\ \
\end{equation}
We have placed the other $3 \times 3$ and also $4\times 4$ matrices in Appendix~\ref{app}.
From eqn.(\ref{HOMFLY}), $[3]$-colored HOMFLY-PT for $W(3,n)$:
\begin{eqnarray*}
\mathcal{H}_{[3]}^{W(3,n)} &=& \frac{1}{S^*_{[3]}}\sum_{\alpha} S^*_{\Xi_{\alpha}} {\rm Tr}_{\Xi_{\alpha}} (\hat{\mathcal{R}}^{\Xi_{\alpha}}_1)(\hat{\mathcal{R}}^{\Xi_{\alpha}}_2)^{-1}\ldots (\hat{\mathcal{R}}^{\Xi_{\alpha}}_1)(\hat{\mathcal{R}}^{\Xi_{\alpha}}_2)^{-1}~,\nonumber\\&=& \frac{1}{S^*_{[3]}}\sum_{\alpha}S^*_{\Xi_{\alpha}} {\rm Tr}_{\Xi_{\alpha}} (\hat{\mathcal{R}}^{\Xi_{\alpha}}_1)(\hat{\mathcal{R}}^{\Xi_{\alpha}}_2)^{-1})^n~,\nonumber\\&=& \frac{1}{S^*_{[3]}} (S^*_{[9]}+ S^*_{[7,1,1]}+S^*_{[5,2,2]} +S^*_{[4,4,1]}+S^*_{[3,3,3]}+S^*_{[4,3,2]}{\rm Tr}_{[4,3,2]} (\hat{\mathcal{R}}^{[4,3,2]})(\hat{\mathcal{R}}^{[4,3,2]}_2)^{-1})^n+\\&&S^*_{[6,2,1]}{\rm Tr}_{[6,2,1]} (\hat{\mathcal{R}}^{[6,2,1]})(\hat{\mathcal{R}}^{[6,2,1]}_2)^{-1})^n+S^*_{[5,4,0]}{\rm Tr}_{[5,4,0]} (\hat{\mathcal{R}}^{[5,4,0]})(\hat{\mathcal{R}}^{[5,4,0]}_2)^{-1})^n+\\&&S^*_{[8,1,0]}{\rm Tr}_{[8,1,0]} (\hat{\mathcal{R}}^{[8,1,0]})(\hat{\mathcal{R}}^{[8,1,0]}_2)^{-1})^n+S^*_{[7,2,0]}{\rm Tr}_{[7,2,0]} (\hat{\mathcal{R}}^{[7,2,0]})(\hat{\mathcal{R}}^{[7,2,0]}_2)^{-1})^n+\\&&S^*_{[6,3,0]}{\rm Tr}_{[6,3,0]} (\hat{\mathcal{R}}^{[6,3,0]})(\hat{\mathcal{R}}^{[6,3,0]}_2)^{-1})^n{\label{HW3}}.
\end{eqnarray*}
Using eqns.($\ref{col3})$ to $(\ref{col34})$, eqn.(\ref{conj}), and Appendix~\ref{app}, we can rewrite the equation(\ref{HW3}) into neat formula
\begin{eqnarray}
\mathcal{H}_{[3]}^{W{(3,n)}} &=& \frac{1}{S^*_{[3]}} (S^*_{[9]}+ S^*_{[7,1,1]}+S^*_{[5,2,2]} +S^*_{[4,4,1]}+S^*_{[3,3,3]} +(S^*_{[6,2,1]}+S^*_{[5,4,0]}) \mathcal{V}_{n,2}[q]\nonumber\\&&+S^*_{[4,3,2]}\mathcal{V}_{n,1}[q]+S^*_{[8,1,0]}\mathcal{V}_{n,3}[q]+S^*_{[7,2,0]}{\rm Tr} (X_1^{[7,2,0]})^n+S^*_{[5,3,1]}{\rm Tr} (X_2^{[5,3,1]})^n\nonumber\\&&
+S^*_{[6,3,0]}{\rm Tr} (X_3^{[6,3,0]})^n{\label{HW33}}.\end{eqnarray}
where the explicit form of $X_3^{[6,3,0]}$, $X_1^{[7,2,0]}$ and $X_2^{[5,3,1]}$ are given in Appendix \ref{app} and the colored HOMFLY-PT polynomials for $W(3,n)$ for color [3] are presented in Appendix~\ref{app1}.
Even though we have explicitly computed $[r]$-colored HOMFLY-PT upto $[r]=3$, the method is straightforward. However, it will be interesting if we can write a closed form expression for arbitrary color $[r]$. This is essential to work on volume conjecture for these hyperbolic knots which we plan to pursue in future. As a piece of evidence that our $[r]$-colored HOMFLY-PT for weaving knots are correct, we work out reformulated invariants and BPS integers in the context of topological string duality in the following section.
\section{Integrality structures in topological strings }\label{s5.int}
Motivated by the AdS-CFT correspondence, Gopakumar-Vafa conjectured that the $SU(N)$ Chern-Simons theory on $S^3$ is dual to closed A-model topological string theory on a resolved conifold $\cal{O}$(-1) + $\cal{O}$ (-1) over $\mathbf P^1$. Particularly, the Chern-Simons free energy $\ln Z[S^3]$ was shown to be closed string partition function on the resolved conifold target space:
\begin{equation}
\ln Z[S^3]=-\sum_g \mathcal{F}_{g}(t) g_{s}^{2-2g},
\end{equation}
where $\mathcal{F}_{g}(t)$ are the genus $g$ topological string amplitude, $g_s=\frac{2 \pi}{k+N}$ denotes the string coupling constant and $t=\frac{2 \pi i N}{k+N}$ denote the ${\rm K\ddot{a}hler}$ parameter of $\mathbf P^1$. Ooguri-Vafa conjectured that the Wilson loop operators in Chern-Simons theory correspond to the following topological string operator on a deformed conifold $T^{*}S^3$:
\begin{eqnarray}\label{uov}
\ln Z(U,V)_{\mathcal{S}^{3}}&=&\sum_{m}\frac{1}{m}Tr_{[1]} U^mTr_{[1]} V^m,
\end{eqnarray}
where $U$ represent the holonomy of the gauge connection $A$ around the knot $\mathcal{K}$ carrying
the fundamental representation($[1]$) in the $U(N)$ Chern-Simons theory on $S^3$, and V is the
holonomy of a gauge field $\tilde{A}$ around the same component knot carrying the fundamental
representation($[1]$) in the $U(M)$ Chern-Simons theory on a Lagrangian sub-manifold $\cal C$ which intersects $S^3$ along the knot $\mathcal{K}$. Gopakumar-Vafa duality require integrating the gauge field $A$ on $S^3$ leading to open topological string amplitude on the resolved conifold background. For unknot, the detailed calculation was performed \cite{OV} giving:
\begin{eqnarray}
\left\langle Z(U,V)\right\rangle_{S^3}&=&\exp{(i \sum_{m=1}^{\infty} \frac{\exp {(\frac{m t}{2})}-\exp {(\frac{-m t}{2})}}{2 m \sin{(\frac{m g_s}{2})}}Tr V^{-m})},\label{gv}
\end{eqnarray}
which was justified using Gopakumar-Vafa duality. Further, Ooguri-Vafa conjectured the generalization of eqn.(~\ref{gv}) for other knots as (also known ${\bf LMOV}$ integrality conjecture):
\begin{eqnarray}
\left\langle Z(U,V)\right\rangle_{S^3}&=&\sum_{{\bf R}} \mathcal{H}^{*\mathcal{K}}_{{\bf R}}(q,{\bf A}) Tr_{{\bf R}} V\nonumber\\
&=&\exp\bigg[\sum_{m=1}^{\infty} \left(\sum\limits_ {R}\frac{1}{m} f^{\mathcal{K}}_{{\bf R}}({A}^m,q^m)Tr_{{\bf R}} V^m\right)\bigg],\label{guv}
\end{eqnarray}
where $f^{\mathcal{K}}_{{\bf R}}({A},q)$, known as reformulated invariant, obeying the following integrality structure:
\begin{eqnarray}
f^{\mathcal{K}}_{{\bf R}} (q, A)&=&\sum_{i,j} \frac{1}{(q-q^{-1})}{\widetilde {\bf N}^{\mathcal{K}}_{{\bf R},i,j}{A}^i q^{j}}\nonumber.
\end{eqnarray}
Here, $R$ denotes the irreducible representation of $U(N)$ and ${\widetilde {\bf N}}^{\mathcal{K}}_{{\bf R}, i,j}$ counts the number of D2-brane intersecting D4-brane (BPS states) where, i and j keeps track of charges and spins respectively\cite{GV1, GV2}. These reformulated invariants can be written in the terms of colored HOMFLY-PT polynomials \ref{guv}. For few lower dimensional representations, the explicit forms are as follows\cite{LMV, LM1, LM2}:
{
\begin{eqnarray}
f^{\mathcal{K}}_{[1]}(q,{ A})&=&\mathcal{H}^{*\mathcal{K}}_{[1]}(q,{A}),\nonumber \\
f^{\mathcal{K}}_{[2]}(q,{ A})&=&\mathcal{H}^{*\mathcal{K}}_{[2]}(q,{ A})-{1\over 2}\Big(\mathcal{H}^{*\mathcal{K}}_{[1]}(q,{ A})^2+\mathcal{H}^{*\mathcal{K}}_{[1]}(q^2,{A}^2)\Big),\nonumber\\
f^{\mathcal{K}}_{[1^2]}(q,{ A})&=&\mathcal{H}^{*\mathcal{K}}_{[1^2]}(q,A)-{1\over 2}\Big( \mathcal{H}^{*\mathcal{K}}_{[1]}(q,A)^2-\mathcal{H}^{*\mathcal{K}}_{[1]}(q^2,A^2)\Big),\nonumber\\
\ldots \nonumber
\end{eqnarray}}
In fact, reformulated invariants obey Ooguri-Vafa conjecture verified for many arborescent knots up to 10 crossings in \cite{Mironov:2017hde}. Moreover, these reformulated invariant can be equivalently written as \cite{LMV}:
\begin{eqnarray}\label{ic}
f^{\mathcal{K}}_{{\bf R}} (q,{ A})=\sum_{m,k\ge 0,s} C_{{\bf R}{\bf S}}\hat {\bf N}^{\mathcal{K}}_{{\bf S},m,k}{ A}^m(q-q^{-1})^{2k-1},
\end{eqnarray}
where $\hat {\bf N}^{\mathcal{K}}_{{\bf S},m,k}$ called refined integers and
\begin{eqnarray}
C_{{\bf R}{\bf S}}={1\over q-q^{-1}}\sum_{\Delta}{1\over z_\Delta}\psi_{{\bf R}}(\Delta)\psi_{{\bf S}}(\Delta){\prod_{i=1}^{l(\Delta)}\Big(q^{\xi_i}-q^{-\xi_i}\Big)}\nonumber.
\end{eqnarray}
Here the sum goes over the Young diagrams $\Delta$ with $l(\Delta)$ lines of lengths $\xi_i$ and the number of boxes $|\Delta|=\sum_{i}^{l(\Delta)} \xi_i$, while $\psi_{{\bf R}}(\Delta)$ denote the characters of symmetric groups at $|R|=|\Delta|$ and $z_{\Delta}$ is the standard symmetric factor of the Young diagram\cite{Fulton}.
Using our colored HOMFLY-PT form for the weaving knot $W(3,n)$ (listed in Appendix ~\ref{app1}), we computed the reformulated invariants for representations upto length $|{\bf R}|=2$. From our analysis, we propose the following:\\
\vskip1cm
\begin{mdframed}[style=sid]
\textbf{Proposition 2.} \emph{Refined BPS integer $\hat{\bf N}^{W(3,n)}_{[ 1],\mp1,k}$ for weaving knot $W(3,n)$ is the coefficient of $z^k$ of polynomial $f^{\mp}_{n}[z]$ of degree $n-1$ i.e
\begin{eqnarray}{\label{CHP}}
f^{\mp}_{n}[z]&=& \pm \frac{2(-1)^n T_{n}(\frac{1+z}{2})+1}{z}~,
\end{eqnarray}}
\end{mdframed}
where $T_{n}(z)$ represents the $nth$ degree Chebyshev polynomial of the first kind at the point z. Rodrigue's formula to obtain $T_n(z)$ is
\begin{equation}
T_n(z) = \frac{(-2)^n n!}{2n!} \sqrt{1-z^2} \frac{d^n} {dz^n} (1-z^2)^{n-1/2}~.
\end{equation}
Here we list the polynomial form for some values of $n$:
For completeness,
\begin{eqnarray*}
f^{-}_{11}[z]&=&11 + 22 z - 66 z^2 - 99 z^3 + 77 z^4 + 154 z^5 + 22 z^6 - 66 z^7 -
44 z^8 - 11 z^9 - z^{10}\\
f^{-}_{10}[z]&=&-10 + 15 z + 60 z^2 - 15 z^3 - 98 z^4 - 35 z^5 + 40 z^6 +
35 z^7 + 10 z^8 + z^{9}\\
f^{-}_{5}[z]&=&5 + 5 z - 5 z^2 - 5 z^3 - z^4\\
f^{-}_{4}[z]&=&-4 + 2 z + 4 z^2 + z^3
\end{eqnarray*}
Unfortunately, we have not managed to write the other integers for fundamental representation $\hat{\bf N}^{W(3,n)}_{[ 1],\pm 3,k}:$ as a closed form. There are other properties of $\hat{\bf N}$ which we have checked
up to the level $|{\bf S}|=2$ for $W(3,n)$ knot. They are
\begin{eqnarray*}
\sum_m \hat{\bf N}^{W(3,n)}_{{\bf S},m,k}&=0&\\
\sum_k \hat{\bf N}^{W(3,n)}_{{\bf [1]
},\mp 1 ,k}&=&\mp \frac{4}{3} T_{n-1}(-1)\sec(\frac{n \pi}{6})^2 \sin(\frac{n \pi}{3})^4 , ~ n\geq 1.
\end{eqnarray*}
here $T_{n-1}(z=-1)$ is the Chebyshev polynomial evaluated at $z=-1$.
We have tabulated below these
refined integers for knot $W(3,4), W(3,5), W(3,10)$, and $W(3,11)$, when $|r=1|$:
\begin{center}\begin{tabular}{cccc}
$\hat{\bf N}^{W[3,4]}_{[ 1]}:$ &
\begin{tabular}{|c|cccc|}
\hline
&&&&\\
$ k \backslash m=$ & -3 & -1 & 1 & 3 \\
&&&&\\
\hline
&&&&\\
0 & 1 & -4 &4 & -1 \\
&&&&\\
1 & -1 &2 & -2 & 1 \\
&&&&\\
2 & -1 & 4 & -4 & 1 \\
&&&&\\
4 & 0 & 1 & -1 & 0 \\
&&&&\\
\hline
\end{tabular}
\end{tabular}, \begin{tabular}{cccc}
$\hat{\bf N}^{W[3,5]}_{[ 1]}:$ &
\begin{tabular}{|c|cccc|}
\hline
&&&&\\
$ k \backslash m=$ & -3 & -1 & 1 & 3 \\
&&&&\\
\hline
&&&&\\
0 & -2 & 5 &- 5 & 2 \\
&&&&\\
1 & -1 & 5 & -5 & +1 \\
&&&&\\
2 & 2 & -5 & 5 & -2 \\
&&&&\\
3 & 1 & -5 & 5 & -1 \\
&&&&\\
4 & 0 & -1 & 1 & 0 \\
&&&&\\
\hline
\end{tabular}
\end{tabular}
\end{center}
\begin{center}\begin{tabular}{cccc}
$\hat{\bf N}^{W[3,10]}_{[ 1]}$ &
\begin{tabular}{|c|cccc|}
\hline
&&&&\\
$ k \backslash m=$ & -3 & -1 & 1 & 3 \\
&&&&\\
\hline
&&&&\\
0 & 3 & -10 &10 & 3\\
&&&&\\
1 & -6& 15 & -15 & 6 \\
&&&&\\
2 & -18 & 60 & -60 & 18 \\
&&&&\\
3 & 11 & -15 & 15 & -11 \\
&&&&\\
4 & 29 & -98 & 98 & -29 \\
&&&&\\
5 & 2 & -35 & 35 & -2 \\
&&&&\\
6 & -14 & 40 & -40 & 14 \\
&&&&\\
7 & -7 & 35 &-35 &7 \\
&&&&\\
8 & -1 & 10 & -10 & 1\\
&&&&\\
9 & 0 & 1 & -1 & 0 \\
&&&&\\
\hline
\end{tabular}
\end{tabular},~\begin{tabular}{cccc}
$\hat{\bf N}^{W[3,11]}_{[ 1]}$ &
\begin{tabular}{|c|cccc|}
\hline
&&&&\\
$ k \backslash m=$ & -3 & -1 & 1 & 3 \\
&&&&\\
\hline
&&&&\\
0&-4 & 11 & -11 & 4 \\
&&&&\\
1&-6 & 22 & -22 & 6 \\
&&&&\\
2&24 & -66 & 66 & -24 \\
&&&&\\
3& 25 & -99 & 99 & -25 \\
&&&&\\
4& -34 & 77 & -77 & 34 \\
&&&&\\
5& -40 & 154 & -154 & 40 \\
&&&&\\
6& 6 & 22 & -22 & -6 \\
&&&&\\
7& 20 & -66 & 66 & -20 \\
&&&&\\
8& 8 & -44 & 44 & -8 \\
&&&&\\
9& 1 & -11 & 11 & -1 \\
&&&&\\
10& 0 & -1 & 1 & 0 \\
&&&&\\
\hline
\end{tabular}
\end{tabular}
\end{center}
The table of refined integers for representations whose length $|R|= 2$ are presented in Appendix ~\ref{app2}.
\section{Conclusion and discussion}
\label{s6.int}
Hybrid weaving knots $\hat W_3(m,n)$ obtained from braid word $\left[\sigma_1^m \sigma_2^{-m}\right]^n$ (see Fig.\ref{GWK}) contains weaving knots $W(3,n)$ as subset which are hyperbolic in nature. Finding a closed form expression for $[r]$-colored HOMFLY-PT for such hybrid weaving knots was attempted using the modified Reshtikhin-Turaev approach \cite{RT1}-\cite{RT2} method.
Using the $\hat{\mathcal{R}_i}$ matrices, we derived the explicit closed form expression of HOMFLY-PT for hybrid weaving knot$\hat{W}_3(m,n)$ (\ref{HWK}). Motivated by the Laurent polynomial structure studied for HOMFLY-PT of weaving knots\cite{RRV}, we proposed such a structure $\mathcal{V}_{n,t}[q]$ (\ref{HW22} and \ref{HW33}) for any $[r]$-colored HOMFLY-PT for the weaving knots. Further we showed that the absolute sum of the coefficients in the Laurent polynomial is related to Fibonacci numbers (see conjecture 1 (\ref{febona})). We have computed the colored HOMFLY-PT for $W(3,n)$ upto $[r]=3$ and presented them in the appendix~\ref{app1}. Clearly, writing the polynomial form is computationally simplified by this modified RT method. Using these knot invariants, we computed reformulated invariants and found some of the refined BPS integers can be written in terms of coefficient of Chebyshev polynomials($T_{n}(x))$ of first kind for $W(3,n)$ (\ref{CHP}).\\
So far, we have have managed to write the closed form expression for trace of 2x2 matrices by introducing the $\mathcal{V}_{n,t}[q]$. For higher dimensional matrices, such a Laurent polynomial structure is not obvious. We have seen a concise form for $[r]$-colored HOMFLY-PT for knot $4_1\equiv W(3,2)$, twist knots and torus knots using $q$-binomial and $q$-Pochammer terms.\\
It will be interesting if we can find a similar expression for weaving knots. Such an expression will help us to address volume conjecture, A-polynomials for these weaving knots. We hope to address these problems in future.
\vspace{0.5cm}
\textbf{Acknowledgements}
VKS would like to acknowledge the hospitality of department of mathematics, IISER, Pune (India) where this work was done during his visit as visiting fellow. PR would like to thank SERB ((MATRICS) MTR/2019/000956 funding.
\bibliographystyle{JHEP}
|
1,116,691,500,853 | arxiv | \section{Different Budgets and Prices are Necessary}\label{app: same budget counter example}
Consider the instance with $3$ agents and $2$ items. Agent $A$ has value $1$ for both items. Agent $B$ has values $1/2$ and $1$ for the first and second item, respectively, while agent $C$ has values $1$ and $1/2$.
It is easy to check that the unique solution that maximizes the product of agents' utilities (i.e. equal budgets in the EG convex program) gives agent $A$ $1/3$ of each item, while agent $B$ gets $2/3$ of item 2 and agent $C$ gets $2/3$ of item $1$. The prices of both items are equal.
In this example, the unique product maximizing solution does not satisfy the CISEF property, as there are two indifference edges from $A$ to $B$ and $A$ to $C$. Therefore, to get an allocation like the one guaranteed by Theorem~\ref{thm: structural} one needs to go beyond equal budgets in the E-G convex program.
\section{Improving on ``EF1 or EF w.h.p.'' Guarantee is Impossible}
\label{app:optimal-fairness-guarantee}
We show these impossibility results under the weakest adversary, where the adversary can give a distribution $D^V$ and item values $v_{it}$ are drawn from $D^V$.
First, we show that it is not always possible to guarantee envy-freeness with high probability. Define $D^V$ to be the uniform distribution over the set $\{ 1 \}$. Note that whenever $T$ is not a multiple of $n$, the allocation will not be envy-free. Next, we show that for any $x$, envy-freeness up to $x$ goods, is not an achievable guarantee. We use the construction of the lower bound in \citet{BKPP18}, which assumes item values bounded within $[0, 1]$.
\begin{lemma}[\citet{BKPP18}]\label{thm: benade et al}
For $n \geq 2$ and $r < 1$, there exists an adversary strategy for setting item values such that any algorithm must have $\mathsf{ENVY}(A) \in \Omega((T/n)^{r/2})$, where $A$ is the allocation $T$ items.
\end{lemma}
In the proof of Lemma~\ref{thm: benade et al}, they use an adversary strategy where all agents other than the first two agents do not value the item. The value of the arriving item to the first two agents depends solely on a state machine, where the states are ``$\ldots, L_2, L_1, 0, R_1, R_2, \ldots$''. The item associated with state $L_i$ has value $(1, \nu_i)$ (so $a_1$ values it at $1$ and $a_2$ values it at $\nu_i$), the item associated with state $0$ has value $(1, 1)$, and the item associated with state $R_i$ has value $(\nu_i, 1)$. The state machine starts at state $0$ and transitions one step left or right after each item arrival, depending on how the previous item was allocated. Thus, the set of all possible item values used for a length $T$ instance is $\{0, 1, \nu_1, \ldots, \nu_T\}$. They define $\nu_i = (i + 1)^r - i^r$.
We now apply Lemma~\ref{thm: benade et al}, setting $r = \frac{1}{2}$. Let $c, T_0$ to be the constants such that for any $T \geq T_0$, the adversary can guarantee $\mathsf{ENVY}(A) \geq c(T/n)^{r/2} = c(T/n)^{1/4}$. We then take any $T' \geq T_0$ where $c(T'/n)^{1/4} > x$ and set $D$ to be the uniform distribution over $\{0, 1, \nu_1, \ldots, \nu_{T'}\}$.
We claim that for any $T \geq T'$, there is a positive probability that the allocation will not be envy-free up to $x$ goods. First, for the given algorithm, there is a positive probability that the first $T$ items drawn will follow the adversary strategy. Then, for the remaining items, there is a positive probability that
all the items will have no value to any agent. Thus, for the final allocation $A$, we will have $\mathsf{ENVY}(A) > x$. Finally, observe that when item values are bounded within $[0, 1]$, $\mathsf{ENVY}(A) > x$ implies that $A$ is not envy-free up to $x$ goods.
\section{Missing Proofs}
\subsection{Proof of Proposition~\ref{prop:util-max-properties}}\label{app: missing proposition}
\begin{proof}[Proof of Proposition~\ref{prop:util-max-properties}]
Allocating to the agent with the highest value clearly generates a Pareto efficient allocation as it maximizes social welfare.
The algorithm first handles the special case where $D$ is a point mass. In this case $v_{it} = v$ for all $i$ and all $t$, therefore allocating the arriving items in a round-robin manner achieves an EF1 allocation, as each agent will have at most one fewer item than other agents.
In the remaining cases, we apply Theorem~\ref{thm: john et al}. The statement and proof of Theorem~\ref{thm: john et al} assume that the distribution is non-atomic, which ensures that $\arg\max_{k \in \mathcal{N}} v_{kt}$ is exactly one agent and makes the utilitarian allocation algorithm well defined. In contrast, our setting assumes atomic distributions. However, the theorem can be easily adapted to our setting by adding random tie-breaking to the utilitarian allocation algorithm as described in Algorithm~\ref{alg:util}. We then rewrite Theorem~\ref{thm: john et al}, replacing [$\arg\max_{k \in \mathcal{N}} v_{kt} = \{ j \}$] with [$j$ receives $t$] and similarly with [$\arg\max_{k \in \mathcal{N}} v_{kt} = \{ i \}$]. This modified theorem can be shown using essentially the same proof as the original.
We now show that the distribution satisfies the two properties. The first property to show is that that for each agent $i \in \mathcal{N}$, $\Pr[i \text{ receives } t] = 1/n$. This property is satisfied by symmetry, as all agent values are drawn from the same distribution and tie-breaking is done randomly. The second property is that for some constant $\Delta > 0$, and for all agents $i, j \in \mathcal{N}$ where $i \neq j$: $\E[v_{it} \mid i \text{ receives } t] - \E[v_{it} \mid j \text{ receives } t] \geq \Delta$.
We largely follow the proof of Lemma 3.2 by \citet{kurokawa2016can} with some simplifications and importantly, handling the case when the distribution $D$ is discrete.
Since agents' value distributions are identical, we can restate this as:
$$\E[v_{it} \mid i \text{ receives } t] - \E[v_{it} \mid i \text{ does not receive t}] \geq \Delta$$
Agents are all identical and the algorithm always allocates the item to an agent with the maximum value for it. This implies that $\E[v_{it} \mid i \text{ receives } t] = \E[\max(v_{1t}, \ldots, v_{nt})]$. Next, if $D$ is not a point mass, we know that $\Var[D] > 0$. From here, we can show that $\E[\max(v_{1t}, \ldots, v_{nt})] > \E[v_{it}]$.
Let $\Var[D] = c$. Let $\bar{X} = \E[X], p = \Pr[ X < \bar{X}]$. Observe that when $\Var[D] > 0, p \in (0, 1)$.
\begin{align*}
c &= \Var[D]\\
&= \E[(X - \bar{X})^2]\\
&\leq \E[\abs{X - \bar{X}}]\\
&=p\E[\bar{X} - X \mid X < \bar{X}] + (1-p)\E[X - \bar{X} \mid X \geq \bar{X}]\\
&=-p\E[X \mid X < \bar{X}] + (1 - p)\E[X \mid X \geq \bar{X}] + (2p - 1)\bar{X}
\end{align*}
From here, we do case-analysis based on $p \leq 1/2$ or $p > 1/2$.
Suppose that $p > 1/2$, we use the substitution $\bar{X} = p\E[X \mid X < \bar{X}] + (1 - p)\E[X \mid X \geq \bar{X}]$ and then we rewrite the above as:
\begin{align*}
c &\leq 2p(\bar{X} - \E[X \mid X < \bar{X}])\\
\implies \frac{c}{2} &\leq \bar{X} - \E[X \mid X < \bar{X}] \leq \E[X \mid X \geq \bar{X}] - \E[X \mid X < \bar{X}]
\end{align*}
Similarly, we $p \leq 1/2$, we go in the other direction and get:
\begin{align*}
c &\leq 2(1-p)(\E[X \mid X \geq \bar{X}] - \bar{X})\\
\implies \frac{c}{2} &\leq \E[X \mid X \geq \bar{X}] - \bar{X} \leq \E[X \mid X \geq \bar{X}] - \E[X \mid X < \bar{X}]
\end{align*}
Finally, we have that:
\begin{align*}
\E[\max(v_{1t}, \ldots, v_{nt})] &\geq (1 - p^n)\E[X \mid X \geq \bar{X}] + p^n \E[X \mid X < \bar{X}]\\
&\geq (1 - p)\E[X \mid X \geq \bar{X}] + p \E[X \mid X < \bar{X}]\\
&\qquad + (p - p^n)(\E[X \mid X \geq \bar{X}] - \E[X \mid X < \bar{X}])\\
&\geq (1 - p)\E[X \mid X \geq \bar{X}] + p \E[X \mid X < \bar{X}] + (p - p^n)\frac{c}{2}\\
&=\E[v_{it}] + (p - p^n)\frac{c}{2}
\end{align*}
Thus, we have that $\E[v_{it} \mid i \text{ receives } t] \geq \E[v_{it}] + (p - p^n)\frac{c}{2}$. From law of total expectation, we know that $\E[v_{it}] = \frac{1}{n}\E[v_{it} \mid i \text{ receives } t] + \frac{n-1}{n}\E[v_{it} \mid i \text{ does not receive } t]$. We can combine and rearrange to also show that $\E[v_{it} \mid i \text{ does not receive } t] \leq \E[v_{it}] - (p - p^n)\frac{c}{2(n-1)}$, which allows us to conclude:
$$\E[v_{it} \mid i \text{ receives } t] - \E[v_{it} \mid i \text{ does not receive } t] \geq (p - p^n)\frac{c}{2(n-1)} + (p - p^n)\frac{c}{2}$$
Setting $\Delta$ to $(p - p^n)\frac{c}{2(n-1)} + (p - p^n)\frac{c}{2}$, which is positive since $p \in (0, 1)$ and $c > 0$, completes the proof.\end{proof}
\subsection{Proof of Lemma~\ref{lem:edge-indifferent}}\label{app: missing lemma proof}
We can rewrite KKT condition~\eqref{eq: KKT 3} as: for all $i \in [n]$, for all $k \in X_i$, $p_j = \frac{v_{ik} \cdot e_i}{\sum_{k' \in X_i} v_{ik'}x_{ik'}} = \frac{v_{ik}}{r_i}$.
Then, for every agent $i$ we have $\sum_{k \in X_i} p_k x_{ik} = \sum_{k \in X_i} \frac{v_{ik} \cdot e_i}{\sum_{k' \in X_i} v_{ik'}x_{ik'}}x_{ik} = e_i$
For the ``only if'' direction, we know for all items $k \in X_j$, $v_{ik} = r_i p_k$. We can substitute this into our previous equation to get:
$$v_i(X_i) = \sum_{k \in X_i} v_{ik} x_{ik} = \sum_{k \in X_i} r_i p_k x_{ik} = r_i e_i = r_i e_j = \sum_{k \in X_j} r_i p_k x_{jk} = \sum_{k \in X_j} v_{ik} x_{jk} = v_i(X_j).$$
For the ``if'' direction, assume that there is an item $k^* \in X_j$ such that $\frac{v_{i k^*}}{p_{k^*}} \neq r_i$. KKT condition~\eqref{eq: KKT 3} implies that $k^*$'s bang-per-buck is at most $r_i$, so we must have $v_{ik^*} < r_ip_{k^*}$. Combined, we get:
\[
v_i(X_i) = \sum_{k \in X_i} v_{ik} x_{ik} = \sum_{k \in X_i} r_i p_k x_{ik} = r_i e_i = r_i e_j = \sum_{k \in X_j} r_i p_k x_{jk} > \sum_{k \in X_j} v_{ik} x_{jk} = v_i(X_j). \qed
\]
\subsection{Proof of Lemma~\ref{lem:optimal-transfers}}\label{app: lemma optimal transfers}
$(\bx, \bp, \be)$ must satisfy the KKT conditions. $\bp$ does not change and $\bx'$ is feasible by definition; it remains to show that $(\bx', \bp, \be')$ satisfies the KKT conditions.
Condition~\eqref{eq: KKT 1} is satisfied since $\sum_{i \in [n]} \Delta_{ik} = 0$ for all items $k$.
Furthermore, notice that $\Delta_{ik} \neq 0$ implies that $\frac{v_{ik}}{p_k} = r_i$: if $\Delta_{ik} > 0$ this fact is implied by the definition of an optimal transfer, while if $\Delta_{ik} < 0$, for $\bx + \Delta$ to be feasible, it must be that $x_{ik} > 0$, so $\frac{v_{ik}}{p_k} = r_i$ is implied by condition~\eqref{eq: KKT 3}.
Thus:
\[ \frac{\sum_{k=1}^m v_{ik}x'_{ik}}{e'_i} = \frac{\sum_{k=1}^m v_{ik}x_{ik} + \sum_{k=1}^m v_{ik}\Delta_{ik}}{e_i + \delta_i} = \frac{\sum_{k=1}^m v_{ik}x_{ik} + \sum_{k=1}^m r_i p_k \Delta_{ik}}{e_i + \delta_i} = \frac{\sum_{k=1}^m v_{ik}x_{ik} + r_i \delta_i}{e_i + \delta_i}, \]
which is equal to $\frac{\sum_{k=1}^m v_{ik}x_{ik}}{e_i},$ since $r_i = v_i(X_i)/e_i$. Therefore, the RHS of condition~\eqref{eq: KKT 2} does not change (the LHS of course didn't change since it only has values and prices), so condition~\eqref{eq: KKT 2} is still satisfied. When $\bx_{ik} > 0$, condition~\eqref{eq: KKT 3} is satisfied by similar reasoning. Finally, it is possible that $\bx_{ik} = 0$ but $\bx'_{ik} > 0$. In this case, we know that $\Delta_{ik} > 0$, and therefore $\frac{v_{ik}}{p_k} = r_i = \frac{\sum_{k=1}^m v_{ik}x_{ik}}{e_i},$ by the definition of an optimal transfer and the definition of $r_j$. Thus, condition~\eqref{eq: KKT 3} is satisfied.
\qed
\subsection{Choosing $b$}\label{app:choosing b}
\paragraph{Operation $1$.}
We first discuss how to choose $b$ in Operation $1$ from Lemma~\ref{lem:const-util-shifts}.
For any pair $(i, j)$, we are concerned about $v_i(A_i) - v_i(A_j)$. Operation $1$ guarantees that $v_i(X_i) = v_i(X'_i)$, so $v_i(A_i) - v_i(A_j)$ can change only when $j \in K$. Define $c = \max_{i \in \mathcal{N},k \in [m]} \frac{v_{ik}}{p_k}$: the maximum bang-per-buck for any agent. Then, we choose $b$ such that:
$$b < \min \left\{ \frac{v_i(A_i) - v_i(A_j)}{c} : i \in \mathcal{N}, j \in K, v_i(A_i) - v_i(A_j) > 0 \right\}$$
Observe that a budget constraint of $b$ ensures that $v_i(A_j)$ will change by at most $b \cdot c < v_i(A_i) - v_i(A_j)$.
\paragraph{Lemma~\ref{lem:budget-change-shifts}.}
Next, we discuss how to choose $b$ in the procedure described in Lemma~\ref{lem:budget-change-shifts}.
We use a similar approach. Unlike before, $v_i(A_i) - v_i(A_j)$ can change when either $i \in S$ or $j \in S$, as $v_i(A_i)$ changes for some $i \in S$. Setting
$$b < \min \left\{ \frac{v_i(A_i) - v_i(A_j)}{2c} : i, j \in \mathcal{N}, i \in S \vee j \in S, v_i(A_i) - v_i(A_j) > 0 \right\}$$
as the budget constraint is sufficient as it ensures that $v_i(A_i)$ and $v_i(A_j)$ can each change by at most $\frac{v_i(A_i) - v_i(A_j)}{2}$.
\subsection{Proof of Theorem~\ref{thm:eff-envy-lb-det}}\label{app: lower bound proof continued}
\begin{proof}[Proof of Theorem~\ref{thm:eff-envy-lb-det} (continued) ]
Suppose that there is an allocation algorithm which guarantees that for all $T$, no matter the instance the adversary selects, $\E[ \mathsf{ENVY}(A^T) ] \leq f(T)$ for some $f(T) \in o(T)$, where the expectation is over the randomness used by the algorithm. We first describe a family of $n$ instances. For $i=1$ to $n$, instance $I_i$'s first $\frac{T}{n}i$ items follow $I$, the instance of the adaptive adversary described above, and the remaining items have no value. Let $g(T)$ be a function such that $g(T)\cdot f(T) \in o(T)$ and $g(T) \in \omega(1)$\footnote{One can think of $g(T) = T^\delta$, for some small $\delta > 0$ that depends on $f(T)$.}.
Again, we bound the number of items the algorithm can allocate to each agent in each segment; this time our bounds will be looser and probabalistic. Consider the behavior of the algorithm when faced with instance $I_1$. At the end of the first segment, i.e. for items $1$ through $\frac{T}{n}$, if the algorithm allocates to some agent $k$ at least $\frac{T}{n^2} + x_1$, for $x_1>0$, with probability at least $\frac{1}{g(T)}$, then the expected envy of some agent $\hat{k}$ (the one who received fewer than $\frac{T}{n^2}$ items) is at least $\varepsilon \cdot x_1$, that is $\E[ \mathsf{ENVY}(A^T) ] \geq \frac{1}{g(T)} \cdot \varepsilon x_1$. Since $\E[ \mathsf{ENVY}(A^T) ] \leq f(T)$, we have that $x_1 \leq \frac{g(T) f(T)}{\varepsilon}$. In other words, with probability $1 - \frac{1}{g(T)}$, each agent receives a number of items within $[\frac{T}{n^2} - x_1, \frac{T}{n^2} + x_1]$. Note that because the first $\frac{T}{n}$ items are identical for all instances, this holds for instances $I_2, \ldots, I_n$.
Similarly, we consider the behavior of the algorithm faced with $I_2$ and look at the end of the second segment. Suppose that conditioned on the algorithm having allocated each agent a number of items within $[\frac{T}{n^2} - x_1, \frac{T}{n^2} + x_1]$ after the first segment, with (conditional) probability at least $\frac{1}{g(T) - 1}$, some agent $k$ receives at least $\frac{T}{n^2} + x_2$ items from the second segment. This translates to an unconditioned probability of occuring of at least $\frac{1}{g(T)}$ and for similar reasons as before, we must have that $x_2 \leq \frac{g(T)f(T)}{\varepsilon} + \frac{2x_1}{\varepsilon} \leq \frac{g(T) f(T)}{\varepsilon} \left( 1 + \frac{2}{\varepsilon} \right)$. Together, we know that with probability $1 - \left(\frac{1}{g(T)} + \frac{1}{g(T) - 1}\left(1 - \frac{1}{g(T)}\right)\right)= 1 - \frac{2}{g(T)}$, each agent receives a number of items within $[\frac{T}{n^2} - x_2, \frac{T}{n^2} + x_2]$. We continue this for larger $i$, for $x_i = \frac{g(T) f(T)}{\varepsilon} \left( 1 + \frac{2}{\varepsilon} \right)^{i-1}$. Finally, we analyze efficiency of the algorithm for instance $I_n$. Each agent receives utility at most $(1 + (n - 1)\varepsilon))(\frac{T}{n^2} + x_n) + \frac{n}{g(T)}T$, where the additional $\frac{n}{g(T)}T$ term accounts for the worst case allocation assuming a deviation.
$$\frac{u_i}{1/n + \varepsilon} < \frac{T}{n} \cdot \frac{1 + (n - 1)\varepsilon}{1 + \varepsilon n} \cdot \left(1 + \frac{n^2}{\varepsilon} \frac{f(T)g(T)}{T}(1 + \frac{2}{\varepsilon})^{n - 1} + n^3g(T)\right).$$
By picking $T$ large enough\footnote{Ensuring $\frac{g(T)f(T)}{T} < \frac{1}{2}\frac{\epsilon}{1+(n-1)\varepsilon} \cdot \frac{\varepsilon}{n^2 (1 + 2/ \varepsilon)^{n - 1} }$ and $g(T) < \frac{\varepsilon}{2n^3}$ will suffice.} the adversary can make sure that the expected utility for every agent $i$ is upper bounded by $\frac{T}{n} \cdot ( 1/n + \varepsilon )$ . On the other hand, the single allocation that gives items from $\frac{T}{n}(i - 1)$ to $\frac{T}{n}i$ to agent $i$ yields utility $u_i = \frac{T}{n}$.
\end{proof}
\section{Strong Envy-Free and Pareto Optimality are not Achievable}
\label{app:sef-po-nonident-cex}
Here, we give an instance where strong envy-freeness and Pareto optimality are not achievable even though agents valuations are not identical up to a multiplicative factor.
\renewcommand{\arraystretch}{1}
\setlength\tabcolsep{2pt}
\begin{table*}[ht]
\begin{center}
\[\begin{array}{ | c | c c c | }
\hline
\text{Item $t$} & g_1 & g_2 & g_3 \\ [0.5ex]
\hline
\text{Value of $g_t$ to Agent 1} & 1 & 1 & 1 \\
\text{Value of $g_t$ to Agent 2} & 0.5 & 1 & 1\\
\text{Value of $g_t$ to Agent 3} & 0.25 & 1 & 1 \\
\hline
\end{array}\]
\caption{Instance with three agents}
\label{table:efpareto-incompatible}
\end{center}
\end{table*}
In Table~\ref{table:efpareto-incompatible}, none of the three agents are identical. However, we claim that in any envy-free and Pareto optimal allocation, agents $2$ and $3$ will be indifferent towards each other's allocation. Intuitively, the problem is that agents $2$ and $3$ have identical valuations over the items they could possibly receive in an envy-free and Pareto efficient allocation, items $2$ and $3$.
Let $X$ be any Pareto efficient and envy-free allocation. Since it is envy-free and agent $1$ is indifferent between the items, we must $X_{11} + X_{12} + X_{13} \geq 1$.
Next, we show that if $X$ is Pareto efficient, then $X_{11} = 1$. If $X_{11} < 1$, then either $X_{12} > 0$ or $X_{13} > 0$. In addition, either $X_{21} > 0$ or $X_{31} > 0$. Without loss of generality, suppose $X_{12} > 0$ and $X_{21} > 0$. Then letting $c = \min(X_{21}, X_{12})$ we give the following allocation $X'$ which Pareto dominates $X$ so $X$ is not Pareto efficient.
\begin{equation*}
X' = X + \begin{bmatrix}
c & -c & 0\\
-c & c & 0\\
0 & 0 & 0
\end{bmatrix}
\end{equation*}
Therefore, we know $X_{11} = 1$ and as a result, $X_{21} = X_{31} = 0$. Finally, because agents $2$ and $3$ have identical valuations for items $2$ and $3$, we have $X_{22} + X_{23} \geq 1$ and $X_{32} + X_{33} \geq 1$. Together, this implies that $X_{22} + X_{23} = X_{32} + X_{33} = 1$ so agents $2$ and $3$ will be indifferent towards each other's allocations.
\section{Deriving KKT Conditions}
\label{app:kkt-derive}
Here, we derive the KKT conditions in order to show that they are both necessary and sufficient conditions for optimal solutions to the Eisenberg-Gale convex program.
We first introduce dual variables $\bp, \bk$ for the first and second inequality constraints. From stationarity, we have:
$$\grad_{\bx}\sum_{i = 1}^n e_i \log \sum_{j = 1}^m v_{ij}x_{ij} = \sum_{j = 1}^m p_j \grad_{\bx} \sum_{i = 1}^n x_{ij} - 1 + \sum_{i = 1}^n\sum_{j = 1}^m - k_{ij} \grad_{\bx} x_{ij}$$
For each $i, j$, we can take the gradient with respect to $x_{ij}$ to get:
\begin{align}
e_i\frac{v_{ij}}{\sum_{j = 1}^m v_{ij}x_{ij}} &= p_j - k_{ij} \nonumber \\
\frac{v_{ij}}{p_j} &= \frac{\sum_{j = 1}^m v_{ij}x_{ij}}{e_i}\left(1 - \frac{k_{ij}}{p_j}\right) \label{ln:stationarity}
\end{align}
The primal and dual feasibility conditions tell us that:
$$-x_{ij} \leq 0, \sum_{i=1}^n x_{ij} - 1 \leq 0, p_j \geq 0, k_{ij} \geq 0$$
Finally, complementary slackness tells us that:
\begin{align}
x_{ij} > 0 \implies k_{ij} = 0 &\text{ and } k_{ij} > 0 \implies x_{ij} = 0 \label{ln:comp-slack}\\
p_j > 0 \implies \sum_{i=1}^n x_{ij} = 1 &\text{ and } \sum_{i=1}^n x_{ij} < 1 \implies p_j = 0 \label{ln:comp-slack-2}
\end{align}
KKT condition~\ref{eq: KKT 1} follows from Equation~\ref{ln:comp-slack-2}.
Meanwhile, we show that KKT conditions~\ref{eq: KKT 2} and~\ref{eq: KKT 3} are equivalent to the stationarity condition plus the first two complementary slackness conditions.
\begin{proposition}
For any $\bx, \bp$ where $p_j > 0$, KKT conditions~\ref{eq: KKT 2} and~\ref{eq: KKT 3} hold if and only if there exists $\bk$ such that Equations~\ref{ln:stationarity} and~\ref{ln:comp-slack} hold.
\end{proposition}
\begin{proof}
Consider any $\bx, \bp$.
We first show the forwards direction. We assume that KKT conditions~\ref{eq: KKT 2} and~\ref{eq: KKT 3} hold and give a $\bk$ such that Equations~\ref{ln:stationarity} and~\ref{ln:comp-slack} hold. Take any $i \in [n], j \in [m]$. We set the value of $k_{ij}$ depending on whether $x_{ij} > 0$. Suppose that $x_{ij} > 0$. Then observe that setting $k_{ij} = 0$ will satisfy both the stationarity and complementary slackness conditions. Otherwise, $x_{ij} = 0$. In this case, the slackness conditions trivially hold and we just need to show that there exists a $k_{ij} \geq 0$ such that Equation~\ref{ln:stationarity} holds. From KKT condition~\ref{eq: KKT 2}, we have:
$$\frac{v_{ij}}{p_j} / \frac{\sum_{j = 1}^m v_{ij}x_{ij}}{e_i} \leq 1$$
Letting $c = \frac{v_{ij}}{p_j} / \frac{\sum_{j = 1}^m v_{ij}x_{ij}}{e_i}$, we solve for $k_{ij}$ which gives $k_{ij} = p_j (1 - c)$, which is non-negative.
Next, we show the reverse direction. Assume that there exists $\bk$ such that Equations~\ref{ln:stationarity} and~\ref{ln:comp-slack} hold. For any $i \in [n], j \in [m]$, because we have that $p_j > 0$ and $k_{ij} \geq 0$, this implies that $1 - \frac{k_{ij}}{p_j} \leq 1$. Combine this with Equation~\ref{ln:stationarity} and we get exactly KKT condition~\ref{eq: KKT 2}.
Next, we show that KKT condition~\ref{eq: KKT 3} holds. Assume that $x_{ij} > 0$. Equation~\ref{ln:comp-slack} tells us that $k_{ij} = 0$. Therefore, $1 - \frac{k_{ij}}{p_j} = 1$ and applying Equation~\ref{ln:stationarity} tells us that $\frac{v_{ij}}{p_j} = \frac{\sum_{j = 1}^m v_{ij}x_{ij}}{e_i}$.
\end{proof}
\section{Improving Guarantee for Independent Agents}
\label{app:independent agents strong ef}
When the adversary is limited to distributions where agents' valuations are independent, under the mild assumption that the distributions $D_i$ are not point masses, we give an ex-post Pareto efficient allocation algorithm that guarantees envy-freeness with high probability. That is, it is possible to improve the ``or EF1'' part of the guarantee of Theorem~\ref{thm:POCR} when agents are independent and the distributions are not point masses.
Our approach builds on our structural result (Theorem~\ref{thm: structural}) obtained for correlated agents, which transforms the online problem into one of finding a fractional allocation for divisible items.
For independent agents, we can find a fractional allocation that is strongly envy-free and Pareto optimal. Under this stronger guarantee, the proof of Theorem~\ref{thm:POCR} gives that when the graph is strongly envy-free, using Algorithm~\ref{alg:POCR} guarantees envy-freeness with high probability.
\subsection{Strong envy-freeness example}
At first, it seems plausible that with independent agents and distributions that are not point masses, any solution to the E-G convex program will be strongly envy-free and Pareto optimal. Unfortunately, this is not the case. In this section, we give an example where there exists a fractional allocation that is both Pareto optimal and strongly envy-free. However, we show that this solution can be missed when choosing an arbitrary solution to the E-G convex program, or even by using the algorithm for finding CISEF and Pareto optimal solutions. We show how the E-G solution can be transformed into a strongly envy-free and Pareto optimal allocation, highlighting the key ideas behind our algorithm.
Consider the following fractional allocation problem for three agents. The agents' value distributions are as follows: For both $X \sim D_1$ and $X \sim D_2$, we have $\Pr[X = 0] = \frac{1}{10}, \Pr[X = 1] = \frac{9}{10}$. For $X \sim D_3$, we have $\Pr[X = 1] = \frac{16}{17}, \Pr[X = 2] = \frac{1}{17}$.
Recall that in the fractional allocation problem, we treat each item type $\gamma_j$ as a divisible item. Each agent $i$ has a value function $v'_i$, where $v'_i(\gamma_j) = v_i(\gamma_j) \cdot f_D(\gamma_j)$. We now present a solution $\bx, \bp$ to the E-G convex program with identical budgets of $1$, shown in Table~\ref{table:independent-agents-nonstrong}.
\setlength\tabcolsep{2pt}
\begin{table*}[ht]
\begin{center}
\[\begin{array}{ | c | c | c | c | c | c | c | c | c | c | c | c | }
\hline
\text{Item type $\gamma_j$} & v_{1t} & v_{2t} & v_{3t} & f_D(\gamma_j) & v'_{1t} & v'_{2t} & v'_{3t} & x_{1t} & x_{2t} & x_{3t} & p_t\\ [0.5ex]
\hline
\gamma_1 & 0 & 0 & 1 & \frac{16}{1700} & 0 & 0 & \frac{16}{1700} & 0 & 0 & 1 & \frac{16}{600}\\
\gamma_2 & 0 & 0 & 2 & \frac{1}{1700} & 0 & 0 & \frac{2}{1700} & 0 & 0 & 1 & \frac{2}{600} \\
\gamma_3 & 0 & 1 & 1 & \frac{144}{1700} & 0 & \frac{144}{1700} & \frac{144}{1700} & 0 & 0 & 1 & \frac{144}{600}\\
\gamma_4 & 0 & 1 & 2 & \frac{9}{1700} & 0 & \frac{9}{1700} & \frac{18}{1700} & 0 & 0 & 1 & \frac{18}{600} \\
\gamma_5 & 1 & 0 & 1 & \frac{144}{1700} & \frac{144}{1700} & 0 & \frac{144}{1700} & 0 & 0 & 1 & \frac{144}{600}\\
\gamma_6 & 1 & 0 & 2 & \frac{9}{1700} & \frac{9}{1700} & 0 & \frac{18}{1700} & 0 & 0 & 1 & \frac{18}{600} \\
\gamma_7 & 1 & 1 & 1 & \frac{1296}{1700} & \frac{1296}{1700} & \frac{1296}{1700} & \frac{1296}{1700} & \frac{75}{162} & \frac{75}{162} & \frac{12}{162} & \frac{1296}{600}\\
\gamma_8 & 1 & 1 & 2 & \frac{81}{1700} & \frac{81}{1700} & \frac{81}{1700} & \frac{162}{1700} & 0 & 0 & 1 & \frac{162}{600} \\
\hline
\end{array}\]
\caption{Fractional allocation problem. All items with positive value to an agent are also MBB items for that agent. The MBB ratio for agents $1$ and $2$ is $\frac{3}{17}$ and for agent $3$ is $\frac{6}{17}$.}
\label{table:independent-agents-nonstrong}
\end{center}
\end{table*}
$\bx, \bp$ are constructed such that each agent spends their entire budget of $1$ on their items. $\bp$ has the property that all items are bang-per-buck items for agent $3$. Agent $3$ has maximum bang-per-buck items $\gamma_5, \gamma_7$ while agent $2$ has MBB items $\gamma_3, \gamma_7$. The allocation $\bx$ satisfies the maximum bang-per-buck constraints. Therefore, $\bx, \bp$ is a solution.
The envy-graph for $\bx$ still contains four edges: $(1, 2), (2, 1), (3, 1), (3, 2)$. Now consider what happens if we apply the algorithm for finding a CISEF and Pareto optimal solution. Procedure $2$, as described in Lemma~\ref{lem:budget-change-shifts}, increases the budget of agent $3$ and gives agent $3$ more of $\gamma_7$. Concretely, we might increase the budget of agent $3$ by $\frac{16}{600}$ and decrease the budget of agents $1$ and $2$ by $\frac{8}{600}$, resulting in a new solution $(\bx' = X', \bp')$ where $\bp' = \bp$ and the only differences between $\bx'$ and $\bx$ are that $\bx'_{17} = \bx'_{27} = \frac{74}{162}$ and $\bx'_{37} = \frac{14}{162}$. The resulting envy-graph is now CISEF but not strongly envy-free, as it contains two edges: $(1, 2), (2, 1)$.
In this example, it is fairly easy to see how to remove the remaining two envy edges and maintain Pareto optimality. Suppose we transfer $\Delta$ worth of $\gamma_7$ from agent $1$ to $3$ and $\Delta$ worth of $\gamma_5$ from agent $3$ to $1$. We do a similar transfer for agents $2$ and $3$ with items $\gamma_7$ and $\gamma_3$. As agent $1$ does not value $\gamma_3$ and agent $2$ does not value $\gamma_5$, performing these two changes will remove the two envy edges. If we choose $\Delta$ small enough, no new envy edges are created.
\subsection{Strong Envy-freeness and Pareto Optimality}
The above approach can be generalized into an algorithm for finding a strongly envy-free and Pareto optimal allocation. For each distribution $D_i$, let $S_i$ be the support of $D_i$. The independent agent adversary can be simulated by the correlated agent adversary by expanding out the individual distributions into a distribution for item types $D$ with support $G_D$ where $G_D = S_1 \times S_2 \times \cdots \times S_n$. In this section, we will use two notations, $\gamma_j$ and $(a_1, \ldots, a_n)$, interchangeably to refer to an item type. As described in the Pareto optimal clique rounding algorithm, we define a new valuation function $v'_i$ for each agent $i$: $v'_i((a_1, \ldots, a_n)) = v_i((a_1, \ldots, a_n))f_D((a_1, \ldots, a_n)) = a_i \cdot f_D((a_1, \ldots, a_n))$.
\begin{theorem}
\label{thm:ind-sef}
Given an instance with items $G_D = S_1 \times S_2 \times \cdots \times S_n$, where for all $i$, $\abs{S_i} \geq 2$, and $n$ agents with valuation functions $v'_i((a_1, \ldots, a_n)) = a_i \cdot f_D((a_1, \ldots, a_n))$, there always exists an allocation that is strongly envy-free and Pareto efficient.
\end{theorem}
Applying Theorem~\ref{thm: structural}, we start with a solution $(\bx = X, \bp)$ to budgets $\be$ where $X$ is CISEF and Pareto optimal. Now suppose in $I(X)$ there exists a clique $K$ with size $k$. Without loss of generality, suppose that these are the first $k$ agents. For agents $i \in K$, let $h_i = \max S_i$ and $l_i = \min S_i$. These are agent $i$'s highest and lowest values for items prior to scaling by the probability of the item. Since $\abs{S_i} \geq 2$, $h_i > l_i$.
For any $(a_1, a_2, \ldots, a_k) \in S_1 \times S_2 \ldots \times S_k$, we denote the subset of items with those values with $G_{(a_1, a_2, \ldots, a_k)} = \{(b_1, b_2, \ldots, b_n) : b_1 = a_1, b_2 = a_2, \ldots, b_k = a_k, (b_1, b_2, \ldots, b_n) \in G_D \}$. In particular, we care about the subsets of items involving the high and low values for each agent. Let $H = (h_1, \ldots, h_k)$ and for each $i \in K$, let $H^i = (l_1, \ldots, l_{i - 1}, h_i, l_{i + 1}, \ldots l_k)$. The corresponding subsets with those values are $G_H$ and $G_{H^i}$.
For a $\gamma_j = (a_1, \ldots, a_n)$, we denote to the item's value to the remaining agents as $R(\gamma_j) = (a_{k+1}, \ldots, a_n)$. In particular, we are interested in the items in $G_H$ and $G_{H^i}$ with values $R(\gamma_j)$. Define $(H^i, R(\gamma_j)) = (l_1, \ldots, l_{i - 1}, h_i, l_{i+1}, \ldots l_k, a_{k + 1}, \ldots, a_n)$ and $(H, R(\gamma_j)) = (h_1, \ldots, h_k, a_{k+1}, \ldots, a_n)$.
Now consider the allocation of the agents in $K$. From the CISEF property, they all have the same allocation, denoted as $X_K$. We first show the following two claims:
\begin{lemma}
\label{lem:high-value-item}
There exists an item $\gamma_j \in G_H \cap X_K$.
\end{lemma}
\begin{lemma}
\label{lem:high-low-item}
For all items $\gamma_j \in X_K$, $\gamma_j \notin G_{H^1} \cup G_{H^2} \cup \cdots \cup G_{H^k}$.
\end{lemma}
\begin{proof}[Proof of Lemma~\ref{lem:high-value-item}]
We first show that for any $\gamma_j$ and Pareto optimal $X$, if $\gamma_j \in X_K$, then $(H, R(\gamma_j)) \in X_K$. Assume for sake of contradiction that $\gamma_j \in X_K$ but $(H, R(\gamma_j)) \notin X_K$. Let $\gamma_{j'} = (H, R(\gamma_j))$ and let agent $i$ be any agent not in $K$ where $\gamma_{j'} \in X_i$. From KKT condition~\eqref{eq: KKT 3}, item $j'$ must maximize agent $i$'s bang-per-buck, meaning:
$$\frac{v_{ij} \cdot f_D(\gamma_{j})}{p_{j}} = \frac{v'_{ij}}{p_{j}} \leq \frac{v'_{ij'}}{p_{j'}} = \frac{v_{ij'} \cdot f_D(\gamma_{j'})}{p_{j'}}$$
However, because $v_{ij} = v_{ij'}$, this implies that $\frac{f_D(\gamma_{j})}{p_{j}} \leq \frac{f_D(\gamma_{j'})}{p_{j'}}$. Meanwhile, item $j$ maximizes agent $1$'s bang-per-buck so:
$$\frac{h_1 \cdot f_D(\gamma_{j'})}{p_{j'}} = \frac{v_{1j'} \cdot f_D(\gamma_{j'})}{p_{j'}} = \frac{v'_{1j'}}{p_{j'}} \leq \frac{v'_{1j}}{p_{j}} = \frac{v_{1j} \cdot f_D(\gamma_{j})}{p_{j}} < \frac{h_1 \cdot f_D(\gamma_{j})}{p_{j}}$$
This implies that $\frac{f_D(\gamma_{j'})}{p_{j'}} < \frac{f_D(\gamma_j)}{p_j}$, contradicting the previous inequality. We know $X_K$ is non-empty, as otherwise the allocation $X$ would not be envy-free. Thus, there exists some $\gamma_j \in X_K$. This implies that $(H, R(\gamma_j)) \in X_K$ and also by definition, $(H, R(\gamma_j)) \in G_{H}$.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:high-low-item}]
From Lemma~\ref{lem:high-value-item}, there exists a $\gamma_j \in G_H \cap X_K$. Now consider an arbitrary $\gamma_{j'} \in G_{H^i}$ for some $i \in K$. As shown earlier in the proof of Lemma~\ref{lem:high-value-item}, if $v_{ij'} = v_{ij}$ and $\gamma_j$ is an MBB item for agent $i$, then $\frac{f_D(\gamma_{j'})}{p_{j'}} \leq \frac{f_D(\gamma_j)}{p_j}$.
However, for all other agents $i' \in K, i' \neq i$:
$$\frac{v'_{i'j'}}{p_{j'}} = \frac{v_{i'j'} \cdot f_D(\gamma_j')}{p_{j'}} \leq \frac{v_{i'j'} \cdot f_D(\gamma_j)}{p_{j}} < \frac{h_{i'} \cdot f_D(\gamma_j)}{p_{j}} = \frac{v'_{i'j}}{p_{j}}$$
In other words, item $j'$ is not a maximum bang-per-buck item for $i'$. As agents in $K$ have the same allocation $X_K$, this means that $\gamma_j \notin X_K$.
\end{proof}
Now, we introduce a third procedure to the indifference edge elimination algorithm:
\paragraph{Procedure $3$.}
Take a clique in the indifference graph and choose an item $\gamma_j \in G_H \cap X_K$, which exists by Lemma~\ref{lem:high-value-item}. We construct $X'$ by performing $k$ transfers. For each $i \in K$, by Lemma~\ref{lem:high-low-item}, there must exist some agent $i' \notin K$ where $(H^i, R(\gamma_j)) \in X_{i'}$. We define an item transfer $\Delta^i$ as: transfer $\delta^i$ worth of $(H^i, R(\gamma_j))$ from agent $i'$ to agent $i$, and $\delta^i$ worth of $\gamma_j$ from $i$ to $i'$.
This transfer has the property that both $\gamma_j$ and $(H^i, R(\gamma_j))$ are MBB items for agent $i$ and $i'$, but $(H^i, R(\gamma_j))$ is not an MBB item for all other agents $i^* \in K, i^* \neq i$. To show these properties, letting $\gamma_{j'} = (H^i, R(\gamma_j))$, we already know that $\gamma_j$ is an MBB item for agent $i$ and $\gamma_{j'}$ is an MBB item for agent $i'$. Because $v_{ij} = v_{ij'}$ and $v_{i'j} = v_{i'j'}$, we can show that $\frac{f_D(\gamma_{j'})}{p_{j'}} \leq \frac{f_D(\gamma_j)}{p_j}$ and also $\frac{f_D(\gamma_{j'})}{p_{j'}} \geq \frac{f_D(\gamma_j)}{p_j}$, implying that $\frac{f_D(\gamma_{j'})}{p_{j'}} = \frac{f_D(\gamma_j)}{p_j}$. Using this equality, we show:
$$\frac{v'_{ij}}{p_j} = \frac{v_{ij} \cdot f_D(\gamma_j)}{p_j} = \frac{v_{ij'} \cdot f_D(\gamma_{j'})}{p_{j'}} = \frac{v'_{ij'}}{p_{j'}}$$
Thus, items $j$ and $j'$ both maximize bang-per-buck for agent $i$. We can show the same for agent $i'$. Meanwhile, for agent $i^*$, the same equality shows $\gamma_{j'} = (H^i, R(\gamma_j))$ is not an MBB item:
$$\frac{v'_{i^*j'}}{p_{j'}} = \frac{v_{i^*j'} \cdot f_D(\gamma_{j'})}{p_{j'}} < \frac{v_{i^*j} \cdot f_D(\gamma_{j})}{p_{j}} = \frac{v'_{i^*j}}{p_{j}}$$
The only remaining condition for $\Delta = \sum_i \Delta^i$ to be an optimal transfer is to set the $\delta^i$ small enough that $X + \Delta$ is feasible.
Next, for any two agents $i, j \in K$, we want to show that this transfer eliminates the edge $(i, j)$. $X' = X + \sum_i \Delta^i$. Transfer $\Delta^i$ will eliminate the edge $(j, i)$ for all $j \in K, j \neq i$ since agent $i$ now owns an item that is not an MBB item for agent $j$. Finally, to ensure no new indifference edges or envy-freeness violations are created, we set $\delta^i = \frac{b}{k}$, where $b$ is chosen using the method discussed in Appendix~\ref{app:choosing b}. The extra $\frac{1}{k}$ factor ensures that we can perform all $k$ transfers safely.
One issue is that $X'$ might not be CISEF. Consider when the recipient in $\Delta^i$, agent $i'$, is part of clique $K'$ with $\abs{K'} > 1$. In this case, we change $X_{i'}$ without changing the allocation for other agents in $K'$. To handle this case, let $X_{K'}$ be the allocation for agents in $K'$. Any transfer of items to and from the agent can instead be spread out across all agents in the clique. Specifically, the final allocation is $X^*$, where $X^*_{K'} = X_{K'} + \frac{1}{\abs{K'}}(X'_{i'} - X_{i'})$ for the relevant $i', K'$ and the same as $X'$ otherwise. This allocation can be shown to be CISEF and Pareto optimal.
\begin{proof}[Proof of Theorem~\ref{thm:ind-sef}]
From Theorem~\ref{thm: structural}, we start with a solution $(\bx = X, \bp)$ to budgets $\be$ where $X$ is CISEF and Pareto optimal. Starting from allocation $X$, we repeat procedure $3$ while the indifference graph still contains cliques (of size greater than $1$). Each time we perform procedure $3$, we eliminate one clique without creating new indifference edges (or envy violations). Since procedure $3$ describes an optimal transfer, the resulting allocation is still Pareto optimal. Once we eliminate all cliques of size greater than $1$, the resulting indifference graph only contains cliques of size $1$ and thus, the allocation is strongly envy-free and Pareto optimal.
\end{proof}
\section{When Fairness and Efficiency are Compatible}\label{sec:compatible}
In this section we go over the first three, weaker adversaries, under which fairness and efficiency are compatible.
We start with the easiest setting, identical agents with i.i.d. items, in Section~\ref{sec:util}. We then proceed, in Sections~\ref{subsec:warm up} and~\ref{subsec:mainalgo}, to our main positive result, an algorithm for correlated agents with i.i.d. items that gives the optimal fairness and efficiency guarantees. This, of course, implies the same result for independent agents with i.i.d. items. In Section~\ref{subsec:warm up} we highlight some key insights, while avoiding some of the technical obstacles, and give an ex-post Pareto efficient algorithm with a slightly weaker fairness guarantee. We give our main algorithm in Section~\ref{subsec:mainalgo}.
\subsection{Identical Agents with i.i.d. Items}
\label{sec:util}
Consider an adversary that picks a single distribution $D$, with support $G_D$ of size $m$, and $v_{it}$ is sampled i.i.d. from $D$, for all agents $i$ and all items $t$. Then, efficiency and fairness can be simultaneously achieved using the following algorithm.
\begin{algorithm}
\eIf{$D$ is a point mass}
{Allocate each arriving item in a round-robin manner.}
{For each arriving item $t$, allocate it to the agent $i$ with the maximum value $v_{it}$, breaking ties uniformly at random.}
\caption{Utilitarian}\label{alg:util}
\end{algorithm}%
\begin{proposition}\label{prop:util-max-properties}
Algorithm~\ref{alg:util} outputs an allocation that is always Pareto efficient. Furthermore, for all $\varepsilon > 0$, there exists $T_0 = T_0(\varepsilon)$, such that if $T \geq T_0$, the output allocation satisfies EF1 or is envy-free with probability at least $1-\varepsilon$.
\end{proposition}
This result is essentially given by~\citet{DickersonGKPS14}, in a different context.~\citet{DickersonGKPS14} show that in a \emph{static} setting with $K$ items and $n$ agents, where the value of agent $i$ for item $t$ is drawn from a distribution $D_{i}$, then under mild conditions on the distributions, an envy-free allocation exists with probability $1$ as $K$ goes to infinity.
\begin{theorem}[\citet{DickersonGKPS14}]\label{thm: john et al}
Assume that for all $i,j \in \mathcal{N}$ and items $t$ the input distributions satisfy (1) $\Pr[ \arg\max_{k \in \mathcal{N}} v_{kt} = \{ i \} ] = 1/n$, and (2) there exist constants $\mu, \mu^*$ such that $$
0 < \E[ v_{it} | \arg\max_{k \in \mathcal{N}} v_{kt} = \{ j \} ] \leq \mu < \mu^* \leq \E[ v_{it} | \arg\max_{k \in \mathcal{N}} v_{kt} = \{ i \} ].$$
Then for all $n \in O( K/ \log K )$, allocating each item to the agent with the highest value is envy-free with probability $1$ as $K \rightarrow \infty$.
\end{theorem}
The first condition implies that each player receives roughly $1/n$ of the goods, and the second condition ensures that each player has higher expected value for each of his own goods compared to goods allocated to other players. Given this result, it is straightforward to prove Proposition~\ref{prop:util-max-properties}. The proof is relegated to Appendix~\ref{app: missing proposition}.
\subsection{Warm-up: Pareto Optimal Rounding}\label{subsec:warm up}
One might ask if we can keep the simplicity of Algorithm~\ref{alg:util} and extend it to work with a stronger adversary, such as when agent's valuations are independent but not necessarily identically distributed. Note that in this case, asking for the probability that agent $i$ has the highest value to be $1/n$ is a fairly strong requirement, so the result of~\citep{DickersonGKPS14} no longer applies. One possible approach is to assign item $t$ to the agent $i$ for whom $F_{D_i}(v_{it})$ is highest, where $F_{D_i}$ is the quantile function for agent $i$'s value distribution. In fact, this approach is fruitful if one focuses solely on fairness, as shown by~\citet{kurokawa2016can}. However, the resulting allocation is not guaranteed to be Pareto efficient. For example, consider a simple scenario with 2 agents: the first agent has values distributed uniformly in $[0,1]$, and the second agent has values uniformly distributed in $[1/2 - \varepsilon, 1/2 + \varepsilon]$ for some small $\varepsilon$. The second agent cares mostly about the number of items she got; items with high quantiles and low quantiles have essentially the same value for her. Therefore, intuitively, exchanging items that agent $2$ won but have large values for agent $1$ (i.e. items where both had high quantiles and $2$ ended up larger) for items that agent $1$ won with low valuation (i.e. items where both had low quantile and $1$ ended up larger) will result in a Pareto improvement.\footnote{For a specific instance, consider the allocation for the following three items: $(0.9, \frac{1}{2} + \varepsilon), (0.1, \frac{1}{2} - \varepsilon), (0.1, \frac{1}{2} - \varepsilon)$. The second agent receives the first item, while the first agent receives the latter two, which is not Pareto optimal.}
All in all, achieving fairness and efficiency simultaneously, beyond identical agents, seems a lot more intricate than fairness or efficiency in isolation. We will skip the independent agent case altogether, and directly study the harder problem of correlated agents: each item $t$ draws its type $\gamma_j$ from a distribution $D$. Items are i.i.d. but agent values can be correlated. Before we present the optimal algorithm we illustrate some key ideas by giving a simple algorithm that achieves ex-post Pareto efficiency and a weaker notion of fairness, namely vanishing envy with high probability. Recall that $f_D(\gamma_j)$ is the probability that an item drawn from $D$ has type $\gamma_j$, and $G_D$, with $|G_D|=m$, is the support of $D$. $v_{i}(\gamma_j)$ is the value of an item of type $\gamma_j$ to agent $i$; note that this is different than $v_{it}$, the value of agent $i$ for the $t$-th item to arrive.
Although the original problem concerns indivisible items, our approach solves an offline divisible item allocation problem as an intermediate step. Rather than each item being assigned to an agent, a fractional allocation is an $n$ by $m$ matrix $X$, where $n$ is the number of agents and $m$ is the support of $D$. For each agent $i$ and item $j$, $X_{ij} \in [0,1]$ represents the proportion of item $j$ allocated to agent $i$. $X$ is constrained so that $\sum_{i \in \mathcal{N}} X_{ij} = 1$ for all items $j$. The $i^{th}$ row of $X$, $X_i$, can be thought of as the fractional allocation of agent $i$. Finally, we assume agents have linear preferences, where for any two agents $i, j$, $v_i(X_j) = \sum_{k \in [m]} v_{ik} X_{jk}$.
\begin{algorithm}
\leavevmode \newline Input: Distribution $D$ over item types, agent valuation functions $v_i$.
\begin{enumerate}
\item Define $v'_i$ for each agent $i$ as follows. For each $\gamma_j \in G_D$, set $v'_i(\gamma_j) = v_i(\gamma_j) f_D(\gamma_j)$.
\item Find the divisible allocation $X$ of $G_D$ that maximizes the product of utilities w.r.t $v'$.
\item In the online setting, allocate the newly arrived item $t$ with type $\gamma_j$ to agent $i$ with probability $X_{ij}$, for all $t = 1 \dots T$.
\end{enumerate}
\caption{Pareto Optimal Rounding}\label{alg:POR}
\end{algorithm}
We first show that Algorithm~\ref{alg:POR} always produces a Pareto optimal allocation. In fact, we show something much stronger: allocating type $\gamma_j$ to agent $i$ with probability $X_{ij}$ always results in a Pareto efficient allocation (ex-post) for \emph{every} Pareto efficient fractional allocation $X$.
\begin{lemma}
\label{lem:por-po}
Given a distribution $D$ over $m$ item types and valuation function $v_i$ for each agent $i$, let $X$ be a fractional and Pareto optimal allocation of $G_D$ under valuation functions $v'_i$, where $v'_i (\gamma_j) = v_i(\gamma_j) \cdot f_D(\gamma_j)$. Let $S$ be a set of $T$ items, where the type of the $t$-th item is drawn from $D$. Let $A = (A_1, \ldots, A_n)$ be any allocation of $S$ where an item with type $\gamma_j$ is allocated to agent $i$ only if $X_{ij} > 0$.
Then $A$ is Pareto optimal under $v$.
\end{lemma}
\begin{proof}
We prove the theorem in two steps. First, we show that $X$ is also Pareto optimal under $v$. Second, we show that if $X$ is Pareto optimal under $v$, then $A$ will be Pareto optimal under $v$.
First, suppose that $X$ is not Pareto optimal under $v$. Let $X'$ be an allocation that dominates $X$ under $v$ and let $\Delta = X' - X$. We construct $\Delta'$, where $\Delta'_{ij} = \Delta_{ij} \cdot \frac{1}{f_D(\gamma_j)}$, so that the change in utilities induced by $\Delta'$ under $v'$ is the same as the change in utilities induced by $\Delta$ under $v$. Therefore, the (possibly infeasible) allocation $X + \Delta'$ dominates $X$ under $v'$. However, for all $c \in [0,1]$, allocation $X + c\Delta$ is feasible and still dominates $X$ under $v$. Setting $c = \min_k f_D(\gamma_k)$, $X + c\Delta'$ is a feasible allocation that Pareto dominates $X$ under $v'$, which contradicts the Pareto optimality of $X$ under $v'$.
Next, let $\mathcal{A}$ be the set of all allocations $A$ (of $T$ items) such that for all $i \in \mathcal{N}$ and $j \in S$ an item with type $\gamma_j$ is allocated to agent $i$ only if $X_{ij} > 0$. We can write $X$ as $\sum_{A \in \mathcal{A}} p_A \cdot A$, such that $p_A > 0$ for all $A$, and $\sum_{A \in \mathcal{A}} p_A = 1$. This can be done by simply giving each item $j \in S$ to agent $i$ with probability $X_{ij}$. It is easy to confirm that the set of all possible allocations that can be outputs by this procedure is precisely $\mathcal{A}$, and the probability that a specific allocation $A$ is generated is exactly $p_A = \prod_{j \in S} \prod_{i \in \mathcal{N}} X_{ij} \cdot \mathds{1}\{ \text{$i$ gets item $j$ in $A$} \}$. Finally, since $X = \sum_{A \in \mathcal{A}} p_A \cdot A$, if an allocation $\hat{A} \in \mathcal{A}$ is Pareto dominated by an allocation $\hat{A}'$ under $v$, by replacing the $p_{\hat{A}} \hat{A}$ term by a $p_{\hat{A}} \hat{A}'$ term we get a fractional allocation $X'$ that dominates $X$, contradicting the Pareto optimality of $X$ under $v$.
\end{proof}
Maximizing the product of utilities leads to a Pareto efficient divisible item allocation. Therefore, Lemma~\ref{lem:por-po} implies that Algorithm~\ref{alg:POR} is ex-post Pareto efficient. Algorithm~\ref{alg:POR} guarantees a notion of fairness slightly weaker than vanishing envy: vanishing envy with high probability.
\begin{theorem}
\label{thm:por-ve}
For all $\varepsilon > 0$, there exists $T_0 = T_0(\varepsilon)$, such that if $T \geq T_0$, Algorithm~\ref{alg:POR} outputs an allocation $A$ such that for all agents $i$, $j$, $\mathsf{ENVY}_{i, j}(A) \in o(T)$ with probability at least $1-\varepsilon$.
\end{theorem}
\begin{proof}
The divisible allocation $X$ that maximizes the product of utilities is envy-free~\cite{varian1974equity}.
Thus, we have $\sum_{k \in [m]} v_i(\gamma_k)f_D(\gamma_k)X_{ik} \geq \sum_{k \in [m]} v_i(\gamma_k)f_D(\gamma_k)X_{jk}$ for all pairs of agents $i, j$.
The value of agent $i$ for the bundle allocated to agent $j$ by Algorithm~\ref{alg:POR}, $v_i(A_j)$, is a random variable depending on randomness in both the algorithm and item draws. Let $I^{k, j}_t$ be an indicator random variable for the event that item $t$ is of type $\gamma_k$ and is assigned to agent $j$. We have for any pair of agents $i, j$: $v_i(A_j) = \sum_{t \in [T]} \sum_{k \in [m]} v_i(\gamma_k) I^{k, j}_t$. Therefore, $\E[v_i(A_j)] = T \cdot \sum_{k \in [m]} v_i(\gamma_k)f_D(\gamma_k)X_{jk}$. From the previous discussion, we then have that $\E[v_i(A_i)] \geq \E[v_i(A_j)]$.
Using Hoeffding's inequality with parameter $\delta = \sqrt{T\log{T}}$ we get: $\Pr[v_i(A_i) - \E[v_i(A_i)]] \leq -\sqrt{T\log{T}}] \leq 2 \exp\left(-\frac{2T\log{T}}{T}\right) = \frac{2}{T^2},$
and similarly for the deviation of $v_i(A_j)$. We conclude that by picking $T_0 = \sqrt{4/\varepsilon}$, then with probability at least $1 - \frac{4}{T^2}\geq 1-\varepsilon$, $\mathsf{ENVY}_{i, j}(A) = \max \{ v_i(A_j) - v_i(A_i), 0 \} \leq 2\sqrt{T \log T} \in o(T)$.
\end{proof}
\subsection{Beyond Vanishing Envy: Optimal Fairness for Correlated Agents}
\label{subsec:mainalgo}
In the proof of Theorem~\ref{thm:por-ve} we use standard tail inequalities to show that, with high probability, the envy between any two agents does not deviate from its expectation by more than $O(\sqrt{T \log{T}})$. Since the divisible allocation we are using to round guarantees envy-freeness, this leads to vanishing envy. If instead we were able to find an allocation $X$ for the divisible item problem that guarantees \textit{strong envy-freeness}, where for every pair of agents $i, j$, $v_i(X_i) > v_i(X_j)$, and used $X$ to guide the online decisions, by a similar calculation the final allocation would be envy-free with high probability, since $\E[v_i(A_i)] - \E[v_i(A_j)]$ decreases much faster than the deviation bound of $\sqrt{T \log {T}}$. Unfortunately, strong envy-free allocations do not always exist, even for divisible items; consider the case of two agents with identical valuations. Interestingly, in isolation this condition is also sufficient: if no two players have identical valuation functions (up to multiplicative factor), there exists a strongly envy-free allocation; see~\citet{barbanel2005geometry}.
However, if we want both Pareto optimality and strong envy-freeness, the condition is no longer sufficient; see Appendix~\ref{app:sef-po-nonident-cex} for an example\footnote{\citet{barbanel2005geometry} gives a sufficient condition for Pareto optimality and strong envy-freeness to be simultaneously achievable. Unfortunately, this condition cannot be satisfied here: it roughly asks for every pair of agents to have a different value for \emph{every} subset (allowing for fractional items) of items. See Theorem 12.36 in \citet{barbanel2005geometry} for the exact statement and proof.}.
Nevertheless, we can still achieve a notion of fairness that is weaker than strong envy-freeness, but still sufficient for our purposes. We say that agent $i$ is indifferent to agent $j$ if $v_i(X_i) = v_i(X_j)$. We then view indifference as a directed graph, where the nodes consist of the $n$ agents and there is an edge from $i$ to $j$ if $i$ is indifferent to $j$. For a divisible item allocation $X$, let $I(X)$ refer to the indifference graph. Note that for an envy-free allocation $X$, lack of an edge from node $i$ to node $j$ in $I(X)$ implies that $v_i(X_i) > v_i(X_j)$, i.e. strong envy-freeness between the two agents.
\begin{definition}\label{def:cisef}
A divisible allocation $X$ is \textit{clique identical strongly envy-free} (CISEF) if $X$ is envy-free and there exists a partition of agents $\mathcal{N}$ into $s$ sets $C_1, \ldots, C_s$ such that: (1) For each $i \in [s]$, $C_i$ is a clique in $I(X)$, (2) for each $i, j \in [s]$ where $i \neq j$, there are no edges between $C_i$ and $C_j$, (3) for each $i \in [s]$, for each pair of agents $j, k \in C_i$, $X_j = X_k$, and (4) there exists positive scalars $r_1, \ldots, r_n$ such that for each $i \in [s]$, pair of agents $j, k \in C_i$ and item $l$ where $X_{jl} > 0$ (or $X_{kl} > 0$), $\frac{v_{jl}}{r_j} = \frac{v_{kl}}{r_k}$.
\end{definition}
\vspace{-2mm}
In other words, an allocation is CISEF if $X$ is envy-free, the graph $I(X)$ is a disjoint union of cliques, agents in a clique have identical allocations, and agents have identical valuations (up to a multiplicative factor) over items allocated to any agent in the clique.
The intuition behind this definition is the following. As discussed in Appendix~\ref{app:sef-po-nonident-cex}, it is impossible to find a strongly envy-free and Pareto optimal allocation in situations where some agents have valuations that are identical for the items that they could receive in a Pareto optimal allocation. In the definition of CISEF, these too similar agents will be given identical allocations and end up in the same clique. However, in all other cases, the allocation is strongly envy-free.
In Theorem~\ref{thm:main-disjoint-cliques} in Section~\ref{sec: CISEF} we prove our main structural result. We will show that for any divisible item problem instance, we can find an allocation $X^*$ that is both Pareto efficient and CISEF. In the remainder of this section we present a slightly modified version of Algorithm~\ref{alg:POR} that, given such a Pareto efficient and CISEF divisible allocation, remains Pareto efficient ex-post and also achieves the target fairness properties.
\begin{algorithm}
Input: Item distribution $D$, agent valuation functions $v_i$.
\begin{enumerate}
\item Define $v'_i$ for each agent $i$ as follows. For each $\gamma_j \in G_D$, set $v'_i(\gamma_j) = v_i(\gamma_j) f_D(\gamma_j)$.
\item Compute a divisible allocation $X^*$ of $G_D$ under valuation functions $v'_i$ that is both Pareto efficient and CISEF (see Section~\ref{sec: CISEF} for an algorithm). Let $C_1, \cdots, C_s$ be the disjoint cliques of the graph.
\item In the online setting, assign the newly arrived item $t$ with type $\gamma_j$ to clique $C_i$ with probability $\sum_{k \in C_i} X^*_{kj}$. After an item is assigned to a clique $C_i$, allocate it among the agents in $C_i$ by giving it to the agent in $C_i$ who has received the least value so far, according to \emph{all} agents in the clique.\footnote{$i$ thinks that $argmin_{j \in \mathcal{N}} v_i(A^t_j)$ has the smallest value so far. We select the agent with the smallest value according to all $i \in \mathcal{N}$, which is unique, up to tie breaking, since all agents agree on the value of all items that have gone to the clique (up to multiplicative factors).}
\end{enumerate}
\caption{Pareto Optimal Clique Rounding}\label{alg:POCR}
\end{algorithm}
\begin{theorem}\label{thm:POCR}
Algorithm~\ref{alg:POCR} is ex-post Pareto efficient and, for all $\varepsilon > 0$, there exists $T_0 = T_0(\varepsilon)$, such that if $T \geq T_0$, for all agents $i, j$, one of the two guarantees holds: either $i$ envies $j$ by at most one item with probability $1$ or $i$ will not envy $j$ with probability at least $1-\varepsilon$.
\end{theorem}
\begin{proof}
Let $A$ be the allocation produced by Algorithm~\ref{alg:POCR} and $X^*$ be the fractional allocation found for the divisible item allocation problem. Pareto efficiency is immediately implied by Lemma~\ref{lem:por-po}, since $X^*$ is Pareto efficient and an item of type $\gamma_j$ is allocated to agent $i$ only if $X^*_{ij} > 0$.
For any two agents $i, j$, there are two cases. In the first case, $i$ and $j$ belong to the same clique $C_k$. Let $S$ be the set of items assigned to $C_k$ during the execution of Algorithm~\ref{alg:POCR}, i.e. $S = \cup_{\ell \in C_k} A_\ell$.
Agents in $C_k$ have identical valuations up to a multiplicative factor for the items that they get with positive probability. Therefore, giving each item to the agent that has received the least value so far (according to any agent, as they rank allocations of $S$ in the same order) ensures that $\mathsf{ENVY}_{i,j}(A)$ is at most the maximum value that $i$ has for any item in $S$.
Now consider the case that $i$ and $j$ belong to different cliques, $C_i$ and $C_j$ respectively. By the definition of a CISEF allocation, we know that $v_i(X^*_i) = v_i(X^*_j) + c$ for some constant $c>0$. Let $\tilde{A}_i$ be the fractional allocation where agent $i$ receives a $1/\abs{C_i}$ fraction of the items assigned to her clique, i.e. $\tilde{A}_{it} = \frac{1}{\abs{C_i}} \mathds{1} \{ t \in A_k \text{ for some agent $k \in C_i$} \}$, and similarly for $\tilde{A}_j$. Since agents in a clique receive identical allocations in $A$ we have that $\E[v_i(\tilde{A}_i) - v_i(\tilde{A}_j)] = T\E[v_i(X^*_i) - v_i(X^*_j)] = cT$.
We can apply Hoeffding's inequality to show that with probability at least $1-\Theta(1/T^2) \geq 1-\varepsilon$, $v_i(\tilde{A}_j) - v_i(\tilde{A}_i) < 2\sqrt{T \log{T}} - cT$, which is negative for sufficiently large $T$. Specifically, we can pick $T$ large enough so that $v_i(\tilde{A}_j) - v_i(\tilde{A}_i) < -2$ with high probability. Finally, observe that $\abs{v_i(A_i) - v_i(\tilde{A}_i)} \leq 1$. This is because $\tilde{A}_i$ is the average allocation of agents in the clique, and, as we argued earlier, the maximum envy for two agents in the same clique is at most $1$, thus $v_i(A_j) - v_i(A_i) < v_i(\tilde{A}_j) - v_i(\tilde{A}_i) + 2$. Combined with the deviation bound, we conclude that $v_i(A_j) - v_i(A_i) \leq 0$ with probability at least $1-\varepsilon$.
\end{proof}
\section{Conclusion and Future Directions}\label{sec: conclusion}
In this paper we study whether fairness and efficiency can be balanced in an online setting under a spectrum of different adversary models. We show tight upper and lower bounds for each setting. For the adaptive adversary, and therefore for all the other, weaker ones as well, the upper bound is given by~\cite{BKPP18}: an algorithm with vanishing envy that gets a $1/n$ approximation of Pareto. We show that this result is essentially tight (Theorem~\ref{thm:eff-envy-lb-det}) for the non-adaptive adversary (and therefore for all the stronger ones) if one wants to balance fairness and efficiency. Our main upper bound (Theorem~\ref{thm:POCR}) is for correlated agents and i.i.d. items and shows how to get Pareto efficiency ex-post and guarantee EF1 ex-post, or envy-freeness with high probability. We argue (Appendix~\ref{app:optimal-fairness-guarantee}) that one cannot hope for a better fairness guarantee even for identical agents, and since the efficiency guarantee is the strongest one, we conclude that this result is also tight. Finally, even though our main upper bound of course holds for identical agents, for this case we show (Proposition~\ref{prop:util-max-properties}) how to adapt the approach of~\citet{DickersonGKPS14} and~\citet{kurokawa2016can} and get the same tight result, but with a simpler algorithm.
\paragraph*{Computation Considerations.}
Given our structural result, Theorem~\ref{thm: structural}, it is clear that all our results run in polynomial time. To get our structural result we assume an exact solution to the E-G program, that we can get in strongly polynomial time via the results of~\citet{orlin2010improved}. Our edge-elimination processes only runs $O(n^2)$ times. The only possible issue is the number of bits in the solution $(\bx, \bp)$ and budgets $\be$, as the item transfers described in Lemma~\ref{lem:const-util-shifts} and~\ref{lem:budget-change-shifts} can both increase the length (in bits) of $\bx$ and $\be$. This increase depends primarily on the budget transfers $b$ (computed in Appendix~\ref{app:choosing b}). We can always choose $b$ such that $b$ is equal to $(v_i(A_i) - v_i(A_j))/4c$ for some $i, j$, where $c$ is a constant that only depends on the instance and the initial solution to the E-G program. Since $v_i(A_i), v_i(A_j)$ are linear functions of $\bx$, their representations are a constant (depending on the $v_{ij}$ and $n$) larger than the elements of $\bx$. In addition, the representation of $b$ is an additive constant larger than the bit length of the min difference $v_i(A_i)- v_i(A_j)$, so performing the transfer of items under budget $b$ will also only increase the bit length of the elements of $\bx, \be$ by a constant.
\paragraph*{Future directions.}
Notice that, with the exception of the non-adaptive adversary, we completely understand what is possible if we want to optimize only fairness or only efficiency. On the other hand, for the non-adaptive adversary, it remains open how well one can do with respect to minimizing the maximum (expected) pairwise envy. Vanishing envy is of course achievable, but we do not know if it's possible to improve the $\tilde{O}(\sqrt{T})$ guarantee, or how big that improvement could be; for instance, we don't even know of a super constant lower bound. Another technical question that remains open is what is possible when the distributions chosen by the adversary can depend on $T$.
Finally, a general direction for future work is balancing fairness and efficiency under different adversaries (for example, one could imagine adversaries weaker than the non-adaptive one, but stronger than correlated agents with i.i.d. items), or under different definitions of ``fair'' and ``efficient''.
\section{When Fairness and Efficiency are Incompatible}\label{sec:incompatible}
In this section we discuss the tradeoff between efficiency and fairness against the latter two, stronger adversaries. Our main conclusion is, informally, that no allocation algorithm with vanishing envy Pareto dominates random allocation.~\citet{BKPP18} show that against an adaptive adversary (and thus against all weaker adversaries), random allocation has vanishing envy. Allocating at random is also $1/n$-Pareto efficient ex-ante.
Here, we show that no algorithm can achieve vanishing envy and expected $(\frac{1}{n} + \varepsilon)$-Pareto efficient for any $\varepsilon > 0$, even against the non-adaptive adversary.
\begin{theorem}
\label{thm:eff-envy-lb-det}
For any $\varepsilon > 0$, no randomized allocation algorithm can achieve both expected envy $f(T)$, where $f(T) \in o(T)$, and $(\frac{1}{n} + \varepsilon)$-Pareto efficient ex-ante against a non-adaptive adversary.
\end{theorem}
\begin{proof}
To build up intuition, we first describe the lower bound for the case of an adaptive adversary. Since against an adaptive adversary randomization does not help, we focus on deterministic algorithms. Consider any vanishing envy algorithm that for all $T$ produces an allocation $A^T$, where $\mathsf{ENVY}(A^T) \leq f(T)$ for some $f(T) \in o(T)$, and assume, for the sake of contradiction, that this algorithm achieves $(\frac{1}{n} + \varepsilon)$-Pareto efficiency for some $\varepsilon > 0$.
Consider the following instance $I$, parameterized by $\varepsilon$ and $T$. For each agent $i$, items $j$ from $\frac{T}{n}(i - 1)$ to $\frac{T}{n}i$ will have value $1$, and all other items $j'$ have value $ \varepsilon$. For all intermediate allocations at time $t \leq T$, we must have $\mathsf{ENVY}(A^t) \leq f(T)$ since an adaptive adversary can make the remaining items valueless to all agents. We start by showing via induction that for all ``segments'' of items $\frac{T}{n}(i - 1)$ to $\frac{T}{n}i$, each agent must receive a number of items in $[\frac{T}{n^2} - x_i, \frac{T}{n^2} + x_i]$, where $x_i = \frac{f(T)}{\varepsilon} \left(1 + \frac{2}{\varepsilon}\right)^{i - 1}$.
For $i=1$, i.e. the first segment, suppose that some agent $k$ receives $\frac{T}{n^2} + y$ items where $y > 0$. Another agent $\hat{k}$ must then receive fewer than $\frac{T}{n^2}$ items. Then, the envy of $\hat{k}$ for $k$ at the end of the first segment, $\mathsf{ENVY}_{\hat{k}, k}(A^{T/n})$ is at least $\varepsilon \cdot y$. But, $\mathsf{ENVY}_{\hat{k}, k}(A^{T/n}) \leq f(T)$, which implies that $y \leq \frac{f(T)}{\varepsilon}$; the lower bound on $y$ is identical. For the inductive step, again suppose that in the segment $\frac{T}{n}(i - 1)$ to $\frac{T}{n}i$ some agent $k$ receives $\frac{T}{n^2} + y$ items, where $y > 0$, and let $\hat{k}$ be the agent who received fewer than $\frac{T}{n^2}$ items. At the start of the segment, to agent $\hat{k}$, the difference in value between agent $k$'s bundle and their bundle is at least $-2 \sum_{i' < i} x_{i'}$, which is if $\hat{k}$ got $\frac{T}{n^2} + x_{i'}$ in each segment, $k$ got $\frac{T}{n^2} - x_{i'}$, and $\hat{k}$ had value $1$ for all items up until $\frac{T}{n}(i - 1)$. Thus, after the $i$-th segment, $\mathsf{ENVY}_{\hat{k}, k}(A^{\frac{T}{n}i}) \geq \varepsilon \cdot y + v_{\hat{k}}\left(A^{\frac{T}{n}(i-1)}_{k}\right) - v_{\hat{k}}\left(A^{\frac{T}{n}(i-1)}_{\hat{k}}\right) \geq \varepsilon \cdot y - 2 \sum_{i' < i} x_{i'},$ which in turn implies that
\[ \textstyle y \leq \frac{1}{\varepsilon}\left(f(T) + 2\sum_{i' < i}x_{i'}\right) = \frac{1}{\varepsilon}\left(f(T) + 2 \sum_{i' < i} \frac{f(T)}{\varepsilon} \left(1 + \frac{2}{\varepsilon}\right)^{i' - 1} \right) = \frac{f(T)}{\varepsilon} \left(1 + \frac{2}{\varepsilon} \right)^{i-1}.
\]
The last equality is obtained by summing the geometric series and simplifying. The bound for $y$ is identical when we consider the case that $y < 0$.
Next, we show that this implies the allocation $A^T$ is not $(\frac{1}{n} + \varepsilon)$-Pareto efficient. First, note that the social welfare maximizing allocation achieves utility $(\frac{T}{n}, \ldots, \frac{T}{n})$, by giving all the items of the $i$-th segment to agent $i$. Meanwhile, since $x_i < x_n$, we have that in $A^T$ each agent gets utility $u_i$ at most $(1 + (n - 1)\varepsilon)(\frac{T}{n^2} + x_n)$. Therefore,
\begin{align*}
\frac{u_i}{1/n + \varepsilon} &< (1 + (n - 1)\varepsilon)\left(\frac{T}{n^2} + x_n\right)\left(\frac{1}{\frac{1}{n} + \varepsilon}\right)= (1 + (n - 1)\varepsilon)\left(\frac{T}{n^2} + \frac{f(T)}{\varepsilon} (1 + \frac{2}{\varepsilon})^{n - 1}\right)\frac{n}{1 + \varepsilon n} \\
&= \frac{1 + (n - 1)\varepsilon}{1 + \varepsilon n} \cdot \left(\frac{T}{n} + n\frac{f(T)}{\varepsilon}(1 + \frac{2}{\varepsilon})^{n - 1}\right) =\frac{T}{n} \cdot \frac{1 + (n - 1)\varepsilon}{1 + \varepsilon n} \cdot \left(1 + \frac{n^2}{\varepsilon} \frac{f(T)}{T}(1 + \frac{2}{\varepsilon})^{n - 1}\right).
\end{align*}
Picking $T$ large enough so that $\frac{f(T)}{T} < \frac{\varepsilon}{1+(n-1)\varepsilon} \cdot \frac{\varepsilon}{n^2 (1 + 2/ \varepsilon)^{n - 1} }$ gives $u_i < \frac{T}{n} \cdot ( 1/n + \varepsilon )$ for every agent $i$. Therefore, $A^T$ is not $(\frac{1}{n} + \varepsilon)$-Pareto efficient, a contradiction. This concludes the proof for an adaptive adversary.
We can use the above result to prove the same for a non-adaptive adversary; details are relegated to Appendix~\ref{app: lower bound proof continued}.
\end{proof}
\section{Introduction}
A set of $T$ indivisible goods has to be allocated to a set of $n$ agents with additive utilities, in a way that is \textit{fair} and \textit{efficient}. A standard fairness concept is envy-freeness, which requires that each agent prefers her own allocation over the allocation of any other agent. Even though envy is clearly unavoidable in this context --- consider the case of a single indivisible good and two agents --- providing \emph{approximately} envy-free solutions is possible~\cite{caragiannis2016unreasonable,lipton2004approximately}. Specifically, an allocation is envy-free up to one item (EF1) if for every pair of agents $i$ and $j$, any envy $i$ has for $j$ can be eliminated by removing at most one good from $j$'s bundle. Recently,~\citet{caragiannis2016unreasonable} show that the allocation that maximizes the product of the agents' utilities (with ties broken based on the number of agents with positive utility) is EF1 and Pareto efficient.
The majority of the literature to date has focused on the case where the items are available to the algorithm upfront.
In many situations of interest, however, items arrive \emph{online}.
A paradigmatic example is that of food banks~\cite{aleksandrov2015online,lee2019webuildai}. Food banks across the world receive food donations they must allocate; these donations are often perishable, and thus allocation decisions must be made quickly, and donations are typically leftovers, leading to uncertainty about items that will arrive in the future. \citet{BKPP18} study this problem, but focus only on fairness. They show that there exists a deterministic algorithm with \textit{vanishing envy}, that is, the maximum pairwise envy (after all $T$ items have been allocated) is sublinear in $T$, when the value $v_{it}$ of agent $i$ for the $t$-th item is normalized to be in $[0,1]$. Specifically, the envy is guaranteed to be at most $O(\sqrt{T \log T / n})$, and this guarantee is tight up to polylogarithmic factors. The same guarantee can be also achieved by the simple randomized algorithm that allocates each item to a uniformly at random chosen agent. These results hold even against an adaptive adversary that selects the value $v_{it}$ after seeing the allocation of the first $t-1$ items. On the other hand, if we focus only on efficiency, our task is much easier. For example, we could simply allocate each item to the agent with the highest value. But, and this brings us to our interest here, the question remains: \emph{How should we make allocation decisions online in a way that is fair to the donation recipients, but also as efficient as possible?}
\subsection{Our Contributions}\label{sec:results}
We study the tradeoff between fairness and efficiency in the following setting. $T$ indivisible items arrive online; item $t$ has value $v_{it}$ for agent $i$ and must be allocated immediately and irrevocably. We investigate this tradeoff under a range of adversary models, each one specifying how the values $v_{it}$ are generated. We consider five adversaries and completely characterize the extent to which fairness and efficiency are compatible under each and every one of them.
The weakest adversary we consider simply selects a distribution $D$ from which each value $v_{it}$ is drawn (independently and identically). For this case, even though the setting studied was otherwise unrelated, the answer is essentially given by~\citet{DickersonGKPS14} (and later simplified and improved by~\citet{kurokawa2016can}): under conditions on $D$, the algorithm that allocates each item to the agent with the highest value, an ex-post Pareto efficient algorithm\footnote{See Section~\ref{sec:prelims} for definitions.}, is envy-free with high probability.
Removing the conditions on $D$ is possible: we prove (Proposition~\ref{prop:util-max-properties}) that a simple variation of this algorithm (that remains ex-post Pareto efficient) either outputs an EF1 allocation, or is envy-free with high probability as the number of items goes to infinity.
Our next, slightly stronger adversary selects a distribution $D_i$ for each agent $i$; agents are independent, but not identical, and items are independent and identical. Unfortunately, as we discuss in Section~\ref{sec:compatible}, it seems very unlikely that the previous greedy approach generalizes even to this adversary. Our first main result is an \emph{optimal} algorithm that works against the even stronger adversary that allows for \emph{correlated} agents, but i.i.d. items: $v_{it}$ can be correlated with $v_{\hat{i}t}$ but not with $v_{i\hat{t}}$. In Theorem~\ref{thm:POCR} we give an ex-post Pareto efficient algorithm that guarantees to every pair of agents $i, j$ that $i$ envies $j$ by at most one item ex-post, or $i$ will not envy $j$ with high probability.
On a high level our approach works as follows. We take the support of the correlated distribution chosen by the adversary\footnote{We assume finite support; see Section~\ref{sec:prelims}.}, scale each item down by the probability it is drawn, and treat this as an offline instance (with $n$ agents and as many items as the support of the joint distribution $D$). We first prove (Lemma~\ref{lem:por-po}) that using a fractional Pareto efficient solution to guide the online decisions (that is, if an $x$ fraction of item $j$ is allocated to agent $i$ in the offline problem, she gets the item with probability $x$ every time it appears) results in \emph{ex-post} Pareto efficient allocations; in fact, applying almost any scaling (or no scaling) doesn't change this. By coupling this with a Pareto efficient and envy-free offline solution, say the fractional allocation that maximizes the product of agents' utilities, we can get an ex-post Pareto efficient algorithm with $\tilde{O}(\sqrt{T})$ envy, since we can treat the pairwise envy between agents $i$ and $j$ contributed by item $t$ as a zero mean random variable with support in $[-1,1]$ (since the $v_{it}$s are in $[0,1]$) and use standard concentration inequalities. If instead the allocation was strongly envy-free, i.e. each agent strictly preferred her own allocation, then these random variables would have negative mean, and the same concentration inequalities would imply negative envy, i.e. envy-freeness, with high probability. This goal is, unfortunately, a bit too optimistic. But, surprisingly, we can provide an allocation with a property (slightly) weaker than strong envy-freeness, which enables, in the online setting, envy-freeness with high probability or EF1 ex-post (the same guarantee as against the weakest adversary!).
Our second main result (Theorem~\ref{thm: structural}) is a structural, and constructive, result about fractional allocations in the offline problem. We give an algorithm that finds a Pareto efficient fractional allocation where agent $i$ either strictly prefers her allocation to the allocation of agent $j$, or, if she is indifferent, then $i$ and $j$ have \emph{identical} allocations and the \emph{same} value (up to multiplicative factors) for all items allocated to them. We do this by giving budgets $\be$ and a solution $(\bx,\bp)$ to the Eisenberg-Gale convex program where $\bx$ has the desired fairness properties. In slightly more detail, we start with the E-G solution where all agents have the same budget, and leverage various properties implied by the KKT conditions in order to iteratively adjust the initial solution and budgets. Our goal is to eliminate edges in the ``indifference graph'', a graph where vertices correspond to agents and there exists a directed edge from $i$ to $j$ if agent $i$ is indifferent between her allocation and the allocation of agent $j$. We end up with an indifference graph that is a disjoint union of cliques, where agents in the same clique have the same allocation and the same value for the items allocated to them (up to multiplicative factors). At this point, the reader familiar with the E-G convex program might be wondering if budget adjustments are even needed.
In Appendix~\ref{app: same budget counter example} we provide an example where in order to get the desired structure, giving a different budget to each agent is necessary.
We believe that our result is of independent interest, and our approach might have further applications.
Before we proceed to even stronger adversaries, we briefly discuss the chosen fairness criteria. Our results focus on the extent to which efficiency is achievable under the restriction that the algorithm must also guarantee the same fairness criteria that can be achieved in isolation. The fairness guarantee of our main positive result consists of a probabilistic and non-probabilistic part. Even in isolation and under the weakest adversary, this is the best fairness guarantee (with respect to envy) that can be achieved, as we cannot improve on the probabilistic or non-probabilistic guarantee. That is, it is impossible to always output an EF1 allocation (ex-post), and it is also impossible to always output an allocation that is envy-free with high probability. One might further ask if we can give a single non-probabilistic guarantee, but, even envy-freeness up to a constant is not achievable in the online setting; see Appendix~\ref{app:optimal-fairness-guarantee} for these impossibility results. Our algorithms do guarantee vanishing envy, but, while not directly comparable to the ``EF1 or EF w.h.p.'' guarantee, qualitatively this guarantee seems much weaker. Finally, one might attempt to restrict the class of distributions considered. In Appendix~\ref{app:independent agents strong ef} we show that if one excludes point masses, it is possible to guarantee EF with high probability (that is, improve the ``or EF1'' part) for the case of independent agents and i.i.d. items. We do not know if the same result is possible for the case of correlated agents.
Our next (and stronger) adversary is a familiar one: a non-adaptive adversary that selects an instance (with $T$ items) after seeing the algorithm's ``code'', but without knowing the random outcomes of coins used by the algorithm. Of course, as in all the previous cases, the result of~\citet{BKPP18} still applies, so $\tilde{O}(\sqrt{T/n})$ envy is still possible via a deterministic algorithm, but also by simply allocating each item uniformly at random. Our third main result (Theorem~\ref{thm:eff-envy-lb-det}) shows that the latter algorithm is essentially optimal in terms of Pareto efficiency among all algorithms with sublinear envy! Specifically, we show that for every $\varepsilon > 0$, there is no algorithm with sublinear envy that achieves $(1/n + \varepsilon)$-Pareto efficiency ex-ante; see Section~\ref{sec:prelims} for definitions, but, intuitively an allocation is $\alpha$-Pareto efficient, for $\alpha \in (0,1]$, if it is not possible to improve all agents' utilities by a factor of $1/\alpha$. This result extends to our last adversary, the fully adaptive one.
See Table~\ref{table:summary} for a summary of our results.
\renewcommand{\arraystretch}{3}
\begin{table}[ht]
\centering
\resizebox{0.9\columnwidth}{!} {
\begin{tabular}{c|c|c}
\multicolumn{1}{c|}{Setting} & \multicolumn{1}{c|}{Lower Bound} & \multicolumn{1}{c}{Upper Bound} \\ \cline{1-3}
\makecell{Identical Agents\\i.i.d. items} & \makecell{(EF1 or EF w.h.p.) impossible \\ \tikzmark{a}{to improve (Appendix~\ref{app:optimal-fairness-guarantee})}} & \makecell{(EF1 or EF w.h.p.) and ex-post PO \\ (Prop.~\ref{prop:util-max-properties}, essentially~\citep{DickersonGKPS14})} \\ \cline{1-3}
\makecell{Independent Agents\\i.i.d. items} & \makecell{\tikzmark{b}{(EF1 or EF w.h.p.) impossible} \\ \tikzmark{c}{to improve}} & \makecell{\tikzmark{f}{(EF1 or EF w.h.p.) and ex-post PO}} \\ \cline{1-3}
\makecell{Correlated Agents\\i.i.d. items} & \makecell{ \tikzmark{d}{(EF1 or EF w.h.p.) impossible} \\ to improve } & \makecell{\tikzmark{e}{(EF1 or EF w.h.p.) and ex-post PO}\\(Theorem~\ref{thm:POCR})} \\ \cline{1-3}
Non-Adaptive & \makecell{Vanishing envy and ex-ante $1/n +\varepsilon$ \\ \tikzmark{g}{Pareto is impossible (Theorem~\ref{thm:eff-envy-lb-det})}} & \makecell{Vanishing envy and\\ \tikzmark{k}{ex-ante $1/n$ Pareto}} \\ \cline{1-3}
Adaptive & \makecell{\tikzmark{t}{Vanishing envy and ex-ante $1/n +\varepsilon$} \\ Pareto is impossible} & \makecell{\tikzmark{l}{Vanishing envy and}\\ ex-ante $1/n$ Pareto \cite{BKPP18}} \\ \cline{1-3}
\end{tabular}
\link{a}{b}
\link{c}{d}
\link{e}{f}
\link{g}{t}
\link{l}{k}}
\caption{Comparing the compatibility of efficiency and envy in each setting.}\label{table:summary}
\vspace{-3mm}
\end{table}
\vspace{-3mm}
\subsection{Related Work and Roadmap}
Our paper is related to the growing literature on \emph{online} or \emph{dynamic} fair division (\cite{BKPP18, heachieving, kash2014no, friedman2015dynamic, friedman2017controlled, li2018dynamic, freeman2018dynamic, aleksandrov2015online, walsh2011online, bogomolnaia2019simple}). The work most closely related to ours is by~\citet{BKPP18}. They study envy-freeness in isolation against a fully adaptive adversary, and design a deterministic algorithm (specifically a derandomization of the simple algorithm that allocates each item uniformly at random) that achieves maximum envy of $\tilde{O}(\sqrt{T/n})$. They also show that this is tight up to polylogarithmic factors. In contrast, here we study the tradeoff between fairness and efficiency under different adversaries; against the fully adaptive adversary we prove that there is no algorithm with sublinear (in $T$) envy that outperforms allocating uniformly at random in terms of efficiency. In the same setting, ~\citet{heachieving} study the number of \emph{adjustments} that are necessary and sufficient in order to maintain an EF1 allocation online.
Very recently,~\citet{bansal2019online} give an algorithm that guarantees envy at most $O(\log T)$ with high probability (efficiency is not considered), for the case of two i.i.d. agents. Notably, as opposed to our positive result here, their result allows for the adversary's distribution to depend on $T$. Even though the setting studied is completely different, the proof that allocating to the agent with the highest value is optimal against the weakest adversary (identical agents and i.i.d. items) is given by~\citet{DickersonGKPS14} (a similar statement also appears in~\citet{kurokawa2016can}), where it is shown that this algorithm outputs an envy-free allocation with probability that goes to $1$ as the number of items goes to infinity; we go over their result in more detail, as well as argue about why it seems unlikely that it generalizes to stronger adversaries, in Section~\ref{sec:compatible}.
For the offline problem, i.e. when all item values are available to the algorithm,~\citet{caragiannis2016unreasonable} show that there is no tradeoff between fairness and efficiency. The (integral) allocation that maximizes the product of the agents' utilities is Pareto efficient and EF1, simultaneously. More recently,~\citet{barman2018finding} show that there always exists an allocation that is EF1 and fractionally Pareto efficient (and also give pseudopolynomial time algorithm for finding allocations that are EF1 and Pareto efficient).
Computing the fractional allocation that maximizes the product of utilities is a special case of the Fisher market equilibrium with affine utility buyers; the latter problem was solved in (weakly) polynomial time by~\citet{devanur2008market}, improved to a strongly polynomial time algorithm by~\citet{orlin2010improved}. Our structural result starts from an exact solution to the Eisenberg-Gale convex program and then uses a polynomial number of operations. Therefore, all our algorithms run in strongly polynomial time; we further comment on this in Section~\ref{sec: conclusion}.
It is worth pointing out a connection between what we call the indifference graph and the so-called maximum bang per buck (MBB) graph. In our indifference graph there is an edge from $i$ to $j$ if $i$ is indifferent between her allocation and the allocation of agent $j$. We show (Lemma~\ref{lem:edge-indifferent}) that this condition is equivalent to $v_{ik}/p_k = v_i(A_i)/e_i$, for all items $k$ that $j$ gets a non-zero fraction of, where $p_k$ is the price of item $k$ and $e_i$ is the budget of agent $i$. On the other hand, the MBB graph is bipartite, one side is agents and the other is items, and there is an edge from agent $i$ to item $k$ if $v_{ik}/p_k = v_i(A_i)/e_i$. Properties of MBB graphs have been crucial in recent algorithmic progress on approximating the Nash Social Welfare, e.g. ~\citet{cole2015approximating,garg2018approximating,chaudhury2018fair}, as well as in computing equilibria in Arrow-Debreu
exchange markets (\citet{garg2019strongly}), but beyond the similarity in the definition we are unaware of any technical overlap.
\paragraph{Roadmap.}
Section~\ref{sec:prelims} poses our model and some preliminaries.
In Section~\ref{sec:compatible} we prove our main positive result: for all i.i.d. distributions over items (with possibly correlated agents), there exists an algorithm that always outputs a Pareto efficient allocation (i.e. this is an ex-post Pareto efficient algorithm) that guarantees that for each pair of agents $i$ and $j$, either $i$ envies $j$ by at most one item, or $i$ does not envy $j$ with high probability. As we mentioned earlier, our algorithm relies on a structural result about divisible allocations; we prove this result in Section~\ref{sec: CISEF}.
In Section~\ref{sec:incompatible} we prove our main negative result: there is a non-adaptive adversary strategy, such that no vanishing envy algorithm has efficiency better than allocating uniformly at random.
We conclude in Section~\ref{sec: conclusion}.
\section{Preliminaries}\label{sec:prelims}
We study the problem of allocating a set of $T$ indivisible items (also referred to as goods) arriving over time, labeled by $\mathcal{G} = \{ 1, 2, \cdots, T \}$, to a set of $n$ agents, labeled by $\mathcal{N} = \{ 1, \dots, n \}$. Agent $i \in \mathcal{N}$ assigns a (normalized) value $v_{it} \in [0,1]$ to each item $t \in \mathcal{G}$. Agents have additive utilities for subsets of items: the value of agent $i$ for a subset of the items $S$ is $v_i(S) = \sum_{t \in S} v_{it}$. An allocation $A$ is a partition of the items into bundles $A_1, \dots, A_n$, where $A_i$ is assigned to agent $i \in \mathcal{N}$.
Items arrive in order, one per round, for a total of $T$ rounds. The agents' valuations for the $t$-th item become available only after the item arrives, and are unknown until then. Let $\mathcal{G}^t = \{ 1, 2, \dots, t \}$ be the set of items that have arrived up until time $t$. We denote an allocation of $\mathcal{G}^t$ by $A^t$.
We would like to allocate the goods to the agents in a way that the final allocation $A^T$ is \textit{fair} and \textit{efficient}.
Before we formally define our notions of fairness and efficiency, we go over the different adversary models, each one specifying how the agents' item values are generated.
\paragraph{Adversary Models.}
One can think of each scenario as a game between the adversary and the algorithm.
For the first three adversaries it will be more intuitive to think of the adversary picking her strategy first, followed by the algorithm, that gets to first see the adversary's strategy. For the last two, it will be more intuitive to think of the algorithm's code being fixed before the adversary picks her strategy. We list our adversaries from weakest to strongest, where a stronger adversary can simulate the strategy of a weaker adversary but not vice versa. All distributions are assumed to be discrete with finite support.
(1) \textbf{Identical agents and i.i.d. items.} The adversary selects a distribution $D$. On round $t$, the value of item $t$ to each agent $i$ is drawn independently from this distribution, i.e. $v_{it} \sim D$. (2) \textbf{Independent agents and i.i.d. items.}
The adversary selects a distribution $D_i$ for each agent $i$. On round $t$, the value of item $t$ to each agent $i$ is drawn independently from their value distribution, that is $v_{it} \sim D_i$. (3) \textbf{Correlated agents and i.i.d. items.}
The adversary specifies a joint distribution $D$ with marginals $D_1, \ldots, D_n$. On round $t$, the vector of values $v_t = (v_{1t},\dots,v_{nt})$ is drawn i.i.d. from $D$. That is, $v_{it}$ can be correlated with $v_{jt}$, but not with $v_{i\hat{t}}$. For simplicity, we treat this setting as follows. Each item $t$ has one of $m$ types. Agent $i$ has value $v_i(\gamma_j)$ for an item of type $\gamma_j$; the type of each item is drawn i.i.d. from a distribution $D$ with support $G_D$, $|G_D|=m$. We write $f_D(\gamma_j)$ for the probability that the $t$-th item has type $\gamma_j$. Our main positive result is for this setting.
(4) \textbf{Non-adaptive adversary.} The adversary selects an instance (with $n$ agents and $T$ items) after seeing the algorithm's code, but doesn't know the outcomes of the random coins flipped by the algorithm. Our main negative result is for this setting.
(5) \textbf{Adaptive Adversary} When the adversary selects the value $v_{it}$ she has access to the algorithm's decisions for the first $t-1$ items. This is the setting studied in~\citet{BKPP18}.
\paragraph{Known vs Unknown $T$.} A subtle point that needs to be addressed is whether $T$ is fixed or not when the adversary picks her strategy. For the latter two adversaries we assume that $T$ is fixed (the results of~\cite{BKPP18} also need this assumption). For the first three adversaries it will be convenient to think of the adversary picking a distribution and then study our algorithm's behavior as $T$ goes to infinity. That is, the distribution picked by the adversary does not depend on $T$; so, it cannot for example have support of size $T$ or variance $1/T$, and so on. This allows us to bypass common issues regarding the dependence on the number of items and existence of fair allocations in probabilistic settings (e.g. the answer could depend on whether $T$ is divisible by $n$); see~\cite{DickersonGKPS14,kurokawa2016can,manurangsi2019envy} for some examples where this dependence is crucial. It is worth noting that the recent result of~\citet{bansal2019online} for the case of two correlated agents allows the agents' distribution to depend on $T$ (but only gets an $O(\log T)$ bound on envy, with high probability, and no guarantees about efficiency).
\vspace{-2mm}
\paragraph{Fairness and Efficiency.}
The utility profile of an allocation $A$ is a vector $u = (u_1, \ldots, u_n)$ where $u_i = v_i(A_i)$.
A utility vector $u$ dominates another utility vector $u'$, denoted by $u \succ u'$, if for all $i \in \mathcal{N}$, $u_i \geq u'_i$ and for some $j \in \mathcal{N}$, $u_j > u'_j$. An allocation that achieves utility $u$ is \textit{Pareto efficient} if there is no allocation with utility vector $u'$ such that $u' \succ u$. Uncertainty about the future when making allocation decisions will make it impossible to provide efficiency and fairness guarantees simultaneously. In order to measure the efficiency of our algorithms we instead use the following notion of approximate Pareto efficiency, initially defined by~\citet{ruhe1990varepsilon}: an allocation with utility profile $u$ is $\alpha$-Pareto efficient, if there is no feasible utility profile in which, compared to $u$, every agent is (strictly) more than $1/\alpha$ times better off.
Since our setting is online, we also need to further specify whether our guarantees are worst-case or average-case with respect to the adversary instance and the randomness of our algorithms. For a worst-case guarantee, we say that an allocation is $\alpha$-Pareto efficient ex-post if it always outputs an $\alpha$-Pareto efficient allocation (that is, for all agent valuations and all possible outcomes of the random coins flipped by the algorithm). On the other hand, an allocation algorithm is $\alpha$-Pareto efficient ex-ante if the expected allocation is $\alpha$-Pareto efficient (where the expectation is with respect to the randomness in the instance and the algorithm). Our main positive result/algorithm will guarantee $1$-Pareto efficiency ex-post, while our main negative result shows that a certain notion of fairness is incompatible with $1/n$-Pareto efficiency ex-ante.
Regarding fairness, in this paper we focus on envy. Given an allocation $A = (A_1, \ldots, A_n)$, let $\mathsf{ENVY}_{i, j}(A) = \max \{ v_i(A_j) - v_i(A_i), 0 \}$ be the pairwise envy between agents $i$ and $j$, and let $\mathsf{ENVY}(A) = \max_{i,j \in \mathcal{N}} \mathsf{ENVY}_{i, j}(A)$ be the maximum envy. Clearly, if $\mathsf{ENVY}(A) = 0$ the allocation is envy-free. An allocation $A$ is EF1, if for all pairs of agents $i,j$ $\mathsf{ENVY}_{i, j}(A) \leq \max_{t \in A_j} v_{it}$. Note that this is a stronger guarantee than $\mathsf{ENVY}(A) \leq 1$, since the highest value an agent has can be less than $1$. An algorithm has vanishing envy if the expected maximum pairwise envy is sublinear in $T$, that is $\E[\mathsf{ENVY}(A)] \in o(T)$, or just $\lim_{T \rightarrow \infty} \E[\mathsf{ENVY}(A)]/T \rightarrow 0$.
\vspace{-2mm}
\section{Achieving Pareto Efficiency and CISEF}\label{sec: CISEF}
In this section we prove our main structural result.
\begin{theorem}\label{thm: structural}
Given any instance with $m$ divisible items and $n$ additive agents, there always exists an allocation that is simultaneously clique identical strongly envy-free (CISEF) and Pareto efficient.
\end{theorem}
We prove Theorem~\ref{thm: structural} by building on a standard approach for finding an envy-free and Pareto efficient allocation, namely solving the Eisenberg-Gale convex program (henceforth E-G program). Recall that the E-G program with ``budgets'' $\be$ is the following.
\begin{align*}
\max & \quad \sum_{i = 1}^n e_i \log \sum_{j = 1}^m v_{ij}x_{ij}, &\text{subject to} & \quad \sum_{i=1}^n x_{ij} \leq 1, \forall j \in [m], \text{ and }\quad x_{ij} \geq 0, \forall i \in [n], j \in [m].
\end{align*}
When $e_i=e_j$ for all $i, j$, the outcome is also known as the competitive equilibrium from equal incomes (CEEI), which is envy-free \cite{varian1974equity}. There exists a solution to the E-G program with primal variables $\bx$ and dual variables $\bp$ (dual variable $p_j \geq 0$ corresponds to the first constraint above) satisfying the following conditions.
\begin{align}
&\textstyle \forall j \in [m] : p_j > 0 \implies \sum_{i=1}^n x_{ij} = 1 \label{eq: KKT 1} \\
&\textstyle \forall i \in [n], j \in [m] : \frac{v_{ij}}{p_j} \leq \frac{\sum_{k=1}^m v_{ik}x_{ik}}{e_i} \label{eq: KKT 2}\\
&\textstyle \forall i \in [n], j \in [m] : x_{ij} > 0 \implies \frac{v_{ij}}{p_j} = \frac{\sum_{k=1}^m v_{ik}x_{ik}}{e_i} = \max_{k \in [m]} \frac{v_{ik}}{p_k}.\label{eq: KKT 3}
\end{align}
These conditions are both necessary and sufficient for a feasible solution to be optimal, and can be derived from the KKT conditions; for completeness we show the derivation in Appendix~\ref{app:kkt-derive}. The standard interpretation is that $e_i$ is the budget of agent $i$ and $p_j$ is the price for item $j$. Then, a solution $\bx, \bp$ consists of prices $\bp$ and allocations $\bx$, such that each agent spends their entire budget $e_i$ on ``optimal'' items and all items are completely sold. We say that an item $j$ is ``optimal'' for agent $i$ given prices $\bp$, when it maximizes the ratio $v_{ij}/p_j$, also known as the \emph{bang-per-buck}.
For the remainder of this section, given a solution $\bx, \bp$, we use $\bx$ and $X$ indistinguishably for the allocation, we write $X_i$ for the allocation of agent $i$, and we say that item $k$ is allocated to $i$, $k \in X_i$, if $x_{ik} > 0$.
We assume without loss of generality that for any solution, $\bx, \bp$, we have $\forall j : p_j > 0$. This holds as long as each item has at least one agent who values it; if this is not the case we can safely drop those items.
We prove the following, which immediately implies Theorem~\ref{thm: structural}\footnote{Pareto efficiency is implied since the objective function is monotone.}.
\begin{theorem}
\label{thm:main-disjoint-cliques}
There exist budgets $\be$ and an optimal solution $(\bx = X, \bp)$ to the E-G convex program with budgets $\be$, such that $X$ is clique identical strongly envy-free.
\end{theorem}
\noindent We start with the optimal solution $X$ to the E-G convex program with identical budgets $e_i = 1$ for all $i$. Then, at a high level, we break the algorithm into two procedures that jointly alter $\bx, \bp$ and $\be$ such that $\bx, \bp$ remains an optimal solution to the convex program with budgets $\be$, while preserving envy-freeness, until $X$ satisfies the desired properties. Specifically, the indifference graph $I(X)$ will end up being a disjoint set of cliques, such that agents in a clique have identical allocations.
\subsection*{Optimal Transfers}
Given an allocation, let $r_i := \frac{v_i(A_i)}{e_i}$ be the maximum bang-per-buck ratio of agent $i$. We say that agent $i$ is indifferent towards any item $k$ for which $\frac{v_{ik}}{p_k} = r_i$. We first give a useful property of solutions $\bx, \bp$.
The proof is relegated to Appendix~\ref{app: missing lemma proof}.
\begin{lemma}
\label{lem:edge-indifferent}
Given a solution $\bx, \bp$ of an E-G program with budgets $\be$, for all agents $i, j$ such that $e_i = e_j$, $v_i(X_i) = v_i(X_j)$ if and only if $\forall k \in X_j : \frac{v_{ik}}{p_k} = r_i$.
\end{lemma}
Lemma~\ref{lem:edge-indifferent} tells us that under equal budgets, if agent $i$ is indifferent to agent $j$'s allocation, then $j$'s items are maximum bang-per-buck items for $i$ as well. This gives some intuition for our approach. Assuming equal budgets, if we move items along indifference edges, we can avoid violating the KKT conditions and the solution remains optimal. We formalize this idea in Lemma~\ref{lem:optimal-transfers}; first we need the following definition. Given a solution $(\bx = X, \bp)$ for budgets $\be$, we consider a change in allocation of items $\Delta$, where $\Delta_{ik}$ is the difference in the allocation of item $k$ for agent $i$.
\begin{definition}[Optimal transfer]
A transfer of items $\Delta$ is an \textit{optimal transfer} if for all items $k$ (1) $\sum_{i \in [n]} \Delta_{ik} = 0$, i.e. the total allocation of item $k$ remains unchanged, (2) $x_{ik} + \Delta_{ik} \in [0,1]$ for all agents $i$, i.e. $\bx + \Delta$ is feasible, and (3) for all agents $i$ such that $\Delta_{ik} > 0$, $\frac{v_{ik}}{p_k} = r_i$, i.e. if agent $i$ is given more of item $k$, then item $k$ maximizes bang-per-buck for agent $i$.
\end{definition}
The proof of the next lemma can be found in Appendix~\ref{app: lemma optimal transfers}.
\begin{lemma}
\label{lem:optimal-transfers}
Let $(\bx = X, \bp)$ be a solution for budgets $\be$ and $\Delta$ be an optimal transfer. Let $\delta$ represent the change in budget where $\delta_i = \sum_{k \in \mathcal{G}} p_k \cdot \Delta_{ik}$. Let $X' = X + \Delta$. Then $(\bx' = A', \bp)$ is a solution for budgets $\be' = \be + \delta$.
\end{lemma}
\subsection*{Indifference Edge Elimination}
For an allocation $X$ and subset of agents $S \subseteq \mathcal{N}$, we overload notation, and let $X_S$ refer to the allocation for agents in $S$. A set of agents $S$ have identical budgets under $\be$ if there exists some $c$ where for all $i \in S$, $e_i = c$.
Recall that for an allocation $X$, $I(X)$ refers to the indifference graph, a graph where we have a vertex for every agent and an edge from (the vertices corresponding to) agent $i$ to agent $j$ if $v_i(X_i) = v_i(X_j)$. In the remainder of this section we refer to agents and vertices interchangeably. Also, recall that for a directed graph $G = (V,E)$, a clique is a subset of vertices $S \subseteq V$ such that for all $v \in S, u \in S$, where $u \neq v$, there is an edge $(u,v) \in E$, and that a weakly connected component (henceforth just component) $S$ is a subset of the agents such that for each pair of agents $i, j \in S$, there is either a path from $i$ to $j$ or a path from $j$ to $i$, and $S$ is a maximal subgraph with this property.
\begin{definition}[Clique acyclic graph]\label{dfn: clique acyclic}
A directed graph $G=(V,E)$ is clique acyclic if the vertices can be partitioned into cliques $C_1, \ldots, C_k$, that is, $C_i \subseteq V$ for all $i$, $\cup_{i=1}^k C_i = V$, and $C_i \cap C_j = \emptyset$ for all $i \neq j$, and where for any cycle $K$ in the graph, $K$ only contains vertices from $C_i$ for some $i$.
\end{definition}
A crucial step towards producing a CISEF allocation will be to find an allocation $X$ such that $I(X)$ is clique acyclic and envy-free, where agents in each clique have the same allocation.
\begin{lemma}
\label{lem:const-util-shifts}
There exists an algorithm that takes as input a solution $(\bx = X, \bp)$ for budgets $\be$ and a component $S$ with identical budgets, and finds a solution $(\bx' = X', \bp)$ for budgets $\be$ where $I(X'_S)$ is clique acyclic and agents in each clique have identical allocations without violating envy-freeness or adding new indifference edges to $I(X)$.
\end{lemma}
Take the allocation $X$ and indifference graph $I(X_S)$. At a high level we attempt to apply one of the following two operations; we only apply Operation $2$ only if Operation $1$ cannot be done.
\begin{itemize}[leftmargin=*]
\item \textbf{Operation 1:} Eliminate every cycle that is not a clique.
\item \textbf{Operation 2:} Partition the graph into cliques by merging cliques and ``re-balancing'' allocations.
\end{itemize}
We explain in detail how these operations work, and prove that they satisfy some basic properties. We proceed to show how to use them to prove the lemma.
\paragraph{Operation $1$.}
Without loss of generality, suppose we have a cycle $K = (1, \ldots, k)$, where edges go from agent $i$ to agent $i + 1$ (and indices wrap around). If there exists an $i$ where there is no edge from $i - 1$ to $i + 1$, we show how to eliminate at least one edge from the cycle, keep $v_i(X_i)$ the same for all agents, and not create any new indifference edges.
To perform this operation notice that Lemma~\ref{lem:edge-indifferent} implies that there exists an item $\ell \in X_{i + 1}$ that is not a bang-per-buck item for agent $i-1$, i.e. $\frac{v_{i-1,\ell}}{p_\ell} < r_{i - 1}$. We construct an optimal transfer $\Delta$, parameterized by a budget $b$, as follows. Agent $i$ will take $b$ worth of item $\ell$ (specifically $b/p_\ell$ units) from agent $i + 1$. All other agents $i' \in K$, $i' \neq i$, will take arbitrary items of total worth $b$ from $i' + 1$. Crucially, for all of the transfers, since there is an $(i',i'+1)$ edge for all $i' \in K$, i.e. $v_{i'}(X_{i'}) = v_{i'}(X_{i'+1})$, Lemma~\ref{lem:edge-indifferent} implies the items taken maximize bang-per-buck for $i'$. Next, notice that we can find a $b > 0$ small enough that $X + \Delta$ is a feasible allocation\footnote{Letting $e_S$ be the budget of agents in $S$, we can choose any $b \leq \min(e_S, x_{i+1, l} \cdot p_l) = x_{i+1, l} \cdot p_l$ (i.e we just need to ensure agent $i$ does not take more than $x_{i+1, l}$ of item $l$).}. Finally, our transfers preserve the total allocation of each item and thus, $\Delta$ is an optimal transfer. Each agent loses and gains $b$ worth of items so there is no change in budget associated with $\Delta$. Therefore, Lemma~\ref{lem:optimal-transfers} implies that $(\bx' = X', \bp)$ is a solution for budgets $\be$. Thus for all agents $i$, $v_i(X_i) = v_i(X'_i)$.
Next, observe that $v_{i - 1}(X'_i) < v_{i - 1}(X_i)$ since it decreases by $b \cdot r_{i - 1}$ but increases by strictly less than $b \cdot r_{i - 1}$. Since $v_{i - 1}(X_{i - 1}) = v_{i - 1}(X'_{i - 1})$, this implies that the indifference edge from agent $i - 1$ to agent $i$ disappears. Finally, we want to ensure that we do not violate envy-freeness or add new indifference edges. There are two cases. The first case is that $i$ is indifferent to $j$ in $X$: we only modify allocations in $S$, and $S$ is a component, so we only consider $i \in S, j \in S$. Because $(\bx', \bp)$ is a solution for budgets $\be$, and agents in $S$ have identical budgets, agent $i$ will still not envy $j$ in $X'$. The second case is that $i$ is not indifferent to $j$ in $X$; we want to find a $b > 0$ small enough such that $i$ will not envy or be indifferent to $j$ in $X'$. We give such a $b$ in Appendix~\ref{app:choosing b}.
We can repeatedly use the above process to eliminate all non-clique cycles. We do this by eliminating cycles in order of increasing size. Suppose that all cycles of size $k$ form a clique. It is then possible to ensure that for all cycles of size $k + 1$, the vertices form a clique. To see why, consider an arbitrary cycle $K$ of size $k + 1$. If there exists an $i$ such that there is no edge from agent $i - 1$ to agent $i + 1$, we can eliminate the cycle. Otherwise, for all $i$, there is an edge from agent $i - 1$ to agent $i + 1$. Then, we know that for all $i$, $K \setminus \{ i \}$ is a cycle of length $k$, and therefore a clique of size $k$, implying the vertices of $K$ form a clique of size $k + 1$. Note that any cycle of size $2$ immediately forms a clique. Therefore, if we repeatedly choose the smallest size non-clique cycle and eliminate it, we will eventually eliminate all cycles that are not a clique.
\paragraph{Operation $2$.}
We construct a set of cliques $C_1, \ldots, C_s$ by starting with each vertex in its own clique and arbitrarily merging cliques if the resulting set of vertices still forms a clique. Suppose we merge to get a clique $C = \{ 1, \ldots, \ell \}$. Lemma~\ref{lem:edge-indifferent} implies that every agent $i \in C$ is indifferent to any item $z \in X_C$, $X_C = X_1 + \ldots + X_{\ell}$, such that $\frac{v_{iz}}{p_z} = r_i$. Thus, we can perform the following re-balance operation to form $X'$: for each agent $i \in C$, set $X'_i = \frac{1}{\abs{C}}X_C$, and for all other agents $i \notin C$, $X'_i = X_i$.
First, note that this is an optimal transfer, by definition. We want to show that we do not violate envy-freeness or add new indifference edges. We separate into cases based on whether agents $i, j$ are in $C$. If both agents are in $C$, nothing changes. If $i \in C$, $j \notin C$, since $v_i(X_i)$ stays the same, nothing changes. If $i \notin C$, $j \in C$, if $i$ was indifferent to all agents in $C$, nothing changes. Otherwise, $i$ is not indifferent to some agent in $C$, in which case after the re-balance, $i$ will lose their indifference edge towards all agents in $C$.
\begin{proof}[Proof of Lemma~\ref{lem:const-util-shifts}]
We start by applying Operation $1$, which eliminates every cycle that is not a clique. Now, notice that eliminating all non-clique cycles does \emph{not} imply that we can partition the graph into cliques in a way that is clique acyclic. To see this most clearly, consider a $5$ vertex instance, where vertices $1$, $2$, $3$ form a clique, and so do vertices $3$, $4$, $5$. All cycles are also cliques, but the graph is still not clique acyclic (the most ``tempting'' partition has the issue that cycle $( 1, 2, 3 )$ contains a vertex that belongs to two cliques). This is where Operation $2$ comes in.
Once we have eliminated all non-clique cycles, if we are not in a clique acyclic graph with the desired properties, we apply Operation $2$. It is possible that during the execution of Operation $2$, an indifference edge will be eliminated. In this case, we stop with Operation $2$ and go back to applying Operation $1$, and so on. Eventually, this process terminates: neither operation creates any new edges, and furthermore, each time we loop we eliminate at least one edge. The two operations preserve the property that $(\bx' = X', \bp)$ is a solution for budgets $\be$, and that $X'$ is envy-free, without adding new indifference edges. In addition, re-balancing in Operation $2$ ensures that agents in each clique have identical allocations. It remains to show that $I(X')$ is clique acyclic.
Assume for sake of contradiction that there exists a cycle that includes vertices in two different cliques $C_1, C_2$. Therefore there exists some edge from $C_1$ to $C_2$ and an edge from $C_2$ to $C_1$. Due to the fact that agents in a clique have identical allocations, this implies that there exists agents $i_1 \in C_1, i_2 \in C_2$ such that $i_1$ has edges to all of $C_2$ and $i_2$ has edges to all of $C_1$. Thus, we can construct a cycle, and thus a clique, containing all agents in $C_1 \cup C_2$. Then, $C_1 \cup C_2$ form a clique, contradicting the fact that no more mergers were possible by Operation $2$.
\end{proof}
Once we have an allocation with the properties of Lemma~\ref{lem:const-util-shifts}, it remains to eliminate the edges between cliques, while preserving the property that agents in a clique have the same allocation.
\begin{lemma}\label{lem:budget-change-shifts}
There exists an algorithm that takes as input an allocation $X$ and component $S$ with identical budgets such that $I(X_S)$ is clique acyclic (with at least one non-clique edge) and agents in each clique have identical allocations, and finds a set of agent budgets $\be'$ and solution $(\bx' =X', \bp)$ for budgets $\be'$, where $I(X'_S)$ consists of $k > 2$ components $S_1, S_2, \cdots, S_k$, each component consists of agents with identical budgets, $I(X'_S)$ has strictly fewer edges than $I(X_S)$, and the new allocation does not violate envy-freeness or add new indifference edges.
\end{lemma}
\begin{proof}
For $I(X_S)$, we view each clique as a vertex. Consider the graph $G$, where each vertex is a clique $C_i$ (of $I(X_S)$), and there is an edge between $C_i$ and $C_j$ if there exists a $v_i \in C_i, v_j \in C_j$ where $(v_i, v_j)$ is an edge in $I(X_S)$. Observe that $G$ forms a DAG, and let $C_s$ be a source vertex in $G$ and let $Sk = \{ C_1, \ldots, C_l \}$ be the sink vertices reachable from $C_s$. We know a source and sink vertex exists because we assume there is at least one non-clique edge.
Now we return to $I(X_S)$. We will create item transfers $\Delta$. We find it intuitive to describe $\Delta$ via a flow in the following graph. Starting from $G$, create a new global source vertex $s$, and add all $(s,v)$ edges for all $v \in C_s$. Create a new global sink vertex $t$, and add a $(v,t)$ edge for all $v \in Sk$. Finally, we let each edge in the graph have infinite capacity. Next, we find a flow of size $b$ from $s$ to $t$, with the additional constraints that the flow from $s$ to each $v \in C_s$ is the same, and for each $C \in Sk$, the flow from each $v \in C$ to $t$ is the same. We show how to construct such a flow, starting from an arbitrary flow of size $b$. Let $f_{ij}$ be the flow along edge $(i, j)$. For the source clique, $C_s$, we can update $f'_{si} = \frac{1}{\abs{C_s}}\sum_{i \in C_s} f_{si}$ so that $f'$ now satisfies the equal flow constraint for the source clique. Then, we can always ensure the flow $f'$ is balanced by updating the flow between vertices in $C_s$: For each $i, j \in C_s, i < j$, set $f'_{ij} = f_{ij} + \frac{1}{\abs{C_s}} \left(f_{sj} - f_{si}\right)$, where negative flows are added as positive flows in the reverse direction. The flow along all other edges remains the same. To show that $f'$ is a feasible flow, we show flows are balanced for vertices in the source clique. Observe for an agent $i \in C_s$:
\[\sum_{j \in \mathcal{N} \cup \{ s \}} f'_{ji} - \sum_{j \in \mathcal{N}} f'_{ij} = \sum_{j \in \mathcal{N}} f_{ji} + \frac{1}{\abs{C_s}}\sum_{j \in C_s} f_{sj} + \frac{1}{\abs{C_s}} \sum_{j \in C_s} \left(f_{si} - f_{sj}\right) - \sum_{j \in \mathcal{N}} f_{ij} = \sum_{j \in \mathcal{N} \cup \{ s \}} f_{ji} - \sum_{j \in \mathcal{N}} f_{ij}.\]
Therefore, if $f$ is a valid flow, so is $f'$. An analogous procedure can be applied to the sink cliques.
We use this flow to guide the item transfers, $\Delta$. For each edge with flow, agent $i$ will take an arbitrary $f_{ij}$ worth of items from agent $j$. The global sink and source vertices are, obviously, excluded. We choose $b$ small enough such that $X + \Delta$ is feasible. We can assume without loss of generality that the sum of flows towards any vertex is at most $b$, as any extra flow must be part of a cycle and can be eliminated. Then if $e_S$ is the budget of agents in $S$, we can choose any $b \leq e_S$. Since we ensure the total allocation of an item is preserved and we only transfer items along indifference edges between agents of the same budget, $\Delta$ is an optimal transfer. Therefore we can apply Lemma~\ref{lem:optimal-transfers}.
Since there are no item transfers to $s$, each agent $v \in C_s$ with an incoming flow has the same increased budget. Similarly, for each sink clique $C$ with a positive flow from $v \in C$ to $t$, each member of $C$ will have the same decreased budget.
Now, if we take each component in the resulting indifference graph, we claim that each agent in the component will have identical budgets.
It is sufficient to show that if $e'_i \neq e'_j$, then there will not be an edge from $v_i$ to $v_j$. Since initially everyone in $S$ had the same budget, and item transfers --- as well as ``budgets'' --- go from sinks to sources, if $e'_i < e'_j$, there was never an edge from $v_i$ to $v_j$ to begin with. Now suppose $e'_i > e'_j$; we have:
\[v_i(X_i) = \sum_{k \in X_i} v_{ik} x_{ik} =^{KKT~\eqref{eq: KKT 3}} \sum_{k \in X_i} r_i p_k x_{ik} = r_i e'_i > r_i e'_j = \sum_{k \in X_j} r_i p_k x_{jk} \geq \sum_{k \in X_j} v_{ik} x_{jk} = v_i(X_j),\]
which implies no $(i,j)$ indifference edge.
Since agents in $C_s$ have a greater budget than all other remaining agents, there will be at least two components.
The final property that we need to prove is that the new allocation does not violate envy-freeness or add new indifference edges. We break into cases. First, if agents $i$ and $j$ were indifferent, as we only modify allocations in component $S$, we only consider when $i \in S, j \in S$. We ensure $v_i(X_i)$ either stays the same or increases, since we transfer items in the opposite direction of indifference edges. For similar reasons, $v_i(X_j)$ either stays the same or decreases. On the other hand, if agent $i$ did not envy $j$ we can set $b$ small enough such that $i$ will not envy $j$ in $X'$. We discuss this final detail in Appendix~\ref{app:choosing b}.
\end{proof}
\subsection*{Putting everything together}
\begin{proof}[Proof of Theorem~\ref{thm:main-disjoint-cliques}]
Our overall algorithm first solves the E-G convex program with budgets $\be$ where $e_i = 1$ for all $i$, to find a solution $(\bx = X, \bp)$. We keep track of the set of components $\calS$ with identical budgets, where initially $\calS = \{ \mathcal{N} \}$. We alternate between applying the algorithm of Lemma~\ref{lem:const-util-shifts} and Lemma~\ref{lem:budget-change-shifts}, henceforth procedure $1$ and procedure $2$. We start by applying the former to each $S \in \calS$. It is possible that after applying procedure $1$, edges are eliminated, resulting in $S$ being split into multiple components. Let $f(S)$ be the set of components formed and update $\calS := \bigcup_{S \in \calS} f(S)$. The ``clique acylic'' and ``identical allocations'' properties are still satisfied by each individual component in $\calS$. We then apply procedure $2$ to each $S \in \calS$. We perform the same update to $\calS$, where $f(S)$ is the set of components with identical budgets found after applying procedure $2$. We repeat until applying the two procedures does not decrease the number of edges in the graph. Finally, we perform the re-balance operation (described in procedure $1$) on each component
Both procedures do not add edges and reduce the number of edges. There are at most $n^2 - n$ initial edges, and thus the algorithm terminates. In addition, both procedures produce allocations and budgets where $(\bx' = A', \bp)$ is a solution to the E-G program for budgets $\be'$. Finally, $X$ is CISEF. First, $X$ is envy-free by construction. Next, consider the final graph $I(X)$. We claim that each component is a clique where agents have identical budgets. The component must be clique-acyclic after procedure $1$ is applied. No edges were removed by procedure $2$, so there could not have been any edges between cliques. Therefore, the component can only consist of one clique. Next, procedure $2$ guarantees that each agent in the clique has the same budget. Lemma~\ref{lem:edge-indifferent} tells us that agents will have identical valuations, up to a multiplicative factor, for items that any agent in the clique receives. The final re-balance operation doesn't alter any of the above properties.
\end{proof}
|
1,116,691,500,854 | arxiv | \section{Introduction}
For a plausible category of quasigroups, it seems that homotopies
between quasigroups, taken as morphisms, are better choice than
homomorphisms (see \cite{Gvaramiya}). However, homomorphisms are
sometimes easier to work with. For example, isotopies do not
preserve units---every quasigroup is isotopic to a loop
(quasigroup with a unit) but is not necessarily a loop itself.
This note is about turning homotopies into homomorphisms.
Smith, \cite{Smith}, proved that there is an adjunction from the
category of semisymmetric quasigroups with homomorphisms to the
category of quasigroups with homotopies. Also, he proved in
\cite{Smith} that the latter category is isomorphic to a
subcategory of the former category, and in \cite{Smith8}, that
every $T$ algebra, for $T$ being the monad defined by the above
adjunction, is isomorphic to the image of a semisymmetric
quasigroup under the comparison functor.
These results, especially the embedding of the category of
quasigroups with homotopies into the category of semisymmetric
quasigroups with homomorphisms, could be of interest to a working
universal algebraist. Our intention is to make them more
accessible to such a reader and to indicate a possible misusing.
Also, we make some alternative proofs and add the fullness of the
comparison functor in order to complete the picture of this
adjoint situation.
At the end of the paper, we show that there is a more economical
way to embed the category of quasigroups with homotopies into the
category of semisymmetric quasigroups with homomorphisms. One
could get an impression, due to \cite{Smith}, that for such an
embedding it is necessary to have a semisymmetrization functor
that is a right adjoint in an adjunction. If one is interested
just in this embedding and not in reflectivity (see the end of
Section \ref{adj}), then this new semisymmetrization suits as any
other.
We assume that the reader is familiar with the notions of
category, functor and natural transformation. If not, we suggest
to consult \cite{ML71} for these notions. All other relevant
notions from Category theory are introduced at the appropriate
places in the text.
\section{Quasigroups}\label{Q}
We start by
recapitulating a few basic facts about quasigroups.
One way to define a \emph{quasigroup} is that it is a grupoid $(Q;
\cdot)$ satisfying:
\begin{equation}
\forall{a b} \exists_{1}x (x \cdot a = b) \qquad \quad \text{ and
} \qquad \quad \forall{a b} {\exists}_1 x (a \cdot x = b) \notag
\end{equation}
Uniqueness of the solution of the equation $x\cdot a = b \;
(a\cdot x = b)$ enables one to define {right (left) division}
operation $x = b \rd a \; (x = a \ld b)$ which is also a
quasigroup (short for: $(Q;\rd)$ is a quasigroup). We can define
three more operations:
$$
x * y = y\cdot x \hspace{2cm} x \rdd y = y \rd x \hspace{2cm} x
\ldd y = y \ld x
$$
dual to $\cdot, \rd, \ld$\/ respectively. They are also
quasigroups.
The six operations $\cdot, \rd, \ld, *, \rdd$ and $\ldd$ are
\emph{parastrophes} of $\cdot$ (and of each other).
A function $f : Q \to R$ between the base sets of quasigroups
$(Q; \cdot)$ and $(R, \cdot)$ is a \emph{homomorphism} iff:
$$f(x) \cdot f(y) = f(x\cdot y)$$
and \emph{isomorphism} if $f$ is a bijection as well.
A triple $\bar{f} = (f_1, f_2, f_3)$ of functions ($f_i : Q \to
R$) is a {homotopy} iff:
$$f_1(x) \cdot f_2(y) = f_3(x\cdot y)$$
which implies (and is implied by any of):
\begin{alignat}{2}
f_3(x) \rd f_2(y) &= f_1(x \rd y) & \qquad \qquad \qquad \qquad f_2(x) \rdd f_3(y) &= f_1(x \rdd y) \notag \\
f_1(x) \ld f_3(y) &= f_2(x \ld y) & \qquad \qquad \qquad \qquad f_3(x) \ldd f_1(y) &=
f_2(x \ldd y) \notag
\end{alignat}
If all three components of $\bar{f}$ are bijections, then
$\bar{f}$ is an {isotopy}.
\begin{center}
***
\end{center}
We can also
define a quasigroup as an algebra $(Q; \cdot, \rd, \ld)$ with three binary operations: multiplication ($\cdot$), right and left division.
The axioms that a quasigroup satisfies are ($xy$ is short for $(x \cdot y)$): \\
\parbox{8cm}{\[
\begin{alignedat}{2}
xy \rd y &= x & \qquad \qquad \qquad \qquad x \ld xy &= y \\
(x \rd y) y &= x & \qquad \qquad \qquad \qquad x (x \ld y) &= y
\end{alignedat}\]}
\hfill
\parbox{3cm}
{
\begin{equation*}
\tag{Q}
\end{equation*}}
\noindent For obvious reasons, such quasigroups are called
\emph{equational, primitive} or \emph{equasigroups}.
Thus, we have the variety of all quasigroups. Another important
variety is the variety of semisymmetric quasigroups, defined by
one of the following five equivalent axioms (in addition to (Q)):
\eqn{x \cdot yx = y}{SS1} \eqn{xy \cdot x = y}{SS2}
\[
x \rd y =yx
\]
\[
x \ld y = yx
\]
\[
x \ld y = x \rd y
\]
Smith, \cite{Smith}, defined a \emph{semisymmetrization} of a
quasigroup $\mathbb{Q} = (Q; \cdot, \rd, \ld)$ as a one--operation
quasigroup $\mathbb{Q}^{\Delta} = (Q^3; \circ)$ where the binary
operation $\circ$ is defined by: \eqn{(x_1,x_2,x_3) \circ
(y_1,y_2,y_3) = (y_3 \rd x_2, y_1 \ld x_3, x_1 y_2)}{SSprod} and
proved that, for any quasigroup $\mathbb{Q}$\/, the
semisymmetrization $\mathbb{Q}^{\Delta}$ of $\mathbb{Q}$ is a
semisymmetric quasigroup.
\section{Twisted quasigroups}\label{twist}
For our purpose, there is a better way to define a quasigroup. In
this definition the \emph{twisted quasigroup} is an algebra $(Q;
\rdd, \ldd, \cdot)$ satisfying appropriate paraphrasing of the
above quasigroup axioms (Q):
\begin{alignat}{2}
\label{ax2}
y \rdd xy &= x & \qquad \qquad \qquad \qquad xy \ldd x &= y \notag \\
(y \rdd x) y &= x & \qquad \qquad \qquad \qquad x (y \ldd x) &= y \notag
\end{alignat}
We have the following symmetry result, lacking for quasigroups
defined as $(Q; \cdot, \rd, \ld)$\/.
\begin{prop}
\label{TQ} An algebra $(Q; \rdd, \ldd, \cdot)$ is a \emph{twisted
quasigroup} iff $(Q; \ldd, \cdot, \rdd)$ is a \emph{twisted
quasigroup} iff $(Q; \cdot, \rdd, \ldd)$ is a \emph{twisted
quasigroup}.
\end{prop}
Analogously, we have the paraphrasing of axioms for \emph{twisted
semisymmetric quasigroups}: (\ref{eq:SS1}),(\ref{eq:SS2}) and
\[
x \rdd y = xy
\]
\[
x \ldd y = xy
\]
\[
x \ldd y = x \rdd y
\]
The last three identities we shorten to symbolic identities:
$\rdd = \cdot, \ldd = \cdot, \ldd = \rdd$\/.
There is also a result corresponding to Proposition~\ref{TQ}:
\begin{prop}
\label{SQ} An algebra $(Q; \rdd, \ldd, \cdot)$ is a
\emph{semisymmetric twisted quasigroup} iff $(Q; \ldd, \cdot,
\rdd)$ is a \emph{semisymmetric twisted quasigroup} iff $(Q;
\cdot, \rdd, \ldd)$ is a \emph{semisymmetric twisted quasigroup}.
\end{prop}
\begin{center}
***
\end{center}
Using twisted quasigroups we can see how a (twisted)
semisymmetrization (defined below), which we call $\nabla$\/,
'works'.
Let us start with three single--operation quasigroups $(Q; \cdot),
(Q; \rdd)$ and $(Q; \ldd)$\/, where $\rdd$ and $\ldd$ are duals of
appropriate division operations of $\cdot$\/. We can define direct
(Cartesian) product
$(Q; \rdd) \times(Q; \ldd) \times (Q; \cdot)$ and an operation $\otimes$ on $Q^3$ such that
\eqn{(x_1,x_2,x_3) \otimes (y_1,y_2,y_3) = (x_1 \rdd y_1, x_2 \ldd
y_2, x_3 y_3)}{Xprod} defines multiplication in the direct
product. Therefore $(Q^3; \otimes)$ is a quasigroup.
Define also a permutation $' : Q^3 \to Q^3$ by $(x_1,x_2,x_3)' =
(x_2,x_3,x_1)$\/. It follows that $(x_1,x_2,x_3)'' =
(x_3,x_1,x_2)$ and $(x_1,x_2,x_3)''' = (x_1,x_2,x_3)$\/. Define
another operation $\nabla_3 : Q^3 \times Q^3\to Q^3$ by $\bar{x}
\nabla_3 \bar{y} = \bar{x}' \otimes \bar{y}''$, where
$\bar{u}=(u_1,u_2,u_3)$\/. The groupoid $(Q^3; \nabla_3)$ is also
a quasigroup, so there are appropriate division operations of
$\nabla_3$ and their duals $\nabla_1$ and $\nabla_2$:
$$\bar{x} \nabla_3 \bar{y} = \bar{z} \mbox{\quad iff \quad}
\bar{y} \nabla_1 \bar{z} = \bar{x} \mbox{\quad iff \quad} \bar{z}
\nabla_2 \bar{x} = \bar{y} .$$ Therefore $(Q^3; \nabla_1,
\nabla_2, \nabla_3)$ is a twisted quasigroup.
Let us calculate $\nabla_1$\/.
\begin{tabbing}
\hspace{1.5em} $\bar{z}$ \= $ = (z_1,z_2,z_3) = \bar{x} \nabla_3
\bar{y} = (x_1,x_2,x_3)' \otimes (y_1,y_2,y_3)''$
\\[1ex]
\> $ = (x_2,x_3,x_1) \otimes (y_3,y_1,y_2) = (x_2 \rdd y_3, x_3
\ldd y_1, x_1 y_2)$\/.
\end{tabbing}
Therefore
\[
\bar{x} = (y_2 \rdd z_3, y_3 \ldd z_1, y_1 z_2) = (y_2,y_3,y_1)
\otimes (z_3,z_1,z_2) = \bar{y}' \otimes \bar{z}'' = \bar{y}
\nabla_3 \bar{z}
\]
i.e. $\nabla_1 = \nabla_3$ (and consequently $\nabla_2 =
\nabla_3$) hence $(Q^3; \nabla_1, \nabla_2, \nabla_3)$ is
semisymmetric twisted quasigroup. So we recognize $\nabla_3$ as a
twisted analogue of Smith's $\circ$ (see identity (\ref{eq:SSprod})). Let
us call $\mathbb{Q}^{\nabla} = (Q^3; \nabla_1, \nabla_2,
\nabla_3)$ a \emph{twisted semisymmetrization} of $\mathbb{Q}$\/.
For $(f_1,f_2,f_3)$ being a homotopy from $\mathbb{Q}$ to
$\mathbb{R}$, we also have:
\begin{tabbing}
\hspace{1.5em}$(f_1\times f_2\times f_3)\,(\bar{x} \nabla_3
\bar{y})$ \= $=(f_1\times f_2\times f_3)\,(\bar{x}' \otimes
\bar{y}'') $
\\[1ex]
\> $= (f_1(x_2 \rdd y_3), f_2(x_3 \ldd y_1), f_3(x_1 \cdot y_2))$
\\[1ex]
\> $= (f_2 x_2 \rdd f_3 y_3, f_3 x_3 \ldd f_1 y_1, f_1 x_1 \cdot
f_2 y_2)$
\\[1ex]
\> $= (f_2 x_2, f_3 x_3, f_1 x_1) \otimes (f_3 y_3, f_1 y_1, f_2
y_2)$
\\[1ex]
\> $ = (f_1 x_1, f_2 x_2, f_3 x_3)' \otimes (f_1 x_1, f_2 x_2,
f_3 x_3)''$
\\[1ex]
\> $ = (f_1\times f_2\times f_3)\,(\bar{x}) \nabla_3 (f_1\times
f_2\times f_3)\,(\bar{y})$\/,
\end{tabbing}
so $f_1\times f_2\times f_3$ is a homomorphism.
\section{Biquasigroups}\label{biq}
\begin{defn}
\label{bi} An algebra $(Q; \rdd, \ldd)$ is a \emph{biquasigroup} iff
$\rdd (\ldd)$ is the dual of the right (left) division operation
of a quasigroup operation $\cdot$\/.
A biquasigroup is semisymmetric iff $\ldd = \rdd$\/.
\end{defn}
\begin{prop}
An algebra $(Q; \rdd, \ldd)$ is a biquasigroup iff $(Q; \ldd,
\cdot)$ is a biquasigroup iff $(Q; \cdot, \ldd)$ is a
biquasigroup.
\end{prop}
\begin{prop}
An algebra $(Q; \rdd, \ldd)$ is a semisymmetric biquasigroup iff $(Q; \ldd,
\cdot)$ is a semisymmetric biquasigroup iff $(Q; \cdot, \ldd)$ is a
semisymmetric biquasigroup.
\end{prop}
\begin{center}
***
\end{center}
Let us start with three single--operation quasigroups $(Q; \cdot),
(Q; \rdd)$ and $(Q; \ldd)$\/, where $\rdd$ and $\ldd$ are duals of
appropriate division operations of $\cdot$\/. We can define direct
(Cartesian) product
$(Q; \rdd) \times(Q; \ldd)$ and an operation $\otimes$ on $Q^2$ such that
\[
(x_1,x_2) \otimes (y_1,y_2) = (x_1 \rdd y_1, x_2 \ldd
y_2)
\]
defines multiplication in the direct product. Therefore $(Q^2;
\otimes)$ is a quasigroup.
Define also a permutation $' : Q^2 \to Q^2$ by $(x_1,x_2)' =
(x_2,x_1)$\/.
Define another operation $\nabla : Q^2\times Q^2 \to Q^2$ by
$\hat{x} \nabla \hat{y} = R_{\hat{y}}(\hat{x}') \otimes
L_{\hat{x}}(\hat{y}')$\/, where $\hat{u}$ is $(u_1,u_2)$,
$L_{\hat{x}}(\hat{y}) = (x_1\cdot y_1, y_2)$ and
$R_{\hat{y}}(\hat{x}) = (x_1, x_2\cdot y_2)$\/. The groupoid
$(Q^2; \nabla)$ is also a quasigroup, moreover a semisymmetric
one.
Therefore $(Q^2; \nabla, \nabla)$ is a semisymmetric
biquasigroup.
Let us define:
$$x \operatorname{Op}{1} y = x \rdd y \qquad\qquad
x \operatorname{Op}{2} y = x \ldd y \qquad\qquad x \operatorname{Op}{3} y = x \cdot y$$ Then
the definition of $\nabla_{12}$, which we abbreviate just by
$\nabla$, is:
$$(x_1,x_2) \nabla_{12} (y_1,y_2) = (x_2 \operatorname{Op}{1} (x_1 \operatorname{Op}{3} y_2), (x_1 \operatorname{Op}{3} y_2) \operatorname{Op}{2} y_1).$$
There are two more alternative semisymmetrizations with corresponding definitions in $(Q^2; \ldd, \cdot)$
(respectively $(Q^2; \cdot, \rdd)$):
$$(x_1,x_2) {\nabla}_{23} (y_1,y_2) = (x_2 \operatorname{Op}{2} (x_1 \operatorname{Op}{1} y_2), (x_1 \operatorname{Op}{1} y_2) \operatorname{Op}{3} y_1)$$
$$(x_1,x_2) {\nabla}_{31} (y_1,y_2) = (x_2 \operatorname{Op}{3} (x_1 \operatorname{Op}{2} y_2), (x_1 \operatorname{Op}{2} y_2) \operatorname{Op}{1} y_2).$$
The indexing of operations is used to emphasize the symmetry.
These semisymmetrizations are object functions of functors related
to a functor explained in details in Section~\ref{new}.
\section{The categories \textbf{Qtp} and \textbf{P}}\label{adj}
This section follows
the lines of \cite{Smith} with some adjustments. The main novelty
is a proof of \cite[Corollary~5.3]{Smith}. We try to keep to the
notation introduced in \cite{Smith}. However, we write functions
and functors to the left of their arguments.
Let \textbf{Qtp} be the category with objects all small quasigroups
$\mathbb{Q}=(Q; \cdot, \rd, \ld)$ and arrows all homotopies. The
identity homotopy on $\mathbb{Q}$ is the triple
$(\mj_Q,\mj_Q,\mj_Q)$, where $\mj_Q$ is the identity function on
$Q$, and the composition of homotopies
\[
(f_1,f_2,f_3)\!:\mathbb{P}\rightarrow \mathbb{Q}\quad\mbox{\rm
and}\quad (g_1,g_2,g_3)\!:\mathbb{Q}\rightarrow \mathbb{R}
\]
is the homotopy
\[
(g_1\circ f_1,g_2\circ f_2,g_3\circ f_3)\!:\mathbb{P}\rightarrow
\mathbb{R}.
\]
Let \textbf{P} be the category with objects all small
semisymmetric quasigroups and arrows all quasigroup homomorphisms.
For every arrow $f\!:\mathbb{Q}\str \mathbb{R}$ of \textbf{P}, the
triple $(f,f,f)$ is a homotopy between $\mathbb{Q}$ and
$\mathbb{R}$.
Let $\Sigma$ be a functor from \textbf{P} to \textbf{Qtp}, which
is identity on objects. Moreover, let $\Sigma f$, for a
homomorphism $f$, be the homotopy $(f,f,f)$.
The category \textbf{P} is a full subcategory of the category
\textbf{Q} with objects all small quasigroups and arrows all
quasigroup homomorphisms. The functor $\Sigma$ is just a
restriction of a functor from \textbf{Q} to \textbf{Qtp}, which is
defined in the same manner.
An \emph{adjunction} is given by two functors, $F\!:\textbf{C}\str
\textbf{D}$ and $G\!:\textbf{D}\str \textbf{C}$, and two natural
transformations, the \emph{unit} $\eta\!:\mj_\textbf{C}\strt GF$
and the \emph{counit} $\varepsilon\!:FG\strt \mj_\textbf{D}$, such
that for every object $C$ of $\textbf{C}$ and every object $D$ of
$\textbf{D}$
\[
G\varepsilon_D \circ \eta_{GD}=\mj_{GD}, \quad \mbox{\rm and}\quad
\varepsilon_{FC}\circ F\eta_C=\mj_{FC}.
\]
These two equalities are called \emph{triangular identities}. The
functor $F$ is a \emph{left adjoint} for the functor $G$, while
$G$ is a \emph{right adjoint} for the functor $F$.
That $\Sigma\!:\textbf{P}\str \textbf{Qtp}$ has a right adjoint is
shown as follows. Let $\rdd$ and $\ldd$ be defined as at the
beginning of Section~\ref{Q}. For $\mathbb{Q}$ a quasigroup, let
$\nabla_3\!:Q^3\times Q^3\to Q^3$ be defined as in
Section~\ref{twist}, i.e.\ for every $\bar{x}=(x_1,x_2,x_3)$ and
$\bar{y}=(y_1,y_2,y_3)$
\[
\bar{x} \nabla_3 \bar{y}=(x_2 \rdd y_3, x_3 \ldd y_1, x_1\cdot
y_2).
\]
That $(Q^3;\nabla_3)$ is a semisymmetric quasigroup follows from
the fact that the structure $(Q^3;\nabla_1,\nabla_2,\nabla_3)$ is
a semisymmetric twisted quasigroup, which is shown in
Section~\ref{twist}. The semisymmetric quasigroup $(Q^3;\nabla_3)$
is the semisymmetrization $\mathbb{Q}^{\Delta}$ of $\mathbb{Q}$
defined at the end of Section~\ref{Q} (see (\ref{eq:SSprod})).
Let $\Delta\!:\textbf{Qtp}\str \textbf{P}$ be a functor, which
maps a quasigroup $\mathbb{Q}$ to the semisymmetric quasigroup
$(Q^3;\nabla_3)$. A homotopy $(f_1,f_2,f_3)$ is mapped by $\Delta$
to the product $f_1\times f_2\times f_3$, which is a homomorphism
as it is shown at the end of Section~\ref{twist}. By the
functoriality of product, we have that $\Delta$ preserves
identities and composition, and it is indeed a functor.
\begin{prop}\label{adjunction}
The functor $\Delta$ is a right adjoint for $\Sigma$.
\end{prop}
\begin{proof}
For $\mathbb{P}$ an object of \textbf{P}, let
$\eta_\mathbb{P}\!:P\str P^3$ be a function defined so that for
every $x\in P$, $\eta_\mathbb{P}(x)=(x,x,x)$. Note that
$\eta_\mathbb{P}$ is a homomorphism from $\mathbb{P}$ to
$\Delta\Sigma\mathbb{P}$, i.e.\ an arrow of \textbf{P}, since
\[
\eta_\mathbb{P}(x\cdot y)=(x\cdot y, x\cdot y, x\cdot y)=(x\rdd
y,x\ldd y,x\cdot y)=\eta_\mathbb{P}(x)\nabla_3\eta_\mathbb{P}(y).
\]
(This is why we consider just a restriction $\Sigma$ of a functor
from \textbf{Q} to \textbf{Qtp}. For an object $\mathbb{P}$ of
\textbf{Q}, the function defined as $\eta_\mathbb{P}$ need not be
a homomorphism.)
Let $\eta$ be the family
\[
\{\eta_\mathbb{P}\mid \mathbb{P}\; \mbox{\rm is an object of }
\textbf{P}\}.
\]
This family is a natural transformation from the identity functor
on \textbf{P} to the composition $\Delta\Sigma$ since for every
arrow $f\!:\mathbb{P}\str \mathbb{Q}$ of \textbf{P} and every
$x\in P$, we have
\[
(\eta_\mathbb{Q}\circ f)(x)=(f(x),f(x),f(x))=(\Delta\Sigma f\circ
\eta_\mathbb{P}) (x).
\]
For $\mathbb{Q}$ an object of \textbf{Qtp} and $i\in\{1,2,3\}$,
let $\pi_i\!:Q^3\str Q$ be the $i$th projection. Let
$\varepsilon_\mathbb{Q}$ be the triple $(\pi_1,\pi_2,\pi_3)$,
which is a homotopy from $\Sigma\Delta \mathbb{Q}$ to
$\mathbb{Q}$, since
\[
\pi_1(\bar{x})\cdot \pi_2(\bar{y})=x_1\cdot
y_2=\pi_3(\bar{x}\:\nabla_3\: \bar{y}).
\]
Hence, $\varepsilon_\mathbb{Q}$ is an arrow of \textbf{Qtp}.
Let $\varepsilon$ be the family
\[
\{\varepsilon_\mathbb{Q}\mid \mathbb{Q}\; \mbox{\rm is an object
of } \textbf{Qtp}\}.
\]
This family is a natural transformation from the composition
$\Sigma\Delta$ to the identity functor on \textbf{Qtp} since for
every arrow $(f_1,f_2,f_3)\!:\mathbb{Q}\str \mathbb{R}$ of
\textbf{Qtp} and every $\bar{x}$ from $\Sigma\Delta \mathbb{Q}$,
we have
\[
(\pi_i\circ (f_1\times f_2\times
f_3))(\bar{x})=f_i(x_i)=(f_i\circ\pi_i)(\bar{x}).
\]
Eventually, we have to show that the following triangular
identities hold for every object $\mathbb{Q}$ of \textbf{Qtp} and
every object $\mathbb{P}$ of \textbf{P}
\[
\Delta(\varepsilon_\mathbb{Q})\circ \eta_{\Delta
\mathbb{Q}}=\mj_{\Delta \mathbb{Q}} \quad\mbox{\rm and}\quad
\varepsilon_{\Sigma \mathbb{P}}\circ
\Sigma(\eta_\mathbb{P})=\mj_{\Sigma \mathbb{P}}.
\]
\begin{tabbing}
For every $\bar{x}\in Q^3$, we have
\\[1.5ex]
\hspace{1.5em}\= $(\Delta(\varepsilon_\mathbb{Q})\circ
\eta_{\Delta \mathbb{Q}})(\bar{x})$ \=
$=\Delta(\varepsilon_\mathbb{Q})(\bar{x},\bar{x},\bar{x})=(\pi_1\times\pi_2\times\pi_3)
(\bar{x},\bar{x},\bar{x})=(x_1,x_2,x_3)$
\\[1ex]
\>\> $=\bar{x}$.
\\[2ex]
For every $x\in P$ and every $i\in\{1,2,3\}$, we have
\\[1.5ex]
\> $(\pi_i\circ \eta_\mathbb{P})(x)=\pi_i(x,x,x)=x$.
\end{tabbing}
\vspace{-4.5ex}
\end{proof}
Moreover, every component of the counit of this adjunction is epi
(i.e.\ right cancellable) and the semisymmetrization is one-one.
This is sufficent for \textbf{Qtp} to be isomorphic to a
subcategory of \textbf{P}. This is one way how to establish this
fact using the previous proposition. It is not clear to us how
this fact is obtained as a corollary of the corresponding
proposition in \cite{Smith}. However, if the goal was just to
establish that \textbf{Qtp} is isomorphic to a subcategory of
\textbf{P}, this adjunction is not necessary at all, which will be
also shown.
A functor $F\!:\textbf{C}\str \textbf{D}$ is \emph{faithful} when
for every pair $f,g\!:A\vdash B$ of arrows of \textbf{C}, $Ff=Fg$
implies $f=g$. An arrow $f\!:A\str B$ of \textbf{C} is \emph{epi}
when for every pair $g,h\!:B\str C$ of arrows of \textbf{C}, the
equality $g\circ f=h\circ f$ implies $g=h$. The following lemmata
will help us to prove that \textbf{Qtp} is isomorphic to a
subcategory of \textbf{P}.
\begin{lem}\label{Dfaith}
The functor $\Delta$ is faithful.
\end{lem}
\begin{proof}
By \cite[IV.3, Theorem~1, Part~(i)]{ML71} (see also
\cite[Section~4, Proposition~4.1]{D14} for an elegant proof of a
related result) it suffices to prove that for every object
$\mathbb{Q}$ of \textbf{Qtp}, the arrow $\varepsilon_\mathbb{Q}$
is epi.
Let $g,h\!:\mathbb{Q}\str \mathbb{R}$ be a pair of arrows of
\textbf{Qtp} such that $ g\circ \varepsilon_\mathbb{Q}=h\circ
\varepsilon_\mathbb{Q}$. This means that for every $i\in\{1,2,3\}$
we have that $g_i\circ\pi_i=h_i\circ \pi_i$. Hence, the function
$g_i$ is equal to the function $h_i$, since the function $\pi_i$
is right cancellable. (However, the homotopy
$\varepsilon_\mathbb{Q}$ need not have a right inverse in
\textbf{Qtp}.)
\vspace{1ex}
The second proof of this lemma is direct and does not rely on
Proposition~\ref{adjunction}. Simply, for homotopies
$(f_1,f_2,f_3)$ and $(g_1,g_2,g_3)$ from $\mathbb{Q}$ to
$\mathbb{R}$, if $f_1\times f_2\times f_3$ and $g_1\times
g_2\times g_3$ are equal as homomorphisms from $\Delta \mathbb{Q}$
to $\Delta \mathbb{R}$ in \textbf{P}, then for every
$i\in\{1,2,3\}$, $f_i=g_i$. Hence, these homotopies are equal
in~\textbf{Qtp}.
\end{proof}
\begin{lem}\label{dif}
If $(Q; \cdot, \rd, \ld)$ and $(Q; \cdot', \rd', \ld')$ are two
different quasigroups, then there are $x,y\in Q$ such that
\[
x\cdot y\neq x\cdot' y.\]
\end{lem}
\begin{proof}
Suppose that for every $x,y\in Q$, $x\cdot y= x\cdot' y$ holds.
Then for every $z,t\in Q$ we have
\[
z\rd t=((z\rd t)\cdot' t)\rd' t=((z\rd t)\cdot t)\rd' t)=z\rd' t.
\]
Analogously, we prove that for every $u,v\in Q$, $u\ld v=u\ld' v$.
Hence, $(Q; \cdot, \rd, \ld)$ and $(Q; \cdot', \rd', \ld')$ are
the same, which contradicts the assumption.
\end{proof}
\begin{lem}
The functor $\Delta$ is one-one on objects.
\end{lem}
\begin{proof}
Suppose that $(Q; \cdot, \rd, \ld)$ and $(Q'; \cdot', \rd', \ld')$
are two different quasigroups. If $Q$ and $Q'$ are different sets,
then $\Delta \mathbb{Q}$ and $\Delta \mathbb{Q}'$ are different.
If $Q=Q'$, then, by Lemma~\ref{dif}, there are $x$ and $y$ in this
set such that $x\cdot y\neq x\cdot' y$. Hence, the operations
$\nabla_3$ for $\Delta \mathbb{Q}$ and $\Delta \mathbb{Q}'$ differ
when applied to $(x,x,x)$ and $(y,y,y)$.
\end{proof}
\vspace{2ex}
As a corollary of these two lemmata we have the following result.
\begin{prop}\label{embedding}
The category {\em{\textbf{Qtp}}} is isomorphic to a subcategory of
{\em{\textbf{P}}}; namely, to its image under the functor
$\Delta$.
\end{prop}
As we have shown by the second proof of Lemma~\ref{Dfaith},
Proposition~\ref{embedding} is independent of
Proposition~\ref{adjunction}. The adjunction, together with this
embedding of \textbf{Qtp} in \textbf{P}, says that the category
\textbf{P} reflects in \textbf{Qtp} in the following sense. A
subcategory \textbf{A} of \textbf{B} is \emph{reflective} in
\textbf{B}, when the inclusion functor from \textbf{A} to
\textbf{B} has a left adjoint called a \emph{reflector}. The
adjunction is called a \emph{reflection} of \textbf{B} in
\textbf{A}.
Propositions \ref{adjunction} and \ref{embedding} say that
\textbf{Qtp} may be considered as a reflective subcategory of
\textbf{P}. The functor $\Sigma$ is a reflector and the adjunction
between $\Sigma$ and $\Delta$ is a reflection of \textbf{P} in
\textbf{Qtp}. However, this does not mean that two quasigroups are
isotopic in \textbf{Qtp} if and only if their semisymmetrizations
are isomorphic in \textbf{P}, which one may conclude from
\cite[first paragraph in the introduction]{ImKoSm}. The reader
should be aware of this potential missusing of these results.
\section{Monadicity of $\Delta$}
For $F\!:\textbf{C}\str \textbf{D}$ a left adjoint for
$G\!:\textbf{D}\str \textbf{C}$, and $\eta$ and $\varepsilon$, the
unit and counit of this adjunction, a $GF$-algebra is a pair
$(C,h)$, where $C$ is an object of \textbf{C} and $h\!: GFC\str C$
is an arrow of \textbf{C} such that the following equalities hold.
\[
h\circ GF h=h\circ G\varepsilon_{FC},\quad\quad h\circ
\eta_C=\mj_C.
\]
A morphism of $GF$-algebras $(C,h)$ and $(C',h')$ is given by an
arrow $f\!:C\str C'$ of \textbf{C} such that $f\circ h=h'\circ
GFf$.
The category $\textbf{C}^{GF}$ has $GF$-algebras as objects and
morphisms of $GF$-algebras as arrows. The \emph{comparison
functor} $K\!:\textbf{D}\str \textbf{C}^{GF}$ is given by
\[
KD=(GD,G\varepsilon_D),\quad\quad Kf=Gf.
\]
In many cases the comparison functor is an isomorphism or an
equivalence (i.e.\ there is a functor from $\textbf{C}^{GF}$ to
\textbf{D} such that both compositions with $K$ are naturally
isomorphic to the identity functors). The right adjoint of an
adjunction or an adjunction are called \emph{monadic} when the
comparison functor is an isomorphism (see \cite[VI.3]{ML71}, also
\cite[Section~4.2]{SR99}). Some other authors (see
\cite[Section~3.3]{BW85}) call an adjunction monadic (tripleable)
when $K$ is just an equivalence.
In the case of adjoint situation involving $\Sigma$ and $\Delta$,
the comparison functor $K\!: \textbf{Qtp}\str
\textbf{P}^{\Delta\Sigma}$ is just an equivalence. To prove this,
by \cite[IV.4, Theorem~1]{ML71} it suffices to prove that $K$ is
full and faithful, and that every $GF$-algebra is isomorphic to
$K\mathbb{Q}$ for some quasigroup $\mathbb{Q}$. The faithfulness
of $K$ follows from \ref{Dfaith} since the arrow function $K$
coincides with the arrow function $\Delta$. That every
$GF$-algebra is isomorphic to $K\mathbb{Q}$ for some quasigroup
$\mathbb{Q}$ is proven in \cite[Section 10, Theorem~33]{Smith8}.
A functor $F\!:\textbf{C}\str \textbf{D}$ is \emph{full} when for
every pair of objects $C_1$ and $C_2$ of $\textbf{C}$ and every
arrow $g\!:FC_1\str FC_2$ of $\textbf{D}$ there is an arrow
$f\!:C_1\str C_2$ of $\textbf{C}$ such that $g=Ff$. It remains to
prove that $K$ is full. For this we use the following lemma.
\begin{lem}\label{lema1}
Every arrow of $\textbf{P}^{\Delta\Sigma}$ from $K\mathbb{Q}$ to
$K\mathbb{R}$ is of the form $f_1\times f_2\times f_3$, for
$(f_1,f_2,f_3)$ a homotopy from $\mathbb{Q}$ to $\mathbb{R}$.
\end{lem}
\begin{proof}
For quasigroups $\mathbb{Q}$ and $\mathbb{R}$ we have that
$K\mathbb{Q}=(\Delta \mathbb{Q},\pi_1\times\pi_2\times\pi_3)$ and
$K\mathbb{R}=(\Delta \mathbb{R},\pi_1\times\pi_2\times\pi_3)$. So,
let
\[
f\!:(\Delta \mathbb{Q},\pi_1\times\pi_2\times\pi_3)\str (\Delta
\mathbb{R},\pi_1\times\pi_2\times\pi_3)
\]
be an arrow of $\textbf{P}^{\Delta\Sigma}$. Since $f$ is a
morphism of $\Delta\Sigma$-algebras, we have that
\[
f\circ(\pi_1\times\pi_2\times\pi_3)=
(\pi_1\times\pi_2\times\pi_3)\circ (f\times f\times f)
\]
as functions from $(Q^3)^3$ to $R^3$.
For $i\in\{1,2,3\}$ and $u\in Q$, let $f_i(u)=\pi_i(f(u,u,u))$.
Let $(x,y,z)$ be an arbitrary element of $Q^3$. Apply the both
sides of the above equality to $((x , x , x ), ( y , y , y ), ( z
, z , z ))\in (Q^3)^3$ in order to obtain
\begin{tabbing}
\hspace{1.5em}\= $f(x,y,z)$ \= $=(\pi_1(f(x,x,x)),
\pi_2(f(y,y,y)), \pi_3(f(z,z,z)))$
\\[1ex]
\>\> $=(f_1(x),f_2(y),f_3(z))$.
\end{tabbing}
Hence, $f=f_1\times f_2\times f_3$ and since it is a homomorphism
from $\Delta \mathbb{Q}$ to $\Delta \mathbb{R}$, we have for every
$\bar{x},\bar{y}\in Q^3$
\[
(f_1\times f_2\times f_3) (\bar{x})\;\nabla_3\; (f_1\times
f_2\times f_3) (\bar{y})=(f_1\times f_2\times
f_3)(\bar{x}\;\nabla_3\; \bar{y}).
\]
By restricting this equality to the third component, we obtain
$f_1(x_1)\cdot f_2(y_2)=f_3(x_1\cdot y_2)$, and hence
$(f_1,f_2,f_3)$ is a homotopy from $\mathbb{Q}$ to $\mathbb{R}$.
\end{proof}
\section{A new semisymmetrization}\label{new}
In this section we introduce a new semisymmetrization functor from
\textbf{Qtp} to \textbf{P}. This leads to another subcategory of
\textbf{P} isomorphic to \textbf{Qtp}. We start with an auxiliary
result.
\begin{lem}\label{homotopy}
The third component $f_3$ of a homotopy is determined by the first
two components $f_1$ and $f_2$.
\end{lem}
\begin{proof}
Let $\mathbb{Q}$ be a quasigroup. For every element $x\in Q$ there
are $y,z\in Q$ such that $x=y\cdot z$ (e.g.\ $x=y\cdot(y\ld x)$).
Hence, $f_3(x)=f_1(y)\cdot f_2(z)$.
\end{proof}
Let $\Gamma\!:\textbf{Qtp}\str \textbf{P}$ be a functor defined on
objects so that $\Gamma \mathbb{Q}$ is a semisymmetric quasigroup
$(Q^2;\nabla)$ (see Section~\ref{biq}) whose elements are pairs
$(x_1,x_2)$, abbreviated by $\hat{x}$, and $\nabla$ is defined so
that
\[
(x_1,x_2)\nabla(y_1,y_2)=(x_2\rdd(x_1\cdot y_2),(x_1\cdot y_2)\ldd
y_1).
\]
(It is straightforward to check that $(\hat{y}\nabla
\hat{x})\nabla \hat{y}=\hat{y}\nabla(\hat{x}\nabla
\hat{y})=\hat{x}$, hence $\Gamma \mathbb{Q}$ is a semisymmetric
quasigroup.)
A homotopy $(f_1,f_2,f_3)$ is mapped by $\Gamma$ to the product
$f_1\times f_2$, which is a homomorphism:
\begin{tabbing}
\hspace{1.5em}$(f_1\times f_2) (\hat{x})\;\nabla\; (f_1\times f_2)
(\hat{y})=$
\\[1ex]
\hspace{11em}\=$=(f_2(x_2)\rdd(f_1(x_1)\cdot
f_2(y_2)),(f_1(x_1)\cdot f_2(y_2))\ldd f_1(y_1))$
\\[1ex]
\>$=(f_1(x_2\rdd(x_1\cdot y_2)),f_2((x_1\cdot y_2)\ldd y_1))$
\\[1ex]
\>$=(f_1\times f_2)(\hat{x}\nabla \hat{y})$.
\end{tabbing}
By the functoriality of product, we have that $\Gamma$ preserves
identities and composition, and it is indeed a functor.
The functor $\Gamma$ is not a right adjoint for $\Sigma$ since a
right adjoint is unique up to isomorphism and $\Gamma \mathbb{Q}$
is not isomorphic to $\Delta \mathbb{Q}$ for every object
$\mathbb{Q}$ of \textbf{Qtp}. However, this adjunction is not
necessary for the faithfulness of $\Gamma$.
\begin{lem}
The functor $\Gamma$ is faithful.
\end{lem}
\begin{proof}
We proceed as in the second proof of Lemma~\ref{Dfaith}. If
$(f_1,f_2,f_3)$ and $(g_1,g_2,g_3)$ are two homotopies from
$\mathbb{Q}$ to $\mathbb{R}$, then
$\Gamma(f_1,f_2,f_3)=\Gamma(g_1,g_2,g_3)$ means that $f_1\times
f_2=g_1\times g_2$. Hence, $f_1=g_1$ and $f_2=g_2$, and by
Lemma~\ref{homotopy}, $f_3=g_3$.
\end{proof}
The functor $\Gamma$, as defined, is not one-one on objects. For
example,
\[
(\{0,1\},+,+,+)\quad \mbox{\rm and} \quad
(\{0,1\},\oplus,\oplus,\oplus),
\]
where $+$ is addition mod 2 and $x \oplus y = x+y+1$\/,
are mapped by $\Gamma$ to
the same object of \textbf{P}. To remedy this matter, one may
redefine $\Gamma$ so that
\[
\Gamma \mathbb{Q}=(Q^2\times\{\mathbb{Q}\},\nabla),
\]
where $\mathbb{Q}$, as the third component of every element,
guarantees that $\Gamma$ is one-one on objects. The operation
$\nabla$ is defined as above, just neglecting the third component.
Hence, \textbf{Qtp} may be considered as another subcategory of
\textbf{P}.
\section*{\centerline{\small Acknowledgements}}
We are grateful to J.D.H.\ Smith for his comments on the previous
version of this note. Work on this paper was supported by the
Ministry of Education and Science of Serbia through grants
ON174008 (first author) and ON174026 (both authors). \small
|
1,116,691,500,855 | arxiv | \section{Solutions in terms of hypergeometric function}
There are well known solutions of the Schr\"{o}dinger equation
\begin{equation}
\left[ -\frac{d^{2}}{dx^{2}}+U\left( x\right) \right] \psi =E\psi
\label{r01}
\end{equation
with the P\"{o}schl-Teller potentia
\begin{equation}
U\left( x\right) =\alpha ^{2}\left[ \frac{\kappa \left( \kappa -1\right) }
\sin ^{2}\left( \alpha x\right) }+\frac{\lambda \left( \lambda -1\right) }
\cos ^{2}\left( \alpha x\right) }\right] ;\ \kappa >1,\ \lambda >1
\label{r02}
\end{equation
in the interval $\left( 0,\frac{\pi }{2\alpha }\right) $, which are
expressed in terms of hypergeometric function $\left. _{2}F_{1}\right.
\left( \alpha ,\beta ;\gamma ;x\right) =\sum_{n=0}^{\infty }\frac{\left(
\alpha \right) _{n}\left( \beta \right) _{n}}{\left( \gamma \right) _{n}
\frac{x^{n}}{n!}$
\begin{equation}
\psi _{n}\left( x\right) =A_{n}\sin ^{\kappa }\left( \alpha x\right) \cos
^{\lambda }\left( \alpha x\right) \left. _{2}F_{1}\right. \left( -n,n+\kappa
+\lambda ;\kappa +\frac{1}{2};\sin ^{2}\left( \alpha x\right) \right) ,\
n\geq 0\ , \label{r03}
\end{equation
$A_{n}$ is a normalization factor, with energy spectrum
\begin{equation}
E_{n}=\alpha ^{2}\left( 2n+\kappa +\lambda \right) ^{2}\ . \label{r04}
\end{equation
These solutions are described, for example, in Ref. \cite{f99}, problem 38
and presented in Ref. \cite{bg90}, Appendix A, problem A.I.6. We consider a
particular form of the P\"{o}schl-Teller potential with values of the
parameters $\kappa =\lambda =2$ and obtain solutions in terms of usual
trigonometric functions in a closed form (not series).
\section{Solutions generated by Darboux transformation method}
In order to obtain the solutions in terms of trigonometric functions we
apply the Darboux transformation (DT) method \cite{matveev-salle}, \cit
{bs95}, \cite{bs97} to the Schr\"{o}dinger equation with the infinite
rectangular well potential in the same interval $\left( 0,\frac{\pi }
2\alpha }\right) $. Then the initial operator of the DT method is the
Hamiltonia
\begin{equation}
\widehat{H}_{0}=-\frac{d^{2}}{dx^{2}},\ x\in \left( 0,\frac{\pi }{2\alpha
\right) ;\ \widehat{H}_{0}\varphi _{k}=\varepsilon _{k}\varphi _{k}\ .
\label{r10}
\end{equation
Its eigenfunctions and energy spectrum are given b
\begin{equation}
\varphi _{k}\left( x\right) =\sqrt{\frac{4\alpha }{\pi }}\sin \left( 2\alpha
kx\right) ,\ \varepsilon _{k}=4\alpha ^{2}k^{2},\ k\geq 1\ . \label{r09}
\end{equation
We note that eigenfunctions $\varphi _{k}$ in Eq. (\ref{r09}) are
orthonormalized. In accordance with the DT method we represent $\widehat{H
_{0}$ in the for
\begin{equation*}
\widehat{H}_{0}=\widehat{L}^{\dagger }\widehat{L}+\omega ^{2}\ ,
\end{equation*
where $\omega ^{2}$ is a real constant, $\widehat{L}$ is the intertwining
operator, and $\widehat{L}^{\dagger }$ is its adjoint:
\begin{equation}
\widehat{L}=\frac{d}{dx}+W\left( x\right) ,\ \widehat{L}^{\dagger }=-\frac{
}{dx}+W\left( x\right) \ . \label{r11}
\end{equation
Applying $\widehat{L}$ to both sides of Eq. (\ref{r10}) one ha
\begin{equation*}
\widehat{L}\widehat{H}_{0}\varphi _{k}=\widehat{L}\left( \widehat{L
^{\dagger }\widehat{L}+\omega ^{2}\right) \varphi _{k}=\left( \widehat{L
\widehat{L}^{\dagger }+\omega ^{2}\right) \widehat{L}\varphi
_{k}=\varepsilon _{k}\widehat{L}\varphi _{k}\
\end{equation*
that can be written a
\begin{equation}
\widehat{H}_{1}\chi _{k}=\varepsilon _{k}\chi _{k},\ \widehat{H}_{1}
\widehat{L}\widehat{L}^{\dagger }+\omega ^{2},\ \chi _{k}=\widehat{L}\varphi
_{k}\ , \label{r12}
\end{equation
where $\chi _{k}$ are eigenfunctions of $\widehat{H}_{1}$. The spectrum
\varepsilon _{k}$ of $\widehat{H}_{1}$ is the same of $\widehat{H}_{0}$,
except maybe some particular values. Using the relations (\ref{r11}), we
writ
\begin{equation}
\widehat{H}_{1}=-\frac{d^{2}}{dx^{2}}+V_{1}\left( x\right) ,\ V_{1}\left(
x\right) =W^{\prime }+W^{2}+\omega ^{2} \label{r14}
\end{equation
Choosing $W\left( x\right) $, $\omega ^{2}$ as follow
\begin{equation}
W\left( x\right) =-\frac{d}{dx}\ln \left( \varphi _{1}\right) =-\frac
\varphi _{1}^{\prime }}{\varphi _{1}},\ \omega ^{2}=\varepsilon _{1}\ ,
\label{r16}
\end{equation
one has for them from Eq. (\ref{r09}
\begin{equation*}
W\left( x\right) =-2\alpha \cot \left( 2\alpha x\right) ,\ \omega
^{2}=4\alpha ^{2}
\end{equation*
and for $V_{1}\left( x\right) $ of Eq. (\ref{r14}
\begin{equation}
V_{1}\left( x\right) =2\left( \frac{\varphi _{1}^{\prime 2}}{\varphi _{1}^{2
}+\varepsilon _{1}^{2}\right) =2\left( 2\alpha \right) ^{2}\left( \cot
^{2}\left( 2\alpha x\right) +1\right) =\frac{8\alpha ^{2}}{\sin ^{2}\left(
2\alpha x\right) }\ . \label{r15}
\end{equation
To obtain the eigenfunctions of $\widehat{H}_{1}$ by means of the DT method
we apply $\widehat{L}$ to the eigenfunctoions $\varphi _{k}$ of the initial
Hamiltonian $\widehat{H}_{0}$ that follows from Eq. (\ref{r12}
\begin{equation}
\chi _{k}=\widehat{L}\varphi _{k}=\frac{d\varphi _{k}}{dx}+W\left( x\right)
\varphi _{k} \label{r21}
\end{equation
\begin{equation*}
=\sqrt{\frac{4\alpha }{\pi }}2\alpha \left[ k\cos \left( 2\alpha kx\right)
-\cot \left( 2\alpha x\right) \sin \left( 2\alpha kx\right) \right] ,\ k\geq
2\ .
\end{equation*
The eigenfunctoions of (\ref{r21}) are not normalized. To normalize them we
use the following properties of an inner product of the eigenfunctions
\begin{equation*}
\left( \chi _{i},\chi _{j}\right) =\left( \widehat{L}\varphi _{i},\widehat{L
\varphi _{j}\right) =\left( \widehat{L}^{\dagger }\widehat{L}\varphi
_{i},\varphi _{j}\right) =\left( \left( \widehat{H}_{0}-\omega ^{2}\right)
\varphi _{i},\varphi _{j}\right)
\end{equation*
\begin{equation*}
=\left( \left( \varepsilon _{i}-\omega ^{2}\right) \varphi _{i},\varphi
_{j}\right) =\left( \varepsilon _{i}-\omega ^{2}\right) \left( \varphi
_{i},\varphi _{j}\right) =\left( \varepsilon _{i}-\omega ^{2}\right) \delta
_{ij}\ .
\end{equation*
Designating the normalized eigenfunctoions by $\widetilde{\chi }_{k}
\widetilde{N}_{k}\chi _{k}$, one ha
\begin{equation*}
\left( \widetilde{\chi }_{k},\widetilde{\chi }_{k}\right) =\left( \widetilde
N}_{k}\chi _{k},\widetilde{N}_{k}\chi _{k}\right) =\widetilde{N
_{k}^{2}\left( \chi _{k},\chi _{k}\right) =\widetilde{N}_{k}^{2}\left(
\varepsilon _{k}-\omega ^{2}\right) =1\ ,
\end{equation*
therefor
\begin{equation*}
\widetilde{N}_{k}=\frac{1}{\sqrt{\varepsilon _{k}-\omega ^{2}}}\ .
\end{equation*
For the choice of $\omega ^{2}$ indicated in Eq. (\ref{r16}) and with use of
Eq. (\ref{r09}) one ha
\begin{equation*}
\widetilde{N}_{k}=\frac{1}{\sqrt{\varepsilon _{k}-\varepsilon _{1}}}=\frac{
}{2\alpha \sqrt{k^{2}-1}}\ .
\end{equation*
Then the normalized eigenfunctions and the energy spectrum ar
\begin{equation}
\widetilde{\chi }_{k}\left( x\right) =N_{k}\left[ k\cos \left( 2\alpha
kx\right) -\cot \left( 2\alpha x\right) \sin \left( 2\alpha kx\right) \right]
,\ N_{k}=\sqrt{\frac{4\alpha }{\pi }}\frac{1}{\sqrt{k^{2}-1}},\ \varepsilon
_{k}=4\alpha ^{2}k^{2},\ k\geq 2\ . \label{r22}
\end{equation
For the ground state energy one ha
\begin{equation}
\varepsilon _{2}=4\alpha ^{2}2^{2}\ . \label{r24}
\end{equation
We note that at the middle of the interval, $x=\frac{\pi }{4\alpha }$
\begin{equation}
\widetilde{\chi }_{k}\left( \frac{\pi }{4\alpha }\right) =N_{k}\left[ k\cos
\left( k\frac{\pi }{2}\right) \right] ,\ \widetilde{\chi }_{k}^{\prime
}\left( \frac{\pi }{4\alpha }\right) =N_{k}2\alpha \left[ 1-k^{2}\right]
\sin \left( k\frac{\pi }{2}\right) \ , \label{r31}
\end{equation
then for even $k=2m$
\begin{equation}
\widetilde{\chi }_{2m}\left( \frac{\pi }{4\alpha }\right) =N_{2m}2m\left(
-1\right) ^{m},\ \widetilde{\chi }_{2m}^{\prime }\left( \frac{\pi }{4\alpha
\right) =0 \label{r32}
\end{equation
and for odd $k=2m+1$
\begin{equation}
\widetilde{\chi }_{2m+1}\left( \frac{\pi }{4\alpha }\right) =0,\ \widetilde
\chi }_{2m+1}^{\prime }\left( \frac{\pi }{4\alpha }\right) =N_{2m+1}2\alpha
\left[ 1-\left( 2m+1\right) ^{2}\right] \left( -1\right) ^{m}\ . \label{r33}
\end{equation}
\section{Correspondence between the solutions}
The potential $V_{1}\left( x\right) $ of Eq. (\ref{r15}) corresponds to the
\"{o}schl-Teller potential $U\left( x\right) $ of Eq. (\ref{r02}) with the
values of the parameters $\kappa =\lambda =2$:
\begin{equation*}
U\left( x\right) =2\alpha ^{2}\left( \frac{1}{\sin ^{2}\left( \alpha
x\right) }+\frac{1}{\cos ^{2}\left( \alpha x\right) }\right) =\frac{2\alpha
^{2}}{\sin ^{2}\left( \alpha x\right) \cos ^{2}\left( \alpha x\right) }
\frac{8\alpha ^{2}}{\sin ^{2}\left( 2\alpha x\right) }\ .
\end{equation*
The eigenfunctions and the energy in the form of Eqs. (\ref{r03}), (\ref{r04
) with $\kappa =\lambda =2$ ar
\begin{eqnarray}
&&\psi _{n}\left( x\right) =A_{n}\sin ^{2}\left( \alpha x\right) \cos
^{2}\left( \alpha x\right) \left. _{2}F_{1}\right. \left( -n,n+4;\frac{5}{2
;\sin ^{2}\left( \alpha x\right) \right) \ , \label{r25} \\
&&E_{n}=\alpha ^{2}\left( 2n+4\right) ^{2}=4\alpha ^{2}\left( n+2\right)
^{2},\ n\geq 0\ . \notag
\end{eqnarray
For the ground state energy one ha
\begin{equation}
E_{0}=4\alpha ^{2}2^{2}\ . \label{r26}
\end{equation
Taking into account Eqs. (\ref{r24}), (\ref{r26}) and Eqs. (\ref{r22}), (\re
{r25}), the following correspondence takes place for energies and the
normalized eigenfunctions
\begin{equation}
E_{n}=\varepsilon _{n+2},\ \psi _{n}\left( x\right) =\widetilde{\chi
_{n+2}\left( x\right) ,\ n\geq 0\ . \label{r27}
\end{equation
where due to Eq. (\ref{r22}) $\widetilde{\chi }_{n+2}\left( x\right) $ is
\begin{eqnarray}
&&\widetilde{\chi }_{n+2}\left( x\right) =N_{n+2}\left[ \left( n+2\right)
\cos \left( 2\left( n+2\right) \alpha x\right) -\cot \left( 2\alpha x\right)
\sin \left( 2\left( n+2\right) \alpha x\right) \right] ,\ \label{r23} \\
&&N_{n+2}=\sqrt{\frac{4\alpha }{\pi }}\frac{1}{\sqrt{\left( n+2\right) ^{2}-
}},\ n\geq 0\ . \notag
\end{eqnarray
We also note that at the middle of the interval, $x=\frac{\pi }{4\alpha }$
\begin{equation}
\psi _{n}\left( \frac{\pi }{4\alpha }\right) =A_{n}\frac{1}{4}\left.
_{2}F_{1}\right. \left( -n,n+4;\frac{5}{2};\frac{1}{2}\right) \ ,
\label{r28}
\end{equation
\begin{equation}
\psi _{n}^{\prime }\left( \frac{\pi }{4\alpha }\right) =A_{n}\frac{\alpha }
10}\left( -n\right) \left( n+4\right) \left. _{2}F_{1}\right. \left(
-n+1,n+5;\frac{7}{2};\frac{1}{2}\right) \ , \label{r29}
\end{equation
where the relatio
\begin{equation*}
\frac{d}{dx}\left. _{2}F_{1}\right. \left( \alpha ,\beta ;\gamma ;x\right)
\frac{\alpha \beta }{\gamma }\left. _{2}F_{1}\right. \left( \alpha +1,\beta
+1;\gamma +1;x\right)
\end{equation*
was used.
\section{Some new properties of the hypergeometric function}
Now we are at position to establish an expression of the hypergeometric
function of the form of Eq. (\ref{r25}) in terms of a composition of
trigonometric functions. We write the normalization factor $A_{n}$ of Eq.
\ref{r25}) as follow
\begin{equation}
A_{n}=C_{n}^{-1}N_{n+2},\ N_{n+2}=\sqrt{\frac{4\alpha }{\pi }}\frac{1}{\sqrt
\left( n+2\right) ^{2}-1}}\ . \label{r30}
\end{equation
Then from Eq. (\ref{r27}) and Eqs. (\ref{r25}), (\ref{r23}) one has
\begin{equation}
\left. _{2}F_{1}\right. \left( -n,n+4;\frac{5}{2};\sin ^{2}\left( \alpha
x\right) \right) =4C_{n}\frac{\left[ \left( n+2\right) \cos \left( 2\alpha
\left( n+2\right) x\right) -\cot \left( 2\alpha x\right) \sin \left( 2\alpha
\left( n+2\right) x\right) \right] }{\sin ^{2}\left( 2\alpha x\right) }\ .
\label{r34}
\end{equation
To determine $C_{n}$ in Eq. (\ref{r34}) we write Eq. (\ref{r27}) for the
eigenfunctions and their derivatives at the point $x=\frac{\pi }{4\alpha }$.
With use of Eq. (\ref{r31}) one ha
\begin{equation}
\widetilde{\chi }_{n+2}\left( \frac{\pi }{4\alpha }\right) =-N_{n+2}\left[
\left( n+2\right) \cos \left( n\frac{\pi }{2}\right) \right] ,\ \widetilde
\chi }_{n+2}^{\prime }\left( \frac{\pi }{4\alpha }\right) =N_{n+2}2\alpha
\left[ \left( n+2\right) ^{2}-1\right] \sin \left( n\frac{\pi }{2}\right) \ .
\label{r35}
\end{equation
Then from Eqs. (\ref{r28}), (\ref{r29}), (\ref{r35}) for even $n=2m$
\begin{equation}
\frac{1}{4}\left. _{2}F_{1}\right. \left( -2m,2m+4;\frac{5}{2};\frac{1}{2
\right) =C_{2m}2\left( m+1\right) \left( -1\right) ^{m+1}\ , \label{r36}
\end{equation
therefor
\begin{equation}
C_{2m}=\frac{\left( -1\right) ^{m+1}}{8\left( m+1\right) }\left.
_{2}F_{1}\right. \left( -2m,2m+4;\frac{5}{2};\frac{1}{2}\right) \ ;
\label{r37}
\end{equation
for odd $n=2m+1$
\begin{equation}
\frac{\alpha }{10}\left( -\left( 2m+1\right) \right) \left( 2m+5\right)
\left. _{2}F_{1}\right. \left( -2m,2m+6;\frac{7}{2};\frac{1}{2}\right)
=C_{2m+1}2\alpha \left[ \left( 2m+3\right) ^{2}-1\right] \left( -1\right)
^{m}\ , \label{r38}
\end{equation
therefor
\begin{equation}
C_{2m+1}=\frac{\left( -1\right) ^{m+1}}{20}\frac{\left( 2m+1\right) \left(
2m+5\right) }{4\left( m+1\right) \left( m+2\right) }\left. _{2}F_{1}\right.
\left( -2m,2m+6;\frac{7}{2};\frac{1}{2}\right) \ . \label{r39}
\end{equation
We also note that due to the properties of the solutions (\ref{r32}), (\re
{r33}) one can conclude tha
\begin{equation}
\left. _{2}F_{1}\right. \left( -\left( 2m+1\right) ,2m+5;\frac{5}{2};\frac{
}{2}\right) =0,\ \left. _{2}F_{1}\right. \left( -2m+1,2m+5;\frac{7}{2};\frac
1}{2}\right) =0,\ m=0,1,2,...\ . \label{r40}
\end{equation
Subsituting Eqs. (\ref{r37}), (\ref{r39}) into Eq. (\ref{r34}), one obtains
explicitl
\begin{eqnarray}
&&\frac{\left. _{2}F_{1}\right. \left( -2m,2m+4;\frac{5}{2};\sin ^{2}\left(
\alpha x\right) \right) }{\left. _{2}F_{1}\right. \left( -2m,2m+4;\frac{5}{2
;\frac{1}{2}\right) }=\frac{\left( -1\right) ^{m+1}}{2\left( m+1\right) }
\label{r41} \\
&&\times \frac{\left[ \left( 2m+2\right) \cos \left( 2\alpha \left(
2m+2\right) x\right) -\cot \left( 2\alpha x\right) \sin \left( 2\alpha
\left( 2m+2\right) x\right) \right] }{\sin ^{2}\left( 2\alpha x\right) }
\notag
\end{eqnarray
an
\begin{eqnarray}
&&\frac{\left. _{2}F_{1}\right. \left( -\left( 2m+1\right) ,2m+5;\frac{5}{2
;\sin ^{2}\left( \alpha x\right) \right) }{\left. _{2}F_{1}\right. \left(
-2m,2m+6;\frac{7}{2};\frac{1}{2}\right) }=\frac{\left( -1\right) ^{m+1}}{20
\frac{\left( 2m+1\right) \left( 2m+5\right) }{\left( m+1\right) \left(
m+2\right) } \label{r42} \\
&&\times \frac{\left[ \left( 2m+3\right) \cos \left( 2\alpha \left(
2m+3\right) x\right) -\cot \left( 2\alpha x\right) \sin \left( 2\alpha
\left( 2m+3\right) x\right) \right] }{\sin ^{2}\left( 2\alpha x\right) }\ .
\notag
\end{eqnarray}
Besides that we present results of some integrals with the combination of
trigonometric functions of Eq. (\ref{r22}) and with the hypergeometric
function of Eq. (\ref{r25}).
1. As far as the eigenfunctions $\widetilde{\chi }_{k}$ are normalized,
\int_{0}^{\frac{\pi }{2\alpha }}\widetilde{\chi }_{k}^{2}dx=1$, from Eq.
\ref{r22}) one ha
\begin{equation*}
\int_{0}^{\frac{\pi }{2\alpha }}\left[ k\cos \left( 2\alpha kx\right) -\cot
\left( 2\alpha x\right) \sin \left( 2\alpha kx\right) \right] ^{2}dx=\frac
\pi }{4\alpha }\left( k^{2}-1\right) \ .
\end{equation*
Passing to integration over the inerval $\left( 0,\pi \right) $ by a change
of variable $2\alpha x\rightarrow x$, one get
\begin{equation}
\int_{0}^{\pi }\left[ k\cos \left( kx\right) -\cot \left( x\right) \sin
\left( kx\right) \right] ^{2}dx=\frac{\pi }{2}\left( k^{2}-1\right) \ .
\label{r51}
\end{equation
Writing the eigenfunctions in terms of the hypergeometric function, Eqs.
\ref{r25}), (\ref{r30}), one ha
\begin{equation*}
\int_{0}^{\frac{\pi }{2\alpha }}\sin ^{4}\left( \alpha x\right) \cos
^{4}\left( \alpha x\right) \left. _{2}F_{1}^{2}\right. \left( -n,n+4;\frac{
}{2};\sin ^{2}\left( \alpha x\right) \right) dx=\frac{\pi }{4\alpha }\left(
\left( n+2\right) ^{2}-1\right) C_{n}^{2}\ .
\end{equation*
Passing to integration over the inerval $\left( 0,\pi /2\right) $ by a
change of variable $\alpha x\rightarrow x$, one get
\begin{equation}
\int_{0}^{\frac{\pi }{2}}\sin ^{4}x\cos ^{4}x\left. _{2}F_{1}^{2}\right.
\left( -n,n+4;\frac{5}{2};\sin ^{2}\left( x\right) \right) dx=\frac{\pi }{4
\left( \left( n+2\right) ^{2}-1\right) C_{n}^{2}\ . \label{r52}
\end{equation
Making a change of variable $z=\sin ^{2}x$ in Eq. (\ref{r52}), one obtain
\begin{equation}
\int_{0}^{1}z^{\frac{3}{2}}\left( 1-z\right) ^{\frac{3}{2}}\left.
_{2}F_{1}^{2}\right. \left( -n,n+4;\frac{5}{2};z\right) dz=\frac{\pi }{2
\left( \left( n+2\right) ^{2}-1\right) C_{n}^{2}\ . \label{r53}
\end{equation
The coefficient $C_{n}$ in Eqs. (\ref{r52}) and (\ref{r53}) must be taken in
dependence of even or odd $n$ a
\begin{eqnarray}
C_{n}^{even} &=&\frac{\left( -1\right) ^{\frac{n}{2}+1}\left.
_{2}F_{1}\right. \left( -n,n+4;\frac{5}{2};\frac{1}{2}\right) }{4\left(
n+2\right) },\ \label{r54} \\
C_{n}^{odd} &=&\frac{\left( -1\right) ^{\frac{n-1}{2}+1}n\left( n+4\right)
\left. _{2}F_{1}\right. \left( -n+1,n+5;\frac{7}{2};\frac{1}{2}\right) }
20\left( \left( n+2\right) ^{2}-1\right) }\ . \notag
\end{eqnarray}
2. One can demostrate that $\widetilde{\chi }_{k}^{2}\left( x\right) $ is
symmetric relative to the middle of the interval $x=\frac{\pi }{4\alpha }$
\begin{equation*}
\widetilde{\chi }_{k}^{2}\left( -\left( x-\frac{\pi }{4\alpha }\right)
\right) =\widetilde{\chi }_{k}^{2}\left( x-\frac{\pi }{4\alpha }\right) \ .
\end{equation*
Then one can write for the expected value of the coordinate $x$ in a state
\widetilde{\chi }_{n}$
\begin{equation}
\left\langle x\right\rangle _{k}=\int_{0}^{\frac{\pi }{2\alpha }}\widetilde
\chi }_{k}^{\ast }\widehat{x}\widetilde{\chi }_{k}dx=\int_{0}^{\frac{\pi }
2\alpha }}\widetilde{\chi }_{k}^{2}\left( x\right) xdx=\int_{-\frac{\pi }
4\alpha }}^{\frac{\pi }{4\alpha }}\widetilde{\chi }_{k}^{2}\left( \xi
\right) \left( \xi +\frac{\pi }{4\alpha }\right) d\xi \label{r61}
\end{equation
\begin{equation*}
=\frac{\pi }{4\alpha }\int_{-\frac{\pi }{4\alpha }}^{\frac{\pi }{4\alpha }
\widetilde{\chi }_{k}^{2}\left( \xi \right) d\xi =\frac{\pi }{4\alpha }\ ,
\end{equation*
where $\xi =x-\frac{\pi }{4\alpha }$. Substituting $\widetilde{\chi
_{k}\left( x\right) $ from Eq. (\ref{r22}) into Eq. (\ref{r61}), one ha
\begin{equation*}
\int_{0}^{\frac{\pi }{2\alpha }}\left[ k\cos \left( 2\alpha kx\right) -\cot
\left( 2\alpha x\right) \sin \left( 2\alpha kx\right) \right] ^{2}xdx=\frac
\pi ^{2}}{16\alpha ^{2}}\left( k^{2}-1\right) \ .
\end{equation*
Passing to integration over the inerval $\left( 0,\pi \right) $ by a change
of variable $2\alpha x\rightarrow x$, one get
\begin{equation}
\int_{0}^{\pi }\left[ k\cos \left( kx\right) -\cot \left( x\right) \sin
\left( kx\right) \right] ^{2}xdx=\frac{\pi ^{2}}{4}\left( k^{2}-1\right) \ .
\label{r62}
\end{equation
For the eigenfunctions in terms of the hypergeometric function, Eqs. (\re
{r25}), (\ref{r30}), one ha
\begin{equation*}
\int_{0}^{\frac{\pi }{2\alpha }}\sin ^{4}\left( \alpha x\right) \cos
^{4}\left( \alpha x\right) \left. _{2}F_{1}^{2}\right. \left( -n,n+4;\frac{
}{2};\sin ^{2}\left( \alpha x\right) \right) xdx=\frac{\pi ^{2}}{16\alpha
^{2}}\left( \left( n+2\right) ^{2}-1\right) C_{n}^{2}
\end{equation*
Passing to integration over the inerval $\left( 0,\pi /2\right) $ by a
change of variable $\alpha x\rightarrow x$, one get
\begin{equation}
\int_{0}^{\frac{\pi }{2}}\sin ^{4}\left( x\right) \cos ^{4}\left( x\right)
\left. _{2}F_{1}^{2}\right. \left( -n,n+4;\frac{5}{2};\sin ^{2}\left(
x\right) \right) xdx=\frac{\pi ^{2}}{16}\left( \left( n+2\right)
^{2}-1\right) C_{n}^{2}\ . \label{r63}
\end{equation
The coefficient $C_{n}$ in Eq. (\ref{r63}) must be taken in dependence of
even or odd $n$ as is given in Eq. (\ref{r54}).
|
1,116,691,500,856 | arxiv | \section{Introduction}
Let $X$ be an abelian surface over a field ${\mathfrak k}$.
Denote by $\operatorname{Coh}(X)$ the category of coherent sheaves on $X$,
by $\bl{D}(X)$ the bounded derived category of $\operatorname{Coh}(X)$
and by $K(X)$ the Grothendieck group of $\bl{D}(X)$.
For $\beta \in \operatorname{NS}(X)_\bb{Q}$ and an ample divisor
$\omega \in \operatorname{Amp}(X)_{\mathbb Q}$,
Bridgeland \cite{Br:3} constructed a stability condition
$\sigma_{\beta,\omega}=(\mathfrak{A}_{(\beta,\omega)},Z_{(\beta,\omega)})$
on $\bl{D}(X)$.
Here $\mathfrak{A}_{(\beta,\omega)}$ is a tilting of $\operatorname{Coh}(X)$,
and $Z_{(\beta,\omega)}:K(X) \to \bb{C}$ is a group homomorphism
called the stability function.
In terms of the Mukai lattice
$(H^*(X,\bb{Z})_{\operatorname{alg}}, \langle \cdot,\cdot \rangle)$,
$Z_{(\beta,\omega)}$ is given by
\begin{align*}
Z_{(\beta,\omega)}(E)=\langle e^{\beta+\sqrt{-1}\omega},v(E) \rangle,
\quad
E \in K(X).
\end{align*}
Here $v(E):=\operatorname{ch}(E)$ is the Mukai vector of $E$.
Hereafter for an object $E \in \bl{D}(X)$, we abbreviately write
$Z_{(\beta,\omega)}(E) := Z_{(\beta,\omega)}([E])$,
where $[E]$ is the class of $E$ in $K(X)$.
For abelian surfaces, these kind of stability conditions forms
a connected component of the space of stability conditions
up to the action of the universal cover
$\widetilde{\GL}^+(2,{\mathbb R})$ of $\GL^+(2,{\mathbb R})$
as stated in \cite[sect. 15]{Br:3}.
Let
$\phi_{(\beta,\omega)}
:\mathfrak{A}_{(\beta,\omega)} \setminus \{ 0\} \to (0,1]$
be the phase function, which is defined by
$$
Z_{(\beta,\omega)}(E)=|Z_{(\beta,\omega)}(E)|
e^{ \sqrt{-1}\pi \phi_{(\beta,\omega)}(E)}
$$
for $0 \ne E \in {\mathfrak A}_{(\beta,\omega)}$.
Let $M_{(\beta,\omega)}(v)$ be the moduli space of
$\sigma_{(\beta,\omega)}$-semi-stable objects $E$ with $v(E)=v$.
It is a projective scheme, if $(\beta,\omega)$ is general
(\cite[Thm. 0.0.2]{MYY:2011:2}).
The stability of objects depends only on a chamber of the parameter
space $\operatorname{NS}(X)_{\mathbb R} \times \operatorname{Amp}(X)_{\mathbb R}$.
For a special chamber,
the stability coincides with Gieseker stability
(\cite{Br:3}, \cite{Toda}, \cite{Bayer}, \cite{MYY:2011:1}).
For the analysis of Gieseker stability, Fourier-Mukai transforms
are very useful tool. However
the Fourier-Mukai transform
does not preserve Gieseker stability in general,
since the category of coherent sheaves is not preserved.
On the other hand, the Fourier-Mukai transform induces an isomorphism
of the moduli spaces of Bridgeland stable
objects as a consequence of \cite[Prop. 10.3]{Br:3}.
So it is very natural to study the moduli of Bridgeland stable
objects and its dependence on the parameter
even for the study of Gieseker stability.
For abelian surfaces, the studies on the dependence of the parameters,
i.e., those of wall and chamber structures, are started by
two groups Maciocia and Meachan \cite{MM},
and Minamide, Yanagida and Yoshioka
\cite{MYY:2011:1}, \cite{MYY:2011:2}.
In this paper, we continue our study on the wall and chamber structures.
In particular, we shall study
the behavior of the structures under Fourier-Mukai
transforms.
We are mainly interested in the most simple case, that is,
the case where $\operatorname{NS}(X)={\mathbb Z}H$.
In this case,
the parameter space is the upper half plane,
the action of Fourier-Mukai transforms
can be described by an arithmetic subgroup
of $\SL(2,{\mathbb R})$ and the action on the upper half plane
is the natural one.
So we can study the structure in detail.
Let us explain the contents of this paper.
We first define the wall and chamber by using the characterization of walls
in \cite[Prop. 4.2.2]{MYY:2011:2}.
Then we present a few basic properties of walls and chambers.
As we explained,
by the paper of Bridgeland (\cite[Prop. 10.3]{Br:3}),
Fourier-Mukai transforms preserve the stability
of objects, that is, it induces an isomorphism of the moduli spaces.
We shall show that the taking dual functor
${\mathcal D}_X$ also preserves
the stability of objects (Theorem \ref{thm:B:3-10.3}).
These are done in sections \ref{sect:walls} and \ref{sect:FM}
In \cite{YY}, we introduced a useful notion
\emph{semi-homogeneous presentations}.
It is a presentation of a coherent sheaf
as the kernel or the cokernel
of a homomorphism $V_{-1} \to V_0$
of semi-homogeneous sheaves $V_{-1}, V_0$ with
some numerical conditions.
Extracting the numerical conditions
from $V_{-1}$ and $V_0$, we also introduced the notion
\emph{numerical solutions}
and constructed moduli spaces
of simple complexes $V_{-1} \to V_0$ associated to numerical solutions.
In \cite{MYY:2011:1}, we found a relation
between these moduli spaces and the wall crossing behavior
for Bridgeland stability conditions.
In this paper, we give a supplement
of this relation.
Thus we shall relate a particular
wall called a {\it codimension 0 wall}
for every numerical solution.
If $(\beta,\omega)$ belongs to a codimension 0 wall, then
for any neighborhood $U$ of $(\beta,\omega)$,
there is no $\sigma_{(\beta,\omega)}$-semi-stable object
which is $\sigma_{(\beta',\omega')}$-semi-stable
for all $(\beta',\omega') \in U$.
In section \ref{sect:restricted-parameter},
we fix an ample divisor $H$ and
study special stability conditions
$\sigma_{(\beta+sH,tH)}$
parametrized by the upper half plane
$\{ (s,t) \mid s,t\in {\mathbb R}, s>0 \}$.
In the $(s,t)$-plane,
the equations of walls are very simple.
They define circles or lines
forming a pencil of circles passing through imaginary
points (Remark \ref{rem:base-point}).
Thus they do not intersect each other.
For a principally polarized abelian surface
with Picard number 1,
these kind of results are obtained by
Maciocia and Meachan \cite{MM}.
Thus our results are generalization of theirs.
We also explain that there are one or two
unbounded chambers which parametrize
Gieseker semi-stable sheaves.
In \cite{Stability}, we showed that Gieseker's stability is
preserved under the Fourier-Mukai transform, if the degree of the stable
sheaf is sufficiently large.
We shall explain the result as an application of this section
(Proposition \ref{prop:asymptotic}).
In section \ref{sect:rho=1},
we assume that
$\operatorname{NS}(X)={\mathbb Z}H$.
We are mainly interested in a Mukai vector $v$
which is a Fourier-Mukai transform of
$1-\ell \varrho_X$.
So we may assume that $v=1-\ell \varrho_X$.
In \cite{YY}, we described the algebraic part of the
Mukai lattice $H^*(X,{\mathbb Z})_{\operatorname{alg}}$
as a lattice in the vector space of
quadratic forms of two variables.
Then we can describe the action
of Fourier-Mukai transforms as a natural $\GL(2,{\mathbb R})$-action.
By using these results, we shall study the structure of walls.
In particular, we shall classify codimension 0 walls by using
our description of Mukai lattice.
We set $n:=(H^2)/2$.
If $\sqrt{\ell/ n} \in {\mathbb Q}$, then there is a unique wall of
codimension 0, since there is one numerical solution.
Assume that $\sqrt{\ell/ n} \not \in {\mathbb Q}$. In this case,
these are infinitely many numerical solutions, thus we have
infinitely many codimension 0 walls.
Let $G_{n,\ell} \subset \GL(2,{\mathbb R})$
be the subgroup
generated by the cohomological action of
(covariant or contravariant) Fourier-Mukai transforms
preserving $\pm v$.
$G_{n,\ell}$ acts on the set of walls.
We show that there are finitely many
$G_{n,\ell}$-orbits of walls and
the set of codimension 0 walls forms an orbit of
$G_{n,\ell}$.
Each orbit has two
accumulation points
$(\pm \sqrt{\ell/n},0)$ in the $(s,t)$-plane.
An observant reader will see that
the main part of section of \ref{sect:rho=1} essentially
appeared in
\cite{YY:preprint} without using Bridgeland stability condition.
In section \ref{sect:example},
we shall study the structure of walls for
$v=1-\ell \varrho_X$, $\ell \leq 4$
on a principally polarized abelian surface $X$.
As an application, we shall classify $M_H(v)$
for a primitive Mukai vector with $\langle v^2 \rangle/2 \leq 4$.
In \cite{Mukai:1979} and \cite[Thm. 3]{Mukai:1980},
Mukai announced that
$M_H(v) \cong X \times \operatorname{Hilb}{\langle v^2 \rangle/2}{X}$
for a primitive Mukai vector with $\langle v^2 \rangle/2=1,2,3$.
Moreover he determined the Fourier-Mukai transform
which induces the isomorphism.
By using the structure of walls,
we give an explanation of Mukai's results for
$\langle v^2 \rangle/2=2,3$.
It is quite surprising that Mukai discovered his results 30 years ago
without using
Bridgeland's stability conditions.
In appendix, we continue to assume that
$X$ is an abelian surface with $\operatorname{NS}(X)={\mathbb Z}H$ and
$(H^2)/2=n$.
We shall identify the period space
with the upper half plane.
Then we show that the action of auto-equivalences
is the action of the modular group $\Gamma_0(n)$.
Finally we would like to mention related works
which appeared during our preparation of this manuscript.
We note that
the examples of Bridgeland stability conditions
in this paper are generalized to
an arbitrary projective surfaces by
Arcara and Bertram \cite{AB}.
For these stability conditions,
Maciocia \cite{Ma} studied the structure of walls.
In particular, he
proved similar results to
section \ref{sect:restricted-parameter}
in a much more general context.
For the stability conditions on principally
polarized abelian surfaces,
Meachan \cite{Meachan} studied the structure of walls
in detail. In particular he
independently found
examples of walls with accumulation points.
\section{Preliminaries on Bridgeland's stability condition}
\label{sect:walls}
As in the introduction,
let $X$ be an abelian surface over a field $\mathfrak{k}$,
and fix an ample divisor $H$ on $X$.
\subsection{Notations for Mukai lattice}
We set $A^*_{\operatorname{alg}}(X)=\oplus_{i=0}^2 A^{i}_{\operatorname{alg}}(X)$
to be the quotient of the cycle group of $X$
by the algebraic equivalence.
Then we have $A^{0}_{\operatorname{alg}}(X)\cong \bb{Z}$,
$A^{1}_{\operatorname{alg}}(X)\cong \operatorname{NS}(X)$ and
$A^{2}_{\operatorname{alg}}(X)\cong \bb{Z}$.
We denote the fundamental class of $A^{2}_{\operatorname{alg}}(X)$ by $\varrho_X$,
and express an element $x\in A^{*}_{\operatorname{alg}}(X)$
by $x=x_0+x_1+x_2 \varrho_X$
with $x_0 \in \bb{Z}$, $x_1 \in \operatorname{NS}(X)$ and $x_2 \in \bb{Z}$.
The lattice structure $\langle \cdot,\cdot\rangle$
of $A^*_{\operatorname{alg}}(X)$ is given
by
\begin{equation}\label{eq:mukai_pairing}
\langle x,y \rangle:= (x_1,y_1)-(x_0 y_2+x_2 y_0),
\end{equation}
where $x=x_0+x_1+x_2 \varrho_X$ and $y=y_0+y_1+y_2 \varrho_X$.
We shall call $(A^*_{\operatorname{alg}}(X), \langle \cdot,\cdot \rangle)$
the Mukai lattice for $X$.
In the case of $\mathfrak{k}=\bb{C}$, this lattice is
sometimes denoted by $H^{*}(X,\bb{Z})_{\text{alg}}$ in literature.
In this paper, we shall use the symbol $H^{*}(X,\bb{Z})_{\text{alg}}$
even when $\mathfrak{k}$ is arbitrary.
The Mukai vector $v(E)\in H^*(X,\bb{Z})_{\operatorname{alg}}$
for $E \in \operatorname{Coh}(X)$ is defined by
\begin{equation*}
v(E):=\operatorname{ch}(E) \sqrt{\operatorname{td}_X}
=\operatorname{ch}(E)
=\operatorname{rk} E+c_1(E)+ \chi(E) \varrho_X.
\end{equation*}
We also use the vectorial notation
$$
v(E)=(\operatorname{rk} E,c_1(E),\chi(E)).
$$
For an object $E$ of ${\bl{D}}(X)$,
$v(E)$ is defined by $\sum_{k}(-1)^k v(E^k)$,
where $(E^k)=(\cdots \to E^{-1} \to E^0 \to E^1 \to \cdots)$
is the bounded complex representing the object $E$.
We take an ample ${\mathbb Q}$-divisor $H$.
For
$v \in H^*(X,\bb{Z})_{\operatorname{alg}}$ and $\beta \in \operatorname{NS}(X)_\bb{Q}$,
we set
\begin{equation}\label{eq:v:rda}
r_\beta(v):=-\langle v,\varrho_X \rangle,\quad
a_\beta(v):=-\langle v,e^\beta \rangle,\quad
d_{\beta,H}(v):=\frac{\langle v,H+(H,\beta)\varrho_X \rangle}{(H^2)}.
\end{equation}
If the choice of $H$ is clear, then we
write $d_\beta(v):=d_{\beta,H}(v)$ for simplicity.
By using \eqref{eq:v:rda}, we have
\begin{equation}\label{eq:v}
v=r_\beta(v)e^\beta+a_\beta(v) \varrho_X +
(d_{\beta,H}(v) H+D_\beta(v))+(d_{\beta,H}(v) H+D_\beta(v),\beta)\varrho_X,\;
D_\beta(v) \in H^{\perp} \cap \operatorname{NS}(X)_\bb{Q}.
\end{equation}
A Mukai vector $v=(r,\xi,a) \ne 0$ is positive, if
(i) $r>0$ or (ii) $r=0$ and $\xi$ is effective, or
(iii) $r=\xi=0$ and $a>0$.
We denote a positive Mukai vector $v$ by
$v>0$.
A Mukai vector $v$ is called isotropic if
$\langle v^2 \rangle =0$.
For $\beta \in \operatorname{NS}(X)_\bb{Q}$,
we define the $\beta$-twisted semi-stability
replacing the usual Hilbert polynomial
$\chi(E(nH))$ by $\chi(E(-\beta+nH))$.
Then $v$ is positive if and only if
$\chi(E(-\beta+nH)) >0$ for $E \in {\bf D}(X)$ with
$v(E)=v$ and
$n \gg 0$.
For a positive Mukai vector $v$,
${\mathcal{M}}_H^\beta(v)^{ss}$ denotes
the moduli stack of $\beta$-twisted semi-stable sheaves $E$ on $X$
with $v(E)=v$.
$\overline{M}_H^\beta(v)$ denotes
the moduli scheme of $S$-equivalence classes of
$\beta$-twisted semi-stable sheaves $E$ on $X$
with $v(E)=v$
and $M_H^\beta(v)$ denotes the open subscheme consisting of
$\beta$-twisted stable sheaves.
If $\beta=0$, then
we write $\overline{M}_H(v):=\overline{M}_H^\beta(v)$.
For a proper morphism $f:Z_1 \to Z_2$,
we denote the derived pull-back ${\bf L}f^*$ and
the derived direct image ${\bf R}f_*$ by $f^*$ and $f_*$
respectively.
For ${\bf E} \in {\bf D}(X \times Y)$,
$\Phi_{X \to Y}^{{\bf E}}:{\bf D}(X) \to {\bf D}(Y)$
denotes the integral functor whose kernel is ${\bf E}$:
\begin{equation}
\Phi_{X \to Y}^{{\bf E}}(E)=p_{Y*}(p_X^*(E) \otimes {\bf E}),\;
E \in {\bf D}(X),
\end{equation}
where $p_X$ and $p_Y$ are projections from $X \times Y$ to
$X$ and $Y$ respectively.
If $\Phi_{X \to Y}^{\bf E}$ is an equivalence, it is
called a \emph{Fourier-Mukai transform}.
If a Fourier-Mukai transform $\Phi_{X \to Y}^{{\bf E}}$ exists and
$X$ is an abelian surface, then $Y$ is also an abelian surface and
$\Phi_{X \to Y}^{\bf E}$ induces an isometry
of Mukai lattices $H^*(X,{\mathbb Z})_{\operatorname{alg}}
\to H^*(Y,{\mathbb Z})_{\operatorname{alg}}$.
We also denote this isometry by $\Phi_{X \to Y}^{\bf E}$.
${\mathcal D}_X(*):={\bf R}{\mathcal H}om_{{\mathcal O}_X}(*,{\mathcal O}_X)$
denotes the taking dual functor.
It is a contravariant functor from ${\bf D}(X)$ to
${\bf D}(X)$.
A contravariant Fourier-Mukai transform is a
composite of a Fourier-Mukai functor and ${\mathcal D}_X$.
If $X$ is an abelian surface, then it is of the form
$\Phi_{X \to Y}^{{\bf E}[2]} \circ {\mathcal D}_X={\mathcal D}_Y \circ
\Phi_{X \to Y}^{{\bf E}^{\vee}}$
with
${\bf E}^{\vee} := {\mathcal D}_{X \times Y}({\bf E})$.
\subsection{Stability conditions and wall/chamber structure}
Let us recall the stability conditions given in
\cite{Br:3} and \cite[\S1]{MYY:2011:1}.
Let
$$
\operatorname{Amp}(X)_{\mathbb R}:=\{x \in \operatorname{NS}(X)_{\mathbb R} \mid (x^2)>0,(x,D)>0 \}.
$$
be the ample cone of $X$,
where $D$ is an effective divisor.
We take $(\beta,\omega) \in \operatorname{NS}(X)_{\mathbb R} \times \operatorname{Amp}(X)_{\mathbb R}$
and $H \in {\mathbb R}_{>0}\omega$.
For $E \in K(X)$ with $v=v(E)$ expressed as \eqref{eq:v}, we have
\begin{align*}
Z_{(\beta,\omega)}(E)=&
\langle e^{\beta+\sqrt{-1}\omega},v(E) \rangle\\
=& -a_\beta(v(E))+\frac{(\omega^2)}{2}r_\beta(v(E))
+d_{\beta,H}(v(E))(H,\omega)\sqrt{-1}.
\end{align*}
Assume that $(\beta,\omega) \in
\operatorname{NS}(X)_{\mathbb Q} \times \operatorname{Amp}(X)_{\mathbb Q}$.
Let $\mathfrak{A}_{(\beta,\omega)}$ be the tilt of $\operatorname{Coh}(X)$
with respect to
the torsion pair $(\mathfrak{T}_{(\beta,\omega)},\mathfrak{F}_{(\beta,\omega)})$
defined by
\begin{enumerate}
\item
$\mathfrak{T}_{(\beta,\omega)}$ is generated by
$\beta$-twisted stable sheaves with $Z_{(\beta,\omega)}(E) \in \bb{H} \cup
\bb{R}_{<0}$.
\item
$\mathfrak{F}_{(\beta,\omega)}$ is generated by
$\beta$-twisted stable sheaves with $-Z_{(\beta,\omega)}(E) \in \bb{H} \cup
\bb{R}_{<0}$,
\end{enumerate}
where $\bb{H} :=\{z \in \bb{C} \mid \mathrm{Im} z>0 \}$
is the upper half plane.
$\mathfrak{A}_{(\beta,\omega)}$ is the abelian category
in \cite{Br:3}
and it depends only on $\beta$ and the ray ${\mathbb Q}_{>0}\omega$.
\begin{NB}
Since we are considering the abelian surface $X$,
the category $\mathfrak{A}_{(\beta,\omega)}$
depends only on $(\beta,{\mathbb Q}_{>0}\omega)$,
as shown in \cite[sect. 1.4]{MYY:2011:1}.
More explicitly, the category is given by
\begin{align*}
\mathfrak{A}_{(\beta,\omega)}=
\{E \in {\bl{D}}(X) \mid H^i(E)=0 \, (i \ne -1, 0),\
\mu_{\max}(H^{-1}(E)) \leq (\beta,H),\ \mu_{\min}(H^0(E))>(\beta,H) \}
\end{align*}
for any $(\eta,\omega) \in \mathfrak{H}_\bb{R}$.
Here $H^*$ is the cohomology with respect to $\operatorname{Coh}(X)$.
\end{NB}
Then the pair
$\sigma_{(\beta,\omega)}=(\mathfrak{A}_{(\beta,\omega)},Z_{(\beta,\omega)})$
satisfies the requirement of stability conditions on $\bl{D}(X)$
\cite{Br:3}.
In particular, the (semi-)stability of objects in
$\mathfrak{A}_{(\beta,\omega)}$ with respect to $Z_{(\beta,\omega)}$
is well-defined.
\begin{defn}\label{defn:phase(v)}
For a non-zero Mukai vector $v \in H^*(X,\bb{Z})_{\operatorname{alg}}$,
we define $Z_{(\beta,\omega)}(v) \in \bb{C}$ and
$\phi_{(\beta,\omega)}(v) \in (-1,1]$ by
\begin{equation*}
Z_{(\beta,\omega)}(v)=
\langle e^{\beta+\sqrt{-1} \omega},v \rangle
= |Z_{(\beta,\omega)}(v)|e^{\pi \sqrt{-1}\phi_{(\beta,\omega)}(v)}.
\end{equation*}
Then
$$
\phi_{(\beta,\omega)}(v(E))
=\phi_{(\beta,\omega)}(E)
$$
for $0 \ne
E \in \mathfrak{A}_{(\beta,\omega)} \cup \mathfrak{A}_{(\beta,\omega)}[-1]$.
\end{defn}
\begin{defn}
$E \in \bl{D}(X)$ is called \emph{semi-stable} of phase $\phi$,
if there is an integer $n$ such that
$E[-n]$ is a semi-stable object of $\mathfrak{A}_{(\beta,\omega)}$
with $\phi_{(\beta,\omega)}(E[-n])=\phi-n$.
If we want to emphasize the dependence on the stability condition,
we say that $E$ is \emph{$\sigma_{(\beta,\omega)}$-semi-stable}.
\end{defn}
\begin{defn}\label{defn:moduli}
For a Mukai vector $v$,
$\mathcal{M}_{(\beta,\omega)}(v)$ denotes the moduli stack of
$\sigma_{(\beta,\omega)}$-semi-stable objects $E$ of
$\mathfrak{A}_{(\beta,\omega)}$ with $v(E)=v$.
$M_{(\beta,\omega)}(v)$ denotes the moduli scheme of
the $S$-equivalence classes of
$\sigma_{(\beta,\omega)}$-semi-stable objects $E$ of
$\mathfrak{A}_{(\beta,\omega)}$ with $v(E)=v$, if it exists.
\end{defn}
\begin{rem}
If $Z_{(\beta,\omega)}(-v) \in {\mathbb H} \cup {\mathbb R}_{<0}$, then we have
\begin{equation}
M_{(\beta,\omega)}(v)=\{E[-1] \mid E \in M_{(\beta,\omega)}(-v) \},
\end{equation}
since $\phi_{(\beta,\omega)}(v) \in (-1,0]$.
\end{rem}
\begin{rem}\label{rem:phase}
If we take the phase of $v$ as
$\phi_{(\beta,\omega)}(v) \in (1,2]$, then
\begin{equation}
M_{(\beta,\omega)}(v)=\{E[1] \mid E \in M_{(\beta,\omega)}(-v) \}.
\end{equation}
More generally, if we take the phase
of $v$ as $\phi_{(\beta,\omega)}(v) \in (n,n+1]$,
then we have
\begin{equation}
M_{(\beta,\omega)}(v)=\{E[n] \mid E \in M_{(\beta,\omega)}((-1)^n v) \}.
\end{equation}
\end{rem}
\begin{NB}
We set $s_0:=d_\beta(v)/\operatorname{rk} v$.
Assume that $\operatorname{rk} v>0$.
Then $d_{\beta+sH}(v)$ is a decreasing function of $s$
with
$Z_{(\beta+s_0 H,tH)}(v) \in {\mathbb R}_{>0}$.
Requiring the continuity of
$\phi_{(\beta+s_0 H,tH)}(v)$ as a function of $s$,
we should have $\phi_{(\beta+s_0 H,tH)}(v):=0$.
Then we have
$\phi_{(\beta+s H,tH)}(v) \in (-1,0]$
for $s \geq s_0$.
Assume that $\operatorname{rk} v<0$.
Then
$Z_{(\beta+s_0 H,tH)}(v) \in {\mathbb R}_{<0}$.
Hence $E=F[1]$, $F \in M_{(\beta+sH,\omega)}(-v)$ for $s<s_0$.
Thus $\phi_{(\beta+sH,\omega)}(F) \in (0,1]$, which implies that
$\phi_{(\beta+sH,\omega)}(E) \in(1,2]$.
\end{NB}
\begin{NB}
\begin{rem}
Let $E$ be an object of ${\bf D}(X)$ with $v(E)=v$.
If $E[n] \in M_{(\beta,\omega)}(v)$ and
$E[n'] \in M_{(\beta',\omega')}(v)$,
$v(E[n])=v(E[n'])=v$. Thus $n$ and $n'$ are even integer.
We note that $\phi_{(\beta,\omega)}(E[n]) \in (-1,1]$.
We first assume that
$\phi_{(\beta,\omega)}(E[n]) \in (0,1]$. Thus we have
$E[n] \in {\mathfrak A}_{(\beta,\omega)}$, which implies that
$H^i(E[n])=0$ for $i \ne -1,0$.
Since $\phi_{(\beta',\omega')}(E[n']) \in (-1,1]$,
$\phi_{(\beta',\omega')}(E[n]) \in (-1+n-n',1+n-n']$.
Hence $n-n'=0$ or $n-n'=-2$.
If $\phi_{(\beta,\omega)}(E[n]) \in (-1,0]$, then $n-n'=0,2$.
c\end{rem}
\end{NB}
\begin{NB}
\begin{lem}
Assume that $E=\oplus_i E_i$ such that
$E_i$ are $\sigma_{(\beta,\omega)}$-stable objects
of ${\mathfrak A}_{(\beta,\omega)}$
with $\phi_{(\beta,\omega)}(E_i)=\phi_{(\beta,\omega)}(E)$.
For a Fourier-Mukai transform $\Phi:{\bf D}(X) \to {\bf D}(Y)$,
we set $F_i=\Phi(E_i)$.
Let $(\beta',\omega')$ be the image of $(\beta,\omega)$
by $\Phi$.
Then $\phi_{(\beta',\omega')}(F_i)=\phi_{(\beta',\omega')}(F)$.
\end{lem}
\begin{proof}
We express $\Phi$ as
$\Phi=\Phi_{X \to Y}^{{\bf G}^{\vee}}$.
We may assume that
${\bf G}_{|X \times \{ y \}} \in {\mathfrak A}_{(\beta,\omega)}$.
If $\phi_{(\beta',\omega')}({\bf G}_{|X \times \{ y \}})>
\phi_{(\beta,\omega)}(E_i)$, then
$\operatorname{Hom}({\bf G}_{|X \times \{ y \}},E_i)=0$ for all $y$.
Hence $H^k(F_i)=0$ except for $k \ne 1,2$.
On the other hand, our assumption
implies that $Z_{(\beta',\omega')}(F_i)=Z_{(\beta',\omega')}(F)$.
By \cite[Prop. 4.2.2]{MYY:2011:2}, we get the claim.
If $\phi_{(\beta',\omega')}({\bf G}_{|X \times \{ y \}})<
\phi_{(\beta,\omega)}(E_i)$, then
$\operatorname{Hom}({\bf G}_{|X \times \{ y \}},E_i[2])=0$ for all $y$.
Hence $H^k(F_i)=0$ except for $k \ne 0, 1$.
Then by the same argument, we get the claim.
If $\phi_{(\beta,\omega')}({\bf G}_{|X \times \{ y \}})=
\phi_{(\beta,\omega)}(E_i)$, then
we see that $F_i[1] \in \operatorname{Coh}(Y)$ or $F_i={\mathfrak k}_y$.
By the same argument, we get the claim.
\end{proof}
\end{NB}
\begin{defn}\label{defn:wall}
Let $v_1 \not \in {\mathbb Q}v$
be a Mukai vector with
$\langle v_1^2 \rangle \geq 0$, $\langle (v-v_1)^2 \rangle \geq 0$
and $\langle v_1,v-v_1 \rangle >0$.
We define a \emph{wall for $v$} by
$$
W_{v_1}:=\{(\beta,\omega) \in \operatorname{NS}(X)_{\mathbb R} \times \operatorname{Amp}(X)_{\mathbb R}
\mid {\mathbb R}Z_{(\beta,\omega)}(v_1)=
{\mathbb R}Z_{(\beta,\omega)}(v) \}.
$$
A connected component of
$\operatorname{NS}(X)_{\mathbb R} \times \operatorname{Amp}(X)_{\mathbb R} \setminus \cup_{v_1} W_{v_1}$
is called a \emph{chamber for $v$}.
\begin{NB}
The equation of $W_{v_1}$ is
$$
\langle e^{\beta+\sqrt{-1}\omega},v \rangle
\langle e^{\beta-\sqrt{-1}\omega},v_1 \rangle
=\langle e^{\beta-\sqrt{-1}\omega},v \rangle
\langle e^{\beta+\sqrt{-1}\omega},v_1 \rangle.
$$
\end{NB}
\end{defn}
Replacing $v$ by $-v$ if necessary, we may assume that
$Z_{(\beta,\omega)}(v) \in {\mathbb H} \cup {\mathbb R}_{<0}$.
Assume that $(\beta,\omega) \in W_{v_1}$.
By \cite[Prop. 4.2.2]{MYY:2011:2},
there are $\sigma_{(\beta,\omega)}$-semi-stable objects
$E_1$ and $E_2$ with $v(E_1)=v_1$ and $v(E_2)=v-v_1$.
Then ${\mathbb R}_{>0}Z_{(\beta,\omega)}(E_1)=
{\mathbb R}_{>0}Z_{(\beta,\omega)}(E_2)$.
Thus there is a properly $\sigma_{(\beta,\omega)}$-semi-stable
object with the Mukai vector $v$.
\begin{prop}[{\cite[Prop. 9.3]{Br:3}}]\label{prop:open}
Let $v$ be a Mukai vector with $\langle v^2 \rangle>0$.
Let ${\mathcal C}$ be a chamber for $v$.
Then ${\mathcal M}_{(\beta,\omega)}(v)$
is independent of $(\beta,\omega) \in {\mathcal C}$.
\end{prop}
\begin{prop}\label{prop:wall:isometry}
Let $\xi:H^*(X,{\mathbb Z})_{\operatorname{alg}} \to H^*(Y,{\mathbb Z})_{\operatorname{alg}}$
be an isometry of Mukai lattices.
Let $W_u$ be a wall for $v$ defined by $u$.
Then $\xi(u)$ defines a wall for $\xi(v)$.
\end{prop}
\begin{proof}
We set $w:=v-u$.
Then $u$ defines a wall for $v$ if and only if
$\langle u^2 \rangle, \langle w^2 \rangle \geq 0$ and
$\langle u,w \rangle>0$.
This condition is preserved under $\xi$.
So the claim holds.
\end{proof}
\begin{NB}
If $\xi$ an orientation preserving isometry, then
the claim will follows from \cite{Br:3}.
\end{NB}
\begin{NB}
Let $\overline{\operatorname{NS}(X)_{\mathbb R} \times {\mathbb R}_{>0}H}
=\operatorname{NS}(X)_{\mathbb R} \times {\mathbb R}_{>0}H \cup \{\infty\}$
be the one point compactification of
$\operatorname{NS}(X)_{\mathbb R} \times {\mathbb R}_{>0}H$.
Then we may regard that
$v({\mathfrak k}_x)$ corresponds to
$(\beta,\omega)=\infty$.
Since
$(r_1 d_0(v)-r d_0(v_1))-\frac{2}{(\omega^2)}
(a_\beta(v_1) d_0(v)-a_\beta(v) d_0(v_1)) \to
(r_1 d_0(v)-r d_0(v_1)),\; (\omega^2) \to \infty$,
$\sigma_{\infty}$-semi-stable objects are
generated by $\mu$-semi-stable sheaves and
$0$-dimensional sheaves.
\end{NB}
Let $v$ be a primitive Mukai vector with $\operatorname{rk} v>0$
and assume that $M_H^\beta(v)$ consists of $\beta$-twisted stable
sheaves (in the sense of Gieseker).
\begin{lem}
Assume that $\langle v^2 \rangle>0$ and
$r:=-\langle v,\varrho_X \rangle \ne 0$.
Let $v_1$ be a Mukai vector such that
$r c_1(v_1)-r_1 c_1(v)=0$,
where we set $r_1 := -\langle v_1,\varrho_X \rangle$.
Then ${\mathbb R}Z_{(\beta,\omega)}(v)={\mathbb R}Z_{(\beta,\omega)}(v_1)$
if and only if $d_\beta(v)=0$.
In particular, if $v_1$ defines a wall, then
the defining equation of the wall $W_{v_1}$ for $v$
is $d_\beta(v)=0$.
\end{lem}
\begin{proof}
We have $r v_1-r_1 v=b \varrho_X$, $0 \ne b \in {\mathbb Z}$.
Hence $rZ_{(\beta,\omega)}(v_1)-r_1 Z_{(\beta,\omega)}(v)=
-b \in {\mathbb R}$.
Then we have
$Z_{(\beta,\omega)}(v_1)=\frac{r_1}{r}Z_{(\beta,\omega)}(v)-\frac{b}{r}$.
Hence the condition is $d_\beta(v)=0$.
\end{proof}
\begin{NB}
\begin{lem}
Assume that $\langle v^2 \rangle>0$ and $r \ne 0$.
Let $v_1$ be a Mukai vector such that
$(r c_1(v_1)-r_1 c_1(v),H)=0$.
If $v_1$ defines a wall, then
the equation is
$d_\beta(v)(r_1 a_\beta(v)-r a_\beta(v_1))=0$.
\end{lem}
\begin{proof}
Since $(r c_1(v_1)-r_1 c_1(v),H)=0$, we have
$(c_1(v_1)-r_1 \beta,H)=\frac{r_1}{r}(c_1(v)-r \beta,H)$.
Thus $d_\beta(v_1)=\frac{r_1}{r}d_\beta(v)$.
Hence we get
$d_\beta(v)(r_1 a_\beta(v)-r a_\beta(v_1))=0$.
\end{proof}
\end{NB}
\begin{NB}
We set $\beta:=\beta_0+\eta$. Then
$r_1 a_\beta(v)-r a_\beta(v_1)=(r_1 a_{\beta_0}(v)-r a_{\beta_0}(v_1))
-(r_1 c_1(v)-r c_1(v_1),\eta)$.
Since $d_\beta(v)=d_{\beta_0}(v)-r(\eta,H)/(H^2)$,
the set consists of two lines.
If $\eta \in {\mathbb R}H$, then
$r_1 a_\beta(v)-r a_\beta(v_1)=r_1 a_{\beta_0}(v)-r a_{\beta_0}(v_1)$
is constant.
\end{NB}
\begin{NB}
\begin{lem}
$d_\beta(v)-r s=0$ is not a wall if and only if
every $\mu$-semi-stable sheaf $E$ with $v(E)=v$ is a
$\mu$-stable vector bundle.
\end{lem}
\begin{proof}
We note that
every $G \in {\mathfrak A}_{(\beta+sH,\omega)}$
with $d_{\beta+sH}(G)=0$ is generated by
${\mathfrak k}_x$ ($x \in X$) and $F[1]$,
where $F$ is a $\mu$-stable vector bundle
with $d_{\beta+sH}(F)=0$.
If $d_{\beta+sH}(v)=d_\beta(v)-rs=0$
is a wall, then
we have a non-trivial decomposition
$-v=n_1 v({\mathfrak k}_x)+\sum_{i=1}^{n_2} v(F_i[1])$ ($n_1, n_2 \geq 0$),
where $F_i$ are $\mu$-stable locally free sheaves with
$d_{\beta+sH}(F_i)=0$.
Then we see that
there is a non-locally free sheaf $F$ with
$v(F)=v$
or there is a properly $\mu$-semi-stable sheaf $F$ with $v(F)=v$.
Conversely if
there is a non-locally free sheaf $F$ with
$v(F)=v$
or there is a properly $\mu$-semi-stable sheaf $F$ with $v(F)=v$, then
there is a non-trivial decomposition
$v(E)=n_1 v({\mathfrak k}_x)+\sum_{i=1}^{n_2} v(F_i[1])$ ($n_1, n_2 \geq 0$).
Then $d_\beta-rs$ defines a wall.
\end{proof}
\end{NB}
\section{Fourier-Mukai transforms.}\label{sect:FM}
\subsection{Stability conditions and Fourier-Mukai transforms.}
We shall recall the (twisted) Fourier-Mukai transform
of $\sigma_{(\beta,\omega)}$ and its relation to
Bridgeland's stability conditions
explained in \cite{MYY:2011:2}.
Let $\Phi:{\bf D}(X) \to {\bf D}^{\alpha_1}(X_1)$ be a
twisted Fourier-Mukai transform such that
$\Phi(r_1 e^\gamma)=-\varrho_{X_1}$ and
$\Phi(\varrho_X)=-r_1 e^{\gamma'}$,
where $\alpha_1$ is a representative of a suitable
Brauer class.
Then we can describe the cohomological Fourier-Mukai transform
as
\begin{equation*}
\Phi(r e^\gamma+a \varrho_X+\xi+(\xi,\gamma)\varrho_X)
=-\frac{r}{r_1} \varrho_{X_1}-r_1 a e^{\gamma'}+
\frac{r_1}{|r_1|}
( \widehat{\xi}+(\widehat{\xi},\gamma')\varrho_{X_1}),
\end{equation*}
where $\xi \in \operatorname{NS}(X)_{\mathbb Q}$ and $
\widehat{\xi}:=
\frac{r_1}{|r_1|}
c_1(\Phi(\xi+(\xi,\gamma)\varrho_X)) \in \operatorname{NS}(X_1)_{\mathbb Q}$.
\begin{rem}
By taking a locally free $\alpha_1$-twisted stable
sheaf $G$ with $\chi(G,G)=0$,
we have a notion of Mukai vector, thus, we have
a map (\cite[Rem. 1.2.10]{MYY:2011:1}):
$$
v_G:{\bf D}^{\alpha_1}(X_1) \to H^*(X_1,{\mathbb Q})_{\operatorname{alg}}.
$$
\end{rem}
We set
\begin{equation}\label{eq:tilde(beta)}
\begin{split}
\widetilde{\omega}:= & -\frac{1}{|r_1|}
\frac{\frac{((\beta-\gamma)^2)-(\omega^2)}{2}\widehat{\omega}-
(\beta-\gamma,\omega)(\widehat{\beta}-\widehat{\gamma})}
{\left(\frac{((\beta-\gamma)^2)-(\omega^2)}{2} \right)^2
+(\beta-\gamma,\omega)^2},\\
\widetilde{\beta}:= & \gamma'-\frac{1}{|r_1|}
\frac{\frac{((\beta-\gamma)^2)-(\omega^2)}{2}(\widehat{\beta}-\widehat{\gamma})
-(\beta-\gamma,\omega) \widehat{\omega}}
{\left(\frac{((\beta-\gamma)^2)-(\omega^2)}{2} \right)^2
+(\beta-\gamma,\omega)^2}.
\end{split}
\end{equation}
\begin{NB}
We set
\begin{equation}
\begin{split}
\xi:=&
\frac{1}{|r_1|}
\frac{2}{((\lambda-s)^2+t^2) (H^2)}(\lambda-s) H,\\
\eta:=&
\frac{1}{|r_1|}
\frac{2}{((\lambda-s)^2+t^2) (H^2)}t H.\\
\end{split}
\end{equation}
\end{NB}
By \cite[sect. 5.1]{MYY:2011:2},
we get the following commutative diagram:
\begin{equation}
\xymatrix{
{\bf D}(X) \ar[r] \ar[d]_{Z_{(\beta,\omega)}}
& {\bf D}^{\alpha_1}(X_1) \ar[d]^{Z_{(\widetilde{\beta},\widetilde{\omega})}} \\
{\mathbb C} \ar[r]_{\zeta^{-1}}
& {\mathbb C}
}
\end{equation}
where
$$
\zeta=-r_1 \left(
\frac{((\gamma-\beta)^2)-(\omega^2)}{2}
+\sqrt{-1}(\beta-\gamma,\omega) \right).
$$
\begin{NB}
If we fix a base point $\beta'$
and write $\beta'=\gamma'-\lambda' \widehat{H}$,
then $\beta'+s' \widehat{H}=\gamma'+(s'-\lambda')\widehat{H}$
and the relation of
$Z_{(\beta+sH,tH)}$ and $Z_{(\beta'+s' \widehat{H},t' \widehat{H})}$
is
$$
s'-\lambda'=\frac{1}{|r_1|}
\frac{2(\lambda-s)}{((\lambda-s)^2+t^2) (H^2)},\;
t'=\frac{1}{|r_1|}
\frac{2t}{((\lambda-s)^2+t^2) (H^2)}.
$$
\end{NB}
Let ${\bf E}$ be a complex such that
$\Phi=\Phi_{X \to X_1}^{{\bf E}^{\vee}[1]}$.
Then
\begin{equation}\label{eq:Phi(phase)}
\phi_{(\widetilde{\beta},\widetilde{\omega})}(\Phi(E))=
\phi_{(\beta,\omega)}(E)-
\phi_{(\beta,\omega)}({\bf E}_{|X \times \{ x_1 \}}).
\end{equation}
\begin{NB}
\begin{proof}
We note that
$$
\phi_{(\widetilde{\beta},\widetilde{\omega})}(\Phi(E)) \equiv
\phi_{(\beta,\omega)}(E)+m \mod 2.
$$
For a $\sigma_{(\beta,\omega)}$-semi-stable
object $E \in {\bf D}(X)$ with
$\phi_{(\beta,\omega)}({\bf E}_{|X \times \{ x_1 \}})+1>
\phi_{(\beta,\omega)}(E)>\phi_{(\beta,\omega)}({\bf E}_{|X \times \{ x_1 \}})$,
we see that $H^i(\Phi(E))=0$ for $i \ne -1,0$.
Hence $\Phi(E) \in {\mathfrak A}_{(\widetilde{\beta},\widetilde{\omega})}[1]
\cup {\mathfrak A}_{(\widetilde{\beta},\widetilde{\omega})} \cup
{\mathfrak A}_{(\widetilde{\beta},\widetilde{\omega})}[-1]$.
In particular,
$\phi_{(\widetilde{\beta},\widetilde{\omega})}(\Phi(E)) \in (-1,2)$.
Since $\Phi({\bf E}_{|X \times \{ x_1 \}})={\mathfrak k}_{x_1}[-1]$,
$\phi_{(\widetilde{\beta},\widetilde{\omega})}
(\Phi({\bf E}_{|X \times \{ x_1 \}}))=0$.
Then
\begin{equation}
\phi_{(\widetilde{\beta},\widetilde{\omega})}(\Phi(E))=
\phi_{(\widetilde{\beta},\widetilde{\omega})}(\Phi(E))
-\phi_{(\widetilde{\beta},\widetilde{\omega})}
(\Phi({\bf E}_{|X \times \{ x_1 \}}))
\equiv \phi_{(\beta,\omega)}(E)-
\phi_{(\beta,\omega)}({\bf E}_{|X \times \{ x_1 \}}) \mod 2{\mathbb Z}.
\end{equation}
Hence $\phi_{(\widetilde{\beta},\widetilde{\omega})}(\Phi(E)) \in (2k,2k+1)$.
Therefore $k=0$ and
\begin{equation}
\phi_{(\widetilde{\beta},\widetilde{\omega})}(\Phi(E))=
\phi_{(\beta,\omega)}(E)-
\phi_{(\beta,\omega)}({\bf E}_{|X \times \{ x_1 \}}).
\end{equation}
\end{proof}
\end{NB}
\begin{NB}
Assume that
$\langle v(E),v({\bf E}_{|X \times \{ x_1 \}})
\rangle \ne 0$.
There is $(\beta,\omega)$
such that ${\mathbb R}Z_{(\beta,\omega)}(v(E))=
{\mathbb R}Z_{(\beta,\omega)}({\bf E}_{|X \times \{ x_1 \}})$.
Assume that $E \in {\mathfrak A}_{(\beta,\omega)}$.
If ${\bf E}_{|X \times \{ x_1 \}}[n] \in {\mathfrak A}_{(\beta,\omega)}$, then
$\operatorname{rk} \Phi(E[1-n])>0$.
In the region
$\phi_{(\beta,\omega)}(E)>
\phi_{(\beta,\omega)}({\bf E}_{|X \times \{ x_1 \}})$,
$1>\phi_{(\widetilde{\beta},\widetilde{\omega})}(\Phi(E[1-n]))>0$.
In particular, $d_{\widetilde{\beta}}(\Phi(E[1-n]))>0$.
\begin{NB2}
$d_{\widetilde{\beta}}(\Phi(E[1-n]))>0$ and $H^i(\Phi(E[1-n]))=0$
for $i \ne -1,0$ is enough to show
$1>\phi_{(\widetilde{\beta},\widetilde{\omega})}(\Phi(E[1-n]))>0$.
Indeed $d_{\widetilde{\beta}}(\Phi(E[1-n]))>0$
implies that
$\phi_{(\widetilde{\beta},\widetilde{\omega})}(\Phi(E[1-n])) \in
(2k,2k+1)$.
Then $H^i(\Phi(E[1-n]))=0$ for $i \ne -2k-1,-2k$.
Therefore $k=0$, which implies the claim.
\end{NB2}
Assume that
$E[1] \in {\mathfrak A}_{(\beta,\omega)}$.
If ${\bf E}_{|X \times \{ x_1 \}}[n] \in {\mathfrak A}_{(\beta,\omega)}$, then
$\operatorname{rk} \Phi(E[2-n])>0$.
In the region
$\phi_{(\beta,\omega)}(E[1])>
\phi_{(\beta,\omega)}({\bf E}_{|X \times \{ x_1 \}})$,
$1>\phi_{(\widetilde{\beta},\widetilde{\omega})}(\Phi(E[2-n]))>0$.
If $\langle v(E),v({\bf E}_{|X \times \{ x_1 \}})
\rangle= 0$. Then there is $(\beta,\omega)$ with
${\mathbb R}Z_{(\beta,\omega)}(v(E))=
{\mathbb R}Z_{(\beta,\omega)}({\bf E}_{|X \times \{ x_1 \}})$
if and only if $v(E) \in {\mathbb Q} v({\bf E}_{|X \times \{ x_1 \}})$.
For ${\bf E}_{|X \times \{ x_1 \}}$,
$\Phi({\bf E}_{|X \times \{ x_1 \}})={\mathfrak k}_{x_1}[-2]$.
Hence $\phi_{(\widetilde{\beta},\widetilde{\omega})}
(\Phi({\bf E}_{|X \times \{ x_1 \}}))=-1$.
If $\phi_{(\beta,\omega)}(E)>
\phi_{(\beta,\omega)}({\bf E}_{|X \times \{ x_1 \}})$,
then $H^i(\Phi(E))=0$ for $i \geq 2$ implies that
$\phi_{(\widetilde{\beta},\widetilde{\omega})}(\Phi(E)) > -2$.
If
$\phi_{(\beta,\omega)}({\bf E}_{|X \times \{ x_1 \}})+1
> \phi_{(\beta,\omega)}(E)$,
then we have $H^i(\Phi(E))=0$ for $i < 0$,
which implies that
$\phi_{(\widetilde{\beta},\widetilde{\omega})}(\Phi(E)) \leq 1$.
\end{NB}
\begin{thm}(cf. \cite[Thm. 2.2.1]{MYY:2011:2})\label{thm:MYY2}
Assume that ${\mathbb R}Z_{(\beta,\omega)}(v)=
{\mathbb R}Z_{(\beta,\omega)}(e^{\gamma})$.
If $r_1 e^\gamma$ does not define a wall, then
$\Phi$ induces an isomorphism
$$
{\mathcal M}_{(\beta,\omega)}(v) \to {\mathcal M}_{\widetilde{\omega}}(u)^{ss},
$$
where $u:=\Phi(v)$.
\end{thm}
As an application of Theorem \ref{thm:MYY2},
we give a different proof of Proposition \ref{prop:open}.
\begin{proof}
\begin{NB}
We take $(\beta_1,\omega_1) \in {\mathcal C}$.
We set $\beta_1:=b_1 H+\eta$, $\eta \in H^{\perp}$.
We can take a curve $(\beta_1+sH,t(s)H)$, $-\epsilon \leq s \leq \epsilon$
such that $t(0)H$ is sufficiently close to $\omega_1$ and
${\mathbb R}Z_{(\beta_1+sH,t(s)H)}(v)=
{\mathbb Z}_{(\beta_1+sH,t(s)H)}(e^{\beta+\lambda H})$ for some
$\lambda \in {\mathbb Q}$.
For a neighborhood $(-\epsilon,\epsilon) \times U$
of $(\beta_1,\omega_1)$ such that
$(-\epsilon,\epsilon) \times U \subset {\mathcal C}$,
\cite[Cor. 4.2.5]{MYY:2011:2} implies that
$M_{(\beta,\omega)}(v)$ is independent of $(\beta,\omega) \in
U \subset {\mathcal C}$.
\end{NB}
For $H \in \operatorname{Amp}(X)_{\mathbb Q}$,
we take $\delta \in \operatorname{NS}(X)_{\mathbb Q}$ such that
$(\delta,H)=1$.
We set $H(\delta):=\{L \in \operatorname{Amp}(X)_{\mathbb R} \mid (\delta,L)=1 \}$. Then
$\{tL \in \operatorname{Amp}(X)_{\mathbb R} \mid L \in H(\delta),\; t \in {\mathbb R}_{>0} \}$
is an open neighborhood of $H$.
Assume that $(\beta_0,\omega_0) \in {\mathcal C}$ and
$\omega_0 \in {\mathbb R}_{>0} H$.
We shall show that $\sigma_{(\beta,\omega)}$-semi-stability
is independent of $(\beta,\omega)$ in a neighborhood of
$(\beta_0,\omega_0)$.
We can take $\gamma \in {\mathbb R}H+\beta_0 \subset \operatorname{NS}(X)_{\mathbb R}$
such that
${\mathbb R} Z_{(\beta_0,\omega_0)}(v)=
{\mathbb R} Z_{(\beta_0,\omega_0)}(e^\gamma)$,
$(\gamma-\beta_0,H)>0$
and $(r \gamma-c_1(v),H) \ne 0$
(\cite[sect. 4.1]{MYY:2011:2}).
Replacing $\omega_0$ if necessary, we may assume that
$\gamma \in \operatorname{NS}(X)_{\mathbb Q}$.
We take a neighborhood $U$ of $H(\delta)$ and $I \subset {\mathbb R}_{>0}$
such that
$\omega_0 \in
\{tL \mid L \in U,\; t \in I \} \subset {\mathcal C}$.
For a fixed $L \in H(\delta)_{\mathbb Q}$,
the semi-stability is independent of $(\beta,L) \in {\mathcal C}$.
Replacing $U$ if necessary, we may assume that the following equation for
$t>0$ has a solution for each $L \in U$ and a neighborhood $V$ of
$\beta_0$:
$$
t^2 \frac{(L^2)}{2}=
\frac{(\langle e^\gamma,e^\beta \rangle(c_1(v)-r \beta)-
\langle v,e^\beta \rangle(\gamma-\beta),L)}{
(r \gamma- c_1(v),L)}.
$$
We set $\omega:=tL$. $\omega$ is a function on
$V \times U$ and
we have ${\mathbb R}Z_{(\beta,\omega)}(v)
={\mathbb R}Z_{(\beta,\omega)}(e^\gamma)$.
\begin{NB}
If $(\gamma-\beta,L)=0$, then
$$
\frac{(\langle e^\gamma,e^\beta \rangle(c_1(v)-r \beta)-
\langle v,e^\beta \rangle(\gamma-\beta),L)}{
(r \gamma- c_1(v),L)}=
\frac{((\beta-\gamma)^2)}{2} \leq 0.
$$
Moreover if the equality holds, then
$\beta=\gamma$.
\end{NB}
For the proof of our claim, it is sufficient to
show the independence of $\sigma_{(\beta,\omega)}$-semi-stability,
where $(\beta,L) \in V \times U$.
We set $X_1:=M_{H}(r_1 e^\gamma)$.
For a universal family ${\bf E}$ on $X \times X_1$
as a twisted object,
we consider the Fourier-Mukai transform
$\Phi_{X \to X_1}^{{\bf E}^{\vee}[1]}$.
Then we have an isomorphism
${\mathcal M}_{(\beta,\omega)}(v) \to {\mathcal M}_{\widetilde{\omega}}(u)^{ss}$,
where $u:=\Phi_{X \to X_1}^{{\bf E}^{\vee}[1]}(v)$.
\begin{NB}
$$
\widetilde{L}:=-\frac{1}{r_1}
\frac{\frac{((\beta-\gamma)^2)-t^2(L^2)}{2}t \widehat{L}-
(\beta-\gamma,tL)(\widehat{\beta}-\widehat{\gamma})}
{\left(\frac{((\beta-\gamma)^2)-t^2(L^2)}{2} \right)^2
+t^2(\beta-\gamma,L)^2}.
$$
We set
$$
\widetilde{\beta}:=\gamma'-\frac{1}{r_1}
\frac{\frac{((\beta-\gamma)^2)-t^2(L^2)}{2}(\widehat{\beta}-\widehat{\gamma})
-(\beta-\gamma,tL)t \widehat{L}}
{\left(\frac{((\beta-\gamma)^2)-t^2(L^2)}{2} \right)^2
+t^2(\beta-\gamma,L)^2}.
$$
\end{NB}
Since $(c_1(u e^{-\widetilde{\beta}}),\widetilde{L})=0$
for all $(\beta,L) \in V \times U$,
if $F \in {\mathcal M}_{\widetilde{L}}(u)^{ss}$ contains
a subsheaf $F_1$ with
$(c_1(F_1(-\widetilde{\beta})),\widetilde{\omega})=0$
and $v(F_1) \not \in {\mathbb Q}u$,
then $\Phi_{X_1 \to X}^{{\bf E}[1]}(F_1)$ defines a wall
for $v$.
\begin{NB}
${\mathbb R}Z_{(\beta,tL)}(\Phi_{X_1 \to X}^{{\bf E}[1]}(F_1))=
{\mathbb R}Z_{(\beta,tL)}(e^\gamma)={\mathbb R}Z_{(\beta,tL)}(v)$.
We set $u_1:=v(F_1)$ and $u_2:=u-u_1$.
Then $\langle u_1,u_2 \rangle \geq
(\langle u_1^2 \rangle+\langle u_2^2 \rangle)/2 \geq 0$.
If the equality holds, then
$\langle u_1 \rangle =\langle u_2^2 \rangle=0$ and
$\frac{c_1(u_1)}{\operatorname{rk} u_1}=\frac{c_1(u_2)}{\operatorname{rk} u_2}$.
Then $u \in {\mathbb Q}u_1$ and $u$ is an isotropic Mukai vector.
\end{NB}
Therefore
for any subsheaf $F_1$ of
$F \in {\mathcal M}_{\widetilde{\omega}}(u)^{ss}$,
$(c_1(F_1(-\widetilde{\beta})),\widetilde{\omega}) \ne 0$
for any $(\beta,L) \in V \times U$ or
we have $v(F_1) \in {\mathbb Q}u$.
For $F \in {\mathcal M}_{\widetilde{\omega_1}}(u)^{ss} \setminus
{\mathcal M}_{\widetilde{\omega_2}}(u)^{ss}$,
there is a subsheaf $F_1$ such that
$(c_1(F_1(-\widetilde{\beta})),\widetilde{\omega_2}) \geq 0$
and
$(c_1(F_1(-\widetilde{\beta})),\widetilde{\omega_1}) \leq 0$.
Since $(c_1(F_1(-\widetilde{\beta})),\widetilde{\omega})$
is a continuous function on $V \times U$, it is a contradiction.
Therefore ${\mathcal M}_{\widetilde{\omega}}(u)^{ss}$
is independent of $(\beta,L)$.
Then we see that ${\mathcal M}_{(\beta,\omega)}(v)$ is independent of
$(\beta,\omega)$.
\end{proof}
\begin{defn}
Let $\sigma_{(\beta,\omega)}$ be a stability condition.
For the contravariant
Fourier-Mukai transform
$\Phi \circ {\mathcal D}_X$, we set $(\beta',\omega'):=(-\beta,\omega)$
and attach the stability condition
$\sigma_{(\widetilde{\beta'},\widetilde{\omega'})}$ associated to
$Z_{(\widetilde{\beta'},\widetilde{\omega'})}$.
We say $\sigma_{(\widetilde{\beta'},\widetilde{\omega'})}$
the stability condition induced by $\Phi \circ {\mathcal D}_X$.
\end{defn}
\begin{lem}\label{lem:dual-phase}
$\phi_{(-\beta,\omega)}(E^{\vee}[1])=-\phi_{(\beta,\omega)}(E)+1$.
\end{lem}
\begin{proof}
For a non-zero object $E \in {\bf D}(X)$, we have
\begin{equation}
Z_{(-\beta,\omega)}(E^{\vee}[1])=
-\langle e^{-\beta+\omega \sqrt{-1}},v(E^{\vee}) \rangle
=-\overline{\langle e^{\beta+\omega \sqrt{-1}},v(E) \rangle}
=|Z_{(\beta,\omega)}(v)|e^{\pi \sqrt{-1} (1-\phi_{(\beta,\omega)}(E))}.
\end{equation}
Hence $\phi_{(-\beta,\omega)}(E^{\vee}[1])=-\phi_{(\beta,\omega)}(E)+1
\mod 2{\mathbb Z}$.
Since $\phi_{(-\beta,\omega)}((E[n])^{\vee}[1])=
\phi_{(-\beta,\omega)}(E^{\vee}[1])-n$, we shall show that
$\phi_{(-\beta,\omega)}(E^{\vee}[1]) \in [0,1)$
for $E \in {\mathfrak A}_{(\beta,\omega)}$.
We note that
${\mathfrak A}_{(\beta,\omega)}$ is generated by
(i) 0-dimensional object $T$, (ii)
$F[1]$ where $F$ is a locally free $\mu$-semi-stable sheaf
with $d_\beta(F) \leq 0$, (iii)
$\mu$-semi-stable sheaf $E$ with $d_\beta(E)>0$ and
(iv)
purely 1-dimensional sheaf $E$.
(i) For a 0-dimensional sheaf $T$,
$T^{\vee}[1] \in {\mathfrak A}_{(-\beta,\omega)}[-1]$.
Thus $\phi_{(-\beta,\omega)}(T^{\vee}[1])=0$.
(ii) For a locally free $\mu$-semi-stable
sheaf $F$ with $d_\beta(F) \leq 0$,
$(F[1])^{\vee}[1]=F^{\vee}$ is a $\mu$-semi-stable sheaf with
$d_{-\beta}((F[1])^{\vee}[1]) \geq 0$.
Hence $\phi_{(-\beta,\omega)}((F[1])^{\vee}[1]) \in [0,1)$.
(iii) Let $E$ be a $\mu$-semi-stable sheaf of
$\operatorname{rk} E>0$ and $d_\beta(E)>0$.
Let $E^{**}$ be the reflexive hull of $E$ and set $T:=E^{**}/E$.
Then we have an exact triangle
$$
T^{\vee}[1] \to (E^{**})^{\vee}[1] \to E^{\vee}[1] \to
T^{\vee}[2].
$$
Since $(E^{**})^{\vee}$ is a locally free $\mu$-semi-stable sheaf
with $d_{-\beta}((E^{**})^{\vee})=-d_\beta(E)<0$ and
$T^{\vee}[2]$ is a 0-dimensional sheaf,
$E^{\vee}[1] \in {\mathfrak A}_{(-\beta,\omega)}$.
(iv) If $E$ is a purely 1-dimensional sheaf, then
$E^{\vee}[1]$ is a purely 1-dimensional sheaf, which implies
$E^{\vee}[1] \in {\mathfrak A}_{(-\beta,\omega)}$.
Therefore the claim holds.
\end{proof}
\subsection{Relations of moduli spaces under Fourier-Mukai
transforms.}
We shall say that a pair
$(\beta,\omega)$ is \emph{general with respect to $v$}
if $(\beta,\omega)$ is not on any wall $W_{v_1}$ for $v$.
\begin{thm}\label{thm:B:3-10.3}
Let $v$ be a Mukai vector with
$\langle v^2 \rangle>0$.
Assume that $(\beta,\omega)$ is general with respect to
$v$.
\begin{enumerate}
\item[(1)]
Any Fourier-Mukai transform
$\Phi_{X \to X'}^{{\bf E}}$
preserves Bridgeland's stability condition.
\item[(2)]
Any contravariant Fourier-Mukai transform
$\Phi_{X \to X'}^{{\bf E}} \circ {\mathcal D}_X$
preserves Bridgeland's stability condition.
\end{enumerate}
\end{thm}
\begin{NB}
The same claims in this section also hold for
the Bridgeland stability of twisted sheaves.
\end{NB}
The first claim is due to Bridgeland (\cite[Prop. 10.3]{Br:3}).
For the proof of (2),
it is sufficient to prove the following claim.
\begin{prop}
Let $v$ be a Mukai vector with
$\langle v^2 \rangle>0$.
Assume that $(\beta,\omega)$
is general with respect to $v$.
If $Z_{(\beta,\omega)}(v) \in {\mathbb H} \cup {\mathbb R}_{<0}$, then
we have an isomorphism
\begin{equation*}
\begin{matrix}
M_{(\beta,\omega)}(v) & \to & M_{(-\beta,\omega)}(-v^{\vee})\\
E & \mapsto & E^{\vee}[1].
\end{matrix}
\end{equation*}
\begin{NB}
If $Z_{(\beta,\omega)}(v) \in {\mathbb R}_{<0}$,
then
we have an isomorphism
\begin{equation}
\begin{matrix}
M_{(\beta,\omega)}(v) & \to & M_{(-\beta,\omega)}(v^{\vee})\\
E & \mapsto & E^{\vee}[2].
\end{matrix}
\end{equation}
Since $Z_{(-\beta,\omega)}(-v^{\vee}) \in {\mathbb R}_{>0}$,
$\phi_{(-\beta,\omega)}(-v^{\vee})=0$.
By Definition \ref{defn:moduli}, $M_{(-\beta,\omega)}(-v^{\vee})=
\{F[-1]|F:=E^{\vee}[2] \in M_{(-\beta,\omega)}(v^{\vee}) \}$.
\end{NB}
\end{prop}
\begin{NB}
\begin{equation}
\deg_{\beta}(E)=-\deg_{-\beta}(E^{\vee})
=\deg_{-\beta}(E^{\vee}[1]).
\end{equation}
$$
Z_{(\beta,\omega)}(-v^{\vee})
=a_{-\beta}(v)-\frac{(\omega^2)}{2}r(v)+d_{-\beta}(v)(H,\omega)\sqrt{-1}.
$$
Hence
$Z_{(\beta,\omega)}(E^{\vee}[1]) \in {\mathbb H} \cup {\mathbb R}_{\geq 0}$
for $E \in {\mathfrak A}_{(-\beta,\omega)}$.
\end{NB}
\begin{NB}
If $(\omega^2) \gg 0$ and $\operatorname{rk} v<0$, then
$M_{(\beta,\omega)}(v)=\{E^{\vee}[1]| E \in M_\omega^{-\beta}(-v^{\vee})\}$
and
$M_{(-\beta,\omega)}(-v^{\vee})=M_\omega^{-\beta}(-v^{\vee})$.
Hence the claim is trivial.
\end{NB}
\begin{proof}
\begin{NB}
We don't need the case $d_\beta(v)=0$, since we can move
$\beta$. However the following is important, since
the general case can be reduced to this special case.
We fitst assume that $d_\beta(v)=0$.
In this case, $\sigma_{(\beta,\omega)}$-semi-stable objects are generated by
$E[1]$ and ${\mathfrak k}_x$, where $E$ is a locally free $\mu$-sermi-stable
sheaves.
Since $E^{\vee}$ and ${\mathfrak k}_x^{\vee}[2]$ are $\mu$-semi-stable
sheaf and a point sheaf respectively,
$(E[1])^{\vee}[2]=E^{\vee}[1]$ and ${\mathfrak k}_x^{\vee}[2]$
are $\sigma_{(\beta,\omega)}$-semi-stable objects of
${\mathfrak A}_{(-\beta,\omega)}$.
Thus the claim holds.
\begin{NB2}
If $(\beta,\omega)$ is general, then
every $\sigma_{(\beta,\omega)}$-semi-stable objects $E$ with $v(E)=v$
are $\mu$-stable.
\end{NB2}
\end{NB}
Let ${\mathcal C}$ be a chamber containing $(\beta,\omega)$.
We may assume that $d_{\beta,\omega}(v)>0$.
Replacing $(\beta,\omega) \in {\mathcal C}$ if necessary,
we can take $\gamma \in \operatorname{NS}(X)_{\mathbb Q}$ such that
${\mathbb R}Z_{(\beta,\omega)}(e^\gamma)={\mathbb R}Z_{(\beta,\omega)}(v)$.
We take a primitive vector $r_1 e^\gamma$ such that
$d_{\beta,\omega}(r_1 e^\gamma)>0$.
We set $X_1:=M_{(\beta,\omega)}(r_1 e^{\gamma})$.
Let
${\bf E}$ be the universal object on $X \times X_1$
as a complex of twisted sheaves.
We set $w:=\Phi_{X \to X_1}^{{\bf E}^{\vee}[1]}(v)$.
By Theorem \ref{thm:MYY2},
$M_{\widetilde{\omega}}^{\alpha_1}(w)$ consists of $\mu$-stable
locally free sheaves and
we have an isomorphism
$$
\Phi_{X \to X_1}^{{\bf E}^{\vee}[1]}:
M_{(\beta,\omega)}(v) \to M_{\widetilde{\omega}}^{\alpha_1}(w),
$$
where $w$ satisfies
$(c_1(w e^{-\widetilde{\beta}}),\widetilde{\omega})=0$.
By taking the dual, we have an isomorphism
$$
M_{\widetilde{\omega}}^{\alpha_1}(w) \to
M_{\widetilde{\omega}}^{-\alpha_1}(w^{\vee}).
$$
We note that ${\bf F}:={\bf E}^{\vee}[1]$ is a family of stable objects
with $v({\bf F}_{|X \times \{x_1 \}})=-r_1 e^{-\gamma}$.
For $\Phi_{X \to X_1}^{{\bf F}^{\vee}[1]}$
we define $(\widetilde{-\beta},\widetilde{\omega})$
by \eqref{eq:tilde(beta)}.
Thus we substitute $(-\gamma,-\gamma',-\beta)$
in \eqref{eq:tilde(beta)}
instead of $(\gamma,\gamma',\beta)$
for the definition of
$(\widetilde{-\beta},\widetilde{\omega})$.
Then we have $
(\widetilde{-\beta},\widetilde{\omega})=
(-\widetilde{\beta},\widetilde{\omega})$.
Since ${\mathbb R}Z_{(-\beta,\omega)}(-r_1 e^{-\gamma})=
{\mathbb R}Z_{(-\beta,\omega)}(-v^{\vee})$,
we have an isomorphism
\begin{equation}
\Phi_{X \to X_1}^{{\bf F}^{\vee}[1]}:
M_{(-\beta,\omega)}(-v^{\vee}) \to
M_{\widetilde{\omega}}^{-\alpha_1}(w^{\vee}).
\end{equation}
By the Grothendieck-Serre duality,
we have
$$
\Phi_{X \to X_1}^{{\bf F}^{\vee}[1]}(E^{\vee}[1])=
\Phi_{X \to X_1}^{{\bf E}}(E^{\vee}[1])=
D_{X_1} \circ \Phi_{X \to X_1}^{{\bf E}^{\vee}[1]}(E).
$$
Hence the claim holds.
\end{proof}
\begin{NB}
\begin{proof}
We first recall the main result in \cite{MYY:2011:2}.
If $\lambda r \ne d_\beta$,
then Lemma \ref{lem:C_v}
implies that
the condition
${\mathbb R}Z_{(\beta+sH,\omega)}(v)=
{\mathbb R}Z_{(\beta+sH,\omega)}(e^{\beta+\lambda H})$
defines a circle
\begin{equation}\label{eq:circle-lambda}
C_{v,\lambda}:\;
t^2=
(\lambda-s)
\left( \frac{a_\beta-d_\beta \lambda \frac{(H^2)}{2}}{\lambda r-d_\beta}
\frac{2}{(H^2)}+s \right).
\end{equation}
Assume that $C_{v,\lambda}$ is not contained in any wall
for $v$.
Then $C_{v,\lambda}$ is contained in a chamber ${\mathcal C}$.
We set $\gamma:=\beta+\lambda H$.
\begin{equation}
\frac{a_\beta-d_\beta \lambda \frac{(H^2)}{2}}{\lambda r-d_\beta}
\frac{2}{(H^2)}+\lambda
=\frac{(\lambda r-d_\beta)^2 (H^2)-
(\langle v^2 \rangle-(D_\beta^2))}
{r(H^2)(\lambda r-d_\beta)}
=\frac{2a_\gamma}{-d_\gamma (H^2)}=
\frac{2\langle e^{\beta+\lambda H},v \rangle}
{d_\gamma (H^2)}
\end{equation}
Then the intersection of $C_{v,\lambda}$ with the
circle
$(s-\lambda)^2+t^2=\frac{2}{|r_1|(H^2)}$
lies on the line
\begin{equation}\label{eq:line}
s=\lambda+\frac{d_\gamma}{a_\gamma |r_1|}.
\end{equation}
We set $X_1:=M_{(\beta+sH,tH)}(r_1 e^{\gamma})$.
Let
${\bf E}$ be the universal object on $X \times X_1$
as a complex of twisted sheaves.
By the Fourier-Mukai transform
$\Phi_{X \to Y}^{{\bf E}^{\vee}[1]}:
{\bf D}(X) \to {\bf D}^{\alpha_1}(X_1)$,
the circle is transformed to
the line \eqref{eq:line}.
\begin{NB2}
$$
s'=\Phi_{X \to Y}^{{\bf E}^{\vee}[1]}(s)=
\lambda+\frac{2}{(H^2)|r_1|}
\frac{s-\lambda}{(s-\lambda)^2+t^2}.
$$
\end{NB2}
Hence the chamber ${\mathcal C}$ containing
the circle $C_{v,\lambda}$ is mapped to a unbounded domain.
We set $w:=\Phi_{X \to X_1}^{{\bf E}^{\vee}[1]}(v)$.
By \cite[Thm. 2.2.1]{MYY:2011:2},
we have a moduli of $\mu$-stable vector bundles
$M_{\widehat{H}}^{\alpha_1}(w)$,
we have an isomorphism
$\Phi_{X \to X_1}^{{\bf E}[1]}:
M_{(\beta+sH,tH)}(v) \to M_{\widehat{H}}^{\alpha'}(w)$
for $(s,t) \in C_{v,\lambda}$.
For $\Phi_{X_1 \to X}^{{\bf E}^{\vee}[2]}$,
we also have an isomorphism
$M_{\widehat{H}}^{-\alpha_1}(w^{\vee}) \to M_{(-\beta+sH,tH)}(-v^{\vee})$,
where $(s,t) \in C_{-v^{\vee},-\lambda}$.
By the Grothendieck-Serre duality,
we have
$$
\Phi_{X \to X_1}^{{\bf E}}(E^{\vee}[1])=
D_{X_1} \circ \Phi_{X \to X_1}^{{\bf E}^{\vee}[1]}(E).
$$
\begin{NB2}
We have an isomorphism
$M_{(\beta+sH,tH)}(r_1 e^{\beta+\lambda H})
\to M_{(-\beta-sH,tH)}(-r_1 e^{-\beta-\lambda H})$
by sending $E$ to $E^{\vee}[1]$.
Thus $M_{(-\beta-sH,tH)}(-r_1 e^{-\beta-\lambda H}) \cong Y$
and
${\bf E}^{\vee}[1]$ is the universal family
on $X \times Y$.
Then $\Phi_{X \to Y}^{({\bf E}^{\vee}[1])^{\vee}[1]}=
\Phi_{X \to Y}^{{\bf E}}$.
\end{NB2}
\begin{NB2}
Assume that $r>0$.
Then
\begin{equation}
\begin{cases}
d_{\beta+sH}(v)=d_\beta(v)-rs>0,& s<\frac{d_\beta}{r},\\
d_{\beta+sH}(-v)=d_\beta(-v)-(-r)s>0,& s>\frac{d_\beta}{r}.
\end{cases}
\end{equation}
$s=\frac{d_\beta}{r}$ is a wall for $v$ if and only if
there is a properly $\mu$-semi-stable sheaf or
there is a non-locally free $\mu$-semi-stable sheaf.
\end{NB2}
If $\lambda r =d_\beta$,
then the condition
${\mathbb R}Z_{(\beta+sH,\omega)}(v)=
{\mathbb R}Z_{(\beta+sH,\omega)}(e^{\beta+\lambda H})$
defines a line $L_{v,\lambda}$
$$
s=\frac{d_\beta}{r}=\lambda.
$$
\begin{NB2}
If $\frac{d_\beta}{r}=\lambda$, then
$a_\beta-\frac{(H^2)}{2}r \lambda^2 \ne 0$.
Indeed if $a_\beta-\frac{(H^2)}{2}r \lambda^2=0$, then
$v=r e^{\beta+\lambda H}+D_\beta+(D_\beta,\beta+\lambda H)\varrho_X$.
Hence $\langle v^2 \rangle =(D_\beta^2) \leq 0$,
which is a contradiction.
\end{NB2}
If $L_{v,\lambda}$ is not contained in any wall for $v$,
then the claim follows from
the fact that $M_{(\beta+\lambda H,tH)}(v)$ and
$M_{(-\beta-\lambda H,tH)}(v)$
consists of $\mu$-stable
vector bundles up to shift.
\end{proof}
\end{NB}
\begin{rem}
By a similar argument, we can prove Theorem
\ref{thm:B:3-10.3} (1):
Let ${\mathcal C}$ be a chamber containing $(\beta,\omega)$.
We may assume that $d_{\beta,\omega}(v)>0$.
Replacing $(\beta,\omega) \in {\mathcal C}$ if necessary,
we can take $\gamma \in \operatorname{NS}(X)_{\mathbb Q}$ such that
${\mathbb R}Z_{(\beta,\omega)}(e^\gamma)={\mathbb R}Z_{(\beta,\omega)}(v)$
and $(\beta,\omega) \in {\mathcal C}$.
We take a primitive vector $r_1 e^\gamma$ such that
$d_{\beta,\omega}(r_1 e^\gamma)>0$.
For any Fourier-Mukai transform $\Phi:{\bf D}(X) \to {\bf D}(Y)$,
let $v_0 \in {\mathbb Q}\Phi(e^{\gamma})$
be a primitive and positive Mukai vector.
If $r_0:=\operatorname{rk} v_0 \ne 0$, then
$v_0=r_0 e^{\widetilde{\gamma}}$,
$\widetilde{\gamma} \in \operatorname{NS}(Y)$.
We first assume that $r_0 \ne 0$.
We set
\begin{equation}
\Phi(e^{\beta+\sqrt{-1} \omega})=
-\langle e^{\beta+\sqrt{-1} \omega}, \Phi^{-1}(\varrho_Y) \rangle
e^{\beta'+\sqrt{-1} \omega'}.
\end{equation}
Then $X_1 \cong M_{\omega'}(v_0)$ and
the universal object ${\bf G}$ on $X_1 \times Y$
satisfies $\Phi_{X_1 \to Y}^{{\bf G}[n]}=
\Phi \circ \Phi_{X_1 \to X}^{{\bf E}}$, $n \in {\mathbb Z}$.
Since
${\mathbb R}Z_{(\beta,\omega)}(e^\gamma)
={\mathbb R}Z_{(\beta,\omega)}(v)$,
we have
${\mathbb R}Z_{(\beta',\omega')}(v_0)
={\mathbb R}Z_{(\beta',\omega')}(\Phi(v))$.
Then
$\Phi_{X_1 \to Y}^{{\bf G}[n]}$
induces an isomorphism
$$
M_{H_1}^{\alpha_1}(w) \to
M_{(\beta',\omega')}(\Phi(v)).
$$
Hence we have an isomorphism
$$
\Phi:M_{(\beta,\omega)}(v) \to
M_{(\beta',\omega')}(\Phi(v)).
$$
If $r_0=0$, then $\Phi$ is induced by isomorphisms
on underlying abelian surfaces and
the action of $\Piczero{X}$.
Hence we also have an isomorphism
$M_{(\beta,\omega)}(v) \to
M_{(\Phi(\beta),\Phi(\omega))}(\Phi(v))$.
If $(\beta,\omega)$ belongs to a wall, then
$\Phi$ also preserves semi-stability. Indeed
assume that
$E$ is $S$-equivalent to
$\oplus_i E_i$, where $E_i$ are $\sigma_{(\beta,\omega)}$-stable
objects with $\phi_{(\beta,\omega)}(E_i)=\phi_{(\beta,\omega)}(E)$.
Then $\Phi(E_i)$ are $\sigma_{(\beta',\omega')}$-stable
objects. By \eqref{eq:Phi(phase)},
$\phi_{(\beta',\omega')}(E_i)$ are the same.
\begin{NB}
Then $\Phi(E_i)$ are $\sigma_{(\widetilde{\beta},\widetilde{\omega})}$-stable
objects. By \eqref{eq:Phi(phase)},
$\phi_{(\widetilde{\beta},\widetilde{\omega})}(E_i)$ are the same.
\end{NB}
Therefore the claim holds.
\begin{NB}
Old argument:
For any Fourier-Mukai fransform $\Phi:{\bf D}(X) \to {\bf D}(Y)$,
let $v_0 \in {\mathbb Q}\Phi(e^{\beta+\lambda H})$
be a primitive and positive Mukai vector.
If $r_0:=\operatorname{rk} v_0 \ne 0$, then
$v_0=r_0 e^{\widetilde{\gamma}}$,
$\widetilde{\gamma} \in \operatorname{NS}(Y)$.
We first assume that $r_0>0$.
We set
\begin{equation}
\Phi(e^{\beta+sH+\sqrt{-1} tH})=
-\langle e^{\beta+sH+\sqrt{-1} tH}, \Phi^{-1}(\varrho_Y) \rangle
e^{\widetilde{\beta}(s)+\sqrt{-1} \widetilde{\omega}(s)}.
\end{equation}
Then $X_1 \cong M_{\widetilde{\omega}(s)}(v_0)$ and
the universal object ${\bf G}$ on $X_1 \times Y$
satisfies $\Phi_{X_1 \to Y}^{{\bf G}[n]}=
\Phi \circ \Phi_{X_1 \to X}^{{\bf E}}$, $n \in {\mathbb Z}$.
For $(s,t) \in C_{v,\lambda}$,
${\mathbb R}Z_{\widetilde{\beta}(s),\widetilde{\omega}(s)}(v_0)
={\mathbb R}Z_{\widetilde{\beta}(s),\widetilde{\omega}(s)}(\Phi(v))$.
Then
$\Phi_{X_1 \to Y}^{{\bf G}[n]}$
induces an isomorphism
$$
M_{H_1}^{\alpha_1}(w) \to
M_{(\widetilde{\beta}(s),\widetilde{\omega}(s))}(\Phi(v)).
$$
Hence we have an isomorphism
$$
\Phi:M_{(\beta+s H,t H)}(v) \to
M_{(\widetilde{\beta}(s),\widetilde{\omega}(s))}(\Phi(v)).
$$
If $r_0=0$, then $\Phi$ is induced by isomorphisms and
the action of $\Piczero{X}$.
Hence we also have an isomorphism
$M_{(\beta+sH,tH)}(v) \to
M_{(\widetilde{\beta}(s),\widetilde{\omega}(s))}(\Phi(v))$.
For each chamber ${\mathcal C}$, we can take
$C_{v,\lambda} \subset {\mathcal C}$.
Hence $\Phi$ preserves the stability condition.
\end{NB}
\end{rem}
\section{Numerical solutions and the walls.}
\subsection{Semi-homogeneous presentations and numerical solutions.}
\label{subsec:intro:sh}
Let us recall the notion of \emph{semi-homogeneous presentations}
introduced in \cite{YY}.
\begin{defn}\label{defn:semi-homog}
A \emph{semi-homogeneous presentation} of $E\in\operatorname{Coh}(X)$ is an exact sequence
\begin{align*}
\begin{matrix}
0\to E \to E_1 \to E_2 \to 0 \quad \mbox{or}\quad
0\to E_1 \to E_2 \to E \to 0,
\end{matrix}
\end{align*}
where $E_i$ ($i=1,2$) are semi-homogeneous sheaves
satisfying the following condition:
if we write $v(E_i)=\ell_i v_i$
with $\ell_i$ positive integers
and $v_i$ primitive Mukai vectors,
then
\begin{align*}
(\ell_1-1)(\ell_2-1)=0,\quad
\langle v_1^2 \rangle=\langle v_2^2 \rangle=0,\quad
\langle v_1,v_2 \rangle=-1.
\end{align*}
We call the first sequence the \emph{kernel presentation}
and the second one the \emph{cokernel presentation}.
\end{defn}
Since semi-homogeneous sheaves on abelian varieties are well-known objects,
semi-homogeneous presentations give useful informations on $E$.
Moreover
the property of having a semi-homogeneous presentation is an open condition,
and describes the birational structure of the moduli spaces $M_H(v)$.
The numerical data appeared in Definition \ref{defn:semi-homog}
are useful to find a semi-homogeneous presentation.
So we introduce the following definition.
\begin{defn
For a Mukai vector $v$, the equation
\begin{align*}
\begin{split}
&v=\pm(\ell_1 v_1-\ell_2 v_2),
\\
&\ell_1, \ell_2 \in {\mathbb{Z}}_{>0},\quad
v_1, v_2\colon \mbox{positive primitive Mukai vectors,}
\end{split}
\end{align*}
is called \emph{the numerical equation of $v$}.
A solution $(v_1,v_2,\ell_1,\ell_2)$ of this equation satisfying
\begin{align*}
(\ell_1-1)(\ell_2-1)=0,\quad
\langle v_1^2 \rangle =\langle v_2^2 \rangle=0,\quad
\langle v_1,v_2 \rangle=-1,
\end{align*}
is called a \emph{numerical solution} of $v$.
\end{defn}
\begin{thm}[{\cite[Thm. 3.9]{YY}}]\label{fct:presentation}
Suppose $\operatorname{NS}(X)=\mathbb{Z} H$ and let $v$ be a Mukai vector with
$\langle v^2 \rangle>0$.
\begin{enumerate}
\item
If $v$ has at least two numerical solutions,
then a general member of $M_H(v)$ has
both kernel presentation and cokernel presentation.
Each presentation is unique.
\item
If $v$ has only one numerical solution,
then a general member of $M_H(v)$ has
either kernel presentation or cokernel presentation.
Such a presentation is unique.
\end{enumerate}
\end{thm}
For each numerical solution
$(v_1,v_2,\ell_1,\ell_2)$ of $v$,
we constructed moduli spaces of simple two-term complexes.
These moduli spaces plays an important role to prove
\cite[Thm. 3.9]{YY}.
We fix an ample divisor $H$ on $X$.
\begin{thm}[{\cite[Thm. 4.9]{YY}}]
\label{fct:moduli}
Let $v$ be a positive Mukai vector with $\mpr{v^2}>0$ and
$(v_1,v_2,\ell_1,\ell_2)$ be a numerical solution of $v$.
\begin{enumerate}
\item
We have the fine moduli space $\mathfrak{M}^{-}(v_1,v_2,\ell_1,\ell_2)$
of simple complexes $V^{\bullet}$ such that
$H^i(V^{\bullet})=0$ ($i \neq -1,0$),
$H^{-1}(V^{\bullet}) \in {\mathcal M}_H(\ell_1 v_1)^{ss}$ and
$H^0(V^{\bullet}) \in {\mathcal M}_H(\ell_2 v_2)^{ss}$.
\item
We have the fine moduli space $\mathfrak{M}^+(v_1,v_2,\ell_1,\ell_2)$
of simple complexes $V^{\bullet}$ such that
$V^{\bullet} \cong [W^{-1} \to W^0]$,
$W^{-1} \in {\mathcal M}_H(\ell_1 v_1)^{ss}$ and
$W^0 \in {\mathcal M}_H(\ell_2 v_2)^{ss}$.
\end{enumerate}
\end{thm}
\begin{rem}
Since
${\mathcal M}_H(\ell_i v_i)^{ss}$ ($i=1,2$) are independent of the choice of $H$,
$\mathfrak{M}^\pm(v_1,v_2,\ell_1,\ell_2)$ are independent of the choice of $H$.
\end{rem}
The relation of $\mathfrak{M}^\pm(v_1,v_2,\ell_1,\ell_2)$
is described as follows:
\begin{prop}[{\cite[Prop. 4.11]{YY}}]
\label{prop:dual}
For $i=1,2$, we denote $Y_i := M_H(v_i)$,
and let $\bl{E}_i$ be a universal family
such that $v(\Phi_{X \to Y_i}^{\bl{E}_i^{\vee}}(v_j))=(1,0,0)$
for $j \ne i$.
For $V^{\bullet} \in \mathfrak{M}^{+}(v_1,v_2,\ell_1,\ell_2)$,
we set
\begin{align*}
\Psi(V^{\bullet}):=
\begin{cases}
\Phi_{Y_1 \to X}^{{\bl{E}}_1[2]}\,\mathcal{D}_{Y_1}\,
\Phi_{X \to Y_1}^{{\bl{E}}_1^{\vee}}(V^{\bullet})
& \ell_1=\langle v^2 \rangle/2,
\\
\Phi_{Y_2 \to X}^{{\bl{E}}_2[1]}\,\mathcal{D}_{Y_2}\,
\Phi_{X \to Y_2}^{{\bl{E}}_2^{\vee}[1]}(V^{\bullet})
& \ell_2=\langle v^2 \rangle/2.
\end{cases}
\end{align*}
Then
$\Psi$ induces an isomorphism
\begin{align*}
\mathfrak{M}^{+}(v_1,v_2,\ell_1,\ell_2) \xrightarrow{\ \sim\ }
\mathfrak{M}^{-}(v_1,v_2,\ell_1,\ell_2).
\end{align*}
\end{prop}
\subsection{Relation with the walls.}
The operation $\Psi$ is first introduced in \cite{Yoshioka:2009:ann}
to construct birational map of moduli spaces,
and it is reformulated as an isomorphism of
$\mathfrak{M}^\pm(v_1,v_2,\ell_1,\ell_2)$
in \cite{YY}.
$\Psi$ plays a fundamental role in these papers.
In \cite[sect. 4]{MYY:2011:1},
we explained its relation with Bridgeland's stability
condition. In particular, we showed that
$\mathfrak{M}^\pm(v_1,v_2,\ell_1,\ell_2)$ are moduli of stable objects
if $v_1$ defines a wall. We shall slightly generalize this fact.
Thus we show the following.
\begin{lem}\label{lem:num.sol:stability}
There is a stability condition
$\sigma_{(\beta,\omega)}=({\mathfrak A}_{(\beta,\omega)},Z_{(\beta,\omega)})$
such that
${\mathfrak M}^\pm(v_1,v_2,\ell_1,\ell_2)$
are the moduli schemes of stable objects
$M_{(\beta,\omega^\pm)}(\pm v)$,
where $\omega^\pm=t^\pm \omega$,
$t^-<1<t^+$, $t^+-t^- \ll 1$.
\end{lem}
\begin{proof}
Replacing $v$ by $-v$, we assume that
$v=\ell_2 v_2-\ell_1 v_1$.
We set
$v=r e^{\gamma}-\frac{\ell}{r}\varrho_X$
and
$v_i=r_i e^{\gamma+\xi_i}$ ($i=1,2$).
Then $r=\pm(r_1 \ell_1-r_2 \ell_2)$,
$r_1 \ell_1 \xi_1=r_2 \ell_2 \xi_2$ and
$\frac{\ell}{r}=
\pm(r_1 \ell_1 \frac{(\xi_1^2)}{2}-r_2 \ell_2 \frac{(\xi_2^2)}{2})$.
Since $0>\langle e^{\gamma+\xi_1},e^{\gamma+\xi_2} \rangle
=-\frac{((\xi_2-\xi_1)^2)}{2}$,
there is an ample divisor $H$ with
$(\xi_1-\xi_2,H) \ne 0$.
We may assume that $(\xi_1,H)<(\xi_2,H)$.
We take $\beta(x):=\gamma+x \xi_1+(1-x) \xi_2$ with
$0<x<1$.
Then
\begin{equation*}
\begin{split}
v_1= & r_1 e^{\beta(x)+(1-x)(\xi_1-\xi_2)}\\
=&
r_1 \left(e^{\beta(x)}+(1-x)(\xi_1-\xi_2+(\xi_1-\xi_2,\beta)\varrho_X)
+\frac{(1-x)^2((\xi_1-\xi_2)^2)}{2}\varrho_X \right),\\
v_2=& r_2 e^{\beta(x)+x(\xi_2-\xi_1)}\\
=&
r_2 \left(e^{\beta(x)}+x(\xi_2-\xi_1+(\xi_2-\xi_1,\beta)\varrho_X)
+\frac{x^2((\xi_1-\xi_2)^2)}{2}\varrho_X \right).
\end{split}
\end{equation*}
If $r_1 \ell_1>r_2 \ell_2$, then
$(\xi_2,H)>(\xi_1,H)$ implies that
$(\xi_2,H)>(\xi_1,H)>0$.
Hence $d_{\beta(x)}(-v_1)>0$ and $d_{\beta(x)}(v_2)>0$.
We have
\begin{equation*}
\frac{d_{\beta(x)}(-v_1)a_{\beta(x)}(v_2)
-d_{\beta(x)}(v_2)a_{\beta(x)}(-v_1)}
{d_{\beta(x)}(-v_1)r_{\beta(x)}(v_2)-d_{\beta(x)}(v_2)r_{\beta(x)}(-v_1)}
=(1-x)x \frac{((\xi_1-\xi_2)^2)}{2}>0.
\end{equation*}
Hence there is a positive number $t$ such that
$Z_{(\beta(x),tH)}(-v_1) \in
{\mathbb R}_{>0}Z_{(\beta(x),tH)}(v_2)$.
Then $(\beta,\omega):=(\beta(x),tH)$ belongs to the wall defined by
$-v_1$.
By \cite[Thm. 4.9]{YY} and \cite[Prop. 4.1.5]{MYY:2011:1},
${\mathfrak M}^\pm(v_1,v_2,\ell_1,\ell_2)=M_{(\beta,\omega^\pm)}(v)$.
If $r_1 \ell_1<r_2 \ell_2$, then
we have $(\xi_1,H)<(\xi_2,H)<0$.
Hence $d_\beta(v_1)>0$ and $d_\beta(-v_2)>0$.
In this case, we also have
\begin{equation}
\frac{d_{\beta(x)}(v_1)a_{\beta(x)}(-v_2)-
d_{\beta(x)}(-v_2)a_{\beta(x)}(v_1)}
{d_{\beta(x)}(v_1)r_{\beta(x)}(-v_2)-d_{\beta(x)}(-v_2)r_{\beta(x)}(v_1)}
=(1-x)x \frac{((\xi_1-\xi_2)^2)}{2}>0.
\end{equation}
Hence there is a positive number $t$ such that
$Z_{(\beta(x),tH)}(v_1) \in
{\mathbb R}_{>0}Z_{(\beta(x),tH)}(-v_2)$.
Then $(\beta,\omega):=(\beta(x),tH)$ belongs to the wall defined by
$v_1$.
By \cite[Thm. 4.9]{YY} and \cite[Prop. 4.1.5]{MYY:2011:1},
${\mathfrak M}^\pm(v_1,v_2,\ell_1,\ell_2)=M_{(\beta,\omega^\pm)}(-v)$.
\end{proof}
\begin{NB}
$\mu_i:=(\xi_i,H)$.
(i)
Assume that $\mu_1,\mu_2>0$.
Since $r_1 \ell_1 \mu_1=r_2 \ell_2 \mu_2$,
$r_1 \ell_1> r_2 \ell_2 $ if and only if $\mu_2>\mu_1$.
(ii)
Assume that $\mu_1,\mu_2<0$.
Since $r_1 \ell_1 \mu_1=r_2 \ell_2 \mu_2$,
$r_1 \ell> r_2 \ell_2$ if and only if $\mu_2<\mu_1$.
(i) Assume that $r>0$.
Then $\mu_1<\mu_2$.
$d_{\beta+sH}(v[1])>0$
for $s-\frac{d_\beta}{r} \in (\mu_1,\mu_2)$.
Assume that $r<0$.
Then $\mu_2 <\mu_1$.
$d_{\beta+sH}(v)>0$ for
$s-\frac{d_\beta}{r} \in (\mu_2,\mu_1)$.
(ii)
Assume that $r>0$.
Then $\mu_2<\mu_1$.
$d_{\beta+sH}(v)>0$
for $s-\frac{d_\beta}{r} \in (\mu_2,\mu_1)$.
Assume that $r<0$.
Then $\mu_2>\mu_1$.
$d_{\beta+sH}(v[1])>0$
for $s-\frac{d_\beta}{r} \in (\mu_1,\mu_2)$.
\end{NB}
\section{Stability conditions on a restricted parameter space.}
\label{sect:restricted-parameter}
\subsection{The structure of walls.}
In this section, we shall partially generalize the structure of walls
in \cite{MM}.
We note that
\begin{equation*}
\{\sigma_{(\xi,\omega)} \mid
\xi \in \operatorname{NS}(X)_{\mathbb R},\ \omega \in \operatorname{Amp}(X)_{\mathbb R} \}=
\bigcup_{(\beta,H) \in \operatorname{NS}(X)_{\mathbb R} \times \operatorname{Amp}(X)_{\mathbb R}}
\{ \sigma_{(\beta+sH ,tH)} \mid s \in {\mathbb R},\ t \in {\mathbb R}_{>0} \}.
\end{equation*}
In this section,
we fix $\beta \in \operatorname{NS}(X)_{\mathbb R}$ and an ample divisor
$H$ on $X$ and
study the wall for the space
of special stability conditions:
$$
\{ \sigma_{(\beta+sH,tH)}\mid s \in {\mathbb R},\ t \in {\mathbb R}_{>0} \}.
$$
\begin{defn}\label{defn:wall2}
Let $v_1$
be a Mukai vector with
$\langle v_1^2 \rangle \geq 0$, $\langle (v-v_1)^2 \rangle \geq 0$
and $\langle v_1,v-v_1 \rangle >0$.
If $(\operatorname{rk} v_1,d_\beta(v_1),a_\beta(v_1)) \not \in
{\mathbb Q}(\operatorname{rk} v,d_\beta(v),a_\beta(v))$,
we define a {\it wall} for $v$ by
$$
W_{v_1}^{\beta,H}:=
\{(s,t) \in {\mathbb R}^2 \mid {\mathbb R}Z_{(\beta+sH,tH)}(v_1)=
{\mathbb R}Z_{(\beta+sH,tH)}(v) \}.
$$
A connected component of
${\mathbb R} \times {\mathbb R}_{>0} \setminus \cup_{v_1}
W_{v_1}^{\beta,H}$
is called a {\it chamber} for $v$.
\end{defn}
\begin{NB}
\begin{equation}
\begin{matrix}
W_{v_1}^{\beta,H} & \to &
W_{v_1} \cap
\{(\beta+sH,tH)
| s \in {\mathbb R}, t \in {\mathbb R}_{>0} \}\\
(s,t) & \mapsto & (\beta+sH,tH)
\end{matrix}
\end{equation}
\end{NB}
For the study of walls, we collect elementary facts
on a family of circles.
\begin{lem}\label{lem:circle}
We take $p \in {\mathbb R}$ and $q \in {\mathbb R}_{>0}$.
For $a \in {\mathbb R}$,
let
$C_a:(x+a)^2+y^2=(a+p)^2-q$ be the circle in $(x,y)$-plane.
We also set
$C_\infty:x=p$.
Thus $C_\infty$ is a line in $(x,y)$-plane.
\begin{enumerate}
\item[(1)]
$C_a \cap C_{a'}=\emptyset$, if $a \ne a'$.
\item[(2)]
$C_a \cap C_\infty=\emptyset$ for $a \in {\mathbb R}$.
\end{enumerate}
\end{lem}
\begin{proof}
(1)
If $C_a \cap C_{a'} \not =\emptyset$, then the intersection
satisfies $x=p$. Then $(p+a)^2+y^2=(a+p)^2-q$, which implies that
$y^2=-q<0$.
Therefore the claim holds.
The proof of (2) is similar.
\end{proof}
\begin{rem}\label{rem:base-point}
By the proof, we see that
$C_a$ $(a \in {\mathbb R})$ forms a pencil
of conics passing through the imaginary points
$\{(p,\pm\sqrt{-q}) \}$.
\end{rem}
\begin{NB}
Applonius circles:
Let $A(a+p-\sqrt{q},0)$ and $B(a+p+\sqrt{q},0)$ be two
points on $(s,t)$-plane.
Then $C_a$ is the locus of $(s,t)$
such that $AP:BP=\sqrt{a+p-\sqrt{q}}:\sqrt{ a+p+\sqrt{q}}$.
Indeed
\begin{equation}
\begin{split}
0=
& (a+p+\sqrt{q})((s-p+\sqrt{q})^2+t^2)-
(a+p-\sqrt{q})((s-p-\sqrt{q})^2+t^2)\\
=&
2\sqrt{q}((s-p)^2+q+t^2)+2(a+p)(s-p)\sqrt{q}.
\end{split}
\end{equation}
Hence the claim holds.
\end{NB}
\begin{lem}\label{lem:circle-center}
Assume that
$C_a:(x+a)^2+y^2=(a+p)^2-q$ with $q>0$
is non-empty.
If $a+p>0$, then
$(p-\sqrt{q},0)$ is contained in $C_a$.
If $a+p<0$, then
$(p+\sqrt{q},0)$ is contained in $C_a$.
\end{lem}
\begin{proof}
Assume that $a+p>0$. Then
$a+p > \sqrt{q}$.
Then
$(a+p)^2-q-(p-\sqrt{q}+a)^2=2\sqrt{q}(a+p-\sqrt{q})>0$.
Hence the claim holds.
If $a+p<0$, then
$a+p < -\sqrt{q}$.
Hence
$(a+p)^2-q-(p+\sqrt{q}+a)^2=-2\sqrt{q}(a+p+\sqrt{q})>0$.
Therefore the claim holds.
\end{proof}
We shall study the structure of walls.
We first assume that $r \ne 0$ and set
\begin{equation*}
\begin{split}
v:= & r+dH+D+b\varrho_X \\
=& r e^{\beta}+(d_\beta H+D_\beta)+(d_\beta H+D_\beta,\beta)\varrho_X
+a_\beta \varrho_X \\
=& r e^{\beta+sH}+(d_\beta-rs)(H+(H,\beta+sH) \varrho_X)
+D_\beta+(D_\beta,\beta)\varrho_X
+\widetilde{a}_\beta \varrho_X\\
\end{split}
\end{equation*}
Then
\begin{equation*}
a_\beta=\frac{d_\beta^2 (H^2)-(\langle v^2 \rangle-(D^2_\beta))}{2r},\;
\widetilde{a}_\beta=
\frac{(d_\beta-rs)^2 (H^2)-(\langle v^2 \rangle-(D_\beta^2))}{2r}.
\end{equation*}
We also set
\begin{equation*}
\begin{split}
v_2:=&
r_2 e^{\beta}+(d_2 H+D_2)+(d_2 H+D_2,\beta)\varrho_X
+a_2 \varrho_X\\
=& r_2 e^{\beta+sH}+(d_2-r_2 s)(H+(H,\beta+sH) \varrho_X)
+D_2+(D_2,\beta)\varrho_X
+\widetilde{a}_2 \varrho_X.
\end{split}
\end{equation*}
\begin{prop}\label{prop:wall}
Assume that $r \ne 0$ and
$\langle v^2 \rangle>0$.
\begin{enumerate}
\item[(1)]
Assume that $rd_2-r_2 d_\beta \ne 0$.
\begin{NB}
$rd_2-r_2 d_\beta$ does not depend on the choice of $\beta$.
\end{NB}
Then
${\mathbb R}Z_{(\beta+sH,tH)}(v)={\mathbb R}Z_{(\beta+sH,tH)}(v_2)$
holds for $(s,t) \in {\mathbb R}^2$ if and only if
\begin{equation}\label{eq:circle}
t^2+\left(s-\frac{a_2 r-a_\beta r_2}{(H^2)(rd_2-r_2 d_\beta)} \right)^2
=\left(\frac{d_\beta}{r}-\frac{a_2 r-a_\beta r_2}{(H^2)(rd_2-r_2 d_\beta)}
\right)^2
-\frac{\langle v^2 \rangle-(D_\beta^2)}{(H^2) r^2}.
\end{equation}
\item[(2)]
Assume that $rd_2-r_2 d_\beta=0$ and $a_2 r-a_\beta r_2 \ne 0$.
Then
${\mathbb R}Z_{(\beta+sH,tH)}(v)={\mathbb R}Z_{(\beta+sH,tH)}(v_2)$
holds for $(s,t) \in {\mathbb R}^2$ if and only if
$$
r s-d_\beta=0.
$$
\begin{NB}
If $r=0$, then $d_\beta=0$.
Since $\langle v^2 \rangle=(D_\beta^2) \leq 0$,
$v=a_\beta \varrho_X$.
\end{NB}
\end{enumerate}
\begin{NB}
Assume that $\langle v^2 \rangle>0$.
Then any pair of different two walls are disjoint
by Lemma \ref{lem:circle} or direct computation.
\end{NB}
\end{prop}
\begin{proof}
(1)
We first note that
$$
a_\beta d_2-a_2 d_\beta=\frac{a_\beta}{r}(rd_2-d_\beta r_2)
-\frac{d_\beta}{r}(a_2 r-a_\beta r_2).
$$
Then
\begin{equation*}
\begin{split}
& (d_2-r_2 s)\widetilde{a}_\beta-(d_\beta-rs)\widetilde{a}_2\\
=& -s^2 \frac{(H^2)}{2}(rd_2-r_2 d_\beta)+s (a_2 r-a_\beta r_2)+
(a_\beta d_2-a_2 d_\beta)\\
=& \frac{(H^2)}{2}\left\{-s^2 (rd_2-r_2 d_\beta)+
s \frac{2}{(H^2)}(a_2 r-a_\beta r_2)+
\frac{2}{(H^2)}
\left(\frac{a_\beta}{r}(rd_2-d_\beta r_2)-
\frac{d_\beta}{r}(a_2 r-a_\beta r_2) \right)
\right\}\\
=& \frac{(H^2)}{2}\left\{-(rd_2-r_2 d_\beta)\left(s-\frac{(a_2 r-a_\beta r_2)}
{(H^2)(rd_2-r_2 d_\beta)} \right)^2+
(rd_2-r_2 d_\beta)
\left(\frac{(a_2 r-a_\beta r_2)}{(H^2)(rd_2-r_2 d_\beta)} \right)^2 \right.\\
& \left.
+\frac{2}{(H^2)}\frac{a_\beta}{r}(rd_2-d_\beta r_2)
-2(rd_2-r_2 d_\beta)\frac{d_\beta}{r}
\frac{(a_2 r-a_\beta r_2)}{(H^2)(rd_2-r_2 d_\beta)}
\right\}\\
=& \frac{(H^2)}{2}\left\{-(rd_2-r_2 d_\beta)
\left(s-\frac{(a_2 r-a_\beta r_2)}
{(H^2)(rd_2-r_2 d_\beta)} \right)^2+
(rd_2-r_2 d_\beta) \left(\frac{d_\beta}{r}-\frac{(a_2 r-a_\beta r_2)}
{(H^2)(rd_2-r_2 d_\beta)} \right)^2 \right.\\
& \left. +(rd_2-r_2 d_\beta)\left(\frac{2}{(H^2)}\frac{a_\beta}{r}-
\left(\frac{d_\beta}{r}\right)^2
\right) \right\}\\
=& \frac{(rd_2-r_2 d_\beta)(H^2)}{2}\left\{-\left(s-\frac{(a_2 r-a_\beta r_2)}
{(H^2)(rd_2-r_2 d_\beta)} \right)^2+
\left(\frac{d_\beta}{r}-\frac{(a_2 r-a_\beta r_2)}
{(H^2)(rd_2-r_2 d_\beta)} \right)^2 -
\frac{\langle v^2 \rangle-(D^2)}{r^2 (H^2)}\right\}.
\end{split}
\end{equation*}
Hence the claim holds.
(2)
If $rd_2-r_2 d_\beta=0$, then the claim follows from
$$
(d_2-r_2 s)\widetilde{a}_\beta-(d_\beta-rs)\widetilde{a}_2=
-s^2 \frac{(H^2)}{2}(rd_2-r_2 d_\beta)+s (a_2 r-a_\beta r_2)+
(a_\beta d_2-a_2 d_\beta).
$$
\end{proof}
By Lemma \ref{lem:circle-center}, we get the following corollary.
\begin{cor}\label{cor:square}
\begin{enumerate}
\item[(1)]
If $
\frac{d_\beta}{r}-\frac{a_2 r-a_\beta r_2}{(H^2)(rd_2-r_2 d_\beta)}>0$,
then
$(s,t)=\left(\frac{d_\beta}{r}-
\sqrt{\frac{\langle v^2 \rangle-(D_\beta^2)}{(H^2) r^2}},0 \right)$
is contained in the circle \eqref{eq:circle}.
\item[(2)]
If $\frac{d_\beta}{r}-\frac{a_2 r-a_\beta r_2}{(H^2)(rd_2-r_2 d_\beta)}<0$,
then $(s,t)=
\left(\frac{d_\beta}{r}+
\sqrt{\tfrac{\langle v^2 \rangle-(D_\beta^2)}{(H^2) r^2}},0 \right)$
is contained in the circle \eqref{eq:circle}.
\end{enumerate}
\end{cor}
\begin{rem}\label{rem:circle-phase}
\begin{enumerate}
\item[(1)]
If $rd_2-r_2 d_\beta<0$, then
$(s,t)$ is surrounded by the circle in Proposition
\ref{prop:wall} if and only if
$\phi_{(\beta+sH,tH)}(v_2) \mod 2{\mathbb Z}$ satisfies
$\phi_{(\beta+sH,tH)}(v)+1>\phi_{(\beta+sH,tH)}(v_2)
>\phi_{(\beta+sH,tH)}(v)$.
\item[(2)]
If $rd_2-r_2 d_\beta>0$, then
$(s,t)$ is surrounded by the circle in Proposition
\ref{prop:wall} if and only if
$\phi_{(\beta+sH,tH)}(v_2) \mod 2{\mathbb Z}$ satisfies
$\phi_{(\beta+sH,tH)}(v)>\phi_{(\beta+sH,tH)}(v_2)
>\phi_{(\beta+sH,tH)}(v)-1$.
\end{enumerate}
\end{rem}
We next treat the case where $r=0$.
\begin{prop}\label{prop:wall:r=0}
Assume that $r=0$ and $\langle v^2 \rangle>0$.
If $rd_2-r_2 d_\beta \ne 0$, then
${\mathbb R}Z_{(\beta+sH,tH)}(v)={\mathbb R}Z_{(\beta+sH,tH)}(v_2)$
holds for $(s,t) \in {\mathbb R}^2$
if and only if
\begin{equation}\label{eq:circle:r=0}
t^2+\left( s-\frac{ a_\beta }{d_\beta (H^2)} \right)^2
=\left(\frac{a_\beta}{d_\beta (H^2)}-\frac{d_2}{r_2} \right)^2
-\frac{\langle v_2^2 \rangle- (D_2^2)}{r_2^2(H^2)}.
\end{equation}
\end{prop}
\begin{proof}
We note that $d_\beta \ne 0$.
Hence $r_2 \ne 0$.
Then the claim follows from the following computation:
\begin{equation*}
\begin{split}
& (d_2-r_2 s)\widetilde{a}_\beta-(d_\beta-rs)\widetilde{a}_2\\
=& -s^2 \frac{(H^2)}{2}(rd_2-r_2 d_\beta)+s (a_2 r-a_\beta r_2)+
(a_\beta d_2-a_2 d_\beta)\\
=& \frac{r_2 d_\beta (H^2)}{2}
\left(s^2-
s \frac{2 a_\beta }{d_\beta (H^2)}+
\frac{2(a_\beta d_2-a_2 d_\beta)}{(H^2) r_2 d_\beta}\right)\\
=& \frac{r_2 d_\beta (H^2)}{2}
\left\{ \left( s-\frac{ a_\beta }{d_\beta (H^2)} \right)^2
-\left(\frac{a_\beta}{d_\beta (H^2)}-\frac{d_2}{r_2} \right)^2
+\frac{\langle v_2^2 \rangle- (D_2^2)}{r_2^2(H^2)}
\right\}.
\end{split}
\end{equation*}
\end{proof}
\begin{cor}\label{cor:finite}
If
$\sqrt{\frac{\langle v^2 \rangle-(D_\beta^2)}{(H^2)}} \in {\mathbb Q}$,
then there are finitely many walls for $v$.
\end{cor}
\begin{proof}
For a fixed $s$,
there are finitely many walls.
If $r=0$, then
$(s,t)=(\frac{a_\beta}{d_\beta (H^2)},0)$ is the center of
\eqref{eq:circle:r=0}. Hence every wall intersects with
$s=\frac{a_\beta}{d_\beta (H^2)}$.
\begin{NB}
If $r=0$, then the assumption always holds.
\end{NB}
If $r \ne 0$,
then Corollary \ref{cor:square}
implies that every wall intersects with
$s=\frac{d_\beta}{r}-
\sqrt{\frac{\langle v^2 \rangle-(D_\beta^2)}{(H^2) r^2}}$
or
$s=\frac{d_\beta}{r}+
\sqrt{\frac{\langle v^2 \rangle-(D_\beta^2)}{(H^2) r^2}}$.
Hence there are finitely many walls for $v$.
\end{proof}
\begin{lem}\label{lem:C_v}
For $v_2=e^{\beta+\lambda H}$,
the condition
${\mathbb R}Z_{(\beta+sH,tH)}(v)={\mathbb R}Z_{(\beta+sH,tH)}(v_2)$
for $(s,t) \in {\mathbb R}^2$
is equivalent to the equation of the circle
$$
C_{v,\lambda}:
t^2+(s-\lambda)
\left(s-
\frac{1}{(r\lambda-d_\beta)}
\left(\lambda d_\beta-\frac{2}{(H^2)}a_\beta \right)
\right)=0
$$
for $r \lambda-d_\beta \ne 0$
and
$$
rs-d_\beta=0
$$
for $r \lambda-d_\beta= 0$.
In particular, the circle $C_{v,\lambda}$
passes the points $(\lambda,0)$
and
$\left(\frac{1}{(r\lambda-d_\beta)}
(\lambda d_\beta-\frac{2}{(H^2)}a_\beta),0 \right)$.
\end{lem}
\begin{proof}
We note that
\begin{equation*}
\begin{split}
v_2= & e^{\beta+\lambda H}=e^\beta+\lambda(H+(H,\beta)\varrho_X)+
\frac{(H^2)}{2}\lambda^2 \varrho_X \\
=& e^{\beta+sH+(\lambda-s) H}=
e^{\beta+sH}+(\lambda-s)(H+(H,\beta)\varrho_X)+
\frac{(H^2)}{2}(\lambda-s)^2 \varrho_X.
\end{split}
\end{equation*}
Then we get
\begin{equation}\label{eq:lambda}
\begin{split}
(d_2-r_2 s)\widetilde{a}_\beta-(d_\beta-rs)\widetilde{a}_2
=& (\lambda-s)\widetilde{a}_\beta-(d_\beta-rs)\frac{(H^2)}{2}(\lambda-s)^2 \\
=& (\lambda-s)(\widetilde{a}_\beta-(\lambda-s)(d_\beta-rs)\frac{(H^2)}{2})\\
=& (\lambda-s) \left((r\lambda-d_\beta)\frac{(H^2)}{2}s-
\left(\lambda d_\beta\frac{(H^2)}{2}-a_\beta \right) \right).
\end{split}
\end{equation}
Assume that $r \lambda-d_\beta \ne 0$.
Then the condition is given by the circle
$$
t^2+(s-\lambda)
\left(s-
\frac{1}{(r\lambda-d_\beta)}\left(\lambda d_\beta-\frac{2}{(H^2)}a_\beta
\right) \right)=0.
$$
In particular, the circle passes the points $(\lambda,0)$
and $\left(\frac{1}{(r\lambda-d_\beta)}
(\lambda d_\beta-\frac{2}{(H^2)}a_\beta),0 \right)$.
Assume that $r \lambda-d_\beta=0$.
Then by \eqref{eq:lambda},
we get
$$
0=(\lambda-s)\left(\lambda d_\beta \frac{(H^2)}{2}-a_\beta \right).
$$
If $a_\beta=\lambda d_\beta \frac{(H^2)}{2}=r\lambda^2 \frac{(H^2)}{2}$,
then we see that
$v=r e^{\beta+\lambda H}+(D_\beta+(D_\beta,\beta+\lambda H))\varrho_X$.
Hence $\langle v^2 \rangle=(D_\beta^2) \leq 0$, which is a contradiction.
Therefore we get
$$
s=\lambda=\frac{d_\beta}{r}.
$$
\end{proof}
\begin{rem}
Assume that $r \ne 0$.
Then
$$
\lambda \ne \frac{1}{r\lambda-d_\beta}
\left(\lambda d_\beta-\frac{2}{(H^2)}a_\beta \right)
\Longleftrightarrow
\lambda \ne
\frac{d_\beta}{r}\pm\sqrt{\frac{\langle v^2 \rangle-(D_\beta^2)}{(H^2)r^2}}.
$$
In particular, if
$\sqrt{\frac{\langle v^2 \rangle-(D_\beta^2)}{(H^2)r^2}} \not \in {\mathbb Q}$,
then $C_{v,\lambda}$ is a circle.
\end{rem}
\begin{NB}
Assume that $v=\ell v_1-v_2$. Then $\mu(v_1)>\mu(v_2)$
or
$\mu(v_1)<\mu(v_2)$.
Assume that $(s,t)$ belongs to the circle
defined by $v_1$,
i.e.,
$s-\frac{d_\beta}{r}$
belongs to the interval whose boundaries
are $\lambda-\frac{d_\beta}{r}$ and
$\frac{\ell}{r n(r\lambda-d_\beta)}$.
If $\mu(v_1)>\mu(v_2)$, then
$Z_{(sH,tH)}(v) \in {\mathbb H}$ for
$d_{\beta+sH}(v_1)>0$ and
if $\mu(v_1)<\mu(v_2)$, then
$Z_{(sH,tH)}(-v) \in {\mathbb H}$ for
$d_{\beta+sH}(v_1)<0$.
\end{NB}
\begin{cor}\label{cor:C_v}
Assume that $v=r e^\beta+a_\beta \varrho_X+d_\beta (H+(H,\beta)\varrho_X)$.
For a numerical solution
$$
(r_1 e^{\beta+\lambda_1 H},r_2 e^{\beta+\lambda_2 H},\ell_1,\ell_2),
$$
we have $C_{v,\lambda_1}=C_{v,\lambda_2}$ and the equation is given by
\begin{equation}
t^2+(s-\lambda_1)(s-\lambda_2)=0.
\end{equation}
\end{cor}
\begin{proof}
Since $v=\pm(\ell_1 r_1 e^{\beta+\lambda_1}-\ell_2 r_2 e^{\beta+\lambda_2})$,
we have $C_{v,\lambda_1}=C_{v,\lambda_2}$.
Since $(\lambda_i,0) \in C_{v,\lambda_i}$ for $i=1,2$
and $\lambda_1 \ne \lambda_2$,
we get the claim.
\end{proof}
\subsection{Relation of stability conditions.}
All walls except $rs=d_\beta$ are disjoint to the line
$rs=d_\beta$.
By Corollary \ref{cor:square},
there are at most two unbounded chambers.
\begin{prop}\label{prop:unbdd}
Let $v$ be a positive and primitive Mukai vector such that
$\langle v^2 \rangle>0$.
Assume that $(s,t) \in {\mathbb R}^2$ belongs to an unbounded chamber.
Then
$$
M_{(\beta+sH,tH)}(v) \cong
\begin{cases}
M_H^\beta(v) & d_{\beta+sH}(v) >0,\\
M_H^{-\beta}(v^{\vee}) & d_{\beta+sH}(v) \leq 0.
\end{cases}
$$
\end{prop}
\begin{proof}
If $d_{\beta+sH}(v)>0$, then
$\operatorname{rk} v \geq 0$.
By \cite[Cor. 2.2.9]{MYY:2011:1}, we get
$M_{(\beta+sH,tH)}(v)=M_H^\beta(v)$.
If $d_\beta(v) \leq 0$,
then $d_{\beta+sH}(-v) \geq 0$ and $\operatorname{rk} (-v) \leq 0$.
If $\operatorname{rk} (-v)=0$, then $d_\beta(-v)=0$ and
$\langle v^2 \rangle \leq 0$,
which is a contradiction.
Thus we have $\operatorname{rk} (-v)< 0$.
Then \cite[Cor. 2.2.9]{MYY:2011:1} implies that we have an isomorphism
$M_H^{-\beta}(v^{\vee}) \to M_{(\beta+sH,tH)}(v)$
via $F \mapsto F^{\vee}$.
\end{proof}
\begin{lem}\label{lem:2-stability}
Let ${\mathcal C}_0$ and ${\mathcal C}_1$ be two chambers such that
${\mathcal C}_0$ is surrounded by ${\mathcal C}_1$.
We take $(s,t_0) \in {\mathcal C}_0$ and $(s,t_1) \in {\mathcal C}_1$.
Let $t_x:=(1-x) t_0+ x t_1$
($0 \leq x \leq 1$) be a segment connecting
$t_0$ and $t_1$.
If $E$ is $\sigma_{(\beta+s H,t_i H)}$-semi-stable
for $i=0,1$, then
$E$ is $\sigma_{(\beta+sH,t_x H)}$-semi-stable
for all $x$.
\end{lem}
\begin{proof}
Assume that $E$ is not $\sigma_{(\beta+sH,t_x H)}$-semi-stable
for some $x \in (0,1)$.
Then there is a subobject $E_1$ of $E$ in
${\mathfrak A}_{(\beta+sH,t_x H)}$
such that $\phi_{(\beta+sH,t_x H)}(E_1)>
\phi_{(\beta+sH,t_x H)}(E)$.
Since $E$ is $\sigma_{(\beta+s H,t_i H)}$-semi-stable
for $i=0,1$,
$\phi_{(\beta+sH,t_i H)}(E_1) \leq
\phi_{(\beta+sH,t_i H)}(E)$.
Then there are two numbers $x_1, x_2$ such that
$0<x_1,x_2<1$ and $\phi_{(\beta+sH,t_{x_i} H)}(E_1)=
\phi_{(\beta+sH,t_{x_i} H)}(E)$.
Since $t_{x_i}$ is uniquely determined by $v(E_2)$,
this does not occur.
Therefore $E$ is
$\sigma_{(\beta+sH,t_x H)}$-semi-stable
for all $x$.
\end{proof}
\begin{defn}
Let $W$ be a wall for $v$ in $(s,t)$-plane.
Let $(\beta,\omega)$ be a point of $W$ and
$(\beta',\omega')$ be a point in an adjacent chamber.
Then we define \emph{the codimension of the wall} $W$ by
\begin{equation}
\operatorname{codim} W:=\min_{v=\sum_i v_i } \left \{ \sum_{i<j}\langle v_i,v_j \rangle
-\left(\sum_i (\dim {\mathcal M}_H^{\beta'}(v_i)^{ss}-
\langle v_i^2 \rangle) \right)
+1 \right\},
\end{equation}
where $v=\sum_i v_i$ are decompositions of $v$
such that $\phi_{(\beta,\omega)}(v)=\phi_{(\beta,\omega)}(v_i)$
and $\phi_{(\beta',\omega')}(v_i)>\phi_{(\beta',\omega')}(v_j)$,
$i<j$.
\end{defn}
By using \cite[Lem. 4.2.4, Rem. 4.2.3]{MYY:2011:1},
we get the following result.
\begin{lem}\label{lem:class-codim0wall}
If $W$ is a codimension 0 wall, then
$W$ is defined by $v_1$ such that
\begin{equation}\label{eq:codim=0}
v=n v_1+v_2,\; \langle v_1,v_2 \rangle=1,\;
\langle v_1^2 \rangle=\langle v_2^2 \rangle=0.
\end{equation}
\end{lem}
If $v_1,v_2$ in \eqref{eq:codim=0} satisfy
$v_1>0,v_2<0$ or $v_1<0$ and $v_2>0$, then $(v_1,-v_2,n,1)$
or $(-v_1,v_2,n,1)$ gives a numerical solution of $v$.
Conversely for a numerical solution $(v_1,v_2,\ell_1,\ell_2)$,
Lemma \ref{lem:num.sol:stability} implies that
for a suitable $\beta$ and $\omega=tH$,
$v_1$ defines a codimension 0 wall.
\begin{prop}
Assume that $\operatorname{NS}(X)={\mathbb Z}H$.
We fix $\beta$.
Then there is a bijective correspondence
between a codimension 0 wall and a numerical solution.
\end{prop}
\begin{proof}
Assume that $\operatorname{NS}(X)={\mathbb Z}H$.
For a numerical solution $(v_1,v_2,\ell_1,\ell_2)$,
$\beta, c_1(v),c_1(v_1),c_1(v_2) \in {\mathbb Q}H$ implies that
$C_{v,\lambda_1}$ in Corollary \ref{cor:C_v}
gives a wall in the $(s,t)$-plane.
Combining Lemma \ref{lem:class-codim0wall},
we get the claim.
\end{proof}
\begin{prop}[{cf. \cite[Prop. 4.2.5]{MYY:2011:1}}]
If $(\beta_1,\omega_1)$ and $(\beta_2,\omega_2)$ are not separated by
any codimension 0 wall, then
$M_{(\beta_1,\omega_1)}(v) \cap M_{(\beta_2,\omega_2)}(v) \ne \emptyset$.
In particular,
$M_{(\beta_1,\omega_1)}(v)$ and $M_{(\beta_2,\omega_2)}(v)$
are birationally equivalent.
\end{prop}
\subsection{Semi-homogeneous presentation and the stability.}
\label{subsect:semi-hom-stab}
For a semi-homogeneous presentation,
Lemma \ref{lem:num.sol:stability} implies that
we can relate a $\sigma_{(\beta,\omega)}$-semi-stability.
We shall study Gieseker semi-stability
of coherent sheaves with two semi-homogeneous presentations.
\begin{lem}\label{lem:s.h-ss}
Let $v$ be a primitive Mukai vector with $r:=\operatorname{rk} v>0$.
Assume that
${\mathcal M}_H^\beta(v)^{ss}$ consists of $\beta$-stable sheaves.
If a simple sheaf $E$ with $v(E)=v$
has two semi-homogeneous presentations
and $d_\beta(v)-r s=0$ is not a codimension 0 wall,
then $E$ is a $\mu$-stable vector bundle.
\end{lem}
\begin{proof}
Since $rs-d_\beta(v)=0$ is not a codimension 0 wall,
two semi-homogeneous presentations
define two circles $C_1$ and $C_2$
which are separated by the line
$rs-d_\beta(v)=0$.
There is a Fourier-Mukai transform $\Phi:=\Phi_{Y \to X}^{{\bf E}}$
and a complex $F$ such that $E=\Phi(F)$ and
$F$ is semi-stable
with respect to two chambers ${\mathcal C}_0$, ${\mathcal C}_1$ such that
${\mathcal C}_0$ is surrounded by ${\mathcal C}_1$
and ${\mathcal C}_i$ are adjacent to $C_i$.
Let ${\mathcal C}$ be an unbounded chamber between $\Phi({\mathcal C}_1)$ and
$\Phi({\mathcal C}_2)$. Then
$F$ is semi-stable with respect to
$\Phi^{-1}({\mathcal C})$ by Lemma \ref{lem:2-stability}.
Hence $E$ is semi-stable with respect to all unbounded
chambers.
We take an element $\omega \in \operatorname{Amp}(X)_{\mathbb Q}$.
Then $E$ is $\sigma_{(\beta+sH,\omega)}$-semi-stable
if $1 \gg d_\beta(v)-rs>0$.
Hence $E$ is $\beta$-twisted semi-stable.
Since $E$ is also $\sigma_{(\beta+sH,\omega)}$-semi-stable
if $1 \gg -(d_\beta(v)-rs)>0$,
$E^{\vee}$ is $(-\beta)$-twisted semi-stable.
Hence $E$ is locally free.
Let $E_1$ be a locally free subsheaf of $E$.
Then there is a generically surjective homomorphism
$E^{\vee} \to E_1^{\vee}$.
Since
$$
\frac{\chi(E_1(-\beta))}{\operatorname{rk} E_1}=
\frac{\chi(E_1^{\vee}(\beta))}{\operatorname{rk} E_1^{\vee}},
$$
$\frac{d_\beta(E_1)}{\operatorname{rk} E_1}=
\frac{d_\beta(E)}{\operatorname{rk} E}$ implies
$\frac{\chi(E_1(-\beta))}{\operatorname{rk} E_1}=
\frac{\chi(E(-\beta))}{\operatorname{rk} E}$.
Thus $E$ is properly $\beta$-twisted semi-stable,
which is a contradiction.
Therefore $E$ is $\mu$-stable.
\end{proof}
\begin{lem}\label{lem:codim0-line}
Let $v$ be a primitive Mukai vector with $r:=\operatorname{rk} v>0$.
Assume that $d_\beta(v)-r s=0$ defines a wall.
\begin{enumerate}
\item[(1)]
$d_\beta(v)-r s=0$ is a codimension 0 wall if and only if
(a)
$v=r e^{\xi}-a \varrho_X$, $\xi \in \operatorname{NS}(X), a \in {\mathbb Z}$,
$(r-1)(a-1)=0$
or (b)
$v=v_1+\ell v_2$,
$v_i=r_i e^{\frac{\xi_i}{r_i}}$,
$r_1,r_2>0$,
$((r_2 \xi_1-r_1 \xi_2)^2)=-r_1 r_2$,
$(r_2 \xi_1-r_1 \xi_2,H)=0$.
\item[(2)]
Assume that $v$ satisfies (a).
We take $(s,t)$ such that $1 \gg d_\beta(v)-rs>0$
and let $E$ be a $\sigma_{(\beta+sH,tH)}$-semi-stable object with
$v(E)=v$. Then
$r=1$ and $E=I_Z(\xi)$ or
$a=1$ and $E=\ker(\oplus_{i=1}^r {\mathcal O}_X(\xi_i) \to {\mathfrak k}_x)$.
\item[(3)]
If $\operatorname{NS}(X)={\mathbb Z}H$, then (b) does not occur.
\end{enumerate}
\end{lem}
\begin{proof}
(1), (3)
By Lemma \ref{lem:class-codim0wall},
we have $v=v_1+\ell v_2$,
$\langle v_i^2 \rangle=0$ and $\langle v_1,v_2 \rangle=1$,
where $\ell=\langle v^2 \rangle/2$.
Assume that $\operatorname{rk} v_1 \operatorname{rk} v_2 \ne 0$.
Then we can set
$v_i:=r_i e^{\frac{\xi_i}{r_i}}$
$((r_2 \xi_1-r_1 \xi_2)^2)=-r_1 r_2$.
Since the wall is $d_\beta-rs=0$,
we have $(r_2 \xi_1-r_1 \xi_2,H)=0$.
By the Hodge index theorem,
$r_1 r_2>0$.
Thus $r_1, r_2>0$.
If $\operatorname{rk} v_1=0$ or $\operatorname{rk} v_2=0$,
then $r_2 d_\beta(v_1)-r_1 d_\beta(v_2)=0$ and
$\langle v_1,v_2\rangle=1$ implies that
$\{v_1, v_2 \}=\{(0,0,1),(-1,0,0) \}$.
Therefore (1) holds.
If $\operatorname{NS}(X)={\mathbb Z}H$ and $(r_2 \xi_1-r_1 \xi_2,H)=0$,
then we have $r_2 \xi_1-r_1 \xi_2=0$.
Hence $r_1 r_2=0$. Thus (b) does not occur.
(2)
By the choice of $(s,t)$,
$\sigma_{(\beta+sH,tH)}$-semi-stability
implies $\beta$-twisted semi-stability.
Thus for $E \in M_{(\beta+sH,tH)}(v)$,
we have
a semi-homogeneous presentation
$$
0 \to E \to E_0 \to E_1 \to 0,
$$
where $(v(E_0),v(E_1))=(e^\xi,\ell \varrho_X)$
or $(v(E_0),v(E_1))=(\ell e^\xi, \varrho_X)$
with $\langle v^2 \rangle=2\ell$.
Hence the claim holds.
\end{proof}
\begin{prop}
Let $X$ be an abelian surface with $\operatorname{NS}(X)={\mathbb Z}H$.
Let $E$ be a simple sheaf on $X$.
If $E$ has two semi-homogeneous presentations,
then $E$ is Gieseker semi-stable.
\end{prop}
\begin{proof}
If $d_\beta(v)-rs=0$ is a codimension 0 wall,
then Lemma \ref{lem:codim0-line} implies the claim.
If $d_\beta(v)-rs=0$ is not a codimension 0 wall,
then the claim follows from Lemma \ref{lem:s.h-ss}.
\end{proof}
\subsection{Relation with the Fourier-Mukai transforms.}
We shall study the Fourier-Mukai transform
on our space of stability conditions.
Let $r_1 e^\gamma$ be a primitive and isotropic Mukai vector.
We set $X_1:=M_{(\beta+sH,tH)}(r_1 e^{\gamma})$.
Let
${\bf E}$ be the universal object on $X \times X_1$
as a complex of twisted sheaves.
Assume that $\gamma=\beta+\lambda H$, $\lambda \in {\mathbb Q}$.
We consider
the Fourier-Mukai transform
$\Phi:=\Phi_{X \to X_1}^{{\bf E}^{\vee}[1]}:
{\bf D}(X) \to {\bf D}^{\alpha_1}(X_1)$.
We set $(\widetilde{\beta+sH},\widetilde{tH})=
(\gamma'+s' \widehat{H},t' \widehat{H})$.
Then
\begin{equation}\label{eq:s'}
\begin{split}
s'=&
\frac{1}{|r_1|}
\frac{2(\lambda-s)}{((\lambda-s)^2+t^2) (H^2)},\\
t'=&
\frac{1}{|r_1|}
\frac{2 t}{((\lambda-s)^2+t^2) (H^2)}.\\
\end{split}
\end{equation}
Since $((s-\lambda)^2+t^2)({s'}^2+{t'}^2)=
\left(\frac{2}{|r_1|(H^2)} \right)^2$,
the image of $(s-\lambda)^2+t^2=\frac{2}{|r_1|(H^2)}$
is ${s'}^2+{t'}^2=\frac{2}{|r_1|(H^2)}$.
\begin{NB}
If we fix a base point $\beta'$
and write $\beta'=\gamma'-\lambda' \widehat{H}$,
then $\beta'+s' \widehat{H}=\gamma'+(s'-\lambda')\widehat{H}$
and the relation of
$Z_{(\beta+sH,tH)}$ and $Z_{(\beta'+s' \widehat{H},t' \widehat{H})}$
is
$$
s'-\lambda'=\frac{1}{|r_1|}
\frac{2(\lambda-s)}{((\lambda-s)^2+t^2) (H^2)},\;
t'=\frac{1}{|r_1|}
\frac{2t}{((\lambda-s)^2+t^2) (H^2)}.
$$
\end{NB}
If $\lambda r \ne d_\beta$,
then Lemma \ref{lem:C_v}
implies that
the condition
${\mathbb R}Z_{(\beta+sH,tH)}(v)=
{\mathbb R}Z_{(\beta+sH,tH)}(e^{\beta+\lambda H})$
defines a circle
\begin{equation}\label{eq:circle-lambda}
C_{v,\lambda}:\;
t^2=
(\lambda-s)
\left( \frac{a_\beta-d_\beta \lambda \frac{(H^2)}{2}}{\lambda r-d_\beta}
\frac{2}{(H^2)}+s \right).
\end{equation}
We have
\begin{equation}\label{eq:a_gamma}
\frac{a_\beta-d_\beta \lambda \frac{(H^2)}{2}}{\lambda r-d_\beta}
\frac{2}{(H^2)}+\lambda
=\frac{(\lambda r-d_\beta)^2 (H^2)-
(\langle v^2 \rangle-(D_\beta^2))}
{r(H^2)(\lambda r-d_\beta)}
=\frac{2a_\gamma}{-d_\gamma (H^2)}=
\frac{2\langle e^{\beta+\lambda H},v \rangle}
{d_\gamma (H^2)}.
\end{equation}
Thus
$C_{v,\lambda}$is
$$
\left(s-\left(\lambda+\frac{a_\gamma}{d_\gamma (H^2)} \right)\right)^2+t^2
=\left(\frac{a_\gamma}{d_\gamma (H^2)} \right)^2.
$$
\begin{lem}\label{lem:circle-lambda-interior}
The image of
\begin{equation}\label{eq:circle-lambda-interior}
t^2 \leq
(\lambda-s)
\left( \frac{a_\beta-d_\beta \lambda \frac{(H^2)}{2}}{\lambda r-d_\beta}
\frac{2}{(H^2)}+s \right)
\end{equation}
by $\Phi$ is
$$
\left \{(s',t') \left| -\tfrac{|r_1| a_\gamma}{d_\gamma}s' \geq 1
\right. \right\}.
$$
\end{lem}
\begin{proof}
By \eqref{eq:a_gamma} and \eqref{eq:s'},
$(s,t)$ satisfies \eqref{eq:circle-lambda-interior}
if and only if
\begin{equation}
\begin{split}
0 \geq & (s-\lambda)^2+t^2-(s-\lambda)\frac{2a_\gamma}{d_\gamma (H^2)} \\
=& \left((s-\lambda)^2+t^2 \right)\left(
1+s' |r_1| \frac{a_\gamma}{d_\gamma} \right).
\end{split}
\end{equation}
Hence the claim holds.
\end{proof}
By Lemma \ref{lem:circle-lambda-interior},
we have the following.
\begin{prop}\label{prop:Phi(C^pm)}
Let ${\mathcal C}^\pm$ be the adjacent chamber of
$C_{v,\lambda}$ such that ${\mathcal C}^-$ is surrounded by
${\mathcal C}^+$.
\begin{enumerate}
\item[(1)]
If $\frac{a_\gamma}{d_\gamma}<0$,
then $\Phi({\mathcal C}^+)$ (resp. $\Phi({\mathcal C}^-)$)
is the unbounded
chamber satisfying
$s' < -\frac{d_\gamma}{|r_1| a_\gamma}$
(resp. $s' > -\frac{d_\gamma}{|r_1| a_\gamma}$).
\item[(2)]
If $\frac{a_\gamma}{d_\gamma}>0$,
then $\Phi({\mathcal C}^-)$ (resp.
$\Phi({\mathcal C}^+)$)
is the unbounded
chamber satisfying
$s' < -\frac{d_\gamma}{|r_1| a_\gamma}$
(resp. $s' > -\frac{d_\gamma}{|r_1| a_\gamma}$).
\end{enumerate}
\end{prop}
\begin{NB}
$\frac{d_\gamma}{|r_1| a_\gamma}=
\frac{d_{\gamma'} (\Phi(v))}{-\operatorname{rk} \Phi(v)}$.
\end{NB}
\begin{NB}
We note that $rs=d_\beta$ does not meet $C_{v,\lambda}$.
Assume that
$d_\gamma=d_\beta-r \lambda>0$.
Then $d_{\beta+sH}(e^\gamma)=\lambda-s \geq 0$ for
$(s,t) \in C_{v,\lambda}$ and $a_\gamma<0$.
We also have $r_1>0$, since we require
$d_{\beta+sH}(r_1 e^\gamma)>0$.
Since $\operatorname{rk} \Phi(v)=-r_1 a_\gamma>0$,
For $E \in M_{(\beta+sH,tH)}(v)$, $(s,t) \in {\mathcal C}^+$,
$\Phi(E)$ is a twisted stable sheaf.
\end{NB}
\begin{NB}
Assume that
$d_\gamma=d_\beta-r \lambda>0$.
Then $d_{\beta+sH}(e^\gamma)=\lambda-s \leq 0$ for
$(s,t) \in C_{v,\lambda}$ and $a_\gamma>0$.
We also have $r_1<0$, since we require
$d_{\beta+sH}(r_1 e^\gamma)>0$.
Since $\operatorname{rk} \Phi(v)=-r_1 a_\gamma>0$,
For $E \in M_{(\beta+sH,tH)}(v)$, $(s,t) \in {\mathcal C}^-$,
$\Phi(E)$ is a twisted stable sheaf.
\end{NB}
\begin{NB}
Wrong arguments (Feb. 18, 2012):
Then the intersection of $C_{v,\lambda}$ with the
circle
$(s-\lambda)^2+t^2=\frac{2}{|r_1|(H^2)}$
lies on the line
\begin{equation}\label{eq:line}
s=\lambda+\frac{d_\gamma}{a_\gamma |r_1|}.
\end{equation}
By $\Phi$,
the circle is transformed to
the line
\begin{equation}
s'=\lambda+\frac{d_\gamma}{a_\gamma |r_1|}.
\end{equation}
\end{NB}
\begin{NB}
$$
s'=\Phi_{X \to Y}^{{\bf E}^{\vee}[1]}(s)=
\lambda+\frac{2}{(H^2)|r_1|}
\frac{s-\lambda}{(s-\lambda)^2+t^2}.
$$
\end{NB}
We set $w:=\Phi_{X \to X_1}^{{\bf E}^{\vee}[1]}(v)$.
Proposition \ref{prop:unbdd} implies
$M_{(\gamma'+s'\widehat{H},t' \widehat{H}) }^{\alpha_1}(w)$
is isomorphic to the moduli space of
semi-stable sheaves.
Then Theorem \ref{thm:B:3-10.3} implies
a generalization of \cite[Thm. 3.3.3]{MYY:2011:2}
for abelian surfaces.
For the preservation of Gieseker's semi-stability, we also
have the following, which is a generalization of
\cite{Stability}.
\begin{prop}\label{prop:asymptotic}
Let $v$ be a positive Mukai vector and
assume that there are walls for $v$.
Let $W^{\max}$ be the wall in the region $rs<d_\beta$ such that
$W^{\max}$ surround all walls in $rs<d_\beta$, that is,
the boundaries of the unbounded chamber is $rs=d_\beta$ and
$W^{\max}$.
We set
$$
W^{\max} \cap \{(s,0) \mid s \in {\mathbb R} \}
=\{(\lambda_1,0), (\lambda_2,0) \},\quad
\lambda_1<\lambda_2.
$$
Let $\Phi:{\bf D}(X) \to {\bf D}(X_1)$ be the Fourier-Mukai
transform as above.
If $\lambda \leq \lambda_1$ or
$\lambda_2 \leq
\lambda<\frac{d_\beta}{r}$, then
$\Phi$ or $\Phi \circ {\mathcal D}_X$
preserves the Gieseker's semi-stability.
\end{prop}
In particular, if $\lambda$ is sufficiently small, we can apply
this proposition, which is nothing but the main result of
\cite{Stability}.
\begin{rem}
Assume that $r:=\operatorname{rk} v>0$.
We set
$$
D:=\min \{((\operatorname{rk} w) c_1(v)-(\operatorname{rk} v) c_1(w),H)>0 \mid
w \in H^*(X,{\mathbb Z})_{\operatorname{alg}} \}
$$
and assume that
$$
D=\min \{(C,H)>0 \mid C \in \operatorname{NS}(X) \}.
$$
We take
$w_0 \in H^*(X,{\mathbb Z})_{\operatorname{alg}}$ such that
\begin{equation*}
((\operatorname{rk} w_0) d_\beta(v)-(\operatorname{rk} v) d_\beta(w_0))(H^2)=
((\operatorname{rk} w_0) c_1(v)-(\operatorname{rk} v) c_1(w_0),H)=D.
\end{equation*}
Replacing $w_0$ by $w_0+k v$ $(k \in {\mathbb Z})$,
we may assume that $\operatorname{rk} v \geq \operatorname{rk} w_0>0$.
Then there is no wall for
$d_\beta(w_0)/\operatorname{rk} w_0 \leq s< d_\beta(v)/\operatorname{rk} v$.
\end{rem}
\section{
The chamber structure for an abelian surface $X$ with
$\operatorname{NS}(X)={\mathbb Z}H$.}
\label{sect:rho=1}
From now on, we assume that $\operatorname{NS}(X)={\mathbb Z}H$.
Let $v$ be a primitive Mukai vector with a numerical solution.
We shall study the walls and chambers for $v$.
By our assumption,
there is an isometry of Mukai lattice sending $v$ to $1-\ell \varrho_X$.
So we may assume that $v=1-\ell \varrho_X$.
Since a generic classification of stable objects
(it induces the birational classification)
is most fundamental,
we are mainly interested in codimension 0 walls.
\subsection{Cohomological Fourier-Mukai transforms}
Let $H_X$ be the ample generator of $\operatorname{NS}(X)$.
We shall describe the action of Fourier-Mukai transforms
on the cohomology lattices in \cite{YY}.
Two smooth projective varieties $Y_1$ and $Y_2$ are said to be
Fourier-Mukai partners if there is an equivalence
$\bl{D}(Y_1)\simeq\bl{D}(Y_2)$.
We denote by $\FM(X)$ the set of Fourier-Mukai partners of $X$.
The set of equivalences between $\bl{D}(X)$ and $\bl{D}(Y)$
is denoted by $\Eq(\bl{D}(X),\bl{D}(Y))$.
For $Y,Z\in\FM(X)$, we set
\begin{align*}
&\Eq_0(\bl{D}(Y),\bl{D}(Z))
:=
\{\Phi_{Y \to Z}^{\bl{E}[2k]}
\in \Eq(\bl{D}(Y),\bl{D}(Z))
\mid \bl{E} \in \operatorname{Coh}(Y \times Z),\, k \in {\bb{Z}}
\},
\\
&\mathcal{E}(Z)
:=
\bigcup_{Y\in\FM(Z)}\Eq_0(\bl{D}(Y),\bl{D}(Z)),
\qquad
\mathcal{E}
:=
\bigcup_{Z\in\FM(X)}\mathcal{E}(Z)
=\bigcup_{Y,Z\in\FM(X)}\Eq_0(\bl{D}(Y),\bl{D}(Z)).
\end{align*}
Note that $\mathcal{E}$ is a groupoid with respect to the composition
of the equivalences.
For $Y \in \FM(X)$, we have $(H_Y^2)=(H_X^2)$.
We set $n:= (H^2_X)/2$.
In \cite[sect. 6.4]{YY}, we constructed an isomorphism of lattices
\begin{align*}
\iota_X:
(H^*(X,\bb{Z})_{\operatorname{alg}},\langle\cdot,\cdot\rangle)
\xrightarrow{\ \sim\ }
(\operatorname{Sym}_2(\bb{Z}, n),B),
\quad
(r,dH_X,a)
\mapsto
\begin{pmatrix}
r & d\sqrt{n} \\ d\sqrt{n} & a
\end{pmatrix},
\end{align*}
where $\operatorname{Sym}_2(\bb{Z}, n)$ is given by
\begin{align*}
\operatorname{Sym}_2(\bb{Z},n):=
\left\{\begin{pmatrix} x &y \sqrt{n} \\ y\sqrt{n}&z\end{pmatrix}\,
\Bigg|\, x,y,z\in \bb{Z}\right\},
\end{align*}
and the bilinear form $B$ on $\operatorname{Sym}_2(\bb{Z},n)$ is given by
\begin{align*}
B(X_1,X_2) := 2ny_1 y_2-(x_1 z_2+z_1 x_2)
\end{align*}
for
$X_i =\begin{pmatrix}x_i & y_i \sqrt{n} \\ y_i \sqrt{n} &z_i \end{pmatrix}
\in\operatorname{Sym}_2(\bb{Z},n)$
($i=1,2$).
Each $\Phi_{X \to Y}$ gives an isometry
\begin{align}
\iota_{Y} \circ \Phi^H_{X \to Y} \circ \iota_{X}^{-1}
\in \operatorname{O}(\operatorname{Sym}_2(\bb{Z}, n)),
\end{align}
where $\operatorname{O}(\operatorname{Sym}_2(\bb{Z}, n))$ is the isometry group
of the lattice $(\operatorname{Sym}_2(\bb{Z}, n),B)$.
Thus we have a map
$$
\eta:
{\mathcal E} \to \operatorname{O}(\operatorname{Sym}_2(\bb{Z}, n))
$$
which preserves the structures of
multiplications.
\begin{defn}\label{defn:G}
We set
\begin{align*}
&\widehat{G} :=
\left\{
\begin{pmatrix} a \sqrt{r} & b \sqrt{s}\\
c \sqrt{s} & d \sqrt{r}
\end{pmatrix}
\Bigg|\,
\begin{aligned}
a,b,c,d,r,s \in \bb{Z},\, r,s>0\\
rs=n, \, adr-bcs = \pm1
\end{aligned}
\right\},
\\
&G := \widehat{G}\cap \SL(2,\bb{R}).
\end{align*}
\end{defn}
We have a right action $\cdot$ of $\widehat{G}$ on the lattice
$(\operatorname{Sym}_2(\bb{Z}, n),B)$:
\begin{align}
\label{eq:action_cdot}
\begin{pmatrix}
r &d \sqrt{n}\\d \sqrt{n}&a \end{pmatrix}
\cdot g :=
{}^t g
\begin{pmatrix}r &d \sqrt{n}\\d \sqrt{n}&a \end{pmatrix}
g,\;
g \in \widehat{G}.
\end{align}
Thus we have an anti-homomorphism:
$$
\alpha:\widehat{G}/\{\pm 1\} \to \operatorname{O}(\operatorname{Sym}_2(n,{\mathbb Z})).
$$
\begin{thm}[{\cite[Thm. 6.16, Prop. 6.19]{YY}}]
Let
$\Phi \in \Eq_0(\bl{D}(Y),\bl{D}(X))$ be an equivalence.
\begin{enumerate}
\item[(1)]
$v_1:=v(\Phi({\mathcal O}_Y))$ and $v_2:=\Phi(\varrho_Y)$
are positive isotropic Mukai vectors with
$\langle v_1,v_2 \rangle=-1$ and we can write
\begin{equation}
\begin{split}
& v_1=(p_1^2 r_1,p_1 q_1 H_Y, q_1^2 r_2),\quad
v_2=(p_2^2 r_2,p_2 q_2 H_Y, q_2^2 r_1),\\
& p_1,q_1,p_2,q_2, r_1, r_2 \in {\mathbb Z},\;\;
p_1,r_1,r_2 >0,\\
& r_1 r_2=n,\;\; p_1 q_2 r_1-p_2 q_1 r_2=1.
\end{split}
\end{equation}
\item[(2)]
We set
\begin{equation}
\theta(\Phi):=\pm
\begin{pmatrix}
p_1 \sqrt{r_1} & q_1 \sqrt{r_2}\\
p_2 \sqrt{r_2} & q_2 \sqrt{r_1}
\end{pmatrix}
\in G/\{\pm 1\}.
\end{equation}
Then $\theta(\Phi)$ is uniquely determined by
$\Phi$ and
we have a map
\begin{equation}
\theta:{\mathcal E} \to G/\{\pm 1\}.
\end{equation}
\item[(3)]
The action of $\theta(\Phi)$ on $\operatorname{Sym}_2(n,{\mathbb Z})$
is the action of $\Phi$ on $\operatorname{Sym}_2(n,{\mathbb Z})$:
\begin{equation}
\iota_X \circ \Phi(v)
=\iota_Y(v)\cdot
\theta(\Phi).
\end{equation}
\begin{NB}
\begin{equation}
\iota_X \Phi\iota_Y^{-1}
\begin{pmatrix}r &d \sqrt{n}\\d \sqrt{n}&a \end{pmatrix}
=
{}^t (\theta(\Phi))
\begin{pmatrix}r &d \sqrt{n}\\d \sqrt{n}&a \end{pmatrix}
\theta(\Phi)
\end{equation}
\end{NB}
Thus we have the following commutative diagram:
\begin{align}
\label{diag:groups}
\xymatrix{
{\mathcal E} \ar[d]_-{\theta} \ar[dr]_{\eta}
&
\\
\widehat{G}/\{\pm 1\} \ar[r]_-{\alpha}
&
\operatorname{O}(\operatorname{Sym}_2(n,{\mathbb Z}))
}
\end{align}
\end{enumerate}
\end{thm}
From now on, we identify the Mukai lattice
$H^*(X,{\mathbb Z})_{\operatorname{alg}}$ with $\operatorname{Sym}_2(n,{\mathbb Z})$
via $\iota_X$.
Then for $g \in \widehat{G}$ and $v \in H^*(X,{\mathbb Z})_{\operatorname{alg}}$,
$v \cdot g$ means
$\iota_X(v \cdot g)=\iota_X(v) \cdot g$.
\begin{NB}
Then the action of the cohomological FMT
$\Phi^H : H^{\operatorname{ev}}(Y,\bb{Z})_{\operatorname{alg}}\to H^{\operatorname{ev}}(X,\bb{Z})_{\operatorname{alg}}$
can be written as follows (\cite[sect. 6.4]{YY}).
Let $\widehat{H}$ be the ample generator of $\operatorname{NS}(Y)$.
For a Mukai vector $v=(r,d\widehat{H},a)\in H^{\operatorname{ev}}(Y,\bb{Z})_{\operatorname{alg}}$,
the image $\Phi^H(v)$ is given by
\begin{align*}
\Phi^H(v)=v\cdot g,\quad
g := \begin{pmatrix}
p_1\sqrt{r_1}&q_1\sqrt{r_2}\\p_2\sqrt{r_2}&q_2\sqrt{r_1}
\end{pmatrix}\in G.
\end{align*}
Here the action $\cdot$ is given by
\begin{align}
\label{eq:action_cdot}
(r,d H,a)\cdot g := (r',d' H,a'),\quad
{}^t g
\begin{pmatrix}r &d \sqrt{n}\\d \sqrt{n}&a \end{pmatrix}
g
=\begin{pmatrix}r'&d' \sqrt{n}\\d' \sqrt{n}&a'\end{pmatrix}.
\end{align}
\end{NB}
We also need to treat the composition of a Fourier-Mukai transform
and the dualizing functor ${\mathcal D}_X$.
For a
Fourier-Mukai transform
$\Phi_{X \to Y}^{\bl{E}^\bullet} \in \Eq(\bl{D}(X),\bl{D}(Y))$,
we set
\begin{align*}
\theta(\Phi_{X \to Y}^{\bl{E}}\mathcal{D}_X)
:=
\begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix}
\theta(\Phi_{X \to Y}^{\bl{E}})
\in \widehat{G}/\{\pm 1\}.
\end{align*}
Then the action of
$\theta(\Phi_{X \to Y}^{\bl{E}}\mathcal{D}_X)$ on
$\operatorname{Sym}_2({\mathbb Z},n)$ is the same as the action of
$\Phi_{X \to Y}^{\bl{E}}\mathcal{D}_X$.
\begin{lem}[{\cite[Lemma 6.18]{YY}}]
\label{fct:matrix}
If $\theta(\Phi_{X \to Y}^{\bl{E}})=
\begin{pmatrix} a & b\\ c & d \end{pmatrix}$,
then
\begin{align*}
\theta(\Phi_{Y \to X}^{\bl{E}})=
\pm\begin{pmatrix}d & b\\c & a \end{pmatrix},\quad
\theta(\Phi_{Y \to X}^{\bl{E}^{\vee}[2]})=
\pm\begin{pmatrix}d & -b\\-c & a \end{pmatrix},\quad
\theta(\Phi_{X \to Y}^{\bl{E}^{\vee}[2]})=
\pm\begin{pmatrix}a & -b\\-c & d \end{pmatrix}.
\end{align*}
\end{lem}
\subsection{The arithmetic group $G$ and numerical solutions
for the ideal sheaf}\label{subsect:solution}
Let $\ell \in \bb{Z}_{>0}$.
We assume that $\sqrt{\ell n}\notin\bb{Z}$.
Our next task is to describe the numerical solution
of the ideal sheaf of 0-dimensional subscheme.
First we introduce an arithmetic group $S_{n,\ell}$.
\begin{defn}
For $(x,y)\in \bb{R}^2$, set
\begin{align*}
P(x,y):=\begin{pmatrix}y&\ell x\\ x&y\end{pmatrix}.
\end{align*}
We also set
\begin{align*}
S_{n,\ell}:=
\left\{\begin{pmatrix}y&\ell x\\ x&y\end{pmatrix}
\,\bigg|\,
\begin{aligned}
x=a \sqrt{r}, y=b \sqrt{s},\;
a,b,r, s \in\bb{Z}\\
r,s>0,\; rs=n,\;
y^2-\ell x^2=\pm 1
\end{aligned}
\right\}.
\end{align*}
\end{defn}
\begin{lem}
\begin{enumerate}
\item[(1)]
$S_{n,\ell}$ is a commutative subgroup of $\GL(2,\bb{R})$.
\item[(2)]
We have a homomorphism
\begin{align*}
\begin{array}{c c c c}
\phi \colon & S_{n,\ell} & \longrightarrow & \bb{R}^\times \\
& P(x,y) & \mapsto & y+x\sqrt{\ell}.
\end{array}
\end{align*}
\item[(3)]
For $\ell>1$, $\phi$ is injective.
For $\ell=1$, we have
\begin{align*}
\Ker\phi=\left\langle\begin{pmatrix} 0&1 \\ 1&0 \end{pmatrix}\right\rangle.
\end{align*}
\item[(4)]
We set a subgroup $G_{n,\ell}$ of $\widehat{G}$
(Definition~\ref{defn:G}) to be
\begin{align*}
G_{n,\ell}:=
\left\{
g \in \widehat{G}
\left|\;
{}^{t} g \begin{pmatrix} 1 & 0 \\ 0 & -\ell \end{pmatrix} g
=\pm\begin{pmatrix} 1 & 0 \\ 0 & -\ell \end{pmatrix}
\right.
\right\}.
\end{align*}
Then
\begin{align*}
G_{n,\ell}
=S_{n,\ell} \rtimes
\left \langle
\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}
\right\rangle.
\end{align*}
\end{enumerate}
\end{lem}
\begin{proof}
The proofs of (1) and (2) are straightforward.
For (3), assume that
$x,y\in \bb{R}$ with $x^2,y^2,x y/\sqrt{n}\in \bb{Q}$
satisfy $y+x\sqrt{\ell}=1$.
Then $(y^2+\ell x^2)+2(x y/\sqrt{n})\sqrt{\ell n}=(y+x\sqrt{\ell})^2=1$.
Our assumptions yields $y^2+\ell x^2=1$ and $x y=0$. If $x=0$, then $y=\pm 1$.
If $y=0$, then $\ell=1$ and $x=1$.
Hence the conclusion holds.
(4) follows from direct computations.
\end{proof}
Then the Dirichlet unit theorem yields the following corollary.
\begin{cor}
If $\ell>1$, then $S_{n,\ell}\cong\bb{Z}\oplus\bb{Z}/2\bb{Z}$.
\end{cor}
\begin{proof}
Let $p_1,\ldots,p_m$ be the prime divisors of $\ell n$ and $\mathfrak{o}$ be
the ring of algebraic integers in $\bb{Q}(\sqrt{p_1},\ldots,\sqrt{p_m})$.
By Dirichlet unit theorem $\mathfrak{o}^\times$
is a finitely generated abelian group whose torsion subgroup is $\{\pm 1\}$.
Hence $\phi(S_{n,\ell})$ is a finitely generated abelian group
whose torsion subgroup is $\{\pm 1\}$.
For $A\in S_{n,\ell}$, we have $\phi(A^2)\in\bb{Z}[\sqrt{\ell n}]$.
Since $\bb{Z}[\sqrt{n\ell}]^\times\cong\bb{Z}\oplus\bb{Z}/2\bb{Z}$,
we get $S_{n,\ell}\cong\bb{Z}\oplus\bb{Z}/2\bb{Z}$.
\end{proof}
\begin{rem}
If $n=1$ and $\ell>1$,
then $S_{1,\ell}$ is the group of units of $\bb{Z}[\sqrt{\ell}]$.
Moreover if $\ell$ is square free and $\ell\equiv 2,3 \pmod{4}$,
then since $\bb{Z}[\sqrt{\ell}]$ is
the ring of the integers of $\bb{Q}[\sqrt{\ell}]$,
a generator of $S_{1,\ell}$ becomes a fundamental unit.
\end{rem}
\begin{lem}\label{lem:numerical-P(p,q)}
For two positive isotropic Mukai vectors $w_0,w_1$
on the fixed abelian surface $X$,
the condition
\begin{align*}
(1,0,-\ell)=\pm (\ell w_0-w_1),\quad \mpr{w_0,w_1}=-1
\end{align*}
is equivalent to
\begin{align*}
w_0=(p^2,-\tfrac{p q}{\sqrt{n}}H,q^2),\
w_1=(q^2,-\tfrac{\ell p q}{\sqrt{n}}H,\ell^2 p^2),\quad
P(p,q)\in S_{n,\ell}.
\end{align*}
\end{lem}
\begin{proof}
If there are isotropic Mukai vectors with the first condition,
then we can write them as $w_0=(r,d H,(r\ell\pm 1))$ and
$w_1=(r \ell\mp 1,d\ell H, r \ell^2)$,
where $d^2(H^2)=2 r (r \ell\mp 1)$.
We set $p:= \sqrt{r}$ and $q:= \sqrt{r \ell \mp 1}$.
Then $w_0=(p^2,-\tfrac{p q}{\sqrt{n}}H,q^2)$,
$w_1=(q^2,-\tfrac{\ell p q}{\sqrt{n}}H,\ell^2 p^2)$
and $\ell p^2-q^2=\pm 1$.
Thus $P(p,q)\in S_{n,\ell}$.
The converse is obvious.
\end{proof}
\begin{cor}
Recall the action $\cdot$ of $\GL(2,\bb{R})$ given in \eqref{eq:action_cdot}.
By the correspondence
\begin{align*}
G_{n,\ell} \ni g \mapsto (w_0,w_1),\quad
w_0:=(0,0,1)\cdot g, \ w_1:=(1,0,0)\cdot g,
\end{align*}
we have a bijective correspondence:
\begin{align*}
\begin{array}{c c c}
G_{n,\ell}
\left/
\left\langle \pm \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix} \right \rangle
\right.
\cong
S_{n,\ell}/\{\pm 1\}
&
\longleftrightarrow
&
\left\{ (w_0,w_1)
\left|
\begin{aligned}
\mpr{w_0,w_1}=-1,\, \mpr{ w_0^2 }=\mpr{w_1^2}=0,\\
w_0,w_1>0,\, (1,0,-\ell)=\pm(\ell w_0-w_1)
\end{aligned}
\right.
\right\}
\end{array}
\end{align*}
\end{cor}
\begin{defn}
\label{defn:ambm}
Assume that $\ell>1$.
Let
$$
A_\ell:=
\begin{pmatrix}
q & \ell p\\
p & q
\end{pmatrix},\;p,q>0
$$
be the generator of $S_{n,\ell}/\{\pm 1\}$.
\begin{NB}
It corresponds to
$z:= q+\sqrt{\ell}p$ ($p,q>0$).
\end{NB}
We set $\epsilon:= q^2-\ell p^2\in\{\pm 1\}$.
For $m \in \bb{Z}$, we set
\begin{align*}
\begin{pmatrix} q & \ell p \\ p & q \end{pmatrix}^m=
\begin{pmatrix} b_m & \ell a_m\\ a_m & b_m \end{pmatrix}.
\end{align*}
\end{defn}
By the definition we have
\begin{align*}
(a_0,b_0)=(0,1),\quad
(a_{-m},b_{-m})=\epsilon^m(-a_m,b_m),\, m\in\bb{Z}_{>0}
\end{align*}
and
\begin{align}
\label{eq:S_{n,l}}
S_{n,\ell}=
\left\{ \left. \pm
\begin{pmatrix} b_m & \ell a_m\\ a_m & b_m \end{pmatrix}
\right| m \in \bb{Z} \right\}.
\end{align}
Next we consider the right action of $\GL(2,\bb{R})$ on $\bb{R}^2$
\begin{align}
\label{eq:quad_action}
(x,y) \mapsto (x,y) X,
\quad X \in \GL(2,\bb{R}).
\end{align}
Then the quadratic map
\begin{align*}
\begin{array}{c c c}
\bb{R}^2 & \to & \operatorname{Sym}_2(\bb{R})\\[3pt]
(x,y) & \mapsto &
\begin{pmatrix} x \\ y \end{pmatrix}
\begin{pmatrix} x & y \end{pmatrix}
\end{array}
\end{align*}
is $\GL(2,\bb{R})$-equivariant.
Using this action, we have the next descriptions of
the topological invariants of fine moduli spaces
$M_H(v)$ of dimension 2.
\begin{align}
\label{eq:top-inv}
\begin{split}
&
\left\{ v
\left|
\begin{aligned}
v \in H^*(X,\bb{Z})_{\operatorname{alg}},\, \mpr{v^2}=0,\, v>0,
\\
\mpr{w,v}=-1,\,\exists w \in H^*(X,\bb{Z})_{\operatorname{alg}}
\end{aligned}
\right.
\right\}
\\
\overset{\varphi_1}{\longleftrightarrow}
&
\left\{
(a\sqrt{r}, b\sqrt{s}) \in \bb{R}^2
\left|
\begin{aligned}
a,b\in\bb{Z},\, r,s\in\bb{Z}_{>0},
\\
r s=n,\, \gcd(a r,b s)=1
\end{aligned}
\right.
\right\}
/\{\pm 1\}
=
\left\{ (0,1) X \mid X \in G \right\}/\{\pm 1\}
\\
\overset{\varphi_2}{\longleftrightarrow}
&
\left\{
\left. \dfrac{b \sqrt{n}}{a r} \in {\bb{P}}^1(\bb{R})
=\bb{R} \cup \{ \infty \}
\right|
r s=n,\; \gcd(a r,b s)=1
\right\},
\end{split}
\end{align}
where we used the correspondences
\begin{align*}
v=(a^2 r, a b H,b^2 s)
\overset{\varphi_1}{\longleftrightarrow}
\pm(a\sqrt{r},b\sqrt{s})
\overset{\varphi_2}{\longleftrightarrow}
\dfrac{\mu(v)}{2\sqrt{n}}=\dfrac{b \sqrt{n}}{a r}.
\end{align*}
Here we used the slope for the Mukai vector
defined by $\mu(v):=(H,c_1(v))/\operatorname{rk} v$.
These correspondences are $\widehat{G}$-equivariant
under the action \eqref{eq:quad_action}.
Lemma \ref{lem:numerical-P(p,q)}, \eqref{eq:S_{n,l}} and \eqref{eq:top-inv}
imply the following one to one correspondence:
\begin{align*}
\begin{array}{c c c}
\left\{
\{ v_1,v_2 \} \left|
\begin{aligned}
&\text{ There is a numerical solution}
\\
&\text{ $(v_1,v_2,\ell_1,\ell_2)$ of $(1,0,-\ell)$ }
\end{aligned}
\right.
\right\}
& \longleftrightarrow &
\left\{
\left.
\left\{\frac{b_m}{a_m}, \frac{\ell a_m}{b_m} \right\}
\subset {\bb{P}}^1(\bb{R})
\right|
m \in \bb{Z}
\right\}
\\
\{ v_1,v_2 \}
& \longleftrightarrow &
\left\{\frac{\mu(v_1)}{2\sqrt{n}},\frac{\mu(v_2)}{2\sqrt{n}} \right\},
\end{array}
\end{align*}
where $(\ell_i,\ell_j)=(\ell,1)$ if and only if
$(\frac{\mu(v_i)}{2\sqrt{n}},\frac{\mu(v_j)}{2\sqrt{n}})=
(\frac{b_m}{a_m},\frac{\ell a_m}{b_m})$.
\begin{defn}
For $m \in {\mathbb Z}$,
we set
\begin{equation}
\begin{split}
u_m:=& a_m^2 e^{\frac{b_m}{a_m \sqrt{n}}H}=
(a_m^2,\tfrac{a_m b_m}{\sqrt{n}}H,b_m^2),\\
u_m':=& b_m^2 e^{\frac{\ell a_m}{b_m \sqrt{n}}H}=
(b_m^2,\tfrac{\ell a_m b_m}{\sqrt{n}}H,\ell^2 a_m^2).
\end{split}
\end{equation}
\end{defn}
\begin{NB}
$u_0=(0,0,1)=\varrho_X$ and $u_0'=(1,0,0)$.
\end{NB}
\subsection{Codimension 0 walls and the action of
$G_{n,\ell}$.}
\begin{defn}
$C_0$ is the wall associated to
the numerical solution
$(1,\varrho_X,1,\ell)$.
For $m \ne 0$,
let $C_m$ be the wall associated to the numerical solution
$(u_m,u_m',\ell,1)$.
\end{defn}
\begin{prop}
\begin{enumerate}
\item[(1)]
$C_0$ is the $t$-axis
and $C_m$ $(m \ne 0)$ is the circle defined by
$$
\left(s-\frac{1}{\sqrt{n}}\frac{b_m}{a_m} \right)
\left(s-\frac{1}{\sqrt{n}}\frac{\ell a_m}{b_m}\right)+t^2=0.
$$
\item[(2)]
$\{C_m \mid m \in {\mathbb Z}\}$ is the set of codimension 0 walls.
\end{enumerate}
\end{prop}
\begin{defn}
\begin{enumerate}
\item[(1)]
For $C_m$ ($m \in {\mathbb Z}$), we define
adjacent chambers $C_m^\pm$ as follows:
\begin{itemize}
\item
$C_m^-$ is surrounded by $C_m^+$ for $m<0$.
\item
$C_0^- \subset \{(s,t) \mid s<0 \}$ and
$C_0^+ \subset \{(s,t) \mid s>0 \}$.
\item
$C_m^+$ is surrounded by $C_m^-$ for $m>0$.
\end{itemize}
\item[(2)]
Let $M_{C_m^\pm}(v)$ be the moduli of stable objects
$M_{(sH,tH)}(v)$ for $(s,t) \in C_m^\pm$.
\end{enumerate}
\end{defn}
Then
we have
$$
M_{C_m^\pm}(v)
=
\begin{cases}
\mathfrak{M}^{\pm}(u_m,u_m',\ell,1), & \frac{b_m}{a_m}<\frac{\ell a_m}{b_m}\\
\mathfrak{M}^{\pm}(u_m',u_m,1,\ell), & \frac{b_m}{a_m}>\frac{\ell a_m}{b_m}
\end{cases}
$$
for $m<0$
and
$$
M_{C_m^\pm}(v)
=
\begin{cases}
\mathfrak{M}^{\mp}(u_m,u_m',\ell,1)[-1], & \frac{b_m}{a_m}<\frac{\ell a_m}{b_m}\\
\mathfrak{M}^{\mp}(u_m',u_m,1,\ell)[-1], & \frac{b_m}{a_m}>\frac{\ell a_m}{b_m}
\end{cases}
$$
for $m \geq 0$.
In particular, we have
\begin{equation}
M_{C_0^-}(v)={\mathfrak M}^+(1,\varrho_X,1,\ell)[-1]=
M_H(1,0,-\ell).
\end{equation}
We note that $\Psi$ in Proposition \ref{prop:dual} satisfies
$\Psi^{-1}=\Psi$. By Proposition \ref{prop:dual}, we have the following
isomorphisms.
\begin{equation*}
\Psi_m:M_{C_m^\pm}(v) \to M_{C_m^\mp}(v).
\end{equation*}
Let $\Phi_{X_1 \to X}^{{\bf E}_m}$ be a Fourier-Mukai
transform such that
$\Phi_{X_1 \to X}^{{\bf E}_m}(\varrho_{X_1})=u_m$
and $\Phi_{X_1 \to X}^{{\bf E}_m}(1)=u_m'$,
where $X_1=M_H(u_m)$.
Then we have
\begin{equation}
\theta(\Phi_{X_1 \to X}^{{\bf E}_m})=
\begin{pmatrix}
b_m & \ell a_m\\
\epsilon^m a_m & \epsilon^m b_m
\end{pmatrix}
=
\begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix}^{\frac{-1+\epsilon^m}{2}}
\begin{pmatrix}
q & \ell p\\
p & q
\end{pmatrix}^m.
\end{equation}
Thus we get
\begin{equation}
\theta(\Psi_m)=
\begin{pmatrix}
q & \ell p\\
p & q
\end{pmatrix}^{-m}
\begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix}
\begin{pmatrix}
q & \ell p\\
p & q
\end{pmatrix}^m.
\end{equation}
We also have
\begin{equation}
\begin{split}
\begin{pmatrix}
q & \ell p\\
p & q
\end{pmatrix}^{m+k}
\theta(\Psi_m)=&
\begin{pmatrix}
q & \ell p\\
p & q
\end{pmatrix}^{m+k}
\begin{pmatrix}
q & \ell p\\
p & q
\end{pmatrix}^{-m}
\begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix}
\begin{pmatrix}
q & \ell p\\
p & q
\end{pmatrix}^m\\
=&\begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix}
\begin{pmatrix}
q & \ell p\\
p & q
\end{pmatrix}^{m-k}.
\end{split}
\end{equation}
Hence
$(\Psi_m(u_{m+k}),\Psi(u_{m+k}'))=(u_{m-k},u_{m-k}')$.
Thus we get the following proposition.
\begin{prop}\label{prop:Psi_m}
$\Psi_m(C_{m+k})=C_{m-k}$ and
$\Psi_m(C_{m+k}^\pm)=C_{m-k}^\mp$.
In particular,
$\Psi_m$ induces an isomorphism
$$
M_{C_{m+k}^\pm}(v) \to M_{C_{m-k}^\mp}(v).
$$
\end{prop}
\begin{rem}\label{rem:2m}
$$
\theta(\Phi_{X_1 \to X}^{{\bf E}_m} \circ
\Phi_{X \to X_1}^{{\bf E}_m})=
\pm A_\ell^{2m}.
$$
Hence $M_H(u_{2m}) \cong X$.
\end{rem}
\begin{prop}\label{prop:fund}
\begin{enumerate}
\item[(1)]
There are finitely many walls between $C_0$ and $C_{-1}$.
\item[(2)]
By the action of $G_{n,\ell}$, every wall is transformed
to a wall between $C_0$ and $C_{-1}$.
\end{enumerate}
\end{prop}
\begin{proof}
(1)
We set
$\lambda_0:=-\frac{q}{p \sqrt{n}}$.
Then for $\beta=\lambda_0 H$, there is finitely many walls.
Since every wall between $C_0$ and $C_{-1}$ intersects with
the line $s=\lambda_0$, the claim holds.
(2)
Let $C$ be a wall between $C_m$ and $C_{m-1}$.
Since $\Psi_0(C)$ is a wall between
$C_{-m}$ and $C_{-m+1}$, we may assume that $m < 0$.
By Proposition \ref{prop:Psi_m}, we see that
$\Psi_m(C)$ is a wall
between $C_{m+1}$ and $C_m$.
Therefore $\Psi_{-1}\circ \cdots \circ \Psi_{m-1}\circ \Psi_m(C)$
is a wall between $C_0$ and $C_{-1}$.
\end{proof}
\begin{rem}
$$
\theta(\Psi_{-1}\circ \cdots \circ \Psi_{m-1}\circ \Psi_m)
=
\begin{cases}
\theta(\Psi_{-1})A_\ell^{-m-1},& 2 \not |m,\\
A_\ell^{-m},& 2 |m.
\end{cases}
$$
\begin{NB}
$\theta(\Psi_{m-1} \circ \Psi_m)=A_\ell^2$.
\end{NB}
\end{rem}
\begin{NB}
\begin{defn}
For the numerical solution $(v_1,v_2,\ell_1,\ell_2)$
corresponding to $\{\frac{b_{-m}}{a_{-m}},\frac{\ell a_{-m}}{b_{-m}} \}
=\{-\frac{b_m}{a_m},-\frac{\ell a_m}{b_m} \}$,
$\mathfrak{M}^{\pm}_m$ denotes the moduli space
$\mathfrak{M}^{\pm}(v_1,v_2,\ell_1,\ell_2)$.
$\mathfrak{M}^{\pm}_m[-1]$ denotes the moduli space
of $V^{\bullet}[-1]$, $V^{\bullet} \in \mathfrak{M}^{\pm}_m$.
\end{defn}
\end{NB}
In Proposition \ref{prop:Psi_m},
we did not specify the correspondence of
complexes. Since $\phi_{(sH,tH)}(v) \mod 2{\mathbb Z}$
is well-defined,
the correspondence is determined up to shift
$[2k]$ ($k \in {\mathbb Z}$).
We next fix the ambiguity of this shift.
We note that $\theta(\Psi_m)=\theta([1] \circ {\mathcal D}_X
\circ \Phi_{X \to X}^{{\bf E}_{2m}^{\vee}[1]})$,
where $[1]$ is the shift functor.
Since
$Z_{(sH,tH)}(v) \in {\mathbb C} \setminus {\mathbb R}_{<0}$,
we take $\phi_{(sH,tH)}(v) \in(-1,1)$
to consider the moduli space
$M_{(sH,tH)}(v)$ as in Definition \ref{defn:phase(v)}.
Then the isomorphisms in Proposition \ref{prop:Psi_m}
are given by the following proposition.
\begin{prop}\label{prop:Psi_m(phase)}
Assume that $m<0$.
$[1] \circ {\mathcal D}_X
\circ \Phi_{X \to X}^{{\bf E}_{2m}^{\vee}[1]}$
induces isomorphisms
\begin{equation*}
\begin{matrix}
M_{C_{m+k}^\pm}(v) & \to & M_{C_{m-k}^\mp}(v)\\
E & \mapsto &
(\Phi_{X \to X}^{{\bf E}_{2m}^{\vee}[1]}(E))^{\vee}[1].
\end{matrix}
\end{equation*}
\end{prop}
\begin{proof}
Since $b_{2m}^2-\ell a_{2m}^2=1$, we have
$\frac{b_{2m}}{a_{2m}}<\frac{\ell a_{2m}}{b_{2m}}<0$
for $m<0$.
For $(s,t) \in C_{2m}$,
$\phi_{(sH,tH)}(({\bf E}_{2m})_{|X \times \{x \}}) \in (-1,0]$.
Hence
$\phi_{(sH,tH)}(({\bf E}_{2m})_{|X \times \{x \}}[1])=
\phi_{(sH,tH)}(E)$
for $(s,t) \in C_{2m}$ and $E \in M_{(sH,tH)}(v)$.
For $E \in M_{(sH,tH)}(v)$,
we also have
$$
\begin{cases}
\phi_{(sH,tH)}(({\bf E}_{2m})_{|X \times \{x \}}[1])>
\phi_{(sH,tH)}(E)>\phi_{(sH,tH)}(({\bf E}_{2m})_{|X \times \{x \}}),\;
&
\text{$(s,t)$ is outside of $C_{2m}$},\\
\phi_{(sH,tH)}(({\bf E}_{2m})_{|X \times \{x \}}[2])>
\phi_{(sH,tH)}(E)>\phi_{(sH,tH)}(({\bf E}_{2m})_{|X \times \{x \}}[1]),\;
&
\text{$(s,t)$ is inside of $C_{2m}$}.
\end{cases}
$$
If $(s,t)$ is outside of $C_{2m}$, then
since $\phi_{(\widetilde{sH},\widetilde{tH})}
(\Phi_{X \to X}^{{\bf E}_{2m}^{\vee}[1]}(({\bf E}_{2m})_{|X \times \{x \}}))
=0$, we have
$\phi_{(\widetilde{sH},\widetilde{tH})}
(\Phi_{X \to X}^{{\bf E}_{2m}^{\vee}[1]}(E)) \in (0,1)$.
Hence
$$
\phi_{(-\widetilde{sH},\widetilde{tH})}
((\Phi_{X \to X}^{{\bf E}_{2m}^{\vee}[1]}(E))^{\vee}[1]) \in (0,1).
$$
If $(s,t)$ is inside of $C_{2m}$, then
$\phi_{(\widetilde{sH},\widetilde{tH})}
(\Phi_{X \to X}^{{\bf E}_{2m}^{\vee}[1]}(E)) \in (1,2)$.
Hence
$$
\phi_{(-\widetilde{sH},\widetilde{tH})}
((\Phi_{X \to X}^{{\bf E}_{2m}^{\vee}[1]}(E))^{\vee}[1]) \in (-1,0).
$$
Therefore the claim holds.
\end{proof}
\begin{rem}\label{rem:Psi_m(phase)}
Assume that $m>0$. Then
$[-1] \circ {\mathcal D}_X
\circ \Phi_{X \to X}^{{\bf E}_{2m}^{\vee}[1]}$
induces isomorphisms
\begin{equation*}
\begin{matrix}
M_{C_{m+k}^\pm}(v) & \to & M_{C_{m-k}^\mp}(v)\\
E & \mapsto &
(\Phi_{X \to X}^{{\bf E}_{2m}^{\vee}[1]}(E))^{\vee}[-1].
\end{matrix}
\end{equation*}
\end{rem}
\begin{NB}
\begin{proof}
Since $r b_{2m}/a_{2m}-d_\beta=b_{2m}/a_{2m}>0$,
we have
$$
\begin{cases}
\phi_{(sH,tH)}(({\bf E}_{2m})_{|X \times \{x \}})>
\phi_{(sH,tH)}(E)>\phi_{(sH,tH)}(({\bf E}_{2m})_{|X \times \{x \}}[-1]),\;
&
\text{$(s,t)$ is outside of $C_{2m}$},\\
\phi_{(sH,tH)}(({\bf E}_{2m})_{|X \times \{x \}}[-1])>
\phi_{(sH,tH)}(E)>\phi_{(sH,tH)}(({\bf E}_{2m})_{|X \times \{x \}}[-2]),\;
&
\text{$(s,t)$ is inside of $C_{2m}$}.
\end{cases}
$$
Then we see that
$\phi_{(\widetilde{sH},\widetilde{tH})}
(\Phi_{X \to X}^{{\bf E}_{2m}^{\vee}[1]}(E)) \in (-2,0)$.
Hence
$$
\phi_{(-\widetilde{sH},\widetilde{tH})}
((\Phi_{X \to X}^{{\bf E}_{2m}^{\vee}[1]}(E))^{\vee}[-1]) \in (-1,1).
$$
\end{proof}
\end{NB}
\begin{NB}
It is NB for Proposition \ref{prop:Psi_m(phase)}.
We note that $\theta(\Psi_m)=\theta([1] \circ {\mathcal D}_X
\circ \Phi_{X \to X}^{{\bf E}_{2m}^{\vee}[1]})$.
\begin{lem}
Assume that $m<0$.
We set $s=\frac{1}{\sqrt{n}}\frac{b_{2m}}{a_{2m}}$.
For $m<k<-m$,
$M_{C_{m+k}^\pm}(v) \subset {\mathfrak A}_{(sH,tH)}$ and
$\Psi_m$ induces isomorphisms
\begin{equation}
M_{C_{m+k}^\pm}(v) \to M_{C_{m-k}^\mp}(v).
\end{equation}
\end{lem}
\begin{proof}
We note that $\theta(\Psi_m)=\theta([1] \circ {\mathcal D}_X
\circ \Phi_{X \to X}^{{\bf E}_{2m}^{\vee}[1]})$.
We have $v(\Phi_{X \to X}^{{\bf E}_{2m}^{\vee}[1]}({\mathfrak k}_x))=
u_{-2m}$.
Hence $\Phi_{X \to X}^{{\bf E}^{\vee}[1]}$ induces an equivalence
${\mathfrak A}_{(sH,tH)} \to {\mathfrak A}_{(-sH,tH)}$.
\end{proof}
We note that $\frac{b_{2m}}{a_{2m}}<\frac{\ell a_{2m}}{b_{2m}}<0$
for $m<0$.
For $(s,t) \in C_{2m}$,
$\phi_{(sH,tH)}(({\bf E}_{2m})_{|X \times \{x \}}) \in (-1,0]$.
Hence
$\phi_{(sH,tH)}(({\bf E}_{2m})_{|X \times \{x \}}[1])=
\phi_{(sH,tH)}(E)$
for $(s,t) \in C_{2m}$ and $E \in M_{(sH,tH)}(v)$.
For $E \in M_{(sH,tH)}(v)$,
we also have
$$
\begin{cases}
\phi_{(sH,tH)}(({\bf E}_{2m})_{|X \times \{x \}}[1])>
\phi_{(sH,tH)}(E)>\phi_{(sH,tH)}(({\bf E}_{2m})_{|X \times \{x \}}),\;
&
\text{$(s,t)$ is outside of $C_{2m}$},\\
\phi_{(sH,tH)}(({\bf E}_{2m})_{|X \times \{x \}}[2])>
\phi_{(sH,tH)}(E)>\phi_{(sH,tH)}(({\bf E}_{2m})_{|X \times \{x \}}[1]),\;
&
\text{$(s,t)$ is inside of $C_{2m}$}.
\end{cases}
$$
If $(s,t)$ is outside of $C_{2m}$, then
since $\phi_{(\widetilde{sH},\widetilde{tH})}
(\Phi_{X \to X}^{{\bf E}_{2m}^{\vee}[1]}(({\bf E}_{2m})_{|X \times \{x \}}))
=0$, we have
$\phi_{(\widetilde{sH},\widetilde{tH})}
(\Phi_{X \to X}^{{\bf E}_{2m}^{\vee}[1]}(E)) \in (0,1)$.
Therefore
$$
\phi_{(-\widetilde{sH},\widetilde{tH})}
((\Phi_{X \to X}^{{\bf E}_{2m}^{\vee}[1]}(E))^{\vee}[1]) \in (0,1).
$$
If $(s,t)$ is inside of $C_{2m}$, then
$\phi_{(\widetilde{sH},\widetilde{tH})}
(\Phi_{X \to X}^{{\bf E}_{2m}^{\vee}[1]}(E)) \in (1,2)$.
Therefore
$$
\phi_{(-\widetilde{sH},\widetilde{tH})}
((\Phi_{X \to X}^{{\bf E}_{2m}^{\vee}[1]}(E))^{\vee}[1]) \in (-1,0).
$$
We note that
$Z_{(sH,tH)}(v) \in {\mathbb C} \setminus {\mathbb R}_{<0}$.
Hence we take $\phi_{(sH,tH)}(v) \in(-1,1)$.
$\phi_{(sH,tH)}(u_{2m})+1>\phi_{(sH,tH)}(v)>\phi_{(sH,tH)}(u_{2m})$
outside of $C_{2m}$.
Hence
$\phi_{(\widetilde{sH},\widetilde{tH})}
(\Phi_{X \to X}^{{\bf E}_{2m}^{\vee}[1]}(E)) \in (0,1)$.
Therefore
$$
\phi_{(-\widetilde{sH},\widetilde{tH})}
((\Phi_{X \to X}^{{\bf E}_{2m}^{\vee}[1]}(E))^{\vee}[1]) \in (0,1).
$$
$\phi_{(sH,tH)}(u_{2m})+2>\phi_{(sH,tH)}(v)>\phi_{(sH,tH)}(u_{2m})+1$
insides of $C_{2m}$.
\end{NB}
\begin{prop}
We set
$$
M_m:=M_{C_{m-1}^+}(v) \cap M_{C_m^-}(v).
$$
\begin{enumerate}
\item[(1)]
$M_m \ne \emptyset$ and $M_m$ is birationally equivalent to
$M_{C_{m-1}^+}(v)$ and $M_{C_m^-}(v)$.
\item[(2)]
We have a sequence of isomorphisms
$$
\cdots \overset{\Psi_{-3}}{\to} M_{-2}
\overset{\Psi_{-2}}{\to} M_{-1}
\overset{\Psi_{-1}}{\to} M_0
\overset{\Psi_{0}}{\to} M_{1}
\overset{\Psi_{1}}{\to} M_{2}
\overset{\Psi_{2}}{\to} \cdots.
$$
\end{enumerate}
\end{prop}
\begin{proof}
(1)
Since there is no codimension 0 wall between
$C_{m-1}$ and $C_m$, $M_m \ne \emptyset$.
Since $M_{C_{m-1}^+}(v)$ and $M_{C_m^-}(v)$ are irreducible,
$M_m$ is birationally equivalent to
$M_{C_{m-1}^+}(v)$ and $M_{C_m^-}(v)$.
(2)
By Proposition \ref{prop:Psi_m}
or Proposition \ref{prop:Psi_m(phase)}
(and Remark \ref{rem:Psi_m(phase)}), we have isomorphisms
\begin{equation}
\begin{split}
\Psi_m:M_{C_{m-1}^+}(v) & \to M_{C_{m+1}^-}(v),\\
\Psi_m:M_{C_{m}^-}(v) & \to M_{C_m^+}(v).
\end{split}
\end{equation}
Hence we have an isomorphism
$$
\Psi_m:M_{C_{m-1}^+}(v) \cap M_{C_m^-}(v) \to
M_{C_m^+}(v) \cap M_{C_{m+1}^-}(v).
$$
Thus the claim holds.
\end{proof}
We note that
$M_0$ is an open subset of
$M_{C_0^-}(v)=\{I_Z \otimes L \mid
I_Z \in \operatorname{Hilb}{\ell}{X}, L \in \Piczero{X}\}$.
Hence
we have two semi-homogeneous presentations of $I_Z \otimes L
\in M_0$:
\begin{equation}
0 \to I_Z \otimes L \to L \to {\mathcal O}_Z \to 0
\end{equation}
and
\begin{equation}
0 \to E_{-1} \to E_0 \to I_Z \otimes L \to 0.
\end{equation}
Starting from these two semi-homogeneous presentations,
we have a sequence of complexes $F_m^\bullet \in M_m$
such that
$F_0^{\bullet}=I_Z \otimes L$ and
$\Psi_m(F_m^\bullet)=F_{m+1}^\bullet$.
Then we have exact triangles
\begin{equation}
\begin{split}
& V_{m-1}^+ \to F_m^{\bullet} \to V_{m-1}^- \to V_{m-1}^+[1]\\
& W_{m}^- \to F_m^{\bullet} \to W_{m}^+ \to W_{m}^-[1]\\
\end{split}
\end{equation}
such that
\begin{itemize}
\item
$(W_0^-,W_0^+)=({\mathcal O}_Z[-1],L)$,
$(V_{-1}^+,V_{-1}^-)=(E_0,E_{-1}[1])$,
\item
$\Psi_m(W_m^\pm)=V_m^\pm$,
$\Psi_m(V_{m-1}^\pm)=W_{m+1}^\pm$,
\item
for $(s,t) \in C_{m-1}$,
$V_{m-1}^\pm$ are $\sigma_{(sH,tH)}$-semi-stable
objects with the same phase
and define the wall $C_{m-1}$,
\item
for $(s,t) \in C_{m}$,
$W_{m}^\pm$ are $\sigma_{(sH,tH)}$-semi-stable
objects with the same phase and
define wall $C_m$.
\end{itemize}
\begin{NB}
For simplicity, assume that $m<0$.
Then
we also see that
$W_m^-[-1], V_m^-[-1] \in {\mathcal M}_H(u)$
$V_m$ and $W_m^\mp$ are semi-homogeneous sheaves
Then $W_m^-[-1]$ and $W_m^+$ are semi-homogeneous
sheaves with
$\{v(W_m^-[-1]),v(W_m^+)\}=\{ \ell u_m,u_m'\}$.
Since $\Psi_m$ is determined by $\{ u_m,u_m'\}$,
\end{NB}
\begin{NB}
Let $E$ be be an object of $M_{(\beta_1,\omega_1)}(v) \cap
M_{(\beta_2,\omega_2)}(v)$.
Let $\Phi$ be a contravariant Fourier-Mukai transform.
Assume that $d_{\widetilde{\beta_1}}(\Phi(E))>0$
and $d_{\widetilde{\beta_2}}(\Phi(E))>0$.
If $\Phi(E)$ is a
$\sigma_{(\widetilde{\beta_1},\widetilde{\omega_1})}$-semi-stable
object of
${\mathfrak A}_{(\widetilde{\beta_1},\widetilde{\omega_1})}$,
then $\Phi(E)$ is also
a $\sigma_{(\widetilde{\beta_2},\widetilde{\omega_2})}$-semi-stable
object of
${\mathfrak A}_{(\widetilde{\beta_2},\widetilde{\omega_2})}$.
\begin{proof}
Since $d_{\widetilde{\beta_2}}(\Phi(E))>0$,
$\phi_{(\widetilde{\beta_2},\widetilde{\omega_2})}(\Phi(E))
\in (2k,2k+1)$.
By our assumption, we have $H^i(\Phi(E))=0$ for $i \ne -1,0$.
Hence $k=0$ and $\Phi(E) \in
{\mathfrak A}_{(\widetilde{\beta_2},\widetilde{\omega_2})}$.
\end{proof}
Assume that $E \in M_{C_{m-1}^+}(v) \cap M_{C_m^-}(v)$.
Since $E \in M_{C_{m-1}}(v)$, we have an exact sequence
$$
0 \to A \to E \to B \to 0
$$
such that $A$ and $B$ are $\sigma_{(\beta,\omega)}$-semi-stable
objects with
$\phi_{(\beta,\omega)}(A)=\phi_{(\beta,\omega)}(B)$.
Then we have an exact sequence
$$
0 \to \Psi_m(B) \to \Psi_m(E) \to \Psi_m(A) \to 0
$$
in ${\mathfrak A}_{(\widetilde{\beta},\widetilde{\omega})}$.
\end{NB}
\subsection{Fourier-Mukai transforms of the families
$F_m^{\bullet}$.}
We first assume $\epsilon=q^2-\ell p^2=-1$.
In this case, the algebraic integers $a_m,b_m$
in Definition~\ref{defn:ambm} satisfy the following relations:
\begin{align*}
\dfrac{b_{2 k - 1}}{a_{2 k - 1}}<\dfrac{\ell a_{2 k}}{b_{2 k}}
<\dfrac{b_{2 k + 1}}{a_{2 k + 1}}<\sqrt{\ell}
<\dfrac{\ell a_{2 k + 1}}{b_{2 k + 1}}<\dfrac{b_{2 k}}{a_{2 k}}
<\dfrac{\ell a_{2 k - 1}}{b_{2 k - 1}}
\quad
(k\in\bb{Z}_{>0}),
\quad
\lim_{k\to\infty}\dfrac{b_k}{a_k}=\lim_{k\to\infty} \dfrac{\ell a_k}{b_k}=
\sqrt{\ell}.
\end{align*}
Thus $\pm \sqrt{\frac{\ell}{n}}$ are the accumulation points
of $\cup_m C_m$.
\begin{NB}
Set $z_m := b_m + \sqrt{\ell} a_m$ and
$\overline{z}_m :=b_m - \sqrt{\ell} a_m$.
Then
$z=z_1=q+\sqrt{\ell}p$,
$\overline{z}_1=q-\sqrt{\ell}p$, and
\begin{align*}
&z_{m} = z_{m-1} z_1=z_1^m,\quad
\overline{z}_m = \overline{z}_{m-1} \overline{z}_1
=\overline{z}_1^m
\quad
(m \in \bb{Z}_{\ge0}).
\end{align*}
If $\epsilon=q^2-\ell p^2=-1$,
then $\overline{z}_1=q-\sqrt{\ell}p<0$.
Thus $\overline{z}_m<0$ if $m$ is odd,
and $\overline{z}_m>0$ if $m$ is even.
Thus
\begin{align*}
\dfrac{b_{2k-1}}{a_{2k-1}}<\sqrt{\ell}<\dfrac{\ell a_{2k-1}}{b_{2k-1}},
\quad
\dfrac{\ell a_{2k}}{b_{2k}}<\sqrt{\ell}<\dfrac{b_{2k}}{a_{2k}}
\quad
(k \in \bb{Z}_{>0}).
\end{align*}
We also have
\begin{align*}
&\ell a_m a_{m-1}-b_m b_{m-1}
=(q^2-\ell p^2) (\ell a_{m-1}a_{m-2}-b_{m-1}b_{m-2})
\\
&\Longleftrightarrow
\dfrac{\ell a_m}{b_m}-\dfrac{b_{m-1}}{a_{m-1}}
=\dfrac{a_{m-2} b_{m-1}}{a_{m-1} b_m}(q^2-\ell p^2)
(\dfrac{\ell a_{m-1}}{b_{m-1}}-\dfrac{b_{m-2}}{a_{m-2}})
\\
&\Longleftrightarrow
\dfrac{\ell a_{m}}{b_{m}}-\dfrac{b_{m+1}}{a_{m+1}}
=\dfrac{a_{m} b_{m-1}}{a_{m+1} b_{m}}(q^2-\ell p^2)
(\dfrac{\ell a_{m-1}}{b_{m-1}}-\dfrac{b_{m}}{a_{m}})
\quad
(m\in \bb{Z}_{\ge2}).
\end{align*}
Note that
$\tfrac{\ell a_1}{b_1}-\tfrac{b_0}{a_0}
=\tfrac{\ell \cdot p}{q}-\tfrac{0}{1}>0$
and
$\tfrac{\ell a_0}{b_0}-\tfrac{b_1}{a_1}
=\tfrac{\ell \cdot 0}{1}-\tfrac{q}{p}>0$.
Therefore if $\epsilon=q^2-\ell p^2=-1$, then we have
\begin{align*}
& \dfrac{b_{2 k - 1}}{a_{2 k - 1}}
<\dfrac{\ell a_{2 k}}{b_{2 k}}
<\dfrac{b_{2 k + 1}}{a_{2 k + 1}},
\quad
\dfrac{\ell a_{2 k + 1}}{b_{2 k + 1}}
<\dfrac{b_{2 k}}{a_{2 k}}
<\dfrac{\ell a_{2 k - 1}}{b_{2 k - 1}}
\quad
(k \in \bb{Z}_{>0}).
\end{align*}
\end{NB}
We regard $(a_m:b_m)$ and $(b_m:\ell a_m)$
as elements of ${\bb{P}}^1({\bb{R}})$.
Then the inhomogeneous coordinates of these points give a sequence
\begin{multline}
\label{eq:interval:case1:050}
-\infty=\dfrac{b_0}{a_0}
<-\dfrac{\ell p}{q}=\dfrac{\ell a_{-1}}{b_{-1}}
<\dfrac{b_{-2}}{a_{-2}}<\cdots<-\sqrt{\ell}<\cdots
<\dfrac{\ell a_{-2}}{b_{-2}}<\dfrac{b_{-1}}{a_{-1}}=-\dfrac{q}{p}
\\
<\dfrac{\ell a_0}{b_0}=0<\dfrac{b_1}{a_1}=\dfrac{q}{p}
<\dfrac{\ell a_2}{b_2}<\cdots<\sqrt{\ell}<\cdots<\dfrac{b_2}{a_2}
<\dfrac{\ell a_1}{b_1}=\dfrac{\ell p}{q}<\dfrac{b_0}{a_0}=\infty,
\end{multline}
where we write the inhomogeneous coordinate
of $(0:1)$ as $\infty$ or $-\infty$.
For a Fourier-Mukai transform
$\Phi_{X \to X'}^{\bl{G}^{\vee}}\colon\bl{D}(X) \to \bl{D}(X')$,
we write $c_1({\bl{G}}_{x'})/\operatorname{rk} {\bl{G}}_{x'}=(\lambda/\sqrt{n}) H$.
If $-\ell p / q<\lambda<-q/p$, then
$\Phi_{X \to X'}^{{\bl{G}^{\vee}}}(F_0^\bullet)$
is not a sheaf for all $F_0^\bullet$.
\begin{defn}
We set
\begin{align*}
\begin{array}{l l l l l l}
I_1&:=&
[0,\tfrac{b_1}{a_1})
\cup
[\tfrac{\ell a_1}{b_1},\infty),
&
I_0&:=&
[-\infty,-\tfrac{\ell a_1}{b_1})
\cup
[-\tfrac{b_1}{a_1},0),
\\[4pt]
I_{2k}&:=&
[\tfrac{b_{2k-1}}{a_{2k-1}},\tfrac{\ell a_{2k}}{b_{2k}})
\cup
[\tfrac{b_{2k}}{a_{2k}},\tfrac{\ell a_{2k-1}}{b_{2k-1}}),
&
I_{-2k}&:=&
[-\tfrac{b_{2k}}{a_{2k}},-\tfrac{\ell a_{2k+1}}{b_{2k+1}})
\cup
[-\tfrac{b_{2k+1}}{a_{2k+1}},-\tfrac{\ell a_{2k}}{b_{2k}}),
\\[4pt]
I_{2k+1}&:=&
[\tfrac{\ell a_{2k}}{b_{2k}},\tfrac{b_{2k+1}}{a_{2k+1}})
\cup
[\tfrac{\ell a_{2k+1}}{b_{2k+1}},\tfrac{b_{2k}}{a_{2k}}),
&
I_{-2k+1}&:=&
[-\tfrac{\ell a_{2k-1}}{b_{2k-1}},-\tfrac{b_{2k}}{a_{2k}})
\cup
[-\tfrac{\ell a_{2k}}{b_{2k}},-\tfrac{b_{2k-1}}{a_{2k-1}}).
\end{array}
\end{align*}
For $I=\coprod_i [s_i, t_i)$, we denote $I^*:=\coprod_i (s_i,t_i]$.
\end{defn}
By \eqref{eq:interval:case1:050}, we have decompositions
$\bb{P}^1({\bb{R}}) \setminus\{\pm\sqrt{\ell}\} =
\coprod_{m \in {\bb{Z}}}I_m=
\coprod_{m \in {\bb{Z}}}I_m^*$.
\begin{thm}\label{thm:csv:1}
(1)
If $\lambda \in I_m$ $(m \leq 0)$, then
$\Phi_{X \to X'}^{{\bl{G}^{\vee}}}(F_m^\bullet)$
is a stable sheaf up to shift.
(2)
If $\lambda \in I_m^*$ $(m \leq 0)$, then
$\mathcal{D}_{X'}\Phi_{X \to X'}^{{\bl{G}^{\vee}}}(F_m^\bullet)=
\Phi_{X \to X'}^{{\bl{G}}[2]}(F_m^{\bullet \vee})$
is a stable sheaf up to shift.
\end{thm}
\begin{proof}
(1)
For a small number $t>0$,
$(\lambda,t)$ belongs to the interior of
the annulus bounded by $C_{-m-1}$ and $C_{-m}$.
By Lemma \ref{lem:2-stability},
$F_m^{\bullet}$ is $\sigma_{(\lambda H,tH)}$-semi-stable.
By Proposition \ref{prop:Phi(C^pm)},
$\Phi_{X \to X'}^{{\bf G}^{\vee}[n]}(F_m^{\bullet})$
is a stable sheaf, where
$n=1$ for $\lambda> -\sqrt{\ell}$ and
$n=2$ for $\lambda<-\sqrt{\ell}$.
The proof of (2) is similar.
\end{proof}
By this theorem,
we have semi-homogeneous presentations
for a general member of $M_H(w)$, $w=\Phi_{X \to X'}^{{\bf G}^{\vee}}(v)$.
\begin{NB}
$(F_m^{\bullet})^{\vee}=F_{-m+1}^{\bullet}$.
\end{NB}
\begin{rem}
\label{fct:sheaf-criterion}
The claim also follows from \cite[Lem. 4.4]{YY}.
\end{rem}
We next
assume that $\epsilon=q^2-\ell p^2=1$.
Then the algebraic integers $a_m,b_m$
in Definition~\ref{defn:ambm} satisfy
\begin{align*}
0<\dfrac{b_m}{a_m} -\sqrt{\ell}<\dfrac{b_{m-1}}{a_{m-1}} -\sqrt{\ell}
\end{align*}
for $m\in\bb{Z}_{>0}$.
We also have the following sequence of inequalities:
\begin{multline*}
-\infty=\dfrac{b_0}{a_0}
<-\dfrac{q}{p}=\dfrac{b_{-1}}{a_{-1}}
<\dfrac{b_{-2}}{a_{-2}}<\cdots<-\sqrt{\ell}<\cdots
<\dfrac{\ell a_{-2}}{b_{-2}}<\dfrac{\ell a_{-1}}{b_{-1}}=-\dfrac{\ell p}{q}
\\
<\dfrac{\ell a_0}{b_1}=0
<\dfrac{\ell a_1}{b_1}=\dfrac{\ell p}{q}
<\dfrac{\ell a_2}{b_2}
<\cdots<\sqrt{\ell}
<\cdots<\dfrac{b_2}{a_2}<\dfrac{b_1}{a_1}=\dfrac{q}{p}
<\dfrac{b_0}{a_0}=\infty.
\end{multline*}
\begin{NB}
Set $z_m := b_m + \sqrt{\ell} a_m$ and
$\overline{z}_m :=b_m - \sqrt{\ell} a_m$.
Then
$z=z_1=q+\sqrt{\ell}p$,
$\overline{z}_1=q-\sqrt{\ell}p$, and
\begin{align*}
&z_{m} = z_{m-1} z_1=z_1^m,\quad
\overline{z}_m = \overline{z}_{m-1} \overline{z}_1
=\overline{z}_1^m
\quad
(m \in \bb{Z}_{\ge0}).
\end{align*}
If $\epsilon=q^2-\ell p^2=+1$,
then $\overline{z}_1=q-\sqrt{\ell}p>0$.
Thus $\overline{z}_m>0$ for any $m\in\bb{Z}_{\ge0}$,
so that
\begin{align*}
\sqrt{\ell}<\dfrac{b_{m}}{a_{m}}
\quad
(m \in \bb{Z}_{>0}).
\end{align*}
We also have
\begin{align*}
& a_m b_{m-1}-a_{m-1}b_m
=(q^2-\ell p^2) (a_{m-1}b_{m-2}-a_{m-2}b_{m-1})
\\
&\Longleftrightarrow
\dfrac{b_{m-1}}{a_{m-1}}-\dfrac{b_m}{a_m}
=\dfrac{a_{m-2} }{a_m}(q^2-\ell p^2)
(\dfrac{b_{m-2}}{a_{m-2}}-\dfrac{b_{m-1}}{a_{m-1}})
\quad
(m\in \bb{Z}_{\ge2}).
\end{align*}
Note also that
$\tfrac{b_1}{a_1}-\tfrac{b_2}{a_2}
=\tfrac{q}{p}-\tfrac{q^2+\ell p^2}{2pq}
=\tfrac{q^2-\ell p^2}{2pq}>0$.
Thus we ahve
\begin{align*}
\dfrac{b_m}{a_m}<\dfrac{b_{m-1}}{a_{m-1}}
\quad
(m\in\bb{Z}_{\ge2}).
\end{align*}
\end{NB}
\begin{defn}
We set
\begin{align*}
\begin{array}{l l l l l l}
I_1 &:=&[0,\tfrac{\ell a_1}{b_1}) \cup [\tfrac{b_1}{a_1},\infty),
&
I_0 &:=&[-\infty,-\tfrac{b_1}{a_1}) \cup [-\tfrac{\ell a_1}{b_1},0),
\\[4pt]
I_{m+1} &:=&[\tfrac{\ell a_m}{b_m},\tfrac{\ell a_{m+1}}{b_{m+1}})
\cup
[\tfrac{b_{m+1}}{a_{m+1}},\tfrac{b_{m} }{a_{m} }),
&
I_{-m}&:=&[-\tfrac{b_m}{a_m},-\tfrac{b_{m+1}}{a_{m+1}})
\cup
[-\tfrac{\ell a_{m+1}}{b_{m+1}},-\tfrac{\ell a_{m}}{b_{m}})\quad
m \geq 1.
\end{array}
\end{align*}
\end{defn}
Then we have
$\bb{P}^1({\bb{R}}) \setminus \{\pm\sqrt{\ell}\}=
\coprod_{m \in \bb{Z}} I_m =\coprod_{m \in \bb{Z}} I_m^*$.
\begin{thm}\label{thm:csv:2}
For a Fourier-Mukai transform
$\Phi_{X \to X'}^{{\bl{G}^{\vee}}}\colon\bl{D}(X) \to \bl{D}(X')$,
we write
$c_1({\bl{G}}_{x'})/\operatorname{rk} {\bl{G}}_{x'}=(\lambda/\sqrt{n}) H$.
(1)
If $\lambda \in I_m$ $(m \leq 0)$, then
$\Phi_{X \to X'}^{\bl{G}^{\vee}}(F_m^\bullet)$ is a stable sheaf
up to shift.
(2)
If $\lambda \in I_m^*$ $(m \leq 0)$, then
$\mathcal{D}_{X'}\, \Phi_{X \to X'}^{\bl{G}^{\vee}}(F_m^\bullet)=
\Phi_{X \to X'}^{{\bl{G}}[2]}\, \mathcal{D}_{X}(F_m^\bullet)$
is a stable sheaf.
\end{thm}
The proof is based on the calculation
in this subsection and is similar to that of Theorem \ref{thm:csv:1}.
We omit the detail.
\section{Examples}\label{sect:example}
Let $X$ be a principally
polarized abelian surface with $\operatorname{NS}(X)={\mathbb Z}H$.
We shall study walls for $v=1-\ell \varrho_X$.
We note that $n:=\frac{(H^2)}{2}=1$.
By using Corollary \ref{cor:square},
it is easy to see that $s=0$ is the unique wall for $\ell=1$.
So we assume that $\ell \geq 2$.
We use the notations in section \ref{sect:rho=1}.
(1)
Assume that $\ell=2$.
In this case,
$S_{1,2}/\{\pm 1 \}$ is generated by
\begin{equation*}
A_2:=
\begin{pmatrix}
1 & 2\\
1 & 1
\end{pmatrix}.
\end{equation*}
Hence we have a numerical solution $v=2(1,-H,1)-(1,-2H,4)$.
We have $u_{-1}=(1,-H,1)$ and
\begin{NB}
Then $C_{-1}=W_{u_{-1}}$ is the circle in the $z$-plane
$$
\left |z+\frac{3}{2} \right|=\frac{1}{2}.
$$
\end{NB}
$C_{-1}=W_{u_{-1}}$ is the circle in the $(s,t)$-plane
$$
\left(s+\frac{3}{2} \right)^2+t^2=\frac{1}{2^2}.
$$
For $s=-1$, we get $c_1(v e^{-sH})=H$.
Thus there is no wall intersecting with $s=-1$.
Then we see that there is no wall between $C_{-1}$ and $C_0:s=0$
(see Figure 1).
Hence we get the following result.
\begin{lem}
$C_n$ $(n \in {\mathbb Z})$ are all the walls for $v=1-2 \varrho_X$.
\end{lem}
\begin{prop}
Let $v$ be a positive and primitive
Mukai vector with $\langle v^2 \rangle=4$.
Then $M_H(v)$ is isomorphic to
$\operatorname{Hilb}{2}{X} \times X$.
\end{prop}
\begin{proof}
There is a Fourier-Mukai transform
$\Phi_{X \to X}^{{\bf E}^{\vee}}:{\bf D}(X) \to {\bf D}(X)$
such that $\Phi_{X \to X}^{{\bf E}^{\vee}}((1,0,-2))=v$.
Then we have an isomorphism
$M_{(\beta,\omega)}(1,0,-2) \to M_H(v)$,
where $\beta=c_1({\bf E}_{|\{ x\} \times X})/\operatorname{rk} {\bf E}_{|\{ x\} \times X}$
and $(\omega^2) \ll 1$.
By Theorem \ref{thm:B:3-10.3} and
Proposition \ref{prop:fund},
$M_{(\beta,\omega)}(1,0,-2) \cong
\operatorname{Hilb}{2}{X} \times X$, which implies the claim.
\end{proof}
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\caption{Walls for $v=1-2\varrho_X$.}
\end{figure}
(2)
Assume that $\ell=3$.
In this case,
$S_{1,3}/\{\pm 1 \}$ is generated by
\begin{equation*}
A_3:=
\begin{pmatrix}
2 & 3\\
1 & 2
\end{pmatrix}.
\end{equation*}
Hence we have a numerical solution $v=(4,-6H,9)-3(1,-2H,4)$.
We have $u_{-1}=(1,-2H,4)$ and
\begin{NB}
$W_{u_{-1}}$ is the circle in $z$-plane defined by
$$
\left|z+\frac{7}{4} \right|=\frac{1}{4}.
$$
\end{NB}
$W_{u_{-1}}$ is the circle in $(s,t)$-plane defined by
$$
\left(s+\frac{7}{4} \right)^2+t^2=\frac{1}{4^2}.
$$
We set $w_{-1}:=(1,-H,1)$.
Then $v=w_{-1}+(0,H,-4)$ and $w_{-2}$ defines a wall
$W_{w_{-1}}$ while the defining equation is
\begin{NB}
$$
\left|z+2 \right|=1.
$$
\end{NB}
$$
\left(s+2 \right)^2+t^2=1.
$$
\begin{lem}
$W_{w_{-1}}$ is the unique wall between $C_0$ and $C_{-1}=W_{u_{-1}}$
(see Figure 2).
\end{lem}
\begin{proof}
Since $W_{u_{-1}}$ passes the point $(s,t)=(-2,0)$,
it is sufficient to classify walls for $s=-2$.
Assume that $w e^{2H}=(r,dH,a)$ defines a wall for $s=-2$.
Since $v e^{2H}=(1,2H,1)$,
we have $d=1$. We set $w':=(v-w)$ and write
$w' e^{2H}=(r',H,a')$.
By the definition of walls,
we have $\langle w^2 \rangle \geq 0, \langle {w'}^2 \rangle \geq 0$
and $\langle w,w' \rangle>0$.
Thus $ra \leq 1$, $r'a' \leq 1$ and $1-r a'-r' a>0$.
Since $r+r'=1$, we may assume that $r>0$ and $r' \leq 0$.
Since $a+a'=1$, we have
$0<1-r a'-r' a=(2a-1)r+(1-a)$.
If $a \leq 0$, then $(2a-1)r+(1-a) \leq (2a-1)+(1-a)=a \leq 0$.
Hence $a \geq 1$.
Since $ra \leq 1$, we have $r=a=1$.
Therefore $w e^{2H}=(1,H,1)$, which implies that
$w=(1,-H,1)$.
\end{proof}
\begin{NB}
Old orgument:
\begin{proof}
For $w=(r,dH,a)$, $c_1(w e^{-sH})=(d-rs)H$.
If $s=-\frac{3}{2}$, then
$$
\{c_1(w e^{-sH})| w \in H^*(X,{\mathbb Z})_{\operatorname{alg}} \}=\frac{1}{2}{\mathbb Z}H.
$$
If $w=(r,dH,a)$ defines a wall for $s=-3/2$, then
$c_1(w e^{-sH})=\frac{1}{2}H, H$.
Since $c_1((v-w)e^{-sH})=\frac{3}{2}H-c_1(w e^{-sH})$, we may assume that
$c_1(w e^{-sH})=\frac{1}{2}H$.
Thus $w=(1+2\lambda,-(1+3\lambda)H,a)$, where
$\lambda \in {\mathbb Z}$.
We set $w':=v-w=(-2\lambda,(1+3\lambda)H,-3-a)$.
Then
$c_1(w e^{-sH})=-((1+s)+(2s+3)\lambda)H$ and
$c_1(w' e^{-sH})=(1+(2s+3)\lambda)H$.
Since $W_w$ contains $W_{u_1}$,
$-((1+s)+(2s+3)\lambda)>0$ and
$(1+(2s+3)\lambda)>0$ for $-2 \leq s \leq -3/2$.
For $s=-2$, we have $1-\lambda>0$ and $1+\lambda>0$.
Hence $\lambda=0$.
Then $W_w$ is
$$
\left(s+\frac{a+3}{2} \right)^2+t^2=\left(\frac{a+3}{2} \right)^2-3.
$$
Since $a \leq 1$, $a=0$ if $a+3 \geq 0$.
If $a+3<0$, then $W_w$ is contained in $s>0$.
Therefore $w=(1,-H,1)$.
\end{proof}
\end{NB}
We have
$u_0=(0,0,1)$, $u_{-1}=(1,-2H,4)$, $u_{-2}=(4^2,-28H,7^2)$
and so on.
We define $w_n \in H^*(X,{\mathbb Z})_{\operatorname{alg}}$ by
$$
w_n:=(a_n^2,a_n b_n H,b_n^2),\;
(a_n,b_n)=(1,-1)A_3^{n-1}.
$$
Thus
$w_{-1}=(1,-H,1)$, $w_{-2}=(9,-15H,5^2)$ and so on.
By Proposition \ref{prop:wall:isometry}
and Proposition \ref{prop:fund}, we get the following.
\begin{lem}
$W_{u_n}$ and $W_{w_n}$ are all the walls for $v$.
\end{lem}
Therefore all moduli spaces are isomorphic
to $\operatorname{Hilb}{3}{X} \times X$.
\begin{NB}
By the action of $A$,
the anulus $W_{u_n} \setminus W_{w_{n+1}}$
and
$W_{w_{n+1}} \setminus W_{u_{n+1}}$
are transformed to
the anulus
$W_{u_{n+1}} \setminus W_{w_{n+2}}$
and
$W_{w_{n+2}} \setminus W_{u_{n+2}}$
respectively.
We set
$$
B_n:=A^{-n}
\begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix}
A^n.
$$
Then
$\theta(\Psi_n)=B_n$ and
by the action of $B_n$,
$W_{w_{n}} \setminus W_{u_{n}}$
are transformed to
$W_{u_{n}} \setminus W_{w_{n+1}}$.
\begin{NB2}
\begin{equation*}
A
\begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix}
=\begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix}
A^{-1}.
\end{equation*}
\begin{equation*}
(0,1)A^{n-k} B_n=(0,1)A^{-k}
\begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix}
A^n
=(0,1)
\begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix}
A^{n+k}
=-(0,1)A^{n+k}.
\end{equation*}
\begin{equation*}
(1,-1)A^{n-k} B_n=(1,-1)A^{-k}
\begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix}
A^n
=(1,-1)
\begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix}
A^{n+k}
=(1,-1)A^{n+k-1}.
\end{equation*}
\end{NB2}
\end{NB}
\begin{prop}
Let $v$ be a positive and primitive
Mukai vector with $\langle v^2 \rangle=6$.
Then $M_H(v)$ is isomorphic to
$\operatorname{Hilb}{3}{X} \times X$.
\end{prop}
\begin{proof}
There is a Fourier-Mukai transform
$\Phi_{X \to X}^{{\bf E}^{\vee}}:{\bf D}(X) \to {\bf D}(X)$
such that $\Phi_{X \to X}^{{\bf E}^{\vee}}((1,0,-3))=v$.
Then we have an isomorphism
$M_{(\beta,\omega)}(1,0,-3) \to M_H(v)$,
where $\beta=c_1({\bf E}_{|\{ x\} \times X})/\operatorname{rk} {\bf E}_{|\{ x\} \times X}$
and $(\omega^2) \ll 1$.
By Theorem \ref{thm:B:3-10.3} and
Proposition \ref{prop:fund},
$M_{(\beta,\omega)}(1,0,-3) \cong
\operatorname{Hilb}{3}{X} \times X$, which implies the claim.
\end{proof}
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\end{picture}%
\caption{Walls for $v=1-3\varrho_X$.}
\end{figure}
(3)
Assume that $\ell=4$.
By Corollary \ref{cor:square},
all walls in $s<0$ intersect with the line $s=-\sqrt{4}=-2$.
Assume that $v_1$ defines a wall for $v$.
Since $v e^{2H}=(1,2H,0)$, $u_1 e^{2H}=(r,H,a)$.
Hence $v_1=(r,(1-2r)H,a-4+4r)$ and
$v-v_1=(1-r,(2r-1)H,-a-4r)$. Replacing $v_1$ by
$v-v_1$ if necessary,
we may assume that $r> 0$.
We have $2n:=\langle v_1^2 \rangle \geq 0$, $\langle (v-v_1)^2 \rangle \geq 0$
and $\langle v_1,v-v_1 \rangle >0$.
Hence $n=1-ra$, $4-2n>a \geq -n$.
Then $n=0,1,2,3$, which implies that $ra=1,0,-1,-2$.
We also have $\left|\frac{a-4+4r+4r}{2(1-2r)}\right| >2$,
which implies that $a(8(2r-1)+a)>0$. In particular, $a \ne 0$.
Then $ra=1,0,-1,-2$ and $r> 0$ implies that
$r=1,2$.
\begin{NB}
We don't need the following argument, since $r>0$:
If $r=0$, then $n=1$ and $2>a \geq -1$. Since $a(a-8)>0$,
$a=-1$. Thus $v_1=(0,H,-5)$.
\end{NB}
If $r=1$, then $a(a+8)>0$ and $ra=1,0,-1,-2$ imply that
$a=1$. Thus $v_1=(1,-H,1)$.
If $r=2$, then $a=-1$ and $a(a+24)>0$, which is impossible.
Therefore $v_1=(1,-H,1)$.
Thus $W_{v_1}$ is the unique wall which is defined by
\begin{NB}
$$
\left|z+\frac{5}{2} \right|=\frac{3}{2}.
$$
\end{NB}
$$
\left(s+\frac{5}{2} \right)^2+t^2=\frac{3^2}{2^2}.
$$
\begin{prop}
Let $v$ be a positive and primitive
Mukai vector with $\langle v^2 \rangle=8$.
Then $M_H(v)$ is isomorphic to
$\operatorname{Hilb}{4}{X} \times X$ or $M_H(0,2H,-1)$.
\end{prop}
\begin{proof}
We first prove that $M_H(1,0,-4) \not \cong
M_H(0,2H,-1)$.
We note that the Hilbert-Chow morphism
of $\operatorname{Hilb}{4}{X}$ induces
a divisorial contraction
of $M_H(1,0,-4)$.
We note that $M_H(3,H,-1)$ has a morphism
to the Uhlenbeck compactification of
the moduli of stable vector bundles,
which contracts a ${\mathbb P}^2$-bundle
over $M_H(3,H,0)$.
Let ${\bf P}$ be the Poincar\'{e} line bundle on
$X \times X$, where we identify
$\Piczero{X}$ with $X$.
Then we have an isomorphism
$M_H(1,0,-4) \cong M_H(1,H,-3) \cong
M_H(3,H,-1)$ by sending $E$ to
$\Phi_{X \to X}^{{\bf P}[1]}(E(H))$
\cite[Prop. 3.5]{Y:7}.
Hence $M_H(1,0,-4)$ has another contraction.
There is no other contraction by a similar
argument in \cite[Example 7.2]{Y:7}.
On the other hand,
$M_H(0,2H,-1)$ has a Lagrangian fibration.
Therefore $M_H(1,0,-4) \not \cong M_H(0,2H,-1)$.
For $s=-2$ and $v=(1,0,-4)$, we have two moduli spaces
$M_{(-2H,t_1 H)}(1,0,-4)$ and
$M_{(-2H,t_2 H)}(1,0,-4)$, where $t_1 >\sqrt{2}$ and
$t_2<\sqrt{2}$.
For $E \in M_{(-2H,t_2 H)}(1,0,-4)$,
$\Phi_{X \to X}^{{\bf P}[1]}(E(2H))
\in M_H(0,2H,-1)$ and we have an isomorphism
$M_{(-2H,t_2 H)}(1,0,-4) \cong M_H(0,2H,-1)$.
By \cite[sect. 7.3]{YY}, every quadratic form
$r x^2+2d xy+a y^2$ is equivalent to
$x^2-4y^2$.
Since $(1,0,-4)^{\vee}=(1,0,-4)$,
there is an auto-equivalence
$\Phi$ such that $\Phi(v)=(1,0,-4)$.
Then $\Phi(M_H(v)) \cong M_{(sH,tH)}(1,0,-4)$
for a suitable $(s,t)$.
Therefore the claim holds.
\end{proof}
(4)
Assume that $\ell=5$.
In this case,
$S_{1,5}/\{ \pm 1 \}$ is generated by
\begin{equation*}
A_5:=
\begin{pmatrix}
2 & 5\\
1 & 2
\end{pmatrix}.
\end{equation*}
Hence we have a numerical solution
$v=5(1,-2H,4)-(4,-10H,25)$.
Then we have $u_{-1}=(1,-2H,4)$ and
$W_{u_{-1}}$ is the circle defined by
$$
\left(s+\frac{9}{4} \right)^2+t^2=\frac{1}{4^2}.
$$
(5)
Assume that $\ell=6$.
In this case,
$S_{1,6}/\{ \pm 1 \}$ is generated by
\begin{equation*}
A_6 :=
\begin{pmatrix}
5 & 12\\
2 & 5
\end{pmatrix}.
\end{equation*}
Hence we have a numerical solution
$v=(5^2,-60H,12^2)-6(4,-10H,5^2)$.
Then we have $u_{-1}=(5^2,-10H,5^2)$ and
$W_{u_{-1}}$ is the circle defined by
$$
\left(s+\frac{49}{20} \right)^2+t^2=\frac{1}{20^2}.
$$
\begin{rem}
Let $X$ be an arbitrary abelian surface and $H$ an ample
divisor on $X$.
Let $v$ be a primitive Mukai vector with
$\langle v^2 \rangle/2=1$.
Then we have an isomorphism
$X \times \Piczero{X} \to M_H(v)$ by sending
$(x, L) \in X \times \Piczero{X}$ to
$T_x^*(E_0) \otimes L$, where $E_0$ is an element
of $M_H(v)$ and $T_x$ is the translation by $x$
(\cite[Cor. 4.3]{Y:nagoya}).
\end{rem}
\section{Appendix}
\subsection{The action of Fourier-Mukai transforms on ${\mathbb H}$.}
Assume that $\operatorname{NS}(X)={\mathbb Z}H$.
Then $\beta+\sqrt{-1}\omega=\frac{z}{ \sqrt{n}}H$ with
$z \in {\mathbb H}$, where
$z:=x+\sqrt{-1}y$ with $x \in {\mathbb R}$ and $y \in {\mathbb R}_{>0}$.
Thus we have an identification of
$\operatorname{NS}(X)_{\mathbb R} \times \operatorname{Amp}(X)_{\mathbb R}$ with
${\mathbb H}$.
We set $Z_z:=Z_{(\beta,\omega)}$.
We study the action of $\Phi$ with $\gamma \in {\mathbb Q}H$.
We write
$$
\gamma'+\widehat{\xi}+\sqrt{-1} \widehat{\eta}=
\frac{z'}{\sqrt{n}}H,\; z' \in {\mathbb H}.
$$
Let $\Phi_{X \to X_1}^{\bf E}:{\bf D}(X) \to
{\bf D}(X_1)$
be a Fourier-Mukai transform
such that ${\bf E}$ is a coherent sheaf.
Then there is
\begin{equation*}
A=\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\in G
\end{equation*}
such that
$\mu(\Phi_{X \to X_1}^{\bf E}({\mathfrak k}_x))=\frac{a}{c} \sqrt{n}$
and
$\mu(\Phi_{X \to X_1}^{\bf E}({\mathcal O}_X))=\frac{b}{d}\sqrt{n}$.
$X_1=M_H(c^2 e^{\frac{d}{c}\frac{H}{\sqrt{n}}})$ and
${\bf E}$ is unique up to the action of $X \times \Piczero{X}$.
Then
$$
\theta(\Phi_{X \to X_1}^{{\bf E}})=
\begin{pmatrix}
d & b \\
c & a
\end{pmatrix}
\in G.
$$
\begin{defn}
For $\Phi_{X \to X_1}^{\bf E}$,
we set
$$
\varphi(\Phi_{X \to X_1}^{\bf E}):=
\pm \begin{pmatrix}
a & b\\
c & d
\end{pmatrix} \in G/\{\pm 1\}.
$$
\end{defn}
For $\Phi=\Phi_{X \to X_1}^{{\bf E}}$,
we have
\begin{equation*}
\begin{split}
r_1 e^{\gamma}= &c^2-\frac{cd}{\sqrt{n}}H+d^2 \varrho_X, \\
r_1 e^{\gamma'}= & c^2+\frac{ac}{\sqrt{n}}\widehat{H}+a^2 \varrho_{X_1}.
\end{split}
\end{equation*}
Hence
\begin{equation*}
\frac{z}{\sqrt{n}}+\frac{d}{c \sqrt{n}}=-\lambda+\sqrt{-1} t,\;
\frac{z'}{\sqrt{n}}-\frac{a}{c \sqrt{n}}=
\frac{\lambda+\sqrt{-1} t}{(\lambda^2+t^2)nc^2}.
\end{equation*}
Thus we get
\begin{equation*}
\begin{split}
\frac{z'}{\sqrt{n}}=& \frac{a}{c \sqrt{n}}-\frac{1}{c^2 n
(\frac{z}{\sqrt{n}}+\frac{d}{c \sqrt{n}})}\\
=& \frac{ac z-(1-ad)}{c(cz+d)\sqrt{n}}
=\frac{a z+b}{\sqrt{n}(c z+d)}.
\end{split}
\end{equation*}
Therefore the action of $A$ on ${\mathbb H}$ is the natural action of
$\SL(2,{\mathbb R})$.
$$
\zeta=-c^2 \frac{(\lambda-t \sqrt{-1})^2 (H^2)}{2}=
-c^2 \frac{(\frac{z}{\sqrt{n}}+\frac{d}{c\sqrt{n}})^2 (H^2)}{2}
=-(cz+d)^2.
$$
\begin{prop}
For $\Phi_{X \to X_1}^{\bf E}$ with
$\varphi(\Phi_{X \to X_1}^{\bf E})=
\begin{pmatrix}
a & b\\
c & d
\end{pmatrix}
\in G$,
$$
-(cz+d)^2 Z_{\frac{az+b}{cz+d}\frac{\widehat{H}}{\sqrt{n}}}
=Z_{z \frac{H}{\sqrt{n}}} \circ (\Phi_{X \to X_1}^{\bf E})^{-1}.
$$
We also have
$$
\Phi_{X \to X_1}^{\bf E}({\mathbb Q}e^{\lambda \frac{H}{{\sqrt{n}}}})
={\mathbb Q}e^{\frac{a\lambda+b}{c\lambda+d}\frac{\widehat{H}}{\sqrt{n}}}.
$$
\end{prop}
We now extend the action of $G$ to $\widehat{G}$.
We set
$$
\Delta:=\begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix}
$$
We note that
$$
\widehat{G}=G \rtimes \left\langle
\Delta
\right\rangle
$$
with
$$
\begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix}
\begin{pmatrix}
a & b\\
c & d
\end{pmatrix}
\begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix}
=
\begin{pmatrix}
a & -b\\
-c & d
\end{pmatrix}.
$$
We define the action of
$\Delta$ on ${\mathbb H}$ as $\Delta(z):=-\overline{z}$.
Then we have
$$
\Delta(A(\Delta(z)))=\frac{az-b}{-cz+d},\;
A=\begin{pmatrix}
a & b\\
c & d
\end{pmatrix}.
$$
Thus we have an action of $\widehat{G}$ on
${\mathbb H}$.
\begin{prop}
We can extend the action of $G$ to
the action of $\widehat{G}$ by
$$
(g,\Delta^n) \cdot z:=
\begin{cases}
g \cdot z, & 2|n,\\
-\overline{g \cdot z},& 2 \not | n,
\end{cases}
$$
where $g \in G$.
\end{prop}
\begin{rem}
In \cite{YY}, we showed that
the cohomological action of
$\Eq_0({\bf D}(X),{\bf D}(X))$
defines a normal subgroup of $G$
which is a conjugate of $\Gamma_0(n)$ in
$\GL(2,{\mathbb R})$.
More precisely,
we set
$G_0:=\theta(\Eq_0({\bf D}(X),{\bf D}(X)))$.
Then
\begin{equation*}
\begin{pmatrix}
\sqrt{n} & 0 \\
0 & 1
\end{pmatrix}^{-1}
G_0
\begin{pmatrix}
\sqrt{n} & 0 \\
0 & 1
\end{pmatrix}
=\Gamma_0(n).
\end{equation*}
We set $\beta+\sqrt{-1}\omega=w H$.
Then $\Eq_0({\bf D}(X),{\bf D}(X))$ acts on $w$-plane
as the action of $\Gamma_0(n)$ on $w$-plane.
\begin{NB}
We shall explain known description of Mukai lattices
\cite[sect. 5]{Ma}.
We set
\begin{equation*}
L_n:=
\left\{
\left.
\begin{pmatrix}
np & q \\
nr & -np
\end{pmatrix}
\in {\mathfrak {sl}}_2({\mathbb R}) \right|
p,q,r \in {\mathbb Z}
\right\}.
\end{equation*}
For $B_1,B_2 \in L_n$, we set
$(B_1,B_2):=\operatorname{tr}(B_1 B_2)/(2n) \in {\mathbb Z}$.
Then we have an isometry
$$
\begin{matrix}
\zeta:&
\operatorname{Sym}_2(n,{\mathbb Z}) & \to & L_n\\
&
\begin{pmatrix}
x & y \sqrt{n} \\
y\sqrt{n} & z
\end{pmatrix}
& \mapsto &
\begin{pmatrix}
ny & -x \\
nz & -ny
\end{pmatrix}.
\end{matrix}
$$
It is easy to see that
the induced action of $G_0$ on $L_n$ is the adjoint action of
$\Gamma_0(n)$. Thus
$$
g
\begin{pmatrix}
ny & -x \\
nz & -ny
\end{pmatrix}
g^{-1},\;
g=
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\in \Gamma_0(n)
$$
corresponds to
$$
{{}^t g_1}
\begin{pmatrix}
x & y \sqrt{n} \\
y \sqrt{n} & z
\end{pmatrix}
g_1,\;
g_1=
\begin{pmatrix}
a & \frac{c}{\sqrt{n}} \\
b \sqrt{n} & d
\end{pmatrix}
\in G_0
$$
via $\zeta^{-1}$.
\end{NB}
\end{rem}
\begin{NB}
We set $w:=\frac{z}{\sqrt{n}}$ and $w':=\frac{z'}{\sqrt{n}}$.
Then
$$
-(c\sqrt{n} w+d)^2 Z_{\frac{aw+\frac{b}{\sqrt{n}}}{c\sqrt{n}w+d}\widehat{H}}
=Z_{w H} \circ \widehat{\Phi}.
$$
\begin{equation*}
\begin{pmatrix}
a & \frac{b}{\sqrt{n}} \\
c\sqrt{n} & d
\end{pmatrix}=
\begin{pmatrix}
\sqrt{n} & 0 \\
0 & 1
\end{pmatrix}^{-1}
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\begin{pmatrix}
\sqrt{n} & 0 \\
0 & 1
\end{pmatrix}
\end{equation*}
and
$$
\Gamma_0(n)=
\begin{pmatrix}
\sqrt{n} & 0 \\
0 & 1
\end{pmatrix}^{-1}
\ker \phi
\begin{pmatrix}
\sqrt{n} & 0 \\
0 & 1
\end{pmatrix}
$$
is the group of the auto-equivalences.
\end{NB}
\begin{NB}
For a rational number $\frac{a}{b}$,
there is an isotropic Mukai vector
$(p^2 n,pq H,q^2)$ such that
$(p n,q)=1$ and $\frac{q}{pn}$ is sufficiently close to
$\frac{a}{b}$.
We take $p_1, q_1 \in {\mathbb Z}$ such that
$(p_1 n,q_1)=1$.
We set $(p,q)=(p_1(1+b q_1 m),q_1(1+an p_1 m))$.
Then $(n,q_1(1+an p_1 m))=1$ for all $m$.
We shall prove that
there are sufficiently large integer $m$
such that $(1+b q_1 m,1+an p_1 m)=1$.
We first assume that $|b q_1| < |an p_1|$.
Then $|1+b q_1 m| < |1+an p_1 m|$ for $|m| \gg 0$.
We can choose $m$ such that $1+an p_1 m$ is prime.
Then $(1+b q_1 m, 1+an p_1 m)=1$.
If $|b q_1| > |an p_1|$, then we also get the claim.
Assume that $|b q_1| = |an p_1|$. Then
$a=q_1 x$ and $b= \pm p_1 n x$.
Hence $\frac{a}{b}=\pm \frac{q_1}{n p_1}$.
So if $(p_1,q_1)$ satisfies
$\frac{a}{b} \ne \pm \frac{q_1}{n p_1}$,
then we can take a suitable $m$.
\end{NB}
\begin{NB}
\subsection{Modular groups.}
\begin{equation}
\begin{pmatrix}
x & y \\
y & z
\end{pmatrix}
\begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix}
=
\begin{pmatrix}
y & -x \\
z & -y
\end{pmatrix}
\in {\mathfrak {sl}}_2({\mathbb R})
\end{equation}
Since
$$
\begin{pmatrix}
d & -c \\
-b & a
\end{pmatrix}=
\begin{pmatrix}
0 & 1 \\
-1 & 0
\end{pmatrix}
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}\begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix},
$$
$$
^t P
\begin{pmatrix}
x & y \\
y & z
\end{pmatrix}
P
\begin{pmatrix}
0 & 1 \\
-1 & 0
\end{pmatrix}
=
(\det P) {^t P}
\begin{pmatrix}
y & -x \\
z & -y
\end{pmatrix}
{^t P^{-1}}
$$
We define an anti-homomorphism
\begin{equation}
\begin{matrix}
\varphi: & \GL_2({\mathbb Q})_+ & \to & \SL_2({\mathbb R})\\
& Q & \mapsto & \frac{^t Q}{\sqrt{\det Q}}
\end{matrix}
\end{equation}
Then
$$
^t \varphi(Q)
\begin{pmatrix}
x & y \\
y & z
\end{pmatrix}
\varphi(Q)
\begin{pmatrix}
0 & 1 \\
-1 & 0
\end{pmatrix}
=
Q
\begin{pmatrix}
y & -x \\
z & -y
\end{pmatrix}
{Q^{-1}}.
$$
Let $M_\sigma, N_\sigma$ be positive integers
such that $M_\sigma N_\sigma=n$ and
$\gcd(M_\sigma, N_\sigma)=1$.
We take integers $a_\sigma, b_\sigma$
such that $a_\sigma N_\sigma-b_\sigma M_\sigma=1$.
For
$$
\gamma_\sigma:=
\begin{pmatrix}
a_\sigma N_\sigma & b_\sigma \\
n & N_\sigma
\end{pmatrix},
$$
$$
\begin{pmatrix}
1 & 0\\
0 & \sqrt{n}
\end{pmatrix}
\varphi(\gamma_\sigma)
\begin{pmatrix}
1 & 0\\
0 & \sqrt{n}
\end{pmatrix}^{-1}
=
\frac{1}{\sqrt{N_\sigma}}
^t \begin{pmatrix}
a_\sigma N_\sigma & b_\sigma \sqrt{n} \\
\sqrt{n} & N_\sigma
\end{pmatrix}
=
{^t \begin{pmatrix}
a_\sigma \sqrt{N_\sigma} & b_\sigma \sqrt{M_\sigma} \\
\sqrt{M_\sigma} & \sqrt{N_\sigma}
\end{pmatrix}}.
$$
$$
\begin{pmatrix}
1 & 0\\
0 & \sqrt{n}
\end{pmatrix}
\varphi
\left(
\begin{pmatrix}
a & b\\
nc & d \\
\end{pmatrix}
\right)
\begin{pmatrix}
1 & 0\\
0 & \sqrt{n}
\end{pmatrix}^{-1}
=
^t \begin{pmatrix}
a & c\sqrt{n} \\
b\sqrt{n} & d
\end{pmatrix}.
$$
We have equivalriant morphisms
\begin{equation}
\begin{matrix}
\operatorname{Sym}_2(n,{\mathbb Z}) & \to & \operatorname{Sym}_2({\mathbb R}) & \to & \mathfrak{sl}_2({\mathbb R}) \\
\begin{pmatrix}
x & \sqrt{n} y\\
\sqrt{n} y & z
\end{pmatrix}
& \mapsto &
\begin{pmatrix}
x & n y\\
n y & nz
\end{pmatrix}
& \mapsto &
\begin{pmatrix}
ny & -x\\
nz & -ny
\end{pmatrix}
\end{matrix}
\end{equation}
\begin{equation}
\begin{pmatrix}
1 & 0\\
0 & \sqrt{n}
\end{pmatrix}
\begin{pmatrix}
x & \sqrt{n} y\\
\sqrt{n} y & z
\end{pmatrix}
\begin{pmatrix}
1 & 0\\
0 & \sqrt{n}
\end{pmatrix}
=
\begin{pmatrix}
x & n y\\
n y & nz
\end{pmatrix}
\end{equation}
\begin{equation}
\begin{pmatrix}
1 & 0\\
0 & \sqrt{n}
\end{pmatrix}^{-1}
\begin{pmatrix}
a & b\\
c & d
\end{pmatrix}
\begin{pmatrix}
1 & 0\\
0 & \sqrt{n}
\end{pmatrix}
=
\begin{pmatrix}
a & b \sqrt{n}\\
\frac{c}{\sqrt{n}} & d
\end{pmatrix}
\end{equation}
\begin{equation}
\begin{pmatrix}
ny & -x\\
n z & -ny
\end{pmatrix}
\begin{pmatrix}
d & -\frac{c}{\sqrt{n}}\\
-b \sqrt{n} & a
\end{pmatrix}
=
\begin{pmatrix}
1 & 0\\
0 & \sqrt{n}
\end{pmatrix}
\begin{pmatrix}
x & \sqrt{n} y\\
\sqrt{n} y & z
\end{pmatrix}
\begin{pmatrix}
a & b\\
c & d
\end{pmatrix}
\begin{pmatrix}
1 & 0\\
0 & \sqrt{n}
\end{pmatrix}
\begin{pmatrix}
0 & -1\\
1 & 0
\end{pmatrix}
\end{equation}
\begin{equation}
\begin{split}
&
\begin{pmatrix}
1 & 0 \\
0 & \sqrt{n}
\end{pmatrix}
{^t \varphi \left(
\begin{pmatrix}
a & b \sqrt{n} \\
c \sqrt{n} & d
\end{pmatrix}
\right)}
\begin{pmatrix}
x & y \sqrt{n} \\
y \sqrt{n} & z
\end{pmatrix}
\varphi \left(
\begin{pmatrix}
a & b \sqrt{n} \\
c \sqrt{n} & d
\end{pmatrix}
\right)
\begin{pmatrix}
1 & 0 \\
0 & \sqrt{n}
\end{pmatrix}
\begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix}\\
=& ^t
\begin{pmatrix}
a & b \\
cn & d
\end{pmatrix}
\begin{pmatrix}
ny & -x \\
nz & -ny
\end{pmatrix}
\begin{pmatrix}
a & b \\
cn & d
\end{pmatrix}^{-1}
\end{split}
\end{equation}
It seems that the cusps of $\Gamma_0(n)$ corresponds to
the Fourier-Mukai partner of $X$.
\end{NB}
|
1,116,691,500,857 | arxiv | \section{Main result and discussion}
\label{intro}
\subsection{Smallest hyperbolic manifolds}
\label{ss:intro-thm}
Let $\mathbf H^n$ be the hyperbolic $n$-space.
By a \emph{hyperbolic $n$-manifold} we mean an orientable manifold
$M = \Gamma\backslash \mathbf H^n$, where $\Gamma$ is a torsion-free discrete subgroup
$\Gamma \subset \mathrm{Isom}^+(\mathbf H^n)$.
The set of volumes of hyperbolic $n$-manifolds being well ordered, it is
natural to try to determine for each dimension $n$ the hyperbolic manifolds
of smallest volume.
For $n=3$ this problem has recently been solved
in \cite{GMM10}, the smallest volume being achieved by a unique compact manifold,
the Weeks manifold. When $n$ is even the volume is proportional to the
Euler characteristic, and this allows to formulate the problem in terms
of finding the hyperbolic manifolds $M$ with smallest $|\chi(M)|$. In
particular this observation solves the problem in the case of surfaces.
For $n>3$, noncompact hyperbolic $n$-manifolds $M$ with $|\chi(M)| = 1$
have been found for $n = 4, 6$ \cite{ERT10}.
In the present paper we consider the case of compact
manifolds of even dimension. In particular, such manifolds have even
Euler characteristic (see \cite[Theorem 1.2]{KelZeh01}). We restrict ourselves to the case of
\emph{arithmetic} manifolds, where Prasad's formula \cite{Pra89} can be used to study
volumes. We complete the proof of the following result.
\begin{theorem}
\label{thm:non-existence}
Let $n>5$. There is no compact arithmetic manifold $M =
\Gamma\backslash\mathbf H^n$ with $|\chi(M)| = 2$.
\end{theorem}
The result for $n > 10$ already follows from the work of Belolipetsky
\cite{Belo04,Belo07}, also based on Prasad's volume formula. More
precisely, Belolipetsky determined the smallest Euler characteristic
$|\chi(\Gamma)|$ for arithmetic orbifold quotients $\Gamma\backslash\mathbf H^n$ ($n$
even). This smallest value grows fast with the dimension $n$, and for
compact quotients we have $|\chi(\Gamma)| > 2$ for $n>10$.
That the result of nonexistence holds
for $n$ high enough is already a consequence of Borel-Prasad's general
finiteness result \cite{BorPra89}, which was the first application of
Prasad's formula. The proof of Theorem \ref{thm:non-existence} for
$n=6, 8, 10$ requires a more precise analysis of the Euler
characteristic of arithmetic subgroups $\Gamma \subset \PO(n,1)$,
and in particular of the special values of Dedekind zeta functions
that appear as factors of $\chi(\Gamma)$.
For $n = 4$, the corresponding problem is not solved, but there is the
following result \cite{Belo07}.
\begin{theorem}[Belolipetsky]
If $M = \Gamma\backslash\mathbf H^4$ is a compact arithmetic manifold with
$\chi(M) \le 16$, then $\Gamma$ arises as a (torsion-free) subgroup of the following
hyperbolic Coxeter group:
\begin{align}
W_1 &= \;\; \Cox
\label{W1}
\end{align}
\label{thm:dim-4}
\end{theorem}
An arithmetic (orientable) hyperbolic $4$-manifold of
Euler characteristic $16$ has been first constructed by
Conder and Maclachlan in \cite{CondMac05}, using the
presentation of $W_1$ to obtain a torsion-free subgroup with the help of
a computer. Further examples with $\chi(M) = 16$ have been obtained by Long
in \cite{Long08} by considering a homomorphism from $W_1$ onto the finite simple
group $\mathrm{PSp}_4(4)$.
\subsection{Hyperbolic homology spheres}
\label{ss:homology-spheres}
Our original motivation for Theorem \ref{thm:non-existence} was the
problem of existence of hyperbolic homology spheres. A \emph{homology
$n$-sphere} (resp. \emph{rational} homology $n$-sphere) is a
$n$-manifold $M$ that possesses the same integral (resp. rational)
homology as the $n$-sphere $S^n$. This forces $M$ to be compact and
orientable.
Rational homology $n$-spheres $M$ have $\chi(M) = 2$ if $n$ is even.
On the other hand, for $M = \Gamma\backslash\mathbf H^n$ with $n = 4k+2$ we have
$\chi(M) < 0$ (cf. \cite[Proposition 23]{Serr71}), and this exclude the possibility of hyperbolic
rational homology spheres for those dimensions. For $n$ even,
Wang's finiteness theorem \cite{Wang72} implies that there is only a
finite number of hyperbolic rational homology $n$-spheres.
Theorem \ref{thm:non-existence} shows the nonexistence of arithmetic rational
homology spheres for $n>5$ even.
For odd dimensions, $\chi(M) = 0$ and \emph{a priori} the volume is not
a limitation for the existence of hyperbolic (rational) homology
spheres. In fact, an infinite tower of covers by hyperbolic integral homology
$3$-spheres has been constructed by Baker, Boileau and Wang in
\cite{BBW01}. In \cite{CalDun06} Calegari and Dunfield constructed an infinite tower of hyperbolic rational
homology $3$-spheres that are arithmetic and obtained by congruence
subgroups. Note that a recent conjecture of Bergeron and Venkatesh
predicts a lot of torsion in the homology groups of such a ``congruence tower'' of arithmetic
$n$-manifolds with $n$ odd \cite{BergVenka}.
\subsection{Locally symmetric homology spheres}
\label{ss:locally-symmetric}
Instead of considering hyperbolic homology spheres, one can more
generally look for homology spheres that are locally isometric to a
given symmetric space of nonpositive nonflat sectional curvature.
Such a symmetric space $X$ is called of noncompact type, and
it is classical that $X$ can be written as $G/K$, where $G$ is a
connected real semisimple Lie group with trivial center with $K \subset G$ a
maximal compact subgroup. Moreover, $G$ identifies as a finite index
subgroup in the group of isometries of $X$ (of index two if $G$ is
simple).
Let us explain why the case $X = \mathbf H^n$ is the main source
of locally symmetric rational homology spheres (among $X$ of noncompact
type). Let $M$ be a compact orientable manifold locally isometric to
$X$. Then $M$ can be written as $\Gamma\backslash X$, where $\Gamma \cong \pi_1(M)$ is
a discrete subgroup of
isometries of $X$. We will suppose that $\Gamma \subset G$, for $G$ as
above. Let $X_u$ be the compact dual of $X$. We have the
following general result (see \cite[Sections 3.2 and 10.2]{Bor74}).
\begin{prop}
\label{prop:inj-Hj}
For each $j$ there is an injective homomorphism $H^j(X_u,\mathbb C) \to
H^j(\Gamma\backslash X,\mathbb C)$.
\end{prop}
In particular, if $\Gamma\backslash X$ is a rational homology sphere, then so is
$X_u$. Note that the compact dual of $X = \mathbf H^n$ is the genuine sphere
$S^n$. By looking at the classification of compact symmetric spaces,
Johnson showed the following in \cite[Theorem 7]{Johns82}.
\begin{cor}
\label{cor:restrict-hyperbolic}
If $M = \Gamma \backslash X$ is a rational homology $n$-sphere with $\Gamma \subset
G$, then $X$ is either the hyperbolic $n$-space $\mathbf H^n$ (with $n \neq
4k+2$), or $X = \PSL_3(\mathbb R)/\PSO(3)$ (which has dimension $5$).
\end{cor}
Proposition \ref{prop:inj-Hj} shows that the correct problem to look at
-- rather than homology spheres -- is the existence of locally symmetric spaces $\Gamma\backslash X$ with the same (rational) homology as the compact dual
$X_u$.
When $X$ is the complex hyperbolic plane $\mathbf H^2_\mathbb C$, the compact
dual is the projective plane $\P_\mathbb C^2$, and the quotients $\Gamma\backslash X$
are compact complex surfaces called \emph{fake projective planes}.
Their classification was recently obtained by the work of Prasad--Yeung
\cite{PraYeu07}, together with Cartwright--Steger \cite{CarSt10} who performed the
necessary computer search. Later, Prasad and Yeung also considered
the problem of the existence of more general arithmetic fake Hermitian spaces
\cite{PraYeu09,PraYeu12}.
The present paper uses the same methodology as in Prasad and Yeung's work,
the main ingredient being the volume formula.
\subsection*{Acknowledgements} It is a pleasure to thank Gopal Prasad,
who suggested this research project.
\section{Proof of Theorem \ref{thm:non-existence}}
\label{sec:proof-thm1}
Let $G = \PO(n,1)^\circ \cong \mathrm{Isom}^+(\mathbf H^n)$,
and consider the universal covering $\phi: \Spin(n,1) \to G$.
For our purpose it will be easier to work with lattices in $\Spin(n,1)$.
A lattice $\overline{\Gamma} \subset G$ is arithmetic exactly
when $\Gamma = \phi^{-1}(\overline{\Gamma})$ is an arithmetic subgroup of
$\Spin(n,1)$.
Since the covering $\phi$ is twofold, we have $\chi(\Gamma) = \frac{1}{2}
\chi(\overline{\Gamma})$, where $\chi$ is the Euler characteristic in
the sense of C.T.C.~Wall. In particular, if $M = \overline{\Gamma}\backslash \mathbf H^n$
is a manifold with $|\chi(M)| = 2$, then $|\chi(\Gamma)| = 1$.
Thus, Theorem \ref{thm:non-existence} is an obvious consequence of the
following proposition. The proof relies on the description of arithmetic
subgroups with the help of Bruhat-Tits theory, as done for instance in
\cite{BorPra89} and \cite{Pra89}. An introduction can be found in
\cite{EmePhD}. We also refer to \cite{Tits79} for the needed facts from
Bruhat-Tits theory.
\begin{prop}
Let $n>4$.
There is no cocompact arithmetic lattice $\Gamma \subset \Spin(n,1)$ such that
$\chi(\Gamma)$ is a reciprocal integer, i.e., such that $\chi(\Gamma) = 1 / q$
for some $q \in \mathbb Z$.
\end{prop}
\begin{proof}
We can assume that $n$ is even.
Let $\Gamma \subset \Spin(n,1)$ be a cocompact lattice.
Clearly, it suffices to prove the proposition for $\Gamma$ maximal. In
this case, $\Gamma$ can be written as the normalizer $\Gamma =
N_{\Spin(n,1)}(\Lambda)$ of some {\em principal} arithmetic subgroup
$\Lambda$ (see \cite[Proposition 1.4]{BorPra89}).
By definition, there exists a number field $k \subset \mathbb R$ and a
$k$-group $\mathbf G$ with $\mathbf G(\mathbb R) \cong \Spin(n,1)$ such that $\Lambda =
\mathbf G(k) \cap \prod_{v \in \V_{\mathrm{f}}} P_v$, for some coherent collection
$(P_v)_{v\in\V_{\mathrm{f}}}$ of
parahoric subgroups $P_v \subset \mathbf G(k_v)$ (indexed by the set $\V_{\mathrm{f}}$ of
finite places of $k$).
It follows from the classification of algebraic groups (cf.
\cite{Tits66}) that $\mathbf G$ is of type $\mathrm{B}_r$ with $r = n / 2\; (> 2)$,
the field $k$ is totally real, and (using Godement's criterion) $k \neq
\mathbb Q$. Let us denote by $d$ the degree $[k:\mathbb Q]$.
Let $T \subset \V_{\mathrm{f}}$ be
the set of places where $P_v$ is not hyperspecial. By Prasad's volume
formula (see \cite{Pra89} and \cite[Section 4.2]{BorPra89}), we have:
\begin{align}
|\chi(\Lambda)| &= 2 |D_k|^{r^2+r / 2}
C(r)^d
\prod_{j=1}^r \zeta_k(2j) \prod_{v \in T} \lambda_v,
\label{eq:volume}
\end{align}
with $D_k$ (resp. $\zeta_k$) the discriminant (resp. Dedekind zeta
function) of $k$; the constant $C(r)$ is given by
\begin{align}
C(r) = \prod_{j=1}^r\frac{(2j-1)!}{(2\pi)^{2j}};
\label{eq-Cr}
\end{align}
and each $\lambda_v$ is given by the formula
\begin{align}
\lambda_v &= \frac{1}{(q_v)^{(\dim\mathcal M_v - \dim \mathbf M_v) / 2}}\;
\frac{|\mathcal M(\mathfrak f_v)|}{|\mathbf M_v(\mathfrak f_v)|},
\label{eq:lambda}
\end{align}
where $\mathfrak f_v$ is the residue field of $k_v$, of size $q_v$, and the
reductive $\mathfrak f_v$-groups $\mathbf M_v$ and $\mathcal M_v$ associated with $P_v$ are those
described in \cite{Pra89}. By definition $\mathcal M_v$ is semisimple of type
$\mathrm{B}_r$.
\begin{table}
\centering
\begin{tabular}{llc}
$\mathbf G / k_v$ & isogeny type of $\mathbf M_v$ & $\lambda_v$\\[9pt] \hline
\mbox{split: } & $\mathrm{B}_{r-1} \times (\mbox{split }\GL_1)$ & $\frac{q^{2r}-1}{q-1}$\\[9pt]
&$\mathrm{D}_i \times \mathrm{B}_{r-i}$ ($i=2,\dots,r-1$) & $
\frac{(q^i+1)\prod_{k=i+1}^r(q^{2k}-1)}{\prod_{k=1}^{r-i}(q^{2k}-1)}$\\[9pt]
&$^1\mathrm{D}_r$ & $q^r + 1$\\[9pt]
\mbox{non-split: } & $\mathrm{B}_{r-1} \times (\mbox{nonsplit } \GL_1)$ & $\frac{q^{2r}-1}{q+1}$\\[9pt]
& $^2\mathrm{D}_{i+1} \times \mathrm{B}_{r-i-1}$ ($i=1,\ldots,r-2$) & $
\frac{(q^{i+1}-1)\prod_{k=i+2}^r(q^{2k}-1)}{\prod_{k=1}^{r-i-1}(q^{2k}-1)}$\\[9pt]
& $^2\mathrm{D}_r$ & $q^r -1$\\[9pt]
\end{tabular}
\caption{$\lambda_v$ for $P_v$ of maximal type}
\label{tab:lambda-fac}
\end{table}
A necessary condition for $\Gamma = N_{\mathbf G(\mathbb R)}(\Lambda)$ to be maximal is that
each $P_v$ defining $\Lambda$ has maximal type in the sense of
\cite{RyzCher97}. We list in Table \ref{tab:lambda-fac} the factors
$\lambda_v$ corresponding to parahoric sugroups $P_v$ of maximal types
(to improve the readability we set $q_v=q$ in the formulas). This list
of maximal type and the formulas for $\lambda_v$ are essentially the same
as in \cite[Table 1]{Belo04}: the only difference is a factor $2$ in the
denominator of some $\lambda_v$, which can be explained from the fact
that Belolipetsky did not work with $\mathbf G$ simply connected.
From \cite[Section 5]{BorPra89} (cf. also \cite[Chapter 12]{EmePhD}) we can
deduce that the index
$[\Gamma : \Lambda]$ of $\Lambda$ in its normalizer has the following
property:
\begin{align}
[\Gamma : \Lambda] \mbox{ divides } h_k 2^{d} 4^{\#T}.
\label{eq:index}
\end{align}
Moreover, a case by case analysis of the possible factor $\lambda_v$
shows that $\lambda_v > 4$, so that $4^{-\#T} \prod_{v \in T} \lambda_v
\ge 1$ (with equality exactly when $T$ is empty).
We thus have the following lower bound for the Euler
characteristic of any maximal arithmetic subgroup $\Gamma \subset
\Spin(n,1)$:
\begin{align}
|\chi(\Gamma)| &\ge \frac{2}{h_k} \left(\frac{C(r)}{2}\right)^d
|D_k|^{r^2 + r / 2 } \zeta_k(2) \cdots \zeta_k(2r)
\label{eq:first-bd-Gamma}
\end{align}
We make use of the following upper bound for the class number (see for
instance \cite[Section 7.2]{BelEme}):
\begin{align}
h_k &\le 16 \left(\frac{\pi}{12}\right)^d |D_k|,
\label{eq:bound-hk}
\end{align}
which together with the basic inequality $\zeta_k(2j) > 1$
transforms \eqref{eq:first-bd-Gamma} into
\begin{align}
|\chi(\Gamma)| &> \frac{1}{8} \left(\frac{6 \cdot C(r)}{\pi}\right)^d
|D_k|^{r^2 + r / 2 -1}.
\label{eq:bd-no-hk}
\end{align}
Moreover, according to \cite[Table 4]{Odl90}, we have that for a degree
$d \ge 5$ the discriminant of $k$ is larger than $(6.5)^d$. With this
estimates we can check that for $r \ge 3$ and $d \ge 5$ we have
$|\chi(\Gamma)| > 1$. For the lower degrees, if we suppose that
$|\chi(\Gamma)| \le 1$, we obtain upper bounds for $|D_k|$ from Equation
\eqref{eq:bd-no-hk}. This upper bounds exclude the existence of such a
$\Gamma$ for $r \ge 6$ (which is already clear from the work of
Belolipetsky \cite{Belo04}). For $r=3$ (where the bounds are the worst) we obtain
the following:
\begin{align*}
d = 2: &\quad |D_k| \le 28;\\
d = 3: &\quad |D_k| \le 134;\\
d = 4: &\quad |D_k| \le 640.
\end{align*}
From the existing tables of number fields (e.g.,
\cite{Bordeaux_data,Qaos}) we can list the possibilities this leaves us
for $k$. We find that no field with $d=4$ can appear, and for $d=2,3$
all possibilities have class number $h_k = 1$. Using Equation
\eqref{eq:bound-hk} with $h_k = 1$ we then improve the upper bounds for
$|D_k|$ and thus shorten the list of possible fileds. For $r=5$ only
$|D_k| = 5$ arises, and for $r=4$ we have $|D_k| \le 11$ (the
possibility $d = 3$ is excluded here). For $r=3$, we are left with
$|D_k| \le 20$ when $d = 2$, and $|D_k| = 49$ or $81$ when $d=3$.
With $h_k = 1$, using the functional equation for $\zeta_k$ and the property
\eqref{eq:index} for the index $[\Gamma:\Lambda]$, we can express the
Euler characteristic of $\Gamma$ as
\begin{align}
|\chi(\Gamma)| &= \frac{1}{2^a} \prod_{v \in T} \lambda_v
\;\prod_{j=1}^r |\zeta_k(1-2j)|
\label{eq:EP-rational}
\end{align}
for some integer $a$. The special values $\zeta_k(1-2j)$, which are
rational by the Klingen-Siegel theorem, can be computed with the software
Pari/GP (cf. Remark \ref{rmk:special-val}).
We list in Table \ref{tab:special-values} the
values we need. We check that for
every field $k$ under consideration a prime factor $>2$ appear in the
numerator of
the product $\prod_{j=1}^r |\zeta_k(1-2j)|$.
A direct computation for $r=3,4,5$ shows that the
formula in Table \ref{tab:lambda-fac} for each factor $\lambda_v$ is actually
given by a polynomial in $q$ (this seems to hold for any $r$).
In particular, we always have $\lambda_v \in \mathbb Z$, and we conclude from
\eqref{eq:EP-rational} that
$|\chi(\Gamma)|$ cannot be a reciprocal integer.
\begin{table}
\centering
\begin{tabular}{ccccccc}
degree & $|D_k|$ & $\zeta_k(-1)$ & $\zeta_k(-3)$ & $\zeta_k(-5)$ &
$\zeta_k(-7)$ & $\zeta_k(-9)$ \\ \hline
$d=2$ & 5 & 1/30 & 1/60 & 67/630 & 361/120 & 412751/1650 \\
&8 & 1/12 & 11/120 & 361/252& 24611/240& \\
&12 & 1/6 & 23/60 & 1681/126& & \\
&13 & 1/6 & 29/60 & 33463/1638 & & \\
&17 & 1/3 & 41/30 & 5791/63 & & \\[2pt]
$d=3$ & 49 & -1/21 & 79/210 & -7393/63 & & \\
&81 & -1/9 & 199/90 & -50353/27 & & \\[8pt]
\end{tabular}
\caption{Special values of $\zeta_k$}
\label{tab:special-values}
\end{table}
\end{proof}
\begin{rmk}
\label{rmk:special-val}
The function \texttt{zetak} in Pari/GP allows to obtain approximate
values for $\zeta_k(1-2j)$. On the other hand the size of the
denominator of the product $\prod_{j=1}^m \zeta_k(1-2j)$ can be bounded
by the method described in \cite[Section 3.7]{Serr71}.
By recursion on $m$, this allows to ascertain that the values
$\zeta_k(1-2j)$ correspond exactly to the fractions given in Table
\ref{tab:special-values}.
\end{rmk}
\begin{rmk}
The fact that for $|D_k| = 5$ the value $\zeta_k(-1) \zeta_k(-3)$ has
trivial numerator explains why the proof fails for
$n = 4$ (i.e., $r=2$). And indeed there is a principal
arithmetic subgroup $\Gamma \subset \Spin(4,1)$ with $|\chi(\Gamma)| =
1 / 14400$ and whose image in $\mathrm{Isom}^+(\mathbf H^4)$ is contained as an index
$2$ subgroup of the Coxeter group $W_1$. On the other hand, for $|D_k| > 5$ the appearance
of a non-trivial numerator in $\zeta_k(-3)$ shows -- at least for the
fields considered in Table \ref{tab:special-values} -- the
impossibility of a $\Gamma$ defined over $k$ with $\chi(\Gamma)$ a
reciprocal integer. This is the first step in Belolipetsky's proof of
Theorem \ref{thm:dim-4}.
\end{rmk}
\bibliographystyle{amsplain}
|
1,116,691,500,858 | arxiv | \section{Introduction}
Whether the radio and the gamma-ray luminosities of blazars are intrinsically correlated is a long-standing debate. The presence or absence of such a correlation could provide insight into blazar emission physics. At radio frequencies low enough that synchrotron emission is self-absorbed on physical scales likely to be associated with gamma-ray emission, measurements of the gamma-ray and radio flux densities typically probe different parts of the blazar jet. If concurrently-measured, time-averaged flux densities at self-absorbed radio frequencies and high-energy ($\geq 100$ MeV) gamma-rays are intrinsically correlated, the implication would be that emission and flaring in different parts of blazar jets are driven by the same disturbances. In this case, further progress on the sequence of events that produce blazar flares can be made through high-cadence monitoring in both wavebands. If on the other hand radio and gamma-ray flux densities can be shown to be uncorrelated (a statement that needs to be carefully distinguished from the absence of evidence for correlation) then it is more likely that, over the timescales used for the flux-averaging, emission regions probed by radio and gamma-ray observations evolve and radiate independently.
Furthermore, should an intrinsic correlation between gamma-ray and radio flux densities be unambiguously demonstrated, radio blazar luminosity functions could be used to establish the shape and normalization of gamma-ray luminosity functions or $\log N-\log S$ distributions (however, proper care should be exercised to account for any significant scatter in the correlation, see, e.g., the discussion in Ackermann et al.~2011). From there, the unresolved blazar contribution to the diffuse gamma-ray background could be estimated (e.g., Stecker \& Salamon 1996; Kazanas \& Perlman 1997; Stecker \& Venters 2010). This is particularly important as blazars constitute a guaranteed background for any search in the diffuse gamma-ray emission for yet-undetected classes of sources such as galaxy clusters, and for signatures of exotic physics.
Strong correlations between radio and gamma-ray luminosities have been claimed based on EGRET data (e.g., Stecker et al.\ 1993; Padovani et al.\ 1993; Stecker \& Salamon 1996). However, these findings have been disputed (e.g., M\"{u}cke et al. 1997; Chiang \& Mukherjee 1998) based on more detailed statistical analyses. The objections against the claimed correlations can be summarized as follows.
First, artificial flux-flux correlations can be induced due to the effect of a common distance modulation of gamma-ray and radio luminosities. Feigelson \& Berg (1983) have argued that in statistically complete surveys of relatively small depth, apparent flux-flux correlations do not appear unless the corresponding luminosities are intrinsically correlated: if luminosities are intrinsically uncorrelated most objects will only have an upper limit rather than a detection in one of the wavebands. However this is not the case in samples that are selected with complex or subjective criteria, samples in which there is clustering around a preferred luminosity value, samples in which detection in both wavebands is one of the selection criteria, or samples in which the luminosity dynamical range is, for any reason, small compared to the distance modulation range. In such cases, the application of a common distance-squared factor to both radio and gamma-ray luminosity will automatically induce an artificial flux-flux correlation.
This effect cannot be avoided simply by searching for correlation in luminosity space, as the danger of inducing an artificial apparent correlation is even greater in this case due to Malmquist bias: in flux-limited (or approximately flux-limited) surveys, most objects are concentrated close to the survey sensitivity at each wavelength. By modulating these limiting fluxes by a common distance factor to return to luminosity space, artificial correlations arise.
Finally, the data used to obtain the claimed correlations were not synchronous. The direction in which non-simultaneity affects any intrinsic correlation is unclear. On the one hand, non-simultaneous data may wash out an intrinsic correlation which might otherwise be found in concurrently measured data. On the other hand, the tendency to detect more flaring objects than objects in a quiescent state in surveys may lead to enhanced correlations, essentially representing peak flux / peak flux correlations of different flares, which although they may be indicative of the overall energetics of flares in a single object, they do not convey any detailed information regarding the time-averaged behavior of the object.
In the {\it Fermi} era, the possibility of a correlation between gamma-ray and radio fluxes of blazars has generated a lot of interest, and the question has been explored using {\it Fermi} Large Area Telescope (LAT) fluxes in combination with archival (Ghirlanda et al.~2010, Mahony et al.~2010, Giroletti et al.~2010; Ackermann et al.~2011), quasi-concurrent (Kovalev et al.~2009) and concurrent (Giroletti et al.~2010; Ackermann et al.~2011, Angelakis et al.~2010; Fuhrmann et al.~2012, in preparation) radio data.
The intrinsic significance of an apparent correlation between radio and gamma-ray flux densities in strictly flux-limited, large datasets is relatively straight-forward to assess, by Monte-Carlo draws from the underlying luminosity functions in both datasets, obeying the same selection criteria as the observed sample of sources (e.g., Bloom 2008). In practice however we frequently encounter the case where a sample of monitored sources has been selected to optimize the likelihood of high-impact observations in individual objects using complex and often subjective criteria, which are difficult to reproduce in a simulation. Although such samples are not ideally configured for unbiased population studies, they may present significant advantages in other respects, such as multi-band coverage, high cadence of observations, and simultaneity between different waveband data. It is thus important to be able to assess as robustly as possible the intrinsic significance of any apparent correlations observed in such samples. Here, we introduce a method for the quantitative assessment of the significance of a correlation in such cases, based on permutations of observed flux densities, while ensuring that the dynamical ranges in luminosity and flux density are kept fixed. When this method is applied in large, statistically complete samples that are split in redshift bins, it asymptotically approaches luminosity-function sampling. For smaller samples, the significances it returns are conservative: existing intrinsic correlations may not be verified, but exaggerated significances are avoided.
Our method has been recently used by the {\it Fermi-}LAT collaboration (Ackermann et al.~ 2011) to study the correlation between GeV and cm radio fluxes (both archival and concurrent, the latter from the Owens Valley Radio Observatory 15 GHz monitoring program, Richards et al.~2011\footnote{Program description and data also available online, at: \\{\tt http://www.astro.caltech.edu/ovroblazars/}}). They have established, at a very high significance level, the existence of a positive correlation ($<10^{-7}$ probability of the correlation arising by chance). Our method is also currently used in studies of multi-frequency concurrent radio observations by the F-GAMMA program (Angelakis et al.~2010; Fuhrmann et al.~2012, in preparation). Here, we discuss in detail the method and its implementation, and we evaluate its performance using both simulated and real ({\it Fermi} and OVRO) data.
We caution the reader that our proposed algorithm assumes perfectly concurrent data and thus does not address any possible effects of non-simultaneity. In addition, we stress that our method cannot compensate for sample selection effects or incompleteness relative to a parent population. For example, if the objects in the examined sample do not constitute a representative sample of the blazar population, even when a statistically significant correlation between radio and gamma-ray flux densities can be established in the objects of the observed sample, it is not possible to generalize this result to the blazar population as a whole. This limitation can only be addressed by more careful sample selection.
This paper is organized as follows. In \S 2 we discuss our method, and in \S 3 we present in detail the implementation of the statistical test we have adopted. Demonstrations of the test and evaluations of its performance are presented in \S 4. We summarize and discuss our conclusions in \S 5.
\section{Method}
\subsection{Small, subjectively selected samples}\label{original}
The purpose of the test is to quantitatively assess the significance of an apparent correlation between concurrent radio and gamma-ray flux densities of blazars in the presence of distance effects and subjective sample selection criteria. We will do so by testing the hypothesis that emission in the two wavebands is intrinsically uncorrelated: we will calculate how frequently a sample of objects similar to the sample at hand, with intrinsically uncorrelated gamma/radio luminosities, will yield an apparent correlation as strong as the one seen in the data, when subjected to the same distance and dynamical-range effects as our actual sample.
In our implementation of the test, the strength of the apparent correlation is quantified by the Pearson product-moment correlation coefficient $r$ (Fisher 1944), defined as
\begin{equation} \label{rcoeff}
r = \frac{\sum_{i=1}^N(X_i-\bar{X})(Y_i-\bar{Y})}
{\sqrt{\sum_{i=1}^N(X_i-\bar{X})^2\sum_{i=1}^N(Y_i-\bar{Y})^2}}\,,
\end{equation}
with $(X_i,Y_i)$ in our case being a pair of the logarithms of the flux densities in each frequency for a single object. The reason for taking the logarithm is two-fold. First, it ensures that, for sources with a power-law distribution of fluxes, there will not be a clustering of most measurements around the low-flux corner of the flux-flux plane, which would then allow single high-flux outliers to induce an artificially high $r$ value. Second, it linearizes any power-law relation between the variables, which improves the behavior of correlation measures that target specifically the linear correlation between variables (such as the Pearson $r$).
This test can also be used with any statistic quantifying correlation strength instead of the Pearson product-moment coefficient, including non-parametric correlation measures (e.g., Siegel \& Castellan 1988; Conover 1999).
Since the sample selection criteria are assumed to be subjective, the challenge in defining our test lies in constructing simulated object samples of intrinsically uncorrelated gamma/radio flux densities, similar in other respects to our actual object sample.
In order to overcome this difficulty, we use only permutations of measured quantities.
Our method is a variation of a classical permutation test for the assessment of a correlation (e.g., Wall \& Jenkins 2003, \S 4.2.3; see also Efron \& Petrosian 1998 for permutation methods for doubly truncated datasets). Its novelty lies in the fact that while we are trying to establish a correlation between flux densities and calculate a distribution of correlation measures in simulated sets of flux density logarithms, we perform permutations in luminosity space (see also Fender \& Hendry 2000 for a similar Monte Carlo approach of evaluating an apparent distance-squared effect and the possible effect of Doppler beaming in the case of radio data of persistent X-ray binaries). In this way, we can simulate the effect of a common distance on intrinsically uncorrelated luminosities, by applying a common redshift to permuted luminosity pairs to return to flux space. By assessing the significance in flux space we avoid Malmquist bias, and we automatically account for the limited flux dynamical ranges in the two frequencies under consideration.
We do so as follows:
\begin{enumerate}
\item From the measured radio and gamma-ray flux densities, we calculate radio and gamma-ray luminosities at a common rest-frame radio frequency and rest-frame gamma-ray energy.
\item We permute the evaluated luminosities, to simulate objects with intrinsically uncorrelated radio/gamma luminosities.
\item We assign a common redshift (one of the redshifts of the objects in our sample, randomly selected) to each luminosity pair, and return to flux-density space. Assigning a common redshift allows us to simulate the common-distance effect on uncorrelated luminosities. Using measured redshifts and luminosities guarantees that the distance and luminosity dynamical range in our simulated samples is also identical to that of our actual sample.
\item To avoid apparent correlations induced by a single very bright or very faint object much brighter or fainter than the objects in our actual sample, we reject any flux-density pairs where one of the flux densities is outside the flux-density dynamical range in our original sample.
\end{enumerate}
Using a randomly selected set of flux density pairs, with number equal to the number of objects in our actual sample, we calculate a value for $r$. We repeat the process a large number of times, and calculate a distribution of $r-$values for intrinsically uncorrelated flux densities. The fraction of $|r|\geq r_{\rm data}$, where $r_{\rm data}$ is the $r-$value for the observed flux densities, is the probability to have obtained an apparent correlation at least as strong as the one seen in the data from a sample with intrinsically uncorrelated gamma-ray/radio emission. This quantifies the statistical significance of the observed correlation.
Formally, the null hypothesis tested with this procedure is
$H_0:$ {\em The radio and gamma-ray luminosity of blazars are independent, and redshift is independent of both luminosities.} We note that in many cases, this is not the hypothesis we would like to be testing, as luminosities depend on redshift in most population models of active galactic nuclei. Ideally, we would like to test for independence between radio and gamma-ray luminosities conditioned on redshift. However, this is not always practically possible due to sample size and redshift span of the sources. For the cases when the sample size is large enough and the sources included in the sample are adequately spread over redshifts, the test discussed in the next subsection will fulfill this requirement. For cases however when sample limitations are prohibitive for such a study, we show that testing $H_0$ {\em with the implementation presented in this work} can provide a conservative alternative to the full problem: if $H_0$ is rejected with high significance, then it is safe to assume that radio and gamma-ray luminosities are also not independent conditioned on redshift. However, if $H_0$ cannot be rejected, no conclusion can be reached for either hypothesis, as absence of evidence for a correlation is not equivalent with evidence for absence of a correlation.
\subsection{Larger samples: splitting the sample in redshift bins}\label{zbinsdisc}
The process of pair rejection discussed in step 4 above
may alter the distribution of luminosities, fluxes, and redshifts of the randomized data and introduce substantial differences from the corresponding distributions of the original dataset.
The cause of this effect is the randomization of redshifts among all sources, and it is straight-forward to understand. Low-luminosity nearby objects, when combined with large redshifts, will result in very faint fluxes which are outside the original flux dynamical range and thus rejected. For this reason, the simulated datasets will have fewer very-low--luminosity objects compared to the original dataset. In addition, rare, high-luminosity, high-redshift objects, when combined with low redshifts, will result in very high fluxes, also outside the original flux dynamical range and thus rejected. For this reason, the simulated datasets will also have fewer very-high--luminosity objects compared to the original dataset. In contrast, the number of intermediate-luminosity objects will be relatively enhanced in simulated datasets. The distributions of redshifts and fluxes of the simulated datasets will also be altered for similar reasons.
If the pair rejection rate is high, the properties of the simulated datasets could be different than the properties of the original dataset, and these biases could affect our estimation of a correlation significance. In small and subjectively selected datasets, this problem is a necessary evil. The effect of these biases is, as we will show below, to worsen the estimated significance of a correlation, rather than induce false positives of enhanced significance. However, in the case of larger samples, there is a simple alteration in the methodology described in \S \ref{original} that can significantly alleviate these biases: splitting the sample in redshift bins.
In this variation of the test, the original sample is split into a number of bins dependent on the available number of objects (as we discuss below, we need about 10 objects or more per bin, and in any case no fewer than 8). We then generate randomized flux-density pairs in each redshift bin with the process described above. Because the range of redshifts that are permuted between objects of different luminosities is much smaller, the likelihood that one of the resulting randomized flux densities will exceed the flux-density dynamical range of the original dataset is much smaller. As a result, the pair rejection rate is decreased, and the luminosity, redshift, and flux distributions of the randomized data pairs resemble more closely those of the original dataset.
The similarity between distributions of the randomized and the original data increases as the size of the sample increases and the width of each redshift bin decreases. If the original dataset is also a statistically complete and flux-limited sample, then the test asymptotically approaches the luminosity-function--sampling test as the size of the original dataset approaches infinity and the size of each redshift bin used approaches zero. This can be understood as follows. In the limit of zero-size redshift bin, all objects within a single redshift bin are at the same distance. Therefore, permuting the luminosities of objects at that distance is equivalent to forming luminosity pairs by randomly sampling each frequency's luminosity function at a specific redshift and with a specific flux-density limit (the limit of the original sample). Repeating the process at all redshift bins is equivalent to sampling the luminosity functions at all redshifts. Then, the ``pool'' of randomized data pairs, from which we draw the mock datasets, could have been equivalently generated through luminosity function sampling.
Formally, the null hypothesis tested with this procedure is
$H_0:$ {\em Conditional on redshift, the radio and gamma-ray luminosity of blazars are independent,} which is the hypothesis that one would generally wish to test. For this reason, this version of the test should be preferred whenever possible.
\section{Implementation}
In this section we describe how the method discussed above can be implemented in practice for small and large datasets.
\subsection{Small, subjectively selected samples}\label{impsimple}
The first step is to convert the blazar gamma-ray fluxes (which are usually reported as integrated photon fluxes $F$ above some fiducial energy $E_0$, usually 100 MeV), to energy flux densities, so that the comparison with radio flux densities can be done on an equal footing\footnote{Other possible choices is to correlate radio flux densities with gamma-ray photon fluxes at some particular energy bin, or with the integrated photon fluxes themselves (see, for example, Abdo et al.~2011). In these cases, Eq. \ref{mockg} should be changed accordingly.}. We do so by assuming that the photon fluxes are power laws, so that the flux (number of photons per unit area-time-energy bin) is
\beq
\frac{dN_{\rm photon}}{dE\, dA\, dt} = F_0\left(\frac{E}{E_0}\right)^{-\Gamma}\,.
\eeq
In this case, the gamma-ray energy flux density $S_\gamma \equiv dE/dE\, dA\, dt $ at $E_0$ is given by $S_\gamma(E_0) =F_0E_0 = F(\Gamma-1)$ and its energy dependence is
\beq\label{fdensity}
S_\gamma (E) = (\Gamma-1)F\left(\frac{E}{E_0} \right)^{-\Gamma+1}\,.
\eeq
The relation between monochromatic flux density $S(\nu)$ and monochromatic luminosity $L(\nu)$ for a source at redshift $z$ is
\begin{equation}\label{basic}
S(\nu)= \frac{L[\nu(1+z)]}{4\pi d^2(1+z)}
\end{equation}
where $d=(c/H_0)\int_0^z dz/\sqrt{\Omega_\Lambda+\Omega_m(1+z)^3}$. Here $H_0$ is the present-day value of the Hubble parameter, and $\Omega_\Lambda$ and $\Omega_m$ are the vacuum energy and matter density parameters. In this work, we have used $\Omega_m = 0.26$ and $\Omega_\Lambda = 1-\Omega_m$, consistent with, e.g., Larson et al. (2011). Note that the value of $H_0$ drops out of the calculation as $d$ in the formalism we describe below appears only in ratios.
If the source has a spectral index $\alpha$ so that $S(\nu) \propto \nu^\alpha$ at the frequency of interest, Eq. (\ref{basic}) implies that the relation between $S(\nu)$ at observer-frame $\nu$ and $L(\nu)$ at rest-frame $\nu$ (the K-correction) is
\beq\label{kc}
L(\nu) = S(\nu)4\pi d^2(1+z)^{1-\alpha}.
\eeq
So if a radio flux density $S_r(\nu)$ (at observer-frame $\nu$) is turned into a luminosity density (at rest-frame $\nu$) using a redshift $z_1$ and a spectral index $\alpha_r$, and this luminosity density is then returned to flux-density--space (at observer-frame $\nu$) using a different redshift $z'$ but the same spectral index $\alpha_r$, we can write
\beq \label{mockr}
S'_r(\nu) = S_r(\nu) \left(\frac{d_1}{d'}\right)^2
\left(\frac{1+z_1}{1+z'}\right)^{1-\alpha_r}\,,
\eeq
where $d_1=d(z_1)$ and $d'=d(z')$.
For the same procedure with gamma-ray flux densities and a source at a redshift $z_2$ we can write
\beq \label{mockg}
S'_\gamma(E_0) = (\Gamma-1)F\left(\frac{d_2}{d'}\right)^2
\left(\frac{1+z_2}{1+z'}\right)^\Gamma\,.
\eeq
In practice, we perform the following steps.
\begin{itemize}
\item[(i)] For each blazar, we use the flux density in radio and gamma-ray frequency to produce monochromatic luminosities at the same (now rest-frame) frequency in the two bands.
\item[(ii)] We construct all possible pairings (excluding the original ones) of radio and gamma-ray luminosities from our observed sample.
\item[(iii)] We assign a common redshift $z'$ to each permuted pair (one of the available redshifts in our sample).
\item[(iv)] We calculate ``mock'' radio and gamma-ray flux densities $S'_r, S'_\gamma$ for each pair using Eq.~(\ref{kc})\footnote{Equivalently, we can use directly Eqs.~(\ref{mockr}) and (\ref{mockg}), without explicitly calculating luminosities first.}.
\item[(v)] We accept the pair if both flux densities are within our original flux-density dynamical range in each band, or reject it otherwise.
\item[(vi)] We randomly select $N$ pairs out of all the possible combinations, where $N$ is equal to the number of our original observations. Each set of $N$ pairs is now a simulated dataset of intrinsically uncorrelated flux/flux observations.
\item[(vii)] For each simulated dataset, we compute $r$ using Eq. (\ref{rcoeff}), where $X_i = \log (S'_{r,i})$ and $Y_i = \log (S'_{\gamma,i})$, with $i$ running from 1 to N.
\item[(viii)] We repeat steps (vi-vii) $m$ times, where $m$ is a sufficiently large number to sample the underlying $|r|$ distribution. In our tests below $m$ is between $10^6 - 10^7$.
\item[(ix)] We calculate the probability for the observed $|r|$ to have occurred through uncorrelated flux densities from the $|r|-$values obtained in step (viii).
\end{itemize}
Our technique can be applied to samples that are very small and still yield a reliable estimate of the distribution of $|r|$. The total number of simulated pairs that we can construct through our permutation technique from $N$ objects is $N_{\rm pairs}=N^2(N-1)$ (where we permute both flux densities as well as redshifts). Only a fraction $N_{\rm surv}$ will survive the low- and high- flux-density cuts that ensure that the flux-density dynamical range remains the same as in the original sample. Assuming a reduction no larger than a factor of 5 (i.e.~$N_{\rm surv} \gtrsim N_{\rm pairs}/5$, shown in practice to be a conservative assumption), the total number of combinations of $N$ pairings different from each other by one or more pairs out of a population of $N_{\rm surv}$ objects then is
\begin{equation}
{\rm pair \,\,\, combinations} = \frac{N_{\rm surv}!}{N!(N_{\rm surv}-N)!} \,,
\end{equation}
which is $\gg 10^7$ for samples with $N \gtrsim 8$. However, in small datasets a statistically significant correlation is harder to establish, even if the distribution of $|r|$ can be estimated with sufficient statistics. In addition, as we will also show in \S \ref{demonstration}, the biases in the luminosity, redshift, and flux distributions of the simulated datasets introduced due to pair rejections (see discussion in \S \ref{zbinsdisc}) tend to worsen the significance that can be established through this test.
\subsection{Splitting larger samples in redshift bins}\label{thesplit}
Whenever the size of the source sample is large enough to allow splitting in more than one redshift bins, this variation of the test is recommended, as the effect of biases introduced through pair rejection decreases with increasing number of redshift bins (decreasing redshift bin size).
To implement this variation of the test, we split the sample in $N_z$ redshift bins. Our choice for the test implementation is to use variable redshift bin size, selected in such a way that the number of sources in each bin is as close to equal as possible, but never fewer than 8. However, other choices are also possible (for example, keeping the redshift bin size approximately equal; or splitting by luminosity distance rather than redshift, and keeping the luminosity distance bin size approximately equal).
For the sources in each one of the $N_z$ bins, we apply steps (i)-(v) of \S \ref{impsimple}. We then combine all accepted simulated data pairs from all redshift bins to generate the ``pool'' of all possible pair combinations. Finally, we apply steps (vi) - (ix) to this combined randomized pair ``pool''.
\section{Demonstrations of the test}\label{demonstration}
In this section we present example applications of our tests, using both real and simulated data, to evaluate the performance of our proposed test and demonstrate several aspects of its implementation.
For the applications on real data, we will use gamma-ray flux measurements from {\it Fermi} LAT and radio flux-density measurements from the OVRO 40 M Monitoring Program (Richards et al.\ 2011).
In addition, we will use simulated data to evaluate the the performance of the method: its effectiveness in rejecting false positives due to common-distance biases in correlation assessments, and its power in establishing significant correlations when such correlations do exist. As a benchmark we will use the face-value estimate of the significance for the Pearson correlation coefficient $r$, which evaluates the probability of a certain (or bigger) value of $r$ to occur by chance in the ``dart-throwing'' scenario (i.e., when pairs are randomly drawn from uncorrelated Gaussian distributions, assuming that no biases exist). In the latter scenario, the significance only depends on the value of $r$ and the sample size $N$. In the null hypothesis (uncorrelated data), the variable
\begin{equation} \label{student}
t = \frac{r\sqrt{N-2}}{1-r^2}
\end{equation}
follows a Student's t-distribution with $N-2$ degrees of freedom. Using Eq.~(\ref{student}) significances (p-values) can be estimated for any given values of $r$ and $N$ by taking the two-tail integral of the appropriate t-distribution.
In general, the variation of the test {\em with} redshift binning is the one which we recommend whenever possible (whenever sample restrictions allow its use), and it is the one which we have used in our simulated datasets.
\subsection{Demonstrations on real data}
\subsubsection{Small sample, no redshift bin splitting}\label{demo-small}
As an example of a relatively small dataset, we use the set of blazars that are included both in the LAT bright AGN source list (Abdo et al.~2009, produced using three months of LAT observations), as well as in the ``complete sample'' of the OVRO 40 M Monitoring Program (Richards et al.\ 2011).
The latter consists of the 1158 sources north of $-20^\circ$ declination in
the Candidate Gamma-Ray Blazar Survey (CGRaBS) sample, which is a sample
of 1625 sources, mostly blazars, selected by their flux and spectral index in radio, and flux in X-rays, to resemble the blazars detected by EGRET (Healey et al. 2008). The 1158 of the ``complete sample'' are observed approximately twice a week at 15 GHz with the Owens Valley Radio Observatory (OVRO) 40 M Telescope.
For this study, we only use sources with known redshifts, and for which a sufficient number of high-quality 15 GHz observations were taken in the same three-month time interval of LAT observations so as to produce a meaningful concurrent 15 GHz average flux density (see Richards et al.\ 2011). This sample contains 38 sources.
\begin{figure}
\includegraphics[width = 3in, clip]{fluxfluxplot2.eps}
\caption{3-month averaged concurrent 15 GHz versus 100 MeV observer-frame flux densities for the 38 blazars in our sample. \label{fluxflux}}
\end{figure}
\begin{figure}
\includegraphics[width = 3in, clip]{histogram_38objects.eps}
\caption{Distribution of $|r|-$values for 38 blazars of the same dynamical range in redshift and radio and gamma-ray flux densities and luminosities as blazars in our sample. The vertical arrow indicates the $r-$value for the actual observations ($r=0.62$). The significance of the correlation is $1.5\times10^{-4}.$\label{rdistfig} }
\end{figure}
Figure \ref{fluxflux} shows 3-month averaged 15 GHz flux densities plotted against 100 MeV observer-frame flux densities obtained by integration over the same time interval for the 38 blazars in our sample. The error bars in this plot are substantially smaller than the scatter of points (see, e.g., Ackermann et al.~2011) and have been omitted for clarity. An apparent correlation between the radio and gamma-ray time-averaged flux densities is obvious, however the statistical significance of an intrinsic correlation between the radio and gamma-ray emission of these objects needs to be quantitatively assessed. To this end, we apply the data randomization analysis we have introduced in \S \ref{impsimple}.
\begin{figure}
\includegraphics[width = 3in, clip]{fluxflux_largesample.eps}
\caption{11-month averaged concurrent 15 GHz versus 100 MeV observer-frame flux densities for the 160 blazars in the larger sample. \label{fluxfluxl}}
\end{figure}
The probability distribution of the values of $|r|$ in our simulated samples with intrinsically uncorrelated radio/gamma luminosities is shown in Fig. \ref{rdistfig}. The vertical arrow in this figure indicates the $r-$value for the observed data, equal to $0.62$. From the 38 objects in our sample a total number of $38^2\times 37 = 53,428$ permuted pairs were generated. Of those, $13,003$ pairs had both gamma-ray and radio flux densities within the dynamical range of the original dataset.
The accepted pairs were used (in $10^7$ randomly drawn sets of 38) to generate the distribution shown in Fig. \ref{rdistfig}. The probability to obtain $|r|\geq 0.62$ from intrinsically uncorrelated flux-density measurements due to the effect of a common distance is $1.5\times 10^{-4}$. For comparison, the significance estimate ignoring any biases and using only Eq.~(\ref{student}) is $3.3\times 10^{-5}$: without a careful analysis, we would evaluate the observed correlation as more significant than we do when accounting for common-distance and flux biases, as these effects are likely to contribute at least part of the observed correlation strength. We will elaborate on the origin and quantitative behavior of this discrepancy in the following sections.
Note that the pair rejection rate is high - only 24\% of the permuted pairs are within the original flux density dynamical range and were accepted; biases introduced in the luminosity, flux, and redshift distributions of the simulated data are therefore a concern. However, as we will show below, were these biases absent, the significance of the correlation would improve.
\subsubsection{Larger sample, behavior of test with increasing number of redshift bins}\label{zbinsapp}
We now turn to a demonstration of the second variation of our test, where the sample is split in redshift bins, and we discuss the alleviation of biases induced through pair rejection, and the improvement of the correlation significance with increasing number of bins.
To allow splitting in enough redshift bins to adequately demonstrate the behavior of the test in the many-bins limit we use the significantly larger sample of 160 sources that: (a) are included in the first year LAT catalog (Abdo et al.~2010); (b) are part of the OVRO 40 M telescope monitored sample; (c) have known redshifts. This same sample has been examined in detail for intrinsic correlations between 15 GHz flux density and LAT gamma-ray fluxes at various energy ranges by Ackermann et al.~(2011), using the test discussed here\footnote{Here, we use for the gamma-ray band data the 100 MeV flux density calculated according to Eq. \ref{fdensity} from integrated photon fluxes for $E>100$ MeV and using the photon index provided in 1LAC, which is different that any of the flux densities or integrated fluxes examined by Ackermann et al.~2011; this is the origin of the small differences in the value of $r$ obtained here for the data.}.
Figure \ref{fluxfluxl} shows 11-month--averaged 15 GHz flux densities plotted against 100 MeV observer-frame flux densities obtained by integration over the same time interval for the 160 blazars in the sample described above. The error bars in this plot are again substantially smaller than the scatter of points (see, e.g., Ackermann et al.~2011) and have been omitted for clarity. Through visual inspection, this sample also appears to feature an apparent correlation between radio and gamma-ray flux densities, with scatter comparable to that of the smaller sample of \S \ref{demo-small}. The correlation coefficient of the data in this case is $r=0.48$.
The biases introduced through pair rejection in our first variation of the test (where the sample is not split in redshift bins) are demonstrated in Figs.~\ref{rlumhist}-\ref{zhist}. The luminosities in these figures are in units of $4\pi(c/H_0)^2S_0$, where $H_0$ is the Hubble parameter, and $S_0 = 1$ Jy for 15 GHz source-frame luminosities and $S_0=10^{-8} {\rm GeV/s-cm^2-GeV}$ for 100 MeV source-frame luminosities.
Figure \ref{rlumhist} shows the fraction of objects in each logarithmic radio luminosity bin for the data (thick black line) and the accepted scrambled pairs (thin lines). Different line colors correspond to different numbers of redshift bins, as in the figure legend. When only one redshift bin is used (thin black line, equivalent to the first variation of our test), the shape of the luminosity distribution of the accepted scrambled pairs has a qualitatively different shape than that of the data: objects in the bins corresponding to the $\sim$3 lowest orders of magnitude in luminosity are significantly underrepresented compared to the original sample, because these low luminosities, corresponding to nearby objects in the data, are frequently rejected when they are combined with high redshifts and produce very low simulated flux densities outside the original flux density dynamical range. When we split the sample in a larger number of bins the effect is alleviated. At 16 redshift bins the radio luminosity distribution of simulated data is very close to that of the original data, and it is essentially converged, as it does not change appreciably when the number of redshift bins is increased to 20.
A very similar behavior for the gamma-ray luminosity distribution is shown in Fig.~\ref{glumhist}.
In the case of the redshift distribution, shown in Fig.~\ref{zhist}, both the very low and the very high redshift bins are underestimated when no data splitting is applied (thin black line). However, at 16 redshift bins the real and simulated data distributions are very similar, and the simulated data distribution is, again, converged. In all distributions, as the number of redshift bins increases, the difference between data and simulated distributions decreases, as a result of the decreasing pair rejection rate which, at 16 redshift bins, is $\lesssim 20\%$ for all bins.
\begin{figure}
\includegraphics[width = 3in, clip]{forpaper_hist_lr.eps}
\caption{Fraction of objects in each logarithm-in radio luminosity bin for the data (thick black line) and the accepted scrambled pairs (thin lines; different colors correspond to different numbers of redshift bins, as in legend). The radio luminosities are in units of $L_{0,\rm radio} = 4\pi(c/H_0)^2S_0$ where $S_0=1$Jy.\label{rlumhist}}
\end{figure}
\begin{figure}
\includegraphics[width = 3in, clip]{forpaper_hist_lg.eps}
\caption{Fraction of objects in each logarithm-in gamma-ray luminosity bin for the data (thick black line) and the accepted scrambled pairs (thin lines; different colors correspond to different numbers of redshift bins, as in legend). The gamma-ray luminosities are in units of $L_{0,\rm \gamma}= 4\pi(c/H_0)^2S_0$ where $S_0=10^{-8} {\rm GeV/s-cm^2-GeV}$.\label{glumhist}}
\end{figure}
\begin{figure}
\includegraphics[width = 3in, clip]{forpaper_hist_z.eps}
\caption{Fraction of objects in each redshift bin for the data (thick black line) and the accepted scrambled pairs (thin lines; different colors correspond to different numbers of redshift bins, as in legend). \label{zhist}}
\end{figure}
\begin{figure}
\includegraphics[width = 3in, clip]{forpaper_multi_hist.eps}
\caption{Distribution of $|r|-$values for randomly selected 160-blazar sets picked from the ensemble of accepted pairs generated through data scrambling. The vertical arrow indicates the $r-$value ($r=0.48$) for the actual observations. Different colors correspond to different numbers of redshift bins, as in legend. \label{multihist}}
\end{figure}
The behavior of the estimated significance as a function of the number of redshift bins is shown in Fig.~\ref{multihist}, where we have plotted the distribution of the absolute values of the correlation coefficients $|r|$ for each test implementation. Again, different colors correspond to different numbers of redshift bins used as in Figs.~\ref{rlumhist}-\ref{zhist}. $10^6$ simulations were used to produce each curve. The $r-$value for the data is shown with the arrow. The significance of the correlation as evaluated with 16 redshift bins is $<10^{-6}$ (if we fit the distribution shown with the blue line in Fig.~\ref{multihist} with a Gaussian, we obtain a significance of $\sim 10^{-7}$).
Again, using Eq.~(\ref{student}) to compare with the simple significance estimate based only on $r$ and $N$, we find that $t=6.88$, for which the two-tailed t-distribution with 158 degrees of freedom yields a much stronger significance of $1.3\times 10^{-10}$. The reason for this substantial difference can be immediately understood qualitatively through inspection of Figs.~\ref{rlumhist} -\ref{zhist}. Both the radio and the gamma-ray luminosity distributions of the data show broad peaks, which means that even if there was no intrinsic correlation between radio and gamma-ray emission and radio and gamma-ray luminosities were simply randomly drawn from these distributions, values of radio and gamma-ray luminosity around the peaks would appear frequently. As a result, pairs of radio/gamma-ray fluxes corresponding to underlying luminosities clustered around likely values would be common. Such pairs, when modulated with a common distance factor, would yield an apparent correlation in flux-flux space by chance, much more frequently than if there was no peak in the luminosity distributions. The unsophisticated significance estimate contains no information about common-distance effects and the behavior of the underlying luminosity distributions, and for this reason {\bf overestimates} the significance of the apparent correlation. However, even when these effects are accounted for using our method, the data show significant intrinsic correlation between radio and gamma-ray fluxes.
We can see that the significance of the correlation monotonically improves with increasing number of bins\footnote{The exact statement is that the significance monotonically improves with decreasing fraction of rejected pairs. Should a particular choice in redshift binning result in increased rejection fraction, the significance would worsen, even if the number of bins was larger.}. The reason for this behavior can be understood from Figs.~\ref{rlumhist} and \ref{glumhist}. The more frequent rejection of pairs at the edges of the luminosity distributions results in the artificial enhancement of the peaks in the luminosity distributions at intermediate luminosities. This stronger peak results in an enhanced incidence of artificial correlations. As a result, the significance of the apparent correlation of the data drops.
This is also the reason for the appearance of a peak at positive values of $|r|$ in the $|r|$ distribution of simulated datasets in Fig.~\ref{multihist} when the number of redshift bins is low. However, it is not guaranteed that a small number of redshift bins will generate such a peak at positive $|r|$ - this depends on the details of the luminosity distribution of the original dataset and of the pair rejection. For example, such a peak does not appear in our smaller dataset example in Fig.~\ref{rdistfig}. Conversely, a large number of redshift bins does not guarantee that such a peak will not appear. If our original dataset is selected in such a way that a certain narrow range of luminosities is over-represented, then such a peak is intrinsic to the dataset and it will appear regardless of number of bins used.
It is also interesting to consider the behavior of the test in the limit of a very large number of redshift bins that could be in principle used if we had a very large sample available for study, and in the case that our sample was a statistically complete, flux-limited set of sources. In this case, each redshift bin could be made very narrow, and all sources within the bin would be located essentially at the same distance. The set of radio luminosities within each bin would then be a representation of the radio luminosity function at a fixed redshift, with a limiting luminosity set by the limiting flux and the bin redshift. The set of gamma-ray luminosities within the same bin would similarly be a representation of the gamma-ray luminosity function. Since all sources would be located at the same distance, data randomization within the bin would never produce fluxes outside the original dynamical range, and no pairs would be rejected. The simulated pairs would then have exactly the same luminosity distribution as the data, and they would continue to be a representation of the luminosity functions at the two frequencies under consideration, as ``fair'' as the original data. As a result, in the limit of the ``perfect sample'' and a large number of redshift bins, our test would yield exactly the same result as a statistical test sampling random radio and gamma-ray fluxes from known luminosity functions. Our test deviates increasingly from this result as the statistical properties and the size of the sample deteriorate.
As shown in Fig.~\ref{multihist}, our proposed test is conservative: a smaller number of redshift bins will generally result in an increased rate of pair rejection and a worse correlation significance. In this way, it is possible that a real, intrinsic correlation cannot be confirmed by this test if a poor sample is used. However, the test will not yield artificially enhanced significances.
\subsection{Demonstrations on simulated data: \\ Uncorrelated datasets}
In this section we discuss the performance of the test when applied to datasets drawn from intrinsically uncorrelated populations. In particular, we evaluate the effectiveness of our test in rejecting false positives that we might have obtained due to common-distance biases had we used the estimate of the significance given by Eq.~(\ref{student}). In \S \ref{dataproduction} we describe how we generate the intrinsically uncorrelated simulated datasets that we use to test the performance of our method, and in \S \ref{robustness} we examine this performance and the robustness of the evaluated significances against common-distance biases.
\subsubsection{Generation of uncorrelated simulated datasets}\label{dataproduction}
To test the performance of our method in the case of intrinsically uncorrelated data, we produce simulated datasets in the following way.
\begin{itemize}
\item We draw a gamma-ray luminosity from a log-normal distribution, with probability density function
\begin{equation}
p(L_\gamma) = \frac{1}{L_\gamma\sqrt{2\pi\sigma_1^2}}
\exp\left[-\frac{\left(\ln L_\gamma -\mu_1\right)^2}{2\sigma_1^2}\right]\,.
\end{equation}
\item We draw a radio luminosity from a log-normal distribution, with probability density function
\begin{equation}
p(L_r) = \frac{1}{L_r\sqrt{2\pi\sigma_2^2}}
\exp\left[-\frac{\left(\ln L_r -\mu_2\right)^2}{2\sigma_2^2}\right]\,.
\end{equation}
\item For this pair, we also draw a common redshift from a uniform distribution with lower limit $z_{\rm low}$ and upper limit $z_{\rm up}$.
\item We evaluate the resulting gamma-ray and radio fluxes, and check whether they reside within an allowed flux dynamical range of three orders of magnitude. If either one does not, we reject the pair and repeat the draw.
\item We repeat the process above until we have 30 pairs within our desired flux dynamical range. This then is our simulated, intrinsically uncorrelated dataset, to which both a common distance factor and a limit in the flux dynamical range have been applied.
\end{itemize}
We anticipate that the effect of the common-distance biases will increase as the luminosity dynamical range decreases and the redshift dynamical range increases. This can be easily understood by considering the extreme limits. Datasets drawn from luminosity delta-functions will always appear perfectly correlated within errors: the spread in fluxes in each waveband is only due to the distance factor, which is the same in each pair, and errors. Conversely, if all sources are at the same redshift, there will be no common-distance effect: the distance factor is always the same, and any observed correlation has to be intrinsic.
To assess when common-distance biases become important, we will use the {\em coefficient of variation} (eg., Frank \& Althoen 1995) of the redshift and luminosity distributions (standard deviation in units of the mean, $c_z$ and $c_L$ respectively) to quantify the dynamical range of each distribution. As we will see in the next section, the importance of common-distance biases is generally dependent on the ratio of the luminosity coefficient of variation to the redshift coefficient variation, $c_L/c_z$, and decreases as this ratio increases. In our simulated datasets we have used radio and gamma-ray luminosity distributions\footnote{Since the flux/flux correlation coefficient is evaluated in logarithmic space, changing the units of the luminosity, or, equivalently, the mean of the luminosity distribution, will only uniformly slide the points along the flux axes and will not affect the apparent correlation strength, as long as the flux limits are also shifted accordingly.} with $\mu_1=\mu_2 = \mu_0$ and $\sigma_1=\sigma_2=\sigma_0$ and, as a result, the same value of $c_L$, but in practice the relevant value of $c_L$ is the one of the more extended of the two distributions. For the distributions we have used here,
\begin{equation}
c_L = \left[\exp\left(\sigma_0^2\right)-1\right]^{1/2}
\end{equation}
and
\begin{equation}
c_z = \frac{z_{\rm up}-z_{\rm low}}{\sqrt{3}(z_{\rm up}+z_{\rm low})}\,.
\end{equation}
\subsubsection{Robustness of the test against common-distance biases}\label{robustness}
To evaluate the robustness of our test against common-distance biases and its ability to reject false positives, we generate, using the procedure described in \S \ref{dataproduction}, simulated datasets with varying values of the ratio $c_L/c_z$ of 30 objects each, and we calculate the significance of the apparent correlation using our method and the simple estimate of Eq.~(\ref{student}).
In practice, we implement the simulated dataset generation for a specific value of $c_L/c_z$ in two distinct ways, and we compare the results as shown in Fig. \ref{sigmaofcc}. First, we keep the redshift distribution fixed to a uniform distribution with lower limit $z_{\rm low} = 0$ and an upper limit $z_{\rm up} = 2$, and we draw the radio and gamma-ray luminosities from identical distributions with $\mu_0=0$ and a varying value of $\sigma_0$. In this way, we derive the black points in Fig. \ref{sigmaofcc}. Next, we keep the luminosity distributions fixed at $\mu_0=0$, $\sigma_0 =1$, and we draw redshifts from uniform distributions with varying upper and lower limits, always symmetric about $z=1$. In this way, we derive the red points in Fig. \ref{sigmaofcc}.
For each dataset, we then evaluate the significance of the apparent correlation using the variation of our test that utilizes redshift binning; these results are shown with the circles/solid lines in Fig.~\ref{sigmaofcc}. We compare these values with the significance estimate of Eq.~(\ref{student}) which does not account for any common-distance bias; these results are shown with the diamonds/dashed lines in Fig.~\ref{sigmaofcc}. For low values of $c_L/c_z$, the simple estimate of Eq.~(\ref{student}) returns false positives with high significance for these intrinsically uncorrelated datasets. Our method however correctly identifies these apparent correlations as artifacts of common-distance biases, and returns a significance value always consistent with no correlation. For higher values of $c_L/c_z$, common-distance biases are less important, and both significance estimates agree, returning a result consistent with no correlation.
The roughly consistent, within noise, behavior of the black and red lines, despite the different method of implementation of the same value of $c_L/c_z$, implies that the $c_L/c_z$ ratio is a good way to quantify the way in which the dynamical ranges in the luminosity and redshift distributions induce common-distance biases in correlations between different wavebands evaluated in flux space. As a rule of thumb, a value of $c_L/c_Z $ smaller than about 5 indicates that common-distance biases may be important, and the simple estimate of Eq.~(\ref{student}) (or, equivalently, permutation methods in flux space alone which do not account for the common distance modulation in each flux pair) should not be trusted as they might yield false positives.
\begin{figure}
\includegraphics[width = 3in, clip]{Sig_of_cc.eps}
\caption{Significance (probability to obtain an $r$ as big or bigger than the data by chance) returned by our method (circles, solid lines) compared to significance returned by the simple estimate of Eq.~(\ref{student}) which does not account for common distance biases, as a function of the ratio of coefficients of variation of the luminosity and redshift distributions. Black points were generated by varying the width of the luminosity distribution while keeping the redshift distribution fixed. Red points were generated by varying the width of the redshift distribution while keeping the luminosity distribution fixed. Our method always succeeds in rejecting artificial correlations induced by common-distance biases. \label{sigmaofcc}}
\end{figure}
\subsection{Demonstrations on simulated data: \\ Correlated datasets}
In the previous section we have shown that our proposed method successfully accounts for common-distance biases and returns results consistent with no correlation even when the simple estimate of Eq.~(\ref{student}) yields very significant false positives. Here we wish to examine whether this robustness against false positives comes at the expense of the power of the test in rejecting the null hypothesis of no correlation when the data are intrinsically correlated. In \S \ref{dataproduction2} we discuss how we generate intrinsically correlated datasets with minimal common-distance biases, and in \S \ref{deponN} we discuss how the power of the test depends on the number of objects in the dataset, $N$, and on the apparent correlation strength, $r$, as well as how these dependencies compare with the simple formula of Eq.~\ref{student}.
\subsubsection{Generation of correlated datasets}\label{dataproduction2}
To generate mock datasets with known intrinsic correlation signals, we assume that the radio and gamma-ray monochromatic luminosities at the frequencies of interest are linearly correlated\footnote{We adopt this assumption in the interest of simplicity for these demonstrations; this does not have to be the case in nature. Nonlinear correlations between the luminosities in the two wavebands will further complicate the relation between intrinsic and apparent correlation strength.}, with some scatter obeying a log-normal distribution:
\begin{equation}
\log L _r = C + \log L_\gamma + \Delta \log L_r
\end{equation}
where $C$ is a normalization constant, and $\Delta \log L_r$ is normally distributed with mean $0$ and standard deviation $\sigma$, i.e. if $\Delta \log L_r =x$ then the probability density of $x$ is given by
\begin{equation}
p(x) = \frac{1}{\sqrt{2\pi} \sigma}\exp\left[-\frac{x^2}{2\sigma^2}\right]\,.
\end{equation}
Using Eq. \ref{kc}, this yields a relationship between radio and gamma-ray flux densities:
\beq\label{mockseq}
\frac{S_r}{S_{r,0}} = \frac{S_\gamma}{S_{\gamma,0}} (1+z)^{\alpha_r+\Gamma - 1} \times 10^{\Delta \log r}\,.
\eeq
The scatter in this intrinsic correlation is quantified by $\sigma$. We normalize the relation assuming that, for $z=\Delta \log r=0$, a 15 GHz radio flux density of $1 {\rm \, Jy}$ corresponds to a gamma-ray flux density of $10^{-8} {\rm GeV cm^{-2} s^{-1} GeV^{-1}}$ at 300 MeV.
We generate mock datasets by starting from the set of 136 sources which (a) are detected by {\it Fermi} LAT at energies between 300 MeV and 1 GeV and are included in the First Fermi Catalog (1LAC, Abdo et al.~2010); (b) are included in the OVRO 40 M monitoring sample; (c) have known redshifts (see Ackermann et al.~2011 for the details of this sample). We use this set to obtain redshifts, gamma-ray fluxes, and gamma-ray spectral indices for our sources; for radio spectral indices, we use the historical values quoted in Ackermann et al.~2011. We then use Eq.~(\ref{mockseq}) to obtain radio fluxes with a known correlation signal, by using the desired value of $\sigma$.
The value of the $c_L/c_z$ ratio in the sample we use is $\sim 4$, so, according to the findings of \S \ref{robustness}, the effect of common-distance biases should be limited, and any apparent correlation between gamma-ray and radio emission should be primarily due to the intrinsic correlations we have imposed in the simulated datasets. In this case, we would expect a well-behaved test to return results that are close to the simple significance estimates of Eq.~(\ref{student}).
\begin{figure}
\includegraphics[width = 3in, clip]{rofshist.eps}
\caption{Distribution of Pearson product-moment correlation coefficients $r$ that arise from random realizations of 80 objects obtained using 80 randomly chosen {\it Fermi} sources from 1LAC and Eq.~\ref{mockseq}, for various values of $\sigma$ as in legend. \label{rofsigma}}
\end{figure}
Figure \ref{rofsigma} shows the distribution of Pearson product-moment correlation coefficients $r$ that arise from random realizations of 80 objects obtained in the manner described above, for various values of the intrinsic correlation scatter $\sigma$. The striking feature of this plot is that the distributions of possible $r$ values of ``observed'' flux/flux correlations arising from different random realizations of the same intrinsic luminosity/luminosity correlation sampled with the same number of points can be quite extended, with its width increasing with increasing $\sigma$.
Even if we assumed that we knew the form of the underlying intrinsic luminosity/luminosity correlation, i.e. if, in our case, we assumed that Eq.~\ref{mockseq} holds exactly, and even with a relatively large sample (80 objects in this case), the observed value of the flux/flux correlation coefficient would only yield a rough and uncertain estimate of the scatter $\sigma$ of the underlying correlation, although the uncertainty of the estimate would improve for increasing values of $r$ (decreasing values of $\sigma$).
\subsubsection{Dependence of significance on number of observations and apparent correlation strength}\label{deponN}
Figure \ref{sofn} shows the dependence of the calculated significance of a correlation of fixed apparent and intrinsic strength (i.e, fixed values of $r$ and $\sigma$) on the number of objects in the sample. In the example presented here, mock datasets of $N$ objects were generated as described in \S \ref{dataproduction2}, using a fixed
intrinsic correlation scatter $\sigma = 0.4$, and requiring an apparent correlation strength of $r=0.55\pm0.001$. For every value of $N$ plotted in Fig.~\ref{sofn}, 10 such mock datasets were produced, and for each dataset the significance was evaluated using the redshift-bin-splitting variation of the test (with the number of redshift bins chosen, for each value of $N$, as discussed in \S \ref{thesplit}.)
The datapoints in Fig.~\ref{sofn} represent the the mean of $\log_{10}$(Significance) for these 10 realizations, and the error bars indicate the standard deviation of the 10 values of $\log_{10}$(Significance). The solid line shows the result of Eq.~(\ref{student}) for $r=0.55$ and varying $N$. Even this modest correlation with appreciable scatter can be established at high significance (better than $\sim 10^{-5}$) with 60 or more objects.
\begin{figure}
\includegraphics[width = 3in, clip]{SigOfN_log.eps}
\caption{Significance of an intrinsic correlation with $\sigma = 0.4$ sampled with $N$ objects and resulting in an apparent correlation strength of $r=0.55\pm0.001$, as a function of $N$. The points indicate the mean and error bars indicate the standard deviation of the calculated significances in 10 random implementations of the correlation. The downwards triangle indicates an upper limit for $N$=100, where the probability of the correlation to arise by chance was always found to be $<10^{-6}$ (none out of $10^7$ scrambled datasets had an $|r|$ at least as big as the ``data''). The solid line shows the result of Eq.~(\ref{student}) for $r=0.55$ and varying $N$.\label{sofn}}
\end{figure}
\begin{figure}
\includegraphics[width = 3in, clip]{SofR_log.eps}
\caption{Significance of an intrinsic correlation with $\sigma = 0.6$ sampled with 20 objects and resulting in an apparent correlation strength of varying $r$, as a function of $r$. The points indicate the mean and error bars indicate the 1$\sigma$ variation of the calculated significances in 10 random implementations of the correlation. The solid line shows the result of Eq.~(\ref{student}) for $N=20$ and varying $r$. \label{sofr}}
\end{figure}
Figure \ref{sofr} shows the dependence of the significance that the redshift-bin-splitting variation of our method yields as a function of the apparent correlation strength (as quantified by $r$), when the underlying, intrinsic correlation and the number of objects are fixed. We have used an intrinsic correlation with relatively large scatter ($\sigma =0.6$), sampled with a relatively small number of objects ($N=20$). As it is obvious from Fig.~\ref{rofsigma} (red line), this large scatter can result in a variety of apparent correlation values. Again, for each value of $r$ plotted in Fig.~\ref{sofr}, we have generated 10 mock datasets as described in \S \ref{dataproduction2}, demanding that their apparent correlation strength is within $0.001$ of the plotted $r$ value. For each dataset the significance was evaluated using the redshift-bin-splitting variation of the test (with the number of redshift bins chosen, for each value of $N$, as discussed in \S \ref{thesplit}.)
The datapoints in Fig.~\ref{sofr} represent the the mean of $\log_{10}$(Significance) for these 10 realizations, and the error bars indicate the standard deviation of the 10 values of $\log_{10}$(Significance). The solid line shows the result of Eq.~(\ref{student}) for $N=20$ and varying $r$.
As we can see in both Fig.~\ref{sofn} and Fig.~\ref{sofr}, the significance returned by our method in the absence of strong common-distance biases and for correlated datasets is consistent with, or only marginally worse than the results of the simple estimate of Eq.~(\ref{student}). We therefore conclude that the robustness of our method against common-distance biases in uncorrelated datasets does not come at the expense of the power of the method in the case of correlated datasets.
\section{Discussion and Conclusions}
In this paper, we have introduced a data-randomization method for assessing the significance of apparent correlations between radio and gamma-ray emission in blazar jets, accounting explicitly for biases introduced through a common distance, small sample size, and complex or subjective sample selection criteria. Our method is designed to be conservative and applicable even to small samples selected with subjective criteria.
An application of this technique to the first {\it Fermi} catalog of point sources (Abdo et al.\ 2010) has been discussed in Ackermann et al.~(2011), which also discusses the dependency of the strength and significance of the radio/gamma flux correlation on gamma-ray photon energy, and on the concurrency of the two datasets. A study of the dependency of correlation strength and significance on radio frequency can be assessed by application of this method to multi-frequency radio monitoring data, such as the results of the F-GAMMA Program (Angelakis et al.~2010; Fuhrmann et al.\ 2012, in preparation).
Using simulated datasets of intrinsically uncorrelated data, we have demonstrated that our proposed method is robust against artificial correlations induced by common-distance biases, and returns results consistent with no correlation even when simple face-value significance estimates that do not account for these biases would have incorrectly claimed highly significant correlations. We have shown that the effect of these biases can be quantified by the ratio of the coefficients of variation of the luminosity distribution (in the waveband which has the widest luminosity distribution) over the redshift distribution. When this ratio is lower than $\sim 5$ false positives are possible when the biases are not accounted for, with their significance increasing with decreasing value of the ratio. In addition, using simulated datasets of intrinsically correlated data, we have shown that our method can establish existing correlations with significance comparable to that of other tests, and thus its robustness against false positives does not come at the expense of its power in rejecting the null hypothesis.
As our method is designed to be applied to astronomical datasets, we have implemented it in such a way to directly address the limited flux dynamical range that is generally encountered in such data. Astronomical datasets are generally expected to have a low-flux limit in each frequency due to the limited sensitivity of any given observing instrument, and a high-flux limit corresponding to the most favorable combination of luminosity/distance that happened to occur given our position in the universe, which is generally determined by chance, and fixed by the observed dataset. Simulated data with one or two of the fluxes outside these limits represent situations possible in nature but impossible to observe in the specific experiment. Such simulated data coud result in apparent correlations in our simulated samples induced by a single very bright or very faint object much brighter or fainter than the objects in our actual sample, while most of the other pairs would be scattered in a limited area of the flux/flux space (the classical case of an artificial correlation seen when an uncorrelated scattered cluster of points is combined with a single point far away from the main cluster). Such configurations would be impossible in our actual observed datasets, but could be frequent in our simulated datasets, thus biasing the simulated datasets toward much higher correlation coefficients, and artificially reducing the significance of any correlation seen in the observed datasets. We exercise care to avoid this bias by limiting the flux dynamical range of our randomized data to that of the observed sample and by rejecting simulated flux pairs outside this range.
We caution the reader that our test does not in any way account for the effects of non-simultaneity. Ideally the test should be applied to data in different frequencies averaged over the same time interval. The spectral indices used in each band to implement the K-correction should also be concurrently measured with the flux averages. Such concurrent spectral indices are straight-forward to obtain in gamma rays, however radio monitoring is routinely performed at a single waveband (as is the case for the OVRO 40 M Monitoring Program), and in practice only archival radio spectral indices are available. It is thus fortunate that our test is robust against small changes in the value of the radio spectral index used in the K-correction. This property of the test can be understood by taking into account that blazars are spectrally flat at radio frequencies so the effect of the K-correction in radio is small to begin with. We have confirmed this by alternatively using archival radio spectral indices measured for each source, or a uniform value of $\alpha_r=-0.5$ across all sources; the evaluated significance in the two cases did not change appreciably (fractional change in the quoted significance less than $0.01$). This result can be explicitly confirmed by using multi-frequency simultaneous data from the F-GAMMA Program where simultaneous radio spectral indices can be obtained. These tests are described in detail in Fuhrmann et al.\ 2012 (in preparation).
In contrast, the test is quite sensitive to the redshift of the sources included in the sample under consideration, as shown in Ackermann et al.~(2011): as the main purpose of the test is to assess the effect of distance biases, the calculations involved are sensitively dependent on said distances. For this reason the test should only be used on samples with known redshifts for all members.
Finally, we stress that this test in itself does not assess the strength of the intrinsic correlation between flux densities of different frequencies. It only addresses its statistical significance, i.e. the probability that an apparent correlation as strong or stronger than the observed one can be obtained from intrinsically uncorrelated data due to observational biases. A correlation may be very weak but picked up at high significance if the dataset is large and the data quality is high; in contrast, a strong correlation in a very small sample may not be very significant. It is similarly important when the test returns a low statistical significance to carefully distinguish between lack of evidence for correlation and evidence for intrinsically uncorrelated data. Intrinsic lack of correlation cannot generally be established. However it is possible to show that a correlation with strength above a certain threshold would have been picked up at a given level of significance by a particular test.
\acknowledgements{We thank Andy Strong and the anonymous referees for insightful comments which improved this manuscript. The OVRO 40 M program is supported in part by NASA grants NNX08AW31G and NNG06GG1G, and NSF grant AST-0808050. Support from the Max-Planck Institut f\"{ur} Radioastronomie for upgrading the OVRO 40 M telescope receiver is also acknowledged. We are grateful to Russ Keeney for his tireless efforts in support of observations at the Owens Valley Radio Observatory. VP acknowledges support for this work provided by NASA through Einstein Postdoctoral Fellowship grant number PF8-90060 awarded by the Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for NASA under contract NAS8-03060, and thanks the Department of Physics at the University of Crete for their hospitality during the completion of part of this work.
WM acknowledges support from the U.S. Department of State and the Comisi\'{o}n Nacional de Investigaci\'{o}n Cient\'{i}fica y Tecnol\'{o}gica (CONICYT) in
Chile for a Fulbright-CONICYT scholarship.
}
|
1,116,691,500,859 | arxiv | \section{The State Space of Quantum Mechanics is Redundant}
The first paper I'd like to discuss is ``The measurement postulates of
quantum mechanics are operationally redundant'' by Masanes, Galley and
M\"uller~\cite{Masanes:2019}. The short version of my spiel is that
they present a condition on \emph{states} that seems more naturally to
me like a condition on \emph{measurement outcomes.} Upon making this
substitution, the Masanes, Galley and M\"uller (MGM) result comes much
closer to resembling Gleason's theorem than they say it does.
I have a sneaky suspicion that a good many other attempted
``derivations of the Born rule'' really amount to little more than
burying Gleason's assumptions under a heap of dubious
justifications. MGM do something more interesting than that. They
start with what they consider the ``standard postulates'' of quantum
mechanics, which in their reckoning are five in number. Then they
discard the last two and replace them with rules of a more qualitative
character. Their central result is that the discarded postulates can
be re-derived from those that were kept, plus the more
qualitative-sounding conditions.
MGM say that the assumptions they keep are about state space, while
the ones they discard are about measurements. But the equations in the
three postulates that they keep could just as well be read as
assumptions about measurements instead. Since they take measurement to
be an operationally primitive notion --- fine by me, anathama to many
physicists! --- this is arguably the better way to go. Then they add a
postulate that has the character of noncontextuality:\ The probability
of an event is independent of how that event is embedded into a
measurement on a larger system. So, they work in the same setting as
Gleason (Hilbert space), invoke postulates of the same nature, and
arrive in the same place. The conclusion, if you take their postulates
about complex vectors as referring to measurement outcomes, is that
``preparations'' are dual to outcomes, and outcomes occur with
probabilities given by the Born rule, thereupon turning into new
preparations.
Let's treat this in a little more detail.
Here is the first postulate of what MGM take to be standard quantum
mechanics:
\begin{quotation}
\noindent To every physical system there corresponds a complex and
separable Hilbert space $\mathbb{C}^d$, and the pure states of the
system are the rays $\psi \in {\rm P}\mathbb{C}^d$.
\end{quotation}
We strike the words ``pure states'' and replace them with ``sharp
effects'' --- an equally undefined term at this point, which can
only gain meaning in combination with other ideas later.
(I spend at least a little of every working day wondering why quantum
mechanics makes use of complex numbers, so this already feels
intensely arbitrary to me, but for now we'll take it as read and press
on.)
MGM define an ``outcome probability function'' as a mapping from rays in
the Hilbert space $\mathbb{C}^d$ to the unit interval $[0,1]$. The
abbreviation OPF is fine, but let's read it instead as \emph{operational
preparation function.} The definition is the same: An OPF is a
function ${\bf f}: {\rm P}\mathbb{C}^d \to [0,1]$. Now, though, it
stands for the probability of obtaining the measurement outcome
$\psi$, for each $\psi$ in the space ${\rm P}\mathbb{C}^d$ of sharp
effects, given the preparation ${\bf f}$. All the properties of OPFs
that they invoke can be justified equally well in this reading. If
${\bf f}(\psi) = 1$, then event $\psi$ has probability 1 of occurring
given the preparation ${\bf f}$. For any two preparations ${\bf f}_1$
and ${\bf f}_2$, we can imagine performing ${\bf f}_1$ with
probability $p$ and ${\bf f}_2$ with probability $1-p$, so the convex
combination $p{\bf f}_1 + (1-p){\bf f}_2$ must be a valid
preparation. And, given two systems, we can imagine that the
preparation of one is ${\bf f}$ while the preparation of the other is
${\bf g}$, so the preparation of the joint system is some composition
${\bf f} \star {\bf g}$. And if measurement outcomes for separate
systems compose according to the tensor product, and this $\star$
product denotes a joint preparation that introduces no correlations,
then we can say that $({\bf f} \star {\bf g})(\psi \otimes \phi) =
{\bf f}(\psi) {\bf g}(\phi)$. Furthermore, we can argue that the
$\star$ product must be associative, ${\bf f} \star ({\bf g} \star
{\bf h}) = ({\bf f} \star {\bf g}) \star {\bf h}$, and everything else
that the composition of OPFs needs to satisfy in order to make the
algebra go.
Ultimately, the same math has to work out, after we swap the words
around, because the final structure is self-dual:\ The same set of rays
${\rm P}\mathbb{C}^d$ provides the extremal elements both of the state
space and of the set of effects. So, if we take the dual of the
starting point, we have to arrive in the same place by the end.
But is either choice of starting point more \emph{natural}?
Beginning with the set of measurement outcomes may help put the
mathematics in conceptual and historical context. For instance, when
proving a no-hidden-variables theorem of the Kochen--Specker type, the
action lies in the choice of measurements, and how the rays that
represent one measurement can interlock with those for
another~\cite{Mermin:1993}. So, from that perspective, putting the
emphasis on the measurements and then deriving the state space is the
more conceptually clean move.
That said, on a deeper level, I don't find either choice all that
compelling. To appreciate why, we need only look again at that arcane
symbol, ${\rm P}\mathbb{C}^d$. That is the setting for the whole
argument, and it is completely opaque. Why the complex numbers? Why
throw away an overall phase? What is the meaning of ``dimension'', and
why does it scale multiplicatively when we compose systems? (A typical
justification for this last point would be that if we have $n$
completely distinct options for the state of one system, and we have
$m$ completely distinct options for the state of a second system, then
we can pick one from each set for a total of $nm$ possibilities. But
what are these options ``completely distinct'' with respect to, if we
have not yet introduced the concept of measurement? Why should
dimension be the quantity that scales in such a nice way, if we have
no reason to care about vectors being orthogonal?) All of this cries
out for a more fundamental understanding~\cite{Stacey:2019}.
\section{Nothing Isn't What It Used To Be}
The second paper I'd like to discuss is by Hossenfelder, originally
titled ``Born's rule from almost
nothing''~\cite{Hossenfelder:2020}. The claim of this paper is nicely
stated in a concise form up front. Let $\ket{\Psi}$ and $\ket{\Phi}$
denote unit-norm elements of a complex vector space $\mathbb{C}^N$.
\begin{quotation}
\noindent \textbf{Claim:}\ The only well-defined and consistent
distribution for transition probabilities $P_N(\ket{\Psi} \to
\ket{\Phi})$ on the complex sphere which is continuous, independent
of~$N$, and invariant under unitary operations is $P_N(\ket{\Psi}
\to \ket{\Phi}) = |\braket{\Psi}{\Phi}|^2$.
\end{quotation}
Here, ``well-defined'' means that $P_N$ always evaluates to a number
in the unit interval, and ``consistent'' means that if $\{\Phi_i\}$ is
an orthonormal basis, then
\begin{equation}
\sum_{i=1}^N P_N(\ket{\Psi} \to \ket{\Phi_i}) = 1 \, ,
\end{equation}
and also
\begin{equation}
P_N(\ket{\Phi_i} \to \ket{\Phi_j}) = \delta_{ij} \, .
\end{equation}
Importantly, we have again an assumption of context-independence,
since the transition probability $P_N$ is posited to be indifferent to
the set in which the final state might be embedded. Whereas in
Gleason's theorem the structure of the possible measurement outcomes was
assumed and the state space derived, here both sets are taken as given
(and indeed identical).
If the initial state is $\ket{\Psi}$ and the possible post-transition
states are $\{\ket{\Phi_i}\}$, then unitary transformations will leave
invariant the 3-vertex Bargmann invariants $\braket{\Psi}{\Phi_i}
\braket{\Phi_i}{\Phi_j} \braket{\Phi_j}{\Psi}$. These will vanish if
the post-transition states are an orthonormal basis, but if we do not
assume Gleason-style context independence, then the Bargmann
invariants can be part of the context and the transition probabilities
can depend upon them. (These invariants can be quite rich
mathematically~\cite{Appleby:2017}, so it is almost a shame that they
don't appear more directly in calculating probabilities!) In that
case, it seems to me, we would have no particular reason to say that
orthogonality should mean zero probability. So, we would have no
particular reason to demand that the post-transition states must form
an orthonormal basis.
Even if we do not wish to incorporate the lovely 3-vertex information,
allowing context dependence opens up the possibilities for
non-Born-rule probability assignments. Let $g$ be any continuous,
nonnegative function with the property that $g(0) = 0$. Then the
generalized Preskill rule~\cite{Fuchs:2011} given by
\begin{equation}
P_N(\ket{\Psi} \to \ket{\Phi_i}|\{\ket{\Phi_j}\})
= \frac{g(|\braket{\Psi}{\Phi_i}|)}
{\sum_j g(|\braket{\Psi}{\Phi_j}|)}
\end{equation}
satisfies the desiderata that $\ket{\Psi}$ transitions to itself with
probability 1 and to an orthogonal $\ket{\Phi_i}$ with probability
0.
Later in the paper, the assumption of continuity is dropped. This
raises further possibilities for contextual, non-Born-rule probability
assignments, even piecewise-continuous ones. For example, suppose that
a measurement corresponds to an orthonormal basis $\{\ket{\Phi_i}\}$,
and let a measurement induce a transition from the initial state
$\ket{\Psi}$ to whichever $\ket{\Phi_i}$ has the largest absolute
overlap $|\braket{\Psi}{\Phi_i}|$. If there is a tie for largest, we
distribute the probability evenly across those outcomes. This rule is
unitarily invariant and independent of the dimension $N$. Moreover, if
one of the $\{\ket{\Phi_i}\}$ is equal to $\ket{\Psi}$, then the state
will remain unchanged with probability 1, and because for any basis at
least one absolute overlap must be greater than zero, a transition
will never occur to an orthogonal state.
\section{Conclusion}
Examining these recent approaches to the Born rule has underlined the
significance of Gleason's noncontextuality assumption. Clearly, it is a
mathematically potent condition. From one perspective, it is
physically rather presumptuous:\ If one takes the traditional view
that going from classical to quantum physics means promoting variables
to Hermitian operators, then why should expectation values not depend
upon \emph{operators,} rather than merely upon eigenvectors taken one
at a time? What conceptual desiderata make a Gleason-style
assumption a \emph{natural} move? Finding the answer, I suspect, will
require beginning before Hilbert spaces and deriving them in turn.
|
1,116,691,500,860 | arxiv | \section{Introduction}
A meticulous measurement of energy and arrival direction of individual primary cosmic ray (CR) nuclei has revealed that the intensity of CRs in the TeV band is remarkably isotropic. The measured degree of deviation from isotropy, about few parts per ten thousand, appears to be much smaller than the implied level of anisotropy by the standard diffusive propagation models of Galactic CRs, which are otherwise effective in reproducing several observables, such as the spectrum and composition \citep[see e.g.][]{Strong2007,Blasi,Pohl2013}. However, the consideration of anisotropy in the spatial diffusion and an inhomogeneous distribution of cosmic-ray sources in the spiral arms can potentially reduce the tension between the observation and the propagation models \citep{David2013,Rahul2014TeV,Rahul2014EeV,EvoliPRL,David2016}.
In recent years, a large number of TeV-band CR events registered by modern observatories have manifested small but significant variation in the flux at all resolved angular scales \citep{Amenomori439,SuperPRD,MilagroPRL,ARGO,HAWC2014}. Although the dipole anisotropy in the measured intensity is understood to be due to asymmetry in the distribution of CR sources, the origin of anisotropy in the intensity at the small angular scales remains unclear. It has been suggested that these patchy structures in the CR intensity skymap may be a combined effect of the prevalent interstellar magnetic turbulence in the vicinity of the Solar system and a spatial inhomogeneity in CR density \citep{Gwenael2012,Ahlers2014,Ahlers2015}. Alternatively, it has been suggested that they could possibly be due to very local alteration of the CR phase-space distribution by heliospheric structures \citep{IbexAnis,Ming2014,Drury2013,Paolo2013,Paolo2016}.
In this paper, we suggest that if the \mpo{CR anisotropy} at small angular scales is mainly due to the local turbulence, then the structures in the CR \mpo{arrival distribution} of a given energy band should be variable on a time-scale determined by gyroradius-scale shift in the position of the Earth with respect to the turbulence. At a velocity of about 20 km s$^{-1}$, the solar system traverses $\sim40$ AU ($\sim 2\, \cdot 10^{-4}$ pc) in approximately 10 years\citep{Witte2004,ibex2009}, which is comparable to the gyroradius of TeV CRs close to our solar system, \mpo{$r_g(E) \sim 3\, \cdot10^{-4}$ pc for protons and 3 $\mu$G as the estimated strength} of the magnetic field in the local interstellar medium \citep{ibexB}. For TeV CRs the variation due to a change in the location of the Solar system is expected to cause a decadal-scale variation.
The HAWC observatory operating in the Northern Hemisphere reports a significant change in the intensity skymap of few-TeV CRs\citep{HAWC2014} in relation to the data accumulated by its predecessor, the Milagro experiment \citep{MilagroPRL}. Specifically, it reports a change in the intensity and location of one of the \mpo{intensity excesses and the} appearance of an entirely new excess region at TeV energies. While the discrepancy is suggested to be due to a difference in the median energy of the \mpo{CR events recorded by} these two observatories, it is likely that part of the discrepancies are of physical origin due to change in the location of the Earth over the operational period of these observatories. In the following we present the numerical method used for estimating CRs flux in differential angular bins at the Earth. Thereafter, we compute skymaps of CRs for slightly different locations of the Solar system to illustrate the variability of \mpo{the CR arrival distribution}.
\section{Numerical method}
We numerically integrate the equation of motion of charged particles in a model turbulent magnetic field to exemplify the chaotic nature of the cosmic-ray trajectories and the origin of small-scale anisotropy. Formally, trajectories of CRs in phase-space are described by a doublet of vectors, $\vec{x}(t)$ and $\vec{p}(t)$, denoting the position and the momentum of a CR at time $t$. The time evolution of the trajectories is governed by $d \hat{p} / dt = \omega_c \hat{p}\times \hat{B}$; and $d\vec{x}/dt =\vec{p}/m$, where $\hat{p}$ and $\hat{B}$ are unit vectors along the momentum and the local magnetic field, respectively, $\omega_c$ is the energy dependent gyro-frequency, and $m$ is the relativistic mass. The three-dimensional \mpo{magnetic turbulence} used in our computation is represented by superposition of several linearly polarized shear Alfv\'en waves of random polarizations and phases \citep{Jokipii1999}. The wavenumbers of Alfv\'en modes are logarithmically spaced with $d \log k=0.01$ between minimum wavenumber $k_{min}=0.5/ l_c $ and maximum wavenumber $k_{max}=5\cdot 10^3/ l_c$, where $l_c$ is the correlation length. The turbulent magnetic field is assumed to follow a \mpo{spatially homogeneous} Kolmogorov scaling. The ratio of mean free path to $r_g$ for CRs of smaller $r_g/l_c$ is larger and requires larger computational time to illustrate random walk of the CR trajectories. {Therefore, majority of our analyses are performed for large rigidity CRs, i.e. $r_g/l_c \simeq 0.1$. However, in section \ref{lowEsec} we illustrate that our results are valid for low rigidity CRs as well.}
Trajectories of charged particles can be decomposed into gyration around a mean magnetic field, $\vec{B_0}$, and \mpo{the variation in pitch angle (the angle between $\vec{p}$ and $\vec{B_0}$) arising from} magnetic fluctuations, $\delta \vec{B}$, perpendicular to $\vec{B}_0$, if $\delta B^2/{B}^2_0$ is much smaller than unity \citep{Eichler1987}.
However, the randomization of trajectories cannot always be quantified in terms of pitch-angle scattering, e.g., in the absence of a coherent large-scale mean magnetic field. More generally, the correlation time of nearby points in phase-space, defined by the Lyapunov exponent, can be used to quantify the chaotic nature of the trajectories. In figure \ref{correlation} we show the time evolution of the spatial separation between few randomly selected pairs of trajectories whose initial momentums are separated by $d\vec{p}=10^{-2}\vec{p}$. The separations between the pair of trajectories first increase nearly exponentially, and the initial correlation in their momentum erodes away. After the trajectories are decorrelated, they \mpo{reflect} independent random walks in phase-space and their separation increases diffusively ($\propto t^{1/2}$). Clearly, the time over which two trajectories remain correlated depends on their initial separation in phase-space. In addition, the numerical integration of the equation of motion at each time step involves an irreducible error characterized by the error tolerance adopted by the numerical scheme.
Therefore, numerically backtracked trajectories of individual CRs cannot be used deterministically to map the phase-space density at two different times by invoking Liouville's theorem. We note that the source of error in mapping the phase-space density is not merely due to numerical discretization, but it could also have a physical origin due to small scale ($\ll r_g$) magnetic fluctuations present in the interstellar medium \mpo{that are} not resolved by the observations or the numerical simulations.
However, correlation in the backtracked locations of multiple CRs (at $t=-T$, say) of known arrival direction at Earth (at $t=0$) can be used to estimate correlation in their spatial origin. Specifically, we consider $n$ CRs whose arrival directions at the Earth are uniformly distributed within a two-dimensional angular bin of solid angle $d \Omega$ pointing in direction $\hat{r}$ in an arbitrary coordinate system. We numerically compute \mpo{the} position of each CR at the backtracked time $t=-T$, denoted as $\vec{x}_i(-T) \, \forall i \in \left\{1,2,...,n\right\}$. The phase-space separation among the CRs increases with the backtracking time, and \mpo{the} initial correlation in their position and momentum is lost after $t\gtrsim \tau_c$.
At the back-tracked time $t=-T$, we view the spatial distribution of their backtracked location $\vec{x}_i(-T)$ as a probability distribution of their origin. We use the centroid of the probability distribution to estimate the differential flux of cosmic rays, $F(\hat{r})$, within the angular bin, i.e. $F(\hat{r},-T) \propto N_{cr}(\vec{x}_c(-T))$, where $N_{cr}$ is a presumed large-scale spatial distribution of CR number density and $\vec{x}_c(-T)$ is an arithmetic mean of all $\vec{x}_i(-T)$.
For simplicity, we consider
that the CR density distribution, $N_{cr}$, follows a uniform spatial gradient.
The inhomogeneity in CR density implies that a variation in $\vec{x}_c$ with the orientation $\vec{r}$ of the angular bins would give rise to variation in the CR \mpo{arrival direction and hence their intensity}.
\begin{figure}
\includegraphics[width=.99\linewidth]{figures/Correlation.pdf}
\caption{The magnitude of spatial separation, $\delta \vec{r}$, between a pair of CR trajectories are shown as a function of time. Initially $\delta \vec{r}=0$ and $d\vec{p}=10^{-2}\vec{p}$. Each curve \mpo{represents a} pair of trajectories starting from the same location but with randomly oriented momentum vector. }
\label{correlation}
\end{figure}
The centroid $\vec{x}_c$ of the probability distribution is not altered by the uncorrelated random walks of the backtracked trajectories (for $t \gtrsim \tau_c$) in a spatially homogeneous and isotropic turbulence.
However, the ensemble of backtracked trajectories would follow any existing large-scale magnetic field whose topology would contribute to the estimation of $\vec{x}_c$. The effect of large-scale magnetic field can be subtracted if the field lines are sufficiently ordered \mpo{for it to} almost equally affect $\vec{x}_c$ for all angular bins regardless of their orientation, $\hat{r}$. Otherwise, the probabilistic approach described here can not be reliably employed to estimate \mpo{the CR intensity}, since the convergence of $\vec{x}_c$ to a limit is not guaranteed for any arbitrary magnetic-field structure. Moreover, the correlation time of trajectories, $\tau_c$, is larger for smaller angular bins, and for a sufficiently small angular bin decorrelation of trajectories due to growth of numerical error may be significant which would bias the convergence of $\vec{x}_c$. In any case, we impose the criteria of convergence for the choice of angular bin size which can be resolved by the numerical calculation and does not depend on the systematic error in the integration of trajectories. Specifically, we ensure that our results converge with respect to the error tolerance adopted in the numerical integration by changing the numerical tolerance and numerical integration method. The results presented here used a fourth-order accurate Bulirsch-Stoer integration scheme with relative error tolerance of $10^{-6}$ per $0.1/ \omega_c$ \citep{Press:2007:NRE:1403886}.
\begin{figure*}
\centering
\subfloat[ $T= 3 / \omega_c$ ]{%
\includegraphics[width=.25\linewidth]{figures/Conv1New.png}%
}\hfill
\subfloat[ $ T= 15 / \omega_c $ ]{%
\includegraphics[width=.25\linewidth]{figures/Conv2New.png}%
}\hfill
\subfloat[ $ T=30/\omega_c $ ]{%
\includegraphics[width=.25\linewidth]{figures/Conv3New.png}%
}\hfill
\subfloat[ $ T=45 /\omega_c $]{%
\includegraphics[width=.25\linewidth]{figures/Conv4New.png}%
}\hfill
\caption{\mpo{CR intensity skymap calculated} by backtracking cosmic-ray trajectories, here shown as a function of backtracking time. The energy distribution of CRs is a Gaussian of median $r_g/l_c=0.1$ with 10 percent \mpo{dispersion} in the energy. The method used to estimate the flux in individual angular bins is described in the text. Here and in the figure \label{skymap} a smoothing filter of 0.1 radian angular width is applied.}
\label{convergence_skymap}
\end{figure*}
To build skymaps, we discretize the entire range of CR arrival directions into two-dimensional angular bins of equal solid angles. The shape of angular bins are determined using the Healpix \citep{healpix} program. We then estimate the CR flux in all angular bins, as described above, by backtracking a large number of CR trajectories. The skymaps of CR \mpo{intensity} constructed from the mean position of backtracked CR trajectories at four different backtracking times are shown in figure \ref{convergence_skymap}. About $10^3$ particles per angular bin of size $4\times 10^{-3}$~sr (Healpix parameter NSIDE=16) are used to estimate the CR flux in each bin. Note that the estimated CR fluxes are additive. That is to say, sum of estimated fluxes in two adjacent bins is the estimated flux in the conjoined bin. The computed intensity in all the bins are scaled such that the amplitude of the dipole anisotropy is unity. It is evident from figure \ref{convergence_skymap} that the structure of the CR \mpo{arrival distribution} as well the amplitude of small-scale anisotropy (in proportion of the dipole) converges to a limit. We consider the \mpo{converged} skymap as representative of the observed CR \mpo{intensity} for an imposed large-scale gradient in CR number density.
\section{Variation in the Skymap }
\begin{figure*}
\centering
\subfloat[ Reference Location ]{%
\includegraphics[width=.25\linewidth]{figures/Fig1aNew.png}%
}\hfill
\subfloat[ $\Delta x=0.1 \, r_g $ ]{%
\includegraphics[width=.25\linewidth]{figures/Fig1bNew.png}%
}\hfill
\subfloat[ $\Delta x=0.5 \, r_g $ ]{%
\includegraphics[width=.25\linewidth]{figures/Fig1cNew.png}%
}\hfill
\subfloat[ $\Delta x= 1.0 \, r_g $]{%
\includegraphics[width=.25\linewidth]{figures/Fig1dNew.png}%
}\hfill
\caption{The simulated skymap of cosmic-ray {($r_g/l_c=0.1$)} intensity for an observer at a randomly chosen reference location is shown in the left most panel. The location of the observer is then changed along a randomly chosen axis, and the simulated skymaps for varying amounts of displacement (noted in the title) along that axis are shown in the other three panels. }
\label{skymap}
\end{figure*}
\begin{figure}
\includegraphics[width=\linewidth]{figures/SpecVsLocation.pdf}
\caption{Power spectra of the simulated cosmic-ray skymaps at different locations of the Earth. The four different locations of the Earth shown here are the same as in the figure \ref{skymap}. }
\label{power_spec}
\end{figure}
\begin{figure}
\includegraphics[width=\linewidth]{figures/DiffMapSpec.pdf}
\caption
\mpo{Power spectra of the variation in cosmic-ray intensity arising from relocation of the observer}, i.e. the power spectra of the difference between a skymap at the position of reference $\Delta x =0$, and three different Earth positions: $\Delta x =0.1$, $0.5$, and 1.0 $r_g$.}
\label{diff_spec}
\end{figure}
{We vary the location of the Earth and estimate the CR flux in each angular bin, as described above. The realization of magnetic turbulence, CR density, and dipole anisotropy, as well as the size and orientation of the angular bins are kept unchanged. In figure \ref{skymap} we show the simulated cosmic-ray skymap at a randomly chosen reference point as well as at three other nearby locations. The shift in the location of the Earth, represented by $\Delta x$, \mpo{is} along a randomly chosen direction. The four example skymaps in the figure illustrate a random and appreciable variation in the observed cosmic-ray skymap for a \mpo{displacement of Earth on the order of $\sim r_g$, assuming} static magnetic turbulence. The size of angular scale affected by the change in observer's location depends on the magnitude of the change.}
{The power spectra of the CR intensity skymap at different locations of Earth are shown in figure \ref{power_spec}. As evident from the figure, the power spectrum shows only a small change due to change in the location of the Earth. The changes in the intensity skymap due to a change in the location of the Earth appears mainly as a change in the angular position of the structures in the skymap. The power spectra suggest that the amplitude of the variation in intensity at a certain angular scales remains nearly independent of our location. That is to say, the maximum variation in the intensity at certain angular scale is about the amplitude of the intensity variation at that particular angular scale in the skymap at any given location. Angular bins differ in intensity because CR in different bins have different scattering history due to interaction with different turbulent eddies with different phases and pitch angles. For a given location of the Earth the entire skymap represents an ensemble of scattering histories. In a homogeneous turbulence, the history of one angular bin is merely replaced by the other as the location of the observer changes. Therefore the power spectrum remains the same even though the intensities in individual angular bins have changed. }
\begin{figure*}
\centering
\subfloat[ Reference Location ]{%
\includegraphics[width=.25\linewidth]{figures/LowE_Fig1a.png}%
}\hfill
\subfloat[ $\Delta x=1 \, r_g $ ]{%
\includegraphics[width=.25\linewidth]{figures/LowE_Fig1b.png}%
}\hfill
\subfloat[ $\Delta y=1 \, r_g $ ]{%
\includegraphics[width=.25\linewidth]{figures/LowE_Fig1c.png}%
}\hfill
\caption{Similar to Figure \ref{skymap}, the simulated cosmic ray intensity skymaps for $r_g=10^{-4} l_c$ are shown for three arbitrary locations of an observer. The three locations are separated by $r_g$ and the global mean magnetic field is along the x-direction. In order to emphasize the change in small angular scale structures, the dipole anisotropy is removed from all the skymaps shown in this figure.}
\label{lowEfigure}
\end{figure*}
\subsection{Temporal variation for low rigidity CRs: $r_g/l_c=10^{-4}$ }
\label{lowEsec}
The changes in the intensity skymap are due to a) alteration of the phase-space structures by the turbulent component of the magnetic field as gyrophase-bunched particles are scattered during transport from one location to the other, and b) rotation of the existing structure in the momentum space due the local mean magnetic field. A combination of both these effects gives rise to $r_g-$scale variability in the skymap. The case of $r_g=0.1 l_c$ correspond to $\sim$PeV cosmic rays and the mean free path of the CRs is only about 10 times larger than $l_c$. In order to illustrate that $r_g-$scale variation holds true for $\sim$TeV energy cosmic rays as well, we have carried out a set of three high-resolution simulation for $r_g=10^{-4} l_c$. In this case $k_{max}=2\cdot 10^5$ and about $10^7$ CR trajectories are computed to produce each skymap. The simulated skymaps for the three different locations of the observer are shown in Figure \ref{lowEfigure}. The figures show that small-scale structures in the CRs intensity skymaps change if position of an observer is changed by about $r_g$. The amplitude of intensity variation at large angular scales $(l\lesssim 5)$ is relatively smaller as compared to the $r_g/l_c=0.1$ case.
\section{Discussion and Conclusions}
The $r_g-$scale variability in the skymap is illustrated for $r_g$ close to the correlation length of the turbulence, and therefore moderately large $\delta B/B_0$. The level of turbulence experienced by TeV particles, though smaller if the turbulent spectrum is Kolmogorov, is nevertheless unknown. It could be larger than predicted by a Kolmogorov spectrum if there are additional sources of it at smaller scales (e.g. stellar winds). In any case, a similar $r_g-$scale variability for TeV energy can be argued as follows: at a given location of the Earth different arrival directions $\hat{r}$ of cosmic rays correspond to different pitch angles and phases $\phi$ with respect to \mpo{the} local magnetic field, $B_0$, and gyroradius-scale fluctuations, $\delta B$. Under the hypothesis that the small-scale anisotropy is due to local interstellar turbulence, the variation in CR \mpo{intensity} with arrival direction \mpo{arises} because trajectories map slightly different CR \mpo{source} regions. The distinction in the history of the observed CRs is determined by their local pitch angle and phase $\phi$. A change in the location of an observer implies that the same arrival direction would correspond to different $\phi$ (possibly even \mpo{a different} pitch angle), and the exact same reason (i.e., different pitch angle and $\phi$ means different flux) would now imply a slightly different flux in the same angular bin. In other words, CRs tracing the same phase-space density have very different arrival direction for observers with gyroradius-scale separation due to the gyration of CRs in the magnetic field.
The power spectrum of the CR intensity skymap suggests that \mpo{the} amplitude of variation in the CR intensity is a steeply increasing function of the angular scale of the anisotropy. The alteration in the relevant characteristic of CRs that determine their history are supposedly a smooth function of the arrival direction. The variation in these characteristics for the same arrival direction is expected to be proportional to the displacement of the Earth with respect to the interstellar turbulence. Therefore, a uniform motion of the Earth would imply that the variation in the CR intensity skymap at smaller angular scales would take place at a
shorter time scale. In figure \ref{diff_spec} we show power spectra of the change in the intensity skymaps to illustrate the proportionality between the angular size of the anisotropy and displacement of the Earth. The change in the intensity in any differential angular bin, at a given location of the Earth, is measured with respect to the intensity skymap at a randomly chosen reference position corresponding to $\Delta x =0 $. In figure \ref{diff_spec} the red line is the power spectrum of the difference between the skymap at the position $\Delta x=0.1 \, r_g$, and the skymap at the reference position $\Delta x=0 \, r_g$. It is evident, by comparison with the power spectrum of the CR intensity skymaps(figure \ref{power_spec}), that the amplitude of fluctuations that represent change in the intensity is much smaller for $l\lesssim 20$. That is to say, for a change of $0.1 \, r_g$ in the Earth position, the largest structures ('spots") observed on the reference map would remain nearly unaffected and only the small-scale structures (\mpo{angular scale smaller than $~10$ degrees}) would be affected (see also figure \ref{skymap}). The power spectrum of the difference in the CR intensity skymap \mpo{is} also shown for $\Delta x=0.5 \, r_g$ (in green) and $\Delta x=1 \, r_g$ (in blue). Similarly, we infer that for $\Delta x= 0.5 \, r_g$ changes in the intensity skymap are observed at \mpo{angular scales up to $~45$ degrees}. Therefore, the time scale determined by $r_g-$scale displacement of the Earth may only be considered as an upper limit on a measurable variation on the intensity skymap.
We \mpo{predict} a measurable change in the skymap of \mpo{anisotropy pattern of CRs of a few TeV in} energy in few decades if the structure are due to magnetic turbulence. Remarkably, CRs of a few TeV can also be affected by the heliosphere but any variation in the \mpo{intensity} due to heliospheric would correlate with the change in the heliosphere and would likely show an angular asymmetry in the flux variation due to the non-spherical shape of the heliospheric boundaries. {Moreover, the effect of the heliosphere should be independent of our location in the ISM and can also be better constrained with further improvements in our understanding of the global structure of the heliosphere.} Also, the structure of turbulent eddies at the gyroscale of CRs may change with time which can lead to faster change in the skymap. An Alfv\'en speed $V_A \sim10 $ km/sec would imply that the turnover time $r_g/V_A $ for eddies at gyroradius scale is comparable to the transit time of the Earth through a $r_g$-scale turbulent eddy.
The IceCube observatory has not measured any significant change in the skymap during a six years period of their operation in the Southern Hemisphere \citep{IceCube2016}. The stability of small scale anisotropy observed by IceCube could be due their higher threshold energy $(\gtrsim 10\, {\rm TeV})$ which would require a longer observational period to measure any measurable change. In any case, the limits on temporal variability of the flux skymap can potentially provide strong constraints for theoretical models concerning the origin of anisotropy. Future observations in combination with archival data \mpo{can likely shed} more light on the origin of small-scale anisotropy as the cosmic ray observatories continue to collect data.\\
We thank D. J. McComas, A. Spitkovsky, and E. J. Zirnstein for helpful discussions. We acknowledge support from the Israel-U.S. Binational Science Foundation, the Israeli Science Foundation (ISF), the ISF-University Grant Commission (India), and the Joan and Robert Arnow Chair of Theoretical Astrophysics. RK was partially supported by the Max-Planck/Princeton Center for Plasma Physics and NSF grant AST-1517638. NG acknowledges support from the Koret Foundation. M.P. acknowledges valuable discussion with the team 'The physical of the very local interstellar medium and its interaction with the heliosphere' at the International Space Science Institute in Bern, Switzerland. The analysis in this paper made use of HEALPIX package \citep{healpix}. Numerical calculations in this paper used computational resources supported by PICSciE-OIT High Performance Computing Center.
|
1,116,691,500,861 | arxiv | \section{Introduction}
Supersymmetry (SUSY) extension is one of the most promising way
to solve the gauge hierarchy problem in the standard model \cite{SUSY}.
However, since any superpartners have not been observed
in current experiments, SUSY should be broken at low energies.
Furthermore, soft SUSY breaking terms are severely constrained
to be almost flavor blind and CP invariant.
Thus, the SUSY breaking has to be mediated to the visible sector
in some clever way not to induce too large CP and flavor violation effects.
Some mechanisms to achieve such SUSY breaking mediations
have been proposed \cite{Luty:2005sn}.
On the other hand, in view of the LHC collider physics,
it is quite important to distinguish if the gauginos are Dirac or Majorana.
So far many studies have carried out based on the assumption
that the gauginos are Majorana particles as in the MSSM.
However, there exist some models which give rise to the gaugino
as Dirac particle which can naturally realized in models
with $U(1)_R$ symmetry \cite{Randall}.
For example, supersoft SUSY breaking scenario \cite{Fox:2002bu, Chacko:2004mi}
which has partly ${\cal N}=2$ supersymmetry in gauge sector
provide a model with Dirac gauginos.
Some other extensions of the model and the related phenomenology
can be found in the literature \cite{Nelson:2002ca, Nomura:2005rj}.
Supersoft SUSY breaking scenario also has some good features in
regards to the SUSY CP problems or the EDMs \cite{Hisano:2006mv}.
In this letter, we extend a model of the Dirac gauginos, which originate from
${\cal N} =2$ SUSY for the gauge sector,
such that the ${\cal N} =2$ SUSY is imposed also to the sfermion sector
but only for the 3rd generation squarks and sleptons.
By this extension, the dominant source of radiative correction
to the Higgs mass squared coming from the stop loop becomes controllable
based on the enhanced symmetry.
Then it becomes a viable model which provides a solution to
the little hierarchy problem in SUSY models.
Even in the original scenario, the Dirac gauginos can be superheavy,
of order $10$ TeV or so, while keeping the scalar masses at the weak scale.
This possibility is phenomenologically interesting because it can suppress
the unwanted flavor changing processes.
And in scope of the LHC, this scenario can have a very distinct signature
related to the exotic sfermions which are accompanied as the ${\cal N} =2$
superpartners as well as the Dirac gauginos.
\section{Dirac gaugino in a warped extra dimension}
\subsection{Introduction to the model of Dirac gaugino}
Here we give a brief description of the Dirac gaugino model based on the ${\cal N} = 2$ SUSY in gauge sector.
At first, an ${\cal N} = 2$ vector multiplet ${\cal V} = (V,~\Sigma)$ contains an ${\cal N} = 1$ vector multiplet,
$V= (A^\mu,~\lambda)$ and an $N = 1$ chiral multiplet in the adjoint representation, $\Sigma = (\phi,~\psi)$.
Basically, in the Dirac gaugino model, the gaugino ($\lambda$) has the Dirac mass terms together with
an adjoint fermion ($\psi$). Suppose the adjoint fermion has
mass term $m_\Sigma$, the mass matrix for these states are given by
\begin{equation}
{\cal L}
=
\left(
\begin{array}{cc}
\lambda & \psi
\end{array}
\right)
\left(
\begin{array}{cc}
0 & M_{D} \\
M_{D} & m_{\Sigma} \\
\end{array}
\right)
\left(
\begin{array}{c}
\lambda \\ \psi
\end{array}
\right) \;.
\end{equation}
Therefore, in the mass eigenstates, there are two copious
gauginos for each gauge group:
$(\tilde{g}_1,~\tilde{g}_2)$ for SU(3),
$(\tilde{W}_1,~\tilde{W}_2)$ for SU(2), and
$(\tilde{B}_1,~\tilde{B}_2)$ for U(1).
Many studies on the nature of the Dirac gauginos
in view of the collider physics have been performed.
Some studies on the Dirac nature of either charginos or neutralinos,
which are some mixed states of both $(\tilde{W}_1,~\tilde{W}_2)$ and $(\tilde{B}_1,~\tilde{B}_2)$,
are given in \cite{Choi:2001ww, Aguilar-Saavedra:2003hw, Choi:2003fs, Hagiwara:2005ym}
and a study on the exotic gluinos $(\tilde{g}_1,~\tilde{g}_2)$ having Dirac mass terms
in \cite{Nojiri:2007jm}.
\subsection{Dirac gaugino masses in a warped extra dimension}
We consider a SUSY model
in the warped five dimensional brane world scenario.
The fifth dimension is compactified on the orbifold $S^1/Z_2$
with two branes, ultraviolet (UV) and infrared (IR) branes,
sitting on each orbifold fixed point.
With an appropriate tuning for cosmological constants
in the bulk and on the branes,
we obtain the warped metric,
\begin{eqnarray}
d s^2 = e^{-2 k r_c |y|} \eta_{\mu \nu} d x^{\mu} d x^{\nu}
- r_c^2 d y^2 \; ,
\end{eqnarray}
for $-\pi\leq y\leq\pi$, where $k$ is the AdS curvature, and
$r_c$ and $y$ are the radius and the angle of $S^1$, respectively.
The most important feature of the warped extra dimension model
is that the mass scale of the IR brane is warped down to
a low scale by the warp factor, $\omega = e^{-k r_c \pi}$,
in four dimensional effective theory.
For simplicity, we take the cutoff of the original five dimensional theory
and the AdS curvature as
$M_5 \simeq k \simeq M_P=2.4 \times 10^{18}$ GeV,
the four dimensional Planck mass,
and so we obtain the effective cutoff scale
as $\Lambda_{\rm IR}= \omega M_P$ in effective four dimensional theory.
Now let us take the warp factor so as for the GUT scale
to be the effective cutoff scale
$ M_{\rm GUT}= \Lambda_{\rm IR}=\omega M_P$,
namely $\omega \simeq 0.01$.
When we start from the warped five dimensional setup,
and introduce an extra U(1) gauge multiplet
which is localized on the IR brane,
the ${\cal N}=2$ vector multiplet consists of
$(A_M,~\lambda_{1,2},~\Sigma)$ where
$\Sigma$ is an adjoint scalar multiplet.
In principle, it is always allowed
to have the following operator that generate the Dirac
gaugino masses:
\begin{equation}
{\cal L}
= \int d^2 \theta \frac{\Sigma}{M_5 \omega}
(W_{U(1)'})^\alpha (W_{\rm MSSM})_\alpha
+ h.c. \;,
\end{equation}
where the warp factor is assigned
for the rescaling of $\Sigma$, $\Sigma \to \Sigma/\omega$.
After developing the VEV of the D-term in $W_{U(1)'}$,
the resultant Dirac gaugino masses at the IR scale
or the GUT scale
after rescaling $\left< D' \right> \to \omega^2 \left< D' \right>$ are given by
\begin{equation}
M_{D_i}= \alpha_{\rm GUT}^{1/2} \frac{\omega \left< D' \right>}{M_5} \;.
\end{equation}
Below the IR scale the Dirac gaugino masses at a given scale
$\mu$ are given by
\begin{equation}
M_{D_i} (\mu) = \left(\frac{\alpha_i(\mu)}{\alpha_{\rm GUT}}
\right)^{\frac{b_i - 2 c_i}{2 b_i}} M_{D_i} \;,
\label{RG}
\end{equation}
where $b_i$ is the beta function and
$c_i$ is the quadratic Casimir of the adjoint representation.
Given a gaugino mass $M_\lambda$, scalar masses
are arisen at the one loop level:
\begin{equation}
m^2_{\rm supersoft} = \frac{c_i \alpha_i}{\pi}
\ln \left(4
\right) M_{D_i}^2 \;,
\end{equation}
where we took a limit of $m_{\Sigma} \ll M_{D_i}$.
Hence the scalar masses are suppressed by
a factor of $\frac{\alpha_i}{\pi}$ compared to the gaugino masses.
For instance, in order to have scalar mass scale of order
$m_{\rm scalar} \sim 100$ GeV,
the corresponding gaugino masses need to be around $M_{D_i} \sim 10$ TeV.
Remarkable feature of the supersoft SUSY breaking scenario is that
the scalar masses generated by gaugino loops are finite, and it never
receive renormalization except for the gaugino mass itself,
though the gaugino masses are renormalized by the amount of gauge
couplings as in Eq. (\ref{RG}).
In additional to the above gaugino mass terms, there exist another
contribution to the gaugino masses from anomaly mediation,
which are the Majorana terms and not the Dirac mass terms.
\begin{equation}
M_i^{\rm AMSB} = \frac{b_i g_i^2}{16 \pi^2} F_\phi \;,
\end{equation}
where $F_{\phi}$ is the F-term of the conpensator multiplet,
$\phi = 1 + \theta^2 F_{\phi} \simeq 1 + \theta^2 m_{3/2}$.
Since the supersoft SUSY breaking contribution gives masses
of order $10$ TeV in order to have a scalar masses,
this contributions can be negligible in most cases.
Note that the gravitino mass is not an independent quantity it is fixed
so as to cancel the cosmological constant.
To obtain the vanishing cosmological constant, we need to have an appropriate
constant superpotential on the UV brane.
The condition for vanishing cosmological constant is described as
\begin{equation}
\left<V \right> = \omega^4 \left< D' \right>^2 - 3 \frac{|W|^2}{M_p^2} = 0 \;,
\end{equation}
where we took into account of the rescaling
$\left< D' \right> \to \omega^2 \left< D' \right>$,
the resultant superpotential is
$|W|^2 = \left(\omega^4 \left< D' \right>^2 M_p^2 \right)/3$,
and then the gravitino mass is given by
\begin{equation}
m_{3/2} = \frac{|W|}{M_p^2} = \frac{\omega^2 \left< D' \right>}{\sqrt{3} M_p}
\simeq \frac{\omega M_{D_i}}{\sqrt{3}}
\sim m_{\rm supersoft}\;,
\end{equation}
where we took a warp factor as $\omega \simeq 0.01$.
Hence the anomaly mediated contribution becomes well negligible.
\subsection{Radiative Electroweak Symmetry Breaking}
In the MSSM, the upper bound on the lightest Higgs mass
at one loop level is given by
\begin{equation}
m_{h}^2 \lesssim
m_Z^2 \cos^2 2\beta + \frac{3 v^2 y_t^4}{4 \pi^2}
\sin^4 \beta
\ln \left(\frac{m_{\tilde{t}}^2}{m_t^2} \right) \;.
\end{equation}
In order to satisfy the LEP-II experimental lower bound on
the Higgs mass $m_h \gtrsim 114$ GeV, we need to push up
the stop mass $m_{\tilde{t}} \gtrsim 1$ TeV which may cause
a destabilization of the gauge hierarchy, that is
the so called little hierarchy problem.
On the other hand, in the supersoft SUSY breaking scenario,
the leading contribution to the negative Higgs boson mass squared
comes from top quark and squark loop, which is given by
\begin{equation}
m_{H_u}^2 \simeq
m_{\tilde{\ell}}^2- \frac{3 y_t^2}{4 \pi^2} m_{\tilde{t}}^2
\ln \left(\frac{M_{D_3}}{m_{\tilde{t}}}\right)\;,
~~m_{H_d}^2 \simeq m_{\tilde{\ell}}^2 \;.
\end{equation}
Hence, the mediation scale in this scenario is not high but just
$M_{\rm mess} = M_{D_3} \cong 10$ TeV, the fine-tuning is relaxed
compared to the minimal SUGRA or any other high scale SUSY breaking
scenario.
Putting the results all together, the negative Higgs boson mass squared
parameter is written by
\begin{equation}
m_{H_u}^2 \simeq
m_{\tilde{\ell}}^2-
\frac{3 y_t^2 \alpha_3^{35/18} \alpha_{\rm GUT}^{-4/9}\ln \left(4 \right)}
{4 \pi^3} \frac{\omega \left< D' \right>}{M_5}
\ln \left(\frac{M_{D_3}}{m_{\tilde{t}}}\right)\;,
~~m_{H_d}^2 \simeq m_{\tilde{\ell}}^2 \;.
\end{equation}
For a given Higgs boson mass squared parameter,
the electroweak symmetry breaking condition is described by
\begin{equation}
\frac{M_Z^2}{2} = - m_{H_u}^2 - |\mu|^2\;.
\end{equation}
So, if we take $|\mu| \cong m_{\tilde{\ell}}$,
the correct electroweak symmetry breaking can be achieved
without requiring fine-tuning of the Higgs mass parameters.
Therefore, the important point to solve the little hierarchy problem
in this scenario is to raise the stop mass
while keeping the slepton mass at the weak scale.
Indeed, there exists a mass hierarchy between the stop
and the slepton according to the gauge couplings or
the gauge quantum numbers.
\begin{equation}
\frac{m_{\tilde{t}}^2}{m_{\tilde{\ell}}^2}
\sim \frac{c_3 \alpha_3}{c_2 \alpha_2}
\times \frac{M_{D_3}^2}{M_{D_2}^2} \;.
\end{equation}
Hence, we can take $m_{\tilde{t}} \simeq 1$ TeV
and $m_{\tilde{\ell}} \simeq 100$ GeV at the same time,
which is needed to solve the little hierarchy problem.
\section{A partly ${\cal N}=2$ extension of the MSSM}
\subsection{Introduction to the model of partly ${\cal N}=2$ extended MSSM}
We extend a model of the Dirac gauginos, which originate from
${\cal N} =2$ SUSY for the gauge sector,
such that the ${\cal N} =2$ SUSY is imposed also to the sfermion sector
but only for the 3rd generation squarks and sleptons.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
& 1st generation & 2nd generation & 3rd generation $\equiv$ $SU(2)_{\cal R}$ doublet \\
\hline \hline
quark doublet & $Q_1$ & $Q_2$ & ${\cal Q} = (q_3,~\overline{q}_3)$ \\
\hline
singlet quark & $u^c_1$ & $u^c_2$ &${\cal U}^c = (u^c_3,~\overline{u}^c_3)$ \\
\hline
singlet quark & $d^c_1$ & $d^c_2$ &${\cal D}^c = (d^c_3,~\overline{d}^c_3)$ \\
\hline \hline
\end{tabular}
\end{center}
\caption{Chiral multiplets in the partly ${\cal N}=2$ extension of the MSSM}
\end{table}
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c|c|}
\hline
gauge group & vector multiplet \\
\hline \hline
SU(3) & ${\cal V}^A = (V^A,~\Sigma^A)~(A=1,\cdots, 8)$ \\
\hline
SU(2) & ${\cal V}^a = (V^a,~\Sigma^a)~(a=1,2,3)$ \\
\hline
U(1) & ${\cal V} = (V,~\Sigma)$ \\
\hline \hline
\end{tabular}
\end{center}
\caption{Vector multiplets in the partly ${\cal N}=2$ extension of the MSSM}
\end{table}
Now we evaluate the effects of the Yukawa interactions between the MSSM matter
in the 3rd generation and the exotic states added in the 3rd generation
to the Higgs mass squared.
In order to have Higgs doublet from the adjoint field, which is almost the same situation
as occurred in the Littlest Higgs Model, we have to extend the SM gauge group,
$SU(3)_c \times SU(2)_L \times U(1)_Y$ to $SU(3)_c \times SU(3)_L \times U(1)^\prime$,
and $SU(3)_L \times U(1)^\prime$ will be broken down to the Standard $SU(2)_L \times U(1)_Y$.
The adjoint field contains two Higgs doublet in addition to the triplet and singlet representations.
\begin{equation}
\Sigma =
\left(
\begin{array}{cc}
T & H_u \\
\overline{H}_u & S \\
\end{array}
\right) \;,
\end{equation}
where $T$ is triplet, $H_u$ and $\overline{H}_u$ are doublet, and $S$ is singlet.
On the other hand, quark doublet is contained in the $SU(3)_L$ fundamental representation
such that
\begin{equation}
Q_3 = \left(
\begin{array}{c}
q_3 \\
u^c_3
\end{array}
\right) \;,
\end{equation}
and also we would have ${\cal N}=2$ partner of it:
\begin{equation}
\overline{Q}_3 = \left(
\begin{array}{c}
\overline{q}_3 \\
\overline{u}^c_3
\end{array}
\right)\;.
\end{equation}
The original interaction which provides a top Yukawa coupling is originated from
gauge interaction because of ${\cal N}=2$ supersymmetry.
\begin{eqnarray}
{\cal L} &=& \int d^4 \theta \,g \left[
\overline{Q}_3^\dag \,\Sigma \,\overline{Q}_3
+ Q_3^\dag \, \Sigma \,Q_3
\right]
\nonumber\\
&=& \int d^2 \theta \, g \,\left[
\left(
\overline{q}_3 H_u \overline{u}^c_3
+ \overline{q}_3 \overline{H}_u \overline{u}^c_3 + \cdots \right)
+ \left(
q_3 H_u u^c_3 + q_3 \overline{H}_u u^c_3 + \cdots \right) \right] \;.
\end{eqnarray}
The effective theory which we analyze below the GUT scale is the MSSM
with the right-handed neutrinos. The low energy effective superpotential in this model is given by
\begin{eqnarray}
W_{\rm eff} &=& \sum_{i,j = 1,2} \left(
Y_{u}^{ij} u^c_i q_j H_u
+ Y_d^{ij} d^c_i q_j H_d
+ Y_{\nu}^{ij} \nu^c_i L_j H_u
+ Y_e^{ij} e^c_i L_j H_d
+ \frac{1}{2} M_{R_{ij}} \nu^c_i \nu^c_j \right)
\nonumber\\
&+& g \,\left[
\left(
\overline{q}_3 H_u \overline{u}^c_3
+ \overline{q}_3 \overline{H}_u \overline{u}^c_3 + \cdots \right)
+ \left(
q_3 H_u u^c_3 + q_3 \overline{H}_u u^c_3 + \cdots \right) \right]
\nonumber\\
&+& \mu H_d H_u \;,
\label{Yukawa4}
\end{eqnarray}
where the 2nd line describes the new interactions which appear
only in the partly ${\cal N}=2$ extension of the MSSM.
Interestingly, in the model of partly ${\cal N}=2$ extension of the MSSM,
in contrast to the usual MSSM, the lightest Higgs mass receives no divergent contribution
from the stops in the loop, and it becomes finite.
That is basically because ${\cal N} = 2$ partners of the top quarks,
$(\overline{q}_3,\,\overline{u}_3^c)$ can play the role to cancel the divergence
from the top quarks in the loop for the radiative corrections to the Higgs mass squared.
In this sense, the Higgs mass squared coming from the stop loop becomes controllable
based on the enhanced symmetry, and it can provides a natural solution to
the little hierarchy problem in the MSSM.
In this setup, the upper bound on the lightest Higgs mass
at one loop level is given by
\begin{equation}
m_{h}^2 \lesssim
m_Z^2 \cos^2 2\beta + \frac{3 v^2 g^2}{4 \pi^2}
\sin^4 \beta
\left[
\ln \left(\frac{m_{\overline{t}_L}^2}{m_{t_L}^2} \right)
+
\ln \left(\frac{m_{\overline{t}_R}^2}{m_{t_R}^2} \right)
\right] \;.
\end{equation}
So, in the limit of exact ${\cal N} = 2$ supersymmetry,
$m_{\overline{t}_L} = m_{t_L}$ and $m_{\overline{t}_R} = m_{t_R}$,
we end up with exactly the finite Higgs boson mass
as expected.
\newpage
\section*{Acknowledgments}
The work of T.K. is supported by the Research
Fellowship of the Japan Society for the Promotion of Science (\#1911329).
|
1,116,691,500,862 | arxiv | \section{Introduction}
Cancer is one of the leading causes of death in the world, with over 80,000
deaths registered in Canada in 2017 (Canadian Cancer Statistics 2017). A
computer-aided system for cancer diagnosis usually involves a pathologist
rendering a descriptive report after examining the tissue glass slides obtained
from the biopsy of a patient. A pathology report contains specific analysis of
cells and tissues, and other histopathological indicators that are crucial for
diagnosing malignancies. An average sized laboratory may produces a large
quantity of pathology reports annually (e.g., in excess of 50,000), but these
reports are written in mostly unstructured text and with no direct link to the
tissue sample. Furthermore, the report for each patient is a personalized
document and offers very high variability in terminology due to lack of
standards and may even include misspellings and missing punctuation, clinical
diagnoses interspersed with complex explanations, different terminology to label
the same malignancy, and information about multiple carcinoma appearances
included in a single report~\cite{GaoHierarchicalattentionnetworks2017}.
In Canada, each Provincial and Territorial Cancer Registry (PTCR) is responsible
for collecting the data about cancer diseases and reporting them to Statistics
Canada (StatCan). Every year, Canadian Cancer Registry (CCR) uses the
information sources of StatCan to compile an annual report on cancer and tumor
diseases. Many countries have their own cancer registry programs. These programs
rely on the acquisition of diagnostic, treatment, and outcome information
through manual processing and interpretation from various unstructured sources
(e.g., pathology reports, autopsy/laboratory reports, medical billing
summaries). The manual classification of cancer pathology reports is a
challenging, time-consuming task and requires extensive training
\cite{GaoHierarchicalattentionnetworks2017}.
With the continued growth in the number of cancer patients, and the increase in
treatment complexity, cancer registries face a significant challenge in manually
reviewing the large quantity of reports
\cite{weegar2017efficient,GaoHierarchicalattentionnetworks2017}. In this
situation, Natural Language Processing (NLP) systems can offer a unique
opportunity to automatically encode the unstructured reports into structured
data. Since, the registries already have access to the large quantity of
historically labeled and encoded reports, a supervised machine learning approach
of feature extraction and classification is a compelling direction for making
their workflow more effective and streamlined. If successful, such a solution
would enable processing reports in much lesser time allowing trained personnel
to focus on their research and analysis. However, developing an automated
solution with high accuracy and consistency across wide variety of reports is a
challenging problem.
For cancer registries, an important piece of information in a pathology report
is the associated ICD-O code which describes the patient's histological
diagnosis, as described by the World Health Organization's (WHO) International
Classification of Diseases for Oncology~\cite{louis20072007}. Prediction of the
primary diagnosis from a pathology report provides a valuable starting point for
exploration of machine learning techniques for automated cancer surveillance. A
major application for this purpose would be ``auto-reporting'' based on analysis
of whole slide images, the digitization of the biopsy glass slides. Structured,
summarized and categorized reports can be associated with the image content when
searching in large archives. Such as system would be able to drastically
increase the efficiency of diagnostic processes for the majority of cases where
in spite of obvious primary diagnosis, still time and effort is required from
the pathologists to write a descriptive report.
The primary objective of our study is to analyze the efficacy of existing
machine learning approaches for the automated classification of pathology
reports into different diagnosis categories. We demonstrate that TF-IDF feature
vectors combined with linear SVM or XGBoost classifier can be an effective
method for classification of the reports, achieving up to 83\% accuracy. We also
show that TF-IDF features are capable of identifying important keywords within a
pathology report. Furthermore, we have created a new dataset consisting of 1,949
pathology reports across 37 primary diagnoses. Taken together, our exploratory
experiments with a newly introduced dataset on pathology reports opens many new
opportunities for researchers to develop a scalable and automatic information
extraction from unstructured pathology reports.
\section{Background}
NLP approaches for information extraction within the biomedical research areas
range from rule-based systems~\cite{kang2012using}, to domain-specific systems
using feature-based classification~\cite{weegar2017efficient}, to the recent
deep networks for end-to-end feature extraction and
classification~\cite{GaoHierarchicalattentionnetworks2017}. NLP has had varied
degree of success with free-text pathology reports~\cite{wieneke2015validation}.
Various studies have acknowledge the success of NLP in interpreting pathology
reports, especially for classification tasks or extracting a single
attribute from a report~\cite{wieneke2015validation,imler2013natural}.
The Cancer Text Information Extraction System (caTIES)~\cite{crowley2010caties}
is a framework developed in a caBIG project focuses on information extraction
from pathology reports. Specifically, caTIES extracts information from surgical
pathology reports (SPR) with good precision as well as recall.
Another system known as Open Registry~\cite{contiero2008comparison} is capable
of filtering the reports with disease codes containing cancer.
In~\cite{d2010evaluation}, an approach called Automated Retrieval Console (ARC)
is proposed which uses machine learning models to predict the degree of
association of a given pathology or radiology with the cancer. The performance
ranges from an F-measure of 0.75 for lung cancer to 0.94 for colorectal cancer.
However, ARC uses domain-specific rules which hiders with the generalization of
the approach to variety of pathology reports.
This research work is inspired by themes emerging in many of the above studies.
Specifically, we are evaluating the task of predicting the primary diagnosis
from the pathology report. Unlike previous approaches, the system does not rely
on custom rule-based knowledge, domain specific features, balanced dataset with
fewer number of classes.
\section{Materials and Methods}
\input{tab-report-dist.tex}
We assembled a dataset of 1,949 cleaned pathology reports. Each report is associated with
one of the 37 different primary diagnoses based on IDC-O codes. The reports are
collected from four different body parts or primary sites from multiple
patients. The distribution of reports across different primary diagnoses and
primary sites is reported in~\autoref{tab:report-distribution}. The dataset was
developed in three steps as follows.
\textbf{Collecting pathology reports:} The total of 11,112 pathology reports
were downloaded from NCI's Genomic Data Commons (GDC) dataset in PDF
format~\cite{grossman2016toward}. Out of all PDF files, 1,949 reports were
selected across multiple patients from four specific primary sites---thymus,
testis, lung, and kidney. The selection was primarily made based on the quality
of PDF files.
\textbf{Cleaning reports:} The next step was to extract the text
content from these reports. Due to the significant time expense of manually re-typing all the
pathology reports, we developed a new strategy to prepare our dataset. We
applied an Optical Character Recognition (OCR) software to convert the PDF
reports to text files. Then, we manually inspected all generated text files
to fix any grammar/spelling issues and irrelevant characters as an artefact produced by the OCR system.
\textbf{Splitting into training-testing data: } We split the cleaned reports
into 70\% and 30\% for training and testing, respectively. This split resulted
in 1,364 training, and 585 testing reports.
\subsection{Pre-Processing of Reports}
We pre-processed the reports by setting their text content to lowercase and
filtering out any non-alphanumeric characters. We used NLTK library to remove
stopping words, e.g., `the', `an', `was', `if' and so on \cite{LoperNLTKNaturalLanguage2002}. We then analyzed the reports to find
common bigrams, such as ``lung parenchyma'', ``microscopic examination'', ``lymph node''
etc. We joined the biagrams with a hyphen, converting them into a single word.
We further removed the words that occur less than 2\% in each of the diagnostic
category. As well, we removed the words that occur more than 90\% across all the
categories. We stored each pre-processed report in a separate text file.
\subsection{TF-IDF features}
TF-IDF stands for \emph{Term Frequency-Inverse Document Frequency}, and it is a useful
weighting scheme in information retrieval and text mining. TF-IDF signifies the
importance of a term in a document within a corpus. It is important to note
that a \emph{document} here refers to a pathology report, a \emph{corpus} refers to the collection of
reports, and a \emph{term} refers to a single word in a report. The TF-IDF weight
for a term $t$ in a document $d$ is given by
\begin{align}
\begin{split}
TF(t, d) &= \frac{\text{Number of times $t$ appears in $d$}}{\text{Total number terms in $d$}},\\
IDF(t) &= log\Big(\frac{\text{Total number of documents}}{\text{Number of documents with $t$}}\Big),\\
TF\text{-}IDF(t, d) &= TF(t, d) * IDF(t)\\
\end{split}
\end{align}
We performed the following steps to transform a pathology report into a feature
vector:
\begin{enumerate}
\item Create a set of vocabulary containing all unique words from all the
pre-processed training reports.
\item Create a zero vector $f_r$ of the same length as the vocabulary.
\item For each word $t$ in a report $r$, set the corresponding index in $f_r$ to
$TF-IDF(t, r)$.
\item The resultant $f_r$ is a feature vector for the report $r$ and it is a
highly sparse vector.
\end{enumerate}
\subsection{Keyword extraction and topic modelling}
The keyword extraction involves identifying important words within reports that summarizes its content. In contrast, the topic modelling allows grouping these keywords using an intelligent scheme, enabling users to further focus on certain aspects of a document. All the words in a pathology report are sorted according to their TF-IDF weights. The
top $n$ sorted words constitute the top $n$ keywords for the report. The $n$
is empirically set to 50 within this research. The extracted keywords are further grouped into
different topics by using latent Dirichlet allocation
(LDA)~\cite{blei2003latent}. The keywords in a report are highlighted using the color theme based on their topics.
\subsection{Evaluation metrics}
Each model is evaluated using two standard NLP metrics---micro and macro
averaged F-scores, the harmonic mean of related metrics precision and recall.
For each diagnostic category $C_j$ from a set of 37 different classes $C$,
the number of true positives $TP_j$, false positives $FP_j$, and false negatives
$FN_j$, the micro F-score is defined as
\begin{align} \label{eq:boom}
\begin{split}
P^{micro} &= \frac{\sum_{C_j}^{C}TP_j}{\sum_{C_j}^{C}(TP_j + FP_j)}\\
R^{micro} &= \frac{\sum_{C_j}^{C}TP_j}{\sum_{C_j}^{C}(TP_j + FN_j)}\\
F^{micro} &= \frac{2P^{micro}R^{micro}}{P^{micro}+R^{micro}},
\end{split}
\end{align}
whereas macro F-score is given by
\begin{equation}
\label{eq:1}
F^{macro} = \frac{1}{|C|}\sum_{C_j}^{C}F(C_j).
\end{equation}
In summary, micro-averaged metrics have class representation roughly proportional
to their test set representation (same as accuracy for classification problem
with a single label per data point), whereas macro-averaged metrics are averaged
by class without weighting by class
prevalence~\cite{QiuDeepLearningAutomated2018}.
\subsection{Experimental setting}
In this study, we performed two different series of experiments: i) evaluating the
performance of TF-IDF features and various machine learning classifiers on the
task of predicting primary diagnosis from the text content of a given report,
and ii) using TF-IDF and LDA techniques to highlight the important keywords
within a report. For the first experiment series, training reports are
pre-processed, then their TF-IDF features are extracted. The TF-IDF features and
the training labels are used to train different classification models. These
different classification models and their hyper-parameters are reported
in \autoref{tab:classifier}. The performance of classifiers is measured
quantitatively on the test dataset using the evaluation metrics discussed in the
previous section. For the second experiment series, a random report is selected
and its top 50 keywords are extracted using TF-IDF weights. These 50 keywords
are highlighted using different colors based on their associated topic, which
are extracted through LDA. A non-expert based qualitative inspection is
performed on the extracted keywords and their corresponding topics.
\section{Results and Discussion}
\subsection{Experiment Series 1}
A classification model is trained to predict the primary diagnosis given the
content of the cancer pathology report. The performance results on this task are
reported in \autoref{tab:results}. We can observe that the XGBoost classifier
outperformed all other models for both the micro F-score metric, with a score of
0.92, and the macro F-score metric, with a score of 0.31. This was an
improvement of 7\% for the micro F-score over the next best model, SVM-L, and a
marginal improvement of 5\% for macro F-score. It is interesting to note that
SVM with linear kernels performs much better than SVM with RBF kernel, scoring
9\% on the macro F-score and 12\% more on the micro F-score. It is suspected
that since words used in primary diagnosis itself occur in some reports, thus enabling the linear
models to outperform complex models.
\input{tab-classifier.tex}
\input{tab-results.tex}
\subsection{Experiment Series 2}
\autoref{fig:keywords} shows the top 50 keywords highlighted using TF-IDF
and LDA. The proposed approach has performed well in highlighting the
important regions, for example the topic highlighted with a red color containing ``presence range tumor necrosis'' provides useful biomarker information to readers.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.95\linewidth]{keywords.png}
\scalebox{0.7}{
\begin{tabular}{ll}
\toprule
\multicolumn{2}{c}{Top 10 Keywords} \\
\midrule
\multicolumn{2}{l}{1. Epithelial (0.377), 2. Presence (0.269), 3. Thymectomy (0.232)} \\
\multicolumn{2}{l}{4. Epithelial cells (0.210), 5. Cells (0.180), 6. Small (0.161), 7. Lobulated (0.161)} \\
\multicolumn{2}{l}{8. Lung parenchyma (0.151), 9. Appear (0.150), 10. Examination (0.139)} \\
\midrule
Topic \# & \multicolumn{1}{c}{Keywords} \\
\midrule
\multirow{3}{*}{\colorbox[HTML]{DCB32D}{Topic 1}} & examination, thymectomy, measuring, resection, showing, \\
& inflammatory, lymph, spaces, lung, immunohistochemistry, \\
& node, modified, complete \\
\multirow{2}{*}{\colorbox[HTML]{8B2E7F}{Topic 2}} & samples, epithelial, proliferation, mixed, CD20, CD5, \\
& histological, according, classification, masaoka \\
\multirow{3}{*}{\colorbox[HTML]{BD4443}{Topic 3}} & lobulated, necrotic, small, right, architecture, \\
& presence, medulla, range, tumor, necrosis, green, appear, \\
& parenchyma, cells, cytokeratin, right, major \\
\bottomrule
\end{tabular}
}
\caption{The top 50 keywords in a report identified using TF-IDF weights. The
keywords are color encoded as per the
abstract ``topics'' extracted using LDA. Each topic is given a separate
color scheme.}
\label{fig:keywords}
\end{figure}
\subsection{Conclusions}
We proposed a simple yet efficient TF-IDF method to extract and corroborate useful keywords from pathology cancer reports. Encoding a pathology report for cancer and tumor surveillance is a
laborious task, and sometimes it is subjected to human errors and variability in the
interpretation. One of the most important aspects of encoding a
pathology report involves extracting the primary diagnosis. This may be very useful for content-based image retrieval to combine with visual information. We used existing classification model and TF-IDF features to predict the
primary diagnosis. We achieved up to 92\% accuracy using XGBoost classifier. The prediction accuracy empowers the adoption of machine learning methods for automated
information extraction from pathology reports.
|
1,116,691,500,863 | arxiv | \section{Introduction}
Realizing non-Abelian statistics in condensed matter systems is an exciting prospect, both from the point of view of fundamental theory, and for possible applications to quantum computing\cite{NayakReview}. Though the possibility of non-abelian statistics is well-established in the context of quantum Hall physics, only recently\cite{KitaevMajorana,OregPRL105.177002,LutchynPRL105.077001} was it realized that 1D superconducting wires with large spin-orbit coupling also have the potential to host particles with non-abelian statistics\cite{MooreRead} known as Majorana zero modes.
Experimental progress towards realizing these systems has been extremely promising: several groups\cite{MourikScience336,DengNanoLett12,DasNatPhys8}
have observed tunneling conductance peaks at zero bias in appropriately engineered nanowires, whose dependence on the applied magnetic field closely matches theoretical predictions.
Observing such zero-bias tunneling peaks at the ends of these 1D topological superconducting (TSC) wires, however, leaves room for doubt as to whether or not the zero-energy excitations causing them have the sought-after non-abelian statistics\cite{RoyTewari}; a direct probe of these statistics would be extremely gratifying for experimentalists and theorists alike. Unfortunately, a bullet-proof demonstration of such statistics requires carrying out a braiding operation, which requires either a tri-junction geometry\cite{AliceaNatPhys7,SauPRB84},
or coupling to Josephson junctions or quantum dots\cite{FuKanePRB79,SauPRA82,FlensbergPRL106,vanHeckNJP14}. Here we propose an alternative experimental protocol, inspired by the principles of measurement-only topological quantum computation\cite{BondersonMeasurement}, which can be realized in a single nanowire. The gist of our proposal is that the effects of non-Abelian statistics can be mimicked by varying in time the coupling between two pairs of Majoranas, in an experimental set-up potentially simpler than is required for a braiding experiment. We will call our proposed experiment the {\it Majorana Parity Resonance} (MPR) experiment.
The key advance required to carry out our experiment is the ability to measure the fermion parity (which encapsulates the collective state of the two Majorana zero modes at the end-points of a TSC segment) simultaneously in two segments of TSC wire. In quantum computing parlance, this means that we must be able to read out the $q$-bit on each wire segment. While no experimental group has yet achieved this, multiple proposals can be found in the literature\cite{BondersonPRL106,HasslerNJP12,HasslerNJP13,Blais2013,YuvalMeasurement} for how to carry out such a measurement. Hence there is reason to be optimistic that it will be possible in the near future to perform such a read-out measurement without dramatically increasing the relaxation rates of the Majorana zero modes.
\begin{figure}
\includegraphics[width=1.0\linewidth]{WireFig1.pdf}
\caption{ Experimental set-up for 1D superconducting wires required for the MPR experiment. Red lines indicate segments of the wire that are in the topological superconducting state; blue lies indicate segments in the normal superconducting state. The red dot at the boundary between normal and topological superconducting regions indicates the candidate Majorana zero mode. The set-up requires two gates: one to tune the two end segments into the TSC regime (shown here in grey) and one that is used to tune the coupling between $\gamma_2$ and $\gamma_3$ (shown here in blue).}
\label{WireFig}
\end{figure}
The experimental set-up is shown in Fig. \ref{WireFig}.
In the MPR experiment, the fermion parity is measured simultaneously in the right-and left-hand wire segments, while the chemical potential in the middle (normal) segment is varied.
Roughly speaking, the experiment proceeds as follows: we initialize the system into a state with a definite, known fermion parity in the two wire segments, and with a very large chemical potential in the middle segment, such that the two wire segments are essentially uncoupled. After some time we apply a brief pulse that lowers the chemical potential in the middle segment, coupling the fermion parities in the two TSC segments. This perturbation causes the two fermion parities to precess, such that an appropriately timed pulse will bring the system from (say) a state where both fermion parities are $1$ to a state where with a high probability we will measure both to be $0$. By measuring the two fermion parities simultaneously shortly before and shortly after the pulse, this oscillation can be detected, giving extremely compelling evidence for the entanglement characteristic of non-abelian statistics.
Our proposal builds on the idea\cite{ParityTime,AguadoPRL} that high-frequency dynamical measurements can harbor signatures characteristic of non-abelian statistics. Indeed, the same set-up\cite{ParityTime} can be used to demonstrate that the fermion parities in the left- and right- wire segments are correlated. The MPR experiment, while technically more challenging, goes one step beyond this: we will show that the fact that the fermion parities precess is the dynamical analogue of a process that exchanges two of the Majoranas through a series of measurements. Because of this connection, the methods described here can also be used to perform high-frequency manipulations of entangled pairs of topological q-bits. (Conversely, our results also highlight the fact that in the regime we consider here, the 1D wire systems do {\it not} behave topologically).
One of the advantages of a dynamical measurement over an actual braiding experiment is that the it can be performed on short time-scales, making the signal much more robust to parity relaxation processes. As we will discuss in more detail below, even using a relatively pessimistic estimate of the relaxation rate\cite{RainisLossPRB85.174533}, we find that for the realistic estimates\cite{RainisPRBB87,DasSarmaSmoking} of the parameters in our model, the minimum pulse width ($\sim 1 \ ns$) is approximately two orders of magnitude faster than the relaxation time ($\sim 100 \ ns$). This suggests that the signal should be measurable provided that the parity can be measured quickly relative to the relaxation time -- a requirement that any experimental probe of non-abelian statistics will have to meet.
It also follows that our approach can be used to coherently rotate pairs of Majorana q-bits on time-scales that can realistically be expected to be fast relative to the predicted relaxation rate.
We note that variants on the experimental geometry shown in Fig. \ref{WireFig} are possible. For example, we can replace the middle, normal superconducting segment of the wire by a superconucting quantum dot, which is itself Josephson-coupled to a bulk superconductor.\cite{vanHeckNJP14} In such a set-up the coupling $t_{23}$ is tuned by the flux through the Josephson junction, rather than by a gate.
The remainder of this work is structured as follows. In Section \ref{MPRSec}, we will describe the experiment in more detail, explaining the origin of the coherent parity rotation in a simple model of the wire system. Section \ref{SomeNumerics} presents numerical results for the dynamics for a range of parameter values, and discusses which regime is germane to current experiments. Section \ref{MeasurementOnlySec} describes the connection between the coherent parity rotation that our experiment probes and the non-abelian statistics of the Majorana zero modes. We conclude with a brief summary of our findings.
\section{The MPR Experiment} \label{MPRSec}
In this section we will describe the MPR experiment in more detail. We present an idealized theoretical model of the system, which consists of $4$ Majorana fermions which we couple to a bath to simulate a finite relaxation time. In the next section we will present numerical results of the dynamics of this system, and discuss the constraints on experimental parameters required in order for a signal to be detected.
Let us begin with a brief review of the TSC system, to clarify our assumptions and notation. In practise, the set-up in Fig. \ref{WireFig} consists of a single nanowire in which superconductivity is induced by proximity to a bulk 3D superconductor. Throughout this work we restrict our attention to energy scales below the induced superconducting gap; in this regime the Hilbert space of the wire consists of the four Majorana zero modes $\gamma_1, ... \gamma_4$ pinned to the endpoints of the two TSC segments.
However, it is important for our purposes that the coherence length of the 3D bulk superconductor is longer than the nanowire; otherwise the required coherence between the pairs of Majorana fermions will not be achieved.
To split our wire into the configuration shown in Fig. \ref{WireFig}, we exploit the fact that the transition between TSC and NTSC wires can be tuned by adjusting either the chemical potential or the magnetic field\cite{OregPRL105.177002,LutchynPRL105.077001}. (Here we will describe our experiment in terms of the former variable, although either could potentially be used.)
Our set-up requires two gates: a two-pronged gate with an arm under each of the two end segments of the wire, and a second gate under the central NTSC segment. The chemical potential of the two-pronged gate is kept fixed at a value that maintains the two end segments in the TSC regime. The chemical potential of the central gate will be pulsed between a high value at which the two TSC segments are effectively decoupled, and a lower value at which the coupling between the Majoranas $\gamma_2$ and $\gamma_3$ is comparable to the coupling between $\gamma_1$ and $\gamma_2$ (which we take to be approximately equal to the coupling between $\gamma_3$ and $\gamma_4$).
At energies well below the superconducting gap, the Hamiltonian for this set-up is:
\begin{equation} \label{HgEq}
H(t) = t_{12} \gamma_1 \gamma_2 + t_{34} \gamma_3 \gamma_4 + t_{23}(t) \gamma_2 \gamma_3
\end{equation}
where $\gamma_i$ are Majorana fermion operators, obeying
\begin{equation}
\gamma_i^\dag =\gamma_i, \ \ \ \ \{ \gamma_i, \gamma_j \} =2 \delta_{ij}
\end{equation}
Any two Majorana fermion operators span a two-state Hilbert space, whose states can be described by a fermion parity $n_f =0,1$.
It is illuminating to re-express the Hamiltonian (\ref{HgEq}) in terms of the fermion creation and anhiliation operators associated with the fermion parity of each topological wire segment:
\begin{eqnarray}
c^\dag_L =& \frac{1}{2} \left( \gamma_1 + i \gamma_2 \right) \ \ \ \ c_L =&\frac{1}{2} \left( \gamma_1 - i \gamma_2 \right) \nonumber \\
c^\dag_R =& \frac{1}{2} \left( \gamma_5 + i \gamma_6 \right) \ \ \ \ c_R =& \frac{1}{2} \left( \gamma_5 - i \gamma_6 \right)
\end{eqnarray}
In the basis of the two fermion parities $n_L$ and $n_R$:
\begin{equation} \label{nlrbasis}
|n_L, n_R \rangle =( |0,0 \rangle, |1,1 \rangle |0,1 \rangle, |1,0 \rangle)^T
\end{equation}
the Hamiltonian is:
\begin{equation} \label{HEqn}
H(t) =
\begin{pmatrix} t_{12} + t_{34} & t_{23}(t) & 0 & 0 \\ t_{23}(t) & -t_{12} - t_{34} & 0 & 0 \\ 0 & 0 & t_{12} - t_{34} & t_{23}(t) \\
0 & 0& t_{23}(t) & - t_{12} + t_{34} \\
\end{pmatrix}
\end{equation}
Let us begin by ignoring any coupling between the wire and the outside world, and consider the dynamics implied by the Hamiltonian (\ref{HEqn}). For the sake of argument let us take $t_{12}, t_{34} >0$, such that for $t_{23} =0$ the ground state of the system in the fermion number basis is $|n_R, n_L \rangle = |11 \rangle$. Let us consider begining the experiment with $t_{23}=0$, and with the system in this ground state. Since $H$ is block-diagonal, we can restrict our attention to the upper block, which is simply a 2-level system analogous to a spin in a magnetic field $\mathbf{B}$. Here $t_{12} + t_{34}$ plays the role of $B_z$, while $t_{23}$ is analogous to $B_x$.
While $t_{23} =0$, our proverbial spin is aligned with this ``magnetic field" -- meaning that the system remains in its ground state $|1,1 \rangle$. Using exactly the same principle as in NMR, we can now turn on the transverse field $B_x$ (a.k.a $t_{23}$) and make our spin precess about the $x$ axis; if the pulse is turned off after an appropriate time interval the spin will be left in its excited state, anti-aligned with $B_z$. In other words, the fermion parities will be rotated from $|1,1 \rangle$ to $|0,0 \rangle$.
Hence a measurement of the fermion parities immediately following the pulse will find the system in its excited state.
In a sense this observation seems mundane; it is simply the standard behavior of a quantum 2-level system. If the levels in question are indeed those of Majorana bound states, however, a positive result demonstrates that turning on $t_{23}$ couples the Majorana qbit (here the fermion number $n_L$) of the left-hand TSC segment coherently to the Majorana qbit $n_R$ of the right-hand TSC segment. In other words, a positive result demonstrates that there exist correlations between the two segments that are associated with the non-abelian statstics of the Majorana zero modes. We will discuss the reasons for this in more detail in Section \ref{MeasurementOnlySec}.
How well can our experiment distinguish between Majorana modes and other types of bound states that might arise at the end-points of the topological wire segments? An important feature of the Majorana system is that tuning the central gate voltage changes the {\it coupling} between the left and right q-bit (in our magnetic analogue, $t_{23}$ is $B_x$). This is fundamentally different from changing the chemical potential for each 2-state system (in which case $t_{23}$ would be analogous to $B_z$), since it produces a coherent rotation rather than a transition that occurs by relaxation. Experimentally the two can be distinguished by verifying that doubling the pulse duration leads to a full precession back to the original state. Thus the MPR experiment essentially tells us whether the bound states on the two wire segments can be coupled by tuning voltages (or, in alternative scenarios, magnetic fields) in accordance with our expectations for Majorana fermions.
Unlike a true braiding experiment, however, it probes a dynamical, non-topological regime, and does not definitively {\it prove} the existence of Majorana statistics; it merely shows that coupling the two wire segments creates a coherent rotation in a 2D subspace of the 4 state Hilbert space, as follows from the non-Abelian statistics if these 4 states arise from Majorana zero modes (see Sect. \ref{MeasurementOnlySec}).
\subsection{Dynamics in the presence of a bath}
We now turn to a more detailed analysis of the system's dynamics, using the approach described in Ref. \onlinecite{ParityTime}. What follows is a discussion of the dynamics of the system when coupled to an external fermionic bath; readers most interested the quantitative outcome of this analysis may skip to section \ref{SomeNumerics}.
In a realistic experimental set-up, the total fermion parity $n_L + n_R$ is not conserved, as the wire cannot be completely isolated from its environment. To account for this, we couple our Majorana wire to a (non-superconducting) fermionic bath. The dynamics of the wire system are then described by the Master equation
\begin{equation} \label{MasterEq}
\dot{\rho} = i \left[ H, \rho \right] + \sum_n \,\Gamma_n\,\left[ L_n \rho L^\dag_n - \frac{1}{2} \left( L^\dag_n L_n \rho + \rho L^\dag_n L_n \right)\right]
\end{equation}
where in the basis (\ref{nlrbasis}) the Linblad operators $L_n$ are given by:
\begin{eqnarray}
L_1 = c^\dag_R& =&
\begin{pmatrix} 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 \\
1 & 0 &0 &0 \\
0&0 &0 & 0 \\
\end{pmatrix} \ \ \ \ \
L_2 = c_R=
\begin{pmatrix} 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 &0 &0 \\
0& -1 &0 & 0 \\
\end{pmatrix} \nonumber \\
L_3 = c^\dag_L& =&
\begin{pmatrix} 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 &0 &0 \\
1& 0 &0 & 0 \\
\end{pmatrix} \ \ \ \ \
L_4 = c_L =
\begin{pmatrix} 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 \\
0 & 1 &0 &0 \\
0& 0 &0 & 0 \\
\end{pmatrix} \nonumber
\end{eqnarray}
If the couplings to the bath are homogeneous and the bath itself is in thermal equilibrium with a constant density of states $\rho$, the rates are given by
\begin{eqnarray} \label{rates}
\Gamma_1 =& \frac{1}{\hbar}\, |\alpha|^2 \rho \, n_F(-2t_{34}) \ \ \ \
\Gamma_2 =& \frac{1}{\hbar}\, |\alpha|^2 \rho \, n_F( 2t_{34}) \nonumber \\
\Gamma_3 =& \frac{1}{\hbar}\, |\alpha|^2 \rho \, n_F(-2t_{12}) \ \ \ \
\Gamma_4 =& \frac{1}{\hbar}\, |\alpha|^2 \rho \, n_F( 2t_{12})
\end{eqnarray}
where $\rho$ is the density of states in the bath, $\alpha$ parametrizes the coupling to the bath, and
\begin{equation}
n_F (\epsilon) = \frac{1}{ 1 + e^{ \beta \epsilon}}
\end{equation}
is the probability that the bath can absorb the energy $ \epsilon$ of the corresponding transition in the wire.
For our Hamiltonian we will of course take (\ref{HEqn}), with a time-dependent voltage applied to the central gate that causes $t_{23}$ to vary in time according to
\begin{equation}
t_{23} (t) = \frac{t_{23}^{(0)}}{2} \left [ \theta \left( \cos \omega( t- \delta t/2) \right) - \theta \left( \cos \omega( t+ \delta t/2) \right) \right]
\end{equation}
This gives a square-wave pulse of amplitude $t_{23}^{(0)}$ and duration $\delta t$, with a time lag of $\pi/\omega - \delta t$ between successive pulses.
It is not difficult to simulate the dynamics described by Eq. (\ref{MasterEq}) numerically at arbitrary temperatures and values of the various couplings; we present the results of these simulations in Sect. \ref{SomeNumerics}. In the limit $T \rightarrow \infty$ (by which in practise we mean that the temperature of the bath is large compared to the couplings $t_{ij}$ between the Majorana fermions -- {\it not} relative to the superconducting gap!), however, the dynamics are sufficiently simple that we can describe them analytically, as we will now do.
In the high-temperature limit the fermi factors in Eq. (\ref{rates}) are all $1/2$, and there is only one effective relaxation rate $\Gamma \equiv 2 \Gamma_i$.
The time-dependent density matrix has the general form:
\begin{equation}
\rho(t) = \begin{pmatrix} \rho_{11}(t) & \rho_{12}(t) & \rho_{13}(t) & \rho_{14}(t) \\
\overline{\rho}_{12}(t) & \rho_{22}(t) & \rho_{23}(t) & \rho_{24}(t) \\
\overline{\rho}_{13}(t) & \overline{\rho}_{23}(t) & \rho_{33}(t) & \rho_{34}(t) \\
\overline{\rho}_{14}(t) & \overline{\rho}_{24}(t) & \overline{\rho}_{34}(t) & \rho_{44}(t) \\
\end{pmatrix}
\end{equation}
where $\rho_{ii}$ are real, and give the probability of finding the system in the corresponding fermion number eigenstate. Evidently the total probability $\sum_i \rho_{ii}$ must equal $1$.
The equations of motion are most simply expressed in terms of the quantities
\begin{eqnarray}
\rho_e^{(S)} =\rho_{11} + \rho_{22} \ \ \ \ \ \rho_e^{(A)} =\rho_{11} - \rho_{22} \nonumber \\
\rho_o^{(S)} =\rho_{33} + \rho_{44} \ \ \ \ \ \rho_o^{(A)} =\rho_{33} - \rho_{44} \nonumber \\
x_{12} = \text{Re} \left( \rho_{12} \right ) \ \ \ \ \ y_{12} = \text{Im} \left( \rho_{12}\right ) \nonumber \\
x_{34} = \text{Re}\left( \rho_{34}\right ) \ \ \ \ \ y_{34} = \text{Im}\left( \rho_{34} \right )
\end{eqnarray}
Here $\rho_{e}^{(S)}(t)$ ($\rho_{o}^{(S)}(t)$) is the probability of finding the system in a state where the total fermion parity $n_L + n_R$ is even (odd) at time $t$. $\rho_{e}^{(A)}(t)$ indicates how much more likely we are to find the system in the state $|0,0 \rangle$ than in the state $|1,1 \rangle$; $\rho_{o}^{(A)}$ determines the relative probabilities of measuring the fermion number combinations $|1,0\rangle$ and $|0,1 \rangle$.
The symmetric components $ \rho_e^{(S)}, \rho_o^{(S)}$ decouple from the remaining components; their dynamics are given by
\begin{eqnarray} \label{Mds}
\dot{\rho}_e^{(S)}(t)&=& - \Gamma \left( \rho_e^{(S)}(t) - \rho_o^{(S)}(t) \right ) \nonumber \\
\dot{\rho}_o^{(S)} (t)&=& - \Gamma \left( \rho_o^{(S)}(t) - \rho_e^{(S)}(t) \right )
\end{eqnarray}
This indicates that $ d/dt \text{Tr}( \rho) =0$, and that the probability of finding the system in the sector of even versus odd fermion parity decays exponentially to $1/2$, as it should at high temperature.
The block off-diagonal components $\rho_{13}, \rho_{14}, \rho_{23}, \rho_{24}$ of the density matrix also decouple from the rest; their dynamics will not be of interest to us here.
The equations of motion for the remaining six components of $\rho$ reduce to two sets of three coupled first order differential equations:
\begin{eqnarray}
\dot{\rho}_e^{(A)}(t)&=& - \Gamma \rho_e^{(A)}(t) -4 t_{23}(t) y_{12}(t) \\
\dot{ x}_{12}(t)&=& - \Gamma x_{12} + 2 \left( t_{34}+ t_{12} \right)y_{12}(t) \nonumber \\
\dot{y}_{12}(t)&=&- \Gamma y_{12} - 2 \left( t_{34}+ t_{12} \right)x_{12}(t) + t_{23}(t) \rho_e^{(A)} \nonumber
\end{eqnarray}
\begin{eqnarray}
\dot{\rho}_o^{(A)}(t)&=& - \Gamma \rho_o^{(A)}(t) - 4 t_{23}(t) y_{34}(t) \\
\dot{x}_{34}(t)&=& - \Gamma x_{34} - 2 \left( t_{34}- t_{12} \right)y_{34}(t) \nonumber \\
\dot{ y}_{34}(t)&=&- \Gamma y_{34} + 2 \left( t_{34}- t_{12} \right)x_{34}(t) + t_{23} (t)\rho_o^{(A)} \nonumber
\end{eqnarray}
where we have temporarily set $\hbar =1$ for notational simplicity.
For $t_{23}$ independent of time, each set can be solved analytically to give:
\begin{widetext}
\begin{eqnarray} \label{Tsols1}
\rho_e^{(A)} (t)\ &=& \frac{ e^{-\Gamma t} }{\epsilon_e^2}\left(\rho_e^{(A)}(0)\left(
t_{23}^2 \cos (2 \epsilon_e t)+ t_{12,34}^2\right)+2 x_{12}(0) t_{23} t_{12,34} \left( 1 -\cos (2 \epsilon_e t) \right)
-2 y_{12}(0) \epsilon_e t_{23} \sin (2 \epsilon_e t) \right )
\nonumber \\
x_{12}(t)&=& \frac{e^{-\Gamma t} }{2 \epsilon_e^2}
\left( 2 x_{12}(0) \left( t_{23}^2 - t_{12,34} ^2\cos (2 \epsilon_e t) \right ) +\rho_e^{(A)}(0)
t_{23} t_{12,34} \left( 1 - \cos (2 \epsilon_e t) \right)+2 y_{12}(0) \epsilon_e t_{12,34} \sin (2 \epsilon_e
t) \right)\nonumber \\
y_{12}(t)&=& \frac{e^{-\Gamma t} }{2 \epsilon_e}\left ( \rho_e^{(A)}(0) t_{23} \sin (2 \epsilon_e t)-2 x_{12}(0)
t_{12,34} \sin (2 \epsilon_e t) +2 y_{12}(0) \epsilon_e \cos (2
\epsilon_e t)\right)
\end{eqnarray}
\end{widetext}
where $t_{12,34} = t_{12} + t_{34}$, and $\epsilon_e = \sqrt{ (t_{12} + t_{34})^2 + t_{23}^2 }$ is the modulus of the band energy in the even-parity sector. (The form of the solution is similar in the odd sector).
We are interested in the behavior of the diagonal component $\rho_e^{(A)}(t)$. For simplicity, let us begin in the ground state for $t_{23} =0$, for which the density matrix has only one non-vanishing component $\rho_{22}$. For $t_{23} =0$, the equations of motion are simply
\begin{eqnarray} \label{}
\rho_e^{(A)} (t)\ &=& e^{-\Gamma t} \rho_e^{(A)}(0)
\nonumber \\
x_{12}(t)&=&
y_{12}(t) =0
\end{eqnarray}
such that the system relaxes at a rate $\Gamma $ towards the equilibrium density matrix $\rho_{ii} = 1/4$. If we turn on $t_{23}$ at time $t= t_0$, however, the new equations of motion are:
\begin{eqnarray}
\rho_e^{(A)} (t)\ &=& \frac{ e^{-\Gamma t } }{\epsilon_e^2}\left(\rho_e^{(A)}(0)\left(
t_{23}^2 \cos (2 \epsilon_e (t- t_0))+ t_{12,34}^2\right) \right )
\nonumber \\
x_{12}(t)&=& \frac{e^{-\Gamma t} }{2 \epsilon_e^2}
\left( \rho_e^{(A)}(0)
t_{23} t_{12,34} \left( 1 - \cos (2 \epsilon_e (t- t_0)) \right) \right)\nonumber \\
y_{12}(t)&=& \frac{e^{-\Gamma t} }{2 \epsilon_e}\left ( -\rho_e^{(A)}(0) t_{23} \sin (2 \epsilon_e (t- t_0)) \right)
\end{eqnarray}
If we return $t_{23}$ to $0$ after a time $\delta t = \pi/(2 \sqrt{ (t_{12,34})^2 + t_{23}^2 })$, then
$\rho_e^{(A)}$ evolves from its initial value of $e^{-\Gamma t_0}$ to a minimal value of
\begin{equation} \label{RhominVal}
\rho_e^{(A)} (t_0 + \delta t) = e^{-\Gamma (t _0 + \delta t) } \frac{ t_{12,34}^2 - t_{23}^2}{t_{12,34}^2+t_{23}^2}\rho_e^{(A)}(0)
\end{equation}
At this point $x_{12} = t_{23} t_{12,34}/ \epsilon_e^2, y_{12} =0$; the ensuing dynamics is given by substituting these initial values into Eq. (\ref{Tsols1}) with $t_{23} =0$. Note that once we set $t_{23}$ to 0 we will not see oscillations in $\rho_e^{(A)}$, though the values of $x_{12}$ and $y_{12}$ will oscillate. If the delay between applications of $t_{23}$ is an integer multiple of $2 \pi/(t_{12}+ t_{34})$, then at the beginning of the second pulse we once again have $x_{12}= t_{23} t_{12,34}/ \epsilon_e^2, y_{12} =0$. In this case, on the second pulse, $\rho_e^{(A)}$ starts at its minimal value
$e^{-\Gamma t } \frac{ t_{12,34}^2 - t_{23}^2}{t_{12,34}^2+t_{23}^2}$,
and after a time $\pi/(2 \epsilon_e)$ has returned to a maximal value of $e^{-\Gamma( t + \delta t)} $.
Note that the maximum contrast is obtained when $t_{23}$ is pulsed to a value that is large compared to $t_{12} + t_{34}$, which corresponds to applying a large $B_x$ field. In this case, in the limit $\Gamma \rightarrow 0$, subsequent pulses cause $\rho_e^{(A)}$ to oscillate between $-1$ and $1$.
\subsection{Finite voltage ramp time} \label{VoltageSect}
In practise, $t_{23}$ cannot be tuned from $0$ to $t_{23}^{(0)}$ instantaneously. In our numerical simulations, we therefore include the effect of a finite voltage ramp time, via
\begin{eqnarray}
t_{23} (t) &=& \frac{1}{\pi} \left | \tan^{-1} \left[ s \cos\left( \omega (t - \delta t/2) + \phi \right) \right ] \right .\nonumber \\
&& \left. - \tan^{-1} \left[ s \cos\left( \omega (t + \delta t/2)+ \phi \right) \right ] \right |
\end{eqnarray}
where $\phi$ determines the time at which the first pulse is applied, $\delta t$ is the duration of the pulse, $\omega$ determines the time between pulses, and $s$ determines the voltage ramp time, which we typically take to be of order $1 \, ns$ unless otherwise specified. Note that in order for the signal to be observable, the voltage ramp time cannot be too long: it is crucial that $t_{23}$ is turned on {\it non}- adiabatically.
\section{Numerical Results} \label{SomeNumerics}
Let us now turn to the question of what the dynamics described in the previous section implies for our experiment.
We will assume that we can prepare the system in the state $|1,1 \rangle$, either by allowing relaxation to the ground state (in the limit that the bath is at low temperature), or by an explicit measurement of the fermion parit
(if the bath is at high temperature). In this case the signal that we are interested in is the probability of finding the system in the state $|0,0 \rangle$ at the end of the sweep, indicating that the qbits in the two TSC wires have been coherently coupled.
This signal can be read off from the time dependence of the diagonal components of the density matrix, which give the probabilities of the four possible measurement outcomes. We will use the following convention when plotting these: the probability of finding the system in the state $|n_L,n_R \rangle = |0,0 \rangle$ is $\rho_{11}$, which will be shown in green. Similarly the probability of finding state $|1,1 \rangle$ is $\rho_{22}$ (red); state $|0,1 \rangle$ is $\rho_{33}$ (purple), and state $|1,0 \rangle$ is $\rho_{44}$ (blue).
\subsection{Experimental parameters}
There are three free parameters that are important in determining the system's behaviour. First, as is apparent from Eq. (\ref{RhominVal}), the minimum value of $\rho_{22} - \rho_{11}$ (which gives the difference in the probabilities of finding the system in the state $|1,1 \rangle$ or in the state $|0,0 \rangle$) is determined by the relative magnitudes of $t_{23}^{(0)}$ and $t_{12} + t_{34}$. Hence the greatest signal will be observed if the maximum value of $t_{23}$ is large compared to the couplings $t_{12}, t_{34}$ between the Majorana zero modes on the same TSC segment. Second, the ratio of the relaxation rate $\Gamma$ to the scale of the couplings $t_{ij}/\hbar$ imposes an absolute limit on the magnitude of the signal, even for $t_{23}^{(0)} \gg t_{12} + t_{34}$; if the relaxation rate is too high the system will reach equilibrium before the pulse can be completed. Third, the temperature $T_{\text{bath}}$ of the bath relative to $|t_{12} + t_{34}|$ determines which relaxation processes can occur. If the bath temperature is high on this scale, then as discussed in the previous section relaxation occurs for all values of $t_{23}$, meaning that the relaxation time must be longer than the total time required to initialize the system, apply the pulse, and take a final measurement. If the bath temperature is low on this scale, on the other hand, relaxation occurs only while the system is in its excited state; initialization is not required, and the limiting factor is the ratio of the relaxation time of the excited states to the duration of the pulse.
Appropriate values for these parameters can be estimated from the literature.
Let us suppose that each of the TSC segments are approximately $1 \mu m$ long. In this case, we expect that by tuning either the chemical potential or Zeeman field, the splitting between the two fermion parity states in each segment is expected to be in the range of approximately $t_{12}, t_{34} \approx 0.5$ to $30 \ \mu eV$\cite{RainisPRBB87,DasSarmaSmoking}. By adjusting the length of the NSC segment, we expect that $t_{23}$ can be tuned along a comparable range of values. (Note, however, that the oscillations as a function of $\mu$ and the Zeeman field expected in $t_{12}$ and $t_{34}$ are often absent in $t_{23}$, as explained in Appendix \ref{WaveFnApp}, such that the voltage pulse applied to the central gate must be relatively large).
This gives a pulse duration in the approximate range of $0.05$ to $2 \ ns$. At these values current experiments, which are performed in the range of $50-75 \ mK \sim 4 -6 \ \mu eV$, can be in the relatively low-temperature or relatively high-temperature regime, depending on whether the $t_{ij}$ are at the high end or the low end of the expected possible range. (In practise, the distribution of quasi-particles in the bulk superconductor is generally not thermal\cite{RainisLossPRB85.174533}, such that the effective temperature may be considerably higher than this due to quasiparticle poisoning).
A pessimistic estimate of the relaxation rate was obtained by Ref. \onlinecite{RainisLossPRB85.174533}, who suggested that in nanowire systems of the type considered here $\Gamma^{-1} \approx 10^{-7} s$. This would be problematic for applications to adiabatic quantum computing, but is nonetheless two orders of magnitude longer than the pulse duration required to perforrm the coherent rotation, suggesting that signatures of Majorana correlations can in principle be detected.
The parameter values used for these estimates, and the important time-scales for the experiment, are summarized in Table \ref{NumbersTab}.
\begin{table}[h!]
\begin{tabular}{|c|c|}
\hline
Parameter & approximate value \\
\hline
$T$ & $4-6 \ \mu eV$ \\
$t_{12}, t_{34} $& $0.5- 30 \ \mu eV$ \\
$t_{23}$ & $0.5 - 30 \ \mu eV$ \\
$\delta t $ &$ \stackrel{<}{\approx} 2 \ ns $ \\
$\Gamma^{-1}$ & $ <100 \ ns$\\
\hline
\end{tabular}
\caption{ \label{NumbersTab} Approximate range of parameter values, based on the experimental set-up of Ref. \onlinecite{MourikScience336}, and parameter estimates of Refs. \onlinecite{RainisPRBB87,DasSarmaSmoking,RainisLossPRB85.174533} for a wire with TSC segments approximately $1 \ \mu m$ long.}
\end{table}
\subsection{Dynamics}
\begin{figure}
\begin{tabular}{ll}
(a)& \\
& \includegraphics[width=0.8\linewidth]{Gamma0Fig.pdf} \\
(b)& \\
& \includegraphics[width=0.8\linewidth]{GammaSmallFig.pdf} \\
\end{tabular}
\caption{Time evolution of the diagonal elements of the density matrix in the limit of extremely slow relaxation rates. $\rho_{11}$ is shown in green; $\rho_{22}$ is red; $\rho_{33}$ is purple, and $\rho_{44}$ is blue. We initialize the system in a state with fermion parity $|1,1 \rangle$ at time $t= 0$, and after some time apply a voltage pulse (indicated by the black line) to the central gate that switches $t_{23}$ from $0$ to $t_{23}^{(0)}$. Both plots are shown for the choice $t_{23}^{(0)}/(2 \pi \hbar) =232 \, MHz, t_{12}/(2 \pi \hbar) = t_{34}/(2 \pi \hbar) =34 \, MHz$, for which $\epsilon_e/(2 \pi \hbar) =242 \, MHz$, or $\epsilon_e = 1 \, \mu eV$.
(a) For $\Gamma= 0$ the pulse rotates the system from the ground state $|1,1 \rangle$ to a state that is predominantly composed of the excited state $|0,0 \rangle$. A second pulse after an appropriate delay returns the system to the ground state.
(b) For $\Gamma = 15 \, MHz$,
the oscillation decays exponentially over several periods. The two states $|1,0 \rangle $ and $|0,1\rangle$ in the odd fermion parity sector are generated with equal probability by relaxation processes.}
\label{LowGamPlots}
\end{figure}
We now discuss the qualitative nature of the dynamics.
Figure \ref{LowGamPlots} shows the situation for long relaxation times, which we expect to be relevant for experiments. If $\Gamma=0$ (Fig. \ref{LowGamPlots} a), the system remains in the sector of even fermion parity indefinitely, and oscillates between being found in the ground state $|1,1 \rangle$ and the excited state $|0,0\rangle$ as subsequent pulses in $t_{23}$ are applied. Tuning the pulse width and inter-pulse delay appropriately results in a stable period 2 oscillation, as shown here. With a more realistic value $\Gamma = 15 \, MHz$ (Fig. \ref{LowGamPlots} b), at high bath temperatures this oscillation is damped over many cycles.
Because we have initialized the system in the sector of even fermion parity, the states $|1,0 \rangle$ and $|0,1 \rangle$ are produced only by relaxation processes. For the choice $t_{12} = t_{34}$ (as is the case in the plots shown here) the two parity odd states are degenerate, and $\rho_{33} = \rho_{44}$ irrespective of $T_{\text{bath}}$.
It is also interesting to consider the case where $\hbar \Gamma$ is of the same order of magnitude as the couplings between the Majorana fermions. This could occur, for example, if the parity measurement has a strong impact on relaxation rates.
In the limit that the bath temperature $T_{\text{bath}}$ is small on the scale of $t_{ij}/\hbar$, the result is shown in Fig. \ref{LowTPlots}.
At low temperature relaxation processes only begin when $t_{23}$ is switched on, and
the lower bound on the relaxation time $\Gamma^{-1}$ for the signal to be visible is that it must not be too much shorter than the duration of the pulse, $\delta t \approx \pi \hbar /(2 \sqrt{ (t_{12}+t_{34})^2 + t_{23}^2 } )$.\footnote{In our analytical discussion of $\delta t$, we did not account for the effect of the finite time required to change $t_{23}$. In our simulations we take this time to be small but finite, resulting in a small correction to the idealized value of $t_{23}$.} Since the system relaxes to its ground state after the pulse is completed, each subsequent pulse begins in the same initial state $|1,1 \rangle$. Hence repeated pulses produce persistent oscillations in the probabilities of finding the system in each of the 4 possible states.
If the time required to make a measurement is fast compared to the pulse duration, we can obtain similar results at high bath temperatures by performing a parity measurement immediately before each pulse, which initializes the system into a state of definite fermion parity.
(Note that if a measurement initializes the system in
either of the states $|1,0 \rangle$ or $|0,1 \rangle$, the ideal pulse duration $\delta t_{o} \approx \pi \hbar/(2 \sqrt{ (t_{12}-t_{34})^2 + t_{23}^2 } )$ is different and the signal will be suppressed.)
\begin{figure}
\includegraphics[width=0.8\linewidth]{T0HighGammaFig.pdf}
\caption{Time evolution of the diagonal elements of the density matrix for low bath temperature and high relaxation rates, shown here for $t_{23}^{(0)}/(2 \pi \hbar) =232 \, MHz, t_{12}/(2 \pi \hbar) = t_{34}/(2 \pi \hbar) =34 \, MHz$, and an effective relaxation rate of $\Gamma=1 \, GHz$ from the excited states. Relaxation occurs only when the system is in an excited state, meaning that the system starts to relax when the pulse in $t_{23}$ is applied. After the end of the pulse the system relaxes to its ground state $|1,1 \rangle$, effectively re-initializing the process. }
\label{LowTPlots}
\end{figure}
Hence if the initialization and measurements themselves can be carried out relatively quickly, even at relaxation rates that are very high compared to the estimate of Ref. \onlinecite{RainisLossPRB85.174533}, our experiment should be able to detect Majorana correlations provided that the duration of the pulse, $\delta t = \hbar \pi/(2 \sqrt{ (t_{12}+t_{34})^2 + t_{23}^2 } )$ is not long relative to the relaxation time $\Gamma^{-1}$. Essentially, this implies that the energy scale for the coupling to the bath must not be large compared to the energy scale of the couplings between the Majorana zero modes. As a rough benchmark, let $\rho_e^{(A)} \equiv \rho_{11} - \rho_{22}$, and imagine that we initialize the system at time $t_0$ such that $\rho_{e}^{(A)} =-1$, meaning that the system is in its ground state $|1,1 \rangle$. We define the magnitude of the signal to be the {\it difference} between the value of $\rho_e^{(A)}$ at time $t_0 + \delta t$ with and without the pulse in $t_{23}$:
\begin{equation} \label{ContrastEq}
S = \frac{1}{2} \left( \rho_e^{(A)} (\delta t)|_{t_{23} = t_{23}^{(0)} } - \rho_e^{(A)} (\delta t)|_{t_{23} = 0 } \right) \ .
\end{equation}
The maximum possible value of $S$ is $1$, which is attained in the limit $t_{23}^{(0)} \gg |t_{12} + t_{34}|$, $\Gamma =0$. The minimum value is $0$, which indicates that the system has relaxed completely before the pulse has been completed.
Fig. \ref{ContrastPlot} shows the magnitude of $S$ for $t_{23}^{(0)}/(2 \pi \hbar) =232 \, MHz, t_{12}/(2 \pi \hbar) = t_{34}/(2 \pi \hbar) =34 \, MHz$, as a function of the relaxation rate $\Gamma$, in the limit $K_B T_{\text{bath}} \gg |t_{12} + t_{34}|, t_{23}^{(0)}$.
\begin{figure}
\includegraphics[width=1.0\linewidth]{ContrastPlot.pdf}
\caption{The magnitude of the signal $S$ (Eq. \ref{ContrastEq}) as a function of the relaxation rate $\Gamma
for $K_B T_{\text{bath}} \gg |t_{12} + t_{34}|, t_{23}^{(0)} $, shown for $t_{12} = t_{34}$, $\epsilon_e/(2 \pi \hbar) = 242 \, MHz$ (corresponding to $\epsilon_e = 1 \, \mu eV$), and for several choices of the ratio $t_{12}/t_{23}^{(0)}$. The $x$ axis here is shown on a log scale, to better display the contrast for the relatively small values of $\Gamma$ expected to be relevant to experiments.
}
\label{ContrastPlot}
\end{figure}
\subsection{Effects of finite voltage ramp time}
In practise, at the frequencies that we are interested in, we must contend with the finite response time required to charge the gate.
We model these numerically as described in Sect. \ref{VoltageSect}.
Here we discuss the quantitative importance of the voltage ramp time.
Intuitively, the limiting factor is that if the system is to be found in the excited state after the pulse, then the charging time of the gate must not be adiabatic at the energy scale of the Majorana Hamiltonian. Indeed the closer the charging time is to the adiabatic limit, the smaller the probability of finding the system in the excited state immediately after the pulse. With the parameter values used here, the energy splittings of the Majorana Hamiltonian vary between approximately $0.2$ and $2 \, \mu eV$, which means that $t_{23}$ must ramp to its maximum value on a time-scale that is fast compared to $\sim 250 MHz$. In practise his requires a charging time of at most a few $ n s$ in order for the a significant signal to be observed.
The change in voltage required to tune $t_{23}$ appropriately is discussed in Appendix \ref{WaveFnApp}. In principle it is possible to tune the couplings $t_{12}, t_{34}$ to be arbitrarily small, requiring smaller maximal values of $t_{23}^{(0)}$ to perform the rotation, as well as implying longer pulse times and allowing the experiment to be performed at longer gate-charging times.
\section{The MPR experiment and braiding via measurement} \label{MeasurementOnlySec}
In the previous section, we showed that pulsing the coupling between the two Majorana fermions $\gamma_2, \gamma_3$ in Fig. \ref{WireFig} produces a coherent rotation of the fermion parities of the two TSC segments.
Here we will discuss in more depth the connection between this coherent rotation and the non-abelian statistics of Majorana zero modes.
To understand this connection, we first observe that
the MPR experiment is closely related to another
experiment, which we call the projective braiding experiment. The set-up for this experiment (Fig. \ref{PBRaidExp}) requires two wires, joined at a `tri-junction' by a segment of normal superconducting wire. The horizontal wire has two topological superconducting segments, joined by a normal superconducting segment; mid-way through this normal segment there is a tri-junction with a vertical TSC wire. The red dots in the Figure indicate the six Majorana fermions localized at the end-points of the three TSC segments.
The projective braiding experiment is performed in the topological regime, where the couplings between all pairs of Majoranas are tiny. It implements the ``measurement only" approach\cite{BondersonMeasurement} to exchanging two Majorana zero modes. Specifically, we will exchange $\gamma_2$ and $\gamma_3$ in Fig. \ref{PBRaidExp} by performing a sequence of measurements of the fermion parity in the TSC wire system. This is equivalent\cite{BondersonPRB035113} to other
proposals\cite{AliceaNatPhys7,SauPRB84} for using tri-junctions to exchange these anyons; such an experiment could therefore be used to detect the non-Abelian statistics of the Majorana zero modes.
\begin{figure}
\includegraphics[width=1.0\linewidth]{WireFigb.pdf}
\caption{ Experimental set-up for 1D superconducting wires required for the projective braiding experiment. Red lines indicate segments of the wire that are in the topological superconducting state; blue lies indicate segments in the normal superconducting state. The red dot at the boundary between normal and topological superconducting regions indicates the candidate Majorana zero mode. }
\label{PBRaidExp}
\end{figure}
We will begin by explaining the principle of the projective braiding experiment\cite{BondersonMeasurement}. Recall that any two Majorana zero modes $\gamma_i, \gamma_j$ comprise a two-state Hilbert space, which can be described by a fermion number $n_f ^{(ij)}=0,1$. The measurement-only protocol relies on the fact that by repeated measurements, one can always project onto the subspace $n_f^{(ij)} =0$, and that such projections can be used to carry out braiding operations. Specifically, as proven in Ref. \onlinecite{BondersonMeasurement}, the following sequence of projections effectively exchanges the Majorana fermions $\gamma_2$ and $\gamma_3$:
(1) Project $(\gamma_5, \gamma_6)$ onto $n_f^{(56)} =0$; \\
(2) project $(\gamma_2, \gamma_5)$ onto $n_f^{(25)}=0$; \\
(3) project $(\gamma_5, \gamma_3)$ onto $n_f^{(53)} =0$; \\
(4)project $(\gamma_5, \gamma_6)$ onto $n_f^{(56)} =0$.\\
Fig. \ref{MeasurementBraids} illustrates how this effectively exchanges the two Majorana zero modes.
\begin{figure}
\includegraphics[width=0.5\linewidth]{MeasBraidFig.pdf}
\caption{ Exchanging $\gamma_2$ and $\gamma_3$ through the sequence of projections described in the text\cite{BondersonMeasurement}. Solid lack lines represent the Majorana zero modes; dashed red lines represent projections onto the state $n_f^{(ij)} =0$. }
\label{MeasurementBraids}
\end{figure}
Now, let us understand how this works in our wire system in practise.
To do so,
it is convenient to re-express these operations in terms of the fermion creation and anhiliation operators associated with the fermion parity of each topological wire segment:
\begin{eqnarray} \label{TheFermionBasis}
c^\dag_L =& \frac{1}{2} \left( \gamma_1 + i \gamma_2 \right) \ \ \ \ c_L =&\frac{1}{2} \left( \gamma_1 - i \gamma_2 \right) \nonumber \\
c^\dag_R =& \frac{1}{2} \left( \gamma_3 + i \gamma_4 \right) \ \ \ \ c_R =& \frac{1}{2} \left( \gamma_3 - i \gamma_4 \right) \nonumber \\
c^\dag_M =& \frac{1}{2} \left( \gamma_5 + i \gamma_6 \right) \ \ \ \ c_M =& \frac{1}{2} \left( \gamma_5 - i \gamma_6 \right)
\end{eqnarray}
The projectors used to carry out the experiment are:
\begin{equation}
P_{n_f^{(ij)} = 0} =(1 - n_f^{(ij)}) = \frac{1}{2} \left( 1 + i \gamma_i \gamma_j \right )
\end{equation}
In terms of the fermion operators defined in Eq. (\ref{TheFermionBasis}), $P_{n_f^{(56)} =0} = 1- n_M$, and the remaining projection operators are:
\begin{eqnarray}
\frac{1}{2} \left( 1 + i \gamma_2 \gamma_5 \right ) =&\frac{1}{2} \left[ 1 + \left( c^\dag_L c^\dag_M + c_M c_L +c^\dag_L c_M + c^\dag_M c_L \right ) \right] \nonumber \\
\frac{1}{2} \left( 1 +i \gamma_5 \gamma_3 \right)=& \frac{1}{2} \left[ 1 - i \left( c^\dag_R c^\dag_M - c_M c_R +c^\dag_R c_M - c^\dag_M c_R \right )\right] \nonumber \\
\end{eqnarray}
Performing steps (2) and (3) amounts to acting with the product operator:
\begin{equation} \label{23Op}
P_{n_f^{(35)} = 0} P_{n_f^{(25)} = 0} = \frac{1}{4} \left( 1 +i \gamma_5 \gamma_3 \right) \left( 1 + i \gamma_2 \gamma_5 \right )
\end{equation}
It is useful to separate the terms in this product into two types: terms with an odd number of $\gamma_5\equiv c^\dag_M + c_M$ operators, which change the eigenvalue of $n_M$, and terms with an even number of $\gamma_5 $, which do not. The former will be anhiliated when the operator (\ref{23Op}) is sandwiched between two applications of $ (1- n_M)$. Hence the operator that performs the exchange-via-measurement operation is
\begin{eqnarray}
\frac{1}{4} P_{n_f^{(56)} = 0} \left( 1 + i \gamma_5 \gamma_3 \right ) \left( 1 + i \gamma_2 \gamma_5 \right ) P_{n_f^{(56)} = 0} \nonumber \\
=\frac{1}{4} P_{n_f^{(56)} = 0} \left( 1 + \gamma_2 \gamma_3 \right ) P_{n_f^{(56)} = 0}
\end{eqnarray}
Since $\gamma_5, \gamma_6$ enter into this expression only in the initial and final projectors, we can restrict our attention to the subspace of the total Hilbert space in which $n_f^{(56)} =0$. In other words, we can remove the vertical wire from Fig. \ref{PBRaidExp} entirely, replacing the 2-level system $\gamma_5, \gamma_6$ of the vertical TSC wire with the vacuum state, since this is the only state that enters into the exchange process.
This results in the single-wire geometry of Fig. \ref{WireFig}, with the exchange process being carried out by the hopping operator $\frac{1}{4}(1 + \gamma_2 \gamma_3)$.
Perhaps the clearest signature of the non-Abelian statistics of Majorana zero modes is the following. Consider two pairs of Majorana zero modes, a left pair with fermion parity $n_L =\frac{1}{2} ( 1 - i \gamma_1 \gamma_2)$ and a right pair with fermion parity $n_R=\frac{1}{2} ( 1 - i \gamma_3 \gamma_4)$. Let us begin in the state $n_L = n_R =0$. We then exchange the $\gamma_2$ and $\gamma_3$ particles twice (aka braiding $\gamma_2$ around $\gamma_3$). After this braiding process, the system will be in the state $n_L = n_R =1$. Performing the same operation again returns the system to its original state $|0,0 \rangle$. In other words, the braiding process coherently rotates $|n_L, n_R \rangle = |0,0 \rangle$ into $|n_L, n_R \rangle = |1,1 \rangle$, and vice versa.
Such a braiding operation is carried out by acting twice with the exchange operator, which gives:
\begin{equation}
\frac{1}{16}(1 + \gamma_2 \gamma_3)^2 = \frac{1}{8} \gamma_2 \gamma_3
\end{equation}
This is just the hopping term with coefficient $t_{23}$ in Eq. (\ref{HgEq}).
This allows us to connect the results of the previous sections to the non-Abelian statistics of the Majorana zero modes: by turning on the coupling $t_{23}$ (and simultaneously turning off the other couplings $t_{12}, t_{34}$) for an appropriate amount of time, we are effectively carrying out a braiding operation of the two Majorana zero modes $\gamma_2$ and $\gamma_3$. The non-abelian statistics dictate that this performs a coherent rotation of the ground state $|1,1 \rangle$ into the excited state $|0,0 \rangle$.
From the topological point of view, this is the underlying reason that our experiment works.
\section{Conclusions}
In this work, we describe a relatively simple experiment which uses a simultaneous measurement of the fermion parity in two segments of a topological superconducting wire to infer correlations that give a strong indication of underlying non-abelian statistics. Our proposed experiment uses an approach very similar in spirit to NMR, in which the system is prepared in its ground state, and coherently rotated into an excited state by applying a ``transverse field" (in this case, by changing a gate voltage under one segment of the wire) for a specified amount of time. The fact that the system is in its excited state after applying the pulse is directly related to the non-abelian statistics of the Majorana zero modes. It also illustrates that there are non-topological ways to coherently manipulate pairs of Majorana q-bits (which are not truly topological unless the couplings between them vanish) -- a fact that must be carefully taken into account in any implementation of truly topological quantum information processing.
Experimentally this approach presents several challenges. First, to maximize the signal one would like to be able to tune the coupling between the two Majoranas $\gamma_2$ and $\gamma_3$ separated by the non-topological wire segment (see Fig. \ref{WireFig}) between a value that is very small, and a value that is larger than the couplings between the Majoranas on the same TSC segment.
For Majorana splittings $t_{ij}$ on the order of $\mu eV$, this requires tuning the voltage on the central gate on a nanosecond time-scale. In the alternative geometry in which this coupling is tuned by a magnetic flux, it is the flux that must be tunable at this rate.
Second, the experiment requires a measurement of the fermion parity of the wire segment, which is also a significant experimental challenge.
However, dynamical experiments of the type described here are nonetheless much easier to carry out than adiabatic braiding experiments of the type discussed in \cite{AliceaNatPhys7,SauPRB84}. In our setup, a signal can be obtained even at relatively unfavourable values of the lifetime, and even if the bath temperature is not small compared to the couplings between the Majorana zero modes. Indeed the limiting factor in our experiment is that the Majoranas must not be more strongly coupled to an external bath than they are to each other -- a limit that is expected to be comfortably satisfied by current experiments. Hence we expect that dynamical experiments of the type described here will give the first evidence of non-Abelian statistics in these systems.
{\bf Acknowledgements} FJB wishes to thank Yuval Oreg, Alexander Shnirman, E. Burnell and Steve Simon for useful conversations, and acknowledges KITP (NSF Grant no. PHY11-25915) for its hospitality.
|
1,116,691,500,864 | arxiv | \section{Introduction}
The field of terrestrial planet formation is gradually approaching a clearer understanding of the appropriate initial conditions to form the terrestrial planets with the aid of progressively high-resolution simulations performed on fast computers. In retrospect, the Classical model \citep{Chambers2001} was built upon the results of early analytical derivations \citep[e.g.][]{Wetherill1980,WetherillStewart1989} and numerical simulations \citep{KokuboIda1995,KokuboIda1996,KokuboIda1998} of the evolution of a self-gravitating solid disc of planetesimals. Several planetesimals in the solid disc will grow much faster than the rest -- so-called runaway growth -- and produce a bimodal population in the disc comprising larger-sized planetary embryos and smaller-sized planetesimals \citep{KokuboIda1998}. These embryos continue growing by accreting nearby planetesimals until they exhaust all the planetesimals in their feeding zones. Eventually it is thought that the embryos collide and merge among themselves to finally form the terrestrial planets. It is commonly known, however, that the Classical model runs into the problem that it cannot account for Mars' relatively small mass. Later proposals to improve on the Classical model include the narrow annulus model \citep{Hansen2009}, the Grand Tack model \citep{Walshetal2011}, the low-mass asteroid belt or depleted disc model \citep{Izidoroetal2014,Izidoroetal2015} and the early outer Solar System instability model \citep{Clementetal2018,Clementetal2019,Clementetal2021}.
The dynamical models mentioned above are based on the conventional understanding that the terrestrial planets evolved to their current size through a series of accretion processes involving increasingly massive planetary bodies. There is now an emerging view, however, which postulates that the growth of the terrestrial planets could in fact be a consequence of the accretion of millimetre- to metre-sized bodies dubbed `pebbles', in direct conflict to the Classical view. Although the {\it pebble accretion} model \citep{OrmelKlahr2010,LambrechtsJohansen2012} was originally developed to resolve a problem related to the formation of the core of the gas giants in the outer Solar System \citep[e.g.][]{Levisonetal2015giantpl}, several recent studies \citep[e.g.][]{Levisonetal2015,Lambrechtsetal2019,Johansenetal2021} have applied this model to the terrestrial planet region and found positive results that encourage further investigations on the plausibility of this formation pathway for the terrestrial planets.
Pebbles, observed to be abundant in protoplanetary discs \citep[e.g.][]{Testietal2003,Wilneretal2005,Rodmannetal2006}, are thought to have formed via the coagulation of smaller-sized dust particles in the outer regions of the disc \citep{LambrechtsJohansen2014,Idaetal2016}. They tend to spiral towards the central star under the influence of gas drag because dynamical friction with the surrounding gas causes them to lose angular momentum \citep{Weidenschilling1977}. During their passage through the protoplanetary disc, these pebbles should encounter planetesimals in the inner Solar System and some of the pebbles will accrete onto the planetesimals.
The work of \citet{Levisonetal2015} investigated the growth and evolution of planetesimals distributed within 2.7~au of the Sun to a flux of inward-drifting pebbles that are continuously generated in the same region over a period of 2~Myr. Since in their simulations the pebbles are formed continuously in the inner Solar System, the surface density of the gas was chosen to be about 5 times the minimum mass solar nebula \citep[MMSN;][]{Hayashi1981}, a value deemed sufficient to sustain continuous pebble formation in the disc. The authors found that the planetesimals closer to the Sun accrete more pebbles and grow to larger sizes compared to planetesimals further away because the pebble accretion cross-section has a semi-major axis dependence resulting from their assumptions on the Stokes number -- a number that quantifies how well the pebbles are coupled to the gas. Mars' smaller size compared to Earth and Venus can therefore be easily reproduced because it formed from the mergers of smaller-sized planetesimals. It is perhaps more accurate to describe the contribution of the pebble accretion mechanism in this model as helping to generate the appropriate initial mass-semi-major axis distribution of planetesimals and planetary embryos that would naturally reproduce the masses of the terrestrial planets via subsequent collisions and mergers. The key assumption of this work is that the pebbles contribute to the mass of planetesimals formed locally in the inner Solar System. This assumption was required because the pebble flux necessary to form the cores of the giant planets \citep{Levisonetal2015giantpl} is too great to form the terrestrial planets. The assumption of local pebble formation over time is, however, peculiar and deviates from the more common assumption used in many studies (including ours) that the pebbles originate in the outer Solar System.
The work of \citet{Lambrechtsetal2019} also looked at the effects of pebble accretion in the inner disc but assumed that the pebbles originate from a region further away. Their simulations start with lunar mass planetary embryos distributed between 0.5 and 3.0~au following an $r^{-2}$ surface density profile. While the gas disc model, pebble accretion formulation and orbital migration prescriptions employed in their work are broadly similar to ours, there are notable differences: the disc temperature and pebble Stokes number $(\mathcal{S}=3\times10^{-3})$ are kept constant. They focused on how the amount of pebble flux flowing through the disc influences the mass-distance distribution of the embryos after gas dissipation and the final planetary system architecture. \citet{Lambrechtsetal2019} demonstrated that there exists a strong dependence of planetary system orbital architecture on the pebble flux. A pebble flux of $\sim 110$ Earth mass $(M_{\rm E})$ per Myr would result in Mars-sized embryos after the gas disc phase and terrestrial planet-like systems after about 100~Myr while a mere doubling of the pebble flux would produce close-in super-Earth-like systems. The authors then suggest that the growth of Jupiter could have restricted the pebble flux to the inner disc and thus put a limit on the final mass of the terrestrial planets.
In recent work, \citet{Johansenetal2021} propose that the planetesimals that went on to become the terrestrial planets (Venus, Earth and Mars) all formed initially near the water snow line, and that these planetesimals subsequently migrated inwards to the current semi-major axis of the terrestrial planets while accreting pebbles along the way. The snow line in their disc model starts between 1.2~au to 2.0~au and migrates inwards with time. The planet formation scenario presented in their work is based on good analytical fits of the growth track equations of \citet{Johansenetal2019}. The masses of Venus and Mars can be reproduced using reasonable values of the disc temperature profile \citep[$T \propto r^{-3/7}$;][]{ChiangGoldreich1997}, Stokes number $(10^{-3}<\mathcal{S}<0.1)$ and pebble flux. The study assumes that these planets' current masses are their masses at the time of gas disc dissipation, and that both Venus and Mars grew entirely from pebbles without giant impacts. For Earth, the maximum mass achieved after the end of the pebble accretion phase (when the gas disc has dissipated) is $0.6~M_{\rm E}$. A purported giant impact with a body of mass $0.4~M_{\rm E}$ is suggested to occur later so that the mass of the Earth can reach its current value. In their model, the terrestrial planets (Venus, Mars and proto-Earth) accreted pebbles of two different compositions in the sequence of (1) non-carbonaceous (specifically, ureilite-like) in the first few Myr, followed by (2) carbonaceous (specifically, CI-like) until the dissipation of the gas disc. Based on earlier studies of calcium and iron isotopes, \citet{Johansenetal2021} assume that $\sim 40\%$ of Earth's composition comes from pebbles that originated in the outer Solar System with an assumed CI-like isotopic composition \citep{Schilleretal2018,Schilleretal2020}. The low mass of Mars can be achieved by assuming that it formed later than the other terrestrial planets or that it stopped accreting earlier \citep{Schilleretal2018}. While the suggestion of planetesimal formation triggered by snow line passage is not new \citep{DrazkowskaAlibert2017}, the only manner in which this can result in planetesimals with a different composition is if the material that condensed at the snow line to become planetesimals has a different composition at a different distance to the Sun, which \citet{Johansenetal2021} argue occurred because of the initial viscous expansion of the disc. Yet, we emphasize here that planetesimal formation at a static snow line is difficult to reconcile with the different isotopic and elemental compositions of the sampled solar system, including Earth and Mars.
The results of prior studies examining the formation of the terrestrial planets in the framework of the pebble accretion model are indeed interesting and they set the stage for further research in this direction. The work of \citet{Levisonetal2015} and \citet{Lambrechtsetal2019} showed that pebble accretion (with a specific value of the pebble mass flux and Stokes number) played a supportive role in the early growth stages of planetesimals in the inner Solar System. In contrast \citet{Johansenetal2021} reported that pebble accretion could potentially be the primary mechanism for the formation of the terrestrial planets in a single stage. This latter hypothesis bypasses the need for an epoch of hierarchical accretion by ever-larger planetesimals and planetary embryos.
Here, we study the effects of pebble accretion in the inner Solar System over the lifetime of the gas disc while simultaneously taking into account the growth of Jupiter. To do this, we run a suite of {\it N}-body simulations with different initial conditions (disc temperature, solid disc mass) with the aim of quantifying the amount of mass delivered to the inner disc by pebble accretion. The pebble accretion formulation we adopt has its foundation on the work of \citet{Idaetal2016} which provides equations for the pebble accretion rate at different locations in a steady state accretion disc while consistently taking into account the various properties of the gas disc (e.g. temperature, surface density, scale height). We chose to use this particular pebble accretion formulation for its simplicity and elegance, and because it does not invoke any additional assumptions \citep[cf.][]{Levisonetal2015,Johansenetal2021}.
The major differences between our work and the studies summarised previously lie in the gas disc model we adopted and the computation and underlying assumptions of the value of the Stokes number. In our simulations, we follow \citet{Idaetal2016} and compute the radius and the Stokes number of the pebbles self-consistently depending on their location in the disc and time. In contrast, the Stokes number used in the works of \citet{Levisonetal2015} and \citet{Lambrechtsetal2019} is fixed while \citet{Johansenetal2021} assume that it lies in a particular range and that the pebbles are mm-sized. Our simulations are in a sense more realistic concerning the treatment of pebbles, but the downside is that a longer computational time is required and that our gas disc model may be too simplistic.
\section{Pebble accretion onto planetesimals}
\subsection{Basic formulation}
\label{sec:pebacc_formula}
Based on the formulation of \citet{OrmelKlahr2010} and \citet{OrmelKobayashi2012} and modified by \citet{Idaetal2016}, the accretion rate of mm to m-sized pebbles onto a planetesimal of mass $M$ and located at distance $r$ from the central star is derived to be
\begin{equation}
\dot{M} = \min \left[1,\frac{C b^2\sqrt{1+4\mathcal{S}^2}}{4\sqrt{2\pi}\mathcal{S}h_{\rm peb}}\left(1 + \frac{3b}{2\chi\eta}\right)\right]{\dot M_{\rm F}},
\label{eq:dotmpl}
\end{equation}
where $C$ is a constant related to the pebble accretion mode (equation~\ref{eq:const2d3d}), $b$ is the normalised (or reduced) impact parameter (equation~\ref{eq:impactparam}), $\mathcal{S}$ is the Stokes number which determines the degree of coupling between the pebbles and the gas (equation~\ref{eq:taus}), $h_{\rm peb}$ is the normalised pebble scale height (equation~\ref{eq:hpeb}), $\chi$ is a function of the Stokes number (equation~\ref{eq:chi}), $\eta$ is the difference between the gas and Keplerian velocities (equation~\ref{eq:eta}), and $\dot{M}_{\rm F}$ is the pebble mass flux (equation~\ref{eq:mdotf}). In the following we describe each of these quantities in more detail.
The flux of pebbles through the disc at a specific time and location from the star $\dot{M}_{\rm F}$ is one of the key quantities that control the pebble accretion rate. Its temporal evolution depends on the gas accretion rate onto the central star and the formation efficiency of pebbles. For our fiducial disc model it is given by \citep{Idaetal2016}
\begin{equation}
\dot{M}_{\rm F} = 10^{-3}\alpha_3^{-1}\dot{M}_{*8}\left(\frac{L_*}{L_\odot}\right)^{2/7}\left(\frac{M_*}{M_\odot}\right)\left(\frac{t+t_{\rm off}}{1~{\rm Myr}}\right)^{-1/3}~M_{\rm E}~{\rm yr}^{-1},
\label{eq:mdotf}
\end{equation}
where $\alpha_3 \equiv \alpha/10^{-3}$, $\alpha$ is the disc viscosity parameter \citep{ShakuraSunyaev1973}, $L_* (L_{\odot})$ is the luminosity of the star (Sun), $M_* (M_{\odot})$ is the mass of the star (Sun) and $t_{\rm off}$ is the timescale at which the planetesimals are assumed to instantaneously form. The other key quantity that influences the pebble accretion rate is the normalised collision impact parameter $b$, which takes the form
\begin{equation}
b = 2\kappa \frac{R_{\rm H}}{r} \mathcal{S}^{1/3}~\min \left(\sqrt{\frac{3R_{\rm H}}{\chi\eta r}}\mathcal{S}^{1/6},1\right),
\label{eq:impactparam}
\end{equation}
where $\kappa$ is the reduction factor accounting for cases where pebbles are weakly coupled to the gas resulting in inefficient accretion, $R_{\rm H} = r(M/3M_*)^{1/3}$ is the Hill radius of the planetesimal. The terms in the parentheses encompass the two possible pebble accretion regimes, namely the Bondi regime (left-hand term) where the relative velocity between the planetesimal and a pebble is dominated by the pebble's drift velocity, and the Hill regime (right-hand term) where the relative velocity is dominated by Keplerian shear. The quantity $\chi$ is defined as
\begin{equation}
\chi = \frac{\sqrt{1+4\mathcal{S}^2}}{1+\mathcal{S}^2}.
\label{eq:chi}
\end{equation}
The Stokes number itself is given by
\begin{equation}
\mathcal{S} = \frac{\rho_{\rm peb}R_{\rm peb}}{\rho_{\rm g} H_{\rm g}}\max\left(1,\frac{4R_{\rm peb}}{9\lambda}\right),
\label{eq:taus}
\end{equation}
where $\rho_{\rm peb}$ and $R_{\rm peb}$ are the bulk density and the physical radius of a pebble, $\rho_{\rm g}$ is the gas midplane density, the disc scale height $H_{\rm g} = c_{\rm s}/\Omega_{\rm K} \propto T^{1/2}$ is a function of the disc temperature $T$, the sound speed $c_{\rm s}$ and the Keplerian orbital frequency $\Omega_{\rm K} = \sqrt{GM_*/r^3}$, and $\lambda$ is the mean free path of the pebble. The gas midplane density $\rho_{\rm g}$ is related to the disc surface density $\Sigma_{\rm g}$ and the disc scale height $H_{\rm g}$ via
\begin{equation}
\rho_{\rm g} = \frac{\Sigma_{\rm g}}{\sqrt{2\pi}H_{\rm g}}.
\end{equation}
The pebble mean free path is given by
\begin{equation}
\lambda = \frac{\mu m_{\rm H}}{\sigma_{\rm H_2} \rho_{\rm g}},
\end{equation}
where $\mu$ is the mean molecular weight of the gas (mostly hydrogen), $m_{\rm H}$ is the mass of a hydrogen atom (mostly that of two protons), and $\sigma_{\rm H_2}$ is the collisional cross section of a hydrogen molecule. The reduction factor $\kappa$ is a function of the Stokes number, expressed as \citep{OrmelKobayashi2012}
\begin{equation}
\ln \kappa = -\left(\frac{\mathcal{S}}{\mathcal{S}^*}\right)^{0.65},
\label{eq:kappa}
\end{equation}
where the quantity $\mathcal{S}^*$ is defined as
\begin{equation}
\mathcal{S}^* = \min \left[2,4 \eta^{-3} \frac{M}{M_*}\right].
\label{eq:S*}
\end{equation}
The quantity $\eta$ is given by
\begin{equation}
\eta = \frac{1}{2}h_{\rm g}^2 \left| \dv{\ln P}{\ln r} \right|,
\label{eq:eta}
\end{equation}
where $h_{\rm g} = H_{\rm g}/r$ is the reduced gas scale height. The remaining quantities in equation~\ref{eq:dotmpl} that we have yet to describe are the constant $C$ that determines the mode of accretion (whether 2D or 3D) and the reduced pebble scale height $h_{\rm peb}$. These quantities are expressed as \citep{Idaetal2016}
\begin{equation}
C = \min\left(\sqrt{\frac{8}{\pi}}\frac{h_{\rm peb}}{b}, 1\right),
\label{eq:const2d3d}
\end{equation}
and
\begin{equation}
h_{\rm peb} = \left(1+\frac{\mathcal{S}}{\alpha}\right)^{-1/2}h_{\rm g}.
\label{eq:hpeb}
\end{equation}
Pebble accretion begins in 3D mode ($C \sim 1$) when the planetesimal is of low mass and then transitions into the 2D mode when the planetesimal grows larger and its Hill radius becomes larger than the scale height of the pebbles $(R_{\rm H} > h_{\rm peb})$.
\subsection{Application to small planetesimals}
According to \citet{OrmelKlahr2010} pebble accretion causes rapid growth onto planetesimals with diameters of $\sim 200~{\rm km}$ or greater. To understand how pebble accretion works on small(er) bodies the question becomes which of the paths in the $\min()$ statements in equations~\ref{eq:dotmpl},~\ref{eq:impactparam},~\ref{eq:S*} and~\ref{eq:const2d3d} above dominate.
We mentioned previously that the two key quantities controlling the pebble accretion rate are the global pebble flux $\dot{M}_{\rm F}$ and the normalised impact parameter $b$. In the following, we will demonstrate that the reduction factor $\kappa$ (equation~\ref{eq:kappa}) turns out to be important as well. The quantity $\mathcal{S}^*$ (equation~\ref{eq:S*}) sets the boundary between the so-called {\it settling regime}, in which the cross section for accretion is very large up to the Hill radius \citep{OrmelKobayashi2012}, and the {\it geometric regime}, in which the cross section is just the surface area of the planetesimal \citep{OrmelKlahr2010}. The two regions are smoothly connected with the parameter $\kappa$, whose functional form is a fit that \citet{OrmelKlahr2010} obtained from numerical experiments. The critical Stokes number that separates the settling regime from the geometric regime results in $\mathcal{S}^* < 2$. At around 1~au, this translates to a critical planetesimal-to-star mass ratio of $M/M_* < \frac{1}{2}\eta^3 \lesssim 1.4\times10^{-9}$ which corresponds to a diameter of $D=1200~{\rm km}$ for a bulk density of $\rho = 3000~{\rm kg~m}^{-3}$ for the planetesimal, assuming nominal disc quantities (see Appendix~\ref{sec:appendix_discmodel} for a description of our disc model). This critical value of $M/M_*$ has a distance dependence. For example, at 3~au $\mathcal{S}^* < 2$ when the planetesimal mass is $M/M_* < 3.5\times10^{-9}$, or equivalent to a diameter of 1700~km. As such, when a planetesimal at 3~au has a diameter smaller than about twice that of Ceres, or that it is roughly 1200~km if it is located at 1~au, the quantity $\mathcal{S}^*$ can become smaller than 2, meaning that for these cases we are no longer in the settling regime and are instead approaching the geometric regime where the accretion cross section reduces to the surface area of the planetesimal. This has important implications with regards to the pebble accretion rate and efficiency of planetesimals below a critical diameter at a given semi-major axis.
In the asteroid belt, the typical value for the pebble Stokes number $\mathcal{S}$ with our model parameters is 0.1 and the quantity $\mathcal{S}^*$ is often equal to 2 \citep{Idaetal2016}. For these values, the reduction factor $\kappa = 0.87$. Suppose now that we want to study accretion onto the parent body of the Howardite-Eucrite-Diogenite (HED) meteorites, which, despite existing controversies, we here assume to be the asteroid 4 Vesta \citep{McCordetal1970,ConsolmagnoDrake1977,BinzelXu1993,Keil2002,McSweenetal2013}. Vesta's diameter is $D \sim 500~{\rm km}$ and $M/M_* = 1.3\times10^{-10}$. For our nominal disc model at 2.5~au (Vesta's modern orbital distance), $\eta \sim 3\times10^{-3}$ and $\mathcal{S}^* \sim 0.02$. If the Stokes number is the typical value of 0.1 then $\kappa = 0.06$, but if the Stokes number is higher then $\kappa$ will be lower, resulting in a much lower rate of accretion than when the impact parameter is almost equal to the Hill radius. Pebble accretion onto the proposed (ordinary chondrite, OC) H-chondrite parent body 6 Hebe \citep{Gaffeyetal1993,GaffeyGilbert1998,Binzeletal2004,Binzeletal2019}, with $D \sim 200~{\rm km}$ and $M/M_* \lesssim 7\times10^{-12}$, is much less efficient yet again because the quantities $\mathcal{S}^*$ and $\kappa$ work out to be $\lesssim 10^{-3}$ and $\lesssim 2\times10^{-9}$, respectively. Thus, because of the $\kappa$ factor, which is related to how the pebbles accrete in the settling regime \citep{OrmelKlahr2010,OrmelKobayashi2012}, accretion onto small objects with $D \sim 200~{\rm km}$ could become {\it extremely inefficient}. However, the impact parameter $b$ must always be equal to or greater than the radius of the planetesimal for it to be physically meaningful. For a planetesimal with bulk density $\rho = 3000~{\rm kg~m}^{-3}$ we have $R/R_{\rm H} \sim 5.7\times10^{-3}$ so that the relative impact parameter in terms of the Hill radius cannot go much below this value. The revised impact parameter therefore should read
\begin{equation}
b=\max \left[2\kappa \frac{R_{\rm H}}{r} \mathcal{S}^{1/3}~\min \left(\sqrt{\frac{3R_{\rm H}}{\chi\eta r}}\mathcal{S}^{1/6},1\right),\frac{R}{r}\right],
\end{equation}
where $R$ is the radius of the planetesimal. This shows that the accretion rate of pebbles cannot become arbitrarily low, but that small planetesimals are still severely disadvantaged in terms of their accretion rate over their larger brethren.
\subsection{The role of the snow line and fragmentation}
\label{sec:pebacc_fragmentation}
\citet{Morbidellietal2016} suggest that the pebbles lose their volatiles at the snow line and could fragment into mm-sized grains akin to chondrules. This fragmentation, if confirmed by mechanical arguments, lowers the Stokes number (equation~\ref{eq:taus}). The Stokes number can either be $\mathcal{S} \propto R_{\rm peb}$, called the Epstein regime, or $\mathcal{S} \propto R_{\rm peb}^2$, called the Stokes regime, depending on the distance to the star and the properties of the gas disc. The transition between the two regimes occurs at \citep{Idaetal2016}
\begin{equation}
r_{\rm ES} = 2.2\rho_{\rm peb}^{-7/26}\left(\frac{L_*}{L_\odot}\right)^{-3/13}\left(\frac{M_*}{M_\odot}\right)^{17/26}\left(\frac{\dot{M}_{\rm F4}^{1/3}\dot{M}_{*8}}{\alpha_3}\right)^{21/52}~{\rm au},
\end{equation}
where we additionally define $\dot{M}_{\rm F4} \equiv \frac{\dot{M}_{\rm F}}{10^{-4}~M_{\rm E}~{\rm yr}^{-1}}$ and $\dot{M}_{*8} \equiv \frac{\dot{M}_*}{10^{-8}~M_\odot~{\rm yr}^{-1}}$. For nominal gas disc temperature and surface density at 1~au, the pebble mean free path $\lambda \sim 2~{\rm cm}$. Inside the snow line, where fragmentation could occur, the pebble radius is assumed to be $R_{\rm peb} \sim 1~{\rm mm}$ \citep{Morbidellietal2016} and so in this case $\mathcal{S} \propto R_{\rm peb}$ because $R_{\rm peb} \ll \lambda$ (equation~\ref{eq:taus}). Assuming the pebble density as $\rho_{\rm peb}=1000~{\rm kg~m}^{-3}$, the Stokes number at 1~au for nominal disc parameters is approximately $\mathcal{S}\sim 0.2$, but after fragmentation it is lowered to $\mathcal{S} \sim 10^{-4}$ \citep{BrasserMojzsis2020}.
With the reduction of the Stokes number at 1~au from $\mathcal{S} \sim 0.2$ to $\mathcal{S} \sim 10^{-4}$ as a consequence of potential fragmentation, pebble accretion only becomes inefficient once $\mathcal{S}^* \lesssim 2\times10^{-5}$, i.e., when $M/M_* \lesssim 10^{-14}$, which corresponds to a planetesimal diameter of $D\lesssim 50~{\rm km}$. This critical diameter is lower than that derived previously for the case where fragmentation is not considered (cf. $D \sim 200~{\rm km}$). Thus, with the inclusion of fragmentation effects the range of planetesimal diameters for which pebble accretion can proceed efficiently becomes wider. Importantly, the accretion rate also depends strongly on the Stokes number, and a lower Stokes number actually results in a much slower overall rate of accretion with all other parameters being equal. Nevertheless, due to the debilitating effect of $\kappa$, accretion onto small planetesimals by fragmented pebbles is still faster than by intact pebbles beyond the snow line. As such, if fragmentation occurred, pebble accretion onto planetesimals with a diameter comparable to the H-chondrite parent body $(D \sim 200~{\rm km};~{\rm 6~Hebe})$ or smaller can still proceed, albeit with low efficiency, but only as long as they remain inside the snow line. To summarise, the temporal evolution of the snow line plays a critical role in the growth rate of planetesimals in the inner disc from the inward spiralling pebbles.
\subsection{Analytical computation of pebble accretion rate}
\label{sec:pebacc_analytical}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{pebacc_contour.pdf}
\caption{Contour plot depicting the log of the mass ratio of the planetesimal $\log (M_{\rm f}/M_{\rm i})$ as a function of its initial mass $M_{\rm i}$ and semi-major axis $a$, with and without pebble fragmentation effects at the snow line. The top row assumes all planetesimals form very soon after the disc at $t_{\rm off} = 0.1~{\rm Myr}$, the middle row assumes planetesimal formation after 0.5~Myr and the bottom row after 1~Myr. Regardless of our choice of parameters (initial planetesimal size, planetesimal semi-major axis, planetesimal formation timing, inclusion/exclusion of fragmentation effects), pebble accretion onto small planetesimals in the asteroid belt is very inefficient. The planetesimal diameter can be computed from $\log (D/{\rm 1~km})= \frac{1}{6}\log (M/1~M_{\rm E})+3.597$, so that $M/M_{\rm E} =10^{-5}$ corresponds to $D=580~{\rm km}$.}
\label{fig:pebacc_contour}
\end{figure*}
We combine all of these aspects to show a global accretion map in Fig.~\ref{fig:pebacc_contour}, which is a contour plot of $\log (M_{\rm f}/M_{\rm i})$, i.e., the log of the final mass of a planetesimal divided by its initial mass. We chose to plot this instead of the final mass to better show the total mass growth. The map was created by integrating equation~\ref{eq:dotmpl} for a single planetesimal located at a fixed semi-major axis as a function of time either for at most 4~Myr \citep[approximate lifetime of the gas disc;][]{Wangetal2017} or until the pebble isolation mass \citep{Lambrechtsetal2014} was reached. The pebble isolation mass of a planetary body is the mass above which pebble accretion stops due to the planetary body causing perturbations in the disc that in turn halt the flow of pebbles \citep{Lambrechtsetal2014}. Each panel of Fig.~\ref{fig:pebacc_contour} is made up of $501 \times 401$ single-planetesimal integrations. For these simulations, we assume the temperature of the gas disc to be 200~K at 1~au. The disc evolution that was used follows that outlined in Appendix~\ref{sec:appendix_discmodel} and the pebble flux is computed using equation~\ref{eq:mdotf}. We also vary the timing of planetesimal formation $t_{\rm off}$. The top row of Fig.~\ref{fig:pebacc_contour} has $t_{\rm off} = 0.1~{\rm Myr}$, the middle row has $t_{\rm off} = 0.5~{\rm Myr}$ and the bottom row has $t_{\rm off} = 1~{\rm Myr}$. These three different starting times account for the delay in the timing of planetesimal formation in the disc. The left column has pebble fragmentation effects included while the right column does not.
It is clear from Fig.~\ref{fig:pebacc_contour} that the mass distribution is bimodal: planetesimals either grow large or they barely grow at all. This bimodal outcome is mainly dependent on the initial diameter of the planetesimals, which in turn determines their pebble accretion efficiency. If their diameter is below a critical value (which has a slight dependence on disc parameters), then they are in the geometric regime and thus accreting pebbles is very inefficient regardless of their distance to the Sun (purple region in the left side of all the panels in Fig.~\ref{fig:pebacc_contour}). If they are larger than the critical diameter, then they are in the settling regime wherein the pebble accretion efficiency is high, and will thus experience more growth. The planetesimal's critical diameter between the geometric and settling regimes depends on the location in the disc, the formation time of the planetesimal, whether pebble fragmentation at the snow line is assumed, and also very likely the temperature of the disc (although this factor is not investigated here directly).
It would appear that a small planetesimal $(M_{\rm i} = 10^{-5}~M_{\rm E}~{\rm for~example})$ attained a larger final mass compared to a large planetesimal $(M_{\rm i} = 10^{-3}~M_{\rm E}~{\rm for~example})$ located at the same semi-major axis, but this is actually not the case: we plot the mass ratio instead of the final mass. Both planetesimals reach the pebble isolation mass at the end of the simulations -- which is of the order of a few Earth masses (with a weak distance dependence). As such, a planetesimal with $M_{\rm i} = 10^{-5}~M_{\rm E}$ growing to $1~M_{\rm E}$ increased its mass by a factor of $\log(1/10^{-5}) = 5$ while the other planetesimal with $M_{\rm i} = 10^{-3}~M_{\rm E}$ increased its mass by a factor of $\log(1/10^{-3}) = 3$.
Without fragmentation (right column of Fig.~\ref{fig:pebacc_contour}), only planetesimals with $M \gtrsim 10^{-5}~M_{\rm E}~(D \gtrsim 580~{\rm km})$ will efficiently accrete pebbles, with a slight dependence on their initial location in the disc (the critical value for planetesimal diameter becomes smaller as we move closer to the Sun); everything smaller will barely accrete anything. Fragmentation (left column of Fig.~\ref{fig:pebacc_contour}), on the other hand, allows for the creation of a narrow annulus of large bodies between $\sim0.8~{\rm au}$ to $\sim1.2~{\rm au}$ whose exact extent depends on the formation time of the planetesimals relative to the evolution of the snow line. In our model the snow line does not reach inside 0.75~au so accretion inside of this distance can be inefficient if planetesimals are predominantly small $(M \lesssim 10^{-7}~M_{\rm E}~{\rm or}~D \lesssim 300~{\rm km})$ and formed late $(t_{\rm off} \gtrsim 1~{\rm Myr})$. In our fiducial model the snow line is at 1.2~au when $t+t_{\rm off}=1~{\rm Myr}$, and it passes 2~au at $t+t_{\rm off}= 0.44~{\rm Myr}$, thus substantial pebble accretion in the terrestrial planet region can only occur if planetesimals form very early, say within 0.5~Myr of the formation of the first solids \citep[e.g.][]{Kruijeretal2014}, and/or form large. In the asteroid belt region accretion generally stops really early on, at a time comparable to the formation age of the iron meteorites, and possibly before most chondrule formation ages -- typically about 1.5~Myr \citep{luuetal2015} -- because of the rapid inward migration of the snow line. Beyond 2~au accretion is generally insubstantial for small planetesimals.
The contour plot, although instructive, only provides information on the evolution of a single planetesimal. To understand the effects of pebble accretion in the early inner Solar System which likely hosted many planetesimals with a range of diameters, it is therefore necessary to turn to {\it N}-body simulations.
\section{Methods}
\label{sec:pebacc_methods}
\subsection{Gas disc model}
In our simulations the gas disc prescription is based on the model of \citet{Idaetal2016}, with the detailed equations given in Appendix~\ref{sec:appendix_discmodel}. The gas disc is assumed to be a steady-state accretion disc with a constant value for the accretion viscosity parameter $\alpha$ \citep{ShakuraSunyaev1973}. The temperature of the disc depends on two heating sources: viscous dissipation or stellar irradiation, which dominate in different regions of the disc. Equations describing the disc temperature profiles (equation~\ref{eq:T_visirr}) due to the aforementioned heating sources are obtained from empirical fits to the disc model by \citet{GaraudLin2007}. The thermal structure of the disc in turn determines the scale height and surface density. In our simulations, the gas disc dissipates away with time and was photoevaporated away in 500~kyr when the stellar accretion rate $\dot{M}_* < 10^{-9}~M_{\odot}~{\rm yr}^{-1}$.
In our simulations, the initial stellar accretion rate $\dot{M}_* = 2.63\times10^{-8}~M_{\odot}~{\rm yr}^{-1}$ and the initial pebble flux $\dot{M}_{\rm F} = 10^{-4}~M_{\rm E}~\rm{yr}^{-1}$. This corresponds to a disc age of $t_0 = 0.5~{\rm Myr}$. This value of $t_0$ in turn determines the subsequent values of $\dot{M}_*(t)$ and $\dot{M}_{\rm F}(t)$ throughout the simulation.
\subsection{Pebble accretion}
The pebbles are assumed to have originated in the outer regions of the protoplanetary disc where most of the solids (dust) are located \citep[e.g.][]{YoudinShu2002,Garaud2007,Birnstieletal2012}. In this region, sub-micron-sized dust grains can grow into pebbles because their growth timescale is much shorter than their migration timescale and the collision rate is low \citep{Birnstieletal2012}. Upon reaching a critical diameter (or a critical Stokes number), the migration timescale becomes comparable to the growth timescale and the pebbles commence their inward migration towards the Sun by virtue of gas drag \citep{Idaetal2016} while their growth practically ceases. Due to the strong radial dependence of the growth timescale, there will be a location in the disc where dust clumps have just reached pebble size and start to migrate inwards. This is known as the pebble formation front and it moves outwards with time until it reaches the outer edge of the protoplanetary disc \citep{LambrechtsJohansen2014,Idaetal2016}, after which the pebble flux is severely reduced because all the solids in the disc have been consumed \citep[e.g][]{Chambers2016,Satoetal2016}.
We implemented the pebble accretion prescriptions of \cite{Idaetal2016} into the {\it N}-body code SyMBA \citep{Duncanetal1998} based on the method presented in \citet{Matsumuraetal2017}. Unlike \citet{Levisonetal2015,Levisonetal2015giantpl} we do not directly compute the growth of planetesimals by accretion of physical pebbles, but instead compute their mass increase based on the pebble flux at their respective locations in the disc. Pebble accretion onto planetesimals occurs outside in, that is, planetesimals at the outer edge of our solid disc are the first to encounter the pebbles. They will accrete a fraction of the pebbles $\dot{M}_{\rm p}$, reducing the pebble flux to $1-\dot{M}_{\rm p}/\dot{M}_{\rm F}$. The planetesimals that are next in line will see the reduced pebble flux and the amount of pebbles they can accrete is computed from the reduced pebble flux. Furthermore, when a planetesimal (or planetary embryo) reaches its pebble isolation mass $M_{\rm p,iso} \sim 1/2(H_{\rm g}/r)^3 M_*$ we assume that its accretion stops and pebbles are prevented from flowing past its orbit to other planetary bodies residing closer than it is to the Sun. At the snow line, we assume that pebbles were fragmented into grains of 1~mm on their path towards the Sun \citep{Morbidellietal2016}. We also study the outcome of simulations excluding this effect.
\subsection{Initial conditions for the N-body simulations}
\begin{figure*}
\centering
\includegraphics[width=0.8\textwidth]{pebacc_initcond_D.pdf}
\caption{Cumulative distribution of planetesimal diameter $D$ for various values of initial solid disc mass $M_{\rm disc,i}$.}
\label{fig:pebacc_initcond_D}
\end{figure*}
\begin{table}
\centering
\caption{Initial solid disc mass $M_{\rm disc,i}$ in units of Earth masses, range of planetesimal diameters $D$ in each disc, and initial number of planetesimals in each disc $N_{\rm plt}$ for the pebble accretion simulations. We repeat the simulations using the same set of initial conditions shown here for different values of initial disc temperature $T_{\rm 1~au} = 200~{\rm K},~250~{\rm K,~and}~300~{\rm K}$.}
\label{tab:pebacc_initcond}
\begin{tabular}{AAA}
\toprule
$M_{\rm disc,i}~(M_{\rm E})$ & $D~({\rm km})$ & $N_{\rm plt}$ \\
\midrule
$3\times10^{-4}$ & $90 - 110$ & $\sim 1200$ \\
$1\times10^{-3}$ & $90 - 300$ & $\sim 1200$ \\
0.05 & $250 - 1200$ & $\sim 2000$ \\
0.10 & $350 - 1200$ & $\sim 2000$ \\
0.25 & $560 - 1200$ & $\sim 2000$ \\
\bottomrule
\end{tabular}
\end{table}
Our planetesimal disc contains planetary bodies with an initial density of $\rho = 2500~{\rm kg~m}^{-3}$ and diameter $D$ assigned following a cumulative size-frequency distribution of $N(>D) \propto D^{-5/2}$ (Fig.~\ref{fig:pebacc_initcond_D}). Their orbital eccentricities $e$ and orbital inclinations $I$ are assigned at random from a uniform distribution with intervals $[0,0.01)$ and $[0^{\circ},1{^\circ})$, respectively. The remaining orbital angles (longitude of ascending node $\Omega$, argument of periapsis $\omega$ and mean anomaly $\mathcal{M}$) are assigned values between $0^{\circ}$ and $360^{\circ}$, also randomly chosen from a uniform distribution. The semi-major axis of the planetesimals ranges from $0.5~{\rm au} < a < 2.0~{\rm au}$ and they are distributed according to an $r^{-1}$ surface density profile. Our choice to limit the outer edge of the solid disc to 2~au is guided by the analytical results in Fig.~\ref{fig:pebacc_contour} and our intent to save computation time (our simulations contain a fairly large number of planetesimals that are all self-gravitating). In Fig.~\ref{fig:pebacc_contour} we showed that planetesimals smaller than $D \lesssim 500~{\rm km}$ located beyond 2~au are not expected to grow by much but smaller planetesimals could if pebble fragmentation is in force. Therefore we chose not to place any planetesimals beyond 2~au in the calculations and focus on the terrestrial planet region. Bodies with a diameter larger than 500~km can experience substantial growth, as was shown by \citet{BrasserMojzsis2020}.
The variables in our simulations are: (1) the initial mass of the solid disc $M_{\rm disc,i}$, (2) the range of planetesimal diameters $D$, and (3) the initial disc temperature at 1~au $(T_{\rm 1~au} =$~200~{\rm K},~250~{\rm K},~{\rm and}~300~{\rm K}) which corresponds to different disc scale heights $h_{\rm g}$ and pebble scale heights $h_{\rm peb}$. The lower mass discs ($M_{\rm disc,i} = 3\times10^{-4}~M_{\rm E},~1\times10^{-3}~M_{\rm E}$) contain $\sim 1200$ planetesimals while the more massive discs contain up to $\sim 2000$ planetesimals. All the planetesimals in our simulations are self-gravitating, that is, they are able to interact with each other via gravity. We summarise the initial conditions in Table~\ref{tab:pebacc_initcond}.
Our choice of planetesimal diameters are based on the results of previous works. The characteristic diameter of early planetesimals that formed via turbulent concentration or streaming instability in a Solar-like protoplanetary disc ranges from $\sim 100~{\rm km}$ up to $\sim 1000~{\rm km}$ \citep[e.g.,][]{Chambers2010,Johansenetal2014,KlahrSchreiber2020}, with a size-frequency distribution that follows a shallow power law $N(>D)\propto D^{-1.8}$ for diameter, corresponding to $N(>M) \propto M^{-0.6}$ for mass \citep[e.g.][]{Johansenetal2015,Simonetal2016,Schaeferetal2017,Abodetal2019}. Models of the collisional evolution of the asteroid belt also suggest that the primordial asteroids were typically 100~km in diameter \citep{Morbidellietal2009}. The combined results of these studies suggest that the initial planetesimal population likely consisted of many small planetesimals and a few massive bodies. Asteroid Vesta could be one of the early-formed massive bodies at the tail end of the distribution.
We opted for a different range of planetesimal diameters for different values of initial disc mass because we wanted to study with high enough resolution how pebble accretion affects both small and large planetesimals. If we were to adopt the same range of planetesimal diameters $(100~{\rm km} \leq D \leq 1200~{\rm km})$ for all disc masses, then we would have $N_{\rm plt} = 10-50$ for $M_{\rm disc,i} = 10^{-4}~M_{\rm E}$, $N_{\rm plt} = 100-400$ for $M_{\rm disc,i} = 10^{-3}~M_{\rm E}$, $N_{\rm plt} = 2500-3200$ for $M_{\rm disc,i} = 10^{-2}~M_{\rm E}$, $N_{\rm plt} \sim 14000$ for $M_{\rm disc,i} = 0.1~M_{\rm E}$, and $N_{\rm plt} \sim 31000$ for $M_{\rm disc,i} = 0.25~M_{\rm E}$. The resolution would be low for the low-mass discs and we could not really study the effect of pebble accretion onto a swarm of small planetesimals because any large planetesimal in the swarm would dominate the growth and potentially scatter the smaller planetesimals away. For the higher mass discs the computing time would take too long: according to our benchmarks of the code, integrating 14000 self-gravitating planetesimals for 4~Myr would take about a year on a 4.1~GHz 32-core AMD Threadripper CPU with OpenMP parallelisation assuming no loss through ejection or collision. For these reasons we chose to fix the maximum number of planetesimals for each disc and then shift the minimum and maximum diameter according to the disc mass. Nevertheless, we have run some simulations with the same planetesimal diameter range for all disc masses whose results are reported in Appendix~\ref{sec:appendix_lowmassdisc}.
In addition to the planetesimal disc, following the work of \citet{BrasserMojzsis2020} our initial set-up also includes a $0.01~M_{\rm E}$ planetary embryo placed at 5.6~au that would eventually become Jupiter. When Jupiter reaches its pebble isolation mass \citep[$20~M_{\rm E}$;][]{Lambrechtsetal2014} in about 1~Myr \citep{Kruijeretal2017jupiter}, it will open a partial gap in the disc around its orbit that prevents pebbles from beyond its orbit to spiral in towards the Sun, effectively shutting off the pebble flux to the inner Solar System \citep{Lambrechtsetal2014}, leaving only very small particles with sizes $\lesssim 100~\mu{\rm m}$ to pass through unobstructed \citep{PaardekooperMellema2006}.
We employ the SyMBA {\it N}-body code \citep{Duncanetal1998} modified to include pebble accretion, the effects of disc-induced orbital migration and orbital eccentricity damping \citep[Appendix~\ref{sec:appendix_migration}; e.g.][]{Tanakaetal2002,Matsumuraetal2017} and gas envelope accretion for massive bodies \citep[Appendix~\ref{sec:appendix_gasaccretion};][]{Ikomaetal2000,Matsumuraetal2017}. We also use the parallelised version from \citet{LauLee2021}. We also simulate the case where orbital migration is excluded to isolate the effects of the accretion of pebbles; these simulations are purely academic. We ran one simulation for each permutation of initial conditions (initial solid disc mass $M_{\rm disc,i}$, disc temperature $T_{\rm 1~au}$, fragmentation on/off, migration on/off). In total we ran 60 simulations, each for 4~Myr and with a time step of 0.01~yr.
We quantify the amount of mass increase in the inner Solar System resulting from the accretion of pebbles by computing the ratio of the final mass of the planetesimal disc versus its initial mass $M_{\rm disc,f}/M_{\rm disc,i}$. For the more massive discs $(M_{\rm disc,i} \geq 0.05~M_{\rm E})$, the larger planetesimals will accrete pebbles efficiently and they can grow to large masses quickly, as we will show in the following section. Therefore, the final disc mass for these cases do not only reflect growth via pebble accretion, but also additional growth due to mergers and gas envelope accretion.
\section{Results}
\subsection{Amount of mass increase in the inner disc}
\begin{figure*}
\centering
\includegraphics[width=0.7\textwidth]{pebacc_mfmi_mdisci.pdf}
\caption{Ratio of final mass in the disc to its initial mass $(M_{\rm disc,f}/M_{\rm disc,i})$ in the region between 0.5~au and 3.0~au as a result of pebble accretion over a period of 4~Myr, plotted with respect to the initial disc mass and disc temperature. The accretion efficiency depends on the sizes of the planetesimals. Large amounts of mass in the form of pebbles can still be accreted by large planetesimals within 2~au before Jupiter reaches its pebble isolation mass.}
\label{fig:pebacc_mfmi_vs_mdisc}
\end{figure*}
We show in Fig.~\ref{fig:pebacc_mfmi_vs_mdisc} the final mass in the disc between 0.5~au and 3.0~au as a fraction of its initial mass $(M_{\rm disc,f}/M_{\rm disc,i})$ for different disc temperatures at the end of the simulations $(t = 4~{\rm Myr})$. We find that the final mass of the solid disc depends mostly on its initial mass. A higher initial disc mass results in a relatively higher final disc mass because these discs have, on average, a higher number of large planetesimals which are more efficient at accreting pebbles. In discs containing mostly small planetesimals with $D \lesssim 300~{\rm km}~(M_{\rm disc,i} \leq 1\times10^{-3})$, the overall increase in mass is very limited as the planetesimals are not in the settling regime in which accretion proceeds efficiently. This outcome is similar to that of the single-planetesimal computations shown earlier in Fig.~\ref{fig:pebacc_contour}, which illustrates that a planetesimal's accretion regime abruptly transitions from the less efficient geometric regime to the more efficient settling regime when their masses (or diameters) are above a critical threshold given a specific set of gas disc parameters.
We also observe a general inverse correlation between the amount of mass increase and the initial temperature of the gas disc, i.e., more growth in cooler discs compared to warmer discs. This is because (1) for cooler discs the pebble scale height is lower and this increases the pebble accretion efficiency of the planetesimals \citep{Idaetal2016}, and (2) the snow line is on average closer to the Sun in cooler discs than in hotter discs. The combined effects of (1) and (2) results in a high accretion efficiency that allows planetesimals to grow to large sizes very quickly, with the extreme case forming gas giants in the asteroid belt when the disc is cool and migration is switched off. This trend with disc temperature does not hold for the cases with $M_{\rm disc,i} \geq 0.05~M_{\rm E}$ when migration is in force (top row of Fig.~\ref{fig:pebacc_mfmi_vs_mdisc}). In these cases, the cold discs appear to record less growth at the end of the simulations compared to warmer discs. The reason for this is that many of the large planetesimals that grow big early in the cold discs migrated towards the Sun and were removed from the simulation.
When comparing the top and bottom rows of Fig.~\ref{fig:pebacc_mfmi_vs_mdisc}, the effect of disc-induced orbital migration is very clear for solid discs containing large planetesimals. When migration effects are turned off as in the control cases (bottom row of Fig.~\ref{fig:pebacc_mfmi_vs_mdisc}), the trend of inverse correlation between mass increase and disc temperature reappears and cold discs record the most increase in disc mass reaching a few thousand times their initial values because some of the planetesimals in these discs attained masses high enough to trigger gas accretion \citep{Ikomaetal2000}. For the case of less massive discs $(M_{\rm disc,i} \leq 1\times10^{-3})$, orbital migration effects do not play a role in influencing the final disc mass because the planetesimals in these discs did not have the opportunity to grow to large sizes for orbital migration to kick in.
Pebble fragmentation at the snow line affects the final disc mass of less-massive and massive discs differently. When comparing the left and right columns of Fig.~\ref{fig:pebacc_mfmi_vs_mdisc}, we observe that when fragmentation effects are included the relative final mass in massive discs $(M_{\rm disc,i} \geq 0.05~M_{\rm E})$ is higher than in the low-mass discs with $M_{\rm disc,i} \leq 1\times10^{-3}~M_{\rm E}$. Fragmentation reduces the Stokes number of the pebbles and allows for planetesimals of smaller sizes to become more efficient at accreting pebbles, as discussed earlier in Section~\ref{sec:pebacc_fragmentation}. Without fragmentation, the discs containing planetesimals with $D\lesssim 110~{\rm km}$ barely grow at all (just a 0.3\% increase) while the discs with $M_{\rm disc,i} \leq 1\times10^{-3}~M_{\rm E}$ containing planetesimals with $D\lesssim 300~{\rm km}$ recorded a growth ranging from 1 -- 2\% for hot discs with migration to a maximum 24\% for cold discs, in contrast to a 15 -- 309\% mass increase when fragmentation is taken into account. These results thus confirm the analytical arguments that small planetesimals have difficulty accreting pebbles efficiently if the pebbles did not break into smaller sizes at the snow line. On the other hand, the lesser degree of growth for massive discs appears paradoxical, but this can be explained by the reduced overall accretion rate of the planetesimals when the Stokes number of the pebbles is reduced.
To summarise this section, our simulation results show that solid discs containing planetesimals larger than $\sim 300~{\rm km}$ in diameter increase their mass by at least a few times when they are exposed to an incoming pebble flux from the outer Solar System over the lifetime of the gas disc. As we have also included Jupiter in the simulations, our results also demonstrate that the more massive planetesimal discs in the inner Solar System are expected to have accreted at least a few times their initial mass in the form of pebbles before Jupiter reaches its isolation mass and shuts off the pebble flux. For lower-mass discs containing mostly planetesimals less than 300~km in diameter the mass increase is not as much as it is for higher-mass discs with more massive planetesimals because of their lower accretion efficiency. However, for the case of cold discs with fragmentation effects considered, the final mass in the disc can be as high as three times the initial mass.
\subsection{Distribution of solids in the disc}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{pebacc_m_vs_a_warmdisc.pdf}
\caption{Distribution of mass in the inner disc at the end of the simulations. Shown here are the outcomes for different initial disc masses with the initial disc temperature at 1~au set to 250~K. The outcomes for discs with initial mass $M_{\rm disc,i} \leq 1\times10^{-3}~M_{\rm E}$ (top row) are representative of the simulations with different disc temperatures and regardless of whether migration and fragmentation are included. For the more massive discs we show the outcome for the case of $M_{\rm disc,i} = 0.10~M_{\rm E}$ (bottom row) when migration effects are included or excluded to illustrate its influence on the final mass distribution in the disc. The grey shaded region represents the initial mass and semi-major axis distribution of the planetesimals.}
\label{fig:pebacc_m_vs_a}
\end{figure}
Having established that the amount of growth in the solid disc is dependent on the size (and mass) of its constituent planetesimals, we move on to report the distribution of solids at the end of the simulations for a few selected examples. We report selectively -- but representative of the results from all initial conditions -- the final mass distribution for warm discs $(T_{\rm 1~au} = 250~{\rm K})$ with fragmentation effects included (Fig.~\ref{fig:pebacc_m_vs_a}). For discs with initial mass $M_{\rm disc,i} \leq 1\times10^{-3}~M_{\rm E}$, the results are similar to the cases with fragmentation and/or migration excluded because (a) the planetesimals do not grow to sizes large enough to be affected by migration, and (b) the planetesimals do not grow much in the case of no fragmentation. Thus for these lower-mass discs, we can clearly see what is happening in the solid disc when the planetesimals are subjected to the pebble flux.
We observe a trend of inside-out growth: a higher mass increase for planetesimals located closer to the Sun compared to their counterparts located further away. This trend is clearest in the top-left panel of Fig.~\ref{fig:pebacc_m_vs_a} where we plot the result for the case where the solid disc contains only planetesimals with diameter less than $\sim 100~{\rm km}$. The minimum mass of the planetesimals does not have a dependence on the distance to the Sun initially (the lower bound of the grey shaded region is flat) but the lower bound of the final mass distribution (black dots) is higher when closer to the Sun but lower further away. This outcome is consistent with what was reported earlier in \citet{BrasserMojzsis2020}. For the case when $M_{\rm disc,i} = 1\times10^{-3}~M_{\rm E}$ (top-right panel of Fig.~\ref{fig:pebacc_m_vs_a}), the final mass distribution in the disc has a lower bound of $3\times10^{-7}~M_{\rm E}$ at 0.5~au while it is $2\times10^{-7}~M_{\rm E}$ at 2~au. The explanation for this inside-out growth outcome is the dependence of the pebble accretion rate on the semi-major axis \citep{Idaetal2016} -- the closer to the Sun, the higher the accretion rate. This outcome can also be understood using a more physical explanation: planetesimals in the region closer to the Sun see a higher concentration of pebbles because the pebbles are confined to an increasingly narrow annulus when they drift inwards, which is effectively an increased surface density in the disc, and thus it is easier to accrete pebbles in this region.
For the case of $M_{\rm disc,i} = 0.10~M_{\rm E}$ (bottom row of Fig.~\ref{fig:pebacc_m_vs_a}), the inside-out growth trend is masked because the planetesimals grow quickly due to their higher accretion efficiency and the inward migration of the snow line. In the example where migration is excluded (bottom-right panel of Fig.~\ref{fig:pebacc_m_vs_a}) we see two bumps, one each near the inner and outer edge of the solid disc, with the former attributed to inside-out growth and the latter caused by the passage of the snow line.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{pebacc_massdist_mdiff_warmdisc.pdf}
\caption{Initial and final mass distribution in each section of the disc (top row) and the corresponding amount of mass increase in each section (bottom row) for initial disc mass $1\times10^{-3}~M_{\rm E}$ and $0.10~M_{\rm E}$ and initial disc temperature of 250~K. The results shown in the left panel are representative of the outcomes for discs with low mass $(M_{\rm disc,i} \leq 1\times10^{-3}~M_{\rm E})$ while the right panel is representative of the general results for discs with high mass $(M_{\rm disc,i} \geq 0.05~M_{\rm E})$.}
\label{fig:pebacc_mdiff}
\end{figure}
The data in Fig.~\ref{fig:pebacc_m_vs_a} can be presented as histograms which shows how different the final distribution is from the initial distribution. We show in the top row of Fig.~\ref{fig:pebacc_mdiff} the results for $M_{\rm disc,i} = 1\times10^{-3}~M_{\rm E}$ and $0.10~M_{\rm E}$ with migration to illustrate the two different possible outcomes depending on the size of planetesimals in the disc. In the less massive disc (left panel of Fig.~\ref{fig:pebacc_mdiff}), the planetesimals increase their mass but remain confined between 0.5 and 2.0~au. In the more massive disc (right panel of Fig.~\ref{fig:pebacc_mdiff}), the planetesimals are distributed in a wider region compared to their initial distribution because smaller planetesimals in the disc have their orbits perturbed by larger planetesimals and migration caused some of them to move inwards a little. If we measure the amount of mass increase in each semi-major axis bin (bottom row of Fig.~\ref{fig:pebacc_mdiff}), we find a smooth gradient in mass for the less massive disc, but not for the more massive disc.
We have shown that pebble accretion can generate a mass gradient in the terrestrial planet region. The mass gradient is only clearly seen in discs where the initial planetesimal diameters are less than $\sim 300~{\rm km}$; the gradient is not obvious if the disc contains many planetesimals larger than 300~km.
\section{Discussion}
\subsection{Jupiter’s capacity as a pebble barrier}
From the observed isotopic dichotomy among non-carbonaceous (NC) and carbonaceous (C) meteorites that sample the inner and outer Solar System, respectively (Fig.~\ref{fig:isotopes}), it is thought that the two reservoirs should have been separated very early and remain separated for at least a few Myr \citep{Kruijeretal2017jupiter}. Using measurements of molybdenum and tungsten isotopes \citet{Kruijeretal2017jupiter} demonstrated that iron meteorites also cluster into two distinct groups similar to the chondritic meteorites, suggesting that the two meteoritic reservoirs co-existed and remain separated over an extended period of time (at least $\sim 3$ to 4~Myr according to Kruijer et al.). The means to sustain the separation of the inner and outer Solar System has been suggested to be the growth of Jupiter \citep{Kruijeretal2017jupiter} which, upon attaining its pebble isolation mass, will create a gap in the disc and prevent solid material from beyond its orbit from entering the inner Solar System \citep{Lambrechtsetal2014}. However, there have been reports \citep{BrasserMojzsis2020,Kleineetal2020} challenging the efficiency of Jupiter as a competent barrier because a substantial amount of pebbles could have drifted into the inner Solar System before Jupiter reached its pebble isolation mass.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{pebacc_isotopes_54Cr50Ti.pdf}
\caption{Measured $\varepsilon^{50}$Ti and $\varepsilon^{54}$Cr isotopic anomalies of known meteorites normalised to the values for the Earth. The meteorites fall into two distinct groups thought to represent the inner and outer Solar System, suggesting that the material in these two regions were kept separate at least until the formation of the chondrites. No data were available at the time of writing for $\varepsilon^{50}$Ti of iron meteorites. Acap. and Lodr. are abbreviations for Acapulcoites and Lodranites, respectively. C stands for carbonaceous and NC stands for non-carbonaceous meteorites. Data sourced from \citet{ShukolyukovLugmair2006,Trinquieretal2007,Trinquieretal2008,Trinquieretal2009,Qinetal2010a,Qinetal2010b,Yamashitaetal2010,Larsenetal2011,Petitatetal2011,Zhangetal2011,Zhangetal2012,YamakawaYin2014}.}
\label{fig:isotopes}
\end{figure}
Previous works have shown that during the growth of Jupiter planetesimals located within its orbit can double their size \citep{BrasserMojzsis2020} or even attain the sizes of the current terrestrial planets \citep{Johansenetal2021}. Our {\it N}-body simulations with a larger number of planetesimals give a similar outcome: our planetesimal discs with initial mass $M_{\rm disc,i} \geq 0.05~M_{\rm E}$ and containing planetesimals with $D > 300~{\rm km}$ at least double their mass before Jupiter shuts off the pebble flux. This outcome holds for the initial disc temperatures we have tested (200 K, 250 K and 300 K), and is also independent of whether fragmentation effects are considered.
Given the substantial increase in mass in the inner Solar System contributed by pebbles despite Jupiter shutting off the pebble flux when it reaches its isolation mass, our results therefore indicate that it is unlikely for Jupiter to be the agent preventing the mixing of material from the two reservoirs. It should be more likely that some other factor(s) is/are at play at separating the inner and outer Solar System, for example, a structural gap in the protoplanetary disc located just within the orbit of Jupiter \citep{BrasserMojzsis2020}.
\subsection{Implications for cosmochemistry}
From mixing models, the maximum contribution of carbonaceous-like material to the composition of the Earth and Mars is capped at around 10\% \citep[e.g.][]{LoddersFegley1997,Sanloupetal1999,Lodders2000,TangDauphas2014,Fitoussietal2016,Dauphas2017,MahBrasser2021}. Based on nucleosynthetic anomalies of several isotopes such as $^{50}$Ti and $^{54}$Cr, the contribution of carbonaceous chondrites towards the making of the Earth is limited to 10 -- 25\% \citep[e.g.][]{Warren2011,Mezgeretal2020}. A mass contribution from outer Solar System material by amounts greater than 10\% poses problems for reconciling the difference in isotopic composition of the terrestrial planets with that of the outer Solar System, and a mass increase beyond 100\% will severely dilute the original composition of the inner disc and mostly homogenise it.
If we take 10\% as the upper limit for the amount of outer Solar System material that could be present in the Earth and Mars, then the successful cases from our numerical simulations are the cases which fulfil all of the following criteria:
\begin{itemize}
\item $M_{\rm disc,i} \leq 1\times10^{-3}~M_{\rm E}$, or if the disc is more massive then all planetesimals have $D \leq 300~{\rm km}$, and
\item $T_{\rm 1~au} \geq 250~{\rm K}$, and
\item pebble fragmentation effects at the snow line are unimportant.
\end{itemize}
For these cases, the overall mass increase in the disc ranges from 0.3 -- 5\%, consistent with the constraint from mixing models and cosmochemistry data. The shortcoming for these cases, however, is that the total mass in the disc after 4~Myr is too little to build the terrestrial planets. To fulfil both requirements for the amount of mass increase $(\lesssim 10\%)$ and disc mass at the end of the pebble accretion phase $(\gtrsim 2~M_{\rm E})$, we should ideally start with a solid disc of $M_{\rm disc,i} = 1.8~M_{\rm E}$ which will grow to $2~M_{\rm E}$ assuming a 10\% mass increase and that the disc only contained planetesimals of diameter $D \lesssim 300~{\rm km}$ and that no mergers occurred during the stage of pebble accretion. However, modelling such a massive disc made up entirely of small planetesimals is at the limit of the capabilities of current hardware and {\it N}-body codes \citep{Wooetal2021}. We therefore reserve this study for future work. If the inner Solar System contained a high number of planetesimals with diameters 300~km and larger, we expect from our results that the isotopic composition of the planetesimals (and the planetary bodies that form from these planetesimals) would be akin to the outer Solar System -- inconsistent with cosmochemistry data.
Will the outcome of the {\it N}-body simulations change if we assume a changing composition of the pebbles? In our study, the implicit assumption is that the pebbles have the same composition, which we leave unspecified but one could assume any of the various carbonaceous chondrites as a proxy. In reality, the pebbles that formed at different times and at different locations in the disc could potentially have different compositions \citep{Lichtenbergetal2021}. The diversity in the isotopic compositions of the carbonaceous chondrites possibly reflects their distinct formation location and/or their formation time \citep{Deschetal2018}, although this is difficult to prove. By looking at the isotopes of O, Ti and Cr, the CI and CO chondrites plot in end-member positions of the carbonaceous meteorite group (see Fig.~\ref{fig:isotopes}), which could mean that there was a compositional gradient in the outer Solar System when these bodies formed \citep[e.g.][]{Trinquieretal2009,Mezgeretal2020}. To model the effect of time-dependent pebble compositions, we need to understand how the compositional gradient scales with heliocentric distance. This information is currently unavailable, although \citet{Deschetal2018} suggested that the CI chondrites formed farthest away from the Sun while CO chondrites formed closer to the Sun, but this suggestion is based mostly on model predictions.
One possible way out of this impasse is to do a similar end-member study by estimating the contribution of CI or CO chondrites to the composition of the terrestrial planets using our isotope mixing models from \citet{MahBrasser2021} -- which is based on the model of \citet{Dauphas2017}. We show in Fig.~\ref{fig:isocomp_mcmc} the computed best-fit composition of the Earth and Mars as a combination of enstatite chondrites, ordinary chondrites and carbonaceous chondrites. The top panels are results previously reported in \citet{MahBrasser2021}. The middle and bottom rows include only CI or CO chondrites respectively. With only CI or CO chondrites, the mixing model returns the maximum contribution of these chondrite types as $\leq 10\%$ (Fig.~\ref{fig:isocomp_mcmc}), similar to the results obtained by including various other types of carbonaceous chondrites \citep[e.g.][]{Dauphas2017,MahBrasser2021}. For Earth and Mars the contribution from H-chondrites is most strongly affected by the choice of either CI or CO. Thus, even if we were to assume a time-dependent variation in the composition of the pebbles from CO-like at the beginning to CI-like at the end (or vice versa), it still remains difficult for our {\it N}-body simulation results to be reconciled with meteorite isotope data of the inner planets: the amount of jovian materials added to the terrestrial planets cannot be more than 10\%.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{pebacc_bestfit_mcmc_isocomp_6panelCICO.pdf}
\caption{Comparison of Monte Carlo mixing model simulation outcomes. Top row shows the best-fit contributions of enstatite chondrites (EL, EH), ordinary chondrites (H, L, LL) and carbonaceous chondrites (CI, CO, CV, CM, CR) to the building blocks of Earth and Mars. The top row is the same plot that was presented in \citet{MahBrasser2021}. The middle and bottom rows show the results if we only include CI or CO chondrites (end-members of the carbonaceous chondrite group for O, Cr and Ti isotopes) in the model. All the results presented are computed from 20 Monte Carlo mixing model simulations based on the work of \citet{Dauphas2017}. The mixing model uses the isotopic anomalies in $\Delta^{17}$O, $\varepsilon^{50}$Ti, $\varepsilon^{54}$Cr, $\varepsilon^{62}$Ni and $\varepsilon^{92}$Mo for the meteorites and Earth and Mars as constraints. The contribution of CI or CO chondrites to the Earth and Mars is limited to $\leq 10\%$.}
\label{fig:isocomp_mcmc}
\end{figure}
Let us consider the possible outcomes if we were to instead assume a specific composition for the planetesimals in the inner Solar System. We note that this is also speculative as it is not known in detail how the nucleosynthetic isotopes are distributed in the inner disc, although there are suggestions that they follow a heliocentric gradient \citep[e.g.][]{Trinquieretal2009,Yamakawaetal2010}. Nevertheless, we can reason that given the vastly distinct isotopic compositions of the NC and C meteorites in O, Ti and Cr for example, a large contribution of jovian material would require the terrestrial material to be initially very depleted in these isotopes so that the final outcome matches what is observed today for the sampled inner Solar System bodies. If the initial composition of the inner Solar System was, for example, ureilite-like then the accretion of pebbles with CI-like composition has been demonstrated to be consistent with the $\varepsilon^{48}$Ca compositions of Earth, Mars and asteroid Vesta \citep{Schilleretal2018}. However, the suggested combination of ureilite + CI (end-members of the NC and C groups) in the same proportion that works for Ca does not hold for other isotopes such as O, Ti and Cr.
The works of \citet{Schilleretal2018,Schilleretal2020} argue for pebble accretion in the inner Solar System based on $\varepsilon^{48}$Ca and $\varepsilon^{54}$Fe isotopes in the Earth and other meteorites. Their combined results argue for the Earth having 40\% CI-chondrite-like material in its mantle. For the ureilite parent body \citet{Schilleretal2018} list $\varepsilon^{48}{\rm Ca} = -1.46\pm0.46$ and for the CI chondrite parent body $\varepsilon^{48}{\rm Ca} = 2.06\pm0.085$. If we denote by $x$ the fraction of CI chondrite in Earth's mantle, then according to \citet{Schilleretal2018} $x$ is the solution to $2.06x-1.46(1-x) = 0$, which yields $x = 0.41\pm0.12~(2\sigma)$; the uncertainties were computed using a Monte Carlo method. If these results are correct, they should also hold for the isotopes of other elements commonly used as tracers, such as the multiple oxygen isotopes, and nucleosynthetic tracers such as $^{50}$Ti and $^{54}$Cr because the Earth can only be made up of a single mixture of sources rather than different mixtures for different elements. For the ureilite parent body $\Delta^{17}{\rm O} = -1.16\pm0.55$ \citep{ClaytonMayeda1996} and $\Delta^{17}{\rm O} = 0.39\pm0.14$ for the CI chondrites \citep{ClaytonMayeda1996}. The required mass fraction of CI chondrite based on oxygen isotopes is $x = 0.75\pm0.23~(2\sigma)$. This fraction deviates significantly compared to the fraction advocated by \citet{Schilleretal2018} based on calcium isotopes (i.e. 40\%). In addition, if molybdenum isotopes are considered then there is no acceptable solution for $x$ as the isotopic anomalies in $\varepsilon^{92}$Mo for both ureilites and CI chondrites have deficits relative to the Earth. We arrive at the same conclusion as well if we were to consider the more abundant isotopes of molybdenum $(\varepsilon^{94,95}{\rm Mo})$ in excess in CI and ureilites compared to Earth.
The above analysis shows that it is difficult to establish whether or not pebble accretion occurred in the inner Solar System when using isotopes alone because the end-member case of ureilite + CI chondrite could be invoked as a mixture to explain the isotopic anomalies of the terrestrial planets (albeit not always in the same proportions). Indeed, by looking at single isotopes independently it is always possible to find a combination of end-member meteoritic reservoirs with the correct mixing proportion to explain the isotopic composition of a planetary body. The difficulty is to find a single mixing ratio that matches all isotopes simultaneously as was done by \citet{Fitoussietal2016} and by \citet{Dauphas2017}. It is therefore necessary to look beyond nucleosynthetic isotopes alone to establish whether the simple model suggested by \citet{Schilleretal2018,Schilleretal2020} and pebble accretion can account for the growth and composition of the terrestrial planets. In the following, we examine the constraints from the major elemental abundances in known meteoritic reservoirs and planetary bodies.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{pebacc_elements_MgSi_AlSi_BSEmodels.pdf}
\caption{Mg/Si versus Al/Si for various chondrites, achondrites, Mars and the Earth. For the Earth, we also show the elemental abundance ratios of the Earth for different amounts of Si in the core. The fractionation lines and pyrolite composition are from \citet{Jagoutzetal1979}. Error bars are computed using the average variations for Mg/Si (5\%) and Al/Si (10\%) determinations for each parent body. APB = angrite parent body; UPB = ureilite parent body. Elemental abundance ratio for the Earth sourced from \citet{McDonoughSun1995,Javoyetal2010,PalmeONeill2014} while the data for meteorites are from \citet{MasonWiik1962,Vonmichaelisetal1968,Ahrensetal1973,ConsolmagnoDrake1977,Dreibusetal1977,Hertogenetal1977,Morganetal1978,Jagoutzetal1979,WattersPrinz1979,Jarosewichetal1987,WassonKallemeyn1988,Jarosewich1990,Kallemeynetal1991,Kallemeynetal1994,Kallemeynetal1996,Kongetal1997,Mittlefehldtetal1998,Goodrich1999,Longhi1999,Brownetal2000,WolfPalme2001,Greenwoodetal2010,Bischoffetal2011,Strackeetal2012,Blinovaetal2014,Hewinsetal2014,Palmeetal2014,CollinetGrove2020,YoshizakiMcDonough2020}.}
\label{fig:elements}
\end{figure}
From the perspective of the major elemental abundances, the combination of chondritic (and perhaps achondritic) meteorites has been reported to be incapable of reproducing the Mg/Si ratio of the terrestrial planets \citep[e.g.][]{DrakeRighter2002,Dauphasetal2015MgSi} because the Earth has an end-member composition that is unlike any of the known meteorites. Following \citet{DrakeRighter2002} we show in Fig.~\ref{fig:elements} the Mg/Si versus Al/Si elemental ratios for the terrestrial planets and the known meteorite parent body compositions. The CI chondrites are similar in elemental composition to the Sun's photosphere \citep[e.g.][]{AndersGrevesse1989,Lodders2003} and there is little variation amongst the carbonaceous chondrites. The non-carbonaceous meteorites, which formed in the same reservoir as the Earth and Mars, all have lower Mg/Si ratios due to nebular and planetary processes \citep{Larimer1979,Alexander2019}; they all have more or less forsterite- or enstatite-rich compositions \citep{Dauphasetal2015MgSi}.
Possible compositions of the bulk Silicate Earth (BSE) computed from models based on terrestrial rocks and chondrites \citep[e.g.][]{McDonoughSun1995,PalmeONeill2014} place it at an end-member location in Fig.~\ref{fig:elements} with an Mg/Si ratio higher than the known meteorites. An exception is the result of \citet{Javoyetal2010} (blue square labelled with J10 in the figure) which is obtained by assuming that the Earth is made up entirely of enstatite chondrites. This result should however be interpreted with caution as a more recent study examining the $\delta^{30}$Si and Mg/Si composition of the Earth showed that it is similar to the enstatite chondrites isotopically but not chemically \citep{Dauphasetal2015MgSi}.
However, the exact composition of the BSE is in fact uncertain. This is partly due to uncertainties in the composition of the lower mantle and the amount of Si in the core. The models of \citet{McDonoughSun1995} and \citet{PalmeONeill2014} assume a homogeneous composition for the whole mantle (pyrolitic, with Mg/Si $\sim1.3$) while the model of \citet{Javoyetal2010} requires that the upper and lower mantle differ in their compositions. Results from seismic tomography observations of subducted crustal slabs \citep{FukaoObayashi2013}, computations and measurements of mineral elasticities \citep{Wangetal2015,Kurnosovetal2018} support a compositionally homogeneous mantle, although there are alternative proposals for a non-pyrolitic and silicon-enriched lower mantle based on its sound-velocity structure \citep{Murakamietal2012,Mashinoetal2020}. Be that as it may, the Mg/Si ratios of the lower mantle derived from these studies ($\sim 1.0$ for \citealt{Murakamietal2012} and 1.14 for \citealt{Mashinoetal2020}) still lie above the Mg/Si ratio of the known meteorites.
The amount of Si fractionated into the Earth's core is estimated to be between 2 wt\% to 7 wt\% based on geochemical and geophysical constraints \citep[e.g.][]{Badroetal2007,Fitoussietal2009,Moynieretal2020}. The Earth's chemical composition is similar within uncertainties to the carbonaceous chondrites if more than 5 wt\% of the Si is in the core. Nevertheless, this still does not change the fact that the Earth's chemical composition cannot be reproduced by any mixture of known planetary materials, not even the combination of ureilites and CI chondrites as was suggested by \citet{Schilleretal2018,Schilleretal2020} because this mixture would lower the Mg/Si ratio and requires the Earth to have more than 7 wt\% Si in the core. Furthermore, the current meteorite repository is likely incomplete and it is therefore an incautious assumption to make that these represent all the possible compositions of the material in the protoplanetary disc. Given the Earth's end-member composition, it has been suggested that a major component of the Earth $(\sim 40\%)$ could originate from the most refractory part of the inner Solar System that we do not have samples from \citep{Morbidellietal2020,Frossardetal2021} and later mixed with less thermally-processed planetesimals such as the known achondrite and chondrite parent bodies (Fig.~\ref{fig:elements}).
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{pebacc_flowchart.pdf}
\caption{Flow chart summarising the two possible outcomes of applying the pebble accretion mechanism to the inner Solar System based on current understanding and without introducing additional ad-hoc assumptions. Our {\it N}-body simulation results show that regardless of the initial mass in the solid disc, there is no combination of initial conditions that can fulfil the constraints from both cosmochemistry and having sufficient disc mass to form the terrestrial planets.}
\label{fig:flowchart}
\end{figure}
It seems unlikely that assuming the initial composition of the inner Solar System to be very depleted in the isotopes that the C group is enriched in (or vice versa) would alter the outcome of the numerical simulations. From the above discussion on the possibility of reconciling our simulation results with the constraints from cosmochemistry, the evidence appears to be against a strong contribution from pebble accretion in the inner Solar System. Fig.~\ref{fig:flowchart} summarises the main message we want to convey from this work. For pebble accretion as we currently understand it to work for the inner Solar System it would thus be necessary to introduce specific assumptions, such as those invoked by \citet{Levisonetal2015} and \citet{Johansenetal2021}. The need to call upon additional assumptions to make pebble accretion work in the inner Solar System in turn raises the question of whether it is absolutely necessary to use this model to explain terrestrial planet formation, for we already have on hand many other planet formation models which can reproduce the features of the inner Solar System by planetesimal mergers.
\section{Conclusions}
We report the outcome of {\it N}-body simulations investigating the growth of a disc of planetesimals with a range of diameters in the terrestrial planet region when they are subjected to a flux of pebbles originating from the outer Solar System. At the end of the simulations $(t = 4~{\rm Myr})$, we find that planetesimal discs made up of many planetesimals with diameters greater than 300 km will increase their mass by at least few times their initial mass, depending on the temperature of the gas disc and if gas-induced migration effects or pebble fragmentation effects at the snow line are included. This is despite Jupiter almost halting the pebble flux when it reaches its isolation mass. Such a large amount of mass increase contributed by outer Solar System pebbles would imply that the isotopic and chemical composition of the planetesimals will be replaced by the signatures of the pebbles. Planetary bodies formed by the subsequent collisions among these planetesimals are expected to have isotopic compositions similar to the outer Solar System bodies, which is inconsistent with the isotopic dichotomy revealed by currently available meteorite samples \citep{Warren2011}.
On the other hand, if the planetesimals in the inner Solar System are mostly bodies with diameters less than 300 km then the amount of mass increase in the disc is limited. The amount of mass increase ranges from 15\% to about 300\%, depending on the initial mass of the planetesimal disc and assuming that the pebbles fragment into smaller-sized grains at the snow line. The amount of mass accreted in the form of pebbles is, however, still considered to be quite high based on the results of mixing models that the maximum contribution of outer Solar System material to the mass of the Earth is about 10\% \citep{Dauphas2017}. In the case where we disregard the fragmentation effects, the mass increase in the disc can be less than 10\% the initial disc mass. However, the more important problem for these discs with small planetesimals is that the final mass in the disc by the time the gas disc dissipates is insufficient to form the terrestrial planets.
It is more likely that the solid disc in the inner Solar System was more massive than $10^{-3}~M_{\rm E}$ and thus should contain more planetesimals larger than 300~km in diameter. Since the growth of these planetesimals proceeds at a faster rate due to their higher accretion efficiency, it is expected that the contribution by pebbles to the mass in the terrestrial planet region will be very large. Even if we were to consider possible end-member compositions for the pebbles and the inner Solar System planetesimals, we still find it difficult for our simulation results to reconcile with cosmochemistry constraints. We therefore suggest that it is rather unlikely for pebble accretion as we currently understand it to play a major role in the formation of the terrestrial planets.
Compared to the work of \citet{Lambrechtsetal2019}, our results for the case of $M_{\rm disc,i} = 0.25~M_{\rm E}$ and without fragmentation (the only case for which a direct comparison can be made) are similar: the planetesimal disc recorded a more than ten times increase in its total mass during the lifetime of the gas disc. \citet{Lambrechtsetal2019} suggested that when Jupiter reached its pebble isolation mass it would stop the inflow of pebbles to the inner Solar System, but we show here that the mass in pebbles that entered the inner Solar System before it was shut off by the growing Jupiter already `contaminates' the inner Solar System with too much outer Solar System material. This comports with the results of \citet{BrasserMojzsis2020}. Our results are, however, at odds with the works of \citet{Levisonetal2015} and \citet{Johansenetal2021} mainly because of the different assumptions made. We assumed that the pebbles formed in the outer Solar System beyond the orbit of Jupiter whereas these previous studies assume for example, that (a) the pebbles form in the inner Solar System \citep{Levisonetal2015}, or that (b) there was a change in the isotopic composition of the pebbles from non-carbonaceous-chondrite-like to carbonaceous-chondrite-like within the lifetime of the gas disc \citep{Johansenetal2021}. As the pebble accretion mechanism is widely studied in the community, our understanding of it is gradually improving and this will provide opportunities for further examinations of the plausibility and validity of the assumptions used by current works.
\section*{Acknowledgements}
J.M. and R.B. thank Tommy Lau and Man Hoi Lee at the University of Hong Kong for their parallelised version of the SyMBA {\it N}-body code. J.M. acknowledges the support of the DFG priority program SPP 1992 ``Exploring the Diversity of Extrasolar Planets'' (BI 1880/3-1). R.B. and S.J.M. thank the Research Centre for Astronomy and Earth Sciences (Budapest, Hungary) for support of the Origins Research Institute (ORI). We thank the reviewer for comments which helped improve the clarity and flow of the manuscript.
\section*{Data Availability}
The data underlying this article will be shared on reasonable request to the corresponding author.
\bibliographystyle{mnras}
|
1,116,691,500,865 | arxiv | \section{Nomenclature}
\vspace{-8 mm}
{\renewcommand\arraystretch{1.0}
\noindent\begin{longtable*}{@{}l @{\quad=\quad} l@{}}
$\mathbf{u}$, $p$ & Fluid velocity and pressure\\
$P$& Airfoil surface pressure time series\\
$C_d, C_l$& Drag and lift coefficients\\
$q$ & Smoothed drag coefficient \\
$\omega$ & Extreme event rate \\
$\tau$ & Load time for prediction of $q$ \\
$m$ & Mass matrix from spectral element grid \\
$w$ & Weights for proper orthogonal decomposition and flow reconstruction \\
$\mathcal{C}$ & chord-length \\
$\mathcal{L}_\bullet$ & Loss function used to train neural network $\bullet$ \\
$dt$ & Time step \\
$Re$ & Reynolds number \\
$u_\infty$ & inlet velocity\\
$\Phi \Sigma \Psi^T$ & components of proper orthogonal decomposition of flow field \\
$\xi$ & Latent space representation of flow field in full-field neural network\\
$r$ & Rank of low dimensional representations of flow\\
$G$ & Network for estimating proper orthogonal decomposition time series \\
$E$ & Encoder network for flow reconstruction \\
$D$ & Decoder network for flow reconstruction \\
$F_{p/\psi / \xi}$ & Non-dynamic models for mapping $P / \psi / \xi$ to $q$ \\
$H_{p/\psi / \xi}$ & Dynamic models for variables $P$, $\psi$, and $\xi$\\
$S, R,F_1,\alpha$ & Precision, recall, $F_1$ score, and area under S-R curve\\
$F_{1,opt}$ & Optimal test set $F_1$ score\\
$\alpha^*$ & Maximum adjusted area under precision-recall curve\\
\end{longtable*}}
\section{Introduction}
Extreme events are common features in engineering and scientific disciplines including climate, ocean engineering, and fluid structure interaction that are characterized by observables of a dynamical system exhibiting heavy tails \cite{sapsis2020statistics}.
The outlier events populating these tails are of particular interest due to their effects on aerodynamics and fatigue, or other potentially adverse consequences.
However, the rarity and intermittency of such events also makes their prediction challenging.
There has been significant recent interest in sampling strategies \cite{mohamad2018sequential,sapsis2020output,blanchard2021bayesian}, optimization schemes \cite{blonigan2019extreme}, and tailored loss functions \cite{guth2019machine,qi2020using} for the prediction of extreme events.
A common goal of many of the past works on extreme events and of the present work is the prediction of extreme events in advance of their occurrence.
The focus of this work is on the two-dimensional incompressible flow around an airfoil at low ($\mathcal{O}(10^4)$) Reynolds number.
Dynamics of the flow around the airfoil at this Reynolds regime are highly nontrivial \cite{lissaman1983low} and have been shown to be characteristically different from those at higher Reynolds \cite{kim2011low}.
Previous works using both experimental and computational tools have found that slow moving airfoils exhibit a large range of wake behaviors, with qualitative changes in the nature of the flow occurring with small changes in angle of attack and Reynolds number \cite{menon2020aerodynamic, gopalakrishnan2017airfoil, wang2014turbulent}.
Similar unstable behavior has been observed in the flow around a cylinder, in the so called transitional regime between ordered and disordered behavior \cite{williamson1996vortex} as well as in vortex-induced-vibrations of flexible cylinders \cite{modarres2011chaotic}.
Due to these instabilities, as well as an apparent lack of fidelity between computational and experimental results, it has been suggested that construction of rigid winged slow flying vehicles may be challenging if not impossible \cite{tank2017possibility}.
Despite apparent challenges, there has long been considerable interest in the study of low Reynolds airfoils \cite{lissaman1983low}.
In particular, recent works have explored the use of machine learning to estimate flow field and aerodynamic data from sensors on the surface of the airfoil.
These include methods for flow reconstruction from limited sensors using neural networks \cite{maulik2020probabilistic, gomez2019unsteady}, filtering based flow estimation \cite{de2017enkf,da2020flow}, as well as prediction of aerodynamic coefficients \cite{dawson2015data}.
Deep learning has also been used for estimating properties of the flow used in low order vortex models.
In \cite{hou2019machine,le2020deep}, authors use neural network based methods to estimate the leading edge suction parameter(LESP).
The model studied in this work has a Reynolds number of 17,500, substantially higher than other computational works which focus on Reynolds number in the range of $\mathcal{O}(10^2) - \mathcal{O}(10^3)$ \cite{le2020deep, de2017enkf, da2020flow, maulik2020probabilistic}.
This is closer to the lower end of the range considered by experimental work \cite{dawson2015data, gomez2019unsteady}.
Unlike some other works \cite{dawson2015data, gomez2019unsteady, hou2019machine, le2020deep}, this work does not study the effects of pitching motions or disturbances in the incident velocity.
We instead focus on prediction of intermittent fluctuations in the aerodynamic coefficients of a stationary airfoil, which have not been the focus of previous works.
This work applies several machine learning techniques to predict fluctuations in the drag coefficient of an airfoil in the transitional regime where we observe chaotic and intermittent behavior. It follows a broader trend of the application of tools from machine learning to problems in fluid dynamics. For a broader view, the interested reader may refer to a number of recent articles outlining and discussing the role of machine learning in fluid dynamics; \cite{brunton2020machine} provides an excellent overview of many of the applications machine learning has seen in fluids, \cite{fukami2020assessment} provides an assessment of several common supervised learning methods applied to flow reconstruction, super-resolution, and coefficient estimation, and \cite{brenner2019perspective} provides a discussion of the role of machine learning in fluids, as well as some pitfalls and concerns.
The paper is organized as follows: In section \ref{sec:problem_description} we formulate the problem of predicting aerodynamic fluctuations.
Section \ref{sec:methods} describes the methodology used in this work, including flow reconstruction methods in section \ref{subsec:flow_reconstruction}, forecasting methods for aerodynamic fluctuations in section \ref{subsec:aero_forecast} and a discussion of performance metrics in section \ref{subsec:error_metrics}. Results are shown in section \ref{sec:results} with comparisons between all methods. In section \ref{sec:discussion} we offer closing thoughts, pitfalls, and potential future directions based on this paper.
\section{Problem description} \label{sec:problem_description}
We consider a NACA 4412 airfoil at chord-length based Reynolds number of $Re=17500$ and $5^\circ$ angle of attack. Flow around the airfoil is simulated using the spectral element code Nek5000 \cite{nek5000-web-page} according to the incompressible Navier-Stokes equations given by,
\begin{equation}
\begin{aligned}
\dfrac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} &= - \nabla p + \dfrac{1}{Re}\nabla^2 \mathbf{u} \\
\nabla \cdot \mathbf{u} &= 0 .
\end{aligned}
\label{eq:incomp_ns}
\end{equation}
The computational grid uses 4368 elements with spectral order 7. Statistics of observables relevant to this work were found to be in agreement with those from a shorter simulation using a more resolved grid having 14144 elements. A convective boundary condition is used for the outflow \cite{dong2015convective}.
The spectral element grid without Gauss-Lobatto Legendre interpolation points and a snapshot of the vorticity are shown in the top row of Figure \ref{fig:grid_vort}.
Further details of the numerical simulation are provided in Appendix A and software for reproducing data used in this work is available online.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{grid_vort}
\caption{Left: Domain of computational problems showing outlines of spectral elements without internal interpolation points. Right: Snapshot of vorticity. Bottom: Sensor placement showing indexing from 0-25 along top and 25-49 along bottom of airfoil. Axes on bottom figure not drawn to scale.}
\label{fig:grid_vort}
\end{figure*}
Pressure recordings along the surface of the airfoil $P(t)$ are taken at a discrete set of 50 points around the airfoil at intervals of $dt=0.01$ throughout the simulation.
These locations are shown in the bottom panel of Fig. \ref{fig:grid_vort}.
Aerodynamic force is computed using the pressure and skin friction.
This decomposes into the streamwise $(x)$ and cross-flow $(y)$ directions, defined by:
\begin{equation}
\vec{F}(t) = \oint \boldsymbol{\tau}(t) - p(t)\mathbf{n} \, ds = D(t) \mathbf{e}_x + L(t) \mathbf{e}_y
\label{eq:aerodynamic_force}
\end{equation}
where $\boldsymbol{\tau}$, $p$, and $\mathbf{n}$ are the skin shear stress, pressure, and wall normal vector, and the integral is taken over the airfoil surface. Forces are then used to compute the non-dimensional drag-coefficient $C_d$ and lift coefficient $C_l$, defined as:
\begin{equation}
C_d(t) = \dfrac{2D(t)}{\rho u_\infty^2 C},\hspace{1 cm} C_l(t) = \dfrac{2L(t)}{\rho u_\infty^2 C},
\label{eq:aerodynamic_coefficients}
\end{equation}
where choord length $C=1$, density $\rho=1$, and free-stream velocity $u_\infty = 1$.
The two dimensional simulation yields a quasi-stable behavior in which intermittent fluctuations are observed in the aerodynamic coefficients, shown in the first two rows of Figure \ref{fig:QOI_fig} alongside the density functions of their absolute deviations.
We note that the density functions clearly exhibit the expected ``heavy tails'' associated with observables of dynamical systems with extreme events \cite{sapsis2021statistics}.
This regime of intermittent fluctuations is persistent for the entirety of the simulation used in this work, but with altered conditions may exhibit mode switching to a state with more regular oscillations.
Further details of this case are given in Appendix C.
In this work we focus solely on the regime where intermittent fluctuations are observed.
The goal of the present work is to predict these intermittent fluctuations in the aerodynamic coefficients using information regarding the surface pressure of the airfoil in advance of their occurrence by some lead time $\tau$.
To focus on non-periodic behavior, we consider predictions on a smoothed time series derived from the drag coefficient.
Specifically,
\begin{equation}
q(t) = \left( C_d (s) \ast \mathcal{N}\left(s \left| 0,\frac{1}{\left(2f_{peak}\right)^2} \right.\right) \right) (t),
\label{eq:q_definition}
\end{equation}
where $f_{peak} \approx 1.44$ is the peak frequency of the drag coefficient.
In practice, the convolution in Eq. \eqref{eq:q_definition} is computed using a compactly supported kernel having width $3/f_{peak}$.
The time series $q(t)$ is also normalized to be mean zero and unit variance.
The quantity $q(t)$ captures the non-periodic behavior of the drag coefficient, in particular maintaining extreme events and the heavy tailed deviation.
The goal of this work is thus concisely stated as learning data-driven models for the prediction problem,
\begin{equation}
P(s \leq t) \to q(t+\tau),
\label{eq:prediction_problem}
\end{equation}
at various $\tau \geq 0$.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{QOI_fig.png}
\caption{Left: Aerodynamic coefficients $C_d$ and $C_l$, and smoothed drag coefficient $q$ as defined in Eq. \eqref{eq:q_definition}. Red dashed lines indicate $\pm 2$ standard deviations away from the mean. Right: Histograms of the deviation of each quantity showing typical heavy tails of systems with extreme events.}
\label{fig:QOI_fig}
\end{figure*}
\section{Methods} \label{sec:methods}
In this section we formulate several data-driven models for Eq. \eqref{eq:prediction_problem}.
We assume knowledge of measurements of pressure at a discrete set of points along the surface of the airfoil up to the current time $t$ and seek to predict the value of the extreme event indicator at time $t+\tau$.
We denote the time series for pressure measurements up to the current time $t$ as $P_t = P(s \leq t)$.
The minimal problem is thus learning a function directly from historical pressure measurements to the $q(t+\tau)$, which may be done using a variety of standard machine learning tools.
\begin{figure*}[h]
\centering
\includegraphics[width=\textwidth]{airfoil_schematic.pdf}
\caption{Schematic of methods for EE prediction showing various flows of information from assumed knowledge of historical pressure measurements $P_t$ to future value of quantity of interest $q(t+\tau)$. Dynamic model for pressure measurements has been omitted.}
\label{fig:airfoil_schematic}
\end{figure*}
A baseline predictor for Eq. \eqref{eq:prediction_problem} is found using a standard recurrent neural network, such as a long-short-term-memory networks (LSTM) \cite{hochreiter1997long}, to interpolate a function directly from $P_t$ to $q(t+\tau)$.
Alternatively, we may try to improve forecasts of $q(t)$ through the offline use of flow field data.
Previous works \cite{blonigan2019extreme} have used modal representations of a flow field combined with adjoint equations to learn precursor states to extreme events.
In this work we discuss two methods for compressing flow field data, sensing expansion coefficients in the compressed basis, and exploiting this knowledge for potentially improved prediction of $q(t+\tau)$: the proper orthogonal decomposition and neural network based flow reconstruction.
In each case we study the predictive capability of initializing a flow solver with the reduced order initial condition, predicting directly from historical representations in the reduced space, and neural network based reduced order models.
We also consider data-driven dynamic models for the pressure measurements.
The methods considered in this work are summarized in Figure \ref{fig:airfoil_schematic}.
Methods are separated into an offline compression stage, a sensing stage where we infer the reduced order state from point pressure measurements, reconstruction of the flow field from the reduced order state, and finally forecast of the quantity of interest, $q(t)$.
Learned functions $G$, $E$, and $D$ are predictors of the POD mode time series, latent space encoder, and flow reconstruction network, respectively.
The letter $F$ has been used to denote LSTM predictions that do not use a dynamic model, with subscript indicating input.
$H$ is used to denote neural network based reduced order models.
In each case, we use time series for pressure at discrete points on the surface of the airfoil as the starting point of the online prediction.
Details on each method are provided in the following sections.
We assume familiarity with common implementations of neural networks and stochastic optimization.
In particular, the work in this manuscript makes frequent use of deep LSTMs \cite{hochreiter1997long} and the Adam method for optimization \cite{kingma2014adam}.
The unfamiliar reader may find an excellent reference in the free online textbook \cite{goodfellow2016deep}.
Further details on the structure and implementation of neural networks is given in Appendix B.
\iffalse
We also note that the term reduced order model (ROM) will be used in a slightly different manner than is usually seen. ROMs are often understood as Markovian models in a reduced space created via projections of the known dynamics \cite{barrault2004empirical,chaturantabut2009discrete,benner2015survey,rowley2004model} or by operator inference \cite{peherstorfer2016data} or other machine learning based approaches. Here, we consider bases that fall far short of being sufficient to capture the full dynamics and forgo attempts to construct Markovian models in the reduced basis. However, we are still able to learn accurate forecasting models in the reduced space via using LSTMs capable of short term forecasts. The use of LSTMs has also been explored for closure modeling and other non-Markovian dynamical systems \cite{wang2020recurrent,gupta2020neural}.
\fi
\subsection{Flow field compression and reconstruction}\label{subsec:flow_reconstruction}
We begin with a discussion of the two offline methods for flow reconstruction: the proper orthogonal decomposition (POD) and an LSTM based encoder-decoder pair.
In this sub-section we provide an overview of the methodology used to form the reduced rank representation for each of these two cases as well as methods for approximating time series associated with each POD mode.
\subsubsection{Proper orthogonal decomposition and sensing}\label{subsub:PODFR}
The proper orthogonal decomposition is a standard tool for decomposing a flow field into spatial modes that are orthogonal with respect to a given inner-product and whose time evolution are also orthogonal \cite{holmes2012turbulence,taira2017modal}. We apply the POD to the velocity field around the airfoil using a weighted inner product. Specifically, the POD finds matrices $\Phi$, $\Sigma$, and $\Psi$ which are discretized solutions to,
\begin{equation}
\mathbf{u} (\mathbf{x},t) - \overline{\mathbf{u}}(\mathbf{x}) = \Phi(\mathbf{x}) \Sigma \Psi(t)^T,
\label{eq:POD}
\end{equation}
where,
\begin{equation}
\begin{aligned}
& \Sigma = diag(\sigma_1, \hdots, \sigma_r), \,\, \sigma_i \geq \sigma_{i+1}\geq 0\\
&\langle \phi_i, \phi_j \rangle_w = \int_\Omega \phi_i(\mathbf{x})^T\phi_j(\mathbf{x})w(\mathbf{x})\,dx = \delta_{i,j} \\
&\langle \psi_i, \psi_j \rangle_w = \int_0^T \psi_i(t)\psi_j(t)\,dt = \delta_{i,j},
\end{aligned}
\label{eq:POD_constraints}
\end{equation}
and $w(\mathbf{x}) > 0$ is a weight used to focus the inner product, and thus variation explained by POD, on regions of interest near the surface of the airfoil. A similar weighted approach was used in \cite{blonigan2019extreme} where a wall focused POD was used as a basis for predicting extreme dissipation events in channel flow. We use the family of weights given by,
\begin{equation}
w(\mathbf{x}) = \frac{1-\epsilon}{1+e^{(d(\mathbf{x}) - l)/\delta}} + \epsilon
\label{eq:weights}
\end{equation}
where $d(\mathbf{x})$ is the distance from $\mathbf{x}$ to the surface of the airfoil. Equation \eqref{eq:weights} describes a smooth sigmoidal curve that decays from 1 (assuming $l \gg \delta$) at the airfoil surface to $\epsilon$ in the far field. For small $\delta$, this transition is localized around $d=l$ and the weights are approximately 1 for $d < l - \delta$ and $\epsilon$ for $d > l + \delta$. We use parameter values $l=1$, $\delta=0.1$, and $\epsilon = 0.1$. The POD is therefore principally focused on describing variation in the velocity field within one chord length of the airfoil surface, with approximately one tenth the weighting for variation outside this region.
In the case where $w(\mathbf{x})=1$ the POD is equivalent to the singular value decomposition of the mean subtracted data, also knwon as principal component analysis.
For non-identity weights, $\Sigma$ and $\Psi(t_j)$, $j=1,\hdots, m$ are given by the eigenvalue decomposition of $\mathbf{U}^T\mathbf{W}_m\mathbf{U}$ where $\bar{\mathbf{U}}\in \mathbb{R}^{n\times m}$ is the mean-subtracted velocity data and $\mathbf{W}_m$ is a diagonal matrix with $w(\mathbf{x}_i) m(\mathbf{x}_i)$ along the diagonal where $m(\mathbf{x}_i)$ is the mass associated with that grid point for the spectral element grid \cite{holmes2012turbulence}.
Modes $\Phi$ are subsequently computed using their definition in Eq. \eqref{eq:POD}.
Alternative methods may compute $\Phi$ before $\Sigma$ and $\Psi$, but these suffer from numerical issues for $\epsilon \ll 1$.
\begin{figure*}[h]
\centering
\includegraphics[width=\textwidth]{POD_modes}
\caption{The mean velocity field and three POD modes. Axes not drawn to scale.}
\label{fig:POD_modes}
\end{figure*}
Applying POD to the airfoil data yields modes $\phi_i$, shown in Fig. \ref{fig:POD_modes}, singular values $\Sigma$, and time series $\psi_i(t)$ corresponding to each mode $\phi_i$. In the online phase of any prediction method, we will only have access to $P_t$, not $\psi_i(t)$. The latter may be estimated from sparse or gappy measurements \cite{venturi2004gappy}. We therefore train a deep LSTM model to estimate the current POD representation of the flow from pressure measurements. Letting $\psi_r(t)$ be the rank $r$ truncation of the time series of the POD expansion we have,
\begin{equation}
\hat{\psi}_r(t) = G(P_t).
\label{eq:G_P_to_Psi}
\end{equation}
The exact form of and training procedure for $G$ is described in greater detail in Appendix B.
True values of POD coefficients as well as there estimates from pressure via Eq. \eqref{eq:G_P_to_Psi} are shown in Fig. \ref{fig:POD_reconstruction}. Note that reconstructed time series are filled in on a much denser grid than true values since they are computed from the finely sampled pressure time series.
Temporal resolution on $\psi(t)$ is limited by the number of output files saved during numerical simulation as well as memory limitations in the computation of the POD.
It is worth noting that the time series $\psi(t)$ are normalized to unit variance and $\Sigma$ is not considered in loss function.
Hence, error in higher modes is treated the same as error in lower modes.
Higher modes did still have higher error, perhaps because they tended to exhibit more chaotic behavior.
The authors did not explore the loss function exhaustively since doing so would be a significant research endeavor on its own.
\begin{figure*}[h]
\centering
\includegraphics[width=\textwidth]{POD_reconstruction}
\caption{Flow field reconstruction using LSTM to predict time series for first 32 POD modes using pressure measurements, $P_t$. Top [Operator $G(P_t)$]: LSTM prediction of POD time series for 1$^{st}$, 10$^{th}$, 20$^{th}$, and 30$^{th}$ modes (red) and sparser true values (blue). Bottom [Operator $\Phi \Sigma G$]: POD reconstruction of streamwise velocity compared to two true fields within test set data.}
\label{fig:POD_reconstruction}
\end{figure*}
After prediction of $\hat{\psi}(t)$ using $G$ one may reconstruct an approximation of the full flow field using Eq. \eqref{eq:POD}.
The lower two rows on Fig. \ref{fig:POD_reconstruction} show the results of this in the streamwise direction as well as the true velocity and absolute error for two snapshots taken from the testing dataset, i.e. snapshots not seen by the optimization algorithm used to learn $G$. These reconstructions suffer from multiple sources of error. Expanding the difference between true and reconstructed fields, we get,
\begin{equation}
\begin{aligned}
\mathbf{u}(t) - \hat{\mathbf{u}}_{POD}(t) &= \Phi\Sigma\psi (t)^T - \Phi_r\Sigma_r G(P_t)^T \\
&= \underbrace{\Phi_{-r}\Sigma_{-r}\psi_{-r} (t)^T}_{\text{Unresolved}} - \Phi_r\Sigma_r (\underbrace{\psi_r(t) - G(P_t)}_{\text{NN Error}} )^T,
\end{aligned}
\label{eq:POD_reconstruction_error}
\end{equation}
where the subscript $-r$ denotes those components not in the first $r$. Error in the neural network increases for modes with high frequency and chaotic behavior, such as mode 30 shown in Fig. \ref{fig:POD_reconstruction}.
However, significant error is also incurred from unresolved modes due to the slow decay of singular values $\Sigma$.
While spatially periodic translational behavior may be represented with pairs of modes (see Eq. (34) in \cite{taira2017modal}) the system studied in the present work tends to shed isolated pairs of vortices.
Moreover, it also exhibits intermittency, which may be difficult to capture in a POD basis.
The authors are not aware of efficient linear methods for representing translation of sparse structures such as the wake vorticies observed in this data.
\subsubsection{Neural network flow reconstruction}\label{subsub:NNFR}
In light of some of the deficiencies of the POD and motivated by the successful applications of neural networks to problems in fluid dynamics \cite{milano2002neural,lu2021learning,raissi2019physics,brunton2020machine,fukami2020assessment} we also consider neural network based approaches to flow reconstruction.
The approach in this work is to use an LSTM-based encoder coupled with a fully connected network predicting the velocity at each grid point.
The fully connected decoder resembles that used in \cite{erichson2020shallow} and \cite{maulik2020probabilistic} though do to the large computational grid we have not used the probabilistic formulation considered in the later.
We note however, that neither of the aforementioned papers included the use of history terms in their prediction of the fluid state, as we will show is done by the LSTM encoder for the networks considered in this work.
For brevity, we will call networks of this type full field neural networks (FFNN), indicating that the output of the network is the values of fluid velocity at each grid point used by a solver.
As in the POD case, we represent the fluid velocity around an airfoil at time $t$ using a low dimensional representation, $\boldsymbol{\xi}(t) \in \mathbb{R}^r$. The time history of the pressure sensors is encoded to the state using an LSTM given by,
\begin{equation}
\boldsymbol{\xi}(t) = E (P_t).
\label{eq:LSTM_encoder}
\end{equation}
Since the full state of the flow field is encoded in $\boldsymbol{\xi}(t)$, one may tune the dimension of $\boldsymbol{\xi}$ to acquire a desired rank for the reduced order representation of $\mathbf{u}$.
We found improvements in reconstruction accuracy up to approximately rank $r=32$, with minimal improvement at high values.
We therefore use $r=32$ for the remainder of this work unless noted otherwise.
The reconstructed velocity field is then given by,
\begin{equation}
\hat{\mathbf{u}}(\mathbf{x}_i, t_m) = D(\boldsymbol{\xi})_i,
\label{eq:grid_decoder}
\end{equation}
where $D$ is a standard fully connected neural network with the final layer being linear. Taken together, Eq. \eqref{eq:LSTM_encoder} and \eqref{eq:grid_decoder} form a recurrent network from pressure sensor time histories to the fluid velocity given by,
\begin{equation}
\hat{\mathbf{u}}(x_i, t_j) = D \circ E (P(t_m))_i,
\label{eq:nn_composition}
\end{equation}
which are trained together using numerical simulation data. Further details on the network structure may be found in Appendix B.
We train the networks using a weighted square-error loss function designed to favor accurate reconstruction near the wall boundary;
\begin{equation}
\begin{aligned}
\mathcal{L}_{E,D} &= \sum_j \int_\Omega \left\| u(\mathbf{x}, t_j) - \hat{u}(\mathbf{x}, \boldsymbol{\xi}) \right\|^2 w(\mathbf{x}) \, dx,
\end{aligned}
\label{eq:cotinuous_net_loss}
\end{equation}
where $\Omega$ is the computational domain and $w(\mathbf{x})$ is as defined in Sec. \ref{subsub:PODFR} using $\epsilon = \delta = 0.1$ and $l = 1$. It is also possible to weigh the loss function to more heavily penalize errors in times immediately preceding an extreme event, but doing so was observed to have little effect on prediction performance. In practice, the integral in Eq. \eqref{eq:cotinuous_net_loss} is approximated using the mass matrix $m$ obtained from the spectral element grid. The expression simplifies to a simple weighted sum of squares error given by,
\begin{equation}
\begin{aligned}
\mathcal{L}_{E,D} &\approx \sum_j \sum_i \left\| \mathbf{u}(\mathbf{x}_i, t_j) - \hat{\mathbf{u}}(\mathbf{x}_i, \boldsymbol{\xi}) \right\|^2 w(\mathbf{x}_i) m(\mathbf{x}_i).
\end{aligned}
\label{eq:discrete_net_loss}
\end{equation}
\begin{figure*}[h]
\centering
\includegraphics[width=\textwidth]{FFNN_reconstruction}
\caption{ [Operator $D \circ E (P_t)$] Reconstruction of streamwise velocity ($u_\infty = 1$ subtracted) using FFNN for flow compression with 32 dimensional latent space and LSTM to map the pressure measurements, $P_t$, to the latent variables, $\boldsymbol{\xi}$. Note maximum error is approximately half that of POD, and regions of high error are significantly more localized.}
\label{fig:FFNN_reconstruction}
\end{figure*}
Snapshots of the true streamwise velocity, the FFNN predicted streamwise velocity, and absolute error are shown in Fig. \ref{fig:FFNN_reconstruction}, which includes the same fields shown in Fig. \ref{fig:POD_reconstruction} to illustrate POD reconstruction.
Note that the maximum absolute error using neural networks is considerably lower than that of POD and that regions of high error are more localized.
This clearly indicates the superior performance of FFNN to POD for flow reconstruction in this particular case.
The choice to use a very large output layer of the neural network makes predictions specific to the particular grid used in training, though interpolation schemes could be used in other cases.
The FFNN decoder's size also makes it highly memory intensive, which limits batch size in training.
It is plausible that similar networks for three dimensional flows would require coordinate descent like algorithms where fractions of the output weights are updated on any given batch.
This is in contrast to operator type networks, where spatial coordinate $x$ is given as input \cite{lu2021learning}.
Methods based on the latter were implemented without physical constraints for flow reconstruction from pressure measurements, but were found to underperform the full-field neural networks discussed in this work.
We note that this could be in part due to using a neural encoder of pressure measurements rather than sparse function evaluations, as used in \cite{lu2021learning}.
It is also possible that the use of physics-informed methods could improve prediction accuracy of the operator type networks and this is noted as a potentially interested research question.
However, we consider such an approach to be outside of the scope of this work.
\subsection{Forecasting aerodynamic fluctuations}\label{subsec:aero_forecast}
We now consider online methods for Eq. \eqref{eq:prediction_problem}.
We separate these methods into two broad categories; those that do not use dynamic models, and those that use dynamic models such as the Navier Stokes equations or data-driven dynamic models.
\subsubsection{Non-dynamic methods}\label{subsec:single_step}
We first consider methods that do not employ any sort of dynamic model. These are simply interpolations a function from the time history of an input quantity to the future value of $q(t)$.
The general form is given by,
\begin{equation}
q(t + \tau) = F_\bullet (\bullet_t),
\label{eq:general_single_step}
\end{equation}
where $\bullet \in [P, \psi, \xi]$ and $F$ is a deep LSTM mapping some time series of historical measurements to the future quantity of interest. In the case where the reduced representation of the flow state is used we have:
\begin{equation}
\begin{aligned}
q(t + \tau) &= F_\psi \circ G (P_t) \\
q(t + \tau) &= F_\xi \circ E (P_t),
\end{aligned}
\label{eq:non_dynamic_feature_maps}
\end{equation}
so the POD sensing network $G$ and neural network encoder $E$ may be considered as feature maps for the forecasting networks $F_{\psi / \xi}$. Each of $F_{P/\psi/\xi}$ are trained using the mean square error loss function. Other loss funcitons more specific to extreme events were considered but found to make negligible difference to the resulting trained network.
Non-dynamic methods of the form given by Eq. \eqref{eq:general_single_step} may be favorable for several reasons.
They are simple, easy to train, and based on the ubiquitous and highly effective LSTM network structure.
They may be of particular interest due to the computational savings offered by avoiding dynamic models.
Approximation of $q(t+\tau)$ may be rapidly computed from the state of the LSTM, which is updated online from streaming data.
However there are also downsides to the non-dynamic approach.
As $\tau$ becomes large, we are approximating larger steps of a chaotic time series.
The problem thus becomes very sensitive to inputs, and balancing sufficient model complexity with overfitting becomes challenging.
Moreover, prediction of $q(t+\tau)$ for any given lead time $\tau$ requires its own trained network.
\subsubsection{Full order dynamical model based prediction methods}\label{subsub:CFD_prediction}
Previous work predicting extreme events for turbulent flows has used a low dimensional representation of the flow as initial condition for a fluid simulation and employed the adjoint to evaluate the gradient of the future quantity of interest with respect to coefficients in the low dimensional expansion \cite{blonigan2019extreme}.
This is possible in cases where the low order initial condition is sufficiently close to the full order state to track its behavior.
Here we evaluate if this is the case for low dimensional initialization from both the POD basis and FFNN.
Numerical simulations are initialized using either the FFNN based reconstructed velocity or the low dimensional reconstruction using the known POD modes and singular values, along with the estimated temporal coefficients $\hat{\psi}(t) = G(P_t)$.
Future values of $q(t)$ are computed using the same convolution as in Eq. \eqref{eq:q_definition};
\begin{equation}
\begin{aligned}
\hat{q}(t+\tau) &= \left( \hat{C}_d(s; \hat{\mathbf{u}}) \ast \mathcal{N}\left(s \left| 0,\frac{1}{\left(2f_{peak}\right)^2} \right.\right) \right) (t+\tau) \\
\text{where, } &\hat{\mathbf{u}} (t) = D(\xi) \text{ or } \hat{\mathbf{u}} (t) = \Phi \Sigma G(\xi),
\end{aligned}
\label{eq:CFD_forecast}
\end{equation}
where $\hat{\mathbf{u}}(s), s\in[t,t+\tau] $ is found via numerical integration of the Navier-Stokes equations.
Since the time series for $\hat{C}_d$ is much shorter than the full simulation used for training neural networks, the common practice of zero padding distorts the values of $\hat{q}$ near the endpoints.
We instead truncate and re-normalize the smoothing kernel.
For valid comparison, the same is done for the true drag coefficient truncated to the interval $[t, t+\tau]$.
Following Eq. \eqref{eq:CFD_forecast}, time series for $q$ and $\hat{q}$ are normalized using the mean and variance of the full length time series defined in Eq. \eqref{eq:q_definition}.
\subsubsection{Data-driven dynamical model based prediction methods}\label{subsec:xi_rom}
We also consider forecasts of either i) the pressure signal, $P(t)$, or ii) the reduced order state of the fluid flow, using data-driven dynamic models.
In the later case we formulate and evaluate reduced order dynamical models for both the reduced POD state and the latent representation learned by the FFNN.
For the POD ROM, we train the dynamic model on the estimated POD coefficients ($\hat{\psi} = G(P_t)$), instead of the true coefficients $\psi$, since the former may be found exactly from airfoil surface pressure.
This eliminates error due to imprecise initial conditions.
Learning a data-driven model of the pressure signal along the surface of the airfoil has, to the best of the authors knowledge, not been used in previous works.
Forecasting models in each case are LSTM networks mapping historical measurements of a given quantity ($P/\hat{\psi}/\xi$) to its value $\kappa$ timesteps in the future. They are represented as,
\begin{equation}
P(t_{j+\kappa}) = H_P (P(t_j), P(t_{j-\kappa}), \hdots)),
\label{eq:basic_lstm}
\end{equation}
with similar networks for $\hat{\psi}$ and $\xi$.
Here $\kappa$ is taken to be three in order to alleviate some of the numerical difficulties with training data-driven dynamic models in the small timestep limit \cite{levine2021framework}.
Thus, the LSTM maps historical measurements of the dynamic quantity $P$ to the value of that quantity three steps in the future: $P(t_{j+3})$.
Since $P$ is measured every $dt=0.01$ time units, the LSTM is effectively a dynamic model for $P$ with timestep equal to $\kappa \, dt = 0.03$.
Networks are trained using the mean square error loss over a prediction window of 20 steps;
\begin{equation}
\mathcal{L}_{H_P} = \sum_i \sum_{j=1}^{20} \| H_P^j(P_i) - P(t_{i+\kappa j})\|^2
\label{eq:lstm_loss}
\end{equation}
where we define composition of $H_P$ with itself by,
\begin{equation}
\begin{aligned}
H_P^1 (P_i) &= H_P (P(t_i), \,P(t_{i-\kappa}), \,P(t_{i-2\kappa}), \,\hdots), \\
H_P^2 (P_i) &= H_P (H_P^1 (P_i), \, P(t_{i}), \,P(t_{i-\kappa}), \,\hdots), \\
H_P^3(P_i)&= H_P (H_P^2 (P_i), \, H_P^1 (P_i), \,P(t_{i}), \,\hdots)\,\,\hdots .
\end{aligned}
\label{eq:lstm_composition}
\end{equation}
The sum over index $i$ in Eq. \eqref{eq:lstm_loss} is taken over all initial times in the training dataset. The same network structure and loss function is used for the full set of 50 pressure measurements as well as the 32 dimensional reduced order representations of the flow field. Layer sizes are scaled to account for the difference in dimension between $P$ and $\psi / \xi$.
For any $\tau \geq 0$, we obtain the estimated forecast,
\begin{equation}
\hat{P}(t+\tau) = H_P^{m_\tau}(P_t),
\label{eq:LSTM_forecast}
\end{equation}
where $m_\tau = \tau / (\kappa dt)$. Following the LSTM based forcast of $P/\hat{\psi}/\xi$ we may use $F_{P/\hat{\psi}/\xi}$ trained for zero lead time to evaluate $\hat{q}(t+\tau) = F_{P} \circ H_{P}^{m_\tau} (P_t)$ and likewise for $\hat{\psi} / \xi$. Since prediction of $q$ with zero lead time is a much simpler problem, we use standard feed-forward neural networks in place of the LSTMs used for non-dynamic predictions with non-zero lead time.
\subsection{Error metrics}\label{subsec:error_metrics}
Each of the neural networks in this work is trained using the mean square error (MSE). The MSE is differentiable and may be evaluated on subsets of the training data, allowing for the use of stochastic optimization schemes run on a graphics processing unit.
However, the MSE may not be a good indicator of success in predicting extreme events.
We have therefore adapted several extreme-event-tailored error metrics to compare the various methods considered in this work.
Specifically, results are compared using the batch relative entropy loss, the maximum adjusted area under the precision-recall curve, and the extreme event rate dependent area under the precision recall curve, and optimal $F_1$ score.
The batch relative entropy loss (BRE) is inspired by the work in \cite{qi2020using}, where authors use a relative entropy loss function to train convolutional neural networks that are capable of making accurate prediction of a system governed by the truncated Korteweg-de Vries (tKdV) equation in regimes with extreme events.
Their work uses empirical partition functions similar to the soft-max activation commonly used in neural networks to transform high-dimensional predictions into probability distribution functions highlighting outlier values.
Loss is subsequently measured loss using the KL-divergence.
This approach was shown to significantly improve prediction accuracy over the MSE on the tKdV problem.
A similar approach was adapted for the present work using partition functions over mini-batches rather than output dimensions.
Specifically, we define the batch relative entropy loss as,
\begin{equation}
BRE = \sum_i z_i \log\left( \dfrac{z_i}{\hat{z}_i} \right)
\end{equation}
where $z_i$, $\hat{z}_i$ are given by the empirical partition functions,
\begin{equation}
z_i = \dfrac{e^{q(t_i)}}{\sum_j e^{q(t_j)}} , \hspace{1cm} \hat{z}_i = \dfrac{e^{\hat{q}(t_i)}}{\sum_j e^{\hat{q}(t_j)}},
\end{equation}
and where the sum is taken over a mini-batch.
We note that this work considers partition functions that weigh positive outliers, as opposed to the symmetric variant used in \cite{qi2020using}, since all events of interest to this work skew positive.
The BRE loss was tested as a means of training the neural networks described in previous sections, but taken over mini-batches was found to perform comparably with the mean square error.
For evaluation, we use the batch relative entropy loss taken over the full testing dataset.
The performance of a predictor of extreme events may also be measured by the ability of that predictor to classify events based on a threshold value of the quantity of interest.
The maximum adjusted area under the precision-recall curve ($\alpha^*$) was proposed in \cite{guth2019machine} as a loss function for predicting extreme events and shown to perform well for predicting extreme dissipation events in Kolmogorov flow and rouge waves in the Majda-McLaughlin-Tabak model.
To define $\alpha^*$ we first introduce the quantity $\omega$ for the extreme event rate and the corresponding threshold $\hat{a}(\omega)$, such that $p(q > \hat{a}(\omega)) = \omega$. Introducing a second threshold $\hat{b}$ for the prediction $\hat{q}$ yields a classifier for which we can compute the precision ($S$ = true positives divided by predicted positives) and recall ($R$ = true positives divided by total positives). Noting that precision is uniquely determined by the extreme event rate and recall, \cite{guth2019machine} computes the area under the precision recall curve,
\begin{equation}
\alpha (\omega) = \int_0^1 S (R,\omega) \, dR = \int_\mathbb{R} S(\hat{b},\omega)\left| \frac{\partial R(\hat{b},\omega)}{\partial \hat{b}} \right| \, d\hat{b},
\label{eq:AUC}
\end{equation}
and the maximum adjusted area under the precision-recall curve as
\begin{equation}
\alpha^* = \underset{\omega}{max} \, \alpha (\omega) - \omega.
\label{eq:alpha_star}
\end{equation}
We compute the integral in Eq. \eqref{eq:AUC} using a finite grid of $\hat{\beta}$ values spread evenly between $min(\hat{q})$ and $max(\hat{q})$ with finite difference scheme to evaluate the derivative term. It's value is bounded between $0$ and $1$ but occasionally falls slightly outside this range due to numerical issues and is subsequently clipped. The maximization in Eq. \eqref{eq:alpha_star} is taken over a discrete set of samples evenly spaced $\omega$ between $0.01$ and $0.25$.
Finally, we consider the extreme event rate dependent optimal $F_1$ score, defined as the $F_1$ score on the testing dataset using the threshold that optimizes the $F_1$ score on the training and validation datasets. That is,
\begin{equation}
F_{1,opt}(q, \hat{q}, \omega) = F_1 \left( (q_{test} > \hat{a}(\omega)), (\hat{q}_{test} > \hat{b}_{opt}) \right)
\label{eq:F1_opt}
\end{equation}
where,
\begin{equation}
\hat{b}_{opt} = \underset{\hat{b}}{arg max} \, F_1 \left( (q_{train/val} > \hat{a}(\omega)), (\hat{q}_{train/val} > \hat{b}) \right).
\end{equation}
Taken together this yields five metrics of predictor performance. Mean square error, batch relative entropy, and $\alpha^*$ are independent of extreme event rate and yield simple scalar metrics of performance, though do not indicate performance at a particular extreme event rate. The extreme event rate dependent $\alpha(\omega)$ and $F_{1,opt}(\omega)$ each seek to measure a balance between precision and recall at variable extreme event rates. The dependency on extreme event rate allows for a more descriptive quantification of error, since we can compare methods for a variety of extreme event rates.
\section{Results} \label{sec:results}
In this section we present results for each of the methods for Eq. \eqref{eq:prediction_problem}.
Results for prediction using the full order model with reduced order initial condition are kept distinct from those using neural networks.
This is due to the weak performance by the former, as well as its considerable computational expense, which limits the number of trials we use to evaluate it.
For predictions of $q(t)$ using dynamic methods, we apply a smoother to remove higher frequency oscillations.
This smoother weights previous predictions with exponentially desceasing weights and does not use any future prediction beyond $\tau$.
Examples of full test set predictions for $\hat{q}(t)$ using each of the six nerual network based methods are shown in Figure \ref{fig:tau7_time_series} for lead time $\tau = 7.0$ and in Appendix D for other lead times.
\subsection{Simulations with reduced order initial conditions}
We first present results showing the failure of approaches taken by the authors to forecast aerodynamic fluctuations using the Navier-Stokes solver and reduced order initial condition.
For each of a variety of ranks, reduced order initial conditions were formed using the estimated POD time series $\hat{\psi}(t)$ and FFNN flow reconstruction $D \circ E (P_t)$ at 50 evenly spaced times throughout the portion on data reserved for testing.
Examples of the resulting smoothed drag coefficient are shown in Figure \ref{fig:CFD_restart_examples}, which compares predicted time series for $\hat{q}(t)$ using several ranks of POD reconstructions (left column) as well as FFNN based reconstructions (right column) to the true value $q(t)$ (black curves).
Each row represents a different initial condition from within the testing dataset.
We note that all the initializations using different reduction methods/orders exhibit poor agreement with the true time series in at least 1 of the three cases shown.
\begin{figure*}[h]
\centering
\includegraphics[width=\textwidth]{CFD_restart_examples}
\caption{Examples of forecasts of $q(t+\tau)$ using Eq. \eqref{eq:CFD_forecast} with various ranks of POD (left column) and FFNN (right column). Examples show prediction from initial conditions are spaced evenly throughout the testing dataset. True value is shown as solid black line. Each row represents a different realization of the flow. }
\label{fig:CFD_restart_examples}
\end{figure*}
\begin{figure*}[h]
\centering
\includegraphics[width=\textwidth]{CFD_restart_error}
\caption{Mean aboslute error, normalized by standard deviations of true time series for the drag coefficient and $q(t)$ using various ranks of reduced order initial conditions from the estimated POD time series and FFNN.}
\label{fig:CFD_restart_error}
\end{figure*}
The mean absolute error in the drag coefficient and smoothed drag coefficient $q$ for each rank are shown in Figure \ref{fig:CFD_restart_error}.
There are several noteworthy features;
the approximated values of $C_d$ for both methods are initially fairly accurate, falling within $10 \%$ of the true values, but very quickly diverge.
By the end of the $10.5$ time unit simulations, the error in each quantity is roughly as large as the standard deviation of the true data.
Initial error in $q(t)$ is higher, as is expected since this quantity includes some information from future estimates of $C_d$.
We also note the lack of noticable correlation between the rank of the latent space used for flow reconstruction and prediction accuracy.
In the POD case, it appears that the lowest rank ($r=8$) performs poorly compared to higher ranks, but $r=64$ is not uniformly better than $r=16$ or $r=32$.
A plausible explanation for this is that as the rank is increased, so too is the difficulty of the sensing problem outlined at the end of Sec. \ref{subsub:PODFR}.
Indeed, using the true (non-reconstructed) $\psi(t)$ we see improved agreement with full order results if $r$ is let to become large.
The connection between rank and accuracy for the FFNN examples is more opaque.
While higher rank initial conditions are slightly more accurate at predicting the initial drag, the difference is small and decays quickly as the simulation progresses.
While the results here are not sufficiently exhaustive to preclude the use of a full order model for predicting extreme events in the flow around an airfoil they at least provide strong evidence of its difficulty.
We include them to show the sensitivity of the problem to small changes in the initial condition (note that the FFNN was able to accurately reconstruct the fluid velocity) and to contrast to other examples in fluids where a representation in low rank basis was found to be an effective precursor to extreme events \cite{blonigan2019extreme}.
\subsection{Comparison of reduced order dynamic methods}
We now consider the data-driven dynamic models for the pressure signals and reduced order representations of the flow field.
Figure \ref{fig:ROM_error_histograms} shows the time evolution of the mean square error in the dynamic quantity and absolute error in prediction of $q$ for each of the three methods $H_{P/ \psi /\xi}$.
The dynamic quantities in each case have been normalized to unit variance so direct comparison of error magnitudes across methods and across indices are meaningful.
Error statistics have been binned for each timestep and normalized such that vertical slices of any subplot are density functions across all examples from the testing dataset. Apparent recurring features in the error plots for $q$ are likely due to the methods missing the same feature from multiple closely sampled initial conditions.
\begin{figure*}[h]
\centering
\includegraphics[width=\textwidth]{ROM_error_histograms}
\caption{Test set empirical probability density functions of the mean square error of the LSTM forecast (left) and absolute error in predicting $q(t+\tau)$ for $50$ dimensional data-driven forecast of pressure measurements (top), and $32$ dimensional reduced order models using POD modes (middle) and FFNN latent space (bottom). Mean error quantities are shown by red curves.}
\label{fig:ROM_error_histograms}
\end{figure*}
In each of the three models, the bulk of the error remains low throughout the forecast window, as is shown by the curves for mean error.
Error statistics for the dynamic model for $P$ remain low for approximately three time units before some trajectories diverge.
Outlier errors for the dynamic model for $\psi$ grow more rapidly than others initially and by the end of the 10.5 convective unit interval are noticeably larger, on average than then other two.
The dynamic model for $\xi$ clearly has both the lowest error in the dynamic variable and has error for $q$ of a similar magnitude to the dynamic model in $P$.
For the models in $P$ and $\psi$ where indices in the dynamic variable carry meaning is also instructive to see what variables accumulate error at what rates.
Figure \ref{fig:ROM_error_index} shows the mean square error for each index in each dynamic variable taken across test set examples.
\begin{figure*}[h]
\centering
\includegraphics[width=\textwidth]{ROM_error_index}
\caption{Time evolution of mean (across test set samples) square error for each of 50 pressure sensors (left), 32 POD modes (middle) and dimensions of FFNN latent space (right) using LSTM based dynamic models.}
\label{fig:ROM_error_index}
\end{figure*}
The sharp discontinuity in the error for pressure sensors is a consequence of their position on the airfoil, shown in the lower panel of Fig. \ref{fig:grid_vort}.
The first 26 pressure sensors are equally placed on the top of the airfoil from front to tail including endpoints and the subsequent 24 on evenly spaced from front to tail on the lower side.
We see that error towards the tail of the airfoil on both the suction and pressure sides is initially lower than towards the front.
However, as time progresses, there is considerable error in pressure sensors towards the rear of the suction edge of the airfoil.
As expected error in the low order POD modes is lower than in the high order, less energetic, modes.
This may be explained by the fact that these low order modes tend to track large scale features of the flow and have slower and less chaotic trajectories.
It is worth noting that, while the left hand column of Fig. \ref{fig:ROM_error_histograms} makes clear that error in $\psi$ grows much more rapidly in the initial forecast than $P$ or $\xi$, the same is not as obvious for low $\tau$ prediction of $q(t+\tau)$ using the dynamic model for $\hat{\psi}$.
This suggests that the value of $q$ may largely be a function of the low order energetic POD modes, which Fig. \ref{fig:ROM_error_index} shows are accurately tracked for longer lead times.
Indices for the FFNN latent space variable $\xi$ do not have meaning and there is no correlation between index and the rate at which error increases.
\subsection{Forecasting aerodynamic fluctuations}
In this section we compare the six neural network based prediction strategies using the metrics outlined in Sec. \ref{subsec:error_metrics}. Examples of predictions for lead time $\tau=7.0$ using all methods are shown in Fig. \ref{fig:tau7_time_series}. Blue curves indicate true values $q(t)$ and red dashed curves show predictions with lead time $\tau=7.0$ using the three non-dynamic and three dynamics methods. Plots for other lead times are shown in Appendix D.
\begin{figure*}[h]
\centering
\includegraphics[width=\textwidth]{tau7_time_series}
\caption{True (blue) and predicted (red) time series for $q(t)$ with lead time $\tau=7$.}
\label{fig:tau7_time_series}
\end{figure*}
Scalar values (MSE, $\alpha^*$, BRE) of prediction accuracy for each method and lead times $\tau$ ranging from $0$ to $10.5$ convective time units in intervals of $0.7 \approx 1/f_{peak}$ are shown in Fig. \ref{fig:MSE_alpha_KL}.
Metrics for prediction via non-dynamic methods are shown as solid lines and those for dynamic models are shown as dashed lines.
\begin{figure*}[h]
\centering
\includegraphics[width=\textwidth]{MSE_alpha_KL}
\caption{Mean square error, maximum adjusted area under the precision-recall curve ($\alpha^*$), and test set relative entropy for each of the six neural network based prediction methods. Evaluation is performed on partition of data reserved for method comparrison (test set).}
\label{fig:MSE_alpha_KL}
\end{figure*}
Figure \ref{fig:MSE_alpha_KL} clearly indicates some qualitative differences between dynamic and non-dynamic methods.
In particular, error statistics for the dynamic models at a particular value of $\tau$ are highly correlated with those for similar lead times.
The curves indicating their performance as a function of lead time are therefore smooth.
The same is not true for non-dynamic models, where predictions for each lead time are performed via their own trained neural network.
It is plausible that more care could be taken to yield consistency across lead times for non-dynamic models but methods for doing so are not immediately apparent.
The scalar error metrics also clearly show that the dynamic models in $P$ and $\xi$, $H_{P}$ and $H_{\xi}$ are the most accurate of the six models. Differences between the other four are less pronounce, though $H_\psi$ does, on average, slightly outperform non-dynamic methods.
This difference is most notable in the MSE, though still apparent for the extreme event specific metrics.
This suggests that a non-negligible portion of the improvement $H_\psi$ has over non-dynamic methods manifests in the non-extreme values of $q$, though the scalar valued metrics do not resolve this feature.
Values for the remaining extreme event rate dependent evaluation metrics are shown in Fig. \ref{fig:F1_and_AUC} for values of $\omega$ between $0.01$ and $0.25$.
The top row shows values for $F_{1,opt}$ and bottom row shows the area under the precision-recall curve.
Within each subplot, values towards the bottom of the image indicate the prediction accuracy of the model for classifying very rare events $(1\%)$ while those at the very top show more common events, with lead time increasing across the horizontal axis.
While differences in values of prediction accuracy using Fig. \ref{fig:F1_and_AUC} may not be immediately clear, the same qualitative features seen in Fig. \ref{fig:MSE_alpha_KL} are again apparent. In particular, errors for the dynamic methods are much smoother in time than those of the non-dynamic methods.
\begin{figure*}[h]
\centering
\includegraphics[width=\textwidth]{F1_and_AUC}
\caption{Extreme event rate dependent area under the precision-recall curve ($\alpha(\omega)$ and $F_{1,opt}$) for each method evaluated on test set.}
\label{fig:F1_and_AUC}
\end{figure*}
Differences between methods become much more apparent when plotted directly. Figure \ref{fig:F1_AUC_comp} shows the differences in values between all methods for both $F_{1,opt}$ (left) and area under the precision-recall curve, $\alpha$ (right). Values within each subfigure indicate the metric evaluated with the method on the corresponding row minus the metric for the method on the corresponding column. Hence, blue (negative) indicates the the method corresponding to that column performed better while red (positive) indicates the method corresponding to the row is better.
\begin{figure*}[h]
\centering
\includegraphics[width=\textwidth]{F1_and_AUC_comp}
\caption{Comparrison of $\alpha$ (left) and $F_{1,opt}$ (right) across various methods, lead times $\tau$, and extreme event rates $\omega$. Each plot shows $\alpha/F_{1,opt}$ for method corresponding to row minus $\alpha/F_{1,opt}$ for method corresponding to column. Blue (negative) indicates column-method outperforms row-method. Hence lower left plot indicates $H_\xi$ has higher $\alpha$ (favorable performance) than $H_\psi$ in most cases. }
\label{fig:F1_AUC_comp}
\end{figure*}
Figure \ref{fig:F1_AUC_comp} shows $H_P$ and $H_\xi$ outperforming other methods across all lead times $\tau$ and extreme event rates $\omega$ except for a small number of datapoints clustered around $\tau=4$ and low $\omega$.
Comparisons between other methods are less easily summarized.
There does not appear to be discernible trends in the comparison between non-dynamic methods using $P$, $\psi$ or $\xi$ as input.
From this it seems reasonable to surmise that the accuracy of non-dynamic methods is not improved in a meaningful manner by exploiting flow field data.
The dynamic model for $\psi$ outperforms non-dynamic methods for mid-range lead times and lower extreme event rates, but under perform in prediction of less rare events (higher $\omega$) at longer lead times.
Differences between methods are largely consistent between $F_{1,opt}$ and $\alpha$.
\section{Discussion}\label{sec:discussion}
We have investigated the feasibility of a variety of methods for forecasting aerodynamic fluctuations occurring in a simulation of two dimensional incompressible flow around a NACA 4412 airfoil using only pressure recordings along the surface of the airfoil.
Some of these methods also exploited, in an offline manner, knowledge of the flow field to construct low dimensional representations that were either used as inputs to machine learned predictors or variables in dynamic models.
Comparison between the methods considered in this work suggests that the complexity of the flow, though low Reynolds and only two dimensional, precludes the use of a full order computational fluid dynamic model with low dimensional input.
Other methods were largely comparable, except for the learned dynamic models of the pressure measurements and of the latent variable of the FFNN based flow reconstruction, which performed notably better than others.
The authors highlight that neural networks allow for tremendous freedom with regards to structure, hyper-parameters, regularization, and other factors that may affect performance.
We have tested several architectures and selected the best result for this paper, but acknowledge that our work falls considerably short of testing across all sizes and training procedures.
Doing so would require an extensive computational resources and results would have no guarantee of generality beyond the problem specifically discussed in this work.
Rather, we sought to investigate the use of various classes of models (non-dynamic, ROM, full order simulation) and representations of data (POD modes, FFNN) for the purpose of forecasting a quantity exhibiting extreme events.
We also note that the work contained in this manuscript has been done without the inclusion of artificial noise, as is common in works applying machine learning to synthetic datasets.
The authors suspect that including noise would not change the results in a meaningful way, since flow reconstruction methods have been shown to be robust to noise \cite{maulik2020probabilistic,erichson2020shallow} and quantities with dynamic models may be estimated with filtering.
Nonetheless, further work studying the effect of noise would be necessary to confirm this.
In Section \ref{sec:results} we suggested that low order initializations of the Navier Stokes solver failed to accurately predict fluctuations, this motivated the use of data-driven dynamic models.
This is not surprising, given the complexity and non-periodic nature of the flow.
It is possible that a mixed strategy of projecting the governing Navier-Stokes equations onto POD modes coupled with an LSTM closure model would outperform both approaches, as suggested in \cite{levine2021framework}.
This approach, however would be highly non-trivial when using the latent space of the FFNN, since it is unclear how one might project the known governing equations.
We also stress the nature of this work as a computational study, rather than one that may be directly applied to engineering.
Numerical simulation for the current work was implemented with a blockage ratio of $3.18\%$.
This falls within the range that might be expected for experiments on bluff bodies \cite{yarusevych2011vortex, west1982effects} but in lower than some works studying the aerodynamic properties of airfoils.
Several works have noted dependence of aerodynamic properties including the drag coefficient, Strouhal frequency certain critical Reynolds numbers on the blockage ratio \cite{turki2003effect}.
This effect was observed in the case used for the present work, but not explored in depth.
We also note that the simulation is of two dimensional flow over a smooth airfoil, which exhibits qualitative differences from three dimensional simulations and experiments \cite{tank2017possibility}.
Thus, the present work should be considered as a study of a computational model of prototypical flow phenomena, rather than experimental or application conditions.
The present work considered the case where pressure measurements are taken at 50 positions around the perimeter of the airfoil.
This is, of course, not practical and future work could consider the use of various sensor placement techniques for determining optimal placement on the airfoil with respect to forecasting fluctuations \cite{le2020deep, de2017enkf}.
We believe this work establishes compelling comparisons and baselines for the prediction of extreme events using measurements on the surface of an airfoil.
Numerous techniques, including several adapted from other works on extreme events were tested and compared.
The results suggest that the use of data-driven dynamic models for quantities subsequently used to predict extreme events outperform those that ignore dynamics.
This work also provides numerous opportunities for future research.
In particular, sensor placement and uncertainty quantification are critical elements of practical engineering that may be explored in the context developed in this work.
\section*{Acknowledgments}
This material is based upon work supported by the Army Research Office (Grant No. W911NF-17-1-0306) and used the Extreme Science and Engineering Discovery Environment (XSEDE) \cite{towns2014xsede}, which is supported by National Science Foundation undder Grant No. ACI-1548562. TPS has also been supported by the Air Force Office of Scientific Research (Grant No. MURI FA9550-21-1-0058). Simulations were performed on Expanse at the San Diego Supercomputing Center through allocation TG-MTH210003.
\small
\begin{spacing}{.5}
\bibliographystyle{plain}
|
1,116,691,500,866 | arxiv | \section{Introduction}
In this paper we study the family of Lieb-Thirring
inequalities\cite{lieb1975bound,lieb1976ineq}, which state that for
any potential $V \in L^{\gamma+\frac d 2}(\R^{d},\R)$ and non-negative
exponent $\gamma$,
\begin{align}
\label{LT}
\tr((-\Delta + V)_{-}^{\gamma}) \leq L_{\gamma,d} \int V^{\gamma + \frac d 2}_{-},
\end{align}
where $V_{-} = \max(-V,0)$ is the negative part of $V$, and $(-\Delta
+ V)_{-}^{\gamma}$ is the $\gamma$-th power of the negative part of
the operator $(-\Delta + V)$, as defined by the functional
calculus. In other words, $\tr((-\Delta + V)_{-}^{\gamma}) = \sum_{i}
|\lambda_{i}|^{\gamma}$ is the $\gamma$-th moment of the negative
eigenvalues of $-\Delta + V$.
This inequality was originally used by Lieb and Thirring in the case
$\gamma = 1, d = 3$ as an important tool to prove the stability of
fermionic matter in \cite{lieb1975bound} (see
\cite{lieb1990stability,liebseiringer} for an overview). The
generalization \eqref{LT} to general $\gamma$ and $d$ has since then
attracted a great deal of attention (see for instance
\cite{laptev2000recent} for a review). Of particular interest are the
values of $\gamma$ and $d$ for which this inequality holds, and the
value of the optimal constants $L_{\gamma,d}$.
Despite the physical significance of the Lieb-Thirring inequality and
the amount of mathematical research on the subject, many questions
remain. This paper aims to investigate some of these numerically,
something that, to our knowledge, has not been done since the work of
Barnes in the appendix of the original paper by Lieb and Thirring
\cite{lieb1976ineq}, in 1976.
We now briefly review what is known about $L_{\gamma,d}$. It is easy
to see that the bound \eqref{LT} cannot hold for $\gamma < 1/2$ when
$d = 1$ by scaling, and $\gamma = 0$ when $d = 2$ (because any
arbitrarily small potential in two dimension creates at least one
bound state). The inequality was proved for $\gamma > 1/2$ in one
dimension and $\gamma > 0$ in other dimensions in the original paper
by Lieb and Thirring\cite{lieb1976ineq}. The proof in the case $\gamma
= 1$ was recently greatly simplified by
Rumin\cite{rumin2011balanced}. The case $\gamma = 0, d \geq 3$
requires completely different methods and was proven independently by
Cwikel, Lieb and
Rosenblum\cite{cwikel1977weak,liebclr,rozenblum1972}. The limit case
$\gamma = 1/2, d = 1$ remained unsolved for twenty years until it was
finally settled by Weidl\cite{weidl1996}, and refined with a sharp
constant by Hundertmark, Lieb and Thomas\cite{hundertmark1998sharp}.
The inequality \eqref{LT} can be interpreted as a comparison of the
energy of the quantum mechanical system given by its Hamiltonian
$-\Delta + V$, with its semiclassical approximation that only involves
integrals of $V$. The semiclassical regime is obtained by letting the
Planck constant $\hbar$ tend to zero in the Hamiltonian $-\hbar\Delta
+ V$. In this paper, we have scaled $\hbar$ away from inequality
\eqref{LT} for convenience, and the corresponding limit is a large (or
extended) $V$. The eigenfunctions become localized in the region of
the phase space defined by $p^{2} + V(x) \leq 0$ and explicit
computations are possible in the limit. More precisely, using the Weyl
asymptotics, one can prove
\begin{align*}
\lim\limits_{\mu \to \infty}\frac{\tr((-\Delta + \mu
V)_{-}^{\gamma})}{\int (\mu V)^{\gamma + \frac d 2}} = L_{\gamma,d,\text{sc}},
\end{align*}
where
\begin{align*}
L_{\gamma,d,\text{sc}} = 2^{-d} \pi^{-d/2}
\frac{\Gamma(\gamma+1)}{\Gamma(\gamma+\frac d 2 + 1)}.
\end{align*}
Therefore, denoting by $L_{\gamma,d}$ the optimal constant in
\eqref{LT}, we have $L_{\gamma,d} \geq L_{\gamma,d,\text{sc}}$, and we
set
\begin{align*}
R_{\gamma,d} = \frac{L_{\gamma,d}}{L_{\gamma,d,\text{sc}}} \geq
1.
\end{align*}
The value of $R_{\gamma,d}$ describes by how much the quantum
mechanical energy can exceed its semiclassical counterpart. This paper
is an investigation of $R_{\gamma,d}$. It aims at providing insight as
to its behavior by trying to solve numerically the variational problem
\begin{align}
\tag{$P_{\gamma,d}$}
\label{variational}
R_{\gamma,d} = \frac 1 {L_{\gamma,d,\text{sc}}}\sup_{\int V^{\gamma + \frac d 2} = 1} \tr((-\Delta +
V)_{-}^{\gamma}),
\end{align}
where we impose the condition $\int V^{\gamma + \frac d 2} = 1$ to
eliminate the scaling invariance of the Lieb-Thirring inequality.
Several important facts about $R_{\gamma,d}$ are known. A scaling
argument by Aizenman and Lieb \cite{aizenman1978semi} shows that
$R_{\gamma,d}$ is decreasing with respect to $\gamma$. Laptev and
Weidl \cite{laptev2000gammathreehalves} showed that $R_{\gamma,d} = 1$
for $\gamma \geq 3/2$. Helffer and Robert \cite{helffer1990riesz}
proved that $R_{\gamma,d} > 1$ for $\gamma < 1$ by expanding
$\tr(-\Delta + V)_{-}^{\gamma}$ for the harmonic potential $V(x) = 1 -
\abs{x}^{2}$ in the semiclassical limit. From these results, we can
deduce that for each dimension $d$ there exists a critical
$\gamma_{c,d}$ such that
\begin{align*}
\begin{cases}
R_{\gamma,d}> 1 & \text{when } \gamma < \gamma_{c,d},\\
R_{\gamma,d} = 1 & \text{when } \gamma \geq \gamma_{c,d},\\
\end{cases}
\end{align*}
with
\begin{align*}
1 \leq \gamma_{c,d} \leq \frac 3 2.
\end{align*}
A trial potential of \cite{lieb1976ineq} provides a lower bound on
$R_{\gamma,d}$ and therefore $\gamma_{c}$. In the appendix of this paper,
Barnes solved numerically the restricted problem
\begin{align}
\label{variational_1}
R_{\gamma,d,1} = \frac 1 {L_{\gamma,d,\text{sc}}}\sup_{\int V^{\gamma + \frac d 2} = 1} (\lambda_{1})_{-}^{\gamma},
\end{align}
where $\lambda_{1}$ is the lowest eigenvalue of $-\Delta + V$. This is
equivalent to restricting \eqref{variational} to the potentials $V$
such that $-\Delta + V$ only has one negative eigenvalue. The solution
$V_{\gamma,d,1}$ of this problem is negative, radial and only has
bound state, \ie $-\Delta + V_{\gamma,d,1}$ only has one negative
eigenvalue. The corresponding $R_{\gamma,d,1}$ is decreasing, and
intersects $1$ at a critical $\gamma_{c,d,1}$. In low dimensions,
$\gamma_{c,1,1} = \frac 3 2$, $\gamma_{c,2,1} \approx 1.165$,
$\gamma_{c,3,1} \approx 0.863$ (our numerical results agree with these
values). Therefore, $\gamma_{c,1} = \frac 3 2$ and $\gamma_{c,2} \geq
1.165$, but nothing can be said about $\gamma_{c,d}$ for $d \geq 3$. A
famous conjecture of Lieb and Thirring states that $R_{\gamma,d} = 1,
$ \ie $\gamma_{c,3} = 1$. This would imply improved bounds on the
energy of a system of fermions, and is of great importance to the
Thomas-Fermi theory\cite{lieb1981thomas}.
Better upper bounds on $R_{\gamma,d}$ have also been derived
recently. For instance, it is proven that $R_{\gamma,d} \leq \frac {2
\pi}{\sqrt 3} \approx 3.63$ for $\gamma \geq \frac 1 2$,
$R_{\gamma,d} \leq \frac \pi {\sqrt 3} \approx 1.82$ for $\gamma \geq
1$ (see \cite{dolbeault}), by generalizing the inequality to
matrix-valued potentials.
Despite these advances on upper bounds, not much is known about lower
bounds for $R_{\gamma,d}$, except for the one-bound state potential
$V_{\gamma,d,1}$ of \cite{lieb1976ineq} and the asymptotic result in
the semiclassical limit of \cite{helffer1990riesz}. In this paper, we
attempt to bridge the gap between these two results by looking
numerically for maximizers of the variational problem
\eqref{variational}. To our knowledge, this is the first numerical
study of the Lieb-Thirring inequalities since the work of
Barnes\cite{lieb1976ineq} (and unpublished recent work by Arnold and
Dolbeault \cite{dolbeaultcomm} in 1D).
The method we use is a finite element discretization of a fixed point
algorithm. We describe this algorithm, and specialize it to the case
of radial potentials. Then we discretize it, and use it to obtain
numerical results. The critical points we obtain are the natural
generalization of the potential with one bound state obtained by
Barnes. They serve as a partial bridge between this potential and
asymptotic results, and yield new lower bounds on $R_{\gamma,d}$.
\section{The optimization algorithm}
\subsection{Optimization scheme}
Let us denote $E(V) = \tr((-\Delta + V)_{-}^{\gamma})$, so that
$$L_{\gamma,d} = \sup_{\int V^{\gamma+\frac d 2} = 1} E(V).$$
The maximization algorithm is based on the following well-known
property:
\begin{proposition}
\label{prop}
For any $V \in L^{\gamma+\frac d 2}(\R^{d},\R)$ and any $\gamma \geq 1$,
\begin{align*}
E(V)^{1/\gamma} = \max_{\substack{\tau \geq 0 \\ \norm{\tau}_{\gamma'} = 1}} - \tr((-\Delta + V)\tau),
\end{align*}
where $\norm{\tau}_{\gamma'} = \lp \tr \lp|\tau|^{\gamma'}\rp\rp^{1/\gamma'}$ is the Schatten norm of the
operator $\tau$ with exponent $\gamma' = \frac {\gamma}{\gamma - 1}$,
the Hölder conjugate of $\gamma$.
\end{proposition}
\begin{proof}
We have
\begin{align*}
-\tr((-\Delta + V)\; \tau) &\leq \tr ((-\Delta + V)_{-} \;\tau)\\
&\leq \norm{(-\Delta + V)_{-}}_{\gamma} \norm{\tau}_{\gamma'}\\
&\leq \tr((-\Delta + V)_{-}^{\gamma})^{1/\gamma}\\
&\leq E(V)^{1/\gamma},
\end{align*}
and the equality is achieved when
\begin{align}
\label{tau_V}
\tau = K_{\tau} (-\Delta + V)_{-}^{\gamma-1},
\end{align}
where $K_{\tau}$ is a normalization constant, chosen to ensure that
$\norm{\tau}_{\gamma'} = 1$.
\end{proof}
From the proposition, we see that when $\gamma \geq 1$,
\begin{align*}
(L_{\gamma,d})^{1/\gamma} = \sup_{\substack{\int V^{\gamma+\frac d 2} = 1,\\ \tau \geq
0, \,\norm{\tau}_{\gamma'} = 1}} -\tr ((-\Delta+V)\tau).
\end{align*}
What we gain from this formulation is that we can explicitely maximize
$-\tr ((-\Delta+V)\tau)$ with respect to $V$ and $\tau$
separately. Indeed, for a fixed $V$, the maximum with respect to
$\tau$ is given by \eqref{tau_V}, and for a fixed $\tau$, the maximum
with respect to $V$ is given by
\begin{align*}
V(x) = -K_{V} \tau(x,x)^{\frac 1 {\gamma + \frac d 2 - 1}},
\end{align*}
where $K_{V}$ is a normalization parameter chosen to ensure that $\int
V^{\gamma + \frac d 2} = 1$, and $\tau(x,y)$ is the integral kernel of
$\tau$.
This suggests the following maximization scheme: Given an approximate
maximum $V_{n}$, we set $\tau_{n} = K_{\tau} (-\Delta +
V_{n})_{-}^{\gamma-1}$ and $V_{n+1}(x) = -K_{V} \tau_{n}(x,x)^{\frac 1
{\gamma + \frac d 2 - 1}}$, and iterate. Explicitely:
\theoremstyle{definition}
\newtheorem{algorithm}{Algorithm}
\begin{algorithm}{Maximization algorithm}
\begin{enumerate}
\item Compute the negative eigenvalues $\lambda_{i}$ and corresponding
eigenvectors $\psi_{i}$ of $-\Delta + V_{n}$
\item Set $\rho_{n} = \sum_{i} (-\lambda_{i})^{\gamma-1} \psi_{i}^{2}$
\item Compute $K_{n} = \norm{\rho_{n}^{\frac 1 {\gamma + \frac d 2 -
1}}}_{\gamma+\frac d 2}^{-1} = \lp\displaystyle\int \rho_{n}^{\frac {\gamma + \frac
d 2} {\gamma + \frac d 2 -
1}}\rp^{- \frac 1 {\gamma+\frac d 2}}$
\item Set $V_{n+1} = - K_{n} \rho_{n}^{\frac 1 {\gamma + \frac d 2 - 1}}$
\end{enumerate}
\end{algorithm}
By construction, this algorithm ensures that
\begin{align*}
E(V_{n+1})^{1/\gamma} &= - \tr((-\Delta+V_{n+1}) \tau_{n+1})\\
&\geq -\tr((-\Delta+V_{n}) \tau_{n+1})\\
&\geq -\tr((-\Delta+V_{n}) \tau_n)\\
&= E(V_{n})^{1/\gamma}
\end{align*}
Therefore, the sequence $E(V_{n})$ is non-decreasing, and since it is
bounded from above, it converges. In general though, $V_{n}$ may not
converge. We give examples in Section \ref{numres} where, because of
the translation invariance of the functional, $V_{n}$ splits into two
bumps that separate from each each other. Even when this behavior is
forbidden, for instance by imposing a finite domain as we do in
numerical computations, a rigorous convergence analysis of the
algorithm is still an open problem.
Note that, even if Algorithm 1 was derived from Proposition
\ref{prop} for $\gamma \geq 1$, it remains a reasonable algorithm when
$\gamma < 1$. In this case, though, the monotonicity of $E(V_{n})$ is
not guaranteed, and indeed was found to be false in numerical
computations. Nevertheless, we did not notice any specific convergence
issue for $\gamma < 1$.
Algorithm 1 can also be seen as a fixed-point scheme for the critical
points of the maximization problem. Indeed, the Euler-Lagrange
equations for \eqref{variational} are
\begin{align}
\label{scf}
V(x) = -K_{V} \left[(-\Delta+V)_{-}^{\gamma-1}(x,x)\right]^{\frac 1 {\gamma + \frac d 2 - 1}},
\end{align}
This is a self-consistent set of equations similar to systems such as
the Hartree-Fock equations of quantum
chemistry\cite{lieb1977hartree}. Our algorithm is similar in spirit to
the Roothaan method\cite{roothaan1951}. In our case, at least for
$\gamma \geq 1$, the scheme monotonically increases the objective
function. Therefore, the oscillatory behavior often seen in the
Hartree-Fock model\cite{cances2000,levitt} cannot occur here. Even for
$\gamma < 1,$ we did not see any such oscillations numerically, and
always observed linear convergence (\ie $\norm{V_{n} - V_{\infty}}
\leq \nu^{n}$ for some $\nu$, $0 < \nu < 1$), or slow separation of
bumps, as will be discussed in Section \ref{numres}.
This algorithm is also used in the context of the Lieb-Thirring
inequalities by Arnold and Dolbeault in 1D\cite{dolbeaultcomm}.
\subsection{Radial algorithm}
\label{radialalg}
Most of our multidimensional computations were done in a radial
setting. To see why this is appropriate, consider the above iteration
for $d \geq 2$, when $V_{n}$ is radial. Then, the Laplacian splits
into $\Delta_{r} + \frac 1 {r^{d-1}} \Delta_{\theta}$, where $\Delta_{r}$
is the radial Laplacian, and $\Delta_{\theta}$ the Laplace-Beltrami
operator on $S^{d-1}$. The eigenvectors of $\Delta_{\theta}$ are
explicitly known to be the functions whose orthoradial part are the
spherical harmonics, labeled by the integers $\ell$ and $m$. Since
$V_{n}$ is radial, $-\Delta_{r} + V_{n}$ commutes with
$\Delta_{\theta}$ and can therefore be diagonalized in the same basis
(separation of variables). We write these eigenvectors of the form
$\psi_{i,\ell,m}(x) = \phi_{i,\ell}(\abs x) J_{\ell,m}(x/\abs x)$, where
$J_{\ell,m}$ is the spherical harmonics of degree $\ell$ and order $m$. The
radial parts $\phi_{i,\ell}$ satisfy the equation
\begin{align}
\label{eigen_l}
- \frac 1 {r^{d-1}} (r^{d-1} \phi_{i,\ell}')' + \frac{\ell(\ell+d-2)}{r^{2}}\phi_{i,\ell} +
V \phi_{i,\ell} = \lambda_{i,\ell} \phi_{i,\ell},
\end{align}
and each $\lambda_{i,\ell}$ has multiplicity
\begin{align*}
h(d,\ell) = \binom{d+\ell-1}{\ell} - \binom{d+\ell-3}{\ell-2}
\end{align*}
(see \cite{stein1971introduction}.)
Therefore, we can obtain all the negative eigenvectors and eigenvalues
by solving only \eqref{eigen_l}. Furthermore, since the lowest
eigenvalue decreases as $\ell$ increases, we can iterate over $\ell$
and stop whenever the lowest eigenvalue becomes positive. Next, we
compute
\begin{align*}
\rho_{n}(x) &= \sum_{\ell} \sum_{i} \sum_{m} (-\lambda_{i,\ell})^{\gamma-1}
\psi_{i,\ell,m}(x)^{2}\\
&= \sum_{\ell} h(d,\ell) \sum_{i} (-\lambda_{i,\ell})^{\gamma-1} \phi_{i,\ell}(r)^{2}
\end{align*}
where the sum only ranges over all negative eigenvalues. This is again
radial, and therefore so is $V_{n+1}$. To summarize:
\begin{algorithm}{Radial maximization algorithm}
\begin{enumerate}
\item Compute the negative eigenvectors $\phi_{i,\ell}$ and
eigenvalues $\lambda_{i,\ell}$ of \eqref{eigen_l} by increasing
$\ell$ until the lowest eigenvalue $\lambda_{0,\ell}$ becomes positive
\item Set $\rho_{n} = \sum_{\ell} \sum_{i} h(d,\ell)
(-\lambda_{i,\ell})^{\gamma-1} \phi_{i,\ell}^{2}$
\item Compute $K_{n} = \norm{\rho_{n}^{\frac 1 {\gamma + \frac d 2 -
1}}}_{\gamma+\frac d 2}^{-1} = \lp\displaystyle\int \rho_{n}^{\frac {\gamma + \frac
d 2} {\gamma + \frac d 2 -
1}}\rp^{- \frac 1 {\gamma+\frac d 2}}$
\item Set $V_{n+1} = - K_{N} \rho_{n}^{\frac 1 {\gamma + \frac d 2 - 1}}$
\end{enumerate}
\end{algorithm}
Note that this algorithm is not an approximation: the iterates
generated by this algorithm coincide with the ones from Algorithm 1
when the initial guess $V_{0}$ is radial.
\section{Discretization}
\subsection{Galerkin basis and weak formulation}
To discretize this algorithm, we use a Galerkin basis on which we
expand $V$ and the eigenvectors. This discretization is variational at
two levels. First, the choice of eigenvectors is restricted to a
finite-dimensional subspace. By the min-max principle, this leads to
overestimation of the eigenvalues and therefore underestimation of
$E(V)$ for a given $V$. Second, the choice of potentials $V$ is also
restricted, decreasing the value of $\max_{\int V^{\gamma+\frac d 2} =
1} E(V)$.
Our choice of discretization, rather than for instance finite
differences, has the advantage that numerical examples can provide
lower bounds on the best constants $R_{\gamma,d}$, with machine
precision as the only source of errors. A disadvantage is that the
algorithm we use involves taking powers of functions. There is no
exact way to do that in a Galerkin basis, and we must use an
approximation, which causes a loss of accuracy in the fixed-point
algorithm.
For the non-radial 1D and 2D cases, we simply use standard finite
elements. The radial case is less standard, and we must derive a weak
formulation of \eqref{eigen_l}. There are several possibilities here. The
simplest is to use a change of variable $\varphi(r) = r^{\frac{d-1}2}
\phi(r)$ to transform the equation back to the more standard form
\begin{align*}
- \varphi'' + \frac {(l-\frac{d-1}{2})(l+\frac{d-3}{2})}{r^{2}}
\varphi + V(r)\varphi = \lambda \varphi.
\end{align*}
We obtain a weak formulation by multiplying by a test function $u$ and
integrating:
\begin{align}
\label{eigen_l_weak_bad}
\int\left[ \phi' u' + \lp\frac {(l-\frac{d-1}{2})(l+\frac{d-3}{2})}{r^{2}}
+ V\rp \phi u\right] = \lambda \int \phi u,
\end{align}
Expanding the function $\phi$ on a Galerkin basis $\phi(r) = \sum_{i}
x_{i} \chi_{i}(r)$, this problem
transforms into the generalized matrix eigenvalue problem
\begin{align}
\label{eigenproblem}
A x = \lambda M x,
\end{align}
where
\begin{align}
\label{stiffness}
A_{ij} &= \int \chi_{i}' \chi_{j}' +
\lp\frac{(l-\frac{d-1}{2})(l+\frac{d-3}{2})}{r^{2}} + V\rp\chi_{i}
\chi_{j},\\
\label{mass}
M_{ij} &= \int \chi_{i}
\chi_{j},
\end{align}
are the stiffness and mass matrices.
When $V$ is expanded on the same basis, we can compute the matrix
elements, solve the eigenvalue problem \eqref{eigenproblem}, and
transform back to $\phi$. However, for $d = 2, l = 0$, $\varphi$
behaves like $\sqrt r$ at $0$, and the singularity of the derivative
prevents an accurate discretization.
Another possibility is to obtain a weak formulation of \eqref{eigen_l}
directly. We have to remember that since the $\phi$ functions are only
radial parts of a $d$-dimensional function, the $L^{2}$ inner product
between two functions $\phi_{1}$ and $\phi_{2}$ is $\int \phi_{1}
\phi_{2} r^{d-1}$. This has to be taken into account in the weak
formulation in order to obtain a self-adjoint equation. Multiplying
\eqref{eigen_l} by $r^{d-1} u$, where $u$ is a test function, and
integrating by parts, we obtain:
\begin{align}
\label{eigen_l_weak}
\int\left[ r^{d-1} \phi' u' + \lp\frac{l(l+d-2)}{r^{2}} + V\rp r^{d-1}\phi u\right] = \lambda \int r^{d-1}\phi u.
\end{align}
Then, as with \eqref{eigen_l_weak_bad}, we can transform this into a
matrix equation and solve it. The disadvantage is that the
integrations required for the assembly step are more involved, and
therefore we only use it for the case $d = 2, l = 0$ where the other
method fails.
\subsection{Finite elements}
We use a Finite Element basis of piecewise linear functions on a grid
of $[0,L]$. The grid we use is a nonuniform grid of $N$ points, with
more points around $0$, to accommodate for the singularity of
\eqref{eigen_l}. $L$ is chosen large enough so that all the
eigenvectors associated to negative eigenvalues of $-\Delta + V$ can
be represented accurately in the basis. But if $\psi$ is an
eigenvector with negative eigenvalue $\lambda$, then, $\psi$ decays as
$\exp(-\sqrt{-\lambda} r)$. This shows that the discretization is
problematic for eigenvalues close to zero, as we shall see in Section
\ref{numres}.
For our piecewise linear basis functions, it is easy to compute the
matrix elements (\ref{stiffness},\ref{mass}) when $V_{n}$ is expanded
on this same basis $\chi_{i}$. We then solve the eigenproblem
\eqref{eigenproblem}, and obtain the eigenvectors. To expand
$\rho^{\frac 1 {\gamma + \frac d 2 - 1}}$ on the basis, we simply
chose the expansion that is exact on the grid points. Then the
normalization can be performed exactly, and the iteration is carried
out.
We present convergence results in \figref{fig:conv_NL}. As expected
from piecewise linear basis functions, we obtain $O(1/N)$ convergence of the
eigenfunctions $\phi$ in $H^{1}$ norm, and $O(1/N^{2})$ convergence of
the eigenvalues $\lambda$ (and therefore of $E(V)$). For a given
stepsize, the discretization error is exponential with respect to $L$,
with a decay rate equal to the decay rate of the eigenfunctions, \ie
$\sqrt{-\lambda}$. Although we used a uniform grid in one dimension in
\figref{fig:conv_NL}, similar results were checked to hold for our
non-uniform grid in $d$ dimensions, using the weak formulation
\eqref{eigen_l_weak_bad}, except for $d = 2, l = 0$, where the
convergence was slower and the weak formulation \eqref{eigen_l_weak} had to
be used to get the same convergence rates.
\begin{figure}[H]
\centering
\resizebox{\textwidth}{!}{ \includegraphics{conv_N.pdf} \includegraphics{conv_L.pdf}}
\caption{Convergence analysis with respect to $N$ and $L$. The
figure on the left illustrates the convergence of the
eigenfunctions and eigenvalues with respect to the number of grid
points $N$, with a fixed $L = 40$, large enough to cause a
negligible error. The slopes, found by linear regression, are
$-1.043$ for the eigenvectors and $-1.999$ for the eigenvalues,
close to their theoretical values of $-1$ and $-2$. The figure on
the right illustrates the convergence of the eigenfunctions and
eigenvalues with respect to the domain size, with constant
stepsize $h = 5 \times 10^{-4}$. The slope for the eigenvectors is
$- 0.5285 \approx \sqrt{-\lambda} = -0.5283$. When the domain size
gets large enough, the error due to $h$ dominates, and causes a
plateau.}
\label{fig:conv_NL}
\end{figure}
\subsection{Diagonalization}
The most computationally challenging step in our algorithm is the
problem of computing all the negative eigenvalues and associated
eigenvectors of a generalized eigenvalue problem $A x = M x$, where
$A$ and $M$ are large sparse symmetric matrices. The spectrum of $A$
consists of $n$ negative eigenvalues, and a large number of positive
eigenvalues, which can be seen as perturbations of the spectrum of the
discrete Laplacian.
In order to compute the $n$ negative eigenvalues, where $n$ is
unknown, we compute the $k$ lowest eigenvalues, check if the $k$-th
one is positive, and repeat the process with a larger $k$ if not. The
computation of the $k$ first eigenvalues can be done by standard
packages such as ARPACK\cite{lehoucq1998arpack}. However, standard
algorithms such as Arnoldi iteration are not suited for the
computation of inner eigenvalues, and one has to solve for the largest
eigenvalues of the shifted and inverted problem $(A-\sigma)^{-1} x =
(\lambda-\sigma)^{-1} x$ instead to ensure reasonably fast
convergence. This requires an adequate shift (one that is close to the
bottom of the spectrum of $A$). Even with this shift-and-invert
strategy, the group of eigenvalues one needs to locate is not
well-separated from the rest of the spectrum, and becomes less and
less so as $L$ increases. This leads to slow convergence for large
$L$.
\subsection{Implementation}
We implemented the algorithm in Python, using the Numpy/Scipy
libraries\cite{scipy}, with the ARPACK eigenvalue
solver\cite{lehoucq1998arpack}.
\section{Numerical results}
\label{numres}
We used the algorithm described above to compute critical points of
the variational problem \eqref{variational}, that is, a solution to
the self-consistent equation \eqref{scf}. Our strategy was to find
a critical point by running the algorithm on a suitable initial
potential $V_{0}$ (for instance, a Gaussian of specified width). Once
convergence is achieved for a specific $\gamma$, we can run the
algorithm for $\gamma + \Delta \gamma$, using as initial potential the
one found for $\gamma$. We have been unable to analytically prove the
correctness of this method, for instance by checking the conditions of
the implicit function theorem. However, we found it reliable enough
for our purpose, as long as $\Delta \gamma$ was chosen small enough.
\subsection{The 1D case}
In one dimension we simply used a grid of size $[-L,L]$ with Dirichlet
boundary conditions. We reproduced the potential with one bound state
$V_{\gamma,1,1}$ of \cite{lieb1976ineq} by using for $V_{0}$ a
Gaussian of relatively small variance (see \figref{fig:conv_1BS}). In
this case, the algorithm converges linearly (\ie $\norm{V_{n+1} -
V_{n}} \approx \nu^{n}$, for some convergence rate $\nu < 1$), and
very quickly (about twenty iterations to achieve machine precision).
\begin{figure}[H]
\centering
\resizebox{\textwidth}{!}{ \includegraphics{conv_1BS.pdf}\includegraphics{conv_1BS_Cn.pdf}}
\caption{Linear convergence from a Gaussian initial data towards
$V_{\gamma,d,1}$. This plot is for $\gamma = 1.2$.}
\label{fig:conv_1BS}
\end{figure}
Contrary to what we observe in higher dimensions, initializing the
algorithm with a Gaussian of larger variance did not make the
algorithm converge to other critical points. Instead, it leads to a
slow divergence where ``bumps'' separate, each bump corresponding to
the potential of $V_{\gamma,1,1}$ (see \figref{fig:div_1BS}).
\begin{figure}[H]
\centering
\resizebox{\textwidth}{!}{ \includegraphics{div_1BS.pdf}\includegraphics{div_1BS_Cn.pdf}}
\caption{Divergence from a sum of Gaussian bumps at $\gamma =
1.2$. Initializing to a single Gaussian of large width yields
similar results. The two bumps separate slowly, until the
finiteness of $L$ forces convergence. Although not displayed
here, the asymptotic logarithmic divergence
\eqref{logarithmicsep} can be checked graphically, e.g. the
spacing between $V_{100}$ and $V_{1000}$ is the same than
between $V_{1000}$ and $V_{10000}$. The asymptotic slope of the
log-log convergence plot is $-1$, which fits with the heuristic
arguments given.}
\label{fig:div_1BS}
\end{figure}
This effect occured regardless of the value of $\gamma$ in the range
$\gamma \in (1/2, 3/2)$. The divergence appears to be
logarithmic. More precisely, a numerical fit showed the asymptotic
relationship for the distance $L_{n}$ between the two bumps
\begin{align}
\label{logarithmicsep}
L_{n} \approx \frac 1 {2 \sqrt{-\lambda}} \log n,
\end{align}
where $\lambda$ is the unique negative eigenvalue of $- \Delta +
V_{\gamma,1,1}$.
This strikingly simple relationship can heuristically be understood by
the fact that, because the eigenfunctions $\phi_{1}$ and $\phi_{2}$
corresponding to the two bumps have exponential decay with decay
constant $\sqrt{-\lambda}$, their interaction is of order
$\exp(-\sqrt{-\lambda} L_{n})$. Due to cancellations, this interaction
leads to a correction of order $\exp(-2\sqrt{-\lambda} L_{n})$ in
$V_{n+1}$, and we have the approximate relationship $L_{n+1} \approx
L_{n} + K \exp(-2\sqrt{-\lambda} L_{n})$, which yields
\eqref{logarithmicsep}. A rigorous explanation of this is an
interesting question.
Based on these results, we conjecture that $V_{\gamma,d,1}$ is, up to
translation, the only maximizer of the functional. This would imply
that
\begin{align*}
R_{\gamma,1} = 2 \lp
\frac{\gamma-1/2}{\gamma+1/2}\rp^{\gamma-1/2}
\end{align*}
for $\gamma \leq \frac 3 2$, in agreement with the original conjecture of
Lieb and Thirring\cite{lieb1976ineq}.
\subsection{The 2D case}
Due to the high cost of accurately solving the maximization problem in
more than one dimension, we only obtained partial results in the
non-radial 2D case. The results indicate that the
algorithm either converges to radial critical points, or form slowly
separating bumps, as in one dimension. We have been unable to find any
example of non-radial critical points, although this does not mean
that they do not exist. An example of separation in bumps is provided
in \figref{fig:2D_sep}.
\begin{figure}[!h]
\centering
\resizebox{0.7\textwidth}{!}{ \includegraphics{2D_sep.pdf}}
\caption{Separation in bumps of 2D non-radial initial data at
$\gamma = 1.2$, computed from Algorithm 1 with standard FEM. The
initial data is taken to be a Gaussian in $r$ multiplied by an
angular factor $(1+\cos(4\theta))$.}
\label{fig:2D_sep}
\end{figure}
In the radial case, we followed branches of critical points of
\eqref{variational} using the continuation method on $\gamma$ we
described in Section \ref{numres}. We found this branch continuation
procedure robust. By varying the shape of the initial data, and in
particular its width, we were able to obtain different branches of
critical points. Generally speaking, as the width of the initial
potential increases, the number of negative eigenvalues of $-\Delta +
V$ increases.
We display the energy of these branches as a function of $\gamma$
in \figref{fig:2D}.
\begin{figure}[H]
\centering
\resizebox{\textwidth}{!}{
\begin{overpic
{2D_R.pdf}
\Large
\put(52,46){1}
\put(34,46){6}
\put(25,46){8}
\put(16,46){11}
\put(90,27){13}
\put(93,28.5){17}
\put(90,29.5){20}
\put(93,30.5){24}
\put(90,32.5){28,33,37,39,54}
\put(90,36){81,112,139}
\end{overpic}
}
\caption{$R(V)$ as a function of $\gamma$ for $d = 2$ using the
branch continuation procedure in the radial setting (Algorithm
2). The branches are labeled by the number of bound states they
have. Based on these results, the maximizer seems to be
$V_{\gamma,d,1}$, up to $\gamma \approx 1.16$. Above this, no
branches were above the semiclassical regime $R = 1$.}
\label{fig:2D}
\end{figure}
\begin{table}[H]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$k$ &1 &4 &6 &8 &11 &13 &17 &28 &54 &81 &139 \\
\hline
$\gamma_{c,2,k}$&1.165&1.150&1.141&1.135&1.126&1.124&1.124&1.119&1.110&1.105&1.100\\
\hline
\end{tabular}
\caption{Values $\gamma_{c,2,k}$ of $\gamma$ at which some branches with $k$ bound
states cross the threshold $R = 1$.}
\label{tab:2Dcrossings}
\end{table}
First, we see that as the number of eigenvalues increases,
$R(V_{\gamma})$ tends to $1$, the semiclassical limit. For a given
$\gamma$, the Lieb-Thirring constant will be given by the supremum of
$R(V_\gamma)$ over such curves. From the branches depicted here, we see
that the supremum is always given by either $V_{\gamma,d,1}$ for
$\gamma < \gamma_{c}^{1} \approx 1.16$, and by the semiclassical limit
for $\gamma > \gamma_{c}^{1}$. All the other curves are below these
two (although not depicted in this zoomed plot, this holds true for
$0.5 < \gamma < 1.5$).
It is important to keep in mind that these branches only represent
critical points of the functional. They are generically local maxima
with respect to radial perturbations, but might not be stable with
respect to non-radial ones. For instance, \figref{fig:2D_sep_8BS}
depicts what happens when the radial potential with eight bound states
is perturbed non-radially by a Gaussian multiplicative noise, with
$\gamma = 1.1$. Because the potential with one bound state has a
higher energy, the potential splits into eight bumps. Performing the
same experiment $\gamma$ larger than about $1.17$, where the potential
with one bound state has a lower energy, when the potential with eight
bound states has a larger energy, we found that no splitting occurs,
which suggests that the potentials are stable beyond their crossing
with the one with one bound state.
\begin{figure}[H]
\centering
\resizebox{0.7\textwidth}{!}{ \includegraphics{2D_sep_8BS.pdf}}
\caption{Separation in bumps of the randomly perturbed radial
potential with eight bound states, $\gamma = 1.1$.}
\label{fig:2D_sep_8BS}
\end{figure}
Based on our numerical results, we conjecture that $\gamma_{c} =
\gamma_{c}^{1} \approx 1.16$. The only way this conjecture could be
false is if some other curve is above the one with one bound state. We
have not been able to find such a curve.
\subsection{The 3D case}
Due to the high cost of computation, we were unable to get meaningful
results in the non-radial setting, and only present our findings in
the radial case. Some of the radial potentials we found are presented
in \figref{fig:3D}, with numerical data about the crossings of the
curves in Table \ref{tab:3Dcrossings}.
\begin{figure}[H]
\centering
\resizebox{\textwidth}{!}{
\begin{overpic
{3D_R.pdf}
\Large
\put(28,58){1}
\put(72,12){4}
\put(40,48){5}
\put(76,20){10}
\put(56,40){14}
\put(91,24){21}
\put(91,28){30,39}
\put(91,30){55}
\put(91,34){111,115,288,341}
\put(64,39){140}
\end{overpic}
}
\caption{$R(V)$ as a function of $\gamma$ for $d = 3$, with the same
methodology as in \figref{fig:2D}. As $\gamma$ increases,
different branches become maximizers, until $\gamma = 1$. The fact
that only some branches are above the threshold $R = 1$ when the
number of bound states increases results from the highly
oscillatory behavior near $\gamma = 1$, as predicted in
\cite{helffer1990riesz}.}
\label{fig:3D}
\end{figure}
\begin{table}[H]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$k$ &1 &4 &5 &10 &14 &21 &30 &55 &111 &140 &341 \\
\hline
$\gamma_{c,3,k}$&0.863&0.852&0.875&0.851&0.880 &0.857&0.862&0.860&0.854&0.891&0.853\\
\hline
\end{tabular}
\caption{Values $\gamma_{c,3,k}$ of $\gamma$ at which some branches with $k$ bound
states cross the threshold $R = 1$.}
\label{tab:3Dcrossings}
\end{table}
Contrary to the dimension 2, here some potentials with a higher number
of bound states have a higher $R$ than $V_{\gamma,d,1}$. The
corresponding curves in the $\gamma - R$ plane are flatter and
flatter, and intersect at a sequence of increasing
$\gamma_{c}^{k}$. This sequence seems to accumulate at $1$, in
accordance to Helffer and Robert's result\cite{helffer1990riesz},
which predicts the existence of similar potentials with a $R$ larger
than $1$ for every $\gamma < 1$ in the nearly semiclassical
regime. However, their study of an harmonic potential of varying width
showed a highly oscillatory behavior of $E(V)$ with respect to the
width of the potential. Although our numerical methods do not permit
us to investigate this regime (it would require a very large domain
size, with a correspondingly large number of points, and a very poor
conditioning of the eigenvalue problem), we expect the same behavior
to occur. This means that the computation of the maximizing branches
(here, those with 1, 5, 14 and 140 bound states) becomes harder and
harder.
The relative energies of the branches display a greater variety than
in the case $d = 2$, reflecting a more complicated energy
landscape. \figref{fig:maximizers} shows the profiles of some critical
potentials. Note that the potentials in dashed lines can be regarded
as ``anomalous'' versions of their similarly extended counterparts,
and have a lower energy. As can be expected, the potential branches
which are maximizers for some $\gamma$ have a profile that is
decreasing in $r$.
\begin{figure}[H]
\centering
\includegraphics[width=0.49\textwidth]{shape_1.pdf}
\includegraphics[width=0.49\textwidth]{shape_45.pdf}
\includegraphics[width=0.49\textwidth]{shape_1014.pdf}
\includegraphics[width=0.49\textwidth]{shape_140.pdf}
\caption{Shape of some critical points of $E(V)$, at $\gamma =
1$. The qualitative shape of the curves does not change much
when varying $\gamma$.}
\label{fig:maximizers}
\end{figure}
Table \ref{tab:eigenvalues} displays the eigenvalue repartitions of a
particular potential, the one with 140 bound states, at $\gamma =
0.88$. Several patterns can be noted. First, for low values of $k$ and
$l$, the approximate relationship $\lambda_{k+1,l} = \lambda_{k,l+2}$
holds. This is because the associated eigenfunctions are localized
close to 0. In this region, the potential can be approximated by a
harmonic potential $V(0) + \frac 1 2 V''(0) x^{2}$, which leads to the
approximation $\lambda_{k,l} = V(0) + \sqrt{V''(0)} (2k+l + \frac 3
2)$, explaining the relationship $\lambda_{k+1,l} \approx
\lambda_{k,l+2}$. This is only valid for small $k$ and $l$, where the
harmonic approximation is valid. When the eigenvalues are close to
zero, another pattern emerges: the last negative eigenvalues have the
same value of $k+l$, leading to a triangular pattern in Table
\ref{tab:eigenvalues}. This seems to be true for the maximizing
branches (1, 5, 14, 140), but not for the others (which tend to have a
different ordering of the eigenvalues close to zero). We do not have
an explanation for this.
\begin{table}[H]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$\lambda_{k,l}$&$k = 1$ &$k = 2$ &$k = 3$ &$k = 4$ &$k =
5$ &$k = 6$ &$k = 7$&$k = 8$\\
\hline
$l = 0$ &-85&-54&-30&-13&-4&-0.7&-0.02&+0.25\\
\hline
$l = 1$ &-69&-41&-20&-7.6&-1.8&-0.1&+0.02&\\
\hline
$l = 2$ &-54&-29&-12&-3.4&-0.3&+0.03&&\\
\hline
$l = 3$ &-39&-18&-5.8&-0.7&+0.04&&&\\
\hline
$l = 5$ &-26&-9&-1.3&+0.06&&&&\\
\hline
$l = 6$ &-14&-2.3&+0.08&&&&&\\
\hline
$l = 7$ &-3.8&+0.1&&&&&&\\
\hline
\end{tabular}
\caption{Eigenvalue repartition for the branch with 140 bound
states, with $\gamma = 0.88$, in units of $10^{-6}$ for readability. As in Section \ref{radialalg},
$\lambda_{k,l}$ is the $k$-th eigenvalue of equation
\eqref{eigen_l}. The positive eigenvalues are finite size effects
and have no physical meaning, therefore we do not write them
beyond the first one.}
\label{tab:eigenvalues}
\end{table}
Every potential we were able to compute was below the $R = 1$
threshold for $\gamma > 1$. These results confirm the Lieb-Thirring
conjecture that $\gamma_{c,3} = 1$. However, the subtle behavior near
$\gamma = 1$ means that there might exist maximizing potentials for
$\gamma > 1$ we were unable to find.
\subsection{The case $d \geq 4$}
The results we obtained in the case $d \geq 4$ are similar to the case
$d = 3$, with $V_{\gamma,d,1}$ as the maximizer for small $\gamma$,
until it is outperformed by branches with a larger number of bound
states, which all fall under the semiclassical limit before $\gamma =
1$. However, the high cost of computation (the higher the dimension,
the more ill-conditioned the eigenvalue problem is) prevents us from
performing a systematic study.
\section{Conclusion}
In this paper, we used a maximization algorithm to investigate the
best constants of the Lieb-Thirring inequalities, an important open
problem of quantum mechanics. Discretizing the problem using finite
elements, we were able to numerically compute critical points of the
functional. Our main findings are the following:
\begin{itemize}
\item In one dimension, the only critical point we found is the
potential with exactly one bound-state $V_{\gamma,1,1}$. For all
initial data, the algorithm seems to either converge to this
potential, or to split in separating bumps. This supports the
conjecture that these potentials are the only maximizers of the
functional for $\frac 1 2 \leq \gamma < \frac 3 2$.
\item In two dimensions, the only critical points we found were
radial. For all initial data, the algorithm seems to either converge
to a radial potential, or to split in diverging bumps. Among the
radial potentials we were able to compute, $V_{\gamma,2,1}$ was the
maximizer for $\gamma < 1.16$. After this point, the maximum
corresponds to the semiclassical regime. The natural conjecture is
that $V_{\gamma,d,1}$ is the maximizer for $\gamma < 1.16$, and
therefore that $\gamma_{c,2} \approx 1.16$.
\item In three dimensions, branches of radial potentials with more
than one bound state outperformed $V_{\gamma,d,1}$. This confirms
the theoretical result from \cite{helffer1990riesz}, and provides
new lower bounds (see \figref{fig:3D}). The
natural conjecture is that there is a non-decreasing sequence of
integers $n_{\gamma}$ with $n_{\gamma} \to \infty$ as $\gamma \to 1$
such that the maximizer of the functional \eqref{variational} has
$n_\gamma$ bound states.
\item In dimension $d \geq 4$, the results are similar to the
three-dimensional case, and we conjecture the same behavior.
\end{itemize}
Beyond these results, the study hinted at a very rich and highly
nonlinear behavior of the maximizers of Lieb-Thirring
inequalities. This study is based on a simple numerical method (finite
element discretization). More involved computations could use a more
appropriate Galerkin basis, such as the Gaussian basis functions used
in quantum chemistry. This could allow for a more accurate computation
of potentials with a larger number of bound states, and a more
detailed exploration of the energy landscape.
On the theoretical side, the properties of the functional
\eqref{variational} remain unexplored. An open question is whether one
can prove the existence of maximizers. The behavior of the
maximization algorithm is also an interesting question, with the
separation in bumps particularly interesting. Finally, we believe that
our method could be adapted to generalizations such as models with
positive temperatures or positive density\cite{frank2011positive},
negative values of $\gamma$\cite{dolbeault2006lieb} or polyharmonic
operators $(-\Delta)^{l} + V$ (see \cite{netrusov1996lieb}).
\section*{Acknowledgments}
I would like to thank Mathieu Lewin and Éric Séré for their help and
advice.
\bibliographystyle{plain}
|
1,116,691,500,867 | arxiv | \section{Missing Proofs in Section \ref{sec:pate}}
\begin{theorem}
Let $\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_{\theta}$ be $\alpha$-fair w.r.t.~$(X,\tilde{A},Y)$ and $h(\cdot)$, and $\tilde{A}$ shares the same support $[m]$ as $A$ where, for $a\in [m], x\in \mathcal{X},
y\in\mathcal{Y}$, the difference between the two conditional distributions $A$ and $\tilde{A}$ given
an event $(X=x, Y=x)$ is upper bounded by $\eta > 0$, i.e.,
\begin{equation}\label{eq:t5-assumption-1}
\left\vert \pr{\tilde{A} = a\mid X=x, Y=y}-\pr{A = a\mid X=x, Y=y}\right\vert \leq \eta.
\end{equation}
Then, $\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_{\theta}$ is $\alpha'$-fair w.r.t.~$(X,A,Y) $ and $h(\cdot)$ with:
\begin{equation}
\alpha' = \frac{\eta\cdot B}{\underset{a\in[m]}{\min} \min\left\{\pr{\tilde{A}=a},\pr{A=a}\right\}}+\alpha\,.
\end{equation}
Furthermore, if the probability of $A$ belonging to any class $a\in [m]$ is strictly greater than $\eta$, i.e.,
\(
\pr{A = a}>\eta,\,\forall~a\in[m],
\)
the fairness bound $\alpha'$ can then be given by
\begin{equation*}
\alpha' = \frac{\eta\cdot B}{\underset{a\in[m]}{\min} \pr{A=a}-\eta}+\alpha\,.
\end{equation*}
\end{theorem}
\begin{proof}
Note that, for any $a\in [m]$,
\begin{align}
&\left\vert \pr{\tilde{A}=a}-\pr{A=a}\right\vert\nonumber\\
=~&\left\vert \int g_{X,Y}(x, y)\cdot\left[\pr{\tilde{A} = a\mid X=x, Y=y}-\pr{A = a\mid X=x, Y=y}\right]dxdy \right\vert\label{eq:t5-1}\\
\leq~& \int g_{X,Y}(x, y)\cdot\left\vert \pr{\tilde{A} = a|X=x, Y=y}-\pr{A = a\mid X=x, Y=y}\right\vert dxdy\nonumber\\
\leq~&\int g_{X,Y}(x, y)\cdot \eta~dxdy\label{eq:t5-2}\\
=~&\eta\nonumber\,,
\end{align}
where the function $g_{X,Y}(\cdot, \cdot)$ in Equation \eqref{eq:t5-1} is the probability density function associated with the joint distribution
$(X,Y)$ and Equation \eqref{eq:t5-2} is due to the assumption in Equation \eqref{eq:t5-assumption-1}. If the probability that the random vector $A$ equals $a$ is
strictly greater than $\eta$ for any $a\in [m]$, it follows that
\begin{equation}\label{eq:t5-a-tilde-a}
\pr{\tilde{A}=a}\geq \pr{A=a}-\eta >0\,, \qquad\forall~a\in[m]\,.
\end{equation}
For any group attribute $a\in [m]$,
\begin{align}
&\left\vert\mathbb{E}_{X,Y|A=a } [ h(\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_{\theta}(X),Y)] - \mathbb{E}_{X,Y}
[ h(\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_{\theta}(X),Y)]\right\vert\nonumber\\
\leq~& \left\vert\mathbb{E}_{X,Y|A=a } [ h(\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_{\theta}(X),Y)] - \mathbb{E}_{X,Y|\tilde{A}=a } [ h(\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_{\theta}(X),Y)]\right\vert +\nonumber\\
&\left\vert\mathbb{E}_{X,Y|\tilde{A}=a } [ h(\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_{\theta}(X),Y)] - \mathbb{E}_{X,Y} [ h(\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_{\theta}(X),Y)]\right\vert\nonumber\\
\leq~&\left\vert\mathbb{E}_{X,Y|A=a } [ h(\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_{\theta}(X),Y)] - \mathbb{E}_{X,Y|\tilde{A}=a } [ h(\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_{\theta}(X),Y)]\right\vert+\alpha\label{eq:t5-main-1}\\
=~&\left\vert\int h(\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_{\theta}(x), y)\cdot \left(g_{(X,Y)\mid A=a}(x, y)-g_{(X,Y)\mid \tilde{A}=a}(x, y)\right)dx dy\right\vert+\alpha\nonumber\\
\leq~& \int h(\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_{\theta}(x), y)\cdot g_{X,Y}(x, y) \left\vert\frac{\pr{A = a\mid X=x, Y=y}}{\pr{A = a}}-
\frac{\pr{\tilde{A} = a\mid X=x, Y=y}}{\pr{\tilde{A} = a}}\right\vert dx dy+\alpha\nonumber\\
\leq ~& \int h(\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_{\theta}(x), y)\cdot g_{X,Y}(x, y) \frac{\eta}{\min\left\{\pr{A = a},\pr{\tilde{A} = a}\right\}}dx dy+\alpha\label{eq:t5-main-3}\\
=~&\frac{\eta\cdot \mathbb{E}_{X,Y} [ h(\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_{\theta}(X),Y)]}{\min\left\{\pr{A = a},\pr{\tilde{A} = a}\right\}}+\alpha\,,\nonumber
\end{align}
where Equation \eqref{eq:t5-main-1} is due to the assumption that $\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}$ is $\alpha$-fair w.r.t. $(X, \tilde{A},Y)$ and $h(\cdot)$.
Equation \eqref{eq:t5-main-3} comes from non-negativity of the functions $h(\cdot)$ and $g(\cdot)$ and the assumption in Equation
\eqref{eq:t5-assumption-1}.
Putting things together, we have the following.
\begin{align*}
&\max_{a\in [m]}~\left\vert\mathbb{E}_{X,Y|A=a } [ h(\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_{\theta}(X),Y)]-\mathbb{E}_{X,Y } [ h(\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_{\theta}(X),Y)]\right\vert\\
\leq ~&\max_{a\in [m]}~\frac{\eta\cdot \mathbb{E}_{X,Y}
[ h(\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_{\theta}(X),Y)]}{\min\left\{\pr{A = a},\pr{\tilde{A} = a}\right\}}+\alpha=\frac{\eta\cdot \mathbb{E}_{X,Y} [ h(\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_{\theta}(X),Y)]}{\underset{a\in[m]}{\min} \min\left\{\pr{A = a},\pr{\tilde{A} = a}\right\}}+\alpha\\
\leq ~&\frac{\eta\cdot B}{\underset{a\in[m]}{\min} \min\left\{\pr{A = a},\pr{\tilde{A} = a}\right\}}+\alpha\,,
\end{align*}
where the last inequality comes from the assumption that $B$ is the supremum of the fairness function $h(\cdot)$.
Suppose that the probability that the random vector $A$ equals $i$ is
strictly greater than $\eta$, for any group attribute $a\in [m]$. It follows that
\begin{align*}
&\max_{a\in [m]}~\left\vert\mathbb{E}_{X,Y|A=a } [ h(\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_{\theta}(X),Y)]-\mathbb{E}_{X,Y} [ h(\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_{\theta}(X),Y)]\right\vert\\
\leq ~&\frac{\eta\cdot B}{\underset{a\in[m]}{\min} \min\left\{\pr{A = a},\pr{\tilde{A} = a}\right\}}+\alpha\\
\leq~& \frac{\eta\cdot B}{\underset{a\in[m]}{\min} \pr{A=a}-\eta}+\alpha\,,
\end{align*}
where the last inequality comes from Equation \eqref{eq:t5-a-tilde-a}.
\end{proof}
\begin{corollary}
Suppose that $\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_{\theta}$ is $\alpha$-fair w.r.t. $(X,\tilde{A}, Y)$, along with $h(\cdot)$,
and $\tilde{A}$ is derived by the randomized response mechanism of $A$,
where, for any $a, a'\in[m]$,
\begin{equation*}
\pr{\tilde{A}=a'\mid A=a}=\begin{cases}
\frac{\exp(\epsilon)}{\exp(\epsilon)+(m-1)}\,,&\text{if } a'=a\,,\\
\frac{1}{\exp(\epsilon)+(m-1)}\,,& \text{otherwise}\,.
\end{cases}
\end{equation*}
Then, $\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_{\theta}$ is $\alpha'$-fair w.r.t. $(X, A, Y) $ and $h(\cdot)$, where
\begin{equation*}
\alpha' = \frac{\eta\cdot B}{\underset{a\in[m]}{\min} \min\{\pr{\tilde{A}=a},\pr{A=a}\}}+\alpha\,,\qquad\text{with }
\eta=\frac{m-1}{\exp(\epsilon)+(m-1)}\,.
\end{equation*}
\end{corollary}
\begin{proof}
In order to derive the argument in this corollary, it suffices to show the following: for any
group attribute $a\in [m]$, $x\in \cX$ and $y\in \cY$,
\begin{equation*}
\left\vert \pr{\tilde{A} = a\mid X=x, Y=y}-\pr{A = a\mid X=x, Y=y}\right\vert
\leq \frac{m-1}{\exp(\epsilon)+(m-1)}\,.
\end{equation*}
It follows that, for any group attribute $a\in[m]$, $x\in\mathcal{X}$ and $y\in\mathcal{Y}$,
\begin{align*}
&\left\vert \pr{\tilde{A} = a\mid X=x, Y=y}-\pr{A = a\mid X=x, Y=y}\right\vert\\
=~& \left\vert\sum_{a'=1}^m \pr{\tilde{A} = a\mid A=a'}\cdot \pr{A = a'\mid X=x, Y=y}-
\pr{A = a\mid X=x, Y=y}\right\vert\\
=~&\left\vert\frac{-(m-1)}{\exp(\epsilon)+(m-1)}\cdot \pr{A = a\mid X=x, Y=y}
+\frac{\pr{A \neq a\mid X=x, Y=y}}{\exp(\epsilon)+(m-1)}\right\vert\\
=~&\left\vert\frac{1-m\cdot \pr{A = a\mid X=x, Y=y}}{\exp(\epsilon)+(m-1)}\right\vert\\
\leq ~&\frac{m-1}{\exp(\epsilon)+(m-1)}\,.
\end{align*}
\end{proof}
\begin{theorem}
The voting mechanism $\tilde{v}_Y(\bm{T})$ is $\alpha'$ fair w.r.t. $(X,A,Y)$ and $h(\cdot)$ with
\begin{equation*}
\alpha' = \eta\cdot B\,.
\end{equation*}
\end{theorem}
\begin{proof}
Consider an arbitrary function $q: \mathbb{R}^K\times \mathbb{R}^{\vert\cY\vert}\mapsto\mathbb{R}$ of the outputs of the $K$ teacher networks.
For notational convenience, let $w_q$ denote the function associated with $q$ such that
\begin{equation*}
w_q(\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}^{(1)}(X),\dots,\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}^{(K)}(X),Y,\tau)\coloneqq h(q(\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}^{(1)}(X),\dots,\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}^{(K)}(X),\tau),Y)\,,
\end{equation*}
where $\tau$ follows a multivariate normal distribution $\cN(0, \sigma^2 I_{\vert\cY\vert})$.
It follows that, for any group attribute $a\in [m]$,
\begin{multline*}
\mathbb{E}_{Z_a,\tau}[w_q(Z_a,\tau)]-\mathbb{E}_{Z,\tau}[w_q(Z,\tau)]=\\
\mathbb{E}_{\tau,X,Y|A=a } [ h(q(\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}^{(1)}(X),\dots,\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}^{(K)}(X),\tau),Y)] - \mathbb{E}_{\tau,X,Y} [ h(q(\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}^{(1)}(X),\dots,\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}^{(K)}(X),\tau),Y)]\,,
\end{multline*}
and
\begin{align}
&\left\vert\mathbb{E}_{Z_a,\tau}[w_q(Z_a,\tau)]-\mathbb{E}_{Z,\tau}[w_q(Z,\tau)]\right\vert\nonumber\\
=~&\left\vert\int w_q(\bm{z},\tau)\cdot
\left(g_{Z_a}(\bm{z})-g_Z(\bm{z})\right) d\bm{z}d\tau\right\vert\nonumber\\
=~&\left\vert\int \left(\int w_q(\bm{z},\tau)d\tau\right)\cdot
\left(g_{Z_a}(\bm{z})-g_Z(\bm{z})\right) d\bm{z}\right\vert\nonumber\\
=~& \left\vert\underbrace{\int_{S_a} r(\bm{z})\cdot
\left(g_{Z_a}(\bm{z})-g_Z(\bm{z})\right) d\bm{z}}_{\text{(I)}}+\underbrace{\int_{\RR^{K}\setminus S_a} r(\bm{z})\cdot
\left(g_{Z_a}(\bm{z})-g_Z(\bm{z})\right) d\bm{z}}_{\text{(II)}}\right\vert\label{eq:t4-main-1}\\
\leq ~& \max\left\{
\int_{S_a} r(\bm{z})\cdot
\left(g_{Z_a}(\bm{z})-g_Z(\bm{z})\right) d\bm{z},
~-\int_{\RR^{K}\setminus S_a} r(\bm{z})\cdot
\left(g_{Z_a}(\bm{z})-g_Z(\bm{z})\right) d\bm{z}
\right\}\label{eq:t4-main-1_5}\\
\leq~& \max\left\{B \int_{S_a}
g_{Z_a}(\bm{z})-g_Z(\bm{z}) d\bm{z},~
B\int_{\RR^{K}\setminus S_a}
g_{Z}(\bm{z})-g_{Z_a}(\bm{z}) d\bm{z}\right\}\label{eq:t4-main-2}\\
\leq ~& B\cdot \eta \label{eq:t4-main-3}\,,
\end{align}
where $r(\bm{z})$ is a shorthand for $\int w_q(\bm{z},\tau)d\tau$
and $S_a$ in Equation \eqref{eq:t4-main-1} is defined as $\{\bm{z}\in\RR^{K+1}\mid g_{Z_a}(\bm{z})
\geq g_{Z}(\bm{z})\}$. Equation \eqref{eq:t4-main-1_5} comes from the fact that the term (I) in Equation \eqref{eq:t4-main-1}
is non-negative while
the term (II) is non-positive. Equation \eqref{eq:t4-main-2} comes from the fact
that $B$ is the supremum of the fairness function $h$ (and equivalently $w_q$). Equation \eqref{eq:t4-main-3}
is due to the assumption that the total variation distance between $Z_a$ and $Z$ is upper bounded by $\eta$.
Therefore,
\begin{align*}
&\max_{a\in[m]}~\Large\vert \mathbb{E}_{\tau,X,Y\mid A=a } [ h(q(\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}^{(1)}(X),\dots,\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}^{(K)}(X),\tau),Y)] \\
& \hspace{20pt} - \mathbb{E}_{\tau,X,Y} [ h(q(\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}^{(1)}(X),\dots,\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}^{(K)}(X),\tau),Y)]
\Large\vert\\
=~&\max_{a\in[m]}~\left\vert\mathbb{E}_{Z_a,\tau}[w_q(Z_a,\tau)]-\mathbb{E}_{Z,\tau}[w_q(Z,\tau)]\right\vert\\
\leq~& B\cdot \eta\,,
\end{align*}
where the aggregated classifier $q(\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}^{(1)}(X),\dots,\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}^{(K)}(X), \tau)$ associated with
an arbitrary operation $q$ is $(B\cdot \eta)$-fair w.r.t.
$(X,A,Y)$ and $h(\cdot)$.
Note that, if $q$ is in the following form:
\begin{multline*}
q(\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}^{(1)}(X),\dots,\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}^{(K)}(X), \tau)\coloneqq \\
\underset{Y \in \mathcal{Y} }{\arg\max}~\#_Y(\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}^{(1)}(X),\dots,\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}^{(K)}(X)) + \tau_Y=\tilde{v}_Y(\bm{T}(X))\,,
\end{multline*}
we end up with the voting mechanism $\tilde{v}_Y(\bm{T}(X))$.
Therefore,
the voting mechanism $\tilde{v}_Y(\bm{T}(X))$ is $(B\cdot \eta)$-fair as well.
\end{proof}
\begin{corollary}
Suppose that the random vector $Z$ is independent of $A$, i.e.,
$\{Z_a\}_{a\in[m]}$ and $Z$ are identically distributed.
Equivalently, the random vector $Z=(\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}^{(1)}(X),\dots,\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}^{(K)}(X),Y)$
is independent of $A$.
Then, the voting mechanism $\tilde{v}_Y(\bm{T})$ is perfectly fair w.r.t. $(X,A,Y)$ and $h(\cdot)$.
\end{corollary}
\begin{proof}
Notice that, for any $a\in[m]$, $Z$ now shares the same distribution with $Z_a$, which indicates that the total
variation distance between $Z$ and $Z_a$, $\tv{Z}{Z_a}$, is $0$. Therefore,
the voting mechanism $\tilde{v}_Y(\bm{T}(X))$ is $0$-fair
(and equivalently perfectly fair) w.r.t. $(X,A,Y)$ and $h(\cdot)$.
\end{proof}
\section{Missing Proofs in Section \ref{sec:privacy_computation}}
\begin{theorem}
For any dataset $D_S$,
\(
\displaystyle \max_{D_S \sim D_S'}\| \#_{A}(\bm{T}, D_S) - \#_{A}(\bm{T}, D_S') \|_2 = \sqrt{2}.
\)
\end{theorem}
\begin{proof}
Since $D_S$ differs with $D_S'$ by at most one single sample and the dataset $D_S$ is partitioned into disjoint
parts, that sample can only belong to a single part of the partition, say $D_i$. As a consequence,
the teacher classifier associated with $D_i$, which is $\mathcal{M}} \newcommand{\cN}{\mathcal{N}^{(i)}(X)$,
can change its prediction from $a \in \cA$ to $a' \in \cA$.
Hence, $ \#_a(\bm{T}, D_S)$ decreases by 1 while $ \#_{a'}(\bm{T}, D_S')$ increases by 1 when switching from $D_S$ to $D_S'$. To sum up, the two vectors of voting counts $ \#_A(\bm{T}(X), D_S) $ and $\#_A(\bm{T}(X), D_S') $ differ by at most two entries, which indicates that the $\ell_2$ sensitivity of reporting voting counts is $\sqrt{2}$.
\end{proof}
With similar arguments, we can show that the sensitivity in reporting voting counts in $\mbox{SF}_T$-PATE is also $\sqrt{2}$.
\begin{theorem}
For any dataset $D_S$,
\(
\displaystyle
\max_{D_S \sim D_S'}\| \#_{Y}(\bm{T}, D_S) - \#_{Y}(\bm{T}, D_S') \|_2 = \sqrt{2}.
\)
\end{theorem}
\begin{proof}
When we change the group attributes of one sample $X \in D$, this can affect up to one fair teacher, say
$\mathcal{M}} \newcommand{\cN}{\mathcal{N}^{(i)}$. Like Theorem \ref{thm:sensitivty_sf_s-pate}, in the worst case,
the fair classifier $\mathcal{M}} \newcommand{\cN}{\mathcal{N}^{(i)}$ can change its prediction from $y \in \cY$ to $y' \in \cY$.
Hence, the vectors of voting counts of $\#_{Y}(\bm{T}, D_S) $ and $\#_{Y}(\bm{T}, D_S') $ differ by at most two entries. Therefore, the $\ell_2$ sensitivity of reporting counts in $\mbox{SF}_T$-PATE is $\sqrt{2}$.
\end{proof}
\section{Experimental Results: Additional Details}
We provide additional details of experimental settings, datasets, and empirical results in this section.
\subsection{Experimental settings}
\smallskip\noindent\textbf{Neural network architectures }
For tabular datasets, the underlying classifiers for all models are feed-forward networks with two hidden layers and nonlinear ReLU activation function. We use the same network architecture for teacher models in both of our approaches.
For vision tasks on UTK-Face, the base classifier in to-be deployed models (student model in PATE-based methods) will be a Resnet 50 with 23M parameters. To speed up training time, we will use a simple convolutional network which consists of two convolutional layers followed by three fully connected linear layer for teacher model in both $\mbox{SF}_S$-PATE and $\mbox{SF}_T$-PATE. Another reason is due to the recent work \cite{cho2019efficacy} which shows student performance might degrade unexpectedly when distilled from oversized teachers. Hence, simpler models are preferred for teacher models.
\smallskip\noindent\textbf{Fair solver settings }
For a fair comparison, all models used in experiments are implemented based on a fair solver which relies on Dual Lagrangian framework \cite{fioretto2020lagrangian}. We set the fairness parameter $\alpha = 0$ for maximum fairness, and train the fair solver in 200 epochs, with learning rate of 1e-3, batch size of 32, and the step-sizes to update multipliers is 1e-2.
\smallskip\noindent\textbf{Hyper-parameters settings }
For both of our proposed models, we set $s$ which is the number of samples in $\bar{D}$ for four tabular datasets and UTK Face to be 200 and 500 respectively. We set the trade-off parameters $\lambda = 1e-3$. We optimize the ensemble size $K$ by using the public evaluation set.
For PF-LD, we optimize the clipping bounds in primal and dual step by using the public evaluation set. We set the noise multipliers in primal and dual step to be 5 and 50 respectively \cite{tran2021differentially}.
\smallskip\noindent\textbf{Computing Infrastructure}
All of our experiments on tabular datasets are performed on a personal desktop machine with 16GB RAM and Core I7.
For experiments on UTKFace and ResNet 50 classifier, we ran experiments on a cluster of 300 GPUs with NVIDIA RTX A6000 graphics cards.
\smallskip\noindent\textbf{Software and Libraries}
All models and experiments were written in Python 3.7. All neural network classifier models in our paper were implemented in Pytorch 1.5.0.
The Tensorflow Privacy package was also employed for computing the privacy loss. We also set privacy parameter $\delta= 1e-4$ in all of our experiments.
\smallskip\noindent\textbf{Datasets}
Our paper evaluates all models based on the following four UCI datasets: \emph{Bank}, \emph{Income}, \emph{Parkinsons}, \emph{Credit card} and UTK Face dataset.
A descriptions of each dataset is reported as follows:
\begin{enumerate}
\item \textbf{Income} (Adult) dataset which has 45K samples of 52 features. The task is to predict if an
individual has low or high income, and the group attributes are defined by race:
\emph{White} vs \emph{Non-White} \cite{UCIdatasets}.
\item \textbf{Bank} dataset which has 11K samples of 17 features. The classification task is to predict if a user subscribes
a term deposit or not and the group attributes are defined by age: \emph{people whose age is less than vs greater than 60 years old} \cite{UCIdatasets}.
\item \textbf{Parkinsons} dataset which has 6K samples of 18 features . The classification task is to predict if
a patient has total UPDRS score that exceeds the median value,
and the group attributes are defined by gender: \emph{female vs male} \cite{article}.
\item \textbf{Credit Card} dataset which has 30K samples of 21 features. The task is to predict if
a customer defaults a loan or not. The group attributes are defined by gender:
\emph{female vs male} \cite{creditdataset}.
\item \textbf{UTKFace} dataset which has more than 20K facial images of 48x48 pixels resolution. The task is to predict one of five predefined races of a given facial image. The group attributes are defined based on the following 9 age ranges: 0-10, 10-15, 15-20, 20-25, 25-30, 30-40, 40-50, 50-60, 60-120. \cite{DBLP:conf/bmvc/HwangPKDB20}.
\end{enumerate}
On each dataset we perform standardization to render all input features with zero mean and unit standard deviation. Each dataset was partitioned into three disjoint subsets: train (70\%), evaluation (5\%) and test set(25\%). We use the evaluation set as an available public dataset to optimize the hyper-parameters such as clipping bounds in PF-LD or ensemble size $K$ in our proposed frameworks for a fair comparison.
\subsection{Accuracy, Privacy, and Fairness trade-off}
We provide additional results regarding the accuracy, fairness, and privacy tradeoffs of proposed models variants SF$_S$ and SF$_T$-PATE against the baseline models PF-LD and M on Bank, Income and Parkinson dataset here. . Figure \ref{fig:generalized_dp} illustrates the accuracy (left subplots) and fairness violations $\xi(\theta, \bar{D})$ (right subplots) at varying of the privacy loss $\epsilon$ (x-axis). These figures again demonstrate that both SF-PATE variants achieve better accuracy/fairness tradeoffs for various privacy parameters than baseline models.
\begin{figure*}
\centering
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width = 1.0\linewidth]{new_comparison_bank_AP.pdf}
\caption{Accuracy parity fairness}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width = 1.0\linewidth]{new_comparison_bank_DP.pdf}
\caption{Demographic parity fairness}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width = 1.0\linewidth]{new_comparison_bank_EO.pdf}
\caption{Equalized odds fairness}
\end{subfigure}
\caption{Bank data}
\label{fig:bank_compare}
\end{figure*}
\begin{figure*}
\centering
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width = 1.0\linewidth]{new_comparison_income_AP.pdf}
\caption{Accuracy parity fairness}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width = 1.0\linewidth]{new_comparison_income_DP.pdf}
\caption{Demographic parity fairness}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width = 1.0\linewidth]{new_comparison_income_EO.pdf}
\caption{Equalized odds fairness}
\end{subfigure}
\caption{Income data}
\label{fig:income_compare}
\end{figure*}
\begin{figure*}
\centering
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width = 1.0\linewidth]{new_comparison_parkinsons_AP.pdf}
\caption{Accuracy parity fairness}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width = 1.0\linewidth]{new_comparison_parkinsons_DP.pdf}
\caption{Demographic parity fairness}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width = 1.0\linewidth]{new_comparison_parkinsons_EO.pdf}
\caption{Equalized odds fairness}
\end{subfigure}
\caption{Parkinson data}
\label{fig:parkinson_compare}
\end{figure*}
\section{Introduction}
A number of decision processes, such as criminal assessment, landing, and hiring, are increasingly being aided by machine learning systems.
A critical concern is that the learned models are prone to report outcomes that are discriminatory against some demographic group, including gender, ethnicity, or age.
These concerns have spurred the recent development of fairness definitions and algorithms for decision-making, focusing attention on the tradeoff between the model accuracy and fairness.
To ensure non-discrimination in learning tasks, knowledge of the \emph{sensitive} attributes is essential. At the same time, legal and ethical requirements often prevent the use of this sensitive data. For example, U.S.~law prevents using racial identifiers in the development of models for consumer lending or credit scoring.
Other requirements may be even more stringent, and prevent the collection of protected user attributes, such as for the case of racial attributes in the E.U.~General Data Protection Regulation (GDPR), or require protection of the consumer data privacy.
In this scenario, an important tension arise between (1) the demand for models to be non-discriminatory, (2) the requirement for such model to use the protected attribute during training, as adopted by common fairness models, and (3) the restriction on the data or protected attributes that can be used.
There is thus a need to provide learning models that can both guarantee non discriminatory decisions and protect the privacy of the individuals' group attributes.
To this end, this paper introduces a novel differential privacy framework to train deep learning models that satisfy several group fairness notions, including \emph{equalized odds}, \emph{accuracy parity}, and \emph{demographic parity} \cite{zafar:17,hardt:16,agarwal:18}, while providing privacy of the protected attributes.
The proposed framework, called \emph{Scalable, Fair, and Private Aggregation of Teacher Ensemble (SF-PATE)} is inspired by the success obtained by teachers ensemble frameworks in private learning tasks \cite{papernot:16}. These frameworks transfer the classification knowledge learned from a pretrained ensemble of models (called teachers) to a target model (called student) via a privacy-preserving aggregation process.
This paper exploit this key idea, but rather than transferring the classification capability of the models, it seek to answer a novel question: {\em can fairness properties of a model ensemble be transferred to a target model}?
In addition to providing an affirmative answer to the question above, the key contributions of this paper can be summarized as follows:
({\bf 1}) It proposes two flavors of SF-PATE. The first, called $\text{SF}_T$-PATE allows transferring fairness properties of the ensemble
while protecting the demographic group attributes; the second, called
$\text{SF}_S$-PATE, in addition, also protects the target labels.
({\bf 2}) In addition to analyze the models privacy, the paper provides an analysis on the fairness properties of SF-PATE and shows that unfairness can be bounded in many practical settings.
({\bf 3}) Importantly, SF-PATE allows the adoption of black-box (non-private) fair models during the knowledge transfer process. This feature is unique to the proposed framework and simplifies the development and adoption of new fairness metrics in privacy-preserving ML.
({\bf 4}) Evaluation on both tabular and image datasets show not only that SF-PATE achieves better accuracy, privacy, and fairness tradeoffs with respect to the current state of the art, but it is also significantly faster. These properties render SF-PATE amenable to train large, overparameterized, models, that ensure privacy, accuracy, and fairness simultaneously.
To the best of the authors knowledge, the proposed framework is the first to enable this important combination, rendering SF-PATE a promising tool for privacy-preserving and fair decision making.
\section{Related work}
Privacy and fairness have mostly been studied in isolation. A few exceptions are
represented by the work of Dwork et al.~\cite{dwork:12}, which is one of
the earliest contribution linking fairness and differential privacy,
showing that individual fairness is a generalization of differential privacy.
Cummings et al.~\cite{cummings:19} study the tradeoff between differential privacy and equal opportunity, a notion of fairness that restricts a classifier to produce equal true positive rates across different groups. The work shows that there is no classifier that achieves $(\epsilon,0)$-differential privacy, satisfies equal opportunity, and has accuracy better than a constant classifier.
A recent line of work has also observed that private models may have a negative impact towards fairness~\cite{bagdasaryan:19,pujol:20}.
Building from these observations, Ekstrand et al.~\cite{ekstrand:18} raise questions about the tradeoff between privacy and fairness and, Jagielski et al.~\cite{jagielski2019differentially} and Mozannar et al.~\cite{mozannar2020fair} propose two simple, yet effective algorithms that satisfy {$(\epsilon, \delta)$ and $\epsilon$- differential privacy, respectively, for equalized odds}. Xu et al.~\cite{xu2019achieving} proposes a private and fair logistic regression model making use of the functional mechanism \cite{JMLR:v12:chaudhuri11a}.
Finally, Tran \cite{tran2021differentially} proposed \emph{PF-LD}, which trains a classifier under differential privacy stochastic gradient descent \cite{abadi:16} and imposes fairness constraints using a privacy-preserving extension of the Dual Lagrangian framework proposed in \cite{fioretto2020lagrangian}.
Although this method was shown having substantial improvements on the privacy and fairness trade-offs over existing baselines, it has a high computational cost, due to the operations required to retain both privacy and fairness during training.
In contrast, this work introduces a semi-supervised framework that relies on transferring privacy and fairness from a model ensemble to construct high-quality private and fair classifiers, while also being scalable.
\section{Problem Settings and Goals}
\label{sec:problem_setting}
The paper considers datasets $D$ consisting of $n$ individual data points $(X_i, A_i, Y_i)$, with $i \in [n]$ drawn i.i.d.~from an unknown distribution $\Pi$. Therein, $X_i \in \mathcal{X}$ is a \emph{non-sensitive} feature vector, $A_i \in \mathcal{A}$, with $\mathcal{A} = [m]$ (for some finite $m$) is a demographic group attribute, and $Y_i \in \mathcal{Y}$ is a class label.
The goal is to learn a classifier $\mathcal{M}} \newcommand{\cN}{\mathcal{N}_\theta : \mathcal{X} \to \mathcal{Y}$, where $\theta$ is a vector of real-valued parameters,
that ensures a specified non-discriminatory notion with respect to $A$ while guaranteeing the \emph{privacy} of the group attribute $A$.
The model quality is measured in terms of a non-negative, and assumed differentiable, \emph{loss function} $\mathcal{L}: \mathcal{Y} \times \mathcal{Y} \to \mathbb{R}_+$, and the problem is that of minimizing the empirical risk function (ERM):
\begin{equation}
\label{eq:erm}
\optimal{\theta} \, = \argmin_\theta J(\mathcal{M}} \newcommand{\cN}{\mathcal{N}_\theta, D) = \underset{\theta}{\arg\min}~
\frac{1}{n} \sum_{i=1}^n
\mathcal{L}(\mathcal{M}} \newcommand{\cN}{\mathcal{N}_\theta(X_i), Y_i).
\tag{P}
\end{equation}
\setcounter{equation}{0}
The paper focuses on learning general classifiers, such as neural networks, that satisfy group fairness (as defined next) and protect the disclosure of the group attributes using the notion of differential privacy.
Importantly, the paper assumes that the attribute $A$ is not part of the model input during inference. This is crucial in the application of interest to this work as the protected attributes cannot be disclosed.
\subsection{Fairness}
\label{sec:fairness}
This work constrains a classifier $\mathcal{M}} \newcommand{\cN}{\mathcal{N}$ to satisfy a given group fairness notion under a distribution over $(X, A, Y)$ for the protected attribute $A$.
The class of fairness constraints considered is specified by the following expression
\begin{equation}\label{eq:main_obj}
\left| \mathbb{E}_{X,Y|A=a } [ h(\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}(X),Y)] - \mathbb{E}_{X,Y} [ h(\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}(X),Y)] \right| \leq \alpha\,, \qquad \forall~a \in [m].
\end{equation}
\noindent
The above compares a property for a group of individuals with respect to the whole population and quantifies its difference.
Therein, $h(\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}(X),Y) \in \RR^{c}$, referred here as \emph{fairness function}, defines a target group fairness notion as depicted in more details below. Parameter $\alpha \in \RR_{+}^c$ represents the \emph{fairness violation} and leads to the following fairness metric.
\begin{definition}[$\alpha$-fairness]\label{def:a-fair}
A model $\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}$ is $\alpha$-fair w.r.t.~a joint distribution $(X,A,Y)$ and fairness function $h(\cdot)$ iff:
\begin{equation}
\label{eq:alpha_fairness}
\xi(D, \theta) =
\max_{a \in [m]}~\left\vert\mathbb{E}_{X,Y|A=a} [ h(\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}(X),Y)] - \mathbb{E}_{X,Y} [ h(\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}(X),Y)]
\right\vert \leq \alpha,
\end{equation}
wehre $\xi(D, \theta)$ is referred to as fairness violation.
\end{definition}
When a classifier $\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}$ is $0$-fair, we also say that it is \emph{perfectly fair}.
While the above capture an ample class of group fairness notions, the paper focuses on the following three popular notions.
\begin{itemize}[leftmargin=*, parsep=0pt, itemsep=0pt, topsep=0pt]
\item[$\bullet$] \emph{Demographic Parity}:
$\mathcal{M}} \newcommand{\cN}{\mathcal{N}$'s predictions are statistically independent of the protected attribute $A$. That is, for all $a \in \cA$, $\hat{y} \in \cY$,
\begin{equation*}
\pr{\mathcal{M}} \newcommand{\cN}{\mathcal{N}(X) = \hat{y} \mid A = a}= \pr{\mathcal{M}} \newcommand{\cN}{\mathcal{N}(X) = \hat{y}},
\end{equation*}
which is expressed by setting $h(\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}(X), Y)$ as $\mathbbm{1}\{\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}(X) =1\}$.
\item[$\bullet$] \emph{Equalized odds}: $\mathcal{M}} \newcommand{\cN}{\mathcal{N}$'s predictions are conditionally independent of the protected attribute $A$ given the label $Y$.
That is, for all $a \in \mathcal{A}$, $\hat{y} \in \mathcal{Y}$, and $y \in \mathcal{Y}$,
\begin{equation*}
\pr{\mathcal{M}} \newcommand{\cN}{\mathcal{N}(X) = \hat{y} \mid A=a, Y=y} =
\pr{\mathcal{M}} \newcommand{\cN}{\mathcal{N}(X) = \hat{y} \mid Y=y},
\end{equation*}
which is expressed by:
\begin{equation*}
h(\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}(X),Y) = \begin{bmatrix}
\mathbbm{1}\{\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}(X) =1 ~ \& ~ Y=0 \} \\
\mathbbm{1}\{\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}(X) =1 ~\&~ Y=1\}
\end{bmatrix}.
\end{equation*}
\item[$\bullet$] \textit{Accuracy parity}:
$\mathcal{M}} \newcommand{\cN}{\mathcal{N}$'s mis-classification rate is conditionally independent of the protected attribute.
That is, for any $a\in \cA$,
\begin{equation*}
\pr{\mathcal{M}} \newcommand{\cN}{\mathcal{N}(X) \neq Y \mid A=a} = \pr{\mathcal{M}} \newcommand{\cN}{\mathcal{N}(X) \neq Y},
\end{equation*}
which is captured with $h(\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}(X),Y)=\mathbbm{1}\{\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}(X) \neq Y \}$.
\end{itemize}
Notice that the fairness function $h(\cdot)$ may not be differentiable w.r.t.~$\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}$. Thus, practically, one considers differentiable surrogates; for example, for the case of demographic parity $h(\cdot) \approx \mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}(X)$, and for accuracy parity, $h(\cdot) \approx \mathcal{L}(\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}(X), Y)$.
\subsection{Differential Privacy}
\label{sec:dp}
Differential privacy (DP) \cite{dwork:06} is a strong privacy notion used to quantify and bound the privacy loss of an individual participation to a computation.
Similarly to \cite{jagielski2019differentially,tran2021differentially}, this work focuses on the instance where the protection is restricted to the group attributes only. A dataset $D \in \mathcal{D} = (\mathcal{X} \times \mathcal{A} \times \mathcal{Y})$ of size $n$ can be described as a pair $(D_P, D_S)$ where
$D_P \in (\mathcal{X} \times \mathcal{Y})^n$ describes the \emph{public} attributes and $D_S \in \mathcal{A}^n$ describes the group attributes.
\emph{The privacy goal is to guarantee that the output of the learning model does not differ much when a single individual group attribute is changed}.
The action of changing a single attribute from a dataset $D_S$, resulting in a new dataset $D_S'$, defines the notion of \emph{dataset adjacency}. Two dataset $D_S$ and $D_S' \in \mathcal{A}^n$ are said adjacent, denoted $D_S \sim D_S'$, if they differ in at most a single entry (e.g., in one individual's group membership).
\begin{definition}[Differential Privacy]
A randomized mechanism $\mathcal{M} : \mathcal{D} \to \mathcal{R}$ with domain $\mathcal{D}$ and range $\mathcal{R}$ is $(\epsilon, \delta)$-differentially private w.r.t.~attribute $A$, if, for any dataset $D_P \in (\mathcal{X} \times \mathcal{Y})^n$, any two adjacent inputs $D_S, D_S' \in \mathcal{A}^n$, and any subset of output responses $R \subseteq \mathcal{R}$:
\[
\pr{\mathcal{M}} \newcommand{\cN}{\mathcal{N}(D_P, D_S) \in R }\leq \exp\left(\epsilon\right)\cdot
\pr{\mathcal{M}} \newcommand{\cN}{\mathcal{N}(D_P, D_S') \in R } + \delta.
\]
\end{definition}
\noindent
When $\delta=0$ the algorithm is said to satisfy $\epsilon$-DP.
Parameter $\epsilon > 0$ describes the \emph{privacy loss} of the algorithm,
with values close to $0$ denoting strong privacy, while parameter
$\delta \in [0,1]$ captures the probability of failure of the algorithm to
satisfy $\epsilon$-DP. The global sensitivity $\Delta_f$ of a real-valued
function $f: \mathcal{D} \to \mathbb{R}^k$ is defined as the maximum amount
by which $f$ changes in two adjacent inputs $D$ and $D'$:
\(
\Delta_f = \max_{D \sim D'} \| f(D) - f(D') \|.
\)
In particular, the Gaussian mechanism, defined by
\(
\mathcal{M}(D) = f(D) + \mathcal{N}(0, \Delta_f^2 \, \sigma^2),
\)
where $\mathcal{N}(0, \Delta_f\, \sigma^2)$ is
the Gaussian distribution with $0$ mean and standard deviation
$\Delta_f\, \sigma^2$, satisfies $(\epsilon, \delta)$-DP for
$\delta > \frac{4}{5} \exp(-(\sigma\epsilon)^2 / 2)$ \cite{dwork:06}.
\section{Learning Private and Fair Models: Challenges}
\label{sec:limitation}
When interpreted as constraints of the form \eqref{eq:alpha_fairness}, fairness properties can be explicitly imposed to problem \eqref{eq:erm}, resulting in a constrained empirical risk problem. The resolution of this problem can be attacked using primal-dual solutions, such as the Lagrangian Dual framework proposed by Tran et al.~\cite{tran2021differentially}. However, the privacy analysis becomes convoluted, often requiring the use of approximate solutions to simplifying it (see \cite{tran2021differentially}). More importantly, calibrating the appropriate amount of noise to enforcing DP requires the use of specialized solutions (e.g., to assess the sensitivity associated with the fairness constraints) and thus prevents the adoption of this framework when new fairness notions arise, as the paper shows in the evaluation, ultimately increasing the adoption barrier.
Finally, these methods are adaptations of celebrated DP-SGD \cite{abadi:16} algorithm, whose private computations rely on sequential clipping operations, and, when used in combination with fairness constraints, produce a very slow training process. Unfortunately, the adoption of fairness constraints inhibits the effectiveness of frameworks like JAX \cite{jax2018github} and Opacus \cite{opacus}, that use vectorization to speed up operations in DP-SGD.
Alternatively, and as proposed in \cite{mozannar2020fair}, one can adopt a pre-processing step to perturb the group attributes and then use a non-private, fair ML model to post-process the private data. This scheme, however has been shown to introduce an unnecessarily large amount of noise, especially for multigroup ($m > 2$) tasks \cite{tran2021differentially}.
The approach proposed in this paper avoids these difficulties by providing a teachers ensemble model that {\bf (1)} provides state-of-the-art accuracy, privacy, and fairness tradeoffs, {\bf (2)} adds negligible computational cost to standard (non-private) training, and {\bf (3)} directly exploits the existence of fair (non-private) algorithms rendering it practical even in the adoption of new fairness notions or when the fairness to be enforced would complicate the privacy analysis.
\section{Scalable, Fair, and Private Aggregation of Teacher Ensembles}
\label{sec:pate}
This section discusses two algorithms to transfer fairness constraints during the private learning process. Both algorithms rely on the presence of an ensemble of \emph{teacher} models $\bm{T} = \{\mathcal{M}} \newcommand{\cN}{\mathcal{N}^{(i)}\}_{i=1}^K$, with each $\mathcal{M}} \newcommand{\cN}{\mathcal{N}^{(i)}$ trained on a non-overlapping portion $D_i$ of the dataset $D$. This ensemble is used to transfer knowledge to a student model $\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_\theta : \cX \to \cY$. The student model $\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}$ is trained using a dataset $\bar{D} \subseteq D_P$ whose samples are drawn i.i.d.~from
the same distribution $\Pi$ considered in Section \ref{sec:problem_setting} but whose protected group attributes are unrevealed.
The framework, and the two variants introduced next, are depicted schematically in Figure \ref{fig:framework}.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{framework.pdf}
\caption{Illustration of the SF-PATE framework. Green (red) colored text and labels depict the SF$_S$ (SF$_T$) version, in which the student (teachers ensemble) is trained under fairness constraints.}
\label{fig:framework}
\end{figure}
\subsection{$\text{SF}_S$-PATE: Semi-supervised Transfer Learning with Fair Student Model}
\label{sec:first_pate}
The first algorithm presented, called $\mbox{SF}_S$-PATE, trains a student model with privacy-preserving group attributes chosen by an ensemble of teachers. Subscript $_S$ in the algorithm's name stands for ``student'' to emphasize that is the student to enforce the fairness constraints during training, as the paper illustrates next.
The teachers ensemble $\bm{T} = \{\mathcal{M}} \newcommand{\cN}{\mathcal{N}^{(i)}\}_{i=1}^K$ is comprised of models $\mathcal{M}} \newcommand{\cN}{\mathcal{N}^{(i)}\!:\!\cX \to \cA$, whose goal is to predict the group attributes (and not the labels, as classically done in classification tasks) given a sample's features.
Note that, importantly, the teacher models are standard classifiers (neither private nor fair).
Their role is to transfer the information of the group attributes associated with the samples $(X_i, Y_i) \in \bar{D}$ provided by the student. We will see next how this information can be shared in a privacy-preserving manner.
The student model $\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_\theta: \cX \to \cY$ is learned by minimizing the following regularized and constrained empirical risk function:
\begin{subequations}
\label{eq:sf_s}
\begin{align}
\label{eq:sf_s_A}
\hat{\theta} = \argmin_\theta &
\sum_{(X_i, Y_i) \in \bar{D}} \mathcal{L}(\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}(X_i), Y_i) +\lambda \| \theta - \theta^* \|^2_2 \\
\label{eq:sf_S_B}
\text{s.t.}\quad
&
\xi \left( \{X_i, \tilde{v}_A\left( \bm{T}(X_i) \right), Y_i \}_{(X_i, Y_i) \in \bar{D}}, \theta \right) \leq \alpha,
\end{align}
\end{subequations}
\noindent
where $\tilde{v}_A: \cA^K \to \cA$ is a privacy-preserving voting scheme returning the group attribute $\tilde{A}$ chosen by the teachers ensemble:
\begin{equation}
\tilde{A} = \tilde{v}_A(\bm{T}(X)) = \argmax_{a \in \cA } \{ \#_a(\bm{T}(X))
+ \mathcal{N}(0, \sigma^2) \},
\label{eq:sf_s_pate_gnmax}
\end{equation}
which perturbs the reported counts $\#_a(\bm{T}(X))\!=\! \left\vert\{k \in [K] \mid \mathcal{M}} \newcommand{\cN}{\mathcal{N}^{(k)}_{\theta}(X) = a\}\right\vert$ associated to group $a \in \cA$ with additive Gaussian noise of zero mean and standard deviation $\sigma$.
Problem \eqref{eq:sf_s} minimizes the standard empirical risk function (first component of \eqref{eq:sf_s_A}), while encouraging
the student model parameters $\theta$ to be close to the optimal, not fair, parameters $\theta^*$ (second component of \eqref{eq:sf_s_A}).
Note that $\theta^*$ is obtained by simply solving problem \eqref{eq:erm} using dataset $\bar{D}$. The regularization parameter $\lambda>0$ controls the trade-off between accuracy (large $\lambda$ values) and fairness (small $\lambda$ values).
The fairness constraints expressed in Equation \eqref{eq:sf_S_B} can be enforced through the adoption of off-the-shelf techniques. In particular, this paper relies on the use of the Lagrangian Dual deep learning framework of Fioretto et al.~\cite{fioretto2020lagrangian} to enforce such constraints.
\begin{wrapfigure}[10]{r}{0.43\textwidth}
\vspace{-20pt}
\includegraphics[width=\linewidth]{new_income_accurac_noisy_z.pdf}
\vspace{-25pt}
\caption{{\small Private group attributes accuracy.}}
\label{fig:accuracy_noisy_z}
\end{wrapfigure}
Figure \ref{fig:accuracy_noisy_z} illustrates the ability of the teachers ensemble to generate high-quality privacy-preserving group attributes: $\tilde{A} = \tilde{v}_A(\bm{T}(X))$, especially under tight privacy regimes. The figure compares the expected accuracy of the private group attributes $\pr{\tilde{A}_i = A_i}$ for different sizes $K$ of the ensemble on the Income dataset (See Appendix). It additionally illustrates the results obtained by a baseline model (\emph{M}) which adds calibrated noise to the group attributes using randomized response, which is the foundation of a state of the art method to train private and fair classifiers \cite{mozannar2020fair}.
\smallskip
The rest of this section provides theoretical arguments to justify this surprising performance of $\mbox{SF}_S$-PATE, as well as shed lights on the fairness guarantees it can provide.
The proofs of all theorems are reported in the Appendix.
Note that $\mbox{SF}_S$-PATE relies on a (non-private) fair classifier to train the student model. While the performance of many such fair classifiers has been established in the literature, a natural question arises when adopting the proposed framework regarding the impact of the privacy-preserving voting scheme to fairness.
The next theorem bounds the fairness violation of the student $\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_{\theta}$ w.r.t.~$(X,A,Y)$, when the group attribute $A$ and its privacy-preserving counterpart $\tilde{A}$ are close enough to each other statistically.
\begin{theorem}\label{thm:fair-bound-approach-1}
Let $\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_{\theta}$ be $\alpha$-fair w.r.t.~$(X,\tilde{A},Y)$ and $h(\cdot)$, and $\tilde{A}$ shares the same support $[m]$ as $A$ where, for $a\in [m], x\in \mathcal{X},
y\in\mathcal{Y}$, the difference between the two conditional distributions $A$ and $\tilde{A}$ given
an event $(X=x, Y=y)$ is upper bounded by $\eta > 0$, i.e.,
$\left\vert \pr{\tilde{A} = a\mid X=x, Y=y}-\pr{A = a\mid X=x, Y=y}\right\vert \leq \eta$.
Then, $\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_{\theta}$ is $\alpha'$-fair w.r.t.~$(X,A,Y) $ and $h(\cdot)$ with:
\begin{equation}
\alpha' = \frac{\eta\cdot B}{\underset{a\in[m]}{\min} \min\left\{\pr{\tilde{A}=a},\pr{A=a}\right\}}+\alpha\,.
\end{equation}
Furthermore, if the probability of $A$ belonging to any class $a\in [m]$ is strictly greater than $\eta$, i.e.,
\(
\pr{A = a}>\eta,\,\forall~a\in[m],
\)
the fairness bound $\alpha'$ is given by
\begin{equation*}
\alpha' = \frac{\eta\cdot B}{\underset{a\in[m]}{\min} \pr{A=a}-\eta}+\alpha\,.
\end{equation*}
\end{theorem}
The next result is associated with private group attribute $\tilde{A}$ generated by the aforementioned randomized response mechanism
as a privacy-preserving step of the baseline model \emph{M} \cite{jagielski2019differentially} and also
bounds fairness guarantees for the baseline model \emph{M}.
\begin{corollary}\label{cor:rr_mechanism}
Suppose that $\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_{\theta}$ is $\alpha$-fair w.r.t. $(X,\tilde{A}, Y)$, along with $h(\cdot)$,
and $\tilde{A}$ is derived by the randomized response mechanism of $A$,
where, for any $a, a'\in[m]$,
\begin{equation*}
\pr{\tilde{A}=a'\mid A=a}=\begin{cases}
\frac{\exp(\epsilon)}{\exp(\epsilon)+(m-1)}\,,&\text{if } a'=a\,,\\
\frac{1}{\exp(\epsilon)+(m-1)}\,,& \text{otherwise}\,.
\end{cases}
\end{equation*}
Then, $\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_{\theta}$ is $\alpha'$-fair w.r.t. $(X, A, Y) $ and $h(\cdot)$, where
\begin{equation*}
\alpha' = \frac{\eta\cdot B}{\underset{a\in[m]}{\min} \min\{\pr{\tilde{A}=a},\pr{A=a}\}}+\alpha\,,\qquad\text{with }
\eta=\frac{m-1}{\exp(\epsilon)+(m-1)}\,.
\end{equation*}
\end{corollary}
Corollary \ref{cor:rr_mechanism} provides a fairness guarantee of the student model
$\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_{\theta}$ w.r.t. the original group attribute $A$, when this model
is trained for achieving a given fairness violation. This is compared with the privacy-preserving counterpart $\tilde{A}$, which is
an output of the randomized response mechanism employed by the baseline model \emph{M}.
This result sheds light on how fairness guarantees
w.r.t. private group attributes can be transferred to those w.r.t. original group attributes.
Furthermore, when compared with the baseline model \emph{M}, $\mbox{SF}_S$-PATE makes the privacy-preserving group attributes $\tilde{A}$ much closer to their original counterparts $A$ under the same privacy budgets, as can be seen in Figure \ref{fig:accuracy_noisy_z}.
Finally, by Theorem \ref{thm:fair-bound-approach-1}, it can be inferred that SF-PATE is likely to achieve better fairness guarantees than those provided by models using randomized response, such as state-of-the-art model \emph{M}.
\subsection{$\mbox{SF}_T$-PATE: Semi-supervised Transfer Learning with Fair Teachers Models}
\label{sec:second_pate}
While the previous algorithm focused on transferring the privacy-preserving group attributes from an ensemble of teachers to a fair student model, this section introduces an SF-PATE variant to transfer fairness {directly}, through the semi-supervised learning scheme.
The proposed algorithm, called $\mbox{SF}_T$-PATE, trains a student model from an ensemble of fair teacher models. Subscript $_T$ in the algorithm's name stands for ``teachers'' to emphasize that the fairness constraints are enforced by the teachers and transferred to the student during training.
The teachers ensemble $\bm{T} = \{\mathcal{M}} \newcommand{\cN}{\mathcal{N}^{(i)}\}_{i=1}^K$ is composed of pre-trained classifiers $\mathcal{M}} \newcommand{\cN}{\mathcal{N}^{(i)}: \cX \to \cY$ that are non-private but fair over their training data $D_i$. Each SF$_T$-PATE teacher solves an empirical risk minimization problem subject to fairness constraint:
\[
\theta_i = \argmin_\theta \sum_{(X, Y) \in D_i} \mathcal{L}(\mathcal{M}} \newcommand{\cN}{\mathcal{N}_\theta(X_i), Y_i) \qquad \text{s.t.} \;\;
\xi(D_i, \theta) \leq \alpha.
\]
The implementation uses the Lagrangian dual method of \cite{fioretto2020lagrangian} to achieve this goal. This is again an important aspect of the SF-PATE framework since, by relying on black-box fair (and not private) algorithms, it decouples the dependency of developing joint private and fair analysis.
The role of the teachers is to transfer fairness via model prediction to the student. The student model $\bar{\mathcal{M}} \newcommand{\cN}{\mathcal{N}}_\theta : \cX \to \cY$ is learned by minimizing a standard empirical risk function:
\begin{equation}
\min_{\theta} \sum_{X \in \bar{D} }
\mathcal{L}\left( \mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}(X), \tilde{v}_Y(\bm{T}(X)\right)
+ \lambda \| \theta - \theta^*\|^2_2,
\label{eq:main_sf_t_pate}
\end{equation}
where $\theta^*$ represents the student model parameters obtained solving the standard classification task (without fairness considerations) \eqref{eq:erm} on $\bar{D}$ and the private voting scheme $\tilde{v}_Y: \cY^K \to \cY$ selects the most voted target label $\tilde{Y}$ by the teachers ensemble:
\begin{equation}
\tilde{Y} = \tilde{v}(\bm{T}(X)) = \argmax_{y \in \cY} \{\#_y(\bm{T}(X) + \mathcal{N}(0, \sigma^2)\}.
\label{eq:sf_t_pate_gnmax}
\end{equation}
Notice that, in addition to protecting the privacy of the sensitive group information $A$, a biproduct of the above scheme is to also protect the labels $Y$ {when $\lambda\!=\!0$}. Thus, SF$_T$-PATE can be adopted in contexts where both the protected group and the labels are sensitive information.
Finally, note that the voting scheme above, emulates the GNMAX framework proposed in \cite{papernot:16}, but the two have fundamentally different goals: GNMAX aims at protecting the participation of individuals into the training data, while SF$_T$-PATE aims at privately transferring the fairness properties of the teachers ensemble to a student model.
\medskip
Next, the paper sheds light on to which extent the fairness properties of the ensemble can be transferred to a student model, and provides theoretical insights to bound the fairness violations induced by the voting mechanism $\tilde{v}_Y$, adopted by SF$_T$-PATE.
For notational convenience, let $B\coloneqq \sup_{y\in \mathcal{Y}, y'\in \mathcal{Y}}h(y, y')$ denote the supremum of the fairness function $h(\cdot)$.\footnote{One can consider $B=1$ for the fairness functions considered in Section \ref{sec:fairness}.}
Let $Z$ be the random vector $(\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}^{(1)}(X),\dots,\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}^{(K)}(X),Y)$ while $Z_a$ the conditional random
vector given the group label $a\in[m]$, i.e.,
$Z_a\coloneqq (\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}^{(1)}(X),\dots,\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}^{(K)}(X),Y)\mid A=a$.
\begin{definition}[Total variation distance]
Let $Z, Z'$ be two probability distributions over a shared probability
space $\Omega$. Then, the total variation distance between them, denoted $\tv{Z}{Z'}$, is given by
\begin{align*}
\tv{Z}{Z'} &\coloneqq \sup_{S\subseteqq \Omega}~\left\vert \pr{Z\in S}-\pr{Z'\in S}\right\vert=\sup_{S\subseteqq \Omega}~\left\vert\int_{S} \left(g_Z(\bm{z})-g_{Z'}(\bm{z})\right)d\bm{z}\right\vert\,,
\end{align*}
where $g_Z(\cdot)$ and $g_{Z'}(\cdot)$ are the probability density functions associated with $Z$ and $Z'$
\end{definition}
The following results assume that, for each group attribute $a\in [m]$, the total variation distance between $Z$ and $Z_a$ is bounded from the above by $\eta$, i.e., $\tv{Z}{Z_a} \leq \eta$.
The following theorem bounds the fairness violation of the voting mechanism $\tilde{v}_Y(\bm{T})$ when the joint distributions of the teachers ensemble and labels $Y$ are roughly similar across the different group classes.
It shed light on the conditions required for the ensemble votes to transfer fairness knowledge accurately.
\begin{theorem}
\label{thm:gen_ensemble}
The voting mechanism $\tilde{v}_Y(\bm{T})$ is $\alpha'$ fair w.r.t. $(X,A,Y)$ and $h(\cdot)$ with
\begin{equation*}
\alpha' = \eta\cdot B\,.
\end{equation*}
\end{theorem}
\noindent
The next corollary is a direct consequence of Theorem \ref{thm:gen_ensemble} and provides a sufficient
condition for perfect fairness of the voting mechanism $\tilde{v}_Y(\bm{T})$.
\begin{corollary}
\label{thm:fair_ensemble}
Suppose that the random vector $Z$ is independent of $A$, i.e.,
$\{Z_a\}_{a\in[m]}$ and $Z$ are identically distributed.
Equivalently, the random vector $Z=(\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}^{(1)}(X),\dots,\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}^{(K)}(X),Y)$
is independent of $A$.
Then, the voting mechanism $\tilde{v}_Y(\bm{T})$ is perfectly fair w.r.t. $(X,A,Y)$ and $h(\cdot)$.
\end{corollary}
\noindent
In summary, the results above provide sufficient conditions for the voting scheme of the ensemble to achieve a required fairness guarantees, which will be transferred to the subsequential student model.
\section{Privacy analysis}
\label{sec:privacy_computation}
The theoretical considerations above relate with the ability of the proposed models to preserve fairness. It remains to be shown that models are also private.
The section first derives the sensitivity of the voting schemes adopted by $\mbox{SF}_S$- and $\mbox{SF}_T$-PATE.
Given sensitive dataset $D_S$, define
$\#_A(\bm{T}, D_S) = \langle \#_a(\bm{T}(X_i) \,|\, X_i \in D_S, a \in \cA \rangle$ and
$\#_Y(\bm{T}, D_S) = \langle \#_y(\bm{T}(X_i) \,|\, X_i \in D_S, y \in \cY \rangle$, as the vectors of vote counts collected by the ensemble in $\mbox{SF}_S$- and $\mbox{SF}_T$-PATE, respectively.
\begin{theorem}
\label{thm:sensitivty_sf_s-pate}
For any dataset $D_S$,
\(
\displaystyle \max_{D_S \sim D_S'}\| \#_{A}(\bm{T}, D_S) - \#_{A}(\bm{T}, D_S') \|_2 = \sqrt{2}.
\)
\end{theorem}
With similar arguments, it can be shown that the sensitivity in reporting voting counts in $\mbox{SF}_T$-PATE is also $\sqrt{2}$.
\begin{theorem}
For any dataset $D_S$,
\(
\displaystyle
\max_{D_S \sim D_S'}\| \#_{Y}(\bm{T}, D_S) - \#_{Y}(\bm{T}, D_S') \|_2 = \sqrt{2}.
\)
\end{theorem}
Both proofs (detailed in the Appendix) rely on bounding the maximal difference of the count vectors $\#_A$ and $\#_Y$ arising across neighboring datasets.
\begin{proposition}
\label{prop:3}
The voting scheme with private Gaussian noise $\mathcal{N}(0, \sigma^2)$ and sensitivity of $\sqrt{2}$ satisfies $(\gamma,\nicefrac{\gamma}{\sigma^2}) $-RDP for all $\gamma \geq 1$.
\end{proposition}
The proof of this proposition follows directly from \cite{mironov2017renyi}. Thus, for fixed standard deviation $\sigma$ and any $X \in \bar{D}$, the private voting schemes $\tilde{v}_{A}(\bm{T}(X))$ and $\tilde{v}_{Y}(\bm{T}(X))$ both satisfy $(\gamma,\nicefrac{\gamma}{\sigma^2}) $-RDP.
Note that the training uses $|\bar{D}| = s$ samples. The following results report composition bounds used to derive the final privacy cost.
\begin{theorem}[Composition for RDP]
\label{app:rdp_comp}
Let $\mathcal{M}} \newcommand{\cN}{\mathcal{N}$ be a mechanism consisting of a sequence of adaptive mechanisms $\mathcal{M}} \newcommand{\cN}{\mathcal{N}_1, \mathcal{M}} \newcommand{\cN}{\mathcal{N}_2, \ldots, \mathcal{M}} \newcommand{\cN}{\mathcal{N}_k$, such that for any $i \in [k]$, $\mathcal{M}} \newcommand{\cN}{\mathcal{N}_i$ guarantees $(\gamma, \epsilon_i)$-RDP. Then $\mathcal{M}} \newcommand{\cN}{\mathcal{N}$ guarantees $(\gamma, \sum_{i=1}^k \epsilon_i)$-RDP.
\end{theorem}
\begin{theorem}[From RDP to DP]
\label{thm:rdp_2_dp}
Let $\mathcal{M}} \newcommand{\cN}{\mathcal{N}$ be an $(\gamma, \epsilon)$-RDP mechanism, then $\mathcal{M}} \newcommand{\cN}{\mathcal{N}$ is also $(\epsilon + \frac{\log \nicefrac{1}{\delta}}{\gamma-1}, \delta)$-DP for any $\delta \in (0,1)$.
\end{theorem}
\noindent
From Theorem \ref{app:rdp_comp}, Proposition \ref{prop:3} and Theorem \ref{thm:rdp_2_dp}, and for fixed noise parameter $\sigma$ used in the voting scheme, the SF-PATE algorithms satisfy $( \nicefrac{s\gamma}{\sigma^2} +\frac{\log \nicefrac{1}{\delta}}{\gamma -1}, \delta)$-DP.
\section{Experiments}
\label{sec:experiments}
This section evaluates the performance of the proposed SF-PATE algorithms against the prior approaches of \cite{tran2021differentially} and \cite{mozannar2020fair}, named {\em PF-LD} and {\em M}, respectively, and representing the current state-of-the-art for learning private and fair classifiers in the context studied in this paper.
In addition to assess the competitiveness of the proposed solutions, the evaluation focuses on two key aspects: (1) It shows that SF-PATE
can naturally handle new fairness notions, even when no viable privacy analysis exists about these notions. (2) It shows that SF-PATE has a low computational overhead
compared to classical (non-private, non-fair) classifiers, rendering it a practical choice for the training of very large models.
These two properties are unique to SF-PATE, enabling it to be applicable to a broad class of challenging decision tasks.
\smallskip\noindent{\textbf{Datasets and Settings}.}
The evaluation is conducted using four UCI tabular datasets: Bank, Parkinson, Income and Credit Card \cite{UCIdatasets,article}, and UTKFace \cite{DBLP:conf/bmvc/HwangPKDB20}, a vision dataset.
The latter is used to demonstrate the scalability of SF-PATE when trained on a ResNet 50 classifier with over 23 million parameters. All experiments are repeated 100 times using different random seeds.
\smallskip\noindent{\textbf{Models and Hyperparameters}.}
To ensure a fair comparison, the experimental analysis uses the same architectures and parameters for all models (including the baselines models \emph{PF-LD} \cite{tran2021differentially} and \emph{M} \cite{jagielski2019differentially}).
For tabular datasets, the underlying classifier is a feedforward neural network with two hidden layers and nonlinear ReLU activation.
The fair, non-private, classifiers adopted by the two SF-PATE variants implement a Lagrangian Dual scheme \cite{fioretto2020lagrangian}, which is also the underlying scheme adopted by baseline models.
For vision tasks on the UTK-Face dataset, the evaluation uses a Resnet 50 classifier.
A more detailed description of these approaches, the settings adopted, and the datasets is deferred to the Appendix.
\vspace{-8pt}
\subsection{Accuracy, Privacy, and Fairness trade-off}
This section compares the accuracy, fairness, and privacy tradeoffs of the proposed models variants SF$_S$- and SF$_T$-PATE against the baseline models \emph{PF-LD} and \emph{M} on the tabular datasets. Figure \ref{fig:credit_card_compare} illustrates the accuracy (top subplots) and fairness violations $\xi(\theta, \bar{D})$ (bottom subplots) at varying of the privacy loss $\epsilon$ (x-axis).
The figures clearly illustrate that both SF-PATE variants achieve better accuracy/fairness tradeoffs for various privacy parameters.
The property of retaining high accuracy while low fairness violation is especially relevant in the tight privacy regime adopted ($\epsilon < 1$).
Additional results on other datasets are deferred to the Appendix, and show similar trends.
\begin{figure*}[tb]
\centering
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width = 1.0\linewidth]{new_comparison_credit_card_AP.pdf}
\caption{Accuracy parity fairness}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width = 1.0\linewidth]{new_comparison_credit_card_DP.pdf}
\caption{Demographic parity fairness}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width = 1.0\linewidth]{new_comparison_credit_card_EO.pdf}
\caption{Equalized odds fairness}
\end{subfigure}
\caption{Credit card data}
\label{fig:credit_card_compare}
\end{figure*}
First, notice that, consistently with the privacy-preserving ML literature showing that teachers ensemble models can outperform, in terms of accuracy, DP-SGD based models \cite{papernot:16,uniyal2021dp}, the SF-PATE models typically outperform the DP-SGD based \emph{FP-LD} model. Remarkably, both SF-PATE variants also report lower fairness violations, for the fairness notions analyzed, indicating the strength of this approach.
Additionally, recall that both $\mbox{SF}_S$-PATE and \emph{M} generate privacy-preserving group features and feed them to a fair model. However, in contrast to \emph{M}, the model ensemble used in $\mbox{SF}_S$-PATE exploits the relationship between the sample features $X$ and its associated group information $A$ to derive more accurate private group information $\tilde{A}$ (as already shown in Figure \ref{fig:accuracy_noisy_z}). This results in student models which are often more accurate than the baseline \emph{M}, while also being fairer. The second SF-PATE variant, $\mbox{SF}_T$-PATE, which operates by privately transferring the fairness knowledge from a teachers ensemble to a student, is found to always outperform both \emph{M} and \emph{PF-LD} on tabular datasets.
Finally, our analysis also shows that the average accuracy of the SF-PATE models is within 2\% of their non-private counterpart.
\vspace{-8pt}
\subsection{Handling new group fairness notions}
Next, the evaluation focuses on the ability of the SF-PATE framework to handle arbitrary fairness notions, even if a privacy analysis is missing, as long as a fair model can be derived. This is due to the framework property to rely on the use of black-box (non-private but fair) models.
This is in sharp contrast with state-of-the-art model \emph{PF-LD}, that requires the development of a privacy analysis for each fairness property to be enforced, in order to calibrate the amount of noise to apply in both primal and dual step.
To demonstrate this benefit, this section first introduces a new fairness notion, which generalizes demographic parity.
\begin{definition}[Generalized demographic parity]
It requires the distribution over the predictions of $\mathcal{M}} \newcommand{\cN}{\mathcal{N}$ to be statistically independent of the protected group attribute A. That is, for all $a \in \cA$, and $\eta \in [0, 1]$
\(
\Pr( \mathcal{M}} \newcommand{\cN}{\mathcal{N}(X) \geq \eta | A = a ) = \Pr( \mathcal{M}} \newcommand{\cN}{\mathcal{N}(X) \geq \eta ).
\)
\end{definition}
Note that the above is a general version of demographic parity, which states that $\pr{\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}(X) \geq 0.5 \mid A= a} = \pr{\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}(X) \geq 0.5}$.
Generalized demographic parity is useful in settings where the decision threshold (e.g, 0.5 above) might not be available at the training time. Matching the distribution of score functions (e.g., credit or income scores) among different groups attributes guarantees demographic parity fairness regardless of the decision threshold adopted. Such fairness constraint can be implemented by equalizing different order statistics of the score functions between group and population level:
$$
\mathbb{E}} \newcommand{\RR}{\mathbb{R}[\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}(X)^h | A=a] = \mathbb{E}} \newcommand{\RR}{\mathbb{R}[\mathcal{M}} \newcommand{\cN}{\mathcal{N}_{\theta}(X)^h] \quad \forall \ a \in [m], h = 1, 2, \ldots, H
$$
Practically, we let H = 2, and use the Lagrangian Dual method of \cite{fioretto2020lagrangian} to enforce these constraints during training.
{\em Notice that it is highly non-trivial to derive a privacy analysis for such fairness notion}--the PF-LD model only does so for $H=1$, and, thus, not viable in this setting.
\begin{figure*}[tb]
\centering
\begin{subfigure}[b]{0.46\textwidth}
\includegraphics[width = 1.0\linewidth]{compare_income.pdf}
\caption{Income dataset}
\end{subfigure}
\begin{subfigure}[b]{0.46\textwidth}
\includegraphics[width = 1.0\linewidth]{compare_bank.pdf}
\caption{Bank dataset}
\end{subfigure}
\begin{subfigure}[b]{0.46\textwidth}
\includegraphics[width = 1.0\linewidth]{compare_parkinsons.pdf}
\caption{Parkinson dataset}
\end{subfigure}
\begin{subfigure}[b]{0.46\textwidth}
\includegraphics[width = 1.0\linewidth]{compare_credit_card.pdf}
\caption{Credit card dataset}
\end{subfigure}
\caption{Comparison between our proposed methods with the baseline model M for generalized demographic parity fairness}
\label{fig:generalized_dp}
\end{figure*}
Figure \ref{fig:generalized_dp} reports the accuracy and fairness violations obtained by the SF-PATE models and the baseline model \emph{M} which adopt, as a post-processing step, a non-private but fair classifier.
Fairness is evaluated in terms of the Wasserstein distance between the score functions of different groups. The smaller the distance the lower the fairness violation. The plots clearly illustrates the advantages of SF-PATE in terms of both accuracy and fairness when compared to model \emph{M}. Remarkably, the fairness violations reported by SF-PATE are often significantly lower than those reported by model \emph{M} for various privacy loss parameters $\epsilon$.
{\em This is significant: The development of private and fair analysis is a non trivial task and framework like the proposed SF-PATE immediately lower the accessibility barrier for non-privacy expert users to develop both private and fair ML models.}
\subsection{Computational Time and Scalability}
\begin{figure}[t]
\centering
\begin{subfigure}[b]{0.8\textwidth}
\includegraphics[width = 1.0\linewidth]{new_ecml_comparison_age_bins_ethnicity.pdf}
\end{subfigure}
\caption{Accuracy/fairness of our models on UTKFace in which the base classifier is ResNet 50}
\label{fig:utk_resnet}
\end{figure}
This section demonstrates another key benefits of SF-PATE models: their ability to scale to large data and perform well on very deep networks. The experiments use a ResNet 50 ($>$ 23M parameters) to illustrate. In these experiments, PF-LD was unable to train even a single epoch ({in 1h}) due to the use of computational expensive per-sample gradient clipping performed during training. This renders such model unusable for many realistic settings. The comparison thus focuses on SF-PATE and \emph{M}.
\begin{wrapfigure}[10]{r}{0.45\textwidth}
\vspace{-0pt}
\centering
\parbox{\linewidth}
{
\resizebox{\linewidth}{!}
{
\begin{tabular}{r | rrrr} \toprule
{\bf Dataset} & $ \mbox{SF}_S$-PATE & $\mbox{SF}_T$-PATE
& \emph{M} & PF-LD \\ \midrule
Bank & 14 & 13 & 31 & 116 \\
Parkinson & 8 & 8 & 17 & 31 \\
Income & 55 & 56 & 129 & 1234 \\
Credit Card & 30 & 31 & 76 & 575 \\
UTK dataset & 1669 & 1662 & 3248 & N/A \\
\bottomrule
\end{tabular}
}
}
\vspace{-6pt}
\makeatletter\def\@captype{table}\makeatothe
\caption{\small Runtime (in sec.) of different models to achieve $\epsilon= 1.0$ across different datasets. The ensemble size for $\mbox{SF}_S$-PATE and $\mbox{SF}_T$-PATE is $300$.}
\label{tab:compare_time}
\end{wrapfigure}
Figure \ref{fig:utk_resnet} illustrates the accuracy and fairness tradeoff at varying of the privacy loss $\epsilon$. The models enforce accuracy parity (similar results can be obtained on other fairness notions). The figure illustrates how both SF-PATE variants bring dramatic accuracy improvements when compared to \emph{M}, at comparable or even lower fairness violations.
Next, Table \ref{tab:compare_time} reports the training time required by the algorithms compared to achieve a private model with parameter $\epsilon = 1.0$ on the benchmarks set analyzed.
Notice the significant training time differences across the models with $\mbox{SF}_S$-PATE being up to three times faster than \emph{M}
and up to two order of magnitude faster than \emph{PF-LD}.
{\em These results illustrate the benefits of the proposed models, rendering it a practical tool for private and fair ML especially for large, overparametetrized models and under stringent privacy regimes.}
\vspace{-8pt}
\subsection{Discussion and Limitations}
While the previous section has shown the advantages of the proposed framework with respect to the state-of-the-art models, this section discusses some key aspects of SF$_S$- and SF$_T$-PATE and provides intuitions and guidelines for when one implementation may be preferred over the other. While both algorithms rely on the same framework, they differ in a fundamental aspect: the fairness enforcing mechanism. SF$_T$-PATE delegates fairness enforcement to its teachers while SF$_S$-PATE to its student.
Notice that, by training an ensemble of fair teachers, SF$_T$-PATE is able to transfer fairness knowledge \emph{directly}, which, as observed in the experimental results, often results in smaller fairness violations than those attained by SF$_S$-PATE. This is notable, especially in the case of ``complex'' fairness constraints; i.e., when multiple concurrent constraints are to be enforced, as in the case of generalized demographic parity which imposes multiple order moments matching.
A shortcoming of this algorithm, however, is that it has to enforce fairness in each teacher model. Thus, one has to ensure that enough data (with large enough representation from all the protected groups) is assigned to each teacher. This may be problematic in very scarce data regimes. On the other hand, by training a \emph{single} fair model (the student) SF$_S$-PATE avoids such potential issue.
Finally, it is worth noting that a limitation of all ensemble models, including those proposed in this paper, is the need to store a model for each of the $K$ teachers. This can be problematic when the models become very large and additional considerations should be taken in such context. This, however, also represents an opportunity for future research to develop effective model storage and model pruning techniques that minimize the loss in accuracy and fairness while retaining privacy.
\section{Conclusions}
This paper was motivated by the need of building models whose outcomes do not discriminate against some demographic groups while also requiring privacy protection of these sensitive attributes. The paper proposed a framework to train deep learning models that satisfy several notions of group fairness while ensuring that the model satisfies differential privacy for the protected attributes.
The proposed framework, called \emph{Scalable, Fair, and Private Aggregation of Teacher Enseble} (SF-PATE) transfer fairness knowledge learned from a pretrained ensemble of models to a target model via a privacy-preserving voting process.
The paper introduced two flavors of SF-PATE. The first, called SF$_T$-PATE, allows transferring properties of the ensemble while protecting the demographic group attribute, and the second, called SF$_S$-PATE, in addition, also protects the target labels.
An important properties of SF-PATE is to allow the adoption of black-box (non-private) fair models during the knowledge transfer process, which ultimately, we believe, will simplify the development and boosts the adoption of new fairness metrics in privacy-preserving ML.
The paper also analyzes the fairness properties of SF-PATE and shows that unfairness can be bounded in many practical settings.
Finally, evaluation on both tabular and image datasets shows not only that SF-PATE achieves better accuracy, privacy, and fairness tradeoffs with respect to the current state-of-the-art, but it is also significantly faster. These properties render SF-PATE amenable to train large, overparameterized, models, that ensure privacy, accuracy, and fairness simultaneously, showing that it may become a practical tool for privacy-preserving and fair decision making.
\section*{Acknowledgement}
Cuong Tran and Keyu Zhu are to be considered both first authors.
This research is partially supported by NSF grant 2133169 and a Google Research Scholar award. Its views and conclusions are those of the authors only.
\bibliographystyle{splncs04}
|
1,116,691,500,868 | arxiv | \section{Introduction}
\label{ag1}
Let $X$ be a smooth manifold, and write $C^\iy(X)$ for the set of
smooth functions $c:X\ra\R$. Then $C^\iy(X)$ is a commutative
$\R$-algebra, with operations of addition, multiplication, and
scalar multiplication defined pointwise. However, $C^\iy(X)$ has
much more structure than this. For example, if $c:X\ra\R$ is smooth
then $\exp(c):X\ra\R$ is smooth, and this defines an operation
$\exp:C^\iy(X)\ra C^\iy(X)$ which cannot be expressed algebraically
in terms of the $\R$-algebra structure. More generally, if $n\ge 0$
and $f:\R^n\ra\R$ is smooth, define an $n$-fold operation
$\Phi_f:C^\iy(X)^n\ra C^\iy(X)$ by
\begin{equation*}
\bigl(\Phi_f(c_1,\ldots,c_n)\bigr)(x)=f\bigl(c_1(x),\ldots,c_n(x)\bigr),
\end{equation*}
for all $c_1,\ldots,c_n\in C^\iy(X)$ and $x\in X$. These operations
satisfy many identities: suppose $m,n\ge 0$, and $f_i:\R^n\ra\R$ for
$i=1,\ldots,m$ and $g:\R^m\ra\R$ are smooth functions. Define a
smooth function $h:\R^n\ra\R$ by
\begin{equation*}
h(x_1,\ldots,x_n)=g\bigl(f_1(x_1,\ldots,x_n),\ldots,f_m(x_1
\ldots,x_n)\bigr),
\end{equation*}
for all $(x_1,\ldots,x_n)\in\R^n$. Then for all $c_1,\ldots,c_n\in
C^\iy(X)$ we have
\e
\Phi_h(c_1,\ldots,c_n)=\Phi_g\bigl(\Phi_{f_1}(c_1,\ldots,c_n),
\ldots,\Phi_{f_m}(c_1,\ldots,c_n)\bigr).
\label{ag1eq1}
\e
A $C^\iy$-{\it ring\/} $\bigl(\fC,(\Phi_f)_{f:\R^n\ra\R\,\,C^\iy}
\bigr)$ is a set $\fC$ with operations $\Phi_f:\fC^n\ra\fC$ for all
$f:\R^n\ra\R$ smooth satisfying identities \eq{ag1eq1}, and one
other condition. Then $C^\iy(X)$ is a $C^\iy$-ring for any manifold
$X$, but there are also many $C^\iy$-rings which do not come from
manifolds, and can be thought of as representing geometric objects
which generalize manifolds.
The most basic objects in conventional algebraic geometry are
commutative rings $R$, or commutative $\mathbb K$-algebras $R$ for
some field $\mathbb K$. The `spectrum' $\Spec R$ of $R$ is an affine
scheme, and $R$ is interpreted as an algebra of functions on $\Spec
R$. More general kinds of spaces in algebraic geometry --- schemes
and stacks --- are locally modelled on affine schemes $\Spec R$.
This paper lays down the foundations of {\it Algebraic Geometry
over\/ $C^\iy$-rings}, in which we replace commutative rings in
algebraic geometry by $C^\iy$-rings. It includes the study of
$C^\iy$-{\it schemes} and {\it Deligne--Mumford\/ $C^\iy$-stacks},
two classes of geometric spaces generalizing manifolds and
orbifolds, respectively.
This is not a new idea, but was studied years ago as part of {\it
synthetic differential geometry}, which grew out of ideas of Lawvere
in the 1960s; see for instance Dubuc \cite{Dubu3} on
$C^\iy$-schemes, and the books by Moerdijk and Reyes \cite{MoRe2}
and Kock \cite{Kock}. However, we have new things to say, as we are
motivated by different problems (see below), and so are asking
different questions.
Following Dubuc's discussion of `models of synthetic differential
geometry' \cite{Dubu1} and oversimplifying a bit, symplectic
differential geometers are interested in $C^\iy$-schemes as they
provide a category $\CSch$ of geometric objects which includes
smooth manifolds and certain `infinitesimal' objects, and all fibre
products exist in $\CSch$, and $\CSch$ has some other nice
properties to do with open covers, and exponentials of
infinitesimals.
Synthetic differential geometry concerns proving theorems about
manifolds using synthetic reasoning involving `infinitesimals'. But
one needs to check these methods of synthetic reasoning are valid.
To do this you need a `model', some category of geometric spaces
including manifolds and infinitesimals, in which you can think of
your synthetic arguments as happening. Once you know there exists at
least one model with the properties you want, then as far as
synthetic differential geometry is concerned the job is done. For
this reason $C^\iy$-schemes have not been developed very far in
synthetic differential geometry.
Recently, $C^\iy$-rings and $C^\iy$-ringed spaces appeared in a very
different context, as part of David Spivak's definition of {\it
derived manifolds\/} \cite{Spiv}. Spivak was a student of Jacob
Lurie, and his goal was to extend parts of Lurie's `derived
algebraic geometry' programme \cite{Luri} to differential geometry.
Spivak's construction is very complex and technical, and his derived
manifolds form a {\it simplicial category}, a kind of $\iy$-category
with $n$-morphisms for all~$n\ge 1$.
In \cite{Joyc3}, the author will develop a theory of `derived
differential geometry' which simplifies, and goes beyond, Spivak's
derived manifolds. Our notion of derived manifolds are called {\it
d-manifolds}, and are built using the theory of locally fair
$C^\iy$-schemes and quasicoherent sheaves upon them of this paper.
They form a 2-category. We also study {\it d-manifolds with
boundary}, and {\it d-manifolds with corners}, and orbifold versions
of all these, {\it d-orbifolds}, which are built using the theory of
locally fair Deligne--Mumford $C^\iy$-stacks and quasicoherent
sheaves upon them of this paper.
Many areas of symplectic geometry involve moduli spaces
$\oM_{g,m}(J,\be)$ of stable $J$-holomorphic curves in a symplectic
manifold $(M,\om)$. The original motivation for the project of
\cite{Joyc3} was to find a good geometric description for the
geometric structure on such moduli spaces $\oM_{g,m}(J,\be)$. In the
Lagrangian Floer cohomology theory of Fukaya, Oh, Ohta and Ono
\cite{FOOO}, moduli spaces $\oM_{g,m}(J,\be)$ are given the
structure of {\it Kuranishi spaces}. Their theory of Kuranishi
spaces seemed to the author to be unsatisfactory and incomplete. In
trying improve it, and making use of ideas from Spivak \cite{Spiv},
the author arrived at the d-manifolds and d-orbifolds theory of
\cite{Joyc3}. The author believes that the `correct' definition of
Kuranishi space in the work of Fukaya et al. \cite{FOOO} should be
that a Kuranishi space is a d-orbifold with corners.
To set up our theory of d-manifolds and d-orbifolds requires a lot
of preliminary work on $C^\iy$-schemes and $C^\iy$-stacks, and
quasicoherent sheaves upon them. That is the purpose of this paper.
We have tried to present a complete, self-contained account which
should be understandable to readers with a reasonable background in
algebraic geometry, and we assume no familiarity with synthetic
differential geometry. We expect this material may have other
applications quite different to those the author has in mind
\cite{Joyc3}, which is why we have written it as a separate paper,
and tried to give a general picture rather than just those parts
needed for~\cite{Joyc3}.
Section \ref{ag2} explains $C^\iy$-rings. The archetypal examples of
$C^\iy$-rings, $C^\iy(X)$ for manifolds $X$, are discussed in
\S\ref{ag3}. Section \ref{ag4} studies $C^\iy$-schemes, \S\ref{ag5}
modules over $C^\iy$-rings, and \S\ref{ag6} sheaves of modules over
$C^\iy$-schemes, quasicoherent sheaves, and coherent sheaves.
Sections \ref{ag7}--\ref{ag9} generalize \S\ref{ag4} and \S\ref{ag6}
to $C^\iy$-stacks. We are particularly interested in {\it
Deligne--Mumford\/ $C^\iy$-stacks}, which are $C^\iy$-stacks locally
modelled on $[\uU/G]$ for $\uU$ an affine $C^\iy$-scheme and $G$
a finite group acting on $\uU$, since Deligne--Mumford
$C^\iy$-stacks generalize orbifolds in the same way that
$C^\iy$-schemes generalize manifolds.
Almost all of \S\ref{ag2}--\S\ref{ag4} is already understood in
synthetic differential geometry, such as Dubuc \cite{Dubu3} and
Moerdijk and Reyes \cite{MoRe2}, except for {\it good\/}
$C^\iy$-rings, which appear to be a new idea, (locally) good
$C^\iy$-schemes, and some facts about the $C^\iy$-rings and
$C^\iy$-schemes of manifolds with boundary or corners. But we
believe it is worthwhile giving a detailed and self-contained
exposition, from our own point of view. Section \ref{ag7} summarizes
material on stacks from \cite{Arti,BEFF,Gome,LaMo,Metz,Nooh}.
Sections \ref{ag5}--\ref{ag6} and \ref{ag8}--\ref{ag9} are new, so
far as the author knows, though they are based on well known
material in algebraic geometry. This paper is surveyed
in~\cite{Joyc2}.
\medskip
\noindent{\it Acknowledgements.} I would like to thank Eduardo Dubuc
and Jacob Lurie for helpful conversations.
\section{$C^\iy$-rings}
\label{ag2}
We begin by explaining the basic objects out of which our theories
are built, $C^\iy$-{\it rings}, or {\it smooth rings}. The
archetypal example of a $C^\iy$-ring is the vector space $C^\iy(X)$
of smooth functions $c:X\ra\R$ for a manifold $X$, and these will be
discussed at greater length in \S\ref{ag3}. The material on {\it
good\/} $C^\iy$-rings in \S\ref{ag25}--\S\ref{ag26} is new, as far
as the author knows, but nearly everything else in this section will
be known to experts in synthetic differential geometry, and much of
it can be found in Moerdijk and Reyes \cite[Ch.~I]{MoRe2}, Dubuc
\cite{Dubu1,Dubu2,Dubu3,Dubu4} or Kock \cite[\S III]{Kock}. We
introduce some new notation for brevity, in particular, our {\it
fair\/} $C^\iy$-rings are known in the literature as `finitely
generated and germ determined $C^\iy$-rings'.
\subsection{Two definitions of $C^\iy$-ring}
\label{ag21}
We first define $C^\iy$-rings in the style of classical algebra.
\begin{dfn} A $C^\iy$-{\it ring\/} is a set $\fC$ together with
operations
\begin{equation*}
\smash{\Phi_f:\fC^n={\buildrel {\!\ulcorner\,\text{$n$ copies
}\,\urcorner\!} \over {\fC\t\cdots \t \fC}}\longra \fC}
\end{equation*}
for all $n\ge 0$ and smooth maps $f:\R^n\ra\R$, where by convention
when $n=0$ we define $\fC^0$ to be the single point $\{\es\}$. These
operations must satisfy the following relations: suppose $m,n\ge 0$,
and $f_i:\R^n\ra\R$ for $i=1,\ldots,m$ and $g:\R^m\ra\R$ are smooth
functions. Define a smooth function $h:\R^n\ra\R$ by
\begin{equation*}
h(x_1,\ldots,x_n)=g\bigl(f_1(x_1,\ldots,x_n),\ldots,f_m(x_1
\ldots,x_n)\bigr),
\end{equation*}
for all $(x_1,\ldots,x_n)\in\R^n$. Then for all
$(c_1,\ldots,c_n)\in\fC^n$ we have
\begin{equation*}
\Phi_h(c_1,\ldots,c_n)=\Phi_g\bigl(\Phi_{f_1}(c_1,\ldots,c_n),
\ldots,\Phi_{f_m}(c_1,\ldots,c_n)\bigr).
\end{equation*}
We also require that for all $1\le j\le n$, defining
$\pi_j:\R^n\ra\R$ by $\pi_j:(x_1,\ldots,x_n)\mapsto x_j$, we have
$\Phi_{\pi_j}(c_1,\ldots,c_n)=c_j$ for
all~$(c_1,\ldots,c_n)\in\fC^n$.
Usually we refer to $\fC$ as the $C^\iy$-ring, leaving the
operations $\Phi_f$ implicit.
A {\it morphism\/} between $C^\iy$-rings $\bigl(\fC,(\Phi_f)_{
f:\R^n\ra\R\,\,C^\iy}\bigr)$, $\bigl({\mathfrak
D},(\Psi_f)_{f:\R^n\ra\R\,\,C^\iy}\bigr)$ is a map
$\phi:\fC\ra{\mathfrak D}$ such that $\Psi_f\bigl(\phi
(c_1),\ldots,\phi(c_n)\bigr)=\phi\ci\Phi_f(c_1,\ldots,c_n)$ for all
smooth $f:\R^n\ra\R$ and $c_1,\ldots,c_n\in\fC$. We will write
$\CRings$ for the category of $C^\iy$-rings.
\label{ag2def1}
\end{dfn}
Here is the motivating example, which we will study at greater
length in~\S\ref{ag3}:
\begin{ex} Let $X$ be a manifold, which may be without boundary, or
with boundary, or with corners. Write $C^\iy(X)$ for the set of
smooth functions $c:X\ra\R$. For $n\ge 0$ and $f:\R^n\ra\R$ smooth,
define $\Phi_f:C^\iy(X)^n\ra C^\iy(X)$ by
\e
\bigl(\Phi_f(c_1,\ldots,c_n)\bigr)(x)=f\bigl(c_1(x),\ldots,c_n(x)\bigr),
\label{ag2eq1}
\e
for all $c_1,\ldots,c_n\in C^\iy(X)$ and $x\in X$. It is easy to see
that $C^\iy(X)$ and the operations $\Phi_f$ form a $C^\iy$-ring.
\label{ag2ex1}
\end{ex}
\begin{ex} Take $X=\{0\}$ in Example \ref{ag2ex1}. Then
$C^\iy(\{0\})=\R$, with operations $\Phi_f:\R^n\ra\R$ given by
$\Phi_f(x_1,\ldots,x_n)=f(x_1,\ldots,x_n)$. This makes $\R$ into the
simplest nonzero example of a $C^\iy$-ring.
\label{ag2ex2}
\end{ex}
Note that $C^\iy$-rings are far more general than those coming from
manifolds. For example, if $X$ is any topological space we could
define a $C^\iy$-ring $C^0(X)$ to be the set of {\it continuous\/}
$c:X\ra\R$ with operations $\Phi_f$ defined as in \eq{ag2eq1}. For
$X$ a manifold with $\dim X>0$, the $C^\iy$-rings $C^\iy(X)$ and
$C^0(X)$ are different.
There is a more succinct definition of $C^\iy$-rings using category
theory:
\begin{dfn} Write $\Euc$ for the full subcategory of $\Man$ spanned
by the Euclidean spaces $\R^n$. That is, the objects of $\Euc$ are
the manifolds $\R^n$ for $n=0,1,2,\ldots$, and the morphisms in
$\Euc$ are smooth maps $f:\R^n\ra\R^m$. Write $\Sets$ for the
category of sets. In both $\Euc$ and $\Sets$ we have notions of
(finite) products of objects (that is, $\R^{n+m}=\R^n\t\R^m$, and
products $S\t T$ of sets $S,T$), and products of morphisms. Define a
({\it category-theoretic\/}) $C^\iy$-{\it ring\/} to be a
product-preserving functor~$F:\Euc\ra\Sets$.
\label{ag2def2}
\end{dfn}
Here is how this relates to Definition \ref{ag2def1}. Suppose
$F:\Euc\ra\Sets$ is a product-preserving functor. Define
$\fC=F(\R)$. Then $\fC$ is an object in $\Sets$, that is, a set.
Suppose $n\ge 0$ and $f:\R^n\ra\R$ is smooth. Then $f$ is a morphism
in $\Euc$, so $F(f):F(\R^n)\ra F(\R)=\fC$ is a morphism in $\Sets$.
Since $F$ preserves products $F(\R^n)=F(\R)\t\cdots\t F(\R)=\fC^n$,
so $F(f)$ maps $\fC^n\ra\fC$. We define $\Phi_f:\fC^n\ra\fC$ by
$\Phi_f=F(f)$. The fact that $F$ is a functor implies that the
$\Phi_f$ satisfy the relations in Definition \ref{ag2def1}, so
$\bigl(\fC,(\Phi_f)_{f:\R^n\ra\R \,\,C^\iy}\bigr)$ is a $C^\iy$
ring.
Conversely, if $\bigl(\fC,(\Phi_f)_{f:\R^n\ra\R\,\,C^\iy}\bigr)$ is
a $C^\iy$-ring then we define $F:\Euc\ra\Sets$ by $F(\R^n)=\fC^n$,
and if $f:\R^n\ra\R^m$ is smooth then $f=(f_1,\ldots,f_m)$ for
$f_i:\R^n\ra\R$ smooth, and we define $F(f):\fC^n\ra\fC^m$ by
$F(f):(c_1,\ldots,c_n)\mapsto\bigl(
\Phi_{f_1}(c_1,\ldots,c_n),\ldots,\Phi_{f_m}(c_1,\ldots,c_n)\bigr)$.
Then $F$ is a product-preserving functor. This defines a 1-1
correspondence between $C^\iy$-rings in the sense of Definition
\ref{ag2def1}, and category-theoretic $C^\iy$-rings in the sense of
Definition~\ref{ag2def2}.
Since all small colimits exist in $\Sets$ \cite[Ex.~V.1.8]{MacL},
regarding $C^\iy$-rings as functors $F:\Euc\ra\Sets$ as in
Definition \ref{ag2def2}, to take small colimits in the category of
$C^\iy$-rings we can take colimits in $\Sets$ object-wise in $\Euc$,
so as in Moerdijk and Reyes \cite[p.~21-22]{MoRe2} we have:
\begin{prop} In the category $\CRings$ of\/ $C^\iy$-rings, all
small colimits exist, and so in particular pushouts and all finite
colimits exist.
\label{ag2prop1}
\end{prop}
In fact Proposition \ref{ag2prop1} holds for the category of models
of any universal-algebra theory. We will write
$\fD\amalg_{\phi,\fC,\psi}\fE$ or $\fD\amalg_\fC\fE$ for the pushout
of morphisms $\phi:\fC\!\ra\!\fD$, $\psi:\fC\!\ra\!\fE$ in
$\CRings$. When $\fC\!=\!\R$, the initial object in $\CRings$,
pushouts $\fD\amalg_\R\fE$ are called {\it coproducts\/} and are
usually written $\fD\ot_\iy\nobreak\fE$. (Note that for
$\R$-algebras $A,B$ the coproduct is the tensor product~$A\ot B$.)
\subsection{$C^\iy$-rings as commutative $\R$-algebras, and ideals}
\label{ag22}
Every $C^\iy$-ring $\fC$ has an underlying commutative $\R$-algebra:
\begin{dfn} Let $\fC$ be a $C^\iy$-ring. Then we may give $\fC$ the
structure of a {\it commutative\/ $\R$-algebra}. Define addition
`$+$' on $\fC$ by $c+c'=\Phi_f(c,c')$ for $c,c'\in\fC$, where
$f:\R^2\ra\R$ is $f(x,y)=x+y$. Define multiplication `$\,\cdot\,$'
on $\fC$ by $c\cdot c'=\Phi_g(c,c')$, where $g:\R^2\ra\R$ is
$f(x,y)=xy$. Define scalar multiplication by $\la\in\R$ by $\la
c=\Phi_{\la'}(c)$, where $\la':\R\ra\R$ is $\la'(x)=\la x$. Define
elements 0 and 1 in $\fC$ by $0=\Phi_{0'}(\es)$ and
$1=\Phi_{1'}(\es)$, where $0':\R^0\ra\R$ and $1':\R^0\ra\R$ are the
maps $0':\es\mapsto 0$ and $1':\es\mapsto 1$. The relations on the
$\Phi_f$ imply that all the axioms of a commutative $\R$-algebra are
satisfied. In Example \ref{ag2ex1}, this yields the obvious
$\R$-algebra structure on the smooth functions~$c:X\ra\R$.
Here is another way to say this. In an $\R$-algebra $A$, the
$n$-fold `operations' $\Phi:A^n\ra A$, that is, all the maps $A^n\ra
A$ we can construct using only addition, multiplication, scalar
multiplication, and the elements $0,1\in A$, correspond exactly to
polynomials $p:\R^n\ra\R$. Since polynomials are smooth, the
operations of an $\R$-algebra are a subset of those of a
$C^\iy$-ring, and we can truncate from $C^\iy$-rings to
$\R$-algebras. As there are many more smooth functions $f:\R^n\ra\R$
than there are polynomials, a $C^\iy$-ring has far more structure
and operations than a commutative $\R$-algebra.
\label{ag2def3}
\end{dfn}
\begin{dfn} An {\it ideal\/} $I$ in $\fC$ is an ideal $I\subset\fC$
in $\fC$ regarded as a commutative $\R$-algebra. Then we make the
quotient $\fC/I$ into a $C^\iy$-ring as follows. If $f:\R^n\ra\R$ is
smooth, define $\Phi_f^I:(\fC/I)^n\ra\fC/I$ by
\begin{equation*}
\bigl(\Phi_f^I(c_1+I,\ldots,c_n+I)\bigr)(x)=f\bigl(c_1(x),\ldots,
c_n(x)\bigr)+I.
\end{equation*}
To show this is well-defined, we must show it is independent of the
choice of representatives $c_1,\ldots,c_n$ in $\fC$ for
$c_1+I,\ldots,c_n+I$ in $\fC/I$. By Hadamard's Lemma there exist
smooth functions $g_i:\R^{2n}\ra\R$ for $i=1,\ldots,n$ with
\begin{equation*}
f(y_1,\ldots,y_n)-f(x_1,\ldots,x_n)=\ts\sum_{i=1}^n(y_i-x_i)
g_i(x_1,\ldots,x_n,y_1,\ldots,y_n)
\end{equation*}
for all $x_1,\ldots,x_n,y_1,\ldots,y_n\in\R$. If $c_1',\ldots,c_n'$
are alternative choices for $c_1,\ab\ldots,\ab c_n$, so that
$c_i'+I=c_i+I$ for $i=1,\ldots,n$ and $c_i'-c_i\in I$, we have
\begin{align*}
f\bigl(c_1'(x),&\ldots,c_n'(x)\bigr)-f\bigl(c_1(x),\ldots,
c_n(x)\bigr)\\
&=\ts\sum_{i=1}^n(c_i'-c_i)g_i\bigl(c'_1(x),\ldots,c'_n(x),
c_1(x),\ldots,c_n(x)\bigr).
\end{align*}
The second line lies in $I$ as $c_i'-c_i\in I$ and $I$ is an ideal,
so $\Phi_f^I$ is well-defined, and clearly $\bigl(\fC/I,(\Phi_f^I)_{
f:\R^n\ra\R\,\,C^\iy}\bigr)$ is a $C^\iy$-ring.
If $\fC$ is a $C^\iy$-ring, we will use the notation $(f_a:a\in A)$
to denote the ideal in $\fC$ generated by a collection of elements
$f_a$, $a\in A$ in $\fC$, in the sense of commutative $\R$-algebras.
That is,
\begin{equation*}
(f_a:a\in A)=\bigl\{\ts\sum_{i=1}^nf_{a_i}\cdot c_i:\text{$n\ge 0$,
$a_1,\ldots,a_n\in A$, $c_1,\ldots,c_n\in\fC$}\bigr\}.
\end{equation*}
\label{ag2def4}
\end{dfn}
\begin{dfn} A $C^\iy$-ring $\fC$ is called {\it finitely
generated\/} if there exist $c_1,\ldots,c_n$ in $\fC$ which generate
$\fC$ over all $C^\iy$-operations. That is, for each $c\in\fC$ there
exists smooth $f:\R^n\ra\R$ with $c=\Phi_f(c_1,\ldots,c_n)$. (Note
that this is a much weaker condition than $\fC$ being finitely
generated as a commutative $\R$-algebra).
By Kock \cite[Prop.~III.5.1]{Kock}, $C^\iy(\R^n)$ is the free
$C^\iy$-ring with $n$ generators. Given such $\fC,c_1,\ldots,c_n$,
define $\phi:C^\iy(\R^n)\ra\fC$ by $\phi(f)=\Phi_f(c_1,\ldots,c_n)$
for smooth $f:\R^n\ra\R$, where $C^\iy(\R^n)$ is as in Example
\ref{ag2ex1} with $X=\R^n$. Then $\phi$ is a surjective morphism of
$C^\iy$-rings, so $I=\Ker\phi$ is an ideal in $C^\iy(\R^n)$, and
$\fC\cong C^\iy(\R^n)/I$ as a $C^\iy$-ring. Thus, $\fC$ is finitely
generated if and only if $\fC\cong C^\iy(\R^n)/I$ for some $n\ge 0$
and ideal $I$ in~$C^\iy(\R^n)$.
An ideal $I$ in a $C^\iy$-ring $\fC$ is called {\it finitely
generated\/} if $I$ is a finitely generated ideal of the underlying
commutative $\R$-algebra of $\fC$ in Definition \ref{ag2def3}, that
is, $I=(i_1,\ldots,i_k)$ for some $i_1,\ldots,i_k\in\fC$. A
$C^\iy$-ring $\fC$ is called {\it finitely presented\/} if $\fC\cong
C^\iy(\R^n)/I$ for some $n\ge 0$, where $I$ is a finitely generated
ideal in~$C^\iy(\R^n)$.
Given such $\fC,n,I$, choose generators $i_1,\ldots,i_k\in
C^\iy(\R^n)$ for $I$. Define $\psi:C^\iy(\R^k)\!\ra\!C^\iy(\R^n)$ by
$\psi(f)(x_1,\ldots,x_n)\!=\!f\bigl(i_1(x_1,\ldots,x_n),\ldots,
i_k(x_1,\ldots,x_n)\bigr)$ for all smooth $f:\R^k\ra\R$ and
$x_1,\ldots,x_n\in\R$. Then $\psi$ is a morphism of $C^\iy$-rings,
and
\e
\begin{gathered}
\xymatrix@R=10pt@C=60pt{ C^\iy(\R^k)
\ar[r]_(0.4){x_1,\ldots,x_k\mapsto 0}
\ar[d]^{\psi} & C^\iy(\{0\})\cong\R \ar[d]_{1\mapsto 1} \\
C^\iy(\R^n) \ar[r]^\phi & \fC}
\end{gathered}
\label{ag2eq2}
\e
is a pushout square in $\CRings$. Conversely, $\fC$ is finitely
presented if it fits into a pushout square~\eq{ag2eq2}.
A difference with conventional algebraic geometry is that
$C^\iy(\R^n)$ is not noetherian, so ideals in $C^\iy(\R^n)$ may not
be finitely generated, and $\fC$ finitely generated does not imply
$\fC$ finitely presented.
\label{ag2def5}
\end{dfn}
Write $\CRingsfg$ and $\CRingsfp$ for the full subcategories of
finitely generated and finitely presented $C^\iy$-rings
in~$\CRings$.
\begin{ex} A {\it Weil algebra\/} \cite[Def.~1.4]{Dubu1} is a
finite-dimensional commutative $\R$-algebra $W$ which has a maximal
ideal $\fm$ with $W/\fm\cong\R$ and $\fm^n=0$ for some $n>0$. Then
by Dubuc \cite[Prop.~1.5]{Dubu1} or Kock \cite[Th.~III.5.3]{Kock},
there is a unique way to make $W$ into a $C^\iy$-ring compatible
with the given underlying commutative $\R$-algebra. This
$C^\iy$-ring is finitely presented \cite[Prop.~III.5.11]{Kock}.
$C^\iy$-rings from Weil algebras are important in synthetic
differential geometry, in arguments involving infinitesimals. See
\cite[\S 2]{BuDu} for a detailed study of this.
\label{ag2ex3}
\end{ex}
\subsection{$C^\iy$-local rings, and localization}
\label{ag23}
\begin{dfn} A $C^\iy$-ring $\fC$ is called a $C^\iy$-{\it local
ring\/} if regarded as an $\R$-algebra, as in Definition
\ref{ag2def3}, $\fC$ is a local $\R$-algebra with residue field
$\R$. That is, $\fC$ has a unique maximal ideal $\m_\fC$
with~$\fC/\m_\fC\cong\R$.
If $\fC,\fD$ are $C^\iy$-local rings with maximal ideals
$\m_\fC,\m_\fD$, and $\phi:\fC\ra\fD$ is a morphism of $C^\iy$
rings, then using the fact that $\fC/\m_\fC\cong\R\cong\fD/\m_\fD$
we see that $\phi^{-1}(\m_\fD)=\m_\fC$, that is, $\phi$ is a {\it
local\/} morphism of $C^\iy$-local rings. Thus, there is no
difference between morphisms and local morphisms.
\label{ag2def6}
\end{dfn}
\begin{rem} We use the term `$C^\iy$-local ring' following Dubuc
\cite[Def.~2.13]{Dubu2}, though they are also called `local
$C^\iy$-rings' in Dubuc \cite[Def.~4]{Dubu3}. Following
\cite{Dubu2,Dubu3}, we include the condition that $\fC$ has residue
field $\R$ in the definition of $C^\iy$-local ring. Moerdijk and
Reyes \cite{MQR,MoRe1,MoRe2} omit this condition. They call local
$C^\iy$-rings with residue field $\R$ {\it pointed local\/
$C^\iy$-rings\/} in \cite[\S I.3]{MoRe2} and {\it Archimedean
local\/ $C^\iy$-rings\/} in~\cite[\S 3]{MQR}.
\label{ag2rem}
\end{rem}
{\it Localizations\/} of $C^\iy$-rings are studied in
\cite{Dubu2,Dubu3,MQR,MoRe1}, see~\cite[p.~23]{MoRe2}.
\begin{dfn} Let $\fC$ be a $C^\iy$-ring and $S$ a subset of $\fC$. A
{\it localization\/} $\fC[s^{-1}:s\in S]$ of $\fC$ at $S$ is a
$C^\iy$-ring $\fD=\fC[s^{-1}:s\in S]$ and a morphism $\pi:\fC\ra\fD$
such that $\pi(s)$ is invertible in $\fD$ for all $s\in S$, with the
universal property that if $\fE$ is a $C^\iy$-ring and
$\phi:\fC\ra\fE$ a morphism with $\phi(s)$ invertible in $\fE$ for
all $s\in S$, then there is a unique morphism $\psi:\fD\ra\fE$ with
$\phi=\psi\ci\pi$.
A localization $\fC[s^{-1}:s\in S]$ always exists --- it can be
constructed by adjoining an extra generator $s^{-1}$ and an extra
relation $s\cdot s^{-1}-1=0$ for each $s\in S$ --- and is unique up
to unique isomorphism. When $S=\{c\}$ we have an exact sequence
$0\ra I\ra \fC\ot_\iy C^\iy(\R)\,{\buildrel\pi \over\longra}\,
\fC[c^{-1}]\ra 0$, where $\fC\ot_\iy C^\iy(\R)$ is the coproduct of
$\fC,C^\iy(\R)$ as in \S\ref{ag21}, with pushout morphisms
$\io_1:\fC\ra\fC\ot_\iy C^\iy(\R)$, $\io_2:C^\iy(\R)\ra\fC\ot_\iy
C^\iy(\R)$, and $I$ is the ideal in $\fC\ot_\iy C^\iy(\R)$ generated
by $\io_1(c)\cdot\io_2(x)-1$, where $x$ is the generator
of~$C^\iy(\R)$.
An $\R$-{\it point\/} $p$ of a $C^\iy$-ring $\fC$ is a $C^\iy$-ring
morphism $p:\fC\ra\R$, where $\R$ is regarded as a $C^\iy$-ring as
in Example \ref{ag2ex2}. By \cite[Prop.~I.3.6]{MoRe2}, a map
$p:\fC\ra\R$ is a morphism of $C^\iy$-rings if and only if it is a
morphism of the underlying $\R$-algebras, as in Definition
\ref{ag2def4}. Define $\fC_p$ to be the localization
$\fC_p=\fC[s^{-1}:s\in\fC$, $p(s)\ne 0]$, with projection
$\pi_p:\fC\ra\fC_p$. Then $\fC_p$ is a $C^\iy$-local ring by
\cite[Lem.~1.1]{MoRe2}. The $\R$-points of $C^\iy(\R^n)$ are just
evaluation at points $p\in\R^n$.
\label{ag2def7}
\end{dfn}
\begin{ex} For $n\ge 0$ and $p\in\R^n$, define $C^\iy_p(\R^n)$ to
be the set of germs of smooth functions $c:\R^n\ra\R$ at $p\in\R^n$,
made into a $C^\iy$-ring in the obvious way. Then $C^\iy_p(\R^n)$ is
a {\it local\/} $C^\iy$-ring in the sense of Definition
\ref{ag2def6}. Here are three different ways to define
$C^\iy_p(\R^n)$, which yield isomorphic $C^\iy$-rings:
\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item[(a)] Defining $C^\iy_p(\R^n)$ as the germs of functions
of smooth functions at $p$ means that points of $C^\iy_p(\R^n)$
are $\sim$-equivalence classes $[(U,c)]$ of pairs $(U,c)$, where
$U\subseteq\R^n$ is open with $p\in U$ and $c:U\ra\R$ is smooth,
and $(U,c)\sim(U',c')$ if there exists $p\in V\subseteq U\cap
U'$ open with $c\vert_V\equiv c'\vert_V$.
\item[(b)] As the localization $\bigl(C^\iy(\R^n)\bigr){}_p=
C^\iy(\R^n)[g\in C^\iy(\R^n):g(p)\ne 0]$. Then points of
$\bigl(C^\iy(\R^n)\bigr){}_p$ are equivalence classes $[f/g]$ of
fractions $f/g$ for $f,g\in C^\iy(\R^n)$ with $g(p)\ne 0$, and
fractions $f/g$, $f'/g'$ are equivalent if there exists $h\in
C^\iy(\R^n)$ with $h(p)\ne 0$ and $h(fg'-f'g)\equiv 0$.
\item[(c)] As the quotient $C^\iy(\R^n)/I$, where $I$ is the ideal
of $f\in C^\iy(\R^n)$ with $f\equiv 0$ near $p\in\R^n$.
\end{itemize}
One can show (a)--(c) are isomorphic using the fact that if $U$ is
any open neighbourhood of $p$ in $\R^n$ then there exists smooth
$\eta:\R^n\ra[0,1]$ such that $\eta\equiv 0$ on an open
neighbourhood of $\R^n\sm U$ in $\R^n$ and $\eta\equiv 1$ on an open
neighbourhood of $p$ in $U$. Any finitely generated $C^\iy$-local
ring is a quotient of some~$C^\iy_p(\R^n)$.
\label{ag2ex4}
\end{ex}
\subsection{Fair $C^\iy$-rings}
\label{ag24}
We now discuss an important class of $C^\iy$-rings, which we call
{\it fair\/} $C^\iy$-rings. Although our term `fair' is new, we
stress that the idea is already well-known, being originally
introduced by Dubuc \cite{Dubu2}, \cite[Def.~11]{Dubu3}, who first
recognized their significance, under the name `$C^\iy$-rings of
finite type presented by an ideal of local character', and in more
recent works would be referred to as `finitely generated and
germ-determined $C^\iy$-rings'. We chose the term `fair' for its
brevity, as we will be using it a lot; the intention is to suggest
that `fair' $C^\iy$-rings are well behaved and nicer to work with
than general $C^\iy$-rings, but `good' $C^\iy$-rings in \S\ref{ag25}
are even better.
\begin{dfn} An ideal $I$ in $C^\iy(\R^n)$ is called {\it fair\/} if
for each $f\in C^\iy(\R^n)$, $f$ lies in $I$ if and only if
$\pi_p(f)$ lies in $\pi_p(I)\subseteq C^\iy_p(\R^n)$ for all
$p\in\R^n$, where $C^\iy_p(\R^n)$ is as in Example \ref{ag2ex4} and
$\pi_p:C^\iy(\R^n)\ra C^\iy_p(\R^n)$ is the natural projection
$\pi_p:c\mapsto[(\R^n,c)]$. A $C^\iy$-ring $\fC$ is called {\it
fair\/} if it is isomorphic to $C^\iy(\R^n)/I$, where $I$ is a fair
ideal. Equivalently, $\fC$ is fair if it is finitely generated and
whenever $c\in\fC$ with $\pi_p(c)=0$ in $\fC_p$ for all $\R$-points
$p:\fC\ra\R$ then $c=0$, using the notation of
Definition~\ref{ag2def7}.
Dubuc \cite{Dubu2}, \cite[Def.~11]{Dubu3} calls fair ideals {\it
ideals of local character}, and Moerdijk and Reyes \cite[I.4]{MoRe2}
call them {\it germ determined}, which has now become the accepted
term. Fair $C^\iy$-rings are also sometimes called {\it germ
determined\/ $C^\iy$-rings}, a more descriptive term than `fair',
but the definition of germ determined $C^\iy$-rings $\fC$ in
\cite[Def.~I.4.1]{MoRe2} does not require $\fC$ finitely generated,
so does not equate exactly to our fair $C^\iy$-rings. By Dubuc
\cite[Prop.~1.8]{Dubu2}, \cite[Prop.~12]{Dubu3} any finitely
generated ideal $I$ is fair, so $\fC$ finitely presented implies
$\fC$ fair. We write $\CRingsfa$ for the full subcategory of fair
$C^\iy$-rings in~$\CRings$.
\label{ag2def8}
\end{dfn}
Since $C^\iy(\R^n)/I$ is fair if and only if $I$ is fair, we have:
\begin{lem} Let\/ $I\subset C^\iy(\R^m),$ $J\subset
C^\iy(\R^n)$ be ideals with\/ $C^\iy(\R^m)/I\cong C^\iy(\R^n)/J$ as
$C^\iy$-rings. Then $I$ is fair if and only if\/ $J$ is fair.
\label{ag2lem1}
\end{lem}
Recall from category theory that if $\cal C$ is a subcategory of a
category $\cal D$, a {\it reflection\/} $R:{\cal D}\ra{\cal C}$ is a
left adjoint to the inclusion ${\cal C}\ra{\cal D}$. That is,
$R:{\cal D}\ra{\cal C}$ is a functor with natural isomorphisms
$\Hom_{\cal C}(R(D),C)\cong\Hom_{\cal D}(D,C)$ for all $C\in{\cal
C}$ and $D\in{\cal D}$. We will define a reflection for
$\CRingsfa\subset\CRingsfg$, following Moerdijk and Reyes
\cite[p.~48-49]{MoRe2} (see also Dubuc~\cite[Th.~13]{Dubu3}).
\begin{dfn} Let $\fC$ be a finitely generated $C^\iy$-ring. Let
$I_\fC$ be the ideal of all $c\in\fC$ such that $\pi_p(c)=0$ in
$\fC_p$ for all $\R$-points $p:\fC\ra\R$ then $c=0$. Then
$\fC/I_\fC$ is a finitely generated $C^\iy$-ring, with projection
$\pi:\fC\ra\fC/I_\fC$. It has the same $\R$-points as $\fC$, that
is, morphisms $p:\fC/I_\fC\ra\R$ are in 1-1 correspondence with
morphisms $p':\fC\ra\R$ by $p'=p\ci\pi$, and the local rings
$(\fC/I_\fC)_p$ and $\fC_{p'}$ are naturally isomorphic. It follows
that $\fC/I_\fC$ is {\it fair}. Define a functor $R_{\rm fg}^{\rm
fa}:\CRingsfg\ra\CRingsfa$ by $R_{\rm fg}^{\rm fa}(\fC)=\fC/I_\fC$
on objects, and if $\phi:\fC\ra\fD$ is a morphism then
$\phi(I_\fC)\subseteq I_\fD$, so $\phi$ induces a morphism
$\phi_*:\fC/I_\fC\ra\fD/I_\fD$, and we set $R_{\rm fg}^{\rm
fa}(\phi)=\phi_*$. It is easy to see $\smash{R_{\rm fg}^{\rm fa}}$
is a {\it reflection}.
If $I$ is an ideal in $C^\iy(\R^n)$, write $\bar I$ for the set of
$f\in C^\iy(\R^n)$ with $\pi_p(f)\in\pi_p(I)$ for all $p\in\R^n$.
Then $\bar I$ is the smallest fair ideal in $C^\iy(\R^n)$ containing
$I$, the {\it germ-determined closure\/} of $I$, and~$R_{\rm
fg}^{\rm fa}\bigl(C^\iy(\R^n)/I\bigr)\cong C^\iy(\R^n)/\bar I$.
\label{ag2def9}
\end{dfn}
\begin{ex} Let $\eta:\R\ra[0,\iy)$ be smooth with $\eta(x)>0$ for
$x\in(0,1)$ and $\eta(x)=0$ for $x\notin(0,1)$. Define $I\subseteq
C^\iy(\R)$ by
\begin{equation*}
I=\bigl\{\ts\sum_{a\in A}g_a(x)\eta(x-a):\text{$A\subset\Z$ is
finite, $g_a\in C^\iy(\R)$, $a\in A$}\bigr\}.
\end{equation*}
Then $I$ is an ideal in $C^\iy(\R)$, so $\fC=C^\iy(\R)/I$ is a
$C^\iy$-ring. The set of $f\in C^\iy(\R)$ such that $\pi_p(f)$ lies
in $\pi_p(I)\subseteq C^\iy_p(\R)$ for all $p\in\R$ is
\begin{equation*}
\bar I=\bigl\{\ts\sum_{a\in\Z}g_a(x)\eta(x-a):g_a\in C^\iy(\R),\;
a\in \Z\bigr\},
\end{equation*}
where the sum $\sum_{a\in\Z}g_a(x)\eta(x-a)$ makes sense as at most
one term is nonzero at any point $x\in\R$. Since $\bar I\ne I$, we
see that $I$ is {\it not fair}, so $\fC=C^\iy(\R)/I$ is {\it not a
fair\/ $C^\iy$-ring}. In fact $\bar I$ is the smallest fair ideal
containing $I$. We have $I_{C^\iy(\R)/I}=\bar I/I$, and~$R_{\rm
fg}^{\rm fa}\bigl(C^\iy(\R)/I)=C^\iy(\R)/\bar I$.
\label{ag2ex5}
\end{ex}
\subsection{Good $C^\iy$-rings}
\label{ag25}
The following class of ideals in $C^\iy(\R^n)$ is defined by
Moerdijk and Reyes \cite[p.~47, p.~49]{MoRe2} (see also Dubuc
\cite[\S 1.7(a)]{Dubu2}), who call them {\it flat ideals\/}:
\begin{dfn} Let $X$ be a closed subset of $\R^n$. Define
$\fm_X^\iy$ to be the ideal of all functions $g\in C^\iy(\R^n)$ such
that $\pd^kg\vert_X\equiv 0$ for all $k\ge 0$, that is, $g$ and all
its derivatives vanish at each $x\in X$. If the interior $X^\ci$ of
$X$ in $\R^n$ is dense in $X$, that is $\smash{\ov{(X^\ci)}}=X$,
then $\pd^kg\vert_X\equiv 0$ for all $k\ge 0$ if and only if
$g\vert_X\equiv 0$. In this case $C^\iy(\R^n)/\fm_X^\iy\cong
C^\iy(X):=\bigl\{f\vert_X:f\in C^\iy(\R^n)\bigr\}$.
\label{ag2def10}
\end{dfn}
Here is an example from Moerdijk and Reyes~\cite[Th.~I.1.3]{MoRe2}.
\begin{ex} Take $X$ to be the point $\{0\}$. If $f,f'\in
C^\iy(\R^n)$ then $f-f'$ lies in $\fm_{\{0\}}^\iy$ if and only if
$f,f'$ have the same Taylor series at 0. Thus
$C^\iy(\R^n)/\fm_{\{0\}}^\iy$ is the $C^\iy$-ring of Taylor series
at 0 of $f\in C^\iy(\R^n)$. Since any formal power series in
$x_1,\ldots,x_n$ is the Taylor series of some $f\in C^\iy(\R^n)$, we
have $C^\iy(\R^n)/\fm_{\{0\}}^\iy\cong\R[[x_1,\ldots,x_n]]$. Thus
the $\R$-algebra of formal power series $\R[[x_1,\ldots,x_n]]$ can
be made into a $C^\iy$-ring.
\label{ag2ex6}
\end{ex}
The following nontrivial result is proved by Reyes and van Qu\^e
\cite[Th.~1]{ReVa}, generalizing an unpublished result of A.P.
Calder\'on in the case $X=Y=\{0\}$. It can also be found in Moerdijk
and Reyes~\cite[Cor.~I.4.12]{MoRe2}.
\begin{prop} Let\/ $X\subseteq\R^m$ and\/ $Y\subseteq\R^n$ be
closed. Then as ideals in $C^\iy(\R^{m+n})$ we
have\/~$(\fm_X^\iy,\fm_Y^\iy)=\fm_{X\t Y}^\iy$.
\label{ag2prop2}
\end{prop}
The next definition is new, so far as the author knows.
\begin{dfn} An ideal $I$ in $C^\iy(\R^n)$ is called {\it good\/} if
it is of the form $I=(f_1,\ldots,f_k,\fm_X^\iy)$ for some
$f_1,\ldots,f_k\in C^\iy(\R^n)$ and closed $X\subseteq\R^n$. A
$C^\iy$-ring $\fC$ is called {\it good\/} if $\fC\cong
C^\iy(\R^n)/I$ for some $n\ge 0$, where $I$ is a good ideal. If $I$
is finitely generated, then taking $X=\es$ shows that $I$ is good.
If $I$ is good then Moerdijk and Reyes \cite[Cor.~I.4.9]{MoRe2}
implies that $I$ is fair.
\label{ag2def11}
\end{dfn}
Good $C^\iy$-rings have the following nice properties:
\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item[(a)] If $X$ is a manifold, possibly with boundary or with
corners, then $C^\iy(X)$ is a good $C^\iy$-ring, as
in~\S\ref{ag3}.
\item[(b)] Good $C^\iy$-rings are closed under finite colimits
in $\CRings$, as in~\S\ref{ag26}.
\item[(c)] The {\it spectrum functor\/} $\Spec$ from
$C^\iy$-rings to affine $C^\iy$-schemes is full and faithful on
good $C^\iy$-rings, as in~\S\ref{ag4}.
\item[(d)] The {\it cotangent module\/} $\Om_\fC$ of a good
$C^\iy$-ring $\fC$ is a finitely presented $\fC$-module, as
in~\S\ref{ag5}.
\end{itemize}
Finitely presented $C^\iy$-rings do not satisfy (a), fair
$C^\iy$-rings do not satisfy (b),(d), and finitely generated or
arbitrary $C^\iy$-rings do not satisfy (c),(d). Write $\CRingsgo$
for the full subcategory of good $C^\iy$-rings in $\CRings$. Then
\begin{equation*}
\CRingsfp\subset\CRingsgo\subset\CRingsfa\subset\CRingsfg
\subset\CRings.
\end{equation*}
Here is an analogue of Lemma~\ref{ag2lem1}.
\begin{prop} Suppose\/ $I\subset C^\iy(\R^m)$ and\/ $J\subset
C^\iy(\R^n)$ are ideals with\/ $C^\iy(\R^m)/I\cong C^\iy(\R^n)/J$ as
$C^\iy$-rings. Then $I$ is good, or finitely generated, if and only
if\/ $J$ is good, or finitely generated, respectively.
\label{ag2prop3}
\end{prop}
\begin{proof} Write $\phi:C^\iy(\R^m)/I\ra C^\iy(\R^n)/J$ for the
isomorphism, and $x_1,\ldots,x_m$ for the generators of
$C^\iy(\R^m)$, and $y_1,\ldots,y_n$ for the generators of
$C^\iy(\R^n)$. Since $\phi$ is an isomorphism we can choose
$f_1,\ldots,f_m\in C^\iy(\R^n)$ with $\phi(x_i+I)=f_i+J$ for
$i=1,\ldots,m$ and $g_1,\ldots,g_n\in C^\iy(\R^m)$ with
$\phi(g_i+I)=y_i+J$ for $i=1,\ldots,n$. Write $\bs
g=(g_1,\ldots,g_n):\R^m\ra\R^n$. Then
\begin{align*}
\phi(x_i+I)&=f_i+J=\Phi_{f_i}(y_1,\ldots,y_n)+J=
\Phi_{f_i}(y_1+J,\ldots,y_n+J)\\
&=\Phi_{f_i}\bigl(\phi(g_1+I),\ldots,\phi(g_n+I)\bigr)=
\phi\bigl(\Phi_{f_i}(g_1,\ldots,g_n)+I\bigr),
\end{align*}
since $\phi$ and the projections $C^\iy(\R^m)\ra C^\iy(\R^m)/I$,
$C^\iy(\R^n)\ra C^\iy(\R^n)/J$ are morphisms of $C^\iy$-rings. So as
$\phi$ is injective we see that
\e
x_i-\Phi_{f_i}(g_1,\ldots,g_n)\in I \quad\text{for $i=1,\ldots,m$.}
\label{ag2eq3}
\e
As $I$ is the kernel of $\phi\ci\pi:C^\iy(\R^m)\ra C^\iy(\R^n)/J$ we
see that
\e
I=\bigl\{e\in C^\iy(\R^m):\Phi_e(f_1,\ldots,f_m)\in J\bigr\}.
\label{ag2eq4}
\e
Let $K$ be the ideal in $C^\iy(\R^m)$ generated by
$x_i-\Phi_{f_i}(g_1,\ldots,g_n)$ for $i=1,\ldots,m$. Then
$K\subseteq I$ by \eq{ag2eq3}. Let $e\in C^\iy(\R^m)$. Then
\begin{align*}
e+K&=\Phi_e(x_1,\ldots,x_m)+K=\Phi_e(x_1+K,\ldots,x_m+K)\\
&=\Phi_e\bigl(\Phi_{f_1}(g_1,\ldots,g_n)+K,\ldots,
\Phi_{f_m}(g_1,\ldots,g_n)+K\bigr)\\
&=\Phi_e\bigl(\Phi_{f_1}(g_1,\ldots,g_n),\ldots,
\Phi_{f_m}(g_1,\ldots,g_n)\bigr)+K\\
&=\Phi_{\Phi_e(f_1,\ldots,f_m)}(g_1,\ldots,g_n)+K.
\end{align*}
So by \eq{ag2eq4}, if $e\in I$ then $e+K=\Phi_h(g_1,\ldots,g_n)+K$
for some $h\in J$. Conversely, if $h\in J$ then $\bs
g^*(h)=\Phi_h(g_1,\ldots,g_n)\in I$. Hence
\e
I=\bigl(x_i-\Phi_{f_i}(g_1,\ldots,g_n),\; i=1,\ldots,m, \text{ and }
\bs g^*(h),\; h\in J\bigr).
\label{ag2eq5}
\e
Suppose $J$ is good, so that $J=(h_1,\ldots,h_l,\m_Y^\iy)$ for some
$h_1,\ldots,h_l\in C^\iy(\R^n)$ and closed $Y\subseteq\R^n$. Define
$X=\bs g^{-1}(Y)$, a closed subset in $\R^n$. Using Proposition
\ref{ag2prop2}, one can show that the ideal $\bigl(\bs
g^*(\m_Y^\iy)\bigr)$ in $C^\iy(\R^n)$ generated by $\bs
g^*(\m_Y^\iy)$ is $\m_X^\iy$. Therefore \eq{ag2eq5} gives
\begin{equation*}
I=\text{$\bigl(x_i-\Phi_{f_i}(g_1,\ldots,g_n)$, $i=1,\ldots,n$,
$\bs g^*(h_j)$, for $j=1,\ldots,l$, and $\m_X^\iy\bigr)$,}
\end{equation*}
so $I$ is good. Conversely, if $I$ is good then $J$ is good. If $J$
is finitely generated we take $Y=\es$, and then $I$ is finitely
generated, and vice versa.
\end{proof}
\begin{ex} The $C^\iy$-local ring $C^\iy_p(\R^n)$ of Example
\ref{ag2ex4} is the quotient of $C^\iy(\R^n)$ by the ideal $I$ of
functions $f$ with $f\equiv 0$ near $p\in\R^n$. For $n>0$ this $I$
is fair, but not good. So $C^\iy_p(\R^n)$ is fair, but not good, by
Proposition~\ref{ag2prop3}.
\label{ag2ex7}
\end{ex}
Moerdijk and Reyes \cite[Cor.~I.4.19]{MoRe2} prove:
\begin{prop} Let\/ $X\subseteq\R^n$ be closed with\/ $X\ne\es,\R^n$.
Then the ideal\/ $\fm_X^\iy$ in $C^\iy(\R^n)$ is not countably
generated.
\label{ag2prop4}
\end{prop}
Hence {\it good ideals need not be finitely generated}. With
Proposition \ref{ag2prop3}, this implies that {\it good\/
$C^\iy$-rings need not be finitely presented}.
\begin{prop} Let\/ $\fC$ be a $C^\iy$-ring, $c\in\fC,$ and
$\fC[c^{-1}]$ be the localization of\/ $\fC$ at\/ $c$. If\/ $\fC$ is
finitely presented, or good, then\/ $\fC[c^{-1}]$ is finitely
presented, or good, respectively.
\label{ag2prop5}
\end{prop}
\begin{proof} There is an exact sequence $0\ra I\ra
C^\iy(\R^n)\,{\buildrel\phi\over\longra}\,\fC\ra 0$ as $\fC$ is
finitely generated in both cases. Write $x_1,\ldots,x_n$ for the
generators of $C^\iy(\R^n)$, and let $c=\phi(e)$ for $e\in
C^\iy(\R^n)$. Then we have an exact sequence $0\ra J\ra
C^\iy(\R^{n+1})\,{\buildrel\psi\over\longra}\,\fC[c^{-1}]\ra 0$,
where if we write the generators of $C^\iy(\R^{n+1})$ as
$x_1,\ldots,x_n,y$, then $J=\bigl(y\,e(x_1,\ldots,x_n)-1$,
$f(x_1,\ldots,x_n):f\in I\bigr)$. If $\fC$ is good then
$I=(f_1,\ldots,f_k$, $\fm_X^\iy)$ for some closed $X\subseteq\R^n$.
Proposition \ref{ag2prop2} then yields
\begin{equation*}
J= \bigl(y\,e(x_1,\ldots,x_n)-1,\;\> f_1(x_1,\ldots,x_n),\ldots,
f_k(x_1,\ldots,x_n),\text{ and }\fm_{X\t\R}^\iy\bigr),
\end{equation*}
where $X\t\R\subseteq\R^{n+1}$ is closed. Hence $J,\fC[c^{-1}]$ are
good. If $\fC$ is finitely presented we take $X=\es$, and then
$\fC[c^{-1}]$ is finitely presented.
\end{proof}
However, the following example taken from Dubuc
\cite[Ex.~7.2]{Dubu4} shows that localizations of fair $C^\iy$-rings
need not be fair:
\begin{ex} Let $\fC$ be the $C^\iy$-local ring $C^\iy_0(\R)$, as in
Example \ref{ag2ex4}. Then $\fC\cong C^\iy(\R)/I$, where $I$ is the
ideal of all $f\in C^\iy(\R)$ with $f\equiv 0$ near 0 in $\R$. This
$I$ is fair, so $\fC$ is fair. Let $c=[(x,\R)]\in\fC$. Then the
localization $\fC[c^{-1}]$ is the $C^\iy$-ring of germs at 0 in $\R$
of smooth functions $\R\sm\{0\}\ra\R$. Taking $y=x^{-1}$ as a
generator of $\fC[c^{-1}]$, we see that $\fC[c^{-1}]\cong
C^\iy(\R)/J$, where $J$ is the ideal of compactly supported
functions in $C^\iy(\R)$. This $J$ is not fair, so by Lemma
\ref{ag2lem1}, $\fC[c^{-1}]$ is not fair.
\label{ag2ex8}
\end{ex}
\subsection{Pushouts of $C^\iy$-rings}
\label{ag26}
Proposition \ref{ag2prop1} shows that pushouts of $C^\iy$-rings
exist. For finitely generated $C^\iy$-rings, we can describe these
pushouts explicitly.
\begin{ex} Suppose the following is a pushout diagram of $C^\iy$-rings:
\begin{equation*}
\xymatrix@C=30pt@R=10pt{ \fC \ar[r]_\be \ar[d]^\al & \fE \ar[d]_\de \\
\fD \ar[r]^\ga & \fF,}
\end{equation*}
so that\/ $\fF=\fD\amalg_\fC\fE$, with $\fC,\fD,\fE$ finitely
generated. Then we have exact sequences
\e
\begin{gathered}
0\ra I\hookra C^\iy(\R^l)\,{\buildrel\phi\over \longra}\,\fC\ra
0,\quad 0\ra J\hookra C^\iy(\R^m)\,{\buildrel\psi\over
\longra}\,\fD\ra 0,\\
\text{and}\quad 0\ra K\hookra C^\iy(\R^n)\,{\buildrel\chi\over
\longra}\,\fE\ra 0,
\end{gathered}
\label{ag2eq6}
\e
where $\phi,\psi,\chi$ are morphisms of $C^\iy$-rings, and $I,J,K$
are ideals in $C^\iy(\R^l),\ab C^\iy(\R^m),\ab C^\iy(\R^n)$. Write
$x_1,\ldots,x_l$ and $y_1,\ldots,y_m$ and $z_1,\ldots,z_n$ for the
generators of $C^\iy(\R^l),C^\iy(\R^m),C^\iy(\R^n)$ respectively.
Then $\phi(x_1),\ldots,\phi(x_l)$ generate $\fC$, and
$\al\ci\phi(x_1),\ldots,\al\ci\phi(x_l)$ lie in $\fD$, so we may
write $\al\ci\phi(x_i)=\psi(f_i)$ for $i=1,\ldots,l$ as $\psi$ is
surjective, where $f_i:\R^m\ra\R$ is smooth. Similarly
$\be\ci\phi(x_1),\ldots,\be\ci\phi(x_l)$ lie in $\fE$, so we may
write $\be\ci\phi(x_i)=\chi(g_i)$ for $i=1,\ldots,l$, where
$g_i:\R^n\ra\R$ is smooth.
Then from the explicit construction of pushouts of $C^\iy$-rings we
obtain an exact sequence with $\xi$ a morphism of $C^\iy$-rings
\e
0\ra L\hookra C^\iy(\R^{m+n})\,{\buildrel\xi\over \longra}\,\fF\ra
0,
\label{ag2eq7}
\e
where we write the generators of $C^\iy(\R^{m+n})$ as
$y_1,\ldots,y_m,z_1,\ldots,z_n$, and then $L$ is the ideal in
$C^\iy(\R^{m+n})$ generated by the elements $d(y_1,\ldots,y_m)$ for
$d\in J\subseteq C^\iy(\R^m)$, and $e(z_1,\ldots,z_n)$ for $e\in
K\subseteq C^\iy(\R^n)$, and $f_i(y_1,\ldots,y_m)-
g_i(z_1,\ldots,z_n)$ for $i=1,\ldots,l$.
For the case of {\it coproducts\/} $\fD\ot_\iy\fE$, with $\fC=\R$,
$l=0$ and $I=\{0\}$, we have
\begin{equation*}
\bigl(C^\iy(\R^m)/J\bigr)\ot_\iy\bigl(C^\iy(\R^n)/K\bigr)=
C^\iy(\R^{m+n})/(J,K).
\end{equation*}
\label{ag2ex9}
\end{ex}
\begin{prop} The subcategories\/ $\CRingsfg,\CRingsfp,\CRingsgo$
are closed under pushouts and all finite colimits in\/~$\CRings$.
\label{ag2prop6}
\end{prop}
\begin{proof} First we show $\CRingsfg,\CRingsfp,\CRingsgo$ are
closed under pushouts. Suppose $\fC,\fD,\fE$ are finitely generated,
and use the notation of Example \ref{ag2ex9}. Then $\fF$ is finitely
generated with generators $y_1,\ldots,y_m,z_1,\ldots,z_n$, so
$\CRingsfg$ is closed under pushouts. Suppose $\fD,\fE$ are good.
Then $J=(d_1,\ldots,d_j,\fm_Y^\iy)$, $K=(e_1,\ldots,e_k,\fm_Z^\iy)$
for $Y\subseteq\R^m$, $Z\subseteq\R^n$ closed. Proposition
\ref{ag2prop2} gives $(\fm_Y^\iy,\fm_Z^\iy)=\fm_{Y\t Z}^\iy$, so by
Example \ref{ag2ex9} we have
\e
\begin{split}
L=\bigl(&d_p(y_1,\ldots,y_m),\; p=1,\ldots,j,\;
e_p(z_1,\ldots,z_n),\; p=1,\ldots,k,\\ &
f_p(y_1,\ldots,y_m)-g_p(z_1,\ldots,z_n),\; p=1,\ldots,l,
\;\>\text{and}\;\>\fm_{Y\t Z}^\iy\bigr).
\end{split}
\label{ag2eq8}
\e
Thus $L$ is good, so $\fF$ is good, and $\CRingsgo$ is closed under
pushouts. For $\fD,\fE$ finitely presented we take $Y=Z=\es$, and
then $L$ is finitely generated, and $\fF$ finitely presented, and
$\CRingsfp$ is closed under pushouts.
Finally, note that $\R$ is an initial object in $\CRings$ and lies
in $\CRingsfg$, $\CRingsfp$ and $\CRingsgo$, and all finite colimits
may be constructed by repeated pushouts possibly involving the
initial object. Hence $\CRingsfg,\ab\CRingsfp,\ab\CRingsgo$ are
closed under finite colimits.
\end{proof}
Here is an example from Dubuc \cite[Ex.~7.1]{Dubu4}, Moerdijk and
Reyes~\cite[p.~49]{MoRe2}.
\begin{ex} Consider the coproduct $C^\iy(\R)\ot_\iy C^\iy_0(\R)$,
where $C^\iy_0(\R)$ is the $C^\iy$-ring of germs of smooth functions
at 0 in $\R$ as in Example \ref{ag2ex4}. Then $C^\iy(\R),\ab
C^\iy_0(\R)$ are fair $C^\iy$-rings, but $C^\iy_0(\R)$ is neither
finitely presented nor good. By Example \ref{ag2ex9},
$C^\iy(\R)\ot_\iy C^\iy_0(\R)=C^\iy(\R)\amalg_\R C^\iy_0(\R)\cong
C^\iy(\R^2)/L$, where $L$ is the ideal in $C^\iy(\R^2)$ generated by
functions $f(x,y)=g(y)$ for $g\in C^\iy(\R)$ with $g\equiv 0$ near
$0\in\R$. This ideal $L$ is not fair, since for example one can find
$f\in C^\iy(\R^2)$ with $f(x,y)=0$ if and only if $\md{xy}\le 1$,
and then $f\notin L$ but $\pi_p(f)\in\pi_p(L)\subseteq
C^\iy_p(\R^2)$ for all $p\in\R^2$. Hence $C^\iy(\R)\ot_\iy
C^\iy_0(\R)$ is not a fair $C^\iy$-ring, by Lemma \ref{ag2lem1}, and
pushouts of fair $C^\iy$-rings need not be fair.
\label{ag2ex10}
\end{ex}
Our next result is referred to in the last part of
Dubuc~\cite[Th.~13]{Dubu3}.
\begin{prop} $\CRingsfa$ is not closed under pushouts in\/
$\CRings$. Nonetheless, pushouts and all finite colimits exist in\/
$\CRingsfa,$ although they may not coincide with pushouts and finite
colimits in~$\CRings$.
\label{ag2prop7}
\end{prop}
\begin{proof} Example \ref{ag2ex10} shows that $\CRingsfa$ is not
closed under pushouts in $\CRings$. To construct finite colimits in
$\CRingsfa$, we first take the colimit in $\CRingsfg$, which exists
by Propositions \ref{ag2prop1} and \ref{ag2prop6}, and then apply
the reflection functor $R_{\rm fg}^{\rm fa}$. By the universal
properties of colimits and reflection functors, the result is a
colimit in~$\CRingsfa$.
\end{proof}
\section{The $C^\iy$-ring $C^\iy(X)$ of a manifold $X$}
\label{ag3}
We now study the $C^\iy$-rings $C^\iy(X)$ of manifolds $X$ defined
in Example \ref{ag2ex1}. We are interested in {\it manifolds without
boundary\/} (locally modelled on $\R^n$), and in {\it manifolds with
boundary\/} (locally modelled on $[0,\iy)\t\R^{n-1}$), and in {\it
manifolds with corners\/} (locally modelled on
$[0,\iy)^k\t\R^{n-k}$). Manifolds with corners were considered by
the author \cite{Joyc1}, and we use the conventions of that paper.
The $C^\iy$-rings of manifolds with boundary are discussed by Reyes
\cite{Reye} and Kock \cite[\S III.9]{Kock}, but Kock appears to have
been unaware of Proposition \ref{ag2prop2}, which makes
$C^\iy$-rings of manifolds with boundary easier to understand.
If $X,Y$ are manifolds with corners of dimensions $m,n$, then
\cite[\S 3]{Joyc1} defined $f:X\ra Y$ to be {\it weakly smooth\/} if
$f$ is continuous and whenever $(U,\phi),(V,\psi)$ are charts on
$X,Y$ then $\psi^{-1}\ci f\ci\phi:(f\ci\phi)^{-1}(\psi(V))\ra V$ is
a smooth map from $(f\ci\phi)^{-1}(\psi(V))\subset\R^m$ to
$V\subset\R^n$. A {\it smooth map\/} is a weakly smooth map $f$
satisfying some complicated extra conditions over $\pd^kX,\pd^lY$ in
\cite[\S 3]{Joyc1}. If $\pd Y=\es$ these conditions are vacuous, so
for manifolds without boundary, weakly smooth maps and smooth maps
coincide. Write $\Man,\Manb,\Manc$ for the categories of manifolds
without boundary, and with boundary, and with corners, respectively,
with morphisms smooth maps.
\begin{ex} Let $0<k\le n$, and consider the closed subset
$\R^n_k=[0,\iy)^k\t\R^{n-k}$ in $\R^n$, the local model for
manifolds with corners. Write $C^\iy(\R^n_k)$ for the $C^\iy$-ring
$\bigl\{f\vert_{\R^n_k}:f\in C^\iy(\R^n)\bigr\}$. Since the interior
$(\R^n_k)^\ci=(0,\iy)^k\t\R^{n-k}$ of $\R^n_k$ is dense in $\R^n_k$,
as in Definition \ref{ag2def10} we have
$C^\iy(\R^n_k)=C^\iy(\R^n)/\smash{\fm_{\R^n_k}^\iy}$. Hence
$C^\iy(\R^n_k)$ is a {\it good\/ $C^\iy$-ring}, by Definition
\ref{ag2def11}. Also, $\smash{\fm_{\R^n_k}^\iy}$ is not countably
generated by Proposition \ref{ag2prop4}, so it is not finitely
generated, and thus $C^\iy(\R^n_k)$ is {\it not a finitely
presented\/ $C^\iy$-ring\/} by Proposition~\ref{ag2prop3}.
Consider the coproduct $C^\iy(\R^m_k)\ot_\iy C^\iy(\R^n_l)$ in
$\CRings$, that is, the pushout $C^\iy(\R^m_k)\amalg_\R
C^\iy(\R^n_l)$ over the trivial $C^\iy$-ring $\R$. By Example
\ref{ag2ex9} and Proposition \ref{ag2prop2} we have
\begin{align*}
C^\iy(\R^m_k)\ot_\iy C^\iy(\R^n_l)&\cong C^\iy(\R^{m+n})/
(\fm_{\R^m_k}^\iy,\fm_{\R^n_l}^\iy)=
C^\iy(\R^{m+n})/\fm_{\R^m_k\t\R^n_l}^\iy\\
&=C^\iy(\R^m_k\t \R^n_l)\cong C^\iy(\R^{m+n}_{k+l}).
\end{align*}
This is an example of Theorem \ref{ag3thm} below, with
$X\!=\!\R^m_k$, $Y\!=\!\R^n_l$ and~$Z\!=\!\{0\}$.
\label{ag3ex1}
\end{ex}
\begin{prop}{\bf(a)} If\/ $X$ is a manifold without boundary then
the $C^\iy$-ring $C^\iy(X)$ of Example\/ {\rm\ref{ag2ex1}} is
finitely presented.
\smallskip
\noindent{\bf(b)} If\/ $X$ is a manifold with boundary, or with
corners, and\/ $\pd X\ne\es,$ then the $C^\iy$-ring $C^\iy(X)$ of
Example\/ {\rm\ref{ag2ex1}} is good, but is not finitely presented.
\label{ag3prop1}
\end{prop}
\begin{proof} Part (a) is proved in Dubuc \cite[p.~687]{Dubu3} and
Moerdijk and Reyes \cite[Th.~I.2.3]{MoRe2} following an observation
of Lawvere, that if $X$ is a manifold without boundary then we can
choose a closed embedding $i:X\hookra\R^N$ for $N\gg 0$, and then
$X$ is a retract of an open neighbourhood $U$ of $i(X)$, so we have
an exact sequence $0\ra I\ra C^\iy(\R^N)\,\smash{{\buildrel
i^*\over\longra}} \,C^\iy(X)\ra 0$ in which the ideal $I$ is
finitely generated, and thus the $C^\iy$-ring $C^\iy(X)$ is finitely
presented.
For (b), if $X$ is an $n$-manifold with boundary, or with corners,
then we can embed $X$ as a closed subset in an $n$-manifold $X'$
without boundary, such that the inclusion $X\hookra X'$ is locally
modelled on the inclusion of $\R^n_k=[0,\iy)^k\t\R^{n-k}$ in $\R^n$
for $k\le n$. We can take $X'$ diffeomorphic to the interior $X^\ci$
of $X$. Choose a closed embedding $i:X'\hookra\R^N$ for $N\gg 0$ as
above, giving $0\ra I'\ra C^\iy(\R^N)\,{\buildrel i^*\over\longra}
\,C^\iy(X')\ra 0$ with $I'$ generated by $f_1,\ldots,f_k\in
C^\iy(\R^N)$. The interior $X^\ci$ of $X$ is open in $X'$, so there
exists an open subset $U$ in $\R^N$ with $i(X^\ci)=U\cap i(X')$.
Therefore $i(X)=\bar U\cap i(X')$.
Let $I$ be the good ideal $(f_1,\ldots,f_k,\fm_{\bar U}^\iy)$ in
$C^\iy(\R^N)$. Since $U$ is open in $\R^N$ and dense in $\bar U$, as
in Definition \ref{ag2def10} we have $g\in\fm_{\bar U}^\iy$ if and
only if $g\vert_{\bar U}\equiv 0$. Therefore the isomorphism
$(i_*)_*:C^\iy(\R^N)/I'\ra C^\iy(X')$ identifies the ideal $I/I'$ in
$C^\iy(X')$ with the ideal of $f\in C^\iy(X')$ such that
$f\vert_X\equiv 0$, since $X=i^{-1}(\bar U)$. Hence
\begin{equation*}
C^\iy(\R^N)/I\!\cong\!C^\iy(X')/\bigl\{f\!\in\! C^\iy(X'):f\vert_X
\!\equiv\!0\bigr\}\!\cong\!\bigl\{f\vert_X\!:\!f\!\in\! C^\iy(X')
\bigr\}\!\cong\! C^\iy(X).
\end{equation*}
As $I$ is a good ideal, this implies that $C^\iy(X)$ is a good
$C^\iy$-ring. If $\pd X\ne\es$ then using Proposition \ref{ag2prop4}
we can show $I$ is not countably generated, so $C^\iy(X)$ is not
finitely presented by Proposition~\ref{ag2prop3}.
\end{proof}
Next we consider the transformation $X\mapsto C^\iy(X)$ as a
functor.
\begin{dfn} Write $(\CRings)^{\rm op}$, $(\CRingsfp)^{\rm op}$,
$(\CRingsgo)^{\rm op}$ for the opposite categories of
$\CRings,\CRingsfp,\CRingsgo$ (i.e.\ directions of morphisms are
reversed). Define functors
\begin{align*}
F_\Man^\CRings:\Man&\longra(\CRingsfp)^{\rm op}\subset (\CRings)^{\rm
op},\\
F_\Manb^\CRings:\Manb&\longra(\CRingsgo)^{\rm op}\subset
(\CRings)^{\rm op},\\
F_\Manc^\CRings:\Manc&\longra(\CRingsgo)^{\rm
op}\subset (\CRings)^{\rm op}
\end{align*}
as follows. On objects the functors $F_{\Man^*}^\CRings$ map
$X\mapsto C^\iy(X)$, where $C^\iy(X)$ is a $C^\iy$-ring as in
Example \ref{ag2ex1}. On morphisms, if $f:X\ra Y$ is a smooth map of
manifolds then $f^*:C^\iy(Y)\ra C^\iy(X)$ mapping $c\mapsto c\ci f$
is a morphism of $C^\iy$-rings, so that $f^*:C^\iy(Y)\ra C^\iy(X)$
is a morphism in $\CRings$, and $f^*:C^\iy(X)\ra C^\iy(Y)$ a
morphism in $(\CRings)^{\rm op}$, and $F_{\Man^*}^\CRings$ map
$f\mapsto f^*$. Clearly $F_\Man^\CRings,F_\Manb^\CRings,
F_\Manc^\CRings$ are functors.
\label{ag3def}
\end{dfn}
If $f:X\ra Y$ is only {\it weakly smooth\/} then $f^*:C^\iy(Y)\ra
C^\iy(X)$ in Definition \ref{ag3def} is still a morphism of
$C^\iy$-rings. From \cite[Prop.~I.1.5]{MoRe2} we deduce:
\begin{prop} Let\/ $X,Y$ be manifolds with corners. Then the map
$f\mapsto f^*$ from weakly smooth maps\/ $f:X\ra Y$ to morphisms
of\/ $C^\iy$-rings $\phi:C^\iy(Y)\ra C^\iy(X)$ is a $1$-$1$
correspondence.
\label{ag3prop2}
\end{prop}
Using the conventions of \cite{Joyc1}, in the category $\Man$ of
manifolds without boundary, the morphisms are weakly smooth maps. So
$F_\Man^\CRings$ is both injective on morphisms (faithful), and
surjective on morphisms (full), as in Moerdijk and Reyes
\cite[Th.~I.2.8]{MoRe2}. But in $\Manb,\Manc$ the morphisms are
smooth maps, a proper subset of weakly smooth maps, so the functors
are injective but not surjective on morphisms. That is:
\begin{cor} The functor $F_\Man^\CRings:\Man\ra(\CRingsfp)^{\rm
op}$ is full and faithful. However, the functors $F_\Manb^\CRings:
\Manb\ra(\CRingsgo)^{\rm op}$ and\/ $F_\Manc^\CRings:
\Manc\ra(\CRingsgo)^{\rm op}$ are faithful, but not full.
\label{ag3cor}
\end{cor}
Of course, if we defined $\Manb,\Manc$ to have morphisms weakly
smooth maps, then $F_\Manb^\CRings,F_\Manc^\CRings$ would be full
and faithful. But this is not what we need for the applications
in~\cite{Joyc3}.
Let $X,Y,Z$ be manifolds and $f:X\ra Z$, $g:Y\ra Z$ be smooth maps.
If $X,Y,Z$ are without boundary then $f,g$ are called {\it
transverse\/} if whenever $x\in X$ and $y\in Y$ with $f(x)=g(y)=z\in
Z$ we have $T_zZ=\d f(T_xX)+\d g(T_yY)$. If $f,g$ are transverse
then a fibre product $X\t_ZY$ exists in~$\Man$.
For manifolds with boundary, or with corners, the situation is more
complicated, as explained in \cite[\S 6]{Joyc1}. In the definition
of {\it smooth\/} $f:X\ra Y$ we impose extra conditions over
$\pd^jX,\pd^kY$, and in the definition of transverse $f,g$ we impose
extra conditions over $\pd^jX,\pd^kY,\pd^lZ$. With these more
restrictive definitions of smooth and transverse maps, transverse
fibre products exist in $\Manc$ by \cite[Th.~6.3]{Joyc1}. The na\"\i
ve definition of transversality is not a sufficient condition for
fibre products to exist. Note too that a fibre product of manifolds
with boundary may be a manifold with corners, so fibre products work
best in $\Man$ or $\Manc$ rather than~$\Manb$.
Our next theorem is given in \cite[Th.~16]{Dubu3} and
\cite[Prop.~I.2.6]{MoRe2} for manifolds without boundary, and the
special case of products $\Man\t\Manb\ra\Manb$ follows from Reyes
\cite[Th.~2.5]{Reye}, see also Kock \cite[\S III.9]{Kock}. It can be
proved by combining the usual proof in the without boundary case,
the proof of \cite[Th.~6.3]{Joyc1}, and Proposition~\ref{ag2prop2}.
\begin{thm} The functors\/ $F_\Man^\CRings,F_\Manc^\CRings$ preserve
transverse fibre products in $\Man,\Manc,$ in the sense of\/
{\rm\cite[\S 6]{Joyc1}}. That is, if the following is a Cartesian
square of manifolds with\/ $g,h$ transverse
\e
\begin{gathered}
\xymatrix@C=30pt@R=10pt{ W \ar[r]_f \ar[d]^e & Y \ar[d]_h \\
X \ar[r]^g & Z,}
\end{gathered}
\label{ag3eq1}
\e
so that\/ $W=X\t_{g,Z,h}Y,$ then we have a pushout square of\/
$C^\iy$-rings
\e
\begin{gathered}
\xymatrix@C=30pt@R=10pt{ C^\iy(Z) \ar[r]_{h^*} \ar[d]^{g^*} &
C^\iy(Y) \ar[d]_{f^*} \\ C^\iy(X) \ar[r]^{e^*} & C^\iy(W),}
\end{gathered}
\label{ag3eq2}
\e
so that\/ $C^\iy(W)=C^\iy(X)\amalg_{g^*,C^\iy(Z),h^*}C^\iy(Y)$.
\label{ag3thm}
\end{thm}
\section{$C^\iy$-ringed spaces and $C^\iy$-schemes}
\label{ag4}
In algebraic geometry, if $A$ is an affine scheme and $R$ the ring
of regular functions on $A$, then we can recover $A$ as the spectrum
of the ring $R$, $A\cong\Spec R$. One of the ideas of synthetic
differential geometry, as in \cite[\S I]{MoRe2}, is to regard a
manifold $M$ as the `spectrum' of the $C^\iy$-ring $C^\iy(M)$ in
Example \ref{ag2ex1}. So we can try to develop analogues of the
tools of scheme theory for smooth manifolds, replacing rings by
$C^\iy$-rings throughout. This was done by Dubuc \cite{Dubu2,Dubu3}.
The analogues of the algebraic geometry notions \cite[\S II.2]{Hart}
of ringed spaces, locally ringed spaces, and schemes, are called
$C^\iy$-ringed spaces, local $C^\iy$-ringed spaces and
$C^\iy$-schemes. Almost nothing in this section is really new,
though we give more detail than our references in places.
\subsection{$C^\iy$-ringed spaces and local $C^\iy$-ringed spaces}
\label{ag41}
\begin{dfn} A {\it $C^\iy$-ringed space\/} $\uX=(X,\O_X)$ is a
topological space $X$ with a sheaf $\O_X$ of $C^\iy$-rings on $X$.
That is, for each open set $U\subseteq X$ we are given a $C^\iy$
ring $\O_X(U)$, and for each inclusion of open sets $V\subseteq
U\subseteq X$ we are given a morphism of $C^\iy$-rings
$\rho_{UV}:\O_X(U)\ra\O_X(V)$, called the {\it restriction maps},
and all this data satisfies the usual sheaf axioms \cite[\S
II.1]{Hart}, \cite[\S 0.3.1]{Grot}.
A {\it morphism\/} $\uf=(f,f^\sh):(X,\O_X)\ra(Y,\O_Y)$ of $C^\iy$
ringed spaces is a continuous map $f:X\ra Y$ and a morphism
$f^\sh:\O_Y\ra f_*(\O_X)$ of sheaves of $C^\iy$-rings on $Y$. That
is, for each open $U\subset Y$ we are given a morphism of
$C^\iy$-rings $f^\sh(U):\O_Y(U)\ra\O_X(f^{-1}(U))$ satisfying the
obvious compatibilities with the restriction maps $\rho_{UV}$ in
$\O_X$ and~$\O_Y$.
A {\it local\/ $C^\iy$-ringed space\/} $\uX=(X,\O_X)$ is a
$C^\iy$-ringed space for which the stalks $\O_{X,x}$ of $\O_X$ at
$x$ are $C^\iy$-local rings for all $x\in X$. As in Remark
\ref{ag2rem}, since morphisms of $C^\iy$-local rings are
automatically local morphisms, morphisms of local $C^\iy$-ringed
spaces $(X,\O_X),(Y,\O_Y)$ are just morphisms of $C^\iy$-ringed
spaces, without any additional locality condition. Moerdijk et al.\
\cite[\S 3]{MQR} call our local $C^\iy$-ringed spaces {\it
Archimedean\/ $C^\iy$-spaces}.
Write $\CRS$ for the category of $C^\iy$-ringed spaces, and $\LCRS$
for the full subcategory of local $C^\iy$-ringed spaces.
For brevity, we will use the notation that underlined upper case
letters $\uX,\uY,\uZ,\ldots$ represent $C^\iy$-ringed spaces
$(X,\O_X),(Y,\O_Y),(Z,\O_Z),\ldots$, and underlined lower case
letters $\uf,\ug,\ldots$ represent morphisms of $C^\iy$-ringed
spaces $(f,f^\sh),\ab(g,g^\sh),\ab\ldots$. When we write `$x\in\uX$'
we mean that $\uX=(X,\O_X)$ and $x\in X$. When we write `$\uU$ {\it
is open in\/} $\uX$' we mean that $\uU=(U,\O_U)$ and $\uX=(X,\O_X)$
with $U\subseteq X$ an open set and~$\O_U=\O_X\vert_U$.
\label{ag4def1}
\end{dfn}
\begin{ex} Let $X$ be a manifold, which may have boundary or
corners. Define a $C^\iy$-ringed space $\uX=(X,\O_X)$ to have
topological space $X$ and $\O_X(U)=C^\iy(U)$ for each open subset
$U\subseteq X$, where $C^\iy(U)$ is the $C^\iy$-ring of smooth maps
$c:U\ra\R$, and if $V\subseteq U\subseteq X$ are open we define
$\rho_{UV}:C^\iy(U)\ra C^\iy(V)$ by~$\rho_{UV}:c\mapsto c\vert_V$.
It is easy to verify that $\O_X$ is a sheaf of $C^\iy$-rings on $X$
(not just a presheaf), so $\uX=(X,\O_X)$ is a $C^\iy$-ringed space.
For each $x\in X$, the stalk $\O_{X,x}$ is the $C^\iy$-local ring of
germs $[(c,U)]$ of smooth functions $c:X\ra\R$ at $x\in X$, as in
Example \ref{ag2ex4}, with unique maximal ideal ${\mathfrak
m}_{X,x}=\bigl\{[(c,U)]\in\O_{X,x}:c(x)=0\big\}$ and
$\O_{X,x}/{\mathfrak m}_{X,x}\cong\R$. Hence $\uX$ is a local
$C^\iy$-ringed space.
Let $X,Y$ be manifolds and $f:X\ra Y$ a weakly smooth map. Define
$(X,\O_X),(Y,\O_Y)$ as above. For all open $U\subseteq Y$ define
$f^\sh(U):\O_Y(U)= C^\iy(U)\ra\O_X(f^{-1}(U))=C^\iy(f^{-1}(U))$ by
$f^\sh(U):c\mapsto c\ci f$ for all $c\in C^\iy(U)$. Then $f^\sh(U)$
is a morphism of $C^\iy$-rings, and $f^\sh:\O_Y\ra f_*(\O_X)$ is a
morphism of sheaves of $C^\iy$-rings on $Y$, so $\uf=(f,f^\sh):
(X,\O_X)\ra(Y,\O_Y)$ is a morphism of (local) $C^\iy$-ringed spaces.
\label{ag4ex1}
\end{ex}
As the category $\Top$ of topological spaces has all finite limits,
and the construction of $\CRS$ involves $\Top$ in a covariant way
and the category $\CRings$ in a contravariant way, using Proposition
\ref{ag2prop1} one may prove:
\begin{prop} All finite limits exist in the category\/~$\CRS$.
\label{ag4prop1}
\end{prop}
Dubuc \cite[Prop.~7]{Dubu3} proves:
\begin{prop} The full subcategory $\LCRS$ of local\/ $C^\iy$-ringed
spaces in $\CRS$ is closed under finite limits in~$\CRS$.
\label{ag4prop2}
\end{prop}
\subsection{Affine $C^\iy$-schemes}
\label{ag42}
We define a functor $\Spec:\CRings^{\rm op}\ra\LCRS$, following
Hartshorne \cite[p.~70]{Hart}, Dubuc \cite{Dubu2,Dubu3}, and
Moerdijk, Qu\^e and Reyes~\cite{MQR}.
\begin{dfn} Let $\fC$ be a $C^\iy$-ring, and use the notation of
Definition \ref{ag2def7}. Write $X_\fC$ for the set of all
$\R$-points $x$ of $\fC$. Then each $c\in\fC$ determines a map
$c_*:X_\fC\ra\R$ by $c_*:x\mapsto x(c)$. Let ${\cal T}_\fC$ be the
smallest topology on $X_\fC$ such that $c_*:X_\fC\ra\R$ is
continuous for all $c\in\fC$. That is, ${\cal T}_\fC$ is generated
by the open sets $(c_*)^{-1}(U)$ for all $c\in\fC$ and open
$U\subseteq\R$. Then $X_\fC$ is a topological space. It is
Hausdorff, since if $x_1\ne x_2\in X_\fC$ then there exists
$c\in\fC$ with $x_1(c)\ne x_2(c)$, and then $c_*:X_\fC\ra\R$ is
continuous and~$c_*(x_1)\ne c_*(x_2)$.
For each open $U\subseteq X_\fC$, define $\O_{X_\fC}(U)$ to be the
set of functions $s:U\ra \coprod_{x\in U}\fC_x$ with $s(x)\in\fC_x$
for all $x\in U$, and such that $U$ may be covered by open sets $V$
for which there exist $c,d\in\fC$ with $x(d)\ne 0$ for all $x\in V$,
with $s(x)=\pi_x(c)\pi_x(d)^{-1}\in\fC_x$ for all $x\in V$. Define
operations $\Phi_f$ on $\O_{X_\fC}(U)$ pointwise in $x\in U$ using
the operations $\Phi_f$ on $\fC_x$. This makes $\O_{X_\fC}(U)$ into
a $C^\iy$-ring. If $V\subseteq U\subseteq X_\fC$ are open, the
restriction map $\rho_{UV}:\O_{X_\fC}(U)\ra\O_{X_\fC}(V)$ mapping
$\rho_{UV}:s\mapsto s\vert_V$ is a morphism of $C^\iy$-rings.
It is then easy to see that $\O_{X_\fC}$ is a sheaf of $C^\iy$-rings
on $X_\fC$. The stalk $\O_{X_\fC,x}$ at $x\in X$ is isomorphic to
$\fC_x$, which is a $C^\iy$-local ring. Hence $(X_\fC,\O_{\fC})$ is
a local $C^\iy$-ringed space, which we call the {\it spectrum\/} of
$\fC$, and write as $\Spec\fC$.
Now let $\phi:\fC\ra\fD$ be a morphism of $C^\iy$-rings. Define
$f_\phi:X_\fD\ra X_\fC$ by $f_\phi(x)=x\ci\phi$. Then $f_\phi$ is
continuous. For $U\subseteq X_\fC$ open define
$f_\phi^\sh(U):\O_{X_\fC}(U)\ra\O_{X_\fD}(f_\phi^{-1}(U))$ by
$f_\phi^\sh(U)s:x\mapsto \phi_x(s(f_\phi(x)))$, where
$\phi_x:\fC_{f_\phi(x)}\ra\fD_x$ is the induced morphism of
$C^\iy$-local rings. Then $f^\sh_\phi:\O_{X_{\fC}}\ra (f_\phi)_*
(\O_{X_\fD})$ is a morphism of sheaves of $C^\iy$-rings on $X_\fC$,
so $\uf_\phi=(f_\phi,f_\phi^\sh): (X_\fD,\O_{\fD})\ra
(X_\fC,\O_{\fC})$ is a morphism of local $C^\iy$-ringed spaces.
Define $\Spec\phi:\Spec\fD\ra\Spec\fC$ by $\Spec\phi=\uf_\phi$. Then
$\Spec$ is a functor $\CRings^{\rm op}\ra\LCRS$, which preserves
limits by Dubuc~\cite[p.~687]{Dubu3}.
The {\it global sections functor\/} $\Ga:\LCRS\ra\CRings^{\rm op}$
acts on objects $(X,\O_X)$ by $\Ga:(X,\O_X)\mapsto\O_X(X)$ and on
morphisms $(f,f^\sh):(X,\O_X)\ra(Y,\O_X)$ by $\Ga:(f,f^\sh)\mapsto
f^\sh(Y)$. As in Dubuc \cite[Th.~8]{Dubu3} or Moerdijk et al.\
\cite[Th.~3.2]{MQR}, $\Ga$ is a {\it left adjoint\/} to $\Spec$,
that is, for all $\fC\in\CRings$ and $\uX\in\LCRS$ there are
functorial isomorphisms
\e
\Hom_\CRings(\fC,\Ga(\uX))\cong \Hom_\LCRS(\uX,\Spec\fC).
\label{ag4eq1}
\e
For any $C^\iy$-ring $\fC$ there is a natural morphism of
$C^\iy$-rings $\Phi_\fC:\fC\ra\Ga(\Spec\fC)$ corresponding to
$\id_\uX$ in \eq{ag4eq1} with~$\uX=\Spec\fC$.
\label{ag4def2}
\end{dfn}
\begin{rem} Our definition of the spectrum $\Spec\fC$ agrees with
Dubuc \cite{Dubu2,Dubu3}, and with the {\it Archimedean spectrum\/}
of \cite[\S 3]{MQR}. Moerdijk et al.\ \cite[\S 1]{MQR} give a
different definition of the spectrum $\Spec\fC$, in which the points
are not $\R$-points, but `$C^\iy$-radical prime ideals'.
\label{ag4rem}
\end{rem}
\begin{ex} Let $X$ be a manifold. It is easy to see that the local
$C^\iy$-ringed space $\uX$ constructed in Example \ref{ag4ex1} is
naturally isomorphic to~$\Spec C^\iy(X)$.
\label{ag4ex2}
\end{ex}
Now suppose $\fC$ is a finitely generated $C^\iy$-ring, with exact
sequence $0\ra I\hookra C^\iy(\R^n)\smash{\,{\buildrel\phi\over
\longra}}\,\fC\ra 0$. Define a map $\phi_*:X_\fC\ra\R^n$ by
$\phi_*:z\mapsto\bigl(z\ci\phi(x_1),\ldots,z\ci\phi(x_n)\bigr)$,
where $x_1,\ldots,x_n$ are the generators of $C^\iy(\R^n)$. Then
$\phi_*$ gives a homeomorphism
\e
\phi_*:X_\fC\,{\buildrel\cong\over\longra}\,X_\fC^\phi=\bigl\{
(x_1,\ldots,x_n) \in\R^n:\text{$f(x_1,\ldots,x_n)=0$ for all $f\in
I$}\bigr\},
\label{ag4eq2}
\e
where the right hand side is a closed subset of $\R^n$. So the
topological spaces in $\Spec\fC$ for finitely generated $\fC$ are
homeomorphic to closed subsets of $\R^n$. Comparing the definitions
of $\Spec$ and the reflection $R_{\rm fg}^{\rm fa}$, we can show:
\begin{prop} Let\/ $\fC$ be a fair\/ $C^\iy$-ring. Then\/
$\Phi_\fC:\fC\ra\Ga(\Spec\fC)$ is an isomorphism. More generally,
if\/ $\fC$ is a finitely generated\/ $C^\iy$-ring then\/ $\Spec\fC$
is naturally isomorphic to $\Spec \smash{R_{\rm fg}^{\rm fa}}(\fC),$
using the notation of Definition\/ {\rm\ref{ag2def9},} and\/
$\Ga(\Spec\fC)$ is naturally isomorphic to\/ $R_{\rm fg}^{\rm
fa}(\fC),$ and\/ $\Phi_\fC:\fC\ra\Ga (\Spec\fC)$ is identified with
the natural surjective projection\/~$\smash{\fC\ra R_{\rm fg}^{\rm
fa}(\fC)}$.
\label{ag4prop3}
\end{prop}
This is a contrast to conventional algebraic geometry, in which
$\Ga(\Spec R)\cong R$ for arbitrary rings $R$, as in
\cite[Prop.~II.2.2]{Hart}. If $\fC$ is not finitely generated then
$\Phi_\fC:\fC\ra\Ga(\Spec\fC)$ need not be surjective, so
$\Ga(\Spec\fC)$ can be larger than $\fC$. Proposition \ref{ag4prop3}
shows that for general $C^\iy$-rings $\fC$ the functor $\Spec$ loses
information about $\fC$, so $\Spec$ is neither full nor faithful,
but for fair $C^\iy$-rings $\Spec$ loses no information up to
isomorphism, so as in \cite[Th.~13]{Dubu3} we have:
\begin{thm} The functor\/ $\Spec:(\CRingsfa)^{\rm op}\ra\LCRS$ is
full and faithful. Hence $\Spec:(\CRingsgo)^{\rm op}\ra\LCRS,$
$\Spec:(\CRingsfp)^{\rm op}\ra\LCRS$ are also full and faithful,
since $(\CRingsfp)^{\rm op}\subset(\CRingsgo)^{\rm op}\subset
(\CRingsfa)^{\rm op}$ are full subcategories.
\label{ag4thm1}
\end{thm}
In the obvious way we define {\it affine\/ $C^\iy$-schemes}.
\begin{dfn} A local $C^\iy$-ringed space $\uX$ is called an {\it
affine\/ $C^\iy$-scheme} if it is isomorphic in $\LCRS$ to
$\Spec\fC$ for some $C^\iy$-ring $\fC$. We call $\uX$ a {\it
finitely presented}, or {\it good}, or {\it fair}, affine
$C^\iy$-scheme if $X\cong\Spec\fC$ for $\fC$ that kind of
$C^\iy$-ring. Write $\ACSch,\ACSchfp,\ACSchgo,\ACSchfa$ for the full
subcategories of affine $C^\iy$-schemes and of finitely presented,
good, and fair, affine $C^\iy$-schemes in $\LCRS$ respectively. Then
Theorem \ref{ag4thm1} implies that $\Spec:(\CRingsfa)^{\rm
op}\ra\ACSchfa$ is an equivalence of categories, and similarly
for~$\ACSchfp,\ACSchgo$.
\label{ag4def3}
\end{dfn}
We did not define {\it finitely generated\/} affine $C^\iy$-schemes,
because they coincide with fair affine $C^\iy$-schemes, as
Proposition \ref{ag4prop3} implies.
\begin{cor} Suppose $\fC$ is a finitely generated\/ $C^\iy$-ring.
Then $\Spec\fC$ is a fair affine $C^\iy$-scheme.
\label{ag4cor1}
\end{cor}
\begin{thm} The full subcategories\/ $\ACSchfp,\ACSchgo,
\ACSchfa,\ab\ACSch$ are closed under all finite limits in\/ $\LCRS$.
Hence, fibre products and all finite limits exist in each of these
subcategories.
\label{ag4thm2}
\end{thm}
\begin{proof} $\ACSch$ is closed under small limits in $\LCRS$
as small limits exist in $(\CRings)^{\rm op}$ by Proposition
\ref{ag2prop2} and $\Spec$ preserves limits by \cite[p.~687]{Dubu3}.
Fibre products and all finite limits exist in $(\CRingsfa)^{\rm op}$
by Proposition \ref{ag2prop7}, although they may not coincide with
fibre products and finite limits in $(\CRings)^{\rm op}$. The
subcategories $(\CRingsfp)^{\rm op},(\CRingsgo)^{\rm op}$ are closed
under fibre products and finite limits in $(\CRings)^{\rm op}$ by
Proposition \ref{ag2prop6}, and hence under fibre products and
finite limits in $(\CRingsfa)^{\rm op}$. By Dubuc
\cite[Th.~13]{Dubu3} the functor $\Spec:(\CRingsfa)^{\rm
op}\ra\LCRS$ preserves limits. (Here we mean limits in
$(\CRingsfa)^{\rm op}$, rather than limits in~$(\CRings)^{\rm op}$.)
Let $\uX,\uY,\uZ$ be finitely presented, or good, or fair, affine
$C^\iy$-schemes, and $\uf:\uX\ra\uZ$, $\ug:\uY\ra\uZ$ be morphisms
in $\LCRS$. Then we have isomorphisms
\e
\uX\cong\Spec\fC,\qquad \uY\cong\Spec\fD, \qquad \uZ\cong\Spec\fE
\quad\text{in $\LCRS$,}
\label{ag4eq3}
\e
where $\fC,\fD,\fE$ are finitely presented, or good, or fair,
$C^\iy$-rings, respectively. Since $\Spec:(\CRingsfa)^{\rm
op}\ra\LCRS$ is full and faithful by Theorem \ref{ag4thm1}, there
exist unique morphisms of $C^\iy$-rings $\phi:\fE\ra\fC$,
$\psi:\fE\ra\fD$ such that \eq{ag4eq3} identifies $\uf$ with
$\Spec\phi$ and $\ug$ with $\Spec\psi$. Then \cite[Th.~13]{Dubu3}
implies that
\begin{equation*}
\uX\t_{\uf,\uZ,\ug}\uY\cong\Spec
\fC\t_{\Spec\phi,\Spec\fE,\Spec\psi}\Spec\fD \cong
\Spec\bigl(\fC\amalg_{\phi,\fE,\psi}\fD\bigr),
\end{equation*}
where $\fC\amalg_\fE\fD$ is the pushout in $\CRingsfa$ rather than
in $\CRings$. Then $\fC\amalg_\fE\fD$ is finitely presented, or
good, or fair, respectively, since $(\CRingsfp)^{\rm
op},\ab(\CRingsgo)^{\rm op}$ are closed under fibre products in
$(\CRingsfa)^{\rm op}$. Hence $\uX\t_\uZ\uY$ is finitely presented,
or good, or fair, respectively, and
$\ACSchfp,\ab\ACSchgo,\ab\ACSchfa$ are closed under fibre products
in $\LCRS$. Since $\Spec\R$ is a terminal object, we see that
$\ACSchfp,\ACSchgo, \ACSchfa$ are also closed under finite limits
in~$\LCRS$.
\end{proof}
\begin{dfn} Define functors
\begin{align*}
F_\Man^\CSch:\Man&\longra\ACSchfp\subset \ACSch,\\
F_\Manb^\CSch:\Manb&\longra\ACSchgo\subset\ACSch,\\
F_\Manc^\CSch:\Manc&\longra\ACSchgo\subset\ACSch,
\end{align*}
by $F_{\Man^*}^\CSch=\Spec\ci F_{\Man^*}^\CRings$, in the notation
of Definitions \ref{ag3def} and \ref{ag4def2}.
By Example \ref{ag4ex2}, if $X$ is a manifold with corners then
$F_\Manc^\CSch(X)$ is naturally isomorphic to the local
$C^\iy$-ringed space $\uX$ in Example~\ref{ag4ex1}.
If $X,Y,\ldots$ are manifolds, or $f,g,\ldots$ are (weakly) smooth
maps, we may use $\uX,\uY,\ldots,\uf,\ug,\ldots$ to denote
$F_\Manc^\CSch(X,Y,\ldots,f,g,\ldots)$. So for instance we will
write $\ul{\R}^n$ and $\ul{[0,\iy)\!}\,$ for $F_\Man^\CSch(\R^n)$
and~$F_\Manb^\CSch\bigl([0,\iy)\bigr)$.
\label{ag4def4}
\end{dfn}
By Corollary \ref{ag3cor}, Theorems \ref{ag3thm} and \ref{ag4thm1}
and $\Spec:(\CRingsfa)^{\rm op}\ra\LCRS$ preserving fibre products,
we find as in Dubuc~\cite[Th.~16]{Dubu3}:
\begin{cor} $F_\Man^\CSch:\Man\hookra\ACSchfp\subset\ACSch$ is a full
and faithful functor, and $F_\Manb^\CSch:\Manb\ra\ACSchgo\subset
\ACSch,$ $F_\Manc^\CSch:\Manc\ra\ACSchgo\subset\ACSch$ are both
faithful functors, but are not full. Also these functors take
transverse fibre products in $\Man,\Manc$ to fibre products in
$\ACSchfp,\ACSchgo$.
\label{ag4cor2}
\end{cor}
In Definition \ref{ag2def5} we saw that a $C^\iy$-ring $\fC$ is
finitely presented if and only if it fits into a pushout square
\eq{ag2eq2} in $\CRings$. Applying $\Spec$, which preserves fibre
products, implies:
\begin{lem} A $C^\iy$-ringed space $\uX$ is a finitely presented
affine\/ $C^\iy$-scheme if and only if it may be written as a fibre
product in $\CRS:$
\begin{equation*}
\xymatrix@R=10pt@C=60pt{ \uX \ar[r] \ar[d] & \ul{\{0\}}
\ar[d]_{\ul{0}} \\ \ul{\R}^n \ar[r]^{\ul{\phi}} & \ul{\R}^k, }
\end{equation*}
where $\phi:\R^n\ra\R^k$ is a smooth map, and\/ $0:\{0\}\ra\R^k$ is
the zero map.
\label{ag4lem1}
\end{lem}
Our next two results show that $\ACSchfp,\ACSchgo$ and $\ACSchfa$
are closed under taking open subsets.
\begin{dfn} Let $(X,\O_X)$ be a $C^\iy$-ringed space, and
$U\subseteq X$ an open subset. A {\it characteristic function\/} for
$U$ is a morphism of $C^\iy$-ringed spaces
$\uf=(f,f^\sh):(X,\O_X)\ra\ul{\R}=(\R,\O_\R)$ such that~$U=\{x\in
X:f(x)\ne 0\}$.
Let $\fC$ be a $C^\iy$-ring, and $c\in\fC$. Define
$\la_c:C^\iy(\R)\ra\fC$ by $\la_c(f)=\Phi_f(c)$ for all smooth
$f:\R\ra\R$. Then $\la_c$ is a morphism of $C^\iy$-rings, so
$\Spec\la_c:\Spec\fC\ra\Spec C^\iy(\R)=(\R,\O_\R)$ is a morphism of
affine $C^\iy$-schemes. Hence if the $C^\iy$-ringed space $(X,\O_X)$
is $\Spec\fC$ then elements $c\in\fC$ generate morphisms of
$C^\iy$-ringed spaces $\uf_c=\Spec(\la_c):(X,\O_X)\ra\ul{\R}$. The
characteristic functions we consider will always be of this form
for~$c\in\fC$.
\label{ag4def5}
\end{dfn}
\begin{prop} Let\/ $(X,\O_X)$ be a fair affine $C^\iy$-scheme. Then
every open\/ $U\subseteq X$ admits a characteristic function.
\label{ag4prop4}
\end{prop}
\begin{proof} By definition $(X,\O_X)\cong\Spec\fC=\smash{(X_\fC,
\O_{X_\fC})}$ for some finitely generated $C^\iy$-ring $\fC$, which
fits into an exact sequence $0\ra I\hookra C^\iy(\R^n)\,
{\buildrel\phi\over\longra}\,\fC\ra 0$. Thus $X$ is homeomorphic to
$X_\fC$, which is homeomorphic to the closed subset $X_\fC^\phi$ in
$\R^n$ given in \eq{ag4eq2}. Let $U\subseteq X$ be open, and $U'$ be
the open subset of $X_\fC^\phi$ identified with $U$ by these
homeomorphisms. As $X_\fC^\phi$ has the subspace topology, there
exists an open $V'\subseteq\R^n$ with~$U'=V'\cap X_\fC^\phi$.
Every open subset in $\R^n$ has a characteristic function,
\cite[Lem.~I.1.4]{MoRe2}. Hence there exists $f'\in C^\iy(\R^n)$
with $V'=\bigl\{x\in\R^n:f'(x)\ne 0\bigr\}$. Definition
\ref{ag4def5} gives a morphism $\Spec\la_{\phi(f')}:
\Spec\fC\ra\ul{\R}$. Let $(f,f^\sh):(X,\O_X)\ra(\R,\O_\R)$ be the
morphism identified with $\Spec\la_{\phi(f')}$ by the isomorphism
$(X,\O_X)\cong\Spec\fC$. Then $f:X\ra\R$ is identified with
$f'\vert_{X_\fC^\phi}: X_\fC^\phi\ra\R$ by the homeomorphisms
$X\cong X_\fC\cong X_\fC^\phi$. As $U$ is identified with
$U'=\bigl\{x\in X_\fC^\phi: f'(x)\ne 0\bigr\}$, it follows that
$U=\{x\in X:f(x)\ne 0\}$, so $(f,f^\sh)$ is a characteristic
function for~$U$.
\end{proof}
\begin{prop} Let\/ $(X,\O_X)$ be a finitely presented, or good, or
fair, affine $C^\iy$-scheme, and\/ $U\subseteq X$ be an open subset.
Then\/ $(U,\O_X\vert_U)$ is also a finitely presented, or good, or
fair, affine $C^\iy$-scheme, respectively.
\label{ag4prop5}
\end{prop}
\begin{proof} As $(X,\O_X)$ is fair, there exists a characteristic
function $\uf:(X,\O_X)\ra\ul{\R}$ for $U$ by Proposition
\ref{ag4prop4}. Consider the fibre product
\e
(X,\O_X)\t_{\uf,\ul{\R},\ui}\ul{\R\sm\{0\}\!}\,,
\label{ag4eq4}
\e
in $\LCRS$, where $i:\R\sm\{0\}\ra\R$ is the inclusion, and
$\ui:\ul{\R\sm\{0\}\!}\,\ra \ul{\R}$ is the image morphism of affine
$C^\iy$-schemes under $F_\Man^\CSch$. Since
$\ul{\R\sm\{0\}\!}\,\cong(\R\sm\{0\},\O_\R \vert_{\R\sm\{0\}})$, it
follows on general grounds that \eq{ag4eq4} is isomorphic to
$(U,\O_X\vert_U)$. But $\R,\R\sm\{0\}$ are manifolds without
boundary, so $\ul{\R},\ul{\R\sm\{0\}\!}\,$ lie in
$\ACSchfp,\ACSchgo,\ACSchfa$, which are closed under fibre products
by Theorem \ref{ag4thm2}. Thus if $(X,\O_X)$ is a finitely
presented, or good, or fair, affine $C^\iy$-scheme then so is
\eq{ag4eq4}, and hence so is~$(U,\O_X\vert_U)$.
\end{proof}
Note that this is better than the situation in conventional
algebraic geometry, where for instance $\C^2$ is an affine
$\C$-scheme, but its open subset $\C^2\sm\{0\}$ is not an affine
$\C$-scheme. This is because characteristic functions need not exist
for open subsets of affine $\C$-schemes. The ideas of the last two
propositions are illustrated by the following expression for the
$C^\iy$-ring $C^\iy(U)$ for open $U\subset\R^n$, proved by Dubuc
\cite[Cor.~1.14]{Dubu2}, \cite[Cor.~15]{Dubu3} and Moerdijk and
Reyes~\cite[Lem.~I.1.4, Cor.~I.2.2]{MoRe2}.
\begin{ex} Let $U$ be an open subset in $\R^n$. Then $U$ has a
characteristic function $f$, that is, there exists $f\in
C^\iy(\R^n)$ such that $U=f^{-1}(\R\sm\{0\})$, and
\begin{equation*}
C^\iy(U)\cong C^\iy(\R^{n+1})/\bigl(x_{n+1}f(x_1,\ldots,x_n)-1\bigr).
\end{equation*}
\label{ag4ex3}
\end{ex}
For general affine $C^\iy$-schemes $(X,\O_X)=\Spec\fC$, open subsets
$(U,\O_X\vert_U)$ need not be affine $C^\iy$-schemes, but we can say
the following. A {\it principal open subset\/} is one of the form
$U_c=\{x\in X:x(c)\ne 0\}$ for some $c\in\fC$. They are closed under
finite intersections, since $U_{c_1}\cap\cdots\cap
U_{c_n}=U_{c_1\cdots c_n}$. Also $(U_c,\O_X\vert_{U_c})\cong
\Spec\fC[c^{-1}]$, so principal open subsets of affine
$C^\iy$-schemes are affine. Since principal open subsets generate
the topology on $X$, every open subset in $X$ is a union of
principal open subsets. Thus we deduce:
\begin{lem} Let\/ $(X,\O_X)$ be an affine $C^\iy$-scheme, and\/
$U\subseteq X$ an open subset. Then\/ $U$ can be covered by open
subsets $V\subseteq U$ such that\/ $(V,\O_X\vert_V)$ is an affine
$C^\iy$-scheme.
\label{ag4lem2}
\end{lem}
Our next result describes the sheaf of $C^\iy$-rings $\O_X$ in
$\Spec\fC$ for $\fC$ a finitely generated $C^\iy$-ring. It is a
version of \cite[Prop.~I.2.2(b)]{Hart} in conventional algebraic
geometry, and reduces to Moerdijk and Reyes
\cite[Prop.~I.1.6]{MoRe2} when~$\fC=C^\iy(\R^n)$.
\begin{prop} Let\/ $\fC$ be a finitely generated\/ $C^\iy$-ring,
write $(X,\O_X)=\Spec\fC,$ and let\/ $U\subseteq X$ be open. By
Proposition\/ {\rm\ref{ag4prop4}} we may choose a characteristic
function\/ $\uf:(X,\O_X)\ra\ul{\R}$ for $U$ of the form
$\uf=\Spec\la_{c}$ for some $c\in\fC$. Then there is a canonical
isomorphism $\O_X(U)\cong R_{\rm fg}^{\rm fa}(\fC[c^{-1}]),$ in the
notation of Definitions\/ {\rm\ref{ag2def7}} and\/
{\rm\ref{ag2def9}}. If\/ $\fC$ is finitely presented, or good,
then\/~$\O_X(U)\cong\fC[c^{-1}]$.
\label{ag4prop6}
\end{prop}
\begin{proof} We have morphisms of $C^\iy$-rings
$\la_c:C^\iy(\R)\ra\fC$ and $i^*:C^\iy(\R)\ra C^\iy(\R\sm\{0\})$,
and $C^\iy(\R),C^\iy(\R\sm\{0\})$ are finitely presented
$C^\iy$-rings by Proposition \ref{ag3prop1}(a). So as $\Spec$
preserves limits in $(\CRingsfg)^{\rm op}$ we have
\begin{equation*}
\Spec\bigl(\fC\amalg_{\la_c,C^\iy(\R),
i^*}C^\iy(\R\sm\{0\})\bigr)\cong \Spec\fC\t_{\uf,
\ul{\R},\ui}\ul{\R\sm\{0\}\!}\cong (U,\O_X\vert_U).
\end{equation*}
But $\fC\amalg_{C^\iy(\R)}C^\iy(\R\sm\{0\})\cong\fC[c^{-1}]$ for
formal reasons. Thus Proposition \ref{ag4prop3} gives $\O_X(U)\cong
\Ga\bigl((U,\O_X\vert_U)\bigr)\cong R_{\rm fg}^{\rm
fa}(\fC[c^{-1}])$. If $\fC$ is finitely presented, or good, then
$\fC[c^{-1}]$ is too by Proposition \ref{ag2prop5}, so $\fC[c^{-1}]$
is fair and $R_{\rm fg}^{\rm fa}\bigl(\fC[c^{-1}]\bigr)
=\fC[c^{-1}]$, and therefore~$\O_X(U)\cong\fC[c^{-1}]$.
\end{proof}
\subsection{Locally finite sums in fair $C^\iy$-rings}
\label{ag43}
We discuss infinite sums in fair $C^\iy$-rings, broadly following
Dubuc~\cite{Dubu4}.
\begin{dfn} Let $\fC$ be a $C^\iy$-ring, and write
$(X,\O_X)=\Spec\fC$. Consider a formal expression $\sum_{a\in
A}c_a$, where $A$ is a (usually infinite) indexing set and
$c_a\in\fC$. We say that $\sum_{a\in A}c_a$ is a {\it locally finite
sum\/} if $X$ can be covered by open sets $U\subseteq X$ such that
for all but finitely many $a\in A$ we have $\pi_x(c_a)=0$ in $\fC_x$
for all $x\in U$, or equivalently, $\rho_{XU}\ci\Phi_\fC(c_a)=0$
in~$\O_X(U)$.
If $\sum_{a\in A}c_a$ is a locally finite sum, we say that $c\in\fC$
is a {\it limit\/} of $\sum_{a\in A}c_a$, written $c=\sum_{a\in
A}c_a$, if $\pi_x(c)=\sum_{a\in A}\pi_x(c_a)$ for all $x\in X$,
where $\sum_{a\in A}\pi_x(c_a)$ makes sense as there are only
finitely many nonzero terms. For general $\fC$ limits need neither
exist, nor be unique. In $C^\iy(\R^n)$, every locally finite sum
$\sum_{i\in I}c_i$ has a unique limit, defined pointwise.
Suppose the topological space $X$ is locally compact. (This is
automatic if $\fC$ is finitely generated, since $X$ is homeomorphic
to a closed subset of $\R^n$.) Then we can express locally finite
sums in terms of a {\it topology\/} on $\fC$. For each $c\in\fC$ and
each compact subset $S\subseteq X$, define ${\cal U}_{c,S}$ to be
the set of $c'\in\fC$ such that $\pi_x(c')=\pi_x(c)$ in $\fC_x$ for
all $x\in S$. We think of ${\cal U}_{c,S}$ as an open neighbourhood
of $c$ in $\fC$. Let $\fC$ have the topology with basis ${\cal
U}_{c,S}$ for all $c,S$. Then $c=\sum_{a=1}^\iy c_a$ is equivalent
to $c=\lim_{N\ra\iy}(\sum_{a=1}^Nc_a)$ in this topology on~$\fC$.
Let $c\in\fC$. Define the {\it support\/ $\supp c$ of\/} $c$ to be
the set of $p\in X$ such that the projection $c_p=\pi_p(c)$ to the
$C^\iy$-local ring $\fC_p$ in Definition \ref{ag2def7} is nonzero.
Then $\supp c$ is closed in $X$. If $U\subseteq X$ is open, we say
that $c\in\fC$ is {\it supported on\/} $U$ if~$\supp c\subseteq U$.
Let $\{U_a:a\in A\}$ be an open cover of $X$. A {\it partition of
unity in\/ $\fC$ subordinate to\/} $\{U_a:a\in A\}$ is
$\{\eta_a:a\in A\}$ with $\eta_a\in\fC$ supported on $U_a$ for $a\in
A$, such that $\sum_{a\in A}\eta_a$ is a locally finite sum in $\fC$
with $\sum_{a\in A}\eta_a=1$ in $\fC$.
If we just say $\{\eta_a:a\in A\}$ {\it is a partition of unity
in\/} $\fC$, we mean that $\sum_{a\in A}\eta_a$ is a locally finite
sum in $\fC$ with~$\sum_{a\in A}\eta_a=1$.
\label{ag4def6}
\end{dfn}
Following Dubuc \cite{Dubu4}, it is now easy to prove:
\begin{prop}{\bf(a)} An ideal\/ $I$ in\/ $C^\iy(\R^n)$ is fair if
and only if it is closed under locally finite sums.
\smallskip
\noindent{\bf(b)} Let\/ $\fC$ be a fair $C^\iy$-ring. Then every
locally finite sum $\sum_{a\in A}c_a$ in $\fC$ has a unique limit.
\smallskip
\noindent{\bf(c)} Let\/ $\fC$ be a fair $C^\iy$-ring,
$(X,\O_X)=\Spec\fC,$ and\/ $\{U_a:a\in A\}$ be an open cover of\/
$X$. Then there exists a partition of unity\/ $\{\eta_a:a\in A\}$
in\/ $\fC$ subordinate to\/~$\{U_a:a\in A\}$.
\label{ag4prop7}
\end{prop}
\subsection{General $C^\iy$-schemes}
\label{ag44}
As in conventional algebraic geometry \cite[\S II.2]{Hart}, we
define a $C^\iy$-{\it scheme\/} to be a local $C^\iy$-ringed space
covered by affine $C^\iy$-schemes.
\begin{dfn} Let $\uX=(X,\O_X)$ be a local $C^\iy$-ringed space. We
call $\uX$ a $C^\iy$-{\it scheme\/} if $X$ can be covered by open
sets $U\subseteq X$ such that $(U,\O_X\vert_U)$ is an affine
$C^\iy$-scheme. We call a $C^\iy$-scheme $\uX$ {\it locally fair},
or {\it locally good}, or {\it locally finitely presented}, if $X$
can be covered by open $U\subseteq X$ with $(U,\O_X\vert_U)$ a fair,
or good, or finitely presented, affine $C^\iy$-scheme, respectively.
We call a $C^\iy$-scheme $\uX$ {\it separated\/} if the underlying
topological space $X$ is Hausdorff. Affine $C^\iy$-schemes are
always separated.
Write $\CSchlf,\CSchlg,\CSchlfp,\CSch$ for the full subcategories of
locally fair, and locally good, and locally finitely presented, and
all, $C^\iy$-schemes, respectively. Our categories of spaces so far
are related as follows:
\begin{equation*}
\xymatrix@C=15pt@R=17pt{\Man \ar[d]^{F_\Man^\CSch} \ar[r]_\subset & \Manb
\ar[d]_{F_\Manb^\CSch} \ar[r]_\subset & \Manc \ar[dl]^(0.2){F_\Manc^\CSch}
\\ \ACSchfp \ar[r]_\subset \ar[d]^\subset & \ACSchgo \ar[r]_\subset
\ar[d]^\subset & \ACSchfa \ar[r]_\subset \ar[d]^\subset & \ACSch
\ar[d]^\subset \ar[dr]^\subset & \CRS \\
\CSchlfp \ar[r]^\subset & \CSchlg \ar[r]^\subset & \CSchlf
\ar[r]^\subset & \CSch \ar[r]^\subset & \LCRS. \ar[u]^\subset }
\end{equation*}
\label{ag4def7}
\end{dfn}
Ordinary schemes are much more general than ordinary affine schemes,
and central examples such as $\CP^n$ are not affine schemes.
However, affine $C^\iy$-schemes are already general enough for many
purposes, and constructions involving affine $C^\iy$-schemes often
yield affine $C^\iy$-schemes. For example:
\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item All manifolds are affine $C^\iy$-schemes.
\item If a $C^\iy$-scheme $\uX$ is Hausdorff and can be covered
by finitely many fair affine $C^\iy$-schemes, one can show $\uX$
is a fair affine $C^\iy$-scheme.
\item Let $\uX$ be a locally fair $C^\iy$-scheme with $X$
Hausdorff and paracompact. Then one can prove $\uX$ is an affine
$C^\iy$-scheme.
\end{itemize}
From Proposition \ref{ag4prop5} and Lemma \ref{ag4lem2} we
immediately deduce:
\begin{prop} Let\/ $(X,\O_X)$ be a locally finitely presented,
locally good, locally fair, or general, $C^\iy$-scheme, and\/
$U\subseteq X$ be open. Then\/ $(U,\O_X\vert_U)$ is also a locally
finitely presented, or locally good, or locally fair, or general,
$C^\iy$-scheme, respectively.
\label{ag4prop8}
\end{prop}
Here is the analogue of Theorem~\ref{ag4thm2}.
\begin{thm} The full subcategories\/ $\CSchlfp,\CSchlg,
\CSchlf$ and\/ $\CSch$ are closed under all finite limits in\/
$\LCRS$. Hence, fibre products and all finite limits exist in each
of these subcategories.
\label{ag4thm3}
\end{thm}
\begin{proof} We first show $\CSchlfp,\ldots,\CSch$ are closed
under fibre products. Let $\uf:\uX\ra\uZ$, $\ug:\uY\ra\uZ$ be
morphisms in one of these categories and $\uW=\uX_{\uf,\uZ,\ug}\uY$
be the fibre product in $\LCRS$, with projections
$\upi_\uX:\uW\ra\uX$, $\upi_\uY:\uW\ra\uY$. Write $\uW=(W,\O_W)$,
$\uf=(f,f^\sh)$, and so on. Let $w\in W$, and set $x=\pi_X(w)\in X$,
$y=\pi_Y(w)\in Y$ and $z=f(x)=g(y)\in Z$. Choose $V\subseteq Z$ with
$z\in V$ and $(V,\O_Z\vert_V)$ in $\ACSchfp,\ldots,\ACSch$
respectively. Then $f^{-1}(V)$ is open in $X$ so
$\bigl(f^{-1}(V),\O_X\vert_{f^{-1}(V)}\bigr)$ lies in
$\CSchlfp,\ldots,\CSch$ by Proposition \ref{ag4prop8}. Thus we may
choose $T\subseteq f^{-1}(V)$ open with $x\in T$ and
$(T,\O_X\vert_T)$ in $\ACSchfp,\ldots,\ACSch$, and similarly we
choose $U\subseteq g^{-1}(V)$ open with $y\in U$ and
$(U,\O_Y\vert_U)$ in~$\ACSchfp,\ldots,\ACSch$.
Let $S=\pi_X^{-1}(T)\cap\pi_Y^{-1}(U)$. Then $S$ is an open
neighbourhood of $w$ in $W$, and $(S,\O_W\vert_S)\!\cong
\!(T,\O_X\vert_T)\t_{(V,\O_Z\vert_V)}(U,\O_Y\vert_U)$. But
$\ACSchfp,\ldots,\ACSch$ are closed under fibre products in $\LCRS$
by Theorem \ref{ag4thm2}, so $(S,\O_W\vert_S)$ lies in
$\ACSchfp,\ACSchgo,\ACSchfa$ or $\ACSch$ respectively. As $W$ can be
covered by $S$, $\uW$ lies in $\CSchlfp,\CSchlg,\CSchlf$ or $\CSch$.
Hence $\CSchlfp,\ldots,\CSch$ are closed under fibre products in
$\LCRS$. They are also closed under finite limits, as in the proof
of Theorem~\ref{ag4thm2}.
\end{proof}
We can generalize the material in \S\ref{ag43} on partitions of
unity for fair $C^\iy$-rings and fair affine $C^\iy$-schemes to
locally fair $C^\iy$-schemes.
\begin{dfn} Let $\uX=(X,\O_X)$ be a $C^\iy$-scheme. Consider a
formal sum $\sum_{a\in A}c_a$, where $A$ is an indexing set and
$c_a\in\O_X(X)$ for $a\in A$. We say $\sum_{a\in A}c_a$ is a {\it
locally finite sum on\/} $\uX$ if $X$ can be covered by open
$U\subseteq X$ such that for all but finitely many $a\in A$ we have
$\rho_{XU}(c_a)=0$ in~$\O_X(U)$.
By the sheaf axioms for $\O_X$, if $\sum_{a\in A}c_a$ is a locally
finite sum there exists a unique $c\in\O_X(X)$ such that for all
open $U\subseteq X$ such that $\rho_{XU}(c_a)=0$ in $\O_X(U)$ for
all but finitely many $a\in A$, we have $\rho_{XU}(c)=\sum_{a\in
A}\rho_{XU}(c_a)$ in $\O_X(U)$, where the sum makes sense as there
are only finitely many nonzero terms. We call $c$ the {\it limit\/}
of $\sum_{a\in A}c_a$, written~$\sum_{a\in A}c_a=c$.
Let $c\in\O_X(X)$. Define the {\it support\/} $\supp c$ of $c$ to be
the set of $x\in X$ such that the projection $c_x$ of $c$ to the
stalk $\O_{X,x}$ of $\O_X$ at $x$ is nonzero. Then $\supp c$ is
closed in $X$. If $U\subseteq X$ is open, we say that $c$ {\it is
supported in\/} $U$ if~$\supp c\subseteq U$.
Let $\{U_a:a\in A\}$ be an open cover of $X$. A {\it partition of
unity on\/ $\uX$ subordinate to\/} $\{U_a:a\in A\}$ is
$\{\eta_a:a\in A\}$ with $\eta_a\in\O_X(X)$ supported on $U_a$ for
$a\in A$, such that $\sum_{a\in A}\eta_a$ is a locally finite sum on
$\uX$ with~$\sum_{a\in A}\eta_a=1$.
\label{ag4def8}
\end{dfn}
\begin{prop} Suppose $\uX$ is a separated, paracompact, locally
fair $C^\iy$-scheme, and\/ $\{\uU_a:a\in A\}$ an open cover of\/
$\uX$. Then there exists a partition of unity $\{\eta_a:a\in A\}$ on
$\uX$ subordinate to\/~$\{\uU_a:a\in A\}$.
\label{ag4prop9}
\end{prop}
\begin{proof} Since $\uX$ is locally fair, each $\uU_a$ is locally
fair, so we can choose an open cover $\{\uU_{ab}:b\in B_a\}$ of
$\uU_a$ for each $a\in A$ such that $\uU_{ab}$ is a fair affine
$C^\iy$-scheme. Let $C=\bigl\{(a,b):a\in A$, $b\in B_a\bigr\}$. Then
$\bigl\{U_{ab}:(a,b)\in C\bigr\}$ is an open cover of $X$, which is
paracompact. Therefore we may choose a {\it locally finite
refinement\/} $\bigl\{V_{ab}:(a,b)\in C\bigr\}$ of
$\bigl\{U_{ab}:(a,b)\in C\bigr\}$. That is, $V_{ab}\subseteq U_{ab}$
is open for all $a,b$, and $\bigcup_{(a,b)\in C}V_{ab}=X$, and each
$x\in X$ has an open neighbourhood $W_x$ in $X$ with $W_x\cap
V_{ab}\ne\es$ for only finitely many~$(a,b)\in C$.
Fix $(a',b')\in C$. Then $X=V_{a'b'}\cup \bigcup_{(a',b')\ne(a,b)\in
C}V_{ab}$. Therefore $X\sm V_{a'b'}$ and $X\sm
\bigl(\bigcup_{(a',b')\ne(a,b)\in C}V_{ab}\bigr)$ are disjoint
closed subsets in $X$. As $X$ is paracompact and Hausdorff it is a
{\it normal\/} topological space, so these disjoint closed sets have
disjoint open neighbourhoods in $X$. Hence we can choose open
$V'_{a'b'}\subseteq X$ such that $X\sm \bigl(\bigcup_{(a',b')\ne
(a,b)\in C}V_{ab}\bigr)\subseteq V'_{a'b'}$ and
$\overline{V'}\!\!_{a'b'}\cap\bigl(X\sm V_{a'b'}\bigr)=\es$, where
$\overline{V'}\!\!_{a'b'}$ is the closure of $V'_{a'b'}$ in $X$.
Taking complements shows that $\overline{V'}\!\!_{a'b'}\subseteq
V_{a'b'}\subseteq U_{a'b'}$ and~$V'_{a'b'}\cup \bigcup_{(a',b')\ne
(a,b)\in C}V_{ab}=X$.
Thus, if we replace $V_{a'b'}$ by $V'_{a'b'}\subseteq V_{a'b'}$,
then $\bigl\{V_{ab}:(a,b)\in C\bigr\}$ is still a locally finite
refinement of $\bigl\{U_{ab}:(a,b)\in C\bigr\}$, which has the extra
property that the closure $\overline{V}\!_{a'b'}$ of $V_{a'b'}$ in
$X$ lies in $U_{a'b'}$. By choosing a well-ordering $\prec$ of $C$
and making $V_{ab}$ smaller for $(a,b)\in C$ in this way in the
order $\prec$, by transfinite induction we see that we can choose
the locally finite refinement $\bigl\{V_{ab}:(a,b)\in C\bigr\}$ such
that $\overline{V}\!_{ab}\subseteq U_{ab}$ for all~$(a,b)\in C$.
Let $(a,b)\in C$. Then $V_{ab}$ is open in the fair affine
$C^\iy$-scheme $\uU_{ab}$, so by Proposition \ref{ag4prop4} there
exists a characteristic function $f_{ab}\in\O_X(U_{ab})$ for
$V_{ab}$. That is, $f_{ab}(x)\ne 0$ for $x\in V_{ab}$ and
$f_{ab}(x)=0$ for $x\in U_{ab}\sm V_{ab}$. The construction also
implies that the support $\supp f_{ab}$ of $f_{ab}$ is
$\overline{V}_{ab}$. Since the closures of $V_{ab}$ in $U_{ab}$ and
$X$ agree, we may extend $f_{ab}$ by zero over $X\sm U_{ab}$ to give
a unique $g_{ab}\in\O_X(X)$ with $\rho_{XU_{ab}}(g_{ab})=f_{ab}$
and~$\supp g_{ab}=\supp f_{ab}=\overline{V}\!_{ab}$.
Consider the sum $\sum_{(a,b)\in C}g_{ab}^2$ in $\O_X(X)$. It is
locally finite as $\bigl\{V_{ab}:(a,b)\in C\bigr\}$ is locally
finite and $\supp g_{ab}^2\subseteq\overline{V}_{ab}$. Thus
$\sum_{(a,b)\in C}g_{ab}^2=c$ for some unique $c\in O_X(X)$. If
$x\in X$ then $x\in V_{ab}$ for some $(a,b)\in C$ as
$\bigcup_{(a,b)\in C}V_{ab}=X$, so $g_{ab}(x)=f_{ab}(x)\ne 0$, and
$g_{ab}(x)^2>0$. Therefore $c(x)=\sum_{(a,b)\in C}g_{ab}(x)^2>0$ for
all $x\in X$. So $c$ is invertible in $\O_X(X)$. Define $\eta_a\in
O_X(X)$ for each $a\in A$ by $\eta_a=c^{-1}\cdot\sum_{b\in
B_a}g_{ab}^2$. This is a locally finite sum, and so well defined. As
$\supp g_{ab}^2\subseteq\overline{V}_{ab}\subseteq U_{ab}\subseteq
U_a$ for all $b\in B_a$, we see that $\supp\eta_a\subseteq U_a$.
Also
\begin{equation*}
\ts\sum_{a\in A}\eta_a=\sum_{a\in A}c^{-1}\cdot\sum_{b\in
B_a}g_{ab}^2=c^{-1}\cdot c=1.
\end{equation*}
Thus $\{\eta_a:a\in A\}$ is a partition of unity on $\uX$
subordinate to~$\{\uU_a:a\in A\}$.
\end{proof}
\section{Modules over $C^\iy$-rings}
\label{ag5}
Next we discuss {\it modules\/} over $C^\iy$-rings. The author knows
of no previous work on these, so all this section may be new,
although much of it is a straightforward generalization of well
known facts about modules over commutative rings.
\subsection{Modules}
\label{ag51}
\begin{dfn} Let $\fC$ be a $C^\iy$-ring. A {\it module\/ $(M,\mu)$
over\/} $\fC$, or $\fC$-{\it module}, is a module over $\fC$
regarded as a commutative $\R$-algebra as in Definition
\ref{ag2def4}. That is, $M$ is a vector space over $\R$ equipped
with a bilinear map $\mu:\fC\t M\ra M$, satisfying $\mu(c_1\cdot
c_2,m)=\mu\bigl(c_1,\mu(c_2,m)\bigr)$ and $\mu(1,m)=m$ for all
$c_1,c_2\in\fC$ and $m\in M$. A {\it morphism\/}
$\al:(M,\mu)\ra(M',\mu')$ of $\fC$-modules $(M,\mu),(M',\mu')$ is a
linear map $\al:M\ra M'$ such that $\al\ci\mu=
\mu'\ci(\id_\fC\t\al):\fC\t M\ra M'$. Then $\fC$-modules form an
{\it abelian category}, which we write as $\fCmod$. Often we write
$M$ for the $\fC$-module, leaving $\mu$ implicit, and write $c\cdot
m$ rather than~$\mu(c,m)$.
Let $W$ be a real vector space. Then we define a $\fC$-module
$(\fC\ot_\R W,\mu_W)$ by $\mu_W(c_1,c_2\ot w)=(c_1\cdot c_2)\ot w$
for $c_1,c_2\in\fC$ and $w\in W$. A $\fC$-module $(M,\mu)$ is called
{\it free\/} if it is isomorphic to $(\fC\ot_\R W,\mu_W)$ in
$\fC$-mod for some $W$. Note as in Example \ref{ag5ex2} below that
if $W$ is infinite-dimensional then free $\fC$-modules $\fC\ot_\R W$
may not be well-behaved, and not a useful idea in some problems.
A $\fC$-module $(M,\mu)$ is called {\it finitely generated\/} if
there is an exact sequence $(\fC\ot_\R\R^n,\mu_{\R^n})\ra (M,\mu)\ra
0$ in $\fC$-mod for some $n\ge 0$. A $\fC$-module $(M,\mu)$ is
called {\it finitely presented\/} if there is an exact sequence
$(\fC\ot_\R\R^m,\mu_{\R^m})\ra(\fC\ot_\R\R^n,\mu_{\R^n})\ra
(M,\mu)\ra 0$ in $\fCmod$ for some $m,n\ge 0$. We write $\fCmodfp$
for the full subcategory of finitely presented $\fC$-modules
in~$\fCmod$.
If $E\ra F\ra G\ra 0$ is an exact sequence in $\fC$-mod with $E,F$
finitely presented (or more generally $E$ finitely generated and $F$
finitely presented) then $G$ is finitely presented. This is because
if $\fC\ot_\R\R^l\ra E\ra 0$ and $\fC\ot_\R\R^m\ra\fC\ot_\R\R^n\ra
F\ra 0$ are exact, we can make an exact sequence
$\fC\ot_\R\R^{l+m}\ra\fC\ot_\R\R^n\ra G\ra 0$. Similarly, if $E,G$
are finitely presented, then $F$ is finitely presented. Hence
$\fCmodfp$ is closed under cokernels and extensions in $\fCmod$. But
it may not be closed under kernels, since $\fC$ may not be
noetherian as a commutative $\R$-algebra.
Now let $\phi:\fC\ra\fD$ be a morphism of $C^\iy$-rings. If
$(M,\mu)$ is a $\fD$-module then
$\phi^*(M,\mu)=\bigl(M,\mu\ci(\phi\t\id_M)\bigr)$ is a $\fC$-module,
and this defines a functor $\phi^*:\fDmod\ra\fCmod$. However,
$\phi^*$ is not very well-behaved, for instance it need not take
finitely generated $\fD$-modules to finitely generated
$\fC$-modules, and we will not use it. If $(M,\mu)$ is a
$\fC$-module then $\phi_*(M,\mu)=\bigl(M\ot_\fC\fD,\mu_\fD\bigr)$ is
a $\fD$-module, where $\mu_\fD=\mu_\fC\t\id_\fD:\fD\t M\ot_\fC\fD
\cong\fC\ot_\fC\fD\t M\ot_\fC\fD\ra M\ot_\fC\fD$, and this induces a
functor $\phi_*:\fCmod\ra\fDmod$ which does take finitely generated
or presented $\fC$-modules to finitely generated or presented
$\fD$-modules.
\label{ag5def1}
\end{dfn}
Vector bundles $E$ over manifolds $X$ give examples of modules
over~$C^\iy(X)$.
\begin{ex} Let $X$ be a manifold, which may have boundary or
corners. Let $E\ra X$ be a vector bundle, and $C^\iy(E)$ the vector
space of smooth sections $e$ of $E$. Define $\mu_E:C^\iy(X)\t
C^\iy(E)\ra C^\iy(E)$ by $\mu_E(c,e)=c\cdot e$. Then
$\bigl(C^\iy(E),\mu_E\bigr)$ is a $C^\iy(X)$-module. If $E$ is a
trivial rank $k$ vector bundle, $E\cong X\t\R^k$, then
$\bigl(C^\iy(E),\mu_E\bigr)\cong\bigl(C^\iy(X)\ot_\R\R^k,
\mu_{\R^k}\bigr)$, so $\bigl(C^\iy(E),\mu_E\bigr)$ is a free
$C^\iy(X)$-module.
Let $E,F\ra X$ be vector bundles over $X$ and $\la:E\ra F$ a
morphism of vector bundles. Then $\la_*:C^\iy(E)\ra C^\iy(F)$
defined by $\la_*:e\mapsto \la\ci e$ is a morphism of
$C^\iy(X)$-modules.
Now let $X,Y$ be manifolds and $f:X\ra Y$ a (weakly) smooth map.
Then $f^*:C^\iy(Y)\ra C^\iy(X)$ is a morphism of $C^\iy$-rings. If
$E\ra Y$ is a vector bundle over $Y$, then $f^*(E)$ is a vector
bundle over $X$. Under the functor $(f^*)_*:C^\iy(Y)$-mod$\,\ra
C^\iy(X)$-mod of Definition \ref{ag5def1}, we see that
$(f^*)_*\bigl(C^\iy(E) \bigr)=C^\iy(E)\ot_{C^\iy(Y)}C^\iy(X)$ is
isomorphic as a $C^\iy(X)$-module to~$C^\iy\bigl(f^*(E)\bigr)$.
\label{ag5ex1}
\end{ex}
If $E\ra X$ is any vector bundle over a manifold then by choosing
sections $e_1,\ldots,e_n\in C^\iy(E)$ for $n\gg 0$ such that
$e_1\vert_x,\ldots,e_n\vert_x$ span $E\vert_x$ for all $x\in X$ we
obtain a surjective morphism of vector bundles $\psi:X\t\R^n\ra E$,
whose kernel is another vector bundle $F$. By choosing another
surjective morphism $\phi:X\t\R^m\ra F$ we obtain an exact sequence
of vector bundles $X\t\R^m\,{\buildrel\phi\over\longra}\,X\t\R^n\,
{\buildrel\psi\over \longra}\,E\ra 0$, which induces an exact
sequence of $C^\iy(X)$-modules $C^\iy(X)\ot_\R\R^m\,{\buildrel
\phi_*\over\longra}\,C^\iy(X)\ot_\R\R^n\, {\buildrel\psi_*\over
\longra}\,C^\iy(E)\ra 0$. Thus we deduce:
\begin{lem} Let\/ $X$ be a manifold, which may have boundary or
corners, and\/ $E\ra X$ be a vector bundle. Then the
$C^\iy(X)$-module $C^\iy(E)$ in Example\/ {\rm\ref{ag5ex1}} is
finitely presented.
\label{ag5lem1}
\end{lem}
\subsection{Complete modules over fair $C^\iy$-rings}
\label{ag52}
We now extend the ideas in \S\ref{ag43} on infinite sums in
$C^\iy$-rings to modules.
\begin{dfn} Let $\fC$ be a fair $C^\iy$-ring, and $M$ a module over
$\fC$. Write $(X,\O_X)=\Spec\fC$. Consider a formal expression
$\sum_{a\in A}m_a$, where $A$ is a (usually infinite) indexing set
and $m_a\in M$. We say that $\sum_{a\in A}a_a$ is a {\it locally
finite sum\/} if $X$ can be covered by open sets $U\subseteq X$ such
that for all but finitely many $a\in A$ we have
$(\id_M\ot\pi_x)(m_a)=0$ in $M\ot_\fC\fC_x$ for all $x\in U$, where
$\id_M\ot\pi_x:M\cong M\ot_\fC\fC\ra M\ot_\fC\fC_x$ is induced by
the projection~$\pi_x:\fC\ra\fC_x$.
If $\sum_{a\in A}m_a$ is a locally finite sum, we say that $m\in M$
is a {\it limit\/} of $\sum_{a\in A}m_a$, written $m=\sum_{a\in
A}m_a$, if $(\id_M\ot\pi_x)(m)=\sum_{a\in A}(\id_M\ot\pi_x)(m_a)$
for all $x\in X$, where the sum makes sense as there are only
finitely many nonzero terms.
We say two locally finite sums $\sum_{a\in A}m_a$, $\sum_{a\in
A'}m'_a$ are {\it equivalent\/} if for all $x\in X$ we have
$\sum_{a\in A}(\id_M\ot\pi_x)(m_a)=\sum_{a\in A'}(\id_M\ot\pi_x)
(m'_a)$ in $\fC_x$. Equivalent locally finite sums have the same
limits.
We call $M$ a {\it complete\/} $\fC$-module if every locally finite
sum in $M$ has a unique limit. Write $\fCmodco$ for the full
subcategory of complete modules in~$\fCmod$.
\label{ag5def2}
\end{dfn}
\begin{ex} Let $\fC$ be a fair $C^\iy$-ring. Consider $\fC$ as a
module over itself. Then Proposition \ref{ag4prop7}(b) implies that
$\fC$ is complete. More generally, $\fC\ot_\R\R^n$ is a complete
$\fC$-module for all $n\ge 0$. However, if $W$ is an
infinite-dimensional vector space then $\fC\ot_\R W$ is in general
{\it not\/} a complete $\fC$-module. The problem is with the notion
of tensor product: by definition, elements of $\fC\ot_\R W$ are {\it
finite\/} sums $\sum_{a=1}^nc_a\ot w_a$, whereas we want to consider
infinite, but locally finite, sums $\sum_{a\in A}c_a\ot w_a$. So, to
obtain a complete module we need to pass to some kind of {\it
completed tensor product\/} $\fC\hat\ot_\R W$, using the topology on
$\fC$ in Definition~\ref{ag4def6}.
When $\fC=C^\iy(\R^n)$ for $n>0$ and $W$ is an infinite-dimensional
vector space, consider the following three sets of maps $\R^n\ra W$:
\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item[(i)] $M_1=\bigl\{$smooth maps $w:\R^n\ra W$ with $w(\R^n)$
contained in a finite-dimensional vector subspace $W'$ of
$W\bigr\}$;
\item[(ii)] $M_2=\bigl\{$smooth $w:\R^n\ra W$ such that $\R^n$ is
covered by open $U\subseteq\R^n$ with $w(U)$ contained in a
finite-dimensional subspace $W'$ of $W\bigr\}$; and
\item[(iii)] $M_3=\bigl\{$all smooth maps $w:\R^n\ra W\bigr\}$.
\end{itemize}
Then $M_1,M_2,M_3$ are $C^\iy(\R^n)$-modules with $M_1\subset
M_2\subset M_3$, where $M_1$ is the tensor product
$C^\iy(\R^n)\ot_\R W$, and $M_2$ is the correct completed tensor
product $C^\iy(\R^n)\hat\ot_\R W$, a complete $C^\iy(\R^n)$-module.
For our purposes $M_3$ is too big. To see this, note when we pass to
germs at $x\in\R^n$ we have
\begin{equation*}
M_1\ot_{C^\iy(\R^n)}C^\iy_x(\R^n)\cong
M_2\ot_{C^\iy(\R^n)}C^\iy_x(\R^n)\cong C^\iy_x(\R^n)\ot_\R W,
\end{equation*}
but $M_3\ot_{C^\iy(\R^n)}C^\iy_x(\R^n)$ is much larger
than~$C^\iy_x(\R^n)\ot_\R W$.
\label{ag5ex2}
\end{ex}
As for $\smash{R_{\rm fg}^{\rm fa}}$ in Definition \ref{ag2def9},
one can show:
\begin{prop} Let\/ $\fC$ be a fair\/ $C^\iy$-ring. Then there is a
reflection functor\/ $R_{\rm all}^{\rm co}:\fCmod\!\ra\!\fCmodco,$
left adjoint to the inclusion\/~$\fCmodco\!\hookra\!\fCmod$.
\label{ag5prop1}
\end{prop}
This can be proved by defining $R_{\rm all}^{\rm co}(M)$ to be the
set of equivalence classes $[\sum_{a\in A}m_a]$ of locally finite
sums $\sum_{a\in A}m_a$, with $\fC$-action $\mu\bigl(c,[\sum_{a\in
A}m_a]\bigr)=\bigl[\sum_{a\in A}\mu(c,m_a)\bigr]$ for $c\in\fC$, and
checking $R_{\rm all}^{\rm co}(M)$ has the required properties.
Alternatively, we can define $R_{\rm all}^{\rm co}$ to be the
functor $\Ga\ci\MSpec$ in \S\ref{ag6} below, and verify it is a
reflection. The correct notion of {\it completed tensor product\/}
$\fC\hat\ot_\R W$ in Example \ref{ag5ex2}, up to isomorphism,
is~$\fC\hat\ot_\R W=R_{\rm all}^{\rm co}(\fC\ot_\R W)$.
To test whether a $\fC$-module $M$ is complete, it is enough to
consider only locally finite sums of the form $\sum_{b\in
B}\eta_b\cdot m_b$ where $\{\eta_b:b\in B\}$ is a partition of unity
in $\fC$ and $m_b\in M$. The proof requires $\fC$ to be fair.
\begin{lem} Let $\fC$ be a fair $C^\iy$-ring, and $M$ a
$\fC$-module. Then every locally finite sum $\sum_{a\in A}m_a$ is
equivalent to one of the form $\sum_{b\in B}\eta_b\cdot m'_b,$ where
$\{\eta_b:b\in B\}$ is a partition of unity in $\fC$ and\/ $m'_b\in
M$ for all\/ $b\in B$. Conversely, all such\/ $\sum_{b\in
B}\eta_b\cdot m'_b$ are locally finite sums.
\label{ag5lem2}
\end{lem}
\begin{proof} Let $\sum_{a\in A}m_a$ be a locally finite sum. Then
$X$ has an open cover of $U$ such that $m_a\vert_U\equiv 0$ for all
but finitely many $a\in A$. Since $\fC$ is fair $X$ is paracompact,
so we can choose a locally finite refinement $\{V_b:b\in B\}$ of
this open cover, and Proposition \ref{ag4prop7}(c) gives a partition
of unity $\{\eta_b:b\in B\}$ in $\fC$ subordinate to $\{V_b:b\in
B\}$. For each $b\in B$, define $m_b'=\sum_{a\in A_b}m_a$ where
$A_b\subseteq A$ is the finite set of $a\in A$ with
$m_a\vert_{V_b}\not\equiv 0$. It is then easy to see that
$\sum_{b\in B}\eta_b\cdot m'_b$ is a locally finite sum equivalent
to $\sum_{a\in A}m_a$. The last part is immediate as $\{\eta_b:b\in
B\}$ is locally finite.
\end{proof}
\begin{prop} Let\/ $\fC$ be a fair\/ $C^\iy$-ring. Then\/ $\fCmodco$
is closed under kernels, cokernels and extensions in\/ $\fCmod,$
that is, $\fCmodco$ is an abelian subcategory of\/~$\fCmod$.
\label{ag5prop2}
\end{prop}
\begin{proof} Let $0\ra M_1\,{\buildrel\al\over\longra}\,M_2
\,{\buildrel\be\over\longra}\,M_3\ra 0$ be an exact sequence in
$\fCmod$. First suppose $M_2,M_3\in\fCmodco$, and $\sum_{a\in A}m_a$
is a locally finite sum in $M_1$. Then $\sum_{a\in A}\al(m_a)$ is
locally finite in $M_2$ which is complete, so $m'=\sum_{a\in
A}\al(m_a)'$ for some unique $m'\in M_2$. As morphisms of modules
preserve limits we have
\begin{equation*}
\be(m')=\ts\sum_{a\in A}\be\ci\al(m_a)'=\sum_{a\in A}0=0,
\end{equation*}
so $\be(m')=0$ as limits in $M_3$ are unique as $M_3$ is complete.
Hence $m'=\al(m)$ for some unique $m\in M_1$ by exactness. This $m$
is the unique limit of $\sum_{a\in A}m_a$, so $M_1$ is complete, and
$\fCmodco$ is closed under kernels.
Now suppose $M_1,M_2\in\fCmodco$. Let $\sum_{a\in A}m_a$ be a
locally finite sum in $M_3$. By Lemma \ref{ag5lem2} we can choose an
equivalent sum $\sum_{b\in B}\eta_b\cdot m'_b$ with $\{\eta_b:b\in
B\}$ a partition of unity. By exactness $m'_b=\be(m_b)$ for some
$m_b\in M_2$, all $b\in B$. Then $\sum_{b\in B}\eta_b\cdot m_b$ is a
locally finite sum in $M_2$, so $\sum_{b\in B}\eta_b\cdot m_b=m$ for
some unique $m\in M_2$. As morphisms preserve limits we have
$\sum_{b\in B}\eta_b\cdot m'_b=\be(m)=\sum_{a\in A}m_a$, so limits
always exist in~$M_3$.
To show limits are unique in $M_3$, it is enough to consider the
zero sequence $\sum 0$. Suppose $m''\in M_3$ is a limit of $\sum 0$.
Then $m''=\be(m')$ for some $m'\in M_2$, and $X$ has an open cover
of $U$ such that $m''\vert_U\equiv 0$. Choose a locally finite
refinement $\{V_b:b\in B\}$ and a subordinate partition of unity
$\{\eta_b:b\in B\}$; we can also arrange that $\eta_b=\ze_b^2$ for
$\ze_b\in\fC$ supported on $V_b$. Then $\ze_b\cdot m''=0$ in $M_3$,
so $\be(\ze_b\cdot m')=0$, and thus $\ze_b\cdot m'=\al(m_b)$ for
$m_b\in M_1$. Hence $\sum_{b\in B}\ze_bm_b$ is a locally finite sum
in $M_1$, so $\sum_{b\in B}\ze_bm_b=m$ for some unique $m\in M_1$ as
$M_1$ is complete. Therefore
\begin{equation*}
\al(m)=\ts\sum_{b\in B}\ze_b\al(m_b)=\sum_{b\in B}\ze_b^2\cdot m'
=\sum_{b\in B}\eta_b\cdot m'=m',
\end{equation*}
and $\al(m)=m'$ by uniqueness of limits in $M_2$, so
$m''=\be\ci\al(m)=0$. Thus limits in $M_3$ are unique, $M_3$ is
complete, and $\fCmodco$ is closed under cokernels. Closedness under
extensions follows by a similar argument.
\end{proof}
As $R_{\rm all}^{\rm co}$ is a reflection, it is an exact functor
between abelian categories. Since every finitely presented
$\fC$-module is the cokernel of a morphism between
$\fC\ot_\R\R^m,\fC\ot_\R\R^n$, which are complete as in Example
\ref{ag5ex2}, we have:
\begin{cor} Let\/ $\fC$ be a fair $C^\iy$-ring. Then every finitely
presented $\fC$-module is complete, that is,
$\fCmodfp\subset\fCmodco$.
\label{ag5cor}
\end{cor}
\subsection{Cotangent modules of $C^\iy$-rings}
\label{ag53}
Given a $C^\iy$-ring $\fC$, we will define the {\it cotangent
module\/} $(\Om_\fC,\mu_\fC)$ of $\fC$. Although our definition of
$\fC$-module only used the commutative $\R$-algebra underlying the
$C^\iy$-ring $\fC$, our definition of the particular $\fC$-module
$(\Om_\fC,\mu_\fC)$ does use the $C^\iy$-ring structure in a
nontrivial way. It is a $C^\iy$-ring version of the {\it module of
relative differential forms\/} or {\it K\"ahler differentials\/} in
Hartshorne~\cite[p.~172]{Hart}.
\begin{dfn} Suppose $\fC$ is a $C^\iy$-ring, and $(M,\mu)$ a
$\fC$-module. A $C^\iy$-{\it derivation} is an $\R$-linear map
$\d:\fC\ra M$ such that whenever $f:\R^n\ra\R$ is a smooth map and
$c_1,\ldots,c_n\in\fC$, we have
\e
\d\Phi_f(c_1,\ldots,c_n)=\ts\sum_{i=1}^n\mu\bigl(\Phi_{\frac{\pd
f}{\pd x_i}}(c_1,\ldots,c_n),\d c_i\bigr).
\label{ag5eq1}
\e
Note that $\d$ is {\it not\/} a morphism of $\fC$-modules. We call
such a pair $(M,\mu),\d$ a {\it cotangent module\/} for $\fC$ if it
has the universal property that for any $\fC$-module $(M',\mu')$ and
$C^\iy$-derivation $\d':\fC\ra M'$, there exists a unique morphism
of $\fC$-modules $\phi:(M,\mu)\ra(M',\mu')$ with~$\d'=\phi\ci\d$.
There is a natural construction for a cotangent module: we take
$(M,\mu)$ to be the quotient of the free $\fC$-module with basis of
symbols $\d c$ for $c\in\fC$ by the $\fC$-submodule spanned by all
expressions of the form $\d\Phi_f(c_1,\ldots,c_n)-\sum_{i=1}^n
\mu\bigl(\Phi_{\frac{\pd f}{\pd x_i}}(c_1,\ldots,c_n),\d c_i\bigr)$
for $f:\R^n\ra\R$ smooth and $c_1,\ldots,c_n\in\fC$. Thus cotangent
modules exist, and are unique up to unique isomorphism. When we
speak of `the' cotangent module, we mean that constructed above. We
may write $(\Om_\fC,\mu_\fC)$, $\d_\fC:\fC\ra\Om_\fC$ for the (or a
choice of) cotangent module for~$\fC$.
Let $\fC,\fD$ be $C^\iy$-rings with cotangent modules
$(\Om_\fC,\mu_\fC),\d_\fC$, $(\Om_\fD,\mu_\fD),\d_\fD$, and
$\phi:\fC\ra\fD$ be a morphism of $C^\iy$-rings. Then the action
$\mu_\fC\ci(\phi\t\id_{\Om_\fD})$ makes $\Om_\fD$ into a
$\fC$-module, and $\d_\fD\ci\phi:\fC\ra\Om_\fD$ is a
$C^\iy$-derivation. Thus by the universal property of $\Om_\fC$,
there exists a unique morphism of $\fC$-modules
$\Om_\phi:\Om_\fC\ra\Om_\fD$ with $\d_\fD\ci\phi=\Om_\phi\ci\d_\fC$.
This then induces a morphism of $\fD$-modules $(\Om_\phi)_*:
\Om_\fC\ot_\fC\fD\ra\Om_\fD$ with $(\Om_\phi)_*\ci
(\d_\fC\ot\id_\fD)=\d_\fD$ as a composition $\fD=\fC\ot_\fC\fD\ra
\Om_\fC\ot_\fC\fD\ra\Om_\fD$. If $\phi:\fC\ra\fD$, $\psi:\fD\ra\fE$
are morphisms of $C^\iy$-rings then~$\Om_{\psi\ci\phi}=
\Om_\psi\ci\Om_\phi:\Om_\fC\ra\Om_\fE$.
\label{ag5def3}
\end{dfn}
\begin{ex} Let $X$ be a manifold. Then the cotangent bundle $T^*X$
is a vector bundle over $X$, so as in Example \ref{ag5ex1} it yields
a $C^\iy(X)$-module $C^\iy(T^*X)$. The exterior derivative
$\d:C^\iy(X)\ra C^\iy(T^*X)$, $\d:c\mapsto \d c$ is then a
$C^\iy$-derivation, since equation \eq{ag5eq1} follows from
\begin{equation*}
\d\bigl(f(c_1,\ldots,c_n)\bigr)=\ts\sum_{i=1}^n\frac{\pd f}{\pd
x_i}(c_1,\ldots,c_n)\,\d c_n
\end{equation*}
for $f:\R^n\ra\R$ smooth and $c_1,\ldots,c_n\in C^\iy(X)$, which
holds by the chain rule. It is easy to show that
$\bigl(C^\iy(T^*X),\mu_{T^*X}\bigr),\d$ have the universal property
in Definition \ref{ag5def3}, and so form a {\it cotangent module\/}
for~$C^\iy(X)$.
Now let $X,Y$ be manifolds, and $f:X\ra Y$ a (weakly) smooth map.
Then $f^*(TY),TX$ are vector bundles over $X$, and the derivative of
$f$ is a vector bundle morphism $\d f:TX\ra f^*(TY)$. The dual of
this morphism is $(\d f)^*:f^*(T^*Y)\ra T^*X$. This induces a
morphism of $C^\iy(X)$-modules $((\d f)^*)_*:C^\iy\bigl(
f^*(T^*Y)\bigr)\ra C^\iy(T^*X)$. This $((\d f)^*)_*$ is identified
with $(\Om_{f^*})_*$ under the natural isomorphism
$C^\iy\bigl(f^*(T^*Y)\bigr)\cong C^\iy(T^*Y)\ot_{C^\iy(Y)}C^\iy(X)$,
where we identify $C^\iy(Y),C^\iy(X),f^*$ with $\fC,\fD,\phi$ in
Definition~\ref{ag5def3}.
\label{ag5ex3}
\end{ex}
The importance of Definition \ref{ag5def3} is that it abstracts the
notion of cotangent bundle of a manifold in a way that makes sense
for any $C^\iy$-ring.
\begin{rem}{\bf(a)} There is a second way to define a cotangent-type module
for a $C^\iy$-ring $\fC$, namely the module ${\rm Kd}_\fC$ of {\it
K\"ahler differentials\/} of the underlying $\R$-algebra of $\fC$.
This is defined as for $\Om_\fC$, but requiring \eq{ag5eq1} to hold
only when $f:\R^n\ra\R$ is a polynomial. Since we impose many fewer
relations, ${\rm Kd}_\fC$ is generally much larger than $\Om_\fC$,
so that ${\rm Kd}_{C^\iy(\R^n)}$ is not a finitely generated
$C^\iy(\R^n)$-module for $n>0$, for instance.
\smallskip
\noindent{\bf(b)} Cotangent modules should be part of a more
complicated story about {\it cotangent complexes}. If $f:A\ra B$ is
a morphism of rings then the {\it cotangent complex\/} $L_f$,
constructed by Illusie \cite{Illu}, is an object in the derived
category of $B$-modules $D^b(B\text{-mod})$. As discussed by Spivak
\cite[\S 7]{Spiv}, presumably the same construction works for
$C^\iy$-rings, so that if $\phi:\fC\ra\fD$ is a morphism of
$C^\iy$-rings then one should obtain a $C^\iy$-{\it cotangent
complex\/} $L_\phi^{C^\iy}$. One can also form the cotangent complex
$L_\phi$ of $\phi$ regarded just as a ring morphism. Then $L_\phi$
and $\smash{L_\phi^{C^\iy}}$ both lie in $D^b(\fD\text{-mod})$, but
will in general be different, since $L_\phi$ ignores the
$C^\iy$-ring structures of $\fC,\fD$ but $\smash{L_\phi^{C^\iy}}$
does not.
If $\fC$ is a $C^\iy$-ring then there is a unique $C^\iy$-ring
morphism $\pi:\R\ra\fC$, since $\R$ is an initial object in
$\CRings$. Thus we can form cotangent complexes $L_\pi^{C^\iy}$ and
$L_\pi$ in $D^b(\fCmod)$. We expect that our cotangent module
$\Om_\fC$ is canonically isomorphic to $H^0(L_\pi^{C^\iy})$, whereas
the module of K\"ahler differentials ${\rm Kd}_\fC$ in (a) is
canonically isomorphic to~$H^0(L_\pi)$.
\label{ag5rem}
\end{rem}
\begin{thm} If\/ $\fC$ is a finitely generated\/ $C^\iy$-ring then
$\Om_\fC$ is a finitely generated\/ $\fC$-module. If\/ $\fC$ is a
fair\/ $C^\iy$-ring then\/ $\Om_\fC$ is complete. If\/ $\fC$ is a
finitely presented, or good, $C^\iy$-ring, then $\Om_\fC$ is
finitely presented.
\label{ag5thm1}
\end{thm}
\begin{proof} If $\fC$ is finitely generated we have an exact
sequence $0\ra I\hookra C^\iy(\R^n)\ab \,\smash{{\buildrel\phi\over
\longra}}\,\ab\fC\ra 0$. Write $x_1,\ldots,x_n$ for the generators
of $C^\iy(\R^n)$. Then any $c\in\fC$ may be written as $\phi(f)$ for
some $f\in C^\iy(\R^n)$, and \eq{ag5eq1} implies that
\begin{equation*}
\d c=\d\Phi_f\bigl(\phi(x_1),\ldots,\phi(x_n)\bigr)
=\ts\sum_{i=1}^n\mu\bigl(\Phi_{\frac{\pd f}{\pd
x_i}}(\phi(x_1),\ldots,\phi(x_n)),\d\ci\phi(x_i)\bigr).
\end{equation*}
Hence the generators $\d c$ of $\Om_\fC$ for $c\in\fC$ are
$\fC$-linear combinations of $\d\ci\phi(x_i)$, $i=1,\ldots,n$, so
$\Om_\fC$ is generated by the $\d\ci\phi(x_i)$, and is finitely
generated.
Now suppose $\fC$ is fair. From the first part we have an exact
sequence $0\ra M\hookra \fC\ot_\R\R^n\,\smash{{\buildrel\al\over
\longra}}\Om_\fC\ra 0$, where $\fC\ot_\R\R^n$ is the free
$\fC$-module with basis $e_1,\ldots,e_n$, and $\al(e_i)=
\d\ci\phi(x_i)$, $i=1,\ldots,n$, and $M$ is the $\fC$-submodule of
$\fC\ot_\R\R^n$ generated by elements
\e
\ts\phi\bigl(\frac{\pd f}{\pd x_1}\bigr)\cdot e_1+\cdots+
\phi\bigl(\frac{\pd f}{\pd x_n}\bigr)\cdot e_n\quad\text{for $f\in
I$.}
\label{ag5eq2}
\e
By Example \ref{ag5ex2} $\fC\ot_\R\R^n$ is complete. We will show
$M$ is complete. Then $\Om_\fC$ is complete, as $\fCmodco$ is closed
under cokernels by Proposition \ref{ag5prop2}. Since locally finite
sums have unique limits in $\fC\ot_\R\R^n\supset M$, it is enough to
show that $M$ is closed under locally finite sums.
Let $\sum_{a\in A}m_a$ be a locally finite sum in $M$. Then by Lemma
\ref{ag5lem2} and \eq{ag5eq2}, $\sum_{a\in A}m_a$ is equivalent to a
locally finite sum of the form
\begin{equation*}
\ts\sum_{b\in B}\phi(\eta_b)\cdot\bigl[\sum_{c\in
C_b}\phi(g_{bc})\cdot \bigl(\ts\phi\bigl(\frac{\pd f_{bc}}{\pd
x_1}\bigr)\cdot e_1+\cdots+ \phi\bigl(\frac{\pd f_{bc}}{\pd
x_n}\bigr)\cdot e_n\bigr)\bigr],
\end{equation*}
where $\{\phi(\eta_b):b\in B\}$ is a partition of unity in $\fC$
which lifts to a locally finite sum $\sum_{b\in B}\eta_b$ in
$C^\iy(\R^n)$, and $C_b$ is a finite indexing set for each $b\in B$,
and $f_{bc}\in I$, $g_{bc}\in C^\iy(\R^n)$ for all $b\in B$
and~$c\in C_b$.
Consider the sum $\sum_{b\in B}\sum_{c\in C_b}\eta_bg_{bc}f_{bc}$ in
$C^\iy(\R^n)$. This is a locally finite sum, as $\sum_{b\in
B}\eta_b$ is locally finite and each $C_b$ is finite, so it has a
unique limit $f$. As $f_{bc}\in I$ for all $b,c$ we have
$\eta_bg_{bc}f_{bc}\in I$, so $f\in I$ as $I$ is closed under
locally finite sums by Proposition \ref{ag4prop7}(a). But
\begin{align*}
\ts\sum_{b\in B}&\phi(\eta_b)\cdot\bigl[\ts\sum_{c\in
C_b}\phi(g_{bc})\cdot \bigl(\ts\phi\bigl(\frac{\pd f_{bc}}{\pd
x_1}\bigr)\cdot e_1+\cdots+ \phi\bigl(\frac{\pd f_{bc}}{\pd
x_n}\bigr)\cdot e_n\bigr)\bigr]\\
&=\ts\sum_{b\in B}\sum_{c\in C_b}\sum_{i=1}^n\phi\bigl(
\frac{\pd(\eta_bg_{bc}f_{bc})}{\pd x_i}\bigr)\cdot e_i
=\sum_{i=1}^n\phi\bigl(\frac{\pd f}{\pd x_i}\bigr)\cdot e_i\in M,
\end{align*}
by \eq{ag5eq2}, where in the first step we use
\begin{align*}
\phi(\eta_b)&\ts\phi(g_{bc})\phi\bigl(\frac{\pd f_{bc}}{\pd
x_1}\bigr)=\phi\bigl(\eta_bg_{bc}\frac{\pd f_{bc}}{\pd
x_1}\bigr)=\phi\bigl(\frac{\pd(\eta_bg_{bc}f_{bc})}{\pd x_i}\bigr)-
\phi\bigl(f_{bc}\cdot\frac{\pd(\eta_bg_{bc})}{\pd x_i}\bigr)\\
&\ts=\phi\bigl(\frac{\pd(\eta_bg_{bc}f_{bc})}{\pd x_i}\bigr)
-\phi(f_{bc})\phi\bigl(\frac{\pd(\eta_bg_{bc})}{\pd x_i}\bigr)
=\phi\bigl(\frac{\pd(\eta_bg_{bc}f_{bc})}{\pd x_i}\bigr),
\end{align*}
since $\phi$ is an algebra morphism and $f_{bc}\in I$ so that
$\phi(f_{bc})=0$. Thus $M$ is closed under finite sums, so $\Om_\fC$
is complete, proving the second part.
For the third part, it is enough to suppose $\fC$ is good. Then we
have an exact sequence $\smash{0\ra I\hookra
C^\iy(\R^n)\ab\,{\buildrel\phi\over \longra}\,\ab\fC\ra 0}$, where
$\phi$ is a morphism of $C^\iy$-rings and the ideal
$I=(f_1,\ldots,f_m,\fm_X^\iy)$ for some closed $X\subseteq\R^n$. We
will define an exact sequence of $\fC$-modules
\e
\xymatrix{ (\fC\ot_\R\R^m,\mu_{\R^m}) \ar[r]^\al &
(\fC\ot_\R\R^n,\mu_{\R^n}) \ar[r]^(0.6)\be & (\Om_\fC,\mu_\fC)
\ar[r] & 0. }
\label{ag5eq3}
\e
Write $(a_1,\ldots,a_m)$, $(b_1,\ldots,b_n)$ for bases of
$\R^m,\R^n$. As $\fC\ot_\R\R^m,\fC\ot_\R\R^n$ are free
$\fC$-modules, the $\fC$-module morphisms $\al,\be$ are specified
uniquely by giving $\al(a_i)$ for $i=1,\ldots,m$ and $\be(b_j)$ for
$j=1,\ldots,n$, which we define to be
\begin{equation*}
\al:a_i\mapsto \ts\sum_{j=1}^n\mu_{\R^n}\bigl(\Phi_{\frac{\pd
f_i}{\pd x_j}}\bigl(\phi(x_1),\ldots,\phi(x_n)\bigr),b_j\bigr)
\quad\text{and}\quad \be:b_j\mapsto \d_\fC\bigl(\phi(x_j)\bigr).
\end{equation*}
Then for $i=1,\ldots,m$ we have
\begin{align*}
\be\ci\al(a_i)&=\ts\sum_{j=1}^n\mu_\fC\bigl(\Phi_{\frac{\pd f_i}{\pd
x_j}}\bigl(\phi(x_1),\ldots,\phi(x_n)\bigr),
\d_\fC\bigl(\phi(x_j)\bigr)\bigr)\\
&=\d_\fC\bigl(\Phi_{f_i}\bigl(\phi(x_1),\ldots,\phi(x_n)\bigr)\bigr)\\
&=\d_\fC\ci\phi\bigl(\Phi_{f_i}(x_1,\ldots,x_n)\bigr)
=\d_\fC\ci\phi\bigl(f_i(x_1,\ldots,x_n))=\d_\fC(0)=0,
\end{align*}
using \eq{ag5eq1} in the second step as $\d_\fC$ is a
$C^\iy$-derivation, $\phi$ a morphism of $C^\iy$-rings in the third,
the definition of $C^\iy(\R^n)$ as a $C^\iy$-ring in the fourth, and
$f_i(x_1,\ldots,x_n)\in I=\Ker\phi$ in the fifth. Hence
$\be\ci\al=0$, and \eq{ag5eq3} is a complex.
Thus $\be$ induces $\be_*:(\fC\ot_\R\R^n)/\al(\fC\ot_\R\R^m)\ra
\Om_\fC$. We will show $\be_*$ is an isomorphism, so that
\eq{ag5eq3} is exact. Define $\d:\fC\ra(\fC\ot_\R\R^n)/
\al(\fC\ot_\R\R^m)$ by
\e
\ts\d\bigl(\phi(h)\bigr)=
\ts\sum_{j=1}^n\mu_{\R^n}\bigl(\Phi_{\frac{\pd h}{\pd
x_j}}\bigl(\phi(x_1),\ldots,\phi(x_n)\bigr),b_j\bigr)
+\al(\fC\ot_\R\R^m).
\label{ag5eq4}
\e
Here every $c\in\fC$ may be written as $\phi(h)$ for some smooth
$h:\R^n\ra\R$ as $\phi$ is surjective. To show \eq{ag5eq4} is
well-defined we must show the right hand side is independent of the
choice of $h$ with $\phi(h)=c$, that is, we must show that the
r.h.s.\ is zero if $h\in I$. It is enough to check this for $h$ in a
set of generators for $I$. If $h=f_i$ for $i=1,\ldots,m$ the r.h.s.\
of \eq{ag5eq4} is zero by definition of $\al$. If $h\in\fm_X^\iy$
then $\frac{\pd h}{\pd x_j}$ also lies in $\fm_X^\iy\subset I$ and
\begin{equation*}
\ts\Phi_{\frac{\pd h}{\pd x_j}}\bigl(\phi(x_1),\ldots,\phi(x_n)
\bigr)=\phi\ci\Phi_{\frac{\pd h}{\pd x_j}}(x_1,\ldots,x_n)
=\phi\bigl(\frac{\pd h}{\pd x_j}\bigr)=0,
\end{equation*}
so the r.h.s.\ of \eq{ag5eq4} is zero. Hence $\d$ in \eq{ag5eq4} is
well-defined.
It is easy to see that $\d$ is a $C^\iy$-{\it derivation}, and that
$\be_*\ci\d=\d_\fC$. So by the universal property of $\Om_\fC$,
there is a unique $\fC$-module morphism $\psi:\Om_\fC\ra(\fC\ot_\R
\R^n)/\al(\fC\ot_\R\R^m)$ with $\d=\psi\ci\d_\fC$. Thus
$\be_*\ci\psi\ci\d_\fC=\be_*\ci\d=\d_\fC=\id_{\Om_\fC}\ci\d_\fC$, so
as $\Im\d_\fC$ generates $\Om_\fC$ as an $\fC$-module we see that
$\be_*\ci\psi=\id_{\Om_\fC}$. Similarly $\psi\ci\be_*$ is the
identity, so $\psi,\be_*$ are inverse, and $\be_*$ is an
isomorphism. Therefore \eq{ag5eq3} is exact, and $\Om_\fC$ is
finitely presented.
\end{proof}
Cotangent modules behave well under localization.
\begin{prop} Let\/ $\fC$ be a $C^\iy$-ring and\/ $c\in\fC,$ with
localization $\Pi^c:\fC\ra\fC[c^{-1}]$ as in Definition\/
{\rm\ref{ag2def7}}. Then the morphism of\/ $\fC[c^{-1}]$-modules
$(\Om_{\Pi^c})_*:\Om_\fC\ot_\fC\fC[c^{-1}] \ra\Om_{\fC[c^{-1}]}$ is
an isomorphism.
\label{ag5prop3}
\end{prop}
\begin{proof} Let $\Om_\fC$ and $\Om_{\fC[c^{-1}]}$ be constructed
as in Definition \ref{ag5def3}. Since $\fC[c^{-1}]$ has an extra
generator $c^{-1}$ and an extra relation $c\cdot c^{-1}=1$, we see
that the $\fC[c^{-1}]$-module $\Om_{\fC[c^{-1}]}$ may be constructed
from $\Om_\fC\ot_\fC\fC[c^{-1}]$ by adding an extra generator
$\d(c^{-1})$ and an extra relation $\d(c\cdot c^{-1}-1)=0$. But
using \eq{ag5eq1} and $c\cdot c^{-1}=1$ in $\fC[c^{-1}]$, we can
show that this extra relation is equivalent to
$\d(c^{-1})=-(c^{-1})^2\d c$. Thus the extra relation exactly
cancels the effect of adding the extra generator, so
$(\Om_{\Pi^c})_*$ is an isomorphism.
\end{proof}
We can also understand how cotangent modules behave under the
reflection functor $\smash{R_{\rm fg}^{\rm fa}}:\CRingsfg
\ra\CRingsfa$ of Definition~\ref{ag2def9}.
\begin{prop} Let\/ $\fC$ be a finitely generated\/ $C^\iy$-ring,
and\/ $\bar\fC=\smash{R_{\rm fg}^{\rm fa}(\fC)}$ its fair
reflection, with surjective projection\/ $\pi:\fC\ra\bar\fC$. Then
there is a canonical isomorphism of\/ $\bar\fC$-modules\/
$\Om_{\bar\fC}\cong R_{\rm all}^{\rm co}(\Om_\fC\ot_\fC\bar\fC)$
identifying\/ $(\Om_\pi)_*:\Om_\fC\ot_\fC\bar\fC\ra\Om_{\bar\fC}$
with the natural surjective morphism\/~$\Om_\fC\ot_\fC\bar\fC\ra
R_{\rm all}^{\rm co}(\Om_\fC\ot_\fC\bar\fC)$.
\label{ag5prop4}
\end{prop}
\begin{proof} We have an exact sequence $0\ra I\ra
C^\iy(\R^n)\,{\buildrel\phi\over\longra}\,\fC\ra 0$. Let $\bar I$ be
the closure of $I$ under locally finite sums in $C^\iy(\R^n)$. Then
$\bar I$ is fair, as in Proposition \ref{ag4prop7}(a), and we obtain
an exact sequence $0\ra\bar I\ra C^\iy(\R^n)\,{\buildrel \bar\phi
\over\longra}\,\bar\fC\ra 0$. There are exact sequences $0\ra M\ra
\fC\ot_\R\R^n\, {\buildrel\al\over\longra}\,\Om_\fC\ra 0$ in
$\fCmod$ and $0\ra\bar M\ra\bar\fC\ot_\R\R^n\,{\buildrel\bar\al\over
\longra}\,\Om_{\bar\fC}\ra 0$ in $\bar\fC$-mod, where we write
$e_1,\ldots,e_n$ for the generators of $\fC\ot_\R\R^n$ and $\bar
e_1,\ldots,\bar e_n$ for the generators of $\bar\fC\ot_\R\R^n$, and
then $\al,\bar\al$ are defined by $\al(e_i)=\d\phi(x_i)$ and
$\bar\al(\bar e_i)=\d\bar\phi(x_i)$, and $M$ is the submodule of
$\fC\ot_\R\R^n$ generated by elements
$\sum_{i=1}^n\phi\bigl(\frac{\pd f}{\pd x_i}\bigr)e_i$ for $f\in I$,
and $\bar M$ is the submodule of $\bar\fC\ot_\R\R^n$ generated by
elements $\sum_{i=1}^n\bar\phi\bigl(\frac{\pd f}{\pd x_i}\bigr)\bar
e_i$ for~$f\in\bar I$.
Thus there is an exact sequence $0\ra\hat M\ra\bar\fC\ot_\R
\R^n\,{\buildrel\bar\al\over\longra}\,\Om_\fC\ot_\fC\bar\fC\ra 0$,
where $\hat M$ is the submodule of $\bar\fC\ot_\R\R^n$ generated by
elements $\smash{\sum_{i=1}^n\bar\phi\bigl(\frac{\pd f}{\pd
x_i}\bigr)\bar e_i}$ for $f\in I$. Given a locally finite sum
$f=\sum_{a\in A}f_a$ in $C^\iy(\R^n)$ with $f_a\in I$ and $f\in\bar
I$, the corresponding sum $\sum_{a\in
A}\bigl[\sum_{i=1}^n\bar\phi\bigl(\frac{\pd f_a}{\pd x_i}\bigr)\bar
e_i\bigr]$ in $\hat M$ is locally finite, with limit
$\smash{\sum_{i=1}^n\bar\phi\bigl(\frac{\pd f}{\pd x_i}\bigr)\bar
e_i}$ in $\bar M$. Hence $\bar M$ is the closure of $\hat M$ under
locally finite sums in $\bar\fC\ot_\R\R^n$. Note too that all
locally finite sums in $\bar\fC\ot_\R\R^n$ have unique limits, as
$\bar\fC$ is fair. The definition of $R_{\rm all}^{\rm co}$ now
implies that
\begin{equation*}
R_{\rm all}^{\rm co}\bigl(\Om_\fC\ot_\fC\bar\fC\bigr)
\cong R_{\rm all}^{\rm co}\bigl((\bar\fC\ot_\R\R^n)/\hat M\bigr)=
(\bar\fC\ot_\R\R^n)/\bar M\cong\Om_{\bar\fC},
\end{equation*}
as we want. The identification of $(\Om_\pi)_*$ with the natural
morphism is clear from the actions of $\al,\bar\al$ on
$e_1,\ldots,e_n$ and~$\bar e_1,\ldots,\bar e_n$.
\end{proof}
Here is a useful exactness property of cotangent modules.
\begin{thm} Suppose we are given a pushout diagram of finitely
generated\/ $C^\iy$-rings:
\e
\begin{gathered}
\xymatrix@C=30pt@R=10pt{ \fC \ar[r]_\be \ar[d]^\al & \fE \ar[d]_\de \\
\fD \ar[r]^\ga & \fF,}
\end{gathered}
\label{ag5eq5}
\e
so that\/ $\fF=\fD\amalg_\fC\fE$. Then the following sequence of\/
$\fF$-modules is exact:
\e
\xymatrix@C=15pt{ \Om_\fC\ot_{\mu_\fC,\fC,\ga\ci\al}\fF
\ar[rrr]^(0.5){\begin{subarray}{l} (\Om_\al)_*\op\\
-(\Om_\be)_*\end{subarray}} &&&
{\begin{subarray}{l}\ts \Om_\fD\ot_{\mu_\fD,\fD,\ga}\fF\,\, \op\\
\ts\Om_\fE\ot_{\mu_\fE,\fE,\de}\fF \end{subarray}}
\ar[rrr]^(0.65){(\Om_\ga)_*\op(\Om_\de)_*} &&& \Om_\fF \ar[r] & 0. }
\label{ag5eq6}
\e
Here\/ $(\Om_\al)_*:\Om_\fC\ot_{\mu_\fC,\fC,\ga\ci\al}\fF\ra
\Om_\fD\ot_{\mu_\fD,\fD,\ga}\fF$ is induced by\/ $\Om_\al:\Om_\fC\ra
\Om_\fD,$ and so on. Note the sign of\/ $-(\Om_\be)_*$
in\/~\eq{ag5eq6}.
\label{ag5thm2}
\end{thm}
\begin{proof} By $\Om_{\psi\ci\phi}=\Om_\psi\ci\Om_\phi$ in
Definition \ref{ag5def3} and commutativity of \eq{ag5eq5} we have
$\Om_\ga\ci\Om_\al=\Om_{\ga\ci\al}=\Om_{\de\ci\be}=\Om_\de\ci
\Om_\be:\Om_\fC\ra\Om_\fF$. Tensoring with $\fF$ then gives
$(\Om_\ga)_*\ci(\Om_\al)_*=(\Om_\de)_*\ci(\Om_\be)_*:
\Om_\fC\ot_\fC\fF\ra\Om_\fF$. As the composition of morphisms in
\eq{ag5eq6} is $(\Om_\ga)_*\ci(\Om_\al)_*-(\Om_\de)_*\ci
(\Om_\be)_*$, this implies \eq{ag5eq6} is a complex.
For simplicity, first suppose $\fC,\fD,\fE,\fF$ are good. Use the
notation of Example \ref{ag2ex9} and the proof of Proposition
\ref{ag2prop6}, with exact sequences \eq{ag2eq6} and \eq{ag2eq7},
where $I=(h_1,\ldots,h_i,\m_X^\iy)\subset C^\iy(\R^l)$,
$J=(d_1,\ldots,d_j,\fm_Y^\iy)\subset C^\iy(\R^m)$ and
$K=(e_1,\ldots,e_k,\fm_Z^\iy)\subset C^\iy(\R^n)$ for
$X\subseteq\R^l$, $Y\subseteq\R^m$, $Z\subseteq\R^n$ closed. Then
$L$ is given by \eq{ag2eq8}. Applying the proof of Theorem
\ref{ag5thm1} to \eq{ag2eq6}--\eq{ag2eq7} yields exact sequences of
$\fF$-modules
\ea
\xymatrix{ \fF\ot_\R\R^i \ar[r]^{\ep_1} & \fF\ot_\R\R^l
\ar[r]^{\ze_1} & \Om_\fC\ot_\fC\fF \ar[r] & 0, }&
\label{ag5eq7}\\
\xymatrix{ \fF\ot_\R\R^j \ar[r]^{\ep_2} & \fF\ot_\R\R^m
\ar[r]^{\ze_2} & \Om_\fD\ot_\fD\fF \ar[r] & 0, }&
\label{ag5eq8}\\
\xymatrix{ \fF\ot_\R\R^k \ar[r]^{\ep_3} & \fF\ot_\R\R^n
\ar[r]^{\ze_3} & \Om_\fE\ot_\fE\fF \ar[r] & 0, }&
\label{ag5eq9}\\
\xymatrix@C=6pt{ \fF\ot_\R\R^{j+k+l} \ar[rr]^(0.3){\ep_4} &&
\fF\ot_\R\R^{m+n} \!=\!\fF\ot_\R\R^m\!\op\!\fF\ot_\R\R^n
\ar[rr]^(0.8){\ze_4} && \Om_\fF \ar[r] & 0, }&
\label{ag5eq10}
\ea
where for \eq{ag5eq7}--\eq{ag5eq9} we have tensored \eq{ag5eq3} over
$\fC,\fD,\fE$ with~$\fF$.
Define $\fF$-module morphisms $\th_1:\fF\ot_\R\R^l\ra\fF\ot_\R\R^m$,
$\th_2:\fF\ot_\R\R^l\ra\fF\ot_\R\R^n$ by
$\th_1(a_1,\ldots,a_l)=(b_1,\ldots,b_m)$,
$\th_2(a_1,\ldots,a_l)=(c_1,\ldots,c_n)$ with
\begin{equation*}
b_q=\sum_{p=1}^l \Phi_{\frac{\pd f_p}{\pd
y_q}}(\xi(y_1),\ldots,\xi(y_m))\cdot a_p, \quad c_r=\sum_{p=1}^l
\Phi_{\frac{\pd g_p}{\pd y_r}}(\xi(z_1),\ldots,\xi(z_n))\cdot a_p,
\end{equation*}
for $a_p,b_q,c_r\in\fF$. Now consider the diagram
\e
\begin{gathered}
\xymatrix@C=15pt@R=15pt{
{\begin{subarray}{l}\ts \fF\ot_\R\R^j\,\,\op \\
\ts \fF\ot_\R\R^k\,\,\op \\
\ts \fF\ot_\R\R^l\end{subarray}}
\ar[rrr]_{\ep_4=\left(\begin{subarray}{l} \ep_2 \,\, 0 \,\,
\phantom{-}\th_1 \\ 0 \,\, \ep_3 \,\, -\th_2\end{subarray}\right)}
\ar[d]^(0.6){\left(0 \,\, 0 \,\, \ze_1\right)} &&&
{\begin{subarray}{l}\ts \fF\ot_\R\R^m\,\,\op \\
\ts \fF\ot_\R\R^n\end{subarray}} \ar[rrr]_(0.55){\ze_4}
\ar[d]^{\left(\begin{subarray}{l} \ze_2 \,\, 0 \\ 0 \,\,
\ze_3\end{subarray}\right)} &&& \Om_\fF \ar[r]
\ar@{=}[d]^{{}\,\,\id_{\Om_\fF}} & 0 \\
\Om_\fC\ot_\fC\fF
\ar[rrr]^(0.5){\left(\begin{subarray}{l} (\Om_\al)_*\\
-(\Om_\be)_*\end{subarray}\right)} &&&
{\begin{subarray}{l}\ts \Om_\fD\ot_\fD\fF\, \op\\
\ts\Om_\fE\ot_\fE\fF \end{subarray}}
\ar[rrr]^(0.62){\left((\Om_\ga)_*\,\, (\Om_\de)_*\right)} &&&
\Om_\fF \ar[r] & 0, }
\end{gathered}
\label{ag5eq11}
\e
using matrix notation. The top line is the exact sequence
\eq{ag5eq10}, where the sign in $-\th_2$ comes from the sign of
$g_p$ in the generators $f_p(y_1,\ldots,y_m)-g_p(z_1,\ldots,z_n)$ of
$L$ in \eq{ag2eq8}. The bottom line is the complex~\eq{ag5eq6}.
The left hand square commutes as $\ze_2\ci\ep_2=\ze_3\ci\ep_3=0$ by
exactness of \eq{ag5eq8}--\eq{ag5eq9} and
$\ze_2\ci\th_1=(\Om_\al)_*\ci\ze_1$ follows from
$\al\ci\phi(x_p)=\psi(f_p)$, and $\ze_3\ci\th_2=(\Om_\be)_*\ci\ze_1$
follows from $\be\ci\phi(x_p)=\chi(g_p)$. The right hand square
commutes as $\ze_4$ and $(\Om_\ga)_*\ci\ze_2$ act on $\fF\ot_\R\R^m$
by $(a_1,\ldots,a_m)\mapsto\sum_{q=1}^ma_q\d_\fF\ci\xi(y_q)$, and
$\ze_4$ and $(\Om_\de)_*\ci\ze_3$ act on $\fF\ot_\R\R^n$ by
$(b_1,\ldots,b_n)\mapsto\sum_{r=1}^nb_r\d_\fF\ci\xi(z_r)$. Hence
\eq{ag5eq11} is commutative. The columns are surjective since
$\ze_1,\ze_2,\ze_3$ are surjective as \eq{ag5eq7}--\eq{ag5eq9} are
exact and identities are surjective. So, since \eq{ag5eq11} is
commutative with surjective columns, exact top row, and bottom row a
complex, it follows that the bottom row is exact by standard facts
about abelian categories. This proves the theorem for
$\fC,\fD,\fE,\fF$ good. For the finitely generated case we can use
the same proof, but allowing $i,j,k$ infinite.
\end{proof}
Here is an example of Theorem \ref{ag5thm2} for manifolds.
\begin{ex} Let $W,X,Y,Z,e,f,g,h$ be as in Theorem \ref{ag3thm}, so
that \eq{ag3eq1} is a Cartesian square of manifolds and \eq{ag3eq2}
a pushout square of $C^\iy$-rings. We have the following sequence of
morphisms of vector bundles on~$W$:
\e
\xymatrix@C=10.5pt{ 0 \ar[r] & (g\ci e)^*(T^*Z)
\ar[rrrr]^(0.45){e^*(\d g^*)\op -f^*(\d h^*)} &&&&
e^*(T^*X)\!\op\!f^*(T^*Y) \ar[rr]^(0.67){\d e^*\op \d f^*} && T^*W
\ar[r] & 0.}\!\!
\label{ag5eq12}
\e
Here $\d g:TX\ra g^*(TZ)$ is a morphism of vector bundles over $X$,
and $\d g^*:g^*(T^*Z)\ra T^*X$ is the dual morphism, and $e^*(\d
g^*):(g\ci e)^*(T^*Z)\ra e^*(T^*X)$ is the pullback of this dual
morphism to $W$. In \S\ref{ag6} and \S\ref{ag9} we will distinguish
between $(g\ci e)^*(T^*Z)$ and $e^*(g^*(T^*Z))$, but here we
identify them for simplicity.
Since $g\ci e=h\ci f$, we have $\d e^*\ci e^*(\d g^*)=\d f^*\ci
f^*(\d h^*)$, and so \eq{ag5eq12} is a complex. As $g,h$ are
transverse and \eq{ag3eq1} is Cartesian, \eq{ag5eq12} is exact. So
passing to smooth sections in \eq{ag5eq12} we get an exact sequence
of $C^\iy(W)$-modules:
\begin{equation*}
\xymatrix@C=5.9pt{ 0\! \ar[r] & \!C^\iy\bigl((g\ci e)^*(T^*Z)\bigr)\!
\ar[rrrr]^(0.46){\begin{subarray}{l}(e^*(\d g^*)\op \\ -f^*(\d h^*))_*
\end{subarray}} &&&& \!C^\iy\bigl(e^*(T^*X)\!\op\!f^*(T^*Y)\bigr)\!
\ar[rr]^(0.68){\begin{subarray}{l}(\d e^*\op\\ \d f^*)_*
\end{subarray}} && \!C^\iy(T^*W)\! \ar[r] & \!0.}
\end{equation*}
The final four terms are the exact sequence \eq{ag5eq6} for the
pushout diagram~\eq{ag3eq2}.
\label{ag5ex4}
\end{ex}
\section{Sheaves of modules on $C^\iy$-schemes}
\label{ag6}
We now develop analogues for $C^\iy$-schemes of sheaves of
$\O_X$-modules, quasicoherent sheaves, and coherent sheaves on a
scheme $X$, following Hartshorne \cite[\S II.5]{Hart} or
Grothendieck \cite[\S 0.3--\S 0.5]{Grot} in conventional algebraic
geometry, and we define cotangent sheaves of $C^\iy$-schemes, based
on sheaves of relative differentials in Hartshorne \cite[\S
II.8]{Hart}. Some issues arise as our $C^\iy$-rings are generally
not noetherian as $\R$-algebras, but in algebraic geometry one
usually only considers coherent sheaves on noetherian schemes. The
author knows of no previous work on all this in the $C^\iy$-scheme
context, so this section may be new.
\subsection{Sheaves of $\O_X$-modules on a $C^\iy$-ringed space
$(X,\O_X)$}
\label{ag61}
We define sheaves of $\O_X$-modules on a $C^\iy$-ringed space,
following~\cite[\S II.5]{Hart}.
\begin{dfn} Let $(X,\O_X)$ be a $C^\iy$-ringed space. A {\it sheaf
of\/ $\O_X$-modules}, or simply an $\O_X$-{\it module}, $\cal E$ on
$X$ assigns a module ${\cal E}(U)=(M_U,\mu_U)$ over the $C^\iy$-ring
$\O_X(U)$ for each open set $U\subseteq X$, and a linear map ${\cal
E}_{UV}:M_U\ra M_V$ for each inclusion of open sets $V\subseteq
U\subseteq X$, such that the following commutes
\e
\begin{gathered}
\xymatrix@R=10pt@C=60pt{ \O_X(U)\t M_U \ar[d]^{\rho_{UV}\t{\cal
E}_{UV}} \ar[r]_{\mu_U} & M_U
\ar[d]_{{\cal E}_{UV}} \\
\O_X(V)\t M_V \ar[r]^{\mu_V} & M_V,}
\end{gathered}
\label{ag6eq1}
\e
and all this data ${\cal E}(U),{\cal E}_{UV}$ satisfies the usual
sheaf axioms~\cite[\S II.1]{Hart}.
A {\it morphism of sheaves of\/ $\O_X$-modules\/} $\phi:{\cal
E}\ra{\cal F}$ assigns a morphism of $\O_X(U)$-modules
$\phi(U):{\cal E}(U)\ra{\cal F}(U)$ for each open set $U\subseteq
X$, such that $\phi(V)\ci{\cal E}_{UV}={\cal F}_{UV}\ci\phi(U)$ for
each inclusion of open sets $V\subseteq U\subseteq X$. Then
$\O_X$-modules form an {\it abelian category}, which we write
as~$\OXmod$.
\label{ag6def1}
\end{dfn}
\begin{rem} Recall that a $C^\iy$-ring $\fC$ has an underlying
commutative $\R$-algebra, and a module over $\fC$ is a module over
this $\R$-algebra, by Definitions \ref{ag2def4} and \ref{ag5def1}.
Thus, by truncating the $C^\iy$-rings $\O_X(U)$ to commutative
$\R$-algebras, regarded as rings, a $C^\iy$-ringed space $(X,\O_X)$
has an underlying ringed space in the usual sense of algebraic
geometry \cite[p.~72]{Hart}, \cite[\S 0.4]{Grot}. Our definition of
$\O_X$-modules are simply $\O_X$-modules on this underlying ringed
space \cite[\S II.5]{Hart}, \cite[\S 0.4.1]{Grot}. Thus we can apply
results from algebraic geometry without change, for instance that
$\OXmod$ is an abelian category, as in~\cite[p.~202]{Hart}.
\label{ag6rem1}
\end{rem}
\begin{dfn} Let $\uf=(f,f^\sh):(X,\O_X)\ra(Y,\O_Y)$ be a morphism
of $C^\iy$-ringed spaces, and $\cal E$ be a sheaf of $\O_Y$-modules
on $(Y,\O_Y)$. Following Grothendieck \cite[\S 4.3.1]{Grot} or
Hartshorne \cite[p.~65, p.~110]{Hart}, define the {\it pullback\/}
$\uf^*({\cal E})$ to be the sheaf of $\O_X$-modules on $(X,\O_X)$
associated to the presheaf $U\mapsto\lim_{V\supseteq f(U)}{\cal
E}(V)\ot_{\O_Y(V)}\O_X(U)$, where $U$ is an open set in $X$, and the
limit is over open sets $V$ in $Y$ containing $f(U)$, and ${\cal
E}(V)\ot_{\O_Y(V)}\O_X(U)$ is computed using the
morphism~$\rho_{f^{-1}(V)U}\ci f^\sh(V):\O_Y(V)\ra \O_X(U)$.
If $\phi:{\cal E}\ra{\cal F}$ is a morphism of sheaves of
$\O_Y$-modules we have an induced morphism of $\O_X$-modules
$\uf^*(\phi):\uf^*({\cal E})\ra\uf^*({\cal F})$. Then
$\uf^*:\O_Y$-mod$\,\ra\O_X$-mod is a {\it functor\/} from
$\O_Y$-modules on $Y$ to $\O_X$-modules on $X$.
\label{ag6def2}
\end{dfn}
\begin{rem} Here is a slightly subtle point. Pullbacks $\uf^*({\cal
E})$ are a kind of fibre product, and may be characterized by a
universal property. So they should be regarded as being {\it unique
up to canonical isomorphism}, rather than unique. One can give an
explicit construction for pullbacks, or use the Axiom of Choice to
choose $\uf^*({\cal E})$ for all $\uf,{\cal E}$, and so speak of
`the' pullback $\uf^*({\cal E})$. However, it may not be possible to
make these choices strictly functorial in $\uf$.
That is, if $\uf:\uX\ra\uY$, $\ug:\uY\ra\uZ$ are morphisms and
$\cE\in\OZmod$ then $(\ug\ci\uf)^*(\cE)$, $\uf^*(\ug^*(\cE))$ are
canonically isomorphic in $\OXmod$, but may not be equal. We will
write $I_{\uf,\ug}(\cE):(\ug\ci\uf)^*(\cE)\ra\uf^*(\ug^*(\cE))$ for
these canonical isomorphisms. Then $I_{\uf,\ug}:(\ug\ci\uf)^*\Ra
\uf^*\ci\ug^*$ is a {\it natural isomorphism\/} of functors. It is
common to ignore this point and identify $(\ug\ci\uf)^*$ with
$\uf^*\ci\ug^*$, but this would cause problems in \cite{Joyc3}.
Vistoli \cite{Vist} makes careful use of natural isomorphisms $(g\ci
f)^*\Ra f^*\ci g^*$ in his treatment of descent theory.
When $\uf$ is the identity $\uid_\uX:\uX\ra\uX$ and $\cE\in\OXmod$
we do not require $\uid^*_\uX(\cE)=\cE$, but as $\cE$ is a possible
pullback for $\uid^*_\uX(\cE)$ there is a canonical isomorphism
$\de_\uX(\cE):\uid^*_\uX(\cE)\ra\cE$, and then
$\de_\uX:\uid^*_\uX\Ra\id_\OXmod$ is a natural isomorphism of
functors.
\label{ag6rem2}
\end{rem}
By Grothendieck \cite[\S 0.4.3.1]{Grot} we have:
\begin{prop} Let\/ $\uX,\uY$ be $C^\iy$-ringed spaces and\/
$\uf:\uX\ra\uY$ a morphism. Then pullback\/ $\uf^*:\OYmod\ra\OXmod$
is a \begin{bfseries}right exact functor\end{bfseries}. That is,
if\/ ${\cal E}\smash{\,{\buildrel\phi\over\longra}\,{\cal F}
\,{\buildrel\psi\over\longra}\,}{\cal G}\ra 0$ is exact in $\OYmod$
then $\uf^*({\cal E})\,{\buildrel\uf^*(\phi)\over
\longra}\,\uf^*({\cal F}) \,{\buildrel\uf^*(\psi)\over\longra}
\,\uf^*({\cal G})\ra 0$ is exact in\/~$\OXmod$.
\label{ag6prop1}
\end{prop}
In general $\uf^*$ is not exact, or left exact, unless
$\uf:\uX\ra\uY$ is {\it flat}.
\subsection{Sheaves on affine $C^\iy$-schemes, and $\MSpec$}
\label{ag62}
In \S\ref{ag42} we defined $\Spec:\CRings^{\rm op}\ra\LCRS$. In a
similar way, if $\fC$ is a $C^\iy$-ring and $(X,\O_X)=\Spec\fC$ we
can define~$\MSpec:\fCmod\ra\OXmod$.
\begin{dfn} Let $(X,\O_X)=\Spec\fC$ for some $C^\iy$-ring $\fC$,
let $(M,\mu)$ be a $\fC$-module, and $U\subseteq X$ be open. Then
$M$ is a representation of $\fC$, regarded as a commutative
$\R$-algebra. We have morphisms of $C^\iy$-rings
$\Phi_\fC:\fC\ra\Ga(\Spec\fC)=\O_X(X)$ and $\rho_{XU}:\O_X(X)\ra
\O_X(U)$, so that $\rho_{XU}\ci\Phi_\fC:\fC\ra\O_X(U)$ is a morphism
of commutative $\R$-algebras. Thus we may form the tensor product
$M\ot_\fC\O_X(U)$ over $\fC$, which is an $\O_X(U)$-module. If
$V\subseteq U\subseteq X$ are open then the algebra morphism
$\rho_{UV}:\O_X(U)\ra\O_X(V)$ induces a morphism
$\id_M\ot\rho_{UV}:M\ot_\fC\O_X(U)\ra M\ot_\fC\O_X(V)$. The
assignment $U\mapsto M\ot_\fC\O_X(U)$ and $(U,V)\mapsto\id_M\ot
\rho_{UV}$ defines a {\it presheaf of\/ $\O_X$-modules\/}
on~$(X,\O_X)$.
Define $\MSpec(M,\mu)$ to be the sheafification of this presheaf. If
$\al:(M,\mu)\ra(M',\mu')$ is a morphism of $\fC$-modules, it induces
a morphism of the associated presheaves, and we define $\MSpec\al:
\MSpec(M,\mu)\ra\MSpec(M',\mu')$ to be the induced morphism of
sheaves. Then $\MSpec:\fCmod\ra\O_X$-mod is a {\it functor}, the
analogue for modules of the functor $\Spec$ in \S\ref{ag42}. Since
sheafification is an exact functor, $\MSpec$ is also an {\it exact
functor}.
When $M=\fC$ the presheaf $U\mapsto\fC\ot_\fC\O_X(U)\cong\O_X(U)$ is
already a sheaf, and $\MSpec\fC\cong\O_X$, regarded as a sheaf of
$\O_X$-modules.
Now suppose $\fC$ is a {\it fair\/} $C^\iy$-ring. Then
$\Phi_\fC:\fC\ra\O_X(X)$ is an isomorphism by Proposition
\ref{ag4prop3}. Define the {\it global sections functor\/}
$\Ga:\OXmod\ra\fCmod$ on objects by $\Ga:{\cal E}\mapsto{\cal
E}(X)$, where the $\O_X(X)$-module ${\cal E}(X)$ is regarded as a
$\fC$-module using $\Phi_\fC^{-1}$, and on morphisms $\al:{\cal
E}\ra{\cal F}$ in $\OXmod$ by $\Ga:\al\mapsto\al(X)$. Then $\Ga$ is
a right adjoint to $\MSpec$, that is, as in \eq{ag4eq1} for all
$M\in\fCmod$ and ${\cal E}\in\OXmod$ there are functorial
isomorphisms
\e
\Hom_\fCmod(M,\Ga({\cal E}))\cong\Hom_\OXmod(\MSpec M,{\cal E}).
\label{ag6eq2}
\e
Taking ${\cal E}=\MSpec M$, we obtain a natural morphism of
$\fC$-modules $\Phi_M:M\ra\Ga(\MSpec M)$ corresponding to
$\id_{\MSpec M}$ in~\eq{ag6eq2}.
\label{ag6def3}
\end{dfn}
Proposition \ref{ag4prop3} showed that $\Ga\ci\Spec:\CRingsfg\ra
\CRingsfa$ and $R_{\rm fg}^{\rm fa}: \CRingsfg\ra\CRingsfa$ are
naturally isomorphic functors. In the same way, for $\fC$ a fair
$C^\iy$-ring the functors $\Ga\ci\MSpec:\fCmod\ra\fCmodco$ and
$R_{\rm all}^{\rm co}:\fCmod\ra\fCmodco$ are naturally isomorphic.
Furthermore, $\MSpec\ci\Ga:\OXmod\ra\OXmod$ is naturally isomorphic
to the identity. In contrast, in conventional algebraic geometry,
both $\Ga\ci\MSpec$ and $\MSpec\ci\Ga$ are naturally isomorphic to
the identity, as in Hartshorne~\cite[Cor.~II.5.5]{Hart}.
Parts (d),(e) below describe the effect of the sheafification used
to define $\MSpec M$ in Definition \ref{ag6def3}. In particular,
when $M$ is finitely presented (e) shows that the presheaf $U\mapsto
M\ot_\fC\O_X(U)$ in Definition \ref{ag6def3} is already a sheaf.
\begin{thm} Let\/ $\fC$ be a fair\/ $C^\iy$-ring, and\/
$(X,\O_X)=\Spec\fC$. Then
\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item[{\rm(a)}] There are natural isomorphisms $\Ga(\MSpec
M)\cong R_{\rm all}^{\rm co}(M)$ for all\/ $M$ in\/ $\fCmod$
which identify\/ $\Phi_M:M\ra\Ga(\MSpec M)$ with the natural
projection\/ $\pi:M\ra R_{\rm all}^{\rm co}(M)$. If\/ $M$ is
complete then\/ $\Phi_M$ is an isomorphism.
\item[{\rm(b)}] If\/ ${\cal E}\in\OXmod$ then\/ $\Ga({\cal E})$
is a complete $\fC$-module, and there is a natural
isomorphism\/~${\cal E}\cong\MSpec\ci\Ga({\cal E})$.
\item[{\rm(c)}] $\MSpec$ and\/ $\Ga$ induce an equivalence of
categories\/~$\fCmodco\sim\OXmod$.
\item[{\rm(d)}] Let\/ $M\in\fCmod$ and\/ $U\subseteq X$ be open.
Then there is a natural isomorphism of\/ $\O_X(U)$-modules
$(\MSpec M)(U)\cong R_{\rm all}^{\rm co}\bigl(M\ot_\fC
\O_X(U)\bigr)$.
\item[{\rm(e)}] Let\/ $M\in\fCmodfp$ and\/ $U\subseteq X$ be open.
Then there is a natural isomorphism of\/ $\O_X(U)$-modules
$(\MSpec M)(U)\cong M\ot_\fC \O_X(U)$.
\end{itemize}
Parts\/ {\rm(d),(e)} also hold if\/ $\fC$ is finitely generated
rather than fair.
\label{ag6thm1}
\end{thm}
\begin{proof} We will first show that the presheaf of
$\O_X$-modules $U\mapsto R_{\rm all}^{\rm co}\bigl(M\ot_\fC
\O_X(U)\bigr)$ on $X$ is actually a sheaf. For the first sheaf
axiom, suppose $U\subseteq X$ is open and $\{V_a:a\in A\}$ is an
open cover of $U$, and $s\in R_{\rm all}^{\rm co}\bigl(M\ot_\fC
\O_X(U)\bigr)$ satisfies $R_{\rm all}^{\rm
co}(\id_M\ot\rho_{UV_a})s=0$ in $R_{\rm all}^{\rm co}\bigl(M\ot_\fC
\O_X(V_a)\bigr)$ for all $a\in A$. We must show $s=0$. Since $\fC$
and hence $\O_X(U)$ are fair, as in \S\ref{ag43} we can choose a
locally finite refinement $\{W_b:b\in B\}$ of $\{V_a:a\in A\}$, and
a partition of unity $\{\eta_b:b\in B\}$ in $\O_X(U)$ subordinate
to~$\{W_b:b\in B\}$.
Let $b\in B$. Then there exists $a\in A$ with $W_b\subseteq V_a$, as
$\{W_b:b\in B\}$ is a refinement of $\{V_a:a\in A\}$. Since $\eta_b$
is supported on $W_b\subset V_a$, one can show using a partition of
unity argument that
\begin{equation*}
\rho_{UV_a}:\bigl\{\eta_b\cdot f:f\in\O_X(U)\bigr\}\longra
\bigl\{\rho_{UV_a}(\eta_b)\cdot f':f'\in\O_X(V_a)\bigr\}
\end{equation*}
is an isomorphism. Thus tensoring over $\fC$ with $M$ shows that
\begin{equation*}
\id_M\!\ot\rho_{UV_a}:\bigl\{\eta_b\cdot m:m\in
M\!\ot_\fC\!\O_X(U)\bigr\}\!\ra\!\bigl\{\rho_{UV_a}(\eta_b)\cdot
m':m'\in M\!\ot_\fC\!\O_X(V_a)\bigr\}
\end{equation*}
is an isomorphism. Therefore applying $R_{\rm all}^{\rm co}$ implies
that
\e
\begin{split}
R_{\rm all}^{\rm co}(\id_M\ot\rho_{UV_a}): \bigl\{\eta_b&\cdot
m:m\in
R_{\rm all}^{\rm co}(M\ot_\fC\O_X(U))\bigr\}\\
&\longra \bigl\{\rho_{UV_a}(\eta_b)\cdot m':m'\in R_{\rm all}^{\rm
co}(M\ot_\fC\O_X(V_a))\bigr\}
\end{split}
\label{ag6eq3}
\e
is an isomorphism. Since
\begin{align*}
R_{\rm all}^{\rm co}(\id_M\ot\rho_{UV_a})(\eta_b\cdot s)=
\rho_{UV_a}(\eta_b)\cdot R_{\rm all}^{\rm co}(\id_M\ot
\rho_{UV_a})(s)=0,
\end{align*}
this shows that $\eta_b\cdot s=0$ in $R_{\rm all}^{\rm
co}\bigl(M\ot_\fC\O_X(U)\bigr)$ for all $b\in B$. But $s=\sum_{b\in
B}\eta_b\cdot s$ as $R_{\rm all}^{\rm
co}\bigl(M\ot_\fC\O_X(U)\bigr)$ is complete and $\{\eta_b:b\in B\}$
is a partition of unity, so $s=0$, as we have to prove.
For the second sheaf axiom, let $U\subseteq X$ be open, $\{V_a:a\in
A\}$ an open cover of $U$, and $s_a\in R_{\rm all}^{\rm
co}\bigl(M\ot_\fC\O_X(V_a)\bigr)$ for $a\in A$ be given such that
$R_{\rm all}^{\rm co}(\id_M\ot\rho_{V_a\, V_a\cap V_{a'}})s_a=
R_{\rm all}^{\rm co}(\id_M\ot\rho_{V_{a'}\, V_a\cap V_{a'}})s_{a'}$
in $R_{\rm all}^{\rm co}\bigl(M\ot_\fC\O_X(V_a\cap V_{a'})\bigr)$
for all $a,a'\in A$. Choose $\{W_b:b\in B\}$, $\{\eta_b:b\in B\}$ as
above, and for each $b\in B$ choose $a_b\in A$ with $W_b\subseteq
V_a$. The argument above with \eq{ag6eq3} an isomorphism shows that
there exists $t_b\in R_{\rm all}^{\rm
co}\bigl(M\ot_\fC\O_X(U)\bigr)$ with
\begin{equation*}
R_{\rm all}^{\rm co}(\id_M\ot\rho_{UV_{a_b}})(\eta_b\cdot t_b)=
\rho_{UV_{a_b}}(\eta_b)\cdot s_{a_b},
\end{equation*}
and moreover $\eta_b\cdot t_b$ is unique. Now define $s=\sum_{b\in
B}\eta_b\cdot t_b$ in $R_{\rm all}^{\rm co}\bigl(M\ot_\fC
\O_X(U)\bigr)$. This is a locally finite sum, as $\{\eta_b:b\in B\}$
is a partition of unity, so $s$ is well-defined as $R_{\rm all}^{\rm
co}\bigl(M\ot_\fC \O_X(U)\bigr)$ is complete. A similar argument to
the first part shows that $R_{\rm all}^{\rm
co}(\id_M\ot\rho_{UV_a})s=s_a$ for $a\in A$, proving the second
sheaf axiom.
The stalk of the sheaf $U\mapsto R_{\rm all}^{\rm co}\bigl(M\ot_\fC
\O_X(U)\bigr)$ at $x\in X$ is $R_{\rm all}^{\rm co}\bigl(M\ot_\fC
\O_{X,x}\bigr)$, where $\O_{X,x}$ is the stalk of $\O_X$ at $x$. But
$R_{\rm all}^{\rm co}\bigl(M\ot_\fC \O_{X,x}\bigr)=M\ot_\fC
\O_{X,x}$, since modules over $C^\iy$-local rings are trivially
complete. But the stalk of the presheaf $U\mapsto M\ot_\fC\O_X(U)$
at $x$ is $M\ot_\fC \O_{X,x}$. Hence the sheaf $U\mapsto R_{\rm
all}^{\rm co}\bigl(M\ot_\fC\O_X(U)\bigr)$ and the presheaf $U\mapsto
M\ot_\fC\O_X(U)$ have the same stalks, so the sheaf is canonically
isomorphic to the sheafification of the presheaf. This proves (d),
and taking $U=X$ proves~(a).
For (e), if $M\in\fCmodfp$ there is an exact sequence
$\fC\ot_\R\R^m\ra\fC\ot_\R\R^n\ra M\ra 0$ in $\fCmod$. Since
$-\ot_\fC\O_X(U)$ is right exact $\O_X(U)\ot_\R\R^m\ra\O_X(U)
\ot_\R\R^n\ra M\ot_\fC\O_X(U)\ra 0$ is exact in $\O_X(U)$-mod, so
$M\ot_\fC\O_X(U)$ is a finitely presented $\O_X(U)$-module. Hence
$M\ot_\fC\O_X(U)$ is complete by Corollary \ref{ag5cor}, and $R_{\rm
all}^{\rm co}\bigl(M\ot_\fC \O_X(U)\bigr)\cong M\ot_\fC\O_X(U)$.
Thus (e) follows from (d). If $\fC$ is finitely generated rather
than fair then $\bar\fC=R_{\rm fg}^{\rm fa}(\fC)$ is fair and the
isomorphism $\Spec\fC\cong\Spec\bar\fC$ identifies $\MSpec M$ with
$\MSpec(M\ot_\fC\bar\fC)$, so (d),(e) for $\fC$ follow from (d),(e)
for $\bar\fC$ as~$(M\ot_\fC\bar\fC)\ot_{\bar\fC} \O_X(U)\cong
M\ot_\fC \O_X(U)$.
For (b), if ${\cal E}\in\OXmod$ then from the sheaf conditions on
(e) it is obvious that every locally finite sum in $\Ga({\cal E})$
has a unique limit, so $\Ga({\cal E})$ is complete. Taking
$M=\Ga({\cal E})$ in \eq{ag6eq2} gives a natural morphism
$\Phi_{\cal E}:\MSpec\ci\Ga({\cal E})\ra{\cal E}$ corresponding to
$\id_{\Ga({\cal E})}$. Let $x\in X$, and suppose $U$ is an open
neighbourhood of $x$ in $X$ and $e\in{\cal E}(U)$. By facts about
smooth functions on $\R^n$ there exists an open neighbourhood $V$ of
$x$ in $U$ and $\eta\in\fC$ such that $\eta$ is supported on $U$,
that is, $\pi_x(\eta)=0$ in $\fC_y$ for all $y\in X\sm U$, and
$\eta\equiv 1$ in $V$. Then $\rho_{XU}(\eta)\cdot e\in{\cal E}(U)$
can be extended by zero on $X\sm U$ to a unique $f\in{\cal
E}(X)=\Ga({\cal E})$ such that $\rho_{XU}(f)=\rho_{XU}(\eta)\cdot
e$. Thus $\rho_{XV}(f)=\rho_{XV}(\eta)\cdot\rho_{UV}(e)=
\rho_{UV}(e)$ as~$\rho_{XV}(\eta)=1$.
Hence, given any $x\in U\subset X$ and $e\in{\cal E}(U)$ we can find
$f\in\Ga({\cal E})$ and open $V$ with $x\in V\subseteq U$ such that
$\rho_{XV}(f)=\rho_{UV}(e)$. Therefore the natural projection
$\Ga({\cal E})\ra{\cal E}_x$ from $\Ga({\cal E})$ to germs of
sections of ${\cal E}$ at $x$ is surjective. Hence $\Phi_{\cal
E}\vert_x:\MSpec\ci\Ga({\cal E})\vert_x\ra{\cal E}_x$ is surjective.
It easily follows that $\Phi_{\cal E}\vert_x$ is an isomorphism, and
as this holds for all $x\in X$, $\Phi_{\cal E}$ is an isomorphism.
This proves (b), and (c) follows from (a) and~(b).
\end{proof}
We can understand pullback $\uf^*$ explicitly in terms of modules
over the corresponding $C^\iy$-rings.
\begin{prop} Let\/ $\fC,\fD$ be\/ $C^\iy$-rings, $\phi:\fD\ra\fC$ a
morphism, $M,N$ be $\fD$-modules, and\/ $\al:M\ra N$ a morphism of\/
$\fD$-modules. Write $\uX\!=\!\Spec\fC,$ $\uY\!=\!\Spec\fD,$
$\uf\!=\!\Spec(\phi):\uX\ra\uY,$ and\/ ${\cal E}\!=\!\MSpec(M),$
${\cal F}\!=\!\MSpec(N)$ in $\OXmod$. Then there are natural
isomorphisms\/ $\uf^*({\cal E})\cong\MSpec(M\ot_\fD\fC)$ and\/
$\uf^*({\cal F})\cong\MSpec(N\ot_\fD\fC)$ in $\OYmod$. These
identify $\MSpec(\al\ot\id_\fC):\MSpec(M\ot_\fD\fC)
\ra\MSpec(N\ot_\fD\fC)$ with\/~$\uf^*(\MSpec\al):\uf^*({\cal
E})\ra\uf^*({\cal F})$.
\label{ag6prop2}
\end{prop}
\begin{proof} Write $\uX=(X,\O_X)$, $\uY=(Y,\O_Y)$ and
$\uf=(f,f^\sh)$. Then $\cal E$ is the sheafification of the presheaf
$V\mapsto M\ot_\fD\O_Y(V)$, and $\uf^*({\cal E})$ is the
sheafification of the presheaf $U\mapsto\lim_{V\supseteq f(U)}{\cal
E}(V)\ot_{\O_Y(V)}\O_X(U)$. These two sheafifications combine into
one, so $\uf^*({\cal E})$ is the sheafification of the presheaf
$U\mapsto\lim_{V\supseteq f(U)}(M\ot_\fD\O_Y(V))
\ot_{\O_Y(V)}\O_X(U)$. But $(M\ot_\fD\O_Y(V))\ot_{\O_Y(V)}\O_X(U)
\cong M\ot_\fD\O_X(U)\cong (M\ot_\fD\fC)\ot_\fC\O_X(U)$, so this is
canonically isomorphic to the presheaf $U\mapsto
(M\ot_\fD\fC)\ot_\fC\O_X(U)$ whose sheafification is
$\MSpec(M\ot_\fD\fC)$. This gives a natural isomorphism $\uf^*({\cal
E})\cong\MSpec(M\ot_\fD\fC)$. The same holds for $N$. The
identification of $\MSpec(\al\ot\id_\fC)$ and $\uf^*(\MSpec\al)$
follows by passing from morphisms of presheaves to morphisms of the
associated sheaves.
\end{proof}
\subsection{Quasicoherent and coherent sheaves on $C^\iy$-schemes}
\label{ag63}
Here is our definition of quasicoherent and coherent sheaves.
\begin{dfn} Let $\uX=(X,\O_X)$ be a $C^\iy$-scheme, and $\cal E$ a sheaf
of $\O_X$-modules. We call $\cal E$ {\it quasicoherent\/} if $X$ can
be covered by open subsets $U$ with $(U,\O_X\vert_U)\cong\Spec\fC$
for some $C^\iy$-ring $\fC$, and under this identification $\cal
E\vert_U$ is isomorphic to $\MSpec M$ for some $\fC$-module $M$. We
call $\cal E$ {\it coherent\/} if furthermore we can take these
$\fC$-modules $M$ to be finitely presented. We call $\cal E$ a {\it
vector bundle of rank\/} $n\ge 0$ if $X$ may be covered by open $U$
such that ${\cal E}\vert_U\cong\O_X\vert_U\ot_\R\R^n$. Vector
bundles are coherent sheaves. Write $\qcoh(\uX)$ and $\coh(\uX)$ for
the full subcategories of quasicoherent and coherent sheaves in
$\OXmod$, respectively.
\label{ag6def4}
\end{dfn}
\begin{rem} Our definition of quasicoherent sheaves follows
Hartshorne \cite[p.~111]{Hart} in conventional algebraic geometry
exactly, replacing schemes by $C^\iy$-schemes. However, our
definition of coherent sheaf is not standard. The $C^\iy$-rings
$\O_X(U)$ we are interested in are generally {\it not noetherian\/}
as commutative $\R$-algebras, and this causes problems with
coherence.
In the non-noetherian case, the notions of coherent sheaf in
Hartshorne \cite[p.~111]{Hart} and Grothendieck \cite[\S
0.5.3]{Grot} are not equivalent. Hartshorne's definition, which
Grothendieck calls sheaves of {\it finite type\/} \cite[\S
0.5.2]{Grot}, requires the $\fC$-modules $M$ to be finitely
generated rather than finitely presented, and is too weak for our
purposes. Grothendieck's definition is too strong: if $X$ is a
manifold of positive dimension and $E\ra X$ a vector bundle of
positive rank, the corresponding $\O_X$-module $\cal E$ over $\uX$
is never coherent in Grothendieck's sense, and even $\O_X$ is not
coherent. Our definition of coherent sheaf, which correspond to {\it
finitely presented quasicoherent sheaves\/} as in \cite[\S
0.5.2.5]{Grot}, is intermediate between those of Hartshorne and
Grothendieck.
\label{ag6rem3}
\end{rem}
We are mainly interested in sheaves on locally fair $C^\iy$-schemes.
In this case Theorem \ref{ag6thm1}(b) implies:
\begin{cor} Let\/ $\uX$ be a locally fair\/ $C^\iy$-scheme. Then
every\/ $\O_X$-module\/ $\cal E$ on\/ $\uX$ is quasicoherent, that
is,~$\qcoh(\uX)=\OXmod$.
\label{ag6cor1}
\end{cor}
The following proposition is elementary, using ideas in \cite[\S
0.4--\S 0.5]{Grot}. The middle part holds as if $\fC$ is a
$C^\iy$-ring then $\fCmodfp$ is closed under cokernels and
extensions in $\fCmod$, but may not be closed under kernels, as in
Definition~\ref{ag5def1}.
\begin{prop} Let\/ $\uX$ be a $C^\iy$-scheme. Then\/ $\qcoh(\uX)$
is closed under kernels, cokernels and extensions in $\OXmod,$ so it
is an \begin{bfseries}abelian category\end{bfseries}. The full
subcategory\/ $\coh(\uX)$ of\/ $\qcoh(\uX)$ is in general
\begin{bfseries}not\end{bfseries} an abelian category, even in the
case when $\uX=F_\Manc^\CSch(X)$ for some manifold\/ $X$ of positive
dimension, because the $C^\iy$-rings\/ $\O_X(U)$ for open
$U\subseteq X$ need not be noetherian as commutative $\R$-algebras.
However, if\/ $0\ra{\cal E}\ra{\cal F}\ra{\cal G}\ra 0$ is exact in
$\qcoh(\uX)$ or\/ $\OXmod$ and\/ ${\cal E},{\cal F}$ are coherent,
then $\cal G$ is coherent, or if\/ ${\cal E},{\cal G}$ are coherent,
then $\cal F$ is coherent. That is, $\coh(\uX)$ is closed under
cokernels and extensions in\/ $\OXmod,$ but may not be closed under
kernels in\/~$\OXmod$.
Suppose\/ $\uf:\uX\ra\uY$ is a morphism of\/ $C^\iy$-schemes. Then
pullback\/ $\uf^*:\OYmod\ra\OXmod$ takes quasicoherent sheaves to
quasicoherent sheaves and coherent sheaves to coherent sheaves. Thus
$\uf^*:\qcoh(\uY)\ra\qcoh(\uX)$ is a right exact functor, by
Proposition\/~{\rm\ref{ag6prop1}}.
\label{ag6prop3}
\end{prop}
As in Godement \cite[\S II.3.7]{Gode} or Voisin
\cite[Def.~4.35]{Vois}, a sheaf of abelian groups $\cE$ on a
topological space $X$ is called {\it fine\/} if for any open cover
of $X$, a subordinate partition of unity exists in the sheaf ${\cal
H}om(\cE,\cE)$. In particular, if $\O_X$ is a sheaf of rings on $X$
for which partitions of unity exist subordinate to any open cover,
then every sheaf of $\O_X$-modules $\cE$ is fine. Therefore by
Proposition \ref{ag4prop9}, if $\uX$ is a separated, paracompact,
locally fair $C^\iy$-scheme, then quasicoherent sheaves on $\uX$ are
fine.
A fundamental property \cite[Prop.~4.36]{Vois} of fine sheaves $\cE$
is that their cohomology groups $H^i(\cE)$ are zero for all $i>0$.
This means that $H^0$ is an exact functor on fine sheaves, rather
than just left exact, since $H^1$ measures the failure of $H^0$ to
be right exact. But $H^0(\cE)=\cE(X)$, and more generally
$H^0(\cE\vert_U)=\cE(U)$ for open $U\subseteq X$. Thus we deduce:
\begin{prop} Suppose\/ $\uX=(X,\O_X)$ is a separated, paracompact,
locally fair $C^\iy$-scheme, and\/ $\cdots \ra\cE^i\,{\buildrel
\phi^i\over\longra}\,\cE^{i+1}\,{\buildrel\phi^{i+1}\over\longra}
\,\cE^{i+2}\ra\cdots$ an exact sequence in $\qcoh(\uX)$. Then
$\cdots\ra\cE^i(U)\,{\buildrel
\phi^i(U)\over\longra}\,\cE^{i+1}(U)\,{\buildrel\phi^{i+1}(U)\over\longra}
\,\cE^{i+2}(U)\ra\cdots$ is an exact sequence of\/ $\O_X(U)$-modules
for each open\/~$U\subseteq X$.
\label{ag6prop4}
\end{prop}
\subsection{Cotangent sheaves of $C^\iy$-schemes}
\label{ag64}
We now define {\it cotangent sheaves}, the sheaf version of
cotangent modules in~\S\ref{ag5}.
\begin{dfn} Let $\uX=(X,\O_X)$ be a $C^\iy$-ringed space. Define
${\cal P}T^*\uX$ to associate to each open $U\subseteq X$ the
cotangent module $(\Om_{\O_X(U)},\mu_{\O_X(U)})$ of Definition
\ref{ag5def3}, regarded as a module over the $C^\iy$-ring $\O_X(U)$,
and to each inclusion of open sets $V\subseteq U\subseteq X$ the
morphism of $\O_X(U)$-modules
$\Om_{\rho_{UV}}:\Om_{\O_X(U)}\ra\Om_{\O_X(V)}$ associated to the
morphism of $C^\iy$-rings $\rho_{UV}:\O_X(U)\ra\O_X(V)$. Then as we
want for \eq{ag6eq1} the following commutes:
\begin{equation*}
\xymatrix@R=10pt@C=60pt{ \O_X(U)\t \Om_{\O_X(U)}
\ar[d]^{\rho_{UV}\t\Om_{\rho_{UV}}} \ar[r]_{\mu_{\O_X(U)}} &
\Om_{\O_X(U)}
\ar[d]_{\Om_{\rho_{UV}}} \\
\O_X(V)\t \Om_{\O_X(V)} \ar[r]^{\mu_{\O_X(V)}} & \Om_{\O_X(V)}.}
\end{equation*}
Using this and functoriality of cotangent modules
$\Om_{\psi\ci\phi}=\Om_\psi\ci\Om_\phi$ in Definition \ref{ag5def3},
we see that ${\cal P}T^*\uX$ is a {\it presheaf of\/ $\O_X$-modules
on\/} $\uX$. Define the {\it cotangent sheaf\/ $T^*\uX$ of\/} $\uX$
to be the sheaf of $\O_X$-modules associated to~${\cal P}T^*\uX$.
If $U\subseteq X$ is open then we have an equality of sheaves of
$\O_X\vert_U$-modules
\e
T^*(U,\O_X\vert_U)=T^*\uX\vert_U.
\label{ag6eq4}
\e
As in Example \ref{ag5ex3}, if $f:X\ra Y$ is a smooth map of
manifolds we have a morphism $(\d f)^*:f^*(T^*Y)\ra T^*X$ of vector
bundles over $X$. Here is an analogue for $C^\iy$-ringed spaces. Let
$\uf:\uX\ra\uY$ be a morphism of $C^\iy$-ringed spaces. Then by
Definition \ref{ag6def2}, $\uf^*\bigl(T^*\uY\bigr)$ is the
sheafification of the presheaf $U\mapsto\lim_{V\supseteq
f(U)}T^*\uY(V)\ot_{\O_Y(V)}\O_X(U)$, and $T^*\uY$ is the
sheafification of the presheaf ${\cal P}T^*\uY$. These two
sheafifications combine into one, so that $\uf^*\bigl(T^*\uY\bigr)$
is the sheafification of the presheaf $\uf^*({\cal P}T^*\uY)$ acting
by
\begin{align*}
U\longmapsto \uf^*({\cal P}T^*\uY)(U)&=\ts
\lim_{V\supseteq f(U)}{\cal P}T^*\uY(V) \ot_{\O_Y(V)}\O_X(U)\\
&=\ts\lim_{V\supseteq f(U)}\Om_{\O_Y(V)}\ot_{\O_Y(V)}\O_X(U).
\end{align*}
Define a morphism of presheaves ${\cal P}\Om_\uf:\uf^*({\cal
P}T^*\uY)\ra{\cal P}T^*\uX$ on $X$ by
\begin{equation*}
({\cal P}\Om_\uf)(U)=\ts\lim_{V\supseteq f(U)}
(\Om_{\rho_{f^{-1}(V)\,U}\ci f^\sh(V)})_*,
\end{equation*}
where $(\Om_{\rho_{f^{-1}(V)\,U}\ci f^\sh(V)})_*:\Om_{\O_Y(V)}
\ot_{\O_Y(V)}\O_X(U)\ra\Om_{\O_X(U)}=({\cal P}T^*\uX)(U)$ is
constructed as in Definition \ref{ag5def3} from the $C^\iy$-ring
morphisms $f^\sh(V):\O_Y(V)\ra \O_X(f^{-1}(V))$ in $\uf$ and
$\rho_{f^{-1}(V)\,U}:\O_X(f^{-1}(V))\ra\O_X(U)$ in $\O_X$. Define
$\Om_\uf:\uf^*\bigl(T^*\uY\bigr)\ra T^*\uX$ to be the induced
morphism of the associated sheaves.
\label{ag6def5}
\end{dfn}
\begin{rem} There is an alternative definition of the cotangent
sheaf $T^*\uX$ of $\uX=(X,\O_X)$ following Hartshorne
\cite[p.~175]{Hart}. We can form the product $\uX\t\uX=(X\t
X,\O_{X\t X})$ in $\CRS$, and there is a natural diagonal morphism
$\ul{\De\!}\,_{\uX}:\uX\ra\uX\t\uX$. Write $\cal I$ for the sheaf of
ideals in $\O_{X\t X}$ associated to $\ul{\De\!}\,_{\uX}$, that is,
for open $U\subseteq X\t X$, ${\cal I}(U)=\Ker\bigl(\De_{\uX}^\sh)
(U):\O_{X\t X}(U)\ra\O_X(\De_\uX^{-1}(U))\bigr)$, as an ideal in
$\O_{X\t X}(U)$. Then $T^*\uX\cong \ul{\De\!}\,^*_\uX({\cal I}/{\cal
I}^2)$. This can be proved using the equivalence of two definitions
of cotangent module in~\cite[Prop.~II.8.1A]{Hart}.
\label{ag6rem4}
\end{rem}
Here are some properties of cotangent sheaves:
\begin{thm}{\bf(a)} Let\/ $\fC$ be a finitely generated\/
$C^\iy$-ring and\/ $\uX=\Spec\fC$. Then there is a canonical
isomorphism\/~$T^*\uX\cong\MSpec\Om_\fC$.
\smallskip
\noindent{\bf(b)} Let\/ $\uX$ be a fair affine\/ $C^\iy$-scheme.
Then\/ ${\cal P}T^*\uX$ in Definition\/ {\rm\ref{ag6def5}} is a
sheaf, so that\/ ${\cal P}T^*\uX\cong T^*\uX$ and\/
$(T^*\uX)(U)\cong\Om_{\O_X(U)}$ for all open\/ $U\subseteq X$.
\smallskip
\noindent{\bf(c)} Let\/ $\uX$ be a locally good\/ $C^\iy$-scheme.
Then\/ $T^*\uX$ is a coherent sheaf.
\smallskip
\noindent{\bf(d)} Suppose\/ $X$ is an $n$-manifold, which may have
boundary or corners, and\/ $\uX=F_\Manc^\CSch(X)$ in the notation of
Definition\/ {\rm\ref{ag4def4}}. Then $T^*\uX$ is a rank\/ $n$
vector bundle on $\uX,$ with\/ $(T^*\uX)(U)\cong C^\iy(T^*X\vert_U)$
for all open\/ $U\subseteq X$. When\/ $X=\R^n_k:=[0,\iy)^k\t \R^{n
-k}$ we have\/~$T^*(\ul{\R^n_k})\cong\O_{\R^n_k}\ot_\R(\R^n)^*$.
\label{ag6thm2}
\end{thm}
\begin{proof} For (a), let $U\subseteq X$ be open. Then by
Proposition \ref{ag4prop4} there exists a characteristic function
$c\in\fC$ for $U$, and Proposition \ref{ag4prop6} gives a canonical
isomorphism $\O_X(U)\cong R_{\rm fg}^{\rm fa}\bigl(\fC[c^{-1}]
\bigr)$. Propositions \ref{ag5prop3} and \ref{ag5prop4} then give
\begin{align*}
\Om_{\O_X(U)}&\cong\Om_{R_{\rm fg}^{\rm fa}(\fC[c^{-1}])}\cong
R_{\rm all}^{\rm co}\bigl(\Om_{\fC[c^{-1}]}\ot_{\fC[c^{-1}]}\O_X(U)
\bigr)\\ &\cong R_{\rm all}^{\rm co}\bigl((\Om_\fC\ot_\fC\fC[c^{-1}])
\ot_{\fC[c^{-1}]}\O_X(U)\bigr)\cong R_{\rm all}^{\rm co}\bigl(\Om_\fC
\ot_\fC\O_X(U)\bigr).
\end{align*}
Hence the presheaf ${\cal P}T^*(X,\O_X)$ in Definition \ref{ag6def5}
is canonically isomorphic to the presheaf $U\mapsto R_{\rm all}^{\rm
co}\bigl(\Om_\fC\ot_\fC \O_X(U)\bigr)$. But by Theorem
\ref{ag6thm1}(d), this presheaf is canonically isomorphic to the
sheaf $\MSpec\Om_\fC$, proving (a). Also, this shows that the
presheaf ${\cal P}T^*(X,\O_X)$ is a sheaf when $\uX=\Spec\fC$ for
$\fC$ fair. Since this only depends on $\uX$ up to isomorphism, part
(b) follows.
For (c), as $\uX$ is locally good it can be covered by open
$U\subseteq X$ with $(U,\O_X\vert_U)\cong\Spec\fC$ for some good
$C^\iy$-ring $\fC$. By (a), this isomorphism identifies
$T^*\uX\vert_U$ with $\MSpec\Om_\fC$. Since $\Om_\fC$ is finitely
presented by Theorem \ref{ag5thm1}, $T^*\uX$ is coherent. For (d),
as $T^*\R^n_k\cong\R^n_k\t(\R^n)^*$ we have
$\Om_{C^\iy(\R^n_k)}\cong C^\iy(\R^n_k)\ot_\R(\R^n)^*$, and thus
$T^*(\ul{\R^n_k})\cong\O_{\R^n_k}\ot_\R(\R^n)^*$ by (a). Any
$n$-manifold $X$ can be covered by open $U$ diffeomorphic to
$\R^n_k$, so that $(U,\O_X\vert_U)\cong \ul{\R^n_k}$ and
$T^*\uX\vert_U\cong\O_X\vert_U\ot_\R(\R^n)^*$ by \eq{ag6eq4}. Hence
$T^*\uX$ is a vector bundle of rank~$n$.
\end{proof}
Here are some important properties of the morphisms $\Om_\uf$ in
Definition \ref{ag6def5}. Equation \eq{ag6eq7} is an analogue of
\eq{ag5eq6} and~\eq{ag5eq12}.
\begin{thm}{\bf(a)} Let\/ $\uf:\uX\ra\uY$ and\/ $\ug:\uY\ra\uZ$ be
morphisms of\/ $C^\iy$-schemes. Then
\e
\Om_{\ug\ci\uf}=\Om_\uf\ci \uf^*(\Om_\ug)\ci I_{\uf,\ug}(T^*\uZ)
\label{ag6eq5}
\e
as morphisms $(\ug\ci\uf)^*(T^*\uZ)\ra T^*\uX$ in $\OXmod$. Here
$\Om_\ug:\ug^*(T^*\uZ)\ra T^*\uY$ in $\OYmod,$ so applying $\uf^*$
gives $\uf^*(\Om_\ug):\uf^*(\ug^*(T^*\uZ))\ra \uf^*(T^*\uY)$ in\/
$\OXmod,$ and\/ $I_{\uf,\ug}(T^*\uZ):(\ug\ci\uf)^*(T^*\uZ)
\ra\uf^*(\ug^*(T^*\uZ))$ is as in Remark\/~{\rm\ref{ag6rem2}}.
\smallskip
\noindent{\bf(b)} Suppose\/ $\uW,\uX,\uY,\uZ$ are locally fair\/
$C^\iy$-schemes with a Cartesian square
\e
\begin{gathered}
\xymatrix@C=30pt@R=10pt{ \uW \ar[r]_\uf \ar[d]^\ue & \uY \ar[d]_\uh \\
\uX \ar[r]^\ug & \uZ}
\end{gathered}
\label{ag6eq6}
\e
in $\CSchlf,$ so that\/ $\uW=\uX\t_\uZ\uY$. Then the following is
exact in $\qcoh(\uW):$
\e
\xymatrix@C=13pt{ (\ug\ci\ue)^*(T^*\uZ)
\ar[rrrr]^(0.47){\begin{subarray}{l}\ue^*(\Om_\ug)\ci
I_{\ue,\ug}(T^*\uZ)\op\\ -\uf^*(\Om_\uh)\ci
I_{\uf,\uh}(T^*\uZ)\end{subarray}} &&&&
\ue^*(T^*\uX)\!\op\!\uf^*(T^*\uY) \ar[rr]^(0.67){\Om_\ue\op \Om_\uf}
&& T^*\uW \ar[r] & 0.}
\label{ag6eq7}
\e
\label{ag6thm3}
\end{thm}
\begin{proof} Combining two or three sheafifications into one as in
the proof of Proposition \ref{ag6prop2}, we see that the sheaves
$T^*\uX,\uf^*(T^*\uY),\uf^*(\ug^*(T^*\uZ))$ and
$(\ug\ci\uf)^*(T^*\uZ)$ on $\uX$ are isomorphic to the
sheafifications of the following presheaves:
\ea
&T^*\uX &\rightsquigarrow & \quad U \longmapsto \Om_{\O_X(U)},
\label{ag6eq8}\\
& \uf^*(T^*\uY) &\rightsquigarrow & \quad
U\longmapsto\lim_{V\supseteq f(U)} \Om_{\O_Y(V)}\ot_{\O_Y(V)}\O_X(U),
\label{ag6eq9}\\
& \uf^*(\ug^*(T^*\uZ)) &\rightsquigarrow & \quad
U\longmapsto\lim_{V\supseteq f(U)} \lim_{W\supseteq g(V)}\!\!\!
\begin{aligned}[t] \bigl(\Om_{\O_Z(W)}\ot_{\O_Z(W)}\O_Y(V)\bigr)
&\\[-3pt] \ot_{\O_Y(V)}\O_X(U),\,\,\,&
\end{aligned}
\label{ag6eq10}\\
& (\ug\ci\uf)^*(T^*\uZ) &\rightsquigarrow & \quad
U\longmapsto\lim_{W\supseteq g\ci f(U)}
\Om_{\O_Z(W)}\ot_{\O_Z(W)}\O_X(U).
\label{ag6eq11}
\ea
Then $\Om_\uf,\Om_{\ug\ci\uf},\uf^*(\Om_\ug),I_{\uf,\ug}(T^*\uZ)$
are the morphisms of sheaves associated to the following morphisms
of the presheaves in \eq{ag6eq8}--\eq{ag6eq11}:
\ea
&\Om_{\uf} &\rightsquigarrow & \quad U \longmapsto \lim_{V\supseteq
f(U)}(\Om_{\rho_{f^{-1}(V)\,U}\ci f^\sh(V)})_*,
\label{ag6eq12}\\
&\Om_{\ug\ci\uf} &\rightsquigarrow & \quad U \longmapsto
\lim_{W\supseteq g\ci f(U)}(\Om_{\rho_{(g\ci f)^{-1}(W)\,U}\ci (g\ci
f)^\sh(W)})_*,
\label{ag6eq13}\\
&\uf^*(\Om_\ug) &\rightsquigarrow & \quad U \longmapsto
\lim_{V\supseteq f(U)} \lim_{W\supseteq g(V)}
(\Om_{\rho_{g^{-1}(W)\,V}\ci g^\sh(W)})_*,
\label{ag6eq14}\\
&I_{\uf,\ug}(T^*\uZ)
&\rightsquigarrow & \quad U \longmapsto \lim_{V\supseteq f(U)}
\lim_{W\supseteq g(V)} I_{UVW},
\label{ag6eq15}
\ea
where $I_{UVW}:\Om_{\O_Z(W)}\ot_{\O_Z(W)}\O_X(U)\ra\bigl(
\Om_{\O_Z(W)}\ot_{\O_Z(W)}\O_Y(V)\bigr)\ot_{\O_Y(V)}\O_X(U)$ is the
natural isomorphism.
Now if $U\!\subseteq\!X$, $V\!\subseteq\! Y$, $W\!\subseteq\! Z$ are
open with $V\!\supseteq\! f(U)$, $W\!\supseteq\! g(V)$ then
\begin{equation*}
\rho_{(g\ci f)^{-1}(W)\,U}\ci(g\ci f)^\sh(W)=\bigl[
\rho_{f^{-1}(V)\,U}\ci f^\sh(V)\bigr]\ci\bigl[\rho_{g^{-1}(W)\,V}\ci
g^\sh(W)\bigr]
\end{equation*}
as morphisms $\O_Z(W)\ra\O_X(U)$, so
$\Om_{\phi\ci\psi}=\Om_\phi\ci\Om_\psi$ in Definition \ref{ag5def3}
implies
\begin{equation*}
(\Om_{\rho_{(g\ci f)^{-1}(W)\,U}\ci(g\ci f)^\sh(W)})_*=
(\Om_{\rho_{f^{-1}(V)\,U}\ci
f^\sh(V)})_*\ci (\Om_{\rho_{g^{-1}(W)\,V}\ci
g^\sh(W)})_*\ci I_{UVW}.
\end{equation*}
Taking limits $\lim_{V\supseteq f(U)}\lim_{W\supseteq g(V)}$ implies
that the presheaves in \eq{ag6eq12}--\eq{ag6eq15} satisfy the
analogue of \eq{ag6eq5}, so passing to morphisms of sheaves
proves~(a).
For (b), first observe that as \eq{ag6eq6} is commutative, by (a) we
have $\Om_\ue\ci\ue^*(\Om_\ug)\ci
I_{\ue,\ug}(T^*\uZ)=\Om_{\ug\ci\ue}=\Om_{\uh\ci\uf}=
\Om_\uf\ci\uf^*(\Om_\uh)\ci I_{\uf,\uh}(T^*\uZ)$, so
$\Om_\ue\ci\bigl(\ue^*(\Om_\ug)\ci I_{\ue,\ug}(T^*\uZ)\bigr)-\Om_\uf
\ci\bigl(\uf^*(\Om_\uh)\ci I_{\uf,\uh}(T^*\uZ)\bigr)=0$, and
\eq{ag6eq7} is a complex. To show it is exact, since exactness is a
local condition it is enough to show that $W$ can be covered by open
sets $W'\subseteq W$ with the restriction of \eq{ag6eq7} to $W'$
exact.
Let $(x,y)\in W$, so that $x\in X$ and $y\in Y$ with $f(x)=g(y)=z\in
Z$. As $\uZ$ is locally fair we can choose an open neighbourhood
$Z'$ of $z\in Z$ with $\uZ'=(Z',\O_Z\vert_{Z'})\cong\Spec\fC$ for
$\fC$ a finitely generated $C^\iy$-ring. As $\uX,\uY$ are locally
fair we can choose open neighbourhoods $X'$ of $x$ in
$f^{-1}(Z')\subseteq X$ and $Y'$ of $y$ in $g^{-1}(Z')\subseteq Y$
with $\uX'=(X',\O_X\vert_{X'})\cong\Spec\fD$,
$\uY'=(Y',\O_Y\vert_{Y'})\cong \Spec\fE$ for $\fD,\fE$ finitely
generated $C^\iy$-rings. Set $W=X'\cap Y'\cap W$, and
$\fF=\fD\amalg_\fC\fE$. Then $\fF$ is a finitely generated
$C^\iy$-ring, and $W'$ is an open neighbourhood of $(x,y)$ in $W$
with $\uW'=(W',\O_W\vert_{W'})\cong\Spec\fF$, since $\Spec$
preserves limits.
Theorem \ref{ag6thm2}(a) now shows that the isomorphisms
$\uW'\cong\Spec\fF,\ldots,\uZ'\cong\Spec\fC$ identify $T^*\uW'\cong
\MSpec\Om_\fF,\ldots,T^*\uZ'\cong \MSpec\Om_\fC$. Theorem
\ref{ag5thm2} gives an exact sequence of $\fF$-modules \eq{ag5eq6}.
Applying the exact functor $\MSpec$ gives an exact sequence in
$\qcoh(\uW')$. Using the identifications above and Proposition
\ref{ag6prop2}, this exact sequence is identified with the
restriction of \eq{ag6eq7} to $W'$. Thus we may cover $W$ by open
subsets $W'$ such that \eq{ag6eq7} is exact on~$W'$.
\end{proof}
\section{Background material on stacks}
\label{ag7}
In \S\ref{ag8} we will study $C^\iy$-{\it stacks}, that is, various
classes of stacks on the site $(\CSch,{\cal J})$ of $C^\iy$-schemes
with the open cover topology. As a preparation for this we now
recall some background we will need on stacks. This section explains
theory for stacks on an arbitrary site $({\cal C},{\cal J})$
satisfying various extra conditions, and \S\ref{ag8} covers material
specific to $C^\iy$-schemes and $C^\iy$-stacks.
Nothing in this section is really new, although our presentation is
not always standard. Where there are several equivalent ways of
presenting something, for instance, defining Grothendieck topologies
using either coverings or sieves, we have chosen one. Our principal
references are Artin \cite{Arti}, Behrend et al.\ \cite{BEFF}, Gomez
\cite{Gome}, Laumon and Moret-Bailly \cite{LaMo}, Metzler
\cite{Metz}, and Noohi~\cite{Nooh}.
The topological and smooth stacks discussed by Metzler and Noohi are
closer to our situation than the stacks in algebraic geometry of
\cite{BEFF,Gome,LaMo}, so we often follow \cite{Metz,Nooh},
particularly in \S\ref{ag75} which is based on Metzler \cite[\S
3]{Metz}. Heinloth \cite{Hein} and Behrend and Xu \cite{BeXu} also
discuss smooth stacks.
\subsection{Grothendieck topologies, sites, prestacks, and stacks}
\label{ag71}
\begin{dfn} A {\it Grothendieck topology\/} $\cal J$ on a category
$\cal C$ is a collection of families $\{\vp_a:U_a\ra U\}_{a\in A}$
of morphisms in $\cal C$ called {\it coverings}, satisfying:
\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item[(i)] If $\vp:V\ra U$ is an isomorphism in $\cal C$, then
$\{\vp:V\ra U\}$ is a covering;
\item[(ii)] If $\{\vp_a:U_a\ra U\}_{a\in A}$ is a covering, and
$\{\psi_{ab}:V_{ab}\ra U_a\}_{b\in B_a}$ is a covering for all
$a\in A$, then $\{\vp_a\ci\psi_{ab}:V_{ab}\ra U\}_{a\in A,\;
b\in B_a}$ is a covering.
\item[(iii)] If $\{\vp_a:U_a\ra U\}_{a\in A}$ is a covering and
$\psi:V\ra U$ is a morphism in $\cal C$ then
$\{\pi_V:U_a\t_{\vp_a,U,\psi}V\ra V\}_{a\in A}$ is a covering,
where the fibre product $U_a\t_UV$ exists in $\cal C$ for all
$a\in A$.
\end{itemize}
A {\it site\/} $({\cal C},{\cal J})$ is a category $\cal C$ with a
Grothendieck topology~$\cal J$.
\label{ag7def1}
\end{dfn}
\begin{dfn} Let $\cal C$ be a category. A {\it category fibred in
groupoids over\/} $\cal C$ is a functor $p_{\cal X}:{\cal X}\ra{\cal
C}$, where $\cal X$ is a category, such that given any morphism
$g:C_1\ra C_2$ in $\cal C$ and $X_2\in{\cal X}$ with $p_{\cal
X}(X_2)=C_2$, there exists a morphism $f:X_1\ra X_2$ in $\cal X$
with $p_{\cal X}(f)=g$, and given commutative diagrams (on the left)
in $\cal X$, in which $g$ is to be determined, and (on the right) in
$\cal C$:
\e
\begin{gathered}
\xymatrix@C=25pt@R=5pt{ X_1 \ar@{..>}[rr]_g \ar[dr]_f && X_2 \ar[dl]^h \\
& X_3 }\end{gathered}\quad {\buildrel p_{\cal X} \over
\rightsquigarrow} \quad \begin{gathered} \xymatrix@C=10pt@R=5pt{
p_{\cal X}(X_1) \ar[rr]_{g'} \ar[dr]_{p_{\cal X}(f)} && p_{\cal
X}(X_2) \ar[dl]^{p_{\cal X}(h)} \\ & p_{\cal X}(X_3), }
\end{gathered}
\label{ag7eq1}
\e
then there exists a unique morphism $g$ as shown with $p_{\cal
X}(g)=g'$ and $f=h\ci g$. Often we refer to $\cal X$ as the category
fibred in groupoids (or prestack, or stack, etc.), leaving $p_{\cal
X}$ implicit.
\label{ag7def2}
\end{dfn}
\begin{dfn} Let $({\cal C},{\cal J})$ be a site, and $p_{\cal X}:{\cal
X}\ra{\cal C}$ be a category fibred in groupoids over $\cal C$. We
call $\cal X$ a {\it prestack\/} if whenever $\{\vp_a:U_a\ra
U\}_{a\in A}$ is a covering family in $\cal J$ and we are given
commutative diagrams in ${\cal X},{\cal C}$ for all $a,b\in A$, in
which $f$ is to be determined:
\e
\begin{gathered}
\xymatrix@R=2pt@C=5pt{ & X_{ab} \ar[dl] \ar@<-1ex>[ddr] \ar[rr] &&
Y_{ab} \ar[dl] \ar[ddr] \\ X_a \ar[ddr]_{x_a} \ar[rr] && Y_a
\ar@<1ex>[ddr]^(0.4){y_a} \\ && X_b \ar[rr] \ar[dl]_(0.4){x_b} &&
Y_b \ar[dl]^(0.4){y_b} \\ & X \ar@{..>}[rr]^(0.7)f && Y}
\end{gathered}
\quad{\buildrel p_{\cal X} \over \rightsquigarrow}\quad
\begin{gathered} \xymatrix@R=2pt@C=1pt{ & \!\!U_a\!\t_U\!U_b
\ar[dl]_(0.7){\pi_{U_a}} \ar[ddr]^(0.4){\pi_{U_b}} \ar@{=}[rr] &&
U_a\!\t_U\!U_b\!\! \ar[dl]^{\pi_{U_a}}
\ar[ddr]^(0.6){\pi_{U_b}} \\
U_a \ar[ddr]_{\vp_a} \ar@{=}[rr] && U_a \ar[ddr]^(0.35){\vp_a} \\ &&
U_b \ar@{=}[rr] \ar[dl]^(0.3){\vp_b} && U_b \ar[dl]^(0.4){\vp_b} \\
& U \ar@{=}[rr] && U,}
\end{gathered}
\label{ag7eq2}
\e
then there exists a unique $f:X\ra Y$ in $\cal X$ with $p_{\cal
X}(f)=\id_U$ making \eq{ag7eq2} commute for all~$a\in A$.
Let $p_{\cal X}:{\cal X}\ra{\cal C}$ be a prestack. We call $\cal X$
a {\it stack\/} if whenever $\{\vp_a:U_a\ra U\}_{a\in A}$ is a
covering family in $\cal J$ and we are given commutative diagrams in
${\cal X},{\cal C}$ for all $a,b,c\in A$, with $X_{ab}=X_{ba}$,
$X_{abc}=X_{bac}=X_{acb}$, etc., in which the object $X$ and
morphisms $x_a$ are be determined:
\begin{small}
\begin{equation}
\begin{gathered}
\xymatrix@R=2pt@C=5pt{ & X_{abc} \ar[dl] \ar@<-1ex>[ddr] \ar[rr]
&& X_{ac} \ar[dl]^(0.4){x_{ac}} \ar[ddr]^(0.4){x_{ca}} \\
X_{ab} \ar[ddr]_{x_{ba}} \ar[rr]_{x_{ab}} && X_a
\ar@{..>}@<1ex>[ddr]^(0.45){x_a} \\ && X_{bc} \ar[rr]^(0.7){x_{cb}}
\ar[dl]_(0.4){x_{bc}} && X_c \ar@{..>}[dl]^(0.4){x_c} \\ & X_b
\ar@{..>}[rr]^(0.7){x_b} && X}
\end{gathered}
\;{\buildrel p_{\cal X} \over \rightsquigarrow}\>\;\> \begin{gathered}
\xymatrix@!0@R=16pt@C=37pt{ & \!\!\!\!\!U_a\!\t_U\!U_b\!\t_U\!U_c \ar[dl]
\ar[ddr] \ar[rr] && U_a\!\t_U\!U_c\!\!\! \ar[dl]
\ar[ddr] \\
\!\!\!U_a\!\t_U\!U_b\!\! \ar[ddr] \ar[rr] && U_a \ar@<1ex>[ddr]^{\vp_a} \\
&& \!\!\!U_b\!\t_U\!U_c \ar[rr] \ar[dl] && U_c \ar[dl]^{\vp_c} \\
& U_b \ar[rr]^{\vp_b} && U,}
\end{gathered}
\label{ag7eq3}
\end{equation}\end{small}then there exists $X\in{\cal X}$ and morphisms
$x_a:X_a\ra X$ with $p_{\cal X}(x_a)=\vp_a$ for all $a\in A$, making
\eq{ag7eq3} commute for all $a,b,c\in A$.
\label{ag7def3}
\end{dfn}
Thus, in a prestack we have a sheaf-like condition allowing us to
glue morphisms in $\cal X$ uniquely over open covers in $\cal C$; in
a stack we also have a sheaf-like condition allowing us to glue
objects in $\cal X$ over open covers in~$\cal C$.
\begin{dfn} Let $({\cal C},{\cal J})$ be a site. A 1-{\it
morphism\/} between stacks or prestacks ${\cal X},{\cal Y}$ on
$({\cal C},{\cal J})$ is a functor $F:{\cal X}\ra{\cal Y}$ with
$p_{\cal Y}\ci F=p_{\cal X}:{\cal X}\ra{\cal C}$. If $F,G:{\cal
X}\ra{\cal Y}$ are 1-morphisms, a 2-{\it morphism\/} $\eta:F\Ra G$
is an isomorphism of functors. That is, for all $X\in{\cal X}$ we
are given an isomorphism $\eta(X):F(X)\ra G(X)$ in $\cal Y$ such
that if $f:X_1\ra X_2$ is a morphism in $\cal X$ then $\eta(X_2)\ci
F(f)=G(f)\ci\eta(X_1):F(X_1)\ra G(X_2)$ in $\cal Y$. With these
definitions, the stacks and prestacks on $({\cal C},{\cal J})$ form
({\it strict\/}) 2-{\it categories}, which we write as $\Sta_{({\cal
C},{\cal J})}$ and~$\Presta_{({\cal C},{\cal J})}$.
A {\it substack\/} $\cal Y$ of a stack $\cal X$ is a strictly full
subcategory $\cal Y$ in $\cal X$ such that $p_{\cal Y}:=p_{\cal
X}\vert_{\cal Y}:{\cal Y}\ra{\cal C}$ is a stack. The inclusion
functor $i_{\cal Y}:{\cal Y}\hookra{\cal X}$ is then a 1-morphism of
stacks.
\label{ag7def4}
\end{dfn}
We will not define 2-categories, but some useful references on
2-categories for stack theory are \cite[App.~B]{BEFF} and
\cite[App.~B]{Gome}. In a 2-category $\bs{\cal C}$, there are three
notions of when objects $X,Y$ in $\bs{\cal C}$ are `the same': {\it
equality\/} $X=Y$, and {\it isomorphism}, that is we have
1-morphisms $f:X\ra Y$, $g:Y\ra X$ with $g\ci f=\id_X$ and $f\ci
g=\id_Y$, and {\it equivalence}, that is we have 1-morphisms $f:X\ra
Y$, $g:Y\ra X$ and 2-isomorphisms $\eta:g\ci f\Ra\id_X$ and
$\ze:f\ci g\Ra\id_Y$. Usually the weakest notion, equivalence, is
the right one to use. We write $X\simeq Y$ to mean $X$ is equivalent
to~$Y$.
If $X,Y$ are objects in $\bs{\cal C}$ and $f,g:X\ra Y$ are
1-morphisms then $f\cong g$ means $f$ is 2-isomorphic to $g$. Write
$\bHom(X,Y)$ for the category with objects 1-morphisms $f,g:X\ra Y$
in $\bs{\cal C}$ and morphisms 2-morphisms $\eta:f\Ra g$
in~$\bs{\cal C}$.
\begin{dfn} Let $({\cal C},{\cal J})$ be a site, and ${\cal X}$ a
prestack on $({\cal C},{\cal J})$, so that $\Sta_{({\cal C},{\cal
J})}$ and $\Presta_{({\cal C},{\cal J})}$ are 2-categories. A {\it
stack associated to\/} $\cal X$, or {\it stackification of\/} $\cal
X$, is a stack $\hat{\cal X}$ with a 1-morphism of prestacks
$i:{\cal X}\ra\hat{\cal X}$, such that for every stack $\cal Y$,
composition with $i$ yields an equivalence of categories
$\bHom(\hat{\cal X},{\cal Y})\,{\buildrel
i^*\over\longra}\,\bHom({\cal X},{\cal Y})$.
\label{ag7def5}
\end{dfn}
As in \cite[Lem.~3.2]{LaMo}, every prestack has an associated stack,
just as every presheaf has an associated sheaf.
\begin{prop} For every prestack\/ $\cal X$ on $({\cal C},{\cal J})$
there exists an associated stack\/ $i:{\cal X}\ra\hat{\cal X},$
which is unique up to equivalence in $\Sta_{({\cal C},{\cal J})}$.
\label{ag7prop1}
\end{prop}
\subsection{Commutative diagrams and fibre products}
\label{ag72}
{\it Commutative diagrams\/} in 2-categories should in general only
commute {\it up to (specified)\/ $2$-isomorphisms}, rather than
strictly. A simple example of a commutative diagram in a 2-category
$\bs{\cal C}$ is
\begin{equation*}
\xymatrix@C=50pt@R=8pt{ & Y \ar[dr]^g \ar@{=>}[d]^\eta \\
X \ar[ur]^f \ar[rr]_h && Z, }
\end{equation*}
which means that $X,Y,Z$ are objects of $\bs{\cal C}$, $f:X\ra Y$,
$g:Y\ra Z$ and $h:X\ra Z$ are 1-morphisms in $\bs{\cal C}$, and
$\eta:g\ci f\Ra h$ is a 2-isomorphism. We define {\it fibre
products\/} in 2-categories, following \cite[\S 3, Def.~B.13]{BEFF}.
\begin{dfn} Let $\bs{\cal C}$ be a 2-category and $f:X\ra Z$,
$g:Y\ra Z$ be 1-morphisms in $\bs{\cal C}$. A {\it fibre product\/}
$X\t_ZY$ in $\bs{\cal C}$ consists of an object $W$, 1-morphisms
$\pi_X:W\ra X$ and $\pi_Y:W\ra Y$ and a 2-isomorphism
$\eta:f\ci\pi_X\Ra g\ci\pi_Y$ in $\bs{\cal C}$ with the following
universal property: suppose $\pi_X':W'\ra X$ and $\pi_Y':W'\ra Y$
are 1-morphisms and $\eta':f\ci\pi_X'\Ra g\ci\pi_Y'$ is a
2-isomorphism in $\bs{\cal C}$. Then there should exist a 1-morphism
$h:W'\ra W$ and 2-isomorphisms $\ze_X:\pi_X\ci h\Ra\pi_X'$,
$\ze_Y:\pi_Y\ci h\Ra\pi_Y'$ such that the following diagram of
2-isomorphisms commutes:
\begin{equation*}
\xymatrix@C=50pt@R=11pt{ f\ci\pi_X\ci h \ar@{=>}[r]_{\eta*\id_h}
\ar@{=>}[d]_{\id_f*\ze_X} & g\ci\pi_Y\ci h
\ar@{=>}[d]^{{}\,\id_g*\ze_Y}
\\ f\ci\pi_X' \ar@{=>}[r]^{\eta'} & g\ci\pi_Y'.}
\end{equation*}
Furthermore, if $\ti h,\ti\ze_X,\ti\ze_Y$ are alternative choices of
$h,\ze_X,\ze_Y$ then there should exist a unique 2-isomorphism
$\th:\ti h\ra h$ with
\begin{equation*}
\ti\ze_X=\ze_X\od(\id_{\pi_X}*\th)\quad\text{and}\quad
\ti\ze_Y=\ze_Y\od(\id_{\pi_Y}*\th).
\end{equation*}
Here we write $g\ci f:X\ra Z$ for the {\it horizontal composition
of\/ $1$-morphisms\/} $f:X\ra Y$, $g:Y\ra Z$, and $\ze\od\eta:f\Ra
h$ for the {\it vertical composition of\/ $2$-morphisms\/}
$\eta:f\Ra g$, $\ze:g\Ra h$, and $\ze*\eta:g\ci f\Ra g'\ci f'$ for
the {\it horizontal composition of\/ $2$-morphisms\/} $\eta:f\Ra
f'$, $\ze:g\ra g'$ in $\bs{\cal C}$. If a fibre product $X\t_ZY$ in
$\bs{\cal C}$ exists then it is unique up to equivalence. Given a
fibre product in $\bs{\cal C}$ as above, the following is called a
2-{\it Cartesian diagram\/}:
\begin{equation*}
\xymatrix@C=50pt@R=7pt{ W \ar[r]_(0.25){\pi_Y} \ar[d]_{\pi_X}
\drtwocell_{}\omit^{}\omit{^\eta} & Y \ar[d]^g \\
X \ar[r]^(0.7)f & Z.}
\end{equation*}
In the 2-category $\Sta_{({\cal C},{\cal J})}$ of stacks on a site
$({\cal C},{\cal J})$, there is a natural construction of fibre
products: if ${\cal X},{\cal Y},{\cal Z}$ are stacks and $F:{\cal
X}\ra{\cal Z}$, $G:{\cal Y}\ra{\cal Z}$ are 1-morphisms, define a
category $\cal W$ to have objects $(X,Y,\al)$, where $X\in{\cal X}$,
$Y\in{\cal Y}$ and $\al:F(X)\ra G(Y)$ is an isomorphism in $\cal Z$
with $p_{\cal X}(X)=p_{\cal Y}(Y)=U$ and $p_{\cal X}(\al)=\id_U$ in
$\cal C$, and for objects $(X_1,Y_1,\al_1),(X_2,Y_2,\al_2)$ in $\cal
W$ a morphism $(f,g):(X_1,Y_1,\al_1)\ra(X_2,Y_2,\al_2)$ in $\cal W$
is a pair of morphisms $f:X_1\ra X_2$ in $\cal X$ and $g:Y_1\ra Y_2$
in $\cal Y$ with $p_{\cal X}(f)=p_{\cal Y}(g)=\vp:U\ra V$ in $\cal
C$ and $\al_2\ci F(f)=G(g)\ci\al_1:F(X_1)\ra G(Y_2)$ in $\cal Z$.
Then $\cal W$ is a stack over $({\cal C},{\cal J})$.
Define 1-morphisms $p_{\cal W}:{\cal W}\ra{\cal C}$ by $p_{\cal
W}:(X,Y,\al)\mapsto p_{\cal X}(X)$ and $p_{\cal W}:(f,g)\mapsto
p_{\cal X}(f)$, and $\pi_{\cal X}:{\cal W}\ra{\cal X}$ by $\pi_{\cal
X}:(X,Y,\al)\mapsto X$ and $\pi_{\cal X}:(f,g)\mapsto f$, and
$\pi_{\cal Y}:{\cal W}\ra{\cal Y}$ by $\pi_{\cal Y}:(X,Y,\al)\mapsto
Y$ and $\pi_{\cal Y}:(f,g)\mapsto g$. Define a 2-morphism
$\eta:F\ci\pi_{\cal X}\Ra G\ci\pi_{\cal Y}$ by $\eta(X,Y,\al)=\al$.
Then ${\cal W},\pi_{\cal X},\pi_{\cal Y},\eta$ is a fibre product
${\cal X}\t_{\cal Z}{\cal Y}$ in~$\Sta_{({\cal C},{\cal J})}$.
The functor $\id_{\cal C}:{\cal C}\ra{\cal C}$ is a {\it terminal
object\/} in $\Sta_{({\cal C},{\cal J})}$, and may be thought of as
a point $*$. {\it Products\/} ${\cal X}\t{\cal Y}$ in $\Sta_{({\cal
C},{\cal J})}$ are fibre products over $*$. If $\cal X$ is a stack,
the {\it diagonal\/ $1$-morphism\/} is the natural 1-morphism
$\De_{\cal X}:{\cal X}\ra{\cal X}\t{\cal X}$. The {\it inertia
stack\/} $I_{\cal X}$ of $\cal X$ is the fibre product ${\cal
X}\t_{\De_{\cal X},{\cal X}\t{\cal X},\De_{\cal X}}{\cal X}$, with
natural {\it inertia\/ $1$-morphism\/} $\io_{\cal X}:I_{\cal
X}\ra{\cal X}$ from projection to the first factor of $\cal X$. Then
we have a 2-Cartesian diagram in~$\Sta_{({\cal C},{\cal J})}$:
\e
\begin{gathered}
\xymatrix@C=45pt@R=7pt{ I_{\cal X} \ar[r] \ar[d]_{\io_{\cal X}}
\drtwocell_{}\omit^{}\omit{^{}} & {\cal X} \ar[d]^{\De_{\cal X}} \\
{\cal X} \ar[r]^(0.25){\De_{\cal X}} & {\cal X}\t{\cal X}.}
\end{gathered}
\label{ag7eq4}
\e
There is also a natural 1-morphism $\jmath_{\cal X}:{\cal X}\ra
I_{\cal X}$ induced by the 1-morphism $\id_{\cal X}$ from $\cal X$
to the two factors $\cal X$ in $I_{\cal X}={\cal X}\t_{{\cal
X}\t{\cal X}}{\cal X}$ and the identity 2-morphism on $\De_{\cal
X}\ci\id_{\cal X}:{\cal X}\ra{\cal X}\t{\cal X}$.
\label{ag7def6}
\end{dfn}
\subsection{Descent theory on a site}
\label{ag73}
The theory of descent in algebraic geometry, due to Grothendieck,
says that objects and morphisms over a scheme $U$ can be described
locally on an open cover $\{U_i:i\in I\}$ of $U$. It is described by
Behrend et al.\ \cite[App.~A]{BEFF}, and at length by Vistoli
\cite{Vist}. We shall express descent as conditions on a general
site~$({\cal C},{\cal J})$.
\begin{dfn} Let $({\cal C},{\cal J})$ be a site. We say that
$({\cal C},{\cal J})$ {\it has descent for objects\/} if whenever
$\{\vp_a:U_a\ra U\}_{a\in A}$ is a covering in $\cal J$ and we are
given morphisms $f_a:X_a\ra U_a$ in $\cal C$ for all $a\in A$ and
isomorphisms $g_{ab}:X_a\t_{\vp_a\ci f_a,U,\vp_b}U_b\ra
X_b\t_{\vp_b\ci f_b,U,\vp_a}U_a$ in $\cal C$ for all $a,b\in A$ with
$g_{ab}=g_{\smash{ba}}^{\smash{-1}}$ such that for all $a,b,c\in A$
the following diagram commutes:
\begin{gather*}
\xymatrix@R=15pt@C=-60pt{ {\begin{subarray}{l}\ts(X_a\t_{\vp_a\ci
f_a,U,\vp_b}U_b)\t_{\pi_U,U,\vp_c}U_c\cong \\ \ts(X_a\t_{\vp_a\ci
f_a,U,\vp_c}U_c)\t_{\pi_U,U,\vp_b}U_b\end{subarray}}
\ar@<.5ex>[rr]^{g_{ab}\t\id_{U_c}}
\ar@<.5ex>[dr]^(0.65){g_{ac}\t\id_{U_b}} &&
{\begin{subarray}{l}\ts(X_b\t_{\vp_b\ci
f_b,U,\vp_c}U_c)\t_{\pi_U,U,\vp_a}U_a\cong \\ \ts(X_b\t_{\vp_b\ci
f_b,U,\vp_a}U_a)\t_{\pi_U,U,\vp_c}U_c\end{subarray}}
\ar@<.5ex>[dl]^(0.35){{}\,\,g_{bc}\t\id_{U_a}}
\ar@<.5ex>[ll]^{g_{ba}\t\id_{U_c}} \\
& {\begin{subarray}{l}\ts(X_c\t_{\vp_c\ci
f_c,U,\vp_a}U_a)\t_{\pi_U,U,\vp_b}U_b\cong \\ \ts(X_c\t_{\vp_c\ci
f_c,U,\vp_b}U_b)\t_{\pi_U,U,\vp_a}U_a,\end{subarray}}
\ar@<.5ex>[ul]^(0.65){g_{ca}\t\id_{U_b}}
\ar@<.5ex>[ur]^(0.3){g_{cb}\t\id_{U_a}\,\,{}} }
\end{gather*}
then there exist a morphism $f:X\ra U$ in $\cal C$ and isomorphisms
$g_a:X_a\ra X\t_{f,U,\vp_a}U_a$ for all $a\in A$ such that
$f_a=\pi_{U_a}\ci g_a$ and the diagram below commutes for
all~$a,b\in A$:
\begin{equation*}
\xymatrix@C=50pt@R=7pt{X_a\t_{\vp_a\ci f_a,U,\vp_b}U_b
\ar[r]_(0.4){g_a\t\id_{U_b}} \ar[dd]^{g_{ab}} &
(X\t_{f,U,\vp_a}U_a)\t_{\vp_a\ci \pi_{U_a},U,\vp_b}U_b \ar[d]^{\cong} \\
& X\t_{f,U,\pi_U}(U_a\t_{\vp_a,U,\vp_b}U_b) \ar[d]^{\cong} \\
X_b\t_{\vp_b\ci f_b,U,\vp_a}U_a &
(X\t_{f,U,\vp_b}U_b)\t_{\vp_b\ci\pi_{U_b},U,\vp_a}U_a.
\ar[l]_(0.6){g_b^{-1}\t\id_{U_a}}}
\end{equation*}
Furthermore $X,f$ should be unique up to canonical isomorphism. Note
that all the fibre products used above exist in $\cal C$ by
Definition~\ref{ag7def1}(iii).
\label{ag7def7}
\end{dfn}
\begin{dfn} Let $({\cal C},{\cal J})$ be a site. We say that
$({\cal C},{\cal J})$ {\it has descent for morphisms\/} if whenever
$\{\vp_a:U_a\ra U\}_{a\in A}$ is a covering in $\cal J$ and $f:X\ra
U$, $g:Y\ra U$ and $h_a:X\t_{f,U,\vp_a}U_a\ra Y\t_{g,U,\vp_a}U_a$
for all $a\in A$ are morphisms in $\cal C$ with $\pi_{U_a}\ci
h_a=\pi_{U_a}$ and for all $a,b\in A$ the following diagram
commutes:
\begin{equation*}
\xymatrix@C=30pt@R=7pt{
(X\t_{f,U,\vp_a}U_a)\t_{\vp_a\ci \pi_{U_a},U,\vp_b}U_b
\ar[d]^{\cong} \ar[r]_{h_a\t\id_{U_b}} &
(Y\t_{g,U,\vp_a}U_a)\t_{\vp_a\ci \pi_{U_a},U,\vp_b}U_b
\ar[d]^{\cong} \\
X\t_{f,U,\pi_U}(U_a\t_{\vp_a,U,\vp_b}U_b)
\ar[d]^{\cong} & Y\t_{g,U,\pi_U}(U_a\t_{\vp_a,U,\vp_b}U_b)
\ar[d]^{\cong} \\
(X\t_{f,U,\vp_b}U_b)\t_{\vp_b\ci\pi_{U_b},U,\vp_a}U_a
\ar[r]^{h_b\t\id_{U_a}} & (Y\t_{g,U,\vp_b}U_b)
\t_{\vp_b\ci\pi_{U_b},U,\vp_a}U_a,}
\end{equation*}
then there exists a unique $h:X\ra Y$ in $\cal C$ with
$h_a=h\t\id_{U_a}$ for all~$a\in A$.
\label{ag7def8}
\end{dfn}
Then \cite[Prop.s A.12, A.13 \& \S A.6]{BEFF} show that descent
holds for objects and morphisms for affine schemes with the fppf
topology, but for arbitrary schemes with the fppf topology, descent
holds for morphisms and fails for objects.
\subsection{Properties of 1-morphisms}
\label{ag74}
Objects $V$ in $\cal C$ yield stacks $\bar V$ on~$({\cal C},{\cal
J})$.
\begin{dfn} Let $({\cal C},{\cal J})$ be a site, and $V$ an object
of ${\cal C}$. Define a category $\bar V$ to have objects $(U,\th)$
where $U\in{\cal C}$ and $\th:U\ra V$ is a morphism in $\cal C$, and
to have morphisms $\psi:(U_1,\th_1)\ra (U_2,\th_2)$ where
$\psi:U_1\ra U_2$ is a morphism in $\cal C$ with
$\th_2\ci\psi=\th_1:U_1\ra V$. Define a functor $p_{\bar V}:\bar
V\ra{\cal C}$ by $p_{\bar V}:(U,\th)\mapsto U$ and $p_{\bar
V}:\psi\mapsto\psi$. Note that $p_{\bar V}$ is {\it injective on
morphisms}. It is then automatic that $p_{\bar V}:\bar V\ra{\cal C}$
is a category fibred in groupoids, since in \eq{ag7eq1} we can take
$g=g'$. It is also automatic that $p_{\bar V}:\bar V\ra{\cal C}$ is
a prestack, since in \eq{ag7eq2} we must have $X_a=Y_a=(U_a,\th_a)$,
$x_a=y_a=\vp_a$, $X=Y=(U,\th)$, etc., and the unique solution for
$f$ is~$f=\id_U$.
The site $({\cal C},{\cal J})$ is called {\it subcanonical\/} if
$\bar V$ is a stack for all objects $V\in{\cal C}$. If descent for
morphisms holds for $({\cal C},{\cal J})$ then $({\cal C},{\cal J})$
is subcanonical. Most interesting sites are subcanonical. Suppose
$({\cal C},{\cal J})$ is a subcanonical site. If $f:V\ra W$ is a
morphism in $\cal C$, define a 1-morphism $\bar f:\bar V\ra\bar W$
in $\Sta_{({\cal C},{\cal J})}$ by $\bar f:(U,\th)\mapsto
(U,f\ci\th)$ and $\bar f:\psi\mapsto\psi$. Then the (2-)functor
$V\mapsto\bar V$, $f\mapsto\bar f$ embeds $\cal C$ as a full
discrete 2-subcategory of~$\Sta_{({\cal C},{\cal J})}$.
\label{ag7def9}
\end{dfn}
\begin{dfn} Let $({\cal C},{\cal J})$ be a subcanonical site. A
stack $\cal X$ over $({\cal C},{\cal J})$ is called {\it
representable\/} if it is equivalent in $\Sta_{({\cal C},{\cal J})}$
to a stack of the form $\bar V$ for some $V\in{\cal C}$. A
1-morphism $F:{\cal X}\ra{\cal Y}$ in $\Sta_{({\cal C},{\cal J})}$
is called {\it representable\/} if for all $V\in{\cal C}$ and all
1-morphisms $G:\bar V\ra{\cal Y}$, the fibre product ${\cal
X}\t_{F,{\cal Y},G}\bar V$ in $\Sta_{({\cal C},{\cal J})}$ is a
representable stack.
\label{ag7def10}
\end{dfn}
\begin{rem} For stacks in algebraic geometry, one often takes a
different definition of representable objects and 1-morphisms:
$({\cal C},{\cal J})$ is a category of schemes with the \'etale
topology, but stacks are called representable if they are equivalent
to an {\it algebraic space\/} rather than a scheme. This is because
schemes are not general enough for some purposes, e.g.\ the quotient
of a scheme by an \'etale equivalence relation may be an algebraic
space but not a scheme.
In our situation, we will have no need to enlarge $C^\iy$-schemes to
some category of `$C^\iy$-algebraic spaces', as $C^\iy$-schemes are
already general enough, e.g.\ the quotient of a locally fair
$C^\iy$-scheme by an \'etale equivalence relation is a locally fair
$C^\iy$-scheme. This is because the natural topology on
$C^\iy$-schemes is much finer than the Zariski or \'etale topology
on schemes, for instance, affine $C^\iy$-schemes are always
Hausdorff.
\label{ag7rem1}
\end{rem}
\begin{dfn} Let $({\cal C},{\cal J})$ be a subcanonical site. Let
$\bs P$ be a property of morphisms in $\cal C$. (For instance, if
$\cal C$ is the category $\Top$ of topological spaces, then $\bs P$
could be `proper', `open', `surjective', `covering map', \ldots). We
say that $\bs P$ is {\it invariant under base change\/} if for all
Cartesian squares in $\cal C$
\begin{equation*}
\xymatrix@C=30pt@R=10pt{ W \ar[r]_f \ar[d]^e & Y \ar[d]_h \\
X \ar[r]^g & Z,}
\end{equation*}
if $g$ is $\bs P$, then $f$ is $\bs P$. We say that $\bs P$ is {\it
local on the target\/} if whenever $f:U\ra V$ is a morphism in $\cal
C$ and $\{\vp_a:V_a\ra V\}_{a\in A}$ is a covering in $\cal J$ such
that $\pi_{V_a}:U\t_{f,V,\vp_a}V_a\ra V_a$ is $\bs P$ for all $a\in
A$, then $f$ is~$\bs P$.
Let $\bs P$ be invariant under base change and local in the target,
and let $F:{\cal X}\ra{\cal Y}$ be a representable 1-morphism in
$\Sta_{({\cal C},{\cal J})}$. If $W\in{\cal C}$ and $G:\bar
W\ra{\cal Y}$ is a 1-morphism then ${\cal X}\t_{F,{\cal Y},G}\bar W$
is equivalent to $\bar V$ for some $V\in{\cal C}$, and under this
equivalence the 1-morphism $\pi_{\bar W}:{\cal X}\t_{F,{\cal
Y},G}\bar W\ra\bar W$ is 2-isomorphic to $\bar f:\bar V\ra\bar W$
for some unique morphism $f:V\ra W$ in $\cal C$. We say that $F$
{\it has property\/} $\bs P$ if for all $W\in{\cal C}$ and
1-morphisms $G:\bar W\ra{\cal Y}$, the morphism $f:V\ra W$ in $\cal
C$ corresponding to $\pi_{\bar W}:{\cal X}\t_{F,{\cal Y},G}\bar
W\ra\bar W$ has property~$\bs P$.
\label{ag7def11}
\end{dfn}
We define {\it surjective\/} 1-morphisms without requiring them
representable.
\begin{dfn} Let $({\cal C},{\cal J})$ be a site, and $F:{\cal
X}\ra{\cal Y}$ be a 1-morphism in $\Sta_{({\cal C},{\cal J})}$. We
call $F$ {\it surjective\/} if whenever $Y\in{\cal Y}$ with $p_{\cal
Y}(Y)=U\in{\cal C}$, there exists a covering $\{\vp_a:U_a\ra
U\}_{a\in A}$ in $\cal J$ such that for all $a\in A$ there exists
$X_a\in{\cal X}$ with $p_{\cal X}(X_a)=U_a$ and a morphism
$g_a:F(X_a)\ra Y$ in $\cal Y$ with~$p_{\cal Y}(g_a)=\vp_a$.
\label{ag7def12}
\end{dfn}
Following \cite[Prop.~3.8.1, Lem.~4.3.3 \& Rem.~4.14.1]{LaMo},
\cite[\S 6]{Nooh}, we may prove:
\begin{prop} Let\/ $({\cal C},{\cal J})$ be a subcanonical site, and
\e
\begin{gathered}
\xymatrix@C=40pt@R=7pt{ {\cal W} \ar[r]_(0.25){f} \ar[d]_{e}
\drtwocell_{}\omit^{}\omit{^\eta} & {\cal Y} \ar[d]^h \\ {\cal X}
\ar[r]^(0.7)g & {\cal Z}}
\end{gathered}
\label{ag7eq6}
\e
be a $2$-Cartesian square in $\Sta_{({\cal C},{\cal J})}$. Let\/
$\bs P$ be a property of morphisms in $\cal C$ which is invariant
under base change and local in the target. Then:
\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item[{\rm(a)}] If\/ $h$ is representable, then\/ $e$ is
representable. If also $h$ is\/ $\bs P,$ then\/ $e$ is\/~$\bs
P$.
\item[{\rm(b)}] If\/ $g$ is surjective, then\/ $f$ is surjective.
\end{itemize}
Now suppose also that\/ $({\cal C},{\cal J})$ has descent for
objects and morphisms, and that\/ $g$ (and hence\/ $f$) is
surjective. Then:
\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item[{\rm(c)}] If\/ $e$ is surjective then\/ $h$ is
surjective, and if\/ $e$ is representable, then\/ $h$ is
representable, and if also $e$ is\/ $\bs P,$ then\/ $h$
is\/~$\bs P$.
\end{itemize}
\label{ag7prop2}
\end{prop}
\subsection{Geometric stacks, and stacks associated to groupoids}
\label{ag75}
The 2-category $\Sta_{({\cal C},{\cal J})}$ of all stacks over a
site $({\cal C},{\cal J})$ is usually too general to do geometry
with. To obtain a smaller 2-category whose objects have better
properties, we impose extra conditions on a stack~$\cal X$:
\begin{dfn} Let $({\cal C},{\cal J})$ be a site. We call a stack
$\cal X$ on $({\cal C},{\cal J})$ {\it geometric\/} if the diagonal
1-morphism $\De_{\cal X}:{\cal X}\ra{\cal X}\t{\cal X}$ is
representable, and there exists $U\in{\cal C}$ and a surjective
1-morphism $\Pi:\bar U\ra{\cal X}$, which we call an {\it atlas} for
$\cal X$. Write $\GSta_{({\cal C},{\cal J})}$ for the full
2-subcategory of geometric stacks in $\Sta_{({\cal C},{\cal J})}$.
Here $\De_{\cal X}$ representable is equivalent to $\Pi$
representable, as $\Pi$ is surjective.
\label{ag7def13}
\end{dfn}
To obtain nice classes of stacks, one usually requires further
properties $\bs P$ of $\De_{\cal X}$ and $\Pi$. For example, in
algebraic geometry with $({\cal C},{\cal J})$ schemes with the
\'etale topology, we assume $\De_{\cal X}$ is quasicompact and
separated, and $\Pi$ is \'etale for Deligne--Mumford stacks $\cal
X$, and $\Pi$ is smooth for Artin stacks~$\cal X$.
The following material is based on Metzler \cite[\S 3.1 \& \S
3.3]{Metz}, Laumon and Moret-Bailly \cite[\S\S 2.4.3, 3.4.3, 3.8,
4.3]{LaMo}, and Lerman~\cite[\S 4.4]{Lerm}.
We can characterize geometric stacks $\cal X$ up to equivalence
solely in terms of objects and morphisms in $\cal C$, using the idea
of {\it groupoid objects\/} in~$\cal C$.
\begin{dfn} A {\it groupoid object\/} $(U,V,s,t,u,i,m)$ in a
category $\cal C$, or simply {\it groupoid\/} in $\cal C$, consists
of objects $U,V$ in $\cal C$ and morphisms $s,t:V\ra U$, $u:U\ra V$,
$i:V\ra V$ and $m:V\t_{s,U,t}V\ra V$ satisfying the identities
\e
\begin{gathered} s\ci u=t\ci u=\id_{U},\;\> s\ci i=t,\;\> t\ci
i=s,\;\> s\ci m=s\ci\pi_2,\;\> t\ci m=t\ci\pi_1,\\
m\ci (i\t\id_{V})=u\ci s,\;\> m\ci (\id_{V}\t i)=u\ci t,\\
m\ci (m\t\id_{V})= m\ci(\id_{V}\t m):
V\t_{U}V\t_{U}V\longra V,\\
m\ci(\id_{V}\t u)=m\ci(u\t\id_{V}):V=V\t_{U}U\longra V,
\end{gathered}
\label{ag7eq5}
\e
where we suppose all the fibre products exist.
Groupoids in $\cal C$ are so called because a groupoid in $\Sets$ is
a groupoid in the usual sense, that is, a category with invertible
morphisms, where $U$ is the set of {\it objects}, $V$ the set of
{\it morphisms}, $s:V\ra U$ the {\it source\/} of a morphism,
$t:V\ra U$ the {\it target\/} of a morphism, $u:U\ra V$ the {\it
unit\/} taking $X\mapsto\id_X$, $i$ the {\it inverse\/} taking
$f\mapsto f^{-1}$, and $m$ the {\it multiplication\/} taking
$(f,g)\mapsto f\ci g$ when $s(f)=t(g)$. Then \eq{ag7eq5} reduces to
the usual axioms for a groupoid.
\label{ag7def14}
\end{dfn}
From a geometric stack with an atlas, we can construct a groupoid
in~$\cal C$.
\begin{dfn} Let $({\cal C},{\cal J})$ be a subcanonical site, and
suppose $\cal X$ is a geometric stack on $({\cal C},{\cal J})$ with
atlas $\Pi:\bar U\ra{\cal X}$. Then $\bar U\t_{\Pi,{\cal X},\Pi}\bar
U$ is equivalent to $\bar V$ for some $V\in{\cal C}$ as $\Pi$ is
representable. Hence we can take $\bar V$ to be the fibre product,
and we have a 2-Cartesian square
\e
\begin{gathered}
\xymatrix@C=42pt@R=7pt{ \bar V \ar[r]_(0.25){\bar t} \ar[d]_{\bar s}
\drtwocell_{}\omit^{}\omit{^\eta} & \bar U
\ar[d]^\Pi \\
\bar U \ar[r]^(0.7)\Pi & {\cal X}}
\end{gathered}
\label{ag7eq7}
\e
in $\Sta_{({\cal C},{\cal J})}$. Here as $({\cal C},{\cal J})$ is
subcanonical, any 1-morphism $\bar V\ra\bar U$ in $\Sta_{({\cal
C},{\cal J})}$ is 2-isomorphic to $\bar f$ for some unique morphism
$f:V\ra U$ in $\cal C$. Thus we may write the projections in
\eq{ag7eq7} as $\bar s,\bar t$ for some unique $s,t:V\ra U$ in~$\cal
C$.
By the universal property of fibre products there exists a
1-morphism $H:\bar U\ra\bar V$ unique up to 2-isomorphism with $\bar
s\ci H\cong\id_{\bar U}\cong \bar t\ci H$. This $H$ is 2-isomorphic
to $\bar u:\bar U\ra\bar V$ for some unique morphism $u:U\ra V$ in
$\cal C$, and then $s\ci u=t\ci u=\id_U$. Similarly, exchanging the
two factors of $U$ in the fibre product we obtain a unique morphism
$i:V\ra V$ in $\cal C$ with $s\ci i=t$ and $t\ci i=s$. In
$\Sta_{({\cal C},{\cal J})}$ we have equivalences
\begin{equation*}
\overline{V\t_{s,U,t}V}\simeq \bar V\t_{\bar s,\bar U,\bar t}\bar V
\simeq (\bar U\t_{\cal X}\bar U)\t_{\bar U}(\bar U\t_{\cal X}\bar U)
\simeq \bar U\t_{\cal X}\bar U\t_{\cal X}\bar U.
\end{equation*}
Let $m:V\t_{s,U,t}V\ra V$ be the unique morphism in $\cal C$ such
that $\bar m$ is 2-isomorphic to the projection
$\overline{V\t_{s,U,t}V}\ra\bar V=\bar U\t_{\cal X}\bar U$
corresponding to projection to the first and third factors of $\bar
U$ in the final fibre product. It is now not difficult to verify
that $(U,V,s,t,u,i,m)$ is a groupoid in~$\cal C$.
\label{ag7def15}
\end{dfn}
Conversely, given a groupoid in $\cal C$ we can construct a
stack~$\cal X$.
\begin{dfn} Let $({\cal C},{\cal J})$ be a site with descent for
morphisms, and $(U,\ab V,\ab s,\ab t,\ab u,\ab i,m)$ be a groupoid
in $\cal C$. Define a prestack $\cal X'$ on $({\cal C},{\cal J})$ as
follows: let $\cal X'$ be the category whose objects are pairs
$(T,f)$ where $f:T\ra U$ is a morphism in $\cal C$, and morphisms
are $(p,q):(T_1,f_1)\ra (T_2,f_2)$ where $p:T_1\ra T_2$ and
$q:T_1\ra V$ are morphisms in $\cal C$ with $f_1=s\ci q$ and $f_2\ci
p=t\ci q$. Given morphisms $(p_1,q_1):(T_1,f_1)\ra (T_2,f_2)$ and
$(p_2,q_2):(T_2,f_2)\ra (T_3,f_3)$ the composition is $(p_2,q_2)\ci
(p_1,q_1)=\bigl(p_2\ci p_1,m\ci (q_1\t (q_2\ci p_2))\bigr)$, where
$q_1\t(q_2\ci p_2):T_1\ra V\t_{t,U,s}V$ is induced by the morphisms
$q_1:T_1\ra V$ and $q_2\ci p_2:T_1\ra V$, which satisfy~$t\ci
q_1=f_2\ci p_1=s\ci(q_2\ci p_2)$.
Define a functor $p_{{\cal X}'}:{\cal X}'\ra{\cal C}$ by $p_{{\cal
X}'}:(T,f)\mapsto T$ and $p_{{\cal X}'}:(p,q)\mapsto p$. Using the
groupoid axioms \eq{ag7eq5} we can show that $p_{{\cal X}'}:{\cal
X}'\ra{\cal C}$ is a category fibred in groupoids. Since $({\cal
C},{\cal J})$ has descent for morphisms, we can also show ${\cal
X}'$ is a prestack. But in general it is not a stack. Let $\cal X$
be the associated stack from Proposition \ref{ag7prop1}. We call
$\cal X$ the {\it stack associated to the groupoid\/}
$(U,V,s,t,u,i,m)$. It fits into a natural 2-commutative diagram
\eq{ag7eq7}.
\label{ag7def16}
\end{dfn}
Groupoids in $\cal C$ are often written $V\rra U$, to emphasize
$s,t:V\ra U$, leaving $u,i,m$ implicit. The associated stack is then
written as~$[V\rra U]$.
Our next theorem is proved by Metzler \cite[Prop.~70]{Metz} when
$({\cal C},{\cal J})$ is the site of topological spaces with open
covers, but examining the proof shows that all he uses about $({\cal
C},{\cal J})$ is that fibre products exist in $\cal C$ and $({\cal
C},{\cal J})$ has descent for objects and morphisms. See also Lerman
\cite[Prop.~4.31]{Lerm}. If fibre products may not exist in $\cal C$
then one must also require the morphisms $s,t$ in $(U,V,s,t,u,i,m)$
to be {\it representable\/} in $\cal C$, that is, for all $f:T\ra U$
in $\cal C$ the fibre products $T_{f,U,s}V$ and $T_{f,U,t}V$ exist
in~$\cal C$.
\begin{thm} Let\/ $({\cal C},{\cal J})$ be a site, and suppose that
all fibre products exist in $\cal C,$ and that descent for objects
and morphisms holds in\/ $({\cal C},{\cal J})$. Then the
constructions of Definitions\/ {\rm\ref{ag7def15}} and\/
{\rm\ref{ag7def16}} are inverse. That is, if\/ $(U,V,s,t,u,i,m)$ is
a groupoid in\/ $\cal C$ and\/ $\cal X$ is the associated stack,
then $\cal X$ is a geometric stack, and the $2$-commutative diagram
\eq{ag7eq7} is $2$-Cartesian, and\/ $\Pi$ in \eq{ag7eq7} is
surjective and so an atlas for $\cal X,$ and\/ $(U,V,s,t,u,i,m)$ is
canonically isomorphic to the groupoid constructed in Definition\/
{\rm\ref{ag7def15}} from the atlas\/ $\Pi:\bar U\ra{\cal X}$.
Conversely, if\/ $\cal X$ is a geometric stack with atlas\/
$\Pi:\bar U\ra{\cal X},$ and\/ $(U,V,s,t,u,i,m)$ is the groupoid
in\/ $\cal C$ constructed from $\Pi$ in Definition\/
{\rm\ref{ag7def15},} and\/ $\ti{\cal X}$ is the stack associated to
$(U,V,s,t,u,i,m)$ in Definition\/ {\rm\ref{ag7def16},} then\/ $\cal
X$ is equivalent to\/ $\ti{\cal X}$ in\/ $\Sta_{({\cal C},{\cal
J})}$. Thus every geometric stack is equivalent to a groupoid stack.
\label{ag7thm}
\end{thm}
In the situation of Theorem \ref{ag7thm} we have 2-Cartesian
diagrams
\e
\begin{gathered}
\xymatrix@C=10pt@R=7pt{ \bar V \ar[rrr]_(0.7){\bar t} \ar[d]_{\bar
s} \drrtwocell_{}\omit^{}\omit{^{}} &&& \bar U \ar[d]^\Pi & \bar V
\ar[rrr]_(0.7){\Pi\ci\bar s} \ar[d]_{\bar s\t\bar t}
\drrtwocell_{}\omit^{}\omit{^{}} &&& {\cal X} \ar[d]^{\De_{\cal X}} \\
\bar U \ar[rrr]^(0.7)\Pi &&& {\cal X}, & \bar U\t\bar U
\ar[rrr]^(0.7){\Pi\t\Pi} &&& {\cal X}\t{\cal X},\\
\!\bar V\!\t_{\bar s\t\bar t,\bar U\t\bar U,\De_{\bar U}}\!\bar U\!
\ar[rrr]_(0.7){\Pi\ci\bar s\t \Pi\ci\bar t} \ar[d]_{\pi_{\bar U}}
\drrtwocell_{}\omit^{}\omit{^{}} &&& I_{\cal X} \ar[d]^{\io_{\cal
X}} & \bar U \ar[d]_{\bar u\t\bar\id_U} \ar[rrr]_(0.7){\Pi}
\drrtwocell_{}\omit^{}\omit{^{}}
&&& {\cal X} \ar[d]^{\jmath_{\cal X}} \\
\bar U \ar[rrr]^(0.7){\Pi} &&& {\cal X}, & \bar V\!\t_{\bar s\t\bar
t,\bar U\t\bar U,\De_{\bar U}}\!\bar U \ar[rrr]^(0.7){\Pi\ci\bar s\t
\Pi\ci\bar t} &&& I_{\cal X}, }
\end{gathered}
\label{ag7eq8}
\e
with surjective rows. So from Proposition \ref{ag7prop2} we deduce:
\begin{cor} In the situation of Theorem\/ {\rm\ref{ag7thm},} let\/
$\bs P$ be a property of morphisms in $\cal C$ which is invariant
under base change and local in the target. Then\/ $\bar\Pi:\bar
U\ra{\cal X}$ is $\bs P$ if and only if\/ $s:V\ra U$ is $\bs P,$
and\/ $\De_{\cal X}:{\cal X}\ra{\cal X}\t{\cal X}$ is\/ $\bs P$ if
and only if\/ $s\times t:V\ra U\t U$ is\/ $\bs P,$ and\/ $\io_{\cal
X}:I_{\cal X}\ra{\cal X}$ is $\bs P$ if and only if\/
$\pi_U:V\t_{s\times t,U\t U,\De_U}U\ra U$ is\/ $\bs P,$ and\/
$\jmath_{\cal X}:{\cal X}\ra I_{\cal X}$ is $\bs P$ if and only if\/
$u\t\id_U:U\ra V\t_{s\times t,U\t U,\De_U}U$ is\/~$\bs P$.
\label{ag7cor1}
\end{cor}
We can describe atlases for fibre products of geometric stacks.
\begin{ex} Suppose $({\cal C},{\cal J})$ is a subcanonical site,
and all fibre products exist in $\cal C$. Let
\e
\begin{gathered}
\xymatrix@C=40pt@R=7pt{ {\cal W} \ar[r]_(0.2){f} \ar[d]_{e}
\drtwocell_{}\omit^{}\omit{^\eta} & {\cal Y} \ar[d]^h \\ {\cal X}
\ar[r]^(0.7)g & {\cal Z}}
\end{gathered}
\label{ag7eq9}
\e
be a 2-Cartesian diagram in $\Sta_{({\cal C},{\cal J})}$, where
${\cal X},{\cal Y},{\cal Z}$ are geometric stacks. Let $\Pi_{\cal
X}:\bar U_{\cal X}\ra{\cal X}$ and $\Pi_{\cal Y}:\bar U_{\cal
Y}\ra{\cal Y}$ be atlases. As $\De_{\cal Z}$ is representable the
fibre product $\bar U_{\cal X}\t_{g\ci\Pi_{\cal X},{\cal
Z},h\ci\Pi_{\cal Y}}\bar U_{\cal Y}$ is represented by an object
$U_{\cal W}$ of $\cal C$. Then we have a 2-commutative diagram,
where we omit 2-morphisms:
\e
\begin{gathered}
\xymatrix@R=2pt@C=15pt{ & \bar U_W:=\bar U_{\cal X}\t_{\cal Z}\bar
U_{\cal Y} \ar@/^2pc/@(ur,ur)@{..>}[rrrd]^{\Pi_{\cal W}}
\ar[rr]^{\pi_1} \ar[dl] \ar[ddr] && {\cal X}\t_{\cal Z}\bar U_{\cal
Y} \ar[dl] \ar[ddl] \ar[dr]^(0.6){\pi_2} \\
\bar U_{\cal X} \ar@<1ex>[ddr] \ar[rr]^{\Pi_{\cal X}} && {\cal X}
\ar[ddl]_g && {\cal W} \ar[ll]^e \ar[ddl]^f \\
&& \bar U_{\cal Y} \ar[dl] \ar[dr]^{\Pi_{\cal Y}} \\
& {\cal Z} && {\cal Y} \ar[ll]_h }
\end{gathered}
\label{ag7eq10}
\e
Here the five squares in \eq{ag7eq10} are 2-Cartesian, although the
triangles are not. Define $\Pi_{\cal W}=\pi_2\ci\pi_1:\bar U_{\cal
W}\ra{\cal W}$, where $\pi_1,\pi_2$ are as in \eq{ag7eq10}.
Proposition \ref{ag7prop2}(a),(b) imply that $\pi_1,\pi_2$ are
representable and surjective, since $\Pi_{\cal X},\Pi_{\cal Y}$ are.
Hence $\Pi_{\cal W}=\pi_2\ci\pi_1$ is also representable and
surjective, so $\cal W$ is a {\it geometric stack}, and $\Pi_{\cal
W}$ is an {\it atlas\/} for $\cal W$. In the same way, if $\bs P$ is
a property of morphisms in $\cal C$ which is invariant under base
change and local in the target and closed under compositions, and
$\Pi_{\cal X},\Pi_{\cal Y}$ are $\bs P$, then $\Pi_{\cal W}$ is~$\bs
P$.
Now let $\bar V_{\cal W}=\bar U_{\cal W}\t_{\cal W}\bar U_{\cal W}$
and complete to a groupoid $(U_{\cal W},V_{\cal W},\ab s_{\cal
W},\ab t_{\cal W},\ab u_{\cal W},\ab i_{\cal W},\ab m_{\cal W})$ in
$\cal C$ as above, with ${\cal W}\simeq[V_{\cal W}\rra U_{\cal W}]$,
and do the same for ${\cal X},{\cal Y}$. Then by a diagram chase
similar to \eq{ag7eq10} we can show that
\e
\bar V_{\cal W}\cong \bar V_{\cal X}\t_{\cal Z}\bar V_{\cal
Y}\quad\text{and}\quad V_{\cal W}\cong (U_{\cal W}\t_{U_{\cal
X}}V_{\cal X})\t_{U_{\cal Y}}V_{\cal Y}.
\label{ag7eq11}
\e
\label{ag7ex}
\end{ex}
\begin{cor} Suppose\/ $({\cal C},{\cal J})$ is a subcanonical site,
and all fibre products exist in\/ $\cal C$. Then the\/
$2$-subcategory\/ $\GSta_{({\cal C},{\cal J})}$ of geometric stacks
is closed under fibre products in\/~$\Sta_{({\cal C},{\cal J})}$.
\label{ag7cor2}
\end{cor}
\section{$C^\iy$-stacks and orbifolds}
\label{ag8}
We now discuss $C^\iy$-{\it stacks}, that is, geometric stacks over
the site $(\CSch,{\cal J})$ of $C^\iy$-schemes with the open cover
topology. The author knows of no previous work on these. We assume
the background material of \S\ref{ag7}. Our principal interest is in
{\it Deligne--Mumford\/ $C^\iy$-stacks}, which are $C^\iy$-stacks
locally modelled on quotients $[\uU/G]$ for $\uU$ an affine
$C^\iy$-scheme and $G$ a finite group. As we explain in
\S\ref{ag88}, {\it orbifolds\/} (which may be without boundary, or
with boundary, or with corners) may be defined as a 2-subcategory of
Deligne--Mumford $C^\iy$-stacks. The author found Metzler
\cite{Metz} and Noohi \cite{Nooh} useful in writing this section.
\subsection{$C^\iy$-stacks}
\label{ag81}
\begin{dfn} Define a Grothendieck topology $\cal J$ on the category
$\CSch$ of $C^\iy$-schemes to have coverings
$\{\ui_a:\uU_a\ra\uU\}_{a\in A}$ where $V_a=i_a(U_a)$ is open in $U$
with $\ui_a:\uU_a\ra(V_a,\O_U\vert_{V_a})$ and isomorphism for all
$a\in A$, and $U=\bigcup_{a\in A}V_a$. Using Proposition
\ref{ag4prop8} we see that up to isomorphisms of the $\uU_a$, the
coverings $\{\ui_a:\uU_a\ra\uU\}_{a\in A}$ of $\uU$ correspond
exactly to open covers $\{V_a:a\in A\}$ of~$U$.
\label{ag8def1}
\end{dfn}
It is a straightforward exercise in sheaf theory to prove:
\begin{prop} The site $(\CSch,{\cal J})$ has descent for objects
and morphisms. Thus it is subcanonical.
\label{ag8prop1}
\end{prop}
The point here is that since coverings of $\uU$ in $\cal J$ are just
open covers of the underlying topological space $U$, rather than
something more complicated like \'etale covers in algebraic
geometry, proving descent is easy: for objects, we glue the
topological spaces $X_a$ of $\uX_a$ together in the usual way to get
a topological space $X$, then we glue the $\O_{X_a}$ together to get
a presheaf of $C^\iy$-rings $\ti{\cal O}_X$ on $X$ isomorphic to
$\O_{X_a}$ on $X_a\subseteq X$ for all $a\in A$, and finally we
sheafify $\ti{\cal O}_X$ to a sheaf of $C^\iy$-rings $\O_X$ on $X$,
which is still isomorphic to $\O_{X_a}$ on $X_a\subseteq X$.
\begin{dfn} A $C^\iy$-{\it stack\/} is a geometric stack on the
site $(\CSch,{\cal J})$. Write $\CSta$ for the 2-category of
$C^\iy$-stacks, $\CSta=\GSta_{(\CSch,{\cal J})}$. If $\uX$ is a
$C^\iy$-scheme then $\bar\uX$ is a $C^\iy$-stack. Write
$\bCSchlfp,\ab\bCSchlg,\ab\bCSchlf,\ab\bCSch$ for the full
2-subcategories of $C^\iy$-stacks $\cal X$ in $\CSta$ which are
equivalent to $\bar\uX$ for $\uX$ in $\CSchlfp,\ab\CSchlg,
\ab\CSchlf$ or $\CSch$, respectively.
Since $(\CSch,{\cal J})$ is a subcanonical site, the embedding
$\CSch\ra\CSta$ taking $\uX\mapsto\bar\uX$, $\uf\mapsto\bar\uf$ is
fully faithful. Hence $\bCSchlfp,\ab\bCSchlg,\ab\bCSchlf,\ab\bCSch$
are equivalent to $\CSchlfp,\ab\CSchlg, \ab\CSchlf,\ab\CSch$,
considered as 2-categories with only identity 2-morphisms. In
practice one usually does not distinguish between schemes and stacks
which are equivalent to schemes, that is, one identifies
$\CSchlfp,\ldots,\CSch$ and $\bCSchlfp,\ldots,\bCSch$.
\label{ag8def2}
\end{dfn}
\begin{rem} Behrend and Xu \cite[Def.~2.15]{BeXu} use `$C^\iy$-stack'
to mean something different, a geometric stack over the site
$(\Man,{\cal J}_\Man)$ of manifolds without boundary with
Grothendieck topology ${\cal J}_\Man$ given by open covers. These
are also called `smooth stacks' or `differentiable stacks' in
\cite{BeXu,Hein,Metz,Nooh}. We will write $\ManSta$ for the
2-category of geometric stacks on~$(\Man,{\cal J}_\Man)$.
The full and faithful embedding $F_\Man^\CSch:\Man\hookra\CSch$ has
${\cal J}_\Man\ab=\ab(F_\Man^\CSch)^*({\cal J})$, as both ${\cal
J},{\cal J}_\Man$ are defined by open covers. Therefore there is a
natural truncation 2-functor $F_\CSta^\ManSta:\CSta\ra\ManSta$,
given on objects by $F_\CSta^\ManSta:{\cal X}\mapsto {\cal
X}\t_{p_{\cal X},\CSch,F_\Man^\CSch}\Man$. See for example Metzler
\cite[Ex.~43, Lem.~44]{Metz} on relating stacks on different sites.
A stack $\cal X$ on $(\CSch,{\cal J})$ encodes all morphisms
$F:\uU\ra{\cal X}$ for $C^\iy$-schemes $\uU$, whereas its image
$F_\CSta^\ManSta({\cal X})$ remembers only morphisms $F:U\ra{\cal
X}$ for manifolds $U$. Thus the truncation functor $F_\CSta^\ManSta$
{\it loses information}, as it forgets morphisms from
$C^\iy$-schemes which are not manifolds.
This includes any information about {\it nonreduced\/}
$C^\iy$-schemes, and the nilpotent parts of $C^\iy$-rings. So, for
example, the point $\Spec\R$ and the double point
$\Spec\bigl(C^\iy(\R)/(x^2)\bigr)$ are different in $\CSta$, but
both are taken to the point in $\ManSta$. For the applications in
\cite{Joyc3} we need this nonreduced information, so it is not
enough to work with stacks on~$(\Man,{\cal J}_\Man)$.
\label{ag8rem1}
\end{rem}
Theorems \ref{ag4thm3} and \ref{ag7thm}, Corollary \ref{ag7cor2} and
Proposition \ref{ag8prop1} imply:
\begin{thm} Let\/ $\cal X$ be a $C^\iy$-stack. Then $\cal X$ is
equivalent to a groupoid stack\/ $[\uV\rra\uU],$ where
$(\uU,\uV,\us,\ut,\uu,\ui,\um)$ is a groupoid in\/ $\CSch$.
Conversely, any groupoid in\/ $\CSch$ defines a $C^\iy$-stack\/
$[\uV\rra\uU]$. All fibre products exist in the\/
$2$-category\/~$\CSta$.
\label{ag8thm1}
\end{thm}
{\it Quotient stacks\/} $[\uU/\uG]$ are a special class of groupoid
stacks~$[\uV\rra\uU]$.
\begin{dfn} A $C^\iy$-{\it group\/} $\uG$ is a $C^\iy$-scheme
$\uG=(G,\O_G)$ equipped with an identity element $1\in G$ and
multiplication and inverse morphisms $\um:\uG\t\uG\ra\uG$,
$\ui:\uG\ra\uG$ in $\CSch$ such that $(\ul{*},\uG,\upi,\upi,
1,\ui,\um)$ is a groupoid object in $\CSch$. Here $\ul{*}=\Spec\R$
is a single point, and $\upi:\uG\ra\ul{*}$ is the unique morphism,
and we regard $1\in G$ as a morphism~$1:\ul{*}\ra\uG$.
Let $\uG$ be a $C^\iy$-group, and $\uU$ a $C^\iy$-scheme. A ({\it
left\/}) {\it action\/} of $\uG$ on $\uU$ is a morphism
$\umu:\uG\t\uU\ra\uU$ such that
\e
\bigl(\uU,\uG\t\uU,\upi_{\uU},\umu,1\t\uid_\uU,
(\ui\ci\upi_{\uG})\t\umu,(\um\ci((\upi_\uG\ci\upi_1)\t
(\upi_\uG\ci\upi_2))) \t(\upi_\uU\ci\upi_2)\bigr)
\label{ag8eq1}
\e
is a groupoid object in $\CSch$, where in the final morphism
$\upi_1,\upi_2$ are the projections from $(\uG\t\uU)\t_{\upi_\uU,
\uU,\umu}(\uG\t\uU)$ to the first and second factors $\uG\t\uU$.
Then define the {\it quotient stack\/} $[\uU/\uG]$ to be the
groupoid stack $[\uG\t\uU\rra\uU]$ for the groupoid \eq{ag8eq1}. It
is a $C^\iy$-stack.
If $\uG=(G,\O_G)$ is a $C^\iy$-group then the underlying space $G$
is a topological group, and is in particular a group, and if
$\uG=(G,\O_G)$ acts on $\uU=(U,\O_U)$ then $G$ acts continuously
on~$U$.
If $G$ is a Lie group then $\uG=F_\Man^\CSch(G)$ is a $C^\iy$-group
in a natural way, by applying $F_\Man^\CSch$ to the smooth
multiplication and inverse maps $m:G\t G\ra G$ and $i:G\ra G$. If a
Lie group $G$ acts smoothly on a manifold $U$ with action $\mu:G\t
U\ra U$ then the $C^\iy$-group $\uG=F_\Man^\CSch(G)$ acts on the
$C^\iy$-scheme $\uU=F_\Man^\CSch(U)$ with action
$\umu=F_\Man^\CSch(\mu):\uG\t\uU\ra\uU$, so we can form the quotient
$C^\iy$-stack~$[\uU/\uG]$.
In particular, any finite group $G$ may be regarded as a Lie group
of dimension 0, with the discrete topology, so it has an associated
$C^\iy$-group, which we will write as $G$ rather than $\uG$. Then an
action of $G$ as a $C^\iy$-group on a $C^\iy$-scheme
$\uU$ is equivalent to an action of $G$ on $\uU$ as a finite group
by isomorphisms.
\label{ag8def3}
\end{dfn}
\subsection{Properties of morphisms of $C^\iy$-stacks}
\label{ag82}
We define some classes of morphisms of $C^\iy$-schemes.
\begin{dfn} Let $\uf=(f,f^\sh):\uX=(X,\O_X)\ra\uY=(Y,\O_Y)$ be a
morphism in $\CSch$. Then:
\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item We call $\uf$ an {\it open embedding\/} if $V=f(X)$ is an open
subset in $Y$ and $(f,f^\sh):(X,\O_X)\ra(V,\O_Y\vert_V)$ is an
isomorphism.
\item We call $\uf$ a {\it closed embedding\/} if $f:X\ra Y$ is a
homeomorphism with a closed subset of $\uY$, and the morphism of
sheaves of $C^\iy$-rings $f^\sh:\O_Y\ra f_*(\O_X)$ is
surjective. Equivalently, $\uf$ is an isomorphism with a closed
$C^\iy$-subscheme of $\uY$. Over affine open subsets
$\uU\cong\Spec\fC$ in $\uY$, $\uf$ is modelled on the natural
morphism $\Spec(\fC/I)\hookra\Spec\fC$ for some ideal $I$
in~$\fC$.
\item We call $\uf$ an {\it embedding\/} if we may write
$\uf=\ug\ci\uh$ where $\uh$ is an open embedding and $\ug$ is a
closed embedding.
\item We call $\uf$ {\it \'etale\/} if each $x\in X$ has an
open neighbourhood $U$ in $X$ such that $V=f(U)$ is open in $Y$
and $(f\vert_U,f^\sh\vert_U):(U,\O_X\vert_U)\ra (V,\O_Y\vert_V)$
is an isomorphism. That is, $\uf$ is a local isomorphism.
\item We call $\uf$ {\it proper\/} if $f:X\ra Y$ is a proper map of
topological spaces, that is, if $S\subseteq Y$ is compact then
$f^{-1}(S)\subseteq X$ is compact.
\item We say that $\uf$ {\it has finite fibres\/} if $f:X\ra Y$
is a finite map, that is, $f^{-1}(y)$ is a finite subset of $X$
for all~$y\in Y$.
\item We call $\uf$ {\it separated\/} if $f:X\ra Y$ is a
separated map of topological spaces, that is, $\De_X=\bigl\{
(x,x):x\in X\bigr\}$ is a closed subset of the topological fibre
product $X\t_{f,Y,f}X=\bigl\{(x,x')\in X\t X:f(x)=f(x')\bigr\}$.
\item We call $\uf$ {\it closed\/} if $f:X\ra Y$ is a
closed map of topological spaces, that is, $S\subseteq X$ closed
implies $f(S)\subseteq Y$ closed.
\item We call $\uf$ {\it universally closed\/} if whenever
$\ug:\uW\ra\uY$ is a morphism then
$\upi_\uW:\uX\t_{\uf,\uY,\ug}\uW\ra\uW$ is closed.
\item We call $\uf$ a {\it submersion\/} if for all $x\in X$ with
$f(x)=y$, there exists an open neighbourhood $U$ of $y$ in $Y$
and a morphism $\ug=(g,g^\sh):(U,\O_Y\vert_U)\ra(X,\O_X)$ with
$g(y)=x$ and~$\uf\ci\ug=\id_{(U,\O_Y\vert_U)}$.
\item We call $\uf$ {\it locally fair}, or {\it locally good}, or
{\it locally finitely presented}, if whenever $\uU$ is a locally
fair, or locally good, or locally finitely presented
$C^\iy$-scheme, respectively, and $\ug:\uU\ra\uY$ is a morphism
then $\uX\t_{\uf,\uY,\ug}\uU$ is locally fair, or locally good,
or locally finitely presented, respectively.
\end{itemize}
\label{ag8def4}
\end{dfn}
\begin{rem} These are mostly analogues of standard concepts in
algebraic geometry, as in Hartshorne \cite{Hart} for instance. But
because the topology on $C^\iy$-schemes is finer than the Zariski
topology in algebraic geometry --- for instance, affine
$C^\iy$-schemes are Hausdorff --- our definitions of \'etale and
proper are simpler than in algebraic geometry. (Open or closed)
embeddings correspond to (open or closed) immersions in algebraic
geometry, but we prefer the word `embedding', as immersion has a
different meaning in differential geometry. Closed morphisms are not
invariant under base change, which is why we define universally
closed. If $X,Y$ are manifolds and $\uX,\uY=F_\Manc^\CSch(X,Y)$,
then $\uf:\uX\ra\uY$ is a submersion of $C^\iy$-schemes if and only
if $\uf=F_\Manc^\CSch(f)$ for $f:X\ra Y$ a submersion of manifolds.
\label{ag8rem2}
\end{rem}
\begin{dfn} Let $\bs P$ be a property of morphisms in $\CSch$. We
say that $\bs P$ {\it is stable under open embedding\/} if whenever
$\uf:\uU\ra\uV$ is $\bs P$ and $\ui:\uV\ra\uW$ is an open embedding,
then $\ui\ci\uf:\uU\ra\uW$ is $\bs P$.
\label{ag8def5}
\end{dfn}
The next proposition is elementary. See Laumon and Bailly \cite[\S
3.10]{LaMo} and Noohi \cite[Ex.~4.6]{Nooh} for similar lists for the
\'etale and topological sites.
\begin{prop} The following properties of morphisms in $\CSch$ are
invariant under base change and local in the target in the site
$(\CSch,{\cal J}):$ open embedding, closed embedding, embedding,
\'etale, proper, has finite fibres, separated, universally closed,
submersion, locally fair, locally good, locally finitely presented.
The following properties are also stable under open embedding: open
embedding, embedding, \'etale, has finite fibres, separated,
submersion, locally fair, locally good, locally finitely presented.
\label{ag8prop2}
\end{prop}
As in \S\ref{ag74}, this implies that these properties are also
defined for representable morphisms in $\CSta$. In particular, if
$\cal X$ is a $C^\iy$-stack then $\De_{\cal X}:{\cal X}\ra{\cal
X}\t{\cal X}$ is representable, and if $\Pi:\bar \uU\ra{\cal X}$ is
an atlas then $\Pi$ is representable, so we can require that
$\De_{\cal X}$ or $\Pi$ has some of these properties.
\begin{dfn} Let $\cal X$ be a $C^\iy$-stack. Following
\cite[Def.~7.6]{LaMo}, we say that $\cal X$ is {\it separated\/} if
the diagonal 1-morphism $\De_{\cal X}:{\cal X}\ra{\cal X}\t{\cal X}$
is universally closed. If ${\cal X}=\bar\uX$ for some $C^\iy$-scheme
$\uX=(X,\O_X)$ then $\cal X$ is separated if and only if $\De_X:X\ra
X\t X$ is closed, that is, if and only if $X$ is Hausdorff.
\label{ag8def6}
\end{dfn}
\begin{prop} Let\/ ${\cal W}={\cal X}\t_{f,{\cal Z},g}{\cal Y}$ be
a fibre product of\/ $C^\iy$-stacks with\/ ${\cal X},{\cal Y}$
separated. Then $\cal W$ is separated.
\label{ag8prop3}
\end{prop}
\begin{proof} We have a 2-commutative diagram with both squares
2-Cartesian:
\e
\begin{gathered}
\xymatrix@!0@C=60pt@R=20pt{ & {\cal W} \ar[rr]_{\De_{\cal W}}
\ar[dl] \ar[dr]^(0.55){\pi_1} && {\cal W}\t{\cal W} \ar[dr] \\
{\cal Z} \ar[dr]_{\jmath_{\cal Z}} && {\cal X}\t_{f\ci\De_{\cal
Z},{\cal Z}\t{\cal Z},g\ci{\cal Z}}{\cal Y} \ar[dl] \ar[dr]
\ar[ur]^(0.45){\pi_2} && {\cal X}\t{\cal X}\t{\cal Y}\t{\cal Y}. \\
& I_{\cal Z} && {\cal X}\t{\cal Y} \ar[ur]_{{}\,\,\,\De_{\cal
X}\t\De_{\cal Y}}}
\end{gathered}
\label{ag8eq2}
\e
Let $[\uV\rra\uU]$ be a groupoid presentation of $\cal Z$, and
consider the fourth 2-Cartesian diagram of \eq{ag7eq8}, with
surjective rows. The left hand morphism $\bar\uu\t\bar\uid_\uU$ has
a left inverse $\pi_\uU$, and so is automatically universally
closed. Hence $\jmath_{\cal Z}$ is universally closed by
Propositions \ref{ag7prop2}(c) and \ref{ag8prop2}, so $\pi_1$ in
\eq{ag8eq2} is universally closed by Propositions \ref{ag7prop2}(a)
and \ref{ag8prop2}. Also $\De_{\cal X},\De_{\cal Y}$ are universally
closed as ${\cal X},{\cal Y}$ are separated, so $\De_{\cal
X}\t\De_{\cal Y}$ in \eq{ag8eq2} is universally closed, and $\pi_2$
is universally closed. Hence $\De_{\cal W}\cong\pi_2\ci\pi_1$ is
universally closed, and $\cal W$ is separated.
\end{proof}
\subsection{Open $C^\iy$-substacks and open covers}
\label{ag83}
\begin{dfn} Let $\cal X$ be a $C^\iy$-stack. A $C^\iy$-{\it
substack\/} $\cal Y$ in $\cal X$ is a substack of $\cal X$, in the
sense of Definition \ref{ag7def4}, which is also a $C^\iy$-stack. It
has a natural inclusion 1-morphism $i_{\cal Y}:{\cal Y}\hookra{\cal
X}$. We call $\cal Y$ an {\it open\/ $C^\iy$-substack\/} of $\cal X$
if $i_{\cal Y}$ is a representable open embedding, a {\it closed\/
$C^\iy$-substack\/} of $\cal X$ if $i_{\cal Y}$ is a representable
closed embedding, and a {\it locally closed\/ $C^\iy$-substack\/} of
$\cal X$ if $i_{\cal Y}$ is a representable embedding. An {\it open
cover\/} $\{{\cal Y}_a:a\in A\}$ of $\cal X$ is a family of open
$C^\iy$-substacks ${\cal Y}_a$ in $\cal X$ with $\coprod_{a\in
A}i_{{\cal Y}_a}:\coprod_{a\in A}{\cal Y}_a\ra{\cal X}$ surjective.
\label{ag8def7}
\end{dfn}
Some properties of $\De_{\cal X},\io_{\cal X},\jmath_{\cal X}$ and
atlases for $\cal X$ can be tested on the elements of an open cover.
The proof is elementary.
\begin{prop} Let\/ $\cal X$ be a\/ $C^\iy$-stack, and\/
$\{{\cal Y}_a:a\in A\}$ an open cover of\/ $\cal X$. Suppose $\bs P$
and\/ $\bs Q$ are properties of morphisms in $\CSch$ which are
invariant under base change and local in the target in $(\CSch,{\cal
J}),$ and that\/ $\bs P$ is stable under open embedding. Then:
\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item[{\rm(a)}] Let\/ $\Pi_a:\bar\uU_a\ra{\cal Y}_a$ be an atlas
for\/ ${\cal Y}_a$ for\/ $a\in A$. Set\/ $\uU=\coprod_{a\in
A}\uU_a$ and\/ $\Pi=\coprod_{a\in A}i_{{\cal
Y}_a}\ci\Pi_a:\bar\uU\ra{\cal X}$. Then $\Pi$ is an atlas for
$\cal X,$ and\/ $\Pi$ is $\bs P$ if and only if\/ $\Pi_a$ is
$\bs P$ for all\/ $a\in A$.
\item[{\rm(b)}] $\De_{\cal X}:{\cal X}\!\ra\!{\cal X}\!\t\!{\cal X}$
is $\bs P$ if and only if\/ $\De_{{\cal Y}_a}:{\cal
Y}_a\!\ra\!{\cal Y}_a\!\t\!{\cal Y}_a$ is $\bs P$ for
all\/~$a\!\in\!A$.
\item[{\rm(c)}] $\io_{\cal X}:I_{\cal X}\ra{\cal X}$ is $\bs Q$
if and only if\/ $\io_{{\cal Y}_a}:I_{{\cal Y}_a}\ra{\cal Y}_a$
is $\bs Q$ for all\/~$a\in A$.
\item[{\rm(d)}] $\jmath_{\cal X}:{\cal X}\ra I_{\cal X}$ is $\bs Q$
if and only if\/ $\jmath_{{\cal Y}_a}:{\cal Y}_a\ra I_{{\cal
Y}_a}$ is $\bs Q$ for all\/~$a\in A$.
\end{itemize}
\label{ag8prop4}
\end{prop}
If ${\cal X}=\bar\uU$ for some $C^\iy$-scheme $\uU=(U,\O_U)$, then
the open $C^\iy$-substacks of $\cal X$ are
$\,\overline{\!(V,\O_U\vert_V)\!}\,$ for all $V\subseteq U$, that
is, they are the images in $\CSta$ of the open $C^\iy$-subschemes of
$U$. We can also describe the open substacks of groupoid
stacks~$[\uV\rra\uU]$:
\begin{prop} Let\/ $(\uU,\uV,\us,\ut,\uu,\ui,\um)$ be a groupoid
in\/ $\CSch$ and\/ ${\cal X}=[\uV\rra\uU]$ the associated\/
$C^\iy$-stack, and write $\uU=(U,\O_U),$ and so on. Then open\/
$C^\iy$-substacks ${\cal X}'$ of\/ $\cal X$ are naturally in $1$-$1$
correspondence with open subsets\/ $U'\subseteq U$ with\/
$s^{-1}(U')=t^{-1}(U'),$ where ${\cal X}'=[\uV'\rra\uU']$ for
$\uU'=(U',\O_U\vert_{U'})$ and\/ $\uV'=(s^{-1}(U'),\O_V
\vert_{s^{-1}(U')})$. If\/ $(\uU,\uV,\us,\ut,\uu,\ui,\um)$ is as in
{\rm\eq{ag8eq1},} so that\/ $\cal X$ is a quotient stack\/
$[\uU/\uG],$ then open\/ $C^\iy$-substacks ${\cal X}'$ of\/ $\cal X$
correspond to\/ $G$-invariant open subsets\/~$U'\subseteq U$.
\label{ag8prop5}
\end{prop}
\begin{proof} From Theorem \ref{ag7thm}, as ${\cal
X}=[\uV\rra\uU]$ we have a natural surjective, representable
1-morphism $\Pi:\bar\uU\ra{\cal X}$. If ${\cal X}'$ is an open
$C^\iy$-substack of $\cal X$ then $\bar\uU\t_{\Pi,{\cal X},i_{{\cal
X}'}}{\cal X}'$ is an open $C^\iy$-substack of $\bar\uU$, and so is
of the form $\,\overline{\!(U',\O_U\vert_{U'})\!}\,$ for some open
$U'\subseteq U$. We have natural 1-isomorphisms
\begin{align*}
&\,\overline{\!(s^{-1}(U'),\O_V\vert_{s^{-1}(U')})\!}=\!
\bar U'\!\t_{i_{\bar U'},\bar U,\bar s}\!\bar V\!\cong\!
{\cal X'}\!\t_{\cal X}\!(\bar U\!\t_{\id_{\bar U},\bar U,\bar s}\!
\bar V)\!\cong\!{\cal X'}\!\t_{i_{\cal X}',\cal X,\pi_{\cal X}}\!\bar V\\
&\qquad\cong {\cal X'}\t_{\cal X}(\bar U\t_{\id_{\bar U},\bar U,\bar t}
\bar V)\cong \bar U'\t_{i_{\bar U'},\bar U,\bar t}\bar V=\,
\overline{\!(t^{-1}(U'),\O_V\vert_{t^{-1}(U')})\!}\,,
\end{align*}
which implies that $s^{-1}(U')=t^{-1}(U')$. Conversely, if
$s^{-1}(U')=t^{-1}(U')$ then defining $\uU',\uV'$ as in the
proposition, it is easy to show that we get a groupoid stack ${\cal
X}'=[\uV'\rra\uU']$ which is naturally an open $C^\iy$-substack of
$\cal X$. When ${\cal X}=[\uU/\uG]$, we see that $s^{-1}(U')=
t^{-1}(U')$ if and only if $U'$ is $G$-invariant.
\end{proof}
\subsection{Deligne--Mumford $C^\iy$-stacks}
\label{ag84}
{\it Deligne--Mumford stacks\/} in algebraic geometry were
introduced in \cite{DeMu} to study moduli spaces of algebraic
curves. As in \cite[Th.~6.2]{LaMo}, Deligne--Mumford stacks are
locally modelled (in the \'etale topology) on quotient stacks
$[X/G]$ for $X$ an affine scheme and $G$ a finite group. This
motivates:
\begin{dfn} A {\it Deligne--Mumford\/ $C^\iy$-stack\/} is a
$C^\iy$-stack $\cal X$ which admits an open cover $\{{\cal Y}_a:a\in
A\}$, as in Definition \ref{ag8def7}, with each ${\cal Y}_a$
equivalent to a quotient stack $[\uU_a/G_a]$ for $\uU_a$ an affine
$C^\iy$-scheme and $G_a$ a finite group. We call $\cal X$ a {\it
locally fair}, or {\it locally good}, or {\it locally finitely
presented}, Deligne--Mumford $C^\iy$-stack if it admits such an open
cover with each $\uU_a$ a fair, or good, or finitely presented,
affine $C^\iy$-scheme, respectively. Write
$\DMCStalf,\DMCStalg,\DMCStalfp$ and $\DMCSta$ for the full
2-subcategories of locally fair, locally good, locally finitely
presented, and all, Deligne--Mumford $C^\iy$-stacks in $\CSta$,
respectively.
\label{ag8def8}
\end{dfn}
\begin{prop} $\DMCStalf,\DMCStalg,\DMCStalfp,\DMCSta$
are closed under taking open\/ $C^\iy$-substacks in\/~$\CSta$.
\label{ag8prop6}
\end{prop}
\begin{proof} Let ${\cal X}$ lie in one of these 2-categories, and
${\cal X}'$ be an open $C^\iy$-substack of $\cal X$. Then $\cal X$
admits an open cover $\{{\cal Y}_a:a\in A\}$ with ${\cal
Y}_a\simeq[\uU_a/G_a]$ with $\uU_a$ affine and $G_a$ finite, and
$\{{\cal Y}_a':a\in A\}$ is an open cover of ${\cal X}'$, where
${\cal Y}_a'={\cal Y}_a\t_{\cal X}{\cal X}'$ is an open
$C^\iy$-substack of ${\cal Y}_a$. Thus ${\cal
Y}_a'\simeq[\uU_a'/G_a]$ by Proposition \ref{ag8prop5}, where
$\uU_a'$ is a $G_a$-invariant open $C^\iy$-subscheme of $\uU_a$. If
the $\uU_a$ are fair, good, or finitely presented then the $\uU_a'$
are too by Proposition \ref{ag4prop5}. Thus
$\DMCStalf,\DMCStalg,\DMCStalfp$ are closed under open subsets.
For $\DMCSta$, as open subsets of affine $C^\iy$-schemes need not be
affine, the $\uU_a'$ need not be affine. We will show that we can
cover $\uU_a'$ by $G_a$-invariant open affine $C^\iy$-subschemes
$\uU_{au}'$. Write $\uU_a'=(U_a',\O_{\uU_a'})$ and
$G_a=(G_a,\O_{G_a})$. Then the finite group $G_a$ acts
continuously on $U_a'$. Let $u\in U_a'$, and $H_u=\{\ga\in G_a:\ga
u=u\}$ be the stabilizer of $u$ in $G_a$. Then the orbit $\{\ga
u:\ga\in G\}\cong G_a/H_u$ of $u$ is a finite set, so as $U_a'$ is
Hausdorff we can choose affine open neighbourhoods $V_{\ga u}$ of
$\ga u$ for each point in the orbit such that $V_{\ga u}\cap
V_{\ga'u}=\es$ if $\ga u\ne\ga'u$. Define $W_u=\bigcap_{\ga\in
G}\ga^{-1}V_{\ga u}$. Then $W_u$ is an $H_u$-invariant open
neighbourhood of $u$ in $U_a'$, and if $\ga\in G_a\sm H_u$ then~$\ga
W_u\cap W_u=\es$.
By Lemma \ref{ag4lem2} we can choose an affine open neighbourhood
$W_u'$ of $u$ in $W_u$. Define $W_u''=\bigcap_{\ga\in H_u}W_u'$, an
$H_u$-invariant open neighbourhood of $u$ in $W_u$. This a finite
intersection of affine open $C^\iy$-subschemes $\uW_u'$ in the
affine $C^\iy$-scheme $\uV_u$, and so is affine, since intersection
is a kind of fibre product, and $\ACSch$ is closed under fibre
products by Theorem \ref{ag4thm2}. Define $U_{au}'=\bigcup_{\ga\in
G_a}W_u''$. Then $U_{au}'$ is a $G_a$-invariant open neighbourhood
of $u$ in $U_a'$. Since $W_u''$ is $H_u$-invariant and $\ga
W_u''\cap W_u''=\es$ if $\ga\in G_a\sm H_u$, we see that $U_{au}'$
is isomorphic to the disjoint union of $\md{G_a}/\md{H_u}$ copies of
$W_u''$. Hence $\uU_{au}'=(U_{au}',\O_{\uU_a'}\vert_{U_{au}'})$ is a
finite disjoint union of affine $C^\iy$-schemes, and is an affine
$C^\iy$-scheme. Therefore we may cover $\uU_a'$ by $G_a$-invariant
open affine $C^\iy$-subschemes $\uU_{au}'$. Using these we obtain an
open cover $\bigl\{{\cal Y}'_{au}:a\in A$, $u\in U_a\bigr\}$ of
${\cal X}'$ with ${\cal Y}'_{au}\simeq[\uU_{au}'/G_a]$, so ${\cal
X}'$ is Deligne--Mumford.
\end{proof}
The proof of Proposition \ref{ag8prop6} only uses
$\uU_a=(U_a,\O_{U_a})$ a $C^\iy$-scheme and $U_a$ Hausdorff, it does
not need $\uU_a$ to be affine. So the same proof yields:
\begin{prop} Any\/ $C^\iy$-stack of the form\/ $[\uX/G]$ with\/
$\uX$ a separated\/ $C^\iy$-scheme and\/ $G$ finite is a separated
Deligne--Mumford\/ $C^\iy$-stack.
\label{ag8prop7}
\end{prop}
However, if $\uX$ is not separated then $[\uX/G]$ need not be
Deligne--Mumford:
\begin{ex} Let $\uX$ be the nonseparated $C^\iy$-scheme
$(\ul{\R}\amalg\ul{\R})/\sim$, where $\sim$ is the equivalence
relation which identifies the two copies of $\ul{\R}$ on
$\,\ul{\!(0,\iy)\!}\,$. Let $G=\Z_2$ act on $\uX$ by exchanging
the two copies of $\ul{\R}$. Let $\cal X$ be the quotient
$C^\iy$-stack $[\uX/G]$. We can think of $\cal X$ as a like copy
of $\R$, where the stabilizer group of $x\in\R$ is $\{1\}$ if
$x\in(-\iy,0]$ and $\Z_2$ if $x\in(0,\iy)$. Using the obvious atlas
$\Pi:\bar{\ul{\R}}\ra{\cal X}$, the third diagram of \eq{ag7eq8}
yields a 2-Cartesian square
\begin{equation*}
\xymatrix@C=15pt@R=10pt{\bar{\ul{\R}}\amalg\,\,\overline{\!
\ul{\!(0,\iy)\!}\!}\,\, \ar[rr] \ar[d]
\drrtwocell_{}\omit^{}\omit{^{}} && I_{\cal X} \ar[d]^{\io_{\cal X}} \\
\bar{\ul{\R}} \ar[rr]^(0.7){\Pi} && {\cal X}. }
\end{equation*}
As the left hand column is not proper, $\io_{\cal X}$ is not proper,
so $\cal X$ is not Deligne--Mumford by Corollary \ref{ag8cor1}
below.
\label{ag8ex1}
\end{ex}
\begin{prop} Suppose $g:[\uX/F]\ra[\uZ/H],$ $h:[\uY/G]\ra[\uZ/H]$
are $1$-morphisms of quotient\/ $C^\iy$-stacks, where $\uX,\uY,\uZ$
are $C^\iy$-schemes and\/ $F,G,H$ are finite groups. Then we have a
$2$-Cartesian square
\e
\begin{gathered}
\xymatrix@C=20pt@R=10pt{ [\uW/(F\t G)] \ar[r]^(0.7){f} \ar[d]_{e}
\drtwocell_{}\omit^{}\omit{^{}} & [\uY/G] \ar[d]^h \\ [\uX/F]
\ar[r]_g & [\uZ/H],}
\end{gathered}
\label{ag8eq3}
\e
where $\Pi_\uX:\bar\uX\ra[\uX/F],\Pi_\uY:\bar\uY
\ra[\uY/G],\Pi_\uZ:\bar\uZ\ra [\uZ/H]$ are the natural atlases and\/
$\bar\uW=\bar\uX\t_{g\ci\Pi_\uX,[\uZ/H],h\ci\Pi_\uY}\bar \uY$. If\/
$\uX,\uY,\uZ$ are separated, or locally fair, or locally good, or
locally finitely presented, then $\uW$ is separated, or locally
fair, or locally good, or locally finitely presented, respectively.
\label{ag8prop8}
\end{prop}
\begin{proof} Write ${\cal W}=[\uX/F]\t_{[\uZ/H]}[\uY/G]$.
Then from the atlases $\Pi_\uX,\Pi_\uY$, Example \ref{ag7ex}
constructs an atlas $\Pi_\uW:\bar\uW\ra\cal W$ for $\cal W$. Since
$[\uX/F]\simeq[F\t\uX\rra\uX]$ and
$[\uY/G]\simeq[G\t\uY\rra\uY]$ it follows from \eq{ag7eq11} that
${\cal W}$ is equivalent to the groupoid stack
$[(F\t G)\t\uW\rra\uW]$ for a natural action of $F\t G$ on
$\uW$. This proves~\eq{ag8eq3}.
If $\uX,\uY,\uZ$ are separated then $[\uZ/H]$ is Deligne--Mumford by
Proposition \ref{ag8prop7}, so $\De_{[\uZ/H]}$ is separated by
Corollary \ref{ag8cor1} below, and thus $\uW$ is separated as
$\uX,\uY$ are and $\bar\uW\cong (\bar\uX\t\bar\uY)\t_{[\uZ/H]\t
[\uZ/H], \De_{[\uZ/H]}}[\uZ/H]$. Form the diagram
\begin{equation*}
\xymatrix@!0@R=13pt@C=50pt{ & \bar\uW' \ar[dl]_{\pi_\uW} \ar@<-1ex>[ddr]
\ar[rr] && \bar\uY' \ar[dl]^(0.3){\pi_\uY} \ar[ddr] \\ \bar\uW
\ar[ddr] \ar[rr] && \bar\uY \ar@<1ex>[ddr]^(0.6){h\ci\Pi_\uY} \\ &&
\bar\uX'
\ar[rr] \ar[dl]_(0.6){\pi_\uX} && \bar\uZ \ar[dl]^(0.4){\Pi_\uZ} \\
& \uX \ar[rr]^(0.7){g\ci\Pi_\uX} && [\uZ/H]}
\end{equation*}
with 2-Cartesian squares, where $\uW',\uX',\uY'$ are
$C^\iy$-schemes. Then $\pi_\uW,\pi_\uX,\pi_\uY$ are \'etale and
surjective, as $\Pi_\uZ$ is. If $\uX,\uY,\uZ$ are locally fair, then
$\uX',\uY'$ are locally fair as $\uX,\uY$ are and $\pi_\uX,\pi_\uY$
are \'etale, so $\uW'\cong\uX'\t_\uZ\uY'$ is locally fair by Theorem
\ref{ag4thm3}, and thus $\uW$ is locally fair as
$\pi_\uW:\uW'\ra\uW$ is \'etale and surjective. The proof for
locally good and locally finitely presented is the same.
\end{proof}
Using this we prove:
\begin{thm} The $2$-subcategories $\DMCSta,\!\DMCStalf\!,\DMCStalg$
and\/ $\DMCStalfp$ are closed under fibre products in\/~$\CSta$.
\label{ag8thm2}
\end{thm}
\begin{proof} Let ${\cal W}={\cal X}\t_{\cal Z}{\cal Y}$ be a
fibre product of Deligne--Mumford $C^\iy$-stacks ${\cal X},{\cal
Y},{\cal Z}$. We must show $\cal W$ is Deligne--Mumford. Now $\cal
Z$ admits an open cover $\{{\cal Z}_c:c\in C\}$ with ${\cal
Z}_c\simeq[\uZ_c/H_c]$ for $\uZ_c$ an affine $C^\iy$-scheme and
$H_c$ finite. For $c\in C$ define ${\cal X}_c={\cal X}\t_{\cal
Z}{\cal Z}_c$ and ${\cal Y}_c={\cal Y}\t_{\cal Z}{\cal Z}_c$, which
are open $C^\iy$-substacks of ${\cal X},{\cal Y}$, and so are
Deligne--Mumford by Proposition \ref{ag8prop6}. Then $\{{\cal
X}_c\t_{{\cal Z}_c}{\cal Y}_c:c\in C\}$ is an open cover of $\cal
W$, so it is enough to prove ${\cal X}_c\t_{{\cal Z}_c}{\cal Y}_c$
is Deligne--Mumford. That is, we may replace $\cal Z$ by ${\cal
Z}_c\simeq[\uZ_c/H_c]$.
Similarly, by choosing open covers of ${\cal X}_c,{\cal Y}_c$ by
substacks equivalent to $[\uX/F],[\uY/G]$, we reduce the problem to
showing $[\uX/F]\t_{[\uZ/H]}[\uY/G]$ is Deligne--Mumford, for
$\uX,\uY,\uZ$ affine $C^\iy$-schemes and $F,G,H$ finite groups. This
follows from Propositions \ref{ag8prop7} and \ref{ag8prop8}, noting
that $\uX,\uY,\uZ$ are separated as they are affine, so $\uW$ is
separated in Proposition \ref{ag8prop8}. This shows $\DMCSta$ is
closed under fibre products. For $\DMCStalf$, $\DMCStalg$,
$\DMCStalfp$ we use the same argument with $\uZ_c,\uZ,\uX,\uY,\uW$
locally fair, and so on.
\end{proof}
\subsection{Characterizing Deligne--Mumford $C^\iy$-stacks}
\label{ag85}
We now explore ways to characterize when a $C^\iy$-stack $\cal X$ is
Deligne--Mumford.
\begin{prop} Let\/ $\cal X$ be a quotient $C^\iy$-stack\/
$[\uU/G]$ for\/ $\uU$ affine and\/ $G$ finite. Then the natural
$1$-morphism $\Pi:\bar\uU\ra{\cal X}$ is an \'etale atlas, and\/
$\De_{\cal X}:{\cal X}\ra{\cal X}\t{\cal X},$ $\io_{\cal X}:I_{\cal
X}\ra{\cal X}$ are universally closed, proper, and separated, with
finite fibres, and\/ $\jmath_{\cal X}:{\cal X}\ra I_{\cal X}$ is an
open and closed embedding.
\label{ag8prop9}
\end{prop}
\begin{proof} As in \eq{ag7eq8} we have 2-Cartesian diagrams with
surjective rows:
\begin{equation*}
\xymatrix@C=20pt@R=7pt{ \bar G\t\bar\uU
\ar[rr]_(0.7){\bar\umu} \ar[d]_(0.6){\bar\upi_\uU}
\drrtwocell_{}\omit^{}\omit{^{}} && \bar\uU \ar[d]_\Pi &
\bar G\t\bar\uU \ar[rr]_(0.7){\Pi\ci\bar\upi_\uU} \ar[d]_{
\bar\upi_\uU\t\bar\umu} \drrtwocell_{}\omit^{}\omit{^{}} && {\cal X}
\ar[d]_{\De_{\cal X}} \\
\bar\uU \ar[rr]^(0.7)\Pi && {\cal X}, & \bar\uU\t\bar\uU
\ar[rr]^(0.7){\Pi\t\Pi} && {\cal X}\t{\cal X}, \\
(\bar G\t\bar\uU)\t_{\bar\uU\t\bar\uU}\bar\uU
\ar[rr]_(0.7){\bar\umu} \ar[d]_(0.6){\bar\upi_\uU}
\drrtwocell_{}\omit^{}\omit{^{}} && I_{\cal X} \ar[d]_{\io_{\cal X}} &
\bar U \ar[d]_{(1\t\id_\uU)\t\id_\uU} \ar[rr]_(0.7)\Pi
\drrtwocell_{}\omit^{}\omit{^{}} && {\cal X} \ar[d]_{\jmath_{\cal X}} \\
\bar U \ar[rr]^(0.7)\Pi && {\cal X} & (\bar G\t\bar\uU)\t_{\bar\uU\t
\bar\uU}\bar\uU \ar[rr]^(0.7){\bar\umu} && I_{\cal X}.}
\end{equation*}
The left column $\bar\pi_\uU$ in the first diagram is \'etale. The
left columns in the second and third diagrams are both universally
closed, proper, and separated, with finite fibres, since $G$ is
finite with the discrete topology, and $U$ is Hausdorff as $\uU$ is
affine. This left column in the fourth is an open and closed
embedding. The result now follows from Propositions
\ref{ag7prop2}(c) and~\ref{ag8prop2}.
\end{proof}
Propositions \ref{ag8prop2}, \ref{ag8prop4} and \ref{ag8prop9} now
imply:
\begin{cor} Let\/ $\cal X$ be a Deligne--Mumford\/ $C^\iy$-stack.
Then $\cal X$ has an \'etale atlas $\Pi:\bar U\!\ra\!{\cal X},$ the
diagonal\/ $\De_{\cal X}:{\cal X}\!\ra\!{\cal X}\!\t\!{\cal X}$ is
separated with finite fibres, and the inertia morphism\/ $\io_{\cal
X}:I_{\cal X}\!\ra\!{\cal X}$ is universally closed, proper, and
separated, with finite fibres, and\/ $\jmath_{\cal X}:{\cal X}\ra
I_{\cal X}$ is an open and closed embedding. If\/ $\cal X$ is
separated then $\De_{\cal X}$ is also universally closed and proper.
\label{ag8cor1}
\end{cor}
The last part holds as then $\De_{\cal X}$ is universally closed
with finite fibres, which implies $\De_{\cal X}$ is proper. Note
that for $\cal X$ not separated we cannot conclude from Proposition
\ref{ag8prop9} that $\De_{\cal X}$ is universally closed or proper,
since these properties are not stable under open embedding. Some of
the conclusions of Corollary \ref{ag8cor1} are sufficient for $\cal
X$ to be separated and Deligne--Mumford.
\begin{thm} Let\/ $\cal X$ be a $C^\iy$-stack, and suppose\/ $\cal
X$ has an \'etale atlas $\Pi:\bar\uU\ra{\cal X},$ and the diagonal\/
$\De_{\cal X}:{\cal X}\ra{\cal X}\t{\cal X}$ is universally closed
and separated. Then\/ $\cal X$ is a separated Deligne--Mumford\/
$C^\iy$-stack.
\label{ag8thm3}
\end{thm}
\begin{proof} Let $(\uU,\uV,\us,\ut,\uu,\ui,\um)$ be the groupoid
in $\CSch$ constructed from $\Pi:\bar\uU\ra{\cal X}$ as in
\S\ref{ag75}, so that ${\cal X}\simeq[\uV\rra\uU]$. Then \eq{ag7eq8}
gives 2-Cartesian diagrams with surjective rows. From the first and
Propositions \ref{ag7prop2}(a) and \ref{ag8prop2} we see that
$\us,\ut$ are \'etale, since $\Pi$ is. From the second
$\us\t\ut:\uV\ra\uU\t\uU$ is universally closed and separated, as
$\De_{\cal X}$ is. Let $p\in U$. Define
\begin{equation*}
H=\bigl\{q\in V:s(q)=t(q)=p\bigr\}\subseteq s^{-1}(\{p\}).
\end{equation*}
It has the discrete topology, as $\us,\ut$ are \'etale.
Suppose for a contradiction that $H$ is infinite. Define a
$C^\iy$-ring
\begin{equation*}
\fC=\bigl\{c:H\amalg\{\iy\}\ra\R:\text{$c(q)=c(\iy)$ for
all but finitely many $q\in H$}\bigr\},
\end{equation*}
with $C^\iy$ operations defined pointwise in $H\amalg\{\iy\}$. Then
$\Spec\fC$ has underlying topological space the one point
compactification $H\amalg\{\iy\}$ of the discrete topological space
$H$. Define $\ug:\Spec\fC\ra\uU\t\uU$ to project $\Spec\fC$ to the
point $(p,p)$. Then the morphism
\e
\upi_{\Spec\fC}:\uV\t_{\us\t\ut,\uU\t\uU,\ug}\Spec\fC\longra\Spec\fC
\label{ag8eq4}
\e
is the projection $H\t(H\amalg\{\iy\})\ra H\amalg\{\iy\}$. The
diagonal in $H$ is closed in $H\t(H\amalg\{\iy\})$, but its image is
$H$, which is not closed in $H\amalg\{\iy\}$. Hence \eq{ag8eq4} is
not a closed morphism, contradicting $\us\t\ut$ universally closed.
So $H$ is finite.
As $(\uU,\uV,\us,\ut,\uu,\ui,\um)$ is a groupoid, $H$ is a {\it
finite group}, with identity $u(p)$, inverse map $i\vert_H$, and
multiplication $m_H=m\vert_{H\t H}$. Since $\us,\ut$ are \'etale, we
can choose small open neighbourhoods $Z_q$ of $q$ in $V$ for all
$q\in H$ such that $\us\vert_{Z_q},\ut\vert_{Z_q}$ are isomorphisms
with open subsets of $\uU$. As $\us\t\ut$ is separated,
$\bigl\{(v,v):v\in V\bigr\}$ is closed in $\bigl\{(v,v')\in V\t
V:s(v)=s(v')$, $t(v)=t(v')\bigr\}$, which has the subspace topology
from $V\t V$. If $q\ne q'\in H$ then $(q,q')$ lies in
$\bigl\{(v,v')\in V\t V:s(v)=s(v')$, $t(v)=t(v')\bigr\}$ but not in
$\bigl\{(v,v):v\in V\bigr\}$, so $(q,q')$ has an open neighbourhood
in $V\t V$ which does not intersect $\bigl\{(v,v):v\in V\bigr\}$.
Making $Z_q,Z_{q'}$ smaller if necessary, we can take this open
neighbourhood to be $Z_q\t Z_{q'}$, and then $Z_q\cap Z_{q'}=\es$.
Thus, we can choose these open neighbourhoods $Z_q$ for $q\in H$ to
be {\it disjoint}.
Define $Y=\bigcap_{q\in H}s(Z_q)$ and $\uY=(Y,\O_U\vert_Y)$. Then
$Y$ is a small open neighbourhood of $p$ in $U$. Making $Y$ smaller
if necessary we can suppose it is contained in an affine open
neighbourhood of $p$ in $U$, and so is Hausdorff. Replace $Z_q$ by
$Z_q\cap s^{-1}(Y)$ for all $q\in H$. Then
$s\vert_{Z_q}:(Z_q,\O_V\vert_{Z_q})\ra \uY$ is an isomorphism for
$q\in H$. Set $Z=\bigcup_{q\in H}Z_q$, noting the union is disjoint,
and $\uZ=(Z,\O_V\vert_Z)$. Then we have an isomorphism
$\uphi=(\phi,\phi^\sh):H\t\uY\ra\uZ$, such that
$\us\ci\uphi=\id_\uY$ and $\phi(q\t Y)=Z_q$ for~$q\in H$.
Now $\uZ$ is open in $\uV$, so $\uZ\t_{\us,\uU,\ut}\uZ$ is open in
$\uV\t_{\us,\uU,\ut}\uV$, and we can restrict the morphism $\um:
\uV\t_{\us,\uU,\ut}\uV\ra\uV$ to $\um\vert_{\uZ\t_\uU\uZ}:\uZ
\t_{\us,\uU,\ut}\uZ\ra\uV$. But
\begin{align*}
\uZ\t_{\us,\uU,\ut}\uZ&\cong(H\t\uY)\t_{\ui_\uY\ci\upi_\uY,\uU,\ut}
\uZ\\ &\cong H\t (Z\cap t^{-1}(Y),\O_V\vert_{Z\cap t^{-1}(Y)})
\subseteq H\t\uZ \cong H\t H\t\uY,
\end{align*}
using $\uphi$ an isomorphism and $\us\ci\uphi=\id_\uY$. Write
$\ul{\Phi}:\uZ\t_{\us,\uU,\ut}\uZ\hookra H\t H\t\uY$ for the induced
open embedding. Define a second morphism $\um':\uZ
\t_{\us,\uU,\ut}\uZ\ra\uV$ by $\um'=\uphi\ci(\um_H\t\id_{\uY})
\ci\ul{\Phi}$, where $\um_H:H\t H\ra H$ is the group multiplication
$m_H:H\t H\ra H$, regarded as a morphism of $C^\iy$-schemes.
Following the definitions we find that $\us\ci(\um\vert_{\uZ
\t_\uU\uZ})=\us\ci\um':\uZ\t_{\us,\uU,\ut}\uZ\ra\uY\subset\uU$. Also
$H\subset Z$, and the definition of $m_H$ from $\um$ implies that
$m\vert_{Z\t_UZ}$ and $m'$ coincide on the finite set $H\t_UH$ in
$Z\t_UZ$. Since $\us$ is \'etale, this implies that
$\um\vert_{\uZ\t_\uU\uZ}$ and $\um'$ must coincide near the finite
set $H\t_UH$ in $Z\t_UZ$. Therefore by making the open neighbourhood
$Y$ of $p$ in $U$ smaller, and hence making $W_q,W,Z$ smaller too,
we can assume that~$\um\vert_{\uZ\t_\uU\uZ}=\um'$.
Let us summarize what we have done so far. We have constructed a
finite group $H$, a Hausdorff open neighbourhood $\uY$ of $p$ in
$\uU$, an open and closed subset $\uZ$ of $\us^{-1}(\uY)$ in $\uZ$
which contains $s^{-1}(p)\cap t^{-1}(p)$, and an isomorphism
$\uphi:H\t\uY\ra\uZ$ with $\us\ci\uphi=\upi_\uY$ which identifies
the groupoid multiplication $\um\vert_{\uZ\t_\uU\uZ}$ with the
restriction to $\uZ\t_\uU\uZ$ of the morphism $\um_H\t\id_\uY:H\t
H\t\uY\ra\uY$ from multiplication in the finite group~$H$.
Consider the morphism $\ut\ci\uphi:H\t\uY\ra\uU\supset\uY$. Roughly
speaking, $\ut\ci\uphi$ is an $H$-action on $\uY$. More accurately,
there should an $H$-action on some open subset of $\uU$ containing
$\uY$, but $\uY$ may not be $H$-invariant, so that $\ut\ci\uphi$
need not map $H\t\uY$ to $\uY$. Replace $Y$ by $Y'=\bigcap_{q\in
H}t(Z_q)$, which is an open subset of $Y$ since when $q$ is the
identity $u(p)$ in $H$ we have $t(Z_{u(p)})=s(Z_{u(p)})=Y$, and
$p\in Y'$ as $p=t(q)\in t(Z_q)$ for $q\in H$. Replace $Z_q$ by
$Z_q'=Z_q\cap s^{-1}(Y)$ and $Z$ by $Z'=\bigcup_{q\in H}Z_q'$. Then
using $\um\vert_{\uZ\t_\uU\uZ}=\um'$ we can show that
$s(Z_q')=t(Z_q')=Y'$ for all $q\in H$, so $Y'$ is an $H$-invariant
open set, and $\ut\ci\uphi$ maps $H\t\uY'\ra\uY'$. Restricting the
groupoid axioms shows that $\ut\ci\uphi$ gives an action of $H$
on~$\uY'$.
Now consider the morphism
\begin{equation*}
\us\t\ut\vert_{s^{-1}(Y')\cap t^{-1}(Y')}:\bigl(s^{-1}(Y')\cap
t^{-1}(Y'),\O_V\vert_{s^{-1}(Y')\cap
t^{-1}(Y')}\bigr)\longra\uY'\t\uY'.
\end{equation*}
This is closed, as $\us\t\ut$ is universally closed. Since $Z'$ is
open and closed in $s^{-1}(Y')\cap t^{-1}(Y')$, its complement is
closed, so its image $\bigl\{(s(v),t(v))\in Y'\t Y':v\in V\sm
Z'\bigr\}$ is closed in $Y'$. But $(p,p)$ does not lie in this
image, since $s^{-1}(p)\cap t^{-1}(p)\subseteq Z'$. Thus, by making
the $H$-invariant open neighbourhood $Y'$ of $p$ in $U$ smaller if
necessary, we can suppose that $s^{-1}(Y')\cap t^{-1}(Y')=Z'$.
The quotient $C^\iy$-stack $[\uY'/H]$ is Deligne--Mumford by
Proposition \ref{ag8prop7}, since $Y'$ is Hausdorff. Thus there
exists an open embedding ${\cal Y}_p\hookra[\uY'/H]$ with ${\cal
Y}_p\simeq[\uU_p/G_p]$ for $\uU_p$ affine and $G_p$ finite, which
includes $p$ in its image. The inclusion morphisms $\uY'\hookra\uU$,
$\uZ'\hookra\uV$ induce a 1-morphism $[\uZ'\rra\uY']\hookra
[\uV\rra\uU]$, which is an open embedding as $\uY'$ is open in
$\uU$, $\uZ'$ is open in $\uV$ and $s^{-1}(Y')\cap t^{-1}(Y')=Z'$ in
$V$. Let $i_{{\cal Y}_p}:{\cal Y}_p\ra{\cal X}$ be the composition
${\cal Y}_p \hookra[\uY'/H] \simeq[\uZ'\rra\uY']\hookra
[\uV\rra\uU]\simeq{\cal X}$. Then $i_{{\cal Y}_p}$ is an open
embedding, as it is a composition of open embeddings and
equivalences. This works for all $p\in U$, and $\{{\cal Y}_p:p\in
U\}$ is an open cover of $\cal X$ with ${\cal Y}_p\simeq[\uU_p/G_p]$
for $\uU_p$ affine and $G_p$ finite. Hence $\cal X$ is
Deligne--Mumford. It is separated as $\De_{\cal X}$ is universally
closed, by assumption.
\end{proof}
Suppose $\uf:\uX\ra\uY$ is a separated morphism of $C^\iy$-schemes
with finite fibres. Then $\uf$ universally closed implies $\uf$
proper. Conversely, if $X,Y$ are compactly generated topological
spaces then $\uf$ proper implies $\uf$ universally closed. If
$\uX,\uY$ are locally fair then $X,Y$ are compactly generated, as
they are locally homeomorphic to closed subsets of $\R^n$. Thus, in
Theorem \ref{ag8thm3}, if $\uU,\uV$ are locally fair then we can
replace $\De_{\cal X}$ universally closed by $\De_{\cal X}$ proper,
yielding:
\begin{thm} Let\/ $\cal X$ be a $C^\iy$-stack, and suppose\/ $\cal
X$ has an \'etale atlas $\Pi:\bar\uU\ra{\cal X}$ with\/ $\uU$
locally fair, and the diagonal\/ $\De_{\cal X}:{\cal X}\ra{\cal
X}\t{\cal X}$ is proper and separated. Then\/ $\cal X$ is a
separated, locally fair Deligne--Mumford\/ $C^\iy$-stack.
\label{ag8thm4}
\end{thm}
The same holds with locally good or locally finitely presented in
place of locally fair. If ${\cal X}\simeq[\uV\rra\uU]$ with $\uU$ a
separated $C^\iy$-scheme then $\uV$ is separated if and only if
$\De_{\cal X}$ is separated. We can always choose $\uU$ separated,
by replacing $\uU$ by the disjoint union of an open cover of $\uU$
by affine open subsets. Thus we can replace the condition that
$\De_{\cal X}$ is separated by $\uU,\uV$ separated. Combining this
and the results above proves:
\begin{thm}{\bf(a)} A\/ $C^\iy$-stack\/ $\cal X$ is separated and
Deligne--Mumford if and only if it is equivalent to a groupoid
stack\/ $[\uV\rra\uU]$ where $\uU,\uV$ are separated\/
$C^\iy$-schemes, $\us:\uV\ra\uU$ is \'etale, and
$\us\t\ut:\uV\ra\uU\t\uU$ is universally closed.
\smallskip
\noindent{\bf(b)} A\/ $C^\iy$-stack\/ $\cal X$ is separated,
Deligne--Mumford and locally fair (or locally good, or locally
finitely presented) if and only if it is equivalent to some\/
$[\uV\rra\uU]$ with\/ $\uU,\uV$ separated, locally fair (or locally
good, or locally finitely presented)\/ $C^\iy$-schemes,
$\us:\uV\ra\uU$ \'etale, and\/ $\us\t\ut:\uV\ra\uU\t\uU$ proper.
\label{ag8thm5}
\end{thm}
\subsection{The underlying topological space of a $C^\iy$-stack}
\label{ag86}
Following Noohi \cite[\S 4.3, \S 11]{Nooh} in the case of
topological stacks, we associate a topological space ${\cal X}_\top$
to a $C^\iy$-stack $\cal X$. In \S\ref{ag87}, if $\cal X$ is
Deligne--Mumford, we will also give ${\cal X}_\top$ the structure of
a $C^\iy$-scheme.
\begin{dfn} Let $\cal X$ be a $C^\iy$-stack. Write $\ul{*}$ for the
point $\Spec\R$ in $\CSch$, and $\bar{\ul{*}}$ for the associated
point in $\CSta$. Define ${\cal X}_\top$ to be the set of
2-isomorphism classes $[x]$ of 1-morphisms $x:\bar{\ul{*}}\ra{\cal
X}$. When $i_{\cal U}:{\cal U}\ra{\cal X}$ is an open
$C^\iy$-substack in $\cal X$, write
\begin{equation*}
{\cal U}_{{\cal X},\top}=\bigl\{[u\ci i_{\cal U}]\in {\cal X}_\top:
\text{$u:\bar{\ul{*}}\ra{\cal U}$ is a 1-morphism}\bigr\}
\subseteq{\cal X}_\top.
\end{equation*}
Define ${\cal T}_{{\cal X}_\top}=\bigl\{{\cal U}_{{\cal X},\top}:
i_{\cal U}:{\cal U}\ra{\cal X}$ is an open $C^\iy$-substack in
${\cal X}\bigr\}$.
We claim that ${\cal T}_{{\cal X}_\top}$ is a {\it topology\/} on
${\cal X}_\top$. To see this, note that taking $\cal U$ to be $\cal
X$ or the empty $C^\iy$-substack gives ${\cal X}_\top,\es\in{\cal
T}_{{\cal X}_\top}$. If $i_{\cal U}:{\cal U}\ra{\cal X}$, $i_{\cal
V}:{\cal V}\ra{\cal X}$ are open $C^\iy$-substacks of $\cal X$ then
${\cal W}={\cal U}\t_{i_{\cal U},{\cal X},i_{\cal V}}{\cal V}$ is an
open $C^\iy$-substack of $\cal X$ with ${\cal W}_{{\cal
X},\top}={\cal U}_{{\cal X},\top}\cap{\cal V}_{{\cal X},\top}$, so
${\cal T}_{{\cal X}_\top}$ is closed under finite intersections. If
$\{{\cal U}_a:a\in A\}$ is a family of open $C^\iy$-substacks in
$\cal X$ then each ${\cal U}_a$ is a subcategory of $\cal X$, so
${\cal V}=\bigcup_{a\in A}{\cal U}_a$ is a subcategory of $\cal X$.
It turns out that ${\cal V}$ is a prestack, and the associated stack
$\hat{\cal V}$ is an open $C^\iy$-substack of $\cal X$ with
$\hat{\cal V}_{{\cal X},\top}=\bigcup_{a\in A}{\cal U}_{a\,{\cal
X},\top}$. So ${\cal T}_{{\cal X}_\top}$ is closed under arbitrary
unions.
Thus $({\cal X}_\top,{\cal T}_{{\cal X}_\top})$ is a topological
space, which we call the {\it underlying topological space\/} of
$\cal X$, and usually write as ${\cal X}_\top$. It has the following
properties. If $f:{\cal X}\ra{\cal Y}$ is a 1-morphism of
$C^\iy$-stacks then there is a natural continuous map $f_\top:{\cal
X}_\top\ra{\cal Y}_\top$ defined by $f_\top([x])=[f\ci x]$. If
$f,g:{\cal X}\ra{\cal Y}$ are 1-morphisms and $\eta:f\Ra g$ is a
2-isomorphism then $f_\top=g_\top$. Mapping ${\cal X}\mapsto{\cal
X}_\top$, $f\mapsto f_\top$ and 2-morphisms to identities defines a
2-functor $F_\CSta^\Top:\CSta\ra\Top$, where the category of
topological spaces $\Top$ is regarded as a 2-category with only
identity 2-morphisms.
If $\uX=(X,\O_X)$ is a $C^\iy$-scheme, so that $\bar\uX$ is a
$C^\iy$-stack, then $\bar\uX_\top$ is naturally homeomorphic to $X$,
and we will identify $\bar\uX_\top$ with $X$. If
$\uf=(f,f^\sh):\uX=(X,\O_X)\ra\uY=(Y,\O_Y)$ is a morphism of
$C^\iy$-schemes, so that $\bar\uf:\bar\uX\ra\bar\uY$ is a 1-morphism
of $C^\iy$-stacks, then $\bar\uf_\top:\bar\uX_\top\ra\bar\uY_\top$
is~$f:X\ra Y$.
For a $C^\iy$-stack $\cal X$, we can characterize ${\cal X}_\top$ by
the following universal property. We are given a topological space
${\cal X}_\top$ and for every 1-morphism $f:\bar\uU\ra{\cal X}$ for
a $C^\iy$-scheme $\uU=(U,\O_U)$ we are given a continuous map
$f_\top:U\ra{\cal X}_\top$, such that if $f$ is 2-isomorphic to
$h\ci\bar\ug$ for some morphism $\ug=(g,g^\sh):\uU\ra\uV$ and
1-morphism $h:\uV\ra{\cal X}$ then $f_\top=h_\top\ci g$. If ${\cal
X}_\top'$, $f_\top'$ are alternative choices of data with these
properties then there is a unique continuous map $j:{\cal X}_\top\ra
{\cal X}_\top'$ with $f_\top'=j\ci f_\top$ for all~$f$.
\label{ag8def9}
\end{dfn}
We can think of a $C^\iy$-stack $\cal X$ as being a topological
space ${\cal X}_\top$ equipped with some complicated extra
geometrical structure, just as manifolds and orbifolds are usually
thought of as topological spaces equipped with extra structure
coming from an atlas of charts. As in Noohi \cite[Ex.~4.13]{Nooh},
it is easy to describe ${\cal X}_\top$ using a groupoid presentation
$[\uV\rra\uU]$ of~$\cal X$:
\begin{prop} Let\/ $\cal X$ be equivalent to a groupoid stack\/
$[\uV\rra\uU]$ from a groupoid\/ $(\uU,\uV,\us,\ut,\uu,\ui,\um)$
in\/ $\CSch,$ where\/ $\uU=(U,\O_U),\us=(s,s^\sh),$ and so on.
Define $\sim$ on\/ $U$ by\/ $p\sim p'$ if there exists $q\in V$
with\/ $s(q)=p$ and\/ $t(q)=p'$. Then $\sim$ is an equivalence
relation on $U,$ so we can form the quotient\/ $U/\!\sim,$ with the
quotient topology. There is a natural homeomorphism\/~${\cal
X}_\top\cong U/\!\sim$.
For a quotient stack\/ ${\cal X}\simeq[\uU/\uG]$ we have\/~${\cal
X}_\top\cong U/G$.
\label{ag8prop10}
\end{prop}
Using this we can deduce properties of ${\cal X}_\top$ from
properties of $\cal X$ expressed in terms of $\uV\rra\uU$. For
instance, if $\cal X$ is separated then $s\times t:V\ra U\t U$ is
(universally) closed, and we can take $U$ Hausdorff. But the
quotient of a Hausdorff topological space by a closed equivalence
relation is Hausdorff, yielding:
\begin{lem} Let\/ $\cal X$ be a separated\/ $C^\iy$-stack. Then the
underlying topological space ${\cal X}_\top$ is Hausdorff.
\label{ag8lem}
\end{lem}
Next we discuss {\it stabilizer groups\/} of $C^\iy$-stacks.
\begin{dfn} Let $\cal X$ be a $C^\iy$-stack, and $[x]\in{\cal
X}_\top$. Pick a representative $x$ for $[x]$, so that
$x:\bar{\ul{*}}\ra{\cal X}$ is a 1-morphism. Then there exists a
$C^\iy$-scheme $\uG=(G,\O_G)$, unique up to isomorphism, with
$\bar\uG=\bar{\ul{*}}\t_{x,{\cal X},x}\bar{\ul{*}}$. Applying the
construction of the groupoid in Definition \ref{ag7def15} with
$\Pi:U\ra{\cal X}$ replaced by $x:\bar{\ul{*}}\ra{\cal X}$, we give
$\uG$ the structure of a $C^\iy$-{\it group}. The underlying group
$G$ has a simple interpretation as the group of
2-morphisms~$\eta:x\Ra x$.
With $[x]$ fixed, this $C^\iy$-group $\uG$ is independent of choices
up to noncanonical isomorphism; roughly, $\uG$ is canonical up to
conjugation in $\uG$. We define the {\it stabilizer group\/} (or
{\it isotropy group}, or {\it orbifold group\/}) $\Iso([x])$ of
$[x]$ to be this $C^\iy$-group $\uG$, regarded as a $C^\iy$-group up
to noncanonical isomorphism.
If $f:{\cal X}\ra{\cal Y}$ is a 1-morphism of $C^\iy$-stacks and
$[x]\in{\cal X}_\top$ with $f_\top([x])=[y]\in{\cal Y}_\top$, for
$y=f\ci x$, then we define $f_*:\Iso([x])\ra \Iso([y])$ by
$f_*(\eta)=\id_f*\eta$. One can show that $f_*$ is a group morphism,
and extends naturally to a morphism of $C^\iy$-groups, and that
$f_*$ is independent of choices of $x\in[x]$ and $y\in[y]$ up to
conjugation in~$\Iso([x]),\Iso([y])$.
\label{ag8def10}
\end{dfn}
If $\cal X$ is a Deligne--Mumford $C^\iy$-stack then $\Iso([x])$ is
a finite group for all $[x]$ in ${\cal X}_\top$, which is discrete
as a $C^\iy$-group. We show that $\cal X$ is modelled near $[x]$ on
a quotient $[\uU/\Iso([x])]$, for $\uU$ an affine $C^\iy$-scheme.
\begin{prop} Let\/ $\cal X$ be a Deligne--Mumford $C^\iy$-stack\/ and
$[x]\in{\cal X}_\top,$ so that\/ $\Iso([x])\cong H$ for some finite
group $H$. Then there exists an open $C^\iy$-substack\/ $\cal U$ in
$\cal X$ with\/ $[x]\in{\cal U}_{{\cal X},\top}\subseteq{\cal
X}_\top$ and an equivalence\/ ${\cal U}\simeq[\uY/H],$ where
$\uY=(Y,\O_Y)$ is an affine $C^\iy$-scheme with an action of\/ $H,$
and\/ $[x]\in{\cal U}_{{\cal X},\top}\cong Y/H$ corresponds to a
fixed point of\/ $H$ in\/~$Y$.
\label{ag8prop11}
\end{prop}
\begin{proof} As $\cal X$ is Deligne--Mumford it is covered by open
$C^\iy$-substacks $\cal V$ equivalent to $[\uV/G]$ for $\uV$ affine
and $G$ finite, so we can choose such $\cal V$ with $[x]\in{\cal
V}_{{\cal X},\top}$. Then $\cal V$ has an \'etale atlas
$\Pi:\bar\uV\ra{\cal V}$ and $\De_{\cal V}$ is universally closed
and separated by Proposition \ref{ag8prop9}, so we can apply the
proof of Theorem \ref{ag8thm3} to $\cal V$ for a point $p\in V$ with
$\Pi_*(p)=[x]$. This constructs an open $C^\iy$-substack $\cal U$ in
$\cal V$ equivalent to $[\uY/H]$, where $\uY$ is affine and
$H=\Iso([x])$, as we want.
\end{proof}
If ${\cal X}=\bar\uX$ for some $C^\iy$-scheme $\uX$ then
$\Iso([x])\cong\{1\}$ for all $[x]\in{\cal X}_\top$. Conversely, a
Deligne--Mumford $C^\iy$-stack with trivial stabilizer groups is a
$C^\iy$-scheme. In conventional algebraic geometry, a
Deligne--Mumford stack with trivial stabilizers is an algebraic
space, but need not be a scheme.
\begin{thm} Suppose $\cal X$ is a Deligne--Mumford\/ $C^\iy$-stack
with\/ $\Iso([x])\cong\{1\}$ for all\/ $[x]\in{\cal X}_\top$. Then\/
$\cal X$ is equivalent to\/ $\bar\uX$ for some\/
$C^\iy$-scheme\/~$\uX$.
\label{ag8thm6}
\end{thm}
\begin{proof} As $\Iso([x])\cong\{1\}$ for all $[x]\in{\cal
X}_\top$, by Proposition \ref{ag8prop11} there is an open cover
$\{{\cal X}_a:a\in A\}$ of $\cal X$ with ${\cal
X}_a\simeq[\uX_a/\{1\}]\simeq\bar\uX_a$ for affine $C^\iy$-schemes
$\uX_a$, $a\in A$. Write $i_a:\bar\uX_a\ra{\cal X}$ for the
corresponding open embedding. As $\De_{\cal X}$ is representable,
for $a,b\in A$ the fibre product $\bar\uX_a\t_{i_a,{\cal X},i_b}\bar
X_b$ is represented by a $C^\iy$-scheme $\uX_{ab}=\uX_{ba}$ with
open embeddings $\ui_{ab}:\uX_{ab}\ra\uX_a$,
$\ui_{ba}:\uX_{ba}\ra\uX_b$ identifying $\uX_{ab}$ with open
$C^\iy$-subschemes of~$\uX_a,\uX_b$.
The idea now is that the $C^\iy$-stack $\cal X$ is made by gluing
the $C^\iy$-schemes $\uX_a$ for $a\in A$ together on the overlaps
$\uX_{ab}$, that is, we identify $\uX_a\supset
\ui_{ab}(\uX_{ab})\cong\uX_{ab}=\uX_{ba}\cong
\ui_{ab}(\uX_{ba})\subset\uX_b$. This is similar to the notion of
descent for objects in \S\ref{ag73}, and it is easy to check that
the natural 1-isomorphisms
\begin{equation*}
\bar\uX_{ab}\t_{\cal X}\bar\uX_c\cong\bar\uX_{bc}\t_{\cal X}
\bar\uX_a\cong\bar\uX_{ca}\t_{\cal X}\bar\uX_b\cong\bar\uX_a
\t_{\cal X}\bar\uX_b\t_{\cal X}\bar\uX_c
\end{equation*}
imply the obvious compatibility conditions of the gluing morphisms
$\ui_{ab}$ on triple overlaps, and that $\uX_{aa}\cong\uX_a$. So by
a minor modification of the proof in Proposition \ref{ag8prop1} that
$(\CSch,{\cal J})$ has descent for objects, we construct a
$C^\iy$-scheme $\uX$ with open embeddings $\uj_a:\uX_a\hookra\uX$
such that $\{\uX_a:a\in A\}$ is an open cover of $\uX$, and
$\uX_a\t_{\uj_a,\uX,\uj_b}\uX_b$ is identified with $\uX_{ab}$ for
$a,b\in A$. Then by descent for morphisms in $(\CSch,{\cal J})$,
there exists a 1-morphism $i:\bar\uX\ra{\cal X}$ with $i_a$
2-isomorphic to $i\ci\bar\uj_a$ for all $a\in A$. This $i$ is an
equivalence, so ${\cal X}\simeq\bar\uX$, as we have to prove.
\end{proof}
We show that $\cal X$ being Deligne--Mumford is essential in
Theorem~\ref{ag8thm6}:
\begin{ex} Let the group $\Z^2$ act on $\R$ by $(a,b):x\mapsto
x+a+b\sqrt{2}$ for $a,b\in\Z$ and $x\in\R$. As $\sqrt{2}$ is
irrational, this is a free action. It defines a groupoid
$\Z^2\t\R\rra\R$ in $\Man$ which is \'etale, but not proper.
Applying $F_\Man^\CSch$ gives a groupoid $\ul{\Z}^2\t\ul{\R}\rra
\ul{\R}$ in $\CSch$, and an associated $C^\iy$-stack ${\cal
X}=[\ul{\R}/\ul{\Z}^2]=[\ul{\Z}^2\t\ul{\R}\rra \ul{\R}]$. The
underlying topological space ${\cal X}_\top$ is~$\R/\Z^2$.
Since each orbit of $\Z^2$ in $\R$ is dense in $\R$, ${\cal X}_\top$
has the indiscrete topology, that is, the only open sets are $\es$
and ${\cal X}_\top$. Thus ${\cal X}_\top$ is not homeomorphic to $X$
for any $C^\iy$-scheme $\uX=(X,\O_X)$, as each point of $X$ has an
affine and hence Hausdorff open neighbourhood. Therefore $\cal X$ is
not equivalent to $\bar\uX$ for any $C^\iy$-scheme $\uX$. So $\cal
X$ is not Deligne--Mumford by Theorem \ref{ag8thm6}. Hence,
$C^\iy$-stacks with finite stabilizer groups need not be
Deligne--Mumford.
\label{ag8ex2}
\end{ex}
\subsection{Coarse moduli $C^\iy$-schemes of $C^\iy$-stacks}
\label{ag87}
Let $\cal X$ be a $C^\iy$-stack. Then we can give the underlying
topological space ${\cal X}_\top$ of \S\ref{ag86} the structure of a
$C^\iy$-ringed space.
\begin{dfn} Let $\cal X$ be a $C^\iy$-stack, and write $X$ for the
underlying topological space ${\cal X}_\top$. Define a presheaf of
$C^\iy$-rings $\O_X'$ on $X$ as follows: if $U\subseteq X$ is open
then $U={\cal U}_{{\cal X},\top}$ for some unique $C^\iy$-substack
$\cal U$ in $\cal X$. (Since our definition of $C^\iy$-substack
makes $\cal U$ a strictly full subcategory of $\cal X$, $\cal U$
really is unique, not just unique up to equivalence.) Define
$\O_X'(U)$ to be the set of 2-isomorphism classes $[c]$ of
1-morphisms $c:{\cal U}\ra\bar{\ul{\R}}$. If $f:\R^n\ra\R$ is smooth
and $[c_1],\ldots,[c_n]\in\O_X'(U)$, define
$\Phi_f\bigl([c_1],\ldots,[c_n]\bigr)=\bigl[\bar\uf\ci(c_1\t\cdots\t
c_n)\bigr]$, using the composition ${\cal U}\, {\buildrel
c_1\t\cdots\t c_n\over\longra}\,\bar{\ul{\R}} \t\cdots
\t\bar{\ul{\R}}\,{\smash{\buildrel\bar\uf\over\longra}}\,\bar{\ul{\R}}$.
Then $\O_X'(U)$ is a $C^\iy$-ring.
If $V\subseteq U\subseteq X$ are open then $U,V$ correspond to
unique open $C^\iy$-substacks ${\cal U},{\cal V}$ in $\cal X$ such
that $\cal V$ is also a $C^\iy$-substack of $\cal U$, with inclusion
$i_{\cal{VU}}:{\cal V}\ra{\cal U}$. Define
$\rho_{UV}:\O_X'(U)\ra\O_X'(V)$ by $\rho_{UV}:[c]\mapsto[c\ci
i_{\cal{VU}}]$. Then $\rho_{UV}$ is a morphism of $C^\iy$-rings. It
is now easy to check that $\O_X'$ is a presheaf of $C^\iy$-rings on
$X$. Let $\O_X$ be the associated sheaf of $C^\iy$-rings. Then
$\uX=(X,\O_X)$ is a $C^\iy$-ringed space, which we call the {\it
underlying $C^\iy$-ringed space\/} of~$\cal X$.
For general $\cal X$ this $\uX$ need not be a $C^\iy$-scheme, or
even a local $C^\iy$-ringed space. If $\uX$ is a $C^\iy$-scheme, we
call $\uX$ the {\it coarse moduli\/ $C^\iy$-scheme\/} of $\cal X$.
Coarse moduli $C^\iy$-schemes have the following universal property:
there is a 1-morphism $\pi:{\cal X}\ra\bar\uX$ called the {\it
structural morphism}, such that $\pi_\top:{\cal X}_\top\ra X$ is a
homeomorphism, and if $f:{\cal X}\ra\bar\uY$ is a 1-morphism for any
$C^\iy$-scheme $\uY$ then $f$ is 2-isomorphic to $\bar\ug\ci\pi$ for
some unique morphism~$\ug:\uX\ra\uY$.
\label{ag8def11}
\end{dfn}
\begin{prop} Let\/ $\fC$ be a fair, or good, or finitely
presented\/ $C^\iy$-ring, and\/ $G$ a finite group acting on $\fC$
by automorphisms. Then\/ $\fC^G$ is a fair, or good, or finitely
presented\/ $C^\iy$-ring.
\label{ag8prop12}
\end{prop}
\begin{proof} Choose a finite set of generators for $\fC$, and by
adding the images of these generators under $G$, extend to a set of
(not necessarily distinct) generators $x_1,\ldots,x_n$ for $\fC$, on
which $G$ acts freely by permutation. This gives an exact sequence
$0\hookra I\ra C^\iy(\R^n)\ra\fC\ra 0$, where $C^\iy(\R^n)$ is
freely generated by $x_1,\ldots,x_n$. Here $\R^n$ is a direct sum of
copies of the regular representation of $G$, and $C^\iy(\R^n)\ra\fC$
is $G$-equivariant. Hence $I$ is a $G$-invariant ideal in
$C^\iy(\R^n)$, which is fair, or good, or finitely generated,
respectively. Taking $G$-invariant parts gives an exact sequence
$0\hookra I^G\ra C^\iy(\R^n)^G\,{\buildrel\pi\over\longra}\,\fC^G\ra
0$, where $C^\iy(\R^n)^G,\fC^G$ are clearly $C^\iy$-rings.
As $G$ acts linearly on $\R^n$ it acts by automorphisms on the
polynomial ring $\R[x_1,\ldots,x_n]$. By results from algebraic
geometry, $\R[x_1,\ldots,x_n]^G$ is a finitely presented
$\R$-algebra, so we can choose generators $p_1,\ldots,p_l$ for
$\R[x_1,\ldots,x_n]^G$, which induce a surjective $\R$-algebra
morphism $\R[p_1,\ldots,p_l]\ra \R[x_1,\ldots,x_n]^G$ with finitely
generated kernel generated by~$q_1,\ldots,q_m\in\R[p_1,\ldots,p_l]$.
One can show that any $G$-invariant smooth function on $\R^n$ may be
written as a smooth function of finitely many $G$-invariant
polynomials on $\R^n$, and so as a smooth function of the generators
$p_1,\ldots,p_l$ for $\R[x_1,\ldots,x_n]^G$. Hence $C^\iy(\R^n)^G$
is generated by $p_1,\ldots,p_l$, giving a surjective morphism
$C^\iy(\R^l)\ra C^\iy(\R^n)^G$. One can also show the kernel of this
morphism is the ideal generated by $q_1,\ldots,q_m$. Thus
$C^\iy(\R^n)^G$ is finitely presented.
Also $\fC^G$ is generated by $\pi(p_1),\ldots,\pi(p_l)$, so $\fC^G$
is finitely generated, and we have an exact sequence $0\hookra J\ra
C^\iy(\R^l)\,{\buildrel\pi \over\longra}\,\fC^G\ra 0$, where $J$ is
the ideal in $C^\iy(\R^l)$ generated by $q_1,\ldots,q_m$ and the
lifts to $C^\iy(\R^l)$ of a generating set for the ideal $I^G$
in~$C^\iy(\R^n)^G\cong C^\iy(\R^l)/(q_1,\ldots,q_m)$.
Suppose now that $I$ is fair. Then for $f\in C^\iy(\R^n)^G$, $f$
lies in $I^G$ if and only if $\pi_p(f)\in\pi_p(I)\subset
C^\iy_p(\R^n)$ for all $p\in\R^n$. If $H$ is the subgroup of $G$
fixing $p$ then $H$ acts on $C^\iy_p(\R^n)$, and $\pi_p(f)$ is
$H$-invariant as $f$ is $G$-invariant, and $\pi_p(I)^H=\pi_p(I^G)$.
Thus we may rewrite the condition as $f$ lies in $I^G$ if and only
if $\pi_p(f)\in\pi_p(I^G)\subset C^\iy_p(\R^n)$ for all $p\in\R^n$.
Projecting from $\R^n$ to $\R^n/G$, this says that $f$ lies in $I^G$
if and only if $\pi_p(f)$ lies in $\pi_p(I^G)\subset
\bigl(C^\iy(\R^n)^G\bigr){}_p$ for all $p\in\Spec\bigl(C^\iy(\R^n)^G
\bigr)\cong\R^n/G$. Since $C^\iy(\R^n)^G$ is finitely presented, it
follows as in \cite[Cor.~I.4.9]{MoRe2} that $J$ is fair, so $\fC^G$
is fair.
Suppose $I$ is finitely generated in $C^\iy(\R^n)$, with generators
$f_1,\ldots,f_k$. As $\R^n$ is a sum of copies of the regular
representation of $G$, so that every irreducible representation of
$G$ occurs as a summand of $\R^n$, one can show that $I^G$ is
generated as an ideal in $C^\iy(\R^n/G)$ by the $n(k+1)$ elements
$f_i^G$ and $(f_ix_j)^G$ for $i=1,\ldots,k$ and $j=1,\ldots,n$,
where $f^G=\frac{1}{\md{G}}\sum_{\ga\in G}f\ci\ga$ is the
$G$-invariant part of $f\in C^\iy(\R^n)$. Therefore $J$ is finitely
generated by $q_1,\ldots,q_m$ and lifts of $f_i^G,(f_ix_j)^G$. Hence
if $\fC$ is finitely presented then $\fC^G$ is finitely presented.
Finally suppose $I$ is good, so $I=(f_1,\ldots,f_k,\m_X^\iy)$ for
some closed $X\subseteq\R^n$. Then as for $I$ finitely generated we
find that $J$ is generated by $q_1,\ldots,q_m$, lifts of
$f_i^G,(f_ix_j)^G$ for $i=1,\ldots,k$ and $j=1,\ldots,n$, and
$\m_Y^\iy$, where $Y\subset\R^l$ is the image of the closed set
$X\subset\R^n$ under the closed map
$\R^n\twoheadrightarrow\R^n/G\hookra\R^l$, so that $Y$ is closed.
Thus $J$ is good.
\end{proof}
\begin{thm} Let\/ $\cal X$ be a locally fair, locally good, or
locally finitely presented Deligne--Mumford\/ $C^\iy$-stack. Then
$\cal X$ has a locally fair, locally good, or locally finitely
presented coarse moduli\/ $C^\iy$-scheme, respectively.
\label{ag8thm7}
\end{thm}
\begin{proof} By definition $\cal X$ can be covered by open
$C^\iy$-substacks $\cal U$ equivalent to $[\uY/G]$ for
$\uY=\Spec\fC$ with $\fC$ a fair, or good, or finitely presented
$C^\iy$-ring and $G$ a finite group acting on $\Spec\fC$. Since
$\Spec$ is fully faithful on fair $C^\iy$-rings by Theorem
\ref{ag4thm1} we have $\Aut\uY\cong\Aut(\fC)$, and the action of $G$
on $\uY$ comes from one on $\fC$. So $\fC^G$ is fair, or good, or
finitely presented, by Proposition~\ref{ag8prop12}.
Use the notation of Definition \ref{ag8def11}, and let $U$ be the
open subset of $X={\cal X}_\top$ corresponding to $\cal U$. Then
$U\cong Y/G$ by Proposition \ref{ag8prop10}. But
$\Spec\fC^G=(Y/G,\O_{Y/G})$, so $\Spec\fC^G$ and $\cal U$ have the
same underlying topological space. Open sets in $Y/G$ are of the
form $Z/G$ for $Z\subseteq Y$ open and $G$-invariant. From
Proposition \ref{ag4prop6} for $G$-invariant open sets $Z$ in $Y$,
using $G$-invariant characteristic functions, one can show that
$(U,\O_X'\vert_U)$ is canonically isomorphic to $\Spec\fC^G$. Thus
$\O_X'\vert_U$ is a sheaf, not just a presheaf, so
$\O_X\vert_U\cong\O_X'\vert_U$. Therefore $X$ can be covered by open
subsets $U$ with $(U,\O_X\vert_U)$ isomorphic to $\Spec\fC^G$ for
$\fC^G$ a fair, or good, or finitely presented $C^\iy$-ring, so
$\uX=(X,\O_X)$ is a locally fair, or locally good, or locally
finitely presented $C^\iy$-scheme, respectively.
\end{proof}
\subsection{Orbifolds as Deligne--Mumford $C^\iy$-stacks}
\label{ag88}
Orbifolds (without boundary) are geometric spaces locally modelled
on $\R^n/G$ for $G$ a finite group acting linearly on $\R^n$, just
as manifolds without boundary are geometric spaces locally modelled
on $\R^n$. Much has been written about orbifolds, and there are
several competing, nonequivalent definitions. We are particularly
interested in the question of whether one regards orbifolds as
forming a 1-category, or as a 2-category. See Lerman \cite{Lerm} for
a discussion of this.
Orbifolds were introduced by Satake \cite{Sata}, who called them
V-manifolds. Satake requires $G$ to act effectively on $\R^n$ in the
local models $\R^n/G$, a condition which we omit. Satake intended
orbifolds to be a 1-category, but there were problems with his
definition(s) of smooth map of orbifolds; it was not clear that
smooth maps could be composed, nor whether one could pull back
orbifold vector bundles by smooth maps. For attempts at fixing the
definition, see Chen and Ruan \cite{ChRu}, Moerdijk \cite{Moer} and
Moerdijk and Pronk \cite{MoPr}. Adem, Leida and Ruan \cite{ALR} is a
book on orbifolds, which follows the groupoid point of view of
\cite{Moer,MoPr}. All these authors regard orbifolds as a 1-{\it
category\/} (an ordinary category).
On the other hand, it has been clear for decades that orbifolds are
the analogue in differential geometry of Deligne--Mumford stacks in
algebraic geometry, but Deligne--Mumford stacks are known to form a
2-category. There are two main routes in the literature to defining
a 2-category of orbifolds $\Orb$. The first, as in Pronk \cite{Pron}
and Lerman \cite[\S 3.3]{Lerm}, is to define orbifolds to be
groupoids $(U,V,s,t,u,i,m)$ in the category $\Man$ such that
$s,t:V\ra U$ are \'etale and $s\times t:V\ra U\t U$ is proper. That
is, orbifolds are considered to be {\it proper \'etale Lie
groupoids}, as in Moerdijk and Pronk \cite{Moer,MoPr}. But to define
1-morphisms and 2-morphisms in $\Orb$ one must do more work: one
makes proper \'etale Lie groupoids into a 2-category $\bf Gpoid$,
and then $\Orb$ is defined as a (weak) 2-category localization of
$\bf Gpoid$ at a suitable class of 1-morphisms.
The second route, as in Behrend and Xu \cite[\S 2]{BeXu}, Lerman
\cite[\S 4]{Lerm} and Metzler \cite[\S 3.5]{Metz}, is to define
orbifolds as a class of Deligne--Mumford stacks on the site
$(\Man,{\cal J}_\Man)$ of manifolds discussed in Remark
\ref{ag8rem1}. The relationship between the two routes is discussed
by Behrend and Xu \cite[\S 2.6]{BeXu}, Lerman \cite{Lerm}, and Pronk
\cite{Pron}, who proves the two approaches give equivalent weak
2-categories. We take a similar approach to the second route, but
defining orbifolds as a class of $C^\iy$-stacks, that is, as stacks
on the site $(\CSch,{\cal J})$ rather than on~$(\Man,{\cal
J}_\Man)$.
\begin{dfn} A $C^\iy$-stack $\cal X$ is called an {\it orbifold
without boundary}, or {\it with boundary}, or {\it with corners}, if
it is equivalent to a groupoid stack $[\uV\rra\uU]$ for some
groupoid $(\uU,\uV,\us,\ut,\uu,\ui,\um)$ in $\CSch$ which is the
image under $F_\Manc^\CSch$ of a groupoid $(U,V,s,t,u,i,m)$ in
$\Man$ or $\Manb$ or $\Manc$ respectively, where $s:V\ra U$ is an
\'etale smooth map, and $s\times t:V\ra U\t U$ is a proper smooth
map. That is, $\cal X$ is the $C^\iy$-stack associated to a {\it
proper \'etale Lie groupoid\/} in $\Man,\Manb$ or $\Manc$. An {\it
orbifold\/} will usually mean an orbifold with corners. Write
$\Orb,\Orbb$ and $\Orbc$ for the full 2-subcategories of orbifolds
without boundary, and with boundary, and with corners, in $\CSta$
respectively; though see Remark \ref{ag8rem3} below on whether
$\Orbb,\Orbc$ should be really be full.
If $\cal X$ is an orbifold without boundary then $U,V$ are manifolds
without boundary, so $\uU,\uV$ above are finitely presented affine
$C^\iy$-schemes as in \S\ref{ag42}, and thus $\cal X$ is a
separated, locally finitely presented Deligne--Mumford $C^\iy$-stack
by Theorem \ref{ag8thm5}(b). If $\cal X$ is an orbifold with
boundary, or with corners, then $\uU,\uV$ are good affine
$C^\iy$-schemes and $\cal X$ is a separated, locally good
Deligne--Mumford $C^\iy$-stack. Thus all these 2-categories satisfy:
\begin{equation*}
\xymatrix@C=8pt@R=17pt{\Man \ar[d]^{F_\Man^\CSch} \ar[r]_\subset
& \Manb \ar[d]_{F_\Manb^\CSch} \ar[r]_\subset & \Manc
\ar[dl]^(0.2){F_\Manc^\CSch} \\
\bCSchlfp \ar[r]_\subset \ar[d]^\subset & \bCSchlg \ar[r]_\subset
\ar[d]^\subset & \bCSchlf \ar[d]^\subset \ar[r]_\subset & \bCSch
\ar[d]^\subset \\
\DMCStalfp \ar[r]^\subset & \DMCStalg \ar[r]^\subset & \DMCStalf
\ar[r]^\subset & \DMCSta \ar[r]^(0.55)\subset & \CSta \\
\Orb \ar[u]_(0.4)\subset \ar[r]^\subset & \Orbb
\ar[u]_(0.4)\subset \ar[r]^\subset & \Orbc. \ar[ul]_(0.25)\subset}
\end{equation*}
\label{ag8def12}
\end{dfn}
\begin{rem} If we define $\Orbb,\Orbc$ as {\it full\/}
2-subcategories of $\CSta$ then 1-morphisms in $\Orbb,\Orbc$
correspond not to smooth maps but to {\it weakly smooth maps\/} of
manifolds with boundary or corners, as in Proposition \ref{ag3prop2}
and Corollary \ref{ag4cor2}. So following the philosophy of
\cite{Joyc1}, arguably one should take 1-morphisms in $\Orbb,\Orbc$
to be the proper subset of {\it smooth\/} 1-morphisms between
orbifolds in $\CSta$, that is, those 1-morphisms which are locally
modelled on smooth maps of manifolds, rather than weakly smooth
maps. Then $\Orbb,\Orbc$ would not be full 2-subcategories.
\label{ag8rem3}
\end{rem}
Our next theorem compares our definition of orbifold with those in
the literature. To prove it, we show that the truncation 2-functor
$F_\CSta^\ManSta$ of Remark \ref{ag8rem1} maps our 2-subcategory
$\Orb$ in $\CSta$ to the 2-subcategory of orbifolds as stacks on
$(\Man,{\cal J}_\Man)$ in \cite[\S 3.4]{Metz} or \cite[\S 4]{Lerm}
(this is obvious, as both 2-subcategories are defined as objects
presented by proper \'etale Lie groupoids), and that
$F_\CSta^\ManSta\vert_\Orb$ is an equivalence of 2-categories (this
follows from $F_\Man^\CSch$ being full and faithful, and orbifolds
being locally modelled on manifolds). Thus our 2-category of
orbifolds is equivalent to those in \cite[\S 3.4]{Metz}, \cite[\S
4]{Lerm}, and the rest of the theorem follows from the references.
\begin{thm} The $2$-category $\Orb$ of orbifolds without boundary
defined above is equivalent to the $2$-categories of orbifolds
considered as stacks on $\Man$ defined in Metzler\/ {\rm\cite[\S
3.4]{Metz}} and Lerman\/ {\rm\cite[\S 4]{Lerm},} and also equivalent
as a weak\/ $2$-category to the weak\/ $2$-categories of orbifolds
regarded as proper \'etale Lie groupoids defined in Pronk\/
{\rm\cite{Pron}} and Lerman\/~{\rm\cite[\S 3.3]{Lerm}}.
Furthermore, the homotopy $1$-category $\Orbho$ of\/ $\Orb$ (that
is, the category whose objects are objects in\/ $\Orb,$ and whose
morphisms are $2$-isomorphism classes of\/ $1$-morphisms in\/
$\Orb$) is equivalent to the $1$-category of orbifolds regarded as
proper \'etale Lie groupoids defined in Moerdijk\/~{\rm\cite{Moer}}.
\label{ag8thm8}
\end{thm}
Since equivalent (2-)categories are considered to be `the same', the
basic moral of Theorem \ref{ag8thm8} is that our notion of orbifold
gives essentially the same geometric objects as those considered by
other recent authors.
We could have taken a different approach: we could instead have
defined a 2-category of orbifolds $\Orb$ following one of the routes
in \cite{Lerm,Metz,Pron} or elsewhere, and then defined an embedding
2-functor $F_\Orb^\CSta:\Orb\ra\CSta$ and shown it was fully
faithful, as we did for $F_\Man^\CSch:\Man\ra\CSch$. We chose not to
do this because as above there are several competing ways to
construct $\Orb$ as a 2-category, and all are rather complicated, so
having already set up a theory of $C^\iy$-stacks, this was the
fastest way to our goal.
From \S\ref{ag86} an orbifold $\cal X$ has an {\it underlying
topological space\/} ${\cal X}_\top$, which is Hausdorff by Lemma
\ref{ag8lem}, and each $[x]\in{\cal X}_\top$ has an orbifold group
$\Iso([x])$, which is a finite group. Also by Theorem \ref{ag8thm7}
$\cal X$ has a {\it coarse moduli\/ $C^\iy$-scheme\/} $\uX$, with
underlying topological space ${\cal X}_\top$, and $\uX$ is locally
finitely presented if $\cal X$ is without boundary, and locally good
if $\cal X$ has boundary or corners.
By Corollary \ref{ag4cor2} $F_\Manc^\CSch$ takes transverse fibre
products in $\Man,\Manc$ to fibre products in $\CSch$. As fibre
products of orbifolds are locally modelled on fibre products of
manifolds, and fibre products of Deligne--Mumford $C^\iy$-stacks are
locally modelled on fibre products of $C^\iy$-schemes, we deduce:
\begin{cor} Transverse fibre products in\/ $\Orb,\Orbc$ involving
smooth rather than weakly smooth\/ $1$-morphisms agree with the
corresponding fibre products in\/~$\CSta$.
\label{ag8cor2}
\end{cor}
The next example illustrates the 2-categorical nature of orbifolds.
\begin{ex} Write $\ul{*}=\Spec\R$ for the point in $\CSch$, and
$\bar{\ul{*}}$ for its image in $\CSta$. Let $H$ be a nontrivial
finite group. Then $H$ acts trivially on $\ul{*}$, so we can form
the quotient $C^\iy$-stack $[\ul{*}/H]$. Both $\bar{\ul{*}}$ and
$[\ul{*}/H]$ are orbifolds, points with stabilizer groups $\{1\}$
and $H$. There is a unique 1-morphism $i:\bar{\ul{*}}\ra[\ul{*}/H]$
corresponding to the group morphism $\{1\}\ra H$. For any
$C^\iy$-stack $\cal X$ there is a unique 1-morphism $\pi:{\cal
X}\ra\bar{\ul{*}}$, as $\bar{\ul{*}}$ is a terminal object
in~$\CSta$.
Consider the fibre product $\bar{\ul{*}}\t_{i,[\ul{*}/H],i}
\bar{\ul{*}}$. In Proposition \ref{ag8prop8} we have
$\uX=\uY=\uZ=\ul{*}$, $F=G=\{1\}$, and $\uW=H$. Thus we have a
2-Cartesian diagram:
\e
\begin{gathered}
\xymatrix@C=40pt@R=7pt{ \bar H \ar[r]_(0.25){\pi} \ar[d]_{\pi}
\drtwocell_{}\omit^{}\omit{^\eta} & \bar{\ul{*}} \ar[d]^i \\
\bar{\ul{*}} \ar[r]^(0.65)i & [\ul{*}/H].}
\end{gathered}
\label{ag8eq5}
\e
That is, $\bar{\ul{*}}\t_{i,[\ul{*}/H],i} \bar{\ul{*}}$ is the
disjoint union of $\md{H}$ copies of the point~$\bar{\ul{*}}$.
Observe that \eq{ag8eq5} only makes sense if orbifolds are a
2-category, not a 1-category. If we regard orbifolds as a
1-category, via the homotopy 1-category $\Orbho$ of Theorem
\ref{ag8thm8}, then the 1-category fibre product
$\bar{\ul{*}}\t_{i,[\ul{*}/H],i} \bar{\ul{*}}$ in $\Orbho$ would be
one point, not $\md{H}$ points. All the nontrivial information in
\eq{ag8eq5} is encoded in the 2-morphism $\eta$. Although there is
only one 1-morphism $i:\bar{\ul{*}}\ra [\ul{*}/H]$, there are
$\md{H}$ different 2-morphisms $\ze:i\Ra i$, which correspond to the
elements of $H$. In \eq{ag8eq5} the 1-morphism
$i\ci\pi:H\ra[\ul{*}/H]$ is the disjoint union of $\md{H}$ copies of
$i:\bar{\ul{*}}\ra[\ul{*}/H]$, and 2-morphism $\eta:i\ci\pi\Ra
i\ci\pi$ is the disjoint union of the $\md{H}$ different
2-morphisms~$\ze:i\Ra i$.
\label{ag8ex3}
\end{ex}
This example illustrates the fact that the underlying topological
space functor $F_\CSta^\Top:\CSta\ra\Top$ from \S\ref{ag86} may not
preserve fibre products, since the fibre product
$F_\CSta^\Top(\bar{\ul{*}})\t_{F_\CSta^\Top([\ul{*}/H])}
F_\CSta^\Top(\bar{\ul{*}})$ is one point, not $\md{H}$ points. In
contrast, the corresponding functor $\smash{F_\CSch^\Top}:
\CSch\ra\Top$ on $C^\iy$-schemes does preserve fibre products.
\section{Sheaves on Deligne--Mumford $C^\iy$-stacks}
\label{ag9}
Finally we discuss sheaves of $\O_{\cal X}$-modules, quasicoherent
sheaves, and coherent sheaves on Deligne--Mumford $C^\iy$-stacks
$\cal X$, generalizing \S\ref{ag6} for $C^\iy$-schemes. Some
references on sheaves on orbifolds or stacks are Behrend and Xu
\cite[\S 3.1]{BeXu}, Deligne and Mumford \cite[Def.~4.10]{DeMu},
Heinloth \cite[\S 4]{Hein}, Laumon and Moret-Bailly \cite[\S
13]{LaMo}, and Moerdijk and Pronk \cite[\S 2]{MoPr}. Our definition
of sheaves on $C^\iy$-stacks is closest to~\cite{Hein,MoPr}.
\subsection{$\O_{\cal X}$-modules, quasicoherent and coherent sheaves}
\label{ag91}
We build our notions of sheaves on Deligne--Mumford $C^\iy$-stacks
from those of sheaves on $C^\iy$-schemes in \S\ref{ag6}, by lifting
to \'etale covers.
\begin{dfn} Let $\cal X$ be a Deligne--Mumford $C^\iy$-stack. Define
a category ${\cal C}_{\cal X}$ to have objects pairs $(\uU,u)$ where
$\uU$ is a $C^\iy$-scheme and $u:\bar\uU\ra{\cal X}$ is an \'etale
morphism, and morphisms $(\uf,\eta):(\uU,u)\ra(\uV,v)$ where
$\uf:\uU\ra\uV$ is an \'etale morphism of $C^\iy$-schemes, and
$\eta:u\Ra v\ci\bar\uf$ is a 2-isomorphism. (Here $\uf$ \'etale is
implied by $u,v$ \'etale and $u\cong v\ci\bar\uf$.) If
$(\uf,\eta):(\uU,u)\ra(\uV,v)$ and $(\ug,\ze):(\uV,v)\ra(\uW,w)$ are
morphisms in ${\cal C}_{\cal X}$ then we define the composition
$(\ug,\ze)\ci(\uf,\eta)$ to be $(\ug\ci\uf,\th):(\uU,u)\ra(\uW,w)$,
where $\th$ is the composition of 2-morphisms across the diagram:
\begin{equation*}
\xymatrix@C=20pt@R=7pt{ \bar\uU \ar[dr]^(0.6){\bar\uf}
\ar@/^/@<1ex>[drrr]_(0.4)u \ar[dd]_{\overline{\ug\ci\uf}}
\dduppertwocell_{}\omit^{}\omit{<-2.5>^{\id}} & \\
& \bar\uV \ar[rr]^(0.5){v} \ar[dl]^(0.45){\bar\ug} &
\ultwocell_{}\omit^{}\omit{^\eta} & {\cal X}. \\
\bar\uW \ar@/_/@<-1ex>[urrr]^(0.35)w && {}
\ultwocell_{}\omit^{}\omit{^\ze} }
\end{equation*}
Define a {\it sheaf of\/ $\O_{\cal X}$-modules\/} $\cal E$, or just
an $\O_{\cal X}$-{\it module\/} $\cal E$, to assign a sheaf of
$\O_U$-modules ${\cal E}(\uU,u)$ on $\uU=(U,\O_U)$ for all objects
$(\uU,u)$ in ${\cal C}_{\cal X}$, and an isomorphism of
$\O_U$-modules ${\cal E}_{(\uf,\eta)}:\uf^*({\cal E}(\uV,v))\ra{\cal
E}(\uU,u)$ for all morphisms $(\uf,\eta):(\uU,u)\ra(\uV,v)$ in
${\cal C}_{\cal X}$, such that for all $(\uf,\eta),(\ug,\ze),
(\ug\ci\uf,\th)$ as above the following diagram of isomorphisms of
sheaves of $\O_U$-modules commutes:
\e
\begin{gathered}
\xymatrix@C=7pt@R=7pt{ (\ug\ci\uf)^*\bigl({\cal E}(\uW,w)\bigr)
\ar[rrrrr]_(0.4){{\cal E}_{(\ug\ci\uf,\th)}}
\ar[dr]_{I_{\uf,\ug}({\cal
E}(\uW,w))\,\,\,\,\,\,\,\,\,\,\,\,{}} &&&&& {\cal E}(\uU,u), \\
& \uf^*\bigl(\ug^*({\cal E}(\uW,w)\bigr) \ar[rrr]^(0.55){\uf^*({\cal
E}_{(\ug,\ze)})} &&& \uf^*\bigl({\cal E}(\uV,v)\bigr)
\ar[ur]_{{}\,\,\,{\cal E}_{(\uf,\eta)}} }
\end{gathered}
\label{ag9eq1}
\e
for $I_{\uf,\ug}({\cal E})$ as in Remark~\ref{ag6rem2}.
A {\it morphism of sheaves of\/ $\O_{\cal X}$-modules\/} $\phi:{\cal
E}\ra{\cal F}$ assigns a morphism of $\O_U$-modules
$\phi(\uU,u):{\cal E}(\uU,u)\ra{\cal F}(\uU,u)$ for each object
$(\uU,u)$ in ${\cal C}_{\cal X}$, such that for all morphisms
$(\uf,\eta):(\uU,u)\ra(\uV,v)$ in ${\cal C}_{\cal X}$ the following
commutes:
\begin{equation*}
\xymatrix@C=40pt@R=7pt{ \uf^*\bigl({\cal E}(\uV,v)\bigr)
\ar[d]_{\uf^*(\phi(\uV,v))} \ar[r]_{{\cal E}_{(\uf,\eta)}} & {\cal
E}(\uU,u) \ar[d]^{\phi(\uU,u)} \\ \uf^*\bigl({\cal F}(\uV,v)\bigr)
\ar[r]^{{\cal F}_{(\uf,\eta)}} & {\cal F}(\uU,u). }
\end{equation*}
We call $\cal E$ {\it quasicoherent}, or {\it coherent}, or a {\it
vector bundle of rank\/} $n$, if ${\cal E}(\uU,u)$ is quasicoherent,
or coherent, or a vector bundle of rank $n$, respectively, for all
$(\uU,u)\in{\cal C}_{\cal X}$. Write $\OcXmod$ for the category of
$\O_{\cal X}$-modules, and $\qcoh({\cal X})$, $\coh({\cal X})$ for
the full subcategories of quasicoherent and coherent sheaves.
\label{ag9def1}
\end{dfn}
\begin{rem}{\bf(a)} Here is a second, different way to define $\O_{\cal
X}$-modules, closer to \cite[\S 3.1]{BeXu}, \cite[Def.~4.10]{DeMu}.
Define a Grothendieck topology ${\cal J}_{\cal X}$ on ${\cal
C}_{\cal X}$ to have coverings
$\bigl\{(\ui_a,\eta_a):(\uU_a,u_a)\ra(\uU,u)\bigr\}{}_{a\in A}$
where $\ui_a:\uU_a\ra\uU$ is an open embedding for all $a\in A$ and
$U=\bigcup_{a\in A}i_a(U_a)$. Then $({\cal C}_{\cal X},{\cal
J}_{\cal X})$ is a site.
We can now use the standard notion of {\it sheaves on a site}, as in
Artin \cite{Arti} or Metzler~\cite[\S 2.1]{Metz}. For all $(\uU,u)$
in ${\cal C}_{\cal X}$, define a $C^\iy$-ring $\O_{\cal
X}(\uU,u)=\O_U(U)$, where $\uU=(U,\O_U)$. For all morphisms
$(\uf,\eta):(\uV,v)\ra(\uU,u)$, define a morphism of $C^\iy$-rings
$\rho_{(\uU,u)(\uV,v)}:\O_{\cal X}(\uU,u)\ra\O_{\cal X}(\uV,v)$ by
$\rho_{(\uU,u)(\uV,v)}=f^\sh(U):\O_U(U)\ra\O_V(V)$. Then $\O_{\cal
X}$ is a {\it sheaf of\/ $C^\iy$-rings on the site\/} $({\cal
C}_{\cal X},{\cal J}_{\cal X})$.
Define a {\it sheaf of\/ $\O_{\cal X}$-modules\/} ${\cal E}'$ to be
a sheaf of modules of $\O_{\cal X}$ on $({\cal C}_{\cal X},{\cal
J}_{\cal X})$. That is, ${\cal E}'$ assigns an $\O_{\cal
X}(\uU,u)$-module ${\cal E}'(\uU,u)$ for all $(\uU,u)$ in ${\cal
C}_{\cal X}$, and a linear map ${\cal E}'_{(\uf,\eta)}:{\cal
E}(\uU,u)\ra{\cal E}(\uV,v)$ for all $(\uf,\eta):(\uV,v)\ra(\uU,u)$
in ${\cal C}_{\cal X}$, such that the analogue of \eq{ag6eq1}
commutes, and the usual axioms for sheaves on a site hold.
If $\cal E$ is as in Definition \ref{ag9def1} then defining ${\cal
E}'(\uU,u)=\Ga\bigl({\cal E}(\uU,u)\bigr)$ gives an $\O_{\cal
X}$-module in the sense of this second definition. Conversely, any
$\O_{\cal X}$-module in this second sense extends to one in the
first sense uniquely up to canonical isomorphism. Thus the two
definitions yield equivalent categories.
\smallskip
\noindent{\bf(b)} As $\O_{\cal X}$-modules are a kind of sheaves of
sets on a site, not sheaves of categories on a site as stacks are,
$\OcXmod$ is a category not a 2-category.
\smallskip
\noindent{\bf(c)} If $\cal X$ is locally fair, or locally good, or
locally finitely presented, then $\uU$ is also locally fair, or
locally good, or locally finitely presented, for all $(\uU,u)$ in
${\cal C}_{\cal X}$, since $u:\bar\uU\ra{\cal X}$ is \'etale.
\smallskip
\noindent{\bf(d)} In Definition \ref{ag9def1} we require the
1-morphisms $u,v,w$ and morphisms $\uf,\ug$ to be {\it \'etale}.
This is important in several places below: for instance, if
$\uf:\uU\ra\uV$ is \'etale then $\uf^*:\O_V$-mod$\,\ra\O_U$-mod is
exact, not just right exact, which is needed to show $\OcXmod$ is
abelian, and also $\Om_\uf:\uf^*(T^*\uV)\ra T^*\uU$ is an
isomorphism, which is needed to define the cotangent sheaf~$T^*{\cal
X}$. We restricted to {\it Deligne--Mumford\/} $C^\iy$-stacks $\cal
X$ in order to be able to use \'etale (1-)morphisms in this way. For
$C^\iy$-stacks $\cal X$ which do not admit an \'etale atlas, the
approach above is inadequate and would need to be modified.
\smallskip
\noindent{\bf(e)} Our notion of vector bundles $\cal E$ over $\cal
X$ correspond to {\it orbifold vector bundles\/} when $\cal X$ is an
orbifold. That is, the stabilizer groups $\Iso([x])$ of $\cal X$ are
allowed to act nontrivially on the vector space fibres ${\cal
E}\vert_x$ of~$\cal E$.
\label{ag9rem1}
\end{rem}
Now $\OcXmod$ is an {\it abelian category}, where $0\ra{\cal
E}\,{\buildrel\phi\over\longra}\,{\cal F}\,{\buildrel\psi\over
\longra}\,{\cal G}\ra 0$ is exact in $\OcXmod$ if and only if
$0\ra{\cal E}(\uU,u)\,{\buildrel\phi(\uU,u)\over\longra}\,{\cal
F}(\uU,u)\,{\buildrel\psi(\uU,u)\over\longra}\,{\cal G}(\uU,u)\ra 0$
is exact in $\O_U$-mod for all $(\uU,u)$ in ${\cal C}_{\cal X}$. To
prove this, note that each $\O_U$-mod in Definition \ref{ag9def1} is
abelian, and the functors $\uf^*$ are exact (not just right exact)
as $\uf$ is \'etale. Thus Corollary \ref{ag6cor1} and Proposition
\ref{ag6prop3} imply:
\begin{prop} Let\/ $\cal X$ be a Deligne--Mumford\/ $C^\iy$-stack.
Then\/ $\OcXmod$ is an abelian category, and\/ $\qcoh({\cal X})$ is
closed under kernels, cokernels and extensions in $\OcXmod,$ so it
is also an abelian category. Also\/ $\coh({\cal X})$ is closed under
cokernels and extensions in\/ $\OcXmod,$ but it may not be closed
under kernels in\/ $\OcXmod,$ so may not be abelian. If\/ $\cal X$
is locally fair then\/~$\qcoh({\cal X})\!=\!\OcXmod$.
\label{ag9prop1}
\end{prop}
\begin{ex} Let $\uX$ be a $C^\iy$-scheme. Then ${\cal X}=\bar\uX$ is
a Deligne--Mumford $C^\iy$-stack. We will define an inclusion
functor ${\cal I}_\uX:\OXmod\ra\OcXmod$ which induces equivalences
between the categories $\OXmod,\qcoh(\uX),\coh(\uX)$ defined in
\S\ref{ag6} and $\OcXmod,\qcoh({\cal X}),\coh({\cal X})$ above. This
shows that our notions of sheaves for $C^\iy$-stacks are good
generalizations of those for $C^\iy$-schemes in~\S\ref{ag6}.
Let $\cal E$ be an object in $\OXmod$. If $(\uU,u)$ is an object in
${\cal C}_{\cal X}$ then $u:\bar\uU\ra{\cal X}=\bar\uX$ is a
1-morphism, so as $\CSch,\bCSch$ are equivalent (2-)categories $u$
is 1-isomorphic to $\bar\uu:\bar\uU\ra\bar\uX$ for some unique
morphism $\uu:\uU\ra\uX$. Define ${\cal E}'(\uU,u)=\uu^*({\cal E})$.
If $(\uf,\eta):(\uU,u)\ra(\uV,v)$ is a morphism in ${\cal C}_{\cal
X}$ and $\uu,\uv$ are associated to $u,v$ as above, so that
$\uu=\uv\ci\uf$, then define
\begin{equation*}
\smash{{\cal E}'_{(\uf,\eta)}= I_{\uf,\uv}({\cal E})^{-1}:\uf^*({\cal
E}'(\uV,v))=\uf^*\bigl(\uv^*({\cal E})\bigr)\ra (\uv\ci\uf)^*({\cal
E})={\cal E}'(\uU,u).}
\end{equation*}
Then \eq{ag9eq1} commutes for all $(\uf,\eta),(\ug,\ze)$, so ${\cal
E}'$ is an $\O_{\cal X}$-module.
If $\phi:{\cal E}\ra{\cal F}$ is a morphism of $\O_X$-modules then
we define a morphism $\phi':{\cal E}'\ra{\cal F}'$ in $\OcXmod$ by
$\phi'(\uU,u)=\uu^*(\phi)$ for $\uu$ associated to $u$ as above.
Then defining ${\cal I}_\uX:{\cal E}\mapsto{\cal E}'$, ${\cal
I}_\uX:\phi\mapsto\phi'$ gives a functor $\OXmod\ra\OcXmod$. There
is a natural inverse construction: if $\ti{\cal E}$ is an object in
$\OcXmod$ then $\ti{\cal E}(\uX,\bar\id_\uX)$ is an object in
$\OcXmod$, and $\ti{\cal E}$ is canonically isomorphic to ${\cal
I}_\uX\bigl(\ti{\cal E}(\uX,\bar\id_\uX)\bigr)$. Using this we can
show ${\cal I}_\uX$ is an equivalence of categories.
\label{ag9ex1}
\end{ex}
\subsection{Writing sheaves in terms of a groupoid presentation}
\label{ag92}
Let $\cal X$ be a Deligne--Mumford $C^\iy$-stack. Then $\cal X$
admits an \'etale atlas $\Pi:\bar\uU\ra{\cal X}$, and as in
\S\ref{ag75} from $\Pi$ we can construct a groupoid
$(\uU,\uV,\us,\ut,\uu,\ui,\um)$ in $\CSch$, with $\us,\ut:\uV\ra\uU$
\'etale, such that $\cal X$ is equivalent to the groupoid stack
$[\uV\rra\uU]$, and we have a 2-Cartesian diagram
\begin{equation*}
\xymatrix@C=40pt@R=7pt{ \bar\uV \ar[r]_(0.2){\bar\ut} \ar[d]_{\bar\us}
\drtwocell_{}\omit^{}\omit{^\eta} & \bar{\uU} \ar[d]^\Pi \\
\bar{\uU} \ar[r]^(0.7)\Pi & {\cal X}.}
\end{equation*}
We can now consider the objects $(\uU,\Pi)$ and $(\uV,\Pi\ci\us)$ in
${\cal C}_{\cal X}$, and the two morphisms
$(\us,\id_{\Pi\ci\us}):(\uV,\Pi\ci\us)\ra(\uU,\Pi)$
and~$(\ut,\eta):(\uV,\Pi\ci\us)\ra(\uU,\Pi)$.
Now let $\cal E$ be an object in $\OcXmod$. Then we have an
$\O_U$-module $E={\cal E}(\uU,\Pi)$, an $\O_V$-module $E'={\cal
E}(\uV,\Pi\ci\us)$, and isomorphisms of $\O_V$-modules ${\cal
E}_{(\us,\id_{\Pi\ci\us})}:\us^*(E)\ra E'$ and ${\cal
E}_{(\ut,\eta)}:\ut^*(E)\ra E'$. Hence $\Phi={\cal
E}_{(\ut,\eta)}^{-1}\ci {\cal E}_{(\us,\id_{\Pi\ci\us})}$ is an
isomorphism of $\O_V$-modules $\Phi:\us^*(E)\ra\ut^*(E)$.
We also have a 2-commutative diagram with all squares 2-Cartesian:
\begin{equation*}
\xymatrix@!0@C=50pt@R=15pt{ & \bar\uW \ar[dl]_(0.6){\bar\upi_1}
\ar[ddr]^(0.4){\bar\upi_2} \ar[rr]^(0.6){\bar\um}
&& \bar\uV \ar[dl]^(0.4){\bar\ut} \ar[ddr]^{\bar\us} \\
\bar\uV \ar[ddr]_{\bar\us} \ar[rr]_(0.4){\bar\ut} && \bar\uU
\ar[ddr]^(0.35){\Pi} \\ && \bar\uV \ar[rr]^(0.6){\bar\us}
\ar[dl]_(0.4){\bar\ut} && \bar\uU \ar[dl]^(0.4){\Pi} \\
& \bar\uU \ar[rr]^(0.6){\Pi} && {\cal X},}
\end{equation*}
omitting 2-morphisms, where $\uW=\uV\t_{\bar\us,\bar\uU,\ut}
\bar\uV$, and $\upi_1,\upi_2:\uW\ra\uV$ are projections to the first
and second factors in the fibre product. So we have an object
$(\uW,\Pi\ci\bar\us\ci\bar\upi_1)$ in ${\cal C}_{\cal X}$, and we
can define $E''={\cal E}(\uW,\Pi\ci\bar\us\ci\bar\upi_1)$. Then we
have a commutative diagram of isomorphisms in $\O_W$-mod:
\e
\begin{gathered}
\xymatrix@!0@C=60pt@R=25pt{ & E'' && \um^*(E') \ar[ll]_(0.4){{\cal
E}_{(\um,\th_3)}}
\\
\upi_1^*(E') \ar[ur]^(0.4){{\cal E}_{(\upi_1,\th_1)}} &&
{\begin{subarray}{l}(\ut\ci\upi_1)^*(E)=\\
{}\quad(\ut\ci\um)^*(E)\end{subarray}}
\ar[ll]^(0.6){\begin{subarray}{l} \upi_1^*({\cal E}_{(\ut,\eta)})\\
\ci I_{\upi_1,\ut}(E)\end{subarray}}
\ar[ur]_(0.7){\begin{subarray}{l} \um^*({\cal E}_{(\ut,\eta)})\\
\ci I_{\um,\ut}(E)\end{subarray}}
\\
&& \upi_2^*(E') \ar@<1ex>[uul]_(0.7){{\cal E}_{(\upi_2,\th_2)}} &&
{\begin{subarray}{l}(\us\ci\upi_2)^*(E)=\\
(\us\ci\um)^*(E)\end{subarray}} \ar[uul]_{\begin{subarray}{l}
\um^*({\cal E}_{(\us,\id_{\Pi\ci\us})})\\
\ci I_{\um,\us}(E)\end{subarray}} \ar[ll]^(0.6){\begin{subarray}{l}
\upi_2^*({\cal E}_{(\us,\id_{\Pi\ci\us})})\\ \ci
I_{\upi_2,\us}(E)\end{subarray}} \ar@{-->}@(d,l)[dlll]_(0.53){\be}
\ar@{-->}[ull]^{\al}
\\
& {\begin{subarray}{l}(\us\ci\upi_1)^*(E)= (\ut\ci\upi_2)^*(E)
\end{subarray}}\ar[uul]^(0.65){\begin{subarray}{l} \upi_1^*({\cal
E}_{(\us,\id_{\Pi\ci\us})})\\ \ci I_{\upi_1,\us}(E)\end{subarray}}
\ar[ur]^(0.4){\begin{subarray}{l} \upi_2^*({\cal E}_{(\ut,\eta)})\\
\ci I_{\upi_2,\ut}(E)\end{subarray}} \ar@{-->}[uur]_(0.6){\ga} }
\end{gathered}
\label{ag9eq2}
\e
Here the morphisms `$\dashra$' are given by $\al=I_{\um,\ut}(E
)^{-1}\ci \um^*(\Phi)\ci I_{\um,\us}(E)$,
$\be=I_{\upi_2,\ut}(E)^{-1}\ci\upi_2^*(\Phi)\ci I_{\upi_2,\us}(E)$
and $\ga=I_{\upi_1,\ut}(E)^{-1} \ci\upi_1^*(\Phi)\ci
I_{\upi_1,\us}(E)$, and as \eq{ag9eq2} commutes we have
$\al=\ga\ci\be$. This motivates:
\begin{dfn} Let $(\uU,\uV,\us,\ut,\uu,\ui,\um)$ be a groupoid in
$\CSch$, with $\us,\ut:\uV\ra\uU$ \'etale, which we write as
$\uV\rra\uU$ for short. Define a $(\uV\rra\uU)$-{\it module\/} to be
a pair $(E,\Phi)$ where $E$ is an $\O_U$-module and
$\Phi:\us^*(E)\ra\ut^*(E)$ is an isomorphism of $\O_V$-modules, such
that
\e
\begin{split}
I_{\um,\ut}(E)^{-1}\ci\um^*(\Phi)\ci I_{\um,\us}(E)=\,
&\bigl(I_{\upi_1,\ut}(E)^{-1}\ci\upi_1^*(\Phi)\ci
I_{\upi_1,\us}(E)\bigr)\ci\\
&\bigl(I_{\upi_2,\ut}(E)^{-1}\ci\upi_2^*(\Phi)\ci
I_{\upi_2,\us}(E)\bigr)
\end{split}
\label{ag9eq3}
\e
in morphisms of $\O_W$-modules
$(\us\ci\um)^*(E)\ra(\ut\ci\um)^*(E)$. Define a {\it morphism of\/
$(\uV\rra\uU)$-modules\/} $\phi:(E,\Phi)\ra(F,\Psi)$ to be a
morphism of $\O_U$-modules $\phi:E\ra F$ such that
$\Psi\ci\us^*(\phi)=\ut^*(\phi)\ci\Phi:\us^*(E)\ra\ut^*(F)$. Then
$(\uV\rra\uU)$-modules form an {\it abelian category\/}
$(\uV\rra\uU)$-mod. The construction above shows that if $\cal X$ is
a Deligne--Mumford $C^\iy$-stack equivalent to $[\uV\rra\uU]$ with
atlas $\Pi:\bar\uU\ra{\cal X}$ then we have a {\it functor\/}
$F_\Pi:\OcXmod\ra(\uV\rra\uU)$-mod defined by $F_\Pi:{\cal
E}\mapsto\bigl({\cal E}(\uU,\Pi),{\cal E}_{(\ut,\eta)}^{-1}\ci {\cal
E}_{(\us,\id_{\Pi\ci\us})}\bigr)$
and~$F_\Pi:\phi\mapsto\phi(\uU,\Pi)$.
Define $\qcoh(\uV\rra\uU)$ and $\coh(\uV\rra\uU)$ to be the full
subcategories of $(E,\Phi)$ in $(\uV\rra\uU)$-mod with $E$
quasicoherent, or coherent, respectively.
\label{ag9def2}
\end{dfn}
\begin{thm} The functor\/ $F_\Pi$ above induces equivalences
between $\OcXmod,\ab\qcoh({\cal X}),\coh({\cal X})$ and
$(\uV\rra\uU)${\rm -mod}$,\qcoh(\uV\rra\uU),\coh(\uV\rra\uU),$
respectively.
\label{ag9thm1}
\end{thm}
\begin{proof} Let $(E,\Phi)$ be an object in $(\uV\rra\uU)$-mod. We
will construct an object $\cal E$ in $\OcXmod$ with $F_\Pi({\cal
E})=(E,\Phi)$, and show $\cal E$ is unique up to canonical
isomorphism. This defines an inverse for $F_\Pi$ up to natural
isomorphism, and so shows $F_\Pi:\OcXmod\ra(\uV\rra\uU)$-mod is an
equivalence. The quasicoherent and coherent sheaf cases are then
immediate.
Suppose $(\uY,y)$ is an object in ${\cal C}_{\cal X}$. We will
construct an $\O_Y$-module ${\cal E}(\uY,y)$, uniquely up to
isomorphism. Now $y:\bar\uY\ra{\cal X}$ is a 1-morphism, and
$\Pi:\bar\uU\ra{\cal X}$ is a surjective 1-morphism. Therefore by
Definition \ref{ag7def12} there exists an open cover $\{\uY_a:a\in
A\}$ of $\uY$ with inclusions $\ui_{\uY_a}:\uY_a\hookra\uY$ and
1-morphisms $\uf_a:\uY_a\ra\uU$ such that $y\ci\bar\ui_{\uY_a}$ is
2-isomorphic to $\Pi\ci\bar\uf_a$ as 1-morphisms $\bar\uY_a\ra{\cal
X}$, for all $a\in A$. Also $\uf_a$ is \'etale as
$\Pi,y,\ui_{\uY_a}$ are.
Thus we have an $\O_{Y_a}$-module $\uf_a^*(E)$ on $\uY_a$ for $a\in
A$. Using $\Phi$ we can construct natural isomorphisms
$\upi_{\uY_a}^*\bigl(\uf_a^*(E)\bigr)\cong
\upi_{\uY_b}^*\bigl(\uf_b^*(E)\bigr)$ of modules on the
intersection/fibre product $\uY_a\cap\uY_b=\uY_{ab}=\uY_a
\t_{\ui_{\uY_a},\uY,\ui_{\uY_b}}\uY_b$ for $a,b\in A$. Using
\eq{ag9eq3} and the groupoid axioms we can show that these
isomorphisms satisfy the natural compatibility condition on triple
overlaps $\uY_a\cap\uY_b\cap\uY_c$ for $a,b,c\in A$. Therefore by a
version of descent for objects for $\O_Y$-modules on $C^\iy$-schemes
$\uY$, there exists an $\O_Y$-module ${\cal E}(\uY,y)$, unique up to
canonical isomorphism, with isomorphisms $\ui_{\uY_a}^*\bigl({\cal
E}(\uY,y)\bigr)\cong\uf_a^*(E)$ of $\O_{Y_a}$-modules for all~$a\in
A$.
This allows us to construct $\O_Y$-modules ${\cal E}(\uY,y)$ for all
$(\uY,y)$ in ${\cal C}_{\cal X}$, up to canonical isomorphism. When
$(\uY,y)=(\uU,\Pi)$ we choose ${\cal E}(\uU,\Pi)=E$. Having chosen
such ${\cal E}(\uY,y)$ for all $(\uY,y)$, a version of descent for
morphisms for $\O_Y$-modules on $C^\iy$-schemes $\uY$ gives us
unique isomorphisms of $\O_Y$-modules ${\cal E}_{(\uf,\eta)}:
\uf^*({\cal E}(\uZ,z))\ra{\cal E}(\uY,y)$ for all morphisms
$(\uf,\eta):(\uY,y)\ra(\uZ,z)$ in ${\cal C}_{\cal X}$, constructed
using compatible open covers for $\uY,\uZ$ and morphisms from them
to $\uU$. Then $\cal E$ is an $\O_{\cal X}$-module with $F_\Pi({\cal
E})=(E,\Phi)$. Uniqueness of ${\cal E}(\uY,y)$ up to canonical
isomorphism and of ${\cal E}_{(\uf,\eta)}$ above implies that $\cal
E$ is unique up to canonical isomorphism.
\end{proof}
For quotient $C^\iy$-stacks $[\uU/G]$ with $G$ a finite group,
so that $\uV=G\t\uU$, a $(\uV\rra\uU)$-module $(E,\Phi)$ is an
$\O_U$-module $E$ with a lift $\Phi$ of the $G$-action on $\uU$ up
to $E$. That is, $(E,\Phi)$ is a $G$-{\it equivariant\/
$\O_U$-module}. Hence, if a Deligne--Mumford $C^\iy$-stack $\cal X$
is equivalent to a quotient $[\uU/G]$ with $G$ finite, then
$\OcXmod,\qcoh({\cal X}),\coh({\cal X})$ are equivalent to the
abelian categories of $G$-equivariant $\O_U$-modules,
quasicoherent and coherent sheaves on~$\uU$.
\begin{ex} Let $\cal X$ be the quotient stack $[\ul{*}/G]$, where
$\ul{*}=\Spec\R$ is a point and $G$ is a finite group. Then
$\OcXmod=\qcoh({\cal X})$ is equivalent to the abelian category of
all $G$-representations over $\R$, and $\coh({\cal X})$ is
equivalent to the abelian category of finite-dimensional
$G$-representations over~$\R$.
\label{ag9ex2}
\end{ex}
\subsection{Pullback of sheaves as a pseudofunctor}
\label{ag93}
In Definition \ref{ag6def2}, for a morphism of $C^\iy$-schemes
$\uf:\uX\ra\uY$ we defined a right exact functor
$\uf^*:\OYmod\ra\OXmod$. As in Remark \ref{ag6rem2}, pullbacks
cannot always be made strictly functorial in $\uf$, that is, we do
not have $\uf^*(\ug^*({\cal E}))=(\ug\ci\uf)^*({\cal E})$ for all
$\uf:\uX\ra\uY$, $\ug:\uY\ra\uZ$ and ${\cal E}\in\OZmod$, but
instead we have canonical isomorphisms~$I_{\uf,\ug}({\cal
E}):(\ug\ci\uf)^*({\cal E})\ra\uf^*(\ug^*({\cal E}))$.
We now generalize this to pullback for sheaves on Deligne--Mumford
$C^\iy$-stacks. The new factor to consider is that we have not only
1-morphisms $f:{\cal X}\ra{\cal Y}$, but also 2-morphisms $\eta:f\Ra
g$ for 1-morphisms $f,g:{\cal X}\ra{\cal Y}$, and we must interpret
pullback for 2-morphisms as well as 1-morphisms.
\begin{dfn} Let $f:{\cal X}\ra{\cal Y}$ be a 1-morphism of
Deligne--Mumford $C^\iy$-stacks, and $\cal F$ be an $\O_{\cal
Y}$-module. A {\it pullback\/} of $\cal F$ to $\cal X$ is an
$\O_{\cal X}$-module $\cal E$, together with the following data: if
$\uU,\uV$ are $C^\iy$-schemes and $u:\bar\uU\ra{\cal X}$ and
$v:\bar\uV\ra{\cal Y}$ are \'etale 1-morphisms, then there is a
$C^\iy$-scheme $\uW$ and morphisms $\upi_\uU:\uW\ra\uU$,
$\upi_\uV:\uW\ra\uV$ giving a 2-Cartesian diagram:
\e
\begin{gathered}
\xymatrix@C=60pt@R=10pt{ \bar\uW \ar[r]_(0.3){\bar\upi_\uV}
\ar[d]_{\bar\upi_\uU}
\drtwocell_{}\omit^{}\omit{^\ze} & \bar\uV \ar[d]^v \\
\bar\uU \ar[r]^(0.7){f\ci u} & {\cal Y}.}
\end{gathered}
\label{ag9eq4}
\e
Then an isomorphism $i({\cal F},f,u,v,\ze):\ab\upi^*_\uU\bigl({\cal
E}(\uU,u)\bigr)\ra \upi^*_\uV\bigl({\cal F}(\uV,v)\bigr)$ of
$\O_W$-modules should be given, which is functorial in $(\uU,u)$ in
${\cal C}_{\cal X}$ and $(\uV,v)$ in ${\cal C}_{\cal Y}$ and the
2-isomorphism $\ze$ in \eq{ag9eq4}. We usually write pullbacks
${\cal E}$ as~$f^*({\cal F})$.
\label{ag9def3}
\end{dfn}
By a similar proof to Theorem \ref{ag9thm1}, but using descent for
objects and morphisms for $\O_Y$-modules on $C^\iy$-schemes $\uY$ in
the \'etale topology rather than the open cover topology on $\uY$,
we can prove:
\begin{prop} Let\/ $f:{\cal X}\ra{\cal Y}$ be a\/ $1$-morphism of
Deligne--Mumford\/ $C^\iy$-stacks, and\/ $\cal F$ be an $\O_{\cal
Y}$-module. Then a pullback\/ $f^*({\cal F})$ exists in $\OcXmod,$
and is unique up to canonical isomorphism.
\label{ag9prop2}
\end{prop}
From now on we will assume that we have {\it chosen\/} a pullback
$f^*({\cal F})$ for all such $f:{\cal X}\ra{\cal Y}$ and $\cal F$.
This could be done either by some explicit construction of
pullbacks, as in the $C^\iy$-scheme case in \S\ref{ag61}, or by
using the Axiom of Choice. As in Remark \ref{ag6rem2} we cannot
necessarily make these choices functorial in~$f$.
\begin{dfn} Choose pullbacks $f^*({\cal F})$ for all 1-morphisms
$f:{\cal X}\ra{\cal Y}$ of Deligne--Mumford $C^\iy$-stacks and all
${\cal F}\in\OcYmod$, as above.
Let $f:{\cal X}\ra{\cal Y}$ be such a 1-morphism, and $\phi:{\cal
E}\ra{\cal F}$ be a morphism in $\OcYmod$. Then $f^*({\cal
E}),f^*({\cal F})\in\OcXmod$. Define the {\it pullback morphism\/}
$f^*(\phi):f^*({\cal E})\ra f^*({\cal F})$ to be the morphism in
$\OcXmod$ characterized as follows. Let $u:\bar\uU\ra{\cal X}$,
$v:\bar\uV\ra{\cal Y}$, $\uW,\upi_\uU,\upi_\uV$ be as in Definition
\ref{ag9def3}, with \eq{ag9eq4} Cartesian. Then the following
diagram of morphisms of $\O_W$-modules commutes:
\begin{equation*}
\xymatrix@C=50pt@R=10pt{ \upi^*_\uU\bigl(f^*({\cal E})(\uU,u)\bigr)
\ar[r]_{i({\cal E},f,u,v,\ze)} \ar[d]_{\pi_\uU^*(f^*(\phi)(\uU,u))} &
\upi^*_\uV\bigl({\cal E}(\uV,v)\bigr) \ar[d]^{\pi_\uV^*(\phi(\uV,v))} \\
\upi^*_\uU\bigl(f^*({\cal F})(\uU,u)\bigr) \ar[r]^{i({\cal
F},f,u,v,\ze)} & \upi^*_\uV\bigl({\cal F}(\uV,v)\bigr).}
\end{equation*}
Using descent for morphisms for $\O_Y$-modules on $C^\iy$-schemes
$\uY$ in the \'etale topology, one can show that there is a unique
morphism $f^*(\phi)$ with this property.
This now defines a {\it functor\/} $f^*:\OcYmod\ra\OcXmod$. By the
last part of Proposition \ref{ag6prop3}, $f^*$ also maps
$\qcoh({\cal Y})\ra\qcoh({\cal X})$ and~$\coh({\cal Y})\ra\coh({\cal
X})$.
Let $f:{\cal X}\ra{\cal Y}$ and $g:{\cal Y}\ra{\cal Z}$ be
1-morphisms of Deligne--Mumford $C^\iy$-stacks, and ${\cal
E}\in\OcZmod$. Then $(g\ci f)^*({\cal E})$ and $f^*(g^*({\cal E}))$
both lie in $\OcXmod$. One can show that $f^*(g^*({\cal E}))$ is a
possible pullback of $\cal E$ by $g\ci f$. Thus as in Remark
\ref{ag6rem2}, we have a canonical isomorphism $I_{f,g}({\cal E}):
(g\ci f)^*({\cal E})\ra f^*(g^*({\cal E}))$. This defines a natural
isomorphism of functors~$I_{f,g}:(g\ci f)^*\Ra f^*\ci g^*$.
Let $f,g:{\cal X}\ra{\cal Y}$ be 1-morphisms of Deligne--Mumford
$C^\iy$-stacks, $\eta:f\Ra g$ a 2-morphism, and ${\cal
E}\in\OcYmod$. Then we have $\O_{\cal X}$-modules $f^*({\cal
E}),g^*({\cal E})$. Let $u:\bar\uU\ra{\cal X}$, $v:\bar\uV\ra{\cal
Y}$, $\uW,\upi_\uU,\upi_\uV$ be as in Definition \ref{ag9def3}. Then
as in \eq{ag9eq4} we have 2-Cartesian diagrams
\begin{equation*}
\xymatrix@C=30pt@R=10pt{ \bar\uW \ar[rrr]_(0.8){\bar\upi_\uV}
\ar[d]_{\bar\upi_\uU} &
\drrtwocell_{}\omit^{}\omit{^{\!\!\!\!\!\!\!\!\!\!\!
\!\!\!\!\!\!\!\!\!\!\!\!\!\ze\od(\eta*\id_{u\ci\bar\upi_\uU})
\,\,\,\,\,\,\,\,\,\,\,\,{}}} && \bar\uV \ar[d]^v & \bar\uW
\ar[rrr]_(0.7){\bar\upi_\uV} \ar[d]_{\bar\upi_\uU}
& \drtwocell_{}\omit^{}\omit{^\ze} && \bar\uV \ar[d]^v \\
\bar\uU \ar[rrr]^(0.8){f\ci u} &&& {\cal Y}, & \bar\uU
\ar[rrr]^(0.7){g\ci u} &&& {\cal Y},}
\end{equation*}
where in $\ze\od(\eta*\id_{u\ci\bar\upi_\uU})$ `$*$' is horizontal
composition and `$\od$' vertical composition of 2-morphisms. Thus we
have isomorphisms of $\O_W$-modules:
\begin{equation*}
\xymatrix@C=100pt@R=0pt{ \upi^*_\uU\bigl(f^*({\cal E})(\uU,u)\bigr)
\ar[dr]^{i({\cal E},f,u,v,\ze\od(\eta*\id_{u\ci\bar\upi_\uU}))}
\ar@{..>}[dd] \\
& \upi^*_\uV\bigl({\cal E}(\uV,v)\bigr). \\
\upi^*_\uU\bigl(g^*({\cal E})(\uU,u)\bigr) \ar[ur]_{i({\cal
E},g,u,v,\ze)}}
\end{equation*}
There is a unique isomorphism `$\dashra$' making this diagram
commute. Taken over all $(\uV,v)$, using descent for morphisms we
can show these isomorphisms are pullbacks of a unique isomorphism
$f^*({\cal E})(\uU,u)\ra g^*({\cal E})(\uU,u)$, and taken over all
$(\uU,u)$ these give an isomorphism of $\O_{\cal X}$-modules
$\eta^*({\cal E}):f^*({\cal E})\ra g^*({\cal E})$. Over all ${\cal
E}\in\OcYmod$, this defines a natural isomorphism~$\eta^*:f^*\Ra
g^*$.
If $\cal X$ is a Deligne--Mumford $C^\iy$-stack with identity
1-morphism $\id_{\cal X}:{\cal X}\ra{\cal X}$ then for each ${\cal
E}\in\OcXmod$, $\cal E$ is a possible pullback $\id_{\cal X}^*({\cal
E})$, so we have a canonical isomorphism $\de_{\cal X}({\cal E}):
\id_{\cal X}^*({\cal E})\ra{\cal E}$. These define a natural
isomorphism~$\de_{\cal X}:\id_{\cal X}^*\Ra\id_{\OcXmod}$.
\label{ag9def4}
\end{dfn}
The proof of the next theorem is long but straightforward. For {\it
pseudofunctors\/} see Borceux \cite[\S 7.5]{Borc} or Behrend et
al.~\cite[\S B.4]{BEFF}.
\begin{thm} Mapping $\cal X$ to $\OcXmod$ for objects\/ $\cal X$ in
$\DMCSta,$ and mapping $1$-morphisms\/ $f:{\cal X}\ra{\cal Y}$ to\/
$f^*:\OcYmod\ra\OcXmod,$ and mapping\/ $2$-morphisms\/ $\eta:f\Ra g$
to\/ $\eta^*:f^*\Ra g^*$ for $1$-morphisms $f,g:{\cal X}\ra{\cal
Y},$ and the natural isomorphisms\/ $I_{f,g}:(g\ci f)^*\Ra f^*\ci
g^*$ for all\/ $1$-morphisms\/ $f:{\cal X}\ra{\cal Y}$ and\/
$g:{\cal Y}\ra{\cal Z}$ in $\DMCSta,$ and\/ $\de_{\cal X}$ for all\/
${\cal X}\in\DMCSta,$ together make up a
\begin{bfseries}pseudofunctor\end{bfseries} $(\DMCSta)^{\rm op}\ra
\mathop{\bf AbCat},$ where $\mathop{\bf AbCat}$ is the\/
$2$-category of abelian categories. That is, they satisfy the
conditions:
\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item[{\bf(a)}] If\/ $f:{\cal W}\ra{\cal X},$ $g:{\cal X}\ra{\cal
Y},$ $h:{\cal Y}\ra{\cal Z}$ are $1$-morphisms in $\DMCSta$
and\/ ${\cal E}\in\OcZmod$ then the following diagram commutes
in\/~$\OcXmod:$
\begin{equation*}
\xymatrix@C=50pt@R=10pt{ (h\ci g\ci f)^*({\cal E})
\ar[r]_{I_{f,h\ci g}({\cal E})} \ar[d]_{I_{g\ci f,h}({\cal E})}
& f^*\bigl((h\ci g)^*({\cal E})\bigr) \ar[d]^{f^*(I_{g,h}({\cal E}))} \\
(g\ci f)^*\bigl(h^*({\cal E})\bigr) \ar[r]^{I_{f,g}(h^*({\cal
E}))} & f^*\bigl(g^*(h^*({\cal E}))\bigr).}
\end{equation*}
\item[{\bf(b)}] If\/ $f:{\cal X}\ra{\cal Y}$ is a $1$-morphism in\/
$\DMCSta$ and\/ ${\cal E}\in\OcYmod$ then the following pairs of
morphisms in\/ $\OcXmod$ are inverse:
\begin{equation*}
\xymatrix@C=8pt@R=10pt{ {\begin{subarray}{l}\ts f^*({\cal E})=\\
\ts (f\!\ci\!\id_{\cal X})^*({\cal E})\end{subarray}}
\ar@/^/[rrr]^{I_{\id_{\cal X},f}({\cal E})} &&& \id_{\cal
X}^*(f^*({\cal E})), \ar@/^/[lll]^{\de_{\cal X}(f^*({\cal E}))} &
{\begin{subarray}{l}\ts f^*({\cal E})=\\ \ts (\id_{\cal Y}\!\ci\!
f)^*({\cal E})\end{subarray}} \ar@/^/[rrr]^{I_{f,\id_{\cal Y}}({\cal
E})} &&& f^*(\id_{\cal Y}^*({\cal E})). \ar@/^/[lll]^{f^*(\de_{\cal
Y}({\cal E}))} }
\end{equation*}
Also $(\id_f)^*(\id_{{\cal E}})=\id_{f^*({\cal E})}:f^*({\cal
E})\ra f^*({\cal E})$.
\item[{\bf(c)}] If\/ $f,g,h:{\cal X}\ra{\cal Y}$ are
$1$-morphisms and\/ $\eta:f\Ra g,$ $\ze:g\Ra h$ are
$2$-morphisms in\/ $\DMCSta,$ so that\/ $\ze\od\eta:f\Ra h$ is
the vertical composition, and\/ ${\cal E}\in\OcYmod,$ then
\begin{equation*}
\ze^*(\cF)\ci\eta^*({\cal E})=(\ze\od\eta)^*({\cal E}):f^*({\cal
E})\ra h^*({\cal E})\quad\text{in\/ $\OcXmod$.}
\end{equation*}
\item[{\bf(d)}] If\/ $f,\ti f:{\cal X}\ra{\cal Y},$ $g,\ti g:{\cal
Y}\ra{\cal Z}$ are $1$-morphisms and\/ $\eta:f\Ra f',$ $\ze:g\Ra
g'$ $2$-morphisms in\/ $\DMCSta,$ so that\/ $\ze*\eta:g\ci f\Ra
\ti g\ci\ti f$ is the horizontal composition, and\/ ${\cal
E}\in\OcZmod,$ then the following commutes in\/~$\OcXmod:$
\begin{equation*}
\xymatrix@C=50pt@R=7pt{ (g\ci f)^*({\cal E})
\ar[rr]_{(\ze*\eta)^*({\cal E})} \ar[d]_{I_{f,g}({\cal E})} &&
(\ti g\ci\ti f)^*({\cal E}) \ar[d]^{I_{\ti f,\ti g}({\cal E})} \\
f^*(g^*({\cal E})) \ar[r]^{\eta^*(g^*({\cal E}))} & \ti
f^*(g^*({\cal E})) \ar[r]^{\ti f^*(\ze^*({\cal E}))} & \ti
f^*(\ti g^*({\cal E})).}
\end{equation*}
\end{itemize}
\label{ag9thm2}
\end{thm}
Here is the analogue of Proposition~\ref{ag6prop1}.
\begin{prop} Let\/ $f:{\cal X}\ra{\cal Y}$ be a $1$-morphism of
Deligne--Mumford\/ $C^\iy$-stacks. Then pullback\/
$f^*:\OcYmod\ra\OcXmod$ is a right exact functor.
\label{ag9prop3}
\end{prop}
\begin{proof} Suppose ${\cal E}\ra{\cal F}\ra{\cal G}\ra 0$ is
exact in $\OcYmod$. Let $u:\bar\uU\ra{\cal X}$, $v:\bar\uV\ra{\cal
Y}$, $\uW,\upi_\uU,\upi_\uV$ be as in Definition \ref{ag9def3}. Then
${\cal E}(\uV,v)\ra{\cal F}(\uV,v)\ra{\cal G}(\uV,v)\ra 0$ is exact
in $\O_V$-mod, so $\upi^*_\uV({\cal E}(\uV,v))\ra\upi^*_\uV({\cal
F}(\uV,v))\ra\upi^*_\uV({\cal G}(\uV,v))\ra 0$ is exact in
$\O_W$-mod by Proposition \ref{ag6prop1}. Thus by the isomorphisms
$i(-,f,u,v,\ze)$, $\upi^*_\uU(f^*({\cal E})(\uU,u))
\ra\upi^*_\uU(f^*({\cal F})(\uU,u))\ra\upi^*_\uV(f^*({\cal
G})(\uU,u))\ra 0$ is exact in $\O_W$-mod. As this is true for all
$(\uV,v)$ we see that $f^*({\cal E})(\uU,u) \ra f^*({\cal
F})(\uU,u)\ra f^*({\cal G})(\uU,u)\ra 0$ is exact in $\O_U$-mod.
Since this holds for all $(\uU,u)$ we see that $f^*({\cal E})\ra
f^*({\cal F})\ra f^*({\cal G})\ra 0$ is exact in $\OcXmod$, as we
have to prove.
\end{proof}
\subsection{Cotangent sheaves of Deligne--Mumford\/ $C^\iy$-stacks}
\label{ag94}
We now develop the analogue of the ideas of~\S\ref{ag64}.
\begin{dfn} Let $\cal X$ be a Deligne--Mumford $C^\iy$-stack. Define
an $\O_{\cal X}$-module $T^*{\cal X}$ called the {\it cotangent
sheaf\/} of $\cal X$ by $(T^*{\cal X})(\uU,u)=T^*\uU$ for all
objects $(\uU,u)$ in ${\cal C}_{\cal X}$ and $(T^*{\cal
X})_{(\uf,\eta)}=\Om_\uf:\uf^*(T^*\uV)\ra T^*\uU$ for all morphisms
$(\uf,\eta):(\uU,u)\ra(\uV,v)$ in ${\cal C}_{\cal X}$, where
$T^*\uU$ and $\Om_\uf$ are as in \S\ref{ag64}. Here as
$\uf:\uU\ra\uV$ is \'etale $\Om_\uf$ is an isomorphism, so
$(T^*{\cal X})_{(\uf,\eta)}$ is an isomorphism of $\O_U$-modules as
required. Also Theorem \ref{ag6thm3}(a) shows that \eq{ag9eq1}
commutes for ${\cal E}=T^*{\cal X}$ for all such
$(\uf,\eta),(\ug,\ze)$. Hence $T^*{\cal X}$ is an $\O_{\cal
X}$-module.
Let $f:{\cal X}\ra{\cal Y}$ be a 1-morphism of Deligne--Mumford
$C^\iy$-stacks. Then $f^*(T^*{\cal Y}),T^*{\cal X}$ are $\O_{\cal
X}$-modules. Define $\Om_f:f^*(T^*{\cal Y})\ra T^*{\cal X}$ to be
the unique morphism characterized as follows. Let $u:\bar\uU\ra{\cal
X}$, $v:\bar\uV\ra{\cal Y}$, $\uW,\upi_\uU,\upi_\uV$ be as in
Definition \ref{ag9def3}, with \eq{ag9eq4} Cartesian. Then the
following diagram of morphisms of $\O_W$-modules commutes:
\begin{equation*}
\xymatrix@C=15pt@R=15pt{ \upi^*_\uU\bigl(f^*(T^*{\cal
Y})(\uU,u)\bigr) \ar[d]_{\pi_\uU^*(\Om_f(\uU,u))}
\ar[rrr]_{i(T^*{\cal Y},f,u,v,\ze)} &&& \upi^*_\uV\bigl((T^*{\cal
Y})(\uV,v)\bigr) \ar@{=}[r] &
\upi^*_\uV(T^*\uV) \ar[d]_{\Om_{\upi_\uV}} \\
\upi^*_\uU\bigl((T^*{\cal X})(\uU,u)\bigr) \ar[rrr]^{(T^*{\cal
X})_{(\upi_\uU,\id_{u\ci\upi_\uU})}} &&& (T^*{\cal
X})(\uW,u\ci\upi_\uU) \ar@{=}[r] & T^*\uW.}
\end{equation*}
This determines $\pi_\uU^*(\Om_f(\uU,u))$ uniquely. Over all
$(\uV,v)$, using descent for morphisms for $\O_U$-modules on
$C^\iy$-schemes $\uU$ in the \'etale topology, this determines the
morphisms $\Om_f(\uU,u)$, and over all $(\uU,u)$ these
determine~$\Om_f$.
\label{ag9def5}
\end{dfn}
From Proposition \ref{ag9prop1} and Theorem \ref{ag6thm2}(c),(d) we
deduce:
\begin{prop} Let\/ $\cal X$ be a Deligne--Mumford\/ $C^\iy$-stack.
If\/ $\cal X$ is locally fair then\/ $T^*{\cal X}$ is quasicoherent.
If\/ $\cal X$ is locally good then\/ $T^*{\cal X}$ is coherent. If\/
$\cal X$ is an orbifold of dimension\/ $n,$ which may have boundary
or corners, then\/ $T^*{\cal X}$ is a vector bundle of rank\/~$n$.
\label{ag9prop4}
\end{prop}
Here is the analogue of Theorem \ref{ag6thm3}. Note the extra
$\eta^*(T^*{\cal Z})$ in~\eq{ag9eq7}.
\begin{thm}{\bf(a)} Let\/ $f:{\cal X}\ra{\cal Y}$ and\/ $g:{\cal
Y}\ra {\cal Z}$ be\/ $1$-morphisms of Deligne--Mumford\/
$C^\iy$-stacks. Then
\e
\Om_{g\ci f}=\Om_f\ci f^*(\Om_g)\ci I_{f,g}(T^*{\cal Z})
\label{ag9eq5}
\e
as morphisms\/ $(g\ci f)^*(T^*{\cal Z})\ra T^*{\cal X}$
in\/~$\OcXmod$.
\smallskip
\noindent{\bf(b)} Let\/ $f,g:{\cal X}\ra{\cal Y}$ be\/ $1$-morphisms
of Deligne--Mumford\/ $C^\iy$-stacks and\/ $\eta:f\Ra g$ a
$2$-morphism. Then $\Om_f=\Om_g\ci\eta^*(T^*{\cal Y}):f^*(T^*{\cal
Y})\ra T^*{\cal X}$.
\smallskip
\noindent{\bf(c)} Suppose\/ ${\cal W},{\cal X},{\cal Y},{\cal Z}$
are locally fair Deligne--Mumford\/ $C^\iy$-stacks with a
$2$-Cartesian square
\e
\begin{gathered}
\xymatrix@C=60pt@R=10pt{ {\cal W} \ar[r]_(0.25)f \ar[d]^e
\drtwocell_{}\omit^{}\omit{^\eta}
& {\cal Y} \ar[d]_h \\
{\cal X} \ar[r]^(0.7)g & {\cal Z}}
\end{gathered}
\label{ag9eq6}
\e
in $\DMCStalf,$ so that\/ ${\cal W}={\cal X}\t_{\cal Z}{\cal Y}$.
Then the following is exact in $\qcoh({\cal W}):$
\e
\xymatrix@C=13pt{ (g\ci e)^*(T^*{\cal Z})
\ar[rrrrrr]^(0.51){\begin{subarray}{l}e^*(\Om_g)\ci I_{e,g}(T^*{\cal Z})\op\\
-f^*(\Om_h)\ci I_{f,h}(T^*{\cal Z})\ci\eta^*(T^*{\cal Z})
\end{subarray}} &&&&&&
{\raisebox{18pt}{$\displaystyle \begin{subarray}{l}\ts e^*(T^*{\cal X})\op\\
\ts f^*(T^*{\cal Y})\end{subarray}$}} \ar[rr]^(0.6){\Om_e\op \Om_f}
&& T^*{\cal W} \ar[r] & 0.}
\label{ag9eq7}
\e
\label{ag9thm3}
\end{thm}
\begin{proof} For (a), let $u:\bar\uU\ra{\cal X}$, $v:\bar\uV\ra{\cal
Y}$ and $w:\bar\uW\ra{\cal Z}$ be \'etale. Then there is a
$C^\iy$-scheme $\uV'$ with $\bar\uV{}'=\bar\uV\t_{g\ci v,{\cal
Z},w}\bar\uW$, and fibre product projections $\upi_\uV:\uV'\ra\uV$,
$\upi_\uW:\uV'\ra\uW$. Define
$v'=v\ci\bar\upi_\uV:\bar\uV{}'\ra{\cal Y}$. Then $v'$ is \'etale,
as $v$ is and $w$ is so $\upi_\uV$ is. Similarly, there is a
$C^\iy$-scheme $\uU'$ with $\bar\uU{}'=\bar\uU\t_{f\ci u,{\cal
Y},v'}\bar\uV{}'$, and fibre product projections
$\upi_\uU:\uU'\ra\uU$, $\upi_{\uV'}:\uU'\ra\uV'$. Define an \'etale
1-morphism $u'=u\ci\bar\upi_\uU:\bar\uU{}'\ra{\cal X}$. Then we have
a 2-commutative diagram
\begin{equation*}
\xymatrix@C=40pt@R=-5pt{ {\cal X} \ar[rr]^f && {\cal Y} \ar[rr]^g
&& {\cal Z} \\
& \bar\uU \ar[ul]_(0.2)u \ar[ur] & \ddtwocell_{}\omit^{}\omit{^\eta}
& \bar\uV \ar[ul]_(0.2)v \ar[ur] & \dltwocell_{}\omit^{}\omit{^\ze}
\\ &&&& \bar\uW \ar[uu]^w \\
&&& \bar\uV{}' \ar[uuul]^{v'} \ar[uu]_{\bar\upi_\uV}
\ar[ur]_{\bar\upi_\uW} \\ && \bar\uU{}' \ar[uuuull]^{u'}
\ar[uuul]_{\bar\upi_\uU} \ar[ur]_{\bar\upi_{\uV'}} }
\end{equation*}
with 2-Cartesian squares. On $\uU'$ and $\uV'$ we have commutative
diagrams:
\ea
\begin{gathered}
\xymatrix@C=12pt@R=15pt{ \upi^*_\uU\bigl(f^*(T^*{\cal
Y})(\uU,u)\bigr) \ar[rrrr]^{i(T^*{\cal Y},f,u,v',\eta)}_\cong
\ar[d]_\cong^(0.45){(f^*(T^*{\cal Y}))_{(\upi_\uU,\id_{u'})}} &&&&
\upi_{\uV'}^*\bigl((T^*{\cal Y})(\uV',v')\bigr) \ar@{=}[r] &
\upi_{\uV'}^*(T^*\uV') \ar[d]_{\Om_{\upi_{\uV'}}} \\
(f^*(T^*{\cal Y}))(\uU',u') \ar[rrrr]^{\Om_f(\uU',u')} &&&&
(T^*{\cal X})(\uU',u') \ar@{=}[r] & T^*\uU' }
\end{gathered}
\label{ag9eq8}\\
\begin{gathered}
\xymatrix@C=12pt@R=15pt{ \upi^*_\uV\bigl(g^*(T^*{\cal
Z})(\uV,v)\bigr) \ar[rrrr]^{i(T^*{\cal Z},g,v,w,\ze)}_\cong
\ar[d]_\cong^(0.45){(g^*(T^*{\cal Z}))_{(\upi_\uV,\id_{v'})}} &&&&
\upi_\uW^*\bigl(T^*{\cal Z}(\uW,w)\bigr) \ar@{=}[r] &
\upi_\uW^*(T^*\uW) \ar[d]_{\Om_{\upi_\uW}}
\\
(g^*(T^*{\cal Z}))(\uV',v') \ar[rrrr]^{\Om_g(\uV',v')} &&&&
(T^*{\cal Y})(\uV',v') \ar@{=}[r] & T^*\uV'. }
\end{gathered}
\label{ag9eq9}
\ea
Applying $\upi_{\uV'}^*$ to \eq{ag9eq9} we make another commutative
diagram on $\uU'$:
\e
\begin{gathered}
\xymatrix@C=90pt@R=15pt{ \upi_{\uV'}^*\bigl(\upi^*_\uV(g^*(T^*{\cal
Z})(\uV,v))\bigr) \ar[r]^{\upi_{\uV'}^*(i(T^*{\cal
Z},g,v,w,\ze))}_\cong
\ar[d]_\cong^(0.45){\upi_{\uV'}^*((g^*(T^*{\cal
Z}))_{(\upi_\uV,\id_{v'})})} &
\upi_{\uV'}^*\bigl(\upi_\uW^*(T^*\uW)\bigr)
\ar[d]_{\upi_{\uV'}^*(\Om_{\upi_\uW})} \\
\upi_{\uV'}^*\bigl((g^*(T^*{\cal Z}))(\uV',v')\bigr)
\ar[r]^{\upi_{\uV'}^*(\Om_g(\uV',v'))}
\ar[d]_\cong^(0.45){(f^*(g^*(T^*{\cal Z})))_{(\upi_\uU,\id_{u'})}} &
\upi_{\uV'}^*(T^*\uV') \ar[d]^\cong_(0.45){(f^*(T^*{\cal
U}))_{(\upi_\uU,\id_{u'})}} \\
\bigl(f^*(g^*(T^*{\cal Z}))\bigr)(\uU',u')
\ar[r]^{(f^*(\Om_g))(\uU',u')} & \bigl(f^*(T^*{\cal
Y})\bigr)(\uU',u'). }
\end{gathered}
\label{ag9eq10}
\e
By Theorem \ref{ag6thm3}(a) the following commutes:
\e
\begin{gathered}
\xymatrix@C=60pt@R=15pt{ (\upi_\uW\ci\upi_{\uV'})^*(T^*\uW)
\ar[r]_{\Om_{\upi_\uW\ci\upi_{\uV'}}}
\ar[d]_{I_{\upi_{\uV'},\upi_\uW}(T^*\uW)}^\cong & T^*\uU' \\
\upi_{\uV'}^*\bigl(\upi_\uW^*(T^*\uW)\bigr)
\ar[r]^{\upi_{\uV'}^*(\Om_{\upi_\uW})} & \upi_{\uV'}^*(T^*\uV').
\ar[u]_{\Om_{\upi_{\uV'}}} }
\end{gathered}
\label{ag9eq11}
\e
Using all this we obtain a commutative diagram on $\uU'$:
\e
\begin{gathered}
\xymatrix@!0@C=77pt@R=30pt{ \bigl((g\ci f)^*(T^*{\cal
Z})\bigr)(\uU',u') \ar[rrr]_{\Om_{g\ci f}(\uU',u')}
\ar[ddd]_\cong^{(I_{f,g}(T^*{\cal Z}))(\uU',u')}
\ar@{<->}[dr]^\cong &&& (T^*{\cal X})(\uU',u') \\
& (\upi_\uW\ci\upi_{\uV'})^*(T^*\uW) \ar[r]
\ar[d]^\cong & T^*\uU' \ar@{=}[ur] \\
& \upi_{\uV'}^*\bigl(\upi_\uW^*(T^*\uW)\bigr)
\ar[r] & \upi_{\uV'}^*(T^*\uV') \ar[u] \\
\bigl(f^*(g^*(T^*{\cal Z}))\bigr)(\uU',u') \ar@{<->}[ur]^\cong
\ar[rrr]^{(f^*(\Om_g))(\uU',u')} &&& \bigl(f^*(T^*{\cal
Y})\bigr)(\uU',u'). \ar@{<->}[ul]^\cong \ar[uuu]^{\Om_f(\uU',u')} }
\end{gathered}\!\!\!\!\!\!\!\!\!\!\!\!
\label{ag9eq12}
\e
Here the right hand quadrilateral of \eq{ag9eq12} comes from
\eq{ag9eq8}, the bottom quadrilateral from \eq{ag9eq10}, the central
square is \eq{ag9eq11}, and the remaining two quadrilaterals are
similar. Thus, the outer square of \eq{ag9eq12} commutes. But this
is just \eq{ag9eq5} evaluated at $(\uU',u')$. If $u:\bar\uU\ra{\cal
X}$, $v:\bar\uV\ra{\cal Y}$ and $w:\bar\uW\ra{\cal Z}$ are \'etale
atlases then $u':\bar\uU{}'\ra{\cal X}$ is also an \'etale atlas,
and \eq{ag9eq5} evaluated on an atlas implies it in general. This
proves part~(a).
Part (b) is immediate from the definitions. For (c), let
$u:\bar\uU\ra{\cal X}$, $v:\bar\uV\ra{\cal Y}$ and
$w:\bar\uW\ra{\cal Z}$ be \'etale. Then $\uU,\uV,\uW$ are locally
fair, as ${\cal X},{\cal Y},{\cal Z}$ are and $u,v,w$ are \'etale.
There are $C^\iy$-schemes $\uU',\uV',$ with
$\bar\uU{}'=\bar\uU\t_{g\ci u,{\cal Z},w}\bar\uW$,
$\bar\uV{}'=\bar\uV\t_{h\ci v,{\cal Z},w}\bar\uW$, and fibre product
projections $\upi_\uU:\uU'\ra\uU$, $\upi_\uW:\uU'\ra\uW$,
$\upi_\uV:\uV'\ra\uV$, $\upi_\uW:\uV'\ra\uW$. Then $\pi_\uU,\pi_\uV$
are \'etale as $w$ is, so $\uU',\uV'$ are locally fair as $\uU,\uV$
are. Define a $C^\iy$-scheme
$\uT=\uU'\t_{\upi_\uW,\uW,\upi_\uW}\uV'$. Then $\uT$ is locally fair
by Theorem \ref{ag4thm3}. The 1-morphisms
$u'\ci\bar\upi_{\uU'}:\bar\uT\ra{\cal X}$ and
$v'\ci\bar\upi_{\uV'}:\bar\uT\ra{\cal Y}$ have a natural
2-isomorphism $g\ci(u'\ci\bar\upi_{\uU'})\Ra
h\ci(v'\ci\bar\upi_{\uV'})$ constructed from the 2-isomorphisms in
the 2-Cartesian squares constructing $\uU',\uV'$. Thus as ${\cal
W}={\cal X}\t_{\cal Z}{\cal Y}$ there is a 1-morphism
$t:\bar\uT\ra{\cal W}$, unique up to 2-isomorphism, such that
$u'\ci\bar\upi_{\uU'}\cong e\ci t$ and $v'\ci\bar\upi_{\uV'}\cong
f\ci t$. Also $t$ is \'etale. This gives a 2-commutative diagram
\begin{equation*}
\xymatrix@!0@C=50pt@R=15pt{ & \bar\uT \ar[dl]_(0.6){\bar\upi_{\uU'}}
\ar[ddr]^(0.4){\bar\upi_{\uV'}} \ar[rr]^(0.6){t}
&& {\cal W} \ar[dl]^(0.4){e} \ar[ddr]^{f} \\
\bar\uU{}' \ar[ddr]_{\bar\pi_\uW} \ar[rr]_(0.4){u'} && {\cal X}
\ar[ddr]^(0.35){g} \\ && \bar\uV{}' \ar[rr]^(0.6){v'}
\ar[dl]_(0.4){\bar\upi_\uW} && {\cal Y} \ar[dl]^(0.4){h} \\
& \bar\uW \ar[rr]^(0.6){w} && {\cal Z},}
\end{equation*}
in which the leftmost and rightmost squares are 2-Cartesian.
Applying Theorem \ref{ag6thm3}(b) to the Cartesian square defining
$\uT$ gives an exact sequence in $\qcoh(\uT)$:
\e
\xymatrix@C=11pt{ (\upi_\uW\!\ci\!\upi_{\uU'})^*(T^*\uW)
\ar[rrrrrr]^(0.52){\begin{subarray}{l}\upi_{\uU'}^*(\Om_{\upi_\uW})\ci
I_{\upi_{\uU'},\upi_\uW}(T^*\uW)\op\\
-\upi_{\uV'}^*(\Om_{\upi_\uW})\ci
I_{\upi_{\uV'},\upi_\uW}(T^*\uW)\end{subarray}} &&&&&&
{\raisebox{18pt}{$\displaystyle \begin{subarray}{l}\ts
\upi_{\uU'}^*(T^*\uU')\op\\
\ts\upi_{\uV'}^*(T^*\uV')\end{subarray}$}}
\ar[rr]^(0.6){\begin{subarray}{l}\Om_{\upi_{\uU'}}\op\\
\Om_{\upi_{\uV'}}\end{subarray}} && T^*\uT \ar[r] & 0.}\!\!
\label{ag9eq13}
\e
By a similar argument to (a), we can use \eq{ag9eq13} to deduce that
\eq{ag9eq7} evaluated at $(\uT,t)$ holds. If $u:\bar\uU\ra{\cal X}$,
$v:\bar\uV\ra{\cal Y}$ and $w:\bar\uW\ra{\cal Z}$ are atlases then
$t:\bar\uT\ra{\cal W}$ is an atlas, so this implies \eq{ag9eq7}, and
proves~(c).
\end{proof}
\baselineskip 11.7pt plus .2pt
|
1,116,691,500,869 | arxiv | \section{Introduction}
Computers analyze massive quantities of data with speed and precision \cite{nvidia2020,intel2019}. At both the hardware and software levels, this performance depends on fixed and precisely engineered protocols for representing and executing basic operations on binary data \cite{intel2019,Neumann1945,Alglave2008}. In contrast, neurobiological systems are characterized by flexibility and adaptability. At the biophysical level, neurons undergo dynamic changes in their composition and patterns of connectivity \cite{Zhang2011,Faulkner2008,Dunn2012,Craik2006}. At the cognitive level, they abstract spatiotemporally complex sensory information to recognize objects, localize spatial position, and even control new virtual limbs through experience \cite{Tacchetti2018,Moser2008,Ifft2013}. Hence, neural systems appear to work on fundamentally different computing principles that are learned, rather than engineered.
To uncover these principles, artificial neural networks have been used to study the representation and manipulation of information. While feed-forward networks can classify input data \cite{Sainath2015}, biological organisms contain recurrent connections that are necessary to sustain short-term memory of internal representations \cite{Jarrell2012}, allowing for more complex functions such as tracking time, distance, and emotional context \cite{Lee2015,Wang2018,Weber2017,Burak2009,Yoon2013}. Further, recurrent neural systems actually manipulate internal representations to simulate the outcome of dynamic processes such as kinematic motion and navigation \cite{Hegarty2004,Kubricht2017,Pfeiffer2013}, and to decide between different actions \cite{Gold2007}. How do recurrent neural systems learn to represent and manipulate complex information?
One promising line of work involves representing static memories as patterns of neural activity, or \emph{attractors}, to which a network evolves over time \cite{Strogatz1994}. These attractors can exist in isolation (e.g. an image of a face) or as a continuum (e.g. smooth translations of a face) using Hopfield or continuous attractor neural networks (CANNs), respectively \cite{Yang2017,Wu2016}. Other studies use a differentiable neural computer (DNC) to read and write information to these attractor neural networks to solve complex puzzles \cite{Graves2016}. For understanding neurobiological systems, these memory networks are limited by requiring specifically engineered patterns of connectivity, and cannot manipulate time-varying memories necessary to plan and produce speech and music \cite{Carroll2004,Fee2010,Donnay2014}. Additionally, DNCs artificially segregate the computing and storage components. Hence, we seek a single neural system that learns to both represent and manipulate temporally complex information by perceiving and replicating examples.
In this work, we use the \emph{reservoir computing framework} \cite{Qiao2017} to obtain such a system (the reservoir), where the complex information is a chaotic attractor that is not static, but evolves in a deterministic yet unpredictable manner through time \cite{Lorenz1963}. Prior work has demonstrated the reservoir's ability to represent and switch between isolated attractors by imitating examples \cite{Jaeger2010,Sussillo2009}. Here, we demonstrate that reservoirs can further learn to interpolate and extrapolate translations, linear transformations, and even bifurcations on their representations of chaotic attractor manifolds simply by imitating examples. Further, we put forth a mechanism of how these computations are learned, providing insights into the set of possible computations, and offering principles by which to design effective networks.
\section{Mathematical Framework}
Neural systems represent and manipulate periodic stimuli through example, such as baby songbirds modifying their song to imitate adult songbirds \cite{Fee2010}. However, they also perform more advanced and original manipulations on aperiodic stimuli with higher-order structure, such as musicians improvising on jazz melodies \cite{Donnay2014}. To model such complex stimuli, we use chaotic attractors that evolve deterministically yet unpredictably along a global structure: a fractional-dimensional manifold. Specifically, we consider the Lorenz attractor defined as
\begin{equation}
\label{eq:lorenz}
\begin{aligned}
\dot{x}_1 &= \sigma(x_2-x_1)\\
\dot{x}_2 &= x_1(\rho - x_3) - x_2\\
\dot{x}_3 &= x_1x_2 - \beta x_3,
\end{aligned}
\end{equation}
and use the parameters $\sigma = 10, \beta = 8/3, \rho = 28$ from the original study \cite{Lorenz1963} (Fig.~\ref{fig:f1}).
\begin{figure}[h!]
\includegraphics[width=90mm]{figure1.pdf}
\caption{\textbf{Chaotic Lorenz manifold.} Lorenz attractor plotted in space (left) and time (right).}
\label{fig:f1}
\end{figure}
Next, we model the neural system as a recurrent neural network driven by our inputs
\begin{align*}
\frac{1}{\gamma} \dot{\bm{r}} = -\bm{r} + \bm{g}\left(A\bm{r} + B\bm{x} + \bm{d}\right),
\end{align*}
where $\bm{r}$ is a real-valued vector of $N$ reservoir neuron states, $A$ is an $N \times N$ matrix of inter-neuron connections, $B$ is an $N \times 3$ matrix of connections from the inputs to the neurons, $\bm{d}$ is an $N \times 1$ bias vector, $\bm{g}$ is a scalar activation function applied entry-wise to its input arguments (hence mapping vectors to vectors), and $\gamma$ is a time constant.
Several prior studies use echo state \cite{Jaeger2010} and FORCE learning \cite{Sussillo2009} which allow reservoirs to predict a chaotic time series by modifying the inter-neuron connections. This modification can be accomplished by using the chaotic time series $\bm{x}(t)$ to drive the reservoir, thereby generating the reservoir time series $\bm{r}(t)$ (Fig.~\ref{fig:f2}a,b). Here, $\bm{x}(t)$ and $\bm{r}(t)$ are $3 \times T$ and $N \times T$ matrices, respectively, from numerically evolving the differential equations over $T$ time steps. By solving for a simple $3 \times N$ readout matrix $W$ that uses linear combinations of reservoir states to approximate the input by minimizing the matrix 2-norm (see Supplement)
\begin{align*}
W = \argmin_W\|W\bm{r}(t) - \bm{x}(t)\|_2,
\end{align*}
the output $\hat{\bm{x}}(t) = W\bm{r}(t)$ mimics the input $\bm{x}(t)$ (Fig.~\ref{fig:f2}c). Finally, we close the feedback loop by substituting the output as the input to create the autonomous reservoir (Fig.~\ref{fig:f2}d)
\begin{align*}
\frac{1}{\gamma} \dot{\bm{r}}' = -\bm{r}' + \bm{g}\left((A+BW)\bm{r}' + \bm{d}\right),
\end{align*}
whose evolution projects to a Lorenz-shaped attractor as $\bm{x}'(t) = W\bm{r}'(t)$ (Fig.~\ref{fig:f2}e). Hence, reservoirs sustain representations of complex temporal information by learning to autonomously evolve along a chaotic attractor from example inputs.
\begin{figure}[h!]
\centering
\includegraphics[width=6.5in]{figure2.pdf}
\caption{\textbf{Representing chaotic attractors with reservoirs.} (\textbf{a}) Time series of a chaotic Lorenz attractor that (\textbf{b}) drives the recurrent neural network reservoir. (\textbf{c}) Weighted sums of the reservoir states are trained to reproduce the original time series. (\textbf{d}) By using these weighted sums of reservoir states to drive the reservoir instead of the inputs, (\textbf{e}) the reservoir autonomously evolves along a trajectory that projects to a Lorenz-shaped chaotic manifold.}
\label{fig:f2}
\end{figure}
To study how reservoirs might perform computations by modifying the position or geometry of these representations in a desired way, we first adapt the framework to include a vector of control parameters $\bm{c}$ that map to the reservoir neurons through matrix $C$ to yield
\begin{align*}
\frac{1}{\gamma} \dot{\bm{r}} = -\bm{r} + \bm{g}\left(A\bm{r} + B\bm{x} + C\bm{c} + \bm{d}\right).
\end{align*}
Such control parameters were also previously used to switch between multiple attractor outputs \cite{Sussillo2009}. The second adaptation is to approximate the reservoir dynamics using a Taylor series to quadratic order around equilibrium values $\bm{r}^*,\bm{x}^*=\bm{0},\bm{c}^*=\bm{0}$, yielding
\begin{align}
\label{eq:reservoirc}
\frac{1}{\gamma}\delta\dot{\bm{r}} = -\delta\bm{r} + U(A\delta\bm{r} + B\bm{x} + C\bm{c}) + V(A\delta\bm{r} + B\bm{x} + C\bm{c})^2.
\end{align}
Here, $\delta\bm{r} = \bm{r}-\bm{r}^*$, $U$, and $V$ are diagonal matrices whose $i$-th entries are the first and half of the second derivatives of $g_i$ evaluated at the fixed point, respectively, and $()^2$ is the entry-wise square of the vector (see Supplement for details). By studying quadratic reservoirs and how they learn to manipulate their representations of chaotic manifolds, we will gain an intuition due to their analytic tractability, and generalizability across many activation functions $\bm{g}$ when driven within a range over which the quadratic expansion is accurate.
\section{Learning a translation operation by example}
Reservoirs learn complex information through simple imitation: approximating the driving inputs using the reservoir states is enough to autonomously represent and evolve about a chaotic manifold. Here we show that this simple scheme is also enough to learn to translate the representation. We begin with a Lorenz time series $\bm{x}_0(t)$, and create shifted copies
\begin{align}
\label{eq:translation}
\bm{x}_c(t) = \bm{x}_0(t) + P\bm{c}.
\end{align}
For the purposes of demonstration, we consider a translation in the $x_1$ direction such that $P = [1; 0; 0]$ is a column vector, and $\bm{c} = 0,1,2,3$ is a scalar. We use these four time series to drive our reservoir according to Eq.~\ref{eq:reservoirc},
thereby generating four reservoir time series $\bm{r}_c(t)$. Numerically, $\bm{x}_c(t)$ and $\bm{r}_c(t)$ are matrices of dimension $3 \times T$ and $N \times T$ over $T$ time steps, which we concatenate along the time dimension into $\bm{x}(t) = [\bm{x}_0(t),\bm{x}_1(t),\bm{x}_2(t),\bm{x}_3(t)]$ and $\bm{r}(t) = [\bm{r}_0(t),\bm{r}_1(t),\bm{r}_2(t),\bm{r}_3(t)]$, respectively. Then, we compute output weights
\begin{align}
\label{eq:training}
W = \argmin_W\|W\bm{r}(t) - \bm{x}(t)\|_2,
\end{align}
such that our output $\hat{\bm{x}} = W\bm{r}(t)$ approximates our input $\bm{x}(t)$ (Fig.~\ref{fig:f3}a--c). Finally, we substitute the output as the input to yield the feedback system (Fig.~\ref{fig:f3}d)
\begin{align}
\label{eq:reservoirfbc}
\frac{1}{\gamma} \delta\dot{\bm{r}}' = -\delta\bm{r}' + U(R\delta\bm{r}' + C\bm{c}) + V(R\delta\bm{r}' + C\bm{c})^2,
\end{align}
where $R = A + BW$ (see Supplement for a discussion on $W\bm{r}_c(t) \approx W\delta\bm{r}_c(t)$).
\begin{figure}[h!]
\centering
\textit{} \includegraphics[width=6.5in]{figure3.pdf}
\caption{\textbf{Learning and extrapolating a translation operation through examples.} (\textbf{a}) Schematic of the time series of the Lorenz and control inputs, beginning with the original Lorenz time series $\bm{x}_0(t)$ at $\bm{c}=0$, followed by three equally spaced shifts in the $x_1$ direction and in the $\bm{c}$ parameter. (\textbf{b}) These inputs generate four reservoir time series $\bm{r}_c(t)$. (\textbf{c}) Next, weighted sums of the reservoir states are used to generate outputs $W\bm{r}_c(t) = \hat{\bm{x}}_c(t) \approx \bm{x}_c(t)$ that mimic the inputs. (\textbf{d}) The outputs $W\bm{r}(t)$ replace the inputs $\bm{x}(t)$ to create a reservoir with a closed feedback loop. (\textbf{e}) Over the course of a single simulation, the reservoir evolves autonomously about a Lorenz-shaped manifold, and translates this representation along $x_1$ by smoothly and continuously changing $\bm{c}$ as a real number over a range much larger than the training range.}
\label{fig:f3}
\end{figure}
As we evolve this autonomous reservoir while varying $\bm{c}$ to extreme values $-40 \leq \bm{c} \leq 40$ both inside and outside of the training values, it has learned to evolve about a Lorenz-shaped manifold that is translated based on the value of $\bm{c}$ (see Supplement for translations in all spatial directions). Hence, by training the network on shifted copies of the input time series, the reservoir has learned a translation operation on the attractor.
\section{Learning a linear transformation operation by example}
In addition to learning a translation operation that does not change the geometry of the representation, here we demonstrate that reservoirs can learn linear transformation using the exact same framework. Similarly, we begin with a Lorenz time series $\bm{x}_0(t)$ generated from Eq.~\ref{eq:lorenz}, and create linearly transformed copies of the time series such that
\begin{align}
\label{eq:transformation}
\bm{x}_c(t) = (I+cP)\bm{x}_0(t),
\end{align}
for $c = 0,1,2,3$, where $P$ is a matrix encoding a transformation (Fig.~\ref{fig:f4}a,c). Specifically, we perform a squeeze along $x_1$ by setting $[P]_{11} = -0.012$ and the remaining elements to 0.
Exactly as before, we drive the reservoir according to Eq.~\ref{eq:reservoirc}, concatenate our input and reservoir time series into $\bm{x}(t)$ and $\bm{r}(t)$ to train the output weights $W$ according to Eq.~\ref{eq:training}, and feed the outputs back as inputs to yield the feedback system Eq.~\ref{eq:reservoirfbc}. This reservoir autonomously evolves about a Lorenz-shaped manifold that stretches based on the parameter $-40 \leq c \leq 40$ (Fig.~\ref{fig:f4}b,d) far outside of the parameters used in the training regime $c = 0,1,2,3$ (see Supplement for more examples). Hence, using the same framework, the reservoir has learned the linear transformation operation on the attractor manifold.
\begin{figure}[h!]
\centering \includegraphics[width=6.5in]{figure4.pdf}
\caption{\textbf{Extrapolating a transformation operation through examples.} (\textbf{a}) 3-dimensional plot of the training data of the Lorenz time series that has been stretched along the $x_1$ direction at $c = 0,1,2,3$. (\textbf{b}) 3-dimensional plot of the feedback reservoir output that autonomously evolves about a Lorenz-shaped manifold that stretches dramatically based on varying $c$ from $-40$ to $40$. We also provide a top view of the (\textbf{c}) training data and (\textbf{d}) predicted output data.}
\label{fig:f4}
\end{figure}
\section{Learning to infer a bifurcation by example}
For both translations and transformations, the reservoir learned a smooth change in its representation of the chaotic manifold. Here we demonstrate that a reservoir can infer, without actually ever having experienced, a much more dramatic change: a bifurcation. In the Lorenz attractor (Eq.~\ref{eq:lorenz} for $\rho > 1,\sigma = 10, \beta = 8/3$), there are two fixed points: one at the center of each wing, which undergo a subcritical Hopf bifurcation when $\rho = \rho^* \approx 24.7$ \cite{Strogatz1994}. When $\rho < \rho^*$, these two fixed points are stable. When $\rho > \rho^*$, the fixed points become unstable, yielding the characteristic wing-shaped flow. Here we demonstrate that a reservoir trained only on stable examples ($\rho < \rho^*$) can accurately predict the unstable flow ($\rho > \rho^*$).
\begin{figure}[h!]
\centering
\includegraphics[width=6.5in]{figure5.pdf}
\caption{\textbf{Extrapolating the bifurcation of the Lorenz.} (\textbf{a}) Two training trajectories for each of the stable Lorenz fixed points at the wings, for $\rho=23$ with $c=0$ (blue) and for $\rho=24$ with $c=1$ (light blue). (\textbf{b}) The predicted trajectory of the feedback reservoir moves towards a stable fixed point for $c=0$, and bifurcates into a Lorenz-shaped manifold as $c$ is increased. (\textbf{c}) Four training examples for one of the stable Lorenz fixed points for $\rho = 21,22,23,24$ with $c = 0,1,2,3$. (\textbf{d}) The predicted trajectory moves towards a stable fixed point for $c=0$, and then bifurcates into a Lorenz-shaped manifold as $c$ is increased.}
\label{fig:f5}
\end{figure}
For the two fixed points $\bm{a}$ and $\bm{b}$, we begin with four training trajectories: $\bm{x}^a_{23}(t)$ and $\bm{x}^b_{23}(t)$ that evolve stably towards the fixed points for $\rho = 23$, and $\bm{x}^a_{24}(t)$ and $\bm{x}^b_{24}(t)$ that evolve stably towards the fixed points for $\rho = 24$ (Fig.~\ref{fig:f5}a). We then drive the reservoir with $\bm{x}^a_{23}(t)$ and $\bm{x}^b_{23}(t)$ while setting $c=0$, and with $\bm{x}^a_{24}(t)$ and $\bm{x}^b_{24}(t)$ while setting $c=1$, and train the output weights. Finally, we evolve the feedback reservoir while changing $c$ from $0$ to $5$, and note that the trajectory bifurcates into a Lorenz-shaped manifold (Fig.~\ref{fig:f5}b).
As a second demonstration, we begin with another set of four training trajectories: $\bm{x}^a_{21}(t),\dotsm,\bm{x}^a_{24}(t)$ that evolve stably towards only one fixed point for $\rho = 21,\dotsm,24$ (Fig.~\ref{fig:f5}c). We then drive the reservoir with $\bm{x}^a_{21}(t),\dotsm,\bm{x}^a_{24}(t)$ while setting $c = 0,\dotsm,3$, and train the output weights. Finally, we evolve the feedback reservoir while changing $c$ from $0$ to $7$, and note that the trajectory again bifurcates into a Lorenz-shaped manifold (Fig.~\ref{fig:f5}d). Hence, after only observing a few stable trajectories before the bifurcation ($\rho < \rho^*$), the reservoir accurately extrapolates the geometry of the Lorenz trajectory after the bifurcation ($\rho > \rho^*$).
\section{Mechanism of how operations are learned}
Now that we have taught reservoirs to manipulate chaotic manifolds, we seek to understand the mechanism. We begin with some intuition by expanding the feedback dynamics
\begin{align*}
\frac{1}{\gamma} \delta\dot{\bm{r}}' = ([U +
\underbrace{2V~\mathrm{diag}(C\bm{c})}_{\mathrm{stretch}}]R - I)\delta\bm{r}' + \underbrace{UC\bm{c}}_{\mathrm{shift}} + V(R\delta\bm{r}')^2+ \underbrace{V(C\bm{c})^2}_{\mathrm{small}},
\end{align*}
and notice that the control parameter can scale the shape of the reservoir's internal dynamics (stretch), and add a constant driving input (shift). For small changes in $\bm{c}$, the quadratic term $C\bm{c}$ is negligible. To formalize this intuition, we consider the time series $\bm{r}'(t) = \bm{r}'_{c=0}(t)$ generated by evolving the autonomous reservoir according to Eq.~\ref{eq:reservoirfbc} at $\bm{c} = \bm{0}$. Next, we take the total differential of Eq.~\ref{eq:reservoirfbc} evaluated at $\bm{r}'(t)$ and $\bm{c}=\bm{0}$ to yield
\begin{align}
\label{eq:differential}
(I-KA)d\bm{r}' + \frac{1}{\gamma}d\dot{\bm{r}}' = K(BWd\bm{r}' + Cd\bm{c}),
\end{align}
where $K = U + 2V~\mathrm{diag}(R\delta\bm{r}'(t))$. Our goal is to write the change in the reservoir state $d\bm{r}'(t)$ that is induced by changing the control parameter by an infinitesimal amount $d\bm{c}$.
\begin{figure}[h!] \includegraphics[width=6.5in]{figure6.pdf}
\caption{\textbf{Changing the control parameter changes the reservoir dynamics to manipulate representations.} (\textbf{a}) Schematic of a reservoir with feedback connections after the output weights $W$ have been trained. (\textbf{b}) Reservoir time series generated by evolving the autonomous reservoir with the original Lorenz input with $\bm{c} = 0$ (dark gold). We also show the predicted time series from solving Eq.~\ref{eq:predict_translate} after training on translated examples and setting $d\bm{c} = \Delta \bm{c} = 20$ (light gold). The output projections of the two time series are shown in blue and green, respectively. (\textbf{c}) The original and predicted reservoir states and their output projections for $\Delta c = -40$ after training on transformed Lorenz inputs by solving Eq.~\ref{eq:predict_transform}. (\textbf{d}) Plot of the real and imaginary components of the two most unstable eigenvalues of the autonomous reservoir trained on two stable Lorenz trajectories (Fig.~\ref{fig:f5}a,b). The reservoir is linearized about its fixed point according to Eq.~\ref{eq:reservoirfblin} as $c$ is slowly changed.}
\label{fig:f6}
\end{figure}
When learning translations, the output weights are trained such that $W\bm{r}_c(t) \approx \bm{x}_c(t) = \bm{x}(t) + P\bm{c}$. For sufficiently nearby training examples (small $P, \bm{c}$), we also implicitly approximate the differential relation $Wd\bm{r}(t) \approx Pd\bm{c}$. Additionally, if the feedback reservoir stabilizes these examples, then $Wd\bm{r}'(t) \approx Pd\bm{c}$. Substituting this relation into Eq.~\ref{eq:differential} yields
\begin{align*}
(I-KA)d\bm{r}' + \frac{1}{\gamma}d\dot{\bm{r}}' \approx K(BP + C)d\bm{c}.
\end{align*}
If we fix $d\bm{c}$, we have $2N$ variables, $d\bm{r}'$ and $d\dot{\bm{r}}'$, but only $N$ equations. By taking the time derivative of the differential relation, we generate another $N$ variables and $N$ equations. Continuing to take time derivatives yields the following system of equations
\begin{align*}
\begin{bmatrix}
H_0 & H_{-1} & 0 & \dotsm\\
H_1 & H_0 & H_{-1} & \dotsm\\
H_2 & 2H_1 & H_0 & \dotsm\\
\vdots & \vdots& \vdots & \ddots
\end{bmatrix}
\begin{bmatrix}
d\bm{r}'\\
d\dot{\bm{r}}'\\
d\ddot{\bm{r}}'\\
\vdots
\end{bmatrix}
\approx
\begin{bmatrix}
K\\
\dot{K}\\
\ddot{K}\\
\vdots
\end{bmatrix}
(BP + C)d\bm{c},
\end{align*}
where $H_{-1} = \frac{1}{\gamma}I$, $H_0 = I-KA$, and $H_i = -K^{(i)}A$ is the $i$-th time-derivative of $KA$. This matrix is a block-Hessenberg matrix, with an analytic solution \cite{Sowik2018} for the first term $d\bm{r}'$. We truncate this solution (see Supplement) to explicitly relate $d\bm{c}$ to $d\bm{r}'$ as follows:
\begin{align}
\label{eq:predict_translate}
d\bm{r}' \approx
-\begin{bmatrix}
\gamma H_0^2 - H_1
\end{bmatrix}^{-1}
\begin{bmatrix}
-\gamma H_0 & ~I
\end{bmatrix}
\begin{bmatrix}
K\\
\dot{K}
\end{bmatrix}
(BP + C)d\bm{c}.
\end{align}
As a demonstration, we pick a finite $\Delta\bm{c} = 20$, and plot the original and predicted change in the reservoir states, and their outputs in spatial coordinates (Fig.~\ref{fig:f6}b). Hence, using only the feedback dynamics Eq.~\ref{eq:reservoirfbc} and sufficiently nearby training examples, changing $\bm{c}$ causes changes in the reservoir states from Eq.~\ref{eq:predict_translate} that map to a translation.
The same approach can be used for linear transformations, where the output weights are trained such that $W\bm{r}_c(t) \approx \bm{x}_c(t) = (I+cP)\bm{x}(t)$. For sufficiently nearby training examples, we implicitly approximate the differential relation $Wd\bm{r}(t) \approx P\bm{x}(t)dc \approx PW\bm{r}(t)dc$, which if properly stabilized, yields $Wd\bm{r}'(t) \approx PW\bm{r}'(t)$. Performing the same time derivatives and solution truncation as in the translation, we get the following relation between $dc$ and $d\bm{r}'$:
\begin{align}
\label{eq:predict_transform}
d\bm{r}' \approx
-\begin{bmatrix}
\gamma H_0^2 - H_1
\end{bmatrix}^{-1}
\begin{bmatrix}
-\gamma H_0 & ~I
\end{bmatrix}
\begin{bmatrix}
K(BPW\bm{r}' + C)\\
\dot{K}(BPW\bm{r}'+C) + KBPW\dot{\bm{r}}'
\end{bmatrix}
dc.
\end{align}
As another demonstration, we set $\Delta c = -40$, and plot the original and predicted change in the reservoir states, and their outputs (Fig.~\ref{fig:f6}c).
Finally, to understand how the reservoir is able to infer a bifurcation, we demonstrate that it learns a smooth translation of eigenvalues. Specifically, at $\rho^*$, the fixed points at the wings of the Lorenz system undergo a Hopf bifurcation, whereby the real component of complex conjugate eigenvalues goes from negative to positive. To track the eigenvalues of the autonomous reservoir, we linearize Eq.~\ref{eq:reservoirfbc} about a fixed point $\delta\bm{r}^*$ such that
\begin{align}
\label{eq:reservoirfblin}
\frac{1}{\gamma}\delta\dot{\bm{r}}' \approx [-I + UR + 2V\mathrm{diag}(R\delta\bm{r}^* + C\bm{c})R](\delta\bm{r}'-\delta\bm{r}^*).
\end{align}
Then, using the output weights trained only on stable Lorenz trajectories (at $c = 0, \rho = 23$ and $c = 1, \rho = 24$; Fig.~\ref{fig:f5}a,b), we track the autonomous reservoir's two most unstable eigenvalues (largest real component) at the fixed point as we vary the control parameter from $c = 0$ to $c = 3$. We find that these eigenvalues are complex conjugates whose real components go from negative to positive (Fig.~\ref{fig:f6}d). Hence, we demonstrate that not only can reservoirs learn smooth translations and transformations by mapping $d\bm{c}$ to $d\bm{r}'$, but they can also perform bifurcations by learning smooth changes in their eigenvalues.
\section{Simultaneous learning of multiple operations}
\begin{figure}[h!]
\centering
\includegraphics[width=6.5in]{figure7.pdf}
\caption{\textbf{Flight of the Lorenz.} A reservoir trained on translated inputs along the $x_1$ and $x_3$ directions evolves autonomously along a Lorenz-shaped chaotic manifold. We can change the $x_1$ and $x_3$ position of its representation by changing control parameters $c_1$ and $c_2$, respectively.}
\label{fig:f7}
\end{figure}
To close, here we demonstrate that reservoirs can easily learn multiple computations by changing multiple control inputs. We train a translation in the $x_1$ direction with control parameter $c_1$, and a translation in the $x_3$ direction with control parameter $c_2$. As before, we begin with a Lorenz time series $\bm{x}_{0,0}(t)$ generated from Eq.~\ref{eq:lorenz}, and created shifted copies
\begin{align*}
\bm{x}_{c_1,c_2}(t) = \bm{x}_{0,0}(t) + c_1\bm{a}_1 + c_2\bm{a}_2,
\end{align*}
where $\bm{a}_1 = [1;0;0]$ corresponds to an $x_1$ shift, and $\bm{a}_2 = [0;0;1]$ corresponds to an $x_3$ shift. We generate 10 shifted inputs, with one unshifted attractor ($c_1 = 0, c_2 = 0$), three shifts in the $x_1$ direction ($c_1 = 1,2,3, c_2 = 0$), three shifts in the $x_3$ direction ($c_1 = 0, c_2 = 1,2,3$), and three shifts in both directions ($c_1 = 1,2,3, c_2 = 1,2,3$). We use these shifted copies along with their corresponding control inputs to drive our reservoir and produce 10 reservoir time series $\bm{r}_{c_1,c_2}(t)$. Then, we concatenate these 10 time series into $\bm{x}(t)$ and $\bm{r}(t)$ to train output weights $W$ according to Eq.~\ref{eq:training}, and perform the feedback according to Eq.~\ref{eq:reservoirfbc} where $\bm{c} = [c_1;~c_2]$ is a vector. By changing parameters $c_1$ and $c_2$, the reservoir evolves about a Lorenz-shaped manifold that is shifted in the $x_1$ and $x_3$ directions (Fig.~\ref{fig:f7}).
\section{Discussion}
In this paper, we teach an RNN to evolve about a Lorenz-shaped manifold, and to control its evolution about a translated, transformed, and bifurcated continua of such manifolds. Our approach contributes to prior work on artificial neural networks in three significant ways \cite{Seung1998learning,Wu2016,Jaeger2010,Sussillo2009}. First, we provide a means by which a neural system can learn continuous interpolated and extrapolated modifications, along with discontinuous bifurcations, of its own representation solely through examples. Second, the learned manifolds are spatially and temporally complex, allowing for potential extensions to learning modifications of time series data such as speech or music with a structured yet unpredictable evolution. Third, we use a randomly generated and arbitrarily connected network that does not need to be artificially engineered to preserve invariance or manipulate information \cite{Wu2016}.
One of the main limitations of this work is the lack of a clear mechanism of how the network connectivity ultimately stabilizes the chaotic manifold. Much progress has been made in tackling this limitation, both by exercising theoretical concepts of generalized synchronization \cite{Rulkov1995}, and by developing tools for controlling chaos \cite{Ott1990}. However, there is insufficient knowledge to guarantee that a set of training and reservoir parameters will always successfully teach the desired computation. Similarly, we are unable to specify exactly how far to space the training examples for the feedback reservoir to successfully learn the linear relationships between the differential of the reservoir states and the control parameter.
A particularly promising area for future work is related to the simple quadratic form of the reservoir. Because all of these results are obtained by driving our reservoir in the quadratic regime, the same results should hold for common neural mass models, such as the Wilson-Cowan model \cite{Wilson1972}. Hence, these results may provide a unifying framework for learning and computing in dynamical neural models. Additionally, these results provide a basis for exploring more complex computations, such as inferring bifurcations in experimental data, and testing the reservoir's ``imagination'' in reconstructing more complex chaotic manifolds using incomplete data. Finally, and perhaps most astonishingly, the reservoir's ability to accurately reconstruct the full nonlinear geometry of the bifurcated Lorenz manifold after only observing pre-bifurcation data implies that it is not only imitating examples, but actually inferring higher-order nonlinear structure. This work therefore provides a starting point for exploring exactly how higher-order structure is learned by neural systems.
\section{Acknowledgments}
We are thankful for the insightful feedback and comments from Harang Ju and Keith Wiley. We gratefully acknowledge support from the John D. and Catherine T. MacArthur Foundation, the Alfred P. Sloan Foundation, the ISI Foundation, the Paul Allen Foundation, the Army Research Laboratory (No. W911NF-10-2-0022), the Army Research Office (Nos. Bassett-W911NF-14-1-0679, Grafton-W911NF-16-1-0474, and DCIST-W911NF-17-2-0181), the Office of Naval Research (ONR), the National Institute of Mental Health (Nos. 2-R01-DC-009209-11, R01-MH112847, R01-MH107235, and R21-M MH-106799), the National Institute of Child Health and Human Development (No. 1R01HD086888-01), National Institute of Neurological Disorders and Stroke (No. R01 NS099348), and the National Science Foundation (NSF) (Nos. DGE-1321851, BCS-1441502, BCS-1430087, NSF PHY-1554488, and BCS-1631550). The content is solely the responsibility of the authors and does not necessarily represent the official views of any of the funding agencies.
\section{Citation Diversity Statement}
Recent work in several fields of science has identified a bias in citation practices such that papers from women and other minorities are under-cited relative to other papers in the field \cite{Dworkin2020}. Here we sought to proactively consider choosing references that reflect the diversity of our field in thought, form of contribution, gender, and other factors. We classified gender based on the first names of the first and last authors, with possible combinations including male/male, male/female, female/male, and female/female. We regret that our current methodology is limited to consideration of gender as a binary variable. Excluding self-citations to the first and senior authors of our present paper, the references contain 50\% male/male, 23.5\% male/female, 11.8\% female/male, and 14.7\% female/female categorizations. We look forward to future work that will help us to better understand how to support equitable practices in science.
\newpage
\section{References}
\section{Methods}
In this section, we describe additional details about the methods and simulations used in the main text. We begin with a more thorough overview of reservoir dynamics and their derivation, followed by specific details of the numerical simulations.
\subsection{Reservoir dynamics}
The reservoir computing framework is a general scheme by which a nonlinear dynamical system (the reservoir) is driven by some input, and a simple linear readout of the reservoir states is trained \cite{Lukosevicius2012}. The reservoir consists of $N$ neural units, where each unit $i$ has a real-valued level of activity over time, $r_i(t)$. We collect this activity into an $N$-dimensional column vector
\begin{align*}
\bm{r}(t) =
\begin{bmatrix}
r_1(t)\\ r_2(t)\\ \vdots \\ r_N(t)
\end{bmatrix},
\end{align*}
that we refer to as the \emph{reservoir state}. These reservoir states are driven by some input time series of $M$ inputs $x_1(t), x_2(t), \dotsm, x_M(t)$, that we collect into the input vector
\begin{align*}
\bm{x}(t) = \begin{bmatrix}
x_1(t)\\ x_2(t)\\ \vdots\\ x_M(t)
\end{bmatrix}.
\end{align*}
In our framework, we add a set of $K$ control inputs $c_1,\dotsm,c_K$ that we collect into the control vector
\begin{align*}
\bm{c} =
\begin{bmatrix}
c_1\\ c_2\\ \vdots\\ c_K
\end{bmatrix}.
\end{align*}
For continuous time systems ($t \in \mathbb{R}_{\geq 0}$), a typical equation for the time-evolution of a reservoir consists of a nonlinear (usually sigmoidal \cite{Lukosevicius2012}) transformation $\bm{g}$ on a linear sum of all inputs and states written as
\begin{align*}
\frac{1}{\gamma} \dot{\bm{r}}(t) = -\bm{r}(t) + \bm{g}(A\bm{r}(t) + B\bm{x}(t) + C\bm{c} + \bm{d}),
\end{align*}
where $\dot{\bm{r}}(t)$ represents the time derivative \cite{Jaeger2010,Sussillo2009}, $A$ is a real-valued matrix of dimension $N \times N$, $B$ is a real-valued matrix of dimension $N \times M$, $C$ is a real-valued matrix of dimension $N \times K$, and $\bm{d}$ is a constant bias vector of dimension $N \times 1$. We can write the dynamics for each reservoir state, $r_i(t)$, as
\begin{align*}
\frac{1}{\gamma}\dot{r}_i(t) = -r_i(t) + g_i\left(\sum_{n=1}^N A_{in}r_n(t) + \sum_{m=1}^M B_{im}x_m(t) + \sum_{k=1}^K C_{ik}c_k + d_i\right).
\end{align*}
If we write $A_{i*}, B_{i*}$, and $C_{i*}$ as the $i$-th row of matrices $A, B$, and $C$, respectively, we can write this equation more concisely as
\begin{align}
\label{eq:idynamics}
\frac{1}{\gamma}\dot{r}_i(t) = -r_i(t) + g_i\left(A_{i*}\bm{r}(t) + B_{i*}\bm{x}(t) + C_{i*}\bm{c} + d_i\right).
\end{align}
We begin by observing that the reservoir states are evolved according to some predetermined input $\bm{x}(t)$ and control input $\bm{c}$ to generate the reservoir state time series $\bm{r}(t)$. Next, linear combinations of the reservoir state are taken to approximate the input $\bm{x}(t)$ by minimizing the matrix 2-norm of the difference in the numerical time series (see Sec. \ref{sec:training})
\begin{align*}
\|W\bm{r}(t) - \bm{x}(t)\|_2,
\end{align*}
where $W$ is the real valued matrix of dimension $M \times N$ that is trained. After training, we perform feedback by replacing the inputs $\bm{x}(t)$ with the trained outputs $W\bm{r}(t)$ to yield the feedback dynamics
\begin{align*}
\frac{1}{\gamma} \dot{\bm{r}}'(t) = -\bm{r}'(t) + \bm{g}(A\bm{r}'(t) + BW\bm{r}'(t) + C\bm{c} + \bm{d}),
\end{align*}
and by factoring the term $R = A+BW$, we obtain
\begin{align*}
\frac{1}{\gamma} \dot{\bm{r}}'(t) = -\bm{r}'(t) + \bm{g}(R\bm{r}'(t) + C\bm{c} + \bm{d}).
\end{align*}
This feedback equation is written element-wise as
\begin{align*}
\frac{1}{\gamma}\dot{r}'_i(t) = -r'_i(t) + g_i\left(\sum_{n=1}^N R_{in}r'_n(t) + \sum_{k=1}^K C_{ik}c_k + d_i\right).
\end{align*}
We note that $R$ is an $N \times N$ matrix.
\newpage
\subsection{Derivation of quadratic reservoir}
While the reservoir computing framework often uses a specific nonlinear sigmoid function for $\bm{g}$, our goal is to find principles of learning relevant to a broad range of functional forms of $\bm{g}$. To achieve this goal, we study the reservoir in the weakly nonlinear regime. By deriving insights in this regime, we aim to make statements about dynamical systems with many different forms of $\bm{g}$ as long they are driven in the same regime. By weakly nonlinear, we mean the \emph{quadratic} regime, where the reservoir evolves nearby some constant stable fixed point attractor $\bm{r}^*$. This regime is explicitly encoded by taking the second-order Taylor series expansion of the dynamics about a steady state in all of the inputs $\bm{r}^*, \bm{x}^*, \bm{c}^*$. For notational convenience, we write $\delta\bm{r} = \bm{r}-\bm{r}^*$, $\delta\bm{x} = \bm{x}-\bm{x}^*$, and $\delta\bm{c} = \bm{c}-\bm{c}^*$, and also omit the notation of time dependence, $(t)$. If we write the full dynamics from Eq.~\ref{eq:idynamics} as
\begin{align*}
\frac{1}{\gamma}\dot{r}_i = f_i(\bm{r},\bm{x},\bm{c}) = -r_i + g_i\left(A_{i*}\bm{r} + B_{i*}\bm{x} + C_{i*}\bm{c} + d_i\right),
\end{align*}
then the Taylor series expansion to second order contains terms
\begin{align*}
T_{i,0} &= f_i|_{\bm{r}=\bm{r}^*,\bm{x}=\bm{x}^*,\bm{c}=\bm{c}^*}\\
T_{i,1} &= \nabla_{\bm{r},\bm{x},\bm{c}} f_i|_{\bm{r}=\bm{r}^*,\bm{x}=\bm{x}^*,\bm{c}=\bm{c}^*}\\
T_{i,2} &= \nabla^2_{\bm{r},\bm{x},\bm{c}} f_i|_{\bm{r}=\bm{r}^*,\bm{x}=\bm{x}^*,\bm{c}=\bm{c}^*},
\end{align*}
where $\nabla$ is the gradient operator with respect to the subscripted variables
\begin{align*}
\nabla_{\bm{r},\bm{x},\bm{c}}
&=
\begin{bmatrix}
\nabla_{\bm{r}}, & \nabla_{\bm{x}}, & \nabla_{\bm{c}}
\end{bmatrix}\\
&=
\begin{bmatrix}
\frac{\partial}{\partial r_1}, & \dotsm, & \frac{\partial}{\partial r_N}, &
\frac{\partial}{\partial x_1}, & \dotsm, &
\frac{\partial}{\partial x_M}, &
\frac{\partial}{\partial c_1}, & \dotsm, &
\frac{\partial}{\partial c_K}
\end{bmatrix},
\end{align*}
yielding a vector of partial derivatives, and $\nabla^2$ yields a matrix of all pairwise second partial derivatives. Then the quadratic dynamics become
\begin{align*}
\frac{1}{\gamma} \delta\dot{r}_i \approx T_{i,0} + T_{i,1} \begin{bmatrix}
\delta\bm{r}\\ \delta\bm{x}\\ \delta\bm{c}
\end{bmatrix}
+
\frac{1}{2}
\begin{bmatrix}
\delta\bm{r}^\top & \delta\bm{x}^\top & \delta\bm{c}^\top
\end{bmatrix}
T_{i,2} \begin{bmatrix}
\delta\bm{r}\\ \delta\bm{x}\\ \delta\bm{c}
\end{bmatrix}.
\end{align*}
As we are evaluating the dynamics about a fixed point, the reservoir does not change its state at this point, such that $f_i|_{\bm{r}=\bm{r}^*,\bm{x}=\bm{x}^*,\bm{c}=\bm{c}^*} = 0$. Hence the term
\begin{align*}
T_{i,0} = 0
\end{align*}
vanishes. The next term, $T_{i,1}$, is a linear approximation of the dynamics $f_i$
\begin{align*}
T_{i,1}
&= \nabla_{\bm{r},\bm{x},\bm{c}}(-r_i) + \nabla_{\bm{r},\bm{x},\bm{c}} g_i\left(A_{i*}\bm{r} + B_{i*}\bm{x} + C_{i*}\bm{c} + d_i\right)\\
&= \begin{bmatrix} 0, & \dotsm, & 0, & -1, & 0, & \dotsm, & 0 \end{bmatrix} + u_i \begin{bmatrix}
A_{i*}, & B_{i*}, & C_{i*}
\end{bmatrix},
\end{align*}
where $u_i = \left. \frac{\partial g_i}{\partial (\bm{r}, \bm{x}, \bm{c})} \right|_{\bm{r}^*,\bm{x}^*,\bm{c}^*}$ is the evaluation of the first derivative of $g_i$ at the fixed point, and the subsequent term $T_{i,2}$ approximates the quadratic curvature of $f_i$ as follows:
\begin{align*}
T_{i,2}
&= \nabla^2_{\bm{r},\bm{x},\bm{c}} f_i|_{\bm{r}=\bm{r}^*,\bm{x}=\bm{x}^*,\bm{c}=\bm{c}^*}\\
&= 2v_i
\begin{bmatrix}
A_{i*}^\top\\
B_{i*}^\top\\
C_{i*}^\top
\end{bmatrix}
\begin{bmatrix}
A_{i*}, & B_{i*}, & C_{i*}
\end{bmatrix},
\end{align*}
where $v_i = \frac{1}{2}\left. \frac{\partial^2 g_i}{\partial (\bm{r}, \bm{x}, \bm{c})^2} \right|_{\bm{r}^*,\bm{x}^*,\bm{c}^*}$ is half of the evaluation of the second derivative of $g_i$ at the fixed point.
Substituting these values back into the quadratic dynamics, we obtain
\begin{align*}
\frac{1}{\gamma} \delta\dot{r}_i \approx
-\delta r_i
+
u_i \begin{bmatrix}
A_{i*}, & B_{i*}, & C_{i*}
\end{bmatrix}
\begin{bmatrix}
\delta\bm{r}\\ \delta\bm{x}\\ \delta\bm{c}
\end{bmatrix} +
v_i
\begin{bmatrix}
\delta\bm{r}^\top & \delta\bm{x}^\top & \delta\bm{c}^\top
\end{bmatrix}
\begin{bmatrix}
A_{i*}^\top\\
B_{i*}^\top\\
C_{i*}^\top
\end{bmatrix}
\begin{bmatrix}
A_{i*}, & B_{i*}, & C_{i*}
\end{bmatrix}
\begin{bmatrix}
\delta\bm{r}\\ \delta\bm{x}\\ \delta\bm{c}
\end{bmatrix},
\end{align*}
and we notice that
\begin{align*}
\begin{bmatrix}
\delta\bm{r}^\top & \delta\bm{x}^\top & \delta\bm{c}^\top
\end{bmatrix}
\begin{bmatrix}
A_{i*}^\top\\
B_{i*}^\top\\
C_{i*}^\top
\end{bmatrix}
=
\begin{bmatrix}
A_{i*}, & B_{i*}, & C_{i*}
\end{bmatrix}
\begin{bmatrix}
\delta\bm{r}\\ \delta\bm{x}\\ \delta\bm{c}
\end{bmatrix},
\end{align*}
to yield
\begin{align*}
\frac{1}{\gamma} \delta\dot{r}_i \approx
-\delta r_i
+
u_i
\begin{bmatrix}
A_{i*}, & B_{i*}, & C_{i*}
\end{bmatrix}
\begin{bmatrix}
\delta\bm{r}\\ \delta\bm{x}\\ \delta\bm{c}
\end{bmatrix}
+
v_i
\left(
\begin{bmatrix}
A_{i*}, & B_{i*}, & C_{i*}
\end{bmatrix}
\begin{bmatrix}
\delta\bm{r}\\ \delta\bm{x}\\ \delta\bm{c}
\end{bmatrix}
\right)^2,
\end{align*}
which can be rewritten as
\begin{align*}
\frac{1}{\gamma}\delta\dot{r}_i \approx -\delta r_i + u_i (A_{i*}\delta\bm{r} + B_{i*}\delta\bm{x} + C_{i*}\delta\bm{c}) + v_i (A_{i*}\delta\bm{r} + B_{i*}\delta\bm{x} + C_{i*}\delta\bm{c})^2.
\end{align*}
Compiling the dynamics of all reservoir nodes $\bm{r}$, we write the compact vector form of the dynamics as
\begin{align*}
\frac{1}{\gamma}\delta\dot{\bm{r}} = -\delta\bm{r} + U(A\delta\bm{r} + B\delta\bm{x} + C\delta\bm{c}) + V(A\delta\bm{r} + B\delta\bm{x} + C\delta\bm{c})^2,
\end{align*}
where $U$ and $V$ are diagonal matrices where the $i$-th elements are $u_i$ and $v_i$ obtained by evaluating the first and second derivatives of $g_i$, respectively. To avoid making any assumptions about the operating point of the input states or control inputs, we linearize about $\bm{x}^* = \bm{0}$ and $\bm{c}^* = \bm{0}$. Further, we notice that $\delta\dot{\bm{r}} = \frac{d}{dt} (\bm{r} - \bm{r}^*) = \dot{\bm{r}}$. Hence, the above vector form of the equation becomes
\begin{align}
\label{eq:quadratic}
\frac{1}{\gamma}\delta \dot{\bm{r}} = -\delta \bm{r} + U(A\delta\bm{r} + B\bm{x} + C\bm{c}) + V(A\delta\bm{r} + B\bm{x} + C\bm{c})^2.
\end{align}
Importantly, we note that the elements of $U$ and $V$ are not allowed to arbitrarily be any value. Instead, their values depend on the first and second derivatives of the activation function $\bm{g}$ evaluated at the fixed points $\bm{r}^*,\bm{x}^*$, and $\bm{c}^*$. For the sake of remaining relevant to the existing literature that frequently uses the hyperbolic tangent function $tanh$ \cite{Lukosevicius2012}, we will restrict the values of $U$ and $V$. First, we note that at the fixed point, we have
\begin{align*}
\bm{0} = -\bm{r}^* + \bm{g}(A\bm{r}^* + B\bm{x}^* + C\bm{c}^*).
\end{align*}
Recall that we evaluate our functions at $\bm{x}^* = \bm{0}$ and $\bm{c}^* = \bm{0}$, such that $\bm{g}(A\bm{r}^*) = \bm{r}^*$, regardless of the form of $\bm{g}$. Next, we consider the specific activation function
\begin{align*}
\bm{g}(\bm{r},\bm{x},\bm{c}) = \tanh(A\bm{r} + B\bm{x} + C\bm{c} + \bm{d}),
\end{align*}
and evaluate its first derivative at $\bm{r}^*, \bm{x}^* = \bm{0}, \bm{c}^* = \bm{0}$ to yield
\begin{align}
\label{eq:U}
\mathrm{d} \bm{g}_{\bm{r}^*,\bm{x}^*=\bm{0},\bm{c}^*=\bm{0}} = \bm{1}-\tanh(A\bm{r}^*)^2 = \bm{1}-\bm{r}^{*2},
\end{align}
where $\bm{1}$ is an $N$-dimensional vector of ones, and the square notation of the vector implies an element-wise square. Hence, $U$ is a diagonal matrix where the $i$-th entry is $1 - r_i^{*2}$. Finally, we take the second derivative
\begin{align}
\label{eq:V}
\mathrm{d}^2 \bm{g}_{\bm{r}^*,\bm{x}^*=\bm{0},\bm{c}^*=\bm{0}} = -2(\tanh(A\bm{r}^*) - \tanh(A\bm{r}^*)^3) = 2(\bm{r}^{*3} - \bm{r}^*),
\end{align}
such that the $i$-th diagonal element of $V$ is given by $r_i^{*3} - r_i^*$. Hence, for the quadratic approximation of $tanh$, it is sufficient to specify a fixed point $\bm{r}^*$ to fully determine the matrices $U$ and $V$.
\newpage
\subsection{Simulation parameters}
For all simulations in the main text of the paper, we used the following parameter choices:
\subsubsection{Global parameters}
\begin{itemize}
\item simulation time step: $\Delta t = 0.001$.
\end{itemize}
\subsubsection{Lorenz training data parameters}
\begin{itemize}
\item dynamical equation: \\
$\dot{x}_1 = \sigma(x_2-x_1) \hspace{3cm} \dot{x}_2 = x_1(\rho-x_3)-x_2 \hspace{3cm} \dot{x}_3 = x_1x_2 - \beta x_3$.
\item parameters: $\sigma = 10, \beta = 8/3, \rho = 28$ (except in the bifurcation example).
\item uniform random initial conditions: $x_1,x_2,x_3 \in [0,10]$.
\item throwaway simulation time (per attractor): $T_{\mathrm{waste}} = 20$.
\item training simulation time (per attractor): $T_{\mathrm{train}} = 200$.
\item translation training shift: $P = \begin{bmatrix} 1\\0\\0\end{bmatrix}$.
\item transformation training stretch: $P = \begin{bmatrix} -.012 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix}$.
\end{itemize}
\subsubsection{Reservoir}
\begin{itemize}
\item dynamical equation: $\frac{1}{\gamma} \dot{\bm{r}} = -\delta\bm{r} + U(A\delta\bm{r}+B\bm{x}+C\bm{c}) + V(A\delta\bm{r}+B\bm{x}+C\bm{c})^2$.
\item reservoir initial condition: $\bm{r}(0) = \bm{0}$.
\item adjacency matrix $A$ has 10\% binary density. We begin with matrix $\tilde{A}$ where each nonzero element is drawn from a uniform random distribution $\tilde{A}_{ij} \in [-1, 1]$. Then the matrix is normalized such that $A = 0.95 \frac{\tilde{A}}{\mathrm{real}(\lambda_{\mathrm{max}})}$, where $\lambda_{\mathrm{max}}$ is the eigenvalue with the largest real value. For a $tanh$ activation function, this normalization is used to ensure the echo-state property in discrete time reservoir systems \cite{Jaeger2010}.
\item number of reservoir neurons: $N = 300$.
\item time constant: $\gamma = 100$.
\item fixed point: each element was drawn from a random uniform distribution $r_i^* \in [-1,-0.8] \cup [0.8,1]$.
\item data input matrix: every row $i$ of $B$ has one non-zero element at index $j$. This index is chosen uniformly at random ($Pr(j=1) = \dotsm = Pr(j=M)$). For examples involving translations and transformations, the value of the element is drawn uniformly from $B_{ij} \in [-0.004,0.004]$. For examples involving bifurcations, the magnitude of the observed data is much smaller (local to the stable fixed point, instead of the full chaotic attractor manifold), and therefore the element is drawn uniformly from $B_{ij} \in [-0.04,0.04]$.
\item control input matrix: every row $i$ of $C$ has one non-zero element at index $j$. This index is chosen uniformly and randomly ($Pr(j=1) = \dotsm = Pr(j=K)$), and the value of the element is drawn uniformly from $C_{ij} \in [-0.002,0.002]$.
\end{itemize}
\subsection{Simulation method}
To simulate both the input and reservoir dynamics, we used a 4-th order Runge-Kutta numerical integration. For the dynamics of the Lorenz attractor,
\begin{align*}
\dot{\bm{x}} = \bm{f}(\bm{x}),
\end{align*}
the Runge-Kutta computes the following values
\begin{align*}
k_{x1} &= \Delta t \cdot \bm{f}\left(\bm{x}(t)\right)\\
k_{x2} &= \Delta t \cdot \bm{f}\left(\bm{x}(t) + \frac{k_{x1}}{2}\right)\\
k_{x3} &= \Delta t \cdot \bm{f}\left(\bm{x}(t) + \frac{k_{x2}}{2}\right)\\
k_{x4} &= \Delta t \cdot \bm{f}\left(\bm{x}(t) + k_{x3}\right),
\end{align*}
and evolves the state forward using
\begin{align*}
\bm{x}(t+\Delta t) = \bm{x}(t) + \frac{1}{6}(k_{x1} + 2k_{x2} + 2k_{x3} + k_{x4}).
\end{align*}
The simulation of the reservoir dynamics requires more careful analysis, because it is a system driven by external inputs. For the original reservoir dynamics given by Eq.~\ref{eq:quadratic}
\begin{align*}
\frac{1}{\gamma}\dot{\bm{r}} = \bm{f}(\bm{r},\bm{x},\bm{c}) = -\delta \bm{r} + U(A\delta\bm{r} + B\bm{x} + C\bm{c}) + V(A\delta\bm{r} + B\bm{x} + C\bm{c})^2,
\end{align*}
the algorithm to update the reservoir states is given by
\begin{align*}
k_{r1} &= \Delta t \cdot \bm{f}\left(\bm{r}(t),\bm{x}(t),\bm{c}(t)\right)\\
k_{r2} &= \Delta t \cdot \bm{f}\left(\bm{r}(t) + \frac{k_{r1}}{2}, \bm{x}(t) + \frac{k_{x1}}{2}, \bm{c}(t) + \frac{k_{c1}}{2}\right)\\
k_{r3} &= \Delta t \cdot \bm{f}\left(\bm{r}(t) + \frac{k_{r2}}{2}, \bm{x}(t) + \frac{k_{x2}}{2}, \bm{c}(t) + \frac{k_{c2}}{2}\right)\\
k_{r4} &= \Delta t \cdot \bm{f}\left(\bm{r}(t) + k_{r3}, \bm{x}(t) + k_{x3}, \bm{c}(t) + k_{c3}\right),
\end{align*}
and the reservoir state evolves forward according to
\begin{align*}
\bm{r}(t+\Delta t) = \bm{r}(t) + \frac{1}{6}(k_{r1} + 2k_{r2} + 2k_{r3} + k_{r4}).
\end{align*}
Hence, when we simulate the Lorenz state $\bm{x}(t)$, we also save the corresponding values $k_{x1},\dotsm,k_{x3}$ to use in the reservoir update algorithm. Finally, we note that in our simulations, we slowly vary the control input $\bm{c}(t)$ over time, requiring us to determine the trajectory of $\bm{c}(t)$ beforehand. However, we require a differential equation that generated $\bm{c}(t)$ to solve for the final parameters $k_{c1},\dotsm,k_{c4}$. We assume the differential equations that generate $\bm{c}$ are constant, such that between time $t$ and $t+\Delta t$, the rate of change of $\bm{c}(t)$ is given by
\begin{align*}
\dot{\bm{c}}(t) = \bm{f}(\bm{c}(t)) = \frac{\bm{c}(t+\Delta t) - \bm{c}(t)}{\Delta t}.
\end{align*}
Such dynamics yield the parameters
\begin{align*}
k_{c1} &= \Delta t \cdot \bm{f}\left(\bm{c}(t)\right) = \bm{c}(t+\Delta t) - \bm{c}(t)\\
k_{c2} &= \Delta t \cdot \bm{f}\left(\bm{c}(t) + \frac{k_{c1}}{2}\right) = \Delta t \cdot \bm{f}\left(\frac{\bm{c}(t+\Delta t) + \bm{c}(t)}{2}\right) = \bm{c}(t+\Delta t) - \bm{c}(t)\\
k_{c3} &= \Delta t \cdot \bm{f}\left(\bm{c}(t) + \frac{k_{c2}}{2}\right) = \Delta t \cdot \bm{f}\left(\frac{\bm{c}(t+\Delta t) + \bm{c}(t)}{2}\right) = \bm{c}(t+\Delta t) - \bm{c}(t).
\end{align*}
The same integration is used with feedback where $\frac{1}{\gamma}\dot{\bm{r}} = \bm{f}(\bm{r},\bm{c}) = -\delta\bm{r} + U(R\delta{\bm{r}} + C\bm{c}) + V(R\delta{\bm{r}} + C\bm{c})^2$.
\newpage
\subsection{Training}
\label{sec:training}
Using the dynamical equations and RK4 integration scheme, we first generated the Lorenz attractor training inputs $\bm{x}(t)$. Each of the single direction translation and transformation examples described in the main text used four Lorenz attractor inputs. The first was the original Lorenz time series $\bm{x}(t)$, and the remaining three were translations or rotations of the original. Each of these four time series were simulated for $T = T_{\mathrm{waste}} + T_{\mathrm{train}} = 20 + 200 = 220$ time. At a time step of $\Delta t = 0.001$, each time series $\bm{x}(t)$ contained $\frac{T}{\Delta t} = 220,000$ simulation time points, stored in data matrix $X_0$ for the original attractor. Because we also kept the 4 outputs of the $RK4$ numerical integration scheme, the data matrix $X_0$ had dimensions $\mathrm{variables} \times \mathrm{time~steps} \times \mathrm{RK4} = 3 \times 220,000 \times 4$. With three additional time series for translation or rotation, $X_1,X_2,X_3$, we concatenated the four time series along the second dimension into the full matrix $X$ with dimension $3 \times 880,000 \times 4$.
Using this Lorenz data matrix $X$, and a corresponding control input data matrix, we drove the reservoir to generate $\bm{r}(t)$, contained in a reservoir data matrix $D$ that was of size $N=300 \times 880,000$. For every $\frac{T}{\Delta t} = 220,000$ time steps, we threw away the first $\frac{T_{\mathrm{waste}}}{\Delta t} = 20,000$ time points, as this simulation allowed both the Lorenz and reservoir systems to forget their initial conditions. The remaining $\frac{T_{\mathrm{train}}}{\Delta t} = 200,000$ time points of each attractor were kept for training. This process yields a Lorenz training matrix $X_{\mathrm{train}}$ of dimension $3 \times 800,000$ (as we throw away the RK4 simulation parameters after driving the reservoir), and a reservoir training matrix $D_{\mathrm{train}}$ of dimension $300 \times 800,000$.
Finally, we seek a training matrix $M$ of dimension $3 \times 300$ that minimizes the matrix 2-norm
\begin{align*}
\|M D_{\mathrm{train}} - X_{\mathrm{train}}\|_2.
\end{align*}
Specifically, we use MATLAB's command \verb|lsqminnorm|, that not only minimizes this norm, but in the event that multiple solutions exist, also minimizes the norm of $M$.
\newpage
\subsection{Training maps the fixed point to 0}
Here, we describe a particular but important methodological nuance to the training. Recall that from Eq.~\ref{eq:quadratic}, our reservoir evolves according to
\begin{align*}
\frac{1}{\gamma}\delta\dot{\bm{r}} = -\delta \bm{r} + U(A\delta\bm{r} + B\bm{x} + C\bm{c}) + V(A\delta\bm{r} + B\bm{x} + C\bm{c})^2,
\end{align*}
where $\delta\bm{r} = \bm{r} - \bm{r}^*$. If our training scheme involved approximating the inputs with the \emph{difference} of the reservoir states $\delta\bm{r}$ such that $W\delta\bm{r}(t) \approx \bm{x}(t)$, then this nuance would be unnecessary, as the feedback dynamics would take the proper form
\begin{align*}
\frac{1}{\gamma}\dot{\bm{r}}'
&= -\delta \bm{r}' + U(A\delta\bm{r}' + BW\delta\bm{r}' + C\bm{c}) + V(A\delta\bm{r}' + BW\delta\bm{r}' + C\bm{c})^2\\
&= -\delta \bm{r}' + U(R\delta\bm{r}' + C\bm{c}) + V(R\delta\bm{r}' + C\bm{c})^2.
\end{align*}
Unfortunately, such a scheme would require the additional assumptions that during training, the neural system was able to accurately retain knowledge of its fixed point $\bm{r}^*$, and that it was also able to take the difference of the neural activity with respect to this fixed point in real time. We avoid these additional assumptions by training on the true reservoir states such that $W\bm{r}(t) \approx \bm{x}(t)$, yielding
\begin{align*}
\frac{1}{\gamma}\delta\dot{\bm{r}}'
&= -\delta \bm{r}' + U(A\delta\bm{r}' + BW\bm{r}' + C\bm{c}) + V(A\delta\bm{r}' + BW\bm{r}' + C\bm{c})^2\\
&= -\delta \bm{r}' + U(R\delta\bm{r}' + BW\bm{r}^* + C\bm{c}) + V(R\delta\bm{r}' + BW\bm{r}^* + C\bm{c})^2.
\end{align*}
In both the linear and quadratic terms, we notice an extra and undesired term $BW\bm{r}^*$. In all of our simulations, the training of matrix $W$ actually maps the fixed point to a small number, such that $W\bm{r}^*$ is on the order of $10^{-6}$, whereas $W\delta\bm{r}(t)$ is on the order of $10^1$. Hence, the matrix $W$ maps the fixed point to values that are 7 orders of magnitude smaller than the magnitude of the inputs, such that $W\bm{r}'(t) = W\delta\bm{r}'(t) + W\bm{r}^* \approx W\delta\bm{r}'(t)$, thereby rendering the undesired term $BW\bm{r}^*$ effectively negligible.
At first, we might be tempted to explain this phenomenon by the fact that the Lorenz attractor $\bm{x}(t)$ is centered around $x_1 = 0$ and $x_2 = 0$. Hence, it would make sense that a constant fixed point $\bm{r}^*$ would map to a value of $0$. However, the third coordinate of the Lorenz system is centered around $x_3 = \rho-1 = 27$, and yet in our simulations training still produces an output matrix M that maps the fixed point to 0, even along the $x_3$ coordinate, across randomly assigned fixed points $\bm{r}*$ and reservoir parameters $A$, $B$, and $C$.
\newpage
\subsection{Truncation of the block-Hessenberg matrix}
To understand the mechanism of learning translations and transformations, we had taken the differential of the reservoir feedback dynamics,
\begin{align*}
(I-KA)d\bm{r}' + \frac{1}{\gamma}d\dot{\bm{r}}' = K(BWd\bm{r}' + Cd\bm{c}).
\end{align*}
If we take time derivatives of the left-hand side of this equation, we obtain
\begin{align*}
\begin{bmatrix}
(I-KA) & & \frac{1}{\gamma}I & & 0 & & 0 & & 0 & \dotsm\\
\downarrow & \searrow & & \searrow\\
-\dot{K}A & & (I-KA) & & \frac{1}{\gamma}I & & 0 & & 0 & \dotsm\\
\downarrow & \searrow & \downarrow & \searrow & & \searrow\\
-\ddot{K}A & & -2\dot{K}A & & (I-KA) & & \frac{1}{\gamma}I & & 0 & \dotsm\\
\downarrow & \searrow & \downarrow & \searrow & \downarrow & \searrow & & \searrow\\
-\dddot{K}A & & -3\ddot{K}A & & -3\dot{K}A & & (I-KA) & & \frac{1}{\gamma}I & \dotsm\\
\vdots & & \vdots & & \vdots & & \vdots & & \vdots & \ddots
\end{bmatrix}
\begin{bmatrix}
d\bm{r}'\\ \\
d\dot{\bm{r}}'\\ \\
d\ddot{\bm{r}}'\\ \\
d\dddot{\bm{r}}'\\
\vdots
\end{bmatrix},
\end{align*}
where the element in the $i$-th row and $j$-th column has a coefficient
\begin{align*}
p_{i,j} = \begin{pmatrix}i-1\\j-1\end{pmatrix} \hspace{1cm} \mathrm{for}~j \leq i,
\end{align*}
according to Pascal's triangle. For the translation examples, we can write the continued time derivatives of the differential relation as
\begin{align*}
\underbrace{
\begin{bmatrix}
H_0 & H_{-1} & 0 & 0 & \dotsm\\
H_1 & H_0 & H_{-1} & 0 & \dotsm\\
H_2 & 2H_1 & H_0 & H_{-1} & \dotsm\\
H_3 & 3H_2 & 3H_1 & H_0 & \dotsm\\
\vdots & \vdots& \vdots & \vdots & \ddots
\end{bmatrix}}_{J}
\begin{bmatrix}
d\bm{r}'\\
d\dot{\bm{r}}'\\
d\ddot{\bm{r}}'\\
d\dddot{\bm{r}}'\\
\vdots
\end{bmatrix}
\approx
\begin{bmatrix}
K\\
\dot{K}\\
\ddot{K}\\
\dddot{K}\\
\vdots
\end{bmatrix}
(BP + C)d\bm{c},
\end{align*}
where $H_{-1} = \frac{1}{\gamma}I$, $H_0 = I-KA$, and $H_i = -K^{(i)}A$ is the $i$-th time-derivative of $KA$. This matrix is a block matrix (each element $H$ is a matrix), and is specifically a block-Hessenberg matrix (zero above the first block-super diagonal). The goal here is to solve for $d\bm{r}'$ with respect to $d\bm{c}$. If we truncate $J$ to a finite-dimensional matrix such that
\begin{align*}
J \simeq
\begin{bmatrix}
J_{11} & J_{12}\\
J_{21} & J_{22}
\end{bmatrix},
\end{align*}
where
\begin{align*}
J_{11} &=
\begin{bmatrix}
H_0\\H_1\\\vdots\\H_{k-1}
\end{bmatrix},
\hspace{2cm}
J_{12} =
\begin{bmatrix}
H_{-1}, & 0, & \dotsm, & 0\\
H_0, & H_{-1}, & \dotsm, & 0\\
\vdots & \vdots & \ddots & \vdots\\
p_{k,2}H_{k-2}, & p_{k,3}H_{k-3}, & \dotsm, & H_{-1}
\end{bmatrix},\\
J_{21} &=
\begin{bmatrix}
p_{k+1,1}H_k
\end{bmatrix},
\hspace{1.6cm}
J_{22} =
\begin{bmatrix}
p_{k+1,2}H_{k-1} & p_{k+1,3}H_{k-2} & \dotsm & H_0
\end{bmatrix},
\end{align*}
Then, the closed form solution for the first $N$ rows of $J^{-1}$ (the first block) can be written \cite{Sowik2018} as
\begin{align}
\label{eq:HTBM}
[J^{-1}]_{(1:N,:)} \simeq
-(J_{22} J_{12}^{-1} J_{11} - J_{21})^{-1}
\begin{bmatrix}
-J_{22} J_{12}^{-1} & I
\end{bmatrix}.
\end{align}
However, in reality, $J$ is not a finite matrix, but an infinite dimensional matrix. An important fact to verify, then, is whether there exists a sufficiently large value of $k$ to yield an accurate inversion. While proving that this inverse converges is outside the scope of this work, we numerically demonstrate in what follows that after $k=1$, successive terms do not perceivably change the results. Specifically, we solve for $d\bm{r}'$ with respect to $d\bm{c}$ for $k = 0, 1, 2, 3$.
As a reference for translation, at $k=0$, the approximation becomes
\begin{align*}
d\bm{r}' \approx H_0^{-1} K(BP+C)d\bm{c},
\end{align*}
and at $k=1$, we obtain the approximation used in the main text. The $0$-th order approximation at $k=0$ yields no change (Fig.~\ref{fig:sf3}a), where the predicted reservoir states (light gold) are identical to the original states (dark gold). The first order approximation at $k=1$ (Fig.~\ref{fig:sf3}b) yields a change in the reservoir states that outputs to the expected translation in spatial coordinates. Taking more terms in the approximation ($k=2$, Fig~\ref{fig:sf3}c; and $k=3$, Fig.~\ref{fig:sf3}d) yields no perceivable change in either the reservoir states or their outputs.
\begin{figure}[h!]
\includegraphics[width=6.5in]{suppfigure3.pdf}
\caption{\textbf{Predicted change in reservoir states given a change in control parameter.} Reservoir time series generated by driving the reservoir with the original Lorenz input with $c = 0$ (dark gold), and the predicted time series from solving for $d\bm{r}'$ after training on translated examples and changing the control parameter $\Delta c = 20$ (light gold), along with their output projections (dark and light red, respectively). These approximations were taken by computing the inverse Eq.~\ref{eq:HTBM} for (\textbf{a}) $k=0$, (\textbf{b}) $k=1$, (\textbf{c}) $k=2$, and (\textbf{d}) $k=3$.}
\label{fig:sf3}
\end{figure}
\newpage
\subsection{Summary}
In sum, we have provided a general form for reservoir dynamics (Eq.~\ref{eq:idynamics}), the derivation for the quadratic form of the reservoir (Eq.~\ref{eq:quadratic}), as well as the dependence of matrices $U$ and $V$ that arise from the choice of fixed point when using $tanh$ as the activation function (Eq.~\ref{eq:U},\ref{eq:V}). We further provide all simulation parameters (time step, Lorenz parameters and initial conditions, reservoir parameters and initial conditions), along with the specific details of our simulation method, data dimensions, and training process. Finally, we provide numerical justification for the truncation of our approximation when deriving the mechanism of learning (Eq.~\ref{eq:HTBM}).
\newpage
\section{Results}
In this section, we provide some additional results to support the generalizability of the framework.
\subsection{Translation in multiple directions}
In the main text, we demonstrated that a reservoir can translate its representation of a Lorenz attractor along the $x_1$ direction. Specifically, we took an untranslated Lorenz time series $\bm{x}_0(t)$, and generated three additional training examples $\bm{x}_c(t)$ for $c = 1,2,3$ such that
\begin{align*}
\bm{x}_c(t) = \bm{x}_0(t) +
c\begin{bmatrix}
1\\0\\0
\end{bmatrix}.
\end{align*}
We then drove the reservoir using these four training examples and an additional control parameter $c$ that we also varied from $c = 0, \dotsm, 4$. Afterwards, we performed the feedback, and translated the reservoir's representation by varying the external control parameter $c$ from $-40$ to $40$. We reproduce this translated representation here (Fig.~\ref{fig:sf1}a). We show the same output of the feedback reservoir trained on four examples translated in the $x_2$ direction ($\bm{x}_c(t) = \bm{x}_0(t) + c[0;1;0]$) and in the $x_3$ direction ($\bm{x}_c(t) = \bm{x}_0(t) + c[0;0;1]$) (Fig.~\ref{fig:sf1}b,c). Hence, we demonstrate that the reservoir can learn these translations in arbitrary directions.
\begin{figure}[h!]
\includegraphics[width=6.5in]{suppfigure1.pdf}
\caption{\textbf{Translation of the Lorenz representation in all three spatial directions.} (\textbf{a}) Output of the feedback reservoir after being trained on 4 time series of a Lorenz attractor translated in the $x_1$ direction at $c = 0,\dotsm,4$. By varying $c$ from $-40$ to $40$, the representation shifts in the $x_1$ direction. (\textbf{b}) The same scheme is employed for translations in the $x_2$ direction, and (\textbf{c}) in the $x_3$ direction.}
\label{fig:sf1}
\end{figure}
\newpage
\subsection{Different types of transformations}
In the main text, we demonstrated that a reservoir trained on the original Lorenz attractor $\bm{x}_0(t)$ and on three transformed examples $\bm{x}_c(t) = (I+cP)\bm{x}_0(t)$ for $c = 1,2,3$, was able to continuously interpolate and extrapolate the transformation on its internal representation, even for control inputs between $-40$ and $40$. Here, we consider a stretch in the $x_3$ direction, a shear in the $x_1$ direction, and a shear in the $x_1$ and $x_2$ directions. Specifically, we use the three matrices
\begin{align*}
P_{\mathrm{stretch,x_3}} =
\begin{bmatrix}
0 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 0.012
\end{bmatrix},
\hspace{.7cm}
P_{\mathrm{shear,x_1}} =
\begin{bmatrix}
0 & 0 & 0\\
0.012 & 0 & 0\\
0 & 0 & 0
\end{bmatrix},
\hspace{.7cm}
P_{\mathrm{shear,x_1,x_2}} =
\begin{bmatrix}
0 & -0.012 & 0\\
0.012 & 0 & 0\\
0 & 0 & 0
\end{bmatrix},
\end{align*}
to train our reservoir for $c = 0,\dotsm,4$. For each transformation, we drive the reservoir with the input Lorenz attractors $\bm{x}_c(t)$ and an additional control input $c$ for $c = 0,\dotsm,4$. We then train the reservoir by applying the feedback method used in the main text. Finally, we drive the autonomous feedback reservoir by varying the control parameter from $c = -40$ to $c = 40$ for these three transformations (Fig.~\ref{fig:sf2}).
\begin{figure}[h!]
\includegraphics[width=6.5in]{suppfigure2.pdf}
\caption{\textbf{Transformation of the Lorenz representation using stretch and shear in several spatial directions.} (\textbf{a}) Output of the feedback reservoir after being trained on 4 time series of a Lorenz attractor stretched in the $x_3$ direction at $c = 0,\dotsm,4$. By varying $c$ from $-40$ to $40$, the representation stretches in the $x_3$ direction. (\textbf{b}) The same scheme is employed for a shear in the $x_2$ direction, and (\textbf{c}) for a shear in the $x_1$ and $x_2$ directions.}
\label{fig:sf2}
\end{figure}
\newpage
\newpage
|
1,116,691,500,870 | arxiv | \section{Introduction}
\label{sec:Intro}
The Advanced Laser Interferometer Gravitational-wave Observatory\footnote{\url{http://www.ligo.caltech.edu/}} (aLIGO; \citealt{Aasietal2015}) and Advanced Virgo\footnote{\url{http://www.ego-gw.it/}} (AdV; \citealt{Acerneseetal2015} have made the first six detections of gravitational waves (GWs) from approximately circular inspiraling binaries \citep{Abbottetal2016c,Abbottetal2016b,Abbottetal2017,LIGOColl2017,Abbottetal2017_2,Abbottetal2017_3} during their first two observing runs, and opened a new window through which the universe can be observed. Two additional GW detectors are planned to join the network of aLIGO and AdV: (i) the Japanese KAGRA\footnote{\url{http://gwcenter.icrr.u-tokyo.ac.jp/en/}} is under construction, with baseline operations beginning in $2018$ \citep{Somiya2012}; while (ii) the proposed LIGO-India\footnote{\url{http://www.gw-indigo.org/}} is expected to become operational in $2022$ \citep{Iyeretal2011,Abbottetal2016a}. These instruments are expected to continue to make detections of GWs and to answer fundamental questions about their astrophysical sources.
One of the most anticipated type of events to be detected with ground-based GW detectors is the circular inspiral and coalescence of compact binaries consisting of neutron stars (NSs) and/or black holes (BHs). The current detections constrain the merger rate density of BH--BH mergers in the universe to $12-213 \, \mathrm{Gpc}^{-3}\mathrm{yr}^{-1}$ \citep{Abbottetal2017}. At a typical detection range of $2 \, \mathrm{Gpc}$, for aLIGO's design sensitivity, this corresponds to a detection rate between $400-7.000 \, \mathrm{yr}^{-1}$. There are various theoretical studies to explain these rates; see \citet{Abadieetal2010} for a partial list of historical compact binary coalescence rate predictions, and \citealt{Dominiketal2013,Kinugawaetal2014,Abbottetal2016g,Abbottetal2016f,Abbottetal2016h,Abbottetal2016d,Abbottetal2016e,Belczynskietal2016,Rodriguezetal2016a,Sasakietal2016,Bartosetal2017,McKernanetal2017,Stoneetal2017,Hoangetal2017} and references therein for recent rate estimates. Because GW emission tends to circularize the orbit as it shrinks the binary separation \citep{Peters1964}, many of the GW sources are expected to have small orbital eccentricity by the time they enter the frequency bands of ground-based GW detectors. For instance, although the eccentricity of the Hulse--Taylor pulsar is currently $0.6171$ \citep{HulseTaylor1975}, it will have an eccentricity of $\sim 10^{-4}$ when it enters the aLIGO band (see Section \ref{subsec:DistIniParam}). However, not all binary sources are circular.
There are multiple reasons to expect a non-negligible eccentricity for some binary sources emitting GWs in the aLIGO band. One mechanism leading to eccentric mergers is the Kozai--Lidov oscillation in hierarchical triple systems \citep{Wen2003}, in which the binary is perturbed by a distant third object. Indeed, high-mass stars, the progenitors of BHs, are commonly found in triples in the galactic field (i.e. more than $25\%$ of massive stars are in triples; see \citealt{Sanaetal2014}). Further, BH binaries can capture triple companions via dynamical encounters in dense stellar systems such as globular clusters (GCs; \citealt{Wen2003,Thompson2011,Aarseth2012,Antoninietal2014,Antoninietal2016,Breiviketal2016,Rodriguezetal2016a,Rodriguezetal2016b}) and in isolated triples in the field \citep{SilsbeeTremaine2017}. BH binaries in orbit around a supermassive BH (SMBH) in galactic nuclei (GNs) also represent hierarchical triples \citep{AntoniniPerets2012,VanLandinghametal2016,PetrovichAntonini2017,Hoangetal2017,RandallXianyu2018,RandallXianyu2017}. In these systems, eccentricity may be increased to a value close to unity during Kozai--Lidov oscillations while the semi-major axis is fixed, and GW emission may then quickly reduce the binary separation and lead to a merger before the orbits can fully circularize. A related channel to form eccentric mergers is through dynamical multibody interactions in dense stellar systems, which can drive the low-eccentricity inner binary to high eccentricity \citep{Gultekinetal2006,OLearyetal2006,Kushniretal2013,AntoniniRasio2016,Rodriguezetal2017}. If primordial BHs comprise dark matter halos, they may form BH binaries in single--single interactions through GW emission and retain some of their initial eccentricity as they enter the frequency bands of ground-based GW detectors \citep{Cholisetal2016}. Finally, binary BHs forming in clusters through binary--single interactions may also produce LIGO/VIRGO sources with non-negligible eccentricities \citep{Samsingetal2014,SamsingRamirezRuiz2017,Samsing2017,Samsingetal2018}.
The expected merger rate densities in the Kozai--Lidov channel are uncertain (as are all other rate estimates), but expectations are $1-1.5 \,{\rm Gpc}^{-3} \, {\rm yr}^{-1}$ for BH binaries forming in nuclear star clusters through multibody interactions \citep{AntoniniRasio2016} and $0.14-6.1 \, {\rm Gpc}^{-3} \, {\rm yr}^{-1}$ in isolated triple systems \citep{SilsbeeTremaine2017}. A merger rate density of order $1-5\, {\rm Gpc}^{-3} \, {\rm yr}^{-1}$ is expected for BH binaries forming via the Kozai--Lidov mechanism in GCs \citep{Antoninietal2014,Antoninietal2016,Rodriguezetal2016a,Rodriguezetal2016b} and GNs \citep{AntoniniPerets2012,Hoangetal2017}, and non-spherical nuclear star clusters may produce BH--BH binary merger rates of up to $15\, {\rm Gpc}^{-3} \, {\rm yr}^{-1}$ \citep{PetrovichAntonini2017}. Smaller-sized GNs with intermediate-mass BHs \citep{VanLandinghametal2016} and binary BHs forming in clusters through binary--single interaction \citep{Samsingetal2014,SamsingRamirezRuiz2017,Samsing2017,Samsingetal2017} may produce higher rates. An attractive feature of these merger channels is that they offer a solution to the so-called ``final AU problem,'' i.e. on how the BHs or their progenitors shrink to a separation below which GW emission can drive the binary to merge within a Hubble time \citep{Stoneetal2017}.
Here, we focus on another channel, which also naturally solves the final AU problem and leads to eccentric mergers. These are close fly-bys between single objects in dense stellar systems, which form binaries due to GW emission. If the velocity dispersion is high in the host environment, as in GNs, then the binary pericenter distance must be small for a binary to form in this way. This implies that the source will form in the LIGO frequency band and/or remain eccentric in the LIGO band until the merger. Despite the required small impact parameter, the merger rate in this channel may be high due to the extremely high number density of BHs in the mass-segregated cusps of GNs.\footnote{Formation rates for these scattering events are proportional to the square of the BH number density.} \citet{OLearyetal2009} showed that the expected aLIGO detection rate at design sensitivity for eccentric mergers forming through GW capture in GNs is higher than $\sim 100 \rm \, yr^{-1}$ if the BH mass function extends to masses above $25 \, M_{\odot}$ \citep[see also][]{KocsisLevin2012}. Since that publication, such heavy BHs have been observed in several LIGO/VIRGO detections \citep{Abbottetal2016f,Abbottetal2017,Abbottetal2017_2}. This also means that the aLIGO detection rate of highly eccentric binaries may be dominated by sources in GNs that form through GW capture.
In this paper, we determine the region of initial orbital parameters including eccentricity, pericenter distance, and masses where eccentric binary black holes (EBBHs) may form, the probability density function (PDF) of EBBH mergers as a function of these parameters, and calculate the expected PDF of initial orbital parameters. However, the initial eccentricity may be difficult to detect for eccentric binaries that form outside of the aLIGO frequency band. In \citet{Gondanetal2017}, we have shown that the GW detector network will be capable of measuring the final eccentricity of the source to much higher accuracy at the LSO. Thus, we also determine the distribution of eccentricity at the LSO. Additionally, we determine the distribution of eccentricity at the time the GW signal enters the aLIGO band.
We also calculate the merger rate distribution as a function of total mass and mass ratio in a single GN in this channel. Recently, \citet{OLearyetal2016} showed that, in GCs, independent of the BH mass function, the likelihood of merger is proportional to $M^4$, where $M$ is the total mass of the binary and comparable component masses are most common. Furthermore, \citet{Kocsisetal2017} found that a different universal relation holds for the likelihood of primordial BH binaries formed in the early universe. Such universal results may be valuable to statistically disentangle different source populations and to determine the BH mass function therein.
In a companion paper, \citet{Gondanetal2017}, we calculate the expected parameter measurement errors of eccentric mergers and show that the initial orbital eccentricity can be measured to sub-percent accuracy with the aLIGO--AdV--KAGRA GW detector network. Thus, the predicted merger rate distributions as functions of source parameters (particularly eccentricity) may be used to confirm or rule out this formation mechanism.
GW signals of EBBHs are very different from those of standard quasi-circular inspirals. At early times, these GW signals in the time domain consist of repeated bursts where the separation between successive bursts shrinks, and their amplitudes change as a function of time, and the waveform transforms into an eccentric quasi-periodic chirp signal before the merger. Eccentricity leads to multiple orbital frequency harmonics in the waveform, pericenter precession causes each harmonic to split into a frequency triplet, and eccentricity also changes the evolution of GW phase. This makes them rich in features and very unique among other signals. Furthermore, these GW signals are more luminous in the aLIGO band than circular sources, which makes them detectable at longer distances \citep{OLearyetal2009}.
Recent studies \citep{BrownZimmerman2010,HuertaBrown2013,Huertaetal2014,Huertaetal2017} have shown that methods based on circular binary templates could miss eccentric signals if the orbital eccentricity exceeds $\sim 0.1$ at the time these GW signals enter the aLIGO band. As we will show in this paper, a significant fraction of EBBHs forming through GW capture in GN hosts have $e > 0.1$ when their GW signals reach the aLIGO band. In such cases, search methods using circular binary templates will be ineffective in finding GW signals of these EBBHs.
So far, three search methods have been developed to date to find the signals of stellar-mass eccentric BH binaries in data streams of GW detectors \citep{Taietal2014,Coughlinetal2015,Tiwarietal2016}. All three methods achieve substantially better sensitivity for eccentric BH binary signals than existing localized burst searches or chirp-like template-based search methods. However, the complex nature of EBBH waveforms makes it necessary to develop more efficient search algorithms to detect these GW signals for moderately small signal-to-noise ratios. Such algorithms would greatly benefit from the detailed prior knowledge of source parameter distributions in the parameter space, particularly the likely range of parameter values, which must be covered with the search pipeline.
Once a source is detected, different algorithms are used to recover its physical parameters. For compact binary coalescence sources on circular inspiral orbits, several algorithms have been developed for this purpose, e.g. \textsc{BAYESTAR} \citep{SingerPrice2016}, \textsc{LALInference} \citep{Veitchetal2015}, \textsc{gstlal} \citep{Cannonetal2012,Priviteraetal2014}, \textsc{Coherent WaveBurst} \citep{Klimenkoetal2016}, \textsc{BayesWave} \citep{CornishLittenberg2015}, and \textsc{LALInferenceBurst} \citep{Veitchetal2015}. The development of algorithms recovering the parameters of compact binaries on eccentric orbits are currently underway. These algorithms will play an important role in the astrophysical interpretation of eccentric sources.
The paper is organized as follows. In Section \ref{GN}, we introduce the adopted models for the GN host environment and stellar populations properties. In Section \ref{sec:Evolution}, we summarize the phases of GW capture-induced binary evolution. In Section \ref{sec:NumMethod}, we describe Monte Carlo (MC) calculations of the parameter distribution of merger rates. In Section \ref{sec:PIOP}, we present an analytical derivation of the rates, which captures the leading-order behavior. In Section \ref{sec:NumRes}, we present our results in two parts. First, we present the distributions of orbital parameters of EBBHs at different stages of their time evolution, then we determine the merger rate distributions and the corresponding aLIGO detection rate distributions. We also identify characteristics of GN host environments to which the results are sensitive. Finally, we summarize the results of the paper and draw conclusions in Section \ref{sec:Summary}. Details of our analytic calculations can be found in Appendices \ref{sec:CalRmin}--\ref{sec:aLIGOeventrate}.
We use $G=1=c$ units in this paper.
\section{Galactic Nuclei and Relaxed Stellar Populations}
\label{GN}
Here, we describe the characteristics of GN host environments (Section \ref{subsec:GNModels}) and stellar populations (Section \ref{subsec:PropStellPop}) that we adopt in this study.
\subsection{Galactic Nuclei}
\label{subsec:GNModels}
GNs are assumed to be relaxed systems of spherically symmetric stellar populations (e.g. white dwarfs or WDs, main sequence stars or MSs, NSs, and BHs) gravitationally bound to a central SMBH. The relaxation of multi-mass stellar populations around an SMBH has been thoroughly investigated \citep{BahcallWolf1977,Freitagetal2006,HopmanAlexander2006b,AlexanderHopman2009,Keshetetal2009,OLearyetal2009,BarOr2013,AharonPerets2016,BarOrAlexander2016,Alexander2017,Baumgardtetal2017}; these studies have found that, within the GN's radius of influence, objects segregate to the central regions and form a power-law number density profile, $n(r) \propto r^{-\alpha}$, in which the $\alpha$ exponent is higher for more massive objects. \footnote{For SMBHs more massive than a few million solar masses, the GN cluster does not have time to reach an equilibrium distribution because the two-body relaxation timescale in that case is larger than a Hubble time \citep{Merritt2010,AntoniniMerritt2012,GualandrisMerritt2012,Antonini2014,DosopoulouAntonini2017}. This might lead to number density profiles different from that obtained for relaxed cusps.} The radius of influence is given as
\begin{equation} \label{eq:rmax}
r_\mathrm{max} = GM_{\rm SMBH}/\sigma_*^2 \, ,
\end{equation}
where $M_{\rm SMBH}$ is the mass of the SMBH, $\sigma_*$ is the velocity dispersion of the underlying stellar populations in the nucleus near the SMBH, and we use the $M_{\rm SMBH}$--$\sigma_*$ relationship
\citep{Tremaineetal2002,Walcheretal2005,MisgeldHilker2011,Norrisetal2014}:
\begin{equation}\label{eq:Msmbhsigma}
M_{\rm SMBH} \simeq 1.3 \times 10^8 \,M_{\odot} \left( \frac{ \sigma_* }{ 200\ \mathrm{km}\ \mathrm{s}
^{-1}} \right) ^4
\end{equation}
to estimate $\sigma$ in Equation (\ref{eq:rmax}). We choose the lower limit of the SMBH mass range to be $10^5 \, M_{\odot}$ \citep{Barthetal2005,GreeneHo2006}. We conservatively set the upper limit of the SMBH mass to $10^7 \, M_{\odot}$, corresponding to relaxed stellar populations; see Appendix \ref{subsec:CritI} for details.
For spherically symmetric and relaxed GNs, we use the \citet{BahcallWolf1976} one-body phase space distribution $f(\mathbf{r}, \mathbf{v})$ generalized for a multi-mass system \citep{OLearyetal2009} to derive the distribution of the magnitude of relative velocity between single objects (Appendices \ref{App:AppGN} and \ref{App:HandRelvel}). For objects with mass $m$, this has the form
\begin{equation} \label{eq:frv}
f_m(\mathbf{r},\mathbf{v}) = C_m E(r,v)^{p_m}
\end{equation}
if $r_{\rm min} \leqslant r \leqslant r_\mathrm{max}$ and $ E(r,v) > 0$, $C_m$ is a normalization constant,
\begin{equation} \label{eq:Erv}
E(r,v) = \frac{ M_{\rm SMBH}}{ r } - \frac{ v^2 }{ 2 }
\end{equation}
is the Keplerian binding energy in the field of the SMBH, and the $p_m$ exponent depends on mass due to mass segregation. For the light stellar components, such as MSs, WDs, and NSs, $p_m\approx 0$, and for the heavier components such as BHs,
\begin{equation} \label{eq:pm}
p_m = p_0 \frac{ m }{ m_\mathrm{BH,max}}\,,
\end{equation}
where $m_\mathrm{BH,max}$ is the highest possible BH mass in the population and $p_0 \approx 0.5 - 0.6$ based on Fokker--Planck simulations \citep{OLearyetal2009}. We consider the standard value of $p_0$ to be $0.5$. Equation (\ref{eq:Erv}) is valid outside the loss cone \citep{ShapiroLightman1976,SyerUlmer1999}. We assume it to be valid between $r_{\rm min} \leqslant r \leqslant r_\mathrm{max}$ if $v$ is less than the escape velocity at radius $r$, where $r_{\rm min}$ is an inner radius where the density cusp exhibits a cutoff. We calculate $r_{\min}$ by requiring that (i) the number density profile of these populations reaches steady state and forms a power-law density cusp within the age of the galaxy; and (ii) the inspiral timescale into the SMBH is larger than the relaxation time (Appendix \ref{sec:CalRmin}).
Because the binding energy is positive for objects bound to the SMBH, $E(r,v) >0$, the velocity at distance $r$ from the SMBH must be less than the escape velocity
\begin{equation}\label{eq:vmax}
v_\mathrm{max}(r)=\sqrt{\frac{2G \, M_{\rm SMBH} }{r}} \, .
\end{equation}
The 3D number density distribution of mass $m$ objects at radius $r$ may be obtained from Equation (\ref{eq:frv}) as
\begin{equation} \label{eq:n(r)}
n_m(r) = \int f_m(\mathbf{r},\mathbf{v}) d^3 v = n_{\rm inf} \left( \frac{r}{r_{\rm max}}
\right)^{-\alpha_{m}} \, ,
\end{equation}
where
\begin{equation} \label{eq:alpham}
\alpha_{m} = \frac{ 3 }{ 2 } + p_m \, ,
\end{equation}
and $n_{\rm inf}$ is the number density of objects at the radius of influence. For MSs of $m = 1 \, M_{\odot}$, this follows from the $M$--$\sigma$ relation (Equation \ref{eq:Msmbhsigma}), assuming that the enclosed stellar mass within the radius of influence $r_{\max}$ is $2M_{\rm SMBH}$ \citep[e.g.][]{OLearyetal2009}
\begin{equation} \label{eq:nrmasMS}
n_{\inf,\rm MS} \equiv n_\mathrm{MS} (r_\mathrm{max}) \ \simeq 1.38 \times 10^5 \mathrm{pc}^{-3}
\sqrt{ \frac{ 10^6 M_{\odot} }{ M_{\rm SMBH} } } \, .
\end{equation}
We define the normalization constants $n_{\inf}$ and the exponent of the number density distribution $\alpha_m$ separately for the different stellar species in Section \ref{subsec:PropStellPop}.
\subsection{Stellar Populations in Galactic Nuclei}
\label{subsec:PropStellPop}
The initial mass functions (IMF), which extend from the brown dwarf boundary $\sim 0.1 \, M_{\odot}$ to $\sim 100 \, M_{\odot}$ (e.g. the \citep{Salpeter1955} IMF and its subsequent refinements, the \citep{MillerScalo1979} and \citep{Kroupa2001} IMFs), result in evolved populations that naturally separate into two mass scales: the $\sim 1 \, M_{\odot}$ scale of low-mass MSs, WDs, and NSs; and the $\sim 10 \, M_{\odot}$ scale of stellar-mass BHs, Wolf--Rayet, O-, and B-stars. In case of a GN, the shape of the present-day mass function (PMF) varies significantly above the scale of $\sim 10 \, M_{\odot}$ \citep{AlexanderHopman2009}. The typical PMF in GNs is not well-understood due to the fact that star formation deep in the potential well of a SMBH can be very different from that of the field. \citet{Bartkoetal2010} find evidence that the stellar disks in the GN of the Galaxy are extremely top-heavy. \citet{Luetal2013} infer that the mass function of these young stars is proportional to $m^{-1.7 \pm 0.2}$, which implies that the number of massive stars is currently higher than
that for a standard Salpeter mass function.
We follow \citet{AlexanderHopman2009} and assume single-mass MS, WD, and NS populations in GNs with component
masses $1 \, M_{\odot}$, $0.6 \, M_{\odot}$, and $1.4 \, M_{\odot}$, respectively.
We normalize the number density distribution for WD, MS, and NS populations in Equations (\ref{eq:n(r)}) as
\begin{equation} \label{eq:nNS}
n_{\inf,i} = n_{\inf, \rm MS} C_{i} \, ,
\end{equation}
where $i$ labels $\{\rm WD, MS, NS\}$, and set the number density exponents to
\begin{equation}
\alpha_{\rm MS}=\alpha_{\rm WD}=1.4 {\rm ~~and~~} \alpha_{\rm NS}=1.5\,
\end{equation}
\citep{HopmanAlexander2006b}. Here, $C_{i}$ represents the number fraction ratios of MSs, WDs, and NSs as $1:0.1:0.01$
for continuous star-forming populations \citep{Alexander2005}, implying that
\begin{equation}
C_{\rm MS}=1\,,\quad C_{\rm WD}=0.1\,,\quad C_{\rm NS} = 0.01 \frac{3 - \alpha_{\rm NS}}{3 - \alpha_{\rm MS}}\,.
\end{equation}
We carry out calculations for three types of BH PMF models: a simple single-component mass distribution, a power-law multi-mass distribution, and a multi-mass distribution given by a population synthesis method, as follows.
\begin{enumerate}
\item In the single-mass BH population model, we assume that all BHs have $m_{\rm BH}=10 \, M_{\odot}$ following \citet{Morris1993}, \citet{MiraldaEscudeGould2000}, and \citet{AlexanderHopman2009} for a $10 \, \mathrm{Gyr}$ old coeval population, implying that the number density exponent in equation (\ref{eq:nrmasMS}) is $\alpha_ \mathrm{BH,s} = 2$ and to determine $C_\mathrm{BH,s}$ we account for BHs segregated into the nucleus. For the Milky Way, as many as $\sim 20,000$ BHs with $10 \, M_{\odot}$ each are expected to have segregated into the nucleus \citep{Morris1993,MiraldaEscudeGould2000,Freitagetal2006,HopmanAlexander2006b}. The number of BHs segregated into the GN is proportional to the mass of the SMBH through the infall rate of BHs (see Equation (9) in \citet{MiraldaEscudeGould2000}), therefore it can be approximated as
\begin{equation} \label{eq:encBH}
N_{\rm BH} \approx 20,000 \times \frac{M_{\rm SMBH}}{M_{\mathrm{SgrA}^*}} \, .
\end{equation}
Here, $M_{\mathrm{SgrA}^*} = 4.3 \times 10^6 \, M_{\odot}$ is the mass of the SMBH, Sgr A$^{*}$, in the Milky Way-size nucleus \citep{Gillessenetal2009}\footnote{We choose $M_{\rm SMBH}=M_{\mathrm{SgrA}^*}$ in our fiducial numerical calculations, but also explore other values.}. The number density distribution of BHs in this single-mass model is of the form
\begin{equation}
n_\mathrm{BH,s} = C_\mathrm{BH,s} n_\mathrm{inf,MS} \left( \frac{ r }{ r_\mathrm{max} } \right)
^{- \alpha_\mathrm{BH,s}} \, ,
\end{equation}
where $C_\mathrm{BH,s}$ can be determined using Equation (\ref{eq:nNS}) together with Equation (\ref{eq:encBH}), which gives
\begin{equation}
C_\mathrm{BH,s} = 0.023 \frac{ 3 - \alpha_\mathrm{BH,s} }{ 3 - \alpha_\mathrm{MS} } \, .
\end{equation}
\item In this multi-mass BH population model, we assume a power-law PMF $m_{\rm BH}^{-\beta}$ following \citet{AlexanderHopman2009} for $10 \, \mathrm{Gyr}$ of continuous star formation. The probability distribution function of BH mass is then
\begin{equation}\label{e:f(m)}
\mathcal{F}(m_{\rm BH})= \frac{ (1-\beta) \, m_{\rm BH}^{-\beta} }{ m_{\rm BH,max}
^{1-\beta} - m_{\rm BH,min}^{1-\beta} } \, .
\end{equation}
We consider $\beta$ in the range $[1,3]$ and arbitrarily choose its standard value to be $\beta \equiv 2.35$, although the exponent for BHs need not be related to the Salpeter IMF of massive stars \citep{Salpeter1955}. This range is consistent with the currently announced GW detections \citep{Abbottetal2017}. We set $m_{\rm BH,min} = 5 \, M_{\odot}$, based on the observations of X-ray binaries \citep{Bailynetal1998,Ozeletal2010,Farretal2011,Belczynskietal2012}, and we set the fiducial value for the highest possible BH mass to be $m_\mathrm{BH,max} = 30 \, M_{\odot}$, but also explore cases between $10 \, M_{\odot}$ and $80 \, M_{\odot}$ \citep{Belczynskietal2010}. Taking into account the mass distribution of BHs, the number density normalization at the radius of influence in Equation (\ref{eq:n(r)}) for BHs in a multi-mass BH population is
\begin{equation} \label{eq:nBHmulti2}
n_\mathrm{BH,m} (m_{\rm BH}) = n_{\rm MS} \, \mathcal{F}(m_{\rm BH}) \, C_{\mathrm{BH},m} \, ,
\end{equation}
where $C_\mathrm{BH,m}$ is set numerically by the total number of BHs inside the GN as
\begin{align}
N_{\rm BH} & = \int_{m_{\rm BH,min}}^{m_{\rm BH,max}} dm_{\rm BH} \, n_\mathrm{BH,m} \, (m_{\rm BH})
\nonumber \\
& \times \int_0^{r_{\max}} 4 \pi r^2 \left(\frac{r}{r_{\max}}\right)^{- \alpha_\mathrm{BH,m}} \, dr
\, .
\end{align}
Here, $\alpha_\mathrm{BH,m}$ is given by combining Equations (\ref{eq:pm}) and (\ref{eq:alpham}).
\item To check the dependence of the result for a qualitatively different mass distribution, we also carried out the calculations for a PMF obtained in a population synthesis study, \citet{Belczynskietal2014_2}, for single low-metallicity stars (see Figure 4 therein). In this case, the PMF ranges between $\sim 3 M_{\odot}$ and $\sim 26 M_{\odot}$ and has three peaks in its distribution, near $7 M_{\odot}$, $14 M_{\odot}$, and $24 M_{\odot}$.
\end{enumerate}
We show in Appendix \ref{subsec:CritII} that the region in single GN where EBBHs are expected to form through GW capture is weakly sensitive to the assumptions on the mass distribution of stellar populations.
\section{Phases of Binary Evolution}
\label{sec:Evolution}
In this section, we introduce the GW capture binary formation mechanism and calculate resulting distribution of initial orbital parameters. Next, we examine the evolution through the eccentric inspiral and the long-term interaction between the formed binary and a third object in simulations.
In the following, we denote the masses of BHs forming the binary as $m_A$ and $m_B$, the symmetric mass ratio of the binary as $\eta = (m_A m_B) /(m_A+m_B)^2$, the total mass as $M_\mathrm{tot} = m_A+m_B$, and the mass ratio as $q = m_A / m_B.$ The reduced mass and symmetric mass ratio satisfy $\mu = \eta M_\mathrm{tot}$ and $\eta=q/(q+1)^2$. Similar to \citet{OLearyetal2009}, we define the dimensionless pericenter distance as $\rho_\mathrm{p} = r_\mathrm{p} / M_\mathrm{tot}$, where $r_\mathrm{p}$ is the pericenter distance changing with each orbit.
\subsection{Formation of Binaries through GW Capture and the Eccentric Inspiral}
\label{subsec:FormBHB}
Two BHs form a binary if they undergo a close encounter and they emit enough energy, in form of GWs, to become bound. Due to the relativistic nature of such events and the low velocity dispersion compared to $c = 1$ in GN hosts, the encounters are almost always nearly parabolic \citep{QuinlanShapiro1987,Lee1993}. Thus, the initial orbit of the formed binary is typically highly eccentric with $e \sim 1$. In this limit, the amount of energy emitted in GWs during the encounter is
\begin{equation}\label{eq:dEGW}
\delta E_\mathrm{GW} =- \frac{ 85 \pi \eta^2 M_\mathrm{tot}^{9/2} }{ 12
\sqrt{2} r_\mathrm{p0}^{7/2} }
\end{equation}
\citep{PetersMathews1963,Turner1977}, where $r_\mathrm{p0}$ is the distance of closest approach (we use
$0$ in the lower index to denote initial values of parameters throughout the paper, i.e. their values at
the time of EBBH formation):
\begin{center}
\begin{equation}\label{eq:rhop0}
r_\mathrm{p0} = \left(\sqrt{\frac{ 1 }{ b^2 } + \frac{M_\mathrm{tot}
^2}{ b^4 w^4 } } + \frac{M_\mathrm{tot}}{b^2 w^2} \right)^{-1}
\end{equation}
\end{center}
\citep{OLearyetal2009}, where $b$ is the impact parameter of the encounter, and $w$ is the magnitude of relative velocity between the BHs forming the EBBH.
The initial properties of the EBBH are determined by the encountering system's final energy, $E_{\rm fin}$, and final angular momentum, $L_\mathrm{fin}$, after the first encounter, where
\begin{equation}\label{eq:Efin}
E_{\rm fin} = E_{\rm kin} + \delta E _\mathrm{GW} \, ,
\end{equation}
\begin{equation}\label{eq:L0}
L_{\rm fin} = M_\mathrm{tot} b \eta w + \delta L \, ,
\end{equation}
where $E_{\rm kin} = \mu w ^2 / 2$ is the kinetic energy in the center of mass system, and $\delta L$ is the amount of angular momentum lost during an encounter \citep{Peters1964}. \citet{OLearyetal2009} found that $\delta L$ is negligible for nearly all first encounters; therefore, we set $\delta L \approx 0$. If the final energy of the system after the encounter is negative, $E_{\rm fin} < 0$, then the system remains
bound with an initial semi-major axis of
\begin{equation}\label{eq:a0}
a_0 = \frac{\eta M_\mathrm{tot}^2 }{2 |E_{\rm fin}|} \, ,
\end{equation}
and with an initial eccentricity of
\begin{equation} \label{eq:e0}
e_0 = \sqrt{1 - \frac{2 |E_{\rm fin}| w ^2 b^2}{M_\mathrm{tot}^3 \eta}} \, .
\end{equation}
Note that the initial pericenter distance of the EBBH system $r_\mathrm{p0}$ can be expressed in terms of $e_0$ and $a_0$ as
\begin{equation} \label{eq:rho0}
r_\mathrm{p0}= a_0 (1 - e_0) \, .
\end{equation}
The criterion $E_{\rm fin} < 0$ for the two BHs to form a bound binary sets an upper limit on the impact parameter of the approach \citep{OLearyetal2009}. Additionally, a lower limit on $b$ is set by the fact that we require the BHs to avoid direct coalescence during the first close approach. Therefore, to leading order, a bound EBBH system can form if $b$ satisfies
\begin{equation}\label{eq:impactlim}
b_\mathrm{min} \equiv \frac{ 4 M_\mathrm{tot} }{ w } < b <
\left( \frac{340 \pi \eta}{ 3 } \right) ^{1/7} \frac{ M_\mathrm{tot}
}{ w ^{9/7}} \equiv b_\mathrm{max} \, .
\end{equation}
Here, the upper limit corresponds to Equations (\ref{eq:dEGW}) and (\ref{eq:Efin}), and the lower limit is valid in the test-particle limit around a Schwarzschild BH \citep[e.g.,][]{Kocsisetal2006,OLearyetal2009}. The leading-order approximation in the allowed range of $b$ values expressed by Equation (\ref{eq:impactlim}) is in a good match with $2.5$ and $3.5$ order post-Newtonian (PN) simulations for relative velocity $w \lesssim 0.01$ \citep[see Figure 1. in ][]{KocsisLevin2012}). Relativistic corrections modify the capture cross section by less than $10\%$ \citep{KocsisLevin2012}
Due to the escape velocity $v_{\max}$ at radius $r$ from the SMBH (Equation~\ref{eq:vmax}), the initial relative velocity is bounded by
\begin{equation}\label{eq:wlim}
0 \leqslant w \leqslant 2 v_{\max}(r) = \sqrt{8M_{\rm SMBH}/r}\, .
\end{equation}
The allowed range of initial binary parameters $\rho_{\mathrm{p}0}$ and $e_0$ may be calculated from the bounds on $b$ and $w$ given by Equations (\ref{eq:impactlim}) and (\ref{eq:wlim}) using Equations (\ref{eq:a0}) and (\ref{eq:rho0}).
After the binary is formed with initial parameters $e_0$ and $\rho_{\rm p0}$, it evolves due to GW radiation reaction. We use the leading-order orbital evolution equation \citep{Peters1964}
\begin{equation}\label{eq:rhoe}
\rho_\mathrm{p} (e) = \frac{c_0}{M_\mathrm{tot}}\frac{ e^{12/19}}{(1+e)}
\left( 1+ \frac{ 121 }{ 304 } e^2 \right)^{ \frac{870}{2299} } \, ,
\end{equation}
where $c_0/M_{\mathrm{tot}}$ may be expressed with $e_0$ and $\rho_\mathrm{p0}$, by solving Equation (\ref{eq:rhoe}) for $\rho_\mathrm{p}=\rho_\mathrm{p0}$ and $e=e_0$.
This evolution equation is valid when the two BHs are relatively far from each other's horizon, for $\rho_\mathrm{p} \gg 2$, up until the binary reaches the LSO. After that, the evolution is no longer quasi-periodic and the BHs quickly coalesce. In the leading-order approximation for infinite mass ratio and zero spins, the eccentricity of the LSO, $e_\mathrm{LSO}$, can be calculated by numerically solving the following equation \citep{Cutleretal1994}
\begin{equation}\label{eq:ELSO}
\rho_\mathrm{p} (e_\mathrm{LSO}) =
\frac{ 6+ 2 e_\mathrm{LSO} }{ 1+e_\mathrm{LSO} }
\end{equation}
for $e_\mathrm{LSO}$ using Equation~(\ref{eq:rhoe}). The initial pericenter distance is set uniquely by $e_\mathrm{LSO}$ and $e_0$ according to
\begin{equation}
\rho_{\mathrm{p}0} = \frac{6+ 2e_\mathrm{LSO}}{1+e_0}
\frac{ e_0^{12/19}}{e_\mathrm{LSO}^{12/19}}
\left[\frac{ 1+ (121/304) e_0^2}{ 1+ (121/304) e_{\mathrm{LSO}}^2} \right]^{ \frac{870}{2299} } \, .
\end{equation}
In the limit $e_0\approx 1$, which we find to be the relevant case (Section \ref{subsec:DistIniParam}),
\begin{equation} \label{eq:rhopeLSO}
\rho_{\mathrm{p}0} \approx \frac{3+ e_\mathrm{LSO}}{e_\mathrm{LSO}^{12/19}}
\left( 1+ \frac{121}{304} e_{\mathrm{LSO}}^2 \right)^{ -\frac{870}{2299} }\,.
\end{equation}
We invert Equation (\ref{eq:rhopeLSO}) numerically to obtain $e_{\rm LSO}(\rho_{\mathrm{p}0})$. The result is well-fitted\footnote{We use the method of least squares to fit the prefactor and exponent. The fit error is $0.001$ for both parameters for the $95 \%$ confidence bounds in the range $\rho_{\mathrm{p}0} \in [8,1000]$. Most EBBHs form within this range in GNs; see Section \ref{subsec:DistIniParam}.} by
\begin{equation} \label{eq:eLSOrhop0}
e_{\rm LSO} = 6.3096 \, \rho_{\mathrm{p}0}^{-1.599} \, .
\end{equation}
This shows that $\rho_{\mathrm{p}0}$ generally is a monotonically decreasing function of $e_{\mathrm{LSO}}$, where $\rho_{\mathrm{p}0} \gtrsim 10$ corresponds to $e_{\mathrm{LSO}} \lesssim 0.19$.
\subsection{Interaction with a Third Object}
\label{sec:EIAPD}
Although the stellar number density is the highest in GNs, \citet{OLearyetal2009} have shown that the binaries, which form by GW capture, are so tight that they typically merge before encountering a third object that would alter their orbital parameters. We conservatively ignore the rare EBBHs in our Monte Carlo (MC) sample that have an encounter with a third object before the merger of the EBBH (Section \ref{subsec:MCCode}), i.e. if the encounter timescale is shorter than the merger timescale,
\begin{equation} \label{eq:conddisruption}
t_{\rm merge}>t_\mathrm{enc}\,.
\end{equation}
The typical timescale for a close encounter between a binary and a single object can be approximated as
\begin{equation}\label{eq:dEGW4}
t_\mathrm{enc} \approx \frac{ w^3 }{ 12 \pi M_{\rm tot}^2 n_{\rm tot} }
\end{equation}
\citep{OLearyetal2009}, where
\begin{equation} \label{eq:ntot}
n_\mathrm{tot}= n_\mathrm{WD}(r) + n_\mathrm{MS}(r) + n_\mathrm{NS}(r) + n_\mathrm{BH}(r)
\end{equation}
is the combined number density of all types of stellar objects at distance $r$ from the SMBH. By substituting Equation (\ref{eq:rhoe}) into Equation (5.7) in \citet{Peters1964}, the time remaining until coalescence is
\begin{equation}\label{eq:tGW}
t_{\rm merge} (\rho_{\rm p0}, m_A, m_B, e_0) =
\frac{15 \, c_0^4(e_0,\rho_{\rm p0}) }{ 304 M_\mathrm{tot}^3 \eta} \mathcal{F}(e_\mathrm{LSO},e_0) \, ,
\end{equation}
where
\begin{equation}\label{eq:FAB}
\mathcal{F}(A,B) \equiv \int _{A} ^{B} de \, \frac{ e^{29/19} \left( 1+ \frac{121}{304} e^2
\right)^{ \frac{1181}{2299}} }{ \left( 1 - e ^2 \right)^{3/2} } \, .
\end{equation}
Note that $\mathcal{F}(e_\mathrm{LSO},e_0) \approx \mathcal{F}(0,e_0)$ within $0.1 \%$ for \mbox{$e_{\rm LSO} \ll e_0 \sim 1$}, and thus $t_{\rm merge}$ can practically be approximated as
\begin{equation}\label{eq:tGW2}
t_{\rm merge} (\rho_{\rm p0}, m_A, m_B, e_0) \approx \frac{15 \, c_0^4(e_0,\rho_{\rm p0})}{
304 M_\mathrm{tot}^3 \eta} \mathcal{F}(0,e_0) \, .
\end{equation}
Because the fraction of binaries that satisfy Equation (\ref{eq:conddisruption}) represents a small fraction of all objects (Section \ref{subsec:DistIniParam}), ignoring this population does not bias the initial binary orbital parameter distributions.
\section{Distributions of Initial Orbital Parameters and EBBH mergers}
\label{sec:NumMethod}
In this section, we derive the distributions of the masses and the initial orbital parameters of EBBHs in GNs.
We parameterize the formation of an EBBH through GW capture via four parameters: the two masses of the BHs involved ($m_A$ and $m_B$), the magnitude of the relative velocity between the two BHs ($w$), and the impact parameter of the approach ($b$). Here, we first treat $m_A$ and $m_B$ as being fixed, while $w$ and $b$ as free parameters. We use the phase space distribution function $f_A(\mathbf{r}_A, \mathbf{v}_A)$ and $f_B (\mathbf{r}_B, \mathbf{v}_B)$, given by Equation (\ref{eq:frv}). The distributions depend on the masses $m_A$ and $m_B$ through the exponents $p_A$ and $p_B$ in Equations (\ref{eq:frv}) and (\ref{eq:pm}) as heavier components form steeper density cusps due to mass segregation.
The differential rate of EBBH mergers involving BHs $A$ and $B$ can be given in the twelve-dimensional phase space of these two objects as
\begin{align} \label{eq:infformrate}
d^{12} \Gamma _{A B} & = \sigma w f_A(\mathbf{r}_A, \mathbf{v}_A) f_B(\mathbf{r}_B, \mathbf{v}_B)
\nonumber \\
& \times d^3 r_A \, d^3 r_B \, d^3 v_A \, d^3 v_B \, ,
\end{align}
where $\sigma\equiv \sigma(w)=\pi b_\mathrm{max}^2(w)-\pi b_\mathrm{min}^2(w)$ is the cross section for two BHs to form an EBBH during their encounter; see Equation (\ref{eq:impactlim}) for $b_\mathrm{min}$ and $b_\mathrm{max}$.
\subsection{Analytic Estimates}
The total merger rates may be obtained from integrating Equation (\ref{eq:infformrate}) over phase space. The partial merger rate distribution as a function of parameters (e.g. radius in the GN, mass, initial eccentricity, and pericenter distance) may be obtained by integrating only over the complementary phase space dimensions. We refer the reader to Appendix \ref{sec:AppA} for details, and present the results of the calculations in Section \ref{sec:NumRes}.
\subsection{The Monte Carlo Code}
\label{subsec:MCCode}
To generate a random sample of EBBHs with component masses $m_A$ and $m_B$, we first generate $\sim 10^3$ random radius values from the galactic center over the radius range \mbox{$r \in [r_{\rm min}^{A,B}, r_\mathrm{max}]$} using Equations (\ref{eq:formrateBHBH_multi}) and (\ref{eq:formrateBHBH_single}). We then randomly draw a pair of $w$ and $b$ for each $r$ from $P_{AB}(w,b)$, using Equations (\ref{APe1}) and (\ref{e:P(w|r)}) in Appendix \ref{sec:AppA}.
EBBHs can form with $w$ as high as $w \simeq 0.05$ near $r_{\rm min}^{A,B}$. The region of validity of Equation (\ref{eq:impactlim}) for \mbox{$b_{\rm min} \propto w^{-1}$} and \mbox{$b_{\rm max} \propto w^{-9/7}$} is as follows. For high $w$, the condition \mbox{$b_{\rm max}(w) > b_{\rm min}(w)$} may be violated over
\begin{equation} \label{eq:wbmax}
w_{{\rm max},b} = \frac{1}{4^3} \sqrt{ \frac{ 85 \pi \eta }{ 3 } } \, .
\end{equation}
These systems cannot form an eccentric binary system, but may only suffer a direct head-on collision. We discard all such $(w,b)$ pairs in our MC sample that do not satisfy $w \leqslant w_{{\rm max},b}$. We find numerically that only $\la 0.01 \%$ ($\la 1\% $) of EBBHs form with $w \ga 0.05$ ($ w \ga w_{\mathrm{max},b}$) over the considered ranges of BH mass, SMBH mass, and BH population parameters; thus, direct head-on collisions represent a negligible fraction of sources.
Furthermore, we keep only those $(w,b)$ pairs that avoid interaction with a third object satisfying Equation (\ref{eq:conddisruption}).
For each random sample of $(w,b)$, we calculate the corresponding initial pericenter distance and eccentricity of the binary $(\rho_\mathrm{p0},e_0)$, using formulae introduced in Section \ref{subsec:FormBHB}.
Given that most of these binaries have $e_0 \sim 1$ (Section \ref{subsec:DistIniParam}), we introduce the new variable
\begin{equation} \label{eq:defeps0}
\epsilon_0 \equiv 1-e_0^2
\end{equation}
in order to accurately represent the PDF of $e_0$ in the vicinity of unity; see Section \ref{sec:NumRes}.
\subsection{Bounds on the Initial Orbital Parameters}
\label{sec:PIOP}
Figure \ref{fig:boundary} shows a $2.5 \times 10^5$ MC sample of EBBHs in the $\epsilon_0$--$\rho_{\mathrm{p}0}$ plane for a Milky Way-sized nucleus with \mbox{$M_{\rm SMBH} = M_{\rm SgrA*}$} and for EBBHs having $m_A = m_B = 30 \, M_{\odot}$. The distribution is bounded by five curves, where one of the conditions of a stable EBBH formation is violated as labeled. We will next discuss these conditions next and derive analytic approximations for each of them.
\begin{figure}
\centering
\includegraphics[width=85mm]{DataBond}
\caption{ \label{fig:boundary} A Monte Carlo realization of the EBBH parameters in the space of initial eccentricity $e_0$ and initial dimensionless pericenter distance \mbox{$\rho_{\rm p0} = r/M_{\rm tot}$} showing the allowed parameter values of EBBHs. The initial eccentricity is typically very close to unity, so we show $\log(1-e_0^2)$ to resolve the deviation from $e_0 = 1$. The allowed region is bounded by curves that represent physical constraints on the binary formation and evolution, as indicated and explained in the text. We assume a Milky Way-sized nucleus with $M_{\rm SgrA*} = 4.3 \times 10^6 \, M_{\odot}$ and show mergers with $m_A = m_B = 10 \, M_{\odot}$ within a BH population with a power-law present-day mass function (PMF) $dN/dm_{\rm BH} \propto m_{\rm BH}^{-2.35}$, a maximum BH mass of $m_\mathrm{BH,max} = 30 \, M_{\odot}$, and a mass segregation parameter $p_0 = 0.5$ (Equation \ref{eq:pm}). We find that $99 \%$ of EBBHs (represented here by green dots) merge before a dynamical encounter may happen with a third object (i.e. $t_{\rm enc} > t_{\rm merge}$ holds), shown by a solid red line. }
\end{figure}
Stable EBBH formation from two single objects bound to the central SMBH requires the following.
\begin{enumerate}[label=(\Roman*),ref=(\Roman*)]
\item\label{i:I} They must avoid direct collision without forming a stable EBBH, $b>b_\mathrm{min}$.
\item\label{i:II} They must also avoid escape to infinity, $b< b_\mathrm{max}$.
\item\label{i:III} The initial kinetic energy must be positive \mbox{$E_{\rm kin} \geqslant 0$}. The two BHs radiate more GW energy during their encounter than the total initial kinetic energy, \mbox{$\delta E_\mathrm{GW} < E_{\rm kin}$}, which implies that $E_{\rm fin} \geqslant \delta E _\mathrm{GW}$.
\item\label{i:IV} EBBHs should not interact with a third object throughout their evolutions, $t_{\rm merge} < t_\mathrm{enc}$.
\item\label{i:V} BHs bound to the central SMBH have an initial velocity less than the escape velocity $v_{\rm max}(r)$, which sets a bound on the maximum relative speed, $w < w_\mathrm{max}(r) \equiv 2 v_{\rm max}(r)$ (Equation \ref{eq:wlim}). Objects exist in the GN within the radius of influence of the SMBH, but sufficiently far to avoid infall into the SMBH due to GW emission, \mbox{$r_{\rm min}^{AB} \leqslant r \leqslant r_{\rm max}$}. This sets a bound on $w$.
\end{enumerate}
The above five criteria define sharp boundaries in the $\epsilon_0 - \rho_\mathrm{p0}$ plane as follows.
\begin{enumerate}[label=(\Roman*),ref=(\Roman*)]
\item[\ref{i:I}] From Equation (\ref{eq:rhop0}), the leading-order approximation for $\rho_
\mathrm{p0}=r_\mathrm{p0}/M_\mathrm{tot}$ in terms of $w$ is
\begin{equation} \label{eq:42}
\rho_\mathrm{p0} \simeq \frac{ b^2 w^2 }{ 2 M_\mathrm{tot} ^2 } \, .
\end{equation}
Using Equation (\ref{eq:42}) and the definition of $b_\mathrm{min}$ given in Equation
(\ref{eq:impactlim}), the criterion $b > b_\mathrm{min}$ can be translated into the
constraint $\rho_ \mathrm{p0}> 8$, to leading post-Newtonian order.
\item[\ref{i:II}] The criterion $b< b_\mathrm{max}$ means that the two encountering BHs must pass each other closely enough to become bound following GW emission, and thus $E_{\rm fin}$ must satisfy $E_{\rm fin}<0$. Using Equation (\ref{eq:Efin}) and the definition of $\epsilon_0$ given in (\ref{eq:defeps0}), this can be translated into a constraint $e_0 < 1$, or equivalently $\epsilon_0 > 0$. Moreover, this criterion must be satisfied for the possible range of $w$ values in the GN, $0 \leqslant w<w_{\rm max}$. Here, $w_\mathrm{max}$ is the highest $w$ occurring in the GN below $w_{{\rm max},b}$. Combining Equations (\ref{eq:Efin}) and (\ref{eq:46}) together with Equation (\ref{eq:dEGW}), $w$ may be expressed in Equation (\ref{eq:dEGW4}) in terms of $\epsilon_0$ and $\rho _\mathrm{p0}$ as
\begin{equation} \label{eq:w_eps_rhop0}
w\simeq \left (\frac{85\pi \eta}{6 \sqrt{2} \rho_\mathrm{p0}^{7/2}} -
\frac{\epsilon_0}{2\rho_\mathrm{p0}} \right )^{1/2} \, .
\end{equation}
Combining Equation (\ref{eq:42}) with Equation (\ref{eq:w_eps_rhop0}), $b$ can be expressed in terms of $\rho_{\rm p0}$ and $\epsilon_0$ as
\begin{equation} \label{eq:b_termsepsrho}
b = \sqrt{2 \rho_{\rm p0}} M_{\rm tot} \left (\frac{85\pi \eta}{6 \sqrt{2}
\rho_\mathrm{p0}^{7/2}} - \frac{\epsilon_0}{2\rho_\mathrm{p0}} \right )^{1/4} \, .
\end{equation}
Conversely, solving for $\rho_{\mathrm{p}0}$ and $\epsilon_0$, gives Equation~(\ref{eq:42}) and
\begin{align}
\epsilon_0 (b,w) &= \frac{ 340 \pi\, \eta M_{\rm tot}^5 }{ 3\, b^5 w^5 } -
\frac{ b^2 w^4}{ M_{\rm tot}^2 } \, .
\end{align}
The constraint on $b$ in Equation (\ref{eq:impactlim}) gives a boundary curve in the $\epsilon_0-
\rho_\mathrm{p0}$ plane.
\item[\ref{i:III}] Using Equations (\ref{eq:Efin}) and (\ref{eq:42}), $E_{\rm fin}$ can be expressed in terms of $\epsilon_0$ and $\rho_\mathrm{p0}$ as
\begin{equation} \label{eq:46}
E_{\rm fin} \simeq - \frac{ M_\mathrm{tot} \eta \epsilon_0 }{ 4 \rho_\mathrm{p0} } \, .
\end{equation}
Finally, using Equation (\ref{eq:dEGW}) together with Equation (\ref{eq:46}), the criterion $\delta E _\mathrm{GW}< E_{\rm fin}$ defines the following upper limit on $\epsilon_0$:
\begin{equation} \label{eq:47}
\epsilon_0 < \frac{85\pi \eta}{3\sqrt{2}}\rho_\mathrm{p0}^{-\frac{5}{2}} \, .
\end{equation}
Note that the $\delta E_\mathrm{GW} = E_{\rm fin}$ case corresponds to the $w=0$ limit; see Equation (\ref{eq:Efin}).
\item[\ref{i:IV}] The boundary curve in the $\epsilon_0-\rho_\mathrm{p0}$ plane defined by $t_{\rm merge} = t_\mathrm{enc}$ can only be constructed numerically by expressing all terms in Equations (\ref{eq:dEGW4}) and (\ref{eq:tGW}) as functions of $\epsilon_0$ and $\rho_\mathrm{p0}$. Here, $a_0$ and $c_0$ can be expressed in terms of $\epsilon_0$ and $\rho_\mathrm{p0}$ using Equations (\ref{eq:rho0}) and (\ref{eq:rhoe}). This boundary must be calculated at each radius in the GN because $n_{\rm tot}$ depends on $r$ (Equation \ref{eq:ntot}). However, an upper boundary can be constructed for the whole EBBH population merging in the GN by choosing $r=r_{\rm max}$, which we use to identify that region of the $\epsilon_0-\rho_\mathrm{p0}$ plane in which EBBHs surely interact with a third object before the merger phase of the EBBH evolution.
\item[\ref{i:V}] BHs A and B should have a relative speed, $w$, lower than $2 v_{\rm max}(r)$ at distance $r$ from the center of the nucleus, i.e. $w<w_\mathrm{max}(r)\equiv 2 v_{\rm max}(r)< w_{\rm max}$. Using Equation (\ref{eq:w_eps_rhop0}), the criterion $w<w_\mathrm{max}$ can be expressed as
\begin{equation} \label{eq:wwmax}
\epsilon_0 > \frac{85\pi \eta}{3 \sqrt{2}} \rho_\mathrm{p0}^{-5/2} - 2 \rho_\mathrm{p0}
w_\mathrm{max}^2(r_{\rm min}^{A,B})
\end{equation}
in the $\epsilon_0$--$\rho_\mathrm{p0}$ plane.
\end{enumerate}
These bounds on $\epsilon_0$ and $\rho_{\rm p0}$ lead to the following scaling relations. The expressions of $b_{\rm min}$ and $b_{\rm max}$ defined by Equation (\ref{eq:impactlim}) may be expressed in the form $M_{\rm tot} w^h \eta^{\alpha}$, where $h=-9/7$ and $-1$ and $\alpha = 0$ and $1/7$, respectively. Thus, the impact parameter lies in the range
\begin{equation} \label{eq:bscaling}
b \propto M_{\rm tot} w^h \eta^{\alpha} \, , \quad -9/7 \leq h \leqslant -1 \, , \quad
0 \leqslant \alpha \leqslant 1/7 \, .
\end{equation}
These relations, together with Equation (\ref{eq:42}), describe $\rho_{\rm p0}$. They may be written as
\begin{equation} \label{eq:rhoscaling}
\rho_{\rm p0} \propto w^{\beta} \eta^{2 \alpha} \, , \quad -4/7 \leqslant \beta \leqslant 0 \, .
\end{equation}
To get a similar expression for $\epsilon_0$, we first examine $E_{\rm fin}$. Combining Equations (\ref{eq:dEGW}) and (\ref{eq:Efin}) together with Equations (\ref{eq:bscaling}) and (\ref{eq:rhoscaling}), $E_{\rm fin}$ can be given as
\begin{equation}
E_{\rm fin} = C_1 \eta M_{\rm tot} w^2 - C_2 M_{\rm tot} w^{-7(h + 1)} \eta^{2-7 \alpha} \,,
\end{equation}
where $C_1$ and $C_2$ are constants. Because $w \ll 1$ for encounters over the considered ranges of BH mass, SMBH mass, and BH population parameters (Section \ref{subsec:MCCode}), therefore $E_{\rm fin}$ is dominated by its second term, and thereby scales as
\begin{equation} \label{eq:EfinScaling}
E_{\rm fin} \propto - M_{\rm tot} w^{-7 (h + 1)} \eta^{2 - 7 \alpha} \, .
\end{equation}
Finally, using Equations (\ref{eq:Efin}), (\ref{eq:bscaling}), and (\ref{eq:EfinScaling}) together yields
\begin{equation} \label{eq:epsscaling}
\epsilon_0 \propto w^{\gamma} \eta^{\kappa} \, , \quad 0 \leqslant \gamma \leqslant 10/7 \, ,
\quad 2/7 \leqslant \kappa \leqslant 1 \, .
\end{equation}
These relations help interpret the results for MC simulations given next.
\section{Results}
\label{sec:NumRes}
We now present the PDF of initial binary parameters describing EBBH events.
\subsection{Radial Distribution of EBBH Mergers in Galactic Nuclei}
\label{subsec:FormRate}
Let us now determine the relative encounter rate as a function of the radius from the center of the GN, $P_{AB}(r)$, for objects with masses $m_A$ and $m_B$.
In Appendix \ref{sec:AppA}, we show that the distribution of merger rates as a function of radius from the center of the GN has the form
\begin{align} \label{eq:radiusPDF}
\frac{\partial^3 \Gamma}{\partial r\partial m_A \partial m_B} & = 4 \pi^2 r^2 n_{m_A}(r) \,
n_{m_B}(r) \, N_{\rm BH}^2\mathcal{F}(m_A) \mathcal{F}(m_B)
\nonumber \\
& \times C_{rA} C_{rB} \left(\zeta_{\rm capt}'(r)- \zeta_{\rm coll}'(r) \right) \, ,
\end{align}
where $n_m(r)$ and $N_{\rm BH}$ are given by Equations (\ref{eq:n(r)}) and (\ref{eq:encBH}), respectively, $C_{rA}$ and $C_{rB}$ are constants defined by Equation (\ref{eq:Cr}), and $\zeta_{\rm capt}'(r)$ and $\zeta_{\rm coll}'(r)$ are terms that follow from the cross section for GW capture and direct collisions, $\pi b_{\max}^2$ and $\pi b_{\min}^2$, respectively, defined by Equations (\ref{eq:captzeta}) and (\ref{eq:collzeta}) in Appendix \ref{sec:AppA}.
\begin{figure}
\centering
\includegraphics[width=80mm]{PAB_Dist}
\caption{ \label{fig:DisPAB} Probability density $\mathrm{log}_{10}(P_\mathrm{AB}(\mathrm{log}_{10}(r)))$ of an EBBH formed by BHs with masses $m_\mathrm{A}$ and $m_\mathrm{B}$ merging at $r$ from the central SMBH of a Milky Way-size nucleus. In this example, the mass of the central SMBH is \mbox{$M_{\rm SgrA*} = 4.3 \times 10^6 \, M_{\odot}$}, and stellar populations surrounding the SMBH are assumed to form spherically symmetric and relaxed populations within the radius of influence of the SMBH, $r_{\rm max}$. We give $P_\mathrm{AB}(r)$, using Equations (\ref{eq:formrateBHBH_multi}) and (\ref{eq:formrateBHBH_single}), for various combinations of $m_\mathrm{A}$ and $m_\mathrm{B}$ values indicated in the figure legend, and for $r\in [r_{\rm min}^{A,B},r_{\rm max}]$ as defined by Equations (\ref{eq_rminAB}) and (\ref{eq:rmax}). Here, $r_{\rm min}^{A,B}$ defines the inner radius at which BHs \emph{A} and \emph{B} can still form an EBBH (see Appendix \ref{sec:CalRmin}). Solid lines correspond to a fiducial multi-mass BH population with a PMF $dN/dm_{\rm BH} \propto m_{\rm BH}^{-2.35}$, $m_\mathrm{BH,max} = 30 \, M_{\odot}$, and $p_0=0.5$ in Equation (\ref{eq:pm}). The dashed-dotted line corresponds to a single-mass BH population of $10 M_{\odot}$. }
\end{figure}
For a multi-mass BH population, the probability distribution of the EBBH events among events with fixed BH masses $m_A$ and $m_B$ as a function of $r$ may be obtained by normalizing Equation (\ref{eq:radiusPDF}) over the radius range $[r_{\rm min}^{A,B}, r_\mathrm{max}]$, where BHs exist in the GN; see Appendix \ref{sec:CalRmin}. We get
\begin{align} \label{eq:formrateBHBH_multi}
P_{AB}(r) = C_1 \left[\left( \frac{ 340 \pi \eta }{ 3 } \right)^{\frac{2}{7}}
\frac{r^{-\frac{3}{14} - p_0 \frac{m_A + m_B}{ m_{\rm BH,max} }}} {M_{\rm SMBH}^
{11/14}} - \frac{ 16 \, r^{-\frac{1}{2} - p_0
\frac{m_A + m_B}{m_{\rm BH,max}}}}{M_{\rm SMBH}^{1/2}}\right] \, ,
\end{align}
where $\eta=q/(1+q)^2$, $q=m_A/m_B$, and $C_1$ is a normalization constant set by the requirement $\int_ {r_{\min}}^{r_{\max}} P_{AB}(r) dr = 1$. For a single-mass distribution, $p_0=0.5$, and $m_A = m_B = \mathrm{max}(m_\mathrm{BH}) = 10 \, M_{\odot}$ (i.e. $q=1$, $\eta=1/4$); thus we get
\begin{equation} \label{eq:formrateBHBH_single}
P_{AB}(r) = C_1 \left[\left( \frac{85 \pi}{3} \right)^{\frac{2}{7}}
\frac{r^{-\frac{17}{14}}}{M_{\rm SMBH}^{11/14}} - \frac{16 \,
r^{-3/2}}{ M_{\rm SMBH}^{1/2}}\right] \, .
\end{equation}
The distribution $P_{AB} (\mathrm{log}_{10}(r))$ is displayed in Figure \ref{fig:DisPAB} for various component masses. For a single-mass BH population, the profile is identical to that of the most massive BHs in a multi-mass BH population when $p_0=0.5$; see Equation \ref{eq:pm}. We find that $P_{AB}(r)$ is steeper for more massive EBBHs as a function of radius, which is due to the steeper density gradient for higher masses (Equation \ref{eq:formrateBHBH_multi}). The profile of $P_{AB}(r)$ is weakly sensitive to the mass ratio for fixed $M_\mathrm{tot}$, i.e. proportional to $\eta^{2/7}$, which is due to the fact that $P_{AB}(r)$ is dominated by its first term in Equation (\ref{eq:formrateBHBH_multi}). Note that the merger rate distribution on a logarithmic radial scale is $d\Gamma_{AB}/d\ln r = r d\Gamma_{AB}/dr \propto r P_{AB}(r)$. By using Equation (\ref{eq:formrateBHBH_single}) for $P_{AB}(r)$ we find that $P_{AB} (\ln r) \propto r^{-3/14}$ for the most massive BHs in the BH population, showing that the merger rate is approximately uniform on a logarithmic scale between $10^{-4}$ and $3 \, \mathrm{pc}$. In fact, roughly $50 \%$ of the most massive EBBHs are formed in the innermost $4 \times 10^{-3} \, \mathrm{pc}$, and $97 \%$ of the events are within $1.2 \, \mathrm{pc}$. However, the least massive EBBHs with $m_A = m_B = 5 \, M_{\odot}$, $p_0=0.5$, and $m_{A,B} \ll m_{\mathrm{ BH,max}}$ are distributed as $P(\ln r) \propto r^{11/14}$. Thus, only $\sim 2 \%$ of them are within $4 \times 10^{-3} \, \mathrm{pc}$, and only roughly $50 \%$ of them are within $1.2 \, \mathrm{pc}$. These low-mass binaries form preferentially further out, near the radius of influence.
Figure \ref{fig:DisPAB} shows that $P_{AB}(r)$ drops off quickly near $r_{\rm min}^{A,B}$. Thus, for fixed component-mass EBBHs, the distributions of orbital parameters do not depend significantly on the exact value of $r_{\rm min}^{A,B}$. For EBBHs with fixed component masses, $r_{\rm min}^{A,B}$ is approximately $2 \times$ lower for the \citet{Belczynskietal2014_2} PMF than for the same BH mass range with a power-law PMF (Appendix \ref{sec:CalRmin}). As a consequence, $P_{AB}(r)$ and the distributions of orbital parameters are very similar for the \citet{Belczynskietal2014_2} and for a power-law PMF.
\subsection{Distributions of Initial Orbital Parameters}
\label{subsec:DistIniParam}
In Appendix \ref{sec:AppA}, we determine the EBBH merger rate distribution as a function of relative velocity and impact parameter. The initial orbital parameters $e_0$ and $\rho_\mathrm{p0}$ may be calculated from these parameters using Equations (\ref{eq:rhop0}) and (\ref{eq:Efin}).
\begin{figure}
\centering
\includegraphics[width=85mm]{FormPlane_10Multi}
\includegraphics[width=85mm]{FormPlane_10Single}
\includegraphics[width=85mm]{FormPlane_30Multi}
\caption{ \label{fig:DistFormPlane10Multi} Two-dimensional histograms of data points generated by the Monte Carlo code. The top and bottom panels correspond to $10 M_{\odot}-10 M_{\odot}$ and $30 \, M_{\odot}-30 \, M_{\odot}$ EBBHs forming in a fiducial multi-mass BH population with a PMF $dN/dm_{\rm BH} \propto m_{\rm BH}^{-2.35}$, $p_0 = 0.5$ and $m_\mathrm{BH,max} = 30 \, M_{\odot}$, and the middle panel corresponds to EBBHs forming in a single-mass BH population. We assumed a Milky Way-size nucleus in all cases. We simulated $10^6$ binaries: those that did not interact with a third object before the merger phase of the EBBH evolution are shown. Histograms were normalized by the total number of the data points. The dashed and dashed-dotted lines correspond to $10 \, \mathrm{Hz}$ and $1 \, \mathrm{Hz}$ characteristic GW frequencies (Equation \ref{eq:rhoaLIGO}), separating the regions detectable by LISA and aLIGO; the intermediate region may be detectable by ET. We find, in panels from top to bottom, that about $(41,80,56) \%$ of EBBHs form within the aLIGO band. The middle and bottom panels are very similar due to the same radial distribution of EBBH merger rate inside the nucleus; see Figure \ref{fig:DisPAB}. From top to bottom panels, respectively, $(1.5, 4.5, 0.2) \%$) of sources form in the LISA band, and $(57.5, 15.5, 43.8) \%$) of sources form in the ET band but outside the aLIGO band. }
\end{figure}
Figure \ref{fig:DistFormPlane10Multi} shows the merger rate distribution as a function of the initial eccentricity and dimensionless pericenter distance. Two-dimensional histograms\footnote{Histograms are normalized by the total number of the data points.} are shown in the \mbox{$\mathrm{log}_{10} (\epsilon_0)$--$\mathrm{log}_{10}(\rho_{\rm p0})$} plane for three choices of the component masses in the three panels, respectively: $m_A = m_B = 10 \, M_{\odot}$ and $m_A = m_B = 30 \, M_{\odot}$ within a fiducial multi-mass BH population (i.e. $p_0=0.5$, $\beta=2.35$, and $m_\mathrm{BH,max}=30 \, M_{\odot}$), and for a single-mass BH population with $m_A = m_B = 10 \, M_{\odot}$. Clearly, the distributions are bounded by the boundaries derived in Figure \ref{fig:boundary} in Section \ref{sec:PIOP}. For low-mass EBBHs forming in a multi-mass BH population, we find that a significant fraction of events form relatively close to the $\delta E_\mathrm{GW} = E_{\rm fin}$ boundary line. For massive EBBHs forming in a multi-mass BH population, as well as for EBBHs forming in a single-mass BH population, we find that a significant fraction of events form at low $\rho_{\rm p0}$ close to the $\delta E_\mathrm{GW} = E_{\rm fin}$ boundary line, which corresponds to an approximately parabolic encounter. The merger rate distribution for a single-mass population is similar to the merger rate distribution for the heaviest components in a multi-mass distribution, as shown by the bottom two panels in Figure \ref{fig:DistFormPlane10Multi}.
The Monte Carlo samples show that EBBHs typically form with high eccentricities ($0.9 \la e_0$), and $\rho_{\rm p0}$ ranges from $\sim 8-9$ up to $\sim 400-900$. The typical range of $\rho_{\mathrm{p}0}$ for GW capture sources can be understood using analytic arguments presented in Appendix \ref{sec:AppA}. Among mergers occuring at a fixed radius $r$, $\rho_{\mathrm{p}0}$ is distributed uniformly for $\rho_{\mathrm{p}0} \leqslant \rho_{\mathrm{p}0, \mathrm{uni}}$
\begin{equation}\label{eq:rhop0uni}
\rho_{\mathrm{p}0, \mathrm{uni}} = \left(\frac{85\pi\eta}{24\sqrt{2}}\right)^{2/7} v_{\max}^{-4/7}
=\left(\frac{85\pi \eta}{48\sqrt{2}}\right)^{2/7} \left(\frac{r}{M_{\mathrm{SMBH}}}\right)^{2/7} \, .
\end{equation}
Note that here the first term is unity ($0.9952$) for equal mass binaries with $\eta=0.25$. If half of the events are located within $0.01\,\mathrm{pc} \sim 5 \times 10^4 \, M_{\rm SgrA*}$, this implies a characteristic cutoff scale at $\rho_{\mathrm{p}0, \mathrm{uni}} \sim 22$. The cutoff scale $\rho_{\mathrm{p}0, \mathrm{uni}}$ depends on the mass of the binary system, due to their different radial merger distributions shown in Figure \ref{fig:DisPAB}. For the most massive BHs in a Milky Way-size nucleus, roughly $50 \%$ of them are located within $ 4 \times 10^{-3} \,\mathrm{pc} \sim 2000 \, M_{\rm SgrA*}$ which implies a cutoff scale at $\rho_{\mathrm{p}0, \mathrm{uni}} \sim 9$, while roughly $50 \%$ of the least massive EBBHs are located within \mbox{$1.2 \,\mathrm{pc} \sim 5.8 \times 10^6 \, M_{\rm SgrA*}$}, which implies a cutoff scale at $\rho_{\mathrm{p}0, \mathrm{uni}}\sim 85$.
The characteristic scale of $\rho_{\mathrm{p}0}$ given by Equation (\ref{eq:rhop0uni}) for a given $v_{\max}$ is applicable for an arbitrary host population. The escape velocity is much lower in GCs than in GN with an SMBH. Plugging in $v_{\max}\sim 60\,\mathrm{km} \, \mathrm{s}^{-1}$ ($30\,\mathrm{km} \, \mathrm{s}^{-1}$) gives a cutoff scale of $\rho_{\mathrm{p}0, \mathrm{uni}} = 160$ (230). Thus, a smoking gun signature of GW capture events in high velocity dispersion environments (i.e. GN) is their small characteristic $\rho_{\mathrm{p}0}$ values below $\sim 100$.
We find that the inner cutoff of $\rho_{\rm p0}$ is insensitive to the component masses of the merging binary, the mass of the SMBH, and the mass distribution of the BH population. This is due to the fact that its leading-order expression gives $\rho_{\rm p0} \geqslant 8$ (Section \ref{sec:PIOP}) independently of system parameters. However, the upper limit of $\rho_{\rm p0}$ is determined by the segment of boundary lines defined by criteria \ref{i:III} and \ref{i:IV} (Figure \ref{fig:boundary}), and it significantly depends on $M_{\rm SMBH}$ and the component masses $m_A$ and $m_B$. Moreover, the results are weakly sensitive to the $p_0$ and $\beta$ parameters of the BH mass distribution, and more sensitive to $m_{\rm BH,max}$.
Figure \ref{fig:Distrhop0} displays the one-dimensional PDF of the initial pericenter distance and eccentricity, respectively, by marginalizing the 2D distribution shown in Figure \ref{fig:DistFormPlane10Multi} over the other parameters for EBBH sources (cf. Figure 4 of \citealt{OLearyetal2009}). Different lines represent the distribution of different component masses for merging binaries. Solid lines show results for a fiducial multi-mass BH population with $dN/dm \propto m^{-2.35}$, $m_\mathrm{BH,max} = 30 \, M_{\odot}$, and $p_0=0.5$.
\begin{figure}
\centering
\includegraphics[width=85mm]{Dist_rho_standard}
\includegraphics[width=85mm]{Dist_eps_standard}
\caption{ \label{fig:Distrhop0} PDF of $\rho_{\rm p0}$ (top panel) and $1-e_0^2$ (bottom panel) for different component-mass EBBHs forming in a Milky Way-size nucleus. Solid lines correspond to EBBHs forming in a fiducial multi-mass BH population with a PMF $dN/dm_{\rm BH} \propto m_{\rm BH}^{-2.35}$, $m_\mathrm{BH,max} = 30 \, M_{\odot}$, $p_0=0.5$ in Equation (\ref{eq:pm}), and dashed-dotted lines correspond to EBBHs forming in a single-mass BH population. We find that more massive EBBHs have lower $\rho_{\rm p0}$ and $e_0$, and EBBHs forming in a single-mass BH populations have lower $\rho_{\rm p0}$ and $e_0$ than similar EBBHs forming in a multi-mass BH population. Moreover, the PDFs of $\rho_{\rm p0}$ and $e_0$ for EBBHs forming in a single-mass BH population are very similar to those of EBBHs formed by the most massive BHs of a multi-mass BH population, which is due to them having the same PDF profile for the EBBH merger rate inside the nucleus (Figure \ref{fig:DisPAB}). We find that both $\rho_{\rm p0}$ and $\epsilon_0$ are lower for binaries of lower $q$ at fixed $M_{\rm tot}$, and the discrepancy between PDFs of different mass ratio EBBHs increases with $M_{\rm tot}$. }
\end{figure}
The main findings of Figures \ref{fig:Distrhop0}--\ref{fig:Distrho_Param} are summarized as follows.
\begin{enumerate}[label=(\Alph*),ref=(\Alph*)]
\item \label{i:A}
At their formation, we find that most sources generally have pericenter distances in the range $\rho_{\mathrm{p}0} \sim 10-80$ and eccentricity $e_0\sim 0.9-0.9999$. More massive EBBHs form with systematically lower $\rho_{\rm p0}$ and with less extreme $e_0$ (e.g. $\rho_{\mathrm{p}0}\lesssim 20$ and $e_0\sim 0.9$ for the heaviest EBBHs). This is due to the fact that, in a multi-mass BH population, more massive EBBHs form closer to the center of the nucleus (Figure \ref{fig:DisPAB}), ergo $w$ extends to higher values $2v_{\rm max}(r)$ (Equation \ref{eq:vmax}), and binary formation by GW capture implies $\rho_{\rm p0} \propto w^{\beta}$ with $-4/7 \leqslant \beta \leqslant 0$ and $\epsilon_0 \propto w^{\gamma}$ with $0 \leqslant \gamma \leqslant 10/7$ (Equations \ref{eq:rhoscaling} and \ref{eq:epsscaling}).\footnote{The encounter cross section is typically dominated by gravitational focusing, which implies a uniform $P(\rho_{\mathrm{p}0})$ distribution for isotropic distribution of relative velocity with a fixed magnitude $w$ \citep{Kocsisetal2006,OLearyetal2009} in the range $0 \leqslant \rho_{\mathrm{p}0} \leqslant k \eta^{2/7} w^{-4/7}$, where $k=(85\pi/6\sqrt 2)^{2/7}$. The decrease of $P(\rho_{\mathrm{p}0})$ is due to the distribution of $w$ in the GN.}
\begin{figure}
\centering
\includegraphics[width=85mm]{Dist_rhop0_mSMBHDep}
\includegraphics[width=85mm]{Dist_eps0_mSMBHDep}
\caption{ \label{fig:SMBHdeprhopaeps0} An example of how different SMBH masses influence the PDF of $\rho_{\rm p0}$ (top panel) and $1-e_0^2$ (bottom panel). Results are shown for $10 \, M_{\odot} - 10 \, M_{\odot}$ EBBHs forming in a fiducial multi-mass BH population with a PMF $dN/dm_{\rm BH} \propto m_{\rm BH}^{-2.35}$, $m_\mathrm{BH,max} = 30 \, M_{\odot}$, and $p_0=0.5$. }
\end{figure}
\item \label{i:B} The maximum BH mass in the distribution $m_{\rm BH,max}$ significantly influences the PDF of $\rho_{\rm p0}$ and $e_0$ for different BH mass mergers. However, the mass segregation parameter (assuming that it is in the range $0.5<p_0<0.6$) and the slope of the BH mass function (assuming the range $1 < \beta < 3$) do not significantly influence these PDFs. This result is consistent with analytic expectations presented in Section \ref{subsec:FormRate}.
\item \label{i:C} Both $\rho_{\rm p0}$ and $\epsilon_0$ are somewhat lower for EBBHs of lower $q$ at fixed $M_{\rm tot}$, which arise due to the Equations (\ref{eq:rhoscaling}) and (\ref{eq:epsscaling}) for $\rho_{\rm p0}$ and $\epsilon_0$. However, because $\rho_{\mathrm{p}0, \mathrm{uni}}\propto \eta^{2/7}$, this dependence is rather weak. Moreover, the difference between the PDFs of EBBHs with different mass ratios increases with $M_{\rm tot}$ due to the $M_{\rm tot}$ dependence of the minimum radius, $r_{\rm min}^{AB}$, and the corresponding increased value of $w$ for increasing $M_{\rm tot}$.
\begin{figure}
\centering
\includegraphics[width=85mm]{rho_Dep_Dist}
\includegraphics[width=85mm]{eps_Dep_Dist}
\caption{ \label{fig:Distrho_Param} An example of how different multi-mass BH population parameters influence the PDF of $\rho_{\rm p0}$ (top panel) and $1-e_0^2$ (bottom panel). Results are shown for $10 \, M_{\odot} - 10 \, M_{\odot}$ EBBHs forming in a Milky Way-size nucleus. Solid thick lines correspond to a fiducial multi-mass BH population with a PMF $dN/dm_{\rm BH} \propto m_{\rm BH}^{-2.35}$, $m_\mathrm{BH,max} = 30 \, M_{\odot}$, and $p_0 =0.5$, while dashed-dotted and dotted lines correspond to modified parameter values for a multi-mass BH population as shown in the legend of the \emph{top panel}. Consistently with analytic expectations, only $m_{\rm BH,max}$ significantly influences the PDF of $\rho_{\rm p0}$ and $1-e_0^2$. }
\end{figure}
\item \label{i:D} In simulations with different $M_{\mathrm{SMBH}}$ (Figure \ref{fig:SMBHdeprhopaeps0}), we find that EBBHs form with lower $\rho_{\rm p0}$ and $e_0$ around more massive SMBHs, independently of the binary component masses and the BH population parameters. This is due to the fact that the average relative velocity with which BHs form EBBHs in the GN increases with $M_{\rm SMBH}$ (Figure \ref{fig:w_Avr_mSMBH}). Thus, by considering Equations (\ref{eq:rhoscaling}) and (\ref{eq:epsscaling}), we find that both $e_0$ and $\rho_{\mathrm{p}0}$ systematically decrease with $M_{\rm SMBH}$.
\item \label{i:E} In a single-mass BH population, $10 \, M_{\odot}-10 \, M_{\odot}$ EBBHs form with lower $\rho_{\rm p0}$ and $e_0$ than those in a multi-mass BH population. Moreover, the PDF of $\rho_{\rm p0}$ and $\epsilon_0$ are very similar for EBBHs forming in a single-mass BH population and EBBHs formed by the most massive BHs in a multi-mass BH population, which is due to the similarity between the corresponding EBBH merger rate distributions and $r_{\rm min}^{A,B}$ values (Figure \ref{fig:DisPAB}).
\end{enumerate}
Eccentric binaries emit a GW signal with a broad spectrum of frequencies. Because the LIGO/VIRGO-type GW instruments are sensitive above $f_{\rm aLIGO} = 10 \, \mathrm{Hz}$, we assume that those EBBHs form within the aLIGO band in the $\epsilon_0$--$\rho_{\rm p0}$ plane for which $f_{\rm GW}>f_{\rm aLIGO}$, where $f_{\rm GW}$ is the peak GW frequency associated with the harmonic. This leads to the maximal emission of GW radiation being estimated as
\begin{equation} \label{eq:fGW}
f_{\rm GW} = \frac{2 (1+e)^{1.1954}}{(1-e^2)^{3/2}} \frac{M_{\rm tot}^{1/2}}{2 \pi a^{3/2}} \, ;
\end{equation}
see \citet{Wen2003}. Here, $f_{\rm GW}$ defines the peak GW frequency. For any initial condition, the function $a\equiv a(e)$ is given by \citet{Peters1964}. Because EBBHs form with $e_0 \sim 1$, in this case, after combining Equation (\ref{eq:fGW}) with Equation (\ref{eq:rho0}), $f_{\rm GW}$ can be expressed as $f_{\rm GW,0} = (2^{0.3046} \pi M_{\rm tot} \rho_{\rm p0}^{3/2})^{-1}$. Thus, those EBBHs form within the aLIGO band for which $\rho_{\rm p0}$ does not exceed the limit
\begin{align} \label{eq:rhoaLIGO}
\rho_{\rm p0,aLIGO} &= (2^{0.3046} \pi M_{\rm tot} f_{\rm aLIGO})^{-2/3} \nonumber\\&= 40.9
\left(\frac{M_{\mathrm{tot}}}{20\,M_{\odot}}\right)^{-2/3} \left( \frac{f_{\rm aLIGO}}{10\,\mathrm{Hz}}
\right)^{-2/3} \, .
\end{align}
Note that this lies in the expected range of the $\rho_{\mathrm{p}0}$ distribution of GW capture sources given in Equation (\ref{eq:rhop0uni}).
Because $\rho_{\mathrm{p}0}$ extends to relatively high values, a significant fraction of EBBHs may form with $f_{\rm GW}$ below the sensitive frequency band of aLIGO. For ground-based GW detectors, the lower bound of the sensitive frequency band is around $\sim 1 \, \mathrm{Hz}$, for the Einstein Telescope\footnote{\url{http://www.et-gw.eu/}} (ET, \citealt{Punturoetal2010}); for space-based GW observatories, the upper bound of the sensitive frequency band for the Laser Interferometer Space Antenna\footnote{\url{https://www.elisascience.org/}} (LISA) is around $\sim 1 \, \mathrm{Hz}$ \citep{AmaroSeoaneetal2013}. Figure \ref{fig:DistFormPlane10Multi} shows the regions where EBBHs form in the ET and LISA frequency bands, respectively.
\begin{table}
\caption{ \label{tab:FracM_SMBH} The fraction of EBBHs having residual eccentricities larger than $0.1$ when their GW signals enter the aLIGO band ($F_{{\rm res} > 0.1}$), of EBBHs forming within the aLIGO band ($F_{\rm aLIGO}$), and of EBBHs having eccentricities larger than $0.1$ at the last stable orbit (LSO, $F_{ {\rm LSO} > 0.1}$). We use \emph{Single} to denote rows corresponding to a single-mass BH population, otherwise a fiducial multi-mass BH population is considered with a PMF $dN/dm_{\rm BH} \propto m_{\rm BH}^{-2.35}$, $m_\mathrm{BH,max} = 30 \, M_{\odot}$, and $p_0 = 0.5$; and $M_{\mathrm{SgrA}^*}$ denotes the mass of the SMBH for a Milky Way-size nucleus. Systems were evolved from their initial orbit using the evolution equations of \citet{Peters1964}. $F_{\rm aLIGO}$ increases with $M_{\rm SMBH}$ at fixed component masses for both type of BH populations, which arises due to the decrease of $\rho_{\rm p0}$ against $M_{\rm SMBH}$. Consequently, a lower-and-lower fraction of EBBHs are needed to evolve from their initial orbit up until their GW frequency $f_{\rm GW}$ \citep{Wen2003} reaches the aLIGO band, which increases $F_{ {\rm res} > 0.1}$, and $F_{e_{\rm LSO} > 0.1}$ with $M_{\rm SMBH}$ as well. Moreover, we find that $F_{ {\rm res} > 0.1}$ ($F_{\rm aLIGO}$, and $F_{ {\rm LSO} > 0.1}$) is higher for EBBHs of lower $q$ at fixed $M_{\rm tot}$, and the discrepancy between $F_{{\rm res} > 0.1}$ ($F_{\rm aLIGO}$, and $F_{ {\rm LSO} > 0.1}$) of different $q$ increases with $M_{\rm tot}$. }
\centering
\begin{tabular}{@{}ccccc}
\hline
$m_A - m_B \, \, (M_{\odot})$ & $M_{\rm SMBH} \, \, (M_{\odot})$ & $F_{ {\rm res} > 0.1}$ &
$F_{\rm aLIGO}$ & $F_{ {\rm LSO} > 0.1}$ \\
\hline\hline
$5 - 5$ & $10^5$ & $88 \%$ & $35 \%$ & $3 \%$ \\
$10 - 10$ & $10^5$ & $69 \%$ & $25 \%$ & $4 \%$ \\
$10 - 10$ (Single) & $10^5$ & $96 \%$ & $78 \%$ & $35 \%$ \\
$5 - 15$ & $10^5$ & $73 \%$ & $27 \%$ & $4 \%$ \\
$10 - 30$ & $10^5$ & $68 \%$ & $32 \%$ & $12 \%$ \\
$20 - 20$ & $10^5$ & $69 \%$ & $31 \%$ & $12 \%$ \\
$30 - 30$ & $10^5$ & $84 \%$ & $51 \%$ & $32 \%$ \\
$5 - 5$ & $10^6$ & $96 \%$ & $48 \%$ & $5 \%$ \\
$10 - 10$ & $10^6$ & $85 \%$ & $35 \%$ & $7 \%$ \\
$10 - 10$ (Single) & $10^6$ & $97 \%$ & $79 \%$ & $36 \%$ \\
$5 - 15$ & $10^6$ & $87 \%$ & $36 \%$ & $6 \%$ \\
$10 - 30$ & $10^6$ & $78 \%$ & $38 \%$ & $15 \%$ \\
$20 - 20$ & $10^6$ & $80 \%$ & $38 \%$ & $16 \%$ \\
$30 - 30$ & $10^6$ & $86 \%$ & $54 \%$ & $36 \%$ \\
$5 - 5$ & $M_{\mathrm{SgrA}^*}$ & $98 \%$ & $58 \%$ & $6 \%$ \\
$10 - 10$ & $M_{\mathrm{SgrA}^*}$ & $91 \%$ & $41 \%$ & $7 \%$ \\
$10 - 10$ (Single) & $M_{\mathrm{SgrA}^*}$ & $98 \%$ & $80 \%$ & $37 \%$ \\
$5 - 15$ & $M_{\mathrm{SgrA}^*}$ & $93\%$ & $44 \%$ & $7 \%$ \\
$10 - 30$ & $M_{\mathrm{SgrA}^*}$ & $81 \%$ & $43 \%$ & $17 \%$ \\
$20 - 20$ & $M_{\mathrm{SgrA}^*}$ & $82 \%$ & $41 \%$ & $17 \%$ \\
$30 - 30$ & $M_{\mathrm{SgrA}^*}$ & $88 \%$ & $56 \%$ & $37 \%$ \\
$5 - 5$ & $10^7$ & $99 \%$ & $64 \%$ & $7 \%$ \\
$10 - 10$ & $10^7$ & $88 \%$ & $45 \%$ & $8 \%$ \\
$10 - 10$ (Single) & $10^7$ & $99 \%$ & $81 \%$ & $38 \%$ \\
$5 - 15$ & $10^7$ & $96 \%$ & $49 \%$ & $9 \%$ \\
$10 - 30$ & $10^7$ & $88 \%$ & $46 \%$ & $18 \%$ \\
$20 - 20$ & $10^7$ & $86 \%$ & $43 \%$ & $18 \%$ \\
$30 - 30$ & $10^7$ & $90 \%$ & $57 \%$ & $38 \%$ \\
\hline\hline
\end{tabular}
\end{table}
\begin{table}
\caption{ \label{tab:FracM_SMBH_2} Same as Table \ref{tab:FracM_SMBH}, but for a source population with $m_\mathrm{BH, max} = 80 \, M_{\odot}$. }
\centering
\begin{tabular}{@{}ccccc}
\hline
$m_A - m_B \, \, (M_{\odot})$ & $M_{\rm SMBH} \, \, (M_{\odot})$ & $F_{ {\rm res} > 0.1}$ &
$F_{\rm aLIGO}$ & $F_{ {\rm LSO} > 0.1}$ \\
\hline\hline
$5 - 5$ & $10^5$ & $89 \%$ & $37 \%$ & $3 \%$ \\
$10 - 10$ & $10^5$ & $68 \%$ & $24 \%$ & $4 \%$ \\
$5 - 15$ & $10^5$ & $70 \%$ & $24 \%$ & $4 \%$ \\
$10 - 30$ & $10^5$ & $51 \%$ & $16 \%$ & $5 \%$ \\
$20 - 20$ & $10^5$ & $50 \%$ & $9 \%$ & $4 \%$ \\
$30 - 30$ & $10^5$ & $43 \%$ & $13 \%$ & $6 \%$ \\
$50 - 50$ & $10^5$ & $41 \%$ & $11 \%$ & $10 \%$ \\
$20 - 80$ & $10^5$ & $56 \%$ & $13 \%$ & $12 \%$ \\
$60 - 60$ & $10^5$ & $50 \%$ & $13 \%$ & $15 \%$ \\
$50 - 70$ & $10^5$ & $49 \%$ & $12 \%$ & $14 \%$ \\
$70 - 70$ & $10^5$ & $67 \%$ & $11 \%$ & $26 \%$ \\
$80 - 80$ & $10^5$ & $63 \%$ & $12 \%$ & $29 \%$ \\
$5 - 5$ & $10^6$ & $96 \%$ & $47 \%$ & $5 \%$ \\
$10 - 10$ & $10^6$ & $81 \%$ & $29 \%$ & $5 \%$ \\
$5 - 15$ & $10^6$ & $86 \%$ & $30 \%$ & $5 \%$ \\
$10 - 30$ & $10^6$ & $66 \%$ & $21 \%$ & $7 \%$ \\
$20 - 20$ & $10^6$ & $63 \%$ & $13 \%$ & $6 \%$ \\
$30 - 30$ & $10^6$ & $55 \%$ & $16 \%$ & $7 \%$ \\
$50 - 50$ & $10^6$ & $58 \%$ & $14 \%$ & $12 \%$ \\
$20 - 80$ & $10^6$ & $51 \%$ & $15 \%$ & $13 \%$ \\
$60 - 60$ & $10^6$ & $57 \%$ & $15 \%$ & $18 \%$ \\
$50 - 70$ & $10^6$ & $55 \%$ & $14 \%$ & $17 \%$ \\
$70 - 70$ & $10^6$ & $62 \%$ & $15 \%$ & $28 \%$ \\
$80 - 80$ & $10^6$ & $68 \%$ & $15 \%$ & $34 \%$ \\
$5 - 5$ & $10^7$ & $99 \%$ & $67 \%$ & $7 \%$ \\
$10 - 10$ & $10^7$ & $94 \%$ & $40 \%$ & $7 \%$ \\
$5 - 15$ & $10^7$ & $96 \%$ & $43 \%$ & $8 \%$ \\
$10 - 30$ & $10^7$ & $79 \%$ & $29 \%$ & $10 \%$ \\
$20 - 20$ & $10^7$ & $78 \%$ & $18 \%$ & $8 \%$ \\
$30 - 30$ & $10^7$ & $69 \%$ & $21 \%$ & $10 \%$ \\
$50 - 50$ & $10^7$ & $61 \%$ & $16 \%$ & $15 \%$ \\
$20 - 80$ & $10^7$ & $67 \%$ & $20 \%$ & $18 \%$ \\
$60 - 60$ & $10^7$ & $68 \%$ & $18 \%$ & $22 \%$ \\
$50 - 70$ & $10^7$ & $63 \%$ & $17 \%$ & $21 \%$ \\
$70 - 70$ & $10^7$ & $66 \%$ & $18 \%$ & $29 \%$ \\
$80 - 80$ & $10^7$ & $72 \%$ & $18 \%$ & $36 \%$ \\
\hline\hline
\end{tabular}
\end{table}
\begin{table}
\caption{\label{tab:Frac_BHPopParam} Same as Table \ref{tab:FracM_SMBH} for $10 \, M_{\odot} -10 \, M_{\odot}$ EBBHs, but for different values of the parameters shown in the first column. We find that only $m_\mathrm{BH,max}$ influences the results significantly: higher maximum BH masses in the cluster lead to much lower LSO eccentricities among mergers with fixed binary component masses. }
\centering
\begin{tabular}{@{}cccccc}
\hline
Parameter & $F_{ {\rm res} > 0.1}$ & $F_{\rm aLIGO}$ & $F_{ {\rm LSO} > 0.1}$ \\
\hline\hline
standard & $91 \%$ & $41 \%$ & $7 \%$ \\
$\beta = 1$ & $91 \%$ & $41 \%$ & $7 \%$ \\
$\beta = 3$ & $91 \%$ & $41 \%$ & $7 \%$ \\
$m_{\rm BH,max} = 10 \, M_{\odot}$ & $98 \%$ & $80 \%$ & $37 \%$ \\
$m_{\rm BH,max} = 15 \, M_{\odot}$ & $94 \%$ & $59 \%$ & $16 \%$ \\
$m_{\rm BH,max} = 80 \, M_{\odot}$ & $90 \%$ & $35 \%$ & $6 \%$ \\
$p_0 = 0.6$ & $92 \%$ & $43 \%$ & $8 \%$ \\
\hline\hline
\end{tabular}
\end{table}
An important question for GW detections is the fraction of objects that are eccentric when they emit GWs in the aLIGO band. In Tables \ref{tab:FracM_SMBH}, \ref{tab:FracM_SMBH_2}, and \ref{tab:FracM_SMBH_Belcz}, we show the fraction of highly eccentric GW capture sources that form with a GW spectrum that peaks above $10 \, \mathrm{Hz}$ ($F_{\rm aLIGO}$), the fraction of EBBHs that have $e> 0.1$ when the GW spectrum first peaks above $10 \, \mathrm{Hz}$ ($F_{ {\rm res} > 0.1}$), and the fraction of EBBHs that have $e > 0.1$ when the binary reaches the LSO ($F_{ {\rm LSO} > 0.1}$) for various binary and SMBH masses. Depending on the component masses and the SMBH mass, we find in source populations with $m_\mathrm{BH, max}=30 \, M_{\odot}$, $80 \, M_{\odot}$, or the \citet{Belczynskietal2014_2} PMF, that the fraction $F_{\mathrm{res} >0.1} = 68 - 99 \%$, $41 - 99 \%$, or $68 - 99 \%$, respectively, if the EBBHs are at least moderately eccentric, with $e > 0.1$ when the spectrum first peaks above $10 \, \mathrm{Hz}$ (see discussion in Section \ref{subsec:DistFGW10Hz}), and $F_{\rm aLIGO} = 25 - 64 \%$, $9 - 67 \%$, or $27 - 86 \%$, respectively, of highly eccentric sources form in the aLIGO band (these sources have $e_0 > 0.9$, see Figure \ref{fig:Dist_fp10}). However, most of these sources circularize to below $e<0.1$ by the time the binary approaches the LSO. We find that the fractions of sources in this formation channel with $e > 0.1$ at the LSO are between $F_{\mathrm{LSO} > 0.1} = 3 - 38 \%$, $3 - 36 \%$, and $3 - 36 \%$, respectively, for EBBHs forming in a source population with $m_{\mathrm{BH,max}} = 30 \, M_{\odot}$, $80 \, M_{\odot}$, and the \citet{Belczynskietal2014_2} PMF as shown in Tables \ref{tab:FracM_SMBH}, \ref{tab:FracM_SMBH_2}, and \ref{tab:FracM_SMBH_Belcz}. In particular, sources that remain eccentric until the LSO have masses comparable to the maximum BH mass in the cluster, while lower-mass binaries are systematically more circular near the LSO. We discuss $F_{\rm aLIGO}$ and $F_{\mathrm{LSO} >0.1}$ separately in Sections \ref{subsec:DistFGW10Hz} and \ref{subsec:DistLSO}.
To check the robustness of these predictions, we show the fractional rates of eccentric sources for different BH mass function exponents $\beta$ and mass segregation parameters $p_0$ in Table \ref{tab:Frac_BHPopParam}. We find that (i) $F_{\rm aLIGO}$ is significantly influenced by $m_\mathrm{BH,max}$ and slightly by $p_0$ and $\beta$; (ii) $F_{\rm aLIGO}$ is higher for EBBHs of lower $q$ at fixed $M_{\rm tot}$; (iii) $F_{\rm aLIGO}$ increases with $M_{\rm SMBH}$; and (iv) $F_{\rm aLIGO}$ is lower for EBBHs forming in a single-mass BH population than similar EBBHs forming in a multi-mass BH population. These results may be explained by arguments listed in \ref{i:B}, \ref{i:C}, \ref{i:D}, and \ref{i:E}, above. However, $F_{\rm aLIGO}$ is not similar for EBBHs forming in a single-mass BH population and EBBHs formed by the most massive BHs in a multi-mass BH population--nor does it systematically increase with the component masses, because $\rho_{\rm p0,aLIGO}$ depends on the component masses as well.
Note, further, that the fraction of EBBHs forming in the LISA band increases with the total mass of the binary up to $\sim 23 \%$ ($\sim 19 \%$, $\sim 15 \%$) for $80 \, M_{\odot} - 80 \, M_{\odot}$ for $10^5 \, M_{\odot}$ ($10^6 \, M_{\odot}$, $10^7 \, M_{\odot}$) GNs. However, we find that these EBBHs emit a negligible fraction of their total signal power in the LISA band. All EBBHs merge within the aLIGO band for the considered ranges of $M_{\rm SMBH}$, $m_{\rm BH}$, and BH populations.
\begin{figure}
\centering
\includegraphics[width=85mm]{Dist_e_fp10_standard}
\caption{ \label{fig:Dist_fp10} The PDF of residual eccentricity for different component-mass EBBHs when their emitted GW signals first peak at $10 \, \mathrm{Hz}$ or when they form at a higher frequency (cf. Figures \ref{fig:Distrhop0} and \ref{fig:DisteLSO} for the initial and final eccentricity distribution). We assume EBBHs forming through GW capture in a Milky Way-size nucleus. Different lines correspond to different component masses in a multi-mass population and in a single-mass population as in Figure \ref{fig:Distrhop0}. The sharp peaks beyond $e_{\rm res} \sim 0.9$ correspond to EBBHs forming within the aLIGO band. For masses $(10 \, M_{\odot} - 10 \, M_{\odot}, 20 \, M_{\odot} - 20 \, M_{\odot}, 30 \, M_{\odot} - 30 \, M_{\odot}, 10 \, M_{\odot} - 30 \, M_{\odot})$, respectively, in a multi-mass BH population, we find that $(41,41,56,43)\%$ of EBBHs form in the aLIGO band. This number is $80\%$ for a single-mass BH population. We find that residual eccentricities at $10 \, \mathrm{Hz}$ are larger than $0.1$ for $(91,82,88,81)\%$ of $(10 \, M_{\odot} - 10 \, M_{\odot}, 20 \, M_{\odot} - 20 \, M_{\odot}, 30 \, M_{\odot} - 30 \, M_{\odot}, 10 \, M_{\odot} - 30 \, M_{\odot})$ EBBHs and $98 \%$ in a single-mass BH population (see Table \ref{tab:FracM_SMBH}). }
\end{figure}
\begin{table}
\caption{ \label{tab:FracM_SMBH_Belcz} Same as Table \ref{tab:FracM_SMBH}, but for a multi-mass BH population with the \citet{Belczynskietal2014_2} PMF and $p_0 = 0.5$. }
\centering
\begin{tabular}{@{}ccccc}
\hline
$m_A - m_B \, \, (M_{\odot})$ & $M_{\rm SMBH} \, \, (M_{\odot})$ & $F_{ {\rm res} > 0.1}$ &
$F_{\rm aLIGO}$ & $F_{ {\rm LSO} > 0.1}$ \\
\hline\hline
$3 - 3$ & $10^5$ & $97 \%$ & $50 \%$ & $3 \%$ \\
$5 - 5$ & $10^5$ & $89 \%$ & $41 \%$ & $4 \%$ \\
$10 - 10$ & $10^5$ & $70 \%$ & $27 \%$ & $5 \%$ \\
$5 - 15$ & $10^5$ & $76 \%$ & $33 \%$ & $6 \%$ \\
$20 - 20$ & $10^5$ & $75 \%$ & $37 \%$ & $14 \%$ \\
$26 - 26$ & $10^5$ & $84 \%$ & $53 \%$ & $31 \%$ \\
$3 - 3$ & $10^6$ & $99.2 \%$ & $67 \%$ & $5 \%$ \\
$5 - 5$ & $10^6$ & $96 \%$ & $51 \%$ & $6 \%$ \\
$10 - 10$ & $10^6$ & $87 \%$ & $41 \%$ & $7 \%$ \\
$5 - 15$ & $10^6$ & $89 \%$ & $40 \%$ & $9 \%$ \\
$20 - 20$ & $10^6$ & $83 \%$ & $47 \%$ & $23 \%$ \\
$26 - 26$ & $10^6$ & $90 \%$ & $55 \%$ & $32 \%$ \\
$3 - 3$ & $M_{\mathrm{SgrA}^*}$ & $99.5 \%$ & $76 \%$ & $6 \%$ \\
$5 - 5$ & $M_{\mathrm{SgrA}^*}$ & $98 \%$ & $64 \%$ & $7 \%$ \\
$10 - 10$ & $M_{\mathrm{SgrA}^*}$ & $93 \%$ & $46 \%$ & $9 \%$ \\
$5 - 15$ & $M_{\mathrm{SgrA}^*}$ & $93 \%$ & $45 \%$ & $10 \%$ \\
$20 - 20$ & $M_{\mathrm{SgrA}^*}$ & $87 \%$ & $52 \%$ & $22 \%$ \\
$26 - 26$ & $M_{\mathrm{SgrA}^*}$ & $91 \%$ & $57 \%$ & $34 \%$ \\
$3 - 3$ & $10^7$ & $99.8 \%$ & $86 \%$ & $7 \%$ \\
$5 - 5$ & $10^7$ & $99 \%$ & $69 \%$ & $9 \%$ \\
$10 - 10$ & $10^7$ & $95 \%$ & $49 \%$ & $10 \%$ \\
$5 - 15$ & $10^7$ & $99 \%$ & $52 \%$ & $12 \%$ \\
$20 - 20$ & $10^7$ & $93 \%$ & $58 \%$ & $27 \%$ \\
$26 - 26$ & $10^7$ & $93 \%$ & $61 \%$ & $36 \%$ \\
\hline\hline
\end{tabular}
\end{table}
The findings of this section for the characteristics of distributions also stand when using the \citet{Belczynskietal2014_2} PMF, which is due to the fact that these characteristics do not depend significantly on $r_{\rm min}^{A,B}$ (Section \ref{subsec:FormRate}). Furthermore, the results presented in Table \ref{tab:FracM_SMBH} are very similar to those presented in Table \ref{tab:FracM_SMBH_Belcz} for similar EBBHs, which arises due the fact that the radial distribution $P_{AB}(r)$ and the BH mass range are very similar for both types of multi-mass BH populations. This is consistent with analytic expectation presented at the end of Section \ref{subsec:FormRate}.
Additionally, our calculations show that only a negligible fraction ($\la 1 \%$) of the EBBHs interact with a third object during the eccentric inspiral. This implies that analytic expressions can be derived to determine the two-dimensional PDFs in the $\epsilon_0$--$\rho_{\rm p0}$ plane (Appendix \ref{TWORPEPS}) and similarly for the marginalized 1D $\rho_{\rm p0}$ and $\epsilon_0$ distributions, (Appendix \ref{PERICDIST} and \ref{ED}).
\subsection{Eccentricity Distribution When the GW Signal First Enters the aLIGO Band}
\label{subsec:DistFGW10Hz}
We evolve systems from their initial orbital parameters using the evolution equations of \citet{Peters1964}. For many systems, the binary has a GW frequency in the aLIGO band at formation. However, systems that form outside of the aLIGO band may also enter the aLIGO band before merger. Figure \ref{fig:Dist_fp10} displays examples for the PDF of residual eccentricity $e_{\rm res}$ when the peak GW signal frequency $f_\mathrm{GW}$ (Equation \ref{eq:fGW}) first reaches $10\,\mathrm{Hz}$. In all cases, we find that (i) a sharp peak occurs in the PDF at $e_{\rm res} \sim 1$, which corresponds to EBBHs forming within the aLIGO band at $f_\mathrm{GW} > 10 \, \mathrm{Hz}$; and (ii) a peak forms at moderately small ($0.05-0.15$) eccentricities. The latter (ii) can be explained as follows: $e_{\rm res}$ can be obtained by solving the equation $f_{\rm aLIGO} = f_{\rm GW}$, which can be rewritten as
\begin{equation} \label{eq:erho_fGW10Hz}
f_{\rm aLIGO} = \left[(1 + e_{\rm res})^{0.3046} \pi M_{\rm tot} \rho_{\rm p}^{3/2}(e_{\rm res},e_0,
\rho_{\rm p0})\right]^{-1} \, ,
\end{equation}
where $\rho_{\rm p}$ is given by Equation (\ref{eq:rhoe}). By definition, $e_{\rm res} \equiv e_0$ for EBBHs with $\rho_{{\rm p}0} \leqslant \rho_\mathrm{p0,aLIGO}$ (Equation \ref{eq:rhoaLIGO}). Solving Equation (\ref{eq:erho_fGW10Hz}) for $e_{\rm res}$ by setting $\rho_{{\rm p}0} \geqslant \rho_\mathrm{p0,aLIGO}$, we find that $e_{\rm res}$ drops off quickly at $\rho_{{\rm p}0} \sim \rho_\mathrm{p0,aLIGO}$ and its tail starts at $e_{\rm res} \sim 0.5$, which implies that a significant fraction of EBBHs with $\rho_{{\rm p}0} \geqslant \rho_\mathrm{p0,aLIGO}$ have $e_{\rm res} \leqslant 0.5$ when they enter the aLIGO band. Furthermore, EBBHs that formed with $\rho_\mathrm{p0} \ga \rho_\mathrm{p0, aLIGO} $ exhibit a peak in the PDF of $e_{\rm res}$ at $e_{\rm res} \sim 0.05-0.15$.
Tables \ref{tab:FracM_SMBH}--\ref{tab:FracM_SMBH_Belcz} present the total fraction of EBBHs with residual eccentricities larger than $0.1$ when their emitted GW signals first exceed $10 \, \mathrm{Hz}$ ($F_{{\rm res} > 0.1}$), for various values of fixed component masses. We find that (i) $F_{{\rm res} > 0.1}$ is significantly influenced by $m_{\rm BH,\max}$, but it is not sensitive to the mass function exponent $\beta$ or the mass segregation parameter $p_0$; (ii) $F_{ {\rm res} > 0.1}$ is higher for EBBHs of lower $q$ at fixed $M_{\rm tot}$, and the difference between $F_{ {\rm res} > 0.1}$ of different $q$ becomes more substantial with increasing $M_{\rm tot}$; (iii) $F_{{\rm res} > 0.1}$ increases with $M_{\rm SMBH}$; and (iv) $F_{ {\rm res} > 0.1}$ is lower for EBBHs forming in a single-mass BH population than for similar EBBHs forming in a multi-mass BH population. These findings arise due to \ref{i:C}, \ref{i:D}, \ref{i:E}, and \ref{i:B} because $\rho_{ \rm p0}$ is the only free parameter that determines $e_{\rm res}$ when the component masses are fixed. However, $F_{ {\rm res} > 0.1}$ is not similar for EBBHs forming in a single-mass BH population versus EBBHs formed by the most massive BHs in a multi-mass BH population; nor does it increase systematically with the component masses, because $e_{\rm res}$ depends on the component masses as well.
The detectability of a compact binary is related to how much energy is radiated in the frequency range where the detector is sensitive. An aLIGO type detector is most sensitive at $\sim 100 \, \mathrm{Hz}$, and the sensitivity decreases rapidly over $\sim 100 \, \mathrm{Hz}$. If a compact binary forms with a peak GW frequency above this limit, then most of the GW energy would be radiated at higher frequencies than $\sim 100 \, \mathrm{Hz}$, which may make such sources difficult to detect. Utilizing the fact that EBBHs are expected to form with $e_0 \sim 1$ and $\rho_{\mathrm{p}0} \geqslant 8$ (Section \ref{subsec:DistIniParam}), Equation (\ref{eq:rhoaLIGO}) can be used to estimate the upper limit for $M_{\rm tot}$ at which EBBHs still form with $f_{\rm GW} \gtrsim 100 \, \mathrm{Hz}$. We find this limit to be $M_{\rm tot} \sim 23 \, M_{\odot}$, and the maximum $\rho_{\mathrm{p}0}$ with which EBBHs with $M_{\rm tot} \lesssim 23 \, M_{\odot}$ form with $f_{\rm GW} \gtrsim 100 \, \mathrm{Hz}$ can be given as
\begin{equation} \label{eq:rho100Hz}
\rho_{\rm p0,100Hz} = 8.816 \left(\frac{M_{\mathrm{tot}}}{20\,M_{\odot}}\right)^{-2/3} \, .
\end{equation}
Because EBBHs with $M_{\rm tot} \sim 23 \, M_{\odot}$ form with generally high $\rho_{\mathrm{p}0}$, a negligible fraction of low-mass EBBHs form with $f_{\rm GW} \gtrsim 100 \, \mathrm{Hz}$ over the considered ranges of $p_0$, $M_{\rm SMBH}$, $m_{\rm BH}$, and the BH PMFs; see Table \ref{tab:FracM_over100Hz} for examples. EBBHs, which form below the aLIGO band, merge within the aLIGO band for $m_{\rm BH}<10^3M_{\odot}$. Therefore, EBBH GW capture sources in GNs are typically within the frequency range of aLIGO-type GW detectors.
Similarly to Section \ref{subsec:DistIniParam}, we find that the PDF of $e_{\rm res}$ is qualitatively very similar for the power-law and \citet{Belczynskietal2014_2} PMF, and the results shown in Table \ref{tab:FracM_SMBH} are very similar to those presented in Table \ref{tab:FracM_SMBH_Belcz} for similar EBBHs.
Overall, we find that, in a source population with $m_\mathrm{BH,max}=30 \, M_{\odot}$, $80 \, M_{\odot}$, or the \citet{Belczynskietal2014_2} PMF, respectively, at least $F_{ {\rm res} > 0.1 } = 65 - 86 \%$, $38 - 58 \%$, and $66 - 89 \%$ of EBBHs have residual eccentricities larger than $0.1$ when their GW signals first reach $10 \, \mathrm{Hz}$ or form with higher frequencies. The actual value within this range is generally set by the binary component masses, the mass segregation parameter, the exponent of the BH mass function, and $M_{\rm SMBH} \in [10^5 \, M_{\odot}, 10^7 \, M_{\odot}]$. Larger values correspond to higher $M_{\rm SMBH}$. Similarly, for a single-mass BH population, we find that the fraction of sources with a residual eccentricity is higher: $F_{ {\rm res} > 0.1 } \geqslant 96 \%$, as shown in Table \ref{tab:FracM_SMBH}.
\begin{table}
\caption{ \label{tab:FracM_over100Hz} The fraction of EBBHs forming with $f_{\rm GW} \gtrsim 100 \, \mathrm{Hz}$ for various BH PMFs and component masses ($M_{\rm tot} \lesssim 23 \, M_{\odot}$, see Section \ref{subsec:DistFGW10Hz} for details). Monte Carlo results shown here correspond to those in Tables \ref{tab:FracM_SMBH}, \ref{tab:FracM_SMBH_2}, and \ref{tab:FracM_SMBH_Belcz}. }
\centering
\begin{tabular}{@{}ccccc}
\hline
$m_A - m_B \, \, [M_{\odot}]$ & $M_{\rm SMBH} \, \, [M_{\odot}]$ & Table \ref{tab:FracM_SMBH} &
Table \ref{tab:FracM_SMBH_2} & Table \ref{tab:FracM_SMBH_Belcz} \\
\hline\hline
$3 - 3$ & $10^5$ & - & - & $7 \%$ \\
$5 - 5$ & $10^5$ & $4 \%$ & $4 \%$ & $4 \%$ \\
$10 - 10$ & $10^5$ & $0.5 \%$ & $0.4 \%$ & $0.5 \%$ \\
$5 - 15$ & $10^5$ & $0.6 \%$ & $0.5 \%$ & $0.9 \%$ \\
$3 - 3$ & $10^6$ & - & - & $10 \%$ \\
$5 - 5$ & $10^6$ & $5 \%$ & $6 \%$ & $6 \%$ \\
$10 - 10$ & $10^6$ & $0.9 \%$ & $1 \%$ & $0.9 \%$ \\
$5 - 15$ & $10^6$ & $0.9 \%$ & $0.7 \%$ & $1.2 \%$ \\
$3 - 3$ & $M_{\mathrm{SgrA}^*}$ & - & - & $12 \%$ \\
$5 - 5$ & $M_{\mathrm{SgrA}^*}$ & $6 \%$ & - & $8 \%$ \\
$10 - 10$ & $M_{\mathrm{SgrA}^*}$ & $1 \%$ & - & $1 \%$ \\
$5 - 15$ & $M_{\mathrm{SgrA}^*}$ & $1 \%$ & - & $1.4 \%$ \\
$3 - 3$ & $10^7$ & - & - & $15 \%$ \\
$5 - 5$ & $10^7$ & $7 \%$ & $8 \%$ & $9 \%$ \\
$10 - 10$ & $10^7$ & $1.2 \%$ & $1.2 \%$ & $1.2 \%$ \\
$5 - 15$ & $10^7$ & $1.1 \%$ & $1.3 \%$ & $1.8 \%$ \\
\hline\hline
\end{tabular}
\end{table}
\subsection{Eccentricity Distribution at the Last Stable Orbit}
\label{subsec:DistLSO}
\begin{figure}
\centering
\includegraphics[width=85mm]{Dist_eLSO_standard}
\includegraphics[width=85mm]{Dist_eLSO_Param}
\includegraphics[width=85mm]{Dist_eLSO_mSMBH}
\caption{ \label{fig:DisteLSO} Top panel: the distribution of eccentricity at the LSO ($e_{\rm LSO}$, cf. Figures \ref{fig:Distrhop0} and \ref{fig:Dist_fp10} for the initial eccentricity distribution and that at $10 \, \mathrm{Hz}$). Different lines correspond to different component masses in a multi-mass BH population and in a single-mass BH population as in Figure \ref{fig:Distrhop0}. In a multi-mass BH population, more massive EBBHs have systematically higher $e_{\rm LSO}$, and EBBHs forming in a single-mass BH populations have higher $e_{\rm LSO}$ than similar EBBHs forming in a multi-mass BH population. Middle panel: the influence of different assumptions on the results. We vary the mass function exponent $\beta$, the maximum mass of the multi-mass BH population $m_\mathrm{BH,max}$, and the mass segregation parameter $p_0$. We find that only $m_{\rm BH,max}$ influences the distribution of $e_{\rm LSO}$ significantly. Bottom Panel: the $M_{\rm SMBH}$ dependence of $e_{\rm LSO}$ distribution (cf. Figure \ref{fig:SMBHdeprhopaeps0} for $e_0$). }
\end{figure}
Let us now examine the distribution of the final eccentricity at LSO shown in Figure \ref{fig:DisteLSO}. These numerical results show that (i) higher $e_{\rm LSO}$ values correspond to higher-mass BHs in the distribution; (ii) the PDF of $e_{\rm LSO}$ is affected significantly by $m_\mathrm{BH,max}$, but it is not influenced significantly by the mass distribution exponent or mass-segregation parameters $\beta$ and $p_0$; (iii) the $M_{\rm tot}$ influences the result more significantly than the mass ratio, a lower $q$ shifts the $e_{\rm LSO}$ distribution to slightly lower values, as shown; (iv) EBBHs form with higher $e_{\rm LSO}$ around more massive SMBHs (Tables \ref{tab:FracM_SMBH}, \ref{tab:FracM_SMBH_2}, and \ref{tab:FracM_SMBH_Belcz}); and (v) EBBHs forming in a single-mass BH populations have an $e_{\rm LSO}$ distribution similar to that of the most massive EBBHs in a multi-mass BH population. Because $e_{\rm LSO}$ is a decreasing function of $\rho_{\mathrm{p}0}$ (Equation \ref{eq:eLSOrhop0}), these results may be explained respectively by arguments for the distribution of $\rho_{\mathrm{p}0}$ listed in \ref{i:A}, \ref{i:B}, \ref{i:C}, \ref{i:D}, and \ref{i:E} above.
The characteristic scale of $e_{\mathrm{LSO}}$ may be understood using the characteristic value for $\rho_{\mathrm{p}0}$ given by Equation (\ref{eq:rhop0uni}) and the relation between $\rho_{\mathrm{p}0}$ and $e_{\mathrm{LSO}}$ given by Equation (\ref{eq:eLSOrhop0}). For mergers at a fixed radius $r$, the distribution of $\rho_{\mathrm{p}0}$ is flat for $\rho_{\mathrm{p}0} \leqslant \rho_{\mathrm{p}0,\mathrm{uni}}$, implying that $P(e_{\rm LSO}) \propto e_{\rm LSO}^{-8/5}$ for $e_{\mathrm{LSO}} \gtrsim e_{\mathrm{LSO,peak}}$, where
\begin{align} \label{eq:eLSOuni}
e_\mathrm{LSO,peak} \approx & 0.2682 \rho_{{\rm p}0,\rm uni}^{8/5}
\nonumber\\ =&
0.0252 \left( 4 \eta \right)^{-16/35}
\left( \frac{v_{\max}}{1,000\,\rm{km/s}}\right)^{32/35}
\nonumber\\ =&
0.2703 \left( 4\eta \right)^{-16/35}
\left( \frac{r}{1000\,M_{\rm SMBH}} \right)^{-16/35} \, .
\end{align}
The distribution on a logarithmic scale follows $P(\ln e_{\mathrm{LSO}}) = e_{\mathrm{LSO}} P(e_{\mathrm{LSO}}) \propto e_{\mathrm{LSO}}^{-3/5}$ for values $e_{\mathrm{LSO}} \gtrsim e_{\mathrm{LSO,peak}}$. The characteristic values of $e_{\mathrm{LSO}}$ may be determined for GW capture events with different masses by combining Equations (\ref{eq:eLSOrhop0}) and (\ref{eq:rhop0uni}). For the most massive EBBHs, we find that roughly $50 \%$ of them are within $r \sim 4 \times 10^{-3} \, \mathrm{pc}$ in a Milky Way-size nucleus, which corresponds to $2000 \, M_{\rm SgrA*}$, where $e_{\mathrm{LSO, peak}}$ is $\sim 0.19$. However, for the least massive EBBHs with $m_A = m_B = 5 \, M_{\odot}$, $50 \%$ of them are within $1.2 \, \mathrm{pc} \sim 5.8 \times 10^6 \, M_{\rm SgrA*}$ for which $e_{\mathrm{LSO,peak}}\sim 0.005$. Thus, the peak of the $e_{\rm LSO}$ distribution is high for heavy members of a given population because they are in the close vicinity of the SMBH, while the low-mass members have a much lower $e_{\rm LSO}$ because they are typically much farther out. Thus, the measurement of the $e_{\rm LSO}$ distribution for different masses may be used to detect the mass segregation of the sources within their host environment.
Conversely, solving Equation (\ref{eq:eLSOuni}) for the velocity dispersion $\sigma \sim v_{\rm max}/\sqrt{2}$ gives
\begin{equation}\label{eq:sigma-eLSO}
\sigma \sim 258\,\frac{\rm km}{\rm s}\,(4\eta)^{1/2} \left(\frac{e_{\rm LSO,peak}}{0.01}\right)^{35/32} \, .
\end{equation}
Thus, the measurement of the peak eccentricity of a GW capture binary at LSO gives an estimate of the typical velocity dispersion of the source environment.
In all of our simulations, we have ignored those encounters in which BHs undergo a direct head-on collision without forming an EBBH (i.e. $\rho_{{\rm p}0} \geqslant 8$ in all cases, see Figure \ref{fig:Distrhop0}). This causes the PDFs of $e_{\rm LSO}$ to cut off at $e_{\rm LSO} \sim 0.23$; see the top panel of Figure \ref{fig:DisteLSO}.
By examining the $e_{\rm LSO}$ distribution for various SMBH mass, BH mass, and BH population parameters, we find that approximately $\lesssim 71 \%$, $\lesssim 14 \%$, and $\lesssim 0.2\%$ of the most massive EBBHs have eccentricities lower than $0.1$, $10^{-2}$, and $10^{-3}$ at LSO, respectively, for the considered ranges of SMBH mass and BH population parameters. The corresponding respective numbers for the lowest-mass EBBHs are $\lesssim 97 \%$, $\lesssim 66 \%$, and $\lesssim 4 \%$. For the \citet{Belczynskietal2014_2} PMF, the corresponding numbers for the highest-mass EBBHS are $\lesssim 69 \%$, $\lesssim 13 \%$, and $\lesssim 0.2 \%$, and for the lowest-mass EBBHs are $\lesssim 97 \%$, $\lesssim 65 \%$, and $\lesssim 5 \%$. In the case of a single-mass BH population, approximately $\lesssim 60 \%$, $\lesssim 16 \%$, and $\lesssim 0.24 \%$ of EBBHs have eccentricities lower than $0.1$, $10^{-2}$, and $10^{-3}$ at LSO, respectively, for SMBH masses between $10^5 \, M_{\odot}$ and $10^7 \, M_{\odot}$.\footnote{Note that these results correspond to $10^5 \, M_{\odot}$ SMBHs because EBBHs with lowest $e_{\rm LSO}$ values form in these SMBHs; see Figure \ref{fig:DisteLSO}. The results are insensitive to the upper bound of the SMBH masses.} Overall, we find that a negligible fraction of EBBHs have eccentricities lower than $10^{-3}$ at LSO over the considered ranges of BH mass, SMBH mass, and BH population parameter ranges. Note that, for initially highly eccentric precessing BH binaries, the expected relative measurement accuracy of $e_{\rm LSO}$ is $\lesssim 5 \%$ for $30 \, M_{\odot} - 30 \, M_{\odot}$ ($10 \, M_{\odot} - 10 \, M_{\odot}$) precessing EBBHs with $\rho_{\mathrm{p}0} \leqslant 50$ ($\rho_{\mathrm{p}0} \leqslant 100$) for the aLIGO-AdV-KAGRA detector network \citep{Gondanetal2017}. This implies that the predicted range of $e_{\rm LSO}$ may typically be distinguished from zero in future GW detections.
Similarly to Section \ref{subsec:DistIniParam}, we find that the PDF of $e_{\rm LSO}$ is qualitatively similar for the \citet{Belczynskietal2014_2} PMF, and results presented in Table \ref{tab:FracM_SMBH} are very similar to those presented in Table \ref{tab:FracM_SMBH_Belcz} for similar EBBHs.
Note that the distribution of $e_\mathrm{LSO}$ has been previously calculated in \citet{OLearyetal2009} for a single-mass population. We find two main differences with respect to those results. First, the PDF peaks at a slightly higher value ($\sim 0.2$ for a single-mass distribution, instead of $\sim 0.1$ as in \citealt{OLearyetal2009}), and it drops off quickly beyond the peak instead of taking values at higher eccentricities. The somewhat higher $e_{\rm LSO}$ arises due to the fact that the minimum radius at which the formation of EBBHs is still considered is about $4.5$ times lower than that in \citet{OLearyetal2009}; this implies higher $w$, and lower $\rho_{\rm p0}$ and consequently larger $e_{\rm LSO}$. Note that we assumed a slightly greater massive SMBH mass than \citet{OLearyetal2009}, which also increases the values of $e_{\rm LSO}$. In our case, the PDF of $e_{\rm LSO}$ drops off quickly beyond the peak of the distribution at $e_{\rm LSO} \sim 0.23$ because we have ignored those encounters for which BHs coalesce before EBBHs form (i.e. $\rho_{\mathrm{p}0} \leqslant 8$ in all cases; see Sections \ref{sec:PIOP} and \ref{subsec:DistIniParam} for details), while those binaries are included in the plotted $e_{\rm LSO}$ distributions in \citet{OLearyetal2009}. However, in a multi-mass distribution, the peak $e_{\mathrm{LSO}}$ values are even lower than in \citet{OLearyetal2009}, as shown in Figure \ref{fig:DisteLSO}.
\begin{figure*}
\centering
\includegraphics[width=85mm]{MergRate_max30beta1}
\includegraphics[width=85mm]{aLIGOEvRateDist_beta1}
\\
\includegraphics[width=85mm]{MergRate_max30beta2p35}
\includegraphics[width=85mm]{aLIGOEvRateDist_beta2p35}
\\
\includegraphics[width=85mm]{MergRate_max30beta3}
\includegraphics[width=85mm]{aLIGOEvRateDist_beta3}
\caption{ \label{fig:NoPMFDist} Left panels show the two-dimensional PDF of merger rates of BH binaries that form due to GW emission in a single nuclear star cluster, as a function of binary mass ratio and total mass. Right panels show the PDF of detection rates for Advanced LIGO at design sensitivity, which includes the mergers from all detectable galaxies in the universe. Different rows of panels correspond to different BH PMFs $dN/dm_{\rm BH}=m_{\rm BH}^{-\beta}$, where $\beta = 1$ (top), 2.35 (middle), and 3 (bottom), respectively. The sharp boundaries are due to the assumption $5 \, M_{\odot} \leqslant m_{\rm BH} \leqslant 30 \, M_{\odot}$ for both BHs forming the binary. The segregation parameter is $p_0 = 0.5$. For different maximum BH mass or $p_0$ assumptions (not shown) the merger rate distribution in a single cluster (left panel) is changed only through a rescaling of the $M_{\rm tot}$ axis (see Equation \ref{eq:P(M,q)}). The merger rates in a single nuclear star cluster do not depend significantly on the SMBH mass.}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=85mm]{MergRate_Belcz_Milky}
\includegraphics[width=85mm]{EvRate_Belcz_Milky}
\caption{ \label{fig:MergRate_BelczModel} Same as Figure \ref{fig:NoPMFDist}, but for the \citet{Belczynskietal2014_2} PMF. The left panel does not depend significantly on the SMBH mass.}
\end{figure*}
\subsection{Total Mass and Mass Ratio Distributions}
\label{subsec:CompMassDepFormRate}
Let us now examine the two-dimensional distribution of the merger rate in the $q$--$M_{\rm tot}$ plane, $\left\langle \partial^2 \Gamma/ \partial M_{\rm tot} \partial q \right\rangle$, which affects the mass-dependent detection rates. This is derived by integrating $ \partial^3 \Gamma / \partial r \partial m_A\partial m_B$ (Equation \ref{eq:radiusPDF}) over the allowed range of $r$, from $r_{\rm min} ^{A,B}$ to $r_{\rm max}$, for each fixed $m_A$ and $m_B$ component mass. In Appendix \ref{sec:MMDGN}, we show that this leads to an analytical expression\footnote{This expression includes GW captures and direct head-on collisions. See Equation (\ref{eq:Ratemulti-mass}) for the case where head-on collision are ignored.}. Up to a constant normalization factor,
\begin{align}
P(m_A,m_B) \approx & \eta^{\frac{2}{7}-\beta} M_{\rm tot}^{2-2\beta}
\left(1 - \frac{2}{3} \overline{M}_{\rm tot} + \frac{4 \eta}{9} \overline{M}_{\rm tot}^2 \right)
\frac{ 1 - \Lambda^{\overline{M}_{\rm tot}-\frac{11}{14}} }{\frac{11}{14}
-\overline{M}_{\rm tot}}
\end{align}
if $m_{\rm BH,min} \leqslant m_{A,B} \leqslant m_{\rm BH,max}$, and zero otherwise, where we introduced the Coulomb logarithm\footnote{In the following expressions, we ignore the possible mass dependence of $\Lambda$.}
\begin{equation} \label{eq:CoulombLog}
\ln \Lambda=\ln\frac{r_{\max}}{r_{\min}^{A,B}} \, ,
\end{equation}
and $\overline{M}_{\rm tot}$ for the dimensionless total mass
\begin{equation}
\overline{M}_{\rm tot}=p_0 \frac{M_{\rm tot}}{m_{\rm BH,max}} \, ,
\end{equation}
where $p_0\sim 0.5$, for which $m_{\rm BH, min}/m_{\rm BH, max} \leqslant \overline{M}_{\rm tot} \leqslant 1$.
The likelihood\footnote{i.e., conditional probability} of merger for any two given BHs in the cluster, with mass $m_A$ and $m_B$ is proportional to $P(m_A,m_B)$ divided by the prior probability proportional to $\mathcal{F}(m_A) \mathcal{F}(m_B)
\propto m_A^{-\beta} m_B^{-\beta} = \eta^{-\beta}M_{\rm tot}^{-2\beta}$, which gives
\begin{equation}
\mathcal{L}(m_A,m_B) \approx \eta^{\frac{2}{7}} M_{\rm tot}^{2} \left(1 - \frac{2}{3}
\overline{M}_{\rm tot} + \frac{4 \eta}{9} \overline{M}_{\rm tot}^2 \right)
\frac{ 1 - \Lambda^{\overline{M}_{\rm tot}-\frac{11}{14}} }{\frac{11}{14}
-\overline{M}_{\rm tot}}
\end{equation}
if $m_{\rm BH,min} \leqslant m_{A,B} \leqslant m_{\rm BH,max}$, and zero otherwise.
Recently \citet{Kocsisetal2017} have introduced a useful indicator\footnote{not to be confused with the number density exponent $\alpha$ in Equation (\ref{eq:alpham})} to distinguish different source populations:
\begin{align} \label{eq:alpha}
\alpha & = -M_{\rm tot}^2 \frac{\partial^2 \ln P(m_A,m_B)}{\partial m_A \partial m_B}
\nonumber \\
& = \frac{10}{7} - \frac{\overline{M}_{\rm tot}^2}{\left(
\overline{M}_{\rm tot}-\frac{11}{14}\right)^2 } + \frac{ \overline{M}_{\rm tot}^2
\Lambda^{\overline{M}_{\rm tot}-\frac{11}{14}} \ln^2 \Lambda}{
\left[ 1 - \Lambda^{\overline{M}_{\rm tot}-\frac{11}{14}}\right]^2}
\end{align}
if $m_{\rm BH,min} \leqslant m_{A,B} \leqslant m_{\rm BH,max}$, and zero otherwise. The $\alpha$ parameter is universal, in that it is independent of the underlying BH mass function. This is a monotonically decreasing function of $\overline{M}_{\rm tot}$ and ranges between $1.43 \geqslant \alpha \geqslant -20.35 + (\Lambda^{-3/28} - \Lambda^{3/28})^{-2} \ln^2 \Lambda$ for $0 \leqslant \overline{M}_{\rm tot} \leqslant 1$. Assuming a Milky Way-size nucleus and a fiducial multi-mass BH population with $dN/dm_{\rm BH} \propto m_{\rm BH}^{-2.35}$, $m_\mathrm{BH,max} = 30 \, M_{\odot}$, and $p_0=0.5$, $\ln \Lambda$ is equal to $10.71$ and $10.17$ for the lightest and heaviest binaries, ergo $\alpha$ ranges between $-5.39 \la \alpha \la 1.36$ from the heaviest to lightest components of the EBBH population. The value of $\alpha$ is different for other source populations: $\alpha \sim 4$ for BH mergers in GCs \citep{OLearyetal2016}, $\alpha = 1$ for primordial BH binaries formed in the early universe \citep{Kocsisetal2017}, and $\alpha = 10/7 = 1.43$ for primordial BHs formed in dark matter halos through GW capture\footnote{Dark matter halos are collisionless systems, where mass segregation does not take place, so $\overline{M}_{\rm tot}=0$ in Equation (\ref{eq:alpha}).} \citep{Birdetal2016}.
Next, we change variables from $m_A$ and $m_B$ to $q$ and $M_{\rm tot}$ using the Jacobi determinant $ \partial^3 \Gamma / \partial r\partial M_{\rm tot}\partial q = J \partial^3 \Gamma / \partial r \partial m_A\partial m_B$, where $J = M_{\rm tot}(1+q)^{-2}$ (see Equation (\ref{eq:mergrate_q_Mtot}) in Appendix \ref{sec:MMDGN})
\begin{align} \label{eq:P(M,q)}
P(M_{\rm tot},q) & \approx q^{\frac{2}{7}-\beta} M_{\rm tot} ^{3-2\beta}
\left(1 - \frac{2}{3} \overline{M}_{\rm tot} + \frac{4 \, q }{9 \, (1+q^2)} \overline{M}_{\rm tot}^2 \right)
\nonumber \\
& \times \frac{ (1+q)^{2\beta - \frac{18}{7}} }{ \left(\frac{11}{14}-\overline{M}_{\rm tot}\right) }
\left[ 1 - \left(\frac{r_{\min}^{A,B}}{r_{\max}}\right)^{\frac{11}{14}-\overline{M}_{\rm tot}} \right]
\end{align}
if $m_{\rm BH,min} \leqslant M_{\rm tot}/(1+q) \leqslant m_{\rm BH,max}$ and $m_{\rm BH,min} \leqslant M_{\rm tot} q/(1+q) \leqslant m_{\rm BH,max}$, and zero otherwise. These analytical results correspond to the distribution of all GW capture events, including direct collisions. The latter represents a small \mbox{($ \la 10 \%$)} fraction of the mergers. In the following, we plot the probability distribution functions for EBBHs that avoid a direct collision.
The distribution of total detection rates from all galaxies is generally different from the merger rate density or the merger rate for a single nuclear star cluster, because the detection horizon $d_{\rm aLIGO}$ is different for different binary parameters. We calculate $d_{\rm aLIGO}$ for an eccentric inspiral waveform by calculating the signal-to-noise ratio following \citet{OLearyetal2009} \citep[see also][]{Gondanetal2017}. The detection rate distribution is then
\begin{align}
\frac{\partial^2 \mathcal{R}_{\rm aLIGO}}{\partial m_A \partial m_B}
& =
\int_{\rho_\mathrm{p0,min}}^{\rho_\mathrm{p0,max}} d\rho_\mathrm{p0} \int _{M_\mathrm{SMBH,min}}
^{M_\mathrm{SMBH,max}} dM_{\rm SMBH} V_{\rm aLIGO} \frac{d n_{\rm gal}}{dM_{\rm SMBH}}
\nonumber \\
& \times \left\langle \frac{\partial^3 \Gamma}{ \partial \rho_\mathrm{p0} \partial m_A
\partial m_B} \right \rangle \, ,
\end{align}
where $d n_{\rm gal}/dM_{\rm SMBH}$ is the number density of galaxies in the universe with nuclear star clusters that host an SMBH of mass $M_{\rm SMBH}$, and $M_\mathrm{SMBH,min}$ and $M_\mathrm{SMBH,max}$ are the lower and upper bounds of the SMBH mass range of interest (Section \ref{subsec:GNModels}). The detectable volume is $V_{\rm aLIGO} = \frac43 \pi d_{\rm aLIGO}^3$, if $d_{\rm aLIGO}$ is much less than the Hubble scale. We substitute Equation (\ref{eq:radiusPDF}) for the partial event rates for different $\rho_\mathrm{p0}$, and change variables from $m_A$ and $m_B$ to $q$ and $M_{\rm tot}$ by using the Jacobian as in Equation (\ref{eq:P(M,q)}).
\begin{figure*}
\centering
\includegraphics[width=85mm]{Chp_Dist_Milky}
\includegraphics[width=85mm]{ChpDist_Belcz}
\\
\includegraphics[width=85mm]{MassRatio_Standard}
\includegraphics[width=85mm]{MassRatioDist_Belcz}
\\
\includegraphics[width=85mm]{MtotDist_smbhMilky}
\includegraphics[width=85mm]{TotMassDist_Belcz}
\caption{ \label{fig:MassFuncs_PowerLaw_Belcz} Left panels: the PDF of chirp mass (top panel), mass ratio (middle panel), and total mass (bottom panel) for merging EBBHs in a single Milky Way-size nucleus. We show how different parameters of a multi-mass BH population influence the PDFs. Solid thick lines correspond to our fiducial multi-mass BH population with a PMF $dN/dm_{\rm BH} \propto m_{\rm BH}^{-\beta}$ with $\beta=2.35$, $p_0=0.5$, and $m_\mathrm{BH,max} = 30 \, M_{\odot}$. Other line styles show the distributions for other parameters as labeled in the legend of the top panel. Right panels: the PDF of chirp mass (top panel), mass ratio (middle panel), and total mass (bottom panel) for merging EBBHs in a single Milky Way-sized nucleus. Solid lines correspond to a multi-mass BH population with the \citet{Belczynskietal2014_2} PMF and $p_0 = 0.5$ (labeled as B04 in the legend). Dashed-dotted lines correspond to a multi-mass BH population with a PMF $dN/dm_{\rm BH} \propto m_{\rm BH}^{-\beta}$ with $\beta=1, \, 2.35, \, 3$, $p_0=0.5$, and the BH mass range corresponds to the \citet{Belczynskietal2014_2} PMF. For all panels the results do not depend significantly on the assumed SMBH mass. }
\end{figure*}
Figures \ref{fig:NoPMFDist} and \ref{fig:MergRate_BelczModel} show the two-dimensional PDF of merging EBBH event rates as a function of $q$ and $M_{\rm tot}$ for power-law BH PMFs and for the \citet{Belczynskietal2014_2} PMF, respectively. The left panels show the distribution in a single Milky Way-size nucleus, and the right panels show the distribution for all galaxies in the universe that are detectable by aLIGO at design sensitivity. The top, middle, and bottom panels correspond to different BH PMF exponents $\beta=1$, $2.35$, and $3$, respectively, where the range of BH masses is assumed to be between $5 \, M_{\odot} \leqslant m_{\rm BH} \leqslant 30 \, M_{\odot}$. Because the two-dimensional PDF of merger rate in a single GN does not depend significantly on the assumed SMBH mass (Appendix \ref{sec:MMDGN}), the left panels display the distribution of the merger rate density for single GNs in the universe. Low-mass binaries dominate the merger rate density if $\beta \gtrsim 3/2$ (see Equation \ref{eq:P(M,q)}) and high-mass binaries close to $M_{\rm tot} \sim 2m_{\rm BH, max}$ dominate the merger rate density for a top-heavy mass function with $\beta=1$. However, because aLIGO is more sensitive to higher-mass mergers with mass ratios closer to 1\footnote{That is, the maximum luminosity distance of detection increases with $M_{\rm tot}$ and $q$ over the considered ranges of $M_{\rm tot}$ and $\rho_{\rm p0}$, see Figure 11 in \citet{OLearyetal2009} or Appendix \ref{sec:aLIGOeventrate}}, the detection rate distributions shown in the right panels are skewed to higher total masses and equal mass ratios. This observational bias causes the detection rates to be highest at $M_{\rm tot} \gtrsim m_{\rm BH, max}$ up to $2 m_{\rm BH, max}$ for $\beta =1$ and $2.35$. For $\beta= 3$, the distribution is peaked near $M_{\rm tot} \sim m_{\rm BH, max}$ with unequal mass ratios. In all cases, equal mass mergers at the low end of the mass distribution $M_{\rm tot} \sim 2 m_{\rm BH, min}$ are highly disfavored for $\beta \leqslant 3$. Note that the mass ratio dependence of the detection rate for fixed $M_{\rm tot}$ is known analytically because $d_{\rm aLIGO} \propto \eta^{1/2} = q^{1/2}(1+q)^{-1}$; see Equation (\ref{eq:dLmax}) or \citet{OLearyetal2009}. Thus, $\partial^2 \mathcal{R}_{\rm aLIGO}/ \partial q \partial M_{\rm tot} \propto q^{(25/14) - \beta} (1+q)^{2\beta - (39/7) }$.
\begin{figure}
\centering
\includegraphics[width=85mm]{ChirpMassDist_aLIGODetRate}
\includegraphics[width=85mm]{MassRatioDist_aLIGODetRate}
\includegraphics[width=85mm]{TotalMassDist_aLIGODetRate}
\caption{ \label{fig:MassDists_DetRate} The 1D marginalized distribution of the merger rate for all galaxies in the universe detectable by aLIGO at design sensitivity, as a function of chirp mass (top panel), mass ratio (middle panel), and binary total mass, respectively. Here, we assumed a fiducial multi-mass BH population with PMFs $dN/dm_{\rm BH} \propto m_{\rm BH}^{-\beta}$, where $\beta = 1,\, 2.35,\, 3$ as labeled in the legend, and $m_\mathrm{BH,max} = 30 \, M_{\odot}$, and $p_0 =0.5$. Furthermore, we also considered the \citet{Belczynskietal2014_2} PMF with $p_0 =0.5$ (black curve). }
\end{figure}
The different rows of panels in Figure \ref{fig:MassFuncs_PowerLaw_Belcz} show the 1D marginalized distribution of the merger rate in a single galaxy in a Milky Way-size nucleus as a function of chirp mass, binary total mass, and mass ratio, respectively. The left and right panels show results for multi-mass BH populations with power-law PMFs and \citet{Belczynskietal2014_2} PMF. The mass ratio distribution of the merger rate density is widely distributed, with comparable component-mass mergers being a factor of $\sim 2$ more likely than mergers with uneven component masses when $\beta \sim 1$, while uneven mass mergers are $20\%$ more likely than comparable mass mergers for $\beta \sim 2.35$. These panels also show how the marginalized merger rate density depends on other model parameters as follows. Varying $m_{\rm BH, \max}$ changes the boundaries of the $P(q,M_{\rm tot})$ distributions. However, Equation (\ref{eq:P(M,q)}) shows that $P(q, M_{\rm tot})$ is independent of $m_{\rm BH, \max}$ and the value of the mass segregation parameter $p_0$, other than for a rescaling of the $M_{\rm tot}$ mass units. The mass distributions are also insensitive to $M_{\rm SMBH}$ (Appendix \ref{sec:MMDGN}).
Finally, Figure \ref{fig:MassDists_DetRate} shows examples for the 1D marginalized distribution of the merger rate for all galaxies in the universe that are detectable by aLIGO at design sensitivity, as a function of chirp mass, binary total mass, and mass ratio, respectively. We used a fiducial multi-mass BH population with PMFs $dN/dm_{\rm BH} \propto m_{\rm BH}^{-\beta}$, where $\beta = 1,\, 2.35,\, 3$; $m_\mathrm{BH,max} = 30 \, M_{\odot}$; and $p_0 =0.5$. The black line shows results for the \citet{Belczynskietal2014_2} PMF with $p_0 =0.5$. The PDFs presented in these panels are skewed to higher total masses and equal mass ratios, compared to those presented in Figure \ref{fig:MassFuncs_PowerLaw_Belcz} because aLIGO is more sensitive to higher-mass mergers with higher mass ratios.
\section{Summary and Conclusion}
\label{sec:Summary}
In this paper, we determined the expected distribution of intrinsic parameters of merging BH binaries that form in GN from initially unbound BHs due to GW emission during close encounters. The main intrinsic parameters describing these binaries are their component masses, the initial impact parameter, and initial relative velocity before the encounter. Due to the large velocity dispersion in GN compared to objects in GCs, the galactic disk, or the halo, the objects in GNs must approach one another at much smaller distances to form a binary. The implication is that the initial pericenter distance of the binary is relatively small when the initial eccentricity is beyond $e_0>0.9$. We extended the study of \citet{OLearyetal2009} by calculating the eccentricity distribution for mergers among different component masses in a multi-mass distribution, and examined the eccentricity distribution at three particular places during the evolution: at formation, at the instant when the peak GW frequency reaches $10 \, \mathrm{Hz}$, and at the LSO. We have also examined how different parameters describing a multi-mass BH distribution affect the distribution of merger rates as a function of masses, eccentricity, and initial pericenter distance. Many of our numerical results and the identified trends are validated by analytical expressions (Appendices \ref{TWORPEPS}--\ref{ED}).
The main findings of this work can be summarized as follows:
\begin{enumerate}[label=(\roman*),ref=(\roman*)]
\item We identified the region in the initial eccentricity--initial dimensionless pericenter distance plane where EBBHs may form through GW capture, and determined the two-dimensional PDF therein. The PDF of initial orbital parameters (Figure \ref{fig:Distrhop0}) shows that EBBHS form with high eccentricities ($0.9 \lesssim e_0 \lesssim 0.9999$), and the distribution of initial pericenter distance drops off quickly beyond $20-80 \, M_{\rm tot}$. The reason is that BHs form EBBHs with high relative velocities in nuclear star clusters around SMBHs (Figure \ref{fig:w_Avr_mSMBH}). The eccentricity makes GW capture sources in these environments qualitatively different from other types of inspiraling GW sources.
\item We determined the radial distribution from the central SMBH for GW capture sources with different component masses. We found that, on a logarithmic radial scale, the merger rate distribution is approximately independent of radius for the most massive components, $P(\ln r) \propto r^{-3/14}$ (Equation \ref{eq:formrateBHBH_multi}), between the radius of influence and the inner boundary set by GW-driven infall into the SMBH (i.e. $\sim 3 \, \mathrm{pc}$ and of order $10^{-4}\, \mathrm{pc}$, respectively, in Milky-Way type galaxies). This implies that roughly $50 \%$ of the high-mass events are formed in the innermost $4 \times 10^{-3} \, {\rm pc}$, and approximately $97 \%$ within $1.2 \, {\rm pc}$. The rate of GW capture sources among the least massive objects within the nuclear star cluster is skewed to higher radii on a logarithmic scale, $P(\ln r) \propto r^{11/14}$, for $m_{\rm BH} \ll m_{\mathrm{BH,max}}$ (Equation \ref{eq:formrateBHBH_multi}). These events are preferentially closer to the radius of influence; only $2 \%$ and roughly $50 \%$ of them are within $4 \times 10^{-3} \, {\rm pc}$ and $1.2 \, {\rm pc}$, respectively.
\item We found that a negligible fraction ($1 \% \la$) of EBBHs interact with a third object during the eccentric inspiral.
\item We determined how the distributions of orbital parameters depend on the mass of the central SMBH, on the binary masses, and on the parameters of the BH population (i.e. the highest possible BH mass, the exponent of the BH mass function, and the mass segregation parameter). We found that the initial pericenter distance systematically decreases with $M_{\rm SMBH}$ and the component masses, while the eccentricity at any stage of the time evolution systematically increases with $M_{\rm SMBH}$ and the component masses. As a consequence, more massive EBBHs around more massive SMBHs reach a $10 \, {\rm Hz}$ frequency limit with higher eccentricity, and they have higher eccentricity at the LSO (see Tables \ref{tab:FracM_SMBH}, \ref{tab:FracM_SMBH_2}, and \ref{tab:FracM_SMBH_Belcz}). The highest possible BH mass of the BH population affects the binary parameters of GW capture sources. Other parameters of the BH population do not influence the eccentricity distribution at formation, at $10\, \mathrm{Hz}$, or at the LSO.
\item We found that almost all EBBHs form below the frequency range where aLIGO type detectors are the most sensitive, and so GW capture sources typically enter the aLIGO type detectors' sensitive frequency band for stellar mass BHs (Table \ref{tab:FracM_over100Hz}).
\item We found that most of the sources become circularized with eccentricities $e_{\mathrm{LSO}} \sim 1 - 10 \%$ when they approach the LSO. The fraction of sources with $e_{\rm LSO} > 0.1$ is $3 - 30 \%$ if the maximum possible BH mass in the population is $30 \, M_{\odot}$: this fraction of eccentric mergers among BHs with masses less than $30 \, M_{\odot}$ is less than $10\%$ if the highest mass in the population reaches $80 \, M_{\odot}$ (Table \ref{tab:FracM_SMBH_2}). Because the current GW detectors are only sensitive to the final part of the signal, the currently known six detections cannot constrain the fraction of sources formed in this initially highly eccentric merger channel. Only 1 in 30 detections with $e_{\rm LSO} > 0.1$ may be expected if the heaviest stellar BH mass in GNs is $80 \, M_{\odot}$. The heaviest BHs in such a population are expected to have higher eccentricities; $29 - 36 \%$ of them have $e_{\rm LSO}>0.1$, but their detection may require algorithms sensitive to non-circular orbit waveform features.
\item The prospects for detecting eccentric mergers are expected to improve significantly with the ongoing detector upgrades. We found that the fraction of sources with an eccentricity larger than 0.1 when the peak GW frequency reaches $10\, \mathrm{Hz}$ is $84 - 90\%$ ($43 - 69\%$) for $30\, M_{\odot} - 30 \, M_{\odot}$ binaries in a BH population with maximum mass of $30M_{\odot}$ ($80M_{\odot}$). For lower-mass binaries, these numbers are higher, e.g. around $68 - 94\%$ of $10 \, M_{\odot} - 10 \, M_{\odot}$ binaries (Tables \ref{tab:FracM_SMBH}, \ref{tab:FracM_SMBH_2}, and \ref{tab:FracM_SMBH_Belcz}). Thus, when Advanced LIGO and VIRGO reach their design sensitivity at $10 \, \mathrm{Hz}$, more than half of GW capture sources in GN are initially eccentric in the detector band. The measurement of the eccentricity and mass distribution will make way to identify the fraction of sources formed in this channel.
\item A negligible fraction of EBBHs have eccentricities at LSO lower than $10^{-3}$ (Figure \ref{fig:DisteLSO}). In a companion paper, \citet{Gondanetal2017}, we have shown that the expected relative measurement accuracy of $e_{\rm LSO}$ for the aLIGO--AdV--KAGRA GW detector network is $\la 5 \%$ when EBBHs form with relatively small $\rho_{{\rm p}0}$ values at a distance of $100 \, \mathrm{Mpc}$\footnote{Eccentric BH binaries form through GW capture in GN with initial dimensionless pericenter distance $\rho_{{\rm p}0}=r_{\rm p0}/M_{\rm tot} \la 50$ ($\rho_{{\rm p}0} \la 100$) for masses $30 \, M_{\odot} -30 \, M_{\odot}$ ($10 \, M_{\odot} -10 \, M_{\odot}$); see Figure \ref{fig:Distrhop0}.}. Other binary formation and evolution channels typically produce binaries with eccentricities at or below $10^{-3}$ when their GW signals enter the aLIGO band \citep{Kowalskaetal2011,Cholisetal2016,Rodriguezetal2016a,Rodriguezetal2016b,SamsingRamirezRuiz2017,SilsbeeTremaine2017,RandallXianyu2018}. The measurement of eccentricity for a BH merger will be a smoking gun signature of sources in high velocity dispersion environments such as in the inner regions of GNs (Equation \ref{eq:sigma-eLSO}).
\item We conclude that Advanced LIGO/VIRGO detections may confirm or rule out this binary formation channel. However, algorithms using circular binary templates will be ineffective, for the majority of sources, in searching for GW signals of EBBHs forming through GW capture in GNs. This especially affects the heaviest BHs because they are the most eccentric among this source population. The current GW detection limits of the heaviest black holes \citep{Abbottetal2017_4,FishbachHolz2017} may be impacted by the inefficiency of the applied search algorithms to eccentric sources.
\item We determined the PDFs of merger rates in single nuclear star clusters in terms of component masses (i.e. total binary mass and mass ratio) and similarly for the detection rates for sources with the Advanced LIGO horizon (Figure \ref{fig:NoPMFDist}). The distributions are sensitive to the BH mass function exponent. The highest possible BH mass in GNs affects the total mass and mass ratio distribution just by a rescaling of the mass unit (Equation \ref{eq:P(M,q)}), but the mass distribution of merger rates are otherwise insensitive to the maximum BH mass in GNs. In particular, mass functions steeper/shallower than $dN/dm_{\rm BH} \propto m_{\rm BH}^{-1.5}$ lead the merger rate density to be dominated by the lowest/highest mass components in the cluster. However, the detection rate distribution is skewed toward higher masses due to Advanced LIGO/VIRGO's higher sensitivity to heavier BHs for mass function exponents shallower than $dN/dm_{\rm BH} \propto m_{\rm BH}^{-3}$.
\item We calculated the value of the universal dimensionless parameter $\alpha = -M_{\rm tot}^2 \partial^2 \ln \mathcal{R} /\partial m_A \partial m_B$, which characterizes the physical origin of a source population independently of the underlying BH mass function \citep{Kocsisetal2017}. We found that, for GW capture sources in a mass-segregated cusp, it follows a monotonically decreasing function of total binary mass (Equation \ref{eq:alpha}) ranging between $1.43 \geqslant \alpha \geqslant -20.35 + (\Lambda^{-3/28} - \Lambda^{3/28})^{-2} \ln^2 \Lambda$, where $\ln \Lambda$ is the Coulomb logarithm (Equation \ref{eq:CoulombLog}). For $\ln \Lambda\sim 10.5$, this parameter is approximately $\alpha \sim -5.39$ and $1.36$ for the heaviest and lightest components, respectively. This is very different from the $\alpha$ of other GW source populations because $\alpha \sim 4$ for BH mergers in GCs \citep{OLearyetal2016}, $\alpha=1$ for primordial BH binaries formed in the early universe \citep{Kocsisetal2017}, and $\alpha=1.43$ for BH binaries that form due to GW capture in otherwise dynamically collisionless systems, such as black holes in dark matter halos \citep{Birdetal2016}.
\item The mass ratio distribution is wide, for mass functions $dN/dm_{\rm BH} \propto m_{\rm BH}^{-1}$ to $dN/dm_{\rm BH} \propto m_{\rm BH}^{-3}$ and for the \citet{Belczynskietal2014_2} mass function (Figures \ref{fig:NoPMFDist} and \ref{fig:MergRate_BelczModel}). For these mass functions, the detection of equal-mass mergers are highly disfavored for low-mass BHs, but equal mass mergers are possible among high-mass BH sources. Unequal mass mergers are possible for a wider range of total masses (Figure \ref{fig:MassFuncs_PowerLaw_Belcz}).
\item Because GW capture sources are formed roughly independent of BH spins and spin directions at pericenter distances exceeding a few tens of $M_{\rm tot}$, the spin direction distribution for these sources is expected to be isotropic.
\end{enumerate}
We worked under the assumption of a relaxed cusp and assumed spherical symmetry. In the Milky Way's center, the distribution of massive young stars does not represent such a relaxed profile, but forms a clockwise disk structure with a number density distribution $n(r) \sim r^{-2.9}$, which is steeper than the model assumed in \citep{Bartkoetal2009,Yeldaetal2014}. The stellar distribution around SMBHs with masses $M_{\rm SMBH}>10^7M_{\odot}$ does not have time to relax within a Hubble time. If BHs in these regions form steeper radial profiles, EBBHs form closer to the center of the GN in higher-velocity dispersion regions. This implies that the eccentricity distribution of mergers is skewed toward higher eccentricities. However, these massive GN are expected to be subdominant, in terms of EBBH rates, because the rates per single GN approximately do not depend on the SMBH mass and low-mass SMBHs are more common (see \citet{Sijackietal2015} and references therein). However, the mass-segregated cusp may have a different radial dependence, even in equilibrium, than that assumed in this paper. \citet{Keshetetal2009} found a steeper number density dependence for the heavy objects if they are subdominant in mass. \citet{Fragioneetal2018} also found a steeper profile due to disrupted binaries. Recently \citet{Szolgyenetal2018} have shown that the heavy objects form a flattened distribution. The mass-segregated radial profile may be different in such configurations \citep{Fouvryetal2018}, leading to a modified EBBH merger rate distribution.
In conclusion, the measurement of the parameter distribution of merging compact objects may be useful to distinguish this formation mechanism from other mechanisms. Search methods targeting EBBH sources forming through GW capture in GN hosts may be optimized for the predicted distribution of binary parameters.
\section*{Acknowledgment}
We thank the anonymous referee for constructive comments, which helped improve the quality of the paper. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 Programme for Research and Innovation under grant agreement ERC-2014-STG No. 638435 (GalNUC), and from the Hungarian National Research, Development, and Innovation Office under grant NKFIH KH-125675. P\'eter Raffai was supported by the \'UNKP-17-4 New National Excellence Program of the Ministry of Human Capacities of Hungary. This work was performed in part at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607761.
|
1,116,691,500,871 | arxiv | \section{Introduction}
\input{sections/1_introduction}
\section{Related Work}
\input{sections/2_related_work}
\section{Dataset Collection Pipeline}
\input{sections/3_data_collection}
\section{The Resulting Data}
\input{sections/4_resulting_data}
\section{Experiments}
\input{sections/5_experiments}
\section{Conclusion}
\input{sections/6_conclusion}
\section*{Limitations}
\input{sections/limitations}
\section*{Ethics Statement}
\input{sections/ethical_considerations}
\section*{Acknowledgements}
This work was partly supported by Institute of Information \& communications Technology Planning \& Evaluation (IITP) grant funded by the Korea government(MSIT) (No. 2022-0-00184, Development and Study of AI Technologies to Inexpensively Conform to Evolving Policy on Ethics). We also would like to thank Dea Adhista for managing the annotators during the validation and manual data collection process. Rifki Afina Putri was supported by Hyundai Motor Chung Mong-Koo Global Scholarship.
\subsection{Automatic Generation}
\label{sec:automatic-gen}
In this stage, we automatically construct unanswerable questions using a Question Generation (QG) model. We use translated SQuAD 2.0 \cite{rajpurkar-etal-2018-know} as the training data of the QG model. In the inference step, we use the answerable questions from TyDiQA-GoldP \cite{clark-etal-2020-tydi} as a starting point to add more unanswerable questions for our dataset. Our QG model architecture is illustrated in Figure \ref{fig:qgen_model}.
\paragraph{Candidate Generation}
We utilize mT5 model \cite{xue-etal-2021-mt5} to generate the unanswerable question candidates. We apply \texttt{generate unans} prefix, followed by \textit{context}, \textit{answerable question}, and \textit{answer} as the input. Then, using top-p and top-k sampling as the decoding method, the model produces several output candidates.
\paragraph{QA Filter}
Since not all output candidates are valid unanswerable questions, we filter out invalid questions using an ensemble of six\footnote{6 was chosen based on related work in Adversarial QA \cite{bartolo-etal-2021-improving}.} Question Answering (QA) models. We fine-tuned XLM-R \cite{conneau-etal-2020-unsupervised} on translated SQuAD 2.0 dataset using different random seeds and used them as the QA models. Based on the prediction of these models, we keep the question if four or more models give an empty answer (i.e., unanswerable) or if four or more models return non-empty answers and all these answers are different.
\paragraph{Similarity Function}
Finally, we apply a similarity function to all remaining output candidates to make sure that the final output is relevant to its corresponding paragraph and answerable question. We calculate similarity between the original answerable question and the remaining question candidates using BLEU score to get the unanswerable question with highest n-gram overlap. We pick the candidate with the highest score as the final output.
\subsection{Validation}
\label{sec:validation}
After obtaining the automatically generated unanswerable questions, we validate them to ensure that they do not have noise or error. We recruit four Indonesian native speakers with 2+ years of experience in Indonesian NLP dataset annotation. Each annotator is asked to give a score to the generated questions with three criteria, adopted from \citet{zhu-etal-2019-learning} and re-defined as follows:
\begin{itemize}
\item \textbf{Unanswerability}: whether the answer can be found in the given paragraph. The score is 1 if the answer cannot be found, 0 otherwise.
\item \textbf{Relevancy}: whether the question is relevant to the paragraph and the answerable question. 3 if the question is relevant to both, 2 if it is only relevant to the paragraph or the answerable question, and 1 if it is not relevant to either.
\item \textbf{Fluency}: whether the question is fluent. 3 if the collective quality of all words in the question is fluent and coherent; 2 if the question is semi-coherent, has a minor typo, or grammatical errors; and 1 if the question is incoherent or incomprehensible.
\end{itemize}
Each question is validated by one annotator, with each annotator validating the same number of questions. Then, we apply cross-checking method to minimize human errors and to ensure consistency of the criteria across the annotators. Suppose that we have four annotators $(a_1, ..., a_4)$, who have evaluated some set of samples $(s_1, ..., s_4)$. Each sample $s_i$ consists of a set of \textit{paragraph}, \textit{answerable}, \textit{unanswerable question}, along with the \textit{unanswerability}, \textit{relevancy}, and \textit{fluency} scores of the unanswerable question. In the cross-checking phase, $a_1$ is assigned to check the scores of $s_2$, $a_2$ is assigned to check the scores of $s_1$, and so on. The disagreement\footnote{Overall, the disagreement percentage is roughly around 10--20\%, with $\sim$84\% of the disagreement are categorized as narrow disagreement (1 vs 2 or 2 vs 3).} is resolved by discussion among the annotators to ensure each annotator has the same level of task understanding and thus resulting in high quality and consistent annotation.
Finally, we keep the questions with perfect unanswerability, relevancy, and fluency scores (i.e., questions with scores of 1, 3, 3). We also keep the questions with scores of (1, 3, 2) and ask the annotators to make minor corrections to those questions. We regard the rest of the automatically generated questions as noise and discard them\footnote{We remove irrelevant questions (questions with relevancy score of 1 or 2) because they are often too far from the context, making the task less challenging. For example, the context describes Ecology definition, the Ans Q is \textit{"what is the definition of \textbf{ecology}?"}, and the UnAns Q is \textit{"what is the definition of \textbf{neo}?"}. We remove this kind of question. Note that we still regard the entity as relevant (and keep the question) if it belongs to the same category, i.e., person name, country, etc.}. From 6,196 generated questions, 3,190 questions are kept in the dataset, with 2,840 questions have a perfect score, and 350 questions have 1, 3, and 2 scores for unanswerability, relevancy, and fluency.
\subsection{Manual Generation}
\label{sec:manual-gen}
In the final stage, we ask human annotators to add more unanswerable questions, especially for the question types that QG model struggles to generate. There are six question types, as listed in Table \ref{tab:data-sample}, and it is important to have a sufficient number of questions for each type. The model generates entity swap questions well (see Figure \ref{fig:test_distribution}), so we request the annotators to write the remaining question types, i.e., \textit{negation}, \textit{antonym}, \textit{question tag swap}, \textit{specific condition}, and \textit{other}. The annotators may also add a new answerable question to be paired with the \textit{negation} question, specifically for the case when the negation word in the answerable question is removed. The annotators are the same as those from the validation stage, and they were assigned to write 500 unanswerable questions each. We also apply the same cross-checking method as the validation stage. In total, we have 2,000 human-written unanswerable questions.
\subsection{Automatic Generation Model Evaluation}
\label{sec:qgen-experiment}
We compare our QG model to these methods:
\begin{itemize}
\item \textbf{TF-IDF}: given an answerable question as the query, unanswerable question is generated by retrieving the most relevant question using TF-IDF features \cite{scikit-learn}. The similarity between the questions are calculated using cosine similarity.
\item \textbf{Rule-based}: we replace the entity in the answerable question with another entity in the context that has the same type, i.e., an entity with type \texttt{PERSON} will be replaced by another entity with type \texttt{PERSON}. If there is no appropriate entity in the question, we randomly swap the question tag with another tag. We extract the entity using XLM-R\textsubscript{BASE} model\footnote{\url{https://huggingface.co/cahya/xlm-roberta-base-indonesian-NER}} and extract the question tag using a simple matching with our predefined question tag list.
\item \textbf{Pair2Seq}: we adapt the pair-to-sequence model by \citet{zhu-etal-2019-learning} with Indonesian FastText \cite{bojanowski-etal-2017-enriching}. We follow the model architecture and the training procedure described in their paper.
\end{itemize}
\paragraph{Dataset}
We use SQuAD 2.0 to train our QG model. First, we align answerable and unanswerable questions with the same plausible answer. For example, if there exists an answerable question such as:
\begin{quote}
\textit{Who ruled the duchy of Normandy?}\newline
Answer: \textit{\textbf{Richard}}
\end{quote}
\noindent
and an unanswerable question such as:
\begin{quote}
\textit{Who ruled the country of Normandy?}\newline
Answer: \texttt{[empty]}\newline
Plausible answer: \textit{\textbf{Richard}}
\end{quote}
\noindent
the above questions will be paired or aligned.
Then, we translate the dataset using Google Translate API v2. Because complex questions tend to have more translation artifacts, we eliminate such questions by removing questions with a conjunction. From this process, we get 14,029 input pairs as the training data and 2,144 as the validation data. We use this dataset to train the question generation model (\S\ref{sec:automatic-gen}).
\paragraph{Implementation}
We implement QG model using SimpleTransformers \cite{rajapakse2019simpletransformers}. We use mT5\textsubscript{BASE} (580M parameters) to generate the questions with the maximum sequence length of 512. We train the model in 5 epochs and a batch size of 8. For the decoding, we use top-k and top-p sampling with a value of 50 and 0.95, respectively. We set the returned sequence number to 10.
\nocite{wolf-etal-2020-transformers}
\paragraph{Evaluation Metric}
We evaluate the models using the existing \textbf{BLEU} score metric\footnote{We use the NLTK version of the BLEU score (\url{https://www.nltk.org/api/nltk.translate.bleu_score.html}). The tokenization is done using the Indonesian tokenizer of Stanza library (\url{https://stanfordnlp.github.io/stanza}).}. However, BLEU is an n-gram based metric, so it can give a high score to the unanswerable questions that are exactly the same as the answerable question. Therefore, we use \textbf{\%diff} to compute the proportion of generated unanswerable questions that is not identical to its corresponding answerable questions. We also propose a new metric called \textbf{Unanswerable BLEU (UBLEU)}, an improved version of BLEU by setting the modified precision ($p_n$) to 0 if the output from the QG model ($q_{out}$) is identical to the paired answerable question ($q_{ans}$), formally defined as:
\[
p_{n} = \frac{
\sum\limits_{C \in \{\textit{Candidates}\}} \sum\limits_{\textit{n-gram} \in C} \alpha \ \textit{Count}_{\textit{clip}}\left( \textit{n-gram} \right)
}{
\sum\limits_{C' \in \{\textit{Candidates}\}} \sum\limits_{\textit{n-gram}' \in C'} \textit{Count}\left( \textit{n-gram}' \right)
}
\]
\noindent where
\[
\alpha = \left\{
\begin{array}{ c l }
0 & \quad \textrm{if } q_{out} = q_{ans} \\
1 & \quad \textrm{otherwise}
\end{array}
\right.
\]
\input{tables/result_qg_auto}
\input{tables/result_qg_human}
Moreover, we conduct human evaluation to further study the performance of the models. We randomly sample 100 questions for each QG models, and ask four annotators to evaluate the questions quality using the same protocol as \S\ref{sec:validation}.
\begin{figure*}[t!]
\begin{center}
\centerline{\includegraphics[width=\linewidth]{figures/data_dist_v2.png}}
\caption{Unanswerable question types distribution of Model Gen, Human Filt, and IDK-MRC test set. The question types are manually labeled by annotators. The bar opacity represents failure rate of the MRC model (XLM-R) in predicting the answer to the questions in each unanswerable question type (lighter is better). Our IDK-MRC dataset has a more balanced question type distribution, resulting in lower failure rate compared to Model Gen and Human Filt dataset.}
\label{fig:test_distribution}
\end{center}
\end{figure*}
\paragraph{Result}
As presented in Table \ref{tab:result-qg-auto} and \ref{tab:result-qg-human}, our QG model shows the best performance in both automatic and human evaluation. Despite a lower \%diff score than TF-IDF and rule-based, our model still achieves better UBLEU and BLEU scores. We also observe a slight reduction of UBLEU and BLEU scores when we add QA filter; however, based on human evaluation, QA Filter can improve the overall quality of the generated questions, especially the percentage of questions with perfect scores.
TF-IDF has a high fluency score because we use the existing answerable questions from different paragraphs, but it results in a low relevancy score. For rule-based, changing the entity in the answerable question to another entity in the context can produce high relevancy. However, it can still generate an answerable question, as shown by a lower unanswerability score. Pair2Seq \cite{zhu-etal-2019-learning} obtains a high unanswerability score but lower relevancy and fluency scores. It suffers from many \texttt{UNK} tokens, displaying the limitation of word embedding representation. Overall, adding QA Filter results in better performance in all evaluation aspects, indicated by a high average score and the number of samples with a perfect score.
\subsection{Automatic vs. Manual Generation}
We now compare the automatic and manual dataset generation from three perspectives: time, cost, and question quality, especially to further analyze whether the automatic generation model can benefit from additional human annotation.
\paragraph{Time and Cost}
For the automatic generation, it takes around 3 hours to train QG model on a single RTX 8000 48GB GPU. After the training has finished, the model takes 30 minutes to generate $\sim$2,000 questions in the inference step. As for the manual process, one person spent 32 hours verifying $\sim$2,000 questions and 10 hours writing $\sim$500 questions (40 hours per 2,000 questions). The cost for one human annotator is about \$7.5/hour, and assuming a GPU price of \$3/hour\footnote{The highest GPU hourly price from \url{https://cloud.google.com/compute/gpus-pricing} (Accessed June 2022).}, automatic generation is certainly more time- and cost-efficient approach than manual generation.
\input{tables/result_qa_combined_nohasans}
\paragraph{Question Quality}
From Figure \ref{fig:test_distribution}, we observe that our automatic QG model manages to generate various unanswerable question types, as shown in Model Gen question types distribution. However, it still produces some noise, i.e., answerable questions (8.51\% of Model Gen test set), even after such questions are discarded by QA Filter model. We also observe that the QG model tends to produce more \textit{entity swap} questions (49.17\% of Model Gen test set). Moreover, many irrelevant or incomprehensible questions are still exist in Model Gen dataset, especially for \textit{negation}, \textit{question tag swap}, and \textit{specific condition} types. This result suggests that even though automatic QG model can generate relatively fluent and valid questions, relying \textit{only} on it for building the dataset may result in noise and imbalance question types distribution. Filtering out the noisy data, automatically or manually, is not enough since the question types distribution is still imbalanced, as can be seen in Human Filt question distribution. The additional human-written questions in IDK-MRC cover this limitation, resulting in a more balanced question type distribution and lower models' failure rate in predicting the answer for each unanswerable question type.
\subsection{Dataset Evaluation on Downstream Task}
Next, we investigate the performance of MRC models trained on our dataset and compare them with Translated SQuAD 2.0 \cite{muis2020seq2seq} and TyDiQA \cite{clark-etal-2020-tydi} datasets.
\paragraph{Implementation}
We pick IndoBERT\textsubscript{BASE} \cite{wilie-etal-2020-indonlu} as the monolingual model and m-BERT\textsubscript{BASE} \cite{devlin-etal-2019-bert}, XLM-R\textsubscript{BASE} \cite{conneau-etal-2020-unsupervised} as the multilingual model. They have 124.5M, 167.4M, and 278.7M parameters, respectively. We implemented the models using SimpleTransformers \cite{rajapakse2019simpletransformers}. We use the standard hyperparameter settings for QA task with maximum sequence length of 384, document stride of 128, and trained the models for 10 epochs, batch size of 8, learning rate of 2e-5 using the Adam optimizer \cite{kingma2014adam}.
\paragraph{Result on IDK-MRC test set}
We tested several models using IDK-MRC test set\footnote{We use IDK-MRC test set since there is no suitable existing dataset to show the performance on Indonesian unanswerable questions. Another option is to test the models on Translated SQuAD 2.0 test set, but we found that many questions have machine translation error or incomplete or wrong ground truth answer (43\% out of 100 randomly sampled questions). Therefore, Translated SQuAD 2.0 is not an adequate dataset to test Indonesian MRC models.} and the result is presented in Table \ref{tab:result-qa}. We observe that the models trained on IDK-MRC dataset perform better than all baselines. Furthermore, even the models trained on our less-cleaned Model Gen dataset can obtain a better result than Translated SQuAD dataset, indicating the importance of a dataset that originates in Indonesian. We also note that the models trained on TyDiQA fail to handle unanswerable questions, as shown by extremely low UnAns scores. This result further highlights the significance of incorporating unanswerable questions in MRC dataset.
\paragraph{Unanswerable failure rate}
To further examine the models' capability in handling unanswerable questions, we also conduct an unanswerability error analysis using CheckList \cite{ribeiro-etal-2020-beyond}, a tool that facilitates behavioral test on many NLP tasks. CheckList provides a list of linguistic capabilities, with each capabilities are divided into different \textit{test types} to further break down the potential \textit{failure} of the linguistic capabilities. In this experiment, we focus on testing the models' capability on predicting the answer to unanswerable questions by dividing the \textit{test types} into the unanswerable questions type listed in Table \ref{tab:data-sample}. The test cases for each test type are automatically generated using CheckList's \texttt{template} function, resulting in 600 test cases for each question type. The template examples are presented in Appendix \ref{appendix:checklist_qtype}.
As shown in last column of Table \ref{tab:result-qa}, we find that our dataset can reduce the average failure rate, further confirming the effectiveness of IDK-MRC compared to the existing dataset. IndoBERT has the most significant failure rate reduction (45.96 to 31.92), followed by m-BERT (26.52 to 13.72) and XLM-R (26.13 to 16.42). Also, it is clear that only relying on the existing TyDiQA dataset is not enough to build a robust model, as shown from very high failure rates.
Overall, most failures occur on \textit{negation} questions, specifically when the negation word appears in the context passage, such as:
\begin{quote}
Context: \textit{Wikia \textbf{tidak} diketuai oleh Ali. (Wikia \textbf{is not} chaired by Ali.)}\newline
Question: \textit{Siapa yang mengetuai Wikia? (Who is Wikia's chair?)}\newline
Predicted Answer: \textit{Ali}\newline
Correct Answer: \texttt{[empty]}
\end{quote}
Besides adding more data samples, we conjecture that some improvement in model architecture or training scheme is needed to solve this problem. It is possible that the model highly correlates "\texttt{who}" question tag with any person's name that appears in the context, and picks it as the answer without considering the meaning of the whole context. Additionally, the high failure rates on IndoBERT model are mainly contributed by the \textit{antonym} and \textit{question tag swap} types, while multilingual models like m-BERT and XLM-R performs significantly better on this type of question. All in all, focusing on better approach to handle the aforementioned question types for future work may further improve the models' performance.
\section{Data Statement}
\subsection{Curation Rationale}
IDK-MRC dataset is built based on the existing paragraph and answerable questions (\texttt{ans}) in TyDiQA-GoldP \cite{clark-etal-2020-tydi}. The new unanswerable questions are automatically generated using the combination of mT5 \cite{xue-etal-2021-mt5} and XLM-R \cite{conneau-etal-2020-unsupervised} models, which are then manually verified by human annotators (\texttt{filtered ans} and \texttt{filtered unans}). We also asked the annotators to manually write additional unanswerable questions as described in \S\ref{sec:manual-gen} (\texttt{additional unans}). Each paragraphs in the final dataset will have a set of \texttt{filtered ans}, \texttt{filtered unans}, and \texttt{additional unans} questions. The illustration of the dataset collection pipeline is shown in Figure \ref{fig:data_collection}.
\subsection{Language Variety}
The texts in IDK-MRC are generated and written using the standard formal style of the Indonesian language.
\subsection{Annotator Demographic}
In our dataset collection pipeline, the annotators are asked to validate the generated unanswerable questions and write a new additional unanswerable questions.
We recruit four annotators with 2+ years of experience in Indonesian NLP annotation using direct recruitment. All of them are Indonesian native speakers who reside in Indonesia (Java Island) and fall under the 18--34 age category. We set the payment to around \$7.5 per hour. Given the annotators' demographic, we ensure that the payment is above the minimum wage rate (as of December 2021). All annotators also have signed the consent form and agreed to participate in this project.
\subsection{Speech Situation}
The paragraphs and answerable questions in IDK-MRC are built based on TyDiQA-GoldP \cite{clark-etal-2020-tydi}, which was originally taken from 2019 snapshots of Indonesian Wikipedia. As for the unanswerable questions, the dataset collection is conducted in December 2021. However, it is generated or written based on the facts or information provided in the existing paragraph in TyDiQA-GoldP dataset.
\subsection{Text Characteristics}
The original texts in IDK-MRC are mainly based from Wikipedia articles covering various topics, such as history, science, biography, and many more\footnote{The complete list of Indonesian Wikipedia article topics can be seen in \url{https://id.wikipedia.org/wiki/Wikipedia:Artikel_pilihan/Topik}}.
\subsection{Provenance Appendix}
As described in the previous section, the paragraphs and answerable questions in IDK-MRC are taken from the existing Indonesian TyDiQA dataset \cite{clark-etal-2020-tydi}. Unfortunately, the authors of TyDiQA did not provide complete data statement information, especially on their annotators demographic. However, since the original source of this dataset is from Wikipedia, we conjecture that the speech situation and text characteristic of this dataset is not far from the one that we have discussed in previous sections.
\section{Unanswerability Analysis by Question Type}
\label{appendix:checklist_qtype}
\input{tables/checklist_templates}
\input{tables/result_checklist}
In this section, we aim to further measure the MRC models' performance on handling each unanswerable question type using CheckList tool.
\paragraph{Test Case Generation}
We utilized \texttt{template} function provided in Checklist to generate the test cases for each unanswerable question type, i.e., negation (in-question and in-context), antonym, entity swap, question tag swap, specific condition, and other. Each question type consists of several question tag, namely \textit{siapa} (who), \textit{apa} (what), \textit{kapan} (when), \textit{di mana} (where), \textit{mengapa} (why), and \textit{berapa} (how long/many/much). Each question tag have 100 test cases, therefore, we have a total of 600 test cases for each unanswerable question type. Some of the template examples are presented in Table \ref{tab:checklist-template}.
\paragraph{Experiment Result}
As shown from Table \ref{tab:checklist}, models trained on our IDK-MRC dataset has a lower failure rate on the \textit{entity swap}, \textit{question tag swap}, \textit{specific condition}, and \textit{other} questions, indicating that adding more examples for these question types can improve the models' unanswerability skills. Also, most models can successfully handle the negation if the negated word exists in the question. When the negated word appears in the context, most models fail to predict the correct answer. Meanwhile, models train on SQuAD has a lowest failure rate on negation case, and we conjecture that it occurs due to the imbalanced question type distribution in the SQuAD training dataset. As reported by \citet{sen-saffari-2020-models}, 85\% of questions containing "\texttt{n't}" and 89\% of questions containing "\texttt{never}" in SQuAD dataset are categorized as unanswerable question. It aligns with our experiment results, which shows that SQuAD has a lowest failure rate on negation question type and a much higher failure rate on the other question types.
\section{Annotation Instruction}
\label{appendix:instruction}
\begin{figure*}[t!]
\begin{center}
\centerline{\includegraphics[width=\linewidth]{figures/instruction_task1.png}}
\caption{Annotation instruction for the validation stage.}
\label{fig:instruction_1}
\end{center}
\end{figure*}
\begin{figure*}[t!]
\begin{center}
\centerline{\includegraphics[width=\linewidth]{figures/instruction_task2.png}}
\caption{Annotation instruction for the manual generation stage.}
\label{fig:instruction_2}
\end{center}
\end{figure*}
\subsection{Validation Stage}
In this stage, human annotators are asked to validate the quality of the model-generated unanswerable questions using criteria as described in \S\ref{sec:validation}. Detailed instruction can be seen in Figure \ref{fig:instruction_1}.
\subsection{Manual Generation Stage}
In this stage, human annotators are asked to write a questions that the question generation model fails to generate as described in \S\ref{sec:manual-gen}. The instruction given to the annotators are shown in Figure \ref{fig:instruction_2}.
|
1,116,691,500,872 | arxiv | \section{Introduction}
A key concept in mathematical finance is that of absence of arbitrage. Informally speaking, an arbitrage opportunity is the possibility of making money out of nothing without taking any risk. Clearly, such strategies should be excluded in order to ensure market efficiency. The mathematical formulation of the no-arbitrage theory has a long history from the first discrete-time result of \cite{harrison1979martingales} and to the characterizations of \cite{delbaen_general_1994} and \cite{delbaen_fundamental_1998} in general semimartingale models. In particular, in \cite{delbaen_general_1994} it is shown that the No Free Lunch With Vanishing Risk (NFLVR) condition is equivalent to the existence of an Equivalent Local Martingale Measure (ELMM), i.e., a new probability measure under which the discounted asset price process is a local martingale.
The NFLVR provides a sound theoretical framework to solve problems of pricing, hedging or portfolio optimization. However, for some applications, requiring total absence of free lunches turns out to be too restrictive. Indeed, it seems reasonable to assume that limited arbitrage opportunities exist in financial markets given that whole desks of ``arbitrageurs'' are working full-time in investment banks to exploit them. This is one of the reasons why market models with arbitrage opportunities have appeared in recent literature, starting with the three-dimensional Bessel process model of \cite{delbaen_arbitrage_1995}. Without relying on the concept of equivalent martingale measure, \cite{platen_benchmark_2006}, see also \cite{platen_various_2006}, developed the Benchmark Approach, a new asset pricing theory under physical measure. In the context of Stochastic Portfolio Theory \citep{karatzas_stochastic_2009}, the NFLVR condition is not imposed and arbitrage opportunities arise in relative sense. These works suggest that NFLVR condition can be replaced by another weaker notion while preserving the solvability of the economics problems mentioned above, and several new concepts have indeed been proposed, see \cite{fontana_weak_2013} for a review.
If one is interested in utility maximization, it has been shown \citep{karatzas_numeraire_2007} that the minimal no free lunch type condition making this problem well posed is the No Unbounded Profit with Bounded Risk (NUPBR) condition. This condition has also been referred to as BK in \cite{kabanov1997ftap} and it is also equivalent to the No Asymptotic Arbitrage of the 1st kind (NAA1) of \cite{kabanov1994large}.
It is known that the NFLVR is equivalent to NUPBR plus the
classical no arbitrage assumption (see Corollary 3.4 and Corollary 3.8 of \cite{delbaen_general_1994} or Proposition 4.2 of \cite{karatzas_numeraire_2007}). This means that markets satisfying only NUPBR may admit arbitrage opportunities.
To benefit from a potential arbitrage, one needs to characterize explicitly the arbitrage strategy, and also to devise a method to compare different strategies, so as to exploit the arbitrage opportunity in the most efficient way.
An important step in this direction was made in \cite{fernholz_optimal_2010}. In this paper, the authors introduce the notion of optimal relative arbitrage with respect to the market portfolio and characterize the optimal relative arbitrage in continuous Markovian market models in terms of the smallest positive solution to a parabolic partial differential inequality. This work is extended in \cite{fernholz_optimal_2011} to market models with uncertainty regarding the relative risk and covariance structure of the assets.
Precise conditions for the existence of both relative arbitrage and strong relative arbitrage opportunities are given in \cite{ruf_optimal_2011}.
The optimal relative arbitrage turns out to be related to the minimal cost needed to superhedge the market portfolio in an almost sure way.
The problem of hedging in markets with arbitrage opportunities is studied in detail in \cite{ruf_hedging_2013}. That paper shows in particular that delta hedging is still the optimal hedging strategy in continuous Markovian markets which admit no equivalent local martingale measure but only a square-integrable market price of risk.
With the exception of the Stochastic Portfolio Theory of Fernholz and Karatzas and the Benchmark Approach of Platen and Heath, the existing literature on markets without arbitrage opportunities remains theoretical, and its findings are rarely used by practitiones to develop actual trading strategies based on arbitrage opportunities. This is related to the fact that examples of market with arbitrage found in the literature are typically ad hoc and are not flexible enough allow calibration to the actual market data. Moreover, it has been shown in \cite{guasoni_fragility_2012} that most of the existing examples of arbitrage in diffusion markets are \emph{fragile}, meaning that they disappear when transaction costs are introduced into the model, however small their value, or when prices are recorded with a small observation error.
The goal of this study is to propose a new methodology for building models admitting optimal arbitrage, with an explicit characterization of the optimal arbitrage strategy. To do so, we start with a probability measure $\mathbb Q$ under which the NFLVR condition holds. We then construct a new probability measure $\mathbb P$, not equivalent to $\mathbb Q$, under which NFLVR no longer holds but NUBPR is still satisfied. This procedure is not new and goes back to the construction of the Bessel process by \cite{delbaen_arbitrage_1995}. However, we extend it in two directions.
Firstly, from the theoretical point of view, we provide a characterization of the superhedging price of a claim under $\mathbb P$ in terms of the superhedging price of a related claim under $\mathbb Q$. This allows us to characterize the optimal arbitrage profit under $\mathbb P$ in terms of the superhedging price under $\mathbb Q$, which is much easier to compute using the equivalent local martingale measures.
Secondly, from the economic point of view, we provide an economic intuition for the new arbitrage model as a model implementing the view of the economic agent concerning the impossibility of certain market events. In other words, if an economic agent considers that a certain event (such as a sovereign default) is impossible, but it is actually priced in the market, our method can be used to construct a new model incorporating this arbitrage opportunity, and to compute the associated optimal arbitrage strategy.
We then combine these two ideas to develop several new classes of examples of models with optimal arbitrage, allowing for a clear economic interpretation, with a special focus on incomplete markets. We also discuss the issue of robustness of these arbitrages to small transaction costs / small observation errors and show that some of our examples are not fragile in the sense of \cite{guasoni_fragility_2012}.
The paper is organized as follows. In Section \ref{section:setting}, we describe the market setting and state the main assumptions. In Section \ref{section:optimal_arbitrage}, optimal arbitrage profit is introduced and related to a superhedging problem. In Section \ref{section:main_result}, we use an absolutely continuous measure change to build markets with optimal arbitrage. Finally, several new examples built using this construction are gathered in Section \ref{section:examples}.
\section{General setting} \label{section:setting}
For the theory of stochastic process and stochastic integration, we refer to \cite{jacod_limit_2002} and \cite{protter_stochastic_2003}.
Let $(\Omega, \mathcal F, \mathbb{F}, \mathbb P)$ be a given fitered probability space, where the filtration $\mathbb{F} = (\mathcal F_t)_{t\geq 0}$ is assumed to satisfy the usual conditions of right-continuity and augmentation by the $\mathbb P$-null sets. For any adapted RCLL process $S$, we denote by $S_-$ its predictable left-continuous version and by $\Delta S:=S - S_-$ its jump process. For a $d$-dimensional semimartingale $S$ and a predictable process $H$, we denote by $ H \cdot S$ the vector stochastic integral of $H$ with respect to $S$. We fix a finite planning horizon $T<\infty$ (a stopping time) and assume that after $T$ all price processes are constant and equal to their values at $T$.
On the stochastic basic $(\Omega, \mathcal F, \mathbb{F}, \mathbb P)$, we consider a financial market with an $\mathbb R^d$-valued nonnegative semimartingale process $S = (S^1,...,S^d)$ whose components model the prices of $d$ risky assets. The riskless asset is denoted by $S^0$ and we assume that $S^0 \equiv 1$, that is, all price processes are already discounted. We suppose that the financial market is frictionless, meaning that there are no trading restrictions, transaction costs, or other market imperfections.
Let $L(S)$ be the set of all $\mathbb{R}^d$-valued $S$-integrable predictable processes. It is the most reasonable class of strategies that investors can choose, but another constraint, which is described below, is needed in order to rule out doubling strategies.
\begin{defi}
Let $x \in \mathbb{R_+}$. An element $H \in L(S)$ is said to be an $x$-admissible strategy if $H_0
= 0$ and $(H \cdot S)_t \ge - x$ for all $t \in [0, T ]$ $\mathbb P$-a.s. An element $H \in L(S)$ is said to be an admissible strategy if it is an $x$-admissible strategy for some $x \in \mathbb{R_+}$.
\end{defi}
For $x \in \mathbb{R_+}$, we denote by $\mathcal{A}_x$ the set of all $x$-admissible strategies and by $\mathcal{A}$ the set of all
admissible strategies. As usual, $H_t$ is assumed to represent the number of units of risky asset that we hold at time $t$. For $(x,H) \in \mathbb{R_+} \times \mathcal{A}$,
we define the portfolio value process $V_t^{x,H} = x + (H \cdot S)_t$. This is equivalent to require that portfolios are only generated by self-financing admissible strategies.
Given the semimartingale $S$, we denote by $\mathcal{K}_x$ the set of all outcomes that one can realize by $x$-admissible strategies starting with zero initial cost:
$$ \mathcal{K}_x = \left\lbrace (H \cdot S)_T| H \text{ is $x$-admissible} \right\rbrace$$
and by $\mathcal{X}_x$ the set of outcomes of strategies with initial cost $x$:
$$\mathcal{X}_x = \left\lbrace x + ( H \cdot S)_T| H \text{ is $x$-admissible} \right\rbrace.$$
Remark that all elements in $\mathcal{X}_x$ are nonnegative. The unions of all $\mathcal{K}_x$ and all $\mathcal{X}_x$ are denoted by $\mathcal{K}$
and $\mathcal{X}$, respectively. All bounded claims which can be superreplicated by admissible strategies are contained in
$$ \mathcal{C} = \left( \mathcal{K} - L^0_+ \right) \cap L^{\infty}.$$
Now, we recall some no-free-lunch conditions, which are studied in the works of \cite{delbaen_general_1994}, \cite{karatzas_numeraire_2007} and \cite{kardaras_market_2012}.
\begin{defi}
\begin{itemize}
\item We say that the market satisfies the No Arbitrage (NA) condition with respect to general admissible integrands if
\[
\mathcal{C} \cap L_+^{\infty} = \left\lbrace 0 \right\rbrace .
\]
\item We say that the market satisfies the No Free Lunch with Vanishing Risk (NFLVR) property, with respect to general admissible integrands, if
$$ \overline{\mathcal{C}} \cap L^{\infty}_+ = \left\lbrace 0 \right\rbrace ,$$
where the bar denotes the closure in the supnorm topology of $L^{\infty}$.
\item There is No Unbounded Profit With Bounded Risk (NUPBR) if the set $ \mathcal{K}_1$ is bounded in $L^0$, that is, if
$$ \lim_{c \to \infty} \sup_{H \in \mathcal{A}_1} \mathbb P\left[ V^{0, H} > c \right] =0$$
holds.
\item The market admits Immediate Arbitrage (IA) if there exists a stopping time $\tau$ such that $\mathbb P\left[ \tau < T \right] >0 $ and a strategy $H = H 1_{]\tau, T] }$ which realizes $\left( H\cdot S \right)_t > 0 $ $\mathbb P$-a.s for all $t \in (\tau, T]$. We say that condition No Immediate Arbitrage (NIA) holds if there exists no immediate arbitrage in the market.
\end{itemize}
\end{defi}
The economic interpretation of no arbitrage type conditions above can be described as follows. Classical arbitrage means that one can make something out of nothing without risk. If there is a FLVR, starting with zero capital, one can find a sequence of wealth processes such that the terminal wealths converge to a nonnegative random variable which is not identical to zero and the risk of the trading strategies becomes arbitrarily small. If an UPBR exists, one can find a sequence of wealth processes with bounded (or indeed arbitrarily small) risk whose terminal wealths are unbounded with a fixed probability.
In this paper, we are interested in financial markets satisfying the following assumption.
\begin{assum}\label{assume:NUPBR} S is locally bounded, the market satisfies
NUPBR but the condition NFLVR fails under the physical measure $\mathbb P$.
\end{assum}
Under the local boundedness assumption, by the Fundamental Theorem of Asset Pricing, the NFLVR condition is equivalent to the existence of a ELMM (see Corollary 1.2 in \cite{delbaen_general_1994}), but for general semimartingales the ELMM must be replaced with an equivalent sigma-martingale measure. So, the limitation to locally bounded processes $S$ allows us to work with local martingales instead of sigma martingales.
When the NFLVR condition fails but the NUBPR condition holds, the ELMM is replaced with a weaker notion of ``deflator''.
\begin{defi}
An equivalent local martingale deflator (ELMD) is a nonnegative process $Z$ with $Z_0=1$ and $Z_T > 0$ such that $ZV$ is a local martingale for all $V \in \mathcal{X}$.
\end{defi}
In particular, an ELMD is a nonnegative local martingale. Fatou's Lemma implies that it is also a supermartingale and its expectation is less or equal to one. Hence, a local martingale density is an ELMD with expectation one. In general, we cannot use an ELMD to define a new probability measure, since the new measure loses mass. It is worth to remark that the situation when the ELMD is a strict local martingale is very different from a market with bubble. Indeed, an asset price is said to be a bubble if it is a strict local martingale under the risk-neutral measure, see \cite{heston_options_2006} , \cite{cox_local_2005}, \cite{jarrow_asset_2007}, \cite{jarrow_asset_2010}, which means that no arbitrage opportunity exists.
The following result has recently been proven in \cite{kardaras_market_2012} in the one dimensional case. An alternative proof in the multidimensional case has been given in \cite{takaoka_condition_2010} by a suitable change of numeraire argument in order to apply the classical results of \cite{delbaen_general_1994}, and in \cite{song_alternative_2013} by only using the properties of the local characteristics of the asset process.
\begin{theorem}
The NUPBR condition is equivalent to the existence of at least one ELMD.
\end{theorem}
\paragraph{Fragile and robust arbitrages} Since real markets have frictions, an arbitrage which disappears in the presence of small transaction costs, or small observation errors, cannot be exploited in practice. This property, known as fragility of arbitrage \citep{guasoni_fragility_2012} is described in the following two definitions.
{
\begin{defi}
For $\varepsilon >0$, two strictly positive processes $S, \tilde{S}$ are $\varepsilon$-close if
$$ \frac{1}{1 + \varepsilon} \leq \frac{\tilde{S}_t}{S_t} \leq 1 + \varepsilon \qquad a.s. \text{ for all } t \in [0,T].$$
\end{defi}
\begin{defi}[Fragility/Robustness]
We say that arbitrage in the $(\mathbb{P},S)$-market is fragile if for every $\varepsilon >0$ there exists a process $\tilde{S}$, which is $\varepsilon$-close to $S$, such that the $(\mathbb{P}, \tilde{S})$-market satisfies NFLVR. If the arbitrage is not fragile we say that it is robust.
\end{defi}
\cite{guasoni_fragility_2012} show that in diffusion settings, if the coefficients of the log-price process are locally bounded, arbitrages are fragile. For instance, when we introduce small frictions in the Bessel example, the arbitrage disappears.
\cite{bender_simple_2012} defines a \emph{simple obvious arbitrage} as a buy and hold strategy, which guarantees to the investor a profit of at least $\varepsilon>0$ if the investor trades at all. It is clear that a simple obvious arbitrage is always robust.
}
\section{Optimal arbitrage} \label{section:optimal_arbitrage}
It is well known that NFLVR holds if and only if both NUPBR and NA hold, see Corollary 3.4 and 3.8 of \cite{delbaen_general_1994} or Proposition 4.2 of \cite{karatzas_numeraire_2007}. Moreover, Lemma 3.1 of \cite{delbaen_existence_1995} shows that if NA fails then either the market admits an immediate arbitrage or an arbitrage that is created by a strategy in $\mathcal{A}_1$. Furthermore, if there exists an immediate arbitrage, the associated strategy is in $\mathcal{A}_0$ and can be freely scaled to produce an unbounded arbitrage. But this situation is not allowed in our market due to Assumption \ref{assume:NUPBR}. Therefore, it is only possible to exploit arbitrages by using strategies in the set $\mathcal{A}_x$ where $x>0$.
This reasoning is formalized by the following lemma.
\begin{lemma} \label{lemma:NUPBR_NIA}
NUPBR implies NIA.
\end{lemma}
\begin{proof}
We will prove that immediate arbitrage implies unbounded profit with bounded risk. Assume $S$ admits immediate arbitrage at stopping time $\tau$ and $\mathbb P(\tau < T) > 0$. There exits a strategy $H$ such that $H = H1_{\left( \tau, T \right] }$ and $(H\cdot S)_t > 0$ for $t \in \left( \tau, T \right]$. We observed that $H^n:= nH \in \mathcal{A}_0$ and the set $\left\lbrace V^{0,H^n}_T \right\rbrace_{n \in \mathbb{N}}$ is not bounded in probability. This means that the market admits unbounded profit with bounded risk.
\end{proof}
For these reasons, arbitrages in our market are limited and the question of optimal arbitrage profit arises naturally.
\begin{defi} \label{defi:optimal_arbitrage}
For a fixed time horizon $T$, we define
$$U(T) : = \sup \left\{ {c > 0:\exists H \in \mathcal{A}_1,V_T^{1,H} \ge c, \mathbb P-a.s} \right\}.$$
If $U(T) > 1$, we call $U(T)$ optimal arbitrage profit.
\end{defi}
The quantity $U(T)$ is the maximum deterministic amount that one can realize at time T starting from unit initial capital. Obviously, this value is bounded from below by 1. This definition goes back to the paper of \cite{fernholz_optimal_2010}. In diffusion setting, these authors characterize the following value
$$ \sup \left\lbrace c > 0: \exists H \in \mathcal{A}_1, V_T^{1,H} \ge c \sum S^i_T, \mathbb P-a.s. \right\rbrace,$$
which is the highest return that one can achieve relative to the market capitalization.
\subsection{Optimal arbitrage and superhedging price}
\begin{defi}\label{defi:superhedge_price}
Given a claim $f \ge 0$, we define
$$SP_+(f) := \inf \left\{ x \ge 0: \exists H \in \mathcal{A}_x ,V_T^{x,H} \ge f, \mathbb P-a.s \right\},$$
that is the minimal amount starting with which one can superhedge $f$ by a nonnegative wealth process.
\end{defi}
A reason for limiting oneself to nonnegative wealth processes is discussed in \cite{ruf_optimal_2011}. If $Z$ is a local martingale deflator and a wealth process $V$ is only restricted to stay above some constant $x < 0$, then $ZV$ may no longer be a supermartingale. Let us compare the definition of $SP_+$-price with the superhedging price in the literature. The superhedging price of a given claim $f$ is commonly defined by
$$SP(f) = \inf \left\{ x \ge 0: \exists H \in \mathcal{A} ,V_T^{x,H} \ge f, \mathbb P-a.s \right\}.$$
In other words, we allow wealth processes (maybe negative) that are uniformly bounded from below. Note that in markets that satisfy NA, $SP_+(f) = SP(f)$. Indeed, if NA holds, for every admissible integrand $H$ we have $ \| (H \cdot S)_t^{-} \|_{\infty} \le \| (H \cdot S)_T^{-} \|_{\infty}$, see Proposition 3.5 in \cite{delbaen_general_1994}. If $ x + (H \cdot S)_T \ge f$ then $(H \cdot S)_T \ge f - x \ge -x$ so that $\| (H \cdot S)_T^{-} \|_{\infty} \le x$. This implies that $ \| (H \cdot S)_t^{-} \|_{\infty} \le x$ or $(H \cdot S)_t \ge -x,$ for all $t \in [0,T]$.
In our market model with arbitrage, $SP(f) \le SP_+(f)$. The difference between the two superhedging prices is discussed in \cite{khasanov_upper_2013}.
The following lemma is simple but useful to our problem.
\begin{lemma} $U(T) = 1/SP_+(1)$.
\end{lemma}
\begin{proof}
$(\le)$ Take any $c>0$ such that there exists a strategy $H \in \mathcal{A}_1$ which satisfies
\begin{itemize}
\item $V_T^{1,H } = 1 + (H\cdot S)_T \ge c, \mathbb P-a.s.$
\item $(H \cdot S)_t \ge -1.$ for all $0 \le t \le T$.
\end{itemize}
Then a simple scaling argument gives us a strategy to hedge 1
\begin{itemize}
\item (superheging) $1/c + 1/c(H\cdot S)_T \ge 1, \mathbb P-a.s.$
\item (admissibility) $1/c(H\cdot S)_t \ge -1/c$ for all $0 \le t \le T$.
\end{itemize}
By Definition \ref{defi:superhedge_price}, one can superhedge 1 at cost $1/c$. Therefore, we get an upper bound for optimal arbitrage profit
\[
U(T) \le \frac{1}{SP_+(1)}.
\]
$(\ge)$ The converse inequality can be proved by the same argument.
\end{proof}
The above lemma has two consequences. First, the knowledge of $SP_+(1)$ is enough to find optimal arbitrage profit. Second, one should find the strategy to superhedge 1 in order to realize optimal arbitrage.
Obviously, $SP_+(1) \le 1$. If $SP_+(1) < 1$, there is optimal arbitrage. If $SP_+(1) = 1$, optimal arbitrage does not exist, but arbitrages may still exist. In Example 8 of \cite{ruf_optimal_2011}, the cheapest price to hold $1$ is $1$, but we can achieve a terminal wealth larger than $1$ with positive probability.
\section{Constructing market models with optimal arbitrage}
\label{section:main_result}
In this section we present two constructions of market models with optimal arbitrage. They both work by starting with a probability measure $\mathbb Q$ under which the price process satisfies NFLVR and making a non-equivalent measure change to construct a new measure $\mathbb P$ allowing for arbitrage. Arbitrage opportunities constructed with an absolutely continuous measure change have been studied in earlier works. The first example of this kind of technique is the Bessel model, which is given in \cite{delbaen_arbitrage_1995}. This technique is generalized in \cite{osterrieder_arbitrage_2006} and \cite{ruf_systematic_2013}. However, we push this idea further by characterizing the superhedging price under $\mathbb P$ in terms of the superhedging price under $\mathbb Q$, which enables us to describe optimal arbitrages.
\subsection{A construction based on a nonegative martingale} \label{section:construction_martingale}
Let $\mathbb Q$ be a probability measure on the filtered measure space $\left( {\Omega ,\mathcal F , \left( \mathcal F_t \right)_{t\geq 0}}\right)$ described in the beginning of Section \ref{section:setting}, and assume that under $\mathbb Q$, the following are true:
\begin{itemize}
\item The risky asset process $S$ is a locally bounded semimartingale which satisfies NFLVR.
\item There exists a nonnegative RCLL martingale $M$ with $M_0 = 1$ and
\begin{equation} \label{eq:condition_on_M}
{\mathbb Q}[\{\tau \le T\} \cap \{ M_{\tau -} > 0 \}] = 0.
\end{equation}
where $\tau = \inf \{t \ge 0: M_t = 0\}$ with the convention that $\inf \emptyset = + \infty.$
\end{itemize}
Since $M$ is right-continuous, this means that $M$ may only hit zero continuously on $[0,T]$.
Using $M$ as a Radon-Nykodym derivative, we define a new probability measure via
$$
\frac{d\mathbb P}{d{\mathbb Q}}\Big|_{\mathcal F_t} = M_{\tau \wedge t}.
$$
Then $\mathbb P$ is only absolutely continuous (but not equivalent) with respect to ${\mathbb Q}$. In fact, $M$ can reach zero under ${\mathbb Q}$ but it is always positive under $\mathbb P$, because $\mathbb P[\tau \le T] = {\mathbb E}^{\mathbb Q}[M_{\tau \wedge T}1_{\tau \le T}] = 0$.
\begin{theorem} \label{theorem:optimal_arbitrage_constrution_martingale}
Under the above assumptions, the $(\mathbb P,S)$-market satisfies NUPBR, and for any $\mathcal F_T$-measurable claim $f \ge 0$, we have $$SP^\mathbb P_+(f) = SP^{\mathbb Q}_+(f1_{M_T > 0}).$$
\end{theorem}
\begin{cor}
Under the assumptions of the theorem let
$$
\sup_{\bar {\mathbb Q} \in ELMM({\mathbb Q},S)} {\mathbb E}^{\bar {\mathbb Q}} [1_{M_T>0}] < 1.
$$
Then the $(\mathbb P,S)$-market admits optimal arbitrage and the optimal arbitrage strategy the superhedging strategy of the claim $1_{M_T > 0}$ in the $(\mathbb{Q},S)$-market.
\end{cor}
\begin{proof}
By the the standard super-replication theorem under absence of arbitrage (Theorem 9 of \cite{delbaen_no-arbitrage_1995}),
$$SP^{\mathbb Q}_+(1_{M_T > 0}) = \sup_{\bar {\mathbb Q} \in ELMM({\mathbb Q},S)} {\mathbb E}^{\bar {\mathbb Q}} [1_{M_T}>0].
$$
\end{proof}
\begin{proof}[Proof of Theorem]Let $\bar {\mathbb Q}$ be a local martingale measure equivalent to $\mathbb Q$, and denote by $\bar Z$ its density with respect to $\mathbb Q$.
\\
Step 1: we prove that the $(\mathbb P,S)$-market satisfies NUPBR by showing that $\bar Z/M$ is an ELMD.
We define
$$\tau_n = \inf\{ t \geq 0: M_t < \frac{1}{n}\}$$
with the convention $\inf \emptyset = + \infty$. Since, by condition (\ref{eq:condition_on_M}), $M$ does not jump to zero, we have that $M_{t\wedge \tau_n}>0$ $\forall t\geq 0$ $\mathbb Q$-a.s.
We remark that ${\mathbb Q} \ll {\mathbb P}$ on $\mathcal F_{t \wedge \tau_n}$. Indeed, take any $A \in \mathcal F_{t \wedge \tau_n}$ such that ${\mathbb P}(A) = 0$, we compute
\begin{equation*}
{\mathbb Q}[A] = {\mathbb E}^{\mathbb Q}\left[ 1_A \frac{M_{t \wedge \tau_n}}{M_{t \wedge \tau_n}} \right] = {\mathbb E}^{\mathbb P}\left[ 1_A \frac{1}{M_{t \wedge \tau_n}} \right] = 0.
\end{equation*}
This means ${\mathbb P}$ is equivalent to ${\mathbb Q}$ on $\mathcal F_{t \wedge \tau_n}$.
By Corollary 3.10, page 168 of \cite{jacod_limit_2002}, to prove that a process $N$ is a $\mathbb P$-local martingale with localizing sequence $(\tau_n)$, we need to prove that $(NM)^{\tau_n}$ is a $\mathbb Q$-local martingale for every $n\geq 1$. Then,
\begin{itemize}
\item $\bar Z/M$ is a $\mathbb P$-local martingale since $\bar Z$ is a $\mathbb{Q}$-local martingale.
\item $\bar Z S/M$ is a $\mathbb P$-local martingale since $\bar ZS$ is a $\mathbb Q$-local martingale.
\item $\bar Z V/M$ is a $\mathbb P$-local martingale for each $\mathbb P$-admissible $V$. Since $\mathbb{P}$ and $\mathbb{Q}$ are equivalent on $\mathcal F_{t \wedge \tau_n}$, we obtain $V^{\tau_n}$ is a $\mathbb{Q}$-admissible wealth process. Thus, for each $n$, we have $\bar Z V^{\tau_n}$ is a $\mathbb{Q}$-local martingale, so is $(\bar Z V)^{\tau_n}$.
\end{itemize}
\noindent Step 2: we prove the equality $SP_+^\mathbb P(f) = SP_+^{\mathbb Q}(f1_{M_T >0}).$\\
($\le$) Take any $x >0$ such that there exists a strategy $H \in \mathcal{A}^{\mathbb Q}_x$ which satisfies $ V_T = x + (H \cdot S)_T \ge f1_{M_T >0}, {\mathbb Q}-a.s.$ Since $\mathbb P \ll {\mathbb Q}$, Theorem 25, page 170 of \cite{protter_stochastic_2003} shows that $H \in L(S)$ under $\mathbb P$ as well and $H_{\mathbb Q} \cdot S = H_{\mathbb P} \cdot S, {\mathbb P}-a.s.$. We also see that $ x + (H \cdot S)_t \ge 0, {\mathbb P}-a.s$ and $ x + (H \cdot S)_T \ge f1_{M_T >0} = f, {\mathbb P}-a.s$. This means
\begin{equation} \label{eq:SPlsSQ}
SP_+^{\mathbb P}(f) \le SP_+^{\mathbb Q}(f1_{M_T > 0}).
\end{equation}
($\ge$) For the converse inequality, take any $x >0$ such that there exists a strategy $H \in \mathcal{A}^{\mathbb P}_x$ and $ V^{\mathbb P}_T = x + (H \cdot S)_T \ge f, {\mathbb P}-a.s.$ We will show that $x \ge SP^{\mathbb Q}_+(f1_{M_T > 0 })$.\\
Define $H^n:=H1_{ t \le \tau_n}$, then $H^n$ is $S$-integrable and $x$-admissible under ${\mathbb Q}$. From Step 1, we see that $\tau_n \wedge T \nearrow T$, ${\mathbb P}$-a.s. and therefore $V^n_T = x + (H^n \cdot S)_T \to V^{\mathbb P}_T$, ${\mathbb P}$-a.s. or
$V^n_T 1_{M_T > 0} \to V^{\mathbb P}_T1_{M_T > 0} \ge f 1_{M_T > 0}$, ${\mathbb Q}$-a.s. The following convergence holds
$$V^n_T - V^n_T 1_{M_T = 0} = V^n_T 1_{M_T >0} \to V^{\mathbb P}_T1_{M_T >0} \ge f 1_{M_T >0}, {\mathbb Q}-a.s.$$
The sequence $V^n_T - x - V^n_T 1_{M_T = 0} = (H^n \cdot S)_T - V^n_T 1_{M_T >0}$ is in the set $\mathcal{K} - L^0_+$ (under ${\mathbb Q}$) and uniformly bounded from below by $-x$. Because the $({\mathbb Q},S)$-market satisfies NFLVR condition, the set $\mathcal{K} - L^0_+$ is Fatou-closed (see Remark after Corollary 1.2 of \cite{delbaen_general_1994}) and we obtain $V^{\mathbb P}_T1_{M_T > 0} - x \in \mathcal{K} - L^0_+$ or $x \ge SP^{\mathbb Q}_+(f1_{M_T > 0})$. In other words,
\begin{equation} \label{eq:SPgtSQ}
SP^{\mathbb P}_+(f) \ge SP^Q_+(f1_{M_T > 0}).
\end{equation}
From (\ref{eq:SPlsSQ}) and (\ref{eq:SPgtSQ}), the proof is complete.
\end{proof}
\subsection{A construction based on a predictable stopping time}\label{section:construction_predictable}
In this section, we apply the construction of Section \ref{section:construction_martingale} to a predictable stopping time. As before, we consider a measure $\mathbb Q$ on the space $(\Omega,\mathcal F, (\mathcal F)_{t\geq 0})$. Let $\sigma$ be a stopping time such that ${\mathbb Q}\left( \sigma > T \right) >0.$ We define a new probability measure, absolutely continuous with respect to ${\mathbb Q}$, by
\begin{align}\left. {\frac{d{\mathbb P}}{d{\mathbb Q}}} \right|_{\mathcal F_t} = \frac{{\mathbb Q}\left[ \sigma > T | \mathcal F_t \right] }{{\mathbb Q}\left[ \sigma > T \right] } := M_t.\label{defprobatau}
\end{align}
Under the new measure, ${\mathbb P}\left( \sigma > T \right) = {\mathbb E}^{\mathbb Q}\left( M_T 1_{\sigma > T} \right) = 1$.\\
This construction has the following economic interpretation. Consider an event (E), such as the default of a company or a sovereign state, whose occurence is characterized by a stopping time $\sigma$. Given a planning horizon $T$, we are interested in the occurence of this event (E) before the planning horizon. Suppose that the market agents have common anticipations of the probability of future scenarios, which correspond to the arbitrage-free probability measure $\mathbb Q$, and that under this probability, the event (E) has nonzero probabilities of occuring both before and after the planning horizon. Consider now an informed economic agent who believes that the event (E) will not happen before the planning horizon $T$. For instance, the agent may believe that the company or the state in question will be bailed out in case of potential default. Our informed agent may then want to construct an alternative model $\mathbb P$, in which the arbitrage opportunity due to mispricing may be exploited and the arbitrage strategy may be constructed in a rigorous manner.\footnote{The ``informed agent'' interpretation of our arbitrage construction hints at possible connections with the research on arbitrage opportunities arising from enlargement of the underlying filtration with additional information, see e.g. \cite{imkeller2001free,fontana2012arbitrages}. The detailed study of these connections is left for further research. } The following corollary provides a method for constructing such a model.
\begin{cor} \label{cor:optimal_arbitrage_predictable}
Assume that the following conditions hold
\begin{itemize}
\item The risky asset process $S$ is a locally bounded semimartingale which satisfies NFLVR under $\mathbb Q$.
\item The filtration $\mathbb{F}$ is quasi-left continuous.
\item $\sigma$ is a predictable stopping time such that for any stopping time $\theta$,
$${\mathbb E}^{\mathbb Q}\left[ 1_{\sigma > T}\right |\mathcal F_{\theta} ] > 0, {\mathbb Q}-a.s. \qquad \text{on } \left\lbrace \sigma > \theta \right\rbrace.$$
\end{itemize}
Then the $({\mathbb P}, S)$-market satisfies NUPBR. Given a $\mathcal F_T$-measurable claim $f \ge 0$, we have
$$SP^{\mathbb P}_+(f) = SP^{\mathbb Q}_+(f1_{\sigma > T}).$$
In addition, if
$$SP^{\mathbb Q}_+(1_{\sigma > T}) = \sup_{\bar {\mathbb Q} \in ELMM({\mathbb Q},S)} {\mathbb E}^{\bar {\mathbb Q}} [1_{\sigma > T}] < 1,$$
then the $({\mathbb P},S)$-market admits optimal arbitrage.
\end{cor}
\begin{proof}
This result will follow from Theorem \ref{theorem:optimal_arbitrage_constrution_martingale} after checking the condition (\ref{eq:condition_on_M}) on $M$. Let $\tau = \inf\{t>0: M_t = 0\}$. By construction, $M_\sigma = 0$ on $\{\sigma \le T\}$ and $M_t>0$ for $t<\sigma$. This means that
$$
\tau= \left\{
\begin{aligned}
&\sigma,\quad &&\sigma \leq T\\
&+\infty, && \text{otherwise.}
\end{aligned}\right.
$$
Since the filtration $\mathbb{F}$ is quasi left continuous and $\sigma$ is a predictable stopping time, $M$ does not jump at $\sigma$ (see \cite{protter_stochastic_2003}, page 190). This means that
$$
\mathbb Q[\{\tau\leq T\}\cap \{M_{\tau-}>0\} ] = 0
$$
and condition \eqref{eq:condition_on_M} is satisfied.
\end{proof}
\section{Examples} \label{section:examples}
\subsection{A complete market example}
Let $W^{\mathbb Q}$ be a Brownian motion and let $\mathbb{F}$ be its completed natural filtration. We assume that the price of a risky asset evolves as follows
$$
S_t = 1 + W^{\mathbb Q}_t
$$
and define a predictable stopping time by $\sigma = \inf\{ t>0: S_t \le 0\}$. Using the law of infimum of Brownian motion, we get
$${\mathbb Q}[\sigma > T] = {\mathbb Q}[(W^{\mathbb Q})^*_T > -1] = 1 - 2\mathcal{N} \left( -\frac{1}{\sqrt{T}} \right) > 0,$$
where $\mathcal N$ denotes the standard normal distribution function.
Next, by Markov property we compute
\begin{align}
{\mathbb E}^{\mathbb Q}[1_{\sigma > T}|\mathcal F_t] = {\mathbb Q}[(W^{\mathbb Q})^*_T > -1|\mathcal F_t] = \left\{ {\begin{array}{ll}
{0} \qquad \text{on } {\sigma \le t} \\
{1 - 2\mathcal{N} \left( -\frac{S_t}{\sqrt{T-t}} \right) > 0} \qquad \text{on }\sigma>t.\\
\end{array}} \right.\label{martrep}
\end{align}
Hence, on $\{\tau > t\}$, we obtain ${\mathbb E}^{\mathbb Q}[1_{\sigma > T}|\mathcal F_t] >0$. This means that the construction of Section \ref{section:construction_predictable} applies and we may define a new measure $\mathbb P$ via \eqref{defprobatau}. Since the $({\mathbb Q},S)$-market is complete and $ELMM({\mathbb Q},S) = \{{\mathbb Q}\}$, the superhedging price of the claim $1_{\sigma > T}$ is
$${\mathbb Q}[\sigma > T] = 1 - 2\mathcal{N} \left( -\frac{1}{\sqrt{T}} \right) < 1,$$
which means that the $\mathbb P$-market admits optimal arbitrage.
Applying the Itô formula to \eqref{martrep}, we get the martingale representation:
\begin{align} \label{example:BM_dynamic_hedging}
{\mathbb E}^{\mathbb Q}[1_{\sigma > T}|\mathcal F_t] = {\mathbb Q}[\sigma > T] + \sqrt{\frac{2}{\pi}} \int\limits_0^{\sigma\wedge t} {\frac{1}{\sqrt{T-s}} e^{-\frac{S_s^2}{2(T-s)}} dW^{\mathbb Q}_s}.
\end{align}
Therefore,
$$ H_t = \sqrt{\frac{2}{\pi}} {\frac{1}{\sqrt{T-t}} e^{-\frac{S_t^2}{2(T-t)}}\mathbf 1_{t\leq \sigma}}$$
is the optimal arbitrage strategy, that is, the hedging strategy for $1_{\sigma > T}$ in the $({\mathbb Q},S)$-market as well as the hedging strategy for 1 in the $({\mathbb P},S)$-market.
Let us now compute the dynamics of $S$ under $\mathbb P$.
By Girsanov's Theorem (see, e.g., Theorem 41 on page 136 of \cite{protter_stochastic_2003}),
$$ W^{\mathbb P}_t = W^{\mathbb Q}_t - \frac{2}{{\mathbb Q}[\sigma > T]\sqrt{2\pi}} \int\limits_0^{\sigma\wedge t} {\frac{1}{M_{s}} e^{-\frac{S_s^2}{2(T-s)}} \frac{1}{\sqrt{T-s}} ds}$$
is a ${\mathbb P}$-Brownian motion. The dynamics of $S$ under ${\mathbb P}$ are therefore given by
\begin{align} \label{eq:example_BM_dynamic_S}
S_t &= 1 + W^{\mathbb P}_t + \frac{2}{{\mathbb Q}[\sigma > T]\sqrt{2\pi}} \int\limits_0^{\sigma\wedge t} \frac{e^{-\frac{S_s^2}{2(T-s)}}}{M_s\sqrt{T-s}} ds\\
& = 1 + W^{\mathbb P}_t + \sqrt{\frac{2}{{\pi}}} \int\limits_0^{\sigma\wedge t} {\frac{1} {1 - 2\mathcal{N} \left( -\frac{S_s}{\sqrt{T-s}} \right) } \frac{e^{-\frac{S_s^2}{2(T-s)}}}{\sqrt{T-s}} ds}
\end{align}
Now, let us discuss the fragility and robustness of the arbitrage in this example in the sense of \cite{guasoni_fragility_2012}. The optimal arbitrage constructed using the predictable stopping time $\sigma = \inf\{ t>0: S_t \le 0\}$ is fragile. Indeed, from (\ref{eq:example_BM_dynamic_S}) we can write the dynamics of $X_t : = \log S_t$ as follows
\begin{equation} \label{eq:example_BM_dynamic_logS}
d X_t = {e^{-X_t}}dW^{\mathbb P}_t + \left[ \frac{2}{{\mathbb Q}[\sigma > T]\sqrt{2\pi}} {e^{-X_t}} \frac{1}{M_{t}} e^{-\frac{S_t^2}{2(T-t)}} \frac{1}{\sqrt{T-t}} - \frac{1}{2}e^{-2X_t} \right] dt
\end{equation}
Since in (\ref{eq:example_BM_dynamic_logS}), the drift is locally bounded and the volatility is continuous and nonsingular, by Theorem 2 of \cite{guasoni_fragility_2012}, we conclude that the arbitrage is fragile: for any $\varepsilon>0$, one can find a process $\widetilde S_t$ which admits an equivalent martingale measure and satisfies $1-\varepsilon \leq \frac{\widetilde S(t)}{S(t)}\leq 1+\varepsilon$ a.s. for all $t\in[0,T]$.
However, we can slightly modify the stopping time $\sigma$ to construct an arbitrage which is not destroyed by small perturbations of the price process as above. More precisely, we choose the predictable stopping time $\sigma$ as the first time when $S_t$ hits a line with positive slope, that is
$$\sigma = \inf\{t \ge 0: S_t \le \alpha t\}$$
with $\alpha > 0$. By Proposition 3.2.1.1 in \cite{jeanblanc_mathematical_2009},
\begin{align*}
{\mathbb Q}[\sigma > T] &= {\mathbb Q} \left[ \inf_{0 \le t \le T} (W^{\mathbb Q}_t - \alpha t) > -1 \right] \\
& = \mathcal{N} \left( \frac{1 - \alpha T }{\sqrt{T}} \right) - e^{2 \alpha} \mathcal{N} \left( \frac{-1 - \alpha T }{\sqrt{T}} \right)\in(0,1),
\end{align*}
and we can define a measure $\mathbb P$ admitting optimal arbitrage via \eqref{defprobatau}.
It is easy to see that $S_T > \alpha T$, ${\mathbb P}$-a.s. This allows to construct a simple buy-and-hold arbitrage strategy.
\begin{itemize}
\item If $\alpha T > 1$, buy one unit of $S$ in the begining and hold it until $T$. This strategy yields a profit of $\alpha T - 1$ with probability $1$.
\item If $\alpha T \leq 1$, introduce the stopping time $\sigma_1 = \inf\{t>0: S_t = \frac{\alpha T}{2}\}$. If $\sigma_1 \leq \frac{T}{2}$, buy one unit of $S$ at $\sigma_1$ and hold it until $T$. Otherwise, do nothing. It is easy to see that $\mathbb P[\sigma_1\leq T/2] = \mathbb Q[\sigma_1\leq T/2, \sigma > T]>0$, which means that this strategy yields a profit greater than $\frac{\alpha T}{2}$ with positive probability.
\end{itemize}
{This strategy is a simple obvious arbitrage in the sense of \cite{bender_simple_2012}, which means that it is robust and not fragile (see discussion at the end of section \ref{section:setting}).
}
We are going to compute the dynamics of $\log S$ in this case and compare to the results of \cite{guasoni_fragility_2012}.
By Markov property and the law of infimum of Brownian motion with drift, we compute the conditional probability
\begin{align*}
{\mathbb E}^{\mathbb Q}[1_{\sigma > T}|\mathcal F_t] = \left\{ {\begin{array}{ll}
{0} \qquad \text{if } {\sigma \leq t} \\
{\mathcal{N} \left( \frac{S_t - \alpha T}{\sqrt{T-t}} \right)- e^{2 \alpha (S_t - \alpha t) } \mathcal{N} \left( \frac{-S_t + 2 \alpha t - \alpha T}{\sqrt{T-t}} \right)} \qquad \text{if } \sigma > t.\\
\end{array}} \right.
\end{align*}
Denoting
$$ Y^1_t = \frac{S_t - \alpha T}{\sqrt{T-t}}, \qquad Y^2_t = \frac{-S_t + 2 \alpha t - \alpha T}{\sqrt{T-t}} $$
and applying the Itô formula, we obtain the dynamics of the conditional law on $\sigma<t$:
\begin{align*}
d{\mathbb E}^{\mathbb Q}[1_{\sigma > T}|\mathcal F_t] &= \left[ \frac{1}{\sqrt{2 \pi}} \frac{e^{-\frac{(Y_t^1)^2}{2}}}{\sqrt{T-t}} + \frac{e^{2\alpha (S_t - \alpha t)}}{\sqrt{2 \pi}} \frac{e^{-\frac{(Y_t^2)^2}{2}}}{\sqrt{T-t}} - \mathcal{N}(Y^2_t) 2 \alpha e^{2\alpha (S_t - \alpha t)} \right] dW^{\mathbb Q}_t,\\
\end{align*}
and the dynamics of $M_t$:
$$
dM_t = \frac{1}{{\mathbb Q}[\sigma > T]} \left[ \frac{1}{\sqrt{2 \pi}} \frac{e^{-\frac{(Y_t^1)^2}{2}}}{\sqrt{T-t}} + \frac{ e^{2\alpha (S_t - \alpha t)}}{\sqrt{2 \pi}} \frac{e^{-\frac{(Y_t^2)^2}{2}}}{\sqrt{T-t}} - \mathcal{N}(Y^2_t) 2 \alpha e^{2\alpha (S_t - \alpha t)} \right] dW^{\mathbb Q}_t.
$$
By Girsanov's Theorem,
$$ dW^{\mathbb P}_t = dW^{\mathbb Q}_t - \frac{1}{M_t{\mathbb Q}[\sigma > T]} \left[ \frac{1}{\sqrt{2 \pi}} \frac{e^{-\frac{(Y_t^1)^2}{2}}}{\sqrt{T-t}} + \frac{e^{2\alpha (S_t - \alpha t)}}{\sqrt{2 \pi}} \frac{e^{-\frac{(Y_t^2)^2}{2}}}{\sqrt{T-t}} - \mathcal{N}(Y^2_t) 2 \alpha e^{2\alpha (S_t - \alpha t)} \right]dt$$
is a ${\mathbb P}$-Brownian motion. Finally, the dynamic of $S$ under ${\mathbb P}$ is
$$ dS_t = dW^{\mathbb P}_t + \frac{1}{M_t{\mathbb Q}[\sigma > T]} \left[ \frac{1}{\sqrt{2 \pi}} \frac{e^{-\frac{(Y_t^1)^2}{2}}}{\sqrt{T-t}} + \frac{e^{2\alpha (S_t - \alpha t)}}{\sqrt{2 \pi}} \frac{e^{-\frac{(Y_t^2)^2}{2}}}{\sqrt{T-t}} - \mathcal{N}(Y_t^2) 2 \alpha e^{2\alpha (S_t - \alpha t)} \right]dt.$$
Applying Itô's formula once again, we see that $X_t = \log S_t$ satisfies
\begin{align}
\label{eq:example_BM_dynamic_logS_drift}
dX_t & = e^{-X_t} dS_t -\frac{1}{2}e^{-2X_t}dt \nonumber\\
& = \frac{e^{-X_t}}{M_t{\mathbb Q}[\sigma > T]} \left[ \frac{1}{\sqrt{2 \pi}} \frac{e^{-\frac{(Y_t^1)^2}{2}}}{\sqrt{T-t}} + \frac{e^{2\alpha (S_t - \alpha t)}}{\sqrt{2 \pi}} \frac{e^{-\frac{(Y_t^2)^2}{2}}}{\sqrt{T-t}} - \mathcal{N}(Y^2_t) 2 \alpha e^{2\alpha (S_t - \alpha t)} \right]dt \nonumber\\
&-\frac{1}{2}e^{-2X_t}dt + {e^{-X_t}}dW^{\mathbb P}_t.
\end{align}
The drift in (\ref{eq:example_BM_dynamic_logS_drift}) can be written as a function of $(t, X_t)$, and it is not locally bounded, for example, $1/M$ is unbounded in a neighborhood of $(t, \log (\alpha t))$. So the result of \cite{guasoni_fragility_2012} breaks down.
\subsection{A robust arbitrage based on the Poisson process}
Another way to ensure robustness of arbitrage with respect to small perturbations is to introduce jumps into the price process dynamics.
Let $N$ be a standard Poisson process under ${\mathbb Q}$ and assume that $\mathbb F = \mathbb F^N$, which is a quasi left-continuous filtration. Then $S_t = 1 + N_t - t$ is a ${\mathbb Q}$-martingale. We define a predictable stopping time
$\tau = \inf \{ t>0: S_t \le 0\}$ and a new probability measure ${\mathbb P} \ll {\mathbb Q}$ via $d{\mathbb P}|_{\mathcal F_t} = S_{t \wedge \tau} d{\mathbb Q}|_{\mathcal F_t}$.
The $({\mathbb P},S)$-market admits optimal arbitrage provided $T>1$, because in this case $SP^{\mathbb Q}(1_{S_T > 0}) = {\mathbb Q}[S_T > 0] < 1$.
{Here, unlike the first example of this section or the Bessel process example discussed in \cite{guasoni_fragility_2012}, we can prove that the arbitrage is not fragile. Indeed, we fix a real number $\varepsilon > 0$ and construct a simple buy-and-hold
arbitrage strategy as follows:
\begin{itemize}
\item if $S$ jumps on $[0, \varepsilon]$ then we do nothing.
\item if $S$ does not jump on $[0, \varepsilon]$, we buy one unit of $S$ at $\varepsilon$ and hold it until the first jump time of $S$.
\end{itemize}
Assuming that $T>1$, the process $N$ must jump before $T$, because $S_t> 0, \mathbb{P}$-a.s. This means that this strategy generates a profit greater than $\varepsilon$ with positive probability. Therefore, this is a simple obvious arbitrage in the sense of \cite{bender_simple_2012}, and so it is not fragile (see discussion at the end of section \ref{section:setting}).
}
\subsection{Extension to incomplete markets}
Assume that $S$ is a nonnegative ${\mathbb Q}$ local martingale with only positive jumps starting at 1. Suppose that the conditions in Corollary \ref{cor:optimal_arbitrage_predictable} are fulfilled. The requirement of quasi-left continuity is not so restrictive, for example, the natural completed filtration of a L\'evy process is
a quasi-left continuous filtration (see Exercise 8 page 148 of \cite{protter_stochastic_2003}).
Let $a$ be a positive number such that $aT/2>1$ and define a predictable stopping time by $\sigma = \inf\{t>0: S_t \leq at\}$.
The measure $\mathbb{P}$ is defined by
\begin{align*}\left. {\frac{d{\mathbb P}}{d{\mathbb Q}}} \right|_{\mathcal F_t} = \frac{{\mathbb Q}\left[ \sigma > T | \mathcal F_t \right] }{{\mathbb Q}\left[ \sigma > T \right] }.
\end{align*}
From the economics point of view, this arbitrage represents a bet that the asset price will remain above the line $\alpha t$. Then for any equivalent local martingale measure $\bar {\mathbb Q}$ such that $S$ is a nonnegative $\bar {\mathbb Q}$ local martingale, we have
\begin{align*}
\bar {\mathbb Q}[\sigma \leq T] &\ge \bar {\mathbb Q}[S_{T/2} \le aT/2] = 1 - \bar {\mathbb Q}[S_{T/2} > aT/2] \\
&\ge 1 - \frac{{\mathbb E}^{\bar {\mathbb Q}} [S_{T/2}]}{aT/2} \ge 1 - \frac{1}{aT/2}.
\end{align*}
The superhedging price is $SP^{\mathbb Q}(1_{\sigma > T} ) = \sup_{\bar {\mathbb Q}} \bar {\mathbb Q}[\sigma > T] \le \frac{1}{aT/2} < 1$. Therefore, the $(\mathbb P,S)$-market admits optimal arbitrage. Since $S_T > aT > 2, {\mathbb P}-a.s.$, the presence of arbitrage is robust with respect to small perturbations of $S$.
\subsection{A variation: building an arbitrage from a bubble}
Let $S$ be a nonnegative ${\mathbb Q}$ local martingale with no positive jumps, satisfying $S_0$ = 1 and $S_t \le \varepsilon < 1$, ${\mathbb Q}$-a.s for $t \ge T$. In other words, the market admits a bubble.
We define
$\sigma = \inf\{t \ge 0: S_t > K\}$ for $K > 1$. In this example, a trader believes that the price of $S$ may not exceed an upper bound $K$. As in previous examples, this trader may construct an arbitrage model $\mathbb{P}$ as in (\ref{defprobatau}), provided that the conditions in Corollary \ref{cor:optimal_arbitrage_predictable} are fulfilled.
Under any ELMM $\bar {\mathbb Q}$, $S_{\sigma \wedge t}$ is a bounded $\bar {\mathbb Q}$ local martingale and hence a $\bar {\mathbb Q}$ martingale. So we get
$$1 = {\mathbb E}^{\bar {\mathbb Q}}[S_{\sigma \wedge T}] = K \bar {\mathbb Q}[\sigma \leq T] + S_T \bar {\mathbb Q}[\sigma > T],$$
and therefore
$$\bar {\mathbb Q}[\sigma \leq T] = \frac{1 - S_T\bar {\mathbb Q}[\sigma > T]}{K} > \frac{1-\varepsilon}{K}.$$
The superhedging price of $1_{\sigma > T}$ is
$$ \sup_{\bar {\mathbb Q} \in ELMM({\mathbb Q},S)} {\mathbb E}^{\bar {\mathbb Q}}[1_{\sigma > T}] < 1 - \frac{1-\varepsilon }{K}.$$
\subsection{A joint bet on an asset and its volatility} \label{example:stochastic_volatility}
Let
$$
S_t = 1 + \int_0^t \sigma^S_udW^{\mathbb Q}_u,
$$
where $W^{\mathbb Q}$ is a Brownian motion and $\sigma^S$ is a volatility process
assumed to be continuous. Let $\underline\sigma< \sigma^S_0$
and define stopping times as follows:
$$
\sigma_1 = \inf\{t>0: S_t \leq 0\},\qquad \sigma_2 = \inf\{t>0: \sigma^S_t
\leq \underline\sigma\},\qquad \sigma = \sigma_1\wedge \sigma_2.
$$
Assume that ${\mathbb Q}[\sigma_1 \leq T, \sigma_2 > T] > 0$ and the conditions in Corollary \ref{cor:optimal_arbitrage_predictable} are fulfilled. This choice of the stopping time represents a bet that $S$ will not hit $0$ and its volatility will stay above $\underline{\sigma}$ up to time $T$. Then, under any equivalent local martingale measure $\bar {\mathbb Q}$, we have
\begin{align*}
\bar {\mathbb Q}[\sigma \leq T] &= \bar {\mathbb Q}[\sigma_1 \leq T, \sigma_2 > T] + \bar {\mathbb Q} [\sigma_2 \leq T] \\ &\geq \bar {\mathbb Q}[B^*_{\underline \sigma^2 T}\leq -1, \sigma_2 > T] + \bar {\mathbb Q}[\sigma_2 \leq T] \geq \bar {\mathbb Q}[B^*_{\underline
\sigma^2 T}\leq -1]>0,
\end{align*}
where $B^*_t = \inf_{0\leq u \leq t} B_u$ and $B$ is the Brownian
motion such that
$$\int\limits_0^t {\sigma^S_udW^{\mathbb Q}_u}= B_{\int_0^t (\sigma^S_u)^2 du}.$$
Because $\bar {\mathbb Q}[\sigma \leq T]$ is bounded from below by the quantity $\bar {\mathbb Q}[B^*_{\underline\sigma^2 T}\leq -1]$. This quantity is positive and does not depend on $\bar {\mathbb Q}$, since $B$ is a Brownian motion under $\bar {\mathbb Q}$. Therefore $\sup_{\bar {\mathbb Q}} \bar Q[\sigma > T]$ is bounded away from one.\\
The superhedging price $SP^{\mathbb Q}(1_{\sigma > T} ) = \sup_{\bar {\mathbb Q}} \bar {\mathbb Q}[\sigma > T] < 1$.
\subsection{A variation: betting on the square bracket}
The construction in Example \ref{example:stochastic_volatility} can be modified as follows. Let $S$ be a nonnegative continuous semimartingale starting at 1 and be a ${\mathbb Q}$ local martingale. We define
$$
\sigma_1 = \inf\{t>0: S_t \le 0\},\quad \sigma_2 = \inf\{t>0:[S]_t \le -a + bt\},\quad \sigma = \sigma_1\wedge \sigma_2,
$$
where $a, b$ are positive constants. Suppose that ${\mathbb Q}[\sigma_1 \leq T, \sigma_2 > T] > 0$ and the conditions in Corollary \ref{cor:optimal_arbitrage_predictable} are fulfilled. Then under any ELMM $\bar {\mathbb Q}$, we compute
\begin{align*}
\bar {\mathbb Q}[\sigma \leq T] &= \bar {\mathbb Q}[\sigma_1 \leq T, \sigma_2 > T] + \bar {\mathbb Q} [\sigma_2 \leq T] \\ &\geq \bar {\mathbb Q}[B^*_{[S]_T}\leq -1, \sigma_2 > T] + \bar {\mathbb Q}[\sigma_2 \leq T] \geq \bar {\mathbb Q}[B^*_{bT-a}\leq -1]>0,
\end{align*}
This implies that the superhedging price
$SP^{\mathbb Q}(1_{\sigma > T} ) = \sup_{\bar {\mathbb Q}} \bar {\mathbb Q}[\sigma > T] < 1$.
|
1,116,691,500,873 | arxiv | \subsection*{Acknowledgements} {#1}
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\begin{document}
\title{Well-posedness and singularity formation for inviscid Keller--{Segel}--fluid system of consumption type
\author{In-Jee Jeong\thanks{Department of Mathematical Sciences and RIM, Seoul National University. E-mail: injee\[email protected]}\and Kyungkeun Kang\thanks{Department of Mathematics, Yonsei University. E-mail: [email protected]}}
\date{\today}
\maketitle
\begin{abstract}
We consider the Keller--Segel system of consumption type coupled with an incompressible fluid equation. The system describes the dynamics of oxygen and bacteria densities evolving within a fluid. We establish local well-posedness of the system in Sobolev spaces for partially inviscid and fully inviscid cases. In the latter, additional assumptions on the initial data are required when either the oxygen or bacteria density touches zero. Even though the oxygen density satisfies a maximum principle due to consumption, we prove finite time blow-up of its $C^{2}$--norm with certain initial data.
\end{abstract}
\section{Introduction}
\subsection{The systems}
In this paper, we study a mathematical model, so called Keller--Segel-fluid system, formulating the dynamics of oxygen, swimming bacteria, and viscous incompressible fluids in two or three dimensions.
Such a model was introduced by Tuval et al. \cite{Tuval2277}, proposing the
dynamical behaviors of bacteria, Bacillus subtilis, living in water and consuming oxygen.
To be more precise, we consider the following Keller--Segel-fluid (KSF) system, in its general form, which describes the evolution of the bacteria density ${\boldsymbol{\rho}}$, the oxygen density ${\bf c}$, and the fluid velocity ${\bf u}$:
\begin{equation} \tag{KSF} \label{eq:KSF}
\left\{
\begin{aligned}
&\partial_t \boldsymbol{\rho} + {\bf u}\cdot\nabla {\boldsymbol{\rho}} = D_\rho \Delta {\boldsymbol{\rho}} -\nabla \cdot (\chi({\bf c}){\boldsymbol{\rho}}\nabla {\bf c}) , \\
&\partial_t {\bf c} + {\bf u}\cdot\nabla {\bf c} = D_c\Delta {\bf c} - k({\bf c}){\boldsymbol{\rho}} , \\
&\partial_t {\bf u} + {\bf u}\cdot\nabla {\bf u} + \nabla {\bf p} = D_u \Delta {\bf u} + {\boldsymbol{\rho}}\nabla \phi , \\
& \mathrm{div}\, {\bf u} = 0.
\end{aligned}
\right.
\end{equation}
Here $\chi(\cdot)$ and $k(\cdot)$ are non-negative and smooth functions denoting chemotactic sensitivity and the oxygen consumption rate, respectively. The main features of the model are that the bacteria moves towards the region of high oxygen concentration and the oxygen is being consumed by the bacteria. Moreover, both the oxygen and bacteria concentrations are being transported by the fluid velocity, and the bacteria density affects the fluid velocity via some potential function $\phi$
(see \cite{Tuval2277} for more details on biological phenomena and \cite{CFKLM} for numerical investigations).
Together with \eqref{eq:KSF}, we shall consider the simplified model obtained by formally taking ${\bf u} \equiv 0$:
\begin{equation} \tag{KS} \label{eq:KS}
\left\{
\begin{aligned}
& \partial_t{\boldsymbol{\rho}} = D_\rho\Delta{\boldsymbol{\rho}} - \nabla\cdot(\chi({\bf c}){\boldsymbol{\rho}}\nabla {\bf c}) ,\\
& \partial_t {\bf c} = D_c\Delta {\bf c} - k({\bf c}){\boldsymbol{\rho}} ,
\end{aligned}
\right.
\end{equation} which is a Keller--Segel-type system (see e.g. \cite{MR2860628}). We shall mainly consider the above systems in the domain $\mathbb T^d$ for some $d\ge 1$, but many of our results carry over to the case of $\mathbb R^k\times\mathbb T^{d-k}$ for $0<k\le d$ and bounded domains of $\mathbb R^d$ with smooth boundary. In this note, we always assume that ${\boldsymbol{\rho}}, {\bf c} \ge 0$, which is preserved by the dynamics of \eqref{eq:KSF} and \eqref{eq:KS}. Observe that the assumption $k \ge 0$ yields a maximum principle for ${\bf c}$: as long as the solution exists, $\nrm{{\bf c}(t)}_{L^\infty}$ decreases with time.
We can compare \eqref{eq:KS} to the classical Keller--Segel (cKS) model
suggested by Patlak \cite{MR81586} and Keller--Segel \cite{MR3925816} to describe the motion of amoeba, Dictyostelium discoideum living in soils, which is given as
\begin{equation} \tag{cKS} \label{eq:cKS}
\left\{
\begin{aligned}
& \partial_t{\boldsymbol{\rho}} = D_\rho\Delta{\boldsymbol{\rho}} - \nabla\cdot(\chi({\bf c}){\boldsymbol{\rho}}\nabla {\bf c}) ,\\
& \tau \partial_t {\bf c} = D_c\Delta {\bf c} +\alpha{\boldsymbol{\rho}} -\beta {\bf c},
\end{aligned}
\right.
\end{equation}
where $\alpha$ and $\beta$ are positive constants indicating production and decaying rate of chemical
substance, respectively.
Here the system \eqref{eq:cKS} consists of biological cell density ${\boldsymbol{\rho}}$ and the concentration of chemical attractant
substance ${\bf c}$ that is produced by ${\boldsymbol{\rho}}$. Compared to \eqref{eq:KS}, the second equation of \eqref{eq:cKS} has a positive coefficient for $\rho$, which seems to cause significantly different behaviors of solutions.
In general, finite time blow-up results of the system \eqref{eq:cKS} have been known for large data in case that $D_\rho>0$, $D_c>0$ and $\tau\ge 0$ (see e.g. \cite{MR1046835}, \cite{MR1627338}, \cite{MR3115832}). There
are also some related results for inviscid models with either $D_\rho=0$ or $D_c=0$ in \eqref{eq:cKS}, but we are not going to list all of the related results here (see e.g. \cite{MR2099126}, \cite{MR3223867}, \cite{MR3473109}, reference therein).
\subsection{Main results}
Much of the mathematical literature concerning the systems \eqref{eq:KSF} and \eqref{eq:KS} was devoted to establishing global well-posedness of solutions, assuming that the dissipation coefficients $D_\rho$, $D_c$, and $D_u$ are strictly positive.
Since our main concerns are focused on partially or fully inviscid case, we shall not try to review all related results, but instead,
just list a few of them related to the issue of global well-posedness.
Ever since mathematical results were made initially in \cite{MR2659745} and \cite{MR2754058} regarding local well-posedness and stability near constant steady state of solutions, the issue on global well-posedness or blow-up in a finite time has received lots of speculation. Among other results, it was shown in \cite{MR2876834} that regular solutions exist globally in time for general data in two dimensions under certain assumptions on $\chi$ and $k$ (we refer to \cite{MR3149063} {and \cite{MR3605965}} for asymptotic behaviors of solutions). We note that the paper \cite{CKL1} establishes global well-posedness of smooth solutions with different conditions on $\chi$ and $k$ in $\mathbb R^2$ (see also \cite{CKL2}, \cite{MR3450942} and \cite{MR3755872} for
blow-up criteria, temporal decay and asymptotic of solutions, { and \cite{MR3542616} for weak solutions}).
Very recently, \cite{MR4185378} obtained global well-posedness with slightly relaxed the conditions on $\chi$ and $k$, compared to those in \cite{MR2876834}.
On the other hand, the goal in this paper is to study local well-posedness of smooth solutions when some or all of the dissipation coefficients vanish. To the best of the authors' knowledge, the only result in this direction is the one given in \cite{CKL2}, where the authors prove global well-posedness in the case $D_c = 0$ and $D_\rho>0$, provided that $L^{N/2}$-norm of $\rho_0$ is sufficiently small.
Our primary motivation for studying the inviscid (or partially inviscid) system is that norm growth for the inviscid system could be responsible for instabilities arising in the slightly viscous case. Indeed, in the original paper \cite{Tuval2277} (see also \cite{CFKLM}) where the system was introduced and instabilities were explored numerically, the dissipation coefficients take values in $10^{-4} \sim 10^{-2}$ after nondimensionalization.
In the presence of local-in-time wellposedness for the inviscid system, one can establish in general that the corresponding viscous problem for the same initial data cannot blow-up \textit{earlier} than the inviscid case; this type of result goes back to a classical work of Constantin \cite{Con2}. On the contrary, if the inviscid equation blows up in finite time, it can be expected that the slightly viscous system should go through a large norm growth, which is divergent as the viscosity constant goes to zero. Similar phenomena should occur in the extreme case of the inviscid system being \textit{ill-posed}, meaning that the Sobolev norm of the solution could blow up in an arbitrary short interval of time. In other words, ``norm inflation'' for the inviscid equation could be responsible for instabilities appearing in the corresponding slightly viscous case. This type of result was recently established for the incompressible Euler and Navier--Stokes equations in \cite{JY, JY2} { (see also \cite{MR3223867}, \cite{MR3782543}, \cite{CKL2}, and \cite{MR3473109} for the case of partially invicid case of Keller--Segel type equations)}.
The conclusion of this paper is that the systems \eqref{eq:KSF} and \eqref{eq:KS} are indeed \textit{locally well-posed} even in the fully inviscid case $D_\rho=D_c=D_u=0$. This is somewhat surprising, since the equation for ${\boldsymbol{\rho}}$ involves two derivatives of ${\bf c}$. Roughly speaking, loss of one derivative can be gained by pairing the derivative of ${\bf c}$ together with ${\boldsymbol{\rho}}$, and the other loss of derivative can be handled by performing a \textit{modified} energy estimate: the key observation is that in the expression of \begin{equation*}
\begin{split}
\frac{d}{dt}\int \frac{k({\bf c} )}{\chi({\bf c})} (\partial^m{\boldsymbol{\rho}})^2 + {\boldsymbol{\rho}} (\partial^{m+1}{\bf c})^2,
\end{split}
\end{equation*} the top order terms completely cancel each other, which allows to close an a priori estimate. Here, the signs $\chi, k\ge 0$ are fundamental: if one of the either sign is reversed, then we expect the resulting systems to be strongly \textit{ill-posed} in sufficiently regular Sobolev spaces. Such an ill-posed behavior could be easily seen from a simple linear analysis which is performed later in Section \ref{sec:inviscid}, Remark \ref{rem:illposed}. This already demonstrates a striking difference between Keller--Segel equations of consumption and production type. {Furthermore, we remark that the modified energy method is robust enough to handle the general flux case (considered very recently in \cite{WinklerIMRN}) where $\nabla\cdot (\chi({\bf c}){\boldsymbol{\rho}} \nabla{\bf c})$ in (KS) is replaced by $\nabla\cdot (\chi({\bf c}){\boldsymbol{\rho}} S \nabla{\bf c})$, under some assumptions on the matrix $S$. See Remark \ref{rem:rotation} below.
}
\subsubsection{Dissipation only on the oxygen concentration}
Our first main results concerns local well-posedness of \eqref{eq:KSF} and \eqref{eq:KS} with smooth initial data when $D_u = 0$ and $D_\rho=0$ but $D_c>0$. While the precise statement is given in Theorem \ref{thm:c-viscous}, we state the main result briefly here, in the simplest setup of $\mathbb T^d$. {We consider $d\ge2$, simply because in the case $d=1$, there are no non-trivial incompressible velocity fields.}
\begin{customthm}{A}[Local well-posedness with dissipation only on ${\bf c}$]\label{MainThm1}
The system \eqref{eq:KSF} in the case $D_{\rho}, D_u \ge 0$ and $D_c >0$ is locally well-posed in $({\boldsymbol{\rho}},{\bf c},{\bf u}) \in (H^m\times H^{m+1}\times H^{m})(\mathbb T^d)$ for any $m>\frac{d}{2}+1$ {with $d\ge2$}.
\end{customthm}
The proof of Theorem \ref{MainThm1} is rather straightforward, the main observation being that the $H^m$-regularity of ${\boldsymbol{\rho}}$ and ${\bf u}$ should be paired with the $H^{m+1}$-regularity of ${\bf c}$. In Theorem \ref{thm:c-viscous}, we actually handle the case of bounded domains with Neumann boundary condition for ${\bf c}$, which gives several technical complications. We defer the detailed discussion to Section \ref{sec:c-viscous}, where we state and prove a precise version of the above result.
\subsubsection{Fully inviscid case}
As we shall demonstrate in Section \ref{sec:inviscid}, a simple linear analysis around trivial steady states of \eqref{eq:KSF} shows that the system could be well-posed even in the absence of dissipation. Let us first consider the simpler case of \eqref{eq:KS}. The key ingredient in the proof of local well-posedness, as illustrated in the above, is to use the modified norms $\nrm{{\bf c}^{\frac{1}{2}}\partial^m{\boldsymbol{\rho}}}_{L^2}$ and $\nrm{{\boldsymbol{\rho}}^{\frac{1}{2}}\partial^{m+1}{\bf c}}_{L^2}$, in the simplest model case of $k(z)=z$ and $\chi(z)=1$. Combining these modified norms, we can guarantee a crucial cancellation of top order terms, which allows one to close an a priori estimate in $({\boldsymbol{\rho}},{\bf c})\in H^{m}\times H^{m+1}$ as long as ${\boldsymbol{\rho}}$ and ${\bf c}$ stay away from zero. With such a modified energy estimate for \eqref{eq:KS}, local well-posedness can be extended without much difficulty to the case of \eqref{eq:KSF}, when the fluid viscosity is strictly positive; that is, $D_u>0$. The fully inviscid assumption $D_u = 0$ gives rise to a serious problem; to explain it in simple terms, the equation for $\partial^{m+1}{\bf c}$ involves $\partial^{m+1}{\bf u}$, and in turn, the equation for $\partial^{m+1}{\bf u}$ involves $\partial^{m+1}{\boldsymbol{\rho}}$. Finally, the equation for $\partial^{m+1}{\boldsymbol{\rho}}$ costs $m+3$ derivatives of ${\bf c}$. The way out of this loop is to consider modified energy estimate for \textit{good variables}; it turns out that the time derivative of the quantity \begin{equation}\label{eq:key}
\begin{split}
\int \frac{k({\bf c})}{\chi({\bf c})} \left| \partial^m{\boldsymbol{\rho}} + \frac{\nabla{\bf c} }{k({\bf c})} \cdot \partial^m{\bf u} \right|^2 + {\boldsymbol{\rho}} \left| \partial^m \nabla {\bf c} - \frac{\nabla{\bf c}}{{\boldsymbol{\rho}} k({\bf c})} \nabla {\bf c} \cdot \partial^m {\bf u} \right|^2 + \left| \partial^m {\bf u} \right|^2
\end{split}
\end{equation} satisfies a good energy structure, for any $m$. Note that when ${\boldsymbol{\rho}}$ and ${\bf c}$ are strictly positive, the above is equivalent with the usual $\dot{H}^m$-norm for $({\boldsymbol{\rho}},\nabla{\bf c},{\bf u})$. This result is briefly stated as follows:
\begin{customthm}{B}[Local well-posedness in the inviscid case with non-vanishing ${\boldsymbol{\rho}}$ and ${\bf c}$]\label{MainThm2}
The system \eqref{eq:KSF} in the case $D_{\rho}=D_u=D_c = 0$ is locally well-posed in $({\boldsymbol{\rho}},{\bf c},{\bf u}) \in (H^m\times H^{m+1}\times H^{m})(\mathbb T^d)$ for any $m>\frac{d}{2}+1$, {$d\ge2$}, and strictly positive ${\boldsymbol{\rho}}_0$ and ${\bf c}_0$.
\end{customthm}
The precise statement of the above is given as Theorem \ref{thm:lwp-away} below. Given this result, it is natural to ask what happens when either ${\boldsymbol{\rho}}$ or ${\bf c}$ vanishes at some points. In this case, the inclusion of velocity (with $D_u=0$) gives rise to a serious issue as the coefficients \begin{equation*}
\begin{split}
\frac{\nabla{\bf c} }{k({\bf c})}, \quad \frac{\nabla{\bf c}}{{\boldsymbol{\rho}} k({\bf c})} \nabla {\bf c}
\end{split}
\end{equation*} are in general \textit{singular} (unbounded when ${\bf c}$ vanishes), which does not allow one to deduce regularity of ${\boldsymbol{\rho}}$ and ${\bf c}$ even when the integral \eqref{eq:key} is finite. Restricting ourselves to the case of \eqref{eq:KS}, we can still ensure existence and uniqueness of smooth solutions, upon imposing some assumptions on the initial data (which we prove to propagate in time) near their zeroes. For now, let us roughly state it as follows:
\begin{customthm}{C}[Local well-posedness in the inviscid case with possibly vanishing ${\boldsymbol{\rho}}$ and ${\bf c}$]\label{MainThm3}
{For any $d\ge1$,} the system \eqref{eq:KS} in the case $D_{\rho}=D_c = 0$ is locally well-posed in sufficiently regular Sobolev spaces with possibly vanishing initial data ${\boldsymbol{\rho}}_0$ and ${\bf c}_0$, under some additional assumptions on the data near their zeroes.
Within this locally well-posed class of solutions, there exist initial data ${\boldsymbol{\rho}}_0, {\bf c}_0 \in C^\infty(\mathbb T^d)$ whose unique solution to \eqref{eq:KS} blows up in finite time; more concretely, there exists some $T^*>0$ such that $\nrm{{\bf c}(t)}_{C^2} \rightarrow+\infty$ as $t \to T^*$.
\end{customthm}
{
\begin{remark}
The terminology ``sufficiently regular'' and the further assumptions that we need to impose in the case of vanishing initial data are somewhat complicated to define and therefore we defer the precise statements to Theorems \ref{thm:lwp-rho-away} and \ref{thm:quadratic}.
\end{remark}
}
As stated in this theorem, we are able to prove singularity formation within this class, which is surprising since ${\bf c}$ satisfies a maximum principle: $\nrm{{\bf c}}_{L^\infty}$ is decreasing with time. A rough heuristics for this singularity formation can be given as follows: arrange initial data for \eqref{eq:KS} so that ${\boldsymbol{\rho}}$ takes its minimum value where ${\bf c}$ reaches its maximum. Then, since the consumption of ${\bf c}$ is proportional to ${\boldsymbol{\rho}}$, the profile of ${\bf c}$ becomes steeper near its maximum. This steepening of ${\bf c}$ is enhanced by the dynamics of ${\boldsymbol{\rho}}$, which concentrates towards the maximum of ${\bf c}$. It will be very interesting to extend this singularity formation to the case of a partially viscous \eqref{eq:KSF} system.
When vanishing of either ${\bf c}$ or ${\boldsymbol{\rho}}$ occurs, it is no longer true that having a bound on the quantity \eqref{eq:key} ensures smoothness. Indeed, the Keller--Segel model in this case should be interpreted as a \textit{degenerate hyperbolic} system, and our main technical tool for local well-posedness is the following weighted Gagliardo--Nirenberg--Sobolev inequalities: \begin{lemma}[Key Lemma] \label{lem:key}
Let $g \ge 0$ on $\mathbb T^d$. For any $\gamma>0$, $m \in \mathbb N$, and a $k$-tuple $(\ell_1, \cdots, \ell_k)$ of $d$-vectors satisfying $m = |\ell_1| + \cdots + |\ell_k|$, we have that\begin{equation}\label{eq:key-1}
\begin{split}
\int \frac{ 1 }{ g^{2k-2+\gamma} } \prod_{1\le i \le k} |\partial^{\ell_i} g|^2 \lesssim_{\gamma,m,d} \int \frac{|\nabla^m g|^2}{ g^{\gamma}} + \int \frac{|\nabla g|^{2m}}{ g^{2m-2+\gamma}},
\end{split}
\end{equation} assuming that the right hand side is finite. In particular, under the same assumption, we obtain that for any $0\le n \le m$,
\begin{equation}
\begin{split}
\nrm{ \nabla^{m-n}( \frac{\nabla^{n} g}{ { g}^{\frac{\gamma}{2}} })}_{L^2}^2 \lesssim_{\gamma,m,d} \int \frac{|\nabla^m g|^2}{ g^{\gamma}} + \int \frac{|\nabla g|^{2m}}{ g^{2m-2+\gamma}}
\end{split}
\end{equation} and for any $k\ge1$
\begin{equation}
\begin{split}
\sup_{x \in \mathbb T^d} \frac{|\nabla g(x)|^{k}}{ g(x)^{k-1+\frac{\gamma}{2}}} \lesssim_{\gamma,k,d} \sum_{m=1}^{\lfloor \frac{d}{2} \rfloor +1 + k} \left( \int \frac{|\nabla^m g|^2}{ g^{\gamma}} + \int \frac{|\nabla g|^{2m}}{ g^{2m-2+\gamma}} \right) .
\end{split}
\end{equation}
\end{lemma} While there exist several versions of Gagliardo--Nirenberg--Sobolev inequalities with a spatial weight, we are not aware of ones in which the weight depends on the function itself. Note that if $g$ is $C^\infty$--smooth and vanishes with sufficiently high order near its zeroes, depending on given $m,\gamma$, the right hand side of \eqref{eq:key-1} is finite. Moreover, for bounded $g$, the right hand side of \eqref{eq:key-1} trivially controls the usual Sobolev norm $H^{m}$. We believe that the above set of inequalities will be a useful tool in the analysis of \textit{degenerate hyperbolic} systems. Let us defer further technical discussion relevant to the statements and difficulties involved with Theorem \ref{MainThm3} to the introduction of Section \ref{sec:inviscid}. For now, let us observe that unlike the other results, Theorem \ref{MainThm3} is very sensitive to the profiles of $\chi$ and $k$ (in the region where ${\bf c}$ is small). Under suitable assumptions on $\chi$ and $k$, this result could be extended to the case of \eqref{eq:KSF}, but we shall not pursue this generalization in this work.
\subsubsection*{Notation and conventions}
Given a function $f : \mathbb R^d \rightarrow \mathbb R $ and an integer $m \ge 0$, $\nabla^mf$ denotes the $d^m$-dimensional vector consisting of all possible partial derivatives of $f$ of order $m$. We define the $H^m$-spaces as
\begin{equation*}
\begin{split}
& \nrm{f}_{H^m}^2 = \sum_{j = 0}^m \nrm{f}_{\dot{H}^j}^2 = \sum_{j = 0}^m \nrm{\nabla^j f}_{L^2}^2
\end{split}
\end{equation*} Next, we define the $L^p$-norm by \begin{equation*}
\begin{split}
& \nrm{f}_{L^p}^p = \int |f|^p \, dx, \quad \nrm{f}_{L^\infty} = \mathrm{esssup}|f(x)|.
\end{split}
\end{equation*}
\section{Local well-posedness with dissipation only on ${\bf c}$}\label{sec:c-viscous}
In this section, we prove local well-posedness for smooth solutions of \eqref{eq:KSF} in the case when there is dissipation only in ${\bf c}$. We shall assume that the equation is posed on a bounded domain $\Omega \subset \mathbb R^d$ with $C^\infty$-smooth boundary. The proof we present easily adapts to the case of domains without boundary, for example to $\mathbb R^k\times\mathbb T^{d-k}$. Presence of the boundary makes the well-posedness proof more technically involved since in general the derivatives of the dependent variables do not satisfy simple boundary conditions.
Without loss of generality, we may normalize $D_c=1$, and the system becomes
\begin{equation} \label{eq:KSF-bdd}
\left\{
\begin{aligned}
&\partial_t \boldsymbol{\rho} + {\bf u}\cdot\nabla {\boldsymbol{\rho}} = -\nabla \cdot ( {\boldsymbol{\rho}}\nabla {\bf c}) , \\
&\partial_t {\bf c} + {\bf u}\cdot\nabla {\bf c} = \Delta {\bf c} - {\bf c}{\boldsymbol{\rho}} , \\
&\partial_t {\bf u} + {\bf u}\cdot\nabla {\bf u} + \nabla {\bf p} = {\boldsymbol{\rho}}\nabla \phi , \\
& \mathrm{div}\, {\bf u} = 0.
\end{aligned}
\right.
\end{equation} Moreover, we shall impose the natural boundary conditions \begin{equation}\label{eq:KSF-bdry}
\begin{split}
{\bf n} \cdot {\bf u} = 0, \quad \partial_{{\bf n}} {\bf c} = 0 \quad \mbox{on}\quad \partial\Omega
\end{split}
\end{equation} where ${\bf n}$ is the outward unit normal vector on $\partial\Omega$. This ensures that the characteristics defined by ${\bf u}$ and ${\bf u} + \nabla{\bf c}$ preserves $\Omega$.
\begin{theorem}\label{thm:c-viscous}
Assume that the initial triple satisfies $$({\boldsymbol{\rho}}_0,{\bf c}_0,{\bf u}_0) \in (H^{m}\times H^{m+1}\times H^m)(\Omega)$$ and the boundary conditions $$\mathrm{div}({\bf u}_0) = 0, \quad \mbox{in}\quad \Omega,$$ $$ {\bf n} \cdot {\bf u}_0 = 0,\quad \mbox{on} \quad \partial\Omega,$$
and
$$\partial_{{\bf n}}(\partial_t^j {\bf c})(t=0) = 0,\quad 0\le j \le \left\lceil \frac{m+1}{2} \right\rceil \quad \mbox{on}\quad \partial\Omega.$$
Then, there exist $T>0$ and a unique solution satisfying
$$({\boldsymbol{\rho}},{\bf c},{\bf u}) \in L^\infty_t([0,T);(H^{m}\times H^{m+1}\times H^m)(\Omega)), \qquad {\bf c} \in L^2_t([0,T);H^{m+2}(\Omega)) $$
of \eqref{eq:KSF-bdd}--\eqref{eq:KSF-bdry} satisfying $({\boldsymbol{\rho}},{\bf c},{\bf u})(t=0) = ({\boldsymbol{\rho}}_0,{\bf c}_0,{\bf u}_0)$. The unique solution blows up at some $0<T^* < + \infty$ if and only if \begin{equation*}
\begin{split}
\lim_{t\nearrow T^*} \left( \nrm{{\bf c}(t)}_{W^{2,\infty}} + \nrm{{\boldsymbol{\rho}}(t)}_{W^{1,\infty}} + \nrm{{\bf u}(t)}_{W^{1,\infty}} \right) = +\infty.
\end{split}
\end{equation*}
\end{theorem}
\begin{remark}
The assumption $\partial_{{\bf n}}(\partial_t^j {\bf c})(t=0) = 0$ is a compatibility condition for the solution to be uniformly smooth up to the boundary, which is easily guaranteed when ${\bf c}_{0}$ is supported away from the boundary. Using the equation for ${\bf c}$, one can rewrite this assumption in terms of $({\boldsymbol{\rho}}_0,{\bf c}_0,{\bf u}_0)$ and their derivatives. {It is already complicated in the case $j=1$: we have \begin{equation*}
\begin{split}
\partial_{{\bf n}} (\partial_t {\bf c})(t=0)= \partial_{{\bf n}}( - {\bf u}_0\cdot\nabla {\bf c}_0 + \Delta {\bf c}_0 - {\bf c}_0{\boldsymbol{\rho}}_0).
\end{split}
\end{equation*} }
\end{remark}
\begin{remark}
{Note that we have taken $\chi=1$ and $k(z)=z$. In the case of Theorem \ref{thm:c-viscous}, extending local well-posedness to the case of general smooth $\chi$ and $k$ is straightforward as additional terms arising from derivatives of $\chi$ and $k$ are not the highest order terms.}
\end{remark}
\begin{proof}
For simplicity, we shall fix $d = 2$ and $m = 3$. There are no essential differences in handling the general case. {To begin with, let us obtain \textit{a priori estimates} for a solution $({\boldsymbol{\rho}},{\bf c},{\bf u})$ of \eqref{eq:KSF-bdd}, which is assumed to be sufficiently smooth and satisfy the boundary conditions, so that the following computations can be justified. Later we shall show how such estimates can be justified, using an approximate solution sequence. We shall divide the proof of a priori estimates into three parts, which correspond to $L^2$, $H^2$, and $H^4$ estimate in terms of ${\bf c}$.
}
\medskip
\noindent \textit{(i) $L^2$ estimate}: To begin with, directly from \eqref{eq:KSF-bdd} we obtain the following $L^2$-estimate: \begin{equation}\label{eq:KSF-bdd-L2}
\begin{split}
\frac{1}{2}\frac{d}{dt}(\nrm{{\boldsymbol{\rho}}}_{L^2}^2 + \nrm{{\bf c}}_{L^2}^2+\nrm{{\bf u}}_{L^2}^2) + \nrm{\nabla{\bf c}}_{L^2}^2 \lesssim \nrm{\Delta{\bf c}}_{L^\infty}\nrm{{\boldsymbol{\rho}}}_{L^2}^2 + \nrm{{\boldsymbol{\rho}}}_{L^\infty} \nrm{{\bf c}}_{L^2}^2 + \nrm{{\boldsymbol{\rho}}}_{L^2}\nrm{{\bf u}}_{L^2}.
\end{split}
\end{equation} We have used the boundary conditions for ${\bf u}$ and ${\bf c}$.
\medskip
\noindent \textit{(ii) $H^2$ estimate}: Next, we consider the set of variables $(\nabla{\boldsymbol{\rho}}, {\bf c}_t,{\nabla{\bf u}})$ where ${\bf c}_t:=\partial_t{\bf c}$: \begin{equation}\label{eq:KSF-bdd-der}
\left\{ \begin{aligned}
&\partial_t\nabla{\boldsymbol{\rho}} + ({\bf u}+\nabla{\bf c})\cdot\nabla (\nabla{\boldsymbol{\rho}}) = -\nabla{\boldsymbol{\rho}} \Delta{\bf c} - {\boldsymbol{\rho}}\cdot\nabla\Delta{\bf c} - (\nabla{\bf u}+\nabla^2{\bf c})\cdot\nabla{\boldsymbol{\rho}} \\
&\partial_t{\bf c}_t + {\bf u}\cdot\nabla{\bf c}_t = \Delta{\bf c}_t - {\boldsymbol{\rho}}{\bf c}_t - {\bf c} {\boldsymbol{\rho}}_t - {\bf u}_t\cdot\nabla{\bf c} \\
&{\partial_t\nabla {\bf u}+ {\bf u}\cdot\nabla \nabla {\bf u}= - \nabla{\bf u} \cdot\nabla{\bf u} - \nabla^2 {\bf p} + \nabla({\boldsymbol{\rho}}\nabla\phi) .}
\end{aligned} \right.
\end{equation}
Here, we are using ${\boldsymbol{\rho}}_t = \partial_t{\boldsymbol{\rho}}$ and ${\bf u}_t = \partial_t{\bf u}$. It is important that (formally) ${\bf c}_t$ satisfies the same boundary conditions with ${\bf c}$. In the equation for $\nabla{\boldsymbol{\rho}}$, using that ${\bf n}\cdot ({\bf u}+\nabla{\bf c}) = 0$ on $\partial\Omega$, we obtain that \begin{equation*}
\begin{split}
\frac{d}{dt} \nrm{\nabla{\boldsymbol{\rho}}}_{L^2}^2 \lesssim \nrm{{\nabla^2}{\bf c}}_{L^\infty}\nrm{\nabla{\boldsymbol{\rho}}}_{L^2}^2 + \nrm{{\boldsymbol{\rho}}}_{L^\infty}\nrm{\nabla\Delta{\bf c}}_{L^2}\nrm{\nabla{\boldsymbol{\rho}}}_{L^2} + \nrm{\nabla{\bf u}}_{L^\infty}\nrm{\nabla{\boldsymbol{\rho}}}_{L^2}^2 .
\end{split}
\end{equation*} Then,\begin{equation}\label{eq:rho-H2}
\begin{split}
\frac{d}{dt} \nrm{\nabla{\boldsymbol{\rho}}}_{L^2}^2 \le \epsilon\nrm{\nabla\Delta{\bf c}}_{L^2}^2 + C_\epsilon (\nrm{{\boldsymbol{\rho}}}_{L^\infty}^2 + \nrm{\Delta{\bf c}}_{L^\infty}+\nrm{\nabla{\bf u}}_{L^\infty} )\nrm{\nabla{\boldsymbol{\rho}}}_{L^2}^2
\end{split}
\end{equation} for any small $\epsilon>0$. We now move on to the estimate of ${\bf c}_t$.
First, we note that
\begin{equation*}
\begin{split}
\nrm{{\bf u}_t}_{L^2} \lesssim \nrm{{\bf u}\cdot\nabla{\bf u}}_{L^2}+\nrm{\nabla{\bf p}}_{L^2} + \nrm{{\boldsymbol{\rho}}\nabla\phi}_{L^2} \lesssim \nrm{\nabla{\bf u}}_{L^\infty} \nrm{{\bf u}}_{L^2} + \nrm{{\boldsymbol{\rho}}}_{L^2}.
\end{split}
\end{equation*} To estimate the pressure term appearing in the right hand side, recall that ${\bf p}$ is the unique solution (up to a constant) to \begin{equation*}
\begin{split}
\Delta {\bf p} & = - \sum_{1 \le i,j \le 2} \partial_i {\bf u}^j \partial_j {\bf u}^i + \nabla\cdot({\boldsymbol{\rho}}\nabla\phi) \mbox{ in } \Omega, \\
\partial_{{\bf n}}{\bf p} & = {\boldsymbol{\rho}} \partial_{{\bf n}} \phi \mbox{ on } \partial\Omega.
\end{split}
\end{equation*} Therefore, ${\bf p}$ satisfies the estimate \begin{equation}\label{eq:pressure}
\begin{split}
\nrm{\nabla {\bf p}}_{H^m} \lesssim \nrm{\nabla{\bf u}}_{L^\infty}\nrm{{\bf u}}_{H^m} + \nrm{\nabla\phi}_{H^m} \nrm{{\boldsymbol{\rho}}}_{H^m}
\end{split}
\end{equation} for each $m\ge0$ (see \cite{GiTr} as well as Step 1 of the proof of Theorem \ref{thm:lwp-away} below). Since we may assume that $\int_\Omega {\bf p} = 0$, we have from Poincar\'{e} inequality that $\nrm{{\bf p}}_{L^2} \lesssim \nrm{\nabla{\bf p}}_{L^2}$.
Next, with an integration by parts, \begin{equation*}
\begin{split}
\int {\bf c} {\boldsymbol{\rho}}_t {\bf c}_t = -\int {\bf c} {\bf u}\cdot\nabla{\boldsymbol{\rho}} {\bf c}_t + \int {\boldsymbol{\rho}}\nabla{\bf c} \cdot \nabla ({\bf c}\bfc_t).
\end{split}
\end{equation*} This gives the bound \begin{equation*}
\begin{split}
\left| \int {\bf c} {\boldsymbol{\rho}}_t {\bf c}_t \right| \lesssim \nrm{{\bf c}}_{L^\infty}\nrm{{\bf u}}_{L^\infty} \nrm{\nabla{\boldsymbol{\rho}}}_{L^2}\nrm{{\bf c}_t}_{L^2} + \nrm{\nabla{\bf c}}_{L^\infty}\nrm{{\boldsymbol{\rho}}}_{L^\infty} \nrm{\nabla{\bf c}}_{L^2}\nrm{{\bf c}_t}_{L^2}+ \nrm{{\boldsymbol{\rho}}}_{L^\infty} \nrm{{\bf c}}_{L^\infty}\nrm{\nabla{\bf c}}_{L^2}\nrm{\nabla{\bf c}_t}_{L^2} .
\end{split}
\end{equation*}
Moreover, using the boundary condition for ${\bf c}_t$, we can use integration by parts in the convection term to obtain \begin{equation*}
\begin{split}
\frac{1}{2}\frac{d}{dt}\nrm{{\bf c}_t}_{L^2}^2 + \nrm{\nabla{\bf c}_t}_{L^2}^2 &\le C \left(\nrm{{\boldsymbol{\rho}}}_{L^\infty}\nrm{{\bf c}_t}_{L^2}^2 + \nrm{{\bf c}}_{L^\infty}\nrm{{\bf u}}_{L^\infty} \nrm{{\bf c}_t}_{L^2}\nrm{\nabla{\boldsymbol{\rho}}}_{L^2} + \nrm{\nabla{\bf c}}_{L^\infty}\nrm{{\boldsymbol{\rho}}}_{L^\infty} \nrm{\nabla{\bf c}}_{L^2}\nrm{{\bf c}_t}_{L^2} \right) \\
&\quad {+ C\nrm{{\bf u}}_{L^\infty}\nrm{\nabla {\bf u}}_{L^\infty} \nrm{\nabla{\bf c}}_{L^2} \nrm{{\bf c}_t}_{L^2} } + \epsilon \nrm{\nabla{\bf c}_t}_{L^2}^2 + C_\epsilon \nrm{{\bf c}}_{L^\infty}^2 \nrm{{\boldsymbol{\rho}}}_{L^\infty}^2 \nrm{\nabla{\bf c}}_{L^2}^2 .
\end{split}
\end{equation*} From the equation for ${\bf c}$, we have \begin{equation*}
\begin{split}
\nrm{\Delta{\bf c}}_{L^2} & \lesssim \nrm{{\bf c}_t}_{L^2} + \nrm{{\bf c}{\boldsymbol{\rho}}}_{L^2} + \nrm{{\bf u}\cdot\nabla{\bf c}}_{L^2} \lesssim \nrm{{\bf c}_t}_{L^2} + \nrm{{\boldsymbol{\rho}}}_{L^\infty} \nrm{{\bf c}}_{L^2} + \nrm{{\bf u}}_{L^\infty}\nrm{{\bf c}}_{L^2}^{\frac{1}{2}} \nrm{{\bf c}}_{H^2}^{\frac{1}{2}}
\end{split}
\end{equation*} which gives \begin{equation*}
\begin{split}
\nrm{{\bf c}}_{H^2} & \lesssim \nrm{{\bf c}_t}_{L^2} + \left(1 + \nrm{{\boldsymbol{\rho}}}_{L^\infty} + \nrm{{\bf u}}_{L^\infty}^2 \right) \nrm{{\bf c}}_{L^2}
\end{split}
\end{equation*} after using $\epsilon$-Young inequality. Similarly, we can estimate \begin{equation}\label{eq:rho-H2-prime}
\begin{split}
\nrm{\nabla\Delta{\bf c}}_{L^2} &\lesssim \nrm{\nabla{\bf c}_t}_{L^2} + (\nrm{{\bf c}}_{L^\infty}+\nrm{{\boldsymbol{\rho}}}_{L^\infty})(\nrm{\nabla{\bf c}}_{L^2}+\nrm{\nabla{\boldsymbol{\rho}}}_{L^2}) + \nrm{\nabla{\bf u}}_{L^\infty}\nrm{\nabla{\bf c}}_{L^2} + \nrm{{\bf u}}_{L^\infty} \nrm{\Delta{\bf c}}_{L^2}.
\end{split}
\end{equation} Next, using $\nrm{\nabla{\bf c}}_{L^2}^2\le\nrm{{\bf c}}_{L^2} \nrm{{\bf c}}_{H^2} $, we obtain \begin{equation}\label{eq:c-H2}
\begin{split}
\frac{1}{2}\frac{d}{dt}\nrm{{\bf c}_t}_{L^2}^2 +\frac{1}{2} \nrm{\nabla{\bf c}_t}_{L^2}^2 &\lesssim \left( \nrm{{\boldsymbol{\rho}}}_{L^\infty} + \nrm{{\bf c}}_{L^\infty}\nrm{{\bf u}}_{L^\infty} + \nrm{\nabla{\bf c}}_{L^\infty}\nrm{{\boldsymbol{\rho}}}_{L^\infty} + \nrm{{\bf c}}_{L^\infty}^2 \nrm{{\boldsymbol{\rho}}}_{L^\infty}^2 \right)\\
&\quad \times \left( \nrm{{\bf c}_t}_{L^2}^2+\nrm{\nabla{\boldsymbol{\rho}}}_{L^2}^2 + (1 + \nrm{{\boldsymbol{\rho}}}_{L^\infty} + \nrm{{\bf u}}_{L^\infty}^2)\nrm{{\bf c}}_{L^2}^2 \right).
\end{split}
\end{equation} On the other hand, in the $\nabla{\bf u}$ equation we easily get from \eqref{eq:pressure} that { \begin{equation}\label{eq:omg-H2}
\begin{split}
\frac{d}{dt} \nrm{\nabla{\bf u}}_{L^2}^2 & \lesssim \left(\nrm{\nabla {\bf u}\cdot\nabla{\bf u} }_{L^2} + \nrm{\nabla^2{\bf p}}_{L^2} + \nrm{\nabla({\boldsymbol{\rho}}\nabla\phi)}_{L^2} \right) \nrm{\nabla{\bf u}}_{L^2} \\
&\lesssim \nrm{\nabla {\bf u}}_{L^\infty} \nrm{{\bf u}}_{H^{1}}^2 + \nrm{{\boldsymbol{\rho}}}_{H^{1}} \nrm{\nabla{\bf u}}_{L^2}.
\end{split}
\end{equation} }
Collecting the estimates \eqref{eq:rho-H2}, \eqref{eq:c-H2}, and \eqref{eq:omg-H2} together with \eqref{eq:rho-H2-prime}, we obtain (upon choosing sufficiently small $\epsilon>0$) \begin{equation}\label{eq:KSF-bdd-H2}
\begin{split}
\frac{d}{dt} \left( \nrm{\nabla{\boldsymbol{\rho}}}_{L^2}^2+\nrm{{\bf c}_t}_{L^2}^2+{\nrm{\nabla{\bf u}}_{L^{2}}^2} \right)\lesssim \left(1 + \nrm{{\bf c}}_{W^{2,\infty}} + \nrm{{\boldsymbol{\rho}}}_{L^\infty} + \nrm{{\bf u}}_{W^{1,\infty}} \right)^4 \left( \nrm{{\boldsymbol{\rho}}}_{H^1}^2+\nrm{{\bf c}_t}_{L^2}^2+ \nrm{{\bf c}}_{L^2} +{\nrm{{\bf u}}_{H^1}^2} \right) .
\end{split}
\end{equation}
\medskip
\noindent \textit{(iii) $H^4$ estimate}: Taking the Laplacian to the equation for $\partial_t\nabla{\boldsymbol{\rho}}$, we obtain
\begin{equation}\label{eq:KSF-bdd-der-lap1}
\begin{split}
\partial_t \nabla\Delta{\boldsymbol{\rho}} + ({\bf u}+\nabla{\bf c})\cdot\nabla (\nabla\Delta{\boldsymbol{\rho}})& = \Delta( -\nabla{\boldsymbol{\rho}} \Delta{\bf c} - {\boldsymbol{\rho}}\cdot\nabla\Delta{\bf c} - (\nabla{\bf u}+\nabla^2{\bf c})\cdot\nabla{\boldsymbol{\rho}} ) \\
&\quad - \Delta({\bf u}+\nabla{\bf c}) \cdot \nabla(\nabla{\boldsymbol{\rho}}) - 2 \sum_i \partial_i({\bf u}+\nabla{\bf c})\cdot\nabla \nabla \partial_i{\boldsymbol{\rho}} .
\end{split}
\end{equation} We shall find the following elementary inequality useful: \begin{equation*}
\begin{split}
\nrm{\Delta f}_{L^4}^2 \lesssim {\nrm{\nabla f}_{H^{2}} \nrm{f}_{W^{1,\infty}}}.
\end{split}
\end{equation*}
This gives us \begin{equation*}
\begin{split}
\left| \int \Delta({\bf u}+ \nabla{\bf c}) \cdot \nabla \partial_i{\boldsymbol{\rho}} \partial_i\Delta{\boldsymbol{\rho}} \right| \lesssim ( \nrm{{\bf u}}_{W^{1,\infty}} + \nrm{{\bf c}}_{W^{2,\infty}} + \nrm{{\boldsymbol{\rho}}}_{W^{1,\infty}} ) ( \nrm{{\bf u}}_{H^3}^2 + \nrm{{\bf c}}_{H^4}^2 + \nrm{{\boldsymbol{\rho}}}_{H^3}^2 ).
\end{split}
\end{equation*} Then, it allows us to bound \begin{equation*}
\begin{split}
&\left|\int ( - \Delta (\nabla{\bf u}+\nabla^2{\bf c})\cdot\nabla{\boldsymbol{\rho}} - \Delta({\bf u}+\nabla{\bf c}) \cdot \nabla(\nabla{\boldsymbol{\rho}}) - 2 \sum_i \partial_i({\bf u}+\nabla{\bf c})\cdot\nabla \nabla \partial_i{\boldsymbol{\rho}} ) \nabla\Delta{\boldsymbol{\rho}} \right| \\
&\quad \lesssim ( \nrm{{\bf u}}_{W^{1,\infty}} + \nrm{{\bf c}}_{W^{2,\infty}} + \nrm{{\boldsymbol{\rho}}}_{W^{1,\infty}} ) ( \nrm{{\bf u}}_{H^3}^2 + \nrm{{\bf c}}_{H^4}^2 + \nrm{{\boldsymbol{\rho}}}_{H^3}^2 ).
\end{split}
\end{equation*} Proceeding similarly, we obtain \begin{equation*}
\begin{split}
\left| \int \Delta(-\nabla{\boldsymbol{\rho}} \Delta{\bf c} - {\boldsymbol{\rho}}\cdot\nabla\Delta{\bf c} ) \cdot \nabla\Delta{\boldsymbol{\rho}} \right| \lesssim
( \nrm{{\bf c}}_{W^{2,\infty}} + \nrm{{\boldsymbol{\rho}}}_{W^{1,\infty}} ) ( \nrm{{\bf c}}_{H^4}^2 + \nrm{{\boldsymbol{\rho}}}_{H^3}^2 ) + \nrm{{\boldsymbol{\rho}}}_{L^\infty} \nrm{\nabla\Delta^2{\bf c}}_{L^2} \nrm{\nabla\Delta{\boldsymbol{\rho}}}_{L^2}
\end{split}
\end{equation*} This gives the estimate \begin{equation}\label{eq:rho-H4}
\begin{split}
\frac{1}{2} \frac{d}{dt} \nrm{\nabla\Delta{\boldsymbol{\rho}}}_{L^2}^2 - \epsilon \nrm{\nabla\Delta^2{\bf c}}_{L^2}^2 & \lesssim ( \nrm{{\bf u}}_{W^{1,\infty}} + \nrm{{\bf c}}_{W^{2,\infty}} + \nrm{{\boldsymbol{\rho}}}_{W^{1,\infty}} ) ( \nrm{{\bf u}}_{H^3}^2 + \nrm{{\bf c}}_{H^4}^2 + \nrm{{\boldsymbol{\rho}}}_{H^3}^2 ) \\
&\quad + C_\epsilon \nrm{{\boldsymbol{\rho}}}_{L^\infty}^2 \nrm{\nabla\Delta{\boldsymbol{\rho}}}_{L^2}^2 .
\end{split}
\end{equation} Let us now estimate ${\boldsymbol{\rho}}_{tt}$ and ${\bf u}_{tt}$. We have: \begin{equation*}
\begin{split}
{\boldsymbol{\rho}}_{tt} = - {\bf u}_t \cdot\nabla{\boldsymbol{\rho}} - {\bf u}\cdot\nabla{\boldsymbol{\rho}}_t - \nabla{\boldsymbol{\rho}}_t\cdot\nabla{\bf c} - \nabla{\boldsymbol{\rho}}\cdot\nabla{\bf c}_t - {\boldsymbol{\rho}}_t\Delta{\bf c} - {\boldsymbol{\rho}}\Delta{\bf c}_t
\end{split}
\end{equation*} recalling that \begin{equation*}
\begin{split}
\nrm{{\bf u}_t}_{L^2}\lesssim \nrm{\nabla{\bf u}}_{L^\infty}\nrm{{\bf u}}_{L^2} + \nrm{{\boldsymbol{\rho}}}_{L^2}
\end{split}
\end{equation*} and observing that \begin{equation*}
\begin{split}
\nrm{{\boldsymbol{\rho}}_t}_{H^1} \lesssim ( \nrm{{\bf u}}_{W^{1,\infty}} + \nrm{{\bf c}}_{W^{2,\infty}} + \nrm{{\boldsymbol{\rho}}}_{W^{1,\infty}} ) (\nrm{ {\boldsymbol{\rho}}}_{H^2} + \nrm{ {\bf c}}_{H^3})
\end{split}
\end{equation*} holds, we can bound \begin{equation*}
\begin{split}
&\nrm{- {\bf u}_t \cdot\nabla{\boldsymbol{\rho}} - {\bf u}\cdot\nabla{\boldsymbol{\rho}}_t - \nabla{\boldsymbol{\rho}}_t\cdot\nabla{\bf c} - \nabla{\boldsymbol{\rho}}\cdot\nabla{\bf c}_t - {\boldsymbol{\rho}}_t\Delta{\bf c} - {\boldsymbol{\rho}}\Delta{\bf c}_t }_{L^2} \\
&\quad \lesssim \left( \nrm{{\bf c}}_{W^{2,\infty}} + \nrm{{\boldsymbol{\rho}}}_{W^{1,\infty}} + \nrm{{\bf u}}_{W^{1,\infty}} \right)^2(\nrm{{\bf u}}_{L^2} + \nrm{{\boldsymbol{\rho}}}_{H^2} + \nrm{{\bf c}}_{H^3} + \nrm{{\bf c}_t}_{H^2}) .
\end{split}
\end{equation*} On the other hand, we have \begin{equation*}
\begin{split}
\nrm{{\bf u}_{tt}}_{L^2} \lesssim \nrm{{\bf u}}_{W^{1,\infty}}^2 \nrm{{\bf u}}_{H^2} + \nrm{{\boldsymbol{\rho}}_t}_{L^2}.
\end{split}
\end{equation*} Moreover, using the equation for ${\bf c}_{tt}$
\begin{equation}\label{eq:KSF-bdd-der-lap2}
\begin{split}
\partial_t {\bf c}_{tt} + {\bf u}\cdot\nabla {\bf c}_{tt} = \Delta {\bf c}_{tt} - {\boldsymbol{\rho}} {\bf c}_{tt} -2 {\boldsymbol{\rho}}_t{\bf c}_t - {\bf c}{\boldsymbol{\rho}}_{tt} - {\bf u}_{tt}\cdot\nabla{\bf c} - {\bf u}_t\cdot\nabla{\bf c}_t ,
\end{split}
\end{equation} we can estimate $\nrm{\nabla\Delta^2{\bf c}}_{L^2}$ in terms of $\nrm{\nabla{\bf c}_{tt}}_{L^2}$ and $\nrm{{\bf c}_t}_{H^2}$, $\nrm{{\bf c}}_{H^4}$ in terms of $\nrm{{\bf c}_{tt}}_{L^2}$ (modulo lower order terms). We omit the details for this, but the proof is parallel to the estimate $\nrm{{\bf c}}_{H^2}\lesssim \nrm{{\bf c}_t}_{L^2} + \cdots$ given in the above. Then, we have \begin{equation}\label{eq:c-H4}
\begin{split}
&\frac{1}{2} \frac{d}{dt} \nrm{{\bf c}_{tt}}_{L^2}^2 + \frac{1}{2} \nrm{\nabla{\bf c}_{tt}}_{L^2}^2 \\
&\quad \lesssim \left( 1 + \nrm{{\bf c}}_{W^{2,\infty}} + \nrm{{\boldsymbol{\rho}}}_{W^{1,\infty}} + \nrm{{\bf u}}_{W^{1,\infty}} \right)^4(\nrm{{\bf u}}_{H^3} + \nrm{{\boldsymbol{\rho}}}_{H^3} + \nrm{{\bf c}_{tt}}_{L^2} + \nrm{{\bf c}}_{L^2}) .
\end{split}
\end{equation}
Next, taking the Laplacian to the both sides of the equation for $\nabla{\bf u}$,
\begin{equation}\label{eq:KSF-bdd-der-lap3}
\begin{split}
{\partial_t \Delta \nabla {\bf u}+ {\bf u}\cdot\nabla \nabla \Delta {\bf u}= - \Delta(\nabla{\bf u} \cdot\nabla{\bf u}) - \Delta\nabla^2 {\bf p} + \Delta\nabla({\boldsymbol{\rho}}\nabla\phi) - \sum_{i} \partial_i {\bf u} \partial_i \nabla\nb{\bf u} - \Delta {\bf u} \cdot\nabla \nabla{\bf u} .}
\end{split}
\end{equation} Then, we may estimate \begin{equation}\label{eq:omg-H4}
\begin{split}
{\frac{d}{dt} \nrm{ \Delta \nabla {\bf u} }_{L^2}^2 \lesssim \nrm{{\boldsymbol{\rho}}}_{H^3} \nrm{{\bf u}}_{H^{3}} + \nrm{{\bf u}}_{W^{1,\infty}} \nrm{{\bf u}}_{H^3}^2. }
\end{split}
\end{equation}
Therefore, from \eqref{eq:rho-H4}, \eqref{eq:c-H4}, and \eqref{eq:omg-H4}, we obtain that \begin{equation}\label{eq:KSF-bdd-H4}
\begin{split}
&\frac{d}{dt} \left( \nrm{\nabla\Delta{\boldsymbol{\rho}}}_{L^2}^2 + \nrm{{\bf c}_{tt}}_{L^2}^2 + \nrm{\Delta\nabla {\bf u}}_{L^2}^2 \right)\\
&\quad \lesssim \left( 1 + \nrm{{\bf c}}_{W^{2,\infty}} + \nrm{{\boldsymbol{\rho}}}_{W^{1,\infty}} + \nrm{{\bf u}}_{W^{1,\infty}} \right)^4(\nrm{{\bf u}}_{H^3} + \nrm{{\boldsymbol{\rho}}}_{H^3} + \nrm{{\bf c}_{tt}}_{L^2} + \nrm{{\bf c}}_{L^2}) .
\end{split}
\end{equation} Then, combining \eqref{eq:KSF-bdd-L2}, \eqref{eq:KSF-bdd-H2}, and \eqref{eq:KSF-bdd-H4}, \begin{equation*}
\begin{split}
&\frac{d}{dt} \left( \nrm{{\boldsymbol{\rho}}}_{H^3}^2 +\nrm{{\bf c}}_{L^2}^2+\nrm{{\bf c}_{t}}_{L^2}^2+ \nrm{{\bf c}_{tt}}_{L^2}^2 + {\nrm{{\bf u}}_{H^3}^2} \right)\\
&\quad \lesssim \left(1+ \nrm{{\boldsymbol{\rho}}}_{H^3}^2 +\nrm{{\bf c}}_{L^2}^2+\nrm{{\bf c}_{t}}_{L^2}^2 + \nrm{{\bf c}_{tt}}_{L^2}^2 + {\nrm{{\bf u}}_{H^3}^2} \right)^6.
\end{split}
\end{equation*} Hence, there exists $T>0$ depending only on $\nrm{{\boldsymbol{\rho}}}_{H^3}^2 +\nrm{{\bf c}}_{L^2}^2+\nrm{{\bf c}_{t}}_{L^2}^2 + \nrm{{\bf c}_{tt}}_{L^2}^2 + {\nrm{{\bf u}}_{H^3}^2} $ at $t = 0$ such that on $[0,T]$, \begin{equation*}
\begin{split}
(\nrm{{\boldsymbol{\rho}}}_{H^3}^2 +\nrm{{\bf c}}_{L^2}^2+\nrm{{\bf c}_{t}}_{L^2}^2 + \nrm{{\bf c}_{tt}}_{L^2}^2 + {\nrm{{\bf u}}_{H^3}^2})(t) {\le 2} (\nrm{{\boldsymbol{\rho}}}_{H^3}^2 +\nrm{{\bf c}}_{L^2}^2+\nrm{{\bf c}_{t}}_{L^2}^2 + \nrm{{\bf c}_{tt}}_{L^2}^2 + {\nrm{{\bf u}}_{H^3}^2})(0).
\end{split}
\end{equation*} This completes the proof of a priori estimates. The blow-up criterion follows immediately from the alternative estimate \begin{equation*}
\begin{split}
&\frac{d}{dt} \left( \nrm{{\boldsymbol{\rho}}}_{H^3}^2 +\nrm{{\bf c}}_{L^2}^2+\nrm{{\bf c}_{t}}_{L^2}^2+ \nrm{{\bf c}_{tt}}_{L^2}^2 + {\nrm{{\bf u}}_{H^3}^2}\right)\\
&\quad \lesssim \left( 1 + \nrm{{\bf c}}_{W^{2,\infty}} + \nrm{{\boldsymbol{\rho}}}_{W^{1,\infty}} + \nrm{{\bf u}}_{W^{1,\infty}} \right)^4\left( \nrm{{\boldsymbol{\rho}}}_{H^3}^2 +\nrm{{\bf c}}_{L^2}^2+\nrm{{\bf c}_{t}}_{L^2}^2 + \nrm{{\bf c}_{tt}}_{L^2}^2 + {\nrm{{\bf u}}_{H^3}^2} \right)^2.
\end{split}
\end{equation*}
\medskip
\noindent {We now deal with the existence of a solution with claimed regularity. Given $({\boldsymbol{\rho}}_0,{\bf c}_0,{\bf u}_0)$ satisfying the assumptions of the theorem, we first mollify it in a way that for each $\epsilon>0$, the triple $({\boldsymbol{\rho}}_0^\epsilon,{\bf c}_0^\epsilon,{\bf u}_0^\epsilon)$ is $C^\infty$--smooth in $\Omega$, satisfies the compatibility conditions, and converge to $({\boldsymbol{\rho}}_0,{\bf c}_0,{\bf u}_0)$ strongly in the norm $H^3\times H^4\times H^3$ as $\epsilon\to 0$. Furthermore, we may assume that \begin{equation*}
\begin{split}
\left( \nrm{{\boldsymbol{\rho}}_0^\epsilon}_{H^3} + \nrm{{\bf c}_0^\epsilon}_{H^4} + \nrm{{\bf u}_0^\epsilon}_{H^{3}} \right) \le 2 \left( \nrm{{\boldsymbol{\rho}}_0}_{H^3} + \nrm{{\bf c}_0}_{H^4} + \nrm{{\bf u}_0}_{H^{3}} \right).
\end{split}
\end{equation*} This mollification is done simply to ensure that all the objects that we are dealing with are $C^\infty$--smooth. Then, we now fix some small $\epsilon>0$ and build a sequence of approximations $({\boldsymbol{\rho}}^{(n,\epsilon)},{\bf c}^{(n,\epsilon)},{\bf u}^{(n,\epsilon)})_{n\ge0}$, which will be shown to be uniformly bounded in a time interval $[0,T]$. In the case $n = 0$, we simply set $({\boldsymbol{\rho}}^{(0,\epsilon)},{\bf u}^{(0,\epsilon)}) = ({\boldsymbol{\rho}}_0^\epsilon,{\bf u}_0^\epsilon)$ for all $t\ge 0$ and ${\bf c}^{(0,\epsilon)}$ be the solution of the heat equation \begin{equation*}
\begin{split}
\partial_t {\bf c}^{(0,\epsilon)} = \Delta {\bf c}^{(0,\epsilon)}, \\
{\bf c}^{(0,\epsilon)}(t=0) = {\bf c}_0^\epsilon,
\end{split}
\end{equation*} with the Neumann boundary condition. Now, given $({\boldsymbol{\rho}}^{(n,\epsilon)},{\bf c}^{(n,\epsilon)},{\bf u}^{(n,\epsilon)})$ for some $n\ge 0$, we define $$({\boldsymbol{\rho}}^{(n+1,\epsilon)},{\bf c}^{(n+1,\epsilon)},{\bf u}^{(n+1,\epsilon)})$$ as the unique solution of the following \textit{linear} system of equations \begin{equation} \label{eq:KSF-approxseq}
\left\{
\begin{aligned}
&\partial_t{\boldsymbol{\rho}}^{(n+1,\epsilon)} + ({\bf u}^{(n,\epsilon)} + \nabla{\bf c}^{(n,\epsilon)})\cdot\nabla{\boldsymbol{\rho}}^{(n+1,\epsilon)} = -\Delta{\bf c}^{(n,\epsilon)} {\boldsymbol{\rho}}^{(n+1,\epsilon)},\\
&\partial_t{\bf c}^{(n+1,\epsilon)} + {\bf u}^{(n,\epsilon)}\cdot\nabla{\bf c}^{(n+1,\epsilon)} = \Delta{\bf c}^{(n+1,\epsilon)} - {\boldsymbol{\rho}}^{(n,\epsilon)} {\bf c}^{(n+1,\epsilon)},\\
&\partial_t {\bf u}^{(n+1,\epsilon)} + {\bf u}^{(n,\epsilon)}\cdot\nabla{\bf u}^{(n+1,\epsilon)} + \nabla {\bf p}^{(n+1,\epsilon)} = {\boldsymbol{\rho}}^{(n,\epsilon)}\nabla\phi, \\
&\nabla \cdot {\bf u}^{(n+1,\epsilon)} = 0,
\end{aligned}
\right.
\end{equation} with initial data $({\boldsymbol{\rho}}_0^\epsilon,{\bf c}_0^\epsilon,{\bf u}_0^\epsilon)$, the Neumann boundary condition for ${\bf c}^{(n+1,\epsilon)}$, and the non-penetration boundary condition for ${\bf u}^{(n+1,\epsilon)}$. The existence of ${\boldsymbol{\rho}}^{(n+1,\epsilon)}$ and ${\bf u}^{(n+1,\epsilon)}$ solving \eqref{eq:KSF-approxseq} can be proved by integrating along the characteristics defined by ${\bf u}^{(n,\epsilon)} + \nabla{\bf c}^{(n,\epsilon)}$ and ${\bf u}^{(n,\epsilon)}$, respectively. Lastly, the existence of a smooth solution ${\bf c}^{(n+1,\epsilon)}$ follows from the theory of parabolic equations (see for instance Ito \cite{Ito}). While these approximate solutions are defined globally in time, we need to obtain Sobolev estimates which are uniform in $n$. The arguments in this step follow closely the proof of a priori estimates we have obtained earlier.
To prove the uniform bound, we proceed by induction: assume that for all $0 \le k\le n$, there exists some $T>0$ such that \begin{equation*}
\begin{split}
M_{k} & := \sup_{0\le t \le T} \left( \nrm{{\boldsymbol{\rho}}^{(k,\epsilon)}}_{H^3} + \nrm{{\bf c}^{(k,\epsilon)}}_{H^4} + \nrm{{\bf u}^{(k,\epsilon)}}_{H^{3}} \right)(t) + \nrm{\nabla {\bf c}^{(k,\epsilon)} (t) }_{L^2([0,T);H^{4})} \\
&\qquad \le 4 \left( \nrm{{\boldsymbol{\rho}}_0}_{H^3} + \nrm{{\bf c}_0}_{H^4} + \nrm{{\bf u}_0}_{H^{3}} \right).
\end{split}
\end{equation*} It is clear that this estimate holds for the base case $k=0$, with any $T>0$. Next, under this induction hypothesis, we can prove the same bound for $k=n+1$ by proceeding similarly as in the proof of a priori estimates, by possibly taking a smaller $T>0$ but in a way independent of $n$. To demonstrate this in the case of ${\boldsymbol{\rho}}^{(n+1,\epsilon)}$, we have \begin{equation*}
\begin{split}
\frac{d}{dt} \nrm{{\boldsymbol{\rho}}^{(n+1,\epsilon)} }_{H^{3}}^{2} \lesssim ( \nrm{{\bf c}^{(n,\epsilon)}}_{H^4} + \nrm{{\bf u}^{(n,\epsilon)}}_{H^{3}} ) \nrm{{\boldsymbol{\rho}}^{(n+1,\epsilon)} }_{H^{3}}^{2} + \nrm{ \Delta{\bf c}^{(n,\epsilon)} }_{H^{3}} \nrm{{\boldsymbol{\rho}}^{(n+1,\epsilon)} }_{H^{3}}^{2}.
\end{split}
\end{equation*} Integrating in time, \begin{equation*}
\begin{split}
\sup_{0\le t \le T} \nrm{{\boldsymbol{\rho}}^{(n+1,\epsilon)} }_{H^{3}} & \le \nrm{{\boldsymbol{\rho}}^{(\epsilon)}_{0} }_{H^{3}}\exp\left( \int_0^T C( \nrm{{\bf c}^{(n,\epsilon)}}_{H^4} + \nrm{{\bf u}^{(n,\epsilon)}}_{H^{3}} + \nrm{ \Delta{\bf c}^{(n,\epsilon)} }_{H^{3}} ) dt \right) \\
& \le \nrm{{\boldsymbol{\rho}}^{(\epsilon)}_{0} }_{H^{3}}\exp\left( CTM_{n} + CT^{\frac12} M_{n} \right) < 2\nrm{{\boldsymbol{\rho}}^{(\epsilon)}_{0} }_{H^{3}}
\end{split}
\end{equation*} by taking $T>0$ smaller if necessary, in a way depending only on the quantity $\nrm{{\boldsymbol{\rho}}_0}_{H^3} + \nrm{{\bf c}_0}_{H^4} + \nrm{{\bf u}_0}_{H^{3}}$. In this way, we obtain that the sequence $({\boldsymbol{\rho}}^{(n,\epsilon)},{\bf c}^{(n,\epsilon)},{\bf u}^{(n,\epsilon)})_{n\ge0}$ is uniformly bounded in $L^\infty([0,T]; H^3\times H^4\times H^3)$. Furthermore, one can see that the sequence $(\partial_t{\boldsymbol{\rho}}^{(n,\epsilon)},\partial_t{\bf c}^{(n,\epsilon)},\partial_t{\bf u}^{(n,\epsilon)})_{n\ge0}$ is uniformly bounded in $L^\infty([0,T]; L^2\times L^2\times L^2)$. Applying the Aubin--Lions lemma and by passing to a subsequence, we can extract a convergent subsequence strongly in $L^\infty([0,T]; H^2\times H^3\times H^2)$. Let us denote the limit by $({\boldsymbol{\rho}}^\epsilon, {\bf c}^\epsilon, {\bf u}^\epsilon)$, which again belongs to the space $L^\infty([0,T]; H^3\times H^4\times H^3)$. The strong convergence in $H^2\times H^3\times H^2$ of $({\boldsymbol{\rho}}^{(n,\epsilon)},{\bf c}^{(n,\epsilon)},{\bf u}^{(n,\epsilon)})_{n\ge0}$ to $({\boldsymbol{\rho}}^\epsilon, {\bf c}^\epsilon, {\bf u}^\epsilon)$ is enough to guarantee pointwise convergence of each term in \eqref{eq:KSF-bdd} as $\epsilon\to 0$. Therefore, it follows that $({\boldsymbol{\rho}}^\epsilon, {\bf c}^\epsilon, {\bf u}^\epsilon)$ is a solution to \eqref{eq:KSF-bdd} with uniform bound \begin{equation*}
\begin{split}
\sup_{0\le t \le T} \left( \nrm{{\boldsymbol{\rho}}^{\epsilon}}_{H^3} + \nrm{{\bf c}^{\epsilon}}_{H^4} + \nrm{{\bf u}^{\epsilon}}_{H^{3}} \right)(t) + \nrm{\nabla {\bf c}^{\epsilon} (t) }_{L^2([0,T);H^{4})} \le 4 \left( \nrm{{\boldsymbol{\rho}}_0}_{H^3} + \nrm{{\bf c}_0}_{H^4} + \nrm{{\bf u}_0}_{H^{3}} \right).
\end{split}
\end{equation*} Then, we can similarly take a convergent subsequence $({\boldsymbol{\rho}}^{\epsilon_{k}}, {\bf c}^{\epsilon_{k}}, {\bf u}^{\epsilon_{k}})$ and denote the corresponding strong limit in $H^2\times H^3\times H^2$ by $({\boldsymbol{\rho}},{\bf c},{\bf u})$. It is not difficult to check that this triple solves \eqref{eq:KSF-bdd} with prescribed initial data and satisfies the claimed regularity in the statement of the theorem. }
\medskip
\noindent {Finally, uniqueness can be proved easily: \textit{assuming} that there are two solutions $({\boldsymbol{\rho}},{\bf c},{\bf u})$ and $(\widetilde{{\boldsymbol{\rho}}},\widetilde{{\bf c}}, \widetilde{{\bf u}})$ belonging to $H^3\times H^4\times H^3$ on $[0,T]$ with the same initial data, one can close an $L^2$ estimate for the difference as in the uniqueness proof of Theorem \ref{thm:lwp-away} below. The $H^3\times H^4\times H^3$--regularity of the solution is sufficient to justify the $L^2$ difference estimate.}
\end{proof}
\section{Local well-posedness in the fully inviscid case}\label{sec:inviscid}
In this section, we consider the Cauchy problem for \eqref{eq:KS} and \eqref{eq:KSF} in the fully inviscid case. We establish sufficient conditions on the initial data which guarantees local well-posedness. Let us explain how the rest of this section is organized. To begin with, in Subsection \ref{subsec:lwp-nonvanishing}, we state and prove Theorem \ref{MainThm2}, which is local well-posedness in the fully inviscid \eqref{eq:KSF} with non-vanishing data. Within this Subsection, we perform a simple linear analysis which motivates our choice of modified good variables. Moreover, some discussion related with removing the non-vanishing assumption is given. Two versions of Theorem \ref{MainThm3}, which are local well-posedness results for fully inviscid \eqref{eq:KS} with vanishing initial data, are stated and proved respectively in Subsections \ref{subsec:v1} and \ref{subsec:v2}.
\subsection{Well-posedness with non-vanishing ${\boldsymbol{\rho}}$ and ${\bf c}$}\label{subsec:lwp-nonvanishing}
We consider the system \eqref{eq:KSF} in $d$-dimensional domains of the form $\Omega = \mathbb T^k \times \mathbb R^{d-k}$ for any $0 \le k \le d$. Our first main result states that, if initially ${\boldsymbol{\rho}}_0$ and ${\bf c}_0$ are bounded away from 0, the fully inviscid system is locally well-posed in sufficiently high Sobolev spaces.
\begin{theorem}[Well-posedness away from 0]\label{thm:lwp-away}
Assume that $D_u = D_\rho = D_c = 0$, \begin{equation*}
\begin{split}
az^\gamma \le \frac{k(z)}{\chi(z)}, \quad 0 \le z \le 1
\end{split}
\end{equation*} for some $a>0, \gamma\ge 0$, and $$\phi \in H^{m+1}(\Omega), \quad \chi, k \in H^{m+1}(\mathbb R_+),$$ for some $2 + \frac{d}{2} < m \le +\infty$.
Assume further that the initial data $({\boldsymbol{\rho}}_0, {\bf c}_0,{\bf u}_0)$ satisfies ${\boldsymbol{\rho}}_0, {\bf c}_0 \in L^\infty(\Omega)$ and \begin{equation*}
\begin{split}
& {\boldsymbol{\rho}}_0(x) \ge \underline{r}_0 > 0 , \quad {\bf c}_0(x) \ge \underline{c}_0 > 0, \quad \mathrm{div}\, {\bf u}_0 = 0
\end{split}
\end{equation*} for some constants $ \underline{r}_0, \underline{c}_0 $ and \begin{equation*}
\begin{split}
\nabla{\boldsymbol{\rho}}_0 \in H^{m-1}(\Omega), \quad {\bf u}_0, \nabla{\bf c}_0 \in H^m(\Omega)
\end{split}
\end{equation*} Then the system \eqref{eq:KSF} is locally-well posed; more precisely, there exists some $T > 0$ depending only on the initial data such that there is a unique solution $({\boldsymbol{\rho}},{\bf c},{\bf u})$ to \eqref{eq:KSF} satisfying the initial condition and \begin{equation*}
\begin{split}
{\boldsymbol{\rho}}, {\bf c} \in L^\infty([0,T); L^\infty(\Omega)), \quad \nabla{\boldsymbol{\rho}} \in C([0,T); H^{m-1}(\Omega)), \quad {\bf u}, \nabla{\bf c} \in C([0,T); H^m(\Omega)).
\end{split}
\end{equation*}
\end{theorem}
\begin{remark}
Taking ${\bf u}_0 = 0$ and $\phi = 0$, the above result gives in particular that the system \eqref{eq:KS} is locally-well posed in the fully inviscid case $(D_\rho = D_c = 0)$ for initial data $({\boldsymbol{\rho}}_0, {\bf c}_0)$ which is away from 0. Moreover, it is not difficult to show that the a priori estimates we obtain for the fully inviscid case carries over to partially viscous cases, namely when some of the viscosity constants $D_u, D_\rho, D_c$ are positive.
\end{remark}
\begin{remark}
A slightly technical issue appears in the case of an unbounded domain, since then the functions ${\boldsymbol{\rho}},{\bf c}$ being bounded away from 0 forces in particular that they do not belong to any $L^p$ for finite $p$. It is then natural to require that there exist time-dependent constants ${\boldsymbol{\rho}}_\infty(t)$ and ${\bf c}_\infty(t)$ such that \begin{equation*}
\begin{split}
{\boldsymbol{\rho}}(t,\cdot) - {\boldsymbol{\rho}}_{\infty}(t) \in H^{m}, \quad {\bf c}(t,\cdot) - {\bf c}_{\infty}(t) \in H^{m+1}.
\end{split}
\end{equation*} Then one can easily guarantee (formally) that the solution must be uniformly bounded in space if the initial data is. Indeed, it is easy to see (by evaluating at infinity) directly from \eqref{eq:KSF} that \begin{equation*}
\begin{split}
\frac{d}{dt} {\boldsymbol{\rho}}_\infty = 0, \quad \frac{d}{dt} {\bf c}_\infty = - k({\bf c}_\infty) {\boldsymbol{\rho}}_\infty
\end{split}
\end{equation*} which determines $({\boldsymbol{\rho}}_\infty(t),{\bf c}_\infty(t))$ in terms of $({\boldsymbol{\rho}}_\infty(0), {\bf c}_\infty(0))$. Hence one may simply work with decaying functions ${\boldsymbol{\rho}}(t,\cdot) - {\boldsymbol{\rho}}_{\infty}(t)$ and $ {\bf c}(t,\cdot) - {\bf c}_{\infty}(t)$. On the other hand, we can simply require ${\bf u}(t,\cdot) \in {L}^\infty$, since the contribution from ${\boldsymbol{\rho}}_\infty$ into $\partial_t{\bf u}$ can be absorbed into the pressure term. A different way to handle this non-decaying issue is to altogether avoid putting ${\boldsymbol{\rho}}, {\bf c}$ in $L^2$ and simply use $L^\infty$ instead. In the following, we shall neglect this issue and just work in $\mathbb T^d$.
\end{remark}
\subsubsection{Discussion}
The non-vanishing assumption on ${\boldsymbol{\rho}}$ and ${\bf c}$ stems naturally from our method of proof, which utilizes the ``modified variables'' $\sqrt{{\bf c}}\partial^m{\boldsymbol{\rho}}$ and $\sqrt{{\boldsymbol{\rho}}} \partial^{m+1}{\bf c}$. Although we expect that such an assumption cannot be entirely omitted, we present two different results in which non-vanishing assumption is relaxed. The first one is simply employing the modified variables $(\sqrt{{\boldsymbol{\rho}}})^{-1} \partial^m{\boldsymbol{\rho}}$ and $(\sqrt{{\bf c}})^{-1} \partial^{m+1}{\bf c}$. A disadvantage in this approach is that, to obtain very regular solutions (i.e. well-posedness for $m$ large), one needs to assume that whenever ${\bf c}_0$ or ${\boldsymbol{\rho}}_0$ vanishes, it must do so with a high order (proportional to $m$). The other approach is to \textit{specify} the profiles of ${\boldsymbol{\rho}}_0$ and ${\bf c}_0$ near the points of vanishing; e.g. ${\boldsymbol{\rho}}_0(x) \simeq A|x-x_0|^2$. There is nothing special about the data being quadratic at its zeroes, and the same method can be applied to smooth data which vanishes with higher order. While the assumption on the initial data is more rigid, propagating $C^\infty$-smoothness is not difficult in this case. Moreover, in this latter setting, we can prove finite-time singularity formation for \eqref{eq:KS}; the $C^2$-norm of ${\bf c}(t,\cdot)$ becomes infinite in finite time.
\subsubsection{Linear analysis}
In the fully inviscid case $D_u = D_\rho = D_c = 0$, there are serious difficulties in closing $H^m$ a priori estimates. To see whether there is a chance of the inviscid system to be well-posed, we first consider \eqref{eq:KS} in the 1D case:
\begin{equation}\label{eq:KS-1D}
\left\{
\begin{aligned}
\partial_t {\boldsymbol{\rho}} & = -\partial_x ( {\boldsymbol{\rho}} \partial_x {\bf c}), \\
\partial_t {\bf c} & = - {\bf c}{\boldsymbol{\rho}} .
\end{aligned}
\right.
\end{equation} Note that we have taken $k(z) = z, \chi(z) = 1$ for simplicity. The term $-\partial_x({\boldsymbol{\rho}}\partial_x{\bf c})$ on the right hand side for the ${\boldsymbol{\rho}}$-equation seems like it incurs loss of derivatives. To see more clearly the effect of this term, we take the linearization approach: while there are no non-trivial steady states (solutions independent of time), the next simplest solutions are given by $(\bar{{\boldsymbol{\rho}}}, \bar{{\bf c}} e^{-\bar{{\boldsymbol{\rho}}}t})$ where $\bar{{\boldsymbol{\rho}}},\bar{{\bf c}}$ are some positive constants. Fixing $\bar{{\boldsymbol{\rho}}} = \bar{{\bf c}} = 1$, writing \begin{equation*}
\begin{split}
{\boldsymbol{\rho}} =1+ \tilde{{\boldsymbol{\rho}}}, \quad {\bf c} = e^{-t} + \tilde{{\bf c}} ,
\end{split}
\end{equation*} and dropping quadratic terms in the perturbation, we arrive at \begin{equation}\label{eq:KS1-lin}
\left\{
\begin{aligned}
\partial_t \tilde{{\boldsymbol{\rho}}} & = -\partial_{xx}\tilde{{\bf c}} ,\\
\partial_t \tilde{{\bf c}} & = - e^{-t} \tilde{{\boldsymbol{\rho}}} - \tilde{{\bf c}}.
\end{aligned}
\right.
\end{equation} This linear system is well-posed: we have that the ``energy'' \begin{equation*}
\begin{split}
\nrm{\tilde{{\boldsymbol{\rho}}}}_{L^2}^2 + e^t\nrm{\partial_x\tilde{{\bf c}}}_{L^2}^2
\end{split}
\end{equation*} is under control: to see this, we compute that \begin{equation*}
\begin{split}
\frac{1}{2}\frac{d}{dt} \left( \nrm{\tilde{{\boldsymbol{\rho}}}}_{L^2}^2 + e^t \nrm{\partial_x\tilde{{\bf c}}}_{L^2}^2 \right) = - \frac{1}{2}e^t \nrm{\partial_x\tilde{{\bf c}}}_{L^2}^2 \le 0 .
\end{split}
\end{equation*} Proceeding similarly for higher derivatives, we see that $\nrm{\partial_x^n\tilde{{\boldsymbol{\rho}}}}_{L^2}^2 + e^t\nrm{\partial_x^{n+1,\epsilon}\tilde{{\bf c}}}_{L^2}^2$ decreases in time for any $n \ge 0$. This suggests that a suitable weighted norm of the solution could be under control for the nonlinear evolution. It is also expected that the weight should be solution-dependent, which naturally gives rise to the non-vanishing assumptions.
Moving on to the case of \eqref{eq:KSF}, we consider the following 1D model system: \begin{equation} \label{eq:KSF-inviscid-1D}
\left\{
\begin{aligned}
&\partial_t{\boldsymbol{\rho}} + {\bf u}\partial_x{\boldsymbol{\rho}} = -\partial_x({\boldsymbol{\rho}}\partial_x{\bf c}) ,\\
&\partial_t{\bf c} + {\bf u}\partial_x{\bf c} = -{\bf c}{\boldsymbol{\rho}},\\
&\partial_t{\bf u} + {\bf u}\partial_x{\bf u} = {\boldsymbol{\rho}} .
\end{aligned}
\right.
\end{equation} From the above, we know that it is natural to rewrite the equation in terms of ${\bf f} = \partial_x{\bf c}$. Taking a derivative in the ${\bf c}$-equation and writing $D_t = \partial_t + {\bf u}\partial_x$ for simplicity, we obtain: \begin{equation*}
\left\{
\begin{aligned}
&D_t {\boldsymbol{\rho}} = -{\boldsymbol{\rho}} \partial_x {\bf f} - {\bf f} \partial_x{\boldsymbol{\rho}} ,\\
&D_t{\bf f} = - {\bf f} \partial_x{\bf u} - {\bf f} {\boldsymbol{\rho}} - {\bf c} \partial_x{\boldsymbol{\rho}} ,\\
&D_t{\bf u} = {\boldsymbol{\rho}}.
\end{aligned}
\right.
\end{equation*} A formal linearization around the state ${\boldsymbol{\rho}}={\bf f}={\bf u} = 1$ gives, after removing terms which do not lose derivatives, \begin{equation} \label{eq:KSF-inviscid-1D-linear}
\left\{
\begin{aligned}
{(\partial_t+\partial_x)}{\boldsymbol{\rho}} & = - \partial_{x}{\bf f} - \partial_x{\boldsymbol{\rho}} , \\
{(\partial_t+\partial_x)} {\bf f} & = - \partial_x{\bf u} - \partial_x{\boldsymbol{\rho}} , \\
{(\partial_t+\partial_x)} {\bf u} & = 0 .
\end{aligned}
\right.
\end{equation} Then it is tempting to rewrite the above as \begin{equation} \label{eq:illposed-1D-good}
\left\{
\begin{aligned}
{(\partial_t+\partial_x)}({\boldsymbol{\rho}} + {\bf u} ) & = - \partial_{x}({\bf f} - {\bf u} ) - \partial_x({\boldsymbol{\rho}} + {\bf u} ) , \\
{(\partial_t+\partial_x)} ({\bf f} - {\bf u} )& = - \partial_x({\boldsymbol{\rho}} + {\bf u} ), \\
{(\partial_t+\partial_x)} {\bf u} & = 0 .
\end{aligned}
\right.
\end{equation} In this form, it is clear that \eqref{eq:KSF-inviscid-1D-linear} is well-posed in Sobolev spaces. This suggests that even the non-linear system could be well-posed for some delicate reason. Of course, in higher dimensions, one needs to take into account the effect of the pressure as well. With this in mind, we shall now give a proof of Theorem \ref{thm:lwp-away}.
\begin{proof}[{\bf Proof of Theorem \ref{thm:lwp-away}}]
For now, we take $\chi(z) = 1$, $k(z) = z$ and $\Omega = \mathbb T^d$ for simplicity and rewrite the inviscid \eqref{eq:KSF} system with ${\bf f} = \nabla {\bf c}$: \begin{equation} \label{eq:KSF-inviscid}
\left\{
\begin{aligned}
&\partial_t \boldsymbol{\rho} + {\bf u}\cdot\nabla {\boldsymbol{\rho}} = -\nabla \cdot ({\boldsymbol{\rho}}{\bf f}) , \\
&\partial_t {\bf f} + {\bf u}\cdot\nabla {\bf f} = -[ \nabla{\bf u} ]^T {\bf f} - {\boldsymbol{\rho}} {\bf f} - {\bf c}\nabla{\boldsymbol{\rho}} , \\
&\partial_t {\bf u} + {\bf u}\cdot\nabla {\bf u} + \nabla {\bf p} = {\boldsymbol{\rho}}\nabla \phi , \\
& \mathrm{div}\, {\bf u} = 0.
\end{aligned}
\right.
\end{equation} Let us divide the proof into several steps.
\medskip
\noindent \textit{(i) Regularity of the pressure}: Taking the divergence of the equation for ${\bf u}$, we obtain that \begin{equation*}
\begin{split}
\Delta {\bf p} = -\sum_{1 \le i,j \le d} \partial_i {\bf u}^j \partial_j {\bf u}^i + \nabla\cdot({\boldsymbol{\rho}} \nabla \phi).
\end{split}
\end{equation*} We then have that \begin{equation*}
\begin{split}
\nabla {\bf p} = \nabla(-\Delta)^{-1} \left(\sum_{1 \le i,j \le d} \partial_i {\bf u}^j \partial_j {\bf u}^i\right) - \nabla(-\Delta)^{-1} \nabla\cdot({\boldsymbol{\rho}}\nabla\phi) .
\end{split}
\end{equation*} We claim that for $m \ge \frac{d}{2} + 1$, \begin{equation*}
\begin{split}
\nrm{\nabla {\bf p}}_{H^m} \lesssim \nrm{{\bf u}}_{H^m}^2 + \nrm{\nabla\phi}_{H^m} \nrm{{\boldsymbol{\rho}}}_{H^m}.
\end{split}
\end{equation*} It is straightforward to bound the second term: since $\nabla(-\Delta)^{-1}\nabla\cdot$ is an operator with bounded Fourier multiplier, \begin{equation*}
\begin{split}
\nrm{\nabla(-\Delta)^{-1} \nabla\cdot({\boldsymbol{\rho}}\nabla\phi) }_{H^m} \lesssim \nrm{{\boldsymbol{\rho}}\nabla\phi}_{H^m} \lesssim \nrm{\nabla\phi}_{H^m} \nrm{{\boldsymbol{\rho}}}_{H^m}.
\end{split}
\end{equation*} On the other hand, for any $n \ge 1$, \begin{equation*}
\begin{split}
\nrm{ \nabla (-\Delta)^{-1} (\partial_i{\bf u}^j \partial_j{\bf u}^i) }_{\dot{H}^n} \lesssim \nrm{\partial_i{\bf u}^j \partial_j {\bf u}^i}_{\dot{H}^{n-1}} \lesssim \nrm{\nabla{\bf u}}_{L^\infty} \nrm{{\bf u}}_{H^n}.
\end{split}
\end{equation*} Lastly, using incompressibility \begin{equation*}
\begin{split}
\nrm{ \nabla (-\Delta)^{-1}\sum_i (\partial_i{\bf u}^j \partial_j{\bf u}^i) }_{L^2} = \nrm{ \nabla (-\Delta)^{-1}\partial_i \sum_i ({\bf u}^j \partial_j{\bf u}^i) }_{L^2} \lesssim \nrm{\nabla{\bf u}}_{L^\infty} \nrm{{\bf u}}_{L^2}.
\end{split}
\end{equation*} The claim follows.
\medskip
\noindent \textit{(ii) Estimate of the infimum}: Assuming for a moment that a sufficiently regular solution exists, we now obtain a simple estimate on the variation of the infimum. For this, let $x^*(t)$ be any time-dependent continuous curve of infimum point of ${\boldsymbol{\rho}}(t)$.
Dividing both sides of the ${\boldsymbol{\rho}}$-equation by ${\boldsymbol{\rho}}^2$, \begin{equation*}
\begin{split}
\partial_t (\frac{1}{{\boldsymbol{\rho}}}) + ({\bf u} + {\bf f}) \cdot\nabla (\frac{1}{{\boldsymbol{\rho}}}) = (\nabla \cdot {\bf f}) (\frac{1}{{\boldsymbol{\rho}}}) .
\end{split}
\end{equation*} Evaluating along the characteristics defined by ${\bf u}+{\bf f}$, we obtain the estimate \begin{equation}\label{eq:rho-inf}
\begin{split}
\left| \frac{d}{dt} \nrm{{\boldsymbol{\rho}}}_{inf}^{-1} \right| \le \nrm{\Delta{\bf c}}_{L^\infty} \nrm{{\boldsymbol{\rho}}}_{inf}^{-1} .
\end{split}
\end{equation} Similarly, from the equation for ${\bf c}$, one can show that \begin{equation}\label{eq:c-inf}
\begin{split}
\left| \frac{d}{dt}\nrm{{\bf c}}_{inf}^{-1} \right| \le \nrm{{\boldsymbol{\rho}}}_{L^\infty}\nrm{{\bf c}}_{inf}^{-1} .
\end{split}
\end{equation}
\medskip
\noindent \textit{(iii) Good variables}: Still proceeding under the assumption of the existence of a sufficiently smooth solution, we introduce the good variables and perform a priori estimates. For each $m\ge 0$, we fix an $m$-th order partial derivative, and define the ``good'' variables as follows: \begin{equation*}
\begin{split}
R_m &:= \partial^m {\boldsymbol{\rho}} + \frac{1}{{\bf c}} {\bf f} \cdot \partial^m {\bf u} ,\\
F_m &:= \partial^m {\bf f} - \frac{1}{{\boldsymbol{\rho}}{\bf c}} {\bf f} ({\bf f} \cdot \partial^m {\bf u}).
\end{split}
\end{equation*} We introduce notation $(\mathcal O_m)^k$ to denote expressions that can be bounded in $L^2$ by a constant multiple of $(1+\nrm{{\bf u}}_{H^m} + \nrm{{\boldsymbol{\rho}}}_{H^m} + \nrm{{\bf c}}_{H^{m+1}} + \nrm{{\boldsymbol{\rho}}}_{inf}^{-1} + \nrm{{\bf c}}_{inf}^{-1} )^k$. From the equation for ${\bf u}$ (using the previous bound for $\nabla{\bf p}$), we have \begin{equation*}
\begin{split}
\partial_t \partial^m{\bf u} + {\bf u} \cdot\nabla \partial^m {\bf u} = (\mathcal O_m)^2.
\end{split}
\end{equation*} Similarly, we may write \begin{equation*}
\begin{split}
\partial_t \partial^m{\boldsymbol{\rho}} + {\bf u} \cdot \nabla \partial^m{\boldsymbol{\rho}} = - {\boldsymbol{\rho}} \nabla \cdot \partial^m{\bf f} - {\bf f} \cdot \nabla \partial^m {\boldsymbol{\rho}} + (\mathcal O_m)^2
\end{split}
\end{equation*} and \begin{equation*}
\begin{split}
\partial_t \partial^m {\bf f} + {\bf u} \cdot \nabla \partial^m{\bf f} = - \nabla ({\bf f} \cdot \partial^m{\bf u}) - {\bf c} \nabla \partial^m{\boldsymbol{\rho}} + (\mathcal O_m)^2.
\end{split}
\end{equation*} We first massage the equation for $\partial_t\partial^m{\boldsymbol{\rho}}$: \begin{equation*}
\begin{split}
\partial_t( \partial^m{\boldsymbol{\rho}} + \frac{1}{{\bf c}} {\bf f} \cdot \partial^m{\bf u}) + ({\bf u}\cdot\nabla)( \partial^m{\boldsymbol{\rho}} + \frac{1}{{\bf c}} {\bf f} \cdot \partial^m{\bf u}) &= \partial_t(\frac{1}{{\bf c}}{\bf f}) \cdot \partial^m{\bf u} - {\boldsymbol{\rho}}\nabla\cdot\partial^m{\bf f} + \frac{1}{{\bf c}}{\bf f} \cdot( ({\bf f}\cdot\nabla) \partial^m{\bf u}) \\
&\quad - ({\bf f}\cdot\nabla) ( \partial^m{\boldsymbol{\rho}} + \frac{1}{{\bf c}} {\bf f} \cdot \partial^m{\bf u}) + {(\mathcal O_m)^6}.
\end{split}
\end{equation*}
Note that we can rewrite the second and third terms on the right hand side as (repeated indices being summed) \begin{equation*}
\begin{split}
- {\boldsymbol{\rho}}\nabla\cdot\partial^m{\bf f} + \frac{1}{{\bf c}}{\bf f} \cdot( ({\bf f}\cdot\nabla) \partial^m{\bf u}) & = -{\boldsymbol{\rho}} \left( \partial^m\partial_j{\bf f}^j - \frac{1}{{\bf c}{\boldsymbol{\rho}}} {\bf f}^\ell {\bf f}^j \partial_j \partial^m {\bf u}^\ell \right) \\
& = -{\boldsymbol{\rho}} \partial_j \left( \partial^m{\bf f}^j - \frac{1}{{\bf c}{\boldsymbol{\rho}}} {\bf f}^j {\bf f}\cdot \partial^m {\bf u} \right) - {\boldsymbol{\rho}} \partial_j(\frac{1}{{\bf c}{\boldsymbol{\rho}}}) {\bf f}^j {\bf f}\cdot\partial^m{\bf u} + (\mathcal O_m)^4.
\end{split}
\end{equation*} That is, we have \begin{equation}\label{eq:R_m}
\begin{split}
D_t R_m = -({\bf f}\cdot\nabla) R_m - {\boldsymbol{\rho}} \nabla \cdot F_m + (\mathcal O_m)^6.
\end{split}
\end{equation} On the other hand, we have \begin{equation}\label{eq:F_m}
\begin{split}
D_t F_m = -{\bf c} \nabla R_m + (\mathcal O_m)^6.
\end{split}
\end{equation} Multiplying both sides of \eqref{eq:R_m} by ${\bf c} R_m$ and integrating, we obtain that \begin{equation*}
\begin{split}
\left| \int {\bf c}{\boldsymbol{\rho}} R_m\nabla \cdot F_m
+ {\frac12} \frac{d}{dt} \nrm{\sqrt{{\bf c}}R_m}_{L^2}^{2} \right| &\lesssim \nrm{\nabla({\bf c}{\bf f})}_{L^\infty} \nrm{R_m}_{L^2}^2 + \nrm{{\bf c}}_{L^\infty}\nrm{R_m}_{L^2}(\mathcal O_m)^6 \\
&\lesssim \nrm{R_m}_{L^2}^2 (\mathcal O_m)^2 + \nrm{R_m}_{L^2} (\mathcal O_m)^7 \\
&\lesssim \nrm{R_m}_{L^2}^2 (\mathcal O_m)^2 + (\mathcal O_m)^{12}.
\end{split}
\end{equation*} {Note that now we are abusing notation to write $(\mathcal O_m)^k$ for quantities bounded by a constant multiple of $(1+\nrm{{\bf u}}_{H^m} + \nrm{{\boldsymbol{\rho}}}_{H^m} + \nrm{{\bf c}}_{H^{m+1}} + \nrm{{\boldsymbol{\rho}}}_{inf}^{-1} + \nrm{{\bf c}}_{inf}^{-1} )^k$.} Similarly, multiplying both sides of \eqref{eq:F_m} by ${\boldsymbol{\rho}} F_m$ and integrating gives \begin{equation*}
\begin{split}
\left| \int {\boldsymbol{\rho}}{\bf c} F_m \nabla R_m + {\frac12} \frac{d}{dt} \nrm{\sqrt{{\boldsymbol{\rho}}} F_m}_{L^2}^2 \right| \lesssim \nrm{F_m}_{L^2}^2 (\mathcal O_m)^2 + (\mathcal O_m)^{12}.
\end{split}
\end{equation*} Combining the above, we have \begin{equation*}
\begin{split}
\frac{d}{dt}\left( \nrm{\sqrt{{\bf c}}R_m}_{L^2}^{2} + \nrm{\sqrt{{\boldsymbol{\rho}}} F_m}_{L^2}^2 \right) \lesssim \left( \nrm{\sqrt{{\bf c}}R_m}_{L^2}^{2} + \nrm{\sqrt{{\boldsymbol{\rho}}} F_m}_{L^2}^2 \right) (\mathcal O_m)^3 + (\mathcal O_m)^{12}.
\end{split}
\end{equation*} Now, from the equation for $\partial_t \partial^m{\bf u}$, we obtain \begin{equation*}
\begin{split}
\frac{d}{dt} \nrm{\partial^m{\bf u}}_{L^2}^2 \lesssim (\mathcal O_m)^3.
\end{split}
\end{equation*} For a stronger reason, we can derive \begin{equation*}
\begin{split}
\frac{d}{dt}\sum_{k=0}^m\left( \nrm{\sqrt{{\bf c}}R_k}_{L^2}^{2} + \nrm{\sqrt{{\boldsymbol{\rho}}} F_k}_{L^2}^2 + \nrm{\partial^k{\bf u}}_{L^2}^2 \right) \lesssim \left( \nrm{\sqrt{{\bf c}}R_m}_{L^2}^{2} + \nrm{\sqrt{{\boldsymbol{\rho}}} F_m}_{L^2}^2 \right) (\mathcal O_m)^3 + (\mathcal O_m)^{12}.
\end{split}
\end{equation*}
Next, it is not difficult to see that by defining \begin{equation*}
\begin{split}
Z_m := \sum_{k=0}^m\left( \nrm{\sqrt{{\bf c}}R_k}_{L^2}^{2} + \nrm{\sqrt{{\boldsymbol{\rho}}} F_k}_{L^2}^2 + \nrm{\partial^k{\bf u}}_{L^2}^2 \right) + \nrm{{\boldsymbol{\rho}}}_{inf}^{-1} + \nrm{{\bf c}}_{inf}^{-1} + \nrm{{\bf f}}_{L^\infty} ,
\end{split}
\end{equation*} we have \begin{equation*}
\begin{split}
\frac{d}{dt} Z_m \lesssim (Z_m)^2 (1 + (Z_m)^2) (\mathcal O_m)^8 + (\mathcal O_m)^{12}.
\end{split}
\end{equation*} We are in a position to close the estimates in terms of $Z_m$: recalling the definitions of $R_m$ and $F_m$, we have \begin{equation*}
\begin{split}
\nrm{{\boldsymbol{\rho}}}_{H^m} + \nrm{{\bf f}}_{H^m} \lesssim \sum_{k=0}^m( \nrm{R_k}_{L^2} + \nrm{F_k}_{L^2} ) + (1+ \nrm{{\boldsymbol{\rho}}}_{inf}^{-1} + \nrm{{\bf c}}_{inf}^{-1})^2 \nrm{{\bf f}}_{L^\infty}^2 \nrm{{\bf u}}_{H^m} \lesssim (1 + Z_m)^5
\end{split}
\end{equation*} which gives $\mathcal O_m \lesssim (1 + Z_m)^5$. Therefore, \begin{equation}\label{eq:apriori-inviscid}
\begin{split}
\frac{d}{dt} Z_m \lesssim ( 1 + Z_m)^{60}.
\end{split}
\end{equation} Therefore, for $Z_m(0)<\infty$, there exists $T>0$ such that $Z_m(t) \le 2Z_m(0)$ for $t<T$.
\medskip
\noindent \textit{(iv) Existence}: Existence of a solution can be shown using viscous approximations; for fixed $\epsilon>0$, we consider the viscous system \begin{equation} \label{eq:KSF-viscous}
\left\{
\begin{aligned}
&\partial_t {\boldsymbol{\rho}}^{(\epsilon)} + {\bf u}^{(\epsilon)}\cdot\nabla {\boldsymbol{\rho}}^{(\epsilon)} = -\nabla \cdot ({\boldsymbol{\rho}}^{(\epsilon)}{\bf f}^{(\epsilon)}) + \epsilon \Delta {\boldsymbol{\rho}}^{(\epsilon)} , \\
&\partial_t {\bf f}^{(\epsilon)} + {\bf u}^{(\epsilon)}\cdot\nabla {\bf f}^{(\epsilon)} = -[ \nabla{\bf u}^{(\epsilon)} ]^T {\bf f}^{(\epsilon)} - {\boldsymbol{\rho}}^{(\epsilon)} {\bf f}^{(\epsilon)} - {\bf c}\nabla{\boldsymbol{\rho}}^{(\epsilon)} + \epsilon \Delta {\bf f}^{(\epsilon)} , \\
&\partial_t {\bf u}^{(\epsilon)} + {\bf u}^{(\epsilon)}\cdot\nabla {\bf u}^{(\epsilon)} + \nabla {\bf p}^{(\epsilon)} = {\boldsymbol{\rho}}^{(\epsilon)}\nabla \phi + \epsilon\Delta{\bf u}^{(\epsilon)} , \\
& \mathrm{div}\, {\bf u}^{(\epsilon)} = 0
\end{aligned}
\right.
\end{equation} with the same initial data $({\boldsymbol{\rho}}_0,{\bf f}_0,{\bf u}_0)$. Existence of a smooth local in time solution to \eqref{eq:KSF-viscous} was established already in \cite{CKL1,CKL2}. Moreover, it is not difficult to show by direct computation that the solution satisfies the a priori estimate \eqref{eq:apriori-inviscid} uniformly in $\epsilon>0$. {Since this step is not entirely obvious, we repeat the infimum estimate for the viscous solutions in the case of ${\boldsymbol{\rho}}^{(\epsilon)}$. (The case of ${\bf c}^{(\epsilon)}$ is similar.) Since ${\boldsymbol{\rho}}^{(\epsilon)}$ is continuous in time, at least for a very small time interval, we have a uniform lower bound ${\boldsymbol{\rho}}^{(\epsilon)}(t,x) \ge c_0>0$ for some constant $c_{0}$ from the positivity of the initial data. Therefore, with the identity \begin{equation*}
\begin{split}
\Delta \frac{1}{{\boldsymbol{\rho}}^{(\epsilon)}} = - \frac{\Delta {\boldsymbol{\rho}}^{(\epsilon)}}{({\boldsymbol{\rho}}^{(\epsilon)})^2} + 2 \frac{\nabla {\boldsymbol{\rho}}^{(\epsilon)}}{{\boldsymbol{\rho}}^{(\epsilon)}} \cdot \nabla \frac{1}{{\boldsymbol{\rho}}^{(\epsilon)}},
\end{split}
\end{equation*}we can compute on this time interval \begin{equation*}
\begin{split}
\partial_t \frac{1}{{\boldsymbol{\rho}}^{(\epsilon)}} + ({\bf u}^{(\epsilon)} + {\bf f}^{(\epsilon)} + 2\epsilon \frac{\nabla {\boldsymbol{\rho}}^{(\epsilon)}}{{\boldsymbol{\rho}}^{(\epsilon)}} ) \cdot\nabla (\frac{1}{{\boldsymbol{\rho}}^{(\epsilon)}}) = (\nabla \cdot {\bf f}^{(\epsilon)}) (\frac{1}{{\boldsymbol{\rho}}^{(\epsilon)}}) + \epsilon \Delta \frac{1}{{\boldsymbol{\rho}}^{(\epsilon)}} .
\end{split}
\end{equation*} Therefore, we obtain the estimate \begin{equation}\label{eq:rho-inf-viscous}
\begin{split}
\left| \frac{d}{dt} \nrm{{\boldsymbol{\rho}}^{(\epsilon)}}_{inf}^{-1} \right| \le \nrm{\Delta{\bf c}^{(\epsilon)}}_{L^\infty} \nrm{{\boldsymbol{\rho}}^{(\epsilon)}}_{inf}^{-1}
\end{split}
\end{equation}
as in the proof of maximum principle for advection-diffusion equations. With a uniform--in--$\epsilon$ bound on $\nrm{\Delta{\bf c}^{(\epsilon)}}_{L^\infty} $, this inequality guarantees that the quantity $\nrm{{\boldsymbol{\rho}}^{(\epsilon)}}_{inf}^{-1} $ is actually uniformly bounded in some common time interval.
Next, the uniform bound on the quantity $Z_m$ guarantees that for all $\epsilon>0$, the solution $({\boldsymbol{\rho}}^{(\epsilon)},{\bf f}^{(\epsilon)},{\bf c}^{(\epsilon)})$ can be extended at least up to time interval $[0,T]$ for some $T>0$. By passing to a weakly convergent subsequence as $\epsilon>0$, one obtains a triple $({\boldsymbol{\rho}},{\bf f},{\bf c})$ with finite $Z_m$ on $[0,T]$. It is not difficult to show that this triple is a solution to \eqref{eq:KSF-inviscid}, with prescribed initial data. }
\medskip
\noindent \textit{(v) Uniqueness}: To prove uniqueness, we assume existence of two solutions $({\boldsymbol{\rho}}_i, {\bf f}_i,{\bf u}_i,{\bf p}_i)$ ($i = 1, 2$) to \eqref{eq:KSF-inviscid} satisfying the properties stated in the theorem with the same initial data. We define \begin{equation*}
\begin{split}
\tilde{\bfrho} = {\boldsymbol{\rho}}_1 - {\boldsymbol{\rho}}_2, \quad \tilde{\bfc} = {\bf c}_1-{\bf c}_2,\quad \tf = {\bf f}_1 - {\bf f}_2, \quad \tu = {\bf u}_1 - {\bf u}_2, \quad \tilde{{\bf p}} = {\bf p}_1-{\bf p}_2.
\end{split}
\end{equation*} Note that $\tilde{\bfrho}$ and $\tu$ respectively satisfies \begin{equation*}
\begin{split}
\partial_t\tilde{\bfrho} + ({\bf u}_1+{\bf f}_1) \cdot\nabla\tilde{\bfrho} + (\tu+\tf)\cdot\nabla{\boldsymbol{\rho}}_2 = - {\boldsymbol{\rho}}_2 \nabla\cdot\tf - \tilde{\bfrho} \nabla\cdot {\bf f}_1 ,
\end{split}
\end{equation*} and \begin{equation*}
\begin{split}
\partial_t\tu + {\bf u}_1 \cdot\nabla\tu + \tu \cdot\nabla{\bf u}_2 + \nabla\tilde{{\bf p}}= \tilde{\bfrho} \nabla\phi .
\end{split}
\end{equation*} From the equations for $\tilde{\bfrho}$ and $\tu$, we obtain that \begin{equation}\label{eq:trho-modified}
\begin{split}
&\partial_t (\tilde{\bfrho} + \frac{{\bf f}_2}{{\bf c}_2}\cdot \tu) + {\bf u}_1\cdot\nabla (\tilde{\bfrho} + \frac{{\bf f}_2}{{\bf c}_2}\cdot \tu) \\
&\quad= -{\bf f}_1\cdot\nabla\tilde{\bfrho} - {\boldsymbol{\rho}}_2\nabla\cdot\tf - (\tu+\tf)\cdot\nabla{\boldsymbol{\rho}}_2 - \tilde{\bfrho} \nabla\cdot {\bf f}_1 - \tu \cdot\nabla{\bf u}_2- \nabla\tilde{{\bf p}} + \tilde{\bfrho} \nabla\phi \\
&\quad\qquad + \partial_t( \frac{{\bf f}_2}{{\bf c}_2} ) \cdot \tu + {\bf u}_1\cdot \nabla ( \frac{{\bf f}_2}{{\bf c}_2} ) \cdot \tu \\
&\quad= -{\bf f}_1\cdot \nabla (\tilde{\bfrho} + \frac{{\bf f}_2}{{\bf c}_2}\cdot \tu) - {\boldsymbol{\rho}}_2 \nabla\cdot (\tf - \frac{{\bf f}_1}{{\boldsymbol{\rho}}_2{\bf c}_2}{\bf f}_2\cdot\tu) - {\boldsymbol{\rho}}_2 \nabla\cdot (\frac{{\bf f}_1}{{\boldsymbol{\rho}}_2}) \frac{{\bf f}_2}{{\bf c}_2}\cdot\tu \\
&\quad\qquad - (\tu+\tf)\cdot\nabla{\boldsymbol{\rho}}_2 - \tilde{\bfrho} \nabla\cdot {\bf f}_1 - \tu \cdot\nabla{\bf u}_2- \nabla\tilde{{\bf p}} + \tilde{\bfrho} \nabla\phi + \partial_t( \frac{{\bf f}_2}{{\bf c}_2} ) \cdot \tu + {\bf u}_1\cdot \nabla ( \frac{{\bf f}_2}{{\bf c}_2} ) \cdot \tu.
\end{split}
\end{equation} On the other hand, $\tilde{\bfc}$ and $\tf$ satisfy \begin{equation*}
\begin{split}
\partial_t\tilde{\bfc} + {\bf u}_1 \cdot\nabla\tilde{\bfc} + \tu \cdot\nabla{\bf c}_2 = -{\boldsymbol{\rho}}_1\tilde{\bfc} - {\bf c}_2 \tilde{\bfrho} ,
\end{split}
\end{equation*} and \begin{equation*}
\begin{split}
\partial_t\tf + {\bf u}_1 \cdot\nabla\tf = -(\nabla{\bf u}_1)^T \tf - {\boldsymbol{\rho}}_1 \tf - \tilde{\bfc} \nabla {\boldsymbol{\rho}}_1 - {\bf c}_2 \nabla\tilde{\bfrho} - \tilde{\bfrho}\nabla{\bf c}_2 - \tu\cdot\nabla{\bf f}_2 - (\nabla\tu)^T {\bf f}_2 .
\end{split}
\end{equation*} From the equation for $\tf$, we have \begin{equation}\label{eq:tf-modified}
\begin{split}
& \partial_t(\tf - \frac{{\bf f}_1}{{\boldsymbol{\rho}}_2{\bf c}_2}{\bf f}_2\cdot\tu ) + {\bf u}_1 \cdot\nabla(\tf - \frac{{\bf f}_1}{{\boldsymbol{\rho}}_2{\bf c}_2}{\bf f}_2\cdot\tu ) \\
& \quad = - {\bf c}_2 \nabla ({\boldsymbol{\rho}} + \frac{{\bf f}_2}{{\bf c}_2}\tu) + {\bf c}_2 \nabla\cdot (\frac{{\bf f}_2}{{\bf c}_2}) \tu \\
& \quad\qquad -\partial_t ( \frac{{\bf f}_1}{{\boldsymbol{\rho}}_2{\bf c}_2}{\bf f}_2 ) \cdot\tu - {\bf u}_1\cdot\nabla ( \frac{{\bf f}_1}{{\boldsymbol{\rho}}_2{\bf c}_2}{\bf f}_2 ) \cdot \tu -(\nabla{\bf u}_1)^T \tf - {\boldsymbol{\rho}}_1 \tf - \tilde{\bfc} \nabla {\boldsymbol{\rho}}_1 - \tilde{\bfrho}\nabla{\bf c}_2 - \tu\cdot\nabla{\bf f}_2 .
\end{split}
\end{equation} Now, from the equation for $\tu$, we easily obtain that \begin{equation*}
\begin{split}
\left|\frac{d}{dt} \nrm{\tu}_{L^2}^2\right| \lesssim \nrm{\tu}_{L^2}^2 + \nrm{ \tilde{\bfrho} + \frac{{\bf f}_2}{{\bf c}_2}\cdot \tu }_{L^2}^2.
\end{split}
\end{equation*} {To derive this estimate, we have used the assumed regularity of the solutions $({\boldsymbol{\rho}}_i, {\bf f}_i,{\bf u}_i,{\bf p}_i)$ for $i = 1$ and $2$: $\tilde{{\bf u}} \in H^m$ ensures that $\partial_{t} \tilde{{\bf u}}$ is defined as a continuous function in space and therefore $\frac12\nrm{\tilde{{\bf u}}}_{L^2}^2$ is differentiable in time with time derivative equal to $\int \tilde{{\bf u}} \partial_t \tilde{{\bf u}}$.} Next, simultaneously using \eqref{eq:trho-modified} and \eqref{eq:tf-modified}, \begin{equation*}
\begin{split}
\left| \frac{d}{dt} \left( \nrm{ \sqrt{{\bf c}_2} (\tilde{\bfrho} + \frac{{\bf f}_2}{{\bf c}_2}\cdot \tu )}_{L^2}^2 + \nrm{ \sqrt{{\boldsymbol{\rho}}_2} (\tf - \frac{{\bf f}_1}{{\boldsymbol{\rho}}_2{\bf c}_2}{\bf f}_2\cdot\tu )}_{L^2}^2 \right) \right|\lesssim \nrm{ \tilde{\bfrho} + \frac{{\bf f}_2}{{\bf c}_2}\cdot \tu }_{L^2}^2 + \nrm{\tf - \frac{{\bf f}_1}{{\boldsymbol{\rho}}_2{\bf c}_2}{\bf f}_2\cdot\tu }_{L^2}^2 + \nrm{\tu}_{L^2}^2.
\end{split}
\end{equation*} Hence, for \begin{equation*}
\begin{split}
\widetilde{X} :=\nrm{ \sqrt{{\bf c}_2} (\tilde{\bfrho} + \frac{{\bf f}_2}{{\bf c}_2}\cdot \tu )}_{L^2}^2 + \nrm{ \sqrt{{\boldsymbol{\rho}}_2} (\tf - \frac{{\bf f}_1}{{\boldsymbol{\rho}}_2{\bf c}_2}{\bf f}_2\cdot\tu )}_{L^2}^2 + \nrm{\tu}_{L^2}^2,
\end{split}
\end{equation*} we have \begin{equation*}
\begin{split}
\left|\frac{d}{dt} \widetilde{X} \right|\lesssim \widetilde{X}.
\end{split}
\end{equation*} Since $\widetilde{X}(t=0) = 0$, we conclude that $\widetilde{X} = 0$ for $t>0$. This finishes the proof of uniqueness.
\medskip
\noindent \textit{(vi) Modification for general $\chi$ and $k$:} In the case of general $\chi$ and $k$, we simply define the good variables by \begin{equation*}
\begin{split}
R_m = \partial^m{\boldsymbol{\rho}} + \frac{\nabla{\bf c}}{k({\bf c})} \partial^m{\bf u}
\end{split}
\end{equation*} and \begin{equation*}
\begin{split}
F_m = \partial^m {\bf f} - \frac{{\bf f}}{{\boldsymbol{\rho}} k({\bf c})} {\bf f} \cdot \partial^m{\bf u} .
\end{split}
\end{equation*} Moreover, instead of $\sqrt{{\bf c}}$, we use the weight \begin{equation*}
\begin{split}
\left( \frac{k({\bf c})}{\chi({\bf c})}\right)^{\frac{1}{2}}
\end{split}
\end{equation*} for $R_m$. The assumption \begin{equation*}
\begin{split}
\frac{k({\bf c})}{\chi({\bf c})} \gtrsim {\bf c}^\gamma
\end{split}
\end{equation*} allows us to bound \begin{equation*}
\begin{split}
\nrm{R_m}_{L^2} \lesssim \nrm{{\bf c}}_{inf}^{-\frac{\gamma}{2}} \nrm{ \left( \frac{k({\bf c})}{\chi({\bf c})}\right)^{\frac{1}{2}} R_m }_{L^2}.
\end{split}
\end{equation*} We omit the details. The proof is now complete.
\end{proof}
{\begin{remark}[Extension to general sensitivity functions]\label{rem:rotation}
In an interesting recent work \cite{WinklerIMRN}, Winkler considered the very general case where the chemotactic sensitivity function is given by a matrix $S$ depending on $x$, ${\boldsymbol{\rho}}$, and ${\bf c}$. In the presence of diffusion, the author established eventual relaxation of Keller--Segel-fluid systems with general bounded and smooth $S$. Here, let us briefly present an extension of Theorem \ref{thm:lwp-away} in the simplest case of $\chi(z) = 1$, $k(z) = z$, and ${\bf u} \equiv 0$: \begin{equation}\label{eq:rot}
\left\{
\begin{aligned}
&\partial_t \boldsymbol{\rho} = -\nabla \cdot ({\boldsymbol{\rho}} S \nabla {\bf c}) , \\
&\partial_t {\bf c} = -{\bf c} {\boldsymbol{\rho}} .
\end{aligned}
\right.
\end{equation} Under the assumption that $\rho$ and ${\bf c}$ are bounded away from 0 at the initial time, we claim that the generalized system \eqref{eq:rot} is locally well-posed if the $n\times n$--matrix $S$ satisfies \begin{equation*}
\begin{split}
S + S^T \mbox{ is a diagonal matrix with strictly positive diagonal entries}.
\end{split}
\end{equation*} That is, $S = (s_{ij})_{1\le i,j \le n}$ satisfies $s_{ij}+s_{ji}=0$ for $i \ne j$ and $s_{jj}>0$. When $S$ is dependent upon $x, {\boldsymbol{\rho}}, {\bf c}$, we just need to require the above condition (together with uniform boundedness and smoothness of $S$) uniformly for each $S(x,{\boldsymbol{\rho}},{\bf c})$. In the case of two spatial dimensions $n=2$, this means that our local well-posedness theory can cover rotation matrices with rotation angle strictly less than $\pi/2$.\footnote{The case of rotation with angle $\pi/2$ results in a very interesting system, for which the question of local well/ill-posedness seems delicate.} To prove this extension, one can notice cancellations between the top order terms in the expression \begin{equation*}
\begin{split}
\frac{d}{dt} \frac12 \int {\bf c} (\partial^m{\boldsymbol{\rho}})^2 + \sum_{j=1}^{n} s_{jj}{\boldsymbol{\rho}} |\partial^{m}\partial_{j}{\bf c}|^2.
\end{split}
\end{equation*} For $m$ large, the top order terms in the above expression are \begin{equation*}
\begin{split}
I = \int {\bf c} {\boldsymbol{\rho}} \nabla (\partial^m{\boldsymbol{\rho}}) \cdot (S \nabla (\partial^m{\bf c})), \qquad II=- \sum_{j=1}^{n} \int {\bf c} {\boldsymbol{\rho}} \partial_j (\partial^m{\bf c}) \partial_j (\partial^m{\boldsymbol{\rho}}).
\end{split}
\end{equation*} Then, in the expression $I$, the contributions from off-diagonal terms of $S$ cancel each other due to the condition $s_{ij} + s_{ji} = 0$ for $i\ne j$, and the contribution from the diagonal entries of $S$ cancel precisely with $II$.
\end{remark}}
\begin{remark}[Ill-posedness in the case of opposite signs] \label{rem:illposed}
We note that having the same signs in the right hand side of \eqref{eq:KS-1D} is crucial for well-posedness. Indeed, consider the following inviscid system obtained by reverting the sign of the right hand side for $\partial_t{\bf c}$:
\begin{equation}\label{eq:KS-1D-prime}
\left\{
\begin{aligned}
\partial_t {\boldsymbol{\rho}} & = -\partial_x ( {\boldsymbol{\rho}} \partial_x {\bf c}), \\
\partial_t {\bf c} & = {\bf c}{\boldsymbol{\rho}} .
\end{aligned}
\right.
\end{equation} Similarly as before, $({\boldsymbol{\rho}},{\bf c}) = (1, e^{t})$ provides a solution to the above. The linearization around this solution is given by \begin{equation}\label{eq:KS-1D-prime-lin}
\left\{
\begin{aligned}
\partial_t \tilde{{\boldsymbol{\rho}}} & = -\partial_{xx}\tilde{{\bf c}}, \\
\partial_t \tilde{{\bf c}} & = e^{t}\tilde{{\boldsymbol{\rho}}} + \tilde{{\bf c}}.
\end{aligned}
\right.
\end{equation} It is not difficult to show that this linear equation is \textit{ill-posed}: there exist smooth data which immediately lose smoothness for $t>0$. Indeed, taking another time derivative, \begin{equation*}
\begin{split}
\partial_{tt}\tilde{{\boldsymbol{\rho}}} = -e^{t}\partial_{xx}\tilde{{\boldsymbol{\rho}}} - \partial_{xx}\tilde{{\bf c}}
\end{split}
\end{equation*} Modulo the coefficient $e^{t}$ and the lower order term $\partial_{xx}\tilde{{\bf c}}$, we have the Laplace equation in the $(t,x)$-coordinates, whose initial value problem is well-known to be illposed. A similar analysis can be repeated for the system \begin{equation}\label{eq:KS-1D-prime2}
\left\{
\begin{aligned}
\partial_t {\boldsymbol{\rho}} & = -\partial_x ( {\boldsymbol{\rho}} \partial_x {\bf c}), \\
\partial_t {\bf c} & = {\boldsymbol{\rho}} .
\end{aligned}
\right.
\end{equation}
\end{remark}
\subsection{Well-posedness with high order vanishing data}\label{subsec:v1}
Let us begin this section with stating a version of Theorem \ref{MainThm3}: for this purpose, we need to introduce
\begin{definition}
We introduce the following function spaces which is defined only for non-negative functions. We say that $g\ge 0$ belongs to $\dot{\mathcal X}^{m,\gamma}$ for some $m\ge 1$ and $0<\gamma<2$ if {$g \in H^{m}$} and the quantity \begin{equation}\label{eq:Xmr}
\begin{split}
\nrm{g}_{\dot{\mathcal X}^{m,\gamma}}^{2} := \nrm{g^{-\frac{\gamma}{2}} |\nabla^mg| }_{L^2}^2 + \nrm{g^{-(1+\frac{\gamma-2}{2m})} \nabla g }_{L^{2m}}^{2m} := \int \frac{|\nabla^m g|^2}{g^{\gamma}} + \int \frac{|\nabla g|^{2m}}{g^{2m+\gamma-2}}
\end{split}
\end{equation} is finite. {To be precise, the integrals in the right hand side are defined by
\begin{equation*}
\begin{split}
\lim_{\epsilon\to 0^+} \left[ \int \frac{|\nabla^m g|^2}{(g+\epsilon)^{\gamma}} + \int \frac{|\nabla g|^{2m}}{(g+\epsilon)^{2m+\gamma-2}} \right] \, .
\end{split}
\end{equation*} For $g \in H^{m}$, the above quantities are well-defined by allowing $+\infty$.} In the case of $m = 0$, we simply set \begin{equation*}
\begin{split}
\nrm{g}_{\dot{\mathcal X}^{0,\gamma}}^{2} := \int g^{2-\gamma},
\end{split}
\end{equation*} and then we define \begin{equation*}
\begin{split}
\nrm{g}_{ {\mathcal X}^{m,\gamma}}^{2} := \sum_{k=0}^{m} \nrm{g}_{\dot{\mathcal X}^{k,\gamma}}^{2} .
\end{split}
\end{equation*}
\end{definition}
We are now ready to precisely state the local well-posedness result for possibly vanishing data.
\begin{theorem}[Well-posedness with vanishing data]\label{thm:lwp-rho-away}
We consider the fully inviscid \eqref{eq:KS} system ($D_\rho, D_c = 0$) on $\mathbb T^d$ under the following assumptions on the coefficients:
\begin{itemize}
\item $\chi$ is $C^\infty$--smooth and uniformly positive,
\item $\phi \in C^\infty(\mathbb T^d)$, and
\item there exist a $C^\infty$--smooth function $F_k : [0,\infty) \rightarrow (0,\infty)$ and $0<\gamma<2$ such that $k(z) = F_k(z^{\gamma})$.
\end{itemize}
Furthermore, we impose the following assumptions on the data: for some $2 + \frac{d}{2} < m $,
\begin{itemize}
\item ${\bf c}_0 \in {{\mathcal X}^{m+1,\gamma}} \cap L^\infty$,
\item ${\boldsymbol{\rho}}_0 \in \mathcal X^{m,1} \cap L^\infty $, and
\item there exists some $\delta$ satisfying \begin{equation*}
\begin{split}
1-\gamma \le \delta \le \min\left\{ \gamma, 2\left(m-1- \lfloor \frac{d}{2} \rfloor \right)\left( 1 - \frac{1-\frac{\gamma}{2}}{ m - \lfloor \frac{d}{2} \rfloor } \right) \right\}
\end{split}
\end{equation*} such that \begin{equation*}
\begin{split}
a \le \frac{{\boldsymbol{\rho}}_0(x)}{{\bf c}_0^{\delta}(x)} \le A
\end{split}
\end{equation*} for some constants $A,a>0$.
\end{itemize} Then the system \eqref{eq:KS} is locally-well posed: there exist some $T > 0$ and a unique solution $({\boldsymbol{\rho}},{\bf c})$ to \eqref{eq:KS} satisfying the initial condition and \begin{equation*}
\begin{split}
{\boldsymbol{\rho}}\in L^\infty([0,T);\mathcal X^{m,1} \cap L^\infty), \quad {\bf c} \in L^\infty([0,T); {{\mathcal X}^{m+1,\gamma}} \cap L^\infty), \quad \frac{{\boldsymbol{\rho}}(x)}{{\bf c}^{\delta}(x)} +\frac{{\bf c}^\delta(x)}{{\boldsymbol{\rho}}(x)} \in L^\infty([0,T);L^\infty).
\end{split}
\end{equation*}
\end{theorem}
\begin{remark}[Examples of initial data]
When a function is bounded away from 0, the $\mathcal X^{m,\gamma}$-norm is simply equivalent with a usual Sobolev norm. In the case when the function has an isolated zero (say the origin), one can ensure that it belongs to $\mathcal X^{m,\gamma}$ as long as the order of vanishing is high enough. To see this, say $g(x) \simeq |x|^{n}$ for some $n\ge 0$ in $\mathbb T^d$. Then, we formally compute that, near $x = 0$, \begin{equation*}
\begin{split}
\int \frac{|\nabla^m g|^2}{g^{\gamma}} \simeq \int |x|^{ (2-\gamma)n - 2m}, \quad \int \frac{|\nabla g|^{2m}}{g^{2m+\gamma-2}} \simeq \int |x|^{ (2-\gamma)n - 2m }.
\end{split}
\end{equation*} Therefore, locally at $x = 0$, $g \in \mathcal X^{m,\gamma}$ if and only if \begin{equation*}
\begin{split}
n > \frac{2m-d}{2-\gamma}.
\end{split}
\end{equation*} (This condition actually forces the condition $\gamma<2$ in the statement of Theorem \ref{thm:lwp-rho-away}.)
\end{remark}
We divide the proof into three parts. First, in \ref{subsubsec:GNS}, we establish some weighted inequalities of Gagliardo--Nirenberg--Sobolev type. We prove a priori estimates for the solution in \ref{subsubsec:apriori}. Finally, we show existence and uniqueness of the solution satisfying the a priori estimates in \ref{subsubsec:eandu}.
\subsubsection{A chain of weighted Gagliardo--Nirenberg--Sobolev inequalities}\label{subsubsec:GNS}
In this section, we shall state a series of inequalities which generalize the well known Gagliardo--Nirenberg--Sobolev inequality to the weighted case. We briefly remark on the notation used: Given an integer $m\ge 0$, we denote $\nabla^m g$ to denote the vector consisting of all partial derivatives for $g$ of order $m$. On the other hand, given some $d$-vector $\ell = (\ell^1,\cdots, \ell^d)$ with integer $\ell^i\ge0$, we define $\partial^\ell g = \partial_{x_1}^{(\ell^1)} \cdots \partial_{x_d}^{(\ell^d)}g$. In particular, $\partial^\ell g$ is an element of $\nabla^{|\ell|} g$, where $|\ell| = \ell^1 + \cdots + \ell^d$. With the above notation, we have the following key lemma:
\begin{lemma}\label{lem:inequality}
Let ${\bf c} \ge 0$ belong to $H^{m}(\mathbb T^d)$. For any integer $m\ge 1$ and a $k$-tuple $(\ell_1, \cdots, \ell_k)$ of $d$-vectors satisfying $m = |\ell_1| + \cdots + |\ell_k|$, we have that\begin{equation}\label{eq:inequality}
\begin{split}
\int_{ \{ {\bf c} > 0 \} } \frac{ 1 }{ {\bf c}^{2k-1} } \prod_{1\le i \le k} |\partial^{\ell_i}{\bf c}|^2 \lesssim_{m,d} \int_{ \{ {\bf c} > 0 \} } \frac{|\nabla^m{\bf c}|^2}{{\bf c}} + \int_{ \{ {\bf c} > 0 \} } \frac{|\nabla{\bf c}|^{2m}}{{\bf c}^{2m-1}}
\end{split}
\end{equation} {as long as the expression on the right hand side is finite.} In particular, under the same assumptions, we obtain that for any $0\le n \le m$,
\begin{equation}\label{eq:inequality-conseq}
\begin{split}
\nrm{ \nabla^{m-n}( \frac{\nabla^{n}{\bf c}}{\sqrt{{\bf c}} })}_{L^2({ \{ {\bf c} > 0 \} })}^2 \lesssim_{m,d} \int_{ \{ {\bf c} > 0 \} } \frac{|\nabla^m{\bf c}|^2}{{\bf c}} + \int_{ \{ {\bf c} > 0 \} } \frac{|\nabla{\bf c}|^{2m}}{{\bf c}^{2m-1}} .
\end{split}
\end{equation}
\end{lemma}
As a corollary, we obtain the following:
\begin{corollary}\label{cor:L-infty}
Assume that for some integer $k\ge 1$, ${\bf c}\ge 0$ satisfies \begin{equation*}
\begin{split}
\int_{ \{ {\bf c} > 0 \} } \frac{|\nabla^m{\bf c}|^2}{{\bf c}} + \int_{ \{ {\bf c} > 0 \} } \frac{|\nabla{\bf c}|^{2m}}{{\bf c}^{2m-1}} < + \infty
\end{split}
\end{equation*} for $m = 1, \cdots, k + 1 + \lfloor \frac{d}{2} \rfloor $. Then
\begin{equation}\label{eq:inequality-infty}
\begin{split}
\sup_{x : {\bf c}(x) >0 } \frac{|\nabla{\bf c}(x)|^{k}}{{\bf c}(x)^{k-\frac{1}{2}}} \le C_d\sum_{m=1}^{\lfloor \frac{d}{2} \rfloor +1 + k} \left( \int_{ \{ {\bf c} > 0 \} } \frac{|\nabla^m{\bf c}|^2}{{\bf c}} + \int_{ \{ {\bf c} > 0 \} } \frac{|\nabla{\bf c}|^{2m}}{{\bf c}^{2m-1}} \right) .
\end{split}
\end{equation}
\end{corollary}
\begin{proof}
The proof is immediate from Lemma \ref{lem:inequality} and the embedding $H^s(\mathbb T^d) \subset L^\infty(\mathbb T^d)$ for $s> \frac{d}{2}$. Indeed, we have that \begin{equation*}
\begin{split}
\frac{|\nabla{\bf c}(x)|^{k}}{{\bf c}(x)^{k-\frac{1}{2}}} \in L^2({ \{ {\bf c} > 0 \} }),
\end{split}
\end{equation*} and for all $s \le 1 + \lfloor \frac{d}{2} \rfloor $, \eqref{eq:inequality} applies to each term in the expression \begin{equation*}
\begin{split}
\nabla^{s} \left( \frac{|\nabla{\bf c}(x)|^{k}}{{\bf c}(x)^{k-\frac{1}{2}}} \right)
\end{split}
\end{equation*} to give an $L^2$ bound, with $m = k + s$. \end{proof}
\begin{proof}[{\bf{Proof of Lemma \ref{lem:inequality}}}]
The proof is based on first establishing
\begin{equation}\label{eq:ineq-GNS}
\begin{split}
\int_{ \{ {\bf c} > 0 \} } \frac{|\nabla^\ell {\bf c}|^{\frac{2m}{\ell}}}{{\bf c}^{\frac{2m}{\ell}-1}} \lesssim_{m,d} \int_{ \{ {\bf c} > 0 \} } \frac{|\nabla^m{\bf c}|^2}{{\bf c}} + \int_{ \{ {\bf c} > 0 \} } \frac{|\nabla{\bf c}|^{2m}}{{\bf c}^{2m-1}}
\end{split}
\end{equation} for every $1 < \ell < m$, {assuming that ${\bf c}\ge0$ is a given function with finite right-hand side. In the arguments below, we shall actually assume that ${\bf c}>0$. This is possible since we may first prove \eqref{eq:ineq-GNS} with ${\bf c}$ replaced by ${\bf c}+\epsilon$ and then take the limit $\epsilon\to0$ first on the right-hand side (using the finiteness assumption for ${\bf c}$ given in Lemma \ref{lem:inequality}). After that, we may take the limit $\epsilon\to0$ on the left-hand side.}
Note that each term in \eqref{eq:ineq-GNS} has $2m$ derivatives of ${\bf c}$ and 1-homogeneity in ${\bf c}$. Once we have this inequality, we immediately deduce \begin{equation*}
\begin{split}
\int \frac{ 1 }{ {\bf c}^{2k-1} } \prod_{1\le i \le k} |\partial^{\ell_i}{\bf c}|^2\le \prod_{1\le i\le k} \left(\int \frac{|\nabla^{|\ell_i|} {\bf c}|^{\frac{2m}{|\ell_i|}}}{{\bf c}^{\frac{2m}{|\ell_i|}-1}}\right)^{\frac{|\ell_i|}{m}} \lesssim_{m,d} \int \frac{|\nabla^m{\bf c}|^2}{{\bf c}} + \int \frac{|\nabla{\bf c}|^{2m}}{{\bf c}^{2m-1}}
\end{split}
\end{equation*} from H\"older's inequality.
\medskip
\noindent {Since there is nothing to prove in the case $m=2$, we shall verify \eqref{eq:ineq-GNS} for some $m>2$. We need to prove that} \begin{equation}\label{eq:goal}
\begin{split}
I_{\ell} \lesssim_{m,d} {I_1} + I_{m}, \quad 1<\ell< m
\end{split}
\end{equation} with \begin{equation*}
\begin{split}
I_{\ell} := \int \frac{|\nabla^\ell {\bf c}|^{\frac{{2m}}{\ell}}}{ {\bf c}^{\frac{{2m}}{\ell}-1}} .
\end{split}
\end{equation*} For a partial derivative $\partial^\ell$ of order $\ell$, \begin{equation*}
\begin{split}
&\int \frac{|\partial^\ell {\bf c}|^{\frac{2m}{\ell}-1}}{{\bf c}^{{\frac{2m}{\ell}-1}}} \mathrm{sgn} (\partial^{\ell}{\bf c}) \partial^\ell {\bf c} \\
&\quad = - ({\frac{2m}{\ell}-1})\int \frac{|\partial^\ell {\bf c}|^{\frac{2m}{\ell}-2}}{{\bf c}^{\frac{2m}{\ell}-1}} \, \partial^{\ell-1} {\bf c} \, \partial^{\ell+1} {\bf c} + ({\frac{2m}{\ell}-1} )\int \frac{|\partial^\ell {\bf c}|^{\frac{2m}{\ell}-1}}{{\bf c}^{\frac{2m}{\ell}}} \, \partial {\bf c} \, \partial^{\ell-1} {\bf c} \, \mathrm{sgn}(\partial^\ell {\bf c}).
\end{split}
\end{equation*} Note that $2<\frac{2m}{\ell} \le m$. Applying H\"older's inequality to the right hand side, we have \begin{equation*}
\begin{split}
I_{\ell} \lesssim_{\ell,m} I_{\ell-1}^{ \frac{\ell-1}{2m}} I_{\ell}^{1-\frac{\ell}{m}} I_{\ell+1}^{\frac{\ell+1}{2m}} + I_1^{\frac{1}{2m}} I_{\ell-1}^{\frac{\ell-1}{2m}} I_{\ell}^{1-\frac{\ell}{2m}}.
\end{split}
\end{equation*} With $\epsilon$-Young inequality, we obtain \begin{equation*}
\begin{split}
I_{\ell} \le \epsilon I_\ell + \epsilon I_{\ell-1} + C_{\epsilon,\ell,m} ( I_1 + I_{\ell+1} ).
\end{split}
\end{equation*} After a suitable weighted summation of the inequality in all of the cases $\ell = 2, \cdots, m-1$, one can obtain \eqref{eq:goal}.
\end{proof}
At this point, one may note that Lemma \ref{lem:key} is a straightforward generalization of the above. The proof can be done in a completely parallel manner with Lemma \ref{lem:inequality}.
{
\begin{remark}
In the special case $m=2$, we have after an integration by parts
\[
\int\frac{{\abs{\nabla {\bf c}}}^4}{{\bf c}^3}=-\int\frac{\abs{\nabla {\bf c}}^2\Delta {\bf c}}{{\bf c}^2}+3\int\frac{{\abs{\nabla {\bf c}}}^4}{{\bf c}^3},
\]
which implies that
\[
2\int\frac{{\abs{\nabla {\bf c}}}^4}{{\bf c}^3}\le \int\frac{\abs{\nabla {\bf c}}^2\Delta {\bf c}}{{\bf c}^2}
\le \int\frac{{\abs{\nabla {\bf c}}}^4}{{\bf c}^3}+\frac{1}{4}\int \frac{\abs{\Delta {\bf c}}^2}{{\bf c}}
\]
by H\"older's inequality. Therefore, the first term on the right-hand side of \eqref{eq:ineq-GNS} is controlled by the other term. Unfortunately, this does not seem to hold for $m>2$.
\end{remark}
}
\subsubsection{A priori estimates}\label{subsubsec:apriori}
{Let us proceed to obtain a priori estimates for the solution, assuming that ${\bf c}, {\boldsymbol{\rho}}$ are sufficiently smooth and non-negative.} Furthermore, we take the simplifying assumption that $k({\bf c}) = {\bf c}^\gamma$ and $\chi({\bf c}) = 1$. Extending the proof to the case of general $k$ and $\chi$ as stated in Theorem \ref{thm:lwp-rho-away} is straightforward.
\medskip
\noindent \textit{(i) Ratio estimates}: We would like to propagate ${\boldsymbol{\rho}}\lesssim {\bf c}^\delta $ and ${\bf c}^\delta \lesssim {\boldsymbol{\rho}}$. To this end, \begin{equation*}
\begin{split}
\frac{\partial}{\partial t} \frac{{\bf c}^\delta}{{\boldsymbol{\rho}}} = \frac{{\bf c}^\delta}{{\boldsymbol{\rho}}}\frac{ \nabla\cdot(\chi({\bf c}){\boldsymbol{\rho}}\nabla{\bf c})}{{\boldsymbol{\rho}}} - \delta k({\bf c}) {\bf c}^{\delta-1}
\end{split}
\end{equation*} so that recalling $\chi = 1$ and $k({\bf c})={\bf c}^{\gamma}$,
\begin{equation*}
\begin{split}
{\frac{d}{dt}} \left(\nrm{ \frac{{\bf c}^\delta}{{\boldsymbol{\rho}}}}_{L^\infty} + \nrm{ \frac{{\boldsymbol{\rho}}}{{\bf c}^\delta}}_{L^\infty}\right) \le C\nrm{ \frac{ \nabla\cdot({\boldsymbol{\rho}}\nabla{\bf c})}{{\boldsymbol{\rho}}}}_{L^\infty} \left(\nrm{ \frac{{\bf c}^\delta}{{\boldsymbol{\rho}}}}_{L^\infty} + \nrm{ \frac{{\boldsymbol{\rho}}}{{\bf c}^\delta}}_{L^\infty}\right) + C\nrm{{\bf c}^{\delta+\gamma-1}}_{L^\infty}.
\end{split}
\end{equation*} From the assumption $\delta+\gamma-1\ge 0$, we have that \begin{equation*}
\begin{split}
\nrm{{\bf c}^{\delta+\gamma-1}}_{L^\infty}\le \nrm{{\bf c}}_{L^\infty}^{\delta+\gamma-1}
\end{split}
\end{equation*} and \begin{equation*}
\begin{split}
\nrm{ \frac{ \nabla\cdot({\boldsymbol{\rho}}\nabla{\bf c})}{{\boldsymbol{\rho}}}}_{L^\infty} \le C( \nrm{\Delta{\bf c}}_{L^\infty} + \nrm{ \frac{{\bf c}^\delta}{{\boldsymbol{\rho}}}}_{L^\infty}^a \nrm{\frac{\nabla{\boldsymbol{\rho}}}{{\boldsymbol{\rho}}^{1-a}}}_{L^\infty} \nrm{\frac{\nabla{\bf c}}{{\bf c}^{\delta a}}}_{L^\infty} ),
\end{split}
\end{equation*} where $0<a<1$ is to be chosen below. From \eqref{eq:inequality-infty} in Corollary \ref{cor:L-infty}, we have that \begin{equation*}
\begin{split}
\nrm{\frac{\nabla{\boldsymbol{\rho}}}{{\boldsymbol{\rho}}^{1-a}}}_{L^\infty} \le C \nrm{{\boldsymbol{\rho}}}_{{\mathcal X}^{m,1}}^{2a}, \quad a := \frac{1}{2(m-1-\lfloor \frac{d}{2} \rfloor)}.
\end{split}
\end{equation*} Hence, we have \begin{equation*}
\begin{split}
\nrm{\frac{\nabla{\bf c}}{{\bf c}^{\delta a}}}_{L^\infty} \le C \nrm{{\bf c}}_{ {\mathcal X}^{m+1,\gamma} }^{\frac{1}{m- \lfloor \frac{d}{2} \rfloor }} \nrm{{\bf c}}_{L^\infty}^{\epsilon}
\end{split}
\end{equation*} for some $\epsilon\ge0$ if \begin{equation*}
\begin{split}
\delta\le 2\left(m-1- \lfloor \frac{d}{2} \rfloor \right)\left( 1 - \frac{1-\frac{\gamma}{2}}{ m - \lfloor \frac{d}{2} \rfloor } \right).
\end{split}
\end{equation*} This, together with $ \nrm{\Delta{\bf c}}_{L^\infty} \lesssim \nrm{c}_{L^\infty}^{\frac{\gamma}{2}} \nrm{{\bf c}}_{ {\mathcal X}^{m+1,\gamma} } $, we have \begin{equation}\label{eq:L-infty}
\begin{split}
\nrm{ \frac{ \nabla\cdot({\boldsymbol{\rho}}\nabla{\bf c})}{{\boldsymbol{\rho}}}}_{L^\infty} \le C(1 + \nrm{{\bf c}}_{L^\infty})^b \left(1 + \nrm{ \frac{{\bf c}^\delta}{{\boldsymbol{\rho}}}}_{L^\infty}\right)^{a} (1 + \nrm{{\bf c}}_{ {\mathcal X}^{m+1,\gamma} } + \nrm{{\boldsymbol{\rho}}}_{ {\mathcal X}^{m,1} } )^2
\end{split}
\end{equation} and then \begin{equation}\label{eq:apriori-1}
\begin{split}
\frac{d}{dt} \left(\nrm{ \frac{{\bf c}^\delta}{{\boldsymbol{\rho}}}}_{L^\infty} + \nrm{ \frac{{\boldsymbol{\rho}}}{{\bf c}^\delta}}_{L^\infty}\right) \le C \left(1 + \nrm{ \frac{{\bf c}^\delta}{{\boldsymbol{\rho}}}}_{L^\infty} + \nrm{ \frac{{\boldsymbol{\rho}}}{{\bf c}^\delta}}_{L^\infty}\right)^{1 + a} (1 + \nrm{{\bf c}}_{L^\infty})^b (1 + \nrm{{\bf c}}_{ {\mathcal X}^{m+1,\gamma} } + \nrm{{\boldsymbol{\rho}}}_{ {\mathcal X}^{m,1} } )^2
\end{split}
\end{equation} follows, with some $b = b(\gamma,\delta,d)>0$. Before we proceed, we note the trivial estimate \begin{equation}\label{eq:apriori-2}
\begin{split}
\frac{d}{dt} \left( \nrm{{\bf c}}_{L^\infty} + \nrm{{\boldsymbol{\rho}}}_{L^\infty} \right) \le C \nrm{\Delta{\bf c}}_{L^\infty}\nrm{{\boldsymbol{\rho}}}_{L^\infty} + C\nrm{\nabla{\boldsymbol{\rho}}}_{L^\infty}\nrm{\nabla{\bf c}}_{L^\infty} \le C \nrm{{\bf c}}_{\mathcal X^{m+1,\gamma}} (\nrm{{\boldsymbol{\rho}}}_{L^\infty} + \nrm{{\boldsymbol{\rho}}}_{\mathcal X^{m,1}} ),
\end{split}
\end{equation} since ${\bf c}$ is pointwise decreasing with time.
\medskip
\noindent \textit{(ii) $\mathcal X^m$--estimates}:
We begin with the system \begin{equation} \label{eq:KS-1st}
\left\{
\begin{aligned}
\partial_t{\bf f} & = -k'({\bf c}){\boldsymbol{\rho}}{\bf f} - k({\bf c})\nabla{\boldsymbol{\rho}} ,\\
\partial_t{\boldsymbol{\rho}} & = - \chi({\bf c}){\bf f} \cdot\nabla{\boldsymbol{\rho}} - \chi({\bf c}){\boldsymbol{\rho}} \nabla\cdot{\bf f} - \chi'({\bf c}){\boldsymbol{\rho}} |{\bf f}|^2 .
\end{aligned}
\right.
\end{equation} We write
\begin{equation*}
\begin{split}
\frac{d}{dt} \frac{1}{2} \int \frac{\chi({\bf c})}{k({\bf c})} |\partial^m{\bf f}|^2 + \frac{1}{{\boldsymbol{\rho}}} |\partial^m{\boldsymbol{\rho}}|^2 = I + II + III + IV,
\end{split}
\end{equation*} with \begin{equation*}
\begin{split}
I = \frac{1}{2} \int\partial_t ( \frac{\chi({\bf c})}{k({\bf c})} )|\partial^m{\bf f}|^2 ,
\end{split}
\end{equation*}\begin{equation*}
\begin{split}
II = \frac{1}{2} \int \partial_t( \frac{1}{{\boldsymbol{\rho}}} )|\partial^m{\boldsymbol{\rho}}|^2 ,
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
III = \int \frac{\chi({\bf c})}{k({\bf c})} \partial^m{\bf f} \cdot \partial^m \left( -k'({\bf c}){\boldsymbol{\rho}}{\bf f} - k({\bf c})\nabla{\boldsymbol{\rho}} \right),
\end{split}
\end{equation*} and \begin{equation*}
\begin{split}
IV = \int \frac{1}{{\boldsymbol{\rho}}}\partial^m{\boldsymbol{\rho}} \partial^m\left( - \chi({\bf c}){\bf f} \cdot\nabla{\boldsymbol{\rho}} - \chi({\bf c}){\boldsymbol{\rho}} \nabla\cdot{\bf f} - \chi'({\bf c}){\boldsymbol{\rho}} |{\bf f}|^2 \right).
\end{split}
\end{equation*}
We estimate \begin{equation*}
\begin{split}
|I| = \frac{\gamma}{2} \int {\bf c}^{\delta+\gamma-1} \frac{{\boldsymbol{\rho}}}{{\bf c}^{\delta}} \frac{1}{{\bf c}^{\gamma}} |\partial^m{\bf f}|^2 \le C \nrm{{\bf c}}_{L^\infty}^{\delta+\gamma-1} \nrm{ \frac{{\boldsymbol{\rho}}}{{\bf c}^{\delta}} }_{L^\infty} \nrm{{\bf c}}_{\mathcal X^{m+1,\gamma}}
\end{split}
\end{equation*} and \begin{equation*}
\begin{split}
|II| =\left| \frac{1}{2} \int \frac{\nabla\cdot({\boldsymbol{\rho}}\nabla{\bf c})}{{\boldsymbol{\rho}}} \frac{|\partial^m{\boldsymbol{\rho}}|^2}{{\boldsymbol{\rho}}} \right| \le C(1 + \nrm{{\bf c}}_{L^\infty})^b (1 + \nrm{ \frac{{\bf c}^\delta}{{\boldsymbol{\rho}}}}_{L^\infty} )^{a} (1 + \nrm{{\bf c}}_{ {\mathcal X}^{m+1,\gamma} } + \nrm{{\boldsymbol{\rho}}}_{ {\mathcal X}^{m,1} } )^2 \nrm{{\boldsymbol{\rho}}}_{ {\mathcal X}^{m,1} }
\end{split}
\end{equation*} recalling $\chi = 1$, $k(z)=z^\gamma$ and using \eqref{eq:L-infty}. It remains to handle $III$ and $IV$. In the case of $IV$, it suffices to estimate integrals of the form \begin{equation*}
\begin{split}
\int \frac{1}{{\boldsymbol{\rho}}} \partial^m{\boldsymbol{\rho}} \partial^{\ell} {\bf f} \partial^{m-\ell+1}{\boldsymbol{\rho}},
\end{split}
\end{equation*} where $0\le \ell \le m+1$. Let us first consider the extreme cases, which are the most troublesome. When $\ell = 0$, with an integration by parts, we have that the corresponding term in $IV$ is given by \begin{equation*}
\begin{split}
\int - \frac{{\bf f}}{{\boldsymbol{\rho}}} \cdot \nabla \partial^m {\boldsymbol{\rho}} \,\partial^m{\boldsymbol{\rho}} = \frac{1}{2}\int \left( \Delta{\bf c} - \frac{{\bf f}\cdot \nabla{\boldsymbol{\rho}} }{{\boldsymbol{\rho}}} \right) \frac{|\partial^m{\boldsymbol{\rho}}|^2}{{\boldsymbol{\rho}}}.
\end{split}
\end{equation*} Recalling the estimate \eqref{eq:L-infty}, we have \begin{equation*}
\begin{split}
\left| \int - \frac{{\bf f}}{{\boldsymbol{\rho}}} \cdot \nabla \partial^m {\boldsymbol{\rho}} \,\partial^m{\boldsymbol{\rho}} \right| \le C (\nrm{\frac{{\bf c}^{\delta}}{{\boldsymbol{\rho}}}}_{L^\infty}\nrm{{\bf c}}_{L^\infty}^{\gamma-\delta} +1 )\nrm{{\bf c}}_{\mathcal X^{m+1,\gamma}} \nrm{{\boldsymbol{\rho}}}_{\mathcal X^{m,1}}.
\end{split}
\end{equation*} On the other hand, when $\ell = m+1$, we are left with \begin{equation*}
\begin{split}
-\int \partial^m{\boldsymbol{\rho}} \nabla\cdot \partial^m{\bf f}.
\end{split}
\end{equation*} The case $0<\ell<m+1$ is easily bounded by Gagliardo--Nirenberg--Sobolev inequality: we take \begin{equation*}
\begin{split}
\left| \int \frac{1}{{\boldsymbol{\rho}}} \partial^m{\boldsymbol{\rho}} \partial^{\ell} {\bf f} \partial^{m-\ell+1}{\boldsymbol{\rho}} \right| \le C\nrm{{\boldsymbol{\rho}}}_{\mathcal X^{m,1}} \nrm{ \partial^{\ell} {\bf f} }_{L^{2p^*}} \nrm{ {\boldsymbol{\rho}}^{-\frac{1}{2}} \partial^{m-\ell+1}{\boldsymbol{\rho}} }_{L^{2p}}
\end{split}
\end{equation*} where \begin{equation*}
\begin{split}
p = \frac{m}{m-\ell+1}, \quad \frac{1}{p} + \frac{1}{p^*} = 1.
\end{split}
\end{equation*} Then, recalling the proof of Lemma \ref{lem:inequality}, we have \begin{equation*}
\begin{split}
\left(\int \frac{(\partial^{m-\ell+1}{\boldsymbol{\rho}})^{2p}}{{\boldsymbol{\rho}}^{2p(\frac{1}{2} + \frac{\ell-1}{2m})}} \right)^{\frac{1}{2p}}\le C \nrm{{\boldsymbol{\rho}}}_{\mathcal X^{m,1}}^{\frac{1}{p}}
\end{split}
\end{equation*} so that \begin{equation*}
\begin{split}
\nrm{ {\boldsymbol{\rho}}^{-\frac{1}{2}} \partial^{m-\ell+1}{\boldsymbol{\rho}} }_{L^{2p}} \le C \nrm{{\boldsymbol{\rho}}}_{L^\infty}^{\frac{\ell-1}{2m}} \nrm{{\boldsymbol{\rho}}}_{\mathcal X^{m,1}}^{\frac{1}{p}}.
\end{split}
\end{equation*} Similarly, we can estimate \begin{equation*}
\begin{split}
\nrm{\partial^{\ell}{\bf f}}_{L^{2p^*}} \le C \nrm{{\bf c}}_{L^\infty}^{b} \nrm{{\bf c}}_{\mathcal X^{m+1,\gamma}}^{\frac{1}{p^*}}
\end{split}
\end{equation*} with some $b = b(\gamma,m,\ell)>0$. Therefore, \begin{equation*}
\begin{split}
\left| IV + \int \partial^m{\boldsymbol{\rho}} \nabla\cdot \partial^m{\bf f} \right| \le C(1 + \nrm{{\bf c}}_{L^\infty} + \nrm{{\boldsymbol{\rho}}}_{L^\infty})^b (1+\nrm{\frac{{\bf c}^{\delta}}{{\boldsymbol{\rho}}}}_{L^\infty} )(1+\nrm{{\bf c}}_{\mathcal X^{m+1,\gamma}})(1+ \nrm{{\boldsymbol{\rho}}}_{\mathcal X^{m,1}})\nrm{{\boldsymbol{\rho}}}_{\mathcal X^{m,1}}.
\end{split}
\end{equation*} with some $b = b(\gamma,m)>0$. Estimating $III$ is similar. Note that \begin{equation*}
\begin{split}
III = -\int \frac{1}{{\bf c}^{\gamma}} \partial^m{\bf f} \cdot \partial^m \nabla ({\bf c}^\gamma{\boldsymbol{\rho}})
\end{split}
\end{equation*} and we need to consider \begin{equation*}
\begin{split}
\int \frac{1}{{\bf c}^{\gamma}} \partial^m{\bf f} \partial^{m+1-\ell} ( {\bf c}^\gamma )\partial^{\ell}{\boldsymbol{\rho}}
\end{split}
\end{equation*} for $0\le\ell\le m+1$. In the case $\ell = m+1$, we simply have \begin{equation*}
\begin{split}
-\int \partial^m{\bf f} \cdot \partial^m \nabla {\boldsymbol{\rho}}.
\end{split}
\end{equation*} Next, when $\ell \le m$, we first rewrite the corresponding integral as \begin{equation*}
\begin{split}
- \int \frac{\partial^{\ell}{\boldsymbol{\rho}}}{{\bf c}^{\gamma}} \partial^m {\bf f} \cdot \partial^{m-\ell} (\gamma{\bf c}^{\gamma-1}{\bf f}).
\end{split}
\end{equation*} We show how to handle the case $\ell = 0$. Extending the estimate to the case $0<\ell \le m$ is simpler. We then need to consider \begin{equation*}
\begin{split}
-\gamma \int \frac{{\boldsymbol{\rho}}}{{\bf c}^{\gamma}} \partial^m{\bf f} \cdot \partial^{j} ( {\bf c}^{\gamma-1} ) \partial^{m-j}{\bf f} = -\gamma \int \frac{{\boldsymbol{\rho}}}{{\bf c}^{\delta}} {\bf c}^{\delta+\gamma-1} \frac{\partial^m{\bf f}}{{\bf c}^{\frac{\gamma}{2}}} \cdot \frac{\partial^{j} ( {\bf c}^{\gamma-1} )}{{\bf c}^{\gamma-1 + \frac{1}{p^*} (\frac{\gamma}{2}-1)}} \frac{\partial^{m-j}{\bf f} }{ {\bf c}^{1-\frac{1}{p} + \frac{\gamma}{2p}} } .
\end{split}
\end{equation*} In the above rewriting, we choose \begin{equation*}
\begin{split}
p = \frac{m+1}{m+1-j} >1, \quad \frac{1}{p}+\frac{1}{p^*} = 1.
\end{split}
\end{equation*} Then, \begin{equation*}
\begin{split}
\left| -\gamma \int \frac{{\boldsymbol{\rho}}}{{\bf c}^{\gamma}} \partial^m{\bf f} \cdot \partial^{j} ( {\bf c}^{\gamma-1} ) \partial^{m-j}{\bf f} \right| & \le C \nrm{\frac{{\boldsymbol{\rho}}}{{\bf c}^{\delta}}}_{L^\infty} \nrm{{\bf c}}_{L^\infty}^{\delta+\gamma-1} \nrm{ \frac{\partial^{j} ( {\bf c}^{\gamma-1} )}{{\bf c}^{\gamma-1 + \frac{1}{p^*} (\frac{\gamma}{2}-1)}} }_{L^{2p^*}} \nrm{ \frac{\partial^{m-j}{\bf f} }{ {\bf c}^{1-\frac{1}{p} + \frac{\gamma}{2p}} } }_{L^{2p}} \\
& \le C\nrm{\frac{{\boldsymbol{\rho}}}{{\bf c}^{\delta}}}_{L^\infty} \nrm{{\bf c}}_{L^\infty}^{\delta+\gamma-1} \nrm{{\bf c}}_{\mathcal X^{m+1,\gamma}}
\end{split}
\end{equation*} where we have used that \begin{equation*}
\begin{split}
\nrm{ \frac{\partial^{m-j}{\bf f} }{ {\bf c}^{1-\frac{1}{p} + \frac{\gamma}{2p}} } }_{L^{2p}} \le C\nrm{{\bf c}}_{\mathcal X^{m+1,\gamma}}^{\frac{1}{p}}
\end{split}
\end{equation*} and \begin{equation*}
\begin{split}
\nrm{ \frac{\partial^{j} ( {\bf c}^{\gamma-1} )}{{\bf c}^{\gamma-1 + \frac{1}{p^*} (\frac{\gamma}{2}-1)}} }_{L^{2p^*}} \le C \nrm{{\bf c}}_{\mathcal X^{m+1,\gamma}}^{\frac{1}{p^*}}.
\end{split}
\end{equation*} This gives \begin{equation*}
\begin{split}
\left| III + \int \partial^m{\bf f} \cdot \partial^m \nabla {\boldsymbol{\rho}} \right| \le C \nrm{\frac{{\boldsymbol{\rho}}}{{\bf c}^{\delta}}}_{L^\infty} \nrm{{\bf c}}_{L^\infty}^{\delta+\gamma-1} \nrm{{\bf c}}_{\mathcal X^{m+1,\gamma}}.
\end{split}
\end{equation*} Collecting the estimates, \begin{equation}\label{eq:calX-m-1}
\begin{split}
\left| I + II + III + IV \right| \le C (1 + \nrm{{\bf c}}_{L^\infty} + \nrm{{\boldsymbol{\rho}}}_{L^\infty})^b (1+\nrm{\frac{{\bf c}^{\delta}}{{\boldsymbol{\rho}}}}_{L^\infty} +\nrm{\frac{{\boldsymbol{\rho}}}{{\bf c}^{\delta}}}_{L^\infty} )(1+\nrm{{\bf c}}_{\mathcal X^{m+1,\gamma}} + \nrm{{\boldsymbol{\rho}}}_{\mathcal X^{m,1}})^3.
\end{split}
\end{equation}
\medskip
\noindent \textit{(iii) $\mathcal X^m$--estimates, part 2}: We now need to estimate \begin{equation*}
\begin{split}
\int \frac{|\nabla{\bf c}|^{2(m+1)}}{{\bf c}^{2(m+1)-2+\gamma}}, \quad \int \frac{|\nabla{\boldsymbol{\rho}}|^{2m}}{{\boldsymbol{\rho}}^{2m-1}}.
\end{split}
\end{equation*} We begin with writing \begin{equation*}
\begin{split}
\int \frac{|\nabla{\bf c}|^{2(m+1)}}{{\bf c}^{2(m+1)-2+\gamma}} = \int \left|\frac{2(m+1)}{2-\gamma}\nabla({\bf c}^{\frac{2-\gamma}{2(m+1)}} )\right|^{2(m+1)} = C_{\gamma,m} \nrm{\nabla ({\bf c}^{\alpha}) }_{L^{2(m+1)}}^{2(m+1)},\quad \alpha = \frac{2-\gamma}{2(m+1)}.
\end{split}
\end{equation*} Then, from \begin{equation*}
\begin{split}
\partial_t {\bf c}^{\alpha} = -\alpha {\bf c}^{\alpha+\gamma-1} {\boldsymbol{\rho}} ,
\end{split}
\end{equation*}we have \begin{equation*}
\begin{split}
\partial_t \nabla ({\bf c}^{\alpha}) = - \alpha {\bf c}^{\gamma+\delta-1} \frac{{\boldsymbol{\rho}}}{{\bf c}^{\delta}} \nabla ({\bf c}^{\alpha}) - \alpha {\bf c}^{\alpha } \nabla({\boldsymbol{\rho}} {\bf c}^{\gamma-1}).
\end{split}
\end{equation*} We then write
\begin{equation*}
\begin{split}
- \alpha {\bf c}^{\alpha } \nabla({\boldsymbol{\rho}} {\bf c}^{\gamma-1}) = -(\gamma-1) {\bf c}^{\gamma+\delta-1} \frac{{\boldsymbol{\rho}}}{{\bf c}^{\delta}} \nabla({\bf c}^\alpha) - \alpha {\bf c}^{\alpha+\gamma-1} \nabla{\boldsymbol{\rho}}
\end{split}
\end{equation*} and furthermore \begin{equation*}
\begin{split}
\nrm{ {\bf c}^{\alpha+\gamma-1} \nabla{\boldsymbol{\rho}} }_{L^{2(m+1)}}^{2(m+1)} = \int {\bf c}^{2-\gamma} {\bf c}^{2(\gamma-1)(m+1)} {\bf c}^{\delta(2m+1-\frac{1}{k})} \frac{{\boldsymbol{\rho}}^{2m+1-\frac{1}{k}}}{{\bf c}^{\delta(2m+1-\frac{1}{k})} } \frac{|\nabla{\boldsymbol{\rho}}|^{2m}}{{\boldsymbol{\rho}}^{2m-1}} \frac{|\nabla{\boldsymbol{\rho}}|^2}{{\boldsymbol{\rho}}^{2-\frac{1}{k}}}, \quad k = m- \lfloor\frac{d}{2}\rfloor - 1\ge 1.
\end{split}
\end{equation*} We have that \begin{equation*}
\begin{split}
2-\gamma + 2(\gamma-1)(m+1) + \delta(2m+1-\frac{1}{k}) \ge (\gamma+\delta-1)(2m+1-\frac{1}{k}) + (2-\gamma) + (\gamma-1)(1+\frac{1}{k})
\end{split}
\end{equation*} and since $\gamma+\delta-1\ge0$, $(2-\gamma) + (\gamma-1)(1+\frac{1}{k})\ge 0$, \begin{equation*}
\begin{split}
\nrm{ {\bf c}^{\alpha+\gamma-1} \nabla{\boldsymbol{\rho}} }_{L^{2(m+1)}}^{2(m+1)} \le C \nrm{{\bf c}}_{L^\infty}^{b} \nrm{\frac{{\boldsymbol{\rho}}}{{\bf c}^\delta}}_{L^\infty}^{2m+1-\frac{1}{k}} \nrm{{\boldsymbol{\rho}}}_{\mathcal X^{m,1}}^2.
\end{split}
\end{equation*} Returning to the equation for $\partial_t \nabla ({\bf c}^\alpha)$ and taking the $L^{2(m+1)}$-norm of both sides, \begin{equation}\label{eq:apriori4}
\begin{split}
\frac{d}{dt} \nrm{ \nabla ({\bf c}^\alpha) }_{L^{2(m+1)}} \le C(1 + \nrm{{\bf c}}_{L^\infty} + \nrm{\frac{{\boldsymbol{\rho}}}{{\bf c}^\delta}}_{L^\infty})^b ( \nrm{ \nabla ({\bf c}^\alpha) }_{L^{2(m+1)}} + \nrm{{\boldsymbol{\rho}}}_{\mathcal X^{m,1}}^{\frac{m}{m+1}} )
\end{split}
\end{equation} for some $b = b(m,\gamma,\delta)\ge 0$. Next, with $\beta = \frac{1}{2m}$, we write
\begin{equation*}
\begin{split}
\partial_t \nabla({\boldsymbol{\rho}}^\beta) = -(1+\beta) \nabla({\boldsymbol{\rho}}^\beta)\Delta{\bf c} - \nabla^2 ({\boldsymbol{\rho}}^\beta)\cdot\nabla{\bf c} .
\end{split}
\end{equation*} With an integration by parts, we estimate \begin{equation}\label{eq:apriori5}
\begin{split}
\frac{d}{dt} \nrm{\nabla({\boldsymbol{\rho}}^\beta)}_{L^{2m}}\le C \nrm{\Delta{\bf c}}_{L^\infty}\nrm{\nabla({\boldsymbol{\rho}}^\beta)}_{L^{2m}}\le C\nrm{{\bf c}}_{\mathcal X^{m+1,\gamma}} \nrm{\nabla({\boldsymbol{\rho}}^\beta)}_{L^{2m}}.
\end{split}
\end{equation}
Combining \eqref{eq:apriori4} and \eqref{eq:apriori5} with \eqref{eq:calX-m-1}, we obtain that \begin{equation}\label{eq:apriori-3}
\begin{split}
\frac{d}{dt} \left( \nrm{{\bf c}}_{\dot{\mathcal X}^{m+1,\gamma}} + \nrm{{\boldsymbol{\rho}}}_{\dot{\mathcal X}^{m,1}} \right) \le C (1 + \nrm{{\bf c}}_{L^\infty} + \nrm{{\boldsymbol{\rho}}}_{L^\infty}+\nrm{\frac{{\bf c}^{\delta}}{{\boldsymbol{\rho}}}}_{L^\infty} +\nrm{\frac{{\boldsymbol{\rho}}}{{\bf c}^{\delta}}}_{L^\infty} )^b(1+\nrm{{\bf c}}_{\mathcal X^{m+1,\gamma}} + \nrm{{\boldsymbol{\rho}}}_{\mathcal X^{m,1}})^3.
\end{split}
\end{equation} Here, $b = b(\gamma,\delta,m)\ge0$. In the left hand side of \eqref{eq:apriori-3}, $\dot{\mathcal X}^{m+1,\gamma}$ and $\dot{\mathcal X}^{m,1}$ can be replaced with ${\mathcal X}^{m+1,\gamma}$ and ${\mathcal X}^{m,1}$, respectively.
\medskip
\noindent \textit{(iv) Concluding the a priori estimate}: Using \eqref{eq:apriori-1}, \eqref{eq:apriori-2}, and \eqref{eq:apriori-3}, we conclude that for \begin{equation}\label{eq:X}
\begin{split}
X := \nrm{{\bf c}}_{\dot{\mathcal X}^{m+1,\gamma}} + \nrm{{\boldsymbol{\rho}}}_{\dot{\mathcal X}^{m,1}} + \nrm{{\bf c}}_{L^\infty} + \nrm{{\boldsymbol{\rho}}}_{L^\infty}+\nrm{\frac{{\bf c}^{\delta}}{{\boldsymbol{\rho}}}}_{L^\infty} +\nrm{\frac{{\boldsymbol{\rho}}}{{\bf c}^{\delta}}}_{L^\infty} ,
\end{split}
\end{equation} we have the inequality \begin{equation}\label{eq:apriori}
\begin{split}
\frac{d}{dt} X \le C(1+X)^b
\end{split}
\end{equation} for some $b = b(\gamma,\delta,m)\ge0$. {Assuming that $X$ is smooth in time, we may take $T>0$ depending only on $X(t=0)$ such that any solution to \eqref{eq:apriori} satisfies \begin{equation}\label{eq:apriori-integrated}
\begin{split}
\sup_{0\le t \le T} X(t) \le 2X(0).
\end{split}
\end{equation}}
\subsubsection{Existence and uniqueness}\label{subsubsec:eandu}
The proof of uniqueness can be done along the lines of the uniqueness proof for Theorem \ref{thm:lwp-away}, by closing an $L^2$ estimate for the difference of two hypothetical solutions. To prove existence, we consider the system \eqref{eq:KS} with initial data ${\boldsymbol{\rho}}_0^{\epsilon} = {\boldsymbol{\rho}}_0 + \epsilon$ and ${\bf c}_0^{\epsilon} = {\bf c}_0+\epsilon$, where $({\boldsymbol{\rho}}_0,{\bf c}_0)$ is the given initial data. Since ${\boldsymbol{\rho}}_0^{\epsilon}$ and ${\bf c}_0^{\epsilon}$ are strictly positive and smooth, there is a corresponding local-in-time solution $({\boldsymbol{\rho}}^{\epsilon},{\bf c}^{\epsilon})$ to \eqref{eq:KS} guaranteed by Theorem \ref{thm:lwp-away}. {We can then define $X^\epsilon(t)$ as in \eqref{eq:X}, simply replacing ${\bf c}$ and ${\boldsymbol{\rho}}$ with ${\bf c}^\epsilon$ and ${\boldsymbol{\rho}}^\epsilon$; \begin{equation*}
\begin{split}
X^{\epsilon}(t) := \left(\nrm{{\bf c}^{\epsilon}}_{\dot{\mathcal X}^{m+1,\gamma}} + \nrm{{\boldsymbol{\rho}}^{\epsilon}}_{\dot{\mathcal X}^{m,1}} + \nrm{{\bf c}^{\epsilon}}_{L^\infty} + \nrm{{\boldsymbol{\rho}}^{\epsilon}}_{L^\infty}
+\nrm{\frac{({\bf c}^{\epsilon})^{\delta}}{{\boldsymbol{\rho}}^{\epsilon}}}_{L^\infty} +\nrm{\frac{{\boldsymbol{\rho}}^{\epsilon}}{({\bf c}^{\epsilon})^{\delta}}}_{L^\infty} \right)(t).
\end{split}
\end{equation*} Then, it is not difficult to see that for each $\epsilon>0$, $X^{\epsilon}$ is $C^1$--in $t$ and the a priori estimate \eqref{eq:apriori} is satisfied for $X^{\epsilon}$ with a (possibly larger) constant $C>0$ which is uniformly bounded for $0<\epsilon\le1$. Furthermore, note that $X^{\epsilon}_0$ is uniformly bounded for $0\le \epsilon\le 1$ and is convergent to $X_0$ as $\epsilon\to 0$. Therefore, for all small $\epsilon>0$, there is a uniform time interval of existence $[0,T]$ with some $T>0$ for the solution $({\boldsymbol{\rho}}^{\epsilon},{\bf c}^{\epsilon})$, on which we have \begin{equation*}
\begin{split}
\sup_{0\le t \le T} X^\epsilon(t) \le 2X^\epsilon(0) \le 4X(0).
\end{split}
\end{equation*}} On the other hand, note that the quantity $X^\epsilon$ controls the Sobolev norm $\nrm{{\bf c}^{\epsilon}}_{H^{m+1}} + \nrm{{\boldsymbol{\rho}}^{\epsilon}}_{H^{m}}$, and therefore we can pass to a weakly convergent subsequence $({\boldsymbol{\rho}}^{\epsilon},{\bf c}^{\epsilon}) \rightarrow ({\boldsymbol{\rho}},{\bf c})$ in $H^{m}\times H^{m+1}$, {which in particular implies uniform pointwise convergence}. Therefore, it is straightforward to verify that the limit $({\boldsymbol{\rho}},{\bf c})$ solves \eqref{eq:KS} with the given initial data and satisfies the {estimate \eqref{eq:apriori-integrated}}. This finishes the proof.
\subsection{Well-posedness and singularity formation with quadratic vanishing}\label{subsec:v2}
In this section, we investigate the dynamics of \eqref{eq:KSF} when the initial data vanishes \textit{quadratically} at some point in the domain. This yields another version of Theorem \ref{MainThm3}, which is stated as Theorem \ref{thm:quadratic} below. As we have explained earlier, Theorem \ref{thm:lwp-rho-away} is unable to handle initial data which vanishes with low order. Let us begin with some discussion which shows that the \eqref{eq:KSF} system could be still wellposed in that case. To avoid complications, we start by considering the simplest possible setting: assume no velocity field (${\bf u} = 0$) and one-dimensional domain. Then, we have the following system
\begin{equation}\label{eq:KS1D}
\left\{
\begin{aligned}
\partial_t {\boldsymbol{\rho}} & = -\partial_x ( {\boldsymbol{\rho}} \partial_x {\bf c}), \\
\partial_t {\bf c} & = - {\bf c}{\boldsymbol{\rho}} .
\end{aligned}
\right.
\end{equation} We now consider the case where ${\boldsymbol{\rho}}_0$ vanishes at some point $x_0$. We further assume that the second derivative of ${\boldsymbol{\rho}}_0$ at $x_0$ is nonzero: that is, \begin{equation}\label{eq:van-assump-rho}
\begin{split}
{\boldsymbol{\rho}}_0(x) \ge a_0(x-x_0)^2, \quad |x-x_0| < \delta_0,
\end{split}
\end{equation} for some $a_0 >0$, $x_0$, and $\delta_0>0$. On the other hand, we keep the assumption that ${\bf c}_0$ is bounded away from zero. Assuming in addition that \begin{equation}\label{eq:van-assump-c}
\begin{split}
\partial_x{\bf c}_0(x_0) = 0,
\end{split}
\end{equation} we have from \eqref{eq:KS1D} that the vanishing point of ${\boldsymbol{\rho}}$ does not change with time. This may be viewed as the simplest scenario in which the initial data touch zero.
\subsubsection*{Linearization against explicit quadratic solutions}
To gain some insight on whether the system \eqref{eq:KS1D} could be locally well-posed in the setup described above, we again take the linearization approach in the regime $|x|\ll1$: taking the approximate solution $(\bar{{\bf c}}, \bar{{\boldsymbol{\rho}}}) = (1-C(t)x^2, R(t)x^2)$ with $C(0) = R(0) = 1$, writing ${\boldsymbol{\rho}} = \bar{{\boldsymbol{\rho}}} + \tilde{{\boldsymbol{\rho}}}$, ${\bf c} = \bar{{\bf c}} + \tilde{{\bf c}}$, and removing terms that are quadratic in the perturbation gives \begin{equation} \label{eq:lin-quad-prim}
\left\{
\begin{aligned}
&\partial_t \tilde{{\bf c}} = - R(t)x^2 \tilde{{\bf c}} - (1 - C(t)x^2)\tilde{{\boldsymbol{\rho}}}, \\
&\partial_t \tilde{{\boldsymbol{\rho}}} = 2C(t)\tilde{{\boldsymbol{\rho}}} + 2C(t)x\partial_x\tilde{{\boldsymbol{\rho}}} - 2xR(t)\partial_x\tilde{{\bf c}} - R(t)x^2 \partial_{xx}\tilde{{\bf c}}.
\end{aligned}
\right.
\end{equation} Then, formally taking $0 < t \ll 1$ and $|x| \ll 1$, we reduce the above into
\begin{equation}\label{eq:lin-quad}
\left\{
\begin{aligned}
\partial_t \tilde{{\bf c}} &=-\tilde{{\boldsymbol{\rho}}} ,\\
\partial_t \tilde{{\boldsymbol{\rho}}} &= -\partial_x (x^2\partial_x\tilde{{\bf c}}) .
\end{aligned}
\right.
\end{equation} We have even removed $ 2C(t)\tilde{{\boldsymbol{\rho}}} + 2C(t)x\partial_x\tilde{{\boldsymbol{\rho}}}$ in the equation for $\tilde{{\boldsymbol{\rho}}}$, which is not important in terms of local well-posedness. Note that for \eqref{eq:lin-quad}, \begin{equation*}
\begin{split}
\frac{1}{2}\frac{d}{dt} \left( \nrm{\tilde{{\boldsymbol{\rho}}}}_{L^2}^2 + \nrm{x\partial_x\tilde{{\bf c}}}_{L^2}^2 \right) = 0.
\end{split}
\end{equation*}
This suggests local well-posedness again in this case, for $\tilde{\bfrho}, \tilde{\bfc} \in H^\infty$. Indeed, $x$-weighted Sobolev norms control the usual Sobolev norm: assume that we have for all $m$, $\nrm{x\partial_x^m{\bf c}}_{L^2} \lesssim_m 1$.
Then from \begin{equation*}
\begin{split}
\nrm{x\partial_x {\bf c}}_{L^2}^2 & = -\int 2x {\bf c}\partial_x {\bf c} - \int x^2 {\bf c} \partial_{xx}{\bf c} \ge \int {\bf c}^2 - \nrm{x{\bf c}}_{L^2}\nrm{x\partial_{xx}{\bf c}}_{L^2},
\end{split}
\end{equation*} we have that \begin{equation*}
\begin{split}
\nrm{{\bf c}}_{L^2}^2 \le \nrm{x{\bf c}}_{L^2}\nrm{x\partial_{xx}{\bf c}}_{L^2} + \nrm{x\partial_x{\bf c}}_{L^2}^2 ,
\end{split}
\end{equation*} and since one can replace ${\bf c}$ with $\partial_x^m{\bf c}$ throughout as well, we conclude that $\nrm{\partial_x^m{\bf c}}_{L^2} \lesssim_m 1$.
\subsubsection*{Singularity formation}
Plugging in the ansatz \begin{equation}\label{eq:ode-sol}
\begin{split}
{\bf c}(t,x) = 1 - C(t)x^2 + O(|x|^4), \quad {\boldsymbol{\rho}}(t,x) = R(t)x^2 + O(|x|^4)
\end{split}
\end{equation} with $C, R \ge 0$ to \eqref{eq:KS1D} gives the system of ODEs \begin{equation}\label{eq:ode}
\left\{
\begin{aligned}
\dot{C}(t) &= R(t),\\
\dot{R}(t) &= 6C(t)R(t)
\end{aligned}
\right.
\end{equation} It is clear that $C, R$ blows up in finite time once $R_0, C_0 > 0$. Indeed, for any $C^2$-initial data $({\bf c}_0,{\boldsymbol{\rho}}_0)$ satisfying \begin{equation*}
\begin{split}
{\bf c}_0(0) = 1, \quad {\partial_{xx}}{\bf c}_0(0) = {-}2C_0, \quad {\boldsymbol{\rho}}_0(0) = 0, \quad {\partial_{xx}}{\boldsymbol{\rho}}_0(0) = 2R_0,
\end{split}
\end{equation*} it can be shown that any $C^2$ solution must blow up in finite time since \begin{equation*}
\begin{split}
C(t) := {-}\frac{1}{2}{\partial_{xx}}{\bf c}(t,0),\quad R(t):= \frac{1}{2}{\partial_{xx}}{\boldsymbol{\rho}}(t,0)
\end{split}
\end{equation*} solves the ODE system \eqref{eq:ode}. To rigorously conclude finite-time singularity formation, we need to establish local in time existence and uniqueness of smooth (at least $C^2$) solutions to \eqref{eq:KS1D} with initial data satisfying \eqref{eq:van-assump-rho} and \eqref{eq:van-assump-c}.
\begin{remark}
We note that the above singularity formation result is similar to the one for the so-called Serre--Green--Naghdi equations obtained in the recent work \cite{BB}. Indeed, their result can be interpreted as a singularity formation for the (KS) system with $k({\bf c})={\bf c}$ and $\chi({\bf c})={\bf c}^{-1}$ (private communication with R. Granero-Belinch\'{o}n).
\end{remark}
\subsubsection*{Local well-posedness and singularity formation}
We consider the system \eqref{eq:KS} in some $d$-dimensional domain $\Omega$. One may take either $\Omega = \mathbb T^k \times \mathbb R^{d-k}$ or some open set in $\mathbb R^d$ with smooth boundary. In the latter case, we do not impose any boundary conditions on ${\boldsymbol{\rho}}$ and ${\bf c}$.
\begin{theorem}\label{thm:quadratic}
Assume that the initial data satisfies \begin{enumerate}
\item (regularity) We have ${\boldsymbol{\rho}}_0, {\bf c}_0 \in H^{\infty}(\Omega)$.
\item (vanishing) There exists $x_0 \in \Omega$ such that \begin{equation}\label{eq:vanishing}
\begin{split}
& {\boldsymbol{\rho}}_0(x_0) = 0, \quad \nabla{\boldsymbol{\rho}}_0(x_0) = {\bf 0}, \quad \nabla{\bf c}_0(x_0) = {\bf 0}.
\end{split}
\end{equation}
\item (non-degeneracy) For some constants $\underline{a}_0, \underline{r}_0, \underline{c}_0, \underline{\delta}_0 > 0$, we have \begin{equation}\label{eq:lowerbound-rho}
\begin{split}
{\boldsymbol{\rho}}_0(x) \ge \begin{cases}
\underline{a}_0|x-x_0|^2 , \quad & |x-x_0| < \delta_0 \\
\underbar{r}_0, \quad & |x-x_0| \ge \delta_0
\end{cases},
\end{split}
\end{equation} and \begin{equation}\label{eq:lowerbound-c}
\begin{split}
{\bf c}_0(x) \ge \underline{c}_0 .
\end{split}
\end{equation}
\end{enumerate} Then, there exist $T > 0$ and a unique solution $({\boldsymbol{\rho}},{\bf c})$ to \eqref{eq:KS} with initial data $({\boldsymbol{\rho}}_0,{\bf c}_0)$ satisfying ${\boldsymbol{\rho}} \in C^0([0,T); H^\infty)$, ${\bf c} \in C^0([0,T); H^{\infty})$, the vanishing conditions \eqref{eq:vanishing} for each $0<t<T$, and the lower bounds \eqref{eq:lowerbound-rho}--\eqref{eq:lowerbound-c} with time-dependent positive constants $\underline{a}(t), \underline{r}(t), \underline{c}(t), \underline{\delta}(t)$.
Moreover, there exists a set of initial data satisfying the above which blows up in finite time: to be more precise, there exists some $T^*>0$ such that the unique solution blows up in $W^{2,\infty}$ \begin{equation*}
\begin{split}
& {\limsup}_{t \rightarrow T^*}\left( \nrm{{\boldsymbol{\rho}}(t)}_{W^{2,\infty}} + \nrm{{\bf c}(t)}_{W^{2,\infty}} \right) = \infty.
\end{split}
\end{equation*}
\end{theorem}
\begin{remark}
If $\Omega$ is unbounded, we can simply modify the regularity condition as $\nabla{\boldsymbol{\rho}}_0, \nabla{\bf c}_0 \in H^{\infty}$, ${\boldsymbol{\rho}}_0, {\bf c}_0 \in L^\infty$.
\end{remark}
\begin{proof}
Without loss of generality, we shall assume that ${\bf 0} \in \Omega$ and $x_0 = {\bf 0}$. We introduce ${\bf f} = \nabla{\bf c}$ and write the inviscid \eqref{eq:KS} in the following slightly more convenient form: \begin{equation} \label{eq:KS-2}
\left\{
\begin{aligned}
\partial_t{\boldsymbol{\rho}} & = -\nabla \cdot ({\boldsymbol{\rho}}{\bf f}) = - {\boldsymbol{\rho}} (\nabla\cdot{\bf f}) - ({\bf f}\cdot\nabla){\boldsymbol{\rho}}, \\
\partial_t{\bf f} & = -\nabla({\bf c}{\boldsymbol{\rho}}) = -{\boldsymbol{\rho}} {\bf f} - {\bf c} \nabla{\boldsymbol{\rho}} .
\end{aligned}
\right.
\end{equation} For simplicity we divide the proof into a few steps.
\medskip
\noindent \textit{Step 1: Propagation of degeneracy}
\medskip
\noindent We assume \textit{formally} that there is a smooth solution (${\boldsymbol{\rho}} \in C^0_tH^m_x, {\bf f} \in C^0_tH^{m}_x$ for some $m>\frac{d}{2}+2$ is sufficient) on some time interval $[0,T)$, and then the vanishing condition propagates in the same time interval. For this we recall \eqref{eq:vanishing} and start with \begin{equation*}
\begin{split}
& \partial_t {\boldsymbol{\rho}} + {\bf f} \cdot \nabla {\boldsymbol{\rho}} = -{\boldsymbol{\rho}} \cdot \nabla {\bf f} ;
\end{split}
\end{equation*} from this it is clear that \begin{equation*}
\begin{split}
& \frac{d}{dt} {\boldsymbol{\rho}}(t,{\bf 0}) = 0
\end{split}
\end{equation*} if ${\bf f}(t,{\bf 0}) = {\bf 0}$. Next, \begin{equation}\label{eq:rho-deriv}
\begin{split}
\partial_t \partial_i {\boldsymbol{\rho}} + {\bf f} \cdot \nabla \partial_i{\boldsymbol{\rho}} = - \partial_i{\bf f} \cdot \nabla {\boldsymbol{\rho}} -\partial_i {\boldsymbol{\rho}} \cdot \nabla {\bf f} - {\boldsymbol{\rho}}\cdot \nabla \partial_i{\bf f}
\end{split}
\end{equation} and we have that \begin{equation*}
\begin{split}
& \frac{d}{dt} \partial_i {\boldsymbol{\rho}}(t,{\bf 0}) = 0
\end{split}
\end{equation*} once $\nabla {\boldsymbol{\rho}}(t,{\bf 0}) = {\bf 0 }$, ${\boldsymbol{\rho}}(t,{\bf 0}) = 0$, and ${\bf f}(t,\bf0) = 0$. Finally, from \begin{equation*}
\begin{split}
& \partial_t {\bf f} = - {\boldsymbol{\rho}} {\bf f} - {\bf c} \nabla {\boldsymbol{\rho}},
\end{split}
\end{equation*} we have \begin{equation*}
\begin{split}
& \frac{d}{dt}{\bf f}(t, {\bf 0}) = {\bf 0}
\end{split}
\end{equation*} if $\nabla {\boldsymbol{\rho}}(t,{\bf 0}) = {\bf 0 }$. Therefore as long as a smooth solution exists, we have \begin{equation*}
\begin{split}
& {\boldsymbol{\rho}}(t,{\bf 0}) = 0, \quad \nabla{\boldsymbol{\rho}}(t,{\bf 0}) = {\bf 0}, \quad \nabla{\bf c}(t,{\bf 0}) = {\bf 0}.
\end{split}
\end{equation*}
\medskip
\noindent \textit{Step 2: Propagation of lower bounds}
\medskip
\noindent We still assume existence of a smooth solution on some time interval $[0,T)$ and prove propagation in time of lower bounds of the form \eqref{eq:lowerbound-rho}--\eqref{eq:lowerbound-c}. We begin with $\underline{c}_0$: from the equation for ${\bf c}$, \begin{equation*}
\begin{split}
& \frac{d}{dt} \nrm{{\bf c}^{-1}}_{L^\infty} \le \nrm{{\boldsymbol{\rho}}}_{L^\infty} \nrm{{\bf c}^{-1}}_{L^\infty}
\end{split}
\end{equation*} and we may simply define $\underline{c}(t) $ as the solution of \begin{equation}\label{eq:def-under-c}
\begin{split}
& \frac{d}{dt} \underline{c}(t) = - \nrm{{\boldsymbol{\rho}}(t)}_{L^\infty} \underline{c}(t) , \quad \underline{c}(0) = \underline{c}_0
\end{split}
\end{equation} so that $\underline{c}^{-1}(t) \ge \nrm{{\bf c}^{-1}}_{L^\infty} $. Next, dividing both sides of the equation for ${\boldsymbol{\rho}}$ by $|x|^2$, we obtain that \begin{equation*}
\begin{split}
& \partial_t \frac{{\boldsymbol{\rho}}}{|x|^2} + ({\bf f} \cdot\nabla) \frac{{\boldsymbol{\rho}}}{|x|^2} = -(\nabla\cdot{\bf f}) \frac{{\boldsymbol{\rho}}}{|x|^2} - \frac{2{\bf f}\cdot x}{|x|^2} \frac{{\boldsymbol{\rho}}}{|x|^2} .
\end{split}
\end{equation*} This shows that along the characteristics defined by ${\bf f}$ we can propagate the lower bound of $ \frac{{\boldsymbol{\rho}}}{|x|^2} $: define the flow $\phi$ via \begin{equation*}
\begin{split}
& \frac{d}{dt}\phi(t,x) = {\bf f}(t,\phi(t,x)) , \quad \phi(0,x) = x
\end{split}
\end{equation*} and then with $R(t,x):= \frac{{\boldsymbol{\rho}}(t,x)}{|x|^2}$, \begin{equation*}
\begin{split}
& \frac{d}{dt} R(t,\phi(t,x)) = -\left[ \nabla\cdot {\bf f} + \frac{2{\bf f}\cdot x}{|x|^2} \right]|_{t,\phi(t,x)} R(t,\phi(t,x))
\end{split}
\end{equation*} so that \begin{equation*}
\begin{split}
R(t,\phi(t,x)) & = R_0(x) \exp\left( \int_0^t -\left[ \nabla\cdot {\bf f} + \frac{2{\bf f}\cdot x}{|x|^2} \right]|_{t',\phi(t',x)} dt' \right) \\
& \ge \frac{{\boldsymbol{\rho}}_0(x)}{|x|^2} \exp\left( {(-2-d)} \int_0^t \nrm{\nabla{\bf f}(t')}_{L^\infty} dt' \right).
\end{split}
\end{equation*} Therefore, for $|x| \le \underline{\delta}_0$, \begin{equation*}
\begin{split}
& \frac{{\boldsymbol{\rho}}(t,\phi(t,x))}{|\phi(t,x)|^2} \ge \underline{a}_0 \exp\left( {(-2-d)} \int_0^t \nrm{\nabla{\bf f}(t')}_{L^\infty} dt' \right).
\end{split}
\end{equation*} Now, \textit{defining} $ \underline{\delta}(t) $ via
\begin{equation}\label{eq:def-under-dlt}
\begin{split}
& \frac{d}{dt} \underline{\delta}(t) = - \nrm{\nabla{\bf f}(t)}_{L^\infty} \underline{\delta}(t), \quad \underline{\delta}(0) = \underline{\delta}_0 ,
\end{split}
\end{equation} we obtain that $\phi(t,B_0(\underline{\delta}_0)) \supset B_0(\underline{\delta}(t))$, simply because the characteristics starting on $\partial B_0(\underline{\delta}(t))$ cannot reach $\partial B_0(\underline{\delta}_0)$ during $[0,t]$. Hence, once we define \begin{equation}\label{eq:def-under-a}
\begin{split}
&\frac{d}{dt} \underline{a}(t) = - 3\nrm{\nabla{\bf f}(t)}_{L^\infty} \underline{a}(t), \quad \underline{a}(0) = \underline{a}_0,
\end{split}
\end{equation} we obtain that \begin{equation*}
\begin{split}
& \frac{{\boldsymbol{\rho}}(t,x)}{|x|^2} \ge \underline{a}(t) , \quad |x| \le \underline{\delta}(t).
\end{split}
\end{equation*} Note that $\underline{\delta}(t)$ can be recovered from $\underline{a}(t)$ and $\underline{\delta}_0$. Hence in the following we may omit the dependence of estimates in $\underline{\delta}(t)$. Finally, defining \begin{equation}\label{eq:def-under-r}
\begin{split}
\frac{d}{dt} \underline{r}(t) = - 3\nrm{\nabla{\bf f}(t)}_{L^\infty} \underline{r}(t), \quad \underline{r}(0) = \underline{r}_0,
\end{split}
\end{equation} we have that \begin{equation*}
\begin{split}
& {{\boldsymbol{\rho}}(t,x)} \ge \underline{r}(t) , \quad |x| > \underline{\delta}(t).
\end{split}
\end{equation*} We omit the proof, which is straightforward.
\medskip
\noindent \textit{Step 3: A priori estimates}
\medskip
\noindent The following lemma allows us to identify the weight $\sqrt{{\boldsymbol{\rho}}}$ as $\frac{|x|}{1+|x|}$: \begin{lemma}\label{lem:equiv}
{Under the hypothesis in Theorem \ref{thm:quadratic},}
we have \begin{equation}\label{eq:equiv}
\begin{split}
\frac{|x|}{1+|x|} \le ( \underline{a}^{-1}(t) + \underline{r}^{-1}(t) )^{\frac{1}{2}} \sqrt{{\boldsymbol{\rho}}(t,x)}
\end{split}
\end{equation} and \begin{equation}\label{eq:equiv2}
\begin{split}
\sqrt{{\boldsymbol{\rho}}(t,x)} \le C\nrm{{\boldsymbol{\rho}}}^{ {\frac{1}{2}}}_{W^{2,\infty}} \frac{|x|}{1+|x|}.
\end{split}
\end{equation}
\end{lemma}
\begin{proof}
We assume $\delta(t) \le 1$ and consider three regions: (i) $0 \le x < \delta(t)$, (ii) $\delta(t) \le x < 1$, (iii) $1 \le x$. (The proof is only simpler in the case $\delta(t) > 1$.) The inequality is trivial for (i) from $\sqrt{{\boldsymbol{\rho}}(t,x)} \ge \sqrt{\underline{a}_0}|x|$. On the other hand, for (ii) and (iii), \begin{equation*}
\begin{split}
\frac{|x|}{1+|x|} < 1 < (\underbar{r}(t))^{-\frac{1}{2}} \sqrt{{\boldsymbol{\rho}}(t,x)} .
\end{split}
\end{equation*} The proof of \eqref{eq:equiv2} is trivial; just consider regions $|x| \le 1$ and $|x|>1$ separately and use Taylor's expansion in the first region.
\end{proof}
\noindent We now fix some $m$-th order derivative $\partial^m$ with $m > d + 4$ and compute \begin{equation*}
\begin{split}
& \frac{d}{dt} \left( {\bf c} (\partial^m {\boldsymbol{\rho}})^2 + \sum_i {\boldsymbol{\rho}} (\partial^m {\bf f}_i)^2 \right) \\
&\quad = -{\bf c} {\boldsymbol{\rho}} (\partial^m{\boldsymbol{\rho}})^2 - 2{\bf c} \partial^m{\boldsymbol{\rho}} \partial^m( \nabla\cdot({\boldsymbol{\rho}}{\bf f}) ) - \sum_i \nabla\cdot({\boldsymbol{\rho}}{\bf f}) (\partial^m{\bf f}_i)^2 - \sum_i 2{\boldsymbol{\rho}} \partial^m{\bf f}_i \partial^m( {\boldsymbol{\rho}}{\bf f}_i + {\bf c}\partial_i{\boldsymbol{\rho}}).
\end{split}
\end{equation*} We then integrate both sides over $\Omega$; the first and third terms are handled as follows: \begin{equation*}
\begin{split}
& \left| \int -{\bf c} {\boldsymbol{\rho}} (\partial^m{\boldsymbol{\rho}})^2 \right| \lesssim \nrm{{\boldsymbol{\rho}}}_{L^\infty} \nrm{\sqrt{{\bf c}} \nabla^m{\boldsymbol{\rho}}}_{L^2}^2,
\end{split}
\end{equation*} and \begin{equation*}
\begin{split}
& \left| - \sum_i \nabla\cdot({\boldsymbol{\rho}}{\bf f}) (\partial^m{\bf f}_i)^2 \right| \lesssim \nrm{\nabla {\bf f}}_{L^\infty} \nrm{\sqrt{{\boldsymbol{\rho}}} \nabla^m{\bf f}}_{L^2}^2 + \nrm{\sqrt{{|({\bf f}\cdot\nabla){\boldsymbol{\rho}}|}} \nabla^m {\bf f}}_{L^2}^2 .
\end{split}
\end{equation*} We further note that \begin{equation*}
\begin{split}
\nrm{\sqrt{|({\bf f}\cdot\nabla){\boldsymbol{\rho}}|} \nabla^m {\bf f}}_{L^2}^2 &\le \nrm{\frac{1+|x|}{|x|} {\bf f}}_{L^\infty} \nrm{\frac{1+|x|}{|x|} \nabla{\boldsymbol{\rho}}}_{L^\infty} \nrm{\frac{|x|}{1+|x|} \nabla^m{\bf f}}_{L^2}^2 \\
&\lesssim ( \underline{a}^{-1}(t) + \underline{r}^{-1}(t) )\nrm{{\bf f}}_{W^{1,\infty}} \nrm{{\boldsymbol{\rho}}}_{W^{2,\infty}} \nrm{\sqrt{{\boldsymbol{\rho}}} \nabla^m{\bf f}}_{L^2}^2
\end{split}
\end{equation*} where we have used the pointwise inequality \eqref{eq:equiv}.
Let us now handle the other two terms; the most difficult terms combine as follows: \begin{equation*}
\begin{split}
& \int -2{\bf c}{\boldsymbol{\rho}} \partial^m{\boldsymbol{\rho}} \partial^m(\nabla\cdot{\bf f}) -\sum_i \int 2{\bf c}{\boldsymbol{\rho}} \partial^m{\bf f}_i \partial^m\partial_i{\boldsymbol{\rho}} = \int 2 \nabla( {\boldsymbol{\rho}}{\bf c}) \cdot \partial^m {\bf f} \partial^m {\boldsymbol{\rho}} .
\end{split}
\end{equation*} In absolute value, this can be bounded by \begin{equation*}
\begin{split}
& \lesssim \nrm{{\bf f}}_{L^\infty} \nrm{{\boldsymbol{\rho}}}_{L^\infty}^{\frac{1}{2}} \nrm{{\bf c}^{-1}}_{L^\infty}^{\frac{1}{2}} \nrm{\sqrt{{\bf c}} \partial^m{\boldsymbol{\rho}}}_{L^2} \nrm{\sqrt{{\boldsymbol{\rho}}} \partial^m {\bf f}}_{L^2} + \nrm{{\bf c}}_{L^\infty}^{\frac{1}{2}} \nrm{\sqrt{{\bf c}} \partial^m{\boldsymbol{\rho}} }_{L^2} {\nrm{|\nabla{\boldsymbol{\rho}}| \partial^m{\bf f}}_{L^2}. }
\end{split}
\end{equation*} Observing the bound \begin{equation*}
\begin{split}
& |\nabla{\boldsymbol{\rho}}(x)| \le C\nrm{{\boldsymbol{\rho}}}_{W^{2,\infty}} \frac{|x|}{1+|x|},
\end{split}
\end{equation*} we conclude the bound \begin{equation*}
\begin{split}
& \left| \int 2 \nabla( {\boldsymbol{\rho}}{\bf c}) \cdot \partial^m {\bf f} \partial^m {\boldsymbol{\rho}} \right| \lesssim ( \underline{a}^{-1}(t) + \underline{r}^{-1}(t) + \underline{c}^{-1}(t) + 1 )(1 + \nrm{{\bf c}}_{W^{1,\infty}} + \nrm{{\boldsymbol{\rho}}}_{W^{2,\infty}} ) \nrm{\sqrt{{\bf c}} \partial^m{\boldsymbol{\rho}}}_{L^2} \nrm{\sqrt{{\boldsymbol{\rho}}} \partial^m {\bf f}}_{L^2}.
\end{split}
\end{equation*} The remaining terms in the fourth term are straightforward to handle: for any {$\frac{m}{2} < k \le m $}, \begin{equation*}
\begin{split}
\left| \int 2{\boldsymbol{\rho}} \partial^m{\bf f} \partial^k{\bf f} \partial^{m-k}{\boldsymbol{\rho}} \right|& \lesssim \nrm{\partial^{m-k}{\boldsymbol{\rho}}}_{L^\infty} \nrm{\sqrt{{\boldsymbol{\rho}}}\partial^m{\bf f}}_{L^2}\nrm{\sqrt{{\boldsymbol{\rho}}}\partial^k{\bf f}}_{L^2} \\
&\lesssim \nrm{ {\boldsymbol{\rho}}}_{H^m} \nrm{\sqrt{{\boldsymbol{\rho}}}\partial^m{\bf f}}_{L^2}\nrm{\sqrt{{\boldsymbol{\rho}}}\partial^k{\bf f}}_{L^2},
\end{split}
\end{equation*} and then for {$1 \le k \le \frac{m}{2}$}, \begin{equation*}
\begin{split}
\left| \int 2{\boldsymbol{\rho}} \partial^m{\bf f} \partial^k{\bf f} \partial^{m-k}{\boldsymbol{\rho}} \right|& \lesssim \nrm{{\boldsymbol{\rho}}}_{L^\infty} \nrm{\partial^{m-k}{\boldsymbol{\rho}}}_{L^2} \nrm{\sqrt{{\boldsymbol{\rho}}}\partial^m{\bf f}}_{L^2}\nrm{\sqrt{{\boldsymbol{\rho}}}\partial^k{\bf f}}_{L^\infty} \\
&\lesssim \nrm{{\boldsymbol{\rho}}}_{L^\infty}^{\frac{3}{2}} \nrm{\partial^{m-k}{\boldsymbol{\rho}}}_{L^2} \nrm{\sqrt{{\boldsymbol{\rho}}}\partial^m{\bf f}}_{L^2}\nrm{ {\bf f}}_{H^{m-1}} ,
\end{split}
\end{equation*}and the case $ k = 0$ can be bounded by \begin{equation*}
\begin{split}
\left| \int 2{\boldsymbol{\rho}} \partial^m{\bf f} \, {\bf f} \partial^m{\boldsymbol{\rho}} \right| \lesssim \nrm{{\boldsymbol{\rho}}}_{L^\infty}^{\frac{1}{2}} \nrm{{\bf f}}_{L^\infty} \nrm{\sqrt{{\boldsymbol{\rho}}}\partial^m{\bf f}}_{L^2}\nrm{\partial^m{\boldsymbol{\rho}}}_{L^2}.
\end{split}
\end{equation*} It only remains to consider for $1 \le k \le m $ the following: \begin{equation*}
\begin{split}
\left| \int 2{\bf c} \, \partial^m{\boldsymbol{\rho}}\, \partial^{m+1-k}{\boldsymbol{\rho}} \, \partial^{k}{\bf f} \right|.
\end{split}
\end{equation*} When $k = m$, we can bound \begin{equation*}
\begin{split}
\lesssim ( \underline{a}^{-1}(t) + \underline{r}^{-1}(t) )^{\frac{1}{2}} \nrm{{\bf c}}_{L^\infty}^{\frac{1}{2}} \nrm{\sqrt{{\bf c}}\, \nabla^m{\boldsymbol{\rho}}}_{L^2} \nrm{{\boldsymbol{\rho}}}_{W^{2,\infty}} \nrm{\sqrt{{\boldsymbol{\rho}}}\,\nabla^m{\bf f}}_{L^2} .
\end{split}
\end{equation*} In the opposite end when $k = 1$, we bound instead as follows: \begin{equation*}
\begin{split}
\lesssim \nrm{{\bf f}}_{W^{1,\infty}} \nrm{\sqrt{{\bf c}}\partial_x^m{\boldsymbol{\rho}}}_{L^2}^{2} .
\end{split}
\end{equation*} To treat the case $2 \le k < m$, we shall prove the following elementary inequality: \begin{equation*}
\begin{split}
\nrm{\partial_x^{k}{\bf f}}_{L^2} \lesssim ( \underline{a}^{-1}(t) + \underline{r}^{-1}(t) )^{\frac{1}{2}} \left(\nrm{\sqrt{{\boldsymbol{\rho}}}\partial_x^k{\bf f}}_{L^2} + \nrm{\sqrt{{\boldsymbol{\rho}}}\partial_x^{k+1}{\bf f}}_{L^2} \right).
\end{split}
\end{equation*} Appealing to Lemma \ref{lem:equiv}, it suffices to establish the following \begin{lemma}
{Suppose that $g\in H^1(\mathbb R^d)$. Then,}
we have \begin{equation}\label{eq:hardy-variant}
\begin{split}
& \nrm{g}_{L^2} \lesssim \nrm{\frac{|x|}{1+|x|} g}_{L^2}+ \nrm{\frac{|x|}{1+|x|} \nabla g}_{L^2}.
\end{split}
\end{equation}
\end{lemma}
\begin{proof} The idea is simply to use Hardy's inequality only in the region $|x| \ll 1$. To be precise, we write \begin{equation*}
\begin{split}
& \nrm{g}_{L^2}^2 = \int \frac{1}{1+ |x|^2} |g(x)|^2 + \int \frac{ |x|^2}{1+ |x|^2} |g(x)|^2
\end{split}
\end{equation*} and it suffices to show that the first term on the right hand side is bounded by \eqref{eq:hardy-variant}. For this we write that \begin{equation*}
\begin{split}
& \int \frac{1}{1+ |x|^2} |g(x)|^2 = \sum_i \frac{1}{d} \int \partial_i(x_i) \frac{1}{1+ |x|^2} |g(x)|^2 \\
&= \frac{2}{d} \int \frac{|x|^2}{(1+|x|^2)^2} |g(x)|^2 + \sum_i \frac{2}{d} \int \frac{-2x_i}{1+ |x|^2} g\partial_i g .
\end{split}
\end{equation*} We consider separately the cases $d=1,2$, and {$d\ge 3$}. In the latter we have \begin{equation*}
\begin{split}
& \int \left[\frac{1}{1+ |x|^2} - \frac{2}{d} \frac{|x|^2}{(1+|x|^2)^2} \right]|g(x)|^2 = \sum_i {\frac{1}{d}} \int \frac{-2x_i}{1+ |x|^2} g\partial_i g \\
&\quad \le {\epsilon} \int \frac{1}{1+|x|^2} |g(x)|^2 + C_\epsilon \int \frac{|x|^2}{1+|x|^2} |\nabla g(x)|^2 .
\end{split}
\end{equation*} The proof is complete in this case since \begin{equation*}
\begin{split}
& \frac{1}{1+ |x|^2} - \frac{2}{d} \frac{|x|^2}{(1+|x|^2)^2} \gtrsim \frac{1}{1+|x|^2}.
\end{split}
\end{equation*} For $d = 1, 2$ we can find large $M_\epsilon$ depending on $\epsilon>0$ such that \begin{equation*}
\begin{split}
& \frac{1}{1+ |x|^2} - \frac{2}{d} \frac{|x|^2}{(1+|x|^2)^2} +\frac{M_\epsilon|x|^2}{1+|x|^2} \gtrsim \frac{2\epsilon}{1+|x|^2}
\end{split}
\end{equation*} for all $|x|$. Then one can similarly conclude the proof.
\end{proof}
\noindent Applying the above lemma, we obtain the bound \begin{equation*}
\begin{split}
&\left| \int 2{\bf c} \, \partial^m{\boldsymbol{\rho}}\, \partial^{m+1-k}{\boldsymbol{\rho}} \, \partial^{k}{\bf f} \right| \lesssim \nrm{{\bf c}}_{L^\infty}^{\frac{1}{2}} \nrm{\sqrt{{\bf c}}\, \partial^m{\boldsymbol{\rho}}}_{L^2} \nrm{\partial^{m+1-k}{\boldsymbol{\rho}}}_{L^\infty} \nrm{ \partial^k{\bf f}}_{L^2} \\
& \lesssim ( \underline{a}^{-1}(t) + \underline{r}^{-1}(t) )^{\frac{1}{2}} {\nrm{{\bf c}}_{L^\infty}} \nrm{\sqrt{{\bf c}}\, \nabla^m{\boldsymbol{\rho}}}_{L^2} \nrm{ {\boldsymbol{\rho}}}_{H^m}\left(\nrm{\sqrt{{\boldsymbol{\rho}}}\nabla^k{\bf f}}_{L^2} + \nrm{\sqrt{{\boldsymbol{\rho}}}\nabla^{k+1}{\bf f}}_{L^2} \right)
\end{split}
\end{equation*} for {$\frac{m}{2} < k < m$. In the case $k \le \frac{m}{2}$}, we can simply estimate \begin{equation*}
\begin{split}
& \left| \int 2{\bf c} \, \partial^m{\boldsymbol{\rho}}\, \partial^{m+1-k}{\boldsymbol{\rho}} \, \partial^{k}{\bf f} \right| \lesssim \nrm{\sqrt{{\bf c}}\, \nabla^m{\boldsymbol{\rho}}}_{L^2}\nrm{\sqrt{{\bf c}}\, \nabla^{m+1-k}{\boldsymbol{\rho}}}_{L^2} \nrm{{\bf f}}_{H^{m-1}}.
\end{split}
\end{equation*} Now we recall the weighted norms \begin{equation*}
\begin{split}
& \nrm{{\boldsymbol{\rho}}}_{Y^m}^2 := \sum_{j=0}^m \nrm{\sqrt{{\bf c}} \nabla^j {\boldsymbol{\rho}}}_{L^2}^2, \quad \nrm{{\bf f}}_{Y^m}^2 := \sum_{j=0}^m \nrm{\sqrt{{\boldsymbol{\rho}}} \nabla^j {\bf f}}_{L^2}^2
\end{split}
\end{equation*} and use \begin{equation*}
\begin{split}
& \nrm{{\boldsymbol{\rho}}}_{H^m} \lesssim \underline{c}^{-\frac{1}{2}}\nrm{{\boldsymbol{\rho}}}_{Y^m}, \quad \nrm{{\bf f}}_{H^{m-1}} \lesssim ( \underline{a}^{-1}(t) + \underline{r}^{-1}(t) )^{\frac{1}{2}} \nrm{{\bf f}}_{Y^m}.
\end{split}
\end{equation*}
Collecting all the estimates, \begin{equation*}
\begin{split}
&\left| \frac{d}{dt} \left( \nrm{{\boldsymbol{\rho}}}_{Y^m}^2 + \nrm{{\bf f}}_{Y^m}^2 \right) \right| \lesssim \left( 1 + \underline{a}^{-1} + \underline{r}^{-1} + \underline{c}^{-1} \right)^2 \left( 1 + \nrm{{\bf c}}_{W^{2,\infty}} + \nrm{{\boldsymbol{\rho}}}_{W^{2,\infty}} \right)^2 \left( \nrm{{\boldsymbol{\rho}}}_{Y^m}^2 + \nrm{{\bf f}}_{Y^m}^2 \right) .
\end{split}
\end{equation*} Strictly speaking, we have shown the above estimate for $\nrm{{\boldsymbol{\rho}}}_{\dot{Y}^m}^2 + \nrm{{\bf f}}_{\dot{Y}^m}^2$ in the left hand side but the same estimate can be shown \textit{a fortiori} for lower norms as well. We now define \begin{equation*}
\begin{split}
& Z_m = \nrm{{\boldsymbol{\rho}}}_{Y^m}^2 + \nrm{{\bf f}}_{Y^m}^2 + 1 + \underline{a}^{-1} + \underline{r}^{-1} + \underline{c}^{-1}
\end{split}
\end{equation*} and note that \begin{equation*}
\begin{split}
& \frac{d}{dt} Z_m \lesssim (Z_m)^4.
\end{split}
\end{equation*} This shows that formally, there exists some time interval $[0,T)$ in which $Z_m$ remains finite. By refining the estimates suitably (see the proof in the previous section), it can be seen that this time interval does not depend on $m$ as long as $m> d + 4$. Hence on the time interval $[0,T)$, the solution belongs to $H^\infty$. Existence and uniqueness can be proved using standard arguments.
\medskip
\noindent \textit{Step 5: Blow-up criterion and finite-time singularity formation}
\medskip
\noindent It only remains to prove finite time singularity formation. Before we proceed, we note that \begin{equation*}
\begin{split}
& \sup_{t \in [0,T)} \nrm{{\boldsymbol{\rho}}(t)}_{W^{2,\infty}} + \nrm{{\bf c}(t)}_{W^{2,\infty}}
\end{split}
\end{equation*} controls blow-up. The proof follows from applying the Gagliardo--Nirenberg--Sobolev inequalities rather than simple product estimates. We consider initial data such that near $x = {\bf 0}$, \begin{equation*}
\begin{split}
& {\boldsymbol{\rho}}_0(x) = \sum_i R_{i,0} x_i^2 + O(|x|^3), \quad {\bf c}_0(x) = 1 - \sum_i C_{i,0} x_i^2 + O(|x|^3).
\end{split}
\end{equation*} Assume towards a contradiction that the solution corresponding to the above initial data is global. We then first see that \begin{equation*}
\begin{split}
& \frac{d}{dt} \partial_i\partial_j {\boldsymbol{\rho}}(t,{\bf 0}) = 0 = \frac{d}{dt} \partial_i\partial_j {\bf c}(t,{\bf 0}) ,\quad \forall i \ne j.
\end{split}
\end{equation*} Hence for all $t\ge 0$, we are guaranteed Taylor expansion of the form \begin{equation*}
\begin{split}
& {\boldsymbol{\rho}}(t,x) = \sum_i R_{i}(t) x_i^2 + O(|x|^3), \quad {\bf c}(t,x) = 1 - \sum_i C_{i}(t) x_i^2 + O(|x|^3).
\end{split}
\end{equation*} {(Here, we note that the coefficient in $O(|x|^3)$ may grow in time, but it is uniformly bounded in any finite time interval.)} Then we obtain by substitution that \begin{equation}\label{eq:ODE-blowup-multiD}
\left\{
\begin{aligned}
\dot{C}_i(t) & = R_i(t) \\
\dot{R}_i(t) & = 2R_i(t)( 2C_i(t) + \sum_{j=1}^d C_j(t) ).
\end{aligned}
\right.
\end{equation} Once we choose $C_i = C > 0$ and $R_i = R > 0$ for some constants $C, R$, the system \eqref{eq:ODE-blowup-multiD} blows up in finite time, which is a contradiction. Therefore there exists some $T^*>0$ such that \begin{equation*}
\begin{split}
& {\limsup}_{t \rightarrow T^*}\left( \nrm{{\boldsymbol{\rho}}(t)}_{W^{2,\infty}} + \nrm{{\bf c}(t)}_{W^{2,\infty}} \right) = \infty.
\end{split}
\end{equation*} The proof is complete. \end{proof}
\begin{remark}
We would like to point out that the above local well-posedness and singularity formation results can be generalized in several ways.
\begin{itemize}
\item The assumption $\nabla{\bf c}_0(x_0)=0$ in \eqref{eq:vanishing} can be dropped. In that case, the model solution in one spatial dimension is replaced by \begin{equation*}
\begin{split}
{\bf c}(t,x) &= 1 - \ell (x-x(t)) - \frac{C(t)}{2} (x-x(t))^2 + O(|x-x(t)|^3), \\
{\boldsymbol{\rho}}(t,x) &= \frac{R(t)}{2}(x-x(t))^2 + O(|x-x(t)|^3).
\end{split}
\end{equation*} Here, $\ell$ is constant in time, $x(t)$ obeys the equation \begin{equation*}
\begin{split}
\dot{x}(t) = \ell R(t) , \quad x(0)=x_0,
\end{split}
\end{equation*} and $(C(t),R(t))$ still solves \eqref{eq:ode}. The local well-posedness argument goes through with $x_0$ replaced by $x(t)$. One can consider the full system involving the velocity as well; now the evolution of $x(t)$ will be affected by the velocity as well.
\item When the velocity becomes involved, it is an interesting problem to see whether finite time singularity formation persists: defining $x(t)$ by \begin{equation*}
\begin{split}
\dot{x}(t) = {\bf u}(t,x(t)), \quad x(0)=x_0
\end{split}
\end{equation*} and assuming \begin{equation*}
\begin{split}
{\boldsymbol{\rho}}_0(x_0) = 0, \quad \nabla{\boldsymbol{\rho}}_0(x_0) = 0, \quad \nabla{\bf c}_0(x_0) = 0,
\end{split}
\end{equation*} the system of equations for the second derivatives $\{ \partial_{i}\partial_j {\bf c}(t,x(t)), \partial_i\partial_j{\boldsymbol{\rho}}(t,x(t)) \}_{1\le i,j\le d}$ is given by \begin{equation} \label{eq:ode-velocity}
\left\{
\begin{aligned}
&\frac{d}{dt} \partial_{i}\partial_j {\bf c}(t,x(t)) = -{\bf c}_0(x_0)\partial_i\partial_j{\boldsymbol{\rho}}(t,x(t)) - \partial_i{\bf u}(t,x(t))\cdot\nabla\partial_j{\bf c}(t,x(t)) - \partial_j{\bf u}(t,x(t))\cdot\nabla\partial_i{\bf c}(t,x(t)),\\
&\frac{d}{dt} \partial_{i}\partial_j {\boldsymbol{\rho}}(t,x(t)) = - \partial_i{\bf u}(t,x(t))\cdot\nabla\partial_j{\boldsymbol{\rho}}(t,x(t)) - \partial_j{\bf u}(t,x(t))\cdot\nabla\partial_i{\boldsymbol{\rho}}(t,x(t)) \\
&\quad+\sum_k\left( \partial_i\partial_k{\boldsymbol{\rho}}(t,x(t))\partial_j\partial_k{\bf c}(t,x(t)) + \partial_j\partial_k{\boldsymbol{\rho}}(t,x(t))\partial_i\partial_k{\bf c}(t,x(t)) + \partial_i\partial_j{\boldsymbol{\rho}}(t,x(t))\partial_k\partial_k{\bf c}(t,x(t)) \right).
\end{aligned}
\right.
\end{equation} To close the system, we need the equation for $\nabla{\bf u}(t,x(t))$: \begin{equation}\label{eq:ode-vel-only}
\begin{split}
\frac{d}{dt} \partial_i{\bf u}(t,x(t)) = -\partial_i{\bf u}(t,x(t))\cdot\nabla{\bf u}(t,x(t)) -\partial_i\nabla p(t,x(t)) + D_u \Delta \partial_i{\bf u}(t,x(t)).
\end{split}
\end{equation} Even in the inviscid case $(D_u=0)$, the system does not close due to the pressure term. Then one can try to remove the pressure term by imposing certain symmetries on the initial data. Such a symmetry assumption should be respected by the system; in particular, it should be compatible with the potential function $\phi$. In the very special case when $d = 2$ and $\phi$ is radial (i.e. function of the variable $x_1^2+x_2^2$), rotational invariance persists for the system \eqref{eq:KSF} and by taking $x(t)=x_0 = 0$, we can remove the pressure term and still conclude finite-time singularity formation.
\item The case of general $\chi, k$ can be considered. Then, the proof of local well-posedness can be shown under mild assumptions on $\chi, k$ (e.g. the one used in Theorem \ref{thm:lwp-away}) and the ODE system \eqref{eq:ode} is now given by \begin{equation} \label{eq:ode2}
\left\{
\begin{aligned}
\dot{C}(t) &= k(1)R(t),\\
\dot{R}(t) &= 6\chi(1)C(t)R(t).
\end{aligned}
\right.
\end{equation}
\item Although we have considered vanishing of ${\boldsymbol{\rho}}$ only, one can treat the case of vanishing ${\bf c}$ as well {in the proof of local well-posedness. However, it is not clear to us whether finite-time singularity formation persists in this class of data.}
\end{itemize}
\end{remark}
\subsection*{Acknowledgments}
IJ has been supported by the New Faculty Startup Fund from Seoul National University, the Science Fellowship of POSCO TJ Park Foundation, and the National Research Foundation of Korea grant No. 2019R1F1A1058486. KK has been supported by NRF-2019R1A2C1084685 and NRF-2015R1A5A1009350. {We are grateful to the anonymous referees for various comments and suggestions, which have significantly improved the manuscript. Especially, we are grateful for pointing to us the possibility of the interesting generalization given in Remark \ref{rem:rotation}.}
\begin{comment}
\section{Growth of solutions in corner domains}
In this section, we investigate the dynamics of \eqref{eq:KSF} in a two-dimensional domain with corners. We are mostly interested in the growth or even finite-time singularity formation of smooth solutions, caused by the presence of corners. For simplicity, we fix the physical domain $\Omega$ to be the (infinite) sector with a $45^\circ$ angle: \begin{equation*}
\begin{split}
&\Omega = \{ (x,y) \in \mathbb R^2 : 0 < x < y \}.
\end{split}
\end{equation*} In the following, we will be able to localize the solutions near the corner, so the only thing relevant in $\Omega$ is that it \textit{locally} looks like a sector of angle $45^\circ$. This particular value of the angle is not so important but this choice will simplify some computations as we will see below.
We consider the system \eqref{eq:KSF} on $\Omega$ with $D_u = D_\rho = 0$ and $D_c > 0$; with a simple rescaling, it can be assumed without loss of generality that $D_c = 1$. Taking for simplicity $\chi(z) = 1$ and $k(z)=z$, we have : \begin{equation} \label{eq:KSF-corner}
\left\{
\begin{aligned}
&\partial_t \boldsymbol{\rho} + {\bf u}\cdot\nabla {\boldsymbol{\rho}} = -\nabla \cdot ( {\boldsymbol{\rho}}\nabla {\bf c}) , \\
&\partial_t {\bf c} + {\bf u}\cdot\nabla {\bf c} = \Delta {\bf c} - {\bf c}{\boldsymbol{\rho}} , \\
&\partial_t {\bf u} + {\bf u}\cdot\nabla {\bf u} + \nabla {\bf p} = {\boldsymbol{\rho}}\nabla \phi , \\
& \mathrm{div}\, {\bf u} = 0.
\end{aligned}
\right.
\end{equation} In this case, we may consider two types of boundary conditions for ${\bf c}$: either simple Neumann condition \begin{equation*}
\begin{split}
&\mbox{type I:}\quad \partial_\gamma {\bf c} = 0 \quad \mbox{on} \quad \partial\Omega ,
\end{split}
\end{equation*} or a mixed-type boundary condition \begin{equation*}
\begin{split}
\mbox{type II:}\quad \begin{cases}
&\partial_\gamma {\bf c} = 0 \quad \mbox{on} \quad \partial\Omega^+, \\
&{\bf c} = 0 \quad \mbox{on} \quad \partial\Omega^-.
\end{cases}
\end{split}
\end{equation*} We see that, upon assuming that ${\bf c}$ is at least of $C^2$, \begin{equation*}
\begin{split}
{\bf c}(x,y) = \begin{cases}
&c_0 + c_2(x^2+y^2) + o(r^2), \quad \mbox{type I}, \\
&c_2xy + o(r^2), \quad \mbox{type II}.
\end{cases}
\end{split}
\end{equation*} for some constants $c_0 \ge 0$ and $c_2$ respectively. Here $r = \sqrt{x^2+y^2}$. For simplicity we shall write ${\bf 0} = (0,0)$.
\subsection{Local well-posedness}
\subsection{Finite-time singularity formation }
We investigate the dynamics of \eqref{eq:KSF-corner} at the corner. To begin with, evaluating the ${\boldsymbol{\rho}}$-equation at the corner gives \begin{equation}\label{eq:rho-corner}
\begin{split}
& \frac{d}{dt} \overline{\rho} = -\overline{\rho} \, \overline{\Delta c}
\end{split}
\end{equation} where $\overline{\rho}(t):= {\boldsymbol{\rho}}(t,{\bf 0})$ and $\overline{\Delta c}(t) := \Delta{\bf c}(t,{\bf 0})$. Next, differentiating the ${\boldsymbol{\rho}}$-equation gives \begin{equation}\label{eq:rho-grad}
\begin{split}
\partial_t\nabla{\boldsymbol{\rho}} + ({\bf u} + \nabla{\bf c}) \cdot \nabla (\nabla{\boldsymbol{\rho}}) = -\Delta{\bf c}\nabla{\boldsymbol{\rho}} - {\boldsymbol{\rho}}\Delta\nabla{\bf c} -(\nabla{\bf u})^T \nabla{\boldsymbol{\rho}} - \nabla^2{\bf c} \nabla{\boldsymbol{\rho}} .
\end{split}
\end{equation} Evaluating at the corner and introducing $Y = \partial_x{\boldsymbol{\rho}}({\bf 0})$ and $Z = \partial_y{\boldsymbol{\rho}}({\bf 0})$ for simplicity, we have \begin{equation}\label{eq:nb-rho-corner}
\begin{split}
& \frac{d}{dt} \begin{pmatrix}
Y \\
Z
\end{pmatrix} = -\overline{\Delta c}\begin{pmatrix}
Y \\
Z
\end{pmatrix} - \bar{\rho} \Delta\nabla{\bf c}({\bf 0}) - \begin{pmatrix}
\partial_x{\bf u}^x({\bf 0}) & \partial_x{\bf u}^y({\bf 0}) \\
\partial_y{\bf u}^x({\bf 0}) & \partial_y{\bf u}^y({\bf 0})
\end{pmatrix} \begin{pmatrix}
Y \\
Z
\end{pmatrix} - \begin{pmatrix}
\partial_{xx}{\bf c}({\bf 0}) & \partial_{xy}{\bf c}({\bf 0}) \\
\partial_{yx}{\bf c}({\bf 0}) & \partial_{yy}{\bf c}({\bf 0})
\end{pmatrix}\begin{pmatrix}
Y \\
Z
\end{pmatrix} .
\end{split}
\end{equation} Next, evaluating the ${\bf c}$-equation at the corner gives \begin{equation}\label{eq:c-corner}
\begin{split}
\frac{d}{dt} \bar{c} = \overline{\Delta c} - \bar{c}\,\bar{\rho}
\end{split}
\end{equation} where $\bar{c}(t) := {\bf c}(t,{\bf 0})$. Moreover, taking the gradient of the ${\bf c}$-equation and using that $\nabla{\bf c}(t,{\bf 0}) = 0$ for all time (in either boundary conditions), we obtain the compatibility relation \begin{equation}\label{eq:nb-c-corner}
\begin{split}
\nabla\Delta {\bf c}(t,{\bf 0}) = {\bf c}(t,{\bf 0}) \nabla {\boldsymbol{\rho}}(t,{\bf 0}).
\end{split}
\end{equation} It will be very convenient to introduce the vorticity $\bfomg := \nabla \times {\bf u}$. Taking the curl of the equation for ${\bf u}$, we obtain \begin{equation}\label{eq:vort2D}
\begin{split}
\partial_t\bfomg + {\bf u}\cdot\nabla \bfomg = \nabla\times({\boldsymbol{\rho}}\nabla\phi) .
\end{split}
\end{equation} Evaluating at the corner gives with $X := \bfomg(t,{\bf 0})$ \begin{equation}\label{eq:vort-corner}
\begin{split}
\frac{d}{dt} X = \partial_y\phi({\bf 0}) Y - \partial_x\phi({\bf 0}) Z.
\end{split}
\end{equation} Now we recall a localization lemma for the singular integral operator $\nabla^2\Delta^{-1}$ in $\Omega$ from \cite{EJB}: \begin{lemma}
Assume that $\bfomg \in C^\alpha(\bar{\Omega})$ for some $0 < \alpha $. Then the unique solution ${\bf u}$ to the system \begin{equation}\label{eq:BS-corner}
\left\{
\begin{aligned}
&\nabla \times {\bf u} = \bfomg, \quad \mbox{on}\quad \Omega \\
&\mathrm{div}\,{\bf u} = 0, \quad \mbox{on}\quad \Omega \\
&\gamma\cdot{\bf u} = 0, \quad \mbox{on}\quad \partial\Omega
\end{aligned}
\right.
\end{equation} satisfies $\nabla u \in C^\alpha(\bar{\Omega})$ with \begin{equation*}
\begin{split}
& \nabla{\bf u} := \begin{pmatrix}
\partial_x{\bf u}^x & \partial_y{\bf u}^x \\
\partial_x{\bf u}^y & \partial_y{\bf u}^y
\end{pmatrix} = \bar{\omega} \begin{pmatrix}
\frac{1}{2} & -1 \\
0 & -\frac{1}{2}
\end{pmatrix} + o(1)
\end{split}
\end{equation*} as $r \rightarrow 0^+$ where $\bar{\omega}:= \bfomg(t,{\bf 0})$.
\end{lemma}
\noindent The above lemma simply says that up to higher-order terms, the velocity field has the Taylor expansion \begin{equation*}
\begin{split}
&{\bf u}(x,y) \simeq -\frac{\bar{\omega}}{2}\nabla^\perp(xy-y^2) ,
\end{split}
\end{equation*} which makes sense in view of \eqref{eq:BS-corner}. Now we can rewrite \eqref{eq:nb-rho-corner} as \begin{equation}\label{eq:nb-rho-corner2}
\begin{split}
\frac{d}{dt} \begin{pmatrix}
Y \\
Z
\end{pmatrix} = -(\overline{\Delta c} + \bar{\rho}\bar{c})\begin{pmatrix}
Y \\
Z
\end{pmatrix} - \frac{X}{2} \begin{pmatrix}
1 & 0 \\
-2 & -1
\end{pmatrix} \begin{pmatrix}
Y \\
Z
\end{pmatrix} - \begin{pmatrix}
\partial_{xx}{\bf c}({\bf 0}) & \partial_{xy}{\bf c}({\bf 0}) \\
\partial_{yx}{\bf c}({\bf 0}) & \partial_{yy}{\bf c}({\bf 0})
\end{pmatrix}\begin{pmatrix}
Y \\
Z
\end{pmatrix} .
\end{split}
\end{equation} Note that given the values of $\overline{\Delta c}$ for all time, $\bar{c}$ and $\bar{\rho}$ are completely determined in terms of their initial values via \eqref{eq:rho-corner} and \eqref{eq:c-corner}. Moreover, if the values of $\nabla^2{\bf c}({\bf 0})$ are also given, then $(X,Y,Z)$ can be solved in terms of the initial data. However in general it is not clear how to get a completely closed system since the equation for $\frac{d}{dt}\overline{\Delta c}$ involves $\Delta^2{\bf c}({\bf 0})$ and so on. To proceed further, we use the boundary condition for ${\bf c}$.
\medskip
\noindent \textbf{Type II boundary condition.} It turns out that the type II case yields a much simpler system, as $\bar{c} = \overline{\Delta c} = 0 $ automatically for all time. Moreover, we denote $C(t) := \partial_{xy}{\bf c}(0) \ge 0$. From \eqref{eq:rho-corner} we obtain that $\bar{\rho}(t) = \bar{\rho}_0$. Next, from \eqref{eq:c-corner}, $\bar{c}(t) = \bar{c}_0 e^{-t\bar{\rho}_0}$. Next, \eqref{eq:nb-rho-corner2} simplifies into \begin{equation}\label{eq:nb-rho-corner-II}
\begin{split}
\frac{d}{dt} \begin{pmatrix}
Y \\
Z
\end{pmatrix} = - \bar{\rho}_0\bar{c}_0e^{-t\bar{\rho}_0} \begin{pmatrix}
Y \\
Z
\end{pmatrix} - \frac{X}{2} \begin{pmatrix}
1 & 0 \\
-2 & -1
\end{pmatrix} \begin{pmatrix}
Y \\
Z
\end{pmatrix} - \begin{pmatrix}
0 & C(t) \\
C(t) & 0
\end{pmatrix}\begin{pmatrix}
Y \\
Z
\end{pmatrix} .
\end{split}
\end{equation} This gives the following result:
\begin{proposition}[Finite time singularity formation under type II boundary condition]
There exists smooth initial data $({\boldsymbol{\rho}}_0,{\bf c}_0,{\bf u}_0)$ for which there is no global in time smooth solution to the system \eqref{eq:KSF-corner}.
\end{proposition}
\begin{proof}
This proof requires $\partial_y\phi({\bf 0}) < 0$ and $\partial_x\phi({\bf 0}) \le 0$. (A nice example is given by $\phi(x,y) = -y$; i.e. the gravitational potential.) We write the $(X,Y,Z)$ system in the form \begin{equation}\label{eq:XYZ-II}
\left\{
\begin{aligned}
\dot{X} & = \partial_y\phi({\bf 0}) Y - \partial_x\phi({\bf 0}) Z, \\
\dot{Y} & = -\epsilon(t)Y - \frac{1}{2}XY - C(t)Z,\\
\dot{Z} & = -\epsilon(t)Z + XY + \frac{1}{2} XZ - C(t)Y.
\end{aligned}
\right.
\end{equation} Take any smooth initial data such that $X_0, Z_0 < 0 < Y_0$. Assume towards contradiction that there is a global smooth solution. Then the above formal computations can be justified for all positive times. For simplicity we even assume that $\epsilon(t) = 0$ but this additional assumption can be easily removed with a simple continuity argument in time. Then, we see that the signs $X(t), Z(t) < 0 < Y(t)$ propagate for all times. Then defining $\tilde{X} = -X$ \begin{equation}\label{eq:blowup-II}
\left\{
\begin{aligned}
\dot{\tilde{X}} & \ge (-\partial_y\phi({\bf 0})) Y, \\
\dot{Y} & \ge \frac{1}{2}\tilde{X}Y
\end{aligned}
\right.
\end{equation} This shows that $\tilde{X}, Y \rightarrow +\infty$ in finite time, which is a contradiction.
\end{proof}
\medskip
\noindent \textbf{Type I boundary condition.} In this case we have $\partial_{xy}{\bf c}({\bf 0}) = 0$; let us denote $\partial_{xx}{\bf c}({\bf 0}) = \partial_{yy} {\bf c}({\bf 0}) = C(t)$. To achieve some additional simplicity we put $\bar{\rho}_0 = 0$. Then $\bar{\rho}(t) = 0$ for all times and $\bar{c}(t) = \bar{c}_0 + 2\int_0^t C(\tau)d\tau$. From the restriction ${\bf c} \ge 0$, we obtain that $\int_0^t C(\tau)d\tau \ge -\frac{1}{2}\bar{c}_0$; that is, the value of $C(t)$ cannot be ``too negative''. Now we may rewrite the $(X,Y,Z)$ system as follows: \begin{equation}\label{eq:XYZ-I}
\left\{
\begin{aligned}
\dot{X} &= \partial_y\phi({\bf 0}) Y - \partial_x\phi({\bf 0}) Z, \\
\dot{Y} & = -\frac{1}{2}XY - C(t)Y, \\
\dot{Z} & = XY + \frac{1}{2}XZ -C(t)Z.
\end{aligned}
\right.
\end{equation} In this case it is interesting to see that there is a damping effect by $C(t)$ (which tends to be positive). In the special case $C(t) = 0$ finite-time blow up can be easily arranged. There does not seem a situation where the solution is guaranteed to blow up for any admissible $C(t)$. We have instead the following conditional blow-up result: \begin{proposition}[Conditional finite time singularity formation under type I boundary condition]
There exists smooth initial data $({\boldsymbol{\rho}}_0,{\bf c}_0,{\bf u}_0)$ for which there is no global in time smooth solution to the system \eqref{eq:KSF-corner} or for any $\epsilon >0$, we have for some time $t > 0$, $C(t) > (\frac{1}{2}-\epsilon)X(t)$. In particular, if we know a priori that $\sup_{t \ge 0} C(t) < \frac{1}{2}X_0$, then there cannot be a global smooth solution.
\end{proposition}
\begin{proof}
This proof requires that $\partial_x\phi({\bf 0}) < 0$. We can take $Y_0 = 0$, which then implies $Y(t) = 0$ for all $t$. Take any smooth initial data with $X_0, Z_0>0$. The statements follow.
\end{proof}
\section{Further issues}
\subsection{Global well-posedness}
\begin{itemize}
\item Global well-posedness in the 2D case with dissipation on only one of the variables (cf. results for the Boussinesq system)
\item Global well-posedness for \eqref{eq:KS}. Case of small perturbations from constant states.
\end{itemize}
\end{comment}
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1,116,691,500,874 | arxiv | \section{Introduction}\label{Introduction}
For a bounded measurable
function $\sigma = \sigma (x, \xi_1, \xi_2)$
on $(\mathbb{R}^n)^{3}$,
the bilinear pseudo-differential operator
$T_{\sigma}$ is defined
by
\[
T_{\sigma}(f_1,f_2)(x)
=\frac{1}{(2\pi)^{2n}}
\int_{(\mathbb{R}^n)^{2}}e^{i x \cdot(\xi_1+\xi_2)}
\sigma (x, \xi_1, \xi_2)
\widehat{f_1}(\xi_1)\widehat{f_2}(\xi_2)
\, d\xi_1 d\xi_2
\]
for $f_1,f_2 \in \mathcal{S}(\mathbb{R}^n)$.
The bilinear H\"ormander symbol class,
$BS_{\rho, \delta}^m=BS_{\rho, \delta}^m(\mathbb{R}^n)$,
$m \in \mathbb{R}$, $0 \leq \delta \leq \rho \leq 1$,
consists of all
$\sigma(x,\xi_1,\xi_2) \in C^{\infty}((\mathbb{R}^n)^{3})$
such that
\[
|\partial^{\alpha}_x\partial^{\beta_1}_{\xi_1}
\partial^{\beta_2}_{\xi_2}\sigma(x,\xi_1,\xi_2)|
\le C_{\alpha,\beta_1,\beta_2}
(1+|\xi_1|+|\xi_2|)^{m+\delta|\alpha|-\rho(|\beta_1|+|\beta_2|)}
\]
for all multi-indices
$\alpha,\beta_1,\beta_2 \in \mathbb{N}_0^n=\{0, 1, 2, \dots \}^n$.
When we state the boundedness of the bilinear operators
$T_{\sigma}$,
we will use the following terminology with a slight abuse.
Let $X_1,X_2$, and $Y$ be function spaces on $\mathbb{R}^n$
equipped with quasi-norms
$\|\cdot \|_{X_1}$, $\|\cdot \|_{X_2}$,
and $\|\cdot \|_{Y}$,
respectively.
If there exist a constant $C$ such that the estimate
\begin{equation*}
\| T_\sigma (f_1,f_2) \|_{Y}
\leq
C
\| f_1 \|_{X_1}
\| f_2 \|_{X_2}
\end{equation*}
holds for all $f_1 \in \mathcal{S} \cap X_1$
and $f_2 \in \mathcal{S} \cap X_2$,
then we simply say that
the operator $T_\sigma$
is bounded from $X_1 \times X_2$ to $Y$.
The interest of this short note is the boundedness
from $L^2 \times L^2$ to $L^1$
for the bilinear operator $T_{\sigma}$
with the symbol $\sigma$ belonging
to the Sj\"ostrand class.
We shall first recall some related boundedness
results on the linear case.
For a bounded measurable
function $\sigma = \sigma (x, \xi)$ on $(\mathbb{R}^n)^{2}$,
the linear pseudo-differential operator
$\sigma(X,D)$ is defined by
\begin{equation*}
\sigma (X,D) f (x)
=
\frac{1}{\, ( 2 \pi )^n \,}
\int_{\mathbb{R}^n} e^{i x \cdot \xi}
\sigma (x , \xi ) \widehat f (\xi ) d\xi.
\end{equation*}
for $f \in \mathcal{S}(\mathbb{R}^n)$.
The (linear) H\"ormander symbol class,
$S_{\rho, \delta}^m=S_{\rho, \delta}^m(\mathbb{R}^n)$,
$m \in \mathbb{R}$, $0 \leq \delta \leq \rho \leq 1$,
consists of all functions
$\sigma \in C^\infty ((\mathbb{R}^n)^{2})$ satisfying
\begin{equation*}
| \partial_x^\alpha \partial_\xi^\beta \sigma (x,\xi) |
\leq C_{\alpha, \beta} (1+|\xi|)^{m + \delta |\alpha| - \rho |\beta|}
\end{equation*}
for all multi-indices $\alpha, \beta \in \mathbb{N}_0^n$.
In \cite{CV}, Calder\'on--Vaillancourt proved that
if symbols belong to the H\"ormander class $S^0_{0,0}$,
the linear pseudo-differential operators are bounded on $L^2$.
Then, Sj\"ostrand \cite{sjostrand 1994} introduced
a new wider class generating $L^2$-bounded pseudo-differential operators
than the class $S^0_{0,0}$.
This new symbol class is today called
as the Sj\"ostrand symbol class,
and is also identified
as the modulation space $M^{\infty,1} ((\mathbb{R}^{n})^2)$
(see also Boulkhemair
\cite{boulkhemair 1995}).
See Section \ref{secmodsp}
for the definition of modulation spaces.
We shall next consider the bilinear case.
Based on the linear case,
one may expect the boundedness
for the bilinear pseudo-differential operators
of the bilinear H\"ormander class $BS^{0}_{0,0}$ and
the Sj\"ostrand class $M^{\infty,1} ((\mathbb{R}^{n})^{3})$.
However, B\'enyi--Torres \cite{BT-2004}
pointed out that bilinear pseudo-differential
operators with symbols in $BS^0_{0,0}$
are not bounded from $L^2 \times L^2 $ to $L^1$.
(See, e.g., \cite{MT-2013} for the boundedness
for the bilinear H\"ormander class.)
Therefore,
since $BS^0_{0,0} \hookrightarrow M^{\infty,1} ((\mathbb{R}^{n})^{3})$,
we never have the boundedness for the Sj\"ostrand class.
On the other hand,
B\'enyi--Gr\"ochenig--Heil--Okoudjou
\cite{benyi grochenig heil okoudjou 2005}
proved that if $\sigma \in M^{\infty,1} ((\mathbb{R}^{n})^3)$,
then $T_\sigma$ is bounded from $L^2 \times L^2$
to the modulation space $M^{1,\infty}$,
whose target space is wider than $L^1$.
Then, B\'enyi--Okoudjou
\cite{benyi okoudjou 2004, benyi okoudjou 2006}
gave that
if $\sigma$ belongs to $M^{1,1} ((\mathbb{R}^{n})^3)$,
embedded into $M^{\infty,1} ((\mathbb{R}^{n})^3)$,
then $T_\sigma$ is bounded from
$L^{2} \times L^{2}$ to $L^{1}$.
According to these results,
in the bilinear case,
we are able to have the boundedness on
$L^2 \times L^2$
for Sj\"ostrand symbol class,
paying some kind of cost.
Very recently,
in \cite{KMT-arXiv},
it was proved that
if $\sigma \in BS_{0,0}^{0}$,
then the operator $T_\sigma$ is bounded from
$H^{s_1} \times H^{s_2}$ to $(L^{2}, \ell^1)$
for $s_1,s_2>0$, $s_1+s_2=n/2$.
Here, $H^s$, $s\in\mathbb{R}$,
is the $L^2$-based Sobolev space
and $(L^{2}, \ell^1)$
is the $L^2$-based amalgam space
(see Section \ref{secpreli}).
The aim of this short note is
to improve this boundedness
for the class $BS_{0,0}^{0}$
to that for the Sj\"ostrand class.
The main result is the following.
\begin{thm}\label{main-thm}
Let $s_1, s_2 \in (0,\infty)$ satisfy $s_1+s_2=n/2$.
Then, if $\sigma \in M^{\infty,1} ((\mathbb{R}^n)^3)$,
the bilinear pseudo-differential operator
$T_{\sigma}$
is bounded from
$H^{s_1} \times H^{s_2}$
to $(L^2,\ell^{1})$.
In particular, all those
$T_{\sigma}$ are
bounded from
$H^{s_1} \times H^{s_2}$
to $L^r $ for all $r\in (1,2]$ and
to $h^1 $.
\end{thm}
We end this section by explaining the organization of this note.
In Section \ref{secpreli}, we will give the basic notations
which will be used throughout this paper
and recall the definitions and properties of some function spaces.
In Section \ref{sectionLemma}, we collect some lemmas
for the proof of Theorem \ref{main-thm}.
In Section \ref{sectionMain}, we show Theorem \ref{main-thm}.
\section{Preliminaries}\label{secpreli}
\subsection{Basic notations}
We collect notations which will be used throughout this paper.
We denote by $\mathbb{R}$, $\mathbb{Z}$, $\mathbb{N}$, and $\mathbb{N}_0$
the sets of real numbers, integers, positive integers,
and nonnegative integers, respectively.
We denote by $Q$ the $n$-dimensional
unit cube $[-1/2,1/2)^n$.
The cubes $\tau + Q$, $\tau \in \mathbb{Z}^n$, are mutually
disjoint and constitute a partition of the
Euclidean space $\mathbb{R}^n$.
This implies integral of a function on $\mathbb{R}^{n}$
can be written as
\begin{equation}\label{cubicdiscretization}
\int_{\mathbb{R}^n} f(x)\, dx =
\sum_{\tau \in \mathbb{Z}^{n}} \int_{Q} f(x+ \tau)\, dx.
\end{equation}
We denote by $B_R$ the closed ball in $\mathbb{R}^n$
of radius $R>0$ centered at the origin.
We write the characteristic function
on the set $\Omega$ as $\mathbf{1}_{\Omega}$.
For $x\in \mathbb{R}^d$, we write
$\langle x \rangle = (1 + | x |^2)^{1/2}$.
For two nonnegative functions $A(x)$ and $B(x)$ defined
on a set $X$,
we write $A(x) \lesssim B(x)$ for $x\in X$ to mean that
there exists a positive constant $C$ such that
$A(x) \le CB(x)$ for all $x\in X$.
We often omit to mention the set $X$ when it is
obviously recognized.
Also $A(x) \approx B(x)$ means that
$A(x) \lesssim B(x)$ and $B(x) \lesssim A(x)$.
We denote the Schwartz space of rapidly
decreasing smooth functions
on $\mathbb{R}^d$
by $\mathcal{S} (\mathbb{R}^d)$
and its dual,
the space of tempered distributions,
by $\mathcal{S}^\prime(\mathbb{R}^d)$.
The Fourier transform and the inverse
Fourier transform of $f \in \mathcal{S}(\mathbb{R}^d)$ are given by
\begin{align*}
\mathcal{F} f (\xi)
&= \widehat {f} (\xi)
= \int_{\mathbb{R}^d} e^{-i \xi \cdot x } f(x) \, dx,
\\
\mathcal{F}^{-1} f (x)
&= \check f (x)
= \frac{1}{(2\pi)^d} \int_{\mathbb{R}^d} e^{i x \cdot \xi } f( \xi ) \, d\xi,
\end{align*}
respectively.
For $m \in \mathcal{S}^\prime (\mathbb{R}^d)$,
the Fourier multiplier operator is defined by
\begin{equation*}
m(D) f
=
\mathcal{F}^{-1} \big[ m \widehat{f}\, \big].
\end{equation*}
We also use the notation $(m(D)f)(x)=m(D_x)f(x)$
when we indicate which variable is considered.
For a measurable subset $E \subset \mathbb{R}^d$,
the Lebesgue space $L^p (E)$, $1 \le p\le \infty$,
is the set of all those
measurable functions $f$ on $E$ such that
$\| f \|_{L^p(E)} =
\left( \int_{E} \big| f(x) \big|^p \, dx \right)^{1/p}
< \infty
$
if $1 \le p < \infty$
or
$\| f \|_{L^\infty (E)}
=
\mathrm{ess}\, \sup_{x\in E} |f(x)|
< \infty$ if $p = \infty$.
We also use the notation
$\| f \|_{L^p(E)} = \| f(x) \|_{L^p_{x}(E)} $
when we want to indicate the variable explicitly.
The uniformly local $L^2$ space, denoted by
$L^2_{ul} (\mathbb{R}^d)$, consists of
all those measurable functions $f$ on
$\mathbb{R}^d$ such that
\begin{equation*}
\| f \|_{L^2_{ul} (\mathbb{R}^d) }
= \sup_{\nu \in \mathbb{Z}^d}
\left( \int_{[-1/2, 1/2)^d}
\big| f(x+\nu)
\big|^2 \, dx
\right)^{1/2}
< \infty
\end{equation*}
(this notion can be found in
\cite[Definition 2.3]{kato 1975}).
Let $\mathbb{K}$ be a countable set.
We define the sequence spaces
$\ell^q (\mathbb{K})$ and $\ell^{q, \infty} (\mathbb{K})$
as follows.
The space $\ell^q (\mathbb{K})$, $1 \le q \le \infty$,
consists of all those
complex sequences $a=\{a_k\}_{k\in \mathbb{K}}$
such that
$ \| a \|_{ \ell^q (\mathbb{K})}
=
\left( \sum_{ k \in \mathbb{K} }
| a_k |^q \right)^{ 1/q } <\infty$
if $1 \leq q < \infty$
or
$\| a \|_{ \ell^\infty (\mathbb{K})}
= \sup_{k \in \mathbb{K}} |a_k| < \infty$
if $q = \infty$.
For $1\le q<\infty$,
the space
$\ell^{ q,\infty }(\mathbb{K})$ is
the set of all those complex
sequences
$a= \{ a_k \}_{ k \in \mathbb{K} }$ such that
\begin{equation*}
\| a \|_{\ell^{q,\infty} (\mathbb{K}) }
=\sup_{t>0}
\big\{ t\,
\sharp \big( \{ k \in \mathbb{K} : | a_k | > t \}
\big)^{1/q} \big\} < \infty,
\end{equation*}
where $\sharp$ denotes the cardinality
of a set.
Sometimes we write
$\| a \|_{\ell^{q}} =
\| a_k \|_{\ell^{q}_k }$ or
$\| a \|_{\ell^{q,\infty}} =
\| a_k \|_{\ell^{q,\infty}_k }$.
If $\mathbb{K}=\mathbb{Z}^n$,
we usually write $ \ell^q $ or $\ell^{ q, \infty }$
for
$\ell^q (\mathbb{Z}^n)$
or $\ell^{q, \infty} (\mathbb{Z}^n)$.
Let $X,Y,Z$ be function spaces.
We denote the mixed norm by
\begin{equation*}
\label{normXYZ}
\| f (x,y,z) \|_{ X_x Y_y Z_z }
=
\bigg\| \big\| \| f (x,y,z)
\|_{ X_x } \big\|_{ Y_y } \bigg\|_{ Z_z }.
\end{equation*}
(Here pay special attention to the order
of taking norms.)
We shall use these mixed norms for
$X, Y, Z$ being $L^p$ or $\ell^p$.
\subsection{Modulation spaces} \label{secmodsp}
We give the definition of modulation spaces which were introduced
by Feichtinger \cite{feichtinger 1983, feichtinger 2006}
(see also Gr\"ochenig \cite{grochenig 2001}).
Let $\varphi \in \mathcal{S}(\mathbb{R}^d)$ satisfy that
$\mathrm{supp}\, \varphi \subset [-1,1]^{d}$ and
$\sum _{k\in \mathbb{Z}^{d}} \varphi (\xi - k) = 1$
for any $\xi \in \mathbb{R}^d$.
Then, for $1 \leq p,q \leq \infty$,
the modulation space $M^{p,q}(\mathbb{R}^d)$
consists of
all $f \in \mathcal{S}'(\mathbb{R}^d)$ such that
\begin{equation*}
\| f \|_{M^{p,q}} =
\big\|\varphi(D-k) f (x) \big\|_{L^{p}_{x}(\mathbb{R}^d) \ell^{q}_{k}(\mathbb{Z}^d) }
< \infty.
\end{equation*}
We note that the definition of modulation spaces
is independent of the choice of the function $\varphi$.
If $p_{1}\leq p_{2}$ and
$q_{1}\leq q_{2}$, $M^{p_{1},q_{1}} \hookrightarrow M^{p_{2},q_{2}}$.
We have $M^{2,2} = L^2$,
and $M^{p,1} \hookrightarrow L^{p} \hookrightarrow M^{p,\infty }$ for $1 \leq p \leq \infty $.
For more details, see also, e.g.,
\cite{kobayashi 2006, wang hudzik 2007}.
\subsection{Local Hardy space $h^1$}
\label{sechardy}
We recall the definition of the local Hardy space
$h^1(\mathbb{R}^n)$.
Let $\varphi \in \mathcal{S}(\mathbb{R}^n)$ be such that
$\int_{\mathbb{R}^n}\varphi(x)\, dx \neq 0$.
Then, the local Hardy space $h^1(\mathbb{R}^n)$
consists of
all $f \in \mathcal{S}'(\mathbb{R}^n)$ such that
$\|f\|_{h^1}= \|\sup_{0<t<1}|\varphi_t*f| \|_{L^1}
<\infty$,
where $\varphi_t(x)=t^{-n}\varphi(x/t)$.
It is known that $h^1(\mathbb{R}^n)$
does not depend on the choice of the function $\varphi$,
and that $h^1(\mathbb{R}^n) \hookrightarrow L^1(\mathbb{R}^n)$.
See Goldberg \cite{goldberg 1979} for more details about
$h^1$.
\subsection{Amalgam spaces}
\label{secamalgam}
For $1 \leq p,q \leq \infty$,
the amalgam space
$(L^p,\ell^q)(\mathbb{R}^n)$ is
defined to be the set of all
those measurable functions $f$ on
$\mathbb{R}^n$ such that
\begin{equation*}
\| f \|_{ (L^p,\ell^q) (\mathbb{R}^n)}
=
\| f(x+\nu) \|_{L^p_x (Q) \ell^q_{\nu}(\mathbb{Z}^n) }
=\left\{ \sum_{\nu \in \mathbb{Z}^n}
\left( \int_{Q} \big| f(x+\nu) \big|^p \, dx
\right)^{q/p} \right\}^{1/q}
< \infty
\end{equation*}
with usual modification when $p$ or $q$ is infinity.
Obviously, $(L^p,\ell^p) = L^p$
and $(L^2, \ell^\infty) = L^2_{ul}$.
If $p_1 \geq p_2$ and $q_1 \leq q_2$,
then
$(L^{p_1}, \ell^{q_1})
\hookrightarrow (L^{p_2},\ell^{q_2}) $.
In particular,
$(L^2,\ell^r) \hookrightarrow L^r$
for $1 \leq r \leq 2$.
In the case $r=1$, the stronger embedding
$(L^2,\ell^1) \hookrightarrow h^1$ holds
(see \cite[Section 2.3]{KMT-arxiv-2}).
See Fournier--Stewart \cite{fournier stewart 1985}
and Holland \cite{holland 1975} for more
properties of amalgam spaces.
We end this subsection with noting the following,
which was proved in \cite[Lemma 2.1]{KMT-arxiv-2}.
\begin{lem}\label{lemequivalentamalgam}
Let $1 \leq p,q \leq \infty$.
If $L> n/ \min (p,q)$ and if
$g$ is a measuarable function on $\mathbb{R}^n$ such that
\begin{equation*}
c \mathbf{1}_{Q} (x) \le |g(x)| \le c^{-1} \langle x \rangle ^{-L}
\end{equation*}
with some positive constant $c$, then
\begin{equation*}
\| f \|_{ (L^p,\ell^q) (\mathbb{R}^{n})}
\approx
\left\|
g(x-\nu ) f(x) \right\|_{L^p_x(\mathbb{R}^{n}) \ell^q_{\nu}(\mathbb{Z}^{n}) }.
\end{equation*}
\end{lem}
\section{Lemmas}\label{sectionLemma}
In this section, we prepare several lemmas.
We denote by $S$ the operator
\begin{equation*}
S (f) (x)
= \int_{\mathbb{R}^n} \frac{ |f(y)| }{ \langle x-y \rangle^{n+1} } dy.
\end{equation*}
We have the following facts,
which was proved in \cite[Lemmas 4.1 and 4.3]{KMT-arxiv-2}.
\begin{lem}\label{propertiesofS}
Let $1 \leq p \leq \infty$.
The following (1)--(3)
hold for all nonnegative measurable
functions $f, g$ on $\mathbb{R}^n$.
\begin{enumerate}
\setlength{\itemindent}{0pt}
\setlength{\itemsep}{3pt}
\item
$S(f \ast g)(x) = \big( S(f) \ast g \big)(x) =
\big( f \ast S(g) \big)(x)$.
\item
$S(f)(x) \approx S(f)(y)$
for $x,y\in \mathbb{R}^n$ such that $|x-y|\lesssim1$.
\item
$\| S(f)(\nu) \|_{\ell^p_\nu} \approx \| S(f)(x) \|_{L^p_x}$.
\item
Let $\varphi$ be a function in $\mathcal{S}(\mathbb{R}^n)$
with compact support.
Then, $| \varphi(D-\nu) f(x) |^{2}
\lesssim S( | \varphi(D-\nu) f |^{2} )(x)$
for any $f \in \mathcal{S}(\mathbb{R}^{n})$, $\nu\in\mathbb{Z}^n$, and $x\in\mathbb{R}^n$.
\end{enumerate}
\end{lem}
The following is proved in \cite[Proposition 3.4]{KMT-arxiv-2}
(see also \cite[Proposition 3.3]{KMT-arXiv}).
\begin{lem}\label{productLweak}
Let $2<p_1,p_2 < \infty$,
$1/p_{1} + 1/p_{2} = 1/2$,
and let
$f_{j} \in \ell^{p_{j}, \infty}(\mathbb{Z}^d)$
be nonnegative sequences
for $j=1,2$.
Then,
\begin{align*}
\sum_{\nu_1, \nu_2 \in \mathbb{Z}^n}
f_1(\nu_1) f_2(\nu_2)
A_0(\nu_1+ \nu_2)
\prod_{j=1,2} A_j(\nu_j)
\lesssim
\|f_{1}\|_{ \ell^{p_{1},\infty} }
\|f_{2}\|_{ \ell^{p_{2},\infty} }
\prod_{j=0,1,2} \|A_j\|_{\ell^{2} }.
\end{align*}
\end{lem}
We end this section by mentioning a lemma which
can be found in Sugimoto \cite[Lemma 2.2.1]{sugimoto 1988 JMSJ}.
The explicit proof is given in \cite[Lemma 4.4]{KMT-arxiv-2}.
\begin{lem}
\label{unifdecom}
There exist the functions $\kappa \in \mathcal{S}(\mathbb{R}^n)$ and $\chi \in \mathcal{S}(\mathbb{R}^n)$ satisfying that
$\mathrm{supp}\, \kappa \subset [-1,1]^n$, $\mathrm{supp}\, \widehat \chi \subset B_1$,
$| \chi | \geq c > 0$ on $[-1,1]^n$
and
\begin{equation*}
\sum_{\nu\in\mathbb{Z}^n} \kappa (\xi - \nu) \chi (\xi - \nu ) = 1, \quad \xi \in \mathbb{R}^n.
\end{equation*}
\end{lem}
\section{Main results}\label{sectionMain}
\subsection{Key proposition}
In this subsection we prove the following key proposition.
The proof is almost the same as in the proof of \cite[Proposition 5.1]{KMT-arxiv-2},
and the essential idea goes back to \cite{boulkhemair 1995}.
\begin{prop}
\label{main-prop}
Let $s_1, s_2 \in (0,\infty)$ satisfy $s_1+s_2=n/2$,
and let
$R_{0}, R_{1}, R_{2} \in [1, \infty)$.
Suppose $\sigma$ is
a bounded continuous function on
$(\mathbb{R}^n)^{3}$ such that
$\mathrm{supp}\, \mathcal{F} \sigma \subset
B_{R_0} \times B_{R_1} \times B_{R_2}$.
Then
\begin{equation*}
\left\| T_{ \sigma } \right\|_{H^{s_1} \times H^{s_2} \to (L^2, \ell^{1})}
\lesssim
\left( R_{0} R_{1} R_{2} \right)^{n/2}
\| \sigma \|_{ L^{2}_{ul} ((\mathbb{R}^{n})^{3})}.
\end{equation*}
\end{prop}
\begin{proof}
We take a function $\theta \in \mathcal{S}(\mathbb{R}^n)$ satisfying that
$|\theta| \geq c >0$ on $Q=[-1/2,1/2)^n$ and
$\mathrm{supp}\, \widehat \theta \subset B_1$.
Then, we have by Lemma \ref{lemequivalentamalgam}
\begin{equation*}
\left\| T_{ \sigma }( f_1, f_2 ) \right\|_{(L^2,\ell^{1})}
\approx
\left\| \theta (x-\mu) T_{ \sigma }( f_1, f_2 )(x) \right\|_{L^2_x(\mathbb{R}^n) \ell^{1}_{\mu}(\mathbb{Z}^n) },
\end{equation*}
and by duality
\begin{equation}
\label{saishodual}
\left\| T_{ \sigma }( f_1, f_2 ) \right\|_{(L^2,\ell^{1})}
\approx
\left\| \sup_{\|g\|_{L^2}=1} \left| \int_{\mathbb{R}^n} \theta (x-\mu)
T_{ \sigma }( f_1, f_2 )(x) \, g(x) \, dx \right|\right\|_{ \ell^{1}_{\mu} }.
\end{equation}
Hence, in what follows we consider
\begin{equation*}
I = \int_{\mathbb{R}^n} \theta (x-\mu) T_{ \sigma }( f_1, f_2 )(x) \, g(x) \, dx
\end{equation*}
for any $\mu\in\mathbb{Z}^n$ and all $g\in L^2(\mathbb{R}^n)$.
Now, we rewrite the integral
$I$ by the two steps below.
Firstly, by using Lemma \ref{unifdecom},
we decompose the symbol $\sigma$ as
\begin{align*}
\sigma ( x, \xi_1, \xi_2)
&=
\sum_{\boldsymbol{\nu}\in(\mathbb{Z}^n)^{2}}
\sigma ( x, \xi_1, \xi_2 )
\kappa (\xi_{1} - \nu_{1}) \chi (\xi_{1} - \nu_{1} )
\kappa (\xi_{2} - \nu_{2}) \chi (\xi_{2} - \nu_{2} )
\\&=
\sum_{\boldsymbol{\nu}\in(\mathbb{Z}^n)^{2}}
\sigma_{\boldsymbol{\nu}} ( x, \xi_1, \xi_2 )
\kappa (\xi_{1} - \nu_{1}) \kappa (\xi_{2} - \nu_{2}),
\end{align*}
where $\boldsymbol{\nu}=(\nu_1,\nu_2) \in (\mathbb{Z}^n)^2$
and we set
\[
\sigma_{\boldsymbol{\nu}} ( x, \xi_1, \xi_2 )
=
\sigma ( x, \xi_1, \xi_2 )
\chi (\xi_{1} - \nu_{1} )
\chi (\xi_{2} - \nu_{2} ).
\]
Denote
the Fourier multiplier operators
$\kappa (D - \nu_{j})$
by $\square_{\nu_{j}}$,
$j=1,2$.
Then,
the integral $I$ is written as
\begin{equation}\label{decomposedI}
I=
\sum_{\boldsymbol{\nu}\in(\mathbb{Z}^n)^{2}}
\int_{\mathbb{R}^n} \theta (x-\mu)
T_{ \sigma_{\boldsymbol{\nu}} } \big( \square_{\nu_1} f_1, \square_{\nu_2} f_2 \big)(x)
g(x)
\, dx .
\end{equation}
The idea of decomposing
symbols by such
$\kappa$ and $\chi$ goes back to
Sugimoto \cite{sugimoto 1988 JMSJ}.
Secondly, in the integral of \eqref{decomposedI},
we transfer the information of the Fourier transform of
$\theta (\cdot-\mu) T_{ \sigma_{\boldsymbol{\nu}} }
( \square_{\nu_1}f_1,\square_{\nu_2}f_2 )$ to $g$.
Observe that
\begin{align*}
&
\mathcal{F} \left[ T_{ \sigma_{\boldsymbol{\nu}} }( \square_{\nu_1}f_1, \square_{\nu_2}f_2 ) \right](\zeta)
\\&=
\frac{1}{(2\pi)^{2n}}
\int_{ (\mathbb{R}^{n})^{2} }
\big( \mathcal{F}_0 \sigma_{\boldsymbol{\nu}} \big) \big( \zeta - (\xi_1 +\xi_2) , \xi_1, \xi_2 \big)
\prod_{j=1,2} \kappa (\xi_{j}-\nu_{j}) \widehat{f_{j}}(\xi_{j})
\, d\xi_1 d\xi_2 .
\end{align*}
Then, combining this with the facts that
$\mathrm{supp}\, \mathcal{F}_{0} \sigma_{\boldsymbol{\nu}}
(\cdot, \xi_1, \xi_2 ) \subset B_{R_0}$
and $\mathrm{supp}\, \kappa(\cdot-\nu_{j})
\subset \nu_{j}+[-1,1]^n$, $j=1,2$,
we see that
\begin{equation*}
\mathrm{supp}\,
\mathcal{F} \left[ T_{ \sigma_{\boldsymbol{\nu}} }( \square_{\nu_1}f_1, \square_{\nu_2}f_2 ) \right]
\subset
\big\{ \zeta \in \mathbb{R}^n : | \zeta-(\nu_1 +\nu_2) | \lesssim R_0 \big\}.
\end{equation*}
Hence, since $\mathrm{supp}\, \widehat \theta \subset B_1$,
\begin{equation*}
\mathrm{supp}\,
\mathcal{F} \left[ \theta (\cdot-\mu)T_{ \sigma_{\boldsymbol{\nu}} }( \square_{\nu_1}f_1, \square_{\nu_2}f_2 ) \right]
\subset
\big\{ \zeta \in \mathbb{R}^n : | \zeta-(\nu_1+\nu_2) | \lesssim R_0 \big\}
\end{equation*}
for any $\mu\in\mathbb{Z}^n$.
Taking a function $\varphi \in \mathcal{S}(\mathbb{R}^n)$
satisfying that $\varphi = 1$
on $\{ \zeta\in\mathbb{R}^n: |\zeta|\lesssim 1\}$,
the integral $I$ given in \eqref{decomposedI}
can be further rewritten as
\begin{equation}\label{decomposedFouriertransI}
I=
\sum_{\boldsymbol{\nu}\in(\mathbb{Z}^n)^2} \int_{\mathbb{R}^n}
\theta (x-\mu) T_{ \sigma_{\boldsymbol{\nu}} }
( \square_{\nu_1}f_1, \square_{\nu_2}f_2 )(x)
\, \varphi \left( \frac{D+\nu_1 + \nu_2}{R_0} \right)
g (x) \, dx .
\end{equation}
Now, we shall actually estimate
the newly rewritten integral $I$
in \eqref{decomposedFouriertransI}.
Since it holds from the facts
$\mathrm{supp}\, \mathcal{F}_{1,2} \sigma (x, \cdot, \cdot)
\subset B_{R_1} \times B_{R_2}$
and $\mathrm{supp}\, \widehat{\chi} \subset B_1$
that
\[
\mathrm{supp}\, \mathcal{F}_{1,2} \sigma_{\boldsymbol{\nu}} (x, \cdot, \cdot)
\subset
B_{2R_1} \times B_{2R_2},
\]
we have
\begin{align*}
&
T_{ \sigma_{\boldsymbol{\nu}} }( \square_{\nu_1}f_1, \square_{\nu_2}f_2 )(x)
\\&=
\frac{1}{(2\pi)^{2n}}
\int_{(\mathbb{R}^n)^2}
\big( \mathcal{F}_{1,2} \sigma_{\boldsymbol{\nu}} \big) (x, y_1-x, y_2-x)
\prod_{j=1,2} \mathbf{1}_{B_{2R_{j}}}(x-y_{j}) \square_{\nu_{j}} f_{j} (y_{j})
\, dy_1 dy_2.
\end{align*}
Then, the Cauchy--Schwarz inequalities
and the Plancherel theorem yield that
\begin{align*}
\left| T_{ \sigma_{\boldsymbol{\nu}} }( \square_{\nu_1}f_1, \square_{\nu_2}f_2 )(x) \right|
\lesssim
\big\| \sigma_{\boldsymbol{\nu}} (x,\xi_1, \xi_2) \big\|_{L^2_{\xi_1, \xi_2}}
\prod_{j=1,2}
\left\{ \left( \mathbf{1}_{B_{2R_{j}}} \ast \big| \square_{\nu_{j}} f_{j} \big|^2 \right) (x) \right\}^{1/2}
\end{align*}
for any ${\boldsymbol{\nu}} = (\nu_1, \nu_2) \in (\mathbb{Z}^n)^2$ and $x \in \mathbb{R}^n$.
From this, the integral $I$ is estimated as
\begin{align}
\label{Ibeforediscretize}
\begin{split}
|I| & \lesssim
\sum_{\boldsymbol{\nu}\in(\mathbb{Z}^n)^2}
\int_{\mathbb{R}^n}|\theta (x-\mu)| \,
\big\| \sigma_{\boldsymbol{\nu}} (x,\xi_1, \xi_2) \big\|_{L^2_{\xi_1, \xi_2}}
\\
&\quad\times
\prod_{j=1,2}
\left\{ \Big( \mathbf{1}_{B_{2R_{j}}} \ast \big|\square_{\nu_{j}} f_{j} \big|^2 \Big) (x) \right\}^{1/2}
\left| \varphi \left( \frac{D+\nu_1 + \nu_2}{R_0} \right) g (x) \right|
\, dx.
\end{split}
\end{align}
Next, we decompose the above integral over $x$
by using \eqref{cubicdiscretization}.
Then the inequality \eqref{Ibeforediscretize}
coincides with
\begin{align*}
|I| &\lesssim
\sum_{\nu_0\in\mathbb{Z}^n}
\sum_{\boldsymbol{\nu}\in(\mathbb{Z}^n)^2}
\int_{Q} |\theta (x+\nu_0-\mu)| \,
\big\| \sigma_{\boldsymbol{\nu}} (x+\nu_0,\xi_1, \xi_2) \big\|_{L^2_{\xi_1, \xi_2}}
\\
&\quad\times
\prod_{j=1,2}
\left\{ \Big( \mathbf{1}_{B_{2R_{j}}} \ast \big|\square_{\nu_{j}} f_{j} \big|^2 \Big) (x+\nu_0) \right\}^{1/2}
\left| \varphi \left( \frac{D+\nu_1 + \nu_2}{R_0} \right) g (x+\nu_0) \right|
\, dx.
\end{align*}
Observe that
$|\theta (x+\nu_0-\mu)| \lesssim \langle \nu_0-\mu \rangle^{-L}$
holds for any $x\in Q$ and some constant $L>0$ sufficiently large,
and
from Lemma \ref{propertiesofS} (4), (1), and (2)
that
\begin{equation*}
\Big( \mathbf{1}_{B_{2R_{j}}} \ast
\big|\square_{\nu_{j}} f_{j}\big|^2 \Big) (x+\nu_0)
\lesssim
S \Big( \mathbf{1}_{B_{2R_{j}}} \ast
\big|\square_{\nu_{j}} f_{j} \big|^2 \Big) (\nu_0)
\end{equation*}
for $x\in Q$.
Then, by the
Cauchy--Schwarz inequality for the integral over $x$,
\begin{align*}
|I| &\lesssim
\sum_{\nu_0\in\mathbb{Z}^n}
\sum_{\boldsymbol{\nu}\in(\mathbb{Z}^n)^2}
\langle \nu_0-\mu \rangle^{-L}
\prod_{j=1,2}
\left\{ S \Big( \mathbf{1}_{B_{2R_{j}}} \ast
\big|\square_{\nu_{j}} f_{j} \big|^2 \Big) (\nu_0) \right\}^{1/2}
\\
&\quad \times
\int_{Q}
\big\| \sigma_{\boldsymbol{\nu}} (x+\nu_0,\xi_1, \xi_2) \big\|_{L^2_{\xi_1, \xi_2}}
\left| \varphi \left( \frac{D+\nu_1 +\nu_2}{R_0} \right) g (x+\nu_0) \right|
\, dx
\\
&\lesssim
\sum_{\nu_0\in\mathbb{Z}^n}
\sum_{\boldsymbol{\nu}\in(\mathbb{Z}^n)^2}
\langle \nu_0-\mu \rangle^{-L}
\prod_{j=1,2}
\left\{ S \Big( \mathbf{1}_{B_{2R_{j}}} \ast \big|\square_{\nu_{j}} f_{j} \big|^2 \Big) (\nu_0) \right\}^{1/2}
\\
&\times
\big\| \sigma_{\boldsymbol{\nu}} (x+\nu_0,\xi_1, \xi_2) \big\|_{L^2_{\xi_1, \xi_2} L^2_x(Q)}
\left\| \varphi \left( \frac{D+\nu_1 + \nu_2}{R_0} \right) g (x+\nu_0) \right\|_{L^2_x(Q)}.
\end{align*}
Here, since the equivalence
\begin{equation*}
\sup_{\nu_0, \boldsymbol{\nu}}
\big\| \sigma_{\boldsymbol{\nu}} (x+\nu_0,\xi_1, \xi_2) \big\|_{L^2_{\xi_1, \xi_2} L^2_x(Q)}
\approx
\| \sigma \|_{L^2_{ul} ((\mathbb{R}^n)^{3})}
\end{equation*}
holds, we have
\begin{align}
\label{Iafterdiscretize}
\begin{split}
|I|
&\lesssim
\| \sigma \|_{L^2_{ul}}
\sum_{\nu_0\in\mathbb{Z}^n}
\sum_{\boldsymbol{\nu}\in(\mathbb{Z}^n)^2}
\langle \nu_0-\mu \rangle^{-L}
\\
&\times
\prod_{j=1,2}
\left\{ S \Big( \mathbf{1}_{B_{2R_{j}}} \ast
\big|\square_{\nu_{j}} f_{j} \big|^2 \Big) (\nu_0) \right\}^{1/2}
\left\| \varphi \left( \frac{D+\nu_1 + \nu_2}{R_0} \right) g (x+\nu_0) \right\|_{L^2_x(Q)}.
\end{split}
\end{align}
In what follows, we write
each summand above by
\begin{align*}
A_j (\nu_j,\nu_0)&=\left\{ S \left( \mathbf{1}_{B_{2R_j}} \ast \big| \square_{\nu_j} f_j \big|^2 \right)(\nu_0) \right\}^{1/2}, \quad
j=1,2,
\\
A_0(\tau,\nu_0)
&=
\left\|\varphi \left( \frac{D+\tau}{R_0} \right) g(x+\nu_0) \right\|_{L^2_x(Q)}.
\end{align*}
Then, the inequality \eqref{Iafterdiscretize}
is written by
\begin{equation}
\label{IwithII}
|I|\lesssim
\| \sigma \|_{L^2_{ul}}
\, I\hspace{-3pt}I
\end{equation}
with
\begin{equation*}
I\hspace{-3pt}I=
\sum_{\nu_0\in\mathbb{Z}^n}
\langle \nu_0-\mu \rangle^{-L}
\sum_{\boldsymbol{\nu}\in(\mathbb{Z}^n)^2}
A_0(\nu_1 + \nu_2, \nu_0)
\prod_{j=1,2} A_j (\nu_j,\nu_0).
\end{equation*}
Now, we shall estimate $I\hspace{-3pt}I$.
By applying Lemma \ref{productLweak}
with $s_1 + s_2 = n/2$
to the sum over $\boldsymbol{\nu}$, we have
\begin{align*}
I\hspace{-3pt}I &=
\sum_{\nu_0\in\mathbb{Z}^n}
\langle \nu_0-\mu \rangle^{-L}
\sum_{\boldsymbol{\nu}\in(\mathbb{Z}^n)^2}
\langle \nu_1 \rangle^{-s_1} \langle \nu_2 \rangle^{-s_2}
A_0(\nu_1 + \nu_2, \nu_0)
\prod_{j=1,2} \langle \nu_j \rangle^{s_j} A_j (\nu_j,\nu_0)
\\&\lesssim
\sum_{ \nu_0 \in \mathbb{Z}^{n} }
\langle \nu_0-\mu \rangle^{-L}
\| A_0 (\tau,\nu_0) \|_{\ell^2_{\tau}}
\prod_{j=1,2}
\| \langle \nu_j \rangle^{s_j} A_j (\nu_j,\nu_0) \|_{\ell^2_{\nu_j}},
\end{align*}
since $\langle \nu_j \rangle^{-s_j}
\in \ell^{n/s_j,\infty}(\mathbb{Z}^n)$, $j=1,2$.
Then, we use the H\"older inequality
to the sum over $\nu_0$ to have
\begin{equation}\label{IIHolder}
I\hspace{-3pt}I
\lesssim
\| A_0 (\tau,\nu_0) \|_{ \ell^2_{\tau} \ell^{2}_{\nu_0} }
\prod_{j=1,2}
\| \langle \nu_0-\mu \rangle^{-L/2} \langle \nu_j \rangle^{s_j} A_j (\nu_j,\nu_0) \|_{ \ell^2_{\nu_j} \ell^{4}_{\nu_0}}.
\end{equation}
Here, the norm of $A_0$ in \eqref{IIHolder}
is estimated by the Plancherel theorem as follows:
\begin{align}\label{A0result}
\begin{split}
&\| A_0 (\tau,\nu_0) \|_{ \ell^2_{\tau} \ell^{2}_{\nu_0} }
=
\left\|\varphi \left( \frac{D+\tau}{R_0} \right) g(x+\nu_0) \right\|_{L^2_x(Q) \ell^2_{\tau} \ell^{2}_{\nu_0}}
\\
&=\left\|\varphi \left( \frac{D+\tau}{R_0} \right) g(x)
\right\|_{L^2_x(\mathbb{R}^n) \ell^2_{\tau}}
\approx
\left\|\varphi \left( \frac{\zeta +\tau}{R_0} \right) \widehat g(\zeta) \right\|_{L^2_\zeta(\mathbb{R}^n) \ell^2_{\tau}}
\approx
R_{0}^{n/2} \| g \|_{L^2},
\end{split}
\end{align}
where we used that
$\| \varphi ( \frac{\zeta +\tau}{R_0} ) \|_{\ell^2_\tau}
\approx R_0^{n/2}$ for any $\zeta \in \mathbb{R}^n$.
Hence, by collecting \eqref{IwithII}, \eqref{IIHolder}, and \eqref{A0result} we have
\begin{equation}
\label{resultofI}
|I|\lesssim
R_{0}^{n/2} \| \sigma \|_{L^2_{ul}}
\, \| g \|_{L^2}
\, \prod_{j=1,2}
\| \langle \nu_0-\mu \rangle^{-L/2} \langle \nu_j \rangle^{s_j} A_j (\nu_j,\nu_0) \|_{ \ell^2_{\nu_j} \ell^{4}_{\nu_0}}.
\end{equation}
We substitute \eqref{resultofI} into \eqref{saishodual},
and then use the H\"older inequality to $\ell^1_{\mu}$.
Then,
\begin{equation}\label{atodual}
\left\| T_{ \sigma }( f_1, f_2 ) \right\|_{(L^2,\ell^1)}
\lesssim
R_{0}^{n/2} \| \sigma \|_{L^2_{ul}}
\, \prod_{j=1,2}
\left\| \langle \nu_0-\mu \rangle^{-L/2} \langle \nu_j \rangle^{s_j} A_j (\nu_j,\nu_0) \right\|_{ \ell^2_{\nu_j} \ell^{4}_{\nu_0} \ell^{2}_{\mu}}.
\end{equation}
To achieve our goal, we shall estimate the norm of $A_j$ in \eqref{atodual}.
We have
\begin{align*}
&
\left\| \langle \nu_0-\mu \rangle^{-L/2} \langle \nu_j \rangle^{s_j} A_j (\nu_j,\nu_0) \right\|_{ \ell^2_{\nu_j} \ell^{4}_{\nu_0} \ell^{2}_{\mu}}
\\
&=
\bigg\| \langle \nu_0-\mu \rangle^{-L}
S \Big( \mathbf{1}_{B_{2R_j}} \ast
\big| \langle \nu_j \rangle^{s_j}
\square_{\nu_{j}} f_{j} \big|^2 \Big)(\nu_0)
\bigg\|_{\ell^{1}_{\nu_{j}}\ell^{2}_{\nu_0} \ell^{1}_{\mu} }^{1/2}
\\
&=
\bigg\| \langle \nu_0 \rangle^{-L}
S \Big( \mathbf{1}_{B_{2R_j}} \ast
\big\| \langle \nu_j \rangle^{s_j} \square_{\nu_{j}}
f_{j} \big\|_{\ell^2_{\nu_{j}}}^2 \Big)(\nu_0+\mu)
\bigg\|_{\ell^{2}_{\nu_0} \ell^{1}_{\mu} }^{1/2}
=(\ast\ast).
\end{align*}
Then, by using the embedding
$\ell^1_{\nu_0} \hookrightarrow \ell^2_{\nu_0}$,
Lemma \ref{propertiesofS} (3),
and the boundedness of the operator
$S$ on $L^1$,
we have
\begin{align*}
(\ast\ast)
\lesssim
\left\|
\mathbf{1}_{B_{2R_j}} \ast
\big\| \langle \nu_j \rangle^{s_j} \square_{\nu_{j}}
f_{j} \big\|_{\ell^2_{\nu_{j}}}^2\right\|_{ L^{1}_{x} }^{1/2}
\lesssim
R_j^{n/2}
\left\| \langle \nu_j \rangle^{s_j} \square_{\nu_{j}}
f_{j} \right\|_{\ell^2_{\nu_{j}} L^{2}_{x} },
\end{align*}
where the first inequality holds if
$L$ is suitably large.
Here, by the Plancherel theorem
\begin{align*}
\left\| \langle \nu_j \rangle^{s_j} \square_{\nu_{j}} f_{j}
\right\|_{\ell^2_{\nu_{j}} L^{2}_{x} }
&\approx
\left\| \langle \nu_j \rangle^{s_j} \kappa(\xi - \nu_{j})
\widehat{f_{j}}(\xi) \right\|_{\ell^2_{\nu_{j}} L^{2}_{\xi} }
\\&\approx
\left\| \langle \xi \rangle^{s_j} \kappa(\xi - \nu_{j})
\widehat{f_{j}}(\xi) \right\|_{\ell^2_{\nu_{j}} L^{2}_{\xi} }
\approx
\| f_{j} \|_{ H^{s_j} },
\end{align*}
where, we used that
$\sum_{\nu_j \in \mathbb{Z}^n} |\kappa (\cdot - \nu_j) |^2 \approx 1$
to have the last equivalence.
Therefore,
\begin{align}\label{resultAj}
&
\left\| \langle \nu_0-\mu \rangle^{-L/2} \langle \nu_j \rangle^{s_j} A_j (\nu_j,\nu_0) \right\|_{ \ell^2_{\nu_j} \ell^{4}_{\nu_0} \ell^{2}_{\mu}}
\lesssim
R_j^{n/2}
\left\| f_{j} \right\|_{ H^{s_j} }.
\end{align}
Substituting \eqref{resultAj} into \eqref{atodual}, we obtain
\begin{equation*}
\left\| T_{ \sigma }( f_1, f_2 ) \right\|_{(L^2,\ell^1)}
\lesssim
\left( R_{0} R_{1} R_{2} \right)^{n/2}
\| \sigma \|_{L^2_{ul}}
\prod_{j=1,2} \left\| f_{j} \right\|_{ H^{s_j} },
\end{equation*}
which completes the proof.
\end{proof}
\subsection{Proof of Theorem \ref{main-thm}}
\label{subsection-Proof-Main-1}
From Proposition \ref{main-prop},
we shall deduce Theorem \ref{main-thm}.
\begin{proof}[Proof of Theorem \ref{main-thm}]
We decompose the symbol $\sigma$
by a partition $\{ \varphi (\cdot - k ) \}_{k\in\mathbb{Z}^n}$
stated in Section \ref{secmodsp}:
\begin{align*}
\sigma (x, \xi_1, \xi_2)
&=
\sum_{ (k_0,k_1,k_2) \in (\mathbb{Z}^n)^{3}}
\varphi (D_x - k_0)
\varphi (D_{\xi_1} - k_1)
\varphi (D_{\xi_2} - k_2)
\sigma (x, \xi_1, \xi_2).
\\&=
\sum_{ \boldsymbol{k} \in (\mathbb{Z}^n)^{3}}
\square_{\boldsymbol{k}}
\sigma (x, \xi_1, \xi_2)
\end{align*}
with
\[
\square_{\boldsymbol{k}}
\sigma (x, \xi_1, \xi_2)
=
\varphi (D_x - k_0)
\varphi (D_{\xi_1} - k_1)
\varphi (D_{\xi_2} - k_2)
\sigma (x, \xi_1, \xi_2),
\]
where $\boldsymbol{k}=(k_0,k_1,k_2) \in (\mathbb{Z}^n)^3$.
Here, we observe that
\begin{equation*}
\varphi (D-k) f (x)
= e^{i x \cdot k} \varphi (D) \left[ M_{k} f \right] (x), \quad
k \in \mathbb{Z}^n,
\end{equation*}
where $M_{k} f (z) = e^{-ik \cdot z} f (z)$.
From this, we see that
\begin{align*}
\square_{\boldsymbol{k}}
\sigma (x, \xi_1, \xi_2)
=
e^{i x \cdot k_0} e^{i \xi_1 \cdot k_1} e^{i \xi_2 \cdot k_2} \,
\square_{(0,0,0)}
\left[ (M_{k_0} \otimes M_{k_1} \otimes M_{k_2} ) \sigma \right] (x,\xi_1, \xi_2).
\end{align*}
Hence,
\begin{align*}
T_{ \square_{\boldsymbol{k}} \sigma } (f_1, f_2)
= e^{i x \cdot k_0} \,
T_{\square_{(0,0,0)}
\left[ (M_{k_0} \otimes M_{k_1} \otimes M_{k_2} ) \sigma \right] }
\big(f_1(\cdot+k_1), f_2(\cdot+k_2)\big).
\end{align*}
Since
\begin{equation*}
\mathrm{supp}\, \mathcal{F} \left[ \square_{(0,0,0)}
\left[ (M_{k_0} \otimes M_{k_1} \otimes M_{k_2} ) \sigma \right] \right]
\subset [-1,1]^{3n},
\end{equation*}
we have
by Proposition \ref{main-prop}
and the translation invariance of the Sobolev space
\begin{align*}
\| T_{ \square_{\boldsymbol{k}} \sigma } \|_{H^{s_1} \times H^{s_2} \to (L^2,\ell^1)}
\lesssim
\left\| \square_{(0,0,0)}\left[ (M_{k_0} \otimes M_{k_1} \otimes M_{k_2} ) \sigma \right] \right\|_{L^2_{ul}}
=
\left\| \square_{\boldsymbol{k}} \sigma \right\|_{L^2_{ul}}.
\end{align*}
Furthermore, we have the equivalence
\begin{equation}\label{L2ulLinfty}
\| \square_{\boldsymbol{k}} \sigma \|_{L^2_{ul}}
\approx
\| \square_{\boldsymbol{k}} \sigma \|_{L^\infty}
\end{equation}
with implicit constants independ of $\boldsymbol{k}$
(see Remark \ref{remL2ulLinfty} below).
Therefore,
\begin{align*}
\| T_\sigma \|_{H^{s_1} \times H^{s_2} \to (L^2,\ell^1)}
&\leq
\sum_{ \boldsymbol{k} \in (\mathbb{Z}^{n})^3 }
\left\| T_{ \square_{\boldsymbol{k}} \sigma } \right\|_{H^{s_1} \times H^{s_2} \to (L^2,\ell^1)}
\\&\lesssim
\sum_{ \boldsymbol{k} \in (\mathbb{Z}^{n})^3 }
\| \square_{\boldsymbol{k}} \sigma \|_{L^2_{ul}}
\approx
\| \sigma \|_{M^{\infty,1}},
\end{align*}
which completes the proof of Theorem \ref{main-thm}.
\end{proof}
\begin{rem}\label{remL2ulLinfty}
The equivalence \eqref{L2ulLinfty}
was already pointed out by Boulkhemair
\cite[Appendix A.1]{boulkhemair 1995}.
However, for the reader's convenience,
we give a proof of \eqref{L2ulLinfty}.
The way of the proof here is essentialy
the same as was given in
\cite[Appendix A.1]{boulkhemair 1995}.
Let $\{ \varphi (\cdot - k ) \}_{k\in\mathbb{Z}^d}$ be
a partition on $\mathbb{R}^d$ stated Section \ref{secmodsp}.
We take a function $\phi \in \mathcal{S}(\mathbb{R}^d)$
satisfying that
$\phi = 1$ on $[-1,1]^d$.
Then,
\begin{align*}
\varphi(D-k) f(x)
&=
\phi(D-k) \varphi(D-k) f(x)
\\&=
\int_{\mathbb{R}^d} e^{i(x-y)\cdot k} \check \phi(x-y)
\, \varphi(D-k) f(y) \,dy .
\end{align*}
By the Cauchy--Schwarz inequality, we have
\begin{align*}
| \varphi(D-k) f(x) |
&\lesssim
\int_{\mathbb{R}^d} \langle x-y \rangle^{-d-1}
| \varphi(D-k) f(y) | \,dy
\\&\lesssim
\left( \int_{\mathbb{R}^d} \langle x-y \rangle^{-d-1}
| \varphi(D-k) f(y) |^2 \,dy \right)^{1/2}
=(\dag)
\end{align*}
for any $k \in \mathbb{Z}^d$ and $x\in\mathbb{R}^d$.
As in \eqref{cubicdiscretization},
we decompose the integral and have
\begin{align*}
(\dag)
&\approx
\left( \sum_{\nu \in \mathbb{Z}^d } \int_{ [-1/2,1/2)^d } \langle x-\nu \rangle^{-d-1}
| \varphi(D-k) f(y+\nu) |^2 \,dy \right)^{1/2}
\\&\leq
\| \varphi(D-k) f \|_{L^2_{ul}(\mathbb{R}^d)}
\left( \sum_{\nu \in \mathbb{Z}^d }\langle x-\nu \rangle^{-d-1} \right)^{1/2}
\approx
\| \varphi(D-k) f \|_{L^2_{ul}(\mathbb{R}^d)}
\end{align*}
for any $k \in \mathbb{Z}^d$ and $x\in\mathbb{R}^d$.
Therefore, we obtain the inequality
$\gtrsim$ in \eqref{L2ulLinfty}.
The opposite inequality is obvious,
so that we have the equivalence \eqref{L2ulLinfty}.
\end{rem}
\if0
\begin{rem}
We shall consider the boundedness
for the linear pseudo-differential operators
with the Sj\"ostrand class symbols.
In \cite[Proposition 5.1]{KMT-arxiv-2},
the following was proved:
Let $R_{0}, R_{1} \in [1, \infty)$ and
$r \in (0, \infty]$.
Suppose $\sigma$ is
a bounded continuous function on
$(\mathbb{R}^n)^{2}$ such that
$\mathrm{supp}\, \mathcal{F} \sigma \subset
B_{R_0} \times B_{R_1}$.
Then
\begin{equation*}
\left\| \sigma (X,D) \right\|_{(L^2,\ell^{r}) \to (L^2, \ell^{r})}
\lesssim
R_{0}^{n/2} R_{1}^{n/\min(2,r)}
\| \sigma \|_{ L^{2}_{ul} ((\mathbb{R}^{n})^{2})}.
\end{equation*}
From this,
following the same lines as in the proof of Theorem \ref{main-thm},
we see that
if $\sigma \in M^{\infty,1} ((\mathbb{R}^n)^2)$,
then the linear pseudo-differential operator
$\sigma(X,D)$ is bounded on $(L^2,\ell^{r})$,
$0 < r \leq \infty$.
Since $(L^2,\ell^2) = L^2$,
this boundedness is the generalization
of the result by Sj\"ostrand.
\end{rem}
\fi
\section*{Acknowledgments}
The author expresses his special thanks
to Professor Akihiko Miyachi and Professor Naohito Tomita
for a lot of fruitful suggestions and encouragement.
|
1,116,691,500,875 | arxiv | \section{Hierarchical graph construction }
The hierarchical graph constructed according to WordTree is shown in Figure \ref{office_graph_65}.
\begin{figure}[!h]
\begin{minipage}[b]{1\linewidth}
\centering
\includegraphics[width=3.4in]{images/office_graph_65.png}
\caption{A hierarchical graph of 87 nodes constructed by the office-home datasets. }
\label{office_graph_65}
\end{minipage
\hfill
\end{figure}
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
\section{Introduction}
The technique of Unsupervised Domain Adaptation (UDA)\cite{zhuang2019comprehensive} is popularly used for (1) handling the scarcity of labelled data in a target domain by leveraging rich labelled data from a related source domain; (2) reducing the cost of training downstream models by enabling models to be reused on other networks.
Recent studies in deep learning reveal that deep networks can disentangle explanatory factors of variations behind domains\cite{yosinski2014transferable,he2016deep}, thus learn more transferable features to improve UDA significantly. A major line of the existing UDA methods formally reduce domain shift and bridge the gap cross-domains by learning domain-invariant feature representations. Generally, these deep domain adaptation methods can be typically categorized into three major categories, including discrepancy-based\cite{ding2018graph}, adversarial-based \cite{cui2020gradually}, and parameter-based domain adaptation\cite{chang2019domain}.
\begin{figure}[t]
\begin{minipage}[b]{1\linewidth}
\centering
\includegraphics[width=2.8in]{images/problem.png}
\caption{Non-shared and Imbalanced UDA. Left parts are the traditional UDA with ignoring the non-shared data and the traditional PDA with removing the non-shared data automatically, respectively. Right parts are using non-shared data and semantic graph reasoning to address NI-UDA.}
\label{problem}
\end{minipage
\hfill
\end{figure}
How to effectively use big data information to transfer to the target specified small sample area to improve the prediction effect is a key issue for reliable big data transfer learning. UDA mainly realizes the inter-domain difference alignment of a specific shared classes by learning the invariant feature representation between domains. However, big data has the characteristics of large scale, diverse types and imbalances, high noise and rapid growth. Traditional UDA either ignores \cite{cao2019learning} or automatically deletes non-shared big data\cite{long2018conditional,Zhang2019}, which can no longer meet the requirements of effective supervision of data utilization and adaptation of massive complex data in the era of big data. As more and more intelligent decision-making tasks of smart devices and services are subdivided by the industry, the problem of the difference between the distribution of source big data with large-scale non-shared and imbalanced classes and the distribution of subdivision task data with specified small-and-imbalanced classes has become increasingly prominent, which has become an urgent need one of the research problem to be solved \cite{Xiao2020}.
For non-shared and imbalanced big data and small application tasks in domain adaptation, we mainly analyze the major needs of two typical industrial requirements.
(1) Catering big data analysis \cite{Xiao2020}, such as domain adaptation from food material big data to small tasks of restaurant identification. The task of automatic identification of food categories by electronic scales has main challenges: 1) transfer from the laboratory to the real environment, and 2) imbalance of food data. (2) Legal big data analysis \footnote{https://wenshu.court.gov.cn/}, such as the domain adaptation from professional big data to the Q\&A community to automatically judge prediction tasks. Main challenges: 1) Imbalanced professional fact description data. In the professional fact description data set, the distribution of various crimes and legal provisions is very uneven. In the crime distribution, the coverage rate of the top 10 most frequent crimes is as high as 78.1\%, while the coverage rate of the last 50 most frequent crimes is only about 0.5\%. 2) Non-shared imbalanced big data is adapted to some imbalanced target domains.
This paper models the domain adaptation process from big data to small tasks, and is committed to simultaneously achieving big data hierarchy knowledge generalization, non-shared and imbalanced domain adaptation under the imbalance of the source domain and the target domain, so as to provide reliable, efficient, generalized, flexible, and scalable intelligent decision-making for large-scale subdivided smart devices and services for big data analysis. The challenges faced stem from two scientific questions: (1) How to construct non-shared and imbalanced reliable domain adaptation for the non-shared and imbalanced characteristics of source big data. (2) How to transfer from non-shared imbalanced big data to small imbalanced target domain to achieve reliable domain adaptation. We aim to tackle the problem in domain adaptation process from big data to small tasks, and model it as \textbf{Non-shared-and-Imbalanced UDA (NI-UDA)}.
The challenges of NI-UDA are: (1) \textbf{Knowledge sharing from non-shared data without negative transfer effect}: How to maximize the aggregation of the structural information brought by these similar non-shared classes while minimizing the negative transfer effect\cite{cao2019learning} caused by these dissimilar non-shared classes? (2) \textbf{Sparse classes transfer}: Sparse shared classes in either domain are hard to produce effect in aligning with their partner classes in UDA\cite{Zhang2019}. Since the small size of sparse classes, the aligning effect of the sparse classes will be ignored in the UDA. For example, 10 images of "wildcat" in target domain will be ignored in UDA, which will led to huge negative false (NF) and negative positive (NP) to the sparse "cat" class for the target prediction.
In this paper, we propose \textbf{GADA} to address the challenges of NI-UDA in adversarial domain adaptation manner, which improve the alignment of adversarial domain adaptation with Hierarchy Graph Reasoning (HGR) and Source Confidence Filter (SCF) to achieve jointly knowledge sharing between non-shared classes and shared classes, and between different domains.
The previous work of TAN\cite{Xiao2020} learned from K-classes in the source domain, ignoring the negative transfer effects brought about by these non-shared classes, that is, error accumulation\cite{Chen2019} and uncertainty propagation\cite{liang2020balanced}. In order to solve the challenge of knowledge sharing from non-shared data without negative transfer effect, we propose a \textbf{Source Classifier Filter (SCF)} mechanism to filter non-shared data with a fixed confidence threshold of source classifier in feature enhance layer. For domain adaptation, SCF can filter 80\% low confidence non-shared data in the initial domain adaptation stage as well as 30\% non-shared data in the converge domain adaptation stage.
The \textbf{motivation example} of HGR to address the challenge of sparse classes transfer is: given the image of sparse "wildcat", suppose the feature representation of "wildcat" is predicted to "wild dog" and "wildcat" with ambiguity scores; then with hierarchy graph reasoning, the feature of sparse "wildcat" will be enhanced by "wildcat"’s parent and sibling nodes (rich "house-cat") or "wild dog"’s parent and sibling nodes ("house dog"), respectively; since the feature representation is more similar to the pattern of "house-cat", the feature representation is enhanced by rich "house-cat", finally initial ambiguity decision of sparse "wildcat" is definitely classified to "wildcat" class. Meanwhile, once the image of rich "house-cat"is well classified, our HGR can speed up the learning for the semantic pattern of its parent and sibling nodes (sparse "wildcat") directly.
In HGR, the prior hierarchy of big source labels is easy to construct based on WordNet\cite{miller1998wordnet} or Wikipedia or Knowledge Graph or human experts. Once the hierarchy of labels is constructed, we can leverage priori hierarchy knowledge to enhance domain adversarial aligned feature representation for sparse classes by the hierarchy graph reasoning of big source labels.
For example, the sparse target "wildcat" can align with the source "wildcat" by graph reasoning of "wildcat" and its sibling rich "house cat".
However it has a difficult to learn semantic patterns on adversarial domain adaptation for sparse classes.
Our HGR has following new characters: (1) our HGR learns semantic patterns directly by the node predictions of sparse classes in self-attention;
(2) mapping the local feature space of sparse classes and semantic graph space non-linearly; (3) adding normalization to graph reason for transferable feature enhancement.
The main contributions are: (1) A novel general framework GADA with HGR and SCF mechanism is proposed to address NI-UDA challenges in imbalanced source scenario and imbalanced target scenario. (2) The key component HGR layer could enhance domain adversarial aligned feature representation of sparse classes by effect and direct semantic pattern learning. (3) GADA consistently improves the SOTA of adversarial UDA on multiple datasets under NI-UDA setting.
\begin{figure*}[t]
\begin{minipage}[b]{1\linewidth}
\centering
\includegraphics[width=7in]{images/workflow.png}
\caption{The work-flow of GADA framework to solve two challenges of NI-UDA with Hierarchy Graph Reasoning (HGR) and Source Classifier Filter (SCF). HGR enhances domain adversarial aligned feature representation of sparse classes in hierarchy graph reasoning: (1) self-attention with hierarchy attention of prediction, (2) non-linearly mapping of two spaces with MLP, (3) normalized graph reasoning for transferable feature enhancement. SCF is to filter non-shared data with a fixed confidence threshold of source classifier for reducing negative transfer effect in HGR layer.}
\label{figure3}
\end{minipage
\hfill
\end{figure*}
\section{Related works}
\textbf{UDA.} UDA can be mainly categorized into three main groups\cite{zhuang2019comprehensive}: (1) Discrepancy-based domain adaptation leverages different criterion to minimize the domain discrepancy, like class criterion\cite{yan2017mind}, architecture criterion\cite{li2018adaptive} and statistic criterion\cite{Zhang2019,ding2018graph}. (2) Adversarial-based domain adaptation that introduce a domain discriminator and exploits the idea of adversarial learning to encourage domain confusion in minmax game \cite{jiang2020implicit,long2018conditional,cui2020gradually}. A domain classifier is trained to tell whether the sample comes from source domain or target domain. The feature extractor is trained to minimize the classification loss and maximize the domain confusion loss. (3) Parameter-based domain adaptation\cite{chang2019domain,li2018adaptive} assumes that models of the source and target domain share the same prior parameters
Some other UDA generalization paradigms are proposed\cite{liang2020we}.\par
\textbf{PDA, ODA and Label Shift.} PDA consider as setting where the label of target domain is a subspace of the source domain with the hard assumption of non-shared classes space is \textbf{unknown}. So these PDA methods are proposed to estimate the non-shared classes space and try to remove the examples in non-shared classes automatically\cite{cao2018pada,cao2019learning,liang2020balanced}.
Different from PDA, Open-Set Domain Adaptation(ODA) \cite{panareda2017open} assumes that target domain has some unknown non-shared classes. Our NI-UDA assume the non-shared classes space is \textbf{known}, but it still has big challenges on transfer learning on sparse shared classes in either domain and how to borrow knowledge from non-shared classes while without the negative transfer effect. Label shift\cite{zhao2019learning} is more challenging than traditional UDA and common for real-world application\cite{Xiao2020}.
The potential label shift, which cause conditional feature distributions alignment become more difficult\cite{tan2019generalized}.
Our NI-UDA can be classified as a special label shift where the distribution of the non-shared classes in target domain is empty.
\textbf{GNNs.} GNNs\cite{zhou2020graph} perform a message passing strategy, where each node update the representation vector by recursively aggregating vectors of neighbor nodes. Recently, GCN\cite{kipf2016semi} and its variants\cite{chien2021adaptive} have emerged in different tasks, demonstrating the ability to capture the inner structure. Liang et al.\cite{liang2018symbolic} performs reasoning over a group of symbolic nodes whose outputs explicitly represent different properties of each semantic in a prior knowledge graph. Ma et al.\cite{Ma2019} proposed GCAN to perform structure-aware alignment by aggregating data structure knowledge for unsupervised domain adaptation. Wang et al.\cite{wang2020learning} leverages the knowledges learned from multiple source domain by combing global prototypes and query samples.
\section{Graph Adversarial Domain Adaptation}
\subsection{Notations and Definitions}
In NI-UDA, the big data domain covers more classes with more training data in each of them. Let $C^K$ and $C^{K'}$ denote the big data label space and specified small shared class spaces, respectively, so we have $C^{K'}\subseteq C^K$ and $K'\ll{K}$ in general. We denote $C^{K'}$ in source domain of big data as shared classes, and $C^K-C^{K'}$ as non-shared classes in big data. Suppose $D^K_s=\{x_s,y_s\}$ and $D^{K'}_s=\{x_s,y_s\}$ represent the training data with respect to $C_s^K$ and $C_s^{K'}$ respectively, in the source domain, and $D^{K'}_t=\{x_t\}$ denotes the training data in the small target domain with respect to $C_t^{K'}$, where $x_s$ and $x_t$ represent the training data while labels $y_s$ are the corresponding categories of $x_s$. So, we also have $D^{K'}_s\subseteq{D^K_s}$. We have two imbalanced scene settings: (1) \textbf{Imbalanced Source Setting}: the data of source domain shared classes is sparsely distributed; (2) \textbf{Imbalanced Target Setting}: The target domain shared classes data is unlabelled and sparsely distributed.
The source and target domains are drawn from different probability distributions $P$ and $Q$ respectively. Beside $ P(D_s^K) \neq Q(D_t^{K'}) $, we further have $P(D_s^{K'}) \neq Q(D_t^{K'})$ in our NI-UDA, where $P(D_s^{K'})$ denotes the distribution of shared-class data in source domains. For such a situation, we want to build a robust deep learning solution to transferring knowledge from $D^K_s$ to $D^{K'}_t$, where $D^K_s\rightarrow{D^{K'}_t}$ is called \textbf{NI-UDA}.
\subsection{GADA method}
\textbf{NI-UDA} problem is addressed by equip the domain adversarial alignment with HGR for sparse classes transfer and SCF for knowledge sharing from non-shared data without negative transfer effect, respectively. The architecture of our proposed GADA is shown in Fig. 2. We train a K-classes classier $f(x)$ to learn discriminative and domain invariant features $HGR(CNN(x))$ by minimizing the overall objective:
\begin{small}
\begin{equation}
\begin{split}
\mathcal{L}(D_s^K,D_s^{K'},D_t^{K'}) = \mathcal{L}_{K'}(D_s^{K'}) \\ + {\lambda}_1 \mathcal{L}_K(D_s^K)
+ {\lambda}_2 \mathcal{L_{A}}(D_s^{K'},D_t^{K'})
\end{split} \label{1}
\end{equation}
\end{small}where ${\lambda}_1, \lambda_2$ are the trade-off factors parameters, the first item is the cross-entropy loss for shared source domain, the second item is the cross-entropy loss for non-shared data with SCF, the last item is the adversarial domain discrepancy between shared source domain and the specified small target domain.
In network configuration, similar to adversarial domain adaptation, we not only use a backbone CNN network $CNN(x)$, two classifier networks $f^1$ and $f^2$ in MDD \cite{Zhang2019}, but also use a feature enhance layer $HGR(CNN(x))$ for sparse classes. The whole GADA network configuration are denoted as:
\begin{small}
\begin{equation}
\begin{split}
p^{1}(x) = \sigma (f^1(CNN(x))), \\
p^{1+}(x) = \sigma (f^1(HGR(CNN(x)))), \\
P^{1++}(x) = p^{1+}(x) \cdot \mathbf{m}, \\
h^1(x) = argmax(p^{1++}); \\
p^{2+}(x) = \sigma (f^2(HGR(CNN(x)))), \\
P^{2++}(x) = p^{2+}(x) \cdot \mathbf{m}, \\
h^2(x) = argmax(p^{2++})
\end{split} \label{2}
\end{equation}
\end{small}where $\mathbf{m}$ is the mask of the one hot representation of $K'$ large-scale and flexible specified target shared classes with all K-classes in the big data domain, $\sigma$ is the softmax operation, and $h^1(x)$ is the argmax operation. Therefore, $p^{1}(x)$ is the K-classes prediction score with backbone feature $CNN(x)$ and the $f^1$ classifier; $p^{1+}(x)$ is the enhanced K-classes prediction score with backbone feature $CNN(x)$, feature enhance layer $HGR(CNN(x))$ for sparse classes and the $f^1$ classifier; to focus on the prediction of specified small target classes, we use the flexible mask $\mathbf{m}$ times $p^{1+}(x)$ to form our target predict score $P^{1++}(x)$; at last, we can use argmax operation $h^1(x)$ to produce the predict label. Note that, we denote $p^{1+}(x)$ and $p^{2+}(x)$ since them use different classifiers $f^1$ and $f^2$ respectively.
\textbf{Small Shared Classifier.} Similar to the source classifier loss in UDA, the classification loss of $K'$ of shared classes is defined as:
\begin{small}
\begin{equation}
\begin{split}
\mathcal{L}_{K'}(D_s^{K'})=\mathbb{E}_{D_s^{K'}} [-log(p^{1++}_{y^s}(x^s)]\label{3}
\end{split}
\end{equation}
\end{small}where $D_s^{K'}$ is the shared source dataset, $p^{1++}_{y^s}(x^s)$ is the target predict score for the example $(x^s, y^s)$, and the cross-entropy loss is used for small shared classifier with feature enhance layer $HGR(CNN(x))$ and the flexible mask $\mathbf{m}$.
\textbf{Big Source Classifier with SCF.} To address the challenge of knowledge sharing from non-shared data without negative transfer effect in NI-UDA, we propose the Source Classifier Filter (SCF) mechanism in big source classifier. The previous work of TAN learned from K-classes in the source domain, ignoring the negative transfer effects brought about by these non-shared classes, that is, error accumulation\cite{Chen2019} and uncertainty propagation\cite{liang2020balanced}. In order to solve this problem, we propose a \textbf{SCF} mechanism. In the backbone feature $CNN(x)$ learning, non-shared classes should be fully learned, thus we have the loss:
\begin{small}
\begin{equation}
\mathcal{L}_K^1(D_s^K)=\mathbb{E}_{D_s^{K}} [-log(p^{1}_{y^s}(x^s)] \label{4}
\end{equation}
\end{small}where $D_s^{K}$ is the big source dataset. However in the enhanced feature learning, hierarchical graph inference will mistakenly learn low-confidence samples, leading to error accumulation. So we propose Sourse Classifier Filter to maximize the aggregation of the structural information while minimize the negative transfer effect in HGR layer. Thus in the feature enhance layer $HGR(CNN(x))$ learning, the enhanced feature learning loss with \textbf{SCF} is defined as:
\begin{small}
\begin{equation}
\begin{split}
\mathcal{L}_K^2(D_s^K)=\mathbb{E}_{D_s^{K}} [-log(p^{1+}_{y^s}(x^s))]\mathbbm{1}(p^1_{y^s}(x^s) > \gamma)
\label{5}
\end{split}
\end{equation}
\end{small}where $\gamma$ is the confidence filter threshold, and $p^{1+}(x)$ is the predictor with enhanced feature $HGR(CNN(x))$, in compare with the origin predictor $p^{1}(x)$ with CNN feature $CNN(x)$. For domain adaptation, SCF can filter 80\% low confidence non-shared data in the initial domain adaptation stage as well as 30\% non-shared data in the converge domain adaptation stage.
The total big source classifier loss is defined as:
$\mathcal{L}_K = \mathcal{L}_K^1 + \mathcal{L}_K^2$. Note that both small shared classifier and big source classifier shared the same classifier network $f^1$, the difference and flexibility is achieved by the flexible mask $\mathbf{m}$.
\textbf{Adversarial Domain Discrepancy.} In the pioneer work\cite{ben2010theory}, the $\mathcal{H}\Delta \mathcal{H}$-divergence of two classifies $f^1$ and $f^2$ is used to measure the distribution discrepancy of two datasets. Thus the adversarial domain discrepancy in GADA is integrated as :
\begin{small}
\begin{equation}
\begin{split}
\mathcal{L_{A}}(D_s^{K'},D_t^{K'})= \sup\limits_{h^1,h^2 \in \mathcal{H}}| \mathbb{E}_{D_t^{K'}} \mathbbm{1}[h^1 \neq h^2 ] - \mathbb{E}_{D_s^{K'}} \mathbbm{1}[h^1 \neq h^2 ]| \label{6}
\end{split}
\end{equation}
\end{small}where $\mathbbm{1}$ of two datasets are the indicator functions to measure the prediction discrepancy of two classifiers $f^1$ and $f^2$ in two datasets $D_s^{K'}$ and $D_t^{K'}$, respectively. Note that both $h^1(x)$ and $h^2(x)$ are based on enhanced feature layer $HGR(CNN(x))$ and the flexible mask $\mathbf{m}$.
In state-of-the-art adversarial domain discrepancy\cite{cui2020gvb,Zhang2019}, Marginal Domain Discrepancy (MDD)\cite{Zhang2019} is integrated as:
\begin{small}
\begin{equation}
\begin{split}
\mathcal{L_{A}^{MDD}}(D_s^{K'},D_t^{K'})= \sup\limits_{f^2 \in \mathcal{F}} \mathbb{E}_{D_t^{K'}}L'(p^{2++}_{h^1(x^t)}(x^t)) \\
- {\lambda}_3 \mathbb{E}_{D_s^{K'}}L(p^{2++}_{h^1(x^s)}(x^s)), \\
L'(p^{2++}_{h^1(x^t)}(x^t)) = log[1-(p^{2++}_{h^1(x^t)}(x^t))],\\
L(p^{2++}_{h^1(x^s)}(x^s)) = -log[p^{2++}_{h^1(x^s)}(x^s)],
\end{split} \label{7}
\end{equation}
\end{small}where $L, L'$ are a disparity for each domain using two different classifiers $f^1$ and $f^2$, the pseudo label with $h^1(x)$, and ${\lambda}_3$ is designed hyper-parameter to balance the generalization and the optimization. Note that both $h^1(x)$ and $p^{2++}(x)$ are based on enhanced feature layer $HGR(CNN(x))$ and the flexible mask $\mathbf{m}$. Except these two components for feature enhance representation for spare classes and the classifier with flexible specified small target task, other components are the same as MDD for domain discrepancy.
\textbf{Hierarchy Graph Reasoning (HGR).} To address the challenge of sparse classes transfer in NI-UDA, our HGR enhances domain adversarial aligned feature representation $HGR(CNN(x))$ through hierarchy graph reasoning for sparse classes: (1) aggregate local feature to semantic node with node prediction in self-attention, (2) maps the local feature space and the semantic space non-linearly with MLP, and (3) adds normalization of the semantic nodes in graph reasoning for transferable feature enhancement.
\subsection{HGR Layer}
\textbf{Graph Construction.}
The key input of HGR layer is the flexible deep hierarchy graph for all the labels of big source domain in priori. For the priori knowledge graph, we assume that the classes have been organized into a flexible deep multi-level hierarchy. There are $K$ leaf nodes corresponding to the $K$ leaf classes, which are organized into $\mathcal{G}=(\mathcal{N},\varepsilon)$, where $\mathcal{N}$ and $\varepsilon$ denote the symbol set and edge set, respectively. In this paper, our priori classes hierarchy graph is constructed by a list of entity classes (e.g. "wildcat", "wild dog"), and its graph edge is a concept of belongings (e.g. "is kind of" or "is part of"). The \textit{K} leaf classes and their super classes consist the symbol set $\mathcal{N}$. The edges are the belongings relation between sub-classes and super-classes. For common-sense hierarchy graph we may follow the WordTree or WordNet\cite{miller1998wordnet}.
\textbf{Hierarchy Attention with Prediction.}
To learn semantic pattern directly for sparse classes, we propose a hierarchy attention mechanism of all semantic nodes with the source classifier prediction by the personalised page rank.
Suppose given a input $x$ ("wildcat"), we get the prediction score with $p^1(x)=\sigma (f^1(CNN(x))) $ where $CNN(X)$ is the CNN feature extractor. Although we get prediction score for leaf nodes for $K$-classes, we have no score for the parent nodes and root node. Therefore personalized page rank algorithm is applied to inference the score for them and enhance the score for sibling nodes. The attention score of graph node sparse "wildcat" will be increased by and its sibling nodes rich "house cat". The workflow is shown in Fig. \ref{ppr}.
\begin{figure*}[!t]
\begin{minipage}[b]{1\linewidth}
\centering
\includegraphics[width=7in]{images/ppr.png}
\caption{The workflow of input $x$(sparse "wildcat") in Personalized PageRank. To illustrate, we set K=6, N=10, where nodes in White are non-shared classes, sparse "wild-cat" and rich "house-cat" belong to the super-class of cats, and sparse "wild dog" and rich "house dog" belong to the super-class of dogs. Through the node prediction and PPR, initial low score of sparse "wildcat" is increased by the rich "house-cat".}
\label{ppr}
\end{minipage
\hfill
\end{figure*}
We then expand the $K$-classes prediction $p^1(x)$ to $\mathbb{N}$-classes prediction $p^{\mathcal{N}}(x)$ with zero padding which is better than the sum of child nodes in practice. After that, We following the classic personalized Page Rank via power iteration and the normalization operation to calculate the hierarchy attention as:
\begin{small}
\begin{equation}
a^{\mathcal{N}}(x) = PPR(p^{\mathcal{N}}(x)) + p^{\mathcal{N}}(x) \label{(8)}
\end{equation}
\end{small}where $PPR$ is the personalized page rank algorithm. See example in Fig. \ref{ppr}. Thus, the word embedding of sparse "wildcat" and its sibling nodes rich "house cat" have been attended with the hierarchy attention score. Assume $\mathcal{S}$ is the word embedding of the $\mathcal{N}$ semantic nodes where $\mathcal{S}\in \mathbb{R}^{\mathcal{N} \times D^S}$ and $D^S$ is the designed size of semantic nodes, the attended word embedding of the $\mathcal{N}$ semantic nodes given an input example $x$ is computed by $ S^{a}(x) = a^{\mathcal{N}}(x) \cdot \mathcal{S} $.
\textbf{Local-to-Graph Self-Attention.}
Meanwhile, given the input $x$ (sparse "wildcat") after all the CNN layers we get local feature tensors $X^l$, our target is to leverage transfer global graph reasoning to form global semantic feature. We aggregate all the local features to each graph node globally with query-key-value attention by the query of attended word embeddings (the word embeddings of sparse "wildcat" and its sibling nodes rich "house cat"). Formally, the feature tensor $X^l\in \mathbb{R}^{H^l\times W^l\times D^l}=CNN(x)$ after $l$-th convolution layer is the module inputs where $H^l$, $W^l$ and $D^l$ are the tensor size. This module aims to produce visual representations $H^{S} \in \mathbb{R}^{|\mathcal{N}| \times D^S}$ of all $|\mathcal{N}|$ symbolic nodes using $X^l$, where $D^S$ is desired feature dimension for each node.
Inspired by \cite{lin2013network}, we transform the local feature space to the semantic space with a two layer MLP as:
\begin{small}
\begin{equation}
X^S = MLP(X^l) \label{(9)}
\end{equation}
\end{small}where $X^S\in \mathbb{R}^{H^l\times W^l\times D^S}$ and $D^S$ is the designed size of semantic nodes. See ablation study.
\textbf{Self Attention} is defined as:
\begin{small}
\begin{equation}
H^S_n=\sum_{i \in H^l \times W^l} a_{i \rightarrow n} X^S_i, \quad a_{i \rightarrow n} = softmax(W^{aT}_nX^S_i) \label{(10)}
\end{equation}
\end{small}where $X^S_i \in \mathbb{R}^{D^S}$ is the local feature in semantic space, $W^a = \{W^{a}_n\} \in \mathbb{R}^{|\mathcal{N}| \times D^c}$ is the self-attention parameters in local feature space, and the output is $H^S=\{H^S_n\} \in \mathbb{R}^{|\mathcal{N}| \times D^S}$. To add transferability, we normalization graph node with Batch Normalization (BN):
\begin{small}
\begin{equation}
H^{s'}=Relu(BN(H^s)) \oplus S^a(x) \label{(11)}
\end{equation}
\end{small}where $S^{a}(x)$ is the \textbf{attended word embedding} after prediction of source classifier, $\oplus$ is the concatenate operation, thus the initial attended semantic feature and self-attended local feature are fused in semantic space.\par
\textbf{Semantic Reasoning.}
The visual global graph node is required to be constraint by human priori knowledge, then we evolve the global graph by semantic reasoning. For example sparse "wildcat"'s node feature is constrained by sibling node feature rich "house cat". Formally, the graph reasoning module performs graph propagation over representations of $H^{s'}$ of all symbolic nodes via the matrix multiplication from, resulting in the evolved features $H^{sr}$.
We reused graph convolutional networks \cite{liang2018symbolic} which is defied as:
\begin{small}
\begin{equation}
H^{sr}=\mathcal{Q'}^{-\frac{1}{2}} A'^g \mathcal{Q'}^{-\frac{1}{2}}
H^{s'} W^g \label{(12)}
\end{equation}
\end{small}where $W^g \in \mathbb{R}^{(D^S+D^S)\times D^S}$ is the trained matrix, $A'^g=A^g+I$ is the adjacency matrix of the graph $\mathcal{G}$ with added self-connections for considering its own representation of each node and $I$ is the identity matrix. $\mathcal{Q}'_{ii}=\sum_j A'^g_{ij}$. Using batch normalization to add transferability:
\begin{small}
\begin{equation}
H^{sr'}=Relu(BN(H^{sr})) \label{(13)}
\end{equation}
\end{small}
\textbf{Graph-to-Local Attention.}
The evolved global representation $H^{sr'}$ of semantic graph nodes (sparse "wildcat" and rich "house cat") can be used to enhance local feature (sparse "wildcat") with attention which is queried by the local features.
We then transform the semantic feature space back to the local feature space with a two layer MLP as:
\begin{small}
\begin{equation}
H^l = MLP(H^{sr'}) \label{(15)}
\end{equation}
\end{small}where $H^l\in \mathbb{R}^{\mathcal{N}\times D^l}$ and $D^l$ is the channel size of the local features.
We use the classic query-key-value attention mechanism to aggregate semantic graph nodes in local feature space to the local features as:
\begin{small}
\begin{equation}
X^{l'} = Attention(X^l, H^l, H^l) \label{(16)}
\end{equation}
\end{small}where $X^l$ is the query, $H^l$ is the key and value, the softmax operation is calculated over the ${\mathcal{N}}$ space.
\textbf{HGR Layer.}
The final output of HGR is the residual of the input local feature $X^l$ and the graph reasoned local feature $X^{l'}$:
\begin{small}
\begin{equation}
X^{l+1} = X^{l} + X^{l'} \label{(17)}
\end{equation}
\end{small}where $X^l$ is original input of HGR layer.
Finally, the HGR layer is embedded in adversarial domain adaptation as the enhanced and aligned feature representation $HGR(CNN(x))$ for sparse classes. To constrain the high level feature map with the HGR, so we add one or more HGR layers after the last CNN layer.
\begin{table*}[!t]
\renewcommand{\arraystretch}{1.0}
\small
\caption{Classification accuracy and f1 value (\%) on Meal-300 dataset in Imbalanced Source Setting }
\label{Tab-meal300-source}
\centering
\setlength{\tabcolsep}{1.1mm}{
\begin{tabular}{c| c c c c c c c c c c c c c}
\hline
& \multirow{3}{*}{Method}
& \multicolumn{2}{c}{M300(\textbf{10})} & \multicolumn{2}{c}{M300(\textbf{20})} & \multicolumn{2}{c}{M300(\textbf{50})} & \multicolumn{2}{c}{M300(\textbf{70})} & \multicolumn{2}{c}{M300(\textbf{100})} & \multirow{3}{*}{Avg-acc} & \multirow{3}{*}{Avg-f1} \\
& & \multicolumn{2}{c}{$\rightarrow$R50(60)} & \multicolumn{2}{c}{$\rightarrow$R50(60)} & \multicolumn{2}{c}{$\rightarrow$R50(60)} & \multicolumn{2}{c}{$\rightarrow$R50(60)} & \multicolumn{2}{c}{$\rightarrow$R50(60)} & \\
\cline{3-12}
& & acc & f1 & acc & f1 & acc & f1 & acc & f1 & acc & f1 & \\
\hline
\multirow{2}{*}{\shortstack{ PDA \\ $300\rightarrow$50}}
& BA$^{3}$US & - & - & - & - & 17.55 & 7.86 & 27.28 & 11.72 & 43.89 & 19.04 & 29.57 & 12.87 \\
\cline{2-14}
& BA$^{3}$US+HGR & - & - & - & -& 19.86 & 8.76 & 28.67 & 12.72 & 46.37 & 20.67 & 31.63 & 14.05 \\
\hline
\multirow{6}{*}{\shortstack{ UDA \\$ M50 \rightarrow$ \\ R50}}
& ResNet & 45.07 & 44.46 & 47.91 & 47.55 & 55.35 & 54.68 & 53.72 & 53.58 & 57.50 & 56.95 & 52.31 & 51.64 \\
& CDAN & 56.31 & 55.48 & 63.11 & 62.87 & 69.30 & 69.09 & 70.94 & 70.62 & 71.89 & 71.60 & 66.31 & 65.93 \\
& GVB-GD & 61.58 & 62.21 & 68.25 & 68.59 & 74.58 & 74.57 & 75.29 & 75.41 & 77.20 & 77.21 & 71.38 & 71.58 \\
& GCAN & 62.57 & 62.67 & 68.96 & 68.91 &73.52& 73.60 & 75.63 & 75.50 & 76.59 &76.40 & 71.45 & 71.41 \\
& MDD & 64.13 & 64.42 & 70.26 & 70.15 & 73.63 & 73.59 & 74.84 & 74.76 & 75.07 & 74.64 & 71.58 & 71.51 \\
\cline{2-14}
& MDD+HGR & 65.87 & 66.09 & 71.24 & 70.39 & 75.36 & 75.29 & 76.26 & 76.11 & 77.08 & 77.06 & 73.16 & 72.98 \\
\hline
\multirow{4}{*}{\shortstack{ NI-UDA \\ $M300 \rightarrow$ \\ $M50 \rightarrow R50$ }}
& TAN\cite{Xiao2020} & 52.88 & 52.03 & 55.29 & 54.94 & 59.00 & 57.66 & 60.50 & 59.80 & 61.48 & 61.36 & 57.83 & 57.16 \\
& GADA(MDD+SGR) & 68.86 & 68.54 & 73.56 & 73.54 & 75.84 & 75.81 & 75.77&75.56 &76.04&75.91 & 74.01& 73.87 \\
& GADA(MDD+HGR) & \bm{$71.92$} & \bm{$71.62$} & \bm{$74.88$} & \bm{$74.94$} & 76.47 & 76.14 & 77.13 & 76.94 & 78.18 & 77.95 & \bm{$75.71$} & \bm{$75.51$} \\
& GADA(GVB+HGR) & 70.33 & 70.10 & 72.77 & 72.62 & \bm{$77.40$} &\bm{$77.22$} & \bm{$77.74$} & \bm{$77.57$} &\bm{$79.48$} &\bm{$79.35$} & 75.54 & 75.37 \\
\hline
\end{tabular}}
\end{table*}
\section{Experiments}
We conducted experiments on two challenge dataset, where one is the real NI-UDA dataset and the other one is benchmark dataset in UDA.
\subsection{Experimental Settings}
\textbf{Dataset.} \textbf{Meal-300} \cite{Xiao2020} is a challenging lab-to-real dataset of food materials in NI-UDA, with two very distinct domains: \emph{Meal(M)}, 53,374 clear and undisturbed images of 300 classes with long-tail data distribution. \emph{Real(R)}, images are collected in the real electronic receiving environment,it has 50 shared classes, each with 60 images.
\textbf{Office-Home}\cite{venkateswara2017deep} consists of 4 different domains: Art(Ar) with 2,427 images, Clipart(Cl) with 4,365 images, Product(Pr) with 4,439 images, and Real-World (Rw) with 4,357 images. Each domain has 65 classes. For NI-UDA, different from \cite{cao2019learning}, the Ar domain has fewer and imbalanced images, thus we choose the top 25 classes with \textbf{the least size} the Ar domain as the shared classes to set up imbalanced scenario. Thus, we form three imbalanced source transfer tasks: $Ar \to (Cl, Pr, Rw)$ and three imbalanced target transfer tasks: $(Cl, Pr, Rw) \to Ar$.
\textbf{Evaluation Protocols.} Similar to \cite{Xiao2020}, we have three different setting scenarios: (1) Imbalanced Source Setting: The average size in source shared classes is set to five different cases, namely 10, 20, 50, 70 and 100 for each class,with five transfer tasks: $M300(\textbf{10})\to R50(60)$, $M300(\textbf{20})\to R50(60)$, $M300(\textbf{50})\to R50(60)$, $M300(\textbf{70})\to R50(60)$ and $M300(\textbf{100})\to R50(60)$. The values in parentheses indicate the average size in the shared class. (2) Imbalanced Target Setting: a) \textbf{Sparse scenario}, we adopt the \emph{Leave-Five-Out }cross-validation method to verify the sparsity of the target domain. For every shared class index $i=1,\ldots,n/5$ in target domain, set these 5 classes with index $i$ to sparse that contains only 10 samples, 60 samples in each of the other 45 non-sparse classes; then, calculate the accuracy and f1 value of 5 sparse class $i$ and 45 non-sparse classes respectively; at the end, average the test results: $Acc_{(n)}=\frac{1}{n}\sum_{1}^{n}acc_{(i)}$, $F1_{(n)}=\frac{1}{n}\sum_{1}^{n}f1_{(i)}$. b) \textbf{Imbalanced scenario}, the first 25 classes are set to sparse, each with 30 images; the last 25 classes set to non-sparse, each with 60 images. Calculate and average the results of 25 sparse classes and 25 non-sparse classes respectively.
\textbf{Baselines.} We compare the proposed GADA with state-of-the-art PDA and adversarial UDA methods, including BA$^3$US\cite{liang2020balanced}, ResNet\cite{he2016deep}, CDAN\cite{long2018conditional}, MDD\cite{Zhang2019}, GVB-GD\cite{cui2020gvb}, GCAN\cite{Ma2019}, TAN\cite{Xiao2020}, CDAN+TAN and SGR\cite{liang2018symbolic}.
\textbf{Implementation details.} We implement all deep methods in PyTorch, and the Imagenet-pre-trained ResNet-50 is used as backbone. Our HGR layer can be inserted between the last layer of CNN and the global pooling layer, which produces 2048 feature maps with $7 \times 7$ size. In the Office-Home dataset, the concept hierarchy graph with 87 symbolic nodes is generated by mapping 65 classes into WordTree, see in appendices. In the Meal-300 datasets, 300 classes are mapped to a conceptual hierarchy graph with 337 symbol nodes according to the provided hierarchical relationship\cite{Xiao2020}. Generally, $\gamma $ is set to 0.7. We adopt the SGD optimizer with learning rate 0.001, nesterov momentum 0.9, and weight decay 0.0005.
\subsection{The Analysis of Experimental Results}
\begin{table*}[!t]
\renewcommand{\arraystretch}{1.0}
\small
\caption{Classification accuracy and f1 value (\%) on Office-Home dataset, Imbalance setting (At least size).}
\label{Tab-office-home}
\centering
\setlength{\tabcolsep}{0.7mm}{
\begin{tabular}{c| c c c c c c c | c c c c c c c c c }
\hline
& \multirow{3}{*}{Method} & \multicolumn{6}{c}{Imbalanced Source Tasks} & \multicolumn{6}{c}{Imbalanced Target Tasks} &
&&
\\
\cline{3-14}
& & \multicolumn{2}{c}{\textbf{Ar}$\rightarrow$\textbf{Cl}} & \multicolumn{2}{c}{\textbf{Ar}$\rightarrow$\textbf{Pr}} & \multicolumn{2}{c}{\textbf{Ar}$\rightarrow$\textbf{Rw}} & \multicolumn{2}{c}{\textbf{Cl}$\rightarrow$\textbf{Ar}} & \multicolumn{2}{c}{\textbf{Pr}$\rightarrow$\textbf{Ar}} & \multicolumn{2}{c}{\textbf{Rw}$\rightarrow$\textbf{Ar}} & \multirow{2}{*}{Avg-acc} & \multirow{2}{*}{Avg-f1} \\
\cline{3-14}
& & acc & f1 & acc & f1 & acc & f1 & acc & f1 & acc & f1 & acc & f1 \\
\hline
\multirow{2}{*}{\shortstack{ PDA \\ $65\rightarrow$25}}
& BA$^3$US & 37.39 & 17.36 & 67.43 & 30.01 & 66.20 & 28.23 & 53.62 & 29.15 & 59.00 & 27.17 & 63.14& 32.17 & 57.79 & 27.34 \\
\cline{2-16}
& BA$^{3}$US+HGR & 41.65 & 18.6 & 74.3 & 35.98 & 70.6 & 30.44 & 56.41 & 29.31 & 59.51 & 29.42 & 63.49 & 31.62 & 60.99 & 29.22 \\
\hline
\multirow{6}{*}{\shortstack{ UDA\\ $S25 \rightarrow$ \\ T25}}
& ResNet & 46.89 & 44.83 & 70.62 & 68.86& 74.80 & 73.18 &49.27 & 46.95 & 55.48 & 54.37 & 66.66 & 66.89& 60.62 & 59.18 \\
& CDAN & 58.40 & 57.37 & 81.05 & 79.39& 82.4 & 81.19& 57.55& 57.37& 63.14& 62.96& 75.15& 74.73& 69.61& 68.83\\
& GVB-GD & 61.20 & 60.54 & 80.11 & 78.64 & 85.06 & 84.24 & 63.14 & 62.74 & 67.28 & 51.34 & 78.88 & 77.57 & 72.61 & 69.17 \\
& GCAN & 62.42 &61.28 & 82.70 & 81.18 & 84.66 & 83.33 & 63.14 & 62.11 & 69.97 & 69.77 & 77.01 & 75.89 & 73.31 & 72.26 \\
& MDD & 62.69 & 61.64 &83.37 & 81.52 & 85.06 & 83.73 & 62.28 & 61.95 & 67.26 & 66.32 & 77.13 & 76.95 & 72.96 & 72.01
\\
\cline{2-16}
&MDD+HGR & 63.88 & 63.02 & 85.11 & 83.64 & 85.53 & 84.34 & 63.56 & 64.21 & 71.22 & 70.86 &77.65 & 77.32 & 74.49 & 73.89 \\
\hline
\multirow{6}{*}{\shortstack{ NI-UDA\\ $S65 \rightarrow$\\ $S25 \rightarrow$ \\ T25}}
& TAN & 52.86 & 50.90 & 81.05 & 78.55 & 82.20 & 80.75 & 59.62 & 59.36 & 64.80 & 63.83 & 73.29 & 72.93 & 68.97 & 67.72 \\
& CDAN+TAN & 59.25 & 58.63 & 83.64 & 81.61 & 83.46 & 82.27 & 59.83 & 60.04 & 65.04 & 64.85 & 74.12 & 73.33 & 70.89 & 70.12 \\
\cline{2-16}
& GADA(CDAN+SGR) & 59.92 & 59.71 & 83.47& 81.44& 83.8 &83.68& 59.00& 58.76& 65.42& 64.79& 76.31& 75.05& 71.32& 70.57 \\
& GADA(CDAN+HGR) & 61.57& 61.24& 84.45& 83.76& 84.76& 84.12& 60.9& 59.17 &65.76 &65.45& 77.01& 75.98& 72.40& 71.62 \\
& GADA(MDD+SGR) & 63.09 & 62.81 & 85.04 & 83.26 & 85.13 & 84.01 & 62.11 & 62.06 & 66.53 & 65.30 & 77.84& 77.46 & 73.29 &72.48 \\
& GADA(MDD+HGR) & \bm{$66.87$} & \bm{$66.34$} & \bm{$87.97$} & \bm{$86.60$} & \bm{$87.53$} & \bm{$86.30$} & \bm{$66.04$} & \bm{$65.06$} & \bm{$73.70$} & \bm{$73.24$} & 78.46 & 78.11 & \bm{$76.76$} & \bm{$75.94$}
\\
& GADA(GVB+HGR)& 65.16 & 64.26 & 85.22 & 83.23 & 85.73 & 84.72 & 64.18 & 63.39 & 69.59 & 68.60 & \bm{$79.08$} & \bm{$78.25$} & 74.82 & 73.74 \\
\hline
\end{tabular}}
\end{table*}
\begin{table*}[!t]
\renewcommand{\arraystretch}{1.0}
\small
\caption{Classification accuracy and f1 value (\%) on Office-Home datasets, Long-tail distribution( in alphabetical order).}
\label{Tab-office-home-alph}
\centering
\setlength{\tabcolsep}{1.2mm}{
\begin{tabular}{ c c c c c c c | c c c c c c c c c }
\hline
\multirow{2}{*}{Method} & \multicolumn{2}{c}{\textbf{Ar}$\rightarrow$\textbf{Cl}} & \multicolumn{2}{c}{\textbf{Ar}$\rightarrow$\textbf{Pr}} & \multicolumn{2}{c}{\textbf{Ar}$\rightarrow$\textbf{Rw}} & \multicolumn{2}{c}{\textbf{Cl}$\rightarrow$\textbf{Ar}} & \multicolumn{2}{c}{\textbf{Pr}$\rightarrow$\textbf{Ar}} & \multicolumn{2}{c}{\textbf{Rw}$\rightarrow$\textbf{Ar}} & \multirow{2}{*}{Avg-acc} & \multirow{2}{*}{Avg-f1} \\
\cline{2-13}
& acc & f1 & acc & f1 & acc & f1 & acc & f1 & acc & f1 & acc & f1 \\
\hline
BA3$^3$US\cite{liang2020balanced} & 60.62 & 26.50 & 83.16 & 45.73& 88.39 & 54.71 & 71.75& 34.84&75.45&35.21&79.25 &39.21 & 76.43 & 39.36 \\
\hline
MDD & 71.16 & 70.18 & 90.51 & 90.12 & 91.32 & 90.37& 79.4 &74.22& 79.49& 75.74& 85.19& 82.52 & 82.84 & 80.52 \\
\hline
GADA(MDD+HGR)& 73.11 & 72.48 & 92.13 & 91.46 & 93.1 & 91.79 & 81.81 & 76.28 & 82.78 & 78.23 & 86.42 & 83.67 &84.89 & 82.31 \\
\hline
\end{tabular}}
\end{table*}
\begin{table*}[!t]
\renewcommand{\arraystretch}{1.0}
\small
\caption{Accuracy (\%) on Office-Home for unsupervised domain adaptation(S65 $\to$ T65).}
\label{Tab-office-home-uda65}
\centering
\setlength{\tabcolsep}{0.6mm}{
\begin{tabular}{ c c c c c c c | c c c c c c c }
\hline
\multirow{1}{*}{Method} & \multicolumn{1}{c}{\textbf{Ar}$\rightarrow$\textbf{Cl}} & \multicolumn{1}{c}{\textbf{Ar}$\rightarrow$\textbf{Pr}} & \multicolumn{1}{c}{\textbf{Ar}$\rightarrow$\textbf{Rw}} & \multicolumn{1}{c}{\textbf{Cl}$\rightarrow$\textbf{Ar}} & \multicolumn{1}{c}{\textbf{Cl}$\rightarrow$\textbf{Pr}} & \multicolumn{1}{c}{\textbf{Cl}$\rightarrow$\textbf{Rw}} &
\multicolumn{1}{c}{\textbf{Pr}$\rightarrow$\textbf{Ar}} &
\multicolumn{1}{c}{\textbf{Pr}$\rightarrow$\textbf{Cl}} &
\multicolumn{1}{c}{\textbf{Pr}$\rightarrow$\textbf{Rw}} &
\multicolumn{1}{c}{\textbf{Rw}$\rightarrow$\textbf{Ar}} &
\multicolumn{1}{c}{\textbf{Rw}$\rightarrow$\textbf{Cl}} &
\multicolumn{1}{c}{\textbf{Rw}$\rightarrow$\textbf{Pr}} &
\multirow{1}{*}{Avg-acc} \\
\hline
MDD\cite{Zhang2019} & 54.9 & 73.7 & 77.8 & 60.0 & 71.4 & 71.8 & 61.2 & 53.6 & 78.1 & 72.5 & 60.2 & 82.3 & 68.1 \\
\hline
GADA(MDD+HGR)& 56.4 & 74.8 & 79.2 & 61.5 & 72.8 & 73.1 & 62.5 & 54.8 & 79.2 & 73.7 & 61.4 & 83.6 & 69.4 \\
\hline
\end{tabular}}
\end{table*}
\begin{table}[!t]
\renewcommand{\arraystretch}{1.0}
\small
\caption{Classification f1 value (\%) on Meal-300 dataset in Imbalanced Target Setting. (A(60): 45 non-sparse classes, A(10): 5 sparse classes; B(60): 25 non-sparse classes, B(30): 25 sparse classes; C(10): all sparse)}
\label{Tab-meal300-target}
\centering
\setlength{\tabcolsep}{0.7mm}{
\begin{tabular}{c|ccc|cc|c}
\hline
& \multirow{2}{*}{Method} & \multicolumn{1}{c}{A(60) } & \multicolumn{1}{c}{A(10)} & \multicolumn{1}{c}{B(60)} & \multicolumn{1}{c}{B(30)} & \multicolumn{1}{c}{C(10)} \\
\cline{3-7}
& & f1 & f1 & f1 & f1 & f1 \\
\hline
\multirow{2}{*}{\shortstack{ PDA\\ 300$\rightarrow$50}}
& BA3US & 40.91& 15.09 & 25.16 & 25.92 & 21.23 \\
\cline{2-7}
& BA$^{3}$US+HGR & 41.22 & 19.48 & 25.50 & 26.02 & 23.17 \\
\hline
\multirow{6}{*}{\shortstack{ UDA\\ $M50 \rightarrow$ \\ R50 }}
& ResNet & 52.73 & 12.61 & 30.20 & 37.14 & 59.07 \\
& CDAN & 55.52 & 12.19 & 31.57 & 37.13 & 71.28 \\
& GVB-GD & 69.13 & 30.04 & 43.98 & 47.64 & 75.18 \\
& MDD & 65.42 & 21.96 & 40.13 & 44.93 & 72.16 \\
\cline{2-7}
& MDD+HGR & 70.32 & 28.72 & 41.65 & 45.36 & 73.11 \\
& GVB+HGR & 69.82 & 30.74 & 44.25 & 48.36 & 75.89 \\
\hline
\multirow{4}{*}{\shortstack{ NI-UDA\\ M300 \\ $\rightarrow$ R50 }}
& TAN & 53.41 & 13.60 & 30.73 & 38.22 & 60.77 \\
\cline{2-7}
& GADA(MDD+SGR) & 67.35 &22.35& 40.29 & 46.01 & 73.80 \\
& GADA(MDD+HGR) & \bm{$72.10$} & 30.84 & 43.31 & 47.23 & 76.08 \\
& GADA(GVB+HGR) & 70.67 & \bm{$32.89$} & \bm{$44.75$} & \bm{$50.90$} & \bm{$77.21$} \\
\hline
\end{tabular}}
\end{table}
\textbf{Meal-300.} Table \ref{Tab-meal300-source} and \ref{Tab-meal300-target} report the detailed comparison of two imbalance settings in NI-UDA on Meal-300 datasets.
(1) \textbf{Imbalanced source scenario}, Our GADA(HGR) improves f1 of the most adversarial UDA methods on average by \textbf{4\%}(MDD) and \textbf{3.79\%}(GVB-GD). Specifically, in the case of high sparsity factor, e.g. $Meal300(10) \rightarrow Real50(60)$, GADA(HGR) can greatly improve transfer performance, \textbf{7.19\% }is improved on MDD, and \textbf{7.89\%} is improved on GVB-GD. In near-balance case, e.g. $Meal300(100) \rightarrow Real50(60)$, GADA(HGR) can still get significantly improvements such as \textbf{3.31\%} and \textbf{2.95\%} points improvement respectively on MDD and GVB-GD. It shows our GADA framework can make full use of non-shared
classes knowledge through hierarchical graph reasoning to solve two challenges of NI-UDA in imbalanced source scenario.
(2) \textbf{Imbalanced target scenario}, GADA(MDD+HGR) improves 1) the f1 value of \textbf{6.68\%} and \textbf{8.88\%} at 45 non-sparse classes and 5 sparse classes, respectively; 2) the f1 value of \textbf{3.18\%} and \textbf{2.3\%} at 25 non-sparse and 25-sparse classes, respectively; 3) \textbf{3.92\% }f1 in the fully sparse target domain, where all the target classes are sparse with 10 samples per class. GADA(GVB-GD+HGR) also has consistency improvement on these three cases, see Table \ref{Tab-meal300-target}. The results demonstrate that GADA can also solve two challenges of NI-UDA in imbalanced target scenario well.
\textbf{Office-Home.} Table \ref{Tab-office-home} shows the comparison results for 3 imbalanced source tasks and 3 imbalanced target tasks under the imbalance setting of Office-home datasets. On average, our GADA(HGR) improves f1 of the most adversarial UDA methods by \textbf{2.79}\%(CDAN) and \textbf{3.93\%}(MDD) and \textbf{4.57\%}(GVB-GD). The experimental results show that our GADA framework can well solve two challenges of NI-UDA in both imbalanced source and target scenarios.
To compare PDA methods, we conduct the experiments on origin PDA setting ($S65\to T25$)\cite{liang2020balanced}, where first 25 classes in alphabetical order are included as shared classes, which is a long-tailed distribution. The comparison results are shown in Table \ref{Tab-office-home-alph}. The BA$^3$US of Imbalanced setting is degraded in compare with PDA setting, it shows Office-Home under imbalanced setting is more challenging and consistent with the NI-UDA problem. GADA still has consistent improvement.
To compare UDA methods, we also conduct the experiments on origin UDA ($S65\to T65$)\cite{Zhang2019}. Table \ref{Tab-office-home-uda65} shows the results of 65 classes in near-balanced UDA settings. GADA still achieves a competitive improvement, which proves the graph reasoning ability of HGR.
In summary, GADA in NI-UDA has achieved a significantly improvement on the Office-Home dataset with a little semantic relatedness. However, GADA's improvement will become large margin on the Meal-300 datasets which has high semantic relatedness. GADA(HGR) shows its superiority power in the aggregation of non-shared knowledge from big data.
\subsection{Ablation Study}
\begin{table}[!t]
\renewcommand{\arraystretch}{1.1}
\small
\caption{GADA Ablation Study}
\label{Tab-gada-ablation}
\centering
\begin{tabular}{c| c c c c c c }
\hline
& \multirow{2}{*}{Methods} & \multicolumn{2}{c}{M300(10) $\rightarrow$ R50} & \multicolumn{2}{c}{Ar $\rightarrow$ Cl} \\
\cline{3-6}
& & acc & f1 & acc & f1 \\
\hline
UDA & MDD & 64.13 & 64.42 & 62.69 & 61.64\\
\hline
\multirow{8}{*}{\shortstack{ NI-UDA \\ GADA}}
& Ours (Baseline) & 68.86 & 68.54 & 63.09 & 62.81 \\
& Ours (w/ BN) & 69.79 & 69.46 & 64.27 & 63.83\\
& Ours (w/ MLP) & 70.43 & 70.17 & 64.80 & 64.54 \\
& Ours (w/ HAP) & 70.94 & 70.50 & 65.45 & 65.16\\
\cline{2-6}
& Ours (w/ HGR)& 71.92 & 71.62 & 66.87 & 66.34 \\
\cline{2-6}
&$L_K^1 \qquad L_K^2$ & \\
\cline{2-6}
& $\checkmark \qquad \quad $ & 71.23 & 71.05 & 66.13 & 65.66 \\
& $\checkmark \qquad \checkmark $ & 71.92 & 71.62 & 66.87 & 66.34 \\
\hline
\end{tabular}
\end{table}
In order to exploit the components of HGR and GADA contribute to the performance, we perform ablation studies on imbalanced task of $Meal300(10) \rightarrow Real50(60), Art \to Clipart $. SGR\cite{liang2018symbolic} is symbolic graph reasoning with feature enhance for classification task and semantic segmentation task. We also propose this general graph reasoning SGR for classification task as a baseline hierarchy graph reasoning component of GADA in NI-UDA. So, we regard the "GADA(MDD+SGR)" model as the baseline. We study the influence of HAP(Hierarchy Attention with Prediction), MLP(Multilayer Perception Mapping), and BN(Batch Normalization) of HGR as show in Table \ref{Tab-gada-ablation}.
In Meal-300 datasets, HAP can promote the baseline by around 1.96\% f1. MLP achieves around 1.63\% f1 improvements. BN apparently boosts the f1 by around 0.92\% from baseline. GADA(MDD+HGR) improves baseline by around 3.08\%. In Office-Home dataset, there is still consistency improvement of imbalanced source task. In compare with baseline GADA(MDD+SGR) in NI-UDA, our method GADA(MDD+HGR) has significantly improvement which shows HGR can borrow the knowledge from non-shared big data sufficiently.
To study the influence of the source domain confidence filtering mechanism of GADA, $L_K^1$ and $L_K^2$ of GADA ablation analysis are demonstrated in last three lines of Table \ref{Tab-gada-ablation}. It shows that the Source Classifier Filter (SCF) mechanism can alleviate the negative transfer effect caused by non-shared classes.
In compare with MDD in UDA, our method GADA(MDD+HGR) has superiority performance in NI-UDA which show non-shared data is useful for domain adaptation and our GADA is sufficient to use these non-shared data while solved the negative transfer challenge of NI-UDA.
\subsection{Parameter sensitivity analysis}
\textbf{Hyper parameters of $\lambda_1,\lambda_2$.}
We investigate a broader scope of our algorithm GADA for source classier and domain discriminator by varying parameters $\lambda_1,\lambda_2$. When $\lambda_1$ is 0, NI-UDA settings will be converted to UDA settings. Generally, the $ (\lambda_1,\lambda_2)$ of GADA(MDD+HGR) on the Office-home and Meal300 datasets are (2, 4) and (2, 3.2) respectively.
\begin{table}[h]
\renewcommand{\arraystretch}{1.2}
\small
\caption{The hyper-parameters of ${\lambda}_1$ ,$ {\lambda}_2$ in the overall loss of GADA}
\label{Tab-hyper-lambda}
\centering
\setlength{\tabcolsep}{2mm}{
\begin{tabular}{ c c c c c c }
\hline
\multirow{2}{*}{$ {\lambda}_1$ } & \multirow{2}{*}{$ {\lambda}_2$ } & \multicolumn{2}{c}{M300(10) $\to$ R50} & \multicolumn{2}{c}{Ar $\to$ Cl} \\
\cline{3-6}
& & acc & f1 & acc & f1 \\
\hline
0 & 1 & 65.87 & 66.09 &63.88 & 63.02 \\
2 & 4 & 71.92 & 71.62 & 66.54 & 65.89\\
2 & 3.2 & 71.34 & 70.92 & 66.87 & 66.34 \\
\hline
\end{tabular}}
\end{table}
\textbf{The dimension of $D^S$.}
we transform the local feature space to the semantic space with a two layer MLP as :\begin{small}
\begin{equation}
X^S = MLP(X^l)
\end{equation}
\end{small}where $X^S\in \mathbb{R}^{H^l\times W^l\times D^S}$ and $D^S$ is the designed size of semantic nodes, i.e., 2048 $\underrightarrow{1 \times1 \, conv}$ 512 $\underrightarrow{1 \times1 \, conv}$ 128. We also have semantic nodes $S^a$ and visual representations $H^{s}$ belongs to $\mathbb{R}^{|\mathcal{N}| \times D^s}$. We study the influence of different sizes of $D^S$, like 256-d and 128-d, in transfer task of $Meal300(10) \to Real(60)$ on GADA(MDD+HGR). \par
As shown in Table \ref{Tab-hyper}, compared with 128-d, 256-d get a slight improvement and more parameters. In fact, we set the dimension of $D^S$ to 128, in order to reduce the training parameters.
\begin{table}[h]
\renewcommand{\arraystretch}{1.2}
\small
\caption{Parameter sensitivity analysis of $D^S$ and num of HGR layer}
\label{Tab-hyper}
\centering
\setlength{\tabcolsep}{1mm}{
\begin{tabular}{c | c c c c }
\hline
\multirow{2}{*}{\shortstack{num of \\ HGR layer}} & \multirow{2}{*}{\shortstack{$X^S\in$ \\ $ \mathbb{R}^{H^l\times W^l\times D^S}$}} & \multirow{2}{*}{\shortstack{ $H^S\in $ \\ $\mathbb{R}^{\mathcal{N} \times D^S}$}} & \multicolumn{2}{c}{M300(10) $\to$ R50} \\
\cline{4-5}
& & & acc & f1 \\
\hline
1 & 128 & 128 & 71.92 & 71.62 \\
1 & 256 & 256 & 72.19 & 71.80 \\
2 & 128 & 128 & 70.73 & 70.59 \\
\hline
\end{tabular}}
\end{table}
\textbf{Nums of HGR layer.}
We also studied the influence of different layers of HGR layer. Table \ref{Tab-hyper} show the result: one layer of HGR has better effect than two layers. \par
\subsection{Visualization}
We utilize t-SNE to visualize the feature distributions of Imbalanced tasks Ar $\to$ Cl on Office-Home datasets(10 classes). As shown in Fig. \ref{fig:tsne}, the imbalanced scenarios of either domain will cause NP, NF problems, which manifests as some samples drifting near the center point of domain invariant clustering. Obviously, GADA model has a better clustering effect. More visualization can be seen in the appendix.
\subsection{Discussion of GADA}
For PDA methods, including SOTAs of ETN\cite{cao2019learning} and BA$^3$US\cite{liang2020balanced} do not analyze the f1 value which is important for our imbalanced setting. From the experimental results, we can draw conclusions: Under the imbalanced setting of PDA, the f1 value will collapse and decrease,and cause a lot of NF, NP problems.
In general, Our GADA has achieved a huge improvement in NI-UDA. Surprisingly, HGR can consistently improve the SOTAs of PDA and UDA. Because GADA can leverage priori hierarchy knowledge to enhance domain adversarial aligned and class structure aligned feature representation. In addition, most UDA methods can be easily extended to NI-UDA. GADA with HGR can consistently improve the SOTAs models of PDA, UDA and NI-UDA, which can be regarded as a general adversarial alignment paradigm. In the future work, we will adopt data augmentation, consistency training, self-training, etc. to further alleviate the challenges of NI-UDA.
\begin{figure}[t]
\subfigure[ MDD] {
\begin{minipage}[t]{0.48\linewidth}
\centering
\includegraphics[width=1.72in]{images/tsne-mdd-ac.png}
\end{minipage}}
\subfigure[ GADA(MDD+HGR)]{
\begin{minipage}[t]{0.48\linewidth}
\centering
\includegraphics[width=2.22in]{images/tsne-gada-ac.png}
\end{minipage}}
\caption{t-SNE visualization of imbalanced tasks Ar $\to$ Cl. }
\label{fig:tsne}
\end{figure}
\section{Conclusion}
In this paper, we present a novel approach (GADA model) for the non-shared-and-imbalanced domain adaptation. The scope of this approach is the assumption of existing sibling relationship between shared classes and non-shared classes. For knowledge sharing from non-shared data without negative transfer effect challenge, our SCF could filter non-shared data with high-confidence non-shared data in feature enhance layer. For sparse classes transfer challenge, HGR could enhance the sibling aligned feature representation for adversarial domain adaptation. We also apply GADA to existing methods and achieve remarkable improvement over original counterparts. |
1,116,691,500,876 | arxiv | \section{CONCLUSIONS}
\label{sec:conclusion}
We presented a novel incremental semiparametric modeling approach for inverse dynamics learning, joining together the advantages of parametric modeling derived from rigid body dynamics equations and of nonparametric Machine Learning methods.
A distinctive trait of the proposed approach lies in its incremental nature, encompassing both the parametric and nonparametric parts and allowing for the prioritized update of both the identified base inertial parameters and the nonparametric weights.
Such feature is key to enabling robotic systems to adapt to mutable conditions of the environment and of their own mechanical properties throughout extended periods.
We validated our approach on the iCub humanoid robot, by analyzing the performances of a semiparametric inverse dynamics model of its right arm, comparing them with the ones obtained by state of the art fully nonparametric and parametric approaches.
\section*{ACKNOWLEDGMENT}
This paper was supported by the FP7 EU projects CoDyCo (No. 600716 ICT-2011.2.1 - Cognitive Systems and Robotics), Koroibot (No. 611909 ICT-2013.2.1 - Cognitive Systems and Robotics), WYSIWYD (No. 612139 ICT-2013.2.1 - Robotics, Cognitive Systems \& Smart Spaces, Symbiotic Interaction), and Xperience (No. 270273 ICT-2009.2.1 - Cognitive Systems and Robotics).
\bibliographystyle{IEEEtran}
\section{INTRODUCTION}
In order to control a robot a model describing the relation between the actuator inputs,
the interactions with the world and bodies accelerations is required. This model is called the \emph{dynamics} model of the robot.
A dynamics model can be obtained from first principles in mechanics, using the techniques of rigid body dynamics (RBD) \cite{reference/robo/FeatherstoneO08},
resulting in a \emph{parametric model} in which the values of physically meaningful parameters
must be provided to complete the fixed structure of the model.
Alternatively, the dynamical model can be obtained from experimental data using Machine Learning techniques, resulting in a \emph{nonparametric model}.
Traditional dynamics parametric methods are based on several assumptions, such as rigidity of links or that friction has
a simple analytical form, which may not be accurate in real systems.
On the other hand, nonparametric methods based on algorithms such as Kernel Ridge Regression (KRR) \cite{hoerl1970ridgeregression,SaundersGV98,cristianini2000introduction}, Kernel Regularized Least Squares (KRLS) \cite{rifkin2003regularized} or Gaussian Processes \cite{rasmussen2006}
can model dynamics by extrapolating the input-output relationship directly from the available data\footnote{Note that KRR and KRLS have a very similar formulation, and that these are also equivalent to the techniques derived from Gaussian Processes, as explained for instance in Chapter 6 of \cite{cristianini2000introduction}.}.
If a suitable kernel function is chosen, then the nonparametric model is a universal approximator
which can account for the dynamics effects which are not considered by the parametric model.
Still, nonparametric models have no prior knowledge about the target function to be approximated. Therefore, they need a sufficient amount of training examples in order to produce accurate predictions on the entire input space.
If the learning phase has been performed offline, both approaches are sensitive to the variation of the mechanical properties over long time spans, which are mainly caused by temperature shifts and wear. Even the inertial parameters can change over time. For example if the robot grasps a heavy object, the resulting change in dynamics can be described by a change of the inertial parameters of the hand.
A solution to this problem is to address the variations of the identified system properties by learning \emph{incrementally}, continuously
updating the model as long as new data becomes available.
In this paper we propose a novel technique that joins parametric and nonparametric model learning in an incremental fashion.
\begin{table}[t]
\caption{Summary of related works on semiparametric or incremental robot dynamics learning.}
\begin{center}
\resizebox{0.48\textwidth}{!}
\begin{tabular}{| c | c c |}
\hline
\rowcolor[gray]{.9}
\textbf{Author, Year} & \textbf{Parametric} & \textbf{Nonparametric} \\
[0.5ex] \hline
Nguyen-Tuong, 2010 \cite{Nguyen-TuongP10} & Batch & Batch \\
\rowcolor[gray]{.9}
Gijsberts, 2011 \cite{GijsbertsM11} & - & Incremental \\
Tingfan Wu, 2012 \cite{conf/iros/wu12} & Batch & Batch \\
\rowcolor[gray]{.9}
De La Cruz, 2012 \cite{conf/ifac/delacruz12} & CAD$^*$ & Incremental \\
Camoriano, 2015 & Incremental & Incremental \\
\hline
\end{tabular}
}
\end{center}
\label{table:soa}
$^*$ In \cite{conf/ifac/delacruz12} the parametric part is used only for initializing the nonparametric model.
\end{table}
Classical methods for physics-based dynamics modeling can be found in \cite{reference/robo/FeatherstoneO08}.
These methods require to identify the mechanical parameters of the rigid
bodies composing the robot \cite{conf/humanoids/Yamane11,conf/humanoids/TraversaroPMNN13,conf/humanoids/OgawaVO14,hollerbach2008model},
which can then be employed in model-based control and state estimation schemes.
In \cite{Nguyen-TuongP10} the authors present a learning technique which
combines prior knowledge about the physical structure of the mechanical
system and learning from available data with Gaussian Process Regression (GPR) \cite{rasmussen2006}.
A similar approach is presented in \cite{conf/iros/wu12}. Both techniques
require an offline training phase and are not incremental, limiting them to scenarios in which the properties of the system do not change significantly over time.
In \cite{conf/ifac/delacruz12} an incremental semiparametric robot dynamics
learning scheme based on Locally Weighted Projection Regression (LWPR) \cite{conf/icml/VijayakumarS00} is presented, that is initialized using a linearized parametric model. However, this approach uses a fixed parametric model,
that is not updated as new data becomes available. Moreover, LWPR has been shown to underperform with respect to other methods (e.g. \cite{GijsbertsM11}).
In \cite{GijsbertsM11}, a fully nonparametric incremental approach for inverse dynamics learning with constant
update complexity is presented, based on kernel methods \cite{schlkopf2002learning} (in particular KRR) and random features \cite{RahimiR07}. The incremental nature of this
approach allows for adaptation to changing conditions in time. The authors also show that the proposed algorithm outperforms other methods such as LWPR, GPR and Local Gaussian Processes (LGP) \cite{lgpr}, both in terms of accuracy and prediction time. Nevertheless, the fully nonparametric nature of this approach undermines the interpretability of the inverse dynamics model.
In this work we propose a method that is incremental with fixed update complexity (as \cite{GijsbertsM11}) and
semiparametric (as \cite{Nguyen-TuongP10} and \cite{conf/iros/wu12}). The fixed update complexity and prediction time are key properties of our method, enabling real-time performances. Both the parametric and nonparametric parts can be updated,
as opposed to \cite{conf/ifac/delacruz12} in which only the nonparametric part is. A comparison between the existing literature and our incremental
method is reported in Table \ref{table:soa}.
We validate the proposed method with experiments performed on an arm of the iCub humanoid robot \cite{Metta2010}.
\begin{figure}[ht!]
\vspace{3mm}
\centering
\includegraphics[width=0.95\linewidth]{figures/icubValidation.jpg}
\caption{iCub learning its right arm dynamics.\label{overflow}}
\end{figure}
The article is organized as follows.
Section \ref{sec:background} introduces the existing techniques for parametric and nonparametric robot dynamics learning.
In Section \ref{sec:semiparametric}, a complete description of the proposed semiparametric incremental learning technique is introduced.
Section \ref{sec:experiments} presents the validation of our approach on the iCub humanoid robotic platform.
Finally, Section \ref{sec:conclusion} summarizes the content of our work.
\section{BACKGROUND}
\label{sec:background}
\subsection{Notation}
The following notation is used throughout the paper.
\begin{itemize}
\item The set of real numbers is denoted by $\mathbb{R}$. Let $\mathbf{u}$ and $\mathbf{v}$ be two $n$-dimensional column vectors of real numbers (unless specified otherwise), i.e. $\mathbf{u},\mathbf{v} \in \mathbb{R}^n$,
their inner product is denoted as $\mathbf{u}^\top \mathbf{v}$, with ``$\top$'' the transpose operator.
\item The Frobenius norm of either a vector or a matrix of real numbers is denoted by $\| \cdot \|$.
\item $I_n \in \mathbb{R}^{n \times n}$ denotes the identity matrix of dimension~$n$;
$0_n \in \mathbb{R}^n$ denotes the zero column vector of dimension~$n$; $0_{n \times m} \in \mathbb{R}^{n \times m}$ denotes the zero matrix of dimension~$n \times m$.
\end{itemize}
\subsection{Parametric Models of Robot Dynamics}
\label{sec:parmod}
Robot dynamics parametric models are used to represent the relation
connecting the geometric and inertial parameters with some dynamic quantities
that depend uniquely on the robot model.
A typical example is obtained by writing the robot inverse dynamics equation in linear form with respect to the
robot inertial parameters $\boldsymbol\pi$:
\begin{equation}
\boldsymbol\tau = M(\mathbf{q})\mathbf{\ddot{q}} + C(\mathbf{q},\mathbf{\dot{q}})\mathbf{\dot{q}} + g(\mathbf{q}) = \ensuremath{\Phi(\mathbf{x})} \boldsymbol\pi,
\end{equation}
where:
$\mathbf{q} \in \mathbb{R}^{n_{dof}}$ is the vector of joint positions,
$\boldsymbol\tau \in \mathbb{R}^{n_{dof}}$ is the vector of joint torques,
$\boldsymbol\pi \in \mathbb{R}^{n_p}$ is the vector of the identifiable (base) inertial parameters \cite{reference/robo/FeatherstoneO08},
$\ensuremath{\Phi(\mathbf{x})} \in \mathbb{R}^{n_{dof} \times n_p}$ is the ``regressor'', i.e. a matrix that depends only
on the robot kinematic parameters. In the rest of the paper we will indicate with $\mathbf{x}$ the triple given by $(\mathbf{q},\mathbf{\dot{q}},\mathbf{\ddot{q}})$.
Other parametric models write different measurable quantities as a product
of a regressor and a vector of parameters, for example
the total energy of the robot \cite{gautier1988identification}, the istantaneous power provided to the robot \cite{gautier1997dynamic}, the sum of all external forces acting on the robot \cite{journal/ayusawa2014} or
the center of pressure of the ground reaction forces \cite{conf/baelemans2013}. Regardless of the choice of the measured variable $\mathbf{y}$,
the structure of the regressor is similar:
\begin{equation}
\label{eq:parametricModel}
\mathbf{y} = \Phi(\mathbf{q},\mathbf{\dot{q}},\mathbf{\ddot{q}}) \boldsymbol\pi = \ensuremath{\Phi(\mathbf{x})} \boldsymbol\pi ,
\end{equation}
where $\mathbf{y} \in \mathbb{R}^{t}$ is the measured quantity.
The $\bm{\pi}$ vector is composed of certain linear combinations of the inertial parameters of the links, the \emph{base inertial parameters} \cite{khalil2004modeling}.
In particular, the inertial parameters of a single body are the mass $m$,
the first moment of mass $m\mathbf{c} \in \mathbb{R}^3$ expressed in a body fixed frame and the inertia matrix $I\in \mathbb{R}^{3 \times 3}$ expressed
in the orientation of the body fixed frame and with respect to its origin.
In \emph{parametric modeling} of robot dynamics, the regressor structure depends on the kinematic parameters
of the robot, that are obtained from CAD models of the robot through kinematic calibration techniques.
Similarly, the inertial parameters $\boldsymbol\pi$ can also be obtained from CAD models of the robot,
however these models may be unavailable (for example) because the manufacturer of the robot does not provide them.
In this case the usual approach is to estimate $\boldsymbol\pi$ from experimental data \cite{hollerbach2008model}.
To do that, given $n$ measures of the measured quantity $\mathbf{y}_i$ (with $i = 1 \dots n$), stacking \eqref{eq:parametricModel} for the
$n$ samples it is possible to write:
\begin{equation}
\label{eq:LSParametric}
\begin{bmatrix}
\ensuremath{\mathbf{y}}_1 \\
\ensuremath{\mathbf{y}}_2 \\
\vdots \\
\ensuremath{\mathbf{y}}_{n}
\end{bmatrix}
=
\begin{bmatrix}
\regressorNum{1} \\
\regressorNum{2} \\
\vdots \\
\regressorNum{n}
\end{bmatrix}
\boldsymbol\pi .
\end{equation}
This equation can then be solved in least squares (LS) sense to find an estimate $\hat{\bm{\pi}}$ of the base inertial
parameters. Given the training trajectories it is possible that not all parameters in $\bm{\pi}$ can be estimated well as the problem in \eqref{eq:LSParametric}
can be ill-posed, hence this equation is usually solved as a \emph{Regularized Least Squares} (RLS) problem. Defining
$$
\overline{\ensuremath{\mathbf{y}}}_{n} = \begin{bmatrix}
\ensuremath{\mathbf{y}}_1 \\
\ensuremath{\mathbf{y}}_2 \\
\vdots \\
\ensuremath{\mathbf{y}}_{n}
\end{bmatrix} ,
\hspace{1em}
\overline{\bm{\Phi}}_{n} = \begin{bmatrix}
\regressorNum{1} \\
\regressorNum{2} \\
\vdots \\
\regressorNum{n}
\end{bmatrix} ,$$
the RLS problem that is solved for the parametric identification is:
\begin{equation}
\label{eq:parametricRLS}
\hat{\bm{\pi}} = \argmin\limits_{\bm{\pi} \in \mathbb{R}^{n_p}} \left( \|\overline{\bm{\Phi}}_{n} \bm{\pi} -\overline{\ensuremath{\mathbf{y}}}_{n}\|^2 + \lambda \| \bm{\pi} \|^2 \right) , \lambda > 0.
\end{equation}
\subsection{Nonparametric Modeling with Kernel Methods}
\label{sec:np_modeling}
Consider a probability distribution $\rho$ over the probability space $\mathcal{X} \times \mathcal{Y}$, where $\mathcal{X} \subseteq \mathbb{R}^d$ is the input space (the space of the $d$ measured attributes) and $\mathcal{Y}\subseteq \mathbb{R}^t$ is the output space (the space of the $t$ outputs to be predicted). In a nonparametric modeling setting, the goal is to find a function $f^*: \mathcal{X} \rightarrow \mathcal{Y}$ belonging to a set of measurable functions $\mathcal{H}$, called \textit{hypothesis space}, such that
\begin{equation}
\label{eq:rm}
f^* = \argmin\limits_{f \in \mathcal{H}}
\underbrace{\int_{\mathcal{X} \times \mathcal{Y}} \ell(f(\mathbf{x}),\mathbf{y}) d\rho(\mathbf{x},\mathbf{y})}_{\mathcal{E}(f)},
\end{equation}
where $\mathbf{x} \in \mathcal{X}$ are row vectors, $\mathbf{y}\in \mathcal{Y}$, $\mathcal{E}(f)$ is called \textit{expected risk} and $\ell(f(\mathbf{x}),\mathbf{y})$ is the \textit{loss function}. In the rest of this work, we will consider the squared loss $\ell(f(\mathbf{x}),\mathbf{y}) = \|f(\mathbf{x})-\mathbf{y}\|^2$.
Note that the distribution $\rho$ is unknown, and that we assume to have access to a discrete and finite set of measured data points $S = \lbrace \mathbf{x}_i,\mathbf{y}_i\rbrace_{i=1}^n$ of cardinality $n$, in which the points are independently and identically distributed (i.i.d.) according to $\rho$.\\
In the context of kernel methods \cite{schlkopf2002learning}, $\mathcal{H}$ is a \textit{reproducing kernel Hilbert space} (RKHS). An RKHS is a Hilbert space of functions such that $\exists k : \mathcal{X} \times \mathcal{X} \rightarrow \mathbb{R}$ for which the following properties hold:
\begin{enumerate}
\item $\forall \mathbf{x} \in \mathcal{X} \quad k_\mathbf{x}(\cdot) = k(\mathbf{x},\cdot) \in \mathcal{H}$
\item $g(\mathbf{x}) = \left\langle g, k_\mathbf{x}\right\rangle _{\mathcal{H}} \forall g \in \mathcal{H}, \mathbf{x} \in \mathcal{X}$ ,
\end{enumerate}
where $\left\langle \cdot , \cdot \right\rangle _{\mathcal{H}} $ indicates the inner product in $\mathcal{H}$. The function $k$ is a \textit{reproducing kernel}, and it can be shown to be symmetric positive definite (SPD).
We also define the kernel matrix $K \in \mathbb{R}^{n \times n} \quad s.t. \quad K_{i,j} = k(\mathbf{x}_i,\mathbf{x}_j)$, which is symmetric and positive semidefinite (SPSD) $\forall \mathbf{x}_{i}, \mathbf{x}_{j} \in \mathcal{X}$, with $i,j \in \lbrace 1, \ldots, n \rbrace, n \in \mathbb{N}^+$.
The optimization problem outlined in \eqref{eq:rm} can be approached empirically by means of many different algorithms, among which one of the most widely used is Kernel Regularized Least Squares (KRLS) \cite{SaundersGV98,rifkin2003regularized}.
In KRLS, a regularized solution $\hat{f}_{\lambda} : \mathcal{X} \rightarrow \mathcal{Y}$ is found solving
\begin{equation}
\label{eq:tik}
\hat{f}_{\lambda} = \argmin\limits_{f \in \mathcal{H}} \left( \sum_{i=1}^n \|f(\mathbf{x}_i) - \mathbf{y}_i\|^2 + \lambda \|f\|_{\mathcal{H}}^2 \right), \lambda > 0 ,
\end{equation}
where $\lambda$ is called \textit{regularization parameter}.
The solution to \eqref{eq:tik} exists and is unique.
Following the representer theorem \cite{schlkopf2002learning}, the solution can be conveniently expressed as
\begin{equation}
\hat{f}_{\lambda}(\mathbf{x}) = \sum_{i = 1}^n \bm{\alpha}_i k(\mathbf{x}_i,\mathbf{x})
\end{equation}
with $\bm{\alpha} = (K+\lambda I_n)^{-1}Y \in \mathbb{R}^{n \times t}$, $\bm{\alpha}_i$ $i$-th row of $\bm{\alpha}$ and $Y = \left[ \mathbf{y}^\top_1, \ldots, \mathbf{y}^\top_n \right]^\top$.
It is therefore necessary to invert and store the kernel matrix $K \in \mathbb{R}^{n \times n}$, which implies $O(n^3)$ and $O(n^2)$ time and memory complexities, respectively.
Such complexities render the above-mentioned KRLS approach prohibitive in settings where $n$ is large, including the one treated in this work. This limitation can be dealt with by resorting to approximated methods such as \textit{random features}, which will now be described.
\subsubsection{Random Feature Maps for Kernel Approximation}
\label{sec:randfeats}
The random features approach was first introduced in \cite{RahimiR07}, and since then is has been widely applied in the field of large-scale Machine Learning. This approach leverages the fact that the kernel function can be expressed as
\begin{equation}
\label{eq:innprod}
k(\mathbf{x},\mathbf{x}') = \left\langle \phi(\mathbf{x}) , \phi(\mathbf{x}') \right\rangle _{\mathcal{H}} ,
\end{equation}
where $\mathbf{x}, \mathbf{x}' \in \mathcal{X}$ are row vectors, $\phi: \mathbb{R}^d \rightarrow \mathbb{R}^p$ is a \textit{feature map} associated with the kernel, which maps the input points from the input space $\mathcal{X}$ to a feature space of dimensionality $p \leq + \infty$, depending on the chosen kernel.
When $p$ is very large, directly computing the inner product as in \eqref{eq:innprod} enables the computation of the solution, as we have seen for KRLS.
However, $K$ can become too cumbersome to invert and store as $n$ grows. A random feature map $\tilde{\phi}: \mathbb{R}^d \rightarrow \mathbb{R}^D$, typically with $D \ll p$, directly approximates the feature map $\phi$, so that
\begin{equation}
k(\mathbf{x},\mathbf{x}') = \left\langle \phi(\mathbf{x}) , \phi(\mathbf{x}') \right\rangle _{\mathcal{H}} \approx \tilde{\phi}(\mathbf{x})\tilde{\phi}(\mathbf{x}')^\top.
\end{equation}
$D$ can be chosen according to the desired approximation accuracy, as guaranteed by the convergence bounds reported in \cite{RahimiR07,4797607}.
In particular, we will use random Fourier features for approximating the Gaussian kernel
\begin{equation}
k(\mathbf{x},\mathbf{x}')=e^{-\frac{\|\mathbf{x}-\mathbf{x}'\|^2}{2\sigma^2}}.
\end{equation}
The approximated feature map in this case is $\tilde{\phi}(\mathbf{x}) = \left[ e^{i\mathbf{x} \bm{\omega}_1}, \ldots, e^{i\mathbf{x}\bm{\omega}_D} \right]$, where
\begin{equation}
\bm{\omega} \sim p(\bm{\omega}) = (2 \pi)^{-\frac{D}{2}}e^{-\frac{\|\bm{\omega}\|^2}{2\sigma^2}},
\end{equation}
with $\bm{\omega} \in \mathbb{R}^d$ column vector. The fundamental theoretical result on which random Fourier features approximation relies is Bochner's Theorem \cite{rudin1990fourier}.
The latter states that if $k(\mathbf{x},\mathbf{x}')$ is a shift-invariant kernel on $\mathbb{R}^d$, then $k$ is positive definite if and only if its Fourier transform $p(\omega)\geq0$.
If this holds, by the definition of Fourier transform we can write
\begin{equation}
k(\mathbf{x},\mathbf{x}') = k(\mathbf{x}-\mathbf{x}') = \int_{\mathbb{R}^d}p(\bm{\omega})e^{i(\mathbf{x}-\mathbf{x}')\bm{\omega}}d\bm{\omega} ,
\end{equation}
which can be approximated by performing an empirical average as follows:
\begin{equation}
\begin{array}{r@{}l}
k(\mathbf{x}-\mathbf{x}') &{}= \mathbb{E}_{\bm{\omega} \sim p} \left[e^{i(\mathbf{x}-\mathbf{x}')\bm{\omega}} \right] \approx \\
&{}\approx \frac{1}{D} \sum_{k=1}^De^{i(\mathbf{x}-\mathbf{x}')\bm{\omega}} = \tilde{\phi}(\mathbf{x}) \tilde{\phi}(\mathbf{x}')^\top.
\end{array}
\end{equation}
Therefore, it is possible to map the input data as $\tilde{\mathbf{x}} = \tilde{\phi}(\mathbf{x}) \in \mathbb{R}^{D}$, with $\tilde{\mathbf{x}}$ row vector, to obtain a nonlinear and nonparametric model of the form
\begin{equation}
\tilde{f}(\mathbf{x}) = \tilde{\mathbf{x}} \tilde{W} \approx \hat{f}_{\lambda}(\mathbf{x}) = \sum_{i = 1}^n \bm{\alpha}_i k(\mathbf{x}_i,\mathbf{x})
\end{equation}
approximating the exact kernelized solution $\hat{f}_{\lambda}(\mathbf{x})$, with $\tilde{W} \in \mathbb{R}^{D \times t}$. Note that the approximated model is nonlinear in the input space, but linear in the random features space. We can therefore introduce the regularized linear regression problem in the random features space as follows:
\begin{equation}
\label{eq:randfeatsminimizationprob}
\tilde{W}^\lambda = \argmin\limits_{\tilde{W} \in \mathbb{R}^{d \times t}} \left( \| \tilde{X} \tilde{W} - Y \|^2 + \lambda \| \tilde{W} \|^2 \right) , \lambda > 0,
\end{equation}
where $\tilde{X} \in \mathbb{R}^{n \times D}$ is the matrix of the training inputs where each row has been mapped by $\tilde{\phi}$.
The main advantage of performing a random feature mapping is that it allows us to obtain a nonlinear model by applying linear regression methods.
For instance, Regularized Least Squares (RLS) can compute the solution $\tilde{W}^\lambda$ of \eqref{eq:randfeatsminimizationprob} with $O(nD^2)$ time and $O(D^2)$ memory complexities.
Once $\tilde{W}^\lambda$ is known, the prediction $\hat{\mathbf{y}} \in \mathbb{R}^{1 \times t}$ for a mapped sample $\tilde{\mathbf{x}} $ can be computed as $\hat{\mathbf{y}} = \tilde{\mathbf{x}} \tilde{W}^\lambda$.
\subsection{Regularized Least Squares}
Let $Z\in \mathbb{R}^{a\times b}$ and
$U \in \mathbb{R}^{a\times c}$ be two matrices of real numbers, with $a,b,c \in \mathbb{N}^+$.
The Regularized Least Squares (RLS) algorithm computes a regularized solution $W^\lambda \in \mathbb{R}^{b\times c}$
of the potentially ill-posed problem $Z W = U $, enforcing its numerical stability.
Considering the widely used Tikhonov regularization scheme, $W^\lambda \in \mathbb{R}^{b\times c}$ is the solution to the following problem:
\begin{equation}
\label{eq:rlsminimizationprob}
W^\lambda = \argmin\limits_{W \in \mathbb{R}^{b \times c}} \underbrace{ \left( \| Z W - U \|^2 + \lambda \| W \|^2 \right)}_{J(W,\lambda)} , \quad \lambda>0 ,
\end{equation}
where $\lambda$ is the regularization parameter. By taking the gradient of $J(W,\lambda)$ with respect to $W$ and equating it to zero, the minimizing solution can be written as
\begin{equation}
\label{eq:rlsSolution}
W^\lambda = (Z^\top Z + \lambda I_{b})^{-1}Z^\top U .
\end{equation}
Both the parametric identification problem \eqref{eq:parametricRLS} and the nonparametric random features problem \eqref{eq:randfeatsminimizationprob} are specific
instances of the general problem \eqref{eq:rlsminimizationprob}.
In particular, the parametric problem \eqref{eq:parametricRLS} is equivalent to \eqref{eq:rlsminimizationprob} with:
$$
W^\lambda = \hat{\boldsymbol\pi}, \hspace{1em} Z = \overline{\Phi}_n, \hspace{1em} U = \overline{\mathbf{y}}_n
$$
while the random features learning problem \eqref{eq:randfeatsminimizationprob} is equivalent to \eqref{eq:rlsminimizationprob} with:
$$
W^\lambda = \tilde{W}^\lambda , \hspace{1em} Z = \tilde{X} , \hspace{1em} U = Y.
$$
Hence, both problems for a given set of $n$ samples can be solved applying \eqref{eq:rlsSolution}.
\subsection{Recursive Regularized Least Squares (RRLS) with Cholesky Update}
\label{sec:recupdate}
In scenarios in which supervised samples become available sequentially, a very useful extension of the RLS algorithm consists in the definition of an update rule for the model which allows it to be incrementally trained, increasing adaptivity to changes of the system properties through time.
This algorithm is called Recursive Regularized Least Squares (RRLS).
We will consider RRLS with the Cholesky update rule \cite{bjoerck_least_squares96}, which is numerically more stable than others (e.g. the Sherman-Morrison-Woodbury update rule).
In adaptive filtering, this update rule is known as the \textit{QR algorithm} \cite{Sayed:2008:AF:1370975}.
Let us define $A = Z^\top Z + \lambda I_{b}$ with $\lambda > 0$ and $B = Z^\top U$. Our goal is to update the model (fully described by $A$ and $B$)
with a new supervised sample $(\mathbf{z}_{k+1}, \mathbf{u}_{k+1})$, with $\mathbf{z}_{k+1} \in \mathbb{R}^b$, $\mathbf{u}_{k+1}\in \mathbb{R}^c$ row vectors.
Consider the Cholesky decomposition $A = R^\top R$. It can always be obtained, since $A$ is positive definite for $\lambda > 0$.
Thus, we can express the update problem at step $k+1$ as:
\begin{equation}
\begin{array}{r@{}l}
A_{k+1} &{}= R^\top_{k+1} R_{k+1}\\
&{}= A_{k} + \mathbf{z}_{k+1}^\top \mathbf{z}_{k+1}\\
&{}= R^\top_{k} R_{k} + \mathbf{z}_{k+i}^\top \mathbf{z}_{k+1} ,
\end{array}
\end{equation}
where $R$ is full rank and unique, and $R_0 = \sqrt{\lambda} I_b$.\\
By defining
\begin{equation}
\tilde{R}_{k} =
\left[ \begin{array}{c}
R_{k} \\
\mathbf{z}_{k+1} \end{array} \right] \in \mathbb{R}^{b+1 \times b},
\end{equation}
we can write $A_{k+1}=\tilde{R}_{k}^\top \tilde{R}_{k}$. However, in order to compute $R_{k+1}$ from the obtained $A_{k+1}$ it would
be necessary to recompute its Cholesky decomposition, requiring $O(b^3)$ computational time. There exists a procedure, based on Givens
rotations, which can be used to compute $R_{k+1}$ from $\tilde{R}_{k}$ with $O(b^2)$ time complexity.
A recursive expression can be obtained also for $B_{k+1}$ as follows:
\begin{equation}
\label{eq:rrlsUpdate}
\begin{array}{r@{}l}
B_{k+1} &{}= Z_{k+1}^\top U_{k+1}\\
&{}= Z_{k}^\top U_{k} + \mathbf{z}_{k+1}^\top \mathbf{u}_{k+1}.
\end{array}
\end{equation}
Once $R_{k+1}$ and $B_{k+1}$ are known, the updated weights matrix $W_k$ can be obtained via back and forward substitution as
\begin{equation}
W_{k+1} = R_{k+1} \setminus (R^\top_{k+1} \setminus B_{k+1}) .
\end{equation}
The time complexity for updating $W$ is $O(b^2)$.
As for RLS, the RRLS incremental solution can be applied to both the parametric \eqref{eq:parametricRLS} and nonparametric with random features \eqref{eq:randfeatsminimizationprob} problems, assuming $\lambda > 0$.
In particular, RRLS can be applied to the parametric case by noting that the arrival of a new sample $\left( \Phi_r , \mathbf{y}_r \right)$ adds
$t$ rows to $Z_k = \overline{\Phi}_{r-1}$ and $U_k = \overline{\mathbf{y}}_{r-1}$. Consequently, the update of $A$ must be decomposed
in $t$ update steps using \eqref{eq:rrlsUpdate}. For each one of these $t$ steps we consider only one row of $\Phi_{r}$ and $\mathbf{y}^\top_r$, namely:
$$
\mathbf{z}_{k+i} = (\Phi_r)_i , \hspace{1em} \mathbf{u}_{k+i} = (\mathbf{y}^\top_r)_i , \hspace{1em} i = 1 \dots t
$$
where $(V)_i$ is the $i$-th row of the matrix $V$.
For the nonparametric random features case, RRLS can be simply applied with:
$$
\mathbf{z}_{k+1} = \tilde{\mathbf{x}}_r , \hspace{1em} \mathbf{u}_{k+1} = \mathbf{y}_r .
$$
where $\left(\tilde{\mathbf x}_r, \mathbf y_r \right)$ is the supervised sample which becomes available at step $r$.
\section{SEMIPARAMETRIC INCREMENTAL DYNAMICS LEARNING}
\label{sec:semiparametric}
\begin{figure*}
\centering
\vspace{3mm}
\begin{overpic}[width=0.9\textwidth]{figures/SemiParametricBlockDiagram8-1.eps}
\put (2.5,16) {$ \mathbf{q} $}
\put (2.5,12.5) {$ \mathbf{\dot{q}} $}
\put (2.5,9) {$ \mathbf{\ddot{q}} $}
\put (2.5,3.3) {$ \bm{f} $}
\put (2.5,1.1) {$ \bm{\tau} $}
\put (9,13.9) {$ \mathbf{x} $}
\put (9,3.9) {$ \mathbf{y} $}
\put (27.7,14.2) {\footnotesize $ \Phi(\mathbf{x}) $}
\put (39.73,24) {$ \hat{\bm{\pi}}$}
\put (54,13.9) {$ \hat{\mathbf{y}} $}
\put (75.2,12.5) {$ \tilde{\mathbf{x}} $}
\put (92.2,12.5) {$ \triangle \tilde{\mathbf{y}} $}
\put (50,4.5) {$ - $}
\put (49,1) {$ + $}
\put (58,4) {$ \triangle \mathbf{y} $}
\put (97.6,11.1) {$ + $}
\put (97.8,24) {$ \tilde{\mathbf{y}}$}
\end{overpic
\caption{Block diagram displaying the functioning of the proposed prioritized semiparametric inverse dynamics estimator. $ \bm{f} $ and $ \bm{\tau} $ indicate measured force and torque components, concatenated in the measured output vector $\mathbf{y}$. The parametric part is composed of the RBD regressor generator and of the parametric estimator based on RRLS. Its outputs are the estimated parameters $\hat{\bm{\pi}}$ and the predicted output $\hat{\mathbf{y}}$. The nonparametric part maps the input to the random features space with the Random Features Mapper block, and the RFRRLS estimator predicts the residual output $\triangle \tilde{\mathbf{y}} $, which is then added to the parametric prediction $\hat{\mathbf{y}}$ to obtain the semiparametric prediction $\tilde{\mathbf{y}}$.}
\label{fig:blockdia}
\end{figure*}
We propose a semiparametric incremental inverse dynamics estimator, designed to have better generalization properties with respect to fully parametric and nonparametric ones, both in terms of accuracy and convergence rates.
The estimator, whose functioning is illustrated by the block diagram in Figure \ref{fig:blockdia}, is composed of two main parts.
The first one is an incremental parametric estimator taking as input the rigid body dynamics regressors $ \Phi(x) $ and computing two quantities at each step:
\begin{itemize}
\item An estimate $\hat{\mathbf{y}}$ of the output quantities of interest
\item An estimate $\hat{\bm{\pi}}$ of the base inertial parameters of the links composing the rigid body structure
\end{itemize}
The employed learning algorithm is RRLS. Since it is supervised, during the model update step the measured output $\mathbf{y}$ is used by the learning algorithm as ground truth.
The parametric estimation is performed in the first place, and it is independent of the nonparametric part.
This property is desirable in order to give priority to the identification of the inertial parameters $\bm{\pi}$.
Moreover, being the estimator incremental, the estimated inertial parameters $\hat{\bm{\pi}}$ adapt to changes in the inertial properties of the links, which can occur if the end-effector is holding a heavy object.
Still, this adaptation cannot address changes in nonlinear effects which do not respect the rigid body assumptions.\\
The second estimator is also RRLS-based, fully nonparametric and incremental. It leverages the approximation of the kernel function via random Fourier features, as outlined in Section \ref{sec:randfeats}, to obtain a nonlinear model which can be updated incrementally with constant update complexity $O(D^2)$, where $D$ is the dimensionality of the random feature space (see Section \ref{sec:recupdate}).
This estimator receives as inputs the current vectorized $\mathbf{x}$ and $\hat{\mathbf{y}}$, normalized and mapped to the random features space approximating an infinite-dimensional feature space introduced by the Gaussian kernel.
The supervised output is the residual $ \triangle \mathbf{y} = \mathbf{y} - \hat{\mathbf{y}}$.
The nonparametric estimator provides as output the estimate $ \triangle \tilde{\mathbf{y}} $ of the residual, which is then added to $\hat{\mathbf{y}}$ to obtain the semiparametric estimate $ \tilde{\mathbf{y}}$.
Similarly to the parametric part, in the nonparametric one the estimator's internal nonlinear model can be updated during operation, which constitutes an advantage in the case in which the robot has to explore a previously unseen area of the state space, or when the mechanical conditions change (e.g. due to wear, tear or temperature shifts).
\section{EXPERIMENTAL RESULTS}
\label{sec:experiments}
\subsection{Software}
For implementing the proposed algorithm we used two existing
open source libraries.
For the RRLS learning part we used GURLS \cite{tacchetti2013gurls},
a regression and classification library based on the Regularized Least Squares (RLS) algorithm, available for
Matlab and C++.
For the computations of the regressors $\Phi(\mathbf{q},\mathbf{\dot{q}},\mathbf{\ddot{q}})$
we used iDynTree%
\footnote{\url{https://github.com/robotology/idyntree}}
, a C++ dynamics library designed for free floating robots. Using SWIG~\cite{beazley1996swig}, iDynTree supports
calling its algorithms in several programming languages, such as Python, Lua and Matlab.
For producing the presented results, we used the Matlab interfaces of iDynTree and GURLS.
\subsection{Robotic Platform}
\begin{figure}[htb]
\begin{overpic}[width=0.48\textwidth,natwidth=1235,natheight=742]{figures/arm3.png}
\put(5,10){FT sensor}
\put(13,13){\vector(1,1){18}}
\put(38,50){Upper arm}
\put(43,49){\vector(0,-1){12}}
\put(65,45){Forearm}
\put(70,44){\vector(-1,-2){7}}
\end{overpic}
\caption{CAD drawing of the iCub arm used in the experiments. The six-axis F/T sensor used for validation is visible in the middle of the upper arm link.}
\label{fig:cadArm}
\end{figure}
iCub is a full-body humanoid with 53 degrees of freedom \cite{Metta2010}. For validating
the presented approach, we learned the dynamics of the right arm of the iCub as measured
from the proximal six-axis force/torque (F/T) sensor embedded in the arm.
The considered output $\mathbf{y}$ is the reading of the F/T sensor, and the inertial
parameters $\boldsymbol\pi$ are the base parameters of the arm~\cite{traversaro2015inertial}.
As $\mathbf{y}$ is not a input variable for the system, the output of the dynamic model is not directly usable for control,
but it is still a proper benchmark for the dynamics learning problem, as also shown in ~\cite{GijsbertsM11}.
Nevertheless, the joint torques could be computed seamlessly from the F/T sensor readings if needed for control purposes, by applying the method presented in \cite{6100813}.
\subsection{Validation}
\label{sec:validation}
\input{results.tex} |
1,116,691,500,877 | arxiv | \section{Introduction}
\
{\it
Two neighbors may agree to drain a meadow, which they possess in common: because it is easy for them to know each other's mind;
and each must perceive, that the immediate consequence of his failing in his part, is the abandoning of the whole project.
But it is very difficult, and indeed impossible, that a thousand persons should agree in any such action; it being difficult
for them to concert so complicated a design, and still more difficult for them to execute it; while each seeks a pretext to free
himself of the trouble and expense, and would lay the whole burden on the others.
}
\
David Hume, A Treatise of Human Nature (London: J. M. Dent, 1952, II, p.239) - From reference~\cite{car07}.
\
Humans show levels of cooperation among non-kin that are unparalleled among other species. This difference becomes striking when facing social dilemmas, i.e., situations in which cooperation is hard to achieve because the best move for an individual does not produce the best outcome for the group. Public goods games (PGG) represent a clear exemplification of this conflict between individual incentives and social welfare. If everybody contributes to the public good, cooperation is the social optimum, but free-riding on others' contributions represent the most rewarding option.
If norms, conventions and societal regulations have been proven effective in preventing the collapse of public goods (for a review, see~\cite{ost90,ost05}), when individuals are faced with unknown strangers, with little or no opportunities for future re-encounters cooperation easily collapses, unless punishment for non-cooperators is provided~\cite{feh00}.
An alternative solution is represented by reputation, through which cheaters can be easily identified and avoided~\cite{now98a,gia12}. Indirect reciprocity supported by reputation~\cite{ale87} can be one of the mechanisms explaining the evolution of cooperation in humans~\cite{mil02}, especially in large groups of unrelated strangers who can, through language, actively communicate about their past experiences with cheaters~\cite{smi10}.
As such, gossip may effectively bypass the ``second-order free-rider problem'', wherein the costs associated with solving one social dilemma produces a new one~\cite{har68,kiy08}. This is the case of punishment: cooperators who do not sustain the costs of punishment are better off than cooperators who also punish. Therefore this solution to social dilemmas itself entails a social dilemma, whereas gossip, being essentially free should not imply such a second-order free-rider problem. In addition to costly punishment and reputation, ostracism of free-riders may represent a third solution. However, the direct effect of ostracizing a member is that the group size decreases, which automatically reduces maximal contribution levels to the public good for all remaining periods. Maier-Rigaud and colleagues show that in the lab PGG with ostracism opportunities increases contribution levels and contrary to monetary punishment, also has a significant positive effect on net earnings~\cite{mai05}.
Models of indirect reciprocity usually take into account dyadic interactions~\cite{now98a}, or group interactions in a mutual aid game~\cite{pan04}, in which providing help has a cost for the helper but it also increases his/her image score, i.e., a publicly visible record of his/her reputation. Image score increases or decreases according to individuals' past behaviors, thus providing a reliable way to discriminate between cheaters and cooperative players. Both in computer simulations~\cite{now98a}, and in lab experiments with humans~\cite{wed00}, cooperation can emerge and be maintained through image score.
When individuals facing a social dilemma can know other players' image score, cooperation can emerge in small groups, as showed by Suzuki and Akiyama~\cite{suz05}. In their work, cooperation can emerge and be maintained for groups of four individuals; though, when group size increases there is a concomitant decrease in the frequency of cooperation. The authors explain this decline as due to the difficulty of observing reputations of many individuals in large communities. This can be true of unstructured communities, but this rarely happens in human societies, characterized by interaction networks. To account for the role of societal structure, we designed a PGG in which players' interactions depend on the kind of network and on the possibility of actively choosing a subset of group members. More specifically, we compare cooperation levels among agents placed on a small-world network~\cite{wat98}, defined by short average path lengths and high clustering, to the performance of agents on a bi-partite graph~\cite{die97,gar11}. The latter is generally used to model relations between two different classes of objects, like affiliation networks linking members and the groups they belong to. This structure is especially interesting for us because it is especially suited for partner selection, as it happens when a club refuses membership to a potential associate.
Here, we are interested in exploring the effect of network structure on the emergence of cooperation in a PGG. We compare two different network topologies and we show that reputation-based partner choice on a bi-partite graph can make cooperation thrive also in large groups of agents. We also show that this effect is robust to number of generations, group size and total number of agents in the system.
\
\
\section{The Model}
We consider a population of $N$ individuals. In each round of the game, $g$ agents are picked up at random to play a PGG among themselves.
Players can cooperate contributing with a cost $c$ to a common pot, or can defect without paying anything. Then, the total amount collected
in the pot is multiplied for a benefit $b$ and equally distributed among all the group members, without taking into account individual
contributions. At the end of each interaction, said $\chi$ the number of contributors in the group, cooperators' payoffs equals $(\chi b/g-1)c$,
whereas defectors' payoffs is $\chi bc/g$. At the collective level, the best outcome is achieved when everyone cooperates, but cheaters are better
off, because defection permits to avoid a loss when the number of cooperators is lower than $gc/b$.
Among the many solutions offered~\cite{feh00}, Suzuki and Akiyama~\cite{suz05} design a modified PGG in which agents can identify cheaters thanks to the so-called image score ~\cite{now98a,now98b}. The basic features of our model are the same of the one by Suzuki and Akiyama: in particular, each player $i$ is characterized by two integer variables:
the image score $s_i\in[-S_{max},S_{max}]$ and
the strategy $k_i\in[-S_{max},S_{max}+1]$, being $S_{max}\geq0$ a parameter of the model. When selected to play a round of the game, an individual
cooperates if the average image score $\langle s\rangle_g$ of its opponents is equal to or higher than its own strategy $k_i$, otherwise
it will not contribute. At the end of the round, the image score of the player is increased by 1 in case of cooperation, otherwise it is
decreased by the same quantity. In any case, $s_i$ remains in the allowed interval $[-S_{max},S_{max}]$: if an agent has an image score of
$S_{max}$ ($-S_{max}$) and contributes (defects), nothing happens to its image score. At the initial stage, all the
image scores and fitness levels are set to zero, whilst the strategies are randomly distributed among the individuals.
The image score is intended to give a quantitative evaluation of the public reputation of an individual in the scope of indirect reciprocity:
if contributing once is rewarded by future contributions by the other individuals, then any cooperative action must be
recognized and considered positively by the entire population; on the other hand, the variability of the strategies describes the different
attitudes and expectations of the single agents~\cite{now98a}.
After $m$ rounds, reproduction takes place. Again, we apply the same evolutionary algorithm used by Suzuki and Akiyama~\cite{suz05}. For $N$ times we
select at random a pair of individuals and with probability $P$ we create a new individual inheriting the strategy of the parent with the highest
fitness. Then parents are put again in the population, and offspring is stored in another pool. When this selection process
has happened $N$ times, the old population is deleted and replaced with the offspring. It is worth noticing that offspring
inherit only the parent's strategy, while their image score and fitness is set equal to zero. Finally, we repeat all the procedure ($m$
rounds followed by the reproduction stage), for an adequate number of generations. The simulation lasts until the system reaches a final
(steady or frozen) configuration.
For sake of clarity, we observe that strategies defined as ($k\leq0$) are the more ``cooperation prone'', with the limit
case of $k=-S_{max}$ which is an absolute cooperator, while the positive ones are the ``cooperation averse'' strategies, with the limit case
of $k=S_{max}+1$ representing an inflexible defector.
Moving from the model described above, we are interested in testing whether two different network structures can promote cooperation for different group size and what effect partner selection can have in such an environment.
\
\
\section{Results}
\
\subsection{Robustness of Suzuki's and Akiyama's results}
Suzuki and Akiyama tested their model for a given set of parameters with the following values: $N=200,\ c=1,\ b=0.85g,\ S_{max}=5,
\ m=800$. Their results show that a cooperative strategy can evolve and invade a population when group size $g$ is small, but it does not survive when groups are large. For medium-sized communities, a coexistence between cooperators and defectors is possible.
\
The first step of the study present in this paper is a check of the robustness of Suzuki and Akiyama results with respect to the values of the
model parameters. A check of the role of $m$ and $N$ is reported already in~\cite{suz05}: it is claimed that the outcome is not relevantly
influenced by the value of these two quantities, so we focus here on $b,\ P$ and $S_{max}$.
The role of $b$ in the PGG is quite clear in literature. Normally it is set to a fixed value larger than one (often 3), independent
from the group size~\cite{hau02}. Using this value,
we found that the final cooperation level decreases sharpenly as $g$ increases, as shown in Fig.~\ref{B3_check}. The fact that in Suzuki's
and Akiyama's work such decreasing is much slower is due to the fact that being $b$ proportional to the group size the number of contributors
needed in order to make cooperation convenient remains constant in $g$ instead of decreasing with it. On the other hand, even though less dramatic,
the decrease is anyway observed, indicating that the negative effect of large groups on cooperation is stronger and it might depend on the PGG
dynamics itself.
\begin{figure}[tbp]
\centering
\includegraphics*[width=0.5\hsize]{B3_check.eps}
\caption{Behaviour of the final frequency of cooperative actions as a function of the group size $g$. All the parameters are the same of
reference~\cite{suz05}, except $b=3$. Each point averaged over 1000 realizations.}
\label{B3_check}
\end{figure}
Concerning the behaviour of the model as a function of the parameter $P$, we tested three different values: $P=0.9$ as in~\cite{suz05},
$P=0.75$ and $P=1.0$. As it can be easily seen in Fig.~\ref{P_check}, there is no fundamental difference due to the exact value of
this parameter.
\begin{figure}[tbp]
\centering
\includegraphics*[width=0.5\hsize]{P_check.eps}
\caption{Behaviour of the frequency of cooperative actions as a function of the number of generations for three different
values of $P$: 0.75, 0.90 and 1.0. The remaining parameters are the same of reference~\cite{suz05}. Each curve averaged over 1000 realizations.}
\label{P_check}
\end{figure}
Finally, changing the value of $S_{max}$, we see that up to $S_{max}\simeq15$, the behaviour of the system is rather homogeneous, as shown
in Fig.~\ref{Smax_check}.
\begin{figure}[tbp]
\centering
\includegraphics*[width=0.5\hsize]{Smax_check.eps}
\caption{Behaviour of the final frequency of cooperative actions as a function of $S_{max}$. The remaining parameters are the same of
reference~\cite{suz05}, the vertical line for $S_{max}=5$ specifies the value utilized in reference~\cite{suz05}. Each point averaged over 1000 realizations.}
\label{Smax_check}
\end{figure}
\
Our results show that the behaviour of the model is actually robust
for a large range of the parameters at stake, thus replicating Suzuky and Akiyama's results.
\
\subsection{Small-world networks}
In order to enlarge the scope of the model, we inserted network structure in it, thus introducing some adaptations into the original model. The first change we made was in the mechanism of assortment. In the original model, every player had the same probability to interact with every other agent, therefore the population is placed on a total connected graph (CG). This configuration is rather unrealistic, especially when we consider
groups bigger than a given size. It is then interesting to test the model behaviour over more realistic, even though still abstract, networks.
The first example we take under consideration is the so-called small-world network (SWN), as conceived by Watts and Strogatz in~\cite{wat98}.
In short, a SWN, is a regular ring with few short-cuts linking originally far away nodes. It is constructed as shown in Fig.~\ref{swn}: we start from
a ring where each node is connected with $2k$ nearest neighbours. Then, with probability $p$, each link is rewired (one of the node is left fixed,
the other is changed), so that it finally leads to the creation of a network with $pNk$ short-cuts. As shown in reference~\cite{wat98}, for $1/Nk<p<1/10$ the network
shows the typical small-world effect: even though at local level the system behaves as a regular lattice, i.e., an individual placed in a
SWN cannot distinguish the network from a regular one just watching his/her neighbours (high clustering coefficient), at a global level the
average distance between two randomly selected individuals is very low (proportional to the logarithm of the system size), unlike the regular
case.
\begin{figure}[tbp]
\centering
\includegraphics*[width=0.75\hsize]{ws_net.eps}
\caption{Construction of SWN according Watts-Strogatz procedure. From reference~\cite{wat98}.}
\label{swn}
\end{figure}
In order to make the model work with this topology, we had to adapt the model dynamics to the specific situation. In particular, instead of extract $g$
agents at each round, we picked up a single player at each round and $g-1$ of its neighbours. In order to be sure that each individual had at least $g-1$
neighbours, we set $k=g-1$. Moreover, at the end of each generation, the offspring was randomly placed on the preexistent network, which is
defined at the beginning and does not change until the end of the simulation. Anyway, averaging over different realizations, each one has
its own networks, so that the averages are also over the topology.
In Fig.~\ref{sw_check1} and~\ref{sw_check2} we see the cooperation frequencies for two values of $g$ and different sizes
of the system. Basically, in particular for $g>2$, the system shows an interesting behaviour: in particular, the dynamics is
always driving the system towards the achievement of complete cooperation, even though the timing can vary: full cooperation is reached
more rapidly for small values of $N$, whilst it can take up to thousands of generations for larger systems. It is worth to notice,
from Fig.~\ref{sw_check2}, that this consensus time seems to reach its limit value already for $N=3200$; in such case we can also distinguish
an initial small decrease of cooperation rate before the final (steep) increase.
In short, these results demonstrate that the small-world topology in itself makes full cooperation possible, although not so fastly as in different
configurations, as we are going to show in the next sections.
\begin{figure}[tbp]
\centering
\includegraphics*[width=0.5\hsize]{SWg02_check.eps}
\caption{Behaviour of the frequency of cooperative actions in a SWN with $p=0.05$ as a function of the number of generations for $g=2$
and different values of $N$ (from top to bottom: 100, 200, 400, 800 and 1600). The remaining parameters are the same of reference~\cite{suz05}.
Each curve averaged over more than 1000 realizations.}
\label{sw_check1}
\end{figure}
\begin{figure}[tbp]
\centering
\includegraphics*[width=0.5\hsize]{SWg04_check.eps}
\caption{Behaviour of the frequency of cooperative actions in a SWN with $p=0.05$ as a function of the number of generations for $g=4$
and different values of $N$ (100, 400 and 1600). The remaining parameters are the same of reference~\cite{suz05}. Each curve averaged
over more than 1000 realizations.}
\label{sw_check2}
\end{figure}
\
\subsection{Bipartite graphs}
Another topological configuration that accounts better for the complexity of real interactions among individuals is the so-called
bipartite graph (BG)~\cite{die97,gar11}.
A bipartite representation contains two types of nodes denoting agents and groups, respectively. It implies that connections can be established only between nodes of different types and no direct connection among individuals is
allowed. Thus, such a bipartite representation preserves the information about the group structure: if two individuals
belong to the same three groups, they are ``more'' connected than two other individuals who are members of the same group. These two pairs would be equally represented in the classical one-mode projected network, while with the bipartite graph
this mesoscopic level of interactions is better depicted, as illustrated in Fig.~\ref{bg}.
\begin{figure}[tbp]
\centering
\includegraphics*[width=0.75\hsize]{bip_graph.eps}
\caption{Structure of a bipartite graph compared with a classical network. From reference~\cite{gar11}.}
\label{bg}
\end{figure}
Also in this case we adapted the original dynamics of the model to make it work on this kind of network. In particular,
the graph has $N$ individuals distributed into $M$ groups, each group composed of $g$ members. At the beginning of each round, the network is built
in this way: given $F\in(0,1)$, we set $gF$ initial members for each groups so that each individual belongs exclusively to one group. For
instance, if $N=150$, $g=20$ and $F=0.75$ (then $M=10$), at this stage we would have 15 agents in the first group, other
15 in the second one and so on until the last 15 in the tenth group. Then, each group must be completed choosing $(1-F)g=5$
individuals from the pool of those which do not belong to the group already. This can be accomplished in two different ways: first, by randomly picking $(1-F)g$ agents among the rest of the population; second, by selecting them according to their reputation, {\it i.e.}, their
image scores. When partner selection is available, an external player is randomly selected by the group, but accepted only if its image score is positive. Only if there is no
player in the whole population with good reputation, a candidate with negative image score is accepted in the group. Alternative ways of implementing partner selection were tested, like for example, accepting candidates with
image score equal or larger than the average strategy of the initial member of the group, but this did not produce any appreciable effects on the outcome of the simulations. Once the network is completely defined, each group plays a round of the game, with the same rules working
on CG and SWN. The procedure (network construction followed by a round of the game of each group) is repeated 10 times, then
the evolution process takes place again following the same rule given of the previous cases.
In Fig.~\ref{bg_check04} and~\ref{bg_check20} we show the behaviour of the model for $N=200$ (or the closest integer compatible with the remaining parameters),
$F=0.75$, with the other parameters equal to the ones utilized by Suzuki and Akiyama~\cite{suz05}.
\begin{figure}[tbp]
\centering
\includegraphics*[width=0.5\hsize]{BG_g04_check.eps}
\caption{Behaviour of the frequency of cooperative actions in a BG with as a function of the number of generations for $g=4,$
and $F=0.75$. The remaining parameters are the same of reference~\cite{suz05}. Each curve averaged
over 1000 realizations.}
\label{bg_check04}
\end{figure}
\begin{figure}[tbp]
\centering
\includegraphics*[width=0.5\hsize]{BG_g20_check.eps}
\caption{Behaviour of the frequency of cooperative actions in a BG with as a function of the number of generations for $g=20$
and $F=0.75$. The remaining parameters are the same of reference~\cite{suz05}. Each curve averaged
over 1000 realizations.}
\label{bg_check20}
\end{figure}
Our results show that the final cooperation level is {\it lower} here then in the CG
case if the added members of the groups are selected at random. However, when reputation-based partner selection is available in a population distributed on a bipartite graph, full cooperation is achieved in a very short amount of time (about ten generations), and this is true also for large groups ($g=20$ in figure). This result does not depend on $F$: even when
partner selection is restricted to a small percentage of agents, it can favour the invasion of the cooperative strategies throughout the system. This effect can be explained by the fact
that, in general, in PGG it is better for individuals to get involved in as many groups as possible in order to maximize their
income~\cite{hau03}. However, if this is not linked to a reputation-based partner selection mechanism, defection is still very profitable and cooperators are driven out of the system. On the contrary, if reputation is used to select group members, having a positive image score has a positive effect on fitness.
\
\subsection{Final strategy distributions}
In the model by Nowak and Sigmund~\cite{now98a,now98b}, based on the same image score mechanism, when the system ends up in a
final configuration of complete cooperation, the only surviving strategy is usually $k=0$, that is, the ``winning'' strategy is a
rather moderately generous one. A similar behaviour appears with our model in CG and SW topologies.
On the other hand, when working on BG topology, the final system configuration, always totally cooperative, presents all the negative
strategies, {\it i.e.} the more cooperative ones, as shown in Fig.~\ref{DISTR_check}. This means that taking account more carefully of the real
properties of the social interactions among idividuals not only enhances hugely cooperation to spread throughout the whole population,
but allows the survival also to the most generous and altruistic strategies.
\begin{figure}[tbp]
\centering
\includegraphics*[width=0.5\hsize]{DISTR_check.eps}
\caption{Final (relative) strategy abundance for a system on BG, same system of Fig.~\ref{bg_check04} ($g=4$) with reputation-based choice
of the added members of each group. Values averaged over 1000 realizations.}
\label{DISTR_check}
\end{figure}
\
\
\section{Discussion}
In a PGG in which the story of agents' past interactions is publicly available as an image score, cooperation can emerge and be maintained for small groups of agents. When we move from a mean-field situation to a small-world network, we observe that cooperation becomes stable after one hundred generations and for $g=4$. The real improvement is achieved thanks to the introduction of a partner choice mechanism on a bi-partite graph, where if a small percentage of group members are chosen on the basis of their reputations, cooperation can thrive.
In a social dilemma the introduction of a reputation mechanism for partner selection on a bipartite graph makes deception unprofitable, thus cooperators can thrive. In such an environment, agents with a positive reputation are more socially desirable, thus they can enter several groups in which their contributions help to achieve the social optimum. On the other hand, defectors with negative reputations are actively avoided, thus driving them to complete extinction after ten generations. Even more striking is the fact that, unlike other models~\cite{suz05,bra03}, full cooperation is maintained even when group size increases.
\
\
\section{Conclusions and perspectives}
The puzzle of the evolution of cooperation in humans can be successfully addressed if we take into account features of human societies that could have paved the way for the emergence of cooperative behaviors, like social networks and reputation. Moving from a replication of Suzuky and Akiyama~\cite{suz05} we showed that cooperation can emerge and be maintained in groups of agents playing a PGG on a network. We used two network topologies with different group and total population size, finding interesting differences especially in terms of the maximum level of cooperation achieved. Our results show that when partner selection is available in an affiliative network, cooperation can be easily reached even in large groups and for large system size.
The importance of social institutions~\cite{ost90} and informal social control~\cite{gia12,bes09} is well known to social scientists who, like Ellickson~\cite{ell91}, have stressed the importance of these features:
{\it ``A close-knit group has been defined as a social network whose members have credible and reciprocal prospects for the application of power against one another and a good supply of information on past and present internal events [\dots].
The hypothesis predicts that departures from conditions of reciprocal power, ready sanctioning opportunities, and adequate information are likely to impair the emergence of welfare-maximizing norms''} (p. 181).
Introducing a small world network does not alter the dynamics of cooperation in a PGG in a fundamental way, and this is also true for a bipartite graph with random partner selection. However, when we model the world as made of groups that can actively select at least one of their members, cooperators outperform free-riders in an easy and fast way. The evolutionary dynamics of our model can be linked to have a proximate explanation in psychological mechanisms for ostracism and social exclusion, two dreadful outcomes for human beings~\cite{abr05,bau95}.
In large groups of unrelated individuals, direct observation is not
possible, and usually records of an individual's past behaviors are
not freely and publicly available. What is abundant and costless is
gossip, i.e., reported information about others' past actions, that
can be used to avoid free-riders, either by refusing to interact with them,
or joining another crew in which free-riders are supposedly absent. For this reason we plan to run simulations in which agents will be able to report private information about their past experiences, thus overcoming the unrealistic limitations posed by image score. We posit that the combination of a bi-partite graph social structure and gossip like exchanges will mimic human societies better and will provide useful insights about the evolution of cooperation in humans.
\
\
\section*{ACKNOWLEDGEMENTS}
\
We gratefully acknowledge support from PRISMA project, within the
Italian National Program for Research and Innovation (Programma Operativo
Nazionale Ricerca e Competitivit\'a 2007-2013. Settore: Smart Cities
and Communities and Social Innovation Asse e Obiettivo: Asse II -
Azioni integrate per lo sviluppo sostenibile).
\
\
|
1,116,691,500,878 | arxiv | \section{Introduction}
Bitcoin is the most popular and successful cryptocurrency. Although Bitcoin has the largest market cap and holds a high volume of transactions per day, its growth as a currency has been gradual compared with its overheated growth as a speculative instrument. Bitcoin offers a lot of peer-to-peer payment opportunities, but its potential in most business use cases has not been fully realized. On the other hand, looking at the blockchain underlying Bitcoin, its technologies have attracted a great deal of attention from businesses, governments, and researchers.
The blockchain, a decentralized data management mechanism featuring tamper resistance, high availability, and data transparency, is expected to be a backbone of various commercial services besides those of the finance sector \cite{beyond}. Smart contracts, a new way to automate the execution of business work flows and their objective validation, has accelerated the evolution of blockchains into distributed application platforms. Smart contracts make blockchains suitable for a wide range of applications such as supply chains \cite{supply}, medical records \cite{medical}, online voting \cite{voting}, IoT platforms \cite{IoT}, transportation systems \cite{transport}, and energy trading \cite{energy}.
Indeed, while blockchain technology has already had a significant impact on society and business, there are many challenges still to be addressed. In particular, its industrial applications have limitations. Even though most enterprise applications require data privacy, transaction scalability, data reversibility, and protocol update-ability, these controls are not implemented on public blockchains such as Bitcoin and Ethereum.
A consortium blockchain (referred to as a ``consortium chain" after this), a type of blockchain whose consensus process is controlled by predetermined authorities, has been developed for tackling the challenges mentioned above. In particular, some groups of financial institutions and enterprises (e.g., Hyperledger Project, Ethereum Enterprise Alliance, and R3) have focused on developing implementations of blockchain platforms and modules for consortia. A number of companies have demonstrated proof-of-concept projects utilizing these platforms for enterprise use cases, to learn about the technology and its potential influence in their marketplaces.
However, there is still a major challenge to overcome before consortium chains can reach mass market levels of penetration \cite{enter}. That is, ``payment" in a consortium chain is an opaque process, and it causes a problem affecting governance of the consortium. This is distinctly different from a public blockchain, which has a settlement function using cryptocurrency. Public blockchains are open for anyone to use and, thus, they gain some network effects by using common tokens for so many entities \cite{OnPub}. These tokens can be used directly for end-to-end payments between parties, and it is possible to exchange different assets efficiently. On the other hand, although consortium chains may also have tokens, in most cases, they hold no market value. One way to add valuable tokens to a consortium chain is to have external authorities (e.g., banks) ensure their value; in such case, final settlement is done via authorities outside the blockchain. The trouble is that the existence of a concentrated authority detracts from the advantages of blockchain technology such as transparency. Here, an ICO (Initial Coin Offering), a type of crowdfunding using cryptocurrency that many startups prefer, is an alternative way to make tokens valuable, but it is challenging for most businesses to make use of one because of legal complications and high costs associated with the process.
Another, more innovative and decentralized way is utilizing a cross-chain protocol to transfer the value of assets in the public blockchain to tokens of the consortium chain. There are several cross-chain solutions to address interoperability and scalability issues. The one type of cross-chain is to place trust in validator nodes such as Federated sidechain\cite{fed}, Cosmos\cite{cosmos}, and Polkadot\cite{polka}: the other is to have individual participants themselves watch for fraud on blockchains, like in AtomicSwap\cite{atomic} and Plasma\cite{Plasma}. In any case, the common premise is that the value of assets in the main chain are transferred to tokens in another chain.
This paper presents Niji, a novel cross-chain protocol that supports atomic and secure payment on a consortium chain without any authorities. The key difference from previous cross-chain solutions is that our protocol does not transfer value to the consortium chain. Instead, by utilizing a payment channel, the protocol enables virtual Bitcoin payments on the consortium chain by verifying payments with smart contracts. Our solution is simple, yet can be easily introduced, and it is practical in the context of the consortium. Since the protocols of each chain are logically decoupled, there is no need to maintain a large and complex system of cooperation, which leads to significant cost savings. The consortium can focus on service operations rather than economic issues. In addition, from the perspective of Layer 2, our contribution scales opportunities to use bitcoins by enabling execution of secure off-chain Bitcoin payments on diverse blockchains. This is different from the approach of forming one network like Lightning, but it will be possible to use Bitcoin payment in the various universes of individual blockchains.
\subsubsection*{Outline}
Section II introduces the elements necessary to understand the rest of this article. Section III contains a detailed description of the basic Niji protocol using a uni-directional payment channel. The extension to a bi-directional channel is shown in section IV. Section V analyzes the security of the Niji protocol and its feasibility through an experimental implementation. We compare Niji with other cross-chain protocols and discuss possible extensions of this protocol in section VI and end this paper in section VII.
\section{Building Blocks}
This section establishes the necessary building blocks for understanding the Niji protocol. In the following, we introduce payment channels, which are techniques for Bitcoin off-chain payment; then we describe the Ethereum Virtual Machine with which the arbitrary code of smart contracts is executed.
\subsection{Payment Channels}
The payment channels achieve end-to-end secure payment for off-blockchain trading. Niji utilizes a payment channel as a sub-protocol on the cross-chain protocol. The simple payment channel that was first discussed by Hearn and Spilman \cite{simp} allows a payer to send a payee numerous payments without committing all of the transactions to the blockchain. The channel is essentially uni-directional; that is, the payer can send bitcoins to the payee, but the payee cannot send in the opposite direction. In a simple payment channel, only two transactions are stored in the blockchain: a {\it funding transaction} and a {\it settlement transaction}. The funding transaction is used to open the channel. The transaction deposits a payer's bitcoins into a 2-of-2 multi-signature account managed by the payer and payee. Conversely, the settlement transaction is used to close the channel. It eventually performs the transfer of funds on the blockchain and settles the balances of the payer and the payee.
While the channel is open, the payer can update his/her payment multiple times within the range of funds deposited in the multi-signature account. For example, in a first-time payment, the payer creates and signs a transaction that transfers 0.1 BTC to the payee from 1 BTC of the multi-signature account. Next, in a second payment to send the remaining 0.4 BTC, the payer creates another transaction that sends 0.5 BTC to the payee and updates the state of the channels. The payer cannot broadcast any of those transactions to the Bitcoin network, since these transactions do not have the required signature of the payee. In this paper, we refer to such an incomplete transaction as an {\it update transaction}, which specifies a funding transaction as input and includes only the signature of the payer. To transfer funds from the multi-signature account, the transaction requires signatures of both the payer and payee. Therefore, only the payee has the right to sign and broadcast the last state of the channel (i.e., settlement transaction with 0.5 BTC) at an arbitrary timing. To protect the payer from the risk that the payee does not respond and does not cooperate by broadcasting any state of the channel, a time-lock that refunds the whole 1.0 BTC to the payer is applied to the output's script of the funding transaction. There are two different types of time-lock; {\it CheckLockTimeVerify} (CLTV), which is an opcode specified in BIP65\cite{bip65}, allows users to create a transaction whose outputs are available until a concrete time in the future. On the other hand, {\it CheckSequenceVerify} (CSV), introduced in BIP112\cite {bip112}, specifies a relative time. When a transaction output including OP\_CSV is stored in the blockchain, it is necessary to wait for the specified block confirmations until the transaction is spendable again.
The simple payment channel described above is substantially uni-directional; that is, the payer can send bitcoins to the payee, but not in the opposite direction. In a bi-directional channel, the flow can go both ways. There are two well-known proposals for Bitcoin bi-directional channels. The first is called Duplex Micropayment Channels, proposed by Decker and Wattenhofer \cite{dmc}. It achieves bi-directional payment channels by using two uni-directional payment channels with a finite lifetime. The second is Lightning Network by Poon and Dryja \cite{ln}, which allows the channel to remain open indefinitely, relying on punishments to promote honesty among parties.
Moreover, regarding the existing solutions using payment channels, there is an interactive process from payment to service provision. For each payment, the payee must confirm whether the channel state is valid. When receiving an update transaction through an off-chain network, the payee must validate the transaction format and verify the included signature. Then, the payee can provide a service, say a WiFi hotspot service. If the payment of the update transaction is invalid, the provision will obviously be refused. Our solution, Niji, can automate this verification process by using smart contracts on the consortium chain.
\subsection{Ethereum Virtual Machine}
We assume that Ethereum Virtual Machine (EVM) is the execution environment of consortium chain smart contract in the Niji protocol implementation. EVM executes ordinary transactions and treats smart contract bytecode as a special transaction. Specifically, each smart contract is given its own storage to keep the state of the contract. Our protocol can run in other environments; however, we should emphasize the utility of EVM from the following aspects. First, EVM has been ported to several private blockchain platforms. The blockchains released by various companies, open source communities, universities, etc., currently have different technologies and frameworks, and market fragmentation is occurring as a result. EVM was originally intended as a runtime environment for smart contracts on the Ethereum platform, and it has since been integrated into several blockchain platforms for enterprise, such as Quorum, Monax, Hyperledger Burrow, and Ethereum on Azure. In addition, go-ethereum, an official Ethereum client of the Go implementation, can build a consortium blockchain using a proof-of-authority algorithm (EIP225\cite{clique}). These facts mean that EVM-based smart contracts can run compatibly on multiple platforms, as described above.
As a second reason, EVM smart contracts are less flexible than other Turing-complete smart contracts like Hyperledger Fabric chaincode. There are trade-offs between the flexibility that a platform provides and the security of the code which developers write for it. EVM's target is basically to provide a ``public" environment where arbitrary code of smart contracts and other operations can be executed on the Ethereum. Therefore, EVM inevitably has more restrictions on executions of smart contracts than does a platform that is secure only for closed environments. For example, the ``gas system" is a restriction to prevent eternal recursion and cycles, and it encourages developers to write efficient code (the gas system is described in section IV-A). Therefore, a protocol designed to work only on EVM may also be able to run on other Turing-complete platforms like Hyperledger Fabric.
\section{Niji Protocol}
\begin{figure*}[ht]
\centering
\includegraphics[width=\textwidth]{figure1.eps}
\caption{Overview of the Niji Protocol. Starting on the left, a user commits to a payment to the bridging contract three times ($i=0,1,2$). Straight solid lines with arrows represent sending transactions to the blockchain networks. Straight dashed lines with arrows represent retrieving data from the blockchains.}
\label{fig_sim}
\vspace{-0.8em}
\end{figure*}
The Niji protocol is essentially based on a Bitcoin payment channel. As described in section II, payment channels come in two types: uni-directional and bi-directional. For an intuitive understanding, we will first explain the uni-directional channel-based protocol (the basic protocol) and then how it is extended to the more practical bi-directional protocol.
\subsection{Overview}
Niji allows Bitcoin payments within the payment channel to trigger automatic execution of the service deployed on the consortium chain. To achieve this, the payee does not locally verify an update transaction received from the payer; instead, the EVM smart contract validates these in a consortium chain. Niji provides an autonomous process from payment to execution of a contract, without any mediators.
To begin with, let us identify the roles representing different functionalities in the Niji system:
\begin{itemize}[\IEEEsetlabelwidth{}]
\item[a)] {\it Networks}: Each node of the Niji system has connections to two blockchain networks: the Bitcoin blockchain and a consortium blockchain. Unlike Bitcoin, the consortium chain network restricts unauthorized access to the network for secure transactions. In this network, nodes do not need to be block generators (i.e., a member of a consortium authority, like a bitcoin miner), but they need to have the right to read and write data (i.e. issue transactions) to the blockchain.
\item[b)] {\it Nodes}: The {\it user} and {\it service provider} represent the parties in this protocol. The user is a payer node which pays bitcoins to the service provider for usage of the services operated on the consortium chain. The user makes Bitcoin payments which trigger execution of a smart contract on the consortium chain while the payment channel is open. The last state of payments is broadcasted as a settlement transaction by the service provider and is eventually stored in the blockchain. The service provider is a payee node with the intention to earn money by providing some kind of service (e.g., sharing a resource like energy for microgrids\cite{energy}) on the consortium chain. The service provider creates a service contract for providing the service and shares it via the consortium chain. It receives a Bitcoin payment from the user as the usage fee of this contract via the Niji protocol.
\item[c)] {\it Smart Contracts}: Two smart contracts are deployed on the consortium blockchain. The {\it bridging contract} has functions to manage the user's payments instead of the service provider doing so. It receives payments and validates them; then, it invokes the service contract's function. The {\it service contract} has an interface to invoke the core functionality of the service, and the interface is opened for the bridging contract. The two contracts are created and deployed on the consortium blockchain before beginning the Niji protocol.
\end{itemize}
\begin{table}[tbp]
\renewcommand{\arraystretch}{1.0}
\caption{Structure of Transaction Template}
\label{timing}
\centering
\begin{tabular}{llll}
\toprule Field & & Value (example)\\
\midrule
Version & & 02000000 \\
Input count & & 01 \\
\cmidrule(l){1-2}\cmidrule(lr){3-3}
Input {[}0{]} & Previous output & \textless{}funding tx hash\textgreater{} \\
& Index & \textless{}index of previous output\textgreater{} \\
& Script & only \textless{}redeemScript of previous \\
& & output\textgreater{} {\bf (no signatures)} \\
& Sequence & FFFFFFFF \\
\cmidrule(l){1-2}\cmidrule(lr){3-3}
Output count & & 02 \\
\cmidrule(l){1-2}\cmidrule(lr){3-3}
Output{[}0{]} & Value & {\bf nil} \\
& Script & OP\_DUP OP\_HASH160 \\
& &\textless{}service provider's pubkey hash\textgreater{}\\
& & OP\_EQUALVERIFY OP\_CHECKSIG \\
Output{[}1{]} & Value & {\bf nil} \\
(change given & Script & OP\_DUP OP\_HASH160 \\
back to user)& & \textless{}user's pubkey hash\textgreater{} \\
& & OP\_EQUALVERIFY OP\_CHECKSIG \\
\cmidrule(l){1-2}\cmidrule(lr){3-3}
Locktime & & 00000000 \\
Hash Type & & 01000000 \\
\bottomrule
\end{tabular}
\end{table}
The protocol itself is designed as follows. The core idea is to automate the process of updating a payment non-interactively. This is achieved using incomplete Bitcoin transactions, which we name {\it transaction templates}, where information on the remittance amount and signatures are missing. The service provider registers a transaction template to the bridging contract during the setup phase and publishes it on the consortium chain in advance. Table I shows an example of a transaction template. For simplicity, the example uses a non-Segwit (Segregated witness) transaction. (Segwit version is described in Appendix A.)
When making a payment, a user creates a signature corresponding to the payment referring to the transaction template and submits only the signature and remittance amount to the bridging contract. The bridging contract verifies the provided signature on behalf of the service provider and validates the payment by using the registered transaction template, user's signature, and remittance amount. If valid, the payment is automatically approved, which means that the service provider accepts the update transaction in the simple payment channel.
\subsection{Protocol details}
Figure 1 shows the three major phases of the Niji protocol from setup to settlement. We describe each phase in the following.
\subsubsection{Setup}
The setup involves opening the Bitcoin simple payment channel and registering a transaction template in the consortium chain. First, to open the typical simple payment channel shown in section II-A, the user and service provider create a 2-of-2 multi-signature address $\alpha_{m}$ and broadcast a funding transaction $T^{f}$ to the Bitcoin network. This transaction $T^{f}$ sends the user's bitcoins to the multi-signature accounts, where the amount is denoted as $\sigma^{f}$. For a refund on $T^{f}$, we use a relative time-lock by applying OP\_CSV \cite{bip65} to the multi-signature accounts; if the time-lock expires, the user can securely return all deposits to his/her wallet. Then, the user broadcasts a transaction ${}^{C}T(setDeposit(\alpha_{m}, \sigma^{f}, \tilde \alpha_{u}))$ as proof of opening the channel, where ${}^{C}T(func)$ is a consortium chain transaction calling a function of the bridging contact. The function $setDeposit(\alpha_{m}, \sigma^{f}, \tilde \alpha_{u})$ stores deposit information including the multi-signature address $\alpha_{m}$, deposit amount $\sigma^{f}$ in the consortium chain, and an address $\tilde \alpha_{u}$. Here, $\tilde \alpha_{u}$ is a special Ethereum-style address which is derived from the Bitcoin public key. How $\tilde \alpha_{u}$ is used for verification is explained in the next subsection.
Next, after making enough confirmations (e.g., six confirmations), the service provider confirms that $T^{f}$ is stored in the Bitcoin blockchain and generates a transaction template $\tilde T$, which is a deformed transaction lacking signatures and the remittance amount. $\tilde T$ specifies $T^{f}$ as input and corresponds to the complete transaction $T^{\sigma}_{us}$, which represents a valid Bitcoin transaction sending $\sigma$ bitcoins from the multi-signature address to the service provider's address. The subscripts $u$ and $s$ represent that $T^{\sigma}_{us}$ has the signatures of the user $u$ and the service provider $s$. $T^{\sigma}_{us}$ is a 5-tuple including $\tilde T$ as follows:
\begin{equation}
T^{\sigma}_{us}=(\tilde T,Sig^{u},Sig^{s},\sigma,\sigma^{c})
\end{equation}
where $Sig^{u}$ and $Sig^{s}$ are the respective signatures of $u$ and $s$, and $\sigma^{c}$ is amount of change returned to $u$.
At the end of setup, the service provider broadcasts a transaction ${}^{C}T(setTmpl(\tilde T))$ to the consortium chain network. A function $setTmpl(\tilde T)$ stores $\tilde T$ in the storage space of the bridging contract on the consortium blockchain.
\subsubsection{Payment}
This phase allows the bridging contract to validate each payment on behalf of the service provider. The bridging contract has a function $getTemplate()$ for retrieving the stored $\tilde T$ and a function $update (Sig, \sigma)$ for updating the signature value and the remittance amount. First, the user obtains $\tilde T$ from the bridging contract by using $getTemplate()$ and validates it if its format is correct. Then, when making a payment, the user provides the user's signature $Sig^{u}$ and $\sigma$ for the bridging contract as a proof of payment using $update (Sig, \sigma)$. The user generates $Sig^{u}$ as follows:
\begin{equation}
modtx = SignatureForm(\tilde T, \sigma)
\end{equation}
\begin{equation}
Sig^{u} = EcdsaSign(sk^{u}, Sha256d(modtx))
\end{equation}
$ modtx $ is the modified transaction form just before it is signed, which is created by $\tilde T$ and $\sigma$, where the redeemScript of $T^{f}$ is placed into $ modtx $'s input and a hash type constant is temporarily appended to the end. To generate signatures, Bitcoin uses ECDSA (the Elliptic Curve Digital Signature Algorithm) over the standard elliptic curve secp256k1, which requires possession of the signing secret key $sk^{u}$ and a sha256 double hash of $modtx$ to be signed \cite{hardB}. ${}^{C}T(update(Sig, \sigma))$ is broadcasted to the consortium blockchain network by the user and stored in the blockchain, and the bridging contract validates its payment automatically. The validation includes confirming whether the value of $\sigma$ is under the value of $\sigma^{f}$ and verifying that the signature $Sig^{u}$ is correct. Algorithm 1 lists the pseudo-code of the payment updating process in the bridging contract.
\begin{algorithm}[tbp]
\caption{Updating a payment}
\label{array-sum}
\begin{algorithmic}[1]
\Procedure {UpdatePayment}{$Sig,\sigma$}
\State Obtain $\tilde T$, $\tilde \alpha_{u}$, $\sigma^{f}$ from contract storage;
\State $fee \gets$ configured bitcoin transaction fee
\State $S_{con} \gets$ deployed service contract
\State $\sigma_{l} \gets 0$ or latest remittance amount
\If{$\sigma_{l}<\sigma$ and $\sigma\leq(\sigma^{f}-fee)$}
\State $modtx \gets SignatureForm(\tilde T, \sigma)$
\State $h \gets Sha256d(modtx)$
\State $(v,r,s) \gets Sig$
\State $pk \gets EcdsaRecover(h, v, r, s)$
\State $addr \gets EthereumAddress(pk)$\\ \Comment {Convert Bitcoin's ECDSA public key to Ethereum-style Address}
\If{$\tilde \alpha_{u} = addr$}
\State $s \gets true$
\State $result \gets S_{con}.invoke()$
\State Save ($Sig$, $\sigma$) in contract storage;
\State \textbf{return} $result$
\EndIf
\EndIf
\State \textbf{return} $fail$
\EndProcedure
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[tbp]
\caption{Generating an update transaction}
\label{array-sum}
\begin{algorithmic}[1]
\Procedure {GetUpdateTx}{$ $}
\State Obtain $\tilde T$, $\sigma_{i=k}$, $Sig^{s}_{i=k}$ from contract storage;\\ \Comment {$k$ is last increment number;}
\State $fee \gets$ configured bitcoin transaction fee
\State $\sigma^{c}\gets\sigma^{f}-\sigma_{k}-fee$
\State $T^{\sigma}_{u\cdot}=(\tilde T,Sig^{u},\cdot,\sigma,\sigma^{c})$
\State \textbf{return} $T^{\sigma}_{u\cdot}$
\EndProcedure
\end{algorithmic}
\end{algorithm}
\subsubsection*{Verification}
Verification of $Sig^{u}$ in the EVM implementation requires a little ingenuity. Since signature verification using the native opcodes of EVM has too much overhead, we use ``precompiled contracts" \cite{pcon}, which allow complex cryptographic computations to be used in EVM. A precompiled contract is a specific pre-defined optimized function that runs outside the EVM, but it can be called from a normal smart contract that runs inside the EVM. $EcdsaRecover$, one such precompiled contract, can verify an elliptic-curve signature and recover its public key $pk$ as follows:
\begin{equation}
EcdsaRecover(h, v, r, s) = pk
\end{equation}
where $h$ is a 32-byte message to be signed, and $(v, r, s)$ is the ECDSA signature for the message (v is the recovery id, a 1 byte value specifying the sign and finiteness of the curve point \cite{pcon}). This signature is compatible with a Bitcoin-style signature which has a strict DER (Distinguished Encoding Rules) format \cite{bip66}; therefore, the public key $pk$ is recovered using $h=Sha256d(modtx)$ and $(v, r, s)^{u}=Sig^{u}$ through $EcdsaRecover$. Finally, the verification is successful if the Ethereum-style address converted from $pk$ matches $\tilde \alpha_{u}$ stored in the bridging contract. Accordingly, $Sig^{u}$ can be verified using $\tilde T$, $\sigma$, and $\alpha_{u}$, which are registered in the bridging contract. Here, the reason for the verification being done via an Ethereum-style address is the specification of Solidity, a high-level language for implementing Ethereum smart contracts. (The verification process in Solidity is detailed in Appendix B.)
\subsubsection*{Update payments}
The total payment amount is updatable multiple times up to the amount of $\sigma^{f}$ that the user first deposited. We denote $\sigma$ and $Sig^{u}$ after update $i$ as $\sigma_{i}$ and $Sig^{u}_{i}$ for $i = 0,\ldots,n$. The user adds a new payment to the current payment amount $\sigma_{i}$ and replaces $\sigma_{i}$ and $Sig^{u}_{i}$ with $\sigma_{i+1}$ and $Sig^{u}_{i+1}$ by broadcasting ${}^{C}T(update(Sig^{u}_{i+1}, \sigma_{i+1}))$. $\sigma_{i}$ must be under $\sigma_{i+1}$ ($\sigma_{i}<\sigma_{i+1}$), since the payment channel is uni-directional. This is checked in the bridging contract.
\subsubsection*{Triggering contract's function}
$invoke()$, a function of the bridging contract, can call services provided by the service contract. After validation of a payment, the bridging contract executes $invoke()$, triggering the service contract's function corresponding to the payment. This process is an internal communication between smart contracts that protects nodes in the procedure. Thus, the procedure of transaction ${}^{C}T(update(Sig, \sigma))$ runs from payment to provision safely and automatically.
\subsubsection{Settlement}
In the settlement phase, the service provider obtains an update transaction from the bridging contract, signs it, and broadcasts it to the Bitcoin network as a settlement transaction. Algorithm 2 lists the pseudo-code of the bridging contract. $getUpdateTx()$, a function of the bridging contract, returns the update transaction $T^{\sigma}_{u\cdot}$ (the service provider's signature is replaced with $\cdot$, which represents “not yet signed”) to the service provider. After signing $T^{\sigma}_{u\cdot}$ with its own secret key, the service provider broadcasts the complete transaction $T^{\sigma}_{us}$, i.e., the settlement transaction for Bitcoin. Finally, the service provider broadcasts a transaction ${}^{C}T(closing())$ that makes any further payments to the bridging contract unacceptable.
\section{Niji with bi-directional channel}
The previous section described the basic specifications of the Niji protocol using the simple payment channel. As mentioned there, the simple payment channel cannot send in the opposite direction, which means that cancellation of payments is not allowed. Security in the simple payment channels is based on the fact that the recipient of payment does not have an incentive to broadcast the old state of the channel; that is, the latest update transaction always brings the maximum benefit to the recipient. However, if a payment in the opposite direction occurs, the recipient has an incentive to broadcast the old state of the channel at any point in time. In the Niji protocol, the service provider can broadcast and settle an old update transaction any time, in spite that the parties have agreed to cancel the payment.
Hence, to achieve secure cancellation, we considered using Duplex Micropayment Channels (DMC) or Lightning Network channels, both bi-directional channels, in the Niji protocol. In DMC, since both parties create two uni-directional channels, the service provider, as well as users, needs to deposit funds. The drawback of using this design is if the service provider has numerous customers, a large amount of money will be tied up. On the other hand, the Lightning Network payment channel requires both parties to exchange their signatures each time a channel state is updated. This interactive process hinders automation from payment to execution of a contract, one of the advantages of Niji. Also, both parties have to monitor not only the consortium chain but also the Bitcoin network to prevent counterparty fraud.
To avoid the above problems, we propose a new non-interactive channel design suitable for the Niji protocol. In particular, to ensure autonomous blockchain cooperation, the bi-directional channel is designed to meet the following requirements.
\begin{itemize}[\IEEEsetlabelwidth{Req.}]
\item[Req. 1.] The user can update their payment without interacting with the service provider node.
\item[Req. 2.] The user is not required to monitor the Bitcoin network and does not have to be on-blockchain at all times.
\end{itemize}
Note that unlike other bi-directional channels, Niji's bi-directional channel takes into account the following special consideration: almost all payments are from the user to the service provider; transactions in the opposite direction occur only a few times (such as cancellation of payment). The central idea of our channel is to introduce this asymmetry in the procedures in each direction. Parties can update normal payments non-interactively, but temporarily collaborate when canceling payments. Our channel is not completely bi-directional like Lightning, but it is adequate for most of the targeted use cases. In following subsections, we describe the separate channel designs for normal payment and cancellation.
\begin{figure}[tbp]
\centering
\includegraphics[trim=0 0 0 0,clip, width=0.36\textwidth]{figure2.eps}
\caption{Examples of update transactions in our bi-directional channel between user and service provider (SP). TL and HL are parameters for unlocking the time-lock and hash-lock, respectively.}
\label{fig_sim}
\end{figure}
\subsection{Payments in the normal direction}
Our channel consists of three type of transaction: funding, update, and settlement. In the normal direction of payment, the channel acts like a simple payment channel. That is, in each payment, the user increases his/her remittance amount and never presents a remittance amount lower than the previous amount. The update transactions have two outputs: the first one specifies the remittance amount, and the second specifies the amount of change given back to the user. The spending output for the remitted amount is restricted by the following additional conditions.
\subsubsection*{Condition 1}
The spending of the output is restricted by a time-lock. The service provider can unlock the script with its signature only after a specified time-lock expires.
\subsubsection*{Condition 2}
The spending of the output is restricted by a hash-lock. The user can unlock the script with the hash's pre-image and its signature.
These conditions are expressed using a Bitcoin script, and either condition is fulfilled by a script in the inputs of the next transaction. Figure 2 shows examples of update transactions that include these conditions in the first output. The examples indicate that the user creates a new update transaction $T^{0.5}_{u\cdot}$ with the remittance amount increased from {0.3BTC} to {0.5BTC}.
The service provider can move the gained remittance to its own wallet by using condition 1 in the first output. However, it is impossible for the service provider to immediately fulfill the condition as it is encumbered with the time-lock. It can obtain the funds only after the time-lock expires. In the example in Figure 2, $TL$ is denoted as the time until the transaction becomes spendable again; if $TL$ = 110, it takes 110 blocks from the point where the transaction occurred. The time-lock gives the user a period for preventing any dishonesty on the part of the service provider.
Condition 2 exists as a user's countermeasure against betrayal by the service provider. The hash-lock requires the subsequent input to include the corresponding pre-image (i.e., $S_{0}$ in Figure 2) of the hash in order to be spendable. A hash is passed to the user beforehand by the service provider, and if the user obtains the pre-image of the hash, he/she can spend the update transaction.
Now, with respect to the hash-lock, it should be noted that the same pre-image $S_{0}$ is reused before and after the update, as shown in Figure 2. This is in stark contrast to the Lightning payment channel, where the pre-images are recreated each time a state is updated. In Lightning, both parties need to exchange information including old pre-images and the hash values of new pre-images to create the next update transaction; thus they are required to interact with one another and to be always online in the meantime. In the proposed channel in Niji, it is not necessary to recreate new images each time the state is updated. This satisfies the first requirement (Req. 1), where interactions with parties are unnecessary, and it enables the service provider to delegate its update processing to the smart contract.
Additionally, the deadline of the time-lock $TL_{u}$ of an update transaction is always set to be after the time-lock $TL_{f}$ of a funding transaction. For example, if $TL_{f}$ is set to 100 blocks, $TL_{u}$ must be set to 101 blocks or later. These time-lock settings definitely guarantee that the service provider can not move the gained funds until the channel expires. Therefore, the user does not have to monitor the Bitcoin network continuously while the channel is alive in order to be sure that the service provider has not committed fraud; therefore, these restrictions result in the fulfillment of the second requirement (Req. 2).
Now, let us describe how the above bi-directional channel for normal payments is applied to the Niji protocol. In particular, it is achieved by adding constraints to the output of the transaction template which is part of an update transaction. Table II shows an example of a whole transaction template and its output[0]'s redeemScript including condition 1 and condition 2 in Figure 3. We will express the transaction template $\tilde T$ as $\tilde T(TL, HL)$; therefore, the template having the redeemScript shown in Figure 3 can be represented as $\tilde T(110, S_{0})$. The procedure for a normal payment is no different from the basic Niji protocol described in section III, except that the redeemScript must be shared with the user; hence, we modify the $setTmpl$ function from $setTmpl(\tilde T)$ to $setTmpl(\tilde T(TL, HL), redeem)$, where $redeem$ is a redeemScript corresponding to $\tilde T(TL, HL)$. Of course, the bridging contract can verify the correctness of $redeem$, which includes whether $redeem$ is the same as the hash of output[0] script in $\tilde T(TL, HL)$, whether the $TL$ included in $redeem$ is longer than the number of blocks specified in $TL_{f}$ of a funding transaction, and so on.
\begin{table}[tbp]
\renewcommand{\arraystretch}{1.0}
\caption{Structure of Transaction Template in Bi-directional Channel}
\label{timing}
\centering
\begin{tabular}{llll}
\toprule Field & & Value \\
\midrule
Version & & 02000000 \\
Input count & & 01 \\
\cmidrule(l){1-2}\cmidrule(lr){3-3}
Input {[}0{]} & Previous output & \textless{}funding tx
hash\textgreater{} \\
& Index & \textless{}index of previous output\textgreater{} \\
& Script & only \textless{}redeemScript of previous \\
& & output\textgreater{} {\bf (no signatures)} \\
& Sequence & FFFFFFFF \\
\cmidrule(l){1-2}\cmidrule(lr){3-3}
Output count & & 02 \\
\cmidrule(l){1-2}\cmidrule(lr){3-3}
Output{[}0{]} & Value & {\bf nil} \\
& Script & OP\_HASH160 \\
& & {\bf \textless{}hash of redeemScript in Fig. 3\textgreater{} }\\
& & OP\_EQUAL \\
Output{[}1{]} & Value & {\bf nil} \\
(change given & Script & OP\_DUP OP\_HASH160 \\
back to user)& & \textless{}user's pubkey hash\textgreater{} \\
& & OP\_EQUALVERIFY OP\_CHECKSIG \\
\cmidrule(l){1-2}\cmidrule(lr){3-3}
Locktime & & 00000000 \\
Hash type & & 01000000 \\
\bottomrule
\end{tabular}
\end{table}
\begin{figure}[tbp]
\renewcommand{\arraystretch}{1.0}
\footnotesize
\begin{tabularx}{\columnwidth}{ll}
\midrule
& OP\_IF \\
& \qquad OP\_HASH160 \\
& \qquad \textless{}hash of $S_0$\textgreater{} \\
& \qquad OP\_EQUALVERIFY \\
& \qquad \textless{}user's pubkey\textgreater{} \\
& OP\_ELSE \\
& \qquad 110 OP\_CSV\\
& \qquad OP\_DROP \\
& \qquad \textless{}service provider's pubkey\textgreater{} \\
& OP\_ENDIF \\
& OP\_CHECKSIG \\
\midrule
\end{tabularx}
\caption{Example of redeemScript in output[0]}
\end{figure}
\subsection{Cancellation in the opposite direction}
Next, we describe how the protocol behaves in the opposite direction, i.e., when payments are canceled. Unlike a normal payment, several agreement operations between parties are required to cancel payments. The user and service provider need to interact with one another and exchange materials for agreement on the cancellation. The key steps are as follows.
\begin{itemize}[\IEEEsetlabelwidth{STEP}]
\item [STEP1.] The user sends a cancel request to the service provider.
\item [STEP2.] The service provider creates a new pre-image and returns the hash value of it.
\item [STEP3.] The user sets the received hash and submits a signature for the new update transaction.
\item [STEP4.] The service provider validates the user's signature and discloses the old pre-image of previous update transaction.
\end{itemize}
\begin{figure}[tbp]
\centering
\includegraphics[width=0.36\textwidth]{figure4.eps}
\caption{Example of update transactions for cancellation of payment in Niji bi-directional channel. Even if the service provider broadcasts an old update transaction, the user can get back the funds by using the disclosed pre-image $S_{0}$.}
\label{fig_sim}
\end{figure}
\begin{figure}[tbp]
\centering
\includegraphics[width=0.5\textwidth]{figure5.eps}
\caption{Process for cancellation of payment in Niji protocol}
\label{fig_sim}
\end{figure}
With the completion of the above four steps, the cancellation is deemed to be agreed upon by the parties. Afterwards, the service provider can no longer broadcast the old update transaction. Figure 4 shows that the old update transaction is replaced with a new update transaction. The remittance amount is reduced from {0.5BTC} to {0.4BTC} with a new pre-image $S_{1}$. If the service provider broadcasts an old update transaction, the disclosed pre-image $S_{0}$ allows the user to get back all funds. (Techniques similar to this can be found in HTLCs or Lightning Network.) Moreover, if the procedure breaks down or terminates before the agreement, it can be regarded that an agreement has not been reached. For example, if the cancellation request is invalid, the service provider can not respond to the user's request.
The cancellation of payment operations is illustrated in Figure 5. Here, it is paramount that all materials required for cancellation are exchanged via the consortium chain. This is achieved by registering a new transaction template which includes the hash value of a new pre-image on the consortium chain (equivalent to STEP 2). $replaceTmpl(\tilde T(110, S_{j+1}), redeem_{j+1})$ replaces the existing $T(110, S_{j})$ and $redeem_{j}$ with a new $T(110, S_{j+1})$ and $redeem_{j+1}$ in the storage space of the bridging contract, where the increment count $j$ represents the number of cancellations. Then, the user makes $Sig_{i+1}$ with the new amount, which is the reduced remittance after cancellation, and registers $Sig_{i+1}$ to the bridging contract via the $update$ function (equivalent to STEP 3). Finally, the service provider discloses $S_{j}$ via the $canceled(S_{j})$, which publishes the pre-image of the previous update transaction on the consortium chain (equivalent to STEP 4).
For secure trading, these procedures can be strictly validated by using a smart contract on the consortium chain, i.e., a bridging contract, which prevents service providers from broadcasting old update transactions. In addition, success of cancellation in the contract can automatically trigger the service contract's functions such as $revoke()$, which reverts the state back to one before the payment was canceled.
\section{Analysis}
The new bi-directional payment channel for the Niji protocol enables autonomous cooperation between blockchains. In the following, we analyze the assumptions of security and evaluate the feasibility of our prototype.
\subsection{Security}
The security of the Niji protocol essentially depends on the safety of the payment channel. Although different channels may be applied to Niji in the future (see section VI-C), we will consider the safety of only the bi-directional channel proposed in this paper. In Niji, since the service provider always broadcasts a settlement transaction, it has more discretion when it comes to making the settlement in comparison with the user. Therefore, the major security risk we consider is an act of betrayal of the service provider. The service provider could potentially behave dishonestly in three ways.
\begin{itemize}[\IEEEsetlabelwidth{}]
\item [1.] No settlement is made at all.
\item [2.] Despite that a cancellation was approved, the service provider broadcasts an old update transaction.
\item [3.] The contract deployed by the service provider does not run properly.
\end{itemize}
In the first case, the service provider abandons the protocol itself and refuses to broadcast a settlement transaction within the channel expiration. To prevent such a deadlock of funds, the output script of a funding transaction allows the user to return all deposits to his/her wallet after the funding transaction's time-lock expires; that is, the user can broadcast the settlement transaction with only the user's signature. The second case is that the service provider does not broadcast the latest channel state that reflects cancellation and instead broadcasts an old channel state which has more incentives. This fraud can be prevented by using the disclosed pre-image at the end of the cancellation procedure, as described in the previous section. With the user's signature and disclosed pre-image, the user can spend the output of a settlement transaction under condition 2 described in section IV-A.
The third case is caused by an issue with the contract itself rather than the operation of the payment channel. All contracts in the Niji protocol, which include Bitcoin scripts and smart contracts on the consortium chain, are provided by the service provider. The user must confirm that the contract is the expected one to ensure safety. Unlike conventional cross-chains using an oracle outside of the blockchain, all of our contracts are on a consortium chain and the entire transaction history is shared among the participants sharing the blockchain; therefore, it is possible for the user to re-validate the state changes and observe whether the contract works properly via the blockchain. In addition, as required by the security level, it is assumed that a healthy consortium chain network continues; hence in the consortium chain network, the block creators are required to be honest and faithful to the protocol; e.g., they are encouraged to use a consensus algorithm with transaction finality like PoA and PBFT.
Incidentally, Plasma \cite{Plasma}, a cross-chain framework that has similar aspects to our approach, has precautions that involve cooperation between blockchains; e.g., a child blockchain is enforced to roll back by submitting proof of fraud to the parent blockchain. In our approach, there is no close cooperation between blockchains to solve the problem of fraud on the consortium chain. Instead, the Bitcoin payment channel provides end-to-end security to users, which is logically separated from the consortium chain. Our approach is simple, but has the flexibility to connect to various consortium chains without being bound by a specific specification. Niji can also be easily combined with existing solutions to improve its abilities.
\subsection{Feasibility}
We evaluated the feasibility of the Niji protocol between the Bitcoin test-net and an EVM-based consortium chain built using the Ethereum client. The prototype was implemented using the bitcoin-core client and go-ethereum client, which supports clique proof-of-authority \cite{clique} as a consensus algorithm for building consortium chains \cite{cons}. We placed four nodes with authority to generate blocks in the Ethereum-based consortium chain and changed the period of proof-of-authority consensus from a default 15 seconds to 1 second to get those blocks mined faster. The Bitcoin client is only used to broadcast Bitcoin transactions between setup and settlement. We confirmed that all Bitcoin transactions in the Niji protocol were accepted into Bitcoin test network.
In this evaluation, we focused on the performance related to consortium chain transactions. Additionally, to clarify only the payment performance of Niji, we excluded measurements related to service provision in the service contract.
\subsubsection{Computational cost}
\begin{table}[tbp]
\renewcommand{\arraystretch}{1.2}
\caption{Gas Cost of Computational Work}
\label{gas_cost}
\centering
\begin{tabular}{lrr}
\toprule Operation & \multicolumn{2}{c}{Gas cost (gas)} \\
\cmidrule(lr){2-3}
& uni-directional & bi-directional \\
\midrule
Deploy bridging contract & 1,785,044 & 2,562,757\\
Set deposit information & 245,922 & 245,988\\
Set transaction template & 285,183 & 431,100\\
Update payment & 455,355 & 468,389\\
Replace transaction template & - & 431,164\\
Disclose pre-image & - & 53,687\\
Close channel & 46,626 & 46,648\\
\bottomrule
\end{tabular}
\end{table}
The bridging contract coded by Solidity, Ethereum’s most popular high-level language, exposes several API calls to receive transactions. These operations consume {\it gas}, the internal pricing used for executing operations in the EVM. Each computation of the contract codes has an associated cost in gas, protecting the blockchain network from denial-of-service attacks with infinite loops and encouraging efficiency in the code. The gas system is also useful on the consortium chain in terms of forcibly terminating harmful code execution, even if the gas itself has no value (i.e., users need not pay for gas with crypto-currency) unlike in the public Ethereum blockchain. In fact, Quorum, the EVM-based consortium chain from JP Morgan, removes pricing of gas, although the gas system itself remains \cite{quor}.
The transaction gas costs of operations through the bridging contract are listed in Table III. These values come within the usual range found in a public Ethereum block, which has a gas limit of approximately 8.0 million (in June 2018). The bi-directional channel tends to require higher gas cost than the uni-directional one for the management of the cancellation procedure. When constructing the consortium chain, the gas limit in each block and the amount of gas included in a transaction should be adjusted with reference to these results. In actual applications, since a payment calls the function of the service contract automatically, the corresponding gas cost for the service is added to the gas value of updating payment shown in Table III.
\subsubsection{Timing analysis}
\begin{table}[tbp]
\renewcommand{\arraystretch}{1.2}
\caption{Time Analysis of Operations in Payment Phase}
\label{timing}
\centering
\begin{tabular}{lcccc}
\toprule Task & Mean & Maximum & Minimum & \shortstack{Std. \\ Deviation}\\
\midrule
Get transaction template & 116 & 454 & 101 & 18.0 \\
Update payment & 1981 & 2664 & 1112 & 252 \\
Total & 2097 & 2778 & 1218 & 253 \\
\end{tabular}
\end{table}
Table IV outlines the timing analysis measurements for the payment phase in Niji with bi-directional channel. All times are in milliseconds. We executed 1000 experimental runs and computed latency statistics including the average (mean), maximum, minimum, and standard deviation. All measurements were performed on a MacBook Pro running OS X 10.13.4 equipped with four cores, 2.3GHz Intel Core i5, and 16 GB memory. One payment operation was divided into two tasks: getting the transaction template and updating payment. ``Total" is the end-to-end response time as defined in one payment. We used Node.js as the measurement environment of the Niji protocol to communicate with go-ethereum and bitcoin-core. Each operation included not only the processes in Node.js but also those in the EVM, such as verifying a signature and parsing a transaction.
In the process of getting a transaction template, the user never broadcasts transactions. Since data is only retrieved from the blockchain, the process runs in a steady rate. On the other hand, payment update, which has time ranges from $1121$ ms to $2664$ ms, is clearly dominated by the consensus time, i.e., the time until the registered data is stored in a new block. In this experiment, we set the parameter of the consensus time to 1 second; therefore, $2094-1000 = 1094$ ms is the overhead for each payment. Although the payment can be updated many times as long as the channel is open, a certain latency as above is required for each payment. Therefore, services using the Niji payment protocol must be designed with this constraint in mind.
Moreover, although scalability is an aspect of performance, it entirely depends on the throughput of the consortium chain itself. The Niji protocol certainly contributes to an improvement of the Bitcoin scalability, but the choice of consortium chain platform may have a great effect on this aspect. In the next section, we discuss possible extensions to Niji, including ones aimed at scalability.
\section{Discussion}
Here, we compare Niji with other cross-chain protocols and discuss possible extensions of the Niji protocol to a wide range of future applications. These extensions could remain within the basic Niji concept or be combined with existing solutions to improve its abilities.
\subsection{Comparison with related work}
Several cross-chain protocols have been proposed for improving scalability and interoperability. These approaches can be classified into two types: utilizing intermediaries like a federation or decentralized networks and directly connecting blockchains with the cooperation of participates. The former way has a premise that the participants place trust in the intermediaries. Federated sidechain\cite{fed} relies on a federation for honest activity and the federation controls multi-signature locks to transfer coins between Bitcoin and the sidechain. As more decentralized approaches, Cosmos\cite{cosmos} and Polkadot\cite{polka} have an inter-blockchain to relay data between blockchains, respectively referred to as Cosmos hub and Relay chain. The inter-blockchain is operated by incentivized nodes, but the fact that participants rely on validators is similar to the federation. Although these approaches certainly need to place trust in intermediaries, their participants are less burdened, as the intermediaries are responsible for the connections and monitoring costs.
On the other hand, in the latter way, i.e., directly connecting blockchains with the cooperation of participates, individual participants themselves need to take precautions to prevent fraud. Atomic swaps \cite{atomic} (or atomic cross-chain trading) is the exchange of one cryptocurrency with another cryptocurrency between two parties, without the need to trust a third party. The tokens are directly traded between users in a trustless atomic manner, but both users must continuously monitor the blockchains to create and broadcast the transaction synchronously together, which places a burden on them. Plasma is a powerful framework that makes smart-contract execution scalable by associating block creations in the child blockchain with the root Ethereum blockchain. This approach also requires periodic commitments, i.e., submission of block headers to the root blockchain. In addition, to prevent fraud, it is necessary for users to constantly monitor both parent and child chains; in general, the latter way tends to burden individual participants.
Although Niji is classified as the latter sort of method, it has a different aspect from other related work. The key idea is that there is no value transfer of bitcoins to tokens of another blockchain. In most of the related work, by bounding funds in a particular contract, you can swap them with another token that holds the same value as the bounded funds in another blockchain. If another blockchain (i.e., consortium chain) fails to work properly due to misconduct or fraud, funds will be not protected at all, unless there is an incentivized and massive mechanism protects the funds like in Plasma. By contrast, in Niji, value transfer of bitcoins does not occur between blockchains. Instead, the value of bitcoins can be used directly and circulated on the consortium chain. Niji has no tight cooperation between blockchains, but it achieves a simple and flexible connection to another blockchain, which has certain security based on the payment channel. Our protocol design reduces the monitoring costs and its concept eliminates the risk of price fluctuations in tokens. As an analogy, the U.S. dollar can be used as legal tender in some countries where either no local currency is issued or both the local currency and US dollar coexist. ``Dollarization" stabilizes the value of the local currency, which leads to stabilization of prices and the economy of the whole country; similarly, ``Bitcoinization" in consortium chains that built by startups or enterprises would bring about stable development and growth for their services.
\subsection{Niji in combination with other platforms}
The Niji implementation described in this paper uses an EVM-based consortium chain platform, and Bitcoin is the only cryptocurrency supported. Of course, the Niji protocol can easily be extended to other platforms, e.g., a combination of Litecoin and Hyperledger Fabric if their requirements can be satisfied. The requirement of available currency is to have a scripting function similar to Bitcoin, that is, the currency has a multi-signature address and the time-lock and hash-lock functionalities. On the other hand, the choice of consortium chain platform has a direct impact on the scalability of Niji payments. Hyperledger Fabric has the ability to handle Turing-complete smart contracts similarly to EVM-based platforms and as well provides high throughput performance \cite{perm}. Since the Niji protocol is designed to run even in restricted environments like EVM, it may work on many consortium platforms including Hyperledger. This would enable appropriate platforms to be selected according to the use case.
\subsection{Possible improvement with eltoo channel}
Decker, Russell, and Osuntokun have proposed the eltoo protocol \cite{eltoo} as another approach to realizing a simpler and more efficient bi-directional channel. Unlike the Lightning Network channel, this novel channel does not require penalty branches. It thus reduces monitoring costs and overcomes the problem of asymmetry of information that endpoints have. The protocol works by overriding the previous state and forces old transactions to be unavailable. eltoo introduces the concept of state numbers, similar to sequence numbers in the original Nakamoto implementation of Bitcoin. While the original sequence numbers of the Nakamoto implementation were not enforceable, eltoo's state numbers enforce replacement of transactions by ensuring that a later state can resume any of the previous states until the last settlement transaction is confirmed. In order to achieve the replacement mechanism, it is necessary to introduce a new sighash flag, SIGHASH\_NOINPUT, which selectively marks transactions that can be tied to previous transactions.
The eltoo protocol can be applied to Niji, because it does not require interactive exchange of something to prevent counterparty fraud when updating a payment. The main benefit of using eltoo would be that it simplifies the cancellation of payment procedure. In Niji, the state number is managed by a smart contract, and the latest state always has priority regardless of whether the money sent to the service provider is increased or a payment is cancelled. The cancellation procedure described in section IV-B requires four steps, but with eltoo, it would be shortened to two steps: the user requests to cancel a payment and then the service provider agrees to it by submitting its signature. In addition, it is also possible for the service provider to refund the user by unilaterally submitting its signature.
In order to enable eltoo protocol on Niji, we need to modify the template transactions slightly. The eltoo channel requires two types of transaction, one for update and one for settlement, and a settlement transaction always requires a new public key pair. These features might entail modifications to the verification process in the bridging contract. More importantly, it is necessary that the eltoo protocol itself become available in the Bitcoin protocol; that is, the new features that compose eltoo, state numbers and SIGHASH\_NOINPUT, must be made available in the Bitcoin blockchain infrastructure in the future.
\section{Conclusion}
We presented Niji, a cross-chain protocol for atomic and secure Bitcoin payment on a consortium blockchain. This protocol provides a means of payment into a consortium chain using the Bitcoin payment channel. The process autonomously runs from payment to service provision with no need for trusted third parties, and with no transfer of bitcoins to other tokens. Our first implementation is designed to work on an EVM-based consortium chain, and our experiments showed the practical feasibility of the protocol. We discussed the security of Niji as well as future extensions that could make use of existing platforms to improve the functionality and scalability of the protocol. These extensions could potentially make payments on the consortium chain more flexible and efficient for a wide range of applications.
|
1,116,691,500,879 | arxiv | \section*{Introduction}
Let $\alg{G}$ be a quasisplit reductive group over a local or global field $F$. Let $n$ be a positive integer and let $\mu_n$ denote the group of $n^{\th}$ roots of unity in $F$. Assume that $\mu_n$ has order $n$. Let $\alg{G}'$ be a central extension of $\alg{G}$ by $\alg{K}_2$, in the sense of Brylinski and Deligne \cite{B-D}. We call the pair $\alg{\tilde G} := (\alg{G}', n)$ a ``degree $n$ cover'' of $\alg{G}$. Fix a separable closure $\bar F / F$ and write $\Gal_F = \Gal(\bar F / F)$. Fix an injective character $\epsilon \colon \mu_n \hookrightarrow \mathbb C^\times$.
The purpose of this article is to provide a robust definition of the L-group ${}^{\mathsf L} \tilde G$ associated to this data. It will be an extension,
$$\tilde G^\vee \hookrightarrow {}^{\mathsf L} \tilde G \twoheadrightarrow \Gal_F,$$
where $\tilde G^\vee$ is a complex reductive group. The reader may replace $\mathbb C$ by any $\mathbb Z[1/n]$-algebra $\Omega$ endowed with $\epsilon \colon \mu_n \hookrightarrow \Omega^\times$, if desired.
We have accumulated evidence that this L-group can be used to parameterize irreducible $\epsilon$-genuine representations (admissible in the local case, automorphic in the global case), in much the same way as the L-group is expected to parameterize representations in the linear case. Our evidence, including cases of split tori, unramified representations (when $\alg{G}$ is unramified and $n$ is coprime to the residual characteristic), and double-covers of semisimple groups over $\mathbb R$, will be forthcoming. Further evidence is provided by the large literature on covering groups: isomorphisms of Hecke algebras, theta correspondences, character lifting, and more, which may be viewed as examples of functoriality if one accepts our proposed L-group.
We devote this entire article to the definition of the L-group ${}^{\mathsf L} \tilde G$, since it requires care to make sure that it is ``well-defined'' by the data given (independent of other choices like Borel subgroup and maximal torus). We strive to give the cleanest definition possible and provide examples along the way. A crucial step involves the construction of a gerbe $\gerb{E}_\epsilon(\alg{\tilde G})$ on $F_{\et}$; the appendix is meant to provide a suitable introduction to gerbes of this sort. Another section defines the ``metaGalois group,'' a canonical extension $\mu_2 \hookrightarrow \widetilde{\Gal}_F \twoheadrightarrow \Gal_F$. The metaGalois group may of intrinsic arithmetic interest, and provides hints of a motivic connection. We do \textbf{not} assume that $F$ contains a primitive $(2n)^{\th}$ root of unity; such an assumption negates the need for the metaGalois group.
Our previous article \cite{MWCrelle} was limited to split reductive groups, and may be viewed as a proof-of-concept by Hopf algebra bludgening. The ``dual group'' $\tilde G^\vee$ here is the same as that from \cite{MWCrelle}. The ``first twist'' of \cite{MWCrelle} is unchanged in substance, but here it is encoded in a 2-torsion element of the center of $\tilde G^\vee$ and the metaGalois group. The ``second twist'' of \cite{MWCrelle} provided the greatest challenge. Among many reformulations that succeed or split groups, we found the gerbe $\gerb{E}_\epsilon(\alg{\tilde G})$, which extends to quasisplit groups.
\subsection*{Philosophies}
A few principles are helpful when considering any putative Langlands program for covering groups.
\begin{enumerate}
\item
There is no $\epsilon$-genuine trivial (or Steinberg) representation for general covers, and so one should not expect a single distinguished splitting of the L-group.
\item
If some set of things is parameterized by cohomology in degree $2$, then that set of things should be viewed as the set of objects in a 2-category.
\item
Things which ``are trivial'' (e.g., extensions, gerbes) can be isomorphic to trivial things in interesting ways.
\end{enumerate}
\subsection*{Acknowledgments}
The ideas of this paper have evolved over the past few years, and I am very grateful for the numerous mathematicians who discussed the constructions in various stages of completeness and correctness. The American Institute of Mathematics hosted a conference at which I spoke with Wee Teck Gan, Gordan Savin, Jeffrey Adams, Sergey Lysenko, Tamotsu Ikeda, Kaoru Hiraga, Tasho Kaletha and others. Their previous work, and our discussions at the conference and elsewhere, have been very helpful. I also appreciate the support of Harvard University during a short visit, where I gained insight from discussions with Dennis Gaitsgory, Dick Gross, and John Tate. During a visit to the University of Michigan, I gained from feedback from Stephen DeBacker, and learned about gerbes from James Milne.
In \cite{GanGao} and \cite{GaoThesis}, Wee Teck Gan and Fan Gao have tested some of the conjectures of this paper, and they have gone further in developing the Langlands program for covering groups. I have greatly appreciated our frequent conversations. Their results provided constraints which kept the constructions of this paper on track.
Pierre Deligne has kindly corresponded with me over the past few years, and his generosity has been incredibly helpful. His ideas led me to a deeper understanding of the crucial questions, and his correspondence motivated me to pursue this project further.
\section*{Notation}
\begin{notedescription}
\item[$F$] A field, typically local or global.
\item[$\bar F$] A separable closure of $F$.
\item[$\mathcal{O}$] The ring of integers in $F$, in the nonarchimedean local case.
\item[$\AA$] The ring of adeles of $F$, in the global case.
\item[${{\mathsf{Fr}}}$] The geometric Frobenius automorphism.
\item[$S$] A connected scheme, typically $\Spec(F)$ or $\Spec(\mathcal{O})$.
\item[$\bar s$] The geometric point of $S$ corresponding to $\bar F$.
\item[$\Gal_S$] The absolute Galois group $\pi_1^{\et}(S, \bar s)$.
\item[$\alg{X}$] An algebraic variety over $S$, or sheaf on $S_{\Zar}$.
\item[$X$ or $X_F$] The $F$-points $\alg{X}(F)$ for such a variety.
\item[$\sch{X}$] A scheme over $\mathbb Z$.
\item[$\sch{G}_m$] The multiplicative group.
\item[$\sch{\mu}_n$] The group scheme over $\mathbb Z$ of $n^{\th}$ roots of unity.
\item[$\mu_n$] The group $\sch{\mu}_n(S)$, assumed to be cyclic of order $n$.
\item[$\sheaf{S}$] A sheaf on $S_{\et}$.
\item[$\sheaf{S}{[U]}$] The sections of $\sheaf{S}$ over $U$ ($U \rightarrow S$ \'etale).
\item[$\dgp{G}$] A local system on $S_{\et}$, of group schemes over $\mathbb Z$.
\item[$\Cat{C}$] A category, with objects $\Obj(\Cat{C})$.
\item[$\gerb{E}$] A gerbe on $S_{\et}$.
\item[$\gerb{E}{[U]}$] The groupoid of sections of $\gerb{E}$ over $U$.
\item[$A$] An abelian group.
\item[$A_{[n]}$] Its $n$-torsion subgroup.
\item[$A_{/n}$] The quotient $A / n A$.
\end{notedescription}
\newpage
\section{Covering groups}
Throughout this article, $S$ will be a scheme in one of the following two classes: $S = \Spec(F)$ for a field $F$, or $S = \Spec(\mathcal{O})$ for a discrete valuation ring $\mathcal{O}$ with fraction field $F$. In the latter case, we assume that $\mathcal{O}$ contains a field, or that $\mathcal{O}$ has finite residue field. We will often fix a positive integer $n$, and we will assume that $\mu_n = \sch{\mu}_n(S)$ is a cyclic group of order $n$. In Section \ref{MetaGaloisSection}, we will place further restrictions on $S$.
\subsection{Reductive groups}
Let $\alg{G}$ be a reductive group over $S$. We follow \cite{SGA3} in our conventions, so this means that $\alg{G}$ is a smooth group scheme over $S$ such that $\alg{G}_{\bar s}$ is a connected reductive group for all geometric points $\bar s$ of $S$. Assume moreover that $\alg{G}$ is \textbf{quasisplit} over $S$.
Let $\alg{A}$ be a maximal $S$-split torus in $\alg{G}$, and let $\alg{T}$ be the centralizer of $\alg{A}$ in $\alg{G}$. Then $\alg{T}$ is a maximal torus in $\alg{G}$, and we say that $\alg{T}$ is a \defined{maximally split} maximal torus. Let $\sheaf{X}$ and $\sheaf{Y}$ be the local systems (on $S_{\et}$) of characters and cocharacters of $\alg{T}$.
Let $\alg{N}$ be the normalizer of $\alg{T}$ in $\alg{G}$. Let $\sheaf{W}$ denote the Weyl group of the pair $(\alg{G}, \alg{T})$, viewed as a sheaf on $S_{\et}$ of finite groups. Then $\sheaf{W}[S] = \alg{N}(S) / \alg{T}(S)$ (see \cite[Expos\'e XXVI, 7.1]{SGA3}). Let $\alg{B}$ be a Borel subgroup of $\alg{G}$ containing $\alg{T}$, defined over $S$. Let $\alg{U}$ be the unipotent radical of $\alg{B}$.
\begin{proposition}
\label{BTConj}
Assume as above that $\alg{G}$ is quasisplit, and $S$ is the spectrum of a field or of a DVR. The group $\alg{G}(S)$ acts transitively, by conjugation, on the set of pairs $(\alg{B}, \alg{T})$ consisting of a Borel subgroup (defined over $S$) and a maximally split maximal torus therein.
\end{proposition}
{\sc Proof:}
As we work over a local base scheme $S$, \cite[Expos\'e XXVI, Proposition 6.16]{SGA3} states that the group $\alg{G}(S)$ acts transitively on the set of maximal split subtori of $\alg{G}$ (defined over $S$).
Every maximally split maximal torus of $\alg{G}$ is the centralizer of such a maximal split torus, and thus $\alg{G}(S)$ acts transitively on the set of maximally split maximal tori in $ \alg{G}$. The stabilizer of such a maximally split maximal torus $\alg{T}$ is the group of $S$-points of its normalizer $\alg{N}(S)$. The Weyl group $\sheaf{W}[S] = \alg{N}(S) / \alg{T}(S)$ acts simply-transitively on the minimal parabolic subgroups containing $\alg{T}$ by \cite[Expos\'e XXVI, Proposition 7.2]{SGA3}. This proves the proposition.
\begin{flushright}Q.E.D \end{flushright}
The roots and coroots (for the adjoint action of $\alg{T}$ on the Lie algebra of $\alg{G}$) form local systems $\Phi$ and $\Phi^\vee$ on $S_{\et}$, contained in $\sheaf{X}$ and $\sheaf{Y}$, respectively. The simple roots (with respect to the Borel subgroup $\alg{B}$) and their coroots form local systems of subsets $\Delta \subset \Phi$ and $\Delta^\vee \subset \Phi^\vee$, respectively. In this way we find a local system on $S_{\et}$ of based root data (cf. \cite[\S 1.2]{BorelCorvallis}),
$$\Psi = \left( \sheaf{X}, \Phi, \Delta, \sheaf{Y}, \Phi^\vee, \Delta^\vee \right).$$
Write $\sheaf{Y}^{\SC}$ for the subgroup of $\sheaf{Y}$ spanned by the coroots.
\subsection{Covers}
In \cite{B-D}, Brylinski and Deligne study central extensions of $\alg{G}$ by $\alg{K}_2$, where $\alg{G}$ and $\alg{K}_2$ are viewed as sheaves of groups on the big Zariski site $S_{\Zar}$. These extensions form a category we call $\Cat{CExt}_S(\alg{G}, \alg{K}_2)$,. Such a central extension will be written $\alg{K}_2 \hookrightarrow \alg{G}' \twoheadrightarrow \alg{G}$ in what follows. We add one more piece of data in the definition below.
\begin{definition}
A degree $n$ \defined{cover} of $\alg{G}$ over $S$ is a pair $\alg{\tilde G} = (\alg{G}', n)$, where
\begin{enumerate}
\item
$\alg{K}_2 \hookrightarrow \alg{G}' \twoheadrightarrow \alg{G}$ is a central extension of $\alg{G}$ by $\alg{K}_2$ on $S_{\Zar}$;
\item
$n$ is a positive integer;
\item
For all scheme-theoretic points $s \in S$, with residue field $\mathfrak{f}(s)$, $\# \sch{\mu}_n( \mathfrak{f}(s) ) = n$.
\end{enumerate}
\end{definition}
Define $\Cat{Cov}_n(\alg{G})$ (or $\Cat{Cov}_{n/S}(\alg{G})$ to avoid confusion) to be the category of degree $n$ covers of $\alg{G}$ over $S$. The objects are pairs $\alg{\tilde G} = (\alg{G}', n)$ as above, and morphisms are those from $\Cat{CExt}_S(\alg{G}, \alg{K}_2)$ (with $n$ fixed).
If $\gamma \colon S_0 \rightarrow S$ is a morphism of schemes, then pulling back gives a functor $\gamma^\ast \colon \Cat{Cov}_{n/S}(\alg{G}) \rightarrow \Cat{Cov}_{n/S_0}(\alg{G}_{S_0})$. Indeed, a morphism of schemes gives inclusions of residue fields (in the opposite direction) and so Condition (3) is satisfied by the scheme $S_0$ when it is satisfied by the scheme $S$.
Central extensions $\alg{K}_2 \hookrightarrow \alg{G}' \twoheadrightarrow \alg{G}$ are classified by a triple of invariants $(Q, \sheaf{D}, f)$. For fields, this is carried out in \cite{B-D}, and the extension to DVRs (with finite residue field or containing a field) is found in \cite{MWIntegral}. The first invariant $Q \colon \sheaf{Y} \rightarrow \mathbb Z$ is a Galois-invariant Weyl-invariant quadratic form, i.e., $Q \in H_{\et}^0(S, \mathscr{S}\mathit{ym}^2(\sheaf{X})^{\sheaf{W}})$. The second invariant $\sheaf{D}$ is a central extension of sheaves of groups on $S_{\et}$, $\sheaf{G}_m \hookrightarrow \sheaf{D} \twoheadrightarrow \sheaf{Y}$. The third invariant $f$ will be discussed later.
A cover $\alg{\tilde G}$ yields a symmetric $\mathbb Z$-bilinear form $\beta_Q \colon \sheaf{Y} \otimes_\mathbb Z \sheaf{Y} \rightarrow n^{-1} \mathbb Z$,
$$\beta_Q(y_1, y_2) := n^{-1} \cdot \left( Q(y_1+y_2) - Q(y_1) - Q(y_2) \right).$$
This defines a local system $\sheaf{Y}_{Q,n} \subset \sheaf{Y}$,
$$\sheaf{Y}_{Q,n} = \{ y \in \sheaf{Y} : \beta_Q(y, y') \in \mathbb Z \text{ for all } y' \in \sheaf{Y} \}.$$
The category of covers is equipped with the structure of a Picard groupoid; one may ``add'' covers via the Baer sum. If $\alg{\tilde G}_1, \alg{\tilde G}_2$ are two covers of $\alg{G}$ of degree $n$, one obtains a cover $\alg{\tilde G}_1 \dotplus \alg{\tilde G}_2 = (\alg{G}_1' \dotplus \alg{G}_2', n)$.
When $\alg{\tilde G} = (\alg{G}', n)$ is a degree $n$ cover of $\alg{G}$, and $\alg{H} \subset \alg{G}$ is a reductive subgroup defined over $S$, write $\alg{\tilde H}$ for the resulting cover of $\alg{H}$. Following \cite{B-D}, the $\alg{G}'$ splits uniquely over any unipotent subgroup $\alg{U} \subset \alg{G}$, so we view $\alg{U}$ as a subgroup of $\alg{G}'$ in this case.
In three arithmetic contexts, a cover $\alg{\tilde G}$ yields a central extension of topological groups according to \cite[\S 10.3, 10.4]{B-D}.
\begin{description}
\item[Global] If $S = \Spec(F)$ for a global field $F$, then $\alg{\tilde G}$ yields a central extension $\mu_n \hookrightarrow \tilde G_\AA \twoheadrightarrow G_\AA$, endowed with a splitting $\sigma_F \colon G_F \hookrightarrow \tilde G_\AA$.
\item[Local] If $S = \Spec(F)$ for a local field $F$, then $\alg{\tilde G}$ yields a central extension $\mu_n \hookrightarrow \tilde G \twoheadrightarrow G$, where $G = \alg{G}(F)$.
\item[Local integral] If $S = \Spec(\mathcal{O})$, with $\mathcal{O}$ the ring of integers in a nonarchimedean local field $F$, then $\alg{\tilde G}$ yields a central extension $\mu_n \hookrightarrow \tilde G \twoheadrightarrow G$, where $G = \alg{G}(F)$, endowed with a splitting $\sigma^\circ \colon G^\circ \hookrightarrow G$.
\end{description}
Fix an injective character $\epsilon \colon \mu_n \hookrightarrow \mathbb C^\times$. This allows one to define $\epsilon$-genuine automorphic representations of $\tilde G_\AA$ in the global context, $\epsilon$-genuine admissible representations of $\tilde G$ in the local context, and $\epsilon$-genuine $G^\circ$-spherical representations of $\tilde G$ in the local integral context.
The purpose of this article is the construction an \defined{L-group} associated to such a $\alg{\tilde G}$ and $\epsilon$. We believe that this L-group will provide a parameterization of irreducible $\epsilon$-genuine representations in the three contexts above.
\subsection{Well-aligned homomorphisms}
Let $\alg{G}_1 \supset \alg{B}_1 \supset \alg{T}_1$ and $\alg{G}_2 \supset \alg{B}_2 \supset \alg{T}_2$ be quasisplit groups over $S$, endowed with Borel subgroups and maximally split maximal tori. Let $\alg{\tilde G}_1 = (\alg{G}_1', n)$ and $\alg{\tilde G}_2 = (\alg{G}_2', n)$ be covers (of the same degree) of $\alg{G}_1$ and $\alg{G}_2$, respectively. Write $\sheaf{Y}_1$ and $\sheaf{Y}_2$ for the cocharacter lattices of $\alg{T}_1$ and $\alg{T}_2$, and $Q_1, Q_2$ for the quadratic forms arising from the covers. These quadratic forms yield sublattices $\sheaf{Y}_{1,Q_1,n}$ and $\sheaf{Y}_{2,Q_2,n}$.
\begin{definition}
A \defined{well-aligned homomorphism} $\tilde \iota$ from $(\alg{\tilde G}_1, \alg{B}_1, \alg{T}_1)$ to $(\alg{\tilde G}_2, \alg{B}_2, \alg{T}_2)$ is a pair $\tilde \iota = (\iota, \iota')$ of homomorphisms of sheaves of groups on $S_{\Zar}$, making the following diagram commute,
\begin{equation}
\label{CDiota}
\begin{tikzcd}
\alg{K}_2 \arrow[hook]{r} \arrow{d}{=} & \alg{G}_1' \arrow[two heads]{r} \arrow{d}{\iota'} & \alg{G}_1 \arrow{d}{\iota} \\
\alg{K}_2 \arrow[hook]{r} & \alg{G}_2' \arrow[two heads]{r} & \alg{G}_2
\end{tikzcd}
\end{equation}
and satisfying the following additional axioms:
\begin{enumerate}
\item
$\iota$ has normal image and smooth central kernel;
\item
$\iota(\alg{B}_1) \subset \alg{B}_2$ and $\iota(\alg{T}_1) \subset \alg{T}_2$. Thus $\iota$ induces a map $\iota \colon \sheaf{Y}_1 \rightarrow \sheaf{Y}_2$;
\item
$(\iota, \iota')$ realizes $\alg{G}_1'$ as the pullback of $\alg{G}_2'$ via $\iota$;
\item
The homomorphism $\iota$ satisfies $\iota(\sheaf{Y}_{1,Q_1,n}) \subset \sheaf{Y}_{2,Q_2,n}$.
\end{enumerate}
\end{definition}
\begin{remark}
Conditions (1) and (2) are inspired by \cite[\S 1.4, 2.1,2.5]{BorelCorvallis}, though more restrictive. By ``normal image,'' we mean that for any geometric point $\bar s \rightarrow S$, the homomorphism $\iota \colon \alg{G}_{1,\bar s} \rightarrow \alg{G}_{2, \bar s}$ has normal image. Condition (3) implies that for all $y \in \sheaf{Y}_1$, $Q_1(y) = Q_2(\iota(y))$. In other words, $Q_1$ is the image of $Q_2$ via the map
$$\iota^\ast \colon H_{\et}^0(S, \mathscr{S}\mathit{ym}^2 (\sheaf{X}_2) ) \rightarrow H_{\et}^0(S, \mathscr{S}\mathit{ym}^2 (\sheaf{X}_1) ).$$
But Condition (3) does not imply Condition (4); one may cook up an example with $\alg{G}_1 = \alg{G}_{\mult}$ and $\alg{G}_2 = \alg{G}_{\mult}^2$ which satisfies (3) but not (4).
\end{remark}
\begin{proposition}
\label{ComposeWellaligned}
The composition of well-aligned homomorphisms is well-aligned.
\end{proposition}
{\sc Proof:}
Suppose that $(\iota_1, \iota_1')$ and $(\iota_2, \iota_2')$ are well-aligned homomorphisms, with $\iota_1 \colon \alg{G}_1 \rightarrow \alg{G}_2$ and $\iota_2 \colon \alg{G}_2 \rightarrow \alg{G}_3$. Conditions (2), (3), and (4) are obviously satisfied by the composition $(\iota_2 \circ \iota_1, \iota_2' \circ \iota_1')$. For condition (1), notice that the kernel of $\iota_2 \circ \iota_1$ is contained in the kernel of $\iota_1$, and hence is central. The only thing left is to verify that $\iota_2 \circ \iota_1$ has normal image. This may be checked by looking at geometric fibres; it seems well-known (cf. \cite[\S 1.8]{KotSTF}).
\begin{flushright}Q.E.D \end{flushright}
Inner automorphisms are well-aligned homomorphisms.
\begin{example}
Suppose that $\alg{\tilde G}$ is a degree $n$ cover of a quasisplit group $\alg{G}$. Suppose that $\alg{B}_0 \supset \alg{T}_0$ and $\alg{B} \supset \alg{T}$ are two Borel subgroups containing maximally split maximal tori. Suppose that $g \in \alg{G}(S)$, and write $\Int(g)$ for the resulting inner automorphism of $\alg{G}$. As noted in \cite[0.N.4]{B-D}, $\Int(g)$ lifts canonically to an automorphism $\Int(g)' \in \Aut(\alg{G}')$. If $\alg{B} = \Int(g) \alg{B}_0$ and $\alg{T} = \Int(g) \alg{T}_0$, then the pair $\left( \Int(g), \Int(g)' \right)$ is a well-aligned homomorphism from $(\alg{\tilde G}, \alg{B}_0, \alg{T}_0)$ to $(\alg{\tilde G}, \alg{B}, \alg{T})$.
\end{example}
While we focus on quasisplit groups in this article, the lifting of inner automorphisms allows one to consider ``pure inner forms'' of covers over a field.
\begin{definition}
Let $\alg{\tilde G} = (\alg{G}', n)$ be a degree $n$ cover of a quasisplit group $\alg{G}$, over a field $F$. Let $\xi \in Z_{\et}^1(F, \alg{G})$ be a 1-cocycle. The image $\Int(\xi)$ in $Z_{\et}^1(F, \alg{Aut}(\alg{G}))$ defines an inner form $\alg{G}_\xi$ of $\alg{G}$. These are called the \defined{pure inner forms} of $\alg{G}$. On the other hand, we may consider the image $\Int(\xi)'$ in $Z_{\et}^1(F, \alg{Aut}(\alg{G}'))$, which by \cite[\S 7.1, 7.2]{B-D} defines a central extension $\alg{G}_\xi'$ of $\alg{G}_\xi$ by $\alg{K}_2$. The cover $\alg{\tilde G}_\xi = (\alg{G}_\xi', n)$ of $\alg{G}_\xi$ will be called a \defined{pure inner form} of the cover $\alg{\tilde G}$.
\end{definition}
We have not attempted to go further in the study of inner forms for covers, but presumably one should study something like strong real forms as in \cite[Definition 1.12]{AdamsBarbaschVogan}, and more general rigid forms as in \cite{Kaletha}, if one wishes to assemble L-packets for covering groups.
The next example of a well-aligned homomorphism is relevant for the study of central characters of genuine representations.
\begin{example}
Let $\alg{\tilde G}$ be a degree $n$ cover of a quasisplit group $\alg{G} \supset \alg{B} \supset \alg{T}$. Let $\alg{H}$ be the maximal torus in the center of $\alg{G}$, with cocharacter lattice $\sheaf{Y}_H \subset \sheaf{Y}$. Let $\alg{C}$ be the algebraic torus with cocharacter lattice $\sheaf{Y}_H \cap \sheaf{Y}_{Q,n}$, and $\iota \colon \alg{C} \rightarrow \alg{G}$ the resulting homomorphism (with central image). Let $\alg{\tilde C}$ denote the pullback of the cover $\alg{\tilde G}$ via $\iota$. Then $\iota$ lifts to a well-aligned homomorphism from $\alg{\tilde C}$ to $\alg{\tilde G}$.
\end{example}
The final example of a well-aligned homomorphism is relevant for the study of twisting genuine representations by one-dimensional representations of $\alg{G}$.
\begin{example}
Let $\alg{H}$ denote the maximal toral quotient of $\alg{G}$. In other words, $\alg{H}$ is the torus whose character lattice equals $\Hom(\alg{G}, \alg{G}_m)$. Let $p \colon \alg{G} \rightarrow \alg{H}$ denote the canonical homomorphism, and write $\iota \colon \alg{G} \rightarrow \alg{G} \times \alg{H}$ for the homomorphism $\Id \times p$. Write $\alg{\tilde G} \times \alg{H}$ for the cover $(\alg{G}' \times \alg{H}, n)$.
The homomorphism $\iota$ realizes $\alg{\tilde G}$ as the pullback via $\iota$ of the cover $\alg{\tilde G} \times \alg{H}$. A Borel subgroup and torus in $\alg{G}$ determines a Borel subgroup and torus in $\alg{G} \times \alg{H}$. In this way, $\iota$ lifts a well-aligned homomorphism of covers from $\alg{\tilde G}$ to $\alg{\tilde G} \times \alg{H}$.
\end{example}
\section{The dual group}
In this section, fix a degree $n$ cover $\alg{\tilde G}$ of a quasisplit group $\alg{G}$ over $S$. Associated to $\alg{\tilde G}$, we define the ``dual group,'' a local system on $S_{\et}$ of affine group schemes over $\mathbb Z$. We refer to Appendix \S \ref{GroupsTorsors}, for background on such local systems. We begin by reviewing the Langlands dual group of $\alg{G}$ in a framework suggested by Deligne (personal communication).
\subsection{The Langlands dual group}
Choose, for now, a Borel subgroup $\alg{B} \subset \alg{G}$ containing a maximally split maximal torus $\alg{T}$. The based root datum of $(\alg{G}, \alg{B}, \alg{T})$ was denoted $\Psi$, and the dual root datum,
$$\Psi^\vee = \left( \sheaf{Y}, \Phi^\vee, \Delta^\vee, \sheaf{X}, \Phi, \Delta \right),$$
is a local system of root data on $S_{\et}$.
This defines a unique (up to unique isomorphism) local system $\dgp{G^\vee}$ on $S_{\et}$ of pinned reductive groups over $\mathbb Z$, called the \defined{Langlands dual group} of $\alg{G}$. The center of $\dgp{G^\vee}$ is a local system on $S_{\et}$ of groups of multiplicative type over $\mathbb Z$, given by
$$\dgp{Z^\vee} = \Spec \left( \mathbb Z [ \sheaf{Y} / \sheaf{Y}^{\SC} ] \right).$$
See Example \ref{LocSpec} for more on local systems and $\Spec$ in this context.
\subsection{The dual group of a cover}
Now we adapt the definition of the dual group to covers. The ideas here are the same as those of \cite{MWCrelle}. The ideas for modifying root data originate in \cite[\S 2.2]{LusztigQuantumGroups} in the simply-connected case, in \cite[Theorem 2.9]{FinkelbergLysenko} for the almost simple case, in \cite[\S 11]{McNamara} and \cite{Reich} in the reductive case. This dual group is also compatible with \cite{ABPTV} and the Hecke algebra isomorphisms of Savin \cite{SavinUnramified}, and the most recent work of Lysenko \cite{LysenkoSatake}.
Associated to the cover $\alg{\tilde G}$ of degree $n$, recall that $Q \colon \sheaf{Y} \rightarrow \mathbb Z$ is the first Brylinski-Deligne invariant, and $\beta_Q \colon \sheaf{Y} \otimes \sheaf{Y} \rightarrow n^{-1} \mathbb Z$ a symmetric bilinear form, and
$$\sheaf{Y}_{Q,n} = \{ y \in \sheaf{Y} : \beta_Q(y, y') \in \mathbb Z \text{ for all } y' \in \sheaf{Y} \} \subset \sheaf{Y}.$$
Define $\sheaf{X}_{Q,n} = \{ x \in n^{-1} \sheaf{X} : \langle x, y \rangle \in \mathbb Z \text{ for all } y \in \sheaf{Y}_{Q,n} \} \subset n^{-1} \sheaf{X}$. For each root $\phi \in \Phi$, define constants $n_\phi$ and $m_\phi$,
\begin{equation}
\label{nm}
n_\phi = \frac{n}{\GCD(n, Q(\phi^\vee))} , \quad m_\phi = \frac{Q(\phi^\vee)}{\GCD(n, Q(\phi^\vee))}.
\end{equation}
Define \defined{modified roots} and \defined{modified coroots} by
$$\tilde \phi = n_\phi^{-1} \phi, \quad \tilde \phi^{\vee} = n_\phi \phi^\vee.$$
These define subsets $\tilde \Phi = \{ \tilde \phi : \phi \in \Phi \} \subset \sheaf{X}_{Q,n}$ and $\tilde \Phi^\vee = \{ \tilde \phi^\vee : \phi^\vee \in \Phi^\vee \} \subset \sheaf{Y}_{Q,n}$, as in \cite{MWCrelle}. Modifying the simple roots and their coroots, we have subsets $\tilde \Delta \subset \tilde \Phi$ and $\tilde \Delta^\vee \subset \tilde \Phi$.
By \cite[Construction 1.3]{MWCrelle}, this defines a local system of based root data $\tilde \Psi$ on $S_{\et}$. Write $\tilde \Psi^\vee$ for its dual,
$$\tilde \Psi^\vee = (\sheaf{Y}_{Q,n}, \tilde \Phi^\vee, \tilde \Delta^\vee, \sheaf{X}_{Q,n}, \tilde \Phi, \tilde \Delta).$$
Write $\sheaf{Y}_{Q,n}^{\SC}$ for the subgroup of $\sheaf{Y}_{Q,n}$ spanned by the modified coroots $\tilde \Phi^\vee$.
Define $\dgp{\tilde G^\vee}$ to be the (unique up to unique isomorphism) local system on $S_{\et}$ of pinned reductive groups over $\mathbb Z$, associated to the local system of based root data $\tilde \Psi^\vee$. Its maximal torus is a local system on $S_{\et}$ of split tori over $\mathbb Z$,
$$\dgp{\tilde T^\vee} = \Spec \left( \mathbb Z[ \sheaf{Y}_{Q,n}] \right).$$
The center of $\dgp{\tilde G^\vee}$ is a local system on $S_{\et}$ of groups of multiplicative type over $\mathbb Z$,
$$\dgp{\tilde Z^\vee} = \Spec \left( \mathbb Z \left[ \sheaf{Y}_{Q,n} / \sheaf{Y}_{Q,n}^{\SC} \right] \right).$$
We call $\dgp{\tilde G^\vee}$ (endowed with its pinning) the \defined{dual group} of the cover $\alg{\tilde G}$.
\begin{proposition}
\label{DualGroupModn}
Suppose that $\alg{\tilde G}_0$ is another cover of $\alg{G}$ of degree $n$, with first Brylinski-Deligne invariant $Q_0$. If $Q \equiv Q_0$ modulo $n$, i.e. $Q(y) - Q_0(y) \in n \mathbb Z$ for all $y \in \sheaf{Y}$, then the resulting modified root data are equal: $\tilde \Psi^\vee = \tilde \Psi_0^\vee$. Thus the dual groups are equal, $\dgp{\tilde G}^ { \raisemath{-3pt}{\vee} } = \dgp{\tilde G}_0^ { \raisemath{-3pt}{\vee} } $.
\end{proposition}
{\sc Proof:}
One checks directly that $\beta_Q \equiv \beta_{Q_0}$ modulo $\mathbb Z$, from which it follows that
$$\sheaf{Y}_{Q,n} = \sheaf{Y}_{Q_0,n}, \quad \sheaf{X}_{Q,n} = \sheaf{X}_{Q_0, n}.$$
Similarly, one checks that the constants $n_\phi$ are equal,
$$\frac{n}{\GCD(n, Q_0(\phi^\vee))} = \frac{n}{\GCD(n, Q(\phi^\vee))}.$$
The result follows.
\begin{flushright}Q.E.D \end{flushright}
The Weyl group of $\dgp{\tilde G^\vee}$ with respect to $\dgp{\tilde T^\vee}$ forms a local system $\sheaf{\tilde W}$ on $S_{\et}$ of finite groups, generated (locally on $S_{\et}$) by reflections $s_{\tilde \phi}$ for every $\tilde \phi \in \tilde \Phi$. The action of $\sheaf{\tilde W}$ on $\sheaf{Y}_{Q,n}$ is given by the formula
$$s_{\tilde \phi}(y) = y - \langle \tilde \phi, y \rangle \tilde \phi^\vee = y - \langle \phi, y \rangle \phi^\vee.$$
This identifies the root reflections $s_{\tilde \phi}$ with the root reflections $s_{\phi}$, and hence identifies the Weyl group $\sheaf{\tilde W}$ with the Weyl group $\sheaf{W}$ of $\alg{G}$ with respect to $\alg{T}$ (where both are viewed as local systems on $S_{\et}$ of finite groups).
The dual group $\dgp{\tilde G^\vee}$ comes with a distinguished 2-torsion element in its center, described here. From the quadratic form $Q \colon \sheaf{Y} \rightarrow \mathbb Z$, observe that $2 Q(y) = n \beta_Q(y,y) \in n \mathbb Z$ for all $y \in \sheaf{Y}_{Q,n}$. Moreover, we have
$$Q(\tilde \phi^\vee) = n_\phi^2 Q(\phi^\vee) = n_\phi m_\phi n \in n \mathbb Z,$$
for all $\phi \in \Phi$. Of course, $Q( n y) \in n \mathbb Z$ as well, for all $y \in \sheaf{Y}_{Q,n}$. We find a homomorphism of local systems of abelian groups on $S_{\et}$,
$$\overline{n^{-1} Q} \colon \frac{\sheaf{Y}_{Q,n}}{\sheaf{Y}_{Q,n}^{\SC} + n \sheaf{Y}_{Q,n}} \rightarrow \tfrac{1}{2} \mathbb Z / \mathbb Z, \quad y \mapsto n^{-1} Q(y) \text{ mod } \mathbb Z.$$
Applying $\Spec$ yields a homomorphism of local systems on $S_{\et}$ of diagonalizable group schemes over $\mathbb Z$,
$$\tau_Q \in \Hom ( \sch{\mu}_2, \dgp{\tilde Z}_{[n]}^ { \raisemath{-3pt}{\vee} } ).$$
Thus $\tau_Q(-1)$ is a distinguished Galois-invariant 2-torsion element in the center of $\dgp{\tilde G^\vee}$. If $n$ is odd, then $\tau_Q(-1) = 1$.
\subsection{Well-aligned functoriality}
\label{WAFDualGroup}
Consider a well-aligned homomorphism $\tilde \iota \colon \alg{\tilde G}_1 \rightarrow \alg{\tilde G}_2$ of covers, each endowed with Borel subgroup and maximally split maximal torus. Here we construct a corresponding homomorphism of dual groups,
$$\iota^\vee \colon \dgp{\tilde G}_2^ { \raisemath{-3pt}{\vee} } \rightarrow \dgp{\tilde G}_1^ { \raisemath{-3pt}{\vee} } .$$
These dual groups are constructed, locally on $S_{\et}$, from root data:
\begin{align*}
\tilde \Psi_1^\vee &= \left( \sheaf{Y}_{1,Q_1,n}, \tilde \Phi_1^\vee, \tilde \Delta_1^\vee, \sheaf{X}_{1,Q_1,n}, \tilde \Phi_1, \tilde \Delta_1 \right); \\
\tilde \Psi_2^\vee &= \left( \sheaf{Y}_{2,Q_2,n}, \tilde \Phi_2^\vee, \tilde \Delta_2^\vee, \sheaf{X}_{2,Q_2,n}, \tilde \Phi_2, \tilde \Delta_2 \right).
\end{align*}
For the construction of $\iota^\vee$, it suffices to work locally on $S_{\et}$, on a finite \'etale cover over which $\alg{G}_1$ and $\alg{G}_2$ split. The well-aligned Condition (4) gives a homomorphism $\iota \colon \sheaf{Y}_{1,Q_1,n} \rightarrow \sheaf{Y}_{2,Q_2,n}$, and its dual homomorphism $\iota^\ast \colon \sheaf{X}_{2,Q_2,n} \rightarrow \sheaf{X}_{1,Q_1,n}$. As $\iota$ has normal image, the coroots from $\Phi_1^\vee$ map to coroots from $\Phi_2^\vee$. Condition (3) implies that the scaled coroots in $\tilde \Phi_1^\vee \subset \sheaf{Y}_{1,Q_1,n}$ map to scaled coroots in $\tilde \Phi_2^\vee \subset \sheaf{Y}_{2,Q_2,n}$. Since the Borel subgroups are aligned, the simple scaled coroots in $\tilde \Delta_1^\vee$ map to simple scaled coroots in $\tilde \Delta_2^\vee$. Dually, the map $\iota^\ast \colon \sheaf{X}_{2,Q_2,n} \rightarrow \sheaf{X}_{1,Q_1,n}$ sends $\tilde \Phi_2$ to $\tilde \Phi_1$ and $\tilde \Delta_2$ to $\tilde \Delta_1$.
This allows us to assemble a homomorphism $\iota^\vee \colon \dgp{\tilde G}_2^ { \raisemath{-3pt}{\vee} } \rightarrow \dgp{\tilde G}_1^ { \raisemath{-3pt}{\vee} } ,$ (cf.~\cite[\S 2.1, 2.5]{BorelCorvallis}). On tori, let $\iota^\vee \colon \dgp{\tilde T}_2^ { \raisemath{-3pt}{\vee} } \rightarrow \dgp{\tilde T}_1^ { \raisemath{-3pt}{\vee} } $ be the homomorphism dual to the map of character lattices $\iota \colon \sheaf{Y}_{1,Q_1,n} \rightarrow \sheaf{Y}_{2,Q_2,n}$. Using the pinnings on $\dgp{\tilde G}_2^ { \raisemath{-3pt}{\vee} } $ and $\dgp{\tilde G}_1^ { \raisemath{-3pt}{\vee} } $, and the map of roots to roots, we obtain a homomorphism from the simply-connected cover $\dgp{\tilde G}_{2,\SC}^ { \raisemath{-3pt}{\vee} } $ of the derived subgroup, $\dgp{\tilde G}_{2, \der}^ { \raisemath{-3pt}{\vee} } $,
$$\iota_{\SC}^\vee \colon \dgp{\tilde G}_{2,\SC}^ { \raisemath{-3pt}{\vee} } \rightarrow \dgp{\tilde G}_1^ { \raisemath{-3pt}{\vee} } .$$
Let $\dgp{\tilde T}_{2,\SC}^ { \raisemath{-3pt}{\vee} } $ be the pullback of $\dgp{\tilde T}_2^ { \raisemath{-3pt}{\vee} } $. The following diagram commutes.
$$\begin{tikzcd}
\dgp{\tilde T}_{2, \SC}^ { \raisemath{-3pt}{\vee} } \arrow{r} \arrow[hook]{d} & \dgp{\tilde T}_2^ { \raisemath{-3pt}{\vee} } \arrow{d}{\iota^\vee} \\
\dgp{\tilde G}_{2, \SC}^ { \raisemath{-3pt}{\vee} } \arrow{r}{\iota_{\SC}^ { \raisemath{-3pt}{\vee} } } & \dgp{\tilde G}_1^ { \raisemath{-3pt}{\vee} }
\end{tikzcd}$$
The homomorphism $\iota_{\SC}^ { \raisemath{-3pt}{\vee} } $ descends to the derived subgroup $\dgp{\tilde G}_{2,\der}^ { \raisemath{-3pt}{\vee} } $, since it is trivial on the kernel of $\dgp{\tilde T}_{2,\SC}^ { \raisemath{-3pt}{\vee} } \rightarrow \dgp{\tilde T}_2^ { \raisemath{-3pt}{\vee} } $. In this way, we have a pair of homomorphisms of groups over $\mathbb Z$,
$$\iota_{\der}^ { \raisemath{-3pt}{\vee} } \colon \dgp{\tilde G}_{2,\der}^ { \raisemath{-3pt}{\vee} } \rightarrow \dgp{\tilde G}_1^ { \raisemath{-3pt}{\vee} } , \quad \iota^\vee \colon \dgp{\tilde T}_{2}^ { \raisemath{-3pt}{\vee} } \rightarrow \dgp{\tilde T}_1^ { \raisemath{-3pt}{\vee} } \subset \dgp{\tilde G}_1^ { \raisemath{-3pt}{\vee} } .$$
These homomorphisms agree on their intersection, giving the desired homomorphism $\iota^\vee \colon \dgp{\tilde G}_2^ { \raisemath{-3pt}{\vee} } \rightarrow \dgp{\tilde G}_1^ { \raisemath{-3pt}{\vee} } $.
Since modified coroots are sent to modified coroots, we find that $\iota^\vee$ sends $\dgp{\tilde Z}_2^ { \raisemath{-3pt}{\vee} } $ to $\dgp{\tilde Z}_1^ { \raisemath{-3pt}{\vee} } $. The quadratic forms $Q_1$ and $Q_2$ induce two group homomorphisms:
$$\overline{n^{-1} Q_1} \colon \frac{\sheaf{Y}_{1,Q_1,n}}{\sheaf{Y}_{1,Q_1,n}^{\SC} + n \sheaf{Y}_{1,Q_1,n}} \rightarrow \tfrac{1}{2} \mathbb Z / \mathbb Z, \quad \overline{n^{-1} Q_2} \colon \frac{\sheaf{Y}_{2,Q_2,n}}{\sheaf{Y}_{2,Q_2,n}^{\SC} + n \sheaf{Y}_{2,Q_2,n}} \rightarrow \tfrac{1}{2} \mathbb Z / \mathbb Z.$$
These define two homomorphisms of group schemes over $\mathbb Z$,
$$\tau_{Q_1} \colon \sch{\mu}_2 \rightarrow \dgp{\tilde Z}_{1, [n]}^ { \raisemath{-3pt}{\vee} } , \quad \tau_{Q_2} \colon \sch{\mu}_2 \rightarrow \dgp{\tilde Z}_{2, [n]}^ { \raisemath{-3pt}{\vee} } .$$
As $Q_2(\iota(y)) = Q_1(y)$ for all $y \in \sheaf{Y}_1$, $\iota^\vee \circ \tau_{Q_1} = \tau_{Q_2}$. In other words, the homomorphism $\iota^\vee \colon \dgp{\tilde G}_2^ { \raisemath{-3pt}{\vee} } \rightarrow \dgp{\tilde G}_1^ { \raisemath{-3pt}{\vee} } $ sends the center to the center, and respects the 2-torsion elements therein, $\iota^\vee(\tau_{Q_2}(-1)) = \tau_{Q_1}(-1)$.
Given a pair of well-aligned homomorphisms,
$$\alg{\tilde G}_1 \xrightarrow{(\iota_1, \iota_1')} \alg{\tilde G}_2 \xrightarrow{(\iota_2, \iota_2')} \alg{\tilde G}_3,$$
their composition is a well-aligned homomrphism $(\iota_3, \iota_3') = (\iota_2 \iota_1, \iota_2' \iota_1')$ from $\alg{\tilde G}_1$ to $\alg{\tilde G}_3$ by Proposition \ref{ComposeWellaligned}.
This gives a commutative diagram of sheaves of abelian groups.
$$\begin{tikzcd}
\sheaf{Y}_{1,Q_1,n} \arrow{r}[swap]{\iota_1} \arrow[bend left=20]{rr}{\iota_3} & \sheaf{Y}_{2,Q_2,n} \arrow{r}[swap]{\iota_2} & \sheaf{Y}_{3,Q_3,n}.
\end{tikzcd}$$
We find such a commutative diagram for dual groups, in the opposite direction.
$$\begin{tikzcd}
\dgp{\tilde G}_3^ { \raisemath{-3pt}{\vee} } \arrow{r}[swap]{\iota_2^\vee} \arrow[bend left=20]{rr}{\iota_3^\vee} & \dgp{\tilde G}_2^ { \raisemath{-3pt}{\vee} } \arrow{r}[swap]{\iota_1^\vee} & \dgp{\tilde G}_1^ { \raisemath{-3pt}{\vee} } .
\end{tikzcd}$$
Let $\Cat{DGp}_S^\ast$ denote the category whose objects are local systems $\dgp{G}^\vee$ on $S_{\et}$ of group schemes over $\mathbb Z$, endowed with central morphisms $\sch{\mu}_2 \rightarrow \dgp{G}^\vee$ (where $\sch{\mu}_2$ is viewed as the constant local system of group schemes). Morphisms in $\Cat{DGp}_S^\ast$ are morphisms of local systems of group schemes over $\mathbb Z$, compatible with the central morphisms from $\sch{\mu}_2$.
Let $\Cat{WAC}_S$ (Well-Aligned-Covers) denote the category whose objects are triples $(\alg{\tilde G}, \alg{B}, \alg{T})$ where $\alg{\tilde G}$ is a cover of a reductive group $\alg{G}$ over $S$, $\alg{B}$ is a Borel subgroup of $\alg{G}$, and $\alg{T}$ is a maximally split maximal torus of $\alg{G}$ contained in $\alg{B}$. Morphisms in $\Cat{WAC}_S$ are well-aligned homomorphisms of covers.
We have proven the following result.
\begin{proposition}
\label{FuncDualGroup}
The construction of the dual group defines a contravariant functor from $\Cat{WAC}_S$ to $\Cat{DGp}_S^\ast$.
\end{proposition}
\begin{thm}
\label{WellDefinedDualGroup}
Let $\alg{\tilde G}$ be a degree $n$ cover of a quasisplit group $\alg{G}$ over $S$. The dual group $\dgp{\tilde G^\vee}$ is well-defined, up to unique isomorphism, by $\alg{\tilde G}$ alone.
\end{thm}
{\sc Proof:}
The construction of the dual group depends on the choice of Borel and torus $\alg{B} \supset \alg{T}$. So it suffices to construct a canonical isomorphism of dual groups $\dgp{\tilde G}^ { \raisemath{-3pt}{\vee} } \rightarrow \dgp{\tilde G}_0^\vee$ for any pair of choices $\alg{B}_0 \supset \alg{T}_0$ and $\alg{B} \supset \alg{T}$. Such ``well-definedness'' is discussed in more detail in \cite[\S 1.1]{DeligneLusztig}.
By Proposition \ref{BTConj}, there exists $g \in \alg{G}(S)$ such that $\Int(g) \alg{T}_0 = \alg{T}$ and $\Int(g) \alg{B}_0 = \alg{B}$. This automorphism $\Int(g)$ lifts to an automorphism of $\alg{G}'$ and defines a well-aligned isomorphism of covers,
$$\begin{tikzcd}
\alg{K}_{2} \arrow[hook]{r} \arrow{d}{=} & \alg{G}' \arrow[two heads]{r} \arrow{d}{\Int(g)'} & \alg{G} \arrow{d}{\Int(g)} & & \alg{B}_{0} \arrow{d} & \alg{T}_{0} \arrow{d} \\
\alg{K}_{2} \arrow[hook]{r} & \alg{G}' \arrow[two heads]{r} & \alg{G} & & \alg{B} & \alg{T}
\end{tikzcd}$$
As a well-aligned isomorphism of covers, this yields an isomorphism (of local systems on $S_{\et}$ of reductive groups over $\mathbb Z$)
$$\Int(g)^\vee \colon \dgp{\tilde G^\vee} \xrightarrow{\sim} \dgp{\tilde G}_0^ { \raisemath{-3pt}{\vee} } .$$
If $g' \in \alg{G}(S)$ also satisfies $\Int(g') \alg{T}_0 = \alg{T}$ and $\Int(g') \alg{B}_0 = \alg{B}$, then $g' g^{-1} \in \alg{N}(S) \cap \alg{B}(S) = \alg{T}(S)$. Thus $g' = t g$ for some $t \in \alg{T}(S)$. Hence $\Int(g') = \Int(t) \Int(g)$, and so by Proposition \ref{FuncDualGroup},
$$\Int(g')^\vee = \Int(g)^\vee \Int(t)^\vee \colon \dgp{\tilde G^\vee} \rightarrow \dgp{\tilde G}_0^ { \raisemath{-3pt}{\vee} } .$$
But $\Int(t)^\vee = \Id$, since $\Int(t)$ leaves all relevant data unchanged. Thus $\Int(g')^\vee = \Int(g)^\vee$. Hence we find a canonical isomorphism $\dgp{\tilde G}^ { \raisemath{-3pt}{\vee} } \xrightarrow{\sim} \dgp{\tilde G}_0^ { \raisemath{-3pt}{\vee} } $.
\begin{flushright}Q.E.D \end{flushright}
\subsection{Change of base scheme}
\label{BaseChangeDualGroup}
Let $\alg{\tilde G}$ be a degree $n$ cover of a quasisplit group $\alg{G} \supset \alg{B} \supset \alg{T}$ over $S$, as before. Let $\gamma \colon S_0 \rightarrow S$ be a morphism of schemes, with $S_0 = \Spec(F_0)$ for some field $F_0$ or $S_0 = \Spec(\mathcal{O}_0)$ for some DVR $\mathcal{O}_0$ (with finite residue field or containing a field, as usual). Then $\gamma$ gives rise to a pullback functor $\gamma^\ast$ from sheaves on $S_{\et}$ to sheaves on $S_{0,\et}$. Pullback via $\gamma$ defines a degree $n$ cover $\alg{\tilde G}_0$ of a quasisplit group $\alg{G}_0 \supset \alg{B}_0 \supset \alg{T}_0$ over $S_0$.
The cocharacter lattice $\sheaf{Y}_0$ of $\alg{T}_0$ is a sheaf on $S_{0,\et}$. It is naturally isomorphic to the pullback $\gamma^\ast \sheaf{Y}$, with $\sheaf{Y}$ the cocharacter lattice of $\alg{T}$. Write $N \colon \gamma^\ast \sheaf{Y} \rightarrow \sheaf{Y}_0$ for the natural isomorphism. The quadratic form $Q \colon \sheaf{Y} \rightarrow \mathbb Z$ pulls back to a quadratic form $\gamma^\ast Q \colon \gamma^\ast \sheaf{Y} \rightarrow \mathbb Z$. The compatibility of Brylinski-Deligne invariants with pullbacks implies that $\gamma^\ast Q = N^\ast Q_0$, with $Q_0$ the first Brylinski-Deligne invariant of $\alg{\tilde G}_0$.
\begin{remark}
This compatibility follows straightforwardly from \cite[\S 3.10]{B-D}; if $\alg{\tilde T}$ arises (after a finite \'etale $U \rightarrow S$) from the cocycle attached to $C \in \sheaf{X} \otimes \sheaf{X}$, then $\alg{\tilde T}_0$ arises from the pullback of this cocycle, i.e., from an element $C_0 \in \sheaf{X}_0 \otimes \sheaf{X}_0$ with $N^\ast C_0 = \gamma^\ast C$. The quadratic form $Q$ is given by $Q(y) = C(y,y)$ and similarly $Q_0(y) = C_0(y,y)$. Since $N^\ast C_0 = \gamma^\ast C$, we find that $N^\ast Q_0 = \gamma^\ast Q$.
\end{remark}
In this way, we find that $N$ restricts to an isomorphism from $\gamma^\ast \sheaf{Y}_{Q,n}$ to $\sheaf{Y}_{0,Q_0,n}$, sending roots and coroots (the sheaves of sets $\gamma^\ast \tilde \Phi$, $\gamma^\ast \tilde \Phi^\vee$ on $S_{0,\et}$) to the corresponding roots and coroots in $\sheaf{Y}_{0,Q_0,n}$. As $\alg{B}_0 = \gamma^\ast \alg{B}$, simple roots and coroots are identified as well.
By our construction of the dual group, we find that $N$ gives an isomorphism of local systems on $S_{0,\et}$ of pinned reductive groups over $\mathbb Z$,
$$N^\vee \colon \gamma^\ast \dgp{\tilde G^\vee} \xrightarrow{\sim} \dgp{\tilde G}_0^ { \raisemath{-3pt}{\vee} } .$$
\subsection{Parabolic subgroups}
\label{ParabolicDualGroup}
We keep the degree $n$ cover $\alg{\tilde G}$ of the quasisplit group $\alg{G} \supset \alg{B} \supset \alg{T}$ over $S$. Let $\alg{P} \subset \alg{G}$ be a parabolic subgroup defined over $S$, containing $\alg{B}$. Suppose that $\alg{P} = \alg{M} \alg{N}$ is a Levi decomposition defined over $S$, with $\alg{N}$ the unipotent radical of $\alg{P}$ and $\alg{M}$ a Levi factor containing $\alg{T}$. Let $\alg{B}_{\alg{M}}$ denote the Borel subgroup $\alg{B} \cap \alg{M}$ of $\alg{M}$. Write $\alg{\tilde M} = (\alg{M}', n)$ for the cover of $\alg{M}$ arising from pulling back $\alg{\tilde G}$.
The first Brylinski-Deligne invariant is the same for $\alg{\tilde G}$ as for $\alg{\tilde M}$, as it depends only upon the cover $\alg{\tilde T}$ of their common maximal torus. Write $\Phi_{\alg{M}}$ and $\Phi_{\alg{M}}^\vee$ for the roots and coroots of $\alg{M}$; these are subsets of $\Phi$ and $\Phi^\vee$, respectively, and by agreement of the first Brylinski-Deligne invariant,
$$\tilde \Phi_{\alg{M}} \subset \tilde \Phi, \quad \tilde \Phi_{\alg{M}}^\vee \subset \tilde \Phi^\vee.$$
As $\alg{B}_{\alg{M}} = \alg{B} \cap \alg{M}$, we have $\tilde \Delta_{\alg{M}} \subset \tilde \Delta$ and $\tilde \Delta_{\alg{M}}^\vee \subset \tilde \Delta^\vee$. We find a pair of local systems on $S_{\et}$ of based root data,
$$\left( \sheaf{Y}_{Q,n}, \tilde \Phi^\vee, \tilde \Delta^\vee, \sheaf{X}_{Q,n}, \tilde \Phi, \tilde \Delta \right), \quad \left( \sheaf{Y}_{Q,n}, \tilde \Phi_{\alg{M}}^\vee, \tilde \Delta_{\alg{M}}^\vee, \sheaf{X}_{Q,n}, \tilde \Phi_{\alg{M}}, \tilde \Delta_{\alg{M}} \right).$$
The first root datum defines a local system $\dgp{\tilde G^\vee}$ on $S_{\et}$ of pinned reductive groups over $\mathbb Z$. The second root datum defines a local system $\dgp{\tilde M^\vee}$ on $S_{\et}$ of pinned reductive Levi subgroups of $\dgp{\tilde G^\vee}$.
Thus the dual group of $\alg{\tilde M}$ is naturally a Levi subgroup of the dual group $\dgp{\tilde G^\vee}$. Moreover, by agreement of the first Brylinski-Deligne invariants, the central 2-torsion element of $\dgp{\tilde M^\vee}$ coincides with the central 2-torsion element $\tau_Q(-1) \in \dgp{\tilde G^\vee}$.
\subsection{Specific cases}
Here we give many specific cases of covers, associated dual groups, and 2-torsion elements in their centers corresponding to $\tau_Q(-1)$.
\subsubsection{Method for simply-connected groups}
Let $\alg{G}$ be a simply-connected semisimple group over $S$, with Borel subgroup $\alg{B}$ containing a maximally split maximal torus $\alg{T}$ over $S$. Let $S' / S$ be a Galois cover over which $\alg{T}$ splits. If $t$ is an integer, there exists a unique Weyl-invariant quadratic form $Q_t$ with value $t$ on all short coroots. This $Q_t$ is a multiple of the Killing form.
Corresponding to $Q_t$, there is a unique, up to unique isomorphism, object $\alg{G}^{(t)} \in \Cat{CExt}_S(\alg{G}, \alg{K}_2)$ with first Brylinski-Deligne invariant $Q_t$ (by \cite[\S 7.3(i)]{B-D} when working over a field or \cite[\S 3.3.3]{MWIntegral} over a DVR). Write $\beta_t$ for the resulting $n^{-1} \mathbb Z$-valued bilinear form. Weyl-invariance of $Q_t$ implies that
\begin{equation}
\beta_t(\phi^\vee, y) = n^{-1} Q_t(\phi^\vee) \langle \phi, y \rangle, \text{ for all } \phi \in \Phi, \text{ and all } y \in \sheaf{Y}.
\end{equation}
(See the proof of Lemma \ref{DeltaM} for a derivation.) It follows that, for any $y \in \sheaf{Y}$,
\begin{equation}
\label{BetaObs}
\beta_t(\phi^\vee, y) \in \mathbb Z \text{ if and only if } \langle \phi,y \rangle \in n_\phi \mathbb Z.
\end{equation}
Let $Y = \sheaf{Y}[S']$, $X = \sheaf{X}[S']$, and $\Delta = \Delta[S'] = \{ \alpha_1, \ldots, \alpha_\ell \}$ a basis of simple roots corresponding to $\alg{B}$. Since we assume $\alg{G}$ is simply connected, $Y = \bigoplus_{i = 1}^\ell \alpha_i^\vee \mathbb Z$. Write $n_i = n_{\alpha_i}$. From \eqref{BetaObs}, we find a characterization of $Y_{Q,n} = \sheaf{Y}_{Q,n}[S']$:
\begin{equation}
\label{YQnCharacterization}
y \in Y_{Q,n} \text{ if and only if } \langle \alpha_i, y \rangle \in n_i \mathbb Z \text{ for all } 1 \leq i \leq \ell.
\end{equation}
This, in turn, can be used to tabulate dual groups. We provide tables here reference, noting that such information can also be found in the examples of \cite[\S 2.4]{FinkelbergLysenko}. But we also include data on the central 2-torsion elements that we have not found in the literature. Our tabulation was greatly assisted by using SAGE \cite{sage}, especially the recently updated package which deftly handles root data.
In what follows, we write $\alg{\tilde G}^ { \raisemath{-3pt}{\vee} } = \dgp{\tilde G}^ { \raisemath{-3pt}{\vee} } [S']$ for the dual group over $S'$ (a pinned reductive group scheme over $\mathbb Z$); it is convenient to view the dual group as a group scheme over $\mathbb Z$ endowed with a $\Gal(S'/S)$-action.
\subsubsection{$\alg{SL}_{\ell + 1}$}
For $\alg{G} = \alg{SL}_{\ell + 1}$, the standard Borel subgroup and maximal torus, and system of roots $\alpha_1, \ldots, \alpha_\ell$, the Dynkin diagram is
\begin{center}
\begin{tikzpicture}
\draw (1,0) -- (3.7,0);
\draw (5.3,0) -- (6,0);
\draw[dotted] (3.7,0) -- (5.3,0);
\foreach \i/\l in {1/1,2/2,3/3,6/\ell}
{
\filldraw[fill=gray, draw=black] (\i,0) circle (0.1) node[below=2pt] {$\alpha_{\l}$};
}
\end{tikzpicture}
\end{center}
Consider the cover $\alg{\tilde G}$ of degree $n$ arising from the quadratic form satisfying $Q(\phi^\vee) = 1$ for all $\phi \in \Phi$. Then $n_i = n$ and $\tilde \alpha_i^\vee = n \alpha_i^\vee$ for all $1 \leq i \leq \ell$. Hence
$$Y_{Q,n}^{\SC} = n Y = n \alpha_1^\vee \mathbb Z + \cdots + n \alpha_\ell^\vee \mathbb Z.$$
The Cartan matrix of $\sch{\tilde G}^ { \raisemath{-3pt}{\vee} } $ has entries $\tilde C_{ij} = \langle \tilde \alpha_i, \tilde \alpha_j^\vee \rangle = \langle \alpha_i, \alpha_j^\vee \rangle$, and so $\sch{\tilde G}^ { \raisemath{-3pt}{\vee} } $ is isogenous to $\sch{SL}_{\ell + 1}$. To determine the dual group up to isomorphism, it suffices to compute the order of the center, since the center of $\sch{SL}_{\ell + 1}$ is $\sch{\mu}_{\ell+1}$. The order of the center is equal to the index $\# Y_{Q,n} / Y_{Q,n}^{\SC}$, and this is computable from \eqref{YQnCharacterization}. The results are given in Table \ref{DualATable}.
\begin{table}[!htbp]
\begin{tabular}{l|ccccc}
& \multicolumn{5}{c}{Group $\alg{G}$} \\
$n$ \phantom{M} & $\alg{SL}_2$ & $\alg{SL}_3$ & $\alg{SL}_4$ & $\alg{SL}_5$ & $\alg{SL}_6$ \\ \hline \hline
1 & $\sch{PGL}_2$ & $\sch{PGL}_3$ & $\sch{PGL}_4$ & $\sch{PGL}_5$ & $\sch{PGL}_6$ \\
2 & ${}^{\ast} \sch{SL}_2$ & $\sch{PGL}_3$ & $\sch{SL}_4 / \sch{\mu}_2$ & $\sch{PGL}_5$ & ${}^{\ast}\sch{SL}_6 / \sch{\mu}_3$ \\
3 & $\sch{PGL}_2$ & $\sch{SL}_3$ & $\sch{PGL}_4$ & $\sch{PGL}_5$ & $\sch{SL}_6 / \sch{\mu}_2$ \\
4 & $\sch{SL}_2$ & $\sch{PGL}_3$ & ${}^\ast \sch{SL}_4$ & $\sch{PGL}_5$ & $\sch{SL}_6 / \sch{\mu}_3$ \\
5 & $\sch{PGL}_2$ & $\sch{PGL}_3$ & $\sch{PGL}_4$ & $\sch{SL}_5$ & $\sch{PGL}_6$ \\
6 & ${}^\ast \sch{SL}_2$ & $\sch{SL}_3$ & $\sch{SL}_4 / \sch{\mu}_2$ & $\sch{PGL}_5$ & ${}^\ast \sch{SL}_6$ \\
\end{tabular}
\caption{Table of dual groups for degree $n$ covers of $\alg{SL}_{\ell+1}$. Groups marked with $\ast$ have $\tau_Q(-1)$ nontrivial.}
\label{DualATable}
\end{table}
The dual groups $\sch{\tilde G}^ { \raisemath{-3pt}{\vee} } $ are consistent with the Iwahori-Hecke algebra isomorphisms found by Savin in \cite[Theorem 7.8]{SavinUnramified}. In other words, the dual group $\sch{\tilde G}^ { \raisemath{-3pt}{\vee} } $ coincides with the Langlands dual group of $\alg{SL}_{\ell + 1} / \alg{Z}_{[n]}$, where $\alg{Z}_{[n]}$ is the $n$-torsion subgroup of the center of $\alg{SL}_{\ell+1}$.
The central 2-torsion elements $\tau_Q(-1)$ follow a somewhat predictable pattern. For covering degree $2$, $\tau_Q(-1)$ is nontrivial for $\alg{SL}_2$, $\alg{SL}_6$, $\alg{SL}_{10}$, $\alg{SL}_{14}$, etc.. In covering degree $4$, $\tau_Q(-1)$ is nontrivial for $\alg{SL}_4$, $\alg{SL}_{12}$, $\alg{SL}_{20}$, etc.. In covering degree $6$, $\tau_Q(-1)$ is nontrivial for $\alg{SL}_2$, $\alg{SL}_6$, $\alg{SL}_{10}$, $\alg{SL}_{14}$, etc.. In covering degree $8$, $\tau_Q(-1)$ is nontrivial for $\alg{SL}_8$, $\alg{SL}_{24}$, etc.. In general, we suspect the following:
\begin{center}
$\tau_Q(-1)$ is nontrivial for a degree $2^e \cdot k$ ($k$ odd) cover of $\alg{SL}_m$ if and only if $m = 2^e \cdot j$ for $j$ odd.
\end{center}
\subsubsection{$\alg{Spin}_{2 \ell + 1}$}
For $\alg{G} = \alg{Spin}_{2 \ell + 1}$, the Dynkin diagram has type B.
\begin{center}
\begin{tikzpicture}
\draw (1,0) -- (3.7,0);
\draw (5.3,0) -- (6,0);
\draw[dotted] (3.7,0) -- (5.3,0);
\draw (6,0.05) -- (7,0.05);
\draw (6,-0.05) -- (7,-0.05);
\draw (6.45,0.125) -- (6.55,0) -- (6.45,-0.125);
\foreach \i/\l in {1/1,2/2,3/3,6/{\ell-1}, 7/\ell}
{
\filldraw[fill=gray, draw=black] (\i,0) circle (0.1) node[below=2pt] {$\alpha_{\l}$};
}
\end{tikzpicture}
\end{center}
Let $\alg{\tilde G}$ be the cover of degree $n$, associated to the quadratic form taking the value $1$ at all short coroots. Thus $Q(\alpha_i^\vee) = 2$ for all long coroots $1 \leq i \leq \ell - 1$. If $n$ is odd, then $n_\alpha = n$ for all coroots $\alpha^\vee$. If $n$ is even, then $n_i = n/2$ for $1 \leq i \leq \ell-1$ and $n_\ell = n$. When $n$ is even, short coroots become long and long become short, after modification. We find that the dual group is isogenous to $\sch{Sp}_{2 \ell}$ if $n$ is odd, and is isogenous to $\sch{Spin}_{2 \ell + 1}$ if $n$ is even.
The centers of $\sch{Sp}_{2 \ell}$ and $\sch{Spin}_{2 \ell + 1}$ are cyclic of order two. Thus the dual group can be identified by the order of its center.
\begin{table}[!htbp]
\begin{tabular}{l|cccccc}
& \multicolumn{6}{c}{Group $\alg{G}$} \\
$n$ \phantom{M} & $\sch{Spin}_7$ & $\sch{Spin}_9$ & $\sch{Spin}_{11}$ & $\sch{Spin}_{13}$ & $\sch{Spin}_{15}$ & $\sch{Spin}_{17}$ \\ \hline \hline
1 & $\sch{PGSp}_{6}$ & $\sch{PGSp}_{8}$ & $\sch{PGSp}_{10}$ & $\sch{PGSp}_{12}$ & $\sch{PGSp}_{14}$ & $\sch{PGSp}_{16}$ \\
2 & $\sch{SO}_{7}$ & $\sch{Spin}_{9}$ & $\sch{SO}_{11}$ & ${}^\ast \sch{Spin}_{13}$ &$ \sch{SO}_{15}$ & $\sch{Spin}_{17}$ \\
3 & $\sch{PGSp}_{6}$ & $\sch{PGSp}_{8}$ & $\sch{PGSp}_{10}$ & $\sch{PGSp}_{12}$ & $\sch{PGSp}_{14}$ & $\sch{PGSp}_{16}$ \\
4 & ${}^\ast \sch{Spin}_{7}$ & $\sch{Spin}_{9}$ & ${}^\ast \sch{Spin}_{11}$ & $\sch{Spin}_{13}$ & ${}^\ast \sch{Spin}_{15}$ & $\sch{Spin}_{17}$ \\
5 & $\sch{PGSp}_{6} $ & $\sch{PGSp}_{8} $ & $\sch{PGSp}_{10} $ & $\sch{PGSp}_{12}$ &$ \sch{PGSp}_{14}$ & $\sch{PGSp}_{16}$ \\
6 & $\sch{SO}_{7}$ & $\sch{Spin}_{9}$ & $\sch{SO}_{11}$ & ${}^\ast \sch{Spin}_{13}$ & $\sch{SO}_{15}$ & $\sch{Spin}_{17}$ \\
\end{tabular}
\caption{Table of dual groups for degree $n$ covers of $\alg{Spin}_{2 \ell+1}$. Groups marked with $\ast$ have $\tau_Q(-1)$ nontrivial.}
\label{DualBTable}
\end{table}
Table \ref{DualBTable} describes the dual groups. Note that, in this case, the isogeny class of the dual group depends on the covering degree modulo $4$. In covering degree $4k + 2$, we find that $\tau_Q(-1)$ is nontrivial for $\alg{Spin}_{8 j + 5}$ (corresponding to rank $4j + 2$) for all positive integers $j$. In covering degree $4k$, we find that $\tau_Q(-1)$ is nontrivial for $\alg{Spin}_{4j + 3}$ for all positive integers $j$.
\subsubsection{$\alg{Sp}_{2 \ell}$}
For $\alg{G} = \alg{Sp}_{2 \ell}$, the Dynkin diagram has type C.
\begin{center}
\begin{tikzpicture}
\draw (1,0) -- (3.7,0);
\draw (5.3,0) -- (6,0);
\draw[dotted] (3.7,0) -- (5.3,0);
\draw (6,0.05) -- (7,0.05);
\draw (6,-0.05) -- (7,-0.05);
\draw (6.55,0.125) -- (6.45,0) -- (6.55,-0.125);
\foreach \i/\l in {1/1,2/2,3/3,6/{\ell-1}, 7/\ell}
{
\filldraw[fill=gray, draw=black] (\i,0) circle (0.1) node[below=2pt] {$\alpha_{\l}$};
}
\end{tikzpicture}
\end{center}
Let $\alg{\tilde G}$ be the cover of degree $n$, associated to the quadratic form taking the value $1$ at all short coroots. As in type B, we find that short coroots become long, and long become short, after modification when $n$ is even. We find that the dual group is isogenous to $\sch{Sp}_{2 \ell}$ if $n$ is even, and is isogenous to $\sch{Spin}_{2 \ell + 1}$ if $n$ is odd. As before, the dual group can be identified by the order of its center.
\begin{table}[!htbp]
\begin{tabular}{l|ccc}
& \multicolumn{3}{c}{Group $\alg{G}$} \\
$n$ \phantom{M} & $\alg{Sp}_6$ & $\alg{Sp}_8$ & $\alg{Sp}_{10}$ \\ \hline \hline
1 & $\sch{SO}_{7}$ & $\sch{SO}_{9}$ & $\sch{SO}_{11}$ \\
2 & ${}^\ast \sch{Sp}_{6}$ & ${}^\ast \sch{Sp}_{8}$ & ${}^\ast \sch{Sp}_{10}$ \\
3 & $\sch{SO}_{7}$ & $\sch{SO}_{9}$ & $\sch{SO}_{11}$ \\
4 & $\sch{Sp}_{6}$ & $\sch{Sp}_{8}$ & $\sch{Sp}_{10}$ \\
5 & $\sch{SO}_{7}$ & $\sch{SO}_{9}$ & $\sch{SO}_{11}$ \\
6 & ${}^\ast \sch{Sp}_{6}$ & ${}^\ast \sch{Sp}_{8}$ & ${}^\ast \sch{Sp}_{10}$ \\
\end{tabular}
\caption{Table of dual groups for degree $n$ covers of $\alg{Sp}_{2 \ell}$. Groups marked with $\ast$ have $\tau_Q(-1)$ nontrivial.}
\label{DualCTable}
\end{table}
As Table \ref{DualCTable} illustrates, the dual group of the degree $n$ cover of $\alg{Sp}_{2 \ell}$ is the simply-connected Chevalley group $\sch{Sp}_{2 \ell}$ when $n$ is even, and the dual group is the adjoint group $\sch{SO}_{2 \ell + 1} = \sch{Spin}_{2 \ell + 1} / \sch{\mu}_2$ when $n$ is odd. The central 2-torsion element $\tau_Q(-1)$ is nontrivial when the covering degree is $4k + 2$ for some non-negative integer $k$. This is consistent (in covering degree $2$) with expectations from the classical theta correspondence for metaplectic groups.
\subsubsection{$\alg{Spin}_{2 \ell}$}
For $\alg{G} = \alg{Spin}_{2 \ell}$, $\ell \geq 4$, the Dynkin diagram has type D.
\begin{center}
\begin{tikzpicture}
\draw (1,0) -- (3.7,0);
\draw (5.3,0) -- (6,0);
\draw[dotted] (3.7,0) -- (5.3,0);
\draw (6,0) -- (7,0.866);
\draw (6,0) -- (7,-0.866);
\filldraw[fill=gray, draw=black] (6,0) circle (0.1) node[right=2pt] {$\alpha_{\ell-2}$};
\filldraw[fill=gray, draw=black] (7,0.866) circle (0.1) node[right=2pt] {$\alpha_{\ell-1}$};
\filldraw[fill=gray, draw=black] (7,-0.866) circle (0.1) node[right=2pt] {$\alpha_{\ell}$};
\foreach \i/\j/\l in {1/1,2/2,3/3}
{
\filldraw[fill=gray, draw=black] (\i,0) circle (0.1) node[below=2pt] {$\alpha_{\l}$};
}
\end{tikzpicture}
\end{center}
Let $\alg{\tilde G}$ be the cover of degree $n$, associated to the quadratic form taking the value $1$ at all coroots. By the same methods as in type A, we find that the dual group is isogenous to $\sch{Spin}_{2 \ell}$.
If $\ell$ is odd, then the center of $\sch{Spin}_{2 \ell}$ is a cyclic group of order $4$. In this case, the dual group is determined by the order of its center.
If $\ell$ is even, then the center of $\sch{Spin}_{2 \ell}$ is isomorphic to $\sch{\mu}_2 \times \sch{\mu}_2$, and so the dual group is not a priori determined by the order of its center. But fortunately, the order of the center of the dual group always equals $1$ or $4$ when $\ell$ is even, and this suffices to identify the dual group.
\begin{table}[!htbp]
\begin{tabular}{l|cccccc}
& \multicolumn{6}{c}{Group $\alg{G}$} \\
$n$ \phantom{M} & $\alg{Spin}_8$ & $\alg{Spin}_{10}$ & $\alg{Spin}_{12}$ & $\alg{Spin}_{14}$ & $\alg{Spin}_{16}$ & $\alg{Spin}_{18}$ \\ \hline \hline
1 & $\sch{PGO}_8$ & $\sch{PGO}_{10}$ & $\sch{PGO}_{12}$ & $\sch{PGO}_{14}$ & $\sch{PGO}_{16}$ & $\sch{PGO}_{18}$ \\
2 & $\sch{Spin}_8$ & $\sch{SO}_{10}$ & ${}^\ast \sch{Spin}_{12}$ & $\sch{SO}_{14}$ & $\sch{Spin}_{16}$ & $\sch{SO}_{18}$ \\
3 & $\sch{PGO}_8$ & $\sch{PGO}_{10}$ & $\sch{PGO}_{12}$ & $\sch{PGO}_{14}$ & $\sch{PGO}_{16}$ & $\sch{PGO}_{18}$ \\
4 & $\sch{Spin}_8$ & ${}^\ast \sch{Spin}_{10}$ & $\sch{Spin}_{12}$ & ${}^\ast \sch{Spin}_{14}$ & $\sch{Spin}_{16}$ & ${}^\ast \sch{Spin}_{18}$\\
5 & $\sch{PGO}_8$ & $\sch{PGO}_{10}$ & $\sch{PGO}_{12}$ & $\sch{PGO}_{14}$ & $\sch{PGO}_{16}$ & $\sch{PGO}_{18}$ \\
6 & $\sch{Spin}_8$ & $\sch{SO}_{10}$ & ${}^\ast \sch{Spin}_{12}$ & $\sch{SO}_{14}$ & $\sch{Spin}_{16}$ & $\sch{SO}_{18}$ \\
\end{tabular}
\caption{Table of dual groups for degree $n$ covers of $\alg{Spin}_{2 \ell}$. Groups marked with $\ast$ have $\tau_Q(-1)$ nontrivial.}
\label{DualDTable}
\end{table}
As Table \ref{DualDTable} illustrates, the dual group of an odd-degree cover of $\alg{Spin}_{2 \ell}$ coincides with the Langlands dual group of the linear group $\alg{Spin}_{2 \ell}$; this dual group is the adjoint group $\sch{PGO}_{2 \ell}$. But the dual group of an even-degree cover of $\alg{Spin}_{2 \ell}$ depends on the parity of $\ell$ and the covering degree modulo $4$. As in type A, these dual groups agree with expectations from the Hecke algebra isomorphisms of Savin \cite{SavinUnramified}.
When the covering degree is a multiple of $4$, the element $\tau_Q(-1)$ is nontrivial for $\alg{Spin}_{4j + 2}$ for all $j \geq 2$. Since $\alg{Spin}_{4j + 2}$ has a unique central element of order two, this suffices to describe $\tau_Q$. When the covering degree has the form $4k +2$, the element $\tau_Q(-1)$ is nontrivial for $\alg{Spin}_{8j + 4}$ for all $j \geq 1$. The center of the group $\sch{Spin}_{8j + 4}$ is isomorphic to $\sch{\mu}_2 \times \sch{\mu}_2$, which has three distinct 2-torsion elements. However, only one of these is invariant under the nontrivial outer automorphism of the pinned Chevalley group $\sch{Spin}_{8j + 4}$. This one must be $\tau_Q(-1)$, since $Q$ is invariant under this outer automorphism.
\subsubsection{Exceptional groups}
Let $\alg{G}$ be a simply-connected split simple group of type $\text{E}_\ell$ (with $\ell = 6, 7, 8$), $\text{F}_4$, or $\text{G}_2$. Let $\alg{\tilde G}$ be the cover of degree $n$, associated to the quadratic form taking the value $1$ at all short coroots (all coroots in type E). As in types A and D, we find that the dual group is semisimple and isogenous to the Chevalley group of the same type as $\alg{G}$. In types $\text{E}_8$, $\text{F}_4$, and $\text{G}_2$, the simply-connected group is centerless, and so $\sch{\tilde G}^ { \raisemath{-3pt}{\vee} } $ coincides with the simply-connected Chevalley group of the same type.
The center of $\sch{E}_6$ has order $3$, and the center of $\sch{E}_7$ has order $2$. Hence the dual group $\sch{\tilde G}^ { \raisemath{-3pt}{\vee} } $ is determined by the order of its center. The dual groups are listed in Table \ref{DualETable}.
\begin{table}[!htbp]
\begin{tabular}{l|ccccc}
& \multicolumn{5}{c}{Group $\alg{G}$} \\
$n$ \phantom{M} & $\alg{E}_6$ & $\alg{E}_7$ & $\alg{E}_8$ & $\alg{F}_4$ & $\alg{G}_2$ \\ \hline
1 & $\sch{E}_6 / \sch{\mu}_3$ & $\sch{E}_7 / \sch{\mu}_2$ & $\sch{E}_8$ & $\sch{F}_4$ & $\sch{G}_2$ \\
2 & $\sch{E}_6 / \sch{\mu}_3$ & ${}^\ast \sch{E}_7$ & $\sch{E}_8$ & $\sch{F}_4$ & $\sch{G}_2$ \\
3 & $\sch{E}_6$ & $\sch{E}_7 / \sch{\mu}_2$ & $\sch{E}_8$ & $\sch{F}_4$ & $\sch{G}_2$ \\
4 & $\sch{E}_6 / \sch{\mu}_3$ & $\sch{E}_7$ & $\sch{E}_8$ & $\sch{F}_4$ & $\sch{G}_2$ \\
5 & $\sch{E}_6 / \sch{\mu}_3$ & $\sch{E}_7 / \sch{\mu}_2$ & $\sch{E}_8$ & $\sch{F}_4$ & $\sch{G}_2$ \\
6 & $\sch{E}_6$ & ${}^\ast \sch{E}_7$ & $\sch{E}_8$ & $\sch{F}_4$ & $\sch{G}_2$ \\
\hline
\end{tabular}
\caption{Table of dual groups for degree $n$ covers of exceptional groups. Groups marked with $\ast$ have $\tau_Q(-1)$ nontrivial.}
\label{DualETable}
\end{table}
In type E, these dual groups agree with expectations from \cite{SavinUnramified}. The central 2-torsion element is nontrivial for $\alg{E}_7$, when the covering degree equals $4j + 2$ for some $j \geq 0$.
\subsubsection{$\alg{GL}_r$}
Suppose that $\alg{G}$ is split reductive, and the derived subgroup of $\alg{G}$ is simply-connected. Let $\alg{T}$ be a split maximal torus in $\alg{G}$ with cocharacter lattice $Y$. Then, for any Weyl-invariant quadratic form $Q \colon Y \rightarrow \mathbb Z$, there exists a cover $\alg{\tilde G}$ with first Brylinski-Deligne invariant $Q$.
For example, when $\alg{G} = \alg{GL}_r$, there is a two-parameter family of such Weyl-invariant quadratic forms. Write $\alg{T}$ for the standard maximal torus of diagonal matrices, and identify $Y = \mathbb Z^r$ in the usual way. For any pair of integers $q,c$, there exists a unique Weyl-invariant quadratic form $Q_{q,c}$ satisfying
$$Q(1,-1,0,\ldots, 0) = q \text{ and } Q(1,0,\ldots, 0) = 1+c.$$
The $n$-fold covers $\widetilde{GL}_r^{(c)}$ studied by Kazhdan and Patterson \cite[\S 0.1]{KazhdanPatterson} can be constructed from Brylinski-Deligne extensions with first invariant $Q_{1,c}$. The proof of following result is left to the reader.
\begin{proposition}
Let $\alg{\tilde G}$ be a degree $n$ cover of $\alg{G} = \alg{GL}_r$ with first Brylinski-Deligne invariant $Q_{1,c}$. If $\GCD(n, 1+r + 2rc) = 1$, then $\sch{\tilde G}^ { \raisemath{-3pt}{\vee} } $ is isomorphic to $\sch{GL}_r$. If $\GCD(n,r) = 1$, then the derived subgroup of $\sch{\tilde G}^ { \raisemath{-3pt}{\vee} } $ is isomorphic to $\sch{SL}_r$ and thus there exists an isogeny $\sch{\tilde G}^ { \raisemath{-3pt}{\vee} } \rightarrow \sch{GL}_r$.
\end{proposition}
This may place the work of Kazhdan and Flicker \cite{KazhdanFlicker} in a functorial context.
\subsubsection{$\alg{GSp}_{2 r}$}
For $\alg{G} = \alg{GSp}_{2r}$, and a standard choice of split maximal torus and Borel subgroup, we write $e_0, \ldots, e_r$ for a basis of $Y$, $f_0, \ldots, f_r$ for the dual basis of $X$, and the simple roots and coroots are
$$\alpha_1 = f_1 - f_2, \ldots, \alpha_{r-1} = f_{r-1} - f_r, \quad \alpha_r = 2 f_r - f_0;$$
$$\alpha_1^\vee = e_1 - e_2, \ldots, \alpha_{r-1}^\vee = e_{r-1} - e_r, \quad \alpha_r^\vee = e_r.$$
The Weyl group is $S_r \ltimes \mu_2^r$, with $S_r$ acting by permutation of indices $1, \ldots, r$ (fixing $e_0$ and $f_0$), and elements $w_j$ (for $1 \leq j \leq r$) of order two which satisfy
$$w_j(e_j) = - e_j, \quad w_j(e_i) = e_i \text{ for } i \neq j,0, \quad w_j(e_0) = e_0 + e_1.$$
Weyl-invariant quadratic forms on $Y$ are in bijection with pairs $(\kappa, \nu)$ of integers. For any such pair, there is a unique Weyl-invariant quadratic form $Q_{\kappa, \nu}$ satisfying
$$Q_{\kappa, \nu}(e_0) = \kappa, \quad Q_{\kappa, \nu}(e_i) = \nu \text{ for } 1 \leq i \leq r.$$
The proof of following result is left to the reader.
\begin{proposition}
Let $\alg{\tilde G}$ be a degree $2$ cover of $\alg{G} = \alg{GSp}_{2r}$, with first Brylinski-Deligne invariant $Q_{0,1}$. Then the dual group $\sch{\tilde G}^ { \raisemath{-3pt}{\vee} } $ is isomorphic to $\sch{GSp}_{2r}$ if $r$ is odd, and to $\sch{PGSp}_{2r} \times \sch{G}_m$ if $r$ is even.
\end{proposition}
We find that that double-covers of $\alg{GSp}_{2r}$ behave differently depending on the parity of $r$; this phenomenon is consistent with the work of Szpruch \cite{Szpruch} on principal series.
\section{The gerbe associated to a cover}
In this section, we construct a gerbe $\gerb{E}_\epsilon(\alg{\tilde G})$ on $S_{\et}$ associated to a degree $n$ cover $\alg{\tilde G}$ of a quasisplit group $\alg{G}$, and a choice of injective character $\epsilon \colon \mu_n \hookrightarrow \mathbb C^\times$. Fix $\alg{\tilde G}$, $\alg{G}$, and $\epsilon$ throughout. Also, choose a Borel subgroup containing a maximally split maximal torus $\alg{B} \supset \alg{T}$; we will see that our construction is independent of this choice (in a 2-categorical sense).
We make one assumption about our cover $\alg{\tilde G}$, which enables our construction and is essentially nonrestrictive.
\begin{assumption}[Odd $n$ implies even $Q$]
\label{OddnEvenQ}
If $n$ is odd, then we assume $Q \colon \sheaf{Y} \rightarrow \mathbb Z$ takes only even values.
\end{assumption}
If $\alg{\tilde G}$ does not satisfy this assumption, i.e., $n$ is odd and $Q(y)$ is odd for some $y \in \sheaf{Y}$, then replace $\alg{\tilde G}$ by $(n+1) \dot \times \alg{\tilde G}$ (its Baer sum with itself $n+1$ times). The first Brylinski-Deligne invariant becomes $(n+1) Q$, which is even-valued. By Proposition \ref{DualGroupModn}, the dual group $\dgp{\tilde G^\vee}$ does not change since $Q \equiv (n+1) Q$ modulo $n$. Moreover, the resulting extensions of groups over local or global fields, e.g., $\mu_n \hookrightarrow \tilde G \twoheadrightarrow G$, remain the same (up to natural isomorphism). Indeed, the Baer sum of $\tilde G$ with itself $n+1$ times is naturally isomorphic to the pushout via the $(n+1)^{\th}$ power map $\mu_n \rightarrow \mu_n$, which equals the identity map.
We work with sheaves of abelian groups on $S_{\et}$, and great care is required to avoid confusion between those in the left column and the right column below. Define
\begin{align*}
\sheaf{\hat T} = \mathscr{H}\mathit{om}(\sheaf{Y}_{Q,n}, \sheaf{G}_m), & \quad \sheaf{\tilde T^\vee} = \mathscr{H}\mathit{om}(\sheaf{Y}_{Q,n}, \mathbb C^\times); \\
\sheaf{\hat T}_{\SC} = \mathscr{H}\mathit{om}(\sheaf{Y}_{Q,n}^{\SC}, \sheaf{G}_m), & \quad \sheaf{\tilde T}_{\SC}^ { \raisemath{-3pt}{\vee} } = \mathscr{H}\mathit{om}(\sheaf{Y}_{Q,n}^{\SC}, \mathbb C^\times); \\
\sheaf{\hat Z} = \mathscr{H}\mathit{om}(\sheaf{Y}_{Q,n} / \sheaf{Y}_{Q,n}^{\SC}, \sheaf{G}_m), & \quad \sheaf{\tilde Z^\vee} = \mathscr{H}\mathit{om}(\sheaf{Y}_{Q,n} / \sheaf{Y}_{Q,n}^{\SC}, \mathbb C^\times).
\end{align*}
Here $\mathbb C^\times$ denotes the constant sheaf on $S_{\et}$. Thus, in the right column, we find the complex points of the dual groups,
$$\sheaf{\tilde T^\vee} = \dgp{\tilde T}^ { \raisemath{-3pt}{\vee} } (\mathbb C), \quad \sheaf{\tilde T}_{\SC}^ { \raisemath{-3pt}{\vee} } = \dgp{\tilde T}_{\SC}^ { \raisemath{-3pt}{\vee} } (\mathbb C), \quad \sheaf{\tilde Z^\vee} = \dgp{\tilde Z}^ { \raisemath{-3pt}{\vee} } (\mathbb C).$$
Composing with $\epsilon$ defines homomorphisms of local systems of abelian groups,
\begin{align*}
\sheaf{\hat T}_{[n]} = \mathscr{H}\mathit{om}(\sheaf{Y}_{Q,n}, \mu_n) & \xrightarrow{\epsilon} \sheaf{\tilde T^\vee}; \\
\sheaf{\hat T}_{\SC, {[n]}} = \mathscr{H}\mathit{om}(\sheaf{Y}_{Q,n}^{\SC}, \mu_n) & \xrightarrow{\epsilon} \sheaf{\tilde T}_{\SC}^ { \raisemath{-3pt}{\vee} } ; \\
\sheaf{\hat Z}_{[n]} = \mathscr{H}\mathit{om}(\sheaf{Y}_{Q,n} / \sheaf{Y}_{Q,n}^{\SC}, \mu_n) & \xrightarrow{\epsilon} \sheaf{\tilde Z^\vee}.
\end{align*}
\subsection{The gerbe associated to a cover of a torus}
Associated to the cover $\alg{\tilde T} = (\alg{T}', n)$, the second Brylinski-Deligne invariant is a central extension of sheaves of groups on $S_{\et}$,
$$\sheaf{G}_m \hookrightarrow \sheaf{D} \twoheadrightarrow \sheaf{Y}.$$
The commutator of this extension is given in \cite[Proposition 3.11]{B-D},
\begin{equation}
\label{CommForm}
\Comm(y_1, y_2) = (-1)^{n \beta_Q(y_1, y_2)}, \text{ for all } y_1, y_2 \in \sheaf{Y}.
\end{equation}
Pulling back via $\sheaf{Y}_{Q,n} \hookrightarrow \sheaf{Y}$, we find an extension of sheaves of groups,
\begin{equation}
\label{BD2Qn}
\sheaf{G}_m \hookrightarrow \sheaf{D}_{Q,n} \twoheadrightarrow \sheaf{Y}_{Q,n}.
\end{equation}
\begin{proposition}
$\sheaf{D}_{Q,n}$ is a commutative extension.
\end{proposition}
{\sc Proof:}
If $n$ is even and $y_1, y_2 \in \sheaf{Y}_{Q,n}$, then $\beta_Q(y_1, y_2) \in \mathbb Z$ and $n \beta_Q(y_1, y_2) \in 2 \mathbb Z$. On the other hand, if $n$ is odd, Assumption \ref{OddnEvenQ} implies that
$$n \beta_Q(y_1, y_2) = Q(y_1 + y_2) - Q(y_1) - Q(y_2) \in 2 \mathbb Z.$$
The commutator formula \eqref{CommForm} finishes the proof.
\begin{flushright}Q.E.D \end{flushright}
Let $\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n})$ denote the sheaf of {\em splittings} of the commutative extension \eqref{BD2Qn}. In other words, $\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n})$ is the subsheaf of $\mathscr{H}\mathit{om}(\sheaf{Y}_{Q,n}, \sheaf{D}_{Q,n})$ consisting of homomorphisms which split \eqref{BD2Qn}.
Over any finite \'etale $U \rightarrow S$ splitting $\alg{T}$, $\sheaf{Y}_{Q,n}$ restricts to a constant sheaf of free abelian groups. Thus $\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n})$ is a $\sheaf{\hat T}$-torsor on $S_{\et}$, which obtains a point over any such $U$. The equivalence class of this torsor is determined by its cohomology class $\left[ \mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n}) \right] \in H_{\et}^1(S, \sheaf{\hat T})$.
Consider the Kummer sequence, $\sheaf{\hat T}_{[n]} \hookrightarrow \sheaf{\hat T}\xtwoheadrightarrow{n} \sheaf{\hat T}$. Write $\Kum$ (for Kummer) for the connecting map in cohomology, $\Kum \colon H_{\et}^1(S, \sheaf{\hat T}) \rightarrow H_{\et}^2(S, \sheaf{\hat T}_{[n]})$. This map in cohomology corresopnds to the functor which sends a $\sheaf{\hat T}$-torsor to its gerbe of $n^{\th}$ roots (see \ref{AppendixGerbeRoots} for details). We write
$\sqrt[n]{\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n})}$ for the gerbe of $n^{\th}$ roots of the $\sheaf{\hat T}$-torsor $\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n})$. It is banded by the local system $\sheaf{\hat T}_{[n]}$ and its equivalence class satisfies
$$\left[ \sqrt[n]{\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n})} \right] = \Kum [\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n})].$$
Finally we push out via the homomorphism of local systems,
$$\epsilon \colon \sheaf{\hat T}_{[n]} = \mathscr{H}\mathit{om}(\sheaf{Y}_{Q,n}, \mu_n) \rightarrow \mathscr{H}\mathit{om}(\sheaf{Y}_{Q,n}, \mathbb C^\times) = \sheaf{\tilde T^\vee}.$$
\begin{definition}
The \defined{gerbe associated to the cover} $\alg{\tilde T}$ is defined by
$$\gerb{E}_\epsilon(\alg{\tilde T}) := \epsilon_\ast \sqrt[n]{\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n})}.$$
It is a gerbe on $S_{\et}$ banded by the local system of abelian groups $\sheaf{\tilde T^\vee}$.
\end{definition}
\begin{example}
\label{SplitTorusGerbeNeutral}
Suppose that $\alg{T}$ is a split torus. Then the exact sequence of sheaves $\sheaf{G}_m \hookrightarrow \sheaf{D}_{Q,n} \twoheadrightarrow \sheaf{Y}_{Q,n}$ splits. Indeed, $\sheaf{Y}_{Q,n}$ is a constant sheaf of free abelian groups, and Hilbert's Theorem 90 gives a short exact sequence
\begin{equation}
\label{SPointsD}
\sheaf{G}_m[S] \hookrightarrow \sheaf{D}_{Q,n}[S] \twoheadrightarrow \sheaf{Y}_{Q,n}[S].
\end{equation}
Since $\sheaf{Y}_{Q,n}[S]$ is a free abelian group, this exact sequence splits, and any such splitting defines an $S$-point of the torsor $\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n})$. An $S$-point of $\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n})$, in turn, neutralizes of the gerbe $\sqrt[n]{\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n})}$.
Thus when $\alg{T}$ is a split torus, the gerbe $\gerb{E}_\epsilon(\alg{\tilde T})$ is trivial. Any splitting of the sequence \eqref{SPointsD} defines a neutralization of $\gerb{E}_\epsilon(\alg{\tilde T})$.
\end{example}
\subsection{The gerbe of liftings}
Recall that $\sheaf{Y}_{Q,n}^{\SC}$ denotes the subgroup of $\sheaf{Y}_{Q,n}$ spanned by the modified coroots $\tilde \Phi^\vee$, and $\sheaf{\hat T}_{\SC} = \mathscr{H}\mathit{om}(\sheaf{Y}_{Q,n}^{\SC}, \sheaf{G}_m)$. The inclusion $\sheaf{Y}_{Q,n}^{\SC} \hookrightarrow \sheaf{Y}_{Q,n}$ corresponds to a surjective homomorphism,
$$p \colon \sheaf{\hat T} \rightarrow \sheaf{\hat T}_{\SC}.$$
The extension $\sheaf{G}_m \hookrightarrow \sheaf{D}_{Q,n} \twoheadrightarrow \sheaf{Y}_{Q,n}$ pulls back via $\sheaf{Y}_{Q,n}^{\SC} \hookrightarrow \sheaf{Y}_{Q,n}$ to an extension,
$$\sheaf{G}_m \hookrightarrow \sheaf{D}_{Q,n}^{\SC} \twoheadrightarrow \sheaf{Y}_{Q,n}^{\SC}.$$
A splitting of $\sheaf{D}_{Q,n}$ pulls back to a splitting of $\sheaf{D}_{Q,n}^{\SC}$, providing a map of torsors,
$$p^\ast \colon \mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n}) \rightarrow \mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n}^{\SC}),$$
lying over $p \colon \sheaf{\hat T} \rightarrow \sheaf{\hat T}_{\SC}$. Taking $n^{\th}$ roots of torsors gives a functor of gerbes,
$$\sqrt[n]{p^\ast}\colon \sqrt[n]{\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n})} \rightarrow \sqrt[n]{\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n}^{\SC})},$$
lying over $p \colon \sheaf{\hat T}_{[n]} \rightarrow \sheaf{\hat T}_{\SC, [n]}$ (see \ref{AppendixGerbeRoots}).
Recall that $\sheaf{\tilde T^\vee} = \mathscr{H}\mathit{om}(\sheaf{Y}_{Q,n}, \mathbb C^\times)$ and $\sheaf{\tilde T}_{\SC}^ { \raisemath{-3pt}{\vee} } = \mathscr{H}\mathit{om}(\sheaf{Y}_{Q,n}^{\SC}, \mathbb C^\times)$. Define $\gerb{E}_\epsilon^{\SC}(\alg{\tilde T}) := \epsilon_\ast \sqrt[n]{\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n}^{\SC})}$ by analogy to $\gerb{E}_\epsilon(\alg{\tilde T}) = \epsilon_\ast \sqrt[n]{\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n})}$. Pushing out via $\epsilon$, the functor $\sqrt[n]{p^\ast}$ yields a functor of gerbes
$$\gerb{p} = \epsilon_\ast \sqrt[n]{p^\ast} \colon \gerb{E}_\epsilon(\alg{\tilde T}) \rightarrow \gerb{E}_\epsilon^{\SC}(\alg{\tilde T}),$$
lying over the homomorphism $p \colon \sheaf{\tilde T^\vee} \rightarrow \sheaf{\tilde T}_{\SC}^ { \raisemath{-3pt}{\vee} } $.
In the next section, we define the {\em Whittaker torsor}, which gives an object $\boldsymbol{w}$ neutralizing the gerbe $\gerb{E}_\epsilon^{\SC}(\alg{\tilde T})$. We take this construction of $\boldsymbol{w}$ for granted at the moment.
\begin{definition}
Define $\gerb{E}_\epsilon(\alg{\tilde G})$ to be the gerbe $\gerb{p}^{-1}(\boldsymbol{w})$ of liftings of $\boldsymbol{w}$ via $\gerb{p}$ (see \ref{AppendixGerbeLiftings}). In other words, $\gerb{E}_\epsilon(\alg{\tilde G})$ is the
category of pairs $(\gerb{e}, j)$ where $\gerb{e}$ is an object of $\gerb{E}_\epsilon(\alg{\tilde T})$ and $j \colon \gerb{p}(\gerb{e}) \rightarrow \boldsymbol{w}$ is an isomorphism in $\gerb{E}_\epsilon^{\SC}(\alg{\tilde T})$. This is a gerbe on $S_{\et}$ banded by $\sheaf{\tilde Z^\vee} = \Ker(\sheaf{\tilde T^\vee} \xrightarrow{p} \sheaf{\tilde T}_{\SC}^ { \raisemath{-3pt}{\vee} } )$.
\end{definition}
The cohomology classes of our gerbes now fit into a sequence
$$\begin{tikzcd}[row sep = 0em]
\left[ \gerb{E}_\epsilon(\alg{\tilde G}) \right] \arrow[mapsto]{r} & \left[ \gerb{E}_\epsilon(\alg{\tilde T}) \right] \arrow[mapsto]{r} & \left[ \gerb{E}_\epsilon^{\SC}(\alg{\tilde T}) \right] = 0 \\
\rotatebox{-90}{$\in$} & \rotatebox{-90}{$\in$}& \rotatebox{-90}{$\in$} \\
H_{\et}^2(S, \sheaf{\tilde Z^\vee}) \arrow{r} & H_{\et}^2(S, \sheaf{\tilde T^\vee}) \arrow{r} &H_{\et}^2(S, \sheaf{\tilde T_{\SC} ^\vee})
\end{tikzcd}$$
\begin{remark}
The construction of this gerbe relies on the (soon-to-be-defined) Whittaker torsor in a crucial way. We view this as a good thing, since any putative Langlands correspondence should also connect the existence of Whittaker models to properties of the Langlands parameter (cf. \cite{VoganLLC}).
\end{remark}
\subsection{The Whittaker torsor}
Now we construct the object $\boldsymbol{w}$ neutralizing the gerbe $\gerb{E}_\epsilon^{\SC}(\alg{\tilde T})$ over $S$. Let $\alg{U}$ denote the unipotent radical of the Borel subgroup $\alg{B} \subset \alg{G}$, and let $\sheaf{U}$ be the sheaf of groups on $S_{\et}$ that it represents. Let $\alg{G}_a$ denote the additive group scheme over $S$, and $\sheaf{G}_a$ the sheaf of groups on $S_{\et}$ that it represents. Recall that $\Delta \subset \Phi$ denotes the subset of simple roots corresponding to the Borel subgroup $\alg{B}$.
For $S' \rightarrow S$ finite \'etale and splitting $\alg{T}$, and $\alpha \in \Delta[S']$, write $\alg{U}_\alpha$ for the one-dimensional root subgroup of $\alg{U}_{S'}$ associated to $\phi$. Let $\sheaf{U}_\alpha$ be the associated sheaf of abelian groups on $S_{\et}'$. Write $\mathscr{H}\mathit{om}^\ast(\sheaf{U}_\alpha, \sheaf{G}_{a})$ for the sheaf (on $S_{\et}'$) of isomorphisms from $\sheaf{U}_\alpha$ to $\sheaf{G}_{a,S'}$. The sheaf $\mathscr{H}\mathit{om}^\ast(\sheaf{U}_\alpha, \sheaf{G}_{a})$ naturally forms a $\sheaf{G}_m$-torsor on $S_{\et}'$, by the formula
$$[h \ast \xi](u) = h^{-1} \cdot \xi(u) \text{ for all } h \in \sheaf{G}_m, \xi \in \mathscr{H}\mathit{om}^\ast(\sheaf{U}_\alpha, \sheaf{G}_a).$$
\begin{definition}
The \defined{Whittaker torsor} is the subsheaf $\mathscr{W}\mathit{hit} \subset \mathscr{H}\mathit{om}(\sheaf{U}, \sheaf{G}_a)$ consisting of those homomorphisms which (locally on $S_{\et}$) restrict to an isomorphism on every simple root subgroup. The sheaf $\mathscr{W}\mathit{hit}$ is given the structure of a $\sheaf{\hat T}_{\SC}$-torsor as follows: for a Galois cover $S' \rightarrow S$ splitting $\alg{T}$, we have
$$\sheaf{\hat T}_{\SC}[S'] = \mathscr{H}\mathit{om}(\sheaf{Y}_{Q,n}^{\SC}, \sheaf{G}_m)[S'] = \mathscr{H}\mathit{om} \left( \bigoplus_{\alpha \in \Delta[S']} \mathbb Z {\tilde \alpha}^\vee, \sheaf{G}_{m} \right) [S'] \equiv \prod_{\alpha \in \Delta[S']} \sheaf{G}_{m}[S'].$$
Similarly, we can decompose the Whittaker sheaf
$$\mathscr{W}\mathit{hit}[S'] \equiv \bigoplus_{\alpha \in \Delta[S']} \mathscr{H}\mathit{om}^\ast(\sheaf{U}_\alpha, \sheaf{G}_{a})[S'].$$
The $\sheaf{G}_m$-torsor structure on $\mathscr{H}\mathit{om}^\ast(\sheaf{U}_\alpha, \sheaf{G}_{a})$ yields (simple root by simple root) a $\sheaf{\hat T}_{\SC}$-torsor structure on $\mathscr{W}\mathit{hit}$. Although we have defined the torsor structure locally on $S_{\et}$, the action descends since the $\Gal(S'/S)$-actions are compatible throughout.
\end{definition}
The third Brylinski-Deligne invariant of $\alg{\tilde G}$ is a homomorphism $f \colon \sheaf{D}_Q \rightarrow \sheaf{D}$ of groups on $S_{\et}$.
$$\begin{tikzcd}
\sheaf{G}_m \arrow[hook]{r} \arrow{d}{=} & \sheaf{D}_Q \arrow[two heads]{r} \arrow[hook]{d}{f} & \sheaf{Y}^{\SC} \arrow[hook]{d} \\
\sheaf{G}_m \arrow[hook]{r} & \sheaf{D} \arrow[two heads]{r} & \sheaf{Y}
\end{tikzcd}$$
Here $\sheaf{D}_Q$ is a sheaf on $S_{\et}$ which depends (up to unique isomorphism) only on the Weyl- and Galois-invariant quadratic form $Q \colon \sheaf{Y}^{\SC} \rightarrow \mathbb Z$. This is reviewed in \cite[\S 1.3, 3.3]{MWIntegral}, and characterized in \cite[\S 11]{B-D} when working over a field.
Consider a Galois cover $S' \rightarrow S$ splitting $\alg{T}$ as before. For any $\eta \in \mathscr{W}\mathit{hit}[S']$, and any simple root $\alpha \in \Delta[S']$, there exists a unique element $e_{\eta, \alpha} \in \sheaf{U}_\alpha[S']$ such that $\eta(e_{\eta,\alpha}) = 1$. From these, \cite[\S 11.2]{B-D} gives elements $[e_{\eta,\alpha}] \in \sheaf{D}_Q[S']$ lying over the simple coroots $\alpha^\vee \in \sheaf{Y}^{\SC}[S']$.
\begin{remark}
When $S = \Spec(F)$ this follows directly from \cite[\S 11.2]{B-D}. When $S = \Spec(\mathcal{O})$, $S' = \Spec(\mathcal{O}')$, $\eta \in \mathscr{W}\mathit{hit}[S']$, and $F'$ is the fraction field of $\mathcal{O}'$, we find elements $e_{\eta,\alpha} \in \sheaf{U}_\alpha[F']$; as $\eta$ gives an isomorphism from $\sheaf{U}_\alpha$ to $\sheaf{G}_a$ (as sheaves of groups on $\mathcal{O}_{\et}'$), it follows that $e_{\eta,\alpha} \in \sheaf{U}_\alpha[\mathcal{O}']$ as well. The map $e \mapsto [e]$ of \cite[\S 11.1]{B-D} similarly makes sense over $\mathcal{O}'$ as well as it does over a field; since we assume $\alg{G}$ is a reductive group over $\mathcal{O}$, split over $\mathcal{O}'$, every root $SL_2$ over $F'$ arises from one over $\mathcal{O}'$. Thus the results of \cite[\S 11.2]{B-D} apply in the setting of $S = \Spec(\mathcal{O})$ as well as in the setting of a field.
\end{remark}
Using the elements $[e_{\eta,\alpha}] \in \sheaf{D}_Q[S']$ lying over the simple coroots $\alpha^\vee$, define
$$\omega(\eta)(\tilde \alpha^\vee) := f([e_{\eta,\alpha}])^{n_\alpha} \in \sheaf{D}_{Q,n}^{\SC}[S'], \text{ lying over } \tilde \alpha^\vee = n_\alpha \alpha^\vee \in \sheaf{Y}_{Q,n}^{\SC}[S'].$$
The map $\omega(\eta) \colon \tilde \alpha^\vee \mapsto f([e_{\eta, \alpha}])^{n_\alpha}$ extends uniquely to a splitting of the sequence
\begin{equation}
\begin{tikzcd}
\sheaf{G}_m[S'] \arrow[hook]{r} & \sheaf{D}_{Q,n}^{\SC}[S'] \arrow[two heads]{r} & \sheaf{Y}_{Q,n}^{\SC}[S'] \arrow[bend right=20]{l}[swap]{\omega(\eta)}.
\end{tikzcd}
\end{equation}
As $(\sheaf{Y}_{Q,n}^{\SC})_{S'}$ is a constant sheaf, this gives an element $\omega(\eta) \in \mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n}^{\SC})[S']$. Allowing $\eta$ to vary, and appyling Galois descent (cf.~\cite[Proposition 11.7]{B-D}), we find a map of sheaves on $S_{\et}$,
$$\omega \colon \mathscr{W}\mathit{hit} \rightarrow \mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n}^{\SC}).$$
To summarize, $\omega$ is the map that sends a nondegenerate character $\eta$ of $\alg{U}$ to the splitting $\omega(\eta)$, which (locally on $S_{\et}$) sends each modified simple coroot $\tilde \alpha^\vee$ to the element $f([e_{\eta, \alpha}])^{n_\alpha}$ of $\sheaf{D}_{Q,n}$.
Both $\mathscr{W}\mathit{hit}$ and $\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n}^{\SC})$ are $\sheaf{\hat T}_{\SC}$-torsors, and the following proposition describes how $\omega$ interacts with the torsor structure.
\begin{proposition}
Let $\nu \colon \sheaf{\hat T}_{\SC} \rightarrow \sheaf{\hat T}_{\SC}$ be the homomorphism corresponding to the unique homomorphism $\sheaf{Y}_{Q,n}^{\SC} \rightarrow \sheaf{Y}_{Q,n}^{\SC}$ which sends $\tilde \alpha^\vee$ to $- n_\alpha Q(\alpha^\vee) \tilde \alpha^\vee$ for all simple roots $\alpha$. Then $\omega$ lies over $\nu$, i.e., the following diagram commutes.
$$\begin{tikzcd}
\sheaf{\hat T_{\SC}} \times \mathscr{W}\mathit{hit} \arrow{r}{\ast} \arrow{d}{\nu \times \omega} & \mathscr{W}\mathit{hit} \arrow{d}{\omega} \\
\sheaf{\hat T_{\SC}} \times \mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n}^{\SC}) \arrow{r}{\ast} & \mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n}^{\SC})
\end{tikzcd}$$
\end{proposition}
{\sc Proof:}
We must trace through the action of $\sheaf{\hat T}_{\SC} = \mathscr{H}\mathit{om}(\sheaf{Y}_{Q,n}^{\SC}, \sheaf{G}_m)$; we work over a finite \'etale cover of $S$ over which $\alg{T}$ splits in what follows. Then, for any simple root $\alpha \in \Delta$, and any element $h \in \sheaf{G}_m$, there exists a unique element $h_\alpha \in \sheaf{\hat T}_{\SC}$ such that for all $\beta \in \Delta$,
$$h_\alpha(\tilde \beta^\vee) = \begin{cases} 1 & \text{ if } \beta \neq \alpha; \\ h & \text{ if } \beta = \alpha. \end{cases}$$
If $\eta \in \mathscr{W}\mathit{hit}$ then $[h_\alpha \ast \eta](e_{h_\alpha \ast \eta, \alpha}) = 1$ and so $\eta(e_{h_\alpha \ast \eta, \alpha}) = h$. Therefore,
$$e_{h_\alpha \ast \eta, \beta} = \begin{cases} e_{\eta, \beta} & \text{ if } \beta \neq \alpha; \\ h \ast e_{\eta, \alpha} & \text{ if } \beta = \alpha. \end{cases}$$
If $\beta \neq \alpha$, then $\omega(h_\alpha \ast \eta)(\tilde \beta^\vee) = f([e_{h_\alpha \ast \eta, \beta}])^{n_\beta} = f([e_{\eta, \beta}])^{n_\beta} = \omega(\eta)(\tilde \beta^\vee)$. On the other hand, in the case $\beta = \alpha$ we compute using \cite[Equation (11.2.1)]{B-D},
\begin{align*}
\omega(h_\alpha \ast \eta)(\tilde \alpha^\vee) = f([e_{h_\alpha \ast \eta, \alpha}])^{n_\alpha} &= f \left( [h \ast e_{\eta, \alpha} ] \right)^{n_\alpha} \\
&= f \left( h^{-Q(\alpha^\vee)} \cdot [e_{\eta, \alpha} ] \right)^{n_\alpha} \\
&= h^{ - n_\alpha Q(\phi_\alpha^\vee) } \cdot f \left( [e_{\eta, \alpha} ] \right)^{n_\alpha} = h^{ - n_\alpha Q(\phi_\alpha^\vee) } \cdot \omega(\eta)(\tilde \alpha^\vee).
\end{align*}
This computation demonstrates that the diagram commutes as desired.
\begin{flushright}Q.E.D \end{flushright}
Now let $\mu \colon \sheaf{\hat T}_{\SC} \twoheadrightarrow \sheaf{\hat T}_{\SC}$ be the homomorphism corresponding to the unique homomorphism $\sheaf{Y}_{Q,n}^{\SC} \hookrightarrow \sheaf{Y}_{Q,n}^{\SC}$ which sends $\tilde \alpha^\vee$ to $- m_\alpha \tilde \alpha^\vee$ for all $\alpha \in \Delta$. As $Q(\alpha^\vee) n_\alpha = m_\alpha \cdot n$, we find that $\nu = n \circ \mu$, where $n$ denotes the $n^{\th}$-power map.
Let $\mu_\ast \mathscr{W}\mathit{hit}$ denote the pushout of the $\sheaf{\hat T}_{\SC}$-torsor $\mathscr{W}\mathit{hit}$, via $\mu$. Since $\nu$ factors through $\mu$, we find that $\omega \colon \mathscr{W}\mathit{hit} \rightarrow \mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n}^{\SC})$ factors uniquely through $\bar \omega \colon \mu_\ast \mathscr{W}\mathit{hit} \rightarrow \mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n}^{\SC})$, making the following diagram commute.
$$\begin{tikzcd}
\sheaf{\hat T_{\SC}} \times \left( \mu_\ast \mathscr{W}\mathit{hit} \right) \arrow{r}{\ast} \arrow{d}{n \times \bar \omega} & \mu_\ast \mathscr{W}\mathit{hit} \arrow{d}{\bar \omega} \\
\sheaf{\hat T_{\SC}} \times \mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n}^{\SC}) \arrow{r}{\ast} & \mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n}^{\SC})
\end{tikzcd}$$
The pair $(\mu_\ast \mathscr{W}\mathit{hit}, \bar \omega)$ is therefore an object of the category $\sqrt[n]{\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n}^{\SC})}[S]$; it {\em neutralizes} the gerbe $\sqrt[n]{\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n}^{\SC})}$. In particular,
$$\left[ \sqrt[n]{\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n}^{\SC})} \right] = 0.$$
Write $\boldsymbol{w} = (\mu_\ast \mathscr{W}\mathit{hit}, \bar \omega)$ for this object. Pushing out via $\epsilon$, we view $\boldsymbol{w}$ as an $S$-object of $\gerb{E}_\epsilon^{\SC}(\alg{\tilde T})$. This completes the construction of the gerbe $\gerb{E}_\epsilon(\alg{\tilde G}) = \gerb{p}^{-1}(\gerb{w})$ associated to the cover $\alg{\tilde G}$ and character $\epsilon$.
\begin{example}
Suppose that $\sheaf{Y}_{Q,n} / \sheaf{Y}_{Q,n}^{\SC}$ is torsion-free and a constant sheaf (equivalently, the center of $\dgp{\tilde G^\vee}$ is connected and constant as a sheaf on $S_{\et}$). Then the following short exact sequence splits:
$$\sheaf{Y}_{Q,n}^{\SC} \hookrightarrow \sheaf{Y}_{Q,n} \twoheadrightarrow \sheaf{Y}_{Q,n} / \sheaf{Y}_{Q,n}^{\SC}.$$
Given such a splitting, write $\sheaf{Y}_{Q,n}^{\cent} \subset \sheaf{Y}_{Q,n}$ for the image of $\sheaf{Y}_{Q,n} / \sheaf{Y}_{Q,n}^{\SC}$ via the splitting. The identification $\sheaf{Y}_{Q,n} = \sheaf{Y}_{Q,n}^{\SC} \oplus \sheaf{Y}_{Q,n}^{\cent}$ corresponds to an isomorphism $\sheaf{\hat T} \xrightarrow{\sim} \sheaf{\hat T}_{\SC} \times \sheaf{\hat Z}$. Let $\sheaf{D}_{Q,n}^{\cent}$ be the pullback of $\sheaf{D}_{Q,n}$ to $\sheaf{Y}_{Q,n}^{\cent}$. From Example \ref{SplitTorusGerbeNeutral}, the short exact sequence $\sheaf{G}_m \hookrightarrow \sheaf{D}_{Q,n}^{\cent} \twoheadrightarrow \sheaf{Y}_{Q,n}^{\cent}$ splits, providing an object of $\sqrt[n]{\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n}^{{\cent}})}$.
Chasing diagrams gives a map of object sets,
$$\sqrt[n]{\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n}^{\SC})}[S] \times \sqrt[n]{\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n}^{{\cent}})}[S] \rightarrow \sqrt[n]{\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n})}[S].$$
A splitting of $\sheaf{D}_{Q,n}^{\cent}$ gives an object of $\sqrt[n]{\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n}^{{\cent}})}[S]$ and $\boldsymbol{w}$ provides an object of $\sqrt[n]{\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n}^{\SC})}[S]$. Hence the gerbe $\gerb{E}_\epsilon(\alg{\tilde G})$ is neutral when $\sheaf{Y}_{Q,n} / \sheaf{Y}_{Q,n}^{\SC}$ is torsion-free and a constant sheaf.
\end{example}
\subsection{Well-aligned functoriality}
\label{WAFGerbe}
Consider a well-aligned homomorphism $\tilde \iota \colon \alg{\tilde G}_1 \rightarrow \alg{\tilde G}_2$ of covers, each endowed with Borel subgroup and maximally split maximal torus, i.e., a morphism in the category $\Cat{WAC}_S$. Fix $\epsilon$ as before. We have constructed gerbes $\gerb{E}_\epsilon(\alg{\tilde G}_1)$ and $\gerb{E}_\epsilon(\alg{\tilde G}_2)$ associated to $\alg{\tilde G}_1$ and $\alg{\tilde G}_2$, banded by $\sheaf{\tilde Z}_1^ { \raisemath{-3pt}{\vee} } $ and $\sheaf{\tilde Z}_2^ { \raisemath{-3pt}{\vee} } $, respectively. We have constructed a homomorphism of dual groups $\iota^\vee \colon \dgp{\tilde G}_2^ { \raisemath{-3pt}{\vee} } \rightarrow \dgp{\tilde G}_1^ { \raisemath{-3pt}{\vee} } $ in Section \ref{WAFDualGroup}, which (after taking $\mathbb C$-points) restricts to $\iota^\vee \colon \sheaf{\tilde Z}_2^ { \raisemath{-3pt}{\vee} } \rightarrow \sheaf{\tilde Z}_1^ { \raisemath{-3pt}{\vee} } $. Here we construct a functor of gerbes $\gerb{i} \colon \gerb{E}_\epsilon(\alg{\tilde G}_2) \rightarrow \gerb{E}_\epsilon(\alg{\tilde G}_1)$, lying over $\iota^\vee \colon \sheaf{\tilde Z}_{2}^ { \raisemath{-3pt}{\vee} } \rightarrow \sheaf{\tilde Z}_{1}^ { \raisemath{-3pt}{\vee} } $.
Well-alignedness give a commutative diagram in which the first row is the pullback of the second.
$$\begin{tikzcd}
\alg{K}_2 \arrow[hook]{r} \arrow{d}{=} & \alg{T}_1' \arrow[two heads]{r} \arrow{d}{\iota'} & \alg{T}_1 \arrow{d}{\iota} \\
\alg{K}_2 \arrow[hook]{r} & \alg{T}_2' \arrow[two heads]{r} & \alg{T}_2
\end{tikzcd}$$
This gives a commutative diagram for the second Brylinski-Deligne invariants. After pulling back to $\sheaf{Y}_{1,Q_1,n}$ and $\sheaf{Y}_{2,Q_2,n}$, we get a commutative diagram of sheaves of abelian groups on $S_{\et}$, in which the top row is the pullback of the bottom.
$$\begin{tikzcd}
\sheaf{G}_m \arrow[hook]{r} \arrow{d}{=} & \sheaf{D}_{1,Q_1,n} \arrow[two heads]{r} \arrow{d}{\iota'} & \sheaf{Y}_{1,Q_1,n} \arrow{d}{\iota} \\
\sheaf{G}_m \arrow[hook]{r} & \sheaf{D}_{2,Q_2,n} \arrow[two heads]{r}& \sheaf{Y}_{2,Q_2,n}
\end{tikzcd}$$
(Assumption \ref{OddnEvenQ} is in effect, so $Q_1$ and $Q_2$ are even-valued if $n$ is odd.)
We have homomorphisms of sheaves of abelian groups,
$$\iota \colon \sheaf{Y}_{1,Q_1,n} \rightarrow \sheaf{Y}_{2,Q_2,n}, \quad \sheaf{Y}_{1,Q_1,n}^{\SC} \rightarrow \sheaf{Y}_{2,Q_2,n}^{\SC}, \quad \frac{\sheaf{Y}_{1,Q_1,n}}{\sheaf{Y}_{1,Q_1,n}^{\SC}} \rightarrow \frac{\sheaf{Y}_{2,Q_2,n}}{\sheaf{Y}_{2,Q_2,n}^{\SC}}.$$
Applying $\mathscr{H}\mathit{om}(\bullet, \sheaf{G}_m)$ yields homomorphisms of sheaves of abelian groups,
$$\hat \iota \colon \sheaf{\hat T}_2 \rightarrow \sheaf{\hat T}_1, \quad \sheaf{\hat T}_{\SC,2} \rightarrow \sheaf{\hat T}_{\SC,1}, \quad \sheaf{\hat Z}_2 \rightarrow \sheaf{\hat Z}_1.$$
A splitting of $\sheaf{D}_{2,Q_2,n}$ pulls back to a splitting of $\sheaf{D}_{1,Q_1,n}$ (call this pullback map $\iota^\ast$), giving a commutative diagram of sheaves on $S_{\et}$.
$$\begin{tikzcd}
\sheaf{\hat T}_2 \times \mathscr{S}\mathit{pl}(\sheaf{D}_{2,Q_2,n}) \arrow{r}{\ast} \arrow{d}{\hat \iota \times \iota^\ast} & \mathscr{S}\mathit{pl}(\sheaf{D}_{2,Q_2,n}) \arrow{d}{\iota^\ast} \\
\sheaf{\hat T}_1 \times \mathscr{S}\mathit{pl}(\sheaf{D}_{1,Q_1,n}) \arrow{r}{\ast} & \mathscr{S}\mathit{pl}(\sheaf{D}_{1,Q_1,n})
\end{tikzcd}$$
This defines a functor of gerbes
$$\sqrt[n]{\iota^\ast} \colon \sqrt[n]{\mathscr{S}\mathit{pl}(\sheaf{D}_{2,Q_2,n})} \rightarrow \sqrt[n]{\mathscr{S}\mathit{pl}(\sheaf{D}_{1,Q_1,n})},$$
lying over $\hat \iota \colon \sheaf{\hat T}_{2,[n]} \rightarrow \sheaf{\hat T}_{1,[n]}$. Pushing out via $\epsilon$ yields a functor of gerbes,
$$\gerb{i} \colon \gerb{E}_\epsilon(\alg{\tilde T}_2) \rightarrow \gerb{E}_\epsilon(\alg{\tilde T}_1).$$
The same process applies to $\iota \colon \sheaf{Y}_{1,Q_1,n}^{\SC} \rightarrow \sheaf{Y}_{2,Q_2,n}^{\SC}$, giving a functor of gerbes, $\gerb{i}^{\SC} \colon \gerb{E}_\epsilon^{\SC}(\alg{\tilde T}_2) \rightarrow \gerb{E}_\epsilon^{\SC}(\alg{\tilde T}_1)$. By pulling back in stages, we find a square of gerbes and functors, and a natural isomorphism $\Fun{S} \colon \gerb{p}_1 \circ \gerb{i} \xRightarrow{\sim} \gerb{i}^{\SC} \circ \gerb{p}_2$ making the diagram 2-commute.
$$\begin{tikzcd}
\gerb{E}_\epsilon(\alg{\tilde T}_2) \arrow{r}{\gerb{i}} \arrow{d}{\gerb{p}_2} & \gerb{E}_\epsilon(\alg{\tilde T}_1) \arrow{d}{\gerb{p}_1} \\
\gerb{E}_\epsilon^{\SC}(\alg{\tilde T}_2) \arrow{r}{\gerb{i}^{\SC}} & \gerb{E}_\epsilon^{\SC}(\alg{\tilde T}_1)
\end{tikzcd}$$
If $\alg{U}_1$ and $\alg{U}_2$ are the unipotent radicals of $\alg{B}_1$ and $\alg{B}_2$, respectively, then pulling back gives a map $\iota^\ast \colon \mathscr{W}\mathit{hit}_2 \rightarrow \mathscr{W}\mathit{hit}_1$. Condition (1) of well-alignedness states that $\Ker(\iota)$ is contained in the center of $\alg{G}_1$, and so simple root subgroups in $\alg{U}_1$ map isomorphically to simple root subgroups in $\alg{U}_2$. Compatibility of quadratic forms $Q_1$ and $Q_2$ implies compatibility in the constants $n_\alpha$ and $m_\alpha$ for simple roots, and so we find a commutative diagram
$$\begin{tikzcd}
(\mu_2)_\ast \mathscr{W}\mathit{hit}_2 \arrow{r}{\bar \omega_2} \arrow{d}{\iota^\ast} & \mathscr{S}\mathit{pl}(\sheaf{D}_{2,Q_2,n}^{\SC}) \arrow{d}{\iota^\ast} \\
(\mu_1)_\ast \mathscr{W}\mathit{hit}_1 \arrow{r}{\bar \omega_1} & \mathscr{S}\mathit{pl}(\sheaf{D}_{1,Q_1,n}^{\SC})
\end{tikzcd}$$
Write $\boldsymbol{w}_1$ for the object $((\mu_1)_\ast \mathscr{W}\mathit{hit}_1, \bar \omega_1)$ of $\gerb{E}_\epsilon^{\SC}(\alg{\tilde T}_1)$, and similarly $\boldsymbol{w}_2$ for the object $((\mu_2)_\ast \mathscr{W}\mathit{hit}_2, \bar \omega_2)$ of $\gerb{E}_\epsilon^{\SC}(\alg{\tilde T}_2)$. The commutative diagram above gives an isomorphism $f \colon \gerb{i}^{\SC}( \boldsymbol{w}_2) \xrightarrow{\sim} \boldsymbol{w}_1$ in $\gerb{E}_\epsilon^{\SC}(\alg{\tilde T}_1)$.
If $(\gerbob{e},j)$ is an object of $\gerb{E}_\epsilon(\alg{\tilde G}_2) = \gerb{p}_2^{-1}(\boldsymbol{w}_2)$, i.e., $\gerbob{e}$ is an object of $\gerb{E}_\epsilon(\alg{\tilde T}_2)$ and $j \colon \gerb{p}_2(\gerbob{e}) \rightarrow \boldsymbol{w}_2$ is an isomorphism, then we find a sequence of isomorphisms,
$$\gerb{i}(j) \colon \gerb{p}_1(\gerb{i}(\gerbob{e})) \xrightarrow{\Fun{S}} \gerb{i}^{\SC}(\gerb{p}_2(\gerbob{e})) \xrightarrow{j} \gerb{i}^{\SC}(\boldsymbol{w}_2) \xrightarrow{f} \boldsymbol{w}_1.$$
In this way $(\gerb{i}(\gerbob{e}),\gerb{i}(j))$ becomes an object of $\gerb{E}_\epsilon(\alg{\tilde G}_1) = \gerb{p}_1^{-1}(\boldsymbol{w}_1)$. This extends to a functor of gerbes, $\gerb{i} \colon \gerb{E}_\epsilon(\alg{\tilde G}_2) \rightarrow \gerb{E}_\epsilon(\alg{\tilde G}_1)$, lying over $\iota^\vee \colon \sheaf{\tilde Z}_2^ { \raisemath{-3pt}{\vee} } \rightarrow \sheaf{\tilde Z}_1^ { \raisemath{-3pt}{\vee} } $.
Given a pair of well-aligned homomorphisms,
$$\alg{\tilde G}_1 \xrightarrow{\tilde \iota_1} \alg{\tilde G}_2 \xrightarrow{\tilde \iota_2} \alg{\tilde G}_2,$$
with $\tilde \iota_3 = \tilde \iota_2 \circ \tilde \iota_1$, we find three functors of gerbes.
$$\begin{tikzcd}
\gerb{E}_\epsilon(\alg{\tilde G}_3) \arrow{r}[swap]{\gerb{i}_2} \arrow[bend left=20]{rr}{\gerb{i}_3} & \gerb{E}_\epsilon(\alg{\tilde G}_2) \arrow{r}[swap]{\gerb{i}_1} & \gerb{E}_\epsilon(\alg{\tilde G}_1).
\end{tikzcd}$$
lying over three homomorphisms of sheaves,
$$\begin{tikzcd}
\sheaf{\tilde Z}_3^ { \raisemath{-3pt}{\vee} } \arrow{r}[swap]{\iota_2^\vee} \arrow[bend left=20]{rr}{\iota_3^\vee} & \sheaf{\tilde Z}_2^ { \raisemath{-3pt}{\vee} } \arrow{r}[swap]{\iota_1^\vee} & \sheaf{\tilde Z}_1^ { \raisemath{-3pt}{\vee} } .
\end{tikzcd}$$
The functoriality of pullbacks, pushouts, taking $n^{\th}$ roots of torsors, etc., yields a \textbf{natural isomorphism} of functors:
\begin{equation}
\label{CompositionWellAligned}
N(\iota_1, \iota_2) \colon \gerb{i}_3 \xRightarrow{\sim} \gerb{i}_1 \circ \gerb{i}_2.
\end{equation}
We can summarize these results using the language of ``weak functors'' \cite[Tag 003G]{Stacks}. We have given all the data for such a weak functor, and a reader who wishes to check commutativity of a few diagrams may verify the following.
\begin{proposition}
The construction of the associated gerbe defines a weak functor from $\Cat{WAC}_S$ (the category of triples $(\alg{\tilde G}, \alg{B}, \alg{T})$ and well-aligned homomorphisms), to the 2-category of gerbes on $S_{\et}$, functors of gerbes, and natural isomorphisms thereof.
\end{proposition}
\subsection{Well-definedness}
The construction of the associated gerbe $\gerb{E}_\epsilon(\alg{\tilde G})$ depended on the choice of Borel subgroup $\alg{B}$ and maximally split maximal torus $\alg{T}$. Now we demonstrate that $\gerb{E}_\epsilon(\alg{\tilde G})$ is well-defined independently of these choices, in a suitable 2-categorical sense.
Consider another choice $\alg{B}_0 \supset \alg{T}_0$. Our constructions, with these two choices of tori and Borel subgroups, yield two dual groups $\dgp{\tilde G}_0^ { \raisemath{-3pt}{\vee} } $ and $\dgp{\tilde G}^ { \raisemath{-3pt}{\vee} } $ and two gerbes $\gerb{E}_\epsilon(\alg{\tilde G})$ and $\gerb{E}_{0,\epsilon}(\alg{\tilde G})$.
Let $\sheaf{Y}_0$ and $\sheaf{Y}$ be the cocharacter lattices of $\alg{T}_0$ and $\alg{T}$, respectively, and $\Phi_0^\vee, \Phi^\vee$ the coroots therein. The Borel subgroups provide systems of simple coroots $\Delta_0^\vee$ and $\Delta^\vee$, respectively (sheaves of sets on $S_{\et}$). Similarly we have character lattices $\sheaf{X}_0, \sheaf{X}$, roots $\Phi_0, \Phi$, and simple roots $\Delta_0, \Delta$. The cover $\alg{\tilde G}$ yields quadratic forms $Q \colon \sheaf{Y} \rightarrow \mathbb Z$ and $Q_0 \colon \sheaf{Y}_0 \rightarrow \mathbb Z$. The second Brylinski-Deligne invariant gives extensions $\sheaf{G}_m \hookrightarrow \sheaf{D} \twoheadrightarrow \sheaf{Y}$ and $\sheaf{G}_m \hookrightarrow \sheaf{D}_0 \twoheadrightarrow \sheaf{Y}_0$.
By Proposition \ref{BTConj}, there exists $g \in \alg{G}(S)$ such that $\Int(g) \alg{T}_0 = \alg{T}$ and $\Int(g) \alg{B}_0 = \alg{B}$. This automorphism $\Int(g)$ lifts to an automorphism of $\alg{G}'$. This defines a well-aligned isomorphism of covers.
$$\begin{tikzcd}
\alg{K}_{2} \arrow[hook]{r} \arrow{d}{=} & \alg{G}' \arrow[two heads]{r} \arrow{d}{\Int(g)'} & \alg{G} \arrow{d}{\Int(g)} & & \alg{B}_{0} \arrow{d} & \alg{T}_{0} \arrow{d} \\
\alg{K}_{2} \arrow[hook]{r} & \alg{G}' \arrow[two heads]{r} & \alg{G} & & \alg{B} & \alg{T}
\end{tikzcd}$$
This well-aligned isomorphism of covers yields an equivalence of gerbes,
$$\gerb{Int}(g) \colon \gerb{E}_\epsilon(\alg{\tilde G}) \rightarrow \gerb{E}_{0,\epsilon}(\alg{\tilde G}),$$
lying over $\Int(g)^\vee \colon \sheaf{\tilde Z^\vee} \rightarrow \sheaf{\tilde Z}_0^ { \raisemath{-3pt}{\vee} } $.
Suppose that $g' \in \alg{G}(S)$ also satisfies $\Int(g') \alg{T}_0 = \alg{T}$ and $\Int(g') \alg{B}_0 = \alg{B}$. As in the proof of Theorem \ref{WellDefinedDualGroup}, $g' = t g$ for a unique $t \in \alg{T}(S)$ and $\Int(g') = \Int(t) \Int(g)$.
This gives a natural isomorphism of functors,
$$N(g', g) \colon \gerb{Int}(g') \xRightarrow{\sim} \gerb{Int}(g) \circ \gerb{Int}(t)$$
Our upcoming Proposition \ref{IntTFunctors} will provide a natural isomorphism $\gerb{Int}(t) \xRightarrow{\sim} \gerb{Id}$. Assuming this for the moment, we find a natural isomorphism $\gerb{Int}(g') \xRightarrow{\sim} \gerb{Int}(g)$. This demonstrates that the gerbe $\gerb{E}_\epsilon(\alg{\tilde G})$ is well-defined in the following 2-categorical sense:
\begin{enumerate}
\item
for each pair $i = (\alg{B}, \alg{T})$ consisting of a Borel subgroup of $\alg{G}$ and a maximally split maximal torus therein, we have constructed a gerbe $\gerb{E}_\epsilon^i(\alg{\tilde G})$;
\item
for each pair $i = (\alg{B}, \alg{T})$, $j = (\alg{B}_0, \alg{T}_0)$, we have constructed a family $P(i,j)$ of gerbe equivalences $\gerb{Int}(g) \colon \gerb{E}_\epsilon^i(\alg{\tilde G}) \rightarrow \gerb{E}_\epsilon^j(\alg{\tilde G})$, indexed by those $g$ which conjugate $i$ to $j$;
\item
for any two elements $g, g'$ which conjugate $i$ to $j$, there is a distinguished natural isomorphism of gerbe equivalences from $\gerb{Int}(g')$ to $\gerb{Int}(g)$.
\end{enumerate}
Once we define the natural isomorphism $\gerb{Int}(t) \xRightarrow{\sim} \gerb{Id}$, we will have defined $\gerb{E}_\epsilon(\alg{\tilde G})$ uniquely up to equivalence, the equivalences being defined uniquely up to unique natural isomorphism (we learned this notion of well-definedness from reading various works of James Milne). Defining the natural isomorphism requires some computation and is the subject of the section below.
\subsubsection{The isomorphism $\gerb{Int}(t) \xRightarrow{\sim} \gerb{Id}$}
For what follows, define $\delta_Q \colon \sheaf{Y} \rightarrow n^{-1} \sheaf{X}$ to be the unique homomorphism satisfying
\begin{equation}
\label{DefinitionOfDelta}
\langle \delta_Q(y_1), y_2 \rangle = \beta_Q(y_1, y_2) \text{ for all } y_1, y_2 \in \sheaf{Y}.
\end{equation}
The constants $m_\phi$ and $n_\phi$ arise in the following useful result.
\begin{lemma}
\label{DeltaM}
For all $\phi \in \Phi$, we have $\delta_Q(n_\phi \phi^\vee) = \delta_Q(\tilde \phi^\vee) = m_\phi \phi$.
\end{lemma}
{\sc Proof:}
For all $\phi \in \Phi$, and all $y \in \sheaf{Y}$, we have
\begin{equation}
\label{DeltaBeta}
\langle \delta_Q(\tilde \phi^\vee), y \rangle = \beta_Q(n_\phi \phi^\vee, y).
\end{equation}
Weyl-invariance of the quadratic form (applying the root reflection for $\phi$) implies
$$\beta_Q(\phi^\vee, y) = \beta_Q(-\phi^\vee, y - \langle \phi, y \rangle \phi^\vee) = - \beta_Q(\phi^\vee, y) + \beta_Q(\phi^\vee, \phi^\vee) \langle \phi, y \rangle.$$
Adding $\beta_Q(\phi^\vee, y)$ to both sides and dividing by two, we find that
$$\beta_Q(\phi^\vee, y) = n^{-1} Q(\phi^\vee) \langle \phi, y \rangle.$$
Substituting into \eqref{DeltaBeta} yields
$$\langle \delta_Q(\tilde \phi^\vee), y \rangle = n_\phi \beta_Q( \phi^\vee, y)= \langle n_\phi n^{-1} Q(\phi^\vee) \cdot \phi, y \rangle.$$
Since this holds for all $y \in \sheaf{Y}$, we have
$$\delta_Q(\tilde \phi^\vee) = n_\phi n^{-1} Q(\phi^\vee) \cdot \phi= m_\phi \cdot \phi.$$
\begin{flushright}Q.E.D \end{flushright}
Consider the homomorphisms of sheaves of abelian groups on $S_{\et}$,
$$\sheaf{T} \xrightarrow{\delta_Q} \sheaf{\hat T} \xtwoheadrightarrow{p} \sheaf{\hat T}_{\SC},$$
obtained by applying $\mathscr{H}\mathit{om}(\bullet, \sheaf{G}_m)$ to the homomorphisms
$$\sheaf{Y}_{Q,n}^{\SC} \hookrightarrow \sheaf{Y}_{Q,n} \xrightarrow{\delta_Q} \sheaf{X}.$$
An object of $\gerb{E}_\epsilon(\alg{\tilde G})$ is a triple $(\sheaf{H}, h, j)$ where
\begin{itemize}
\item
$\sheaf{H} \xrightarrow{h} \mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n})$ is an $n^{\th}$ root of the $\sheaf{\hat T}$-torsor $\sheaf{D}_{Q,n}$.
\item
$j \colon p_\ast \sheaf{H} \rightarrow \mu_\ast \mathscr{W}\mathit{hit}$ is an isomorphism in the gerbe $\gerb{E}_\epsilon^{\SC}(\alg{\tilde T}) = \epsilon_\ast \sqrt[n]{\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n}^{\SC})}$. Thus $j = j_0 \wedge \tau^\vee$ where $j_0 \colon p_\ast \sheaf{H} \rightarrow \mu_\ast \mathscr{W}\mathit{hit}$ is an isomorphism in the gerbe $\sqrt[n]{\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n}^{\SC})}$ and $\tau^\vee \in \sheaf{\tilde T}_{\SC}^ { \raisemath{-3pt}{\vee} } $.
\end{itemize}
Given $\sheaf{H} \xrightarrow{h} \mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n})$ and $\hat t \in \sheaf{\hat T}$, write $\hat t \circ h \colon \sheaf{H} \rightarrow \mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n})$ for the map obtained by composing $h$ with the automorphism $\hat t$ of the $\sheaf{\hat T}$-torsor $\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n})$.
Similarly, given $j = j_0 \wedge \tau^\vee$, with $p_\ast \sheaf{H} \xrightarrow{j_0} \mu_\ast \mathscr{W}\mathit{hit}$, $\tau^\vee \in \sheaf{\tilde T}_{\SC}^{ { \raisemath{-3pt}{\vee} } }$, and given $\hat t_{\SC} \in \sheaf{\hat T}_{\SC}$, write $\hat t_{\SC} \circ j$ for $(\hat t_{\SC} \circ j_0) \wedge \tau^\vee$. Here $\hat t_{\SC} \circ j_0 \colon p_\ast \sheaf{H} \rightarrow \mu_\ast \mathscr{W}\mathit{hit}$ is the map obtained by composing $j$ with the automorphism $\hat t_{\SC}$ of the $\sheaf{\hat T}_{\SC}$-torsor $\mu_\ast \mathscr{W}\mathit{hit}$.
The following result describes the functor $\gerb{Int}(t)$ explicitly.
\begin{lemma}
For all $t \in \alg{T}(S)$, the equivalence of gerbes $\gerb{Int}(t) \colon \gerb{E}_\epsilon(\alg{\tilde G}) \rightarrow \gerb{E}_\epsilon(\alg{\tilde G})$ sends an object $(\sheaf{H}, h,j)$ to the object $(\sheaf{H}, \delta_Q(t)^{-n} \circ h, p(\delta_Q(t))^{-1} \circ j)$.
\end{lemma}
{\sc Proof:}
We work locally on $S_{\et}$ throughout the proof. The action $\Int(t)$ of $t$ on the extension $\sheaf{G}_m \hookrightarrow \sheaf{D}_{Q,n} \twoheadrightarrow \sheaf{Y}_{Q,n}$ is given by \cite[Equation 11.11.1]{B-D}, which states (in different notation) that
$$\Int(t) d = d \cdot \delta_Q(y)(t)^{-n} = d \cdot y \left( \delta_Q(t)^{-n} \right),$$
for all $d \in \sheaf{D}_{Q,n}$ mapping to $y \in \sheaf{Y}_{Q,n}$. We find that $\Int(t)$ is the automorphism of the extension $\sheaf{D}_{Q,n}$ determined by the element $\delta_Q(t)^{-n} \in \sheaf{\hat T} = \mathscr{H}\mathit{om}(\sheaf{Y}_{Q,n}, \sheaf{G}_m)$. Hence the functor $\gerb{Int}(t) \colon \gerb{E}_\epsilon(\alg{\tilde T}) \rightarrow \gerb{E}_\epsilon(\alg{\tilde T})$ sends $(\sheaf{H}, h)$ to $(\sheaf{H}, \delta_Q(t)^{-n} \circ h)$. It remains to see how $\gerb{Int}(t)$ affects the third term in a triple $(\sheaf{H}, h, j)$.
Conjugation by $t$ gives a map of torsors $\mathscr{W}\mathit{hit} \rightarrow \mathscr{W}\mathit{hit}$. Let $j_0'$ be the map which makes the following triangle commute:
$$\begin{tikzcd}
p_\ast \sheaf{H} \arrow{r}{j_0} \arrow{dr}[swap]{j_0'} & \mu_\ast \mathscr{W}\mathit{hit} \arrow{d}{\Int(t)} \\
\phantom{a} & \mu_\ast \mathscr{W}\mathit{hit}
\end{tikzcd}$$
We have $\gerb{Int}(t) (\sheaf{H}, h, j_0 \wedge \tau^\vee) = (\sheaf{H}, \delta_Q(t)^{-n} \circ h, j_0' \wedge \tau^\vee)$ and must describe $j_0'$.
Take an element $\eta= (\eta_\alpha : \alpha \in \Delta) \in \mathscr{W}\mathit{hit}$ (decomposed with respect to the basis of simple roots $\Delta$). Conjugation by $t$ yields a new element $\Int(t) \eta$ satisfying
$$\Int(t) \eta(u) = \eta(t^{-1} u t), \quad \text{ for all } u \in \sheaf{U}.$$
Decomposing along the simple root subgroups,
$$\Int(t) \eta = (\alpha(t) \ast \eta_\alpha : \alpha \in \Delta) \in \mathscr{W}\mathit{hit},$$
where $[a \ast \eta_\alpha](e) = \eta_\alpha(a^{-1} e)$ for any $a \in \sheaf{G}_m$.
Comparing to the action of $\sheaf{\hat T}_{\SC}$ on $\mathscr{W}\mathit{hit}$, we find that
$$\Int(t) \eta= \theta(t) \ast \eta,$$
where $\theta \colon \sheaf{T} \rightarrow \sheaf{\hat T}_{\SC}$ denotes the homomorphism dual to the homomorphism of character lattices sending $\tilde \alpha^\vee \in \sheaf{Y}_{Q,n}^{\SC}$ to $\alpha \in \sheaf{X}$.
In the quotient $\sheaf{\hat T}_{\SC}$-torsor $\mu_\ast \mathscr{W}\mathit{hit}$, the action is given by $\Int(t) \bar \eta = \mu(\theta(t)) \cdot \bar \eta$, for all $\bar \eta \in \mu_\ast \mathscr{W}\mathit{hit}$. More explicitly, $\mu \circ \theta \colon \sheaf{\hat T}_{\SC} \rightarrow \sheaf{T}$ is the homomorphism dual to the map of character lattices sending $\tilde \alpha^\vee$ to $-m_{\alpha} \alpha$. By Lemma \ref{DeltaM}, we have $\mu(\theta(t)) = p(\delta_Q(t))^{-1}$ (recall that $p$ corresponds to the inclusion $\sheaf{Y}_{Q,n}^{\SC} \hookrightarrow \sheaf{Y}_{Q,n}$). It follows that $j_0' = \Int(t) \circ j_0 = p(\delta_Q(t))^{-1} \circ j_0$. This yields the result:
$$\gerb{Int}(t) (\sheaf{H}, h, j) = (\sheaf{H}, \delta_Q(t)^{-n} \circ h, p(\delta_Q(t))^{-1} \circ j).$$
\begin{flushright}Q.E.D \end{flushright}
\begin{proposition}
\label{IntTFunctors}
Let $(\sheaf{H}, h, j)$ be an object of the category $\gerb{E}_\epsilon(\alg{\tilde G})$. Then, for all $t \in \alg{T}(S)$, the morphism $\delta_Q(t)^{-1} \colon \sheaf{H} \rightarrow \sheaf{H}$ defines an isomorphism from $\gerb{Int}(t) (\sheaf{H}, h, j)$ to $(\sheaf{H}, h, j)$. As objects vary, this defines a natural isomorphism $\Fun{A}(t) \colon \gerb{Int}(t) \Rightarrow \gerb{Id}$ of functors from $\gerb{E}_\epsilon(\alg{\tilde G})$ to itself. For a pair of elements $t_1, t_2 \in \alg{T}(S)$, we have a commutative diagram of functors and natural isomorphisms.
$$\begin{tikzcd}
\gerb{Int}(t_1 t_2) \arrow[Rightarrow]{r}{=} \arrow[Rightarrow]{d}{\Fun{A}(t_1 t_2)} & \gerb{Int}(t_1) \circ \gerb{Int}(t_2) \arrow[Rightarrow]{d}{\Fun{A}(t_1) \circ \Fun{A}(t_2)} \\
\gerb{Id} \arrow[Rightarrow]{r}{=} & \gerb{Id}
\end{tikzcd}$$
\end{proposition}
{\sc Proof:}
As $\sheaf{H} \xrightarrow{h} \mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n})$ is an $n^{\th}$ root of the torsor $\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n})$, we have $\delta_Q(t)^{-n} \circ h = h \circ \delta_Q(t)^{-1}$. As $j = j_0 \wedge \tau$, with $j_0$ an isomorphism in the gerbe $\sqrt[n]{\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n}^{\SC})}$, we have $p(\delta_Q(t)^{-1}) \circ j = j \circ p(\delta_Q(t)^{-1})$.
Hence the isomorphism $\delta_Q(t)^{-1} \colon \sheaf{H} \rightarrow \sheaf{H}$ defines an isomorphism
\begin{align*}
\gerb{Int}(t) (\sheaf{H}, h, j) &= (\sheaf{H}, \delta_Q(t)^{-n} \circ h, p(\delta_Q(t)^{-1}) \circ j) \\
&= (\sheaf{H}, h \circ \delta_Q(t)^{-1}, j \circ p(\delta_Q(t)^{-1}) ) \\
& \xrightarrow{\delta_Q(t)^{-1}} (\sheaf{H}, h, j).
\end{align*}
As their definition depends only on $t$, these isomorphisms $\delta_Q(t)^{-1}$ form a natural isomorphism $\Fun{A}(t) \colon \gerb{Int}(t) \Rightarrow \gerb{Id}$. As $\delta_Q(t_1 t_2)^{-1} = \delta_Q(t_1)^{-1} \delta_Q(t_2)^{-1}$, we find an {\em equality} of functors $\gerb{Int}(t_1 t_2) = \gerb{Int}(t_1) \circ \gerb{Int}(t_2)$, and the commutative diagram of the proposition.
\begin{flushright}Q.E.D \end{flushright}
\subsection{Change of base scheme}
\label{BaseChangeGerbe}
Let $\alg{\tilde G}$ be a degree $n$ cover of a quasisplit group $\alg{G} \supset \alg{B} \supset \alg{T}$ over $S$, as before. Let $\gamma \colon S_0 \rightarrow S$ be a morphism of schemes as in Section \ref{BaseChangeDualGroup}. Then $\gamma$ gives rise to a pullback functor $\gamma^\ast$ from sheaves on $S_{\et}$ to sheaves on $S_{0,\et}$, and from gerbes on $S_{\et}$ to gerbes on $S_{0, \et}$.
Assuming that $\sch{\mu}_n(S)$ and $\sch{\mu}_n(S_0)$ are cyclic groups of order $n$, we may identify these groups via $\gamma \colon \sch{\mu}_n(S) \rightarrow \sch{\mu}_n(S_0)$, and a character $\epsilon \colon \mu_n \hookrightarrow \mathbb C^\times$ corresponds to a character $\epsilon_0 \colon \sch{\mu}_n(S_0) \hookrightarrow \mathbb C^\times$.
Pullback via $\gamma$ defines a degree $n$ cover $\alg{\tilde G}_0$ of a quasisplit group $\alg{G}_0 \supset \alg{B}_0 \supset \alg{T}_0$ over $S_0$. We have constructed gerbes $\gerb{E}_\epsilon(\alg{\tilde G})$ and $\gerb{E}_{\epsilon_0}(\alg{\tilde G}_0)$ associated to this data, banded by $\sheaf{\tilde Z}^ { \raisemath{-3pt}{\vee} } $ and $\sheaf{\tilde Z}_0^ { \raisemath{-3pt}{\vee} } $, respectively. We also consider the pullback gerbe $\gamma^\ast \gerb{E}_\epsilon(\alg{\tilde G})$, banded by $\gamma^\ast \sheaf{\tilde Z}^ { \raisemath{-3pt}{\vee} } $, and recall from Section \ref{BaseChangeDualGroup} that there is a natural isomorphism $N^\vee \colon \gamma^\ast \sheaf{\tilde Z}^ { \raisemath{-3pt}{\vee} } \rightarrow \sheaf{\tilde Z}_0^ { \raisemath{-3pt}{\vee} } $.
Similarly, we find isomorphisms of sheaves on $S_{0,\et}$,
\begin{enumerate}
\item $\hat N \colon \gamma^\ast \sheaf{\hat T} \xrightarrow{\sim} \sheaf{\hat T}_0$ (from $N \colon \gamma^\ast \sheaf{Y}_{Q,n} \xrightarrow{\sim} \sheaf{Y}_{0,Q_0,n}$);
\item $N^\vee \colon \gamma^\ast \sheaf{\tilde T}_{\SC}^ { \raisemath{-3pt}{\vee} } \xrightarrow{\sim} \sheaf{\tilde T}_{0,\SC}^ { \raisemath{-3pt}{\vee} } $;
\item $N \colon \gamma^\ast \sheaf{D}_{Q,n} \xrightarrow{\sim} \sheaf{D}_{0,Q_0,n}$ (the construction of the second Brylinski-Deligne invariant is compatible with pullbacks);
\item $N \colon \gamma^\ast \mathscr{W}\mathit{hit} \xrightarrow{\sim} \mathscr{W}\mathit{hit}_0$ (since $\alg{B}_0 = \gamma^\ast \alg{B}$).
\end{enumerate}
From these observed isomorphisms, we find that the pullback of \'etale sheaves from $S_{\et}$ to $S_{0,\et}$ defines a functor
$$\gerb{N}' \colon \gerb{E}_\epsilon(\alg{\tilde G}) \rightarrow \gamma_\ast \gerb{E}_{\epsilon_0}(\alg{\tilde G}_0)),$$
given on objects by $\gerb{N}'(\sheaf{H}, h, j) = (\gamma^\ast \sheaf{H}, \gamma^\ast h, \gamma^\ast j)$. For example, if $\sheaf{H}$ is a $\sheaf{\hat T}$-torsor, then $\gamma^\ast \sheaf{H}$, a priori a $\gamma^\ast \sheaf{\hat T}$-torsor, becomes a $\sheaf{\hat T}_0$-torsor via $\hat N$.
From \cite[Chapitre V, Proposition 3.1.8]{Giraud}, such an equivalence of gerbes $\gerb{N}'$ determines a unique, up to unique natural isomorphism, equivalence of gerbes,
$$\gerb{N} \colon \gamma^\ast \gerb{E}_\epsilon(\alg{\tilde G}) \rightarrow \gerb{E}_{\epsilon_0}(\alg{\tilde G}_0),$$
lying over the natural isomorphism of bands $N^\vee \colon \gamma^\ast \sheaf{\tilde Z}^ { \raisemath{-3pt}{\vee} } \rightarrow \sheaf{\tilde Z}_0^ { \raisemath{-3pt}{\vee} } $. In this way, the construction of the gerbe associated to a cover is compatible with change of base scheme.
\subsection{Parabolic subgroups}
\label{ParabolicGerbe}
Return to the degree $n$ cover $\alg{\tilde G}$ of a quasisplit group $\alg{G} \supset \alg{B} \supset \alg{T}$ over $S$. Let $\alg{P} \subset \alg{G}$ be a parabolic subgroup defined over $S$, containing $\alg{B}$. As in Section \ref{ParabolicDualGroup}, we consider a Levi decomposition $\alg{P} = \alg{M} \alg{N}$ and resulting cover $\alg{\tilde M}$. Fix $\epsilon \colon \mu_n \hookrightarrow \mathbb C^\times$ as before.
Consider the gerbes $\gerb{E}_\epsilon(\alg{\tilde G})$ and $\gerb{E}_\epsilon(\alg{\tilde M})$. An object of $\gerb{E}_\epsilon(\alg{\tilde G})$ is a triple $(\sheaf{H}, h, j)$, where $(\sheaf{H}, h)$ is an $n^{\th}$ root of the torsor $\mathscr{S}\mathit{pl}(\sheaf{D}_{Q,n})$, $j = j_0 \wedge \tau$, $\tau \in \sheaf{\tilde T}_{\SC}^ { \raisemath{-3pt}{\vee} } $, and $j_0 \colon p_\ast\sheaf{H} \rightarrow \mu_\ast \mathscr{W}\mathit{hit}$ is an isomorphism of $\sheaf{\hat T}_{\SC}$-torsors.
Restriction of characters via $\alg{U} \supset \alg{U}_{\alg{M}} = \alg{U} \cap \alg{M}$ provides a homomorphism of sheaves from $\mathscr{W}\mathit{hit}$ to $\mathscr{W}\mathit{hit}_{\alg{M}}$ (the Whittaker torsor for $\alg{M}$). The inclusion $\sheaf{Y}_{\alg{M}}^{\SC} \subset \sheaf{Y}^{\SC}$ of coroot lattices provides a homomorphism of sheaves of abelian groups $\sheaf{\hat T}_{\SC} \twoheadrightarrow \sheaf{\hat T}_{\alg{M},\SC}$, where
$$\sheaf{\hat T}_{\SC} = \mathscr{H}\mathit{om}(\sheaf{Y}^{\SC}, \sheaf{G}_m), \quad \sheaf{\hat T}_{\alg{M},\SC} = \mathscr{H}\mathit{om}(\sheaf{Y}_{\alg{M}}^{\SC}, \sheaf{G}_m).$$
Similarly, it provides a homomorphism $\sheaf{\tilde T}_{\SC}^ { \raisemath{-3pt}{\vee} } \twoheadrightarrow \sheaf{\tilde T}_{\alg{M}, \SC}^ { \raisemath{-3pt}{\vee} } $, where
$$\sheaf{\tilde T}_{\SC}^ { \raisemath{-3pt}{\vee} } = \mathscr{H}\mathit{om}(\sheaf{Y}^{\SC}, \mathbb C^\times), \quad \sheaf{\tilde T}_{\alg{M},\SC}^ { \raisemath{-3pt}{\vee} } = \mathscr{H}\mathit{om}(\sheaf{Y}_{\alg{M}}^{\SC}, \mathbb C^\times).$$
Define $\tau_{\alg{M}}^\vee$ to be the image of $\tau^\vee$ under this homomorphism.
The constants defining $\mu \colon \sheaf{\hat T}_{\SC} \rightarrow \sheaf{\hat T}_{\SC}$ are the same as those defining the corresponding map, $\mu_{\alg{M}} \colon \sheaf{\hat T}_{\alg{M},\SC} \rightarrow \sheaf{\hat T}_{\alg{M},\SC}$. We find a commutative diagram.
$$\begin{tikzcd}
\sheaf{\hat T}_{\SC} \times \mu_\ast \mathscr{W}\mathit{hit} \arrow{r}{\ast} \arrow{d} & \mu_\ast \mathscr{W}\mathit{hit} \arrow{d} \\
\sheaf{\hat T}_{\alg{M},\SC} \times (\mu_{\alg{M}})_\ast \mathscr{W}\mathit{hit}_{\alg{M}} \arrow{r}{\ast} & (\mu_{\alg{M}})_\ast \mathscr{W}\mathit{hit}_{\alg{M}}
\end{tikzcd}$$
Composing $j_0 \colon p_\ast \sheaf{H} \rightarrow \mu_\ast \mathscr{W}\mathit{hit}$ with the map to $(\mu_{\alg{M}})_\ast \mathscr{W}\mathit{hit}_{\alg{M}}$ defines a map
$$j_{\alg{M},0} \colon \sheaf{H} \rightarrow (\mu_{\alg{M}})_\ast \mathscr{W}\mathit{hit}_{\alg{M}}.$$
This defines a functor of gerbes,
$$\gerb{i} \colon \gerb{E}_\epsilon(\alg{\tilde G}) \rightarrow \gerb{E}_\epsilon(\alg{\tilde M}), \quad (\sheaf{H}, h, j_0 \wedge \tau^\vee) \mapsto (\sheaf{H}, h, j_{\alg{M},0} \wedge \tau_{\alg{M}}^\vee),$$
lying over $\iota^\vee \colon \sheaf{\tilde Z}^ { \raisemath{-3pt}{\vee} } \hookrightarrow \sheaf{\tilde Z}_{\alg{M}}^ { \raisemath{-3pt}{\vee} } $.
\section{The metaGalois group}
\label{MetaGaloisSection}
We now specialize to three classes of base scheme of arithmetic interest.
\begin{description}
\item[Global] $S = \Spec(F)$ for a global field $F$;
\item[Local] $S = \Spec(F)$ for a local field $F$;
\item[Local integral] $S = \Spec(\mathcal{O})$ for the ring of integers $\mathcal{O}$ in a nonarchimedean local field $F$.
\end{description}
Choose a separable closure $\bar F / F$ in all three cases, and write $\Gal_F = \Gal(\bar F / F)$ for the resulting absolute Galois group. In the local integral case, the separable closure $\bar F / F$ provides a geometric base point $\bar s$ for $\Spec(\mathcal{O})$ as well, and define $\Gal_\mathcal{O} = \pi_1^{\et}(\Spec(\mathcal{O}), \bar s)$. This is a profinite group, topologically generated by a geometric Frobenius element ${\mathsf{Fr}}$; thus we write $\Gal_\mathcal{O} = \langle {\mathsf{Fr}} \rangle_{\prof}$.
When $S = \Spec(F)$ for a local or global field, or $S = \Spec(\mathcal{O})$ for the ring of integers in a nonarchimedean local field, write $\Gal_S$ for $\Gal_F$ or $\Gal_\mathcal{O}$ accordingly.
\subsection{Construction of the metaGalois group}
The metaGalois group will be a profinite group fitting into a central extension,
$$\mu_2 \hookrightarrow \widetilde{\Gal}_S \twoheadrightarrow \Gal_S.$$
When $F$ has characteristic two, we {\em define} the metaGalois group $\widetilde{\Gal}_F$ to be the trivial extension $\Gal_F \times \mu_2$. The metaGalois group $\widetilde{\Gal}_\mathcal{O}$ will {\em not be defined} when $\mathcal{O}$ has residual characteristic $2$, reflecting the idea that metaGalois representations cannot be ``unramified at $2$'' (though one might propose an alternative notion of ``minimally ramified'').
\subsubsection{Local fields}
When $F$ is a local field (with $2 \neq 0$) the quadratic Hilbert symbol defines a symmetric nondegenerate $\mathbb Z$-bilinear form
$$\Hilb_2 \colon F^\times_{/2} \times F^\times_{/2} \rightarrow \mu_2.$$
The abelianized Galois group $\Gal_F^{\ab}$ is defined, up to unique isomorphism, from $F$ alone. When $F$ is nonarchimedean, we normalize the valuation so that $\val(F^\times) = \mathbb Z$, and we normalize the reciprocity map of local class field theory to send a geometric Frobenius element to an element of valuation $1$. The reciprocity map gives a surjective homomorphism
$$\rec_{F/2} \colon \Gal_F^{\ab} \twoheadrightarrow F^\times_{/2}.$$
Composing the Hilbert symbol with $\rec_{F/2}$ defines a function
$$h \colon \Gal_F^{\ab} \times \Gal_F^{\ab} \rightarrow \mu_2,$$
and it is straightforward to verify that $h$ is a (bimultiplicative) continuous symmetric $2$-cocycle. This incarnates a commutative extension
$$\mu_2 \hookrightarrow \widetilde{\Gal}_F^{\ab} \twoheadrightarrow \Gal_F^{\ab}$$
of profinite groups. Concretely, $\widetilde{\Gal}_F^{\ab} = \Gal_F^{\ab} \times \mu_2$ as sets, and
$$(\gamma_1, \epsilon_1) \cdot (\gamma_2, \epsilon_2) := \left( \gamma_1 \gamma_2, \epsilon_1 \epsilon_2 \cdot h(\gamma_1, \gamma_2) \right).$$
The pullback of this extension to $\Gal_F$ will be called the \defined{metaGalois group of $F$}, written $\widetilde{\Gal}_F$. It is a central extension of $\Gal_F$ by $\mu_2$,
$$\mu_2 \hookrightarrow \widetilde{\Gal}_F \twoheadrightarrow \Gal_F.$$
\subsubsection{The local integral case}
Suppose that $F$ is a nonarchimedean \textbf{nondyadic} (i.e., $\val(2) = 0$) local field. Then the quadratic Hilbert symbol satisfies
$$\Hilb_2(u,v) = 1 \text{ for all } u,v \in \mathcal{O}^\times.$$
Write $\mathcal{I} \subset \Gal_F$ for the inertial subgroup, so that $\rec_F(\mathcal{I}) = \mathcal{O}^\times$. The cocycle $h$ is trivial when restricted to $\mathcal{I} \times \mathcal{I}$. Thus $\gamma \mapsto (\gamma, 1)$ gives a canonical splitting $\sigma^\circ \colon \mathcal{I} \hookrightarrow \widetilde{\Gal}_F$. The natural map $\Gal_F \twoheadrightarrow \Gal_\mathcal{O}$ identifies $\Gal_\mathcal{O}$ with $\Gal_F / \mathcal{I}$. Define $\widetilde{\Gal}_\mathcal{O} = \widetilde{\Gal}_F / \sigma^\circ(\mathcal{I})$ to obtain a commutative diagram with exact rows.
$$\begin{tikzcd}
\mu_2 \arrow{d}{=} \arrow[hook]{r} & \widetilde{\Gal}_F \arrow{r} \arrow[two heads]{d} & \Gal_F \arrow[two heads]{d} \\
\mu_2 \arrow[hook]{r} & \widetilde{\Gal}_\mathcal{O} \arrow{r} & \Gal_\mathcal{O}
\end{tikzcd}$$
We call $\widetilde{\Gal}_\mathcal{O}$ the \defined{metaGalois group of $\mathcal{O}$}. If $\gamma \in \Gal_F$ lifts $\Fr$, then every element of $\widetilde{\Gal}_\mathcal{O}$ is equal to $(\gamma^{\hat n}, \pm 1)$ (mod $\sigma^\circ(\mathcal{I})$) for some $\hat n \in \hat \mathbb Z$. In this way, a Frobenius lift provides an isomorphism from $\widetilde{\Gal}_\mathcal{O}$ to the group with underlying set $\langle \gamma \rangle_{\prof} \times \mu_2$ and multiplication given by
$$(\gamma^{\hat n_1}, \epsilon_1) \cdot (\gamma^{\hat n_2}, \epsilon_2) = \left( \gamma^{\hat n_1 + \hat n_2}, \epsilon_1 \epsilon_2 \cdot (-1)^{ \hat n_1 \hat n_2 (q-1)/2} \right),$$
where $q$ is the cardinality of the residue field of $\mathcal{O}$.
\subsubsection{Global fields}
When $F$ is a global field (with $2 \neq 0$ as before), the Hilbert symbol defines a symmetric $\mathbb Z$-bilinear form,
$$\Hilb_2 \colon \AA^\times_{/2} \times \AA^\times_{/2} \rightarrow \mu_2,$$
obtained as the product of local Hilbert symbols. This defines a continuous symmetric 2-cocycle, from which we get a commutative extension,
\begin{equation}
\label{ExtensionAdelic}
\mu_2 \hookrightarrow \widetilde{\AA^\times_{/2}} \twoheadrightarrow \AA^\times_{/2}.
\end{equation}
Global quadratic reciprocity for the Hilbert symbol ($\Hilb_2(u,v) = 1$ for all $u,v \in F^\times$) provides a canonical splitting $\sigma_F \colon F^\times_{/2} \hookrightarrow \widetilde{\AA^\times_{/2}}$. Taking the quotient yields a commutative extension,
\begin{equation}
\label{QuotientExtension}
\mu_2 \hookrightarrow \frac{\widetilde{\AA^\times_{/2}}}{\sigma_F(F^\times_{/2})} \twoheadrightarrow \frac{ \AA^\times_{/2} }{F^\times_{/2}}.
\end{equation}
The global reciprocity map of class field theory gives an surjective homomorphism,
$$\rec_{F/2} \colon \Gal_F^{\ab} \twoheadrightarrow \AA^\times_{/2} / F^\times_{/2} \equiv (\AA^\times / F^\times)_{/2}.$$
Pulling back \eqref{QuotientExtension} via $\rec_{F/2}$ yields a commutative extension,
$$\mu_2 \hookrightarrow \widetilde{\Gal}_F^{\ab} \twoheadrightarrow \Gal_F^{\ab}.$$
Pulling back via $\Gal_F \twoheadrightarrow \Gal_F^{\ab}$ defines the \defined{metaGalois group of $F$},
$$\mu_2 \hookrightarrow \widetilde{\Gal}_F \twoheadrightarrow \Gal_F.$$
\subsubsection{Compatibilities}
\label{CompatibilityMetaGalois}
If $v \in \mathcal V$ is a place of a global field $F$, then an embedding $\bar F \hookrightarrow \bar F_v$ of separable closures determines an injective homomorphism $\iota_v \colon \Gal_{F_v} \hookrightarrow \Gal_F$. As the global Hilbert symbol is the product of local ones, we find a homomorphism $\tilde \iota_v$ realizing $\widetilde{\Gal}_{F_v}$ as the pullback of the extension $\widetilde{\Gal}_F$.
$$\begin{tikzcd}
\mu_2 \arrow[hook]{r} \arrow{d}{=} & \widetilde{\Gal}_{F_v} \arrow{r} \arrow[hook]{d}{\tilde \iota_v} & \Gal_{F_v} \arrow[hook]{d}{\iota_v} \\
\mu_2 \arrow[hook]{r} & \widetilde{\Gal}_F \arrow{r} & \Gal_F
\end{tikzcd}$$
For local and global fields $F$, a choice of separable closure $\bar F / F$ entered the construction of the metaGalois group. Suppose that $\bar F_0$ is another separable closure of $F$. Every $F$-algebra isomorphism $\iota \colon \bar F_0 \xrightarrow{\sim} \bar F$ yields an isomorphism $\iota \colon \Gal(\bar F_0 / F) \xrightarrow{\sim} \Gal(\bar F / F)$. The resulting isomorphism $\Gal(\bar F_0 / F)^{\ab} \xrightarrow{\sim} \Gal(\bar F / F)^{\ab}$ does not depend on $\iota$.
The separable closure $\bar F_0$ yields a cocycle $h_0 \colon \Gal(\bar F_0 / F)^{\ab} \times \Gal(\bar F_0 / F)^{\ab} \rightarrow \mu_2$, and thus a metaGalois group $\widetilde{\Gal}(\bar F_0 / F)$. Since the defining cocycles $h_0$ and $h$ factor through abelianized Galois groups, the isomorphism $\iota \colon \Gal(\bar F_0 / F) \rightarrow \Gal(\bar F / F)$ lifts canonically to an isomorphism of metaGalois groups.
$$\begin{tikzcd}
\mu_2 \arrow{d}{=} \arrow[hook]{r} & \widetilde{\Gal}(\bar F_0 / F)\arrow[two heads]{r} \arrow{d}{\tilde \iota} & \Gal(\bar F_0 / F)\arrow{d}{\iota} \\
\mu_2 \arrow[hook]{r} & \widetilde{\Gal}(\bar F / F) \arrow[two heads]{r} & \Gal(\bar F / F)
\end{tikzcd}$$
\subsection{The Brauer class}
When $F$ is a local or global field, the metaGalois group is an extension $\mu_2 \hookrightarrow \widetilde{\Gal}_F \twoheadrightarrow \Gal_F$. As such, it has a cohomology class in the Brauer group.
$$\left[ \widetilde{\Gal}_F \right] \in H_{\et}^2(F, \mu_2) = \Br(F)_{[2]}.$$
This Brauer class is often trivial -- the metaGalois group often splits, though it rarely has a canonical splitting.
\begin{proposition}
\label{OddRCTrivialBr}
Suppose that $F$ is a \textbf{nondyadic} (i.e., $\val(2) = 0$) nonarchimedean local field. Then $\left[ \widetilde{\Gal}_F \right]$ is the trivial class.
\end{proposition}
{\sc Proof:}
The projection $\widetilde{\Gal}_F \rightarrow \widetilde{\Gal}_\mathcal{O}$ identifies the metaGalois group of $F$ with the pullback of the metaGalois group of $\mathcal{O}$.
$$\begin{tikzcd}
\mu_2 \arrow{r} \arrow{d}{=} & \widetilde{\Gal}_F \arrow{r} \arrow{d} & \Gal_F \arrow{d} \\
\mu_2 \arrow{r} & \widetilde{\Gal}_\mathcal{O} \arrow{r} & \Gal_\mathcal{O} = \langle {\mathsf{Fr}} \rangle_{\prof}
\end{tikzcd}$$
But every extension of $\hat \mathbb Z$ by $\mu_2$ splits (though not canonically); hence the metaGalois group splits and its Brauer class is trivial.
\begin{flushright}Q.E.D \end{flushright}
\begin{proposition}
\label{RealNontrivBr}
Over $\mathbb R$, the metaGalois group is a nonsplit extension, so $\left[ \widetilde{\Gal}_\mathbb R \right]$ is the unique nontrivial class in the Brauer group $\Br(\mathbb R)$.
\end{proposition}
{\sc Proof:}
Let $\sigma$ denote complex conjugation, $\Gal_\mathbb R = \Gal(\mathbb C / \mathbb R) = \{ \Id, \sigma \}$. The metaGalois group is a cyclic group of order $4$ sitting in an extension
$$\mu_2 \hookrightarrow \widetilde{\Gal}_\mathbb R \twoheadrightarrow \Gal_\mathbb R.$$
Indeed, the cocycle $h$ satisfies $h(\sigma, \sigma) = \Hilb_2(-1,-1) = -1$. Thus
$$(\sigma,1) \cdot (\sigma,1) = (\Id, -1) \in \widetilde{\Gal}_\mathbb R.$$
Hence $(\sigma,1)$ is an element of order $4$ and $\left[ \widetilde{\Gal}_\mathbb R \right]$ is nontrivial.
\begin{flushright}Q.E.D \end{flushright}
\begin{proposition}
Let $F_2$ be a \textbf{dyadic} nonarchimedean local field of characteristic zero. Then $\left[ \widetilde{\Gal}_{F_2} \right]$ is trivial if $[F_2 : \mathbb Q_2]$ is even, and is nontrivial if $[F_2 : \mathbb Q_2]$ is odd.
\end{proposition}
{\sc Proof:}
Let $d = [F_2 : \mathbb Q_2]$. By approximation, there exists a global field $F$ such that $F \otimes_\mathbb Q \mathbb Q_2$ is isomorphic to $F_2$ as an $F$-algebra. The global metaGalois group $\widetilde{\Gal}_F$ has a Brauer class $\beta_F$ with local components $\beta_{F,v}$ satisfying
\begin{itemize}
\item
$\beta_{F,v}$ is trivial when $F_v$ has odd residual characteristic (by Proposition \ref{OddRCTrivialBr});
\item
$\beta_{F,2} = \left[ \widetilde{\Gal}_{F_2} \right]$ at the unique place of even residual characteristic;
\item
$\beta_{F,v}$ is nontrivial at all real places (by Proposition \ref{RealNontrivBr});
\item
(Parity condition) $\beta_{F,v}$ is nontrivial at a set of places of even cardinality.
\end{itemize}
We have $d = [F_2 : \mathbb Q_2] = [F : \mathbb Q] = r_1 + 2 r_2$, where $r_1$ is the number of real places, and $r_2$ the number of complex places. It follows that $d$ is even if and only if $r_1$ is even. The parity condition on the global Brauer class implies that $r_1$ is even if and only if $\beta_{F,2}$ is the trivial class.
\begin{flushright}Q.E.D \end{flushright}
\begin{corollary}
Let $F$ be a global field, with $2 \neq 0$ in $F$. Then the Brauer class of $\widetilde{\Gal}_F$ is that of the unique quaternion algebra which is ramified at all real places and all dyadic places of odd degree over $\mathbb Q_2$.
\end{corollary}
{\sc Proof:}
This follows directly from the previous three propositions, and the local-global compatibility of the metaGalois group.
\begin{flushright}Q.E.D \end{flushright}
In particular, the Brauer class $\left[ \widetilde{\Gal}_\mathbb Q \right]$ is that of the quaternion algebra $\frac{(-1,-1)}{\mathbb Q}$ ramified only at $2$ and $\infty$. If $F$ is a global field of characteristic $p \neq 2$, then $\left[ \widetilde{\Gal}_F \right]$ is the trivial class.
\subsection{Splitting by additive characters}
The metaGalois group may be a nonsplit extension of $\Gal_S$ by $\mu_2$, and even when it splits, it rarely splits canonically. However, an additive character suffices to split the metaGalois group after pushing out via $\mu_2 \hookrightarrow\mu_4$. In the three cases of interest, define a $\sheaf{G}_m[S]$-torsor $\Psi_S$ as follows.
\begin{itemize}
\item
When $F$ is local, let $\Psi_F$ be the set of nontrivial continuous homomorphisms from $F$ to $\mathbb C^\times$. If $u \in F^\times$, $\psi \in \Psi_F$, write $[u \ast \psi](x) = \psi(u^{-1} x)$. In this way, $\Psi_F$ is a $F^\times$-torsor.
\item
When $F$ is global, let $\Psi_F$ be the set of nontrivial continuous homomorphisms from $\AA / F$ to $\mathbb C^\times$. If $u \in F^\times$, $\psi \in \Psi_F$, write $[u \ast \psi](x) = \psi(u^{-1} x)$. In this way, $\Psi_F$ is a $F^\times$-torsor.
\item
When $F$ is local nonarchimedean, with ring of integers $\mathcal{O}$, let $\Psi_\mathcal{O}$ be the set of nontrivial continuous homomorphisms from $F / \mathcal{O}$ to $\mathbb C^\times$. If $u \in \mathcal{O}^\times$, $\psi \in \Psi_\mathcal{O}$, write $[u \ast \psi](x) = \psi(u^{-1} x)$. In this way, $\Psi_\mathcal{O}$ is a $\mathcal{O}^\times$-torsor.
\end{itemize}
Define here $\mu_4 = \sch{\mu}_4(\mathbb C) = \{ 1,-1, i, -i \}$. When $F$ is a local field (with $2 \neq 0$), and $\psi \in \Psi_F$, the \defined{Weil index} is a function $\mathbf{w}_F(\bullet, \psi) \colon F^\times_{/2} \rightarrow \mu_4$ which satisfies
\begin{equation}
\label{WeilHilbert}
\frac{\mathbf{w}_F(uv, \psi)}{\mathbf{w}_F(u, \psi) \mathbf{w}_F(v,\psi)} = \Hilb_2(u,v).
\end{equation}
Our $\mathbf{w}_F(u, \psi)$ is defined to be $\gamma(u x^2) / \gamma(x^2)$ in Weil's notation from \cite[\S 29]{WeilActa} and is written $\gamma_F(u, \psi)$ in \cite[\S A.3]{RangaRao} and elsewhere. From \cite[Proposition A.11]{RangaRao}, the local Weil indices are trivial on $\mathcal{O}^\times$ at all nondyadic places.
When $F$ is a global field and $\psi \in \Psi_F$, the \defined{Weil index} is the function
$$\mathbf{w}_F(\bullet, \psi) \colon \AA^\times_{/2} \rightarrow \mu_4$$
defined as the product of local Weil indices. The global Weil index is trivial on $F^\times$ by \cite[\S II.30, Proposition 5]{WeilActa}. As the global Hilbert symbol $\Hilb_2 \colon \AA^\times \times \AA^\times \rightarrow \mu_2$ is defined as the product of local Hilbert symbols, the formula \eqref{WeilHilbert} holds in the global setting too.
Write $\widetilde{\Gal}_S^{(4)}$ for the pushout of $\widetilde{\Gal}_S$ via the inclusion $\mu_2 \hookrightarrow \mu_4$ (when $S = \Spec(F)$ or $S = \Spec(\mathcal{O})$ as usual).
$$\begin{tikzcd}[column sep = 4em]
\mu_2 \arrow[hook]{r} \arrow[hook]{d}{\iota} & \widetilde{\Gal}_S \arrow[two heads]{r} \arrow[hook]{d} & \Gal_S \arrow{d}{=} \\
\mu_4 \arrow[hook]{r} & \widetilde{\Gal}_S^{(4)} \arrow[two heads]{r} & \Gal_S \arrow[bend right=20, dashed]{l}[swap]{s(\psi)?}
\end{tikzcd}$$
The splittings of $\widetilde{\Gal}_S^{(4)}$, if they exist, form a $\Hom(\Gal_S, \mu_4)$-torsor. In what follows, if $u \in \sheaf{G}_m[S]$, define $\chi_u \colon \Gal_S \rightarrow \mu_2$ to be the quadratic character associated to the \'etale extension $F[\sqrt{u}]$ (in the local or global case) or $\mathcal{O}[\sqrt{u}]$ (in the nondyadic local integral case).
\begin{proposition}
\label{WeilSplitsMetaGalois}
For each additive character $\psi \in \Psi_S$, the Weil index provides a splitting $s(\psi) \colon \Gal_S \rightarrow \widetilde{\Gal}_S^{(4)}$. Moreover, this system of splittings satisfies
$$s(u \ast \psi) = \chi_u \ast s(\psi) \text{ for all } u \in \sheaf{G}_m[S].$$
\end{proposition}
The splittings $s(\psi)$ are described in three cases below.
\subsubsection{Local fields}
When $F$ is a local field, the pushout $\widetilde{\Gal}_S^{(4)}$ can be identified with the product $\Gal_F \times \mu_4$ as a set, with multiplication given by
$$(\gamma_1, \zeta_1) \cdot (\gamma_2, \zeta_2) = \left( \gamma_1 \gamma_2, \zeta_1 \zeta_2 \cdot \Hilb_2(\rec_{F/2}(\gamma_1), \rec_{F/2}(\gamma_2) ) \right).$$
For $\psi \in \Psi_F$, \eqref{WeilHilbert} provides a splitting $s(\psi) \colon \Gal_F \rightarrow \widetilde{\Gal}_F^{(4)}$,
$$s(\psi)(\gamma) = \left( \gamma, \mathbf{w}_F( \rec_{F/2}(\gamma), \psi) \right), \text{ for all } \gamma \in \Gal_F.$$
If $u \in F^\times$, then \cite[Corollary A.5]{RangaRao} states that $\mathbf{w}_F(a, u \ast \psi) = \Hilb_2(a,u) \cdot \mathbf{w}_F(a, \psi)$. Since $\Hilb_2(\rec_{F/2} \gamma, u) = \chi_u(\gamma)$, we find $s(u \ast \psi) = \chi_u \ast s(\psi)$.
\subsubsection{The local integral case}
When $F$ is a nonarchimedean, nondyadic local field, the local Weil index is trivial on $\mathcal{O}^\times$. Given a character $\psi \in \Psi_\mathcal{O} \subset \Psi_F$, the splitting $s(\psi) \colon \Gal_F \rightarrow \widetilde{\Gal}_F^{(4)}$ coincides with the canonical splitting $\sigma^\circ$ on intertia,
$$s(\psi) (\gamma) = \sigma^\circ(\gamma) = (\gamma, 1), \text{ for all } \gamma \in \mathcal{I}.$$
It follows that $s(\psi)$ descends to a splitting of $\widetilde{\Gal}_\mathcal{O}^{(4)}$ at nondyadic places.
$$\begin{tikzcd}[column sep = 4em]
\mu_4 \arrow[hook]{r} & \widetilde{\Gal}_\mathcal{O}^{(4)} \arrow[two heads]{r} & \Gal_\mathcal{O}. \arrow[bend right=20]{l}[swap]{s(\psi)}
\end{tikzcd}$$
If $u \in \mathcal{O}^\times$, write $\bar u$ for its image in the residue field $\mathbb F_q$. As before, we have $s(u \ast \psi) = \chi_u \cdot s(\psi)$. But now, the quadratic character $\chi_u$ is restricted to $\Gal_\mathcal{O} = \langle \Fr \rangle_{\prof}$; we have
$$\chi_u(\Fr) = \bar u^{(q-1) / 2} \in \mu_2.$$
In other words, $\chi_u$ is the character of $\Gal_\mathcal{O}$ which sends $\Fr$ to the Legendre symbol of the reduction of $u$.
\subsubsection{Global fields}
In the global setting, pushing out via $\mu_2 \hookrightarrow \mu_4$ gives a short exact sequence
\begin{equation}
\label{Adelic4}
\mu_4 \hookrightarrow \widetilde{\AA^\times_{/2}}^{(4)} \twoheadrightarrow \AA^\times_{/2}.
\end{equation}
The middle term is given by $\widetilde{\AA^\times_{/2}}^{(4)} = \AA^\times_{/2} \times \mu_4$ as a set, with multiplication given by
$$(u_1, \zeta_1) \cdot (u_2, \zeta_2) = (u_1 u_2, \zeta_1 \zeta_2 \cdot \Hilb_2(u_1, u_2)).$$
A character $\psi \in \Psi_F$ provides a splitting of the extension \eqref{Adelic4},
$$s_\AA(\psi)(u) = (u, \mathbf{w}_F(u, \psi) ) \text{ for all } u \in \AA^\times_{/2}.$$
Since $\mathbf{w}_F(u, \psi) = 1$ for all $u \in F^\times$, this splitting restricts to the canonical splitting $\sigma_F \colon F_{/2}^\times \rightarrow \widetilde{\AA_{/2}^\times}$. Thus $s_\AA(\psi)$ descends and pulls back to a splitting $s(\psi) \colon \Gal_F \rightarrow \widetilde{\Gal}_F^{(4)}$. If $u \in F^\times$, then our local results and local-global compatibility imply that $s(u \ast \psi) = \chi_u \cdot s(\psi)$.
\subsection{Restriction}
Suppose that $F' / F$ is a finite separable extension with $F' \subset \bar F$. In the local integral case, suppose that $F' / F$ is unramified and let $\mathcal{O}'$ be the ring of integers in $F'$. Write $S' = \Spec(F')$ in the cases of local or global fields, and write $S' = \Spec(\mathcal{O}')$ in the local integral case. We have defined metaGalois groups for $S$ and $S'$.
\begin{equation}
\label{MGalInc}
\begin{tikzcd}
\mu_2 \arrow[hook]{r} \arrow{d}{=} & \widetilde{\Gal}_{S'} \arrow[two heads]{r} \arrow[dashed]{d}{?}& \Gal_{S'} \arrow[hook]{d} \\
\mu_2 \arrow[hook]{r} & \widetilde{\Gal}_S \arrow[two heads]{r} & \Gal_S
\end{tikzcd}
\end{equation}
The inclusion $F' \subset \bar F$ gives an inclusion of Galois groups $\Gal_{S'} \hookrightarrow \Gal_S$, but a natural inclusion of metaGalois groups is not obvious. In particular, the cocycle defining $\widetilde{\Gal}_S$ does not restrict to the cocycle defining $\widetilde{\Gal}_{S'}$.
Fortunately, a beautiful insight of Wee Teck Gan gives such an inclusion of metaGalois groups, using a ``lifting theorem'' of Edward Bender \cite{Bender}. We explain this insight here.
In the case of local fields, consider a nonzero element $u \in F'$, and the ``trace form'' (cf. \cite{SerreTrace}) $F' \rightarrow F$ given by $x \mapsto \Tr_{F' / F}(u x^2)$. Viewing this as a quadratic form on a finite-dimensional $F$-vector space $F'$, it has a Hasse-Witt invariant (an element of $\{ \pm 1 \}$). Define
$$\HW(u) = \frac{ \text{Hasse-Witt invariant of } x \mapsto \Tr_{F'/F}(u x^2) }{ \text{Hasse-Witt invariant of } x \mapsto \Tr_{F'/F}(x^2) }.$$
This function depends only on the square class of $u$.
Bender's theorem \cite[Theorem 1]{Bender} states that
$$\Hilb_{F',2}(u,v) = \frac{ \HW(u) \HW(v) }{ \HW(uv) } \cdot \Hilb_{F,2} ( \Norm_{F'/F} u, \Norm_{F'/F} v).$$
Let $\iota \colon \Gal_{F'} \hookrightarrow \Gal_F$ be the canonical inclusion, so that $\rec_{F}( \iota(\gamma)) = \Norm_{F'/F} \rec_F(\gamma)$ for all $\gamma \in \Gal_{F'}$.
\begin{proposition}
Let $F$ be a local field (with $2 \neq 0$ as usual). Then the function $\tilde \iota \colon \widetilde{\Gal}_{F'} \hookrightarrow \widetilde{\Gal}_F$, given by
$$\tilde \iota (\gamma, \pm 1) = \left( \iota(\gamma), \pm \HW( \rec_{F'} \gamma) \right)$$
is a group homomorphism completing the commutative diagram \eqref{MGalInc}.
\end{proposition}
{\sc Proof:}
Consider any $\gamma_1, \gamma_2 \in \Gal_{F'}$ and define $u_1 := \rec_{F'}(\gamma_1)$, $u_2 := \rec_{F'}(\gamma_2)$. Thus $\rec_{F}( \iota(\gamma_1)) = \Norm_{F'/F} u_1$ and $\rec_{F}( \iota(\gamma_2)) = \Norm_{F'/F} u_2$. For all $\epsilon_1, \epsilon_2 \in \{ \pm 1 \}$, we compute
\begin{align*}
\tilde \iota \left( (\gamma_1, \epsilon_1) \cdot (\gamma_2, \epsilon_2) \right) &= \tilde \iota \left( \gamma_1 \gamma_2, \epsilon_1 \epsilon_2 \Hilb_{F',2} \left( \rec_{F'}(\gamma_1), \rec_{F'}(\gamma_2) \right) \right) \\
& = \left( \iota(\gamma_1 \gamma_2), \epsilon_1 \epsilon_2 \Hilb_{F',2} (u_1, u_2) \cdot \HW(u_1 u_2) \right) \\
& = \left( \iota(\gamma_1) \iota(\gamma_2), \epsilon_1 \epsilon_2 \Hilb_{F,2} (\Norm_{F' / F} u_1, \Norm_{F' / F} u_2) \HW(u_1) \HW(u_2) \right) \\
& = \left( \iota(\gamma_1), \epsilon_1 \HW(u_1) \right) \cdot \left( \iota(\gamma_2), \epsilon_2 \HW(u_2) \right) \\
&= \tilde \iota( \gamma_1, \epsilon_1) \cdot \tilde \iota(\gamma_2, \epsilon_2).
\end{align*}
\begin{flushright}Q.E.D \end{flushright}
In the local integral case, when $\mathcal{O}$ is the ring of integers in a nondyadic nonarchimedean field, $\HW(u) = 1$ for all $u \in \mathcal{O}^\times$. From this it follows that $\tilde \iota \colon \widetilde{\Gal}_{F'} \hookrightarrow \widetilde{\Gal}_{F}$ descends to an injective homomorphism.
$$\begin{tikzcd}
\mu_2 \arrow[hook]{r} \arrow{d}{=} & \widetilde{\Gal}_{\mathcal{O}'} \arrow[two heads]{r} \arrow{d}{\tilde \iota}& \Gal_{\mathcal{O}'} \arrow[hook]{d}{\iota} \\
\mu_2 \arrow[hook]{r} & \widetilde{\Gal}_\mathcal{O} \arrow[two heads]{r} & \Gal_\mathcal{O}
\end{tikzcd}$$
In the global case, when $F$ is a number field, we note that $\prod_{v} \HW_v(u) = 1$ for all $u \in F^\times$ (here $\HW_v$ denotes the invariant as the place $v$). From this it follows that the injective homomorphisms $\tilde \iota_v \colon \widetilde{\Gal}_{F_v'} \hookrightarrow \widetilde{\Gal}_{F_v}$ yield a injective homomorphism globally.
$$\begin{tikzcd}
\mu_2 \arrow[hook]{r} \arrow{d}{=} & \widetilde{\Gal}_{F'} \arrow[two heads]{r} \arrow{d}{\tilde \iota}& \Gal_{F'} \arrow[hook]{d}{\iota} \\
\mu_2 \arrow[hook]{r} & \widetilde{\Gal}_F \arrow[two heads]{r} & \Gal_F
\end{tikzcd}$$
Taken together, these inclusions $\tilde \iota \colon \widetilde{\Gal}_{S'} \hookrightarrow \widetilde{\Gal}_S$ allow one to canonically ``restrict'' metaGalois representations (representations of $\widetilde{\Gal}_S$).
\section{L-groups, parameters, L-functions}
\subsection{L-groups}
We use the term ``L-group'' to refer to a broad class of extensions of Galois groups by complex reductive groups. Unlike Langlands, Vogan, and others, we do not assume that our L-groups are endowed with a conjugacy class of splittings. Our L-groups are more closely related to the ``weak E-groups'' of \cite[Definition 3.24]{VoganLLC}. But we maintain the letter ``L'' since our L-groups are still connected to L-functions.
The other difference between our L-groups and those in the literature is that (for reasons which will become clear) we consider our L-groups as objects of a 2-category. A base scheme $S$, of the three classes discussed in the previous section, will be fixed.
\begin{definition}
An \defined{L-group} is a pair $(G^\vee, {}^{\mathsf L} G)$, where $G^\vee$ is a complex linear algebraic group (not necessarily connected) and ${}^{\mathsf L} G$ is an extension of locally compact groups
$$G^\vee \hookrightarrow {}^{\mathsf L} G \twoheadrightarrow \Gal_S,$$
for which the conjugation action of any element of ${}^{\mathsf L} G$ on $G^\vee$ is complex-algebraic.
\end{definition}
\begin{remark}
For complex linear algebraic groups, we do not distinguish between the underlying variety and its $\mathbb C$-points. Thus we say $G^\vee$ is a complex linear algebraic group, and also view $G^\vee$ as a locally compact group.
\end{remark}
Of course, Langlands' L-group ${}^{\mathsf L} G = \Gal_F \ltimes G^\vee$ (associated to a reductive group $\alg{G}$ over a field $F$) is an example. When $W$ is a finite-dimensional complex vector space, the direct product $\Gal_S \times GL(W)$ is an L-group. Since we don't assume $G^\vee$ to be connected, our metaGalois group $\widetilde{\Gal}_S$ is another example.
\begin{definition}
Given two L-groups,
$$G_1^\vee \hookrightarrow {}^{\mathsf L} G_1 \twoheadrightarrow \Gal_S, \quad G_2^\vee \hookrightarrow {}^{\mathsf L} G_2 \twoheadrightarrow \Gal_S,$$
an \defined{L-morphism} ${}^{\mathsf L} \rho \colon {}^{\mathsf L} G_1 \rightarrow {}^{\mathsf L} G_2$ will mean a continuous group homomorphism lying over $\Id \colon \Gal_S \rightarrow \Gal_S$, which restricts to a complex algebraic homomorphism $\rho^\vee \colon G_1^\vee \rightarrow G_2^\vee.$ An \defined{L-equivalence} will mean an invertible L-morphism.
\end{definition}
In other words, an L-morphism fits into a commutative diagram, with the middle column continuous and the left column complex-algebraic.
$$\begin{tikzcd}
G_1^\vee \arrow[hook]{r} \arrow{d}{\rho^\vee}& {}^{\mathsf L} G_1 \arrow[two heads]{r} \arrow{d}{{}^{\mathsf L} \rho} & \Gal_S \arrow{d}{=} \\
G_2^\vee \arrow[hook]{r} & {}^{\mathsf L} G_2 \arrow[two heads]{r} & \Gal_S
\end{tikzcd}
$$
An \defined{L-representation} of ${}^{\mathsf L} G$ will mean a pair $({}^{\mathsf L} \rho, W)$ where $W$ is a finite-dimensional complex vector space and ${}^{\mathsf L} \rho \colon {}^{\mathsf L} G \rightarrow \Gal_S \times GL(W)$ is an L-morphism. If we project further, from $\Gal_S \times GL(W)$ to $GL(W)$, then an L-representation yields a continuous homomorphism $\rho \colon {}^{\mathsf L} G \rightarrow GL(W)$, whose restriction to $G^\vee$ is complex algebraic. Such a pair $(\rho, W)$ determines an L-representation $({}^{\mathsf L} \rho, W)$, and so we often abuse notation slightly and say $(\rho, W)$ is an L-representation of ${}^{\mathsf L} G$.
\begin{definition}
Given two L-morphisms ${}^{\mathsf L} \rho, {}^{\mathsf L} \rho' \colon {}^{\mathsf L} G_1 \rightarrow {}^{\mathsf L} G_2$, a natural isomorphism ${}^{\mathsf L} \rho \xRightarrow{\sim} {}^{\mathsf L} \rho'$ will mean an element $a \in Z_2^\vee$ (the center of $G_2^\vee$) such that
$${}^{\mathsf L} \rho'(g) = a \cdot {}^{\mathsf L} \rho(g) \cdot a^{-1} \text{ for all } g \in {}^{\mathsf L} G_1.$$
In particular, note that $\rho$ and $\rho'$ coincide on $G_1^\vee$ when they are naturally isomorphic.
This defines the \defined{2-category of L-groups}, L-morphisms, and natural isomorphisms of L-morphisms.
\end{definition}
In many cases of interest (e.g., when ${}^{\mathsf L} G_2$ arises as the L-group of a split reductive group), the only natural isomorphism is the identity. However, in some nonsplit cases, e.g., ${}^{\mathsf L} G_2 = \Gal_S \ltimes SL_3(\mathbb C)$, the Langlands L-group of a quasisplit $\alg{G} = \alg{PGU}_3$, a nontrivial element $a \in Z_2^\vee$ does not lie in the center of ${}^{\mathsf L} G_2$. Such an element $a$ may determine a nonidentity natural isomorphism.
\subsection{Parameters}
Write $\mathcal{W}_S$ for the \defined{Weil group}. When $S = \Spec(F)$ for a local or global field, this Weil group $\mathcal{W}_S$ is $\mathcal{W}_F$ defined as in \cite{ArtinTate}; when $S = \Spec(\mathcal{O})$, we define $\mathcal{W}_S$ to be the free cyclic group $\langle \Fr \rangle \cong \mathbb Z$ generated by a geometric Frobenius $\Fr$. In all cases, the Weil group is endowed with a continuous homomorphism $\mathcal{W}_S \rightarrow \Gal_S$ with dense image.
Let $G^\vee \hookrightarrow {}^{\mathsf L} G \twoheadrightarrow \Gal_S$ be an L-group. A \defined{Weil parameter} is a continuous homomorphism $\phi \colon \mathcal{W}_S \rightarrow {}^{\mathsf L} G$ lying over $\mathcal{W}_S \rightarrow \Gal_S$, such that $\phi(w)$ is semisimple for all $w \in \mathcal{W}_S$ (see \cite[\S 8.2]{BorelCorvallis}). The reader may follow \cite{BorelCorvallis} and \cite{GrossReeder} to define Weil-Deligne parameters in this general context, when working over a local field.
Write $\Par(\mathcal{W}_S, {}^{\mathsf L} G)$ for the set of ${}^{\mathsf L} G$-valued Weil parameters. It is endowed with an action of $G^\vee$ by conjugation: if $g \in G^\vee$ and $\phi$ is a parameter, then define
$${}^ g \phi(w) = \phi(g^{-1} w g).$$
Two parameters are called \defined{equivalent} if they are in the same $G^\vee$-orbit (this makes sense in the local or global context).
Composition with an L-morphism ${}^{\mathsf L} \rho \colon {}^{\mathsf L} G_1 \rightarrow {}^{\mathsf L} G_2$ defines a map,
$${}^{\mathsf L} \rho \colon \Par(\mathcal{W}_S, {}^{\mathsf L} G_1) \rightarrow \Par(\mathcal{W}_S, {}^{\mathsf L} G_2).$$
Moreover, this map is equivariant, in the sense that for all $g_1 \in G_1^\vee$ and all parameters $\phi \in \Par({}^{\mathsf L} G_1)$, we have
$${}^{\mathsf L} \rho \left( {}^{g_1} \phi \right) = {}^{\rho^\vee(g_1)} \left( {}^{\mathsf L} \rho(\phi) \right).$$
Thus the L-morphism $\rho$ descends to a well-defined map of equivalence classes
$${}^{\mathsf L} \rho \colon \frac{ \Par(\mathcal{W}_S, {}^{\mathsf L} G_1)}{G_1^\vee-\text{conjugation} } \rightarrow \frac{\Par(\mathcal{W}_S, {}^{\mathsf L} G_2)}{G_2^\vee-\text{conjugation} }.$$
Next, consider a natural isomorphism of L-morphisms $\rho \xRightarrow{\sim} \rho'$, with $\rho, \rho' \colon {}^{\mathsf L} G_1 \rightarrow {}^{\mathsf L} G_2$. We find two maps of parameter spaces,
$${}^{\mathsf L} \rho, {}^{\mathsf L} \rho' \colon \Par(\mathcal{W}_S, {}^{\mathsf L} G_1) \rightarrow \Par(\mathcal{W}_S, {}^{\mathsf L} G_2),$$
and an element $a \in Z_2^\vee$ such that ${}^{\mathsf L} \rho'$ is obtained from ${}^{\mathsf L} \rho$ by conjugation by $a$.
It follows that ${}^{\mathsf L} \rho$ and ${}^{\mathsf L} \rho'$ induce the {\em same} map on equivalence classes,
$${}^{\mathsf L} \rho = {}^{\mathsf L} \rho' \colon \frac{ \Par(\mathcal{W}_S, {}^{\mathsf L} G_1)}{G_1^\vee-\text{conjugation} } \rightarrow \frac{\Par(\mathcal{W}_S, {}^{\mathsf L} G_2)}{G_2^\vee-\text{conjugation} }.$$
Suppose that an L-group ${}^{\mathsf L} G$ is defined up to L-equivalence, and the L-equivalence defined up to unique natural isomorphism. Then the set of equivalence classes of parameters
$$\frac{\Par(\mathcal{W}_S, {}^{\mathsf L} G)}{G^\vee-\text{conjugation}}$$
is uniquely defined up to unique isomorphism.
Refinements of the Langlands parameterization suggest that one should look not only at equivalence classes of (Weil or Weil-Deligne) parameters, but also irreducible representations of the component group of the centralizer of a parameter. Or, following Vogan \cite{VoganLLC}, one can look at $G^\vee$-equivariant perverse sheaves on a suitable variety of parameters. The fact that conjugation by $a \in Z_2^\vee$ commutes with the conjugation action of $G_2^\vee$ implies that conjugation by $a$ preserves not only the equivalence class of a Weil parameter for ${}^{\mathsf L} G_2$, but also the equivalence class of such a refined parameter. If an L-group is defined up to L-equivalence, and the L-equivalence defined up to unique natural isomorphism, then the set of equivalence classes of refined parameters is uniquely defined up to unique isomorphism.
\subsection{L-functions}
Let $G^\vee \hookrightarrow {}^{\mathsf L} G \twoheadrightarrow \Gal_S$ be an L-group, and $\phi \colon \mathcal{W}_S \rightarrow {}^{\mathsf L} G$ a Weil parameter (or we may take $\phi$ to be a Weil-Deligne parameter in the local case). Let $(\rho, W)$ be an L-representation of ${}^{\mathsf L} G$. Then
$$\rho \circ \phi \colon \mathcal{W}_S \rightarrow GL(W)$$
is a Weil representation (or Weil-Deligne representation in the local case). As such we obtain an L-function (as defined by Weil and discussed in \cite[\S 3.3]{TateCorvallis}),
$$L(\phi, \rho, s) := L(\rho \circ \phi, s).$$
Choosing an additive character $\psi$ as well gives an $\epsilon$-factor (see \cite[\S 3.4]{TateCorvallis}, based on work of Langlands and Deligne),
$$\epsilon(\phi, \rho, \psi, s) := \epsilon(\rho \circ \phi, \psi, s).$$
In the local integral case $S = \Spec(\mathcal{O})$, we have $\mathcal{W}_S = \langle \Fr \rangle$, and we define the L-functions and $\epsilon$-factors to be those coming from the unramified representation of $\mathcal{W}_F$ by pullback.
In the setting of Langlands L-groups, a zoo of L-representations arises from complex algebraic representations of $G^\vee$, yielding well-known ``standard'' L-functions, symmetric power and exterior power L-functions, etc..
But in our very broad setting, we limit our discussion to {\em adjoint} L-functions, as these play an important role in representation theory and their definition is ``internal.'' Consider any L-morphism $\rho \colon {}^{\mathsf L} H \rightarrow {}^{\mathsf L} G$ of L-groups.
$$\begin{tikzcd}
H^\vee \arrow[hook]{r} \arrow{d}{\rho^\vee} & {}^{\mathsf L} H \arrow[two heads]{r} \arrow{d}{ {}^{\mathsf L} \rho} & \Gal_S \arrow{d}{=} \\
G^\vee \arrow[hook]{r} & {}^{\mathsf L} G \arrow[two heads]{r} & \Gal_S
\end{tikzcd}$$
For example, we might consider the case where $H^\vee$ is a Levi subgroup of $G^\vee$ (as arises in the Langlands-Shahidi method, \cite{Shahidi}).
Let $\Lie{g}^\vee$ be the complex Lie algebra of $G^\vee$. The homomorphism ${}^{\mathsf L} \rho$ followed by conjugation gives an adjoint representation:
$$Ad_\rho \colon {}^{\mathsf L} H \rightarrow GL\left( \Lie{g}^\vee \right).$$
Suppose we have a decomposition of $\Lie{g}^\vee$ as a representation of ${}^{\mathsf L} H$,
\begin{equation}
\label{AdSummands}
\Lie{g}^\vee = \bigoplus_{i = 0}^h \Lie{g}_{i}^\vee.
\end{equation}
For example, when $H^\vee$ is a Levi subgroup of a parabolic $P^\vee \subset G^\vee$, we may decompose $\Lie{g}^\vee$ into $\Lie{h}^\vee$ and the steps in the nilradical of the Lie algebra of $P^\vee$ and its opposite.
A decomposition \eqref{AdSummands} gives representations $Ad_{i} \colon {}^{\mathsf L} H \rightarrow GL\left( \Lie{g}_{i}^\vee \right)$. When $\phi \colon \mathcal{W}_S \rightarrow {}^{\mathsf L} H$ is a Weil parameter, we obtain L-functions
$$L(\phi, Ad_i, s) := L(Ad_i \circ \phi, s).$$
In particular, when ${}^{\mathsf L} H = {}^{\mathsf L} G$, and $\rho = \Id$, we write $Ad$ for the adjoint representation of ${}^{\mathsf L} G$ on $\Lie{g}^\vee$. This yields \defined{the adjoint L-function} $L(\phi, Ad, s)$ for any Weil parameter $\phi \colon \mathcal{W}_S \rightarrow {}^{\mathsf L} G$. When $H^\vee$ is a Levi subgroup of a parabolic in $G^\vee$, and $Ad_i$ arises from a step in the nilradical of the parabolic, we call $L(\phi, Ad_i, s)$ a \defined{Langlands-Shahidi L-function}.
\begin{remark}
The importance of such L-functions for covering groups is suggested by recent work of D. Szpruch \cite{Szp}, who demonstrates that the Langlands-Shahidi construction of L-functions carries over to the metaplectic group. But it is not clear how to extend the Langlands-Shahidi method to other covering groups, where uniqueness of Whittaker models often fails. The thesis work of Gao Fan \cite{GaoThesis} takes some promising steps in this direction. The general machinery of adjoint L-functions also suggests an analogue, for covering groups, of the Hiraga-Ichino-Ikeda conjecture \cite[Conjecture 1.4]{HII} on formal degrees (see Ichino-Lapid-Mao \cite{IchinoLapidMao}). It is also supported by the simpler observation that theta correspondence for the metaplectic group $\widetilde{Sp}_{2n}$ provides a definition of adjoint L-functions independently of choices of additive characters.
\end{remark}
\subsection{The L-group of a cover}
Now we define the L-group of a cover. Let $\alg{\tilde G}$ be a degree $n$ cover of a quasisplit group $\alg{G}$ over $S$. Fix an injective character $\epsilon \colon \mu_n \hookrightarrow \mathbb C^\times$. Choose a separable closure $\bar F / F$, yielding a geometric base point $\bar s \rightarrow S$ and the absolute Galois group $\Gal_S = \pi_1^{\et}(S, \bar s)$.
Recall the constructions of the previous three sections.
\begin{itemize}
\item
$\dgp{\tilde G^\vee}$ denotes the dual group of $\alg{\tilde G}$, a local system on $S_{\et}$ of pinned reductive groups over $\mathbb Z$, with center $\dgp{\tilde Z^\vee}$. It is endowed with a homomorphism $\tau_Q \colon \sch{\mu}_2 \rightarrow \dgp{\tilde Z^\vee}$.
\item
$\gerb{E}_\epsilon(\alg{\tilde G})$ is the gerbe associated to $\alg{\tilde G}$, a gerbe on $S_{\et}$ banded by $\sheaf{\tilde Z^\vee} = \dgp{\tilde Z^\vee}(\mathbb C)$.
\item
$\mu_2 \hookrightarrow \widetilde{\Gal}_S \twoheadrightarrow \Gal_S$ is the metaGalois group.
\end{itemize}
Define $\tilde Z^\vee = \sheaf{\tilde Z}_{\bar s}^ { \raisemath{-3pt}{\vee} } = \dgp{\tilde Z}_{\bar s}^ { \raisemath{-3pt}{\vee} } (\mathbb C)$. This is the center of the complex reductive group $\tilde G^\vee = \sheaf{\tilde G}_{\bar s}^ { \raisemath{-3pt}{\vee} } = \dgp{\tilde G}_{\bar s}^ { \raisemath{-3pt}{\vee} } (\mathbb C)$.
Pushing out $\widetilde{\Gal}_S$ via $\tau_Q \colon \mu_2 \rightarrow \tilde Z^\vee$ defines an L-group,
\begin{equation}
\label{Twist1}
\tilde Z^\vee \hookrightarrow (\tau_Q)_\ast \widetilde{\Gal}_S \twoheadrightarrow \Gal_S.
\end{equation}
From Theorem \ref{AppxFundGpDefined}, the fundamental group of the gerbe $\gerb{E}_\epsilon(\alg{\tilde G})$ is an L-group, well-defined up to L-equivalence, and the L-equivalence well-defined up to unique natural isomorphism,
\begin{equation}
\label{Twist2}
\tilde Z^\vee \hookrightarrow \pi_1^{\et}(\gerb{E}_\epsilon(\alg{\tilde G}), \bar s) \twoheadrightarrow \Gal_S.
\end{equation}
\begin{remark}
The extensions \eqref{Twist1} and \eqref{Twist2} play the role of the first and second twist in \cite{MWCrelle}. In fact \eqref{Twist1} is canonically isomorphic to the first twist in the split case; the extension \eqref{Twist2} may not coincide with the second twist under some circumstances, and the construction here is more general than \cite{MWCrelle} in both cases.
\end{remark}
The Baer sum of \eqref{Twist1} and \eqref{Twist2} is an L-group which will be called ${}^{\mathsf L} \tilde Z$,
$$\tilde Z^\vee \hookrightarrow {}^{\mathsf L} \tilde Z \twoheadrightarrow \Gal_S.$$
\defined{The L-group} of $\alg{\tilde G}$ is defined to be the pushout of ${}^{\mathsf L} \tilde Z$ via the inclusion $\tilde Z^\vee \hookrightarrow \tilde G^\vee$,
$$\tilde G^\vee \hookrightarrow {}^{\mathsf L} \tilde G \twoheadrightarrow \Gal_S.$$
More explicitly, this pushout is in the $\Gal_S$-equivariant sense. In other words,
$${}^{\mathsf L} \tilde G = \frac{\tilde G^\vee \rtimes {}^{\mathsf L} \tilde Z}{ \langle (z, z^{-1}) : z \in \tilde Z^\vee \rangle },$$
where the semidirect product action ${}^{\mathsf L} \tilde Z \rightarrow \Aut(\tilde G^\vee)$ is given by projection ${}^{\mathsf L} \tilde Z \rightarrow \Gal_S$ followed by the action $\Gal_S \rightarrow \Aut(\tilde G^\vee)$ (the Galois group acts by pinned automorphisms on the dual group).
By construction, ${}^{\mathsf L} G$ is well-defined by $\alg{\tilde G}$ and $\epsilon$ up to L-equivalence, and the equivalence well-defined up to unique natural isomorphism.
\subsubsection{Local-global compatibility}
Suppose that $\gamma \colon F \hookrightarrow F_v$ is the inclusion of a global field $F$ into its localization at a place. Let $\alg{\tilde G}$ be a degree $n$ cover of a quasisplit group $\alg{G}$ over $F$, and let $\epsilon \colon \mu_n \hookrightarrow \mathbb C^\times$ be an injective character. Let $\bar F \hookrightarrow \bar F_v$ be an inclusion of separable closures, inducing an inclusion $\Gal_{F_v} \hookrightarrow \Gal_F$ of absolute Galois groups. Write $S = \Spec(F)$ and $S_v = \Spec(F_v)$, and $\bar s \rightarrow S$ and $\bar s_v \rightarrow S_v$ for the geometric base points arising from $\bar F \hookrightarrow \bar F_v$.
Write $\alg{\tilde G}_v$ for the pullback of $\alg{\tilde G}$ via $\Spec(F_v) \rightarrow \Spec(F)$. Similarly, write $Q_v$ for its Brylinski-Deligne invariant. The results of Section \ref{BaseChangeDualGroup} identify $\tilde G^\vee = \dgp{\tilde G}_{\bar s}^ { \raisemath{-3pt}{\vee} } (\mathbb C)$ with the corresponding dual group for $\alg{\tilde G}_v$ (relative to the separable closure $\bar F_v$). Thus we simply write $\tilde G^\vee$ for their dual groups and $\tilde Z^\vee$ for the centers thereof.
The results of Sections \ref{BaseChangeDualGroup} and \ref{CompatibilityMetaGalois} together provide an L-morphism, unique up to unique natural isomorphism.
\begin{equation}
\label{LGTwist1}
\begin{tikzcd}
\tilde Z^\vee \arrow[hook]{r} \arrow{d}{=} & (\tau_{Q_v})_\ast {}^{\mathsf L} \widetilde{\Gal}_{F_v} \arrow[two heads]{r} \arrow[hook]{d} & \Gal_{F_v} \arrow[hook]{d} \\
\tilde Z^\vee \arrow[hook]{r} & (\tau_Q)_\ast {}^{\mathsf L} \widetilde{\Gal}_F \arrow[two heads]{r} & \Gal_F
\end{tikzcd}
\end{equation}
The results of Section \ref{BaseChangeGerbe} and following Theorem \ref{AppxFundGpDefined} give an L-morphism, unique up to unique natural isomorphism.
\begin{equation}
\label{LGTwist2}
\begin{tikzcd}
\tilde Z^\vee \arrow[hook]{r} \arrow{d}{=} & \pi_1^{\et}(\gerb{E}_\epsilon(\alg{\tilde G}_v), \bar s_v) \arrow[two heads]{r} \arrow[hook]{d} & \Gal_{F_v} \arrow[hook]{d} \\
\tilde Z^\vee \arrow[hook]{r} & \pi_1^{\et}(\gerb{E}_\epsilon(\alg{\tilde G}), \bar s) \arrow[two heads]{r} & \Gal_F
\end{tikzcd}
\end{equation}
Applying the Baer sum to \eqref{LGTwist1} and \eqref{LGTwist2}, and pushing out via $\tilde Z^\vee \hookrightarrow \tilde G^\vee$, yields an L-morphism, unique up to unique natural isomorphism.
$$\begin{tikzcd}
\tilde G^\vee \arrow[hook]{r} \arrow{d}{=} & {}^{\mathsf L} \tilde G_v \arrow[two heads]{r} \arrow[hook]{d} & \Gal_{F_v} \arrow[hook]{d} \\
\tilde G^\vee \arrow[hook]{r} & {}^{\mathsf L} \tilde G \arrow[two heads]{r} & \Gal_F
\end{tikzcd}$$
\subsubsection{Parabolic subgroups}
Return to a degree $n$ cover $\alg{\tilde G}$ of a quasisplit group $\alg{G}$ over $S$, and let $\alg{P} \subset \alg{G}$ be a parabolic subgroup defined over $S$. As before, consider a Levi decomposition $\alg{P} = \alg{M} \alg{N}$ and the resulting cover $\alg{\tilde M}$. Fix $\epsilon \colon \mu_n \hookrightarrow \mathbb C^\times$.
Compatibility of the dual groups from Section \ref{ParabolicDualGroup} gives inclusions
$$ \tilde Z^\vee \hookrightarrow \tilde Z_M^\vee \hookrightarrow \tilde M^\vee \hookrightarrow \tilde G^\vee,$$
where $\tilde Z^\vee$ denotes the center of $\tilde G^\vee$, and $\tilde Z_M^\vee$ denotes the center of $\tilde M^\vee$. As these inclusions are compatible with the 2-torsion elements in $\tilde Z^\vee$ and $\tilde Z_M^\vee$, we find an L-morphism.
\begin{equation}
\label{PSTwist1}
\begin{tikzcd}
\tilde Z^\vee \arrow[hook]{r} \arrow[hook]{d} & (\tau_Q)_\ast {}^{\mathsf L} \widetilde{\Gal}_F \arrow[two heads]{r} \arrow[hook]{d} & \Gal_S \arrow{d}{=} \\
\tilde Z_M^\vee \arrow[hook]{r} & (\tau_{Q_{\alg{M}}})_\ast {}^{\mathsf L} \widetilde{\Gal}_F \arrow[two heads]{r} & \Gal_S \\
\end{tikzcd}
\end{equation}
Section \ref{ParabolicGerbe} provided a functor of gerbes $\gerb{i} \colon \gerb{E}_\epsilon(\alg{\tilde G}) \rightarrow \gerb{E}_\epsilon(\alg{\tilde M})$, lying over $\sheaf{\tilde Z}^ { \raisemath{-3pt}{\vee} } \hookrightarrow \sheaf{\tilde Z}_{\alg{M}}^ { \raisemath{-3pt}{\vee} } $. The \'etale fundamental groups give an L-morphism.
\begin{equation}
\label{PSTwist2}
\begin{tikzcd}
\tilde Z^\vee \arrow[hook]{r} \arrow[hook]{d} & \pi_1^{\et}(\gerb{E}_\epsilon(\alg{\tilde G}), \bar s) \arrow[two heads]{r} \arrow[hook]{d} & \Gal_S \arrow{d}{=} \\
\tilde Z_M^\vee \arrow[hook]{r} & \pi_1^{\et}(\gerb{E}_\epsilon(\alg{\tilde M}), \bar s) \arrow[two heads]{r} & \Gal_S \\
\end{tikzcd}
\end{equation}
Applying the Baer sum to \eqref{PSTwist1} and \eqref{PSTwist2} yields an L-morphism, unique up to unique natural isomorphism.
$$\begin{tikzcd}
\tilde Z^\vee \arrow[hook]{r} \arrow[hook]{d} & {}^{\mathsf L} \tilde Z \arrow[two heads]{r} \arrow[hook]{d} & \Gal_S \arrow{d}{=} \\
\tilde Z_M^\vee \arrow[hook]{r} & {}^{\mathsf L} \tilde Z_M \arrow[two heads]{r} & \Gal_S \\
\end{tikzcd}$$
The universal property of pushouts yields an L-morphism.
$$\begin{tikzcd}
\tilde M^\vee \arrow[hook]{r} \arrow[hook]{d}& {}^{\mathsf L} \tilde M \arrow[two heads]{r} \arrow[hook]{d} & \Gal_S \arrow{d}{=} \\
\tilde G^\vee \arrow[hook]{r} & {}^{\mathsf L} \tilde G \arrow[two heads]{r} & \Gal_S
\end{tikzcd}$$
This L-morphism is well-defined up to conjugation by $\tilde Z_M^\vee$.
|
1,116,691,500,880 | arxiv | \section{Introduction}
One of the main weaknesses of stellar physics remains our poor description
of thermal convection. It is true that the
widely used mixing-length treatment permits to
construct models which represent fairly well the gross properties of stars,
but it fails when
one attempts to apply it to more subtle processes, such as convective
penetration, the differential rotation of the
Sun, or its magnetic activity.
The situation is rapidly changing however. Over the two past decades significant
progress has been achieved through numerical simulations of
increasingly `turbulent' convection in a stratified medium.
These have shown that fully compressible
convection is highly intermittent, displaying strong, long-lived,
downwards directed flows, which contrast with the slower, random upward motions
(Hurlburt et al. 1986; Cattaneo et al. 1991;
Nordlund et al. 1992; Muthsam et al. 1995; Singh et al. 1995; Brummel et al. 1996;
Stein \& Nordlund 1998). These coherent structures are called {\it plumes}
(`panaches' in french), by analogy with those observed in the Earth atmosphere. They
originate in the upper boundary layer, where they are initiated by the strong
temperature and density fluctuations, which arise there in the steep superadiabatic
gradient.
True, these numerical results are still
difficult to apply as such to the
convection zone of a star, because of
the huge gap in the relevant control parameters,
the Reynolds and the Prandtl numbers.
But they suggest that
coherent structures may play an important role in the dynamics of a
turbulent convective layer, and thus open the possibility of a different
description of stellar convection.
And indeed these plumes have now been detected in the upper layers of the solar
convection zone, thanks to time-distance tomography. This powerful helioseismic
technique reveals the presence of downdrafts apparently associated with the
supergranular network, in which the temperature is strongly correlated with the
vertical velocity (D'Silva et al. 1996; Duvall et al. 1997). Fig. \ref{tomofig}
shows an example of such tomographies; it can been downloaded from the SOHO gallery
on the Web.
\begin{figure}
\centerline{\psfig{figure=tomofig.ps,width=9cm}}
\caption{\small Time-distance tomography with the MDI instrument on board of~SOHO;
the field shown measures 8 Mm in depth and 150 Mm in width.
(http://sohowww.nascom.nasa.gov/gallery/MDI/mdi009.gif)}
\label{tomofig}
\end{figure}
Turbulent plumes are commonly
observed in the Earth atmosphere, where they rise above concentrated heat sources
(chimneys, nuclear plants,
volcanos, etc.). Their theoretical interpretation is still based on the entrainment
hypothesis,
which was first proposed by G.I. Taylor (cf. Morton et al. 1956); we recall it
below. In astrophysics, plumes have been invoked by Schmitt et
al. (1984) as responsible for the penetration into the stable domain
beneath the solar convection zone. Somewhat later an estimate of that
penetration has been given by Zahn (1991), which is also
based on a crude plume model.
More recently, Rieutord and Zahn (1995) moved a step forwards, by letting the plumes
develop through the whole convection zone. They adapted to a medium which
is highly stratified, in density and temperature, the
treatment which is used in geophysical fluid dynamics. Furthermore,
they took into account the backflow which is induced by the confinement
of the plumes in a limited volume, and which determines the depth the plumes may
reach. This review recalls the main properties of that model; it will also address
the role the plumes may play in convective penetration and in the generation of gravity waves.
\section{One single plume in an isentropic stratification}
We shall first examine the basic properties of a turbulent plume
which resides in an isentropic, plane-parallel atmosphere, assuming
that the plume is fed steadily by a cold layer of fluid lying on the top
of the atmosphere.
A schematic view
of the flow in such a plume is given in fig.~\ref{plume}.
\begin{figure}
\centerline{\psfig{figure=plumefig.ps,width=6cm}}
\caption{\small The plume flow in an adiabatic atmosphere
(after Rieutord \& Zahn 1995; courtesy A\&A).}
\label{plumfig}
\end{figure}
\subsection{Governing equations}
The equations governing the structure of turbulent plumes have been
established in the original work of Morton et al. (1956).
One assumes that the flow is stationary, that the plumes are axisymmetric about
the vertical, and that all horizontal variations,
namely those of the vertical velocity $v_z$,
of the density excess $\delta\rho$ and of the excess of specific enthalpy
$\delta h$, have the same gaussian profile:
\begin{equation}\left\{ \begin{array}{l}
\disp{v_z(r,z) = V(z)\exp(-s^2/b^2)}\\
\disp{\delta\rho(r,z) = \Delta\rho(z)\exp(-s^2/b^2)}\\
\disp{\delta h(r,z) = \Delta h(z)\exp(-s^2/b^2)}
\end{array}\right. \end{equation}
where $b\equiv b(z)$ is the effective radius of the plume. For
convenience, we take the vertical coordinate $z$ pointing downward;
the density
and the specific enthalpy of the isentropic atmosphere will be designated by
$\rho_0 (z)$ and $h_0(z)$.
The equations describing the vertical profile of the plume are derived
from the three basic equations of fluid mechanics expressing
the conservation of mass, momentum and
energy.
\subsubsection{Conservation of mass.}
In the steady flow of the plume, the mass conservation implies
\[ \hbox{div}(\rho\vec{v}) = \frac{1}{s}{\partial \over \partial s} \left({s\rho
v_s} \right) + {\partial \over \partial z} \left({\rho v_z} \right) =0 \ ; \]
when integrated over $s$, this equation becomes
\begin{equation} \dnz{}\int_0^{+\infty}\!\rho v_z sds\; +\; [s\rho v_s]_0^{+\infty} = 0 .
\label{mass} \eeq
The entrainment hypothesis made by G.I. Taylor postulates that the radial
inflow of matter into the plume is proportional to the central vertical velocity:
\begin{equation} \lim_{s \rightarrow +\infty} sv_s = -\alpha b(z)|v_z(0,z)|
\label{tayl}\eeq
where $\alpha$ is the entrainment constant. The absolute value
guarantees that the plume is always accreting matter, whether it is directed
upwards or downwards.
Using the gaussian profile of $v_z$ and
$\delta\rho$, one casts the mass equation in its final form
\begin{equation}
\dnz {}\left[ \rho_0b^2 V \right] =
2\alpha\rho_0 bV \, ,
\label{masse1}
\eeq
in the limit of vanishing density contrast $\Delta\rho/\rho_0 \ll 1$.
\subsubsection{Conservation of momentum.}
Ignoring the viscous stresses, the flow obeys
the steady Euler equation:
\[ \partial_j(\rho v_jv_i + \delta p\, \delta_{ij}) =
\delta\!\rho\, g_i \, , \]
\noindent where the static equilibrium has been subtracted. We assume that
the plume is in pressure
balance with the surrounding medium; it leads us to
\[ \frac{1}{s}{\partial \over \partial s} \left({s\rho v_s v_z} \right) +
{\partial \over \partial z} \left({\rho v_z^2} \right) = \delta\!\rho\, g \]
($g$ is the gravity, which we assume constant). After integration as before, we get
\begin{equation}
\dnz{}\left[ \rho_0b^2 V^2\right] = 2gb^2\Delta\rho \, .
\label{moment1}
\eeq
\subsubsection{Conservation of energy.}
We start from the steady energy equation
\[ \hbox{div}(\rho e\vec{v})=\hbox{div}(\chi\vec{\nabla} T) - p\hbox{div}\vec{v} \ ,\]
\noindent where $e$ is the specific internal energy and $\chi$ the radiative
conductivity, and rewrite it by
using the momentum equation and the definition of the specific enthalpy
$dh \equiv de + d(p/\rho)$:
\[ \hbox{div} \left[ (h+{\frac{1}{2}} v^2)\rho\vec{v} - \chi\vec{\nabla} T \right] =
\rho\vec{v}\cdot{\vec{g}} . \]
This equation may be further simplified by remembering that
\[ {\vec{g}} = \vec{\nabla} h_0 \]
\noindent in an isentropic atmosphere (see below).
Finally, the conservation of the energy flux is expressed in conservative form:
\begin{equation} \hbox{div} \left[ \left( \delta h+{\frac{1}{2}} v^2 \right) \rho\vec{v} -
\chi\vec{\nabla} T \right] = 0 ,\eeq
where we recognize respectively the enthalpy, kinetic and radiative
fluxes.
To make contact with previous work, we first neglect the radiative flux.
Proceeding as before, we
integrate in the radial direction, and obtain
\begin{equation} {\frac{1}{2}} \rho_0b^2 V\Delta h +
\frac{1}{6} \rho_0b^2 V^3 = -F/\pi \ ,
\eeqn{ener}
where $F$ designates the total flux (enthalpy plus kinetic) carried
by one plume.
From now on, we shall drop the subscript $_0$, since there will be no ambiguity
anymore between the local values of density and temperature, and their
counterparts in the isentropic atmosphere.
\subsubsection{Thermodynamics.}
For simplicity, we shall assume that the fluid is
a perfect
monatomic gas, with $\gamma = 5/3$ being the adiabatic exponent. We thus
complete the system of governing equations with the two thermodynamic
relations:
\[ \Delta h = c_p \Delta T \]
\[ {\Delta \rho \over \rho} + {\Delta T \over T} = 0 \]
\noindent where the latter expresses the pressure balance between the
plume and the surrounding medium ($c_p$ is the heat capacity at constant pressure).
\subsubsection{The isentropic atmosphere.}
As is well known, the isentropic atmosphere is a polytrope of
index $q=1/(\gamma - 1)$. In plane-parallel geometry,
the density and the temperature vary with depth $z$ as:
\begin{equation}\left\{ \begin{array}{l}
\disp{\rho (z) = \rho_{i}(z/z_i)^q} \\
\\
\disp{T(z) = T_{i}(z/z_i) }.
\egreqn{poly}
The origin of the vertical coordinate $z$ is taken at the `surface', where
pressure, density and temperature all vanish, and the
subscript $_{i}$ designates the reference level for which we choose here the base
of the atmosphere. The reference temperature $T_{i}$ is related to the
reference depth by $z_i = c_p T_{i}/ g$.
\subsection{Asymptotic behavior}
The equations governing the plume flow are nonlinear and in most cases
it is necessary to
integrate them numerically, with boundary conditions imposed at the start of
the plume, for instance on $\Delta \rho$, $b$ and $V$. However, the system
has asymptotic solutions which can easily be obtained in analytic form.
The one which presents most interest corresponds to
the developed phase of the downwards directed plume, when it is fully controlled
by entrainment.
The solution then obeys the power laws
\begin{equation} b = b_i \left( {z / z_i} \right)^p, \quad V = V_i \left( {z / z_i} \right)^r,
\quad \Delta \rho = \Delta \rho_i \left( {z / z_i} \right) ^m \label{asreg}\eeq
where the exponents take the values:
\begin{equation}\left\{ \begin{array}{l}
p = 1 \\
\\
\disp{r = -\frac{q+2}{3} = - \frac {2 \gamma -1} {3 (\gamma - 1)} }\\
\\
\disp{m = -\frac{2q+7}{3} = - \frac {7 \gamma - 5} {3 (\gamma - 1)} \, .}
\end{array}\right. \end{equation}
A direct consequence of Taylor's
entrainment hypothesis (\ref{tayl}) is that the plume
radius increases linearly with depth. The half-angle $\beta=$atan$(b_i/z_i)$ of
the cone is related to the adiabatic exponent:
\begin{equation}
\tan \beta = \frac{3\alpha}{q+2} = \frac{3\alpha (\gamma -1)}{2\gamma -1}
\, . \label{beta}
\eeq
With $\gamma = 5/3$ (perfect monatomic gas), we have $\tan \beta = 6\alpha/7$,
whereas $\tan \beta = 6\alpha/5$ in the absence of stratification. The
entrainment coefficient $\alpha$ has been determined experimentally in various
situations, and the value 0.083 is widely adopted (Turner 1986); we shall assume
that this value also applies to our fully ionized gas.
For given initial conditions imposed at the top of the plume, the solutions
rapidly reach their asymptotic regime: the plumes start contracting while they
accelerate, and then they take their characteristic conical shape (cf. Rieutord
\& Zahn 1995).
Note that one does not retrieve the
incompressible case by letting the polytropic index tend to zero,
$q \rightarrow 0$. The
reason for this singular limit is the assumption made in the classical
treatment (Morton et al. 1956) that the surrounding medium is isothermal,
while in our stratified atmosphere the temperature grows linearly with depth.
Let us emphasize however that
our asymptotic solution (\ref{asreg}) is rather special, since
its focal point $z = 0$ coincides with the `surface' of the
atmosphere.
\subsection{Turbulent plumes carry upwards a net energy flux}
An interesting property of this asymptotic regime is the strict
proportionality between the flux of kinetic energy and that of enthalpy:
\begin{equation} R = \frac{\disp{\int {\frac{1}{2}}\rho V^3 sds}}
{\disp{\int\rho c_p \, \delta T \, V sds}}
= cst . \eeq
With the gaussian profile we have adopted for the plume, this ratio
is given by
\begin{equation} R = - \frac{2}{q+2} = - \frac{2\gamma-2}{2\gamma -1} \; . \eeq
Its value is 1 for the Boussinesq fluid (no density stratification)
but it is 4/7 in the isentropic atmosphere (for the perfect monatomic gas), meaning
that the net energy flux is directed upwards.
In their numerical experiment Cattaneo et al. (1991) observed to
their surprise that the strong downdrafts transported a flux of kinetic energy
which very nearly canceled the flux of enthalpy. The reason is that their
plumes were not turbulent, but laminar flows which satisfied approximately
Bernoulli's theorem. In recent simulations, which have been achieved with better
spatial resolution, there is more entrainment, and the plumes do carry a net
energy flux upwards.
\section{Towards a more realistic model; application to the solar convection zone}
Up to now, we have considered the case of a single plume
in an unlimited isentropic plane layer. To progress towards a more realistic
situation, we must take into
account the spherical geometry of the fluid layer and include the variations of
gravity with depth.
Here we shall apply the model to the solar convection zone (SCZ); the convective
core with its rising plumes has been treated by Lo and Schatzman (1997).
We neglect the mass of the envelope, so that the
density and temperature field of the background are given by
\begin{equation}\left\{ \begin{array}{l}
\disp{T(z) = T_i \; \frac{\eta}{1-\eta} \left( \frac{1}{x}-1 \right)}\\
\\
\disp{\rho(z) = \rho_i \left({T(z)\over T_i} \right)^q} \\
\\
\disp{x = 1 - z}
\egreqn{atmos2}
where $x=r/R_\odot$ is the radial coordinate scaled by the
radius, and $\eta=r_{\rm cz}/R_\odot$ its value at the base of the SCZ (our reference
level).
We shall also take into account the
radiative flux in the energy equation, thus (\ref{ener})
now reads
\begin{equation} {\frac{1}{2}} \rho b^2 V\Delta h + \frac{1}{6}\rho b^2 V^3
= -(F_{\rm tot}- F_{\rm rad})/\pi \ . \label{ener2}
\eeq
With our assumption that all the
convective energy is transported by the plumes, whose number is $N$, the total
flux amounts to the luminosity of the star:
\begin{equation} N \, F_{\rm tot} = L_\odot . \eeq
The radiative contribution to the flux carried by one plume is $F_{\rm rad}$, where
\[ N \, F_{\rm rad} = L_{\rm rad} =
- 4 \pi r^2 \, \chi \left( \dz{T} \right)_{\rm ad}, \quad \chi= \frac{16\sigma
T^3}{3 \kappa \rho}\, , \]
with $\chi$ being the radiative
conductivity, $\kappa$ the opacity, and $\sigma$ the Stefan constant.
\subsection{Plumes diving in a rising counterflow}
We shall now take into account that there
are numerous plumes and that they share a finite volume. This
has two consequences: the plumes will interact and possibly merge, and they
will move in a rising counterflow.
Let us first estimate the
maximum number of plumes which may coexist in the SCZ, assuming that they
all originate at the top and that all reach its bottom without being hindered by the
counterflow. This number is given by the ratio of the area of the base of the SCZ to
the section of a single plume at this depth:
\[ N = 4 \left[{\eta \over (1-\eta) \tan \beta }\right]^2 \]
where $\beta$ is given by (\ref{beta}), and $\eta = r_{\rm cz}/R_\odot$ is the
scaled radius of the base of the SCZ. Taking $\eta = 0.7$, the result is
$ N_{\rm max} \sim 4300$.
We see that even if the plumes were closely packed, they would not be very
numerous. But they will be actually less, because as soon as their occupy more
than half of the total area, at depth $z_i/ \sqrt{2}$, the velocity of
the backflow exceeds that of the plumes, and this will affect their dynamics.
To model the effect of this counterflow,
we shall assume that the entrainment of mass into the
plume is proportional to the relative velocity of the
plume with respect to the surrounding fluid. This yields the
mass conservation equation
\begin{equation}
\dnz{}\left[ \rho b^2 V\right] = 2\alpha \,\rho b(V-V_u)
\label{masse2}\eeq
where $V_u$ is the upward velocity of the surrounding fluid; since
$V_u<0$, entrainment is enhanced. The momentum equation needs
also be completed by a term taking into account the entrainment of
momentum into the plume. We thus transform (\ref{moment1}) into
\begin{equation}
\dnz{}\left[ \rho b^2 V^2\right] = 2gb^2\Delta\rho
+ 4 \alpha \, \rho b V_u(V-V_u) \, .
\label{moment2}
\eeq
Equations (\ref{masse2}) and (\ref{moment2}) are completed by
\begin{equation} (4\pi r^2 -N\pi b^2)\rho V_u + N\pi b^2\rho V = 0 \label{flux0}\eeq
which expresses the global conservation of mass.
The consequence of these additional effects is that plumes are able to reach
the bottom of the convection zone only if they are strong enough, and not
too numerous. From the numerical integration of these equations
it turns out that only about $1000$
plumes are able to reach the bottom of the SCZ (see Rieutord \& Zahn 1995).
\subsection{Plume coalescence}
At the top of the SCZ, the number of
plumes is probably of the same order as that of the granules, which is
considerably larger than 1000. Hence a drastic
reduction of the plumes number must occur as depth increases.
The plumes are stopped at a level
which depends on their strength: only the 1000 strongest are able to reach the
bottom. The others will either dissolve in the
(turbulent) backflow and disappear as such, or they will merge with a stronger
companion.
Such merging is observed in the numerical simulations
(Stein \& Nordlund 1989; Spruit et al. 1990):
two neighboring plumes are advected by each other, since they both
accrete surrounding
fluid, and they are pulled to each other until they finally coalesce.
The merging depth
can be estimated by assuming the
flow be in the asymptotic regime; the result is
\begin{equation} z_m = \frac{d}{\sqrt{2\alpha\beta}} \eeqn{mdepth}
where $d$ is the initial separation of the two plumes, which we
assumed here to be identical.
Using the value of $\beta$ for the monatomic ideal gas and
$\alpha=0.083$, we get
$ z_m \sim 9.2 \, d \ .$
Similar coalescences will then repeat at increasing depths, until the survivors
reach the base of the convection zone, in a sort of inverse cascade
with a large scale flow building up from smaller scales.
\section{Penetration at the base of a convection zone}
When the plumes reach the bottom of the unstable domain, they still possess
a finite velocity which enables them to penetrate some distance
into the stable, subadiabatic region, where they establish a nearly
adiabatic stratification by releasing their entropy when they come to rest.
A first attempt to estimate the extent of penetration of such plumes
was made by Schmitt et al. (1984).
However, they did not include the return flow,
and they had to impose both the velocity $V$ and the filling factor $f$ at
the base of the unstable domain because they did not solve the plume equations
above that level. They found empirically that the penetration depth varies as
$f^{1/2} V^{3/2}$, and this scaling can be easily explained.
\begin{figure}
\centerline{\psfig{figure=penfig.ps,width=8cm}}
\caption{\small Schematic structure of the base of a convective envelope
(after Zahn 1991; courtesy A\&A).}
\label{penfig}
\end{figure}
\subsection{An estimate of the penetration penetration depth}
The stratification at the base of the a convective envelope is sketched in
fig. (\ref{penfig}). (A) designates the unstable region, where the
temperature gradient is maintained close to adiabatic by the convective
motions. One thus assumes that the radiative leaks are negligible compared to the
advective transport of heat, which is the case when the P\'eclet number is much larger
than unity:
\begin{equation}
Pec = {v \ell \over K} \gg 1, \quad K = {\chi \over \rho c_p},
\eeq
where $K$ is the thermal diffusivity and $\chi$ the radiative conductivity.
$v$ and $\ell$ are the velocity and size characterizing the convective motions,
i.e. here the central velocity $V$ and the width $b$ of our plumes. At the base
of the SCZ, $Pec \approx 10^6$.
Due to the steady increase
with depth of the radiative conductivity $\chi$, the radiative flux
$F_{\rm rad}=\chi (dT/dz)_{\rm ad}$ rises until it equals the total flux
$F_{\rm total}$ at the level $z=z_i$, where also the radiative gradient equals the
adiabatic gradient: $(dT/dz)_{\rm rad} = (dT/dz)_{\rm ad}$. If there were
no convective penetration, this would be the edge of the convection zone, as predicted
by the Schwarzschild criterion, and the temperature gradient would thereafter decrease
as $dT/dz = F_{\rm total}/\chi$.
But the motions penetrate into the stable region (B) and they
render it nearly adiabatic over some distance $d_{\rm pen}$, while being decelerated by
the buoyancy force. When the P\'eclet number has dropped below unity, the temperature
gradient settles from adiabatic to radiative in a thermal boundary layer (C).
Since the penetration depth is rather small, one may
simplify the problem by ignoring the variation with depth of most quantities
(density, width of the plume, etc.), and by neglecting the kinetic energy flux
and the turbulent entrainment (or detrainment). One keeps of course the
variation with $z$ of the conductivity; in the vicinity of $z_i$,
the radiative flux is approximated by
\begin{equation}
F_{\rm rad} = \chi \left({dT \over dz} \right)_{\rm ad} = F_{\rm total} \left[ 1 + \left({d \chi
\over dz} \right)_i \left( z-z_i \right) \right] \, .
\label{fluxrad}
\eeq
Therefore the convective flux varies as
\begin{equation}
F_{\rm conv} = - F_{\rm total} \left({d \chi \over dz} \right)_i
\left( z-z_i \right) \, .
\label{fluxconv}
\eeq
This enthalpy flux may be expressed in terms of the vertical velocity and of
the temperature contrast in the plumes, as in (\ref{ener}):
\begin{equation}
F_{\rm conv} = - f \rho c_p V \Delta T \, ,
\label{fluxconv1}
\eeq
where we have introduced the filling factor $f$ of the plumes, defined as the
fractional area covered by them: $f = N b^2 /R^2_{\rm cz}$.
To estimate the penetration depth, we follow the plumes from $z=z_i$, where
their velocity is $V_i$, until they stop at
$z = z_i + d_{\rm pen}$. Their deceleration is described to first order by
\begin{equation}
{1 \over 2} {d V^2 \over dz} = g {\Delta \rho \over \rho} =
- g {\Delta T \over T} .
\label{decel}
\eeq
After elimination of $\Delta T$ with (\ref{fluxconv1}), the integration of
(\ref{decel}) yields the following expression for the penetration depth
$d_{\rm pen}$:
\begin{equation}
d_{\rm pen}^2 = {3 \over 5} H_p H_\chi \left\{f {\rho V_i^3 \over F_{\rm total}}
\right\} = {3 \over 5} H_p H_\chi Q \,
, \label{pendepth}
\eeq
where $H_p$ is the scale-height of the pressure and $H_\chi$ that of the
radiative conductivity. This is the scaling obtained empirically by Schmitt et
al. (1984).
At that point one may argue that the term $Q$ in curly brackets is determined by
the dynamics in the convection zone, and that it will not depend much on its
depth, provided this depth is large enough and that convection is sufficiently
adiabatic. $Q$ is probably smaller than unity: in the mixing length treatment,
$Q = (1/10) \, \Lambda / H_p$ in the bulk of the convection zone. This
figure, with $\Lambda / H_p \approx 1.5$ and $H_\chi \approx H_p/2$, would yield a
penetration depth of the order of $1/5$ of the pressure scale-height at the base of
the solar convection zone.
\subsection{Penetration of plumes originating at the top of the convection zone}
Rieutord and Zahn (1995) computed the penetration in a more consistent way,
by integrating the governing equations from the top of
the convection zone until the plumes come
to rest. When neglecting the backflow, the penetration extends to about one half
of the pressure scale-height, independent on the number of plumes. But the picture
changes drastically when one includes the interaction of the plumes with the
backflow. Then the penetration depth depends on the number of plumes: it decreases
almost linearly with $N$, from one pressure scale-height for $N \rightarrow 0$ to $0$
for $N \approx 1200$. The reason for this trend is that the backflow increases with
the number of plumes, and that it slows down their motion by loading them with
upward momentum.
Unfortunately, this model is unable to predict the number of plumes. Moreover, a much
more sophisticated model is needed to describe the termination of the plume flow,
which would include the loss of mass by the plume in this region, or
`detrainment', and its transfer to the upward motion.
\subsection{Penetration vs. overshooting}
What we have called convective {\it penetration} is that part of the excursion of
convective motions into the stable region below where they enforce an almost
adiabatic stratification ($\nabla_{\rm ad} - \nabla \ll 1$). It corresponds to
region (B) in fig. \ref{penfig}, in which the P\'eclet number is substantially
larger than unity (we recall that at the base of the SCZ, $Pec \approx 10^6$).
In comparison, the thermal adjustment layer (C) is very thin: about 1 km in the
Sun, assuming that all plumes have the same strength (Zahn 1991).
The situation is very different in stars which possess a shallow convection zone.
For instance, at the base of the convection zone of an A-type star, $Pec \approx
1/100$, which means that the convective motions which enter the stable layer below
rapidly settle in thermal equilibrium with their surroundings, and that
they hardly feel the buoyancy force (Toomre et al. 1976; Freytag et al. 1996).
The region (B) has disappeared -- all what remains is region (C). To insist on the
contrast with the almost adiabatic penetration at high P\'eclet number, we prefer to
call this convective {\it overshoot} (or undershoot).
This is not only semantic matter. To take one example, numerical simulations which
are performed to represent A-type stars cannot be used to predict the amount of
penetration below the solar convection zone, as was attempted by Bl\"ocker et al.
(1998). In fact, for lack of spatial resolution, present day simulations are unable
to achieve a P\'eclet number which would realistically describe convective
penetration; for instance, in the calculations made by Hurlburt et al. (1994),
regions (B) and (C) are of comparable size, and a substantial fraction of the
extension into the stable region is due to overshoot.
\subsection{Generation of internal gravity waves}
In the laboratory, convective penetration is observed to generate internal
waves, which propagate in the stable adjacent layer (Townsend 1966; Adrian 1975). the
same is also expected at the boundary of stellar convection zones. Such waves have
indeed been observed in numerical simulations of penetrative convection,
where they seem to be produced by the pronounced downdrafts (Hurlburt et al.
1986; Hurlburt et al. 1994), which are the 2-dimensional analogues of
3-dimensional plumes. In these calculations one observes a strong feedback of
the waves on the downdrafts, because the waves are reflected back by the lower
boundary of the computational domain, a property which is not expected from the
solar gravity waves produced by the plumes.
The generation of gravity waves by turbulent plumes is presently being
investigated by M. Kiraga, in 2- and 3-dimensional simulations. He takes great care
of avoiding the reflexion of the waves by the lower boundary of his computational
domain, by implementing in its vicinity a strong viscous damping. The first results
have been presented at the Granada workshop on convection (Kiraga et al. 1999); they
look promising, and will be used to test the prescriptions which have been proposed
for estimating the flux of gravity waves.
Our interest in these gravity waves is motivated by the fact that they may extract
angular momentum from the radiative interior of solar-type stars (see Kumar \&
Quataert 1997; Zahn et al. 1997; Kumar et al. 1999).
\section{Summarizing the properties of turbulent plumes}
There is little doubt that the convection zone of solar-like stars exhibits
strong spatial intermittency, with downwards directed plumes carrying a
substantial fraction of the energy flux. Such plumes are seen in all numerical
simulations, and they have now been observed in the uppermost part of the solar
convection zone.
If Taylor's parametrization of the entrainment of surrounding fluid into
turbulent plumes holds in the hot plasma of stellar interiors, and we see no
reason why it should not, then we expect such plumes to
traverse the whole convection zone.
According to the crude model presented above in \S2, these plumes behave much as if
they were traversing an isentropic atmosphere, with no feed-back at all, and
ignoring the radiation flux.
Shortly after their start,
they reach an asymptotic regime with their size increasing linearly with
depth. Due to the density stratification, the cone angle $\beta$ is somewhat smaller
than in the Boussinesq case treated so far: $\tan \beta =6\alpha /7$ instead
of $6\alpha /5$, with $\alpha$ being the entrainment constant.
These plumes carry kinetic energy downwards, i.e. in the `wrong'
direction. But their enthalpy flux always exceeds
the kinetic flux: the ratio between the two is constant in the asymptotic
regime, and its value is 4/7, if one assumes that the horizontal profile of the
plume is gaussian.
The maximum number of plumes that may reach the base
of the solar convection zone, when taking the counterflow into account, is around
1000, which means that the spacing between plumes is about 60 Mm. Note however
that this evaluation is based on a very strong
assumption: namely that the totality of the convective
energy flux (enthalpy plus kinetic) is carried by the plumes, thus
neglecting the contribution of the interstitial medium.
Finally, under the same assumption, the extent of penetration, at
the base of the convection zone, depends on the number of
plumes $N$ reaching that level: it varies from $0$
for $N \approx 12000$ to one pressure scale-height for $N \rightarrow 0$.
Unfortunately, the number of plumes cannot be predicted with the simple model
presented above. Let us recall that helioseismology yields a value of about
$0.1 H_p$, with the assumption that there is sharp transition from adiabatic to
radiative slope at the edge of the penetration layer (Roxburgh \& Vorontsov 1994).
With plumes of different strength, the transition would be smoother and more
difficult to diagnostic.
All these results are independent of the top boundary conditions on the
mass flux and on the initial momentum flux carried by the plumes.
What determines the flow is the convective energy flux near the surface,
which would be conserved along the plume if there were no radiative transport.
Unfortunately the simple model presented above is unable to render the complex
dynamics of these layers, where important radiative exchanges occur. They can
only been described by full-fledged 3-dimensional hydrodynamical simulations, such
as performed by \AA. Nordlund, R. Stein and their collaborators. However we cannot
rule out that a sophisticated closure model, well beyond what has been described
above, may succeed in better capturing the main properties of such turbulent
plumes (see V. Canuto's contribution to this workshop).
How do these plumes change our picture of stellar convection zones?
Plumes presumably exert an important effect on the thermal structure -- that is,
on the small deviations from the isentropic stratification which are known to induce
meridional circulation. The reason for this is that the heat transport by the plumes
is of advective nature, whereas in most models so far, as for example in the
mixing-length treatment, it has been represented by a diffusive process.
They will also play a major role in the dynamics of a convection zone,
and in determining its rotation profile. Since the ambiant angular momentum will be
entrained into the plumes, these will spin up with depth, and they will concentrate
both vorticity and helicity. Owing to their robustness, such vortices could well be
the cause of the observed differential rotation, especially of its peculiar
property of being quasi-independent of depth (Brown et al. 1989; Kosovichev et
al. 1997).
Furthermore, one can easily imagine that plumes play a major role in the
generation and in the confinement of magnetic fields. Since they are sites of high
helicity, they will be the seat of strong alpha-effect. The dynamo simulations
of Brandenburg et al. (1996) have shown that laminar diving plumes are
indeed playing an important part in the field generation and its storage
at the base of the convective layer. In more recent simulations, they have been
clearly observed to `pump down' the magnetic field, which is definitely an advective
process (Dorch 1998).
To conclude, we have yet to assess all the consequences of the presence of
plumes in stellar convection zones. For that we have to rely mainly on numerical
simulations, since the stratification is much stronger in stars than in the
Earth atmosphere, and since it cannot be reproduced in the laboratory. These
simulations have not reached yet a sufficient resolution to adequately describe
turbulent plumes, but they are progressing at fast pace, and their results can be
checked through helioseismology.
|
1,116,691,500,881 | arxiv | \section{Introduction}
\label{intro}
Galaxy mergers are commonly thought to be the main mechanism driving the
coevolution of galaxies and their central supermassive black holes (SMBH). An
SMBH binary is likely to form in the centre of the merged galaxy and
subsequently accrete from a circumbinary disc. It is reasonable to expect that
the angular momentum of the binary and that of the accreting gas are
uncorrelated, and that each has no prior knowledge of the other. We can
therefore expect a random distribution of orientations for such circumbinary
discs.
Understanding the evolution of a misaligned circumbinary disc is needed if we
are to understand the evolution of SMBH binaries. In a recent paper
\citep{Nixonetal2011a} we showed that a retrograde circumbinary disc can be
very efficient in extracting angular momentum from the binary orbit. This may
offer a solution to the final parsec problem (\citealt{Begelmanetal1980};
\citealt{MM2001}).
In another recent paper \citep*[][hereafter NKP]{Nixonetal2011b} we showed
that the dominant effect of the binary potential, on a misaligned circumbinary
disc, is to induce radially--dependent precessions of the misaligned disc
particle orbits. This differential precession is known to induce warping of
the disc as rings of gas dissipate energy through viscosity. The precession
vanishes only when the orbit is in the plane of the binary (either prograde or
retrograde) and thus we expect that both prograde and retrograde orbits are
possible equilibria for the gas. NKP showed that the evolution of a misaligned
circumbinary disc is formally similar to the evolution of a misaligned disc
around a spinning compact object. In this case the evolution is driven by the
Lense--Thirring effect (e.g. \citealt{BP1975}; \citealt{Pringle1992};
\citealt{SF1996}; \citealt{LP2006}; \citealt{NK2012}). This implies that the
analysis of \citet{Kingetal2005}, which calculates the conditions for co-- or
counter--alignment, holds for circumbinary discs as well. The disc co-- or
counter--aligns depending on the magnitudes and directions of $\mathbi{J}_{\rm
b}$ and $\mathbi{J}_{\rm d}$, the angular momentum of the binary and disc
respectively. The whole disc counteraligns with the binary if the initial
inclination angle of the disc, $\theta$, and the magnitudes of the disc and
binary angular momentum, $J_{\rm d}$ and $J_{\rm b}$ respectively, satisfy
\begin{equation}
\cos\theta < -\frac{J_{\rm d}}{2J_{\rm b}}.
\label{criterion}
\end{equation}
NKP only considered the zero-frequency (azimuthally symmetric $m=0$) term in
the binary potential. This is a reasonable approach as all other terms induce
oscillatory effects which cancel out on long timescales.
It is natural to assume that coaligned discs are stable, however in the past
counteraligned discs have been incorrectly found to be unstable
\citep{SF1996}. I therefore use a full three dimensional hydrodynamic approach
to check the assumptions in NKP are valid and to confirm that co-- and
counter--alignment are stable. In \sref{cocounter} I discuss the possibility
of both co-- and counter--alignment of the same disc (cf \citealt{LP2006}). In
\sref{sims} I report a simulation of a counteraligning disc and give my
conclusions in \sref{conc}.
\section{Simultaneous co-- {\it and} counter--alignment?}
\label{cocounter}
\citet{LP2006} considered the alignment of a disc and a spinning black
hole. They showed that for a disc where $\theta>\pi/2$ and $J_{\rm d} > 2
J_{\rm h}$, initial counteralignment of the inner disc occurs, followed by
subsequent coalignment of the outer disc. During the alignment of the outer
disc the inner disc retains its retrograde nature and so a warp of significant
amplitude is achieved ($\Delta \theta \sim \pi$). This scenario produces a
disc which is simultaneously counteraligned (in the inner parts) and coaligned
(in the outer parts). For the subsequent evolution of such a disc see
\citet*{Nixonetal2012}.
If however the angular momenta are such that $J_{\rm d} \lesssim J_{\rm h}$
then the disc can only wholly co-- or counter--align and the above process is
impossible. In this section we discuss whether it is feasible to have
circumbinary discs such that both co-- and counter--alignment are
simultaneously possible. This possibility is constrained as the mass of the
disc is limited by self--gravitational collapse and the radius of the disc is
limited by the gravitational sphere of influence of the binary. Therefore
there must be a maximum feasible disc angular momentum.
The disc simultaneously co-- and counter--aligns if (and only if)
$\theta>\pi/2$ and the condition (\ref{criterion}) does not hold. So for
randomly aligned discs, simultaneous co-- and counter--alignment is only
possible if $J_{\rm d} \gtrsim 2 J_{\rm b}$ (assuming $\cos\theta \sim
-1$). Thus I derive the condition for $J_{\rm d} \gtrsim 2 J_{\rm b}$. The
disc angular momentum is
\begin{equation}
J_{\mathrm{d}} \sim M_{\mathrm{d}}\sqrt{GMR_{\mathrm{d}}}
\label{Jd}
\end{equation}
where $M_{\mathrm{d}}$ is the mass in the disc, $M$ is the total binary mass, $R_{\mathrm{d}}$
is a characteristic radius for the disc and $G$ is the gravitational
constant. We note that the definition of $J_{\rm d}$ in this case is simply
the total angular momentum in the disc. The exact definition of $J_{\rm d}$ is
usually more subtle because the disc takes time to communicate its angular
momentum.
The angular momentum of a binary with eccentricity $e$ is
\begin{equation}
J_{\mathrm{b}} = \mu\sqrt{GMa\left(1-e^{2}\right)}
\label{Jb}
\end{equation}
where $M$ is the total mass of the binary and $\mu$ is the reduced mass ($\mu
\sim M_2$ for $M_2 \ll M_1$).
So for simultaneous co-- and counter--alignment we require
\begin{equation}
M_{\mathrm{d}}\sqrt{GMR_{\mathrm{d}}} \gtrsim 2\mu\sqrt{GMa\left(1-e^{2}\right)}.
\label{Jd_JB}
\end{equation}
Self--gravity limits $M_{\mathrm{d}} \lesssim (H/R)M$, so we get
\begin{equation}
(H/R)M \sqrt{GMR_{\mathrm{d}}} \gtrsim 2\mu\sqrt{GMa\left(1-e^{2}\right)}.
\label{Jd_Jb2}
\end{equation}
After a bit of algebra this tells us that we can only get simultaneous co--
and counter--alignment if
\begin{equation}
\frac{R_{\mathrm{d}}}{a} \gtrsim 4\left(\frac{R}{H}\right)^{2} \left(\frac{M_2}{M}
\right)^2\left(1-e^{2}\right)
\label{crit}
\end{equation}
For the most optimistic parameters we have $a\sim 0.1$ pc and the sphere of
influence of the binary $\sim 10$ pc. This gives the LHS as $\lesssim 100$.
For typical disc thickness ($H/R \sim 10^{-3}$) it is clear that binaries with
a low eccentricity ($e \approx 0$) we cannot get simultaneous co-- and
counter--alignment unless the mass ratio is extreme ($M_2/M_1 \lesssim
10^{-2}$). However if the disc is very thick then this is possible. For small
binary mass--ratios the disc mass may be greater than the mass of the
secondary, therefore the hydrodynamic drag on the binary may become
significant \citep{Ivanovetal1999}. If the binary is significantly eccentric,
as predicted for binaries that have accreted through a retrograde disc
\citep{Nixonetal2011a} then simultaneous co-- and counter--alignment may be
possible. I shall return to these possibilities in future work. However these
arguments suggest that it is reasonable to assume an SMBH binary must wholly
co-- or counter--align the disc with the binary plane.
\section{Simulation}
\label{sims}
To confirm the stable counteralignment of a circumbinary disc I perform one
simple simulation. I use the SPH code \textsc{phantom}, a low-memory, highly
efficient SPH code optimised for the study of non-self-gravitating
problems. This code has performed well in related simulations. For example
\citet{LP2010} simulated warped accretion discs and found
excellent agreement with the analytical work of \citet{Ogilvie1999} on the
nature of the internal accretion disc torques.
The implementation of accretion disc $\alpha-$viscosity \citep{SS1973} in
\textsc{phantom} is described in \citet{LP2010}. Specifically, we use the
`artificial viscosity for a disc' described in Sec. 3.2.3 of \citet{LP2010},
similar to earlier SPH accretion disc calculations
\citep[e.g.][]{Murray1996}. The main differences compared to standard SPH
artificial viscosity are that the disc viscosity is applied to both
approaching and receding particles and that no switches are used. The
implementation used here differs slightly from \citet{LP2010} in that the
$\beta^{\rm AV}$ term in the signal velocity is retained in order to prevent
particle interpenetration. The disc viscosity in \textsc{phantom} was
extensively calibrated against the 1D thin $\alpha-$disc evolution in
\citet{LP2010} (c.f. Fig.~4 in that paper) and the disc scale heights employed
here are similar. We use $\alpha^{\rm AV} = 1$ and $\beta^{\rm AV} = 2$ which
at the employed resolution (see below) corresponds to a physical viscosity
$\alpha_{\rm SS} \approx 0.05$. Note that the initial viscosity is slightly
smaller than this, however during the simulation the disc spreads and
particles are accreted and thus the spatial resolution decreases which
increases the viscosity (see eq.~38 of \citealt{LP2010}).
\subsection{Setup}
The simulation has two equal mass sink particles representing the binary (of
total mass unity in code units), on a circular orbit with separation $0.5$ (in
code units). Initially the circumbinary gas disc is flat and composed of $1$
million SPH particles, in hydrostatic equilibrium from $1.0$ to $2.0$ in
radius, and surface density distribution $\Sigma \propto R^{-1}$, set up by
means of the usual Monte--Carlo technique. The vertical hydrostatic
equilibrium corresponds to $H/R = 0.05$ at $R=1$. The equation of state for
the gas is isothermal. The disc is initially tilted at $170^{\circ}$ to the
binary plane. Any gas which falls within a radius of 0.5 from the binary
centre of mass is removed as it no longer has any effect of the alignment of
the circumbinary disc. The disc mass is negligible in comparison to the binary
mass and thus the gravitational back--reaction of the gas on the binary is not
included.
\subsection{Geometry}
The binary is taken to orbit in the $x$--$y$ plane with binary angular
momentum vector in the $z$--direction. We define the `tilt' and the `twist' of
the disc with respect to the binary using Euler angles (e.g. \citealt{BP1975};
\citealt{Pringle1996}), where the unit angular momentum vector in the disc is
described at any radius by
\begin{equation}
\boldsymbol{\ell} = \left(\cos\gamma \sin\beta, \sin\gamma \sin\beta,
\cos\beta \right)
\label{geometry}
\end{equation}
with $\beta\left(R,t\right)$ the local angle of disc tilt with respect to the
$z$--axis and $\gamma\left(R,t\right)$ the local angle of disc twist measured
from the $x$--axis.
\subsection{Stable counteralignment}
As predicted by NKP the dominant effect of the binary on the disc is to induce
precessions in the gas orbits. As the radial range of the disc in this
simulation is only a factor of two, the precession rate changes little between
the inner and outer parts of the disc. However there is still a differential
precession across the disc, which leads to a twist. This causes dissipation
between rings of gas and so a small amplitude warp in the disc
(cf. Fig.~\ref{fig:tilt}). The disc angular momentum vector precesses around
the binary angular momentum vector for the duration of the simulation. In
Fig.~\ref{fig:twist} the twist angle in the disc (at unit radius) is plotted
against time. As the twist angle is defined between $\pm 180^{\circ}$ the plot
has a `saw--tooth' shape, as the disc precesses until the twist is
$-180^{\circ}$ which is then equivalent to $180^{\circ}$.
\begin{figure}
\begin{flushleft}
\includegraphics[angle=0,width=0.75\columnwidth]{tilt.eps}
\caption{The tilt angle in the disc as a function of radius. This is plotted
at six different times corresponding to (in units of the dynamical time at
$R=1$) $t=0$ (black), $t=500$ (red), $t=1000$ (green), $t=1500$ (blue),
$t=2000$ (grey), $t=2500$ (purple). Initially the disc is flat with a
global tilt of $170^{\circ}$. In time the disc spreads and
counteraligns wrt the binary plane. }
\label{fig:tilt}
\end{flushleft}
\end{figure}
\begin{figure}
\begin{flushleft}
\includegraphics[angle=0,width=0.75\columnwidth]{twist.eps}
\caption{The twist angle (angle from $x$--axis to the line of nodes) at
$R=1$ in the disc as a function of time. Initially the line of nodes is in
the $y$--direction and hence the angle is $90^{\circ}$. The binary
potential induces precession of the gas orbits. The twist is calculated
between $\pm 180^{\circ}$ which generates the `saw--tooth' structure in
the plot.}
\label{fig:twist}
\end{flushleft}
\end{figure}
Figs.~\ref{fig:xyrender}~\&~\ref{fig:xzrender} show the disc structures at
various times. Fig.~\ref{fig:xyrender} shows the disc column density viewed
face on, and Fig.~\ref{fig:xzrender} shows it edge on. In
Fig.~\ref{fig:xyrender} we see the usual viscous spreading of the disc, and
also the lack of any resonances in the disc. This is expected in a
counterrotating circumbinary accretion disc (\citealt{PP1977};
\citealt{Nixonetal2011a}). In Fig.~\ref{fig:xzrender} we can see the tilt of
the disc. Initially the disc is tilted by $170^{\circ}$, but then
precesses and counteraligns with the binary plane.
\begin{figure*}
\begin{center}
\subfigure{
\includegraphics[width=0.3\linewidth]{t0_xy.eps}
\hspace{0.25 cm}
\includegraphics[width=0.3\linewidth]{t1_xy.eps}
\hspace{0.25 cm}
\includegraphics[width=0.3\linewidth]{t2_xy.eps}}
\subfigure{
\includegraphics[width=0.3\linewidth]{t3_xy.eps}
\hspace{0.25 cm}
\includegraphics[width=0.3\linewidth]{t4_xy.eps}
\hspace{0.25 cm}
\includegraphics[width=0.3\linewidth]{t5_xy.eps}}
\end{center}
\caption{Face--on column density rendering of the disc at various times in
the simulation. Each component of the binary is represented by a red
filled circle. Over time the disc spreads and precesses around the
binary. The precession induces dissipation in the disc which aligns the
disc with the binary plane.}
\label{fig:xyrender}
\end{figure*}
\begin{figure*}
\begin{center}
\subfigure{
\includegraphics[width=0.3\linewidth]{t0_xz.eps}
\hspace{0.25 cm}
\includegraphics[width=0.3\linewidth]{t1_xz.eps}
\hspace{0.25 cm}
\includegraphics[width=0.3\linewidth]{t2_xz.eps}}
\subfigure{
\includegraphics[width=0.3\linewidth]{t3_xz.eps}
\hspace{0.25 cm}
\includegraphics[width=0.3\linewidth]{t4_xz.eps}
\hspace{0.25 cm}
\includegraphics[width=0.3\linewidth]{t5_xz.eps}}
\end{center}
\caption{Edge--on column density rendering of the disc at various times in
the simulation. Each component of the binary is represented by a red
filled circle. Over time the disc spreads and precesses around the
binary. The precession induces dissipation in the disc which aligns the
disc with the binary plane.}
\label{fig:xzrender}
\end{figure*}
I have also performed simulations (not illustrated) of a moderately eccentric
binary ($e=0.3$) and an unequal mass ratio binary ($M_2/M_1=0.5$) which both display
the same behaviour.
\section{Conclusions}
\label{conc}
In this paper I have shown that for realistic parameters a circumbinary disc
must usually wholly co-- {\it or} counter--align with a binary. However for
extreme mass ratios or high eccentricities the binary may be dominated by the
disc angular momentum (cf. eq.~\ref{crit}). In this case evolution similar to
Fig.~9 of \citet{LP2006} is expected with simultaneous co-- and
counter--alignment of the disc.
I have shown that a disc with an initial
inclination of $170^{\circ}$ to the binary plane stably
counteraligns. Circumbinary discs, with $J_{\rm d}
\ll J_{\rm b}$ and an initial misalignment angle of $> 90^{\circ}$
counteralign with respect to the binary. If the disc angular momentum is not
negligible, \citet{Kingetal2005} showed that the condition for
counteralignment is
\begin{equation}
\cos\theta < -\frac{J_{\rm d}}{2J_{\rm b}}.
\end{equation}
A counteraligned circumbinary disc is efficient at shrinking the binary as it
directly absorbs negative angular momentum when capturing gas into
circumprimary or circumsecondary discs \citep{Nixonetal2011a}. This
interaction increases the binary eccentricity. \citet{Nixonetal2011a} show
that the timescale to increase the eccentricity from zero to unity is $\sim
M_2/{\dot M}$ where ${\dot M}$ is the mass inflow rate through the retrograde
circumbinary disc. Once the eccentricity is high enough gravitational wave
losses will drive the binary to coalescence.
\section*{Acknowledgments}
\label{acknowledgments}
I thank Andrew King and Daniel Price for providing useful feedback on the
manuscript. I also thank the referee for useful suggestions. CJN holds an STFC
postgraduate studentship. Research in theoretical astrophysics at Leicester is
supported by an STFC Rolling Grant. I acknowledge the use of \textsc{splash}
\citep{Price2007} for the rendering of the figures. This research used the
ALICE High Performance Computing Facility at the University of Leicester.
Some resources on ALICE form part of the DiRAC Facility jointly funded by STFC
and the Large Facilities Capital Fund of BIS.
\bibliographystyle{mn2e}
|
1,116,691,500,882 | arxiv | \section{Introduction}
Graph representation learning has been widely adopted to improve the performance of numerous scenarios, including traffic prediction \cite{DBLP:conf/ijcai/WuPLJZ19}, recommendation systems \cite{DBLP:conf/sigir/0001DWLZ020}, and drug discovery \cite{DBLP:conf/nips/LiuABG18}. As a powerful strategy for learning low-dimensional embeddings, they can preserve graph attributive and structural features without the need for manual annotation. Among these methods, graph contrastive learning (GCL) \cite{DBLP:conf/nips/SureshLHN21,DBLP:conf/www/0001XYLWW21,DBLP:conf/nips/YouCSCWS20,DBLP:journals/corr/abs-2006-04131,DBLP:conf/iclr/VelickovicFHLBH19} has been the most representative one, which learns node embeddings through maximizing the mutual information between representations of two augmented graph views. Under such principle, namely InfoMax \cite{DBLP:journals/computer/Linsker88}, the learned node representations through GCL have achieved great success when applied in downstream tasks.
To explore better generalization from GCL to downstream tasks, previous methods heuristically or experimentally define data augmentation or pretext tasks. Specifically, \citet{DBLP:conf/www/0001XYLWW21} leverage domain knowledge to remove unimportant structures as augmentations. \citet{DBLP:conf/nips/YouCSCWS20} conduct various experiments and draw conclusions indicating that difficult pretext tasks can improve generalization ability. Similar conclusions have also been explored in computer vision field, \citet{DBLP:journals/corr/abs-2111-06377} find experimentally that harder pretext tasks, i.e., masking an input image with a high percentage, would make the model generalize better. However, these works are still heuristic. As a result, the generalization ability of GCL and its theoretical principle are still less reported.
In this paper, we propose a metric for GCL generalization ability named GCL-GE inspired by the generalization error of supervised learning \cite{DBLP:journals/corr/abs-2106-10262,DBLP:conf/nips/XuR17,DBLP:conf/aistats/RussoZ16}. However, this metric can hardly be optimized directly due to the agnostic downstream task. To combat this, we theoretically prove a mutual information (MI) upper bound for it from an information-theory perspective. Such upper bound indicates that reducing MI when fitting pretext tasks is beneficial to improve the generalization ability on downstream tasks. In that case, there is a contradiction between our findings and the InfoMax principle of GCL, which reveals that traditional GCL will lead to poor generalization by simply maximizing MI.
Guided by the discovery aforementioned, we design a new GCL framework named InfoAdv with enhanced generalization ability. Specifically, to achieve high generalization ability via minimizing the MI upper bound, we first decompose the bound into two sub-parts based on the Markov data processing inequality \cite{DBLP:journals/tit/Raginsky16}: (1) MI between original graph and its augmentation view; (2) MI between the augmentation view and the representations. Such decomposition allows the new framework to be flexibly adapted to current GCL
methods based on data augmentation. We propose a view generator
and a regularization term based on KL divergence to achieve the goal of two sub-parts respectively. To retain the ability of capturing the characteristics of graph, our framework is attached to current GCL methods to jointly optimize the generalization metric and InfoMax via an adversarial style, striking the right balance between pretext tasks fitting and generalization ability on downstream tasks.
To demonstrate the effectiveness of our framework, we empirically validate our theoretical findings on a number of representative benchmarks, and experimental results demonstrate that our model achieves state-of-the-art performance. Moreover, extensive ablation studies are conducted to evaluate each component of our framework.
\section{Preliminaries}
In this section, we briefly introduce the framework of GCL and the generalization error under supervised learning.
\subsection{Graph Contrastive Learning}
An attributed graph is represented as $\mathcal{G}=(\mathcal{V}, \mathcal{E})$ , where $\mathcal{V}=$ $\left\{v_{1}, v_{2}, \cdots, v_{N}\right\}$ is the node set, and $\mathcal{E} \subseteq \mathcal{V} \times \mathcal{V}$ is the edge set. $\boldsymbol{X} \in \mathbb{R}^{N \times D}$ is the feature matrix, where $N$ is the number of nodes and $D$ is the feature dimension. $\boldsymbol{A} \in\{0,1\}^{N \times N}$ is the adjacency matrix, where $\boldsymbol{A}_{i j}=1$ if $\left(v_{i}, v_{j}\right) \in \mathcal{E}$.
The aim of GCL is obtaining a GNN encoder $f_{\theta}$ which takes $\mathcal{G}$ as input and outputs low-dimensional node embeddings. GCL achieves this goal by mutual information maximization between proper augmentations $t_1$ and $t_2$, namely InfoMax principle \cite{DBLP:journals/computer/Linsker88}. Specifically, taking the pair consisting of the anchor node $v_{i}$ on augmentations $t_1$ and itself $v_{i}^{+}$ on augmentation $t_2$ as a positive pair, and the pair consisting of it and other node $\boldsymbol{v^{-}}=\left\{ v_{j} \mid i \neq j \right\}$ as a negative sample, we need to maximize MI of positive pair $(v_{i}, v_{i}^{+})$, while minimize MI of negative pair $(v_{i},v_{j})$. The learned representations via the trained encoder $f_{\theta}$ is finally used in downstream tasks $\mathcal{T}$ where a new model $q_{\psi}$ is trained under supervision of ground truth label $y_i$ for node $v_i$. We denote the loss function of GCL pretext task and downstream task as $\ell_{un}$ and $\ell_{sup}$. The $f_{\theta}$ and $q_{\psi}$ is optimized as Eq.(\ref{gcl}). In Appendix \ref{Related Work}, we further describe related SOTA GCL methods, and provide detailed comparisons with our method.
\begin{equation}
\begin{split}
f_{\theta^{*}} &= \underset{f_{\theta}}{\arg \min } \sum_{i=1}^{N} \ell_{un} (v_i, v_{i}^{+},\boldsymbol{v^{-}},f_{\theta})\\
q_{\psi^{*}} &=\underset{q_{\psi}}{\arg \min } \sum_{i=1}^{N} \ell_{sup} ( f_{\theta}(\mathcal{G}_{v_{i}}), y_i, q_{\psi}) \label{gcl}
\end{split}
\end{equation}
\subsection{Generalization Error of Supervised Learning}
Define a training dataset $S=\left\{(\boldsymbol{x}_{s}^{i} , {y}_{s}^{i}) \mid i \in N \right\} \in (\mathcal{X} \times \mathcal{Y})^{N}$, which is composed of $N$ i.i.d. samples generated from an unknown data distribution $\mu=P(X, Y)$. $X$ and $Y$ are the random variables of data and labels, and the training dataset $\boldsymbol{X}_S \in \mathbb{R}^{N \times D}, \boldsymbol{Y}_S \in \mathbb{R}^{N}$. Given a learning algorithm $P_{W \mid S}$ taking the training dataset $S$ as input, the output is a hypothesis $w$ of space $\mathcal{W}=\{w: \mathcal{X} \rightarrow \mathcal{Y}\}$ whose random variable is denoted as $W$. The loss function is $\ell: \mathcal{X} \times \mathcal{Y} \times \mathcal{W} \rightarrow \mathbb{R}^{+}$ \cite{DBLP:journals/corr/abs-2106-10262,DBLP:conf/nips/XuR17}.
According to Shwartz and David \cite{DBLP:books/daglib/0033642}, the empirical risk of a hypothesis $\boldsymbol{w} \in \mathcal{W}$ is defined on the training dataset $S$ as $L_{S}(\boldsymbol{w})$ in Eq.(\ref{eq:er_pr_sl1}). The population risk of $\boldsymbol{w}$ is defined on unknown distribution $\mu$ as $L_{\mu}(\boldsymbol{w})$ in Eq.(\ref{eq:er_pr_sl2}). The generalization error is the difference between the population risk and its empirical risk over all hypothesis $W$, which can be defined as $\operatorname{gen}\left(\mu, P_{W \mid S}\right)$ in Eq.(\ref{eq:gen_sl}). In supervised learning, generalization error is a measure of how much the learned model suffers from overfitting, which can be bounded by the input-output MI of the learning algorithm $I(S ; W)$ \cite{DBLP:conf/nips/XuR17}.
\begin{gather}
L_{S}(\boldsymbol{w}) \triangleq \frac{1}{N} \sum_{i=1}^{N} \ell\left( \boldsymbol{x}_{s}^{i}, y_{s}^{i},\boldsymbol{w}\right) \label{eq:er_pr_sl1}\\
L_{\mu}(\boldsymbol{w}) \triangleq \underset{(\boldsymbol{x}, y) \sim \mu}{\mathbb{E}} \ell(\boldsymbol{x}, y,\boldsymbol{w}) \label{eq:er_pr_sl2} \\
\operatorname{gen}\left(\mu, P_{W \mid S}\right) \triangleq \mathbb{E}\left[L_{\mu}(W)-L_{S}(W)\right] \label{eq:gen_sl}
\end{gather}
\section{MI Upper Bound for GCL-GE}
In this section, we define the generalization error metric for graph contrastive learning (GCL) and prove a mutual information upper bound for it based on information theory.
\subsection{Generalization Error Definition} \label{Error_Definition_Section}
In graph representation learning, the generalization ability is used to measure the model performance difference between pretext task and downstream task, and the model is trained on pretext task. The pretext task is denoted as $\mathcal{P}$ with loss function $\ell_{un}$; The downstream task is denoted as $\mathcal{T}$ with loss function $\ell_{sup}$. Given a GCL algorithm $P_{F \mid \mathcal{P}}$ fitting pretext task $\mathcal{P}$, the output is a hypothesis $f_{\theta}$ of space $\mathcal{F}$ whose random variable is denoted as $F$. We define the GCL's empirical risk of a hypothesis $f_{\theta}$ as $L_{\mathcal{P}}(\mathcal{G},f_{\theta})$ in Eq.(\ref{eq:er}), measuring how precisely does the pretext task fit. The population risk of the same hypothesis is defined as $L_{\mathcal{T}}(\mathcal{G},f_{\theta})$ in Eq.(\ref{eq:pr}), measuring how well the fixed encoder trained by pretext task is adapted to the downstream task.
\begin{align}
L_{\mathcal{P}}(\mathcal{G},f_{\theta}) &\; \triangleq \frac{1}{N} \sum_{i=1}^{N} \ell_{un} (v_i, v_{i}^{+},\boldsymbol{v^{-}},f_{\theta}) \label{eq:er}\\
L_{\mathcal{T}}(\mathcal{G},f_{\theta}) &\; \triangleq \frac{1}{N} \sum_{i=1}^{N} \ell_{sup} ( f_{\theta}(\mathcal{G}_{v_{i}}), y_i, q_{\psi}) \label{eq:pr}
\end{align}
Inspired by the generalization of supervised learning, the GCL generalization error is the difference between the population risk and the empirical risk over all hypothesis $F$. Considering that the two losses may have different scales, an additional constant $s$ related to pretext task is needed to adjust the scale. Thus, the GCL generalization error, using \textbf{\textit{GCL-GE}} as shorthand, are defined as $\operatorname{gen}\left(\mathcal{T}, P_{F \mid \mathcal{P}}\right)$ in Eq.(\ref{eq:ge}).
\begin{equation}
\operatorname{gen}\left(\mathcal{T}, P_{F \mid \mathcal{P}}\right) \; \triangleq \; \mathbb{E}\left[L_{\mathcal{T}}(\mathcal{G},F)-s \cdot L_{\mathcal{P}}(\mathcal{G},F)\right] \label{eq:ge}
\end{equation}
Overall, the GCL-GE in Eq.(\ref{eq:ge}) is the difference between the population risk and the empirical risk with some scaling transform, which captures the generalization ability of GCL. Recalling that in supervised learning, the goal is to transfer a hypothesis from seen training data to unseen testing data. For GCL, correspondingly, the goal is to transfer a hypothesis from pretext task to downstream task. However, it is worth knowing that the intuitive definition of GCL-GE in Eq.(\ref{eq:ge}) is intractable compared to supervised learning, because of the following two main difficulties.
(1) Task difficulty: the task of training and testing are the same in supervised learning ($\ell$ in Eq.(\ref{eq:er_pr_sl1}) and Eq.(\ref{eq:er_pr_sl2})), but pretext task and downstream task are different in GCL ($\ell_{un}$ in Eq.(\ref{eq:er}) and $\ell_{sup}$ in Eq.(\ref{eq:pr})) . This problem makes their scales not guaranteed to be the same, making it difficult to measure the difference between population risk and empirical risk.
(2) Model difficulty: instead of using the same model to train and test in supervised learning, there will be a new model to handle downstream task in GCL, i.e., $q_{\psi}$ in Eq.(\ref{eq:pr}). The new model fits its own parameters based on the representations to complete the downstream task, making it difficult to evaluate the performance of the representation alone.
\subsection{Generalization Upper Bound} \label{MI_Bound_Section}
In this section, we are devoted to make GCL-GE defined above optimizable, whose first step is to solve the above two difficulties. For this purpose, we introduce the definitions of Latent Class and Mean Classifier, and therefore prove an upper bound of GCL-GE to guide its optimization. The two definitions are as follows: (1) the definition "Latent Class" follows the contrastive framework proposed by \citet{DBLP:conf/icml/SaunshiPAKK19} and we extend it to graph scenarios.
Such assumption considers there exists a set of latent classes in GCL pretext task learning and the classes are allowed to overlap and/or be fine-grained. Meanwhile, the classes in downstream tasks are a subset of latent classes. The formal definitions and details are provided in Appendix \ref{2diff}. Such assumption provides a path to formalize how capturing similarity in unlabeled data can lead to quantitative guarantees on downstream tasks. (2) the second definition "Mean Classifier" is leveraged as the downstream model to solve the model difficulty, where the parameters are not learned freely. Instead, they are fixed to be the mean representations of nodes belonging to the same label in downstream tasks, and classifies the nodes in the test set by calculating the distance between each node and each class. With the restrictions on the freedom of downstream model, the model difficulty can be solved. Note that Mean Classifier is a simplification of commonly used downstream models, e.g., Linear regression (LR), and LR has the ability to learn Mean Classifier. The formal definitions and proof of simplification are provided in Appendix \ref{2diff}.
Solving the above two difficulties, We first correlate the pretext task and downstream task, and prove the numerical relationship between their loss (two core elements in GCL-GE) in Lemma 1. Then we prove a mutual information upper bound for GCL-GE (See Theorem 2) from an information-theoretic point of view. This bound is only related to pretext task and no longer depends on downstream task. Guided by such bound, the GCL-GE can be optimized to improve the generalization performance.
\paragraph{Lemma 1}(Proved in Appendix \ref{Proof_for_Lemma}) Numerical relationship between GCL pretext task loss and downstream task loss. Let $L_{\mathcal{P}}$ denotes the loss function of GCL pretext task, and $L_{\mathcal{T}}^{\mu}$ denotes the loss function of downstream task under Mean Classifier. For any $f_{\theta} \in \mathcal{F}$, if $f_{\theta}(\mathcal{G})$ is $\sigma^{2}$-subgaussian random vector in every direction for every node class $c$, inequality Eq.(\ref{eq:lemma}) holds.
\begin{equation}
L_{\mathcal{P}}(\mathcal{G},f_{\theta}) \leq \gamma(1-\tau) L_{\mathcal{T}}^{\mu}(\mathcal{G},f_{\theta}) + \beta \tau + \epsilon \label{eq:lemma}
\end{equation}
The Eq.(\ref{eq:lemma}) reveals that the scaling parameter $s$ in Eq.(\ref{eq:ge}) is $\gamma(1-\tau)$, which is used to adjust the scale and make the two losses comparable. Noting that, $\gamma=1+c^{\prime} R \sigma \sqrt{\log \frac{R}{\epsilon}}$ and $\beta = 1+c^{\prime} \sigma R$, where $c^{\prime}$ is some constant and $R=\max _{v \in \mathcal{V}}\|f_{\theta}(\mathcal{G}_v)\|$. $\tau$ is the probability of sampling a same latent class twice in the set of all latent classes $\mathcal{C}$ with distribution $\rho$. $\epsilon$ is a noise term.
\begin{equation}
\tau \triangleq \underset{c \pm \sim \rho^{2}}{\mathbb{E}} \mathbf{1}\left[c^{+}=c^{-}\right]=\sum_{c \in \mathcal{C}}[\rho(c)]^{2} .
\end{equation}
\paragraph{Theorem 2}(Proved in Appendix \ref{Proof_for_MIUB}) Mutual Information upper bound for GCL-GE. If the learned representation $F(\mathcal{G})$ is $\sigma^{2}$-subgaussian random vector in every direction for every node class, then the generalization error $\operatorname{gen}\left(\mathcal{T}, P_{F \mid \mathcal{P}}\right)$ is bounded by
$I(\mathcal{G} ; F)$, MI between the original graph and the GCL learned hypothesis $F$, i.e.
\begin{equation}
\operatorname{gen}\left(\mathcal{T}, P_{F \mid \mathcal{P}}\right) \leq \frac{\sqrt{2\sigma^{2}I(\mathcal{G}, F)}-(\beta \tau +\epsilon)}{\gamma(1-\tau)} \label{eq:final}
\end{equation}
\subsection{Contradiction with InfoMax} \label{Contradiction_Section}
The upper bound we proved in section \ref{MI_Bound_Section} provides an information-theoretic understanding of GCL generalization. It reveals that the key to decrease GCL generalization error, i.e., improve generalization performance, is to reduce the MI between original graph and GCL learned hypothesis, i.e., minimize $I(\mathcal{G}, F)$. However, the objective of traditional GCL is to maximize MI between the representation of two views, which can be further proved to have relation with the MI between original graph and GCL learned hypothesis $I(\mathcal{G}, F)$ (Proved in Appendix \ref{Proof_for_Contradiction}).
Overall, there exists the contradiction between our generalization bound and the traditional GCL's InfoMax principle. In fact, the InfoMax principle of traditional GCL concentrates on pretext task fitting to learn hypothesis, while our bound aims to improve its generalization ability. These two principles involve a trade-off between them, with which there are similar concepts in supervised learning \cite{DBLP:conf/nips/XuR17,DBLP:conf/aistats/RussoZ16}. To improve the performance on downstream task, the key is to capture these two principles of GCL simultaneously. In other word, striking the right balance between pretext task fitting and downstream task generalization by controlling the MI is necessary.
\begin{figure*}[tbp]
\centering
\includegraphics[width=0.95\linewidth]{pic/InfoAdv.png}
\captionsetup{font=small}
\caption{The key modules of InfoAdv are: (1) a view generator $t_{\phi}$ in data augmentation part, (2) a target encoder $f_{\theta}$ in representation learning part, (3) the learning objectives. $t_{\phi}$ minimizes KL divergence loss to generate augmented views. $f_{\theta}$ minimizes both a KL regularization term and InfoNCE to learn representations. Optimizing $t_{\phi}$ and $f_{\theta}$ jointly can not only capture the characteristics of the graph, but also improve the generalization.} \label{InfoAdv}
\end{figure*}
\section{Methodology: InfoAdv for Generalizable GCL}
In this section, we present our mutual information adversarial framework (\textbf{\textit{InfoAdv}}) in detail, starting with theoretical motivation based on previous sections, followed by the overall framework of InfoAdv, and the last is the instantiation details.
\subsection{Theoretical Motivation}
InfoAdv is designed for both downstream task generalization and pretext task fitting. Pretext task fitting aims to preserve attribute and structural information in node representations, while downstream task generalization hopes to make the learned representations perform well on downstream tasks. These two goals need to be achieved with the help of both the generalization principle and the InfoMax principle. To make our method can be flexibly adapted to the current GCL framework and thus improve their generalization ability, we implement these two principles based on traditional GCL framework. The generalization principle is achieved from two perspectives: data augmentation and representation learning \cite{DBLP:journals/corr/abs-2102-10757}, and the InfoMax principle is reached by traditional GCL losses. In the following two sections, we discuss them in detail.
\subsubsection{Generalization Principle}
For the purpose of generalization, we need to reduce the MI between the original graph and the learned hypothesis. Thus the question turns to that: How to decrease $I(\mathcal{G};f_{\theta})$ under the framework of graph contrastive learning?
The framework of graph contrastive learning can be divided into two parts: (1) generating a new view through data augmentation; (2) learning the representations via a contrastive loss function defined on the original graph and the generated view \cite{DBLP:journals/corr/abs-2102-10757}. Thus we focus on this two core components: the view generator and the contrastive loss function, to reduce MI. We leverage Data Processing Inequality theorem (See Theorem 3) proposed in \cite{DBLP:conf/nips/XuR17,DBLP:journals/tit/Raginsky16} to further achieve our goal:
\paragraph{Theorem 3} For all random variables $X, Y, Z$ satisfying the Markov relation $X \rightarrow Y \rightarrow Z$, the inequality holds as:
\begin{equation}
I(X ; Z) \leq \min \{I(X ; Y), I(Y ; Z)\}
\end{equation}
For GCL, the Markov relation is $\mathcal{G} \rightarrow \mathcal{G}_{view} \rightarrow f_{\theta}$, the first arrow represents process of data augmentation, and the second arrow represents GNN's representation learning, which is directed by the contrastive loss function. Since $I(\mathcal{G},f_{\theta}) \leq \min \{I(\mathcal{G};\mathcal{G} _{view}),I(\mathcal{G} _{view};f_{\theta})\}$, decreasing $I(\mathcal{G};\mathcal{G} _{view})$ or $I(\mathcal{G} _{view};f_{\theta})$ both can decrease $I(\mathcal{G},f_{\theta})$ to improve GCL generalization performance. To achieve this goal, we propose a view generator to decrease $I(\mathcal{G};\mathcal{G} _{view})$ and a regularization term in the contrastive loss function to decrease $I(\mathcal{G}_{view};f_{\theta})$ simultaneously. These two components bring two perspectives for us to improve generalization. We achieve them by the following way:
\begin{itemize}
\item [(1)] View Generator: a view generator is introduced to minimize $I(\mathcal{G} ;\mathcal{G} _{view})$ by optimizing KL divergence, which is a upper bound \cite{DBLP:conf/icml/PooleOOAT19,DBLP:conf/icml/AlemiPFDS018,DBLP:journals/corr/KipfW16a,agakov2004algorithm} of MI between original graph and its augmented view. The purpose of this optimization is to find the view that minimizes MI.
\item [(2)] Loss Regularization: another KL regularization term is introduced to minimize $I(\mathcal{G} _{view};f_{\theta})$. The KL term is an upper bound of MI between input data and its representations, pushing representations to be close to a prior-distribution. The purpose of this optimization is to learn the representation that minimizes MI.
\end{itemize}
\subsubsection{InfoMax Principle}
For the purpose of InfoMax, we just preserve the traditional objective of GCL, that is, optimizing contrastive loss function termed InfoNCE, which is frequently used for contrastive learning. The InfoNCE has been proved to be a lower bound of MI between the representation of two views \cite{DBLP:conf/iclr/TschannenDRGL20,DBLP:conf/icml/PooleOOAT19,DBLP:journals/corr/abs-1807-03748}.
\subsection{Overall Framework}
The overall framework is shown in Figure \ref{InfoAdv}. There are two learnable components contained in InfoAdv, a \textbf{\textit{View Generator}} $t_{\phi}$ and a contrast-based \textbf{\textit{Target Encoder}} $f_{\theta}$. The view generator only concentrates on the generalization principle, while the target encoder concerns both the generalization and the InfoMax. The objective function can be written as Eq.(\ref{eq:whole_objective}), where $t_{\phi}$ and $f_{\theta}$ play a two-player max-min game. We iteratively optimize these two components by iteratively fixing one to optimize the another. Specifically, if the target encoder is fixed, the objective of view generator is shown as Eq.(\ref{eq:t}). Reversely if the view generator is fixed, the objective of target encoder is shown as Eq.(\ref{eq:f}).
\begin{gather}
V(t_{\phi},f_{\theta}) = \max _{f_{\theta}} \min _{t_{\phi}} I\left(f_{\theta}(\mathcal{G}) ; f_{\theta}(t_{\phi}(\mathcal{G}))\right)- \lambda \cdot KL(f_{\theta} \| \mathcal{N}(0,1)) \label{eq:whole_objective}\\
t_{\phi}^{*} = \arg \min_{t_{\phi}} I\left(f_{\theta}(\mathcal{G}) ; f_{\theta}(t_{\phi}(\mathcal{G}))\right)\label{eq:t}\\
f_{\theta}^{*} = \arg \max _{f_{\theta}} I\left(f_{\theta}(\mathcal{G}) ; f_{\theta}(t_{\phi}(\mathcal{G}))\right) - \lambda \cdot KL(f_{\theta} \| \mathcal{N}(0,1))\label{eq:f}
\end{gather}
We can see that, the first term of target encoder's objective defined in Eq.(\ref{eq:f}) is the standard InfoMax of GCL, which does not take generalization into consideration. Relying solely on InfoMax will lead to an infinite increase in mutual information. Therefore, a KL regularization term within the target encoder Eq.(\ref{eq:f}) and a view generator Eq.(\ref{eq:t}) are added to prevent MI from increasing unlimitedly to improve the generalization performance. The jointly optimization of InfoMax and generalization form an adversarial framework. Comparing to the traditional GCL optimized only by InfoMax principle and may suffer from poor generalization, InfoAdv can not only capture the characteristics of the graph, but also improve the generalization.
\subsection{Instantiation Details}
This subsection describes the instantiation details of InfoAdv. The algorithm for InfoAdv can be found in Appendix \ref{algor}, Algorithm \ref{InfoAdv_Algorithm}.
We start with the part of data augmentation as shown in Figure \ref{InfoAdv}, where a view generator is optimized to generate augmented views. Our model does not restrict the type of augmentation, without loss of generality, we choose the most representative edge-dropping as our instantiation. Specifically, The edge-dropping view generator consists of three components: (1) a GNN encoder; (2) an edge score MLP; (3) a probabilistic edge sampler. The GNN encoder firstly takes original graph as input and produces the node embedding matrix $\boldsymbol{H} = \text{GNN} \left(\boldsymbol{X},\boldsymbol{A} \right) $, where node $v_i$ has embedding $\boldsymbol{h}_{i}$.
Then the embeddings of two nodes connected by an edge are concatenated and sent to MLP to get an edge score $\boldsymbol{s}_{ij} = \text{MLP}\left(\left[\boldsymbol{h}_{i} ; \boldsymbol{h}_{j} \right] \right)$.
We normalize the score so that it belongs to the interval $\left[a,b\right]$ representing the edge dropping probability, where $0 \leq a \leq b \leq 1$.
Next, a discrete Bernoulli edge sampler using Gumbel-Max reparameterization trick \cite{DBLP:journals/jmlr/BinghamCJOPKSSH19,DBLP:conf/iclr/MaddisonMT17} is utilized to return an edge-dropping mask $\widetilde{\boldsymbol{R}} \sim \mathcal{B}\left(\boldsymbol{s} \right)$. The mask is then applied to original graph to conduct edge-dropping and get augmented adjacency matrix $\widetilde{\boldsymbol{A}} = \boldsymbol{A} \otimes \widetilde{\boldsymbol{R}}$. Eventually, the view generator generates augmentation views for the following target encoder.
Next to be discussed is the representation learning part, where the target encoder aims to implement both the KL regularization term and the InfoMax. Regularization term aims to push representations to be close to a prior-distribution and we take the uniform Gaussian distribution as the prior distribution. Thus we need to obtain the mean and the variance of representations. As a result, the target encoder $f_{\theta}$ consists of two GNNs, $\text{GNN}^{\mu}$ and $\text{GNN}^{\sigma}$ to learn the mean and the variance respectively. And these two GNNs share all layer parameters except the last layer. We denote the first view of graph as $(\widetilde{\boldsymbol{X}}_1,\widetilde{\boldsymbol{A}}_1)$, the second view as $(\widetilde{\boldsymbol{X}}_2,\widetilde{\boldsymbol{A}}_2)$. Taking the first view whose embeddings denoted as $\boldsymbol{U}$ as example, the view are sent into $f_{\theta}$ to get their mean embeddings $\boldsymbol{U}^{\mu}=\text{GNN}^{\mu}(\widetilde{\boldsymbol{X}}_1,\widetilde{\boldsymbol{A}}_1)$ and variance embeddings $\boldsymbol{U}^{\sigma}=\text{GNN}^{\sigma}(\widetilde{\boldsymbol{X}}_1,\widetilde{\boldsymbol{A}}_1)$. Then reparameterization trick\cite{DBLP:journals/corr/KingmaW13} is utilized for modeling the distributed representation $\boldsymbol{U}=\boldsymbol{U}^{\mu} + \epsilon \; \boldsymbol{U}^{\sigma}, \epsilon \sim \mathcal{N}(0,1)$. The second view whose embeddings denoted as $\boldsymbol{V}$ follows the same procedure.
Finally, as shown in the learning objective part of Figure \ref{InfoAdv}, the loss function of view generator and target encoder are defined in Eq.(\ref{eq:loss_func_f_f}).
The InfoMax term of target encoder's loss function is adapted from GRACE \cite{DBLP:journals/corr/abs-2006-04131}. The second term of target encoder's loss function $\mathcal{J}_2$ is the KL term which serves as an upper bound of MI.
\begin{scriptsize}
\begin{gather}
t_{\phi}^{*} = \arg \min_{t_{\phi}} \mathcal{J}_2^{\prime}, \quad
f_{\theta}^{*} = \arg \min_{f_{\theta}} \mathcal{J}_1+ \lambda \mathcal{J}_2 \label{eq:loss_func_f_f} \\
\mathcal{J}_1 =\frac{1}{2 N} \sum_{i=1}^{N}\left[\ell_{1}\left(\boldsymbol{u}_{i}^{\mu}, \boldsymbol{v}_{i}^{\mu}, \boldsymbol{u}_{i} \right)+\ell_{1}\left(\boldsymbol{v}_{i}^{\mu}, \boldsymbol{u}_{i}^{\mu}, \boldsymbol{v}_{i}\right)\right] \\
\mathcal{J}_2 =\frac{1}{2 N} \sum_{i=1}^{N} \left[\ell_{2}(\boldsymbol{u}^{\mu}_{i},\boldsymbol{u}^{\sigma}_{i})+\ell_{2}(\boldsymbol{v}^{\mu}_{i},\boldsymbol{v}^{\sigma}_{i})\right] \;
\mathcal{J}_2^{\prime} =\frac{1}{N} \sum_{i=1}^{N} \ell_{2}(\boldsymbol{u}^{\mu}_{i},\boldsymbol{u}^{\sigma}_{i})\\
\ell_{1}\left(\boldsymbol{u}_{i}^{\mu}, \boldsymbol{v}_{i}^{\mu}, \boldsymbol{u}_{i}\right)=-\log \frac{e^{\theta\left(\boldsymbol{u}_{i}^{\mu}, \boldsymbol{v}_{i}^{\mu} \right) / \tau}}{e^{\theta\left(\boldsymbol{u}_{i}^{\mu}, \boldsymbol{v}_{i}^{\mu} \right) / \tau}+NEG} \\
NEG=\sum_{k=1}^{N} \mathbbm{1}_{[k \neq i]} e^{\theta\left(\boldsymbol{u}_{i}, \boldsymbol{v}_{k}^{\mu} \right) / \tau}+\sum_{k=1}^{N} \mathbbm{1}_{[k \neq i]} e^{\theta\left(\boldsymbol{u}_{i}^{\mu}, \boldsymbol{u}_{k}^{\mu}\right) / \tau}\\
\ell_{2}(\boldsymbol{u}_{i}^{\sigma},\boldsymbol{u}_{i}^{\mu}) = \frac{1}{2}\left[{(\boldsymbol{u}_{i}^{\sigma})}^{2}+{(\boldsymbol{u}_{i}^{\mu})}^{2}-2\log \left( \boldsymbol{u}_{i}^{\sigma} \right)-1\right]
\end{gather}
\end{scriptsize}
\section{Experiments and Analysis}
This section is devoted to empirically evaluating our proposed instantiation of InfoAdv through answering the following questions. We begin with a brief introduction of the experimental setup, and then we answer the questions according to the experimental results and their analysis.
\begin{itemize}
\item [(Q1)] Does InfoAdv improve generalization performance?
\item [(Q2)] How does each generalization component affect InfoAdv's performance?
\item [(Q3)] How does each key hyperparameter affect InfoAdv?
\item [(Q4)] How does the update frequency ratio affect InfoAdv?
\end{itemize}
\subsection{Datasets and Setups}
We use two kinds of eight widely-used datasets to study the performance on node classification task and link prediction task, they are: (1) Citation networks including Cora \cite{DBLP:journals/ir/McCallumNRS00}, Citeseer \cite{DBLP:conf/dl/GilesBL98}, Pubmed \cite{DBLP:journals/aim/SenNBGGE08}, Coauthor-CS \cite{DBLP:journals/corr/abs-1811-05868}, Coauthor-Phy \cite{DBLP:journals/corr/abs-1811-05868}. (2) Social networks including WikiCS \cite{DBLP:journals/corr/abs-2007-02901}, Amazon-Photos \cite{DBLP:journals/corr/abs-1811-05868}, Amazon-Computers \cite{DBLP:journals/corr/abs-1811-05868}. Details of datasets and experimental settings can be found in Appendix \ref{datasets} and Appendix \ref{setting}.
\begin{table*}
\centering
\captionsetup{font=small}
\caption{Summary of performance on node classification in terms of $F_1$-micro score in percentage with standard deviation. The highest performance is highlighted in boldface.}
\label{Node Classification}
\scriptsize
\setlength{\tabcolsep}{3mm}{
\begin{tabular}{lllllllll}
\toprule
Datasets & Cora & Citeseer & Pubmed & WikiCS & A-Photo & A-Com & Co-CS & Co-Phy \\
\midrule
RawFeat & 64.23$\pm$1.09 & 64.47$\pm$0.72 & 84.70$\pm$0.25 & 71.98$\pm$0.00 & 87.37$\pm$0.48 & 78.80$\pm$0.17 & 91.87$\pm$0.11 & 94.44$\pm$0.17 \\
DeepWalk & 75.16$\pm$0.98 & 52.67$\pm$1.34 & 79.91$\pm$0.33 & 74.35$\pm$0.06 & 89.44$\pm$0.11 & 85.68$\pm$0.06 & 84.61$\pm$0.22 & 91.77$\pm$0.15 \\
DW+Feat & 77.53$\pm$1.44 & 62.50$\pm$0.61 & 84.85$\pm$0.45 & 77.21$\pm$0.03 & 90.05$\pm$0.08 & 86.28$\pm$0.07 & 87.70$\pm$0.04 & 94.90$\pm$0.09 \\
\cmidrule(r){1-9}
GAE & 81.63$\pm$0.72 & 67.35$\pm$0.29 & 83.72$\pm$0.32 & 70.15$\pm$0.01 & 91.62$\pm$0.13 & 85.27$\pm$0.19 & 90.01$\pm$0.71 & 94.92$\pm$0.07 \\
VGAE & 82.08$\pm$0.45 & 67.85$\pm$0.65 & 83.82$\pm$0.14 & 75.63$\pm$0.19 & 92.20$\pm$0.11 & 86.37$\pm$0.21 & 92.11$\pm$0.09 & 94.52$\pm$0.00 \\
\cmidrule(r){1-9}
DGI & 82.60$\pm$0.40 & 68.80$\pm$0.70 & 86.00$\pm$0.10 & 75.35$\pm$0.14 & 91.61$\pm$0.22 & 83.95$\pm$0.47 & 92.15$\pm$0.63 & 94.51$\pm$0.52 \\
GRACE & 83.70$\pm$0.68 & 72.39$\pm$0.21 & 86.35$\pm$0.17 & 79.56$\pm$0.02 & 92.79$\pm$0.11 & 87.20$\pm$0.32 & 93.01$\pm$0.06 & 95.56$\pm$0.05 \\
GCA & 83.81$\pm$0.91 & 72.85$\pm$0.17 & 86.61$\pm$0.12 & 80.07$\pm$0.12 & 93.10$\pm$0.07 & 87.70$\pm$ 0.43 & 93.08$\pm$0.06 & 95.61$\pm$0.01 \\
AD-GCL & 83.93$\pm$0.55 & 72.78$\pm$0.23 & 85.26$\pm$0.41 & 78.06$\pm$0.35 & 91.36$\pm$0.16 & 85.96$\pm$ 0.37 & 93.17$\pm$0.02 & 95.66$\pm$0.06 \\
\cmidrule(r){1-9}
\textbf{InfoAdv} & \textbf{84.82$\pm$0.83} & \textbf{73.00$\pm$0.42} & \textbf{86.87$\pm$0.28} & \textbf{81.41$\pm$0.03} & 93.54$\pm$0.04 & \textbf{88.27$\pm$0.45} & 93.20$\pm$0.05 & \textbf{95.73$\pm$0.05} \\
\; \textbf{(w/o) Gen} & 84.40$\pm$0.99 & 72.51$\pm$0.48 & 86.67$\pm$0.23 & 79.63$\pm$0.48 & 93.26$\pm$0.12 & 88.06$\pm$0.39 & \textbf{93.23$\pm$0.14} & 95.67$\pm$0.09 \\
\; \textbf{(w/o) Reg} & 84.36$\pm$0.54 & 72.74$\pm$0.27 & 86.82$\pm$0.25 & 80.97$\pm$0.19 & \textbf{93.80$\pm$0.12} & 87.61$\pm$0.11 & 93.04$\pm$0.06 & 95.65$\pm$0.04 \\
\cmidrule(r){1-9}
GCN & 82.69$\pm$1.08 & 72.09$\pm$0.78 & 84.81$\pm$0.40 & 77.19$\pm$0.12 & 92.42$\pm$0.22 & 86.51$\pm$0.54 & 93.03$\pm$0.31 & 95.65$\pm$0.16 \\
GAT & 83.60$\pm$0.29 & 72.56$\pm$0.44 & 85.27$\pm$0.15 & 77.65$\pm$0.11 & 92.56$\pm$0.35 & 86.93$\pm$0.29 & 92.31$\pm$0.24 & 95.47$\pm$0.15 \\
\bottomrule
\end{tabular}}
\end{table*}
\begin{table*}[h]
\captionsetup{font=small}
\centering
\caption{ Summary of performance on link prediction in terms of Area Under Curve (AUC) and Average Precision (AP) in percentage. The highest performance is highlighted in boldface.}
\label{Link Prediction}
\scriptsize
\setlength{\tabcolsep}{1.8mm}{
\begin{tabular}{llllllllllllllllll}
\toprule
\multirow{2}{*}{Datasets} & \multicolumn{2}{c}{Cora} & \multicolumn{2}{c}{Citeseer} & \multicolumn{2}{c}{Pubmed} & \multicolumn{2}{c}{WikiCS} & \multicolumn{2}{c}{A-Photo}& \multicolumn{2}{c}{A-Com} & \multicolumn{2}{c}{Co-CS} & \multicolumn{2}{c}{Co-Phy} & \\
\cmidrule(r){2-18}
& AUC & AP & AUC & AP & AUC & AP & AUC & AP & AUC & AP & AUC & AP & AUC & AP & AUC & AP & \\
\midrule
DGI & 83.45 & 82.95 & 85.36 & 86.94 & 95.24 & 94.53 & 78.18 & 79.30 & 87.47 & 86.93 & 85.25 & 85.22 & 89.99 & 89.01 & 84.35 & 83.80 & \\
GRACE & 80.65 & 81.64 & 87.81 & 89.52 & 94.99 & 94.68 & 95.28 & 94.78 & 93.99 & 92.60 & \textbf{94.53} & \textbf{93.87} & 93.82 & 93.44 & 87.95 & 86.50 & \\
GCA & 83.93 & 83.92 & 88.81 & 89.54 & 95.22 & 94.50 & 94.27 & 94.38 & 96.07 & 95.23 & 93.13 & 93.00 & 94.25 & 93.83 & 85.33 & 81.63 & \\
AD-GCL & 80.99 & 82.15 & 89.13 & 89.89 & 96.20 & 95.57 & 94.12 & 93.43 & 85.06 & 82.65 & 86.07 & 85.31 & 93.39 & 93.04 & 88.35 & 87.04 & \\
\cmidrule(r){1-18}
\textbf{InfoAdv} & \textbf{84.39} & \textbf{85.96} & \textbf{93.67} & \textbf{94.15} & 96.43 & 96.10 & \textbf{96.71} & \textbf{97.02} & \textbf{97.25} & \textbf{96.73} & 93.46 & 92.94 & \textbf{95.02} & \textbf{94.57} & 90.82 & 89.40 & \\
\; \textbf{(w/o) Gen} & 81.91 & 83.28 & 91.77 & 91.38 & \textbf{96.69} & \textbf{96.28} & 96.45 & 96.66 & 93.67 & 91.98 & 94.17 & 93.49 & 94.80 & 94.24 & \textbf{92.28} & \textbf{91.33} &\\
\; \textbf{(w/o) Reg} & 82.31 & 83.90 & 82.28 & 85.19 & 95.13 & 94.85 & 95.64 & 95.81 & 95.77 & 94.81 & 93.07 & 92.42 & 93.43 & 92.94 & 88.19 & 86.40 & \\
\bottomrule
\end{tabular}}
\end{table*}
\subsection{Performance on Downstream Tasks (AQ1)}
In this section we empirically evaluate our proposed instantiation of InfoAdv on the following three tasks.
\subsubsection{Generalization Performance on Node Classification Task}
We firstly evaluate InfoAdv on node classification using eight open benchmark datasets against SOTA methods. In each case, InfoAdv learns node representations in a fully self-supervised manner, followed by linear evaluation of these representations. We compare InfoAdv with both unsupervised methods and supervised methods\cite{DBLP:conf/iclr/KipfW17,DBLP:conf/iclr/VelickovicCCRLB18}. The unsupervised methods further include traditional graph embedding methods\cite{DBLP:conf/kdd/PerozziAS14} and SOTA self-supervised learning methods\cite{DBLP:conf/www/0001XYLWW21,DBLP:conf/nips/SureshLHN21,DBLP:journals/corr/abs-2006-04131,DBLP:conf/iclr/VelickovicFHLBH19,DBLP:journals/corr/KipfW16a} (both generative and contrastive). The results is summarized in Table \ref{Node Classification}, we can see that InfoAdv outperforms all methods mentioned above, which verifies the effectiveness of our proposed generalization bound and its principle that a balance should be struck between InfoMax and generalization.
\subsubsection{Generalization Performance on Link Prediction Task}
Considering that our method follow the GCL framework, We then evaluate InfoAdv on link prediction using eight public benchmark datasets against the SOTA graph contrastive methods. methods. In each case, we split edges in original graph into training set, validation set, and testing set, where InfoAdv is only trained in the training set. We introduce negative edge samples and calculate the link probability via the inner product of the representations. The results are summarized in Table \ref{Link Prediction}, we can see that InfoAdv outperforms all of the baseline methods, also verifying its generalization capability.
\subsubsection{Generalization Performance with Increasing Noise Rate}
We finally conduct generalization analysis for InfoAdv over noisy graph with increasing noise rate. Previous GCL methods fit the pretext task under single InfoMax principle, i.e., to capture as much information as possible from the original graph. However, to improve generalization performance, only the downstream task related information is required, whereas the irrelevant information or the noise information will decrease the generalization performance on the downstream task. Based on this, we verify the generalization ability of GCL model by adding noise, indicating that the generalizable model is more resistant to noise.
As the results shown in Figure \ref{rub}, we have two key observations: (1) with the rising noise rate, InfoAdv and its variants (w/o Gen and w/o Reg) perform stable, compared to GRACE \cite{DBLP:journals/corr/abs-2006-04131} and GCA\cite{DBLP:conf/www/0001XYLWW21} who drop sharply. (2) Within the different generalization components of InfoAdv, the performance of InfoAdv outperforms InfoAdv w/o Gen and InfoAdv w/o Reg. The two observation support that, our method shows powerful generalization ability in the noisy graph, which is more challenging for generalization ability.
\begin{figure*} [h]
\centering
\includegraphics[width=0.9\linewidth,height=4cm]{pic/rub.png}
\captionsetup{font=footnotesize}
\caption{The performance of InfoAdvs and baselines on two datasets with increasing noise rate. The dotted line is the mean scores, the color band is the standard deviation. The x-axis is the hyperparameter for introducing noise, and the noise rate increases synchronously with the hyperparameter. The y-axis is the node classification $F_1$-micro score.} \label{rub}
\end{figure*}
\subsection{Ablation Studies (AQ2)}
In this section, we study the impact of each generalization component to answer Q2. "(w/o) Gen" denotes without view generator. "(w/o) Reg" denotes without the KL regularization term in target encoder. The results are presented in the bottom of Table \ref{Node Classification} and Table \ref{Link Prediction}, where we can see that both KL term and view generator improve model performance consistently. In addition, the combination of the two generalization components further benefit the performance.
\subsection{Key Hyperparameter Sensitivity (AQ3)}
In this section, we study the sensitivity of two critical hyperparameters: the balance for KL regularization term $\lambda$, and the edge-drop probability of the view learned from view generator $p_{ea}$.
We show the model sensitivity on the transductive node classification task on Cora, under the perturbation of these two hyperparameters, that is, varying $\lambda$ from $1e-07$ to $1e+04$ and $p_{ea}$ from 0.1 to 0.9 to plot the 3D-surface for performance in terms of $F_1$-micro.
The result is shown in Figure \ref{hyper_lamb}, where four key conclusions can be observed,
\begin{itemize}
\item [(1)] The model is relatively stable when $\lambda$ is not too large, as the model performance remains steady when $\lambda$ varies in the range of $1e-07$ to $1e+02$.
\item [(2)] The model stays stable even when $\lambda$ is very small, since the model maintains a high level of performance even $\lambda$ reachs as small as $1e-07$.
\item [(3)] Higher edge-drop probability of the view learned from view generator helps model reach better performance, as the best performance occurs when $p_{ea}=0.7$.
\item [(4)] The KL regularization component does work. If we do not use the KL term, that is to set $\lambda=0$, the performance drops sharply.
\end{itemize}
\begin{figure*}[!htbp]
\centering
\includegraphics[width=0.85\linewidth]{./pic/hyper_lamb2.png}
\captionsetup{font=small}
\caption{The \textbf{left} side is the 3D-surface for the performance on Cora dataset, whose three axis are the balance for KL regularization term $\lambda$, the edge-drop probability of the view learned from view generator $p_{ea}$, and the performance $F_1$-micro score. The \textbf{middle} is the colorbar for the performance, indicating that the warmer the color is, the better the model performs. The \textbf{right} side is the heatmap for the 3D-surface when it is observed from the top, i.e., $F_1$-micro axis.} \label{hyper_lamb}
\end{figure*}
\subsection{Update Frequency Ratio (AQ4)}
This section is devoted to analysis update frequency ratio between view generator and decoder.
As is shown in the left side of Figure \ref{fr}, we can see that, at the beginning, with the increasing of the update frequency ratio, i.e., view generator update more and more frequently than decoder, the performance would as well improve and the convergence speed would become faster. when ratio increases up to '10:1', the performance reach the highest. However, when the view generator update too frequently, e.g., up to '100:1' the performance would drop soon.
As is shown in the right side of Figure \ref{fr}, we can see that, with view generator update with more and more high frequency, the edge preserving rate drops as well, indicating that more edges would be removed and making the task more difficult. However, when ratio increases too high up to '100:1', at first, the edge preserving rate dose keep in the lowest level, but with the training continuing, its preserving rate would begin to increase even higher than '10:1'.
These phenomenon may indicates that, the appropriate increase of the update frequency ratio would help to gain better generalization performance.
\begin{figure*}[h]
\centering
\includegraphics[width=0.9\linewidth]{./pic/fr.png}
\captionsetup{font=footnotesize}
\caption{The \textbf{left} side is the performance curve of 4 different update frequency ratio, whose x axis is the training epoch and the y axis is the $F_1$ score; The \textbf{right} side is the edge preserving rate curve of the same 4 different update frequency ratio, whose x axis is the training epoch and the y axis is the edge preserving rate} \label{fr}
\end{figure*}
\section{Related Work}
In the field of GCL, many work are concentrated on how to do better graph augmentations. Some of them are heuristical, e.g., GCA, GraphCL and JOAO \cite{DBLP:conf/www/0001XYLWW21,DBLP:conf/nips/YouCSCWS20,DBLP:conf/icml/YouCSW21}. Specially, JOAO \cite{DBLP:conf/icml/YouCSW21} dose learn augmentations with an adversarial optimization framework, but their objective is the combination of augmentation types, not the augmentation view. Most recently, AutoGCL \cite{yin2022autogcl} propose a data augmentation strategy. SAIL \cite{yu2022sail} consider the discrepancy of node proximity measured by graph topology and node feature. There are also works focus on the generalization ability of GCL. \cite{trivedi2022understanding} theoretically find that GCL generalization error bound with graph augmentations is linked to the average graph edit distance between classes. AF-GCL \cite{wang2022augmentation} consider GCL generalization for both homophilic graph and heterophilic graph.
Contrastive information bottleneck \cite{DBLP:conf/nips/SureshLHN21,DBLP:conf/nips/XuCLCZ21,DBLP:conf/nips/Tian0PKSI20}, which is proposed recently dose shares some ideas with our InfoAdv. The common idea of us is the MI minimization. However, there are several fundamental differences as follows.(1) The main contribution of this paper, as well as the original motivation of InfoAdv framework, is the generalization error metric of GCL and its MI upper bound, which, neither AD-GCL \cite{DBLP:conf/nips/SureshLHN21}, InfoGCL \cite{DBLP:conf/nips/XuCLCZ21} nor \citet{DBLP:conf/nips/Tian0PKSI20} involve. (2) As for self-supervised version of Tian et al. and AD-GCL, (despite of the field differences between CV and Graph) they focus only on the MI between two views for contrastive learning, while we concern the MI between GCL input-output. Our method can theoretically cover theirs due to the data processing inequality. (3) Specifically for AD-GCL, they leverage an MI lower bound to conduct MI minimization. However in terms of MI minimization, a single MI lower bound can not guarantee MI is truly minimized. Our method consider MI upper bound, which is more reasonable and could ensure the right MI minimization. (4) As for semi-supervised version of Tian et al. and InfoGCL, they are in need of knowledge from downstream task, which can hardly be satisfied in self-supervised learning. (5) AD-GCL, InfoGCL and Tian et al. define their MI between contrastive views, whereas we define MI between input data and output hypothesis, which is not bound with contrastive learning and can be easily extended to other learning paradigm such as predictive(generative) learning.\cite{DBLP:journals/corr/abs-2102-10757,DBLP:journals/corr/abs-2006-08218}.
Specificcally, for AD-GCL, we provide a more detailed comparison: Theoretically, (1) We provide a metric termed GCL-GE to bridge the generalization gap between GCL upstream and downstream , while AD-GCL does not connect this gap. (2) We prove a realizable bound for GCL-GE optimization even for agnostic downstream task, while AD-GCL does not consider. Methodologically, (1)AD-GCL leverages an MI lower bound (i.e., InfoNCE) to conduct MI minimization. However in terms of MI minimization, a single MI lower bound can not guarantee MI is truely minimized. On the contrary, our method consider MI upper bound (i.e., KL regularization ), which is more reasonable and could ensure the right MI minimization. (2) AD-GCL only improve GCL via data augmentation, while our method consider two components, both data augmentation and representation learning. Experimentally, We modify AD-GCL to node-level classification task and the results show that our InfoAdv outperforms AD-GCL on all eight popular datasets as below. (See Section 5, Table \ref{Node Classification} and Table \ref{Link Prediction})
There are plenty of work done for the generalization error of supervised learning, among which, traditional works draw bounds according to VC dimension, Rademacher complexity \cite{DBLP:books/daglib/0033642} and PAC-Bayesian \cite{DBLP:journals/ml/McAllester99}. Recently, the MI bounds based on information theory \cite{DBLP:conf/aistats/RussoZ16,DBLP:conf/nips/XuR17,DBLP:conf/nips/AsadiAV18,DBLP:conf/colt/SteinkeZ20,DBLP:journals/corr/abs-2106-10262} for generalization error has aroused widespread interests. For the generalization error of contrastive learning, Arora et al. \cite{DBLP:conf/icml/SaunshiPAKK19} propose a learning framework and come up with a Rademacher complexity bounds for its generalization error. Moreover, Nozawa et al. \cite{DBLP:conf/uai/NozawaGG20} adopt PAC bounds based on Arora's framework. Our different with those work are as below: (1) We prove a MI upper bound, which is a better bound compared to Arora et al. and Nozawa et al. (2) We propose a generalization error metric explicitly inspired by supervised learning, which is not mentioned in Arora et al. and Nozawa et al. (3) We extend Arora's framework to the graph(GNN) scenarios.
\section{Conclusion}
In this paper, we first propose GCL-GE, a metric for GCL generalization ability. Considering the intractability of the metric itself due to the agnostic downstream task, we theoretically prove a mutual information upper bound for it from an information-theoretic perspective. Then, guided by the bound, we design a GCL framework named InfoAdv with enhanced generalization ability, which jointly optimizes the generalization metric and InfoMax to strike the right balance between pretext tasks fitting and the generalization ability on downstream tasks. Finally, We empirically validate our theoretical findings on a number of representative benchmarks, and experimental results demonstrate that our model achieves state-of-the-art performance. Our InfoAdv principle is suitable not only for contrastive learning, but also other learning paradigm like predictive(generative) learning, which will be left to the further exploration.
\newpage
\bibliographystyle{unsrtnat}
\section{Introduction}
Graph representation learning has been widely adopted to improve the performance of numerous scenarios, including traffic prediction \cite{DBLP:conf/ijcai/WuPLJZ19}, recommendation systems \cite{DBLP:conf/sigir/0001DWLZ020}, and drug discovery \cite{DBLP:conf/nips/LiuABG18}. As a powerful strategy for learning low-dimensional embeddings, they can preserve graph attributive and structural features without the need for manual annotation. Among these methods, graph contrastive learning (GCL) \cite{DBLP:conf/nips/SureshLHN21,DBLP:conf/www/0001XYLWW21,DBLP:conf/nips/YouCSCWS20,DBLP:journals/corr/abs-2006-04131,DBLP:conf/iclr/VelickovicFHLBH19} has been the most representative one, which learns node embeddings through maximizing the mutual information between representations of two augmented graph views. Under such principle, namely InfoMax \cite{DBLP:journals/computer/Linsker88}, the learned node representations through GCL have achieved great success when applied in downstream tasks.
To explore better generalization from GCL to downstream tasks, previous methods heuristically or experimentally define data augmentation or pretext tasks. Specifically, \citet{DBLP:conf/www/0001XYLWW21} leverage domain knowledge to remove unimportant structures as augmentations. \citet{DBLP:conf/nips/YouCSCWS20} conduct various experiments and draw conclusions indicating that difficult pretext tasks can improve generalization ability. Similar conclusions have also been explored in computer vision field, \citet{DBLP:journals/corr/abs-2111-06377} find experimentally that harder pretext tasks, i.e., masking an input image with a high percentage, would make the model generalize better. However, these works are still heuristic. As a result, the generalization ability of GCL and its theoretical principle are still less reported.
In this paper, we propose a metric for GCL generalization ability named GCL-GE inspired by the generalization error of supervised learning \cite{DBLP:journals/corr/abs-2106-10262,DBLP:conf/nips/XuR17,DBLP:conf/aistats/RussoZ16}. However, this metric can hardly be optimized directly due to the agnostic downstream task. To combat this, we theoretically prove a mutual information (MI) upper bound for it from an information-theory perspective. Such upper bound indicates that reducing MI when fitting pretext tasks is beneficial to improve the generalization ability on downstream tasks. In that case, there is a contradiction between our findings and the InfoMax principle of GCL, which reveals that traditional GCL will lead to poor generalization by simply maximizing MI.
Guided by the discovery aforementioned, we design a new GCL framework named InfoAdv with enhanced generalization ability. Specifically, to achieve high generalization ability via minimizing the MI upper bound, we first decompose the bound into two sub-parts based on the Markov data processing inequality \cite{DBLP:journals/tit/Raginsky16}: (1) MI between original graph and its augmentation view; (2) MI between the augmentation view and the representations. Such decomposition allows the new framework to be flexibly adapted to current GCL
methods based on data augmentation. We propose a view generator
and a regularization term based on KL divergence to achieve the goal of two sub-parts respectively. To retain the ability of capturing the characteristics of graph, our framework is attached to current GCL methods to jointly optimize the generalization metric and InfoMax via an adversarial style, striking the right balance between pretext tasks fitting and generalization ability on downstream tasks.
To demonstrate the effectiveness of our framework, we empirically validate our theoretical findings on a number of representative benchmarks, and experimental results demonstrate that our model achieves state-of-the-art performance. Moreover, extensive ablation studies are conducted to evaluate each component of our framework.
\section{Preliminaries}
In this section, we briefly introduce the framework of GCL and the generalization error under supervised learning.
\subsection{Graph Contrastive Learning}
An attributed graph is represented as $\mathcal{G}=(\mathcal{V}, \mathcal{E})$ , where $\mathcal{V}=$ $\left\{v_{1}, v_{2}, \cdots, v_{N}\right\}$ is the node set, and $\mathcal{E} \subseteq \mathcal{V} \times \mathcal{V}$ is the edge set. $\boldsymbol{X} \in \mathbb{R}^{N \times D}$ is the feature matrix, where $N$ is the number of nodes and $D$ is the feature dimension. $\boldsymbol{A} \in\{0,1\}^{N \times N}$ is the adjacency matrix, where $\boldsymbol{A}_{i j}=1$ if $\left(v_{i}, v_{j}\right) \in \mathcal{E}$.
The aim of GCL is obtaining a GNN encoder $f_{\theta}$ which takes $\mathcal{G}$ as input and outputs low-dimensional node embeddings. GCL achieves this goal by mutual information maximization between proper augmentations $t_1$ and $t_2$, namely InfoMax principle \cite{DBLP:journals/computer/Linsker88}. Specifically, taking the pair consisting of the anchor node $v_{i}$ on augmentations $t_1$ and itself $v_{i}^{+}$ on augmentation $t_2$ as a positive pair, and the pair consisting of it and other node $\boldsymbol{v^{-}}=\left\{ v_{j} \mid i \neq j \right\}$ as a negative sample, we need to maximize MI of positive pair $(v_{i}, v_{i}^{+})$, while minimize MI of negative pair $(v_{i},v_{j})$. The learned representations via the trained encoder $f_{\theta}$ is finally used in downstream tasks $\mathcal{T}$ where a new model $q_{\psi}$ is trained under supervision of ground truth label $y_i$ for node $v_i$. We denote the loss function of GCL pretext task and downstream task as $\ell_{un}$ and $\ell_{sup}$. The $f_{\theta}$ and $q_{\psi}$ is optimized as Eq.(\ref{gcl}). In Appendix \ref{Related Work}, we further describe related SOTA GCL methods, and provide detailed comparisons with our method.
\begin{equation}
\begin{split}
f_{\theta^{*}} &= \underset{f_{\theta}}{\arg \min } \sum_{i=1}^{N} \ell_{un} (v_i, v_{i}^{+},\boldsymbol{v^{-}},f_{\theta})\\
q_{\psi^{*}} &=\underset{q_{\psi}}{\arg \min } \sum_{i=1}^{N} \ell_{sup} ( f_{\theta}(\mathcal{G}_{v_{i}}), y_i, q_{\psi}) \label{gcl}
\end{split}
\end{equation}
\subsection{Generalization Error of Supervised Learning}
Define a training dataset $S=\left\{(\boldsymbol{x}_{s}^{i} , {y}_{s}^{i}) \mid i \in N \right\} \in (\mathcal{X} \times \mathcal{Y})^{N}$, which is composed of $N$ i.i.d. samples generated from an unknown data distribution $\mu=P(X, Y)$. $X$ and $Y$ are the random variables of data and labels, and the training dataset $\boldsymbol{X}_S \in \mathbb{R}^{N \times D}, \boldsymbol{Y}_S \in \mathbb{R}^{N}$. Given a learning algorithm $P_{W \mid S}$ taking the training dataset $S$ as input, the output is a hypothesis $w$ of space $\mathcal{W}=\{w: \mathcal{X} \rightarrow \mathcal{Y}\}$ whose random variable is denoted as $W$. The loss function is $\ell: \mathcal{X} \times \mathcal{Y} \times \mathcal{W} \rightarrow \mathbb{R}^{+}$ \cite{DBLP:journals/corr/abs-2106-10262,DBLP:conf/nips/XuR17}.
According to Shwartz and David \cite{DBLP:books/daglib/0033642}, the empirical risk of a hypothesis $\boldsymbol{w} \in \mathcal{W}$ is defined on the training dataset $S$ as $L_{S}(\boldsymbol{w})$ in Eq.(\ref{eq:er_pr_sl1}). The population risk of $\boldsymbol{w}$ is defined on unknown distribution $\mu$ as $L_{\mu}(\boldsymbol{w})$ in Eq.(\ref{eq:er_pr_sl2}). The generalization error is the difference between the population risk and its empirical risk over all hypothesis $W$, which can be defined as $\operatorname{gen}\left(\mu, P_{W \mid S}\right)$ in Eq.(\ref{eq:gen_sl}). In supervised learning, generalization error is a measure of how much the learned model suffers from overfitting, which can be bounded by the input-output MI of the learning algorithm $I(S ; W)$ \cite{DBLP:conf/nips/XuR17}.
\begin{gather}
L_{S}(\boldsymbol{w}) \triangleq \frac{1}{N} \sum_{i=1}^{N} \ell\left( \boldsymbol{x}_{s}^{i}, y_{s}^{i},\boldsymbol{w}\right) \label{eq:er_pr_sl1}\\
L_{\mu}(\boldsymbol{w}) \triangleq \underset{(\boldsymbol{x}, y) \sim \mu}{\mathbb{E}} \ell(\boldsymbol{x}, y,\boldsymbol{w}) \label{eq:er_pr_sl2} \\
\operatorname{gen}\left(\mu, P_{W \mid S}\right) \triangleq \mathbb{E}\left[L_{\mu}(W)-L_{S}(W)\right] \label{eq:gen_sl}
\end{gather}
\section{MI Upper Bound for GCL-GE}
In this section, we define the generalization error metric for graph contrastive learning (GCL) and prove a mutual information upper bound for it based on information theory.
\subsection{Generalization Error Definition} \label{Error_Definition_Section}
In graph representation learning, the generalization ability is used to measure the model performance difference between pretext task and downstream task, and the model is trained on pretext task. The pretext task is denoted as $\mathcal{P}$ with loss function $\ell_{un}$; The downstream task is denoted as $\mathcal{T}$ with loss function $\ell_{sup}$. Given a GCL algorithm $P_{F \mid \mathcal{P}}$ fitting pretext task $\mathcal{P}$, the output is a hypothesis $f_{\theta}$ of space $\mathcal{F}$ whose random variable is denoted as $F$. We define the GCL's empirical risk of a hypothesis $f_{\theta}$ as $L_{\mathcal{P}}(\mathcal{G},f_{\theta})$ in Eq.(\ref{eq:er}), measuring how precisely does the pretext task fit. The population risk of the same hypothesis is defined as $L_{\mathcal{T}}(\mathcal{G},f_{\theta})$ in Eq.(\ref{eq:pr}), measuring how well the fixed encoder trained by pretext task is adapted to the downstream task.
\begin{align}
L_{\mathcal{P}}(\mathcal{G},f_{\theta}) &\; \triangleq \frac{1}{N} \sum_{i=1}^{N} \ell_{un} (v_i, v_{i}^{+},\boldsymbol{v^{-}},f_{\theta}) \label{eq:er}\\
L_{\mathcal{T}}(\mathcal{G},f_{\theta}) &\; \triangleq \frac{1}{N} \sum_{i=1}^{N} \ell_{sup} ( f_{\theta}(\mathcal{G}_{v_{i}}), y_i, q_{\psi}) \label{eq:pr}
\end{align}
Inspired by the generalization of supervised learning, the GCL generalization error is the difference between the population risk and the empirical risk over all hypothesis $F$. Considering that the two losses may have different scales, an additional constant $s$ related to pretext task is needed to adjust the scale. Thus, the GCL generalization error, using \textbf{\textit{GCL-GE}} as shorthand, are defined as $\operatorname{gen}\left(\mathcal{T}, P_{F \mid \mathcal{P}}\right)$ in Eq.(\ref{eq:ge}).
\begin{equation}
\operatorname{gen}\left(\mathcal{T}, P_{F \mid \mathcal{P}}\right) \; \triangleq \; \mathbb{E}\left[L_{\mathcal{T}}(\mathcal{G},F)-s \cdot L_{\mathcal{P}}(\mathcal{G},F)\right] \label{eq:ge}
\end{equation}
Overall, the GCL-GE in Eq.(\ref{eq:ge}) is the difference between the population risk and the empirical risk with some scaling transform, which captures the generalization ability of GCL. Recalling that in supervised learning, the goal is to transfer a hypothesis from seen training data to unseen testing data. For GCL, correspondingly, the goal is to transfer a hypothesis from pretext task to downstream task. However, it is worth knowing that the intuitive definition of GCL-GE in Eq.(\ref{eq:ge}) is intractable compared to supervised learning, because of the following two main difficulties.
(1) Task difficulty: the task of training and testing are the same in supervised learning ($\ell$ in Eq.(\ref{eq:er_pr_sl1}) and Eq.(\ref{eq:er_pr_sl2})), but pretext task and downstream task are different in GCL ($\ell_{un}$ in Eq.(\ref{eq:er}) and $\ell_{sup}$ in Eq.(\ref{eq:pr})) . This problem makes their scales not guaranteed to be the same, making it difficult to measure the difference between population risk and empirical risk.
(2) Model difficulty: instead of using the same model to train and test in supervised learning, there will be a new model to handle downstream task in GCL, i.e., $q_{\psi}$ in Eq.(\ref{eq:pr}). The new model fits its own parameters based on the representations to complete the downstream task, making it difficult to evaluate the performance of the representation alone.
\subsection{Generalization Upper Bound} \label{MI_Bound_Section}
In this section, we are devoted to make GCL-GE defined above optimizable, whose first step is to solve the above two difficulties. For this purpose, we introduce the definitions of Latent Class and Mean Classifier, and therefore prove an upper bound of GCL-GE to guide its optimization. The two definitions are as follows: (1) the definition "Latent Class" follows the contrastive framework proposed by \citet{DBLP:conf/icml/SaunshiPAKK19} and we extend it to graph scenarios.
Such assumption considers there exists a set of latent classes in GCL pretext task learning and the classes are allowed to overlap and/or be fine-grained. Meanwhile, the classes in downstream tasks are a subset of latent classes. The formal definitions and details are provided in Appendix \ref{2diff}. Such assumption provides a path to formalize how capturing similarity in unlabeled data can lead to quantitative guarantees on downstream tasks. (2) the second definition "Mean Classifier" is leveraged as the downstream model to solve the model difficulty, where the parameters are not learned freely. Instead, they are fixed to be the mean representations of nodes belonging to the same label in downstream tasks, and classifies the nodes in the test set by calculating the distance between each node and each class. With the restrictions on the freedom of downstream model, the model difficulty can be solved. Note that Mean Classifier is a simplification of commonly used downstream models, e.g., Linear regression (LR), and LR has the ability to learn Mean Classifier. The formal definitions and proof of simplification are provided in Appendix \ref{2diff}.
Solving the above two difficulties, We first correlate the pretext task and downstream task, and prove the numerical relationship between their loss (two core elements in GCL-GE) in Lemma 1. Then we prove a mutual information upper bound for GCL-GE (See Theorem 2) from an information-theoretic point of view. This bound is only related to pretext task and no longer depends on downstream task. Guided by such bound, the GCL-GE can be optimized to improve the generalization performance.
\paragraph{Lemma 1}(Proved in Appendix \ref{Proof_for_Lemma}) Numerical relationship between GCL pretext task loss and downstream task loss. Let $L_{\mathcal{P}}$ denotes the loss function of GCL pretext task, and $L_{\mathcal{T}}^{\mu}$ denotes the loss function of downstream task under Mean Classifier. For any $f_{\theta} \in \mathcal{F}$, if $f_{\theta}(\mathcal{G})$ is $\sigma^{2}$-subgaussian random vector in every direction for every node class $c$, inequality Eq.(\ref{eq:lemma}) holds.
\begin{equation}
L_{\mathcal{P}}(\mathcal{G},f_{\theta}) \leq \gamma(1-\tau) L_{\mathcal{T}}^{\mu}(\mathcal{G},f_{\theta}) + \beta \tau + \epsilon \label{eq:lemma}
\end{equation}
The Eq.(\ref{eq:lemma}) reveals that the scaling parameter $s$ in Eq.(\ref{eq:ge}) is $\gamma(1-\tau)$, which is used to adjust the scale and make the two losses comparable. Noting that, $\gamma=1+c^{\prime} R \sigma \sqrt{\log \frac{R}{\epsilon}}$ and $\beta = 1+c^{\prime} \sigma R$, where $c^{\prime}$ is some constant and $R=\max _{v \in \mathcal{V}}\|f_{\theta}(\mathcal{G}_v)\|$. $\tau$ is the probability of sampling a same latent class twice in the set of all latent classes $\mathcal{C}$ with distribution $\rho$. $\epsilon$ is a noise term.
\begin{equation}
\tau \triangleq \underset{c \pm \sim \rho^{2}}{\mathbb{E}} \mathbf{1}\left[c^{+}=c^{-}\right]=\sum_{c \in \mathcal{C}}[\rho(c)]^{2} .
\end{equation}
\paragraph{Theorem 2}(Proved in Appendix \ref{Proof_for_MIUB}) Mutual Information upper bound for GCL-GE. If the learned representation $F(\mathcal{G})$ is $\sigma^{2}$-subgaussian random vector in every direction for every node class, then the generalization error $\operatorname{gen}\left(\mathcal{T}, P_{F \mid \mathcal{P}}\right)$ is bounded by
$I(\mathcal{G} ; F)$, MI between the original graph and the GCL learned hypothesis $F$, i.e.
\begin{equation}
\operatorname{gen}\left(\mathcal{T}, P_{F \mid \mathcal{P}}\right) \leq \frac{\sqrt{2\sigma^{2}I(\mathcal{G}, F)}-(\beta \tau +\epsilon)}{\gamma(1-\tau)} \label{eq:final}
\end{equation}
\subsection{Contradiction with InfoMax} \label{Contradiction_Section}
The upper bound we proved in section \ref{MI_Bound_Section} provides an information-theoretic understanding of GCL generalization. It reveals that the key to decrease GCL generalization error, i.e., improve generalization performance, is to reduce the MI between original graph and GCL learned hypothesis, i.e., minimize $I(\mathcal{G}, F)$. However, the objective of traditional GCL is to maximize MI between the representation of two views, which can be further proved to have relation with the MI between original graph and GCL learned hypothesis $I(\mathcal{G}, F)$ (Proved in Appendix \ref{Proof_for_Contradiction}).
Overall, there exists the contradiction between our generalization bound and the traditional GCL's InfoMax principle. In fact, the InfoMax principle of traditional GCL concentrates on pretext task fitting to learn hypothesis, while our bound aims to improve its generalization ability. These two principles involve a trade-off between them, with which there are similar concepts in supervised learning \cite{DBLP:conf/nips/XuR17,DBLP:conf/aistats/RussoZ16}. To improve the performance on downstream task, the key is to capture these two principles of GCL simultaneously. In other word, striking the right balance between pretext task fitting and downstream task generalization by controlling the MI is necessary.
\begin{figure*}[tbp]
\centering
\includegraphics[width=0.95\linewidth]{pic/InfoAdv.png}
\captionsetup{font=small}
\caption{The key modules of InfoAdv are: (1) a view generator $t_{\phi}$ in data augmentation part, (2) a target encoder $f_{\theta}$ in representation learning part, (3) the learning objectives. $t_{\phi}$ minimizes KL divergence loss to generate augmented views. $f_{\theta}$ minimizes both a KL regularization term and InfoNCE to learn representations. Optimizing $t_{\phi}$ and $f_{\theta}$ jointly can not only capture the characteristics of the graph, but also improve the generalization.} \label{InfoAdv}
\end{figure*}
\section{Methodology: InfoAdv for Generalizable GCL}
In this section, we present our mutual information adversarial framework (\textbf{\textit{InfoAdv}}) in detail, starting with theoretical motivation based on previous sections, followed by the overall framework of InfoAdv, and the last is the instantiation details.
\subsection{Theoretical Motivation}
InfoAdv is designed for both downstream task generalization and pretext task fitting. Pretext task fitting aims to preserve attribute and structural information in node representations, while downstream task generalization hopes to make the learned representations perform well on downstream tasks. These two goals need to be achieved with the help of both the generalization principle and the InfoMax principle. To make our method can be flexibly adapted to the current GCL framework and thus improve their generalization ability, we implement these two principles based on traditional GCL framework. The generalization principle is achieved from two perspectives: data augmentation and representation learning \cite{DBLP:journals/corr/abs-2102-10757}, and the InfoMax principle is reached by traditional GCL losses. In the following two sections, we discuss them in detail.
\subsubsection{Generalization Principle}
For the purpose of generalization, we need to reduce the MI between the original graph and the learned hypothesis. Thus the question turns to that: How to decrease $I(\mathcal{G};f_{\theta})$ under the framework of graph contrastive learning?
The framework of graph contrastive learning can be divided into two parts: (1) generating a new view through data augmentation; (2) learning the representations via a contrastive loss function defined on the original graph and the generated view \cite{DBLP:journals/corr/abs-2102-10757}. Thus we focus on this two core components: the view generator and the contrastive loss function, to reduce MI. We leverage Data Processing Inequality theorem (See Theorem 3) proposed in \cite{DBLP:conf/nips/XuR17,DBLP:journals/tit/Raginsky16} to further achieve our goal:
\paragraph{Theorem 3} For all random variables $X, Y, Z$ satisfying the Markov relation $X \rightarrow Y \rightarrow Z$, the inequality holds as:
\begin{equation}
I(X ; Z) \leq \min \{I(X ; Y), I(Y ; Z)\}
\end{equation}
For GCL, the Markov relation is $\mathcal{G} \rightarrow \mathcal{G}_{view} \rightarrow f_{\theta}$, the first arrow represents process of data augmentation, and the second arrow represents GNN's representation learning, which is directed by the contrastive loss function. Since $I(\mathcal{G},f_{\theta}) \leq \min \{I(\mathcal{G};\mathcal{G} _{view}),I(\mathcal{G} _{view};f_{\theta})\}$, decreasing $I(\mathcal{G};\mathcal{G} _{view})$ or $I(\mathcal{G} _{view};f_{\theta})$ both can decrease $I(\mathcal{G},f_{\theta})$ to improve GCL generalization performance. To achieve this goal, we propose a view generator to decrease $I(\mathcal{G};\mathcal{G} _{view})$ and a regularization term in the contrastive loss function to decrease $I(\mathcal{G}_{view};f_{\theta})$ simultaneously. These two components bring two perspectives for us to improve generalization. We achieve them by the following way:
\begin{itemize}
\item [(1)] View Generator: a view generator is introduced to minimize $I(\mathcal{G} ;\mathcal{G} _{view})$ by optimizing KL divergence, which is a upper bound \cite{DBLP:conf/icml/PooleOOAT19,DBLP:conf/icml/AlemiPFDS018,DBLP:journals/corr/KipfW16a,agakov2004algorithm} of MI between original graph and its augmented view. The purpose of this optimization is to find the view that minimizes MI.
\item [(2)] Loss Regularization: another KL regularization term is introduced to minimize $I(\mathcal{G} _{view};f_{\theta})$. The KL term is an upper bound of MI between input data and its representations, pushing representations to be close to a prior-distribution. The purpose of this optimization is to learn the representation that minimizes MI.
\end{itemize}
\subsubsection{InfoMax Principle}
For the purpose of InfoMax, we just preserve the traditional objective of GCL, that is, optimizing contrastive loss function termed InfoNCE, which is frequently used for contrastive learning. The InfoNCE has been proved to be a lower bound of MI between the representation of two views \cite{DBLP:conf/iclr/TschannenDRGL20,DBLP:conf/icml/PooleOOAT19,DBLP:journals/corr/abs-1807-03748}.
\subsection{Overall Framework}
The overall framework is shown in Figure \ref{InfoAdv}. There are two learnable components contained in InfoAdv, a \textbf{\textit{View Generator}} $t_{\phi}$ and a contrast-based \textbf{\textit{Target Encoder}} $f_{\theta}$. The view generator only concentrates on the generalization principle, while the target encoder concerns both the generalization and the InfoMax. The objective function can be written as Eq.(\ref{eq:whole_objective}), where $t_{\phi}$ and $f_{\theta}$ play a two-player max-min game. We iteratively optimize these two components by iteratively fixing one to optimize the another. Specifically, if the target encoder is fixed, the objective of view generator is shown as Eq.(\ref{eq:t}). Reversely if the view generator is fixed, the objective of target encoder is shown as Eq.(\ref{eq:f}).
\begin{gather}
V(t_{\phi},f_{\theta}) = \max _{f_{\theta}} \min _{t_{\phi}} I\left(f_{\theta}(\mathcal{G}) ; f_{\theta}(t_{\phi}(\mathcal{G}))\right)- \lambda \cdot KL(f_{\theta} \| \mathcal{N}(0,1)) \label{eq:whole_objective}\\
t_{\phi}^{*} = \arg \min_{t_{\phi}} I\left(f_{\theta}(\mathcal{G}) ; f_{\theta}(t_{\phi}(\mathcal{G}))\right)\label{eq:t}\\
f_{\theta}^{*} = \arg \max _{f_{\theta}} I\left(f_{\theta}(\mathcal{G}) ; f_{\theta}(t_{\phi}(\mathcal{G}))\right) - \lambda \cdot KL(f_{\theta} \| \mathcal{N}(0,1))\label{eq:f}
\end{gather}
We can see that, the first term of target encoder's objective defined in Eq.(\ref{eq:f}) is the standard InfoMax of GCL, which does not take generalization into consideration. Relying solely on InfoMax will lead to an infinite increase in mutual information. Therefore, a KL regularization term within the target encoder Eq.(\ref{eq:f}) and a view generator Eq.(\ref{eq:t}) are added to prevent MI from increasing unlimitedly to improve the generalization performance. The jointly optimization of InfoMax and generalization form an adversarial framework. Comparing to the traditional GCL optimized only by InfoMax principle and may suffer from poor generalization, InfoAdv can not only capture the characteristics of the graph, but also improve the generalization.
\subsection{Instantiation Details}
This subsection describes the instantiation details of InfoAdv. The algorithm for InfoAdv can be found in Appendix \ref{algor}, Algorithm \ref{InfoAdv_Algorithm}.
We start with the part of data augmentation as shown in Figure \ref{InfoAdv}, where a view generator is optimized to generate augmented views. Our model does not restrict the type of augmentation, without loss of generality, we choose the most representative edge-dropping as our instantiation. Specifically, The edge-dropping view generator consists of three components: (1) a GNN encoder; (2) an edge score MLP; (3) a probabilistic edge sampler. The GNN encoder firstly takes original graph as input and produces the node embedding matrix $\boldsymbol{H} = \text{GNN} \left(\boldsymbol{X},\boldsymbol{A} \right) $, where node $v_i$ has embedding $\boldsymbol{h}_{i}$.
Then the embeddings of two nodes connected by an edge are concatenated and sent to MLP to get an edge score $\boldsymbol{s}_{ij} = \text{MLP}\left(\left[\boldsymbol{h}_{i} ; \boldsymbol{h}_{j} \right] \right)$.
We normalize the score so that it belongs to the interval $\left[a,b\right]$ representing the edge dropping probability, where $0 \leq a \leq b \leq 1$.
Next, a discrete Bernoulli edge sampler using Gumbel-Max reparameterization trick \cite{DBLP:journals/jmlr/BinghamCJOPKSSH19,DBLP:conf/iclr/MaddisonMT17} is utilized to return an edge-dropping mask $\widetilde{\boldsymbol{R}} \sim \mathcal{B}\left(\boldsymbol{s} \right)$. The mask is then applied to original graph to conduct edge-dropping and get augmented adjacency matrix $\widetilde{\boldsymbol{A}} = \boldsymbol{A} \otimes \widetilde{\boldsymbol{R}}$. Eventually, the view generator generates augmentation views for the following target encoder.
Next to be discussed is the representation learning part, where the target encoder aims to implement both the KL regularization term and the InfoMax. Regularization term aims to push representations to be close to a prior-distribution and we take the uniform Gaussian distribution as the prior distribution. Thus we need to obtain the mean and the variance of representations. As a result, the target encoder $f_{\theta}$ consists of two GNNs, $\text{GNN}^{\mu}$ and $\text{GNN}^{\sigma}$ to learn the mean and the variance respectively. And these two GNNs share all layer parameters except the last layer. We denote the first view of graph as $(\widetilde{\boldsymbol{X}}_1,\widetilde{\boldsymbol{A}}_1)$, the second view as $(\widetilde{\boldsymbol{X}}_2,\widetilde{\boldsymbol{A}}_2)$. Taking the first view whose embeddings denoted as $\boldsymbol{U}$ as example, the view are sent into $f_{\theta}$ to get their mean embeddings $\boldsymbol{U}^{\mu}=\text{GNN}^{\mu}(\widetilde{\boldsymbol{X}}_1,\widetilde{\boldsymbol{A}}_1)$ and variance embeddings $\boldsymbol{U}^{\sigma}=\text{GNN}^{\sigma}(\widetilde{\boldsymbol{X}}_1,\widetilde{\boldsymbol{A}}_1)$. Then reparameterization trick\cite{DBLP:journals/corr/KingmaW13} is utilized for modeling the distributed representation $\boldsymbol{U}=\boldsymbol{U}^{\mu} + \epsilon \; \boldsymbol{U}^{\sigma}, \epsilon \sim \mathcal{N}(0,1)$. The second view whose embeddings denoted as $\boldsymbol{V}$ follows the same procedure.
Finally, as shown in the learning objective part of Figure \ref{InfoAdv}, the loss function of view generator and target encoder are defined in Eq.(\ref{eq:loss_func_f_f}).
The InfoMax term of target encoder's loss function is adapted from GRACE \cite{DBLP:journals/corr/abs-2006-04131}. The second term of target encoder's loss function $\mathcal{J}_2$ is the KL term which serves as an upper bound of MI.
\begin{scriptsize}
\begin{gather}
t_{\phi}^{*} = \arg \min_{t_{\phi}} \mathcal{J}_2^{\prime}, \quad
f_{\theta}^{*} = \arg \min_{f_{\theta}} \mathcal{J}_1+ \lambda \mathcal{J}_2 \label{eq:loss_func_f_f} \\
\mathcal{J}_1 =\frac{1}{2 N} \sum_{i=1}^{N}\left[\ell_{1}\left(\boldsymbol{u}_{i}^{\mu}, \boldsymbol{v}_{i}^{\mu}, \boldsymbol{u}_{i} \right)+\ell_{1}\left(\boldsymbol{v}_{i}^{\mu}, \boldsymbol{u}_{i}^{\mu}, \boldsymbol{v}_{i}\right)\right] \\
\mathcal{J}_2 =\frac{1}{2 N} \sum_{i=1}^{N} \left[\ell_{2}(\boldsymbol{u}^{\mu}_{i},\boldsymbol{u}^{\sigma}_{i})+\ell_{2}(\boldsymbol{v}^{\mu}_{i},\boldsymbol{v}^{\sigma}_{i})\right] \;
\mathcal{J}_2^{\prime} =\frac{1}{N} \sum_{i=1}^{N} \ell_{2}(\boldsymbol{u}^{\mu}_{i},\boldsymbol{u}^{\sigma}_{i})\\
\ell_{1}\left(\boldsymbol{u}_{i}^{\mu}, \boldsymbol{v}_{i}^{\mu}, \boldsymbol{u}_{i}\right)=-\log \frac{e^{\theta\left(\boldsymbol{u}_{i}^{\mu}, \boldsymbol{v}_{i}^{\mu} \right) / \tau}}{e^{\theta\left(\boldsymbol{u}_{i}^{\mu}, \boldsymbol{v}_{i}^{\mu} \right) / \tau}+NEG} \\
NEG=\sum_{k=1}^{N} \mathbbm{1}_{[k \neq i]} e^{\theta\left(\boldsymbol{u}_{i}, \boldsymbol{v}_{k}^{\mu} \right) / \tau}+\sum_{k=1}^{N} \mathbbm{1}_{[k \neq i]} e^{\theta\left(\boldsymbol{u}_{i}^{\mu}, \boldsymbol{u}_{k}^{\mu}\right) / \tau}\\
\ell_{2}(\boldsymbol{u}_{i}^{\sigma},\boldsymbol{u}_{i}^{\mu}) = \frac{1}{2}\left[{(\boldsymbol{u}_{i}^{\sigma})}^{2}+{(\boldsymbol{u}_{i}^{\mu})}^{2}-2\log \left( \boldsymbol{u}_{i}^{\sigma} \right)-1\right]
\end{gather}
\end{scriptsize}
\section{Experiments and Analysis}
This section is devoted to empirically evaluating our proposed instantiation of InfoAdv through answering the following questions. We begin with a brief introduction of the experimental setup, and then we answer the questions according to the experimental results and their analysis.
\begin{itemize}
\item [(Q1)] Does InfoAdv improve generalization performance?
\item [(Q2)] How does each generalization component affect InfoAdv's performance?
\item [(Q3)] How does each key hyperparameter affect InfoAdv?
\item [(Q4)] How does the update frequency ratio affect InfoAdv?
\end{itemize}
\subsection{Datasets and Setups}
We use two kinds of eight widely-used datasets to study the performance on node classification task and link prediction task, they are: (1) Citation networks including Cora \cite{DBLP:journals/ir/McCallumNRS00}, Citeseer \cite{DBLP:conf/dl/GilesBL98}, Pubmed \cite{DBLP:journals/aim/SenNBGGE08}, Coauthor-CS \cite{DBLP:journals/corr/abs-1811-05868}, Coauthor-Phy \cite{DBLP:journals/corr/abs-1811-05868}. (2) Social networks including WikiCS \cite{DBLP:journals/corr/abs-2007-02901}, Amazon-Photos \cite{DBLP:journals/corr/abs-1811-05868}, Amazon-Computers \cite{DBLP:journals/corr/abs-1811-05868}. Details of datasets and experimental settings can be found in Appendix \ref{datasets} and Appendix \ref{setting}.
\begin{table*}
\centering
\captionsetup{font=small}
\caption{Summary of performance on node classification in terms of $F_1$-micro score in percentage with standard deviation. The highest performance is highlighted in boldface.}
\label{Node Classification}
\scriptsize
\setlength{\tabcolsep}{3mm}{
\begin{tabular}{lllllllll}
\toprule
Datasets & Cora & Citeseer & Pubmed & WikiCS & A-Photo & A-Com & Co-CS & Co-Phy \\
\midrule
RawFeat & 64.23$\pm$1.09 & 64.47$\pm$0.72 & 84.70$\pm$0.25 & 71.98$\pm$0.00 & 87.37$\pm$0.48 & 78.80$\pm$0.17 & 91.87$\pm$0.11 & 94.44$\pm$0.17 \\
DeepWalk & 75.16$\pm$0.98 & 52.67$\pm$1.34 & 79.91$\pm$0.33 & 74.35$\pm$0.06 & 89.44$\pm$0.11 & 85.68$\pm$0.06 & 84.61$\pm$0.22 & 91.77$\pm$0.15 \\
DW+Feat & 77.53$\pm$1.44 & 62.50$\pm$0.61 & 84.85$\pm$0.45 & 77.21$\pm$0.03 & 90.05$\pm$0.08 & 86.28$\pm$0.07 & 87.70$\pm$0.04 & 94.90$\pm$0.09 \\
\cmidrule(r){1-9}
GAE & 81.63$\pm$0.72 & 67.35$\pm$0.29 & 83.72$\pm$0.32 & 70.15$\pm$0.01 & 91.62$\pm$0.13 & 85.27$\pm$0.19 & 90.01$\pm$0.71 & 94.92$\pm$0.07 \\
VGAE & 82.08$\pm$0.45 & 67.85$\pm$0.65 & 83.82$\pm$0.14 & 75.63$\pm$0.19 & 92.20$\pm$0.11 & 86.37$\pm$0.21 & 92.11$\pm$0.09 & 94.52$\pm$0.00 \\
\cmidrule(r){1-9}
DGI & 82.60$\pm$0.40 & 68.80$\pm$0.70 & 86.00$\pm$0.10 & 75.35$\pm$0.14 & 91.61$\pm$0.22 & 83.95$\pm$0.47 & 92.15$\pm$0.63 & 94.51$\pm$0.52 \\
GRACE & 83.70$\pm$0.68 & 72.39$\pm$0.21 & 86.35$\pm$0.17 & 79.56$\pm$0.02 & 92.79$\pm$0.11 & 87.20$\pm$0.32 & 93.01$\pm$0.06 & 95.56$\pm$0.05 \\
GCA & 83.81$\pm$0.91 & 72.85$\pm$0.17 & 86.61$\pm$0.12 & 80.07$\pm$0.12 & 93.10$\pm$0.07 & 87.70$\pm$ 0.43 & 93.08$\pm$0.06 & 95.61$\pm$0.01 \\
AD-GCL & 83.93$\pm$0.55 & 72.78$\pm$0.23 & 85.26$\pm$0.41 & 78.06$\pm$0.35 & 91.36$\pm$0.16 & 85.96$\pm$ 0.37 & 93.17$\pm$0.02 & 95.66$\pm$0.06 \\
\cmidrule(r){1-9}
\textbf{InfoAdv} & \textbf{84.82$\pm$0.83} & \textbf{73.00$\pm$0.42} & \textbf{86.87$\pm$0.28} & \textbf{81.41$\pm$0.03} & 93.54$\pm$0.04 & \textbf{88.27$\pm$0.45} & 93.20$\pm$0.05 & \textbf{95.73$\pm$0.05} \\
\; \textbf{(w/o) Gen} & 84.40$\pm$0.99 & 72.51$\pm$0.48 & 86.67$\pm$0.23 & 79.63$\pm$0.48 & 93.26$\pm$0.12 & 88.06$\pm$0.39 & \textbf{93.23$\pm$0.14} & 95.67$\pm$0.09 \\
\; \textbf{(w/o) Reg} & 84.36$\pm$0.54 & 72.74$\pm$0.27 & 86.82$\pm$0.25 & 80.97$\pm$0.19 & \textbf{93.80$\pm$0.12} & 87.61$\pm$0.11 & 93.04$\pm$0.06 & 95.65$\pm$0.04 \\
\cmidrule(r){1-9}
GCN & 82.69$\pm$1.08 & 72.09$\pm$0.78 & 84.81$\pm$0.40 & 77.19$\pm$0.12 & 92.42$\pm$0.22 & 86.51$\pm$0.54 & 93.03$\pm$0.31 & 95.65$\pm$0.16 \\
GAT & 83.60$\pm$0.29 & 72.56$\pm$0.44 & 85.27$\pm$0.15 & 77.65$\pm$0.11 & 92.56$\pm$0.35 & 86.93$\pm$0.29 & 92.31$\pm$0.24 & 95.47$\pm$0.15 \\
\bottomrule
\end{tabular}}
\end{table*}
\begin{table*}[h]
\captionsetup{font=small}
\centering
\caption{ Summary of performance on link prediction in terms of Area Under Curve (AUC) and Average Precision (AP) in percentage. The highest performance is highlighted in boldface.}
\label{Link Prediction}
\scriptsize
\setlength{\tabcolsep}{1.8mm}{
\begin{tabular}{llllllllllllllllll}
\toprule
\multirow{2}{*}{Datasets} & \multicolumn{2}{c}{Cora} & \multicolumn{2}{c}{Citeseer} & \multicolumn{2}{c}{Pubmed} & \multicolumn{2}{c}{WikiCS} & \multicolumn{2}{c}{A-Photo}& \multicolumn{2}{c}{A-Com} & \multicolumn{2}{c}{Co-CS} & \multicolumn{2}{c}{Co-Phy} & \\
\cmidrule(r){2-18}
& AUC & AP & AUC & AP & AUC & AP & AUC & AP & AUC & AP & AUC & AP & AUC & AP & AUC & AP & \\
\midrule
DGI & 83.45 & 82.95 & 85.36 & 86.94 & 95.24 & 94.53 & 78.18 & 79.30 & 87.47 & 86.93 & 85.25 & 85.22 & 89.99 & 89.01 & 84.35 & 83.80 & \\
GRACE & 80.65 & 81.64 & 87.81 & 89.52 & 94.99 & 94.68 & 95.28 & 94.78 & 93.99 & 92.60 & \textbf{94.53} & \textbf{93.87} & 93.82 & 93.44 & 87.95 & 86.50 & \\
GCA & 83.93 & 83.92 & 88.81 & 89.54 & 95.22 & 94.50 & 94.27 & 94.38 & 96.07 & 95.23 & 93.13 & 93.00 & 94.25 & 93.83 & 85.33 & 81.63 & \\
AD-GCL & 80.99 & 82.15 & 89.13 & 89.89 & 96.20 & 95.57 & 94.12 & 93.43 & 85.06 & 82.65 & 86.07 & 85.31 & 93.39 & 93.04 & 88.35 & 87.04 & \\
\cmidrule(r){1-18}
\textbf{InfoAdv} & \textbf{84.39} & \textbf{85.96} & \textbf{93.67} & \textbf{94.15} & 96.43 & 96.10 & \textbf{96.71} & \textbf{97.02} & \textbf{97.25} & \textbf{96.73} & 93.46 & 92.94 & \textbf{95.02} & \textbf{94.57} & 90.82 & 89.40 & \\
\; \textbf{(w/o) Gen} & 81.91 & 83.28 & 91.77 & 91.38 & \textbf{96.69} & \textbf{96.28} & 96.45 & 96.66 & 93.67 & 91.98 & 94.17 & 93.49 & 94.80 & 94.24 & \textbf{92.28} & \textbf{91.33} &\\
\; \textbf{(w/o) Reg} & 82.31 & 83.90 & 82.28 & 85.19 & 95.13 & 94.85 & 95.64 & 95.81 & 95.77 & 94.81 & 93.07 & 92.42 & 93.43 & 92.94 & 88.19 & 86.40 & \\
\bottomrule
\end{tabular}}
\end{table*}
\subsection{Performance on Downstream Tasks (AQ1)}
In this section we empirically evaluate our proposed instantiation of InfoAdv on the following three tasks.
\subsubsection{Generalization Performance on Node Classification Task}
We firstly evaluate InfoAdv on node classification using eight open benchmark datasets against SOTA methods. In each case, InfoAdv learns node representations in a fully self-supervised manner, followed by linear evaluation of these representations. We compare InfoAdv with both unsupervised methods and supervised methods\cite{DBLP:conf/iclr/KipfW17,DBLP:conf/iclr/VelickovicCCRLB18}. The unsupervised methods further include traditional graph embedding methods\cite{DBLP:conf/kdd/PerozziAS14} and SOTA self-supervised learning methods\cite{DBLP:conf/www/0001XYLWW21,DBLP:conf/nips/SureshLHN21,DBLP:journals/corr/abs-2006-04131,DBLP:conf/iclr/VelickovicFHLBH19,DBLP:journals/corr/KipfW16a} (both generative and contrastive). The results is summarized in Table \ref{Node Classification}, we can see that InfoAdv outperforms all methods mentioned above, which verifies the effectiveness of our proposed generalization bound and its principle that a balance should be struck between InfoMax and generalization.
\subsubsection{Generalization Performance on Link Prediction Task}
Considering that our method follow the GCL framework, We then evaluate InfoAdv on link prediction using eight public benchmark datasets against the SOTA graph contrastive methods. methods. In each case, we split edges in original graph into training set, validation set, and testing set, where InfoAdv is only trained in the training set. We introduce negative edge samples and calculate the link probability via the inner product of the representations. The results are summarized in Table \ref{Link Prediction}, we can see that InfoAdv outperforms all of the baseline methods, also verifying its generalization capability.
\subsubsection{Generalization Performance with Increasing Noise Rate}
We finally conduct generalization analysis for InfoAdv over noisy graph with increasing noise rate. Previous GCL methods fit the pretext task under single InfoMax principle, i.e., to capture as much information as possible from the original graph. However, to improve generalization performance, only the downstream task related information is required, whereas the irrelevant information or the noise information will decrease the generalization performance on the downstream task. Based on this, we verify the generalization ability of GCL model by adding noise, indicating that the generalizable model is more resistant to noise.
As the results shown in Figure \ref{rub}, we have two key observations: (1) with the rising noise rate, InfoAdv and its variants (w/o Gen and w/o Reg) perform stable, compared to GRACE \cite{DBLP:journals/corr/abs-2006-04131} and GCA\cite{DBLP:conf/www/0001XYLWW21} who drop sharply. (2) Within the different generalization components of InfoAdv, the performance of InfoAdv outperforms InfoAdv w/o Gen and InfoAdv w/o Reg. The two observation support that, our method shows powerful generalization ability in the noisy graph, which is more challenging for generalization ability.
\begin{figure*} [h]
\centering
\includegraphics[width=0.9\linewidth,height=4cm]{pic/rub.png}
\captionsetup{font=footnotesize}
\caption{The performance of InfoAdvs and baselines on two datasets with increasing noise rate. The dotted line is the mean scores, the color band is the standard deviation. The x-axis is the hyperparameter for introducing noise, and the noise rate increases synchronously with the hyperparameter. The y-axis is the node classification $F_1$-micro score.} \label{rub}
\end{figure*}
\subsection{Ablation Studies (AQ2)}
In this section, we study the impact of each generalization component to answer Q2. "(w/o) Gen" denotes without view generator. "(w/o) Reg" denotes without the KL regularization term in target encoder. The results are presented in the bottom of Table \ref{Node Classification} and Table \ref{Link Prediction}, where we can see that both KL term and view generator improve model performance consistently. In addition, the combination of the two generalization components further benefit the performance.
\subsection{Key Hyperparameter Sensitivity (AQ3)}
In this section, we study the sensitivity of two critical hyperparameters: the balance for KL regularization term $\lambda$, and the edge-drop probability of the view learned from view generator $p_{ea}$.
We show the model sensitivity on the transductive node classification task on Cora, under the perturbation of these two hyperparameters, that is, varying $\lambda$ from $1e-07$ to $1e+04$ and $p_{ea}$ from 0.1 to 0.9 to plot the 3D-surface for performance in terms of $F_1$-micro.
The result is shown in Figure \ref{hyper_lamb}, where four key conclusions can be observed,
\begin{itemize}
\item [(1)] The model is relatively stable when $\lambda$ is not too large, as the model performance remains steady when $\lambda$ varies in the range of $1e-07$ to $1e+02$.
\item [(2)] The model stays stable even when $\lambda$ is very small, since the model maintains a high level of performance even $\lambda$ reachs as small as $1e-07$.
\item [(3)] Higher edge-drop probability of the view learned from view generator helps model reach better performance, as the best performance occurs when $p_{ea}=0.7$.
\item [(4)] The KL regularization component does work. If we do not use the KL term, that is to set $\lambda=0$, the performance drops sharply.
\end{itemize}
\begin{figure*}[!htbp]
\centering
\includegraphics[width=0.85\linewidth]{./pic/hyper_lamb2.png}
\captionsetup{font=small}
\caption{The \textbf{left} side is the 3D-surface for the performance on Cora dataset, whose three axis are the balance for KL regularization term $\lambda$, the edge-drop probability of the view learned from view generator $p_{ea}$, and the performance $F_1$-micro score. The \textbf{middle} is the colorbar for the performance, indicating that the warmer the color is, the better the model performs. The \textbf{right} side is the heatmap for the 3D-surface when it is observed from the top, i.e., $F_1$-micro axis.} \label{hyper_lamb}
\end{figure*}
\subsection{Update Frequency Ratio (AQ4)}
This section is devoted to analysis update frequency ratio between view generator and decoder.
As is shown in the left side of Figure \ref{fr}, we can see that, at the beginning, with the increasing of the update frequency ratio, i.e., view generator update more and more frequently than decoder, the performance would as well improve and the convergence speed would become faster. when ratio increases up to '10:1', the performance reach the highest. However, when the view generator update too frequently, e.g., up to '100:1' the performance would drop soon.
As is shown in the right side of Figure \ref{fr}, we can see that, with view generator update with more and more high frequency, the edge preserving rate drops as well, indicating that more edges would be removed and making the task more difficult. However, when ratio increases too high up to '100:1', at first, the edge preserving rate dose keep in the lowest level, but with the training continuing, its preserving rate would begin to increase even higher than '10:1'.
These phenomenon may indicates that, the appropriate increase of the update frequency ratio would help to gain better generalization performance.
\begin{figure*}[h]
\centering
\includegraphics[width=0.9\linewidth]{./pic/fr.png}
\captionsetup{font=footnotesize}
\caption{The \textbf{left} side is the performance curve of 4 different update frequency ratio, whose x axis is the training epoch and the y axis is the $F_1$ score; The \textbf{right} side is the edge preserving rate curve of the same 4 different update frequency ratio, whose x axis is the training epoch and the y axis is the edge preserving rate} \label{fr}
\end{figure*}
\section{Related Work}
In the field of GCL, many work are concentrated on how to do better graph augmentations. Some of them are heuristical, e.g., GCA, GraphCL and JOAO \cite{DBLP:conf/www/0001XYLWW21,DBLP:conf/nips/YouCSCWS20,DBLP:conf/icml/YouCSW21}. Specially, JOAO \cite{DBLP:conf/icml/YouCSW21} dose learn augmentations with an adversarial optimization framework, but their objective is the combination of augmentation types, not the augmentation view. Most recently, AutoGCL \cite{yin2022autogcl} propose a data augmentation strategy. SAIL \cite{yu2022sail} consider the discrepancy of node proximity measured by graph topology and node feature. There are also works focus on the generalization ability of GCL. \cite{trivedi2022understanding} theoretically find that GCL generalization error bound with graph augmentations is linked to the average graph edit distance between classes. AF-GCL \cite{wang2022augmentation} consider GCL generalization for both homophilic graph and heterophilic graph.
Contrastive information bottleneck \cite{DBLP:conf/nips/SureshLHN21,DBLP:conf/nips/XuCLCZ21,DBLP:conf/nips/Tian0PKSI20}, which is proposed recently dose shares some ideas with our InfoAdv. The common idea of us is the MI minimization. However, there are several fundamental differences as follows.(1) The main contribution of this paper, as well as the original motivation of InfoAdv framework, is the generalization error metric of GCL and its MI upper bound, which, neither AD-GCL \cite{DBLP:conf/nips/SureshLHN21}, InfoGCL \cite{DBLP:conf/nips/XuCLCZ21} nor \citet{DBLP:conf/nips/Tian0PKSI20} involve. (2) As for self-supervised version of Tian et al. and AD-GCL, (despite of the field differences between CV and Graph) they focus only on the MI between two views for contrastive learning, while we concern the MI between GCL input-output. Our method can theoretically cover theirs due to the data processing inequality. (3) Specifically for AD-GCL, they leverage an MI lower bound to conduct MI minimization. However in terms of MI minimization, a single MI lower bound can not guarantee MI is truly minimized. Our method consider MI upper bound, which is more reasonable and could ensure the right MI minimization. (4) As for semi-supervised version of Tian et al. and InfoGCL, they are in need of knowledge from downstream task, which can hardly be satisfied in self-supervised learning. (5) AD-GCL, InfoGCL and Tian et al. define their MI between contrastive views, whereas we define MI between input data and output hypothesis, which is not bound with contrastive learning and can be easily extended to other learning paradigm such as predictive(generative) learning.\cite{DBLP:journals/corr/abs-2102-10757,DBLP:journals/corr/abs-2006-08218}.
Specificcally, for AD-GCL, we provide a more detailed comparison: Theoretically, (1) We provide a metric termed GCL-GE to bridge the generalization gap between GCL upstream and downstream , while AD-GCL does not connect this gap. (2) We prove a realizable bound for GCL-GE optimization even for agnostic downstream task, while AD-GCL does not consider. Methodologically, (1)AD-GCL leverages an MI lower bound (i.e., InfoNCE) to conduct MI minimization. However in terms of MI minimization, a single MI lower bound can not guarantee MI is truely minimized. On the contrary, our method consider MI upper bound (i.e., KL regularization ), which is more reasonable and could ensure the right MI minimization. (2) AD-GCL only improve GCL via data augmentation, while our method consider two components, both data augmentation and representation learning. Experimentally, We modify AD-GCL to node-level classification task and the results show that our InfoAdv outperforms AD-GCL on all eight popular datasets as below. (See Section 5, Table \ref{Node Classification} and Table \ref{Link Prediction})
There are plenty of work done for the generalization error of supervised learning, among which, traditional works draw bounds according to VC dimension, Rademacher complexity \cite{DBLP:books/daglib/0033642} and PAC-Bayesian \cite{DBLP:journals/ml/McAllester99}. Recently, the MI bounds based on information theory \cite{DBLP:conf/aistats/RussoZ16,DBLP:conf/nips/XuR17,DBLP:conf/nips/AsadiAV18,DBLP:conf/colt/SteinkeZ20,DBLP:journals/corr/abs-2106-10262} for generalization error has aroused widespread interests. For the generalization error of contrastive learning, Arora et al. \cite{DBLP:conf/icml/SaunshiPAKK19} propose a learning framework and come up with a Rademacher complexity bounds for its generalization error. Moreover, Nozawa et al. \cite{DBLP:conf/uai/NozawaGG20} adopt PAC bounds based on Arora's framework. Our different with those work are as below: (1) We prove a MI upper bound, which is a better bound compared to Arora et al. and Nozawa et al. (2) We propose a generalization error metric explicitly inspired by supervised learning, which is not mentioned in Arora et al. and Nozawa et al. (3) We extend Arora's framework to the graph(GNN) scenarios.
\section{Conclusion}
In this paper, we first propose GCL-GE, a metric for GCL generalization ability. Considering the intractability of the metric itself due to the agnostic downstream task, we theoretically prove a mutual information upper bound for it from an information-theoretic perspective. Then, guided by the bound, we design a GCL framework named InfoAdv with enhanced generalization ability, which jointly optimizes the generalization metric and InfoMax to strike the right balance between pretext tasks fitting and the generalization ability on downstream tasks. Finally, We empirically validate our theoretical findings on a number of representative benchmarks, and experimental results demonstrate that our model achieves state-of-the-art performance. Our InfoAdv principle is suitable not only for contrastive learning, but also other learning paradigm like predictive(generative) learning, which will be left to the further exploration.
\newpage
\bibliographystyle{unsrtnat}
|
1,116,691,500,883 | arxiv | \section{Introduction}\label{sec:sec1}
Droplet impact on solid or liquid interfaces is a ubiquitous and fascinating phenomenon in nature \cite{Martin1993DropletImpact, Yarin2005Review}, and has a wide range of applications; these include inkjet-printing, spray-painting, spray-cooling, internal combustion engines, fire suppression, deposition of solder bumps on printed circuit boards, surface-cleaning, and cell-printing. The outcome after a droplet impacts on a surface is a fascinating problem and has received considerable attention in the literature dating back to over a century ago \cite{Worthington1908Book}. Various impacting substrates have been employed, ranging from dry \cite{Kolinski2014PRL} and pre-wetted surfaces \cite{Cossali1997}, surfaces with physical \cite{Range1998RoughImpact} and chemical heterogeneities \cite{Chen2010wettabilityImpact, Kannan2008hydrophobicImpact}, thin \cite{Wang2000thinLiquidFilm} and deep liquid layers \cite{Adomeit2000PIV, Blanchette2009DropletCoal}, porous media \cite{Ding2011PorousJFM}, heated \cite{Bernardin1997, Tran2012PRLLeidenfrost}, sublimating \cite{Antonini2013SublimationImpact}, and electrically-conducting \cite{Deng2010Electric},
elastic membranes \cite{Pepper2008ElaticMembrane}, and granular materials \cite{Marston2010OnPowder}. Non-Newtonian \cite{Bergeron2000NaturePolymerAdditive, Smith2010PolynerAdditivesPRL}, surfactant-laden \cite{Gatne2009Surfactant}, and liquid-metallic droplets \cite{Aziz2000LiquidMetal, Dykhuizen1994LiquidMetal} have also been studied. Different outcomes of impact have been observed, such as bouncing, coalescence, partial coalescence, splashing. The transitions among different regimes have been investigated \cite{Cossali1997, Rioboo2003, Wang2000thinLiquidFilm}.
Depending upon the choice of impacting surfaces and droplet physical properties, the impact dynamics can be extremely complex. Thus, analysing the accompanying phenomena during the impact process, and understanding the physics behind the phenomena are of critical importance. Generally, the impact dynamics of droplets is dominated by the interplay between surface tension, inertia, and viscous forces; this may be modified by the presence of non-Newtonian and/or physico-chemical and/or thermal effects. The relative importance of these forces can be quantified using dimensionless numbers, such as the Ohnesorge and Weber numbers. These dimensionless numbers are used widely in the investigation of droplet impact, e.g., in defining the regimes of impact \cite{Pan2007} and in developing empirical correlations \cite{Scheller1995, Cossali1997}.
Although droplet impact phenomena have been investigated for more than a century, it is the recent development of high-speed imaging that allows researchers to unveil the fast evolving dynamic features of the phenomena \cite{Thoroddsen2008HighSpeedPhotography}. Many phenomena occurring during impact remain poorly understood, and continue to attract interest. These phenomena include the collapse of the gas layer beneath the droplet during the impact \cite{deRuiter2012GasLayer, Driscoll2011GasLayerInterference, Tran2013GasLayerJFM}, the entrapment of air bubbles when a droplet impacts on a liquid surface \cite{Bouwhuis2012BubbleTrapPRL, Hasan1990JFM}, and the ejection of liquid sheets during the impact and its subsequent breakup into secondary droplets \cite{Thoraval2012LiquidSheetPRL, Thoroddsen2012LiquidSheetJFM}. Bird et al.\ \cite{bird2009inclinedSplashing} found that during the impact of droplets on angled or moving surfaces, the asymmetry leads to an azimuthal variation of the ejected rim, and the tangential component of impact can act to enhance or suppress a splash. \v{S}ikalo et al.\ \cite{Sikalo2005Inclined} studied the impact of droplets on inclined dry and wetted surfaces, and found the bouncing on dry smooth or wetted surfaces, and no bouncing on rough surfaces. Gilet and Bush \cite{Bush2012WetInclinedImpact} studied the bouncing of droplets on an inclined substrate with a thin layer of high-viscosity fluid, and analysed the contact time and the conversion of energy. The liquid layer with high viscosity was used to ensure a smooth impact surface which facilitates droplet bouncing. Despite the large number of studies on droplet impact, the impact of droplets on flowing liquid films, to the best of our knowledge, has not been studied. This is surprising since it is a common phenomenon occurring in many two-phase flows, such as annular flows \cite{Hewitt1978book}, spray-cooling \cite{Kim2007SprayCoolingReview}, and spray-painting. The interaction between the droplet and the flowing liquid film could produce unique dynamic features distinct from the impact phenomena on quiescent liquids. This will be the subject of the present paper.
A falling liquid film is used as the flowing liquid film for droplet impact in this study. Falling films are common in nature, important to many industrial applications, and are characterised by rich nonlinear dynamics \cite{Chang1994FallingFilmReview, Chang2002FFbook, CrasterMatar2009RMP, Kalliadasis2011FFbook}. The pioneering study of Kapitza was performed half a century ago \cite{Kapitza1948FallingFilm}. Since then, the flow in falling liquid films has been studied extensively through experimental measurements \cite{Adomeit2000PIV, Zhou2009FFthickness}, low-dimensional modelling \cite{Scheid2006FallingFilm3DJFM, Shkadov1967Model}, and full numerical simulations \cite{Gao2003FFVOF}. The effects of many factors that influence the flow in falling films have been investigated, such as the effects of thermocapillarity \cite{Frank2006FFThermocapillary}, electric fields \cite{Jayaratne1964, Tseluiko2006FFElec}, and surfactants \cite{Ji1994FFsurfactant, Strobel1969FFsurfactant}. Different processes that may be involved in falling films have also been studied, such as heat transfer \cite{Scheid2008FallingFilm3DTemperatureEPL}, mass transfer \cite{Yang1992FFadsorption}, chemical reactions \cite{Dabir1996FFreaction}, and phase change \cite{Palen1994FFevaporation}. Even though the flow of falling liquid films has been widely studied in the literature, it still remains an important research topic and attracts the attention of researchers of various disciplines. In this study, we use high-speed imaging to examine the impact of droplets of controlled diameter, angle of incidence and speed, on isothermal falling films in order to explore how the film dynamics influence the impact phenomena. In addition to the Ohnesorge and Weber numbers, this system is also parameterised by the film Reynolds number, which is based on the film mean thickness and velocity. The results of our work show that the film dynamics have a profound effect on the impact process, giving rise to phenomena that are not observed in the quiescent film case.
The rest of this paper is organised as follows. The experimental system is described in Section \ref{sec:sec2}, including the falling film rig, the droplet generation unit, and the high-speed imaging system. The experimental results are presented in Section \ref{sec:sec3}. A regime map of droplet impact is created, and the effects of the Weber, Ohnesorge, and Reynolds numbers are studied by varying the impact speeds, the droplet sizes, and the flow rates of the falling film. A simple model based on lubrication theory is used to explain the bouncing phenomenon. Finally, concluding remarks are provided in Section \ref{sec:sec4}.
\section{Experimental method}\label{sec:sec2}
\subsection{Falling liquid film}
The experimental setup used to study droplet impact on flowing liquid films is shown in Figure \ref{fig:fig01}. The falling film rig comprised a rigid and impermeable plate 41 cm long and 28 cm wide made of a titanium foil with a 50 $\mu$m thickness inclined at 45$^\circ$ to the horizontal. The liquid used was deionised water of density $10^3$ kg/m$^3$, viscosity $10^{-3}$ Pa$\cdot$s, and surface tension 72.8 mN/m, and the flow rate was controlled by a valve and a variac transformer for the pump. An ultrasonic flow metre (Flownetix 100 series, Flownetix, UK) was used to measure the flow rate of the falling film, which was in the range 0.9$\pm$0.1--10.7$\pm$0.3 l/min. The lower limit of the flow rate of the liquid film is the minimum flow rate that can ensure the whole substrate is wetted. The mean film thicknesses $h$ and the mean flow speed $\bar{u}$ were estimated using the Nusselt solution \cite{Kalliadasis2011FFbook} to be in the range 0.29--0.65 mm, and 0.19--0.98 m/s, respectively. The Reynolds numbers of the falling films $\text{Re}\equiv \rho q/(w \mu) $ were therefore in the range 54--636, where $\rho$ and $\mu$ are the density and the viscosity of the liquid, $q$ and $w$ are the flow rate and the width of the falling liquid film, respectively. Since laminar-turbulence transition in falling films happens at $\text{Re}>1000$ \cite{Chang1994FallingFilmReview}, the flow in the falling liquid film studied here is laminar.
\begin{figure}
\centering
\includegraphics[scale=0.6]{figure01}
\caption{Schematic diagram of the experimental setup for droplet impact on falling liquid films which comprises a falling film rig, a droplet-generation unit, and a high-speed imaging system. }\label{fig:fig01}
\end{figure}
\subsection{Droplet generation unit}
Droplets of uniform size were generated from the tip of blunt syringe needles. Deionised (DI) water was pumped into the needle using a syringe pump (Braintree Scientific Inc., UK) at constant flow rates. The droplets broke up at the tip of the syringe needle in the dripping mode, fell under their own weights, and impacted obliquely onto the inclined liquid film. The dripping mode and the gravity-driven acceleration of the droplets ensure that the process is highly reproducible. To study the effect of droplet size on the impact process, the droplet size was varied via a change in the diameter of the needle. Three droplet diameters were studied, 3.3, 4.0, and 5.2 mm with standard deviation less than 0.1 mm, while the corresponding Ohnesorge number,$\text{Oh}\equiv \mu /\sqrt{\rho \gamma d}$, were 0.0021, 0.0019, and 0.0016, respectively. Here, $\rho $ is the density of the liquid, $\gamma$ is the surface tension, $\mu$ is the dynamic viscosity, and $d$ is the droplet diameter before impact. To study the effect of the droplet speed on the impact process, the impact speeds of the droplets were varied by changing the vertical distance of the needle towards the falling liquid film from 1 to 50 cm. The corresponding speeds of the droplet $v_0$ were in the range 0.30$\pm$0.02--2.99$\pm$0.09 m/s, and the corresponding Weber numbers, $\text{We}\equiv \rho dv_{0}^{2}/\gamma $, were in the range of 4--630 for different droplet sizes. The impact point on the liquid film is 34 cm from the inlet of the falling film, which is larger than 500 times of the film thickness of the Nusselt solution \cite{Kalliadasis2011FFbook} at the highest flow rate studied, and allows the development of the film flow on the substrate. The details of the flow in the falling film will be discussed in Subsection \ref{sec:sec3_1}.
\subsection{high-speed imaging system}
The droplet impact process was recorded using a high-speed camera (Olympus i-SPEED 3) at a frame rate of 5000 frames per second. A macro lens (Sigma EX 105mm) was used to capture the details during the impact. Two halogen lamps (500W) were used for illumination. To minimize the heating effect of the halogen lamps, the lamps were kept on only when the camera was recording. Side-view images were captured at a resolution of 804$\times$396 pixels (about 40 $\mu$m/pixel) with 10$^\circ$ deviation from the plane of the film, while top view images were capture at a resolution of 528$\times$600 pixels perpendicular to the film (about 40 $\mu$m/pixel). The high-speed images were analysed using customised Matlab routines. The speed and the size of the droplets were measured from the high-speed images.
\section{Results and discussion}\label{sec:sec3}
In this section, we present a discussion of our results starting with a brief description of the wavy falling film dynamics in the absence of droplet impact; this provides an indication of the expected state of the interface prior to droplet impact at various film $\text{Re}$. A detailed investigation of the wavy flow in the liquid film is beyond the scope of this study. We then describe the main phenomena associated with the impact process, before presenting a flow regime diagram as a function of system parameters. The results of a simple model based on lubrication theory are also discussed, and used to explain the trends associated with a subset of the phenomena observed.
\subsection{Wavy flow in liquid films}\label{sec:sec3_1}
The wavy flow in the falling liquid films was analysed from the top-view and side-view images near the point of the droplet impact, as shown in Figure \ref{fig:fig02}. At low $\text{Re}$ values, the liquid films are characterised by interfaces that exhibit mild deformations with occasional solitary waves. These waves are formed with coherent precursor waves, as shown in Figure \ref{fig:fig02}a. As $\text{Re}$ increases, the waves become more frequent and less coherent. The evolution of the waves can be seen from the time-space plot of a line along the flow direction, which is obtained from the top-view images, as shown in Figure \ref{fig:fig03}.
In a time-space plot, the (spatial) wave length is the vertical distance of consecutive waves, and the (temporal) frequency of the waves is the reciprocal of the horizontal distance of consecutive waves. The amplitude of the waves could not be measured in this experiment quantitatively due to the limitation of the experimental facility. The variation of the film thickness, however, is represented by the light intensity. Previous studies of falling liquid films \cite{Chang1994FallingFilmReview,Chang2002FFbook,CrasterMatar2009RMP,Kalliadasis2011FFbook} have shown that the amplitude of the waves is usually smaller or comparable to the mean film thickness, and the wave lengths are much larger than the mean film thickness. The surface curvature of the liquid film is very small as compared to the droplet curvature due to the small aspect ratio of the liquid film (thickness vs film length).
At low $\text{Re}$, the solitary wave in Figure \ref{fig:fig03}a travels downstream at a constant speed. With increasing $\text{Re}$, the waves interact with adjacent waves as they propagate downstream, as shown in Figure \ref{fig:fig03}b-c. The speed of the waves can be estimated from the gradient of the line of the waves in the time-space plot, which shows that the wave speed increases with $\text{Re}$.
In the following subsections, the impact processes on falling films at different conditions of $\text{We}$, $\text{Re}$, and $\text{Oh}$ numbers are analysed. Various combinations of the parameters result in different phenomena as observed in the experiments: bouncing, partial coalescence, total coalescence, and splashing, as will be detailed below. Because the waves on the liquid films are formed naturally and stochastically, the drop may impact on different parts of the waves. Therefore, the outcome of the impact may vary, and the results are presented in a probability manner.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{figure02}
\caption{Typical images of the wavy flow in the falling liquid films at different Reynolds numbers: (a) $\text{Re}= 54$, (b) $\text{Re}=245$, and (c) $\text{Re}=636$. The flow rates are 0.9, 4.1, and 10.7 l/min, respectively.}\label{fig:fig02}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{figure03}
\caption{Time-space plots showing the evolution of the wavy flow in the falling liquid films at different Reynolds numbers: (a) $\text{Re}= 54$, (b) $\text{Re}=245$, and (c) $\text{Re}=636$. The flow rates are 0.9, 4.1, and 10.7, l/min respectively. The speeds of the waves can be estimated from the gradient of the dashed lines of the waves. }\label{fig:fig03}
\end{figure}
\subsection{Bouncing}\label{sec:sec3_2}
`Bouncing' refers to situations wherein a droplet approaching the falling film, does not coalesce with it, but instead bounces back, as shown in Figure \ref{fig:fig04}, which was generated for $\text{Oh}=0.0021$, $\text{We}=4$, and $\text{Re}=636$. The coalescence of a droplet with a liquid film involves three stages: (i) droplet approach towards the film; (ii) the thinning of the thin, intervening gas layer between the droplet and the film; and (iii) the rupture of the gas layer. The first stage is driven by the momentum of the droplet gained from gravity, the second stage is resisted by the viscous force in the thin gas layer, and the third stage is dominated by capillary force. As the gas layer becomes thinner, the resistance becomes larger, according to lubrication theory \cite{Davis1989Lubrication, Deen1998TransportPhenomena}. Therefore, the gas layer could potentially stabilise the droplet against coalescence. Rupture of the gas layer occurs when its thickness decreases below a certain level so that van der Waals forces come into operation to destabilise the layer \cite{Hahn1985CoalescenceVanDerWaals}.
Prior to impact, the droplet is almost spherical (see Figure \ref{fig:fig04}a), as the capillary force minimizes the surface area and, in turn, the surface energy. Upon impact, the droplet progressively adopts a pyramidal (see Figure \ref{fig:fig04}e) and then a pancake shape (see Figure \ref{fig:fig04}g).
The latter increases the contact area of the droplet with the film and further retards the thinning of the gas layer, according to lubrication theory \cite{Davis1989Lubrication, Deen1998TransportPhenomena}.
The surface energy of the droplet, $E_\gamma$, can be estimated through image analysis by assuming a surface of revolution, ${{E}_{\gamma }}\equiv S\gamma $, where $S$ is the surface area of the droplet. The result shows that from the undistorted droplet shape (see Figure \ref{fig:fig04} c) to the maximum deformation (see Figure 4g), the surface energy increases from approximately $2.5\times10^{-6}$ J to $3.4\times10^{-6}$ J. Meanwhile, the kinetic energy before the impact, ${{E}_{v}}\equiv \frac{1}{2}m{{v}_{0}}^{2}$, and the loss of potential energy, ${{E}_{g}}\equiv mg\Delta z$, are approximately $8\times10^{-7}$ J and $5\times10^{-7}$ J, respectively. Here, $m$ is the mass of the droplet, and $\Delta z$ is the vertical distance that the droplet travels from the instant of apparent contact to the instant of maximum deformation. Therefore, the relative conversion of energy from kinetic and potential energy to surface energy during the process can be calculated from
\begin{equation}\label{eq:eq_Econversion}
\varepsilon \equiv \frac{\Delta {{E}_{\gamma }}}{{{E}_{v}}+{{E}_{g}}},
\end{equation}
which is estimated to be about 70\%. Such high ratio of energy conversion has also been found for droplet bouncing on horizontal and inclined wet surfaces \cite{Bush2012WetInclinedImpact}.
The Weber number is a key parameter that affects the thinning of the gas layer prior to possible rupture: lower Weber numbers (lower impact speeds) lead to longer gas layer drainage times, and, therefore, an increased probability of droplet bouncing. The momentum of the falling film, on the other hand, for a fixed inclination angle, increases with the film Reynolds number (the liquid flow rate). Therefore, droplets have a greater chance of bouncing on liquid films with higher Reynolds numbers. This is consistent with the phenomenon of droplet levitation at the hydraulic jump point of a liquid film produced from a vertical jet impinging on a horizontal surface \cite{Sreenivas1999LevitationJFM}. The details will be explained in detail in Section \ref{sec:sec3_7a}.
\begin{figure}
\centering
\includegraphics[scale=0.5]{figure04}
\caption{`Bouncing' of a droplet on a falling liquid film. The dimensionless parameters are $\text{Oh}=0.0021$, $\text{We}=4$, and $\text{Re}=636$. The droplet speed is 0.3 m/s, the flow rate of the falling film is 10.7 l/min, and the droplet diameter is 3.3 mm. See Ref.\ \cite{RefSM}, Movie 1.
}\label{fig:fig04}
\end{figure}
\subsection{Partial coalescence}\label{sec:sec3_3}
For a range of system parameters, the droplets were observed to undergo `partial coalescence' with the falling film. This corresponds to situations wherein, upon impacting the film, a fraction of the droplet coalesces with the film, but immediately after impact, a small daughter droplet or a cascade of daughter droplets \cite{Blanchette2006PartialCoalescence, Gilet2007PartialCoalescence} are created; an example of partial coalescence is shown in Figure \ref{fig:fig05}. It occurs as a consequence of small Weber numbers and small film Reynolds numbers.
The rupture of the gas layer starts with the formation of a liquid bridge connecting the droplet and the liquid film. Due to the large curvature at the liquid bridge and the corresponding large capillary force, the liquid bridge immediately grows, and produces capillary waves on the surface of the droplet and on the liquid film (see Figure \ref{fig:fig05}g and Figure \ref{fig:fig06} for the formation and the propagation of the capillary waves). The waves on the droplet then propagate rapidly towards the apex of the droplets (see Figure \ref{fig:fig06}c). Immediately after the collapse of the gas layer, the liquid at the bottom of the droplet is significantly affected by the capillary waves and is pulled towards the liquid film. However, the liquid at the droplet apex is less affected, and there is no sufficient time to catch up with that near the bottom of the droplet. Therefore, the droplet becomes a long thin liquid thread (see Figure \ref{fig:fig05}l), which eventually breaks up into two parts (see Figure \ref{fig:fig05}o) due to the inward momentum of the collapsing neck \cite{Blanchette2006PartialCoalescence}. The lower part mixes with the liquid in the falling film, while the upper part gradually falls on the liquid film due to gravity as a daughter droplet. The daughter droplet tends to be more stable against coalescence because of its smaller speed than the primary droplet before partial coalescence, as shown in Figure \ref{fig:fig05}p.
The surface energy after pinch-off of the daughter droplet (see Figure 5n) can be estimated through image analysis to be $E_{\gamma} \sim 6 \times 10^{-7}$ J. It is much smaller than the initial surface energy before the impact (see Figure 5a), $E_{\gamma} \sim 2.5 \times 10^{-6}$ J. This indicates significant energy loss due to viscous dissipation and the formation of
capillary waves during the process.
\begin{figure}
\centering
\includegraphics[scale=0.5]{figure05}
\caption{`Partial coalescence' of a droplet on a falling liquid film. The dimensionless parameters are $\text{Oh}=0.0021$, $\text{We}=4$, and $\text{Re}=54$. The droplet speed is 0.3 m/s, the flow rate of the falling film is 0.9 l/min, and the droplet diameter is 3.3 mm. See Ref.\ \cite{RefSM}, Movie 2.
}\label{fig:fig05}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.6]{figure06}
\caption{Formation and propagation of capillary waves immediately after the collapse of the gas layer between the droplet and the liquid film. The dimensionless parameters are $\text{Oh}=0.0021$, $\text{We}=4$, and $\text{Re}=54$. The droplet speed is 0.3 m/s, the flow rate of the falling film is 0.9 l/min, and the droplet diameter is 3.3 mm.
}\label{fig:fig06}
\end{figure}
\subsection{Total coalescence}\label{sec:sec3_4}
In contrast to the situations considered above, `total coalescence' refers to complete coalescence of the droplet with the falling film upon impact without forming smaller, secondary droplets; an example of total coalescence is shown in Figure \ref{fig:fig07} generated for $\text{Oh}=0.0021$, $\text{We}=76$, and $\text{Re}=636$. If the Weber number is above a certain level, the upper part of the droplet, follows the lower part of the droplet and coalesces with the liquid film (see Figure \ref{fig:fig07}f), without breaking up into a daughter droplet, as in the case of partial coalescence. In addition, the high-momentum liquid in the droplet pushes away the liquid in the flowing film and produces a crater (see Figure \ref{fig:fig07}g). The waves produced by this event propagate away from the crater and dissipate gradually (see Figure \ref{fig:fig07}h).
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{figure07}
\caption{`Total coalescence' of a droplet on a falling film. The dimensionless parameters are $\text{Oh}=0.0021$, $\text{We}=76$, and $\text{Re}=636$. The droplet speed is 1.3 m/s, the flow rate of the falling film is 10.7 l/min, and the droplet diameter is 3.3 mm. See Ref.\ \cite{RefSM}, Movie 3.
}\label{fig:fig07}
\end{figure}
\subsection{Splashing}\label{sec:sec3_5}
If the momentum of the droplet is sufficiently large, characterised by high $\text{We}$, the droplet drives the liquid in the film to form a `crown', which then disintegrates into many droplets, as shown in Figure \ref{fig:fig08}. This `splashing' phenomenon occurs because the gas layer separating the impacting droplet and the falling film undergoes rapid rupture, and capillary forces are not sufficiently strong so as to balance the large force per unit area at high Weber numbers. Thus, following impact, the liquid undergoes severe deformation leading to the formation of a cylindrical sheet of liquid around the droplet (see Figure \ref{fig:fig08}e), which then assumes a crown shape (see Figure \ref{fig:fig08}h). The crown rim then undergoes an Rayleigh-Plateau instability \cite{Zhang2010CrownRPInstability} that results in the creation and subsequent ejection of small droplets (see Figure \ref{fig:fig08}k). The crown formation and the ejection of small droplets for the impact of droplets on quiescent liquids has been well studied in the literature \cite{Martin1993DropletImpact, Wang2000thinLiquidFilm, Yarin2005Review, Thoraval2012LiquidSheetPRL, Thoroddsen2012LiquidSheetJFM}. Being different from impact on quiescent liquids, the wavy interface of the liquid film results in a more irregular crown structure and different sizes of the small droplets. The crown is also not symmetric due to the oblique impact angle \cite{bird2009inclinedSplashing}: the rim on the downstream side is much higher than that on the upstream side of the crown, which leads to the ejection of more droplets on the downstream side. The details will be discussed in Subsection \ref{sec:sec3_8}.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{figure08}
\caption{`Splashing' during droplet impact on a falling film. The dimensionless parameters are $\text{Oh}=0.0021$, $\text{We}=316$, and $\text{Re}=636$. The droplet speed is 2.67 m/s, the flow rate of the falling film is 10.7 l/min, and the droplet diameter is 3.3 mm. See Ref.\ \cite{RefSM}, Movie 4.
}\label{fig:fig08}
\end{figure}
\subsection{Regime map}\label{sec:sec3_6}
A regime map for droplet impact on falling liquid films is created by varying the flow rate of the falling film and the impact speed of the droplets; this is presented in Figure \ref{fig:fig09} as a $\text{Re}$ vs $\text{We}$ plot which shows the regime transitions as these parameters are varied for $\text{Oh}=0.0021$. For a given $\text{Re}$, large $\text{We}$ values lead to splashing, while moderate $\text{We}$ values lead to total coalescence. At the lowest $\text{We}$ studied, multiple phenomena are observed, namely bouncing, partial coalescence, and total coalescence, and their probability of occurrence in repeated experiments are plotted in Figure \ref{fig:fig10}.
As shown in Figure \ref{fig:fig10}, partial coalescence always occurs for the lowest $\text{Re}$ studied. As $\text{Re}$ increases, the probability of partial coalescence decreases, while that of total coalescence rises. This is because a thick liquid film facilitates the flow of liquid in the direction perpendicular to the film and allows the liquid near the top region of the droplet to catch up with that near the bottom region, thereby reducing the probability of formation of secondary droplets following impact. The probability of bouncing also increases with the film Reynolds number. This is consistent with the phenomenon of droplet levitation at the hydraulic jump point of a liquid film produced from a vertical jet impinging on a horizontal surface \cite{Sreenivas1999LevitationJFM}, which corresponds to a large $\text{Re}$ and negligible $\text{We}$ in Figure \ref{fig:fig09}.
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{figure09}
\caption{Regime map for droplet impact on falling liquid films. Here, `S', `T', and `M' designate `splashing', `total coalescence', and multiple phenomena as described in Figure \ref{fig:fig10}. The Ohnesorge number is fixed at $\text{Oh}=0.0021$. The droplet speed is in the range of 0.30--2.99 m/s, the flow rate of the falling film is in the range of 0.9--10.7 l/min, and the droplet diameter is 3.3 mm. The lines are used only to guide readers' eyes.
}\label{fig:fig09}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.46]{figure10}
\caption{Probability of occurrence of different phenomena as a function of $\text{Re}$ with $\text{Oh}=0.0021$ and $\text{We}=4$. The droplet speed is 0.30 m/s, the flow rate of the falling film is in the range of 0.9--10.7 l/min, and the droplet diameter is 3.3 mm.
}\label{fig:fig10}
\end{figure}
\subsection{Lubrication gas layer and force analysis}\label{sec:sec3_7a}
To explain the phenomena presented above, the gas layer between the droplet and the liquid film and the forces involved are analysed. Prior to the rupture of the gas layer between the droplet and the liquid film, the gas layer drains gradually. The drainage of the gas layer is controlled by the competition between the inertia of the approaching droplet and the lubrication force in the gas layer. The former serves as the driving force and is indicated by the normal component of the droplet speed to the falling film, while the latter serves as the resistance and is affected by the flow speed in the film. Thus, droplet bouncing is expected only at low normal speeds of impact.
Even though the thickness of the gas layer is too thin to be determined from the observation in Figure \ref{fig:fig04}, measurements of the film thickness for droplet impact on solid surfaces in the literature \cite{VeenDrainageInterferometry2012, deRuiter2012GasLayer} have shown that the thickness of the gas layer can be down to several micrometres. To estimate the lubrication force acted on the droplet by the gas layer during the impact process, two complementary methods are presented. The first method is through an analysis of the flow in the gas layer, the liquid film, and the droplet, while the second method relies on the analysis of the droplet deformation shown on the high-speed images.
\subsubsection{Estimation from the flow in the gas layer, the liquid film, and the droplet}
Due to the small thickness of the gas layer and the liquid film relative to the drainage length of the gas layer in the tangential direction, the flow in the gas layer and in the liquid film can be approximated using lubrication theory. The approximation is also applied in a penetration depth $h_d$ in the droplet, and the velocity gradient vanishes and the velocity approaches the droplet tangential speed at the penetration depth \cite{Li1996PenitrationDepth, Yeo2001233}, as shown in Figure \ref{fig:figA1}. We also neglect deformations of the film-gas interface in the present, approximate analysis:
\begin{equation}\label{eq:eqA1}
{{\mu }_{g}}\frac{{{\partial }^{2}}{{u}_{g}}}{\partial {{z}^{2}}}=\frac{\partial {{p}_{g}}}{\partial x},
\end{equation}
\begin{equation}\label{eq:eqA2}
{{\mu }_{l}}\frac{{{\partial }^{2}}{{u}_{l}}}{\partial {{z}^{2}}}=\frac{\partial {{p}_{l}}}{\partial x},
\end{equation}
\begin{equation}\label{eq:eqA3}
{{\mu }_{l}}\frac{{{\partial }^{2}}{{u}_{d}}}{\partial {{z}^{2}}}=\frac{\partial {{p}_{d}}}{\partial x},
\end{equation}
where the subscript $g$ indicates the gas layer, and $l$ the liquid film, and $d$ the droplet. Then the velocity profiles along the $z$ direction are parabolic:
\begin{equation}\label{eq:eqA4}
{{u}_{g}}={{A}_{g}}{{z}^{2}}+{{B}_{g}}z+{{C}_{g}},
\end{equation}
\begin{equation}\label{eq:eqA5}
{{u}_{l}}={{A}_{l}}{{z}^{2}}+{{B}_{l}}z+{{C}_{l}},
\end{equation}
\begin{equation}\label{eq:eqA6}
{{u}_{d}}={{A}_{d}}{{z}^{2}}+{{B}_{d}}z+{{C}_{d}},
\end{equation}
where the coefficients ${{A}_{g}}$, ${{B}_{g}}$, ${{C}_{g}}$, ${{A}_{l}}$, ${{B}_{l}}$, ${{C}_{l}}$, ${{A}_{d}}$, ${{B}_{d}}$, ${{C}_{d}}$ can be obtained by satisfying the following boundary conditions and closure equations. A no-slip boundary condition is imposed at the substrate, and continuity of velocity is imposed at the gas/liquid-film interface and at the gas/droplet interface,
\begin{equation}\label{eq:eqA7}
{{\left. {{u}_{l}} \right|}_{z=0}}=0,
\end{equation}
\begin{equation}\label{eq:eqA8}
{{\left. {{u}_{g}} \right|}_{z={{h}_{l}}}}={{\left. {{u}_{l}} \right|}_{z={{h}_{l}}}},
\end{equation}
\begin{equation}\label{eq:eqA9}
{{\left. {{u}_{g}} \right|}_{z={{h}_{0}}}}={{\left. {{u}_{d}} \right|}_{z={{h}_{0}}}},
\end{equation}
where ${{h}_{0}}\equiv {{h}_{l}}+{{h}_{g}}$. The matching of shear stress at the gas/liquid-film interface and at the gas/droplet interface yields
\begin{equation}\label{eq:eqA10}
{{\left. {{\mu }_{g}}\frac{\partial {{u}_{g}}}{\partial z} \right|}_{z={{h}_{l}}}}={{\mu }_{l}}{{\left. \frac{\partial {{u}_{l}}}{\partial z} \right|}_{z={{h}_{l}}}},
\end{equation}
\begin{equation}\label{eq:eqA11}
{{\left. {{\mu }_{g}}\frac{\partial {{u}_{g}}}{\partial z} \right|}_{z={{h}_{0}}}}={{\mu }_{l}}{{\left. \frac{\partial {{u}_{d}}}{\partial z} \right|}_{z={{h}_{0}}}}.
\end{equation}
The shear stress at the penetration depth of the droplet vanishes, and the speed approaches the tangential speed of the droplet with respect to the substrate, ${{U}_{0}}$:
\begin{equation}\label{eq:eqA12}
{{\left. {{u}_{d}} \right|}_{z={{h}_{2}}}}={{U}_{0}},
\end{equation}
\begin{equation}\label{eq:eqA13}
{{\mu }_{l}}{{\left. \frac{\partial {{u}_{d}}}{\partial z} \right|}_{z={{h}_{2}}}}=0,
\end{equation}
where ${{h}_{2}}\equiv {{h}_{l}}+{{h}_{g}}+{{h}_{d}}$. The flow rate of the liquid film is given by
\begin{equation}\label{eq:eqA14}
w\int_{0}^{{{h}_{l}}}{{{u}_{l}}dz}=q.
\end{equation}
The pressure gradients in the liquid film and in the gas layer are assumed to be the same because
film deformations are neglected, and normal viscous stresses correspond to high-order terms within lubrication theory; thus, Eqs.\ (\ref{eq:eqA1}) and (\ref{eq:eqA2}) yield
\begin{equation}\label{eq:eqA15}
{{\mu }_{g}}\frac{{{\partial }^{2}}{{u}_{g}}}{\partial {{z}^{2}}}={{\mu }_{l}}\frac{{{\partial }^{2}}{{u}_{l}}}{\partial {{z}^{2}}}.
\end{equation}
Therefore, the coefficients in Eqs.\ (\ref{eq:eqA4})--(\ref{eq:eqA5}) can be obtained:
\begin{widetext}
\begin{equation}\label{eq:eqA16}
\begin{aligned}
&{{A}_{g}}=-\frac{3{{\mu }_{l}}{{D}_{2}}}{{{h}_{l}}^{2}{{\mu }_{g}}{{D}_{1}}}, \quad
&{{B}_{g}}=\frac{2{{\mu }_{l}}{{D}_{3}}}{{{h}_{l}}^{2}{{\mu }_{g}}{{D}_{1}}}, \quad
&{{C}_{g}}=-\frac{\left( {{\mu }_{g}}-{{\mu }_{l}} \right)D_4}{{{h}_{l}}{{\mu }_{g}}{{D}_{1}}}, \quad
&{{A}_{l}}=-\frac{3{{D}_{2}}}{{{h}_{l}}^{2}{{D}_{1}}}, \quad
&{{B}_{l}}=\frac{2{{D}_{3}}}{{{h}_{l}}^{2}{{D}_{1}}}, \quad \\
&{{C}_{l}}=0, \quad
&{{A}_{d}}=\frac{D_5}{{{h}_{l}}^{2}\left( {{h}_{0}}-{{h}_{2}} \right){{D}_{1}}}, \quad
&{{B}_{d}}=-2{{A}_{d}}{{h}_{2}}, \quad
&{{C}_{d}}={{A}_{d}}h_{2}^{2}+{{U}_{0}},\quad
\end{aligned}
\end{equation}
where
\begin{equation*}\label{eq:eqA17}
\begin{aligned}
& {{D}_{1}}=3{{h}_{0}}^{2}{{\mu }_{g}}-3{{h}_{0}}^{2}{{\mu }_{l}}-{{h}_{l}}^{2}{{\mu }_{g}}+{{h}_{l}}^{2}{{\mu }_{l}}-3{{h}_{0}}{{h}_{2}}{{\mu }_{g}}-{{h}_{0}}{{h}_{l}}{{\mu }_{g}}+{{h}_{2}}{{h}_{l}}{{\mu }_{g}}+2{{h}_{0}}{{h}_{l}}{{\mu }_{l}} \\
& {{D}_{2}}=\left( {{h}_{0}}{{\mu }_{g}}-{{h}_{2}}{{\mu }_{g}}-2{{h}_{0}}{{\mu }_{l}}-2{{h}_{l}}{{\mu }_{g}}+2{{h}_{l}}{{\mu }_{l}} \right){q}/{w}\;+{{U}_{0}}{{h}_{l}}^{2}{{\mu }_{g}} \\
& {{D}_{3}}=\left( 3{{h}_{0}}^{2}{{\mu }_{g}}-3{{h}_{0}}^{2}{{\mu }_{l}}-3{{h}_{l}}^{2}{{\mu }_{g}}+3{{h}_{l}}^{2}{{\mu }_{l}}-3{{h}_{0}}{{h}_{2}}{{\mu }_{g}} \right){q}/{w}\;+{{U}_{0}}{{h}_{l}}^{3}{{\mu }_{g}} \\
& D_4= \left( 6{{h}_{0}}^{2}{{\mu }_{l}}-6{{h}_{0}}^{2}{{\mu }_{g}}+6{{h}_{0}}{{h}_{2}}{{\mu }_{g}}+3{{h}_{0}}{{h}_{l}}{{\mu }_{g}}-3{{h}_{2}}{{h}_{l}}{{\mu }_{g}}-6{{h}_{0}}{{h}_{l}}{{\mu }_{l}} \right){q}/{w}\;+{{h}_{l}}^{3}{{\mu }_{g}}{{U}_{0}} \\
& D_5=\left( 3{{h}_{0}}^{2}{{\mu }_{l}}-3{{h}_{l}}^{2}{{\mu }_{g}}+3{{h}_{l}}^{2}{{\mu }_{l}}+6{{h}_{0}}{{h}_{l}}{{\mu }_{g}}-6{{h}_{0}}{{h}_{l}}{{\mu }_{l}} \right){q}/{w}\;+\left( {{h}_{l}}^{3}{{\mu }_{g}}-3{{h}_{0}}{{h}_{l}}^{2}{{\mu }_{g}} \right){{U}_{0}}
\end{aligned}
\end{equation*}
\end{widetext}
Although the exact value of pressure could not be known due to the complex structure at the rim of the gas layer, the pressure difference between the gas layer and the ambient can be estimated from the pressure gradient,
\begin{equation}\label{eq:eqA18}
{{p}_{g}}-{{p}_{a}}\sim -\frac{d}{2}\frac{\partial {{p}_{g}}}{\partial x}=-\frac{d}{2}{{\mu }_{g}}\frac{{{\partial }^{2}}{{u}_{g}}}{\partial {{z}^{2}}}=-{{\mu }_{g}}{{A}_{g}}d,
\end{equation}
where we have made use of Eqs.\ (\ref{eq:eqA1}), (\ref{eq:eqA4}), (\ref{eq:eqA15}), and (\ref{eq:eqA16}). Therefore, the lubrication force can be estimated as
\begin{equation}\label{eq:eqA19}
{{F}_{\text{lubrication}}}=\int\limits_{A}{\left( {{p}_{g}}-{{p}_{a}} \right)dA}\sim -\frac{\pi }{4}{{d}^{3}}{{\mu }_{g}}{{A}_{g}}.
\end{equation}
The liquid film thickness ${{h}_{l}}$ can be estimated from the Nusselt solution for the falling liquid film \cite{Kalliadasis2011FFbook},
\begin{equation}\label{eq:eqA20}
{{h}_{l}}={{\left( \frac{3\mu q}{w\rho g\cos \theta } \right)}^{1/3}}.
\end{equation}
The penetration depth in the droplet is assumed to be $h_d=4h_g$ as in Ref.\ \cite{Yeo2001233}. Therefore, during the drainage of the gas layer, the magnitude of the lubrication force ${{F}_{\text{lubrication}}}$ can be estimated at different thicknesses of the gas layer, as shown in Figure \ref{fig:figA2}.
The lubrication force (i.e., the resistant force) being large than the normal component of the gravitational force (i.e., the driving force) is a minimum requirement of droplet bouncing. Therefore, the gravitational force of the droplet in the direction normal to the falling film, ${{F}_{\text{gravity}}}=\frac{1}{6}\pi {{d}^{3}}\rho g\cos \left( \theta \right)$, is used as a reference to scale the estimated lubrication force, where $\theta$ is the inclination angle of the falling film to the horizontal. Figure \ref{fig:figA2} shows that as the gas layer becomes thinner, the lubrication force becomes larger. It also shows that the lubrication force increases with increasing the film flow rate. Since a large lubrication force indicates a large resistance of the droplet, a slow drainage of the gas layer, and a high chance of droplet bouncing, the result in Figure \ref{fig:figA2} is consistent with the experimental observation shown in Figure \ref{fig:fig10}.
\begin{figure}
\centering
\includegraphics[scale=0.9]{figureA1}
\caption{Schematic diagram of the flow in the gas layer and the liquid film beneath the droplet during the impact on flowing liquid film.
}\label{fig:figA1}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.65]{figureA2}
\caption{Estimation of the lubrication force normalised by the gravitational force normal to the film, ${{F}_{\text{gravity}}}=\frac{1}{6}\pi {{d}^{3}}\rho g\cos \left( \theta \right)$, as a function of the flow rate of the liquid film at different thickness of the gas layer.
}\label{fig:figA2}
\end{figure}
\subsubsection{Estimation from the droplet deformation in high-speed images}
The force applied on the droplet by the falling liquid film via the gas layer can also be estimated from the droplet deformation in the high-speed images. When the droplet reaches the maximum deformation, it has a pancake shape, as shown in Figure \ref{fig:fig04}g. The pressure in the droplet can be estimated through the surface curvature of the droplet,
\begin{equation}\label{eq:eqL2}
{{p}_{d}}\sim \gamma \left( \frac{1}{R}+\frac{1}{r} \right)+{{p}_{a}}
\end{equation}
where the $R$ and $r$ are two principle radii of the curved droplet interface measured on the side of the non-spherical droplet, as shown Figure \ref{fig:fig10a}. Since the base of the droplet is almost flat and does not deform rapidly during the later stage of the drainage of the gas layer, the pressure in the droplet and in the gas layer can be regarded to be in equilibrium, and the pressure in the gas layer, ${{p}_{g}}$, can be approximated by the pressure in the droplet, ${{p}_{d}}$,
\begin{equation}\label{eq:eqL3}
{{p}_{g}}\sim \gamma \left( \frac{1}{R}+\frac{1}{r} \right)+{{p}_{a}}
\end{equation}
The lubrication force applied on the droplet by the liquid film via the gas layer can be estimated by multiplying the pressure in the gas layer and the base area of the droplet, ${{S}_{b}}\sim \pi {{\left( R-r \right)}^{2}}$. Therefore, the lubrication force is
\begin{equation}\label{eq:eqL4}
{{F}_{\text{lubrication}}}\sim \gamma \left( \frac{1}{R}+\frac{1}{r} \right)\pi {{\left( R-r \right)}^{2}}
\end{equation}
where the effect of the ambient pressure ${{p}_{a}}$ in Eq.\ (\ref{eq:eqL3}) is cancelled by the ambient pressure on the upper side of the droplet.
For the bouncing event shown in Figure \ref{fig:fig04}, image analysis shows that the ratio of the lubrication force to the gravitational force ${{{F}_{\text{lubrication}}}}/{{{F}_{\text{gravity}}}}\sim 10$. Therefore, the lubrication force by the gas layer on the droplet can be much larger than the gravitational force of the droplet, and can result in bouncing.
\begin{figure}
\centering
\includegraphics[scale=0.7]{figureA3}
\caption{Schematic diagram of the droplet deformation and the pressure.
}\label{fig:fig10a}
\end{figure}
At impact, both the gas-droplet and gas-film interfaces undergo deformation \cite{Tran2013GasLayerJFM} whose severity depends on the system parameters. An estimate of the liquid film deformation is obtained by comparing inertial, capillary, and viscous forces. The inertial force tends to deform the interface, while the capillary and viscous forces tend to resist this deformation. Their relative importance can be quantified by their ratios represented by the Weber number and the Reynolds number for film deformation, respectively given by:
\begin{equation}\label{eq:wedeform}
\text{We}_{\text{deformation}}=\frac{\rho \,{{\left( {{v}_{0}}\cos \theta \right)}^{2}}\,h}{\gamma }
\end{equation}
\begin{equation}\label{eq:redeform}
\text{Re}_{\text{deformation}}=\frac{\rho \,\left( {{v}_{0}}\cos \theta \right)\,h}{\mu }
\end{equation}
The characteristic velocity is the normal velocity of droplet towards the plate, ${{v}_{0}}\cos \theta $, while the characteristic length is the film thickness $h$ as calculated from the Nusselt solution \cite{Kalliadasis2011FFbook}. At an impact speed of 0.30 m/s, for instance, ${\text{We}_{\text{deformation}}}=0.18$, and ${\text{Re}_{\text{deformation}}}=62$, which indicate that the surface tension is very important in resisting the deformation of the film interface. Equations (\ref{eq:wedeform}--\ref{eq:redeform}) also indicate that thicker films which correspond to high film $\text{Re}$ are easy to deform. In addition, the relative importance between the viscous and capillary forces is quantified by
\begin{equation}\label{eq:cadeform}
{\text{Ca}_{\text{deformation}}}=\frac{{\text{We}_{\text{deformation}}}}{{\text{Re}_{\text{deformation}}}}=\frac{\mu {{v}_{0}}\cos \theta }{\gamma }
\end{equation}
which is in the range of 0.003--0.03 for different impact speeds (0.30--2.99 m/s), indicating that the capillary force is much stronger than the viscous force.
During impact, at larger $\text{Re}$ values of the liquid films, the average liquid films are thicker and more deformable since the restoring capillary force becomes progressively weaker. The deformation of the gas/liquid-film interface delays the drainage of the gas layer, preventing its rupture. In addition, the force exerted by the film on the droplet at high film $\text{Re}$, which is transmitted through the gas layer, pushes the droplet away from the impact region. These factors also could increase the probability of droplet bouncing.
To further understand the collapse of the gas layer, the lifetime of the gas layer was studied, and a histogram of this lifetime is plotted in Figure \ref{fig:fig11}. The starting time was determined from the high-speed images, which show noticeable capillary waves on the surfaces of the liquid film upon apparent impact. The ending time was determined from the images which show the rupture of the gas layer indicated by the capillary waves on the droplet and on the liquid film, as explained in Figure \ref{fig:fig06}.
For the bouncing phenomenon, the lift-off of the droplet away from the film was considered as the ending time. Due to this distinctive feature and the fast propagation of the capillary waves in the high-speed images, the life time of the gas layer can be determined at an accuracy of about 0.2--0.4 ms at the filming frame rate 5000 fps. As shown in Figure \ref{fig:fig11}, the bouncing and total coalescence phenomena correspond to long and short gas layer lifetimes, respectively. This indicates that delaying (accelerating) the rupture of the gas layer results in bouncing (total coalescence). This is because the speed of the liquid at the top region of the droplet can be retarded via droplet deformation during the thinning of the gas layer. Following rapid rupture of the gas layer, the fluid near the droplet apex has a higher chance of catching up with the fluid near the bottom of the droplet, preventing the formation of secondary droplets, and leading to total coalescence.
\begin{figure}
\centering
\includegraphics[scale=0.4]{figure11}
\caption{Histogram of the lifetime of the gas layer for three impact phenomena: bouncing, partial coalescence and total coalescence. The dimensionless parameters are $\text{Oh}=0.0021$, $\text{We}=4$, and $\text{Re}$ in the range of 54--636. The droplet speed is 0.30 m/s, the flow rate of the falling film is in the range of 0.9--10.7 l/min, and the droplet diameter is 3.3 mm.
}\label{fig:fig11}
\end{figure}
\subsection{Dimple formation and droplet ejection for large droplets}\label{sec:sec3_7}
The Ohnesorge number can be varied by changing the droplet size. We consider a case shown in Figure \ref{fig:fig12} characterised by $\text{Oh} = 0.0016$, $\text{We} = 6.5$, and $\text{Re} = 54$. These parameters are similar to those that were used to generate Figure \ref{fig:fig05}, which depicted a case of partial coalescence; the decrease in $\text{Oh}$ and the slight increase in $\text{We}$ were brought about by the increase in the droplet diameter from 3.3 mm to 5.2 mm. The case presented in Figure \ref{fig:fig11} illustrates an impact accompanied by the droplet surface switching from a convex to a concave shape, followed by the ejection of a small droplet. Inspection of Figure \ref{fig:fig11}a shows that the droplet shape prior to impact is far from spherical, as expected for large droplet size, i.e., large $\text{Oh}$ numbers. Upon apparent contact with the liquid film (see Figure \ref{fig:fig12}b), capillary waves are formed and propagate along the surface of the droplet towards the top of the droplet and also on the surface of the liquid film (see Figure \ref{fig:fig12}c-d). The liquid in the top region of the droplet is less affected by the impact than that near the bottom of the droplet and continues to move towards the film by inertia (see Figure \ref{fig:fig12}e-f). Following rupture of the gas layer, capillary waves form and propagate on the surface of the droplet towards its apex. The interaction of these waves with the downwards-moving fluid from the apex leads to the formation of a hat-shaped droplet (see Figure \ref{fig:fig12}g-h) followed by a dimple at the droplet apex (see Figure \ref{fig:fig12}i-k). The formation of the dimple is facilitated by the relative weakness of capillary forces for such large droplets and their consequent inability to act as effective restoring forces to counterbalance severe interfacial deformation. For the concave interface associated with the dimple, capillarity pulls the liquid at the rim of the dimple causing it to shrink (see Figure \ref{fig:fig12}l-n). The collision of the interface at the rim of the dimple results in the ejection of a small droplet (see Figure \ref{fig:fig12}o-p). Due to the flow in the liquid film, the droplet is not symmetric, and the small droplet is ejected in a direction which is not perpendicular to the substrate but deviated in the streamwise direction. The liquid eventually settles down as the energy from the impact is dissipated gradually (see Figure \ref{fig:fig12}q-r).
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{figure12}
\caption{Dimple formation and droplet ejection during the impact of a large droplet on a falling film. The dimensionless parameters are $\text{Oh}=0.0016$, $\text{We}=6.5$, and $\text{Re}=54$. The droplet speed is 0.3 m/s, the flow rate of the falling film is 0.9 l/min, and the droplet diameter is 5.2 mm. See Ref.\ \cite{RefSM}, Movie 5.
}\label{fig:fig12}
\end{figure}
\subsection{The asymmetric splashing process}\label{sec:sec3_8}
We devote the concluding part of this section to quantitatively examining the details of
asymmetric
splashing at different film flow rates, droplet speeds, and droplet sizes. The asymmetric splashing process was studied to consider the effects of relevant parameters and quantified by the propagation of the baseline of the droplet, as shown in Figure \ref{fig:fig13}a. The streamwise direction of the falling film is defined as the positive direction of the movement of the baseline, and the initial contact point of the droplet on the falling liquid film is defined as the origin. The initial contact time is determined by interpolating the interface position from the image sequence, which provides a higher precision of the initial contact time than the time delay between two consecutive frames. The length of the baseline of the droplet, and the position of the front and rear point is normalized by the diameter of the droplet, and the time is normalized by the speed and the diameter of the droplet:
\begin{equation}\label{eq:x}
\hat{x}=x/d
\end{equation}
\begin{equation}\label{eq:t}
\hat{t}=t{{v}_{0}}/d
\end{equation}
Since the droplet impact in the present study is different from the perpendicular impact of droplets on quiescent films, the propagation of the front and rear of the baselines are asymmetric, as shown in Figure \ref{fig:fig13}. The degree of asymmetry is influenced by the oblique impact of the droplet and the flow rate of the liquid film. Upon impact, the front point of the baseline moves along the flow direction, and the rear point of the baseline moves backwards. In addition, the front point moves at a higher speed than the rear point.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{figure13}
\caption{
Effects of the droplet size, the droplet speed, and the film flow rate on the propagation of the baseline during droplet splashing. (a) Schematic diagram of the definition of the coordinate system. (b) Effect of the droplet size. The droplet speed is 1.87 m/s, the film flow rate is 0.9 l/min, and correspondingly, $\text{Re}=54$ and $\text{We}$ in the range of 156--254. (c) Effect of the droplet speed. The film flow rate is 4.1 l/min, the droplet diameter is 3.3 mm, and correspondingly, $\text{Re}=245$ and $\text{Oh}=0.0021$. (d) Effect of the film flow rate. The droplet speed is 2.67 m/s, and the droplet radius is 3.3 mm, and correspondingly, $\text{Oh}=0.0021$ and $\text{We}= 316$. The inset shows two images of crowns at different film flow rates.
}\label{fig:fig13}
\end{figure}
The effect of the droplet size on the asymmetric splashing process is shown in Figure \ref{fig:fig13}b. The curves of the baselines for different droplet sizes collapse, even though the dimensional data show clear difference at different droplet sizes (see the inset plot of Figure \ref{fig:fig13}b). For smaller droplets, the baselines propagate much more slowly at both the front and the rear of the droplets, and the splashing process is much shorter than that for large droplets. This is because the mass and the momentum introduced into the film by small droplets are much less than those by large droplets.
The propagation of the baseline for droplet impact at different droplet speeds is shown in Figure \ref{fig:fig13}c. The propagation of the baseline appears to be weakly dependent on the droplet speed. The dimensional plot (see the inset plot of Figure \ref{fig:fig13}c) shows that with increasing the droplet speed, both the front and the rear points move faster in the downstream direction. This is mainly due to the fact that droplets with higher speeds (larger momentum) can overcome the resistance of impact and move faster downstream than droplets with lower speeds.
The effect of the film flow rate on the propagation of the baseline is shown in Figure \ref{fig:fig13}d. At a small flow rate, the liquid film is thin, and the liquid sheet of the crown is ejected close to the substrate. In contrast, at a high flow rate, the liquid film is thicker, and the liquid sheet of the crown is ejected close to the direction perpendicular to the substrate, as shown in the inset images of Figure \ref{fig:fig13}d. Similar phenomena also exist for droplet impact on quiescent liquid films \cite{Wang2000thinLiquidFilm}. Therefore, at a low film flow rate, the movement of the liquid sheet of the crown can promote the movement of the droplet baseline. With increasing the film flow rate, the film thickness increases, and the motion enhancing effect of the crown decrease. Consequently, the speeds of the front and rear points reduce. If the film flow rate increases further, the flow speed in the falling film also increases to the point where the film simply transports the droplet downstream so that both the front and rear points move in the streamwise direction.
\section{Conclusion}\label{sec:sec4}
In this paper, the oblique impact of droplets on inclined falling liquid films was studied using high-speed imaging. The results of droplet impact on flowing liquid films showed unique features different from the impact process on quiescent liquids. Under different impact conditions, various impact phenomena were observed: droplet bouncing, partial coalescence, total coalescence, and splashing. An impact regime map was generated for the impact of droplets on falling liquid films. Splashing phenomena occur at large Weber numbers, $\text{We}$, while other phenomena manifest themselves at low $\text{We}$. Low Reynolds numbers of the liquid films, $\text{Re}$, are associated with partial coalescence, while the probability of bouncing and total coalescence increases with $\text{Re}$. Dimple formation and droplet ejection were observed for the impact of large droplets. Splashing processes were quantified through the propagation of the droplet baselines during impact, and the effects of the droplet size, the droplet speed, and the flow rate of the falling liquid film were analysed.
The analysis of the lubrication force during the impact process shows that a higher flow rate in the liquid film produces a larger lubrication force, slows down the drainage process, and increases the probability of droplet bouncing.
The study mainly focused on the fluid dynamics of the droplet impact on falling liquid films. There are many open questions related to droplet impact that are worthy of investigation, such as the direct measurement of the flow field during the impact process, the evolution of the gas layer, and the effects of droplet impact on the flow in the liquid films. In various relevant applications, such as annular flow, spray cooling, and spray painting, the impact processes are much more complex. For example, the impact of swarms of droplets on liquid films involves the interaction of numerous impact processes; the flow of the gas phase may induce turbulence in the liquid films and droplet entrainment on the film surface. This investigation will deepen our understanding of the physics involved in the phenomena of droplet impact, leading to a marked improvement in processes and applications that rely on these phenomena.
\section*{Acknowledgements}
We would like to acknowledge the support of the Engineering and Physical Sciences Research Council, UK, through the Programme Grant, MEMPHIS (EP/K003976/1).
|
1,116,691,500,884 | arxiv | \subsubsection{#1}}
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\title[Dislocations in Multilattices]{Theoretical Study of Elastic Far-Field Decay from Dislocations in Multilattices}
\author{Derek Olson$^1$ and Christoph Ortner$^2$}
\date{\today}
\thanks{
$^1$Mathematical Sciences, Amos Eaton 301, Rensselaer Polytechnic Institute, 110 8th St., Troy, NY 12180, [email protected] \\
\indent $^2$Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK, [email protected] }
\begin{document}
\begin{abstract}
We extend recent results of~\cite{Ehrlacher2013} characterising the \textit{decay} of elastic fields generated by defects in crystalline materials to dislocations in multilattices. Specifically, we establish that the elastic field generated by a dislocation in a multilattice can be decomposed into a continuum field predicted by linearised elasticity, and a discrete and nonlinear core corrector representing the defect core. We discuss the consequences of this result for cell size effects in numerical simulations.
\end{abstract}
\maketitle
\section{Introduction}
A key approximation in all numerical simulations of crystalline defects is the boundary condition emulating the crystalline far-field. The ``quality'' of this boundary condition has a significant consequence for the severity of cell-size effects in such simulations. The study of these cell-size effects, such as~\cite{Ehrlacher2013}, and the development of new boundary conditions, including~\cite{2017-bcscrew}, begins with a characterisation of the elastic field surrounding the defect core, which was initiated in \cite{Ehrlacher2013}. Such a characterisation is also interesting in other contexts, e.g., in the study of defect interactions~\cite{2014-dislift}. In the present work, we extend results concerning the decay of elastic far-fields generated by defects in crystalline materials to {\em dislocations in multilattices} (also referred to as complex crystals).
While results of this kind have been known in the materials science community from computational experiments and justified by associated continuum results dating back to Volterra~\cite{volterra}, we fill a gap in the existing literature by producing the a rigorous proof of such decay estimates for dislocations in multilattices modeled via classical empirical potentials, i.e., in a discrete and fully nonlinear setting.
Our results are vital in establishing convergence of numerical methods for simulating crystal defects including direct atomistic simulations (c.f.~\cite{Ehrlacher2013}) as well as multiscale atomistic-to-continuum methods (c.f.~\cite{multiBlendingArxiv}). Indeed, such direct atomistic computational methods date back decades to~\cite{chang1966edge} which investigated computational methods for dislocations in iron (a simple crystal) and later works including~\cite{nandedkar1987atomic,Nandedkar1990} which investigated dislocations in silicon and other diamond cubic lattices (a multilattice). Thus, the extension of these decay results from simple lattices to multilattices is vital for physical applications as it enlarges the pool of admissible materials to include all physical crystals including graphene, silicon, and germanium (to name but a few).
A particularly important application that we have in mind are dislocations in ionic crystals, which provides two significant challenges compared to the setting in \cite{Ehrlacher2013}: multi-lattice structure and long-range (Coulomb) interaction. In the present work, we take the first step, establishing the necessary techniques for multi-lattices, but restrict ourselves to short-range interactions only. In future work, we plan to extend our theory to ionic interactions, which represent a significant additional technical challenge.
Not surprisingly, the decay estimates for the strain and strain gradients generated by a dislocation in the multilattice setting match those of the simple lattice. However, there is a perhaps striking difference in the proof. A fundamental tool that is used to prove decay of the elastic fields in the simple lattice setting is algebraic decay of the residual atomistic forces evaluated at a continuum elasticity predictor displacement. These residual forces are shown to decay at a rate of $|r|^{-3}$ in~\cite{Ehrlacher2013} where $r$ is the distance from the dislocation core, but perhaps surprisingly, these same forces do not in general decay at $|r|^{-3}$ in the multilattice setting. What is true in this setting, however, is that the {\em net force on a multi-lattice site} (the sum of the forces on all species of atoms at a single multilattice site) decay at $|r|^{-3}$. This fact, along with a strong localisation of multilattice shifts then turns out to be sufficient to prove the expected decay of the strain fields for multilattice dislocations.
This paper is organized as follows. First, we describe the multilattice structure and the specific material models admitted in our analysis of dislocations. The main task accomplished in this technical section is defining the potential energy for straight-line dislocations over an appropriate admissible space of multilattice displacements. We show this potential energy is well-defiend (finite) under a set of physically motivated assumptions and then state our main result concerning decay of the elastic fields generated by the dislocation. The proof is broken up in a number of separate steps in the following section which closely mimic the results~\cite{Ehrlacher2013} and~\cite{olsonOrtner2016}. Finally, we end with a straightforward consequence of the elastic decay rates: convergence of a direct atomistic method for a lattice statics simulation of a dislocation in an infinite crystal, followed by a numerical example of an edge dislocation in silicon confirming our theoretical predictions.
\section{Model Problem}
We begin by letting $\mathcal{L}$ denote a Bravais lattice:
\[
\mathcal{L} = {\sf B} \mathbb{Z}^3, \qquad {\sf B} \in \mathbb{R}^{3 \times 3}_+.
\]
A multilattice, or complex crystal, is obtained by taking a union of shifted lattices,
\[
\mathcal{M} = \bigcup_{\alpha = 0}^{S-1} \mathcal{L} + p^{\rm ref}_\alpha,
\]
where each $p^{\rm ref}_\alpha \in \mathbb{R}^3$ represents a shift vector. Without loss of generality, we shall always assume $p^{\rm ref}_\alpha = 0$. We refer to $\xi \in \mathcal{L}$ as a \textit{lattice site}, while $\xi+p^{\rm ref}_\alpha$ represents an actual atom location.
Hence, $\mathcal{L} + p^{\rm ref}_\alpha$ represents the lattice locations for the $\alpha$th species of atom.
We adopt several equivalent descriptions of the kinematics of the multilattice. We denote a deformation field by $y_\alpha(\xi)$ and a displacement field by $u_\alpha(\xi)$ for each species $\alpha$. Note carefully that the argument of both of these fields is a lattice site $\xi \in \mathcal{L}$. The relationship between the two is
\[
y_\alpha(\xi) = \xi + p^{\rm ref}_\alpha + u_\alpha(\xi).
\]
We collect the set of all deformations (and displacements) at a single lattice site into a tuple which we denote by
\[
\bm{y}(\xi) = \big(y_0(\xi), \ldots, y_{S-1}\big), \qquad \bm{u}(\xi) = \big(u_0(\xi), \ldots, u_{S-1}(\xi)\big).
\]
We now specify our model to a situation to model straight dislocations, mimicking the setup of~\cite{Ehrlacher2013}. For convenience, we shall assume that the lattice is oriented so that the dislocation direction is parallel to the $x_3$ direction and the Burger's vector is of the form
\[
b = (b_1, 0, b_3) \in \mathcal{L}.
\]
We further assume, without loss of generality, that the displacement fields are independent of the $x_3$-direction and thus only functions of $x_1$ and $x_2$. We denote the resulting two-dimensional reference lattice by
\[
\Lambda = {\sf A} \mathbb{Z}^2 := \{ (\ell_1, \ell_2) \,:\, \ell \in \mathcal{L} \}.
\]
Yet another way of describing the multilattice kinematics, motivated by the definition of the multilattice, is by using displacement-shift notation, $(U, \bm{p})$. Here, we set
\[
U(\ell) = u_0(\ell), \qquad p_\alpha(\ell) = u_\alpha(\ell) - u_0(\ell), \qquad \bm{p}(\ell) = \big( p_0(\ell), \ldots, p_{S-1}(\ell)\big),
\]
though other choices are available and may in fact be preferable depending on symmetries of the underlying multilattice (see, e.g.,~\cite{koten2013} for one such example).
Having described the multilattice kinematic variables, we now describe the basic assumptions on the potential energy of the multilattice that we require for our analysis. These assumptions will be quite general so as to allow wide-ranging applicability to any classical interatomic potential including (but not limited to) multi-body potentials, pair functionals, bond-order potentials, etc.
The foremost assumption we make is that the atomistic energy may be written as a sum over empirical \textit{site} potentials. Hence the energy of the lattice must be decomposable as a sum:
\begin{equation}\label{at_free}
\mathcal{E}^{{\rm a}}(\bm{u}) = \sum_{\ell \in \Lambda} V_\ell(D\bm{u}(\ell)),
\end{equation}
where $V_\ell: \big(\mathbb{R}^3\big)^{|\mathcal{R}|} \to \mathbb{R}$ is the site potential at site $\ell$,
\[
\mathcal{R} = \left\{(\rho \alpha\beta): \rho \in \Lambda, \alpha,\beta = 0, \ldots, S-1\right\}
\]
is an interaction range allowing us to index pairs of interacting atoms of species $\alpha$ and $\beta$ whose sites are connected by a vector $\rho$, and
\[
D\bm{u}(\ell) := \big(D_{(\rho \alpha\beta)} \bm{u}(\ell)\big)_{\rho \in \mathcal{R}}, \qquad D_{(\rho \alpha\beta)} \bm{u}(\ell) := u_\beta(\ell+\rho) - u_\alpha(\ell)
\]
is the interaction stencil of finite differences needed to compute the energy at site $\ell \in \Lambda$. We assume $\mathcal{R}$ is finite (this assumption will be justified shortly) and satisfies the conditions
\begin{align}
& {\rm span}\{ \rho \,|\, (\rho\alpha\alpha) \in \mathcal{R} \} = \mathbb{R}^2 \text{ for all $\alpha \in \mathcal{S}$}, \label{cond1} \\
& (0\alpha\beta) \in \mathcal{R} \quad \text{for all $\alpha \neq \beta \in \mathcal{S}$ } \label{cond2}.
\end{align}
These two conditions, as well as a further condition encoding slip invariance (see condition~\eqref{cond3}) of the site potential may always be met by enlarging the interaction range if necessary. We set
\[
\mathcal{R}_1 = \{\rho \in \Lambda : (\rho \alpha\beta) \in \mathcal{R}\}
\]
and use $r_{\rm cut}$ to denote the cut-off distance for the potential so that if $|\rho| > r_{\rm cut}$, then $(\rho \alpha\beta) \notin \mathcal{R}$.
The site potential, $V_\ell(\cdot)$, is defined by
\begin{equation}\label{ell_potential}
V_\ell(D\bm{u}(\ell)) = V(D\bm{u}^0(\ell) + D\bm{u}(\ell)) - V(D\bm{u}^0(\ell)),
\end{equation}
where in this equation, $V$ represents a site potential for a defect-free lattice, and $\bm{u}^0$ represents a predictor displacement derived from solving a linear elastic model of a dislocation. (This predictor shall be defined in~\eqref{cond3}).
Having introduced the site potential for the defect and defect-free lattice, we briefly pause to introduce notation for the derivatives of the site potential. As arguments of $V$ and $V_\ell$ are finite differences, $D_{(\rho \alpha\beta)} \bm{u}$, for $(\rho \alpha\beta) \in \mathcal{R}$, we will write derivatives of the site potentials with respect to each of these arguments as follows. Let $\bm{g} = (g_{(\rho \alpha\beta)})_{(\rho \alpha\beta) \in \mathcal{R}}$. Then
\begin{align*}
[V_{\ell,(\rho \alpha\beta)}(\bm{g})]_{i} :=~& \frac{\partial V_\ell(\bm{g})}{\partial g_{(\rho \alpha\beta)}^i }, \quad i = 1,\ldots, 3, \\
V_{\ell,(\rho \alpha\beta)}(\bm{g}) :=~& \frac{\partial V_\ell(\bm{g})}{\partial g_{(\rho \alpha\beta)}}, \\
[V_{\ell,(\rho \alpha\beta)(\tau\gamma\delta)}(\bm{g})]_{ij} :=~& \frac{\partial^2 V_\ell(\bm{g})}{\partial g_{(\tau\gamma\delta)}^j \partial g_{(\rho \alpha\beta)}^i}, \quad i,j = 1,\ldots, 3, \\
V_{\ell,(\rho \alpha\beta)(\tau\gamma\delta)}(\bm{g}) :=~& \frac{\partial^2 V_\ell(\bm{g})}{\partial g_{(\tau\gamma\delta)} \partial g_{(\rho \alpha\beta)}},
\end{align*}
and so on with analogous definitions for $V(\cdot)$ except that we will drop the usage of the comma as a subscript when writing derivatives of $V(\cdot)$.
We now further define the potential energy for the defect-free (or homogeneous lattice) as
\begin{equation}\label{energy_hom}
\mathcal{E}^{{\rm a}}_{\rm hom}(\bm{u}) = \sum_{\ell \in \Lambda} V(D\bm{u}) .
\end{equation}
The relevant function space we introduce, over which $\mathcal{E}^{\rm a}_{\rm hom}$ and $\mathcal{E}^{\rm a}$ will be defined, is a quotient space of multilattice displacements whose (semi-)norm,
\[
\| \bm{u}\|_{{\rm a}_1}^2 := \sum_{\ell \in \Lambda} |D\bm{u}(\ell)|_\mathcal{R}^2,
\qquad \text{where } |D\bm{u}|_{\mathcal{R}}^2 := \sum_{(\rho \alpha\beta) \in \mathcal{R}} |D_{(\rho \alpha\beta)} \bm{u}(\ell)|^2,
\]
is finite. Because this is only a semi-norm, we form the quotient with the kernel of the semi-norm, which is the set of constant multilattice displacements, to obtain the spaces
\begin{align*}
\bm{\mathcal{U}} := \mathcal{U}/ \mathbb{R}^3, \qquad \text{where}
\quad \mathcal{U} := \left\{\bm{u}:\mathcal{L}^S \to \mathbb{R}^3, \|\bm{u}\|_{{\rm a}_1} < \infty \right\}.
\end{align*}
It is then proven in~\cite{olsonOrtner2016} that $\bm{\mathcal{U}}_0$, defined by
\begin{align*}
\mathcal{U}_0 :=~& \left\{\bm{u} \in \mathcal{U} : Du_0, u_\alpha - u_0 \,
\mbox{have compact support for each $\alpha$} \right\}, \\
\bm{\mathcal{U}}_0 :=~& \mathcal{U}_0 / \mathbb{R}^3,
\end{align*}
is dense in $\bm{\mathcal{U}}$ and subsequently that
\begin{theorem}[Olson and Ortner 2017 \cite{olsonOrtner2016}]\label{well_defined}
If the homogeneous multilattice reference configuration, $\bm{u} = \bm{0}$, is an equilibrium of the defect free energy, that is,
\begin{equation}\label{ostrich1}
\sum_{\ell \in \Lambda} \sum_{(\rho \alpha\beta) \in \mathcal{R}} V_{(\rho \alpha\beta)}(\bm{0}) \cdot D\bm{v}(\ell) = 0, \quad \forall \, \bm{v} \in \bm{\mathcal{U}}_0,
\end{equation}
and if the homogeneous site potential is ${\rm C}^4$ with uniformly bounded derivatives, then the energy functional, $\mathcal{E}^{\rm a}_{\rm hom}(\bm{u})$, is well-defined and ${\rm C}^3$ on $\bm{\mathcal{U}}$.
\end{theorem}
\medskip
A further assumption on the site potential and multilattice is that the defect-free multilattice be a stable equilibrium of the defect free energy. In essence, this amounts to saying that our model is physically reasonable and is equivalent to the usual assumption of phonon-stability made in the solid-state physics community. For a discussion of this, see~\cite{olsonOrtner2016,weinan2007cauchy}. Mathematically, this amounts to a discrete ellipticity condition and allows us to make use of a wide variety of estimates for elliptic systems (in particular for Green's functions for elliptic equations) of equations after proper translation.
\begin{assumption}\label{coercive}
The perfect multilattice reference configuration, $\bm{u} = \bm{0}$,
is a stable equilibrium of $\mathcal{E}^{\rm a}_{\rm hom}$ in the sense that there exists $\gamma_{\rm a} > 0$
such that
\begin{equation} \label{eq:stab-hom}
\<\delta^2\mathcal{E}^{\rm a}_{\rm hom}(\bm{0})\bm{v},\bm{v}\> \geq~ \gamma_{\rm a}\|\bm{v}\|_{{\rm a}_1}^2 , \quad \forall \, \bm{v} \in \bm{\mathcal{U}}.
\end{equation}
\end{assumption}
\medskip
Having rigorously defined the defect-free energy and function space over which it is defined, we are now in a position to rigorously define the dislocation energy~\eqref{at_free} and the admissible space of displacements over which it is defined. To that end, we must {\bfseries(1)} encode slip invariance of the atomistic energy into the site potential $V$, {\bfseries(2)} define the admissible function space as a proper subset of $\bm{\mathcal{U}}$ allowing us to employ a finite interaction range in the reference configuration, and {\bfseries(3)} define the predictor displacement $\bm{u}^0$ utilized in~\eqref{ell_potential}.
\subsection{Slip Invariance and Admissible Function Space}
To accomplish these three tasks, we assume a dislocation core position $\hat{x} = (\hat{x}_1, \hat{x}_2) \notin \Lambda$ and assume the dislocation branch cut is given by
\[
\Gamma = \left\{(x_1,x_2) \in \mathbb{R}^2 : x_1 \geq \hat{x}_1\right\},
\]
which is consistent with our assumption that the Burgers vector is parallel to
the $x_1$-direction.
Our final assumption on the homogeneous site potential, $V$, is that it it is slip invariant: if we define the operator, $S_0$, acting on multilattice displacements by ($b_{12}$ represents the projection of the Burger's vector to the $(x_1,x_2)$ plane)
\[
(S_0w)_\alpha(x) = \begin{cases}
&w_\alpha(x) ,\qquad \qquad x_2 > \hat{x}_2 \\
&w_\alpha(x-b_{12}) - b, \qquad x_2 < \hat{x}_2,
\end{cases}
\]
then
\begin{equation}\label{cond3}
\begin{split}
V(DS_0w(\ell)) = V(Dw(\ell)), \qquad \forall \ell_2 > \hat{x}_2 \\
V(DS_0w(\ell+b_{12})) = V(Dw(\ell)), \qquad \forall \ell_2 < \hat{x}_2.
\end{split}
\end{equation}
This condition ensures that the energy of the lattice remains invariant under crystallographic slip by a lattice vector. However, as noted in~\cite{Ehrlacher2013}, in order for this condition to not invalidate our assumption of a finite atomistic interaction range, we must define the admissible displacement space for the dislocation energy~\eqref{at_free} as a proper subset of $\bm{\mathcal{U}}$.
For this, we define a continuous, piecewise linear, nodal interpolant of a lattice function $u$ by $Iu$. This can be done by creating a partition of the domain whose vertices are exactly the lattice sites $\ell \in \Lambda$ and taking $I$ to be the standard $\mathcal{P}_1$ interpolant. We can then choose a global bound, $m_{\mathcal{A}}$, for $\|\nabla I u_\alpha\|_{L^\infty}, \|I p_\alpha\|_{L^\infty}$ and a radius $r_{\mathcal{A}}$ large enough so that
\begin{equation}\label{admissible}
\begin{split}
\mathcal{A} := \big\{\bm{u} \in \bm{\mathcal{U}} : \|\nabla Iu_\alpha \|_{L^\infty}, \|I p_\alpha \|_{L^\infty} < m_{\mathcal{A}}, \, \mbox{and} \, |\nabla Iu_\alpha(x) |, |Ip_\alpha(x) | < 1/2, \, x > &r_{\mathcal{A}}, \, \forall \, \alpha \big\},
\end{split}
\end{equation}
contains the possible minimizers of the dislocation energy. Arguing as in~\cite[Appendix B]{Ehrlacher2013}, the principle idea here is that the finite energy criterion on $\| \cdot \|_{{\rm a}_1}$ in the definition of $\bm{\mathcal{U}}$ implies $\nabla I u_\alpha \to 0$ as $|x| \to \infty$ and similarly for $Ip_\alpha \to 0$. Thus, $m_{\mathcal{A}}$ and $r_{\mathcal{A}}$ may always be chosen large enough so that a \textit{particular} local minimum of the dislocation energy is in $\mathcal{A}$. But then we may always increase these parameters so that all elements of $\bm{\mathcal{U}}$ within some ball of arbitrary radius about this minimum are also contained in $\mathcal{A}$, which permits us to perform our local calculus arguments. Full details may be found in~\cite[Appendix B]{Ehrlacher2013}.
We may then formulate the slip invariance condition by defining a mapping, $S$, of both lattice and multilattice functions by
\[
(Su)(\ell) = \begin{cases} &u(\ell), \qquad\qquad \ell_2 > \hat{x}_2 \\
&u(\ell-b_{12}), \qquad \ell_2 < \hat{x}_2
\end{cases}, \qquad (S\bm{u})_\alpha(\ell) = \begin{cases} &u_\alpha(\ell), \qquad\qquad \ell_2 > \hat{x}_2 \\
&u_\alpha(\ell-b_{12}), \qquad \ell_2 < \hat{x}_2
\end{cases}
\]
and a mapping, $R = S^*$, of lattice displacements by
\[
Ru(\ell) = \begin{cases} &u(\ell), \qquad \qquad \ell_2 > \hat{x}_2 \\
&u(\ell+b_{12}), \qquad \ell_2 < \hat{x}_2.
\end{cases}
\]
The slip invariance condition can now be expressed (using the same notation as~\cite{Ehrlacher2013}) as
\begin{equation}\label{slip}
V\big(D(\bm{u}^0 + \bm{u})(\ell)\big) = V\big(RDS_0(\bm{u}^0 + \bm{u})(\ell)\big), \qquad \forall \ell \in \Lambda, \bm{u} \in \mathcal{A}.
\end{equation}
We next fix a dislocation core radius $\hat{r}$ (which is defined in Lemma~\ref{ml_thm_vs}) and upon defining
\[
\Omega_\Gamma = \{(x_1, x_2) \in \mathbb{R}^2 : x_1 \geq \hat{x}_1 \} \setminus B_{\hat{r}+b_{1}}(\hat{x}),
\]
we (likewise to~\cite{Ehrlacher2013}) define the ``elastic strains''
\begin{equation}\label{ml_strain}
e(\ell) := \big(e_{(\rho \alpha\beta)}(\ell)\big)_{(\rho \alpha\beta) \in \mathcal{R}}, \qquad e_{(\rho \alpha\beta)}(\ell) = \begin{cases} &RD_{(\rho \alpha\beta)}S_0\bm{u}^0(\ell), \qquad \ell \in \Omega_{\Gamma} \\
&D_{(\rho \alpha\beta)}\bm{u}^0(\ell), \qquad \ell \notin \Omega_\Gamma,
\end{cases}
\end{equation}
and
\begin{equation}\label{fancy_D}
\tilde{D}\bm{u}(\ell) := \big(\tilde{D}_{(\rho \alpha\beta)}\bm{u}(\ell)\big)_{(\rho \alpha\beta) \in \mathcal{R}}, \qquad \tilde{D}_{(\rho \alpha\beta)}\bm{u}(\ell) = \begin{cases} &RD_{(\rho \alpha\beta)}S\bm{u}(\ell), \qquad \ell \in \Omega_{\Gamma} \\
&D_{(\rho \alpha\beta)}\bm{u}(\ell), \qquad \ell \notin \Omega_\Gamma.
\end{cases}
\end{equation}
Using this notation, the slip invariance condition~\eqref{slip} may be written as
\begin{equation}\label{slip_strain}
V\big(D(\bm{u}^0 + \bm{u})(\ell)\big) = V\big(e(\ell) + \tilde{D}\bm{u}(\ell)\big).
\end{equation}
Moreover, we have the identity
\begin{equation}\label{slip_fd_id}
\tilde{D}_{(\rho \alpha\beta)} \bm{u}(\ell) = \tilde{D}_\rho u_0 + \tilde{D}_\rho p_\beta(\ell) + p_\beta(\ell) - p_\alpha(\ell),
\end{equation}
which is intuitive, but can be proven via tedious algebraic manipulations and considering the cases {\bfseries{(1)}} $\ell \notin \Omega_\Gamma$ and {\bfseries{(2)}} $\ell \in \Omega_\Gamma$. We have included these manipulations in Appendix~\ref{tedious}.
\subsubsection{Continuum-Elasticity Dislocation Predictor}
Having recalled the well-posedness of the defect-free energy, $\mathcal{E}^{\rm a}_{\rm hom}$, in Theorem~\ref{well_defined} and having described the fundamental assumptions on the energy (smoothness of the site potential and the coercivity condition of Assumption~\ref{coercive}) and the slip invariance condition on the site potential (condition~\eqref{slip_strain}), we are now in a position to complete the definition of the dislocation defect energy first alluded to in~\eqref{at_free}. Specifically, it remains to define the predictor $\bm{u}^0 = (U^0, \bm{p}^0)$ utilized in~\eqref{ell_potential}.
As is done in the simple lattice case~\cite{Ehrlacher2013}, this will be accomplished by a slight modification of the solution of a linearized, elastic problem, where the elasticity tensor is taken from the linearized Cauchy--Born~\cite{cauchy,born1954} model. Thus, we shall define $(U^{\rm lin})$ by
\begin{equation}\label{lin_elastic}
\begin{split}
\nabla \cdot (\mathbb{C} \nabla U^{\rm lin}) =~& 0, \\
U^{\rm lin}(x+) - U^{\rm lin}(x-) =~& -b, \quad \mbox{on $\Gamma$,} \\
\nabla_{e_2} U^{\rm lin}(x+) - \nabla_{e_2} U^{\rm lin}(x-) =~& 0, \qquad \mbox{on $\Gamma$,}
\end{split}
\end{equation}
where $\mathbb{C}$ is the linearized Cauchy-Born (see~\cite{cauchy,born1954}) tensor for a multilattice defined by
\begin{equation}\label{elastic_tensor}
\begin{split}
W({\sf F}, \bm{p}) =~& V\Big(\big({\sf F} \rho + p_\beta - p_\alpha\big)_{(\rho \alpha\beta) \in \mathcal{R}}\Big) \\
\mathbb{C}_{ijkl} =~& \frac{\partial^2}{\partial {\sf F}_{ij}\partial {\sf F}_{kl}} \min_{\bm{p}} W({\sf F}, \bm{p}).
\end{split}
\end{equation}
It was shown in~\cite[Equation 3.11]{olsonOrtner2016} that Assumption~\ref{coercive} implies that linearized Cauchy--Born tensor satisfies a Legendre-Hadamard condition, and therefore the first set of three equations in~\eqref{lin_elastic} is solvable by the classical techniques of Hirth and Lothe~\cite{hirth1982theory}. As we are working with a multilattice, we then must obtain a corresponding set of shift fields, $\bm{p}^{\rm lin}$. In the Cauchy--Born theory for multilattices, the shift fields are obtained by minimization of the energy density~\cite{born1954}:
\begin{equation}\label{p_min}
\bm{p} =~ {\rm argmin}_{\bm{p}} V\Big(\big(\nabla_\rho U^{\rm lin} + p_\beta - p_\alpha\big)_{(\rho \alpha\beta) \in \mathcal{R}}\Big),
\end{equation}
but we define $\bm{p}^{\rm lin}$ by
\begin{equation}\label{p_lin_elastic}
\frac{\partial^2}{\partial \bm{p} \partial \bm{p} } \big[V\big(\bm{0}\big)\big]\bm{p}^{\rm lin} = -\partial_{\bm{p}{\sf F}} \big[V\big(\bm{0} \big) \big] \nabla U^{\rm lin}.
\end{equation}
which is equivalent to performing one step of Newton's method to the minimization problem~\eqref{p_min} where the initial shift fields are taken to be the reference shifts in the perfect multilattice.
We have carefully not yet defined the predictor, $\bm{u}^0 = (U^0, \bm{p}^0)$, to be exactly $(U^{\rm lin},\bm{p}^{\rm lin})$ as the precise definition requires the introduction of a smooth transition function, $\eta:\mathbb{R}\to\mathbb{R}$, which satisfies $\eta(x) = 1$ for $x \geq 1$, $\eta(x) = 0$ for $x \leq 0$, and $\eta'(x) > 0$ for $0 < x < 1$; and an argument function $\arg:\mathbb{R}^2 \to [0, 2\pi)$ where $\arg(x)$ is the angle between the positive $x$-axis and $x$. We then set our predictor displacement and shift fields to be
\begin{equation}\label{predictor_def}
\begin{split}
U^0(x) =~& U^{\rm lin}(\zeta^{-1}(x)), \qquad \zeta(x) := x - b_{12}\eta(|x-\hat{x}|/\hat{r}) \arg(x-\hat{x}),
\end{split}
\end{equation}
and $\bm{p}^0(x)$ is obtained from
\begin{equation}\label{p_predictor_def}
\frac{\partial^2}{\partial \bm{p} \partial \bm{p} } \big[V\big(\bm{0} \big)\big]\bm{p}^0 = -\partial_{\bm{p}{\sf F}} \big[V\big(\bm{0} \big)\big] \nabla U^0.
\end{equation}
We note that the modification of $U^{\rm lin}$ in~\eqref{predictor_def} is due to our need to obtain a smooth predictor in the far-field as well as the defect core as alluded to in~\cite{Ehrlacher2013}. The function $\zeta$ is defined verbatim from~\cite{Ehrlacher2013}, and hence all of the estimates of \cite[Lemma 3.1]{Ehrlacher2013} apply equally to $U^0$ in the present work. Moreover, it is straightforward to deduce corresponding estimates for each $p^0_\alpha$ from equation~\eqref{p_predictor_def} and the estimates for $U^0$. Summarizing these key results from~\cite[Lemma 3.1]{Ehrlacher2013}, we have
\begin{theorem}{(Multilattice version of~\cite[Lemma 3.1]{Ehrlacher2013})}\label{ml_thm_vs}
There exists a solution, $U^{\rm lin}$, to~\eqref{lin_elastic} and $\hat{r}$ such that $\zeta$ is bijective on $\mathbb{R}^2\setminus\Gamma$. Furthermore $U^0$ is then defined and a solution, $\bm{p}^0$, to~\eqref{p_predictor_def} exists. These functions satisfy $\nabla^j S_0 u^0_\alpha(x+) = \nabla^j S_0 u^0_\alpha(x-)$ and $\nabla^j p^0_\alpha(x+) = \nabla^j p^0_\alpha(x-)$ for all $\alpha$, nonnegative integers $j$, and $x \in \Gamma$, and
\[
|\nabla^j u^0_\alpha(x) - \nabla^j u^{\rm lin}(\zeta^{-1}(x))| \lesssim |x|^{-j-1}, \qquad |\nabla^j p^0_\alpha(x) - \nabla^j p^{\rm lin}(\zeta^{-1}(x))| \lesssim |x|^{-j-2}.
\]
\end{theorem}
\begin{proof}
As all results concerning displacements are proven in~\cite{Ehrlacher2013}, we shall only concern ourselves with the results for the shifts. Once we establish existence of $\bm{p}$, the corresponding estimates are immediate from the definition of $\bm{p}^0$ and $\bm{p}^{\rm lin}$ in~\eqref{p_lin_elastic} and~\eqref{p_predictor_def} and the corresponding results for $U^0$ and $U^{\rm lin}$.
For existence of a solution to~\eqref{p_predictor_def}, we need only note that it was shown in~\cite[Theorem 3.7]{olsonOrtner2016} that the atomistic stability assumption, Assumption~\ref{coercive}, implies a corresponding estimate on stability of the Cauchy-Born model. In particular, $\frac{\partial^2}{\partial \bm{p} \partial \bm{p} } \big[V\big(\bm{0} \big)\big]$ was shown to be invertible.
\end{proof}
It now follows that $(S_0 U^0, \bm{p}^0)$ are smooth, which allows us to perform Taylor expansions of finite differences using the $\tilde{D}$ operator:
\begin{lemma}\label{diff_grad}
With $\bm{u}^0 = (U^0, \bm{p}^0)$ and $\tilde{D}$ as defined above and $|\ell|$ large enough,
\begin{align}
\big|\tilde{D}_{(\rho \alpha\beta)} \bm{u}^0(\ell) - \big(\nabla_{\rho} U^0(\ell) + p^0_\beta(\ell) - p^0_\alpha(\ell)\big)\big| \lesssim~& |\ell|^{-2}, \quad \ell \notin \Omega_\Gamma \label{zero_diff}\\
\big|\tilde{D}_{(\rho \alpha\beta)} \bm{u}^0(\ell) - \big(R\nabla_{\rho} S_0U^0(\ell) + p^0_\beta(\ell) - p^0_\alpha(\ell)\big)\big| \lesssim~& |\ell|^{-2}, \quad \ell \in \Omega_\Gamma
, \label{first_diff} \\
\big| e(\ell)\big| \lesssim~& |\ell|^{-1}, \label{second_diff} \\
\big|\tilde{D}_{-\rho} e(\ell)\big| \lesssim~& |\ell|^{-2} \label{third_diff}.
\end{align}
\end{lemma}
\begin{proof}
We first prove~\eqref{zero_diff} and consider the two cases as to whether $\ell \in \Omega_\Gamma$. If $\ell \notin \Omega_\Gamma$, then $\tilde{D} = D$, $(U^0,\bm{p}^0)$ is smooth, and we have a straightforward Taylor expansion
\begin{align*}
&\tilde{D}_{(\rho \alpha\beta)} \bm{u}^0(\ell) =~ D_{(\rho \alpha\beta)} \bm{u}^0(\ell) =~ D_{\rho}u^0(\ell) + D_{\rho}p^0_\beta(\ell) + p^0_\beta(\ell) - p^0_\alpha(\ell) \\
&=~ \nabla_{\rho} U^0(\ell) + \nabla_{\rho} p^0_\beta(\ell) + p^0_\beta(\ell) - p^0_\alpha(\ell) + \mathcal{O}(\|\nabla^2 U^0\|_{L^\infty(B_{r_{\rm cut}}(\ell))}) + \mathcal{O}(\|\nabla p^0_\beta\|_{L^\infty(B_{r_{\rm cut}}(\ell))}),
\end{align*}
and now from~\cite{hirth1982theory}, we know that $|\nabla^2 U^{\rm lin}| \lesssim~ |\ell|^{-2}$ and thus $|\nabla^2 U^{0}| \lesssim~ |\ell|^{-2}$ from~Theorem~\ref{ml_thm_vs} which further implies $|\nabla \bm{p}^0(\ell)| \lesssim~ |\ell|^{-2}$ from \eqref{p_lin_elastic}. Hence
\[
\big|\tilde{D}_{(\rho \alpha\beta)} \bm{u}^0(\ell) - \big(\nabla_{\rho} U(\ell) + p_\beta(\ell) - p_\alpha(\ell)\big)\big| \lesssim~ |\ell|^{-2}.
\]
If $\ell \in \Omega_\Gamma$, then $\tilde{D} = RDS$,
\begin{align*}
&\tilde{D}_{(\rho \alpha\beta)} \bm{u}^0(\ell) =~ (\tilde{D}_\rho U^0)(\ell) + (\tilde{D}_\rho p^0_\beta)(\ell) + p^0_\beta(\ell) - p^0_\alpha(\ell) \\
&=~ (RD_\rho S_0 U^0)(\ell) + p^0_\beta(\ell_1) + p^0_\beta(\ell_2) - p^0_\beta(\ell) - p^0_\alpha(\ell)
\end{align*}
for $\ell_1 = \ell+\rho\pm b_{12}$, $\ell_2 = \ell \pm b_{12}$ by definition of $\tilde{D}$ applied to $p^0$. But now both $S_0U^0$ and $\bm{p}^0$ are smooth so we again have a Taylor expansion of
\begin{align*}
&\tilde{D}_{(\rho \alpha\beta)} \bm{u}^0(\ell) \\
&=~ R\nabla_{\rho}S_0U^0(\ell) + p^0_\beta(\ell) - p^0_\alpha(\ell) + \mathcal{O}(\|\nabla^2 S_0U^0\|_{L^\infty(B_{r_{\rm cut}}(\ell))} + \mathcal{O}(\|\nabla p^0_\beta\|_{L^\infty(B_{r_{\rm cut}}(\ell))}) \\
&=~ R\nabla_{\rho}S_0U^0(\ell) + p^0_\beta(\ell) - p^0_\alpha(\ell) + \mathcal{O}(\|\nabla^2 S_0U^0\|_{L^\infty(B_{r_{\rm cut}}(\ell))} + \mathcal{O}(\|\nabla p^0_\beta\|_{L^\infty(B_{r_{\rm cut}}(\ell))}),
\end{align*}
which, because $S_0$ simply represents a shift by one Burger's vector, implies
\begin{align*}
&\big|\tilde{D}_{(\rho \alpha\beta)} \bm{u}^0(\ell) - \big(R\nabla_{\rho}S_0U^0 (\ell) + p_\beta(\ell) - p_\alpha(\ell)\big)\big| \\
&\lesssim~ \mathcal{O}(\|\nabla^2 U^0\|_{L^\infty(B_{r_{\rm cut}+b_1}(\ell))}) + \mathcal{O}(\|\nabla p^0_\beta\|_{L^\infty(B_{r_{\rm cut}+b_1}(\ell))}) \\
&\lesssim~ \mathcal{O}(\|\nabla^2 U^0\|_{L^\infty(B_{r_{\rm cut}+b_1}(\ell))} + \mathcal{O}(\|\nabla p^0_\beta\|_{L^\infty(B_{r_{\rm cut}+b_1}(\ell))}) \\
&\lesssim~ |\ell|^{-2}.
\end{align*}
In order to prove~\eqref{second_diff}, we consider only the case $\ell \notin \Omega_\Gamma$ as the other case is analogous:
\begin{align*}
\big|e(\ell)\big| \leq~& \big|\tilde{D}_{(\rho \alpha\beta)} \bm{u}^0(\ell) - \big(\nabla_{\rho} U^0(\ell) + p^0_\beta(\ell) - p^0_\alpha(\ell)\big)\big| + \big|\nabla_{\rho} U^0(\ell) + p^0_\beta(\ell) - p^0_\alpha(\ell)\big| \\
\lesssim~& |\ell|^{-2} + \big|\nabla_{\rho} U^0(\ell) + p^0_\beta(\ell) - p^0_\alpha(\ell)\big| \\
\leq~& |\ell|^{-2} + \big|\nabla_{\rho} U^0(\ell) + p^0_\beta(\ell) - p^0_\alpha(\ell)\big|.
\end{align*}
From~\cite{hirth1982theory}, we know that $|\nabla U^{\rm lin}| \lesssim~ |\ell|^{-1}$ and thus $|\nabla U^{0}| \lesssim |\ell|^{-1}$from Theorem~\ref{ml_thm_vs} which further implies $|\bm{p}^0(\ell)| \lesssim~ |\ell|^{-1}$ from \eqref{p_lin_elastic}). Thus
\[
\big|e(\ell)\big| \lesssim~ |\ell|^{-1}.
\]
For~\eqref{third_diff}, we apply Taylor remainder estimates analogous to those proven in~\eqref{zero_diff} and~\eqref{first_diff}, and for brevity, we consider only the case $\ell \in \Omega_\Gamma$:
\begin{align*}
\big|\tilde{D}_{-\rho} e(\ell)\big| =~& \big| \tilde{D}_{-\rho}\big( RD_\rho S_0 U^0 + \tilde{D}_\rho p^0_\beta + p_\beta^0 - p_\alpha^0\big) \big| \\
=~& \big| RD_{-\rho}D_{\rho}S_0U^0 + RD_{-\rho}D_{\rho} S_0 p^0_\beta + \tilde{D}_{-\rho}p_\beta^0 - \tilde{D}_{-\rho}p_\alpha^0 \big| \\
=~& \big| R\nabla^2_\rho S_0U^0 + \nabla_{-\rho} p_\beta^0 - \nabla_{-\rho}p_\alpha^0 \big| + \mathcal{O}(\|\nabla^3 S_0U^0\|_{L^\infty(B_{r_{\rm cut}}(\ell))}) \\
&\quad + \mathcal{O}(\|\nabla^2 p^0_\beta\|_{L^\infty(B_{r_{\rm cut}}(\ell))}) + \mathcal{O}(\|\nabla^2 p^0_\alpha\|_{L^\infty(B_{r_{\rm cut}}(\ell))})
\end{align*}
We may again utilize~\cite{hirth1982theory}, to see that $|\nabla^2 U^{\rm lin}| \lesssim |\ell|^{-2}$ and $|\nabla^3 U^{\rm lin}| \lesssim |\ell|^{-3}$ and thus $|\nabla^2 S_0U^{0}| \lesssim |\ell|^{-2}$ and $|\nabla^3 S_0U^{0}| \lesssim |\ell|^{-3}$ from~Theorem~\ref{ml_thm_vs}. Obtaining corresponding estimates on the shifts from \eqref{p_lin_elastic} gives~\eqref{third_diff}.
\end{proof}
Next, we estimate the decay of the residual forces from the homogeneous energy when evaluated at the continuum dislocation predictor; these decay estimates will turn out to be vital in proving the decay estimates for the dislocation strain fields themselves. For ease of notation and visual clarity, throughout the proof, we will use the notation to $\|\cdot\|_{L^\infty}$ to represent $\|\cdot\|_{L^\infty(B_{r_{\rm cut}+b_1}(\ell))}$, and $\displaystyle\sum^\gamma$ will represent a summation over which $\gamma$ is held fixed.
\begin{lemma}\label{at_force}
Suppose that $(U,\bm{p})$ are smooth. The force on an atomistic degree of freedom $(\eta,\gamma)$ for $\eta \in \Lambda$ and $\gamma \in \{0, \ldots, S-1\}$ and $\ell$ large enough is given by
\begin{equation}\label{at_force_eq}
\begin{split}
\frac{\partial \mathcal{E}_{\rm hom}^{\rm a}}{\partial u_\gamma(\eta)}|_{(U,\bm{p})} =~& \overbrace{\sum_{(\tau\iota\chi)} \sum_{(\rho\alpha\gamma)}^\gamma V_{,(\rho\alpha\gamma)(\tau\iota\chi)}(0) \Big[\nabla_{\tau,-\rho}^2 U(\eta) - \nabla_{\rho}p_\chi(\eta) + \nabla_{\rho} p_\iota(\eta) \Big]}^{1} \\
&+~ \overbrace{\sum_{(\tau\iota\chi)} \Big\{\sum_{(\rho\alpha\gamma)}^\gamma V_{,(\rho\alpha\gamma)(\tau\iota\chi)}(0) - \sum_{(\rho\gamma\alpha)}^\gamma V_{,(\rho\gamma\alpha)(\tau\iota\chi)}(0)\Big\}\big( \nabla_\tau U + p_\chi(\eta) - p_\iota(\eta) \big)}^{2a} \\
&+~ \underbrace{\sum_{(\tau\iota\chi)} \Big\{\sum_{(\rho\alpha\gamma)}^\gamma V_{,(\rho\alpha\gamma)(\tau\iota\chi)}(0) - \sum_{(\rho\gamma\alpha)}^\gamma V_{,(\rho\gamma\alpha)(\tau\iota\chi)}(0)\Big\}\big(\frac{1}{2}\nabla^2_{\tau}U + \nabla_\tau p_\chi(\eta) \big)}_{2b} \\
&+~ \mathcal{O}\big(\|\nabla^2 U\|_{L^\infty}\|\nabla u\|_{L^\infty} + \|\nabla \bm{p}\|_{L^\infty}\|\bm{p}\|_{L^\infty} + \|\nabla^3 U\|_{L^\infty} + \|\nabla^2 \bm{p}\|_{L^\infty} \big).
\end{split}
\end{equation}
In particular, if $(U, \bm{p}) = (U^0, \bm{p}^0)$, and if $\ell$ is large enough and $\ell \notin \Omega_\Gamma$, then
\begin{equation}\label{at_force_lin}
\begin{split}
\sum_{\gamma} \frac{\partial \mathcal{E}^{\rm a}_{\rm hom}}{\partial u_\gamma(\eta)}|_{(U,\bm{p})} =~& \mathcal{O}\big(|\ell|^{-3}\big),
\end{split}
\end{equation}
and if $\ell \in \Omega_\Gamma$ and if $(U, \bm{p}) = (SU^0, S\bm{p}^0)$, then
\begin{equation}\label{at_force_lin_s}
\begin{split}
\sum_{\gamma} \frac{\partial \mathcal{E}^{\rm a}_{\rm hom}}{\partial u_\gamma(\eta)}|_{(U,\bm{p})} =~& \mathcal{O}\big(|\ell|^{-3}\big),
\end{split}
\end{equation}
\end{lemma}
\begin{proof}
First, we establish the expression in~\eqref{at_force_eq}. Observe that
\begin{align*}
\frac{\partial \mathcal{E}^{\rm a}}{\partial u_\gamma(\eta)}|_{(U,\bm{p})} =~& \sum_{(\rho\alpha\gamma)}^\gamma V_{,(\rho\alpha\gamma)}(D\bm{u}(\eta-\rho)) - \sum_{(\rho\gamma\alpha)}^\gamma V_{,\rho\gamma\alpha}(D\bm{u}(\eta)),
\end{align*}
which we now Taylor expand about $\bm{0}$ to obtain
\begin{equation}\label{taylor1}
\begin{split}
&\frac{\partial \mathcal{E}^{\rm a}}{\partial u_\gamma(\eta)}|_{(U,\bm{p})}\\
&=~ \sum_{(\rho\alpha\gamma)}^\gamma V_{,(\rho\alpha\gamma)}(\bm{0}) - \sum_{(\rho\gamma\alpha)}^\gamma V_{,(\rho\gamma\alpha)}(\bm{0}) \\
&~+ \sum_{(\rho\alpha\gamma)}^\gamma\sum_{(\tau\iota\chi)} V_{,(\rho\alpha\gamma)(\tau\iota\chi)}(\bm{0})\big(D_{(\tau\iota\chi)}\bm{u}(\eta-\rho)\big) - \sum_{(\rho\gamma\alpha)}^\gamma\sum_{(\tau\iota\chi)} V_{,(\rho\gamma\alpha)(\tau\iota\chi)}(\bm{0})\big(D_{(\tau\iota\chi)}\bm{u}(\eta)\big) \\
&~ +\frac{1}{2}\int_0^1 (1-t)^2\sum_{(\rho\gamma\alpha)}^\gamma\sum_{(\tau\iota\chi)}\sum_{(\sigma\mu\nu)} V_{,(\rho\gamma\alpha)(\tau\iota\chi)(\sigma\mu\nu)}(tD\bm{u})\big(D_{(\tau\iota\chi)}\bm{u}(\eta-\rho)\big)\big(D_{(\sigma\mu\nu)}\bm{u}(\eta-\rho)\big) \, dt \\
&-~ \frac{1}{2}\int_0^1 (1-t)^2\sum_{(\rho\gamma\alpha)}^\gamma\sum_{(\tau\iota\chi)}\sum_{(\sigma\mu\nu)} V_{,(\rho\gamma\alpha)(\tau\iota\chi)(\sigma\mu\nu)}(tD\bm{u})\big(D_{(\tau\iota\chi)}\bm{u}(\eta)\big)\big(D_{(\sigma\mu\nu)}\bm{u}(\eta)\big) \, dt \\
&=~ \sum_{(\rho\alpha\gamma)}^\gamma\sum_{(\tau\iota\chi)} V_{,(\rho\alpha\gamma)(\tau\iota\chi)}(\bm{0})\big(D_{(\tau\iota\chi)}\bm{u}(\eta-\rho)\big) - \sum_{(\rho\gamma\alpha)}^\gamma\sum_{(\tau\iota\chi)} V_{,(\rho\gamma\alpha)(\tau\iota\chi)}(\bm{0})\big(D_{(\tau\iota\chi)}\bm{u}(\eta)\big) \\
&~ + \mathcal{O}\big(\|\nabla^2 U\|_{L^\infty}\|\nabla u\|_{L^\infty} + \|\nabla \bm{p}\|_{L^\infty}\|\bm{p}\|_{L^\infty}\big),
\end{split}
\end{equation}
where in obtaining the last line we have used that $\partial p_\gamma W(0) = 0$ in the equilibrated reference configuration, which implies $0 = \sum_{(\rho\alpha\gamma)}^\gamma V_{,(\rho\alpha\gamma)}(0) - \sum_{(\rho\gamma\alpha)}^\gamma V_{,(\rho\gamma\alpha)}(0)$~\cite[A.4]{olsonOrtner2016}, and we Taylor expanded the finite differences in the remainder term. Next, we rewrite this as
\begin{equation}\label{taylor2}
\begin{split}
&\frac{\partial \mathcal{E}^{\rm a}}{\partial u_\gamma(\eta)}|_{(U,\bm{p})} \\
&=~ \sum_{(\tau\iota\chi)} \Big\{ \sum_{(\rho\alpha\gamma)}^\gamma V_{,(\rho\alpha\gamma)(\tau\iota\chi)}(\bm{0})\big(D_{(\tau\iota\chi)}\bm{u}(\eta-\rho) - D_{(\tau\iota\chi)}\bm{u}(\eta) \big) + \sum_{(\rho\alpha\gamma)}^\gamma V_{,(\rho\alpha\gamma)(\tau\iota\chi)}(0)\big(D_{(\tau\iota\chi)}\bm{u}(\eta) \big) \\
&~- \sum_{(\rho\gamma\alpha)}^\gamma V_{,(\rho\gamma\alpha)(\tau\iota\chi)}(\bm{0}) \big(D_{(\tau\iota\chi)}\bm{u}(\eta) \big)
\Big\} + \mathcal{O}\big(\|\nabla^2 U\|_{L^\infty}\|\nabla u\|_{L^\infty} + \|\nabla \bm{p}\|_{L^\infty}\|\bm{p}\|_{L^\infty}\big) \\
&=~ \overbrace{\sum_{(\tau\iota\chi)} \Big\{ \sum_{(\rho\alpha\gamma)}^\gamma V_{,(\rho\alpha\gamma)(\tau\iota\chi)}(\bm{0})\big(D_{(\tau\iota\chi)}\bm{u}(\eta-\rho) - D_{(\tau\iota\chi)}\bm{u}(\eta) \big) \Big\}}^{A_1} \\
&\qquad+~ \underbrace{\sum_{(\tau\iota\chi)} \Big\{\sum_{(\rho\alpha\gamma)}^\gamma V_{,(\rho\alpha\gamma)(\tau\iota\chi)}(0) - \sum_{(\rho\gamma\alpha)}^\gamma V_{,(\rho\gamma\alpha)(\tau\iota\chi)}(\bm{0})\Big\}\big(D_{(\tau\iota\chi)}\bm{u}(\eta) \big)}_{A_2} \\
&\qquad\qquad+~ \mathcal{O}\big(\|\nabla^2 U\|_{L^\infty}\|\nabla u\|_{L^\infty} + \|\nabla \bm{p}\|_{L^\infty}\|\bm{p}\|_{L^\infty}\big) .
\end{split}
\end{equation}
We then rewrite term $A_1$ and then Taylor expand (keeping only the lowest order error terms) to produce
\begin{equation}\label{taylor25}
\begin{split}
A_1
&=~ \sum_{(\tau\iota\chi)} \sum_{(\rho\alpha\gamma)}^\gamma V_{,(\rho\alpha\gamma)(\tau\iota\chi)}(\bm{0}) \Big[D_{-\rho} \big(U(\eta+\tau) - U(\eta) + p_\chi(\eta+\tau) - p_\iota(\eta) \big) \Big] \\
&=~ \sum_{(\tau\iota\chi)} \sum_{(\rho\alpha\gamma)}^\gamma V_{,(\rho\alpha\gamma)(\tau\iota\chi)}(\bm{0}) \Big[D_{-\rho} \big(\nabla_\tau U(\eta) + p_\chi(\eta+\tau) - p_\chi(\eta) + p_\chi(\eta) - p_\iota(\eta) \big) \Big] \\
&\qquad + \mathcal{O}\big(\|\nabla^3 U\|_{L^\infty} \big) \\
&=~ \sum_{(\tau\iota\chi)} \sum_{(\rho\alpha\gamma)}^\gamma V_{,(\rho\alpha\gamma)(\tau\iota\chi)}(\bm{0}) \Big[D_{-\rho} \big(\nabla_\tau U(\eta) + \nabla_\tau p_\chi(\eta) + p_\chi(\eta) - p_\iota(\eta) \big) \Big] \\
&\qquad + \mathcal{O}\big(\|\nabla^3 U\|_{L^\infty} + \|\nabla^2 \bm{p}\|_{L^\infty}\big) \\
&=~ \sum_{(\tau\iota\chi)} \sum_{(\rho\alpha\gamma)}^\gamma V_{,(\rho\alpha\gamma)(\tau\iota\chi)}(\bm{0}) \Big[D_{-\rho} \big(\nabla_\tau U(\eta) + p_\chi(\eta) - p_\iota(\eta) \big) \Big] + \mathcal{O}\big(\|\nabla^3 U\|_{L^\infty} + \|\nabla^2 \bm{p}\|_{L^\infty}\big)\\
&=~ \sum_{(\tau\iota\chi)} \sum_{(\rho\alpha\gamma)}^\gamma V_{,(\rho\alpha\gamma)(\tau\iota\chi)}(\bm{0}) \Big[\nabla_{\tau,-\rho}^2 U(\eta) - \nabla_{\rho}p_\chi(\eta) + \nabla_{\rho} p_\iota(\eta) \Big] + \mathcal{O}\big(\|\nabla^3 U\|_{L^\infty} + \|\nabla^2 \bm{p}\|_{L^\infty}\big).
\end{split}
\end{equation}
Focusing now on term $A_2$, we note that
\begin{equation}\label{taylor3}
\begin{split}
&A_2 =~ \sum_{(\tau\iota\chi)} \Big\{\sum_{(\rho\alpha\gamma)}^\gamma V_{,(\rho\alpha\gamma)(\tau\iota\chi)}(\bm{0}) - \sum_{(\rho\gamma\alpha)}^\gamma V_{,(\rho\gamma\alpha)(\tau\iota\chi)}(\bm{0})\Big\}\big(D_{(\tau\iota\chi)}\bm{u}(\eta) \big) \\
&=~ \sum_{(\tau\iota\chi)} \Big\{\sum_{(\rho\alpha\gamma)}^\gamma V_{,(\rho\alpha\gamma)(\tau\iota\chi)}(\bm{0}) - \sum_{(\rho\gamma\alpha)}^\gamma V_{,(\rho\gamma\alpha)(\tau\iota\chi)}(\bm{0})\Big\}\big(U(\tau+\eta) - U(\eta) + p_\chi(\eta+\tau) - p_\iota(\eta) \big) \\
&=~ \sum_{(\tau\iota\chi)} \Big\{\sum_{(\rho\alpha\gamma)}^\gamma V_{,(\rho\alpha\gamma)(\tau\iota\chi)}(\bm{0}) - \sum_{(\rho\gamma\alpha)}^\gamma V_{,(\rho\gamma\alpha)(\tau\iota\chi)}(\bm{0})\Big\}\big( \nabla_\tau U + \frac{1}{2}\nabla^2_{\tau}U + p_\chi(\eta+\tau) - p_\chi(\eta) \\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+ p_\chi(\eta) - p_\iota(\eta) \big) + \mathcal{O}\big(\|\nabla^3 U\|_{L^\infty} \big)\\
&=~ \sum_{(\tau\iota\chi)} \Big\{\sum_{(\rho\alpha\gamma)}^\gamma V_{,(\rho\alpha\gamma)(\tau\iota\chi)}(\bm{0}) - \sum_{(\rho\gamma\alpha)}^\gamma V_{,(\rho\gamma\alpha)(\tau\iota\chi)}(\bm{0})\Big\}\big( \nabla_\tau U + \frac{1}{2}\nabla^2_{\tau}U + \nabla_\tau p_\chi(\eta) \\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+ p_\chi(\eta) - p_\iota(\eta) \big) + \mathcal{O}\big(\|\nabla^3 U\|_{L^\infty} + \|\nabla^2 \bm{p}\|_{L^\infty}\big) \\
&=~ \overbrace{\sum_{(\tau\iota\chi)} \Big\{\sum_{(\rho\alpha\gamma)}^\gamma V_{,(\rho\alpha\gamma)(\tau\iota\chi)}(\bm{0}) - \sum_{(\rho\gamma\alpha)}^\gamma V_{,(\rho\gamma\alpha)(\tau\iota\chi)}(\bm{0})\Big\}\big( \nabla_\tau U + p_\chi(\eta) - p_\iota(\eta) \big)}^{A_{21}} \\
&\qquad +~ \underbrace{\sum_{(\tau\iota\chi)} \Big\{\sum_{(\rho\alpha\gamma)}^\gamma V_{,(\rho\alpha\gamma)(\tau\iota\chi)}(\bm{0}) - \sum_{(\rho\gamma\alpha)}^\gamma V_{,(\rho\gamma\alpha)(\tau\iota\chi)}(\bm{0})\Big\}\big(\frac{1}{2}\nabla^2_{\tau}U + \nabla_\tau p_\chi(\eta) \big)}_{A_{22}} \\
& \qquad\qquad+ \mathcal{O}\big(\|\nabla^3 U\|_{L^\infty} + \|\nabla^2 \bm{p}\|_{L^\infty}\big).
\end{split}
\end{equation}
Inserting the expressions~\eqref{taylor25} and~\eqref{taylor3} into~\eqref{taylor2} yields the desired expression for $\frac{\partial \mathcal{E}^{\rm a}_{\rm hom}}{\partial u_\gamma(\eta)}$ in~\eqref{at_force_eq}.
To establish~\eqref{at_force_lin}, we observe that from Appendix~\ref{app:cb_linear}, $(U, \bm{p})$ is a solution to the linear elastic Cauchy-Born model provided
\begin{equation}\label{lin_cb_weak}
\begin{split}
&\<\partial_{{\sf G} {\sf G}}W(\bm{0})\nabla U, \nabla V \> + \sum_{\nu} \< \partial_{{\sf G} p_\nu}W(\bm{0}) \nabla U, q_\nu \> + \sum_{\mu}\< \partial_{ p_\mu {\sf G}}W(\bm{0}) p_\mu, \nabla V \> \\
&\qquad+ \sum_{\mu,\nu}\< \partial_{p_\mu p_\nu}W(\bm{0}) p_\mu, q_\nu \> = 0, \quad \forall (W,\bm{q}) \in {\rm C}^\infty_0.
\end{split}
\end{equation}
In strong form, this reads
\begin{equation}\label{lin_cb_strong}
\begin{split}
\nabla \cdot (\partial_{{\sf G} {\sf G}}W(\bm{0})\nabla U) + \sum_{\mu} \nabla \cdot(\partial_{ p_\mu {\sf G}}W(\bm{0}) p_\mu) =~& 0, \\
\partial_{{\sf G} p_\nu}W(\bm{0}) \nabla U + \sum_{\mu } \partial_{p_\mu p_\nu}W(\bm{0}) p_\mu =~& 0, \qquad \mbox{for each species $\nu$.}
\end{split}
\end{equation}
It is then a simple calculus exercise to compute (see also~\cite{olsonOrtner2016} for the same expressions)
\begin{equation}\label{deriv_express}
\begin{split}
\partial^2_{{\sf G}_{mn} \bm{p}_\mu^l} W(\bm{0}) =~& \sum_{(\rho\alpha\mu) \in \mathcal{R}}^\mu \sum_{(\tau\iota\chi)\in\mathcal{R}} V_{,(\rho\alpha\mu)(\tau\iota\chi)}^{lm}(\bm{0})\tau_n - \sum_{(\rho\mu\alpha) \in \mathcal{R}}^\mu \sum_{(\tau\iota\chi)\in\mathcal{R}} V_{,(\rho\mu\alpha)(\tau\iota\chi)}^{lm}(\bm{0})\tau_n \\
\partial^2_{{\sf G}_{mn} {\sf G}_{rs}} =~& \sum_{(\rho\alpha\mu)\in \mathcal{R}} \sum_{(\tau\iota\chi) \in \mathcal{R}} V_{,(\rho\alpha\mu)(\tau\iota\chi)}^{mr}(\bm{0})\rho_n\tau_s \\
\partial^2_{\bm{p}_\mu^k\bm{p}_\gamma^l} W(\bm{0}) =~& \sum_{(\tau\iota\mu) \in \mathcal{R}}^\mu \sum_{(\rho\alpha\gamma) \in \mathcal{R}}^\gamma V^{kl}_{,(\rho\alpha\gamma)(\tau\iota\mu) }\big( \bm{0} \big) - \sum_{(\tau\iota\mu) \in \mathcal{R}}^\mu \sum_{(\rho\gamma\alpha) \in \mathcal{R}}^\gamma V^{kl}_{,(\rho\gamma\alpha)(\tau\iota\mu)}\big(\bm{0} \big) \\
&\quad + \sum_{(\tau\mu\iota) \in \mathcal{R}}^\mu \sum_{(\rho\gamma\alpha) \in \mathcal{R}}^\gamma V^{kl}_{,(\rho\gamma\alpha)(\tau\mu\iota) }\big( \bm{0} \big) - \sum_{(\tau\mu\iota) \in \mathcal{R}}^\mu \sum_{(\rho\alpha\gamma) \in \mathcal{R}}^\gamma V^{kl}_{,(\rho\alpha\gamma)(\tau\mu\iota)}\big(\bm{0} \big),
\end{split}
\end{equation}
and therefore
\begin{equation}\label{div1}
\begin{split}
-\big[\nabla \cdot (\partial_{{\sf G}\mG}W(\bm{0}) \nabla U)\big]_{m} =~& -\sum_{(\rho\alpha\mu)(\tau\iota\chi)} V_{,(\rho\alpha\mu)(\tau\iota\chi)}^{mr} \rho_n\tau_s u_{r,sn} \\
=~& \sum_{(\rho\alpha\mu)(\tau\iota\chi)} V_{,(\rho\alpha\mu)(\tau\iota\chi)}^{mr} \nabla^2_{-\rho\tau}u_r,
\end{split}
\end{equation}
and
\begin{equation}\label{div2}
\begin{split}
&\big[\nabla \cdot (\partial_{{\sf G} p_\mu}W(\bm{0}) p_\mu)\big]_{m} \\
&=~ \sum_{(\rho\alpha\mu) \in \mathcal{R}}\sum_{(\tau\iota\chi)\in\mathcal{R}} V_{,(\rho\alpha\mu)(\tau\iota\chi)}^{lm}(\bm{0})\tau_np_{\mu,n}^{l} - \sum_{(\rho\mu\alpha) \in \mathcal{R}}\sum_{(\tau\iota\chi)\in\mathcal{R}} V_{,(\rho\mu\alpha)(\tau\iota\chi)}^{lm}(\bm{0})\tau_np_{\mu,n}^{l} \\
&=~ \sum_{(\rho\alpha\mu) \in \mathcal{R}}\sum_{(\tau\iota\chi)\in\mathcal{R}} V_{,(\rho\alpha\mu)(\tau\iota\chi)}^{lm}(\bm{0})(\nabla_\tau p_{\mu})^{l} - \sum_{(\rho\mu\alpha) \in \mathcal{R}}\sum_{(\tau\iota\chi)\in\mathcal{R}} V_{,(\rho\mu\alpha)(\tau\iota\chi)}^{lm}(\bm{0})(\nabla_\tau p_{\mu})^{l} \\
&=~ \sum_{(\tau\iota\chi) \in \mathcal{R}}\sum_{(\rho\alpha\mu)\in\mathcal{R}} V_{,(\tau\iota\chi)(\rho\alpha\mu)}^{lm}(\bm{0})(\nabla_\rho p_{\chi})^{l} - \sum_{(\tau\chi\iota) \in \mathcal{R}}\sum_{(\rho\alpha\mu)\in\mathcal{R}} V_{,(\tau\chi\iota)(\rho\alpha\mu)}^{lm}(\bm{0})(\nabla_\rho p_{\chi})^{l}
\end{split}
\end{equation}
where in obtaining the last line we have merely relabeled $\tau$ to $\rho$, $\iota$ to $\alpha$ and $\chi$ to $\mu$.
We use~\eqref{div1} and~\eqref{div2} to compute the summation over $\gamma$ of {\textbf{(1)}} in~\eqref{at_force_eq}, which is valid since $U^{\rm lin}$ is smooth in this region:
\begin{equation}\label{sum1}
\begin{split}
&\sum_{(\tau\iota\chi)} \sum_{(\rho\alpha\gamma)} V_{,(\rho\alpha\gamma)(\tau\iota\chi)}(0) \Big[\nabla_{\tau,-\rho}^2 U(\eta) - \nabla_{\rho}p_\chi(\eta) + \nabla_{\rho} p_\iota(\eta) \Big] \\
&= -\nabla \cdot (\partial_{{\sf G}\mG}W(0) \nabla U) + \sum_{(\tau\iota\chi)} \sum_{(\rho\alpha\gamma)} V_{,(\tau\iota\chi)(\rho\alpha\gamma)}(0)\Big[- \nabla_{\rho}p_\chi(\eta) + \nabla_{\rho} p_\iota(\eta) \Big] \, \mbox{by Clairaut's Thm.} \\
&= -\nabla \cdot (\partial_{{\sf G}\mG}W(0) \nabla U) - \sum_\mu \nabla \cdot (\partial_{{\sf G} p_\mu}W(0) p_\mu) = 0 \quad \mbox{by~\eqref{lin_cb_strong}}.
\end{split}
\end{equation}
When summing over $\gamma$ in terms {\textbf{(2a)}} and {\textbf{(2b)}} of~\eqref{at_force_eq}, the terms in braces disappear, which when combined with~\eqref{sum1} yields the second component of~\eqref{at_force_lin}. Moreover, even if $\gamma$ is not summed over, then term {\textbf{(2a)}} vanishes at the linear elastic solution due to the second constraint of~\eqref{lin_cb_strong} and due to the expressions~\eqref{deriv_express}: indeed, we may observe that
\begin{equation}\label{nonDiv1}
\begin{split}
&[\partial_{p_\mu {\sf G}} W(\bm{0}) \nabla U]_{l} \\
&=~ \sum_{(\rho\alpha\mu) \in \mathcal{R}}^\mu\sum_{(\tau\iota\chi)\in\mathcal{R}} V_{,(\rho\alpha\mu)(\tau\iota\chi)}^{lm}(\bm{0})(\nabla_\tau U)^m - \sum_{(\rho\mu\alpha) \in \mathcal{R}}^\mu\sum_{(\tau\iota\chi)\in\mathcal{R}} V_{,(\rho\mu\alpha)(\tau\iota\chi)}^{lm}(\bm{0}))(\nabla_\tau U)^m,
\end{split}
\end{equation}
while
\begin{equation}\label{nonDiv2}
\begin{split}
&(2a) = \sum_{(\tau\iota\chi)} \Big\{\sum_{(\rho\alpha\gamma)}^\gamma V_{,(\rho\alpha\gamma)(\tau\iota\chi)}(\bm{0}) - \sum_{(\rho\gamma\alpha)}^\gamma V_{,(\rho\gamma\alpha)(\tau\iota\chi)}(\bm{0})\Big\}\big( p_\chi(\eta) - p_\iota(\eta) \big) \\
&~=~ \sum_{(\tau\iota\chi)} \sum_{(\rho\alpha\gamma)}^\gamma V_{,(\rho\alpha\gamma)(\tau\iota\chi)}(\bm{0}) p_\chi(\eta) - \sum_{(\tau\iota\chi)}\sum_{(\rho\gamma\alpha)}^\gamma V_{,(\rho\gamma\alpha)(\tau\iota\chi)}(\bm{0}) p_\chi(\eta) \\
&\quad- \sum_{(\tau\iota\chi)} \sum_{(\rho\alpha\gamma)}^\gamma V_{,(\rho\alpha\gamma)(\tau\iota\chi)}(\bm{0}) p_\iota(\eta) + \sum_{(\tau\iota\chi)}\sum_{(\rho\gamma\alpha)}^\gamma V_{,(\rho\gamma\alpha)(\tau\iota\chi)}(\bm{0}) p_\iota(\eta) \\
&~=~ \sum_{(\tau\iota\chi)} \sum_{(\rho\alpha\gamma)}^\gamma V_{,(\rho\alpha\gamma)(\tau\iota\chi)}(\bm{0}) p_\chi(\eta) - \sum_{(\tau\iota\chi)}\sum_{(\rho\gamma\alpha)}^\gamma V_{,(\rho\gamma\alpha)(\tau\iota\chi)}(\bm{0}) p_\chi(\eta) \\
&\quad- \sum_{(\tau\chi\iota)} \sum_{(\rho\alpha\gamma)}^\gamma V_{,(\rho\alpha\gamma)(\tau\chi\iota)}(\bm{0}) p_\chi(\eta) + \sum_{(\tau\chi\iota)}\sum_{(\rho\gamma\alpha)}^\gamma V_{,(\rho\gamma\alpha)(\tau\chi\iota)}(\bm{0}) p_\iota(\eta) \,\, \mbox{relabeling $\chi$ to $\iota$ and $\iota$ to $\chi$} \\
&~=~ \sum_\chi \partial_{\bm{p}_\chi \bm{p}_\gamma}W(\bm{0}) p_\chi.
\end{split}
\end{equation}
We may now replace the linear elastic solution with $(U^0, \bm{p}^0)$ and make an at most $|\ell|^{-3}$ error according to Lemma~\ref{diff_grad} in $(1)$ and $2b$ which yields~\eqref{at_force_lin} when combined with decay estimates for the remainder terms used previously in Lemma~\ref{diff_grad}. Moreover, even though $S\bm{p}^0$ is not smooth, we are still able to use Taylor expansions of $\tilde{D}_\rho p^0$ just as we did in Lemma~\ref{diff_grad} so that we may replace $(U^0, \bm{p}^0)$ with $(SU^0, S\bm{p}^0)$ to obtain~\eqref{at_force_lin_s}.
\end{proof}
We have now fully defined all ingredients in the dislocation energy~\eqref{at_free},
\[
\mathcal{E}^{{\rm a}}(\bm{u}) = \sum_{\ell \in \Lambda} V_\ell(D\bm{u}(\ell)),
\]
and we can further show this energy is well-defined and continuously Frechet differentiable over the admissible displacements, $\mathcal{A}$, defined in~\eqref{admissible}.
\begin{theorem}[Dislocation Energy is well-defined]\label{energy_thm}
Under the hypotheses on the site potential and coercivity of the defect-free energy, the atomistic energy function, $\mathcal{E}^{\rm a}$, is well defined and belongs to ${\rm C}^3(\mathcal{A})$, provided the site potential is $\rm{C}^4$. More generally, a ${\rm C}^{k+1}$ site potential yields a ${\rm C}^k$ energy.
\end{theorem}
It should come as no surprise that the proof makes heavy use of techniques previously used in~\cite{Ehrlacher2013, olsonOrtner2016}.
\begin{proof}[Proof of Theorem~\ref{energy_thm}]
We proceed as in the proof of~\cite[Lemma 3.2]{Ehrlacher2013}. Namely, we recall the definition of the atomistic energy as
\begin{align*}
\mathcal{E}^{\rm a}(\bm{u}) =~& \sum_{\ell \in \Lambda} V_{\ell}(D\bm{u}(\ell)),
\end{align*}
and note that if $\bm{u}$ belongs to the space $\bm{\mathcal{U}}_0$, then
\begin{align*}
\<\delta \mathcal{E}^{\rm a}(0),\bm{u}\> = \frac{d}{dt}\big[ \mathcal{E}^{\rm a}(0+ tu)\big]_{t = 0} = \sum_{\ell \in \Lambda} \< \delta V_\ell(0), D\bm{u}(\ell) \>,
\end{align*}
as this becomes a finite sum (and hence we can differentiate term-by-term), and thus
\[
\mathcal{E}^{\rm a}(\bm{u}) = \sum_{\ell \in \Lambda}\big[ V_{\ell}(D\bm{u}(\ell)) - \< \delta V_\ell(0), D\bm{u}(\ell) \>\big] + \<\delta \mathcal{E}^{\rm a}(0),\bm{u}\>
\]
for these displacements $\bm{u}$. If we can show that {\textbf{(1)}}
\[
\sum_{\ell \in \Lambda}\big[ V_{\ell}(D\bm{u}(\ell)) - \< \delta V_\ell(0), D\bm{u}(\ell) \>\big]
\]
is well-defined for displacements having finite energy and is differentiable and {\textbf{(2)}} that
\[
\<\delta \mathcal{E}^{\rm a}(0),\bm{u}\>
\]
is a bounded linear functional, then we will have that $\mathcal{E}^{\rm a}$ agrees with a ${\rm C}^k$ functional on the dense subset, $\bm{\mathcal{U}}_0$, of $\bm{\mathcal{U}}$, and hence may be uniquely extended to a ${\rm C}^k$ functional on $\mathcal{A}$.
Showing that the function
\[
\sum_{\ell \in \Lambda}\big[ V_{\ell}(D\bm{u}(\ell)) - \< \delta V_\ell(0), D\bm{u}(\ell) \>\big]
\]
is well defined and continuously differentiable can be done verbatim to~\cite[Theorem 2.1]{olsonOrtner2016} by simply replacing the homogeneous site potential, $V$, in that work with $V_\ell$ in the present work so we omit the details.
To show that $\<\delta \mathcal{E}^{\rm a}(0),\bm{u}\> = \sum_{\ell \in \Lambda} \< \delta V_\ell(0), D\bm{u}(\ell)\>$ is a bounded functional, we recall first the identity~\eqref{slip_fd_id}
\[
\tilde{D}\bm{u}(\ell) = \tilde{D}_\rho u_0 + \tilde{D}_\rho p_\beta(\ell) + p_\beta(\ell) - p_\alpha(\ell).
\]
Using this identity and the definition of $e(\ell)$ from~\eqref{ml_strain}, we have
\begin{align*}
\sum_{\ell \in \Lambda} \< \delta V_\ell(0), D\bm{u}(\ell)\> =~& \sum_{\ell \in \Lambda} \< \delta V(e(\ell)), \tilde{D}\bm{u}(\ell)\> \\
=~&\sum_{\ell \in \Lambda} \sum_{(\rho \alpha\beta)}V_{(\rho \alpha\beta)} \big(e(\ell)\big)\cdot \tilde{D}_{(\rho \alpha\beta)}\bm{u}(\ell) \\
=~& \sum_{\ell \in \Lambda} \sum_{(\rho \alpha\beta)}V_{(\rho \alpha\beta)} \big(e(\ell)\big)\cdot\big[\tilde{D}_\rho u_0(\ell) + \tilde{D}_\rho p_\beta(\ell) + p_\beta(\ell) - p_\alpha(\ell)\big] \\
=~& \overbrace{\sum_{\ell \in \Lambda} \sum_{(\rho \alpha\beta)}V_{(\rho \alpha\beta)} \big(e(\ell)\big)\cdot\big[\tilde{D}_\rho u_0(\ell) + \tilde{D}_\rho p_\beta(\ell) \big]}^{T_1} \\
&~\qquad+ \underbrace{\sum_{\ell \in \Lambda} \sum_{(\rho \alpha\beta)}V_{(\rho \alpha\beta)} \big(e(\ell)\big)\cdot\big[p_\beta(\ell) - p_\alpha(\ell)\big]}_{T_2}.
\end{align*}
For term $T_2$, we have
\begin{align*}
T_2 =~& \sum_{\ell\in\Lambda}\sum_{(\rho \alpha\beta)}V_{(\rho \alpha\beta)} \big(e(\ell)\big)\cdot\big[p_\beta(\ell) - p_\alpha(\ell)\big] \\
=~& \sum_{\ell\in\Lambda}\sum_{(\rho \alpha\beta)}\big[V_{(\rho \alpha\beta)} \big(e(\ell)\big) - V_{(\rho \alpha\beta)}(\bm{0})\big]\cdot\big[p_\beta(\ell) - p_\alpha(\ell)\big] \\
&\qquad + \sum_{\ell\in\Lambda}\sum_{(\rho \alpha\beta)}\big[V_{(\rho \alpha\beta)}(\bm{0})\big]\cdot\big[p_\beta(\ell) - p_\alpha(\ell)\big] \\
=~& \sum_{\ell\in\Lambda}\sum_{(\rho \alpha\beta)}\big[V_{(\rho \alpha\beta)} \big(e(\ell)\big) - V_{(\rho \alpha\beta)}(\bm{0})\big]\cdot\big[p_\beta(\ell) - p_\alpha(\ell)\big] \\
&\qquad + \sum_{\ell\in\Lambda}\sum_{(\rho \alpha\beta)}\big[V_{(\rho \alpha\beta)}(\bm{0}) - V_{(\rho\beta\alpha)}(\bm{0})\big]\cdot\big[p_\beta(\ell)\big] \\
=~& \sum_{\ell\in\Lambda}\sum_{(\rho \alpha\beta)}\big[V_{(\rho \alpha\beta)} \big(e(\ell)\big) - V_{(\rho \alpha\beta)}(\bm{0})\big]\cdot\big[p_\beta(\ell) - p_\alpha(\ell)\big] \qquad \mbox{by~\cite[Appendix A.4 ]{olsonOrtner2016}} \\
=~& \sum_{\ell\in\Lambda}\sum_{(\rho \alpha\beta),(\tau\gamma\delta)}\big[V_{(\rho \alpha\beta)(\tau\gamma\delta)} \big(\bm{0}\big) e_{(\tau\gamma\delta)}(\ell) + \mathcal{O}(\|e_{(\tau\gamma\delta)}\|_{L^\infty}^2)\big]\cdot\big[p_\beta(\ell) - p_\alpha(\ell)\big] \\
=~& \sum_{\ell\in\Lambda}\sum_{(\tau\gamma\delta)}\big[\sum_{(\rho \alpha\beta)}V_{(\rho \alpha\beta)(\tau\gamma\delta)} \big(\bm{0}\big)- \sum_{(\rho\beta\alpha)}V_{(\rho\beta\alpha)(\tau\gamma\delta)} \big(\bm{0}\big) +\mathcal{O}(|\ell|^{-2})\big]e_{(\tau\gamma\delta)}(\ell) \cdot\big[p_\beta(\ell)\big] \\
=~& \sum_{\ell\in\Lambda}\sum_{(\tau\gamma\delta)}\big[\sum_{(\rho \alpha\beta)}V_{(\rho \alpha\beta)(\tau\gamma\delta)} \big(\bm{0}\big)- \sum_{(\rho\beta\alpha)}V_{(\rho\beta\alpha)(\tau\gamma\delta)} \big(\bm{0}\big) +\mathcal{O}(|\ell|^{-2}) \big]\big[\nabla_\tau U^0 + p_\delta - p_\gamma \\
&\hspace{3.7in}+ \mathcal{O}(|\ell|^{-2})\big] \cdot\big[p_\beta(\ell)\big],
\end{align*}
where we have used Lemma~\ref{diff_grad} in the obtaining the final line. Now by Lemma~\ref{at_force}, term (2a) vanishes, which is exactly the term immediately above up to an $\mathcal{O}(|\ell|^{-2})$ error. But this is then summable so that
\[
|T_2| \lesssim~ \|\bm{p}\|_{\ell^2} \lesssim~ \|\bm{u}\|_{{\rm a}_1}.
\]
For term $T_1$, we utilize~\cite[Lemma 5.7]{Ehrlacher2013} which allows to write ``integrate $\tilde{D}$ by parts:''
\begin{equation}\label{t1}
\begin{split}
T_1 &=~\sum_{\ell \in \Lambda} \sum_{(\rho \alpha\beta)}V_{(\rho \alpha\beta)} \big(e(\ell)\big)\cdot\big[\tilde{D}_\rho u_0(\ell) + \tilde{D}_\rho p_\beta(\ell) \big] \\
&=~ \sum_{\ell \in \Lambda} \sum_{(\rho \alpha\beta)}\tilde{D}_{-\rho}\big(V_{(\rho \alpha\beta)} \big(e(\ell)\big)\big)\cdot\big[u_0(\ell) + p_\beta(\ell) \big] \\
&=~ \sum_{\ell \in \Lambda} \sum_{(\rho \alpha\beta)}\tilde{D}_{-\rho}\big(V_{(\rho \alpha\beta)} \big(e(\ell)\big)\big)\cdot u_0(\ell) + \sum_{\ell \in \Lambda} \sum_{(\rho \alpha\beta)}\tilde{D}_{-\rho}\big(V_{(\rho \alpha\beta)} \big(e(\ell)\big)\big)\cdot p_\beta(\ell) .
\end{split}
\end{equation}
Next, observe that by Lemma~\ref{at_force}, if $\ell \notin \Omega_\Gamma$,
\begin{equation}\label{force_elastic1}
\Big|\sum_{(\rho \alpha\beta)}\tilde{D}_{-\rho}\big(V_{(\rho \alpha\beta)} \big(e(\ell)\big)\big)\Big| = \Big|\sum_{(\rho \alpha\beta)}D_{-\rho}\big(V_{(\rho \alpha\beta)} \big(D\bm{u}^0\big)\big)\Big| =~ \Big|\sum_{\gamma}\frac{\partial \mathcal{E}^{\rm a}}{\partial u_\gamma(\ell)}|_{(U^0,\bm{p}^0)}\Big| \lesssim~ |\ell|^{-3} ,
\end{equation}
and if $\ell \in \Omega_\Gamma$,
\begin{equation}\label{force_elastic2}
\begin{split}
\Big|\sum_{(\rho \alpha\beta)}\tilde{D}_{-\rho}\big(V_{(\rho \alpha\beta)} \big(e(\ell)\big)\big)\Big| =~& \Big|\sum_{(\rho \alpha\beta)}RD_{-\rho}S\big(V_{(\rho \alpha\beta)} \big(RDS_0\bm{u}^0\big)\big)\Big| \\
=~& \Big|\sum_{(\rho \alpha\beta)}RD_{-\rho}SR\big(V_{(\rho \alpha\beta)} \big(DS_0\bm{u}^0\big)\big)\Big| \\
=~& \Big|\sum_{(\rho \alpha\beta)}RD_{-\rho}\big(V_{(\rho \alpha\beta)} \big(DS_0\bm{u}^0\big)\big)\Big| \\
=~& \Big|\sum_{\gamma}\frac{\partial \mathcal{E}^{\rm a}}{\partial u_\gamma(\ell)}|_{(S_0U^0,S\bm{p}^0)}\Big| \lesssim~ |\ell|^{-3}.
\end{split}
\end{equation}
Moreover,
\begin{equation}\label{force_elastic3}
\Big|\tilde{D}_{-\rho}\big(V_{(\rho \alpha\beta)} \big(e(\ell)\big)\big)\Big| \lesssim~ |\ell|^{-2},
\end{equation}
since
\begin{align*}
\Big|\tilde{D}_{-\rho}\big(V_{(\rho \alpha\beta)} \big(e(\ell)\big)\big)\Big| =~& \Big|\tilde{D}_{-\rho}\big(V_{(\rho \alpha\beta)} \big(\bm{0}) + \sum_{(\tau\gamma\delta)} V_{(\rho \alpha\beta)(\tau\gamma\delta)}(\bm{0}) e_{(\tau\gamma\delta)}(\ell)\big)\big| + \mathcal{O}(|e_{(\tau\gamma\delta)}(\ell)|^2)\\
\lesssim~& |\ell|^{-2}
\end{align*}
by~\eqref{second_diff} and~\eqref{third_diff}.
Inserting estimates~\eqref{force_elastic1},~\eqref{force_elastic2}, and~\eqref{force_elastic3} into~\eqref{t1}, we obtain
\begin{equation}\label{t12}
|T_1| \lesssim~ \Big|\sum_{\ell \in \Lambda} f(\ell)\cdot u_0(\ell) \Big|+ \sum_{\ell \in \Lambda}\sum_{(\rho \alpha\beta)}\big|1+|\ell|\big|^{-2}|p_\beta(\ell)|
\end{equation}
where $|f(\ell)| \lesssim~ |\ell|^{-3}$ for sufficiently large $|\ell|$. We may then apply~\cite[Corollary 5.2]{Ehrlacher2013} to deduce the existence of $\bm{g}: \Lambda \to (\mathbb{R}^3)^{\mathcal{R}_1}$ such that
\[
\sum_{\ell \in \Lambda} f(\ell)\cdot u_0(\ell) = \sum_{\ell \in \Lambda} g_\rho(\ell) D_\rho u_0(\ell)
\]
where $|g_\rho(\ell)| \lesssim |\ell|^{-2}$ for sufficiently large $|\ell|$. This can then be inserted into~\eqref{t12} to yield
\[
|T_1| \lesssim \sum_{\ell \in \Lambda}\sum_{\beta \in \mathcal{S}} \big|1+|\ell|\big|^{-2}\big|D_\rho u_0(\ell) + p_\beta(\ell)\big| \lesssim~ \|\bm{u}\|_{{\rm a}_1},
\]
where we have applied the Cauchy-Schwarz inequality and summability of $|\ell|^{-2}$ in the final inequality. Combining our estimates for $T_1$ and $T_2$ shows $\<\delta \mathcal{E}^{\rm a}(0),\cdot\>$ is a bounded linear functional and thus completes the proof.
\end{proof}
This concludes our introductory section defining all of the perquisites necessary to state our main result concerning decay of the elastic far-fields in a multilattice generated by a straight dislocation. These decay rates are phrased in terms of finite differences (alternatively, they could be written as derivatives of smooth interpolants of lattice functions) using the notation
\begin{align*}
D_{\rho}u_\alpha(\xi) &:= u_\alpha(\xi + \rho) - u_\alpha(\xi)
\qquad \text{for }
\rho \in \Lambda, \quad \alpha \in \mathcal{S}, \qquad \text{and} \\
D_{\bm{\rho}} u_\alpha(\xi) &:= D_{\rho_1}D_{\rho_2}\cdots D_{\rho_k} u_\alpha(\xi)
\qquad \text{for } \bm{\rho} = (\rho_1, \dots, \rho_k) \in \Lambda^k.
\end{align*}
\begin{theorem}[Decay of Dislocation far-fields]\label{decay_thm}
Let $\bm{u}^\infty = (U^\infty, \bm{p}^\infty) \in \mathcal{A}$ be a local minimizer of $\mathcal{E}^{\rm a}(\bm{u})$, and suppose the site potential is ${\rm C}^k$ with $k \geq 4$, satisfies the slip invariance condition, and Assumption~\ref{coercive} is satisfied. Then for all $|\ell|$ large enough,
\begin{equation}\label{decay1}
\begin{split}
\big|\tilde{D}_{{\rho}} U^\infty(\ell)\big| \lesssim~& |\ell|^{-2}\log |\ell|,
\quad \forall {\rho} \in \mathcal{R}_1, \quad \text{and} \\
\big| p_\alpha^\infty(\ell)\big| \lesssim~& |\ell|^{-2}\log |\ell|,
\quad \forall \rho \in \mathcal{R}_1.
\end{split}
\end{equation}
\end{theorem}
\begin{remark}\label{decay_remark}
Though we have only stated the decay of the elastic strains, brute-force computations may also be used to show that corresponding decay estimates also hold for strain gradients (and the corresponding shift gradients) in the sense that
\begin{equation}\label{decay2}
\begin{split}
\big|\tilde{D}_{\bm{\rho}} U^\infty(\ell)\big| \lesssim~& |\ell|^{-1-j}\log |\ell|,
\quad \forall \bm{\rho} \in (\mathcal{R}_1)^j, 1 \leq j \leq k-2, \quad \text{and} \\
\big|D_{\bm{\rho}} p_\alpha^\infty(\ell)\big| \lesssim~& |\ell|^{-2-j}\log |\ell|,
\quad \forall \bm{\rho} \in (\mathcal{R}_1)^j, 0 \leq j \leq k-3.
\end{split}
\end{equation}
We remark on how to prove such a result at the end of the proof of Theorem~\ref{decay_thm}, but choose not to state this as part of the theorem as we ourselves do not go through a rigorous proof. These decay rates may also be validated by numerical computations.
\end{remark}
\section{Proof of Theorem~\ref{decay_thm}}\label{proof}
The main idea of proving Theorem~\ref{decay_thm} is to show that $\bm{u}^\infty$ solves a linearized problem whose Green's function may be estimated in terms of existing Green's function estimates developed in~\cite{olsonOrtner2016} for point defects in multilattices. The residual terms found in this linearization process are estimated in close analogy to~\cite{Ehrlacher2013}, and then the two estimates are combined to yield the theorem. It is with this breakdown in mind that we split this section into separate subsections. In Section~\ref{lin_res}, we derive the linearized problem and corresponding estimates on the residual; in Section~\ref{greeny} we recall the needed properties of the Green's function (matrix). Finally, in Section~\ref{pure_analysis}, we combine these results in a ``pure'' analysis problem to derive the estimates~\eqref{decay1}.
\subsection{Linearized Problem and Residual Estimates}\label{lin_res}
Our goal here is to establish
\begin{lemma}\label{lin_lemma}
There exists $\bar{f}: \Lambda \to \big(\mathbb{R}^3\big)^{\mathcal{R}}$ such that
\begin{align*}
\sum_{\ell \in \Lambda} \<\delta^2 V(\bm{0}) \tilde{D}\bm{u}^\infty, \tilde{D}\bm{v}\> =~& \sum_{\ell \in \Lambda} \sum_{(\rho \alpha\beta) \in \mathcal{R}} \bar{f}_{(\rho \alpha\beta)}(\ell)\cdot \tilde{D}_{(\rho \alpha\beta)}\bm{v}(\ell) - \<\delta \mathcal{E}^{\rm a}(0),\bm{v}\>
\end{align*}
where $\bar{f}_{(\rho \alpha\beta)}$ satisfies
\begin{align*}
|\bar{f}_{(\rho \alpha\beta)}(\ell)| \lesssim~ |\ell|^{-2} + |\tilde{D}\bm{u}^\infty(\ell)|^{2}.
\end{align*}
\end{lemma}
\begin{proof}
As $\bm{u}^\infty$ solves the atomistic Euler-Lagrange equations,
\[
0 = \<\delta \mathcal{E}^{\rm a}(\bm{u}^\infty),\bm{v} \>, \, \forall \bm{v} \in \bm{\mathcal{U}}_0,
\]
we may simply Taylor expand (using~\eqref{slip_strain})
\begin{align*}
0 =~& \sum_{\ell \in \Lambda} \<\delta V(e(\ell) + \tilde{D}\bm{u}^\infty(\ell)), \tilde{D}\bm{v}\> \\
=~& \sum_{\ell \in \Lambda} \big[\<\delta V(e(\ell) + \tilde{D}\bm{u}^\infty(\ell)) - \delta V(e(\ell)) - \delta^2 V(e(\ell))\tilde{D}\bm{u}^\infty, \tilde{D}\bm{v}\> \big] \\
&+~ \sum_{\ell \in \Lambda} \<\delta^2 V(e(\ell)) - \delta^2 V(0)\tilde{D}\bm{u}^\infty(\ell), \tilde{D}\bm{v}\> \\
&+~ \sum_{\ell \in \Lambda} \<\delta^2 V(0)\tilde{D}\bm{u}^\infty(\ell), \tilde{D}\bm{v}\> + \sum_{\ell \in \Lambda} \<\delta V(e(\ell)), \tilde{D}\bm{v}\>.
\end{align*}
We then rearrange terms to arrive at
\begin{equation}\label{rearr}
\begin{split}
\sum_{\ell \in \Lambda} \<\delta^2 V(0)\tilde{D}\bm{u}^\infty(\ell), \tilde{D}\bm{v}\> =~& -\sum_{\ell \in \Lambda} \big[\<\delta V(e(\ell) + \tilde{D}\bm{u}^\infty(\ell)) - \delta V(e(\ell)) - \delta^2 V(e(\ell))\tilde{D}\bm{u}^\infty, \tilde{D}\bm{v}\> \big] \\
&- \sum_{\ell \in \Lambda} \<\delta^2 V(e(\ell)) - \delta^2 V(0)\tilde{D}\bm{u}^\infty(\ell), \tilde{D}\bm{v}\> \\
&- \sum_{\ell \in \Lambda} \<\delta V(e(\ell)), \tilde{D}\bm{v}\>.
\end{split}
\end{equation}
Upon defining
\begin{align*}
\bar{f}_{(\rho \alpha\beta)} =~& -V_{(\rho \alpha\beta)}\big(e(\ell) + \tilde{D}\bm{u}^\infty(\ell)\big) + V_{(\rho \alpha\beta)}(e(\ell)) + \sum_{(\tau\gamma\delta)} V_{(\rho \alpha\beta)(\tau\gamma\delta)}(e(\ell))\tilde{D}_{(\tau\gamma\delta)}\bm{u}^\infty(\ell) \\
&- \sum_{(\tau\gamma\delta)} \big(V_{(\rho \alpha\beta)(\tau\gamma\delta)}(e(\ell))\tilde{D}_{(\tau\gamma\delta)}\bm{u}^\infty(\ell) - V_{(\rho \alpha\beta)(\tau\gamma\delta)}(0)\tilde{D}_{(\tau\gamma\delta)}\bm{u}^\infty(\ell) \big),
\end{align*}
it only remains to establish the given estimate on $\bar{f}_{(\rho \alpha\beta)}$. However, this is a straightforward consequence of Taylor's Theorem and the aforementioned decay estimates on $e_{(\tau\gamma\delta)}(\ell)$ stated in Lemma~\ref{diff_grad}.
\end{proof}
Next, we must estimate the term $\<\delta \mathcal{E}^{\rm a}(0),\bm{v}\> = \sum_{\ell \in \Lambda} \<\delta V(e(\ell)), \tilde{D}\bm{v}\>$.
\begin{theorem}\label{lin_prob_thm}
There exists $\bar{f}: \Lambda \to \big(\mathbb{R}^3\big)^{\mathcal{R}}$, $\bm{g}: \Lambda \to \big(\mathbb{R}^3\big)^{\mathcal{R}_1}$ and $\bm{k}:\Lambda \to \big(\mathbb{R}^3\big)^{\mathcal{S}}$ such that for $\bm{v} = (v_0, \bm{q}) \in \bm{\mathcal{U}}_0$,
\begin{equation*}\begin{split}
&\<\tilde{H}\bm{u}^\infty,\bm{v}\> := \sum_{\ell \in \Lambda} \<\delta^2 V(\bm{0}) \tilde{D}\bm{u}^\infty, \tilde{D}\bm{v}\> = \sum_{\ell \in \Lambda}\big( \sum_{(\rho \alpha\beta) \in \mathcal{R}}\bar{f}_{(\rho \alpha\beta)}(\ell)\cdot \tilde{D}_{(\rho \alpha\beta)}\bm{v}(\ell) + \<\bm{g}(\ell), Dv_0(\ell)\> \\
&\hspace{4.5in}+ \<\bm{k}(\ell),\bm{q}(\ell)\>\big) ,
\end{split}
\end{equation*}
where for sufficiently large $|\ell|$, $f_{(\rho \alpha\beta)}, g_{\rho}, k_\gamma$ satisfy
\begin{align*}
|\bar{f}_{(\rho \alpha\beta)}(\ell)| \lesssim~& |\ell|^{-2} + |\tilde{D}\bm{u}^\infty(\ell)|^{2} \\
|g_{\rho}(\ell)| \lesssim~& |\ell|^{-2} \\
|k_{\gamma}(\ell)| \lesssim~& |\ell|^{-2}.
\end{align*}
\end{theorem}
\begin{proof}
From Lemma~\ref{lin_lemma},
\begin{align*}
\sum_{\ell \in \Lambda} \<\delta^2 V(\bm{0}) \tilde{D}\bm{u}^\infty, \tilde{D}\bm{v}\> =~& \sum_{\ell \in \Lambda} \sum_{(\rho \alpha\beta) \in \mathcal{R}} \bar{f}_{(\rho \alpha\beta)}(\ell)\cdot \tilde{D}_{(\rho \alpha\beta)}\bm{v}(\ell) -\sum_{\ell \in \Lambda} \<\delta V(\tilde{D}\bm{u}^{0}), \tilde{D}\bm{v}\>,
\end{align*}
and
\begin{equation}\label{e1}
\begin{split}
&\sum_{\ell \in \Lambda} \<\delta V(\tilde{D}\bm{u}^{0}), \tilde{D}\bm{v}\> =~ \sum_{\ell \in \Lambda} \sum_{(\rho \alpha\beta)} V_{(\rho \alpha\beta)}(\tilde{D}\bm{u}^{0})\cdot \tilde{D}_{(\rho \alpha\beta)} \bm{v} \\
&=~ \sum_{\ell \in \Lambda} \sum_{(\rho \alpha\beta)} V_{(\rho \alpha\beta)}(\tilde{D}\bm{u}^{0})\cdot \big(\tilde{D}_\rho v_0 + \tilde{D}_\rho q_\beta + q_\beta - q_\alpha\big)\\
&=~ \sum_{\ell \in \Lambda} \sum_{(\rho \alpha\beta)} V_{(\rho \alpha\beta)}(\tilde{D}\bm{u}^{0})\cdot \tilde{D}_\rho v_0 + \sum_{\ell \in \Lambda} \sum_{(\rho \alpha\beta)} V_{(\rho \alpha\beta)}(\tilde{D}\bm{u}^{0})\cdot \tilde{D}_\rho q_\beta \\
&\quad+ \sum_{\ell \in \Lambda} \sum_{(\rho \alpha\beta)} V_{(\rho \alpha\beta)}(\tilde{D}\bm{u}^{0})\cdot \big( q_\beta - q_\alpha\big)\\
&=:~ B_1 + B_2 + B_3.
\end{split}
\end{equation}
We observe that $B_3$ was exactly the term, $T_2$, estimated in Theorem~\ref{energy_thm}, where we saw that
\[
B_3 = \<\bm{k}_1,\bm{q}\>, \qquad |k_1(\ell)| \lesssim |\ell|^{-2}.
\]
Meanwhile, $B_2$ is the second term of $T_1$ in~\eqref{t1}, which we saw could be estimated by
\begin{align*}
B_2 =~& \sum_{\ell \in \Lambda} \sum_{(\rho \alpha\beta)} V_{(\rho \alpha\beta)}(\tilde{D}\bm{u}^{0})\cdot \tilde{D}_\rho q_\beta \\
=~&\sum_{\ell \in \Lambda} \tilde{D}_{-\rho}\sum_{(\rho \alpha\beta)} V_{(\rho \alpha\beta)}(e(\ell))\cdot q_\beta = \<\bm{k}_2,\bm{q}\>, \qquad |k_2(\ell)| \lesssim |\ell|^{-2}.
\end{align*}
Thus,
\begin{equation}\label{hequation}
B_2 + B_3 = \<\bm{k}, \bm{q}\>, \qquad |k(\ell)| \lesssim |\ell|^{-2}.
\end{equation}
Next, we observe $B_1$ is precisely the first term estimated in $T_1$ in~\eqref{t1} so, as in Theorem~\ref{energy_thm}, there exists $\bm{g}: \Lambda \to (\mathcal{R}^d)^{\mathcal{R}_1}$ with $|g(\ell)| \lesssim~ |\ell|^{-2}$ and
\begin{align*}
B_1 =~&
\sum_{\ell \in \Lambda} \sum_{\rho \in \mathcal{R}_1} g_\rho (\ell)\cdot D_\rho v_0(\ell).
\end{align*}
\end{proof}
\subsection{Green's Function}\label{greeny}
Having defined our linearized problem and estimated the residual in Theorem~\ref{lin_prob_thm}, we will proceed to estimate the decay of $\bm{u}^\infty$ in the current section. Doing that will require comparing the atomistic Green's matrix for the homogeneous energy derived in~\cite{olsonOrtner2016} for point defects to the Green's matrix for the dislocation solution. Thus, we will introduce the homogeneous Green's function and the corresponding estimates for its decay here.
From~\cite{olsonOrtner2016}, we let $\xi \in \mathcal{B}$, where $\mathcal{B}$ is the first Brillouin zone associated to the atomic lattice, denote the Fourier variable and set
\begin{align*}
\hat{H}_{00}(\xi) :=~& \sum_{(\rho \alpha\beta)\in \mathcal{R}}\sum_{(\tau\gamma\delta)\in \mathcal{R}} (e^{-2\pi i \xi \cdot \tau}-1)V_{,(\rho \alpha\beta)(\tau\gamma\delta)}(0)(e^{2\pi i \xi \cdot \rho}-1), \\
[\hat{H}_{0\bm{p}}(\xi)]_\beta :=~& \sum_{\rho\in\mathcal{R}_1}\sum_{\alpha = 0}^{S-1}\sum_{(\tau\gamma\delta)\in \mathcal{R}} \big[(e^{-2\pi i \xi\cdot \tau}-1)V_{,(\rho\alpha\beta)(\tau\gamma\delta)}(0)(e^{2\pi i \xi\cdot \rho}) \\
&\hspace{3in}- (e^{-2\pi i \xi\cdot \tau}-1)V_{,(\rho\beta\alpha)(\tau\gamma\delta)}(0)\big], \\
[\hat{H}_{\bm{p}0}(\xi)]_\delta :=~& \sum_{\tau\in\mathcal{R}_1}\sum_{\gamma = 0}^{S-1}\sum_{(\rho \alpha\beta)\in \mathcal{R}} \big[(e^{-2\pi i \xi\cdot \tau}) V_{,(\rho\alpha\beta)(\tau\gamma\delta)}(0)(e^{2\pi i \xi\cdot \rho} - 1) \\
&\hspace{3in}- V_{,(\rho\alpha\beta)(\tau\delta\gamma)}(0)(e^{2\pi i \xi\cdot \rho} - 1)\big], \\
[\hat{H}_{\bm{p}\bm{p}}(\xi)]_{\beta\delta} :=~& \sum_{\rho,\tau \in \mathcal{R}_1}\sum_{\alpha,\gamma = 0}^{S-1}\left[e^{-2\pi i \xi\cdot \tau} V_{,(\rho\alpha\beta)(\tau\gamma\delta)}(0) e^{2\pi i \xi\cdot \rho} + V_{,(\rho\beta\alpha)(\tau\delta\gamma)}(0) - e^{-2\pi i \xi\cdot \tau}V_{,(\rho\beta\alpha)(\tau\gamma\delta)}(0) \right. \\
&\hspace{3in}\left. - e^{2\pi i \xi \cdot \rho} V_{,(\rho\alpha\beta)(\tau\delta\gamma)}(0)\right],
\end{align*}
and define the {\em dynamical matrix}~\cite{wallace1998},
\begin{equation}\label{dynam_matrix}
\hat{H}(\xi) := \begin{bmatrix} \hat{H}_{00}(\xi) & \hat{H}_{0\bm{p}}(\xi) \\ \hat{H}_{\bm{p}0}(\xi) & \hat{H}_{\bm{p}\bm{p}}(\xi)\end{bmatrix}.
\end{equation}
It has the property that
\begin{equation}\label{planch}
\<H\bm{u}^\infty, \bm{v} \> := \<\delta^2 \mathcal{E}^{\rm a}_{\rm hom}(\bm{0})\bm{u}^\infty, \bm{v}\> =~ \int_{\mathcal{B}} \begin{bmatrix} \hat{Z}(\xi) \\ \hat{\bm{q}}(\xi)\end{bmatrix}^* \hat{H}(\xi) \begin{bmatrix} \hat{U}^\infty(\xi) \\ \hat{\bm{p}}^\infty(\xi) \end{bmatrix}\, d\xi, \quad \forall \bm{v} = (Z,\bm{q}) \in \bm{\mathcal{U}}.
\end{equation}
The atomistic Green's matrix for the homogeneous energy is then defined by
\begin{equation}\label{green_mat}
\mathcal{G} := (\hat{H}^{-1})^{\vee},
\end{equation}
where $\vee$ denotes the inverse Fourier transform. It follows from Assumption~\ref{coercive} that $H$ is an invertible operator (see also~\cite{olsonOrtner2016}). Upon partitioning $\mathcal{G}$ as
\[
\begin{pmatrix}
G_{00} &G_{0\bm{p}}\\
G_{\bm{p}0} &G_{\bm{p}\bm{p}}
\end{pmatrix},
\]
several important decay estimates for the individual blocks were established in~\cite[Theorem 4.4]{olsonOrtner2016}: for $\bm{\rho} \in (\mathcal{R}_1)^t$, $t \geq 0$ and
$|\bm{\rho}| := t \in \mathbb{Z}$,
\begin{align}
\left|D_{\bm{\rho}}\mathcal{G}_{00}(\ell)\right| \lesssim~& (1+|\ell|)^{-d-|\bm{\rho}|+2} \qquad |\bm{\rho}| \geq 1, \label{hessianDecay1} \\
\left|D_{\bm{\rho}}\mathcal{G}_{0\bm{p}}(\ell)\right| \lesssim~& (1+|\ell|)^{-d-|\bm{\rho}|+1} \qquad |\bm{\rho}| \geq 0, \label{hessianDecay2}\\
\left|D_{\bm{\rho}}\mathcal{G}_{\bm{p}\bm{p}}\right| \lesssim~& (1+|\ell|)^{-d-|\bm{\rho}|} \qquad \quad |\bm{\rho}| \geq 0. \label{hessianDecay3}
\end{align}
\subsection{Analysis Problem}\label{pure_analysis}
We fix $\ell \in \Lambda$ and define the lattice function $\bm{v} = (Z,\bm{q})$ by $\bm{v} = \sum_{m}\mathcal{G}(k-\ell)\bm{e}_m$ where $\hat{e}_m$ is the $m$th standard basis vector. Using the relation~\eqref{planch}, we may write our infinite-domain dislocation solution as
\begin{equation}\label{conv}
\begin{pmatrix}
U^\infty(\ell) \\
\bm{p}^\infty(\ell)
\end{pmatrix}
=
\<H\bm{u}^\infty, \bm{v}\> = \sum_m\int_{\mathcal{B}} e^{2\pi i \xi \cdot \ell} \begin{bmatrix} \hat{U}^\infty(\xi) \\ \hat{\bm{p}}^\infty(\xi) \end{bmatrix}_m \hat{e}_m\, d\xi
\end{equation}
where $H$ is defined in~\eqref{planch} and $\mathcal{G}$ is the associated Green's matrix defined in~\eqref{green_mat}.
Thus,
\begin{equation}\label{conv_var}
\begin{split}
\begin{pmatrix}
D_\tau U^\infty(\ell) \\
\bm{p}^\infty(\ell)
\end{pmatrix}
=~& \sum_m\left<H(k)\begin{pmatrix}
U^\infty(k) \\
\bm{p}^\infty(k)
\end{pmatrix},\begin{pmatrix}
D_\tau G_{00}(k-\ell) &G_{0\bm{p}}(k-\ell)\\
D_\tau G_{\bm{p}0}(k-\ell) &G_{\bm{p}\bm{p}}(k-\ell)
\end{pmatrix} \hat{e}_m \right>
\end{split}
\end{equation}
As in~\cite{Ehrlacher2013}, we consider two cases depending on the location of $\ell$ in the lattice:
\subsubsection{Case 1} $B_{3/4|\ell|}(\ell) \cap \Gamma = \emptyset$
In this case we take a bump function $\eta(x)$ from~\cite{Ehrlacher2013}: define $s_1 := 1/2|\ell|-r_{\rm cut}, s_2 := 1/2|\ell|, \eta = 1$ in $B_{s_1/2}(\ell)$, $\eta = 0$ outside of $B_{s_1}(\ell)$, and $|\nabla \eta(x)| \lesssim~ |\ell|^{-1}$. We then make the substitution
\begin{equation}\label{eta_sub}
\begin{split}
\begin{pmatrix}
D_\tau U^\infty(\ell) \\
\bm{p}^\infty(\ell)
\end{pmatrix}
=~& \sum_m\left< H(k)\begin{pmatrix}
U^\infty(k) \\
\bm{p}^\infty(k)
\end{pmatrix},
\eta(k-\ell)\begin{pmatrix}
D_\tau G_{00}(k-\ell) &G_{0\bm{p}}(k-\ell)\\
D_\tau G_{\bm{p}0}(k-\ell) &G_{\bm{p}\bm{p}}(k-\ell)
\end{pmatrix}\hat{e}_m \right> \\
&~+ \left<H(k)\begin{pmatrix}
U^\infty(k) \\
\bm{p}^\infty(k)
\end{pmatrix},(1-\eta(k-\ell))\begin{pmatrix}
D_\tau G_{00}(k-\ell) &D_\tau G_{0\bm{p}}(k-\ell)\\
G_{\bm{p}0}(k-\ell) &G_{\bm{p}\bm{p}}(k-\ell)
\end{pmatrix}\hat{e}_m \right>
\end{split}
\end{equation}
Where $\eta \neq 0$, $D = \tilde{D}$ in this case so that from Theorem~\ref{lin_prob_thm}
\begin{equation}
\begin{split}
&\sum_m\left< H(k)\begin{pmatrix}
U^\infty(k) \\
\bm{p}^\infty(k)
\end{pmatrix},
\eta(k-\ell)\begin{pmatrix}
D_\tau G_{00}(k-\ell) &G_{0\bm{p}}(k-\ell)\\
D_\tau G_{\bm{p}0}(k-\ell) &G_{\bm{p}\bm{p}}(k-\ell)
\end{pmatrix}\hat{e}_m \right> \\
&=
\sum_m\left< \tilde{H}(k)\begin{pmatrix}
U^\infty(k) \\
\bm{p}^\infty(k)
\end{pmatrix},
\eta(k-\ell)\begin{pmatrix}
D_\tau G_{00}(k-\ell) &G_{0\bm{p}}(k-\ell)\\
D_\tau G_{\bm{p}0}(k-\ell) &G_{\bm{p}\bm{p}}(k-\ell)
\end{pmatrix}\hat{e}_m \right> \\
&= \sum_m\sum_{\ell \in \Lambda}\sum_{(\rho \alpha\beta)\in\mathcal{R}} D_{(\rho \alpha\beta)}\left(\eta(k-\ell)
\begin{pmatrix}
D_\tau G_{00}(k-\ell) & G_{0\bm{p}}(k-\ell)\\
D_\tau G_{\bm{p}0}(k-\ell) &G_{\bm{p}\bm{p}}(k-\ell)
\end{pmatrix}\hat{e}_m \right)\cdot \bar{f}_{(\rho \alpha\beta)}(k) \\
&~+~ \sum_m\sum_{\ell \in \Lambda}\sum_{\rho\in\mathcal{R}_1} D_{\rho}\Big(\eta(k-\ell)
\begin{pmatrix}
D_\tau G_{00}(k-\ell) &
G_{0\bm{p}}(k-\ell)
\end{pmatrix}_m\Big)\cdot g_\rho(k) \quad \mbox{$(\cdot)_m$ is $m$th column} \\
&~+~\sum_{m}\sum_{\ell \in \Lambda} \eta(k-\ell)
\begin{pmatrix}
D_\tau G_{0\bm{p}}(k-\ell) &
G_{\bm{p}\bm{p}}(k-\ell)
\end{pmatrix}_m \cdot \bm{k}(k).
\end{split}
\end{equation}
We may now utilize the decay estimates for $\mathcal{G}$ in~\eqref{hessianDecay1},~\eqref{hessianDecay2},~\eqref{hessianDecay3}, for $\eta$, and those for $\bar{f},g,\bm{k}$ in Theorem~\ref{lin_prob_thm} to obtain
\begin{equation}
\begin{split}
&\sum_m\left<H(k)\begin{pmatrix}
U^\infty(k) \\
\bm{p}^\infty(k)
\end{pmatrix},\eta(k-\ell)\begin{pmatrix}
D_\tau G_{00}(k-\ell) & G_{0\bm{p}}(k-\ell)\\
D_\tau G_{\bm{p}0}(k-\ell) &G_{\bm{p}\bm{p}}(k-\ell)
\end{pmatrix}\hat{e}_m \right> \\
&\lesssim \sum_{k \in B_{s_2}(\ell)} \left( |k|^{-2} + |\tilde{D}\bm{u}^\infty(k)|^2 \right)|\ell-k|^{-2} \\
&=~ \sum_{k \in B_{s_2}(\ell)} \left( |k|^{-2} + |D\bm{u}^\infty(k)|^2 \right)|\ell-k|^{-2}.
\end{split}
\end{equation}
Next, we estimate
\begin{equation}
\begin{split}
&\left<H(k)\begin{pmatrix}
U^\infty(k) \\
\bm{p}^\infty(k)
\end{pmatrix},(1-\eta(k-\ell))\begin{pmatrix}
D_\tau G_{00}(k-\ell) & G_{0\bm{p}}(k-\ell)\\
D_\tau G_{\bm{p}0}(k-\ell) &G_{\bm{p}\bm{p}}(k-\ell)
\end{pmatrix}\hat{e}_m \right>\\
&=~ \sum_{k \in \ell}\<\delta^2 V(0)D\bm{u}^\infty,D\Big((1-\eta(k-\ell))\begin{pmatrix}
D_\tau G_{00}(k-\ell) & G_{0\bm{p}}(k-\ell)\\
D_\tau G_{\bm{p}0}(k-\ell) &G_{\bm{p}\bm{p}}(k-\ell)
\end{pmatrix}\hat{e}_m \Big)\>.
\end{split}
\end{equation}
Using the product rule for finite differences, decay estimates on $\mathcal{G}$ and $|\nabla \eta(x)| \lesssim |x|^{-1}$, it is then straightforward to estimate
\begin{equation}
\begin{split}
&\left<H(k)\begin{pmatrix}
U^\infty(k) \\
\bm{p}^\infty(k)
\end{pmatrix},(1-\eta(k-\ell))\begin{pmatrix}
D_\tau G_{00}(k-\ell) & G_{0\bm{p}}(k-\ell)\\
D_\tau G_{\bm{p}0}(k-\ell) &G_{\bm{p}\bm{p}}(k-\ell)
\end{pmatrix}\hat{e}_m \right> \\
&\lesssim~ \sum_{k \in \Lambda: |k-\ell| \gtrsim s_1/4 } |D\bm{u}(k)|^\infty|k-\ell|^{-2}.
\end{split}
\end{equation}
We now remark that these estimates are set up to be identical to those used to obtain~\cite[Equation 6.31]{Ehrlacher2013}, and thus we likewise obtain for sufficiently large $\ell$
\begin{equation}\label{641}
\max\{|DU^{\infty}(\ell)|, |\bm{p}^\infty(\ell)|\} \lesssim~ |\ell|^{-1} + \|(1+|\ell-k|)^{-1}D\bm{u}^\infty(k)\|_{\ell^2(\Lambda \cap B_{s_2}(\ell))}.
\end{equation}
\subsubsection{Case 2} $B_{3/4|\ell|}(\ell) \cap \Gamma \neq \emptyset$
We argue as in~\cite{Ehrlacher2013} by using a reflection argument whereby the branch cut $\Gamma = \{(x_1, x_2) : x_2 = \hat{x}_2, x_1 > \hat{x}_1\}$ is replaced by $\Gamma_S = \{(x_1, x_2) : x_2 = \hat{x}_2, x_1 < \hat{x}_1\}$ and the energy is replaced by
\[
\mathcal{E}_S(\bm{u}) = \sum_{\ell \in \Lambda} V(D(S_0u_0 + u)).
\]
For this energy, we have that $\<\delta \mathcal{E}_S(S\bm{u}^\infty), \bm{v}\> = 0$ since
\begin{align*}
&\<\delta \mathcal{E}_S(S\bm{u}^\infty), \bm{v}\> \\
&=~ \sum_{\ell \in \Lambda} \sum_{(\rho \alpha\beta)}V_{(\rho \alpha\beta)}(D(S_0u_0 + Su^\infty))D_{(\rho \alpha\beta)}\bm{v} \\
&=~ \sum_{\ell \in \Lambda} \sum_{(\rho \alpha\beta)}V_{(\rho \alpha\beta)}(D(S_0u_0 + Su^\infty))\cdot(D_{\rho}v_0(\ell) + q_\beta(\ell+\rho) - q_\alpha(\ell)) \\
&=~ \sum_{\ell \in \Lambda} \sum_{(\rho \alpha\beta)}SRV_{(\rho \alpha\beta)}(D(S_0u_0 + Su^\infty))\cdot(D_{\rho}v_0(\ell) +q_\beta(\ell+\rho) - q_\beta(\ell)+ q_\beta(\ell) - q_\alpha(\ell) ) \\
&=~ \sum_{\ell \in \Lambda} \sum_{(\rho \alpha\beta)}V_{(\rho \alpha\beta)}(RD(S_0u_0 + Su^\infty))\cdot(R(D_{\rho}v_0(\ell) +q_\beta(\ell+\rho) - q_\beta(\ell)+ q_\beta(\ell) - q_\alpha(\ell))) \\
&=~ \sum_{\ell \in \Lambda} \sum_{(\rho \alpha\beta)}V_{(\rho \alpha\beta)}(e(\ell) + \tilde{D}u^\infty))\cdot(RD_{\rho}v_0(\ell) + RD_\rho q_\beta(\ell) + Rq_\beta(\ell) - Rq_\alpha(\ell) )) \\
&=~ \sum_{\ell \in \Lambda} \sum_{(\rho \alpha\beta)}V_{(\rho \alpha\beta)}(e(\ell) + \tilde{D}u^\infty))\cdot(RD_{\rho}Sw_0(\ell) + RD_{\rho}Sr_\beta(\ell) + r_\beta(\ell) - r_\alpha(\ell) )) \\
&=~ \sum_{\ell \in \Lambda} \sum_{(\rho \alpha\beta)}V_{(\rho \alpha\beta)}(e(\ell) + \tilde{D}u^\infty))\cdot \tilde{D}\bm{w}(\ell) \\
&=~ \<\delta \mathcal{E}(\bm{u}^\infty), \bm{w}\> = 0,
\end{align*}
where $w_0 = Rv_0$, $r_\gamma = Rq_\gamma$, and $w_\gamma = w_0 + r_\gamma$. As we have only reflected the branch cut, this problem is identical to the previous case in the sense that an estimate for $S\bm{u}^\infty$ now exists in the $x \geq \hat{x}_1$ plane (where there is no branch cut):
\[
\max\{|DSU^{\infty}(\ell)|, |S\bm{p}^\infty(\ell)|\} \lesssim~ |\ell|^{-1} + \|(1+|\ell-k|)^{-1}DS\bm{u}^\infty(k)\|_{\ell^2(\Lambda \cap B_{s_2}(\ell))}.
\]
As $R$ simply represents a translation operation by one Burger's vector, we then also have
\[
\max\{|RDSU^{\infty}(\ell)|, |RS\bm{p}^\infty(\ell)|\} \lesssim~ |\ell|^{-1} + \|(1+|\ell-k|)^{-1}RDS\bm{u}^\infty(k)\|_{\ell^2(\Lambda \cap B_{s_2}(\ell))},
\]
or
\[
\max\{|\tilde{D}U^{\infty}(\ell)|, |\bm{p}^\infty(\ell)|\} \lesssim~ |\ell|^{-1} + \|(1+|\ell-k|)^{-1}\tilde{D}\bm{u}^\infty(k)\|_{\ell^2(\Lambda \cap B_{s_2}(\ell))}.
\]
It is now immediate from the proof of~\cite[Lemma 6.7]{Ehrlacher2013} that
\begin{equation}\label{sub_optimal}
\max\{|\tilde{D}U^{\infty}(\ell)|, |\bm{p}^\infty(\ell)|\} \lesssim~ |\ell|^{-1}.
\end{equation}
\subsubsection{Proof of Theorem~\ref{decay_thm} }
Thus far, we have carefully set everything up so that we may use the techniques and analysis of~\cite{Ehrlacher2013} to complete the proof of Theorem~\ref{decay_thm}. The primary idea is to obtain suboptimal estimates as in~\eqref{sub_optimal}, which translates into higher regularity of the residual in Theorem~\ref{lin_prob_thm}, which can then in turn be used to prove higher regularity of $\tilde{D}\bm{u}^\infty$.
We return to equation~\eqref{conv_var} and set
\[
\bm{v} = \sum_m \begin{pmatrix}
D_\tau G_{00}(k-\ell) & G_{0\bm{p}}(k-\ell)\\
D_\tau G_{\bm{p}0}(k-\ell) &G_{\bm{p}\bm{p}}(k-\ell)
\end{pmatrix}\hat{e}_m
\]
to write
\begin{equation}\label{conv_diff2}
\begin{split}
&\begin{pmatrix}
D_\tau U^\infty(\ell) \\
\bm{p}^\infty(\ell)
\end{pmatrix} \\
&=
\sum_{k \in \Lambda} \<\delta^2 V(\bm{0}) D\bm{u}^\infty, D\bm{v}\> +
\sum_{k \in \Lambda} \<\delta^2 V(\bm{0}) \tilde{D}\bm{u}^\infty, \tilde{D}\bm{v}\> -\sum_{k \in \Lambda} \<\delta^2 V(\bm{0}) \tilde{D}\bm{u}^\infty, \tilde{D}\bm{v}\> \\
&= \sum_{k \in \Lambda} \Big(\sum_{(\rho \alpha\beta) \in \mathcal{R}} \bar{f}_{(\rho \alpha\beta)}(k)\cdot \tilde{D}_{(\rho \alpha\beta)}\bm{v}(k) + \<\bm{g}(k), Dv_0(k)\> + \<\bm{k}(k),\bm{q}(k)\>\Big) \quad \mbox{by Theorem~\ref{lin_prob_thm}} \\
&\quad+
\sum_{k \in \Lambda} \<\delta^2 V(\bm{0}) D\bm{u}^\infty, D\bm{v}\> - \sum_{k \in \Lambda} \<\delta^2 V(\bm{0}) \tilde{D}\bm{u}^\infty, \tilde{D}\bm{v}\>.
\end{split}
\end{equation}
It follows from Equation~\eqref{sub_optimal} that $|\bar{f}_{(\rho \alpha\beta)}(k)| \lesssim (1+ |k|)^{-2}$ and from the decay estimates on $\mathcal{G}$ that
\begin{equation}\label{next_eq}
\begin{split}
\big|\sum_{k \in \Lambda} \Big(\sum_{(\rho \alpha\beta) \in \mathcal{R}} \bar{f}_{(\rho \alpha\beta)}(k)\cdot \tilde{D}_{(\rho \alpha\beta)}\bm{v}(k) + \<\bm{g}, Dv_0\> + \<\bm{k},\bm{q}\>\Big)\big| \lesssim~& \sum_{k \in \Lambda} (1+ |k|)^{-2}(1+|\ell-k|)^{-2} \\\
\lesssim~& |\ell|^{-2}\log|\ell|.
\end{split}
\end{equation}
Next, we assume $\ell_1 < \hat{x}_1$ so that we have $\tilde{D} = D$ for $\ell$ large enough, which means
\begin{equation*}
\begin{split}
\sum_{k \in \Lambda} \<\delta^2 V(\bm{0}) D\bm{u}^\infty, D\bm{v}\> - \sum_{k \in \Lambda} \<\delta^2 V(\bm{0}) \tilde{D}\bm{u}^\infty, \tilde{D}\bm{v}\>
\end{split}
\end{equation*}
is nonzero for \textit{only} those $k \in \Lambda$ within $r_{\rm cut}$ of the branch cut, $\Gamma$. If we let $U_\Gamma$ (as in~\cite{Ehrlacher2013}) denote the set of all such $k$, we have
\begin{equation}\label{sec_eq}
\begin{split}
\sum_{k \in \Lambda} \<\delta^2 V(\bm{0}) D\bm{u}^\infty, D\bm{v}\> - \sum_{k \in \Lambda} \<\delta^2 V(\bm{0}) \tilde{D}\bm{u}^\infty, \tilde{D}\bm{v}\> \\
&=~ \sum_{k \in U_\Gamma} \<\delta^2 V(\bm{0}) D\bm{u}^\infty, D\bm{v}\>- \sum_{k \in U_\Gamma} \<\delta^2 V(\bm{0}) \tilde{D}\bm{u}^\infty, \tilde{D}\bm{v}\>.
\end{split}
\end{equation}
In this case we can simply use the suboptimal bounds directly on $\bm{u}^\infty$ and the decay estimates for $\mathcal{G}$ to get
\begin{equation}\label{third_eq}
\begin{split}
\sum_{k \in \Lambda} \<\delta^2 V(\bm{0}) D\bm{u}^\infty, D\bm{v}\> - \sum_{k \in \Lambda} \<\delta^2 V(\bm{0}) \tilde{D}\bm{u}^\infty, \tilde{D}\bm{v}\> \\
&\lesssim~ \sum_{k \in U_\Gamma} (1+|k|)^{-1}(1 + |\ell-k|)^{-2}.
\end{split}
\end{equation}
As $U_\Gamma$ is simply a strip, this summation is now effectively one dimensional and can be bounded by noting $|\ell - k| \gtrsim |\ell| + |k|$ when $\ell_1 < \hat{x}_1$ and $k \in U_\Gamma$ so
\begin{equation}\label{fourth_eq}
\begin{split}
\sum_{k \in U_\Gamma} (1+|k|)^{-1}(1 + |\ell-k|)^{-2} \lesssim~& \int_{|\ell|}^\infty |\ell - k|^{-1}|k|^{-2}\, dk \\
\lesssim~& \int_{|\ell|}^\infty \frac{1}{|\ell||k|^{1} + |k|^{2}}\, dk \lesssim~ |\ell|^{-2}\log|\ell|.
\end{split}
\end{equation}
In the case $\ell_1 > \hat{x}_1$, then we may simply perform another reflection argument by placing the branch cut in the left-half plane. It therefore follows from~\eqref{fourth_eq} and~\eqref{next_eq} that in fact
\begin{equation}\label{optimal}
\max\{|\tilde{D}U^{\infty}(\ell)|, |\bm{p}^\infty(\ell)|\} \lesssim~ |\ell|^{-2}\log|\ell|,
\end{equation}
for \textit{all} large enough $|\ell|$. This completes the proof of Theorem~\ref{decay_thm}.
As for the case for higher order derivatives (differences) alluded to in Remark~\ref{decay_remark}:
\[
\max\{|\tilde{D}^jU^{\infty}(\ell)|, |D^{j-1}\bm{p}^\infty(\ell)|\} \lesssim~ |\ell|^{-1-j}\log|\ell|,
\]
we simply give a high-level view of the argument as it is nearly identical to that given in~\cite[Proof of Theorem 3.6]{Ehrlacher2013} but with our usual modifications to extend to the multilattice case. The principal idea is to again write
\begin{equation}\label{conv_diff3}
\begin{split}
\begin{pmatrix}
D_\tau U^\infty(\ell) \\
\bm{p}^\infty(\ell)
\end{pmatrix}
=~&
\sum_m\< H(k)\begin{pmatrix}
U^\infty(k) \\
\bm{p}^\infty(k)
\end{pmatrix} , \begin{pmatrix}
D_\tau G_{00}(k-\ell) & G_{0\bm{p}}(k-\ell)\\
D_\tau G_{\bm{p}0}(k-\ell) &G_{\bm{p}\bm{p}}(k-\ell)
\end{pmatrix}\hat{e}_m \>
\end{split}
\end{equation}
and take higher finite differences in succession. At each stage, the estimates on the residual can be improved by taking into account the decay proven in the previous order finite differences, and thus improved estimates can be obtained rigorously via an induction argument.
\section{Applications and Numerical Examples}
Having proven our main result, in this section, we use it to prove a result concerning convergence of a numerical method to the true defect solution, $\bm{u}^\infty$. We illustrate this with a numerical example of an edge dislocation in silicon.
\subsection{Algorithm}
A simple, yet effective algorithm for approximating a single, straight-line dislocation in an infinite crystal is to fix some finite computational domain $\Omega \subset \Lambda$ and define the approximation space
\[
\bm{\mathcal{U}}_{\Omega} := \{\bm{u} \in \mathcal{A} : u_\alpha = 0, \, \forall \ell \notin \Omega\}.
\]
In this situation, all displacements in $\bm{\mathcal{U}}_{\Omega}$ are given a boundary condition which is just the continuum-elasticity predictor $\bm{u}^0 = (U^0, \bm{p}^0)$. If we parametrize the ``size'' of $\Omega$ by a radius
\[
\Omega(R) = \sup \{ r > 0 : B_r(\xi) \subset \Omega, \xi \in \Omega\},
\]
it is straightforward to prove
\begin{theorem}\label{alg_theorem}
Suppose that $\bm{u}^\infty \in \mathcal{A}$ is a solution to the atomistic Euler-Lagrange equations $\<\delta \mathcal{E}^{\rm a}(\bm{u}^\infty),\bm{v}\> = 0$ for all $\bm{v} \in \bm{\mathcal{U}}_0$ and that Assumption~\ref{coercive} holds. Then there exists $R_0$ such that for all domains $\Omega$ with $\Omega(R) \geq R_0$, there exists a solution $\bm{u}_\Omega$ to
\[
\bm{u}_\Omega = {\rm argmin}_{\bm{\mathcal{U}}_{\Omega}} \mathcal{E}^a(\bm{u})
\]
which satisfies
\begin{equation}\label{strain_est}
\|\bm{u}_\Omega - \bm{u}^\infty\|_{{\rm a}_1} \lesssim R^{-1}\log R.
\end{equation}
\end{theorem}
\begin{proof}
For simplicity we assume that $\Omega$ is chosen so that
\[
\Omega(R) = \sup \{ r > 0 : B_r(0) \subset \Omega \}.
\]
As in any Galerkin method, the key here is in estimating the best approximation error of $\bm{u}^\infty$ in the space $\bm{\mathcal{U}}_\Omega$. This was accomplished in the work~\cite[Lemma A.1]{olsonOrtner2016} and~\cite[Lemma 12]{multiBlendingArxiv} where an approximation, $\Pi \bm{u}^\infty \in \bm{\mathcal{U}}_\Omega$, to $\bm{u}^\infty$ was defined, and it was shown that
\[
\|\bm{u}^\infty - \Pi \bm{u}^\infty\|_{{\rm a}_1} \lesssim \|\nabla I U\|_{L^2(\mathbb{R}^2\setminus B_{R/2}(0))} + \|I\bm{p}\|_{L^2(\mathbb{R}^2\setminus B_{R/2}(0))},
\]
where we recall $I$ was a piecewise linear interpolation operator. Using the decay estimates in Theorem~\ref{decay_thm}, we may then estimate these last two terms and have
\[
\|\bm{u}^\infty - \Pi \bm{u}^\infty\|_{{\rm a}_1} \lesssim~ R^{-1}\log R.
\]
By continuity, Assumption~\ref{coercive} implies
\[
\<\delta^2 \mathcal{E}^{\rm a}(\Pi \bm{u}^\infty)\bm{v},\bm{v} \> \gtrsim \|\bm{v}\|_{{\rm a}_1}^2, \quad \forall \bm{v} \in \bm{\mathcal{U}}_0,
\]
(with a possibly different implied constant). Moreover,
\begin{align*}
\<\delta \mathcal{E}^{\rm a}(\Pi \bm{u}^\infty), \bm{v} \> =~& \<\delta \mathcal{E}^{\rm a}(\Pi \bm{u}^\infty), \bm{v} \> - \<\delta \mathcal{E}^{\rm a}(\bm{u}^\infty), \bm{v} \> \\
\lesssim~& \|\Pi \bm{u}^\infty - \bm{u}^\infty\|_{{\rm a}_1} \|\bm{v}\|_{{\rm a}_1}
\end{align*}
so we may apply the inverse function theorem to deduce the existence of a solution $\bm{u}_\Omega \in \bm{\mathcal{U}}_\Omega$ to
\[
\<\delta \mathcal{E}^{\rm a}(\bm{u}), \bm{v}\> = 0, \quad \forall \bm{v} \in \bm{\mathcal{U}}_\Omega,
\]
which satisfies
\[
\|\bm{u}_\Omega - \Pi \bm{u}^\infty\|_{{\rm a}_1} \lesssim~ R^{-1}\log R.
\]
But then
\[
\|\bm{u}_\Omega - \bm{u}^\infty\|_{{\rm a}_1} \lesssim~ R^{-1}\log R
\]
as well.
\end{proof}
\subsection{Edge Dislocation in Silicon}
The structure of silicon is two interpenetrating FCC lattices, where in the definition of a multilattice, we may take:
\[
{\sf B} = \frac{1}{2} \begin{pmatrix} 0 &1 &1 \\
1 &0 &1 \\
1 &1 &0 \end{pmatrix}, \quad p_0 = \bm{0}, \quad p_1 = \begin{pmatrix}1/4 \\ 1/4 \\ 1/4 \end{pmatrix}.
\]
We consider an edge dislocation with Burger's vectors $\bm{b} = \frac{1}{2}\<\bar{1} 1 0\>$ in the $\{\bar{1} \bar{1} 1\}$ slip plane such that the dislocation line has direction $\<1 1 2\>$, which is also the system analyzed in~\cite{nandedkar1987atomic}. (Here, we have chosen units so that the lattice constant is unity.)
We choose to model silicon with a Stilinger-Weber type potential~\cite{stillinger1985}, and the analytical solution to the continuum elasticity predictor problem may then be obtained from~\cite{hirth1982theory} after which the shifts may be solved for.
We now perform a self-convergence study in the $\|\cdot\|_{{\rm a}_1}$ norm for increasing values of $R$ and plot the results in the blue curve of Figure~\ref{fig1}, which shows very good $R^{-1}$ decay as expected from the theory. We additionally plot several other curves in the figure, which are discussed below and are obtained by using various other continuum elasticity predictors for $\bm{u}^0 = (U^0, \bm{p}^0)$.
\begin{figure}[htp!]
\caption{Energy Error for Silicon Edge Dislocation using Stilinger-Weber potential.}
\label{fig1}
{\resizebox{3.5in}{3.0in}{\includegraphics{sw_edge_err2}}}
\end{figure}
The red and green curves are derived by using isotropic and anisotropic continuum elasticity solutions~\cite{hirth1982theory} where the elasticity constants in $\mathbb{C}$ are obtained from empirical tables~\cite{hirth1982theory} and not Cauchy--Born theory. This is for example the approach taken in~\cite{nandedkar1987atomic}. We then solve for the continuum displacement, $U$, and subsequently, the shifts before carrying out our numerical algorithm described above. We note that a lower order convergence appears, but at the moment, we have no explanation as to why these models still appear to be converging, though one possible explanation would be a pre-asymptotic effect in the highly symmetric diamond cubic lattice. Alternatively, there may very well be some form of convergence occurring that deserves further investigation. The final purple curve uses a slightly modified Cauchy-Born predictor using the modified Cauchy-Born energy of~\cite{koten2013}. It is only applicable to those multilattice crystals which are single-species two lattices (which silicon is as it is two carbon FCC lattices). Though the convergence matches the regular Cauchy-Born rate, the symmetrized Cauchy-Born energy has the remarkable result that the individual residual forces decay at a higher rate, and would thus lead to simplified proof of our results. Our main interest in this energy was to see if any benefits could be seen, but this simplified setting did not appear to show a significant difference.
|
1,116,691,500,885 | arxiv | \section{Introduction}
The origin of flavor has been a significant puzzle in particle physics
since the discovery of the muon. The replication of fermion generations
and the strongly hierarchical pattern of their masses and mixing angles
is left unexplained in the Standard Model. In most theories of flavor,
new symmetries are introduced at mass scales that are large compared to
the electroweak scale. To stabilize this hierarchy of scales, it is
natural to work in the framework of the supersymmetric standard model
(SSM). However, this complicates the flavor problem by introducing a
new sector of particles whose masses and mixing angles must also be
understood. While no superpartner has yet been observed, the acceptable
spectrum is constrained by low-energy processes. Most notably, a high
degree of degeneracy is required among the light generation squarks to
suppress dangerous flavor-changing effects \cite{DG}, unless there is a strong
alignment of quark and squark eigenstates \cite{NS,LNS}. The challenge
in a supersymmetric theory of flavor is to simultaneously explain both
the suppression of flavor changing effects from the scalar sector and the
hierarchical pattern of the quark Yukawa couplings.
In this talk, I will advocate imposing a discrete, gauged non-Abelian family
symmetry on the SSM to obtain the desired degree of sfermion
degeneracy \cite{hallmur}. This choice is reasonable given that global
continuous symmetries may be broken by quantum gravitational effects \cite{qg},
while gauged continuous symmetries may generate $D$-term contributions to
the squark and slepton masses that are nonuniversal \cite{Dterm}. In
Ref.~\cite{hallmur}, it was demonstrated that the non-Abelian discrete group
$(S_3)^3$ is a promising choice for this flavor symmetry. The group $S_3$
has both a doublet (${\bf 2}$) and a non-trivial singlet representation
(${\bf 1}_A$) into which the three generations of fermions can be
embedded. In order to construct a viable model for the quarks, three
separate $S_3$ factors are required, for the left-handed doublet fields $Q$,
and the right-handed singlet fields $U$ and $D$. The first and second
generation fields transform as doublets, which ensures the degeneracy among
the light generation squarks. The third generation fields must then transform
as ${\bf 1}_A$s so that the theory is free of discrete gauge anomalies.
While the group $S_3$ acts identically on three objects, the representation
structure distinguishes between the generations. Thus, it is possible to
choose the quantum numbers for the Higgs fields so that only the top
Yukawa coupling is allowed in the symmetry limit. The hierarchical
structure of Yukawa matrices can then be understood as a consequence of the
sequential breaking of the flavor symmetry group.
After reviewing the basic scenario in the quark sector \cite{hallmur},
I show how we may extend the $(S_3)^3$ model to the leptons \cite{CHM1}.
This work was done in collaboration with Larry Hall and Hitoshi Murayama.
I assume that the fundamental sources of flavor symmetry breaking are
gauge singlet fields $\phi$ that transform in the same way as the
irreducible ``blocks'' of the quark Yukawa matrices. I will call these
fields `flavons' below. I assume that the fermion Yukawa matrices arise
from higher dimension operators involving the $\phi$ fields,
that are present at the Planck scale. With flavor symmetry breaking
originating only from the Yukawa matrices, I estimate the
contributions to lepton flavor violation and proton decay. The latter
originates from non-renormalizable operators that conserve $R$-parity,
but violate baryon and lepton number, that are also presumably
generated at the Planck scale. I show that the flavor symmetry is
sufficient to suppress these operators to an acceptable level. In
addition, I show that the dominant proton decay modes in the $(S_3)^3$
model are of the form $p \rightarrow K l$, where $l$ is a {\em first}
generation lepton. This is never the case in either supersymmetric or
non-supersymmetric grand unified theories. The prediction of these
rather unique modes is exciting since the total decay rate is likely to
be within the reach of the SuperKamiokande experiment.
\section{The Basic Model}
In the $(S_3)^3$ model of Ref.~\cite{hallmur}, the quark chiral
superfields $Q$, $U$, and $D$ are assigned to ${\bf 2}+{\bf 1_A}$
representations of $S_3^Q$, $S_3^U$ and $S_3^D$, respectively. The
first two generation fields are embedded in the doublet, for the reasons
described in the Introduction. The Higgs fields both transform
as $({\bf 1}_A, {\bf 1}_A, {\bf 1}_S)$'s, so that the top quark Yukawa
coupling is invariant under the flavor symmetry group. The
transformation properties of the Yukawa matrices are:
\begin{equation}
Y_U \sim \left(
\begin{array}{cc|c}
\multicolumn{2}{c|}{
({\bf \tilde{2}},{\bf \tilde{2}},{\bf 1}_S) }
& ({\bf \tilde{2}},{\bf 1}_S,{\bf 1}_S) \\ \hline
\multicolumn{2}{c|}{({\bf 1}_S,{\bf \tilde{2}},{\bf 1}_S)} &
({\bf 1}_S,{\bf 1}_S,{\bf 1}_S) \end{array} \right)
\end{equation}
\begin{equation}
Y_D \sim \left(\begin{array}{cc|c}
\multicolumn{2}{c|}{
({\bf \tilde{2}},{\bf 1}_A,{\bf 2})}
& ({\bf \tilde{2}},{\bf 1}_A,{\bf 1}_A) \\ \hline
\multicolumn{2}{c|}{({\bf 1}_S,{\bf 1}_A,{\bf 2})} &
({\bf 1}_S,{\bf 1}_A,{\bf 1}_A) \end{array}\right)
\label{eq:transp}
\end{equation}
where I use the notation ${\bf \tilde{2}} \equiv {\bf 2} \otimes {\bf
1}_A$,\footnote{${\bf \tilde{2}} = (a,b)$ is equivalent to ${\bf 2} =
(b, -a)$.}. Note that these matrices involve at most 7 irreducible multiplets
of $(S_3)^3$. In Ref.~\cite{hallmur}, $(S_3)^3$ was broken by only four types
of flavons: $\phi({\bf\tilde{2}}, {\bf 1}_S, {\bf 1}_S)$, $\phi({\bf
\tilde{2}}, {\bf\tilde{2}}, {\bf 1}_S)$, $\phi({\bf 1}_S, {\bf 1}_A, {\bf
1}_A)$, and $\phi({\bf \tilde{2}}, {\bf 1}_A, {\bf 2})$, the minimal number
which leads to realistic masses and mixings \cite{CM3}:
\begin{equation}
Y_U = \left( \begin{array}{cc|c}
h_u & h_c \lambda & - h_t V_{ub}\\
0 & h_c & - h_t V_{cb} \\ \hline
0 & 0 & h_t
\end{array} \right) ,
\end{equation}
\begin{equation}
Y_D = \left( \begin{array}{cc|c}
h_d & h_s \lambda & 0\\
0 & h_s & 0 \\ \hline 0 & 0 & h_b
\end{array} \right),
\end{equation}
with $\lambda \simeq 0.22$. The form of the Yukawa matrices presented
above can be understood as a consequence of a sequential breaking of
the flavor symmetry. I will assume that the $2\times 2$ blocks
of $Y_U$ and $Y_D$ are each generated by two flavon fields that
acquire vevs at different stages of the symmetry breaking. Thus,
$Y_D = Y_1 + Y_2$, and $Y_U=Y'_1+Y'_2$ where
\begin{equation}
Y_1 = \left( \begin{array}{cc}
0 & a h_s \lambda \\ 0 & h_s
\end{array} \right) \,\,\, , \,\,\,
Y_2 = \left( \begin{array}{cc}
h_d & 0 \\ 0 & 0
\end{array} \right).
\label{eq:y}
\end{equation}
and
\begin{equation}
Y'_1 = \left( \begin{array}{cc}
0 & a' h_c \lambda \\ 0 & h_c
\end{array} \right) \,\,\, , \,\,\,
Y'_2 = \left( \begin{array}{cc}
h_u & 0 \\ 0 & 0
\end{array} \right).
\label{eq:y'}
\end{equation}
Note that $a$ and $a'$ are order one constants, with $a-a'=1$.
\section{Incorporating Lepton Sector}
Three principles determine the precise transformation
properties of the lepton fields:
\begin{enumerate}
\item There are no new flavor symmetries (e.g. new $S_3$ factors)
that arise only in the lepton sector. The only flavor symmetry in
the theory is $S_3^Q \times S_3^U \times S_3^D$.
\item The transformation properties of the lepton fields are chosen
so that the charged lepton Yukawa matrix is similar to that of the
down quarks.
\item The most dangerous dimension-five operator
that contributes to proton decay, $(QQ)(QL)$, is forbidden in the $(S_3)^3$
symmetry limit.
\end{enumerate}
As we will see below, these principles are sufficient to completely
determine the transformation properties of the lepton fields.
Let us first consider the consequences of the first two conditions.
The down-quark Yukawa matrix is a coupling between the left-handed quark
fields $Q \sim ({\bf 1}_A+{\bf 2}, {\bf 1}_S, {\bf 1}_S)$ and the right-handed
down quark fields $D \sim ({\bf 1}_S, {\bf 1}_S, {\bf 1}_A + {\bf 2})$.
We know that the Yukawa matrix of the charged leptons is quite similar
to that of the down quarks, up to factors of order
three \cite{GJ} at high scales:
\begin{equation}
m_b \simeq m_\tau, \,\,\,\,\,\,\,\,\, m_s \simeq \frac{1}{3} m_\mu,
\,\,\,\,\,\,\,\,\, m_d \simeq 3 m_e.
\label{eq:3factors}
\end{equation}
Therefore, we look for an assignment of lepton transformation properties
that leads automatically to this observed similarity. There are only
two possibilities:
\begin{equation}
\begin{array}{c|cc}
& S_3^Q & S_3^D \\ \hline L & {\bf 1}_A+{\bf 2} &
{\bf 1}_S \\ E & {\bf 1}_S & {\bf 1}_A + {\bf 2}
\end{array}
\mbox{ or }
\begin{array}{cc}
S_3^Q & S_3^D\\ \hline {\bf 1}_S & {\bf
1}_A+{\bf 2} \\ {\bf 1}_A + {\bf 2} & {\bf 1}_S
\end{array}
\label{true}
\end{equation}
The third condition above allows us to distinguish between these
two alternatives. In the first assignment, the operator
$(Q_i Q_i) (Q_j L_j)$ is allowed by the $(S_3)^3$ symmetry, and
we have proton decay at an unacceptable rate. Therefore, only the second
assignment in Eq.~(\ref{true}) satisfies all three criteria listed
above.
The remaining question that we need to answer is how the factors of
three in Eq.~(\ref{eq:3factors}) enter in the Yukawa matrices.
One plausible explanation is that they originate from fluctuations
in the order one coefficients that multiply the $(S_3)^3$ breaking
parameters. Thus, we assume $Y_D=Y_1+Y_2$, while
$Y_l = 3 Y_1 + \frac{1}{3} Y_2$.
\subsection{Lepton Flavor Violation}
The strongest constraint on lepton flavor violation comes from the
non-observation of the $\mu \rightarrow e \gamma$ decay mode. In our
model, the contribution of the off-diagonal term in the purely
left-handed slepton mass matrix (the LL matrix) is small
enough ($\sim h_s^2 \lambda \sim 1 \times 10^{-7}$) to avoid the experimental
constraint for any value of $m_{\tilde{l}}$ above the LEP bound. The
stringent limits come from the purely right-handed slepton mass matrix (RR)
and the left-right (LR) matrix, which we discuss in this section. For
simplicity, we work in the approximation where the exchanged neutralino is a
pure bino state.
The one-loop slepton and bino exchange diagram that picks up the off-diagonal
(2,1) component in LR mass matrix generates the operator
\begin{equation}
\frac{e}{2} F_2(M_1^2,(m_{LR}^2)_{21},m^2_R, m^2_L)
\,\,\, \bar{e}_R i\sigma^{\mu\nu} \mu_L
F_{\mu\nu} \, ,
\label{eq:mego}
\end{equation}
where $F_{\mu\nu}$ is the electromagnetic field strength and the
function $F_2$ is defined in Ref.~\cite{CHM1}.
The decay width is given by
\begin{equation}
\Gamma( \mu \rightarrow e \gamma) = \frac{\alpha}{4} m_\mu^3 |F_2|^2 \,,
\end{equation}
and the bound Br$( \mu \rightarrow e \gamma) < 4.9 \times 10^{-11}$ implies
$|F_2| < 2.6 \times 10^{-12}$~GeV$^{-1}$. In order to compare this bound to
the prediction of our model, let us take $m_R = m_L = m = 300$~GeV and
$M_1 = 100$~GeV as a representative case. We obtain
\begin{equation}
\frac{(m^2_{LR})_{21}}{m^2} < 1.0 \times 10^{-5}
\label{eq:cons}
\end{equation}
for this choice of parameters. In our model, the (2,2) and (1,2)
elements in $Y_l$ belong to the same irreducible multiplet, and
diagonalization of $Y_l$ also diagonalizes LR mass matrix at $O(h_s
\lambda)$. The term which may not be simultaneously diagonalizable
comes from the piece $Y_2 \sim h_d$, and hence
\begin{equation}
(m^2_{LR})_{21} \sim m_d \lambda A,
\end{equation}
where $m_d$ is the down quark mass evaluated at the Planck scale $m_d
\simeq 10~\mbox{MeV}/ 3$, and $A$ is a typical trilinear coupling. If
we take $A \sim 100$~GeV, then $(m^2_{LR})_{21}/m^2 \sim 0.8 \times
10^{-6}$ and the constraint (\ref{eq:cons}) is easily satisfied.
The (1,2) element in the LR mass matrix contributes
in exactly the same way as the (2,1) entry, except that the chiralities
of the electron and muon in Eq.~(\ref{eq:mego}) are flipped. Hence,
the (1,2) element is subject to the same constraint, which again is
clearly satisfied in our model.
The RR mass matrix also contributes to the operator in (\ref{eq:mego}).
For the bino and slepton masses chosen earlier, we
obtain the bound
\begin{equation}
\frac{(m^2_{RR})_{12}}{m^2} < 0.023,
\label{eq:mrrc}
\end{equation}
while in our model
\begin{equation}
(m^2_{RR})_{12}/m^2 \simeq h_t V_{cb} \lambda \sim 0.009 \,\,.
\end{equation}
Thus, the bound on the (1,2) element of the RR matrix also satisfied.
Note that the contributions to $\mu\rightarrow e\gamma$ involving
mixing to the third generation scalars are comparable to those
in the minimal SO(10) model \cite{BHS}, and hence they are
phenomenologically safe.
\subsection{Proton Decay}
Since we have assumed that all possible
nonrenormalizable operators are generated at the Planck scale,
the task of studying proton decay in our model is a simple one.
We first write down all the possible dimension-five operators
that contribute to proton decay and identify their transformation
properties under $(S_3)^3$. The coefficients can be estimated as the
product of Yukawa couplings that will produce the desired symmetry
breaking effect. A list of possible operators and their
coefficients is given in Ref.~\cite{CHM1}.
The leading contribution to proton decay comes from the operator
$$
(Q_i Q_i)(Q_i L_i)/ M_*
,$$
where $M_* \equiv M_{Pl}/\sqrt{8\pi}$
is the reduced Planck mass, and where parentheses indicate a contraction
of SU(2) indices. This operator transforms as
a $({\bf 2}, {\bf 1}_S, {\bf 2})$ under $(S_3)^3$, and therefore has a
coefficient of order $h_b h_s$. The coefficients for all four components of
this operator are
given by
\[
\frac{c}{2} \frac{h_b}{M_*} \left[ h_s (Q_2 Q_2) (Q_1 L_1)
- h_s \lambda (Q_1 Q_1) (Q_2 L_1) \right. \]\begin{equation}
- \left. {\cal O}(h_d) (Q_2 Q_2)(Q_1 L_2)
+ h_d (Q_1 Q_1) (Q_2 L_2) \right]
\label{dimen5}
\end{equation}
where $c$ is an unknown coefficient of ${\cal O}(1)$. The striking feature
of this multiplet is that the operators involving first generation lepton
fields $L_1$ have larger coefficients than those involving second generation
fields $L_2$. Thus, our model favors proton decay to $\nu_e$ and $e$
over decay to $\nu_\mu$ and $\mu$. This result is in striking contrast
to the situation in grand unified theories, where the amplitude is
proportional to the Yukawa coupling of the final-state lepton.
In our case, however, there is a residual $Z_2$ flavor symmetry in the
limit where the first generation Yukawa couplings are set to zero.
Under this symmetry, the $U$, $D$, and $L$ fields of the second and
third generation are odd, while all other matter fields are even.
Thus, when the first generation Yukawa couplings are set to zero,
dimension-five operators of the form $QQQL$ containing $L_2$ or $L_3$
are forbidden by this $Z_2$, while those involving $L_1$ are invariant.
The predicted nucleon decay modes are obtained from Eq.~(\ref{dimen5})
by `dressing' the two-scalar-two-fermion operators with wino
exchange. We obtain\footnote{In the
following discussion, we assume that the Cabibbo mixing originates from
the down sector, i.e. $a=1$, $a'=0$ in eq.~(\ref{eq:y}) and (\ref{eq:y'}).
However we checked that all the results remain the
same even when the Cabibbo mixing comes from both the down and the up
sectors, or even solely from the up sector.}
\begin{eqnarray}
{\cal L} & = & \frac{\alpha_W}{2\pi} \frac{c h_b h_s}{M_*} \left[
- \lambda (su) (d\nu_e) + \lambda (su) (ue) \right] \nonumber \\
& \times & ( f(c,e) + f(c, d) ) \,\, ,
\label{eq:sumres}
\end{eqnarray}
where parentheses now indicate the contraction of spinor indices.
All the fields in (\ref{eq:sumres}) are in the mass eigenstate basis, and
terms of higher orders in $\lambda$ have been neglected. The function
$f$ is the ``triangle diagram factor'' \cite{NA}, a function of the
wino and scalar masses; above we have used the fact $f(c,d) = f(u,d)$
to good accuracy. The ratios of decay widths can be estimated using
the chiral Lagrangian technique \cite{CWH,CD,HMY}. We find
\[
\Gamma(p\rightarrow K^+ \bar{\nu}_e) : \Gamma(p\rightarrow K^0 e^+)
: \Gamma(n \rightarrow K^0 \bar{\nu}_e) \]\begin{equation}
= 0.4 : 1 : 2.7 \,\,\, .
\label{eq:ratios}
\end{equation}
Note that the proton's charged lepton decay mode dominates over the
neutrino mode. This is a consequence of the cancellation of the
operators $(du)(s\nu_e)$ between the first two terms of Eq.~(\ref{dimen5}).
The dominance of $p\rightarrow K^0 e^+$ over
$p\rightarrow K^+\overline{\nu}_e$ is rarely the case in SUSY-GUTs.
Finally, we come to the overall rate. We find
{\samepage
\[
\tau(n \rightarrow K^0 \bar{\nu}_e)
= \frac{4 \times 10^{31} \mbox{ yrs}}{c^2} \left(
\frac{0.003\mbox{ GeV}^3}{\xi} \right. \]\begin{equation}
\left. \;\;\;
\frac{0.81}{A_S}
\frac{5}{(1+\tan^2 \beta)} \,
\frac{\mbox{TeV}^{-1}}{f(c,e)+f(c,d)} \right)^2 \, .
\label{eq:rate}
\end{equation}}
This result includes the effect of running the dimension-five operator
between the Planck scale and $m_Z$, (a factor of $A_S = 0.81$ in the
amplitude if $m_t = 175$~GeV, $\tan \beta = 2$, $\alpha_s (m_Z) = 0.12$)
and between $m_Z$ and $m_n$ (a factor of $0.22$ in the amplitude). In
the expression above, $\xi$ is the hadronic matrix element of the
four-fermion operator evaluated between nucleon and kaon states; its exact
value is rather uncertain, but is estimated to be within the range
$\xi = 0.003$--0.03~GeV$^3$. If we take that $M_2 \sim
100$~GeV and $m_{\tilde{q}} \sim 700$~GeV, and $m_{\tilde{l}} \sim 300$~GeV,
then the triangle functions $f(c,e)+f(c,d) \sim (1.8\mbox{ TeV})^{-1}$.
Thus, if $c=1$ and $\xi=0.003$ GeV$^3$, we obtain a mean lifetime
$12.7\times 10^{31}$ years, which can be compared to the experimental bound,
$\tau(n \rightarrow K^0 \bar{\nu}_e) > 8.6 \times 10^{31}$~years.
It is interesting to note that the coefficient $4 \times 10^{31}$
in (\ref{eq:rate}) would be the same in the minimal SU(5) SUSY-GUT
with an extremely large color-triplet Higgs mass $M_{H_C} = 10^{17}$~GeV.
Thus, the rate in our model is roughly comparable. Overall,
the $(S_3)^3$ symmetry gives us just enough suppression of dimension-five
operators to evade the current bounds, so the model is phenomenologically
viable. Since the SuperKamiokande experiment is expected to extend
Kamiokande's current reach by another factor or 30, there is a very good
chance that the $n\rightarrow K^0\overline{\nu}_e$ mode may be seen. It
is an exciting prediction of this model that the $p \rightarrow K^0 e^+$
and $K^+ \bar{\nu}_e$ modes are likely to be seen at the same time
because their rates are close to each other, as we saw
in eq.~(\ref{eq:ratios}).
\section{Conclusions}
I have shown that the discrete flavor group $(S_3)^3$ can
account for the masses and mixing angles of the standard model
while simultaneously solving the supersymmetric flavor changing
problem. The most striking prediction that emerged
from the analysis is the dominance of proton decay to final states
involving first generation lepton fields, unlike the case in SUSY GUTs.
I showed that the ratios of decay widths for the largest modes
$n \rightarrow K^0 \bar{\nu}_e$, $p\rightarrow K^+ \bar{\nu}_e$,
and $p\rightarrow K^0 e^+$ are approximately 0.4 :: 1 :: 2.7. Given
the estimate of the total rate, I pointed out that all three modes may
be within the reach of the SuperKamiokande experiment and could well be
discovered simultaneously.
\begin{center}
{\bf Acknowledgments}
\end{center}
I am grateful to Lawrence Hall and Hitoshi Murayama for an enjoyable
collaboration. {\em This work was supported by the Director, Office of
Energy Research, Office of High Energy and Nuclear Physics, Division of
High Energy Physics of the U.S. Department of Energy under
Contract DE-AC03-76SF00098.}
|
1,116,691,500,886 | arxiv | \section{Introduction}
Cooperative diversity by using relays in wireless networks allows increasing the total throughput of the network while (possibly) relying on single antenna nodes. Fundamental principles and the main idea of cooperative communications can be found in \cite{b1}, where a three-terminal Relay Channel
is studied. If a limited feedback control channel is available from the destination to the relaying nodes, the throughput can be increased by using a cooperative version of Hybrid Automatic Repeat reQuest (HARQ) protocol \cite{b2}. We investigate the performance of different flavors of cooperative HARQ protocols for the Multiple Access Multiple Relay Channel (MAMRC), denoted by ($M$,$L$,$1$)-MAMRC where $M$ is the number of sources, $L$ the number of relays. User cooperation is included in our model which means that the number of relays that can help a given source is $L+M-1$. Each source listens to the other node transmissions with the final goal to maximize its number of correctly decoded source messages. The performance metric is the average spectral efficiency. Transmissions are orthogonal in time. During the first phase, each source transmits in turn its message in consecutive time slots. During the second phase (retransmission phase) the destination schedules a relay or a source to transmit for each time slot. All nodes are half-duplex, i.e., they can not transmit and receive at the same time. All the links are subject to slow-fading and Additive White Gaussian Noise (AWGN). For that reason, we use the outage information-theory tool to analyze the performance of the different protocols. Each node can cooperate with its successfully decoded source messages or decoding set. Indeed, contrary to the classical Decode and Forward (DF) approach where a relay need to wait until it decodes all the source messages correctly, here, a node can cooperate as soon as its decoding set is not empty. This relaying behavior is called Selective Decode-and-Forward (SDF) relaying function. By receiving a Channel Distribution Information (CDI) from all sources and relays (average SNR of all links), the destination can perform slow-link adaptation. It consists in allocating a rate among a discrete Modulation and Coding Scheme (MCS) family to each source in order to maximize the average spectral efficiency. For each possible $M$-tuple of source rates, the optimal slow link adaption algorithm exhaustively check which one achieves the best metric by performing Monte-Carlo simulations over a sufficient number of channel outcomes. The rate allocation is conveyed by a slow limited control channel from destination to sources prior to the HARQ protocol. Note that the optimal algorithm based on exhaustive search for finding the $M$-tuple of rates is quickly becoming intractable for either a large MCS family and/or an increasing number of sources. We designed a low complexity search algorithm for these cases whose performance gets very close to the optimal one. Its detailed presentation is out of the scope of this paper. In the following, we always assume that a slow link adaption rate allocation takes place before any source transmission. Since the CDI variations are much slower than the channel variations, the slow link adaptation keeps valid for many channel outcomes (actually our simulation are performed for a fixed CDI). There exist a limited feedback broadcast control channel from the destination towards sources and relays (e.g., to carry the scheduling decision of the destination) and multiple unicast forward coordination control channels from sources and relays towards the destination (to help the destination to take its scheduling decision). Particular care is paid to minimize the control overhead in this paper. Among the three proposed HARQ protocols, one consists in sending incremental redundancies on all the messages \rv{from the scheduled node decoding set} (Multi-User encoding) while the other one helps a single source (Single User encoding) chosen randomly. The latter is particularly attractive since its implementation can reuse state-of-art rate compatible punctured codes such as low density parity check codes or turbo codes. The third one is of the chase combining (CC) type, where the selected node repeats the transmission (including modulation and coding scheme) of one source chosen randomly \stecer{from} its decoding set. It allows Maximal Ratio Combining (MRC) at the destination of all the transmissions related to a given source. We can expect that such a protocol behaves poorly in general compared to the IR-type of HARQ.
In \cite{b3} and \cite{b4}, the performance of different HARQ protocols is investigated for the single source, single relay and single destination case. Both CC and IR types of HARQ protocol were analyzed. In \cite{b4}, it is shown that IR-type of HARQ performs better than CC-type of HARQ in terms of system outage probability, average number of retransmissions and average transmission rates. The advantage of using relay selection (with limited feedback) over distributed space-time block coded transmissions in multiple relay networks is shown in \cite{b5}. User co-operation is included as in our paper. In \cite{b6}, Multi-User relay channel consisting of two sources, one relay and a destination is shown to take benefit from Multi-User encoding (network coding). In this work, a feedback channel is assumed to be available from both the relay and the destination towards the sources. For the multiple-source multiple-relay channel, a relay ordering algorithm based on finite field network coding has been proposed in \cite{b7}. An outage analysis has been done for that protocol, where Separate Network Channel Coding (SNCC) is used in combination with the DF relaying protocol. In \cite{b8}, the relay selection strategies that aim to maximize the long-term aggregate throughput are studied for slow-fading MAMRC, where SDF relaying protocol is applied under the JNCC/JNCD framework (Multi-User encoding at the relays with Multi-User iterative joint decoding at the destination). A proper comparison between the two IR-type of HARQ and the CC-type of HARQ has not been performed by the previously mentioned works. Our goal in this paper is to identify the most efficient cooperative HARQ protocol, i.e., the one that achieves the best complexity-performance tradeoff keeping in mind that Single User encoding and decoding is well mastered in terms of code construction and, clearly, less complex than Multi-User encoding and iterative joint decoding. On the other hand, the Chase Combining approach can be considered as having a similar complexity \stecer{to} Single User IR-HARQ. As a result, the HARQ protocol comparison comes down to a performance comparison where information theory outage analysis is particularly relevant.
The remainder of the paper is organized as follows. The system model is detailed in section \ref{sec:sys_model}. In section \ref{sec:harq} the performance metric, the outage event definitions, as well as the three different HARQ protocols together with the proposed node selection strategy are described. Numerical results are presented in section \ref{sec:numerical_results}. Finally, we conclude the paper in section \ref{sec:conclusion}.
\section{System Model}
\label{sec:sys_model}
In this paper, we investigate OMAMRC under slow-fading assumption. $M$ sources, belonging to the set $\mathcal{S}=\{s_1,\dots,s_M\}$, transmit independent messages $\textbf{u}_s \in \mathbb{F}_2^{K_s}$ of $K_s$ information bits towards a common destination. The length of a source message depends on the selected MCS by the destination, where the decision about the selection is conveyed over the error-less limited feedback broadcast control channel. $L$ relays, that operate in half-duplex mode and that belong to the set $\mathcal{R}=\{r_1,\dots,r_L\}$, help the destination in decoding the sources' messages. They overhear the messages from sources due to the broadcast property of wireless medium, and apply SDF relaying protocol. Relays do not have their own messages to transmit. Additionally, user-cooperation is performed, i.e. when not transmitting, sources listen to other sources and relays transmissions and help the decoding at the destination by applying the SDF relaying protocol (see Fig. \ref{fig:fig_1}). Moreover, HARQ protocol is used, which is either of type Incremental Redundancy, or Chase Combining. In the case of IR-type of HARQ protocol, two types of encoding are considered: \stecer{S}ingle \stecer{U}ser encoding and \stecer{M}ulti-\stecer{U}ser encoding, depending on the number of sources that the node performing the relaying functions will help during its transmission. We define the set of all source and relay nodes as $\mathcal{N}=\mathcal{S} \cup \mathcal{R}$.
\begin{figure}[!t]
\centering
\includegraphics[scale=0.3]{figure1}
\caption{Cooperative Orthogonal Multiple Access Multiple Relay Channel (OMAMRC) with feedback.}
\label{fig:fig_1}
\end{figure}
CSI is available only at the receiver side of each link and is assumed perfect. Hence, the destination only has the perfect knowledge of CSI of source-to-destination (S-D) links, $\textbf{h}_{\textrm{S,D}}=[ h_{s_1,d},\dots,h_{s_M,d} ]$, and of relay-to-destination (R-D) links $\textbf{h}_{\textrm{R,D}}=[ h_{r_1,d},\dots,h_{r_L,d} ] $. On the other hand, the CSI of source-to-source (S-S), source-to-relay (S-R) and relay-to-relay (R-R) links are unknown to it.
Transmission of source messages is split into frames, during which exactly one message from each source is sent, as well as the retransmissions related to those messages. Slow (block) fading is assumed, where within one frame the radio-links between the different nodes are considered to be fixed, while they change independently from frame to frame. Furthermore, we consider that during a certain number of frames $N_f>>1$, the probability distribution of the quality of each link remains constant. That means that the quality of the given link in the given frame represents one realization of the associated probability distribution. The choice of the MCS for each source by the destination takes place in the ``initial phase'' by applying the slow-link adaptation algorithm. That phase occurs before any transmission, and is repeated whenever the probability distributions of different channels change (see Fig. \ref{fig:fig_2}). CDI of each link in the network is needed as an input to the slow-link adaptation algorithm. For S-S, S-R and R-R links, sources and relays convey the information about CDI to the destination over forward coordination control channels that are assumed to be errorless. The destination can track the CDI of S-D and R-D links by itself. The information about the selected MCSs is conveyed from the destination to all nodes over limited feedback control channel. The source rates are kept fixed between two occurrences of the initial phase.
\begin{figure}[!t]
\centering
\includegraphics[scale=0.3]{figure2}
\caption{Transmission of a frame: initial, first and second phase.}
\label{fig:fig_2}
\end{figure}
Transmission frame is split into two phases. The first phase consists of $M$ time-slots made of $N_1$ channel uses each, where each one of the $M$ sources transmits in turn. User co-operation being used, when one source transmits a message, both relays and non-transmitting sources listen and try to decode that message relying on a Cyclic Redundancy Check (CRC) code for error detection. The second phase consists of maximum of $T$ time-slots of duration of $N_2$ channel uses each, called also ``retransmission rounds'' in the following. $T$ is a system design parameter chosen by the destination, which depends on the latency requirements. In each retransmission round the destination selects one relay or a source to transmit, where a source can either retransmit its own message or act as a relay for other sources. In \cite{b8}, the scheduling strategy that consists in selecting the node whose link to the destination has the best quality among the nodes that can help the destination (their decoding sets contain at least one message that the destination has not been able to decode at the end of the previous round) is shown to achieve a performance close to the optimal (exhaustive) one. Taking into account the teaching of \cite{b8}, we propose a low overhead control signaling exchanges between the destination and the other nodes as follows:
\begin{itemize}
\item The destination broadcasts $M$ bits that indicate its decoding set $\mathcal{S}_{d,t-1}$ after round $t-1$ over the control channel.
\item If the decoding set of the destination consists of all source messages, a new frame begins and the sources transmit new messages while the relays and destination empty their memory buffers. Otherwise, each cooperating source and each relay which was able to decode at least one source message that is not included in the decoding set of the destination \rv{sends a signal on a dedicated unicast control channel. Each cooperating source or relay which did not decode any message needed by the destination, i.e., any message that is not included in the decoding set of the destination after round $t-1$, remains silent (ON-OFF modulation).}
\item Using the adopted node selection strategy, the destination can make the scheduling decision about the node to select for transmission. Its decision is broadcasted using a control channel.
\item Selected node transmits applying the appropriate type of HARQ protocol.
\end{itemize}
Note that the end of the first phase is considered as the end of the round zero. The non-selected nodes in a given retransmission round can benefit from the transmission of the scheduled node as well, and update their decoding sets accordingly. The number of retransmission rounds used in the second phase $T_{\mathit{used}} \in \{1,\dots,T\}$ depends on the success of the decoding process at the destination. Each node in the network is equipped with one antenna only and transmits with the same power. In the rest of the paper, the following notations are used:
\begin{itemize}
\item $x_{a,k}\in \mathbb{C}$ is the coded modulated symbol \rv{whose power is normalized to unity} for channel use $k$, sent from node $a\in \mathcal{S}\cup \mathcal{R}$.
\item $y_{a,b,k}$ is a received signal at node $b\in \mathcal{S}\cup \mathcal{R}\cup\{d\} \setminus \{a\}$, originating from node $a$.
\item $\gamma_{a,b}$ is the average signal-to-noise ratio (SNR) that captures both path-loss and shadowing effects.
\item $h_{a,b}$ are the channel fading gains, which are independent and follow a zero-mean circularly symmetric complex Gaussian distribution with variance $\gamma_{a,b}$.
\item $n_{a,b,k}$ are independent and identically distributed AWGN samples, which follow a zero-mean circularly-symmetric complex Gaussian distribution with unit variance.
\end{itemize}
Using the previous notation, we can represent the received signal at node $b\in \mathcal{S}\cup \mathcal{R}\cup\{d\} \setminus \{a\}$ which originates from node $a\in \mathcal{S}\cup \mathcal{R}$ as:
\begin{equation}
y_{a,b,k}=h_{a,b}x_{a,k}+n_{a,b,k},
\end{equation}
where $k$ denotes a current channel use, taking a value $k \in \{1,\dots,N_1\}$ during the first phase, and $k \in \{1,\dots,N_2\}$ during the second phase.
\section{Cooperative HARQ protocols}
\label{sec:harq}
\subsection{Performance metric and outage events}
Let us denote with $R_s=K_s/N_1$ the initial transmission rate of a source $s$ in bit per complex dimension or bit per channel use [b.c.u]. We can define a long-term transmission rate $\bar{R}_s$ per source as the fraction of the number of transmitted information bits over the total number of channel uses spent, for a number of frames that tends to infinity:
\begin{equation}
\bar{R}_s=\frac{R_s}{M+\alpha\mathbb{E}(T_{\text{used}})},
\label{eq:first}
\end{equation}
where $\mathbb{E}(T_{\text{used}})=\sum_{t=1}^{T}t\textrm{Pr}\{T_{\text{used}}=t\}$ is the average number of retransmission rounds used in the second phase, and $\alpha=N_2 / N_1$.
A performance metric that we use throughout the paper is the average spectral efficiency, which can be defined as:
\begin{equation}
\eta=\sum_{i=1}^{M}\bar{R}_{s_i}(1-\textrm{Pr}\{\mathcal{O}_{s_i,T}\}),
\label{eq:third}
\end{equation}
where $\mathcal{O}_{s,T}$ is the ``individual outage event of source $s$ after round $T$'', which is the event that source $s$ is not decoded correctly at the destination after round $T$.
Before defining it analytically for different HARQ protocols in the following subsections, we should emphasize that the individual outage event of source $s$ after round $t$, $\mathcal{O}_{s,t}(a_t,\mathcal{S}_{a_t,t-1} |\textbf{h}_{\textrm{S,D}},\textbf{h}_{\textrm{R,D}},\mathcal{P}_{t-1})$, directly depends on the choice of a transmitting node $a_t\in \mathcal{N}$ in round $t$ and its associated decoding set $\mathcal{S}_{a_t,t-1}$. Furthermore, it is conditional on the knowledge of $\textbf{h}_{\textrm{S,D}}$, $\textbf{h}_{\textrm{R,D}}$ and $\mathcal{P}_{t-1}$, the last one denoting the set which collects the nodes $\hat{a}_k$ selected in rounds $k \in \{ 1,\dots,t-1 \}$ prior to round $t$ together with their associated decoding sets $\mathcal{S}_{\hat{a}_k,k-1}$, and the decoding set of the destination $\mathcal{S}_{d,t-1}$. The same holds for the ``common outage event after round $t$'' $\mathcal{E}_t(a_t,\mathcal{S}_{a_t,t-1} |\textbf{h}_{\textrm{S,D}},\textbf{h}_{\textrm{R,D}},\mathcal{P}_{t-1})$, which is the event that at least one source is not decoded correctly at the destination at the end of the round $t$. If $\mathbb{E}\{.\}$ is the expectation operator, and $\textbf{1}_{\{\mathcal{V}\}}$ is the function having a value $1$ if the event $\mathcal{V}$ is true, and $0$ otherwise, we can define the probability of the individual outage event of source $s$ after round $t$ for a candidate node $a_t$ as $\mathbb{E}\{\textbf{1}_{\{\mathcal{O}_{s,t}(a_t,\mathcal{S}_{a_t,t-1} |\textbf{h}_{\textrm{S,D}},\textbf{h}_{\textrm{R,D}},\mathcal{P}_{t-1})\} } \}$. We can define the probability of the common outage event in a similar way. In order to simplify the notation in the rest of the paper, we will omit the condition on $\textbf{h}_{\textrm{S,D}}$, $\textbf{h}_{\textrm{R,D}}$ and $\mathcal{P}_{t-1}$ when \stecer{recalling} the individual and common outage events.
\subsection{IR-type of HARQ protocol with \stecer{M}ulti-\stecer{U}ser \stecer{en}coding}
\label{MU}
In this part, we assume that in given round $t$, the selected node $a_t$ sends incremental redundancies on all the messages in its decoding set. If the decoding set at the destination after round $t-1$ is given by $\mathcal{S}_{d,t-1}$, we define the set of non-successfully decoded sources at the destination as $\bar{\mathcal{S}}_{d,t-1}=\mathcal{S} \setminus {\mathcal{S}}_{d,t-1}$. First, we want to analytically define the common outage event $\mathcal{E}_{t,\mathcal{B}}^{\text{IR,MU}}(a_t,\mathcal{S}_{a_t,t-1})$ after round $t$ for a candidate node $a_t$ of some subset $\mathcal{B}$ of the set of non-successfully decoded sources at the destination $\mathcal{B}\subseteq \bar{\mathcal{S}}_{d,t-1}$. Since in a given round the transmitted incremental redundancies potentially contain multiple source messages, the destination has no choice but to decode the source messages jointly, i.e., considering the received transmissions as part of a joint codeword on all the source messages. As a result, we resort to Multiple Access Channel (MAC) framework, where the event $\mathcal{E}_{t,\mathcal{B}}^{\text{IR,MU}}(a_t,\mathcal{S}_{a_t,t-1})$ is true if the vector of rates of sources contained in $\mathcal{B}$ lies outside of the corresponding MAC capacity region.
We can express this event as:
\begin{equation}
\begin{split}
&\mathcal{E}_{t,\mathcal{B}}^{\text{IR,MU}}(a_t,\mathcal{S}_{a_t,t-1})=\bigcup_{\mathcal{U}\subseteq \mathcal{B}} \Big\{ \sum_{s \in \mathcal{U}}R_s > \sum_{s \in \mathcal{U}} I_{s,d}\\
&+ \sum_{l=1}^{t-1} \alpha I_{\hat{a}_l,d} \textbf{1}_{\{\mathcal{C}_{\hat{a}_l}^{\text{IR,MU}} \}} + \alpha I_{a_t,d} \textbf{1}_{\{ \mathcal{C}_{a_t}^{\text{IR,MU}} \}} \Big\},
\end{split}
\label{eq:reduced_mac1}
\end{equation}
where $I_{a,b}$ denotes the mutual information between the nodes $a$ and $b$, the sources contained in the set $\mathcal{I}=\bar{\mathcal{S}}_{d,t-1}\setminus \mathcal{B}$ are considered as interference and $\mathcal{C}_{\hat{a}_l}^{\text{IR,MU}}$ and $\mathcal{C}_{a_t}^{\text{IR,MU}}$ have the following definitions:
\begin{equation}
\label{eq:IR_MU_C_com}
\begin{split}
&\mathcal{C}_{\hat{a}_l}^{\text{IR,MU}}=\Big\{ \{\mathcal{S}_{\hat{a}_l,l-1}\cap \mathcal{U} \neq \emptyset\}\wedge \{\mathcal{S}_{\hat{a}_l,l-1}\cap \mathcal{I}=\emptyset\} \Big\},\\
&\mathcal{C}_{a_t}^{\text{IR,MU}}=\Big\{ \{\mathcal{S}_{a_t,t-1}\cap \mathcal{U} \neq \emptyset\}\wedge \{\mathcal{S}_{a_t,t-1}\cap \mathcal{I}=\emptyset\} \Big\},
\end{split}
\end{equation}
with $\wedge$ standing for the logical and. Since IR-type of HARQ is used, we basically compare the sum-rate of sources contained in each subset $\mathcal{U}\subseteq \mathcal{B}$ with the accumulated mutual information at the destination that originates from: (1) the transmissions during the first phase; (2) the transmissions of previously activated nodes in rounds $1, \dots, t-1$; and (3) the transmission of the candidate node $a_t$. Node $\hat{a}_k$ for $k=\{1,\dots,T\}$ is involved in the calculation only if it was able to successfully decode at least one source from the set $\mathcal{U}$ while its decoding set does not contain any interference, i.e., source message that \stecer{is} outside $\mathcal{B}$. Multiplication by $\alpha$ serves as a normalization before adding two mutual information originating from two different phases, where the transmission uses $N_1$ and $N_2$ time slot channel uses in the first and second phase, respectively. If one or more MAC inequalities associated to the sum-rate of sources in different sets $\mathcal{U}$ is not respected, the common outage event of the set $\mathcal{B}$ is proclaimed.
By similar reasoning, the individual outage event of the source $s$ after round $t$ can be defined as:
\begin{equation}
\begin{split}
&\mathcal{O}_{s,t}^{\text{IR,MU}}(a_t,\mathcal{S}_{a_t,t-1})=\bigcap_{\mathcal{I}\subset \bar{\mathcal{S}}_{d,t-1}} \bigcup_{\mathcal{U}\subseteq \bar{\mathcal{I}}:s\in \mathcal{U}}\Big\{ \sum_{s \in \mathcal{U}}R_s > \sum_{s \in \mathcal{U}} I_{s,d}\\
&+ \sum_{l=1}^{t-1} \alpha I_{\hat{a}_l,d} \textbf{1}_{\{\mathcal{C}_{\hat{a}_l,s}\}} + \alpha I_{a_t,d} \textbf{1}_{\{\mathcal{C}_{a_t,s}\}} \Big\},
\label{eq:Ost_JNCC}
\end{split}
\end{equation}
where $\bar{\mathcal{I}}=\bar{\mathcal{S}}_{d,t-1}\setminus \mathcal{I}$, and $\mathcal{C}_{\hat{a}_l,s}^{\text{IR,MU}}$ and $\mathcal{C}_{a_t,s}^{\text{IR,MU}}$ have the following definitions:
\begin{equation}
\label{eq:IR_MU_C_ind}
\begin{split}
&\mathcal{C}_{\hat{a}_l,s}^{\text{IR,MU}}=\Big\{ \{s\in \mathcal{S}_{\hat{a}_l,l-1}\cap \mathcal{U}\}\wedge \{\mathcal{S}_{\hat{a}_l,l-1}\cap \mathcal{I}=\emptyset\} \Big\},\\
&\mathcal{C}_{a_t,s}^{\text{IR,MU}}=\Big\{ \{s\in \mathcal{S}_{a_t,t-1}\cap \mathcal{U}\}\wedge \{\mathcal{S}_{a_t,t-1}\cap \mathcal{I}=\emptyset\} \Big\},
\end{split}
\end{equation}
\subsection{IR-type of HARQ protocol with \stecer{S}ingle \stecer{U}ser encoding}
\label{SU}
As stated in the introduction, Single User encoding is particularly attractive since its implementation can reuse state-of-art rate compatible punctured codes such as low density parity check codes or turbo codes. Here, a selected node in retransmission round $t$ of the second phase cooperates with a single source from its decoding set, i.e., it transmits incremental redundancies for a single source. The choice of the source that the selected node will help is random, but among all sources which the destination has not successfully decoded up until that round. That information is available to each node in the network due to the control information exchange mechanism described in section \ref{sec:sys_model}.
Let us denote with $s_{\hat{a}_k}$ a randomly chosen source by the node $\hat{a}_k$ in round $k \in \{1, \dots, T\}$ from its decoding set under the previously described condition. In this case, since the selected nodes during the second phase do not apply \stecer{M}ulti-\stecer{U}ser encoding anymore and since the transmission is orthogonal in time, there is no need to use the MAC framework. The individual outage event of the source $s$ after round $t$ for the selected node $a_t$ which cooperates with the source $s_{a_t}$ can be simply defined as:
\begin{equation}
\begin{split}
&\mathcal{O}_{s,t}^{\text{IR,SU}}(a_t,s_{a_t})=\Big\{ R_s > I_{s,d} + \sum_{l=1}^{t-1} \alpha I_{\hat{a}_l,d} \textbf{1}_{\{s=s_{\hat{a}_l}\}}\\
&+ \alpha I_{a_t,d} \textbf{1}_{\{s=s_{a_t}\}} \Big\},
\label{eq:Ost_JNCC_SU}
\end{split}
\end{equation}
To find the common outage event of sources contained in the set $\mathcal{B}\subseteq \bar{\mathcal{S}}_{d,t-1}$ after round $t$, for the selected node $a_t$ which cooperates with the source $s_{a_t}$, we simply check if the individual outage event of any source $s$ contained in $\mathcal{B}$ is true:
\begin{equation}
\mathcal{E}_{t,\mathcal{B}}^{\text{IR,SU}}(a_t,s_{a_t})=\bigcup_{s \in \mathcal{B}} \mathcal{O}_{s,t}^{\text{IR,SU}}(a_t,s_{a_t}).
\label{eq:reduced_mac2}
\end{equation}
\subsection{CC-type of HARQ protocol}
\label{CC}
In this type of protocol, the selected node $a_t$ in round $t$ in the second phase apply the exact same MCS as source $s$ whose message is randomly selected from the decoding set of the destination $\bar{\mathcal{S}}_{d,t-1}$. It implies the constraint that $N_1=N_2$ or $\alpha=1$. At the destination, Maximal Ratio Combining (MRC) (at symbol or coded bit level) is used after each round in order to decode the message of a given source. \rv{By doing so, we obtain the highest achievable SNR, denoted $\gamma_{\text{MRC}}$, for a given source at the destination which is equal to the summation of individual SNRs from the previous rounds. This kind of protocol offers less complexity in decoding then the protocol based on \stecer{M}ulti-\stecer{U}ser encoding. The individual outage event of the source $s$ after round $t$ for the selected node $a_t$ which cooperates with the source $s_{a_t}$ is defined in this case as:
\begin{equation}
\mathcal{O}_{s,t}^{\text{CC}}(a_t,s_{a_t})=\Big\{ R_s > I(\gamma_{\text{MRC}}(a_t,s_{a_t}))\Big\}
\end{equation}
where
\begin{equation}
\begin{split}
&\gamma_{\text{MRC}}(a_t,s_{a_t}) = |h_{s,d}|^2 + \sum_{l=1}^{t-1}|h_{\hat{a}_l,d}|^2 \textbf{1}_{\{s=s_{\hat{a}_l}\}} \\
&+ |h_{a_t,d}|^2 \textbf{1}_{\{s=s_{a_t}\}}
\end{split}
\label{eq:Ost_JNCC_CC}
\end{equation}
}
The common outage event of sources contained in the set $\mathcal{B}\subseteq \bar{\mathcal{S}}_{d,t-1}$ after round $t$, for the selected node $a_t$ which cooperates with the source $s_{a_t}$ is, just as in the previous case, defined as:
\begin{equation}
\mathcal{E}_{t,\mathcal{B}}^{\text{CC}}(a_t,s_{a_t})=\bigcup_{s \in \mathcal{B}} \mathcal{O}_{s,t}^{\text{CC}}(a_t,s_{a_t}).
\label{eq:reduced_mac3}
\end{equation}
\subsection{Node selection strategy used during the second phase}
\label{node_selection}
In \cite{b8}, it is shown that the node selection strategy which offers the best trade-off between the performance and computational complexity is the one where in a given round of the second phase the node with the highest mutual information between itself and the destination is selected among all nodes that were able to decode at least one source from the set of non-successfully decoded sources at the destination after round $t-1$:
\begin{equation}
\hat{a}_t=\argmax_{a_t\in\mathcal{S}\cup\mathcal{R}}\{ I_{a_t,d} \textbf{1}_{\{ \bar{\mathcal{S}}_{d,t-1}\cap \mathcal{S}_{a_t,t-1} \neq \emptyset \}}\}.
\label{eq:strategy1}
\end{equation}
Namely, it is demonstrated by performing Monte-Carlo simulations that such a strategy performs close to the upper-bound given by the strategy based on the exhaustive search for the best activation sequence, which requires the knowledge of the CSI of each link in the network and is much more complex. By observing the expressions for the individual outage probability, it is clear that such a node selection strategy can be equally applied to the IR-type of HARQ with the \stecer{S}ingle \stecer{U}ser encoding and the CC-type of HARQ.
\section{Numerical Results}
\label{sec:numerical_results}
In this Section, we want to evaluate the performance of the three types of HARQ protocols described in Sections \ref{MU}, \ref{SU}, \ref{CC} in terms of the average spectral efficiency by performing Monte-Carlo simulations. The node selection strategy described in subsection \ref{node_selection} is assumed to be used in the second phase. Also, we assume the presence of \stecer{the optimal} slow-link adaptation algorithm \stecer{conditional on the chosen node selection strategy}. A discrete MCS family whose rates belong to $\{$0.5$,$1$,$1.5$,$2$,$2.5$,$3$,$3.5$\}$ [b.c.u] is used for the initial rates. Independent Gaussian distributed channel inputs are assumed (with zero mean and unit variance) where $I_{a,b}=\log_2(1+|h_{a,b}|^2)$. There are some other formulas that could be also used for the calculation of $I_{a,b}$which take into account, for example, discrete entries, finite length of the codewords, non-outage achieving \stecer{Multi-User encoding/iterative joint decoding} architectures etc. Although the calculation would be different for each type of HARQ protocol, the basic concept of \stecer{the} work would stay the same.
In the first part of the simulations, we consider ($3$,$3$,$1$)-OMAMRC with $\alpha^{\text{IR}}=0.5$ and $T^{\text{IR}}=4$ for IR-types of HARQ protocol, and $\alpha^{\text{CC}}=1$ and $T^{\text{CC}}=2$ for CC-type of HARQ protocol. The asymmetric link configuration is assumed, where the average SNR of each link is in the range $\{-15\textrm{dB}, \dots, 20\textrm{dB}\}$, where the source $s_1$ is set on purpose to be in the best propagation condition, while the source $s_3$ is in the worst one. Concretely, the network is configured as follows: (1) the average SNR of the links between source $s_1$ and each relay, as well as the link between source $s_1$ and the destination, is set to $\gamma$;
(2) the average SNR of the links between source $s_2$ and each relay, as well as the link between source $s_2$ and the destination, is set to $\gamma-4\textrm{dB}$;
(3) the average SNR of the links between source $s_3$ and each relay, as well as the link between source $s_3$ and the destination, is set to $\gamma-7\textrm{dB}$;
(4) the average SNR of the links between all relays, as well as the links between each relay and the destination is set to $\gamma$;
(5) the average SNR of the links between all sources are set according to the Tab. \ref{tab:tab1}.
\begin{table}[!t]
\caption{Average SNR of the links between all sources}
\label{tab:tab1}
\centering
\begin{tabular}{!{\vrule width 0.8pt} c !{\vrule width 0.8pt} c | c | c !{\vrule width 0.8pt}}
\noalign{\hrule height 0.8pt}
$\gamma_{x,y}[\textrm{dB}]$ & $s_1$ & $s_2$ & $s_3$\\
\noalign{\hrule height 0.8pt}
$s_1$ & N.A. & $\gamma-1\textrm{dB}$ & $\gamma-2\textrm{dB}$\\
\hline
$s_2$ & $\gamma-1\textrm{dB}$ & N.A. & $\gamma-5\textrm{dB}$\\
\hline
$s_3$ & $\gamma-2\textrm{dB}$ & $\gamma-5\textrm{dB}$ & N.A.\\
\noalign{\hrule height 0.8pt}
\end{tabular}
\end{table}
As a result, the initial rates associated to all sources are asymmetric. They are shown on Fig. \ref{fig:figs1} as a function of $\gamma$, which is the average SNR \stecer{of the link} between source $s_1$ and the destination. On that figure, IR-type of HARQ protocol with \stecer{M}ulti-\stecer{U}ser \stecer{en}coding is labeled as ``IR-HARQ MU'', IR-type of HARQ protocol with \stecer{S}ingle \stecer{U}ser \stecer{en}coding as ``IR-HARQ SU'' while CC-type of HARQ protocol is labeled as ``CC-HARQ''. Fig. \ref{fig:figs2} shows the average spectral efficiency of the network as a function of $\gamma$. We observe that the IR-type of HARQ protocol with \stecer{M}ulti-\stecer{U}ser \stecer{en}coding provides the highest average spectral efficiency. This result was expected since the selected nodes in the second phase may help the decoding of multiple sources at the same time. IR-type of HARQ protocol with \stecer{S}ingle \stecer{U}ser \stecer{en}coding performance is not far behind, providing slightly lower average spectral efficiency. It can be explained by the fact that ``only'' three sources are present in the network, so there is often a case where the selected node in the second phase cooperates with exactly one source\stecer{, even if \stecer{M}ulti-\stecer{U}ser \stecer{en}coding is employed}. Naturally, CC-type of HARQ has a \stecer{noticeably worse} performance compared with two IR based protocols.
\begin{figure}[!t]
\centering
\includegraphics[scale=0.37]{fig1_sim}
\caption{Allocated rates to sources for different HARQ protocols for asymmetric link configuration in ($3$,$3$,$1$)-OMAMRC.}
\label{fig:figs1}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[scale=0.37]{fig2_sim}
\caption{Average spectral efficiency obtained by using different HARQ protocols for asymmetric link configuration in ($3$,$3$,$1$)-OMAMRC.}
\label{fig:figs2}
\end{figure}
Fig. \ref{fig:figs3} and Fig. \ref{fig:figs4} show the same comparison but for ($4$,$3$,$1$)-OMAMRC and ($5$,$3$,$1$)-OMAMRC, respectively. The average SNR of the links between source $s_4$ and each relay, as well as the link between source $s_4$ and the destination, is set to $\gamma-9\textrm{dB}$, while the average SNR of the links between source $s_5$ and each relay, as well as the link between source $s_5$ and the destination, is set to $\gamma-10\textrm{dB}$. The average SNR of the link between sources $s_4$ and $s_5$ is set to $\gamma-9.5\textrm{dB}$, while the same parameter for the links between sources $s_4$ and $s_5$ and all other sources is set to $\gamma$ reduced by a value from the set $[0\textrm{dB}, \dots, 9\textrm{dB}]$, following the similar logic as in the case of ($3$,$3$,$1$)-OMAMRC. We observe that the performance ordering of the different protocols remains the same. But, as the number of sources in the network grows, we notice that \stecer{for IR-type of HARQ }the difference in performance between the \stecer{M}ulti-\stecer{U}ser and \stecer{S}ingle \stecer{U}ser \stecer{encoding} slowly grows. Indeed, a scheduled node has all the more chances to have more than one source in its decoding set as the number of sources increases.
\begin{figure}[!t]
\centering
\includegraphics[scale=0.37]{fig3_sim}
\caption{Average spectral efficiency obtained by using different HARQ protocols for asymmetric link configuration in ($4$,$3$,$1$)-OMAMRC.}
\label{fig:figs3}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[scale=0.37]{fig4_sim}
\caption{Average spectral efficiency obtained by using different HARQ protocols for asymmetric link configuration in ($5$,$3$,$1$)-OMAMRC.}
\label{fig:figs4}
\end{figure}
As a general conclusion, we can argument that for the OMAMRC with relatively small number of sources the IR-type of HARQ with \stecer{S}ingle \stecer{U}ser \stecer{en}coding offers the best compromise between performance and complexity.
It is also interesting to observe that the average spectral efficiency decreases for all three types of the HARQ protocol when the number of sources increases. There are two reasons from our understandings. The first one is that by adding more sources that are progressively in worse conditions than the previous ones, the probability that the added source will be successfully decoded decreases. The other reason is that the number of retransmission rounds in the second phase is fixed to $T^{\text{IR}}=4$ and $T^{\text{CC}}=2$, so by adding more sources, even if they are all in the same conditions in average, \stecer{it may happens that }there are not enough available retransmissions for helping them all \stecer{efficiently}.
For that reason, in the last part of simulations, we consider the symmetric link configuration where the average SNR of each link is equal to $\gamma$, and where the number of possible retransmission rounds in the second phase varies with the number of sources. Namely, we try to keep a constant ratio between the number of time-slots in the first phase and the number of possible retransmissions in the second phase. Let $M_1=3$ be the number of sources in ($3$,$3$,$1$)-OMAMRC, with $T_1^{\text{IR}}=4$ the number of retransmissions in the second phase for IR-type of HARQ, and with $T_1^{\text{CC}}=2$ the same number, but for CC-type of HARQ. In ($4$,$3$,$1$)-OMAMRC, $M_2=4$, $T_2^{\text{IR}}=\lceil \frac{T_1}{M_1}M_2 \rceil=6$, and $T_2^{\text{CC}}=\frac{T_2^{\text{IR}}}{2}=3$. In the case of ($5$,$3$,$1$)-OMAMRC, by similar reasoning and forcing the $T_3^{\text{CC}}$ to be the round number, we choose $T_3^{\text{IR}}=8$ and $T_3^{\text{CC}}=4$. Fig. \ref{fig:figs5} shows the comparison of the average spectral efficiency for all $M_1$, $M_2$ and $M_3$ where we see that in this case the more sources there are in the network, the higher the average spectral efficiency is. For the clarity of the figure only the range $\gamma \in \{0\textrm{dB}, \dots, 15\textrm{dB}\}$ is shown.
\begin{figure}[!t]
\centering
\includegraphics[scale=0.37]{fig5_sim}
\caption{Average spectral efficiency obtained by using different HARQ protocols for symmetric link configuration and different OMAMRC.}
\label{fig:figs5}
\end{figure}
\section{Conclusion}
\label{sec:conclusion}
In this paper, we investigate the performance of three different cooperative HARQ protocols for slow-fading MAMRC. Among the three proposed HARQ protocols, two follow the \stecer{I}ncremental \stecer{R}edundancy (IR) approach. One consists in sending incremental redundancies \rv{on all the source messages decoded correctly by the scheduled node} (Multi-User encoding) while the other one helps a single source (Single User encoding) chosen randomly. The third one is of the \stecer{C}hase \stecer{C}ombining (CC) type, where the selected node repeats the transmission (including modulation and coding scheme) of one source chosen randomly in its correctly decoded source message set (its decoding set). It allows Maximal Ratio Combining (MRC) at the destination of all the transmissions related to a given source. Single User encoding and decoding is well mastered in terms of code construction (state of the art rate compatible punctured codes) and, clearly, less complex than Multi-User encoding and iterative joint decoding. On the other hand, the Chase Combining approach can be considered as having a similar complexity \stecer{to} Single User IR-HARQ. To identify the most efficient cooperative HARQ protocol, i.e., the one that achieves the best complexity-performance tradeoff, we resort to \rv{information theory outage based average spectral efficiency performance comparison}. We conclude that IR-type of HARQ with \stecer{S}ingle \stecer{U}ser encoding offers the best trade-off between performance and complexity for a small number of sources in our setting.
|
1,116,691,500,887 | arxiv | \section{Introduction}
In the recent years there emerged a large number of articles devoted to scalar fields, which are necessary for an explanation of a secondary acceleration of the Universe. It is reasonable to take models with minimal interaction, in which interaction of a scalar field with an <<ordinary matter>> is implemented by means of a gravitational interaction. Also it is reasonable to take models with non-minimal interaction, in which interaction of a scalar field with an <<ordinary matter>> is implemented by means of some connection. This connection could be realized through the factor depending on the scalar field potential in the Lagrange matter function (potential connection) or on the factor which depends on the scalar field's derivatives (kinetic connection). These models are intended to describe to some extent at least qualitative behavior of the Universe at large and small times. However, being essentially highly phenomenological, these models are not able to satisfy aesthetical requirements of theorists and unlikely are long-living. From the other hand, there exist theoretical models based on the gravitation theory's modifications, which are able to describe a kinematics of the cosmological extension.
These are, primarily, $f(R)$ gravitation models of A.Starobinsky \cite{Starobinsky}, and also other modifications of the gravitation theory such as Poincare calibration theories of gravi\-ta\-tion\footnote{see e.g., A.V.Minkevich \cite{Minkevich}.}. By no means negating a possibility of modification of the Einstein theory we nonetheless consider a cosmological model constructed on the fundamental scalar interaction since the scalar field being highly symmetrical one, from one hand can be the foundation of the elementary particle physics and, from the other hand, can realize a concept of physical vacuum. In the article we consider fantom fields, i.e. fields possessing a <<kinetic>> energy.
In contrast to specified nonminimal models of scalar interaction we consider statistical systems of scalar charged particles in which some
sort of particles can directly interact with a scalar field by means of fundamental \textit{scalar charge}. From the other hand, statistical system
possessing non-zero scalar charge and being itself a source of a scalar field, can efficiently influence on scalar field managing its behavior.
Such a scalar interaction was introduced in the general relativistic theory in 1983 by the Author \cite{Ignatev1,Ignatev2,Ignatev3,Ignatev4}
and a bit later - by G.G.Ivanov \cite{Ivanov}. In particular, in articles
\cite{Ignatev2,Ignatev3} based on the kinetic theory
the self-consistent set of equations describing a statistical system of particles with scalar interaction was obtained. In \cite{kuza} there were investigated group properties of equilibrium statistical configurations with scalar interaction and in \cite{Ignat_Popov} it was established a tight connection between the variable mass particle dynamics and the dynamics of scalar charge particle. In recent Author's articles \cite{YuNewScalar1,YuNewScalar2,YuNewScalar3}\footnote{see also monographes \cite{Yubook1,Yubook2}.} the macroscopic theory of statistical systems with scalar interaction was significantly improved and extended to the case of fantom scalar fields\footnote{see also review \cite{YuSTFI}}.
As opposed to the cited above nonminimal models of scalar interaction, the considered strict dynamic model of interaction of a scalar field with elementary particles
is based on the Hamilton mi\-cro\-sco\-pic equations of motion and following statistical averaging procedure. It turns out that
an interaction of a scalar field with particles at that could be introduced by a single way; consequently the ma\-cro\-sco\-pic equations of matter and scalar field
are obtained in a single way by means of standard averaging procedures. Thereby a tight connection between micro and macro levels of a scalar field description is established. Naturally enough that obtained in that way model of scalar interaction should have more complex construction than similar phenomenological models at the same time it should reveal more rich possibilities of behavior\footnote{see e.g. \cite{YuMif}}.
\section{The Canonical Equations Of Motion}
The canonical equations of relativistic particle motion relative to a pair of canonically conjugated dynamic variables $x^{i} $ (coordinates) and $P_{i} $ (generalized momentum) have a form (see e.g. \cite{Ignatev2})\footnote{Here and then the generic unit system is accepted $G=c=\hbar =1$}:
\begin{equation} \label{EQ1}
\frac{dx^{i} }{ds} =\frac{\partial H}{\partial P_{i} } ;\quad \quad \frac{dP_{i} }{ds} =-\frac{\partial H}{\partial x^{i} } ,
\end{equation}
where $H(x,P)$ is a relativistically invariant Hamilton function. Calculating the full derivative of the dynamic variables $\psi (x^{i} ,P_{k} )$, with an account of \eqref{EQ1} we obtain:
\begin{equation} \label{EQ2}
\frac{d\psi }{ds} =[H,\psi ],
\end{equation}
where the invariant Poisson brackets are introduced:
\begin{equation} \label{EQ3}
[H,\psi ]=\frac{\partial H}{\partial P_{i} } \frac{\partial \psi }{\partial x^{i} } -\frac{\partial H}{\partial x^{i} } \frac{\partial \psi }{\partial P_{i} } \; .
\end{equation}
As a result of \eqref{EQ3} the Hamilton function is an integral of particle motion:
\begin{equation} \label{EQ4}
\frac{dH}{ds} =[H,H]=0,\Rightarrow H= {\rm Const}.
\end{equation}
The relation \eqref{EQ4} could be named the normalization ratio.
The invariant Hamilton function is determined ambiguously. Indeed, in consequence of \eqref{EQ3}
if $H(x,P)$ is a Hamilton function, then any continuously dif\-fe\-ren\-ti\-able function $f(H)$ is also a Hamilton function.
The sole possibility of introduction of the invariant Hamilton function quadratic by generalized particle momentum at presence of only gravitational and scalar fields
is:
\begin{equation} \label{EQ7}
H(x,P)=\frac{1}{2} \left[\psi(x)(P,P)-\varphi(x) \right],
\end{equation}
where $(a,b)$ are here and then a scalar product of 4-dimensional vectors $a$ and $b$, and $\psi(x)$ and $\varphi(x)$ are certain scalar functions.
Choosing a non-zero normalization of the Hamilton function \cite{Ignat_Popov,YuNewScalar1},
\begin{equation}\label{EQ7a}
H(x,P)=\frac{1}{2} \left[\psi(x)(P,P)-\varphi(x) \right]=0.
\end{equation}
we obtain:
\begin{equation}\label{EQ7b}
(P,P)=\frac{\varphi}{\psi},
\end{equation}
and from the first group of the canonical equations of motion(\ref{EQ1})
we obtain the relation between the generalized momentum and a particle velocity vector:
\begin{equation} \label{EQ10a}
u^{i} \equiv \frac{dx^{i} }{ds} =\psi P^{i} \Rightarrow P^{i} =\psi^{-1} u^{i} ,
\end{equation}
Substituting the last relation to the normalization ratio (\ref{EQ7b}) and requiring the unit normalization of particle velocity vector
\begin{equation} \label{EQ11}
(u,u)=1.
\end{equation}
we obtain:
\[\psi\varphi=1 \Rightarrow \psi=\varphi^{-1},\]
thereby particle's invariant Hamilton function could be defined by only one scalar function $\varphi(x)$. Taking into account the last relation, let us write down the Hamilton function in the final form:
\begin{equation}\label{EQ7 }
H(x,P)=\frac{1}{2} \left[\varphi^{-1}(x)(P,P)-\varphi(x) \right]=0,
\end{equation}
and from the canonical equations \eqref{EQ1} we obtain the relation between the generalized momentum and particle velocity vector:
\begin{equation}
\label{EQ10} P^i=\varphi \frac{dx^i}{ds}.
\end{equation}
From the definition \eqref{EQ7} it follows that the ge\-ne\-ra\-li\-zed momentum vector is timelike:
\begin{equation}
\label{EQ8} (P,P)=\varphi^2.
\end{equation}
Let us notice useful for future reasoning a relation which is a consequence of \eqref{EQ3}, \eqref{EQ7} and \eqref{EQ8}:
\begin{equation} \label{EQ9}
[H,P^{k} ]=\nabla ^{k} \varphi \equiv g^{ik} \partial _{i} \varphi.
\end{equation}
\subsection{The Lagrange Equations Of Motion}
From the second group of the canonical equations \eqref{EQ1} we obtain the equations of motion in Lagrange forulation \cite{Yubook1}:
\begin{equation} \label{EQ12}
\frac{d^{2} x^{i} }{ds^{2} } +\Gamma _{jk}^{i} \frac{dx^{j} }{ds} \frac{dx^{k} }{ds} =\partial _{,k} \ln |\varphi|{\rm {\mathcal P}}^{ik} ,
\end{equation}
where:
\begin{equation} \label{EQ13}
{\rm {\mathcal P}}^{ik} ={\rm {\mathcal P}}^{ki} =g^{ik} -u^{i} u^{k}
\end{equation}
is a tensor of an orthogonal projection to the direction $u$, such that:
\begin{equation} \label{EQ14}
{\rm {\mathcal P}}^{ik} u_{k} \equiv 0;\quad {\rm {\mathcal P}}^{ik} g_{ik} \equiv 3.
\end{equation}
From these relations and the Lagrange equations \eqref{EQ12} it follows the strict consequence of velocity and acceleration vectors orthogonality:
\begin{equation} \label{EQ15}
g_{ik} u^{i} \frac{du^{k} }{ds} \equiv 0.
\end{equation}
Let us note that the Lagrange equations of motion (\ref{EQ12}) are invariant relative to a sign of the scalar function $\varphi(x)$:
\begin{equation} \label{EQ16a}
\varphi(x)\rightarrow -\varphi(x).
\end{equation}
The Hamilton function (\ref{EQ7 }) at its zero normalization $H\rightarrow -H$ is also invariant relative to the trans\-for\-ma\-ti\-on (\ref{EQ16a}). Therefore from the relations \eqref{EQ8}, \eqref{EQ10}, and the Lagrange equations \eqref{EQ12} it follows that the quadrate of $\varphi $ scalar has a meaning of a quadrate of \textit{ the efficient inert mass of particle, $m_{*} $, in a scalar field}:
\begin{equation} \label{EQ16}
\varphi^2 =m_{*}^2 .
\end{equation}
Let us note that a following action function formally coinciding with the Lagrange function of a rela\-ti\-vis\-tic particle
with a rest mass $\varphi$ in a gravitational field corresponds to the cited choice of the Hamilton function:
\begin{equation} \label{EQ17}
S=\int m_* ds,
\end{equation}
where an effective mass $m_*$ due to invariance of the Lagrange equations relative to the transformation (\ref{EQ16a}) can be defined as $m_*=\varphi$, or $m_*=|\varphi|$.
In this article we resolve the relation (\ref{EQ16}) by next form:
\begin{equation}\label{phi=m}
m_*=\varphi,
\end{equation}
as opposed to how it wa proposed in the previous articles: $m_*=|\varphi|$. Thus at (\ref{phi=m}) an effective mass of a particle in general could be negative.
\subsection{Integrals Of Motion}
Let us now find conditions of existence of canonical equations' of state \eqref{EQ1} line integral which is con\-nec\-ted to particle's total energy and momentum. For that let us calculate the scalar product's $(\xi ,P)$ full derivative by the canonical parameter. Using the canonical equations of motion \eqref{EQ1}, the normalization ratio \eqref{EQ8}, and the relation of the generalized mo\-men\-tum to the kinematic one \eqref{EQ10}, we find:
\begin{equation} \label{EQ18}
\frac{d(\xi ,P)}{ds} =\frac{1}{m_{*} } P^{i} P^{k} \mathop{L}\limits_{\xi } g_{ik} +\mathop{L}\limits_{\xi } m_{*} ,
\end{equation}
where $\mathop{L}\limits_{\xi } $ is a Lie derivative by the direction $\xi $ \footnote{see e.g. \cite{Petrov}.}. Assuming further
\begin{equation} \label{EQ19}
\frac{d(\xi ,P)}{ds} =0\Leftrightarrow (\xi ,P)={\kern 1pt} {\rm Const}{\kern 1pt} ,
\end{equation}
with an account of arbitrariness of the generalized momentum's vector and its normalization ratio we obtain the conditions of the equation fulfillment:
\begin{equation} \label{EQ20}
\mathop{L}\limits_{\xi } g_{ik} =\sigma g_{ik} \Rightarrow \sigma =-\mathop{L}\limits_{\xi } \ln |m_{*} |,
\end{equation}
where $\sigma$ is a arbitrary scalar. Substituting this result back to the relation \eqref{EQ18}, we find the necessary and sufficient conditions of existence of the canonical equations' line integral (see e.g. \cite{Yubook1}):
\begin{equation} \label{EQ21}
\mathop{L}\limits_{\xi } m_{*} g_{ik} =0.
\end{equation}
Thus \textit{ to exist the line integral, it is enough and sufficient that conformal corresponding space with metrics $m_{*} g_{ik} $ allows a group of motions with the Killing vector $\xi $}. Let us note that line integrals \eqref{EQ19} have the meaning of total momentum (at spacelike vector $\xi $) or total energy (at timelike vector $\xi $).
\subsection{The Choice Of mass Function}
There emerges the question of choice of function $m_{*} (\Phi )$. Let us highlight one important circumstance yet not concretizing this function. Let us consider statistical fields $g_{ik} $ and $\Phi $, allowing timelike Killing vector $\xi ^{i} =\delta _{4}^{i} $, when particle's total energy $P_{4} $ is conserved. Further let us consider the reference frame in which $g_{\alpha 4} =0$, so that coordinate $x^{4} $ coincides with a world time $t$. Then from the relations of connection between the kinematic ve\-lo\-city $u^{i} $ and vector of particle's total momentum $P_{i} $ \eqref{EQ10} if follows:
\begin{equation} \label{EQ22}
P_{4} ds=m_{*} dt,
\end{equation}
where $P_{4} =E_{0} ={\kern 1pt} {\rm Const}{\kern 1pt} >0$ is a full energy of a charged particle. Therefore if we want to conserve the same orientation of world and proper time (i.e. $dt/ds>0$), it is required to choose such a mass function which always stay nonnegative. In the article we nonetheless remove the requirement of positiveness of an effective mass of particle.\footnote{It should be noted that as early as A.A. Vlasov in his fundamental work \cite{Vlasov} supposed a possibility of discontinuity of a standard connection between these times.}
Further, from one hand, in absence of a scalar field or, more precisely, in a constant scalar field, mass function should
change to particle rest mass, $m\ge 0$.
From the other hand, the Lagrange equations \eqref{EQ12} in case of a weak scalar field and small velocities should change to classical equations of motion in a scalar field:
\begin{equation}\label{weak_F}
m\frac{d^2x^\alpha}{dt^2}=-q\partial_\alpha\Phi.
\end{equation}
Hence following from the correspondence principle we should have:
\begin{equation} \label{EQ25}
\varphi(0)=m;\quad \lim\limits_{\Phi=0}\left( \frac{\varphi_{,k}}{\varphi}\right)=q\Phi _{,k} ,
\end{equation}
where $\varphi=\varphi(\Phi)$.
Supposing $q\Phi=f$, let us rewrite \eqref{EQ25} in a equivalent form:
\begin{equation}\label{m_0}
\varphi(0)=m;\quad \lim\limits_{f=0}\frac{d\varphi}{df}=1,
\end{equation}
whence \eqref{EQ25} mean that at small values of the scalar potential $\Phi $ function $m_{*} (\Phi )$ should have an ex\-pan\-sion of form:
\begin{equation} \label{EQ26}
m_{*} (\Phi ){\rm \simeq } m+q\Phi+\ldots.
\end{equation}
To this condition corresponds the linear function
\begin{equation}\label{m_*}
m_{*} (\Phi )=m+q\Phi,
\end{equation}
as well as function used in the cited papers:
\begin{equation}\label{|m|}
m_{*} (\Phi )=|m+q\Phi|.
\end{equation}
Further we will choose particle's effective mass function in form (\ref{m_*}). Then according to(\ref{EQ10}) we obtain the relation between
the particle's ge\-ne\-ra\-lized mo\-men\-tum and the kinematic velocity:
\begin{equation}\label{Pu}
P^i=m_*u^i=(m+q\Phi)u^i;
\end{equation}
the normalization ratio for the generalized mo\-men\-tum vector takes form
\begin{equation}\label{norm_m}
(P,P)=m_*^2=(m+q\Phi)^2
\end{equation}
and has an identical form for mass function (\ref{m_*}), as well as for mass function (\ref{|m|}).
To refine the meaning of introduced kinematic and dynamic values let us consider a problem of scalar charged particle's movement in the Minkovsky space in a statistical scalar field, which potential depends only on one coordinate, $x^{1} =x$, and let for the simplicity $m_{*} =m+q\Phi =x$. This we have 3 Killing vectors - one of which is timelike and two are spacelike:
\def\stackunder#1#2{\mathrel{\mathop{#2}\limits_{#1}}}
$$\stackunder{1}{\xi}^i=\delta _{4}^{i};\; \stackunder{2}{\xi}^i=\delta _{2}^{i} ;\; \stackunder{3}{\xi}^i =\delta _{3}^{i} .$$
Correspondingly to these vectors there exist three linear integrals of motion:
\begin{eqnarray} \label{EQ30}
P_{2} =P_{2}^{0} = {\rm Const} ;\; P_{3} =P_{3}^{0} = {\rm Const};\nonumber\\
P_{4} =P_{4}^{0} = {\rm Const}.
\end{eqnarray}
Let for the simplicity $P_{2} =P_{3} =0$. Then with an account of the normalization ratio \eqref{norm_m}
we obtain from the canonical equations of motions the non\-tri\-vial one:
\begin{eqnarray} \label{EQ31}
\frac{P^1}{m_*}=u^1\equiv\frac{dx}{ds} =\mp \frac{\sqrt{(P^0_{4})^{2} -x^{2}}}{x }.\\
\label{EQ31a}
u^4\equiv\frac{dt}{ds}\equiv \frac{P_4}{m_*}=\frac{P^0_4}{x}\Rightarrow
\end{eqnarray}
\begin{equation}\label{Eqx}
\frac{dx}{dt}=\mp \frac{1}{P^0_4}\sqrt{(P^0_4)^2-x^2},
\end{equation}
where signs $\mp$ correspond to a movement to the left or to the right.
Integrating (\ref{Eqx}) with the initial condition $x(0)=0$,
we find the solution:
\begin{equation}
x=\mp P^0_4\sin(t/P^0_4).
\end{equation}
Thus a particle performs harmonic oscillations by clocks of the world time with an amplitude $P^0_4$ and frequency $P^0_4$.
However at that, following (\ref{EQ31a})
kinematic velocity's components $u^i$ in moments of time $t=\pi k$ undergo discontinuities of the second kind and change the sign.
However components of the generalized momentum, $P_i$,
as well as com\-po\-nents of the kinematic momentum, $p^i$, being the observed variables defined by the standard procedure as
\begin{equation}\label{p_k}
p^i=m_*\frac{dx^i}{ds}\equiv P^i,
\end{equation}
remain continuous functions and kinematic momentum's component $p^4$ does not change the sign, remaining constant. Nevertheless, the sign of microscopic proper time's relation to the world time is changed. Let us notice, that this outcome is not dependent on the massive function's choice method; the same outcome has been obtained in \cite{YuNewScalar1}, where positively defined mass function had been being used. Therefore reasonings cited in more recent Author's papers in favor of nonnegativeness of the mass function and formula (\ref{|m|}) on the basis of the Lagrange equations' of motion analysis, are not strict. Apparently, cited example of particle interaction with a scalar field presents the realization of A.Vlasov \cite{Vlasov} concept about the possibility of discontinuity of the relation of microscopic proper time of the particle with the world macroscopic time.
\subsection{Quantum equations}
Let us note that with a use of the standard procedure of quantum equations' obtainment
from the classical Hamilton function it is required to make a change in the function:\footnote{In
this place we temporarily depart from universal system of units, in which $\hbar=1$.}
\begin{equation}\label{quant_trans}
P_i\rightarrow i\hbar \frac{\partial}{\partial x^i},
\end{equation}
As a result of covariant generalization, Hamilton operator can be obtained from the Hamilton function:
\begin{equation}\label{Hamiltonian}
\hat{{\rm H}}=-m_*^{-1}(\hbar^2 g^{ik}\nabla_i\nabla_k + m^2_*).
\end{equation}
Thus for a free massive scalar field we could have obtained the wave equations in form of standard Klein-Gordon equations with the only difference that
bosons rest mass should be changed by the effective mass:
\begin{equation}\label{free_bozon}
(\square+m_*^2/\hbar^2)\Psi=0,
\end{equation}
and for free fermions we could have obtained co\-rres\-pon\-ding Dirac equations:
\begin{equation}\label{Dirac_eq}
(\hbar\gamma^i\nabla_i+m_*)\Psi=0,
\end{equation}
where $\gamma$ are spinors.
Let us note that from (\ref{free_bozon}) having substituted $\Psi=\Phi$ and chosen simplest mass function
$m_*=|q\Phi|$ it right away follows an equation of the free scalar field with
a cubic nonlinearity:
\begin{equation}\label{Phi3}
\square\Phi+q^2/\hbar^2\Phi^3=0.
\end{equation}
Thus constant of self-action in a scalar field's cubic equation takes quite defined meaning:
$$\lambda=\frac{q^2}{\hbar^2}.$$
|
1,116,691,500,888 | arxiv | \section{Introduction}\label{Intro}
The properties of single polymer chains at surfaces have received considerable
attention in recent years. Much of this has been spurred by new experimental
techniques such as atomic force microscopy (AFM) and optical/magnetic tweezers
\cite{Smith} which allow one to manipulate single polymer chains. Study of
single polymer molecules at surfaces, such as mica or self-assembled monolayers,
by Atomic Force Microscopy (AFM) method provides a great scope for
experimentation \cite{Senden,Hugel,Seitz,Rohrbach,Gaub_2,Friedsam,Gaub_1}.
Applications range from sequential unfolding of collapsed biopolymers over
stretching of coiled synthetic polymers to breaking individual covalent
bonds\cite{Rief,Ortiz,Grandbois}.
In these experiments it is customary to anchor a polymer molecule with one end
to the substrate whereas the other end is fixed on the AFM cantilever. The
polymer molecule can be adsorbed on the substrate while the cantilever recedes
from the substrate. In so doing one can prescribe the acting force in AFM
experiment whereas the distance between the tip and the surface is measured.
Conversely, it is also possible to fix the distance and measure the
corresponding force, a method which is actually more typical in AFM-experiments.
From the standpoint of statistical mechanics these two cases could be qualified
as $f$-ensemble (force is fixed while fluctuating the height of the chain end is
measured) and as $h$-ensemble ($h$ is fixed while one measures the fluctuating
$f$). Recently these two ways of descriptions as well as their interrelation
were discussed for the case of a phantom polymer chain by Skvortsov et al.
\cite{Skvortsov}. In recent papers \cite{PRER,Bhattacharya} we studied
extensively the desorption of a single tethered {\em self-avoiding} polymer from
a flat solid substrate with an external force applied to a free chain's end in
the $f$-ensemble.
In the present paper we consider the detachment process of a single
self-avoiding polymer chain, keeping the distance $h$ between the free chain's
end and the substrate as the control parameter. We derive analytical results for
the main observables which characterize the detachment process. The mean
value as well as the probability distribution function (PDF) of the order
parameter are presented in close analytical expressions using the
Grand Canonical Ensemble (GCE) method\cite{Bhattacharya}. The basic force-height
relationship which describes the process of polymer detachments by pulling, that
is, the relevant equation of state for this system is also derived both for
extendible as well as for rigid bonds and shown to comply well with our Monte
Carlo simulation results. We demonstrate also that a number of properties behave
differently in the vicinity of the phase transition, regarding which of the two
equivalent ensembles is used as a basis for the study of systems's behavior.
\section{Single chain adsorption: using distance as a control parameter}
\label{Theory}
\subsection{Deformation of a tethered chain}
Before considering the adsorption-desorption behavior of a polymer in terms of
the chain end distance $h$, we first examine how a chain tethered to a solid
surface responds to stretching. This problem amounts to finding the chain free
end probability distribution function (PDF) $P_{N}(h)$ where $N$ is the chain
length, i.e., the number of beads. The partition function of such a chain at
fixed distance $h$ of the chain end from the anchoring plane is given by
\begin{eqnarray}
\Xi_{\rm tail}(N, h) = \frac{\mu_{3}^{N}}{N^{\beta}} \: l_0 P_{N} (h)
\label{Partition_Tail}
\end{eqnarray}
where $\beta = 1 - \gamma_1$ and the exponent $\gamma_1 =
0.680$\cite{Vanderzande}. Here $\mu_3$ is a model dependent connective constant
(see e.g. ref.\cite{Vanderzande}). In Eq.~(\ref{Partition_Tail}) $l_0$ denotes a
short-range characteristic length which depends on the chain model. Below we
discuss the chain deformation within two models: the bead-spring (BS) model for
elastic bonds and the freely jointed bond vectors (FJBV) model where the bonds
between adjacent beads are considered rigid.
\subsubsection{Bead-spring model}
The form of $P_N(h)$ has been discussed earlier \cite{Binder,Eisenriegler} and
later used in studies of the monomer density in polymer brushes \cite{Kreer}.
Here we outline this in a way which is appropriate for our purposes. The
average end-to-end chain distance scales as $R_N = l_0 N^{\nu}$, where $l_0$ is
the mean distance between two successive beads on a chain and $\nu \approx
0.588$ is the Flory exponent \cite{Grosberg}. The short distance behavior, $h
\ll R_N$, is given by
\begin{eqnarray}
P_N(h) \propto \left(\frac{h}{R_N}\right)^{\zeta}
\label{Short}
\end{eqnarray}
where the exponent $\zeta \approx 0.8$. For the long distance
behavior, $h/R_N \gg 1$, we assume,
following Ref.\cite{Kreer}, that the PDF of the end-to-end vector ${\bf r}$ is
given by des Cloizeaux's expression \cite{Cloiseaux} for a chain in the bulk :
$P_N({\bf r})=(1/R_N) \: F({\bf r}/R_N)$ where the scaling function $F(x)
\propto x^t \: \exp [-D x^\delta]$, and the exponents $t=(\beta-d/2+\nu
d)/(1-\nu)$ , $\delta=1/(1-\nu)$. Here and below $d$ denotes the space
dimensionality. One should emphasize that the presence of a surface is
manifested only by the replacement of the universal exponent $\gamma$ with another
universal exponent $\gamma_1$ (as compared to the pure bulk case!). By
integration of $P_N({\bf r})$ over the $x$ and $y$ coordinates while $h$ is
measured along the $z-$coordinate, one obtains $P_N(h) \propto
(h/R_N)^{2+t-\delta} \:\: \exp [-D(h/R_N)^{\delta}]$. As the long distance
behavior is dominated mainly by the exponential function while the short
distance regime is described by Eq.~(\ref{Short}), we can approximate the
overall behavior as
\begin{eqnarray}
P_N(h) = \frac{A}{R_N} \: \left(\frac{h}{R_N}\right)^{\zeta} \: \exp \left[-D
\left(\frac{h}{R_N}\right)^{\delta}\right]
\label{Overall}
\end{eqnarray}
\vspace*{0.9cm}
\begin{figure}[htb]
\includegraphics[scale=0.34]{P_h.eps}
\caption{Probability distribution $P_N(h)$ of chain end positions $h$ above the
grafting plane for a polymer with $N = 128$ monomers at zero strength of the
adsorption potential $\epsilon = 0.0$. In the inset the MC data for $P_N(h)$
(solid black line) is compared to the theoretic result, Eq.~(\ref{Overall}).
Dashed line denotes the expected slope of $\zeta \approx 0.78$ of the
probability distribution for small heights.}
\label{pdf_ends}
\end{figure}
A comparison of the distribution, Eq.~(\ref{Overall}), with our simulation data
is shown in Fig.~\ref{pdf_ends}. The constants $A$ and $D$ in
Eq.~(\ref{Overall}) can be found from the conditions: $\int P_N (h) d h =1$ and
$\int h^2 P_N (h) d h =R_N^2$. This leads to:
\begin{eqnarray}
A = \delta
\left[\Gamma\left(\frac{1+\zeta}{\delta}\right)\right]^{-(1+\zeta)/2}
\left[\Gamma\left(\frac{3+\zeta}{\delta}\right)\right]^{-(1-\zeta)/2}
\label{A}
\end{eqnarray}
and
\begin{eqnarray}
D = \left[\Gamma\left(\frac{3+\zeta}{\delta}\right)\right]^{\delta/2}
\left[\Gamma\left(\frac{1+\zeta}{\delta}\right)\right]^{-\delta/2}
\label{D}
\end{eqnarray}
where $\delta \approx 2.43$ and $\zeta \approx 0.8$. One gets thus the estimates
$A\approx 2.029$ and $D \approx 0.670$.
The free energy of the tethered chain with a fixed distance $h$ takes on the
form $F_{\rm tail}(N, h) = - k_BT \ln \Xi_{\rm tail} (N, h)$ where $k_B$
denotes the Boltzmann constant. By making use of Eqs. (\ref{Partition_Tail}) and
(\ref{Overall}) the expression for the force $f_N$, acting on the end-monomer
when kept at distance $h$ is given by
\begin{eqnarray}
f_N = \frac{\partial}{\partial h} \: F_{\rm tail}(N, h)
= \frac{k_BT}{R_N} \:\left[ \delta D \left( \frac{h}{R_N}\right)^{\delta-1} -
\zeta \left( \frac{R_N}{h}\right)\right]
\label{Deformation}
\end{eqnarray}
One should note that at $h/R_N \gg 1$ we have $h \propto R_N (R_N
f_N/k_BT)^{1/(\delta-1)}$ which, after taking into account that $\delta^{-1} =
1-\nu$, leads to the well known Pincus deformation law: $h \propto l_0 N (l_0
f_N/k_BT)^{1/\nu -1}$ \cite{de Gennes}. Within the framework of this
approximation the (dimensionless) elastic energy reads $U_{\rm el}/k_BT = - N
(l_0 f_{N}/k_BT)^{1/\nu}$. In result the corresponding free energy of the chain
tail is given by
\begin{eqnarray}
\frac{F_{\rm tail}}{k_BT} = - N \left(\frac{l_0 f_N}{k_BT}\right)^{1/\nu} - N
\ln \mu_3
\label{Free_Energy_BS}
\end{eqnarray}
Eq.~(\ref{Deformation}) indicates that there
exists a height $h_0 = (\zeta/\delta D)^{1/\delta} R_N$ over the surface where
the force $f_N$ changes sign and becomes negative (that is, an entropic
repulsion dominates). According to Eq.~(\ref{Deformation}) the force diverges as
$f_N \propto - k_BT/h$ upon further decrease of the distance $h$.
\subsubsection{Freely jointed chain}
It is well known \cite{Grosberg} that the Pincus law, Eq. (\ref{Deformation}),
describes the deformation of a linear chain at intermediate force strength,
$1/N^{\nu} \ll l_0 f_N/k_BT \leq 1$. Direct Monte Carlo simulation results
indicate that, depending on the model, deviations from Pincus law emerge at
$h/R_N \ge 3$ (bead-spring off-lattice model) \cite{Lai}, or $h/R_N \ge 6$ (Bond
Fluctuation Model \cite{Wittkop}). In such ``overstretched'' regime (when the
chain is stretched close to its contour length) one should take into account
that the chain bonds cannot expand indefinitely. This case could be treated,
therefore, within the simple freely jointed bond vectors (FJBV) model
\cite{Lai,Schurr} where the bond length $l_0$ is fixed. In this model the force
- deformation relationship is given by
\begin{eqnarray}
f_N = \frac{k_BT}{l_0} \: {\cal L}^{-1} \left( \frac{h}{l_0 N}\right)
\label{L}
\end{eqnarray}
where ${\cal L}^{-1}$ denotes the inverse Langevin function ${\cal L}(x) =
\coth(x) - 1/x$ and $l_0$ is the fixed bond vector length. We discuss the main
results pertaining to the FJBV model in Appendix A. The elastic deformation
energy reads $U_{\rm el}/k_BT = - (l_0 f_N/k_BT) \sum_{i=1}^{N} <\cos \theta_i>
= - N (l_0 f_N/k_BT) {\cal L}(l_0 f_N/k_BT)$, where $\theta_i$ is the average
polar angle of the $i$-th bond vector (see Appendix A). Thus the corresponding
free energy of the chain tail for the FJBV model reads
\begin{eqnarray}
\frac{F_{\rm tail}}{k_BT} = - N {\cal G}\left(\frac{l_0 f_N}{k_BT} \right) -
N \ln \mu_3
\label{Free_Energy_FJBV}
\end{eqnarray}
where we have used the notation ${\cal G}(x) = x {\cal L}(x) = x \coth (x) - 1$.
Now we are in a position to discuss the pulling of the adsorbed chain controlled
by the chain height $h$.
\subsection{Pulling controlled by the chain end position}
Consider now an adsorbed chain when the adsorption energy per monomer is
sufficiently large, $\varepsilon \geq \varepsilon_c$, where $\varepsilon_c$
denotes a corresponding {\it critical} energy of adsorption. Below we will also
use the notation $\epsilon = \varepsilon/k_B T$ for the dimensionless adsorption
energy. The problem of force-induced polymer desorption could be posed as
follows: how is the process of polymer detachment governed by the chain end
position $h$? Figure \ref{Tearing}a gives a schematic representation of such a
system, and the situation in a computer experiment, as shown in the snapshot
Fig.~\ref{Tearing}b, is very similar.
\begin{figure}[bht]
\hspace{-2.0cm}
\includegraphics[scale=0.6]{ads_desorp.eps}
\hspace{0.5cm}
\includegraphics[scale=0.3]{pull2.eps}
\caption{ (a) Schematic graph of an adsorbed polymer chain, partially detached
from the plane by an external force which keeps the last monomer at height $h$.
The total chain is built up from a tail of length $M$ and an adsorbed part
of length $N - M$. The force $f_M$ acting on the chain end is conjugated to $h$,
i.e., $f_M = \partial F_{\rm tail}/\partial h$. (b) A snapshot from the MC
simulation: $N = 128,\; h = 25.0,\; \epsilon = 4.0$ and $\langle f \rangle =
6.126$. }
\label{Tearing}
\end{figure}
As is evident from Fig. \ref{Tearing}a, the system is built up from a tail of
length $M$ and an adsorbed portion of length $N-M$. The adsorbed part can be
treated within the GCE approach \cite{Bhattacharya}. In our earlier treatment
\cite{Bhattacharya} it was shown that the free energy of the adsorbed
portion is $F_{\rm ads} = k_BT (N - M)\ln z^{*}(\epsilon)$, where the fugacity
per adsorbed monomer $z^{*}(\epsilon)$ depends on $\epsilon$ and can be found
from the basic equation
\begin{eqnarray}
\Phi(\alpha, \mu_3 z^{*}) \: \Phi (\lambda, \mu_2 w z^{*})= 1
\label{Basic}
\end{eqnarray}
The so called {\it polylog function} in Eq. (\ref{Basic}) is defined as $\Phi
(\alpha, z) = \sum_{n=1}^{\infty} z^n/n^{\alpha}$ and the connective constants
$\mu_3$, $\mu_2$ in three and two dimensional space have values which are model
dependent \cite{Vanderzande}. The exponents $\alpha = 1+ \phi$ and $\lambda =
1-\gamma_{d=2}$ where $\phi \approx 0.5$ is the {\it crossover exponent} which
governs the polymer adsorption at criticality, and in particular, the fraction
of adsorbed monomers at the critical adsorption point (CAP) $\epsilon =
\epsilon_{c}$. The constant $\gamma_{d=2} = 1.343$ \cite{Vanderzande}. Finally
$w = \exp(\epsilon)$ is the additional statistical weight gained by each
adsorbed segment.
In equilibrium, the force conjugated to $h$, that is, $f_M = \partial F_{\rm
tail}/\partial h$, should be equal to the {\it chain resistance force to
pulling} $f_{\rm p} = (k_BT/l_0) {\cal F}(\epsilon)$ (where ${\cal F}(\epsilon)$
is a scaling function depending only on $\epsilon$), i.e.,
\begin{eqnarray}
f_M = \begin{cases}\frac{k_BT}{R_M} \:\left[ \delta D \left(
\frac{h}{R_M}\right)^{\delta-1} -
\zeta \left( \frac{R_M}{h}\right)\right] = f_{\rm p} &\mbox{, for BS-model}\\
\\
\frac{k_BT}{l_0} \: {\cal L}^{-1}\left(\frac{h}{l_0 M}\right) = f_{\rm p}
&\mbox{,
for FJBV-model}
\end{cases}
\label{Mech_Euqilibrium}
\end{eqnarray}
The resisting force $f_{\rm p}$ holds the last adsorbed monomer on the adhesive
plane (see again Fig.\ref{Tearing}a whereby this monomer is shown to experience
a force $f_{M}$). One should emphasize that the force $f_{\rm p}$ {\em stays
constant} in the course of the pulling process as long as one monomer, at least,
is adsorbed on the surface. Thus $f_{\rm p}$ corresponds to a {\it plateau} on
the deformation curve (force $f$ vs. chain end position $h$). The adsorbed
monomer (see Fig. \ref{Tearing}) has a chemical potential, $\mu_{\rm ads}=\ln
z^{*}$, which in equilibrium should be equal to the chemical potential of a
desorbed monomer in the tail, $\mu_{\rm des}= \partial (F_{\rm
tail}/k_BT)/\partial N$. The expression for $F_{\rm tail}$ depends on the model
and is given either by Eq.(\ref{Free_Energy_BS}) for the BS-model or by
Eq.(\ref{Free_Energy_FJBV}) in the case of FJBV-model. Taking this into account
the condition $\mu_{\rm ads}=\mu_{\rm des}$ leads to the following ``plateau
law'' relationship
\begin{eqnarray}
\frac{l_0 \: f_{\rm p}}{k_BT} = \begin{cases}\left|\ln [\mu_3
z^{*}(\epsilon)]\right|^{\nu} &\mbox{, for BS-model}\\
\\
{\cal G}^{-1}\left(\left|\ln [\mu_3 z^{*} (\varepsilon)]\right|\right) &\mbox{,
for
FJVB-model}
\end{cases}
\label{Local_Equilibrium}
\end{eqnarray}
where ${\cal G}^{-1}(x)$ stands for the inversion of the function ${\cal G}(x) =
x \coth (x) - 1$. One should note that Eq.(\ref{Local_Equilibrium}) coincides
with Eq.(3.16) in Ref. \cite{Bhattacharya} which determines the detachment line
in the pulling process controlled by the applied force. Close to the critical
point $\epsilon_c$, the plateau force $f_{\rm p}$ goes to zero. Indeed, since in
the vicinity of the critical point $\ln [\mu_3 z^{*}(\epsilon)] \propto -
(\epsilon - \epsilon_c)^{1/\phi}$ (see ref.\cite{Bhattacharya}), and ${\cal
G}^{-1} (x)\approx (3 x)^{1/2}$, one may conclude that $f_{\rm p} \propto
(\epsilon - \epsilon_c)^{\nu/\phi}$ for the BS-model and $f_{\rm p} \propto
(\epsilon - \epsilon_c)^{1/2 \phi}$ for the FJVB-model.
One can solve Eq.(\ref{Mech_Euqilibrium}) with respect to $M$ (taking into
account that $h \gg R_M$), and arrive at an expression for the tail length
\begin{eqnarray}
M(h, \epsilon) = \begin{cases} \frac{h}{l_0} \: \left(\frac{k_BT}{l_0
f_{\rm p}}\right)^{1/\nu - 1} &\mbox{, for BS-model}\\
\\
\frac{h}{l_0} \:\left[{\cal L}\left(\frac{l_0
f_{\rm p}}{k_BT}\right)\right]^{-1}&\mbox{, for
FJVB-model}
\end{cases}
\label{M_vs_h}
\end{eqnarray}
where the force at the plateau, $f_{\rm p}$, is described by Eq.
(\ref{Local_Equilibrium}). If for the degree of adsorption one uses as an order
parameter the fraction of chain contacts with the plane, $n = N_s/N$, where
$N_s$ is the number of monomers on the surface, one can write
\cite{Bhattacharya}
\begin{eqnarray}
n = - \frac{1}{k_BT N} \frac{\partial}{\partial \epsilon} \left(F_{\rm ads} +
F_{\rm tail}\right)
\label{Fraction}
\end{eqnarray}
where $F_{\rm ads}$ and $F_{\rm tail}$ are free energies of the adsorbed and
desorbed portions of the chain respectively. The free energy $F_{\rm ads} =
k_BT [ N - M(h, \epsilon)] \ln z^{*}(\epsilon)$ whereas $F_{\rm tail} = k_BT
\mu_{\rm des} M(h, \epsilon)$ (recall that $\mu_{\rm des}$ is the chemical
potential of a desorbed monomer). After substitution of these expressions in eq.
(\ref{Fraction}) and taking into account that in equilibrium $\mu_{\rm ads} =
\mu_{\rm des}$ (the sequence of operations is important: taking the derivative
with respect to $\epsilon$ is to be followed by the condition $\mu_{\rm ads}
=\mu_{\rm des}$) so one gets
\begin{eqnarray}
n = - \left[1 - \frac{M(h, \epsilon)}{N}\right] \frac{\partial \ln
z^{*}(\epsilon)}{\partial
\epsilon}
\label{Order_Parameter}
\end{eqnarray}
i.e. the order parameter $n$ is defined by the product of monomer fraction in
the adsorbed portion, $1 - M/N$, and the fraction of surface contacts in this
portion, $ - \partial \ln z^{*}/\partial \epsilon$.
The expressions for the order parameter can be recast in the form
\begin{eqnarray}
n = \left|\frac{\partial \ln
z^{*}(\epsilon)}{\partial
\epsilon}\right| \times \begin{cases}
1 - \frac{h}{c_1 l_0 N} \left(\frac{k_BT}{l_0 f_{\rm
p}}\right)^{1/\nu - 1} &\mbox{, BS-model}\\
\\
1 - \frac{h}{c_2 l_0 N} \left[{\cal L} \left(\frac{l_0 f_{\rm
p}}{k_BT}\right)\right]^{-1}
&\mbox{, FJVB-model}
\end{cases}
\label{Order_Parameter1}
\end{eqnarray}
Here $c_1$ and $c_2$ are some constants of the order of unity.
As one can see from Eq.~(\ref{Order_Parameter1}), the order parameter decreases
linearly and steadily with $h/N$. This behavior is qualitatively different from
the abrupt jump of $n$ when the pulling force $f$ is changed as a control
parameter. In Section \ref{MC_results} we will show that this predictions is in
a good agreement with our MC - findings. The transition point on the $n$ vs. $h$
curve corresponds to total detachment, $n = 0$. The corresponding distance $h$
will be termed ``detachment height'' $h_D$. The dependence of $h_D$ on the
adsorption energy $\epsilon$ can be obtained from Eq.(\ref{Order_Parameter1})
where $n$ is set to zero, i.e.
\begin{eqnarray}
\frac{h_D}{l_0 N} = \begin{cases}
\left( \frac{l_0 f_{\rm p}}{k_BT}\right)^{1/\nu -1} &\mbox{, BS-model}\\
\\
{\cal L}\left( \frac{l_0 f_{\rm p}}{k_BT}\right) &\mbox{, FJVB-model}
\end{cases}
\label{Detachment_Line}
\end{eqnarray}
where again $f_{\rm p}$ as a function of $\epsilon$ is given by Eq.
(\ref{Local_Equilibrium}).
The line given by Eq. (\ref{Detachment_Line}), is named ``detachment line''. It
corresponds to an adsorption - desorption polymer transition which appears as of
{\it second order} since this order parameter $n$ goes to zero continuously as
$h$ increases. One should emphasize, however, that this ``detachment''
transition has the same nature as the force-induced desorption transition
\cite{Bhattacharya} in the $f$-ensemble where the pulling force $f$, rather than
the distance $h$, is fixed and used as a control parameter. This phase
transformation is known to be of {\it first order}.
It is easy to understand (cf. with Eq.(\ref{M_vs_h}) ) that the condition $M(h,
\varepsilon) = N$ corresponds to the detachment line as well as
to a terminal point of the force plateau. It can be seen in the MC -
simulation results in Section\ref{MC_results}, Fig.~\ref{force}.
\section{Probability distribution $P(K)$ of the number of adsorbed
monomers}\label{Theory2}
The grand canonical ensemble (GCE) method, which has been used in our recent
paper \cite{Bhattacharya}, is a good starting point to calculate the probability
distribution function $P(K)$ of the adsorbed monomers number $K$. According to
this approach, the GCE-partition function of an adsorbed chain has the form
\begin{equation}
\Xi (z, w) = \sum_{N=1}^{\infty} \: \sum_{K=0}^{\infty} \Xi_{N, K} \: z^N \:
w^K = \dfrac{V_0 (wz) \: Q (z)}{1 - V(w z)\: U(z)}
\label{GCE}
\end{equation}
where $z$ and $w = \exp(\epsilon)$ are the fugacities conjugated to chain length
$N$ and to the number of adsorbed monomers $K$, respectively. In Eq.(\ref{GCE})
$U(z)$, $V(wz)$ and $Q(z)$ denote the GCE partition functions for loops, trains
and tails, respectively. The building block adjacent to the tethered chain end
corresponds to $V_0 (wz) = 1 + V (wz)$. It has been shown \cite{Bhattacharya}
that the functions $U(z)$, $V(wz)$ and $Q(z)$ can be expressed in terms of
polylog functions, defined in the paragraph after Eq. (\ref{Basic}), as $U (z) =
\Phi (\alpha, \mu_3 z)$, $V (wz) = \Phi (1 - \gamma_{d=2}, \mu_2 w z)$ and $Q
(z) = 1 + \Phi (1 - \gamma_{1}, \mu_3 z)$, where $\mu_3$ and $\mu_2$ are $3d$-
and $2d$ - connective constants respectively. By making use of the inverse
Laplace transformation of $\Xi (z, w)$ with respect to $z$ (see, e.g.
\cite{Rudnick}) the (canonical with respect to the chain length $N$) partition
function is obtained as
\begin{equation}
\Xi_{N}(w) = \sum_{K=0}^{\infty} \: \Xi_{N,K} \: {w}^{K} =\exp[-N
\ln z^{*} (w)]
\label{GC}
\end{equation}
where $z^{*}(w)$ is a simple pole of $\Xi (z, w)$ in complex $z$-plane given by
equation $V(w z^*) U(z^*) = 1$, i.e. by Eq.(\ref{Basic}).
The (non-normalized) probability for the chain to have $K$ adsorbed monomers is
$P(K) \propto \Xi_{N,K} \: {\rm e}^{\epsilon K} = \exp[- {\cal F}(K)/k_BT]$,
where ${\cal F}(K)$ is the free energy at given $K$. It is convenient to
redefine the fugacity $w$ as $w \rightarrow \xi w$, (as well as $w_c
\rightarrow \xi w_c$) where $\xi$ is an arbitrary complex variable. Then the
probability $P(K)$ can be found as the coefficient of $\xi^K$ in the function
$\Xi_{N} (\xi w)$ expansion in powers of $\xi$. Therefore
\begin{equation}
P(K) = \exp[- {\cal F}(K)/k_BT] = \frac{1}{2\pi i} \: \displaystyle \oint
\dfrac{\exp[- N \ln z^{*} (\xi \: w)]}{\xi^{K+1}} \: d \xi
\label{Integral}
\end{equation}
where the contour of integration is a closed path in the complex $\xi$ plane
around $\xi =0$. (see e.g. \cite{Rudnick}). To estimate the integral in
Eq.(\ref{Integral}) we use the {\it steepest descent method} \cite{Rudnick}.
For large $N$ the main contribution to the integral in Eq. (\ref{Integral}) is
given by the saddle point $\xi = \xi_0$ of the integrand which is defined by the
extremum of the function $g(\xi) = - \ln z^{*}(\xi \: w) -
[(K+1)/N] \ln \xi$, i.e., by the condition
\begin{equation}
\dfrac{K+1}{\xi_0} = - \left. N \: \dfrac{\partial \ln z^{*}}{\partial
\xi}\right|_{\xi = \xi_0}
\label{SP}
\end{equation}
The integral is dominated by the term $\exp [- N \ln z^{*}(\xi_0 w) - (K+1)
\ln \xi_0]$. Another contribution comes from the
integration along the steepest descent line. As a result one obtains
\begin{eqnarray}
P(K) \propto \dfrac{\exp [- N \ln z^{*}(\xi_0 \: w) - (K+1) \ln
\xi_0]}{\sqrt{N\left[\left(\dfrac{N}{K}\right)
\left[\dfrac{\partial\ln z^{*}}{\partial \xi}\right]_{\xi = \xi_0}^2 -
\left.\dfrac{\partial^2 \ln z^{*}}{\partial \xi^2}\right|_{\xi = \xi_0}\right]}}
\label{Method}
\end{eqnarray}
The validity of the steepest descent method is ensured by the condition of
the second derivative $N g''(\xi_0)$ being large, which yields
\begin{eqnarray}
N g''(\xi_0) = N\left[\left(\dfrac{N}{K}\right)
\left[\dfrac{\partial\ln z^{*}}{\partial \xi}\right]_{\xi = \xi_0}^2 -
\left.\dfrac{\partial^2 \ln z^{*}}{\partial \xi^2}\right|_{\xi = \xi_0}\right]
\gg 1
\label{Valid}
\end{eqnarray}
A more explicit calculation whithin this method can be performed in the
vicinity of the critical point $\epsilon = \epsilon_c$.
\subsection{PDF of the number of adsorbed monomers close to the critical point
of adsorption}
In this case the explicit form of $\ln z^{*}$ is known \cite{Bhattacharya} and
after the redefinition of the fugacity, $w \rightarrow \xi w$, it reads
\begin{eqnarray}
\ln z^{*} (\xi w) = - a_1 (w - w_c)^{1/\phi} \xi^{1/\phi} - \ln \mu_3
\label{Z_Star}
\end{eqnarray}
where $a_1$ is a constant of the order of unity. The critical adsorption
fugacity $w_c = {\exp (\epsilon_c)}$ is defined by the equation
\begin{eqnarray}
\zeta (\alpha)\: \Phi (1 - \gamma_{d=2}, \mu_2 w_c/\mu_3) = 1
\label{Critical}
\end{eqnarray}
with $\zeta (x)$ denoting the Riemann zeta-function.
By using Eq.(\ref{Z_Star}) in Eq.(\ref{SP}) one arrives at the expression for
the saddle point
\begin{eqnarray}
\xi_0 = \left( \dfrac{K}{N}\right)^{\phi} \dfrac{a_2}{w - w_c}
\label{SP_Solution}
\end{eqnarray}
where $a_2=(\phi/a_1)^{\phi}$. Using Eq. (\ref{Z_Star}) and Eq.
(\ref{SP_Solution}) in
Eq. (\ref{Method}) then yields the expression for PDF
\begin{eqnarray}
P(K) \propto [(w - w_c) {\rm e}^{a_2}]^{K} \left(\dfrac{K}{N}\right)^{- \phi K
- 1/2} \: \dfrac{\mu_{3}^{N}}{N^{1/2}}
\label{Distr1}
\end{eqnarray}
For reasonably large $K$ and after normalization one arrives at the final
expression for the PDF
\begin{eqnarray}
P(K) = \dfrac{\eta^K}{{\cal C}(\eta) K^{\phi K} }
\label{Distr3}
\end{eqnarray}
where we have introduced the usual adsorption scaling variable $\eta = b_1 (w -
w_c)N^\phi$ ($b_1$ is a constant of the order of unity; see e.g. \cite{Hsu}) as
well as the normalization constant
${\cal C}$:
\begin{eqnarray}
{\cal C}(\eta) = \sum_{K=1}^{N} \: \dfrac{\eta^K}{K^{\phi K}}
\label{Normalization}
\end{eqnarray}
One can readily see that the width of the distribution increases with $w$
or with $\epsilon$. To this end one may directly calculate the fluctuation
variance as follows:
\begin{eqnarray}
\overline{(K - \overline{K})^2} = - N \dfrac{\partial^2 \ln z^{*}(w)}{\partial
(\ln w )^2} \propto N w (w/\phi - w_c) (w - w_c)^{1/\phi - 2}
\label{Variance_1}
\end{eqnarray}
where the expression for $\ln z^{*}(w)$ given by Eq.(\ref{Z_Star})
(where also $\xi=1$ ) has been used. Taking into account that $\phi \approx
0.5$, it becomes clear that the variance really grows with $w$.
The validity of the steepest descent method is ensured by the condition
Eq.(\ref{Valid}). Using Eq.(\ref{Z_Star}) and Eq. (\ref{SP_Solution}), one can
verify that this criterion holds when $N (w - w_c)^2 (K/N)^{1 - 2\phi} \gg 1$.
We recall that $\phi \approx 0.5$, so that $(w - w_c) N^{1/2} \gg 1$. In
result the criterion becomes
\begin{eqnarray}
N^{-1/2} \ll (w - w_c) \ll 1
\label{Criterion}
\end{eqnarray}
In the deep adsorption regime this condition might be violated. Nevertheless,
the steepest descent method could still be used there, provided that the
appropriate solution for $z^{*}(w)$ (see Eq.(\ref{Basic})) is chosen.
\subsection{The regime of deep adsorption}
In the deep adsorption regime one should use the the solution for $z^{*} (w)$
which was also discussed in ref. \cite{Bhattacharya}. Namely, in this case
\begin{eqnarray}
z^{*} (w) \approx \dfrac{1}{\mu_2 w} \left[1 - \left( \dfrac{\mu_3}{\mu_2
w}\right)^{1/(1-\lambda)} \right]
\label{Deep_Ads}
\end{eqnarray}
With Eq.~(\ref{Deep_Ads}) the mean value $\overline{K} = - N \partial \ln
z^{*}(w)/\partial \ln w$ can be written as
\begin{eqnarray}
\dfrac{\overline{K}}{N} = 1 - \dfrac{1}{1 - \lambda} \left(
\dfrac{\mu_3}{\mu_2 w}\right)^{1/(1-\lambda)}
\label{K}
\end{eqnarray}
thus $\overline{K}$ tends to $N$ with $w$ growing as it should be. The
variance of the fluctuations within the GCE then becomes
\begin{eqnarray}
\overline{(K - \overline{K})^2} = - N \dfrac{\partial^2 \ln
z^{*}(w)}{\partial (\ln w )^2} = \dfrac{N}{(1 - \lambda)^2}
\left(\dfrac{\mu_3}{\mu_2 w}\right)^{1/(1-\lambda)},
\label{Variance_2}
\end{eqnarray}
i.e., the fluctuations decrease when the adsorption energy grows. Comparison
of this result with the result given by Eq. (\ref{Variance_1}) leads to the
important conclusion that the fluctuations of the number of adsorbed monomers
first grow with $\epsilon$, attain a maximum, and finally decrease with
increasing surface adhesion $\epsilon$. The position of the maximum reflects the
presence of finite-size effects, and, as the chain length $N\to \infty$, this
maximum occurs at the CAP.
Consider now the steepest descent treatment for the deep adsorption regime. With
Eq.(\ref{Deep_Ads}) (after the rescaling $w \rightarrow \xi w$) in
Eq.(\ref{SP}), the saddle point becomes
\begin{eqnarray}
\xi_0 = b_2 \left(\dfrac{\mu_3}{\mu_2 w} \right) \left(1 -
\dfrac{K}{N}\right)^{-(1 - \lambda)}
\label{SP_Deep}
\end{eqnarray}
where $b_2 = (1 - \lambda)^{-(1 - \lambda)}$ (recall that $1-\lambda =
\gamma_{d=2} = 1.343$). The main contribution comes from the exponential term
in Eq.(\ref{Method}) which is given by
\begin{eqnarray}
\exp\left[-N \ln z^{*} (w \xi_0) - (K+1) \ln \xi_0\right] = (\mu_2 w)^{N}
\dfrac{\left(b_2 \mu_3 {\rm e}^{1-\lambda} N^{1-\lambda}/\mu_2
w\right)^{N-K}}{\left(N - K\right)^{(1-\lambda)(N-K)}}
\label{Exponential}
\end{eqnarray}
For $N-K \gg 1$ the expression for the non-normalized PDF takes on the form
\begin{eqnarray}
P(K) \propto \dfrac{\left(b_3 \mu_3 N^{1-\lambda}/\mu_2
w\right)^{N-K} }{(N-K)^{(1-\lambda) (N - K)} }
\end{eqnarray}
where $b_3$ is a constant of the order of unity. Normalization of this
distribution yields
\begin{eqnarray}
P(K) = \dfrac{\chi^{N-K}
}{{\cal R}(\chi) (N-K)^{(1-\lambda)(N - K)} }
\label{PDF_Deep}
\end{eqnarray}
where the parameter $\chi = b_3 (\mu_3/\mu_2 w) N^{1-\lambda}$ and the
normalization
constant
\begin{eqnarray}
{\cal R}(\chi) = \sum_{K=1}^{N-1} \dfrac{\chi^{N-K}
}{(N-K)^{(1-\lambda)(N - K)} }.
\label{R}
\end{eqnarray}
The validity condition, Eq.(\ref{Valid}), in the deep adsorption regime
(after substitution of Eq. (\ref{Deep_Ads}) into Eq.(\ref{Valid})) requires
\begin{eqnarray}
(N - K)^{3/2 - \lambda} \gg (\mu_3/\mu_2 w) N^{1 - \lambda},
\label{Valid_Deep}
\end{eqnarray}
i.e., $K$ should not be very close to $N$. In Fig.~\ref{pdf_theory}a we show the
PDF of the number of chain contacts, $P(K)$, for a free chain without pulling
and several adsorption strengths of the substrate. One can readily verify that
visually the shape of $P(K)$ resembles very much a Gaussian distribution at
moderate values of $\epsilon_c\approx 1.7 < \epsilon < 6.0 $. The PDF
variance goes through a sharp maximum at $\epsilon \gtrsim \epsilon_c$ and then
declines, as expected from Eq.~(\ref{Variance_2}).
One should note that in the $f$-ensemble (where the force $f$ and not the
distance $h$ acts as a controll parameter \cite{Bhattacharya}) the order
parameter $n = \overline{K}/N$ undergoes a jump at the detachment adsorption
energy $\epsilon_D$. This means that $N (\partial^2 \ln z^{*}/\partial
\epsilon^2)_{\epsilon_D} = - (\partial \overline{K}/\partial
\epsilon)_{\epsilon_D} \rightarrow - \infty$. Thus at the detachment point the
variance of the fluctuations $\overline{(K - \overline{K})^2} = - N (\partial^2
\ln z^{*}/\partial \epsilon^2)_{\epsilon_D} \rightarrow \infty$, which
practically means that for chains of a finite length the distribution at
$\epsilon = \epsilon_D$ becomes very broad, in sharp contrast to
Eq.~(\ref{Variance_2}). This has indeed been observed in our MC-simulation
results (see Fig.12 in ref. \cite{Bhattacharya}).
\subsection{$P(K)$ distribution in the subcritical regime $w < w_c$ of
underadsorption}
In the subcritical regime, $w < w_c$, the fraction of adsorbed points (order
parameter) $n = \overline{K}/N = 0$, in the thermodynamic limit. Nevertheless,
$\overline{K} \neq 0$ and one can examine the form of the PDF $P(K)$. At $w <
w_c$ the solution for $z^* (w)$ (the simple pole of $\Xi (z, w)$ in the complex
$z$-plane) does not exist because $V(w z) U(z) < 1$ (see Eq.(\ref{GCE})).
However, the tail GCE-partition function $Q(z) = 1 + \Phi (1- \gamma_1, \mu_3
z) \propto \Gamma (\gamma_1)/(1 - \mu_3 z)^{\gamma_1}$ has a {\em branch point}
at $z = 1/\mu_3$ (see Eq. (A 11) in ref. \cite{Bhattacharya}) which governs the
coefficient at $z^N$, i.e., the partition function $\Xi_{N}(w)$. The calculation
(following Section 2.4.3 in ref. \cite{Rudnick}) yields
\begin{eqnarray}
\Xi_{N}(w) = \dfrac{1 + \Phi (\lambda, \mu_2 w/\mu_3)}{1 - \zeta
(\alpha) \: \Phi (\lambda, \mu_2 w/\mu_3)} \: \mu_3^N \: N^{\gamma_{1}
- 1}
\label{Subcritical_GC}
\end{eqnarray}
where $\lambda = 1 - \gamma_{d=2}$ and we have also used that at $z = 1/\mu_3$
the loop and train GCE-partition
functions are $U(1/\mu_3)= \Phi (\alpha, 1) = \zeta
(\alpha)$ and $V(w/\mu_3) = \Phi (\lambda, \mu_2 w/\mu_3)$
respectively.
This expression has a pole at $w = w_c$ (cf. Eq. (\ref{Critical})) which yields
the coefficient of $w^K$ , i.e. $\Xi_{N,K}$. Recall that $\Xi_N(w) =
\sum_{K=0}^{\infty} \Xi_{N,K} w^K$, so that $P(K) \propto \Xi_{N,K} w^K$.
Expansion of the denominator in Eq. (\ref{Subcritical_GC}) around $w = w_c$
reveals the simple pole as follows
\begin{eqnarray}
\Xi_{N}(w) = \dfrac{[1 + 1/\zeta(\alpha)] \: \mu_3^N N^{\gamma_1 -
1}}{\zeta(\alpha) \: \Phi (-\gamma_{d=2}, \mu_2 w_c/\mu_3)} \left(
\dfrac{w_c}{w_c - w}\right).
\label{Pole}
\end{eqnarray}
In (\ref{Pole}) we have used the relationship $x (d/d x) \Phi (1-\gamma_{d=2},
x) = \Phi(- \gamma_{d=2}, x)$. The coefficient of $w^K$, i.e., $\Xi_{N,K}$, is
proportional to $w_c^{- K}$. Therefore $P(K) \propto (w/w_c)^{K} = \exp
[-(\epsilon_c - \epsilon)K]$. Taking the normalization condition
$\sum_{K=0}^{N} P(K) = 1$ into account, the final expression for $P(K)$ can be
recast in the form
\begin{eqnarray}
P(K) = \dfrac{1 - {\exp}[-(\epsilon_c - \epsilon)]}{1 - {\exp}[- N
(\epsilon_c - \epsilon)]} \: {\exp}[- K(\epsilon_c - \epsilon)],
\label{Exp}
\end{eqnarray}
i.e., $P(K)$ has a simple exponential form. The calculation of the average
$\overline{K} = \sum_{K=0}^{N} K P(K)$ leads to the simple result $\overline{K}
= [\exp (\epsilon_c - \epsilon) - 1]^{-1} \approx 1/(\epsilon_c - \epsilon)$,
i.e., $\overline{K} \rightarrow \infty$ at $\epsilon \rightarrow \epsilon_c$. On
the other hand we know that $\overline{K} = N^{\phi}$ at $\epsilon =
\epsilon_c$. In order to prevent a divergency at $\epsilon = \epsilon_c$, one
should incorporate an appropriate cutoff in the PDF given by Eq.(\ref{Exp}).
With this the distribution is given by
\begin{eqnarray}
P(K) = \dfrac{1 - {\exp}[-(\epsilon_c - \epsilon + 1/N^{\phi})]}{1 - {\exp}[- N
(\epsilon_c - \epsilon + 1/N^{\phi})]} \: {\exp}[- K(\epsilon_c - \epsilon +
1/N^{\phi})].
\label{Exp_Cutoff}
\end{eqnarray}
Thus the expression for the average number of adsorbed monomer has the correct
limit behavior, i.e.,
\begin{eqnarray}
\overline{K} = \dfrac{1}{\epsilon_c - \epsilon + 1/N^{\phi}}
\label{Average_Cutoff}
\end{eqnarray}
\subsection{Probability distribution function $P(K)$ in the $h$-ensemble}
Eventually we examine how the fixed chain-end hight $h$ affects the PDF of the
number of contacts $K$. To this end we refer again to Fig. \ref{Tearing}a where
an adsorbed chain with a fixed height $h$ of the last monomer is depicted. The
adsorbed chain consists of a tail of length $M$ and of an adsorbed part with $N
- M$ beads. One should bear in mind that $M$ is function of the control
parameters $h$ and of $w = \exp (\epsilon)$ given by
eqs.(\ref{Local_Equilibrium})
and (\ref{M_vs_h}).
The partition function of the adsorbed part is then given by
\begin{eqnarray}
\Xi_{N} (w) = \sum_{K=0}^{N} \: \Xi_{N, K} \: w^K = \exp\left\lbrace - [N -
M(h, w)] \ln z^{*} (w)\right\rbrace
\label{Adsorbed_Part}
\end{eqnarray}
where we took into account that the free energy of the adsorbed portion is given
as $F_{\rm ads} = k_B T [N - M(h, w)] \ln z^{*}(w)$ (see Sec. II B).
\begin{figure}[bht]
\hspace{-2.0cm}
\includegraphics[scale=0.3, angle=0]{PDF_theor.eps}
\hspace{0.50cm}
\includegraphics[scale=0.3, angle=0]{PDF_pull.eps}
\caption{\label{pdf_theory} (a) Probability Distribution Function of the number
of chain contacts with the grafting plane, $K$, for different degrees of
adhesion $\epsilon$ and no pulling force $f = 0$ as follows from eqs.
(\ref{Distr3}) and (\ref{PDF_Deep}). The variance of $P(K)$ is
shown in the inset for different $\epsilon$. (b) The same as in (a) for
$\epsilon = 1.75$ and different heights of the chain end $h$. The change of the
variance $\langle K^2\rangle - \langle K\rangle^2$ with $h$ is shown in the
inset.}
\end{figure}
As mentioned above, the PDF $P(K) \propto \Xi_{N, K} \: w^K $, so that by
means of rescaling $w \rightarrow \xi w$ and $w_c \rightarrow \xi w_c$ the PDF
can be found as the coefficient of $\xi^K$, i.e.,
\begin{equation}
P(K) = \frac{1}{2\pi i} \: \displaystyle \oint
\dfrac{\exp\left\lbrace - [N -
M(h, \xi w)] \ln z^{*} (\xi w)\right\rbrace}{\xi^{K+1}} \: d
\xi
\label{Coefficient}
\end{equation}
As before, the steepest descent method can be used to calculate the
integral in Eq. (\ref{Coefficient}). However, in this case the calculations are
more complicated and we have relegated most of them to Appendix B. As may be
seen there, the saddle point equation can not be solved analytically in
the general case but could be treated iteratively for relatively small heights
$h$.
One can readily see that for $h = 0$ Eq. (\ref{PDF_Pull}) reduces to
Eq.(\ref{Distr3}). The PDF, following from Eq. (\ref{PDF_Pull}) is shown in
Fig. \ref{pdf_theory}b for $\epsilon = 1.75$ and several values of the height
$h$. It can be seen that the curve for $h = 0$ coincides with the curve for
$\epsilon = 1.75$ in Fig. \ref{pdf_theory}a as it should be. Evidently, both
the mean value $\overline{K}$ and $\overline{K^2} - \overline{K}^2$ decline
with growing $h$.
\section{Monte Carlo Simulation Model}\label{MC_model}
We use a coarse grained off-lattice bead-spring model \cite{MC_Milchev} which
has proved rather efficient in a number of polymers studies so far. The system
consists of a single polymer chain tethered at one end to a flat impenetrable
structureless surface. The surface interaction is described by a square well
potential,
\begin{equation}\label{ads_pot}
U_w(z) = \begin{cases}
\epsilon, & z < r_c \\
0, & z \ge r_c
\end{cases}
\end{equation}
The strength $\epsilon$ is varied from $2.0$ to $5.0$ while the
interaction range $r_c = 0.125$. The effective bonded interaction is described
by the FENE (finitely extensible nonlinear elastic) potential:
\begin{equation}
U_{FENE}= -K(1-l_0)^2ln\left[1-\left(\frac{l-l_0}{l_{max}-l_0} \right)^2 \right]
\label{fene}
\end{equation}
with $K=20$, and the fully stretched-, mean-, and minimum bond lengths
$l_{max}=1, l_0 =0.7, l_{min} =0.4$. The nonbonded interactions between monomers
are described by the Morse potential:
\begin{equation}\label{Morse}
\frac{U_M(r)}{\epsilon_M} =\exp(-2\alpha(r-r_{min}))-2\exp(-\alpha(r-r_{min}))
\end{equation}
with $\alpha =24,\; r_{min}=0.8,\; \epsilon_M/k_BT=1$. In few cases, needed to
clarify the nature of the polymer chain resistance to stretching, we have taken
the nonbonded interactions between monomers as purely repulsive by shifting the
Morse potential upward by $\epsilon_M$ and removing its attractive branch for
$r \ge r_{min}$.
We employ periodic boundary conditions in the $x-y$ directions and impenetrable
walls in the $z$ direction. The lengths of the studied polymer chains are
typically $64$, and $128$. The size of the simulation box was chosen
appropriately to the chain length, so for example, for a chain length of $128$,
the box size was $256 \times 256 \times 256$ . All simulations were carried out
for constant position of the last monomer $z$-coordinate, that is, in the fixed
height ensemble. The the fluctuating force $f$, exerted on the last bead by the
rest of the chain was measured and average over about $2000$ measurements.
The standard Metropolis algorithm was employed to govern the moves with self
avoidance automatically incorporated in the potentials. In each Monte Carlo
update, a monomer was chosen at random and a random displacement attempted with
$\Delta x,\;\Delta y,\;\Delta z$ chosen uniformly from the interval $-0.5\le
\Delta x,\Delta y,\Delta z\le 0.5$. If the last monomer was displaced in $z$
direction, there was an energy cost of $-f\Delta z$ due to the pulling force.
The transition probability for the attempted move was calculated from the change
$\Delta U$ of the potential energies before and after the move was performed as
$W=exp(-\Delta U/k_BT)$. As in a standard Metropolis algorithm, the attempted
move was accepted, if $W$ exceeds a random number uniformly distributed in the
interval $[0,1)$.
As a rule, the polymer chains have been originally equilibrated in the MC method
for a period of about $ 5 \times 10^5$ MCS after which typically $500$
measurement runs were performed, each of length $2 \times 10^6$ MCS. The
equilibration period and the length of the run were chosen according to the
chain length and the values provided here are for the longest chain length.
\section{Monte Carlo simulation results}\label{MC_results}
In order to verify the theoretical predictions, outlined in Section
\ref{Theory}, we carried out extensive Monte Carlo simulations with the
off-lattice model, defined in Section \ref{MC_model}. In these simulations we
fix the end monomer of the polymer chain at height $h$ above the adsorbing
surface, and measure the (fluctuating) force, needed to keep the last bead at
distance $h$, as well as the corresponding fraction of adsorbed monomers $n$.
These computer experiments are performed at different strengths $\epsilon$ of
the adsorption potential, Eq.~(\ref{ads_pot}).
\vspace{0.5cm}
\begin{figure}[bht]
\hspace{-2.0cm}
\includegraphics[scale=0.33]{op_theor.eps}
\hspace{0.50cm}
\includegraphics[scale=0.34]{op_MC.eps}
\caption{\label{OP} (a) Order parameter (fraction of adsorbed monomers) $n$
variation with changing height $h/l_0N$ of the fixed chain-end for polymers of
length $N = 128$ and different adsorption strength $\varepsilon/k_BT$. (b)
Variation of $n$ with $\varepsilon/k_BT$ for different fixed positions of the
chain-end $h/l_0N$ as it is seen from MC-data. Insets show the resulting $n-
\epsilon$ relationship at
several fixed heights $h$.}
\end{figure}
In Figs.~\ref{OP}a,b we compare the predicted dependence of the order parameter
$n$ on the (dimensionless) height $h/l_0N$ at several values of $2.0 \le
\epsilon \le 5.0$ with the results from MC simulations. Note, that the critical
point of adsorption $\epsilon_c \approx 1.69$ so we take our measurements above
the region of critical adsorption. Typically, both in the analytic results,
Fig.~\ref{OP}a, and
in the MC-data, Fig.~\ref{OP}b, for $N=128$, one recovers the predicted linear
decrease of $n$ with growing $h$. Finite-size effects lead to some rounding of
the simulation data (in Fig.~\ref{OP}b these effects are seen to be larger for
$N=64$ than for $N=128$) when $n\to 0$ so that the height of detachment $h_D$ is
determined from the intersection of the tangent to $n(h)$ and the $x-$axis where
\begin{figure}[htb]
\includegraphics[scale=0.4]{phase.eps}
\caption{Phase diagram showing the dependence of the critical height of polymer
detachment from the substrate, $h_D/l_0 N$, with the relative strength of
adsorption $(\varepsilon - \varepsilon_{c})/k_BT $, where $\varepsilon_{c}/k_BT$
is the critical point of adsorption at zero force. The theoretical curves
follow Eq. (\ref{Detachment_Line}).}
\label{phase_diag}
\end{figure}
$n=0$. Evidently, with growing adsorption strength, $\epsilon$, larger height
$h_D$ is needed to detach the polymer from the substrate. Thus, one may
construct a phase diagram for the desorption transition, which we show in
Fig.~\ref{phase_diag}. The theoretical prediction is given by eq.
(\ref{Detachment_Line}).
In the insets of Fig.~\ref{OP}a,b we also show the variation of the fraction of
adsorbed segments $n$ with adsorption strengths $\epsilon$ for several heights
$20 \le h \le 50$ of a chain with $N = 128$. It is evident that, apart from the
rounding of the MC data at $n \to 0$, one finds again good agreement between the
behavior, predicted by Eq.~(\ref{Order_Parameter}), and the simulation results.
The gradual change of $n$ in the whole interval of possible variation of $h$
suggests a pseudo-continuous phase transition, as pointed out in the end of
Section \ref{Theory}, Eq.~(\ref{Order_Parameter1}). Of course, if $h$ is itself
expressed in terms of the measured pulling force, one would again find that $n$
changes abruptly with varying $f$ at some threshold value $f_D$, indicating a
first order transformation from an adsorbed into a desorbed state of the polymer
chain.
\vspace{0.9cm}
\begin{figure}[htb]
\includegraphics[scale=0.4]{op_raw.eps}
\caption{PDF of the order parameter (fraction of contacts with the plane) for
different adsorption strength $\epsilon$ at zero force. The critical adsorption
point (CAP)
$\epsilon_{c} \approx 1.67$. The change of the variance $\langle n^2\rangle -
\langle n\rangle^2$ with varying $\epsilon$ is displayed in the inset.
}
\label{pdf_MC}
\end{figure}
It has been pointed out earlier by Skvortsov et al.\cite{Skvortsov} that, while
both the fixed-force and the fixed-height ensembles are equivalent as far as the
mean values of observables such as the fraction of adsorbed monomers and other
related quantities are concerned, this does not apply to some more detailed
properties like those involving fluctuations. Therefore, it is interesting to
examine the fluctuations of the order parameter, $n$, for different values of
our control parameter $h$, and compare them to theoretical predictions for
$P(K)$ from Section \ref{Theory}. First we compare the order parameter
distribution $P(n)$ for zero force, Fig.~\ref{pdf_MC}, obtained from our
computer experiment, to that, predicted by Eqs.~(\ref{Distr3}),
(\ref{PDF_Deep}), (\ref{Exp_Cutoff}), and displayed in Fig.~\ref{pdf_theory}a.
It is evident from Fig.~\ref{pdf_MC} that for free chains at different strengths
of adhesion there is a perfect agreement between analytical and simulational
results. For rather weak adsorption $\epsilon = 1.3\pm 1.6 < \epsilon_c=1.67$ in
the subcritical regime, one can verify from Fig.~\ref{pdf_MC} that $P(n)$
gradually transforms from nearly Gaussian into exponential distribution, as
expected from Eq.~(\ref{Exp_Cutoff}). For $\epsilon > \epsilon_c$ the
distribution width $\langle (n - \langle n \rangle)^2\rangle$ grows and goes
through a sharp maximum in the vicinity of $\epsilon_c$, and then drops as
$\epsilon$ increases further - compare insets in Fig.~\ref{pdf_MC} and
Fig.~\ref{pdf_theory}a.
\vspace{0.6cm}
\begin{figure}[htb]
\hspace{-2.0cm}
\includegraphics[scale=0.33]{ophistogram.eps}
\hspace{0.50cm}
\includegraphics[scale=0.33]{moments.eps}
\caption{\label{PDF_OP} (a) Probability distribution $P(n)$ of the order
parameter $n$ (i.e., the fraction of adsorbed monomers) for $N = 128$ and
$\epsilon = 3.0$ at different heights of the chain-end $h$ over the grafting
plane. In the inset we show $P(n)$ at the detachment line $h_D = 54.3$. (b)
Variation of the second- and third central moments of $P(n)$ with $h$. The
maximum of $\langle (n - \langle n \rangle)^3\rangle$ is reached at $h = h_D$.}
\end{figure}
Let us consider now PDF in the presence of pulling.
In Fig.~\ref{PDF_OP}a we display the distribution $P(n)$ measured in the MC
simulations for different heights $h$ and constant adsorption energy $\epsilon =
3.0$. One can readily verify from our results that far enough from the
detachment line, $h < h_D$, the shape of $P(n)$ looks like Gaussian and that the
second moment, $\langle (\Delta n)^2\rangle$, remains unchanged with varying
height $h$. Of course, when $h \to h_D$, the maximum of $P(n)$ shifts to lower
values of $n$. Only in the immediate vicinity of $h_D$, where $n \to 0$ and the
fluctuations strongly decrease, one observes a significant deviation from the
Gaussian shape - cf. the inset in Fig.~\ref{PDF_OP}a. The latter is illustrated
in more detail in Fig.~\ref{PDF_OP}b where we show the measured variation of the
second moment, $\langle (\Delta n)^2\rangle$, and that of the third moment,
$\langle (n - \langle n \rangle)^3\rangle$ with increasing height $h$. The
deviation from Guassinity in $P(n)$, measured by the
deviation of the third moment from zero, is localized in the vicinity of the
detachment height $h_D$. The corresponding theoretical prediction for the
relatively weak adsorption strength is depicted in Fig. \ref{pdf_theory}b. It
can be seen that with increasing $h$ the almost Gaussian distribution tends to
Poisson-like one. Also the fluctuations decrease with $h$ in accordance with
MC-findings.
The force $f$, exerted by the chain on the end-monomer, when the latter is kept
at height $h$ above the surface, is one of the main properties which can be
measured in experiments carried out within the fixed-height ensemble. Note that
$f$ has the same magnitude and opposite sign, regarding the force, applied by
the experimentalist. The variation of the force $f$ with increasing height $h$
is shown in Fig.~\ref{force}a for several values of the adsorption potential
$2.0 \le \epsilon \le 5.0$. In Fig.~\ref{force}a we distinguish between two
contributions to the total force $f$, acting on the end bead. The first stems
from the quasi-elastic forces of the bonded interaction (FENE) whereas the
second contribution is due to the short-range (attractive) interactions between
non-bonded monomers (in our model - the Morse potential). A typical feature of
the $f - h$ relationship, namely, the existence of a broad interval of heights
$h$ where the force remains constant (a plateau in the force) is readily seen in
Fig.~\ref{force}a. With growing strength of adsorption $\epsilon$ the length of
this plateau as well as the magnitude of the plateau force increase. Note, that
for $\epsilon = 0$ no plateau whatsoever is found. Upon further extension (by
increasing $h$) of the chain, the plateau ends and the measured force starts
to grow rapidly in magnitude - an effect, caused by a change of the chain
conformation itself in the entirely desorbed state.
A closer inspection of Fig.~\ref{force}a reveals that the non-bonded
contribution to $f$, which is generally much weaker than the bonded one,
behaves differently, depending on whether the forces between non-nearest
neighbors along the backbone of the chain are purely repulsive, or contain an
attractive branch. While for strong adsorption, $\epsilon \ge 3.0$, a plateau
is observed even for attractive non-bonded interactions, for weak adsorption,
$\epsilon \le 2.0$, an increase of the non-bonded contribution at $h/l_0N
\approx 0.35$, (seen as a {\em minimum} in Fig.~\ref{force}a) is observed. This
effect is entirely missing in the case of purely repulsive nonbonded
\begin{figure}[bht]
\hspace{-2.0cm}
\includegraphics[scale=0.30]{f_h_plateau.eps}
\hspace{0.50cm}
\includegraphics[scale=0.40]{forceb_b.eps}
\caption{\label{force} (a) Variation of the two components to the total force,
exerted by the chain on the end-monomer which is fixed at (dimensionless) height
$h/l_0N$ for different adsorption potentials $2.0 \le \epsilon/k_BT \le 5.0$:
bonding interactions (full symbols) and non-bonding Morse interactions (empty
symbols). In the inset the same is shown for a neutral plane $\epsilon = 0.0$
for purely repulsive monomers (triangles) and for such with a weak Morse
attraction (circles). (b) Variation of the total force (plateau hight) exerted
by the AFM tip on the chain-end for chain length $N=128$ with the
relative strength of adsorption $(\varepsilon - \varepsilon_c)/k_BT$.}
\end{figure}
interactions - see the inset in Fig.~\ref{force}a where the contributions
from bonded and non-bonded interactions are shown for a neutral surface
$\epsilon = 0$. If one plots the magnitude of the measured force at the
plateau against the corresponding value of the the adsorption potential,
$\epsilon$, one may check the theoretical result, Eq.~(\ref{Local_Equilibrium})
- Fig.~\ref{force}b. Evidently, the theoretical predictions about $f_{\rm p}$ agree
well with the observed variation of the detachment force in the MC simulation,
both within the BS- or FJBV models, as long as only the excluded volume
interactions in the MC data are taken into account. If the total contribution to
$f_{\rm p}$, including also attractive non-bonded interactions in the chain, is
depicted - black triangles in Fig.~\ref{force}b - then the agreement with the
theoretical curves deteriorates since the latter do not take into account the
possible presence of attractive non-bonded interactions.
\vspace{0.6cm}
\begin{figure}[bht]
\hspace{-2.0cm}
\includegraphics[scale=0.32]{h_f_Pincus.eps}
\hspace{0.90cm}
\includegraphics[scale=0.31]{force_B.eps}
\caption{\label{f-h} (a) Variation of the total applied force $f$ with growing
height of the end monomer in terms of Pincus reduced variables, $f l_0
N^{\nu}/k_B T$ versus $h/l_0N^\nu$, for a polymer with purely repulsive
nonbonded forces for $N=64,\;128$. (b) The same as in (a) but in terms of
reduced units $f l_0/k_B T$ versus $h/l_0N$ for purely repulsive (empty
symbols)
as well as for usual Morse potential (full symbols) of nonbonded interactions
between monomers. The FJBV-model results, Eq.~(\ref{L}), is
shown by a solid line. Arrows indicate the unperturbed gyration radius positions
$R_g/N$ for $N=64,\;128$}
\end{figure}
The $f - h$ relationship, which gives the equation of state of the stretched
polymer, may be derived within one of the different theoretical models, e.g.,
that of BS-, Eq.~(\ref{Deformation}), or FJBV-model, Eq.~(\ref{L}), as
mentioned in Section \ref{Theory}. Which of these theoretical descriptions is
the more adequate can be decided by comparison with experiment. In
Fig.~\ref{f-h}a,b, we present such comparison by plotting our simulation data
using different normalization for the height $h$. From Fig.~\ref{f-h}a it
becomes evident that the data from our computer experiment for $N=64$ and
$N=128$ collapse on a single curve, albeit this collapse only holds as long as
$h/l_0N^\nu \le 3.0$ for the BS-model while it fails for stronger
stretching. In contrast, this collapse works well for all values of $h$,
provided the height is scaled with the contour length of the chain $N$, rather
than with $N^\nu$, as in the FJBV model - Fig.~\ref{f-h}b - regardless of
whether a purely repulsive, or the full Morse potential (which includes also an
attractive part) of interactions is involved. The analytical expression,
Eq.~(\ref{L}), is found to provide perfect agreement with the simulation data
for strong stretching, $h/l_0N \ge 0.4$. From the simulation data on
Fig.~\ref{f-h} one may even verify that the force $f$ goes through zero at some
height $h > 0$ and then turns negative, provided one keeps the chain end very
close to the grafting surface (cf. eq.(\ref{Deformation})).
\section{Summary}
In the present work we have treated the force-induced desorption of a
self-avoiding polymer chain from a flat structureless substrate both
theoretically and by means of Monte Carlo simulation within the constant-height
ensemble. The motivation for this investigation has been the necessity to
distinguish between results obtained in this ensemble and results, derived in
the constant-force ensemble, considered recently\cite{Bhattacharya}, as far as
both ensembles could in principle be used by experimentalists. We demonstrate
that the observed behavior of the main quantity of interest, namely, the
fraction of adsorbed beads $n$ (i.e., the order parameter of the phase
transition) with changing height $h$ differs qualitatively from the variation of
the order parameter when the pulling force is varied. In the constant-height
ensemble one observes a steady variation of $n$ with changing $h$ whereas in
the constant-force ensemble one sees an abrupt jump of $n$ at a particular
value of $f_D$, termed a detachment force. However, this should not cast doubts
on the genuine first-order nature of the phase transition which can be recovered
within the constant-height ensemble too, provided one expresses the control
parameter $h$ in terms of the average force $f$. This equivalence has been
studied extensively for Gaussian chains by Skvortsov et al. \cite{Skvortsov} who
noted that ensemble equivalence does not apply to fluctuations of the pertinent
quantities too.
Indeed, in our earlier study\cite{PRER} we found diverging variance of the PDF
$P(n)$ at $f_D$ whereas in our present study the fluctuations of the order
parameter are observed to stay finite at the transition height $h_D$. These
findings confirm theoretical predictions based on analytic results which we
derive within the GCE-approach. Within this approach we have explored two
different theoretical models for the basic force - extension relationship,
namely, the bead-spring (BS) model as well as that of a Freely-Jointed
Bond-Vectors (FJBV) model. Our simulation results indicate a good agreement
between theory and computer experiment.
\section*{Acknowledgments}
We are indebted to A. Skvortsov for useful discussions during the preparation
of this work. A.~Milchev thanks the Max-Planck Institute for Polymer Research in
Mainz, Germany, for hospitality during his visit at the institute. A.~Milchev
and V.~Rostiashvili acknowledge support from the Deutsche Forschungsgemeinschaft
(DFG), grant No. SFB 625/B4.
\begin{appendix}
\section{Freely jointed bond vectors model}
The deformation law in the overstretched regime (when the chain deformation
is close to its saturation) could be treated better within the FJBV model.
Consider a tethered chain of length $N$ with one end anchored at the
origin of the coordinates and an external force $f_N$ acting on the free end of
the chain. The corresponding deformation energy reads
\begin{eqnarray}
U_{\rm ext } = - f_N \: r_{N}^{\perp} = - f_N \: \sum_{i=1}^{N} \: b_i
\cos \theta_i
\end{eqnarray}
where $r_{N}^{\perp}$ is the $z$-coordinate (directed perpendicular to the surf
ace) of the chain end, $b_i$ and $\theta_i$ are the length and the polar angle
of the $i$-th bond vector respectively. The corresponding partition function of
the FJBV model is given by
\begin{eqnarray}
Z_{N} (f_N) &=& \int \prod_{i=1}^{N} d \phi_i \sin \theta_i d \theta_i
\exp\left(\frac{f_N}{k_B T} \sum_{i=1}^{N} b_i \cos \theta_i\right)\nonumber\\
&=& (4\pi)^{N} \prod_{i=1}^{N} \left(\frac{k_B T}{b_i f_N} \right) \cosh
\left(\frac{b_i f_N}{k_B T} \right)
\label{Partition_Func}
\end{eqnarray}
The average orientation of the $i$-th bond vector can be calculated as
\begin{eqnarray}
<\cos \theta_i> = \left(\frac{k_B T}{f_N}\right) \: \frac{\partial}{\partial
b_i}
\ln Z_N (f_N) = \coth \left(\frac{b_i f_N}{k_B T}\right) - \left(\frac{k_B
T}{b_i f_N
}\right)
\label{Cos}
\end{eqnarray}
From Eq.(\ref{Cos}) the chain end mean distance from the surface, $h$, is given
by
\begin{eqnarray}
h = \sum_{i=1}^{N} b_i <\cos \theta_i> = b N {\cal L} \left(\frac{b f_N }{k_B T}
\right)
\label{H_vs_Force}
\end{eqnarray}
where we have taken into account that the lengths of all bond vectors are of
equal length, $b_i = b$, and ${\cal L} (x) = \coth (x) - 1/x$ is the Langevin
function. This leads to the force - distance relationship
\begin{eqnarray}
\frac{b f_N}{k_B T} = {\cal L}^{-1}\left(\frac{h}{b N}\right) = \begin{cases}
\frac{1}{1-h/b
N} & \text{, at} \:\: h/b N \leq 1\\
\frac{2 h}{b N} & \text{, at} \:\: h/b N \ll 1
\end{cases}
\label{Inversion}
\end{eqnarray}
which we use in Sec.II. The notation ${\cal L}^{-1}(x)$ stands for the inverse
Langevin function.
\section{Calculation of PDF in the $h$-ensemble}
Using Eq.(\ref{Z_Star}) for $\ln z^{*} (w)$ as well as Eqs.
(\ref{Local_Equilibrium}) and (\ref{M_vs_h}) (for the BS-model), one obtains
an expression for the tail length
\begin{equation}
M(h, \xi \: w) = \dfrac{h/l_0}{\left[a_1^\phi (w - w_c)
\: \xi \right]^{(1-\nu)/\phi} }
\label{Tail_Length}
\end{equation}
The saddle point (SP) equation in this case reads (cf. Eq.(\ref{SP}))
\begin{equation}
\dfrac{K+1}{\xi_0} = - \left. \left[N - M(h, \xi_0 w) \right] \:
\dfrac{\partial \ln z^{*}}{\partial
\xi}\right|_{\xi = \xi_0} + \left. \dfrac{\partial M}{\partial \xi} \right|_{\xi
= \xi_0} \: \ln z^{*}(\xi_0 w)
\label{SP_General}
\end{equation}
Taking into account Eqs. (\ref{Z_Star}) and (\ref{Tail_Length}), after
introducing the notation $y = (w - w_c) \xi_0$, the SP-equation can be recast
into
\begin{equation}
\left( \dfrac{K}{N}\right) = y^{1/\phi} - B_1 \left( \dfrac{h}{l_0 N}\right)
y^{\nu/\phi} - B_2 \left( \dfrac{h}{l_0 N}\right) \dfrac{\ln
\mu_3}{y^{(1-\nu)/\phi}}\;,
\label{SP_Recast}
\end{equation}
where $B_1$ and $B_2$ are constants of the order of unity. In the particular
case $h = 0$ Eq.(\ref{SP_Recast}) goes back, as expected, to
Eq.(\ref{SP_Solution}). Eq. (\ref{SP_Recast}) can be solved iteratively for $h
\ll l_0 N$ as
\begin{equation}
y = \left[ \left( \dfrac{K}{N}\right) + B_1 \left( \dfrac{K}{N}\right)^\nu
\left( \dfrac{h}{l_0 N}\right) + B_2 \ln \mu_3 \left(
\dfrac{N}{K}\right)^{1-\nu} \left( \dfrac{h}{l_0 N}\right) \right]^{\phi}
\label{Iteration}
\end{equation}
As before, the main contribution in the integral given by Eq.
(\ref{Coefficient}) reads
\begin{eqnarray}
P(K) &\propto& \exp \left\lbrace- \left[N - M(h, \xi_0 w)\right] \ln
z^{*}(\xi_0 w) - (K+1) \ln \xi_0 \right\rbrace \nonumber\\
&=& \left\lbrace (w -
w_c) N^{\phi} \exp \left[1 - \dfrac{(h/l_0 N)}{(K/N)^{1-\nu} \left[\rho
(K/N, h/l_0 N)\right]^{1-\nu}}\right] \rho (K/N, h/l_0 N)\right\rbrace^K \:
\left[K \rho (K/N, h/l_0 N) \right]^{-\phi K}
\end{eqnarray}
where one introduces the notation
\begin{eqnarray}
\rho \left(K/N, h/l_0 N \right) \equiv 1 + B_1 \dfrac{h/l_0
N}{(K/N)^{1-\nu}} + B_2 \ln \mu_3 \: \dfrac{(h/l_0 N)}{(K/N)^{2-\nu}}
\label{Notation}
\end{eqnarray}
After normalization, the final expression for the PDF reads
\begin{eqnarray}
P(K) = \dfrac{1}{{\cal W}(\eta, h/l_0 N)} \: \left\lbrace \eta \: \exp
\left[1 - \dfrac{(h/l_0 N)}{(K/N)^{1-\nu} \left[\rho
(K/N, h/l_0 N)\right]^{1-\nu}}\right] \rho (K/N, h/l_0 N)\right\rbrace^K \:
\left[K \rho (K/N, h/l_0 N) \right]^{-\phi K}
\label{PDF_Pull}
\end{eqnarray}
where $\eta = (w - w_c) N^\phi$ as before and the normalization constant reads
\begin{eqnarray}
{\cal W}(\eta, h/l_0 N) = \sum_{K=1}^{N} \: \left\lbrace \eta \: \exp
\left[1 - \dfrac{(h/l_0 N)}{(K/N)^{1-\nu} \left[\rho
(K/N, h/l_0 N)\right]^{1-\nu}}\right] \rho (K/N, h/l_0 N)\right\rbrace^K \:
\left[K \rho (K/N, h/l_0 N) \right]^{-\phi K}
\label{Normalization_Pull}
\end{eqnarray}
Again one can readily see that for $h = 0$ Eq.(\ref{PDF_Pull}) reduces to
Eq.(\ref{Distr3}). The PDF, following from Eq.~(\ref{PDF_Pull}) is shown in
Fig.~\ref{pdf_theory}b for several values of the height $h$. Evidently, both
the mean value $\overline{K}$ and the variance $\overline{K^2}-\overline{K}^2$
decline with growing $h$.
\end{appendix}
|
1,116,691,500,889 | arxiv | \section{introduction}
Trisecting an arbitrary angle and doubling the cube by ruler and compass are two of the famous problems posed by Greeks which are known to be unsolvable. The first proof of the impossibility for the geometric constructions is attributed to Wantzel (1837). For history on this subject, we refer the reader to \cite[pp. 25-26]{S}.
It is also well known that the trisection and duplication are possible if it is allowed to use one or more conic sections in addition to ruler and compass(\cite{V}). The old constructions used hyperbolas and parabolas as conic sections. Recently, Hummel \cite{H} and Gibbins--Smolinsky \cite{GS} independently found the constructions using ellipses.
Gibbins and Smolinsky, at the end of their paper, asked if one can reduce the number of types of ellipses for the involved constructions. More specifically Hummel, in the final remark of his paper, asked if all points constructible from ellipses are constructible using the same ellipse in addition to a ruler and a compass. The goal of this paper is to show that the answer is Yes. More generally, we prove that every conic-constructible point can be obtained using a ruler and a compass, together with a single fixed non-degenerate conic different from a circle (Theorem \ref{main}). It can be pointed out that our result resembles the Poncelet-Steiner's theorem which states that any construction with a ruler and a compass can be accomplished by a ruler together with a fixed circle and its center(see \cite[\S3.6]{E}).
The precise meaning of the conic-constructibility will be reviewed in the next section. In \S3, we will state the main result and give a proof.
\section{Conic-constructible points}
The point $(x,y)$ in the plane $\mathbb{R}^2$ is identified with a complex number $x + i y \in \mathbb{C}$.
Starting from the initial set $P=P_0$ of points, perform the drawings:
(i) Given two points in $P$, draw a line through the two points using a ruler.
(ii) Given two points $z_1$ and $z_2$ in $P$, draw a circle centered at $z_1$ with radius $|z_1-z_2|$ using a compass.
Let $P_1$ be the set of points by adjoining all the intersections of lines and circles from the above drawings. Now, replacing $P_0$ by $P_1$ and running the above process, we get the set $P_2$. Repeating this inductively, we get a sequence of sets
\[
P= P_0 \subset P_1 \subset P_2 \subset \cdots.
\]
A point is called \textit{constructible from $P$} if it is inside $P_\infty := \bigcup_{i=0}^\infty P_i$. A point is called \textit{constructible} if it is constructible from the initial set $\{ 0,1 \}$ in $\mathbb{C}$. It is well known that a point is constructible if and only if it lies in a subfield $F$ of $\mathbb{C}$ which has a finite sequence of subfields starting from rational numbers:
\[
\mathbb{Q} = F_0 \subset F_1 \subset F_2 \subset \ldots \subset F_n = F
\]
such that $[F_{i+1}: F_i] = 2$ for each $i$.
Now in addition to the above drawings (i) and (ii), consider another one:
(iii) Draw all parabolas, ellipses, and hyperbolas having foci in $P$, directrix lines which are obtained from (i), and eccentricities equal to the length of a segment $\overline{z_1 z_2}$ for some $ z_1, z_2 \in P$.
Let $P=Q_0$ and let $Q_1$ be the set of points by adjoining all the intersections of lines and circles and conics from the drawings (i), (ii), and (iii). Replacing $Q_0$ by $Q_1$ and running the same kind of process, we get $Q_2$. Again by induction, we get a sequence of sets
\[
P= Q_0 \subset Q_1 \subset Q_2 \subset \cdots.
\]
A point is called \textit{conic-constructible from $P$} if it is inside $Q_\infty := \bigcup_{i=0}^\infty Q_i$. A point is called \textit{conic-constructible} if it is conic constructible from the initial set $\{ 0,1 \}$ in $\mathbb{C}$.
Note that for any $P$, both the set of constructible points derived from $P$ and the set of conic-constructible points derived from $P$ are subfields of $\mathbb{C}$, because the process (i) and (ii) are already enough to produce the complex numbers given by the operations $+, -, \times, \div$. The $x$-coordinates of the constructible (or conic-constructible) points derived from $P$ again form a subfield of $\mathbb{R}$, which are called \textit{constructible (or conic-constructible) numbers derived from $P$}.
Videla found a useful criterion:
\begin{proposition} \label{videla} {\rm (\cite[Theorem 1, 2]{V})} \\
(1) The set of conic-constructible points forms the smallest subfield of $\mathbb{C}$ containing 0, 1, and $i$ which is closed under conjugation, square roots and cube roots.\\
(2) A point in $ \mathbb{C}$ is conic-constructible if and only if it is contained in a subfield of $\mathbb{C}$ of the form $\mathbb{Q}(\alpha_1, \alpha_2, \ldots, \alpha_n)$, where $\alpha_1^{k_1} \in \mathbb{Q}$ and $\alpha_i^{k_i} \in \mathbb{Q}(\alpha_1, \ldots, \alpha_{i-1})$ for $2 \le i \le n$, where $k_i \in \{2,3\}$ for each $i = 1, 2, \ldots, n$. \qed
\end{proposition}
The point of Proposition \ref{videla} (2) is that taking square and cube roots of a given complex number are precisely the constructions required to obtain all the conic-constructible points. In other words, providing the constructions for real cube roots and trisections of angles are sufficient to get all conic-constructible points.
Let $K$ be a subfield of $\mathbb{R}$ such that every positive number $x \in K$ has a square root in $K$. A line passing through two points of $K^2$ is called a \textit{line in $K$}. A circle is called a \textit{circle in $K$} if its center is in $K^2$ and it passes through a point in $K^2$. Similarly, a conic (parabola, ellipse, or hyperbola) is called a \textit{conic in $K$} if its foci are in $K^2$, its directrix line is in $K$, and its eccentricity is in $K$.
Recall the following useful fact.
\begin{proposition} \label{useful} \ Let $E$ be a non-degenerate conic (different from a circle), defined by the equation $ax^2 + bxy +cy^2 +dx+ ey+f = 0$. Then $E$ is in $K$ if and only if the coefficients $a,b, c, d, e, f$ are in $K$ (after suitable rescaling).
\label{fact} \end{proposition}
\begin{proof}
The proof for the case of ellipse is given in \cite[Proposition 3]{GS} and the argument can be modified to work also for parabolas and hyperbolas.
\end{proof}
\section{result}
We introduce two more notions: $C$-constructible points and $e$-constructible points. Let $C$ be a non-degenerate conic in the field of constructible numbers. For the initial set $P \subset \mathbb{C}$, a point is \textit{$C$-constructible from $P$} if it is obtained by the same process as conic-constructible points except that the conics in step (iii) are confined to the fixed one: $C$.
We call an ellipse or a hyperbola of eccentricity $e>0$ to be ``regular'' if it is given by the equation
\[
(1-e^2) (x-a)^2 + (y-b)^2 = \lambda^2
\]
for some $a, b, \lambda \in \mathbb{R}$. Also, we call a parabola to be regular if it is of the form
\[
x = \lambda(y-a)^2 +b
\]
for some $a, b, \lambda \in \mathbb{R}$.
Let $e>0$ be a constructible number. A point is \textit{$e$-constructible from $P$} if it is obtained by the same process as conic-constructible points except that
(1) the conics in step (iii) are confined to the regular ones of eccentricity $e$, and
(2) the intersections of two conics, neither of which is a circle, are not adjoined. \\
In other words, (1) a conic is drawn only when it is regular with eccentricity $e$, and (2) for any drawn conic different from a circle, only the intersections with lines and circles are counted for the next stage.
\\
As before when $P = \{0,1 \}$, we say $C$-constructible and $e$-constructible respectively, for short. By definition, if $C$ is a regular conic with eccentricity $e$, then
\begin{center}
$C$-constructible \ $\Rightarrow$ \ $e$-constructible \ $\Rightarrow$ \
conic-constructible.
\end{center}
\begin{lemma} \label{e}
Let $e>0$ be any constructible number.
Every conic-constructible point is $e$-constructible.
\end{lemma}
\begin{proof}
Let $z \in \mathbb{C}$ be a conic-constructible point. By Proposition \ref{videla}, $z \in \mathbb{Q} (\alpha_1, \alpha_2, \ldots, \alpha_n)$, where either $\alpha_i^2$ or $\alpha_i^3$ is contained in $ \mathbb{Q} (\alpha_1, \ldots, \alpha_{i-1})$ for each $i$.
We will show that $z$ is $e$-constructible by induction on $n$. Suppose that all the points in $H:= \mathbb{Q} (\alpha_1, \alpha_2, \ldots, \alpha_{n-1})$ are $e$-constructible.
If $\alpha_n^2 \in H$, then $\alpha_n$ is constructible from $H$.
Now suppose $\alpha_n^3 = re^{i \theta} \in H$. Let $q = \cos \theta$ and let $K$ be the field of constructible numbers derived from $0,1$ and $r, q$.
First consider the intersection of the following circle and conic:
\[
x^2 + y^2 - rx - y = 0 \ \ \text{and} \ \ (1-e^2)x^2 + y^2 - rx - (1-e^2)y = 0 .
\]
By Proposition \ref{fact}, these are a circle and a conic in $K$.
From these two equations, we get $x^4 - rx = 0$. Hence the $x$-coordinate of the intersection points other than the origin corresponds to the cube root of $r$.
Next for $q = \cos \theta$, by triple-angle formula, $\cos \left( {\theta}/{3} \right)$ is a real solution to the equation
\[
4x^3 - 3x - q = 0.
\]
To get this equation, we intersect the following circle and conic:
\[
x^2 + y^2 - \frac{q}{4}x - \frac{7}{4}y = 0 \ \ \text{and} \ \ (1-e^2)x^2 + y^2- \frac{q}{4}x - \left(\frac{7}{4} - e^2 \right)y = 0 .
\]
Again, these are a circle and a conic in $K$.
Note that the conics in the above are regular of eccentricity $e$.
From these intersections, we constructed the numbers $\sqrt[3]{r}$ and $ \cos (\theta/3)$, and eventually the point $\alpha_n$. Thus so far we have shown that $\alpha_n$ is an $e$-constructible point derived from 0, 1, and $r, q$. Since we assumed that all the points in $H$ are $e$-constructible and $\alpha_n^3 = re^{i \theta} \in H$, $r$ and $q= \cos \theta$ are also $e$-constructible.
Therefore, $\alpha_n$ is $e$-constructible and thus all the points in $\mathbb{Q}(\alpha_1, \alpha_2, \ldots, \alpha_{n})$ are $e$-constructible.
\end{proof}
Now comes our main result: Every conic-constructible point can be constructed by using a single fixed conic in addition to a ruler and a compass.
\begin{theorem} \ Let $C$ be any non-degenerate conic, different from a circle, in the field of constructible numbers.
Then every conic-constructible point is $C$-constructible.
\label{main} \end{theorem}
\begin{proof} Let $e>0$ be any constructible number.
By Lemma \ref{e}, it suffices to prove that every $e$-constructible point is $C$-constructible.
We first introduce a hierarchy on the $e$-constructible points starting from the field of constructible points $\mathbb{F}_0$:
\[
\mathbb{F}_0 \subset \mathbb{F}_1 \subset \mathbb{F}_2 \subset \cdots .
\]
Draw all the regular conics in $\mathbb{F}_0$ of eccentricity $e$. Let $Q_{1}^e$ be the set of points by adjoining to $\mathbb{F}_0$ all the intersections of lines and circles with any of the drawn regular conics. Let $\mathbb{F}_1$ be the field of constructible points derived from $Q_1^e$.
Inductively, for each $k \ge 0$,
let $Q_{k+1}^e$ be the set of points by adjoining to $\mathbb{F}_k$ all the intersections of lines and circles with any of the drawn regular conics in $\mathbb{F}_k$ of eccentricity $e$. And define $\mathbb{F}_{k+1}$ as the field of constructible points derived from $Q_{k+1}^e$. Then the set of $e$-constructible points coincides with $\bigcup_{k=0}^\infty \mathbb{F}_k$.
Now we prove that every $e$-constructible point is $C$-constructible by induction on $k$. Certainly the constructible points $\mathbb{F}_0$ are $C$-constructible. Now assume that all the points in $\mathbb{F}_k$ are $C$-constructible. Take any point $z \in \mathbb{F}_{k+1} \setminus \mathbb{F}_k$ which is obtained as an intersection of a circle $R_k$ in $\mathbb{F}_k$ and a regular conic $C_k$ in $\mathbb{F}_k$ of eccentricity $e$.
We may assume that $C$ is regular. Indeed, we may rotate $C$ by an angle $\theta$ where $\cos \theta$ and $\sin \theta$ are constructible, to send it to a regular conic, because $C$ is in the field of constructible numbers. From now on, we assume that $C$ is regular.
Since both $C$ and $C_k$ are in $\mathbb{F}_k$ of the same eccentricity, $C_k$ can be obtained from $C$ by a magnification by a factor $\lambda >0$ and a translation in $a+bi$, where $a,b$ and $\lambda$ are in $\mathbb{F}_k$. Therefore, the intersection point $z \in R_k \cap C_k$ can be obtained as follows:
(a) Translate the circle $R_k$ by a point in $\mathbb{F}_k$ and magnify by a factor in $\mathbb{F}_k$ to get $R_k'$.
(b) Intersect $R_k'$ with $C$ to get the corresponding intersection point $z'$.
(c) Reverse the process (a) to get $z$ from $z'$.
Since we assumed that all of the points in $\mathbb{F}_k$ are $C$-constructible, the intersection $z' \in R_k' \cap C$ is a $C$-constructible point, because $R_k'$ is a circle in $\mathbb{F}_k$. We conclude that $z$ is a $C$-constructible point.
Since every point in $\mathbb{F}_{k+1}$ is constructible from $\mathbb{F}_k$ and the intersection points $z \in R_k \cap C_k$ obtained as in the above, it is $C$-constructible. This completes the proof.
\end{proof}
\vspace{0.5cm}
\noindent \textit{Acknowledgment.} This paper grew out from an R\&E project ``A study on the numbers constructible from conic sections'', which was supported by Seoul Science High School during March-December 2012.
\vspace{0.5cm}
|
1,116,691,500,890 | arxiv |
\section{Introduction}
Pre-trained language models such as ELMo~\cite{peters2018deep}, GPT~\cite{radford2018improving}, BERT~\cite{devlin2018bert}, and ERNIE~\cite{sun2019ernie} have proved to be effective for improving the performances of various natural language processing tasks including sentiment classification \cite{socher2013recursive}, natural language inference \cite{bowman2015large}, text summarization \cite{hu2015lcsts}, named entity recognition \cite{sang2003introduction} and so on. In general, pre-trained language models are learned on a large amount of text data in a self-supervised manner, and then fine-turned on downstream tasks or directly deployed through zero/few-shot learning without task-specific fine-tuning. Such pre-trained language models have become the new paradigm for natural language processing tasks.
In the past year or two, one of the important trends of pre-trained language models is their increasing model size, which leads to lower perplexity in pre-training and better performances on downstream tasks. Megatron-LM \cite{Megatron-LM}, with one billion parameters, is proposed for language understanding using a simple but efficient intra-layer model parallel approach, which achieves the state-of-the-art results on several datasets. T5 \cite{T5} explores the limits of pre-trained models with 10 billion parameters, but soon the record was broken by the GPT-3 model~\cite{gpt-3} with 175 billion parameters which has a good performance under the few-shot or even zero-shot settings. Soon afterwards, Switch-Transformer~\cite{switch-transformer} is proposed as the world's first trillion-parameter pre-trained language model.
However, these large-scale pre-trained language models with hundreds of billions of parameters are trained on plain texts. For example, the 175-billion-parameter GPT-3 is trained on a corpus with 570GB filtered texts from Common Crawl. Such raw texts lack explicit representation of knowledge such as linguistic knowledge and world knowledge. In addition, most large-scale models are trained in an auto-regressive way, but \cite{devlin2018bert} shows that such models demonstrate poorer performance with traditional fine-tuning when adapting to downstream language understanding tasks.
In this work, to solve the problem caused by a single auto-regressive framework and to explore the performance of knowledge enhanced pre-trained models with large-scale parameters, we propose a unified framework called ERNIE 3.0 to train large-scale knowledge enhanced models on a 4TB corpus consisting of plain texts and a large-scale knowledge graph by fusing the auto-regressive network and the auto-encoding network. The proposed ERNIE 3.0 can handle both natural language understanding tasks and natural language generation tasks through zero-shot learning, few-shot learning or fine-tuning. Furthermore, the proposed framework supports the introduction of various customized tasks at any time. These tasks share the same encoding networks and are trained through multi-task learning. This method makes the encoding of lexical, syntactic and semantic information across different tasks possible. Moreover, when given a new task, our framework could incrementally train the distributed representations based on the previous training parameters, with no need to train them from scratch.
In summary, our contributions are as follows:
\begin{itemize}
\item We propose a unified framework ERNIE 3.0, which combines auto-regressive network and auto-encoding network so that the trained model can handle both natural language understanding and generation tasks through zero-shot learning, few-shot learning or fine-tuning.
\item We pre-train large-scale knowledge enhanced models with 10 billion parameters and evaluate them with a series of experiments on both natural language understanding and natural language generation tasks. Experimental results show that ERNIE 3.0 consistently outperforms the state-of-the art models on 54 benchmarks by a large margin and achieves the first place on the SuperGLUE~\cite{wang2019superglue} benchmark.
\end{itemize}
\section{Related Work}
\subsection{Large-scale Pre-trained Models}
Since BERT \cite{devlin2018bert} is proposed as a powerful language model for natural language understanding, pre-trained language models have attracted more and more attention and become the new paradigm for natural language processing. One of the research trends is increasing model size, which leads to lower perplexity and better performance~\cite{kaplan2020scaling}. As a result, many large-scale pre-trained models have been proposed in the past two years. T5 model \cite{T5} is proposed to push the performance for both natural language understanding and natural language generation tasks with 11 billion parameters. The T5 model converts all text-based language tasks into a text-to-text format by a unified framework and fully explores the effectiveness of pre-training objectives, architectures, unlabeled datasets, transfer approaches, and other factors. After the T5 model, GPT-3 \cite{gpt-3}, which includes 175 billion parameters, is proposed to achieve an amazing performance on a wide range of tasks under the few-shot and zero-shot settings. Specifically, GPT-3 is an auto-regressive language model, 10x more than its predecessor, GPT-2, proposed by \cite{radford2019language}. However, GPT-3 shows a lack of common sense, exists biases and privacy issues in the tests \cite{rise_risk_gpt3}. \cite{switch-transformer} have proposed a 1 trillion parameters model named Switch Transformer with simplifying MoE~\cite{jacobs1991adaptive, jordan1994hierarchical} routing algorithm to improve model with less communication and computational costs, and \cite{switch-transformer} also proposed a large scale distributed training solution to tackle the problem of training complexity, communication costs, and training instability.
Besides the models mentioned above, more non-English large models have been proposed recently. \cite{zhang2020cpm} released a 2.6 billion parameters Chinese Pre-trained Language Model (CPM) with generative pre-training on large-scale Chinese training data and the model structure was inspired by \cite{gpt-3}. \cite{zhang2021cpm} have released a 11 billion parameters model CPM-2. To accelerate the pre-training based on existing PLMs instead of training models from scratch, the knowledge inheritance techniques have been introduced and during the fine-tuning stage, prompt tuning is involved to better exploit the knowledge within the pre-trained model. \cite{lin2021m6} have proposed a cross-modal pre-training method called M6({M}ulti-{M}odality to {M}ulti-{M}odality {M}ultitask {M}ega-Transformer) including 100 billion parameters for unified pre-training on multiple modalities data. \cite{zeng2021pangu} proposed a 200 billion parameters auto regressive language model named PangGu-$\alpha$ which is trained on a cluster of 2048 Ascend 910 AI processors with distributed training techniques including data parallelism, op-level model parallelism, pipeline model parallelism, optimizer model parallelism and re-materialization. Except for those Chinese large-scale models, a Korean 204 billion parameters language model named HyperCLOVA \cite{HyperClova} has been proposed, and its volume of machine-learned data in Korean was 6,500 times larger than GPT-3’s. From what has been discussed above, observations now suggest that large-scale pre-trained models have attracted more and more attention from industry and academia.
\subsection{Knowledge Enhanced Models}
Pre-trained language models capture syntactical and semantic knowledge from large-scale corpus, but lack world knowledge. Recently, several works have attempted to incorporate world knowledge in pre-trained language models. The typical form of world knowledge is a knowledge graph. Many works (\cite{zhang2019ernie, peters2019knowledge, he2019integrating}) integrate entity and relation embedding from knowledge graph in pre-trained language models. WKLM~\cite{xiong2019pretrained} replaced entity mentions in the original documents with names of other entities of the same type and train the models to distinguish the correct entity mention from randomly chosen ones. KEPLER~\cite{wang2021kepler} optimized the models with knowledge embedding and mask language model objectives to align the world knowledge and language representation into the same semantic space. CoLAKE~\cite{sun2020colake} integrated the language context and the knowledge context in a word-knowledge graph and jointly learned contextualized representation for language and knowledge with the extended mask language model objective. Another existing form of world knowledge is the extra annotation of large-scale data. ERNIE 1.0~\cite{sun2019ernie} introduced phrase masking and named entity masking and predicts the whole masked phrases and named entities to help the model learn the dependency information in both local contexts and global contexts. CALM~\cite{zhou2020pre} teached models to detect and revised a corrupted sentence with the incorrect ordering of concepts and to distinguish truth sentences from less plausible ones via two kinds of self-supervised pre-training tasks. K-Adapter\cite{wang2020k} utilized adapters trained on different knowledge sources with extra annotations to distinguish where the knowledge comes from.
\section{ERNIE 3.0}
\begin{figure*}[ht]
\centering
\includegraphics[width=1.0\textwidth]{ernie-framework.png}
\caption{The framework of ERNIE 3.0.} \label{label:ernie 3.0 framework}
\end{figure*}
A significant improvement has been achieved on various natural language processing tasks for knowledge enhanced pre-trained models with the base or large model size, such as ERNIE, ERNIE 2.0 and SpanBERT~\cite{joshi2020spanbert}, in which the base/large model size represent 12/24 layers Transformer respectively. In order to explore the effectiveness of knowledge enhanced large-scale pre-trained model, we propose the ERNIE 3.0 framework to pre-train model on massive unsupervised corpus including plain texts and knowledge graph. Furthermore, we employ various types of pre-training tasks to enable the model to learn the different levels of knowledge consisting of valuable lexical, syntactic and semantic information more effectively, in which the pre-training tasks spread three task paradigms, that is natural language understanding, natural language generation and knowledge extraction. Therefore, ERNIE 3.0 innovatively designs a \textbf{Continual Multi-Paradigms Unified Pre-training Framework} to enable the collaborative pre-training among multi-task paradigms. The explicit introduction of ERNIE 3.0 will be explained in the following sections.
\subsection{Overview of ERNIE 3.0 Framework}
The Framework of the ERNIE 3.0 is shown in Figure \ref{label:ernie 3.0 framework}, which can be widely used for pre-training, fine-tuning and zero/few-shot learning. Unlike the prevalent unified pre-training strategy of employing a shared Transformer network for different well-designed cloze tasks and utilizing specific self-attention masks to control what context the prediction conditions on, ERNIE 3.0 designs a new \textbf{\textit{Continual Multi-Paradigms Unified Pre-training Framework}}. We believed that the different task paradigms of natural language processing depend on identical underlying abstract features consistently, such as lexical information and syntactic information, but the requirements of top-level concrete features are incompatible, in which the natural language understanding tasks have the disposition to learn the semantic coherence while natural language generation tasks expect further contextual information. Therefore, inspired by the classical model architecture of multi-task learning, in which the lower layers are shared across all tasks while the top layers are task-specific, we proposed the ERNIE 3.0 to enable the different task paradigms to share the underlying abstract features learned in a shared network and utilizing the task-specific top-level concrete features learned in their own task-specific network respectively. Furthermore, in order to help the model efficiently learn the lexical, syntactic and semantic representations, ERNIE 3.0 exploits the continual multi-task learning framework introduced in ERNIE 2.0~\cite{sun2020ernie}. As for the application of different kinds of downstream tasks, we will first initialize the ERNIE 3.0 with the combination of parameters of a pre-trained shared network and corresponding task-specific networks for different task paradigms, and then execute the corresponding follow-up procedure using data from specific tasks.
We refer to the backbone shared network and task-specific networks as the \textbf{Universal Representation Module} and \textbf{Task-specific Representation Module}s in ERNIE 3.0. Specifically, the universal representation network plays the role of universal semantic features extractor (for example, it can be a multi-layer Transformer), in which the parameters are shared across all kinds of task paradigms, including natural language understanding, natural language generation and so on. And the task-specific representation networks undertake the function of extracting the task-specific semantic features, in which the parameters are learned by task-specific objectives. ERNIE 3.0 not only enables the model to distinguish the task-specific semantic information among different task paradigms, but also mitigates the dilemma that large-scale pre-trained models are difficult to implement with limited time and hardware resources, in which ERNIE 3.0 permits the models to only update the parameters of a task-specific representation network during the fine-tuning phase. Specifically, ERNIE 3.0 employs the collaborative architecture of a Universal Representation Module and two Task-specific Representation Modules, namely natural language understanding (NLU) specific representation module and natural language generation (NLG) specific representation module.
\subsubsection{Universal Representation Module}
ERNIE 3.0 uses a multi-layer Transformer-XL \cite{dai2019transformer} as the backbone network like other pre-trained models such as XLNet \cite{yang2019xlnet}, Segatron \cite{DBLP:journals/corr/abs-2004-14996} and ERNIE-Doc \cite{ding2020ernie}, in which Transformer-XL is similar to Transformer but introduces an auxiliary recurrence memory module to help modelling longer texts. We refer to the backbone as Universal Representation Module and it is shared across all the task paradigms. Proverbially, the Transformer can capture the contextual information for each token in the sequence via self-attention and generate a sequence of contextual embedding. It is evident that the larger the scale of Transformer model, the stronger its capacity to capture and store up various semantic information with different levels. Therefore, ERNIE 3.0 sets the universal representation module with a larger size to enable the model to effectively capture universal lexical and syntactic information from training data by learning various pre-training tasks of different paradigms. And what needs special attention is that the memory module is only valid for natural language generation tasks while controlling the attention mask matrices.
\subsubsection{Task-specific Representation Module}
Similar to the basic shared representation module, the task-specific representation module is also a multi-layer Transformer-XL, which is used to capture the top-level semantic representations for different task paradigms. ERNIE 3.0 sets the task-specific representation module to a manageable size, that is a base model size, instead of the multi-layer perceptron or shallow Transformer commonly used in multi-task learning, which will produce three obvious benefits, the first is that the base network has a stronger ability to capture semantic information than multi-layer perceptron and shallow Transformer; the second is that the task-specific networks with base model size enable ERNIE 3.0 to distinguish the top-level semantic information among different task paradigms without significantly increasing the parameters of a large-scale model; finally, the smaller model size of a task-specific network than a shared network would lead to realizable practical applications for large scale pre-trained model when only fine-tuning on the task-specific representation module. ERNIE 3.0 constructs two task-specific representation modules, that is NLU-specific representation module and NLG-specific representation module, in which the former is a bi-directional modeling network while the latter is a uni-directional modeling network.
\subsection{Pre-training Tasks}
\label{sec:pre-training tasks}
We construct several tasks for various task paradigms to capture different aspects of information in the training corpora and make the capacity of understanding, generation and reasoning available to pre-trained model.
\subsubsection{Word-aware Pre-training Tasks}\label{sec:Word-aware pretrain-task}
\textbf{Knowledge Masked Language Modeling} \,\,
ERNIE 1.0~\cite{sun2019ernie} proposed an effective strategy to enhance representation through knowledge integration, namely Knowledge Integrated Masked Language Modeling task. It introduced phrase masking and named entity masking that predict the whole masked phrases and named entities to help the model learn the dependency information in both local contexts and global contexts.
\textbf{Document Language Modeling} \,\,
Generative pre-training models usually utilize traditional language model (such as GPT~\cite{radford2018improving}, GPT-2~\cite{radford2019language}) or sequence-to-sequence language model (such as BART~\cite{lewis2020bart}, T5~\cite{T5}, ERNIE-GEN~\cite{xiao2020erniegen}) as the pre-training task, the latter trains on the network with an auxiliary decoder structure. ERNIE 3.0 opt for traditional language model as the pre-training task to abate the network complexity and heighten the effectiveness of unified pre-training. In addition, to enable the NLG network of ERNIE 3.0 to model longer text, we introduce the Enhanced Recurrence Memory Mechanism proposed in ERNIE-Doc~\cite{ding2020ernie}, which can model a larger effective context length than traditional recurrence Transformer by changing the shifting-one-layer-downwards recurrence to the same-layer recurrence.
\subsubsection{Structure-aware Pre-training Tasks}\label{sec:Structure-aware pretrain-task}
\textbf{Sentence Reordering} \,\,
Sentence reordering task, which is introduced in ERNIE 2.0~\cite{sun2020colake}, aims to train the model to learn the relationship between sentences by reorganizing permuted segments. At length, a given paragraph is randomly split into 1 to m segments during pre-training and all of the combinations are shuffled by a random permuted order. Then, the pre-trained model is asked to reorganize these permuted segments, modeled as a k-class classification problem where \begin{math} k=\sum_{n=1}^{m} n!\end{math}.
\textbf{Sentence Distance} \,\,
Sentence distance task, an extension of traditional next sentence prediction (NSP) task, is widely used in various pre-trained models to enhance their ability to learn the sentence-level information, which can be modeled as a 3-class classification problem. The three categories represent that the two sentences are adjacent, nonadjacent but in the same document and from two different documents respectively.
\begin{figure*}[ht]
\centering
\includegraphics[width=1.0\textwidth]{ernie-knowledge.pdf}
\caption{Universal Knowledge-Text Prediction.}
\end{figure*}
\subsubsection{Knowledge-aware Pre-training Tasks}\label{sec: pretrain-task}
\textbf{Universal Knowledge-Text Prediction} \,\,
To incorporate knowledge into one pre-trained language model, we introduce universal knowledge-text prediction (UKTP) task, which is an extension of knowledge masked language modeling. While knowledge masked language modeling only requires unstructured texts, universal knowledge-text prediction task requires both unstructured texts and knowledge graphs. The universal knowledge-text prediction task is illustrated in Figure 2. Given a pair of triple from knowledge graph and the corresponding sentence from encyclopedia, we randomly mask relation in triple or words in a sentence. To predict the relation in the triple, the model needs to detect mentions of head entity and tail entity and determine semantic relationship that holds between them in the corresponding sentence. The essence of this process is similar to the distant supervision algorithm~\cite{mintz2009distant} in relation extraction tasks. The distant supervision algorithm assume that if two entities participate in a relation, any sentence that contain those two entities might express that relation. Meanwhile, to predict words in the corresponding sentence, the model not only considers the dependency information in the sentence, but also logical relationship in the triple. Specifically, the procedure of obtaining pairs of a triple and this corresponding sentence is as follows: given a document from encyclopedia, we first find the candidate triples in the knowledge graph whose mentions of head entity or tail entity is title of the document, and then select triples from candidate triples whose mentions of head entity and tail entity are mentioned in the same sentence in the document.
ERNIE 3.0 trains the NLU network through knowledge masked language modeling to improve the capacity of capturing the lexical information, trains the sentence reordering task and the sentence distance discerning task to strengthen the ability of capturing the syntactic information, and finally optimizes the model with the universal knowledge-text prediction task to improve knowledge memorization and reasoning. Meanwhile, ERNIE 3.0 trains the NLG network with the document language modeling task to enable various generation styles.
\subsection{Pre-training Process}
\subsubsection{Pre-training Algorithm}
Progressive training was originally proposed to improve stability, which starts from an efficient and small model and gradually increase the capacity \cite{simonyan2014very}. Recent study leverages this paradigm to accelerate model training. As large-scale pre-training keeps advancing the state-of-the-art(\cite{devlin2018bert}, \cite{radford2018improving}), their overwhelming computational consumption becomes the major burden towards further developing more powerful models(\cite{radford2019language}). Preliminary application of progressive training has been made on Transformer pre-training. BERT(\cite{devlin2018bert}) designs a two-stage training with a reduced sequence length for the first 90\% of updates. \cite{radford2019language} also gradually increase the batch size linearly from a small value to the full value. \cite{efficientnetv2} also notice that changing the regularization factors (e.g. \cite{srivastava2014dropout}, \cite{zhang2017mixup}) stage-wise with
respect to the input size can speed up training networks. To further improve convergence speed of the training process, we propose to adjust the training regularization factors in a more comprehensive and smooth way by progressively and simultaneously increasing the training factors including the input sequence length, the batch size, the learning rate and the dropout rate. In fact, it is common that Transformer models adopts the learning rate warm-up strategy to increase training stability and our improved progressive learning strategy is compatible to the existing strategy.
\subsubsection{Pre-training Data}
\begin{table}[]
\centering
\resizebox{1\textwidth}{!}{
\begin{tabular}{@{}lccccccccccc@{}}
\toprule \toprule
\textbf{Corpus} & \textbf{ERNIE 2.0} & \textbf{Search} & \textbf{Web} & \textbf{QA-long} & \textbf{QA-short} & \textbf{Novel} & \textbf{Poetry\&Couplet} & \textbf{Medical} & \textbf{Law} & \textbf{Fin} & \textbf{KG} \\ \midrule
\# of tokens & 17.8B & 42.4B & 314.7B & 33.8B & 0.1B & 96.4B & 46.5M & 17.8B & 16.2B & 0.6B & 0.7B \\
multiplier & 20 & 7 & 1 & 3 & 40 & 1 & 20 & 1 & 1 & 10 & 10 \\ \midrule
\multicolumn{12}{l}{\# tokens of context length in each percentile using ERNIE-3.0 wordpiece tokenizer} \\ \midrule
50\% & 135 & 75 & 793 & 184 & 15 & 2063 & 30 & 314 & 1162 & 843 & 16 \\
95\% & 1257 & 827 & 2757 & 1168 & 22 & 3652 & 88 & 983 & 4587 & 1572 & 44 \\ \bottomrule \bottomrule
\end{tabular}}
\caption{Statistics of Pre-training Datasets.}
\label{tab:pre-dataset}
\end{table}
To ensure the success of the pre-training of ERNIE 3.0, we construct a large-scale, wide-variety and high-quality Chinese text corpora amounting to 4TB storage size in 11 different categories. To our best knowledge, this is currently the largest Chinese pre-training corpora compared with CLUECorpus2020~\cite{xu2020clue} (100GB), Chinese multi-modal pre-training data~\cite{lin2021m6} (300GB), WuDaoCorpus2.0 used by CPM-2~\cite{zhang2021cpm} (2.3TB Chinese data and 300GB English data) and PanGu Corpus~\cite{zeng2021pangu} (1.1TB).
In detail, we build the corpus for ERNIE 3.0 based on that from ERNIE 2.0 (including baike, wikipedia, feed and etc), Baidu Search (including Baijiahao, Zhidao, Tieba, Experience), Web text, QA-long, QA-short, Poetry~\footnote{\url{https://www.luge.ai/text-generation/chinese-poetry.html\#_1-chinese-poetry}}\&Couplet~\footnote{\url{https://github.com/v-zich/couplet-clean-dataset}}, Domain-specific data from medical, law and financial area and Baidu knowledge graph with more than 50 million facts. To improve the data quality, we adopt the following pre-processing strategies:
\begin{itemize}
\item Deduplication is conducted on different granularities including character level, paragraph level and document level. On the character level, we replace consecutive identical characters (i.e., spaces, tabs, exclamation mark, question mark and etc) with one single character. One the paragraph level, we replace two identical consecutive paragraphs consisting of $N$ sentences with one single paragraph where $0<N<100$. The two aforementioned deduplication strategies are critical for ERNIE 3.0 to generate non-repeating contents. At last, we adopted Message Digest Algorithm5 (MD5) to filter duplicate documents by comparing the sum of the MD5 of top-3 longest sentences from each document.
\item Sentences with less than 10 words are filtered since they may be problematic or incomplete ones which contains limited semantic information for model pre-training.
\item We further conduct sentence segmentation using regular expressions and word segmentation based on Baidu's word segmentation tool. This helps ERNIE 3.0 to learn better sentence boundary and named entity knowledge during pre-training.
\end{itemize}
Then, each dataset is multiplied by a user-defined multiplier number to increase the data diversity after truncating the data for NLU-network pre-training.
\subsubsection{Pre-training Settings}
Both the universal representation module and the task-specific representation modules of ERNIE 3.0 uses the Transformer-XL\cite{dai2019transformer} structure as the backbone. For the universal representation module, we adopt a structure with 48 layers, 4096 hidden units and 64 heads. For the task-specific representation modules, we adopt a structure with 12 layers, 768 hidden units and 12 heads. The total parameter of universal representation module and task-specific representation modules is 10 billion. The activation function used is GeLU\cite{hendrycks2016gaussian}.
The maximum sequence length of context and the memory length of language generation is set to 512 and 128, respectively.
The total batch size of all pre-training tasks is set to 6144.
We use Adam\cite{kingma2014adam} with learning rate of 1e-4, $\beta_1=0.9$, $\beta_2=0.999$, L2 weight decay of 0.01, learning rate warmup over the first 10,000 steps and linear decay of the learning rate.
In the first 10,000 steps, we also use the progressive learning to speedup convergence in the initial stage of pre-training. The model is trained for a total of 375 billion tokens with 384 NVDIA v100 GPU cards and is implemented on PaddlePaddle framework.
By virtue of parameter sharding used in \cite{rajbhandari2020zero,ramesh2021zero}, we manage to reduce the memory usage of our model and address the problem of the total parameter of model exceeding the memory of a single GPU card.
\section{Experiments}
We compare the performance of ERNIE 3.0 with the state-of-the-art \footnote{the previous state-of-the-art results are all from the public single model that we can find.} pre-training models through fine-tuning on both natural language understanding tasks (in Sec.~\ref{sec:nlu-tasks}) and natural language generation tasks (in Sec.~\ref{sec:nlg-tasks}), and zero-shot learning (in Sec.~\ref{sec:zero-shot-tasks})\footnote{The previous SoTA results of ERNIE 2.0 and RoBERTa-wwm-ext on corresponding datasets are reproduced by ourselves, except for the datasets that already have released pre-trained results.}.
\subsection{Evaluation Tasks}\label{sec:evaluation-tasks}
We executed extensive experiments on 54 NLP tasks to evaluate the fine-tuning and zero-shot learning performances of the models.
\subsubsection{Natural Language Understanding Tasks}
45 datasets belonging to 14 kinds of natural language understanding tasks are used in our experiments, as follows:
\begin{itemize}
\item \textbf{Sentiment Analysis}: NLPCC2014-SC~\footnote{\url{http://tcci.ccf.org.cn/conference/2014/pages/page04_dg.html}}, SE-ABSA16\_PHNS~\footnote{\url{http://alt.qcri.org/semeval2016/task5/}}, SE-ABSA16\_CAME, BDCI2019~\footnote{\url{https://www.datafountain.cn/competitions/350}}.
\item \textbf{Opinion extraction}: COTE-BD~\cite{li2018cote}, COTE-DP~\cite{li2018cote}, COTE-MFW~\cite{li2018cote}.
\item \textbf{Natural Language Inference}: XNLI~\cite{conneau2018xnli}, OCNLI~\cite{xu2020clue}, CMNLI~\cite{xu2020clue}.
\item \textbf{Winograd Schema Challenge} CLUEWSC2020~\cite{xu2020clue}.
\item \textbf{Relation Extraction}: FinRE~\cite{li2019finre}, SanWen~\cite{xu2017discourse}.
\item \textbf{Event Extraction}: CCKS2020~\footnote{\url{http://sigkg.cn/ccks2020/?page_id=69}}.
\item \textbf{Semantic Similarity}: AFQMC~\cite{xu2020clue}, LCQMC \cite{liu2018lcqmc}, CSL~\cite{xu2020clue}, PAWS-X~\cite{yang2019paws}, BQ Corpus~\cite{chen2018bq}.
\item \textbf{Chinese News Classification}: TNEWS~\footnote{\url{https://github.com/aceimnorstuvwxz/toutiao-text-classfication-dataset}}, IFLYTEK~\cite{co2019iflytek}, THUCNEWS~\footnote{\url{http://thuctc.thunlp.org/}}, CNSE~\cite{liu2018matching}, CNSS~\cite{liu2018matching}.
\item \textbf{Closed-Book Question Answering}: NLPCC-DBQA~\footnote{\url{http://tcci.ccf.org.cn/conference/2016/dldoc/evagline2.pdf}}, CHIP2019, cMedQA~\cite{zhang2017cmedqa}, cMedQA2~\cite{zhang2018multi}, CKBQA~\footnote{\url{https://github.com/pkumod/CKBQA}}, WebQA~\cite{li2016webqa}.
\item \textbf{Named Entity Recognition}: CLUENER~\cite{xu2020clue}, Weibo~\cite{peng2015weibo}, OntoNotes~\cite{weischedel2011ontonotes}, CCKS2019~\footnote{\url{https://www.biendata.xyz/competition/ccks_2019_1/}}.
\item \textbf{Machine Reading Comprehension}: CMRC 2018 \cite{DBLP:journals/corr/abs-1810-07366}, CMRC2019~\cite{cui2020sentence}, DRCD \cite{shao2018drcd}, DuReader \cite{he2017dureader}, Dureader$_\text{robust}$~\cite{tang2020dureaderrobust}, Dureader$_\text{checklist}$, Dureader$_\text{yesno}$~\footnote{\url{https://aistudio.baidu.com/aistudio/competition/detail/49/?isFromLUGE=TRUE}}, C$^3$~\cite{sun2020c3}, CHID~\cite{zheng2019chid}.
\item \textbf{Legal Documents Analysis}: CAIL2018-Task1~\cite{xiao2018cail2018}, CAIL2018-Task2~\cite{xiao2018cail2018}.
\item \textbf{Cant Understanding}: DogWhistle Insider, DogWhistle Outsider\cite{xu-etal-2021-blow}.
\item \textbf{Document Retrieval}: Sogou-log \cite{sogou-log}.
\end{itemize}
\subsubsection{Natural Language Generation Tasks}
9 datasets belonging to 7 kinds of natural language generation tasks are used in our experiments, as follows:
\begin{itemize}
\item \textbf{Text Summarization}: LCSTS~\cite{hu2015lcsts}
\item \textbf{Question Generation}:KBQG~\footnote{\url{http://tcci.ccf.org.cn/conference/2017/dldoc/taskgline05.pdf}}, DuReader-QG~\cite{he2017dureader}, DuReader$_{\text{robust}}$-QG~\cite{tang2020dureaderrobust}.
\item \textbf{Closed-Book Question Answering}: MATINF-QA~\cite{xu2020matinf}.
\item \textbf{Math}: Math23K~\cite{wang2017math}.
\item \textbf{Advertisement Generation}: AdGen~\cite{shao2019adgen}.
\item \textbf{Translation}: WMT20-enzh~\cite{barrault2020wmt}.
\item \textbf{Dialogue Generation}: KdConv~\cite{zhou2020kdconv}.
\end{itemize}
\begin{table}[]
\small
\resizebox{1\textwidth}{!}{
\begin{tabular}{clcclcc}
\hline
\hline
\textbf{ID} & \textbf{Task} & \textbf{Dataset} & \textbf{Metric} & & \textbf{Previous SoTA Model} & \textbf{ERNIE 3.0} \\ \hline
\multirow{5}{*}{1} & \multirow{5}{*}{Sentiment Analysis} & NLPCC2014-SC & Acc. & \multicolumn{1}{l|}{Test} & 83.53 (SKEP) & \textbf{86.00} \\
& & \multicolumn{1}{l}{SE-ABSA16\_PHNS} & Acc. & \multicolumn{1}{l|}{Test} & 82.91 (SKEP) & \textbf{93.95} \\
& & SE-ABSA16\_CAME & Acc. & \multicolumn{1}{l|}{Test} & 90.06 (SKEP) & \textbf{96.05} \\
& & \multirow{2}{*}{BDCI2019} & \multirow{2}{*}{Acc.} & \multicolumn{1}{l|}{Dev} & - & 96.83 \\
& & & & \multicolumn{1}{l|}{Test} & 96.26 (ERNIE 2.0) & \textbf{97.70} \\ \hline
\multirow{3}{*}{2} & \multirow{3}{*}{Opinion Extraction} & COTE-BD & F1 & \multicolumn{1}{l|}{Test} & 84.50 (SKEP) & \textbf{90.23} \\
& & COTE-DP & F1 & \multicolumn{1}{l|}{Test} & 86.30 (SKEP) & \textbf{92.75} \\
& & COTE-MFW & F1 & \multicolumn{1}{l|}{Test} & 87.90 (SKEP) & \textbf{89.90} \\ \hline
\multirow{3}{*}{3} & \multirow{3}{*}{Natural Language Inference} & OCNLI & Acc. & \multicolumn{1}{l|}{Dev} & 78.80 (RoBERTa*) & \textbf{82.75} \\
& & \multirow{2}{*}{XNLI} & \multirow{2}{*}{Acc.} & \multicolumn{1}{l|}{Dev} & 83.25 (Zen 2.0) & \textbf{84.42} \\
& & & & \multicolumn{1}{l|}{Test} & 83.09 (Zen 2.0) & \textbf{83.77} \\ \hline
4 & Winograd Schema Challenge & WSC2020 & Acc. & \multicolumn{1}{l|}{Dev} & 69.70 (RoBERTa*) & \textbf{95.40} \\ \hline
\multirow{4}{*}{5} & \multirow{4}{*}{Relation Extraction} & \multirow{2}{*}{FinRE} & \multirow{2}{*}{F1} & \multicolumn{1}{l|}{Dev} & 63.33 (ERNIE 2.0) & \textbf{64.87} \\
& & & & \multicolumn{1}{l|}{Test} & 60.60 (ERNIE 2.0) & \textbf{62.88} \\
& & \multirow{2}{*}{SanWen} & \multirow{2}{*}{F1} & \multicolumn{1}{l|}{Dev} & 79.92 (ERNIE 2.0) & \textbf{81.32} \\
& & & & \multicolumn{1}{l|}{Test} & 77.97 (ERNIE 2.0) & \textbf{82.59} \\ \hline
\multirow{2}{*}{6} & \multirow{2}{*}{Event Extraction} & \multirow{2}{*}{CCKS2020} & \multirow{2}{*}{F1} & \multicolumn{1}{l|}{Dev} & 60.64 (ERNIE 2.0) & \textbf{61.70} \\
& & & & \multicolumn{1}{l|}{Test} & 61.34 (ERNIE 2.0) & \textbf{64.33} \\ \hline
\multirow{8}{*}{7} & \multirow{8}{*}{Semantic Similarity} & AFQMC & Acc. & \multicolumn{1}{l|}{Dev} & 74.92 (RoBERTa*) & \textbf{77.02} \\
& & \multirow{2}{*}{LCQMC} & \multirow{2}{*}{Acc.} & \multicolumn{1}{l|}{Dev} & - & 90.29 \\
& & & & \multicolumn{1}{l|}{Test} & 89.16 (CPM-2) & \textbf{90.38} \\
& & CSL & Acc. & \multicolumn{1}{l|}{Dev} & 82.17 (RoBERTa*) & \textbf{84.50} \\
& & \multirow{2}{*}{PAWS-X} & \multirow{2}{*}{Acc.} & \multicolumn{1}{l|}{Dev} & 86.25 (ERNIE 2.0) & \textbf{87.00} \\
& & & & \multicolumn{1}{l|}{Test} & 86.35 (ERNIE 2.0) & \textbf{87.10} \\
& & \multirow{2}{*}{BQ Corpus} & \multirow{2}{*}{Acc.} & \multicolumn{1}{l|}{Dev} & 87.11 (ZEN 2.0) & \textbf{87.41} \\
& & & & \multicolumn{1}{l|}{Test} & 85.99 (ZEN 2.0) & \textbf{86.10} \\ \hline
\multirow{8}{*}{8} & \multirow{8}{*}{Chinese News Classification} & TNEWS & Acc. & \multicolumn{1}{c|}{Dev} & 58.32 (RoBERTa*) & \textbf{69.94} \\
& & IFLYTEK & Acc. & \multicolumn{1}{l|}{Dev} & 62.75 (RoBERTa*) & \textbf{63.45} \\
& & \multirow{2}{*}{THUNCEWS} & \multirow{2}{*}{Acc.} & \multicolumn{1}{l|}{Dev} & 97.7 (RoBERTa*) & \textbf{98.33} \\
& & & & \multicolumn{1}{l|}{Test} & 97.6 (RoBERTa*) & \textbf{98.66} \\
& & \multirow{2}{*}{CNSE} & \multirow{2}{*}{Acc.} & \multicolumn{1}{l|}{Dev} & 85.64 (RoBERTa*) & \textbf{88.94} \\
& & & & \multicolumn{1}{l|}{Test} & 85.57 (RoBERTa*) & \bf{88.92} \\
& & \multirow{2}{*}{CNSS} & \multirow{2}{*}{Acc.} & \multicolumn{1}{l|}{Dev} & 93.06 (ERNIE 2.0) & \textbf{93.84} \\
& & & & \multicolumn{1}{l|}{Test} & 92.73 (ERNIE 2.0) & \bf{93.76} \\ \hline
\multirow{7}{*}{9} & \multirow{7}{*}{Closed-Book Question Answering} & \multirow{2}{*}{NLPCC-DBQA} & \multirow{2}{*}{MRR/F1} & \multicolumn{1}{c|}{Dev} & 96.04/85.69 (Zen 2.0) & \textbf{96.71/87.57} \\
& & & & \multicolumn{1}{l|}{Test} & 96.11/86.47 (Zen 2.0) & \textbf{96.50/88.49} \\
& & CHIP2019 & Acc. & \multicolumn{1}{l|}{Test} & 89.22 (ERNIE 2.0) & \textbf{89.90} \\
& & \multirow{2}{*}{cMedQA} & \multirow{2}{*}{Acc.} & \multicolumn{1}{l|}{Dev} & 78.6 (BERT\_BiGRU*) & \textbf{84.60} \\
& & & & \multicolumn{1}{l|}{Test} & 78.2 (BERT\_BiGRU*) & \textbf{82.65} \\
& & \multirow{2}{*}{cMedQA2} & \multirow{2}{*}{Acc.} & \multicolumn{1}{l|}{Dev} & 81.3 (BERT\_BiGRU*) & \textbf{83.48} \\
& & & & \multicolumn{1}{l|}{Test} & 82.2 (BERT\_BiGRU*) & \textbf{83.68} \\ \hline
\multirow{6}{*}{10}& \multirow{6}{*}{Named Entity Recognition} & CLUENER & F1 & \multicolumn{1}{c|}{Dev} & 80.42 (RoBERTa*) & \textbf{81.23} \\
& & \multirow{2}{*}{Weibo} & \multirow{2}{*}{F1} & \multicolumn{1}{l|}{Dev} & - & 70.06 \\
& & & & \multicolumn{1}{l|}{Test} & 67.60 (Glyce+BERT) & \textbf{69.23} \\
& & \multirow{2}{*}{OntoNotes} & \multirow{2}{*}{F1} & \multicolumn{1}{l|}{Dev} & - & 79.59 \\
& & & & \multicolumn{1}{l|}{Test} & 81.63 (Glyce+BERT) & \textbf{82.64} \\
& & CCKS2019 & F1 & \multicolumn{1}{l|}{Test} & 81.58 (ERNIE 2.0) & \textbf{82.70} \\ \hline
\multirow{4}{*}{11}& \multirow{4}{*}{Cant Understanding} & \multicolumn{1}{l}{\multirow{2}{*}{DogWhistle Insider}} & \multirow{2}{*}{Acc.} & \multicolumn{1}{c|}{Dev} & 75.4 (ALBERT) & \textbf{79.06} \\
& & \multicolumn{1}{l}{} & & \multicolumn{1}{l|}{Test} & 76.1 (ALBERT) & \textbf{79.22} \\
& & \multicolumn{1}{l}{\multirow{2}{*}{DogWhistle Outsider}} & \multirow{2}{*}{Acc.} & \multicolumn{1}{l|}{Dev} & 34.6 (ALBERT) & \textbf{38.68} \\
& & \multicolumn{1}{l}{} & & \multicolumn{1}{l|}{Test} & 34.6 (ALBERT) & \textbf{38.22} \\ \hline\hline
\end{tabular}
}
\end{table}
\begin{table}[]
\resizebox{1\textwidth}{!}{
\begin{tabular}{clcclcc}
\hline\hline
\textbf{ID} & \textbf{Task} & \textbf{Dataset} & \textbf{Metric} & & \textbf{Previous SoTA Model} & \textbf{ERNIE 3.0} \\ \hline
\multirow{14}{*}{12} & \multirow{14}{*}{Machine Reading Comprehension} & CMRC2018 & EM/F1 & \multicolumn{1}{c|}{Dev} & 74.3/90.5 (ERNIE-Gram) & \textbf{75.30/92.29} \\
& & CRMC2019 & QAC/PAC & \multicolumn{1}{c|}{Dev} & 82.6/23.3 (RoBERTa*) & \textbf{92.53/57.33} \\
& & \multirow{2}{*}{DRCD} & \multirow{2}{*}{EM/F1} & \multicolumn{1}{c|}{Dev} & 90.8/95.3 (MacBERT) & \textbf{91.54/96.45} \\
& & & & \multicolumn{1}{c|}{Test} & 90.9/95.3 (MacBERT) & \textbf{91.41/95.84} \\
& & DuReader & EM/F1 & \multicolumn{1}{c|}{Dev} & 64.2/77.3 (ERNIE 2.0) & \textbf{67.69/79.66} \\
& & \multirow{2}{*}{DuReader$_\text{robust}$} & \multirow{2}{*}{EM/F1} & \multicolumn{1}{c|}{Dev} & 75.23/86.77 (ERNIE 2.0) & \textbf{77.27/88.54}\\
& & & & \multicolumn{1}{c|}{Test} & 51.20/67.96 (ERNIE 2.0) & \textbf{60.87/75.63} \\
& & \multirow{2}{*}{DuReader$_\text{checklist}$} & \multirow{2}{*}{EM/F1} & \multicolumn{1}{c|}{Dev} & 55.66/64.12 (ERNIE 2.0) & \textbf{61.33/70.59} \\
& & & & \multicolumn{1}{c|}{Test} & 59.11/48.79 (ERNIE 2.0) & \textbf{64.87/53.82} \\
& & \multirow{2}{*}{DuReader$_\text{yesno}$} & \multirow{2}{*}{Acc.} & \multicolumn{1}{c|}{Dev} & 88.69 (ERNIE 2.0) & \textbf{89.95} \\
& & & & \multicolumn{1}{c|}{Test} & 88.82 (ERNIE 2.0) & \textbf{89.64} \\
& & \multirow{2}{*}{C3} & \multirow{2}{*}{Acc.} & \multicolumn{1}{c|}{Dev} & - & 87.63 \\
& & & & \multicolumn{1}{c|}{Test} & 86.1 (CPM-2) & \textbf{86.69} \\
& & CHID & Acc. & \multicolumn{1}{c|}{Dev} & 85.81 (RoBERTa*) & \textbf{91.67} \\ \hline
\multirow{4}{*}{13} & \multirow{4}{*}{Legal Document Analysis} & \multirow{2}{*}{CAIL2018 Task1} & \multirow{2}{*}{F1-macro/F1-micro} & \multicolumn{1}{c|}{Dev} & 83.85/91.50 (ERNIE 2.0) & \textbf{88.64/93.11} \\
& & & & \multicolumn{1}{c|}{Test} & 80.40/89.94 (ERNIE 2.0) & \textbf{86.83/91.82} \\
& & \multirow{2}{*}{CAIL2018 Task2} & \multirow{2}{*}{F1-macro/F1-micro} & \multicolumn{1}{c|}{Dev} & 78.58/89.46 (ERNIE 2.0) & \textbf{82.62/90.93} \\
& & & & \multicolumn{1}{c|}{Test} & 75.35/86.97 (ERNIE 2.0) & \textbf{81.10/88.52} \\
\hline
\multirow{1}{*}{14} & \multirow{1}{*}{Document Retrieval} & \multirow{1}{*}{Sogou-log} & \multirow{1}{*}{MRR/NDCG@1} & \multicolumn{1}{c|}{Test} & 36.3/35.5 (CPM-2) & \textbf{38.20/37.24} \\
\hline\hline
\end{tabular}
}
\caption{Results on Natural Language Understanding Tasks. We compare ERNIE 3.0 with 10 previous SoTA baselines including CPM-2\cite{zhang2021cpm}, ERNIE 2.0\cite{sun2020ernie}, ERNIE-Gram\cite{xiao2020erniegram}, SKEP\cite{tian2020skep}, RoBERTa-wwm-ext-large\cite{cui2019pre} (marked as RoBERTa*), ALBERT\cite{lan2019albert}, MacBERT\cite{cui2020revisiting}, Zen 2.0\cite{song2021zen}, Glyce\cite{meng2019glyce} and crossed BERT siamese BiGRU\cite{cui2020chinese} (marked as BERT\_BiGRU*).}\label{tab:result-understanding}
\end{table}
\subsection{Experiments on Fine-tuning Tasks}\label{sec:finetune-tasks}
\subsubsection{Fine-tuning on Natural Language Understanding Tasks}\label{sec:nlu-tasks}
The results of natural language understanding tasks are reported in Table~\ref{tab:result-understanding}.
\noindent\textbf{Sentiment Analysis}. Sentiment Analysis is a classification task aiming to determine whether a sentence is positive, negative, or neutral. We consider 4 datasets from different domains, including shopping (NLPCC2014-SC), electronics (SE-ABSA16\_PHNS, SE-ABSA16\_CAM), and financial (BDCI2019). ERNIE 3.0 achieves a substantial improvement on all four datasets.
\noindent\textbf{Opinion Extraction}. Similar to the sentiment analysis task, opinion extraction requires the model to mine the opinion of a sentence. We use 3 sub-datasets from Chinese Customer Review (COTE). Experiment results show that ERNIE 3.0 also outperforms the current SoTA system by a great margin.
\noindent\textbf{Natural Language Inference}. Natural Language Inference is the task to determine whether a given premise semantically entails another hypothesis. We use OCNLI and XNLI datasets. The results indicate that ERNIE 3.0 has achieved 3.9 and 0.7 accuracy improvement on two datasets, respectively. The improvement on the XNLI dataset is quite limited, and this may be due to the poor quality of the dataset since the XNLI dataset is translated from English.
\noindent\textbf{Winograd Schemas Challenge}. WSC2020 is an anaphora resolution task where the model is asked to decide whether a pronoun and a noun in a sentence co-refer, ERNIE 3.0 achieves a significant improvement of 25.7 points.
\noindent\textbf{Relation Extraction}. The task of relation extraction is to identify the relationship between different entities like persons and organizations. We consider FinRE and SanWen -- two relation extraction datasets for financial news and Chinese literature respectively. ERNIE 3.0 outperforms the previous SoTA model by 2.46 points on average.
\noindent\textbf{Event Extraction}. Similar to relation extraction, the event extraction task aims to identify the event entities and classify them into different categories. We choose CCKS2020 -- a text-level event subject extraction dataset of financial field. ERNIE 3.0 has 3 points of improvement on the test set.
\noindent\textbf{Semantic Similarity}. Semantic Similarity is a classic NLP task that determines the similarity between various terms such as words, sentences, documents. In this work, we focus on sentence-level similarity tasks. We test ERNIE 3.0 on several datasets in varied fields including AFQMC, LCQMC, CSL, PAWS-X, and BQ. Experiment results show that ERNIE 3.0 outperforms the baseline models by a remarkable margin. Especially, under comparable number of parameters, ERNIE 3.0 surpasses CPM-2 with 1.2 points on LCQMC dataset.
\noindent\textbf{Chinese News Classification}. We also evaluate ERNIE 3.0 on Chinese news classification. We consider 6 datasets including news title (TNEWS), app descriptions (IFLYTEK), and news stories (THUCNEWS, CNSE, CNSS). Under different types of classification tasks, ERNIE 3.0 can consistently achieve better accuracy with 2.8 points improvement on average.
\noindent\textbf{Closed-Book Question Answering}. Closed-Book Question Answering aims to directly answer the questions without any additional references or knowledge. We select a general QA dataset NLPCC-DBQA and three medical field datasets -- CHIP2019, cMedQA, and cMedQA2 to test the ability of ERNIE 3.0. Experiment results show that ERNIE 3.0 performs better on all QA tasks, we believe knowledge enhanced pre-training methods do bring benefits to the closed-book QA task.
\noindent\textbf{Cant Understanding}. Cant, also known as doublespeak, is an advanced language usage for humans. However, it is rather difficult for machines to understand this type of language. We test the cant understanding ability of ERNIE 3.0 on DogWhistle -- a dataset based on \emph{Decrypto} game. The model is required to select the right answer with the guidance of the corresponding cant. ERNIE 3.0 gets the best result and shows its potential for understanding much more difficult languages.
\noindent\textbf{Named Entity Recognition}. Named Entity Recognition is a classical NLP task of extracting and classifying entities in text. We select widely used OntoNotes, CLUENER, Weibo, and a domain-specific dataset CCKS2019. From the results, ERNIE 3.0 performs better than the baseline models across all datasets.
\noindent\textbf{Machine Reading Comprehension}. We comprehensively evaluate the ability of ERNIE 3.0 on machine reading comprehension in different aspects, including span-predict reading comprehension (CMRC2018, DuReader, DRCD, DuReader$_\text{checklist}$), multiple-choice reading comprehension (C3, DuReader$_\text{yesno}$), cloze and completion (CHID, CMRC2019), and robustness test (Dureader$_\text{robust}$). With the help of knowledge enhanced pre-training, ERNIE 3.0 surpasses the baseline models with significant enhancements on all types of tasks. To be more specific, ERNIE 3.0 achieve at least 1.0 points of EM improvement on 5 span-predict tasks and 0.89 accuracy improvement on multiple-choice tasks on average. Also, under comparable number of parameters, ERNIE 3.0 outperforms CPM-2 with 0.6 points on C3 dataset. For the robustness test, ERNIE 3.0 also performs best on the test set with over-sensitivity and over-stability samples.
\noindent\textbf{Legal Documents Analysis}. Next, we test the ability of ERNIE 3.0 on document analysis, we choose two domain-specific tasks of law. These two datasets from CAIL2018 are both multi-label document classification tasks. ERNIE 3.0 outperforms ERNIE 2.0 with remarkable increment.
\noindent\textbf{Document Retrieval}. Document retrieval aims to match documents given queries. We evaluate the retrieval ability of ERNIE 3.0 on Sogou-Log. Following previous work \cite{zhang2021cpm}, we report NDCG@1 performance on the test-same test set and MRR performance on the test-raw test set and ERNIE 3.0 outperforms CPM-2.
\subsubsection{Fine-tuning on Natural Language Generation Tasks}\label{sec:nlg-tasks}
The results of natural language generation tasks are reported in Table~\ref{tab:result-generation}.
\noindent\textbf{Text Summarization}.
We consider a Large Scale Chinese Short Text Summarization (LCSTS) dataset which requires a model to understand the text and refine the key information to generate
coherent, informative summaries. LCSTS is a classic
Chinese text summarization dataset which consists of 2 million real Chinese short texts with short summaries from Sina Weibo.
ERNIE 3.0 achieves 48.46\% Rouge-L score which outperforms CPM-2 with comparable number of parameters (11B) and current SoTA ProphetNet-zh.
\noindent\textbf{Question Generation}. Question Generation is the reverse task of Machine Reading Comprehension (MRC) which requires the model to understand a document and generate a
reasonable question based on a given short answer. We use a suite of three datasets including knowledge base question generation (KBQG), two MRC datasets named Dureader and Dureader${_\text{robust}}$. ERNIE 3.0 performs best on these three datasets compared to the baselines.
\noindent\textbf{Math}. To test ERNIE 3.0's ability to perform simple arithmetic operations, we consider the Math23K dataset which contains 23,161 real math word problems for elementary school students with problem descriptions, structured equations and answers. ERNIE 3.0 is fine-tuned to generate the postfix expression of the structured equation given the problem description, then the final answer can be calculated using the Python \textit{eval()} function (note that the `[' and `]' should be replaced with `(' and `)' respectively, also the `\%' should be replaced with `*0.01' to avoid the failed solutions using Python \textit{eval()} function). It shows that ERNIE 3.0 is a great math solver which achieves high accuracy 75\% compared to CPM-2 69.37\%.
\begin{table}[]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{@{}llccccccc@{}}
\toprule \toprule
\textbf{Task} & \textbf{Dataset} & \textbf{Metric} & \textbf{RoBERTa-Large} & \textbf{ERNIE 2.0-Large} & \textbf{ProphetNet-zh} & \textbf{mT5} & \textbf{CPM-2} & \textbf{ERNIE 3.0} \\ \midrule
Text Summarization & LCSTS & \multicolumn{1}{c|}{ROUGE-L} & 40.98 & 41.38 & 37.08 & 34.8 & 35.88 & \textbf{48.46} \\ \midrule
\multirow{3}{*}{Question Generation} & KBQG & \multicolumn{1}{c|}{BLEU-4} & - & 57.40 & - & - & - & \textbf{64.70} \\
& DuReader-QG & \multicolumn{1}{c|}{BLEU-4} & 32.29 & 34.15 & - & - & - & \textbf{48.36} \\
& DuReader$_\text{robust}$-QG & \multicolumn{1}{c|}{BLEU-4} & 37.10 & 39.30 & - & - & - & \textbf{41.70} \\ \midrule
Closed-Book Question Answering & MATINF-QA & \multicolumn{1}{c|}{ROUGE-L} & - & - & 15.47 & - & - & \textbf{17.33} \\ \midrule
Math & Math23K & \multicolumn{1}{c|}{Acc.} & - & - & - & 61.60 & 69.37 & \textbf{75.00} \\ \midrule
Advertisement Generation & AdGen & \multicolumn{1}{c|}{BLEU-4} & - & - & - & 9.82 & 10.60 & \textbf{30.16} \\ \midrule
Translation & WMT20-enzh & \multicolumn{1}{c|}{BLEU} & - & - & - & 23.98 & 26.21 & \textbf{26.80} \\ \midrule
Dialogue Generation & KdConv & \multicolumn{1}{c|}{BLEU-4} & 15.75 & 13.94 & - & - & - & \textbf{23.85} \\ \bottomrule \bottomrule
\end{tabular}%
}
\caption{Results on Natural Language Generation Tasks. We reported the results on the test set.}
\label{tab:result-generation}
\end{table}
\noindent\textbf{Advertisement Generation}. We consider AdGen which consists of 119K pairs of advertising text and clothing specification tables from a Chinese e-commerce platform.
It requires the model to generate a long advertising text that covers all given attribute-value pairs for a piece of clothing.
An attribute-value pair is joined with a colon, and several attribute-value pairs are concatenated sequentially using a `|' according to their segment number.
Then we take the structural attribute-value pairs string as input for ERNIE 3.0. It shows that ERNIE 3.0 is capable to generate a coherent and intriguing long advertising text by extracting information from a structural input with 19.56 percent point improvement w.r.t BLEU-4 compared to CPM-2.
\noindent\textbf{Translation}. For ERNIE 3.0, we mainly consider the pre-training on Chinese corpus. To test its multilingual ability, we expand our vocabulary to include extra 10K English subwords. On a classic multilingual dataset WMT20-enzh, we fine-tuned ERNIE 3.0 to translate English to Chinese.
Compared to mT5-xxLarge and CPM-2, ERNIE 3.0~\footnote{Due to the large size of the training dataset of WMT20-enzh, ERNIE 3.0 is not fully trained to convergence. We reported the BLEU score at 1.5 epoch checkpoint using SacreBLEU project~\cite{post-2018-call}.} is the best and presents superior multilingual ability.
\noindent\textbf{Dialogue Generation}. Next, we evaluate ERNIE 3.0 on Dialog Generation task. We consider a Chinese multi-domain knowledge-driven conversation dataset that contains
4.5K conversations from three domains (film, music, and travel). We train and test ERNIE 3.0 on the fused set of data from aforementioned three domains by only giving dialogue history to generate the current utterance.
Knowledge triplets are excluded from inputs, so it's suitable to test a model's ability to model multi-turn conversations by leveraging inherent knowledge during pre-training.
Compared to baselines, ERNIE 3.0 improves the performance a lot by 8.1 percent point, and we believe the knowledge graph enhanced pre-training attributes a lot.
\begin{table}[]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{@{}lllcccc@{}}
\toprule \toprule
\textbf{Task Paradigm} & \textbf{Task} & \textbf{Dataset} & \textbf{Metric} & \textbf{RoBERTa-Large} & \textbf{ERNIE 2.0-Large} & \textbf{ERNIE 3.0} \\ \midrule
\multirow{3}{*}{\textbf{NLU}} & Sentiment Analysis & NLPCC14-SC & Acc. & 83.56 & 84.36 & \textbf{86.00} \\
& Machine Reading Comprehension & DuReader$_\text{robust}$ & EM/F1 & 51.10/67.18 & 51.20/67.96 & \textbf{60.87/75.63} \\
& Semantic Similarity & LCQMC & Acc. & 87.40 & 87.90 & \textbf{90.38} \\ \midrule
\multirow{3}{*}{\textbf{NLG}} & Question Generation & DuReader$_\text{robust}$-QG & BLEU-4 & 37.10 & 39.30 & \textbf{41.70} \\
& Text Summarization & LCSTS & Rouge-L & 40.98 & 41.38 & \textbf{48.46} \\
& Dialogue Generation & KdConv & BLEU-4 & 15.75 & 13.94 & \textbf{23.85} \\ \midrule
\multicolumn{4}{l}{\textbf{Average}} & 53.99 & 54.41 & \textbf{59.77} \\ \bottomrule \bottomrule
\end{tabular}%
}
\caption{Results on the LUGE benchmark. We reported the results on the test set.}
\label{tab:result-luge}
\end{table}
\begin{table}[]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{@{}lcccccc@{}}
\toprule \toprule
\textbf{Task Type} & \textbf{Dataset} & \textbf{Metric} & \textbf{CPM-1} & \textbf{PanGu-$\alpha$-2.6B} & \textbf{PanGu-$\alpha$-13B} & \textbf{ERNIE 3.0} \\ \midrule
\multirow{2}{*}{Chinese News Classification} & TNEWS & \multicolumn{1}{c|}{Acc.} & 65.44 & 60.95 & 60.26 & \textbf{68.40} \\
& IFLYTEK & \multicolumn{1}{c|}{Acc.} & 68.91 & 74.26 & 73.80 & \textbf{75.34} \\ \midrule
\multirow{2}{*}{Semantic Similarity} & AFQMC & \multicolumn{1}{c|}{Acc.} & 66.34 & 59.29 & 65.76 & \textbf{68.99} \\
& CSL & \multicolumn{1}{c|}{Acc.} & 52.30 & 50.50 & 49.30 & \textbf{55.63} \\ \midrule
\multirow{2}{*}{Natural Language Inference} & OCNLI & \multicolumn{1}{c|}{Acc.} & 44.20 & 42.61 & 41.53 & \textbf{44.31} \\
& CMNLI & \multicolumn{1}{c|}{Acc.} & 49.10 & 47.56 & 49.29 & \textbf{49.41} \\ \midrule
Winograd Schema Challenge & WSC2020 & \multicolumn{1}{c|}{Acc.} & 73.68 & 73.36 & 75.00 & \textbf{78.38} \\ \midrule
\multirow{6}{*}{Cloze and completion} & CHID & \multicolumn{1}{c|}{Acc.} & 68.62 & 68.73 & 70.64 & \textbf{77.78} \\
& PD & \multicolumn{1}{c|}{Acc.} & 35.73 & 38.47 & 43.84 & \textbf{66.07} \\
& CFT & \multicolumn{1}{c|}{Acc.} & 38.99 & 42.39 & 46.60 & \textbf{49.30} \\
& CMRC2017 & \multicolumn{1}{c|}{Acc.} & 24.60 & 37.83 & 38.90 & \textbf{56.66} \\
& CMRC2019 & \multicolumn{1}{c|}{Acc.} & 47.69 & 61.93 & 68.19 & \textbf{75.00} \\
& WPLC & \multicolumn{1}{c|}{PPL} & - & 48.98 & 45.85 & \textbf{17.03} \\ \midrule
\multirow{4}{*}{Machine Reading Comprehension} & C3 & \multicolumn{1}{c|}{Acc.} & 49.81 & 53.42 & \textbf{54.47} & 52.62 \\
& CMRC2018 & \multicolumn{1}{c|}{EM/F1} & 0.59/10.12 & 1.21/16.65 & 1.46/19.28 & \textbf{7.61/25.61} \\
& DRCD & \multicolumn{1}{c|}{EM/F1} & 0.00/4.62 & 0.80/9.99 & 0.66/10.55 & \textbf{10.58/26.29} \\
& DuReader & \multicolumn{1}{c|}{EM/F1} & 16.63 & 21.07 & 24.46 & \textbf{29.79} \\ \midrule
\multirow{2}{*}{Closed-book Question Answering} & WebQA & \multicolumn{1}{c|}{EM/F1} & 6.00/12.59 & 4.43/13.71 & 5.13/14.47 & \textbf{22.53/38.95} \\
& CKBQA & \multicolumn{1}{c|}{Acc.} & 13.40 & 14.61 & 14.21 & \textbf{20.64} \\ \bottomrule \bottomrule
\end{tabular}%
}
\caption{Results on zero-shot learning tasks.}
\label{tab:zero-shot-result}
\end{table}
\subsubsection{LUGE benchmark}\label{sec:luge-tasks}
In order to further evaluate the capabilities of different models comprehensively and conveniently, we conduct experiments on the Language Understanding and Generation Evaluation Benchmarks(LUGE))~\footnote{\url{https://www.luge.ai/}}. We use six representative tasks (see Tab.~\ref{tab:result-luge}) from LUGE. ERNIE 3.0 delivers an average 5.36 percent improvement over leading pre-trained models such as ERNIE 2.0 and RoBERTa.
\subsection{Experiments on Zero-shot Learning}\label{sec:zero-shot-tasks}
We have demonstrated that ERNIE 3.0 is superior to previous SoTA methods on both NLU and NLG tasks following the pretraining-then-finetuning
paradigm. In this section, we conduct various types of tasks with the zero-shot setting where a model is applied without any gradient updates
or fine-tuning. ERNIE 3.0 achieves strong performance compared to recently proposed large-scale language models such as CPM-1 (2.6B), PanGu-$\alpha$-2.6B
and PanGu-$\alpha$-13B on most downstream tasks.
At last, we show that ERNIE 3.0 can generate more coherent, natural and accurate responses rated on our manually collected 450 cases across 13 different tasks.
\subsubsection{Evaluation}
The evaluation methods can be classified into two categories, namely perplexity-based method and generation-based method.
\begin{itemize}
\item \textbf{Perplexity-based Method}. On tasks that choose one single correct answer from multiple candidates such as CHID and CMRC2017, we compare the per-token perplexity score~\footnote{The perplexity score of a sample is normalized by the number of tokens.} when filling each answer into the blank of the context. The one with lower per-token perplexity score will be the predicted as the correct answer. On tasks that require binary or multiple classification, we assign each label with a more semantically meaningful name and use a prompt to formalize the context and the label as a human-readable text. Then, this kind of tasks can be treated as multi-choice tasks. The prompts we used are similar to that in CPM-1 and PanGu-$\alpha$.
\item \textbf{Generation-based Method}. On tasks with free-form completion such as Closed-book QA, we use beam search with a beam width of 8 and no length penalty. The maximum generated length of a completion is limited by a pre-defined number based on 95\% percentile point of answers' length on the dataset. Then metrics such as exact match (EM), F1 and Rouge-1 are used. On tasks with restrained completion such as extractive MRC, we use restrained beam search with the same parameters as before. A Trie-Tree is constructed for each sample to
efficiently and effectively restrain the space of generation and only generate completion occurred in a given text.
\end{itemize}
\subsubsection{Results}
\noindent\textbf{Chinese News Classification}. For the TNEWS and IFLYTEK datasets, there are 15 and 119 categories respectively. We randomly sample three candidates as negative labels for each sample and compare the per-token perlexity score among these four choices.
This sampling strategy is aligned with CPM-1's and PanGu-$\alpha$'s to reduce the total computational cost since we need to calculate per-token perlexity score
for each candidate separately. ERNIE 3.0 performs well on TNEWS even reaching competitiveness with prior state-of-the-art fine-tuning approaches and performs slightly well on IFLYTEK.
\noindent\textbf{Semantic Similarity}. We consider AFQMC and CSL datasets. ERNIE 3.0 outperforms baselines at a large margin.
However, the accuracy is slightly above than a random-guess model.
This may be partly attributed to the sub-optimal selection of the prompt (like \textsc{The following two sentences have the same/different semantics: \$SENT\_A. \$SENT\_B.}).
\noindent\textbf{Natural Language Inference}. ERNIE 3.0 is evaluated on two NLI datasets, namely OCNLI and CMNLI where CMNLI consists of XNLI and MNLI by translating English to Chinese. We use the prompt as \textsc{\$SENT\_A? No/Yes/Maybe, \$SENT\_B}. The performance of ERNIE 3.0 is comparable to baselines, it shows that there is still a large room for improvement for pre-trained models on zero-shot NLI task.
\noindent\textbf{Winograd Schema Challenge}: We formalize the WSC2020 dataset as a multi-choice completion task where a pronoun is replaced with each candidates to calculate the per-token perplexity of a sample. ERNIE 3.0 improves the performance by 3.38 percent point compared to PanGu-$\alpha$-13B.
\noindent\textbf{Cloze and completion}. On the CHID dataset, we split each sentence that contains only one blank word as a sample, and formalize as a multi-choice task. ERNIE 3.0 achieves the best score among baselines. For Chinese Word Prediction with Long Context (Chinese WPLC), a sample consists of a masked text and a correct word. Following PanGu-$\alpha$, we replace the mask token with the correct word and calculate the perplexity score of a whole sentence. Compared to PanGu-$\alpha$, ERNIE 3.0 achieves much lower perplexity score. On the CMRC2019 dataset, we randomly sample three negative candidates for each blank from the original candidates, then beam search is applied to calculate the optimal path for a sample. We also formalize the PD, CFT and CMRC2017 as multi-choice tasks where the text before the blank is taken as the input, and the multiple choices are the words the appear in the whole text. ERNIE 3.0 surpassed the baselines with a large margin.
\noindent\textbf{Machine Reading Comprehension}. We consider four MRC datasets. On C3, a multi-choice machine reading comprehension tasks,
we use the prompt as \textsc{Question: \$Question? Answer: \$Choice. The answer is in the following document: \$Document}. For CMRC2018, DRCD and DuReader, we evaluate ERNIE 3.0 using generation-base method and the prompt is \textsc{Document: \$Document. Question: \$Question? Answer:}.
ERNIE 3.0 outperforms baselines with a large margin on CMRC2018, DRCD and DuReader dataset.
\noindent\textbf{Closed-book Question Answering}. We evaluated ERNIE 3.0 on two Closed-book Question Answering datasets which require the model to generate answers using its inherent knowledge learned during pre-training. WebQA is a large scale real-word QA dataset from Baidu Zhidao. We only provide ERNIE 3.0 with the question without additional evidence. The prompt is similar to MRC's but without document input (\textsc{Question: \$Question? Answer:}). ERNIE 3.0 achieves better performance compared to baselines. We presented the detailed analysis about CKBQA dataset in Section.~\ref{sec:analysis}.
\subsubsection{Case Study}
\begin{table}[]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{@{}llcccc@{}}
\toprule \toprule
\textbf{Type} & \textbf{Task (\# of cases)} & \textbf{CPM-1} & \textbf{PLUG} & \textbf{PanGu-$\alpha$} & \textbf{ERNIE 3.0} \\ \midrule
\multirow{3}{*}{Question Answering} & Factual QA (30) & \textbf{1.67/1.50/1.03} & 1.23/0.83/0.27 & 1.60/1.07/0.60 & \textbf{1.67/1.50/1.03} \\
& Opinion QA (30) & 1.27/0.80/- & 1.43/1.13/- & 1.60/1.23/- & \textbf{1.67/1.33/-} \\
& Reasoning (30) & 1.20/0.83/\textbf{0.27} & 1.03/0.83/0.07 & 1.03/0.83/0.00 & \textbf{1.70/1.60/}0.23 \\ \midrule
\multirow{2}{*}{Interpretation} & Interpretation of Terms (30) & 1.23/0.73/0.70 & 1.50/0.97/0.80 & 1.57/0.97/0.70 & \textbf{1.83/1.60/1.33} \\
& Reverse Dictionary (30) & 0.11/0.11/0.07 & 1/0.86/0.36 & 1.32/1.00/\textbf{1.00} & \textbf{1.43/1.32}/0.93 \\ \midrule
\multirow{2}{*}{Dialogue} & Single-Turn Dialogue (30) & 1.63/\textbf{0.90}/- & 1.37/0.17/- & 1.40/0.87/- & \textbf{1.83}/0.70/- \\
& Multi-Turn Dialogue (50) & 1.10/0.83/- & 0.80/0.87/- & 1.10/1.03/- & \textbf{1.43/1.33/-} \\ \midrule
\multirow{5}{*}{Text Generation} & Recipe Generation (30) & 0.80/0.63/- & \textbf{1.67}/1.03/- & 1.40/1.03/- & 1.30/\textbf{1.10}/- \\
& Novel Generation (50) & 0.87/0.93/- & 1.20/1.00/- & 1.23/1.03/- & \textbf{1.27/1.13/-} \\
& Professional Manuscripts Generation (50) & 0.97/0.83/- & \textbf{1.37}/1.07/- & 1.23/0.83/- & 1.33/\textbf{1.10}/- \\
& Couplet Generation (30) & 0.73/0.60/- & 0.77/0.86/- & 1.10/0.90/- & \textbf{1.50/1.47/-} \\
& Poetry Generation (30) & 1.80/1.60/- & 1.17/1.00/- & 1.833/1.07/- & \textbf{1.87/1.30/-} \\ \midrule
Summarization & Chinese News Summarization (30) & 1.21/1.10/- & 0.93/0.86/- & 1.24/1.03/- & \textbf{1.41/1.31/-} \\ \midrule
\multicolumn{2}{l}{\textbf{Average}} & 1.03/0.81/0.52 & 1.21/0.95/0.375 & 1.38/1.00/0.58 & \textbf{1.54/1.34/0.88} \\ \bottomrule \bottomrule
\end{tabular}%
}
\caption{The zero-shot generation performance manually evaluated on our collected 450 cases. (we reported the average score of \textit{coherence}, \textit{fluency}, and \textit{accuracy} respectively on a scale of
{[}0, 1, 2{]})}
\label{tab:zero-shot-generation}
\end{table}
\begin{table}[]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{@{}ll@{}}
\toprule \toprule
\textbf{Input (Factural QA)} & \begin{CJK}{UTF8}{gbsn}问题:发明加减消元法的数学家所在国家有哪些品牌汽车？ 答: \end{CJK} \\ \midrule
\textbf{Generation} & \parbox{0.8\textwidth}{\begin{CJK}{UTF8}{gbsn}有很多，比如说奔驰、宝马、奥迪、大众、保时捷。\end{CJK}} \\ \midrule
\textbf{Input (Opinion QA)} & \begin{CJK}{UTF8}{gbsn}问题:宝宝眼睛流泪是怎么回事？ 答:\end{CJK} \\ \cmidrule(l){2-2}
\textbf{Generation} & \parbox{0.8\textwidth}{\begin{CJK}{UTF8}{gbsn}这种情况可能有三个原因：一是宝宝泪腺的炎症。二是宝宝的眼睛有倒睫毛或者睫毛刺激眼球引起流泪。三是宝宝有结膜炎。所以如果宝宝眼睛流泪，首先要排除宝宝倒睫毛刺激眼球引起的流泪，如果不是倒睫毛引起的流泪，还要考虑有没有泪腺炎症或者结膜炎的可能。\end{CJK}} \\ \midrule
\textbf{Input (Reverse Dictornary)} & \begin{CJK}{UTF8}{gbsn}问题:描写"人的品质优良"的词语有哪些？ 答:\end{CJK} \\ \cmidrule(l){2-2}
\textbf{Generation} & \begin{CJK}{UTF8}{gbsn}品质优良,品德高尚,诚实,忠厚,老实\end{CJK} \\ \midrule
\textbf{Input (Novel Generation)} & \parbox{0.8\textwidth}{\begin{CJK}{UTF8}{gbsn}江西孟龙潭与朱孝廉客都中，偶涉一兰若，殿宇禅舍，俱不甚弘敞，惟一老僧挂褡其中。见客入，肃衣出迓，导与随喜。殿中塑志公像，两壁画绘精妙，人物如生。东壁画散花天女，内一垂髫者，拈花微笑，樱唇欲动，眼波将流。\end{CJK}} \\ \cmidrule(l){2-2}
\textbf{Generation} & \parbox{0.8\textwidth}{\begin{CJK}{UTF8}{gbsn}西壁画一小沙弥，面目奇古，双手合十，似在祈祷。客大骇，请问之。僧曰：“此乃佛经也。”客曰：“此何所谓？”僧曰：“此有佛性，无所不有。”客曰：“何以见之？”僧曰：“佛性即是佛性。佛性无所不有，所以不能言说。”客大悟，遂去。\end{CJK}} \\ \midrule
\textbf{Input (Couplet Generation)} & \begin{CJK}{UTF8}{gbsn}上联:口十心思，思乡，思友，思父母 下联:\end{CJK} \\ \cmidrule(l){2-2}
\textbf{Generation} & \begin{CJK}{UTF8}{gbsn}眼四面观，观景，观人，观自己\end{CJK} \\ \midrule
\textbf{Input (News Summarization)} & \parbox{0.8\textwidth}{\begin{CJK}{UTF8}{gbsn}花盆里有黑色小飞虫，这应该是夏天大家都比较头疼的问题。灭小飞虫方法很多，给大家分享几种。电蚊拍电，粘虫板粘，杀虫剂喷，烟蒂水灌根，诱杀法，其他异味水灌根。消灭花盆里的小黑飞虫，可以把烟蒂水灌根和电蚊拍拍打结合起来，坚持一段时间就能彻底消灭。大蒜香烟泡一泡，用这一碗水，小虫去无踪。文章标题是《\end{CJK}} \\ \cmidrule(l){2-2}
\textbf{Generation} & \parbox{0.8\textwidth}{\begin{CJK}{UTF8}{gbsn}花盆里有黑色小飞虫怎么办？》\end{CJK}} \\ \bottomrule \bottomrule
\end{tabular}%
}
\caption{Illustrations of zero-shot generations from ERNIE 3.0.}
\label{tab:gen-examples}
\end{table}
We manually collected 450 cases to evaluate the zero-shot generation ability of current large-scale pre-trained models on 13 tasks from 5 different types including Question Answering, Interpretation, Dialogue, Text Generation and Summarization. In human evaluation, the annotators are asked to score the generation quality on a scale of $[0, 1, 2]$. We reported the average score of \textit{coherence}, \textit{fluency}, and \textit{accuracy} in Tab.~\ref{tab:zero-shot-generation}, and showed some zero-shot generations of ERNIE 3.0 in Tab.~\ref{tab:gen-examples}. ERNIE 3.0 can generate the most coherent, fluent and accurate texts on average as compared to CPM-1, PLUG, PanGu-$\alpha$~\footnote{We use the implementation of CPM-1 in \url{https://github.com/jm12138/CPM-Generate-Paddle}, PLUG in \url{https://nlp.aliyun.com/portal?/BigText_chinese\#/BigText_chinese} and PanGu-$\alpha$ in \url{https://git.openi.org.cn/PCL-Platform.Intelligence/PanGu-Alpha}}. The introduction of three scoring metrics are listed as follows, and the scoring details are provided in Tab.~\ref{tab:score-detail}.
\begin{itemize}
\item \textbf{Coherence} measures whether the generation is relevant and consistent with the context.
\item \textbf{Fluency} evaluates whether the generated text is natural or readable. A fluent text should have no semantic contradiction among the generated text.
\item \textbf{Accuracy} is a metric to evaluate whether the generated text is the same as the ground truth.
\end{itemize}
\begin{table}[]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{@{}c|lll@{}}
\toprule \toprule
\textbf{Score} & \textbf{Coherence} & \textbf{Fluency} & \textbf{Accuracy} \\ \midrule
0 & \begin{tabular}[c]{@{}l@{}}The generation is not related to the context.\\ The generation has obvious conflicts with the context.\end{tabular} & \begin{tabular}[c]{@{}l@{}}The generation is unnatural.\\ There are contradictions in the generated text.\end{tabular} & The answer is wrong. \\ \midrule
1 & \begin{tabular}[c]{@{}l@{}}The generation is weakly related to the context.\\ The generation has minor logic conflicts with the context.\end{tabular} & \begin{tabular}[c]{@{}l@{}}The generation has minor influent part.\\ The generation slightly influences the reading.\end{tabular} & The answer is partly correct. \\ \midrule
2 & \begin{tabular}[c]{@{}l@{}}The generation is strongly related to the context.\\ The logic in the generation is aligned with the context.\end{tabular} & \begin{tabular}[c]{@{}l@{}}The generation is semantically complete and fluent.\\ There are no contradictions in the generated text.\end{tabular} & The answer is correct. \\ \bottomrule \bottomrule
\end{tabular}%
}
\caption{Scoring details for zero-shot generation.}
\label{tab:score-detail}
\end{table}
\subsection{Experiments on SuperGLUE}
\input{super_glue.tex}
As a multi-task benchmark for natural language understanding, SuperGLUE~\cite{wang2019superglue} is usually used to evaluate the performance of pre-training models. We also test the performance of ERNIE 3.0 on SuperGLUE, which covers a diverse range of NLP datasets as follows.
\begin{itemize}
\item BoolQ (Boolean Questions, \cite{chen2018bq}) is a QA task where each example consists of a short passage and a yes/no question about the passage. The task is to answer the questions with YES or NO, and the metric of this task is accuracy.
\item CB (Commitment Bank, \cite{demarneffe:cb}) is an imbalanced corpus of natural language inference task. The task is evaluated using accuracy and macro-F1.
\item COPA (Choice of Plausible Alternatives \cite{roemmele2011choice}) is a causal reasoning task based on common sense knowledge. The data are curated from blogs and a photography-related encyclopedia. Following the original work, we evaluate this task using accuracy.
\item MultiRC (Multi-Sentence Reading Comprehension \cite{khashabi2018looking}) is a QA task where each example consists of a context paragraph, a question about that paragraph, and a list of possible answers. The system must predict which answers are true and which are false. The evaluation metrics are F1 over all answer-options (F$1_{a}$) and exact match of each question’s set of answers (EM).
\item ReCoRD (Reading Comprehension with Commonsense Reasoning Dataset, \cite{zhang2018record}) is a multiple-choice QA task. It requires the model to pick an entity to complete the answer, given a context of news article and a Cloze-style question. This task is evaluated with max (over all mentions) token-level F1 and exact match.
\item RTE (Recognizing Textual Entailment \cite{dagan2006pascal}) dataset comes from a series of annual competitions on textual entailment. It is a natural language inference corpus and evaluated with accuracy.
\item WiC (Word-in-Context \cite{pilehvar2018wic}) is a word sense disambiguation task cast as binary classification of sentence pairs using accuracy as the evaluation metrics.
\item WSC (Winograd Schema Challenge \cite{levesque2011winograd}) is a coreference resolution task in which examples consist of a sentence with a pronoun and a list of noun phrases from the sentence as choices. The system must select the correct referent of the pronoun from the provided choices. This task is evaluated with accuracy.
\end{itemize}
Similar to the pre-training corpus used in RoBERTa~\cite{liu2019roberta} and DeBERTa~\cite{he2020deberta}, we compiled the English pre-training corpus for ERNIE 3.0 including English Wikipedia, BookCorpus~\cite{zhu2015aligning}, CC-News~\cite{ccnews}, OpenWebText~\cite{openwebtext}, Stories~\cite{trinh2018simple}. As shown in the Table \ref{tab:superglue}, ERNIE 3.0 surpasses T5~\cite{T5} and DeBERTa~\cite{he2020deberta} and obtains a score of 90.6, taking the first place in SuperGLUE Benchmark.
\section{Analysis}\label{sec:analysis}
\begin{wrapfigure}{r}{0.4\textwidth}
\begin{center}
\includegraphics[width=0.4\textwidth]{ablation-twobranch-vs-singlebranch.pdf}
\end{center}
\caption{Perplexity variation of the NLG pre-training task with respect to training steps.}
\label{fig:ablation-branch}
\end{wrapfigure}
\paragraph{The Effectiveness of the Task-specific Representation Modules}
To verify the effectiveness of the task-specific networks, we compare our proposed structure with those which share parameters under various pre-training tasks. For the ablation test, we choose understanding and generation as two different training paradigms and utilize the corresponding tasks mentioned in Section \ref{sec:pre-training tasks}. The unified network follows the base model settings (12 layers, 768 dims, 12 attention heads), and the task-specific networks for each task paradigms are set to 3 layers, 256 dims, and 4 attention heads. For the contrast model, the task-specific network is shared across different task paradigms. Figure \ref{fig:ablation-branch} illustrates the perplexity variation of the NLG task during the pre-training process.
As shown in Figure \ref{fig:ablation-branch}, the model with its own task-specific network for different task paradigms reaches a higher convergence speed. Furthermore, as training progresses, the performance gap becomes bigger compared to the model with a shared task-specific network. The experimental result shows the effectiveness of the proposed task-specific networks and demonstrates the necessity of distinguishing different tasks.
\paragraph{Universal Knowledge-Text Prediction}
A group of ablation experiments is conducted to evaluate the performance of the universal knowledge-text prediction task. The relation extraction task is a typical knowledge-driven task, aiming to predict the relationship between two entities mentioned in a given sentence. Specifically, we add four special tokens, \textsc{[HD]}, \textsc{[/HD]}, \textsc{[TL]} and \textsc{[/TL]} to identify the mention of a head entity and a tail entity respectively, then the relation classification is performed on the sum of the final representations of the aforementioned four special tokens. We construct the experiments on SanWen and FinRE datasets and as shown in Table \ref{tab:uktp}, the knowledge enhancement strategy achieves impressive empirical performance on the relation extraction task.
\begin{wraptable}{r}{0.45\textwidth}
\centering
\begin{tabular}{lll}
\toprule \toprule
\textbf{Dataset} & \textbf{ERNIE$_\text{Base}$} & \textbf{ERNIE$_\text{Base}$+UKTP} \\
\midrule
SanWen & 75.56 &77.36(+1.80) \\
FinRE & 58.19 &59.75(+1.56) \\
\bottomrule \bottomrule
\end{tabular}
\caption{Ablation experiments of universal knowledge-text prediction task.}
\label{tab:uktp}
\end{wraptable}
In addition, the zero-shot generation experiment on CKBQA also confirms the effectiveness of the universal knowledge-text prediction task. Specifically, the knowledge-based question answering (KBQA) task requires a model to search and reason for correct answers based on a knowledge graph. It's suitable to measure the knowledge learning capability of the pre-trained languages models using the KBQA task. We use the "QUESTION: \$QUESTION? ANSWER:" as the prompt for zero-shot learning and then compare the performance of our proposed model with several state-of-the-art pre-trained language models on the CKBQA dataset. As shown in Table \ref{tab:zero-shot-result}, ERNIE 3.0 significantly outperforms PanGu-$\alpha$ and CPM-1 in the CKBQA dataset which indicates that ERNIE 3.0 has the ability to memorize and learn more knowledge.
\paragraph{Progressive Learning to Speed up Convergence}
We record the training convergence speed on two architecture settings including $\mathrm{ERNIE_{Base}}$ and $\mathrm{ERNIE_{1.5B}}$, in which the architecture settings of $\mathrm{ERNIE_{Base}}$ follows \cite{sun2019ernie} and
\begin{wraptable}{r}{0.45\textwidth}
\centering
\begin{tabular}{ll}
\toprule \toprule
\textbf{Method} & \textbf{Training Time} \\
\midrule
ERNIE$_\text{Base}$ & 11h30m \\
+Progressive Learning & 4h(-65.21\%) \\
\midrule
ERNIE$_\text{1.5B}$ & 5h55m \\
+Progressive Learning & 3h4m(-48.2\%) \\
\bottomrule \bottomrule
\end{tabular}
\caption{Progressive Learning To Speedup Training.}
\label{tab:4dpro}
\end{wraptable}
$\mathrm{ERNIE_{1.5B}}$ model consists of 48 layers with a hidden size of 1,536 and 24 attention heads. As shown in Tab.~\ref{tab:4dpro}, we record the time for the loss value of the model converges to the same as that of the ERNIE 3.0. For the $\mathrm{ERNIE_{Base}}$ model, the convergence time is reduced by 65.21\% from 11 hours to 4 hours, and for the $\mathrm{ERNIE_{1.5B}}$, the convergence time is reduced by 48\%. For both two settings, we carry out pre-training on 8 NVIDIA Tesla V100 GPUs. For $\mathrm{ERNIE_{Base}}$, we increased the batch size from 8 to 2048 and the sequence length from 128 to 512, the learning rate increases linearly from 0 to 1e-4, and the dropout keeps 0 in the progressive warmup stage. For $\mathrm{ERNIE_{1.5B}}$, we gradually increase the batch size from 8 to 8192, The learning rate increases from 0 to 6e-4, the dropout also keeps 0. The rest settings for the experiment remain as same as \cite{sun2019ernie}. For $\mathrm{ERNIE_{1.5B}}$, to achieve the peak batch size within the constraint of GPU memory, the gradient accumulation strategy is used during the pre-training.
\section{Conclusion}
we proposed the ERNIE 3.0 framework to pre-train a knowledge enhanced 10-billion parameter model on a 4TB corpus including plain texts and a knowledge graph. In order to handle both language understanding and generation tasks with zero-shot learning, few-shot learning and fine-tuning, ERNIE 3.0 designs a unified pre-training framework that integrates both auto-encoder networks and auto-regressive networks. We construct extensive experiments on various datasets from different task paradigms and fields, and the results demonstrate the effectiveness of ERNIE 3.0 as compared to the previous state-of-the-art pre-trained models.
\bibliographystyle{unsrt}
|
1,116,691,500,891 | arxiv |
\section{CDF II detector and trigger} \label{sec:ss-det}
CDF~II is a multipurpose detector, equipped with a charged particle
spectrometer and a finely segmented calorimeter. In this section, we
describe the detector components that are relevant to this analysis.
The description of these subsystems can be found in
Refs.~\cite{det1,det2,det3_0,det3,det4_0,det4,det5,det6,det7,det8}.
Two devices inside the 1.4 T solenoid are used for measuring the momentum
of charged particles: the silicon vertex detector (SVXII and ISL) and the
central tracking chamber (COT). The SVXII detector consists of
microstrip sensors arranged in six cylindrical shells with radii between
1.5 and 10.6 cm, and with a total $z$ coverage~\footnote{
In the CDF coordinate system, $\theta$ and $\phi$ are the polar and
azimuthal angles of a track, respectively, defined with respect to the
proton beam direction, $z$. The pseudorapidity $\eta$ is defined as
$-\ln \;\tan (\theta/2)$. The transverse momentum of a particle is
$p_T= p \; \sin (\theta)$. The rapidity is defined as
$y=1/2 \cdot \ln ( (E+p_z)/(E-p_z) )$, where $E$ and $p_z$ are the
energy and longitudinal momentum of the particle associated with the track.}
of 90 cm. The first SVXII layer, also referred to as the L00 detector,
is made of single-sided sensors mounted on the beryllium beam pipe.
The remaining five SVXII layers are made of double-sided sensors and
are divided into three contiguous five-layer sections along the beam
direction $z$. The vertex $z$-distribution for $p\bar{p}$ collisions is
approximately described by a Gaussian function with a rms of 28 cm.
The transverse profile of the Tevatron beam is circular and has a rms
spread of $\simeq 25\; \mu$m in the horizontal and vertical directions.
The SVXII single-hit resolution is approximately $11\; \mu$m and allows
a track impact parameter
resolution of approximately $35\; \mu$m, when
also including the effect of the beam transverse size. The two additional
silicon layers of the ISL help to link tracks in the COT to hits in the
SVXII. The COT is a cylindrical drift chamber containing 96 sense wire
layers grouped into eight alternating superlayers of axial and stereo
wires. Its active volume covers $|z| \leq 155$ cm and 40 to 140 cm in
radius. The transverse momentum resolution of tracks reconstructed using
COT hits is $\sigma(p_T)/p_T^2 \simeq 0.0017\; [\gevc]^{-1}$. The trajectory
of COT tracks is extrapolated into the SVXII detector, and tracks are
refitted with additional silicon hits consistent with the track extrapolation.
The central muon detector (CMU) is located around the central electromagnetic
and hadronic calorimeters, which have a thickness of 5.5 interaction lengths
at normal incidence. The CMU detector covers a nominal pseudorapidity range
$|\eta| \leq 0.63$ relative to the center of the detector, and is segmented
into two barrels of 24 modules, each covering 15$^\circ$ in $\phi$. Every
module is further segmented into three submodules, each covering
4.2$^\circ$ in $\phi$ and consisting of four layers of drift chambers.
The smallest drift unit, called a stack, covers a 1.2$^\circ$ angle in
$\phi$. Adjacent pairs of stacks are combined together into a tower.
A track segment (hits in two out of four layers of a stack) detected in
a tower is referred to as a CMU stub. A second set of muon drift chambers
(CMP) is located behind an additional steel absorber of 3.3 interaction
lengths. The chambers are 640 cm long and are arranged axially to form a
box around the central detector. The CMP detector covers a nominal
pseudorapidity range $|\eta| \leq 0.54$ relative to the center of the
detector. Muons which produce a stub in both the CMU and CMP systems are
called CMUP muons. The CMX muon detector consists of eight drift chamber
layers and scintillation counters positioned behind the hadron calorimeter.
The CMX detector extends the muon coverage to $|\eta| \leq 1$ relative to
the center of the detector.
The luminosity is measured using gaseous Cherenkov counters (CLC) that
monitor the rate of inelastic $p\bar{p}$ collisions. The inelastic
$p\bar{p}$ cross section at $\sqrt{s}=1960$ GeV is scaled from measurements
at $\sqrt{s}=1800$ GeV using the calculations in Ref.~\cite{sigmatot}.
The integrated luminosity is determined with a 6\% systematic
uncertainty~\cite{klimen}.
CDF uses a three-level trigger system. At Level 1 (L1), data from every
beam crossing are stored in a pipeline capable of buffering data from 42
beam crossings. The L1 trigger either rejects events or copies them into
one of the four Level 2 (L2) buffers. Events that pass the L1 and L2
selection criteria are sent to the Level 3 (L3) trigger, a cluster of
computers running speed-optimized reconstruction code.
For this study, we select events with two muon candidates identified by the
L1 and L2 triggers. The L1 trigger uses tracks with $p_T \geq 1.5 \; \gevc$
found by a fast track processor (XFT). The XFT examines COT hits from the
four axial superlayers and provides $r-\phi$ information in azimuthal
sections of 1.25$^\circ$. The XFT passes the track information to a set of
extrapolation units that determine the CMU towers in which a CMU stub
should be found if the track is a muon. If a stub is found, a L1 CMU
primitive is generated. The L1 dimuon trigger requires at least two CMU
primitives, separated by at least two CMU towers. The L2 trigger
additionally requires that at least one of the muons also has a CMP stub
matched to an XFT track with $p_T \geq 3 \;\gevc$. All these trigger
requirements are emulated by the detector simulation on a run-by-run basis.
The L3 trigger requires a pair of CMUP muons with invariant mass larger
than $5 \; {\rm GeV/}c^2$, and $|\delta z_0| \leq 5$ cm, where $z_0$ is the $z$
coordinate of the muon track at its point of closest approach to the beamline
in the $r-\phi$ plane. These requirements define the dimuon trigger
used in this analysis.
\section{Present understanding of the dimuon sample composition}
\label{sec:ss-expsit}
The value of
$\sigma_{b\rightarrow\mu,\bar{b}\rightarrow \mu}$
and $\sigma_{c\rightarrow\mu,\bar{c}\rightarrow \mu}$, the correlated cross sections
for producing pairs of central heavy-flavored quarks that decay semileptonically, is
derived in Ref.~\cite{bbxs} by
fitting the
impact parameter~\cite{d0} distribution of the primary muons with the
expected shapes from all sources believed to be significant:
semileptonic heavy flavor decays, prompt
quarkonia decays, Drell-Yan production, and instrumental backgrounds
due to punchthrough of prompt or heavy-flavored hadrons which mimic a muon signal~\cite{strange}.
In the following, the sum of
these processes
will be referred to as the prompt plus heavy flavor ($P+HF$) contribution.
The notation $K^{puth} \rightarrow \mu$ and
$\pi^{puth} \rightarrow \mu$ will be used to indicate muon signals mimicked
by punchthrough of kaons and pions, respectively.
In order to properly model the data with the templates of the
various $P+HF$ sources, the study in Ref.~\cite{bbxs} has used strict
selection criteria, referred to as tight SVX selection in the following,
by requiring muon tracks with hits in the two innermost
layers of the SVX
detector, and in at least two of the next four outer layers.
The tight SVX requirements select events in
which both muons arise from parent particles that have decayed within
a distance of $\simeq 1.5$ cm from the $p\bar{p}$ interaction primary
vertex in the plane transverse to the beamline.
This requirement suppresses the yield of primary muons due to
in-flight-decays of pions and kaons, in the following referred to as
$\pi^{ifd} \rightarrow \mu$ and $K^{ifd} \rightarrow \mu$, respectively.
This type of contribution to the dimuon dataset prior to any SVX requirement
was considered negligible
in previous~\cite{2mucdf,bmix} and recent~\cite{williams} studies by the CDF and
D0 collaborations.
As shown by Fig.~\ref{fig:figbb_6}, the tight SVX
sample is well modeled by fits using the prompt and heavy flavor
contributions~\cite{bbxs}.
The sample composition determined by the fit and corrected for the
appropriate efficiency of the tight SVX requirements~\footnote{
The efficiency of the tight SVX selection has been measured~\cite{a0disc} to be $0.257 \pm 0.004$ for
prompt dimuons and $0.237\pm0.001$ for dimuons produced by heavy flavor decays
by using control
samples of data from various sources
($J/\psi\rightarrow \mu^+\mu^-$, $B^\pm \rightarrow \mu^+ \mu^- K^\pm$,
$B \rightarrow \mu D^0$, and $\Upsilon\rightarrow \mu^+\mu^-$).}
is listed in
the first two columns of Table~\ref{tab:tab_1}.
\begin{table}[htp]
\caption[]{Number of events attributed to the different dimuon sources by
the fit to the muon impact-parameter distribution.
The fit parameters $BB$, $CC$, and $PP$ represent
the $b\bar{b}$, $c\bar{c}$, and prompt dimuon contributions,
respectively. The component $BC$ represents events containing
$b$ and $c$ quarks. The fit parameter $BP$ ($CP$) estimates
the number of events in which there is only one $b$ ($c$) quark
in the detector acceptance and the second muon is produced by
prompt hadrons in the recoiling jet that mimic a muon signal.
Real muons are muons from semileptonic decay of heavy flavors,
Drell-Yan production or quarkonia decays.
The data correspond to an
integrated luminosity of 742 pb$^{-1}$. The dimuon data set consists of
743006 events.}
\begin{center}
\begin{ruledtabular}
\begin{tabular}{lccc}
Component & No. of Events & No. of real $\mu-\mu$ & No. and type of misidentified $\mu$ \\
$BB$ & $230308 \pm 2861$ & $R_{bb} \times BB $ &
$ R_{bb} \times [7902 \; K^{puth} \rightarrow \mu $ \\
& & & +8145 $ \pi^{puth} \rightarrow \mu$ ] \\
$CC$ & $103198 \pm 6603$ & $ R_{cc} \times CC$ &
$ R_{cc} \times [17546 \; K^{puth} \rightarrow \mu$ \\
& & & + 9535 $\pi^{puth} \rightarrow \mu$] \\
$PP$ & $161696 \pm 2533$ & $\Upsilon=51680 \pm 649$ & 4400 $K^{puth} \rightarrow \mu\; K^{puth} \rightarrow \mu$ \\
& & + $DY=54200 \pm 5420$ & +30000 $\pi^{puth} \rightarrow \mu\; \pi^{puth}\rightarrow \mu$ \\
& & & + 23000 $K^{puth} \rightarrow \mu\; \pi^{puth} \rightarrow \mu$ \\
$BP$ & $~43096 \pm 3087$ & &11909 $K^{puth} \rightarrow \mu$ + 29253 $\pi^{puth} \rightarrow \mu $ \\
$CP$ & $~41582 \pm 5427$ & & 16447 $K ^{puth} \rightarrow \mu$ + 35275 $\pi ^{puth} \rightarrow \mu $ \\
$BC$ & $~~9135 \pm 2924$ & & \\
$P+HF$ & $589015 \pm 5074$ & & \\
\end{tabular}
\end{ruledtabular}
\end{center}
\label{tab:tab_1}
\end{table}
\begin{figure}[htp]
\begin{center}
\vspace{-0.2in}
\leavevmode
\includegraphics*[width=0.5\textwidth]{fig_6.eps}
\caption[]{The projection of the two-dimensional impact parameter
distribution of muon pairs onto one of the two axes is
compared to the fit result (histogram).}
\label{fig:figbb_6}
\end{center}
\end{figure}
The difference between the total number of dimuons and the
$P+HF$ component indicates the presence of an important source of dimuons
produced beyond 1.5 cm which is
suppressed by the tight SVX requirements.
Because unnoticed by previous experiments, this
source was whimsically referred to as the ghost contribution.
The relative size of the ghost and $P+HF$ contributions depends upon the type
of SVX requirement applied to the trigger muons.
Reference~\cite{a0disc} shows that neglecting the presence of ghost events
affected previous measurements of
$\sigma_{b\rightarrow\mu,\bar{b}\rightarrow \mu}$~\cite{2mucdf,d0b2}
and of $\bar{\chi}$~\cite {bmix} at the Tevatron.
Finally, the ghost sample is shown
to be the source of the dimuon invariant mass discrepancy
observed in Ref.~\cite{dilb}.
Reference~\cite{a0disc} has studied a number of potential sources
of muons originating beyond the beam pipe.
Contrary to what assumed by previous experiments, the one source
found to contribute significantly arises
from in-flight-decays of pions and kaons.
Based upon a generic QCD simulation,
that study estimates a contribution of 57000 events.
A smaller
contribution ($ 12052\pm 466$ events) from $K^0_S$ and hyperon decays in
which the punchthrough of a hadronic prong mimics a muon signal was estimated
using the data. Secondary inelastic interactions in the tracking volume were
found to be a negligible source of ghost events.
The final estimate of the size of possible sources of ghost events
underpredicts
the observed number by approximately a factor of two (154000
observed and 69000 accounted for), but the
difference was not considered significant because
of the simulation uncertainty.
The present study uses events selected with the tight SVX requirements to tune the
QCD simulation. Since these data are well modeled by the impact parameter templates of the
$P+HF$ components, misidentified muons can only arise from the punchthrough of prompt hadrons or
hadrons produced by heavy-flavor decays. The numbers of misidentified muons
in the data are derived
by subtracting the expected number of real muons, listed in the third
column of Table~\ref{tab:tab_1}, from the corresponding
components in the second column. We then compare
these differences to the rate of $K^{puth} \rightarrow \mu$ and
$\pi^{puth} \rightarrow \mu$ misidentifications predicted by the simulation, and
listed on the fourth column of the same table.
The simulation is tuned by adjusting the predicted rate of pions and kaons
to reproduce the observed number of
muon misidentifications. Then, the tuned simulation
is used to predict the number of muons due to in-flight-decays with 5\% accuracy.
\section{Rates of misidentified muons in the data and simulation} \label{sec:ss-anal}
We make use of three different samples of simulated events
generated with the {\sc herwig} parton-shower Monte-Carlo program~\cite{herwig},
the settings of which are described in Appendix~A
of Ref.~\cite{bbxs}.
We use option 1500 of the {\sc herwig} program to generate final states
produced by hard scattering of partons with transverse momentum larger
than 3 $\gevc$ (sample A=generic QCD).
Hadrons with heavy flavors are subsequently
decayed using the {\sc evtgen} Monte Carlo program~\cite{evtgen}.
The detector response to particles produced by the above generators
is modeled with the CDF~II detector simulation that in turn is based on
the {\sc geant} Monte Carlo program~\cite{geant}.
The values of the heavy flavor cross sections
predicted by the generator are scaled to the measured values
$\sigma_{b\rightarrow\mu,\bar{b}\rightarrow \mu}= 1549 \pm 133$ pb
and $\sigma_{c\rightarrow\mu,\bar{c}\rightarrow \mu}= 624 \pm 104$ pb~\cite{bbxs}.
The next simulated sample (sample B=single $b$ + single $c$) is extracted from A by
requiring the presence of at least a
trigger muon generated from heavy-flavor semileptonic decays.
The simulated sample C=$b\bar{b}$ + $c\bar{c}$ is extracted from B by requiring the presence of at least two
trigger muons generated from heavy flavor decays. This sample has been used
to construct impact parameter templates and estimate
kinematic acceptances in Ref.~\cite{bbxs}.
In the various simulations, we evaluate the number of dimuons from heavy flavor decays
and the number of pairs of tracks of different type that pass the same kinematic selection.
The ratios of these numbers are listed in Tables~\ref{tab:tab_2} to~\ref{tab:tab_4}.
The rate of pairs of tracks of different type predicted by the simulation are normalized to the data by
multiplying these ratios by the number of dimuons from $b\bar{b}$ or $c\bar{c}$ production
observed in the data.
\begin{table}[htp]
\caption[]{Ratio of the numbers of $\pi\pi$, $KK$, and $K\pi$ pairs to that of
primary dimuons from $b\bar{b}$ decays (221096 pairs) in the generic QCD
simulation (sample A).}
\begin{center}
\begin{ruledtabular}
\begin{tabular}{lccc}
Process& $R_{KK}$ & $R_{K\pi}$ & $R_{\pi\pi}$ \\
generic QCD & 867 & 8935 & 22913 \\
\end{tabular}
\end{ruledtabular}
\end{center}
\label{tab:tab_2}
\end{table}
\begin{table}[htp]
\caption[]{Ratio of the numbers of $\mu-K(\pi)$ combinations to that of
primary dimuons from $b\bar{b}$ and $c\bar{c}$ production in the single-$b$
and single-$c$ simulated samples (221096 and 83590 dimuons, respectively).}
\begin{center}
\begin{ruledtabular}
\begin{tabular}{lcc}
Process & $R_K$ & $R_\pi$ \\
single $b$ & 11.1 & ~54.4 \\
single $c$ & 40.7 & 173.7 \\
\end{tabular}
\end{ruledtabular}
\end{center}
\label{tab:tab_3}
\end{table}
\begin{table}[htp]
\caption[]{Ratio of the numbers of $\mu-K(\pi)$ combinations to that of
primary dimuons from heavy flavor production in the $b\bar{b}$
and $c\bar{c}$ simulated samples (221096 and 83590 dimuons, respectively).}
\begin{center}
\begin{ruledtabular}
\begin{tabular}{lcc}
Process & $R_K$ & $R_\pi$ \\
$b\bar{b}$ & ~7.4 & 15.2 \\
$c\bar{c}$ & 43.5 & 46.9 \\
\end{tabular}
\end{ruledtabular}
\end{center}
\label{tab:tab_4}
\end{table}
The probability $P^{puth}_{K(\pi)}$ that a kaon (pion) is not contained by the calorimeter
and mimics a muon signal
has been measured in Ref.~\cite{bbxs} by using
kaons and pions from $D^{*\pm} \rightarrow \pi^\pm D^0$ with $D^0 \rightarrow K^+ \pi^-$ decays.
The probability that kaon (pion) in-flight-decays mimic a trigger
muon, $P^{ifd}_{K(\pi)}$, has been derived in Ref.~\cite{a0disc} by using
the simulated sample C. These probabilities depend on the particle transverse momentum.
Table~\ref{tab:tab_5} lists the average probabilities that kaons (pions) mimic
a primary muon when applying the $P^{puth}_{K(\pi)}$ and $P^{ifd}_{K(\pi)}$
probabilities to
simulated kaon (pion) tracks
with $p_T \geq 3 \; \gevc$ and $|\eta|<0.7$.
\begin{table}[htp]
\caption[]{ Average probabilities (\%) that punchthroughs or in-flight decays result into
a primary muon. The $p_T$ distribution of kaons and pions in the different simulations are
almost indistinguishable.}
\begin{center}
\begin{ruledtabular}
\begin{tabular}{cccc}
$<P^{puth}_K> $ & $<P^{puth}_{\pi}>$& $<P^{ifd}_K>$ & $<P^{ifd}_\pi>$ \\
$0.483\pm 0.003$ & $0.243 \pm.0.004$ & $0.345\pm 0.005$ & $0.0727\pm 0.0016$ \\
\end{tabular}
\end{ruledtabular}
\end{center}
\label{tab:tab_5}
\end{table}
By weighting simulated pion (kaon) tracks that
pass the muon kinematic selection with the corresponding
$P^{puth}_{K(\pi)}$ probability, we obtain the prediction
of misidentified primary muons for the various $P+HF$ components that is
listed in the fourth column of Table~\ref{tab:tab_1}.
The third column of the same table lists the number of real muons for the various
$P+HF$ contributions.
The sum of real plus misidentified muon pairs is in general agreement
with the data listed in the second column of the table.
Therefore, it is reasonable to use the observed rate of dimuons, the
knowledge of the fraction of real dimuons due to semileptonic decay of heavy flavors,
Drell-Yan or $\Upsilon$ mesons, and the knowledge of the $P^{puth}_{K(\pi)}$ probabilities
to normalize the absolute yields of pions and kaons
predicted by the simulation. The simulation fitted to the data is then used to predict
the rate of events due to in-flight-decay misidentifications
by weighting simulated tracks with the $P^{ifd}_{K(\pi)}$ probabilities,
the average of which is listed in Table~\ref{tab:tab_5}.
In addition, the total rate of $K \rightarrow \mu= K^{puth} \rightarrow \mu + K^{ifd} \rightarrow \mu$
misidentifications predicted by the simulation
can be further constrained with data.
This is done in the next section
by using the number of primary muons
due to misidentification of $K^{*0}$, $K^{*\pm}$, and
$K^0_S$ decays.
We first describe the evaluation of the content of real muons in the various $P+HF$ components
and the function used to fit the simulation to the data.
Reference~\cite{bbxs}
estimates that the fraction $R_{bb}=0.96 \pm 0.04$ of the $BB$
component is due to real muons from $b$-quark semileptonic decays whereas
the remaining 4\% is due to muons mimicked by the punchthrough of hadrons produced by
heavy flavor decays.
Similarly, the fraction $R_{cc}=0.81 \pm 0.09$ of the $CC$
component is due to real muons from $c$-quark semileptonic decays whereas
the remaining 19\% is due to muons mimicked by
the punchthrough of hadrons produced by heavy flavor decays.
The uncertainty of the fraction of real muons due to $b\bar{b}$ ($c\bar{c}$)
production is accounted for by multiplying $R_{bb}$ ($R_{cc}$) by
the fit parameter $f_{bb}$ ($f_{cc}$) constrained to 1 with a 4\% (11\%) Gaussian error.
The number of $\Upsilon$ mesons contributing to the $PP$ component ($\Upsilon=51680 \pm 649$ candidates)
has been determined
in Ref.~\cite{bbxs} by fitting the dimuon invariant mass spectrum with
three Gaussian functions to model the signal
and a straight line to model the combinatorial background. The Drell-Yan contribution is evaluated as
$DY=\Upsilon \times \sigma_{DY}/\sigma_{\Upsilon}$. The cross section $\sigma_{DY}$
in the $5-80\, {\rm GeV/}c^2$ mass range is evaluated with a NLO calculation~\cite{drelly}, and we use
the measured value of $\sigma_{\Upsilon}$~\cite{yxs}.
The ratio $\sigma_{DY}/\sigma_{\Upsilon}$ is 1.05 with a 10\% error mostly due to the
measurement in Ref.~\cite{yxs}.
To account for the uncertainty, we weight the $DY$ contribution with the fit parameter
$f_{dy}$ constrained to 1 with a 10\% Gaussian error.
The magnitude of the $BP$ ($CP$) component, predicted with the single-$b$ (single-$c$)
simulation, with respect to that of the $BB$ ($CC$) contribution
depends on the ratio of NLO to LO
terms evaluated by the {\sc herwig} generator. Because of the dependence on the renormalization
and factorization scales, the uncertainty of the single-$b\;(c)$ cross section to that of the
$b\bar{b}\; (c\bar{c})$ cross section is estimated~\cite{mnr} to be $\simeq$ 20 (30)\%~\footnote{
However, the study in Ref.~\cite{ajets} shows that the {\sc herwig} generator
predicts the observed single and correlated
heavy-flavor cross sections to better than 10\%.}.
We account for this uncertainty by
weighting the rate of pion and kaon tracks predicted by the single-$b$ (single-$c$)
simulation with the additional fit parameter $f_{sb}$ ($f_{sc}$)
constrained to 1 with a 20\% (30\%) Gaussian error.
The simulation prediction of the number of muons mimicked by the punchthrough of pions (kaons)
is weighted with the fit free parameter $f_{\pi}$ ($f_{K}$).
These fit parameters provide the absolute normalization of
the pion (kaon) rate predicted by the simulation
including the uncertainties of the punchthrough probabilities.
\section{Measurement of the $K \rightarrow \mu$ contribution}
\label{sec:ss-tkaon}
The small rate of $K \rightarrow \mu = K^{puth} \rightarrow \mu + K^{ifd} \rightarrow \mu$
misidentifications is measured using a higher
statistics sample of dimuon events corresponding to an integrated luminosity of 3.9 fb$^{-1}$.
The number of $K \rightarrow \mu$ misidentification, $N_K$,
is derived from $N_{K^{*0}}$, the number of identified $K^{*0} \rightarrow K^+\pi^-$ decays
with $K^+\rightarrow \mu^+$ (and charge-conjugate states).
The number $N_{K^{*0}}$ is related to $N_K$ by
$$N_{K^{*0}}=N_K \cdot \epsilon_0 \cdot R(K^{*0}),$$
where $R(K^{*0})$ is the fraction of kaons that result from $K^{*0} \rightarrow K^+\pi^-$ decays
and $\epsilon_0$ is the efficiency to reconstruct the pion.
We also select $K^0_S \rightarrow \pi^+ \pi^-$ with $\pi \rightarrow \mu$
candidates and reconstruct $K^{*\pm} \rightarrow K^0_S \pi^\pm$ decays.
The number of $K^{*\pm}$ is related to that of $K^0_S$ by
$$N_{K^{*\pm}}=N_{K^0_S} \cdot \epsilon_1 \cdot R(K^{*\pm}),$$
where $R(K^{*\pm})$ is the fraction of $K^0_S$ resulting from $K^{*\pm} \rightarrow K^0_S \pi^\pm$
decays and $\epsilon_1$ is the efficiency to reconstruct the additional pion.
We use isospin invariance to set $R(K^{*\pm})= R(K^{*0})$. Since the additional pion
used to search for the $K^{*\pm}$ and $K^{*0}$ candidates is selected with
the same kinematic requirements, we set
$\epsilon_0=\epsilon_1$. It follows that
$$N_K = N_{K^0_S}/N_{K^{*\pm}} \times N_{K^{*0}}.$$
We search for $K^{*0}$ decays
by combining primary muons, assumed to be kaons, with all opposite charge tracks,
assumed to be pions, with $p_T \geq 0.5 \; \gevc$
and in an angular cone with $\cos \theta \geq 0.6$ around
the direction of the primary muon.
We require tracks with at least 10 axial and 10 stereo COT hits.
We constrain the pair to arise from a common three-dimensional point, and
reject combinations if the probability of the vertex-constrained fit is smaller than 0.001.
The invariant mass spectrum of the selected $K^{*0} \rightarrow K^+\pi^-$
candidates is shown in Fig.~\ref{fig:fig_2}.
\begin{figure}[htp]
\begin{center}
\vspace{-0.2in}
\leavevmode
\includegraphics*[width=0.5\textwidth]{m1_1.eps}
\caption[]{Invariant mass distribution of $K^{*0} \rightarrow K^+\pi^-$ candidates
passing our selection criteria. The line represents the fit described in the text.}
\label{fig:fig_2}
\end{center}
\end{figure}
We fit the invariant mass distribution with a Breit-Wigner function smeared by the
detector mass resolution to model the signal.
We fix the mass and width
of the Breit-Wigner function to 896 and 51 MeV/$c^2$~\cite{pdg}, respectively~\footnote{
The mass resolution due to the track reconstruction in
simulated events which include kaon in-flight-decays
is 4.9 MeV/$c^2$, and is negligible compared to the resonance width.}.
We use a fourth order polynomial to model the combinatorial background under
the signal, and the fitted range of invariant mass is
conveniently chosen to yield a fit with 50\% probability.
The size of the signal is not affected by the arbitrary
choice of the function used to model the combinatorial
background or of the fitted mass range, and is solely determined by the accurate knowledge of the
signal shape.
The fit yields $N_{K^{*0}}=87471 \pm 2217$ $K^{*0}$ mesons.
We search for
$K^0_S \rightarrow \pi^+ \pi^-$ with a $\pi \rightarrow \mu$ misidentification
by combining primary muons
with tracks passing the same requirements as those used in the $K^{*0}$ search.
In this case, both tracks are assumed to be pions.
As in the previous case, we select pairs consistent with arising from a common three-dimensional vertex.
We take advantage of the $K^0_S$ long lifetime to suppress the combinatorial background.
We further require that the distance between the $K^0_S$ vertex and the event primary vertex,
corrected by the $K^0_S$ Lorentz boost, corresponds to $ct>0.1$ cm.
The invariant mass of the $K^0_S$ candidates is shown in Fig.~\ref{fig:fig_3}.
\begin{figure}[htp]
\begin{center}
\vspace{-0.2in}
\leavevmode
\includegraphics*[width=0.5\textwidth]{v1_1.eps}
\caption[]{Invariant mass distribution of $K^0_S$ candidates
passing our selection criteria. The line represents the fit described in the text.}
\label{fig:fig_3}
\end{center}
\end{figure}
We fit the signal with two Gaussian functions and the combinatorial background with a straight line
in the mass range $0.4-0.6\; {\rm GeV/}c^2$.
Having fixed the peak of the Gaussian functions at 0.497 $ {\rm GeV/}c^2$~\cite{pdg}, the fit returns an averaged
$\sigma$ of 8.4 MeV/$c^2$, consistent with what is expected from simulated events~\footnote{
Because of the $K^0_S$ long decay path, reconstructed track segments may be shorter than the
available tracking detector length. When $K^0_S$ mesons decay before entering the
COT volume, the mass resolution is 4 MeV/$c^2$.},
and a signal of $32445 \pm 421$ $K^0_S$ mesons
in the mass range $0.474-0.522 \; {\rm GeV/}c^2$.
We search for $K^{*\pm}$ by combining $K^0_S$ candidates with mass between 0.474 and 0.522 $ {\rm GeV/}c^2$
and $ct>0.1$ cm with any additional track, assumed to be a pion, that pass the same selection as
pion tracks used to find $K^{*0}$ candidates. We constrain the $K^0_S$ mass to 0.497 $ {\rm GeV/}c^2$ and
require that the $K^0_S$ candidate and the pion track are consistent with
arising from a common three-dimensional vertex.
The invariant mass distribution of $K^{*\pm}$ candidates is shown in Fig.~\ref{fig:fig_4}.
\begin{figure}[htp]
\begin{center}
\vspace{-0.2in}
\leavevmode
\includegraphics*[width=0.5\textwidth]{t1.eps}
\caption[]{Invariant mass distribution of $K^{*\pm} \rightarrow K^0_S \pi^\pm$ candidates
passing our selection criteria. The line represents the fit described in the text.}
\label{fig:fig_4}
\end{center}
\end{figure}
We fit the invariant mass distribution with a Breit-Wigner function to model the signal
and a fourth order polynomial to model the combinatorial background. We fix the mass and width
of the Breit-Wigner function to 892 and 51 MeV/$c^2$~\cite{pdg}, respectively~\footnote{
In simulated events, when constraining the $K^0_S$ mass to the PDG value, the
mass-constrained $K^0_S$ momentum is measured as accurately as that of
a track corresponding to a $K \rightarrow \mu$ decay.
The resulting $K^{*\pm}$ mass resolution is approximately 5 MeV/$c^2$.}.
The fit returns a signal of $3326 \pm 246$ $K^{*\pm}$ mesons.
The signals obtained by analyzing the 3.9 fb$^{-1}$ sample are rescaled
to estimate the number of $K \rightarrow \mu$ misidentifications present in the 742 pb$^{-1}$
dataset. After rescaling,
we obtain $N_K = N_{K^0_S}/N_{K^{*\pm}} \times N_{K^{*0}}=164769\pm 13067$.
This number is used to constrain the total
number $N_K^{\rm sim}$ of $ K \rightarrow \mu$ misidentifications
predicted by the simulation.
\section{Fit of the simulation prediction to the data}
\label{sec:ss-fit}
We fit the simulation prediction with a $\chi^2$-minimization method~\cite{minuit}.
The $\chi^2$ function is defined as
$$\chi^2= \sum_{i=1}^{5} (D[i] -P[i])^2/ED[i]^2 + (N_K^{\rm sim} -164769)^2/13067^2\; ,$$
where $D[i]$ and $ED[i]$ are the size and error of the component listed in the second column
and i-th row of Table~\ref{tab:tab_1}.
The term $P[i]$ is the sum of the real muon contribution and the
punchthrough contribution predicted by the simulation in the third and forth columns
of the same table, respectively.
These contributions are weighted with the fit parameters described in the previous section,
and the terms read
$$ P[1]= f_{bb}\; (0.96 \cdot 230308 + f_k \; 7902 + f_\pi \; 8145),$$
$$ P[2]= f_{cc}\; (0.81 \cdot 103198 + f_k \; 17546 + f_\pi \; 9535),$$
$$ P[3]= \Upsilon + f_{dy} \; DY + f_K^2 \; 4400 + f_\pi^2 \; 30000 +f_K \; f_\pi \; 23000, $$
$$P[4]=f_{sb}\; (f_K \; 11909 +f_\pi \; 29253),\; {\rm and}$$
$$P[5]=f_{sc}\; (f_K \; 16447 +f_\pi \; 35275).$$
In addition, the sum $\sum_{i=1}^{5} D[i]$
is constrained to the observed number of $P+HF-BC$
events within its error.
The fit results are shown in Tables~\ref{tab:tab_6} to~\ref{tab:tab_8}.
In Table~\ref{tab:tab_6}, the fit parameters that tune the
various cross sections predicted by the {\sc herwig} generator
are very close to their nominal values indicating that the
default simulation provides a quite accurate modeling of the data.
\begin{table}[htp]
\caption[]{Parameter values returned by the fit described in the text.
The fit yields $\chi^2=2.5$ for 5 DOF.}
\begin{center}
\begin{ruledtabular}
\begin{tabular}{lc}
$f_{bb}$ & $0.97 \pm 0.01$ \\
$f_{cc}$ & $ 0.95 \pm 0.04$ \\
$f_{dy}$ & $ 1.01 \pm 0.09$ \\
$f_\pi$ & $ 0.97 \pm 0.08$ \\
$f_K$ & $ 1.01 \pm 0.08$ \\
$f_{sb}$ & $1.05 \pm 0.08$ \\
$f_{sc}$ & $0.82 \pm 0.10$ \\
\end{tabular}
\end{ruledtabular}
\end{center}
\label{tab:tab_6}
\end{table}
\begin{table}[htp]
\caption[]{Parameter correlation coefficients returned by the fit.}
\begin{center}
\begin{ruledtabular}
\begin{tabular}{lcccccc}
Fit parameter & $f_{bb}$ & $f_{cc}$ & $f_{dy}$ & $f_{\pi}$ & $f_{K}$ & $f_{sb}$ \\
$f_{cc}$ & $-0.11$ & & & & & \\
$f_{dy}$ & $~0.17$ & $~0.10$ & & & & \\
$f_{\pi} $ & $-0.16$ & $-0.10$ & $-0.84$ & & & \\
$ f_{K}$ & $-0.08$ & $-0.17$ & $~0.71$ & $-0.48$ & & \\
$f_{sb} $ & $-0.07$ & $-0.03$ & $~0.50$ & $-0.51$ & $-0.01$ & \\
$f_{sc} $ & $-0.02$ & $-0.21$ & $~0.36$ & $-0.35$ & $-0.12$ & $0.15$ \\
\end{tabular}
\end{ruledtabular}
\end{center}
\label{tab:tab_7}
\end{table}
\begin{table}[htp]
\caption[]{Number of events due to different production mechanisms
are compared to the result of the present fit.}
\begin{center}
\begin{ruledtabular}
\begin{tabular}{lcc}
Component & No. of Events & Fit result \\
$BB$ & $230308 \pm 2861$ & 230607 \\
$CC$ & $103198 \pm 6603$ & 104463 \\
$PP$ & $161696 \pm 2533$ & 161387\\
$BP$ & $~43096 \pm 3087$ & 42490 \\
$CP$ & $~41582 \pm 5427$ & 41822 \\
$P+HF$ & $589015 \pm 5074$ & 589905 \\
\end{tabular}
\end{ruledtabular}
\end{center}
\label{tab:tab_8}
\end{table}
The fit returns 163501 $K \rightarrow \mu$ candidates ($164769 \pm 13067$ are measured in the data),
51\% of which are due to punchthrough and 49\%
to in-flight-decays. We verify this result by measuring the fraction of
$K\rightarrow \mu$ decays in identified $K^{*0} \rightarrow K^+\pi^-$ decays
that pass the tight SVX requirements.
The efficiencies of the tight SVX requirement applied to primary muons
are $0.356 \pm 0.002$ for muons due to punchthrough of
prompt and heavy-flavored hadrons and $0.166 \pm 0.005$ for
muons arising from in-flight decays~\footnote{
In Ref.~\cite{a0disc}, which uses 0.74 fb$^{-1}$ of data,
these efficiencies have been measured to be 0.45 and 0.21,
respectively. In 3.9 fb$^{-1}$ of data, by using $\Upsilon$ candidates, we
measure a smaller efficiency of the tight SVX selection.
The efficiency loss comes from periods of data taking in which the pedestals
of the L00 channels were miscalibrated.
}.
Based on the kaon composition
returned by the fit, we estimate the efficiency of the tight SVX requirement
applied to $K\rightarrow \mu$ misidentifications to be $0.263 \pm 0.008$,
where the error includes the uncertainty of the efficiencies and that of the
kaon composition returned by the fit.
Figure~\ref{fig:fig_5} shows the invariant mass distribution of $K^{*0}$ candidates
after applying the tight SVX requirement.
\begin{figure}[htp]
\begin{center}
\vspace{-0.2in}
\leavevmode
\includegraphics*[width=0.5\textwidth]{k_svx.eps}
\caption[]{Invariant mass distribution of $K^{*0}$ candidates
in which the $K\rightarrow \mu$ misidentification
passes the tight SVX requirement. The line represents the fit described in the text.}
\label{fig:fig_5}
\end{center}
\end{figure}
We fit the invariant mass distribution with
the same function used to fit the $K^{*0}$ mass distribution in Fig.~\ref{fig:fig_2}.
The fit returns $22689 \pm 985$ $K^{*0}$ candidates to be
compared with $87741 \pm 2219$ $K^{*0}$ candidates
before applying the tight SVX requirement. The resulting
efficiency of the tight SVX requirement is $0.253 \pm 0.013$, in agreement
with what expected ($0.263 \pm 0.008$) using the
composition of the kaon sample returned by the fit.
Continuing with the analysis of the results returned by the fit,
the total number of $\pi \rightarrow \mu$ misidentifications is 240915, 64\% of which
are due to punchthrough and 36\% to in-flight-decays.
The fractional composition of the $K \rightarrow \mu$ and $\pi \rightarrow \mu$
misidentifications is summarized in Table~\ref{tab:tab_9}.
\begin{table}[htp]
\caption[]{Contributions (\%) of various processes to pion or kaon
misidentifications.}
\begin{center}
\begin{ruledtabular}
\begin{tabular}{lcc}
Type & $K^{puth} \rightarrow \mu$ & $\pi^{puth} \rightarrow \mu$ \\
All sample & 51 & 64 \\
generic QCD & 20 & 33 \\
single $b$ + $b\bar{b}$ + & & \\
single $c$ + $c\bar{c}$ & 31 & 31\\
Type & $K^{ifd} \rightarrow \mu$ & $\pi^{ifd} \rightarrow \mu$ \\
All sample & 49 & 36 \\
generic QCD & 27 & 27 \\
single $b$ + $b\bar{b}$ + & & \\
single $c$ + $c\bar{c}$ & 22 & 9\\
\end{tabular}
\end{ruledtabular}
\end{center}
\label{tab:tab_9}
\end{table}
The total fraction of misidentified muons in the dataset is 27\%.
The number of misidentified muons due in-flight-decays of pions and kaons (ghost events)
is $113613 \pm 5332$.
Since the number of muons from in-flight-decays is derived from that of muons mimicked by
hadron punchthrough using the fake probabilities listed in Table~\ref{tab:tab_5}, the
uncertainty of these probabilities yields an additional error of 3845 events.
After adding the $ 12052\pm 466$ events from $K^0_S$ and hyperon decays, we predict
$125665 \pm 5351$ ghost events, whereas the dimuon dataset
contains $153991 \pm 5074$ events of this type. The number of
unaccounted events ($28326 \pm 7374$) is
($12.8 \pm 3.2$)\% of the $b\bar{b}$ production and $(18.3 \pm 4.7)$\% of the ghost
sample.
\section{Revised estimate of the rate of additional real muons in ghost events}
\label{sec:ss-multimu}
As a cross-check of its $b\bar{b}$ content,
Reference~\cite{a0disc} has investigated the rate of sequential semileptonic
decays of single $b$ quarks in the dimuon sample.
We provide here a summary of that study and its conclusions.
That study searches for additional muons with $p_T \geq 2 \; \gevc$ and $|\eta| \leq 1.1$
in a dimuon sample corresponding to an integrated
luminosity of 1426 pb$^{-1}$. The sample of 1426571 events
contains $1131090 \pm 9271$ $P+HF$ events in which both muons originate inside the beam pipe
and $295481 \pm 9271$ ghost events in which at least one muon is produced outside.
The study selects pairs of primary and additional muons with opposite charge ($OS$) and invariant mass
smaller than 5 $ {\rm GeV/}c^2$.
In the case of Drell-Yan or quarkonia production, which was not simulated, the rate of same-charge pairs ($SS$)
is a measure of the fake muon contribution since misidentified muons arise from the underlying event
which has no charge correlation with primary muons.
The rate of additional muons mimicked by hadronic punchthrough is also estimated with a
probability per track derived by using
kaons and pions from $D^{*\pm} \rightarrow \pi^\pm D^0$ with $D^0 \rightarrow K^+ \pi^-$ decays.
This misidentification probability is approximately
ten times larger than that for primary muons that have to penetrate twice as many
interaction lengths~\footnote{ Therefore, the contribution of in-flight-decays
to additional muons is negligible in comparison with the punchthrough contribution.}.
The punchthrough probabilities for pions and kaons differ by a factor of two.
In addition, in simulated events due to heavy flavor production,
the pion to kaon ratio depends on the invariant mass and the
charge of the muon-hadron pairs. Therefore, for $P+HF$ events,
the rate of $OS-SS$ pairs is compared to that predicted by the heavy flavor simulation
in which pions and kaons are weighted with the corresponding probabilities of mimicking
a muon signal. In $P+HF$ events, the number of sequential semileptonic decay-candidates
($29262 \pm 850$) is correctly modeled by the rate of sequential decays
of single $b$-quarks predicted by the simulation ($29190 \pm 1236$).
This number is 2.5\% of the $P+HF$ total contribution and ($6.9\pm 0.4$)\% of the $b\bar{b}$ contribution
($424506 \pm 18454$ events).
In the remaining $295481 \pm 9271$ ghost events, the number of additional muons in an angular cone
with $\cos \theta \geq 0.8$ around a primary muon is $49142 \pm 519$.
In the absence of a simulation of ghost events, that study assumes
that tracks in ghost events are a 50-50\% mixture of pion and kaons,
and estimates the number of misidentified additional muons to be $20902 \pm 284$.
The resulting number of unaccounted ghost events with three or more muons is
$27970 \pm 538$, ($9.5 \pm 0.4$)\% of the ghost events.
As shown by Table~\ref{tab:tab_9}, one half of the ghost events
arise from heavy flavor production acquired with a misidentified muon.
This type of event should contain an appreciable fraction of additional
muons due to semileptonic decays of heavy quarks.
This contribution was not included in the estimate of Ref.~\cite{a0disc}.
We estimate this contribution using
$K^0_S$ and $K^{*0}$ candidates due to $\pi \rightarrow \mu$ and $K \rightarrow \mu$
misidentification, respectively.
As shown in Sec.~\ref{sec:ss-tkaon}, there are $32445 \pm 421$ and
$87471 \pm 2217$ candidates, respectively.
We measure the fraction of these candidates in which at least one of the primary muons is accompanied
by an additional muon in a $\cos \theta \geq 0.8$ angular cone around its direction.
We also estimate the contribution of fake additional muon by weighting all hadronic tracks
that pass the additional muon selection criteria, assumed to be a 50-50\% mixture of pions and kaons,
with the corresponding misidentification probabilities~\cite{a0disc}.
Figure~\ref{fig:fig_6} (\ref{fig:fig_7}) shows the invariant mass distribution of
$K^0_S$ ($K^{*0}$) candidates when at least one primary muon is accompanied
by an additional muon or a predicted misidentified muon.
\begin{figure}[htp]
\begin{center}
\leavevmode
\includegraphics*[width=0.5\textwidth]{k0s2.eps}\includegraphics*[width=0.5\textwidth]{k0s3.eps}
\caption[]{Invariant mass distribution of $K^0_S$ candidates
accompanied by (left) an additional muon and (right) a misidentified muon.
Lines represent the fits described in the text.}
\label{fig:fig_6}
\end{center}
\end{figure}
\begin{figure}[htp]
\begin{center}
\vspace{-0.2in}
\leavevmode
\includegraphics*[width=0.5\textwidth]{kstar2.eps}\includegraphics*[width=0.5\textwidth]{kstar3.eps}
\caption[]
{Invariant mass distribution of $K^{*0}$ candidates
accompanied by (left) an additional muon and (right) a misidentified muon.
Lines represent the fits described in the text.}
\label{fig:fig_7}
\end{center}
\end{figure}
As previously done, we fit the $K_S^0$ distributions with two Gaussian functions
to model the signal and a straight line to model the background.
The $K^{*0}$ distribution is fitted with a Breit-Wigner function plus a fourth order
polynomial.
The fits return $4572 \pm91$ and $1954 \pm 109$ events in which a $K_S^0$ meson
is accompanied by an additional muon and by a fake muon, respectively.
The fits return $10176 \pm 739$ and $5230 \pm 493$ events in which a $K^{*0}$ meson
is accompanied by an additional muon and by a fake muon, respectively.
As shown in Fig.~\ref{fig:fig_8}
for events triggered by $K_S^0$ misidentifications, sometimes the additional muon is
contributed by the second prong of the $K_S^0$ decay.
Figure~\ref{fig:fig_8} shows the invariant mass distribution
of primary and additional muons that pass the analysis selection.
The usual fit yields $403 \pm33$ events in which the additional muon is mimicked by the second leg
of the $K^0_S$ decay that also produced the primary muon. We remove this contribution
to evaluate the fraction of real muons accompanying $K \rightarrow \mu$ misidentifications.
We will add it for the fraction of events triggered by misidentified $K_S^0$ decays.
\begin{figure}[htp]
\begin{center}
\vspace{-0.2in}
\leavevmode
\includegraphics*[width=0.5\textwidth]{k0s_puth.eps}
\caption[]
{Invariant mass distribution of $K^0_S$ candidates reconstructed using primary and additional muons.
The line represents the fit described in the text.}
\label{fig:fig_8}
\end{center}
\end{figure}
After removing the predicted numbers of fake muons, the fraction of events with an identified
$K_S^0$ meson that contain additional real muons is ($6.83 \pm 0.45$)\%.
It is ($5.6 \pm 1.05$)\% for events with an identified
$K^{*0}$ meson.
The average of the two fractions is ($6.6 \pm 0.4$)\%. We multiply this fraction
by the number of ghost events due to ordinary sources ($241507 \pm 10284$) to
predict the number of real muons in events due to heavy flavor that are classified as ghost events
because one of the primary muons was produced by a pion or kaon in-flight-decay.
This procedure yields a slight overestimate because,
according to the simulation tuned with the data, ($53.4 \pm 0.4$)\% of the $K \rightarrow \mu$ misidentifications are due to
events with heavy flavors, whereas ($51.3 \pm 0.6$)\% of the ghost events arise from heavy flavor production.
As shown by Table~~\ref{tab:tab_10},
the improved estimate still does not account for $12169 \pm 1319$ ghost events
with additional real muons.
\begin{table}[htp]
\caption[]{Number of additional muons in ghost events are compared
to the number of expected fake muons and real muons from heavy flavor decays.
The data correspond to an integrated luminosity of 1426 pb$^{-1}$.}
\begin{center}
\begin{ruledtabular}
\begin{tabular}{cccc}
Data & Fakes & $K_S^0$ second leg & Heavy Flavor \\
$49142 \pm 519$ & $20902 \pm 284$ & $147 \pm 15$ & $15924 \pm 1179$ \\
\end{tabular}
\end{ruledtabular}
\end{center}
\label{tab:tab_10}
\end{table}
\section{Conclusions}\label{sec:ss-concl}
This article reports an improved undestanding of the dimuon samples
acquired by the CDF experiment.
One dataset, corresponding to an integrated luminosity of 742 pb$^{-1}$,
consists of 743006 events
containing two central ($|\eta|<0.7$) primary (or trigger) muons, each with transverse
momentum $p_T \geq 3 \; \gevc$, and with invariant mass larger than
5 $ {\rm GeV/}c^2$ and smaller than 80 $ {\rm GeV/}c^2$.
These data are split into two subsets: one, referred to as $P+HF$, consisting of
$589015\pm 5074$ events in which both muons
originate inside the beam pipe of radius 1.5 cm; and one, referred to as ghost,
consisting of $153991 \pm 5074$ events in which at least one muon
originates beyond the beam pipe.
The study in Ref.~\cite{bbxs} shows that the number and properties of $P+HF$ events
are correctly modeled by the expected contributions of
semileptonic heavy flavor decays, prompt
quarkonia decays, Drell-Yan production, and instrumental backgrounds
due to punchthrough of prompt or heavy-flavored hadrons which mimic a muon signal.
A previous study~\cite{a0disc} has investigated significant
sources of ghost events, such as in-flight-decays of pions and kaons and
hyperon decays. That study could account for approximately half of the ghost events
but was unable to asses the uncertainty of the in-flight-decay prediction.
The present study shows that the {\sc herwig} parton-shower generator
provides an accurate model of the data. The large discrepancy in the previous
study was generated by not including the contribution of final states in which
a $b(c)$ hadron decays semileptonically and the second muon is produced by
the in-flight-decay of a particle in the recoiling jet.
After tuning by a few percent the pion and kaon rates predicted by the simulation
with a fit to the data, we show that ordinary sources, mostly in-flight-decays,
account for $125665 \pm 5351$ of the
$153991 \pm 5074$ ghost events isolated in the sample of 743006 dimuons.
For comparison, a D0 study has used a similar dimuon sample to set a limit~\cite{williams}
of $(0.4\pm0.26\; {\rm stat}\pm0.53\; {\rm syst})$\% to the fraction of muons
produced at a distance larger than 1.6 cm from the beamline including pion and kaon
in-flight-decays.
This appears to be in contradiction with the present result, and also with a recent
estimate~\cite{d0cp} of the fraction of
$K \rightarrow \mu$ and $\pi \rightarrow \mu$
contributions in the D0 subset of same-charge dimuons ($\simeq 40$\%).
The present study also improves a previous estimate~\cite{a0disc}
of the content of additional muons
with $p_T \geq 2\; \gevc$ and $|\eta| \leq 1.1$ in ghost events.
We find that ($23 \pm 6$)\% of the unaccounted ghost events contain additional
real muons. For comparison, the fraction of $b\bar{b}$ events that
contain additional muons due to sequential semileptonic
decays is ($6.9 \pm 0.4$)\%.
Both results presented in this article have implications for measurements
derived in dimuon datasets
without properly accounting for the presence of ghost events. As an example,
the measurement of the dimuon charge asymmetry performed by the D0 experiment~\cite{d0cp}
estimates the fraction of $K \rightarrow \mu$ and $\pi \rightarrow \mu$ misidentifications
with a similar method.
After removing this background,
the remaining muon pairs with same charge
are attributed to $b\bar{b}$ production. The present study shows that, after removing
this type of misidentified muons,
the data set still contains an additional component that cannot be accounted for
with ordinary sources. The size of this component, equally
split in opposite and same sign pairs~\cite{a0disc}, is
($12.8 \pm 3.2$)\% of the total number of dimuons due to $b\bar{b}$ production.
\section{Acknowledgments}
We thank the Fermilab staff and the technical staffs of the participating institutions for their vital contributions.
This work was supported by the U.S. Department of Energy and National Science Foundation;
the Italian Istituto Nazionale di Fisica Nucleare; the Ministry of Education,
Culture, Sports, Science and Technology of Japan;
the Natural Sciences and Engineering Research Council of Canada; the National Science Council of the Republic of China;
the Swiss National Science Foundation; the A.P. Sloan Foundation;
the Korean World Class University Program, the National Research Foundation of Korea;
the Science and Technology Facilities Council and the Royal Society, UK;
the Institut National de Physique Nucleaire et Physique des Particules/CNRS; the Russian Foundation for Basic Research;
the Ministerio de Ciencia e Innovaci\'{o}n, and Programa Consolider-Ingenio 2010, Spain;
the Slovak R\&D Agency; the Academy of Finland; and the Australian Research Council (ARC).
\input{bibliography_10242.tex}
\end{document}
|
1,116,691,500,892 | arxiv | \section{Introduction}
Self-organization is the spontaneous formation of structure via the interactions of their constituent particles \cite{whitesides_self-assembly_2002}.
The resulting structures from a self-organized system can be found in equilibrium and non-equilibrium states; the latter provides a pathway towards active and adaptive matter.
While adaptive materials are common in living systems such as cells, it is possible to artificially introduce dynamic forces which can externally deliver energy to a system while mediating self-organization \cite{grzybowski_self-assembly_2009}.
Dynamic self-organization is relatively new but active field of self-assembly with the goal of extending our knowledge of equilibrium thermodynamics onto living systems.
Some examples include the dynamic assembly of non-biological components including active colloids \cite{ginot_aggregation-fragmentation_2018,wang_adaptive_2015,howse_self-motile_2007}, rotating discs \cite{grzybowski_dynamic_2000,grzybowski_dynamics_2002}, and magnetic swimmers \cite{kokot_dynamic_2017,tierno_controlled_2008,snezhko_magnetic_2011}.
Optical binding is a long-range light-induced force which causes dielectric particles to interact mutually through light-scattering \cite{burns_optical_1989}.
In theory, optical binding can be felt between polarizable molecules ($\sim$ 1 nm) \cite{thirunamachandran_intermolecular_1980} up to biological cells \cite{bai_lateral_2018} ($\sim$ 5 $\upmu$m).
In practice, because the force is relatively weak and requires a strong scattering response, the effects of optical binding are observed most strongly for objects that are on the order of the wavelength of light in size.
Moreover, optical binding can give rise to non-equilibrium forces because the optical field applied to the system is also a constant supply of external momentum which can contribute to particle motion \cite{li_non-hermitian_2021}, although this effect has been largely unexplored in previous experimental work.
In principle optical binding forces can be explained using existing approaches -- forces can be exactly predicted using a multiple scattering calculations -- but nonetheless it remains difficult to predict and explain the behavior of systems composed of many particles \cite{li_non-hermitian_2021}.
Numerous past studies have illustrated the complexity of the optical binding force.
Two-particle studies have shown that the force can contain multiple stable points defined by multiples of the light wavelength \cite{burns_optical_1989, dapasse_optical_1994,mohanty_optical_2004}.
By increasing the number of particles, previous experimental and computational studies have shown the self-organization of 1-dimensional chains \cite{tatarkova_one-dimensional_2002}, static and drifting 2-dimensional lattices exhibiting stable and quasi-stable behaviors \cite{ng_photonic_2005}, and observations of bistability \cite{metzger_observation_2006} and multistability \cite{karasek_long-range_2008} in the equilibrium positions of optically bound particles \cite{dholakia_colloquium_2010}.
The complexity that arises from optical binding in multi-particle systems can largely be attributed to feedback in the form of interference with the incoming field through multiple scattering effects \cite{dholakia_colloquium_2010}.
One often neglected fact is that optical binding forces are sensitive to $N$-body effects, and can not be treated as strictly pair-wise interactions \cite{rodriguez_optical_2008}.
These $N$-body interactions can lead to highly correlated and emergent behaviors but can consequently be difficult to model.
More recent studies have described complex dynamical assemblies of metallic and dielectric nanoparticles
\cite{forbes_optical_2020, nan_dissipative_2018,brzobohaty_complex_2020}.
These studies suggest that highly nonlinear and dynamic interactions can occur for smaller particles, where interactions are highly pair-wise, because of the complexity of the potential landscapes generated by optical scattering alone \cite{yan_potential_2014}.
The presence of thermal noise in a complex potential landscape can disrupt stability.
Thus, in studying a system which $N$-body forces may be strong \emph{and} potential landscapes complex, we aim to untangle the effects between the two.
In this manuscript we study optical binding of large numbers of particles ($N>$ 20) free to move in three-dimensional space.
We assess which features of the assembly can be described by pairwise interactions verses features which emerge from higher-order effects.
By comparing strong and weak scatterers, we share new insights on the relationship between refractive index, scattering strength, non-conservative forces, and non-linear optical binding.
We present observations of unexpected emergent driven behaviors which appear to manifest from the deviation from pairwise forces.
We also use the coupled dipole method (CDM) \cite{draine_discrete-dipole_1994,hoekstra_radiation_2001}, also known as the discrete dipole approximation (DDA), to explain the origin of these effects.
\section{Experimental Design}
\label{sec:headings}
\begin{figure*}[t!]
\centering
\includegraphics{figure1.pdf}
\caption{(a) Experimental setup. The laser beam is split into two paths that are aligned to counter propagate through the colloidal sample. Both beams are adjusted so that they are totally internally reflected at the coverslip. The polarization and incident angle of each beam can be adjusted individually. The camera, electronically tunable lens, and illumination source are all synchronized to a wave-function generator for the acquisition of 3-dimensional videos. (b) Depth profile of sample cell, we define z-axis as direction along the sample depth and $\Vh k$ as the axis corresponding to the beam propagation axis. (c) image still of HIPs in the optical field. (d) image still of LIPs in the optical field.}
\label{fig:opticalpath}
\end{figure*}
\subsection{Experimental parameters}
There are a number of relevant parameters which can alter how an object scatters light and consequentially alter the properties of optical binding interactions.
In practice, the most important quantities are the size parameter, $ka$, and the relative refractive index $m = n_p / n_0$ (where $k$ is the wavenumber of the incident field in the background medium, $a$ is the particle radius, and $n_p/n_0$ is the particle/background medium index of refraction).
Tuning the parameters will not necessarily yield linear responses, which is why we must be careful about generalizing optical binding behavior.
For instance, the size parameter alone can span three different regimes: the Rayleigh limit, $ka\ll$ 1, the ray-optics limit, $ka\gg$ 1, and the Mie scattering regime, $ka \sim$ 1, each of which are separated by drastically different assumptions including the relevance of multiple scattering, scattering modes, and how the strength of the forces scale.
Our current study is focused in the Mie regime.
The relative refractive index, $m$, affects the scattering strength of a single particle, and so modulates the importance of multiple-scattering events \cite{ashkin_observation_1986}.
We use this study as an opportunity to contrast the behaviors of strong and weak scatterers.
We perform a side-by-side analysis of two commonly available colloidal microspheres which differ in refractive index.
We refer to these as high-index particles (HIPs, m=1.2, made of polystyrene) and low-index particles (LIPs, m=1.1, made of SiO$_2$).
Unique to field-driven self-organization, direction and geometry become particularly important.
Because optical scattering is directional, optical binding is an interaction which breaks symmetry along the axis of the field direction.
As a result, experimental and numerical studies have often focused on either two distinct optical binding geometries: lateral \cite{burns_optical_1989, dapasse_optical_1994,mohanty_optical_2004,wei_lateral_2016,ng_photonic_2005,li_non-hermitian_2021} and longitudinal\cite{karasek_analysis_2006,karasek_long-range_2008,karasek_longitudinal_2009,singer_self-organized_2003,tatarkova_one-dimensional_2002} optical binding.
Lateral optical binding describes binding that occurs between particles with a displacement perpendicular to the direction of propagation, $\Vh k$, while longitudinal optical binding describes binding of particles displaced along $\Vh k$.
In the current study, we implement a novel experimental approach to present the self-organization of many particles free to interact in all directions.
We believe this configuration is particularly illuminating for understanding potential bulk behaviors.
Finally, previous studies have focused on how the stability of optically bound matter is affected by damping conditions \cite{ng_photonic_2005,li_non-hermitian_2021} suggesting that the low damping can lead to instabilities due to the systems inability to remove energy.
Because we use sub-micron particles suspended in a fluid, our study is in an over-damped regime.
Our estimated Reynold's number is on the order of $Re = 10^{-6}$ (assuming 500 nm diameter particles in water with a characteristic velocity of 1 $\upmu$m/s).
\begin{figure}[t!]
\centering
\includegraphics[scale=.65]{figure2.pdf}
\caption{Strength of optical forces for a range of size-parameters ($ka$). The optical gradient force (blue) was numerically simulated by placing a single particle in a node of a standing wave, the optical scattering force (orange) was numerically simulated by placing a single particle in a propagating plane-wave. The optical binding force was obtained by placing to particles laterally in a counter-propagating field at a separation of $2\lambda$. The reference force, $F_0$, is also plotted. In all simulations we assume a light intensity of 1 mW/$\upmu$m$^2$.
}
\label{fig:scale}
\end{figure}
\subsection{Optical setup}
The experiments are performed on a custom designed inverted microscope (\figref{opticalpath} (a)).
Two Gaussian beams ($\lambda$ = 532 nm in vacuum, or 400 nm in solution) are aligned and focused through the sample tube to generate a counter-propagating crossed polarized optical field ($w_0 $ = 27.0 $\pm$ 0.3 $\upmu $m).
The polarizations of each of the two beams can be adjusted individually which allows us to switch between a counter-propagating standing wave and a crossed polarized configuration.
In this specific setup, a portion of beam power is redirected back towards the laser source which is removed using an optical isolator placed between the laser head and the beam splitter.
The final focusing of the beam allows us to further modulate the beam power density.
The orientation of the beam through the sample is at an angle of approximately $\upphi_g$ = 45\degree{} through the glass layers and $\upphi$ = 36\degree{} through the water.
We achieve total-internal reflection at the glass-air interface at the bottom surface of the coverslip depicted in \figref{opticalpath} (c).
The sample tube depth is 10 $\pm$ 1 $\upmu$m.
We collect 3-dimensional video microscopy data using a combination of a microscope objective (40X, NA = 0.75) and an electronically tunable lens (ETL).
The scanning of the ETL is synchronized with the imaging camera acquisition to capture 10 volumes per second with 50 frames per volume.
By adjusting the focal length of the ETL, one can adjust the z-depth of the imaging plane shown in \figref{opticalpath} (c).
By driving the ETL approximately 6 $\%$ of its full range, we scan more than the full 10 $\mu m$ of the depth of the sample.
The signal output to the ETL is in the form of a periodic sawtooth pattern (driven at 10 Hz) which increases linearly for 70 $\%$ of the total period and quickly decreases linearly for the remaining 30 $\%$.
As such, the first 70 $\%$ of the period is used to create the full volume.
The 3D scanning method gave us a spatial resolution of 0.8 $\upmu$m/pixel in the z-direction and a time resolution of 10 Hz which was sufficient to reliably locate particles within the volume.
The colloidal particles in the sample exhibit only Brownian motion until they enter the beam region.
Particles that diffuse into the region remain confined within the Gaussian beam but can move thermally in all 3-dimensions within the area of confinement.
Radiation pressure forces, which can be a strong optical force in the direction of the field propagation \cite{ashkin_observation_1986}, are carefully balanced by the two counter-propagating beams.
As nearby colloids drift into the beam, the density of the particles increase slowly.
While there is no direct control to stop diffusive colloids from drifting into the beam, the variability in the number of particles over the entire 2 mins of observation are between 1 and 5 particles ($\sim$ 3--7 \% of total particles).
The light intensity is a parameter which plays a linear role in the strength of the optical binding force and can be tuned quite easily.
For the current study, we adjust the peak light intensity of the Gaussian beam up to $\sim$ 2.5 mW/$\upmu \text{m}^2$.
For the numerical simulations we use a reference intensity of $I_0$ = 1 mW/$\upmu \text{m}^2$.
Subsequently we define a reference force, $F_0$, which is the total optical momentum impinging on a single particle in a plane wave of intensity $I_0$:
\begin{equation}
F_0 = \frac{I_0 \pi r^2}{c},
\end{equation}
where $r$ is the radius of the particle and $c$ is the speed of light.
The value approximately 650 fN for 500 nm diameter particles.
It is also useful to compare this to thermal excitation.
For an order of magnitude estimate of how the thermal activity compares to the strength of the optical forces in the experiment, we can convert the thermal energy, $k_bT$, to an equivalent force by including a length scale.
Given that the optical binding force has spatial oscillations with the wavelength of the incoming light, the approximate scale of the equivalent thermal force is given by $F_T \sim k_bT / \lambda \sim 10$ fN.
We performed a numerical analysis to predict suitable light intensity and particle parameters which should allow us to observe optical binding effects (\figref{scale}).
By comparing the scaling of optical forces -- including binding -- with an estimated force of thermal excitation, we show that the size parameter is important in determining the strength of optical binding effects relative to thermal motion.
The particles used in this study have a size parameter of $ka = 4$; as can be seen in \figref{scale}, a power density of 1 mW/$\upmu$m$^2$ should ensure that all optical forces are comparable to or larger than the effective thermal forces.
Potentially more important than the strength of the force is the nature of the interaction.
We note that the optical binding force is a combination of second-order gradient and scattering forces.
In \figref{scale}, we plot the relative strength of first-order gradient and scattering forces for HIPs and LIPs to approximate their relative contributions to the binding force.
We show here that the size parameter, $ka$, has a strong effect on the relative balance of scattering and gradient forces, thus we draw a distinction between recent studies done on dielectric nanoparticles \cite{brzobohaty_complex_2020}.
Note that the optical binding force is computed only for a pair of particles spaced by 2$\lambda$; for larger numbers of closely spaced particles it can be 1--2 orders of magnitude stronger.
We observe for $ka = 4$ that the HIPs have a strong scattering force response, relative to the gradient force.
Because scattering forces are generally non-conservative, we expect that it is the stronger presence of scattering forces in optical binding for the HIPs that give them a more dynamic self-organization behavior as opposed to the LIPs.
\subsection{Sample preparation}
Colloidal samples are diluted in water ($\sim$ 0.01 \text{w/v} \%) and placed into thin rectangular Borosilicate tubes (100 {$\upmu$}m wide $\times$ 10 {$\upmu$}m thick $\times$ 50 mm long) purchased from \emph{VitroCom}.
We performed experiments with polystyrene (`HIPs', diameter $d$ = $518 \pm 10$ nm, relative index $m$ = $1.20$ \cite{zhang_complex_2020,hale_optical_1973}) and silicon dioxide (`LIPs', diameter $d$ = $518 \pm 20$ nm, relative index $m$ = $1.10$ \cite{malitson_interspecimen_1965,hale_optical_1973}) purchased from \emph{microParticles GmbH}.
In the experiments, the number of colloidal particles are in the range of 30-50 for HIP experiments and 40-100 for the LIP experiments.
We found that LIPs were more readily collected into the beam, resulting in higher effective particle densities compared to the HIPs even when the initial density is the same.
The tubes allow us to keep a precise sample depth of $10 \pm 1 \upmu$m.
This tube depth is important for confining particles to a range in the z-direction.
The tubes are coated with index matching fluid and placed in-between a coverslip and microscope slide.
Finally, the coverslip is glued to the microscope slide, using UV curable Epoxy (\emph{Norland Optical Adhesive NOA61}), which seals the tube and index matching fluid.
The samples are fixed onto the \emph{Mad City Labs - RM21} microscopy base allowing for micro-precision movement of the sample in the $x$-$y$ plane.\\
\begin{figure*}[t!]
\centering
\includegraphics[scale=1]{figure3.pdf}
\caption{\textbf{(a,b)} Two-body force map of optical binding forces for HIPs and LIPs ($ka = 4$) placed in counter-propagating crossed-polarized plane wave ($\lambda = 0.4 \upmu \text{m}$ in solution) propagating in the $+ \Vh k = +z'$ and $- \Vh k = -z'$ direction.
Maps are obtained by placing two particles in a CDM simulation and calculating the force applied on each particle at various displacements which fill the map space.
The forces are provided in a unit-less scale normalized by the reference force, $F_0$, described by the single scattering force on a single particle of the same size and material. The direction of the arrows represent the direction of the force, while the size represents the relative strength.
\textbf{(c-f)} 2D pair-correlation functions for HIPs in the x-y plane at various light intensities.
\textbf{(g-j)} 2D pair-correlation functions for LIPs in the x-y plane at various light intensities.
The color bar in the pair-correlation functions gives the 2D probability density of the particles, and has units of $\upmu$m$^{-2}$.
}
\label{fig:static}
\end{figure*}
\section{Results}
\subsection{Observations of self-organization behavior}
Optical binding effects are immediately apparent between particles within the beam region.
For instance, spatial ordering is immediately observable within the beam area -- effects which can be enhanced by increasing the light intensity.
The most apparent feature is the generation of multiple chains of particles aligned along the beam propagation direction (\figref{opticalpath} (c)).
Once collected into a chain, particles within the chain can be observed to move collectively together within the beam region.
The density of particles is large enough to observe the formation of multiple chains at a time.
Other features of the formed structures arise on further examination.
For instance, the HIPs are found to self-organize into extended chains which extend along the sample depth--or--small clusters of typically (3-5) particles long.
Both HIP structures can be observed in \figref{opticalpath} (c) and \dag ESI Movie S1.
Unlike the HIPs, the LIPs tend to form extremely long and close packed chains of particles (rods) (\dag ESI Movie S2).
Two LIP rods can be observed in \figref{opticalpath} (d).
We note that because the chains extend along the entire available depth of the sample, that the sample depth plays a relevant role as a boundary for the system.
\subsection{Comparing observations to two-body forces}
To understand the dominating interactions which are leading to the general behavior of both systems, we numerically generated two-body force maps for the two particle types (\figref{static} (a,b)).
The force maps were created using a CDM based simulation \cite{draine_discrete-dipole_1994,hoekstra_radiation_2001} (see supplementary material for details).
Comparing force maps for HIP ($m=1.2$) and LIP ($m=1.1$) particles reveals surprisingly little difference, apart from the overall strength.
The strength differences can be observed in \figref{scale} by comparing how the optical forces are generally stronger for HIPs.
The strength can be found to scale approximately like the reflectivity of a dielectric plane, which scales like $\sim(m - 1)^2$.
Despite this, the two-body force maps can be used to describe some of the overall structures observed in the system.
The long range alignment along the beam propagation for the HIPs and LIPs are qualitatively consistent with the two-body force maps (\figref{static}), which suggest strong forces can tend to pull particles onto the axis of propagation ($\Vh k$).
The $\Vh k$ axis, as shown in \figref{opticalpath} (b) is rotated 54\degree{} from $z$.
For comparison, we approximate the energy $F_0 \lambda \sim 65 k_bT$, thus the HIPs can experience $ \sim 20 k_bT$ of energy keeping them aligned in the $\Vh k$ axis in the beam at 1 mW/$\upmu$m$^2$.
The LIPs experience $ \sim 5 k_bT$ at 1 mW/$\upmu$m$^2$.
The HIPs force-map shows a repulsive force between particles that are aligned along the field propagation (\figref{static} (a)).
This fits with the observation of HIPs forming extended particle chains (\figref{opticalpath} (c)), rather than tightly packing.
What the force-map fails to predict for the HIPs are the presence of the smaller tightly packed HIP clusters also found in the experiment.
While it is expected that we observe structures that cannot be predicted from two-body optical binding interactions, here we observe an emergence of a force which is not only strongly contributing, but can act in the opposite direction of the two-body force.
For the HIPs, the two-body force is repulsive along the \Vh{k} axis but we observe experimentally an unexpected close-range attractive force suddenly drives particles together into small clusters.
\subsection{Pair-correlation function}
To map the time-averaged structure created by optical binding forces, we compute a 2D pair correlation function (2D PCF).
Given all the 2D particle positions, $\vec r_i$, we can compute all relative displacements $\vec \Delta_{ij} = r_i - r_{j \neq i}$; the 2D pair correlation function is then the histogram of these displacements, averaged over all frames in the data set and normalized by the bin size (so that the result is expressed as a 2D density).
The form of the pair-correlation function also gives an approximation for an effective potential through which the particles interact.
We collected the data for the pair-correlation function by recording the positions of colloidal particles all located within the same weakly focused beam of light.
The beam is fixed at a constant light intensity for a duration of 2 mins during which we collected 3-dimensional imaging data.
3-dimensional volumes were used to locate particles over the entire 10 $\upmu$m in depth.
However, as the optical configuration results in poor resolution in z, we summed over the z-axis to generate the 2-dimensional plots.
As such, the \Vh{k} axis is projected along the x-axis.
The particles were located using the \emph{trackpy} implementation of Crocker and Grier \cite{crocker_methods_1996, allan_trackpy_2015}.
While the particle speeds and dense particle clustering made it challenging to track particles through time, we found that the algorithms were quite efficient at locating particles within a frame.
We found relatively low fluctuations in the total number of particles from frame to frame suggesting that the particle locating algorithms were performing consistently.
Inter-particle displacements were determined by particle locations given for each time-step, thus the frequencies of each displacement over the entire run were available.
The 2D pair-correlation function (PCF) for the HIPs suggest an increase in spatial order with the increase in light intensity \figref{static} (e-h).
The form of the PCF shows the dominating feature that the HIPs tend to align along the beam propagation.
At higher light intensities, $I >$ 1 mW/$\upmu$m$^2$, multiple lines off-axis begin to appear.
This not only suggests that particles are interacting to form long range structures, but that there are optical binding interactions occurring in multiple directions within the field.
For example, at high powers there is clear evidence of preferred inter-chain transverse spacing of $\sim 1\ \upmu$m.
This can be explained by the computed pairwise force diagrams, which have a converging force in the y direction at these separations.
The two-body force maps (\figref{static} (a-b)) can be used to help explain the multiple off-axis lines that appear in the PCF at higher light intensities, as we can observe multiple off-axis lines in which the force arrows converge.
The 2D PCF for the LIPs are very consistent with the observation that the particles collapse into tightly bound rods (\figref{static} (i-l)).
One feature that becomes prominent are secondary off-axis lines at higher light intensities.
The secondary lines are evidence of multiple rods interacting to form a long-range regular spacing over time.
\begin{figure}[t!]
\centering
\includegraphics[scale=1.2]{figure4.pdf}
\caption{\textbf{(a)} Distribution of relative FMSD values over 2 min duration for various intensities. For the high light intensity case (2.3 mW/$\upmu$m$^2$) we share a comparative distribution of FMSD values for the same system in a counter-propagating standing wave. \textbf{(b)} The average relative FMSD value over 2 minutes for various light intensities. The shaded regions represent the variability in the relative FMSD values over the 2 minutes. Values over the dotted line represent the average motion of the particles greater than what would be observed in a purely diffusive system. \textbf{(g)} Distribution of relative FMSD values over 2 min duration for various light intensities. For the higher light intensity case (2.3 mW/$\upmu$m$^2$) we share a comparative distribution of FMSD values for the same system in a counter-propagating standing wave. \textbf{(h)} The average relative FMSD value over 2 minutes for various light intensities. The shaded regions represent the variability in the relative FMSD values over the 2 minutes. Values over the dotted line represent the average motion of the particles greater than what would be observed in a purely diffusive system.
}
\label{fig:dynamic}
\end{figure}
\subsection{Volume Mean Squared Difference (VMSD)}
\begin{figure}[t!]
\centering
\includegraphics[scale=0.8]{figure5.pdf}
\caption{\textbf{(a-c)} Three snapshots of the optically bound HIPs over a 0.2 s duration. Two clusters are circled: (red) cluster of 5 particles ballistically moving (>80 $\upmu \text{m/s}$) through the sample and (blue) isolated cluster of 4 particles remaining relatively still.}
\label{fig:img_stills}
\end{figure}
We use the mean squared displacement of particles to determine their dynamic behavior.
Due to high-density particle clustering and the high velocities of clusters evidenced in \figref{img_stills}, it is difficult to reliably track individual particles through time.
This is especially true for close-packed clusters, which -- as we shall show later -- appear to drive motion in the HIPs.
Instead of attempting to directly track the particles in time, we use a proxy for the mean squared displacement by obtaining the mean squared difference between two subsequent image volumes.
We found that the volume mean squared difference (VMSD) between subsequent frames gives us more consistent measures of motion while remaining highly correlated with the true mean squared displacement (details of the method are described in the supplementary material).
The VMSD, $\Delta$, is given by:
\begin{equation}
\Delta(t) = \frac{1}{N_p(t)} \left[\sum_{i,j,k}^{volume}(p_{i,j,k,t} - p_{i,j,k,t-dt})^2 - \Delta_{0}\right]
\end{equation}
where $p_{i,j,k,t}$ is the pixel value at a given location and time, $dt = 0.1$ is the time between frames, and $\Delta_0$ is the background VMSD caused primarily by camera noise, and is computed for when there are no moving particles in the frame.
As shown in the supplemental materials, $\Delta$ is proportional to the squared particle displacement provided the particles average motion between frames is smaller to or comparable the particle size.
In practice, we normalize this value relative to one obtained by turning the optical binding laser off, in which case the particles experience only Brownian motion.
The distribution average particle motion for the HIPs is provided in \figref{dynamic} (a).
At lower light intensities, we find narrow distributions of the average particle motion and average values lower than what is found in a Brownian system, reflecting the fact that the particles are being confined by the optical binding forces.
At higher light intensities, we find that the distributions shift to higher average values and the size of the fluctuations are greater, indicating the presence of a non-conservative driving force.
Surprisingly, comparing the pair-correlation function to the VMSD values suggests that the average motion is increasing even as the particles are becoming more ordered.
This would not be expected for a conservative pair-wise force; in this case ordering will result in weaker fluctuations.
We do indeed observe this for lower power levels (<0.4 mW/$\upmu$m$^2$).
Above these power levels, the increasing motion suggests that we are forming collections of particles which experience additional non-conservative forces from the optical field.
We found that the behavior can be altered by aligning the polarizations of the counter-propagating beams to generate a standing wave pattern.
In this configuration, the optical binding area includes multiple planes of high light intensity perpendicular the the propagation axis and separated by $\lambda/2$.
While the overall average motion is suppressed in the standing wave, the system is still observed to fluctuate strongly between low average motion ($\sim$ 0.5 $\times$ Brownian motion) and high average motion ($\sim$ 2.5 $\times$ Brownian motion).
At the same light intensities to the previous configuration, the standing wave had the effect of dramatically suppressing the overall motion (\dag ESI Movie 3).
Using the same range of intensities, the same analysis was done for the LIP system (\figref{dynamic} (b,d)).
At lower light intensities, we find narrow distributions of the average particle motion and mean values lower than what is found in a Brownian system.
At higher light intensities, the average values do not exceed what is found in the Brownian system; however the distributions are far less Gaussian.
The distribution suggests that the average motion is low with occasional rare events that lead to high motion.
The LIPs in the standing wave do not show significant differences than particles in the cross-polarized counter-propagating beam.
Comparing the dynamic behavior between the HIPs and LIPs, there are clear differences between the two systems.
The optical binding force when increased is shown to significantly increase the average kinetic energy of the system of HIPs.
This implies that optical binding is a source of non-conservative motion that is especially present in the HIP case.
We believe that non-conservative second-order scattering forces, expected to be stronger for the HIPs, is contributing to the higher overall motion.
The LIPS are instead dominated by second-order gradient forces thus pack into the rod structures until density limitations require the formation of new rods.
Where the standing-wave generates first-order gradient forces which compete with the second-order scattering forces to suppress the overall motion, the same standing-wave does not show much of an effect on the LIPs which are already dominated by gradient forces.
\subsection{Pair-correlation function difference}
\begin{figure*}[t!]
\centering
\includegraphics{figure6.pdf}
\caption{\textbf{(a)} HIP PCF for frames with higher (top 30\%) FMSD values. \textbf{(b)} HIP PCF for frames with lower (bottom 30\%) FMSD values. \textbf{(c)} The difference between the high FMSD and low FMSD pair-correlation functions for HIPs. \textbf{(d)} LIP PCF for frames with higher (top 30\%) FMSD values. \textbf{(e)} LIP PCF for frames with lower (bottom 30\%) FMSD values. \textbf{(f)} The difference between the high FMSD and low FMSD pair-correlation functions for LIPs. Displacements found more often in the higher FMSD frames are positive (blue) while displacements found more often in the lower FMSD frames are negative (red). The PCFs are normalized by dividing the 2D particle density by the number of particles.}
\label{fig:PCF}
\end{figure*}
We see clear dynamic differences between the HIPs and LIPs; in particular, the HIPs are observed to experience higher average kinetic motion at high powers and a wider range of fluctuations of that behavior.
In each case, we can use the fact that the kinetic motion does fluctuate to subset the data into high motion frames and low motion frames.
Doing so gives us a method to compare particle configurations which may be correlated with driving instability in the system.
To do so, we sort frames from a single video by the FMSD value which represents the average kinetic motion.
We can use each subset of frames to generate PCFs for the top 30\% and the bottom 30\% FMSD values.
Because of the wide distribution of FMSD values, we focus on particles in a high powered (2.3 mW/$\upmu$m$^2$) standing wave.
Comparing the PCFs for the high FMSD frames (\figref{PCF} (a)) versus the low FMSD frames (\figref{PCF} (b)) for the HIPs shows subtle differences.
Primarily, we find more defined features for the low FMSD PCF.
This suggest that the system fluctuates between slower moving ordered states to faster moving disordered states.
Indeed, we observe sudden collapses and regeneration into order states in these optically bound systems.
Secondly, we observe that there is a higher distribution of particles in close proximity for the faster moving system.
We illuminate this subtle difference by subtracting the low FMSD PCF from the high FMSD PCF shown in \figref{PCF} (c).
We not only find that that the overall system moves faster when HIPs are generally closer together, we also find that the correlation is strongest for small clusters.
We propose that multiple scattering effects which are stronger when particles are at close-range could be driving the collapse of stability and increase in average kinetic motion of this system.
For the LIPs, we find that the high FMSD PCF (\figref{PCF} (d)) versus the low FMSD PCF (\figref{PCF} (e)) are both structurally similar.
Not until taking the difference between the two as shown in \figref{PCF}(f), do we see that there are subtle differences.
For the low FMSD PCF, we observe that particles tend to be distributed among multiple lines: a single line that passes the origin and multiple off-axis lines.
These lines are evidence of the self-organization into multiple rods.
The center line represent displacements between particles that belong to the same rod and the off-axis lines represent displacements between particles that are located in neighboring rods.
Interestingly, the differences between the high and low PCFs suggest that inter-rod distance may play a role in the average kinetic motion of this system.
For instance, one can observe in \figref{PCF}(f) that the the first off-axis line is closer to the center line for the higher FMSD frames.
Comparing the HIPs and the LIPs is useful for understanding how multiple scattering is effecting these systems at different scales.
For the HIPs, where scattering is much stronger, we find that the presence of small clusters are generally correlated with higher kinetic motion and less structural order of the overall system.
We propose that HIP clusters can be treated as an emergent species which can alter the system dynamics.
On the other hand, the LIPs can generate much larger stable structures before the system fluctuates dynamically.
\begin{figure*}[t!]
\centering
\includegraphics[height=8.5cm,width=16cm]{figure7.pdf}
\caption{ (a,b) 1-dimensional simulation of forces on particle chains. Particles are placed into a equidistant configuration aligned with the beam axis of propagation (\Vh{k} = \Vh{z}') with inter-particle distance, $z_s'$. At each inter-particle spacing, the total outward force is calculated. A positive force represents a net-force pushing particles away from each other while a negative force represents a net-force in which particles are attracted. We performed the simulation for (a) HIPs ($m$=1.2, $ka$ = 4) and (b) LIPs ($m$=1.1, $ka$ = 4) for various cluster sizes (N) (c,d) 1-dimensional simulation of forces between clusters and neighboring particles. A cluster of N touching spheres is aligned along the axis of propagation with a single particle and the absolute sum over the absolute difference of force on the cluster and the force on the particle are plotted for various particle-cluster spacings, $z'_p$. Simulations were performed for (c) HIP ($m=1.2$, $ka = 4$) and (d) LIP ($m=1.1$, $ka = 4$) particles.}
\label{fig:numerical}
\end{figure*}
We see clear dynamic differences between the HIPs and LIPs; in particular, the HIPs are observed to experience higher average kinetic motion at high powers and a wider range of fluctuations of that behavior.
In each case, we can use the fact that the kinetic motion does fluctuate to subset the data into high motion frames and low motion frames.
Doing so gives us a method to compare particle configurations which may be correlated with driving instability in the system.
To do so, we sort frames from a single video by the VMSD value which represents the average kinetic motion.
We can use each subset of frames to generate PCFs for the top 30\% and the bottom 30\% VMSD values.
Because of the wide distribution of VMSD values, we focus on particles in a high powered (2.3 mW/$\upmu$m$^2$) standing wave.
Comparing the PCFs for the high VMSD frames (\figref{PCF} (a)) versus the low VMSD frames (\figref{PCF} (b)) for the HIPs shows subtle differences.
Primarily, we find more defined features for the low VMSD PCF.
This suggest that the system fluctuates between slower moving ordered states to faster moving disordered states.
Indeed, we observe sudden collapses and regeneration into order states in these optically bound systems.
Secondly, we observe that there is a higher distribution of particles in close proximity for the faster moving system.
We illuminate this subtle difference by subtracting the low VMSD PCF from the high VMSD PCF shown in \figref{PCF} (c).
We propose that multiple scattering effects which are stronger when particles are at close-range could be driving the collapse of stability and increase in average kinetic motion of this system.
For the LIPs, we find that the high VMSD PCF (\figref{PCF} (d)) versus the low VMSD PCF (\figref{PCF} (e)) are both structurally similar.
Not until taking the difference between the two as shown in \figref{PCF} (f), do we see that there are subtle differences.
For the low VMSD PCF, we observe that particles tend to be distributed among multiple lines: a single line that passes the origin and multiple off-axis lines.
These lines are evidence of the self-organization into multiple rods.
The center line represent displacements between particles that belong to the same rod and the off-axis lines represent displacements between particles that are located in neighboring rods.
Interestingly, the differences between the high and low PCFs suggest that inter-rod distance may play a role in the average kinetic motion of this system.
For instance, one can observe in \figref{PCF} (f) that the the first off-axis line is closer to the center line for the higher VMSD frames.
Comparing the HIPs and the LIPs is useful for understanding how multiple scattering is effecting these systems at different scales.
For the HIPs, where scattering is much stronger, we find that the presence of small clusters are generally correlated with higher kinetic motion and less structural order of the overall system.
We propose that HIP clusters can be treated as an emergent species which can alter the system dynamics.
On the other hand, the LIPs can generate much larger stable structures before the system fluctuates dynamically.
\subsection{Pathway to cluster formation}
The two-body force maps cannot describe how a cluster or rod forms; in fact, the two-body force maps suggest a weak repulsive force among particles that are aligned along the beam propagation.
As a result, we believe the cluster formation can only be explained by considering a complete $N$-body force which we compute using the CDM.
We find numerically that when the separation between aligned particles is sufficiently large ($\gtrsim 0.25 \upmu$m) the total force is always repulsive for HIP chains and increases in strength with the number of particles (\figref{numerical} (a)).
This suggests that a chain of particles aligned along the propagation axis should remain spread apart.
This is consistent with some of the structures found in the experiment: the lower density HIP chains are often found extending from the lower to upper boundary of the sample.
The strong repulsion at long range could also act as a barrier to cluster formation at very high light intensity and low particle density.
The trend is similar for LIP chains $\upmu$m (\figref{numerical} (b)).
At smaller distances, we observe a decrease in repulsive strength for all HIP chains.
For HIP chains of more than 3 particles, we observe an overall change in sign of the force (\figref{numerical} (a)).
This result suggests that the two-body repulsive force (\figref{static} (a)) dominates as long as the particles are far away; however, when many HIPs become in close range, $N$-body forces are strong enough to overcome two-body repulsion and switch the sign of the force entirely.
The same analysis of the LIP chains of particles shows a similar reduction of the repulsive force at shorter distances; however, an attractive force does not appear until the number of particles exceeds $N=14$ (\figref{numerical} (b)).
The initial reduction of the force, even at small $N$, is indicative of the presence of non-pairwise forces; however the effects are clearly weaker in comparison to the HIPs.
The numerical results provide a potential pathway to the previously unexpected formation of clusters.
The results also highlight the unique nature of the $N$-body optical binding forces.
These effects are shown to be comparatively much stronger for the HIPs.
This supports the hypothesis that a major factor driving the differences in behavior between the HIP and LIP systems is a stronger presence of $N$-body forces due to the higher scattering strength of HIPs.
\subsection{Non-conservative forces on clusters}
We simulated the effects of a particle cluster as it grows in size and interacts with neighboring particles.
We simulate a simplified configuration of N-particle closed packed cluster in line with a single lone particle at varying distance.
We then compute the total force on the clustered particles, $F_c$, as well as the force on the lone particle, $F_p$.
To compare the relative presence of conservative and non-conservative forces, we use the following quantity:
\begin{equation}
\gamma = \frac{|F_c + F_p|}{|F_c-F_p|}
\end{equation}
If the forces are equal and opposite -- indicating a conservative interaction -- we will obtain $\gamma = 0$.
In contrast, if the system is equal but in the same direction, this would be indicative of a system which would be driven ($\gamma \rightarrow \infty$).
Observing $\gamma$ for the interaction between the HIP cluster and particle, we find that the value increases exponentially as the cluster size increases from 2-6 particles (\figref{numerical} (c)).
At $\gamma > 1$ the force between the cluster and the particle are in the same direction, while for $\gamma \gg 1$ the forces are highly non-conservative.
For HIPs, there are configurations which could lead to uni-directional interactions first appearing for clusters of size 4.
For clusters of sizes $> 5$ particles, $\gamma > 1$ at long ranges.
This result is in good agreement with our observations of driven HIP clusters.
At certain cluster sizes, non-conservative forces appear to allow clusters of particles to propel in an otherwise symmetric initial field.
Despite the particles being identical in the simulation, the cluster-particle interaction adds an asymmetry which can drive the system into unidirectional motion in some cases.
Here we study a limited set of configurations due to their common observable presence in the experimental data; however, we expect there are a number of alternative configurations that can also exhibit highly propelled motion.
We expect that these strongly non-conservative interactions are cause for the increased average motion in the high light intensity binding experiments (\figref{dynamic} (a,b)).
For a similar set of cluster sizes in the LIP case, we find $\gamma < 0.1$ regardless of cluster size (\figref{numerical} (d)).
Indeed, we observed in the experiments that LIPs can build themselves into extremely long rods without exhibiting strongly non-conservative responses at small cluster sizes, as we found for the HIPs.
It is likely that non-conservative forces won't have a strong presence for LIPs until they've assembled into elongated rods for which we do observe fluctuations in the VMSD values.
\section{Conclusion}
Optical binding of many particles in 3D gives rise to complex behavior that can not be predicted from two-body interactions alone.
A simplified view of the interaction does, however, predict some features of the resulting structures: in particular the formation of extended chains of particles along the axis of light propagation.
However, comparing low to high index particles -- whose two-body interaction is nearly identical apart from overall strength -- already demonstrates differences that must arise from $N$-body interactions.
Our results stress how the relative strength of gradient and scattering forces play a role in these systems.
For instance, $ka \ll 1$ particles have been shown to experience interesting conservative interactions on complex potential landscapes; however exploring larger $ka$ affords the capability of controlling non-negligible contributions of non-conservative binding forces.
It is at this size regime that we are able to find considerable differences between the LIPs and HIPs.
In particular, the lower index particles form close packed (or nearly closed packed) chains, while the higher index particles are aligned with the beam, but usually spaced out by several particle diameters.
More strikingly, the dynamic behavior of these particles demonstrates effects that can only be explained by considering non-conservative $N$-body interactions.
This is most dramatically demonstrated by the temporary formation of close-packed clusters of 4-5 high index particles, which appear to drive an instability of the system as a whole.
Our results suggest that although optical binding can be used to guide the self-organization of colloidal particles, the effects of $N$-body forces are critical for the assembly of many-particle systems
when their scattering strength is sufficiently strong.
Notably, this is distinct from nano-particle assembly with optical binding, in which pairwise forces are generally a good approximation \cite{forbes_optical_2020}.
Potentially, this more complex force landscape offer new possibilities: with proper system design these forces could be used to promote or evade specific configurations or produce driven arrangements with dynamic behavior.
Unfortunately, predicting the behavior of large scale optically bound systems remains a difficult task without a quick and accurate method of computing the forces for a fully coupled $N$-body system.
Efforts on this front can be aided by further experimentation which can validate assumptions regarding $N$-body and non-conservative effects over a larger parameter space than the current study.
\bibliographystyle{unsrtnat}
\section{Optical scattering, gradient, and binding forces}
The analysis in the manuscript was guided by comparing two types of scattering bodies, Polystyrene (PS) and Silica (SiO$_2$) particles, denoted as HIP and LIP.
Initial calculations were done comparing the relative strengths of three optical forces: the scattering force, gradient force, and binding force.
The scattering force is a first-order optical force which arises from the back-reflection of incident photons \cite{ashkin_acceleration_1970}.
The scattering force is non-conservative and is dependent on the direction of light propagation.
We determine the strength of the force through simulating a single particle in single plane wave.
The gradient force is a first-order optical force due to the induced dipole moments generated in the scattering object by the incident light \cite{ashkin_observation_1986}.
The force is completely conservative and is highly depended on the gradients in the electric field.
We determine the strength of the force through simulating a single particle in a standing wave generated by two counter-propagating plane waves.
The two counter-propagating waves negate scattering forces, the standing wave pattern generates gradients in the optical field which apply gradient forces onto the single object.
The optical binding force is a second-order optical force due to the modification of an incident field on an object by the scattered fields of other objects \cite{burns_optical_1989}.
Optical binding is a combination of conservative and non-conservative forces.
We simulate the force by placing two objects separated in cross-polarized counter-propagating plane waves.
The counter-propagation negates the first-order scattering forces while cross-polarizing removes the standing wave suppressing the first-order gradient force.
Remaining forces must be due to optical binding effects.
At the size range of interest we find that PS particles have relatively stronger scattering forces while LIPs have relatively stronger gradient forces.
The conclusion made is that HIPs are stronger scatterers.
\section{Numerical Simulations}
The numerical simulations were achieved using a Coupled Dipole Method (CDM) based scattering code \cite{hoekstra_radiation_2001,draine_discrete-dipole_1994}.
In each case, we ensure 10 dipoles per wavelength is used.
We use an iterative solver in which the number of iterations are set until the root mean square of the dipole moments converge to $< 1 \%$.
The 2-dimensional maps were generated by running a stand-alone simulation at each particle displacement.
A single particle was fixed at an origin while a second particle was moved along a 2-dimensional space on the plane in the immediate vicinity of the fixed particle.
\section{Force map generation}
The force maps were generated using CDM based simulation.
By placing two particles in a cross-polarized counter-propagating field, we calculate the radiation forces placed on each particle.
Filling out the 2-dimensional space is done by fixing one object at an origin and repeating the simulation for various displacements.
\section{Correlating volume mean squared difference (VMSD) with mean squared displacement (MSD)}
\begin{figure*}[t!]
\centering
\includegraphics{sup_figure1.pdf}
\caption{A simulated PS particle displaced along the x, y, and z axis plotted against a corresponding squared volume difference.}
\label{fig:ref_fmsd}
\end{figure*}
\begin{figure}[h]
\centering
\includegraphics{sup_figure2.pdf}
\caption{VMSD values compared to 2-dimensional mean squared displacement (MSD) in the xy-plane for (a) HIP (PS) and (b) LIP (SiO$_2$) particles.}
\label{fig:msd_VMSD}
\end{figure}
3-Dimensional videos were taken of particles undergoing Brownian motion.
19 individual particles located in a single volume were averaged to generate a reference particle.
We used the reference particle to simulate particle motion by shifting it a known amount in a generated noisy background.
The results of the simulated displacements along different axes are provided in \figref{ref_fmsd}.
We can correlate the size of the simulated step to the value of taking the sum squared volume difference (VSD) of the array before and after the step.
We found that larger displacements correspond to a larger VSD in general.
One limitation is that the motion in the z-direction is underrepresented compared to motion in the x-y plane due to the inherent asymmetry of the particles imaging profile along the z-axis.
This limitation does not allow us to place an absolute estimate of how fast the objects are moving in physical units; however we can still measure the relative motion under the assumption that the motion in the compared systems share similar axial preferences.
Observations of the data support this assumption.
The second limitation is that the VSD value is capped at displacements that are slightly larger than the particle diameter.
At the volume rate of 10 volumes/s, particles undergoing Brownian motion are moving slow enough; however we have observed that particles undergoing optical binding forces can be propelled much faster.
One way to alleviate the limitation is to record at a higher volume rate; however, it is currently unfeasible to cover the full range of possible particle speeds.
This limitation is also present in traditional tracking methods.
Although the motion of the fastest particles are underrepresented in the VSD values, the VMSD method still allows a maximum value to be assigned.
Traditional tracking methods would not be able to assign a velocity value at all.
We also note that underrepresenting the motion of the faster moving particles only acts to underrepresent the strength of the findings.
The VMSD at any given time is determined by subtracting two subsequent volumes from each other, removing the contribution from the background, and dividing by the total number of particles in the frame:
\begin{equation}
\Delta(t) = \frac{1}{N_p(t)} \left[\sum_{i,j,k}^{volume}(p_{i,j,k,t} - p_{i,j,k,t-dt})^2 - \Delta_{0}\right]
\end{equation}
The background contribution, $\Delta_{0}$, is determined in each video by monitoring the squared difference of sections that have zero particles.
We find a non-zero constant value which can be subtracted to remove contributions that are due to noise.
The number of particles, $N_p$, is estimated by the following:
\begin{equation}
N_p = c_0 \sum_{i,j,k}^{volume}{|p_{i,j,k,t} - b_{i,j,k}|} + c_1
\end{equation}
Where $b_{i,j,k}$ is belongs to a volume representing the sample devoid of particles.
$c_0$ and $c_1$ are fitting parameters determined by linearly fitting the number of particles found by traditional tracking methods to $\sum_{i,j,k}^{volume}{|p_{i,j,k,t} - b_{i,j,k}|}$ for a system of particles undergoing Brownian motion.
Some results in \figref{msd_VMSD}.
For the HIPs, we show that for diffusive particles and particles interacting in a weak field, there is a clear correlation between the VMSD and the true MSD (\figref{msd_VMSD}).
For the LIPs, we also find correlation between the VMSD and true MSD for the diffusive particles; however in a weak field, the correlation is reduced.
We found that in a weak field, the Lower-index particles tended to pack into tight formations which effected the individual tracking capabilities.
Thus we utilize the purely diffusive correlation to justify the use of VMSD in a difficult tracking atmosphere.
We represent the motion in terms of the VMSD values relative to the Brownian Motion case.
Doing so allows us to compare the HIP and LIP systems in how their motion changes relative to diffusive motion.
\section{Electronic Supplementary Information}
All movies are 2 mins played in real-time.
\subsection{Movie S1}
HIP (Polystyrene) particles in cross polarized counter-propagating beam (peak power density - 1.5 mW/$\mu m^2$).
\subsection{Movie S2}
LIP (SiO$_2$) particles in cross polarized counter-propagating beam (peak power density - 1.5 mW/$\mu m^2$).
\subsection{Movie S2}
HIP (Polystyrene) particles in standing wave (peak power density - 1.5 mW/$\mu m^2$).
\bibliographystyle{unsrtnat}
|
1,116,691,500,893 | arxiv | \section{Introduction}
In flavor physics, the two ratios $R_D$ and $R_{D^*}$ are among
several long-term tensions between the Standard Model (SM) and the
related experimental measurements. These ratios are defined by
\begin{align}
R(\{D^*,D\}) \equiv \frac{\mathcal{B}(B_q\rightarrow \{D^*,D\}
\tau\nu)}{\mathcal{B}(B_q\rightarrow \{D^*,D\} l\nu )}\,,
\end{align}
where $l=e,\mu$. The processes $B_q\rightarrow \{D^*,D\} \ell \nu$
with $\ell=l,\tau$ arise at tree-level from the four fermion
transition $b\rightarrow c\tau\nu$. Upon combining different
experimental data by BaBar Collaboration
\cite{Lees:2012xj,Lees:2013uzd}, Belle Collaboration
\cite{Huschle:2015rga,Hirose:2016wfn,Hirose:2017dxl,Belle:2019rba}
and LHCb Collaboration
\cite{Aaij:2015yra,Aaij:2017deq,Aaij:2017uff}, the Heavy Flavor
Averaging Group (HFLAV) obtained the average of $R_D$ and
$R_{D^*}$ for 2021 as~\cite{HFLAV},
\begin{align}
R_D &= 0.339 \pm 0.026 \pm 0.014\,, \label{eq:current-data-2}\\
R_{D^*} &= 0.295 \pm 0.010 \pm 0.010\,.
\label{eq:current-data-1}
\end{align}
The corresponding averages of SM predictions as reported by
HFLAV~\cite{HFLAV} based on the predictions obtained in Refs.
~\cite{Bigi:2016mdz,Gambino:2019sif,Bordone:2019vic} are given by
\begin{align}
R_D^{\mathrm{SM}} &= 0.298 \pm 0.003\,, \\
R_{D^*}^{\mathrm{SM}} &= 0.252 \pm 0.005\,.
\end{align}
Clearly, the listed results of $R_D$ and $R_{D^*}$ in
Eqs.(\ref{eq:current-data-2},\ref{eq:current-data-1}) exceed the
SM predictions given above, by $1.4\sigma$ and $2.9\sigma$
respectively. Moreover, as reported by HFLAV~\cite{HFLAV}, the
difference with the SM predictions reported above, corresponds to
about $3.4\sigma [R_D-R_{D^*}]$ combination. This may serve as a
hint for a violation of lepton-flavor universality.
In the literature, many theoretical models have been investigated
aiming to explain the $R_D$ and $R_{D^*}$ anomalies. Based on the
remark that the measured ratio $R_{D^*}$ is higher than its value
in the SM, attempts in many models have focused on enhancing the
rate of $b\to c\tau \nu$ transition through extra contributions
arising from the new particles mediating the transition. This
turns to be much easier than reducing the rate of $b\to c (e,\mu)
\nu$ transitions, given the much more severe restraints on the
couplings of new physics to muons and electrons
\cite{Asadi:2018wea}.
The new particles mediating the $b\to c\tau \nu$ can be spin-0 or
spin-1. In addition, they can either carry baryon and lepton
number (leptoquarks) or be $B/L$ neutral (charged Higgs and $W'$).
As reported in Ref. \cite{Asadi:2018wea}, the existing models can
be classified into three main categories :
\begin{itemize}
\item \textit{Models with charged Higgs}
\cite{Tanaka:2010se,Fajfer:2012jt,Crivellin:2012ye,Celis:2012dk}.
In these class of models, integrating out the charged Higgs, that
mediates $b\to c\tau \nu$ transition at tree-level results in
scalar-scalar operators contributing to decay rates. The
contributions of these additional operators can be constrained
using the measured $B_c$ lifetime. This leads to the upper limits
${\rm BR}(B_c^{-} \to \tau^{-} \bar{\nu}_\tau) \leq 40$
\cite{Celis:2016azn}, ${\rm BR}(B_c^{-} \to \tau^{-}
\bar{\nu}_\tau) \leq 30$ \cite{Alonso:2016oyd} and a much stronger
bound ${\rm BR}(B_c^{-} \to \tau^{-} \bar{\nu}_\tau) \leq 10 \%$
was obtained from the LEP data taken at the $Z$ peak
\cite{Akeroyd:2017mhr}. Later on, the bounds using the measured
$B_c$ lifetime had been critically investigated and relaxed upper
limit of $\leq 39 \%$~\cite{Bardhan:2019ljo} and $\leq 60
\%$~\cite{Blanke:2018yud,Blanke:2019qrx} were obtained.
\item \textit{Models with Heavy charged vector bosons}
\cite{He:2012zp,Boucenna:2016qad}. In such class of models,
integrating out $W'$'s yields new contributions to the
vector-vector operators. Simultaneous explanation of both $R_D$
and $R_{D^{*}}$ with left-handed neutrinos requires non zero
values for the associated new contributions. It should be noted
that, these class of models are subjected to constraints
originating from the presence of an accompanying $Z'$ mediator.
Thus, the vertex $Z'b_Ls_L$ can be inferred from the $W'b_Lc_L$
vertex through the $SU(2)$ invariance. The $Z'b_Ls_L$ term in the
Lagrangian can lead to tree-level flavor-changing neutral currents
(FCNCs) and hence there is a need to some mechanism to subdue the
contributions of this term. One way to do that is to assume
minimal flavor violation (MFV) as considered in Refs.
\cite{Greljo:2015mma,Faroughy:2016osc}. However, adopting this
choice in general can not suppress $Z'bb$ and $Z'\tau\tau$
vertices. Consequently, LHC direct searches for $Z'\to\tau\tau$
resonances can result in stringent restraints on the related
parameters of the models. However, as shown in Refs.
\cite{Faroughy:2016osc,Crivellin:2017zlb}, selecting unnaturally
high $Z'$ widths helps to avoid these severe constraints.
\item \textit{Models with Leptoquarks}
\cite{Fajfer:2012jt,Tanaka:2012nw}. Either scalar and vector
leptoquarks can be potentially favored as an explanations for the
$R_{D^{(*)}}$ anomaly \cite{Crivellin:2017zlb,Calibbi:2017qbu}.
These models are subjected also to many important constraints
including those from $b\rightarrow s \nu\nu$ restraints
\cite{Crivellin:2017zlb}, searches at the LHC for $\tau\tau$
resonances \cite{Faroughy:2016osc,Crivellin:2017zlb}, and the
measured $B_c$ life-time \cite{Alonso:2016oyd, Akeroyd:2017mhr}.
However, these constraints turn to be weaker compared to the
corresponding ones imposed on the alternative models described
above \cite{Asadi:2018wea}.
\end{itemize}
In this study, we aim to derive the effective Hamiltonian
resulting from the four fermion transition $b\to c\tau \nu$ in the
minimal left-right model with an inverse-seesaw (LRIS). At
tree-level, the transition is mediated by the exchanging of the
charged Higgs boson in the model. Having the effective
Hamiltonian, we will proceed to show the dominant Wilson
coefficients contributing to the ratios $R_D$ and $R_{D^*}$.
Consequently, we will display the dependency of the ratios on the
parameters of the model to determine the most relevant ones.
Moreover, we reexamine the relation between the observable
$R_{D^*}$ and ${\rm BR}(B_c \to \tau \bar{\nu}_\tau)$, reported in
Refs.~\cite{Alonso:2016oyd,Akeroyd:2017mhr}, in light of Belle
combination~\cite{Belle:2019rba} and LHCb~\cite{Aaij:2017deq}
experimental results, and the projection at Belle II experiment
with an integrated luminosity of 50 $\rm ab^{-1}$
\cite{Belle-II:2018jsg}. This turns to be very important because
the constraint on ${\rm BR}(B_c \to \tau\bar{\nu}_\tau)$ affects
substantially the contributions from scalar
operators~\cite{Bardhan:2019ljo,Blanke:2018yud,Blanke:2019qrx}.
Finally, we will also investigate the possibility of resolving the
anomalies after including the new contributions to the Wilson
coefficients of the total effective Hamiltonian in the presence of
the LRIS model.
This paper is organized as follows. In Sec.~\ref{model}, we give a
brief review of the Left-Right Model with inverse seesaw. In the
review, we discuss the gauge structure and the particle content
of the model. We also discuss the fermion interactions with
charged Higgs and with the charged $W,W'$ gauge bosons. These
interactions are required to derive the effective Hamiltonian
governs the processes contributing to $R_D$ and $R_{D^*}$.Then, in
Sec.~\ref{higgs}, we present the effective Hamiltonian describing
$b\to c\tau \nu$ transition in the presence of a general New
Physics (NP) beyond the SM. Particularly, we derive the analytic
expressions of the Wilson coefficients up to one loop level
originating from the charged Higgs mediation in LRIS model under
study in this paper. Based on the Hamiltonian, we show the total
expressions of $R_D$ and $R_{D^*}$. In Sec.~\ref{nums}, we give
our estimation of $R_D$ and $R_{D^*}$ and their dependency on the
parameter space. Finally, in Sec.~\ref{sec:conclusion}, we give
our conclusion.
\section{\label{model}Left-Right Model with inverse seesaw (LRIS)}
The gauge sector of the minimal left-right model with an
inverse-seesaw (LRIS) is based on the symmetry $SU(3)_C\times
SU(2)_L\times SU(2)_R\times U(1)_{B-L}$. The particle content of
the model includes the usual particles in the SM in addition to
extra fermions, scalars and gauge bosons. Regarding the fermion
content of the model, it is the same as its counterpart in the
conventional left-right
models~\cite{Mohapatra:1974gc,Senjanovic:1975rk,Mohapatra:1977mj,Deshpande:1990ip,Aulakh:1998nn,Maiezza:2010ic,Borah:2010zq,Nemevsek:2012iq}.
The implementation of the IS mechanism for neutrino masses can be
carried out via introducing three SM singlet fermions $S_1$ with
$B-L$ charge $=+2$ and three singlet fermions $S_2$ with $B-L$
charge $=-2$. It should be remarked that, this particular choice
of the $B-L$ charges of the pair $S_{1,2}$ assures that the
$U(1)_{B-L}$ anomaly is free.
The scalar sector of the LRIS model contains the SM Higgs
doublet, an extra new $SU(2)_R$ scalar doublet $\chi_R$ and a
scalar bidoublet $\phi$. The extra doublet and bidoublet are
essential for breaking the symmetries of the model upon having
nonvanishing VEVs $\langle \zeta\rangle \neq 0$ for
$\zeta=\phi,\chi_R$. We define $\langle \chi_R\rangle
=v_R/\sqrt{2}$ and assume that $v_R$ of an order TeV to break the
right-handed electroweak sector together with $B-L$. Regarding the
VEV of the scalar bidoublet $\phi$, we use the parametrization
$\langle \phi \rangle =\text{diag}(k_1/\sqrt{2},k_2/\sqrt{2})$
where $k_1=v s_{\beta},~k_2=v c_{\beta}$ and hence $v^2 =k_1^2 +
k_2^2 $ with $v$ of an order $\mathcal{O}(100)~\text{GeV}$ to
break the electroweak symmetry of the SM. Here and after we use
the definitions $s_x=\sin x,~c_x=\cos x$, $t_x=\tan x$ and
$ct_x=\cot x$.
The scalar-fermion interactions in the LRIS model can be inferred
from the left-right symmetric Yukawa Lagrangian which can be
generally expressed as
\begin{equation}
\mathcal{L}_{\text{Y}}=\sum_{i,j=1}^3 y_{ij}^L \bar{L}_{Li}\phi
L_{Rj} +\tilde{y}_{ij}^L \bar{L}_{Li} \tilde{\phi} L_{Rj} +
y_{ij}^Q \bar{Q}_{Li}\phi Q_{Rj} +\tilde{y}_{ij}^Q \bar{Q}_{Li}
\tilde{\phi} Q_{Rj} +y^s_{ij} \bar{L}_{Ri} \tilde{\chi}_R
S^c_{2j}+H.c., \label{Lag}\end{equation} here $i$ and $j$ are
family indices that run from $1\ldots3$, $y^Q$ and $\tilde y^Q$
represent the quark Yukawa couplings while $y^L$ and $\tilde{y}^L$
stand for the lepton Yukawa couplings. It should be noted that a
$Z_2$ discrete symmetry is implemented in order to forbid a mixing
mass term $M \bar{S^c_1} S_{2}$ and thus preserving the IS
mechanism, as discussed in~\cite{Khalil:2010iu}. Under this
symmetry, only $S_1$ has an odd charge while the rest of other
particles have even charges. The fields in Eq.(\ref{Lag}) are
defined as
\begin{equation} \chi_R=
\begin{pmatrix}\chi_R^+
\\\chi_R^0 \\\end{pmatrix},\,\,\,\, \phi=
\begin{pmatrix}\phi_1^0 &\phi_1^+\\\phi_2^-
&\phi_2^0\\\end{pmatrix},\,\,\,\, Q_{A}=\begin{pmatrix}u_{A}
\\d_{A} \\\end{pmatrix},\,\,\, L_{A}=\begin{pmatrix}\nu_{A} \\e_{A}
\\\end{pmatrix},\,\,\,\, for\,\,\, A=L,R.\end{equation}
The dual bidoublet $\widetilde{\phi}$ and doublet
$\widetilde{\chi}_{R}$ are defined as
$\widetilde{\phi}=\tau_2\phi^*\tau_2$ and
$\widetilde{\chi}_{R}=i\tau_2\chi^*_{R}$ respectively. Clearly,
from the scalar sector of the model and before symmetry breaking,
there are $12$ scalar degrees of freedom: $4$ of $\chi_R$ and $8$
of $\phi$. After symmetry breaking, six scalars out of these $12$
degrees of freedom remain as physical Higgs bosons while the other
degrees of freedom are eaten by the neutral gauge bosons: $Z_\mu$
and $Z'_\mu$ and the charged gauge bosons: $W^\pm_\mu$ and
$W'^\pm_\mu$ to acquire their masses. The physical Higgs bosons
are two charged Higgs bosons, one pseudoscalar Higgs boson, and
the remaining three are $CP$-even neutral Higgs bosons. For a
detailed discussion about the Higgs sector of the LRIS model we
refer to Ref.\cite{Ezzat:2021bzs}.
We turn now to the neutrino sector of LRIS model. After breaking
the $B-L$ symmetry, a nonrenormalizable term of dimension seven
$\propto \chi_R^4 \bar{S^c_2}S_2/M^3$ yields a small mass term
$\mu_s \bar{S^c_2} S_2$. In turn, this implies that
$\mu_s=\lambda_s v_R^4/M^3 \raise0.3ex\hbox{$\;<$\kern-0.75em\raise-1.1ex\hbox{$\sim\;$}}
\mathcal{O}(1)~\text{KeV}$~\cite{Abdallah:2011ew}. Here
$\lambda_s$ stands for an interaction effective dimensionless
coupling. With the adopted choice
$v_R\sim\mathcal{O}(10^{3})~\text{GeV}$, one can see that
$\mu_s\sim\mathcal{O}(10^{-7})~\text{GeV}$ corresponding to
$\lambda_s/M^3\sim\mathcal{O}(10^{-19})~\text{GeV}$. For a choice
$\lambda_s\sim\mathcal{O}(1)$, the nonrenormalizable scale $M$ is
an intermediate scale, namely
$M\sim\mathcal{O}(10^{3})~\text{TeV}$ whereas for
$\lambda_s\sim\mathcal{O}(10^{-3})$, one obtains
$M\sim\mathcal{O}(10)~\text{TeV}$. Consequently, the Lagrangian of
neutrino masses can be expressed as \begin{equation} {\cal L}_m^\nu = M_D
\bar{\nu}_L \nu_R + M_R \bar{\nu}^c_R S_2 + \mu_s \bar{S^c_2}
S_2+H.c., \end{equation} where $M_{R}$ and $M_D$ are $3\times 3$ matrices
related to $v_R$ and Dirac neutrino masses respectively via
$M_{R}=y^s v_R/\sqrt{2}$ and $M_D=v(y^L s_{\beta}+\tilde{y}^L
c_{\beta})/\sqrt{2}$ in the standard neutrino IS
matrices~\cite{Mohapatra:1986aw,Mohapatra:1986bd,GonzalezGarcia:1988rw,Weiland:2013xia}.
In the flavor basis $\psi=(\nu_L^c,\nu_R,S_2)$, we have a term
$\bar{\psi}^c\mathcal{M}_\nu\psi$ where $\mathcal{M}_\nu$ is the
$9\times 9$ neutrino mass matrix given by
\begin{equation}
\mathcal{M}_\nu=
\begin{pmatrix}
0 & M_D & 0 \\
M_D^T & 0 & M_{R}\\
0 & M_{R}^T & \mu_s
\end{pmatrix}.
\end{equation}
Following the standard procedure, the diagonalization of
$\mathcal{M}_\nu$ results in the light and heavy neutrino mass
eigen states $\nu_{\ell_i},\nu_{h_j}$ with the mass eigenvalues
given by:
\begin{align}
\label{nulmass}
m_{\nu_{\ell_i}}&=M_D M_{R}^{-1} \mu_s (M_{R}^T)^{-1} M_D^T,\quad i=1\ldots3,\\
\label{nuhmass} m_{\nu_{h_j}}^2&=M_{R}^2+M_D^2,\quad j=1\ldots6.
\end{align}
The light neutrino mass matrix in Eq.(\ref{nulmass}) must be
diagonalized by the physical neutrino
mixing matrix $U_{\!M\!N\!S}$ \cite{mns}, {\it i.e.}, %
\begin{equation}%
U_{\!M\!N\!S}^T m_{\nu_{\ell_i}} U_{\!M\!N\!S} =
m_{\nu_{\ell_i}}^{\rm d} \equiv
{\rm diag}\{m_{\nu_e}, m_{\nu_\mu}, m_{\nu_\tau}\}.%
\end{equation}
Thus, one can easily show that the Dirac neutrino mass
matrix can be defined as :%
\begin{equation} %
M_D=U_{\!M\!N\!S}\, \sqrt{m_{\nu_{\ell_i}}^{\rm d}}\, R\, \sqrt{\mu^{-1}_s}\, M_R, %
\label{mD}
\end{equation} %
where $R$ is an arbitrary orthogonal matrix. Clearly, for $M_R$ at
TeV scale and $\mu_s \ll M_R$ the light neutrino masses can be of
order eV. The $9\times 9$ neutrino mass matrix ${\cal M}_\nu$, can
be diagonalized with the help of the matrix $V$ satisfying $V^T
{\cal
M}_\nu V = {\cal M}_\nu^{\rm diag}$. The matrix can be expressed as \cite{Dev:2009aw}%
\begin{equation} V=
\left(%
\begin{array}{cc}
V_{3\times 3} & V_{3\times6}\\
V_{6\times 3} & V_{6\times6} \\
\end{array}%
\right).%
\end{equation}%
As a good approximation, the matrix $V_{3\times3}$ can be given a %
\begin{equation}%
V_{3\times3} \simeq \left(1-\frac{1}{2} F F^T \right)
U_{\!M\!N\!S} \, . %
\end{equation}%
where $F = M_D M^{-1}_R$. Generally $F$, as one can see from the
expression of $V_{3\times 3}$, is not unitary matrix. This
unitarity violation, {\it i.e.,~}, the deviation from the standard
$U_{\!M\!N\!S}$ matrix, depends on the size of $\frac{1}{2} F F^T$
\cite{Abdallah:2011ew}. The
matrix $V_{3\times6}$ is given by%
\begin{equation}%
V_{3\times6}=\left({\bf 0}_{3\times3},F \right) V_{6\times6}. %
\label{V36}
\end{equation} %
Finally, the matrix $V_{6\times 6}$ is the one that diagonalize
the $\{\nu_R, S_2\}$ mass matrix.
The process under study can receive dominant contributions from
the charged Higgs mediation at tree level as we will show in the
following. Thus, below, we list the relevant charged Higgs
couplings related to our study following Ref.\cite{Ezzat:2021bzs}.
In the flavor basis $(\phi_1^{\pm},\phi_2^{\pm},\chi_R^{\pm})$,
the charged Higgs bosons symmetric mass matrix takes the form
\begin{equation}\label{mhpm2}
M_{H^\pm}^2=\frac{\alpha_{32}}{2}\begin{pmatrix}
\frac{v_R^2 s_{\beta}^2}{c_{2\beta}} & \frac{v_R^2 s_{2\beta}}{2c_{2\beta}} & -v v_R s_{\beta} \\
. & \frac{v_R^2 c_{\beta}^2}{c_{2\beta}} & -v v_R c_{\beta} \\
. & . & v^2 c_{2\beta}
\end{pmatrix}.
\end{equation}
The above matrix can be diagonalized by the unitary matrix,
\begin{equation}
Z^{H^\pm}=\begin{pmatrix}
\frac{v c_{2\beta}}{\sqrt{v^2c_{2\beta}^2+v_R^2s_{\beta}^2}} & 0 & \frac{v_R s_{\beta}}{\sqrt{v^2c_{2\beta}^2+v_R^2s_{\beta}^2}} \\
-\frac{\frac{1}{2}v_R^2
s_{2\beta}}{\sqrt{(v^2c_{2\beta}^2+v_R^2s_{\beta}^2)(v^2c_{2\beta}^2+v_R^2)}}
&
\sqrt{\frac{v^2c_{2\beta}^2+v_R^2s_{\beta}^2}{v^2c_{2\beta}^2+v_R^2}}
& \frac{v v_R
c_{\beta}c_{2\beta}}{\sqrt{(v^2c_{2\beta}^2+v_R^2s_{\beta}^2)(v^2c_{2\beta}^2+v_R^2)}}
\\
-\frac{v_R s_{\beta}}{\sqrt{v^2c_{2\beta}^2+v_R^2}} & -\frac{v_R
c_{\beta}}{\sqrt{v^2c_{2\beta}^2+v_R^2}} & \frac{v
c_{2\beta}}{\sqrt{v^2c_{2\beta}^2+v_R^2}}
\end{pmatrix}.
\end{equation}
The mass eigenstates basis can be obtained using the rotation
$(\phi_1^{\pm},\phi_2^{\pm},\chi_R^{\pm})^T=Z^{H^\pm
T}(G_1^{\pm},G_2^{\pm},H^{\pm})^T$ such that
$Z^{H^\pm}M_{H^\pm}^2Z^{H^\pm T}=\text{diag}(0,0,m_{H^\pm}^2)$.
Here $G_1^{\pm}$ and $G_2^{\pm}$ represent the massless charged
Goldstone bosons and $H^{\pm}$ is a massive physical charged
Higgs boson. The massless Goldstone bosons are eaten by the
charged gauge bosons $W_\mu$ and $W'_\mu$ to acquire their masses
via the familiar Higgs mechanism. On the other hand, the mass of
the charged Higgs boson $H^{\pm}$ is given by:
\begin{equation}
m_{H^{\pm}}^2 = \frac{\alpha_{32}}{2}\left(\frac{
v_R^2}{c_{2\beta}}+v^2c_{2\beta}\right),\label{chHm}
\end{equation}
where $\alpha_{32}=\alpha_3-\alpha_2$. Clearly, the charged Higgs
boson mass can be of the order of hundreds~\text{GeV} if we pick
out the values $v_R\sim\mathcal{O}(\text{TeV})$ and
$\alpha_{32}\sim\mathcal{O}(10^{-2})$. It is clear to see that,
the physical charged Higgs boson is a linear combination of the
flavor basis fields $\phi_1^{\pm},\phi_2^{\pm},\chi_R^{\pm}$,
namely given as
\begin{equation}
H^\pm=Z^{H^\pm}_{13}\phi_1^\pm+Z^{H^\pm}_{23}\phi_2^\pm+Z^{H^\pm}_{33}\chi_R^\pm.
\end{equation}
The effective Lagrangian, in the LRIS model, describing the
charged Higgs and the charged gauge bosons $W$ and $W'$ couplings
to quarks and leptons can be expressed as \begin{equation} \mathcal{L}_{eff}=
\mathcal{L}^{\bar q q' H^\pm}_{eff}+\mathcal{L}^{\bar \nu \ell
H^\pm}_{eff}+\mathcal{L}^{ W,W'}_{eff}\end{equation} The Lagrangians
$\mathcal{L}^{\bar q q' H^\pm}_{eff}$ and $\mathcal{L}^{\bar \nu
\ell H^\pm}_{eff}$ can be obtained from expanding
$\mathcal{L}_{\text{Y}}$, given in Eq.(\ref{Lag}), and rotating
the fields to their corresponding ones in the mass eigenstates
basis. It is direct to obtain
\begin{eqnarray}
\mathcal{L}^{\bar q q' H^\pm}_{eff}&=& \bar{u}_i {\Gamma_{u_i d_j
}^{H^\pm\,LR\,\rm{eff} } }P_R d_j H^\pm\,+ \bar{u}_i {\Gamma_{u_i
d_j }^{H^\pm\,RL\,\rm{eff} } }P_L d_j H^\pm+ h.c.\, ,
\label{Higgs-vertex}
\end{eqnarray}
where
\begin{eqnarray} {\Gamma_{u_i d_j
}^{H^\pm\,LR\,\rm{eff} } } &=&
\sum_{a=1}^3 (V^R_{CKM})^*_{j a} y^Q_{i a} Z_{{3 2}}^{H^{\pm}} + \sum_{a=1}^3
(V^R_{CKM})^*_{j a} \tilde{y}^Q_{i a} Z_{{3 1}}^{H^{\pm}}
\nonumber\\
{\Gamma_{u_i d_j }^{H^ \pm\,RL\,\rm{eff} } } &=&
\sum_{a=1}^{3}(V^L_{CKM})^*_{j a} y^{Q^*}_{a i} Z_{{3
1}}^{H^{\pm}} + \sum_{a=1}^{3}(V^L_{CKM})^*_{j a}
\tilde{y}^{Q^*}_{a i} Z_{{3 2}}^{H^{\pm}}.
\label{Higgsv1}
\end{eqnarray}
In the LRIS model and after electroweak symmetry breaking, quarks
and charged leptons acquire their masses via Higgs mechanism.
Consequently, we can express the quark Yukawa couplings in terms
of the quark masses and CKM matrices in the left and right sectors
as \begin{eqnarray} y^Q &=&\frac{1}{(t_\beta-ct_\beta)}\bigg(\frac{M^{diag}_u
V_{\text{CKM}}^{R \dag}}{v_d} -\frac{V^L_{\text{CKM}} M^{diag}_d}{v_u} \bigg)\nonumber\\
\tilde y^Q &=&\frac{1}{(ct_\beta-t_\beta)}\bigg(\frac{M^{diag}_u
V_{\text{CKM}}^{R \dag}}{v_u} - \frac{V^L_{\text{CKM}} M^{diag}_d
}{v_d} \bigg),\label{yuQ} \end{eqnarray} with $v=246$ GeV, $v_u= \frac{v
s_{\beta}}{\sqrt{2}}$, $v_d= \frac{v c_{\beta}}{\sqrt{2}}$. In the
above equation, $M^{diag}_{u} (M^{diag}_{d})$ is the diagonal up
(down) quark mass matrix, $V^L_{\text{CKM}}$ and
$V^R_{\text{CKM}}$ are the CKM matrices in the left and right
sectors respectively. The mixing matrices for left and right
quarks result in the CKM matrices in the left and right sectors
$V_{\text{CKM}}^{L,R}=V^{u\dag}_{L,R} V^d_{L,R}$. We can choose
the bases where $V^u_L=I$ and as a result, in this basis,
$V_{\text{CKM}}^L=V^d_L$. Turning now to the right sector, we
follow Ref.\cite{Kiers:2002cz}, where the matrix
$V^R_{\text{CKM}}$ is given by \begin{equation} V^R_{\text{CKM}} = K ~
V^L_{\text{CKM}}\left(\theta^R_{12}, \theta^R_{23},\theta^R_{13},
\delta_R \right) ~ \tilde{K}^+, \end{equation} here the diagonal matrices $K$
and $\tilde{K}$ contain five of six non-removable phases in
$V^R_{\text{CKM}}$. It was emphasized that CP violation and FCNC
in the right-handed sector can be under control if the
$V^R_{\text{CKM}}$ is of the form \begin{equation} V^R_{\text{CKM}}=
\left(\begin{array}{ccc}
1 & 0 & 0 \\
0 & c_{\theta^R_{13}} & s_{\theta^R_{13}} e^{i \alpha} \\
0 & s_{\theta^R_{13}} & -c_{\theta^R_{13}} e^{i\alpha}
\end{array} \label{VRCKMp}\right),
\end{equation} where $c_{\theta^R_{13}}= \cos(\theta^R_{13})$ and
$s_{\theta^R_{13}}= \sin(\theta^R_{13})$ and as we will see that,
in the following, the phase $\alpha$ in the third column plays an
important role in increasing the values of $R_{D^*}$. It is also
possible to have a non-vanishing phase in the second column, which
turns out to be irrelevant and has no effect on the $R_D$ or
$R_{D^*}$ results. Therefore, we set it to zero. We work in the
bases where $V^d_R=I$ and thus $V_{\text{CKM}}^R=V^u_R$, similar
to the left-handed quark sector.
We proceed now to the lepton-neutrinio charged Higgs interactions
in the model under study. These interactions contribute to the
effective Lagrangian $\mathcal{L}^{\bar \nu \ell H^\pm}_{eff}$ and
generally, can be written as
\begin{eqnarray}
\mathcal{L}^{\bar \nu \ell H^\pm}_{eff}&=& \bar\nu_i
{\Gamma_{\nu_i \ell_j }^{H^\pm\,LR\,\rm{eff} } }P_R \ell_j
H^\pm\,+ \bar\nu_i {\Gamma_{\nu_i \ell_j }^{H^\pm\,RL\,\rm{eff} }
}P_L \ell_j H^\pm + h.c.\, ,
\label{Higgs-vertex}
\end{eqnarray}
with
\begin{eqnarray} {\Gamma_{\nu_i \ell_j }^{H^\pm\,LR\,\rm{eff} } }
&=&\sum_{a=1}^{3} \tilde y^L_{a j} V_{{i a}} Z_{{3 1}}^{H^{\pm}}
- \sum_{a=1}^{3}y^L_{a j} V_{{i a}} Z_{{3 2}}^{H^{\pm}}
-\sum_{a=1}^{3}y^{s *}_{j a} V_{{i 6 + a}} Z_{{3 3}}^{H^{\pm}}
\nonumber\\
{\Gamma_{\nu_i \ell_j }^{H^ \pm\,RL\,\rm{eff} } } &=& -
\sum_{a=1}^{3} V^{*}_{{i 3 + a}} y^{L *}_{j a} Z_{{3
1}}^{H^{\pm}} + \sum_{a=1}^{3} V^{*}_{{i 3 + a}} \tilde y^{L *}_{j
a} Z_{{3 2}}^{H^{\pm}}.
\label{Higgsv2}
\end{eqnarray}
The lepton Yukawa couplings $y^L$ and $\tilde{y}^L$ can be
expressed as
\begin{align}
\label{yukmat} y^L
&=\frac{t_{2\beta}}{\sqrt{2}\,v_u}\bigg(M_\ell-t_\beta M_D \bigg)\,,\\
\tilde{y}^L&=\frac{t_{2\beta}}{\sqrt{2}\,v_u}\bigg(M_D -t_\beta
M_\ell\bigg),
\end{align}
where $M_\ell$ is charged lepton diagonal mass matrix. Finally,
the leptons and quarks gauge interactions related to our processes
at tree-level can be deduced from the effective Lagrangian
$\mathcal{L}^{ W,W'}_{eff}$ given by
\begin{eqnarray}
\mathcal{L}^{ W,W'}_{eff}&=& \frac{1}{\sqrt{2}} g_2 \cos\phi_W
(V_{\text{CKM}}^L)^*_{ji} W_\mu \bar u_i \gamma_{\mu} P_L d_j -
\frac{1}{\sqrt{2}} g_R \sin\phi_W (V_{\text{CKM}}^R)^*_{ji} W_\mu
\bar u_i \gamma_{\mu} P_R d_j\nonumber\\&+&\frac{1}{\sqrt{2}} g_2
\sin\phi_W (V_{\text{CKM}}^L)^*_{ji} W'_\mu \bar u_i \gamma_{\mu}
P_L d_j - \frac{1}{\sqrt{2}} g_R \cos\phi_W
(V_{\text{CKM}}^R)^*_{ji} W'_\mu \bar u_i \gamma_{\mu} P_R d_j
\nonumber\\&+&\frac{1}{\sqrt{2}} g_2 \cos\phi_W V_{{i j}} W_\mu
\bar \nu_i \gamma_{\mu} P_L e_j - \frac{1}{\sqrt{2}} g_R
\sin\phi_W V^{*}_{{i 3 + j}} W_\mu \bar \nu_i \gamma_{\mu} P_R
e_j \nonumber\\&+&\frac{1}{\sqrt{2}} g_2 \sin\phi_W V_{{i j}}
W'_\mu \bar \nu_i \gamma_{\mu} P_L e_j - \frac{1}{\sqrt{2}} g_R
\cos\phi_W V^{*}_{{i 3 + j}} W'_\mu \bar \nu_i \gamma_{\mu} P_R
e_j+ h.c.\, ,
\label{Higgs-vertex}
\end{eqnarray}
where $\phi_W$ is the mixing angle between $W_L$ and $W_R$, which
is of order $10^{-3}$. We have now all the ingredient required for
deriving the effective Hamiltonian contributing to the transition
$b\to c\tau\nu$. In next section, we derive this Hamiltonian and
list the expressions of the Wilson coefficients corresponding to
this Hamiltonian.
\section{\label{higgs} The effective Hamiltonian relevant to the processes in the LRIS}
In the presence of NP beyond SM, the effective Hamiltonian governs
$\vert \Delta c \vert=1$ B decays transition relevant to our processes up to one-loop level
derived from the diagrams in Fig.\ref{ZBs}, can be expressed as %
\begin{eqnarray} {\mathcal H}_{eff} &=& -\frac{4 G_F}{\sqrt{2}}V_{cb}\bigg(
(1+g_V^{LL})\,(\bar{c} \gamma_\mu P_L b) (\bar \ell_j \gamma^\mu
P_L \nu_i)+ g_V^{RR}(\bar{c} \gamma_\mu P_R b) (\bar \ell_j
\gamma^\mu P_R \nu_i) + g_V^{LR}(\bar{c} \gamma_\mu P_L b) (\bar
\ell_j \gamma^\mu P_R \nu_i)\nonumber\\&+& g_V^{RL} (\bar{c}
\gamma_\mu P_R b) (\bar \ell_j \gamma^\mu P_L
\nu_i)+g_S^{LL}(\bar{c} P_L b) (\bar\ell_j P_L \nu_i) +
g_S^{RR}(\bar{c} P_R b) (\bar\ell_j P_R \nu_i)+g_S^{LR}(\bar{c}
P_L b) (\bar\ell_j P_R \nu_i)\nonumber\\&+&g_S^{RL}(\bar{c} P_R b)
(\bar\ell_j P_L \nu_i)+ g_T^{LL}(\bar{c}\sigma_{\mu \nu } P_L b)
(\bar \ell_j \sigma^{\mu \nu
}P_L{\nu}_i)+g_T^{RR}(\bar{c}\sigma_{\mu \nu } P_R b) (\bar
\ell_j\sigma^{\mu \nu } P_R{\nu}_i)\bigg),\label{Hef}\end{eqnarray} where
$V_{cb}$ is the $cb$ Cabibbo-Kobayashi-Maskawa (CKM) matrix
element, $P_{L,R}=\frac{1}{2}(1\mp \gamma_5)$, $\sigma_{\mu\nu}=
\frac{i}{2} [\gamma_\mu, \gamma_\nu]$ and as mentioned in
Ref.\cite{Tanaka:2012nw} the tensor operator with chirality
$(\bar{c}\sigma_{\mu \nu } P_R b) (\bar \ell_j \sigma^{\mu \nu }
P_L{\nu}_i)$ vanishes. In case of $\ell =\tau$ we find that
\begin{figure}[tbhp]
\includegraphics[width=4cm,height=4cm]{treeH.eps}
\hspace{0.2cm}
\includegraphics*[width=4cm,height=4cm]{TreeV1.eps}
\hspace{0.2cm}
\includegraphics*[width=4cm,height=4cm]{TreeV3.eps}
\hspace{0.2cm}
\includegraphics*[width=4cm,height=4cm]{TreeV2.eps}
\hspace{0.2cm}
\includegraphics*[width=4cm,height=4cm]{box20b1.eps}
\hspace{0.2 cm}
\includegraphics[width=4cm,height=4cm]{box21b1.eps}
\hspace{0.2cm}
\includegraphics*[width=4cm,height=4cm]{box2b.eps}
\hspace{0.2 cm}
\includegraphics*[width=4cm,height=4cm]{box22b1.eps}
\medskip
\caption{ Diagrams contributing to ${\mathcal H}_{eff}$, in
Eq.(\ref{Hef}) up to one loop-level due to charged Higgs mediation
with $V=Z,\gamma$. \label{ZBs}}
\end{figure}
\begin{eqnarray}
g_V^{LL} &=& (\cos^2\phi_W + \frac{m^2_W}{m^2_{W'}} \sin^2\phi_W ) Z_{M,{i 3}}^{v} ,\nonumber\\
g_V^{RR} &=& \frac{1}{V_{cb}} (\sin^2\phi_W + \frac{m^2_W}{m^2_{W'}} \cos^2\phi_W )(V_{\text{CKM}}^R)^*_{32} Z^{v,*}_{M,{i 6}} ,\nonumber\\
g_V^{LR} &=& - (1 + \frac{m^2_W}{m^2_{W'}} ) Z^{v,*}_{M,{i 6}}\cos\phi_W \sin \phi_W ,\nonumber\\
g_V^{RL} &=& -\frac{1}{V_{cb}} (1 + \frac{m^2_W}{m^2_{W'}} )
(V_{\text{CKM}}^R)^*_{32} Z_{M,{i3}}^{v}\cos\phi_W \sin \phi_W ,\nonumber\\
g_S^{LL} &=& -\frac{\sqrt{2}}{4 G_F V_{cb} m^2_{H^\pm}} {\Gamma_{u_2 d_3 }^{H^\pm\,RL\,\rm{eff} } }{(\Gamma_{\nu_i \ell_3 }^{H^\pm\,LR\,\rm{eff}})^* },\nonumber\\
g_S^{RR} &=& -\frac{\sqrt{2}}{4 G_F V_{cb}
m^2_{H^\pm}}{\Gamma_{u_2 d_3 }^{H^\pm\,LR\,\rm{eff} }
}{(\Gamma_{\nu_i\ell_3}^{H^\pm\,RL\,\rm{eff} })^* },\nonumber\\
g_S^{LR} &=& -\frac{\sqrt{2}}{4 G_F V_{cb} m^2_{H^\pm}}{\Gamma_{u_2 d_3 }^{H^\pm\,RL\,\rm{eff} } } {(\Gamma_{\nu_i
\ell_3}^{H^\pm\,RL\,\rm{eff} })^* },\nonumber\\
g_S^{RL} &=& -\frac{\sqrt{2}}{4 G_F V_{cb} m^2_{H^\pm}}
{\Gamma_{u_2 d_3 }^{H^\pm\,LR\,\rm{eff} } } {(\Gamma_{\nu_i \ell_3
}^{H^\pm\,LR\,\rm{eff} })^* },\nonumber\\
g_T^{LL} &=& -\frac{\sqrt{2} \alpha_E}{96\pi G_F V_{cb}
m^2_{H^\pm}} {\Gamma_{u_2 d_3 }^{H^\pm\,RL\,\rm{eff}
}}{(\Gamma_{\nu_i \ell_3 }^{H^\pm\,LR\,\rm{eff} })^* }\bigg(
f(x_c,x_\tau)+\frac{1}{2} f(x_b,x_\tau)\bigg)
+\Delta g^{LL\,Z,Z'}_{{T}},\nonumber\\
g_T^{RR} &=& -\frac{\sqrt{2} \alpha_E}{96 \pi G_F
V_{cb}m^2_{H^\pm}} {\Gamma_{u_2 d_3 }^{H^\pm\,LR\,\rm{eff} } }
{(\Gamma_{\nu_i \ell_3 }^{H^\pm\,RL\,\rm{eff} })^* }\bigg(
f(x_c,x_\tau)+\frac{1}{2} f(x_b,x_\tau)\bigg)+\Delta
g^{RR\,Z,Z'}_{ T},\label{WiC}
\end{eqnarray}
where $i$ refers to the neutrino flavor, $ x_k
=\frac{m_k^2}{m^2_H}$ and $ f(x_i,x_j)=
\frac{1}{(x_i-x_j)}\Big(\frac{x_i}{1-x_i}log
x_i-(x_i\leftrightarrow x_j)\Big)$.
As can be seen from Eq.(\ref{WiC}), the coefficients $
g_S^{LL}$ and $g_S^{RL}$ have the same lepton vertex
$(\Gamma_{\nu_i \ell_3 }^{H^\pm\,LR\,\rm{eff}})^*$. Therefore,
they are expected to have different values due to receiving
different contributions of the quark vertices $\Gamma_{u_i d_j
}^{H^ \pm\,RL\,\rm{eff} }$ and $\Gamma_{u_i d_j
}^{H^\pm\,LR\,\rm{eff} }$ expressed in Eq.(\ref{Higgsv1}) in terms
of $y^Q$ and $\tilde y^Q$. Furthermore, Eq.(\ref{yuQ}) shows that
the contributions from the terms proportional to $V^L_{CKM}$ to
both $y^Q$ and $\tilde y^Q$ are suppressed by the small down quark
masses appear in the matrix $M^{diag}_d$.
In the processes under consideration, the corresponding
Wilson coefficient $ g_S^{LL}$ ($g_S^{RL}$) receives contributions only
from the elements in the second column (row) of the
matrix $M^{diag}_u V_{\text{CKM}}^{R \dag}$, which are present in both of
$y^Q$ and $\tilde y^Q$. With the texture of $V_{\text{CKM}}^{R}$
given previously, we find that
\begin{equation} M^{diag}_u V_{\text{CKM}}^{R \dag} = \left(%
\begin{array}{ccc}
m_u & 0 & 0 \\
0 & m_c\, c_{\theta^R_{13}} & m_c\, s_{\theta^R_{13}} \\
0 & m_t\,s_{\theta^R_{13}} e^{-i\alpha} & m_t\,c_{\theta^R_{13}} e^{-i\alpha} \\
\end{array}%
\right).
\end{equation}
In this regard, one can show that $R_D$, $R_{D^*}$ and ${\rm
BR}(B^-_c \to \tau^- \bar{\nu}_\tau)$ receive contributions from
the terms proportional to $m_t\,s_{\theta^R_{13}} e^{-i\alpha}$
which originate from $ g_S^{LL}$ only and not from $g_S^{RL}$. The
other contributions generated from the Wilson coefficient
$g_S^{RL}$ are proportional to the charm quark mass which is small
compared to the top quark mass.
Finally in Eq.(\ref{WiC}), the quantities $\Delta
g^{LL(RR)\,Z,Z'}_{ T}$ refer to the suppressed contributions,
comparing to photon ones, originating from $Z$ and $Z'$ mediating
the diagrams. Upon neglecting the small contributions $\Delta
g^{LL,RR\,Z,Z'}_{T }$ we find the following relations
$g_T^{LL} = - \frac{\alpha_E}{24\pi} \bigg(
f(x_c,x_\tau)+\frac{1}{2} f(x_b,x_\tau)\bigg) g_S^{LL} $ and
$g_T^{RR} = - \frac{\alpha_E}{24\pi} \bigg(
f(x_c,x_\tau)+\frac{1}{2} f(x_b,x_\tau)\bigg) g_S^{RR}$. For a
charged Higgs of a mass $430\,GeV$ we find that $g_T^{LL} \simeq
1.5 \times 10^{-3} g_S^{LL}$ and $g_T^{RR} \simeq 1.5 \times
10^{-3} g_S^{RR}$. Clearly the tensor contributions to the
processes under study can be safely neglected.
We will assume that NP effects are only present in the
third generation of leptons ($\tau, \nu_\tau$). This assumption is
motivated by the absence of deviations from the SM for light
lepton modes $\ell= e$ or $\mu$. The ratios $R(M)$ ($M=D,D^\ast$
can be written in terms of the scalar $g_{S}^{LL}, g_{S}^{RL},
C_{S}^{LR},$ and
$C_{S}^{RR}$~\cite{Iguro:2018vqb,Asadi:2018wea,Asadi:2018sym}. The
numerical expressions for these contributions
are~\cite{Iguro:2018vqb,Asadi:2018wea,Asadi:2018sym}:
\begin{eqnarray}
R_D &=& R_D^{\rm SM} \Big[1+ 1.49 \ \text{Re}(g_{S}^{RL}+ g_{S}^{LL})^\ast + 1.02 \big( |g_{S}^{RL}+ g_{S}^{LL}|^2 + |g_{S}^{LR}+ g_{S}^{RR}|^2 \big)\Big], \\
R_{D^*} &=& R_{D^*}^{\rm SM} \Big[1+ 0.11 \
\text{Re}(g_{S}^{RL}- g_{S}^{LL})^\ast + 0.04 \big( |g_{S}^{RL}-
g_{S}^{LL}|^2 + |g_{S}^{LR}- g_{S}^{RR}|^2 \big) \Big].
\end{eqnarray}
The tree-level charged Higgs boson exchange also modifies the
branching ratio of the tauonic decay $B_c^- \to \tau^-
\bar{\nu}_{\tau}$ as
follows~\cite{Iguro:2018vqb,Asadi:2018wea,Asadi:2018sym}
\begin{eqnarray} \label{BRBc_TypeII}
{\rm BR}(B_c^- \to \tau^- \bar{\nu}_{\tau}) &=& {\rm BR}(B_c^- \to
\tau^- \bar{\nu}_{\tau})_{\text{SM}} \ \left[ \bigg| 1
+ \frac{m_{B_c}^2}{m_\tau(m_b+m_c)} (g_{S}^{RL}- g_{S}^{LL})\bigg|^2 +
\bigg|\frac{m_{B_c}^2}{m_\tau(m_b+m_c)} (g_{S}^{LR}- g_{S}^{RR})\bigg|^2\right], \nonumber \\
\end{eqnarray}
\noindent where $m_{B_c}^2/m_\tau(m_b+m_c) = 4.065$ and
\begin{eqnarray}
\label{SM_leptonic} {\rm BR}(B^-_c \to \tau^-
\bar{\nu}_\tau)_{\rm SM} &=&\tau_{B_c} \dfrac{G_F^2}{8\pi}
|V_{cb}|^2 f_{B_c}^2 m_{B_c} m_{\tau}^2 \Big(1-
\dfrac{m_{\tau}^2}{m_{B_c}^2}\Big)^2 , \end{eqnarray}
where $V_{cb}$ stands for the CKM matrix element,
$\tau_{B_c}$ and $f_{B_c}$ denote the $B^-_c$ meson lifetime and
decay constant, respectively. The SM prediction of ${\rm BR}(B^-_c
\to \tau^- \bar{\nu}_\tau)_{\rm SM} = (2.25 \pm 0.21)\times
10^{-2}$ \cite{Fleischer:2021yjo}. Unfortunately, no direct
constraints from upper bounds on the leptonic $B_c$ branching
ratios are available from the LHC. In view of this, an estimate of
a bound on ${\rm BR}(B^-_c \to \tau^- \bar{\nu}_\tau)$ has been
derived from LEP data at the Z peak in Ref.
\cite{Akeroyd:2017mhr}. The bound turns to be strong ${\rm
BR}(B_c^{-} \to \tau^{-} \bar{\nu}_\tau) \leq 10 \%$. Later on,
the bounds using the measured $B_c$ lifetime had been critically
investigated and relaxed upper limit of $\leq 39
\%$~\cite{Bardhan:2019ljo} and $\leq 60
\%$~\cite{Blanke:2018yud,Blanke:2019qrx} were obtained. Thus we
will follow Ref. \cite{Fleischer:2021yjo} and take in our
analysis the bound: ${\rm BR}(B^-_c \to \tau^- \bar{\nu}_\tau)< 60\% $
\section{Numerical results and analysis \label{nums}}
It is worth to recall that, due to the requirement of having
light neutrino masses we found that the contributions of the right
neutrino sector to the the corresponding Wilson coefficients $
g_S^{RR}$ and $g_S^{LR}$ are very small and thus can be safely
ignored, leaving us with only $ g_S^{LL}$ and $g_S^{RL}$.
Furthermore, $g_S^{RL}$ is about one order of magnitude smaller
than $ g_S^{LL}$; thus for real $ g_S^{LL}$ and $g_S^{RL}$ ({\it
i.e.}, $ Re(g_S^{AB}) = g_S^{AB}$), one finds
\begin{eqnarray}
R_D &\simeq& R_D^{\rm SM} \Big[1+ 1.49 g_{S}^{LL}+ 1.02 |g_{S}^{LL}|^2 \Big], \\
R_{D^*} &\simeq& R_{D^*}^{\rm SM} \Big[1- 0.11 g_{S}^{LL} + 0.04
|g_{S}^{LL}|^2 \Big].
\end{eqnarray}
This expression clearly shows that enhancing the values of
$R_{D^*}$ to be in the range of the given experimental results,
while keeping the limit $ -1\raise0.3ex\hbox{$\;<$\kern-0.75em\raise-1.1ex\hbox{$\sim\;$}} g_S^{LL} \raise0.3ex\hbox{$\;<$\kern-0.75em\raise-1.1ex\hbox{$\sim\;$}} 1$ in mind is
possible for a range of negative values of $g_S^{LL}$ as can be
seen from the left plot in Fig.\ref{RegsLL}. However, as can be
remarked from the right plot in Fig.\ref{RegsLL}, these negative
values reduce $R_D$ below their allowed $2\sigma$ range of the
experimental results shown by the horizontal orange and blue lines
in the plot. Clearly, we deduce that the phase $\alpha$ of the
mixing matrix $V^R_{CKM}$ is crucial to solve the concerned
anomalies for the processes under consideration.
\begin{figure}[tbhp]
\includegraphics[width=7cm,height=6cm]{RegsLLDs.EPS}
\hspace{0.2cm}
\includegraphics[width=7cm,height=6cm]{RegsLLD.EPS}
\medskip
\caption{ Left (right) $R_{D^*}$ ($R_D$) variation with
$g_{S}^{LL}$ where the horizontal orange and blue lines represent
the allowed $2\sigma$ range of their experimental values.
\label{RegsLL}}
\end{figure}
The attainable values of the Wilson coefficients $ g_S^{LL}$ and
$g_S^{RL}$ are affected by the charged Higgs mass. As a result, to
enhance these coefficients and thus the values of $R_D$ and
$R_{D^*}$ while adhering to direct search constraints, the charged
Higgs masses should not be too heavy, namely of the order of
hundreds GeV. According to Eq.(\ref{chHm}), this can be
accomplished by considering $v_R\sim\mathcal{O}(\text{TeV})$,
$\alpha_{32}\sim\mathcal{O}(10^{-2})$ and $t_\beta$ less than one.
Remarkably, from the pre factor $1/(t_\beta-ct_\beta)$ in
Eq.(\ref{yuQ}), it is direct to see that small $t_\beta$ values
can enhance also the quark Yukawa couplings and hence together
with the angle $\theta^R_{13}$ and the complex phase $\alpha$ of
the right-quark mixing $V^R_{CKM}$, defined in Eq.(\ref{VRCKMp}),
play crucial roles in increasing the values of $R_D$ and
$R_{D^*}$, and allow them to take values that are compatible with
the limit of the experiments at the same time. In our scan of the
parameter space we take $\alpha_{32} \in [0.007,0.016]$, $t_\beta
\in [0.15,0.25]$, $c_{\theta^R_{13}} \in [-1,1]$, $\alpha \in
[0,\pi]$ and $v_R=6413$ GeV. With this in hand, we show below our
results corresponding to the scanned points in the parameter space
respecting the bound ${\rm BR}(B^-_c \to \tau^- \bar{\nu}_\tau)<
60\% $. Moreover, we have checked that, for these parameters in
the chosen ranges and values, the contribution of the $g_S^{RL}$
are irrelevant and can be safely neglected. This confirms our
previous conclusion that only the Wilson coefficient $g_S^{LL}$
plays the major role through the terms proportional to
$m_t\,s_{\theta^R_{13}} e^{-i\alpha}$. In the following, we
present a set of elucidative plots of $R_D$ and $R_{D^*}$ versus
some selected relevant parameters of the model.
\begin{figure}[ht]
\includegraphics[width=9.5cm,height=5.5cm]{RDandRDstarvsMHplot.eps}
\caption{$R_D$ and $R_{D^*}$ as function of the charged Higgs mass
where the parameters as stated in the text are chosen as follows:
$c_{\theta^R_{13}} \in [-1,1]$, $\alpha \in [0,\pi]$, $\alpha_{32}
\in [0.007,0.016]$ and $t_\beta \in [0.15,0.25]$.} \label{charged
Higgs}
\end{figure}
\begin{figure}[ht]
\includegraphics[width=8.5cm,height=5.5cm]{RDandRDstarvsSin.eps}
\hspace{0.2cm}
\includegraphics[width=8.5cm,height=5.5cm]{alphavsRDandRDstarplot.eps}
\caption{$R_D$ and $R_{D^*}$ as function of the sin of mixing
angle $\theta^R_{13}$ left and the phase $\alpha$ of the
$V^R_{CKM}$ matrix right and other parameters are fixed as in the
previous figure. In the left plot we show the points corresponding
to $\theta^R_{13} \in [0,\frac{\pi}{2}]$ as $\theta^R_{13}$ close
to zero are favored for resolving the $R_D$ and $R_{D^*}$
anomalies.} \label{arg VR}
\end{figure}
In Fig. \ref{charged Higgs}, we display the variation of $R_D$ and
$R_{D^*}$ with the charged Higgs mass. As can be seen from the
figure, it is possible to account for the experimental results of
$R_D$ and $R_{D^*}$ within $2\sigma$ range while respecting the
${\rm BR}(B^-_c \to \tau^- \bar{\nu}_\tau)$ constraints with
charged Higgs masses $m_{H^\pm}$ can be chosen of order 500 GeV.
On the other hand, the dependence of $R_D$ and $R_{D^*}$ on the
sin of the mixing angle $\theta^R_{13}$ and the phase $\alpha$ of
the $V^R_{CKM}$ matrix is depicted in Fig. \ref{arg VR}. It is
clear from left plot in the figure that, small values of
$s_{\theta^R_{13}}$ are preferable to satisfy the experimental
results of $R_D$ and $R_{D^*}$ within $2\sigma$ range while
respecting the ${\rm BR}(B^-_c \to \tau^- \bar{\nu}_\tau)$
constraints. However, this is not the case regarding the phase
$\alpha$ as in this case large phases are favored as can be seen
from the right plot in the figure.
We can obtain the regions in the $(\theta^R_{13},\alpha)$
parameter space in which the anomalies are satisfied through
varying $\theta^R_{13}$ and $\alpha$ while assigning fixed values
of the other parameters. As an example, we take the fixed values
$t_\beta\simeq 0.21$, $\alpha_{32} = 0.015$ which result in
$m_{H^\pm} = 580$ GeV and the other parameters are fixed as
before. In Fig.\ref{ZBs11} left, we show the experimentally
allowed $2\sigma$ regions in the $(\theta^R_{13},\alpha)$ plane of
$R_{D^*}$ in blue and $R_D$ in red without imposing the
constraints from ${\rm BR}(B^-_c \to \tau^- \bar{\nu}_\tau) <
60\%$. As shown in the plot, the regions in which the $R_{D^*}$
and $R_D$ anomalies are satisfied correspond to values of
$\theta^R_{13}\in [1.9,\pi]$ and approximately $\alpha \in
[0.35.0.9]$ or $\alpha \in [2.2.2.8]$. However, not all these
ranges of $\theta^R_{13}$ and $\alpha$ survive if we impose the
${\rm BR}(B^-_c \to \tau^- \bar{\nu}_\tau) < 60\%$ constraint
which is represented in yellow color in the right plot of
Fig.\ref{ZBs11}. In this case, the allowed ranges are
$\theta^R_{13}\in [1.9.2.4]$ and approximately $\alpha \in
[0.35.0.55]$ or $\alpha \in [2.6.2.8]$ as can be noted from the
plot. Clearly, the ${\rm BR}(B^-_c \to \tau^- \bar{\nu}_\tau)$
constraint has a sensible effect on the parameter space satisfying
the anomalies.
\begin{figure}[tbhp]
\includegraphics[width=7cm,height=7cm]{RDRDs.EPS}
\hspace{0.2cm}
\includegraphics[width=7cm,height=7cm]{RDRDsBRB.EPS}
\medskip
\caption{ Left (right) the allowed regions in the
($(\theta^R_{13},\alpha)$) plane by the $2\sigma$ experimental
results of $R_{D^*}$ in blue and $R_D$ in red without (with)
imposing the constraints from ${\rm BR}(B^-_c \to \tau^-
\bar{\nu}_\tau) < 60\%$ which is given in yellow color in the
right plot for $m_{H^\pm} = 580$ GeV for $t_\beta\simeq 0.21$,
$\alpha_{32} = 0.015$ which result in $m_{H^\pm} = 580$ GeV and
the other parameters are fixed as before. \label{ZBs11}}
\end{figure}
Finally we present in Fig. \ref{correlation} the correlation
between $R_D$ and $R_{D^*}$ for the same set of the parameter
space considered in the scan over the values and ranges mentioned
in the beginning of this section. Only points that satisfy the
${\rm BR}(B^-_c \to \tau^- \bar{\nu}_\tau)$ constraints are
included here. It is remarkable from this figure that both $R_D$
and $R_{D^*}$ are satisfied for a larger region of parameter
space, thanks to the complex mixing of right-handed quarks.
\begin{figure}[tbhp]
\includegraphics[width=8.5cm,height=6.5cm]{RDvsRDstar.eps}
\medskip
\caption{ The correlation between $R_D$ and $R_{D^*}$ for the
same set of parameter space considered in Fig. \ref{charged
Higgs}. \label{correlation}}
\end{figure}
\section{Conclusion \label{sec:conclusion}}
In this work we have explored the possibility of resolving the
tension between the SM prediction and the experimental results of
the $R_D$ and $R_{D^*}$ ratios using a low scale left-right
symmetric model based on $SU(3)_C\times SU(2)_L\times
SU(2)_R\times U(1)_{B-L}\times Z_2$. The scalar sector of the
model contains charged Higgs boson with masses that can be chosen
in the order of hundreds GeV without any conflict with direct
search constraints. We have shown that integrating out the charged
Higgs mediating the tree-level diagrams generates a set of non
vanishing scalar Wilson coefficients contributing to the effective
Hamiltonian governing the transition $b\to c\tau\bar\nu$ and hence
to the ratios $R_{D^*}$ and $R_D$.
The dependency of the scalar Wilson coefficients on the matrix
elements of the quark mixing angle in the right sector turns to
be important. We emphasized that the mixing element $(V^R_{CKM})_{23}$ should be complex in order to satisfy both $R_D$ and
$R_{D^*}$. We have also shown the complex phase associated with this mixing element
is essential to accommodate the experimental results of the
ratios for charged Higgs masses of order 500 GeV while respecting the
constraints from ${\rm BR}(B^-_c \to \tau^- \bar{\nu}_\tau) $.
\section*{acknowledgements}
The work of K. E. and S. K. is partially supported by Science, Technology $\&$ Innovation Funding Authority (STDF) under grant number 37272.
|
1,116,691,500,894 | arxiv | \section{Introduction}
Interference alignment, in a variety of forms \cite{Jafar_FnT}, has produced some of the most novel \cite{Cadambe_Jafar_int}, albeit fragile \cite{Arash_Jafar_GC15, Arash_Jafar_IC} solutions for circumventing the interference barrier in wireless interference networks. Remarkably, the same ideas have found robust applications beyond wireless settings --- from distributed storage \cite{Cadambe_Jafar_Maleki_Ramchandran_Suh, Shah_IA} and network coding \cite{PBNA, Han_Wang_Shroff} to index coding \cite{Maleki_Cadambe_Jafar, Sun_Jafar_nonshannon}, hat-guessing \cite{Riis_Hat} and topological interference management problems \cite{Jafar_TIM, Yi_Gesbert}. In this paper we report the discovery of another robust application of interference alignment outside the context of wireless networks. Specifically, we show how the idea of Blind Interference Alignment (BIA) that was originally introduced to exploit the diversity of coherence patterns in wireless interference networks, translates into new solutions for the Private Information Retrieval (PIR) problem previously studied by computer scientists.
\subsection{Blind Interference Alignment}
The idea of blind interference alignment was introduced in \cite{Jafar_corr} to take advantage of the diversity of coherence intervals that may arise in a wireless network. For instance, different channels may experience different coherence times and coherence bandwidths. A diversity of coherence patterns can also be artificially induced by the switching of reconfigurable antennas in pre-determined patterns. As one of the simplest examples of BIA, consider a $K$ user interference channel, where the desired channels have coherence time $1$, i.e., they change after every channel use, while the cross channels (which carry interference) have coherence time $2$, i.e., they remain unchanged over two channel uses. The transmitters are aware of the coherence times but otherwise have no knowledge of the channel coefficients. The BIA scheme operates over two consecutive channel uses. Over these two channel uses, each transmitter repeats its information symbol, and each receiver simply calculates the difference of its received signals. Since the transmitted symbols remain the same and the cross channels do not change, the difference of received signals from the two channel uses eliminates all interference terms. However, because the desired channels change, the desired information symbols survive the difference at each receiver. Thus, one desired information symbol is successfully sent for each message over 2 channel uses, free from interference, achieving $\frac{1}{2}$ DoF per message. Remarkably, with only the knowledge of channel coherence patterns and no knowledge of the actual values of channel coefficients, the BIA scheme is able to achieve the same DoF as possible with perfect knowledge of channel coefficients at all transmitters. Applications of BIA extend well beyond this simple example \cite{Wang_Gou_Jafar, BIA_Dynamic, BIA_IC, BIA_Z}. For instance, in the $X$ channel comprised of $M$ transmitters and $K$ receivers, using only the knowledge of suitable channel coherence patterns, BIA schemes achieve $\frac{MK}{M+K-1}$ DoF, which cannot be improved upon even with perfect channel knowledge \cite{Jafar_corr, Wang_Gou_Jafar}.
\subsection{Private Information Retrieval}
In private information retrieval \cite{PIRfirst}, the goal is to (as efficiently as possible) allow a user to retrieve his desired information from data providers without disclosing anything about his interests. Note that the user can hide his interests trivially by requesting all the information available to the data providers, but that could be very inefficient (expensive). The goal of the PIR problem is to find the most efficient solution. Here is the problem description. We have $N \geq 2$ non-communicating databases\footnote{It turns out that the setting of $N=1$ is not interesting. It is proved in \cite{PIRfirst} that in order to be private in this case, the user has to download all the messages. Motivated by this result, \cite{PIRfirst} initiates the study of the benefit of non-communicating databases, and we adopt this model in this paper.}, each stores a set of $K \geq 2$ independent messages $W_1, \cdots, W_K$. A user wants one of the messages, say $W_i, i \in \{1,2, \cdots, K\}$, but requires each database to learn absolutely nothing (in the information theoretic sense)\footnote{There is another line of research, where privacy needs to be satisfied only for computationally bounded databases \cite{William, Yekhanin, CPIR}.} about the retrieved message index, $i$. To do so, the user generates $N$ queries $Q_1, \cdots, Q_N$ and sends $Q_j, j \in \{1,2,\cdots, N\}$ to the $j$-th database. After receiving query $Q_j$, the $j$-th database returns an answering string $A_j$ to the user. The user must be able to obtain the desired message $W_i$ from all the answers $A_1, \cdots, A_N$. To be private, each query $Q_j$ and each answer $A_j$ must be independent of the desired message index, $i$.
For example, suppose we have $N = 2$ databases and $K$ messages. To retrieve $W_1$ privately, the user first generates a random length-$K$ vector $[h_1, h_2, \cdots, h_K]$, where each element is uniformly distributed over a finite field. Then the user sends $Q_1 = [h_1, h_2, \cdots, h_K]$ to the first database and $Q_2 = [h_1+1, h_2, \cdots, h_K]$ to the second database. Each database generates the answering string by taking a inner product of the query vector and the message symbols,
\begin{eqnarray*} \label{exeia}
\left[
\begin{array}{c}
A_1 \\
A_2
\end{array}
\right]
= \left[
\begin{array}{c}
h_1 \\
h_1 + 1
\end{array}
\right] W_1 +
h_2 \left[
\begin{array}{c}
1 \\
1
\end{array}
\right] W_2 + \cdots +
h_K \left[
\begin{array}{c}
1 \\
1
\end{array}
\right] W_K.
\end{eqnarray*}
The user can obtain $W_1$ by subtracting $A_1$ from $A_2$. Privacy is guaranteed as the query vectors $Q_1$, $Q_2$ are marginally random and independent of the desired message index, such that each database by itself can not tell which message is requested. The observant readers might have noticed the similarity between this PIR scheme and the earlier example of BIA, where over two channel uses (across two databases), the channel (corresponding scalar in the query vector) for the desired message changes while the channels for the interfering messages remain the same. Note that the scalars in the query vector correspond to the channel coefficients, and the answering strings correspond to the received signal equations.
The cost of a PIR scheme is traditionally measured by the total amount of communication between the user and the databases, i.e., the sum of lengths of each query string (upload) and each answering string (download).
Prior work in theoretical computer science society studies the coding theoretic setting, where each message is one bit \cite{PIRfirst, YekhaninPhd, 2PIR}.
In this paper, we consider the information theoretic formulation, where the size of each message goes to infinity. In this setting, the upload cost is negligible compared to the download cost\footnote{Similar observation is also made in \cite{Chan_Ho_Yamamoto}.
The justification argument (traces back to Proposition 4.1.1 of \cite{PIRfirst}) is that the upload cost does not scale with the message size. This is because we can reuse the original query functions for each part of the message.}.
Therefore we concentrate only on the download cost, measured in unit of the message size. For the above example, each message is one symbol and we download 2 equations in total, so that the download cost is 2.
We aim to characterize the optimal download cost, as an exact function of $N$ and $K$.
\subsection{Overview of Contribution}
The main contribution of this paper is the discovery of an interesting connection between PIR and BIA such that a good BIA scheme translates into a good PIR protocol.
Let us see how to obtain a PIR protocol from a BIA scheme. The number of users in the BIA problem translates into the number of messages in the PIR problem.
The received signals for user $i$ in BIA, translate into the answering strings when message $W_i$ is the desired message in the PIR problem. The channel vectors associated with user $i$ in the BIA problem translate into the query vectors for desired message $W_i$ in the PIR problem. The privacy requirement of the PIR scheme takes advantage of the observation that in BIA, over each channel use, the received signal at each receiver is statistically equivalent, because the transmitter does not know the channel values and the channel to each receiver has the same distribution. The most involved aspect of translating from BIA to PIR is that in BIA, the knowledge of the channel realizations across channel uses reveals the switching pattern, which in turn reveals the identity of the receiver. To remove this identifying feature of the BIA scheme, the channel uses are divided into subgroups such that the knowledge of the switching pattern within each group reveals nothing about the identity of the receiver. Each sub-group of channel uses is then associated with a different database. Since the databases are not allowed to communicate with each other, and each sub-group of queries (channel uses) reveals nothing about the message (user), the resulting scheme guarantees privacy. Finally, the symmetric degrees of freedom (DoF) value per user in BIA is the ratio between the number of desired message symbols and the number of channel uses (received signal equations), and the download cost $\eta$ in PIR is the ratio between the total number of equations in all answering strings and the number of symbols of the desired message. In this way, the DoF value achieved with BIA translates into the reciprocal of the efficiency of the corresponding PIR protocol, i.e., $\eta = 1/\mbox{DoF}$. We summarize these connections in the following table.
\begin{table}[h]
\centering
\scalebox{1}{
\begin{tabular}{c | c}
PIR & BIA \\
\hline
Message & Receiver \\
Queries & Channel Coefficients \\
Answers & Received Signals \\
Efficiency & 1/DoF
\end{tabular}
}
\end{table}
As an application of the connection between BIA and PIR, we find the optimal efficiency $\eta$ for the PIR problem when we have $K = 2$ messages and the number of databases, $N$, is arbitrary. We show that for this case, $\eta=1 + \frac{1}{N}$. Recall that with BIA the $X$ channel with $N$ transmitters and $K=2$ receivers, achieves its optimal DoF value $1/(1+\frac{1}{N})=1/\eta$ \cite{Jafar_corr}. Indeed, for this setting the optimal PIR scheme for $K=2$ messages, $N$ databases, is found by translating from the BIA scheme for the $X$ channel with $K=2$ receivers and $N$ transmitters. A matching information theoretic outer bound is obtained to establish optimality. We note that the cost is strictly decreasing in $N$, meaning that having more databases strictly decreases the download cost, and when the number of databases becomes large, the cost approaches 1 (we just download what is needed).
The best previously known scheme achieves the download cost of $1 + \frac{1}{N-1}$ \cite{Shah_Rashmi_Kannan}, which is strictly improved upon here.
\section{System model}\label{sec:model}
We define the system model for the PIR problem as follows. There are $K$ messages $W_1, \cdots, W_K$. We assume that the messages are independent and of the same size, i.e.,
\begin{eqnarray}
&& H(W_1, \cdots, W_K) = H(W_1) + \cdots + H(W_K), \label{h1}\\
&& H(W_1) = \cdots = H(W_K). \label{h2}
\end{eqnarray}
There are $N$ databases, and each database stores all the messages $W_1, \cdots, W_K$. A user wants to retrieve $W_i, i \in \{1, \cdots, K\}$ privately, i.e., without revealing anything about the message identity $i$ to any of the databases.
To retrieve $W_i$ privately, the user first generates $N$ queries $Q_1^{[i]}, \cdots, Q_N^{[i]}$, where the superscript denotes the desired message index. Each query $Q_j^{[i]}, j \in \{1,\cdots, N\}$ is a random variable with finite support. The queries are independent of the messages,
\begin{eqnarray}
I(W_1, \cdots, W_K; Q_1^{[i]}, \cdots, Q_N^{[i]}) = 0 \label{qwind}.
\end{eqnarray}
The user then sends query $Q_j^{[i]}$ to the $j$-th database. After receiving $Q_j^{[i]}$, the $j$-th database generates an answering string $A_j^{[i]}$, which is a deterministic function of $Q_j^{[i]}$ and the data stored (i.e., all messages $W_1, \cdots, W_K$),
\begin{eqnarray}
H(A_j^{[i]} | Q_j^{[i]}, W_1, \cdots, W_K) = 0. \label{ansdet}
\end{eqnarray}
Each database returns to the user its answer $A_j^{[i]}$. From all answers $A_1^{[i]}, \cdots, A_N^{[i]}$, the user can decode the desired message $W_i$,
\begin{eqnarray}
\mbox{[Correctness]} ~H(W_i | A_1^{[i]}, \cdots, A_N^{[i]}, {\color{black} Q_1^{[i]}, \cdots, Q_N^{[i]}}) = 0. \label{corr}
\end{eqnarray}
To satisfy the privacy constraint that each database learns nothing about the desired message index $i$ information theoretically, each query $Q_j^{[i]}$ must be distributed independently of $i$,
\begin{eqnarray}
\mbox{[Privacy]} ~~~ I(Q_j^{[i]}; i) = 0, \forall j. \label{qi}
\end{eqnarray}
As the answering string is a deterministic function of the query and all messages, the answering string must be independent of $i$ as well,
\begin{eqnarray}
I(A_j^{[i]}; i) = 0, \forall j. \label{ai}
\end{eqnarray}
{\color{black} For compact notation, we may sometimes ignore the desired message index superscript and simplify the notation as $Q_j/A_j$, when it would cause no confusion.}
The metric that we study in this paper is the download cost (efficiency) $\eta$. It is defined as the ratio between the total download cost and the size of the desired message,
\begin{eqnarray}
\eta \triangleq \frac{\sum_{j=1}^N H(A_j^{[i]})}{H(W_i)}. \label{eta_def}
\end{eqnarray}
We aim to characterize the optimal (minimum) download cost $\eta^*$, over all private information retrieval schemes.
\section{Main Result}\label{sec:main}
We present our main result in the following theorem.
\begin{theorem}\label{thm:download}
For the private information retrieval problem with $2$ messages and $N \geq 2$ databases, the optimal download cost $\eta^*$ is $1 + \frac{1}{N}$.
\end{theorem}
Before presenting the proof for the general case of $N$ databases in Section \ref{sec:proof}, let us consider the $N=2$ setting.
\subsection{$N=2$ databases, $\eta^*=\frac{3}{2}$}\label{sec:dw1}
\subsubsection{Achievability}
The scheme follows from the BIA scheme of the $X$ channel with 2 transmitters and 2 receivers, where each transmitter has an independent message for each of the two receivers. The transmission takes place over 3 channel uses. User 1 desires symbol $a_1$ from Transmitter 1 and symbol $a_2$ from Transmitter 2, and his channel changes from the first channel use to the second channel use and then remains the same from the second channel use to the third channel use, i.e., the channel pattern for User 1 is ${\bf h}^{[1]}(1), {\bf h}^{[1]}(2), {\bf h}^{[1]}(2)$. User 2 desires symbol $b_1$ from Transmitter 1 and symbol $b_2$ from Transmitter 2, and his channel remains the same from the first channel use to the second channel use and then changes from the second channel use to the third channel use, i.e., the channel pattern for User 2 is ${\bf h}^{[2]}(1), {\bf h}^{[2]}(1), {\bf h}^{[2]}(2)$. The ${\bf h}^{[i]}(j)$ are random and statistically equivalent realizations of the $1\times 2$ channel vector from the two transmitters to Receiver $i$. In this channel, suppose the symbols are repeated by the transmitters over those channel uses where desired user's channel changes and the undesired user's channel remains the same, i.e., $a_1, a_2$ are repeated over the first two channel uses and $b_1,b_2$ are repeated over the second and third channel uses.
Suppressing noise, the signal equations seen at the two receivers are
\begin{eqnarray}
\allowdisplaybreaks
{\bf y}^{[1]} =
\left[ \begin{array}{c}
{\bf h}^{[1]}(1) \\
{\bf h}^{[1]}(2) \\
0
\end{array}
\right]
\left[ \begin{array}{c}
a_1 \\
a_2
\end{array}
\right] +
\left[ \begin{array}{c}
0 \\
{\bf h}^{[1]}(2) \\
{\bf h}^{[1]}(2)
\end{array}
\right]
\left[ \begin{array}{c}
b_1 \\
b_2
\end{array}
\right], \\
{\bf y}^{[2]} =
\left[ \begin{array}{c}
{\bf h}^{[2]}(1) \\
{\bf h}^{[2]}(1) \\
0
\end{array}
\right]
\left[ \begin{array}{c}
a_1 \\
a_2
\end{array}
\right] +
\left[ \begin{array}{c}
0 \\
{\bf h}^{[2]}(1) \\
{\bf h}^{[2]}(2)
\end{array}
\right]
\left[ \begin{array}{c}
b_1 \\
b_2
\end{array}
\right].
\end{eqnarray}
Thus, each user sees two different linear combinations of desired symbols and only one linear combination (repeated) of interfering symbols.
Subtracting the interference from its repeated observation allows each user access to two resolvable interference-free linear combinations of desired symbols. Thus, 2 symbols are successfully sent to each receiver over 3 channel uses, achieving $\frac{2}{3}$ DoF per receiver.
Now let us translate this BIA scheme into a PIR scheme, wherein $W_1 = \{a_1, a_2\}, W_2 = \{b_1, b_2\}$. To achieve download cost $\frac{3}{2}$, the total number of answering strings from the 2 databases must correspond to $3$ equations. The queries are the channels in the BIA problem. However, the channel uses must be grouped in such a way that the switching pattern does not reveal the identity of the receiver. For example, if the first two channel uses are in the same group then the identity of the receiver would be revealed (if the channel changes then it is Receiver 1 and if it does not then it is Receiver 2). However, if the first and third channel uses are placed in the same group, then we note that the channel changes within this group from one channel use to the next regardless of the receiver, thus revealing nothing about the identity of the receiver. So this is the grouping we choose. Channel uses 1 and 3 are placed in one group and associated with Database 1, and channel use 2 is placed in the other group, associated with Database 2. Based on this grouping, the resulting queries $Q^{[i]}_j$ from the $j^{th}$ database when message $W_i$ is desired, are summarized in the following table.
\begin{table}[h]
\centering
\begin{tabular}{|c|c|c|}\hline
$Q^{[i]}_j$ & Message $W_i=W_1$ & Message: $W_i=W_2$ \\
\hline
Database: $j=1$&$ {\bf h}^{[1]}(1)$, ${\bf h}^{[1]}(2)$ &$ {\bf h}^{[2]}(1)$, ${\bf h}^{[2]}(2)$\\ \hline
Database: $j=2$&$ {\bf h}^{[1]}(2)$ &$ {\bf h}^{[2]}(1)$\\ \hline
\end{tabular}
\end{table}
The answering strings in the PIR problem are the received signals of the BIA scheme. In correspondence with the queries, the answering string from the first database, $A_1$, is the received signal over the first and third channel uses and the answering string from the second database, $A_2$, is the received signal over the second channel use.
\begin{eqnarray}
A_1^{[1]} &=&
\left[ \begin{array}{c}
{\bf h}^{[1]}(1)\\
0
\end{array}
\right]
\left[ \begin{array}{c}
a_1 \\
a_2
\end{array}
\right] +
\left[ \begin{array}{c}
0\\
{\bf h}^{[1]}(2) \\
\end{array}
\right]
\left[ \begin{array}{c}
b_1 \\
b_2
\end{array}
\right]
\\
A_1^{[2]} &=&
\left[ \begin{array}{c}
{\bf h}^{[2]}(1)\\
0
\end{array}
\right]
\left[ \begin{array}{c}
a_1 \\
a_2
\end{array}
\right] +
\left[ \begin{array}{c}
0\\
{\bf h}^{[2]}(2) \\
\end{array}
\right]
\left[ \begin{array}{c}
b_1 \\
b_2
\end{array}
\right]
\\
A_2^{[1]} &=&
\left[ \begin{array}{c}
{\bf h}^{[1]}(2)
\end{array}
\right]
\left[ \begin{array}{c}
a_1 \\
a_2
\end{array}
\right] +
\left[ \begin{array}{c}
{\bf h}^{[1]}(2)
\end{array}
\right]
\left[ \begin{array}{c}
b_1 \\
b_2
\end{array}
\right] \\
A_2^{[2]} &=&
\left[ \begin{array}{c}
{\bf h}^{[2]}(1)
\end{array}
\right]
\left[ \begin{array}{c}
a_1 \\
a_2
\end{array}
\right] +
\left[ \begin{array}{c}
{\bf h}^{[2]}(1)
\end{array}
\right]
\left[ \begin{array}{c}
b_1 \\
b_2
\end{array}
\right]
\end{eqnarray}
Clearly, from $A_1^{[1]}, A_2^{[1]}$, we can decode $W_1$, and from $A_1^{[2]}, A_2^{[2]}$, we can decode $W_2$, because the answering strings are the same as the received signals in the BIA scheme.
Narrowing down to specific choices, several simplifications are readily noted. First, note that for the user to be able to recover his desired information, it suffices to set ${\bf h}^{[1]}(i)={\bf h}^{[2]}(i)={\bf h}(i)$, $i=1,2$ and ensure that ${\bf h}(1)$ is linearly independent of ${\bf h}(2)$. Note that this means the query to Database 1 is the same regardless of the message. Now we just need two linearly independent vectors, say $e_1=[1, 0]$ and $e_2=[0,1]$. In order to achieve privacy, it suffices if the user privately tosses a fair coin to decide if ${\bf h}(1)=e_1, {\bf h}(2)=e_2$ or ${\bf h}(1)=e_2, {\bf h}(2)=e_1$.
The PIR scheme thus obtained can be specified explicitly as follows.
\begin{table}[h]
\centering
\begin{tabular}{|c|c|c|c|c|}\hline
&\multicolumn{2}{c}{Prob. 1/2} \vline&\multicolumn{2}{c}{Prob. 1/2} \vline\\ \cline{2-5}
& Want $W_1 =(a_1,a_2)$ &Want $W_2=(b_1,b_2)$& Want $W_1=(a_1,a_2)$ &Want $W_2=(b_1,b_2)$ \\
\hline
Database $1$&$a_1$, $b_2$ &$a_1$, $b_2$& $a_2$, $b_1$&$a_2$, $b_1$\\ \hline
Database $2$& $a_2+b_2$ &$a_1+b_1$&$a_1+b_1$&$a_2+b_2$\\ \hline
\end{tabular}
\caption{The information that the user downloads from each database depending on his desired message and the outcome of a private fair coin toss.}
\end{table}
Thus, the user is equally likely to ask Database 1 for either ($a_1$, $b_2$) or ($a_2$, $b_1$), regardless of his desired message. Similarly, the user is equally likely to ask Database 2 for either $a_1+b_1$ or $a_2+b_2$, regardless of his desired message. Thus, from the perspective of each database, the information downloaded is independent of the desired message, guaranteeing privacy. Also note that in each case, the user has enough information to decode his desired message. This completes the achievability proof for $N=2$.
{\it Remark: We note that the PIR scheme presented above is asymmetric in the sense that we download $2$ equations from the first database and $1$ equation from the second database. This should not raise any concern, because without loss of generality, any PIR scheme can be made symmetric so that we download an equal number of equations from every database. To this end, we replicate the PIR scheme, so that now each message consists of 4 symbols, i.e., $W_1 = \{a_1, a_2, a_1', a_2'\}$, $W_2 = \{b_1, b_2, b'_1, b'_2\}$. For symbols $a_1, a_2, b_1, b_2$, we use the original PIR protocol. For symbols $a'_1, a'_2, b'_1, b'_2$ we also use the original PIR protocol (by generating another independent set of channels ${\bf h'}^{[1]}(1), {\bf h'}^{[1]}(2), {\bf h'}^{[2]}(1), {\bf h'}^{[2]}(2)$), but switch the roles of Database 1 and Database 2, so that for these symbols, we download $2$ equations from Database $2$ and one equation from Database 1. This produces a symmetric PIR scheme.}
\subsubsection{Converse}
Since the queries and answering strings for any given database are marginally independent of the desired message index, let us assume without loss of generality that $A^{[1]}_1=A^{[2]}_1=A_1, Q^{[1]}_1= Q^{[2]}_1=Q_1$, i.e., the query and answering string for the first database is the same, regardless of which message is requested. Then, from $A_1, A^{[1]}_2, A^{[2]}_2$ one can decode both messages, and from Fano's inequality we have,
\allowdisplaybreaks
\begin{eqnarray}
&& 2H(W_1) = H(W_1,W_2| Q_1, Q^{[1]}_{2}, Q^{[2]}_{2})\\
&=&I(A_1, A^{[1]}_2, A^{[2]}_2; W_1, W_2| Q_1, Q^{[1]}_{2}, Q^{[2]}_{2}) \\
&=& H(A_1, A^{[1]}_2, A^{[2]}_2 | Q_1, Q^{[1]}_{2}, Q^{[2]}_{2}) \label{f1} \\
&=& H(A_1, A_2^{[2]} | Q_1, Q_2^{[1]}, Q_2^{[2]}) + H(A_2^{[1]} | A_1, A_2^{[2]}, Q_1, Q_2^{[1]}, Q_2^{[2]}) \\
&\leq& H(A_1, A_2^{[2]}) + H(A_2^{[1]} | A_1, A_2^{[2]}, Q_1, Q_2^{[1]}, Q_2^{[2]}, W_2) \label{f2} \\
&\leq& H(A_1) + H(A_2^{[2]}) + H(A_2^{[1]} | A_1, Q_1, Q_2^{[1]}, W_2) \label{f3}\\
&=& \eta H(W_1) + H(A_1, A_2^{[1]} | Q_1, Q_2^{[1]}, W_2) - H(A_1 | Q_1, Q_2^{[1]}, W_2) \label{fx} \\
&=& \eta H(W_1) + H(A_1, A_2^{[1]} | Q_1, Q_2^{[1]}, W_2) \notag \\
&&~- 1/2 H(A_1 | Q_1, Q_2^{[1]}, W_2) - 1/2 H(A_2^{[1]} | Q_1, Q_2^{[1]}, W_2) \label{f5} \\
&\leq& \eta H(W_1) + H(A_1, A_2^{[1]} | Q_1, Q_2^{[1]}, W_2) \notag \\
&& - 1/2 H(A_1 | Q_1, Q_2^{[1]}, W_2) - 1/2 H(A_2^{[1]} | A_1, Q_1, Q_2^{[1]}, W_2) \notag \\
&=& \eta H(W_1) + 1/2 H(A_1, A_2^{[1]} | Q_1, Q_2^{[1]}, W_2) \\
&=& \eta H(W_1) + 1/2 H(W_1) \label{fff}
\end{eqnarray}
where (\ref{f1}) is due to the fact that the answering strings are deterministic functions of the messages and queries, i.e., $H(A_1, A^{[1]}_2, A^{[2]}_2 | W_1, W_2, Q_1, Q^{[1]}_{2}, Q^{[2]}_{2}) = 0$, and (\ref{f2}) follows from the correctness condition that $W_2$ is a deterministic function of $A_1, A_2^{[2]}$. (\ref{f3}) follows from the property that dropping conditioning does not reduce entropy. In (\ref{fx}), we plug in the definition of $\eta$. In (\ref{f5}), we use a symmetric argument without loss of generality, so that $H(A_1 | Q_1, Q_2^{[1]}, W_2) = H(A_2^{[1]} | Q_1, Q_2^{[1]}, W_2)$ (this argument is proved in Lemma \ref{sym} in the next section). (\ref{fff}) follows from the fact that from $A_1, A_2^{[1]}$, we can decode $W_1$, and applying Fano's inequality. Rearranging (\ref{fff}), we have that $\eta \geq \frac{3}{2}$ and the outer bound proof is complete.
\section{Proof of Theorem \ref{thm:download}}\label{sec:proof}
\subsection{Outer Bound}
We define $A_{j:l}^{[i]} \triangleq \{A_{j}^{[i]}, A_{j+1}^{[i]}, \cdots, A_{l}^{[i]} \}, 1 \leq j \leq l \leq N, i \in \{1, 2\}$.
Similarly, $Q_{j:l}^{[i]} \triangleq \{Q_{j}^{[i]}, Q_{j+1}^{[i]}, \cdots, Q_{l}^{[i]} \}$.
Without loss of generality, we assume that $A^{[1]}_1=A^{[2]}_1=A_1$, and that the PIR scheme is symmetrized by combining all permutations of any given PIR scheme as explained earlier. A consequence of this symmetry is formalized in the following lemma.
\begin{lemma}\label{sym}
[Symmetry]
Without loss of generality, we have
\begin{eqnarray}\label{syma}
&& H(A_1^{[1]}| Q_{1:N}^{[1]}, W_{2}) = \cdots = H(A_N^{[1]}| Q_{1:N}^{[1]}, W_{2}) \label{syma} \\
&& H(A_1^{[1]}| Q_{1:N}^{[1]}, W_{2}) \geq \frac{1}{N} H(W_1) \label{symaa}
\end{eqnarray}
\end{lemma}
{\it Proof:} We first prove (\ref{syma}). It follows from a permutation argument. Given a retrieval scheme described by a set of query and answer functions, we replicate the scheme $N!$ times and use one copy for each permutation of the databases. As a result, for the replicated scheme, the conditional entropy of the answer from each database is the same, and (\ref{syma}) follows.
Next we proceed to prove (\ref{symaa}). Note that from $A_{1:N}^{[1]}$, we can decode $W_1$. From Fano's inequality, we have
\begin{eqnarray}
H(W_1)
= I(A_{1:N}^{[1]}; W_1 | Q_{1:N}^{[1]}, W_2) = H(A_{1:N}^{[1]} | Q_{1:N}^{[1]}, W_2) \notag \\
\leq \sum_{j=1}^N H(A_{j}^{[1]}| Q_{1:N}^{[1]}, W_{2}) = N H(A_{1}^{[1]}| Q_{1:N}^{[1]}, W_{2}) \notag
\end{eqnarray}
where the first line follows from the fact that the answers are deterministic functions of the messages and queries, and the second line is due to the property that dropping conditioning does not reduce entropy, and (\ref{syma}). Thus, (\ref{symaa}) is proved
\hfill\mbox{\rule[0pt]{1.5ex}{1.5ex}}
As in the $N=2$ case, from $A_1, A_{2:N}^{[1]}, A_{2:N}^{[2]}$, we can decode both messages $W_1, W_2$.
From Fano's inequality, we have
\begin{eqnarray}
2H(W_1) &=& H(W_1, W_2 | Q_{1:N}^{[1]}, Q_{2:N}^{[2]}) \\
&=& H(A_1, A_{2:N}^{[1]}, A_{2:N}^{[2]} | Q_{1:N}^{[1]}, Q_{2:N}^{[2]}) \label{oo1} \\
&\leq& H(A_1, A_{2:N}^{[2]}) + H(A_{2:N}^{[1]} | A_1, A_{2:N}^{[2]}, Q_{1:N}^{[1]}, Q_{2:N}^{[2]}) \\
&=&\eta H(W_1)
+H(A_{2:N}^{[1]} | A_1, A_{2:N}^{[2]}, Q_{1:N}^{[1]}, Q_{2:N}^{[2]},W_2) \label{o2} \\
&\leq& \eta H(W_1) +H(A_{2:N}^{[1]} | A_1, Q_{1:N}^{[1]},W_2) \label{oo2} \\
&=& \eta H(W_1) + H(A_{1:N}^{[1]} | Q_{1:N}^{[1]}, W_{2}) - H(A_1 | Q_{1:N}^{[1]}, W_{2}) \\
&\leq& \eta H(W_1) + H(W_1) - H(W_1)/N \label{oo3}
\end{eqnarray}
where (\ref{oo1}) is due to the fact that the answering strings are deterministic functions of the messages and queries, (\ref{o2}) is due to the definition of $\eta$ and $W_2$ is a deterministic function of $A_1, A_{2:N}^{[2]}$, and in (\ref{oo3}) the second term follows from the fact that from $A_{1:N}^{[1]}$ we can decode $W_1$ and applying Fano's inequality and the last term is due to (\ref{symaa}). Rearranging (\ref{oo3}) produces the desired outer bound and the proof is complete.
An intuitive understanding of the outer bound proof is as follows. In total, we download $\eta H(W_1)$ amount of information, out of which $H(W_1)$ is the desired message and the remaining $(\eta - 1) H(W_1)$ is interference. When we request $W_2$, the interference would be $W_1$, and $(\eta - 1) H(W_1)$ would be the amount of the information about $W_1$ contained in the answering strings. From (\ref{symaa}), we know that this interference is at least $\frac{1}{N} H(W_1)$, so that $(\eta - 1) H(W_1) \geq \frac{1}{N} H(W_1)$, and $\eta \geq 1+ \frac{1}{N}$.
\subsection{Inner Bound}
The achievable scheme follows from the BIA scheme of the $X$ channel with $N$ transmitters and 2 receivers, which achieves $N/(N+1)$ DoF per receiver \cite{Jafar_corr}.
The scheme operates over $N^2 -1$ channel uses. Transmitter $i \in \{1,\cdots, N\}$ wishes to send $N-1$ symbols $a_{1,i}, a_{2,i}, \cdots, a_{N-1,i}$ to User 1 and another $N-1$ symbols $b_{1,i}, b_{2,i}, \cdots, b_{N-1,i}$ to User 2.
The channel coherence pattern is given by the following grid, where ${\bf h}^{[i]}(j)$ are random and statistically equivalent realizations of the $1 \times N$ channel vector from the $N$ transmitters to Receiver $i$.
\begin{table}[h]
\centering
\scalebox{1}{
\begin{tabular}{|c|c|c|c|c|c|}
\hline
${\bf h}^{[2]}(1)$ & 1 & 2 & $\cdots$ & $N-1$ & $N$ \\
\hline
${\bf h}^{[2]}(2)$ & $N+1$ & $N+2$ & $\cdots$ & $2N-1$ & $2N$ \\
\hline
$\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ \\
\hline
${\bf h}^{[2]}(N-1)$ & $N^2 - 2N + 1$ & $\cdots$ & $\cdots$ & $N^2 - N-1$ & $N^2 - N$ \\
\hline
${\bf h}^{[2]}(N)$ & $N^2 - N + 1$ & $\cdots$ & $\cdots$ & $N^2-1$ & \\
\hline
& ${\bf h}^{[1]}(1)$ & ${\bf h}^{[1]}(2)$ & $\cdots$ & ${\bf h}^{[1]}(N-1)$ & ${\bf h}^{[1]}(N)$ \\
\hline
\end{tabular}
}
\end{table}
In this channel, suppose symbols are repeated by the transmitters over channel uses where desired users' channel changes and the undesired users' channel remains the same, i.e., $a_{p, i}, p \in \{1, \cdots, N-1\}$ are repeated by Transmitter $i$ over the channel uses where the channel to User $2$ remains ${\bf h}^{[2]}(p)$, and $b_{q,i}, q \in \{1, \cdots, N-1\}$ are repeated by Transmitter $i$ over the channel uses where the channel to User 1 remains ${\bf h}^{[1]}(q)$. Thus, the received signal at each receiver has the property that the desired signal has rank $N^2 - N$ and the interference has rank $N-1$ \cite{Jafar_corr}.
Further, the desired signal does not overlap with the interference, and each user is able to achieve DoF $N/(N+1)$.
Now let us translate the BIA scheme to a PIR scheme, wherein each message consists of $N^2 -N$ symbols. To achieve download cost $\frac{N+1}{N} = \frac{N^2 - 1}{H(W_1)}$, the answering strings from all $N$ databases must consist of $N^2 - 1$ equations. The queries are the channels in the BIA problem. To ensure that the switching pattern does not reveal the identity of the receiver, the channel uses are grouped as follows.
Channel uses at the diagonal of the grid, i.e., channel uses $1, N+2, 2N+3, \cdots, N^2 - N-1$ ($N-1$ in total), are placed in the first group and associated with Database 1. Channel uses at the diagonal of the grid shifted cyclicly to the right by 1, i.e., channel uses $2, N+3, \cdots, N^2 - N, N^2 - N + 1$ ($N$ in total), are placed in the second group and associated with Database 2. Channel uses at the diagonal of the grid shifted cyclicly to the right by 2 are placed in the third group, associated with Database 3, and so on.
As a result, the channel changes within this group from one channel use to the next regardless of the receiver, thus revealing nothing about the identity of the receiver.
When message $W_i$ is desired, the query vector sent to the $j^{th}$ database is comprised of the channels to Receiver $i$ in the $j^{th}$ group.
The answering string from each database is comprised of the received signal equations corresponding to the channel uses in each query vector. When message $W_i$ is desired, the answering string is the received signal at the Receiver $i$. As the answering strings are the same as the received signal in the BIA scheme, we can decode the desired message.
Similar to the proof for $N = 2$ presented in the previous section, let us specify the choices. To ensure decodability, we set ${\bf h}^{[1]}(i)={\bf h}^{[2]}(i)={\bf h}(i)$, $i=1,\cdots, N$ and ensure ${\bf h}(i)$ are linearly independent. For such a purpose, we just need $N$ vectors $e_0, \cdots, e_{N-1}$, where $e_{i-1}$ is an $N \times 1$ vector, where the $i^{th}$ entry is one and all other entries are zero. To be private, it suffices if the user privately draw a number $l$ uniformly from the set $\{0, 1, \cdots, N-1\}$ and set ${\bf h}(i) = e_{i-1 + l \mod N}$.
This completes the achievability proof.
\section{Discussion} \label{sec:disc}
We identified an interesting connection between PIR and BIA, which ensures that a good BIA scheme can be translated into a good PIR protocol. Based on this insight we found the optimal download cost for the $K = 2$ messages setting with $N\geq 2$ databases.
An immediate open problem is to find the optimal download cost when $K \geq 3$. Here the best known scheme achieves the cost of $1+ \frac{1}{N-1}$ \cite{Shah_Rashmi_Kannan}. Interestingly, the scheme proposed in \cite{Shah_Rashmi_Kannan} can also be translated from a BIA scheme. Consider a $K$ user MISO interference channel, where each transmitter is equipped with $N-1$ antennas and each receiver is equipped with a single antenna. The BIA scheme achieves $1/(1+ \frac{1}{N-1}) = \frac{N-1}{N}$ DoF per receiver, where each transmitter sends $N-1$ symbols to its desired receiver over $N$ channel uses. Over these $N$ channel uses, the desired channel changes after every channel use, the cross channels remain unchanged, and each transmitter repeats its information symbols (one from each antenna). Thus, each receiver sees only one repeated linear combination of interfering symbols and he can project the received signal equations along the direction of the nullspace of the all 1 vector to extract the $N-1$ desired symbols. Following the general connection between BIA and PIR, this BIA scheme can be easily translated into a PIR scheme, where each message consists of $N-1$ symbols and we download one equation from each of the $N$ databases. Each channel use in BIA is associated with one database in PIR, so that when $W_i, i \in \{1,\cdots, K\}$ is desired, the query to the $j^{th}, j \in \{1,\cdots, N\}$ database in PIR is the channel vector from all transmitters to the $i^{th}$ receiver over the $j^{th}$ channel use in BIA and the answering string from the $j^{th}$ database in PIR is the received signal equation at the $i^{th}$ receiver over the $j^{th}$ channel use in BIA. This completes the description of the PIR scheme with download cost $1 + \frac{1}{N-1}$. However, the cost of $1+ \frac{1}{N-1}$ does not match the converse provided by direct extensions of the techniques introduced in this paper. This leaves open the possibility of another BIA scheme that may provide a better efficiency. Notably, in the regime where the number of messages becomes large, i.e., $K \rightarrow \infty$, the download cost of $1+ \frac{1}{N-1}$ is asymptotically optimal. The converse can be established as follows, which is a simple extension of our outer bound proof. Without loss of generality, we assume $A_1^{[1]} = \cdots = A_1^{[K]} = A_1$. Then, from $A_1, A_{2:N}^{[2]}, \cdots, A_{2:N}^{[K]}$ one can decode all $K$ messages $W_1, \cdots, W_K$, so that the entropy of $A_1, A_{2:N}^{[2]}, \cdots, A_{2:N}^{[K]}$ is no less than $KH(W_1)$. By a simple extension of Lemma \ref{sym} to $K$ messages setting, we know that by symmetry, the entropy of each term in $A_1, A_{2:N}^{[2]}, \cdots, A_{2:N}^{[K]}$ is the same and is equal to $\eta H(W_1)/N$. As there have $1 + (K-1)(N-1)$ terms in total, we have
\begin{eqnarray*}
\eta \geq \frac{NK}{1 + (K-1)(N-1)} = \frac{N}{\frac{1}{K} + (1- \frac{1}{K})(N-1)}
\end{eqnarray*}
and when $K \rightarrow \infty$, $\eta \geq 1 + 1/(N-1)$.
\bibliographystyle{IEEEtran}
|
1,116,691,500,895 | arxiv | \section{Introduction}
Wannier functions (WF) were introduced by Wannier in 1937 \cite{W} as bases
in subspaces of states corresponding to energy bands in solids, bases
consisting of exponentially localized functions (localized orbitals).
For periodic crystals they are defined as Fourier transform of Bloch
functions of the corresponding bands. Since then WF proved to be a key tool
in quantum theory of solids as they provide a tight binding description of
the electronic band structure of solids. At the conceptual level they lay at
the foundation of all effective mass type theories
e.g the famous Peierls-Onsager substitution describing the dynamics of Bloch
electrons in the presence of an external magnetic field (see e.g.\cite{N2}and
references therein). At the quantitative level, especially after the seminal
paper by Marzari and Vanderbilt \cite{MV}, WF become an effective tool in {\em ab initio}
computational studies of electronic properties of materials. Moreover during the last decades
WF proved to be an essential ingredient in the study of low dimensional nanostructures such as
linear chains of atoms, nanowires, nanotubes etc (see e.g. \cite{CMSN},\cite{CBRM}).
In particular WF are essential for most formulations of transport phenomena using real
space Green's function method based on Landauer-B\"uttiker formalism both at
rigorous \cite{CJM} and computational levels \cite{N},\cite{CMSN}.
A few remarks are in order here. The first one is that realistic low dimensional
systems are not strictly one (two) dimensional but rather quasi one (two) dimensional
and one has to take into account the (restricted) motion along perpendicular directions.
This adds specific features as for example the screw symmetry \
in nanotubes and nanowires absent in strictly one dimensional systems.
The second one is that realistic systems, due to the presence of defects, boundaries,
randomness etc, do not have usually full translation symmetry and this ask for a theory
of WF not based on Bloch formalism. Finally let us remind that contrary to a widespread
opinion (see e.g. the discussion in \cite{N2}) that WF always exist for isolated band in
solids this is not true. More precisely, in more than one dimension there are subtle
topological obstructions and these are related to the QHE \cite{Th}, \cite{CNP}, \cite{BPCMM}: a band
for which WF are known to exist gives no contribution to the quantum Hall current. It is
then crucial to have rigorous proofs of the existence of exponentially localized WF.
For one dimensional periodic systems the existence of exponentially localized WF has been
proved by Kohn in his classic paper \cite{K} about analytic structure of Bloch functions.
An extension of Kohn analysis to quasi one dimensional systems has been done recently by
Prodan \cite{Pr}. As for higher dimensions
it was known since the work by des Cloizeaux \cite{dC1} \cite{dC2} that
there are obstructions to the existence of exponentially localized WF and that these
obstructions are of topological origin (more precisely as explicitly stated in \cite{N1}
these obstructions are connected to the topology of a vector bundle of orthogonal projections).
The fact that for simple bands of time reversal invariant systems the obstructions are absent
was proved by des Cloizeaux \cite{dC1} \cite{dC2} under the additional condition of the existence of centre of
inversion and by Nenciu \cite{N1} in the general case. While the proofs in \cite{dC1} \cite{dC2},
\cite{N1} did not use the vector bundle theory it was suggested in \cite{N2},\cite{NN2}
that the characteristic classes theory in combination with some deep results in the theory of
analytic functions of several complex variables (Oka principle) can be
used to give alternative proof of the above results and to extend them to composite bands
of time reversal symmetric systems.
This has been substantiated recently in \cite{P}, \cite{BPCMM} where the existence of
exponentially localized Wannier functions has been proved for composite bands of time
reversal symmetric systems in two and three dimensions settling in the affirmative a long
standing conjecture. In conclusion the situation is
satisfactory as far as periodic time reversal symmetric Hamiltonians are considered
(as already mentioned for Hamiltonians which are not time reversal symmetric
exponentially localized Wannier functions might not exists).
As already said above both the theory and applications of Wannier
functions boosted since Marzari and Vanderbilt \cite{MV},
introduced studied and proposed methods to compute the so called
maximally localized Wannier functions (MLWFs) defined by the fact
that they minimize the position mean square deviation. It was
conjectured in \cite{MV} that they can be chosen to be real
functions and that they have "optimal" exponential localization in
the sense that they have the same exponential localization as the
integral kernel of the projection operator of the corresponding
band.
MLWFs proved to be an
invaluable tool in the theory of electronic properties of periodic
media especially in the modern theory of electronic
polarizability (see e.g. \cite{WDRV} and references therein).
In the one dimensional case the theory of MLWFs is much more
developed. It is known \cite{MV} that MLWFs are identical to the
eigenfunctions of the "band position" operator and then they are
unique (up to uninteresting phases) and can be chosen to be real
functions. Moreover the phases of the corresponding Bloch
functions are related to the parallel transport procedure
\cite{MV}, \cite{BW}. Recently a detailed study of Wannier
functions, including their exponential decay, emphasizing the
difference between the cases with and without inversion symmetry
appeared in \cite{BN}. In the same paper there are pointed out situations in which the
Wannier functions could decay slower than the kernel of the projector, which shows that choosing the optimal phase is not a trivial task. Our results show
that that by choosing the right phase one must always obtain an optimal decay.
Motivated by the great interest in nonperiodic structures much effort has been devoted
to extend the results about existence of exponentially localized bases for isolated bands
in nonperiodic systems. The basic difficulty stems from the fact that for nonperiodic
systems one cannot define Wannier functions as Fourier transforms of the Bloch functions.
One way out of the difficulty is to start from the periodic case or tight-binding limit
where the Wannier functions are known to exist and and use perturbation or ``continuity''
arguments. The basic idea is that since the obstructions are of topological origin the
existence of exponentially localized WF is stable against perturbations. Indeed along
these lines it has been possible to prove the existence of (generalized) WF for a
variety of nonperiodic systems \cite{KO}, \cite{RK}, \cite{GK}, \cite{N2}, \cite{NN2}.
Since in the periodic case the obstructions to the existence of exponentially localized WF
are absent \cite{dC1},\cite{dC2},\cite{N1} in one dimension it was naturally to
conjecture \cite{NN2},\cite{Ni} that in one dimension WF exist for all isolated bands
irrespective of periodicity properties.
The first problem to be solved was to find an alternative definition of WF. The basic
idea goes back to Kivelson \cite{Ki}, who proposed to {\em define} the generalized WF
as eigenfunctions of the ``band position'' operator.
To substantiate the idea one has to prove that the band position operator is self-adjoint,
has discrete spectrum and its eigenfunctions are exponentially localized.
For the particular case of a periodic one dimensional crystal with one defect Kivelson proved
that the eigenfunctions of the band position operator are indeed exponentially localized
and asked for a general proof. In the general case, by a bootstrap argument, Niu \cite{Ni} argued
that the
eigenfunctions of the band position operator (if they exist) are at least polynomially localized.
In full generality the fact that for all isolated parts of the spectrum the band position
operator is self-adjoint, has discrete spectrum and its eigenfunctions are exponentially
localized has been proved in \cite{NN3}.
In this paper we extend the results in \cite{NN3} to quasi
one-dimensional systems i.e. three dimensional systems for which
the motion extends to infinity only in one direction. In addition
we add the result (which is new even in the strictly one
dimensional case) that (see Theorem \ref{main} below for details)
the ``density" of WF is uniformly bounded. While the main ideas
of the proof are the same as in \cite{NN3} there are major
differences both at the technical and physical level. In
particular for quasi one dimensional systems with screw symmetry
the constructed WF inherits this symmetry a property which is very
useful in computational applications. Finally let us point out
that as in the periodic case, generalized WF defined as
eigenfunctions of the band position operator have very nice
properties e.g. they are (up to uninteresting phases) uniquely
defined and for real (i.e. time reversal invariant) Hamiltonians
they can be chosen to be real functions and this solves for the
general quasi one dimensional case the ``strong conjecture'' in
Section V. of \cite{MV}. As for their exponential localization we
have the following "optimality" result (see Proposition \ref{optimal}
for a precise statement)
which seems to be new even
in the one dimensional periodic case: the
eigenfunctions of the band position operator have
the same exponential localization as the integral kernel of the projection
operator of the corresponding band.
\section{The results}
Consider in $L^{2}(\mathbb{R}^{3})$ the following Hamiltonian describing a
particle subjected to a scalar potential $V$:
\begin{equation}\label{ham}
H=\mathbf {P}^{2}+V,
\; \; \; {\bf P}=-i\nabla,\; \;\;
\sup_{{\bf x} \in {\mathbb{R} }^3} \int_{\vert {\bf x-y} \vert\leq 1}
\vert V({\bf y}) \vert^2 d{\bf y} < \infty
\end{equation}
which, as is well known (see \cite {RS}), is essentially self-adjoint
on $C_0^{\infty}({\mathbb{R}}^3)$. We have already said in the introduction
that we are interested in potentials $V$ which tend to zero as the
distance from the $Ox_{1}$ axis tends to infinity. Let us now be more
precise. The notation $\mathbf{x} = (x_{1},\mathbf{x}_{\perp})$ will be used
throughout the paper. For any $R>0$, define:
\begin{equation}\label{l2locu}
I_{V}(R):=\sup_{x_{1}\in \mathbb{R} ,|\mathbf{x}_{\perp}| \geq R} \int_{\vert {\bf x-y} \vert\leq 1}
\vert V({\mathbf{y}}) \vert^2 d{\mathbf{y}}.
\end{equation}
The decay assumption for $V$ will be:
\begin{equation}\label{confv}
\lim_{R\rightarrow \infty}I_{V}(R)=0.
\end{equation}
It is easy to see that $[0,\infty)\subset \sigma(H)$ (using a Weyl
sequence argument), thus the only region where $H$ might have an
isolated spectral island is below zero.
Now suppose that $\sigma_0$ is such an isolated part of the spectrum
and define:
\begin{equation}\label{E+}
-E_{+}:= \sup \{ E:\; E \in \sigma_{0}\} < 0.
\end{equation}
If $\Gamma$ is a positively oriented contour of finite
length enclosing $\sigma_0$, then the spectral subspace corresponding
to $\sigma_0$ is:
\begin{equation}\label{subs}
{\cal K}\;:={\rm Ran}(P_0),\quad P_0= \frac{i}{2\pi}
\int_{\Gamma}(H-z)^{-1}dz.
\end{equation}
At a heuristic level, due to the fact that the wave packets from
$\mathcal{K}$ cannot propagate in the classically forbidden region (see
\eqref{E+} and \eqref{confv}), at negative energies the motion is
confined near the $Ox_{1}$ axis, i.e. the system has a quasi one
dimensional behavior.
\subsection{The technical results}
The following proposition states the "localization"
properties of $P_0$. On one hand, this give a precise meaning to the
previously discussed quasi one dimensional character,
and on the other hand it provides some key ingredients to the proof of
exponential localization of eigenfunctions of the band position
operator.
Let $a\in \mathbb{R}$, and let $\langle X_{\parallel, a}\rangle$ be the
multiplication operator corresponding to:
\begin{equation}\label{ga}
g_a(\mathbf{x}):=\sqrt{(x_1-a)^2 +1},
\end{equation}
and $\langle X_{\perp}\rangle$ be multiplication operator given by:
\begin{equation}\label{gp}
g_{\perp}(\mathbf{x}):=\sqrt{|\bx_{\perp}|^2 +1}.
\end{equation}
\begin{proposition}\label{locP0}
There exist $\alpha_{\parallel}>0$, $\alpha_{\perp}>0$, $M<\infty$ such
that:
\begin{equation}\label{expP0}
\sup_{a\in \mathbb{R}} \parallel e^{\alpha_{\parallel} \langle X_{\parallel, a}\rangle}P_0 e^{-\alpha_{\parallel} \langle X_{\parallel, a}\rangle}
\parallel \leq M,\quad {\rm and}
\end{equation}
\begin{equation}\label{confP0}
\parallel e^{\alpha_{\perp} \langle X_{\perp}\rangle}P_0 e^{\alpha_{\perp} \langle X_{\perp}\rangle}
\parallel \leq M.
\end{equation}
\end{proposition}
\noindent The proof of Proposition \ref{locP0} will also give values for $\alpha_{\parallel}$
and $\alpha_{\perp}$. In particular $\alpha_{\perp}$ can be any number
strictly smaller than $\sqrt{E_+}$.
\vspace{0.5cm}
We now can formulate the main technical result of this paper. To emphasize its
generality we stress that its proof only uses the decay condition
(\ref{confv}) and the existence of an isolated part of the
spectrum satisfying (\ref{E+}).
\begin{theorem}\label{main}
Let $X_{\parallel}$ be the operator of multiplication with $x_1$ in
$L^2(\mathbb{R}^3)$ and
consider in $\mathcal{K}$ the operator
\begin{equation}\label{hX}
\hat{X}_{\parallel}:=P_{0}X_{\parallel}P_{0}
\end{equation}
defined on
$
{\cal D}(\hat{X}_{\parallel})={\cal D}(X_{\parallel})\cap{\cal K}.
$
Then
\noindent {\rm i}. $\hat{X}_{\parallel}$ is self-adjoint on ${\cal D}(\hat{X})$;
\noindent {\rm ii}. $\hat{X}_{\parallel}$ has purely discrete spectrum;
\noindent {\rm iii}. Let $g \in G := \sigma(\hat{X}_{\parallel})$ be an eigenvalue,
$m_g$ its multiplicity, and $\{W_{g,j}\}_{1\leq j\leq
m_g}$ an orthonormal basis in the eigenspace of
$\hat{X}$ corresponding to $g$. Then for all $\beta \in [0,1]$,
there exists $ M_1<\infty$ independent of $ g$, $j $ and
$\beta $ such that:
\begin{equation}\label{locW}
\int_{\mathbb{R}^3}e^{2(1-\beta) \alpha_{\parallel} \vert x_1-g\vert}
e^{2\beta \alpha_{\perp} \vert \bx_{\perp}\vert}\vert
W_{g,j}(\mathbf{x})\vert ^2
d\mathbf{x} \leq M_1,
\end{equation}
where $\alpha_{\parallel}$ and $\alpha_{\perp}$ are the same exponents as those provided
by the proof of Proposition \ref{locP0};
\noindent {\rm iv}. Let $a \in \mathbb{R}$ and $L \geq 1$.
Denote by $N(a,L)$ the total multiplicity of the spectrum of $\hat{X}_{\parallel}$
contained in $[a-L,a+L]$.
Then there exists $M_2 < \infty$ such that
\begin{equation}\label{density}
N(a,L) \leq M_2 \cdot L.
\end{equation}
\end{theorem}
Finally, we turn to the question of optimal localization properties of
our Wannier functions. Theorem \ref{main} provides an optimal
exponential decay on the transverse direction, but in the
parallel direction it only implies a decay which is bound by the
maximal decay of the
resolvent in the gap. The conjecture on optimal exponential decay, as
stated in Section V of \cite{MV}, is whether
the $ W_{g,j}$'s have the same exponential decay as the integral kernel
$\mathcal{P}_0(\mathbf{x},\mathbf{y})$ of $P_0$ (which can be larger than the
maximal decay of the resolvent in the gap; we are
indebted to one of the referees for pointing this to us).
Concerning this issue, we have the following result showing the
optimality of the "parallel" decay of $
W_{g,j}$ at the exponential level.
\begin{proposition}\label{optimal}
Assume that for all $\alpha<\alpha_0$ we are given an {\rm a priori} bound
\begin{equation}\label{decP}
\sup_{a\in \mathbb{R}} \parallel e^{\alpha \langle X_{\parallel, a}\rangle}P_0 e^{-\alpha \langle X_{\parallel, a}\rangle}
\parallel <\infty.
\end{equation}
Then
for all $\alpha<\alpha_0$ there exists $M_1(\alpha)$, independent
of $g$ and $j$, such that
\begin{equation}\label{optW}
\int_{\mathbb{R}^3}e^{2 \alpha \vert x_1-g\vert} \vert
W_{g,j}(\mathbf{x})\vert ^2
d\mathbf{x} \leq M_1(\alpha).
\end{equation}
\end{proposition}
\vspace{0.5cm}
\noindent{\bf Remark}. Here $\alpha_0$ is the "exact" exponential
decay of $\mathcal{P}_0(\mathbf{x},\mathbf{y})$. In certain particular periodic cases
one might obtain a power-like asymptotic behavior of
$e^{\alpha_0|x_1-y_1|}\mathcal{P}_0(\mathbf{x},\mathbf{y})$ in the variables $x_1,y_1$.
We cannot say anything about an eventual asymptotic behavior of
$e^{\alpha_0 \vert x_1-g\vert}W_{g,j}(\mathbf{x})$. But due to the generality
of the setting, we consider our result to be optimal.
\subsection{Further properties of the Wannier basis}
We come now to the case when $V$ (hence $H$) has additional
symmetries. The point here is that although the Wannier functions are
not eigenfunctions of $H$, one would like them to inherit in some sense
the symmetries of $H$. The reason is that usually the Wannier basis is
used in order to write down an effective Hamiltonian in $\mathcal{K}$, and one would
like this effective Hamiltonian to inherit as much as possible the
symmetries of $H$.
First we comment on time reversal invariance.
Since $V(\mathbf{x}) $ is real, $H$ commutes with the anti-unitary operator
induced by complex conjugation. It follows (see (\ref{subs})) that
$P_0$ and $\hat{X}_{\parallel}$ are also real, thus the eigenfunctions of $\hat{X}_{\parallel}$ can be
chosen to be real. Hence Theorem \ref{main} provides us with a Wannier basis
which is time reversal invariant.
Second we consider the so called "screw-symmetry" along the $Ox_1$-axis,
of much interest in the physics of carbon nanotubes. Namely, writing
\begin{equation}\label{cil}
\bx_{\perp} =(r,\theta),\quad r\geq 0,\; \theta \in [0,2\pi),
\end{equation}
one assumes that for some $\theta_0 \in [0,2\pi)$ we have:
\begin{equation}\label{screwv}
V(x_1,r,\theta )=V(x_1+1,r,\theta+\theta_0 ).
\end{equation}
Here $\theta+\theta_0$ has to be understood modulo $2\pi$.
Defining the screw-symmetry operators $T_n^{\theta_0}$ by:
\begin{equation}\label{Tn}
(T_n^{\theta_0} f)(x_1,r,\theta ):=f(x_1-n,r,\theta-n\theta_0 ),
\end{equation}
one has a (unitary!) representation of $\mathbb{Z}$ in $L^2(\mathbb{R}^3)$.
Taking into account (\ref{screwv}) and the fact that $[-\Delta,T_n^{\theta_0}]=0$
(use cylindrical coordinates to prove this), one
obtains:
\begin{equation}\label{HTn}
[H,T_n^{\theta_0}]=0,
\end{equation}
and then from functional calculus and (\ref{subs}):
\begin{equation}\label{P0Tn}
[P_0,T_n^{\theta_0}]=0.
\end{equation}
In particular, this implies that the family $\{T_n^{\theta_0}\}_{n\in
\mathbb{Z}}$ induces a unitary representation of $\mathbb{Z}$ in
$\mathcal{K}$. Moreover, from (\ref{hX}) and
\ref{P0Tn}) one obtains:
\begin{equation}\label{TnX}
[T_n^{\theta_0},\hat{X}_{\parallel}]=nT_n^{\theta_0}.
\end{equation}
Let $p <\infty$ be the number of eigenvalues of $\hat{X}_{\parallel}$ in the interval
$[0,1)$, and let $\{g_j\}_{j=1}^p$ be the distinct eigenvalues (each
with multiplicity $m_j <\infty$). We have:
\begin{equation}\label{Wuc}
\hat{X}_{\parallel} W_{g_j,\alpha_j}=g_j W_{g_j,\alpha_j}, \;\alpha_j= 1,2,...,m_{g_j}.
\end{equation}
From (\ref{TnX} ) and (\ref{Wuc}) one obtains that for all
$g_j,\;\alpha_j,\;n \in \mathbb{Z}$:
\begin{equation}\label{Wn}
\hat{X}_{\parallel} T_n^{\theta_0} W_{g_j,\alpha_j}= (g_j+n)T_n^{\theta_0} W_{g_j,\alpha_j}.
\end{equation}
Conversely, for every other $g \in \sigma (\hat{X}_{\parallel} )$, choose an
eigenvector $W_{g}$. We can find $n\in \mathbb{Z}$ such that $g+n\in
[0,1)$. Since $\hat{X}_{\parallel} T_n^{\theta_0} W_{g}= (g+n)T_n^{\theta_0} W_{g}$, it means that $g+n$
must be one of the $g_j$'s considered above. Therefore we proved the
following corollary:
\begin{corollary}\label{screwW}
The spectrum of $\hat{X}_{\parallel}$ consists of a union of $p$ ladders:
\begin{equation}\label{spectX}
G=\cup_{j=1}^p G_j, \quad G_j=\{g:\; g=g_j+n,\;n\in \mathbb{Z} \},\quad j\in\{1,2,...,p\},
\end{equation}
and an orthonormal basis in $\mathcal{K}$ can be chosen as:
\begin{align}\label{Wbasis}
&W_{n,g_j,\alpha_j}:= W_{g_j+n,\alpha_j}:=T_n^{\theta_0} W_{g_j,\alpha_j}, \\
& n \in \mathbb{Z},\; j\in \{1,2,...,p\},\;\alpha_j\in\{
1,2,...,m_{g_j}\}.\nonumber
\end{align}
\end{corollary}
\vspace{0.5cm}
It is interesting to express the effective Hamiltonian
$P_0HP_0$ as an infinite matrix with the help of the Wannier basis.
For notational simplicity we relabel the pair $(g_j,\;\alpha_j)$ as
$l\in \{1,2,...,N_c=\sum_{j=1}^p m_{g_j}\}$ and write the Wannier basis as
$\{W_{n,l}\}_{n \in \mathbb{Z},\;l\in \{1,2,...,N_c\}}$. Note that
$N_c$ is nothing that the number of Wannier functions per unit cell $[0,1)$. Let
\begin{equation}
h_{l,k}^{\theta_0}(m,n):=\langle W_{m,l},HW_{n,k}\rangle.
\end{equation}
The important fact is that in spite of a
rotation with an angle $\theta_0$ for which it might happen that
$\frac{\theta_0}{2\pi}$ to be irrational, from \eqref{HTn} and
(\ref{Wbasis}) one obtains (with the usual abuse of notation):
\begin{equation}\label{transl}
h_{l,k}^{\theta_0}(m,n)=h_{l,k}^{\theta_0}(m-n).
\end{equation}
Then a standard computation gives the effective Hamiltonian as an
operator in $(l^2)^{N_c}$ which is of {\em standard translation invariant}
tight binding type:
\begin{equation}\label{heff}
(h_{eff}^{\theta_0}\phi)_l(m):=\sum_{k,n}h_{l,k}^{\theta_0}(m-n)\phi_k(n).
\end{equation}
This is another consequence of the quasi one-dimensional character of the
motion for negative energies. More precisely, it reflects the fact that for arbitrary
values of $\theta_0$, since $T_n^{\theta_0}$ is a unitary
representation of $\mathbb{Z}$, one can still develop a Bloch type analysis but with
a more complicated form of "Bloch" functions:
\begin{equation}\label{bloch}
\Psi_k(\mathbf{x}) = e^{ikx_1}u_k (\mathbf{x}),\; u_k (\mathbf{x})=T_n^{\theta_0} u_k (\mathbf{x}).
\end{equation}
However, due to the complicated symmetry of the resulting Bloch functions (which does
not allow to represent the fiber Hamiltonian as a differential
operator on the unit cell with "simple" boundary conditions), the
analysis gets much harder. The Bloch analysis reduces to the standard
one (with a larger unit cell) for rational
values of $\frac{\theta_0}{2\pi}$.
\section{Proofs}
This section is devoted to the proof of Proposition \ref{locP0},
Theorem \ref{main} and Proposition \ref{optimal}. A certain number
of unimportant finite positive constants appearing during the
proof will be denoted by $M$.
One of the key ingredients in the proofs is the exponential decay
of the integral kernel of the resolvent of Schr\"odinger
operators. This is an elementary result in the Combes-Thomas-Agmon
theory of weighted estimates. We summarize the needed result in:
\begin{lemma}\label{CTAL}
Let $W$ be a potential such that
$\sup_{{\bf x} \in {\bf R }^3} \int_{\vert {\bf x-y} \vert\leq 1}
\vert W({\bf y}) \vert^2 d{\bf y} < \infty$. Define $K:=\mathbf {P} ^{2}+
W(\mathbf{x})$ as an operator sum, and let $h$ be a real function
satisfying:
\begin{align}\label{prima1}
h\in C^{\infty}(\mathbb{R}^3),\quad \sup_{\mathbf{x}\in\mathbb{R}^3}\{
|\nabla h(\mathbf{x})|+ |\Delta h(\mathbf{x})|\}= m <
\infty.
\end{align}
Fix $z\in \rho(H)$. Then there exists $\alpha_{z }>0$ such that
\begin{equation}\label{prima11}
\Vert e^{\alpha_{z}h}
(K-z)^{-1}e^{-\alpha_{z}h}\Vert \leq M,
\end{equation}
\begin{equation}\label{adoua11}
\Vert e^{\alpha_{z}h}
P_j(K-z)^{-1}e^{-\alpha_{z}h}\Vert \leq M,
\end{equation}
where $P_j=-i\frac{\partial}{\partial x_j }$, $j\in\{1,2,3\}$.
\end{lemma}
Without giving the details of the proof of Lemma \ref{CTAL}, for later
use we write down a key identity in \eqref{CTA}: under the condition
\begin{equation}\label{CTAcond}
1+\alpha_{z} (\pm i\mathbf {P}\cdot \nabla h \pm i \nabla h\cdot \mathbf {P}
-\alpha_{z}|\nabla h|^2)
(K-z)^{-1} \quad {\rm invertible}
\end{equation}
one has
\begin{align}\label{CTA}
& e^{\pm \alpha_{z}h}
(K-z)^{-1}e^{\mp \alpha_{z}h}\\
&= (K-z)^{-1}[1+\alpha_{z} (\pm i\mathbf {P}\cdot
\nabla h \pm i \nabla h\cdot \mathbf {P}
-\alpha_{z}|\nabla h|^2)
(K-z)^{-1}]^{-1}. \nonumber
\end{align}
Then (\ref{CTAcond}) holds true if for example $\alpha_{z}>0$ is small
enough.
\vspace{0.5cm}
\subsection{Proof of Proposition \ref{locP0}}
Take $\Gamma$ in
(\ref{subs}) a
contour of finite length enclosing $\sigma_0$ and satisfying
\begin{equation}\label{Gamma}
\dist (\Gamma, \sigma (H))=\frac{1}{2}\dist (\sigma_0 , \sigma
(H)\setminus \sigma_0 ).
\end{equation}
Then since $|\nabla g_a | \leq 1$, $|\Delta g_a |^2 \leq 2$, the
estimate (\ref{expP0}) follows directly from Lemma \ref{CTAL} by taking
$\alpha_{\parallel}$ sufficiently small such that for all $z \in\Gamma $:
$$
\Vert \alpha_{\parallel} (i\mathbf {P}\cdot \nabla g_a +i \nabla g_a \cdot \mathbf {P}
-\alpha_{\parallel} |\nabla g_a|^2)
(K-z)^{-1} \Vert \leq b <1.
$$
We now prove (\ref{confP0}). If $R>0$, define:
\begin{equation}\label{HR}
H_R= -\Delta +(1-\chi_R)V,
\end{equation}
where
\begin{equation}\label{carR}
\chi_R(\mathbf{x} )=\left\{\begin{array}{rlc}
1&\mbox{for}&\vert \bx_{\perp} \vert \leq R\\
0&\mbox{for}&\vert \bx_{\perp} \vert > R\end{array}\right. .
\end{equation}
From (\ref{confv}) it follows that
\begin{displaymath}
\lim_{R\rightarrow \infty} \inf \sigma (H_R) =0.
\end{displaymath}
In particular, for sufficiently large $R$, $(H_R-z)^{-1}$ is analytic
inside $\Gamma$. Since $H-H_R=\chi_R V$, then using resolvent
identities we obtain:
\begin{align}\label{HHR}
&(H-z)^{-1}=(H_R-z)^{-1}\\
&-(H_R-z)^{-1}\chi_R V(H_R-z)^{-1}+
(H_R-z)^{-1}\chi_R V(H-z)^{-1}\chi_R V(H_R-z)^{-1}.\nonumber
\end{align}
From (\ref{subs}), (\ref{HHR}) and the fact that $(H_R-z)^{-1}$ is analytic
inside $\Gamma$ one has
\begin{equation}\label{P0R}
P_0= \frac{i}{2\pi}
\int_{\Gamma}(H_R-z)^{-1}\chi_R V(H-z)^{-1}\chi_R V(H_R-z)^{-1}.
\end{equation}
Notice that for all $\alpha >0$:
\begin{equation}\label{RV}
\sup_{{\bf x} \in {\bf R }^3} \int_{\vert {\bf x-y} \vert\leq 1}
\vert (e^{\alpha g_{\perp}}\chi_R V)({\bf y}) \vert^2 d{\bf y} < \infty.
\end{equation}
Take now $\alpha_{\perp} >0$ such that \ref{CTAcond} holds true for all $z
\in \Gamma$, $K=H_R$, $h=g_{\perp}$ and $\alpha_z=\alpha_{\perp} $. That is let
us suppose that
\begin{equation}\label{acincea6}
1+\alpha_{\perp} (\pm i\mathbf {P}\cdot \nabla g_{\perp} \pm i \nabla g_{\perp}\cdot \mathbf {P}
-\alpha_{\perp} |\nabla g_{\perp}|^2)
(H_R-z)^{-1} \quad {\rm is}\:\: {\rm invertible}
\end{equation}
uniformly on $\Gamma$. Then we can rewrite
$P_0$ as:
\begin{align}\label{prima6}
P_0&=e^{-\alpha_{\perp} \langle X_{\perp}\rangle}\left \{\frac{i}{2\pi}\int_{\Gamma}
\left [e^{\alpha_{\perp} \langle X_{\perp}\rangle}(H_R-z)^{-1}e^{-\alpha_{\perp} \langle X_{\perp}\rangle}\right
]\right . \nonumber \\
&\left [e^{\alpha_{\perp} g_{\perp}}\chi_R V(H-z)^{-1}\right ]\;
\left [e^{\alpha_{\perp} g_{\perp}}\chi_R V
(H_R-z)^{-1}\right ] \\
&\left . \left [1+\alpha_{\perp} (-i\mathbf {P}\cdot \nabla g_{\perp} -i \nabla g_{\perp}\cdot \mathbf {P}
-\alpha_{\perp} |\nabla g_{\perp}|^2)
(H_R-z)^{-1}\right ]^{-1}dz\right \} e^{-\alpha_{\perp} \langle X_{\perp}\rangle}.\nonumber
\end{align}
Due to \eqref{RV} the operator under the integral sign is uniformly bounded in
$z$ and the proof of Proposition \ref{locP0} is finished provided we
can show why we can choose $\alpha_{\perp}$
as close to $\sqrt{E_+}$ as we want. The argument is as follows. Choose
$0\leq \alpha_{\perp} <\sqrt{E_+}$. Choose a contour $\Gamma$ which is very close
to $\sigma_0$, at a distance $\delta>0$, infinitesimally small.
Using the spectral theorem (or in this
case the Plancherel theorem), there exists $\delta$ small enough such that the
following estimates hold true:
\begin{align}\label{adoua6}
\sup_{z\in\Gamma}\left \Vert (\mathbf {P}^2-z)^{-1}\right \Vert \leq {\rm const},\quad \sup_{z\in\Gamma}\max_{j\in\{1,2,3\}}\left \Vert P_j(\mathbf {P}^2-z)^{-1}\right \Vert \leq
{\rm const}.
\end{align}
Hence we can find $\delta$ small enough and $R$
large enough such that the operator in \eqref{acincea6} is invertible
if
\begin{equation}\label{asasea6}
1+\alpha_{\perp} (\pm i\mathbf {P}\cdot \nabla g_{\perp} \pm i \nabla g_{\perp}\cdot \mathbf {P}
-\alpha_{\perp} |\nabla g_{\perp}|^2)
(\mathbf {P}^2-\Re(z))^{-1}\quad {\rm is}\:\: {\rm invertible}
\end{equation}
uniformly on $\Gamma$.
Now the operator in \eqref{asasea6} is invertible if
\begin{align}\label{atreia6}
1&\pm i\alpha_{\perp} (\mathbf {P}^2-\Re(z))^{-\frac{1}{2}}
(\mathbf {P}\cdot \nabla h +\nabla h\cdot
\mathbf {P})(\mathbf {P}^2-\Re(z))^{-\frac{1}{2}} \nonumber \\
&-\alpha_{\perp}^2(\mathbf {P}^2-\Re(z))^{-\frac{1}{2}}|\nabla h|^2
(\mathbf {P}^2-\Re(z))^{-\frac{1}{2}}
\end{align}
is invertible (by a resummation of the Neumann series and analytic
continuation). Now assume that uniformly on $\Gamma$ we have:
$$0<\alpha_{\perp}^2(\mathbf {P}^2-\Re(z))^{-\frac{1}{2}}|\nabla h|^2
(\mathbf {P}^2-\Re(z))^{-\frac{1}{2}}\leq \frac{\alpha_{\perp}^2}{-\Re(z)}<1,$$
which can be achieved if $\alpha_{\perp}^2<E_+$ and $\delta$ is chosen to be
small enough. Define
$$S:=\left (1- \alpha_{\perp}^2(\mathbf {P}^2-\Re(z))^{-\frac{1}{2}}|\nabla h|^2
(\mathbf {P}^2-\Re(z))^{-\frac{1}{2}}\right)^{-\frac{1}{2}},$$
and
$$T=T^*:=S(\mathbf {P}^2-\Re(z))^{-\frac{1}{2}}
(\mathbf {P}\cdot \nabla h +\nabla h\cdot
\mathbf {P})(\mathbf {P}^2-\Re(z))^{-\frac{1}{2}}S.$$
Then the operator in \eqref{atreia6} is invertible if $1\pm i\alpha_{\perp}
T$ is invertible, which is always the case:
$$(1\pm i\alpha_{\perp} T)^{-1}=(1\mp i\alpha_{\perp}
T)(1+\alpha_{\perp}^2T^2)^{-1}.$$
Therefore Proposition \ref{locP0} is proved. \qed
\subsection{ Proof of Theorem \ref{main}}
\noindent {\it Proof of} (i). First we recall an older result
(see e.g. \cite{A, N2, NN1}), according to which the commutator
$[X_{\parallel},P_0]$
defined on ${\cal D}(X_{\parallel})$
has a bounded closure on $L^2 (\mathbb{R}^3)$. We seek an approximate
resolvent of $\hat{X}_{\parallel}$ by defining for $\mu > 0$ the operator
\begin{equation}\label{ares}
\hat{R}_{\pm \mu}=P_0 (X_{\parallel} \pm i\mu)^{-1}P_0.
\end{equation}
Since one can rewrite $\hat{R}_{\pm \mu}$ as
$$
\hat{R}_{\pm \mu}=(X_{\parallel} \pm i\mu)^{-1}P_0+
(X_{\parallel} \pm i\mu)^{-1}[X_{\parallel},P_0]
(X_{\parallel} \pm i\mu)^{-1}P_0
$$
it follows that
$\hat{R}_{\pm \mu}{\cal K}\subset \it D (\hat{X}_{\parallel})$
and by a straightforward computation (as operators in ${\cal K}$)
\begin{equation}
(\hat{X}_{\parallel} \pm i\mu)\hat{R}_{\pm \mu}=
P_0 (X_{\parallel} \pm i\mu)P_0(X_{\parallel} \pm i\mu)^{-1}P_0=
1_{\cal K} + \hat{A}_{\pm \mu}
\end{equation}
with
\begin{equation}
\hat{A}_{\pm \mu}=P_0[X_{\parallel},P_0 ] (X_{\parallel} \pm i\mu)^{-1}P_0.
\end{equation}
Since $[X_{\parallel},P_0]$ is bounded and $\Vert (X_{\parallel} \pm
i\mu)^{-1}\Vert \leq \frac{1}{\mu}$, it follows that for
sufficiently large $\mu$:
\begin{equation}
\Vert\hat{A}_{\pm \mu}\Vert \leq \frac{1}{2}.
\end{equation}
Then again as operators in ${\cal K}$:
\begin{equation}\label{surj}
(\hat X \pm i\mu)\hat{R}_{\pm \mu}
(1_{\cal K} + \hat{A}_{\pm \mu})^{-1}=1_{\cal K}
\end{equation}
This implies that $\;\;\hat X \pm i\mu\;\;$is surjective
on $\;\;\hat{R}_{\pm \mu}
(1_{\cal K} + \hat{A}_{\pm \mu})^{-1}{\cal K}\subset {\it D}(\hat X)\;$.
By the fundamental criterion of self-adjointness \cite{RS}
$\hat X$ is self-adjoint in ${\cal K}$ on ${\cal D}(\hat X)$.
In addition, from \eqref{surj} one obtains the following formula for the
resolvent of $\hat{X}_{\parallel}$:
\begin{equation}\label{hXres}
(\hat{X}_{\parallel} \pm i\mu)^{-1}=\hat{R}_{\pm \mu}
(1_{\cal K} + \hat{A}_{\pm \mu})^{-1}.
\end{equation}
\vspace{0.5cm}
\noindent {\it Proof of} (ii).
We will show that $\hat{R}_{\pm \mu}$ is compact in ${\cal K}$ which
implies (see \eqref{hXres}) that $\hat{X}_{\parallel}$ has compact resolvent, thus
purely discrete spectrum. Consider a cut-off function $\phi_N$ which
equals $1$ if $|\mathbf{x}|\leq N$ and is zero if $|\mathbf{x}|\geq 2N$.
For $N\geq 1 $ we can decompose:
\begin{equation}\label{prima10}
\hat{R}_{\pm \mu}=P_0 (X_{\parallel} \pm i\mu)^{-1}\phi_NP_0+
P_0 (X_{\parallel} \pm i\mu)^{-1}(1-\phi_N)P_0.
\end{equation}
Writing $$\phi_NP_0=\{\phi_N(\mathbf {P}^2+1)^{-1}\}\{(\mathbf {P}^2+1)P_0\}$$ we see
that $\phi_NP_0$ is compact (even Hilbert-Schmidt)
in $L^2(\mathbb{R}^3)$ (the first factor is Hilbert-Schmidt while the
second one is bounded). Now if $0<\alpha$ is small enough, we know
that $e^{\alpha g_\perp}P_0$ is bounded (see \eqref{confP0}). Since
$$\lim_{N\to \infty}\left \Vert (X_{\parallel} \pm i\mu)^{-1}(1-\phi_N)
e^{-\alpha g_\perp}\right \Vert =0,$$
we have shown:
$$\lim_{N\to \infty}\left \Vert \hat{R}_{\pm \mu}-
P_0 (X_{\parallel} \pm i\mu)^{-1}\phi_NP_0\right \Vert =0,$$
thus $\hat{R}_{\pm \mu}$ equals the norm limit of a sequence of compact
operators, therefore it is compact. Accordingly, since the
self-adjoint operator $\hat{X}_{\parallel}$ has compact resolvent
it has purely discrete spectrum \cite{RS}:
\begin{equation}
\sigma(\hat{X}_{\parallel})=\sigma_{disc}(\hat{X}_{\parallel})=: G,
\end{equation}
and the proof of the second part of Theorem \ref{main} is finished.
\vspace{0.5cm}
\noindent{\it Proof of} (iii). Now we will consider the exponential
localization of eigenfunctions of $\hat{X}_{\parallel}$. Let $g \in G$ be an eigenvalue,
$m_g$ its multiplicity and $W_{g,j},\;\; 1\leq j\leq m_g$ be
an orthonormal basis in the eigenspace of $\hat{X}_{\parallel}$ corresponding to
$g$. We shall prove that uniformly in $g$
and $j$
\begin{align}\label{parW}
\Vert e^{\alpha_{\parallel} \langle X_{\parallel, g}\rangle}W_{g,j} \Vert & \leq M \quad {\rm and}
\\ \label{perW}
\Vert e^{\alpha_{\perp} \langle X_{\perp}\rangle}W_{g,j} \Vert & \leq M.
\end{align}
Taking \eqref{parW} and \eqref{perW} as given, one can easily
obtain \eqref{locW} by a simple convexity argument: the function
$ f(x)= a^{1-x}b^{x}$ ; $a,b>0$ is convex on $\mathbb{R}$, and for $0\leq \beta
\leq 1$ one has:
\begin{equation}
\beta e^{2 \alpha_{\parallel} g_a(\mathbf{x})} +(1-\beta )e^{2 \alpha_{\perp} g_{\perp} }
\geq e^{2(1- \beta)\alpha_{\parallel} g_a(\mathbf{x})}e^{2 \beta \alpha_{\perp} g_{\perp} },
\end{equation}
which together with \eqref{parW} and \eqref{perW} it proves
\eqref{locW} with $M_1=M^2$. Since \eqref{perW} follows directly from
\eqref{confP0} and $W_{g,j}=P_0 W_{g,j}$ we are left with
the proof of \eqref{parW}.
Although the proof of \eqref{parW} mimics closely the proof in the one
dimensional case \cite{NN3}, we give it here for completeness. In order
to emphasize the main idea of the proof let us remind one of the
simplest proofs of the exponential decay of eigenfunctions of Schr\"odinger
operators corresponding to discrete eigenvalues (assuming that the
potential $V$ is bounded and has compact support). Namely assume that for
some $E>0$ we have $(-\Delta +V+E)\Psi =0$,
which can be rewritten as
\begin{equation}\label{Svp}
\Psi =- (-\Delta +E)^{-1}V\Psi.
\end{equation}
Since for $|\alpha|<\sqrt{E}$, $e^{\alpha |\cdot|} (-\Delta
+E)^{-1}e^{-\alpha |\cdot|}$ and $e^{\alpha |\cdot|}V$ are bounded:
$$
\Psi =- e^{-\alpha |\cdot|}\left \{e^{\alpha |\cdot|} (-\Delta
+E)^{-1}e^{-\alpha |\cdot|}\right \}(e^{\alpha |\cdot|}V)\Psi
$$
which proves the exponential localization of $\Psi$. The main idea in
proving
\eqref{parW} is to rewrite the eigenvalue equation for $\hat{X}_{\parallel}$ in a form
similar to \eqref{Svp} and and then to use \eqref{expP0}.
Let us start with some notation. If $b>0$ (sufficiently large) and
$a\in {\mathbb{R}}$, define:
\begin{equation}\label{adoua10}
f_{a,b}(\mathbf{x}):=b\;f\left (\frac{x_1-a}{b}\right )
\end{equation}
where $f$ is a real $C_0^{\infty}(\mathbb{R})$ cut-off function satisfying
$0\leq f(y)\leq 1$ and
\begin{displaymath}
f(y)=\left\{\begin{array}{rlc}
1&\mbox{for}&\vert y \vert \leq \frac{1}{2}\\
0&\mbox{for}&\vert y \vert \geq 1\end{array}\right. .
\end{displaymath}
Define the function $h_{a,b}$ by:
\begin{equation}\label{hab}
h_{a,b}(\mathbf{x}):=x_1-a+if_{a,b}(\mathbf{x}).
\end{equation}
Note that by construction, $ h_{a,b}$ only depends on $x_1$, and obeys:
\begin{equation}\label{invh}
\vert h_{a,b}(\mathbf{x}) \vert \geq \frac{b}{2}.
\end{equation}
Moreover, its first two derivatives are uniformly bounded:
\begin{equation}\label{derh}
\sup_{\mathbf{x} \in\mathbb{R}^3}\sup_{a\in {\mathbb{R}}}\sup_{b\geq 1} \lbrace \vert
\nabla h_{a,b}(\mathbf{x})\vert + \vert \Delta h_{a,b}(\mathbf{x})
\vert \rbrace = K< \infty.
\end{equation}
The eigenvalue equation for $W_{g,j}$ reads as
$P_0 (\hat{X}_{\parallel}-g)P_0 W_{g,j} = 0$. Using \eqref{hab} it can be rewritten as:
\begin{equation}\label{meee}
P_0 h_{g,b}P_0 W_{g,j}=iP_0 f_{g,b}P_0 W_{g,j}.
\end{equation}
We now prove that $P_0 h_{g,b}P_0$ is invertible.
Like in the proof self-adjointness of $\hat{X}_{\parallel}$ we compute
\begin{equation}\label{par}
P_0 h_{g,b}^{-1}P_0 P_0 h_{g,b}P_0 =
1_{\cal K}+P_0 h_{g,b}^{-1}\left[P_0, h_{g,b}\right]P_0.
\end{equation}
The key remark is that $\left[P_0, h_{g,b}\right]$ is bounded. Indeed
we have the identity:
\begin{align}\label{P0h}
\left[P_{0},h_{g,b}\right]&=
-\frac{1}{2\pi }\int_{\Gamma}(H-z)^{-1}
\left\{ \mathbf {P} \cdot \nabla h_{g,b} +
\nabla h_{g,b}\cdot \mathbf {P} \right\}(H-z)^{-1}dz\nonumber \\
&=
-\frac{1}{2\pi }\int_{\Gamma}(H-z)^{-1}
\left\{ -i\Delta h_{g,b} +
2\nabla h_{g,b}\cdot \mathbf {P} \right\}(H-z)^{-1}dz.
\end{align}
It follows that
$\left[P_0, h_{g,b}\right]$ is uniformly bounded in $g\in{\mathbb{R}}$ and
$b\geq 1$ (see \eqref{derh}).
Taking into account \eqref{invh} one obtains that the operator
\begin{equation}\label{B}
\hat{B}_{g,b}=P_0 h_{g,b}^{-1}\left[P_0, h_{g,b}\right]P_0
\;\;:\;\;{\cal K}\rightarrow{\cal K}
\end{equation}
satisfies
\begin{equation}\label{B12}
\Vert\hat{B}_{g,b}\Vert \leq \frac{1}{2}
\end{equation}
if $b\geq b_0$ for some large enough $b_0 <\infty$.
It follows that $1+\hat{B}_{g,b}$ is invertible and then the
eigenvalue equation (see \eqref{meee}, \eqref{par} and \eqref{B})
takes the form
\begin{equation}\label{eef}
W_{g,j}=i \left(1+\hat{B}_{g,b}\right)^{-1}
P_0 h_{g,b}^{-1}P_0 f_{g,b}P_0 W_{g,j}
\end{equation}
which is the analog of \eqref{Svp}. By construction (see the
definition of $f_{g,b}$ in \eqref{adoua10}):
\begin{displaymath}
\Vert e^{\alpha_{\parallel} \langle X_{\parallel, g}\rangle}f_{g,b}\Vert \leq
be^{\alpha_{\parallel} (b+1)}.
\end{displaymath}
Moreover,
$$
e^{\alpha_{\parallel} \langle X_{\parallel, g}\rangle}P_0 h_{g,b}^{-1}P_0
e^{-\alpha_{\parallel} \langle X_{\parallel, g}\rangle}=
\left \{e^{\alpha_{\parallel} \langle X_{\parallel, g}\rangle}P_0 e^{-\alpha_{\parallel} \langle X_{\parallel, g}\rangle}\right \}
h_{g,b}^{-1}\left \{ e^{\alpha_{\parallel} \langle X_{\parallel, g}\rangle}P_0
e^{-\alpha_{\parallel} \langle X_{\parallel, g}\rangle}\right \}
$$
is bounded due to \eqref{expP0}. Thus the only thing it remains to be
proved is the existence of a $b$ large enough such that the following
bound holds:
\begin{equation}\label{atreia10}
\sup_{g\in \mathbb{R}}\left \Vert e^{\alpha_{\parallel} \langle X_{\parallel, g}\rangle} \left(1+\hat{B}_{g,b}\right)^{-1}
e^{-\alpha_{\parallel} \langle X_{\parallel, g}\rangle}\right \Vert <\infty.
\end{equation}
Using the Neumann series for $\left( 1+\hat{B}_{g,b}\right)^{-1}$,
it follows that it suffices to prove that
\begin{equation}\label{B=0}
\lim_{b \rightarrow \infty}\sup_{g\in\mathbb{R}}\left \Vert
e^{\alpha_{\parallel} \langle X_{\parallel, g}\rangle} \hat{B}_{g,b} e^{-\alpha_{\parallel} \langle X_{\parallel, g}\rangle}\right \Vert=0.
\end{equation}
Since (see \eqref{invh}) $\lim_{b \rightarrow \infty}\Vert
h_{g,b}^{-1}\Vert=0$
(uniformly in $g \in \mathbb{R}$), for \eqref{B=0} to holds true it is
sufficient to show:
\begin{equation}\label{apatra11}
\sup_{g\in\mathbb{R}}\left \Vert e^{\alpha_{\parallel} \langle X_{\parallel, g}\rangle}\left[P_{0},h_{g,b}\right] e^{-\alpha_{\parallel}
\langle X_{\parallel, g}\rangle}\right \Vert \leq {\rm const}.
\end{equation}
But this easily follows from \eqref{P0h}, \eqref{derh},
\eqref{prima11} and \eqref{adoua11} where we take $K=H$, $\alpha_z=\alpha_{\parallel}$
and $ h=g_g$. The proof of (iii) is concluded.
\vspace{0.5cm}
\noindent {\it Proof of} (iv). We start with a technical result:
\begin{lemma}\label{proiectorHS}
Fix $0\leq \alpha_{\perp}<\sqrt{E_+}$. Then there exists a bounded operator $D$ such that
\begin{equation}\label{asasea11}
P_0= e^{-\alpha_{\perp} \langle X_{\perp}\rangle}(\mathbf {P}^2 +1)^{-1}D
\end{equation}
\end{lemma}
\noindent{\it Proof}. We use the notation and ideas of Proposition \ref{locP0}, and
we rewrite $P_0$ in a convenient form.
First, for $R>0$ we have
$$
(H-z)^{-1}=
(H_R-z)^{-1}-(H_R-z)^{-1}\chi_R V(H-z)^{-1}.
$$
Second, choose $\Gamma$ close enough to $\sigma_0$ and $R$ large
enough, such that $(H_R-z)^{-1}$ becomes analytic
inside $\Gamma$ and \eqref{acincea6} holds true for all $z
\in \Gamma$. Then we can write:
\begin{align}\label{P01}
&P_0=-e^{-\alpha_{\perp} \langle X_{\perp}\rangle}\frac{i}{2\pi}\int_{\Gamma} \\
&(H_R-z)^{-1}[1+\alpha_{\perp} (i\mathbf {P}\cdot \nabla g_{\perp} +i \nabla g_{\perp}\cdot \mathbf {P}
-\alpha_{\perp} |\nabla g_{\perp}|^2)
(H_R-z)^{-1}]^{-1}e^{\alpha_{\perp} g_{\perp}}\chi_R V(H-z)^{-1} dz.\nonumber
\end{align}
Now by the closed graph theorem we have that $(\mathbf {P}^2+1)(H_R+1)^{-1}$ is
bounded (here $R$ is large enough such that $(-\infty, -1/2)\subset
\rho(H_R)$),
and together with the spectral
theorem:
$$\sup_{z\in\Gamma}\Vert (\mathbf {P}^2+1)(H_R-z)^{-1}\Vert <\infty.$$
Use this in \eqref{P01} and we are done.
\qed
\vspace{0.5cm}
We now have all the necessary ingredients for proving the last
statement of our theorem. For every $L>0$ and $a\in \mathbb{R}$, denote by
$\chi_{L,a}$ the characteristic function of the slab
$\{ \mathbf{x}:\; |x_1 -a| \leq L \}$. Then define the operator
$B:=\chi_{L,a}P_0$. Using \eqref{asasea11} let us show that $B$ is
Hilbert-Schmidt, and moreover, uniformly in $a\in
\mathbb{R}$ we have:
\begin{equation}\label{prima100}
\Vert B \Vert _2^2 \leq M\cdot L,
\end{equation}
for some $M< \infty$. Indeed, since $
B= \chi_{L,a}e^{-\alpha_{\perp} \langle X_{\perp}\rangle}(-\Delta
+1)^{-1}D$,
a direct computation using the explicit formula for the integral
kernel of the free Laplacian gives:
$$
\Vert \chi_{L,a}e^{-\alpha_{\perp} \langle X_{\perp}\rangle}(\mathbf {P}^2
+1)^{-1}\Vert ^2_2 \leq {\rm const}\cdot L.
$$
It follows that the operator
$
\chi_{L,a}P_0 \chi_{L,a}= BB^*
$
is trace class and
\begin{equation}\label{LaP}
\left \vert {\rm Tr} (\chi_{L,a}P_0\chi_{L,a})\right \vert \leq \Vert B\Vert_2^2\leq
M\cdot L
\end{equation}
for some $M< \infty$ independent of $L$ and $a$.
Now let $P_0^{L,a}$ be the orthogonal projection onto the subspace spanned
by those $W_{g,j}$ for which $g\in [a-L,a+L]$:
\begin{equation}\label{adoua100}
P_0^{L,a}:= \sum_{|g-a|\leq L}\sum_{j=1}^{m_g}\langle \cdot,
W_{g,j}\rangle W_{g,j}.
\end{equation}
We can choose $A$ sufficiently large such that \eqref{parW} implies:
\begin{equation}\label{apatra100}
\int _{|x_1-a| \geq A} | W_{g,j}(\mathbf{x} )|^2 d\mathbf{x} \leq \frac{1}{2},
\end{equation}
uniformly in $a$ and $g\in [a-L,a+L]$. Since $P_0 \geq P_0^{L,a}$, from \eqref{LaP} one obtains:
\begin{align}\label{atreia100}
M\cdot (L+A)&\geq {\rm Tr} (\chi_{L+A,a} P_0 \chi_{L+A,a})\geq
{\rm Tr} (\chi_{L+A,a} P_0^{L,a} \chi_{L+A,a})\nonumber \\
&=\sum_{|g-a|\leq L}\sum_{j=1}^{m_g}\int_{\mathbb{R}^3}\chi_{L+A,a}(\mathbf{x})\vert
W_{g,j}(\mathbf{x})\vert ^2 d\mathbf{x} \nonumber \\
&\geq
\sum_{|g-a|\leq L}\sum_{j=1}^{m_g}\frac{1}{2}=\frac{1}{2}N(a, L),
\end{align}
where in the last inequality we used \eqref{apatra100}.
In particular, if $L \geq 1$, then uniformly in $a\in
\mathbb{R}$ we have
$$
N(a, L) \leq 2M\cdot (1+A)L
$$
and the proof is finished.
\subsection{Proof of Proposition \ref{optimal}}
The only thing we have to prove is that (\ref{apatra11}) holds
true for $\alpha_{\parallel}$ replaced by any $\alpha<\alpha_0$, where
$\alpha_0$ is the a-priori given, "exact" exponential localization.
We introduce the multiplication operator given by
$\{e^{\alpha|\cdot-t|}f\}(\mathbf{x}):=e^{\alpha|x_1-t|}f(\mathbf{x})$.
We start by noticing that due to the bound $
e^{\pm \alpha(\sqrt{s^2 +1}-\vert s \vert)} \leq e^{\alpha}$ we can
replace \eqref{decP} with:
\begin{equation}\label{decPP}
\sup_{t\in \mathbb{R}}\Vert e^{\alpha|\cdot-t|} P_0 e^{-\alpha |\cdot -
t|}\Vert <\infty.
\end{equation}
The same replacement can be done in \eqref{apatra11}. Now the integral
kernel $\mathcal{A} (\mathbf{x},\mathbf{y})$ of the operator $ A:=e^{\alpha |\cdot -g|}\left[P_{0},h_{g,b}\right]
e^{-\alpha |\cdot-g|}$ equals
\begin{equation}\label{ultima10}
\mathcal{A} (\mathbf{x},\mathbf{y})=
\mathcal{P}_0(\mathbf{x},\mathbf{y})e^{\alpha(|x_1-g| -|y_1-g|)}(h_{g,b}(\mathbf{y})-h_{g,b}(\mathbf{x})).
\end{equation}
We consider $A$ as an operator on $L^2(\mathbb{R}^3)=\bigoplus_{p\in
\mathbb{Z}}L^2([p,p+1]\times \mathbb{R}^2)$. Let $\chi_p$ be the characteristic
function of the slab $[p,p+1]\times \mathbb{R}^2$. We have that $A_{pp'}:=\chi_pA\chi_{p'}$ is a
bounded operator between $L^2([p',p'+1]\times \mathbb{R}^2)$ and $L^2([p,p+1]\times \mathbb{R}^2)$, and we can write
$A=\{A_{pp'}\}_{p,p'\in \mathbb{Z}}$. We will bound the norm of $A$ with a
Schur-Holmgren type estimate (see below Lemma \ref{schurholmgren}):
\begin{equation}\label{margy222}
||A||\leq
\left (\sup_{p'\in \mathbb{Z}}\sum_{p\in\mathbb{Z}}||A_{pp'}||\right )^{\frac{1}{2}} \left (\sup_{p\in
\mathbb{Z}}\sum_{p'\in\mathbb{Z}}||A_{pp'}||\right )^{\frac{1}{2}} .
\end{equation}
For $0\leq x_1,y_1\leq 1$, the kernel of $A_{pp'}$ can be written as:
\begin{align}\label{margy223}
\mathcal{A}_{pp'} (\mathbf{x},\mathbf{y})&=\mathcal{P}_0(x_1+p,\bx_{\perp};y_1+p',\by_{\perp})e^{\alpha(|x_1+p-g|
-|y_1+p'-g|)}
(h_{g,b}(y_1+p')-h_{g,b}(x_1+p))\nonumber \\
&=\mathcal{P}_0(x_1+p,\bx_{\perp};y_1+p',\by_{\perp})e^{\alpha(|x_1+p-g|
-|y_1+p'-g|)}
(h_{g,b}(p')-h_{g,b}(p))\nonumber \\
&+\mathcal{P}_0(x_1+p,\bx_{\perp};y_1+p',\by_{\perp})e^{\alpha(|x_1+p-g|
-|y_1+p'-g|)}
(h_{g,b}(y_1+p')-h_{g,b}(p'))\nonumber \\
&+\mathcal{P}_0(x_1+p,\bx_{\perp};y_1+p',\by_{\perp})e^{\alpha(|x_1+p-g|
-|y_1+p'-g|)}
(-h_{g,b}(x_1+p)+h_{g,b}(p))\nonumber\\
&=:\mathcal{A}_{pp'}^{(1)} (\mathbf{x},\mathbf{y})+\mathcal{A}_{pp'}^{(2)} (\mathbf{x},\mathbf{y})+\mathcal{A}_{pp'}^{(3)} (\mathbf{x},\mathbf{y}).
\end{align}
The last two kernels can be analyzed with the same methods as the
first one, thus we only estimate the norm of $A_{pp'}^{(1)}$. The
crucial observation is that we can write this operator as a product of
three operators having the corresponding kernels:
\begin{align}\label{margy224}
&\mathcal{A}_{pp'}^{(1)} (\mathbf{x},\mathbf{y})=e^{\alpha(|x_1+p-g|-|p-g|)}\nonumber \\
&\cdot e^{\alpha(|p-g|
-|p'-g|)} \mathcal{P}_0(x_1+p,\bx_{\perp};y_1+p',\by_{\perp})(h_{g,b}(p')-h_{g,b}(p))\nonumber \\
&\cdot e^{-\alpha(|y_1+p'-g|-|p'-g|)}.
\end{align}
The kernel in the middle corresponds to the operator
$\chi_pP_0\chi_p'$ times some coefficients depending on $p,p'$.
Using the triangle inequality to bound the exponentials, and
(\ref{derh}) in order to write $
\vert h_{g,b}(\mathbf{y})-h_{g,b}(\mathbf{x})\vert \leq K \vert x_1-y_1 \vert
$, we have:
$$ ||A_{pp'}^{(1)}||\leq K e^{2\alpha} e^{\alpha |p-p'|}|p-p'|\cdot
||\chi_pP_0\chi_p'||.$$
Using $t=p'$ and $(\alpha+\alpha_0)/2$ in \eqref{decPP} we obtain
$$||\chi_pP_0\chi_p'||\leq C e^{-(\alpha+\alpha_0)|p-p'|/2},$$
thus
$$||A_{pp'}^{(1)}||\leq C' |p-p'|e^{-(\alpha_0-\alpha)|p-p'|/2}$$
which is summable in the sense of \eqref{margy222}.
The same strategy can be applied in the case of $A_{pp'}^{(2)}$ and
$A_{pp'}^{(3)}$. The last thing to be done is to prove the
Schur-Holmgren estimate:
\begin{lemma}\label{schurholmgren}
The estimate \eqref{margy222} holds true.
\end{lemma}
\begin{proof}
Let $\psi\in L^2(\mathbb{R}^3)$ with compact support and $||\psi||=1$. We write:
\begin{align}\label{kkjja}
||A\psi||^2=\sum_{p\in \mathbb{Z}}||\chi_pA\psi||^2.
\end{align}
But
\begin{align}\label{kkjja2}
||\chi_pA\psi||&\leq \sum_{p'\in
\mathbb{Z}}\sqrt{||A_{pp'}||}\;\sqrt{||A_{pp'}||} \;||\chi_{p'}\psi||\leq \left \{ \sum_{p'\in
\mathbb{Z}}||A_{pp'}||\right\}^{\frac{1}{2}} \left \{ \sum_{p'\in
\mathbb{Z}}||A_{pp'}||\; ||\chi_{p'}\psi||^2\right\}^{\frac{1}{2}}\nonumber
\\
&\leq \left \{ \sup_{s\in \mathbb{Z}}\sum_{t\in
\mathbb{Z}}||A_{st}||\;\right\}^{\frac{1}{2}}\left \{ \sum_{p'\in
\mathbb{Z}}||A_{pp'}||\; ||\chi_{p'}\psi||^2\right\}^{\frac{1}{2}}
\end{align}
where in the second inequality we used Cauchy-Schwarz with respect to
$p'$. Introduce this in \eqref{kkjja} and the bound follows after the
use of $\sum_{p'\in
\mathbb{Z}}||\chi_{p'}\psi||^2=1$.
\end{proof}
\vspace{1cm}
\noindent{\bf Acknowledgements.} Part of this work was done during a visit of G. Nenciu
at the Department of Mathematical Sciences, Aalborg University;
both hospitality and financial support are gratefully acknowledged.
H. Cornean acknowledges support from Danish F.N.U. grant
{\it Mathematical Physics and Partial Differential Equations}.
A. Nenciu and G. Nenciu were partially supported by CEEX Grant 05-D11-45/2005.
We also thank the first referee for his/hers most valuable comments regarding Proposition \ref{optimal}.
|
1,116,691,500,896 | arxiv | \section{Introduction}
\label{sec:introduction}
\setcounter{footnote}{0}
Early numerical investigations of the formation of the first stars in metal-free
cosmic halos arrived at the conclusion that the stars were isolated and rather massive, with masses
$\gtrsim 100\,M_\odot$ \citep{Bromm:99,Bromm:02,Abel:00,Abel:02,Gao:07,OShea:07}.
This was a theoretically attractive possibility, given that such
massive stars would have unique radiative, nucleosynthetic, and other
signatures. For example, a metal-free $200\,M_\odot$ star could emit
$\sim 2.6\times10^{50}$ ionizing photons per second
\citep[e.g.,][]{BKL:01,Schaerer:02}, which would photoionize not only
the star's parent cosmic minihalo, but a larger region of
the primordial intergalactic medium (IGM). The star could explode as a
$\sim 10^{53}\,\textrm{ergs}$ pair-instability supernova (PISN), which
could synthesize its own mass worth
of metals. The baryonic content of the halo would
be disrupted \citep{Bromm:03b} and the metals would be dispersed far
outside, thus enriching a relatively large region of the surrounding
medium \citep[e.g.,][]{Wise:08b,Greif:10,Maio:10,Wise:12}. While this
enrichment would have only a minor effect on the ultimate metal budget of
the IGM \citep{Scannapieco:02,Yoshida:04}, it would
have the potential to raise the metallicity of larger collapsing halos
above a ``floor'' thought to be necessary for fragmentation into
low-mass stars \citep{Bromm:03a,Smith:08}.
Much of the theoretical work on the formation of the
first stars and galaxies \citep[e.g.,][]{Wise:08a,Wise:12,Greif:10}, and on strategies for detecting metal-free
stellar populations in the observations of galaxies and supernovae \citep[e.g.,][]{Schaerer:03,Johnson:09,Raiter:10,Zackrisson:11}, has postulated the massive
character of metal-free stars.
Recent numerical investigations, however, have shown that angular-momentum-induced
fragmentation \citep{Stacy:10,Clark:11} and radiative termination of
accretion \citep[][see, also, \citealt{McKee:08}]{Hosokawa:11,Stacy:12} during protostellar accretion
could limit the masses of the first stars to as low as
a few tens of solar masses, and could yield initial protostellar mass functions covering the
entire range $0.1-10\,M_\odot$ \citep{Greif:11}. The final stellar mass range, at the
end of protostellar accretion, could be
$\sim 1-50\,M_\odot$ with a characteristic mass of the
order of $10\,M_\odot$. The cosmic imprint of such
moderate mass first stars should be different than that of the very massive
ones:
the ionizing luminosities could be $\lesssim
\textrm{few}\times 10^{49}\,\textrm{photons}\,\textrm{s}^{-1}$ \citep{Schaerer:02}, supernova energies
could be $\sim10^{51}\,\textrm{ergs}$, metal yields could be
at most a few solar masses per supernova, and the characteristic nucleosynthetic
return could potentially be more consistent with currently available
observational data
\citep[e.g.,][]{Tumlinson:06,Vangioni:11}.
The evolution of
minihalos containing only moderate mass stars should differ in critical ways from
that of objects containing very massive Population III
stars. With one-dimensional hydrodynamic simulations,
\citet{Whalen:08} showed that the \ion{H}{2} region of a
moderate mass star at the center of the minihalo may remain confined
within the minihalo. They calculated that the blastwave of a $10^{51}\,\textrm{ergs}$
supernova in $\lesssim 10^6\,M_\odot$ minihalos, does, after $\sim 10\,\textrm{Myr}$,
reach the virial radius of the halo, and concluded that this results in
the ``disruption'' of the baryonic component of the halo. They also
considered high-mass $\sim 10^7\,M_\odot$ minihalos in which a strong
photodissociating background had prevented star formation
\citep[e.g.,][]{OShea:08}, and found that there, the \ion{H}{2} region
may not form and the supernova blastwave may be
confined at very small radii. They speculated that the subsequent
prompt fallback of metal-enriched supernova ejecta into the center of the
halo could lead to the formation of a globular-cluster-like stellar
system, similar to the suggestion of \citet{Wada:03} that a relatively
small
number of supernovae exploding in a more massive, $\sim10^8\,M_\odot$
cosmic halo could also lead to a fallback and prompt star
cluster formation. This is a particularly interesting possibility, in view of the
unexplained origin of metal-poor globular clusters \citep{Brodie:06}: do they form in
the centers of their own low-mass dark matter halos \citep[e.g.,][]{Peebles:84,Rosenblatt:88,BrCl:02,Boley:09,Griffen:10}, or do
they form
off-center in the merging and fragmentation of more massive
protogalactic disks \citep[e.g.,][and references therein]{Kravtsov:05}?
The aim of this
paper is to pursue implications of the hypothesis that the first
stars had moderate masses---masses that are
insufficient to produce extraordinarily extended \ion{H}{2} regions and ultra-energetic
supernovae. It will be necessary to
understand the entire evolution of supernova ejecta, from the point of
injection in the supernova explosion until some fraction of the
ejecta has become incorporated in new stars. Existing treatments of
the initial phases, including metal injection in a supernova, the subsequent
excursion into the environment, and a possible fallback in the host
halo, have entailed one-dimensional simulations
\citep{Kitayama:05,Whalen:08} or relatively low-resolution
three-dimensional, grid-based or smoothed particle hydrodynamic
simulations \citep{Nakasato:00,Bromm:03b,Greif:07,Wise:08b}. These treatments
point to either a prompt and widespread dispersal of the supernova
ejecta into the primordial IGM---assuming that the
supernovae were extraordinarily energetic---or, in the case of particularly
high-mass halos hosting moderate mass stars, to a possible suppression of \ion{H}{2}
region breakout and the subsequent trapping and prompt fallback of the
ejecta.
It is important to
realize, however, that the fate of the supernova ejecta may be
influenced by dynamical processes operating on relatively small
spatial scales and strongly violating spherical symmetry. In addition
to the likely intrinsic asphericity of the explosion itself, one
such process is the Rayleigh-Taylor instability of the decelerating contact
discontinuity after the ejecta have started to interact with the
ambient medium. The ambient medium into which the supernova expands is
bound to be inhomogeneous, especially if the luminosity of the
progenitor stars and other hot stars that have formed with it is
insufficient to completely photoionize the densest clumps remaining
from the progenitor gas cloud. On
the somewhat larger $\sim100\,\textrm{pc}$ scales of the host
minihalo, a pronounced departure from spherical symmetry will arise from
the filamentary geometry of the cosmic web. Such
asymmetry can interact with the expanding blastwave. Further, as an aging
supernova remnant forms a thin, momentum-conserving snowplow shell, the
decelerating shell is itself unstable to the Rayleigh-Taylor
instability and forms fingers. Finally, while the blastwave
instabilities lead to partial mutual entrainment of the ejecta and the shocked
primordial gas, any ejecta-enriched gas falling back toward the center
of the halo, where it may engage in a new round of star formation, may
not necessarily be chemically homogeneous (full chemical mixing requires at
least a few turbulent eddy turnover times which may not be available in
the blastwave).
In view of these complexities, here we report the first results from
a program to study the detailed transport of the nucleosynthetic output
of the first stars, from the point of injection in supernovae, until
at least a fraction of
the new metals have recondensed in a descendent cosmic
halo where they may contribute to the formation of second-generation,
metal-enriched stars. The character of the new star
formation in the descendent halo must depend on the rate of gas inflow
$\dot M_{\rm inf}$
into the nucleus of the halo, where the gas can rapidly cool and
fragment into stars. The cooling rate, and the compressibility of the turbulence that may
be present, are sensitive to the metallicity of the inflowing gas
$Z_{\rm inf}$ and to the degree of metallicity homogeneity in the
gas. We present the results of a cosmological simulation specifically
designed to measure these parameters and thus shed light on the
conditions for the formation of metal-enriched stars in the aftermath
of a Population III supernova, assuming that it exploded with only
$10^{51}\,\textrm{ergs}$ of energy. The simulation shares
some similarities with that of \citet{Boley:09}
and its setup has many common elements with that of
\citet{Wise:12}. The latter work followed the evolution of
a cosmological region of the same size as ours but to much lower redshifts,
with a focus on the statistics of star formation and metal enrichment
across a large number of star-forming halos. Our focus, instead, is on the
detailed, smaller-scale character of metal transport in the aftermath
of a single supernova.
The paper is organized as follows. In Section \ref{sec:simulations},
we describe our numerical algorithm. In Section
\ref{sec:results}, we present our results. In Section
\ref{sec:discussion}, we discuss implications for the formation of metal-enriched
star clusters in the early universe, and in Section \ref{sec:conclusions},
we present our main conclusions. We adopt the cosmological parameters
consistent with the {\it Wilkinson Microwave Anisotropy Probe}
seven-year data \citep{Komatsu:11}. We explicitly indicate whether
comoving or physical length units are utilized; all other quantities,
such as density, are expressed in physical units.
\section{Simulating a Supernova in a Minihalo}
\label{sec:simulations}
To investigate the fate of the first metals injected
into the universe by a moderate mass Population III star, we carry out a
cosmological simulation large enough not only to permit the formation
of such a star, but also to allow the subsequent growth of the
star's host halo, by accretion and merging, into a larger object that
is the potential site of second-generation star formation. The
metals must be inserted early on, before the freely expanding supernova remnant has
started to interact with the primordial circumstellar medium; this is possible thanks
to our ability to drastically increase the local mesh resolution at a specific
time in the simulation---just prior to supernova explosion---and then
gradually degrade the resolution as the remnant expands. To reproduce
the remnant's dynamics correctly, during the progenitor star's life on
the main sequence, we calculate the transfer of the star's
ionizing radiation and the response of the circumstellar medium by the
associated photoheating in the \ion{H}{2} region. The simulation, which we
proceed to describe, is designed to resolve the rudimentary time and
length scales required to assess the impact of the supernova on the
host halo and the nature of the metal-enriched flows in the aftermath
of the supernova: how quickly do baryons return into the center of the
halo (or one of its descendent halos), and how do the
baryonic accretion rate and the metallicity of the accreted baryons evolve?
The description of the simulation is organized as follows.
Section \ref{sec:initial_conditions_gravity} describes the procedure
by which the initial conditions were generated and the gravitational
potential computation. Section \ref{sec:chemistry}
describes the chemical and thermodynamic calculations, with and
without an ionizing source. Section \ref{sec:amr} describes the
strategies we employed to control the mesh refinement in the
simulations. Section \ref{sec:ionization} describes our algorithm for
transfer of ionizing radiation from the first star. Finally, Section
\ref{sec:supernova} describes the insertion of the supernova remnant
and the subsequent tracking of the metal transport in the simulation.
\subsection{Cosmological Initial Conditions and Gravity}
\label{sec:initial_conditions_gravity}
We initialize our simulation in a periodic cosmological box of volume
$1\,\textrm{Mpc}^3$ (comoving) at redshift $z=146$. The initial
conditions are generated with the multiscale cosmological initial
conditions package {\tt GRAPHIC2}
\citep{Bertschinger:01} with two levels of nested refinement on
top of the $128^3$ base grid. The highest refined region had an effective
resolution of $512^3$, corresponding to a dark matter particle mass of
$230\,M_\odot$. While we will focus on the dynamics of an initially $\sim
10^6\,M_\odot$ minihalo at redshift $z\sim 20$, the region of high refinement was centered on the
overdensity that first collapses to form a more massive
$\sim 10^8\,M_\odot$ halo at a lower redshift. The time integration was carried out
with the multipurpose astrophysical adaptive mesh refinement (AMR)
code {\tt FLASH} \citep{Fryxell:00}, version 3.3, employing the direct
multigrid Poisson gravity solver of \citet{Ricker:08}. The AMR
refinement criteria, which we describe in Section \ref{sec:amr} below,
are set to ensure that relevant baryonic processes are sufficiently
well resolved in the simulation. In collapsing regions where baryons
dominate the gravitational potential, the AMR resolution length quickly becomes smaller
than the dark matter particle separation. There, we employ a
mass- and momentum-conserving quadratic-kernel-based softening
procedure developed in
\citet{SafranekShrader:12} which is applied to the dark matter
density variable during mapping onto the computational mesh, to render the dark matter particle
contribution to the gravitational potential smooth on the scale of the
local AMR grid.\\
\subsection{Chemistry and Cooling}
\label{sec:chemistry}
Prior to the initialization of the \ion{H}{2} region, we integrate the full
nonequilibrium chemical network for hydrogen, helium, deuterium, and
their chemical derivatives. The chemical and radiative cooling
updates, which
include, among other processes, the cooling or heating by the cosmic microwave background (CMB),
are operator-split from the hydrodynamic update and are
subcycled within each cell. The chemical
reaction rates are the same as we have
summarized in \citet{SafranekShrader:10}, except that the chemical and
radiative cooling update is isochoric in the present simulation. The
network is integrated with the Bulirsch-Stoer-type semi-implicit
extrapolation method of \citet{Bader:83}. We test
the thermodynamic evolution of the gas in
\citet{SafranekShrader:12}.
After turning on the ionizing source,
which we describe in Section \ref{sec:ionization} below, we
continue to integrate the nonequilibrium chemical network in the
neutral gas, but in the ionized gas, we adopt a scheme in which the
gas temperature and chemical abundances relax exponentially toward a target photoionization
equilibrium. The equilibrium gas temperature and chemical abundances
are functions of the photoionization parameter $\xi\equiv 4\pi F/n_{\rm H}$ only,
where $F$ is the radiative energy flux and $n_{\rm H}$ is the number
density of hydrogen nuclei and are precomputed with
the same chemical network that we integrate in the neutral gas. The
relaxation time scale is chosen to be the true, nonequilibrium thermal time scale
$t_{\rm therm}\equiv (d\ln T/dt)^{-1}$,
defined by the instantaneous local radiative cooling and
photoionization heating rates.
After inserting the supernova remnant, the
gas can contain an arbitrary admixture of metals in addition to the
primordial elements. The radiative point source is removed, thus
discontinuing the
photoionization
equilibrium calculation. We revert to an integration of the nonequilibrium
chemical network for the primordial species and calculate the
ionization state of the metals and their contribution to the cooling
by interpolation from precomputed tables. At temperatures above $8000\,\textrm{K}$,
the collisional ionization equilibrium state of each of the metal species,
including C, N, O, Ne, Mg, Si, S, and Fe, is calculated separately
using methods described in \citet{Gnat:07} and \citet{Gnat:11}. Below
$8000\,\textrm{K}$ but above $200\,\textrm{K}$, we ignore the metallic
contribution to the free electron abundance
and tabulate a single density-independent cooling function (such that
the volumetric radiative loss rate scales with the square of the density), appropriate to a gas at
relatively low densities $\lesssim 10^3\,\textrm{cm}^{-3}$, using the
code {\tt CLOUDY} \citep{Ferland:98}. The tabulated data extend to a
minimum metallicity of $10^{-3}\,Z_\odot$; below this metallicity, we
set the metal cooling rate in this range of temperatures to zero.
Below $200\,\textrm{K}$, we
disable cooling by metal lines altogether; however, because
low-temperature thermodynamics is not the focus of this work, this has
no effect on the forthcoming results. The treatment of primordial species
takes into account the electrons provided by the metals and vice
versa, but we do not include collisional charge exchange reactions between the
primordial species and the metal species. We assume that the metals have solar abundance
ratios. This is an arbitrary, unnatural choice given that our metals
are being produced by a core-collapse supernova, but this choice
should have only a very minor effect on the thermodynamic evolution of
the supernova remnant.
\subsection{Adaptive Mesh Refinement Strategies}
\label{sec:amr}
The simulation is set to adjust the {\tt FLASH} refinement level $\ell$ based on the local baryon
density $\rho_{\rm b}$ to ensure that denser regions are more highly
refined.
The grid separation (cell size) is related to the
refinement level via $\Delta x = 2^{-\ell+1-3} L$, where $L$ is the
size of the computational box.\footnote{The factor of $2^{-3}$ arises from
the subdivision in {\tt FLASH} of each computational block relative to which the
refinement level is defined into $8^3$ cells.} As
gravitational collapse drives the local fluid density $\rho_{\rm b}$
to increasingly higher densities, the local refinement level is
adjusted to satisfy $\rho_{\rm b}< 3\bar \rho_{\rm b} \,
2^{3(1+\phi)(\ell-\ell_{\rm base})}$ at all times, where $\ell_{\rm
base}=5$ is the refinement level corresponding to the base grid of size
$128=2^{\ell_{\rm base}+2}$, and consistent with
\citet{SafranekShrader:12},
we choose the scaling parameter to equal $\phi=-0.3$. We insert the
star particle when the hydrogen number density first exceeds
$1000\,\textrm{cm}^{-3}$; this happens at redshift $z\approx19.7$ and refinement level $\ell=14$,
corresponding to cell size $\Delta x\approx 0.74\,\textrm{pc}$ (physical).
At the end of the star's life and just prior to inserting the young
supernova remnant, we force the simulation to refine further to level
$\ell=21$ at the location of the star particle, corresponding to cell size
$\Delta x\approx 0.006\,\textrm{pc}$ (physical). This allows us to
insert the supernova remnant well within the free-expansion phase,
before the ejecta have started to interact with the circumstellar
environment. Then, we disable automatic derefinement, and allow the
simulation to refine additional computational cells, up to the maximum
level $\ell_{\rm max}$ initially equal to $21$, based on the standard
second derivative test in {\tt FLASH}, but in this case, the second
derivative test is computed using the cell metallicity. This
ensures that the entire region enriched with metals will be resolved at the
refinement level $\ell_{\rm max}$. As the supernova remnant expands,
we gradually and cautiously lower $\ell_{\rm max}$, and thus coarsen the resolution
of the metal-enriched region, to allow us to make ever longer time
steps and integrate the simulation for $40\,\textrm{Myr}$
past the insertion of the supernova.
\subsection{The First Star and Ionizing Radiation Transfer}
\label{sec:ionization}
When the hydrogen number density first exceeds $1000\,\textrm{cm}^{-3}$ near
the center of the collapsing minihalo, we insert a collisionless star
particle.\footnote{This threshold density
for star formation is ad hoc, but consistent with the choice in,
e.g., \citet{Wise:12}.}
The star is free to move in the combined gravitational
potential of the dark matter and baryons but does not contribute to the
potential, i.e., it is treated as massless. We assume that it is an isotropic
source of hydrogen-ionizing radiation and treat the radiation as monochromatic
with photon energy $h\nu=16\,\textrm{eV}$ and photon emission rate
$Q({\rm H})=6\times10^{49}\,\textrm{photons}\,\textrm{s}^{-1}$. This ionizing
photon rate
is about three times as high as has been estimated for a
$40\,M_\sun$ metal-free star without mass loss, $Q({\rm
H},40\,M_\odot)\approx
2\times10^{49}\,\textrm{photons}\,\textrm{s}^{-1}$
\citep{Schaerer:02}. The ionizing photon energy is calibrated to the
latter lower luminosity star.
We deliberately overestimate the ionizing luminosity to allow for the
presence of additional, lower-mass, longer-lived stars also producing additional
ionizing photons in a compact cluster around the primary $40\,M_\odot$
star. Also, the artificially high ionizing luminosity allows the \ion{H}{2}
region to break out more easily at the moderate spatial resolution at which
we insert the star particle and perform ionizing radiation transfer.
\begin{figure}
\includegraphics[width=0.45\textwidth]{f1.pdf}
\caption{The halo mass $M_{200}$ and radius $R_{200}$ as a function of
redshift, over the course of $50\,\textrm{Myr}$. The dotted line marks the formation of the star, and dashed
line the supernova explosion. The steps are an artifact of the method
we use to estimate $R_{200}$.\label{fig:200}}
\end{figure}
\begin{figure*}
\includegraphics[width=0.33\textwidth]{f2a.pdf}
\includegraphics[width=0.33\textwidth]{f2b.pdf}
\includegraphics[width=0.33\textwidth]{f2c.pdf}
\caption{Density-temperature phase plot for the gas within
$1\,\textrm{kpc}$ (physical) from the center of the minihalo at
redshifts, from left to right, $z=19.7$ (right before the insertion
of the star), $z=19.5$ (in the Sedov-Taylor phase of the supernova,
$8\,\textrm{kyr}$ after the explosion),
and at $z=17$ (end of the simulation, $40\,\textrm{Myr}$ after the explosion). By the end of the
simulation, the gas heated by photoionization and the supernova has
cooled and the halo has resumed central gravitational collapse,
albeit at four times higher halo mass and now a nonzero
metallicity. The dashed line shows the CMB temperature.\label{fig:denstemp}}
\end{figure*}
To carry out ionizing radiation transfer, we center a spherical coordinate
system on the current position of the star particle, and pixelize the angular
coordinate using the {\tt HEALPix} scheme of \citet{Gorski:05} with $N_{\rm
pix}=3072$ pixels. Within each angular pixel, we split the radial
coordinate in the range $4\times10^{20}\,\textrm{cm}\leq r < 10^{24}\,\textrm{cm}$
(comoving), corresponding to $0.7\,\textrm{pc}\lesssim r
<17\,\textrm{kpc}$ (physical), into $N_{\rm rad}=103$ logarithmic radial bins. For
each volume element $E_{p,b}$ defined by an angular {\tt HEALPix}
pixel $p$ and
a radial bin $b$, and for each mesh cell $E_c$ defined by the
cell index $c$, we compute the volume of the intersection of the two
elements $V_{p,b;c}\equiv {\rm vol}(E_{p,b} \cap E_c)\geq 0$. In general,
only a fraction of the cell lies within $E_{p,b}$. In this case,
we recursively split the cell into sub-cells via an
octree subdivision procedure and compute the contribution of the subcells lying
within $E_{p,b}$ to the intersection volume; the recursion is continued until a
desired accuracy is achieved. Armed with the intersection volumes, we
compute the average case-$B$ hydrogen recombination rate $\dot
n_{{\rm rec},p,b}$ evaluated under the assumption of full ionization (as in the
standard Str\"omgren calculation)
within each $E_{p,b}$ via
\begin{equation}
\dot n_{{\rm rec},p,b} = \frac{1}{V_{p,b}}\sum_{c} \alpha_{{\rm
rec},B}(T_c) n_c^2 V_{p,b;c} ,
\end{equation}
where $V_{p,b}={\rm vol}(E_{p,b})\approx \sum_c V_{p,b;c}$, $T_c$ and
$n_c$, are, respectively the temperature and density of hydrogen
nuclei within the cell, and $\alpha_{{\rm
rec},B}(T)$
is the hydrogen
recombination coefficient. Within each bin, we compute the
approximate integral of $4\pi r^2 \dot n_{{\rm rec},p}(r)$ along the radial
direction (now replacing the bin index $b$ with the continuous radial
coordinate $r$) and determine the pixel-specific Str\"omgren radius
$R_{{\rm S},p}$ that solves the equation
\begin{equation}
Q ({\rm H}) = \int_0^{R_{{\rm S},p}} 4\pi r^2 \dot n_{{\rm rec},p}(r)
dr .
\end{equation}
We assume that no ionizing radiation penetrates to radii $r\geq
R_{{\rm S},p}$. At smaller radii, we compute the local ionizing
photon number flux (in the units of
$\textrm{photons}\,\textrm{cm}^{-2}\,\textrm{s}^{-1}$) using
\begin{equation}
f_r(r) = \frac{1}{4\pi r^2} \left[Q ({\rm H}) - \int_0^{r}
4\pi r^2 \dot n_{{\rm rec},p}(r') dr' \right] .
\end{equation}
The local ionizing flux is then used to compute the local
photoionization parameter for cell $c$ lying within the pixel $p$ with
a center at a radius $r_c$ via $\xi_p(r)=4\pi h\nu f_p (r_c) /n_c$. Given a
value of the
photoionization parameter, the photoionization
equilibrium temperature of the gas and the chemical abundances within
the cell are evaluated via table lookup (see Section
\ref{sec:chemistry}). The computation is fully parallel and is
repeated at every hydrodynamic time step. We do not compute the radiation pressure from Thomson
scattering and photoionizations. This radiative transfer scheme is not
explicitly photon conserving and is not designed to correctly
reproduce the ionization
front kinematics, but it is sufficiently accurate in the present
context where the sole interest is in the hydrodynamic response of the
gas to photoionization heating and its impact on supernova blastwave dynamics.
\subsection{Supernova and Metal Transport}
\label{sec:supernova}
We insert the supernova in the free-expansion phase, when the remnant
is about $t-t_{\rm SN}=35\,\textrm{yr}$ old. The ejecta are
cold and have a total mass $M_{\rm SN}=40\,M_\odot$ and a total kinetic energy of
$E_{\rm SN}=10^{51}\,\textrm{ergs}$.
The metal yield is assumed to be $Z_{\rm SN}=0.1$ and uniformly premixed in the
ejecta, implying $4\,M_\odot$ of metals. The
velocity of the ejecta is linear in the distance from the center of
the explosion and the density is uniform. The initial radius of the
remnant is chosen such that its radius is $10\%$ of the radius of the
remnant at which the swept up mass equals the ejecta mass, or in this case,
$R_{\rm SN}=0.075\,\textrm{pc}$. The simulation carries out advection
of the absolute metallicity $Z$ by treating it as a passive mass scalar
quantity. This advection is subject to undesirable numerical
diffusion, which is unavoidable in Eulerian solvers of this type
\citep[see, e.g.,][]{Plewa:99}. Therefore, in parallel with passive
scalar advection, we
also insert a number $N_{\rm part}=5\times10^5$ of passive
Lagrangian tracer particles, originally uniformly distributed in the
ejecta. The simulation advances particle positions by
integrating the first-order ordinary differential equation $d{\mathbf
X}_i/dt={\bf v}({\mathbf X}_i)$, where ${\mathbf X}_i$ is the location
of particle $i$, and ${\mathbf v}({\mathbf X}_i)$ is the baryon fluid
velocity, quadratically interpolated from the computational grid, at the location of
the particle. We do not take into account the possibility that
the supernova may be a source of molecules and dust. Both are undoubtedly produced in
the free-expansion phase, but their survival of the reverse shock
hinges on small-scale clumpiness of the ejecta and on other complex, incompletely-understood
physics \citep[see, e.g.,][]{Cherchneff:09,Cherchneff:10}.
\begin{figure*}
\includegraphics[width=0.49\textwidth]{f3a.pdf}
\includegraphics[width=0.472\textwidth]{f3b.pdf}
\caption{Characteristic radii corresponding to Lagrangian mass coordinates
enclosing baryonic masses of, from top to bottom,
$(2.1,\,1.0,\,0.5,\,\cdot\cdot\cdot)\times10^6\,M_\odot$ (left
panel) and ejecta mass fractions, as tracked by passive tracer particles, of $(100\%,\,50\%,\,25\%,\,\cdot\cdot\cdot)$
(right panel), both
as a function of time
since the explosion. The dotted line
shows the virial radius $R_{200}$ of
the host halo. The baryonic radii are computed from the gravitational
potential minimum of the halo. Supernova-driven expansion is evident
in the mass coordinates inside the virial radius, but all mass
coordinates return to infall owing in part to a persistent
filamentary streaming
of fresh, cold baryons from the cosmic web into the halo center. For
ejecta radii, at times
$t<0.02\,\textrm{Myr}$, the location
of the progenitor star is taken as the coordinate
origin; at later times, the origin
is the location of the gravitational
potential minimum; the change of origin is visible as a discontinuity
in the curves. By the end of the simulation, a half of the ejecta
has turned around and is falling back toward the center of the halo. \label{fig:lagrangian_radii}}
\end{figure*}
\section{Results}
\label{sec:results}
\subsection{Thermodynamical Properties}
In Figure \ref{fig:200}, we show the redshift evolution of the radius
$R_{200}$ within which the average halo density equals $200$ times the
critical density, as well as the corresponding enclosed halo mass
$M_{200}$. The star particle is inserted at $z\approx19.7$ and the
supernova at $z\approx 19.5$. The halo radius doubles from $R_{\rm
200}\approx 160\,\textrm{pc}$ to $300\,\textrm{pc}$ in the course of the
simulation, and the halo mass quadruples from $M_{\rm 200}\approx
10^6\,M_\odot$ at star insertion to $4\times10^6\,M_\odot$ at the end
of the simulation.
Figure \ref{fig:denstemp} shows the density-temperature phase diagram
for gas within $1\,\textrm{kpc}$ (physical) from the center of the
halo, immediately prior to the formation of the star particle, after
the supernova explosion, and finally at the end of the simulation when
the halo mass has quadrupled.
Comparison of the thermodynamic profile before stellar insertion and
at the end of the simulation illustrates how the halo relatively
quickly ``recovers'' from the radiative and kinetic impact of the first star and
resumes central gravitational collapse.
Prior to
the photoionization by the star, the archetypical thermodynamic
structure of a minihalo in which gas cools by rovibrational
transitions of molecular hydrogen is evident. Photoionization and
the supernova blastwave temporarily disrupt this structure. The photoionization raises
the temperature of the photoionized gas to $\approx 10^4\,\textrm{K}$, while the explosion
introduces hot gas that cools both adiabatically and radiatively. Because of the
low energy and compactness of the remnant, the cooling by inverse
Compton scattering of the CMB is relatively unimportant, different from the
case of ultra-energetic Population~III supernovae, where this cooling channel dominates the early evolution \citep{Bromm:03b}.
It is worth noting that an
examination of the three-dimensional structure of the thermodynamic
state of the gas shows that the dense, cold gas inside the
filamentary structure feeding from the nearby cosmic web into the
center of the minihalo is self-shielding and remains unaffected by
photoionization.\footnote{The simulation does not properly resolve the
length scales involved in the photoevaporation of dense cloud cores, thus
we are not sure that the simulation has converged in this respect.}
The blastwave also largely avoids these dense
filaments, and they remain a repository of low-entropy gas.
This is easily seen in Figure \ref{fig:projection},
which shows mass-weighted
projected density, temperature, and metallicity $8.5\,\textrm{Myr}$
after the explosion.
Outside of the filaments,
after $40\,\textrm{Myr}$, the effects of photoionization and
the blastwave have largely been erased; the thermal phase structure
within the halo is similar to that of a higher-mass unperturbed
minihalo. The appearance of warmer
($\lesssim 10^4\,\textrm{K}$) gas at low densities reflects the
higher virial mass of the halo at the end of the simulation and
the residual entropy left by the fossil \ion{H}{2} region.
The presence of metals at local metallicities $Z\gtrsim
10^{-5}$ and an enhanced abundance of hydrogen-deuteride
now facilitate more rapid cooling toward low temperatures.
The enhanced cooling is now especially apparent at high densities
$n_{\rm H}\gtrsim 100\,\textrm{cm}^{-3}$ that are found in the central
region. The dense gas is insufficiently well resolved at the end of the
simulation; the resolution limit hinders compression of the fluid to
densities $n_{\rm H}\gtrsim 1000\,\textrm{cm}^{-3}$.
Nevertheless, it is clear that the dense gas resides at temperatures a
factor of a few lower than the gas at the same densities prior to the
insertion of the ionizing source. We do not observe the anticipated
gas cooling to the CMB temperature, either through metal line cooling
or the cooling by hydrogen-deuteride \citep[e.g.,][]{Johnson:06},
because we enforced an artificial
temperature floor at $200\,\textrm{K}$.
\begin{figure*}
\includegraphics[width=\textwidth]{f4.pdf}
\caption{Baryon mass-weighted projection of hydrogen density,
temperature, and fluid-based metallicity in units of the solar
metallicity $8.5\,\textrm{Myr}$ after the
explosion. The circle shows the virial radius of the halo, which is here
$180\,\textrm{pc}$ (physical).\label{fig:projection}}
\end{figure*}
\subsection{Blastwave Evolution and Fallback}
An insight in the rate of recovery of the halo can be obtained from
Figure \ref{fig:lagrangian_radii}, which shows the Lagrangian
radii that track the radial
location of a grid of mass coordinates
of the spherically averaged baryon and
supernova ejecta mass as a function of time from the explosion. It is
worth keeping in mind that the underlying flow is complex and contains simultaneous
inflows and outflows that are averaged over; each of the radii
reflects only the net
displacement of the mass coordinate. The
baryonic Lagrangian radii show expansion, driven by the supernova
blastwave, starting at $\approx 20\,\textrm{kyr}$ at the innermost
radius shown, and appearing at progressively larger mass coordinates
until outward-directed motion becomes evident near the virial radius at $\approx 2\,\textrm{Myr}$ after
the explosion. Each of the radii corresponding to mass coordinates inside
the halo, however, shows a reversal from a net outflow to a net inflow.
The reversal commences at $\approx 400\,\textrm{kyr}$ at the innermost
radius shown, and reaches progressively larger radii until $\approx
20\,\textrm{Myr}$ when the flow at the virial radius
turns around into an inflow. Outside of the virial radius, cosmic infall
persists largely unabated throughout the remnant's history.
An
examination of the evolution of the actual three-dimensional structure
of the baryonic flow (see also the projection in Figure \ref{fig:projection})
reveals that the prompt reversal can only in part be
attributed to a fallback of the mass that the blastwave has swept up
and to which it has
transferred momentum. Instead, the filamentary inflow of dense, cold,
molecular gas from the nearby cosmic web directly into the center of the
minihalo contributes a large fraction of the mass inflow rate. The baryonic
filaments are seen diagonally from top left to bottom right in Figure
\ref{fig:metal_fallback}, which shows a slice of the density field.
Because the filaments survive photoionization and the explosion
largely intact, they can sustain a net inflow into the center of the
halo even when a majority of the ejecta and the swept up mass are
moving outward.
\begin{figure*}
\includegraphics[width=\textwidth]{f5.pdf}
\caption{Metal dispersal and fallback at the end of the simulation,
$40\,\textrm{Myr}$ after the supernova explosion, at $z=17$. The left panel is
$1.1\,\textrm{kpc}$ (physical) wide and centered on the gravitational
potential minimum; the right panel is a $360\,\textrm{pc}$ detail.
The black points are the metal particles in projection, while the
color shows a slice through the physical hydrogen density $n_{\rm H}$.
Metal-bearing Rayleigh-Taylor fingers have a positive radial
velocity and have breached the virial radius of the halo, while a
majority of the metal mass is falling back into the halo center and
remains incompletely mixed with the primordial gas. Density in the
central unresolved core exceeds coverage of the color scale and is $\sim10^4\,\textrm{cm}^{-3}$.\label{fig:metal_fallback}}
\end{figure*}
The Lagrangian radii for the supernova ejecta in Figure
\ref{fig:lagrangian_radii}, right panel, show first signs of
interaction of the ejecta with the circumstellar medium around
$1-10\,\textrm{kyr}$ after the explosion. What follows is a similar
picture of expansion eventually reversing into fallback. Fractions of
$(3.125\%,\, 6.25\%,\, 12.5\%,\, 25\%,\, 50\%)$ of the ejecta mass
turn around and start falling back at around $ (0.2,\,1.0,
\,4.0,\, 8.0,\, 3.0)\,\textrm{Myr}$ after the explosion. The
corresponding turnaround radii are
$(3,\,8,\,30,\,60,\,200)\,\textrm{pc}$.
The outer somewhat less than $\sim 50\%$ of the ejecta passes the virial radius and continues to travel outward until the
end of the simulation. The net outflow of ejecta at $r\gtrsim
R_{200}$ can be contrasted with the net inflow of baryons at the same
radii, underscoring the three-dimensional nature of the flow and an
inadequacy of one-dimensional integrations
in treating the long-term dynamics of supernova remnants in cosmic
minihalos. The spherical symmetry implicit in one-dimensional
integrations forces the ejecta and the swept up circumstellar medium
into a spherical thin shell at late times. It does not allow for the
presence of the dense, low-entropy clouds that enter the halo by
infall from
the filaments of the cosmic web. The expanding shell may pass
and leave behind such clouds.
The spherical symmetry also does not allow radial segregation of the
ejecta as a result of the instability of the decelerating thin
shell.
Rayleigh-Taylor fingering is evident in Figure \ref{fig:projection} at
$8.5\,\textrm{Myr}$ after the explosion. The long term evolution of
the fragmented shell is seen in Figure
\ref{fig:metal_fallback}, left panel, which shows the ejecta tracer particles in
projection, superimposed on a slice of the density field through the
center of the halo. It is also clear in these figures that the blastwave
has expanded biconically perpendicular to the sheet-like baryonic
overdensity deriving from the cosmic web; this pristine gas, unpolluted by
the ejecta, is still able to stream into the halo center. In the right
panel of Figure \ref{fig:metal_fallback}, showing the central
$360\,\textrm{pc}$, pristine baryonic streams arrive diagonally
into the central few tens of parsecs, where they form a two-armed
spiral and join a
self-gravitating central
core. At this point in the simulation, the spatial resolution is
insufficient to resolve the internal structure of the core.
Strikingly, we observe filamentary accretion of ejecta-enriched gas
perpendicular to the pristine sheet. This is the return of the
ejecta from the supernova remnant back into the center of the halo. A
closer examination of the time evolution of the ejecta kinematics
shows that in the snowplow phase, the ejecta are compressed into a thin
shell, as expected. The thickness of the shell is itself not
well resolved, thus we are not in the position to explore its
susceptibility to direct gravitational fragmentation \citep[e.g.,][]{Salvaterra:04,Machida:05,Chiaki:12}.
The Rayleigh-Taylor instability buckles and redistributes momentum in
the shell. The highest momentum fragments continue semi-ballistically
outward, consistent with the observation by \citet{Whalen:08} that the
blastwave of
even a moderate $10^{51}\,\textrm{ergs}$ explosion has enough momentum to
escape a standard minihalo. The low-momentum fragments, however, do not have
sufficient momentum to escape; they turn around, fall back, and
eventually join the central unresolved core. The falling ejecta make
thin multiply-folded streams and filaments, indicating what seems to
be a certain lack of mixing with the
halo gas. The pristine gas accreting through the filaments of the
cosmic web, and the ejecta-enriched gas accreting from the supernova
remnant, meet in the central core. If the core is turbulent---poor
resolution at this stage of the simulation would not allow us to
detect such turbulence---turbulent mixing of the ejecta and some swept
up pristine gas with a much
larger mass of pristine gas would ensue.\\
\subsection{Metal Enrichment}
In Figure \ref{fig:mdot}, we show the net rates of baryon and
metal outflow from or inflow into the central $20\,\textrm{pc}$ of the
halo. Some baryonic inflow is already evident early on, after
$50\,\textrm{kyr}$ from the explosion, and after $\approx 1.5\,\textrm{Myr}$, baryons
transition to permanent inflow. After $\approx 5\,\textrm{Myr}$, the
inflow rate settles at $\dot M_{\rm b}\approx
0.002\,M_\odot\,\textrm{yr}^{-1}$. The metal flow transitions from
outflow to inflow at $\approx 4\,\textrm{Myr}$ after the explosion,
and fluctuates in the range $\dot M_Z\sim
(0.5-5)\times10^{-7}\,M_\odot\,\textrm{yr}^{-1}$. The average asymptotic
absolute metallicity of the inflowing material is
\begin{equation}
\label{eq:metallicity_accreting}
Z_{\rm inflow}= \frac{\dot M_Z}{\dot M_{\rm b}} \sim
10^{-4}\left(\frac{Z_{\rm SN}}{0.1}\right) \sim 0.005 \,Z_\odot
\left(\frac{Z_{\rm SN}}{0.1}\right) ,
\end{equation}
where we are referring to the total metal mass
ignoring fractional abundances of individual metal species.
We proceed to examine the evolution of the metal-mass-weighted
metallicity probability density function (PDF). The metallicity PDF is
computed in three different ways.
In the first, ``fluid-fluid'' method, the metallicity is
obtained by summing the amplitudes of the passive
scalars defining metal abundances
on the computational grid. The metal mass at
metallicity $Z$ is then calculated by computing a volume integral of
$Z\rho$ over the cells with metallicities
in a narrow logarithmic bin containing $Z$. The shortcoming of this
method is that it yields, as a consequence of a well-known shortcoming
of passive scalar transport schemes, nonvanishing, small metallicities even in
many cells that ejecta could not have reached on hydrodynamic
grounds.\footnote{The spurious diffusion can easily be understood by
considering a step-function
compositional discontinuity $Z(x)=H(x)$ in an otherwise uniform medium traveling
in the $x$-direction with velocity $v_x>0$, and let $\Delta x$ be the
grid cell size. The code approximates the solution to the transport
equation $\partial Z/\partial t+v\partial Z/\partial x=0$ by computing
cell boundary fluxes with the piecewise-parabolic method; here, we adopt a simpler
scheme after assuming that just to the right of the discontinuity, the
metallicity of every cell is much smaller than that of the cell
immediately to the left. Let $\Delta t$ denote the computational
time step and $\delta\equiv v \Delta t/\Delta x$ the
dimensionless Courant
parameter for fluid velocity, and assume that the discontinuity is slow, $\delta\ll 1$.
Then it is straightforward to show that after $t/\Delta t$ steps,
the metal-free region to the left of the traveling discontinuity will
develop a spurious tail of nonzero metallicities of
the asymptotic form
\begin{equation}
Z(x) \sim { t/\Delta t \choose x/\Delta x } \, \delta^{x/\Delta x} ,
\end{equation}
where the first factor is the binomial coefficient. The tail
establishes itself with an unphysical speed $\sim \Delta x/\Delta t \gg v$.} With this in mind,
the first method is more accurate where the metallicity is near its
peak value, and is highly inaccurate elsewhere. Our remaining two methods
for computing the metallicity PDF are designed to alleviate the impact
of spurious diffusion.
In the second,
``fluid-particle'' method, the metallicity $Z$ of a computational cell is calculated in the same
way as in the first method, but the metal mass is calculated by summing
the mass of the ejecta tracer particles in the cell,
and multiplying by the ejecta absolute metallicity $Z_{\rm SN}=0.1$. The tracer
particle mass inside a cell is calculated with the cloud-in-cell method, with
the weighting function taken to correspond to spreading the particle
mass over a cubical kernel identical to the host cell, but centered on
the particle.
In the second method, only the cells with tracer particle presence
contribute to the metallicity PDF. It should be kept in mind,
however, that a metal particle displaced from the center of its host
cell contributes mass to the neighboring cell that may be separated
from the host cell by a shock transition or a contact discontinuity.
\begin{figure}
\includegraphics[width=0.49\textwidth]{f6.pdf}
\caption{Metal (lower curve; $\dot{M}_Z$) and total baryon (upper
curve; $\dot{M}_{\rm b}$) net mass
flow rate through a sphere of radius $20\,\textrm{pc}$ centered on
the gravitational potential minimum.
Dashed lines indicate outflows and solid lines inflows. The total mass of ejecta in the
simulation is
$M_{\rm SN}=40\,M_\odot$ and the metal mass is $M_Z=Z_{\rm SN}M_{\rm
SN}=4\,M_\odot$. Net outflow reverses into an inflow earlier in the
baryons because of the presence of cold filaments delivering
metal-free gas from the cosmic web into the halo center.\label{fig:mdot}}
\end{figure}
\begin{figure*}
\includegraphics[width=0.49\textwidth]{f7a.pdf}
\includegraphics[width=0.49\textwidth]{f7b.pdf}
\includegraphics[width=0.49\textwidth]{f7c.pdf}
\includegraphics[width=0.49\textwidth]{f7d.pdf}
\caption{Absolute metallicity distribution at four different times after the
supernova. The histograms show the fraction of total metal mass as
a function of absolute metallicity calculated in three different
ways: by computing the metallicity and metal mass from the fluid
variables defined on the AMR grid (black curve), by computing
metallicity from the fluid variables and metal mass from the
Lagrangian particles tracing the supernova ejecta assuming ejecta
metallicity of $Z_{\rm SN}=0.1$ (red curve), and by computing both
metallicity and metal mass with the Lagrangian particles (blue
curve). The blue column in the first histogram shows the initial
metallicity of the ejecta. The peak at $Z\approx (2,\,0.7,\,0.3)\times10^{-4}$ at
$t-t_{\rm SN}=(1.187,\,13.367,\,40.127)\,\textrm{Myr}$ contains the fluid that
has become incorporated in the central unresolved core (please see text for additional
explanation). \label{fig:histograms}}
\end{figure*}
In
the third, ``particle-particle'' method, the metallicity inside the cell is itself computed
by summing up the metal mass inside the cell (again, as in the second
method, computed by adding up tracer particle mass inside the cell
and multiplying by $Z_{\rm SN}$), and dividing this by the sum of the metal
mass and the fluid hydrogen and helium masses in the cell. Since
the combined abundance of hydrogen and helium is near unity, the
third method completely ignores the fluid metallicity as defined by
the passive scalar advected on the computational grid. Ejecta
particles as tracers of metallicity obviously do not suffer from the
same spurious diffusion as the fluid metallicity, but they are
affected by systematic issues of their own. In convergent flows such
as near the insufficiently well resolved snowplow shell, metal particles can get trapped in discontinuous
flow structures, which can produce a spurious \emph{increase} of local
metallicity as computed by the third method. Thus, while the fluid
metallicity is bounded from above by its maximum value, $Z_{\rm
SN}=0.1$, at the point of injection, the particle-based metallicity
can exceed this value and even approach unity.
In Figure \ref{fig:histograms}, we show the metal-mass-weighted
metallicity PDF at four different
times: first, in the Sedov-Taylor phase, and, then at approximately $1$, $13$, and
$40\,\textrm{Myr}$ after the explosion. It is immediately clear that
already at early times, the metallicity PDF exhibits a tail $dM_{\rm
Z}/d\ln Z\propto Z$ extending
toward very low metallicities, where differences between the PDFs computed
with the three methods became more severe. Spurious diffusion
across compositional discontinuities in the fluid-based methods, and
cloud-in-cell smearing in particle-based methods, both contribute
to the tail. We also notice that the particle-particle method has
developed a tail extending into the forbidden region, $Z>Z_{\rm SN}$,
a clear signature of spurious particle trapping.
As the supernova remnant ages, metal mixing in the complex
interior flow, which is numerical in the simulation but can be facilitated
by turbulence in nature \citep[e.g.,][]{Pan:11},
broadens the metallicity peak from its initial
location at $Z\approx Z_{\rm SN}$ to an approximate range $10^{-5}\lesssim Z
\lesssim Z_{\rm max}(t)$, where $Z_{\rm max}(t)$ is a time- and
PDF-extraction-method-dependent maximum absolute metallicity. For fluid-based
metallicity PDFs, we find that $Z_{\rm max}\approx
(0.07,\,0.02,\,0.002)$ at $t-t_{\rm SN}\approx
(1,\,13,\,40)\,\textrm{Myr}$. We also notice a prominent narrow
metallicity peak, built from cells belonging to the central,
unresolved core that is accreting both from the pristine filaments of
the cosmic web and from the ejecta-enriched supernova remnant
fallback. The peak is migrating slowly toward lower metallicities, covering
$Z_{\rm core}\approx (2,\,0.7,\,0.3)\times10^{-4}$ in the same three epochs. The
metallicity of the unresolved core is consistent with the average
metallicity of the accreting fluid given in equation (\ref{eq:metallicity_accreting}).
Finally, we turn to the energetics of the metal-enriched material. In
Figure \ref{fig:energy}, we show the evolution of the total kinetic and
internal (thermal) energy of fluid elements having absolute
metallicity $Z\geq 10^{-6}$. At $1\,\textrm{kyr}$ after the
explosion,
the kinetic energy
decreases below its initial value when the ejecta start interacting with the ambient
medium and the well-resolved reverse shock facilitates kinetic-to-internal
conversion. Adiabatic and radiative cooling both contribute to the
rapid decrease of internal energy at $0.01-1.0\,\textrm{\rm Myr}$
after the explosion. The renewed rise of both energies
in the old remnant reflect the fallback of ejecta-enriched fluid in
the halo center.
\begin{figure}
\includegraphics[width=0.49\textwidth]{f8.pdf}
\caption{The kinetic (blue curve), internal (green curve), and combined
kinetic and internal (red curve) energy of the metal enriched fluid, defined
as the fluid with absolute metallicity $Z\geq 10^{-6}$. \label{fig:energy}}
\end{figure}
\section{Discussion}
\label{sec:discussion}
\subsection{Metal-Enriched Star Formation: Continuous, Bursty, or Self-Limiting?}
The prompt resumption of baryonic infall into the
center of the cosmic minihalo suggests that star formation
could resume on similarly short, $\sim 1-5\,\textrm{Myr}$ time scales. In the simulation
presented here, the metals that
the supernova has synthesized begin returning to the halo center
$\sim4\,\textrm{Myr}$ after the explosion. The mean
metallicity of the inflow, averaged over the
primordial streams and the metal-enriched collapsing supernova remnant streams
converging
in the center of the halo,
is $\sim 0.005\,Z_\odot$ assuming a net supernova metal yield of $4\,M_\odot$. If the
metal-rich gas successfully mixes with the primordial gas, then,
according to a prevailing belief,
the metallicity of the combined streams is well sufficient to ensure that new
star formation in this gas should be producing normal, low-metallicity
Population II stars. The Population III to II transition cannot be
ascertained in the present simulation because the simulation does
not adequately resolve the cold gas at densities $n_{\rm H}\gtrsim
100\,\textrm{cm}^{-3}$. This gas belongs to the central core and is
colder than primordial gas at the same densities preceding the
insertion of the first star. It is at unresolved densities that the
metallicity of the gas strongly affects the thermodynamic
evolution and thus governs the outcome of fragmentation
\citep[e.g.,][assuming low metallicities,
$Z\lesssim 0.01\,Z_\odot$]{Omukai:05,Santoro:06,Schneider:06,Smith:09,SafranekShrader:10}.
The gaseous core in the center of the minihalo that is fed by metal-enriched streams
can be compared to the star-forming core of a
giant molecular cloud (GMC). We attempt such comparison with an eye
on the overall hydrodynamics of the gas flow, putting aside any differences in chemical
composition, dust content, CMB temperature, etc., in the two systems. Dense GMC cores grow by accreting from
larger-scale, lower-density cloud environments which are supersonically
turbulent. Smaller cores and
their embedded forming star clusters merge with each other to form larger
associations. Star formation in the cores
is thought to be self-regulated, with the ``feedback'' from ongoing star
formation controlling the amount of star-forming
gas available and eventually leading to GMC dissolution
\citep[e.g.,][]{Murray:10,Murray:11,Goldbaum:11,Lopez:11}. The
feedback entails, e.g., the pressure of the \ion{H}{2}
regions and the pressure of
direct stellar radiation. While the baryonic mass of a cosmic minihalo is perhaps smaller than that of a
GMC, the centrally concentrated underlying dark matter
halo renders the gravitational potential depth in a minihalo's central
gaseous core similar to that in a
GMC's star-forming core. The underlying dark matter gravitational
potential may make a cosmic minihalo less susceptible to star-formation-induced dissolution
than a GMC, assuming that the two systems are forming stars at the same rate.
GMCs are
thought to sustain star formation over tens of millions of years, in
spite of the \ion{H}{2} regions, supernovae, and other feedback activity that commences
quickly following the formation of the first massive stars. The
feedback's impact preceding the eventual dissolution of the cloud
complex
may be limited to reducing the average star
formation rate. We do not
know whether
star formation in metal-enriched minihalos and their descendent $\gtrsim
10^7\,M_\odot$ halos can be sustained as in GMCs in spite of the
feedback, or is the feedback from second-generation stars so
effective as to evacuate baryons from the halo and prevent further star
formation on cosmological time scales \citep[see, e.g.,][]{Mori:02}. It seems that these two
scenarios can be evaluated only with attention to the detailed,
small-scale mechanics of star formation and its feedback. A
corollary of the first scenario would be the presence of multiple,
chemically distinct
stellar populations in the stellar system that forms this way.
The statistics of low-luminosity stellar systems
in the Local Group in conjunction with theoretical modeling of
structure formation in a $\Lambda$CDM universe
suggests that star formation must have, to some
extent, been suppressed in low-mass cosmic halos \citep[e.g.,][]{Susa:04a,Ricotti:05,Gnedin:06,Moore:06,Madau:08,Bovill:09,Bovill:11,Koposov:09,Munoz:09,Salvadori:09,Busha:10,Gao:10,Griffen:10,Maccio:10,Okamoto:10,Font:11,Lunnan:11,Rashkov:11}. This suppression, however,
could have arisen both internally, as in GMCs,
and externally, where the suppression is driven by the heating of the
IGM by the background
ionizing radiation coming from other, more massive, less common halos
\citep[e.g.,][]{Thoul:96,Kepner:97,Barkana:99,Shapiro:04,Susa:04b,Mesinger:06,Mesinger:08,Okamoto:08}. The
heating could have shut off the inflow of fresh gas into the halo,
eventually stunting star formation. The prevalence of either
suppression mechanism
as a function of halo properties cannot be
understood without first understanding the detailed
mechanics of star formation and feedback following the initial
fallback of Population III supernova ejecta.
\subsection{The Implications of Uncertainties Related to the Nature of First Stars}
A source of uncertainty regarding the
formation and nature of the first metal-enriched star clusters is the
incomplete understanding of the precise character of metal-free
Population III stars, and of the energetics and
nucleosynthetic output of the associated supernovae. Theoretical
investigations of Population III star formation have not yet
converged and the precise functional form of the resulting initial mass function is not yet known. The
evolution of metal-free stars is
very sensitive to their poorly-understood internal rotational structure
starting with the
protostellar phase \citep[see, e.g.,][]{Stacy:11}.
Similarly, the explosion mechanism in metal-poor stars is not known.
Each of the suggested mechanisms, including the standard
delayed-neutrino mechanism hypothesized to drive Type II explosions,
and the pair-instability and pulsational
pair-instability mechanisms expected in relatively massive metal-free
stars \citep[see, e.g.,][and references therein]{Chatzopoulos:12,Yoon:12},
as well as the more exotic ``hypernova''-type mechanisms involving
a (possibly magnetized) outflow from a central compact object, implies
its own characteristic explosion energy and nucleosynthetic
imprint.
Many investigations of the formation of first
stars and galaxies and their role in reionization have explored implications of
the hypothesis that Population III stars had high masses, explosion energies,
and metal yields. We find that the opposite limit of moderate
masses, explosion energies, and metal yields might imply sharply
divergent outcomes,
not only for the parent cosmic minihalos, but also
for the overall pace of transformation of the early universe. A related issue
is the nature of the first galaxies \citep[for a review, see][]{BrYo:11}.
Their assembly process is influenced
by the feedback from the Population~III
stars that formed in the minihalo progenitor systems, and this feedback
in turn sensitively depends on the mass scale of the first stars.
In the case of extremely massive stars, negative feedback is very disruptive,
completely sterilizing the minihalos after only one episode of star
formation. Only after $\sim 10^8\,\textrm{yr}$ did the hot gas sufficiently cool
to enable the recollapse into a much more massive dark matter halo, where
a second-generation starburst would be triggered \citep{Wise:08b,Greif:10}. This separation into
two distinct stages of star formation would be smoothed out in the
case studied here. Indeed, the first galaxies may then already have
commenced their assembly in massive minihalos, as opposed to the atomic cooling halos \citep[e.g.,][]{OhH:02}
implicated in the previous scenario.\\ \\
\section{Conclusions}
\label{sec:conclusions}
We have carried out a cosmological hydrodynamical simulation designed
to investigate the evolution of a cosmic minihalo in the aftermath of
the formation of the first, Population III star, assuming that the
star has a moderate mass and explodes as a moderate-energy Type II-like
supernova, consistent with the recent downward revision of Population
III star mass estimates \citep{Stacy:10,Stacy:12,Clark:11,Greif:11,Hosokawa:11}.
We analyzed the dynamics of supernova ejecta and gas flow
inside the minihalo. These can be compared to similar studies
involving massive Population III stars exploding as ultra-energetic
PISNe. Our main conclusions are as follows.
The moderate-mass star and the moderate-energy supernova it produces
inflict significantly smaller damage to the host minihalo than that
inflicted by a massive star exploding as an ultra-energetic supernova.
The star only partially photoionizes the host
halo. The densest gas flowing from the filaments of the cosmic web
into the star's parent cloud remains neutral and survives
the supernova blastwave. This dense gas resumes accretion into the
center of the minihalo only $\sim1\,\textrm{Myr}$ after the
explosion. After $\sim 20\,\textrm{Myr}$, all spherically-averaged
baryonic
mass coordinates are moving inward and the accretion rate at
the center of the halo is $\sim 0.002\,M_\odot\,\textrm{yr}^{-1}$.
Following instability of the blastwave during the
snowplow phase, a fraction of the supernova ejecta starts falling back to the center of the
halo. The ejecta fallback accretion rate reaches a steady state after
$\sim 5\,\textrm{Myr}$. The accreting ejecta are incompletely mixed with the
primordial gas and are confined in thin sheets and filaments.
Less than a half of the ejecta escape the virial radius; the escaping
ejecta can be traced back to the outermost Rayleigh-Taylor fingers.
The average metallicity of the accreting matter is in the range
$0.001-0.01\,Z_\odot$. This result depends on the assumed supernova
energy and metal yield, but in this study, we did not pursue parameter
space exploration. The metallicity of the accreted gas
is sufficient to ensure that new stars forming in the central core,
unresolved in the simulation, will have even
lower characteristic masses.
These results bring into focus metal-enriched star formation in cosmic minihalos and
their immediate descendent halos. The character of post-supernova
metal transport and gravitational fragmentation of the
metal-enriched gas, and of the impact of subsequent star formation and
supernovae, are both worth addressing in follow-up investigations.
The early universe likely exhibited a range of different explosion settings,
from modest to ultra-energetic. It will be intriguing to see how these
high-redshift systems are matched with possible local fossils, such as
the ultra-faint dwarf galaxies
\citep[e.g.,][]{Bovill:09,Salvadori:09,Tumlinson:10} and metal-poor
globular clusters \citep[e.g.,][]{Brodie:06}. From what we have already learned, Population~III supernova feedback
appears to play a key role in shaping early cosmic history and the
nature of relic stellar systems.
\acknowledgements
We gratefully acknowledge stimulating discussions with J.\
Scalo and helpful technical consultation with A.\ Dubey, C.\ Daley,
T.\ Plewa, P.\ Ricker, and M.\ Ruszkowski. We thank R.\ Banerjee for
providing part of the software used to visualize this work.
This research was supported by NSF grants AST-0708795 and AST-1009928
and NASA ATFP grant NNX09AJ33G. The authors acknowledge the Texas
Advanced Computing Center (TACC) at The University of Texas at Austin
for providing computing and visualization resources that have contributed to the research results reported within this paper. The software used in this work was in part developed by the DOE NNSA-ASC OASCR Flash Center at the University of Chicago.
|
1,116,691,500,897 | arxiv |
\section{Introduction}
The operations performed by message passing Graph Neural Networks (GNNs) emulate the structure of the input graph. While this property has clear computational advantages, it brings with it a series of fundamental limitations. As observed by \citet{GIN} and \citet{morris2019weisfeiler} the local neighbourhood aggregations used by GNNs are at most as powerful as the Weisfeiler-Lehman (WL) test \citep{weisfeiler1968reduction} in distinguishing non-isomorphic graphs. Therefore, GNNs fail to detect certain higer-order meso-scale structures such as cliques or (induced) cycles~\citep{ARVIND202042, chen2020can}, which are particularly important in applications dealing with social and biological networks or molecular graphs. At the same time, many such layers have to be stacked to make long-range interactions in the graph possible. Besides the computational burden this incurs, deep GNNs typically come with additional problems such as over-smoothing \citep{li2018deeper} and over-squashing \citep{alon2020bottleneck} of the node representations.
To address these problems, we propose a novel message passing procedure based on (regular) cell complexes, also known as CW complexes\footnote{We use these terms interchangeably. For the latter, the C stands for ``closure-finite'', and the W for ``weak'' topology. The term was coined by
\citet{bams/1183513543}.}, topological objects that form the building block of algebraic topology \citep{hatcher_book}. When paired with a theoretically-justified ``lifting'' transformation augmenting the graph with higher-dimensional constructs called ``cells'', our method results in a multi-dimensional and hierarchical message passing procedure over the input graph. Our approach generalises and subsumes the recently proposed Message Passing Simplicial Networks (MPSNs) \citep{bodnar2021weisfeiler}, which operate on simplicial complexes (SCs), topological generalisations of graphs. However, SCs have a rigid combinatorial structure that significantly limits the range of lifting transformations one could use to meaningfully modulate the message passing procedure. In contrast, we show that cell complexes, which in turn generalise simplicial complexes and come with additional flexibility, allow one to construct new and better ways of decoupling the input and computational graphs.
\paragraph{Main Contributions} To summarise, we propose a message passing scheme operating on regular cell complexes. We call this family of models CW Networks (CWNs) and study their expressive power using a cellular version of the WL test. We show that for an entire class of ``lifting'' transformations CWNs are at least as powerful as the WL test. Furthermore, we prove that for some of the maps in this class, CWNs can be strictly more powerful than WL, Simplicial WL (SWL) and also not less powerful than 3-WL. We also express the fundamental symmetries of these models and show how they can be seen as generalised convolutional operators on cell complexes. Experimentally, we focus our attention on a particular ``lifting'' map based on induced cycles. When applied to molecular graphs, it leads to an intuitive hierarchical message passing procedure involving the atoms, the bonds between them and the chemical rings of the molecules. We demonstrate that this provably powerful approach obtains state-of-the-art results on popular large-scale molecular graph datasets and other related tasks. To the best of our knowledge, this is the first work proposing a cell complex representation for molecules. Our code is available at \url{https://github.com/twitter-research/cwn}.
\section{Background}
\begin{definition}[\citet{HGh19}]
A \define{regular cell complex} (Figure \ref{fig:cell_def}) is a topological space $X$ together with a partition $\{X_\sigma\}_{\sigma\in P_X}$ of subspaces $X_\sigma$ of $X$ called \define{cells}, and such that
\begin{enumerate}[leftmargin=*,topsep=0pt,itemsep=-0.5ex]
\item For each $x\in X$ there exists an open neighbordhood of $x$ that intersects finitely many cells.
\item For all $\sigma,\tau$ we have that $X_\tau\cap \overline{X_\sigma}\ne \emptyset $ iff $X_\tau\subseteq \overline{X_\sigma}$, where $\overline{X_\sigma}$ is the closure of a cell.
\item Every cell is homeomorphic to $\mathbb{R}^n$ for some $n$.
\item ({\em Regularity}) For every $\sigma\in P_X$ there is a homeomorphism $\phi$ of a closed ball in $\mathbb{R}^{n_\sigma}$ to $\overline{X_\sigma}$ such that the restriction of $\phi$ to the interior of the ball is a homeomorphism onto $X_\sigma$.
\end{enumerate}
\end{definition}
\begin{wrapfigure}{r}{0.35\textwidth}
\centering
\vspace{-15pt}
\begin{subfigure}[b]{1.0\linewidth}
\centering
\includegraphics[width=\textwidth]{figures/definition.jpeg}
\label{fig:y equals x}
\end{subfigure}\hspace{3mm}
\vspace{-23pt}
\caption{A cell complex $X$ and the corresponding homeomorphisms to the closed balls for three cells of different dimensions in the complex.}
\label{fig:cell_def}
\vspace{-20pt}
\end{wrapfigure}
We note that by condition $(2)$ the indexing set $P_X$ has a poset structure $\tau\leq \sigma \Leftrightarrow X_\tau\subseteq \overline{X_\sigma}$,
while condition $(4)$ guarantees that this poset structure encodes all the topological information about $X$. Thus, we can identify a regular cell complex $X$ with this poset, called \define{face poset} of $X$. We also use $\tau < \sigma$ for the strict version of this partial order.
Intuitively, one constructs a cell complex through a hierarchical gluing procedure. One starts with a set of vertices (0-cells). Then edges (1-cells) are attached to these by gluing the endpoints of closed line segments to them. We have now only described a (multi) graph. However, one can generalise this even further by taking a two-dimensional closed disk and glue its boundary (i.e. a circle) to any simple cycle in the (multi) graph previously built as in Figure \ref{fig:gluing}. While we are generally not concerned with dimensions above two, this can be further generalised by gluing the boundary of $n$-dimensional balls to certain $(n-1)-$cells in the complex.
\begin{figure}
\centering
\begin{minipage}{.63\textwidth}
\includegraphics[width=0.48\linewidth]{figures/glue_disks.jpeg}
\hfill
\includegraphics[width=0.48\linewidth]{figures/glued_disks.jpeg}
\caption{Closed two-dimensional disks are glued to the boundary of the rings present in the graph (left). The result is a 2D regular cell complex (right).}
\label{fig:gluing}
\end{minipage}
\hfill
\begin{minipage}{0.33\textwidth}
\centering
\includegraphics[width=0.35\linewidth]{figures/sphere.jpeg}
\hspace{10pt}
\includegraphics[width=0.35\linewidth]{figures/empty_tetrahderon.jpeg}
\caption{A sphere and an empty tetrahedron. The latter is also a simplicial complex.}
\label{fig:1}
\end{minipage}
\vspace{-10pt}
\end{figure}
Consider the examples in Figure \ref{fig:1}. The shown sphere is a cell complex obtained from two 0-cells (i.e.\ vertices), to which two 1-cells (i.e.\ edges), which form the equator, were attached. The boundary of two 2-dimensional disks (i.e.\ the two hemispheres) were glued to the equator to form a sphere. The second example is a tetrahedron with empty interior. It is a particular type of cell complex called a \emph{simplicial complex} (SC). The only 2-cells it allows are triangle-shaped. More generally, the $n$-dimensional cells of SCs are $n$-simplices, which makes them slightly more rigid structures.
\begin{definition}
The $\mathbf{k}$\textbf{-skeleton} of a cell complex $X$, denoted $X^{(k)}$, is the subcomplex of $X$ consisting of cells of dimension at most $k$.
\end{definition}
This definition is useful for referring for certain parts of the complex. For instance, $X^{(0)}$ contains the vertices in the complex, while $X^{(1)}$ contains the vertices \emph{and} the edges (i.e. the underlying graph).
The combinatorial structure of the complex can be more compactly described by an incidence relation we call the \emph{boundary relation}, whose reflexive and transitive closure gives the partial order defined above. The boundary relation describes what cells are on the boundary of other cells. For instance, the edges of the sphere in Figure \ref{fig:1} are on the boundary of the 2-cells forming the two hemispheres.
\begin{definition}
\label{def:boundary_rel}
We have the \textbf{boundary relation} $\sigma \prec \tau$ iff $\sigma < \tau$ and there is no cell $\delta$ such that $\sigma < \delta < \tau$.
\end{definition}
We can use this to define the four types of (local) adjacencies present in cell complexes. These adjacencies will be the fundamental building block of our message passing procedure. To explain these in more familiar terms, for each adjacency, we exemplify how it shows up in graphs.
\begin{definition}[Cell complex adjacencies]
\label{def:adj}
For a cell complex $X$ and a cell $\sigma \in P_X$, we define:
\begin{enumerate}[leftmargin=*, topsep=0pt,itemsep=-0.5ex]
\item The boundary adjacent cells ${\mathcal{B}}(\sigma) = \{ \tau \mid \tau \prec \sigma \}$. These are the lower-dimensional cells on the boundary of $\sigma$. For instance, the boundary cells of an edge are its vertices.
\item The co-boundary adjacent cell ${\mathcal{C}}(\sigma) = \{ \tau \mid \sigma \prec \tau \}$. These are the higher-dimensional cells with $\sigma$ on their boundary. For instance, the co-boundary cells of a vertex are the edges it is part of.
\item The lower adjacent cells ${\mathcal{N}}_{\downarrow}(\sigma) = \{ \tau \mid \exists \delta$ such that $\delta \prec \sigma$ and $\delta \prec \tau \}$. These are the cells of the same dimension as $\sigma$ that share a lower dimensional cell on their boundary. The line graph adjacencies between the edges are a classic example of this.
\item The upper adjacent cells ${\mathcal{N}}_{\uparrow}(\sigma) = \{ \tau \mid \exists \delta$ such that $\sigma \prec \delta$ and $\tau \prec \delta \}$. These are the cells of the same dimension as $\sigma$ that are on the boundary of the same higher-dimensional cell as $\sigma$. The typical graph adjacencies between vertices are the canonical example here.
\end{enumerate}
\end{definition}
\section{Cellular Weisfeiler Lehman}
\paragraph{Overview} The results in this section show how one can transform graphs into higher-dimensional cell complexes in such a way that performing colour refinement on the resulting cell complexes makes it easier to test their isomorphism. The message passing model from Section \ref{sec:CWN_MMP} will take advantage of these theoretical results. All proofs can be found in Appendix \ref{app:proofs_cellular}.
\begin{definition}
Let $c$ be a colouring of the cells in a complex $X$ with $c_\sigma$ denoting the colour assigned to cell $\sigma \in P_X$. Define ${\mathcal{B}}(\sigma, \tau) := {\mathcal{B}}(\sigma) \cap {\mathcal{B}}(\tau)$ and ${\mathcal{C}}(\sigma, \tau) := {\mathcal{C}}(\sigma) \cap {\mathcal{C}}(\tau)$. We define the following multi-sets of colours:
\begin{enumerate}[leftmargin=*, topsep=0pt,itemsep=-0.5ex]
\item The colours of the boundary cells of $\sigma$: $c_{\mathcal{B}}(\sigma) = \{\mskip-5mu\{ c_\tau \mid \tau \in {\mathcal{B}}(\sigma) \}\mskip-5mu\}$.
\item The colours of the co-boundary cells of $\sigma$: $c_{\mathcal{C}}(\sigma) = \{\mskip-5mu\{ c_\tau \mid \tau \in {\mathcal{C}}(\sigma) \}\mskip-5mu\}$.
\item The lower adjacent colours of $\sigma$: $c_\downarrow(\sigma) = \{\mskip-5mu\{ (c_\tau, c_\delta) \mid \tau \in {\mathcal{N}}_{\downarrow}(\sigma)$ and $\delta \in {\mathcal{B}}(\sigma, \tau) \}\mskip-5mu\}$.
\item The upper adjacent colours of $\sigma$: $c_\uparrow(\sigma) = \{\mskip-5mu\{ (c_\tau, c_\delta) \mid \tau \in {\mathcal{N}}_{\uparrow}(\sigma)$ and $\delta \in {\mathcal{C}}(\sigma, \tau) \}\mskip-5mu\}$.
\end{enumerate}
\end{definition}
Note that unlike in graphs and simplicial complexes, the sets ${\mathcal{B}}(\sigma, \tau)$ and ${\mathcal{C}}(\sigma, \tau)$ can have more than one element. For instance, two (closed) 2-cells might intersect in more than one edge (e.g. the two hemispheres in Figure \ref{fig:1}), and conversely, two edges might be on the boundary of the same two 2-cells. This illustrates the more flexible combinatorial structure of cell complexes.
\paragraph{Cellular WL (CWL)} We consider CWL, a colour refinement scheme for cell complexes that generalises the Simplicial WL~\citep{bodnar2021weisfeiler} and WL~\citep{weisfeiler1968reduction} tests. We use $c_\sigma^t$ to refer to the colour assigned by CWL to cell $\sigma$ at iteration $t$ of the algorithm. When the input is a simplicial complex, this recovers the SWL algorithm. A step of the algorithm is graphically depicted in Figure \ref{fig:CWL} for a single cell.
\begin{enumerate}[leftmargin=*]
\item Given a regular cell complex $X$, all the cells $\sigma$ are initialised with the same colour.
\item Given the colour $c_\sigma^t$ of cell $\sigma$ at iteration $t$, we compute the colour of cell $\sigma$ at the next iteration $c_\sigma^{t+1}$ by injectively mapping the multi-sets of colours belonging to the adjacent cells of $\sigma$ using a perfect HASH function: $c_\sigma^{t+1} = \mathrm{HASH}\big(c_\sigma^t, c_{\mathcal{B}}^t(\sigma), c_{\mathcal{C}}^t(\sigma), c_\downarrow^t(\sigma), c_\uparrow^t(\sigma)\big). $
\item The algorithm stops when a stable colouring is reached. Two cell complexes are considered non-isomorphic if their colour histograms are different. Otherwise, the test is inconclusive.
\end{enumerate}
First, we state the following theorem from \citet{bodnar2021weisfeiler} involving SWL and simplicial complexes. This theorem shows that on simplicial complexes, certain adjacencies can be pruned without affecting the non-isomorphic SCs that can be distinguished. This has important computational implications.
\begin{theorem}
SWL without coboundary and lower-adjacencies has the same expressive power in distinguishing non-isomorphic simplicial complexes as SWL with the complete set of adjacencies.
\end{theorem}
It is not immediately clear whether an equivalent theorem would also hold for cell complexes. This is because cells, unlike simplices, can have widely different shapes and, as described above, the adjacencies between them take more complicated forms. Nevertheless, we show that a positive result can be obtained.
\begin{theorem}
\label{thm:sparse_cwl}
CWL without coboundary and lower-adjacencies has the same expressive power in distinguishing non-isomorphic cell complexes as CWL with the complete set of adjacencies.
\end{theorem}
We note this does not mean that the removed adjacencies are completely redundant in practice. Even if they are not needed from a (theoretical) colour refinment perspective, they might still include important inductive biases that make them suitable for certain tasks.
\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{figures/CWL.png}
\caption{The CWL colouring procedure for the yellow edge of the cell complex. All cells have been assigned unique colours to aid the visualisation of the adjacencies. Note that the yellow edge aggregates long-range information from the light green edge.}
\label{fig:CWL}
\end{figure}
We are now interested in examining various procedures for mapping, or ``lifting'', graphs into the space of regular cell complexes. Such a procedure can be used to test the isomorphism of two graphs by performing colour refinement on the cell complexes they are mapped to. The hope is that CWL applied to these cell complexes is more powerful than WL applied to the initial graphs. We will later show that for a wide range of transformations, this is indeed the case. We start by rigorously defining what we mean by a ``lifting''.
\begin{definition}
A \textbf{cellular lifting map} is a function $f: {\mathcal{G}} \to {\mathcal{X}}$ from the space of graphs ${\mathcal{G}}$ to the space of regular cell complexes ${\mathcal{X}}$ with the property that two graphs $G_1, G_2$ are isomorphic iff the cell complexes $f(G_1), f(G_2)$ are isomorphic.
\end{definition}
This property ensures that testing the isomorphism of the two cell complexes is equivalent to testing the isomorphism in the input graphs. This would not be the case if two non-isomoprhic graphs were mapped to the same cell complex.
\begin{example}
\label{ex:clique_complex}
It can be verified that the function mapping each graph to its clique complex (i.e.\ every $(k+1)$-clique in the graph becomes a $k$-simplex) is a cellular lifting map.
\end{example}
The clique complex lifting map from Example \ref{ex:clique_complex} has been used by \citet{bodnar2021weisfeiler} to show that SWL is strictly more powerful than WL. We restate this result:
\begin{theorem}
SWL with clique complex lifting is strictly more powerful than WL.
\end{theorem}
A natural question is what other lifting transformations make CWL strictly more powerful than WL? We first describe a space of lifting transformations that make CWL at least as powerful as WL.
\begin{definition}
A lifting map is \textbf{skeleton-preserving} if for any graph $G$, the $1$-skeleton of $f(G)$ and $G$ are isomorphic as (multi) graphs.
\end{definition}
Intuitively, skeleton-preserving liftings ensure that the additional structure added by the lifting map comes from attaching cells of dimension at least two to the graph. These mappings keep the 0-cells and 1-cells intact and are, therefore, restricted from making modifications to the input graph structure. An important remark is that for simplicial complexes, attaching simplices based on cliques present in the graph is the only possible skeleton preserving transformation. Once again, this illustrates the limitations of simplicial complexes for adding useful higher-dimensional structures to the graph.
\begin{wrapfigure}[9]{r}{0.35\textwidth}
\begin{subfigure}[t!]{1.0\linewidth}
\centering
\vspace{-.3cm}
\includegraphics[width=1.0\textwidth]{figures/lifting_map_example.jpeg}
\vspace{-4mm}
\end{subfigure}
\caption{A graph, its clique complex and the graph with duplicated edges. The first map is skeleton-preserving, while the second is not.}
\label{fig:sk_example}
\end{wrapfigure}
\begin{example}
\label{ex:skeleton-lift}
The function from Example \ref{ex:clique_complex} is also skeleton-preserving because the 1-skeleton of the clique complex of a graph is trivially isomorphic to the graph. A lifting function mapping each graph to a multi-graph where each edge is doubled by a parallel edge is not skeleton-preserving (Figure \ref{fig:sk_example}).
\end{example}
We now show that all the maps in the skeleton-preserving class have the following desirable property:
\begin{theorem}
\label{thm:skeleton}
Let $f$ be a skeleton-preserving lifting map. Then CWL($f$) (i.e. CWL using lifting $f$) is at least as powerful as WL in distinguishing non-isomorphic graphs.
\end{theorem}
To prove that some of these make CWL strictly more powerful than WL, it is sufficient to find a pair of graphs that cannot be distinguished by WL, but can be distinguished by CWL. The following result gives examples of such maps.
\begin{definition}
Let $k$-$\mathrm{CL}$, $k$-$\mathrm{IC}$, $k$-$\mathrm{C}$ be the lifting maps attaching cells to all the cliques, induced cycles and simple cycles, respectively, of size at most $k$.
\end{definition}
\begin{corollary}
\label{cor:WL_lifting_maps}
For all $k \geq 3$, CWL($k$-$\mathrm{CL}$), CWL($k$-$\mathrm{IC}$) and CWL$(k$-$\mathrm{C})$ are strictly more powerful than WL.
\end{corollary}
We note that this is not a complete list. For instance, the result can also be extended to combinations of the above or other transformations. We can also relate CWL to the higher-order 3-WL test.
\begin{theorem}
\label{thm:lifting3WL}
There exists a pair of graphs indistinguishable by 3-WL but distinguishable by CWL($k$-$\mathrm{CL}$) with $k \geq 4$, CWL($k$-$\mathrm{IC}$) with $k \geq 4$ and CWL$(k$-$\mathrm{C})$ with $k \geq 8$.
\end{theorem}
Finally, we conclude this section by showing how CWL can achieve a superior expressive power compared to SWL. This result is proven by Corollary \ref{cor:cwl_better_than_swl} in the Appendix.
\begin{theorem}
Let $k$-$\mathrm{CL}$ $\cup$ $k$-$\mathrm{IC}$ and $k$-$\mathrm{CL}$ $\cup$ $k$-$\mathrm{C}$ denote combined liftings attaching cells to the union of the specified substructures. CWL($k_1$-$\mathrm{CL} \cup k_2$-$\mathrm{IC}$) and CWL($k_1$-$\mathrm{CL} \cup k_2$-$\mathrm{C}$) are strictly more powerful than SWL($k_1$-$\mathrm{CL}$) for all $k_2 \geq 5$.
\end{theorem}
\section{Molecular Message Passing with CW Networks}
\label{sec:CWN_MMP}
We now describe CW Networks with an applied focus on molecular graphs to ground the discussion. Therefore, from now on we assume the use of the skeleton-preserving lifting transformation that attaches 2-cells to all the induced cycles (i.e. chordless cycles) in the graph as in Figure \ref{fig:gluing}. This leads to a message passing procedure involving atoms (vertices / 0-cells), the bonds between atoms (edges / 1-cells) and chemical rings (induced cycles / 2-cells). Additionally, in virtue of Theorem \ref{thm:sparse_cwl}, we consider only the boundary and upper adjacencies between these cells without sacrificing the expressive power. The equations for the other adjacencies, which we do not use, can be found in Appendix \ref{app:proofs_cellular}. We note however, that the theoretical results in this section are general and not particular to these specific choices of adjacencies and lifting transformation.
\paragraph{Molecular Message Passing} The cells in our CW Network receive two types of messages:
\begin{align}
&m_{{\mathcal{B}}}^{t+1}(\sigma) = \text{AGG}_{\tau \in {\mathcal{B}}(\sigma)}\Big(M_{{\mathcal{B}}}\big(h_{\sigma}^{t}, h_{\tau}^{t}\big)\Big) \quad
&m_{\uparrow}^{t+1}(\sigma) = \text{AGG}_{\tau \in {\mathcal{N}}_{\uparrow}(\sigma), \delta \in {\mathcal{C}}(\sigma, \tau)}\Big(M_{\uparrow}\big(h_{\sigma}^{t}, h_{\tau}^{t}, h_{\delta}^t\big)\Big). \nonumber
\end{align}
The first specifies messages from atoms to bonds and from bonds to rings. The second type of message, specifies messages between atoms connected by a bond and messages between bonds that are part of the same ring (Figure \ref{fig:message_passing}). Note that for the second type of adjacency, when two atoms communicate, we include the features of the bond between them. Similarly, when two bonds communicate, we include the features of the ring they communicate through.
The update operation takes into account these two types of incoming messages and updates the features of the cells:
\begin{equation}\label{eq:cwn_update}
h_{\sigma}^{t+1} = U\Big(h_{\sigma}^{t}, m_{{\mathcal{B}}}^{t}(\sigma), m_{\uparrow}^{t+1}(\sigma) \Big) .
\end{equation}
To obtain a global embedding for a cell complex $X$ from a model with $L$ layers, the readout function takes as input the separate multi-sets of features corresponding to the atoms, bonds and the rings:
\begin{equation}
h_X = \text{READOUT}(\{\mskip-5mu\{ h_{\sigma}^L \}\mskip-5mu\}_{dim(\sigma)=0}, \{\mskip-5mu\{ h_{\sigma}^L \}\mskip-5mu\}_{dim(\sigma)=1}, \{\mskip-5mu\{ h_{\sigma}^L \}\mskip-5mu\}_{dim(\sigma)=2}).
\end{equation}
\begin{wrapfigure}[]{r}{0.22\textwidth}
\begin{subfigure}[t!]{1.0\linewidth}
\centering
\vspace{-10pt}
\includegraphics[width=1.0\textwidth]{figures/message_passing.png}
\end{subfigure}
\caption{Hierarchical depiction of the message passing procedure. \textcolor{orange}{Orange} arrows indicate boundary messages received by cells $\sigma$ and $\tau$, while \textcolor{BlueViolet}{blue} ones show upper messages received by cells $\tau$ and $\delta$.}
\label{fig:message_passing}
\vspace{-40pt}
\end{wrapfigure}
\paragraph{Expressivity} Naturally, the ability of CWNs to distinguish non-isomorphic regular cell complexes is bounded by CWL. Similarly to GNNs and WL, CWNs can also be shown to be as powerful as CWL as long as they are equipped with a sufficient number of layers and the parametric local aggregators they use can learn to be injective. Multiple such multi-set aggregators \citep{GIN, Corso2020_PNA} are known to exist and can be directly employed in our model.
\begin{theorem}
\label{thm:CWandCWN}
CW Networks are at most as powerful as CWL. Additionally, when using injective neighbourhood aggregators and a sufficient number of layers, CWNs are as powerful as CWL.
\end{theorem}
Corollary~\ref{cor:WL_lifting_maps} states that CWL is strictly more powerful than the standard WL when the lifting procedure attaches 2-cells to induced cycles of size $k \geq 3$. As a consequence of Theorem~\ref{thm:CWandCWN}, this result also holds for molecular message passing CWNs equipped with injective aggregators. In practice, $k$ is to be considered as a standard hyperparameter, and its choice can either be driven by validation set performance, or by domain knowledge (if available).
\paragraph{Symmetries} Given a graph $G$ with adjacency matrix $\bm{A}$ and feature matrix $\bm{X}$, a function $f$ is (node) permutation equivariant if ${\bm{P}} f(\bm{A}, \bm{X}) = f({\bm{P}} \bm{A} {\bm{P}}^T, {\bm{P}} \bm{X})$, for any permutation matrix ${\bm{P}}$. GNN layers respect this equation, which ensures
they compute the same functions up to a permutation (i.e. relabeling) of the nodes. Similarly, it can be shown that CW Networks are equivariant with respect to permutations of the cells and corresponding permutations of the boundary relations $\sigma \prec \tau$ between cells. We define this notion of equivariance more formally in Appendix \ref{app:symmetries}.
\begin{theorem}
\label{thm:CWequivariant}
CW Network layers are cell permutation equivariant.
\end{theorem}
\paragraph{Long-Range Interactions}
Several graph-related tasks require the ability to capture long-range interactions between nodes. For instance, certain molecular properties depend on atoms placed on the opposite sides of a ring \citep{gilmer2017neural, ramakrishnan2014quantum}.
As a consequence of the coupling between the input and computational graphs, $L$ message passing operations are necessary in GNNs to let a node receive information from an $L$-hops distant node.
In contrast, our hierarchical message passing scheme requires \emph{at most} $L$ layers since $2$-cells create shortcuts.
For example, a constant number of CWN layers ($3$) is enough to capture dependencies between atoms on the opposite sides of a ring, independently of the ring size. In Section~\ref{ss:synth} we verify this in a controlled scenario. Additional experiments on real world graphs in Section~\ref{ss:real} confirm that it can achieve state-of-the-art performance with a limited number of layers.
\paragraph{Anisotropic Filters}
Due to the lack of a canonical ordering between neighbours, many common GNNs use symmetric convolutional kernels, resulting in isotropic filters treating neighbours equally. Recent works have proposed to address this limitation by employing additional structural information~\citep{beaini2020directional, bouritsas2020improving}. CWNs also implicitly achieve this form of anisotropy by integrating information from the higher-order cells and their associated substructures into the message passing procedure. For instance, bond features can learn to encode their membership to a ring and also communicate directly with other bonds present in the ring. Consequently, the messages between atoms connected through these bonds are modulated by the presence of the ring as well as by the presence of other nodes and bonds part of that ring.
\paragraph{CWNs as Generalised Convolutions}
Our message passing scheme can be seen a (non-linear) generalisation of linear diffusion operators on cell complexes.
Recent works~\citep{ebli2020simplicial, bunch2020simplicial} have introduced convolutional operators on SCs by employing the Hodge Laplacian \citep{schaub2020random},
a generalisation of the graph Laplacian.
By leveraging on the cellular Sheaf Laplacian \citep{HGh19}, a similar construction can be extended to cell complexes to define cellular convolutional operators. In Appendix~\ref{app:convs} we discuss this approach and show that our cellular message passing scheme subsumes it. This represents a promising avenue for studying CWNs from a spectral perspective, an endeavour we leave for future work.
\paragraph{Computational Complexity} When considering cells of a constant maximum dimension and boundary size, the computational complexity of the message passing scheme is linear in the size of the input complex. For the molecular applications we are interested in, the average number of rings per molecule is upper bounded by a small constant (e.g. three for MOLHIV), so the size of the complex is approximately the same as the size of the graph. Therefore, in this setting, the computational complexity of the model is similar to that of message passing GNNs. Separately of this, the one-time preprocessing step of computing the lifting of the graphs should also be considered. The $C$ induced cycles in a graph can be listed in ${\mathcal{O}}\big((|E| + |V|C) \text{ polylog } |V|\big)$ time \citep{ferreira2014amortized}. Again, given that $C$ is upper bounded by a small constant for the molecular datasets of interest in this work, the complexity of the lifting procedure is also almost linear in the size of the graph. A more detailed analysis backed up by wall-clock time experiments is given in Appendix \ref{app:complexity}.
\section{Experiments}
\label{sec:results}
In this section we validate the theoretical and empirical properties of our proposed message passing scheme in controlled scenarios as well as in real-world graph classification problems, with a focus on large scale molecular benchmarks. For simplicity, in all experiments we employ a model which stacks CWN layers with local aggregators as in GIN~\citep{GIN}. We name our architecture ``Cell Isomorphism Network'' (CIN). $0$-cells are always endowed with the original node features; higher-dimensional cells are populated in a benchmark specific manner. See Appendix~\ref{app:results} for details on feature initialisation, message passing and readout operations, hyperparameters, implementation and benchmark statistics.
\begin{figure}[t]
\centering
\begin{subfigure}[t!]{0.44\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/ring_transfer.pdf}
\caption{RingTransfer Results. Accuracy is over $5$ balanced classes. A score of $0.2$ is equivalent to a random guess. Error bars show the min and max. Our model obtains high-scores in average even for large rings despite using only three layers.}
\label{fig:rt}
\end{subfigure}
\hfill
\begin{subfigure}[t!]{0.53\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/sr_exp_rings_sns.pdf}
\caption{Failure rates on the SR isomorphism task, \emph{the smaller the better} (mean and std-error over $5$ runs). In parantheses, for each model, the maximum size $k$ of rings lifted to $2$-cells.}
\label{fig:sr_iso}
\end{subfigure}
\caption{Results on the RingTransfer and SR synthetic benchmarks.}
\label{fig:synth}
\vspace{-12pt}
\end{figure}
\subsection{Synthetic Benchmarks}\label{ss:synth}
\paragraph{CSL} Circular Skip Link dataset was first introduced in~\cite{pmlr-v97-murphy19a}
and has been recently adopted as a reference benchmark to test the expressivity of GNNs~\citep{dwivedi2020benchmarkgnns}. It consists of 150 $4$-regular graphs from 10 different isomorphism classes, which we need to predict.
Unsolvable by the WL test and message passing approaches \citep{pmlr-v97-murphy19a, Chen2019_ring_gnn}, we use it to validate the expressive power of CWNs.
\begin{wraptable}[7]{l}{0.425\textwidth}
\centering
\vspace{-13pt}
\begin{minipage}[t]{1.0\linewidth}
\caption{Classification accuracy on CSL.}\label{tab:csl}
\resizebox{\columnwidth}{!}{
\begin{tabular}{l ccc}
\toprule
Method &
Mean &
Min &
Max \\
\midrule
MP-GNNs &
10.000$\pm$0.000 &
10.000 &
10.000 \\
RingGNN &
10.000$\pm$0.000 &
10.000 &
10.000 \\
3WLGNN &
97.800$\pm$10.916 &
30.000 &
100.000 \\
\midrule
CIN (Ours) &
100.000$\pm$0.000 &
100.000 &
100.000 \\
\bottomrule
\end{tabular}%
}
\end{minipage}
\end{wraptable}
We follow the same evaluation setting as ~\citet{dwivedi2020benchmarkgnns}: $5$-fold cross validation procedure and $20$ different random weight initialisations. For our model, we set the maximum ring size $k=8$. In Table~\ref{tab:csl} we follow the common practice on this dataset and report the mean, minimum and maximum test accuracy obtained by CIN over the $100$ runs, along with the results by the baselines presented in~\citet{dwivedi2020benchmarkgnns}.
MP-GNNs,
that is classic message passing GNNs (GAT~\citep{velivckovic2017graph}, MoNet~\citep{Monti_etal2017}, GIN~\citep{GIN}, etc.), and RingGNN~\citep{Chen2019_ring_gnn} perform as random guessers. In contrast, our model is able to identify the isomorphism class of each test graph in every run while featuring only a fraction of the computational complexity of 3WLGNN, the best performing reference baseline~\citep{dwivedi2020benchmarkgnns, maron2019provably}.
\paragraph{SR} Similarly to~\citet{bodnar2021weisfeiler} and~\cite{bouritsas2020improving}, we consider Strongly Regular graphs within the same family as hard examples of non-isomorphic graphs we seek to distinguish. Any pair of graphs within the same family cannot provably be distinguished by 3-WL test~\citep{bodnar2021weisfeiler, bouritsas2020improving}. We reproduce the same experimental setting of~\citet{bodnar2021weisfeiler}. In particular, we consider $9$ distinct SR families\footnote{Data available at: \url{http://users.cecs.anu.edu.au/~bdm/data/graphs.html}.} and run our model untrained on the cell complex lifting of each graph, with $k = 4, 5, 6$. $0$-cells (nodes) are initialised with a constant unitary signal, while $1$- and $2$-cells are initialised with the sum of the contained $0$-cells. We additionally run an MLP baseline with sum readout to appreciate the contribution of message passing. We report the percentage of non-distinguished pairs in Figure~\ref{fig:sr_iso}.
Contrary to 3-WL, both CIN and the MLP baseline are able to distinguish many pairs across all families, with better performance attained for larger $k$. For $k = 6$, we observed CIN to disambiguate all pairs in all families ($0.0$\% failure rate). Despite the strong results achieved by the baseline, we found CIN to always distinguish a larger number of non-isomorphic pairs for the same values of $k$, this confirming the importance of cellular message passing.
\paragraph{RingTransfer} In order to empirically validate the ability of CIN to capture long-range node dependencies, we additionally design a third synthetic benchmark dubbed as `RingTransfer'. Graphs in this dataset are chordless cycles (rings) of size $k$. In each graph we mark two special nodes as \textbf{target} and \textbf{source}, always placed at distance $\lfloor \frac{k}{2} \rfloor$. The task is for \textbf{target} to output the one-hot encoded label assigned to \textbf{source}. All other nodes in the ring are assigned a unitary constant feature vector. A model has to learn to transfer the information contained in \textbf{source} to the opposite side of the ring, where \textbf{target} resides. We initialise $1$- and $2$-dimensional cells with a null signal. In Figure~\ref{fig:rt} we show the performance of a $3$-layer CIN as a function of the ring size $k$, along with that of GIN~\citep{GIN} baselines equipped with $\lfloor \frac{k}{2} \rfloor$ stacked layers. We observe that our model learns to solve the task with only $3$ computational steps, independent of $k$. As for GIN, we observed degradation in the performance for $k \geq 24$, up to complete failure. We hypothesise this to be due to the difficulties of training such a deep GNN ($\geq 12$ layers). We further verify the (theoretically expected) failure of GIN (not included) when endowed with less than $\lfloor \frac{k}{2} \rfloor$ layers.
\subsection{Real-World Graph Benchmarks}\label{ss:real}
\begin{table}[!t]
\centering
\caption{TUDatasets. The first section of the table includes the accuracy of graph kernel methods, while the second includes GNNs. The top three are highlighted by \textbf{\textcolor{red}{First}}, \textbf{\textcolor{violet}{Second}}, \textbf{Third}.}
\label{tab:tud_datasets}
\resizebox{\linewidth}{!}{%
\begin{tabular}{l | lllll | lll}
\toprule
Dataset &
MUTAG &
PTC &
PROTEINS &
NCI1 &
NCI109 &
IMDB-B &
IMDB-M &
RDT-B \\
\midrule
RWK \citep{gartner2003graph} &
79.2$\pm$2.1 &
55.9$\pm$0.3 &
59.6$\pm$0.1 &
$>$3 days &
N/A &
N/A &
N/A &
N/A \\
GK ($k=3$) \citep{shervashidze2009efficient} &
81.4$\pm$1.7 &
55.7$\pm$0.5 &
71.4$\pm$0.3 &
62.5$\pm$0.3 &
62.4$\pm$0.3 &
N/A &
N/A &
N/A \\
PK \citep{neumann2016propagation} &
76.0$\pm$2.7&
59.5$\pm$2.4 &
73.7$\pm$0.7 &
82.5$\pm$0.5 &
N/A &
N/A &
N/A &
N/A \\
WL kernel \citep{shervashidze2011weisfeiler} &
90.4$\pm$5.7 &
59.9$\pm$4.3 &
75.0$\pm$3.1 &
\first{86.0}$\pm$1.8 &
N/A &
73.8$\pm$3.9 &
50.9$\pm$3.8 &
81.0$\pm$3.1 \\
\midrule
DCNN \citep{DCNN_2016} &
N/A&
N/A &
61.3$\pm$1.6 &
56.6$\pm$1.0 &
N/A &
49.1$\pm$1.4 &
33.5$\pm$1.4 &
N/A \\
DGCNN \citep{zhang2018end} &
85.8$\pm$1.8 &
58.6$\pm$2.5 &
75.5$\pm$0.9 &
74.4$\pm$0.5 &
N/A &
70.0$\pm$0.9 &
47.8$\pm$0.9 &
N/A \\
IGN \citep{maron2018invariant} &
83.9$\pm$13.0 &
58.5$\pm$6.9 &
\third{76.6}$\pm$5.5 &
74.3$\pm$2.7 &
\third{72.8}$\pm$1.5 &
72.0$\pm$5.5 &
48.7$\pm$3.4 &
N/A \\
GIN \citep{GIN} &
89.4$\pm$5.6 &
64.6$\pm$7.0 &
76.2$\pm$2.8 &
82.7$\pm$1.7 &
N/A &
75.1$\pm$5.1 &
52.3$\pm$2.8 &
\first{92.4}$\pm$2.5 \\
PPGNs \citep{maron2019provably} &
\third{90.6}$\pm$8.7 &
66.2$\pm$6.6 &
\first{77.2}$\pm$4.7 &
83.2$\pm$1.1 &
\second{82.2}$\pm$1.4 &
73.0$\pm$5.8 &
50.5$\pm$3.6 &
N/A \\
Natural GN \citep{de2020natural} &
89.4$\pm$1.6 &
\third{66.8}$\pm$1.7 &
71.7$\pm$1.0 &
82.4$\pm$1.3 &
N/A &
73.5$\pm$2.0 &
51.3$\pm$1.5 &
N/A\\
GSN \citep{bouritsas2020improving} &
\second{92.2} $\pm$ 7.5 &
\first{68.2} $\pm$ 7.2 &
\third{76.6} $\pm$ 5.0 &
\third{83.5} $\pm$ 2.0 &
N/A &
\first{77.8} $\pm$ 3.3 &
\first{54.3} $\pm$ 3.3 &
N/A \\
SIN \citep{bodnar2021weisfeiler} &
N/A &
N/A &
76.4 $\pm$ 3.3 &
82.7 $\pm$ 2.1 &
N/A &
\second{75.6} $\pm$ 3.2 &
\third{52.4} $\pm$ 2.9 &
\third{92.2} $\pm$ 1.0 \\
\midrule
{{\bf CIN} (Ours)} &
\first{92.7} $\pm$ 6.1 &
\first{68.2} $\pm$ 5.6 &
\second{77.0} $\pm$ 4.3 &
\second{83.6} $\pm$ 1.4 &
\first{84.0} $\pm$ 1.6 &
\second{75.6} $\pm$ 3.7 &
\second{52.7} $\pm$ 3.1 &
\first{92.4} $\pm$ 2.1 \\
\bottomrule
\end{tabular}
}
\end{table}
\paragraph{TUD} We test our model on $8$ TUDataset benchmarks \citep{morris2020tudataset} with small and medium sizes from biology (\textbf{PROTEINS} \citep{dobson2003distinguishing,borgwardt2005protein}), chemistry (i.e. molecules -- \textbf{MUTAG} \citep{kazius2005derivation,riesen2008iam}, \textbf{PTC}, \textbf{NCI1} and \textbf{NCI109} \citep{wale2008comparison}) to social networks (\textbf{IMDB-B}, \textbf{IMDB-M}, \textbf{RDT-B}).
We consider induced cycle of size up to $k=6$ for our graph lifting procedure. We initialise node (and $0$-cell) features as described in~\citet{GIN}, and higher dimensional cells by averaging or summing the features of the included $0$-cells. The training setting and evaluation procedure follow those in \citet{GIN}.
We report the results in Table~\ref{tab:tud_datasets}. CIN compares more than favourably with the baselines, displaying strong empirical performance on all benchmarks. The mean accuracy of CIN ranks top on four out of eight datasets. On the remaining datasets, CIN achieves the second place. We observe that the best results are on datasets from the biological and chemical domains, where rings play a relevant role.
\paragraph{ZINC}
We study the effectiveness of cellular message passing on larger scale molecular benchmarks from the ZINC database \citep{ZINCdataset}. \textbf{ZINC} (12k graphs) and \textbf{ZINC-FULL} (250k graphs) \citep{dwivedi2020benchmarkgnns,jin2018junction,you2018graph,gomez2018automatic} are two graph regression task datasets for drug constrained solubility prediction. In these experiments, we consider rings up to size $k=18$. We follow the training and evaluation procedures in~\citep{dwivedi2020benchmarkgnns}. Our experiments encompass different scenarios, examine the impact of ablating edge features and of constraining the parameter budget of the architecture to $100$k. All results are illustrated in Table~\ref{tab:mol_dataset} where we also include the results for \textbf{ZINC-FULL} obtained by the same exact architectures. Our model exhibits particularly strong performance on these benchmarks: it attains state-of-the-art results on both the two dataset variants, outperforming other models by a significant margin. CIN attains strong results even when constrained by the parameter budget. It still achieves state-of-the-art performance on \textbf{ZINC} and is on-par with the best unconstrained baseline under edge-feature ablation.
\paragraph{Mol-HIV} We additionally test our model on the molecular \textbf{ogbg-molhiv} dataset from the Open Graph Benchmark~\citep{hu2020open} ($41$k graphs). The task is to predict the capacity of compounds to inhibit HIV replication. Rings of size up to $k=6$ are considered as $2$-cells. We take the architecture in~\cite{Fey2020_himp} as reference and replicate the same hyperparameter setting in our model, including the use of only $2$ message passing layers. We report the mean of test ROC-AUC metrics at the epoch of best validation performance for $10$ random weight initialisations. Similarly to ZINC, we experiment with a ``small'' model whose number of parameters is constrained in the order of $100$k. Table~\ref{tab:mol_dataset} displays the results. CIN significantly outperforms other strong GNN baselines, even when constrained by the parameter budget. Consistently with~\cite{Fey2020_himp}, we observe that only two layers are sufficient when performing hierarchical message passing across meso-scale structures such as rings.
\begin{table}[!t]
\centering
\begin{minipage}[t]{0.80\textwidth}
\centering
\caption{ZINC (MAE), ZINC-FULL (MAE) and Mol-HIV (ROC-AUC).}
\label{tab:mol_dataset}
\resizebox{\columnwidth}{!}{
\begin{tabular}{l cccc}
\toprule
\multirow{2}{*}{Method} &
\multicolumn{2}{c}{ZINC $\downarrow$} &
ZINC-FULL $\downarrow$&
MOLHIV $\uparrow$\\
&
No Edge Feat. &
With Edge Feat. &
All methods &
All methods \\
\midrule
GCN \citep{kipf2017graph} &
0.469$\pm$0.002 &
N/A &
N/A &
76.06$\pm$0.97 \\
GAT \citep{velivckovic2017graph} &
0.463$\pm$0.002 &
N/A &
N/A &
N/A \\
GatedGCN \citep{bresson2017residual} &
0.422$\pm$0.006 &
0.363$\pm$0.009 &
N/A &
N/A \\
GIN \citep{GIN} &
0.408$\pm$0.008 &
0.252$\pm$0.014 &
0.088$\pm$0.002 &
77.07$\pm$1.49 \\
PNA \citep{Corso2020_PNA} &
0.320$\pm$0.032 &
0.188$\pm$0.004 &
N/A &
79.05$\pm$1.32 \\
DGN \citep{beaini2020directional} &
0.219$\pm$0.010 &
0.168$\pm$0.003 &
N/A &
79.70$\pm$0.97 \\
HIMP \citep{Fey2020_himp} &
N/A &
0.151$\pm$0.006 &
0.036$\pm$0.002 &
78.80$\pm$0.82\\
GSN \citep{bouritsas2020improving} &
0.139$\pm$0.007 &
0.108$\pm$0.018 &
N/A &
77.99$\pm$1.00 \\
\midrule
\textbf{CIN-small} (Ours) &
0.139$\pm$0.008 &
0.094$\pm$0.004 &
0.044$\pm$0.003 &
80.55$\pm$1.04 \\
\textbf{CIN} (Ours) &
\textbf{0.115}$\pm$\textbf{0.003} &
\textbf{0.079}$\pm$\textbf{0.006} &
\textbf{0.022}$\pm$\textbf{0.002} &
\textbf{80.94}$\pm$\textbf{0.57} \\
\bottomrule
\end{tabular}%
}
\end{minipage}
\vspace{-11pt}
\end{table}
\section{Related Work, Discussion and Conclusion}
\label{sec:conclusion}
\paragraph{Cell complex models} Recent works have proposed the generalisation of GNNs to simplicial complexes \citep{ebli2020simplicial, bunch2020simplicial, glaze2021principled, hajij2021simplicial}. All these
simplicial methods are subsumed by the model in~\citet{bodnar2021weisfeiler}, which CWNs in turn subsume.
To the best of our knowledge, \citet{hajij2020cell} is the only other example of message passing on cell complexes, but this work does not study the expressive power of the proposed scheme, neither it experimentally validates its performance. In contrast, our work comprehensively characterises the expressiveness of cellular message passing, and introduces a theoretically grounded and empirically effective framework to apply it on graph structured data in a way to address several limitations of standard Graph Neural Networks.
\paragraph{Molecular substructures} A few other works have extended GNNs to account for
molecular substructures. Junction Trees (JT), which conveniently represent singletons, bonds and rings as supernodes in a tree, have been used in molecular graph generation~\citep{jin2018junction, jin2019learning}. JTs are also used in the recent work of~\citet{Fey2020_himp}, who employs them to design a hierarchical message passing scheme based on the tree structure. However, this hierarchy has a different configuration than the one cell complexes provide. Information about cycles is also used in GSNs~\citep{bouritsas2020improving} to augment the node features, but the model retains the usual message passing procedure of GNNs. These last two models are of particular relevance to the present work, since they utilise information about chemical rings. It is important to remark that CWNs compare favourably with both of them in all our benchmarks.
\paragraph{Higher-order GNNs} A related line of work has studied lifting graphs into $k$-dimensional tensor representations that can be processed by provably expressive $k$-GNNs \citep{maron2018invariant, maron2019provably, azizian2021expressive}. With higher values of $k$, these models achieve higher-expressivity, but due to the computational complexity this incurs, values of $k \geq 3$ are of little use in practice. Therefore, unlike CWNs, these models cannot explicitly represent in practice chemical rings of common sizes (e.g. five or six). Furthermore, by being upper-bounded by 3-WL, the 2-GNN models cannot count the number of induced cycles of size greater than four (see Appendix \ref{app:proofs_cellular} for details). In contrast, CWNs can easily count these important chemical substructures through the readout operation it performs on the 2-cells.
\paragraph{Limitations}
The main limitations of the model are of computational nature. While the computational complexity of the message passing procedure and its preprocessing step is suitable for molecular and geometric graphs, the number of rings (and more generally simple cycles) in general graphs can be exponential in the number of nodes. In that case, one has to resort to smaller 2-cells like triangles, which can be found efficiently in general graphs. Moreover, one has to typically use weights specific for each dimension of the cell complex, increasing the number of parameters compared to GNNs. However, we have shown that our model can compensate this increase with a reduced number of layers and still achieve state-of-the-art results on some of the molecular benchmarks.
From a theoretical point of view, this work is concerned only with \emph{regular} cell complexes. Adopting this restriction is useful from multiple perspectives: regular cell complexes are easier to analyse, their combinatorial structure completely describes their topology and convolutions can be defined on them through the Sheaf Laplacian (see Appendix \ref{app:convs}). Nonetheless, some of our theoretical results could be extended to non-regular complexes, which could be obtained by lifting transformations not studied in this work, such as attaching 2-cells to paths in the graph. We leave addressing non-regular complexes and their trade-offs to future developments of this work.
\paragraph{Societal Impacts} Most of our paper is theoretical in
nature and we do not see immediate direct negative societal impacts. Within the scope of social network applications, we do not yet have sufficient evidence of performance improvement on related benchmarks to justify obvious adoption in such a domain. In contrast, the empirical performance on molecular benchmarks suggests it may have a positive impact on applications of immediate interest in pharmaceutics, such as drug discovery~\citep{drug_discovery}.
\paragraph{Conclusion} We have proposed a provably powerful message passing procedure on cell complexes motivated by a novel colour refinement algorithm to test their isomorphism. This allows us to consider flexible lifting operations on graphs to implement more expressive architectures which benefit from decoupling the computational and input graphs. Our methods show excellent performance on diverse synthetic and real-world molecular benchmarks.
\section*{Funding and Acknowledgements}
YW and GM acknowledge support from the ERC under the EU's Horizon 2020
programme (grant agreement n\textsuperscript{o} 757983). MB is supported in part by ERC Consolidator grant n\textsuperscript{o} 724228 (LEMAN). The authors declare no competing interests. We are also grateful to Ben Day, Gabriele Corso and Nikola Simidjievski for their helpful feedback. We would also like to thank Vijay P. Dwivedi and Chaitanya K. Joshi for clarifying certain aspects of their Benchmarking GNNs \citep{dwivedi2020benchmarkgnns} work, and to Muhammet Balcilar for signalling a numerical precision issue in early SR graphs experiments.
\bibliographystyle{plainnat}
|
1,116,691,500,898 | arxiv | \section*{Introduction}
\label{0}
In \cites{LZa,LZ}, we developed a theory of Grothendieck's six operations for
\'etale cohomology of Artin stacks and prove several desired properties
including the base change theorem. In the article, we develop the
corresponding adic formalism and establish adic analogues of results of
\cite{LZ}. This extends all previous theories on the subject, including SGA~5
\cite{SGA5}, Deligne \cite{WeilII}, Ekedahl \cite{Ekedahl} (for schemes),
Behrend \cite{Behrend} and Laszlo--Olsson \cite{LO2}. We prove, among other
things, the base change theorem in derived categories, which was previous
known only on the level of sheaves \cite{LO2} (and under other restrictions).
Another limitation of the existing theories, including those for schemes, is
the constructibility assumption. This assumption is not often met, for
example, when considering morphisms between Artin stacks that are only
locally of finite type. By contrast, the adic formalism developed in this
article applies to unrestricted derived categories.
In addition, we define perverse $t$-structures on Artin stacks for general
perversity, extending the work of Gabber \cite{Gabber} for schemes and the
work of Laszlo and Olsson \cite{LO3} for the middle perversity.
As in our preceding article, the approach we are taking is different from
all the previous theories. We work in the world of $\infty$-categories in
the sense of Lurie \cites{Lu1,Lu2}, by enhancing all the derived categories,
functors, and natural transforms to the level of $\infty$-categories. At
this level, we may use some new machineries among which the most important
ones are gluing objects, Adjoint Functor Theorem, $\infty$-categorical
descent, all in \cites{Lu1,Lu2}, and some other techniques developed in
\cite{LZ}. In particular, we obtain several other special descent properties
for the derived category of lisse-\'{e}tale sheaves.
\subsection{Results}
\label{0ss:result}
In this section, we will state our constructions and results only in the
classical setting of Artin stacks on the level of usual derived categories
(which are homotopy categories of the derived $\infty$-categories), among
other simplification. We will provide the precise references of the complete
results in later chapters, for higher Deligne--Mumford stacks and higher
Artin stacks, stated on the level of stable $\infty$-categories. We refer
the reader to \cite{LZ}*{\Sec 0.1} for our convention on Artin stacks.
Let $\cX$ be an Artin stack and let $\lambda$ be a ringed diagram, that is,
a functor from a partially ordered set to the category of unital commutative
rings. Recall that $\rD_{\cart}(\cX_{\liset},\lambda)$ is the full
subcategory of $\rD(\cX_{\liset},\lambda)$ spanned by complexes whose
cohomology sheaves are all Cartesian. We define in \Sec
\ref{1ss:adic_complexes} a strictly full subcategory
$\rD(\cX,\lambda)_{\adic}$ of $\rD_{\cart}(\cX_{\liset},\lambda)$ consisting
of \emph{adic complexes}, possessing the property that the inclusion
$\rD(\cX,\lambda)_{\adic}\to \rD_{\cart}(\cX_{\liset},\lambda)$ admits a
right adjoint functor $R_{\cX}$. Let $f\colon \cY\to\cX$ be a morphism of
Artin stacks. We then define operations:
\begin{gather*}
\underline{f}^*\colon
\rD(\cX,\lambda)_{\adic}\to\rD(\cY,\lambda)_{\adic},\quad
\underline{f}_*\colon \rD(\cY,\lambda)_{\adic}\to\rD(\cX,\lambda)_{\adic};\\
-\underline{\otimes}_{\cX}-\colon
\rD(\cX,\lambda)_{\adic}\times\rD(\cX,\lambda)_{\adic}\to\rD(\cX,\lambda)_{\adic},\\
\underline{\HOM}_{\cX}\colon
\rD(\cX,\lambda)_{\adic}^{op}\times\rD(\cX,\lambda)_{\adic}\to\rD(\cX,\lambda)_{\adic}.
\end{gather*}
The pairs $(\underline{f}^*,\underline{f}_*)$ and
$(-\underline{\otimes}\s{K},\underline{\HOM}(\s{K},-))$ for every
$\sK\in\rD(\cX,\lambda)_{\adic}$ are pairs of adjoint functors.
We fix a nonempty set $\sf{L}$ of rational primes. If $\cX$ is
$\sf{L}$-coprime, $f\colon \cY\to \cX$ is locally of finite type, and
$\lambda$ is $\sf{L}$-torsion ringed diagram, then we have another pair of
adjoint functors:
\begin{align*}
\underline{f}_!\colon \rD(\cY,\lambda)_{\adic}\to\rD(\cX,\lambda)_{\adic},\quad
\underline{f}^!\colon \rD(\cX,\lambda)_{\adic}\to\rD(\cY,\lambda)_{\adic}.
\end{align*}
These operations satisfy the similar properties as in the non-adic version
(see Propositions \ref{1pr:operation_property} and \ref{1pr:poincare}). In
particular, we have the following theorem.
\begin{theorem}[Base Change]
Let $\lambda$ be an $\sf{L}$-torsion ringed diagram, and let
\begin{align*}
\xymatrix{
\c{W} \ar[d]_-{q} \ar[r]^-{g} & \c{Z} \ar[d]^-{p} \\
\cY \ar[r]^-{f} & \cX }
\end{align*}
be a \emph{Cartesian} square of $\sf{L}$-coprime Artin stacks where $p$ is
locally of finite type. Then we have a natural isomorphism of functors:
\begin{align*}
f^*\circ p_!\simeq q_!\circ g^*\colon \rD_{\cart}(\c{Z},\lambda)_{\adic}\to\rD_{\cart}(\cY,\lambda)_{\adic}.
\end{align*}
\end{theorem}
The adic formalism introduced above does \emph{not} assume the
constructibility at the first place. In other words, we are free to talk
about adic complexes for any sheaves. In particular, in terms of
Grothendieck's fonctions-faisceaux dictionary, we make sense of divergent
integrals on stacks over finite fields, those appear for example in
\cite{FN}.
In \Sec \ref{3}, we introduce a special case of the adic formalism, namely,
the $\fm$-adic formalism on which there is a good notion of constructibility.
Such formalism is enough for most applications. Let $\Lambda$ be a ring and
$\fm\subseteq \Lambda$ be a principal ideal, satisfying the conditions in
Definition \ref{3de:pring}. The typical example is that $\Lambda$ is a
$1$-dimensional valuation ring and $\fm$ is a proper ideal. The pair
$(\Lambda,\fm)$ corresponds to a ringed diagram $\Lambda_{\bullet}$ with the
underlying category $\NN=\{0\to1\to2\to\cdots\}$ and
$\Lambda_n=\Lambda/\fm^{n+1}$. We call this setup as the $\fm$-adic
formalism. Now we fix a pair $(\Lambda,\fm)$ as above such that $\Lambda$ is
Noetherian and $\Lambda/\fm$ is $\sf L$-torsion. Let $\sS$ be either a
quasi-excellent finite-dimensional scheme or a regular scheme of dimension
$\leq1$, that is $\sf{L}$-coprime. Consider Artin stacks that are locally of
finite type over $\sS$. In this setup, we define the intersection
$\rD_{\cons}(\cX_{\liset},\Lambda_{\bullet})\cap\rD(\cX,\Lambda_{\bullet})_{\adic}$
of constructible complexes and adic complexes as the category of
constructible adic complexes. We denote this category by
$\rD_{\cons}(\cX,\Lambda_{\bullet})_{\adic}$, which is a full subcategory of
$\rD(\cX,\Lambda_{\bullet})_{\adic}$. In \Sec
\ref{3ss:constructible_adic}\footnote{We actually denote the
($\infty$-)category of constructible adic complexes by
$\underline{\cD}_{\cons}(\cX,\Lambda_{\bullet})$ in \Sec
\ref{3ss:constructible_adic}. See \Sec \ref{0ss:deligne} for an
explanation.}, we show that the usual $t$-structure on
$\rD(\cX,\Lambda_{\bullet})_{\adic}$ restricts to the one on
$\rD_{\cons}(\cX,\Lambda_{\bullet})_{\adic}$. Moreover, the six operations
mentioned previously restrict to the following refined ones:
\begin{gather*}
\underline{f}^*\colon
\rD_{\cons}(\cX,\Lambda_{\bullet})_{\adic}\to\rD_{\cons}(\cY,\Lambda_{\bullet})_{\adic},\quad
\underline{f}^!\colon \rD_{\cons}(\cX,\Lambda_{\bullet})_{\adic}\to\rD_{\cons}(\cY,\Lambda_{\bullet})_{\adic};\\
-\underline{\otimes}_\cX-\colon
\rD_{\cons}^{(-)}(\cX,\Lambda_{\bullet})_{\adic}\times\rD_{\cons}^{(-)}(\cX,\Lambda_{\bullet})_{\adic}
\to\rD_{\cons}^{(-)}(\cX,\Lambda_{\bullet})_{\adic},\\
\underline{\HOM}_\cX\colon
\rD_{\cons}^{(b)}(\cX,\Lambda_{\bullet})_{\adic}^{op}\times\rD_{\cons}^{(+)}(\cX,\Lambda_{\bullet})_{\adic}
\to\rD_{\cons}^{(+)}(\cX,\Lambda_{\bullet})_{\adic}.
\end{gather*}
If $f$ is \emph{quasi-compact and quasi-separated}, then we have
\begin{align*}
\underline{f}_*\colon \rD_{\cons}^{(+)}(\cY,\Lambda_{\bullet})_{\adic}\to\rD_{\cons}^{(+)}(\cX,\Lambda_{\bullet})_{\adic},\quad
\underline{f}_!\colon \rD_{\cons}^{(-)}(\cY,\Lambda_{\bullet})_{\adic}\to\rD_{\cons}^{(-)}(\cX,\Lambda_{\bullet})_{\adic}.
\end{align*}
In \Sec \ref{3ss:compatibility}, we show that our theory of constructible
adic formalism coincides with Laszlo--Olsson \cite{LO3} under their
assumptions.
In \Sec \ref{2}, we define the perverse $t$-structure, in both non-adic and
adic setting, for general ``perversity'' for (higher) Artin stacks, while in
all previous theory only middle perversity is considered \cite{LO3}. We
define a \emph{perversity smooth (resp.\ \'{e}tale) evaluation} $\sf p$ on an
Artin (resp.\ a Deligne--Mumford) stack $\cX$ (Definition
\ref{2de:perversity_evaluation}) to be an assignment to each atlas $u\colon
X\to \cX$ a weak perversity function $\sf p_u$ on $X$ in the sense of Gabber
\cite{Gabber}, satisfying certain compatibility condition. In particular,
when $\cX$ is a scheme, a perversity \'{e}tale evaluation is same as a weak
perversity function.
\begin{theorem}[(Adic) perverse $t$-structure, \Secs \ref{2ss:perverse_t}, \ref{2ss:adic_perverse_t}]
Let $\cX$ be an $\sf{L}$-coprime Artin stack equipped with a perversity
smooth evaluation $\sf p$ and $\lambda$ be an $\sf{L}$-torsion ringed
diagram.
\begin{enumerate}
\item There is a unique up to equivalence $t$-structure
$(\TS{\tp}{\rD}{\leq0}(\cX,\lambda),\TS{\tp}{\rD}{\geq0}(\cX,\lambda))$
on $\rD(\cX,\lambda)=\rD_{\cart}(\cX_{\liset},\lambda)$, called the
\emph{perverse $t$-structure}, such that for every atlas $u\colon
X\to \cX$,
$u^*\TS{\tp}{\rD}{\leq0}(\cX,\lambda)=\TS{\tp_u}{\rD}{\leq0}(X,\lambda)$
and
$u^*\TS{\tp}{\rD}{\geq0}(\cX,\lambda)=\TS{\tp_u}{\rD}{\geq0}(X,\lambda)$,
where the corresponding $t$-structure on the scheme $X$ is defined
by Gabber \cite{Gabber}.
\item If $f\colon\cY\to\cX$ is a smooth morphism, then $f^*$ is
perverse $t$-exact with respect to compatible perversity smooth
evaluations $\sf p$ on $\c X$ and $\sf q$ on $\cY$.
\item We have above results in the adic setting, where
$\TS{\tp}{\rD}{\leq0}(\cX,\lambda)_{\adic}=\TS{\tp}{\rD}{\leq0}(\cX,\lambda)\cap\rD(\cX,\lambda)_{\adic}$.
\item Moreover, the classical description of the perverse $t$-structure
via cohomology on stalks again holds (Propositions
\ref{2pr:perverse_stalk} and \ref{2pr:adic_perverse_stalk}).
\end{enumerate}
\end{theorem}
In particular, when $\tp=0$, we recover the usual $t$-structure in the
non-adic case and obtain the similar usual $t$-structure in the adic case.
When $\tp$ is the middle perversity evaluation, we generalize the classical
notion of middle perverse $t$-structure for schemes, in both non-adic and
adic cases.
In \Sec \ref{3ss:constructible_adic_perverse_t}, we show that under certain
conditions on $(\Lambda,\fm)$ and the perversity smooth evaluation $\sf p$,
the adic perverse $t$-structure restricts to the one on
$\rD_{\cons}(\cX,\Lambda_{\bullet})_{\adic}$. In particular, when $\sf p$ is
the middle perversity smooth evaluation (that is, the middle perversity
function in the case of schemes), the corresponding (adic) perverse
$t$-structure coincides with the one defined by Laszlo--Olsson \cite{LO3},
under their further restrictions on $(\Lambda,\fm)$ and $\cX$.
In \Sec \ref{4}, we prove several additional $\infty$-categorical descent
properties of derived $\infty$-categories and their adic version we have
constructed. In particular, we have the following theorem, which is the
incarnation on the level of usual derived categories of the main result in
\Sec \ref{4ss:hyperdescent}.
\begin{theorem}[Proposition \ref{4pr:hyperdescent_stack}]\label{0th:hyperdescent}
Let $f\colon \cY\to \cX$ be morphism of Artin stacks and $y\colon
\cY^+_0\to\cY$ be a smooth surjective morphism. Let $\cY^+_{\bullet}$ be a
smooth hypercovering of $\cY$ with the morphism $y_n:\cY^+_n\to
\cY^+_{-1}=\cY$. Put $f_n=f\circ y_n:\cY^+_n\to \cX$.
\begin{enumerate}
\item For every complex $\sK$ in $\rD^{\geq0}(\cY,\lambda)$ (resp.\
$\rD^{\geq0}(\cY,\lambda)_{\adic}$), we have a convergent spectral
sequence
\[\r E_1^{p,q}=\r{H}^q(f_{p*}y_p^*\sK)\Rightarrow \r{H}^{p+q} f_* \sK\quad
(\text{resp.\ }\underline{\r E}_1^{p,q}=\r{H}^q(\underline{f}_{p*}\underline{y}_p^*\sK)\Rightarrow \r{H}^{p+q} \underline{f}_* \sK).\]
\item If $\cX$ is $\sf L$-coprime; $\lambda$ is $\sf L$-torsion, and
$f$ is locally of finite type, then for every complex $\sK$ in
$\rD^{\leq0}(\cY,\lambda)$ (resp.\
$\rD^{\leq0}(\cY,\lambda)_{\adic}$), we have a convergent spectral
sequence
\[\tilde{\r E}_1^{p,q}=\r{H}^q (f_{-p!}y_{-p}^! \sK)\Rightarrow \r{H}^{p+q} f_!\sK\quad
(\text{resp.\ }\underline{\tilde{\r E}}_1^{p,q}
=\r{H}^q(\underline{f}_{-p!}\underline{y}_{-p}^! \sK)\Rightarrow \r{H}^{p+q}\underline{f}_!\sK).\]
\end{enumerate}
\end{theorem}
Finally, we would like to emphasize that all conventions and notation from
\cite{LZ}, especially those in \Sec 0.5 there, will be continually adopted in
the current article, unless otherwise specified.
\subsection{Refinement of Deligne's construction}
\label{0ss:deligne}
Let $\Lambda$ be the ring of integers of a finite extension of $\d{Q}_\ell$,
$\fm$ be the maximal ideal of $\Lambda$. Let $X$ be a Noetherian separated
algebraic space over $\Z[1/\ell]$. Deligne \cite{WeilII}*{1.1.2} defines a
category $\rD^-(X,\Lambda_{\bullet})_{\adic}$ (which he denotes by
$\rD^-(X,\Lambda)$) as the $2$-limit of $\rD^-(X,\Lambda_n)$, for $n\in\NN$,
where the transition functors are the derived tensor $-\overset{\r
L}{\otimes}_{\Lambda_n}\Lambda_m$ for $m\to n$, in the $2$-category of
categories. Moreover, he defines a subcategory
$\rD^\rb_\cons(X,\Lambda_{\bullet})$ as the $2$-limit of
$\rD^\rb_{\cons,\ftd}(X,\Lambda_n)$, where $\ftd$ stand for finite
Tor-dimension. If $\Hom(K_n,L_n)$ is finite for all $K,L\in
\rD^\rb_\cons(X,\Lambda_\bullet)$ (this is the case if $X$ is of finite type
over a finite field or an algebraically closed field), then
$\rD^\rb_{\cons}(X,\Lambda_\bullet)$ is a triangulated category.
We provide a refinement of Deligne's construction via limit in (part of)
\Secs \ref{1ss:natural_transformation}, \ref{1ss:evaluation_functor} and
\ref{1ss:adic_complexes}. Indeed, if we consider the enhancement
$\cD(\cX,\lambda)_{\adic}$, which is a presentable stable $\infty$-category,
of $\rD(\cX,\lambda)_{\adic}$, then $\cD(\cX,\lambda)_{\adic}$ is naturally
the $\infty$-categorical limit of the diagram $\xi\to\cD(\cX,\Lambda(\xi))$,
where the transition functors are the ($\infty$-categorical) derived tensor
$-{\otimes}_{\Lambda(\xi')}\Lambda(\xi)$ for $\xi\to\xi'$. Here,
$\lambda=(\Xi,\Lambda)$ is an arbitrary ringed diagram and $\cX$ is an Artin
stack (see Corollary \ref{1co:enhanced_adic_limit}).
\subsection{Acknowledgments}
We thank Ofer Gabber, Luc Illusie, Aise Johan de~Jong, Jo\"el Riou, Shenghao
Sun, and Xinwen Zhu for useful conversations. Part of this work was done
during a visit of the first author to the Morningside Center of Mathematics,
Chinese Academy of Sciences, in Beijing. He thanks the Center for its
hospitality. The first author was partially supported by NSF grant
DMS--1302000. The second author was partially supported by China's
Recruitment Program of Global Experts; National Natural Science Foundation of
China Grant 11321101; National Center for Mathematics and Interdisciplinary
Sciences and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of
Sciences.
\section{The adic formalism}
\label{1}
In this chapter, we provide the adic formalism. In \Sec
\ref{1ss:adic_object}, we work in the general nonsense and define the
abstract ``adic objects'' in a system assigning each coefficient a (diagram
of) $\infty$-category. We then define the adic category to be the full
subcategory spanned by adic objects. In \Sec
\ref{1ss:natural_transformation}, we study a natural and fundamental relation
between the adic category and certain limit category constructing from the
same system, refining a consideration of Deligne. In \Sec
\ref{1ss:evaluation_functor}, we record some simple but important properties
of the system arising from algebraic geometry, namely, those
$\infty$-categories that are derived categories of (lisse-)\'{e}tale sheaves
on schemes or Artin stacks. In this geometric case, we may identify the adic
category and the limit category mentioned previously, which is proved in \Sec
\ref{1ss:adic_complexes}. In \Sec \ref{1ss:enhanced_six}, we construct the
enhanced adic operation maps and study the usual $t$-structure on the adic
categories. \Sec \ref{1ss:adic_dualizing}, we introduce the adic dualizing
complex and prove the biduality property, which will be used later to prove
the compatibility between our theory and Laszlo--Olsson's \cites{LO,LO2}
under their restrictions.
\subsection{Adic objects}
\label{1ss:adic_object}
Let $\cR\subseteq \Ring$ be a full subcategory. We denote by $\RindR\subseteq
\Rind$ the full subcategory spanned by those ringed diagrams $(\Xi,\Lambda)$
such that $\Lambda\colon \Xi^{op}\to \Ring$ factorizes through $\cR$. In
particular, if $\c{R}$ is the subcategory spanned by torsion (resp.\
$\sf{L}$-torsion) rings, then $\RindR$ is $\Rind_{\tor}$ (resp.\
$\Rind_{\ltor}$).
Let $\cC\colon\N(\RindR)^{op}\to\PSL$ be a functor. We denote by
$\cC(\Xi,\Lambda)$ the image of $(\Xi,\Lambda)$ under $\cC$. Fix an object
$(\Xi,\Lambda)$ of $\RindR$. For every morphism $\varphi\colon \xi\to\xi'$
in $\Xi$, there is a commutative diagram in $\RindR$ of the form
\begin{align*}
\xymatrix{
(\Xi,\Lambda) \ar[d]_-{=} & (\Xi_{/\xi},\Lambda_{/\xi})
\ar[d]^-{i_{\varphi}} \ar[l]_-{i_{\xi}} \ar[r]^-{p_{\xi}}
& (\{\xi\},\Lambda(\xi)) \ar[d]^-{\tilde{\varphi}} \\
(\Xi,\Lambda) & (\Xi_{/\xi'},\Lambda_{/\xi'}) \ar[l]_-{i_{\xi'}} \ar[r]^-{p_{\xi'}} & (\{\xi'\},\Lambda(\xi')), }
\end{align*}
which induces the following diagram in $\PSL$ by applying $\cC$:
\begin{align*}
\xymatrix{
\cC(\Xi,\Lambda) \ar[r]^-{i_{\xi}^*} & \cC(\Xi_{/\xi},\Lambda_{/\xi})
& \cC(\{\xi\},\Lambda(\xi)) \ar[l]_-{p_{\xi}^*} \\
\cC(\Xi,\Lambda) \ar[r]^-{i_{\xi'}^*} \ar[u]^-{=} & \cC(\Xi_{/\xi'},\Lambda_{/\xi'})
\ar[u]_-{i_{\varphi}^*} & \cC(\{\xi'\},\Lambda(\xi')) \ar[l]_-{p_{\xi'}^*} \ar[u]_-{\tilde{\varphi}^*}, }
\end{align*}
where $-^*=\cC(-)$. Let $p_{\xi*}$ (resp.\ $p_{\xi'*}$) be a right adjoint
of $p_{\xi}^*$ (resp.\ $p_{\xi'}^*$) and $\alpha_{\varphi}\colon
\tilde{\varphi}^*p_{\xi'*}\to p_{\xi*}i_{\varphi}^*$ be the natural
transformation.
\begin{definition}[Adic object]\label{1de:adic_object}
An object $\s{K}$ of $\cC(\Xi,\Lambda)$ is \emph{adic} (with respect to
$\cC$) if the natural morphism
\begin{align*}
\tilde{\varphi}^*p_{\xi'*}i_{\xi'}^*\s{K}\xrightarrow{\alpha_{\varphi}(i_{\xi'}^*\s{K})}
p_{\xi*}i_{\varphi}^*i_{\xi'}^*\s{K}
\end{align*}
is an equivalence for every morphism $\varphi\colon \xi\to\xi'$ in $\Xi$.
The target of $\alpha_{\varphi}(i_{\xi'}^*\s{K})$ is equivalent to
$p_{\xi*}i_{\xi}^*\s{K}$. It is clear that adic objects are stable under
equivalence.
\end{definition}
Let $\cC\colon\N(\RindR)^{op}\to\PSL$ be a functor and $(\Xi,\Lambda)$ be an
object of $\RindR$. We denote by
$\cC(\Xi,\Lambda)_{\adic}\subseteq\cC(\Xi,\Lambda)$ the (strictly) full
subcategory spanned by adic objects. It is clearly a stable $\infty$-category
and the inclusion is an exact functor. We emphasize that the full subcategory
$\cC(\Xi,\Lambda)_{\adic}$ depends on the functor
$\cC\colon\N(\RindR)^{op}\to\PSL$, not just on $\cC(\Xi,\Lambda)$. If $\cC'$
is another such functor, then $(\cC\times \cC')(\Xi,\Lambda)_\adic$ is
equivalent to $\cC(\Xi,\Lambda)_\adic\times \cC'(\Xi,\Lambda)_\adic$.
\begin{definition}\label{1de:topological}\leavevmode
\begin{enumerate}
\item A functor $\cC\colon \N(\RindR)^{op}\to\Cat$ is
\emph{topological} if
\begin{enumerate}
\item $\cC$ factorizes through $\PSL$.
\item For every object $(\Xi,\Lambda)$ of $\RindR$ such that
$\Xi$ admits a final object, $\cC(\Xi,\Lambda)_{\adic}$
is presentable.
\item For every morphism $(\Gamma,\gamma)\colon
(\Xi^0,\Lambda^0)\to(\Xi^1,\Lambda^1)$ of $\RindR$ that
carries the final object $\xi^0$ of $\Xi^0$ to the final
object $\xi^1$ of $\Xi^1$, the following diagram
\[\xymatrix{
\cC(\Xi^1,\Lambda^1) \ar[d] & \cC(\{\xi^1\},\Lambda^1(\xi^1)) \ar[d]\ar[l]_-{p_{\xi^1}^*} \\
\cC(\Xi^0,\Lambda^0) & \cC(\{\xi^0\},\Lambda^0(\xi^0))
\ar[l]_-{p_{\xi^0}^*} }\] is right adjointable. Moreover, its right
adjoint is in $\PSL$.
\end{enumerate}
\item A morphism $\delta\colon\cC^0\to\cC^1$ in
$\Fun(\N(\RindR)^{op},\Cat)$ is \emph{topological} if
\begin{enumerate}
\item Both $\cC^0$ and $\cC^1$ are topological.
\item For every object $(\Xi,\Lambda)$ of $\RindR$ with the
final object $\xi$, the following diagram
\[\xymatrix{
\cC^0(\Xi,\Lambda) \ar[d] & \cC^0(\{\xi\},\Lambda(\xi)) \ar[d]\ar[l]_-{p_{\xi}^{0*}} \\
\cC^1(\Xi,\Lambda) & \cC^1(\{\xi\},\Lambda(\xi)) \ar[l]_-{p_{\xi}^{1*}}
}\] is right adjointable.
\end{enumerate}
\item We denote by $\Fun^{\top}(\N(\RindR)^{op},\Cat)$ (resp.\
$\Fun^{\top}(\N(\RindR)^{op},\PSL)$) the subcategory of
$\Fun(\N(\RindR)^{op},\Cat)$ (resp.\ $\Fun(\N(\RindR)^{op},\PSL)$)
spanned by topological functors and topological morphisms. In other
words, an $n$-cell of $\Fun(\N(\RindR)^{op},\Cat)$ (resp.\
$\Fun(\N(\RindR)^{op},\PSL)$) belongs to
$\Fun^{\top}(\N(\RindR)^{op},\Cat)$ (resp.\
$\Fun^{\top}(\N(\RindR)^{op},\PSL)$) if all its vertices are
topological functors and edges are topological morphisms.
\end{enumerate}
\end{definition}
\begin{lem}\label{1le:preserve_adic}
We have
\begin{enumerate}
\item Let $\cC\colon \N(\RindR)^{op}\to\Cat$ be a topological functor.
Then for every morphism
$(\Gamma,\gamma)\colon(\Xi^0,\Lambda^0)\to(\Xi^1,\Lambda^1)$ in
$\RindR$, the functor $\cC(\Gamma,\gamma)\colon
\cC(\Xi^1,\Lambda^1)\to\cC(\Xi^0,\Lambda^0)$ preserves adic
objects, that is, it carries adic objects to adic objects.
\item Let $\delta\colon \cC^0\to\cC^1$ be a topological morphism. Then
for every object $(\Xi,\Lambda)$ of $\RindR$, the functor
$\delta(\Xi,\Lambda)\colon \cC^0(\Xi,\Lambda)\to\cC^1(\Xi,\Lambda)$
preserves adic objects.
\end{enumerate}
\end{lem}
\begin{proof}
We prove (2), and (1) is similar. Let $\varphi\colon\xi\to\xi'$ be a morphism
in $\Xi$. Consider the following diagram with all solid squares commutative
(up to obvious homotopies):
\begin{align*}
\xymatrix{
& \cC^0(\Xi,\Lambda) \ar[dl]_-{\delta(\Xi,\Lambda)} \ar[rr]^-{i^{0*}_{\xi}}
& & \cC^0_{/\xi} \ar[dl]_-{\delta_{/\xi}} \ar@{-->}[rr]^-{p_{\xi*}^0}
& & \cC^0(\xi) \ar[dl]^-{\delta(\xi)} \\
\cC^1(\Xi,\Lambda) \ar[rr]^<>(0.3){i^{1*}_{\xi}}
& & \cC^1_{/\xi} \ar@{-->}[rr]^-<>(0.3){p_{\xi*}^1}
& & \cC^1(\xi) & \\
& \cC^0(\Xi,\Lambda) \ar[dl]_-{\delta(\Xi,\Lambda)}
\ar[rr]^<>(0.3){i^{0*}_{\xi'}}|(.55)\hole \ar[uu]^<(0.3){=}|\hole
& & \cC^0_{/\xi'} \ar[dl]^<<{\delta_{/\xi'}}
\ar@{-->}[rr]^<>(0.3){p_{\xi'*}^0}|\hole \ar[uu]_<>(0.3){i^{0*}_{\varphi}}|\hole
& & \cC^0(\xi') \ar[dl]^-{\delta(\xi')} \ar[uu]_<>(0.3){\tilde{\varphi}^{0*}} \\
\cC^1(\Xi,\Lambda) \ar[rr]^-{i^{1*}_{\xi'}} \ar[uu]^<>(0.3){=}
& & \cC^1_{/\xi'} \ar@{-->}[rr]^-{p_{\xi'*}^1} \ar[uu]^<>(0.3){i^{1*}_{\varphi}}
& & \cC^1(\xi') \ar[uu]_<>(0.3){\tilde{\varphi}^{1*}}, & }
\end{align*}
where we have written $\delta_{/\xi}\colon \cC^0_{/\xi}\to\cC^1_{/\xi}$
instead of $\delta(\Xi_{/\xi},\Lambda_{/\xi})\colon
\cC^0(\Xi_{/\xi},\Lambda_{/\xi})\to\cC^1(\Xi_{/\xi},\Lambda_{/\xi})$ and
$\delta(\xi)\colon \cC^0(\xi)\to\cC^1(\xi)$ instead of
$\delta(\{\xi\},\Lambda(\xi))\colon
\cC^0(\{\xi\},\Lambda(\xi))\to\cC^1(\{\xi\},\Lambda(\xi))$ for short. By
definition, we need to show that for every adic object $\s{K}$ of
$\cC^0(\Xi,\Lambda)$, the natural morphism
\begin{align*}
\tilde{\varphi}^{1*}p^1_{\xi'*}i_{\xi'}^{1*}\delta(\Xi,\Lambda)\s{K}
\xrightarrow{\alpha^1_{\varphi}(i_{\xi'}^{1*}\delta(\Xi,\Lambda)\s{K})}
p^1_{\xi*}i_{\xi}^{1*}\delta(\Xi,\Lambda)\s{K}
\end{align*}
is an equivalence. But we have the following commutative diagram in
$\H\cC^1(\xi)$
\begin{align*}
\xymatrix{
\delta(\xi)\tilde{\varphi}^{0*}p^0_{\xi'*}i_{\xi'}^{0*}\s{K}
\simeq\tilde{\varphi}^{1*}\delta(\xi')p^0_{\xi'*}i_{\xi'}^{0*}\s{K}
\ar@<-10ex>[d]^-{\delta(\xi)(\alpha^0_{\varphi}(i_{\xi'}^{0*}\s{K}))} \ar[r]
& \tilde{\varphi}^{1*}p^1_{\xi'*}\delta_{/\xi'}i_{\xi'}^{0*}\s{K}\simeq
\tilde{\varphi}^{1*}p^1_{\xi'*}i_{\xi'}^{1*}\delta(\Xi,\Lambda)\s{K}
\ar@<10ex>[d]_-{\alpha^1_{\varphi}(i_{\xi'}^{1*}\delta(\Xi,\Lambda)\s{K})} \\
\delta(\xi)p^0_{\xi*}i_{\phi}^{0*}i_{\xi'}^{0*}\s{K}
\simeq\delta(\xi)p^0_{\xi*}i_{\xi}^{0*}\s{K} \ar[r]
& p^1_{\xi*}\delta_{/\xi}i_{\xi}^{0*}\s{K} \simeq p^1_{\xi*}i_{\xi}^{1*}\delta(\Xi,\Lambda)\s{K}, }
\end{align*}
where the horizontal arrows are equivalences since $\delta$ is topological.
Therefore, $\alpha^1_{\varphi}(i_{\xi'}^{1*}\delta(\Xi,\Lambda)\s{K})$ is an
equivalence since $\alpha^0_{\varphi}(i_{\xi'}^{0*}\s{K})$ is.
\end{proof}
\begin{lem}\label{1le:limit_topological}
The $\infty$-category $\Fun^{\top}(\N(\RindR)^{op},\PSL)$ admits small limits
and those limits are preserved under the inclusion
$\Fun^{\top}(\N(\RindR)^{op},\PSL)\subseteq\Fun(\N(\RindR)^{op},\PSL)$.
\end{lem}
\begin{proof}
We will frequently use \cite{Lu2}*{6.2.3.18} in this article.
First, we show that for a limit diagram $\cC\colon
K^{\triangleleft}\to\Fun(\N(\RindR)^{op},\PSL)$ such that $\cC\res K$
factorizes through $\Fun^{\top}(\N(\RindR)^{op},\PSL)$, then the limit
functor $\cC^{-\infty}$ is topological. To check the requirement (1c), we
pick a morphism $(\Gamma,\gamma)\colon (\Xi^0,\Lambda^0)\to(\Xi^1,\Lambda^1)$
of $\RindR$ that sends the final object $\xi^0$ of $\Xi^0$ to the final
object $\xi^1$ of $\Xi^1$. By assumption, the functor $(\cC\res
K)\res(\Delta^1\times\Delta^1)^{op}$, where $\Delta^1\times\Delta^1$ is the
diagram
\[\xymatrix{
(\Xi^0,\Lambda^0) \ar[d]_-{(\Gamma,\gamma)}\ar[r]^-{p_{\xi^0}} & (\{\xi^0\},\Lambda^0(\xi^0)) \ar[d] \\
(\Xi^1,\Lambda^1) \ar[r]^-{p_{\xi^1}} & (\{\xi^1\},\Lambda^1(\xi^1)), }\] can
be viewed as a functor $K\times\Delta^1\to\Fun^{\r{RAd}}(\Delta^1,\Cat)$. By
\cite{Lu2}*{6.2.3.18} and the fact that $\PSL\subseteq\CAT$ preserves small
limits, the limit functor $\cC^{-\infty}$ satisfies (c), and the limit
diagram factorizes through $\PSL$. In particular, for every object $k$ of
$K$, the natural morphism $\cC^{-\infty}\to\cC^k=\cC(k)$ satisfies condition
(2b). To check that $\cC^{-\infty}$ satisfies condition (1b), we only need to
show that the diagram $\cD\colon K^\triangleleft \to
\Fun(\N(\RindR^{op}),\CAT)$ that carries $k$ to $(\Xi,\Lambda)\mapsto
\cC^k(\Xi,\Lambda)_{\adic}$ is a limit diagram. Let
$\cC^{\infty}(\Xi,\Lambda)'$ be the limit $\lim_K\cC^k(\Xi,\Lambda)_{\adic}$,
which is naturally a strictly full subcategory of $\cC^{\infty}(\Xi,\Lambda)$
and contains $\cC^{\infty}(\Xi,\Lambda)_{\adic}$ by \cite{LZ}*{3.1.5} and the
proof of Lemma \ref{1le:preserve_adic} (2). We need to prove
$\cC^\infty(\Xi,\Lambda)'\supseteq \cC^\infty(\Xi,\Lambda)_{\adic}$. By
definition, we may assume $\Xi$ admits a final object $\xi'$ and need to show
that for every other object $\xi\in\Xi$, the diagram
\[\xymatrix{
\cC^{\infty}(\Xi_{/\xi},\Lambda_{/\xi})' & \cC^{\infty}(\{\xi\},\Lambda(\xi)) \ar[l]_-{p_{\xi}^*} \\
\cC^{\infty}(\Xi,\Lambda)' \ar[u]^-{i_{\varphi}^*} & \cC^{\infty}(\{\xi'\},\Lambda(\xi')) \ar[u]_-{\tilde\varphi^*} \ar[l]_-{p_{\xi'}^*}
}
\]
induced by the map $\phi\colon \xi\to \xi'$ is right adjointable. This
follows from the fact that, for every $k\in K$, the corresponding diagram
with $\cC^\infty(-)$ replaced by $\cC^k(-)$ and $\cC^\infty(-)'$ replaced by
$\cC^k(-)_{\adic}$ is right adjointable.
Second, we show that for an arbitrary diagram $\cC\colon
K^{\triangleleft}\to\Fun^{\top}(\N(\RindR)^{op},\PSL)$, the induced morphism
$\cC^{-\infty}\to\lim(\cC\res K)$ in $\Fun(\N(\RindR^{op}),\CAT)$ is
topological. To check condition (2b), we pick an object $(\Xi,\Lambda)$ of
$\RindR$ with a final object $\xi$ of $\Xi$. Then this again follows from
\cite{Lu2}*{6.2.3.18}.
\end{proof}
\begin{lem}\label{1le:pullback}
Let $\cD$ be a full subcategory of an $\infty$-category $\cC$ and $f\colon \cD\to
\cC$ be the inclusion. Then the pullback of $f$ in the category $\Sset$ by
any functor $g\colon \cC'\to \cC$ with source in $\CAT$ is a pullback in $\CAT$.
\end{lem}
\begin{proof}
This follows immediately from \cite{LZ}*{3.1.5} applied to the pullback of
$\id_\cC$ by $g$.
\end{proof}
\begin{proposition}\label{1pr:adic_inclusion}
Let $\cC\colon \N(\RindR)^{op}\to\PSL$ be a topological functor. For every
object $(\Xi,\Lambda)$, the inclusion
$\cC(\Xi,\Lambda)_{\adic}\to\cC(\Xi,\Lambda)$ is a morphism in $\PSL$.
\end{proposition}
By the proposition, the inclusion
$\cC(\Xi,\Lambda)_{\adic}\to\cC(\Xi,\Lambda)$ admits a right adjoint
$R\colon\cC(\Xi,\Lambda)\to\cC(\Xi,\Lambda)_{\adic}$ which we call the
\emph{colocalization functor}.
\begin{proof}
By definition, the inclusion
$\cC(\Xi,\Lambda)_{\adic}\subseteq\cC(\Xi,\Lambda)$ fits into the following
diagram
\begin{align*}
\xymatrix{
\cC(\Xi,\Lambda)_{\adic} \ar[rr]\ar[d] && \prod_{\xi\in\Ob(\Lambda)}\cC(\Xi_{/\xi},\Lambda_{/\xi})_{\adic} \ar[d] \\
\cC(\Xi,\Lambda) \ar[rr]^-{\prod_{\xi\in\Ob(\Lambda)}i_{\xi}^*} &&
\prod_{\xi\in\Ob(\Lambda)}\cC(\Xi_{/\xi},\Lambda_{/\xi}), }
\end{align*}
which is a pullback diagram in $\CAT$ by the above lemma. Since $p_{\xi*}$
commutes with small colimits by condition (1c) of Definition
\ref{1de:topological}, $\cC(\Xi_{/\xi},\Lambda_{/\xi})_{\adic}$ admits small
colimits and the inclusion into $\cC(\Xi_{/\xi},\Lambda_{/\xi})$ preserves
such colimits. By condition (1b) of Definition \ref{1de:topological}, the
source is presentable (and stable), so the above inclusion is a morphism in
$\PSL$. Therefore, the right vertical arrow is a morphism in $\PSL$ since
$\Lambda$ is small. Moreover, the functor
$\prod_{\xi\in\Ob(\Lambda)}i_{\xi}^*$ preserves small colimits since each
$i_{\xi}^*$ does and $\Lambda$ is small. Therefore, the inclusion
$\cC(\Xi,\Lambda)_{\adic}\to\cC(\Xi,\Lambda)$ is a morphism in $\PSL$,
because the inclusion $\PSL\subseteq \CAT$ preserves small limits.
\end{proof}
\begin{corollary}\label{1co:adic_right_adjoint}
Let $\cC\colon \N(\RindR)^{op}\to\PSL$ be a topological functor and
$(\Xi,\Lambda)$ be an object of $\RindR$. Let $F\colon
\cC(\Xi,\Lambda)\to\cC(\Xi,\Lambda)$ be a functor admitting right adjoints
such that the set $F(\cC(\Xi,\Lambda)_{\adic})$ is contained in
$\cC(\Xi,\Lambda)_{\adic}$. Then the induced functor
$F\res\cC(\Xi,\Lambda)_{\adic}\colon
\cC(\Xi,\Lambda)_{\adic}\to\cC(\Xi,\Lambda)_{\adic}$ admits right adjoints
as well.
\end{corollary}
\begin{proof}
By the above proposition, $\cC(\Xi,\Lambda)_{\adic}$ is presentable (without
assuming that $\Xi$ admits the final object). Therefore, by Adjoint Functor
Theorem, we only need to show that $F\res\cC(\Xi,\Lambda)_{\adic}$ preserves
small colimits. This then follows from the fact that the inclusion
$\cC(\Xi,\Lambda)_{\adic}\subseteq\cC(\Xi,\Lambda)$ preserves small colimits,
and the assumptions on $F$.
\end{proof}
To conclude this section, we introduce another formalism, which makes use of
a limit construction instead of adic objects. This can be seen as a
refinement of Deligne's construction \cite{WeilII}*{1.1.2}.
\begin{notation}
Let $\cC\colon \N(\RindR)^{op}\to\PSL$ be a topological functor. For every
object $\lambda=(\Xi,\Lambda)$ of $\RindR$, there is a tautological functor
$\lambda^*\colon \Xi\to\RindR$ sending $\xi$ to $(*,\Lambda(\xi))$. Composing
with $\cC$, we obtain a diagram $\cC\circ\N(\lambda^*)^{op}\colon
\N(\Xi)^{op}\to\N(\RindR)^{op}\to\PSL$. Let $\underline{\cC}(\Xi,\Lambda)$ be
the limit $\lim\cC\circ\N(\lambda^*)^{op}$, viewed as an object of $\PSL$. We
will construct a natural functor
$\cC(\Xi,\Lambda)_{\adic}\to\underline{\cC}(\Xi,\Lambda)$ in \Sec
\ref{1ss:natural_transformation} (in a coherent way) and show in \Sec
\ref{1ss:adic_complexes} that it is an equivalence if $\cC$ comes from the
derived $\infty$-categories of (higher) stacks.
\end{notation}
\subsection{The natural transformations $\alpha_{\c{R}}$ and $\alpha_{\c{R}}^{\otimes}$}
\label{1ss:natural_transformation}
\begin{notation}
We denote by
\[\Fun^{\top}(\N(\RindR)^{op},\Mon_{\Pf}^{\PSL}(\Cat))\subseteq
\Fun(\N(\RindR)^{op},\Mon_{\Pf}^{\PSL}(\Cat))\] the subcategory spanned by
cells that factorize through $\Fun^{\top}(\N(\RindR)^{op},\Cat)$. Here we
view $\Fun(\N(\RindR)^{op},\Mon_{\Pf}^{\PSL}(\Cat))$ as a subcategory of
$\Fun(\Pfo,\Fun(\N(\RindR)^{op},\Cat))$.
\end{notation}
We construct two endofunctors
\begin{align}\label{1eq:adicr}
\AdicR,\AdR\colon \Fun^{\top}(\N(\RindR)^{op},\Mon_{\Pf}^{\PSL}(\Cat))
\to\Fun^{\top}(\N(\RindR)^{op},\Mon_{\Pf}^{\PSL}(\Cat)),
\end{align}
and a natural transformation $\alpha_{\c{R}}\colon\AdicR\to\AdR$. If we
represent objects of the functor category by their compositions with the
restriction functor $\rG_\zeta$ in \cite{LZ}*{1.5.8 (4)}, then $\AdicR$
(resp.\ $\AdR$) sends an object $\cC'\times\cC
\xrightarrow{-\otimes_{\cC'}f^*-} \cC'$ to
\[\cC'_{\adic}\times\cC_{\adic}\xrightarrow{-\otimes_{\cC'}f^*-}
\cC'_{\adic}\quad(\text{resp.\
}\underline{\cC}'\times\underline{\cC}\xrightarrow{-\otimes_{\cC'}f^*-}
\underline{\cC}').
\]
Here, $\cC_{\adic}$ (resp.\ $\underline{\cC}$) takes the value
$\cC_{\adic}(\Xi,\Lambda)$ (resp.\ $\underline{\cC}(\Xi,\Lambda)$) at
$(\Xi,\Lambda)$, which justifies the notation.
By definition, there is a tautological functor $\RindR\to\cat$ sending
$(\Xi,\Lambda)$ to $\Xi$. Applying Grothendieck's construction, we obtain an
op-fibration $\pi\colon \RindR^{\univ}\to\RindR$. The objects of
$\RindR^{\univ}$ are pairs $((\Xi,\Lambda),\xi)$ where $(\Xi,\Lambda)$ is an
object of $\RindR$ and $\xi$ is an object of $\Xi$. Moreover, there are
functors $p^{\univ}\colon \RindR^{\univ}\times[1]\to\RindR$ sending
$((\Xi,\Lambda),\xi)\times[1]$ to $p_{\xi}\colon
(\Xi_{/\xi},\Lambda_{/\xi})\to(\{\xi\},\Lambda(\xi))$, and $i^{\univ}\colon
\RindR^{\univ}\diamond_{\RindR}\RindR\to\RindR$ sending the morphism
$((\Xi,\Lambda),\xi)\to(\Xi,\Lambda)$ to $i_{\xi}\colon
(\Xi_{/\xi},\Lambda_{/\xi})\to(\Xi,\Lambda)$.
Let
\[\cC^{\univ}\colon \N(\RindR)^{op}\to\Fun(\Fun^{\top}(\N(\RindR)^{op},\Mon_{\Pf}^{\PSL}(\Cat))\times\Pfo,\Cat)\]
be the universal functor, and let $\cC^{\univ}_{\adic}$ be the functor with
the same source and target as $\cC^{\univ}$ and carrying $(\Xi,\Lambda)$ to
$(\cC,X)\mapsto (\rG_X\circ \cC)_{\adic}(\Xi,\Lambda)$. Here $\rG_X\colon
\Mon_{\Pf}^{\PSL}(\Cat)\to\PSL$ is the evaluation functor at $X$ defined in
\cite{LZ}*{1.5.8 (4)}. By the definition of adic objects, we can apply
partial right adjunction to $\cC^{\univ}_{\adic}\circ\N(p^{\univ})^{op}$ with
respect to direction 2 to obtain a functor
\[\N(\RindR^{\univ})^{op}\times\Delta^1\to\Fun(\Fun^{\top}(\N(\RindR)^{op},\Mon_{\Pf}^{\PSL}(\Cat))\times\Pfo,\Cat).\]
Composing this with $\cC^{\univ}_{\adic}\circ\N(i^{\univ})^{op}$, we obtain a
functor
\[F\colon \N(\RindR)^{op}\diamond_{\N(\RindR)^{op}}\N(\RindR^{\univ})^{op}
\to\Fun(\Fun^{\top}(\N(\RindR)^{op},\Mon_{\Pf}^{\PSL}(\Cat))\times\Pfo,\Cat),\]
whose restriction to $\N(\RindR)^{op}$ is (equivalent to)
$\cC^{\univ}_{\adic}$ and restriction to $\N(\RindR^\univ)^{op}$ is
(equivalent to) $\cC^{\univ}_{\adic}\circ \N(t)^{op}$, where $t\colon
\RindR^\univ\to \RindR$ is the functor carrying $((\Xi,\Lambda),\xi)$ to
$(\{\xi\},\Lambda(\xi))$. Let $F_0$ be the right Kan extension of
$F\res\N(\RindR^{\univ})^{op}$ along the inclusion
$\N(\RindR^{\univ})^{op}\subseteq\N(\RindR)^{op}\diamond_{\N(\RindR)^{op}}\N(\RindR^{\univ})^{op}$.
Then the restriction of $F\to F_0$ to $\N(\RindR)^{op}$ provides the desired
natural transformation $\alpha_{\c{R}}\colon\AdicR\to\AdR$. To see this, the
only nontrivial point is the following: By construction and the definition
of right Kan extensions, for every object $\lambda$ of $\RindR$,
$F_0(\lambda)$ is the limit of the diagram
\begin{align*}
&\N(\RindR)^{op}_{\lambda/}\times_{\N(\RindR)^{op}}\N(\RindR^{\univ})^{op}\to\N(\RindR^{\univ})^{op}\\
&\xrightarrow{F}\Fun(\Fun^{\top}(\N(\RindR)^{op},\Mon_{\Pf}^{\PSL}(\Cat))\times\Pfo,\Cat),
\end{align*}
which is equivalent to the limit of the diagram
\begin{align*}
&\{\lambda\}\times_{\N(\RindR)^{op}}\N(\RindR^{\univ})^{op}\to\N(\RindR^{\univ})^{op}\\
&\xrightarrow{F}\Fun(\Fun^{\top}(\N(\RindR)^{op},\Mon_{\Pf}^{\PSL}(\Cat))\times\Pfo,\Cat),
\end{align*}
by \cite{Lu1}*{4.1.2.10, 4.1.2.15}. The latter diagram, restricted to each
fixed object $(\cC,X)$ of
$\Fun^{\top}(\N(\RindR)^{op},\Mon_{\Pf}^{\PSL}(\Cat))\times\Pfo$, is nothing
but $\rG_X\circ \cC\circ\N(\lambda^*)^{op}$.
\begin{notation}
We denote by $\Fun^{\top}(\N(\RindR)^{op},\PSLM))$ the subcategory of
$\Fun(\N(\RindR)^{op},\PSLM)$ spanned by cells that factorize through
$\Fun^{\top}(\N(\RindR)^{op},\Cat)$ after viewing
$\Fun(\N(\RindR)^{op},\PSLM)$ as a subcategory of
$\Fun(\N(\Fin),\Fun(\N(\RindR)^{op},\Cat))$.
\end{notation}
We construct two endofunctors
\begin{align}\label{1eq:adicrt}
\AdicRT,\AdRT\colon \Fun^{\top}(\N(\RindR)^{op},\PSLM)\to\Fun^{\top}(\N(\RindR)^{op},\PSLM),
\end{align}
and a natural transformation $\alpha^{\otimes}_{\c{R}}\colon\AdicRT\to\AdRT$,
such that $\AdicRT$ (resp.\ $\AdRT$) sends an object $\cC^{\otimes}$ to
$\cC_{\adic}^{\otimes}$ (resp.\ $\underline{\cC}^{\otimes}$), where
$\(\rG\circ\cC^{\otimes}_\adic\)(\Xi,\Lambda)=\(\rG\circ\cC^{\otimes}\)(\Xi,\Lambda)_{\adic}$
(resp.\
$\(\rG\circ\underline{\cC}^{\otimes}\)(\Xi,\Lambda)=\underline{\(\rG\circ\cC^{\otimes}\)}(\Xi,\Lambda)$).
Here $\rG\colon \PSLM\to \PSL$ is the forgetful functor defined in
\cite{LZ}*{1.5.3}. The construction of $\alpha^{\otimes}_{\c{R}}$ is exactly
the same as that of $\alpha_{\c{R}}\colon \AdicR\to\AdR$.
\begin{remark}\label{1re:coef_compatibility}
We have the following properties of compatibility.
\begin{enumerate}
\item If $\c{R}'\subseteq\c{R}$, then the following diagram
\[\xymatrix{
\Fun^{\top}(\N(\RindR)^{op},\Mon_{\Pf}^{\PSL}(\Cat))\ar[r]\ar[d]_-{\AdicR} &
\Fun^{\top}(\N(\Rind_{\c{R}'})^{op},\Mon_{\Pf}^{\PSL}(\Cat)) \ar[d]^-{\Adic_{\c{R}'}} \\
\Fun^{\top}(\N(\RindR)^{op},\Mon_{\Pf}^{\PSL}(\Cat)) \ar[r] &
\Fun^{\top}(\N(\Rind_{\c{R}'})^{op},\Mon_{\Pf}^{\PSL}(\Cat))
}\] commutes up to homotopy. The same holds for $\AdR$ and $\alpha_{\c{R}}$.
\item The map $\pf$ \cite{LZ}*{(1.1)} induces a map
\begin{align*}
\Fun(\N(\RindR)^{op},\pf)&\colon\Fun((\Delta^1)^{op},\Fun^{\top}(\N(\RindR)^{op},\PSLM))\\
&\to\Fun^{\top}(\N(\RindR)^{op},\Mon_{\Pf}^{\PSL}(\Cat)).
\end{align*}
Moreover, by construction, the following diagram
\[\xymatrix{
\Fun((\Delta^1)^{op},\Fun^{\top}(\N(\RindR)^{op},\PSLM)) \ar[rrr]^-{\Fun(\N(\RindR)^{op},\pf)} \ar[d]_-{\Fun((\Delta^1)^{op},\AdicRT)}
&&& \Fun^{\top}(\N(\RindR)^{op},\Mon_{\Pf}^{\PSL}(\Cat)) \ar[d]^-{\AdicR} \\
\Fun((\Delta^1)^{op},\Fun^{\top}(\N(\RindR)^{op},\PSLM)) \ar[rrr]^-{\Fun(\N(\RindR)^{op},\pf)}
&&& \Fun^{\top}(\N(\RindR)^{op},\Mon_{\Pf}^{\PSL}(\Cat))
}\] commutes up to homotopy. We have the same phenomena for $\AdR$ and $\alpha_{\c{R}}$.
\end{enumerate}
\end{remark}
The following lemma is similar to Lemma \ref{1le:limit_topological} and will
be used in \Sec \ref{1ss:evaluation_functor}.
\begin{lem}\label{1le:monlimit_topological}
We have
\begin{enumerate}
\item The $\infty$-category
$\Fun^{\top}(\N(\RindR)^{op},\Mon_{\Pf}^{\PSL}(\Cat))$ admits small
limits and those limits are preserved under the inclusion
\[\Fun^{\top}(\N(\RindR)^{op},\Mon_{\Pf}^{\PSL}(\Cat))\subseteq\Fun(\N(\RindR)^{op},\Mon_{\Pf}^{\PSL}(\Cat)).\]
\item The $\infty$-category $\Fun^{\top}(\N(\RindR)^{op},\PSLM))$
admits small limits and those limits are preserved under the
inclusion
$\Fun^{\top}(\N(\RindR)^{op},\PSLM))\subseteq\Fun(\N(\RindR)^{op},\PSLM))$.
\end{enumerate}
\end{lem}
\begin{proof}
Both follow from Lemma \ref{1le:limit_topological}, (the dual of)
\cite{Lu1}*{5.1.2.3}, and \cite{Lu2}*{3.2.2.5}.
\end{proof}
\subsection{Evaluation functors}
\label{1ss:evaluation_functor}
Recall that we have maps
\begin{align*}
\EO{}{\Chparl}{}{}&\colon \delta^*_{2,\{2\}}\Fun(\Delta^1,\Chparl)^{\cart}_{F^0,A^{\to}}
\to\Fun(\N(\Rind_{\ltor})^{op},\Mon_{\Pf}^{\PSL}(\Cat));\\
\EO{}{\Chpdm{}}{}{}&\colon \delta^*_{2,\{2\}}\Fun(\Delta^1,\Chpdm{})^{\cart}_{F^0,A^{\to}}
\to\Fun(\N(\Rind_{\tor})^{op},\Mon_{\Pf}^{\PSL}(\Cat));\\
\EO{}{\Chpar{}}{*}{\otimes}&\colon(\Chpar{})^{op}\to\Fun(\N(\Rind)^{op},\PSLM).
\end{align*}
For an object $(\Xi,\Lambda)$ of $\Rind$ admitting the final object
$\xi\in\Xi$, we have both the projection
$(\Xi,\Lambda)\xrightarrow{p_{\xi}}(\{\xi\},\Lambda(\xi))$ and the inclusion
$(\{\xi\},\Lambda(\xi))\xrightarrow{s_{\xi}}(\Xi,\Lambda)$. Now if
$\cC=\EO{}{\Chpar{}}{*}{}(X)$, where $X$ is a higher Artin stack, then the
induced functors
$p_{\xi*},s_{\xi}^*\colon\cC(\Xi,\Lambda)\to\cC(\{\xi\},\Lambda(\xi))$ are
canonically and coherently equivalent in a sense that we now explain.
Let $\Rind^{\r{fin}}\subseteq\Rind$ be the subcategory spanned by objects
$(\Xi,\Lambda)$ admitting final objects and morphisms preserving final
objects. We have natural functors $\Rind^{\r{fin}}\to\Fun([1],\Rind)$
sending $(\Xi,\Lambda)$ to
$(\Xi,\Lambda)\xrightarrow{p_{\xi}}(\{\xi\},\Lambda(\xi))$ and
$(\{\xi\},\Lambda(\xi))\xrightarrow{s_{\xi}}(\Xi,\Lambda)$, respectively,
where $\xi$ is the final object of $\Xi$.
Apply the functor $\EO{}{\Chpar{}}{*}{\otimes}$ and the forgetful functor $
\PSLM \to \Cat^\otimes$. On the one hand, we obtain a functor
\[\EO{p}{\Chpar{}}{*}{\otimes}\colon(\Chpar{})^{op}\times\N(\Rind^{\r{fin}})^{op}\to\Fun(\Delta^1,\Cat^\otimes)\]
sending $(X,(\Xi,\Lambda))$ to
$\cD(X,(\{\xi\},\Lambda(\xi)))\xrightarrow{p_{\xi}^*}\cD(X,(\Xi,\Lambda))$.
On the other hand, we have
\[\EO{s}{\Chpar{}}{*}{\otimes}\colon(\Chpar{})^{op}\times\N(\Rind^{\r{fin}})^{op}\to\Fun(\Delta^1,\Cat^\otimes)\]
sending $(X,(\Xi,\Lambda))$ to
$\cD(X,(\Xi,\Lambda))\xrightarrow{s_{\xi}^*}\cD(X,(\{\xi\},\Lambda(\xi)))$.
We view these functors as
\[(\Chpar{})^{op}\times\N(\Rind^{\r{fin}})^{op}\times\N(\Fin)\to\Fun(\Delta^1,\Cat).\]
\begin{lem}\label{1le:evaluation}
The functor $\EO{p}{\Chpar{}}{*}{\otimes}$ factorizes through
$\Fun^{\r{RAd}}(\Delta^1,\Cat)$ and $\EO{s}{\Chpar{}}{*}{\otimes}$ factorizes
through $\Fun^{\r{LAd}}(\Delta^1,\Cat)$. Moreover,
$\EO{p}{\Chpar{}}{*}{\otimes}$ is equivalent to $\epsilon\circ
\EO{s}{\Chpar{}}{*}{\otimes}$, where $\epsilon$ is the equivalence
$\Fun^{\r{LAd}}(\Delta^1,\Cat) \to \Fun^{\r{RAd}}(\Delta^1,\Cat)$ in
\cite{Lu2}*{6.2.3.18 (2)}.
\end{lem}
In particular, $p_{\xi}^*$ is a left adjoint of $s_{\xi}^*$.
\begin{proof}
When $k=-2$, we have an equivalence between the two functors
$(\Schqcs)^{op}\times(\Rind)^{op}\to\Ab$ sending $(X,(\Xi,\Lambda))$ to
\emph{exact and monoidal} functors $p_{\xi*},s_{\xi}^*\colon
\Mod(X,(\Xi,\Lambda))\to\Mod(X,\Lambda(\xi))$, respectively. In this case
the lemma follows from the construction in \cite{LZ}*{2.2}.
In general, we first show by induction on $k$ that the functor
$\EO{p}{\Chpar{k}}{*}{\otimes}$ factorizes through
$\Fun^{\r{RAd}}(\Delta^1,\Cat)$. This holds by descent. Next we prove by
induction on $k$ that $\EO{p}{\Chpar{k}}{*}{\otimes}$ and $\epsilon\circ
\EO{s}{\Chpar{k}}{*}{\otimes}$ are equivalent. This holds by
\cite{LZ}*{4.1.1} since smooth surjective morphisms are of both
$\EO{p}{\Chpar{}}{*}{\otimes}$ and $\EO{s}{\Chpar{}}{*}{\otimes}$-descent.
\end{proof}
\begin{remark}\label{1re:evaluation}
In general, if $(\Xi,\Lambda)$ is an object of $\Rind$ and $\xi\in\Xi$, then
we have successive inclusions
$e_{\xi}\colon(\{\xi\},\Lambda(\xi))\xrightarrow{s_{\xi}}(\Xi_{/\xi},\Lambda_{/\xi})\xrightarrow{i_{\xi}}(\Xi,\Lambda)$
which induce the \emph{evaluation functor (at $\xi$)}
\[e_{\xi}^*\colon\cD(X,(\Lambda,\Xi))\to\cD(X,\Lambda(\xi))\] for a higher Artin
stack $X$. By the above lemma, $e_{\xi}^*$ and $p_{\xi*}\circ i_{\xi}^*$ are
canonically and coherently equivalent. For brevity, we sometimes also write
$\sK_{\xi}$ for $e_{\xi}^*\sK$ for an object $\sK\in\cD(X,(\Xi,\Lambda))$.
The functor
\[\prod_{\xi\in\Xi}e_{\xi}^*\colon\cD(X,(\Lambda,\Xi))\to\prod_{\xi\in\Xi}\cD(X,\Lambda(\xi))\]
is conservative. This is obvious when $X$ is in $\Schqcs$. The general case
follows, because simplicial limits of conservative functors are
conservative.
\end{remark}
\begin{lem}\label{1le:scheme_adic}
Let $X$ be a scheme in $\Schqcs$,
$\cC=\EO{}{\Chpar{}}{*}{}(X)\colon\N(\Rind)^{op}\to\PSL$. Let
$(\Xi,\Lambda)$ be an object of $\Rind$ such that $\Xi$ admits a final
object $\xi$. Then the image of the natural map
$p_{\xi}^*\colon\cC(\{\xi\},\Lambda(\xi))\to\cC(\Xi,\Lambda)$ is contained
in $\cC(\Xi,\Lambda)_{\adic}$. Moreover, the induced map
$p_{\xi}^*\colon\cC(\{\xi\},\Lambda(\xi))\to\cC(\Xi,\Lambda)_{\adic}$ is an
equivalence of $\infty$-categories. In particular,
$\cC(\Xi,\Lambda)_{\adic}$ is presentable.
\end{lem}
\begin{proof}
By the definition of adic objects, to show the first assertion, it suffices
to show that the unit transformation $\id_{\cC(\{\xi\},\Lambda(\xi))}\to
p_{\xi*}p_\xi^*$ is an equivalence. This follows from condition (2b) of
Definition \ref{1de:topological} applied to $p_\xi$, which is a special case
of Lemma \ref{1le:evaluation}. For the second assertion, we only need to show
that for every adic complex $\sK\in\cD(\Xi,\Lambda)_{\adic}$, the adjunction
map $p_{\xi}^*p_{\xi*}\sK\to\sK$ is an equivalence. Since
$\prod_{\xi'\in\Xi}e_{\xi'}^*$ is conservative, this is equivalent to showing
that $b\colon e_{\xi'}^*p_{\xi}^*p_{\xi*}\sK\to e_{\xi'}^*\sK$ is an
equivalence for every object $\xi'\in\Xi$. Let $\phi$ be the map $\xi'\to
\xi$. Since $\cK$ is adic, the composite
\[{\tilde \phi}^* p_{\xi*}\sK \xrightarrow{a} p_{\xi'*}p_{\xi'}^* {\tilde \phi}^* p_{\xi*}\sK \simeq
p_{\xi'*}i_\phi^* p_\xi^* p_{\xi*} \sK \xrightarrow{b} p_{\xi'*} i_\phi^* \sK
\]
is an equivalence. Moreover, we have shown that $a$ is an equivalence.
Therefore, $b$ is an equivalence.
\end{proof}
\begin{proposition}
We have
\begin{enumerate}
\item The map $\EO{}{\Chparl}{}{}$ (resp.\ $\EO{}{\Chpdm{}}{}{}$)
defined above factorizes through the subcategory
$\Fun^{\top}(\N(\Rind_{\ltor})^{op},\Mon_{\Pf}^{\PSL}(\Cat))$
(resp.\
$\Fun^{\top}(\N(\Rind_{\tor})^{op},\Mon_{\Pf}^{\PSL}(\Cat))$).
\item The map $\EO{}{\Chpar{}}{*}{\otimes}$ factorizes through the
subcategory $\Fun^{\top}(\N(\Rind)^{op},\PSLM)$.
\end{enumerate}
\end{proposition}
\begin{proof}
By the construction of the above maps by descent and Lemma
\ref{1le:monlimit_topological}, we only need to show for (1) that
$\EO{}{\Schqcs}{}{}$ factorizes though
$\Fun^{\top}(\N(\Rind_{\tor})^{op},\Mon_{\Pf}^{\PSL}(\Cat))$, and for (2)
that $\EO{}{\Schqcs}{*}{\otimes}$ factorizes through
$\Fun^{\top}(\N(\Rind)^{op},\PSLM)$. We check the conditions in Definition
\ref{1de:topological}: (1a) is automatic; (1b) follows from Lemma
\ref{1le:scheme_adic}; (1c) and (2b) follow from Lemma \ref{1le:evaluation}
and (2a) is then automatic.
\end{proof}
\begin{remark}
The arguments of this section show that the functor \cite{LZ}*{(2.3)}
\[ \EO{}{\PTopos}{*}{\otimes}\colon\N(\PTopos)^{op}\to\Fun(\N(\Rind)^{op},\PSLM).
\]
$\EO{}{\PTopos}{*}{\otimes}$ factorizes through
$\Fun^{\top}(\N(\Rind)^{op},\PSLM)$.
\end{remark}
\subsection{Adic complexes as limits}
\label{1ss:adic_complexes}
We put $\Adic^{\otimes}=\Adic^\otimes_{\Ring}$,
$\Ad^{\otimes}=\Ad^\otimes_{\Ring}$, and
$\alpha^\otimes=\alpha^\otimes_{\Ring}$.
\begin{definition}[Enhanced adic operation map]\label{1de:enhanced_adic}
Define the following four groups of data. Each contains two functors and a
natural transformation between them.
\begin{align*}
\EO{\adic}{\Chparl}{}{}&=\Adic_{\Ring_{\ltor}}\circ\EO{}{\Chparl}{}{}\colon \delta^*_{2,\{2\}}\Fun(\Delta^1,\Chparl)^{\cart}_{F^0,A^{\to}}\\
&\to\Fun^{\top}(\N(\Rind_{\ltor})^{op},\Mon_{\Pf}^{\PSL}(\Cat)),\\
\EOA{}{\Chparl}{}{}&=\Ad_{\Ring_{\ltor}}\circ\EO{}{\Chparl}{}{}\colon \delta^*_{2,\{2\}}\Fun(\Delta^1,\Chparl)^{\cart}_{F^0,A^{\to}} \\
&\to\Fun^{\top}(\N(\Rind_{\ltor})^{op},\Mon_{\Pf}^{\PSL}(\Cat)),\\
\EO{\alpha}{\Chparl}{}{}&=\alpha_{\Ring_{\ltor}}\circ(\EO{}{\Chparl}{}{}\times\id_{\Delta^1});
\end{align*}
\begin{align*}
\EO{\adic}{\Chpdm{}}{}{}&=\Adic_{\Ring_\tor}\circ\EO{}{\Chpdm{}}{}{}\colon \delta^*_{2,\{2\}}\Fun(\Delta^1,\Chpdm{})^{\cart}_{F^0,A^{\to}}\\
&\to\Fun^{\top}(\N(\Rind_{\tor})^{op},\Mon_{\Pf}^{\PSL}(\Cat)),\\
\EOA{}{\Chpdm{}}{}{}&=\Ad_{\Ring_\tor}\circ\EO{}{\Chpdm{}}{}{}\colon \delta^*_{2,\{2\}}\Fun(\Delta^1,\Chpdm{})^{\cart}_{F^0,A^{\to}}\\
&\to\Fun^{\top}(\N(\Rind_{\tor})^{op},\Mon_{\Pf}^{\PSL}(\Cat)),\\
\EO{\alpha}{\Chpdm{}}{}{}&=\alpha_{\Ring_\tor}\circ(\EO{}{\Chpdm{}}{}{}\times\id_{\Delta^1});
\end{align*}
\begin{align*}
\EO{\adic}{\Chpar{}}{*}{\otimes}&=\Adic^{\otimes}\circ\EO{}{\Chpar{}}{*}{\otimes}\colon
(\Chpar{})^{op}\to\Fun^{\top}(\N(\Rind)^{op},\PSLM),\\
\EOA{}{\Chpar{}}{*}{\otimes}&=\Ad^{\otimes}\circ\EO{}{\Chpar{}}{*}{\otimes}\colon
(\Chpar{})^{op}\to\Fun^{\top}(\N(\Rind)^{op},\PSLM),\\
\EO{\alpha}{\Chpar{}}{*}{\otimes}&=\alpha^{\otimes}\circ(\EO{}{\Chpar{}}{*}{\otimes}\times\id_{\Delta^1});
\end{align*}
\begin{align*}
\EO{\adic}{\PTopos}{*}{\otimes}&=\Adic^{\otimes}\circ\EO{}{\PTopos}{*}{\otimes}\colon
(\PTopos)^{op}\to\Fun^{\top}(\N(\Rind)^{op},\PSLM),\\
\EOA{}{\PTopos}{*}{\otimes}&=\Ad^{\otimes}\circ\EO{}{\PTopos}{*}{\otimes}\colon
(\PTopos)^{op}\to\Fun^{\top}(\N(\Rind)^{op},\PSLM),\\
\EO{\alpha}{\PTopos}{*}{\otimes}&=\alpha^{\otimes}\circ(\EO{}{\PTopos}{*}{\otimes}\times\id_{\Delta^1}).
\end{align*}
By construction and Remark \ref{1re:coef_compatibility}, the six functors in
first three groups satisfy (P4).
For objects $X$ of $\PTopos$ (resp.\ $\Chpar{}$) and $\lambda=(\Xi,\Lambda)$
of $\Rind$, we write $\cD(X,\lambda)^{\otimes}_{\adic}$ for
$\EO{\adic}{\PTopos}{*}{\otimes}(X,\lambda)$ (resp.\
$\EO{\adic}{\Chpar{}}{*}{\otimes}(X,\lambda)$), which is a full (symmetric
monoidal) subcategory of $\cD(X,\lambda)^{\otimes}$. Similarly, we write
$\underline{\cD}(X,\lambda)^{\otimes}$ for
$\EOA{}{\Chpar{}}{*}{\otimes}(X,\lambda)$ (resp.\
$\EOA{}{\PTopos}{*}{\otimes}(X,\lambda)$). Recall that $\Xi$ is by
definition a partially ordered set.
\end{definition}
\begin{proposition}\label{1pr:ptopos_adic}
The natural transformation $\EO{\alpha}{\PTopos}{*}{\otimes}$ is a natural
equivalence. In other words, for every object $T$ of $\PTopos$ and every
object $\lambda=(\Xi,\Lambda)$ of $\Rind$, the symmetric monoidal functor
\[\EO{\alpha}{\PTopos}{*}{\otimes}(T,\lambda)\colon\cD(T,\lambda)^{\otimes}_{\adic}\to\underline{\cD}(T,\lambda)^{\otimes}
\simeq\lim_{\Xi^{op}}\cD(T,\Lambda(\xi))^{\otimes}
\]
is an equivalence.
\end{proposition}
\begin{proof}
By definition $\cD(T,\lambda)_{\adic}$ is a full subcategory of
$\cD(T,\lambda)=\cD(\Mod(T_{\et}^{\Xi},\Lambda))$. We analyze the
construction of the functor
$\alpha\colon\cD(T^{\Xi},\Lambda)_{\adic}\to\lim_{\Xi^{op}}\cD(T,\Lambda(\xi))$.
First, we have a functor $\Delta^1\times\N(\Xi)\to\PSL$ sending
$\Delta^1\times(\varphi\colon\xi\to\xi')$ to the square
\[\xymatrix{
\cD(T^{\Xi_{/\xi}},\Lambda_{/\xi}) \ar[r]^-{p_{\xi*}} \ar[d]_-{i_{\varphi*}} & \cD(T,\Lambda(\xi)) \ar[d]^-{\tilde{\varphi}_*} \\
\cD(T^{\Xi_{/\xi'}},\Lambda_{/\xi'}) \ar[r]^-{p_{\xi'*}} &
\cD(T,\Lambda(\xi')). }\]This corresponds to a projectively fibrant
simplicial functor $\cF\colon \fC[\N(D)]\to \Mset$, where $D=[1]\times\Xi$.
Let $\phi_D\colon \fC[\N(D)]\to D$ be the canonical equivalence of
simplicial categories and put $\cF'=(\Fibr^D \circ \St^+_{\phi_D^{op}} \circ
\Un^+_{\N(D)^{op}})\cF\colon D\to \Mset$. We write $\cF'$ in the form
$\cF'\colon[1]\to(\Mset)^{\Xi}$. Applying the marked unstraightening functor
$\Un^+_{\phi}$ for the weak equivalence of simplicial categories
$\phi:\fC[\N(\Xi)^{op}]\to\Xi^{op}$, we obtain a morphism
$\tilde{\alpha}:F_1\to F_2$ of Cartesian fibrations in the category
$(\Mset)_{/\N(\Xi)^{op}}$. Moreover, by \cite{Lu1}*{5.2.2.5}, both $F_1$ and
$F_2$ are \emph{coCartesian} fibrations as well, but $\tilde{\alpha}$ does
\emph{not} send coCartesian edges to coCartesian ones in general. By a
similar argument, we have a map
\[\cD(T^{\Xi},\Lambda)\to\Map^{\r{coCart}}_{\N(\Xi)^{op}}(\N(\Xi)^{op},F_1):=\Map^\flat_{\N(\Xi)^{op}}((\N(\Xi)^{op})^\sharp,(F_1,\cE)),\]
where $\cE$ is the set of coCartesian edges of $F_1$. Composing with the
obvious inclusion and $\Map_{\N(\Xi)^{op}}(\N(\Xi)^{op},\tilde{\alpha})$, we
obtain a map
$\alpha':\cD(T^{\Xi},\Lambda)\to\Map_{\N(\Xi)^{op}}(\N(\Xi)^{op},F_2)$. We
have the equivalence
$\Map^{\r{coCart}}_{\N(\Xi)^{op}}(\N(\Xi)^{op},F_2)\simeq\lim_{\Xi^{op}}\cD(T,\Lambda(\xi))$,
and the following pullback diagram
\[\xymatrix{
\cD(T^{\Xi},\Lambda)_{\adic} \ar[r]^-{\alpha} \ar[d] & \Map^{\r{coCart}}_{\N(\Xi)^{op}}(\N(\Xi)^{op},F_2) \ar[d] \\
\cD(T^{\Xi},\Lambda) \ar[r]^-{\alpha'} &
\Map_{\N(\Xi)^{op}}(\N(\Xi)^{op},F_2), }\]by the definition of adic objects,
where vertical arrows are inclusions. We also note that $\alpha'$ commutes
with small colimits by \cite{Lu1}*{5.1.2.2}.
Recall that for every object $\xi$ of $\Xi$, we have an exact evaluation
functor $e_{\xi}^*\colon\Mod(T^{\Xi},\Lambda)\to\Mod(T,\Lambda(\xi))$ (on the
level of Abelian categories) which admits a (right exact) left adjoint
$e_{\xi!}\colon\Mod(T,\Lambda(\xi))\to\Mod(T^{\Xi},\Lambda)$. We define a
truncation functor
$t_{\leq\xi}\colon\Mod(T^{\Xi},\Lambda)\to\Mod(T^{\Xi},\Lambda)$ such that
for a sheaf $F_{\bullet}\in\Mod(T^{\Xi},\Lambda)$,
\[(t_{\leq\xi}F_{\bullet})_{\xi'}=\begin{cases}
F_{\xi'} & \text{if }\xi'\leq\xi,\\
0 & \text{otherwise,}
\end{cases}\] which is exact and admits a right adjoint. Let $\del_{/\Xi}$ be the
category of simplices of $\N(\Xi)$ of dimension $\le 1$. Then all $n$-cells
of $\N(\del_{/\Xi})$ are degenerate for $n\geq2$. Define a functor
\[\beta'\colon\N(\del_{/\Xi}^{op})\to\Fun(\Map_{\N(\Xi)^{op}}(\N(\Xi)^{op},F_2),\cD(T^{\Xi},\Lambda))\]
sending a typical subcategory $\xi\to(\xi\to\xi')\leftarrow\xi'$ of
$\del_{/\Xi}$ to
\[\xymatrix{
\bL e_{\xi!}\circ \epsilon_{\xi} & t_{\leq\xi}\circ\bL e_{\xi'!}\circ
\epsilon_{\xi'} \ar[l]\ar[r] & \bL e_{\xi'!}\circ \epsilon_{\xi'}, }\]
where
$\epsilon_{\xi}\colon\Map_{\N(\Xi)^{op}}(\N(\Xi)^{op},F_2)\to\cD(T,\Lambda(\xi))$
is the restriction to the fiber at $\xi$. The functor
$\Fun(\alpha',\cD(T^\Xi,\Lambda))\circ\beta'$ extends to a functor
$\N(\del_{/\Xi})^{\triangleright}\to\Fun(\cD(T^{\Xi},\Lambda),\cD(T^{\Xi},\Lambda))$
carrying $(\xi\to(\xi\to\xi')\leftarrow\xi')^\triangleleft$ to
\[\xymatrix{
\bL e_{\xi!}\circ \epsilon_{\xi}\circ \alpha' \ar[rd] & t_{\leq\xi}\circ\bL e_{\xi'!}\circ
\epsilon_{\xi'}\circ \alpha' \ar[l]\ar[r]\ar[d] & \bL e_{\xi'!}\circ \epsilon_{\xi'}\circ\alpha'\ar[ld] \\
& \r{id},
}\]
which induces a natural transformation $\colim
(\beta'\circ\alpha')\simeq(\colim \beta')\circ\alpha'\to\id$. One checks
that the restriction $\beta=\colim
\beta'\res\Map^{\r{coCart}}_{\N(\Xi)^{op}}(\N(\Xi)^{op},F_2)$ takes values
in $\cD(T^{\Xi},\Lambda)_{\adic}$. We prove that $\beta\circ\alpha\to\id$ is
an equivalence. Pick an object $\sK$ of $\cD(T^{\Xi},\Lambda)_{\adic}$. We
need to show that the diagram
$\beta^{\triangleright}_{\sK}\colon\N(\del_{/\Xi}^{op})^{\triangleright}\to\cD(T^{\Xi},\Lambda)$,
depicted as
\[\xymatrix{
\bL e_{\xi!}\sK_{\xi}\ar[dr] & t_{\leq\xi}\bL e_{\xi'!} \sK_{\xi'} \ar[l]\ar[r]\ar[d] &
\bL e_{\xi'!} \sK_{\xi'}\ar[dl] \\
& \sK, }
\]
is a colimit diagram. We only need to check this after applying $e_{\xi_0}^*$
for every $\xi_0\in\Xi$, since $e_{\xi_0}^*$ commutes with colimits. The
composite functor $e_{\xi_0}^*\circ\beta^{\triangleright}_{\sK}$ has value
(equivalent to) $\sK_{\xi_0}$ on the cone point, vertices $\{\xi\}$,
$(\xi\to\xi')$ of $\del_{/\Xi}$ for $\xi\geq \xi_0$ and $0$ otherwise, with
all morphisms being either identities on $\sK_{\xi_0}$ or $0$, or the zero
morphism $0\to\sK_{\xi_0}$. It is clear that
$e_{\xi_0}^*\circ\beta^{\triangleright}_{\sK}$ induces an equivalence
$\colim(e_{\xi_0}^*\circ\beta^{\triangleright}_{\sK}\res\N(\del_{/\Xi}^{op}))\simeq\sK_{\xi_0}$
in $\cD(T,\Lambda(\xi_0))$.
For the other direction, the functor $\Fun_{\N(\Xi)^{op}}(\N(\Xi)^{op},F_2),
\alpha')\circ \beta'$ also extends to a functor
\[\N(\del_{/\Xi})^{\triangleright}\to\Fun(\Map_{\N(\Xi)^{op}}(\N(\Xi)^{op},F_2),\Map_{\N(\Xi)^{op}}(\N(\Xi)^{op},F_2))\]
carrying $(\xi\to(\xi\to\xi')\leftarrow\xi')^\triangleleft$ to
\[\xymatrix{
\alpha'\circ\bL e_{\xi!}\circ \epsilon_{\xi} \ar[rd] & \alpha'\circ t_{\leq\xi}\circ\bL e_{\xi'!}\circ
\epsilon_{\xi'} \ar[l]\ar[r]\ar[d] & \alpha'\circ \bL e_{\xi'!}\circ \epsilon_{\xi'}\ar[ld] \\
& \r{id},
}\] which induces a natural transformation $\colim (\alpha'\circ\beta')\simeq\alpha'\circ(\colim
\beta')\to\id$, where the equivalence of two functors is due to the fact
that $\alpha'$ commutes with colimits. Restricting to
$\Map^{\r{coCart}}_{\N(\Xi)^{op}}(\N(\Xi)^{op},F_2)$, one obtains a map
$\alpha\circ\beta\to\id$ which is an equivalence by an argument similar to
the above. Therefore, $\alpha$ is an equivalence and the proposition
follows.
\end{proof}
\begin{corollary}\label{1co:enhanced_adic_limit}
The three natural transformations $\EO{\alpha}{\Chparl}{}{}$,
$\EO{\alpha}{\Chpdm{}}{}{}$, and $\EO{\alpha}{\Chpar{}}{*}{\otimes}$ are all
natural equivalences. In particular, for every object $X$ of $\Chpar{}$ and
every object $\lambda=(\Xi,\Lambda)$ of $\Rind$, the symmetric monoidal
functor
\[\EO{\alpha}{\Chpar{}}{*}{\otimes}(X,\lambda)\colon\cD(X,\lambda)^{\otimes}_{\adic}\to\underline{\cD}(X,\lambda)^{\otimes}
\simeq\lim_{\Xi^{op}}\cD(X,\Lambda(\xi))^{\otimes}\]
is an equivalence.
\end{corollary}
\begin{proof}
It suffices to show the second assertion. Since smooth surjective morphisms
are of both $\EO{\adic}{\Chpar{}}{*}{\otimes}$- and
$\EOA{}{\Chpar{}}{*}{\otimes}$-descent, we may assume that $X$ is an object
of $\Schqcs$, that is, a disjoint union of quasi-compact and separated
schemes. In this case, it suffices to apply Proposition \ref{1pr:ptopos_adic}
to $T=X_{\et}$.
\end{proof}
\begin{remark}\label{1re:constant_coef_limit}
In the special case where $\Lambda\colon\Xi^{op}\to\Ring$ is a constant
functor with value $\Lambda$ (by abuse of notation), we have an equivalence
\begin{align}\label{1eq:constant_coef_limit}
\cD(X,\lambda)^{\otimes}_{\adic}\xrightarrow{\sim}\underline{\cD}(X,\lambda)^{\otimes}
\simeq\prod_{\pi_0(\Xi)}\cD(X,\Lambda)^{\otimes}
\end{align}
given by the product functor $\prod_{\pi_0(\Xi)}e_{\xi_i}^*$, where we have
arbitrarily chosen an object $\xi_i$ in each connected component $\Xi_i$ of
$\Xi$. The resulting functor is independent of such choices up to
equivalence.
Assume $\Xi$ is connected for simplicity. Let $\pi\colon
(\Xi,\Lambda)\to(*,\Lambda)$ be the projection. Then
$\pi^*\colon\cD(X,\Lambda)^{\otimes}\to\cD(X,\lambda)_{\adic}^{\otimes}$ is
an equivalence since its composition with \eqref{1eq:constant_coef_limit} is
the identity. In particular, the right adjoint functor (between the
underlying $\infty$-categories) $\b R\pi_*\res\cD(X,\lambda)_{\adic} \colon
\cD(X,\lambda)_{\adic}\to\cD(X,\Lambda)$ is an equivalence as well. The
special case $\Xi=\NN$ was proved in \cite{LO2}*{2.2.5} \emph{assuming the
finiteness condition on cohomological dimension}.
\end{remark}
\subsection{Enhanced six operations and the usual $t$-structure}
\label{1ss:enhanced_six}
Recall that \cite{Lu2}*{1.4.4.12}, for a presentable stable
$\infty$-category $\cD$, a $t$-structure is \emph{accessible} if the full
subcategory $\cD^{\leq0}$ is presentable. For a topos $X\in \PTopos$ or a
scheme $X\in\Schqcs$, the usual $t$-structure on $\cD(X,\lambda)$ is
accessible by \cite{Lu2}*{1.3.5.21}. For a higher Artin stack $X$, the usual
$t$-structure on $\cD(X,\lambda)$ is accessible by construction
\cite{LZ}*{4.3.7} (Part (2) of (P6)).
Let $\cD^{\leq n}(X,\lambda)_{\adic}=\cD^{\leq
n}(X,\lambda)\cap\cD(X,\lambda)_{\adic}$ and $\cD^{\geq
n}(X,\lambda)_{\adic}=\cD^{\leq n-1}(X,\lambda)^{\perp}_{\adic}$. Recall by
Proposition \ref{1pr:adic_inclusion} that the inclusion
$\cD(X,\lambda)_{\adic}\subseteq\cD(X,\lambda)$ admits a right adjoint, that
is, the colocalization functor
$R_X\colon\cD(X,\lambda)\to\cD(X,\lambda)_{\adic}$. By Lemma
\ref{1le:pullback} and \cite{Lu1}*{5.5.3.12}, $\cD^{\leq
n}(X,\lambda)_{\adic}$ is presentable. The inclusion $\cD^{\leq
n}(X,\lambda)_{\adic}\subseteq\cD(X,\lambda)$ preserves all small colimits
and $\cD^{\leq n}(X,\lambda)_{\adic}$ is closed under extension. By
\cite{Lu2}*{1.4.4.11 (1)}, the pair $(\cD^{\leq
n}(X,\lambda)_{\adic},\cD^{\geq n}(X,\lambda)_{\adic})$ define an accessible
$t$-structure, called the usual $t$-structure, on $\cD(X,\lambda)_{\adic}$.
Now we define six operations for adic complexes and study their behavior
under the above $t$-structure. It is clear that $-\otimes-$ preserves the
subcategory $\cD(X,\lambda)_{\adic}$. Therefore, we have the induced
(derived) tensor product
\begin{description}
\item[3\underline{L}]
$-\underline{\otimes}-=-\underline{\otimes}_X-\colon\cD(X,\lambda)_{\adic}\times
\cD(X,\lambda)_{\adic}\to\cD(X,\lambda)_{\adic},$
\end{description}
that is left $t$-exact with respect to the above $t$-structure. Moving the
first factor of the source
$\cD(X,\lambda)_{\adic}\times\cD(X,\lambda)_{\adic}$ to the target side, we
can write the functor $-\underline{\otimes}-$ in the form
$\cD(X,\lambda)_{\adic}\to\FL(\cD(X,\lambda)_{\adic},\cD(X,\lambda)_{\adic})$,
because the tensor product on $\cD(X,\lambda)_{\adic}$ is closed. Taking
opposites and applying \cite{Lu1}*{5.2.6.2}, we obtain a functor
$\cD(X,\lambda)_{\adic}^{op}\to\FR(\cD(X,\lambda)_{\adic},\cD(X,\lambda)_{\adic})$,
which can be written as
\begin{description}
\item[3\underline{R}]
$\underline{\HOM}(-,-)=\underline{\HOM}_X(-,-)\colon
\cD(X,\lambda)_{\adic}^{op}\times\cD(X,\lambda)_{\adic}
\to\cD(X,\lambda)_{\adic}$.
\end{description}
Moreover, we have
\[\underline{\HOM}_X(-,-)\simeq R_X\circ
\HOM_X(-,-)\res\cD(X,\lambda)_{\adic}^{op}\times\cD(X,\lambda)_{\adic}.\]
For every morphism $\pi\colon \lambda'\to \lambda$ of $\Rind$, $\pi^*$
preserves adic objects by Lemma \ref{1le:preserve_adic} (1), so that we have
\[\underline\pi^*=\pi^*\res \cD(X,\lambda)\colon \cD(X,\lambda)_{\adic}\to \cD(X,\lambda')_{\adic},\]
which admits a right adjoint
\[\underline\pi_*\simeq R_{X} \circ \pi_*\res \cD(X,\lambda')\colon \cD(X,\lambda')_{\adic}\to \cD(X,\lambda)_{\adic}.\]
Let $f\colon Y\to X$ be a morphism of higher Artin stacks. Then $f^*$
preserves adic objects by Lemma \ref{1le:preserve_adic} (2). Therefore, we
have the induced functor
\begin{description}
\item[1\underline{L}] $\underline{f}^*=f^*\res
\cD(X,\lambda)_{\adic}\colon\cD(X,\lambda)_{\adic}\to\cD(Y,\lambda)_{\adic},$
\end{description}
which admits a right adjoint, denoted by
\begin{description}
\item[1\underline{R}] $\underline{f}_*\simeq R_X\circ
f_*\res\cD(Y,\lambda)_{\adic}\colon\cD(Y,\lambda)_{\adic}\to\cD(X,\lambda)_{\adic}.$
\end{description}
It follows from the definition and the corresponding properties of $f^*$ and
$f_*$, that $\underline{f}^*$ is right $t$-exact (i.e.\ preserves $\cD^{\le
0}(-)_{\adic}$), and $\underline{f}_*$ is left $t$-exact (i.e.\ preserves
$\cD^{\ge 0}(-)_{\adic}$). For $f$ surjective and $\sK\in \cD(X,\lambda)$,
$\sK$ is adic if and only if $f^*\sK$ is adic.
If $f$ is a morphism, locally of finite type, of higher Artin stacks in
$\Chparl$ (resp.\ of higher Deligne--Mumford stacks, resp.\ locally
quasi-finite of higher Deligne--Mumford stacks) and $\lambda$ is an object
of $\Rind_{\ltor}$ (resp.\ $\Rind_{\tor}$, resp.\ $\Rind$), then $f_!$
preserves adic objects. Therefore, we have the induced functor
\begin{description}
\item[2\underline{L}]$\underline{f}_!=f_!\res
\cD(Y,\lambda)_{\adic}\colon\cD(Y,\lambda)_{\adic}\to\cD(X,\lambda)_{\adic},$
\end{description}
which admits a right adjoint, denoted by
\begin{description}
\item[2\underline{R}]$\underline{f}^!\simeq R_Y\circ
f^!\res\cD(X,\lambda)_{\adic}\colon\cD(X,\lambda)_{\adic}\to\cD(Y,\lambda)_{\adic}.$
\end{description}
It follows from the definition and the corresponding properties of $f_!$ and
$f^!$ \cite{LZ}*{Lemma 6.1.13}, that $\underline{f}_![2d]$ (resp.\
$\underline{f}^![-2d]$) is right (resp.\ left) $t$-exact if
$d=\dim^+(f)<\infty$.
If $f\colon Y\to X$ is a morphism, locally of finite type, of locally
Noetherian and locally finite-dimensional higher Artin stacks in $\Chparl$,
and if $\lambda$ is an object of $\Rind_{\ltor}$, then $f_*$ and $f^!$
preserve adic objects by \cite{LZ}*{Proposition 6.2.1}, so that
$\underline{f}_*$ and $\underline{f}^!$ are restrictions of $f_*$ and $f^!$.
The $t$-structure on $\cD(X,\lambda)$ naturally induces a $t$-structure on
$\underline{\cD}(X,\lambda)$. More precisely, for every integer $n$, an
object $\sK$ of
$\underline{\cD}(X,\lambda)=\lim_{\Xi^{op}}\cD(X,\Lambda(\xi))$ is in
$\underline{\cD}^{\leq n}(X,\lambda)$ if and only if $\sK_{\xi}$ is in
$\cD^{\leq n}(X,\Lambda(\xi))$ for all objects $\xi$ of $\Xi$. Here,
$\sK_{\xi}$ is the image of $\sK$ under the natural functor
$\lim_{\Xi^{op}}\cD(X,\Lambda(\xi))\to\cD(X,\Lambda(\xi))$. Let
$\underline{\cD}^{\geq n}(X,\lambda)=\underline{\cD}^{\leq
n-1}(X,\lambda)^{\perp}$. We have induced functors and properties similar to
those for $\cD(X,\lambda)_{\adic}$. The equivalence
$\EO{\alpha}{\Chpar{}}{*}{}(X,\lambda)$ is compatible with the two
$t$-structures on $\cD(X,\lambda)_{\adic}$ and $\underline{\cD}(X,\lambda)$,
as well as the various induced functors.
In what follows, we will identify $\cD(X,\lambda)_{\adic}$ and
$\underline{\cD}(X,\lambda)$ (as stable $\infty$-categories with
$t$-structures) and the corresponding functors. We will use the
\emph{underlined} version in all the notation. In particular, we will view
$\underline{\cD}(X,\lambda)$ as a (strictly) full subcategory of
$\cD(X,\lambda)$.
\begin{proposition}\label{1pr:operation_property}
The six operations and the usual $t$-structure in the above adic setting
have all the properties stated in \cite{LZ}*{Propositions 6.1.1, 6.1.2,
6.1.3, 6.1.4, 6.1.11, 6.1.12, Corollary 6.1.10}.
\end{proposition}
\begin{proof}
(The analogues of) Propositions \cite{LZ}*{6.1.1, 6.1.2} are encoded in
$\EOA{}{\Chparl}{}{}\simeq\EO{\adic}{\Chparl}{}{}$ and
$\EOA{}{\Chpdm{}}{}{}\simeq\EO{\adic}{\Chpdm{}}{}{}$. Propositions
\cite{LZ}*{6.1.3, 6.1.4} follow from the previous two propositions.
Proposition \cite{LZ}*{6.1.11} follows by restricting to the full subcategory
spanned by adic objects, since $f_*$ preserves adic objects in this case.
Proposition \cite{LZ}*{6.1.12} follows from property (P4) of
$\EOA{}{\Chparl}{}{}$ and $\EOA{}{\Chpdm{}}{}{}$. Corollary
\cite{LZ}*{6.1.10} follows from Proposition \cite{LZ}*{6.1.1} and Proposition
\ref{1pr:poincare} (2).
\end{proof}
With slight modification as follows, Proposition \cite{LZ}*{6.1.9} also
holds in the adic setting by restriction to the full subcategory of adic
objects.
\begin{proposition}[Poincar\'{e} duality]\label{1pr:poincare}
Let $f\colon \cY\to\cX$ be a flat (resp.\ flat and locally quasi-finite)
morphism of $\Chparl$ (resp.\ $\Chpdm{}$), locally of finite presentation.
Let $\lambda$ be an object of $\Rind_{\ltor}$ (resp.\ $\Rind$). Then
\begin{enumerate}
\item There is a natural transformation $\underline{u}_f\colon
\underline{f}_!\circ \underline{f}^*\langle
d\rangle\to\r{id}_{\cX}$ for every integer $d\geq\dim^+(f)$.
\item If $f$ is moreover smooth, then the induced natural
transformation $\underline{u}_f\colon \underline{f}_!\circ
\underline{f}^*\langle\dim f\rangle\to\r{id}_{\cX}$ is a counit
transformation, so that the induced map $\underline{f}^*\langle\dim
f\rangle\to \underline{f}^!$ is a natural equivalence of functors
$\underline{\cD}(\cX,\lambda)\to\underline{\cD}(\cY,\lambda)$.
\end{enumerate}
\end{proposition}
\begin{remark}\label{1re:star_pullback_exact}
By Poincar\'{e} duality, $\underline f^*$ is $t$-exact if $f$ is a smooth
morphism of higher Artin stacks in $\Chparl$ and $\lambda$ is an object of
$\Rind_{\ltor}$ or if $f$ is an \'etale morphism of higher Deligne--Mumford
stacks. We will show in \Sec \ref{3ss:finiteness_condition} that this holds
for more general morphisms $f$ under a finiteness assumption.
\end{remark}
\begin{lem}\label{1le:adic_hom}
Let $(\Xi,\Lambda)$ be an object of $\Rind$ and $\xi$ be an object of $\Xi$,
$\sK$ be an adic object of $\cD(X,(\Xi,\Lambda))$. The following diagram
\[\xymatrix{
\cD(X,(\Xi,\Lambda)) \ar[d]_-{e_{\xi}^*} && \cD(X,(\Xi,\Lambda)) \ar[d]^-{e_{\xi}^*}\ar[ll]_-{-\otimes\sK} \\
\cD(X,\Lambda(\xi)) && \cD(X,\Lambda(\xi)) \ar[ll]_-{-\otimes e_{\xi}^*\sK}
}
\]
is right adjointable and its transpose is left adjointable. In other words,
the natural morphisms $e_{\xi!}(\sL\otimes e_\xi^*\sK)\to
(e_{\xi!}\sL)\otimes \sK$ and $e_{\xi}^*\HOM(\sK,\sL')\to
\HOM(e_{\xi}^*\sK,e_{\xi}^*\sL')$ are equivalences for objects $\sL$ of
$\cD(X,\Lambda(\xi))$ and $\sL'$ of $\cD(X,(\Xi,\Lambda))$.
\end{lem}
\begin{proof}
By \cite{LZ}*{6.1.7}, we may assume that $\xi$ is the final object of $\Xi$.
In this case, $e_{\xi}^*$ can be identified with $\pi_*$, where $\pi\colon
(\Xi,\Lambda)\to (\{\xi\},\Lambda(\xi))$. Since $\sK$ is adic, the morphism
$\pi^* e_\xi^*\sK\to \sK$ is an equivalence. A left adjoint of the transpose
of the above diagram is then given by the diagram
\[\xymatrix{
\cD(X,(\Xi,\Lambda)) \ar[d]_-{-\otimes\sK} && \cD(X,\Lambda(\xi)) \ar[ll]_-{\pi^*}\ar[d]^-{-\otimes e_{\xi}^*\sK} \\
\cD(X,(\Xi,\Lambda)) && \cD(X,\Lambda(\xi)) \ar[ll]_-{\pi^*}.
}
\]
\end{proof}
\subsection{Adic dualizing complexes}
\label{1ss:adic_dualizing}
Let $X$ be an object of $\Chpar{}$ and $\lambda=(\Xi,\Lambda)$ be an object
of $\Rind$. Let $\sO$ be an objects of $\cD(X,\lambda)$ (resp.\
$\underline{\cD}(X,\lambda)$). By adjunction of the pair of functors
$-\otimes\sK$ and $\HOM(\sK,-)$ (resp.\ $-\underline{\otimes}\sK$ and
$\underline{\HOM}(\sK,-)$), we have a natural transformation
$\delta_{\sO}\colon\id\to\H\HOM(\H\HOM(-,\sO),\sO)$ (resp.\
$\underline{\delta}_{\sO}\colon\id\to\H\underline{\HOM}(\H\underline{\HOM}(-,\sO),\sO)$)
between endofunctors of $\H\cD(\cX,\lambda)$ (resp.\
$\H\underline{\cD}(\cX,\lambda)$), which is called the \emph{biduality
transformation}\footnote{In fact, $\delta_{\sO}$ can be enhanced to a
natural transformation
$\tilde{\delta}_{\sO}\colon\id\to\HOM(\HOM(-,\sO),\sO)$ (resp.\
$\underline{\tilde{\delta}}_{\sO}\colon\id\to\underline{\HOM}(\underline{\HOM}(-,\sO),\sO)$)
between endofunctors of $\cD(\cX,\lambda)$ (resp.\
$\underline{\cD}(\cX,\lambda)$), that is,
$\H\tilde{\delta}_{\sO}=\delta_{\sO}$. We omit the details here since we do
not need such enhancement in what follows.}.
In the remainder of this section, we fix an $\sf{L}$-coprime base scheme
$\sS$ that is a disjoint union of \emph{excellent} schemes\footnote{A scheme
is \emph{excellent} if it is quasi-compact and admits a Zariski open cover
by spectra of excellent rings \cite{EGAIV}*{7.8.2}.}, \emph{endowed with a
global dimension function}. Let $\Rind_{\sf{L}\text{-}\r{dual}}$ be the full
subcategory of $\Rind_{\ltor}$ spanned by ringed diagrams
$\Lambda\colon\Xi^{op}\to\Ring$ such that $\Lambda(\xi)$ is an
($\sf{L}$-torsion) Gorenstein ring of dimension 0 for every object $\xi$ of
$\Xi$.
\begin{definition}[Potential dualizing complex]\label{1de:dualizing_complex}
Let $\lambda=(\Xi,\Lambda)$ be an object of $\Rind_{\sf{L}\text{-}\r{dual}}$.
For an object $f\colon X\to\sf{S}$ of $\Chpars$ with $X$ in $\Schqcs$, we say
a complex $\sO$ in $\cD(X,\lambda)$ is a \emph{pinned/potential dualizing
complex} (on $X$) if
\begin{enumerate}
\item $\sO$ is adic, and
\item for every object $\xi$ of $\Xi$,
$\sO_{\xi}=e_{\xi}^*\sO\in\cD(X,\Lambda(\xi))$ is a
pinned/potential dualizing complex.
\end{enumerate}
For a general object $f\colon X\to\sf{S}$ of $\Chpars$, we say a complex
$\sO$ in $\cD(X,\lambda)$ is a \emph{pinned/potential dualizing complex} if
for every atlas $u\colon X_0\to X$ with $X_0$ in $\Schqcs$, $u^!\sO$ is a
\emph{pinned/potential dualizing complex} on $X_0$.
\end{definition}
\begin{proposition}\label{1pr:dualizing_complex}
Let $f\colon X\to\sf{S}$ be an object of $\Chpars$ and $\lambda$ be as above.
The full subcategory of $\cD(X,\lambda)$ spanned by all pinned/potential
dualizing complexes is equivalent to the nerve of an ordinary category
consisting of only one object $\s O$ with $\Hom(\s O,\s
O)=\(\lim_{\xi\in\Xi}\Lambda(\Xi)\)^{\pi_0(X)}$. Moreover, pinned/potential
dualizing complexes are constructible and compatible under extension of
scalars.
\end{proposition}
In the proof, we will use the following observation which is essentially
\cite{Lu1}*{A.3.2.27}. Let $\cC\colon K^{\triangleleft}\to\Cat$ be a functor
that is a limit diagram. Let $X,Y$ be two objects in the limit
$\infty$-category $\cC_{-\infty}$ and write $X_k,Y_k$ the natural images in
$\cC_k$ for every vertex $k$ of $K$. Then $\Map_{\cC_{-\infty}}(X,Y)$ is
naturally the homotopy limit (in the $\infty$-category $\cH$ of spaces) of a
diagram $K\to\cH$ sending $k$ to $\Map_{\cC_k}(X_k,Y_k)$.
\begin{proof}
When $\Xi=*$ is a singleton and $X$ is in $\Schqcs$, the proposition is
proved in \cite{Ill} (see \cite{LZ}*{6.3.3 (1)}). We also note that if
$\sO_{\sf{S}}$ is a pinned dualizing complex on $\sf{S}$, then
$f^!\sO_{\sf{S}}$ is a pinned dualizing complex on $X$. First, we prove by
induction on $k$ that for an object $f\colon X\to\sf{S}$ of $\Chpars$ with
$X$ in $\Chpar{k}$,
\begin{enumerate}
\item For any two pinned dualizing complexes $\sO$ and $\sO'$,
$\Map_{\cD(X,\Lambda)}(\sO,\sO')$ is discrete;
\item There is a unique distinguished equivalence $o\colon\sO\to\sO'$
such that for every atlas $u\colon X_0\to X$ with $X_0$ in
$\Schqcs$, $u^!o$ is the one preserving pinning.
\end{enumerate}
It is clear that once the equivalence $o$ in (2) exists, it is compatible
under $f^!$ for every smooth morphism $f$. Choose an atlas $u\colon Y\to X$
(with $Y$ in $\Chpar{(k-1)}$). Since $u$ is of universal
$\EO{}{\Chparl}{!}{}$-descent, both (1) and (2) follow from the induction
hypothesis, the above observation, and the fact that limit of $k$-truncated
spaces is $k$-truncated (which follows from \cite{Lu1}*{5.5.6.5}). Second, we
show that
$\Map_{\cD(X,\Lambda)}(\sO,\sO)\simeq\pi_0\Map_{\cD(X,\Lambda)}(\sO,\sO)$ is
isomorphic to $\Lambda^{\pi_0(X)}$. Without loss of generality, we assume
that $X$ is connected. Choose an atlas $u=\coprod_Iu_i\colon \coprod_IY_i\to
X$ with $Y_i$ in $\Schqcs$ and is connected. We have the following
commutative diagram
\[\xymatrix{
\Lambda \ar[r]^-{\alpha} \ar@{=}[d] & \pi_0\Map_{\cD(X,\Lambda)}(\sO,\sO) \ar[d]^-{\beta} \\
\Lambda \ar[r] & \bigoplus_I\pi_0\Map_{\cD(Y_i,\Lambda)}(u_i^!\sO,u_i^!\sO).
}\] Since $u^!$ is conservative, $\beta$ is injective. Since
$\Lambda\to\pi_0\Map_{\cD(Y_i,\Lambda)}(u_i^!\sO,u_i^!\sO)$ is an isomorphism
for every $i\in I$, $\alpha$ is injective. If we write elements of
$\bigoplus_I\pi_0\Map_{\cD(Y_i,\Lambda)}(u_i^!\sO,u_i^!\sO)$ in the
coordinate form $(\dots,\lambda_i,\dots)$ with respect to the basis
consisting of distinguished equivalences, then the image of $u^!$ must belong
to the diagonal since $X$ is connected. Therefore, $\alpha$ is an
isomorphism. The fact that pinned dualizing complexes are constructible and
compatible under extension of scalars follows from the case of schemes.
Now we consider the general coefficient $\lambda=(\Xi,\Lambda)$. First, we
construct a pinned dualizing complex $\sO_{\sf{S},\lambda}$ on the base
scheme $\sf S$. Let $\del_{/\Xi}$ be the category of simplices of $\Xi$.
Then all $n$-simplices of $\N(\del_{/\Xi})$ are degenerate for $n\geq2$. For
every object $\xi$ of $\Xi$, denote by $\sO_{\sf S,\xi}$ the pinned
dualizing complex in $\cD(\sf S,\Lambda(\xi))$. We will use the functors
$e_{\xi!}$ and $t_{\leq\xi}$ introduced in the proof of Proposition
\ref{1pr:ptopos_adic}. Define a functor
$\delta\colon\N(\del_{/\Xi})\to\cD(\sf S,\lambda)$ sending a typical
subcategory $\xi\leftarrow(\xi\leq\xi')\rightarrow\xi'$ of $\del_{/\Xi}$ to
\[\xymatrix{
\bL e_{\xi!}\sO_{\sf S,\xi} & \bL e_{\xi!}\r E_{\xi\leq\xi'}\sO_{\sf
S,\xi'}\simeq t_{\leq\xi}\bL e_{\xi'!}\sO_{\sf S,\xi'} \ar[l]\ar[r] & \bL
e_{\xi'!}\sO_{\sf S,\xi'}, }\] where the left arrow is given by the
distinguished equivalence $\r E_{\xi\leq\xi'}\sO_{\sf
S,\xi'}\xrightarrow{\sim}\sO_{\sf S,\xi}$. It is easy to see that $\sO_{\sf
S,\lambda}=\lim\delta$, viewed as an element in $\cD(\sf S,\lambda)$,
satisfies the two requirements in Definition \ref{1de:dualizing_complex},
hence is a pinned dualizing complex. For an object $f\colon X\to\sf{S}$ of
$\Chpars$, let $\sO_{f,\lambda}=f^!\sO$. Then it is a pinned dualizing
complex on $X$. The rest of the proposition follows from the fact that
$\sO_{f,\lambda}$ is adic, Proposition \ref{1co:enhanced_adic_limit}, the
observation before the proof, and the same assertion when $\Xi$ is a
singleton.
\end{proof}
In what follows, we write $\cD=\cD_X=\HOM(-,\Omega_{X,\lambda})$,
$\rD=\rD_X=\r{h}\cD_X=\r{h}\HOM(-,\Omega_{X,\lambda})$, and similarly for
$\underline{\cD}$, $\underline{\rD}$.
\begin{proposition}
Let $\sK\in\underline{\cD}(X,\lambda)$ such that
$\delta_{\Omega_{X,\Lambda(\xi)}}(e_{\xi}^*\sK)$ is an equivalence for every
object $\xi$ of $\Xi$. Then $\underline{\delta}_{\Omega_{X,\lambda}}(\sK)$ is
an equivalence as well.
\end{proposition}
The assumption of the proposition is satisfied for
$\sK\in\underline{\cD}(X,\lambda)\cap \cD^{(\r{b})}_\cons(X,\lambda)$.
\begin{proof}
We need to show that the natural morphism
$\sK\to\underline{\rD}\:\underline{\rD}\sK$ is an isomorphism (in the
homotopy category of $\underline{\cD}(X,\lambda)$). By definition,
\begin{align*}
\underline{\rD}\:\underline{\rD}\sK
&=\r{h}\underline{\HOM}(\sK,\r{h}\underline{\HOM}(\sK,\Omega_{X,\lambda}))\\
&\simeq\r{h}R_X\r{h}\HOM(\sK,\r{h}R_X \r{h}\HOM(\sK,\Omega_{X,\lambda})) \\
&\simeq\r{h}R_X\r{h}\HOM(\sK,\r{h}\HOM(\sK,\Omega_{X,\lambda})).
\end{align*}
It suffices to show that
$\delta_{\Omega_{X,\lambda}}(\sK)\colon\sK\to\r{h}\HOM(\sK,\r{h}\HOM(\sK,\Omega_{X,\lambda}))$
is an equivalence. In fact, since $\sK$ is adic, we have
\begin{align*}
e_{\xi}^*\r{h}\HOM(\sK,\r{h}\HOM(\sK,\Omega_{X,\lambda}))&\simeq
\r{h}\HOM(e_{\xi}^*\sK,\r{h}\HOM(e_{\xi}^*\sK,e_{\xi}^*\Omega_{X,\lambda}))\\
&\simeq\r{h}\HOM(e_{\xi}^*\sK,\r{h}\HOM(e_{\xi}^*\sK,\Omega_{X,\Lambda(\xi)}))
\end{align*}
for every object $\xi\in\Xi$ by Lemma \ref{1le:adic_hom}, which is equivalent
to $e_{\xi}^*\sK$ by the assumption.
\end{proof}
\section{The $\fm$-adic formalism and constructibility}
\label{3}
In this chapter, we introduce a special case of the adic formalism, namely,
the $\fm$-adic formalism on which there is a good notion of
constructibility. Such formalism is enough for most applications. The basic
notion of the $\fm$-adic formalism is given in \Sec \ref{3ss:madic}. In \Sec
\ref{3ss:finiteness_condition}, we introduce some finiteness conditions
under which we may refine the construction of the usual $t$-structure. Then
we define the category of constructible adic complexes in this setting in
\Sec \ref{3ss:constructible_adic}, and on which the constructible adic
perverse $t$-structure in \Sec \ref{3ss:constructible_adic_perverse_t}. The
last section \Sec \ref{3ss:compatibility} is dedicated to proving the
compatibility between our theory and Laszlo--Olsson \cites{LO2,LO3} under
their restrictions.
\subsection{The $\fm$-adic formalism}
\label{3ss:madic}
\begin{definition}\label{3de:pring}
Define a category $\PRing$ as follows. The objects are pairs
$(\Lambda,\fm)$, where $\Lambda$ is a small ring and $\fm\subseteq\Lambda$
is a principal ideal, such that
\begin{itemize}
\item $\fm$ is generated by an element that is not a zero divisor;
\item the natural homomorphism $\Lambda\to\lim_{n}\Lambda_n$ is an
isomorphism, where $\Lambda_n=\Lambda/\fm^{n+1}$ ($n\in\NN$).
\end{itemize}
A morphism from $(\Lambda',\fm')$ to $(\Lambda,\fm)$ is a ring homomorphism
$\phi\colon \Lambda'\to\Lambda$ satisfying $\phi^{-1}(\fm)=\fm'$. We denote
by $\PRing_{\ltor}\subseteq\PRing$ the full subcategory spanned by
$(\Lambda,\fm)$ such that $(\NN,\Lambda_{\bullet})\in\Rind_{\ltor}$.
We have a natural functor $\PRing\to\Fun([1],\Rind)$ sending $(\Lambda,\fm)$
to $(\NN,\Lambda_{\bullet})\xrightarrow{\pi}(\ast,\Lambda)$. In what
follows, we simply write $\Lambda_{\bullet}$ for the ringed diagram
$(\NN,\Lambda_{\bullet})$.
\end{definition}
Let $(\Lambda,\fm)$ be an object of $\PRing$. Let $X$ be a higher Artin
stack. We have a pair of adjoint functors
\[\bL\pi^*=\bL\pi_X^*\colon\cD(X,\Lambda)\to\cD(X,\Lambda_{\bullet});\qquad
\bR\pi_*=\bR\pi_{X*}\colon\cD(X,\Lambda_{\bullet})\to\cD(X,\Lambda).
\]
As $\pi$ is perfect in the sense of \cite{LZ}*{2.2.8}, $\bL\pi^*$ admits a
left adjoint \cite{LZ}*{6.1.6}.
\begin{definition}[Normalized complex]
A complex $\sK\in\cD(X,\Lambda_{\bullet})$ is called \emph{normalized} if the
cofiber\footnote{The underlying object in the ordinary triangulated category
is a cone \cite{Lu2}*{1.1.2.10}.} \cite{Lu2}*{1.1.1.6} of the adjunction map
$\bL\pi^*\bR\pi_*\sK\to\sK$ is $0$. We denote by
$\cD_{\r{n}}(X,\Lambda_{\bullet})$ the full subcategory of
$\cD(X,\Lambda_{\bullet})$ spanned by normalized.
\end{definition}
The subcategory $\cD_{\r{n}}(X,\Lambda_{\bullet})\subseteq
\cD(X,\lambda_\bullet)$ is a stable subcategory stable under small limits.
Note that $\cD(X,\Lambda)=\underline{\cD}(X,\Lambda)$, so that the image of
$\bL\pi^*=\underline \pi^*$ is contained in
$\underline{\cD}(X,\Lambda_{\bullet})$. In particular, we have
$\cD_{\r{n}}(X,\Lambda_{\bullet})\subseteq\underline{\cD}(X,\Lambda_{\bullet})$.
For the other direction, we have the following result. We define
$\underline{\cD}^{(+)}(X,\Lambda_{\bullet})=\underline{\cD}(X,\Lambda_{\bullet})\cap\cD^{(+)}(X,\Lambda_{\bullet})$.
\begin{lem}\label{3le:adic_normalize}
We have
$\underline{\cD}^{(+)}(X,\Lambda_{\bullet})\subseteq\cD_{\r{n}}(X,\Lambda_{\bullet})$.
\end{lem}
\begin{proof}
The proof is similar to \cite{Zh}*{4.13}.
\end{proof}
As $\otimes$, $f^*$, $f_!$ preserve adic complexes, these operations
preserve normalized complexes in $\cD^{(+)}$. Next we examine effects of
$\HOM$, $f_*$, $f^!$ on normalized complexes, which imply that the
restrictions of $\underline{\HOM}$, $\underline{f}_*$, $\underline{f}^!$ to
$\underline{\cD}^{(+)}$ coincide with $\HOM$, $f_*$, $f^!$.
\begin{proposition}\label{3pr:hom_normalize}
Let $X$ be a higher Artin (resp.\ higher Deligne--Mumford) stack and let
$(\Lambda,\fm)$ be an object of $\PRing_{\ltor}$ (resp.\ $\PRing$). For
$\sK,\sL\in \cD_{\r{n}}(X,\Lambda_\bullet)$, and more generally for
$\sK,\sL$ in the essential image of $\b{L}\pi^*$, $\HOM(\sK,\sL)$ is adic.
In particular, $\HOM$ restricts to
\[\underline{\HOM}\colon
\underline{\cD}^{(+)}(X,\Lambda_\bullet)^{op}\times
\underline{\cD}^{(+)}(X,\Lambda_\bullet)\to
\underline{\cD}(X,\Lambda_\bullet).\]
\end{proposition}
\begin{proof}
By Poincar\'e duality, we reduce to the case of schemes. This case is
essentially proved in \cite{Zh}*{4.18}.
\end{proof}
\begin{proposition}\label{3pr:star_pushforward_normalize}
Let $f\colon Y\to X$ be a morphism of higher Artin stacks and let
$(\Lambda,\fm)$ be an object of $\PRing$. Then $f_*\colon
\cD(Y,\Lambda_\bullet)\to \cD(X,\Lambda_\bullet)$ preserves normalized
complexes. In particular, $f_*$ restricts to
\[\underline{f}_*\colon \underline{\cD}^{(+)}(Y,\Lambda_\bullet)\to
\underline{\cD}^{(+)}(X,\Lambda_\bullet).\]
\end{proposition}
\begin{proof}
This follows from the fact that $f_*$ commutes with $\b{L}\pi^*$
\cite{LZ}*{6.1.6}.
\end{proof}
\begin{proposition}\label{3pr:shrink_pullback_normalize}
Let $f\colon Y\to X$ be a morphism locally of finite type of $\Chparl$ and
let $(\Lambda,\fm)$ be an object of $\PRing_{\ltor}$. Then $f^!\colon
\cD(X,\Lambda_\bullet)\to \cD(Y,\Lambda_\bullet)$ preserves normalized
complexes in $\cD^{(+)}$. In particular, $f^!$ restricts to
\[\underline{f}^!\colon \underline{\cD}^{(+)}(X,\Lambda_\bullet)\to
\underline{\cD}^{(+)}(X,\Lambda_\bullet).\]
\end{proposition}
\begin{proof}
By Poincar\'e duality applied to atlases, we are reduced to the case of a
closed immersion of schemes, which follows from the fact that $f^!$ commutes
with $\b{L}\pi^*$ \cite{LZ}*{6.1.7} (with no restriction on $\Lambda$ in
this case).
\end{proof}
The truncation functors $\tau^{\le n}$, $\tau^{\ge n}$ do not preserve
normalized complexes in general. In the rest of \Sec \ref{3ss:madic}, we
study the effects of the truncation functors on normalized complexes.
Let $\cA$ be an Abelian category. An object $M_{\bullet}$ in
$\Fun(\NN^{op},\cA)$ is called \emph{essentially null} if for each $n\in\NN$,
there is an $r\in\NN$ such that $M_{r+n}\to M_n$ is the zero morphism. If
$\cA$ admits sequential limits, then we have a left exact functor $\lim\colon
\Fun(\NN^{op},\cA)\to\cA$. Given a topos $T$, we have a pair of adjoint
functors
\[\pi^*\colon \Mod(T,\Lambda)\to\Mod(T^{\NN},\Lambda_{\bullet});\qquad
\pi_*\colon \Mod(T^{\NN},\Lambda_{\bullet})\to\Mod(T,\Lambda)\] induced by
the morphism $\pi\colon (\NN,\Lambda_{\bullet})\to(*,\Lambda)$. Then
$\pi_*=\lim\circ\nu$, where $\nu\colon
\Mod(T^{\NN},\Lambda_{\bullet})\to\Fun(\NN^{op},\Mod(T,\Lambda))$ is the
obvious forgetful functor, which is exact.
\begin{lem}\label{3le:essentially_null}
Let $F_{\bullet}$ be a module in $\Mod(T^{\NN},\Lambda_{\bullet})$ such that
$\nu F_{\bullet}$ is essentially null. Then $\r{R}^n\pi_*F_{\bullet}=0$ for
all $n\geq0$.
\end{lem}
\begin{proof}
Note that $\r{R}^n\pi_*F_\bullet$ is the sheaf associated to the presheaf
$(U\mapsto \r{H}^n(U^\NN,F_\bullet))$, where $U$ runs over objects of $T$.
Let $a\colon (U^\NN,\Lambda_\bullet)\to (*^\NN,\Lambda_\bullet)$ be the
morphism of ringed topoi. Since $\r{R}^q a_*F_\bullet$ is essentially null
for all $q$, we have $\r{R}\Gamma(U^\NN,F_\bullet)\simeq
\r{R}\lim\r{R}a_*F_\bullet=0$.
\end{proof}
Let $\cD_0(X,\Lambda_{\bullet})$ be the full subcategory of
$\cD(X,\Lambda_{\bullet})$ spanned by complexes whose cohomology sheaves are
all essentially null. Let
$\cD^{(+)}_0(X,\Lambda_{\bullet})=\cD^{(+)}(X,\Lambda_{\bullet})\cap\cD_0(X,\Lambda_{\bullet})$.
Both are stable subcategories.
\begin{lem}\label{3le:pi_vanish}
For $\sK\in\cD^{(+)}_0(X,\Lambda_{\bullet})$, we have $\bR\pi_*\sK=0$.
\end{lem}
\begin{proof}
We may assume that $X$ is a scheme and $\sK\in\cD^+_0(X,\Lambda_{\bullet})$.
Then the statement follows from Lemma \ref{3le:essentially_null}.
\end{proof}
\begin{lem}
The cohomological amplitude of $\pi^*$ is contained in $[-1,0]$.
\end{lem}
\begin{proof}
By assumption, $\fm=(\lambda)$, where $\lambda\in \Lambda$ is not a zero
divisor. Since $\Lambda\xrightarrow{\times \lambda^{n+1}} \Lambda$ is a
resolution of $\Lambda_n$, the tor-dimension of the $\Lambda$-module
$\Lambda_n$ is $\le 1$.
\end{proof}
\begin{definition}
A complex $\sK\in\cD(X,\Lambda_{\bullet})$ is called \emph{essentially
normalized} if the cofiber of the adjunction map $\bL\pi^*\bR\pi_*\sK\to\sK$
is in $\cD^{(+)}_0(X,\Lambda_{\bullet})$. We denote by
$\cD_{\r{en}}(X,\Lambda_{\bullet})$ the full subcategory of
$\cD(X,\Lambda_{\bullet})$ spanned by essentially normalized complexes, which
is a stable subcategory.
\end{definition}
\begin{lem}\label{3le:essentially_normalize}
The image of the functor
$\bL\pi^*\circ\bR\pi_*\res\cD_{\r{en}}(X,\Lambda_{\bullet})$ is contained in
$\cD_{\r{n}}(X,\Lambda_{\bullet})$. Moreover, the induced functor
$\cD_{\r{en}}(X,\Lambda_\bullet)\to \cD_{\r{n}}(X,\Lambda_\bullet)$ is right
adjoint to the inclusion
$\cD_{\r{n}}(X,\Lambda_{\bullet})\subseteq\cD_{\r{en}}(X,\Lambda_{\bullet})$.
\end{lem}
\begin{proof}
For the fist assertion, we need to show that
$\bL\pi^*\bR\pi_*\bL\pi^*\bR\pi_*\sK\to\bL\pi^*\bR\pi_*\sK$ is an
equivalence for $\sK\in\cD_{\r{en}}(X,\Lambda_{\bullet})$. By definition,
the cofiber of $\bL\pi^*\bR\pi_*\sK\to\sK$ is contained in
$\cD^{(+)}_0(X,\Lambda_{\bullet})$. The assertion then follows from Lemma
\ref{3le:pi_vanish}.
For the second assertion, we need to show that the natural transformation
$\bL\pi^*\circ\bR\pi_*\to\id$ induces a homotopy equivalence (i.e.\ an
equivalence in $\cH$)
\[\Map_{\cD_{\r{n}}(X,\Lambda_{\bullet})}(\sK,\bL\pi^*\bR\pi_*\sL)\to\Map_{\cD_{\r{en}}(X,\Lambda_{\bullet})}(\sK,\sL),\]
for every object $\sK$ (resp.\ $\sL$) of $\cD_{\r{n}}(X,\Lambda_{\bullet})$
(resp.\ $\cD_{\r{en}}(X,\Lambda_{\bullet})$). By definition, the cofiber
$\sL'$ of $\bL\pi^*\bR\pi_*\sL\to\sL$ is in
$\cD^{(+)}_0(X,\Lambda_{\bullet})$, and $\sK$ is equivalent to
$\bL\pi^*\bR\pi_*\sK$. Therefore, the assertion follows from the fact that
\[\Map_{\cD_{\r{en}}(X,\Lambda_{\bullet})}(\bL\pi^*\bR\pi_*\sK,\sL')\simeq
\Map_{\cD(X,\Lambda)}(\bR\pi_*\sK,\bR\pi_*\sL')\simeq \{*\}.
\]
Here in the second equivalence we have used the fact that $\bR\pi_*\sL'=0$,
which follows from Lemma \ref{3le:pi_vanish}.
\end{proof}
\begin{lem}\label{3le:hom_vanish}
Let $\sK\in \underline{\cD}(X,\Lambda_\bullet)\cap
\cD^{(-)}(X,\Lambda_\bullet)$, $\sL\in \cD^{(+)}_0(X,\Lambda_\bullet)$. Then
$\HOM(\sK,\sL)\in \cD^{(+)}_0(X,\Lambda_\bullet)$ and $\Hom(\sK,\sL)=0$.
\end{lem}
\begin{proof}
The proof is similar to \cite{Zh}*{4.19}.
\end{proof}
\begin{lem}\label{3le:normalize_truncation}
For every $\sK\in\cD_{\r{n}}(X,\Lambda_{\bullet})$ and $n\in\ZZ$,
$\tau^{\geq n}\sK$ is in $\cD_{\r{en}}(X,\Lambda_{\bullet})$. Moreover, the
fiber of the adjunction map $\bL\pi^*\bR\pi_*\tau^{\geq n}\sK\to\tau^{\geq
n}\sK$ is concentrated in degree $n-1$.
\end{lem}
\begin{proof}
This is essentially proved in \cite{Zh}*{4.14}. Let us recall the arguments.
The fiber of the map $a\colon \bL\pi^* \tau^{\ge n}\bR\pi_*\sK \to \tau^{\ge
n} \bL\pi^* \bR\pi_* \sK\simeq \tau^{\ge n}\sK$ is concentrated in degree
$n-1$ and belongs to $\cD_0(X,\Lambda_\bullet)$. Consider the diagram
\[\xymatrix{\bL\pi^*\bR\pi_*\bL\pi^*\tau^{\ge n}\bR\pi_*\sK \ar[d]_{\bL\pi^*\bR\pi_*a}\ar[r]^-b&\bL\pi^*\tau^{\ge n} \bR \pi_* \sK\ar[d]^a\\
\bL\pi^*\bR\pi_*\tau^{\ge n} \sK\ar[r]^-c &\tau^{\ge n}\sK.}
\]
By Lemma \ref{3le:essentially_null}, $\bL\pi^*\bR\pi_*a$ is an equivalence.
By Proposition \ref{3le:adic_normalize}, $b$ is an equivalence. Therefore,
the fiber of $c$ is equivalent to the fiber of $a$.
\end{proof}
We denote by $\Mod_{\r{en}}(X,\Lambda_{\bullet})$ the full subcategory of
$\cD_{\r{en}}(X,\Lambda_{\bullet})$ spanned by complexes that are
concentrated at degree $0$, and $\Mod_0(X,\Lambda_{\bullet})$ the full
subcategory of $\Mod_{\r{en}}(X,\Lambda_{\bullet})$ spanned by essentially
null modules. Then $\Mod_0(X,\Lambda_{\bullet})$ is closed under sub-objects,
quotients and extensions.
\begin{proposition}\label{3pr:essentially_normalize_t}
For $n\in\d Z$, let $\cD^{\leq n}_{\r{en}}(X,\Lambda_{\bullet})=\cD^{\leq
n}(X,\Lambda_{\bullet})\cap\cD_{\r{en}}(X,\Lambda_{\bullet})$ and $\cD^{\geq
n}_{\r{en}}(X,\Lambda_{\bullet})=\cD^{\geq
n}(X,\Lambda_{\bullet})\cap\cD_{\r{en}}(X,\Lambda_{\bullet})$. Then
$(\cD^{\leq 0}_{\r{en}}(X,\Lambda_{\bullet}),\cD^{\geq
0}_{\r{en}}(X,\Lambda_{\bullet}))$ defines a $t$-structure on
$\cD_{\r{en}}(X,\Lambda_{\bullet})$ with heart
$\Mod_{\r{en}}(X,\Lambda_{\bullet})$, and
$\Mod_{\r{en}}(X,\Lambda_{\bullet})$ is (equivalent to the nerve of) a full
subcategory of $\Mod(X,\Lambda_{\bullet})$, closed under kernels, cokernels
and extensions.
\end{proposition}
\begin{proof}
We only need to show that $\tau^{\leq 0}$ and $\tau^{\geq 0}$ preserve the
full subcategory $\cD_{\r{en}}(X,\Lambda_{\bullet})$. Since
$\cD_{\r{en}}(X,\Lambda_{\bullet})$ is a stable full subcategory, we only
need to prove this for $\tau^{\geq 0}$, that is, the cofiber of the
adjunction map $\bL\pi^*\bR\pi_*\tau^{\geq0}\sK\to\tau^{\geq0}\sK$ is in
$\cD^{(+)}_0(X,\Lambda_{\bullet})$ for every object $\sK$ of
$\cD_{\r{en}}(X,\Lambda_\bullet)$. Consider the diagram
\[\xymatrix{\bL\pi^*\bR\pi_*\tau^{\ge 0} \bL\pi^*\bR\pi_*\sK\ar[r]^-b\ar[d]_{\bL\pi^*\bR\pi_*a} & \tau^{\ge 0}\bL\pi^*\bR\pi_*\sK\ar[d]^a\\
\bL\pi^*\bR\pi_*\tau^{\ge 0}\sK \ar[r]^-c & \tau^{\ge 0}\sK.}
\] By
definition, the cofiber of $\bL\pi^*\bR\pi_*\sK\to\sK$ is in
$\cD^{(+)}_0(X,\Lambda_{\bullet})$, so that the cofiber of $a$ is in
$\cD^{(+)}_0(X,\Lambda_{\bullet})$. It follows that $\bL\pi^*\bR\pi_*a$ is
an equivalence, by Lemma \ref{3le:pi_vanish}. By Lemma
\ref{3le:essentially_normalize},
$\bL\pi^*\bR\pi_*\sK\in\cD_{\r{n}}(X,\Lambda_{\bullet})$. Thus, by the first
part of Lemma \ref{3le:normalize_truncation}, the cofiber of $b$ is in
$\cD^{(+)}_0(X,\Lambda_{\bullet})$. Therefore, by the octahedral axiom, the
cofiber of $c$ is in $\cD^{(+)}_0(X,\Lambda_{\bullet})$ as well.
\end{proof}
\begin{corollary}\label{3co:normalize_t}
The essential image of $\bL\pi^*\circ\bR\pi_*\res \cD_{\r{en}}^{\geq
n}(X,\Lambda_{\bullet})$ is right perpendicular to the full subcategory
$\cD_{\r n}(X,\Lambda_{\bullet})\cap\cD^{<n}(X,\Lambda_{\bullet})$ of
$\cD_{\r n}(X,\Lambda_{\bullet})$.
\end{corollary}
\subsection{Constructible adic complexes}
\label{3ss:constructible_adic}
Let $(\Lambda,\fm)$ be an object of $\PRing$ such that $\Lambda/\fm^{n+1}$
is Noetherian for all $n$. For a higher Artin stack $X$, we define
\begin{gather*}
\underline{\cD}_{\cons}(X,\Lambda_\bullet)=\underline{\cD}(X,\Lambda_\bullet)\cap \cD_{\cons}(X,\Lambda_\bullet),\\
\underline{\cD}_{\cons}^{(+)}(X,\Lambda_\bullet)=\underline{\cD}(X,\Lambda_\bullet)\cap \cD_{\cons}^{(+)}(X,\Lambda_\bullet),
\qquad \underline{\cD}_{\cons}^{(-)}(X,\Lambda_\bullet)=\underline{\cD}(X,\Lambda_\bullet)\cap \cD_{\cons}^{(-)}(X,\Lambda_\bullet).
\end{gather*}
The following is an immediate consequence of the definitions and
\cite{LZ}*{6.2.3}.
\begin{lem}\label{3le:constructible_adic}
Let $f\colon Y\to X$ be a morphism of higher Artin stacks. Then $f^*$ and
$-\otimes-$ restrict to the following:
\begin{description}
\item[1L$'$] $\underline{f}^*\colon
\underline{\cD}_{\cons}(X,\Lambda_\bullet)\to\underline{\cD}_{\cons}(Y,\Lambda_\bullet)$;
\item[3L$'$] $-\underline{\otimes}_X-\colon
\underline{\cD}^{(-)}_{\cons}(X,\Lambda_\bullet)\times\underline{\cD}^{(-)}_{\cons}(X,\Lambda_\bullet)
\to\underline{\cD}^{(-)}_{\cons}(X,\Lambda_\bullet)$.
\end{description}
In particular, $\underline{\cD}_{\cons}^{(-)}(X,\lambda)^\otimes$
\cite{Lu2}*{2.2.1} is a symmetric monoidal category.
\end{lem}
As in \cite{LZ}*{6.2}, to state the results for the other operations, we
work in a relative setting. Let $\sS$ be an $\sf{L}$-coprime higher Artin
stack. Assume that there exists an atlas $S\to \sS$, where $S$ is either a
quasi-excellent scheme or a regular scheme of dimension $\le 1$. Combining
\cite{LZ}*{6.2.4} and Propositions \ref{3pr:hom_normalize},
\ref{3pr:star_pushforward_normalize}, \ref{3pr:shrink_pullback_normalize},
we have the following.
\begin{proposition}\label{3pr:constructible_adic}
Let $f\colon Y\to X$ be a morphism of $\Chpars$. Assume
$(\Lambda,\fm)\in\PRing_{\ltor}$. Then $f_*$, $f_!$, $f^!$, $\HOM$ restrict
to the following:
\begin{description}
\item[1R$'$] $\underline{f}_*\colon
\underline{\cD}^{(+)}_{\cons}(Y,\Lambda_\bullet)\to\underline{\cD}^{(+)}_{\cons}(X,\Lambda_\bullet)$,
if $f$ is quasi-compact and quasi-separated, and
$\underline{f}_*\colon
\underline{\cD}_{\cons}(Y,\Lambda_\bullet)\to\underline{\cD}_{\cons}(X,\Lambda_\bullet)$
if $\sS$ is locally finite-dimensional and $f$ is quasi-compact,
quasi-separated, and $0$-Artin;
\item[2L$'$] $\underline{f}_!\colon
\underline{\cD}^{(-)}_{\cons}(Y,\Lambda_\bullet)\to\underline{\cD}^{(-)}_{\cons}(X,\Lambda_\bullet)$,
if $f$ is quasi-compact and quasi-separated;
\item[2R$'$] $\underline{f}^!\colon
\underline{\cD}^{(+)}_{\cons}(X,\Lambda_\bullet)\to\underline{\cD}^{(+)}_{\cons}(Y,\Lambda_\bullet)$,
and, if $\sS$ is locally finite-dimensional, $\underline{f}^!\colon
\underline{\cD}_{\cons}(X,\Lambda_\bullet)\to\underline{\cD}_{\cons}(Y,\Lambda_\bullet)$;
\item[3R$'$] $\underline{\HOM}_X(-,-)\colon
\underline{\cD}^{(\r{b})}_{\cons}(X,\Lambda_\bullet)^{op}\times\underline{\cD}^{(+)}_{\cons}(X,\Lambda_\bullet)
\to\underline{\cD}^{(+)}_{\cons}(X,\Lambda_\bullet)$.
\end{description}
\end{proposition}
\subsection{Finiteness conditions and the usual $t$-structure}
\label{3ss:finiteness_condition}
Let $X$ be a higher Artin stack and $(\Lambda,\fm)$ be an object of
$\PRing$. Recall that $\cD_{\r{n}}(X,\Lambda_{\bullet})\subseteq
\underline{\cD}(X,\Lambda_{\bullet})$.
\begin{definition}
The pair $(X,(\Lambda,\fm))$ is said to be \emph{admissible} if
$\underline{\cD}(X,\Lambda_{\bullet})\subseteq
\cD_{\r{n}}(X,\Lambda_{\bullet})$ (so that
$\underline{\cD}(X,\Lambda_{\bullet})= \cD_{\r{n}}(X,\Lambda_{\bullet})$),
that is, for every $\sK\in\underline{\cD}(X,\Lambda_{\bullet})$, the
adjunction map $\bL\pi^*\bR\pi_*\sK\to\sK$ is an equivalence.
\end{definition}
\begin{proposition}\label{3pr:adic_t}
Let $(X,(\Lambda,\fm))$ be an admissible pair. Then
$R_X\res\cD_{\r{en}}(X,\Lambda_{\bullet})\simeq\bL\pi^*\circ\bR\pi_*$, and
$\underline{\cD}^{\geq n}(X,\Lambda_{\bullet})$ is the essential image of
$\bL\pi^*\circ\bR\pi_*\res\cD_{\r{en}}^{\geq n}(X,\Lambda_{\bullet})$.
\end{proposition}
\begin{proof}
The first assertion follows from Lemma \ref{3le:essentially_normalize}. We
denote by $\cD'$ the essential image of
$\bL\pi^*\circ\bR\pi_*\res\cD_{\r{en}}^{\geq n}(X,\Lambda_{\bullet})$ in the
second assertion. Then $\cD'\subseteq \underline{\cD}^{\geq
n}(X,\Lambda_{\bullet})$ by Corollary \ref{3co:normalize_t}. Moreover, since
$\bL\pi^*\colon \cD(X,\Lambda)\to \underline{\cD}(X,\Lambda_\bullet)$ is left
$t$-exact, $\bR\pi_*\res \underline{\cD}(X,\Lambda_\bullet) \colon
\underline{\cD}(X,\Lambda_\bullet)\to \cD(X,\Lambda)$ is right $t$-exact.
Thus, for $\sK\in \underline{\cD}^{\ge n}(X,\Lambda_\bullet)$, the fiber of
$\sK\to \tau^{\ge n}\sK$ (where $\tau^{\ge n}$ is the truncation functor in
$\cD(X,\Lambda_\bullet)$), equivalent to the fiber of $\bL\pi^*\bR\pi_*
\sK\simeq\bL\pi^* \tau^{\ge n}\bR\pi_*\sK\to \tau^{\ge n}\bL\pi^*\bR\pi_*
\sK$, is concentrated in degree $n-1$ and belongs to
$\cD_0(X,\Lambda_\bullet)$. Therefore, $\sK\simeq \bL\pi^*\bR\pi_* \sK\simeq
\bL\pi^*\bR\pi_* \tau^{\ge n}\sK$, which belongs to $\cD'$ by Lemma
\ref{3le:normalize_truncation}.
\end{proof}
\begin{remark}\label{3re:t}
Let $\Mod_{\r{en}}'(X,\Lambda_{\bullet})$ be the full subcategory of
$\Mod_{\r{en}}(X,\Lambda_{\bullet})$ spanned by complexes (modules) $\sK$
such that $\r H^i\bL\pi^*\bR\pi_*\sK=0$ for $i>0$, which is an exact
category. Then the projection functor
$\underline{\cD}^{\heartsuit}(X,\Lambda_{\bullet})\to
\Mod_{\r{en}}'(X,\Lambda_{\bullet})/\Mod_0(X,\Lambda_{\bullet})$ from the
heart of the usual $t$-structure to the full subcategory of
$\Mod_{\r{en}}(X,\Lambda_{\bullet})/\Mod_0(X,\Lambda_{\bullet})$ spanned by
the image of $\Mod_{\r{en}}'(X,\Lambda_{\bullet})$ is an equivalence of
categories. In fact, $\bL\pi^*\circ\bR\pi_*\res
\Mod_{\r{en}}'(X,\Lambda_{\bullet})$ induces a quasi-inverse.
\end{remark}
\begin{proposition}\label{3pr:right_exact}
Let $f\colon Y\to X$ be a morphism of higher Artin stacks.
\begin{enumerate}
\item Let $(\Lambda,\fm)$ be an object of $\PRing$ such that
$(X,(\Lambda,\fm))$ is admissible. Then
$\underline{f}^*\colon\underline{\cD}(X,\Lambda_{\bullet})\to\underline{\cD}(Y,\Lambda_{\bullet})$
is $t$-exact with respect to the usual $t$-structures.
\item Assume $f$ is locally of finite type (resp.\ locally of finite type,
resp.\ a locally quasi-finite morphism of Deligne--Mumford stacks).
Let $(\Lambda,\fm)$ be an object of $\PRing_{\ltor}$ (resp.\
$\PRing_{\tor}$, resp.\ $\PRing$) such that $(Y,(\Lambda,\fm))$ is
admissible. Then
$\underline{f}_!\colon\underline{\cD}(Y,\Lambda_{\bullet})\to\underline{\cD}(X,\Lambda_{\bullet})$
is right $t$-exact with respect to the usual $t$-structures.
\end{enumerate}
\end{proposition}
\begin{proof}
(1) We only need to show the right $t$-exactness of $\underline{f}^*$. Let
$\sK\in\underline{\cD}^{\geq n}(X,\Lambda_{\bullet})$, $\sL\in
\underline{\cD}^{\le n-1}(Y,\Lambda_\bullet)$. Consider the fiber sequence
$\tau^{\le n-1} \sK \to \sK \to \tau^{\ge n}\sK$ in
$\cD(X,\Lambda_\bullet)$. It induces a fiber sequence $f^*\tau^{\le n-1} \sK
\to \underline{f}^* \sK \to f^*\tau^{\ge n} \sK$ in
$\cD(Y,\Lambda_\bullet)$. By assumption and Proposition \ref{3pr:adic_t},
$\tau^{\le n-1}\sK\in \cD^{(+)}_0(X,\Lambda_\bullet)$. Thus $f^*\tau^{\le
n-1}\sK\in \cD^{(+)}_0(Y,\Lambda_\bullet)$, and $\Hom(\sL,f^*\tau^{\le
n-1}\sK)=0$ by \ref{3le:hom_vanish}. It follows that
$\Hom(\sL,\underline{f}^*\sK)=0$. Therefore,
$\underline{f}^*\sK\in\underline{\cD}^{\geq n}(Y,\Lambda_{\bullet})$.
(2) Similar to (1).
\end{proof}
Let $f\colon Y\to X$ be a smooth surjective morphism in $\Chparl$ (resp.\
$\Chpdm{}$), $(\Lambda,\fm)$ be an object of $\PRing_{\ltor}$ (resp.\
$\PRing$). By Poincar\'e duality, if $(Y,(\Lambda,\fm))$ is admissible, then
$(X,(\Lambda,\fm))$ is locally admissible. This applies in particular to the
case where $Y$ is an algebraic space. In this case, admissibility is related
to the following finiteness condition on cohomological dimension.
\begin{definition}\label{3de:bounded}
Let $X$ be a higher Artin (resp.\ Deligne--Mumford) stack, $R$ be a ring. We
say $X$ is \emph{locally $R$-bounded}, if there exists an atlas (resp.\
\'{e}tale atlas) $\coprod_{i\in I}X_i\to X$ with $X_i$ algebraic spaces such
that for every $i\in I$, and every scheme $U$ \'{e}tale and of finite
presentation over $X_i$,
\[\r{cd}_R(U):=\max\{n\res \r{H}^n(U,F)\neq 0\text{ for some }F\in\Mod(U,R)\}<\infty.\]
\end{definition}
\begin{proposition}\label{3pr:five_condition}
Let $X$ be an algebraic space, $(\Lambda,\fm)$ be an object of $\PRing$.
Consider the following conditions:
\begin{enumerate}
\item The pair $(X,(\Lambda,\fm))$ is admissible.
\item For every $\sK\in\underline{\cD}(X,\Lambda_{\bullet})$,
$\bR\pi_*(F_{\bullet}\otimes_{\Lambda_{\bullet}}\sK)=0$, where
$F_{\bullet}\in\Mod(X,\Lambda_{\bullet})\simeq\cD^{\heartsuit}(X,\Lambda_{\bullet})$
is
\[\cdots\xrightarrow{0}\Lambda/\fm\xrightarrow{0}\cdots\xrightarrow{0}\Lambda/\fm.\]
\item $\bR\pi_*\sK=0$ for every $\sK\in\cD_0(X,\Lambda_{\bullet})$.
\item There exists an \'etale cover $\coprod_{i\in I} X_i\to X$ by
algebraic spaces such that, for every $i\in I$, the cohomological
dimension of
$\pi_*\colon\Mod(X_{i,\et}^{\NN},\Lambda_{\bullet})\to\Mod(X_{i,\et},\Lambda)$
is finite.
\item The algebraic space $X$ is locally $(\Lambda/\fm)$-bounded.
\end{enumerate}
We have $(5)\Rightarrow(4)\Rightarrow(3)\Rightarrow(2)\Leftrightarrow(1)$.
\end{proposition}
\begin{proof}
$(5)\Rightarrow(4)$: By \'{e}tale base change, we can assume that for every
scheme $U$ \'{e}tale and of finite type over $X$,
$\r{cd}_{\Lambda/\fm}(U)=N<\infty$. Since for $n\in\NN$, every
$\Lambda_n=\Lambda/\fm^{n+1}$-module is a successive extension of
$\Lambda/\fm$-modules, we have $\r{cd}_{\Lambda_n}(U)=N$. For a sheaf
$F_{\bullet}\in\Mod(X_{\et}^{\NN},\Lambda_{\bullet})$,
$\r{R}^i\pi_*F_{\bullet}$ is the sheaf associated to the presheaf
$U\mapsto\r{H}^i(U_{\et}^{\NN},F_{\bullet})$. Thus, from the exact sequence
\[\xymatrix{
0 \ar[r] & \r{R}^1\lim_n\r{H}^{i-1}(U_{\et},F_n) \ar[r] & \r{H}^i(U_{\et}^{\NN},F_{\bullet})
\ar[r] & \lim_n\r{H}^{i}(U_{\et},F_n) \ar[r] & 0,
}\] we know that $\r{R}^i\pi_*F_{\bullet}=0$ for $i>N+1$.
$(4)\Rightarrow(3)$: We can assume $X$ to be quasi-compact. Then this
follows from Lemma \ref{3le:pi_vanish} and the following standard
observation. Let $f\colon\cB\to \cA$ be a left exact additive functor of
Grothendieck Abelian categories, such that $\r{R}^if=0$ for $i>d$ where $d$
is a non-negative integer. Then $\r{R}f$ sends $\rD^{\leq n}(\cB)$ to
$\rD^{\leq n+d}(\cA)$. In fact, let $X$ be an element of $\rD^{\le n}(\cB)$.
By \cite{KS}*{14.3.4}, we can compute $\r{R}fX$ by $fY$, where $Y$ is
\emph{any} resolution of $X$ with $f$-acyclic components. We can take $Y$ to
be $\tau^{\le n+d+1}$ of a homotopically injective resolution with injective
components (fibrant replacement) of $X$. Then $Y$ belongs to $\r{C}^{\le
n+d+1}(\cB)$. This shows that $\r{R}f$ sends $\rD^{\le n}(\cB)$ to $\rD^{\le
n+d+1}(\cA)$. It follows that $\r{R}f$ sends $\rD^{\le n}(\cB)$ to $\rD^{\le
n+d}(\cA)$ by truncation.
$(3)\Rightarrow(2)$: In fact, for every $\sK\in\cD(X,\Lambda_{\bullet})$,
$F_{\bullet}\otimes_{\Lambda_{\bullet}}\sK$ belongs to
$\cD_0(X,\Lambda_{\bullet})$.
$(2)\Leftrightarrow(1)$: Let $\sK\in \underline{\cD}(X,\Lambda_\bullet)$. We
need to show that $\bR\pi_*(F_{\bullet}\otimes_{\Lambda_{\bullet}}\sK)=0$ if
and only if the adjunction map $\bL\pi^*\bR\pi_*\sK\to\sK$ is an equivalence.
Since $\prod_{n\in\NN}e_n^*$ is conservative, the latter is equivalent to the
condition that the morphism $\epsilon \colon
\Lambda_n\overset{\r{L}}{\otimes}_{\Lambda}\r R\pi_*\sK\to\sK_n:=e_n^*\sK$ is
an isomorphism in the derived category $\rD(X,\Lambda_n)$ for all $n\in\NN$.
The morphism $\epsilon$ can be decomposed as
\begin{align*}
\Lambda_n\overset{\r{L}}{\otimes}_{\Lambda}\r{R}\pi_*\sK
&\xrightarrow{\alpha} \r{R}\pi_*(\r{L}\pi^*
\Lambda_n\overset{\r{L}}{\otimes}_{\Lambda_\bullet}\sK)
\xrightarrow{\beta}\r{R}\pi_*(\pi^* \Lambda_n\overset{\r{L}}{\otimes}_{\Lambda_\bullet}\sK)\\
&\xrightarrow{\gamma}\r{R}(\pi_{\ge n})_*(\pi_{\ge n}^*
\Lambda_n\overset{\r{L}}{\otimes}_{\Lambda_{\bullet,\ge n}}\sK_{\ge n})
\xrightarrow{\delta}\r{R}(\pi_{\ge n})_*(e_{n*} \sK_n)\simeq \sK_n,
\end{align*}
where $\pi_{\ge n}\colon (\NN_{\ge n},\Lambda_{\bullet,\ge n})\to
(*,\Lambda)$, $e_n\colon (\{n\},\Lambda)\to (\NN_{\ge
n},\Lambda_{\bullet,\ge n})$. Here $\NN_{\ge n}\subseteq \NN$ is the full
subcategory spanned by integers $\ge n$.
By assumption, $\fm$ is generated by an element $\lambda$ that is not a zero
divisor. Thus we have a finite free resolution
$[\Lambda\xrightarrow{\times\lambda^{n+1}}\Lambda]$ of $\Lambda_n$ as an
$\Lambda$-module. Therefore, $\r{L}\pi^*\Lambda_n$ is represented by the
complex of $\Lambda_\bullet$-modules
$[\Lambda_{\bullet}\xrightarrow{\times\lambda^{n+1}}\Lambda_{\bullet}]$ (in
degrees $-1$ and $0$). This implies that
$\r{L}\pi^*\Lambda_n\overset{\r{L}}{\otimes}_{\Lambda_{\bullet}}\sK$ is
represented by the mapping cone of
$\sK\xrightarrow{\times\lambda^{n+1}}\sK$, which is a fibrant object. Then
$\Lambda_n\overset{\r{L}}{\otimes}_{\Lambda}\r R\pi_*\sK$ and
$\r{R}\pi_*(\r{L}\pi^*\Lambda_n\overset{\r{L}}{\otimes}_{\Lambda_{\bullet}}\sK)$
are both represented by
$\pi_*\sK'\xrightarrow{\times\lambda^{n+1}}\pi_*\sK'$, where $\sK'$ is a
fibrant replacement of $\sK$, and $\alpha$ is represented by the identity.
Consider the diagram
\[\xymatrix{(\NN_{\ge n},\Lambda_{\bullet,\ge n})\ar[r]^-{j}\ar[d]
&(\NN,\Lambda_\bullet)\ar[d]\\
(\NN_{\ge n},\lambda_{\ge n}) \ar[r]^-{j'}\ar[rd]_{\pi'_{\ge n}} & (\NN,\lambda)\ar[d]^{\pi'}\\
&\Lambda,}
\]
where $\lambda$ is the constant ring with value $\Lambda$. By the cofinality
of $\NN_{\ge n}$ in $\NN$, the natural transformation $\pi'_* \to (\pi'_{\ge
n})_*\circ {j'}^*$ is an isomorphism. Since $j'_*$ admits an \emph{exact}
left adjoint, it follows that $\r{R}\pi'_* \to \r{R}(\pi'_{\ge n})_*\circ
{j'}^*$ is an isomorphism. Thus the natural transformation $\r{R}\pi_*\to
\r{R}\pi_{\ge n}\circ j^*$ is an isomorphism. Therefore, $\gamma$ is an
isomorphism.
The morphism $\delta$ is induced by the morphism $\pi_{\ge
n}^*\Lambda_n\overset{\r{L}}{\otimes}_{\Lambda_{\bullet,\ge n}}\sK_{\ge n}
\to e_{n*}e_n^*(\pi_{\ge
n}^*\Lambda_n\overset{\r{L}}{\otimes}_{\Lambda_{\bullet,\ge n}}\sK_{\ge
n})\simeq e_{n*}\sK_n$, which is an isomorphism since $\sK$ is adic.
Therefore, $\epsilon$ is an isomorphism if and only if $\beta$ is an
isomorphism.
By the above resolution of $\Lambda$, the cone of
$\r{L}\pi^*\Lambda_n\to\Lambda_n$ is $G^n_\bullet[-2]$, where
$G^n_m=\Lambda/\fm^{\min(m,n)+1}$ and the transition maps are multiplication
by $\lambda$, so that $G^0_\bullet =F_{\bullet}$. Thus, if $\beta$ is an
isomorphism for $n=0$, then
$\bR\pi_*(F_{\bullet}\otimes_{\Lambda_{\bullet}}\sK)=0$. For $n\ge 1$,
$G^n_\bullet$ is an extension of $F_\bullet$ by $G^{n-1}_{\bullet+1}$. Thus,
if $\bR\pi_*(F_{\bullet}\otimes_{\Lambda_{\bullet}}\sK)=0$, then, by the
above, $\beta$ is an isomorphism for all $n\in \NN$.
\end{proof}
\subsection{Unbounded constructible adic complexes}
\label{3ss:un_constructible_adic}
Let $(\Lambda,\fm)$ be an object of $\PRing_{\ltor}$ such that
$\Lambda/\fm^{n+1}$ is Noetherian for all $n$. Let $\sS$ be an
$\sf{L}$-coprime higher Artin stack. Assume that there exists an atlas $S\to
\sS$, where $S$ is either a quasi-excellent scheme or a regular scheme of
dimension $\le 1$.
\begin{proposition}\label{3pr:un_constructible_adic}
Let $f\colon Y\to X$ be a morphism of $\Chpars$. Then $f^!$ and $\HOM$
restrict to the following:
\begin{description}
\item[2R$'$] $\underline{f}^!\colon
\underline{\cD}_{\cons}(X,\Lambda_\bullet)\to\underline{\cD}_{\cons}(Y,\Lambda_\bullet)$,
if $(X,(\Lambda,\fm))$ and $(Y,(\Lambda,\fm))$ are admissible.
\item[3R$'$] $\underline{\HOM}_X(-,-)\colon
\underline{\cD}^{(-)}_{\cons}(X,\Lambda_\bullet)^{op}\times\underline{\cD}^{(+)}_{\cons}(X,\Lambda_\bullet)
\to\underline{\cD}^{(+)}_{\cons}(X,\Lambda_\bullet)$, if
$(X,(\Lambda,\fm))$ is admissible.
\end{description}
\end{proposition}
Let $X$ be a scheme in $\Schqcs$. Recall that a complex
$\sK\in\cD(X,\Lambda_{\bullet})$ is a $\lambda$-complex \cite{LO2}*{3.0.6}
if $\r H^n\sK$ is constructible and almost adic. In particular,
$\sK\in\cD_\cons(X,\Lambda_{\bullet})$. The proofs of the following
statements are similar to \cite{Zh}.
\begin{lem}\label{3le:lambda_complex}
Let $X$ be a scheme in $\Schqcs$ such that Condition (3) in Proposition
\ref{3pr:five_condition} holds for the pair $(X,(\Lambda,\fm))$. Let
$\cD_{\cons}'(X,\Lambda_{\bullet})$ be the full subcategory of
$\cD_{\r{en}}(X,\Lambda_{\bullet})$ spanned by $\lambda$-complexes. We have
\begin{enumerate}
\item $\cD_{\cons}'(X,\Lambda_{\bullet})$ is closed under the
truncation functors $\tau^{\geq n}$ and $\tau^{\leq n}$.
\item The essential image of
$\bL\pi^*\bR\pi_*\cD_{\cons}'(X,\Lambda_{\bullet})$ coincides with
\[\underline{\cD}_{\cons}(X,\Lambda_\bullet)=\underline{\cD}(X,\Lambda_{\bullet})\cap\cD_{\cons}(X,\Lambda_{\bullet}).\]
\end{enumerate}
\end{lem}
\begin{proposition}
Let the assumptions be as in the above lemma. Put
\[\underline{\cD}_{\cons}^{\leq n}(X,\Lambda_{\bullet})=
\underline{\cD}^{\leq
n}(X,\Lambda_{\bullet})\cap\cD_{\cons}(X,\Lambda_{\bullet}).\] Then the right
perpendicular full subcategory $\underline{\cD}_{\cons}^{\geq
n}(X,\Lambda_{\bullet})$ of $\underline{\cD}_{\cons}^{\leq
n-1}(X,\Lambda_{\bullet})^{\perp}$ in
$\underline{\cD}_{\cons}(X,\Lambda_{\bullet})$ is the essential image of
$\bL\pi^*\bR\pi_*(\cD_{\cons}'(X,\Lambda_{\bullet})\cap\cD_{\r{en}}^{\geq
n}(X,\Lambda_{\bullet}))$. Moreover, the truncation functors
$\underline\tau^{\leq n}\simeq\bL\pi^*\circ\bR\pi_*\circ\tau^{\leq n}$ and
$\underline\tau^{\geq n}\simeq\bL\pi^*\circ\bR\pi_*\circ\tau^{\geq n}$.
\end{proposition}
\begin{corollary}\label{3co:constructible_adic_t}
Let $X$ be a higher Artin stack that is locally $(\Lambda/\fm)$-bounded.
Then the full subcategory $\underline{\cD}_{\cons}(X,\Lambda_\bullet)$ is
preserved under the truncation functors $\underline\tau^{\leq n}$ and hence
$\underline\tau^{\geq n}$ on $\underline{\cD}(X,\Lambda_\bullet)$.
\end{corollary}
\subsection{Compatibility with Laszlo--Olsson}
\label{3ss:compatibility}
We prove the compatibility between our adic formalism and Laszlo--Olsson's
\cite{LO2}, under their assumptions.
Let $\sf{L}=\{\ell\}$. Let $\sS$ be an $\sf{L}$-coprime scheme satisfying
that
\begin{enumerate}
\item It is affine excellent and finite-dimensional;
\item For every $\sS$-scheme $X$ of finite type, there exists an
\'{e}tale cover $X'\to X$ such that
$\r{cd}_{\ell}(Y)<\infty$\footnote{According to our notation,
$\r{cd}_{\ell}$ is nothing but $\r{cd}_{\d{F}_{\ell}}$.} for every
scheme $Y$ \'{e}tale and of finite type over $X'$;
\item It admits a global dimension function and we fix such a function
(see \cite{LZ}*{6.3.1}).
\end{enumerate}
Fix a complete discrete valuation ring $\Lambda$ with the maximal ideal
$\f{m}$ and residue characteristic $\ell$ such that
$\Lambda=\lim_n\Lambda_n$, where $\Lambda_n=\Lambda/\f{m}^{n+1}$, as in
\cite{LO2}. In particular, $(\Lambda,\f{m})$ is an object of $\PRing$. For
every stack $\cX$ in $\ChpsLMB$, the pair $\cX$ is locally
$(\Lambda/\f{m})$-bounded.
From the definition of $\underline{\cD}_{\cons}(\cX,\Lambda_{\bullet})$,
which is the full subcategory of $\cD(\cX,\Lambda_{\bullet})$ spanned by
constructible adic complexes, \cite{LO2}*{3.0.10, 3.0.14, 3.0.18}, and
\cite{LZ}*{5.3.6}, we have a canonical equivalence between categories
\begin{align}\label{3eq:compatibility}
\r{h}\underline{\cD}_{\cons}(\cX,\Lambda_{\bullet})\simeq\b{D}_c(\cX,\Lambda),
\end{align}
where the latter one is defined in \cite{LO2}*{3.0.6}.
\begin{proposition}
For a morphism $f\colon\cY\to\cX$ of finite type in $\ChpsLMB$, there are
natural isomorphisms between functors:
\begin{gather*}
\r{h}\underline{f}^*\simeq
\r{L}f^*\colon\b{D}_c(\cX,\Lambda)\to\b{D}_c(\cY,\Lambda),\qquad
\r{h}\underline{f}_*\simeq
\r{R}f_*\colon\b{D}_c^{(+)}(\cY,\Lambda)\to\b{D}_c^{(+)}(\cX,\Lambda);\\
\r{h}\underline{f}_!\simeq
\r{R}f_!\colon\b{D}_c^{(-)}(\cY,\Lambda)\to\b{D}_c^{(-)}(\cX,\Lambda),\qquad
\r{h}\underline{f}^!\simeq
\r{R}f^!\colon\b{D}_c(\cX,\Lambda)\to\b{D}_c(\cY,\Lambda);\\
\r{h}(-\underline{\otimes}-)\simeq(-)\overset{\b{L}}{\otimes}(-)\colon\b{D}_c^{(-)}(\cX,\Lambda)
\times\b{D}_c^{(-)}(\cX,\Lambda)\to\b{D}_c^{(-)}(\cX,\Lambda),\\
\r{h}\underline{\HOM}(-,-)\simeq\b{\s{R}hom}_{\Lambda}(-,-)\colon\b{D}_c^{(-)}(\cX,\Lambda)^{\r{opp}}
\times\b{D}_c^{(+)}(\cX,\Lambda)\to\b{D}_c^{(+)}(\cX,\Lambda).
\end{gather*}
that are compatible with \eqref{3eq:compatibility}.
\end{proposition}
By Lemma \ref{3le:constructible_adic} and Proposition
\ref{3pr:constructible_adic}, the six operations on the left side in the
above proposition do have the correct range.
\begin{proof}
The isomorphisms for tensor product, internal Hom and $f^*$ simply follow
from the same definitions here and in \cite{LO2}*{\Secs 4, 6}. The
isomorphism for $f_*$ follows from the adjunction and that for $f^*$
(\cite{LZ}*{6.3.2}). The isomorphism for $f_!$ will follows from the
adjunction and that for $f^!$ which will be proved below.
By the compatibility of dualizing complexes and the isomorphisms for
internal Hom, we have natural isomorphisms
$\underline{\rD}_{\cX}\simeq\rD_{\cX}$ and
$\underline{\rD}_{\cY}\simeq\rD_{\cY}$. Moreover,
$\cD(\cX,\Lambda_{\bullet})_{\r{dual}}$ contains
$\cD_{\r{cons}}(\cX,\Lambda_{\bullet})$. Therefore, by \cite{LO2}*{9.1}, to
prove the isomorphism for $f^!$, we only need to show that our functors
satisfy
\[\r{h}\underline{f}^!\simeq\underline{\rD}_{\cY}\circ\r{h}\underline{f}^*\circ
\underline{\rD}_{\cX}.\] In fact, by the biduality isomorphism, we have
\begin{align*}
\r{h}\underline{f}^!&\simeq\r{h}\underline{f}^!\circ\underline{\rD}_{\cX}\circ\underline{\rD}_{\cX}(-)\\
&=
\r{h}\underline{f}^!\circ\r{h}\underline{\HOM}_{\cX}(\r{h}\underline{\HOM}_{\cX}(-,\Omega_{\cX}),\Omega_{\cX}) \\
&\simeq \r{h}R_{\cY}\circ\r{h}f^!\circ\r{h}\HOM_{\cX}(\r{h}\underline{\HOM}_{\cX}(-,\Omega_{\cX}),\Omega_{\cX}) \\
&\simeq \r{h}R_{\cY}\circ\r{h}\HOM_{\cY}(\r{h}f^*\circ\r{h}\underline{\HOM}_{\cX}(-,\Omega_{\cX}),f^!\Omega_{\cX}) \\
&\simeq \r{h}\underline{\HOM}_{\cY}(\r{h}\underline{f}^*\circ\r{h}\underline{\HOM}_{\cX}(-,\Omega_{\cX}),\Omega_{\cY})\\
&=\underline{\rD}_{\cY}\circ\r{h}\underline{f}^*\circ\underline{\rD}_{\cX}.
\end{align*}
\end{proof}
\begin{remark}
In view of the above compatibility, we prove all the expected properties of
the six operations, in particular the Base Change Theorem, in the adic case
of Laszlo--Olsson \cite{LO2}.
\end{remark}
\section{Perverse $t$-structures}
\label{2}
In \Sec \ref{2ss:perversity_evaluation}, we define a general notion of
perversity, which we call \emph{perversity smooth/\'{e}tale evaluation} for
higher Artin/Deligne--Mumford stacks. We then construct the perverse
$t$-structure for a perversity evaluation on an Artin stack in \Sec
\ref{2ss:perverse_t} using descent. In \Sec \ref{2ss:adic_perverse_t}, we
define the adic perverse $t$-structure. In both cases, we provide
descriptions of the $t$-structures in terms of cohomology on stalks as in
the classical situation.
\subsection{Perversity evaluations}
\label{2ss:perversity_evaluation}
\begin{definition}
Let $X$ be a scheme in $\Schqcs$.
\begin{enumerate}
\item Following \cite{Gabber}*{\Sec 1}, a \emph{weak perversity
function} on $X$ is a function $p\colon |X|\to\ZZ\cup\{+\infty\}$
such that for every $n\in\ZZ$, the set $\{x\in|X|\res p(x)\geq n\}$
is ind-constructible.
\item An \emph{admissible perversity function} on $X$ is a weak
perversity function $p$ such that for every $x\in |X|$, there is an
open dense subset $U\subseteq\overline{\{x\}}$ satisfying the
condition that for every $x'\in U$, $p(x')\leq p(x)+2\codim(x',x)$.
\item A \emph{codimension perversity function} on $X$ is a function
$p\colon |X|\to\ZZ\cup\{+\infty\}$ such that for every immediate
\'etale specialization $x'$ of $x$, $p(x')=p(x)+1$.
\end{enumerate}
\end{definition}
\begin{remark}\leavevmode
\begin{enumerate}
\item A weak perversity function on a locally Noetherian scheme is
locally bounded from below.
\item An admissible perversity function on a scheme that is locally
Noetherian and of finite dimension is locally bounded from above.
\item A codimension perversity function on a scheme is \emph{not}
necessarily a weak perversity function.
\item A codimension perversity function that is also a weak perversity
function is an admissible perversity function. If $X$ is locally
Noetherian, then a codimension perversity function is a weak
perversity function and hence an admissible perversity function.
\item A codimension perversity function is the opposite of a dimension
function in the sense of \cite{PS}*{2.1.8}. If $X$ is locally
Noetherian and admits a dimension function, then $X$ is universally
catenary by \cite{PS}*{2.2.6}. In this case, immediate \'etale
specializations coincide with immediate Zariski specializations
\cite{PS}*{2.1.4}.
\item If $p$ is a weak (resp.\ admissible, resp.\ codimension)
perversity function on $X$ and $d\colon \lvert X\rvert \to \Z\cup
\{+\infty\}$ is a locally constant function, then $p+d$ is a weak
(resp.\ admissible, resp.\ codimension) perversity function on $X$.
\end{enumerate}
\end{remark}
\begin{definition}
A function $q\colon \NN\to \Z$ or $\Z\to \Z$ is called \emph{admissible} if
$q$ and $\b 2-q$ are both increasing, where $\b 2(x)=2x$ and similarly for
$\b 0$ and $\b 1$, which will be used below.
\end{definition}
Let $f\colon Y\to X$ be a smooth morphism of schemes in $\Schqcs$, and let
$p\colon \lvert X\rvert\to\ZZ\cup\{+\infty\}$, $q\colon \NN\to \Z$ be
functions. We define the pullback $f^*_qp\colon \lvert Y\rvert
\to\ZZ\cup\{+\infty\}$ by $(f^*_qp)(y)=p(f(y))-q(\trdeg[k(y):k(f(y))])$ for
every point $y\in|Y|$. In particular, $f_{\b 0}^*=p\circ f$.
\begin{lem}\label{2le:perverse_zero}
Let $f\colon Y\to X$ be a morphism (resp.\ \'etale morphism, resp.\ \'etale
morphism) of schemes in $\Schqcs$. If $p$ is a weak (resp.\ admissible,
resp.\ codimension) perversity function on $X$, then $f^*_{\b 0}p$ is a weak
(resp.\ admissible, resp.\ codimension) perversity function on $Y$.
\end{lem}
\begin{proof}
We have $f^*_{\b 0}p=p\circ f$. If $p$ is a weak perversity function, then
\[\{y\in \lvert Y\rvert \mid f^*_{\b 0}p(y)\ge n\}=f^{-1}(\{x\in \lvert X \rvert \mid p(x) \ge n\})\]
is ind-constructible by \cite{EGAIV}*{1.9.5 (vi)}. The other two cases
follow from the trivial fact that $\codim(y',y)= \codim (f(y'),f(y))$ for
every specialization $y'$ of $y$ on $Y$.
\end{proof}
\begin{lem}\label{2le:perverse_pullback}
Let $f\colon Y\to X$ be a morphism of locally Noetherian schemes in
$\Schqcs$, locally of finite type.
\begin{enumerate}
\item Let $p$ be a weak perversity function on $X$, and let $q\colon
\NN\to \Z$ be an increasing function. Then $f^*_qp$ is a weak
perversity function on $Y$.
\item Let $p$ be an admissible perversity function on $X$, and let
$q\colon \NN\to \Z$ be an admissible function. Then $f^*_qp$ is an
admissible perversity function on $Y$.
\item Let $p$ be a codimension perversity function on $X$. Then $f^*_{\b
1}p$ is a codimension perversity function on $Y$.
\end{enumerate}
\end{lem}
\begin{proof}
(1) For a locally closed subset $Z$ of a scheme $X$, we endow it with the
reduced induced subscheme structure. For every point $y\in|Y|$, let
$U_y\subset \overline{\{y\}}$ be a nonempty open subset such that the
induced morphism $f_y\colon \overline{\{y\}}\to \overline{\{f(y)\}}$ is
flat. Such an open subset exists by \cite{EGAIV}*{6.9.1}. For $y'\in U_y$,
we have
\[\delta(y',y):=\trdeg[k(y):k(f(y))]-\trdeg[k(y'):k(f(y'))]= \codim(y',U_y\times_{\overline{f(y)}}\{f(y')\})\ge 0\]
by \cite{EGAIV}*{14.3.13} because $f_y$ is universally open
\cite{EGAIV}*{2.4.6}. Therefore, for $n\in\ZZ$,
\[\{y\in|Y|\res f^*_qp(y)\geq n\}=\bigcup_{y\in|Y|}f^{-1}\left\{x\in|X|\mid p(x)\geq n+q(\trdeg[k(y):k(f(y))])\right\}
\cap U_y,\] is a union of ind-constructible subsets, and hence is itself
ind-constructible. In other words, $f^*p$ is a weak perversity function.
(2) Let $y\in|Y|$ be a point, $x=f(y)$, and let $U_x\subset
\overline{\{x\}}$ be a dense open subset such that $p(x')\le
p(x)+2\codim(x',x)$. We prove that for $y'\in U_y\cap f^{-1}(U_x)$,
$f^*_qp(y')\leq f^*_qp(y)+2\codim(y',y)$. We may assume that $p(x)\in \Z$.
Let $x'=f(y')$. We have $f^*_qp(y)=p(x)-q(\trdeg[k(y):k(x)])$ and
$f^*_qp(y')=p(x')-q(\trdeg[k(y'):k(x')])$. Moreover, by
\cite{EGAIV}*{6.1.2},
\[\delta(y',y)=\codim(y',y)-\codim(x',x).\]
Therefore,
\begin{align*}
f^*_qp(y')-f^*_qp(y)&=p(x')-p(x)+q(\trdeg[k(y):k(x)])-q(\trdeg[k(y'):k(x')])\\
&\leq2\codim(x',x)+2\delta(y',y)=2\codim(y',y).
\end{align*}
In other words, $f^*p$ is an admissible perversity function on $Y$.
(3) This is essentially proved in \cite{PS}*{2.5.2}.
\end{proof}
\begin{definition}[Pointed schematic neighborhood]
Let $X$ be a higher Artin (resp.\ Deligne--Mumford) stack. A \emph{pointed
smooth (resp.\ \'etale) schematic neighborhood} of $X$ is a triple
$(X_0,u_0,x_0)$ where $u_0\colon X_0\to X$ is a smooth (resp.\ an \'{e}tale)
morphism with $X_0$ in $\Schqcs$, and $x_0\in |X_0|$ is a scheme-theoretical
point. A morphism $v\colon (X_1,u_1,x_1)\to(X_0,u_0,x_0)$ of pointed smooth
(resp.\ \'etale) schematic neighborhoods is a smooth (resp.\ an \'{e}tale)
morphism $v\colon X_1\to X_0$ such that there is a triangle
\begin{align}\label{2eq:schematic_neighborhood}
\xymatrix{
X_1 \ar[rd]_-{u_1} \ar[rr]^-{v} & & X_0 \ar[ld]^-{u_0} \\
& X }
\end{align}
and $v(x_1)=x_0$. We say $(X_1,u_1,x_1)$ \emph{dominates} $(X_0,u_0,x_0)$ if
there is such a morphism. The category of pointed smooth (resp.\ \'etale)
schematic neighborhoods of $X$ is denoted by $\Vosm(X)$ (resp.\ $\Voet(X)$).
\end{definition}
\begin{lem}
Let $X$ be a higher Artin stack, and let $v\colon
(X_1,u_1,x_1)\to(X_0,u_0,x_0)$ be a morphism of pointed smooth schematic
neighborhoods of $X$. Then the codimension of $x_1$ in the base change
scheme $X_{1,x_0}=X_1\times_{X_0}\{x_0\}$ depends only on the source and the
target of $v$.
\end{lem}
We will denote by $\delta^{(X_1,u_1,x_1)}_{(X_0,u_0,x_0)}$ this codimension.
It is clear that
$\delta^{(X_2,u_2,x_2)}_{(X_0,u_0,x_0)}=\delta^{(X_2,u_2,x_2)}_{(X_1,u_1,x_1)}+\delta^{(X_1,u_1,x_1)}_{(X_0,u_0,x_0)}$
if $(X_2,u_2,x_2)$ dominates $(X_1,u_1,x_1)$. Moreover, if $v$ is \'etale,
$\delta^{(X_1,u_1,x_1)}_{(X_0,u_0,x_0)}=0$.
\begin{proof}
Note that $\codim(x_1,X_{1,x_0})=\dim_{x_1} (v)-\trdeg[k(x_1):k(x_0)]$. It is
clear that $\dim_{x_1} v=\dim_{x_1} u_1-\dim_{x_0} u_0$ does not depend on
$v$. We will show that $\trdeg[k(x_1):k(x_0)]$ does not depend on $v$ either.
Let $f\colon Y\to X$ be an atlas of $X$ with $Y$ a scheme in $\Schqcs$. Let
\[\xymatrix{
Y_1 \ar[rr]^-{v'} \ar[dr]_-{u'_1} && Y_0 \ar[dl]^-{u'_0} \\
& Y }\] be the base change of \eqref{2eq:schematic_neighborhood}, $f_0\colon Y_0\to
X_0$ and $f_1\colon Y_1\to X_1$. Let $w_0\colon Y'_0\to Y_0$ be an atlas
with $Y'_0$ a scheme in $\Schqcs$, and let
\[\xymatrix{
Y'_1 \ar[r]^-{v''} \ar[d]_-{w_1} & Y'_0 \ar[d]^-{w_0} \\
Y_1 \ar[r]^-{v'} & Y_0 }\] be the base change. Then $v''$ is a smooth
morphism of schemes in $\Schqcs$. Since $f_0\circ w_0\colon Y'_0\to X_0$ is
smooth and surjective, the base change scheme
$Y'_{0,x_0}=Y'_0\times_{X_0}\{x_0\}$ is nonempty and smooth over the residue
field $k(x_0)$ of $x_0$. Similarly, we have a nonempty scheme $Y'_{1,x_1}$,
smooth over $k(x_1)$. Choose a generic point $y'_1$ of $Y'_{1,x_1}$. Then
its image $y'_0$ in $Y'_{0,x_0}$ is a generic point. Let $y$ be the image of
$y'_0$ in $Y$. Then
$\r{tr.deg}[k(x_1):k(x_0)]=\r{tr.deg}[k(y'_1):k(y)]-\r{tr.deg}[k(y'_0):k(y)]$,
which does \emph{not} depend on $v$.
\end{proof}
For a higher Artin (resp.\ Deligne--Mumford) stack $X$ and a function
$\tp\colon \Ob(\Vosm(X))\to\ZZ\cup\{+\infty\}$ (resp.\ $\tp\colon
\Ob(\Voet(X))\to \ZZ\cup\{+\infty\}$), we have, by restriction, the function
$\tp_{u_0}\colon |X_0|\to\ZZ\cup\{+\infty\}$ for every smooth (resp.\
\'{e}tale) morphism $u_0\colon X_0\to X$ with $X_0$ in $\Schqcs$. If
$f\colon Y\to X$ is a smooth (resp.\ an \'{e}tale) morphism of higher Artin
(resp.\ Deligne--Mumford) stacks, then composition with $f$ induces a
functor $f\colon \Vosm(Y)\to\Vosm(X)$ (resp.\ $f\colon \Voet(Y)\to
\Voet(X)$), and we let $f^*\tp=\tp\circ f$.
\begin{definition}[(admissible/codimension) perversity evaluations]\label{2de:perversity_evaluation}
Let $X$ be a higher Artin stack. A \emph{smooth evaluation} on $X$ is a
function $\tp\colon \Ob(\Vosm(X))\to\ZZ\cup\{+\infty\}$ such that for
$(X_1,u_1,x_1)$ dominating $(X_0,u_0,x_0)$, we have $\tp(X_0,u_0,x_0) \le
\tp(X_1,u_1,x_1)\le
\tp(X_0,u_0,x_0)+2\delta^{(X_1,u_1,x_1)}_{(X_0,u_0,x_0)}$.
A \emph{perversity smooth evaluation} (resp.\ \emph{admissible perversity
smooth evaluation}, \emph{codimension perversity smooth evaluation}) on $X$
is an evaluation $\tp$ such that for every $(X_0,u_0,x_0)\in\Ob(\Vosm(X))$,
$\tp_{u_0}$ is a weak perversity function (resp.\ admissible perversity
function, codimension perversity function) on $X_0$.
Similarly, we define \'etale evaluations and (admissible/codimension)
perversity \'etale evaluations on a higher Deligne--Mumford stack $X$ using
$\Voet(X)$.
We say that a smooth (resp.\ \'etale) evaluation $\tp$ is \emph{locally
bounded} if for every smooth (resp.\ \'etale) morphism $u_0\colon X_0\to X$
with $X_0$ a quasi-compact separated scheme, $\tp_{u_0}$ is bounded.
\end{definition}
\begin{remark}
If $X$ is a scheme in $\Schqcs$, then the map from the set of \'etale
evaluations on $X$ to the set of functions $\lvert X\rvert\to
\ZZ\cup\{+\infty\}$, carrying $\tp$ to $\tp_{\id_X}$, is bijective. Under
this bijection, the notions of (weak) perversity, admissible perversity, and
codimension perversity coincide. If $f\colon Y\to X$ is a morphism of schemes
in $\Schqcs$, then $f^*$ for \'etale evaluations coincide with $f^*_{\b 0}$
for functions.
\end{remark}
\begin{example}\leavevmode
\begin{enumerate}
\item Let $X$ be a higher Artin (resp.\ Deligne--Mumford) stack. Any
constant smooth (resp.\ \'etale) evaluation is an admissible
perversity smooth (resp.\ \'etale) evaluation.
\item Let $f\colon Y\to X$ be a morphism of higher Deligne--Mumford
stacks, locally of finite type, let $\tp$ be an \'etale evaluation
on $X$, and let $q\colon \NN\to \Z$ be a function. We define an
\'etale evaluation $f^*_q \tp$ on $Y$ as follows. For any object
$(Y_0,v_0,y_0)$ of $\Voet(Y)$, there exists a morphism
$(Y_1,v_1,y_1)\to (Y_0,v_0,y_0)$ in $\Voet(Y)$ such that there
exists a diagram
\[\xymatrix{Y_1\ar[r]^{v_1}\ar[d]_{f_0} & Y\ar[d]^f\\
X_0\ar[r]^{u_0} & X,}
\]
where $X_0$ is in $\Schqcs$ and $u_0$ is \'etale. We put
$f^*_q\tp(Y_0,v_0,y_0)=\tp(X_0,u_0,f_0(y_1))-q(\trdeg[k(y_1):k(f_0(y_1))])$.
This clearly does not depend on choices. If $\tp$ is a perversity
\'etale evaluation, then $f^*_{\b 0}\tp$ is a perversity \'etale
evaluation by Lemma \ref{2le:perverse_zero}.
\item Let $f\colon Y\to X$ be a morphism of higher Artin stacks with $X$
being a higher Deligne--Mumford stack, locally of finite type, let
$\tp$ be an \'etale evaluation on $X$, and let $q\colon \Z\to \Z$ be
an admissible function. We define a smooth evaluation $f^*_q\tp$ on
$Y$ by $(f^*_q\tp)(Y_0,v_0,y_0)=(v_0\circ f)^*_{q'}(y_0)$ for every
object $(Y_0,v_0,y_0)$ of $\Vosm(Y)$, where $q'\colon \NN\to \Z$ is
the function $q'(n)=q(n-\dim_{y_0} v_0)$. If $\tp$ is a perversity
\'etale evaluation, then $f^*_{\b 0}\tp$ is a perversity smooth
evaluation. If $X$ is locally Noetherian and $\tp$ is a perversity
(resp.\ admissible perversity, resp.\ codimension perversity) \'etale
evaluation, then $f^*_{q}\tp$ (resp.\ $f^*_q\tp$, resp.\ $f^*_{\b
1}\tp$) is a perversity (resp.\ admissible perversity, resp.\
codimension) smooth evaluation by Lemma \ref{2le:perverse_pullback}.
If $X$ is an object of $\Schqcs$ and $p\colon \lvert X\rvert\to
\Z\cup \{+\infty\}$ is a function, we denote by $f^*_q p$ the smooth
evaluation $f^*_q \tp$ on $Y$, where $\tp$ is the perversity \'etale
evaluation corresponding to $p$.
\end{enumerate}
\end{example}
\subsection{Perverse $t$-structures}
\label{2ss:perverse_t}
Let $\cC$ be a stable $\infty$-category equipped with a $t$-structure. We say
that $\cC$ is \emph{weakly left complete} (resp.\ \emph{weakly right
complete}) if $\cC^{\le -\infty}=\bigcap_n \cC^{\le -n}$ (resp.\ $\cC^{\ge
\infty}=\bigcap_n \cC^{\ge n}$) consists of zero objects. The family $(\r
H^i)_{i\in \Z}$ is conservative if and only if $\cC$ is weakly left complete
and weakly right complete (cf.\ \cite{BBD}*{1.3.7}). The following
generalizes \cite{Lu2}*{1.2.1.19}.
\begin{lem}\label{2le:wlc}
Let $\cC$ be a stable $\infty$-category equipped with a $t$-structure.
Consider the following conditions
\begin{enumerate}
\item The $\infty$-category $\cC$ is left complete (see the definition
preceding \cite{Lu2}*{1.2.1.18}).
\item The $\infty$-category $\cC$ is weakly left complete.
\end{enumerate}
Then (1) implies (2). Moreover, if $\cC$ admits countable products and there
exists an integer $a$ such that countable products of objects of $\cC^{\le
0}$ belong to $\cC^{\le a}$, then (2) implies (1).
\end{lem}
\begin{proof}
The first assertion is obvious since the image of $\cC^{\le -\infty}$ under
the functor $\cC\to \hat\cC$ consists of zero objects.
To show the second assertion, it suffices to replace $f(n-1)$ by $f(n-a-1)$
in the proof of \cite{Lu2}*{1.2.1.19}.
\end{proof}
Let $X$ be a scheme $X$ in $\Schqcs$, let $p\colon \lvert X\rvert \to
\Z\cup\{+\infty\}$ be a function, and let $\lambda=(\Xi,\Lambda)$ be an
object of $\Rind$. Following Gabber \cite{Gabber}*{\Sec 2}, we define full
subcategories $\TS{p}{\cD}{\leq 0}(X,\lambda),\TS{p}{\cD}{\geq
0}(X,\lambda)\subseteq \cD(X,\lambda)$ as follows. For $\sK$ in
$\cD(X,\lambda)$,
\begin{itemize}
\item $\sK$ belongs to $\TS{p}{\cD}{\le 0}(X,\lambda)$ if and only if
$i^*_{\overline x}j^*_{\overline x}\sK\in \cD^{\le p(x)}(\overline
x,\lambda)$ for all $x\in X$;
\item $\sK$ belongs to $\TS{p}{\cD}{\ge 0}(X,\lambda)$ if and only if
$\sK\in \cD^{(+)}(X,\lambda)$ and $i^!_{\overline x}j^*_{\overline
x}\sK\in \cD^{\ge p(x)}(\overline x, \lambda)$ for all $x\in X$.
\end{itemize}
Here $\overline x$ is a geometric point above $x$, $i_{\overline x}\colon
\overline x\to X_{(\overline x)}$, $j_{\overline x}\colon X_{(\overline
x)}\to X$. We will omit $j_{\overline x}^*$ from the notation when no
confusion arises. For an immersion $i\colon Z\to X$ in $\Schqcs$, $z\in Z$,
we have an equivalence $i^!_{\overline z}\simeq i^!_{\overline z}i^!$ of
functors $\cD^{(+)}(X,\lambda)\to \cD^+(\overline z, \lambda)$.
Gabber showed in \cite{Gabber} that if $p$ is a weak perversity function and
$\Xi=*$, then $(\TS{p}{\cD}{\leq 0}(X,\lambda),\TS{p}{\cD}{\geq
0}(X,\lambda))$ is a $t$-structure on $\cD(X,\lambda)$. This generalizes
easily to the case of general $\Xi$ as follows. By \cite{Lu2}*{1.4.4.11},
there exists a $t$-structure $(\TS{p}{\cD}{\leq 0}(X,\lambda),\cD')$ on
$\cD(X,\lambda)$. For $\sK\in \TS{p}{\cD}{\leq 0}(X,\lambda)$, $\sL\in
\TS{p}{\cD}{\geq 0}(X,\lambda)$, we have $a_*\HOM(\sK,\sL[1])\in \cD^{\ge
1}(\lambda)$, so that $\Hom(\sK,\sL[1])=\r{H}^0(\Xi,a_*\HOM(\sK,\sL[1]))=0$,
where $a\colon X_{\et}\to *$ is the morphism of topoi. Thus
$\TS{p}{\cD}{\geq 0}(X,\lambda)\subseteq \cD'$. For every $\xi\in \Xi$, the
functor $\bL e_{\xi!}\colon \cD(X,\Lambda(\xi))\to \cD(X,\lambda)$ is left
$t$-exact for the $t$-structures $(\TS{p}{\cD}{\leq
0}(X,\Lambda(\xi)),\TS{p}{\cD}{\geq 0}(X,\Lambda(\xi)))$ and
$(\TS{p}{\cD}{\leq 0}(X,\lambda),\cD')$. It follows that $e_\xi^*$ is right
$t$-exact for the same $t$-structures. Therefore $\cD'\subseteq
\TS{p}{\cD}{\geq 0}(X,\lambda)$.
Thus $(\TS{p}{\cD}{\leq 0}(X,\lambda),\TS{p}{\cD}{\geq 0}(X,\lambda))$ is a
$t$-structure on $\cD(X,\lambda)$. By definition, this $t$-structure is
accessible, and, if $p$ takes values in $\Z$, weakly left complete. By
\cite{Gabber}*{3.1}, this $t$-structure is weakly right complete, thus right
complete. By Lemma \ref{2le:wlc}, the above $t$-structure is left complete
if $p$ is locally bounded and every quasi-compact closed open subscheme of
$X$ is $\lambda$-cohomologically finite. We say that a scheme $Y$ is
\emph{$\lambda$-cohomologically finite} if there exists an integer $n$ such
that, for every $\xi\in \Xi$, the $\lambda(\xi)$-cohomological dimension of
the \'etale topos of $Y$ is at most $n$.
By definition, for any morphism $f\colon Y\to X$ of schemes in $\Schqcs$,
$f^*\colon \cD(X,\lambda)\to \cD(Y,\lambda)$ carries $\TS{p}{\cD}{\le
0}(X,\lambda)$ to $\TS{f^*_{\b 0}p}{\cD}{\le 0}(Y,\lambda)$. Moreover, if $f$
is \'etale, then $f^*$ carries $\TS{p}{\cD}{\ge 0}(X,\lambda)$ to
$\TS{f^*_{\b 0}p}{\cD}{\ge 0}(Y,\lambda)$. We will prove an analogue of this
for smooth morphisms in Proposition \ref{2pr:perverse_t}.
\begin{lem}
Let $f\colon Y\to X$ be a smooth morphism in $\Schqcs_{\sf L}$, let
$\lambda$ be an object of $\Rind_{\ltor}$, and let $y$ be a point of $Y$,
$x=f(y)$. Then there is an equivalence $i_{\overline y}^!\circ f^!\simeq
g^*\circ i_{\overline x}^!\langle d\rangle$ of functors
$\cD^{(+)}(X,\lambda)\to \cD^{+}(\overline y,\lambda)$, where $g\colon
\overline y\to \overline x$, $d=\trdeg[k(y): k(x)]$.
\end{lem}
\begin{proof}
Consider the diagram with Cartesian squares
\[\xymatrix{\overline y\ar[rrd]_g\ar[r]^{i_{\overline y}} & V\ar[r]^{j} & Y_{\overline x}\ar[d]_{f_{\overline
x}}\ar[r]^{i'_{\overline x}} & Y_{\overline{\{x\}}} \ar[r]^{i'}\ar[d]^{f_{\overline{\{x\}}}}
& Y\ar[d]^f \\
&&\overline x \ar[r]^{i_{\overline x}} & \overline{\{x\}}\ar[r]^i & X}
\]
where $V$ is a regular integral subscheme of $Y_{\overline x}$ such that the
image of $\overline y$ in $V$ is a generic point. We have a sequence of
equivalences of functors
\begin{align*}
i_{\overline y}^!\circ f^! &\simeq i_{\overline y}^*\circ j^! \circ {i'_{\overline x}}^* \circ {i'}^!\circ f^!
\simeq i_{\overline y}^*\circ j^! \circ {i'_{\overline x}}^* \circ f_{\overline{\{x\}}}^! \circ i^!\\
&\simeq i_{\overline y}^*\circ j^! \circ f_{\overline x}^! \circ i_{\overline x}^* \circ i^! &&\text{by Poincar\'e duality}\\
&\simeq i_{\overline y}^*\circ (f_{\overline x}\circ j)^! \circ i_{\overline x}^!\\
&\simeq i_{\overline y}^*\circ (f_{\overline x}\circ j)^* \circ i_{\overline x}^! \langle d\rangle &&\text{by Poincar\'e duality}\\
&\simeq g^*\circ i_{\overline
x}^!\langle d\rangle.
\end{align*}
\end{proof}
\begin{lem}\label{2le:perverse_smooth_pullback}
Let $\lambda$ be an object of $\Rind_{\ltor}$, let $f\colon Y\to X$ be a
smooth morphism of schemes in $\Schqcs_{\sf L}$, and let $p\colon \lvert
X\rvert \to \Z\cup \{+\infty\}$ be a function. Then $f^!$ carries
$\TS{p}{\cD}{\ge 0}(X,\lambda)$ to $\TS{f^*_{\b 2}p}{\cD}{\ge 0}(Y,\lambda)$.
Moreover, if $p$ is a weak perversity function on $X$ and $q$ is a weak
perversity function on $Y$ satisfying $f^*_{\b 0} p \le q \le f^*_{\b
2}p+2\dim f$, then $f^!\colon \cD(X,\lambda)\to\cD(Y,\lambda)$ is $t$-exact
with respect to the $t$-structures associated to $p$ and $q$.
\end{lem}
\begin{proof}
The first assertion follows from the above lemma. The second assertion
follows from the first assertion and Poincar\'e duality $f^!\simeq
f^*\langle\dim f\rangle$.
\end{proof}
Let $X$ be an $\sf{L}$-coprime higher Artin (resp.\ a higher
Deligne--Mumford) stack equipped with a perversity evaluation $\tp$, and let
$\lambda$ be an object of $\Rind_{\ltor}$ (resp.\ $\Rind$). For an atlas
(resp.\ \'{e}tale atlas) $u\colon X_0\to X$ with $X_0$ a scheme in
$\Schqcs$, we denote by
$\TS{\tp}{\cD}{\leq0}_u(X,\lambda)\subseteq\cD(X,\lambda)$ (resp.\
$\TS{\tp}{\cD}{\geq0}_u(X,\lambda)\subseteq\cD(X,\lambda)$) the full
subcategory spanned by complexes $\sK$ such that $u^*\sK$ is in
$\TS{\tp_u}{\cD}{\leq0}(X_0,\lambda)$ (resp.\
$\TS{\tp_u}{\cD}{\geq0}(X_0,\lambda)$).
\begin{lem}\label{2le:perverse_t_indep}
The pair of subcategories
$(\TS{\tp}{\cD}{\leq0}_u(X,\lambda),\TS{\tp}{\cD}{\geq0}_u(X,\lambda))$ do
not depend on the choice of $u$.
\end{lem}
In what follows, we will write
$(\TS{\tp}{\cD}{\leq0}(X,\lambda),\TS{\tp}{\cD}{\geq0}(X,\lambda))$ for
$(\TS{\tp}{\cD}{\leq0}_u(X,\lambda),\TS{\tp}{\cD}{\geq0}_u(X,\lambda))$.
\begin{proposition}\label{2pr:perverse_t}
Let $X$ be an $\sf{L}$-coprime higher Artin (resp.\ a higher
Deligne--Mumford) stack equipped with a perversity smooth (resp.\ \'{e}tale)
evaluation $\tp$, and let $\lambda$ be an object of $\Rind_{\ltor}$ (resp.\
$\Rind$). Then
\begin{enumerate}
\item The pair of subcategories
$(\TS{\tp}{\cD}{\leq0}(X,\lambda),\TS{\tp}{\cD}{\geq0}(X,\lambda))$
determine a right complete accessible $t$-structure on
$\cD(X,\lambda)$, which is weakly left complete if $\tp$ takes
values in $\Z$. This $t$-structure is left complete if $\tp$ is
locally bounded and if for every smooth (resp.\ \'etale) morphism
$X_0\to X$ with $X_0$ a quasi-compact separated scheme, $X_0$ is
$\lambda$-cohomologically finite.
\item If $f\colon Y\to X$ is a smooth (resp.\ \'etale) morphism, then
$f^*\colon \cD(X,\lambda)\to\cD(Y,\lambda)$ is $t$-exact with
respect to the $t$-structures associated to $\tp$ and $f^*\tp$.
\end{enumerate}
\end{proposition}
\begin{proof}[Proof of Lemma \ref{2le:perverse_t_indep} and Proposition \ref{2pr:perverse_t}]
There exists $k\ge 2$ such that $X$ and $Y$ are in $\Chpar{k}$ (resp.\
$\Chpdm{k}$). We proceed by induction on $k$. The case $k=-2$ follows from
Gabber's theorem and Lemma \ref{2le:perverse_smooth_pullback}. The induction
step follows from the proofs of \cite{LZ}*{4.3.7, 4.3.8}.
\end{proof}
\begin{remark}
We call the $t$-structure in Proposition \ref{2pr:perverse_t} the
\emph{perverse $t$-structure} with respect to $\tp$ and denote by
$\TS{\tp}{\tau}{\leq0}$ and $\TS{\tp}{\tau}{\geq0}$ the corresponding
truncation functors respectively. The for every (\'{e}tale) atlas $u\colon
X_0\to X$ with $X_0$ a scheme in $\Schqcs$,
$u^*\circ\TS{\tp}{\tau}{\leq0}\simeq \TS{\tp_u}{\tau}{\leq0}\circ u$ and
$u^*\circ\TS{\tp}{\tau}{\geq0}\simeq \TS{\tp_u}{\tau}{\geq0}\circ u$.
If $\tp=0$, then we recover the usual $t$-structure. If $X$ is a higher
Deligne-Mumford stack and $\tp$ is a perversity smooth evaluation, then the
$t$-structure associated to $\tp$ coincides with the $t$-structure associated
to $\tp\res \Voet(X)$. If $X$ is in $\Schqcs$, then the $t$-structure
associated to $\tp$ coincides with the $t$-structure defined by Gabber
associated to the function $\tp_{\id_X}$.
\end{remark}
By definition, the perverse $t$-structure can be described as follows.
\begin{proposition}\label{2pr:perverse_stalk}
Let $X$ be an $\sf{L}$-coprime higher Artin stack (resp.\ a higher
Deligne--Mumford stack) equipped with a perversity smooth (resp.\ \'etale)
evaluation $\tp$, and let $\lambda$ be an object of $\Rind_{\ltor}$ (resp.\
$\Rind$). Let $\sK$ be a complex in $\cD(X,\lambda)$.
\begin{enumerate}
\item $\sK$ belongs to $\TS{\tp}{\cD}{\leq n}(X,\lambda)$ if and only
if for every pointed smooth (resp.\ \'etale) schematic neighborhood
$(X_0,u_0,x_0)$ of $X$,
$i_{\overline{x_0}}^*u_0^*\sK\in\cD^{\leq\tp(X_0,u_0,x_0)+n}(\overline{x_0},\lambda)$.
\item $\sK$ belongs to $\TS{\tp}{\cD}{\geq n}(X,\lambda)$ if and only
if $\sK\in\cD^{(+)}(X,\lambda)$ and for every pointed smooth
(resp.\ \'etale) schematic neighborhood $(X_0,u_0,x_0)$ of $X$,
$i_{\overline{x_0}}^!u_0^*\sK\in\cD^{\geq\tp(X_0,u_0,x_0)+n}(\overline{x_0},\lambda)$.
\end{enumerate}
\end{proposition}
At the end of the section, we study the restriction of perverse
$t$-structures constructed above to various subcategory of constructible
complexes. We fix an $\sf{L}$-coprime base scheme $\sS$ that is a disjoint
union of excellent schemes, endowed with a global dimension function.
\begin{proposition}
Let $\lambda=(\Xi,\Lambda)$ be an object of
$\Rind_{\sf{L}\text{-}\r{dual}}$. Let $f\colon X\to\sf{S}$ be an object of
$\Chpars$ equipped with an \emph{admissible} perversity smooth evaluation
$\tp$. Then the truncation functors $\TS{\tp}{\tau}{\leq0}$,
$\TS{\tp}{\tau}{\geq0}$ preserve $\cD_{\cons}^{(\r b)}(X,\lambda)$.
Moreover, if $\tp$ is a locally bounded, then $\TS{\tp}{\tau}{\leq0}$,
$\TS{\tp}{\tau}{\geq0}$ preserve $\cD_{\cons}^{?}(X,\lambda)$ for
$?=(+),(-)$ or empty.
\end{proposition}
\begin{proof}
We reduce easily to the case of a scheme. In this case, the result is
essentially \cite{Gabber}*{8.2}.
\end{proof}
\subsection{Adic perverse $t$-structures}
\label{2ss:adic_perverse_t}
For the adic formalism, we define
\[\TS{\tp}{\underline{\cD}}{\leq
n}(X,\lambda)=\TS{\tp}{\cD}{\leq n}(X,\lambda)\cap\underline{\cD}(X,\lambda),\quad
\TS{\tp}{\underline{\cD}}{\geq n}(X,\lambda)=\TS{\tp}{\underline{\cD}}{\leq
n-1}(X,\lambda)^{\perp}\subseteq\underline{\cD}(X,\lambda).\] Then the pair $(\TS{\tp}{\underline{\cD}}{\leq
0}(X,\lambda),\TS{\tp}{\underline{\cD}}{\geq 0}(X,\lambda))$ defines a
$t$-structure, called the \emph{adic perverse $t$-structure} with respect to
$\tp$, on $\underline{\cD}(X,\lambda)$. Denote
$\TS{\tp}{\underline{\tau}}{\leq0}$ and $\TS{\tp}{\underline{\tau}}{\geq0}$
the corresponding truncation functors respectively. We first have the
following.
\begin{lem}\label{2le:adic_perverse_t_indep}
Let $X$ be an $\sf{L}$-coprime higher Artin stack (resp.\ a higher
Deligne--Mumford stack) equipped with a perversity smooth (resp.\ \'etale)
evaluation $\tp$, and $\lambda$ be an object of $\Rind_{\ltor}$ (resp.\
$\Rind$). Let $\sK$ be a complex in $\underline{\cD}(X,\lambda)$. Let
$u\colon X_0\to X$ be an atlas (resp.\ \'{e}tale atlas) with $X_0$ a scheme
in $\Schqcs$. Then $\sK$ belongs to $\TS{\tp}{\underline{\cD}}{\leq
n}(X,\lambda)$ (resp.\ $\TS{\tp}{\underline{\cD}}{\geq n}(X,\lambda)$) if and
only if $\underline{u}^*\sK$ belongs to $\TS{\tp_u}{\underline{\cD}}{\leq
n}(X_0,\lambda)$ (resp.\ $\TS{\tp_u}{\underline{\cD}}{\geq n}(X_0,\lambda)$).
\end{lem}
\begin{proof}
We only need to show that $\underline{u}^*$ is $t$-exact. By definition, we
obviously have $\underline{u}^*\TS{\tp}{\underline{\cD}}{\leq
n}(X,\lambda)\subseteq\TS{\tp_u}{\underline{\cD}}{\leq n}(X_0,\lambda)$. For
the other direction, assume $\sK\in\TS{\tp}{\underline{\cD}}{>n}(X,\lambda)$,
that is, $\Hom(\sL,\sK)=0$ for all
$\sL\in\underline{\cD}(X,\lambda)\cap\TS{\tp}{\cD}{\leq n}(X,\lambda)$. By
Poincar\'{e} duality, we only need to show that for
$\sL'\in\underline{\cD}(X_0,\lambda)\cap\TS{\tp_u}{\cD}{\leq n-2\dim
u}(X_0,\lambda)$, $\Hom(\sL',u^!\sK)=0$, or equivalently,
$\Hom(u_!\sL',\sK)=0$. This follows from the fact that $u_!$ preserves adic
objects and $u_!\sL'\in\TS{\tp}{\cD}{\leq n}(X,\lambda)$.
\end{proof}
We have the following description in terms of the cohomology on stalks, that
is similar to the Proposition \ref{2pr:perverse_stalk}.
\begin{proposition}\label{2pr:adic_perverse_stalk}
Let $X$ be an $\sf{L}$-coprime higher Artin stack (resp.\ a higher
Deligne--Mumford stack) equipped with a perversity smooth (resp.\ \'etale)
evaluation $\tp$, and $\lambda$ be an object of $\Rind_{\ltor}$ (resp.\
$\Rind$). Let $\sK$ be a complex in $\underline{\cD}(X,\lambda)$.
\begin{enumerate}
\item $\sK$ belongs to $\TS{\tp}{\underline{\cD}}{\leq n}(X,\lambda)$
if and only if for every pointed smooth (resp.\ \'etale) schematic
neighborhood $(X_0,u_0,x_0)$ of $X$,
$\underline{i_{\overline{x_0}}}^*\underline{u_0}^*\sK\in\underline{\cD}^{\leq\tp(X_0,u_0,x_0)+n}(\overline{x_0},\lambda)$.
\item Assume $\tp$ is \emph{locally bounded}. Then $\sK$ belongs to
$\TS{\tp}{\underline{\cD}}{\geq n}(X,\lambda)$ if and only if
$\sK\in\underline{\cD}^{(+)}(X,\lambda)$ and for every pointed
smooth (resp.\ \'etale) schematic neighborhood $(X_0,u_0,x_0)$ of
$X$,
$\underline{i_{\overline{x_0}}}^!\underline{u_0}^*\sK\in\underline{\cD}^{\geq\tp(X_0,u_0,x_0)+n}(\overline{x_0},\lambda)$.
\end{enumerate}
\end{proposition}
\begin{proof}
For (1), by definition, $\sK$ belongs to $\TS{\tp}{\underline{\cD}}{\leq
n}(X,\lambda)$ if and only if $\sK\in\TS{\tp}{\cD}{\leq n}(X,\lambda)$,
viewed as object of $\cD(X,\lambda)$. By Proposition \ref{2pr:perverse_stalk}
(1), it is equivalent to say that for $(Y,u,y)$ of $X$,
$i_{\overline{y}}^*u^*\sK\in\cD^{\leq\tp(Y,u,y)+n}(\overline{y},\lambda)$. We
only need to notice that $i_{\overline{y}}^*u^*\sK$ is adic.
For (2), by Lemma \ref{2le:adic_perverse_t_indep}, we may assume
$X\in\Schqcs$ is quasi-compact and $\tp=p$ is a bounded weak perversity
function. Then $\sK\in\TS{p}{\underline{\cD}}{\geq n}(X,\lambda)$ is
equivalent to that for every $\sL\in\TS{p}{\underline{\cD}}{<n}(X,\lambda)$,
$\HOM(\sL,\sK)\in\cD^{>0}(X,\lambda)$, which is then equivalent to that
$\HOM(\sL,\sK)\in\cD^+(X,\lambda)$ and $i_{\overline x}^!\HOM(\sL,\sK)\simeq
\HOM(i_{\overline x}^*\sL,i_{\overline x}^!\s K)\simeq\HOM(i_{\overline
x}^*\sL,\underline{i_{\overline x}}^!\s K)\in\cD^{>0}(\overline x,\lambda)$
for every geometric point $\overline x$ of $X$. Assume $\alpha<p<\beta$. Then
$\TS{p}{\underline{\cD}}{<n}(X,\lambda)\supseteq\underline{\cD}^{<\alpha+n}(X,\lambda)$.
Therefore, $\sK\in\TS{p}{\underline{\cD}}{\geq n}(X,\lambda)$ implies
$\sK\in\underline{\cD}^{\geq\alpha+n}(X,\lambda)$ and
$\underline{i_{\overline x}}^!\s K\in\underline{\cD}^{\geq p(x)+
n}(\overline{x},\lambda)$ for every geometric point $\overline x$ of $X$.
Conversely, assume $\sK\in\underline{\cD}^+(X,\lambda)$, say in
$\underline{\cD}^{\geq \gamma}(X,\lambda)$, and $\underline{i_{\overline
x}}^!\s K\in \underline{\cD}^{\geq p(x)+ n}(\overline{x},\lambda)$ for every
geometric point $\overline x$ of $X$. We only need to show that
$\HOM(\sL,\sK)\in\cD^+(X,\lambda)$. In fact, we have
$\HOM(\sL,\sK)\in\cD^{\geq \gamma-\beta-n}(X,\lambda)$.
\end{proof}
\begin{remark}\label{2re:perverse_truncation}
Let $\tp,\tq$ be two perversity smooth (resp.\ \'etale) evaluations on an
$\sf{L}$-coprime higher Artin stack (resp.\ a higher Deligne--Mumford stack)
$X$. Let $\lambda$ be an object of $\Rind_{\ltor}$ (resp.\ $\Rind$).
\begin{enumerate}
\item Since $\TS{\tp}{\tau}{\geq0}$ (resp.\
$\TS{\tp}{\underline{\tau}}{\geq0}$) preserves
$\cD^{(+)}(X,\lambda)$ (resp.\ $\underline{\cD}^{(+)}(X,\lambda)$),
so does $\TS{\tp}{\tau}{\leq0}$ (resp.\
$\TS{\tp}{\underline{\tau}}{\leq0}$). Therefore, the intersection
of
$(\TS{\tp}{\cD}{\leq0}(X,\lambda),\TS{\tp}{\cD}{\geq0}(X,\lambda))$
(resp.\
$(\TS{\tp}{\underline{\cD}}{\leq0}(X,\lambda),\TS{\tp}{\underline{\cD}}{\geq0}(X,\lambda))$)
with $\cD^{(+)}(X,\lambda)$ (resp.\
$\underline{\cD}^{(+)}(X,\lambda)$)induces a $t$-structure on the
latter $\infty$-category.
\item If $\tp\leq\tq$, then
\begin{enumerate}
\item $\TS{\tp}{\tau}{\leq0}$ (resp.\
$\TS{\tp}{\underline{\tau}}{\leq0}$) preserves
$\TS{\tq}{\cD}{\leq0}(X,\lambda)$ (resp.\
$\TS{\tq}{\underline{\cD}}{\leq0}(X,\lambda)$) since
$\TS{\tp}{\cD}{\leq0}(X,\lambda)\subseteq\TS{\tq}{\cD}{\leq0}(X,\lambda)$
(resp.\
$\TS{\tp}{\underline{\cD}}{\leq0}(X,\lambda)\subseteq\TS{\tq}{\underline{\cD}}{\leq0}(X,\lambda)$);
\item $\TS{\tq}{\tau}{\geq0}$ (resp.\
$\TS{\tq}{\underline{\tau}}{\geq0}$) preserves
$\TS{\tp}{\cD}{\geq0}(X,\lambda)$ (resp.\
$\TS{\tp}{\underline{\cD}}{\geq0}(X,\lambda)$) since
$\TS{\tq}{\cD}{\geq0}(X,\lambda)\subseteq\TS{\tp}{\cD}{\geq0}(X,\lambda)$
(resp.\
$\TS{\tq}{\underline{\cD}}{\geq0}(X,\lambda)\subseteq\TS{\tp}{\underline{\cD}}{\geq0}(X,\lambda)$);
\item $\TS{\tp}{\tau}{\geq0}$ (resp.\
$\TS{\tp}{\underline{\tau}}{\geq0}$) is equivalent to
the identity function when restricted to
$\TS{\tq}{\cD}{\geq0}(X,\lambda)$ (resp.\
$\TS{\tq}{\underline{\cD}}{\geq0}(X,\lambda)$) since
$\TS{\tq}{\cD}{\geq0}(X,\lambda)\subseteq\TS{\tp}{\cD}{\geq0}(X,\lambda)$
(resp.\
$\TS{\tq}{\underline{\cD}}{\geq0}(X,\lambda)\subseteq\TS{\tp}{\underline{\cD}}{\geq0}(X,\lambda)$);
\item $\TS{\tq}{\tau}{\leq0}$ (resp.\
$\TS{\tq}{\underline{\tau}}{\leq0}$) is equivalent to
the identity function when restricted to
$\TS{\tp}{\cD}{\leq0}(X,\lambda)$ (resp.\
$\TS{\tp}{\underline{\cD}}{\leq0}(X,\lambda)$) since
$\TS{\tp}{\cD}{\leq0}(X,\lambda)\subseteq\TS{\tq}{\cD}{\leq0}(X,\lambda)$
(resp.\
$\TS{\tp}{\underline{\cD}}{\leq0}(X,\lambda)\subseteq\TS{\tq}{\underline{\cD}}{\leq0}(X,\lambda)$);
\item $\TS{\tp}{\tau}{<0}$ (resp.\
$\TS{\tp}{\underline{\tau}}{<0}$) is equivalent to the
null function when restricted to
$\TS{\tq}{\cD}{\geq0}(X,\lambda)$ (resp.\
$\TS{\tq}{\underline{\cD}}{\geq0}(X,\lambda)$);
\item $\TS{\tq}{\tau}{>0}$ (resp.\
$\TS{\tq}{\underline{\tau}}{>0}$) is equivalent to the
null function when restricted to
$\TS{\tp}{\cD}{\leq0}(X,\lambda)$ (resp.\
$\TS{\tp}{\underline{\cD}}{\leq0}(X,\lambda)$);
\end{enumerate}
\item By 2 (a), if $\tp$ is locally bounded, then the
intersection of
$(\TS{\tp}{\cD}{\leq0}(X,\lambda),\TS{\tp}{\cD}{\geq0}(X,\lambda))$
(resp.\
$(\TS{\tp}{\underline{\cD}}{\leq0}(X,\lambda),\TS{\tp}{\underline{\cD}}{\geq0}(X,\lambda))$)
with $\cD^{(-)}(X,\lambda)$ or $\cD^{(\r b)}(X,\lambda)$
(resp.\ $\underline{\cD}^{(-)}(X,\lambda)$ or
$\underline{\cD}^{(\r b)}(X,\lambda)$)induces a
$t$-structure on the latter $\infty$-category.
\item By 2 (e) and (f), if $X$ is quasi-compact and $\tp$ is bounded,
then there exist constant integers $\alpha<\beta$ such that
$\TS{\tp}{\r H}{0}=\TS{\tp}{\r H}{0}\circ\tau^{[\alpha,\beta]}$
(resp.\ $\TS{\tp}{\underline{\r H}}{0}=\TS{\tp}{\underline{\r
H}}{0}\circ\underline{\tau}^{[\alpha,\beta]}$).
\end{enumerate}
\end{remark}
\subsection{Constructible adic perverse $t$-structures}
\label{3ss:constructible_adic_perverse_t}
We fix an $\sf{L}$-coprime base scheme $\sS$ that is a disjoint union of
schemes that are excellent, quasi-compact, finite dimensional, and admit a
global dimension function for which we fix one. We fix also an object
$(\Lambda,\fm)$ of $\PRing_{\ltor}$ such that $\Lambda/\fm^{n+1}$ is an
($\sf{L}$-torsion) Gorenstein ring of dimension 0 for every $n\in\NN$ and
$\sS$ is locally $(\Lambda/\fm)$-bounded.
\begin{proposition}
For an object $f\colon X\to\sf{S}$ of $\Chpars$ equipped with an
\emph{admissible} perversity evaluation $\tp$, the truncation functors
$\TS{\tp}{\underline{\tau}}{\leq0}$, $\TS{\tp}{\underline{\tau}}{\geq0}$
preserve $\underline{\cD}_{\cons}^{?}(X,\Lambda_{\bullet})$ for
$?=(+),(-),(\r b)$ or empty.
\end{proposition}
\begin{proof}
By Lemma \ref{2le:adic_perverse_t_indep}, we may assume that $X$ is a
quasi-compact, separated (and excellent, finite dimensional) scheme that is
$(\Lambda/\fm)$-bounded, and $\tp=p$ is an admissible perversity function on
$X$. In particular, $p$ is bounded. We prove by Noetherian induction. We may
further assume $X$ is irreducible. For a complex
$\sK\in\underline{\cD}_{\cons}(X,\Lambda_{\bullet})$, we may assume
$\sK\in\underline{\cD}_{\cons}^{\r b}(X,\Lambda_{\bullet})\subseteq\cD^{\r
b}(X,\Lambda_{\bullet})$ by Remark \ref{2re:perverse_truncation} (3). Choose
a dense open subset $U$ of $X$ such that
\begin{itemize}
\item $U$ is essentially smooth;
\item $p(x)\leq p(\eta)+\codim_X(x)$ for $x\in|U|$, where $\eta$ is the
unique generic point of $X$;
\item $p(x)\geq p(\eta)$ for $x\in |U|$;
\item The complex $\sK_U:=\sK\res U$, viewed as an element of $\cD^{\r
b}(U,\Lambda_{\bullet})$, has smooth almost adic cohomology
sheaves.
\end{itemize}
Then the perverse truncation for $\sK_U$ is simply the usual truncation (up
to a shift by $p(\eta)$), which preserves constructibility by Corollary
\ref{3co:constructible_adic_t}.
\end{proof}
Our definition of the constructible adic perverse $t$-structure coincides
with Laszlo--Olsson \cite{LO3} under their restrictions, where in particular
$X$ is a locally Noetherian ($1$-)Artin stack over a field $k$ (that is, $\sf
S=\Spec k$) with $\r{cd}_{\ell}(k)<\infty$, and $\tp$ is the middle
perversity smooth evaluation, that is, the unique perverse smooth evaluation
such that for every atlas $u\colon X_0\to X$ with $X_0$ a scheme in
$\Schqcs$, $\tp_u=(f\circ u)_{\b 1}^*p_0$, where $f\colon X\to \sf S$ is the
structure morphism and $p_0$ is the zero perverse function on $\sf S=\Spec
k$.
\section{Descent properties}
\label{4}
In \Sec \ref{4ss:hyperdescent}, we prove that our construction of derived
$\infty$-categories, as well as those of adic complexes, of (higher) Artin
stacks satisfies not only the smooth descent, but also the smooth
\emph{hyperdescent}.
\subsection{Hyperdescent}
\label{4ss:hyperdescent}
The \'etale $\infty$-topos of an affine scheme is not hypercomplete in
general. By contrast, the stable $\infty$-categories we constructed satisfy
smooth hyperdescent. We begin with a general definition.
\begin{definition}[$F$-descent]
Let $\cC$, $\cD$ be $\infty$-categories, let $F\colon \cC^{op}\to \cD$ be a
functor, and let $X_\bullet^+\colon \N(\del_+)^{op}\to \cC$ be an augmented
simplicial object of $\cC$.
\begin{enumerate}
\item We say that $X_\bullet^+$ is an \emph{augmentation of $F$-descent}
if $F\circ (X_\bullet^+)^{op}$ is a limit diagram in $\cD$.
\item Assume $\cC$ admits pullbacks. We say that $X_\bullet^+$ is a
\emph{hypercovering for universal $F$-descent} if $X^+_q \to
(\cosk_{q-1} (X^+_{\bullet}/X^+_{-1}))_q$ is a morphism of universal
$F$-descent for all $q\ge 0$.
\end{enumerate}
\end{definition}
By definition, a morphism of $\cC$ is of $F$-descent \cite{LZ}*{3.1.1} if
and only if its \v Cech nerve is an augmentation of $F$-descent. We now give
several criteria for $(2)\Rightarrow(1)$.
\begin{proposition}\label{4pr:n_category}
Let $\cC$ be an $\infty$-category admitting pullbacks, let $\cD$ be an
$n$-category admitting finite limits for an integer $n\ge 0$, and let
$F\colon \cC^{op}\to \cD$ be a functor. Then every hypercovering
$X_\bullet^+$ for universal $F$-descent is an augmentation of $F$-descent.
\end{proposition}
To prove Proposition \ref{4pr:n_category}, we need a few lemmas.
\begin{lem}\label{4le:induction}
Let $\cC$, $\cD$ be $\infty$-categories such that $\cC$ admits finite
limits, let $F\colon \cC^{op}\to \cD$ be a functor, let $e$ be a final
object of $\cC$, and let $f_\bullet\colon U_\bullet\to V_\bullet$ be a
morphism of simplicial objects of $\cC$ such that $V_\bullet \to e$ is an
augmentation of $F$-descent and $f_q$ is a morphism of $F$-descent for all
$q$. Assume that there exists an integer $n\ge 0$ such that $U_\bullet$ is
$n$-coskeletal, $V_\bullet$ is $(n-1)$-coskeletal, and $f_q$ is an
equivalence for $q<n$. Then $U_\bullet\to e$ is an augmentation of
$F$-descent.
\end{lem}
\begin{proof}
We may assume $F(e)$ is an initial object of $\cD$. Let $W_+\colon
\N(\del_{+}\times \del)^{op}\to $ be a \v{C}ech nerve of $f_\bullet$,
$W=W_+\res \N(\del\times \del)^{op}$. For every $q\ge 0$, $W_+\res
\N(\del_{+}\times \{[q]\})^{op}$ is a \v{C}ech nerve of $f_q$, which is a
morphism of $F$-descent by assumption. It follows that $F\circ W_+^{op}\res
\N(\del_{+}\times \{[q]\})$ is a limit diagram. We may thus identify the
limit of $F\circ W^{op}$ with the limit $F\circ W_+^{op}\res
\N(\{[-1]\}\times \del_{s})$. Since $W_+\res \N(\{[-1]\}\times \del)^{op}$
can be identified with $V_\bullet$, the limit of $F\circ W^{op}$ can be
identified with $F(e)$. Let $D_\bullet=W\circ \delta$, where $\delta\colon
\N(\del)^{op}\to \N(\del\times \del)^{op}$ is the diagonal map. Since
$\N(\del)^{op}$ is sifted \cite{Lu1}*{5.5.8.4}, the limit of $F\circ
D_\bullet^{op}$ can be identified with $F(e)$. The proof of
\cite{Lu1}*{6.5.3.9} exhibits $U_\bullet\res \N(\del_s)^{op}$ as a retract of
$D_\bullet\res \N(\del_s)^{op}$. It follows that the limit of $F\circ
U_\bullet^{op}$ is a retract of $F(e)$, and hence is $F(e)$.
\end{proof}
\begin{lem}\label{4le:coskeletal}
Let $\cC$, $\cD$ be $\infty$-categories such that $\cC$ admits pullbacks,
let $F\colon \cC^{op}\to \cD$ be a functor, and let $X_\bullet^+$ be an
$n$-coskeletal hypercovering for universal $F$-descent for an integer $n\ge
-1$. Then $X_\bullet^+$ is an augmentation of $F$-descent.
\end{lem}
\begin{proof}
Since morphisms of universal $F$-descent are stable under pullback and
composition, $\cosk_m (X_\bullet^+/X_{-1}^+)\to
\cosk_{m-1}(X_\bullet^+/X_{-1}^+)$ satisfies the assumptions of Lemma
\ref{4le:induction}. It follows by induction that $\cosk_n
(X^+_\bullet/X^+_{-1})$ is an augmentation of $F$-descent.
\end{proof}
\begin{lem}\label{4le:n_category}
Let $n\ge -1$ be an integer, let $\cD$ be an $n$-category admitting finite
colimits, and let $f_\bullet \colon Y_\bullet \to X_\bullet$ be a morphism
of semisimplicial (resp.\ simplicial) objects of $\cD$ such that $Y_q\to
X_q$ is an equivalence for $q\le n$. Then the induced morphism between
geometric realizations $\lvert f_\bullet \rvert \colon\lvert Y_\bullet
\rvert \to \lvert X_\bullet \rvert$ is an equivalence in $\cD$.
\end{lem}
The existence of the geometric realizations is guaranteed by
\cite{Lu2}*{1.3.3.10}.
\begin{proof}
The semisimplicial case follows from the simplicial case by taking left Kan
extensions. The simplicial case follows from the proof of
\cite{Lu2}*{1.3.3.10}.
\end{proof}
\begin{proof}[Proof of Proposition \ref{4pr:n_category}]
It suffices to apply the dual version of Lemma \ref{4le:n_category} to
$h\colon X^+_\bullet \to \cosk_n (X^+_\bullet/X_{-1}^+)$ and Lemma
\ref{4le:coskeletal}.
\end{proof}
The following can be used to deduce Gabber's hyper base change theorem
\cite{Org}*{Th\'eor\`eme 2.2.5} (see \cite{Zhhyper}*{Remark 2.3}).
\begin{proposition}\label{4pr:t_descent}
Let $\cC$ be an $\infty$-category admitting pullbacks, let $\cD$ be a stable
$\infty$-category endowed with a weakly right complete $t$-structure that
either admits countable limits or is right complete, let $F\colon \cC^{op}\to
\cD$ be a functor, and let $X_\bullet^+\colon \N(\del_+)^{op}\to \cC$ be a
hypercovering for universal $F$-descent such that $F\circ (X_\bullet^+)^{op}$
factorizes through $\cD^{\ge 0}$. Then $X_\bullet^+$ is an augmentation of
$F$-descent.
\end{proposition}
\begin{proof}
Let $n\ge 0$. By Lemma \ref{4le:coskeletal},
$Y_\bullet^+=\cosk_n(X_\bullet^+/X_{-1}^+)$ is an augmentation of
$F$-descent, so that it suffices to show that the morphism $h_\bullet\colon
X_\bullet^+\to Y_\bullet^+$ induces an isomorphism $c\colon K=\lim_{p\in
\del} F(X_p) \to \lim_{p \in \del} F(Y_p)=L$ is an isomorphism. By
\cite{Lu2}*{1.2.4.4, 1.2.4.5}, we have a morphism of converging spectral
sequences
\[\xymatrix{\r E_1^{pq}=\rH^q F(X_p)\ar@{=>}[r]\ar[d]_-{c_1^{pq}} & \rH^{p+q} K\ar[d]^-{\rH^{p+q} c}\\
\r E'{}_1^{pq}=\rH^q F(Y_p)\ar@{=>}[r] & \rH^{p+q} L,}
\]
concentrated in the first quadrant. For $p\le n$, since $h_p$ is an
equivalence, $c_1^{pq}$ is an isomorphism for all $q$. It follows that
$c_r^{pq}$ is an isomorphism for $p+q\le n-1$, and $\tau^{\le n-1}c$ is an
equivalence. Since $n$ is arbitrary and $\cD$ is weakly right complete, $c$
is an equivalence.
\end{proof}
We denote by $\PSLt$ (resp.\ $\PSRt$) the $\infty$-category defined as
follows.
\begin{itemize}
\item Objects of $\PSLt$ (resp.\ $\PSRt$) are presentable stable
$\infty$-categories $\cC$ equipped with a $t$-structure.
\item Morphisms of $\PSLt$ (resp.\ $\PSRt$) are $t$-exact functors
admitting right (resp.\ left) adjoints.
\end{itemize}
The $\infty$-categories $\PSLt$ (resp.\ $\PSRt$) admit small limits, and
those limits are preserved by the forgetful functor $\PSLt\to \PSL$ (resp.\
$\PSRt\to \PSR$). For a diagram $K\to \PSLt$ or $K\to \PSRt$, $(\lim
\cC_k)^{\le 0}$ (resp.\ $(\lim \cC_k)^{\ge 0}$) is the full subcategory of
$\lim \cC_k$ spanned by objects whose image in $\cC_k$ is in $\cC_k^{\le 0}$
(resp.\ $\cC_k^{\ge 0}$). For an interval $I\subseteq \Z$, we have an
equivalence $(\lim \cC_k)^{\in I}\to \lim \cC_k^{\in I}$.
We denote by $\PSLt[,\wrc]$ (resp.\ $\PSRt[,\rc,\wlc]$) the full subcategory
of $\PSLt$ (resp.\ $\PSRt$) spanned by those $\cC$ that are weakly right
complete (resp.\ right complete and weakly left complete). These full
subcategories are stable under small limits in $\PSRt$.
\begin{proposition}\label{4pr:hyperdescent}
Consider a diagram
\[\xymatrix{(\cD')^{op}\ar[d]_-{j^{op}}\ar[r]^-F & \PSRt[,\rc,\wlc]\ar[d]^-{P}\\
\cD^{op}\ar[r]^-{G} & \Cat}
\]
of $\infty$-categories, where $\cD$ admits pullbacks, $j$ is an inclusion
satisfying the right lifting property with respect to $\partial
\Delta^n\subseteq \Delta^n$ for $n\ge 2$, $P$ is the forgetful functor.
Assume that the arrows in $\cD'$ are stable under pullbacks in $\cD$ by
arrows in $\cD'$. Let $X_\bullet^+\colon \N(\del_+)^{op}\to \cD$ be a
hypercovering for universal $G$-descent such that $X_\bullet^+\res
\N(\del_{s+})^{op}$ factorizes through $j$. Then $X_\bullet^+$ is an
augmentation of $G$-descent.
\end{proposition}
\begin{proof}
By the right completeness of $F(X^+_p)$ for $p\ge -1$, it suffices to show
$(F\circ (X_\bullet^+)^{op}\res \N(\del_{s+}))^{\le 0}$ is a limit diagram.
Let $f_!$ be a left adjoint of the $t$-exact functor $f^*\colon
F(X^+_{-1})\to \lim F\circ (X_\bullet^+)^{op}\res \N(\del_s)=\cC$. The
restrictions of these provide adjoint functors $(f_!)^{\le 0}$ and
$(f^*)^{\le 0}\colon F(X^+_{-1})^{\le 0}\to \cC^{\le 0}$. Let us first show
that $a\colon f_! f^*K \to K$ is an equivalence for all $K\in
F(X^+_{-1})^{\le 0}$, namely that $(f^*)^{\le 0}$ is fully faithful. This is
similar to Proposition \ref{4pr:t_descent}. Let $n\ge 0$. The morphism
$h_\bullet\colon X^+_\bullet \to \cosk_n (X^+_\bullet/X^+_{-1})=Y^+_\bullet$
induces a diagram
\[\xymatrix{f_!f^* K \ar[rr]^-{c}\ar[rd]_-{a} && g_!g^* K\ar[dl]^-{b}\\
&K}
\]
where $g_!$ is a left adjoint of the $t$-exact functor $g^*\colon
F(X^+_{-1})\to \lim F\circ (Y^+_\bullet)^{op}\res \N(\del_s)$. By Lemma
\ref{4le:coskeletal}, $Y^+_\bullet$ is an augmentation of $G$-descent, so
that $b$ is an equivalence. Moreover, $c=\colim (f_{p!} f_p^* K\to
g_{p!}g_p^*K)$, where $f_{p!}$ is a left adjoint of $f_p^*\colon
F(X^+_{-1})\to F(X^+_{p})$, $g_{p!}$ is a left adjoint of $g_p^*\colon
F(Y^+_{-1})\to F(Y^+_{p})$, $f_{p!}f_p^*\to g_{p!}g_p^*$ is induced by $h_p$.
By \cite{Lu2}*{1.2.4.6, 1.2.4.7}, we have a morphism of converging spectral
sequences
\[\xymatrix{
\r E_1^{pq}=\rH^q(f_{-p!}f_{-p}^* K)\ar@{=>}[r]\ar[d]_-{c_1^{pq}} & \rH^{p+q} f_! f^* K\ar[d]^-{\rH^{p+q} c}\\
\r E'{}_1^{pq}=\rH^q(g_{-p!}g_{-p}^* K)\ar@{=>}[r] & \rH^{p+q} g_! g^* K,}
\]
concentrated in the third quadrant. For $p\ge -n$, since $h_p$ is an
equivalence , $c_1^{pq}$ is an isomorphism for all $q$. It follows that
$c_r^{pq}$ is an isomorphism for $p+q\ge 1-n$, and $\tau^{\ge 1-n}c$ is an
equivalence. Therefore, $\tau^{\ge 1-n} a$ is an equivalence. Since $n$ is
arbitrary and $F(X^+_{-1})$ is weakly left complete, $a$ is an equivalence.
It remains to show that $d\colon L\to f^*f_! L$ is an equivalence for every
$L\in \cC^{\le 0}$. Since $\cC$ is weakly left complete, it suffices to show
that $\tau^{\ge 1-n} d$ is an equivalence for every $n \ge 1$. For this, we
may assume $L\in \cC^{[1-n,0]}$. We will show that $L$ is in the essential
image of $(f^*)^{\le 0}$. Since $(f^*)^{\le 0}$ is fully faithful, this
proves that $d$ is an equivalence. Let $H\colon \PSRt[,\rc,\wlc]\to
\c{C}\r{at}_n$ be the functor sending $\cF$ to $\cF^{[1-n, 0]}$. It suffices
to show that $H\circ F\circ (X^+_\bullet)^{op}\res \N(\del_{s+})$ is a limit
diagram. Since $\c{C}\r{at}_n$ is an $(n+1)$-category, we may assume that
$X^+_\bullet/X^+_{-1}$ is $(n+1)$-coskeletal by Lemma \ref{4le:n_category}
applied to $X^+_\bullet \to \cosk_{n+1} (X^+_\bullet/X^+_{-1})$. In this
case, $F\circ (X^+_\bullet)^{op}\res \N(\del_{s+})$ is a limit diagram by
Lemma \ref{4le:coskeletal}.
\end{proof}
The following variant of Proposition \ref{4pr:hyperdescent} will be used to
establish proper hyperdescent. To state it conveniently, we introduce a bit
of terminology. Let $\cC$ be an $\infty$-category admitting pullbacks,
$F\colon\cC^{op}\to \Cat$ be a functor. We say that a morphism $f$ of $\cC$
is \emph{$F$-conservative} if $F(f)$ is conservative. We say that $f$ is
\emph{universally $F$-conservative} if every pullback of $f$ in $\cC$ is
$F$-conservative. We say that an augmented simplicial object $X_\bullet^+$ of
$\cC$ is a \emph{hypercovering for universal $F$-conservativeness} if
$X^+_n\to (\cosk_{n-1}(X^+_\bullet/X^+_{-1}))_n$ is universally
$F$-conservative for all $n\ge 0$.
\begin{proposition}\label{4pr:hyperdescent2}
Let $\cC$ be an $\infty$-category admitting pullbacks, let $F\colon
\cC^{op}\to \PSLt[,\wrc]$ be a functor, let $a$ be an integer, let $G\colon
\PSLt[,\wrc]\to \Cat$ be the functor sending $\cC$ to $\cC^{\ge a}$ (resp.\
$\cC^+=\bigcup_n \cC^{\ge n}$), and let $X_\bullet^+$ be a hypercovering for
universal $(G\circ F)$-descent (resp.\ and universal $(P\circ
F)$-conservativeness, where $P\colon \PSLt[,\wrc]\to \Cat$ is the forgetful
functor). Then $X_\bullet^+$ is an augmentation of $(G\circ F)$-descent.
\end{proposition}
\begin{proof}
The proof of the case of $\cC^{\ge a}$ is similar to the proof of Proposition
\ref{4pr:hyperdescent}. In the case of $\cC^{+}$, the conservativeness
implies that $G(\lim F\circ (X^+_\bullet)^{op})\to \lim G\circ F\circ
(X_\bullet^+)^{op}$ is an equivalence. The rest of the proof is similar.
\end{proof}
Consider the functors \cite{LZ}*{\Sec 6.4}
\begin{align*}
\EO{}{\Chpar{}}{*}{\otimes}&\colon\N(\Chpar{})^{op}\to\Fun(\N(\Rind)^{op},\PSLM),\\
\EO{}{\Chparl}{}{!}&\colon\N(\Chparl)
\to\Fun(\N(\Rind_{\ltor})^{op},\PSL),
\end{align*}
and their adic version in \Sec \ref{1ss:adic_complexes}
\begin{align*}
\EOA{}{\Chpar{}}{*}{\otimes}&\colon\N(\Chpar{})^{op}\to\Fun(\N(\Rind)^{op},\PSLM),\\
\EOA{}{\Chparl}{}{!}&\colon\N(\Chparl)
\to\Fun(\N(\Rind_{\ltor})^{op},\PSL).
\end{align*}
We say that an augmented simplicial object $X_\bullet^+$ in $\Chpar{}$ (or
$\Chpdm{\infty}$ introduced below) is a \emph{(P) hypercovering} for a
property (P) on morphisms if $X^+_q \to (\cosk_{q-1}
(X^+_{\bullet}/X^+_{-1}))_q$ is \emph{surjective} and satisfies (P) for all
$q\ge 0$.
\begin{proposition}\label{4pr:hyperdescent_stack}
Every smooth hypercovering in $\Chpar{}$ (resp.\ $\Chparl$) is an
augmentation of $\EO{}{\Chpar{}}{*}{\otimes}$-descent (resp.\
$\EO{}{\Chparl}{op}{!}$-descent) and $\EOA{}{\Chpar{}}{*}{\otimes}$-descent
(resp.\ $\EOA{}{\Chparl}{op}{!}$-descent).
\end{proposition}
Together with \cite{Lu2}*{1.2.4.5} and its dual version, Proposition
\ref{4pr:hyperdescent_stack} implies Theorem \ref{0th:hyperdescent}.
\begin{proof}
Let $X_\bullet^+$ be an augmented simplicial object of $\Chpar{}$ (resp.\
$\Chparl$). It suffices to apply Proposition \ref{4pr:hyperdescent} to the
full subcategory $\Chpar{}_{\r{sm}/X_{-1}}\subseteq \Chpar{}_{/X_{-1}}$
spanned by higher Artin stacks smooth over $X_{-1}$. In the notation of
Proposition \ref{4pr:hyperdescent}, $F$ associates the usual $t$-structure
(resp.\ the usual $t$-structure shifted by twice the relative dimension over
$X_{-1}$). This proof applies to both the non-adic case and the adic case.
The adic case can also be deduced from the non-adic case by taking limits.
\end{proof}
\begin{definition}
The $\infty$-category of \emph{$\infty$-DM stacks} $\Chpdm{\infty}$ is the
$\infty$-category $\mathrm{Sch}(\cG_{\et}(\Z))$ of
$\mathcal{G}_{\et}(\Z)$-schemes in the sense of \cite{LuV}*{2.3.9, 2.6.11}.
\end{definition}
Using Proposition \ref{4pr:hyperdescent}, we can adapt the DESCENT program
\cite{LZ}*{\Sec 4} to define enhanced operation maps for $\infty$-DM stacks:
\begin{align*}
\EO{}{\Chpdm{\infty}}{}{}&\colon \delta^*_{2,\{2\}}\Fun(\Delta^1,\Chpdm{\infty})^{\cart}_{F^0,A^{\to}}
\to\Fun(\N(\Rind_{\tor})^{op},\Mon_{\Pf}^{\PSL}(\Cat)),\\
\EO{}{\Chpdm{\infty}}{*}{\otimes}&\colon
(\Chpdm{\infty})^{op}\to\Fun(\N(\Rind)^{op},\PSLM),
\end{align*}
compatible with the enhanced operation maps constructed for higher DM stacks
\cite{LZ}*{\Sec 6.5}. Applying Definition \ref{1de:enhanced_adic}, we obtain
similarly the enhanced adic operation maps for $\infty$-DM stacks:
\begin{align*}
\EOA{}{\Chpdm{\infty}}{}{}&\colon \delta^*_{2,\{2\}}\Fun(\Delta^1,\Chpdm{\infty})^{\cart}_{F^0,A^{\to}}
\to\Fun(\N(\Rind_{\tor})^{op},\Mon_{\Pf}^{\PSL}(\Cat)),\\
\EOA{}{\Chpdm{\infty}}{*}{\otimes}&\colon
(\Chpdm{\infty})^{op}\to\Fun(\N(\Rind)^{op},\PSLM),
\end{align*}
compatible with the enhanced adic operation maps constructed for higher DM
stacks in \Sec \ref{1ss:adic_complexes}. By restriction, we obtain functors:
\begin{align*}
\EO{}{\Chpdm{\infty}}{}{!},\EOA{}{\Chpdm{\infty}}{}{!}\colon (\Chpdm{\infty})_F
\to\Fun(\N(\Rind_{\tor})^{op},\PSL).
\end{align*}
We obtain the following result.
\begin{proposition}
Every smooth hypercovering in $\Chpdm{\infty}$ is an augmentation of
$\EO{}{\Chpdm{\infty}}{*}{\otimes}$-descent and (resp.\
$\EO{}{\Chpdm{\infty}}{op}{!}$-descent) and
$\EOA{}{\Chpdm{\infty}}{*}{\otimes}$-descent (resp.\
$\EOA{}{\Chpdm{\infty}}{op}{!}$-descent).
\end{proposition}
\subsection{Proper descent}
\label{4ss:proper_descent}
The following lemma is an immediate consequence of \cite{Lu2}*{Proposition
1.2.4.5}.
\begin{lem}\label{4le:real}
Let $\cC$, $\cD$ be stable $\infty$-categories equipped with left complete
$t$-structures. Let $F\colon \cC\to \cD$ be a conservative $t$-exact
functor. Then $\cC^{\le 0}$ admits geometric realizations, and geometric
realizations are preserved by $F$.
\end{lem}
\begin{lem}\label{4le:split}
Let $\cC$, $\cD$, $\cE$ be stable $\infty$-categories equipped with
$t$-structures such that $\cC$ and $\cD$ are left complete and right
complete. Let $F\colon \cC\to \cD$, $G\colon \cC\to \cE$ be conservative
$t$-exact functors. Then $\cC$ admits $G$-split geometric realizations, and
those geometric realizations are preserved by $F$.
\end{lem}
\begin{proof}
Let $X_\bullet$ be a $G$-split simplicial object of $\cC$, $Y_\bullet\colon
\N(\del_{-\infty})^{op}\to \cD$ be an extension of $G\circ X_\bullet$. Then
the unnormalized cochain complex
\[\dots \to \rH^q Y_2\to \rH^q Y_1 \to \rH^q Y_0\to \rH^q Y_{-1} \to 0\]
is acyclic. It follows that the unnormalized cochain complex
\[\dots \to \rH^q X_2\to \rH^q X_1 \xrightarrow{\theta^q} \rH^q X_0\]
is an acyclic resolution of the object $A^q=\coker(\theta^q)$ in the heart of
$\cC$. The lemma then follows from \cite{Lu2}*{Corollary 1.2.4.12}.
\end{proof}
Consider the functors
\begin{align*}
\EO{\ge 0}{\Chparl}{*}{}&\colon
\N(\Chparl)^{op}\to\Fun(\N(\Rind_{\ltor})^{op},\Cat),\\
\EO{\ge 0}{\Chpdm{\infty},\tor}{*}{}&\colon
\N(\Chpdm{\infty})^{op}\to\Fun(\N(\Rind_{\tor})^{op},\Cat)
\end{align*}
sending $X$ to $\lambda\mapsto \cD^{\ge 0}(X,\lambda)$. Note that here we
have put extra conditions on the categories of ringed diagrams.
\begin{proposition}\label{4pr:proper_descent}
Let $\sf{S}$ be an $\sf{L}$-coprime \emph{locally Noetherian} higher Artin
stack (that is, there exists an atlas $S\to \sf{S}$ where $S$ is a locally
Noetherian scheme).
\begin{enumerate}
\item For every object $\lambda$ of $\Rind_{\ltor}$ and every
Cartesian square
\[\xymatrix{
W \ar[r]^-{g} \ar[d]_-{q} & Z \ar[d]^-{p} \\
Y \ar[r]^-{f} & X }
\]
in $\Chparl$ (resp.\ $\Chpars$) with $p$ proper of finite diagonal (resp.\
proper, $1$-Artin, of separated diagonal), the induced square
\[
\xymatrix{
\cD^{\ge 0}(Z,\lambda) \ar[d]_-{g^*} & \cD^{\ge 0}(X,\lambda) \ar[l]_-{p^*} \ar[d]^-{f^*}\\
\cD^{\ge 0}(W,\lambda) & \cD^{\ge 0}(Y,\lambda) \ar[l]_-{q^*}
}
\]
is right adjointable.
\item Every proper finite-diagonal hypercovering in $\Chparl$ (resp.\
proper, $1$-Artin, separated-diagonal hypercovering in $\Chpars$) is
an augmentation of $\EO{\ge 0}{\Chparl}{*}{}$-descent.
\end{enumerate}
\end{proposition}
A similar result holds for $\EO{\ge 0}{\Chpdm{\infty},\tor}{*}{}$.
\begin{proof}
Let us first show that (1) implies (2). By Proposition
\ref{4pr:hyperdescent2}, to show (2), it suffices to show that every
surjective morphism proper of finite diagonal (resp.\ proper, $1$-Artin, of
separated diagonal) is of $\EO{\ge 0}{\Chparl}{*}{}$-descent. For this, we
apply \cite{LZ}*{3.3.6}. Assumption (1) of \cite{LZ}*{3.3.6} follows from the
dual of Lemma \ref{4le:real}. Assumption (2) of \cite{LZ}*{3.3.6} is (1).
To show (1), applying \cite{LZ}*{4.3.6} and smooth base change, we are
reduced to the case where $X$ and $Y$ are quasi-compact quasi-separated
schemes. In this case, there exists a finite cover \cite{Rydh}*{Theorem~B}
(resp.\ proper cover \cite{OlChow}*{1.1}) $r_0\colon X_0\to X$. Since (1) is
known in the case of algebraic spaces, $r_0$ is of descent, so that every
object of $\cD^{\ge 0}(X,\lambda)$ has the form $\lim_{n\in \del}
r_{n*}r_n^*\sK$, where $r_\bullet$ is the \v Cech nerve of $r_0$. We then
conclude by Lemma \ref{4le:real}.
\end{proof}
The above result can be extended to $\cD(X,\lambda)$ under cohomological
finiteness conditions. For an object $\lambda$ of $\Rind$, Consider the
functors
\begin{align*}
\EO{}{\Chpar{},\lambda}{*}{\otimes}&\colon
\N(\Chpar{})^{op}\to \Cat,\\
\EO{}{\Chpdm{\infty},\lambda}{*}{\otimes}&\colon
\N(\Chpdm{\infty})^{op}\to\Cat
\end{align*}
sending $X$ to $\lambda\mapsto \cD(X,\lambda)^\otimes$.
\begin{proposition}
Let $\sf{S}$ be an $\sf{L}$-coprime \emph{locally Noetherian} higher Artin
stack (that is, there exists an atlas $S\to \sf{S}$ where $S$ is a locally
Noetherian scheme).
\begin{enumerate}
\item Let $\lambda$ be an object of $\Rind_{\ltor}$ and consider a
Cartesian square
\[\xymatrix{
W \ar[r]^-{g} \ar[d]_-{q} & Z \ar[d]^-{p} \\
Y \ar[r]^-{f} & X }
\] in $\Chparl$ (resp.\ $\Chpars$) with $p$ proper of finite diagonal (resp.\
proper, $1$-Artin, of separated diagonal). Assume that for every
morphism $U\to X$ locally of finite type with $U$ an affine scheme,
$X_0$ is $\lambda$-cohomologically finite. Then the induced square
\[
\xymatrix{
\cD(Z,\lambda) \ar[d]_-{g^*} & \cD(X,\lambda) \ar[l]_-{p^*} \ar[d]^-{f^*}\\
\cD(W,\lambda) & \cD(Y,\lambda) \ar[l]_-{q^*}
}
\] is right adjointable.
\item Let $X^+_\bullet$ be a proper finite-diagonal hypercovering in
$\Chparl$ (resp.\ proper, $1$-Artin, separated-diagonal
hypercovering in $\Chpars$). Assume that for every morphism $U\to
X^+_{-1}$ locally of finite type with $U$ an affine scheme, $X_0$
is $\lambda$-cohomologically finite. Then $X^+_\bullet$ is an
augmentation of $\EO{}{\Chpar{},\lambda}{*}{\otimes}$-descent.
\end{enumerate}
\end{proposition}
A similar result holds for $\EO{}{\Chpdm{\infty},\lambda}{*}{\otimes}$ for
$\lambda$ in $\Rind_\tor$.
\begin{proof}
It suffices to repeat the proof of Proposition \ref{4pr:proper_descent} with
Proposition \ref{4pr:hyperdescent2} replaced by Proposition
\ref{4pr:hyperdescent} and Lemma \ref{4le:real} replaced by Lemma
\ref{4le:split}.
\end{proof}
\subsection{Flat descent}
\label{4ss:flat_descent}
Consider the functors
\begin{align*}
\EO{\ge 0}{\Chpar{}}{*}{}&\colon
\N(\Chpar{})^{op}\to\Fun(\N(\Rind)^{op},\Cat),\\
\EO{\ge 0}{\Chpdm{\infty}}{*}{}&\colon
\N(\Chpdm{\infty})^{op}\to\Fun(\N(\Rind)^{op},\Cat)
\end{align*}
sending $X$ to $\lambda\mapsto \cD^{\ge 0}(X,\lambda)$.
\begin{proposition}
Any flat and locally finitely presented hypercovering of higher Artin stacks
(resp.\ $\infty$-DM stacks) is an augmentation of $\EO{\ge
0}{\Chpar{}}{*}{}$-descent (resp.\ $\EO{\ge
0}{\Chpdm{\infty}}{*}{}$-descent).
\end{proposition}
This is an analogue of flat cohomological descent \cite{SGA4}*{Vbis
Proposition 4.3.3 c)}.
\begin{proof}
By Proposition \ref{4pr:hyperdescent2}, we are reduced to show that any
surjective flat and locally finitely presented morphism $f\colon Y\to X$ of
higher Artin stacks (resp.\ $\infty$-DM stacks) is of descent. By
\cite{LZ}*{3.1.2} and smooth (resp.\ \'etale) descent, we are reduced to the
case of schemes. Let $X'$ be a disjoint union of strict localizations of $X$,
such that the morphism $g\colon X'\to X$ is surjective. By
\cite{EGAIV}*{17.16.2, 18.5.11}, there exists a finite surjective morphism
$g'\colon Z\to X'$, such that the composite map $Z\to X$ factorizes through
$f$. Since $g$ and $g'$ are both of universal $\EO{\ge 0}{\Sch}{*}{}$-descent
by \cite{LZ}*{3.3.6} and the dual of Lemma \ref{4le:real}, it follows from
\cite{LZ}*{3.1.2} that $f$ is of universal $\EO{\ge 0}{\Sch}{*}{}$-descent.
\end{proof}
Again this result can be extended to $\cD(X,\lambda)$ under cohomological
finiteness conditions.
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|
1,116,691,500,899 | arxiv | \section{Introduction}
One of the most important sectors of the standard model is Yang-Mills theory
which describes QCD and the electro-weak theory. The degrees of freedom of a
gauge theory are encoded in the connection$\ (A_{\mu})^{a}$, which is a Lie
algebra valued one form. The action functional is invariant under gauge
transformation
\begin{equation}
A_{\mu}\rightarrow U^{-1}A_{\mu}U+U^{-1}\partial_{\mu}U
\label{gaugetranformation
\end{equation}
whereas the physical observables are invariant under proper gauge
transformations. The latter has to be everywhere smooth and it has to decrease
at infinity in a suitable way (see, for instance, \cite{Benguria:1976in}).
Since it is still unknown how to use in practice gauge invariant variable in
Yang-Mills case\footnote{In the cases of topological field theories in 2+1
dimensions \cite{wittenjones} this goal has been partially achieved.}, the
most convenient practical choices for the gauge fixing are the Coulomb gauge
and the Landau gauge\footnote{The axial and light-cone gauge fixing choices
are affected by quite non-trivial problems (see, for instance, \cite{DeW03
).}. The standard approach to fix the gauge and to use Feynman expansion
around the trivial vacuum $A_{\mu}=0$ provides one with excellent results when
the coupling constant is small.
As it was for the first time discovered in \cite{Gri78}, a \textit{proper
gauge fixing} is not possible globally due to the presence of Gribov
copies\footnote{Furthermore, it has been shown by Singer \cite{singer}, that
if Gribov ambiguities occur in Coulomb gauge, they occur in all the gauge
fixing conditions involving derivatives of the gauge field (See also
\cite{jackiw0}). Other gauge fixings (such as the axial gauge, the temporal
gauge, and so on) free from gauge fixing ambiguities are possible but these
choices have their own problems (see, for instance, \cite{DeW03}).}. In the
QCD case, this effect is very important in the non-perturbative regime. Even
if one chooses a gauge free of Gribov copies, the effects of Gribov
ambiguities in other gauges generate a non-perturbative breaking of the BRST
symmetry \cite{Fuj}.
It has been suggested to exclude from the domain of the path-integral gauge
potentials $A_{\mu}$ which generate zero modes of the FP operator (see, in
particular, \cite{Gri78} \cite{Zw82} \cite{Zw89} \cite{DZ89}\ \cite{Zwa96}
\cite{Va92}; two nice reviews are \cite{SS05} \cite{EPZ04}). In this
framework, which is called the (refined) \textit{Gribov-Zwanziger approach} to
QCD, The region $\Omega$ around $A_{\mu}=0$ in which the FP operator is
positive is called Gribov region
\begin{align}
& \Omega\overset{def}{=}\left\{ \left. A_{\mu}\right\vert \ \partial^{\mu
}A_{\mu}=0\ \ and\ \ \det\partial^{\mu}D\left( A\right) _{\mu}>0\right\}
,\label{grireg}\\
D\left( A\right) _{\mu} & =\partial_{\mu}+\left[ A_{\mu,\cdot}\right]
\ ,\label{covder
\end{align}
where $D\left( A\right) _{\mu}$ is the covariant derivative corresponding to
the field $A_{\mu}$. In the case in which the space-time metric is flat and
the \textit{topology is trivial} this approach coincides with usual
perturbation theory when the gauge field $A_{\mu}$ is close to the origin
(with respect to a suitable functional norm \cite{Va92}). At the same time,
this framework takes into account the infra-red effects related to the
partial\footnote{The condition to have a positive Faddeev-Popov operator is
not enough to completely eliminate Gribov copies in the Coulomb and Landau
gauges. It can be shown \cite{Va92} that there exist a smaller region (called
the \textit{modular region}) contained in the Gribov region which is free of
gauge fixing ambiguities. However, it is still not clear how to implement the
restriction to the modular region in practice.}\ elimination of the Gribov
copies \cite{Zw89} \cite{MaggS} \cite{Gracey}. When one takes into account the
presence of suitable condensates \cite{SoVae2} \cite{SoVar} \cite{SoVar2}
\cite{SoVar3} \cite{soreprl} the agreement with lattice data is
excellent\footnote{Recently, in \cite{criticism}, some criticism of this
approach has been proposed. However, this argument is only valid in the
perturbative framework (for instance, the gluons are assumed to be asymptotic
states) while the changes of the Gribov-Zwanziger approach are
non-perturbative in nature.} \cite{DOV} \cite{CucM}. Within the (refined)
\textit{Gribov-Zwanziger approach} to QCD, the \textit{copy-free neighborhood}
of the trivial vacuum $A_{\mu}=0$ \textit{has to be identified with}
\textit{perturbative region of the theory}. In this way, within the
perturbative region, the standard recipe to fix the gauge and perform the
perturbative expansion makes sense thanks to the lack of any overcounting. The
appearance of copies of the vacuum satisfying the strong boundary conditions
is one of the worst pathologies of perturbation theory.
On the other hand, it is a well established fact by now that Yang-Mills theory
may have knotted excitations \cite{knotted} \cite{knotted1} \cite{knotted2}
(the simplest non-trivial example corresponding to a closed tube: the "donut"
or \textit{unknot} with topology $S^{1}\times D^{2}$). Moreover, as it is well
known (see, for instance, \cite{infinitevolume}), non-trivial topologies such
as
\mathbb{R}
\times T^{2}$ and
\mathbb{R}
\times T^{3}$ are very important to understand the infinite volume limit of
Yang-Mills theory as well as chiral symmetry breaking generated by the Casimir
force (a related reference is \cite{casimir1}). In order to describe these
situations one has to be able to define consistently Yang-Mills theory inside
bounded regions with the non-trivial topologies.
The main goal of the present paper is to analyze whether or not Gribov copies
of the vacuum can actually exist inside a space-time region of the topology
such as $S^{1}\times D^{2}$ (but more general topologies relevant, for
instance, for the infinite volume limit of Yang-Mills as well as lattice
QCD-such as $T^{3}$ and
\mathbb{R}
\times T^{2}$,
\mathbb{R}
\times T^{3}$ as well as the interior of an ellipse-will be considered). The
absence of Gribov copies of the vacuum (satisfying strong boundary conditions)
is a necessary condition for the existence of a perturbative region in the
functional space around $A_{\mu}=0$. It will be shown that such requirement
implies non-trivial restrictions on the possible sizes and shapes of the
corresponding regions. In a sense, these results are more surprising than the
ones obtained in \cite{CGO} \cite{ACGO} \cite{CGO2} in which it has been shown
that the pattern of appearance of Gribov copies strongly depends on the
space-time metric. All the examples of non-trivial copies of the vacuum will
be in flat metrics (but with non-trivial topologies). The present analysis
will also show that the Gribov phenomenon strongly depends on the shapes and
sizes of the bounded regions where one wants to study gauge theories.
Unlike the equation for the zero-modes of the Faddeev-Popov operator, the
issue of the appearance of Gribov copies of the vacuum is non-linear in nature
and therefore can be quite complicated when non-standard topologies are
considered. The technical tool necessary in order to analyze such an issue is
a novel (to the best of authors knowledge) generalization of the
\textit{hedgehog ansatz} beyond spherical symmetry for the non-linear
sigma-model. Such a generalization is quite interesting in itself since it
allows to reduce the non-linear system of coupled partial differential
equations corresponding to the equations of motion of the non-linear sigma
model to a single non-linear partial differential equation which can be
analyzed with the tools of solitons theory.
The paper is organized as follows. In section two the notion of strong
boundary conditions with non-trivial topologies will be analyzed. In section
three, the relations between the Gribov copies equation and the non-linear
sigma-model will be discussed. In section four, the generalized hedgehog
ansatz will be constructed. In sections five and six various non-trivial
examples of copies of the vacuum will be described. Some conclusions will be
drawn in the last section. In the Appendix a novel way to\ implement both
spherical and elliptical symmetries (which could be useful in the context of
calorons) will be presented.
\section{Strong boundary conditions with non-trivial topologies}
Here, we will first review the definition of strong boundary conditions in the
case in which the metric is flat and the topology trivial. In the present
paper we will mainly analyze the $SU(2)$ case but many of the present results
also extend to other Lie groups. A useful starting point is the definition of
non-Abelian charges $Q^{a}$ (see \cite{Benguria:1976in} \cite{Gri78}
\cite{abbdeser}, \cite{jackiw}; see, for a detailed review, \cite{lavelle})
\begin{equation}
Q^{\left( a\right) }=\int_{M}d^{3}x\partial_{i}E^{ia}=\lim_{R\rightarrow
\infty}\int_{\Sigma_{R}}d^{2}\Sigma\left( n_{i}E^{ia}\right) \ ,
\label{charge1
\end{equation}
where $M$ is the constant-time hypersurface in four-dimensional Minkowski
space, $\Sigma_{R}$ is the two-dimensional sphere of radius $R$, $n_{i}$ is
the outer pointing unit normal to $\Sigma_{R}$, the indices $i$, $j$, $k$
refers to space-like directions while $a$, $b$, $c$,...are internal $SU(2)$ indices.
Thus, one can define a \textit{proper gauge transformation} $U$ as a smooth
gauge transformation which does not change the value of the charge as a
surface integral at infinity
\begin{equation}
\lim_{R\rightarrow\infty}\int_{\Sigma_{R}}d^{2}\Sigma\left( n_{i
E^{ia}\right) =\lim_{R\rightarrow\infty}\int_{\Sigma_{R}}d^{2}\Sigma\left(
n_{i}\left( U^{-1}EU\right) ^{ia}\right) \ . \label{proper1
\end{equation}
Therefore, $U$ is proper if it is smooth and approaches to an element of the
center of the gauge group at spatial infinity.
A Gribov copy $U$ on a flat and topologically trivial space-time
satisfies\textit{ the strong boundary} conditions if $U$ is proper. A copy of
this type is particularly problematic since it would represent a failure of
the whole gauge fixing procedure. Indeed, if the vacuum $A_{\mu}=0$ possesses
a copy fulfilling the strong boundary conditions, not even usual perturbation
theory leading to the standard Feynman rules in the Landau or Coulomb gauge
would be well defined.
\subsection{Strong boundary conditions on $S^{1}\times D^{2}$}
The simplest topology corresponding to a closed knotted tube is $\Upsilon
=S^{1}\times D^{2}$: this case is relevant in the analysis of non-trivial
topological excitations of glue-balls. In particular, there is a sound
evidence supporting the existence of excited glue-ball states with the
topology of $\Upsilon=S^{1}\times D^{2}$ (see, in particular, \cite{knotted1}
\cite{knotted2}). In this case, the spatial metric describing the region
$\Upsilon$ reads
\begin{align}
ds^{2} & =\left( d\phi\right) ^{2}+\left( dr^{2}+r^{2}d\theta^{2}\right)
\ ,\label{metr1}\\
0 & \leq\phi\leq2\pi\ ,\ \ 0\leq\theta\leq2\pi\ ,\ \ 0\leq r\leq
R\ ,\nonumber
\end{align}
where the coordinate $\phi$\ corresponds to the $S^{1}$ circle of $\Upsilon$,
while the coordinates $r$ and $\theta$ describe the disk $D^{2}$ of $\Upsilon$.
The radius of the disk is $R$ and the boundary of $\Upsilon$ is
\[
\partial\Upsilon=S^{1}\times\partial D^{2}\ .
\]
Thus, in this case a smooth gauge transformation $U$ is proper i
\[
\left. U\right\vert _{S^{1}\times\partial D^{2}}\in Z_{2}\ ,
\]
where $Z_{2}$ is the center\footnote{Obviously, the center of $SU(2)$ is made
of two elements: $\pm\mathbf{1}$.} of $SU(2)$. In other words, a smooth gauge
transformation $U=U(\phi,r,\theta)$ is proper i
\begin{align}
U(\phi+2m\pi,r,\theta+2n\pi) & =U(\phi,r,\theta)\ ,\ \ m,n\i
\mathbb{Z}
\ ,\label{tub1}\\
U(\phi,R,\theta) & \in Z_{2}\ ,\ \forall\ \phi,\ \theta\ . \label{tub2
\end{align}
\subsection{strong boundary conditions on $T^{3}$}
In the cases in which the spatial topology is $T^{3}$ (which, for instance, is
relevant in the case of lattice QCD), the flat spatial metric describing
$T^{3}$ reads
\begin{align}
ds^{2} & =\sum_{i=1}^{3}\lambda_{i}^{2}\left( d\phi_{i}\right)
^{2}\ ,\ \ \lambda_{i}\i
\mathbb{R}
\ ,\label{metr2}\\
0 & \leq\phi_{i}\leq2\pi\ ,\nonumber
\end{align}
where the coordinate $\phi_{i}$\ corresponds to the $i$-th factor $S^{1}$ in
$T^{3}$ while $\lambda_{i}$ represents the size of the $i$-th factor $S^{1}$
in $T^{3}$. In this case, due to the fact that $\partial T^{3}=0$, a smooth
gauge transformation $U=U(\phi_{1},\phi_{2},\phi_{3})$ is proper i
\begin{equation}
U(\phi_{1}+2m_{1}\pi,\phi_{2}+2m_{2}\pi,\phi_{3}+2m_{3}\pi)=U(\phi_{1
,\phi_{2},\phi_{3})\ ,\ \ m_{i}\i
\mathbb{Z}
\ . \label{tub3
\end{equation}
\subsection{strong boundary conditions on
\mathbb{R}
\times T^{2}$}
The cases in which the spatial topology is
\mathbb{R}
\times T^{2}$ may correspond to situations in which the bounded region in
which we want to define perturbative Yang-Mills theory is much longer in one
direction (the $x$ axis) with respect to the other two.
It is worth emphasizing here that, in the Euclidean theory, the topology
\mathbb{R}
\times T^{3}$ is also very important in order to understand both the infinite
volume limit in QCD (see, for instance, \cite{infinitevolume}) and effects of
chiral symmetry breaking in QCD (see, for instance, \cite{casimir1}). In
particular, it has been shown in \cite{casimir1} the relevance of the
Casimir\footnote{Indeed, Casimir force is a genuine finite-size effect which,
obviously, is not visible if the theory is analyzed in unbounded regions.}
force to understand chiral symmetry breaking. Hence, the importance of
finite-size effects to explain these phenomena and the necessity to have a
well-defined perturbation theory in such bounded regions with non-trivial
topology require a deeper understanding of the Gribov problem in bounded
region as well. As it will be shown in the following, the present results on
the Coulomb gauge can be trivially extended to the case of the Landau gauge
defined on
\mathbb{R}
\times T^{3}$. Thus, the gauge-fixing pathologies arising in this case have to
be carefully taken into account.
The flat spatial metric describing
\mathbb{R}
\times T^{2}$ reads
\begin{align}
ds^{2} & =dx^{2}+\sum_{i=1}^{2}\lambda_{i}^{2}\left( d\phi_{i}\right)
^{2}\ ,\ \ \lambda_{i}\i
\mathbb{R}
\ ,\label{metr2.5}\\
0 & \leq\phi_{i}\leq2\pi\ ,\nonumber
\end{align}
where the coordinate $\phi_{i}$\ corresponds to the $i$-th factor $S^{1}$ in
$T^{2}$ while $\lambda_{i}$ represents the size of the $i$-th factor $S^{1}$
in $T^{2}$. In this case, $\partial\left(
\mathbb{R}
\times T^{2}\right) $ is non-trivial and can be identified with the two limit
point $x\rightarrow\pm\infty$ of
\mathbb{R}
$. Thus, a smooth gauge transformation $U=U(\phi_{1},\phi_{2},x)$ is proper i
\begin{align}
U(\phi_{1}+2m_{1}\pi,\phi_{2}+2m_{2}\pi,x) & =U(\phi_{1},\phi_{2
,x)\ ,\ \ m_{i}\i
\mathbb{Z}
\ ,\label{tub3.5}\\
\underset{x\rightarrow\pm\infty}{\lim}U(\phi_{1},\phi_{2},x) & \in
Z_{2}\ ,\ \forall\ \phi_{1},\ \phi_{2}\ . \label{tub3.75
\end{align}
\section{Gribov copies and non-linear sigma model}
In the present paper we will analyze the copies of the vacuum in the Coulomb
gauge but many of the present results can be easily extended to the Landau
gauge. Since the Gribov copies of the vacuum in the Coulomb gauge can be seen
as the Euler-Lagrange equations corresponding to a non-linear sigma model, we
will first review some basic features of this model. The non-linear sigma
model Lagrangian in $D$ space-like dimensions can be written in terms of group
valued scalar field $U$. In the following we will consider the $SU(2)$ group,
so that the corresponding Lagrangian is
\begin{align}
S & =\frac{\kappa^{2}}{2}\int\sqrt{-g}d^{D}xTr\left[ R^{i}R_{i}\right]
\ ,\label{symh}\\
R_{i} & =U^{-1}\partial_{i}U\ , \label{skyrme2
\end{align
\begin{align*}
R_{i} & =R_{i}^{a}t_{a}\ ,\\
t^{a}t^{b} & =-\delta^{ab}\mathbf{1}-\varepsilon^{abc}t^{c}\ ,\ \\
\left[ R_{i},R_{j}\right] ^{c} & =C_{ab}^{c}R_{i}^{a}R_{j}^{b
\ ,\ C_{ab}^{c}=-2\varepsilon_{cab}\ ,\\
\varepsilon^{abc}\varepsilon^{mnc} & =\left( \delta_{a}^{m}\delta_{b
^{n}-\delta_{a}^{n}\delta_{b}^{m}\right) \ ,
\end{align*}
where $D$ is the number of space dimensions, $\kappa$ is the sigma-model
coupling constants, $g$ is the determinant of the space-like metric,
$\mathbf{1}$ is the identity $2\times2$ matrix and $t^{a}$ the generator of
$SU(2)$ (where the Latin letters $a$, $b$, $c$ and so on corresponds to group
indices). The equation characterizing the appearance of Gribov copies of the
vacuum in the Coulomb gauge (which corresponds to the Euler-Lagrange equations
of the action in Eq. (\ref{symh})) read
\begin{equation}
\nabla^{i}R_{i}=\nabla^{i}\left( U^{-1}\partial_{i}U\right) =0\ ,
\label{nonlinearsigma1
\end{equation}
where $\nabla^{i}$ is the Levi-Civita covariant derivative corresponding to
the metric (which, in the present case will have one of the forms in Eqs.
(\ref{metr1}), (\ref{metr2}) and (\ref{metr2.5})). The case of the Landau
gauge is analogous.
The following standard parametrization of the $SU(2)$-valued functions
$U(x^{i})$ is useful
\begin{align}
U(x^{i}) & =Y^{0}\mathbf{1}+Y^{a}t_{a}\ ,\ U^{-1}(x^{i})=Y^{0
\mathbf{1}-Y^{a}t_{a}\ ,\label{standard1}\\
Y^{0} & =Y^{0}(x^{i})\ ,\ \ \ Y^{a}=Y^{a}(x^{i})\ ,\label{standard2}\\
\left( Y^{0}\right) ^{2}+Y^{a}Y_{a} & =1\ , \label{standard3
\end{align}
where, of course, the sum over repeated indices is understood also in the case
of the group indices (in which case the indices are raised and lowered with
the flat metric $\delta_{ab}$). Therefore, the $R_{i}$ in Eq. (\ref{skyrme2})
can be written as follows
\begin{equation}
R_{i}^{c}=\varepsilon^{abc}Y_{a}\partial_{i}Y_{b}+Y^{0}\partial_{i}Y^{c
-Y^{c}\partial_{i}Y^{0}\ . \label{standard4
\end{equation}
\section{Generalized Hedgehog ansatz}
Due to the intrinsic non-linear nature of the Gribov copies of the vacuum, it
is necessary to introduce a suitable technical tool which allows to study such
phenomenon with non-trivial topologies.
Here we will first discuss the geometrical interpretation of \textit{hedgehog
ansatz} as an effective tool to reduce the field equations of the non-linear
sigma model (which, generically, are a \textit{system of coupled non-linear
partial differential equations}) to a single scalar non-linear partial
differential equation. The geometrical analysis of such very important feature
of the hedgehog ansatz allows to construct the natural generalization of the
hedgehog ansatz for the non-linear sigma model. This section is interesting in
itself since, to the best of authors knowledge, this is the first systematic
reduction of the equations of motion of the non-linear sigma model (which is a
system of coupled non-linear partial differential equations) to a single
non-linear scalar equation beyond spherical symmetry. In terms of the group
element $U$ the standard spherically symmetric hedgehog ansatz read
\begin{align}
U & =\mathbf{1}\cos\alpha(r)+\widehat{n}^{a}t_{a}\sin\alpha(r)\ ,\ \ U^{-1
=\mathbf{1}\cos\alpha(r)-\widehat{n}^{a}t_{a}\sin\alpha
(r)\ ,\label{hedgehog1.1}\\
\ \widehat{n}^{1} & =\sin\theta\cos\phi\ ,\ \ \ \widehat{n}^{2}=\sin
\theta\sin\phi\ ,\ \ \ \widehat{n}^{3}=\cos\theta\ , \label{hedgehog1.111
\end{align}
where $r$, $\theta$ and $\phi$ are spherical coordinates of the flat Euclidean
metric, the $\widehat{n}^{a}$ are normalized with respect to the internal
metric $\delta_{ab}$. In terms of the variables $Y^{0}$ and $Y^{a}$ the
hedgehog ansatz corresponds to the following choice
\begin{equation}
Y^{0}=\cos\alpha(r)\ ,\ \ Y^{a}=\widehat{n}^{a}\sin\alpha(r)\ ,\ \ \delta
_{ab}\widehat{n}^{a}\widehat{n}^{b}=1\ . \label{hedgehog1.2
\end{equation}
Thus, the expression for $R_{i}^{a}$ in Eq. (\ref{standard4}) read
\begin{equation}
R_{i}^{c}=\delta^{dc}\left( \sin^{2}\alpha\right) \varepsilon_{abd
\widehat{n}^{a}\partial_{i}\widehat{n}^{b}+\widehat{n}^{c}\partial_{i
\alpha+\frac{\sin\left( 2\alpha\right) }{2}\partial_{i}\widehat{n}^{c}\ .
\label{hedgehog1.5
\end{equation}
The equations of motion for the non-linear sigma model in Eq.
(\ref{nonlinearsigma1}) corresponding to this ansatz rea
\begin{equation}
\left( \nabla^{i}\partial_{i}\alpha\right) +L\sin2\alpha=0\ ,
\label{sinegordon1
\end{equation}
where $L$ is a suitable function of the radial coordinate $r$ (see below).
Indeed, this is a very important and non-trivial characteristic of the
spherically symmetric hedgehog ansatz which reduces a system of coupled
non-linear partial differential equations to a single scalar equation. It is
easy to see that in the case of the flat three-dimensional Euclidean metric on
\mathbb{R}
^{3}$ in spherical coordinates the above reduces to the standard Gribov
pendulum of the vacuum (see \cite{Gri78} \cite{SS05}).
The previous analysis suggests to analyze which are the geometrical conditions
which allow to reduce a system of non-linear coupled Partial Differential
Equations to a single scalar PDE. Let us consider the following generalization
of the hedgehog ansatz
\begin{align}
U & =\mathbf{1}\cos\alpha+\widehat{n}^{a}t_{a}\sin\alpha\ ,\ \ U^{-1
=\mathbf{1}\cos\alpha-\widehat{n}^{a}t_{a}\sin\alpha\ ,\label{hedgehog1.75}\\
\delta_{ab}\widehat{n}^{a}\widehat{n}^{b} & =1\ ,\ \ \alpha=\alpha\left(
x^{i}\right) \ ,\ \ \widehat{n}^{a}=\widehat{n}^{a}\left( x^{i}\right) \ ,
\label{hedgehog1.95
\end{align}
such tha
\begin{align}
\left( \partial_{i}\alpha\right) \left( \nabla^{i}\widehat{n}^{a}\right)
& =0\ ,\label{hedgehogcondition3}\\
\left( \nabla^{i}\partial_{i}\right) \widehat{n}^{c} & =2L\widehat{n
^{c}\ , \label{hedgehogcondition4
\end{align}
where $L$ (which has to be the same for all the non-vanishing\footnote{Note
that, in order for the solution to be non-trivial, at least two of the three
$\widehat{n}^{c}$\ have to be non-vanishing.} $\widehat{n}^{c}$) may depend on
the space-time coordinates.
The important point is that, when the conditions (\ref{hedgehogcondition3})
and (\ref{hedgehogcondition4})\ are satisfied, the function $\alpha$ (which in
the usual spherically symmetric hedgehog ansatz can depend only on the radial
coordinate $r$) can now depend on any set of coordinates and the functions
$\widehat{n}^{a}$ (which in the usual case have to coincide with the unit
radial vector) can be adapted to cases in which there is no spherical symmetry
(and, therefore, no natural radial coordinate). Indeed, the expression for
$R_{i}^{a}$ in Eq. (\ref{standard4}) read
\begin{equation}
R_{i}^{c}=\delta^{dc}\left( \sin^{2}\alpha\right) \varepsilon_{abd
\widehat{n}^{a}\partial_{i}\widehat{n}^{b}+\widehat{n}^{c}\partial_{i
\alpha+\frac{\sin\left( 2\alpha\right) }{2}\partial_{i}\widehat{n}^{c}\ .
\label{hedgehog2
\end{equation}
Hence, provided Eqs. (\ref{hedgehogcondition3}) and (\ref{hedgehogcondition4})
are fulfilled and taking into account that the expressio
\[
\left( \nabla^{i}\widehat{n}^{a}\right) \left( \partial_{i}\widehat{n
^{b}\right) =g^{ij}\left( \partial_{i}\widehat{n}^{a}\right) \left(
\partial_{j}\widehat{n}^{b}\right)
\]
is symmetric under the exchange of $a$ and $b$ (so that its contraction with
$\varepsilon_{abd}$\ vanishes) the coupled system of non-linear Partial
Differential Equations (PDE) corresponding to the equations of motion of the
non-linear sigma model reduces to the single scalar non-linear partial
differential equation of sine-Gordon type
\begin{equation}
\left( \nabla^{i}\partial_{i}\alpha\right) +L\sin2\alpha=0\ .
\label{sinegordon1.5
\end{equation}
Indeed, it is easy to see that in the usual spherically symmetric flat ansatz
in Eqs. (\ref{hedgehog1.1}) and (\ref{hedgehog1.111}) satisfies the conditions
in Eqs. (\ref{hedgehogcondition3}) and (\ref{hedgehogcondition4}) with
$L=-1/r^{2}$ and that, correspondingly, Eq. (\ref{sinegordon1.5}) reduces to
the usual Gribov pendulum (see \cite{Gri78} \cite{SS05}). However, it is worth
noting here that the present derivation is far simpler (just three lines of
computation) and, furthermore, it discloses in a very clear way the geometry
behind the hedgehog ansatz.
The usual Gribov pendulum equation on flat and topologically trivial
three-dimensional spaces (see \cite{Gri78} \cite{SS05}) does not coincide with
the sine-Gordon equation due to the radial dependence of the determinant of
the three-dimensional Euclidean metric in spherical coordinates. For these
reasons, there are no analytic examples of copies of the vacuum in this case.
On the other hand, using the present generalized hedgehog ansatz, one can
construct many analytic examples of copies of the vacuum. These examples are
very useful since they allow to express the requirement of absence of copies
of the vacuum in terms of explicit constraints on the shapes and sizes of the
corresponding bounded regions where one wants to define a perturbative region.
\subsection{Regularity conditions}
Another important requirement to satisfy the strong boundary conditions is the
regularity of the pure gauge field $U^{-1}dU$ everywhere. In the standard
spherically-symmetric case, the only problematic point is the origin of the
coordinates in which the spherical coordinates system is not well defined. One
way to understand this condition in general is to analyze the behavior of the
one-form
\begin{equation}
R^{c}=R_{i}^{c}\wedge dx^{i} \label{regularity1
\end{equation}
(where $R_{i}^{c}$ is given by (\ref{hedgehog2}) and $\alpha$ satisfies
(\ref{sinegordon1.5})) when one changes coordinates from the Cartesian to a
coordinates system which is singular somewhere. Indeed, in the following we
will need to use non-Cartesian coordinates systems which are adapted to the
symmetry of the problem (such as the elliptic coordinates systems in the next sections).
In the case of Cartesian coordinates, the regularity at the origin can be read
by looking at the components of the one-form $R_{i}^{c}$ since the coordinates
system is regular everywhere. On the other hand, if one is in a spherical or
elliptical coordinates system, the angular coordinates are not well defined at
the origin and some extra care is required. For instance, to disclose the
singularity of the one forms $d\theta$ and $d\phi$ at the origin one can
analyze the Jacobian of the transformation from Cartesian coordinates
($x^{\prime i}$) to spherical coordinates ($x^{j}$)
\begin{equation}
dx^{\prime i}=J_{j}^{i}dx^{j}\ . \label{regularity2
\end{equation}
Due to the regularity of the Cartesian coordinates, the Jacobian itself
encodes the informations of the singularity at the origin. In the spherically
symmetric case, for example, if one takes into account the Jacobian, the
regularity condition at the origin coincides\footnote{In the spherical case
the factor $\sin\theta$ which appears in the denominator of many components of
the Jacobian is canceled out by a similar factor in the numerator of the
spherically symmetric hedgehog ansatz in the normalized internal vector
$\widehat{n}^{i}$.} with the usual one derived in Cartesian coordinates
\cite{SS05}.
The requirement for regularity can be described as follows. Let's call $x_{s}$
a point in which the coordinates are singular and let's consider the following
behavior for $\alpha$ when $x\rightarrow x_{s}$
\begin{equation}
\alpha\left( x\right) \underset{x\rightarrow x_{s}}{\rightarrow}n\pi+\beta
f\left( x\right) \label{regularity3
\end{equation}
with $f\left( x\right) \underset{x\rightarrow x_{s}}{\rightarrow}0$ and
$\beta$ an arbitrary constant. Then, a copy of the form (\ref{hedgehog2})
satisfy
\begin{equation}
R_{i}^{c}\underset{x\rightarrow x_{s}}{\rightarrow}\beta^{2}\delta^{dc
f^{2}\varepsilon_{abd}\widehat{n}^{a}\partial_{i}\widehat{n}^{b}+\beta
\widehat{n}^{c}\partial_{i}f+\beta f\partial_{i}\widehat{n}^{c}\ .
\label{regularity4
\end{equation}
Then using (\ref{regularity2}) we see that $f(x)$ must be such tha
\begin{equation}
R_{i}^{c}\left( J^{-1}\right) _{j}^{i} \label{regularity5
\end{equation}
(where $\left( J^{-1}\right) _{j}^{i}$ is the inverse Jacobian of the
transformation from Cartesian to the curvilinear coordinates system of
interest) are regular functions in $x=x_{s}$.
\section{Explicit examples of vacuum copies in non-trivial topologies}
Here we will discuss many example of vacuum copies.
\subsection{$S^{1}\times D^{2}$ topology}
Here we will consider the following ansatz corresponding to the metric in Eq.
(\ref{metr1}):
\[
\alpha=\alpha(r)\ ,\ \ \widehat{n}^{1}=\cos\left( \theta\right)
\ ,\ \ \widehat{n}^{2}=\sin\left( \theta\right) \ ,\ \widehat{n}^{3}=0\ .
\]
It is easy to see that the above ansatz satisfies the conditions in Eqs.
(\ref{hedgehogcondition3}) and (\ref{hedgehogcondition4}) with the following
$L$
\begin{align*}
\left( \nabla^{i}\partial_{i}\right) \widehat{n}^{1} & =-\frac{1}{r^{2
}\widehat{n}^{1}\ ,\ \ \left( \nabla^{i}\partial_{i}\right) \widehat{n
^{2}=-\frac{1}{r^{2}}\widehat{n}^{2}\ \Rightarrow\\
L & =-\frac{1}{2r^{2}}\ .
\end{align*}
Thus, in this case the equation (\ref{sinegordon1.5}) for the copy of the
vacuum reduces t
\begin{equation}
r\partial_{r}\left( r\partial_{r}\alpha\right) -\gamma\sin2\alpha
=0\ \Leftrightarrow\label{caso1.01
\end{equation
\begin{align}
\frac{\partial^{2}\alpha}{\partial\tau^{2}} & =\gamma\sin2\alpha
\ ,\ \ \gamma=\frac{1}{2}\ ,\ \ \tau-\tau_{0}=\log r\ ,\label{caso1.1}\\
E & =\frac{1}{2}\left[ \left( \frac{\partial\alpha}{\partial\tau}\right)
^{2}+\gamma\cos2\alpha\right] \ ,\label{caso1.2}\\
\tau-\tau_{0} & =\pm\int_{\alpha(\tau_{0})}^{\alpha(\tau)}\frac{dy
{\sqrt{2E-\gamma\cos2y}}\ , \label{caso1.3
\end{align}
where $\tau_{0}$ and $E$ are integration constants.
In terms of the radial coordinate $r$ one has to require that the copy is
regular at the origin and that it approaches an element of the center when
$r=R$ which is the boundary of $D^{2}$. Without loss of generality one can
require (see Eq. (\ref{hedgehog1.75}))
\begin{equation}
\alpha\underset{r\rightarrow0}{\rightarrow}0\ ,\ \ \alpha(R)=n\pi\ ,\ \ \ n\i
\mathbb{Z}
\ . \label{condition1
\end{equation}
In terms of the variable $\tau$ the above conditions rea
\begin{align}
& \alpha\underset{\left( \tau-\tau_{0}\right) \rightarrow-\infty
}{\rightarrow}0\ ,\ \ \alpha\underset{\left( \tau-\tau_{0}\right)
\rightarrow\tau^{\ast}}{\rightarrow}n\pi\ ,\label{condition2}\\
\tau^{\ast} & =\log R\ .\nonumber
\end{align}
Hence, we have to analyze under which conditions on the parameter $\gamma$ and
on the integration constants $E$ and $\tau_{0}$ Eq. (\ref{caso1.2}) can have
solutions satisfying the conditions in Eq. (\ref{condition2}). The simplest
way to answer to this question is to interpret Eq. (\ref{caso1.2}) as the
energy conservation of the following one-dimensional problem in which $\tau$
plays the role of the effective time
\begin{align}
\overline{E} & =\frac{1}{2}\left( \frac{\partial A}{\partial\tau}\right)
^{2}+V\left( A\right) \ ,\label{1Deffective1}\\
V(A) & =2\gamma\cos A\ ,\ \ \ A=2\alpha\ ,\ \ \ \overline{E}=4E\ .
\label{1Deffective2
\end{align}
The boundary conditions in Eq. (\ref{condition2}) in terms of $A$ read (we
will consider the case $n=1$
\begin{equation}
A\underset{\left( \tau-\tau_{0}\right) \rightarrow-\infty}{\rightarrow
}0\ ,\ \ A\underset{\left( \tau-\tau_{0}\right) \rightarrow\tau^{\ast
}{\rightarrow}2\pi\ . \label{1Deffectiveboundcond1
\end{equation}
The above conditions mean that $0$ and $2\pi$ have to be two turning points
and that the corresponding period has to diverge
\begin{equation}
\frac{1}{\sqrt{2}}\int_{0}^{2\pi}\frac{dy}{\sqrt{\overline{E}-2\gamma\cos y
}\rightarrow\infty\ . \label{caso1.4
\end{equation}
In order to satisfy this condition, it is enough to choos
\begin{equation}
\overline{E}=2\gamma=1\ , \label{csao1.5
\end{equation}
the explicit form for the copy bein
\begin{equation}
\alpha=2\arctan\left[ \frac{r}{\bar{r}_{0}}\right] \label{spaccspacc
\end{equation}
with $\bar{r}_{0}$ an integration constant (whose role will be described in a moment).
In terms of the original radial variable $r$, the condition in Eq.
(\ref{condition1}) can only be satisfied\footnote{The reason is that the copy
profile in Eq. (\ref{spaccspacc}) is an increasing function from $0$ to
$\infty$ and its maximum is $\pi$. Note that, however, the copy profile
approaches the asymptotic value very rapidly.} if $R\rightarrow\infty$. In
physical terms, this means that the condition in Eq. (\ref{condition1}) can be
fulfilled when $R$ is very large compared to $\bar{r}_{0}$:
\[
R\gg\bar{r}_{0}.
\]
Obviously, in the present case the only two natural lengths are the radius of
the disk $R$ and the perimeter of the $S^{1}$ factor (which has been set to
$2\pi$) of the donut $S^{1}\times D^{2}$ and consequently $\bar{r}_{0}$
represents the size of $S^{1}$. Thus, the previous analysis tells that when
the radius of the disk is much larger than the perimeter of $S^{1}$ the donut
is on the verge of supporting smooth copies of the vacuum satisfying the
strong boundary conditions. Hence, in order to avoid this pathology the donut
cannot be too "fat".
It is easy to check that such copy is regular at the origin. The components of
$U^{-1}dU$ in Cartesian coordinates for this case are given b
\[
R_{i}^{c}\left( J^{-1}\right) _{j}^{i}\text{ \ },\text{ \ }i=r,\theta\text{
\ },\text{ \ }j=1,2
\]
Wit
\[
J^{-1}=\left(
\begin{array}
[c]{cc
\cos(\theta) & \sin(\theta)\\
-\frac{\sin(\theta)}{r} & \frac{\cos(\theta)}{r
\end{array}
\right)
\]
For $r\rightarrow0$ copies must decay a
\begin{equation}
\alpha\underset{r\rightarrow0}{\rightarrow}\beta r\text{ }, \label{ec
\end{equation}
for some $\beta\i
\mathbb{R}
$ and the profile in Eq. (\ref{spaccspacc}) satisfies this condition.
\subsection{$T^{3}$ topology}
As it has been already emphasized, the $T^{3}$ topology is relevant, for
instance, in the analysis of the thermodynamical limit. Here we will consider
the following ansatz corresponding to the metric in Eq. (\ref{metr2}):
\begin{align}
\alpha & =\alpha(\phi_{1})\ ,\ \ \widehat{n}^{1}=\cos\left( p\phi_{2
+q\phi_{3}\right) \ ,\ \ \widehat{n}^{2}=\sin\left( p\phi_{2}+q\phi
_{3}\right) \ ,\label{caso2}\\
\widehat{n}^{3} & =0\ ,\ \ p,q\i
\mathbb{Z}
\ .\nonumber
\end{align}
It is easy to see that the above ansatz satisfies the conditions in Eqs.
(\ref{hedgehogcondition3}) and (\ref{hedgehogcondition4}) with the following
$L$
\[
L=-\frac{1}{2}\left( \left( \frac{p}{\lambda_{2}}\right) ^{2}+\left(
\frac{q}{\lambda_{3}}\right) ^{2}\right) \ .
\]
Thus, in this case the equation (\ref{sinegordon1.5}) for the copy of the
vacuum reduces to a flat elliptic sine-Gordon equation
\begin{align}
\left( \frac{\partial^{2}}{\partial\phi_{1}^{2}}\right) \alpha &
=\gamma\sin2\alpha\ ,\label{caso2.1}\\
\gamma & =\frac{\lambda_{1}^{2}}{2}\left( \left( \frac{p}{\lambda_{2
}\right) ^{2}+\left( \frac{q}{\lambda_{3}}\right) ^{2}\right) \ .
\label{caso2.11
\end{align}
In order for a solution of Eq. (\ref{caso2.1}) to define a copy of the vacuum
satisfying the strong boundary condition (see Eqs. (\ref{hedgehog1.75}) and
(\ref{tub3})) it is necessary to requir
\begin{equation}
\alpha(\phi_{1}+2m\pi)=\alpha(\phi_{1})\ ,\ \ m\i
\mathbb{Z}
\ , \label{caso2.2
\end{equation}
where we will consider $m=1$ in the following. As in the previous subsection,
Eq. (\ref{caso2.1}) can be reduced to a first order conservation law
\begin{align}
E & =\frac{1}{2}\left[ \left( \frac{\partial\alpha}{\partial\phi_{1
}\right) ^{2}+\gamma\cos2\alpha\right] \ ,\label{caso2.22}\\
\phi_{1}-\phi_{0} & =\pm\int_{\alpha(\phi_{0})}^{\alpha(\phi_{1})}\frac
{dy}{\sqrt{2E-\gamma\cos2y}}\ ,\nonumber
\end{align}
where $\phi_{0}$ and $E$ are integration constants. Hence, we have to analyze
under which conditions on the parameter $\gamma$ and on the integration
constants $E$ and $\phi_{0}$ Eq. (\ref{caso2.22}) can have solutions
satisfying the conditions in Eq. (\ref{caso2.2}). As in the previous
subsection, it is possible to interpret Eq. (\ref{caso2.22}) as the energy
conservation of a one-dimensional problem in Eqs. (\ref{1Deffective1}) and
(\ref{1Deffective2}) in which $\phi_{1}$ plays the role of the effective time
$\tau=\phi_{1}$. However, in this case the boundary conditions in Eq.
(\ref{caso2.2}) become
\begin{equation}
A\left( \tau+2\pi\right) =A\left( \tau\right) \ .
\label{1Deffectiveboundcond2
\end{equation}
Thus, given two consecutive turning points $A_{0}$ and $A_{1}$
\[
A_{0}=\arccos\frac{\overline{E}}{2\gamma}\ ,\ \ \ A_{1}=2\pi-A_{0}\ ,
\]
one has to require that the the time needed to go from $A_{0}$ to $A_{1}$ is
half of the period in Eq. (\ref{1Deffectiveboundcond2}), namely
\[
\tau-\tau_{0}=\int_{A_{0}}^{A_{1}}\frac{dy}{\sqrt{2(\overline{E}-2\gamma\cos
y)}}=\pi\ .
\]
The time for a particle to go from $A_{0}$ to $A_{1}$ runs from $0$ to
infinity as $\overline{E}$ runs from $-2\gamma$ to $2\gamma$. This means that,
given a $\gamma$, there is always an $\overline{E}$ such that $\tau-\tau
_{0}=\pi$. Then, for $-2\gamma<$ $\overline{E}<2\gamma,$ it is always possible
to construct copies of the vacuum satisfying the strong boundary conditions.
Furthermore, it is easy to see that the norm of the copy is finite
\begin{align*}
\left\Vert U^{-1}dU\right\Vert & =\int_{T^{3}}d^{3}x\sqrt{g}tr\left(
U^{-1}dU\right) ^{2}\\
& =\left( 2\pi\right) ^{2}\lambda_{1}\lambda_{2}\lambda_{3}\int_{T^{3
}d\phi_{1}\left( \left( \frac{\partial_{\phi_{1}}\alpha}{\lambda_{1
}\right) ^{2}+\sin^{2}\alpha\left( \left( \frac{p}{\lambda_{2}}\right)
^{2}+\left( \frac{q}{\lambda_{3}}\right) ^{2}\right) \right) <\infty\ .
\end{align*}
On the other hand, it worth emphasizing that while there is a common agreement
on the importance of the strong boundary conditions, it is not clear yet
whether or not it is mandatory to require the finite norm condition. For
instance, it is possible to construct configurations of the Yang-Mills gauge
potential which have infinite norm but finite energy and/or action (see, for a
recent discussion, \cite{SorellaNorm}).
It is worth emphasizing that, at a first glance, one could think that the
presence of Gribov copies of the vacuum prevents one from using the Gribov
semi-classical approach whose aim is, of course, to eliminate Gribov copies
from a suitable neighborhood of the vacuum itself. However, the copies which
can be eliminated using the Gribov semiclassical approach are "small" copies,
namely zero-modes of the Faddeev-Popov operator (which is a linear condition).
On the other hand, the vacuum copies which have been constructed here are
solutions of the full non-linear equation for the copies and they would
disappear if one would only consider the linearized equation (which reduces
just to the Laplace equation in all the cases discussed in the present paper).
Therefore, the Gribov semiclassical approach can be applied, for instance, to
the $R\times T^{3}$ or $T^{4}$ cases. For instance, in the $T^{4}$ case in the
Landau gauge, the inverse Gribov propagator $\left( \Delta_{G}\right) ^{-1}$
would have, schematically, the usual for
\[
\left( \Delta_{G}\right) ^{-1}=\square+\gamma\square^{-1}\ ,
\]
where $\gamma$ is the non-perturbative Gribov mass parameter determined by the
usual gap equation \cite{Gri78} and $\square$ is the Laplacian on $T^{4}$.
Hence, the only technical difference with respect the usual case would be
that, in order to invert the Laplacian one should use the Fourier series
instead of the Fourier transform. Indeed, in this way one can eliminate, from
a suitable neighborhood of the vacuum $A_{\mu}=0$, zero modes of the
Faddeev-Popov operator but the copies of the vacuum discussed above cannot be
eliminated with this procedure. The physical consequences of this fact are
very interesting also from the point of view of lattice QCD and are actually
under investigation (for the relations between the continuous analysis of
Gribov copies and the lattice see also \cite{new1}, \cite{new2} and
\cite{new3}).
\subsection{
\mathbb{R}
\times T^{2}$ topology}
As it has been already explained, this topology\textbf{ }is very important,
for instance, in relation with the infinite volume limit of Yang-Mills theory
as well as chiral symmetry breaking.
Actually, as far as the infinite volume limit and chiral symmetry breaking are
concerned (see, for instance, \cite{infinitevolume} \cite{casimir1}), the
setting corresponds to the Euclidean theory defined on
\mathbb{R}
\times T^{3}$. Thus, in this case the Landau gauge is to be preferred.
However, the present construction trivially extends to this situation as well.
One has just to interpret the coordinate $x$ in the metric in Eq.
(\ref{metr2.5}) as the Euclidean time in order to apply the present analysis
to this case.
Here we will consider the following ansatz corresponding to the metric in Eq.
(\ref{metr2.5}):
\begin{align}
\alpha & =\alpha(x)\ ,\ \ \widehat{n}^{1}=\cos\left( p\phi_{1}+q\phi
_{2}\right) \ ,\ \ \widehat{n}^{2}=\sin\left( p\phi_{1}+q\phi_{2}\right)
\ ,\label{caso3.1}\\
\widehat{n}^{3} & =0\ ,\ \ p,q\i
\mathbb{Z}
\ .\nonumber
\end{align}
It is easy to see that the above ansatz satisfies the conditions in Eqs.
(\ref{hedgehogcondition3}) and (\ref{hedgehogcondition4}) with the following
$L$
\[
L=-\frac{1}{2}\left( \left( \frac{p}{\lambda_{2}}\right) ^{2}+\left(
\frac{q}{\lambda_{3}}\right) ^{2}\right) \ .
\]
Thus, in this case the equation (\ref{sinegordon1.5}) for the copy of the
vacuum reduces to a flat elliptic sine-Gordon equation
\begin{align}
\left( \frac{\partial^{2}}{\partial x^{2}}\right) \alpha & =\gamma
\sin2\alpha\ ,\label{caso3.2}\\
\gamma & =\frac{\lambda_{1}^{2}}{2}\left( \left( \frac{p}{\lambda_{2
}\right) ^{2}+\left( \frac{q}{\lambda_{3}}\right) ^{2}\right) \ .
\label{caso3.3
\end{align}
In order for a solution of Eq. (\ref{caso3.2}) to define a copy of the vacuum
satisfying the strong boundary condition (see Eqs. (\ref{hedgehog1.75}),
(\ref{tub3.5}) and (\ref{tub3.75})) it is necessary to requir
\begin{equation}
\underset{x\rightarrow+\infty}{\lim}\alpha(x)=m\pi\ ,\ \ \underset
{x\rightarrow-\infty}{\lim}\alpha(x)=n\pi\ ,\ \ m,n\i
\mathbb{Z}
\ . \label{caso3.4
\end{equation}
Similarly to the previous subsections, Eq. (\ref{caso3.2}) can be reduced to
\begin{align}
E & =\frac{1}{2}\left[ \left( \frac{\partial\alpha}{\partial x}\right)
^{2}+\gamma\cos2\alpha\right] \ ,\label{caso3.33}\\
x-x_{0} & =\pm\int_{\alpha(x_{0})}^{\alpha(x)}\frac{dy}{\sqrt{2E-\gamma
\cos2y}}\ ,\nonumber
\end{align}
where $x_{0}$ and $E$ are integration constants. Hence, we have to analyze
under which conditions on the parameter $\gamma$ and on the integration
constants $E$ and $x_{0}$ Eq. (\ref{caso3.33}) can have solutions satisfying
the conditions in Eq. (\ref{caso3.4}) for some $m$ and $n$. As in the previous
subsections, (\ref{caso3.33}) has the form of energy conservation of a
one-dimensional problem (\ref{1Deffective1}) with the identifications
(\ref{1Deffective2}). In this case the strong boundary conditions in Eq.
(\ref{caso3.4}) become
\begin{equation}
\underset{x\rightarrow+\infty}{\lim}A(x)=2m\pi\ ,\ \ \underset{x\rightarrow
-\infty}{\lim}A(x)=2n\pi\ ,\ \ m,n\i
\mathbb{Z}
\ . \label{caso3.5
\end{equation}
For $m=1$ and $n=0$ this means that $0$ and $2\pi$ have to be two turning
points with infinite period. As in (\ref{caso1.4}), this condition is ensured
for $\overline{E}=2\gamma$. Furthermore, in this case as well it is easy to
see that the norm of the copy is finite
\[
\left\Vert U^{-1}dU\right\Vert =\int_{T^{2}\time
\mathbb{R}
}\sqrt{g}d^{3}xtr\left( U^{-1}dU\right) ^{2}<\infty\ .
\]
wher
\[
\alpha\left( x\right) =2\arctan\left[ \exp\left( x-\bar{x}_{0}\right)
\right] \ ,
\]
($\bar{x}_{0}$ being an integration constant) is the solution of Eq.
(\ref{caso3.33}).
\section{Elliptic Gribov Pendulum}
In this section, we will analyze the elliptic generalization of the Gribov
pendulum equation. It is easy to see that if one would try a naive
generalization of the spherical Gribov pendulum \cite{Gri78} to elliptic
coordinates (which reduces to the usual case in the spherical limit) then the
system of equations for the Gribov copies of the vacuum do not reduce to a
single scalar equation as it happens in the spherical case and, consequently,
it would be extremely difficult to analyze the corresponding system of
equations and the corresponding boundary conditions. On the other hand, within
the present framework, one is lead to the two ansatzs in Eqs.
(\ref{prolansatz}) and (\ref{oblansatz}) which \textit{do reduce} the
equations for the Gribov copies of the vacuum to a \textit{single scalar
equation}: this allows to study the strong boundary conditions in the usual
way. To the best of authors knowledge, this is the first non-trivial elliptic
generalization of the Gribov pendulum.
This analysis is particularly important since it sheds light on how sensible
the Gribov phenomenon is with respect to deformation of the spherical
symmetry. Even if the usual Gribov pendulum equation\footnote{Actually, the
whole issue of gauge copies is usually analyzed only in unbounded region.} is
analyzed in the unbounded region $r\in\left[ 0,\infty\right[ $ \cite{Gri78}
\cite{SS05}, one implicitly assumes that similar results hold in a spherical
bounded region. The reason is that, of course, experimentally gluons can live
only within baryons and glueballs which are bounded regions. Therefore, it
makes sense to study the arising of gauge copies of the vacuum in bounded
region as well. Indeed, the results that no copy of the vacuum appear in the
case of the usual Gribov pendulum also holds in the case of a bounded
spherical region: as it will be now discussed, the elliptic case is more complicated.
\subsection{Prolate Spheroid}
\bigskip The line element for of flat three-dimensional Euclidean space in
prolate spheroidal coordinates is given b
\begin{equation}
ds^{2}=a^{2}\left( \sinh^{2}\mu+\sin^{2}\nu\right) \left( d\mu^{2}+d\nu
^{2}\right) +a^{2}\sinh^{2}\mu\sin^{2}\nu d\phi^{2
\end{equation}
For a prolate spheroidal bounded region the coordinate ranges are given by
$0\leq\mu<R$, $\nu\in\left[ 0,\pi\right] $, $\phi\in\lbrack0,2\pi)$. The
coordinate $\mu$ represents the elliptic radius since $\mu=const$ surfaces are
ellipses. Such $\mu=const$ ellipses have different eccentricities: large
eccentricities (namely, large deformations from spherical symmetries)
correspond to small $\mu$ while small eccentricities (namely, small deviations
from spherical symmetry) correspond to large $\mu$. Thus, in the above
coordinates system, if one wants to analyze the limit of "large deformations
from spherical symmetry" then one has to consider the small $\mu$ region. On
the other hand, if one wants to consider the almost spherical case, then the
large $\mu$ limit has to be considered. As it will be shown in the following,
the large deformation limit is the most interesting case.
The Laplacian in this coordinates is given b
\[
\nabla^{i}\partial_{i}=\frac{1}{a^{2}\left( \sinh^{2}\mu+\sin^{2}\nu\right)
}\left[ \partial_{\mu}^{2}+\partial_{\nu}^{2}+\coth\mu\partial_{\mu}+\cot
\nu\partial_{\nu}\right] +\frac{1}{a^{2}\sinh^{2}\mu\sin^{2}\nu
\partial_{\phi}^{2
\]
Then, the following ansatz satisfies the conditions in Eqs.
(\ref{hedgehogcondition3}) and (\ref{hedgehogcondition4}
\begin{equation}
\alpha=\alpha\left( \mu,\nu\right) \text{ \ },\text{ \ }n^{1}=\cos\phi\text{
\ },\text{ \ }n^{2}=\sin\phi\text{ \ },\text{ \ }n^{3}=0\ , \label{prolansatz
\end{equation}
with the following $L$
\[
L=-\frac{1}{2a^{2}\sinh^{2}\mu\sin^{2}\nu}\ .
\]
\subsubsection{The strong boundary conditions}
Due to th fact that $\mu$ represents the elliptic radius, the strong boundary
conditions in an unbounded region corresponds to the following conditions on
$\alpha$
\begin{align}
& \alpha\left( \mu,\nu\right) \underset{\mu\rightarrow0}{\rightarrow
n\pi+f(\mu,\nu)\ ,\label{probo1}\\
& \alpha\left( \mu,\nu\right) \underset{\mu\rightarrow\infty}{\rightarrow
}m\pi\ ,\ \ \ n,m\i
\mathbb{Z}
\ . \label{probo2
\end{align}
where $f(\mu,\nu)$ ensures regularity of the copies. On the other hand, if one
is analyzing the theory in a bounded region of elliptic radius $R$ then the
condition in Eq. (\ref{probo2}) has to be replaced b
\begin{equation}
\alpha\left( \mu,\nu\right) \underset{\mu\rightarrow R}{\rightarrow
m\pi\ ,\ \ \ m\i
\mathbb{Z}
\ , \label{probo3
\end{equation}
since the boundary of the region is the surface $\mu=R$ and one has to require
that the gauge copy belongs to the center of the gauge group on the boundary
of the region itself.
\subsubsection{Regularity conditions}
In the prolate spheroidal case, the inverse of the Jacobian in the
transformation (\ref{regularity2}) reads
\begin{equation}
J^{-1}=\left( \frac{\partial(\mu,\nu,\phi)}{\partial(x,y,z)}\right)
=\frac{1}{a}\left(
\begin{array}
[c]{ccc
\frac{\cosh\mu\sin\nu\cos\phi}{\sinh^{2}\mu+\sinh^{2}\nu} & \frac{\cosh\mu
\sin\nu\sin\phi}{\sinh^{2}\mu+\sinh^{2}\nu} & \frac{\sinh\mu\cos\nu}{\sinh
^{2}\mu+\sinh^{2}\nu}\\
\frac{\sinh\mu\cos\nu\cos\phi}{\sinh^{2}\mu+\sinh^{2}\nu} & \frac{\sinh\mu
\cos\nu\sin\phi}{\sinh^{2}\mu+\sinh^{2}\nu} & -\frac{\cosh\mu\sin\nu
{\sinh^{2}\mu+\sinh^{2}\nu}\\
-\frac{\sin\phi}{\sinh\mu\sin v} & \frac{\cos\phi}{\sinh\mu\sin\nu} & 0
\end{array}
\right) \label{probo4
\end{equation}
Thus, singularities could appear for $\mu=0$ and $\nu=0$. Following the
prescription in Eq. (\ref{regularity4}), near points with $\mu=0$ and/or
$\nu=0$ copies behave a
\begin{align*}
R_{\mu}^{c} & \rightarrow\beta\widehat{n}^{c}\partial_{\mu}f\\
R_{\nu}^{c} & \rightarrow\beta\widehat{n}^{c}\partial_{\nu}f\\
R_{\phi}^{c} & \rightarrow\beta^{2}\delta^{dc}f^{2}\varepsilon_{abd
\widehat{n}^{a}\partial_{\phi}\widehat{n}^{b}+\beta f\partial_{\phi
\widehat{n}^{c
\end{align*}
where $f$ must be such that functions (\ref{regularity5}) are regular. A
sufficient condition which ensures regularity is
\[
f(\mu,\nu)\underset{\mu,\nu\rightarrow0}{\rightarrow}\mu^{3}\nu^{3
\]
Besides the singularity at $\mu=0$, there are singularities for $\nu=0$ and
$\nu=\pi$ as well (which are similar to the $1/\sin\theta$ singularity in the
spherical case). However, in the elliptic case, the $\sin\nu$ factor is not
automatically canceled by the internal vector $\widehat{n}^{i}$ of the
hedgehog ansatz (which only depends on $\phi$ in the present case). Therefore
the profile function $\alpha(\mu,\nu)$ has to take care of this divergence.
\subsubsection{\bigskip Prolate pendulum}
Eq. (\ref{sinegordon1.5}) in the prolate elliptic coordinates reduces to the
following elliptic prolate Gribov pendulum
\begin{equation}
\frac{1}{\sinh^{2}\mu+\sin^{2}\nu}\left[ \partial_{\mu}^{2}+\partial_{\nu
}^{2}+\coth\mu\partial_{\mu}+\cot\nu\partial_{\nu}\right] \alpha=\frac
{1}{2\sinh^{2}\mu\sin^{2}\nu}\sin2\alpha\ . \label{prolate equation
\end{equation}
Now we consider the limits $\mu\rightarrow0$ and $\mu\rightarrow\infty$ which
correspond to the cases of large and small deformation from spherical symmetry respectively.
\begin{itemize}
\item For $\mu\rightarrow0$, equation (\ref{prolate equation}) takes the form
\begin{equation}
\left[ \partial_{\mu}^{2}+\partial_{\nu}^{2}+\coth\mu\partial_{\mu}+\cot
\nu\partial_{\nu}\right] \alpha=\frac{1}{2\sinh^{2}\mu}\sin2\alpha\ ,
\label{prolate2
\end{equation}
Interestingly enough, in this limit the ansatz $\alpha=\alpha\left(
\mu\right) $ is consistent and the above equation reduces t
\begin{equation}
\frac{d^{2}\alpha}{d\mu^{2}}+\coth\mu\frac{d\alpha}{d\mu}=\frac{1}{2\sinh
^{2}\mu}\sin2\alpha\ , \label{prolate3
\end{equation}
With the following change of coordinat
\begin{align}
\tau & =\ln\left\vert \tanh\left( \frac{\mu}{2}\right) \right\vert
\label{spaccrpolate}\\
\mu & =2\operatorname{arcth}e^{\tau}\\
\sinh\left( \operatorname{arcth}x\right) & =\frac{x}{\sqrt{1-x^{2}}
\end{align}
the equation (\ref{prolate3}) can be written a
\[
\frac{d^{2}\alpha\left( \tau\right) }{d\tau^{2}}=\frac{1}{2}\sin
2\alpha\left( \tau\right) \ .
\]
Of course, in this case, the natural boundary conditions correspond to the
analysis \ within a bounded region in Eqs. (\ref{probo1}) and (\ref{probo3})
in which, in suitable units, the radius $R$ is very smal
\[
R\ll1.
\]
In terms of the coordinates $\tau$ in Eq. (\ref{spaccrpolate}) Eqs.
(\ref{probo1}) and (\ref{probo3}
\[
\alpha\left( \tau\right) \underset{\tau\rightarrow-\infty}{\rightarrow
2n\pi\ ,
\
\[
\alpha\left( \tau\right) \underset{\tau\rightarrow\tau^{\ast}}{\rightarrow
}2m\pi\ ,\ \ \ n,m\i
\mathbb{Z}
\ .
\]
In the limit of large prolate deformations, the equation can be integrated
analytically
\begin{align}
E & =\frac{1}{2}\left[ \left( \frac{d\alpha}{d\tau}\right) ^{2}+\frac
{1}{2}\cos2\alpha\right] \ ,\label{profinal1}\\
\tau-\tau_{0} & =\pm\int_{\alpha(\tau_{0})}^{\alpha(\tau)}\frac{dy
{\sqrt{2E-\frac{1}{2}\cos2y}}\ . \label{profinal2
\end{align}
This result discloses in a very clear way how sensible the Gribov phenomenon
is not only to topology but also to the shapes of the region where it is
analyzed. Indeed, even if the present analytic solution cannot satisfy the
regularity conditions in $\nu=0$ and $\nu=\pi$ (which are the north and south
poles of the ellipse) since it is $\nu-$independent, a small deformation of
the ellipse at the poles could eliminate the necessity to require regularity
of the solution at $\nu=0$ and $\nu=\pi$ and consequently could give rise to
the sudden appearance of a copy of the vacuum. Hence, the present analysis
strongly suggests that large prolate deformations from spherical symmetry can
be quite pathological. These results can be relevant in the cases in which one
is analyzing Yang-Mills theory in bounded regions.
\end{itemize}
\bigskip
\begin{itemize}
\item For $\mu\rightarrow\infty,$ equation (\ref{prolate equation}) reduces
to
\[
\left[ \partial_{\mu}^{2}+\partial_{\nu}^{2}+\partial_{\mu}+\cot\nu
\partial_{\nu}\right] \alpha=\frac{1}{2}\csc^{2}\nu\sin2\alpha\ .
\]
Since the large $\mu$ limit corresponds to small deviations from spherical
symmetry, in this case one should recover the standard results on the absence
of vacuum copies.
\end{itemize}
\subsection{Oblate Spheroid}
The line element for a flat three-dimensional Euclidean space in oblate
spheroidal coordinates is given b
\[
ds^{2}=a^{2}\left( \sinh^{2}\mu+\sin^{2}\nu\right) \left( d\mu^{2}+d\nu
^{2}\right) +a^{2}\cosh^{2}\mu\cos^{2}\nu d\phi^{2
\]
For a oblate spheroidal bounded region the coordinate ranges are given by
$0\leq\mu<\mu^{\ast},\nu\in\left[ 0,\pi\right] ,\phi\in\lbrack0,2\pi)$. Also
in this case $\mu$ is the elliptic radius since $\mu=const$ surfaces are
ellipses with eccentricities which decrease with $\mu$. As in the prolate
case, if one wants to analyze the limit of "large deformations" then one has
to consider the small $\mu$ region. On the other hand, if one wants to
consider the almost spherical case, then the large $\mu$ limit has to be considered.
The Laplacian in this coordinates read
\[
\nabla^{i}\partial_{i}=\frac{1}{a^{2}\left( \sinh^{2}\mu+\sin^{2}\nu\right)
}\left[ \partial_{\mu}^{2}+\partial_{\nu}^{2}+\tanh\mu\partial_{\mu}+\tan
\nu\partial_{\nu}\right] +\frac{1}{a^{2}\cosh^{2}\mu\cos^{2}\nu
\partial_{\phi}^{2
\]
Thus, the following ansatz satisfies the conditions in Eqs.
(\ref{hedgehogcondition3}) and (\ref{hedgehogcondition4}
\begin{equation}
\alpha=\alpha\left( \mu,\nu\right) \text{ \ },\text{ \ }n^{1}=\cos\phi\text{
\ },\text{ \ }n^{2}=\sin\phi\text{ \ },\text{ \ }n^{3}=0 \label{oblansatz
\end{equation}
with the following $L$
\[
L=-\frac{1}{2a^{2}\cosh^{2}\mu\cos\nu}\ .
\]
\subsubsection{Strong boundary conditions}
As in the prolate case, the strong boundary conditions in an unbounded region
corresponds to the following conditions on $\alpha$
\begin{align}
& \alpha\left( \mu,\nu\right) \underset{\mu\rightarrow0}{\rightarrow
2n\pi\ ,\label{oblobo1}\\
& \alpha\left( \mu,\nu\right) \underset{\mu\rightarrow\infty}{\rightarrow
}2m\pi\ ,\ \ \ n,m\i
\mathbb{Z}
\ . \label{oblobo2
\end{align}
On the other hand, if one is analyzing the theory in a bounded region of
elliptic radius $R$ then the condition in Eq. (\ref{oblobo2}) has to be
replaced b
\begin{equation}
\alpha\left( \mu,\nu\right) \underset{\mu\rightarrow R}{\rightarrow
2m\pi\ ,\ \ \ n,m\i
\mathbb{Z}
\ . \label{oblobo3
\end{equation}
\paragraph{Regularity conditions}
As it has been explained in the previous subsection, the informations about
the singularities at the origin are encoded in the Jacobian. Also in this
case, besides the singularity for $\mu=0$, there are singularities for $\nu=0$
and $\nu=\pi$ as well Since the $\sin\nu$ factor is not automatically canceled
by the internal vector $\widehat{n}^{i}$ of the hedgehog ansatz, the profile
function $\alpha(\mu,\nu)$ has to take care of this divergence: a sufficient
condition to ensure regularity is
\[
f(\mu,\nu)\underset{\mu,\nu\rightarrow0}{\rightarrow}\mu^{3}\nu^{3
\]
\subsubsection{Oblate pendulum}
Hence, the equation for copies of the vacuum (\ref{sinegordon1.5}) in the
oblate case now takes the form of the following oblate Gribov pendulu
\begin{equation}
\frac{1}{\sinh^{2}\mu+\sin^{2}\nu}\left[ \partial_{\mu}^{2}+\partial_{\nu
}^{2}+\tanh\mu\partial_{\mu}+\tan\nu\partial_{\nu}\right] \alpha=\frac
{1}{2\cosh^{2}\mu\cos^{2}\nu}\sin\left( 2\alpha\right) \ .
\label{oblate equation
\end{equation}
Now we consider the limits $\mu\rightarrow0$ and $\mu\rightarrow\infty$
corresponding to large and small deformation from spherical symmetry respectively.
\begin{itemize}
\item \bigskip For $\mu\rightarrow0$ (\ref{oblate equation}) reduces t
\[
\left[ \partial_{\mu}^{2}+\partial_{\nu}^{2}+\tan\nu\partial_{\nu}\right]
\alpha=\frac{1}{2}\tan^{2}\nu\sin\left( 2\alpha\right) \ .
\]
Unlike the prolate case, in the present case it is not possible to find
analytic solutions in the limit of large deformations.
\item For $\mu\rightarrow\infty$ (\ref{oblate equation}) reduces t
\begin{equation}
\left[ \partial_{\mu}^{2}+\partial_{\nu}^{2}+\partial_{\mu}+\tan\nu
\partial_{\nu}\right] \alpha=\frac{\tanh^{2}\mu}{2\cos^{2}\nu}\sin\left(
2\alpha\right)
\end{equation}
\end{itemize}
Also in this case, since the large $\mu$ limit corresponds to small deviations
from spherical symmetry, one should recover the standard results on the
absence of vacuum copies.\bigskip
It is worth emphasizing that in the large deformation limit the prolate and
oblate Gribov pendulum equations (\ref{prolate equation}) and
(\ref{oblate equation}) differ significantly. In particular, unlike the
spherical or oblate cases, in the prolate case the Gribov pendulum equation
can be integrated exactly. This strongly suggests that prolate deformations
from spherical symmetry are more pathological than oblate deformations.
Indeed, the present results strongly suggest that the strong deformation limit
of Eq. (\ref{prolate equation}) may support copies of the vacuum. This
analysis appears to be quite relevant as far as the issue of gauge copies in
bounded region is concerned.
\section{Conclusions and further comments}
In this paper the arising of Gribov copies in regions with non-trivial
topologies (such as closed tubes $S^{1}\times D^{2}$, or
\mathbb{R}
\times T^{3}$) but flat metric has been analyzed. The technical tool has been
a generalization of the hedgehog ansatz beyond spherical symmetry. Such a
generalization of the hedgehog ansatz is very interesting in itself since, in
the case of the non-linear sigma-model, it provides one with a geometrical
recipe to reduce the the field equations of the non-linear sigma model (which
are a \textit{system of coupled non-linear partial differential equations}) to
a single scalar non-linear partial differential equation even when there is no
spherical symmetry. This ansatz allows to construct many analytic examples of
Gribov copies of the vacuum. Moreover, the elliptic Gribov pendulum has also
been derived (to the best of authors knowledge, for the first time) both in
the prolate and oblate cases. Our results suggest that large prolate
deformations from spherical symmetry are likely to be more pathological than
the oblate deformations. The requirement of absence of Gribov copies of the
vacuum satisfying the strong boundary conditions implies geometrical
constraints on the topology, on the shapes and on the sizes of the regions
with non-trivial topologies (such as upper bounds on the deviations from
spherical symmetry or constraint on the shape of the donut $S^{1}\times D^{2
$). Moreover, we have shown that in the case of a flat metric \textit{but with
the topology of} $T^{3}$ it is possible to construct copies of the vacuum
satisfying the strong boundary conditions and with finite norm. The present
results are interesting in relations with the infinite volume limit of
Yang-Mills theory (which is related to the
\mathbb{R}
\times T^{3}$\ topology). Indeed, one of the main point of the present
analysis has been to show that when the $T^{3}$ (as well as $T^{4}$) topology
\textit{explodes} to $R^{3}$ (or to $R^{4}$) the vacuum copies disappear since
neither.$R^{3}$ nor $R^{4}$ support copies of the vacuum. Hence, as far as the
Gribov copies are concerned, such limit is not smooth and should be studied
more carefully. This analysis is also relevant in all the cases in which
gluons are confined in regions of finite sizes with non-trivial topologies
such as in the cases of knotted flux tubes, lattice QCD and so on. Due to the
close relation between Gribov ambiguity and confinement the issue of the
Gribov copies in bounded regions (both on flat and on curved space-times) is
very important and worth to be investigated.
\section{Acknowledgments}
We thank Silvio Sorella for useful comments. This work is partially supported
by FONDECYT grant 1120352, and by the \textquotedblleft Southern Theoretical
Physics Laboratory\textquotedblright\ ACT-91 grant from CONICYT. The Centro de
Estudios Cient\'{\i}ficos (CECs) is funded by the Chilean Government through
the Centers of Excellence Base Financing Program of CONICYT. F. C. is also
supported by Proyecto de Inserci\'{o}n CONICYT 79090034, and by the Agenzia
Spaziale Italiana (ASI).
\section*{Appendix: \bigskip Landau gauge}
In this Appendix we will present two applications of the present generalized
hedgehog ansatz which lead to a novel realization of spherical and elliptical
symmetries respectively. The new way to implement spherical and elliptical
symmetry corresponds to a configuration in which the internal vectors $n^{i}$
of the generalized hedgehog ansatz depend only on the Euclidean time. This
realization of the spherical symmetry is only possible within the present
"generalized hedgehog" framework and it could be useful in the context of
calorons which are instantons which are periodic in Euclidean time. In both
cases we will construct the Landau gauge pendulum.
\subsection*{\bigskip A novel spherical Case}
Let us consider the following line element corresponding to a four-dimensional
euclidean spac
\[
ds^{2}=d\tau^{2}+dr^{2}+r^{2}\left( d\theta^{2}+\sin^{2}\theta d\phi
^{2}\right) \ ,\ \ \ 0\leq\tau\leq2\pi\ ,
\]
where $\tau$ plays the role of euclidean time (which is a periodic coordinate
of period $2\pi$) and the spatial section is written in spherical coordinates.
This situation is relevant in the cases in which one wants to describe finite
temperature effects. An ansatz satisfying conditions (\ref{hedgehogcondition3
) and (\ref{hedgehogcondition4}) suited to deal this situation i
\[
\alpha=\alpha\left( r\right) \text{ \ },\text{ \ }n^{1}=\cos\left(
\omega\tau\right) \text{ \ },\text{ \ }n^{2}=\sin\left( \omega\tau\right)
\text{ \ },\text{ \ }n^{3}=0
\]
with
\[
L=-\frac{\omega^{2}}{2
\]
Then, the equation for Gribov copies (\ref{sinegordon1.5}) takes the for
\begin{equation}
\partial_{r}^{2}\alpha+\frac{2}{r}\partial_{r}\alpha=\frac{\omega^{2}}{2
\sin\left( 2\alpha\right) \ , \label{novelspherical
\end{equation}
which obviously \textit{is not equivalent} to the usual spherically symmetric
Gribov pendulum \cite{Gri78} \cite{SS05}. Defining $x=\ln r$ we can write the
above equation a
\[
\partial_{x}^{2}\alpha+\partial_{x}\alpha=\frac{\omega^{2}}{2}e^{2x
\sin\left( 2\alpha\right) \ ,
\]
another useful form is, defining $y=-1/r$, the following
\[
\partial_{y}^{2}\alpha=\frac{\omega^{2}}{2y^{4}}\sin\left( 2\alpha\right)
\]
The regularity condition at the origin can be analyzed as in the previous sections.
\subsection*{Prolate Spheroidal Case}
\bigskip Let us consider the line element of four-dimensional Euclidean
space-time (in which the Euclidean time has been compactified to describe
finite-temperature effects) in prolate elliptic coordinate
\[
ds^{2}=d\tau^{2}+a^{2}\left( \sinh^{2}\mu+\sin^{2}\nu\right) \left(
d\mu^{2}+d\nu^{2}\right) +a^{2}\sinh^{2}\mu\sin^{2}\nu d\phi^{2
\]
An ansatz satisfying conditions (\ref{hedgehogcondition3}) and
(\ref{hedgehogcondition4}) suited to deal this situation i
\[
\alpha=\alpha\left( \mu,\nu\right) \text{ \ },\text{ \ }n^{1}=\cos\left(
\omega\tau\right) \text{ \ },\text{ \ }n^{2}=\sin\left( \omega\tau\right)
\text{ \ },\text{ \ }n^{3}=0
\]
with
\[
L=-\frac{\omega^{2}}{2
\]
Then, Eq. (\ref{sinegordon1.5}) reduces t
\[
\frac{1}{a^{2}\left( \sinh^{2}\mu+\sin^{2}\nu\right) }\left[ \partial_{\mu
}^{2}+\partial_{\nu}^{2}+\tanh\mu\partial_{\mu}+\tan\nu\partial_{\nu}\right]
\alpha=\frac{\omega^{2}}{2}\sin\left( 2\alpha\right)
\]
Also in this case, the regularity conditions can be studied as in the previous sections.
|
1,116,691,500,900 | arxiv | \section{Introduction}
Denning's seminal work \cite{Denning76} proposed \textit{complete lattices} as the appropriate mathematical framework for questions regarding \textit{secure information flow} (SIF).
An information flow model (IFM) is characterised as
$\langle N, P, SC, \sqcup, \sqsubseteq \rangle$ where:
\textit{Storage objects} in $N$ are assigned \textit{security levels} drawn from a (finite) complete lattice $SC$. $P$ is a set of processes (also assigned security classes as clearances).
The partial ordering $\sqsubseteq$ represents \textit{permitted flows} between classes; reflexivity and transitivity capture intuitive aspects of information flow; antisymmetry helps avoid redundancies in the framework, and the join operation $\sqcup$ succinctly captures the combination of information belonging to different security classes in arithmetic, logical and computational operations.
This lattice model provides an abstract uniform framework that identifies the commonalities of the variety of analyses for different applications -- \textit{e.g.}, confidentiality and trust -- whether at the \textit{language} level or a \textit{system} level.
In the ensuing decades, the vast body of secure information flow analyses has been built on these mathematical foundations, with the development of a plethora of static and dynamic analysis techniques for programming languages \cite{sabelfeld2003language,myers1999jflow,Pottier2003-FlowCaml,liu2017fabric,roy2009laminar,Lourenco2015-ug}, operating systems \cite{Krohn2007-aa,zeldovich2006-osdi,cheng2012aeolus,efstathopoulos2005asbestos,roy2009laminar}, databases \cite{schultz2013ifdb}, and hardware architectures \cite{ferraiuolo2018hyperflow, zhang2015secVerilog-asplos}, etc.
The soundness of this lattice model was expressed in terms of semantic notions of system behaviour, for instance, as properties like non-interference \cite{DBLP:conf/sp/GoguenM82a} by Volpano \textit{et al} \cite{DBLP:journals/jcs/VolpanoIS96} and others.
Alternative semantic notions of security such as safety properties have been proposed as well, \textit{e.g.}, \cite{Boudol2008SIFasSafety}, but for brevity we will not explore these further.
The objective of this paper is to propose a simple way in which large-scale distributed secure systems can be built by connecting components in a secure and modular manner.
Our work begins with the observation that large information systems are not monolithic: Different organisations define their own information flow policies independently, and subsequently collaborate or federate with one another to exchange information.
In general, the security classes and the lattices of any two organisations may be quite different --- \textit{there is no single universal security class lattice}.
Moreover, \textit{modularity} and \textit{autonomy} are important requirements since each organisation would naturally wish to retain control over its own security policies and the ability to redefine them.
Therefore, fusing different lattices by taking their union is an unsatisfactory approach, more so since the security properties of application programs would have to be re-established in this possibly enormous lattice.
When sharing information, most organisations limit the cross-domain communications to a limited set of security classes (which we call \textit{transfer} classes).
In order to ensure that shared data are not improperly divulged, two organisations usually negotiate agreements or memorandums of understanding (MoUs), promising that they will respect the security policies of the other organisation.
We argue that a good notion of secure connection should require reasoning only about those flows from only the transfer classes mentioned in a MoU.
Usually, cross-domain communication involves downgrading the security class of privileged information to public information using primitives such as encryption, and then upgrading the information to a suitable security class in the other domain.
Such approaches, however, do not gel well with correctness notions such as non-interference.
Indeed the question of how to translate information between security classes of different lattices is interesting \cite{deng2017secchisel}.
\paragraph{Contributions of this paper.} \ \
In this paper, we propose a simple framework and sufficient conditions under which secure flow guarantees can be enforced without exposing the complexities and details of the component information flow models.
The framework consists of (1) a way to connect security classes of one organisation to those in another while satisfying intuitive requirements; (2) a simple language that extends the operations within an organisation with primitives for transferring data between organisations; and (3) a type system and operational model for these constructs, which we use to establish that the framework conserves security.
In \S\ref{sec:Bi-DirFlow}, we first identify, using intuitive examples,
violations in secure flow that may arise when two secure systems are permitted to exchange information in both directions.
Based on these lacunae, we formulate \textit{security} and \textit{precision} requirements for secure bidirectional flow.
We then propose a framework that guarantees the absence of such policy violations, without impinging on the autonomy of the individual systems, without the need for re-verifying the security of the application procedures in either of the domains, and confining the analysis to only the transfer classes involved in potential exchange of data.
Our approach is based on \textit{monotone functions} and an elegant theory of \textit{connections} \cite{MELTON1994lagoisconnections} between the security lattices.
Theorem \ref{thm:secureconnection} shows that\textit{ Lagois connections} between the security lattices satisfy the security and precision requirements.
We present, in \S\ref{sec:model}, a minimal operational language consisting of a small set of \textit{atomic primitives} to effect the transfer of data between domains.
The framework is simple and can be adapted for establishing secure connections between distributed systems at any level of abstraction (language, system, database, ...).
We assume each domain uses \textit{atomic transactional operations} for object manipulation and intra-domain computation.
The primitives of our model include reliable communication between two systems, transferring object data in designated \textit{output} variables of one domain to designated \textit{input} variables of a specified security class in the other domain.
We also assume a generic set of operations in each domain for copying data from input variables to domain objects, and from domain objects to output variables.
To avoid interference between inter-domain communication and the computations within the domains, we assume that the sets of designated input and output variables are all mutually exclusive of one another, and also with the program/system variables used in the computations within each domain.
Thus by design we avoid the usual suspects that cause interference and insecure transfer of data.
The operational description of the language consists of the primitives
together with their execution rules (\S\ref{sec:operations}).
The correctness of our framework is demonstrated by expressing soundness (with respect to the operational semantics) of a type system (\S\ref{sec:typing}), stated in terms of the security lattices and their connecting functions.
In particular, Theorem \ref{thm:soundness} shows the standard semantic property of \textit{non-interference} in both domains holds of all operational behaviours.
We adapt and \textit{extend} the approach taken by Volpano \textit{et al} \cite{DBLP:journals/jcs/VolpanoIS96} to encompass systems coupled using the Lagois connection conditions (and assuming atomicity of the data transfer operations), to show that security is conserved.
We believe that our formulation is general enough to be applicable to other behavioural notions of secure information flow as a safety property \cite{Boudol2008SIFasSafety}.
In \S\ref{sec:related}, we briefly review some related work.
We conclude in \S\ref{sec:conclusion} with a discussion on our approach and directions for future work.
\section{Lagois Connections and All That}\label{sec:Bi-DirFlow}
\paragraph{Motivating Examples.} \ \
Consider a university system in which students study in semi-autonomously administered colleges (one such is $\ld{C}$) that are affiliated to a university ($\rid{U}$). The university also has ``university professors'' with whom students can take classes.
We assume each institution has established the security of its information flow mechanisms and policies.
We first observe that formulating an agreement \textit{between} the institutions that respects the flow policies within the institutions is not entirely trivial.
Consider an arrangement where the college \textit{Faculty} and \textit{University Faculty} can share information (say, course material and examinations), and the \textit{Dean of Colleges} in the University can exchange information (e.g., students' official grade-sheets) with the college's \textit{Dean of Students}.
Even such an apparently reasonable arrangement suffers from insecurities, as illustrated in Fig. \ref{fig:Bi-breach-equiv-classes} by the flow depicted using dashed red arrows, where information can flow from the college's \textit{Faculty} to the college's \textit{Dean of Students}. (Moral: internal structure of the lattices matters.)
\begin{figure}[t
\begin{minipage}[t]{.45\textwidth}
\begin{tikzpicture}[framed,->,node distance=0.9cm,on grid]
\title{W and D}
\node(T2) {\scriptsize$\top 2$};
\node(T1) [xshift=-3cm] {\scriptsize$\top 1$};
\node(Dir1) [below of = T1] {\scriptsize$CollegePrincipal$};
\node(D1) [below left of = Dir1] {\scriptsize$Dean\ (F)$};
\node(F1) [below of = D1] {\scriptsize$Faculty$};
\node(DS1) [xshift=-2.3cm,yshift=-1.7cm] {\scriptsize$Dean\ (S)$};
\node(S1) [below right of = F1] {\scriptsize$Student$};
\node(B1) [below of=S1] {\scriptsize$\bot 1$};
\node(Sec2) [below of = T2] {\scriptsize$Chancellor$};
\node(AS2) [below of=Sec2] {\scriptsize$Vice\ Chancellor$};
\node(Dir2) [below of=AS2] {\scriptsize$Dean(Colleges)$};
\node(E2) [below of=Dir2] {\scriptsize$Univ.Fac.$};
\node(B2) [below of=E2] {\scriptsize$\bot 2$};
\draw [Fuchsia, ->] (Dir1) to (T1);
\draw [Fuchsia, ->] (D1) to (Dir1);
\draw [Fuchsia, ->] (F1) to (D1);
\draw [Fuchsia, ->] (S1) to (F1);
\draw [Fuchsia, ->] (S1) to (DS1);
\draw [Fuchsia, ->] (DS1) to (Dir1);
\draw [Fuchsia, ->] (B1) to (S1);
\draw [Fuchsia, ->] (B2) to (E2);
\draw [Fuchsia, ->] (E2) to (Dir2);
\draw [Fuchsia, ->] (Dir2) to (AS2);
\draw [Fuchsia, ->] (AS2) to (Sec2);
\draw [Fuchsia, ->] (Sec2) to (T2);
\draw [OliveGreen, thick, <->](F1) [bend right = 5] to (E2);
\draw [OliveGreen, thick, <->](DS1) to (Dir2);
\draw [OliveGreen, thick, ->](Dir1) [bend left = 10] to (AS2);
\draw [red, thick, dashed] (F1) [bend left = 7] to (E2);
\draw [red, thick, dashed] (E2) [bend left = 35] to (Dir2);
\draw [red, thick, dashed] (Dir2) [bend left = 15] to (DS1);
\draw (-3,-2.2) ellipse (1.4cm and 2.4cm);
\draw (0,-2.2)[red] ellipse (1.2cm and 2.5cm);
\end{tikzpicture}
\caption{\small Green arrows represent permitted flows according to the information exchange arrangement between a college and a university. Red dashed arrows highlight a \textit{new} flow that is a security violation. \label{fig:Bi-breach-equiv-classes}}
\end{minipage}
\quad \quad
\begin{minipage}[t]{.45\textwidth}
\begin{tikzpicture}[framed,->,node distance=0.9cm,on grid]
\title{W and D}
\node(T2) {\scriptsize$\top 2$};
\node(T1) [xshift=-3cm] {\scriptsize$\top 1$};
\node(Dir1) [below of = T1] {\scriptsize$CollegePrincipal$};
\node(D1) [below left of = Dir1] {\scriptsize$Dean\ (F)$};
\node(F1) [below of = D1] {\scriptsize$Faculty$};
\node(DS1) [xshift=-2.3cm,yshift=-1.7cm] {\scriptsize$Dean\ (S)$};
\node(S1) [below right of = F1] {\scriptsize$Student$};
\node(B1) [below of=S1] {\scriptsize$\bot 1$};
\node(Sec2) [below of = T2] {\scriptsize$Chancellor$};
\node(AS2) [below of=Sec2] {\scriptsize$Vice\ Chancellor$};
\node(Dir2) [below of=AS2] {\scriptsize$Dean(Colleges)$};
\node(E2) [below of=Dir2] {\scriptsize$Univ.Fac.$};
\node(B2) [below of=E2] {\scriptsize$\bot 2$};
\draw [blue, thick] (S1) to (E2);
\draw [green, thick] (F1) to (E2);
\draw [green, thick] (B1) to (B2);
\draw [green, thick] (T1) to (T2);
\draw [blue, thick] (Dir1) [bend left = 10] to (Dir2);
\draw [green, thick] (D1) [bend right = 15] to (Dir2);
\draw [green, thick] (DS1) to (Dir2);
\draw [Fuchsia, ->] (Dir1) to (T1);
\draw [Fuchsia, ->] (D1) to (Dir1);
\draw [Fuchsia, ->] (F1) to (D1);
\draw [Fuchsia, ->] (S1) to (F1);
\draw [Fuchsia, ->] (S1) to (DS1);
\draw [Fuchsia, ->] (DS1) to (Dir1);
\draw [Fuchsia, ->] (B1) to (S1);
\draw [Fuchsia, ->] (B2) to (E2);
\draw [Fuchsia, ->] (E2) to (Dir2);
\draw [Fuchsia, ->] (Dir2) to (AS2);
\draw [Fuchsia, ->] (AS2) to (Sec2);
\draw [Fuchsia, ->] (Sec2) to (T2);
\draw (-3,-2.2) ellipse (1.4cm and 2.4cm);
\draw (0,-2.2)[red] ellipse (1.2cm and 2.5cm);
\end{tikzpicture}
\caption{\small Unidirectional flow: If the blue arrows denote identified flows connecting important classes, then the green arrows are constrained by monotonicity to lie between them. \label{fig:WnD}}
\end{minipage}
\label{fig:first-combined}
\end{figure}
As long as information flows \textit{unidirectionally} from colleges to the University, \textit{monotone functions} from the security classes of the college lattice $\ld{C}$ to university security lattice $\rid{U}$ \textit{suffice} to ensure secure information flow.
A function $\rid{\alpha}: \ld{C} \rightarrow \rid{U}$ is called \textit{monotone} if whenever $\ld{sc_1} ~\ld{\sqsubseteq}~ \ld{sc_2}$ in $\ld{C}$ then $\rid{\alpha}(\ld{sc_1}) ~\rd{\sqsubseteq}~ \rid{\alpha}(\ld{sc_2})$ in $\rid{U}$.\footnote{Note that it's not necessary to make the function $\rid{\alpha}$ total or onto.}
Monotonicity also constrains possible flows between classes of the two domains, once certain important flows between certain classes have been identified (see Fig. \ref{fig:WnD}).
Moreover, since monotone functions are closed under composition, one can chain them to create secure \textit{unidirectional} information flow connections through a series of administrative domains.
Monotonicity is a basic principle adopted for information flow analyses, e.g. \cite{liu2017fabric}.
\begin{figure}[t]
\begin{minipage}[t]{.45\textwidth}
\centering
\begin{tikzpicture}[framed,->,node distance=0.9cm,on grid]
\title{W and D}
\node(T2) {\scriptsize$\top 2$};
\node(T1) [xshift=-3cm] {\scriptsize$\top 1$};
\node(Dir1) [below of = T1] {\scriptsize$CollegePrincipal$};
\node(D1) [below left of = Dir1] {\scriptsize$Dean\ (F)$};
\node(F1) [below of = D1] {\scriptsize$Faculty$};
\node(DS1) [xshift=-2.3cm,yshift=-1.7cm] {\scriptsize$Dean\ (S)$};
\node(S1) [below right of = F1] {\scriptsize$Student$};
\node(B1) [below of=S1] {\scriptsize$\bot 1$};
\node(Sec2) [below of = T2] {\scriptsize$Chancellor$};
\node(AS2) [below of=Sec2] {\scriptsize$Vice\ Chancellor$};
\node(Dir2) [below of=AS2] {\scriptsize$Dean(Colleges)$};
\node(E2) [below of=Dir2] {\scriptsize$Univ.Fac.$};
\node(B2) [below of=E2] {\scriptsize$\bot 2$};
\draw [blue, thick] (S1) [bend right = 5] to (E2);
\draw [green, thick] (F1) [bend right = 5] to (E2);
\draw [green, thick] (B1) to (B2);
\draw [green, thick, ->] (T1) to (T2);
\draw [brown, thick, ->] (T2) [bend right=10] to (T1);
\draw [blue, thick] (Dir1) [bend left = 10] to (Dir2);
\draw [green, thick] (D1) to (Dir2);
\draw [green, thick] (DS1) to (Dir2);
\draw [Fuchsia, ->] (Dir1) to (T1);
\draw [Fuchsia, ->] (D1) to (Dir1);
\draw [Fuchsia, ->] (F1) to (D1);
\draw [Fuchsia, ->] (S1) to (F1);
\draw [Fuchsia, ->] (S1) to (DS1);
\draw [Fuchsia, ->] (DS1) to (Dir1);
\draw [Fuchsia, ->] (B1) to (S1);
\draw [Fuchsia, ->] (B2) to (E2);
\draw [Fuchsia, ->] (E2) to (Dir2);
\draw [Fuchsia, ->] (Dir2) to (AS2);
\draw [Fuchsia, ->] (AS2) to (Sec2);
\draw [Fuchsia, ->] (Sec2) to (T2);
\draw [red, thick] (Dir2) [bend right = 10] to (F1);
\draw [brown, thick] (B2) [bend left=20] to (B1);
\draw [brown, thick] (E2) [bend right=10] to (F1);
\draw [brown, thick] (AS2) to (T1);
\draw [brown, thick] (Sec2) to (T1);
\draw (-3,-2.2) ellipse (1.4cm and 2.4cm);
\draw (0,-2.2)[red] ellipse (1.2cm and 2.5cm);
\end{tikzpicture}
\caption{\small
The blue/green and brown/red arrows define monotone functions in each direction.
However, the red arrow highlights a flow that is a security violation.
\label{fig:Bi-WnD}}
\end{minipage}
\quad \quad
\begin{minipage}[t]{.45\textwidth}
\centering
\begin{tikzpicture}[framed,->,node distance=0.9cm,on grid]
\title{W and D}
\node(T2) {\scriptsize$\top 2$};
\node(T1) [xshift=-3cm] {\scriptsize$\top 1$};
\node(Dir1) [below of = T1] {\scriptsize$CollegePrincipal$};
\node(D1) [below left of = Dir1] {\scriptsize$Dean\ (F)$};
\node(F1) [below of = D1] {\scriptsize$Faculty$};
\node(DS1) [xshift=-2.3cm,yshift=-1.7cm] {\scriptsize$Dean\ (S)$};
\node(S1) [below right of = F1] {\scriptsize$Student$};
\node(B1) [below of=S1] {\scriptsize$\bot 1$};
\node(Sec2) [below of = T2] {\scriptsize$Chancellor$};
\node(AS2) [below of=Sec2] {\scriptsize$Vice\ Chancellor$};
\node(Dir2) [below of=AS2] {\scriptsize$Dean(Colleges)$};
\node(E2) [below of=Dir2] {\scriptsize$Univ.Fac.$};
\node(B2) [below of=E2] {\scriptsize$\bot 2$};
\draw [green, thick] (S1) [bend right = 10] to (E2);
\draw [green, thick] (F1) [bend right = 10] to (Dir2);
\draw [green, thick] (B1) to (B2);
\draw [green, thick, ->] (T1) [bend right=10] to (T2);
\draw [brown, thick, ->] (T2) [bend right=10] to (T1);
\draw [green, thick] (Dir1) [bend right= 3] to (Sec2);
\draw [green, thick] (D1) [bend right=15] to (AS2);
\draw [green, thick] (DS1) [bend right=10] to (AS2);
\draw [Fuchsia, ->] (Dir1) to (T1);
\draw [Fuchsia, ->] (D1) to (Dir1);
\draw [Fuchsia, ->] (F1) to (D1);
\draw [Fuchsia, ->] (S1) to (F1);
\draw [Fuchsia, ->] (S1) to (DS1);
\draw [Fuchsia, ->] (DS1) to (Dir1);
\draw [Fuchsia, ->] (B1) to (S1);
\draw [Fuchsia, ->] (B2) to (E2);
\draw [Fuchsia, ->] (E2) to (Dir2);
\draw [Fuchsia, ->] (Dir2) to (AS2);
\draw [Fuchsia, ->] (AS2) to (Sec2);
\draw [Fuchsia, ->] (Sec2) to (T2);
\draw [brown, thick] (B2) [bend left=10] to (S1);
\draw [brown, thick] (Sec2) to (T1);
\draw [brown, thick] (AS2) to (Dir1);
\draw [brown, thick] (E2) [bend left=10] to (F1);
\draw [brown, thick] (Dir2) [bend left=10] to (D1);
\draw (-3,-2.2) ellipse (1.4cm and 2.4cm);
\draw (0,-2.2)[red] ellipse (1.2cm and 2.5cm);
\end{tikzpicture}
\caption{\small
The arrows define a secure and precise connection.
However, the security classification escalates quickly in a few round-trips, when information can flow in both directions.
\label{fig:Bi-WnD-yCC}}
\end{minipage}
\end{figure}
However, when there is ``blowback'' of information, mere monotonicity is \textit{inadequate} for ensuring SIF.
Consider the bidirectional flow situation in Fig. \ref{fig:Bi-WnD}, where data return to the original domain.
Monotonicity of both functions $\rid{\alpha}: \ld{C} \rightarrow \rid{U}$ and $\ld{\gamma}: \rid{U} \rightarrow \ld{C}$ does \textit{not} suffice for security because the composition $\ld{\gamma} \circ \rid{\alpha}$ may \textit{not} be non-decreasing.
In Fig. \ref{fig:Bi-WnD}, both $\rid{\alpha}$ and $\ld{\gamma}$ are monotone but their composition
can lead to information leaking from a higher class, e.g., \textit{College Principal}, to a lower class, e.g., \textit{Faculty} within $\ld{C}$ --- an outright violation of the college's security policy.
Similarly, composition $\rid{\alpha} \circ \ld{\gamma}$ may lead to violation of the University's security policy.
\begin{figure*}[htb]
\centering
\begin{tikzpicture}[->,node distance=1cm,on grid]
\title{Sec cond. 1}
\node(T2) {$\rid{m_2}$};
\node(T1) [xshift=-1.5cm] {$\ld{\gamma}(\rid{m_2})$};
\node(B1) [below of=T1] {$\ld{l_1}$};
\node(B2) [below of=T2] {$\rid{\alpha}(\ld{l_1})$};
\draw [blue, thick] (B1) to (B2);
\draw [Fuchsia, thick] (B2) to [bend right=20] (T2);
\draw [blue, thick] (T2) to (T1);
\draw (-1.5,-0.5) ellipse (0.55cm and 1cm);
\draw (0,-0.5) ellipse (0.55cm and 1cm);
\node(T3) [xshift=2cm] {$\ld{l_2}$};
\node(T4) [xshift=3.5cm] {$\rid{\alpha}(\ld{l_2})$};
\node(B3) [below of=T3] {$\ld{\gamma}(\rid{m_1})$};
\node(B4) [below of=T4] {$\rid{m_1}$};
\draw [blue, thick] (B4) to (B3);
\draw [Fuchsia, thick] (B3) to [bend left=20] (T3);
\draw [blue, thick] (T3) to (T4);
\draw (2,-0.5) ellipse (0.55cm and 1cm);
\draw (3.5,-0.5) ellipse (0.55cm and 1cm);
\end{tikzpicture}
\caption{\small Secure flow conditions: (\textbf{sc1}) $\ld{l_1} \ld{\sqsubseteq} \ld{\gamma}(\rid{m_2})$ ~~(\textbf{sc2}) $\rid{m_1} \rd{\sqsubseteq} \rid{\alpha}(\ld{l_2})$.\label{fig:sec11}}
\end{figure*}
\paragraph{Requirements.} \ \
We want to ensure that any ``round-trip'' flow of information, e.g., from a domain $\ld{L}$ to $\rid{M}$ and back to $\ld{L}$, is a permitted flow in the lattice $\ld{L}$, from where the data originated.
Thus we require the following (tersely stated) ``security conditions'' \textbf{SC1} and \textbf{SC2} on $\rid{\alpha}: \ld{L} \rightarrow \rid{M}$ and $\ld{\gamma}: \rid{M} \rightarrow \ld{L}$, which preclude any violation of the security policies of both the administrative domains
(see Fig. \ref{fig:sec11}):
\[
\textbf{SC1}~~ \lambda \ld{l}.\ld{l} ~\ld{\sqsubseteq}~
\ld{\gamma} \circ \rid{\alpha}
~~~~~~\hfill~~~~~~
\textbf{SC2} ~~ \lambda \rid{m}.\rid{m} ~ \rid{\sqsubseteq}~
\rid{\alpha} \circ \ld{\gamma}
\]
We also desire \textit{precision}, based on a principle of least privilege escalation ---
if data are exchanged between the two domains without any computation done on them, then the security level should not be needlessly raised.
Precision is important for meaningful and useful analyses.
\[
\begin{array}{c}
\textbf{PC1}~~\rid{\alpha}(\ld{l_1}) = \rid{\bigsqcup} ~ \{\rid{m_1} ~|~ \ld{\gamma}(\rid{m_1}) = \ld{l_1} \}, \; \; \forall \ld{l_1} \in \ld{\gamma}[\rid{M}]\\
\textbf{PC2} ~~\ld{\gamma}(\rid{m_1}) = \ld{\bigsqcup} ~ \{\ld{l_1} ~|~ \rid{\alpha}(\ld{l_1}) = \rid{m_1}\}, \; \; \forall \rid{m_1} \in \rid{\alpha}[\ld{L}]
\end{array}
\]
Further, if the data were to go back and forth between two domains more than once, the security classes to which data belong should not become increasingly restrictive after consecutive bidirectional data sharing (See Fig. \ref{fig:Bi-WnD-yCC}, which shows monotone functions that keep climbing up to the top).
This convergence requirement may be stated informally as conditions \textbf{CC1} and \textbf{CC2}, requiring \textit{fixed points} for the compositions $\ld{\gamma} \circ \rid{\alpha}$ and $\rid{\alpha} \circ \ld{\gamma}$.
Since security lattices are finite, \textbf{CC1} and \textbf{CC2} necessarily hold -- such fixed points exist, though perhaps only at the topmost elements of the lattice.
We would therefore desire a stronger requirement, where fixed points are reached
as low in the orderings as possible. \\
\vspace{-0.37cm}
\begin{figure}
\begin{minipage}[t]{.45\textwidth}
\centering
\begin{tikzpicture}[framed,->,node distance=1cm,on grid]
\title{W and D}
\node(T2) {\scriptsize$\top 2$};
\node(T1) [xshift=-3cm] {\scriptsize$\top 1$};
\node(Dir1) [below of = T1] {\scriptsize$CollegePrincipal$};
\node(D1) [below left of = Dir1] {\scriptsize$Dean\ (F)$};
\node(F1) [below of = D1] {\scriptsize$Faculty$};
\node(DS1) [xshift=-2.3cm,yshift=-1.7cm] {\scriptsize$Dean\ (S)$};
\node(S1) [below right of = F1] {\scriptsize$Student$};
\node(B1) [below of=S1] {\scriptsize$\bot 1$};
\node(Sec2) [below of = T2] {\scriptsize$Chancellor$};
\node(AS2) [below of=Sec2] {\scriptsize$Vice\ Chancellor$};
\node(Dir2) [below of=AS2] {\scriptsize$Dean(Colleges)$};
\node(E2) [below of=Dir2] {\scriptsize$Univ.Fac.$};
\node(B2) [below of=E2] {\scriptsize$\bot 2$};
\draw [green, thick] (S1) [bend left = 5] to (Dir2);
\draw [red, thick] (Dir2) [bend left = 5] to (S1);
\draw [red, thick] (F1) to (Dir2);
\draw [green, thick] (B1) to (B2);
\draw [green, thick, ->] (T1) [bend right=10] to (T2);
\draw [brown, thick, ->] (T2) [bend right=10] to (T1);
\draw [green, thick] (Dir1) [bend left= 10] to (AS2);
\draw [red, thick] (D1) to (Dir2);
\draw [red, thick] (DS1) to (Dir2);
\draw [Fuchsia, ->] (Dir1) to (T1);
\draw [Fuchsia, ->] (D1) to (Dir1);
\draw [Fuchsia, ->] (F1) to (D1);
\draw [Fuchsia, ->] (S1) to (F1);
\draw [Fuchsia, ->] (S1) to (DS1);
\draw [Fuchsia, ->] (DS1) to (Dir1);
\draw [Fuchsia, ->] (B1) to (S1);
\draw [Fuchsia, ->] (B2) to (E2);
\draw [Fuchsia, ->] (E2) to (Dir2);
\draw [Fuchsia, ->] (Dir2) to (AS2);
\draw [Fuchsia, ->] (AS2) to (Sec2);
\draw [Fuchsia, ->] (Sec2) to (T2);
\draw [brown, thick] (B2) [bend left=20] to (B1);
\draw [brown, thick] (Sec2) to (T1);
\draw [brown, thick] (AS2) to (Dir1);
\draw [brown, thick] (E2) [bend left=10] to (S1);
\draw (-3,-2.5) ellipse (1.4cm and 2.7cm);
\draw (0,-2.5)[red] ellipse (1.2cm and 2.7cm);
\end{tikzpicture}
\caption{\small
The arrows define a Galois Connection.
However, the red arrows highlight flow security violations when information can flow in both directions.
\label{fig:Bi-WnD-yNotGC}}
\end{minipage}
\quad \quad
\begin{minipage}[t]{.45\textwidth}
\centering
\begin{tikzpicture}[framed,->,node distance=1cm,on grid]
\title{W and D}
\node(T2) {{\color{blue}\scriptsize$\top 2$}};
\node(T1) [xshift=-3cm] {{\color{blue} \scriptsize$\top 1$}};
\node(Dir1) [below of = T1] {{\color{blue}\scriptsize$CollegePrincipal$}};
\node(D1) [below left of = Dir1] {\scriptsize$Dean\ (F)$};
\node(F1) [below of = D1] {\scriptsize$Faculty$};
\node(DS1) [xshift=-2.3cm,yshift=-1.7cm] {\scriptsize$Dean\ (S)$};
\node(S1) [below right of = F1] {{\color{blue}\scriptsize$Student$}};
\node(B1) [below of=S1] {{\color{blue}\scriptsize$\bot 1$}};
\node(Sec2) [below of = T2] {\scriptsize$Chancellor$};
\node[align=center] (AS2) [below of=Sec2] {\scriptsize $Vice$ \\ \scriptsize $Chancellor$};
\node(Dir2) [below of=AS2] {{\color{blue}\scriptsize$Dean(Colleges)$}};
\node(E2) [below of=Dir2] {{\color{blue}\scriptsize$Univ.Fac.$}};
\node(B2) [below of=E2] {{\color{blue}\scriptsize$\bot 2$}};
\draw [black, thick, <->] (F1) to (E2);
\draw [OliveGreen, ->] (S1) to (E2);
\draw [black, thick,<->] (B1) to (B2);
\draw [black, thick, <->] (T1) to (T2);
\draw [black, thick, <->] (Dir1) [bend left = 10] to (Dir2);
\draw [OliveGreen] (D1) to (Dir2);
\draw [OliveGreen] (DS1) to (Dir2);
\draw [OliveGreen] (AS2) to (T1);
\draw [OliveGreen] (Sec2) to (T1);
\draw [Fuchsia, ->] (Dir1) to (T1);
\draw [Fuchsia, ->] (D1) to (Dir1);
\draw [Fuchsia, ->] (F1) to (D1);
\draw [Fuchsia, ->] (S1) to (F1);
\draw [Fuchsia, ->] (S1) to (DS1);
\draw [Fuchsia, ->] (DS1) to (Dir1);
\draw [Fuchsia, ->] (B1) to (S1);
\draw [Fuchsia, ->] (B2) to (E2);
\draw [Fuchsia, ->] (E2) to (Dir2);
\draw [Fuchsia, ->] (Dir2) to (AS2);
\draw [Fuchsia, ->] (AS2) to (Sec2);
\draw [Fuchsia, ->] (Sec2) to (T2);
\draw (-3,-2.5) ellipse (1.4cm and 2.7cm);
\draw (0,-2.5)[red] ellipse (1.2cm and 2.7cm);
\draw [dotted] (0,-1.4) ellipse (0.8cm and 1.2cm);
\draw [dotted] (-3,-1.7) ellipse (1.35cm and 0.3cm);
\draw [dotted] (-3,-3.2) ellipse (0.5cm and 0.55cm);
\end{tikzpicture}
\caption{\small A useful increasing Lagois connection for sharing data.
Black arrows define permissible flows between buds. \label{fig:Bi-L-WnD}}
\end{minipage}
\end{figure}
\paragraph{Galois connections aren't the answer.} \ \
Any discussion on a pair of partial orders linked by a pair of monotone functions suggests the notion of a Galois connection, an elegant and ubiquitous mathematical structure that finds use in computing, particularly in static analyses.
However, Galois connections are not the appropriate structure for bidirectional informational flow control.
Let $\ld{L}$ and $\rid{M}$ be two complete security class lattices, and $\rid{\alpha}: \ld{L} \rightarrow \rid{M}$ and $\ld{\gamma}: \rid{M} \rightarrow \ld{L}$ be two monotone functions such that $(\ld{L}, \rid{\alpha}, \ld{\gamma}, \rid{M})$ forms a Galois connection.
Recall that a Galois connection satisfies the condition
\[
\textbf{GC1}~~
\forall \ld{l_1} \in \ld{L}, \rid{m_1} \in \rid{M},\ \ \ \rid{\alpha}(\ld{l_1}) ~\rd{\sqsubseteq}~ \rid{m_1} ~\iff~
\ld{l_1} ~\ld{\sqsubseteq}~ \ld{\gamma}(\rid{m_1})
\]
So in a Galois connection we have
$\rid{\alpha}(\ld{\gamma}(\rid{m_1})) ~ \rd{\sqsubseteq}~ \rid{m_1} \iff \ld{\gamma}(\rid{m_1}) ~\ld{\sqsubseteq}~ \ld{\gamma}(\rid{m_1})$.
Since $\ld{\gamma}(\rid{m_1}) ~\ld{\sqsubseteq}~ \ld{\gamma}(\rid{m_1})$ holds trivially, we get $\rid{\alpha}(\ld{\gamma}(\rid{m_1}))~ \rd{\sqsubseteq}~ \rid{m_1}$.
If $\rid{\alpha}(\ld{\gamma}(\rid{m_1})) ~\rd{\neq}~ \rid{m_1}$ then $ \rid{\alpha}(\ld{\gamma}(\rid{m_1})) ~\rd{\sqsubset}~ \rid{m_1}$ (strictly), which would violate secure flow requirement \textbf{SC2}. Fig. \ref{fig:Bi-WnD-yNotGC} illustrates such a situation.
\paragraph{Why not Galois insertions?}
Now suppose $\ld{L}$ and $\rid{M}$ are two complete security class lattices, and $\rid{\alpha}: \ld{L} \rightarrow \rid{M}$ and $\ld{\gamma}: \rid{M} \rightarrow \ld{L}$ be two monotone functions such that $(\ld{L}, \rid{\alpha}, \ld{\gamma}, \rid{M})$ forms a \textit{Galois insertion}.
Then the flow of information permitted by $\rid{\alpha}$ and $\ld{\gamma}$ is guaranteed to be secure.
However, Galois insertions mandate conditions on the definitions of functions $\rid{\alpha}$ and $\ld{\gamma}$ that are much too strong, i.e.,
\begin{itemize}
\item $\ld{\gamma}: \rid{M} \rightarrow \ld{L}$ is \textit{injective}, i.e.,
$\forall \rid{m_1,m_2} \in \rid{M} : \ld{\gamma}(\rid{m_1}) ~\ld{=}~ \ld{\gamma}(\rid{m_2}) \implies \rid{m_1} = \rid{m_2}$
\item $\rid{\alpha}: \ld{L} \rightarrow \rid{M}$ is \textit{surjective}, i.e.,
$\forall \rid{m_1} \in \rid{M}, \exists \ld{l_1} \in \ld{L}: \rid{\alpha}(\ld{l_1}) = \rid{m_1}$.
\end{itemize}
Typically data are shared only from a few security classes of any organisation.
Organisations rarely make public their entire security class structure and permitted flow policies.
Organisations also typically do not want any external influences on some subsets of its security classes.
Thus, if not all elements of $\rid{M}$ are transfer classes, it may be impossible to define a Galois insertion
$(\ld{L}, \rid{\alpha}, \ld{\gamma}, \rid{M})$ because we cannot force $\rid{\alpha}$ to be surjective.
\paragraph{Lagois Connections.} \ \
Further, the connection we seek to make between two domains should allow us to transpose them.
Fortunately there is an elegant structure, i.e., \textit{Lagois Connections} \cite{MELTON1994lagoisconnections}, which exactly satisfies this as well as the requirements of security and bidirectional sharing (\textbf{SC1, SC2, PC1, PC2, CC1} and \textbf{CC2}).
They also conveniently generalise Galois Insertions.
\begin{definition}[Lagois Connection \cite{MELTON1994lagoisconnections}]
If $L = (\ld{L},\ld{\sqsubseteq})$ and $M = (\rid{M},\rd{\sqsubseteq})$ are two partially ordered sets, and $\rid{\alpha}: \ld{L} \rightarrow \rid{M}$ and $\ld{\gamma}: \rid{M} \rightarrow \ld{L}$ are order-preserving functions, then we call the quadruple $(\ld{L}, \rid{\alpha}, \ld{\gamma}, \rid{M})$ an {\em increasing} Lagois connection, if it satisfies the following properties:
\[
\begin{array}{llcll}
\textbf{LC1}~~ & \lambda \ld{l}.\ld{l} ~\ld{\sqsubseteq}~
\ld{\gamma} \circ \rid{\alpha}
& ~~~~~~~~~~~ &
\textbf{LC2}~~ & \lambda \rd{m}.\rd{m} ~\rd{\sqsubseteq}~
\rid{\alpha} \circ \ld{\gamma} \\
\textbf{LC3}~~ & \rid{\alpha} \circ \ld{\gamma} \circ \rid{\alpha} = \rid{\alpha}
& ~~~~~~~~~~~ &
\textbf{LC4}~~ & \ld{\gamma} \circ \rid{\alpha} \circ \ld{\gamma} = \ld{\gamma}
\end{array}
\]
\end{definition}
\textbf{LC3} ensures that $\ld{\gamma}(\rid{\alpha}(\ld{c_1}))$ is the least upper bound of all security classes in $\ld{C}$ that are mapped to the same security class, say $\rid{u_1} = \rid{\alpha}(\ld{c_1})$ in $\rid{U}$.
The main result of this section is that if the negotiated monotone functions $\rid{\alpha}$ and $\ld{\gamma}$ form a Lagois connection between the security lattices $\ld{L}$ and $\rid{M}$, then information flows permitted are secure and precise.
\begin{theorem}\label{thm:secureconnection}
Let $\ld{L}$ and $\rid{M}$ be two complete security class lattices, $\rid{\alpha}: \ld{L} \rightarrow \rid{M}$ and $\ld{\gamma}: \rid{M} \rightarrow \ld{L}$ be two monotone functions.
Then the flow of information permitted by $\rid{\alpha}$, $\ld{\gamma}$ satisfies conditions \textbf{SC1}, \textbf{SC2}, \textbf{PC1}, \textbf{PC2}, \textbf{CC1} and \textbf{CC2}
if $(\ld{L}, \rid{\alpha}, \ld{\gamma}, \rid{M})$ is an increasing Lagois connection.
\end{theorem}
\begin{proof}
Condition \textbf{SC1} holds because
if $\rid{\alpha}(\ld{l_1}) ~\rd{\sqsubseteq}~ \rid{m_2}$,
by monotonicity of $\ld{\gamma}$,
$\ld{\gamma}(\rid{\alpha}(\ld{l_1})) ~\ld{\sqsubseteq}~ \ld{\gamma}(\rid{m_2})$.
But by \textbf{LC1},
$\ld{l_1} ~\ld{\sqsubseteq}~ \ld{\gamma} (\rid{\alpha}(\ld{l_1}))$.
So $\ld{l_1} ~\ld{\sqsubseteq}~\ld{\gamma}(\rid{m_2})$.
(A symmetric argument holds for \textbf{SC2}.)
Conditions \textbf{PC1} and \textbf{PC2} are shown in Proposition 3.7 of \cite{MELTON1994lagoisconnections}.
Conditions \textbf{CC1} and \textbf{CC2} hold since the compositions $\ld{\gamma} \circ \rid{\alpha}$ and $\rid{\alpha} \circ \ld{\gamma}$ are \textit{closure} operators, i.e., idempotent, extensive, order-preserving endo-functions on $\ld{L}$ and $\rid{M}$.
\end{proof}
In fact, Lagois connections ensure that information in a security class in the original domain remains accessible even after doing a round-trip from the other domain (Proposition 3.8 in \cite{MELTON1994lagoisconnections}):
\begin{align}
\ld{\gamma}(\rid{\alpha}(\ld{l})) ~=~
\ld{\sqcap} \{ \ld{l^*} \in \ld{\gamma}[\rid{M}] ~|~
\ld{l} ~\ld{\sqsubseteq}~ \ld{l^*} \}, \label{TIGHT1} \\
\rid{\alpha}( \ld{\gamma}(\rid{m})) ~=~
\rid{\sqcap} \{ \rid{m^*} \in \rid{\alpha}[\ld{L}] ~|~
\rid{m} ~\rd{\sqsubseteq}~ \rid{m^*} \}.
\label{TIGHT2}
\end{align}
We list various propositions about Lagois connections, which illustrate some of their important properties.
In particular, the two functions $\ld{\gamma}$ and $\rid{\alpha}$ uniquely determine each other. Moreover, there are largest members of their pre-images, which act as representatives for the equivalence classes of the equivalence relations $\rid{\thicksim_M}$ and $\ld{\thicksim_L}$ induced by these functions.
Further, since our security domains are complete lattices, these distinguished points are closed under meets.
\begin{proposition}[Proposition 3.7 in \cite{MELTON1994lagoisconnections}]
Let $(\ld{L}, \rid{\alpha}, \ld{\gamma}, \rid{M})$ be a Lagois connection and let $\rid{m} \in \rid{\alpha}[\ld{L}]$ and $\ld{l} \in \ld{\gamma}[\rid{M}]$. The $\rid{\alpha}^{-1}(\rid{m})$ has a largest member, which is $\ld{\gamma}(\rid{m})$, and $\ld{\gamma}^{-1}(\ld{l})$ has a largest member, which is $\rid{\alpha}(\ld{l})$.
\end{proposition}
This means that for all $\rid{m} \in \rid{\alpha}[\ld{L}]$ and $\ld{l} \in \ld{\gamma}[\rid{M}]$, $\ld{\gamma}(\rid{m})$ and $\rid{\alpha}(\ld{l})$ exist, and are defined by largest members in their pre-image. Also, the images $\ld{\gamma}[\rid{M}]$ and $\rid{\alpha}[\ld{L}]$ are isomorphic lattices.
$\rid{M^*} = \rid{\alpha}[\ld{\gamma}[\rid{M}]] = \rid{\alpha}[\ld{L}]$ defines a system of representatives for $\rid{\thicksim_M}$. Then, $\rid{\alpha}(\ld{\gamma}(\rd{m}))$ is the representative of the equivalence class $[\rd{m}]$ of $\rd{m}$ that lies in $\rid{M^*}$, called \textit{budpoint}, such that,
\begin{align}
\textit{if}~ \rid{m}\in \rid{M} ~\textit{and}~ \rid{m^*} \in \rid{M^*} ~\textit{with}~ \rid{m} ~\rid{\thicksim_M}~ \rid{m^*}~\textit{then}~ \rid{m} ~\rd{\sqsubseteq}~ \rid{m^*}
\end{align}
Symmetrically, $\ld{L^*} = \ld{\gamma}[\rid{\alpha}[\ld{L}]] = \ld{\gamma}[\rid{M}]$ defines a system of representatives for $\ld{\thicksim_L}$.
\begin{proposition}[Proposition 3.9 in \cite{MELTON1994lagoisconnections}]
If $(\ld{L}, \rid{\alpha}, \ld{\gamma}, \rid{M})$ is a Lagois connection, then the functions $\rid{\alpha}$ and
$\ld{\gamma}$ uniquely determine each other; in fact
\begin{align}
\ld{\gamma}(\rid{m}) ~=~ \ld{\bigsqcup}~ \rid{\alpha}^{-1}~[~\rid{\sqcap} \{~\rid{m^*} \in \rid{\alpha}[\ld{L}]~|~ \rid{m} ~\rd{\sqsubseteq}~ \rid{m^*}\} ~]\\
\rid{\alpha}(\ld{l}) ~=~
\rid{\bigsqcup}~ \ld{\gamma}^{-1} [~\ld{\sqcap} \{~ \ld{l^*} \in \ld{\gamma}~[~\rid{M}] ~|~
\ld{l} ~\ld{\sqsubseteq}~ \ld{l^*} \} ~]
\end{align}
\end{proposition}
\begin{proposition}[Proposition 3.11 in \cite{MELTON1994lagoisconnections}]
\label{prop:meet-existence}
If $(\ld{L}, \rid{\alpha}, \ld{\gamma}, \rid{M})$ is a Lagois connection and $\ld{A} \subseteq \ld{\gamma}[\rid{M}]$, then
\begin{enumerate}
\item the meet of $\ld{A}$ in $\ld{\gamma}[\rid{M}]$ exists if and only if the meet of $\ld{A}$ in $\ld{L}$ exists, and whenever either exists, they are equal.
\item the join $\ld{\hat{a}}$ of $\ld{A}$ in $\ld{\gamma}[\rid{M}]$ exists if the join $\ld{\check{a}}$ of $\ld{A}$ in $\ld{L}$ exists, and in this case $\ld{\hat{a}} = \ld{\gamma}(\rid{\alpha}(\ld{\check{a}}))$
\end{enumerate}
\end{proposition}
\section{An Operational Model}\label{sec:model}
\subsection{Computational Model.}\label{sec:operations}
Let us consider two different organisations $\ld{L}$ and $\rid{M}$ that want to share data with each other.
We start with the assumptions that the two domains comprise storage objects $\ld{Z}$ and $\rd{Z}$ respectively, which are manipulated using their own sets of \textit{atomic} transactional operations, ranged over by $\ld{t}$ and $\rd{t}$ respectively.
We further assume that these transactions within each domain are internally secure with respect to their flow models, and have no insecure or interfering interactions with the environment.
Thus, we are agnostic to the level of abstraction of the systems we aim to connect securely, and since our approach treats the application domains as ``black boxes'', it is readily adaptable to any level of discourse (language, system, OS, database) found in the security literature.
We extend these operations with a minimal set of operations to transfer data between the two domains.
To avoid any concurrency effects, interference or race conditions arising from inter-domain transfer, we augment the storage objects of both domains with a fresh set of \textit{export} and \textit{import} variables into/from which the data of the domain objects can be copied \textit{atomically}.
We designate these sets $\ld{X}, \rd{X}$ as the respective \textit{export} variables, and $\ld{Y}, \rd{Y}$ as the respective \textit{import} variables, with the corresponding variable instances written as $\ld{x_i}$, $\rd{x_i}$ and $\ld{y_i}$, $\rd{y_i}$.
These export and import variables form mutually disjoint sets, and are distinct from any extant domain objects manipulated by the applications within a domain.
These variables are used exclusively for transfer, and are manipulated atomically.
We let $\ld{w_i}$ range over all variables in $\ld{N} ~=~\ld{Z} \cup \ld{X} \cup \ld{Y}$ (respectively $\rd{w_i}$ over $\rd{N} ~=~ \rd{Z} \cup \rd{X} \cup \rd{Y}$).
Domain objects are copied \textit{to export} variables and \textit{from import} variables by special operations $\ld{rd}(\ld{z}, \ld{y})$ and $\ld{wr}(\ld{x}, \ld{z})$ (and $\rd{rd}(\rd{z}, \rd{y})$ and $\rd{wr}(\rd{x}, \rd{z})$ in the other domain).
We assume \textit{atomic transfer} operations (\textit{trusted by both domains}) $T_{RL}, T_{LR}$ that copy data from the export variables of one domain to the import variables of the other domain as the only mechanism for inter-domain flow of data.
Let ``phrase'' $p$ denote a command in either domain or a transfer operation, and let $s$ be any (empty or non-empty) sequence of phrases.
\[
\begin{array}{c}
\text{(command)} ~~~
\ld{c} ::= \ld{t} ~|~ \ld{rd}(\ld{z}, \ld{y}) ~|~ \ld{wr}(\ld{x}, \ld{z}) ~~~~~~
\rd{c} ::=
\rd{t} ~|~ \rd{rd}(\rd{z}, \rd{y}) ~|~ \rd{wr}(\rd{x}, \rd{z}) \\
\text{(phrase)} ~~~ p ::= T_{RL}(\rd{x},\ld{y}) ~|~ T_{LR}(\ld{x},\rd{y}) ~|~
\ld{c} ~|~ \rd{c} ~~~\hfill~~~ \text{(seq)}~~~ s ::= \epsilon ~|~ s_1 ; p \\
\end{array}
\]
\begin{figure}
\[
\begin{array}{c}
\inferrule* [Left = \ld{T}]{\ld{\mu} \vdash \ld{t} \Rightarrow \ld{\nu}
}{\langle \ld{\mu},\rd{\mu} \rangle \vdash \ld{t} \Rightarrow \langle \ld{\nu}, \rd{\mu} \rangle}
~~~~\hfill~~~~
\inferrule* [Left = \rd{T}]{\rd{\mu} \vdash \rd{t} \Rightarrow \rd{\nu}
}{\langle \ld{\mu},\rd{\mu} \rangle \vdash \rd{t} \Rightarrow \langle \ld{\mu}, \rd{\nu} \rangle} \\[1ex]
\inferrule*[Left = \ld{Wr}]
}{\langle \ld{\mu},\rd{\mu} \rangle \vdash \ld{wr}(\ld{x},\ld{z}) \Rightarrow \langle \ld{\mu}[\ld{x} := \ld{\mu}(\ld{z})], \rd{\mu} \rangle}
\\
\inferrule*[Left = \rd{Wr}]
}{\langle \ld{\mu},\rd{\mu} \rangle \vdash \rd{wr}(\rd{x},\rd{z}) \Rightarrow \langle \ld{\mu}, \rd{\mu}[\rd{x} := \rd{\mu}(\rd{z})] \rangle}
\\[1ex]
\inferrule*[Left = \ld{Rd}]
}{\langle \ld{\mu},\rd{\mu} \rangle \vdash \ld{rd}(\ld{z},\ld{y}) \Rightarrow \langle \ld{\mu}[\ld{z} := \ld{\mu}(\ld{y})], \rd{\mu} \rangle} \\
\inferrule*[Left = \rd{Rd}]
}{\langle \ld{\mu},\rd{\mu} \rangle \vdash \rd{rd}(\rd{z},\rd{y}) \Rightarrow \langle \ld{\mu}, \rd{\mu}[\rd{z} := \rd{\mu}(\rd{y})] \rangle} \\[1ex]
\inferrule*[Left = Trl]
}{\langle \ld{\mu},\rd{\mu} \rangle \vdash T_{RL}(\ld{y},\rd{x}) \Rightarrow
\langle \ld{\mu}[\ld{y} := \rd{\mu}(\rd{x})],\rd{\mu} \rangle} \\
\inferrule*[Left = Tlr]
}{\langle \ld{\mu},\rd{\mu} \rangle \vdash T_{LR}(\rd{y},\ld{x}) \Rightarrow
\langle \ld{\mu},\rd{\mu}[\rd{y} := \ld{\mu}(\ld{x})] \rangle} \\[1ex]
\inferrule* [Left = Seq0]{}{
\langle \ld{\mu},\rd{\mu} \rangle \vdash \epsilon \Rightarrow^* \langle \ld{\mu},\rd{\mu} \rangle} \\
\inferrule* [Left = Seqs]{\langle \ld{\mu},\rd{\mu} \rangle \vdash s_1 \Rightarrow^* \langle \ld{\mu_1},\rd{\mu_1} \rangle, ~~~
\langle \ld{\mu_1},\rd{\mu_1} \rangle \vdash p \Rightarrow \langle \ld{\mu_2},\rd{\mu_2} \rangle}{\langle \ld{\mu},\rd{\mu} \rangle \vdash s_1 ; p \Rightarrow^* \langle \ld{\mu_2},\rd{\mu_2} \rangle}
\end{array}
\]
\caption{Execution Rules}
\label{fig:evalrules11}
\end{figure}
A \textit{store} (typically $\ld{\mu}, \ld{\nu}, \rd{\mu}, \rd{\nu}$) is a finite-domain function from variables to a set of values (not further specified).
We write, e.g., $\ld{\mu}(\ld{w})$ for the contents of the store $\ld{\mu}$ at variable $\ld{w}$, and $\ld{\mu}[\ld{w} := \rd{\mu}(\rd{w})]$ for the store
that is the same as $\ld{\mu}$ everywhere except at variable $\ld{w}$,
where it now takes value $\rd{\mu}(\rd{w})$.
The rules specifying execution of commands are given in Fig. \ref{fig:evalrules11}.
Assuming the specification of intradomain transactions of the form
$\ld{\mu} \vdash \ld{t} \implies \ld{\nu}$ and $\rd{\mu} \vdash \rd{t} \implies \rd{\nu}$, our rules allow us to specify judgments of the form
$\langle \ld{\mu}, \rd{\mu} \rangle \vdash p \implies \langle \ld{\nu}, \rd{\nu} \rangle$ for phrases, and (the reflexive-transitive closure) for sequences of phrases.
Note that phrase execution occurs \textit{atomically}, and the intra-domain transactions, as well as copying to and from the export/import variables affect the store in only one domain, whereas the \textit{atomic transfer} is only between export variables of one domain and the import variables of the other.
\subsection{Typing Rules}\label{sec:typing}
Let the two domains have the respective different IFMs:
\[ FM_L = \langle \ld{N}, \ld{P}, \ld{SC}, \ld{\sqcup}, \ld{\sqsubseteq} \rangle ~~~\hfill~~~ FM_M = \langle \rd{N}, \rd{P}, \rd{SC}, \rid{\sqcup}, \rd{\sqsubseteq} \rangle, \]
such that the flow policies in both are defined over different sets of security classes $\ld{SC}$ and $\rd{SC}$.\footnote{Without loss of generality, we assume that $\ld{SC} \cap \rd{SC} = \emptyset$, since we can suitably rename security classes.}
The (security) types of the core language are as follows.
Metavariables $\ld{l}$ and $\rd{m}$ range over the sets of security classes, $\ld{SC}$ and $\rd{SC}$ respectively, which are partially ordered by $\ld{\sqsubseteq}$ and $\rd{\sqsubseteq}$.
A type assignment $\ld{\lambda}$ is a finite-domain function from variables $\ld{N}$ to $\ld{SC}$ (respectively, $\rd{\lambda}$ from $\rd{N}$ to $\rd{SC}$).
The important restriction we place on $\ld{\lambda}$
and $\rd{\lambda}$ is that they map export and import variables $\ld{X}, \rid{X}, \rd{Y}, \rid{Y}$ only to points in the security lattices $\ld{SC}$ and $\rd{SC}$ respectively which are in the domains of $\ld{\gamma}$ and $\rid{\alpha}$, i.e., these points participate in the Lagois connection.
Intuitively, a variable $w$ mapped to security class $\ld{l}$ can store information of security class $\ld{l}$ or lower.
The type system works with respect to given type assignment. Given the security level, e.g., $\ld{l}$, the typing rules track for each command \textit{within that domain} whether all written-to variables in that domain are of security classes ``above'' $\ld{l}$, and additionally for transactions within a domain, they ensure ``simple security'', i.e., that all variables which may have been read belong to security classes ``below'' $\ld{l}$.
We assume for the transactions within a domain, e.g., $\ld{L}$, we have a type system that will give us judgments of the form $\ld{\lambda} \vdash \ld{c}: \ld{l}$.
The novel extension of our approach is to extend this framework to work over two connected domains, i.e., given implicit security levels of the contexts in the respective domains.
Cross-domain transfers will require pairing such judgments, and thus our type systems will have judgments of the form
\[ \langle \ld{\lambda}, \rd{\lambda} \rangle \vdash p: \langle
\ld{l}, \rd{m} \rangle \]
We introduce a syntax-directed set of typing rules for the core language, given in Fig. \ref{fig:syntax-type-rules11}.
In many of the rules, the type for one of the domains is not constrained by the rule, and so any suitable type may be chosen as determined by the context, e.g., $\rd{m}$ in the rules \ld{\sc Tt},
\ld{\sc Trd}, \ld{\sc Twr} and $\ld{TT_{RL}}$,
and both $\ld{l}$ and $\rd{m}$ in {\sc Com0}.
\begin{figure}[t]
\[
\begin{array}{l}
\inferrule*[Left = \ld{Tt}]{}{
\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash \ld{t}: \langle \ld{l}, \rd{m} \rangle ~\text{if for all $\ld{z}$ assigned in $\ld{t}$, }
\ld{l} ~\ld{\sqsubseteq}~ \ld{\lambda}(\ld{z}) \\
~~~~~~~~~~~~~~~~~~~~\hfill
\text{ \& for all } \ld{z_1} \text{ read in $\ld{t}$, }
\ld{\lambda}(\ld{z_1}) ~\ld{\sqsubseteq}~ \ld{l}}
\\
\inferrule*[Left = \rd{Tt}]{}{
\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash \rd{t}: \langle \ld{l}, \rd{m} \rangle ~\text{if for all $\rd{z}$ assigned in $\rd{t}$, }
\rd{m} ~\rd{\sqsubseteq}~ \rd{\lambda}(\rd{z}) \\
~~~~~~~~~~~~~~~~~~~~\hfill
\text{ \& for all } \rd{z_1} \text{ read in $\rd{t}$, }
\rd{\lambda}(\rd{z_1}) ~\rd{\sqsubseteq}~ \rd{m}}
\\[1ex]
\inferrule*[Left = \ld{Trd}]{\ld{\lambda}(\ld{y}) ~\ld{\sqsubseteq}~ \ld{\lambda}(\ld{z})}{
\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash \ld{rd}(\ld{z},\ld{y}):
\langle \ld{\lambda}(\ld{z}), \rd{m} \rangle
} \\
\inferrule*[Left = \rd{Trd}]{\rd{\lambda}(\rd{y}) ~\rd{\sqsubseteq}~ \rd{\lambda}(\rd{z})}{
\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash \rd{rd}(\rd{z},\rd{y}):
\langle \ld{l}, \rd{\lambda}(\rd{z}) \rangle
} \\[1ex]
\inferrule*[Left = \ld{Twr}]{\ld{\lambda}(\ld{z}) ~\ld{\sqsubseteq}~ \ld{\lambda}(\ld{x})}{
\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash \ld{wr}(\ld{x},\ld{z}):
\langle \ld{\lambda}(\ld{x}), \rd{m} \rangle} \\
\inferrule*[Left = \rd{Twr}]{\rd{\lambda}(\rd{z}) ~\rd{\sqsubseteq}~ \rd{\lambda}(\rd{x})}{
\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash \rd{wr}(\rd{x},\rd{z}):
\langle \ld{l}, \rd{\lambda}(\rd{x}) \rangle} \\[1ex]
\inferrule*[Left = \ld{$TT_{RL}$}]{
\ld{\gamma}(\rd{\lambda}(\rd{x})) ~\ld{\sqsubseteq}~ \ld{\lambda}(\ld{y})
}{
\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash T_{RL}(\ld{y},\rd{x}):
\langle \ld{\lambda}(\ld{y}), \rd{\lambda}(\rd{x}) \rangle} \\
\inferrule*[Left = \rd{$TT_{LR}$}]{
\rid{\alpha}(\ld{\lambda}(\ld{x})) ~\rd{\sqsubseteq}~ \rd{\lambda}(\rd{y})
}{
\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash T_{LR}(\rd{y},\ld{x}):
\langle \ld{\lambda}(\ld{x}), \rd{\lambda}(\rd{y}) \rangle }\\[1ex]
\inferrule*[Left = Com0]{}{
\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash \epsilon: \langle \ld{l}, \rd{m} \rangle} \\
\inferrule*[Left = ComP]{
\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash p: \langle \ld{l_1}, \rd{m_1} \rangle
~~~~~\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash s: \langle \ld{l}, \rd{m} \rangle
}{\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash s; p:
\langle \ld{l_1} \ld{\sqcap} \ld{l} , \rd{m_1} \rid{\sqcap} \rd{m} \rangle}
\end{array}
\]
\caption{Typing rules}
\label{fig:syntax-type-rules11}
\end{figure}
For transactions e.g., $\ld{t}$ entirely within domain $\ld{L}$, the typing rule \ld{\sc Tt} constrains the type in the left domain to be at a level $\ld{l}$ that dominates all variables read in
$\ld{t}$, and which is dominated by all variables written to in $\ld{t}$, but places no constraints on the type $\rd{m}$ in the other domain $\rid{M}$.
In the rule \ld{\sc Trd}, since a value in import variable $\ld{y}$ is copied to the variable $\ld{z}$, we have $\ld{\lambda}(\ld{y}) ~\ld{\sqsubseteq}~ \ld{\lambda}(\ld{z})$, and the type in the domain $\ld{L}$ is $\ld{\lambda}(\ld{z})$ with no constraint on the type $\rd{m}$ in the other domain.
Conversely, in the rule \ld{\sc Twr}, since a value in variable $\ld{z}$ is copied to the export variable $\ld{x}$, we have $\ld{\lambda}(\ld{z}) ~\ld{\sqsubseteq}~ \ld{\lambda}(\ld{x})$, and the type in the domain $\ld{L}$ is $\ld{\lambda}(\ld{x})$ with no constraint on the type $\rd{m}$ in the other domain.
In the rule $\ld{TT_{RL}}$, since the contents of
a variable $\rd{x}$ in domain $\rid{M}$ are copied into a variable $\ld{y}$ in domain $\ld{L}$, we require $\ld{\gamma}(\rd{\lambda}(\rd{x})) ~\ld{\sqsubseteq}~ \ld{\lambda}(\ld{y})$, and constrain the type in domain $\ld{L}$ to $\ld{\lambda}(\ld{y})$.
The constraint in the other domain is unimportant (but for the sake of convenience, we peg it at $\rd{\lambda}(\rd{x})$).
Finally, for the types of sequences of phrases, we take the meets of the collected types in each domain respectively, so that we can guarantee that no variable of type lower than these meets has been written into during the sequence.
Note that Proposition \ref{prop:meet-existence} ensures that these types have the desired properties for participating in the Lagois connection.
\subsection{Soundness} \label{sec:soundness}
We now establish soundness of our scheme by showing a non-interference theorem with respect to operational semantics and the type systems built on the security lattices.
This theorem may be viewed as a conservative adaptation (to a minimal secure data transfer framework in a Lagois-connected pair of domains) of the main result of Volpano \textit{et al} \cite{DBLP:journals/jcs/VolpanoIS96}.
We assume that underlying base transactional languages in each of the domains have the following simple property (stated for $\ld{L}$, but an analogous property is assumed for $\rid{M}$):
Within each transaction $\ld{t}$, for each assignment of an expression $\ld{e}$ to any variable $\ld{z}$, the following holds:
If $\ld{\mu}$, $\ld{\nu}$ are two stores such that
for all $\ld{w} \in vars(\ld{e})$, we have $\ld{\mu}(\ld{w}) = \ld{\nu}(\ld{w})$, then after executing the assignment, we will get $\ld{\mu}(\ld{z}) = \ld{\nu}(\ld{z})$.
That is, if two stores are equal for all variables appearing in the expression $\ld{e}$, then the value assigned to the variable $\ld{z}$ will be the same.
This assumption plays the r\^{o}le of ``Simple Security'' of expressions in \cite{DBLP:journals/jcs/VolpanoIS96} in the proof of the main theorem.
The type system plays the r\^{o}le of ``Confinement''.
We start with two obvious lemmas about the operational semantics, namely preservation of domains, and a ``frame'' lemma:
\begin{lemma}[Domain preservation]\label{lemma:equaldomainoneval}
If $\langle \ld{\mu}, \rd{\mu} \rangle \vdash s \Rightarrow^* \langle \ld{\mu_1}, \rd{\mu_1} \rangle$, then $dom(\ld{\mu}) = dom(\ld{\mu_1})$, and $dom(\rd{\mu}) = dom(\rd{\mu_1})$.
\end{lemma}
\begin{proof}
By induction on the length of the derivation of $\langle \mu, \rd{\mu} \rangle \vdash s \Rightarrow^* \langle \mu_1, \rd{\mu_1} \rangle$.
\end{proof}
\begin{lemma}[Frame]\label{lemma:notassigned}
If $\langle \mu, \rd{\mu} \rangle \vdash s \Rightarrow^* \langle \mu_1, \rd{\mu_1} \rangle, w \in dom(\mu) \cup dom(\rd{\mu})$, and $w$ is \textit{not} assigned to in $s$, then $\ld{\mu}(w) = \ld{\mu_1}(w)$ and
$\rd{\mu}(w) = \rd{\mu_1}(w)$.
\end{lemma}
\begin{proof}
By induction on the length of the derivation of $\langle \ld{\mu}, \rd{\mu} \rangle \vdash s \Rightarrow^* \langle \ld{\mu_1}, \rd{\mu_1} \rangle$.
\end{proof}
The main result of the paper assumes an ``adversary'' that operates at a security level $\ld{l}$ in domain $\ld{L}$ and at security level $\rd{m}$ in domain $\rid{M}$.
Note however, that these two levels are interconnected by the monotone functions $\rid{\alpha}: \ld{L} \rightarrow \rid{M}$ and
$\ld{\gamma}: \rid{M} \rightarrow \ld{L}$, since these levels are connected by the ability of information at one level in one domain to flow to the other level in the other domain.
\begin{theorem}[Type Soundness]\label{thm:soundness}
Suppose $\ld{l}, \rd{m}$ are the ``adversarial'' type levels in the respective domains, which satisfy the condition $\ld{l} = \ld{\gamma}(\rd{m})$ and
$\rd{m} = \rid{\alpha}(\ld{l})$.
Let
\begin{enumerate}[label=(\alph*)]
\item \label{assume:1}$\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash s:
\langle \ld{l_0}, \rd{m_0}\rangle$;
\item \label{assume:2}$\langle \ld{\mu}, \rd{\mu} \rangle \vdash s \Rightarrow^* \langle \ld{\mu_f}, \rd{\mu_f} \rangle$;
\item \label{assume:3}$\langle \ld{\nu}, \rd{\nu} \rangle \vdash s \Rightarrow^* \langle \ld{\nu_f}, \rd{\nu_f} \rangle$;
\item \label{assume:4}$dom(\ld{\mu}) = dom(\ld{\nu}) = dom(\ld{\lambda})$ and
$dom(\rd{\mu}) = dom(\rd{\nu}) = dom(\rd{\lambda})$;
\item \label{assume:5} $\ld{\mu}(\ld{w}) = \ld{\nu}(\ld{w})$ for all
$\ld{w}$ such that $\ld{\lambda}(\ld{w}) ~\ld{\sqsubseteq}~ \ld{l}$, and
$\rd{\mu}(\rd{w}) = \rd{\nu}(\rd{w})$ for all
$\rd{w}$ such that $\rd{\lambda}(\rd{w}) ~\rd{\sqsubseteq}~ \rd{m}$.
\end{enumerate}
Then
$\ld{\mu_f}(\ld{w}) = \ld{\nu_f}(\ld{w})$ for all
$\ld{w}$ such that $\ld{\lambda}(\ld{w}) ~\ld{\sqsubseteq}~ \ld{l}$, and
$\rd{\mu_f}(\rd{w}) = \rd{\nu_f}(\rd{w})$ for all
$\rd{w}$ such that $\rd{\lambda}(\rd{w}) ~\rd{\sqsubseteq}~ \rd{m}$.
\begin{comment}
Then, $\langle \nu_f, \rd{\nu_f}] (w'') = [\mu_f, \rd{\mu_f}] (w'')$ for all w'' such that $[\lambda, \rd{\lambda}] (w'') \sqsubseteq (\tau \sqcup\ \gamma(\rd{\tau}))$ or $[\lambda, \rd{\lambda}] (w'') \sqsubseteq (\rd{\tau} \sqcup \alpha(\tau))$, where $w'' \in x,y,z,\rd{x},\rd{y},\rd{z}$
\end{comment}
\end{theorem}
\begin{proof}
By induction on the length of sequence $s$.
The base case is vacuously true.
We now consider a sequence $s_1; p$.
$\langle \ld{\mu},\rd{\mu} \rangle \vdash s_1 \Rightarrow^* \langle \ld{\mu_1},\rd{\mu_1} \rangle$ and
$\langle \ld{\mu_1},\rd{\mu_1} \rangle \vdash p \Rightarrow \langle \ld{\mu_f},\rd{\mu_f} \rangle$
and
$\langle \ld{\nu},\rd{\nu} \rangle \vdash s_1 \Rightarrow^* \langle \ld{\nu_1},\rd{\nu_1} \rangle$ and
$\langle \ld{\nu_1},\rd{\nu_1} \rangle \vdash p \Rightarrow \langle \ld{\nu_f},\rd{\nu_f} \rangle$
By induction hypothesis applied to $s_1$, we have
$\ld{\mu_1}(\ld{w}) = \ld{\nu_1}(\ld{w})$ for all
$\ld{w}$ such that $\ld{\lambda(\ld{w})} ~\ld{\sqsubseteq}~\ld{l}$,
and
$\rd{\mu_1}(\rd{w}) = \rd{\nu_1}(\rd{w})$ for all
$\rd{w}$ such that $\rd{\lambda}(\rd{w}) ~\rd{\sqsubseteq}~ \rd{m}$.
Let $\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash s_1: \langle \ld{l_s}, \rd{m_s} \rangle$, and $\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash p: \langle \ld{l_p}, \rd{m_p} \rangle$.
We examine four cases for $p$ (the remaining cases are symmetrical). \\
\textbf{Case} $p$ is $\ld{t}$: \ \ \
Consider any $\ld{w}$ such that
$\ld{\lambda}(\ld{w}) ~\ld{\sqsubseteq}~\ld{l}$.
If $\ld{w} \in \ld{X} \cup \ld{Y}$ (i.e., it doesn't appear in $\ld{t}$), or if $\ld{w} \in \ld{Z}$ but is not assigned to in $\ld{t}$, then
by Lemma \ref{lemma:notassigned} and the induction hypothesis, $\ld{\mu_f}(\ld{w}) =
\ld{\mu_1}(\ld{w}) = \ld{\nu_1}(\ld{w}) =
\ld{\nu_f}(\ld{w})$. \\
Now suppose $\ld{z}$ is assigned to in $\ld{t}$.
From the condition $\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash p: \langle \ld{l_p}, \rd{m_p} \rangle$, we know that
for all $\ld{z_1}$ assigned in $\ld{t}$, $\ld{l_p} ~\ld{\sqsubseteq}~ \ld{\lambda}(\ld{z_1})$ and
for all $\ld{z_1}$ read in $\ld{t}$, $\ld{\lambda}(\ld{z_1}) ~\ld{\sqsubseteq}~ \ld{l_p}$.
Now if $\ld{l} \ld{~\sqsubseteq}~ \ld{l_p}$, then since in $\ld{t}$ no variables $\ld{z_2}$ such that
$\ld{\lambda}(\ld{z_2}) ~\ld{\sqsubseteq}~\ld{l}$ are assigned to.
Therefore by Lemma \ref{lemma:notassigned}, $\ld{\mu_f}(\ld{w}) = \ld{\mu_1}(\ld{w}) = \ld{\nu_1}(\ld{w}) = \ld{\nu_f}(\ld{w})$, for all $\ld{w}$ such that $\ld{\lambda}(\ld{w}) ~\ld{\sqsubseteq}~ \ld{l}$. \\
If $\ld{l_p} \ld{~\sqsubseteq}~ \ld{l}$, then for all $\ld{z_1}$ read in $\ld{t}$, $\ld{\lambda}(\ld{z_1}) ~\ld{\sqsubseteq}~ \ld{l_p}$.
Therefore, by assumption on transaction $\ld{t}$, if any variable $\ld{z}$ is assigned an expression $\ld{e}$,
since $\ld{\mu_1}$, $\ld{\nu_1}$ are two stores such that
for all $\ld{z_1} \in \ld{Z_e} = vars(\ld{e})$, $\ld{\mu_1}(\ld{z_1}) = \ld{\nu_1}(\ld{z_1})$, the value of $\ld{e}$ will be the same.
By this simple security argument, after the transaction $\ld{t}$, we have $\ld{\mu_f}(\ld{z}) = \ld{\nu_f}(\ld{z})$.
Since the transaction happened entirely and atomically in domain $\ld{L}$, we do not have to worry ourselves with changes in the other domain
$\rid{M}$, and do not need to concern ourselves with
the adversarial level $\rd{m}$.\\[1ex]
\textbf{Case} $p$ is $\ld{rd}(\ld{z},\ld{y})$: \ \ \
Thus $\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash \ld{rd}(\ld{z},\ld{y}):
\langle \ld{\lambda}(\ld{z}), \rd{m} \rangle$,
which means $\ld{\lambda}(\ld{y}) ~\ld{\sqsubseteq}~ \ld{\lambda}(\ld{z})$.
If $\ld{l} ~\ld{\sqsubseteq}~ \ld{\lambda}(\ld{z})$,
there is nothing to prove (Lemma \ref{lemma:notassigned}, again).
If $\ld{\lambda}(\ld{z}) ~\ld{\sqsubseteq}~ \ld{l}$, then since by I.H., $\ld{\mu_1}(\ld{y}) = \ld{\nu_1}(\ld{y})$, we have $\ld{\mu_f}(\ld{z}) =
\ld{\mu_1}[\ld{z} := \ld{\mu_1}(\ld{y})](\ld{z}) =
\ld{\nu_1}[\ld{z} := \ld{\nu_1}(\ld{y})](\ld{z}) =
\ld{\nu_f}(\ld{z})$. \\[1ex]
\textbf{Case} $p$ is $\ld{wr}(\ld{x},\ld{z})$: \ \ \
Thus $\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash \ld{wr}(\ld{x},\ld{z}):
\langle \ld{\lambda}(\ld{x}), \rd{m} \rangle$,
which means $\ld{\lambda}(\ld{z}) ~\ld{\sqsubseteq}~ \ld{\lambda}(\ld{x})$.
If $\ld{l} ~\ld{\sqsubseteq}~ \ld{\lambda}(\ld{x})$,
there is nothing to prove (Lemma \ref{lemma:notassigned}, again).
If $\ld{\lambda}(\ld{x}) ~\ld{\sqsubseteq}~ \ld{l}$, then since by I.H., $\ld{\mu_1}(\ld{z}) = \ld{\nu_1}(\ld{z})$, we have $\ld{\mu_f}(\ld{x}) =
\ld{\mu_1}[\ld{x} := \ld{\mu_1}(\ld{z})](\ld{x}) =
\ld{\nu_1}[\ld{x} := \ld{\nu_1}(\ld{z})](\ld{x}) =
\ld{\nu_f}(\ld{x})$. \\[1ex]
\textbf{Case} $p$ is $T_{RL}(\ld{y},\rd{x})$: \ \ \
So $\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash T_{RL}(\ld{y},\rd{x}):
\langle \ld{\lambda}(\ld{y}), \rd{\lambda}(\rd{x}) \rangle$, and
$\ld{\gamma}(\rd{\lambda}(\rd{x})) ~\ld{\sqsubseteq}~ \ld{\lambda}(\ld{y})$.
If $\ld{l} ~\ld{\sqsubseteq}~ \ld{\lambda}(\ld{y})$,
there is nothing to prove (Lemma \ref{lemma:notassigned}, again).
If $\ld{\lambda}(\ld{y}) ~\ld{\sqsubseteq}~\ld{l}$,
then by transitivity, $\ld{\gamma}(\rd{\lambda}(\rd{x})) ~\ld{\sqsubseteq}~ \ld{l}$.
By monotonicity of $\rid{\alpha}$:
$\rid{\alpha}(\ld{\gamma}(\rd{\lambda}(\rd{x}))) ~\rd{\sqsubseteq}~ \rid{\alpha}(\ld{l}) ~=~ \rd{m}$
(By our assumption on $\ld{l}$ and $\rd{m}$).
But by \textbf{LC2}, $\rd{\lambda}(\rd{x}) ~\rd{\sqsubseteq}~ \rid{\alpha}(\ld{\gamma}(\rd{\lambda}(\rd{x})))$.
So by transitivity, $\rd{\lambda}(\rd{x}) ~\rd{\sqsubseteq}~ \rd{m}$.
Now, by I.H., since $\rd{\mu_1}(\rd{x}) = \rd{\nu_1}(\rd{x})$, we have
$\ld{\mu_f}(\ld{y}) =
\ld{\mu_1}[\ld{y} := \rd{\mu_1}(\rd{x})](\ld{y}) =
\ld{\nu_1}[\ld{y} := \rd{\nu_1}(\rd{x})](\ld{y}) =
\ld{\nu_f}(\ld{y})$.
\end{proof}
\section{Related Work}\label{sec:related}
The notion of Lagois connections \cite{MELTON1994lagoisconnections} has surprisingly not been employed much in computer science.
The only cited use of this idea seems to be the work of
Huth \cite{huth1993equivalence} in establishing the correctness of programming language implementations.
To our knowledge, our work is the only one to propose their use in secure information flow control.
Abstract Interpretation and type systems \cite{cousot1997types-as-ai} have been used in secure flow analyses, e.g., \cite{cortesi2015datacentricsemantics, cortesi2018} and \cite{zanotti2002sectypingsbyai}, where security types are defined using Galois connections employing, for instance, a standard collecting semantics.
Their use of two domains, concrete and abstract, with a Galois connection between them, for performing static analyses \textit{within a single domain} should not be confused with our idea of secure connections between independently-defined security lattices of two organisations.
There has been substantial work on SIF in a distributed setting at the systems level.
DStar \cite{zeldovich2008-nsdi} for example, uses sets of opaque identifiers to define security classes.
The DStar framework extends a \textit{particular} DIFC model \cite{Krohn2007-aa,zeldovich2006-osdi} for operating systems to a distributed network.
The only partial order that is considered in DStar's security lattice is subset inclusion.
So it is not clear if DStar can work on general IFC mechanisms such as FlowCaml \cite{Pottier2003-FlowCaml}, which can use any partial ordering.
Nor can it express the labels of JiF \cite{myers1999jflow} or Fabric \cite{liu2017fabric} completely.
DStar allows bidirectional communication between processes $R$ and $S$ only if $L_R \sqsubseteq_{O_R} L_S$ and $L_S \sqsubseteq_{O_S} L_R$, i.e., if there is an order-isomorphism between the labels.
Our motivating examples indicate such a requirement is far too restrictive for most practical arrangements for data sharing between organisations.
Fabric \cite{liu2009fabric,liu2017fabric} adds \textit{trust relationships} directly derived from a principal hierarchy to support federated systems with mutually distrustful nodes and allows dynamic delegation of authority.
Most of the previous DIFC mechanisms \cite{myers1999jflow, zeldovich2006-osdi, Krohn2007-aa, efstathopoulos2005asbestos, roy2009laminar, cheng2012aeolus} including Fabric are susceptible to the vulnerabilities illustrated in our motivating examples, which we will mention in the concluding discussion.
\section{Conclusions and Future Work}\label{sec:conclusion}
Our work is similar in spirit to Denning's motivation for proposing lattices, namely to identify a simple and mathematically elegant structure in which to frame the construction of scalable secure information flow in a modular manner that preserved the autonomy of the individual organisations.
From the basic requirements, we identified the elegant theory of Lagois connections as an appropriate structure.
Lagois connections provide us a way to connect the security lattices of two (secure) systems in a manner that does not expose their entire internal structure and allows us to reason only in terms of the interfaced security classes.
We believe that this framework is also applicable in more intricate information flow control formulations such as decentralised IFC \cite{myers-phd-tr-award} and models with declassification, as well as formulations with data-dependent security classes \cite{Lourenco2015-ug}.
We intend to explore these aspects in the future.
In this paper, we also proposed a minimal operational model for the transfer of data between the two domains.
This formulation is spare enough to be adaptable at various levels of abstraction (programming language, systems, databases), and is intended to illustrate that the Lagois connection framework can \textit{conserve} security, using non-interference as the semantic notion of soundness.
The choice of non-interference and the use of a type system in the manner of Volpano \textit{et al.} \cite{DBLP:journals/jcs/VolpanoIS96} was to illustrate in familiar terms how those techniques (removed from a particular language formulation) could be readily adapted to work in the context of secure connections between lattices.
In this exercise, we made suitable assumptions of atomicity and the use of fresh variables for communication, so as to avoid usual sources of interference.
We believe that the Lagois connection framework for secure flows between systems is readily adaptable for notions of semantic correctness other than non-interference, though that is an exercise for the future.
In the future we intend to explore how the theory of Lagois connections constitutes a robust framework that can support the discovery, decomposition, update and maintenance of secure MoUs for exchanging information.
In this paper, we concerned ourselves only with two domains and bidirectional information exchange.
Compositionality of Lagois connections allows these results to extend to chaining connections across several domains.
In the future, we also intend to explore how one may secure more complicated information exchange arrangements than merely chains of bidirectional flow.
We close this discussion with a reminder of why it is important to have a framework in which secure flows should be treated in a modular and autonomous manner.
Consider Myer's DIFC model described in \cite{myers-phd-tr-award}, where a principal can delegate to others the capacity to act on its behalf.
We believe that this notion does not scale well to
large, networked systems since a principal may repose different levels of trust in the various hosts in the network.
For this reason, we believe that frameworks such as Fabric \cite{liu2009fabric, liu2017fabric} may provide more power than mandated by a principle of least privilege.
In general, since a principal rarely vests unqualified trust in another in all contexts and situations, one should confine the influence of the principals possessing delegated authority to only specific domains.
A mathematical framework that can deal with localising trust and delegation of authority in different domains and controlling the manner in which information flow can be secured deserves a deeper study.
We believe that algebraic theories such as Lagois connections can provide the necessary structure for articulating these concepts.
\paragraph{Acknowledgments.} The second author thanks Deepak Garg for insightful discussions on secure information flow. Part of the title is stolen from E.M. Forster.
|
1,116,691,500,901 | arxiv | \section{INTRODUCTION}
In recent years, many new charmed mesons have been discovered experimentally, including lots of orbitally high excited states. For example, in 2004, the FOCUS Collaboration \cite{Focus2004} and the Belle Collaboration \cite{Belle2004} observed the $D_0^{*}$, which is the $1P$ scalar and has been studied widely and carefully
\cite{Close2005,Godfrey2005,zhaoqiang2008}. In 2013, the LHCb collaboration announced several new charmed structures, including the $D_J(3000)$ and $D_J^*(3000)$ \cite{LHCb2013}. The $D_J(3000)$ was observed in the $D^{*}\pi$ mass spectrum. Its mass and width are $2971.8\pm 8.7 \ \mathrm{MeV}$ and $188.1 \pm 44.8\ \mathrm{MeV}$, respectively. Spin analysis indicates that $D_J(3000)$ has an unnatural parity, and the assignments of $2P(1^+)$, $3S(0^-)$ and $1F(3^+)$ etc. have been discussed\cite{Sun2013,Lu2014,YuGL2015,Godfrey2016}. Our previous study favored the broad $2P(1^+)$ assignments\cite{lsc2018}.
The $D_J^*(3000)$ is observed in the $D\pi$ mass spectrum, whose mass and width are
\begin{eqnarray}
\begin{aligned}
M&_{D_J^*(3000)}=3008.1\pm 4.0 \ \mathrm{MeV}, \\
\Gamma &_{D_J^*(3000)}=110.5 \pm 11.5\ \mathrm{MeV}.
\end{aligned}
\end{eqnarray}
The parity of this particle is still uncertain in present experiments. From its decay mode of $D\pi$, many authors treat it as a natural parity particle. Considering that its mass is around $3000$ MeV, the assignments of $2^3P_0$, $1^3F_4$, $3^3S_1$, $1^3F_2$ and $2^3P_2$ are possible \cite{Zhu2017}. Different models give the theoretical predictions of their masses and we summarized them in Table \ref{massspectrum}. The OZI-allowed strong decays with these possible assignments also have been studied by several models, and the results are summarized in Table \ref{decayduibi}.
\begin{table}[htb!]
\renewcommand\arraystretch{0.95}
\caption[massspectrum]{Several natural parity candidates of $D_J^*(3000)^0$ (MeV) }\label{massspectrum}
\vspace{0.3em}\centering
\begin{tabular}{ccccccc}
\toprule[1.5pt]
{$J^P$} & {$n^{2S+1}J_L$} & Godfrey1985\cite{Godfrey1985} & Pierro2001\cite{Pierro2001} & Ebert2009\cite{Ebert2009} & Sun2013\cite{Sun2013} & Godfrey2016\cite{Godfrey2016} \\
\midrule[1pt]
\multirow{2}{*}{$0^+$} & $1 ^3P_0$ & 2400 & 2377 & 2466 & 2398 & 2399 \\
{ } & $2 ^3P_0$ & - & 2949 & 2919 & 2932 & 2931 \\
{$1^-$} & $3 ^3S_1$ & - & 3226 & 3096 & 3111 & 3110 \\
{$2^+$} & $2 ^3P_2$ & - & 3035 & 3012 & 2957 & 2957 \\
\multirow{2}{*}{$3^-$} & $1 ^3D_3$ & 2830 & 2799 & 2863 & 2833 & 2833 \\
{ } & $2 ^3D_3$ & - & - & 3335 & 3226 & 3226 \\
{$4^+$} & $1 ^3F_4$ & 3110 & 3091 & 3187 & 3113 & 3113 \\
\bottomrule[1.5pt]
\end{tabular}
\end{table}
\begin{table}[htb!]
\renewcommand\arraystretch{0.95}
\caption[decayduibi]{Decay widths of $D_J^*(3000)^0$ with different assignments (MeV) }\label{decayduibi}
\vspace{0.3em}\centering
\begin{tabular}{cccccccc}
\toprule[1.5pt]
{\quad $n^{2S+1}L_J$ \quad} & Mode&\quad Sun\cite{Sun2013} \quad & \quad Yu\cite{YuGL2015} \quad & \quad L{\"{u}}\cite{Lu2014} \quad & \quad Song\cite{Song2015} \quad & Godfrey\cite{Godfrey2016} \ \\
\midrule[1pt]
\multirow{4}{*}{$3 ^3S_1$} & $D\pi$ & 0.91 & 5.45 & 14.0 & 13.5 & 3.21 \\
{ } & $D^*\pi$ & 3.5 & 4.85 & 19.4 & 25.7 & 5.6 \\
{ } & Total & 18.0 & 87.2 & 158.0 & 103.0 & 80.4 \\
\midrule[0.7pt]
\multirow{4}{*}{$2 ^3P_0$} & $D\pi$ & 49 & 35.9 & 83.5 & 72.5 & 25.4 \\
{ } & $D^*\pi$ & - & - & - & - & - \\
{ } & Total & 194 & 224.5 & 639.3 & 298.4 & 190 \\
\midrule[0.7pt]
\multirow{4}{*}{$2 ^3P_2$} & $D\pi$ & 1.8 & 5.0 & 1.92 & 1.46 & 5.0 \\
{ } & $D^*\pi$ & $8.1\times 10^{-3}$ & 17.8 & 11.89 & 0.12 & 17.1 \\
{ } & Total & 47.0 & 174.5 & 110.5 & 68.9 & 114 \\
\midrule[0.7pt]
\multirow{4}{*}{$1 ^3F_2$} & $D\pi$ & 16 & 18.8 & 28.6 & 26.1 & 23.1 \\
{ } & $D^*\pi$ & 13 & 15.7 & 21.0 & 18.8 & 18.5 \\
{ } & Total & 136 & 116.4 & 342.9 & 222.0 & 243 \\
\midrule[0.7pt]
\multirow{4}{*}{$1 ^3F_4$} & $D\pi$ & 1.2 & 21.3 & 9.96 & 4.97 & 15.8 \\
{ } & $D^*\pi$ & 1.8 & 14.1 & 9.41 & 5.31 & 15.2 \\
{ } & Total & 39 & 102.3 & 103.9 & 94.5 & 129 \\
\bottomrule[1.5pt]
\end{tabular}
\end{table}
Since the parity is conserved in strong decays, the $D^{*}\pi$ channel is forbidden for the $^3P_0$ states. In Table \ref{decayduibi}, all assignments except $2^3P_0$ have both $D\pi$ and $D^{*}\pi$ decay modes and most calculations give the similar decay widths of these two channels. However, $D_J(3000)$ was only found in $D^{*}\pi$ spectrum, while $D_J^*(3000)$ only in $D\pi$ spectrum \cite{LHCb2013} in LHCb experiment. The theoretical results that $D_J^*(3000)$ has similar decay widths of $D\pi$ and $D^{*}\pi$ modes are not consistent with present experimental data. Thus, the assignment of $2^3P_0$ for $D_J^*(3000)$ is more reasonable and some recent researches also favor this assignment \cite{Gupta2018}.
We also note that the theoretical predictions for the total widths of $D_J^*(3000)$ as the $2^3P_0$ state are larger than the experimental data. It can be explained that the estimated decay width by calculating the OZI-allowed strong decays is sensitive to its mass and there are divergences of the mass values between the preliminary detection of the $D_J^*(3000)$ with the present theoretical predictions. In our previous work, we have found that the excited states have large relativistic corrections, so non-relativistic or semi-relativistic models may give large uncertainties. This conclusion can be obtained from the results in Table \ref{decayduibi}: all the assignments of $D_J^*(3000)$ are highly excited states and The corresponding results vary from different methods. For example, the total width for the $3^3S_1$ case ranges from $18$ to $158$ MeV, which shows large divergences between different methods.
Thus, we treat $D_J^*(3000)$ as the second excited state of P-wave scalar meson ($2^3P_0$), and calculate its OZI-allowed two-body strong decays, trying to find out if it is consistent with the LHCb results.
We use the improved Bethe-Salpeter (BS) method \cite{BS1951,Salpeter1952} which contains the relativistic corrections \cite{WangGL2004,Wangth2013,LiQ2017}. In all possible channels, there is a light meson in the final state. We use the reduction formula, Partially Conserved Axial-vector Current(PCAC) relation, and low-energy theorem to deal with the case when the light final meson is a pseudo-scalar. This approach cannot be applied to the channels containing a light vector meson. So, we also adopt the effective Lagrangian method \cite{Wangth2017}.
The rest content of this paper is organized as follows. In Sec.~\uppercase\expandafter{\romannumeral2}, we derive the form of transition amplitudes with BS method and show the details of the effective Lagrangian method. In Sec.~\uppercase\expandafter{\romannumeral3}, we give the numerical results of OZI-alowed two-body strong decays of $D_0^*(2400)$ and $D_J^*(3000)$, and compare them with other researches. Summary and conclusion are presented in Sec.~\uppercase\expandafter{\romannumeral4}.
\section{Two-body Strong Decay}
We take the channel $D_0^*(2400)^0 \to D^+ \pi ^- $ as an example to illustrate the calculation details. The Feynman diagram of this process is shown in Fig.~\ref{feynman1}.
\begin{figure}[htb]
\centering
\includegraphics[scale=0.41]{Feynman1_1-eps-converted-to.pdf}
\caption[fig1]{Feynman diagram for the decay channel $D_0^*(2400)^0 \to D^+ \pi ^- $.}
\label{feynman1}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[scale=0.42]{Feynman2_2-eps-converted-to.pdf}
\caption[fig1]{Feynman diagram for $D_0^*(2400)^0 \to D^+ \pi ^- $ (with the low-energy approximation).}
\label{feynman2}
\end{figure}
By using the reduction formula, the transition matrix element can be written as
\begin{equation}\label{smatrix}
\begin{split}
T=&\langle D^+(P_{f1})\pi ^-(P_{f2})\left| D_0^*(P_i)\right. \rangle \\
=&\int {\rm{d}} ^4x {\rm{e}} ^{ {\rm{i}} P_{f2}\cdot x} (M_{f2}^2 -P_{f2}^2)\langle D^+(P_{f1})\left| \phi _{\pi}(x)\right| D_0^*(P_i) \rangle ,
\end{split}
\end{equation}
where, $\phi_{\pi}$ is the light pseudo-scalar meson field. By using the PCAC relation, the field can be expressed as \cite{WangGL2005}
\begin{equation}\label{axialvector}
\phi_{\pi}(x)=\frac{1}{M_{f2}^2 f_{\pi}}\partial ^{\mu}(\overline{u}\gamma _{\mu} \gamma_{5}d),
\end{equation}
where $M_{f2}$ is the mass of $\pi$, and $f_{\pi}$ is its decay constant.
Inserting Eq.~(\ref{axialvector}) into Eq.~(\ref{smatrix}), the transition matrix can be written as
\begin{equation}
\begin{split}
T=&\frac{M_{f2}^2-P_{f2}^2}{M_{f2}^2 f_{\pi}}\int {\rm{d}} ^4 x {\rm{e}} ^{iP_{f2}\cdot x} \langle D^+(P_{f1}) \left| \partial ^{\mu}(\overline{u}\gamma _{\mu} \gamma_{5}d) \right| D_0^*(P_i)\rangle \\
=&\frac{-{\rm{i}} P_{f2}^{\mu}(M_{f2}^2-P_{f2}^2)}{M_{f2}f_{\pi}}\int {\rm{d}} ^4 x {\rm{e}} ^{iP_{f2}\cdot x} \langle D^+(P_{f1})\left| \overline{u}\gamma _{\mu} \gamma_{5}d \right| D_0^*(P_i)\rangle .
\end{split}
\end{equation}
According to the low energy theorem \cite{WangGL2005}, the momentum of the light meson is much smaller than its mass and can be ignored. Then the Feynman diagram turns to Fig.~\ref{feynman2} and the amplitude can be written as
\begin{equation}\label{smatrix2}
\begin{split}
T \approx &-{\rm{i}} \frac{P_{f2}^{\mu}}{f_{\pi}}\int {\rm{d}} ^4 x {\rm{e}} ^{{\rm{i}} P_{f2}\cdot x} \langle D^+(P_{f1})\left| \overline{u}\gamma _{\mu} \gamma_{5}d \right| D_0^*(P_i)\rangle \\
=&-{\rm{i}} \frac{P_{f2}^{\mu}}{f_{\pi}} (2\pi)^4 \delta ^4 (P_i -P_{f1}-P_{f2}) \langle D^+(P_{f1})\left| \overline{u}\gamma _{\mu} \gamma_{5}d \right| D_0^*(P_i)\rangle .
\end{split}
\end{equation}
Besides using the PCAC rule and low energy theorem, we also use the effective Lagrangian method to get the transition amplitude of this process and the results of these two approaches are consistent. The Lagrangian is introduced by \cite{Zhong2010el,Wangth2017,lsc2018},
\begin{equation}
\mathcal{L}_{qqP}=\frac{g}{\sqrt{2}f_h} \bar{q}_{i} \gamma^{\xi} \gamma^5 q_j \partial_{\xi}\phi_{ij},
\end{equation}
where
\begin{equation}
\phi_{ij}=\sqrt{2}
\left[
\begin{matrix}
\frac{1}{\sqrt{2}} \pi^0 + \frac{1}{\sqrt{6}}\eta & \pi^+ & K^+ \\
\pi^- & \-\frac{1}{\sqrt{2}}\pi^0 + \frac{1}{\sqrt{6}}\eta & K^0 \\
K^- & K^0 & -\frac{2}{\sqrt{6}}\eta
\end{matrix}
\right]
\end{equation}
is the chiral field of the pseudoscalar meson. The quark-meson coulping constant $g$ is taken to be unity and $f_h$ is the decay constant.
Within Mandelstam formalism \cite{Mandelstam1955}, we can write the hadronic transition amplitude as the overlap integral over the relativistic wave functions of the initial and final mesons \cite{Chang2006}
\begin{equation}\label{feynmanamp}
\begin{split}
\mathcal{M}=& -{\rm{i}} \frac{P_{f2}^{\mu}}{f_{\pi}}\langle D^+(P_{f1})\left| \overline{u}\gamma _{\mu} \gamma_{5}d \right| D_0^*(P_i)\rangle \\
=&-{\rm{i}} \frac{P_{f2}^{\mu}}{f_{\pi}} \int \frac{{\rm{d}} ^3 q}{(2 \pi)^3} \mathrm{Tr}\left[\overline{\varphi} _{P_{f1}}^{++}({q}_{f1\perp})\frac{\slashed{P}_i}{M_i}\varphi_{P_i}^{++}({q}_{\perp})\gamma _{\mu}\gamma _{5}\right],
\end{split}
\end{equation}
where $q$ and ${q}_{f1}$ are the relative momenta between quark and anti-quark in initial and final meson, respectively. For the initial meson $D^*_0(c\bar u)$, $q=p_c-\frac{m_c}{m_u+m_c}P_i=\frac{m_u}{m_u+m_c}P_i-p_u$, where $m_u$, $m_c$ are the quark masses and $p_u$ and $p_c$ are the quark momenta. And for the final meson $D^+(c\bar{d})$, due to the conservation law of momentum, its internal relative momentum is related to that of the initial meson by $q_{f1}=q-\frac{m_c}{m_c+m_d}P_{f1}$. Then, only the BS wave functions in the transition amplitude need to be figured out.
The BS equation of two-body bound state can read in momentum space as \cite{BS1951,LiQ2017}
\begin{equation}
S_1^{-1}\chi_P(q) S_2^{-1} = {\rm{i}} \int \frac{{\rm{d}} ^4 k}{(2\pi)^4}I(P;q,k)\chi_P(k) ,
\end{equation}
where $\chi_P(q)$ is the four-dimensional BS wave function; $I(P;q,k)$ is the interaction kernel; $S_1$ and $S_2$ are the propagators for the quark and anti-quark respectively.
We follow Salpeter \cite{Salpeter1952} to take the instantaneous approximation $I(P;q,k) \approx I(q_{\perp}-k_{\perp})$
The three-dimensional salpeter wave function $\psi(q_{\perp})$ is defined by
\begin{equation}
\psi(q_{\perp}) = {\rm{i}} \int \frac{{\rm{d}} q_P}{2\pi}\chi_P(q),\quad \chi_P(q)=S_1(p_1) \int \frac{{\rm{d}} ^3 k}{(2\pi)^3} I(q_{\perp}-k_{\perp})\psi_P(k_{\perp}) S_2(p_2)
\end{equation}
In this work, we adopt the Cornell potential as the interaction kernel $I(r)$ as follow form \cite{WangGL2004,LiQ2017}
\begin{equation}
I(r) = V_s(r)+V_0+\gamma_0 \otimes \gamma^0 V_v(r) = \frac{\lambda}{\alpha}(1-{\rm{e}}^{-\alpha r}) + V_0 - \frac{4}{3}\frac{\alpha _s}{r}{\rm{e}} ^{-\alpha r},
\end{equation}
where $\lambda$ is the string constant, $\alpha_s(r)$ is the running strong coupling constant and $V_0$ is an adjustable parameter fixed by the meson's mass. In momentum space, the potential can read as
\begin{equation}
I(\vec{q})= -\left( \frac{\lambda}{\alpha} +V_0 \right) (2\pi)^3 \delta^3(\vec{q})+\frac{\lambda}{\pi^2}\frac{1}{(\vec{q}^2 + \alpha^2)^2} - \frac{2}{3\pi^2}\frac{\alpha_s(\vec{q})}{(\vec{q}^2)+\alpha^2},
\end{equation}
where the coupling constant $\alpha_s(\vec{q})$ is defined by:
\begin{equation}
\alpha_s(\vec{q})=\frac{12\pi}{27}\frac{1}{\log(\alpha+\frac{\vec{q}^2}{\Lambda_{QCD}^2})}.
\end{equation}
In the above process, we take the instantaneous approximation in the interaction kernel, where we omit the retardation effect. According to the results of paper \cite{Qiao1996,Qiao1999,Ebert2000}, this effect affects much on the light mesons, but has limited influence on the heavy-flavor mesons, because these mesons have larger mass values.
In addition, retardation effect mainly affects the mass spectra prediction. When we calculate the decay width, we adjust the $V_0$ to match the experimental data, which further reduces this effect. The results of our previous work \cite{FuHF2011,LiQ2017} are agree with experimental data very well, so the instantaneous approximation is applicable for heavy-light mesons.
Then, we express the relativistic wave function of a scalar meson with instantaneous approximation ($P_i\cdot q=0$) as
\begin{equation}
\varphi_{0^+}(q_{\perp})=M \left[
\frac{\slashed{q}_{\perp}}{M}f_{a1}(q_{\perp})+\frac{\slashed{P}\slashed{q}_{\perp}}{M^2} f_{a2}(q_{\perp})+f_{a3}(q_{\perp})+\frac{\slashed{P}}{M}f_{a4}(q_{\perp})
\right],
\end{equation}
where $f_{ai}$($i=1,2,3,4$) are the functions of $q_{\perp}^2$ and their value can be obtained by solving the full Salpeter equations. It is notable that $\varphi_{0^+}(q_{\perp})$ is a general form for $J^P=0^+$ states and the items containing $q$ are the high order relativistic corrections.
Within BS method, the four wave functions $f_{ai}$ are not independent, they have the following relations \cite{WangGL2009}
\begin{equation}
\begin{split}
&f_{a3}=\frac{q_{\perp}^2(\omega_1+\omega_2)}{M(m_1 \omega_2+m_2 \omega_1)}f_{a1}, \\
&f_{a4}=\frac{q_{\perp}^2(\omega_1-\omega_2)}{M(m_1\omega_2+m_2\omega_1)}f_{a2},
\end{split}
\end{equation}where $m_1=m_c$, $m_2=m_u$, $\omega_1=\sqrt{m_1^2-{q}_{\perp}^2}$, and $\omega_2=\sqrt{m_2^2-{q}_{\perp}^2}$.
In our calculation, we only keep the positive energy parts $\varphi_{P_i}^{++}({q}_{i\perp})$ of the relativistic wave functions because the negative energy part contributes too small \cite{Wangth2017}.
The positive energy part of the wave function can be written as
\begin{equation}\label{0+}
\varphi_{0^+}^{++}(q_{\perp})=A_1+A_2\frac{\slashed{P}}{M}+A_3\frac{\slashed{q}_{\perp}}{M}+A_4\frac{\slashed{P}\slashed{q}_{\perp}}{M^2},
\end{equation}
where
\begin{equation}
\begin{split}
A_1=&\;\frac{(\omega_1+\omega_2)q^2_{\perp}}{2(m_1\omega_2+m_2\omega_1)}\left(
f_{a1}+\frac{m_1+m_2}{\omega _1+\omega_2}f_{a2}
\right), \\
A_2=&\ \frac{(m_1-m_2)q^2_{\perp}}{2(m_1\omega_2+m_2\omega_1)}\left(
f_{a1}+\frac{m_1+m_2}{\omega _1+\omega_2}f_{a2}
\right), \\
&A_3=\frac{M}{2}\left(
f_{a1}+\frac{m_1+m_2}{\omega_1+\omega_2}f_{a2}
\right), \\
&A_4=\frac{M}{2}\left(
\frac{\omega_1+\omega_2}{m_1+m_2}f_{a1}+f_{a2}
\right).
\end{split}
\end{equation}
To calculate the values of wave functions, we should determine the parameters' values in the interaction kernel. We try to fix $V_0$ by the mass of the ground state. In this case, the theoretical mass of $D_J^*(3000)$ is much less than the present experimental data. Thus, we adjust $V_0$ to make its mass value be equal to the experimental data, then get the wave functions.
In this work, besides the wave function for $0^+$ state, we also need the wave functions of $0^-$, $1^-$, $1^+$, etc., which are presented in the appendix.
After finishing the integral, we can get the amplitude of $0^+ \to 0^-0^-$ as follow
\begin{equation}
\begin{split}
\mathcal{M}_{(0^+ \to 0^- 0^-)}=-{\rm{i}} \frac{P_{f2}^{\mu}}{f_{\pi}}
(P_{\mu} n_1 + P_{f1 \mu} n_2) ,
\end{split}
\end{equation}
where $n_1$ and $n_2$ are the form factors. They are the overlap integral over the wave functions of the initial and final states.
If the final light meson is $\eta$ or $\eta '$, the $\eta-\eta '$ mixing should be considered
\begin{equation}
\left(
\begin{array}{c}
\eta \\
\eta '
\end{array}
\right)
=
\left(
\begin{array}{cc}
\cos \theta_P & -\sin \theta_P \\
\sin \theta_P & \cos \theta_P
\end{array}
\right)
\left(
\begin{array}{c}
\eta_8 \\
\eta_1
\end{array}
\right),
\end{equation}
where $\eta_1= (u\bar{u}+d\bar{d}+s\bar{s})/{\sqrt{3}}$ and $\eta_8= {(u\bar{u}+d\bar{d}-2s\bar{s})}/{\sqrt{6}}$, we choose the mixing angle $\theta_P=-11.4^ \circ$ \cite{PDG2016}. Then, we get the transition amplitude with an extra coefficient after considering the mixing
\begin{equation}
\begin{split}
\mathcal{M}(\eta)=- {\rm{i}} P_{f2}^{\mu} M^2_{\eta} \left(
\frac{\cos \theta_P}{\sqrt{6}f_{\eta_8}M^2_{\eta_8}}-\frac{\sin \theta_P}{\sqrt{3}f_{\eta_1}M^2_{\eta_1}}
\right) \langle D^0(P_{f1})\left| \overline{u}\gamma _{\mu} \gamma_{5}u \right| D_0^*(P_i)\rangle, \\
\mathcal{M}(\eta ')=- {\rm{i}} P_{f2}^{\mu} M^2_{\eta '}\left(
\frac{\sin \theta_P}{\sqrt{6}f_{\eta_8}M^2_{\eta_8}}+\frac{\cos \theta_P}{\sqrt{3}f_{\eta_1}M^2_{\eta_1}}
\right) \langle D^0(P_{f1})\left| \overline{u}\gamma _{\mu} \gamma_{5}u \right| D_0^*(P_i)\rangle.
\end{split}
\end{equation}
In the case when heavy-light $1^+$ state is involved, if we use the $S$-$L$ coupling, the $^3P_1$ and $^1P_1$ states cannot describe the physical states. Within the heavy quark limit($m_Q\to \infty$), its spin decouples and the properties of the heavy-light $1^+$ state are determined by those of the light quarks. So $j$-$j$ coupling should be used instead. The orbital angular momentum $\vec{L}$ couples with the light quark spin $\vec{s}_q$, which is $\vec{j}_l=\vec{L}+\vec{s}_q$. Then $1^+$ state can be grouped into a doublet by the total angular momentum of the light quark($|j_l=1/2\rangle$ and $|j_l=3/2\rangle$). The relation between the two descriptions are \cite{Matsuki2010mixingangle,Barnes2005highercharmonia}
\begin{equation}
\left(
\begin{array}{c}
|J^P=1^+,j_l={3}/{2} \rangle \\
|J^P=1^+,j_l={1}/{2} \rangle \\
\end{array}
\right)
=
\left(
\begin{array}{cc}
\cos \theta & \sin \theta \\
-\sin \theta & \cos \theta \\
\end{array}
\right)
\left(
\begin{array}{c}
|{^1}P_1 \rangle \\
|{^3}P_1 \rangle \\
\end{array}
\right).
\end{equation}
In our method, we solve the Salpeter equations for $^3P_1$ and $^1P_1$ states individually, and use these mixing relations to calculate the contributions of two physical $1^+$ states.
We list some mixing states related to our work
\begin{equation}
\left(
\begin{array}{c}
D_1(2420) \\
D_1(2430) \\
\end{array}
\right)
=
\left(
\begin{array}{cc}
\cos \theta & \sin \theta \\
-\sin \theta & \cos \theta \\
\end{array}
\right)
\left(
\begin{array}{c}
D (1{^1}P_1) \\
D (1{^3}P_1) \\
\end{array}
\right),
\end{equation}
\begin{equation}
\left(
\begin{array}{c}
D_{s1}(2536) \\
D_{s1}(2460) \\
\end{array}
\right)
=
\left(
\begin{array}{cc}
\cos \theta & \sin \theta \\
-\sin \theta & \cos \theta \\
\end{array}
\right)
\left(
\begin{array}{c}
D_s (1 {^1}P_1) \\
D_s (1 {^3}P_1 )\\
\end{array}
\right).
\end{equation}
In our calculation, for these doublets, we choose the ideal mixing angle $\theta=35.3^{\circ}$ in the heavy quark limit.
For the $^3P_1(1^{++})$ and $^1P_1(1^{+-})$ states, the corresponding hadronic transition amplitudes are
\begin{equation}
\begin{split}
\mathcal{M}_{(0^+\to 1^{++} 0^-)}= \frac{-{\rm{i}}}{f_{\pi}} \varepsilon_{1\mu}P^{\mu} t_1, \\
\mathcal{M}_{(0^+\to 1^{+-} 0^-)}= \frac{-{\rm{i}}}{f_{\pi}} \varepsilon_{1\mu}P^{\mu} t_2,
\end{split}
\end{equation}
where $\varepsilon$ is the polarization vector of the $1^+$ state; $t_1$ and $t_2$ are the form factors. Then, the form factors of the physical states are
\begin{equation}
\begin{split}
t_{D_1(2420),D_{s1}(2536)}= t_2 \cos \theta + t_1 \sin \theta, \\
t_{D_1(2430),D_{s1}(2460)}=-t_2 \sin \theta + t_1 \cos \theta.
\end{split}
\end{equation}
The PCAC rule can only be applied to light pseudo-scalar mesons and it is not valid for light vector meson.
If $\rho$ or $\omega$ meson appears in the final states, we choose the effective Lagrangian method to calculate the transition amplitude. The Lagrangian of quark-meson coupling can be expressed as \cite{Zhong2010el,Wangth2017,lsc2018}
\begin{equation}\label{efl}
\mathcal{L}_{qqV}= \bar{q}_i(a \gamma_{\mu}+\frac{{\rm{i}} b}{2 M_{P_{f2}}}\sigma_{\mu \nu}P_{f2}^\nu )V_{ij}^{\mu}q_j,
\end{equation}
where $V^{\mu}_{ij}$ is the field of the light vector meson; $q_i$ and $\bar q_j$ are its constitute quarks. And we choose the parameters $a=-3$ and $b=2$ which represent the vector and tensor coupling strength \cite{Wangth2017}, respectively. Then we use Eq.~(\ref{efl}) to derive the light-vector meson's vertex and get the transition amplitude
\begin{equation}\label{feynmanampefl}
\begin{split}
\mathcal{M}=-{\rm{i}} \int \frac{{\rm{d}} ^3 q}{(2 \pi)^3} \mathrm{Tr}\left[\overline{\varphi} _{P_{f1}}^{++}({q}_{f1\perp})\frac{\slashed{P}_i}{M_i}\varphi_{P_i}^{++}({q}_{\perp})(a \gamma_{\mu}+\frac{{\rm{i}} b}{2 M_{f2}}\sigma_{\mu \nu}P_{f2}^\nu )\varepsilon_{2}^{\mu}\right] .
\end{split}
\end{equation}
After finishing the trace and integral, the transition amplitudes can be expressed as
\begin{equation}
\begin{split}
\mathcal{M}_{(0^+ \to 1^- 1^{-})}=\varepsilon_{1\mu} \varepsilon_2^{\mu} t_1
+\varepsilon_{1\mu}P^{\mu} \varepsilon_{2\nu} P^{\nu} t_2,
\end{split}
\end{equation}
where $\varepsilon_{1\mu}$ and $\varepsilon_{2\nu}$ are the polarization vectors of final heavy vector meson and the light vector meson, respectively; $t_1$, $t_2$ and $t_3$ are the form factors.
Then, the two-body decay width can be expressed as
\begin{equation}
\begin{split}
\Gamma=\frac{1}{8 \pi} \frac{\vert \vec{P}_{f1} \vert}{M^2_i}|\mathcal{M}|^2 ,
\end{split}
\end{equation}
where $\vec{P}_{f1}$ is the three-dimensional momentum of the final charmed meson
\begin{equation}
\vert \vec{P}_{f1} \vert = \sqrt{\left(\frac{M_i^2+M_{f1}^2-M_{f2}^2}{2M_i}\right)^2-M_{f1}^2} .
\end{equation}
\section{Results and Discussions}
In this paper, the masses of constituent quarks that we adopt are listed as follows: $m_u=0.305{\ \rm{GeV}},\ m_d=0.311{\ \rm{GeV}},\ m_s=0.50{\ \rm{GeV}},\ {\rm and}~ m_c=1.62{\ \rm{GeV}}$ \cite{Wangth2013}. Other parameters are $\alpha=0.060{\ \rm{GeV}}$, $\lambda=0.210{\ \rm{GeV}}^2$, $\Lambda_{QCD}=0.270 {\ \rm{GeV}}$, $f_{\pi}=0.1304{\ \rm{GeV}}$, $f_K=0.1562{\ \rm{GeV}}$ \cite{PDG2016}, $f_{\eta_1}=1.07f_{\pi}$, $f_{\eta_8}=1.26f_{\pi}$, $M_{\eta_1}=0.923{\ \rm{GeV}}$, and $M_{\eta_8}=0.604{\ \rm{GeV}}$ \cite{Wangth2017}. The masses of other involved mesons are shown in Table \ref{canshu}.
\begin{table}[htb]
\renewcommand\arraystretch{1.01}
\caption[canshu]{Masses of involved mesons ($ \mathrm{GeV} $) \cite{PDG2016}. }
\label{canshu}
\vspace{0.5em}\centering
\begin{tabular}{cccc}
\toprule[1.5pt]
{\quad $ m_{D_0^*(2400)^0}=2.318 $\quad } & $ m_{D_0^*(2400)^+}=2.351 $ & \quad {$ m_{D_J^*(3000)^{(0,+)}}=3.008 $} & \quad {$ m_{D_{s}^+}=1.968 $} \quad \\
$ m_{D_1(2420)^0}=2.421 $ & $ m_{D_1(2420)^+}=2.423 $ & \quad $ m_{D_1(2430)^{(0,+)}} =2.427 $ & \quad {$ m_{D_s^{*+}}=2.112 $}\\
\bottomrule[1.5pt]
\end{tabular}
\end{table}
We first calculate the the decay widths of the $1P$ states. It only have two OZI-allowed decay channels and the results are presented in Table \ref{d2400_0}. In the case of $D_0^*(2400)^0$, the decay width of $D^+\pi^-$ is almost twice as that of $D^0\pi^0$. Because there is a factor $1/\sqrt{2}$ in the constitute quarks of $\pi^0$. Other decays that involve $\rho^0$ and $\omega^0$ have similar relation too.
\begin{table}[htb]
\renewcommand\arraystretch{0.90}
\caption[tab1]{$D_0^*(2400)^{0,+}$ strong decay widths (MeV). Ref. \cite{zhaoqiang2008} adopts Chiral Quark Model, Ref. \cite{Close2005} adopts the $^3P_0$ Model and Ref. \cite{Godfrey2005} adopts the Pseudoscalar Emission Model.}
\label{d2400_0}
\vspace{0.5em}\centering
\begin{tabular}{ccccccc}
\toprule[1.5pt]
\multicolumn{2}{c}{Chanel} & \quad Ours\quad & \quad Ref. \cite{zhaoqiang2008} \quad & \quad Ref. \cite{Close2005} \quad & \quad Ref. \cite{Godfrey2005} & \quad Exp. \cite{PDG2016} \\
\midrule[1pt]
\multirow{2}{*}{$D_0^*(2400)^0\to $} & $D^+ \pi^-$ & \quad 151.5 \quad & \multirow{2}{*}{266} & \multirow{2}{*}{283} & \multirow{2}{*}{277} & \multirow{2}{*}{$267\pm 40$} \\
{ } & $ D^0 \pi^0$ &\quad 74.8 \quad& { } & & & \\
\multirow{2}{*}{$D_0^*(2400)^+\to $} & $ D^+ \pi^0$ & \quad 81.6 \quad& \multirow{2}{*}{$\Box$} & \multirow{2}{*}{$\Box$} &\multirow{2}{*}{$\Box$} & \multirow{2}{*}{$230\pm17$} \\
{ }& $ D^0 \pi^+$ & \quad 164.3 \quad & { }& & & \\
\bottomrule[1.5pt]
\end{tabular}
\end{table}
The total decay width of $D_0^*(2400)^+$ is larger than that of $D_0^*(2400)^0$ in our calculation, which are 245.9 MeV and 226.3 MeV, respectively. According to the present experimental data, the charged $D_0^*(2400)^{+}$ is heavier than the neutral $D_0^*(2400)^{0}$. The different phase spaces may result in this discrepancy. We also notice that the estimated decay widths are sensitive to the mass of the initial meson. Considering the experimental mass values have errors ($m_{D_0^*(2400)^0}=2318\pm 29 \ \mathrm{MeV},\ m_{D_0^*(2400)^{\pm}}=2351\pm \ 7 \mathrm{MeV}$ \cite{PDG2016}) and these experimental masses value have divergence with different theoretical predictions\cite{Godfrey1985,Pierro2001,Ebert2009,Sun2013,Godfrey2016}, we give the two-body decay width changing along with the initial meson mass from 2300 MeV to 2420 MeV, which is shown in Fig.~\ref{width_mass_2400}. The neutral one's total decay width changes from 214.0 to 287.2 MeV, and the charged one's is from 212.7 to 289.0 MeV . We believe that these OZI-allowed decays happen around the mass threshold, which results in such sensitivity of decay width to the initial $1P$ state mass.
In Table \ref{d2400_0}, we also list the results from other models \cite{Godfrey2005,zhaoqiang2008,Close2005} as well as the experimental results for comparison. According to Table \ref{d2400_0} and Fig.~\ref{width_mass_2400}, we conclude that our results of the $1P$ states are consistent with experimental data, which means we can apply the same method to study the $2P$ states.
\begin{table}[htb!]
\renewcommand\arraystretch{0.99}
\caption[D3000duibi]{Two-body strong decay widths (MeV) of $ D_J^*(3000)^0 $ as the $2P(0^+)$ state. ``-'' means the channel is forbidden, ``$\Box$'' means the channel is not included by this method. Ref. \cite{YuGL2015} uses $^3P_0 $ Model; Ref. \cite{Sun2013} uses QPC Model; Ref. \cite{Godfrey2016} uses Relativistic quark model and Ref. \cite{Gupta2018}. uses effective Lagrangian approach. }\label{D3000results}
\vspace{0.3em}\centering
\begin{tabular}{llccccc}
\toprule[1.5pt]
{Chanel }\qquad \quad \quad & Final States \quad &\quad Ours \quad & \quad Ref. \cite{YuGL2015}\ \ & \quad Ref. \cite{Sun2013} &\quad
Ref. \cite{Godfrey2016} &\quad Ref. \cite{Gupta2018}\\
\midrule[1pt]
\multirow{2}{*}{ $ D(^1S_0) \pi $ } & $ D^+ \pi^- $ & 11.6 & 23.94 & \multirow{2}{*}{ 49 } & \multirow{2}{*}{25.4} & 66.2 \\
& $ D^0 \pi^0 $ & 6.1 & {11.97} & { } & { } \vspace{0.3em} & 33.3\\
\multirow{2}{*}{$ D(2 ^1S_0)\pi$ }& $D(2550)^+ \pi^-$& 6.9 & \multirow{2}{*}{$\Box$} & \multirow{2}{*}{ $\Box$ } & \multirow{2}{*}{ 18.6} & \multirow{2}{*}{ $\Box$ }\\
& $D(2550)^0 \pi^0 $ & 3.3 & { } & { } & { } \vspace{0.6em} & \\
{$ D \eta $} & $ D^0 \eta^0 $ & 0.51 & 4.26 & 8.8 & 1.53 & 10.8 \\
{$ D \eta ' $}& $ D^0 \eta '^0$ & 6.0 & 1.07 & 2.7 & 4.94 & { $\Box$ } \\
{$ D_s K$} & $ D_s^+ K^- $ & \~{}$10^{-3}$ & 2.85 & 6.6 & 0.76 & 54.2 \vspace{0.6em} \\
\multirow{2}{*}{$ D_1(2420)\pi$ }& $ D_1(2420)^0 \pi^0 $ & 18.7 & 26.20 & \multirow{2}{*}{38} & \multirow{2}{*}{96.1$(^1P_1)$} & \multirow{2}{*}{ $\Box$ }\\
& $ D_1(2420)^+ \pi^- $ & 36.8 & $\Box$ & {} & & \\
{$ D_1(2420) \eta$}& $ D_1(2420)^0 \eta^0 $ & 0.85 & 1.37 & 1.1 & $\Box$ & $\Box$ \\
\multirow{2}{*}{$ D_1(2430) \pi$}& $ D_1(2430)^0 \pi^0 $ & 2.1 & 6.69 & \multirow{2}{*}{30} & \multirow{2}{*}{ $\Box$ } & \multirow{2}{*}{ $\Box$ } \\
{ } & $ D_1(2430)^+ \pi^- $ & 4.1 & { $\Box$ } & { } & & \\
{$ D_1(2430) \eta$}& $ D_1(2430)^0 \eta ^0 $ & 0.12 & 0.35 & 0.91 & $\Box$ & $\Box$ \\ \vspace{0.6em}
{$ D_{s}(2460) K$}& $ D_{s1}(2460)^+ K^- $ & 1.2 & 12.81 & 1.5 & { $\Box$} \vspace{0.3em} & $\Box$ \\
\multirow{2}{*}{$ D^{*} \rho$}& $ D^{*}(2007)^0 \rho ^0 $& 7.0 & 31.60 & \multirow{2}{*}{41} & \multirow{2}{*}{32} & \multirow{2}{*}{ $\Box$ } \\
& $D^{*}(2010)^+ \rho ^-$ & 13.3 & 62.01 &{ } \vspace{0.6em} & \\
{$ D^{*} \omega$}& $ D^{*}(2007)^0 \omega ^0 $ & 7.5 & 29.91 & 13 & 10.2 & $\Box$ \\
{$ D_s^{*} K^*$}& $ D_s^{*+} K^*(892)^- $ & 4.1 & 3.06 & 1.0 &$\Box$ & $\Box$ \\
{$ D_s(2536) K^-$}& $D_{s1}(2536)^+ K^-$ & - & 6.40 & - & - \vspace{0.6em} & - \\
\midrule[1pt]
\multicolumn{2}{c}{Total} & 130.2 & 224.5 & 193.6 & 189.5 & 164.5\\
\multicolumn{2}{c}{Experimental value} & \multicolumn{4}{c}{$110.5\pm 11.5$} & \\
\bottomrule[1.5pt]
\end{tabular}
\end{table}
\begin{figure}[htb]
\centering
\subfigure[$\Gamma_{ D_0^*(2400)^{(0,+)}}$ versus the mass]{
\label{width_mass_2400}
\includegraphics[width=0.45\textwidth]{width_mass_D0star0_plus-eps-converted-to.pdf}}
\hspace{0.2cm}
\subfigure[$\Gamma_{ D_J^*(3000)^{(0,+)}}$ versus the mass]{
\label{width_mass_3000}
\includegraphics[width=0.45\textwidth]{width_mass_spectrum_sum-eps-converted-to.pdf}}
\label{width_mass}
\caption{Total decay widths of $D_0^*(2400)$ and $D_J^*(3000)$ change with the masses.}
\end{figure}
\begin{figure}[htbp]
\centering
\subfigure[$ 0^+ $($1P$) state $ D_0^*(2400)^0$]{
\label{bse1p0+}
\includegraphics[width=0.45\textwidth]{0plus_wvf_plot_line-eps-converted-to.pdf}}
\hspace{0.2cm}
\subfigure[$ 0^+ $($ 2P $) state $ D_J^*(3000)^0 $]{
\label{bse2p0+}
\includegraphics[width=0.45\textwidth]{2p0plus_wvf_plot_line-eps-converted-to.pdf}}
\subfigure[$D_J^*(3000)^0 $ and $D_1(2420)$($^3P_1 $ state)]{
\label{bse1p1++}
\includegraphics[width=0.45\textwidth]{3p11plusplus_2P0plus_wvf_plot_line-eps-converted-to.pdf}}
\hspace{0.2cm}
\subfigure[$D_J^*(3000)^0 $ and $D_1(2420)$($^1P_1$ state)]{
\label{bse1p1+-}
\includegraphics[width=0.45\textwidth]{1p11plusminus_2P0plus_wvf_plot_line-eps-converted-to.pdf}}
\subfigure[$D_J^*(3000)^0 $ and $D^0$($^1S_0$ state)]{
\label{bse1s0-}
\includegraphics[width=0.45\textwidth]{2p_1s_plot_line-eps-converted-to.pdf}}
\hspace{0.2cm}
\subfigure[$D_J^*(3000)^0 $ and $D^0$($2^1S_0$ state)]{
\label{bse2s0-}
\includegraphics[width=0.45\textwidth]{2p_2s_plot_line-eps-converted-to.pdf}}
\caption{Several examples of wave functions for some states}
\label{bseplot}
\end{figure}
Under the assumption of $0^+$($2P$) state, the results of ours and other models are shown in Table \ref{D3000results}. The total width of our calculation is 130.2 MeV, which is smaller than the results of other models and close to the upper limit of experimental value. Though the $2P$ state $D_J^*(3000)^0$ has larger phase space and more decay channels than those of $1P$ state $D_0^*(2400)^0$, why we get a narrower full width? The reason is the different structures of wave functions. The numerical values of the wave functions $f_{a1}$ and $f_{a2}$ for $1P$ state as the function of internal momentum $|\vec{q}|$ are all positive (Fig.~\ref{bse1p0+}), while the wave functions of the $2P$ state have a node (Fig.~\ref{bse2p0+}). The wave function values after the node become negative and it makes contrary contribution to the positive part, which will cause cancellation in the overlap integral between these two parts. The node structure reduces the decay width of the $2P$ state.
From our results, the channels of $D \pi,\ D_1(2420)\pi ,\ D\rho$ give large contribution to the full decay width, and $D_1(2420)\pi$ channel is dominant.
As shown in Eq.~(\ref{feynmanamp}), large transition amplitude means that these three channels have large overlap integrals. For example, we draw the wave functions of final state $D_1(2420)(^3P_1 \& ^1P_1)$ and initial $D_J^*(3000)$ in Fig.~\ref{bse1p1++} and \ref{bse1p1+-}. When the recoil momentum of the final meson is small (ignoring the difference between internal momenta of the initial and final states here), the peak values of the initial and final wave functions are coincident. Thus, we obtain large overlap integral values of the wave functions before the $2P$ node. However, the part after the node gives small cancellation since the corresponding values of the $1P$ wave function are small at this time. As a result, we obtain a large decay width of $D_1(2420)\pi$ channel.
Other examples of $D(1^1S_0)$ and $D(2^1S_0)$ are shown in Fig.~\ref{bse1s0-} and \ref{bse2s0-}. For the channel of $D(1^1S_0)\pi$, when compared with $D_1(2420) \pi$, the part after the node gives negative contribution.
Because the peak values of the those wave functions are not coincident, the positive contribution of the part before the node is not dominant. The negative part after the node changes the sign of the overlap integral, which cause the width of $D(1^1S_0)\pi$ is narrower than that of $D_1(2420)\pi$.
For the $D(2^1S_0)\pi$ channel, both wave functions have node structure. Compared with $D(1^1S_0)\pi$ channel, the contribution from the part before the $2P$ node is obviously smaller for $D(2^1S_0)\pi$. But the contribution after the $2P$ node becomes positive again because both wave functions ($2S$ and $2P$ states) are negative at this time. Therefore, although the phase space is narrow, the decay width of $D(2^1S_0)\pi$ channel is not very small.
It has been mentioned in Sec.~\uppercase\expandafter{\romannumeral1} that the relativistic corrections are large for the excited states. We can explain this argument according to the figures of the wave functions. If contribution from large internal momentum $q$ is significant, we can conclude that relativistic corrections are considerable. In Fig.~\ref{bse1s0-}, for the $1S$ state, the peak value of the wave function appear in the region of small $q$, while for the $1P$ state, in Fig.~\ref{bse1p1++} and \ref{bse1p1+-}, the peak values appear in the region of middle $q$. This means the $1P$ state has larger relativistic correction than that of the $1S$ ground state. When comparing Fig.~\ref{bse1s0-} with \ref{bse2s0-}, wave functions after the node give sizable contribution, which happens in large $q$ region. This means higher excited states have larger relativistic corrections. So we conclude that a relativistic model is needed to deal with the excited state problem.
In our study, we also calculate the decay widths of $ D_J^*(3000)^+ $, shown in Table \ref{D3000plusresults}. All channels are similar to $D_J^*(3000)^0$, and the full width is 131.3 MeV.
\begin{table}[htb!]
\renewcommand\arraystretch{1.0}
\caption[D3000duibi]{Two-body strong decay widths (MeV) of $ D_J^*(3000)^+ $ as the $2P(0^+)$ state. }\label{D3000plusresults}
\vspace{0.3em}\centering
\begin{tabular}{llccllc}
\toprule[1.5pt]
{\ Chanel }\qquad \quad \quad & Final States \quad &\ \ \ Width \ \ \ & {\quad} & {Chanel }\qquad \quad \quad & Final States \quad &\ \ \ Width \ \ \\
\midrule[1pt]
\multirow{2}{*}{ $ D(^1S_0) \pi $ } & $ D^+ \pi^0 $ & 6.5 & & \multirow{2}{*}{ $ D(2 ^1S_0)\pi$ } & $ D^0 \pi^+ $ & 3.8 \\
& $ D^0 \pi^+ $ & 13.5 & & { } & $ D^0 \pi^+ $ & \ {7.7} \vspace{0.3em}\\
{$ D \eta $} & $ D^0 \eta^0 $ & 0.56 & & {$ D \eta ' $}& $ D^0 \eta '^0$ & \ 5.7 \vspace{0.3em} \\
\multirow{2}{*}{$ D(2420)\pi$ }& $ D_1(2420)^+ \pi^0 $ & 18.3 & & \multirow{2}{*}{$ D(2430) \pi$}& $ D_1(2430)^+ \pi^0 $ & 2.1 \\
& $ D_1(2420)^0 \pi^+ $ & 37.4 & & & $D_1(2430)^0 \pi^+$ & 4.3 \\
{$ D(2420) \eta$}& $ D_1(2420)^+ \eta^0 $ & 0.77 & & {$ D(2430) \eta$}& $ D_1(2430)^+ \eta ^0 $ & 0.11 \\
\multirow{2}{*}{$ D^{*} \rho$}& $ D^{*}(2010)^+ \rho ^0 $& 6.1 & & {$ D^{*} \omega$}& $ D^{*}(2010)^+ \omega ^0 $ & 6.5 \\
& $D^{*}(2007)^0 \rho ^-$ & 12.9 & & {$ D_{s}(2460) K$}& $ D_{s1}(2460)^+ K^0 $ & 1.2 \\
{$ D_s K$} & $ D_s^+ K^0 $ & 0.05 & & {$ D_s^{*} K^*$}& $ D_s^{*+} K^{*}(892)^0 $ & 3.8 \\
\midrule[1pt]
\multicolumn{3}{c}{Total} &\multicolumn{3}{c}{131.3} \\
\bottomrule[1.5pt]
\end{tabular}
\end{table}
Considering many theoretical prediction of the mass are lower than 3000 MeV \cite{Godfrey1985,Pierro2001,Ebert2009,Sun2013,Godfrey2016} and the properties of these states could be revised after more experimental data collected, we also calculate the total width changing with the mass from 2900 to 3020 MeV, which is shown in Fig.~\ref{width_mass_3000}. The total width of the neutral one ranges from 135.6 to 126.3 MeV, while the charged one's result changes from 136.8 to 127.7 MeV. The full width of the $2P$ state becomes narrower along with the phase space increasing, which is opposite to that of the $1P$ case. The reason is that the recoil momentum becomes more considerable when phase space is larger for the $2P$ state. This results in greater contribution from the part after the nodes, so the decay width gets smaller.
We also notice that there is a rise at the tail of the curve. It is because some new channels open when the mass of $D_J^*(3000)$ increase to 3000 MeV, such as $D_s^*K^*$, $D_1(2420)\eta$, $D_1(2430)\eta$. Thus, the total widths have the sudden rise.
\section{SUMMARY}
In this work, we study the two-body strong decay properties of two orbitally excited scalar $D$ mesons by the improved BS method.
Our results of the $D_0^*(2400)$, as the $0^+(1P)$ states, are consistent with the present experimental data, which shows the suitability of our method. However, the sensitivity of decay width to its mass means more precise measurements are needed.
For the $0^+(2P)$ assignment of $D_J^*(3000)$, the full decay width is about 131 MeV, which is a little higher but close to the present experimental data. Besides the $D\pi$ mode, we find $D\rho$ and $D_1(2420)\pi$ channels also contribute much to the full width, and they can be helpful in the further investigation. Considering the theoretical uncertainties from relativistic corrections of highly excited states and the preliminary experimental data at present, $D_J^*(3000)$ is still a strong candidate for the $2^3P_0$ state. We expect more experimental and theoretical efforts on this newly discovered resonance.
\section*{ACKNOWLEDGEMENTS}
This work was supported in part by the National Natural Science
Foundation of China (NSFC) under Grant No.~11575048, No.~11405037, No.~11505039, No.~11447601, No.~11535002 and No.~11675239.
We thank the HPC Studio at Physics Department of Harbin Institute of Technology for access to computing resources through [email protected].
|
1,116,691,500,902 | arxiv | \section{Introduction}
Partial Differential Equations with nonlocal or fractional operators are widely used to model scientific problems in mechanics, physics, signal processing and other subjects, see for example \cite{signal} and references therein. We consider in this paper a nonlocal conservation law which appears in the formation and dynamics of sand structures such as dunes and ripples \cite{fowler2,lagree}. Namely, Fowler (\cite{fowler2,fowler3,fowler1}) introduced the following equation \label{fowler1}
\begin{equation}
\begin{cases}
\partial_t u(t,x) + \partial_x\left(\frac{u^2}{2}\right) (t,x) + \mathcal{I} [u(t,\cdot)] (x) -
\partial^2_{xx} u(t,x) = 0 & t \in (0,T), x \in \mathbb{R}, \\
u(0,x)=u_{0}(x) & x \in \mathbb{R},
\end{cases}
\label{fowlereqn}
\end{equation}
where $u=u(t,x)$ represents the dune height, $u_0 \in L^{2}(\mathbb{R})$ is an initial condition, $T$ is any given positive time and $\mathcal{I}$
is a nonlocal operator defined as follows: for any
Schwartz function $\varphi \in \mathcal{S}(\mathbb{R})$ and any $x \in \mathbb{R}$,
\begin{equation}
\mathcal{I} [\varphi] (x) := \int_{0}^{+\infty} |\zeta|^{-\frac{1}{3}}
\varphi''(x-\zeta) d\zeta . \label{nonlocalterm}
\end{equation}
Equation \eqref{fowlereqn} is valid for a river flow over an erodible bottom $u(t,x)$ with slow variation. The nonlocal term appears after a subtle modeling of the basal shear stress.
This operator appears also in the work of Kouakou \& Lagr\'ee \cite{kouakou,lagree}. \\
The nonlocal term $\mathcal{I}$ can be seen as a fractional power of order $2/3$ of the Laplacian with the bad sign. Indeed, it has been proved \cite{alibaud}
\begin{equation}
\mathcal{F} \left( \mathcal{I} [\varphi]-\varphi''\right) (\xi) = \psi_{\mathcal{I}}(\xi) \mathcal{F}
\varphi (\xi)
\label{fourier}
\end{equation}
where
\begin{equation}
\psi_{\mathcal{I}}(\xi)=4 \pi^2 \xi^2-a_{\mathcal{I}}
|\xi|^{\frac{4}{3}} + i \: b_{\mathcal{I}} \xi |\xi|^{\frac{1}{3}},
\label{psiexpression}
\end{equation}
with $a_\mathcal{I}, b_\mathcal{I}$ positive constants
and $\mathcal{F}$ denotes the Fourier transform. One simple way to establish this fact is the derivation of a new formula for the operator $\mathcal{I}$ \cite{alibaud}
\begin{equation}
\mathcal{I} [\varphi](x)= C_{\mathcal{I}} \int_{-\infty}^{0}
\frac{\varphi(x+z)-\varphi(x)-\varphi'(x) z}{|z|^{7/3}} \:
dz,\label{intformula}
\end{equation}
with $C_\mathcal{I}=\frac{4}{9}$. \\
The operator $\mathcal{I}[u]$ is a weighted mean of second derivatives of $u$ with the bad sign and has therefore an anti-diffusive effect and creates instabilities which are controlled by the diffusive operator $-\partial_{xx}^2$. This remarkable feature enabled to apply this model for signal processing. Indeed, the diffusion is used to reduce the noise whereas the nonlocal anti-diffusion is used to enhance the contrast \cite{signal}. \\
\begin{remark}
For causal functions (i.e. $\varphi(x) = 0 $ for $x<0$), this operator is
up to multiplicative constant, the Riemann-Liouville fractional derivative operator which is defined as follows \cite{pod}
\begin{equation}
\frac{1}{\Gamma(2/3)} \int_0^{+\infty} \frac{\varphi^{''}(x-\xi)}{|\xi|^{1/3}} d\xi= \frac{d^{-2/3}}{d x^{-2/3}} \varphi'' (x) =
\frac{d^{4/3}}{d x^{4/3}} \varphi (x),
\end{equation}
where $\Gamma$ denotes the Euler function.
\end{remark}
Recently, some results regarding this equation have been obtained, namely, existence of travelling-waves $u_{\phi}(t,x) = \phi(x - ct)$ where $\phi \in C^1_{b}(\mathbb{R})$ and $c \in \mathbb{R}$ represents wave velocity, the global well-posedness for $L^2$-initial data, the failure of the maximum principle, the local-in-time well-posedness in a subspace of $C^1_{b}$ \cite{alibaud, alvarez} and
the global well-posedness in a $L^2$-neighbourhood of $C^1_{b}$, solving namely for $u = u_{\phi} +v$, with $v \in L^2(\mathbb{R})$ \cite{afaf}.
For this purpose, the following Cauchy problem solved by the perturbation $v$ has been considered:
\begin{equation}
\begin{cases}
\partial_t v(t,x) + \partial_x(\frac{v^2}{2}+ u_{\phi} v)(t,x) + \mathcal{I}[v(t, \cdot)](x) -
\partial^2_{xx} v(t,x) = 0 & t \in (0,T), x \in \mathbb{R}, \\
v(0,x)=v_{0}(x) & x \in \mathbb{R},
\end{cases}
\label{fowlermodif}
\end{equation}
where $v_0 \in L^{2}(\mathbb{R})$ is an initial perturbation and $T$ is any given positive time. \\
Let us note also that any constant is solution of the Fowler equation. We shall prove below that these solutions are unstable. \\
To prove that a solution $u_\phi$ is unstable when $\phi$ is constant, we introduce the notion of \emph{mild solution} (see Definition \ref{mild}) based on Duhamel's formula \eqref{duhamel}. We also give some numerical results that illustrate this fact. \\
The remaining of this paper is organized as follows: in the next section, we define the notion of mild solution and we give some results.
Section \ref{nostability} contains the proof of the instability. We introduce in section \ref{numerik} an explicit finite difference scheme for which we give numerical simulations to illustrate the theory of the previous section.
\section{Duhamel formula and some results}
\begin{definition}
\label{mild}
Let $u_\phi$ be a constant, $T>0$ and $v_{0} \in L^{2}(\mathbb{R})$. We say that $v \in L^\infty((0,T);L^2(\mathbb{R})) $ is a \textbf{mild solution} to \eqref{fowlermodif} if for any $t \in \left(0,T\right)$,
\begin{equation}
v(t,x)=K(t, \cdot)\ast v_{0}(x)- \int^{t}_{0} \partial_{x}K(t-s,\cdot)\ast \left(\frac{v^{2}}{2}\right)(s,\cdot)(x) \: ds
\label{duhamel}
\end{equation}
where $K(t,x)=\mathcal{F}^{-1}\left(e^{-t \phi_{\mathcal{I}}(\cdot)}\right)(x)$
is the kernel of the operator $\mathcal{I}-\partial^2_{xx} + u_\phi \partial_x$ and $\phi_\mathcal{I}(\xi) = 4 \pi^2 \xi^2-a_{\mathcal{I}}
|\xi|^{\frac{4}{3}} + i \: b_{\mathcal{I}} \xi |\xi|^{\frac{1}{3}} + i\: 2 \pi u_\phi \xi $
\end{definition}
The expression \eqref{duhamel} is the Duhamel formula and is obtained using the spatial Fourier transform. The use of this formula allows to prove the local-in-time existence with the help of the contracting fixed point theorem. The global existence is obtained thanks to $L^2$ a priori estimate. We refer to \cite{alibaud,afaf} for the proof. \\
\begin{lemma}\label{kernel2}
Let $t>0$. Then, $\partial_x K(t,\cdot) \in L^2(\mathbb{R})$ and satisfies
\begin{equation}
||\partial_x K(t, \cdot) ||_{L^2(\mathbb{R})} \leq C\left(t^{-3/4} + e^{\alpha t}\right),
\end{equation}
where $C$ is a positive constant independent of $t$.
\end{lemma}
\begin{proof}
By Plancherel formula, we have
\begin{eqnarray*}
||\partial_x K(t, \cdot)||^2_{L^2(\mathbb{R})} &=& ||\mathcal{F}\left( \partial_x K(t, \cdot)\right) ||^2_{L^2(\mathbb{R})} = ||\xi \mapsto 2 i \pi \xi e^{-t\phi_\mathcal{I}(\xi)} ||^2_{L^2(\mathbb{R})}, \\
&=& 2 \int_0^{+\infty} 4 \pi^2 \xi^2 e^{-2t(4\pi^2 \xi^2- a_\mathcal{I} \xi^{4/3})} \, d\xi.
\end{eqnarray*}
Let $\xi_0>0$ such that for all $\xi>\xi_0 $,
$$ 4\pi^2 \xi^2- a_\mathcal{I} \xi^{4/3} \geq \xi^2.$$
Then, if we denote by $\alpha = - \min Re(\phi_\mathcal{I})$ (see Figure \ref{psifigure}), we have
\begin{eqnarray*}
||\partial_x K(t, \cdot)||^2_{L^2(\mathbb{R})}
&\leq & 8 \pi^2 \int_0^{\xi_0} \xi^2 e^{-2t(4\pi^2 \xi^2- a_\mathcal{I} \xi^{4/3})} \, d\xi + 8 \pi^2 \int_{\xi_0}^{+\infty} \xi^2 e^{-2t(4\pi^2 \xi^2- a_\mathcal{I} \xi^{4/3})} \, d\xi, \\
&\leq & 8 \pi^2 \int_0^{\xi_0} \xi^2 e^{2t \alpha} \, d\xi + 8 \pi^2 \int_{0}^{+\infty} \xi^2 e^{-2t\xi^2} \, d\xi, \\
&\leq & 8 \pi^2 \frac{\xi_0^3}{3} e^{2t \alpha} + \frac{4\pi^2 }{\sqrt{2}} t^{-3/2} \int_{0}^{+\infty} \xi^2 e^{-\xi^2} \, d\xi, \\
&\leq & C \left(t^{-3/2} + e^{2\alpha t} \right),
\end{eqnarray*}
which completes the proof of this lemma.
\end{proof}
\begin{remark}
\label{remarkL2}
Using again Plancherel formula, we have for any initial data $v_0$ with values in $L^2(\mathbb{R})$, the following $L^2$-estimate \cite{alibaud}:
\begin{equation*}
||v(t, \cdot)||_{L^2(\mathbb{R})} \leq e^{\alpha t} ||v_0||_{L^2(\mathbb{R})}.
\end{equation*}
\end{remark}
\section{Instability\label{nostability}}
In this section, we give the proof of the instability. \\
Let us first note that a fundamental property of the kernel $K$ is the non-positivity \cite{alibaud}, see Figure \ref{kernelfig}. This feature enabled to prove the failure of the maximum principle. We use again this property to show that the constant solutions of the Fowler equation are unstable.
\begin{figure}[htp]
\begin{center}
\includegraphics[width=9cm,height=5cm]{kernelb.eps}
\caption{Evolution of the kernel $K$ for $t=0.1$ and $t=0.5$ s }\label{kernelfig}
\end{center}
\end{figure}
\begin{theorem}
Any constant solution $u_\phi$ of the Fowler equation is unstable, i.e.
$\exists \varepsilon >0, \forall \delta >0, \exists v_0 \in L^2(\mathbb{R})$ \mbox{ an initial data with } $||v_0||_{L^2(\mathbb{R})} \leq \delta$ \mbox{ and } $t_0>0$ \mbox{ such that } $||v(t_0, \cdot)||_{L^2(\mathbb{R})} > \varepsilon$,
\label{theoreme2}
where $v(t, \cdot)$ is the solution of \eqref{fowlermodif} with $v(0,\cdot)=v_0$.
\end{theorem}
\begin{proof}
We adapt the strategy used by De Bouard in \cite{anne}, making it more elementary in our setting. \\
We denote by $S(t)$ the linear semigroup associated with equation \eqref{fowlermodif} i.e. $S(t)w = K(t, \cdot)\ast w$. Hence, we have
\begin{equation}
T(t)v_0 := v(t, \cdot) = S(t)v_0 - \frac{1}{2} \int_0^t \partial_x K(t-s, \cdot) \ast v^2(s, \cdot) \, ds,
\end{equation}
where $T$ denotes the nonlinear semigroup associated with equation \eqref{fowlermodif}.\\
First, by Young inequality, Lemma \ref{kernel2} and Remark \ref{remarkL2}, we have
\begin{eqnarray}
||T(t)v_0-S(t)v_0 ||_{L^2(\mathbb{R})} &\leq& \frac{1}{2} \int_0^t ||\partial_x K(t-s, \cdot)||_{L^2(\mathbb{R})} ||v(s, \cdot)||_{L^2(\mathbb{R})}^2 \, ds, \nonumber \\
&\leq & \frac{C}{2} \int_0^t \left[ (t-s)^{-3/4} + e^{\alpha(t-s)} \right] e^{2\alpha s} ||v_0||_{L^2(\mathbb{R})}^2 \, ds, \nonumber \\
&\leq & b(t) ||v_0||_{L^2(\mathbb{R})}^2,
\end{eqnarray}
where $b(t) = \frac{C}{2} e^{2\alpha t} (4 t^{1/4} + \frac{1}{\alpha}e^{\alpha t})>0$ for all $t\in (0,T)$. \\
Hence, we obtain for $t_0>0$ fixed,
\begin{equation}
||T(t_0)v_0-S(t_0)v_0 || \leq b_0 ||v_0||_{L^2(\mathbb{R})}^2,
\label{difft0}
\end{equation}
where $b_0=b(t_0)$. \\
Assume now that the constant $u_\phi$ is a stable solution of the Fowler equation i.e,
$$\forall \varepsilon >0, \exists \eta >0, ||v_0|| < \eta \Rightarrow ||v(t,\cdot)||_{L^2(\mathbb{R})} < \varepsilon, $$
for all positive $t$.
Let then
\begin{equation*}
0<\varepsilon < \frac{e^{\alpha t_0} - 1}{16 b_0},
\end{equation*}
and let $\eta>0$ be such that $||v_0||_{L^2(\mathbb{R})} < \eta \Rightarrow ||v(t, \cdot)|| \leq \varepsilon$, for all positive $t$. \\
Let $N$ be an integer large enough such that
\begin{equation}
e^{\alpha t_0 N} \geq \frac{e^{\alpha t_0} - 1 }{4 \eta b_0}.
\label{Nchoisi}
\end{equation}
Let us now consider the following initial data: $v_0 = \delta w_0$, where $\mathcal{F}(w_0) = \frac{1}{\sqrt{d-c}} 1_{[c,d]}$, with $0<c<d$ satisfying (see Figure \ref{psifigure}):
\begin{itemize}
\item $\mbox{Re}\left( \phi_{\mathcal{I}}\right)(c)<\mbox{Re}\left( \phi_{\mathcal{I}}\right) (d)=-\beta<0$,
\item for all $\xi \in [c,d], \mbox{Re}\left( \phi_{\mathcal{I}}\right)(\xi) \leq - \beta$,
\item $\beta = \alpha - \gamma$ where $ 0< \gamma < \frac{\ln(2)}{Nt_0}$.
\end{itemize}
And $\delta$ is defined as
\begin{equation}
\delta = e^{-\alpha N t_0} \frac{e^{\alpha t_0} - 1 }{4 b_0}.
\label{definitiondelta}
\end{equation}
The shape of $w_0$ is given by
\begin{equation*}
w_{0}(x) = \frac{1}{\sqrt{d-c}} \left\{
\begin{array}{ll}
2\frac{\sin(\frac{d-c}{2}x)}{x}e^{-i \frac{d+c}{2} x} & \mbox{if } x \neq 0, \\
d-c & \mbox{otherwise.}
\end{array}
\right.
\end{equation*}
\begin{figure}[h!]
\centering
\includegraphics[width=8cm,height=55mm]{phi.eps}
\caption{Behaviour of $ \mbox{Re}\left( \phi_\mathcal{I}\right) $ }
\label{psifigure}
\end{figure}
It follows from \eqref{Nchoisi} and \eqref{definitiondelta} that $||v_0||_{L^2(\mathbb{R})}=\delta \leq \eta$, and thus for this initial data, we get that $||v(t,\cdot)||_{L^2(\mathbb{R})} \leq \varepsilon$ for all $t>0$. \\
For $n \in \left\lbrace 0, \cdots, N \right\rbrace $, we have
\begin{eqnarray*}
v(nt_0) &=& S(nt_0)v_0 + \sum_{k=0}^{n-1} S\left( (n-1-k)t_0\right)\left[v((k+1)t_0)-S(t_0)v(kt_0)\right], \\
&=& S(nt_0)v_0 + \sum_{k=0}^{n-1} S\left( (n-1-k)t_0\right)\left[T(t_0)v(kt_0)-S(t_0)v(kt_0)\right].
\end{eqnarray*}
Hence, we get using \eqref{difft0} and the $L^2$-estimate given in Remark \ref{remarkL2}
\begin{eqnarray*}
||v(nt_0) - S(nt_0)v_0||_{L^2(\mathbb{R})} &\leq & \sum_{k=0}^{n-1} e^{\alpha t_0(n-k-1)} ||v((k+1)t_0)-S(t_0)v(kt_0)||_{L^2(\mathbb{R})}, \\
&\leq & \sum_{k=0}^{n-1} e^{\alpha t_0(n-k-1)} b_0||v(kt_0)||^2_{L^2(\mathbb{R})}, \\
&\leq & \sum_{k=0}^{n-1} e^{\alpha t_0(n-k-1)} b_0 e^{2\alpha k t_0} ||v_0||_{L^2(\mathbb{R})}^2, \\
&=& \delta^2 b_0 e^{\alpha t_0 (n-1)} \frac{e^{\alpha n t_0}-1}{e^{\alpha t_0}-1}, \\
&\leq& \frac{\delta}{4} e^{\alpha t_0 n}.
\end{eqnarray*}
Moreover, by Plancherel formula, we have
\begin{eqnarray*}
\vert \vert S(t)v_0\vert \vert_{L^{2}(\mathbb{R})}^2 = \vert \vert K(t,\cdot) \ast v_0 \vert \vert_{L^{2}(\mathbb{R})}^2 &=& \vert \vert \mathcal{F}(K(t,\cdot) \ast v_0) \vert \vert_{L^{2}(\mathbb{R})}^2, \\
&= & \int_\mathbb{R} |\mathcal{F}(K(t,\cdot))(\xi) \mathcal{F}(v_0)(\xi)|^2 d\xi, \\
&=& \int_c^d \frac{\delta^2 }{d-c} e^{-2t \, \mbox{Re} (\phi_{\mathcal{I}})(\xi)} d\xi, \\
&\geq& e^{2 \beta t} \vert \vert v_0 \vert \vert_{L^{2}(\mathbb{R})}^2.
\end{eqnarray*}
We finally infer that
\begin{eqnarray*}
||v(Nt_0)||_{L^2(\mathbb{R})} &\geq& ||S(Nt_0)v_0||_{L^2(\mathbb{R})} - ||v(Nt_0) - S(Nt_0)v_0||_{L^2(\mathbb{R})}, \\
&\geq & \delta e^{\beta N t_0} - \frac{\delta}{4} e^{\alpha t_0 N}, \\
&\geq& \frac{e^{\alpha t_0}-1}{16 b_0} > \varepsilon,
\end{eqnarray*}
which gives us a contradiction and completes the proof of this theorem.
\end{proof}
\begin{remark}
We can give a physical interpretation of this result: a flat profile $u_\phi=constant$ is unstable under the morphodynamics described by the Fowler model.
\end{remark}
\section{Numerical simulations\label{numerik}}
The spatial discretization is given by a set of points ${x_{j}; j=1,...,N}$ and the discretization in time is represented by a sequence of times $t^0=0<...<t^n<...<T$. For the sake of simplicity we will assume constant step size $\delta x$ and $\delta t$ in space and time, respectively. The discrete solution at a point will be represented by $u^n_j \approx u(t^n,x_j)$.
The schemes consist in computing approximate values $u^n_j$ of solution to \eqref{fowlereqn} on $[n\delta t, (n+1) \delta t[\times [j \delta x, (j+1) \delta x[$ for $n \in \mathbb{N}$ and $j \in \mathbb{N}$ thanks to the following relation:
\begin{equation}
\label{FDscheme}
\frac{u_{j}^{n+1}-u_{j}^{n}}{\delta t}+ \frac{1}{2} \frac{(u_{j}^{n})^2-(u_{j-1}^{n})^2}{\delta x}- \frac{u_{j+1}^{n}-2u_{j}^{n}+u_{j-1}^{n}}{\delta x^{2}}+ \mathcal{I}_{\delta x}[u^{n}]_{j}=0,
\end{equation}
where $\mathcal{I}_{\delta x}$ is the discretization of the nonlocal term $\mathcal{I}$.
We use a basic quadrature rule to approximate $\mathcal{I}$ given by \eqref{nonlocalterm}
\begin{equation} \label{discretization14}
\mathcal{I}_{\delta x}[\varphi]_{j}=\delta x^{-4/3} \sum^{+ \infty}_{l=1} l^{-1/3} \left(\varphi_{j-l+1}-2\varphi_{j-l}+ \varphi_{j-l-1}\right).
\end{equation}
Let us note that for compactly supported initial datum the sum is a finite sum.
In this section, we want to simulate the instability stated previously. To this aim we must be careful to distinguish between the instability which stems from the model and numerical instabilities, which could be caused by a careless discretization. \\
The numerical stability ensures that the difference between the approximate solution and the exact solution remains bounded for increasing $t$ for $\delta x, \delta t$ given. \\
As noticed, the Fowler model amplifies slowly the low frequencies whereas the high frequencies are quickly dampened. Thus, the classical notion of $A$-stability (strong stability) and $C$-stability are not suitable nor desirable. A new definition of stability for this model has been considered in \cite{azaf} and numerical stability criteria have been obtained.\\
In the following numerical test, we have to take care to choose $\delta x$ and $\delta t$ accurately following the stability condition, see \cite{azaf}. We expose in Figure \ref{rides} the evolution of an initial flat bottom disturbed by a small bump for different times. As we can see the perturbation creates small ripples that grow with time without bound. This result proves that the perturbed solution goes away from the non-perturbed solution. This
illustrates the instability of constant solutions for the Fowler model \eqref{fowlereqn}. Figure \ref{logevo} shows the log-plot for the $L^2$-norm evolution of this perturbed bottom, which confirms the validity of the estimate given in Remark \ref{remarkL2}.
\begin{figure}[h!]
\centering
\subfigure[ Evolution of the solution disturbed. ]
{\includegraphics[scale = 0.2]{simulationb.eps} \label{rides} }
\subfigure[ Evolution of $\log( ||v(t, \cdot)||_{L^2})$ ]
{\includegraphics[scale=0.4]{lognorme2.eps} \label{logevo} }
\caption{ Evolution of a flat bottom disturbed by a small bump. \label{instability}
}
\end{figure}
\section{Acknowledgements}
We thank Pascal Azerad and Cl\'ement Gallo for helpful comments. The author is supported by the ANR MATHOCEAN ANR-08-BLAN-0301-02
|
1,116,691,500,903 | arxiv | \section{Introduction}\label{sec:intro}
\begin{comment}
Thermodynamics was built on experimental grounds, in order to tame the power of fire~\cite{1824Carnot}. Initially centered on the notions of heat and temperature, this phenomenological approach has given rise to numerous applications, from heat engines to cryogenic methods, to name but a few. As a matter of fact, `thermodynamics' is somehow a misnomer, because dynamical features are not accounted for: the emphasis is on relationships between physical quantities for a given transformation. This approach was formalized during the 19th century and after the 1860s, it was completed by statistical physics, in which it takes its roots. It was thus possible to shed light on the variables of macroscopic thermodynamics, and thus to refine our understanding. For instance, an adiabatic transformation was associated to a transformation that preserves the occupation probabilities of the energy levels~\cite{Callen}. Subsequent developments have focused on non-equilibrium statistical physics, within the growing field of stochastic thermodynamics that accounts for the individual and fluctuating trajectories of out-of-equilibrium systems~\cite{Sekimoto,2021Peliti}. It provides a theoretical framework that makes it possible to generalize the concepts of work, heat and entropy to single trajectories
(see Appendix~\ref{app:stoch-frame})
and to describe small size systems in contact with a thermal reservoir.
By introducing the time variable into the description of these concepts, it became possible to formulate new questions and in particular, that of the control and optimization of thermodynamic transformations and engine cycles at an elementary scale. This field has experienced a particular upsurge in recent years, in the wake of quantum control methods~\cite{2019Guery-Odelin}, and in particular the field of Shortcuts To Adiabiticity~\cite{Adiabatic}. We explore here different control methods worked out in stochastic thermodynamics, and most often inspired by some quantum counterpart. In the next paragraphs, we provide a brief overview of a few quantum frameworks and features that have been particularly inspiring for control ideas in stochastic thermodynamics. These methods form the breadcrumb trail in our review article.
\end{comment}
Thermodynamics originated in efforts to tame the motive power of fire~\cite{1824Carnot}.
Originally concerned with notions of heat and temperature, the field was formalized during the 19th century into a set of universal principles that
govern the properties of macroscopic systems in thermal equilibrium, as well as transformations between equilibrium states~\cite{Callen}.
Beginning in the 1860s, statistical physics revealed the microscopic roots of thermodynamics, greatly enhancing its power to predict and explain systems' material properties.
Reversible transformations have traditionally played a central role in the foundations of thermodynamics.
Such transformations occur in an idealized adiabatic (infinitely slow~\cite{Adiabatic}) limit, in which a system's dynamical behavior is essentially irrelevant.
More recent developments, however, have focused on nonequilibrium, finite-time processes, where dynamics become important.
In particular the growing field of {\it stochastic thermodynamics}~\cite{Sekimoto,2012Seifert,2021Peliti}
extends the concepts of heat, work and entropy production to individual trajectories of microscopic systems, evolving under stochastic equations of motion.
By introducing time into the description of thermodynamic processes, it becomes possible to formulate new questions, and in particular to investigate the control and optimization of finite-time thermodynamic transformations, see Fig.~\ref{fig:rabbit}.
Recent years have seen a surge of activity in this area.
In this review, we survey finite-time control methods that have been developed within the framework of stochastic thermodynamics.
Many of these methods (though not all!) have been inspired by developments in the control of quantum systems, particularly in the field of {\it shortcuts to adiabaticity}~\cite{2019Guery-Odelin}.
For this reason, we begin with a brief overview of quantum frameworks and features that are especially relevant for control ideas in stochastic thermodynamics.
The goal of quantum shortcuts to adiabaticity is to steer a system to evolve from an eigenstate $\vert n(0)\rangle$ of an initial Hamiltonian $\widehat H(0)$ to the corresponding eigenstate $\vert n(t_{\rm f})\rangle$ of a final Hamiltonian $\widehat H(t_{\rm f})$. For infinitely slow driving, this is achieved automatically by virtue of the quantum adiabatic theorem~\cite{2004Griffiths}, which guarantees that the system remains in the instantaneous eigenstate $\vert n(t)\rangle$ of $\widehat H(t)$ at all times. For rapid driving, three broad approaches for achieving the above-mentioned goal have emerged: {\it inverse engineering}~\cite{2011Chen,2014Torrontegui}, {\it transitionless} or {\it counterdiabatic driving}~\cite{2003Demirplak,2005Demirplak,2009Berry}, and {\it fast-forward} methods~\cite{2010Masuda,2012Torrontegui}.
In inverse engineering methods, instead of deducing a system's evolution under a given driving protocol (as is the customary approach in physics), one seeks to engineer a driving protocol that produces the desired evolution. This is accomplished by exploiting the equations of motion that govern the system's dynamics. In quantum mechanics, inverse engineering methods have been applied to a variety of dynamical frameworks, including the Schr\"odinger equation, the evolution operator, the dynamical invariants, and the density matrix formalism \cite{2019Guery-Odelin}.
In the counterdiabatic approach, for a given time-dependent reference Hamiltonian $\widehat H(t)$, one seeks a Hamiltonian $\widehat H_{CD}(t)$ with the following property: if the system evolves unitarily under $\widehat H + \widehat H_{CD}$ from an initial eigenstate $\vert n(0)\rangle$, then throughout the process the system remains in the $n$'th eigenstate of $\widehat H(t)$.
In other words the system follows the adiabatic trajectory $\vert n(t)\rangle$, even when the driving is rapid.
Explicit expressions for $\widehat H_{CD}$ are given by Eqs.~\eqref{eq:HCD-q} and Eq.~\eqref{eq:HLCD-q} below~\cite{2003Demirplak,2005Demirplak,2009Berry,2017Patra}.
Both results steer the system exactly along the adiabatic trajectory $\vert n(t)\rangle$, hence the solution to the counterdiabatic problem is not always unique.
In the fast-forward approach~\cite{2010Masuda,2012Torrontegui}, a potential $\widehat U_{FF}(t) = U_{FF}(\widehat{\bm{x}},t)$ is designed such that, if the system evolves under $\widehat H + \widehat U_{FF}$ from an initial eigenstate $\vert n(0)\rangle$, then at $t=t_{\rm f}$ the system arrives at the desired final state $\vert n(t_{\rm f})\rangle$. At intermediate times the state of the system takes the form $\psi(\bm{x},t) = e^{iS(\bm{x},t)/\hbar} \langle \bm{x}\vert n(t) \rangle$, where $S(\bm{x},t)$ is real. Note that while $\psi(\bm{x},t)$ itself is not an eigenstate of $\widehat H(t)$, its coordinate-space probability distribution coincides with the eigenstate probability distribution: $\vert\psi(\bm{x},t)\vert^2 = \vert\langle \bm{x}\vert n(t)\rangle\vert^2$.
In the shortcuts described above, the goal is to make the system arrive rapidly at a destination it would have reached naturally had the process been carried out quasistatically. In the quantum case, the desired destination is an energy eigenstate, but as we shall see the same goal can be reformulated for classical systems governed by Hamilton's equations (Sec.~\ref{sec:isolated-systems}), and for stochastic systems evolving under overdamped and underdamped Brownian dynamics (Secs.~\ref{sec:inverse-engineering} - \ref{sec:fast-forward}).
In fact, not only is the goal the same---rapid evolution to a quasistatic outcome---but there are close similarities between the various quantum, classical and stochastic shortcuts designed to achieve this goal.
A number of strategies for constructing shortcuts can be unified within a framework organized around the continuity equation~\cite{2017Patra}. In each case the strategy involves identifying a velocity field $v(x,t)$, or else the corresponding acceleration field $a(x,t)$, then using this field to construct the counterdiabatic or fast-forward Hamiltonian or potential---see Eqs.~\eqref{eq:HLCD}, \eqref{eq:uffdef}, \eqref{eq:HLCD-q} and \eqref{eq:UCD-stoch}.
\begin{figure*
\begin{center} \includegraphics[width=0.8\textwidth]{wonderlandv3.png}
\end{center}
\caption{The White Rabbit is doing his best to overcome his unpunctuality,
which is not an easy task in Wonderland. In this review, we study a
related type of question, and first address the possibility of finding a path from A to B (A and B are fixed).
When several such paths may exist, a second level of question
deals with searching for the optimal one; for the White Rabbit, this amounts to finding the fastest, and arrive on time.
\label{fig:rabbit}
}
\end{figure*}
Although the term shortcuts to adiabaticity is widely used in the context of quantum and Hamiltonian classical dynamics, for stochastic systems other expressions such as {\it engineered swift equilibration}~\cite{2016Martinez} and {\it shortcuts to isothermality}~\cite{2017Li} have been introduced.
While these terms are descriptive within particular contexts, they do not fully capture the broad scope of methods that have been developed.
In this review we will use the general terminology, \textit{swift state-to-state transformations}, and the acronym {\it SST}, to embrace the entire catalogue of shortcutting methods, see Appendix \ref{app:acronyms}
for a summary of acronyms used throughout.
This review focuses primarily on systems evolving under stochastic dynamics, but we begin in Sec.~\ref{sec:isolated-systems} by describing shortcuts developed for classical mechanical systems governed by Hamiltonian dynamics. Mainly, we describe the generalisation to classical mechanics of the counterdiabatic---both global and local---driving and the fast-forward driving. These methods, both in their quantum and classical versions, are based on the time manipulation of the system Hamiltonian. Also, we briefly describe other shortcuts---for instance in the framework of the Boltzmann equation. Section~\ref{sec:systems-with-bath} is devoted to shortcuts for systems in contact with a thermal bath and thus described by stochastic dynamics, codified for instance by the Fokker-Planck equation---focusing on the connection between equilibrium states. Therein, we further generalise three main STA quantum approaches (inverse engineering,
counterdiabatic method,
and fast-forward) to the stochastic framework, by time manipulating the potential energy, i.e. by applying a suitably chosen external force. In addition, we consider a new possibility of tuning the evolution of these systems: engineering the thermal environment.
Loosely speaking, it may argued that the finite-time driving associated with SST, which involves the exploration of non-equilibrium states, requires more resources than an infinitely slow adiabatic connection---for which the system is at equilibrium for all times. To be concrete, let us think of an isothermal process. For adiabatic connection, the work done along the process is minimum and equals the free energy difference between the final and initial states. For SST, there emerges an irreversible contribution to the work that depends on the path swept by the system. It is therefore natural to explore how to design protocols that optimally use the resources available by minimising some ``cost'' function, e.g. the irreversible work for the isothermal connection. Also, the system parameters that become time-dependent in SST\ must often verify certain constraints. The question of minimising a certain cost function that is a functional of the trajectory, with constrained parameters, is the central problem of optimal control theory~\cite{1987Pontryagin,2012Liberzon}---a collection of tools that has been employed in the applied mathematics and engineering literature for a long time, but only recently in physics.
In Sec. \ref{sec:OCT}, we analyze how these tools translate to the setting of stochastic thermodynamics, including a brief discussion of information geometry ideas and the emergence of the so-called classical speed limits. Then, in Sec. \ref{sec:beyond-equilibrium}, we extend previously discussed techniques to transitions between non-equilibrium states. Section~\ref{sec:heat_engine_and_beyond} deals with applications of stochastic shortcuts, including heat engines. Finally, perspectives and conclusions are drawn in Sec.~\ref{sec:conclusions}.
\section{Shortcuts for isolated classical systems}
\label{sec:isolated-systems}
\subsection{Background and Setup}
The study of classical adiabatic invariants traces back more than a century~\cite{1895LeCornu,1902Rayleigh} to the problem of a simple pendulum whose length $\ell$ varies with time.
If $\ell$ changes slowly, then so too do both the pendulum frequency $\nu$
and its energy $E$.
However, in the harmonic regime of small oscillations the ratio $E/\nu$ remains fixed in the limit of infinitely slow variation of the pendulum length~\footnote{
Note the correspondence between the classical adiabatic invariant, $E/\nu$, and the quantum number $n$: since the $n$'th eigenenergy of a harmonic oscillator is given by
$E_n = h \nu \left( n + \frac{1}{2} \right)$,
the quantum adiabatic invariance of $n$ matches the classical adiabatic invariance of $E/\nu$.}.
More generally, consider a classical particle of mass $m$ in one degree of freedom, described by a Hamiltonian
\begin{equation}
\label{eq:Hzt}
H(z,t) = \frac{p^2}{2m} + U(x,t) ,
\end{equation}
where $z=(x,p)$ denotes a point in phase space.
For any fixed value of $t$, we take $U(x,t)$ to be a confining potential, with closed (periodic) orbits in phase space.
We assume that $U(x,t)$ has a single minimum whose location may depend on $t$~\footnote{For potentials with two or more local minima, the adiabatic invariance of the action breaks down when phase space separatrices are crossed~\cite{1986Tennyson}.
To exclude this complicating feature, we assume only a single minimum.};
that $U(x,t)$ varies with time only during the interval $t_{\text{i}} = 0\le t\le t_{\text{f}}$;
and that the time-dependence of the potential is turned on and off smoothly at $t=0$ and $t=t_{\text{f}}$, respectively---more precisely, $U(x,t)$ is twice differentiable with respect to time.
Under these conditions, if we treat $t$ as fixed parameter then every trajectory evolving under $H(z,t)$ is closed (i.e.\ periodic).
If we instead let $t$ denote the running time then the {\it action}
\begin{equation}
\label{eq:defI}
I(E,t) = \oint_E dx \, p
\end{equation}
is an adiabatic invariant~\cite{1980Goldstein}.
The right side indicates a clockwise line integral around the {\it energy shell} $E$, that is the level set $H(z,t) = E$, which forms a closed loop in phase space (see Fig.~\ref{fig:shellsAndLoops}).
We will use the notation ${\cal E}(t,I)$ to denote the energy shell of $H(z,t)$ whose action is $I$.
Imagine a classical trajectory $z(t)$ that evolves under Hamilton's equations of motion as $H(z,t)$ is varied extremely slowly over a long time interval $0\le t\le t_{\text{f}}$.
Let $E(t) \equiv H(z(t),t)$ denote the slowly evolving energy of this trajectory.
In stating that the action $I$ is an adiabatic invariant, we mean that in the limit of infinitely slow driving its value remains constant along this trajectory:
\begin{equation}
\label{eq:adiabaticInvarianceAtAllTimes}
I(E(t),t) = I(E(0),0) \equiv I_{\text{i}} \quad,\quad 0\le t \le t_{\text{f}}
\end{equation}
even though, in general, $E(t) \ne E(0)$.
For a harmonic oscillator with time-dependent frequency $\nu(t)$, the action defined by Eq.~\eqref{eq:defI} is equal to $I(E,t) = E/\nu$, in agreement with the discussion of the simple pendulum, above.
The integral in Eq.~\eqref{eq:defI} gives the phase space volume~\footnote{By convention we use the term ``volume'' rather than ``area'', even though phase space is two-dimensional.} enclosed by the energy shell, hence we can equally well write
\begin{equation}
\label{eq:defI-alt}
I(E,t) = \int dz \, \theta\left[ E - H(z,t) \right]
\end{equation}
where $\theta(\cdot)$ is the unit step function, and $\int dz = \int dx \int dp$ denotes integration over phase space.
Thus the adiabatic invariance of the action can be described as follows:
if a trajectory is initially located on an energy shell ${\cal E}(0,I_{\text{i}})$ that encloses phase space volume $I_{\text{i}}$, then for any $t \in [0,t_{\text{f}}]$ the trajectory will be located on the energy shell ${\cal E}(t,I_{\text{i}})$, which encloses the same amount of phase space.
Classical shortcuts to adiabaticity (SST) are concerned with the situation in which the variation of the potential $H(z,t)$ is not slow.
Consider again the Hamiltonian given by Eq.~\eqref{eq:Hzt}, only now imagine that the interval $[0,t_{\text{f}}]$ over which $H$ varies with time is not particularly long---indeed, it can be arbitrarily short, though finite.
In this situation, the action $I$ is generally not invariant:
\begin{equation}
I(E(t),t) \ne I(E(0),0) \, .
\end{equation}
We will view the non-invariance of the action as a deficiency, to be corrected using tools similar to those devised for quantum SST / STA.
In what follows, the term {\it slow driving} refers to the adiabatic limit, while {\it fast driving} denotes non-adiabatic time-dependence of $H(z,t)$.
While the discussion above has focused on the evolution of a single trajectory $z(t)$, the invariance of the action for slow driving and the breaking of that invariance for rapid driving are conveniently visualized in terms of closed {\it loops} evolving in phase space; see Fig.~\ref{fig:shellsAndLoops}.
Imagine, at $t=0$, a collection of infinitely many initial conditions distributed over a single energy shell of the initial Hamiltonian, with action $I_{\text{i}}$.
These initial conditions define a loop ${\cal L}_{\text{i}}$ that coincides with the energy shell, ${\cal L}_{\text{i}} = {\cal E}(0,I_{\text{i}})$, as depicted schematically in Fig.~\ref{fig:initialShellAndLoop}.
From each of these initial conditions a trajectory $z(t)$ evolves under $H(z,t)$.
At any later time, $t>0$, a snapshot of these trajectories defines a new closed loop ${\cal L}(t)$.
Under slow driving, Eq.~\eqref{eq:adiabaticInvarianceAtAllTimes} implies that the loop ${\cal L}(t)$ ``clings'' at all time to the instantaneous shell whose action is $I_{\text{i}}$, that is, ${\cal L}(t) = {\cal E}(t,I_{\text{i}})$ for all $t \in [0,t_{\text{f}}]$.
In particular, the initial loop ${\cal L}_{\text{i}}={\cal L}(0)$ is mapped onto a final loop ${\cal L}_{\text{f}} \equiv {\cal L}({t_{\text{f}}}) = {\cal E}(t_{\text{f}},I_{\text{i}})$ that coincides with an energy shell of the final Hamiltonian (see the gray loop in Fig.~\ref{fig:finalShellAndLoop}).
In terms of the initial and final energies of the trajectory, we have
\begin{equation}
\label{eq:I0I1}
I(E_{\text{i}},0) = I(E_{\text{f}},t_{\text{f}}) = I_{\text{i}} \, ,
\end{equation}
where $E_{\text{i}}=E(0)$ and $E_{\text{f}}=E(t_{\text{f}})$.
Under fast driving, by contrast, the loop ${\cal L}(t)$ strays away from the energy shell ${\cal E}(t,I_{\text{i}})$ as illustrated for $t=t_{\text{f}}$ in Fig.~\ref{fig:finalShellAndLoop}.
\begin{figure}[htp]
\begin{center}
\subfigure{
\label{fig:initialShellAndLoop}
}
\subfigure{
\label{fig:finalShellAndLoop}
}
\includegraphics[width=0.49\textwidth]{test2.png}
\end{center}
\caption{In (a), the gray loop depicts an energy shell ${\cal E}(0,I_{\text{i}})$ of $H(z,0)$, and the red loop ${\cal L}_{\text{i}}={\cal L}(0)$ depicts initial conditions for a set of trajectories that subsequently evolve under $H(z,t)$, where $z=(x,p)$.
The two loops are identical at $t=0$, as shown. In (b), the gray loop depicts the energy shell ${\cal E}(t_{\text{f}},I_{\text{i}})$ of $H(z,t_{\text{f}})$, while the red loop ${\cal L}_{\text{f}} = {\cal L}(t_{\text{f}})$ depicts the final conditions for the set of trajectories. In the adiabatic limit the two would coincide, ${\cal L}_{\text{f}}={\cal E}(t_{\text{f}},I_{\text{i}})$, as the action $I$ becomes invariant. For any choice of $t_{\text{f}}$, the phase space volume enclosed by both loops in (b) is equal to that enclosed by the loops in (a), by Liouville's theorem, even though the loop ${\cal L}_{\text{f}}$ is not itself an energy shell.
}
\label{fig:shellsAndLoops}
\end{figure}
The goal of classical SST\ can be stated as follows: for a rapidly driven Hamiltonian $H(z,t)$ and an action $I_{\text{i}}$, devise a strategy that evolves the loop ${\cal L}_{\text{i}} = {\cal E}(0,I_{\text{i}})$ to the loop ${\cal L}_{\text{f}} = {\cal E}(t_{\text{f}},I_{\text{i}})$, under Hamiltonian evolution.
This is analogous to the situation in quantum SST, where the goal is to use unitary evolution to evolve a wavefunction from an eigenstate of an initial Hamiltonian to the corresponding eigenstate of the final Hamiltonian.
In the classical case, we attempt to meet this goal by designing an auxiliary term $H_{\rm aux}(z,t)$ such that the desired evolution is generated by the Hamiltonian
\begin{equation}
\label{eq:hamsum}
H_{\rm SST}(z,t) = H(z,t) + H_{\rm aux}(z,t) \, .
\end{equation}
The auxiliary term effectively steers the evolving loop ${\cal L}(t)$ to the desired target, namely the final energy shell ${\cal E}(t_{\text{f}},I_{\text{i}})$.
While the dynamics are generated by $H_{\rm SST}(z,t)$, we emphasize that the energy shells ${\cal E}(t,I)$ are always defined with respect to the original Hamiltonian $H(z,t)$.
We now identify three different flavors of this problem, which differ from one another in how ambitiously the above-mentioned goal is addressed.
\begin{itemize}
\item In one version of the problem, $H_{\rm aux}$ is independent of the choice of action $I_{\text{i}}$, and the action remains invariant throughout the entire process, i.e.\ Eq.~\eqref{eq:adiabaticInvarianceAtAllTimes} is satisfied at all times and for any choice of $E_{\text{i}}=E(0)$.
This represents the strongest version of the problem, and we will refer to it as {\it global counterdiabatic (GCD)} driving.
\item In a somewhat more relaxed version, we continue to insist that the action be invariant throughout the process (Eq.~\eqref{eq:adiabaticInvarianceAtAllTimes}), but now we allow $H_{\rm aux}$ to depend on the choice of $I_{\text{i}}$, equivalently on the choice of $E_{\text{i}}$; we will call this {\it local counterdiabatic (LCD)} driving.
\item In the most relaxed version of the problem, we allow $H_{\rm aux}$ to depend on the choice of $I_{\text{i}}$ (as with LCD driving), and we further allow the invariance of the action to be broken at intermediate times $0<t<t_{\text{f}}$, insisting only that $I(E_{\text{f}},t_{\text{f}})=I(E_{\text{i}},0)$ (Eq.~\eqref{eq:I0I1}).
We refer to this version as {\it fast-forward (FF)} driving.
\end{itemize}
The designations global and local reflect the distinction that in one case (GCD) a single $H_{\rm aux}$ must succeed for all energy shells, i.e.\ globally, while in the other (LCD) we are free to design $H_{\rm aux}$ based on the energy shell under consideration, i.e.\ locally.
The terms counterdiabatic and fast-forward are taken from the literature on quantum shortcuts---the former signifies the suppression of non-adiabatic excitations~\cite{2003Demirplak}, while the latter evokes a mechanism for rapidly arriving at a desired final destination, regardless of the intermediate path taken to get there~\cite{2010Masuda}.
All three versions of the classical SST\ problem---as defined above, for one degree of freedom---have been solved, in the sense that in each version an explicit recipe has been devised for constructing an auxiliary Hamiltonian that achieves the desired steering of trajectories.
As described in greater detail below, for GCD driving the auxiliary Hamiltonian is generally a complicated, non-linear function of both position ($x$) and momentum ($p$).
For LCD driving, the auxiliary Hamiltonian is simpler, taking the form $H_{\rm aux}(z,t) = pv(x,t)$, with $v(x,t)$ given by Eq.~\eqref{eq:vdef} below.
Finally, for FF driving, the auxiliary Hamiltonian does not depend on momentum at all, in other words the desired goal can be satisfied by adding a time dependent potential energy function $U_{\rm FF}(x,t)$ to the original Hamiltonian.
Perhaps not surprisingly, as we relax our demands on the performance of the auxiliary Hamiltonian, from global to local to fast-forward, the form of $H_{\rm aux}$ becomes simpler, at least in its dependence on momentum: $H_{\rm aux}(x,p,t) \rightarrow pv(x,t) \rightarrow U_{\rm FF}(x,t)$.
LCD and FF driving are closely related: the function $v(x,t)$ that appears in the LCD auxiliary Hamiltonian (Eq.~\eqref{eq:HLCD}) is also used to construct the fast-forward potential $U_{\rm FF}$, as elaborated in Sec.~\ref{subsec:recipes}.
In general, GCD driving is not closely related to the other two.
However, for a particular class of driving protocols that go by the name of {\it scale-invariant driving}~\cite{2014Deffner}, GCD and LCD driving are identical, giving the same $H_{\rm aux}(z,t)$ (see Eq.~\eqref{eq:HCD-si} below).
Scale-invariant protocols are those for which the potential $U(x,t)$ in Eq.~\eqref{eq:Hzt} has the form~\cite{2013delCampo,2014Deffner}
\begin{equation}
\label{eq:scaleInvart}
U(x,t) = \frac{1}{\sigma^2} U_0 \left( \frac{x-\mu}{\sigma} \right)
\end{equation}
where $\sigma = \sigma(t)$ and $\mu = \mu(t)$ are functions of time.
By varying $\mu$ we translate the potential, while varying $\sigma$ stretches or squeezes the potential along the $x$ axis, and rescales its magnitude, without otherwise altering its profile.
Potentials of this form give rise to convenient scaling properties of both the classical dynamics and the quantum energy eigenstates~\cite{2014Deffner}.
Power-law potentials $U(x,t) = \alpha \, [x/L(t)]^c$, with $c=2, 4, \cdots$, offer an illustrative example of scale-invariant driving~\cite{2013Jarzynski}.
The harmonic oscillator with time-dependent stiffness ($c=2$) and the particle-in-a-box, with time-dependent box length ($c\rightarrow\infty$) belong to this class.
We note in passing that if $U(x,t)$ is scale-invariant \eqref{eq:scaleInvart} for some $\sigma(t)$ and $\mu(t)$, then so is $U(x,t)+b/(x-\mu)^2$ for any real constant $b$.\footnote{
This follows trivially, since $b/y^2 = (1/\sigma^2) \, b/(y/\sigma)^2$, with $y=x-\mu$.}
Potentials of this form will arise in Sec.~\ref{subsec:boltzmann}.
\subsection{Recipes for classical shortcuts to adiabaticity}
\label{subsec:recipes}
We now describe how to construct $H_{\rm aux}(z,t)$ for the three situations just outlined.
For the derivations of these methods and further details, we refer to the original papers, cited below.
In Secs.~\ref{subsubsec:gcd}-\ref{subsubsec:ff}, we will assume that $U(x,t)$ is twice differentiable with respect to time (see comments after Eq.~\eqref{eq:Hzt}), hence both $U$ and $\partial_t U$ are continuous functions of time.
Since $\partial_t U=0$ outside the interval $[0,t_{\text{f}}]$, this assumption implies the boundary conditions
\begin{equation}
\label{eq:bc}
\frac{\partial U}{\partial t}(x,0^+) = \frac{\partial U}{\partial t}(x,t_{\text{f}}^-) = 0 .
\end{equation}
In other words the time-dependence of $U$ is turned on and off smoothly rather than abruptly.
In Sec.~\ref{subsubsec:bc} we will briefly consider the implications of relaxing this assumption.
\subsubsection{Global counterdiabatic driving}
\label{subsubsec:gcd}
Classical GCD mirrors the quantum approach developed by Demirplak and Rice~\cite{2003Demirplak} and Berry~\cite{2009Berry}.
In the quantum case, the auxiliary or {\it counterdiabatic} Hamiltonian is given as a sum over energy eigenstates of the original Hamiltonian $\widehat H(t)$:
\begin{equation}
\label{eq:HCD-q}
\widehat H_{\rm GCD}(t) = i\hbar \sum_m \left( \vert \dot m\rangle \langle m \vert - \langle m \vert \dot m\rangle \vert m\rangle \langle m\vert \right) \, .
\end{equation}
Here $\vert m\rangle = \vert m(t)\rangle$ denotes the $m$'th eigenstate of $\widehat H(t)$, and $\vert \dot m\rangle = \partial_t \vert m(t)\rangle$ is its time derivative.
We are interested in constructing the classical counterpart of $\widehat H_{\rm GCD}(t)$.
While Eq.~\eqref{eq:HCD-q} may not seem well-suited to this end, the quantum operator defined (uniquely) by that equation can equivalently be defined by~\cite{2013Jarzynski}:
\begin{subequations}
\label{eq:HCD-q2}
\begin{eqnarray}
\label{eq:HCD-q2a}
\left[ \widehat H_{\rm GCD},\widehat H \right] &=& i\hbar \left( \partial_t\widehat H - {\rm diag} \, \partial_t\widehat H \right) \\
\label{eq:HCD-q2b}
\langle n \vert \widehat H_{\rm GCD} \vert n\rangle &=& 0 \qquad \forall \, n
\end {eqnarray}
\end{subequations}
where ${\rm diag} \, \widehat A \equiv \sum_m \vert m\rangle \langle m\vert\widehat A\vert m\rangle \langle m\vert$.
Defining $\widehat H_{\rm GCD}$ in this manner is convenient, as the correspondence between the quantum commutator $[ \widehat A,\widehat B ]$ and the classical Poisson bracket $\{ A,B \}$,
together with the correspondence between quantum energy eigenstates and classical energy shells, suggest a natural classical analogue:
\begin{subequations}
\label{eq:HCD-c2}
\begin{eqnarray}
\label{eq:HCD-c2a}
\left\{ H_{\rm GCD},H \right\} &=& \partial_t H - \langle \partial_t H\rangle_H \\
\label{eq:HCD-c2b}
\langle H_{\rm GCD} \rangle_E &=& 0 \qquad \forall \, E
\end {eqnarray}
\end{subequations}
The notation
\begin{equation}
\langle A\rangle_E = \frac{ \int dz \, \delta\left(E-H\right) A(z) }{ \int dz \, \delta\left(E-H\right) }
\end{equation}
denotes a microcanonical average of an observable $A(z)$ over the energy shell $E$ of $H(z,t)$.
The left side of Eq.~\eqref{eq:HCD-c2a} is evaluated at a phase point $z$, and the notation $\langle \partial_t H\rangle_H$ on the right indicates that the average is taken over the energy shell containing $z$;
see Ref.~\cite{2013Jarzynski} for further details, as well as Ref.~\cite{2013Deng} where the same result is derived by means of classical generating functions, and Ref.~\cite{2017Kolodrubetz}, where $\widehat{H}_{\rm GCD}$ and $H_{\rm GCD}$ are framed as {\it adiabatic gauge potentials}.
Yet another interesting approach is taken in Ref.~\cite{2017Okuyama}, where $H_{\rm GCD}$ is constructed from the dispersionless Korteweg-de Vries hierarchy, building on earlier work in the quantum context~\cite{2016Okuyama}.
We have introduced Eq.~\eqref{eq:HCD-c2} as the classical analogue of Eq.~\eqref{eq:HCD-q2}.
It can be verified directly from classical analysis~\cite{2013Jarzynski} that the function $H_{\rm GCD}(z,t)$ defined by Eq.~\eqref{eq:HCD-c2} has the counterdiabatic property we seek: along a trajectory $z_G(t)$ evolving under $H + H_{\rm GCD}$, the action is preserved exactly:
\begin{equation}
\label{eq:GCDtraj}
\frac{d}{dt} I\left[H(z_G(t),t),t\right] = 0
\end{equation}
Hence the solution to a quantum problem, Eq.~\eqref{eq:HCD-q2}, combined with semiclassical reasoning, has yielded the exact solution to the analogous classical problem, Eq.~\eqref{eq:HCD-c2}.
To translate Eq.~\eqref{eq:HCD-c2} into an explicit function $H_{\rm GCD}(z,t)$, note that Eq.~\eqref{eq:HCD-c2a} implies the following relation, for any points $z_a$ and $z_b$ on an energy shell $E$ of $H(z,t)$:
\begin{equation}
\label{eq:zazb}
\begin{split}
H_{\rm GCD}(z_b,t) &- H_{\rm GCD}(z_a,t) \\
&= \int_a^b ds \, \left[ \partial_t H(z(s),t) - \langle \partial_t H\rangle_E \right] \, ,
\end{split}
\end{equation}
where $t$ is treated here as a parameter, and $z(s)$ is a trajectory of energy $E$ that evolves with time $s$ under fixed $H(z,t)$, from $z(a)=z_a$ to $z(b)=z_b$.
This relation determines $H_{\rm GCD}(z,t)$ for all points on the energy shell $E$, up to an additive constant whose value is in turn determined by Eq.~\eqref{eq:HCD-c2b}.
This additive constant can depend on time, but it has no dynamical relevance.
Equation \eqref{eq:HCD-c2} thus provides an explicit recipe for constructing $H_{\rm GCD}$.
For scale-invariant driving (see Eq.~\eqref{eq:scaleInvart}) this recipe leads to the particularly simple expression~\cite{2014Deffner}
\begin{equation}
\label{eq:HCD-si}
H_{\rm GCD}(z,t) \, = \, \frac{\dot\sigma}{\sigma} (x-\mu) p \,+\,
\dot \mu \, p
\end{equation}
where the dots denote derivatives with respect to time.
In the special cases of the harmonic oscillator and the particle-in-a-box, Eq.~\eqref{eq:HCD-si} reproduces results derived previously by other means~\cite{2010Muga,2013diMartino,2013Jarzynski}.
For non-scale-invariant driving, it is difficult to obtain closed-form expressions for $H_{\rm GCD}$, and one must resort to solving Eq.~\eqref{eq:HCD-c2} or \eqref{eq:zazb} numerically.
An exception is the case of the ``tilted piston'', which involves a particle inside a one-dimensional box with hard walls separated by a distance $L$, with a potential inside the box that is linear in $x$, with slope $\alpha$.
Either $L$ or $\alpha$, or both, can be made time-dependent.
Even for this relatively modest extension of the particle-in-a-box, the exact expression for $H_{\rm GCD}$ is complicated and non-linear in $p$~\cite{2017Patra-a}.
Finally, although Eq.~\eqref{eq:HCD-c2} uniquely defines $H_{\rm GCD}$, Eq.~\eqref{eq:GCDtraj} remains satisfied for a trajectory $z_G(t)$ evolving under $H+H_{\rm GCD}+f(H)$, for any differentiable function $f(\cdot)$.
As a result, to construct a globally counterdiabatic auxiliary Hamiltonian we need only satisfy Eq.~\eqref{eq:HCD-c2a} and not necessarily Eq.~\eqref{eq:HCD-c2b} -- imposing the latter (i.e.\ setting $f=0$) amounts to a kind of gauge choice.
Analogous comments apply in the quantum case to Eq.~\eqref{eq:HCD-q2}.
\subsubsection{Local counterdiabatic driving}
\label{subsubsec:lcd}
In the case of local counterdiabatic driving, we select a value of action, $I_{\text{i}}$.
For any $t\in[0,t_{\text{f}}]$, ${\cal E}(t,I_{\text{i}})$ is the energy shell of $H(z,t)$ with the same action $I_{\text{i}}$; we refer to this shell as the {\it adiabatic energy shell} for our choice of $I_{\text{i}}$.
This energy shell represents the desired evolution we wish to generate: if a trajectory begins on the energy shell ${\cal E}(0,I_{\text{i}})$ at $t=0$, we want to guarantee that it will be found on the shell ${\cal E}(t,I_{\text{i}})$ at all $t\in [0,t_{\text{f}}]$.
Unlike with GCD, this goal can be accomplished with an auxiliary Hamiltonian that is linear in momentum, $p$.
The LCD auxiliary Hamiltonian involves a function $v(x,t)$ that is constructed as follows.
At time $t$, the shell ${\cal E}(t,I_{\text{i}})$ forms a loop in phase space, with left and right turning points, $x_L(t)$ and $x_R(t)$.
Let
\begin{equation}
\bar p_\pm(x,t) = \pm \left[ 2m ({\cal E}(t,I_{\text{i}}) - U(x,t)) \right]^{1/2}
\end{equation}
denote the upper and lower branches of this loop, and let
\begin{equation}
{\cal S}(x,t) = 2 \int_{x_L(t)}^x dx^\prime \, \bar p_+(x^\prime,t)
\end{equation}
denote the volume of phase space enclosed by ${\cal E}(t,I_{\text{i}})$ between the left turning point and a vertical line located at position $x \in [x_L,x_R]$.
(To avoid confusion, we stress that ${\cal S}$ is {\it not} Hamilton's principal function, which appears in the Hamilton-Jacobi equation~\cite{1980Goldstein}.)
Since ${\cal S}$ increases monotonically with $x$ we can invert it, writing $x=x({\cal S},t)$, with ${\cal S}\in[0,I_{\text{i}}]$.
We then define a velocity field
\begin{equation}
\label{eq:vdef}
v(x,t) = \frac{\partial}{\partial t}x({\cal S},t)
= - \frac{\partial{\cal S}/\partial t}{\partial{\cal S}/\partial x}
\end{equation}
using the cyclic identity of partial derivatives.
From Eq.~\eqref{eq:bc} it follows that $v(x,0)=v(x,t_{\text{f}})= 0$.
The LCD auxiliary Hamiltonian is then given by
\begin{equation}
\label{eq:HLCD}
H_{\rm LCD}(x,p,t) = pv(x,t).
\end{equation}
Adding $H_{\rm LCD}$ to the original Hamiltonian $H$ (as per Eq.~\eqref{eq:hamsum}) gives us
\begin{equation}
\label{eq:hSTAlcd}
H_{\rm SST}(z,t) = \frac{p^2}{2m} + U(x,t) + pv(x,t)
\end{equation}
which generates the equations of motion
\begin{equation}
\label{eq:hameqlcd}
\dot x = \frac{p}{m} + v(x,t) \quad,\quad
\dot p = -\frac{\partial U}{\partial x} -p \frac{\partial v}{\partial x}(x,t) .
\end{equation}
Under these equations, the action is conserved exactly for any trajectory $z_L(t)$ (where $L$ is short for $LCD$) launched from the initial adiabatic energy shell~\cite{2017Jarzynski}.
That is,
\begin{equation}
\label{eq:LCDtraj}
z_L(t) \in {\cal E}(t,I_{\text{i}})
\end{equation}
for all $t\in[0,t_{\text{f}}]$.
For such a trajectory, the first terms appearing on the right in Eq.~\eqref{eq:hameqlcd} generate motion along the instantaneous energy shell, while the second terms (involving $v$) force the trajectory to remain attached to the evolving energy shell.
For scale-invariant driving, Eq.~\eqref{eq:vdef} gives
\begin{equation}
\label{eq:vsi}
v(x,t) = \frac{\dot\sigma}{\sigma}(x-\mu) + \dot \mu
\end{equation}
As mentioned shortly before Eq.~\eqref{eq:scaleInvart}, in this special case the GCD (Eq.~\eqref{eq:HCD-c2}) and LCD (Eq.~\eqref{eq:HLCD}) prescriptions yield the same auxiliary Hamiltonian, Eq.~\eqref{eq:HCD-si}.
\subsubsection{Fast-forward driving}
\label{subsubsec:ff}
Having defined $v(x,t)$ above, it is straightforward to construct the FF auxiliary potential $U_{\rm FF}(x,t)$.
We first introduce an acceleration field
\begin{equation}
\label{eq:adef}
a(x,t) = \frac{\partial v}{\partial x} v + \frac{\partial v}{\partial t} = \frac{\partial^2}{\partial t^2} x({\cal S},t) .
\end{equation}
$U_{\rm FF}$ is then defined, up to an arbitrary function of time, by
\begin{equation}
\label{eq:uffdef}
-\frac{\partial U_{\rm FF}}{\partial x} = ma(x,t) .
\end{equation}
Adding $U_{\rm FF}$ to $H$ gives
\begin{equation}
\label{eq:hSTAff}
H_{\rm SST}(z,t) = \frac{p^2}{2m} + U(x,t) + U_{\rm FF}(x,t)
\end{equation}
which generates the equations of motion
\begin{equation}
\label{eq:hameqff}
\dot x = \frac{p}{m} \quad,\quad
\dot p = -\frac{\partial U}{\partial x} +ma(x,t) .
\end{equation}
If two trajectories are launched from identical initial conditions on the energy shell ${\cal E}(0,I_{\text{i}})$, and one of them, $z_L(t)$, evolves under Eq.~\eqref{eq:hameqlcd}, while the other, $z_F(t)$, evolves under Eq.~\eqref{eq:hameqff} (where $F$ is short for $FF$), then the two are related by~\cite{2017Jarzynski}
\begin{equation}
\label{eq:FFboost}
x_F(t) = x_L(t)
\quad,\quad
p_F(t) = p_L(t) + mv(x_L(t),t) \, .
\end{equation}
The two trajectories reunite at $t=t_{\text{f}}$, as $v(x,t_{\text{f}})=0$.
Thus the fast-forward trajectory $z_F(t)$ starts on the energy shell ${\cal E}(0,I_{\text{i}})$ at $t=0$, then strays from ${\cal E}(t,I_{\text{i}})$ at intermediate times, but ultimately arrives at the final adiabatic energy shell:
\begin{equation}
\label{eq:FFtraj}
z_F(t_{\text{f}}) = z_L(t_{\text{f}}) \in {\cal E}(t_{\text{f}},I_{\text{i}})
\end{equation}
In Sec. 2.5 of Ref.~\cite{2017Kolodrubetz}, the close relationship between LCD and FF driving is described in terms of canonical gauge transformations that map Hamiltonians of the form given by Eq.~\eqref{eq:hSTAlcd} into those given by Eq.~\eqref{eq:hSTAff}.
A similar approach was used in Ref.~\cite{2014Deffner}, and analogous unitary transformations were earlier introduced in the quantum context in Ref.~\cite{2012Ibanez}.
For scale-invariant driving, Eqs.~\eqref{eq:vsi} - \eqref{eq:uffdef} lead to the following fast-forward potential~\cite{2014Deffner}:
\begin{equation}
\label{eq:siuff}
U_{\rm FF}(x,t) = -\frac{m}{2}\frac{\ddot\sigma}{\sigma}(x-\mu)^2 - m\ddot \mu x .
\end{equation}
For the special cases of the harmonic oscillator and particle-in-a-box, equivalent results were obtained using an inverse engineering approach, in Refs.~\cite{2010Chen, 2012delCampo}.
Let us focus briefly on the harmonic oscillator, for which the original Hamiltonian is given by
\begin{equation}
\label{eq:ho}
H(z,t) = \frac{p^2}{2m} + \frac{m}{2}\omega^2(t) x^2 ,
\end{equation}
with $\dot\omega(0)=\ddot\omega(0)=\dot\omega(t_{\text{f}})=\ddot\omega(t_{\text{f}})=0$.
The potential $U(x,t)$ in Eq.~\eqref{eq:ho} can be cast into scale-invariant form (see Eq.~\eqref{eq:scaleInvart}) by setting $U_0(x)=mx^2/2$, $\sigma = \omega^{-1/2}$ and $\mu=0$.
Using Eq.~\eqref{eq:siuff}, we can then combine $U$ and $U_{\rm FF}$ into a single quadratic potential (see Eq.~\eqref{eq:hSTAff}):
\begin{equation}
H_{\rm SST}(z,t) = \frac{p^2}{2m} + \frac{m}{2}\Omega^2(t) x^2
\end{equation}
with
\begin{equation}
\label{eq:Omega2}
\Omega^2 = \omega^2 - \frac{3}{4}\frac{\dot\omega^2}{\omega^2} + \frac{1}{2} \frac{\ddot\omega}{\omega} \, .
\end{equation}
Rewriting the right side of Eq.~\eqref{eq:Omega2} in terms of $\sigma$ rather than $\omega$, and re-ordering terms, we get
\begin{equation}
\label{eq:ermakov}
\ddot\sigma + \Omega^2\sigma = \sigma^{-3} \, .
\end{equation}
This is the Ermakov equation, which was used in Ref.~\cite{2010Chen} as the starting point for designing a SST\ protocol for the harmonic oscillator.
When $\Omega$ is constant, this equation admits a simple solution, since $\sigma^2$ follows a harmonic oscillator
equation, see Eq. \eqref{eq:ermakovSolution} below.
\subsubsection{Boundary conditions in time}
\label{subsubsec:bc}
To this point, we have assumed that the time-dependence of $U$ is turned on and off smoothly; see Eq.~\eqref{eq:bc}.
For GCD and LCD driving this assumption is unnecessary: as long as $U$ is {\it once}-differentiable with respect to time, the results described in Secs.~\ref{subsubsec:gcd} and \ref{subsubsec:lcd} remain valid.
A discontinuity in $\partial_t U$ at $t=0$ or $t=t_{\text{f}}$ merely implies that $H_{\rm aux}$ is turned on or off abruptly.
With FF driving the situation is subtler.
If $\partial_t U$ is discontinuous at $t=0$, then so is the velocity field $v(x,t)$, hence the acceleration field $a(x,t)$ (Eq.~\eqref{eq:adef}) is undefined.
In this situation, the classical fast-forward method of Sec.~\ref{subsubsec:ff} can be salvaged if we apply the following {\it impulsive} potential at $t=0$:
\begin{equation}
U_{imp}(x,t) = -m \, \delta(t) \int^x dx^\prime \, v(x^\prime,0^+) .
\end{equation}
(The choice of the lower limit of integration is arbitrary.)
This potential causes a trajectory with initial phase space conditions $(x,p)$ at $t=0^-$ to jump suddenly to $(x,p+mv(x,0^+))$ at $t=0^+$.
If $\partial_t U$ is discontinuous at $t=t_{\text{f}}$, then another impulse is required:
\begin{equation}
U_{imp}(x,t) = +m \, \delta(t-t_{\text{f}}) \int^x dx^\prime \, v(x^\prime,t_{\text{f}}^-) .
\end{equation}
Once these impulses are included, the FF driving works as described in Sec.~\ref{subsubsec:ff}.
In particular all points located on the initial energy shell ${\cal E}(0,I_{\text{i}})$ at $t=0^-$ evolve to points on the final energy shell ${\cal E}(t_{\text{f}},I_{\text{i}})$ at $t=t_{\text{f}}^+$.~\footnote{If $\partial_t U$ is discontinuous at intermediate times $t_1, t_2, \cdots$, then similar impulses at those times can ``rescue'' the fast-forward method.}
For an illustration of the effects of such impulses in the context of the particle-in-a-box under fast-forward driving, see Sec.\ III.A of Ref.~\cite{2014Deffner}.
\subsubsection{Beyond one degree of freedom}
\label{subsubsec:beyond1dof}
It is natural to ask whether the classical GCD, LCD and FF recipes discussed above can be generalized to systems with $n>1$ degrees of freedom.
This question has largely remained unexamined in the literature.
When $n>1$, the definition of the adiabatic invariant itself depends on the dynamics generated by the system Hamiltonian $H(\bm{x},\bm{p},t)$.
If the dynamics are integrable at all values of $t$, then the phase space coordinates can be written in terms of action-angle variables $(\bm{I},\bm{w})$, and the action variables $(I_1,\cdots I_n)$ are adiabatic invariants ~\cite{1980Goldstein}.
It is plausible that the methods described above could be applied to construct a separate auxiliary Hamiltonian $H_{\rm aux}^k$ for each action-angle pair $(I_k,w_k)$, such that evolution under $H+\sum_k H_{\rm aux}^k$ would preserve the value of each invariant.
If the dynamics are ergodic over the energy shell at all values of $t$, then the phase space volume
\begin{equation}
\Omega(E,t) = \int d^n\bm{x} \int d^n\bm{p} \,\, \theta\left[ E - H(\bm{x},\bm{p},t) \right]
\end{equation}
is the sole adiabatic invariant~\cite{Hertz1910a,Hertz1910b,Anosov1960,Kasuga1961a,Kasuga1961b,Kasuga1961c,Ott1979,Lochak1988}.
In the special case when the ergodic Hamiltonians $H(\bm{x},\bm{p},t)$, with $t\in[0,t_{\text{f}}]$, constitute a {\it canonical family}~\footnote{$H(\bm{x},\bm{p},t)$ is a canonical family if $H(t_1)$ can be mapped to $H(t_2)$ by a canonical transformation, for any $t_1, t_2 \in [0,t_{\text{f}}]$~\cite{1994Robbins}.}, the $n$-dimensional analogue of Eq.~\eqref{eq:HCD-c2} is known to have a solution~\cite{1995Jarzynski}, suggesting that $H_{\rm GCD}$ can be constructed for canonical families of ergodic Hamiltonians.
However, it is not clear how to extend this approach to the typical situation in which $H(\bm{x},\bm{p},t)$ is not a canonical family.
Finally, integrable and ergodic Hamiltonians are themselves somewhat special: a generic classical Hamiltonian in $n>1$ degrees of freedom has a mixed phase space, with some trajectories evolving regularly and others irregularly (chaotically)~\cite{1993Ott}.
Adiabatic invariants for such systems are not as unambiguously defined as for integrable or ergodic Hamiltonians, hence the very notion of what constitutes a shortcut to adiabaticity becomes murky.
\subsection{Classical shortcuts and microcanonical ensembles}
The discussion in Sec.~\ref{subsec:recipes} focused on the evolution of individual trajectories (Eqs.~\eqref{eq:GCDtraj}, \eqref{eq:LCDtraj}, \eqref{eq:FFtraj}).
It is also useful to analyze this problem at the statistical level.
In this section, we consider a microcanonical ensemble of initial conditions on the energy shell ${\cal E}(0,I_{\text{i}})$.
This ensemble can be understood as a probability density that is distributed over the loop ${\cal L}_{\text{i}}$ shown in Fig.~\ref{fig:initialShellAndLoop}.
Specifically, if we use the angle $w \in [0,2\pi)$ of action-angle variables $(I,w)$~\cite{1980Goldstein} to label points around the loop, then the microcanonical distribution is uniform in $w$.
We emphasize that the microcanonical distribution represents a particular choice of initial conditions for the ensemble; it does not arise from coupling with a thermal bath.
We now describe what happens to this ensemble, as trajectories evolve from these initial conditions under global counterdiabatic, local counterdiabatic, and fast-forward dynamics.
In the GCD case, the ensemble of trajectories clings to the adiabatic energy shell ${\cal E}(t,I_{\text{i}})$ during the process, as the action is preserved (Eq.~\eqref{eq:GCDtraj}).
Moreover the ensemble remains microcanonical at all times: for any $t\in[0,t_{\text{f}}]$, a snapshot of the ensemble of trajectories would show them to be distributed uniformly, with respect to the angle variable $w$, on the adiabatic energy shell.
For LCD driving, the ensemble of trajectories also clings to the adiabatic energy shell (Eq.~\eqref{eq:LCDtraj}) only now the distribution does not necessarily remain microcanonical.
In particular, a snapshot at $t=t_{\text{f}}$ would reveal a collection of final conditions that are distributed non-uniformly with respect to $w$ -- see Fig. 3c of Ref.~\cite{2017Jarzynski} for an illustration.
The result that GCD driving preserves the microcanonical distribution while LCD driving (in general) does not, can be traced back to the fact that in the former case the auxiliary Hamiltonian is the same for all $I_{\text{i}}$, whereas in the latter case $H_{\rm aux}$ generally depends on $I_{\text{i}}$,
as discussed in Ref.~\cite{2017Jarzynski}, Appendix D.
For FF driving, the trajectories depart from the adiabatic energy shell at intermediate times but return to that shell at the final time.
From Eq.~\eqref{eq:FFtraj} it follows that the final conditions are generally distributed non-uniformly (i.e.\ non-microcanonically), as they coincide with the final conditions achieved under LCD driving.
At intermediate times the LCD and FF phase space distributions differ.
However, the projections of these distributions onto the $x$-axis are identical, by Eq.~\eqref{eq:FFboost}.
An analogous situation holds in the quantum case: the LCD and FF wavefunctions differ for $t \in (0,t_{\text{f}})$, but their $x$-space probability distributions coincide, $\vert\psi_{FF}(x,t)\vert^2 = \vert\psi_{LCD}(x,t)\vert^2 = \vert\langle x\vert n(t) \rangle\vert^2$.\footnote{
This conclusion follows by comparing Eqs.(6) and (22) of Ref.~\cite{2017Patra}, which give the exact time-dependent wavefunctions for LCD and FF dynamics, respectively.
}
Under scale-invariant driving, GCD and LCD trajectories are identical, as mentioned earlier.
In this special situation, all three flavors of shortcuts (GCD, LCD, FF) map an initial microcanonical distribution to a final microcanonical distribution.
\subsection{Shortcuts in classical kinetic theory}
\label{subsec:boltzmann}
We now discuss a situation in which classical shortcuts arise in the context of kinetic theory.
The discussion so far has focused on a single particle in a one-dimensional potential, Eq.~\eqref{eq:Hzt}.
The situation becomes more complicated when dealing with $N>1$ mutually interacting particles.
Under appropriate conditions, however, SST-inspired tools have found interesting applications to many-body, interacting systems.
Ref.~\cite{2014Guery-Odelin} considers a dilute gas of identical particles of mass $m$, governed by the Boltzmann equation
\begin{equation}
\label{eq:Boltzmann}
\frac{\partial f}{\partial t} + \bm{v}\cdot\nabla_{\bm{r}} f + \frac{1}{m}\bm{F}(\bm{r},t)\cdot\nabla_{\bm{v}} f = I_{\rm coll}[\bm{v}\vert f,f]
\end{equation}
where $f(\bm{r},\bm{v},t)$ is the single-particle density, $\bm{r}$ and $\bm{v}$ denote three-dimensional position and velocity, $\bm{F}=-\nabla_{\bm{r}}U(\bm{r},t)$ is a conservative force, and $I_{\rm coll}$ is a term that models the effect of two-body collisions.
When $I_{\rm coll}=0$ we get the collisionless equation
\begin{equation}
\label{eq:collisionless}
\partial_t f + \bm{v}\cdot\nabla_{\bm{r}} f + \frac{\bm{F}}{m}\cdot\nabla_{\bm{v}} f = 0 \, ,
\end{equation}
describing a gas of non-interacting particles.
If the potential $U$ is time-independent, then under Eq.~\eqref{eq:Boltzmann} the gas generically evolves to a canonical distribution, by Boltmann's H theorem~\cite{1988Cercignani,2010Kremer}.
However, Boltzmann realized already in the 1870's that when $U\propto r^2$ (with $r=|\bm{r}|$),
oscillatory ``breathing mode'' solutions of Eq.~\eqref{eq:Boltzmann} also exist.
It is known that $I_{\rm coll}=0$ when $f$ has the form
\begin{equation}
\label{eq:nullifyCollisions}
f(\bm{r},\bm{v},t) = \exp\left( -\alpha-\eta v^2 - \boldsymbol{\gamma}\cdot\bm{v}\right)
\end{equation}
where $\alpha$, $\eta$ and $\boldsymbol{\gamma}$ are arbitrary functions of position and time~\cite{2014Guery-Odelin}.
In other words the distribution $f$ given by Eq.~\eqref{eq:nullifyCollisions} belongs to the kernel of $I$.
Thus any solution of the collisionless equation (Eq.~\eqref{eq:collisionless}) that has the form given by Eq.~\eqref{eq:nullifyCollisions}, is also a solution of the Boltzmann equation (Eq.~\eqref{eq:Boltzmann}).
Building on this insight, the authors of Ref.~\cite{2014Guery-Odelin} discovered novel exact solutions of the Boltzmann equation.
In particular, they constructed time-dependent potentials of the form
\begin{equation}
\label{eq:2014G-Opotential}
U(\bm{r},t) = \frac{m}{2}\omega^2(t) r^2 + \frac{b}{r^2} \, ,
\end{equation}
with appropriately engineered $\omega(t)$, that rapidly steer the dilute gas from an initial canonical distribution at temperature $T_{\text{i}}$, to a final canonical distribution at temperature $T_{\text{f}}$, under the assumption that the evolution of the gas is accurately described by the Boltzmann equation.
Here, $b\ge 0$ should be time independent.
We now extend these results to include scale-invariant potentials as well as time-periodic driving.
Consider the time-dependent, spherically symmetric potential
\begin{equation}
\label{eq:tdssp}
\overline{U}(\bm{r},t) = \frac{1}{\sigma^2} U_0\left( \frac{r}{\sigma} \right) + \frac{b}{r^2} \, ,
\end{equation}
where $\sigma(t) > 0$ is a twice-differentiable function of time, and $b\ge 0$ is a constant.
Next, let $\beta(0)>0$ denote an inverse temperature, and define
\begin{equation}
\label{eq:betadef}
\beta(t) = \frac{\sigma^2(t)}{\sigma^2(0)} \beta(0)
\quad,\quad
\bar{\bm{v}}(\bm{r},\bm{v},t) = \bm{v} - \frac{\dot\sigma(t)}{\sigma(t)} \bm{r}
\end{equation}
and
\begin{equation}
\label{eq:BoltzPot}
U(\bm{r},t) = \overline{U}(\bm{r},t) - \frac{m}{2} \frac{\ddot\sigma(t)}{\sigma(t)}r^2
\end{equation}
(compare with Eqs.~\eqref{eq:vsi}, \eqref{eq:siuff}).
We claim that the time-dependent distribution
\begin{equation}
\label{eq:Boltzmann_exactf}
f(\bm{r},\bm{v},t) = \frac{1}{Z_0} e^{-\beta(t) [ m\bar v^2/2 + \overline{U}(\bm{r},t) ]}
\end{equation}
is an exact solution of the Boltzmann equation (Eq.~\eqref{eq:Boltzmann}), when the force $\bm{F}(\bm{r},t)$ is obtained from the potential given by Eq.~\eqref{eq:BoltzPot}~\footnote{The time-{\it in}-dependence of the normalization constant $Z_0$ follows from the form of $\overline{U}(\bm{r},t)$, as can be shown by inspection.}.
To establish this claim, note that Eq.~\eqref{eq:Boltzmann_exactf} has the form given by Eq.~\eqref{eq:nullifyCollisions}, therefore $I_{\rm coll}=0$.
Also, it follows from direct substitution that Eq.~\eqref{eq:Boltzmann_exactf} satisfies the collisionless equation, Eq.~\eqref{eq:collisionless}.
Thus Eq.~\eqref{eq:Boltzmann_exactf} solves Eq.~\eqref{eq:Boltzmann} exactly.
We can extend these results further by replacing Eq.~\eqref{eq:tdssp} with
\begin{equation}
\label{eq:BoltzPot2}
\overline{U}(\bm{r},t) = \frac{1}{\sigma^2} \sum_{j=1}^3 U_{0j}\left( \frac{x_j}{\sigma} \right) + \, U_h(x_1,x_2,x_3)
\end{equation}
where the $x_j$'s are the Cartesian components of $\bm{r}$, the $U_{0j}$'s are three (generally unrelated) potential functions, and $U_h$ is a homogeneous function of degree -2:
\begin{equation}
U_h(x_1,x_2,x_3) \,=\, \frac{1}{(x_1x_2x_3)^{2/3}} \, \psi\left(\frac{x_1}{x_2},\frac{x_1}{x_3} \right),
\end{equation}
where $\psi$ is an arbitrary function.
This form for $U_h$ includes a wide range of potentials, including as special cases $b/r^2$ (see Eq.~\eqref{eq:tdssp}) and $a_1/x_1^{2} + a_2/x_2^{2} + a_3/x_3^{2}$. The potential $U_h$ should bear no explicit time dependence, unlike $\overline{U}$, which depends explicitly on time through $\sigma$.
If we again define $U(\bm{r},t) = \overline{U} - m\ddot\sigma r^2/2\sigma$ (see Eq.~\eqref{eq:BoltzPot}), then $f(\bm{r},\bm{v},t)$ given by Eq.~\eqref{eq:Boltzmann_exactf} remains an exact solution of the Boltzmann equation \eqref{eq:Boltzmann}.
This result can be established as in the previous paragraph.
If we set $U_0(x)=U_{0j}(x)=\kappa x^2/2$, for some fixed $\kappa >0$ and for $j=1,2,3$, and $U_h = b/r^2$ for some $b\ge 0$, then Eqs.~\eqref{eq:tdssp} and \eqref{eq:BoltzPot2} describe the same potential, and $U(\bm{r},t)$ has the form given by Eq.~\eqref{eq:2014G-Opotential}.
If $U_{0j}(x)=\kappa_jx^2/2$ for different fixed values $\kappa_1, \kappa_2, \kappa_3>0$, and $U_h=0$, then $\overline{U}(\bm{r},t)$ is an anisotropic harmonic potential of the form considered in Ref.~\cite{2015Papoular}, where exact solutions of the Boltzmann equation were derived for such potentials.
Note that, in general, even if the three $U_{0j}$'s are identical functions, the resulting potential $\overline{U}(\bm{r},t)$ (Eq.~\eqref{eq:BoltzPot2}) is not spherically symmetric.
We can use these results to design a protocol for driving the gas from a canonical distribution at initial inverse temperature $\beta(0)^{-1}$ to a canonical distribution at final inverse temperature $\beta(t_{\text{f}})^{-1}$, in an arbitrary, finite time $t_{\text{f}}$.
To do this, we simply choose $\sigma(t)$ to satisfy the boundary conditions $\dot\sigma=\ddot\sigma=0$ at $t=0$ and $t=t_{\text{f}}$, and
\begin{equation}
\sigma(t_{\text{f}})/\sigma(0)= \sqrt{\beta(t_{\text{f}})/\beta(0)} \, .
\end{equation}
This protocol can be used with either of the potential forms given by Eqs.~\eqref{eq:BoltzPot} or \eqref{eq:BoltzPot2}.
In the isotropic and anisotropic harmonic cases discussed in the previous paragraph, these protocols reduce to the ones obtained in Refs.~\cite{2014Guery-Odelin} and \cite{2015Papoular}, respectively.
Alternatively, we can design a driven breathing mode of the Boltzmann equation by choosing the protocol
\begin{equation}
\sigma(t) = \bar\sigma + \Delta\sigma \cos^2(\omega t) \quad, \quad 0 < \Delta\sigma < \bar\sigma
\end{equation}
again using either Eq.~\eqref{eq:BoltzPot} or Eq.~\eqref{eq:BoltzPot2}.
The exact solution given by Eq.~\eqref{eq:Boltzmann_exactf} then oscillates periodically in time, with frequency $2\omega$.
Finally, let us consider how the above results relate to the {\it undriven} breathing modes discovered by
Boltzmann \cite{2014Guery-Odelin}.
In Eq.~\eqref{eq:tdssp}, take $U_0(x) = mx^2/2$ and $b=0$ so that $\overline{U}(\bm{r},t)$ becomes a spherically symmetric harmonic oscillator with a time-dependent stiffness $\sigma^{-4}(t)$.
By Eq.~\eqref{eq:BoltzPot} this choice leads to the total potential
\begin{equation}
U(\bm{r},t) = \frac{m}{2} \left( \frac{1}{\sigma^4} - \frac{\ddot\sigma}{\sigma} \right) r^2 \equiv \frac{m}{2}\Omega^2(t) r^2 \, .
\end{equation}
Note that the relationship between $\sigma$ and $\Omega$ here is described by the Ermakov equation \eqref{eq:ermakov}.
If we now take $\Omega(t)$ to be constant rather than time-dependent, and we solve for $\sigma(t)$ (see Eq.~\eqref{eq:ermakovSolution} below), then Eq.~\eqref{eq:Boltzmann_exactf} becomes a time-periodic solution of the Boltzmann equation for a dilute gas in a fixed harmonic potential $U(\bm{r},t) = m\Omega^2 r^2/2$.
This class of solutions coincides with Boltzmann's breathing modes.
A related question pertains to the type of static confining potential for which a breathing solution can exist.
It is addressed in Appendix \ref{app:breathers}.
When $\Omega$ is constant, the general solution of the Ermakov equation
\eqref{eq:ermakov} is given by
\begin{equation}
\label{eq:ermakovSolution}
\sigma(t) = \left[ \tau - \sqrt{\tau^2-\frac{1}{\Omega^2}} \cos(2\Omega t + \phi) \right]^{1/2}
\end{equation}
where $\phi$ and $\tau\ge 1/\Omega$ are constants.
Substituting this result into Eq.~\eqref{eq:betadef} for $\beta(t)$ reveals that the effective inverse temperature $\beta(t)$ oscillates harmonically with frequency $2\Omega$.
Moreover, taking the first three derivatives of $\beta(t)$ with respect to time, we straightforwardly obtain
\begin{equation}
\dddot\beta(t) + 4\Omega^2 \dot\beta(t) = 0
\label{eq:PRL2014_9b}
\end{equation}
which is equivalent (for static $\Omega$) to Eq. (9b) of Ref.~\cite{2014Guery-Odelin}.
\subsection{Relations to quantum shortcuts to adiabaticity}
\label{subsec:classical-quantum}
The classical shortcuts described above have quantum counterparts as we now briefly discuss.
We have already seen this correspondence in the global counterdiabatic case: just as the classical term $H_{\rm GCD}(z,t)$ defined by Eq.~\eqref{eq:HCD-c2} generates the desired shortcut for any choice of initial energy shell (Eq.~\eqref{eq:GCDtraj}), so too its quantum counterpart $\widehat H_{\rm GCD}(t)$ given by Eq.~\eqref{eq:HCD-q2} generates the desired shortcut for any energy eigenstate~\cite{2003Demirplak,2009Berry}.
For local counterdiabatic driving, the auxiliary term $H_{\rm LCD}=pv(x,t)$ is designed for a specific choice of initial energy shell, as discussed in Sec.~\ref{subsubsec:lcd}.
In the quantum case, let us instead choose an initial energy eigenstate $\vert n(0)\rangle$ and construct the cumulative distribution
\begin{equation}
{\cal F}(x,t) = \int_{-\infty}^x dx^\prime \, \left\vert \phi_n(x^\prime,t) \right\vert^2
\quad,\quad
\phi_n(x,t) \equiv \langle x \vert n(t) \rangle.
\end{equation}
Now define a velocity field $v(x,t)$ in a manner analogous to Eq.~\eqref{eq:vdef}, but with ${\cal F}(x,t)$ playing the role of ${\cal S}(x,t)$:
\begin{equation}
\label{eq:qvdef}
v(x,t) = -\frac{\partial_t{\cal F}}{\partial_x{\cal F}} \, .
\end{equation}
Using this field we construct a quantum auxiliary term
\begin{equation}
\label{eq:HLCD-q}
\widehat H_{\rm LCD}(t) = \frac{\widehat p\widehat v + \widehat v\widehat p}{2}
\quad , \quad
\widehat v \equiv v(\widehat x,t)
\end{equation}
(compare with Eq.~\eqref{eq:HLCD}).
If the system begins in the state $\vert n(0)\rangle$ and then evolves under $\widehat H(t) + \widehat H_{\rm LCD}(t)$, it will remain in the instantaneous eigenstate $\vert n(t)\rangle$ (up to an overall time-dependent phase) for all $t\in [0,t_{\text{f}}]$~\cite{2017Patra}.
Using the velocity field $v(x,t)$ given by Eq.~\eqref{eq:qvdef}, we next construct an acceleration field $a(x,t) = v\partial_x v + \partial_t v$ and a corresponding fast-forward potential $U_{\rm FF}(x,t)$ via $-\partial_x U_{\rm FF} = ma$ (see Eqs.~\eqref{eq:adef} and \eqref{eq:uffdef}).
If the system evolves under $\widehat H(t) + \widehat U_{\rm FF}(x,t)$, from an initial state $\vert n(0)\rangle$, then at the final time $t=t_{\text{f}}$ it will arrive at the eigenstate $\vert n(t_{\text{f}})\rangle$ (up to a phase), though at intermediate times it generally will not be in the state $\vert n(t)\rangle$~\cite{2017Patra}.
The potential $U_{\rm FF}(x,t)$ obtained in this manner -- using ${\cal F}$ rather than ${\cal S}$ to construct $v$ -- is equivalent to the fast-forward potential originally derived by Masuda and Nakamura~\cite{2010Masuda} and further studied in Refs.~\cite{2012Torrontegui,2016Martinez-Garaot,2015Takahashi}.
It is interesting to note that both quantum and classical auxiliary terms are constructed using a velocity field determined from a cumulative function, either ${\cal F}(x,t)$ or ${\cal S}(x,t)$.
As we shall see later, this pattern applies as well to stochastic shortcuts to adiabaticity, where the cumulative function ${\cal F}(x,t)$ is given in terms of a canonical probability distribution.
The discussion in the previous three paragraphs requires a rather strong caveat.
For excited states $n>0$, the field $v(x,t)$ given by Eq.~\eqref{eq:qvdef} generically diverges at the nodes of the eigenstate, i.e.\ where $\phi_n(x,t) = 0$, leading to ill-behaved auxiliary terms $\widehat H_{\rm LCD}$ and $\widehat U_{\rm FF}$.
Thus in general the quantum LCD and FF approaches described above are limited to ground states $n=0$, which have no nodes;
scale-invariant systems are an exception to this statement~\cite{2014Deffner}.
Ref.~\cite{2021Patra} develops a semiclassical fast-forward approach that avoids the problem posed by the nodes of $\phi_n(x)$.
In this approach, the fields $v$ and $a$ are constructed using ${\cal S}$ (Eq.~\eqref{eq:vdef}) rather than ${\cal F}$ (Eq.~\eqref{eq:qvdef}).
The resulting auxiliary term $\widehat U_{\rm FF}$ is well-behaved, but no longer guides the wavefunction {\it exactly} to the desired final state $\vert n(t_{\text{f}})\rangle$.
Instead it provides an approximate shortcut that is expected to work well in the semiclassical regime of large $n$.
Numerical simulations support this expectation~\cite{2021Patra}.
\section{Shortcuts for classical systems in contact with a thermal bath}
\label{sec:systems-with-bath}
In the previous section, we have obtained finite-time protocols
for driving an isolated system, with the same final state
as with an infinitely long driving. In the remainder,
our interest goes to systems that are (strongly) coupled to an environment; the systems we have in mind are epitomized by a Brownian object such as a colloid in a
fluid. The latter plays the role of a thermal bath \cite{2017Ciliberto}, and necessitates a stochastic/probabilistic description,
a new feature compared to the treatment in section \ref{sec:isolated-systems}.
This goes with a point-of-view-change,
from single trajectories to probability distributions.
From an experimental perspective, two routes can be proposed,
where the question, formulated in a probabilistic fashion, becomes meaningful.
a) One may be interested in repetitions of an experiment
involving a Brownian object, such as a macromolecule,
or a nano device; statistics is then gathered by automatizing
the protocol, as e.g. in~\cite{2016Martinez_b} or~\cite{2020Besga};
b) A single experiment can allow for measuring a distribution function, provided it involves a collection of Brownian objects,
manipulated simultaneously. When they do not interact,
or when interactions are weak such as in a
low density colloidal system, routes a) and b) are equivalent.
In a nutshell, what we now aim at suitably driving with an external force
is a probability density, which shall be denoted $\rho(x,t)$:
What should the drive be to evolve the system from an initial density
$\rho(x,t_\text{i})$ at time $t_\text{i}$ to a target distribution $\rho(x,t_\text{f})$ at time $t_\text{f}$?
Such a question of fast driving a system from a given equilibrium state to another appeared under different names in the literature:
Engineered Swift Equilibration \cite{2016Martinez}~\footnote{Also variants such as Engineered Swift Relaxation for the connection of non-equilibrium states~\cite{2021Prados}, see also Sec.~\ref{sec:beyond-equilibrium}.}, or Shortcuts to Isothermality \cite{2017Li,comment11}.
It has been investigated experimentally for small systems (e.g. colloids \cite{2016Martinez} and AFM tips \cite{2016Le-Cunuder}) in contact with a thermostat. There, one searches for the proper variation of the control parameters to ensure a transfer from the initial state at time $t_\text{i}$ to the desired final state at time $t_{\text{f}}$.
\subsection{Inverse Engineering}\label{sec:inverse-engineering}
A method that often proves efficient in practice is the commonly-called inverse engineering technique. This approach has been successfully applied to ordinary differential equations in classical and quantum physics \cite{2019Guery-Odelin}. If we denote by $\bm X$ the dynamical variables (state vector of size $n\times 1$), the control problem is encapsulated in a set of coupled differential equations
\begin{equation}
\dot{ \bm X} = {\bm f} ( {\bm X}, {\bm \lambda}(t) ).
\label{controlequation}
\end{equation}
There is no general statement for the reachability of the desired state under the driving provided by the control vector $ {\bm \lambda}(t)$ of size $r\times 1$ except when $ {\bm f}$ is a linear function with time-invariant coefficients, for which the Kalman rank condition applies \cite{1963Kalman,1987Pontryagin} \footnote{For a linear system of the type ${\bm f} ({\bm X},{\bm \lambda(t)})=A{\bm X}(t) + B{\bm \lambda}(t)$, we define the $n\times nr$ controllability matrix $C=[B\;AB\;...\;A^{n-1}B]$. The system is controllable if $C$ has $n$ linearly independent columns, i.e. rank$(C)=n$ \cite{1963Kalman,1987Pontryagin}.}. When the desired state belongs to the reachable set of solutions, there exists a wide variety of possible protocols to reach the final state.
The inverse engineering method provides a convenient way to find out a proper driving. It amounts to impose the evolution of the dynamical variable ${\bm X}$ and to infer from Eq.~\eqref{controlequation}, the expression for $ {\bm \lambda}(t)$. However, this inverse use of the differential equation that governs the dynamics is not always easy to handle. The mathematical property which allows for such an inverse use of the dynamical (including nonlinear) equations is known as the flatness property, and can be considered as an extension of the Kalman's controllability criterion \cite{1995FLIESS}. This strategy has been successfully used to transport a particle in a moving harmonic potential, both in classical and quantum physics \cite{2011Torrontegui,2014Guery-Odelin_b,2021Zhang}, or to shuttle the particle, i.e. to set a given velocity to the particle---see \cite{2021Qi} and references therein.
For a system in contact with a thermal bath, the dynamical variables obey stochastic differential equations (Langevin-like) that cannot be directly written as a set of continuous equations such as that of Eqs.~(\ref{controlequation}). For Brownian motion, this difficulty has been circumvented in two steps. First, as emphasized above,
the discussion is made not on the individual trajectories but on the density distribution that obeys the Smoluchowski, or Fokker-Planck (FP), equation (see Appendix \ref{app:stoch-frame} for a crash recapitulation of the essential aspects of the Langevin and frameworks
for stochastic processes).
Second, an ansatz depending on a set of a few effective dynamical variables, such as the moments of the PDF, is usually proposed to get a finite set of equations in the form of Eqs.~(\ref{controlequation}).
The coupled equations on the moments can be alternatively derived from the Langevin equation.
Consider the overdamped motion of a bead of micron size immersed in a fluid and trapped by a general confining potential $U(x,t)$. The density $\rho(x,t)$ obeys the overdamped ---or Smoluchowski---equation
\begin{equation}
\label{eq:FP}
\gamma \partial_t\rho(x,t) = \partial_x \left[ \partial_xU(x,t) \rho(x,t) \right] + \beta^{-1} \partial^2_{x}\rho (x,t),
\end{equation}
where $\gamma$ is the friction coefficient and $\beta=(k_BT)^{-1}$ refers to the inverse of the temperature, $k_B$ being Boltzmann's constant---see Appendix~\ref{app:stoch-frame} for a detailed discussion of the stochastic dynamics framework.
Particular attention has been paid to the harmonic potential, which models paradigmatic systems such as optical tweezers. Equation \eqref{eq:FP} then becomes
\begin{equation}
\label{eq:FP-harm}
\gamma \partial_t \rho(x,t) = \partial_x \left[ \kappa x \rho(x,t) \right] + \beta^{-1} \partial^2_{x}\rho(x,t).
\end{equation}
In the harmonic problem, the control (time-dependent)
parameters are {\it a priori} the temperature $T$ (see
section \ref{sec:engineering-thermal-env}) and the stiffness $\kappa$ of the potential. Here, the statistics of the bead's position remains Gaussian
if it is initially, with a standard deviation $\sigma(t)$ that
plays the role of an effective dynamical variable and obeys
\begin{equation}
\label{eq:standard-dev-harmonic}
\dot \sigma = - \frac{\kappa(t)}{\gamma}\sigma + \frac{k_BT}{\gamma}\frac{1}{\sigma} .
\end{equation}
For both time-independent temperature and stiffness, the system approaches the canonical equilibrium distribution, for which the variance of the position is
\begin{equation}
\label{eq:sigma-eq}
\sigma_{eq}^2=\frac{k_B T}{\kappa},
\end{equation}
i.e. the equilibrium equation of state.
Let us assume for instance that the temperature is constant and we vary the stiffness of the trap. The initial state corresponds to thermal equilibrium state with the initial stiffness $\kappa_{\text{i}}=\kappa(t_\text{i})$. The objective is to reach the equilibrium state with the desired final value of $\kappa_{\text{f}}=\kappa(t_{\text{f}})$ in a chosen amount of time $t_{\text{f}}$. Those boundaries conditions define the values $\sigma_{\text{i}}=(k_BT/\kappa_{\text{i}})^{1/2}$ and $\sigma_{\text{f}}=(k_BT/\kappa_{\text{f}})^{1/2}$ at initial and final time. The inverse engineering technique involves the choice of an interpolation function $\sigma(t)$ between those two values, and to subsequently infer the time-dependent stiffness $\kappa(t)$ to be applied, directly from Eq.~\eqref{eq:standard-dev-harmonic},
since both $\sigma$ and $\dot\sigma$ are then known. For the very same problem in the underdamped regime, one shall solve the Kramers equation for the phase space distribution. In this latter case, the effective dynamical system boils down to a set of 3 coupled linear equations for the time evolution of the 3 moments $\langle x^2 \rangle$, $\langle v^2 \rangle$, and $\langle x v \rangle$. As a result, the strategy to extract $\kappa(t)$ is slightly more involved \cite{2018Chupeau_b}.
The inverse engineering method applied here
is specific to the manipulation of Gaussian states. As already commented, a remarkable property is that such an initial condition, under time-dependent harmonic forcing, remains Gaussian
at all times, and thus preserves its shape \footnote{This can be viewed as a consequence of the central limit theorem, and dwells on the fact that the solution to Langevin equation is then linear in the noise history.}.
Yet, while it is difficult to compute analytically the potential required
to connect two arbitrary non-Gaussian states, it turns out that if both have the same shape, a simple solution can be found, therefore generalizing the aforementioned
Gaussian result. We here impose the following shape for the PDF $\rho(x,t)$:
\begin{equation}
\label{eq:sh-pres}
\rho(x,t) = \frac{1}{\sigma(t)Z_{\text{i}}} \exp \left[ -\beta U_{\text{i}}\left( \frac{x-\mu(t)}{\sigma(t)}\right)\right] , \end{equation}
with $\mu(t)$ and $\sigma(t)$ continuous real functions, such that $\mu(t_\text{i})=0$ and $\sigma(t_\text{i})=1$, which accounts for shifting and re-scaling the space dependence. Note that, although we used similar ideas in the scale-invariant protocols introduced in Eq.~\eqref{eq:scaleInvart}---justifying the same notation for the parameters, processes preserving the shape are different from those scale-invariant protocols~\footnote{Specifically, one can be mapped onto the other if $U_\text{i}(x)$ is proportional to a power of $x$. For example, if we consider the harmonic potential $U_\text{i} (x)=k_\text{i} x^2/2$, the potential $U$ remains harmonic for both cases, shape-preserving and scale-invariant, but with a different stiffness, $k_\text{i}/\sigma^2$ and $k_\text{i}/\sigma^4$ respectively. For general $U_\text{i}$, the map is not guaranteed.}. The quantity $Z_{\text{i}}=\int_{\cal D} dx\, e^{-\beta U_{\text{i}}(x)}$ is the partition function that guarantees the correct normalization of the distribution. Such distributions only connect states that belong to the same family of potentials, whatever this family is. The final potential then reads \cite{2021Plata}
\begin{equation}
U_{\text{f}} (x)=U_{\text{i}}\left[ \frac{x-\mu(t_{\text{f}})}{\sigma(t_{\text{f}})}\right].
\end{equation}
The time-dependent driving potential required to ensure the shape preservation is found by introducing the ansatz \eqref{eq:sh-pres} into the equation. One finds \cite{2021Plata}
\begin{align}
\label{eq:driv-sh-pres-U}
U(x,t) & = U_{\text{i}} \left[ \frac{x - \mu(t)}{\sigma(t)}\right] \nonumber \\ & -\gamma \frac{2\dot{\mu}(t)\sigma(t)[x-\mu(t)]+[x-\mu(t)]^2\dot{\sigma}(t)}{2\sigma(t)}.
\end{align}
Such a driving potential involves two different contributions: the shape-bearing potential itself and an additional harmonic potential, whose stiffness and center are determined by certain combinations of the shift and scaling functions $\mu(t)$ and $\sigma(t)$. The extra time-dependent harmonic potential is nothing but the counterdiabatic term, to be discussed further below. To enforce the smoothness of the potential, one can add the following extra conditions on the parameters at initial and final time: $\dot{\mu}(t_\text{i})=\dot{\mu}(t_{\text{f}})=0$ and $\dot{\sigma}(t_\text{i})=\dot{\sigma}(t_{\text{f}})=0$.
Inverse engineering techniques have also been employed to address the underdamped situation~\cite{2018Chupeau_b}.
The authors worked out eligible conservative, velocity-independent, drivings $U(x,t)$. However, the problem becomes involved and limitations appear, presumably inherent to the functional forms chosen for constructing explicit solutions.
Related in spirit to inverse engineering are stochastic methods
that generate Brownian paths conditioned to start and end at prescribed (ensemble of) point
~\cite{2015Majumdar}.
The conditioning, that endows these paths with precise statistical properties, can be of various types: a bridge, meaning a path that starts at some $x_\text{i}$ at $t=t_\text{i}$, and ends at a given $x_{\text{f}}$ at $t=t_{\text{f}}$;
an excursion, meaning a bridge with $x_{\text{f}}=x_\text{i}$ that is furthermore constrained to lie
at all times to the right of $x_\text{i}$, etc.
These methods can be used to generate computationally
the constrained paths in an efficient manner; a naive variant would amount to pruning an ensemble of unconstrained paths, keeping only those
trajectories that fulfil the imposed constraints, a highly inefficient way of proceeding. For overdamped Langevin dynamics,
it was shown on general grounds that irrespective of $x_\text{i}$,
addition of an external harmonic force centered at $x_{\text{f}}$ and with
stiffness $\gamma /(t_{\text{f}}-t) $, which thus diverges for $t\to t_{\text{f}}$,
generates a {\it bona fide} bridge \cite{2015Majumdar}. More generally, as might be anticipated from the previous example,
the gist of the approach is to add a time-dependent entropic potential of the form
$-2 k_BT \log Q(x,t)$; $Q$, encoding the constraints considered (such as remaining in the
allowed half-space for an excursion), is the probability density to be at $x_{\text{f}}$ at time $t_{\text{f}}$,
having started at point $x$ at time $t<t_{\text{f}}$
\cite{2015Majumdar,1957Doob}:
\begin{equation}
Q(x,t) \equiv P(x_{\text{f}},t_{\text{f}}| x,t) .
\end{equation}
While this result can be obtained by inverse engineering,
it should be stressed that it does feature an important
difference with short-cutting ideas discussed here, in the sense
that there does not exist an infinitely slow process that
is being accelerated in some form.
Note that $Q$ introduced above fulfills the backwards equation, and is as such intimately related to first passage problems~\cite{2001Redner}.
\subsection{Counterdiabatic method}
\label{sec:counterdiabatic}
In this section, we extend the counterdiabatic method (see Secs.~\ref{sec:intro} and \ref{sec:isolated-systems}) to systems in contact with a thermal bath.
As above, we model the evolution of such systems with overdamped Langevin dynamics at the single-trajectory level, and with the FP equation at the ensemble level, and we restrict ourselves to systems with a single degree of freedom, $x$.
Because momentum is ignored in the overdamped limit, we will use the generic notation $U(x)$ rather than $H(x,p)$ to denote the system's energy function.
This potential, used to drive the system, can be viewed as a Hamiltonian.
In the isolated quantum and classical cases described in Secs.~\ref{sec:intro} and \ref{sec:isolated-systems}, counterdiabatic driving aims to preserve an adiabatic invariant under rapid driving.
In the present context the role of the adiabatic invariant is played by the functional form of the probability distribution function (PDF), $\rho(x,t)$.
Specifically, when the potential $U$ is driven very slowly, its PDF evolves through a continuous sequence of equilibrium states $\rho_{eq}(x,t)$,
\begin{align}
\rho_{eq}(x,t)&= e^{\beta[F(t)-U(x,t)]}, \nonumber \\ F(t)&=-\beta^{-1} \ln \left[ \int dx \,e^{-\beta U(x,t)} \right].
\label{eq:rho_eq_preserved}
\end{align}
In the counterdiabatic method we seek to construct a term $U_{CD}(x,t)$ such that under the full driving potential $U_{\rm SST}=U+U_{CD}$, the system evolves through the same equilibrium states $\rho_{eq}(x,t)$, even when the time dependence of $U(x,t)$ is not slow.
Note that, although the fine details may be different, the idea behind the counterdiabatic method could be cast under the umbrella of inverse engineering techniques. That is, we want to find an auxiliary $H_{CD}$ that enforces the conservation of the adiabatic invariant. The separation into categories of the techniques analyzed in this review has to be understood as a choice made more by pedagogical purposes than by a rigorous categorization.
Consider an overdamped Brownian particle whose statistical state $\rho(x,t)$ obeys the FP equation~\eqref{eq:FP}---see Appendix~\ref{app:stoch-frame} for details---with the potential $U_{\rm SST}(x,t)=U(x,t)+U_{CD}(x,t)$. The instantaneous equilibrium distribution associated with $U$,
$\rho_{eq}$ given by Eq.~\eqref{eq:rho_eq_preserved},
plays the role of adiabatic invariant. In other words, we aim to steer the system so that the equilibrium PDF of the unperturbed potential $U(x,t)$ is maintained throughout the process.
When $U(x,t)$ varies at finite rate, $U_{CD}\neq 0$ is needed to preserve the prescribed evolution $\rho_{eq}$. Specifically, the counterdiabatic term escorting the adiabatic evolution is given by~\cite{2017Li}
\begin{equation}
- \partial_x U_{CD}(x,t) = -\gamma \frac{\int^x dx' \partial_t \rho_{eq}(x',t)}{\rho_{eq}(x,t)}.
\label{eq:UCD_FP}
\end{equation}
This expression can be derived from the cumulative distribution
\begin{equation}\label{eq:cumulative-distr}
\mathcal{F}(x,t)= \int_{-\infty}^{x} dx' \rho(x',t)
\end{equation}
and its velocity field
\begin{equation}\label{eq:veloc-field-CD}
v(x,t)=\partial_t x(\mathcal {F},t) = - \frac{\partial_t \mathcal{F}(x,t)}{\partial_x \mathcal{F}(x,t)},
\end{equation}
where $x(\mathcal {F},t)$ is the function obtained by inverting $\mathcal{F}(x,t)$ for fixed $t$. Specifically, using the velocity field, we obtain
\begin{equation}
\label{eq:UCD-stoch}
- \partial_x U_{CD}(x,t) = \gamma v(x,t).
\end{equation}
This approach is particularly appealing since it has been shown to be useful as a unified procedure to derive counterdiabatic terms in quantum, classical and thermal systems \cite{2017Patra}---as in Eqs.~\eqref{eq:vdef} and \eqref{eq:HLCD}.
The counterdiabatic force given in Eq. \eqref{eq:UCD_FP} allows to
recover the inverse engineering results derived for shape preserving
potentials in section~\ref{sec:inverse-engineering}. Indeed,
injecting relation \eqref{eq:sh-pres}
into \eqref{eq:UCD_FP} (with $\rho$ playing the role of $\rho_{eq}$) yields, rather unexpectedly,
a counterdiabatic force that does not depend on $U$ and thus holds irrespective of the functional form chosen for the PDF \cite{2021Plata}, provided this form is conserved
(and is thus a preserved ``adiabatic invariant''):
\begin{equation}
\label{eq:driv-sh-pres}
\partial_x U_{CD}(x,t) \,=\, -\gamma \, \frac{\dot{\mu}(t)\sigma(t)+[x-\mu(t)]\dot{\sigma}(t)}{\sigma(t)},
\end{equation}
which is consistent with the potential given in Eq.
\eqref{eq:driv-sh-pres-U}, remembering
$U_{\rm SST}=U+U_{CD}$.
The counterdiabatic method has been also employed in the underdamped situation~\cite{2014Tu,2017Li,2021Li_b}. At variance with the conservative drivings $U(x,t)$ derived with the inverse engineering technique~\cite{2018Chupeau_b}, the counterdiabatic potential contains a term depending on the momentum of the particle. This entails serious difficulties for the experimental implementation of such a counterdiabatic driving.
Finally, we present in appendix~\ref{app:fluctuation-relation} a derivation of the work fluctuation relation~\cite{1997Jarzynski} from counterdiabatic type of arguments.
\subsection{Fast-forward}\label{sec:fast-forward}
Now, let us consider another strategy for the swift connection of equilibrium states: the so-called fast-forward procedure. In the quantum case~\cite{2008Masuda,2010Masuda}, fast-forward refers to a protocol that makes it possible to reach a desired final state, independently of the path swept to do so. For the isolated classical systems analysed in Sec.~\ref{sec:isolated-systems} of this review, fast-forward has been employed in a similar sense. Therein, the fast-forward protocol led the system to the target state, with the same value of the adiabatic invariant as the initial state, by adding a velocity-independent potential $U_{FF}(x,t)$. The price of such a procedure was the adiabatic invariant no longer preserved at intermediate times. Derivation of fast-forward protocols are unified in both quantum and classical mechanics by using the acceleration flow field~\cite{2017Patra}, as discussed in Sec.~\ref{subsubsec:ff}.
Here, we consider the extension of the fast-forward idea to the context of systems with stochastic dynamics, again described by the overdamped FP equation~\eqref{eq:FP}. In contrast to the counterdiabatic driving just described, there is no underlying adiabatic transformation over which the shortcut is built. The idea is the following: one considers a certain reference process---not necessarily slow---that connects two given states and then searches for a tailor-made external potential that accelerates this reference process~\cite{2021Plata}. In this way, the ``frames" of the ``movie" are fixed, given by the reference process, but are played at a higher rate in the fast-forward protocol. As explained below, the same idea allows for reproducing the frames at a lower rate (slow-forward) or even play the movie backwards, at a higher or a lower rate (fast-backward or slow-backward, respectively)---somehow generalizing the shortcuts described in this review.
Going into specifics, and following Ref.~\cite{2021Plata}, we consider a reference solution $\rho_r(x,t)$ of Eq.~\eqref{eq:FP} under a reference potential $U_r(x,t)$. Then, we introduce a time distortion $\Lambda(t)$ of the reference, i.e., $\rho(x,t)=\rho_r(x,\Lambda(t))$, and look for the potential $U(x,t)$ required to drive the evolution following the given prescription $\rho_r(x,\Lambda(t))$. It is important to remark that, at variance with the classical case, the connecting path is fixed and given by $\rho_r(x,\Lambda(t))$.
The FP equation~\eqref{eq:FP} is rewritten as a continuity equation
\begin{subequations}\label{eq:FP-continuity}
\begin{eqnarray}
\partial_t \rho&=&-\partial_x \left(\rho v\right), \\
\label{eq:veloc-field-FP}
v(x,t)&=& -\gamma^{-1} \left[ \partial_x U(x,t) +\beta^{-1} \partial_x \ln\rho(x,t) \right]
\end {eqnarray}
\end{subequations}
where $v(x,t)$ is a velocity field~\footnote{This velocity field is exactly the same defined in the counterdiabatic driving, since spatial integration of the FP equation, written in the form \eqref{eq:FP-continuity}, from $-\infty$ to $x$ leads precisely to Eq.~\eqref{eq:veloc-field-CD}.}. The derivation of the fast-forward protocol is based on the relation between the velocity fields
in the reference and the manipulated process,
\begin{equation}
v(x,t)=\dot{\Lambda}(t) v_r(x,\Lambda(t)).
\end{equation}
Hence, the driving potential can be solved. For the sake of clarity, we display the solution for the auxiliary potential,
\begin{align}
-\partial_x U_{FF}(x,t) = &[1-\dot{\Lambda}(t)] \nonumber \\
& \times [\partial_x U_r(x,\Lambda(t))+ \beta^{-1} \partial_x \ln \rho_r(x,\Lambda(t)) ] .
\end{align}
The total driving potential is $U(x,t)=U_r(x,\Lambda(t))+U_{FF}(x,t)$. The solution is given in terms of the reference process and the time map $\Lambda(t)$. As expected, when the reference and manipulated dynamics coincide, i.e.,
$\Lambda(t)=t$, one simply has $\partial_x U(x,t) =\partial_x U_r(x,t)$ or, equivalently, $\partial_x U_{FF}(x,t) = 0$.
This type of driving not only allows for acceleration
($\dot\Lambda>1$) but also for deceleration ($0<\dot\Lambda<1$) and even for the inversion of time's arrow ($\dot\Lambda<0$, meaning that the reference dynamics can be ``played backwards''). Combining simple reference processes, it is possible to build up an operational welding protocol that connects arbitrary states \cite{2021Plata}. Specifically, one can produce a welding connection between an initial state $\rho_\text{i}(x)$ and the target state $\rho_\text{f}(x)$ through an intermediate state $\rho_\text{int}(x)$, distorting two consecutive reference processes. This construction relies on the acceleration of a first reference relaxation process from $\rho_\text{i}(x)$ to $\rho_\text{int}(x)$; and accelerating and reversing a second relaxation process from $\rho_\text{f}(x)$ to $\rho_\text{int}(x)$---see Fig.~\ref{fig:welding}. If one seeks an operating time $t_\text{f}$
(with a starting time $t_\text{i}=0$), one can assign a time duration
$t_\text{f}/2$ for each of the two steps, but other choices are
possible.
\begin{figure}
\centering
\includegraphics[width=0.35\textwidth]{welding.png}
\caption{Sketch of the welding strategy to connect two arbitrary distributions $\rho_{\text{i}}$ and $\rho_{\text{f}}$. The connection is made in two steps. In the first step, a reference process from $\rho_{\text{i}}$ to $\rho_{\text{int}}$ is submitted to a fast-forward evolution. In the second step, a reference process starting at $\rho_{\text{f}}$ and finishing at $\rho_{\text{int}}$ is time-reversed and accelerated (fast backward).
In doing so, one achieves the desired connection in a chosen time.}
\label{fig:welding}
\end{figure}
Note that the counterdiabatic method could be understood as a limit of the fast-forward protocol presented above. Let us consider a reference process of duration $t_r$ and take the limit $t_r\to\infty$, so that the reference process becomes quasistatic. Hence, the limit reference process would sweep equilibrium states.
In this way, the resulting limit of the fast-forward protocol would converge to the counteradiabatic method.
\subsection{Engineering the thermal environment}\label{sec:engineering-thermal-env}
In previous developments, we addressed Brownian objects in some environment at thermal equilibrium, meaning the temperature
entering the Langevin and FP equations
is fixed. For colloidal systems, the environment is usually
water, and it may be difficult to impose a chosen
time dependence for its temperature,
especially if a massive heating is sought \cite{2013Martinez}.
Yet, it is possible
to ``fool'' the colloidal beads, subjecting them to a random
forcing that will emulate an effective
temperature exceeding several thousand kelvins \cite{2013Martinez,2017Ciliberto}.
The method is quite robust, and essentially requires that the
forcing frequency be large compared to the bead inverse relaxation time.
It is then possible to finely control the time dependence
of the effective temperature,
by playing on the amplitude of the forcing,
which opens new means for driving the system
\cite{2018Chupeau_b}, and paves the way towards the more general goal
of reservoir engineering.
Following this idea, a micrometric silica
sphere has been driven in \cite{2018Chupeau_b} by the joint monitoring of a
harmonic trap stiffness ($\kappa$), and the point of zero force
($x_0$).
In other words, the confining potential is of the form
\begin{equation}
U(x,t) \,=\, \frac{1}{2} \kappa(t) \left( x-x_0(t)
\right)^2 ,
\end{equation}
and the idea is to impose the proper time dependence
jointly on both $\kappa$ and $x_0$.
Compared to the more usual situation where $x_0$ is fixed,
a new contribution $\kappa(t) x_0(t)$ arises in the force balance.
It is important that a) $x_0$ remain small compared to the
bead size, in order not to affect the effective stiffness
and b) that the correlation time of the signal $x_0(t)$ be small compared to the protocol
duration (itself by construction smaller than the intrinsic relaxation time). Then, the forcing of $x_0$ results in an effective
heat bath for the colloidal degree of freedom: this forcing
has a time-dependent
amplitude and, for practical purposes, can be viewed as
delta-correlated in time. In Ref. \cite{2018Chupeau_b}, the control of $x_0$ was achieved with an acousto-optic deflector,
and the bath engineering made it possible to quickly deconfine a
colloidal state. Should one be able to play only on the
stiffness $\kappa(t)$, equivalent transformations would require
transiently expulsive forces with $\kappa<0$, which
represent an experimental challenge \cite{2020Albay_b,2021Bayati}.
A limitation of the approach is that the
extra stochastic forcing applied results in enhanced
Brownian fluctuations, and in an effective temperature increase. Other techniques would have to be applied
when it comes to cooling the center-of-mass motion of trapped
beads, such as feedback-based approaches \cite{2011Li}.
Recently, the effective heating of optically trapped object
allowed to devise finite-time adiabatic processes \cite{2020Plata_b}.
Here, we stress that ``adiabatic'' is understood in its
usual thermodynamics meaning, of heat-exchange free~\cite{Adiabatic}. A Brownian object is inherently fluctuating, and strongly coupled
to its environment. Consider the compression at fixed temperature $T$
of a colloidal bead,
where for instance the stiffness $\kappa$ of a harmonic potential
is increased. When the bead has relaxed, the internal energy difference
$\Delta U$ vanishes on average between the initial and final states,
which have the same temperature.
The first principle of thermodynamics~\cite{2021Peliti}
states that, since the bead received work from the confining
force, heat flew towards the bath, on average~\cite{Sekimoto}.
If one seeks a vanishing heat exchange on average,
it is mandatory that the environment temperature increase.
In the quasi-static limit, this increase has to be proportional
to $\sqrt{\kappa}$. This can be seen as a consequence of Laplace relation
between temperature $T$ and volume $V$ for the adiabatic reversible transformation of an ideal gas: $T V^{2/3}= \text{const}$ for a monoatomic gas. Here, the confinement length is
$\sigma_{eq}$, so that the role
of the volume is played by $\sigma_{eq}^3$. Since $\sigma^2_{eq}=k_BT/\kappa$, this yields
a Laplace condition $T\sigma_{eq}^2=\text{const}$ or, equivalently, $T^2/\kappa = \text{const}$. This can be viewed as the statement that $n\Lambda^3 =\text{const}$, where $n$ is the typical density and $\Lambda$ is De Broglie wavelength, which guarantees that volume in phase space is conserved. This volume is computed from the (cubed) product of the typical length in real space, $\sigma$, times the typical velocity, scaling like $\sqrt{T}$. This discussion also illustrates that the quasi-static criterion $T^2/\kappa = \text{const}$ is space-dimension independent~\cite{2013Bo,2015Martinez_b}.
For finite-time adiabatic processes, not only does the mean heat released to the thermal environment vanish between the initial and final states of the transformation, but it also does at any time in between. The operating time of these irreversible adiabats can be optimized by jointly controlling the potential and the temperature. The condition of zero heat involves the kinetic contribution to the energy: the only assumption being, consistently with the overdamped description, that the velocity degree of freedom is always at equilibrium with the time-dependent value of the temperature. Some general results emerge~\cite{2020Plata_b}, like (i) forbidden regions, i.e., final states that cannot be reached adiabatically, and (ii) a speed limit, the existence of a minimum, in general nonvanishing, time
$t_\text{f}^*$
for the adiabatic connection.
For the specific case of a harmonic confining potential, it was shown that $T\sigma^2$ is nondecreasing over the adiabats. This implies that $T_{\text{f}}/T_{\text{i}} \geq \sqrt{\kappa_{\text{f}}/\kappa_{\text{i}}}$, taking into account that the system is at equilibrium at the initial and final times---see Eq.~\eqref{eq:sigma-eq}. It is only in the quasistatic limit that $T\sigma^2$ remains constant and, moreover, the equilibrium equation of state \eqref{eq:sigma-eq} holds for all times, which leads to recover Bo and Celani's result of constant $T^2/\kappa$~\cite{2013Bo}.
Advantage was taken of these finite-time adiabatic transformations to construct an irreversible Carnot engine featuring interesting efficiency properties~\cite{2020Plata}, see also Sec.~\ref{sec:heat-engines}. In Fig.~\ref{fig:adia-cexample}, the fastest possible adiabatic connection is illustrated for a $20\%$ compression, $\sigma_{\text{f}}=0.8\sigma_{i}$.
\begin{figure
\begin{center}
\includegraphics[width=2.7in]{plotk2.png}
\includegraphics[width=2.7in]{plotT4.png}
\includegraphics[width=2.7in]{plotextra2.png}
\end{center}
\caption{Fastest control and evolution for an adiabatic (in the sense of zero average heat) $20\%$ compression of a harmonically trapped particle. Time $\tau=t/t_{\text{f}}^*$ has been made dimensionless with the shortest possible duration of the process $t_{\text{f}}^*$, which is reached for a linear evolution of $\sigma_x^2$. The fastest connection requires the stiffness to be discontinuous at the initial and final times. The example shown corresponds to $T_{\text{f}}/T_{\text{i}}=16$, $\kappa_{\text{f}}/\kappa_{\text{i}}=25$ and
entails an acceleration
of a factor around $5.7$ with respect to the relaxation time scale $t_{\text{rel}}=\gamma / \kappa_{\text{f}}$.
See Sec.~\ref{subsec:OCT-other-fig-merit} and Ref.~\cite{2020Plata_b} for further details.
\label{fig:adia-cexample}
}
\end{figure}
\section{Optimal control theory}\label{sec:OCT}
Hitherto, this review has focused on techniques that allow for connecting two given states. Naturally, having established that going from A to B in a finite time is feasible raises the question of what the best path is. Answering this kind of question is the main goal of optimal control theory~\cite{1987Pontryagin,2012Liberzon}, which combines well with inverse engineering problems. However, the choice of the best pathway depends
on the quantity to be optimized
(time, some kind of cost function\ldots)
that should thus be defined in the first place.
(See for instance Refs.~\cite{2021Zhang,2021Prados} for specific examples, ranging from ultracold atoms to granular systems.)
\subsection{Minimization of the mean work}\label{sec:irr-work-opt}
\subsubsection{Harmonic connections}\label{sec:irr-work-opt-harmonic}
Although optimal connection problems in the context of finite-time thermodynamics date back to the 1970's and 1980's~\cite{1975Curzon,1982Band,1984Andresen}, the first solution of an optimal connection problem in the ``modern'' context of stochastic thermodynamics and SST\ is due to Schmiedl and Seifert~\cite{2007Schmiedl}. They considered an overdamped Brownian particle submitted to harmonic trapping, where either the position or the stiffness of the trap is controlled. We recall that experiments with colloidal particles are usually well described by the overdamped FP, or Smoluchowski, equation~\eqref{eq:FP}~\cite{2017Ciliberto}---see also Appendix~\ref{app:stoch-frame}.
In Ref.~\cite{2007Schmiedl}, the optimal control needed to minimize the mean work for an isothermal process
\begin{align}
\mean{W}&= \int_0^{t_{\text{f}}} \!\!dt \int_{-\infty}^{+\infty} \!\!\! dx \, \partial_t U(x,t) \,\rho(x,t) \nonumber \\
& =\int_0^{t_{\text{f}}} \!\!dt \int_{-\infty}^{+\infty} \!\!\! dx\, \dot{\bm{\lambda}}(t) \cdot \partial_{\bm{\lambda}} U(x,\bm{\lambda}(t)) \,\rho(x,t) .
\label{eq:mean-work}
\end{align}
was derived. Note that we are assuming that the potential $U$ depends on time through some externally controlled parameters $\bm{\lambda}$~\footnote{See Appendix~\ref{app:stoch-frame} for a brief account of the definitions of work and heat in the context of stochastic thermodynamics.}. Dealing with the variation of the work, an Euler-Lagrange equation for the control parameter was obtained and analytically solved. The assumption of harmonic potential, and therefore Gaussian states, has much to do with the fact that the problem is analytically solvable. The dynamics of the system, in principle codified in the FP equation, can be simplified to an ordinary differential equation for the only relevant moment of the distribution---e.g. its standard deviation, see Eq.~\eqref{eq:standard-dev-harmonic}.
It is natural to try to transpose optimal protocols to actual experiments, for example the compression or decompression of a harmonically trapped Brownian particle---by controlling the stiffness of the trap. Still, the experimental implementation of the optimal protocols presents some difficulties. Specifically, the negative values of the stiffness needed for decompression---for short enough connecting times \cite{comment101}---are experimentally challenging. A step forward to solve this issue has been made by employing an optical feedback trap~\cite{2020Albay_b}. Although the connection considered therein is not optimal, it is neatly shown that it is possible to decompress the Brownian particle in a finite-time with a potential that becomes repulsive---i.e., with negative stiffness---inside a certain time window. Also, the optimal control---the stiffness of the trap---possesses finite discontinuities at the initial and final times in the overdamped limit. These discontinuities have the same formal origin as in the classic problems of finite-time thermodynamics~\cite{1982Band}, the linearity of the Lagrangian in its highest derivative~\cite{1955Newman}.
Discontinuities in the ``control functions'' are better rationalized in the context of Pontryagin's maximum principle of optimal control theory than within the framework of variational calculus~\cite{1987Pontryagin,2012Liberzon}. In optimal control theory, the control function only has to be piecewise continuous, and thus discontinuities in the control like those appearing in the stiffness of the harmonic trap are treated in a natural and mathematical rigorous way, see e.g. Ref.~\cite{2019Plata}. More general potentials, beyond the harmonic case, represent a challenge because solving the FP equation cannot be mapped onto solving an ordinary differential equation. However, numerical minimization of the mean work has been carried out, which shows that the predicted discontinuities are robust features of the optimal control~\cite{2008Then}. In the underdamped case, the discontinuities of the control become harsher, they do not involve finite jumps but delta peaks~\cite{2008Gomez-Marin}.
Optimal harmonic connections considering the limitation stemming from bounded stiffness have also been investigated in the overdamped case \cite{2019Plata}. Specifically, the stiffness has been assumed to be bounded between 0 and a maximum value, $0<\kappa(t)<\kappa_{\text{max}}$. Pontryagin's principle provides the adequate framework to solve such constrained optimal problem. The time evolution of the stiffness turns out to be built by two pieces. In the first one, the equations for the protocol are similar to those coming out from the unconstrained problem~\cite{2007Schmiedl} whereas, in the second piece, the control is kept fixed and equal to one of its limiting values. These two pieces smoothly match, in the sense that the dynamical variable---the variance of the position of the Brownian particle---is continuous and has continuous time derivative. As a consequence of the bounds, and depending on the target value of the stiffness and the desired connection time, the target state may become inaccessible. When the connection is possible, the minimum work is greater than the minimum one found for the unconstrained case, the difference between them becomes large in some situations. Similarly to the unconstrained case, the control develops finite jumps at the initial and final times.
\subsubsection{Beyond the harmonic case}\label{sec:irr-work-opt-nonharmonic}
After the problem of optimally connecting Gaussian states---in the sense of minimizing the mean work---our interest goes to the analytical derivation of the optimal connection for an arbitrary nonlinear potential---still in the isothermal case. This challenging problem has been first addressed in a series of related papers~\cite{2011Aurell,2012Aurell,2012Aurell_b,2013Muratore,2013Muratore_b,2014Muratore}.
Starting from the Langevin description, instead of the equivalent FP equation for the PDF, they show that the above question has quite a general answer in the overdamped limit. Building on long-established relations between dynamical systems and stochastic control theory~\cite{1981Accardi,1983Guerra}, the minimization of the average heat released to the reservoir can be mapped onto an optimal mass transport problem, ruled by the Burgers equation, and explicitly solved for several physical situations~\cite{2011Aurell,2012Aurell}. Since optimizing average heat and work is the same problem~\footnote{Note that the first principle states that $\mean{Q}+\mean{W}=\Delta U$. As the initial and final states of the SST\ are fixed, $\Delta U$ is also fixed and minimizing $\mean{W}$ thus entails minimizing $-\mean{Q}$, i.e. the average heat released to the thermal bath.}, for the particular case of a harmonic trap the optimal protocol that minimizes the mean work in Refs.~\cite{2007Schmiedl,2008Schmiedl} is recovered. Once more, the optimal control presents discontinuities at the initial and final times~\footnote{The authors addressed the regularization of these discontinuities by introducing a penalty for the current acceleration~\cite{2012Aurell_b}. A different ``surgery'' procedure, which also introduces boundary layers, has been considered to avoid these jumps~\cite{2019Plata}.}. It is worth stressing that the minimization of heat is shown to be equivalent to the minimization of entropy production, providing a refined version of the second law of thermodynamics~\cite{2012Aurell}. In general, this line of research evidences that differential geometry concepts can also be useful to investigate optimization problems in stochastic thermodynamics~\cite{2013Muratore_b,2012Sivak}.
The generalization of the above results to Markov jump processes, governed by a master rather than a FP equation, has also been carried out~\cite{2013Muratore}. In the continuum limit, the results converge to those previously described. Furthermore, the possibility of extending the above results to the underdamped, Langevin-Kramers, case has been investigated~\cite{2014Muratore}. Therein, the emergence of singularities and also of momentum dependence in the optimal driving potential makes the situation less clear-cut than that found in the overdamped situation. Recently, the minimisation of the work for the particular case of a counterdiabatic connection in the underdamped regime has been investigated~\cite{2021Li_b}, but the driving is once more velocity-dependent. In fact, the derivation of the optimal conservative driving potential $U(x,t)$ that provides the minimum irreversible work in the underdamped situation is, to the best of our knowledge, an open question.
In the overdamped limit, the general problem for the optimization of the mean work performed during the connection between arbitrary states has also been worked in the FP framework~\cite{2019Zhang,2020Zhang}. Specifically, the starting point is the FP equation for the probability density $\rho(x,t)$, written as a continuity equation, Eq.~\eqref{eq:FP-continuity}. Still, the main role is played by the cumulative distribution introduced in Eq.~\eqref{eq:cumulative-distr}.
Making use of the method of characteristics, the general results for the optimal connecting potential obtained from the Langevin equation~\cite{2011Aurell} are recovered. Here, we give the main results for deriving the minimum work and the associated optimal protocol---for a more detailed derivation thereof, see Appendix~\ref{app:Aurell-Zhang-opt-work}.
For a quasi-static process in which the system remains at equilibrium for all times, the average work $\mean{W}$ equals the free energy difference $\Delta F$ between the final and initial states. For a finite-time process, the second principle implies $\mean{W}>\Delta F$ and the irreversible (or excess) work is defined as
\begin{equation}
W_{\text{irr}}\equiv \mean{W}-\Delta F \geq 0.
\end{equation}
Starting from Eq.~\eqref{eq:mean-work} for the mean work, repeated use of integration by parts and the FP equation leads to
\begin{subequations}
\begin{align}
W_{\text{irr}}=&\int_0^{t_{\text{f}}} dt\, P_{\text{irr}}(t), \\
P_{\text{irr}}(t)=&\gamma \int_{-\infty}^{+\infty} \!\!\! dx \, v^2(x,t) \rho(x,t) \geq 0,
\end{align}
\end{subequations}
i.e. $P_{\text{irr}}$ stands for the irreversible power in the considered finite-time process and $v$ is given in \eqref{eq:veloc-field-FP}. Note that, consistently with our discussion above, $W_{\text{irr}}$ (or $P_{\text{irr}}$) vanishes for a reversible process only: $v(x,t)$ then identically equals zero---and thus the PDF has the equilibrium shape $\rho(x,t)\propto e^{-\beta U(x,t)}$---for all $(x,t)$.
The problem of minimizing the average work $\mean{W}$ is then equivalent to finding the fields $\rho(x,t)$ and $v(x,t)$ such that $W_{\text{irr}}$ becomes minimum, while verifying the FP equation~\eqref{eq:FP-continuity}. This can be done by employing the method of Lagrange multipliers, introducing an auxiliary field $\psi(x,t)$ and seeking the unconstrained minimum of
\begin{equation}
J[\rho,v,\psi]\equiv \int_0^{t_{\text{f}}}\! dt \!\int_{-\infty}^{+\infty} \!\!\! dx\, L, \; L=v^2\rho+\psi\left[ \partial_t\rho + \partial_x (\rho v )\right].
\end{equation}
The Euler-Lagrange equations for this problem read
\begin{equation}
\partial_t\psi + v\partial_x \psi=v^2, \quad \partial_x \psi=2 v,
\end{equation}
plus Eq.~\eqref{eq:FP-continuity}. Combining them, one gets the closed equation $\partial_t\psi+\frac{1}{4}\left( \partial_x\psi\right)^2=0$, which is nothing but the Burgers equation for the auxiliary field $\psi(x,t)$~\cite{2011Aurell}. Equivalently, one may write the Burgers equation in terms of the velocity field $v(x,t)$,
\begin{equation}\label{eq:Burgers}
\partial_t v+ v \partial_x v=0.
\end{equation}
In order to find the optimal profiles for $\rho(x,t)$ and $v(x,t)$---note that the latter provides us with the driving potential $U(x,t)$, the FP equation \eqref{eq:FP-continuity} and the Burgers equation \eqref{eq:Burgers} must be simultaneously solved. The solution is
\begin{subequations}\label{eq:v-rho-sol-main}
\begin{align}
v= & \varphi(x-vt),
\label{eq:v-sol-main}\\
\rho(x,t)= & \frac{\rho_{\text{i}}(x-vt)}{1+t\varphi'(x-vt)},
\label{eq:rho-sol-main}
\end{align}
\end{subequations}
where
\begin{equation}
\rho_{\text{i}}(x)\equiv \rho(x,t=0)
\end{equation}
is the initial distribution and $\varphi$ is---for the time being---an arbitrary function.
The function $\varphi(x)$ is determined by the system reaching the target distribution at the final time $t=t_{\text{f}}$, i.e. $\rho(x,t_{\text{f}})=\rho_{\text{f}}(x)$. This condition is easier to implement by employing the cumulative distribution introduced in Eq.~\eqref{eq:cumulative-distr}. Specifically, the initial and final cumulative distributions $\mathcal{F}_{\text{i},\text{f}}$, corresponding to the initial and final probability distributions $\rho_{\text{i},\text{f}}$, are needed. After defining
\begin{equation}
\label{eq:Xi-def}
\Xi(x)\equiv \mathcal{F}_{\text{i}}^{-1}\left(\mathcal{F}_{\text{f}}(x)\right),
\end{equation}
one gets
\begin{equation}\label{eq:phi-Xi}
\varphi(x)=\frac{\Xi^{-1}(x)-x}{t_{\text{f}}},
\end{equation}
where $A^{-1}(x)$ stands for the inverse function of $A$, i.e. $A^{-1}(A(x))=x$. Note that finding $\varphi$ makes it possible to obtain the driving potential, making use of Eq.~\eqref{eq:v-rho-sol-main} and the definition of the velocity field $v(x,t)$, Eq.~\eqref{eq:veloc-field-FP},
\begin{equation}
\partial_x U=-\gamma v -k_B T\partial_x\ln\rho.
\end{equation}
The irreversible power over the optimal protocol is shown to be
\begin{align}\label{eq:Pirr-opt}
P_{\text{irr}}^* \,= \,
\frac{\gamma}{t_{\text{f}}^2}\int_{-\infty}^{+\infty} \!\!\! dx\, \rho_{\text{i}}(x) \left[\Xi^{-1}(x)-x\right]^2.
\end{align}
Note that, as emphasized by our notation, the optimal irreversible power does not depend on time; it is a constant proportional to $t_{\text{f}}^{-2}$. The irreversible work immediately follows,
\begin{align}\label{eq:Wirr*}
W_{\text{irr}}^*=t_{\text{f}} P_{\text{irr}}^*=\frac{\gamma}{t_{\text{f}}}\int_{-\infty}^{+\infty} \!\!\! dx\, \rho_{\text{i}}(x) \left[\Xi^{-1}(x)-x\right]^2,
\end{align}
which is then proportional to $t_{\text{f}}^{-1}$. Quite expectedly, it vanishes in the limit $t_{\text{f}}\to\infty$, where the optimal process tends to be quasi-static.
In general, the optimal potential stemming from the optimal velocity field, by employing Eq.~\eqref{eq:veloc-field-FP}, has discontinuities at both the initial and final times. For the harmonic case, this approach leads to the optimal connections that can be worked out by using simpler methods, as described in Sec.~\ref{sec:irr-work-opt-nonharmonic}. For the general non-harmonic case, finding the optimal driving potential in an explicit closed form can only be done in a few examples~\cite{2019Zhang,2020Zhang}. The main difficulty stems from the expression for the mapping $\Xi(x)$ defined in Eq.~\eqref{eq:Xi-def}: only in simple cases is it possible to calculate explicitly the cumulative distributions $\mathcal{F}_{\text{i}}$ and $\mathcal{F}_{\text{f}}$, and to invert $\mathcal{F}_{\text{i}}$ and compose it with $\mathcal{F}_{\text{f}}$.
The above difficulties limit the usefulness of the exact optimal protocol for practical implementations. Non-optimal driving potentials but with values of $W_{\text{irr}}$ close to the optimal one, which can be expressed in closed form and do not present discontinuities, can be derived with the welding procedure introduced in Ref.~\cite{2021Plata}---see also Sec.~\ref{sec:fast-forward}. Therein, the authors look for the (sub)optimal connection belonging to fast-forward protocols that leads to minimal work. Remarkably, such a protocol conserves the property of the global optimum of delivering work at constant power.
Linear response theory~\cite{2008Marconi} is a standard framework for understanding nonequilibrium fluctuations. In fact, the problem above---i.e. optimization of the mean work performed during the connection between arbitrary states---has also been addressed using linear response theory~\cite{2012Sivak,2014Bonanca}. Sivak and Crooks obtained the optimal protocol minimizing the irreversible work in the linear regime, making use of information geometry concepts---see Sec.~\ref{sec:speed-limits} for further details. Once more, this approach leads to an optimal work with constant power~\cite{2007Crooks}. However, the assumption of linear response prevents this analysis from capturing the discontinuities of the optimal potential at the initial and final times. Bonan\ifmmode \mbox{\c{c}}\else \c{c}\fi{}a and Deffner deepened in the linear response approach, thoroughly discussing its range of validity and comparing its predictions with exact results~\cite{2014Bonanca}.
\subsection{Optimization of other figures of merit}
\label{subsec:OCT-other-fig-merit}
In the previous subsection, we have gone over the literature related to the optimization of average work. Nonetheless, the optimal approach in the context of SST\ is not limited to the minimization of the average work. Herein, we present different studies where the optimization of other relevant quantities has been examined.
The minimization of the statistical error of the free energy---an estimator of the difference between the free energy obtained after averaging over a certain number of individual measurements and the real free energy change---has been addressed \cite{2010Geiger}. Studying this quantity makes it possible to estimate the number of experiments needed to attain a certain accuracy when performing free energy measurements. Analytical results are not available even for simple (harmonic) cases, but numerical optimization provides step-like protocols with a significant reduction of the statistical error. Interestingly, this problem is intimately related to the reverse process~\cite{2007Lechner,2020Albay}.
Minimization of the average work carries no insight on work fluctuations. Hence, interest aroused in the analysis of alternative figure of merits, combining information of both average and fluctuations. Solon and Horowitz studied the minimization of an objective function, which is a linear combination of the work average $\langle W \rangle$ and the work standard deviation $\sigma_W$, specifically $\mathcal{J}_\alpha= \alpha \langle W \rangle + (1-\alpha) \sigma_W$, \cite{2018Solon}. By varying the coefficient $\alpha$ from $0$ to $1$, the weight of work fluctuations (mean work) is reduced (increased) in the search of the optimum protocol. The notion of Pareto-optimal solutions is applied to classify all possible optimal protocols. Due to the mathematical complexity of the optimization problem, this is carried out numerically by tuning the value of the control in a finite set of times. A first-order phase transition is found when illustrating the optimization procedure above in a quantum dot.
The physical approach to information and memory storage~\cite{1991Landauer,2012Berut,2015Parrondo} may also give rise to optimization problems in the general framework described in this review. Landauer's principle states the minimum heat cost exhausted to erase one random bit stored in a memory device~\cite{1961Landauer,2021Dago}. Such a bound refers to a quasistatic process. Finite-time processes have been considered as well, posing new relevant problems, not only from the theoretical point of view but also from the technological one, where fast computational operations with memory devices are necessary~\cite{2018Chiuchiu,2020Proesmans,2020Touzo,2021Zhen,2021Lucero}.
All previous instances belong in the optimization of energetic observables. Remarkably, optimal problems involving time
have also been investigated. The minimization of time related observables is closely linked to information geometry concepts and the so-called speed limits, which are discussed in Sec.~\ref{sec:speed-limits}. Below, we briefly report some results that have been obtained outside the general framework of information geometry ideas.
The optimal static external potential required to minimize the escape time $\tau$, of a Brownian particle confined in a box has been investigated~\cite{2012Palyulin,2020Chupeau}. The escape time is defined here as the mean first passage time to the end of the box, and it is considered that the external potential does not introduce any bias between the starting and final points. Rather surprisingly, in the overdamped regime, the escape time
can approach zero arbitrarily close \cite{2012Palyulin}, which requires
divergent and strongly ``squeezing'' potentials \cite{2012Palyulin,2020Chupeau}.
However, when some constraints are considered (e.g., the maximal potential difference is below a certain threshold $\Delta U$) an expression, which reminds Heisenberg's time-energy uncertainty principle, is obtained $\tau_{opt} \Delta U=\text{const}$. Related to this, the optimization problem for static external potential minimizing the first passage time to a certain target, distributed according to a certain symmetric probability distribution with respect the initial position, has been also worked out~\cite{2017Kusmierz}. On a different note, in the context of stochastic resetting, optimal first passage time is a hot topic~\cite{2011Evans,2020Evans,2014Kusmierz,2020Besga,2021Stanislavsky}. Usually, in those studies, the external potential is fixed and the resetting rate is the object that plays the role of the external control.
Another relevant time optimization problem is the minimization of the connection time between the initial and target states. For the SST\ between equilibrium states, the minimum connection time is zero for the unconstrained problem---similarly to the situation described above for the escape time of the Brownian particle. A different situation arises when the connection problem has additional restrictions, for example when the protocol has to be adiabatic in the thermodynamic sense of zero average heat. Therein, the adiabatic constraint together with the second principle gives rise to the emergence of a speed limit, i.e. the emergence of a minimum time for the adiabatic connection~\cite{2020Plata_b}. In general, the instantaneous adiabat does not exist, and moreover there appear forbidden regions beyond the quasi-static curve $T^2/\kappa=\text{const}$ at which the minimum connection time diverges---as discussed in Sec.~\ref{sec:engineering-thermal-env}.
\subsection{Information thermodynamics and speed limits}\label{sec:speed-limits}
In the previous sections, we have discussed the optimization of the work and other figures of merit over SST\ protocols---including the minimization of the connection time for some specific situations. Here, we focus on the emergence of the so-called classical speed limits, which are closely related to information thermodynamics concepts, and their relevance in the context of SST.
The acceleration of the connection entailed by SST\ protocols comes at a price: for example, we have already discussed that there appears a non-vanishing irreversible contribution to the average work, which only vanishes for infinite connection time, for the SST\ connecting equilibrium states at constant temperature. In fact, the minimum irreversible work, as given by Eq.~\eqref{eq:Wirr*}, is proportional to $t_{\text{f}}^{-1}$ and then blows up for very short connecting times. Therefore, a natural question arises, whether or not there exists a speed limit for SST, i.e. a minimum value for the connection time $t_{\text{f}}$.
The existence of a speed limit in the context of quantum mechanics dates back to the 1940s~\cite{1991Tamm}. It is related to the time-energy uncertainty relation; for review of the subject, see Ref.~\cite{2017Deffner}. The simplest situation is that of the time evolution of a pure state in a conservative system. In this case, there appear two inequalities for the time $t_{\text{f}}$ necessary to evolve from an initial state $\rangle{\psi(0)}$ to an orthogonal target state $\rangle{\psi(t_{\text{f}})}$, $\braket{\psi(0)}{\psi(t_{\text{f}})}=0$. First, the Mandelstamm-Tamm bound~\cite{1991Tamm}, $t_{\text{f}}\Delta H\geq \pi \hbar/2$, and, second, the Margolus-Levitin bound~\cite{1998Margolus}, $t_{\text{f}}\expval{H}\geq \pi \hbar/2$. In these expressions, $\expval{H}$ and $\Delta H$ are the (time-independent) expectation value and standard deviation of the energy, respectively. In fact, it has been proven that the combination of the Mandelstamm-Tamm and Margolus-Levitin inequalities gives the tightest bound for $t_{\text{f}}$~\cite{2009Levitin}, i.e.
\begin{equation}
t_{\text{f}}\geq\max\left(\frac{\pi\hbar}{2\Delta H},\frac{\pi\hbar}{2\expval{H}}\right).
\end{equation}
This inequality expresses the existence of a \textit{natural} time scale, which cannot be beaten for the evolution of a quantum system with time-independent Hamiltonian $H$.
In statistical mechanics, only more recently the possible existence of inequalities resembling the quantum time-energy uncertainty relation---and the possible associated emergence of a speed limit---has been investigated~\cite{2012Sivak,2018Okuyama,2018Shanahan,2018Ito,2018Nicholson,2018Shiraishi,2020Ito,2020Nicholson,2020Falasco,2020Plata_b,2020Rosales-Cabara,2021Prados,2021Nakazato,2022Yan}. One of the first derivations of an inequality for the connection time is probably that of Ref.~\cite{2012Sivak}, within the linear response regime. For a review of these ideas, see Ref.~\cite{2020Deffner}. Let us consider a general system with control functions $\bm{\lambda(t)}$, and a protocol that drives the system from an initial state with $\bm{\lambda}(t=0)=\bm{\lambda}_{\text{i}}$ to the target state with $\bm{\lambda}(t_{\text{f}})=\bm{\lambda}_{\text{f}}$; the initial and final states must be close enough in order to use linear response theory. The irreversible power can be written as a bilinear function of the time derivative of the control functions $\bm{\lambda}(t)$, with coefficients provided by the---positive definite---friction tensor $\bm{\zeta}(\bm{\lambda})$,
\begin{equation}\label{eq:Pirr-zeta-rel}
P_{\text{irr}}(t)=\frac{d\bm{\lambda^T(t)}}{dt} \cdot \bm{\zeta}(\bm{\lambda(t)}) \cdot
\frac{d\bm{\lambda(t)}}{dt},
\end{equation}
where the superindex $T$ stands for transpose. This expression naturally introduces a metric in the problem and, in fact, a \textit{statistical length} $\mathcal{L}_{\text{lin}}$~\footnote{The subindex ``$\text{lin}$'' stresses that this definition of statistical length is valid in linear response, see below for a more general definition based on the Fisher information.} is defined as
\begin{equation}
\mathcal{L}_{\text{lin}}=\int_{0}^{t_{\text{f}}} dt \sqrt{P_{\text{irr}}(t)}.
\end{equation}
This length somehow measures the distance swept by the system in parameter space along the SST\ path from the initial to the target state. The connection time $t_{\text{f}}$ and the irreversible work $W_{\text{irr}}$ are shown to verify the inequality
\begin{equation}\label{eq:sl-Sivak-Crooks}
t_{\text{f}}\, W_{\text{irr}} \geq \mathcal{L}_{\text{lin}}^2.
\end{equation}
The equality only holds when $P_{\text{irr}}$ is constant, as it was for the protocol that minimizes the irreversible work---but note that Eq.~\eqref{eq:Pirr-opt} holds for arbitrary initial and final equilibrium states, not necessarily close, and thus it is not restricted to linear response~\footnote{A perturbative approach to solve the FP equation has been introduced with the aim of extending the range of validity of Sivak and Crook's results~\cite{2021Wadia}.}.
Speed limits have been discussed in a broader context, beyond the linear response regime~\cite{2018Ito,2018Shiraishi,2018Nicholson,2019Hasegawa,2019Dechant,2020Ito,2020Falasco,2020Plata_b,2020Rosales-Cabara,2020Nicholson,2020Ito,2021Prados,2021Nakazato,2022Yan}. Here, we focus on processes that involve a net transformation of states; for time-periodic or stationary processes, there are specific inequalities showing that the entropy production rate bounds the rate at which physical processes can be carried out~\cite{2020Falasco}---which has been experimentally checked~\cite{2022Yan}. On the one hand, the first results were derived under the assumption of Markovian dynamics~\cite{2018Ito,2018Shiraishi} and are thus restricted to system with dynamics described by master, for discrete variables, and FP (or Langevin) equations, for continuous variables. On the other hand, approaches based on information geometry concepts hold for general dynamics~\cite{2020Nicholson,2020Ito}, not necessarily Markovian. Key to the latter results is the Fisher information $I(t)$,
\begin{equation}
I(t)\equiv \mean{\left(\partial_t \ln P\right)^2}=\int d\bm{x} \frac{\left(\partial_t \rho(\bm{x},t)\right)^2}{\rho(\bm{x},t)},
\end{equation}
which is the curvature of the Kullback-Leibler divergence,
\begin{equation}
\int\! d\bm{x} \rho(\bm{x},t+dt) \ln\! \left[\! \frac{\rho(\bm{x},t+dt)}{\rho(\bm{x},t)}\right]\!=\frac{1}{2}(dt)^2 I(t)+O(dt)^3.
\end{equation}
The Fisher information can be connected with entropy production and the so-called thermodynamic uncertainty relations~\cite{2018Nicholson,2018Ito,2019Hasegawa,2020Ito}, and it is directly linked with the thermodynamic length $\mathcal{L}$, first introduced in the 1980s~\cite{1983Salamon,1985Salamon,1985Feldmann}. For equilibrium systems, this relation was addressed in a pioneering work by Crooks~\cite{2007Crooks}, which showed that
\begin{equation}
\mathcal{L}=\int_0^{t_{\text{f}}} dt\, \sqrt{I(t)}.
\end{equation}
Note that, in general, $\mathcal{L}\ne\mathcal{L}_{\text{lin}}$.
Also relevant to our discussion is the divergence---also called the thermodynamic cost~\cite{2020Ito}---of the path,
\begin{equation}
\mathcal{C}=\frac{1}{2}\int_0^{t_{\text{f}}} dt\, I(t).
\end{equation}
The Cauchy-Schwarz inequality makes it possible to establish the following lower bound for the connection time,
\begin{equation}\label{eq:sl-Ito-Dechant}
2 t_{\text{f}} \,\mathcal{C} \geq \mathcal{L}^2.
\end{equation}
Interestingly, Eq.~\eqref{eq:sl-Ito-Dechant} is implicitly written in Ref.~\cite{2007Crooks} [see Eq.~(9) therein], although it has not been explicitly stated as establishing a speed limit for finite-time SST\ until recently---see Ref.~\cite{2018Ito} for the case of Markovian dynamics and Ref.~\cite{2020Ito} for arbitrary dynamics. At variance with the quantum case, it must be remarked that the tightness of the bound in Eq.~\eqref{eq:sl-Ito-Dechant} has not been proven, to the best of our knowledge. Indeed, Eq.~\eqref{eq:sl-Ito-Dechant} is valid for an arbitrary dynamics---see Ref.~\cite{2021Prados} for an analysis thereof for granular fluids described at the kinetic level, where the PDF
obeys the non-linear (inelastic) Boltzmann equation---but there may exist a larger bound for the connection time.
Equations~\eqref{eq:sl-Sivak-Crooks} and \eqref{eq:sl-Ito-Dechant} provide us with two inequalities that must hold in SST. In principle, one could argue that Eq.~\eqref{eq:sl-Sivak-Crooks} was derived under the framework of linear response but it is a direct consequence of the definition of irreversible work and the Cauchy-Schwarz inequality---it is linking $P_{\text{irr}}(t)$ with a Riemannian metric that linear response ensures. A comparison between the predictions of both inequalities for the harmonic case is given in Appendix~\ref{app:inequal-comparison}.
\section{Beyond equilibrium: connecting NESS or arbitrary states}\label{sec:beyond-equilibrium}
In the statistical mechanics context, SST\ have been mainly employed to connect equilibrium states~\cite{2007Schmiedl,2011Aurell,2012Aurell,2012Aurell_b,2013Muratore,2013Muratore_b,2014Muratore,2013Martinez,2016Martinez,2016Le-Cunuder,2017Ciliberto,2017Li,2018Chupeau_b,2018Chupeau,2020Albay_b,2020Plata_b,2021Bayati,2021Plata,2019Plata,2008Then,2008Gomez-Marin}. The analysis of accelerated connection between NESSs has been initiated in \cite{2020Baldassarri,2021Prados}. The delay in the engineering of SST\ connections between NESSs stems from some difficulties that are inherent to the initial and final states being non-equilibrium, as explained below.
For the sake of concreteness, we revisit the already addressed paradigmatic example of a colloidal Brownian particle trapped in a one-dimensional potential $U(x,t)$ and immersed in a fluid at equilibrium with temperature $T$. In the overdamped regime, the distribution function $\rho(x,t)$ obeys the FP equation \eqref{eq:FP} and the typical SST\ problem is the connection between two equilibrium states, those corresponding to the initial potential $U_{\text{i}}(x)$ and the target one $U_{\text{f}}(x)$. An advantage of the equilibrium situation is our perfect knowledge of the initial and target distributions, $\rho_{\text{i}}(x)\propto\exp[-\beta U_{\text{i}}(x)]$ and $\rho_{\text{f}}(x)\propto\exp[-\beta U_{\text{f}}(x)]$.
An especially simple case is that of harmonic confinement, in which the potential $U(x,t)$ is harmonic for all times. As described in Sec.~\ref{sec:inverse-engineering}, the PDF is Gaussian for all times therein, and thus completely determined by its average and variance. Moreover, the evolution equations of the average and variance are analytically solvable in closed form. Therefore, although the intermediate states between the initial and target PDFs are indeed non-equilibrium ones, SST\ connections can be exactly worked out, both non-optimal~\cite{2016Martinez} and optimal in some sense~\cite{2007Schmiedl,2008Then,2008Gomez-Marin,2019Plata}---as already analysed in Sec.~\ref{sec:irr-work-opt-harmonic}.
In principle, SST\ methods are transposable to situations in which the initial and final states are NESSs, instead of equilibrium states. Still, some problems emerge because there is not a general form, playing the role of the canonical distribution, for the PDF corresponding to any NESS. As a consequence, the initial and target states are not perfectly known in general; this constitutes a first limitation for the catalogue of NESS that can be considered as candidates to be SST -connected.
The SST\ connection between two NESS of the Brownian gyrator is the subject of study in Ref.~\cite{2020Baldassarri}. The Brownian gyrator is an overdamped particle moving in a two-dimensional potential
\begin{equation}
U(\bm{x},t)=\frac{1}{2}\kappa_x(t) x^2+\frac{1}{2}\kappa_y(t) y^2+ u(t) x y, \quad \bm{x}\equiv (x,y),
\end{equation}
where $\kappa_x$ and $\kappa_y$ are both positive and $\kappa_x \kappa_y-u^2>0$, in order to have a confining potential. The gyrator is coupled to two heat baths with temperatures $T_x$ and $T_y$ in the $x$ and $y$ directions, respectively. The two-dimensional position $\bm{x}$ is a Markov process and the FP equation for its PDF $\rho(\bm{x},t)$ reads
\begin{align}
\gamma \partial_t \rho=\partial_x \left(\rho\, \partial_x U\right)+\partial_y \left(\rho\,\partial_y U\right)+k_B \left(T_x \partial_x^2 \rho+T_y \partial_y^2 \rho\right).
\end{align}
If $T_x=T_y$, and $(\kappa_x,\kappa_y,u)$ are time-independent parameters, the stationary solution of this FP equation is the canonical PDF at temperature $T=T_x=T_y$ corresponding to the static potential $U(\bm{x})$. If $T_x\ne T_y$, the stationary solution of this FP equation can be exactly computed and is Gaussian, although it does not have the canonical shape~\cite{2020Baldassarri}, and induces a current which is rotational.
The approach of Ref.~\cite{2020Baldassarri} is quite similar to that of Ref.~\cite{2016Martinez} for the engineered swift equilibration of a Brownian particle moving in a one-dimensional harmonic potential. A non-optimal SST\ connection of two NESS of the Brownian gyrator, corresponding to different values of the triplet $(\kappa_x,\kappa_y,u)$, can be built in a simple way, because the PDF is Gaussian, not only in the initial and target states, but for all times. Therefore, one can write
\begin{equation}
\rho(\bm{x},t)=\frac{e^{-\frac{1}{2}\nu_1(t) x^2-\frac{1}{2}\nu_2(t) y^2-\nu_3(t) x y}}{2\pi \left[\nu_1(t) \nu_2(t)-\nu_3^2(t)\right]^{-1/2}}.
\end{equation}
The functions $\bm{\nu}(t)\equiv(\nu_1(t),\nu_2(t),\nu_3(t))$ obey a closed system of first-order ODEs, in which $\bm{\lambda}\equiv(\kappa_x,\kappa_y,u)$ play the role of control functions and appear linearly. Thus, the controls $\bm{\lambda}$ can be explicitly written as $\bm{A}(\bm{\nu})+\bm{B}(\bm{\nu})\dot{\bm{\nu}}$, where $\bm{A}$ and $\bm{B}$ are certain functions of $\bm{\nu}$---the exact expression for which is not relevant here and can be found in Ref.~\cite{2020Baldassarri}. In this way, the ``equilibrium swift equilibration'' technique introduced in Ref.~\cite{2016Martinez} is generalized to the connection of two NESS, taking advantage of the simplicity of the mathematical problem for the Brownian gyrator. Although the problem is two-dimensional and the initial and target states are NESS, the main characteristic features that facilitates the SST\ connection of a harmonically trapped particle still hold: perfect knowledge of the initial and target distributions, Gaussianity of the PDF for all times, and simple enough evolution equations for the relevant variables (making it possible to obtain exact analytical solutions thereof).
Another physical situation in which the SST\ connection between two NESS has been considered is the uniformly heated granular fluid~\cite{2021Prados}. This case is more involved than the Brownian gyrator, due to the intrinsically dissipative character of the dynamics. The evolution equation for the PDF $P(\bm{v},t)$ has two terms: a diffusive, FP, term stemming from the stochastic forcing and an inelastic Boltzmann collision term, which makes the evolution equation non-linear in the PDF.
In the long-time limit, the granular fluid reaches a homogeneous NESS due to the balance---in average---of the energy loss in collisions and the energy input by the applied stochastic forcing. Even for this homogeneous NESSs, the velocity PDF is non-Gaussian and thus is not completely characterized by the variance---the granular (kinetic) temperature. In addition, nor is the velocity PDF perfectly known: it is necessary to resort to approximate schemes,
keeping track of non-Gaussianities that are essential.
The simplest way of doing so is through the so-called first Sonine approximation, in which the granular fluid is assumed to be described by its granular (kinetic) temperature $T$ and the excess kurtosis $a_2$ (the fourth cumulant of the velocity),
\begin{equation}\label{eq:P1-Sonine}
P(\bm{v};t)=\frac{e^{-w^{2}}}{\left[\pi v_{T}^2(t)\right]^{d/2}}\left[1+a_{2}(t)
S_{2}\left(w^{2}\right)\right], \quad \bm{w}\equiv\frac{\bm{v}}{v_{T}(t)}.
\end{equation}
The thermal velocity $v_{T}$ is defined by $v_{T}^{2}\equiv 2T/m$ in terms of the granular temperature, and $S_{2}(x)=x^{2}/2-(d+2)x/2+d(d+2)/8$ is the second Sonine polynomial where $d$ is space dimension.
Within this approximation, the evolution equations of $T$ and $a_2$ constitute a system of two coupled ODES, in which the control function is the intensity of the thermostat $\chi(t)$. The problem of finding the protocol that minimises the connection time between two NESSs corresponding to different values of the driving intensity, $\chi_{\text{i}}$ and $\chi_{\text{f}}$, has been analysed in Ref.~\cite{2021Prados}. This control problem is, at first, non-trivial, since the evolution equations in the Sonine approximation are non-linear in $(T,a_2)$ and then not analytically solvable. Still, the control function $\chi(t)$ enters linearly in the evolution equations, and this simplifies the mathematical problem: the optimal controls are of bang-bang type. That is, the optimal control comprises two time-windows, inside each of them $\chi(t)$ is constant and equal to either its maximum possible value $\chi_{\max}$ or its minimum possible value $\chi_{\min}$, with one switching between these extreme values at a certain intermediate time~\footnote{For evolution equations that are linear both in the controls and in the variables, there exist rigorous theorems that ensure that the optimal control is of bang-bang type, with $n-1$ switchings for a system with $n$ variables. For evolution equations that are non-linear in the variables but linear in the controls, Pontryagin's maximum principle implies that a similar situation is expected and, since here $n=2$, there is one switching~\cite{2021Prados}.}. For a full-power thermostat, $\chi_{\min}=0$ and $\chi_{\max}=\infty$, the optimal control problem can be exactly solved---the bangs with $\chi_{\max}=\infty$ are instantaneous whereas the bangs with $\chi_{\min}=0$ correspond to time windows where the granular fluid freely cools. Both the thermodynamic length and the information geometry cost can be evaluated over the optimal protocols, being consistent with the recently derived general inequalities that impose the existence of a speed limit for arbitrary dynamics, not necessarily Markovian~\cite{2020Ito,2020Nicholson}---see also Sec.~\ref{sec:speed-limits}. The case of a more realistic thermostat, where $\chi_{\min}>0$ and $\chi_{\max}<\infty$, has been addressed in the linear response regime~\cite{2022Ruiz-Pino}.
\section{Applications: heat engines and beyond}
\label{sec:heat_engine_and_beyond}
\subsection{Heat engines}\label{sec:heat-engines}
SST\ have been employed in the last decade to design irreversible heat-engines~\cite{2008Schmiedl,2010Esposito,2012Blickle,2013Bo,2013Deng,2014Holubec,2014Sheng,2014Tu,2015Muratore,2016Holubec,2016Martinez_b,2017Polettini,2017Lee,2018Abah,2019Abah,2019Albay,2020Albay_b,2020Nakamura,2020Plata,2021Li,2021Tu}. Loosely speaking, these heat engines can be considered as the mesoscopic counterparts of the classical, macroscopic, heat engines, in which the branches of the corresponding cycle last for a finite time. Note that we focus on classical heat engines, described at the mesoscopic level by nonequilibrium classical statistical mechanics---their quantum counterparts are thus not addressed in this review.
Let us go back to the colloidal particle trapped in a harmonic potential of stiffness $\kappa$, and immersed in a fluid at equilibrium with temperature $T$, thus described by the Smoluchowski equation~\eqref{eq:FP-harm}. Recall that both the stiffness of the trap and the bath temperature can be time-dependent, with their time-dependence being externally controlled. This system is experimentally realizable, see for example Refs.~\cite{2012Blickle} and \cite{2016Martinez_b} for experimental implementations of Stirling-like and Carnot-like cycles, respectively.
A difficulty arises in the designing of these mesoscopic heat engines when they involve, as is the case of the Otto or the Carnot cycles, adiabatic---in the sense of zero heat, not in the sense of infinitely slow employed before \cite{Adiabatic}---branches. Complete decoupling of the system from the heat bath is not possible, since the interaction between the mesoscopic object---i.e., the Brownian particle---and the heat bath cannot be switched off. In addition, zero-heat (in average, fluctuations are unavoidable) and isoentropic processes are not equivalent for finite-time protocols like those in SST. This makes the definition of adiabatic processes a subtle issue in the mesoscopic world~\cite{2008Schmiedl,2013Bo,2016Martinez_b,2019Plata,2020Nakamura,2020Plata}.
In a pioneering work~\cite{2008Schmiedl}, Schmiedl and Seifert analyzed an irreversible Carnot-like heat engine, built on an overdamped Brownian particle confined in a harmonic trap. This heat engine works cyclically, with two isothermal and two pseudoadiabatic branches~\cite{2008Schmiedl}---like the original Carnot engine---that connects equilibrium states. However, unlike with the original Carnot engine, all the branches are irreversible. More specifically, during the hot (cold) isotherm, the system is in contact with a heat bath at temperature $T_h$ ($T_c$) for a certain finite time $t_h$ ($t_c$). In the \textit{instantaneous pseudoadiabats}, the temperature of the system suddenly jumps from $T_h$ to $T_c$ (or vice versa) while keeping the probability distribution for the position of the Brownian particle unchanged. Therefore, what remains constant over the adiabats is the configurational contribution to the entropy or, equivalenty, the ratio $T/\kappa$. As already noted in Ref.~\cite{2008Schmiedl}, this means that there appears a nonzero heat---and a nonzero entropy increment---along the pseudoadiabatic branches associated with the instantaneous change of the kinetic contribution to the energy. This is the reason why we term these branches pseudoadiabatic, to differentiate them from the actual adiabats, over which the heat vanishes in average, as described below.
This heat engine is optimized in the following way: keeping the isotherm times $t_h$ and $t_c$ fixed, one can minimize the irreversible work, i.e., maximize the output work $-W$ (which is a function of $t_h$ and $t_c$). Afterwards, one can maximize the output power $P=-W/(t_h+t_c)$ and obtain the corresponding efficiency at maximum power $\eta^*$, which reads
\begin{equation}\label{eq:etaSS}
\eta_{\text{SS}}=\frac{2\eta_{\text{C}}}{4-\eta_{\text{C}}},
\end{equation}
where $\eta_{\text{C}}$ is Carnot's efficiency,
\begin{equation}\label{eq:etaC}
\eta_C=1-\frac{T_c}{T_h}.
\end{equation}
The study of efficiency at maximum power is a classic problem in the field of finite-time thermodynamics~\cite{1975Curzon,1977Andresen,1985DeVos,1985Hoffmann,1989Chen,1994Chen}. The so-called Curzon-Ahlborn efficiency at maximum power is
\begin{equation}\label{eq:etaCA}
\eta_{\text{CA}}=1-\sqrt{\frac{T_c}{T_h}},
\end{equation}
which was obtained for an endoreversible heat engine~\cite{1975Curzon}---an engine that operates reversibly but for the irreversible coupling to the heat-baths~\footnote{A microscopic theory for the Curzon-Ahlborn cycle has been recently proposed~\cite{2021Chen}.}. There is no general proof ensuring that the efficiency of any arbitrary heat engine at maximum power is bounded by the Curzon-Ahlborn value. Still, $\eta_{\text{SS}}<\eta_{\text{CA}}$, although in the limit of small temperature difference, for which $\eta_C\ll 1$, both efficiencies coincide up to second order in $\eta_C$,
\begin{equation}\label{eq:etaCA-SS-comparison}
\eta_{\text{CA}}=\frac{\eta}{2}+\frac{\eta^2}{8}+O(\eta^3), \quad \eta_{\text{SS}}-\eta_{\text{CA}}=O(\eta^3).
\end{equation}
Nevertheless, the universality of the efficiency at maximum power up to quadratic order in $\eta_{\text{C}}$ is quite a general feature~\cite{2009Esposito_b,2010Esposito,2014Sheng,2014Holubec,2015Sheng,2016Holubec,2020Nakamura,2020Plata}, which can be shown to hold for heat engines with left-right symmetry (switching of the baths entail an inversion of the fluxes)~\cite{2009Esposito}. Less strict conditions for this universality have been derived in Refs.~\cite{2014Sheng,2015Sheng}, which also apply to heat engines without left-right symmetry such as the Feynman ratchet~\cite{2008Tu} or the Curzon-Ahlborn endoreversible heat engine~\cite{1975Curzon}. Besides, this kind of universality has been extended to other figures of merit, beyond the maximum power regime~\cite{2010Salas}.
Different variants of this Carnot-like heat engine have been investigated in the literature. Holubec considered an overdamped Brownian particle moving in a log-harmonic potential~\cite{2014Holubec}. The cycle is exactly the same as in Ref.~\cite{2008Schmiedl}, with two optimal isotherms and two instantaneous pseudoadiabats, adapted to the different binding potential. The efficiency at maximum power also agrees with $\eta_{\text{CA}}$ up to second order in $\eta_{\text{C}}$. Bo and Celani analyzed a Brownian particle immersed in a fluid with inhomogeneous temperature~\cite{2013Bo}. The linear temperature profile makes it necessary to consider the underdamped description, because of the so-called \textit{entropic anomaly} stemming from the temperature gradient~\cite{2012Celani}. In this work, the authors also derive the condition to obtain quasistatic (reversible) adiabatic branches: $T^2/\kappa$
is the quantity that must be kept constant, see section \ref{sec:engineering-thermal-env}. Martinez et al. employed these quasistatic adiabatic branches to realize experimentally an irreversible Carnot-like engine~\cite{2016Martinez_b}, with a charged colloidal particle immersed in water and optically trapped. The charge of the particle allows for adding a noisy electrostatic force: in this way, the effective temperature felt by the particle can be varied from room temperature to thousands of kelvins. Over the ``adiabats'', there is a nonzero fluctuating heat that must be taken into account when defining the efficiency of such an engine. The authors analyzed the probability distribution function of the efficiency as a function of the number of cycles and observed the appearance of super-Carnot efficiencies, even far from the quasistatic regime. Tu also investigated a Carnot-like engine built with a Brownian particle in the underdamped description, but with the temperature of the thermal bath being homogeneous~\cite{2014Tu}. Suitable shortcuts are introduced along the isothermal branches, whereas the Brownian particle is decoupled from the thermal bath and submitted to a velocity-dependent force during the ``adiabatic'' branches. Both the decoupling from the thermal bath and the velocity-dependent force limit the experimental feasibility of such an engine, but makes it possible to exactly derive the Curzon-Ahlborn value for the efficiency at maximum power.
As the discussion above suggests, the construction of the adiabatic branches of the Carnot-like engine is a subtle issue. Recently, Plata et al. showed how to build finite-time actually adiabatic---in the sense of zero average heat---branches for an overdamped Brownian particle trapped in an arbitrary nonlinear potential~\cite{2020Plata_b}---see also Secs.~\ref{sec:engineering-thermal-env} and \ref{subsec:OCT-other-fig-merit}. Figure \ref{fig:cycles} illustrates the difference between the classical quasistatic Carnot cycle and its optimal finite-time counterpart. Nakamura et al. also thoroughly studied the adiabatic connection---again in the sense of zero average heat---for the specific case of harmonic confinement within the underdamped description of the dynamics~\cite{2020Nakamura}. Extending ideas of the fast-forward protocol---already discussed in section~\ref{sec:fast-forward}, they built a Carnot-like heat engine with fast-forwarded isothermal and adiabatic branches. On the one hand, similarly to the situation in Ref.~\cite{2014Tu}, the extra potential contains terms involving the velocity; the efficiency at maximum power then equals the Curzon-Ahlborn bound for ``weak dissipation" (in the ``strongly'' underdamped regime). On the other hand, for strong dissipation (in the overdamped limit), the efficiency at maximum power is smaller than $\eta_{\text{SS}}$, as given by \eqref{eq:etaSS}. Nakamura et al. claimed that the overdamped approximation cannot be used to describe the full cycle of a Carnot-like stochastic heat engine. Still, Plata et al. built an irreversible Carnot-like engine with an overdamped colloidal particle trapped in a harmonic potential~\cite{2020Plata}. The main novelty with regard to Ref.~\cite{2008Schmiedl} is the authors' employing of the finite-time adiabatic branches derived in Ref.~\cite{2020Plata_b}, instead of the instantaneous pseudoadiabats that involve a nonvanishing heat. In addition to minimum work (maximum work output) isotherms, minimum time adiabats are employed to construct the Carnot-like cycle. Further optimization of the cycle shows that the efficiency at maximum power of this overdamped Carnot-like heat engine is very close to the Curzon-Ahlborn value throughout the whole range of temperature ratios $T_c/T_h$.
\begin{figure}
\centering
\includegraphics[width=0.35\textwidth]{cycle-quasi.png}
\includegraphics[width=0.35\textwidth]{cycle-fin.png}
\caption{Sketch of a Carnot cycle with a overdamped harmonic oscillator. The cycle is made by two isothermal branches (red lines) and two adiabatic ones (green lines). On the one hand, in the traditional Carnot cycle (top panel), the states are swept quasistatically, and the system keeps at equilibrium at all times. On the other hand, the optimal finite-time version (bottom panel) involves non-equilibrium evolution between operation equilibrium points. See Ref.~\cite{2020Plata_b} for further details.}
\label{fig:cycles}
\end{figure}
Other cycles, of the Stirling or Otto types, have been studied in the literature. In fact, the first experimental realization of a mesoscopic heat engine was achieved with the Stirling cycle~\cite{2012Blickle}, which comprises two isothermal and two isochoric branches. A Brownian particle (a single melamine bead of diameter close to $3\mu$m) suspended in water was trapped in a two-dimensional harmonic potential created with optical tweezers. The stiffness of the trap linearly varies with time along the isothermal branches, whereas it is kept constant in the almost instantaneous processes that connect the isotherms---which are thus isochoric, the average work vanishes. The determined efficiency at maximum power is, within experimental error, in agreement with the Curzon-Ahlborn value. Optimal protocols for the overdamped Stirling engine have been derived in Ref.~\cite{2015Muratore} by a mapping to an optimal mass transport problem, with an efficiency at maximum power given by Eq.~\eqref{eq:etaSS}. Also, a Stirling engine with active baths has been investigated~\cite{2016Krishnamurthy}. Therein, it seems that non-Gaussianities are responsible for the improved efficiency of the active engine.
SST\ for the Otto cycle have been considered in Ref.~\cite{2013Deng}. In this paper, it is shown that the accelerated connection intrinsic to SST\ does not only increase power output of the engine but also may increase its efficiency, both in the quantum and the classical case. Abah et al. further investigated the quantum Otto engine, establishing the energetic cost of driving with SST\ and finding that the efficiency at maximum power is very close to the Curzon-Ahlborn bound~\cite{2019Abah}.
It is worth mentioning that several works have addressed the problem of attaining the Carnot efficiency at finite power~\cite{2013Allahverdyan,2015Shiraishi,2016Campisi,2016Koning,2017Lee,2017Polettini,2017Holubec,2018Holubec,2021Cangemi}. Mostly, attaining---or even exceeding~\cite{2021Cangemi}---Carnot's efficiency is connected with the antiadiabatic limit of infinitely fast transformations~\cite{2016Campisi,2017Lee,2017Polettini,2017Holubec}.
Recently, the optimization of quite general heat engines has also been considered~\cite{2020Zhang,2021Frim_b}. In Ref.~\cite{2020Zhang}, the cycle comprises two isothermal branches connected by the instantaneous \textit{pseudoadiabats} described above. The driving potential that minimizes irreversible work or heat dissipation is obtained, which in relevant cases leads to results previously derived in the framework of optimal mass transport~\cite{2011Aurell,2012Aurell,2013Bo_b}---see also Sec.~\ref{sec:OCT} of this review. In Ref.~\cite{2021Frim_b}, the full space of non-equilibrium thermodynamic cycles is explored but within the linear response regime. Therein, the authors employ information geometry ideas for deriving an upper bound for the efficiency and building finite-time heat engines that outperform---in terms of efficiency---other recent proposals.
\subsection{Other Applications}\label{ssec:otherApplications}
The idea to control the time required to reach a given transformation,
or perform a certain task has a respectable history in the field of
engineering \cite{1958Smith}, where a common situation is that pertaining to
crane driving (trolley displacement and rope length). Yet, the similarities
between such a macroscopic system and the dynamical equations ruling
the transport of ultracold
ions or neutral atoms in effectively one-dimensional traps
with time-dependent controllable parameters (as used in
advances towards scalable quantum-information
processing), opens the way for a transfer of method between the two
fields \cite{2017Gonzalez}.
Besides, SST\ allow to circumvent some shortcomings of
existing manipulation techniques. In particular, optical manipulations
of colloids are plagued by the impracticality of creating time-dependent
expulsive confinements, that are required when seeking to deconfine
a colloidal state \cite{2016Martinez,2018Chupeau}. While feedback protocols offer a way out \cite{2020Albay_b}, we seek a purely feedforward technique, without applying
any retroaction on the system.
Taking advantage of diffusiophoresis, {i.e.} the migration of colloids induced by solute gradients, one can drive the system by
a proper time-dependent control of the salt concentration
in a buffer in contact with the solution.
A fast decompression can thereby be achieved
\cite{2021Bayati}, from a joint optical and diffusiophoretic driving.
To remain in the field of soft matter, a related
proposal of driving a bulk system from its boundaries was put
forward in \cite{2019PalaiaPhD}: by monitoring the potential
(or the charge) difference between the two plates of a nanocapacitor,
one can accelerate significantly electrolyte relaxation
following a charging of the system.
A rather unexpected application of SST\ has emerged in the context of biological evolution~\cite{2021Iram}.
The setting is provided by the Wright-Fisher
model~\cite{1932Wright}, describing the evolution of a population of organisms through a space of $M$ possible genotypes.
The state of the population is characterized by a time-dependent probability distribution $\rho(\bm{x},t)$, where $\bm{x}\equiv (x_1,\cdots x_M)$ is a frequency vector whose component $x_m$ gives the fraction of the population found in genotype $m\in[1,M]$ at time $t$.
The Wright-Fisher model describes the diffusive evolution of $\rho(\bm{x},t)$ due to mutations that induce a random walk among genotypes.
It is additionally assumed that there exists an externally controlled environmental parameter $\lambda$, corresponding for example to a drug concentration that affects the fitness of a pathogen.
If $\lambda$ is held fixed, then under the Wright-Fisher model the population evolves to an evolutionary equilibrium state $\rho^{\rm eq}(\bm{x},\lambda)$.
If the parameter is varied with time, then the actual state of the population $\rho(\bm{x},t)$ lags behind the instantaneous equilibrium state $\rho^{\rm eq}(\bm{x},\lambda(t))$.
Applying the tools of SST, Ref.~\cite{2021Iram} shows how to construct a counter-diabatic control protocol that eliminates this lag.
This opens the possibility of controlling biological evolution to drive a population from an initial equilibrium state to a desired final equilibrium state in finite time.
SST\ have also been used to develop strategies for implementing computational tasks rapidly and accurately, and to study the inherent thermodynamic costs associated with these tasks~\cite{2021Boyd}.
Specifically, protocols were designed to create, erase and transfer a single bit of information stored in a double-well potential.
These tasks dissipate energy into the bit's thermal surroundings.
Ref.~\cite{2021Boyd} explored how this dissipated work scales with the rate of computation, the characteristic length scale of the system, and the robustness of the (inherently metastable) information storage.
A key finding of Ref.~\cite{2021Boyd} was that although Landauer's principle~\cite{1961Landauer,2015Parrondo} establishes the minimum dissipated energy needed to erase a bit of information, there is no similar bound on the amount of time needed to erase the bit.
Using SST, a bit can be erased arbitrarily rapidly and with arbitrarily high fidelity, provided one is willing to pay the requisite thermodynamic cost in dissipated energy.
The results of Ref.~\cite{2021Boyd} illustrate, in a computational thermodynamic setting, the refined second law of thermodynamics obtained in Ref.~\cite{2012Aurell}. Generalizing these
approaches to underdamped systems is challenging
\cite{2021Dago}.
To conclude, we emphasize that the key ideas behind SST,
and inverse engineering, can be
illustrated in an early undergraduate course, with RC, LC or RLC circuits. An experimental demonstration was proposed in
\cite{2019Faure}, that furthermore provides an original
approach to the venerable teaching of differential
equations.
\section{Perspectives and conclusions}
\label{sec:conclusions}
Shortcuts to adiabaticity (STA) are finite-time protocols that produce the same state as
an infinitely slow (so-called adiabatic) driving. We have here generalized this
approach to finite-time protocols connecting arbitrary states, referred to under the terminology of
SST\ (Swift State-to-state Transformations). For both Hamiltonian and stochastic dynamics, we have classified SST\ into different types, inverse-engineering, counterdiabatic (CD), and fast-forward (FF), mirroring the usual categorization of quantum STA. This categorization is indeed useful for presentation purposes but not so clear-cut as it may seem: in the stochastic case, CD protocols can be thought both as a special case of inverse-engineering ones and as a certain limit of FF protocols---which somehow smears the boundaries between the different categories. One of the goals of our review is to highlight the common fundamentals behind all of them. To conclude the paper, we offer below some perspectives that emerge.
For isolated Hamiltonian systems (Sec.~\ref{sec:isolated-systems}) we foresee three topics of potential future investigation. In systems with one degree of freedom, one could drop the assumption that the steering potential $U(x,t)$ is a confining potential with a single minimum, and instead consider a confining potential with a double-well structure, with a local maximum separating the two wells. If $U_{\rm max}$ gives the value of the potential at this local maximum, then each energy shell with $E>U_{\rm max}$ forms a single closed loop in phase space, while each energy shell $E<U_{\rm max}$ forms a pair of non-intersecting closed loops. The shell $E=U_{\rm max}$ is a {\it separatrix}, with the characteristic shape of a ``figure-eight'' closed loop. The adiabatic invariance of the action $I(E,t)$ breaks down in the vicinity of the separatrix~\cite{1986Tennyson} and this in turn leads to subtle topological effects~\cite{Lu2015}, which may have interesting implications for the design of shortcuts for such systems. Next, as already noted in Sec.~\ref{subsubsec:beyond1dof}, it is plausible that Hamiltonian shortcuts can be generalized to integrable systems with $N>1$ degrees of freedom, as such systems effectively decompose into $N$ independent one-degree-of-freedom systems. Finally, as also suggested in Sec.~\ref{subsubsec:beyond1dof}, it would be worthwhile to investigate the design of SST\ for ergodic and chaotic Hamiltonian systems that constitute {\it canonical families}. These are systems with the (non-generic) property that $H(\bm{x},\bm{p},t)$ and $H(\bm{x},\bm{p},t^\prime)$ are related by a canonical transformation, for any $t,t^\prime \in [0,t_\text{f}]$. Progress on any of these three questions may in turn provide insight into quantum shortcuts for corresponding systems.
Most SST\ have been developed for one-dimensional systems. As a rule, the generalization of these protocols---either inverse engineering, counterdiabatic, fast-forward, or thermal engineering---to higher-dimensional systems is not straightforward. Still, inverse engineering protocols, akin to those originally devised for a colloidal particle in a one-dimensional harmonic trap~\cite{2016Martinez}, have been worked out for an arbitrary curved configuration space~\cite{2021Frim}. This shows the feasibility of extending SST\ to arbitrary geometries, but engineering optimal protocols---in the sense of minimizing work, heat, entropy production, connection time or any other relevant physical property over them---remains a challenge in this case. On a related note, the protocols detailed in our review are mostly designed in the overdamped regime. Future theoretical developments in SST\ are expected in the underdamped regime to accompany the development of new experimental setups having a tunable damping rate \cite{2017rondin}.
The irreversible work unavoidably appearing in SST\ has been linked to information geometry ideas in linear response~\cite{2012Sivak}. In the information geometry approach, a key role is played by the Fisher information, which at equilibrium is related to derivatives of the distribution function and, in certain specific situations, to fluctuations of relevant quantities~\cite{1983Salamon,1984Andresen,2007Crooks,2012Crooks}. This theoretical framework also makes it possible to establish a speed-limit inequality for SST, Eq.~\eqref{eq:sl-Sivak-Crooks}, which resembles the quantum speed limit (the time-energy uncertainty relation~\cite{1991Tamm,1998Margolus}). Recently, the information geometry approach has been employed beyond linear response. This has led to the emergence of several versions of classical speed-limits inequalities---also resembling the quantum time-energy uncertainty relation---for systems described by stochastic dynamics~\cite{2018Shanahan,2018Okuyama,2018Ito,2018Shiraishi,2020Nicholson,2020Ito,2021Nakazato}. Nevertheless, the tightness of the obtained classical bounds, for instance that given by Eq.~\eqref{eq:sl-Ito-Dechant}, is not well established---unlike in the quantum case~\cite{2009Levitin,2017Deffner}. Clarifying this specific issue and, on a more general note, linking information geometry concepts and physical quantities beyond linear response are open questions. Answering them constitutes an enticing perspective for future work, and may provide a novel way to tackle and move forward non-equilibrium statistical mechanics and, more specifically, the field of SST.
We have exposed here several strategies for the control of systems in contact with a thermostat. Closed-loop control protocols, i.e. those with feedback, have not been discussed since they are intrinsically subject to the system's
own time scales. However, such approaches have the advantage of not requiring a perfect modeling of the system under study. This is to be contrasted with open-loop control protocols detailed in this
review article, that are of the feed-forward type.
They provide a wide range of strategies to accelerate the transition from a given state to another over an {\it a priori} arbitrarily short amount of time. The methods presented here
provide such solutions, despite the intrinsic randomness associated to the coupling with a reservoir, and even propose to engineer the randomness.
\begin{acknowledgments}
We would like to thank A. Baldassarri, M. Baldovin, L. Bellon, B. Besga, A. Chepelianskii, S. Ciliberto, S. Dago, T. de Guillebon, S.N. Majumdar, I. Mart\'inez, J.G. Muga, I. Palaia, A. Patr\'on, A. Petrossian, A. Puglisi, D. Raynal, L. Rondin, N. Ruiz-Pino. C.A.P. and A.P. acknowledge financial support from Grant PGC2018-093998-B-I00 funded by MCIN/AEI/10.13039/501100011033/ and by ERDF ``A way of making Europe.'' C.A.P. also acknowledges financial support from Junta de Andaluc\'{\i}a and European Social Fund through the program PAIDI-DOCTOR. This work was supported by the `Agence Nationale de la Recherche' grant No. ANR-18-CE30-0013. C.J. acknowledges financial support from the Simons Foundation through Award No.\ 681566, and from the U.S. National Science Foundation under Grant No.\ 2127900.
\end{acknowledgments}
|
1,116,691,500,904 | arxiv | \subsection{Graphs with large damage throttling number}\label{sec:largedamagethrot}
In this subsection, we show that there exists connected graphs with high cop number and high damage number. Since $c(G) \leq \operatorname{th}_c(G)$ for any graph $G$, if it were true that $\operatorname{th}_c(G) = O(\sqrt{n})$ for all connected graphs, Meyniel's conjecture would follow. However, graphs have been found that have cop throttling number asymptotically larger than $\sqrt{n}$.
\begin{cor}\cite[Corollary 2.3]{CopThrot2}\label{cor:HighCopThrot}
There exist connected graphs of order $n$ with cop throttling number $\Omega(n^{2/3})$.
\end{cor}
We have shown in Propositions \ref{prop:c(G)leqthd(G)} and \ref{prop:INEQthcthd} that $c(G) \leq \operatorname{th}_d(G) \leq \operatorname{th}_c(G) - 1$ for all connected graphs $G$, so it is natural to ask whether $\operatorname{th}_d(G) = O(\sqrt{n})$ as this would also imply Meyniel's conjecture. Note that the argument used to prove Corollary \ref{cor:HighCopThrot} relies on the fact that capture of the robber is required to achieve a finite cop throttling number. However, this argument is not sufficient for damage throttling since capture is not required to achieve a finite damage throttling number. To that end, we prove the next lemma which is used in the main result of this subsection. It may also be of independent interest, since it strengthens the result that there exist connected graphs of order $n$ with cop number $\Omega(\sqrt{n})$.
\begin{lem}\label{dmg_lem}
For any $n$ sufficiently large and constant $c < 1$, there exists a connected graph of order $n$ where the robber can safely damage $\Omega(n)$ vertices and evade capture forever against at most $c \sqrt{n}$ cops.
\end{lem}
\begin{proof}
We modify the Moore graph $H$ of degree $d = \left\lfloor \sqrt{n-1} \right\rfloor$ and diameter $2$, which is a regular graph of girth $5$ and order $d^2+1$. This graph is known to have cop number $d$ (in general $c(G) \geq \delta(G)$ for any $G$ of girth at least $5$). Let the graph $G$ of order $n$ be obtained from $H$ by picking a single vertex $v$ of $H$ and adding a path $P$ of length $n-1-d^2$ that is connected to $v$ at an endpoint.
If $k \leq c \sqrt{n}$, then there exists a constant $r > 0$ such that $d-k > r \sqrt{n}$ for $n$ sufficiently large since $c < 1$. The robber uses the following strategy to safely damage $\Omega(n)$ vertices of $H$ in $G$ and evade capture forever against $k$ cops.
After the cops make their initial placements, the robber views any cops on the path $P$ as being on $v$ instead, ignoring the path. Since $d-k > r \sqrt{n}$, the number of vertices occupied by or adjacent to a cop at any time is at most $d-1+(d-1)d < d^2+1$, so the robber can choose an initial position not occupied by any cop or adjacent to any cop.
Call a vertex \emph{guarded} if it is occupied by or adjacent to a cop. Since the Moore graph has diameter $2$, every vertex in $H$ is within distance $2$ of the robber's current vertex. Since the Moore graph avoids $C_3$ and $C_4$, each cop guards at most one neighbor of the robber. During every \emph{odd} round, the robber goes to its unguarded neighbor with the fewest number of damaged neighbors. The robber always has at least $r \sqrt{n}$ unguarded neighbors.
During every \emph{even} round, the robber goes to any unguarded undamaged neighbor. If in any even round $2i$ the robber has no unguarded undamaged neighbor, then every unguarded neighbor of the robber's vertex in the previous odd round $2i-1$ had at least $r \sqrt{n}$ damaged neighbors. The closed neighborhoods of the unguarded neighbors of the robber's vertex $u$ have no intersection besides $u$ since $H$ avoids $C_3$ and $C_4$, so there must be at least $(r \sqrt{n}) (r \sqrt{n}-1) = \Omega(n)$ damaged vertices if the robber is ever forced to go to a damaged vertex in an even round. Thus the robber can safely damage $\Omega(n)$ vertices in $G$.
\end{proof}
The next theorem has almost the same proof as the corresponding theorem in \cite{CopThrot2}, with the main difference being that in this construction we use the graphs from Lemma \ref{dmg_lem} instead of arbitrary graphs with $\Omega(\sqrt{n})$ cop number, and the robber uses a specific evasion strategy that safely damages many vertices instead of an arbitrary evasion strategy.
\begin{thm}
For $n$ sufficiently large, there exist connected graphs $X$ of order $n$ with $\operatorname{th}_d(X) = \Omega(n^{2/3})$.
\end{thm}
\begin{proof}
We construct a connected graph $X$ with $\operatorname{th}_d(X) = \Omega(n^{2/3})$ by starting with a spider of order $n$ with $\left\lfloor n^{1/3}\right\rfloor$ legs of length approximately $n^{2/3}$, and then replacing the half of each leg farthest from the center vertex by a copy of one of the graphs $G$ from Lemma \ref{dmg_lem} with the same order as the replaced vertices. If the number of cops does not already give the bound $\operatorname{th}_d(X) = \Omega(n^{2/3})$, then the robber will start on the copy of $G$ on the leg of the spider with the fewest cops. It will take at least $\Omega(n^{2/3})$ rounds for any cops on other legs to reach the robber's leg, so the robber can use the strategy in Lemma \ref{dmg_lem} to safely visit a new undamaged vertex in every even round until it has damaged at least $\Omega(n^{2/3})$ vertices.
\end{proof}
\begin{subsection}{Graph families such that $\operatorname{th}_d(G)=\operatorname{th}_c(G)-1$}\label{subsec:boundtight}
Although Corollary \ref{cor:characterization} provides a complete characterization of graphs $G$ that satisfy $\operatorname{th}_d(G)=\operatorname{th}_c(G)-1$, the given conditions are not easy to verify. Therefore, further study of this equality is useful. In order to find several families of graphs that achieve this equality, we now turn our attention to the $k$-radius of a graph and use the following result.
\begin{prop}\cite{BPPR17}\label{prop:radkCopThrot}
If $G$ is a connected graph on $n$ vertices and $1 \leq k \leq n$ is an integer, then $\operatorname{capt}_k(G)\geq \operatorname{rad}_k(G).$
\end{prop}
The proof of Proposition \ref{prop:radkCopThrot} uses a stationary robber, but such a strategy is not optimal for damage. We now prove the analogous result using a different robber strategy.
\begin{prop}\label{prop:radk}
If $G$ is a connected graph on $n$ vertices and $1 \leq k \leq n$ is an integer, then $\operatorname{dmg}_k(G)\geq \operatorname{rad}_k(G) - 1.$
\end{prop}
\begin{proof}
First, note that if $\operatorname{rad}_k(G) \leq 1$, then $\operatorname{dmg}_k(G) \geq 0$ is trivially satisfied. Next, suppose $\operatorname{rad}_k(G) \geq 2$ and consider an arbitrary initial placement of $k$ cops on a subset $S\subseteq V(G)$ of vertices. Choose a vertex $x\in V(G)$ such that $d(S,x)$ is maximum. Choose $u\in S$ such that $d(u,x) = d(S, x) $. Let $P$ be a shortest path in $G$ from $u$ to $x$. Place the robber on the vertex $y$ in $V(P)$ such that $d(y,u)=2$. If the robber moves towards $x$ along the path $P$ in each round, then $|V(P)|-2$ vertices are damaged. Since $|V(P)|-1\geq \operatorname{rad}_k(G)$, this means $\operatorname{dmg}_k(G)\geq \operatorname{rad}_k(G)-1$.
\end{proof}\medskip
As noted in \cite[Remark 3.2]{CRthrottle}, Proposition \ref{prop:radkCopThrot} yields $\operatorname{th}_c(G) \geq \underset{1 \leq k \leq n}{\min}\{k+\operatorname{rad}_k(G)\}$ as an immediate corollary. Proposition \ref{prop:radk} leads us to the following analogous result for $\operatorname{th}_d(G)$.
\begin{cor}\label{cor:thdANDradk}
For any graph $G$, $\operatorname{th}_d(G) \geq \underset{1 \leq k \leq n}{\min}\{k + \operatorname{rad}_k(G)\} - 1.$
\end{cor}
We have established in Proposition \ref{prop:INEQthcthd} that for any nontrivial graph $G$, $\operatorname{th}_d(G) \leq \operatorname{th}_c(G) -1$. While we are also interested in graphs where $\operatorname{th}_d(G) < \operatorname{th}_c(G) -1$ (see Section \ref{sec:diff}), we now turn our attention to instances when this bound is an equality. Using our previous results about $k$-radius, we show the desired equality holds for several classes of graphs.
\begin{prop}\label{prop:minradk}
If $\operatorname{th}_c(G)=\underset{1 \leq k \leq n}{\min}\{k + \operatorname{rad}_k(G)\}$, then $\operatorname{th}_d(G)=\operatorname{th}_c(G) - 1$.
\end{prop}
\begin{proof}
Suppose $\operatorname{th}_c(G) = \underset{1 \leq k \leq n}{\min}\{k + \operatorname{rad}_k(G)\}.$ By Corollary \ref{cor:thdANDradk}, \[\operatorname{th}_d(G) \geq \underset{1 \leq k \leq n}{\min}\{k + \operatorname{rad}_k(G)\} -1 = \operatorname{th}_c(G)-1.\] Since $\operatorname{th}_d(G) \leq \operatorname{th}_c(G)-1$ by Proposition \ref{prop:INEQthcthd}, it follows that $\operatorname{th}_d(G) =\operatorname{th}_c(G)-1.$
\end{proof}\medskip
Next, we apply known results about graphs $G$ for which $\operatorname{th}_c(G)=\underset{1 \leq k \leq n}{\min}\{k + \operatorname{rad}_k(G)\}$ in order to show that $\operatorname{th}_d(G)=\operatorname{th}_c(G)-1$ for these graphs. Recall that a chordal graph is a graph in which every induced cycle is a $C_3$.
\begin{prop}\cite{CRthrottle}
For any tree or cycle $G$ on $n$ vertices, $\operatorname{th}_c(G)=\underset{1 \leq k \leq n}{\min}\{k + \operatorname{rad}_k(G)\}$.
\end{prop}
\begin{cor}\label{cor:treesAndCycles}
For any tree or cycle $G$, $\operatorname{th}_d(G)=\operatorname{th}_c(G)-1$.
\end{cor}
\begin{prop}\cite{CopThrot2}\label{chordRadk}
For any connected chordal graph $G$ and integer $1 \leq k \leq |V(G)|$, $\operatorname{capt}_k(G)=\operatorname{rad}_k(G)$.
\end{prop}
\begin{cor}\label{cor:chordalThrottling}
For any connected chordal graph $G$, $\operatorname{th}_d(G)=\operatorname{th}_c(G)-1$.
\end{cor}
It is worth noting that the converse of Proposition \ref{prop:minradk} does not hold; that is, there exist graphs such that $\operatorname{th}_d(G)=\operatorname{th}_c(G)-1$ and $\operatorname{th}_c(G)>\underset{1 \leq k \leq n}{\min}\{k + \operatorname{rad}_k(G)\}$. The Petersen graph $P$ provides such an example, which we will examine next. First, recall that a graph $G$ is $\operatorname{SRG}(n, k, \lambda, \mu)$ if $|V(G)| = n$, $G$ is $k$-regular, every pair of adjacent vertices in $G$ has $\lambda$ common neighbors, and every pair of non-adjacent vertices in $G$ has $\mu$ common neighbors. The well-known fact that $P$ is $\operatorname{SRG}(10, 3, 0, 1)$ is particularly useful for determining $\operatorname{th}_d(P)$.
\begin{thm}
For the Petersen graph $P$, $\underset{1 \leq k \leq n}{\min}\{k + \operatorname{rad}_k(P)\}=3$, $\operatorname{th}_c(P)=4$, and $\operatorname{th}_d(P) = 3$.
\end{thm}
\begin{proof}
In order to find $\underset{1 \leq k \leq n}{\min}\{k + \operatorname{rad}_k(P)\}$, note that $\operatorname{rad}(P) = 2$ and $\gamma(P) = 3$. This gives the following values of $\operatorname{rad}_k(P)$:
\[
\operatorname{rad}_k(P)=
\begin{cases}
2 &\text{if } k=1,2; \\
1 &\text{if } 3\leq k\leq 9; \\
0 &\text{if } k=10.
\end{cases}
\]
So we see that $\underset{1 \leq k \leq n}{\min}\{k + \operatorname{rad}_k(P)\} = 3$.
Next, note that $c(P) = 3$ \cite{AF84} and since $c(P) = \gamma(P)$, we have the following capture times:
\[
\operatorname{capt}_k(P)=
\begin{cases}
\infty &\text{if } k=1,2; \\
1 &\text{if } 3\leq k\leq 9; \\
0 &\text{if } k=10.
\end{cases}
\]
These capture times imply that $\operatorname{th}_c(P) = 4$.
Finally, we calculate $\operatorname{th}_d(P)$ by considering all possible damage numbers. We know that $\operatorname{dmg}_1(P)=5$ \cite{CS19} and since $\gamma(P) = 3$, $\operatorname{dmg}_k(P)=0$ for all integers $3\leq k\leq 10$. We now prove that $\operatorname{dmg}_2(P)=2$ by showing that the robber can always damage two vertices, and that the cops can always prevent the robber from damaging a third vertex. Since $P$ is $\operatorname{SRG}(10,3,0,1)$, each pair of adjacent vertices has no common neighbors and each pair of non-adjacent vertices has exactly one common neighbor. Thus, while one cop dominates four vertices, two adjacent cops dominate six vertices and two non-adjacent cops dominate seven.
To show that the robber can always damage two vertices, we note that regardless of cop placement, the robber will always be adjacent to a vertex not dominated by either of the cops. Otherwise, the cops could move such that one of them dominates two of the robber's neighbors, which contradicts $P$ being $\operatorname{SRG}(10,3,0,1)$. So, in round $1$, the robber moves to an adjacent non-dominated and undamaged vertex, thus damaging their starting vertex. In round $2$, the robber damages the vertex they occupy and moves to an adjacent non-dominated vertex. Note that the vertex the robber moves to may be its original starting vertex. Thus, $\operatorname{dmg}_2(P)\geq 2$.
To show that the cops can prevent a third damaged vertex, place the cops on non-adjacent vertices so that they dominate seven vertices. Playing according to the robber strategy above, at the start of round $2$, the robber is on a non-dominated vertex. Further, the robber is adjacent to three vertices, namely $u$, the now-damaged starting vertex; $v$, an undamaged but dominated vertex; and $w$, which is undamaged and may or may not be dominated.
If $w$ is dominated, the cops can stay still in round $2$, which forces the robber back to $u$. This damages a second vertex, but the robber is now on a previously damaged vertex. If $w$ is not dominated, then a cop that dominates $v$ stays still, while the other cop moves to dominate $w$. This move is possible since every vertex not adjacent to $w$ has a common neighbor with $w$. Thus, the only non-dominated vertex adjacent to the robber is $u$, and so the robber must move to $u$. By repeating this strategy, the cops restrict the robber's movement to these two damaged vertices. Thus, $\operatorname{dmg}_2(P)\leq 2$.
Therefore, $\operatorname{dmg}_2(P)=2$ and we have the following:
\[
\operatorname{dmg}_k(P)=
\begin{cases}
5 &\text{if } k=1; \\
2 &\text{if } k=2; \\
0 &\text{if } 3\leq k\leq 10.
\end{cases}
\]
These damage numbers imply that $\operatorname{th}_d(P)=3$.
\end{proof}\medskip
\end{subsection}
\begin{subsection}{Graphs on few vertices}\label{subsec:fewvertices}
In this subsection, we explore the gap between damage and cop throttling in graphs with few vertices. As we have seen previously, when $\gamma(G)$ cops play optimally on a graph $G$, the robber is captured in the first round and no vertices are damaged. As such, we can expect that for graphs with small enough domination numbers, we will be able to restrict the gap between damage and cop throttling to just $1$.
\begin{lem}\label{lem:domNum2}
If $G$ is a nontrivial connected graph with $\gamma(G)\leq 2$, then $\operatorname{th}_d(G)=\operatorname{th}_c(G)-1$.
\end{lem}
\begin{proof}
Suppose $G$ is a graph on $n$ vertices. If $\gamma(G)=1$, then $\operatorname{capt}_k(G)=1$ and $\operatorname{dmg}_k(G)=0$ for all $k<n$; this gives $\operatorname{th}_d(G)= 1 = \operatorname{th}_c(G)-1$. If $\gamma(G)=2$, then $\operatorname{capt}_1(G)\geq2$ and $\operatorname{capt}_k(G)=1$ for all $2\leq k<n$. Furthermore, we know $\operatorname{dmg}_1(G)\geq1$ and $\operatorname{dmg}_k(G)=0$ for all $2\leq k<n$. Together, this implies that $\operatorname{th}_d(G)=2 = \operatorname{th}_c(G)-1.$
\end{proof}
Most nontrivial graphs with order at most $6$ have a domination number of $2$. Thus, Lemma \ref{lem:domNum2} and additional consideration for those graphs with $\gamma(G)=3$ allow us to classify all nontrivial connected graphs on at most $6$ vertices as exhibiting a difference of $1$ between damage and cop throttling.
\begin{prop}\label{prop:order6Gap}
If $G$ is a connected graph of order $2\leq n\leq6$, then $\operatorname{th}_d(G)=\operatorname{th}_c(G)-1$.
\end{prop}
\begin{proof}
It is well-known that $\gamma(G) \leq \frac{n}{2}$ for all connected graphs $G$ with $|V(G)| \geq 2$. If $|V(G)| \leq 5$, then $\gamma(G) \leq 2$ and by Lemma \ref{lem:domNum2}, $\operatorname{th}_d(G)=\operatorname{th}_c(G)-1$.
Suppose now that $|V(G)|=6$; then, $\gamma(G) \leq \frac{6}{2} = 3$. If $\gamma(G)\leq 2$, then $\operatorname{th}_d(G)=\operatorname{th}_c(G)-1$ by Lemma \ref{lem:domNum2}. Using the \textit{Sage} code in \cite{code_domnum3}, we find that the only two graphs of order $6$ with $\gamma(G)=3$ are those illustrated in Figure \ref{fig:order6dom3}.
\begin{figure}[H] \begin{center}
\scalebox{.7}{\includegraphics{ordersixdomthreegraphs.pdf}}\\
\caption{The two order-$6$ graphs $G$ with $\gamma(G) = 3$.}\label{fig:order6dom3}
\end{center}
\end{figure}
Observe that the graph on the left in Figure \ref{fig:order6dom3} is a tree and the graph on the right is chordal. Therefore, by Corollary \ref{cor:treesAndCycles} and Corollary \ref{cor:chordalThrottling} respectively, these two graphs also have $\operatorname{th}_d(G)=\operatorname{th}_c(G)-1$ as desired. \qedhere
\end{proof}
Thus, any nontrivial connected graph of at most $6$ vertices will exhibit the lowest possible gap of $1$ between its damage and cop throttling numbers. This motivates us to consider what the minimum order is of graphs with a difference of more than $1$ between $\operatorname{th}_d(G)$ and $\operatorname{th}_c(G)$. More generally, we are interested in finding graphs that exhibit this larger gap of at least $2$.
\end{subsection}
\end{section}
\begin{section}{Differences between damage and cop throttling}\label{sec:diff}
In this section, we turn to examining various graphs for which the gap between damage throttling and cop throttling is at least two. While these are harder to find, infinite families of graphs that realize this gap do exist; in particular, we demonstrate two infinite families in which $\operatorname{th}_c(G)$ and $\operatorname{th}_d(G)$ remain constant and one in which they grow without bound. A graph exhibiting a gap of three is also presented.
First, we continue our discussion of graphs on few vertices by showing that the smallest graphs exhibiting a gap of two consist of only $7$ vertices.
\begin{prop}
The connected graphs of smallest order such that $\operatorname{th}_d(G)\leq\operatorname{th}_c(G)-2$ have $7$ vertices; there are thirteen of them in total.
\end{prop}
\begin{proof}
By Proposition \ref{prop:order6Gap}, $\operatorname{th}_d(G)=\operatorname{th}_c(G)-1$ for any connected graphs of order $6$, so it suffices to consider graphs on order $7$.
For any $7$-vertex graph $G$, $\gamma(G)\leq\frac72$, so $\gamma(G)\in\{1,2,3 \}$. By Lemma \ref{lem:domNum2}, if $G$ has domination number $1$ or $2$, then $\operatorname{th}_d(G)=\operatorname{th}_c(G)-1$. Thus, we need only consider graphs $G$ of order $7$ with $\gamma(G)=3$. To find these, we will algorithmically check every $7$-vertex graph to see if it can be dominated by three vertices. Using the \textit{Sage} code in \cite{code_domnum3}, we find forty-two such graphs that have $\gamma(G)=3$. Of these, we will show that the twenty-nine graphs displayed in Figure \ref{fig:29GapOfOne} have a gap of only one, but the thirteen graphs in Figure \ref{fig:13GapOfTwo} have the desired gap of two.
\begin{figure}[ht]
\centering
\begin{tabular}{|c|m{12.5cm}|}
\hline
trees & \includegraphics[scale=.3]{DN3GraphsTrees.png} \\
\hline
chordal graphs & \includegraphics[scale=.3]{DN3GraphsChordal.png}\\
\hline
cycle & \includegraphics[scale=.3]{DN3GraphsCycle.png}\\
\hline
$\operatorname{dmg}_1(G)>1$ & \includegraphics[scale=.3]{DN3GraphsDamage.png}\\
\hline
\end{tabular}
\caption{The twenty-nine order-$7$ graphs $G$ with $\gamma(G) = 3$ but $\operatorname{th}_d(G)=\operatorname{th}_c(G)-1$.}
\label{fig:29GapOfOne}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[scale=.45]{ordersevengapoftwo.pdf}
\caption{The thirteen order-$7$ graphs $G$ with $\gamma(G) =3$ and $\operatorname{th}_d(G)=\operatorname{th}_c(G)-2$, with a vertex indicated in each in black on which the cop can place to ensure $\operatorname{dmg}_1(G)=1$.}
\label{fig:13GapOfTwo}
\end{figure}
Observe that of the graphs in Figure \ref{fig:29GapOfOne}, five are trees, fifteen are chordal, and one is a cycle; by Corollaries \ref{cor:treesAndCycles} and \ref{cor:chordalThrottling}, these twenty-one graphs have $\operatorname{th}_d(G)=\operatorname{th}_c(G)-1$.
For each of the remaining eight graphs in Figure \ref{fig:29GapOfOne} and thirteen graphs in Figure \ref{fig:13GapOfTwo}, we will directly calculate $\operatorname{th}_c(G)$ and $\operatorname{th}_d(G)$. First, by the Python code in \cite{code_dismantlable}, none of these graphs are dismantlable, and so by \cite{CRbook}, they are not cop-win. Since $\gamma(G)=3$ for each of these graphs, $\operatorname{capt}_2(G)\geq 2$ and $\operatorname{capt}_k(G)=1$ for $3\leq k\leq 6$. Therefore $\operatorname{th}_c(G)=4$.
We now calculate $\operatorname{th}_d(G)$. Since $\gamma(G)=3$, we have $\operatorname{dmg}_2(G)\geq 1$ and $\operatorname{dmg}_k(G)=0$ for $3\leq k\leq 7$. Observe that if $\operatorname{dmg}_1(G)=1$, then $G$ will have $\operatorname{th}_d(G)=2$ and achieve the desired gap; otherwise, if $\operatorname{dmg}_1(G)>1$, then $\operatorname{th}_d(G)=3$ and the gap is only one. We thus wish to characterize which of these twenty-one graphs have $\operatorname{dmg}_1(G)=1$.
Note that in each of the thirteen graphs in Figure \ref{fig:13GapOfTwo}, if the cop places on the black vertex, then by passing in each subsequent round, the cop restricts the robber on only damaging one vertex. Thus, for these graphs, $\operatorname{dmg}_1(G)=1$ and so $\operatorname{th}_d(G)=\operatorname{th}_c(G)-2$, as desired. The eight graphs in the last row of Figure \ref{fig:29GapOfOne} do not contain such a vertex the cop can place on; this implies the robber can move to a new vertex in round $1$ without getting captured in round $2$. As such, $\operatorname{dmg}_1(G)>1$ and $\operatorname{th}_d(G)=\operatorname{th}_c(G)-1$ for these eight graphs.
\end{proof}
If we consider disconnected graphs, the gap between damage and cop throttling can grow arbitrarily far apart, as exhibited by the following infinite family.
\begin{obs}
For $n \geq 2$, $\operatorname{th}_d(\overline{K_n}) = 2$ and $\operatorname{th}_c(\overline{K_n}) - 1 = n - 1$. Thus, as $n$ increases, $\operatorname{th}_d(\overline{K_n})$ and $\operatorname{th}_c(\overline{K_n}) - 1$ get arbitrarily far apart.
\end{obs}
However, we would like to find examples of infinite families of connected graphs that exhibit the desired gap. While graphs with a dominating vertex $v$ will not realize such a gap, we can carefully reduce the number of vertices that $v$ dominates to restrict the robber's motion and lower the $k$-damage numbers. This motivates the following definitions which are illustrated in Figure \ref{fig:gearaccordion}.
\begin{defn}
Recall the definition of the wheel graph $W_n$ of order $n$ as the graph obtained by adding a dominating vertex, known as the hub, to the cycle $C_{n-1}$. For each integer $\ell \geq 2$, the \emph{gear graph} of order $2\ell + 1$ is denoted $G_{2 \ell + 1}$ and is obtained from $W_{2 \ell + 1}$ by deleting every other edge incident to the hub.
\end{defn}
\begin{defn}
Further, recall the definition of the fan graph $F_n$ as the graph obtained by adding a dominating vertex to a the path $P_{n-1}$. We can now analogously define the \emph{accordion graph} for each integer $\ell \geq 2$, denoted $A_{2 \ell}$, as the graph obtained from $F_{2\ell}$ by deleting every other edge incident to the dominating vertex.
\end{defn}
\vspace{-.5 cm}
\begin{figure}[H] \begin{center}
\scalebox{.7}{\includegraphics{gearAcc.pdf}}\\
\caption{The gear and accordion graphs are shown on the left and right respectively.}\label{fig:gearaccordion}
\end{center}
\end{figure}
Next, we show that the gear and accordion graphs are infinite families of connected graphs for which $\operatorname{th}_d(G) = \operatorname{th}_c(G) - 2$.
\begin{thm} For each integer $\ell\geq 4$, if $G \in \{G_{2\ell + 1}, A_{2 \ell}\}$, then $\operatorname{th}_d(G)=\operatorname{th}_c(G)-2$. Thus,
there exist infinitely many connected graphs $G$ such that $\operatorname{th}_d(G) < \operatorname{th}_c(G) - 1$.
\end{thm}
\begin{proof}
Consider $G_{2\ell+1}$ for any $\ell\geq 4$. By placing and remaining on the hub vertex, one cop can restrict the robber to damaging only a single vertex. Since $G_{2\ell+1}$ does not have a dominating vertex, the robber can always damage at least one vertex, and so we have $\operatorname{dmg}_1(G_{2\ell+1})=1$. This implies $\operatorname{th}_d(G_{2\ell+1})\leq2$. Since $\operatorname{th}_d(G_{2\ell+1})=1$ only when there is a dominating vertex, $\operatorname{th}_d(G_{2\ell+1})=2$.
Observe that $G_{2\ell+1}$ is not cop-win, so consider $k=2$. Place both cops on the hub vertex. Assuming the robber places optimally on a vertex non-adjacent to the hub, the cops can move to either side of the robber in round $1$ and capture them in round $2$, giving $\operatorname{capt}_2(G_{2\ell+1})\leq 2$. Further, $\operatorname{capt}_2(G_{2\ell+1})>1$, since $\gamma(G_{2\ell+1})>2$ for $\ell\geq 4$. Thus, $\operatorname{capt}_2(G_{2\ell + 1})= 2$ and $\operatorname{th}_c(G_{2\ell+1})\leq 4$. Since $\operatorname{capt}_3(G_{2\ell+1})>0$, we see that $\operatorname{th}_c(G_{2\ell+1})>3$ and conclude that $\operatorname{th}_c(G_{2\ell+1})=4$.
Using the same argument, we can show for the accordion graph $A_{2\ell}$ that $\operatorname{th}_d(A_{2\ell})=2$ and $\operatorname{th}_c(A_{2\ell})=4$ for all $\ell\geq 4$. Since $G_{2\ell+1}$ and $A_{2\ell}$ are infinite families, there are infinitely many graphs such that $\operatorname{th}_d(G)<\operatorname{th}_c(G)-1$, as desired.
\end{proof}\medskip
Notice that for each $G \in \{G_{2\ell+1}, A_{2\ell}\}$, $c(G)=\operatorname{th}_d(G)$. The next proposition shows that for graphs $G$ with this property, we can more easily determine whether $\operatorname{th}_d(G)<\operatorname{th}_c(G)-1$.
\begin{prop}\label{prop:copGammaThrot}
If $c(G)=\operatorname{th}_d(G)$, then one of the following is true:
\begin{enumerate}
\item $\gamma(G)=c(G)$
\item $\operatorname{th}_d(G)<\operatorname{th}_c(G)-1$
\end{enumerate}
\end{prop}
\begin{proof}
Let $c(G)=\operatorname{th}_d(G)$ and assume $\operatorname{th}_d(G)\not<\operatorname{th}_c(G)-1$. If $G$ is trivial, $c(G)=\operatorname{th}_d(G)=\gamma(G)$. If $G$ is non-trivial, then by Proposition \ref{prop:INEQthcthd}, $\operatorname{th}_d(G)=c(G)=\operatorname{th}_c(G)-1$ and so either $\operatorname{capt}_{c(G)}(G)=1$ or $\operatorname{capt}_{c(G)+1}(G)=0$.
If $\operatorname{capt}_{c(G)}(G)=1$, then by definition, some arrangement of $c(G)$ cops dominate the graph and $\gamma(G)\leq c(G)$. However, any dominating set of vertices in a graph form an initial cop placement which catches the robber, so clearly $c(G)\leq \gamma(G)$. Therefore, $\gamma(G)=c(G)$.
If $\operatorname{capt}_k(G)=0$, then $k=n$. Thus, if $\operatorname{capt}_{c(G)+1}(G)=0$, we must have $c(G)=n-1$ and so G has at least one edge. Recall that $c(G)\leq \gamma(G)$ and for graphs with at least one edge, $\gamma(G)\leq n-1$. Therefore $n-1=\gamma(G)$ and this implies the graph has at most one edge. The only graphs with one edge are the disjoint union of $K_2$ with some number of isolated vertices. However, we can easily observe that for this class of graphs, $\gamma(G)=c(G)$.
\end{proof}\medskip
Thus far, we have considered several graphs $G$ that satisfy $\operatorname{th}_d(G) = \operatorname{th}_c(G) - 2$. We now find a graph that exhibits a larger difference between the cop and damage throttling numbers.
\begin{thm}\label{thm:GapOf3Graph}
There exists a connected graph $G$ with $\operatorname{th}_d(G) \leq \operatorname{th}_c(G) - 3$.
\end{thm}
\begin{proof}
We will show that the graph $G$ in Figure \ref{fig:GapOf3Graph} has $\operatorname{th}_c(G)=6$ and $\operatorname{th}_d(G)=3$.
\hspace{-.5 cm}
\begin{figure}[H] \begin{center}
\scalebox{.75}{\includegraphics{gapofthreegraph.pdf}}\\
\caption{A graph $G$ with $\operatorname{th}_c(G) - \operatorname{th}_d(G) = 3$.}\label{fig:GapOf3Graph}
\end{center}
\end{figure}
\vspace{-.5 cm}
We show first that $\gamma(G)=5$. Note that $\{1,4,9,10,14 \}$ is a five-vertex dominating set. Suppose that a dominating set $S$ of only four vertices exists. In order to dominate vertices $1$, $7$, $8$, and $14$, $S$ must include a vertex from each of $\{1,2,3\}$, $\{5,7,9\}$, $\{6,8,10\}$, and $\{12,13,14\}$, respectively. Since the vertices in these four sets each have degree 2, every vertex in $S$ dominates three vertices and so $S$ can dominate at most twelve vertices. Therefore no such set $S$ of size four exists.
Observe that $G$ is not cop-win; so, we consider $k\geq 2$. By domination, $\operatorname{capt}_k(G)=1$ for $5\leq k\leq 13$ and $\operatorname{capt}_k(G)\geq 2$ for $2\leq k\leq 4$. It remains to show that $\operatorname{capt}_2(G)\geq 4$ and $\operatorname{capt}_3(G)\geq 3$.
Now consider $k=2$. We will show for any cop placement, the robber evades capture for at least four rounds. Given a cop placement, if there exists a vertex that is at least distance $4$ away from both cops, then the robber can stay on that vertex, and will evade capture until at least the fourth round.
So, assume there is no vertex of distance at least $4$ from both cops. We can algorithmically check over all possible cop placements to determine which have this property; for a Python implementation of this, see \cite{code_gapof3}. In total, there are thirty-five such placements, which can be reduced to twelve cases when considering horizontal, vertical, and rotational symmetries of the graph. In each case, the robber's strategy to avoid capture until at least the fourth round is to initially place on
\begin{enumerate}[label=(\alph*)]
\item vertex 14, if the cop set is \{1,9\}, \{1,11\}, \{2,9\}, \{2,10\}, \{2,11\}, \{4,9\}, \{4,11\}, \{5,9\}, \{5,10\}, or a set symmetric to one of these;
\item vertex 13, if the cop set is \{1,12\}, \{2,12\}, or a set symmetric to either of these; or
\item vertex 12, if the cop set is \{2,13\} or a set symmetric to this.
\end{enumerate}
It is straightforward to verify that the robber cannot be caught in fewer than four rounds given these initial placements. Thus, $\operatorname{capt}_2(G)\geq4$.
For three cops, we proceed similarly to show that no matter where the three cops place, the robber always has a strategy to avoid capture until at least the third round. As before, given an initial cop placement, if there exists a vertex that is at least a distance $3$ from all three cops, then the robber can place on that vertex, and can only be caught in the third round or later.
So assume there is no vertex at least distance $3$ away from all cops. Once again, we will check all possible cop placements to determine which have this property using the Python code in \cite{code_gapof3}. In total, there are sixty-eight such placements, reducible to nineteen cases when accounting for symmetries of the graph. In each case, the robber's strategy to avoid capture until at least the third round is to initially place on
\begin{enumerate}[label=(\alph*)]
\item vertex 14, if the cop set is \{1,2,11\}, \{1,4,11\}, \{1,5,11\}, \{2,2,11\}, \{2,3,11\}, \{2,4,11\}, \{2,5,11\}, \{2,6,11\}, \{2,7,11\}, \{2,8,11\}, \{4,4,11\}, \{4,5,11\}, \{4,7,11\}, or a set symmetric to one of these;
\item vertex 13, if the cop set is \{1,4,12\}, \{2,4,12\}, or a set symmetric to either of these;
\item vertex 12, if the cop set is \{2,4,13\} or a set symmetric to this; or
\item vertex 3, if the cop set is \{2,9,11\}, \{2,10,11\}, \{2,11,11\}, or a set symmetric to one of these.
\end{enumerate}
It is easy to check that the robber cannot be caught in fewer than three rounds using these initial placements. Thus, $\operatorname{capt}_3(G)\geq3$. We obtain that $\operatorname{th}_c(G) = 6$ from the following:
\[
\operatorname{capt}_k(G)
\begin{cases}
=\infty &\text{if } k=1; \\
\geq4 &\text{if } k=2; \\
\geq3 &\text{if } k=3; \\
\geq2 &\text{if } k=4; \\
=1 &\text{if } 5\leq k\leq 13; \\
=0 &\text{if } k=14.
\end{cases}
\]
Now, considering damage, note that $\operatorname{dmg}_k(G)=0$ for $5\leq k\leq 14$. For $k=2$, by placing cops on vertices $4$ and $11$, we restrict the robber to placing and thereafter staying on one of $\{1,7,8, 14\}$. This gives $\operatorname{dmg}_2(G)=1$ and implies that $\operatorname{dmg}_k(G) = 1$ for each $k \in \{3,4\}$. Lastly, if $k=1$, the robber can always place such that they are distance $4$ away from the cop since $\operatorname{rad}(G) = 4$. As such, the robber can damage at least two vertices. In summary:
\[
\operatorname{dmg}_k(G)
\begin{cases}
\geq2 &\text{if } k=1; \\
=1 &\text{if } 2\leq k\leq 4; \\
=0 &\text{if } 5\leq k\leq 14.
\end{cases}
\]
Thus, $\operatorname{th}_d(G)=3$, giving a gap of three between the cop and damage throttling numbers.
\end{proof}
\begin{subsection}{The family $H_n$}\label{subsec:hn}
When searching for graphs with $\operatorname{th}_d(G)<\operatorname{th}_c(G)-1$, it is natural that graphs with high capture time are worth investigating. For each integer $n\geq 7$, let $H_n$ denote the graph illustrated in Figure \ref{fig:HnGraph}. In $\cite{G10}$, it was shown that $\operatorname{capt}(H_n) = n -4$, which can achieved by placing the cop on vertex $3$ and moving the cop along vertices $3,2,4,7,8, \ldots, n$. Furthermore, this is the highest possible $1$-capture time for cop-win graphs on at least seven vertices. Thus, we dedicate this section to the infinite family $H_n$.
\begin{figure}[H] \begin{center}
\scalebox{1}{\includegraphics{HnGraph.pdf}}\\
\caption{The graph $H_n$ for $n \geq 7$.}\label{fig:HnGraph}
\end{center}
\end{figure}
We refer to the vertices in $\{8,9, \ldots, n\}$ as the \emph{tail} of $H_n$ when $n \geq 8$. As seen with many graphs so far, it is useful to consider the domination number before determining the $k$-capture time and $k$-damage number. This is also the case with $H_n$.
\begin{prop}\label{prop:domHn}
For each integer $n \geq 7$, the domination number of $H_n$ is \[\gamma(H_n) = \left\lceil \frac{n-8}{3} \right\rceil + 2.\]
\end{prop}
\begin{proof}
For $n=7$ and $n=8$, a dominating set is $\{1,7 \}$, so $\gamma(H_n)\leq2$ in these cases. Since there is no dominating vertex in $H_n$, $\gamma(H_n)=2=\lceil \frac{n-8}{3} \rceil + 2$ for $n \in \{7,8\}$. For $n=9$, a dominating set is $\{1,7,9\}$, so $\gamma(H_9)\leq 3$. Suppose we can dominate $H_9$ with only two vertices. To dominate vertex $9$, a dominating set must include vertex $8$ or $9$. However, we cannot dominate vertices $1$ through $6$ with only one vertex, so $\gamma(H_9)>2$. Thus, $\gamma(H_9)=3=\lceil \frac{9-8}{3} \rceil + 2$.
For $n\geq 10$, let $S$ be the set of vertices consisting of $1$, $7$, and every third vertex on the tail of $H_n$ starting at $10$; note that this is $\lceil \frac{n-8}{3} \rceil + 2$ vertices. The set $S$ dominates $H_n$ and so $\gamma(H_n)\leq \lceil \frac{n-8}{3} \rceil + 2$. Suppose that $H_n$ can be dominated by $\lceil \frac{n-8}{3} \rceil + 1$ vertices. In order to dominate the tail, we must include at least every third vertex in a dominating set $S$, which requires $\lceil \frac{n-8}{3} \rceil$ vertices. At most, this will dominate vertices $7, 8, \dots,n$. So, only one more vertex can be added to $S$ to dominate vertices $1, 2, \dots,6$, but no such vertex exists. Therefore, $\gamma(H_n)= \lceil \frac{n-8}{3} \rceil + 2$.
\end{proof}
In our investigation of $H_n$, the next step is computing $\operatorname{capt}_k(H_n)$ and $\operatorname{th}_c(H_n)$ as follows.
\begin{lem}\label{lem:kCaptValues}
If $n$ and $k$ are integers with $n \geq 7$ and $1 \leq k \leq n$, then \[\operatorname{capt}_k(H_n) = \begin{cases} \lceil \frac{n-3-k}{2k-1} \rceil & \text{if }1 \leq k < \lceil \frac{n-8}{3} \rceil + 2;\\ 1 & \text{if }\lceil \frac{n-8}{3} \rceil + 2 \leq k < n;\\ 0 & \text{if }k=n. \end{cases}\]
\end{lem}
\begin{proof}
Recall that the domination number of $H_n$ is $\gamma(H_n) = \left\lceil \frac{n-8}{3} \right\rceil + 2$ by Proposition \ref{prop:domHn}, $\operatorname{capt}_k(H_n) = 1$ when $\gamma(H_n) \leq k < n$, and $\operatorname{capt}_ n(H_n) = 0$. Next, consider $2 \leq k < \gamma(H_n)$ and the following strategy for $k$ cops. Place one cop on vertex 3 and distribute the remaining cops on the path $P$ with $V(P) = \{3,2,4,7,8, \ldots , n\}$ to optimize cop throttling on $P$. Let $\mathcal{C}$ be this set of $k$ vertices on which the cops initially place. Note that for any vertex $x \in V(P)$, the distance between $x$ and the nearest cop is at most $\lceil \frac{n-3-k}{2k-1} \rceil$. Let $y$ be the initial position of the cop that is closest to vertex $3$. If the robber places to the left of $y$, then the robber is guaranteed to be caught in at most $\lceil \frac{n-3-k}{2k-1} \rceil$ rounds. Otherwise, the cop on vertex $3$ can push the robber towards the tail of $H_n$ by moving along the vertices $2,4,7,8,$ and so on. This also guarantees capture of the robber in at most $\lceil \frac{n-3-k}{2k-1} \rceil$ rounds.
For the lower bound, we argue that any initial cop placement other than $\mathcal{C}$ yields a capture time greater than or equal to that of $\mathcal{C}$. Then, assuming the cops initially place on $\mathcal{C}$, we produce a strategy for the robber that avoids capture for at least $\lceil \frac{n-3-k}{2k-1} \rceil$ rounds. First, observe that from the perspective of the cops, it is always optimal for at most two cops to initially place in the set $\{1, 2, \ldots, 7\}$. This is because $\gamma(H_7) = 2$. Suppose two cops place in the set $\{1, 2, \ldots, 7\}$. Since vertex $7$ is the vertex in $\{1, 2, \ldots 7\}$ that is closest to the tail of $H_n$, it is optimal for the two cops in $H_7$ to place on vertices $7$ and $3$. In this case, if $n \geq 8$, the pigeonhole principle guarantees that there is a vertex $z \in \{8,9, \ldots n\}$ that is at least $\lceil \frac{n-3-k}{2k-1} \rceil$ vertices away from the nearest cop. So the robber can stay on vertex $z$ and avoid capture for at least $\lceil \frac{n-3-k}{2k-1} \rceil$ rounds. Note that if $n = 7$, then $k =2$ and the robber is caught in one round anyway.
Now, suppose no cops place in the set $\{1, 2, \ldots 7\}$ and thus, $n \geq 9$. Then by staying on $V(P)$, the robber can similarly avoid capture for $\lceil \frac{n-3-k}{2k-1} \rceil$ rounds. In the case that exactly one cop places in $\{1, 2, \ldots 7\}$, it is optimal for that cop to start on vertex $3$ so that they can push the robber towards the tail of $H_n$ as quickly as possible. Recall that this is accomplished when that cop moves along the path $P$. In this case, it is optimal for the remaining cops to place so that the capture time on $P$ is minimized. Let $S$ be the set of vertices in a longest subpath of $P$ that does not contain a cop. If $S$ does not contain vertex $2$, then the robber can avoid capture for $\lceil \frac{n-3-k}{2k-1} \rceil$ rounds by placing at the optimal vertex in $S$ as waiting to be captured. If $S$ does contain vertex $2$, then the robber can avoid capture for $\lceil \frac{n-3-k}{2k-1} \rceil$ rounds by starting at vertex $5$ and moving along vertices $7,8,9,$ and so on. In this case, the cop on $3$ must move along $P$ to force the robber into the tail of $H_n$. Note that the capture time is less if the robber starts anywhere else on $S$ because the cop on vertex $3$ can take a shortcut by moving to vertex $4$ in the first round. We can now conclude that $\operatorname{capt}_k(H_n) = \lceil \frac{n-3-k}{2k-1} \rceil$ if $2 \leq k < \gamma(H_n)$. Finally, observe that if $k = 1$, $\operatorname{capt}_k(H_n) = n - 4 = \lceil \frac{n-3-k}{2k-1} \rceil$.\end{proof}
\begin{thm}
For all integers $n \geq 7$, $\operatorname{th}_c(H_n) \geq \lceil \sqrt{2n-7}\rceil.$
\end{thm}
\begin{proof}
We wish to minimize $k+\operatorname{capt}_k(H_n)$ over $1\leq k\leq n$. By Lemma \ref{lem:kCaptValues}, $k+\operatorname{capt}_k(H_n)=k+1$ for any $k$ satisfying $\lceil \frac{n-8}{3} \rceil + 2 \leq k < n$; thus, for such values of $k$, we have that $k+\operatorname{capt}_k(H_n)\geq \lceil \frac{n-8}{3} \rceil + 3$. Further, when $k=n$, we have that $k+\operatorname{capt}_k(H_n)=n$.
For $1 \leq k < \lceil \frac{n-8}{3} \rceil + 2 = \gamma(H_n)$, we know $\operatorname{capt}_k(H_n)=\lceil \frac{n-3-k}{2k-1} \rceil$ by Lemma \ref{lem:kCaptValues}. Then,
\[
\min_{1\leq k<\gamma(H_n)}\left\{ k+ \left\lceil \frac{n-3-k}{2k-1} \right\rceil \right\} \geq \min_{1\leq k<\gamma(H_n)}\left\{ k+ \frac{n-3-k}{2k-1} \right\}.
\]
We now determine the minimum value of $f(k):=k+ \frac{n-3-k}{2k-1}$. Note that
\begin{eqnarray*}
f'(k)
&=& \frac{(2k-1)^2-2n+7}{(2k-1)^2}
\end{eqnarray*}
and so $f'(k)=0$ if $k=\frac{1\pm \sqrt{2n-7}}{2}$. Since $k\geq1$, we obtain $k = \frac{1+ \sqrt{2n-7}}{2}$ as our critical point. For $k> \frac{1}{2}$, the function $f(k)$ is concave-up, so $f(\frac{1+ \sqrt{2n-7}}{2})$ is the minimum. Note that
\begin{eqnarray*}
f\left( \frac{1 + \sqrt{2n-7}}{2} \right) &=& \frac{1 + \sqrt{2n-7}}{2} + \frac{n-3-\frac{1 + \sqrt{2n-7}}{2} }{2(\frac{1 + \sqrt{2n-7}}{2})-1} \\
&=& \frac{1 + \sqrt{2n-7}}{2} + \frac{n-3-\frac{1 + \sqrt{2n-7}}{2} }{\sqrt{2n-7}} \\
&=& \frac{\sqrt{2n-7}+2n-7+2n-7-\sqrt{2n-7}}{2\sqrt{2n-7}} \\
&=& \sqrt{2n-7}.
\end{eqnarray*}
Since $\operatorname{th}_c(H_n)$ is an integer,
\[
\min_{1\leq k<\gamma(H_n)}\left\{ k+ \left\lceil \frac{n-3-k}{2k-1} \right\rceil \right\} \geq \lceil \sqrt{2n-7} \rceil.
\]
Considering all possible values of $k$, we now have that
\[
\operatorname{th}_c(H_n) \geq \min\left\{ \lceil \sqrt{2n-7} \rceil, \left\lceil \frac{n-8}{3} \right\rceil+3, n \right\}.
\]
First, observe that $\left\lceil \frac{n-8}{3} \right\rceil+3\leq \frac{n-8}{3} +4\leq n$ for all $n\geq 2$. So it remains to show that $\lceil \sqrt{2n-7} \rceil\leq \left\lceil \frac{n-8}{3} \right\rceil+3$ for all $n\geq 7$. To do this, we note $\left\lceil \frac{n-8}{3} \right\rceil+3=\left\lceil \frac{n+1}{3} \right\rceil$. Every $n\geq 7$ can be written as $n=3a-1$, $n=3a-2$, or $n=3a-3$ for some integer $a\geq 3$. In each of these cases, $\left\lceil \frac{n+1}{3} \right\rceil=a$.
If $n=3a-1$, $\lceil \sqrt{2n-7} \rceil=\lceil \sqrt{6a-9} \rceil$; if $n=3a-2$, $\lceil \sqrt{2n-7} \rceil=\lceil \sqrt{6a-11} \rceil$; and if $n=3a-3$, $\lceil \sqrt{2n-7} \rceil=\lceil \sqrt{6a-13} \rceil$. Since $\lceil \sqrt{6a-13} \rceil\leq \lceil \sqrt{6a-11} \rceil\leq \lceil \sqrt{6a-9} \rceil$, we only need to consider when $\lceil \sqrt{6a-9} \rceil\leq a$. Because $a$ is an integer, $\lceil \sqrt{6a-9} \rceil\leq a$ if and only if $6a-9\leq a^2$. Since this holds for $a\geq 3$, then $\lceil \sqrt{2n-7} \rceil\leq a= \left\lceil \frac{n-8}{3} \right\rceil+3$ for all $n\geq 7$.
Thus, for all $n\geq 7$,
\[
\operatorname{th}_c(H_n) \geq \min\left\{ \lceil \sqrt{2n-7} \rceil, \left\lceil \frac{n-8}{3} \right\rceil+3, n \right\}= \lceil \sqrt{2n-7} \rceil.
\qedhere\]
\end{proof}
Now, we consider $\operatorname{dmg}_k(H_n)$ and $\operatorname{th}_d(H_n)$.
\begin{lem}\label{lem:kDMGUpperBound}
If $n$ and $k$ are integers with $n>10$ and $1 \leq k \leq n$, then \[\operatorname{dmg}_k(H_n) \leq \begin{cases} \lfloor\frac{n-3}{2}\rfloor -1 & \text{if } k=1;\\ \lceil \frac{n-5-k}{2k-1} \rceil - 1 & \text{if }2 \leq k < \lceil \frac{n-4}{3}\rceil;\\
1 & \text{if } \lceil \frac{n-4}{3}\rceil\leq k< \lceil\frac{n-8}{3}\rceil + 2;\\
0 & \text{if }\lceil\frac{n-8}{3}\rceil + 2\leq k\leq n. \end{cases}\]
\end{lem}
\begin{proof}
Recall that by \cite{CS19}, $\operatorname{dmg}_1(H_n) = \lfloor\frac{n-3}{2}\rfloor -1$. Furthermore, note that $\operatorname{dmg}_k(H_n) = 0$ when $\gamma(H_n) = \lceil\frac{n-8}{3}\rceil + 2\leq k\leq n.$ For $2 \leq k < \lceil \frac{n-4}{3} \rceil$, we provide a cop strategy and argue that given this strategy, the robber can visit a limited number of unique vertices. In this case, place one cop on vertex $4$. Additionally, place the remaining $k-1$ cops on the tail as follows: starting at the end of the tail, divide the path into subpaths of length $2\lceil \frac{n-5-k}{2k-1} \rceil +1$ and place a cop at the center of each subpath. Thus, in each of these subpaths every vertex is distance at most $\lceil \frac{n-5-k}{2k-1} \rceil$ from a cop. Since
\[n-(k-1)\left(2\left \lceil \frac{n-5-k}{2k-1} \right \rceil +1\right)\leq 6+\left\lceil\frac{n-5-k}{2k-1}\right\rceil,\]
any remaining tail vertices not in a subpath are within $\lceil \frac{n-5-k}{2k-1} \rceil$ of the cop on vertex $4$. So if the robber places on any tail vertex, the two closest cops will move towards the robber and capture will occur in at most $\lceil \frac{n-5-k}{2k-1} \rceil$ steps. In this case, damage is at most $\lceil \frac{n-5-k}{2k-1} \rceil -1$.
Otherwise, the robber must place on vertex $1$ since vertex $4$ dominates vertices $2,3,5$, and $6$. By remaining still, the cop on vertex $4$ limits the robber to damaging only vertex $1$. So in this case, the robber can always choose to place on the tail to maximize damage unless the tail is dominated by cops. The tail is dominated if $k-1\geq \lceil \frac{n-7}{3}\rceil$, i.e., $k\geq \lceil \frac{n-4}{3}\rceil$. Since we assumed $2 \leq k<\lceil \frac{n-4}{3}\rceil$, we have $\operatorname{dmg}_k(H_n)\leq \lceil \frac{n-5-k}{2k-1} \rceil - 1$, as desired.
Finally, if $\lceil \frac{n-4}{3}\rceil\leq k< \gamma(H_n)$, using the strategy above, the cops dominate the tail of $H_n$. This means that in order to damage a vertex, the robber must place on vertex $1$. Again, the cop on vertex $4$ can prevent any further damage. Thus, $\operatorname{dmg}_k(H_n) =1$ in this case.
\end{proof}
\begin{thm}
For each integer $a \geq 3$, if $n = 2a^2 - 2a + 6$, then $\operatorname{th}_d(H_n) \leq \sqrt{2n - 11} - 1$.
\end{thm}
\begin{proof}
Let $k= \lceil \frac{1+ \sqrt{2n-11}}{2} \rceil$. Note that since $n=2a^2 -2a+6$,
\begin{eqnarray*}
k&=& \left\lceil \frac{1+ \sqrt{2(2a^2 -2a+6)-11}}{2} \right\rceil\\
&=& \left\lceil \frac{1+ \sqrt{(2a-1)^2}}{2} \right\rceil\\
&=& a.
\end{eqnarray*}
By Lemma \ref{lem:kDMGUpperBound}, $\operatorname{th}_d(H_n) \leq k+\lceil \frac{n-5-k}{2k-1} \rceil -1$ for $2\leq k< \lceil \frac{n-4}{3} \rceil$. In order to verify that $2\leq \lceil \frac{1+ \sqrt{2n-11}}{2} \rceil< \lceil \frac{n-4}{3}\rceil$, observe that $a\leq \frac{2a^2-2a+6}{3}\leq \lceil \frac{n-4}{3}\rceil$ for all $a\geq 3$. Thus,
\[\operatorname{th}_d(H_n) \leq a + \left\lceil \frac{n-5-a} {2a-1}\right\rceil - 1.\]
Since $n=2a^2-2a+6$,
\begin{eqnarray*}
\operatorname{th}_d(H_n)&\leq & a + \left\lceil \frac{2a^2-2a+6-5-a} {2a-1}\right\rceil-1\\
&=& a + \left\lceil \frac{(2a-1)(a-1)} {2a-1}\right\rceil - 1\\
&=& 2a-2.
\end{eqnarray*}
Finally, note that $\sqrt{2n-11}= \sqrt{(2a-1)^2} = 2a-1$ and so,
\[\operatorname{th}_d(H_n) \leq \sqrt{2n-11} -1 . \qedhere\]
\end{proof}
\begin{lem}
If $a\geq 3$ is an integer and $n = 2a^2 - 2a + 6$, then \[\left \lceil \sqrt{2n-7} \right \rceil - (\sqrt{2n-11} - 1) \geq 2.\]
\end{lem}
\begin{proof}
Note that $4a^2 -4a +5 > (2a -1)^2 = 4a^2 -4a +1 \geq 0$.
Then, we have
\begin{align*}
\sqrt{4a^2 -4a +5} > (2a -1) &\Rightarrow \sqrt{4a^2 -4a +5} - (2a - 1) > 0\\
&\Rightarrow \left \lceil \sqrt{4a^2 -4a +5} - (2a -1) \right \rceil \geq 1\\
&\Rightarrow \left \lceil \sqrt{4a^2 -4a +5} \right \rceil -(2a-1) \geq 1\\
&\Rightarrow \left \lceil \sqrt{2n-7} \right \rceil - (\sqrt{2n-11} -1) \geq 2. \qedhere
\end{align*}
\end{proof}
\begin{cor}
For every integer $a \geq 3$, if $n = 2a^2 - 2a + 6$, then $\operatorname{th}_d(H_n) < \operatorname{th}_c(H_n) - 1$.
\end{cor}
\end{subsection}
Hence, we have obtained an infinite family of graphs $G$ for which $\operatorname{th}_d(G) \leq \operatorname{th}_c(G) -2$.
\end{section}
\begin{section}{Further investigations of damage numbers}\label{sec:damageNumbers}
In this section, we further explore the $k$-damage number and its implications for damage throttling. First, we discuss a necessary condition for $c(G)=k+\operatorname{dmg}_k(G)$ and characterize graphs for which $c(G)=1+\operatorname{dmg}_1(G)$. Then we provide an upper bound for $\operatorname{dmg}_k(G)$ in terms of maximum degree, which is a generalization of a bound for $\operatorname{dmg}_1(G)$ in $\cite{CS19}$.
In the proof of Proposition \ref{prop:c(G)leqthd(G)}, we showed $c(G)\leq k+\operatorname{dmg}_k(G)$ for each $1\leq k\leq n$. Proposition \ref{prop:copGammaThrot} considers the case where $c(G)=\operatorname{th}_d(G)$ and shows that for such graphs, either $c(G)=\gamma(G)$ or $\operatorname{th}_d(G)<\operatorname{th}_c(G)-1$. It is easy to observe that $c(G)=\operatorname{th}_d(G)$ if and only if there exists an integer $\ell$ such that $c(G)=\ell+\operatorname{dmg}_{\ell}(G)$. So characterizing instances where $c(G)= k+\operatorname{dmg}_k(G)$ for some $k$ is useful, as it produces a class of graphs which are good candidates for achieving a gap of two or greater between $\operatorname{th}_c(G)$ and $\operatorname{th}_d(G)$. First, we prove a helpful lemma.
\begin{lem}\label{c(G)leq k+dmg_k(G)-1}
Suppose $G$ is a connected graph. For each integer $1 \leq k \leq n$, if $\operatorname{dmg}_k(G) \geq 2$, then $c(G) \leq k +\operatorname{dmg}_k(G) -1$.
\end{lem}
\begin{proof}
By Proposition \ref{prop:c(G)leqthd(G)}, we know that $c(G) \leq \operatorname{th}_d(G)$. Apply the same cop strategy used in Proposition \ref{prop:c(G)leqthd(G)}, but with $\operatorname{dmg}_k(G)-1$ undercover cops. As $G$ is a connected graph, the $\operatorname{dmg}_k(G)-1$ undercover cops can catch the robber because once these cops are deployed, every vertex in the damaged area will either be occupied by a cop, or adjacent to only vertices containing cops.
\end{proof}\medskip
Applying Lemma \ref{c(G)leq k+dmg_k(G)-1}, we see the following necessary condition for $c(G)=k+\operatorname{dmg}_k(G)$.
\begin{obs}
For any connected graph $G$ and integer $1\leq k\leq n$, if $c(G)=k+\operatorname{dmg}_k(G)$, then $\operatorname{dmg}_{k}(G)\leq 1$. Furthermore, either $\operatorname{dmg}_{c(G)}(G)=0$ or $\operatorname{dmg}_{c(G)-1}(G)=1$.
\end{obs}
For the case where $k=1$, we now prove a complete characterization of connected graphs $G$ which achieve $c(G)=1+\operatorname{dmg}_1(G)$.
\begin{prop}\label{prop:c(G)=dmg(G) +1}
If $G$ is a connected graph, then $c(G) = 1+ \operatorname{dmg}_1(G)$ if and only if G has a dominating vertex, or $c(G)=2$ and there exists some $v\in V(G)$ such that $E(G-N[v])=\emptyset$.
\end{prop}
\begin{proof}
If $c(G)=\operatorname{dmg}_1(G)+1$, then by Lemma \ref{c(G)leq k+dmg_k(G)-1}, $\operatorname{dmg}_1(G) \leq 1$, and therefore $c(G) \leq 2$. If $c(G)=1$, then $\operatorname{dmg}_1(G)=0$ and $G$ must have a dominating vertex. Otherwise, if $c(G)=2$, then $\operatorname{dmg}_1(G)=1$. If the robber is ever able to move without being captured in the same round, then $\operatorname{dmg}_1(G)>1$. Therefore, there must exist an initial placement of one cop which restricts the robber to a single ``safe" vertex; in other words, there exists some $v\in V(G)$ such that $E(G-N[V])=\emptyset$.
Conversely, if $G$ has a dominating vertex, then $\operatorname{dmg}_1(G)=0$ and $c(G)=1$, which means $c(G)=\operatorname{dmg}_1(G)+1$. If $c(G)=2$ and there exists some $v\in V(G)$ such that $E(G-N[v])=\emptyset$, then the cops can prevent damage to all but one vertex. So $\operatorname{dmg}_1(G)=1$ and therefore, $c(G)=\operatorname{dmg}_1(G)+1$.
\end{proof}\medskip
It is worth noting that the set of graphs $G$ for which $c(G)=2$ and some $v \in V(G)$ exists such that $E(G-N[v]) = \emptyset$ is not empty. In fact, there is an infinite family of graphs that satisfy these conditions. For example, we can add an edge between a vertex $v \in V(C_4)$ and every vertex in an arbitrary graph $H$. Furthermore, we can add any number of leaves to the two neighbors of $v$ in the cycle. This family is illustrated in Figure \ref{fig:4cycleplus}.
\begin{figure}[H] \begin{center}
\scalebox{.6}{\includegraphics{copvsdmg.pdf}}\\
\caption{An infinite family of graphs satisfying the conditions in Proposition \ref{prop:c(G)=dmg(G) +1}.}\label{fig:4cycleplus}
\end{center}
\end{figure}
In $\cite{CS19}$, the authors prove the following upper bound for $\operatorname{dmg}_1(G)$.
\begin{prop}\cite{CS19}
For a graph $G$ on $n$ vertices, $\operatorname{dmg}_1(G)\leq n-\Delta(G) - 1$.
\end{prop}
We now prove an analogous upper bound for $\operatorname{dmg}_k(G)$ and apply it to provide an upper bound for $\operatorname{th}_d(G)$. For a graph $G$, let $S_k$ be the set of $k$-vertex subsets $S$ of $V(G)$ such that the vertices in $S$ have the $k$ largest degrees of $G$.
\begin{prop}\label{GenUppBndDmgk}
For all graphs $G$ on $n$ vertices, \[\operatorname{dmg}_k(G)\leq \min_{S\in S_k} \{n-|N[S]|\}.\]
\end{prop}
\begin{proof}
Place $k$ cops on the $k$ vertices of $S\in S_k$. By remaining still, a cop placed on $v\in S$ protects the vertices in $N[v]$ from being damaged. Therefore, using this cop placement, $|N[S]|$ vertices remain undamaged.
\end{proof}
\begin{cor}\label{GenUppBndThd}
For all graphs $G$ on $n$ vertices, \[\operatorname{th}_d(G)\leq \min_{1\leq k\leq n}\left\{k+ \min_{S\in S_k} \{n-|N[S]|\}\right\}.\]
\end{cor}
\end{section}
\begin{section}{Concluding Remarks}\label{sec:conclusion}
As shown in Section \ref{sec:diff}, there are infinite families of graphs $G$ such that $\operatorname{th}_d(G)\leq \operatorname{th}_c(G)-2$. However, we were not able to verify the existence of an infinite family of graphs $G$ satisfying $\operatorname{th}_d(G)=\operatorname{th}_c(G)-a$ for any $a\geq 3$ (we do provide a single graph for which $\operatorname{th}_d(G)=\operatorname{th}_c(G)-3$ in Theorem \ref{thm:GapOf3Graph}). Despite this, we believe such families exist and it would be interesting to find them.
In \cite{CS19}, the authors ask the question: which graphs $G$ satisfy
$\operatorname{dmg}_1(G)=n-\Delta(G)-1$? We observe that graphs for which $\Delta(G)=n-1$ or $\Delta(G)=n-2$ achieve this equality, but this does not characterize all such graphs. For example, the graphs in Figure \ref{fig:jellyfish} achieve equality in this bound, but have $\Delta(G)=n-3$.
\begin{figure}[H] \begin{center}
\scalebox{.7}{\includegraphics{jellyfish.pdf}}\\
\caption{An infinite family of graphs $G$ with $\operatorname{dmg}_1(G) = |V(G)| - \Delta(G) - 1$, $\operatorname{rad}(G) = 2$, and $\Delta(G) = |V(G)| - 3$.}\label{fig:jellyfish}
\end{center}
\end{figure}
We now propose similar questions for the generalized bounds in Proposition \ref{GenUppBndDmgk} and Corollary \ref{GenUppBndThd}.
\begin{quest}\label{quest:GenBounds}
Which graphs $G$ satisfy $\operatorname{dmg}_k(G)=\underset{S\in S_k}{\min} \{n-|N[S]|\}$ for some integer $k$ and furthermore, which graphs satisfy $\operatorname{th}_d(G)= \underset{1\leq k\leq n}{\min}\left\{k+ \underset{S\in S_k}{\min} \{n-|N[S]|\}\right\}$?
\end{quest}
Observe that in Question \ref{quest:GenBounds}, a graph $G$ that realizes the first equality does not necessarily realize the second. However, a graph that satisfies the second equality must satisfy the first for some integer $k$.
\end{section}
|
1,116,691,500,905 | arxiv | \section*{#1}%
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\begin{document}
\title{Quantitative Quasiperiodicity}
\author{Suddhasattwa Das\footnotemark[1], \and Yoshitaka Saiki\footnotemark[2],
\and Evelyn Sander\footnotemark[4], \and James A Yorke\footnotemark[3]}
\footnotetext[1]{Department of Mathematics, University of Maryland, College Park}
\footnotetext[2]{Graduate School of Commerce and Management, Hitotsubashi University}
\footnotetext[3]{University of Maryland, College Park}
\footnotetext[4]{Department of Mathematical Sciences, George Mason University}
\date{\today}
\maketitle
\renewcommand*\contentsname{Contents}
\begin{abstract}
{ The Birkhoff Ergodic Theorem concludes that time averages, that is,
Birkhoff averages, $\Sigma_{n=1}^N f(x_n)/N$ of a function $f$ along an
ergodic trajectory $(x_n)$ of a function $T$ converges to the space
average $\int f d\mu$, where $\mu$ is the unique invariant probability
measure.
Convergence of the time average to the space average is slow.
We introduce a modified average of $f(x_n)$ by giving very small weights to the
``end'' terms when $n$ is near $0$ or $N$.
When $(x_n)$ is a trajectory
on a quasiperiodic torus and $f$ and $T$ are $C^\infty$, we show that
our weighted Birkhoff averages converge ``super'' fast to $\int f d\mu$,
{\em i.e.} with error smaller than every polynomial of $1/N$. Our goal
is to show that our weighted Birkhoff average is a powerful
computational tool, and this paper illustrates its use for several
examples where the quasiperiodic set is one or two dimensional. In
particular, we compute rotation numbers and conjugacies (i.e. changes
of variables) and their Fourier series, often with 30-digit precision.}
\end{abstract}
\section{Introduction}
Quasiperiodicity is a key type of observed
dynamical behavior in a diverse set of applications. Tori with
quasiperiodic motion persist for small perturbations by the
Kolmogorov-Arnold-Moser theory, but such behavior is also observed for
non-conservative systems well beyond this restricted regime. We believe
that quasiperiodicity is one of only three types of dynamical behaviors
occurring in basic sets of typical systems. See~\cite{sander:yorke:15}
for the statement of our formal conjecture of this basic set
triumvirate. For example, quasiperiodicity occurs in a system of weakly
coupled oscillators, in which there is an invariant smooth attracting
torus in phase space with behavior described exclusively by the phase
angles of rotation of the system. Indeed, it is the property of the
motion being described using only a set of phase angles that always
characterizes quasiperiodic behavior. In a now classical set of papers,
Newhouse, Ruelle, and Takens demonstrated a route to chaos through a
region with quasiperiodic behavior, causing a surge in the study of the
motion~\cite{newhouse:ruelle:takens:78}. There is active current
interest in development of a systematic numerical and theoretical
approach to bifurcation theory for quasiperiodic systems.
Our goal in this paper is to present a fast numerical method for the
fast calculation of the limit of Birkhoff averages in quasiperiodic systems, allowing us to compute
various key quantities. If $f$ is
integrable and the dynamical system is ergodic on the set in which the trajectory lives, then the Birkhoff Ergodic Theorem asserts that the Birkhoff average $ \Sigma_{n=1}^N f(x_n)/N $ of a function $f$ along an ergodic
trajectory $(x_n)$ converges to the space average $\int f d\mu$ as
$N\to\infty$ for $\mu$-almost every $x_0$, where $\mu$ is the unique invariant probability measure.
We develop a numerical method for calculating the limit of such averages, where
instead of weighting the terms $f(x_n)$ in the average equally, we
weight the early and late terms of the set $\{1,\dots,N\}$ much less
than the terms with $n\sim N/2$ in the middle. That is, rather than
using the equal weighting $(1/N)$ in the Birkhoff average, we use a
weighting function $w(n/N)$, which will primarily be the following well
known $C^\infty$ function that we will call the {\bf exponential
weighting function}, $w_{\exp}(t) = \exp(1/(t(t-1))$. In a companion
paper~\cite{Das-Yorke}, it is rigorously shown that for quasiperiodic systems and $C^\infty$ $f$, this weighting
function leads to super convergence with respect to $N$, meaning faster
than any polynomial in $N^{-1}$. This super convergence arises from the
fact that we are taking advantage of the quasiperiodic nature of the map
or flow. In particular, our method uses the underlying structure of a
quasiperiodic system, and would not give improved convergence results
for chaotic systems. We demonstrate the method and its convergence rate
by computing rotation numbers, conjugacies, and their Fourier series in
dimensions one and two. We will refer to a 1D quasiperiodic curve as a
\emph{(topological) circle}.
Other authors have considered related numerical methods before,
in particular~\cite{seara:villanueva:06, luque:villanueva:14}, which we will
compare to when we introduce our method. See also~\cite{Simo1, Simo2,
Durand:2002ug,Baesens:1990vj,Broer:1993uv,Vitolo:2011it,Hanssmann:2012jy,
Sevryuk:2012dv,Broer:1990ip,Kuznetsov:2015dl}.
Our paper proceeds as follows: In Section~\ref{sec:Quasi}, we give the formal
definition of quasiperiodicity, rotation, and the conjugacy map.
In Section~\ref{sec:Method_Rot}, we describe our
numerical method in detail. We illustrate our method for
a series of four examples, including an example of a two-dimensionally quasiperiodic map.
In all cases, we get fast convergence and are in most cases able to
give results with thirty digits of precision. For convenience of the reader,
have summarized our numerical findings in Table~\ref{table:summary}.
We start by describing our results for a key example of
quasiperiodicity: the (circular) restricted three-body problem. This is an
idealized model of the motion of the planet, a large moon, and a spacecraft
governed by Newtonian mechanics, in a model studied by
Poincar{\'e}~\cite{ThreeBody2,ThreeBody1}. In particular, we consider a
planar three-body problem consisting of two massive bodies (``planet" and
``moon") moving in circles about their center of mass and a third body
(``spacecraft") whose mass is infinitesimal, having no effect on the
dynamics of the other two.
We assume that the moon has mass $\mu$ and the planet mass is $1-\mu$ where $\mu = 0.1$, and
writing equations in rotating coordinates around the center of mass. Thus the planet remains fixed at
$(-0.1,0)$, and the moon is fixed at $(0.9,0)$. In these coordinates, the satellite's
location and velocity are given by the {\em generalized position vector} $(q_1,q_2)$
and {\em generalized velocity vector}
$(p_1,p_2)$.
The equations of motion are as follows (see \cite{ThreeBody1}).
\begin{equation}
{\displaystyle
\begin{array}{rcl}\label{eqn:ThreeBody}
dq_1/dt &=& p_1+q_2, \\
dq_2/dt &=& p_2-q_1, \\
dp_1/dt &=& p_2- \mu(q_1-1+\rho) d_{moon}^{-3} -(1-\mu)(q_1+ \mu) d_{planet}^{-3},\\
dp_2/dt &=& -p_1-\mu q_2 d_{moon}^{-3}-(1-\mu)q_2 d_{planet}^{-3},
\end{array}
}
\end{equation}
where
\[ d_{moon} = ((q_1-1+\mu)^2+q_2^2)^{0.5} \mbox{ and } d_{planet} = ((q_1+\mu)^2+q_2^2)^{0.5}.
\]
The following function $H$ is a Hamiltonian for this system
\begin{equation}\label{eqn:Hamiltonian}
H=[(p_1^2+p_2^2)/2]+[q_2p_1-q_1p_2]-[\mu~d_{planet}^{-1} + (1-\mu)~d_{moon}^{-1}].
\end{equation}
The terms in the square brackets are resp. the kinetic energy, angular moment and the angular potential.
For fixed $H$, Poincar{\'e} reduced this problem to the study of the Poincar{\'e} return map for a fixed value of $H$, only considering
a discrete trajectory of the values of $(q_1,p_1)$ on the section $q_2=0$ and $\frac{dq_2}{dt}>0$.
Thus we consider a map in two dimensions rather than a flow in four dimensions.
Figure \ref{fig:3B3} shows one possible motion of the spacecraft for the full flow. The
orbit is spiraling on a torus. The black circle shows the corresponding trajectory on the
Poincar{\'e} return map. Fig.~\ref{fig:3B1} shows the Poincar{\'e} return map
for the spacecraft for a variety of starting points.
A variety of orbits are shown, most of which are quasiperiodic invariant circles.
An exception is A-trajectory in Fig. \ref{fig:3B_global}(a), which is an invariant recurrent set consisting of $42$
circles. Each circle is an invariant quasiperiodic circle under the 42-nd iterate of the map.
Using our numerical method for
Birkhoff averages, in the itemized list given below we summarize our results for the quasiperiodic orbit labeled $B_1$.
The error in the quasiperiodicity computations using weighted Birkhoff averages decreases exponentially
in the number of iterates $N$ (see Fig.~\ref{fig:3B2}c). This speed of convergence
means the accuracy of our solutions is the limit of numeric precision.
In particular, we have computed trajectories for the Poincar{\'e} return map using an 8-th order Runge-Kutta
method with time step $10^{-5}$, in quadruple precision, meaning our results given
below are computed up to thirty digits of accuracy.
Section~\ref{sec:Quasi} formally defines the computed values given in this list.
\begin{enumerate}
\item The rotation number is given in Table~\ref{table:summary}, computed to 30 digits of accuracy.
\item We can compute the Fourier series of up to 200 terms. There is a conjugacy map $h$ between the
first return map and a rigid rotation on the circle. Evaluating the Fourier series allows us to reconstruct the conjugacy map ({\em cf.} Fig.~\ref{fig:3B2}a).
\item The exponential decay of the coefficients in the Fourier series is a strong indication
of the analyticity of conjugacy function ({\em cf.} Fig.~\ref{fig:3B2}b).
\end{enumerate}
\begin{figure}[t]
\centering
\subfigure[ ]{\includegraphics[width = .33\textwidth]{ThreeB_global_1.png}}
\subfigure[ ]{\includegraphics[width = .33\textwidth]{ThreeB_global_2.png}}
\subfigure[ ]{\includegraphics[width = .33\textwidth]{Solvability.png}}
\caption{ {\bf Poincar{\'e}-return map for the restricted three-body problem.} \label{fig:3B1}
All three parts show a projection to the $q_1-p_1$ plane. Figures (a) and (b) show various quasiperiodic trajectories
on the Poincar{\'e} section $q_2=0$, of the restricted three-body problem (\ref{eqn:ThreeBody}). Note that
the planet is fixed at the point $(-0.1,0)$ and the planet at $(0.9,0)$. Thus some trajectories orbit
both the planet-moon system and some only orbit the planet or the moon. Each time the flow hits
$q_2=0$ and $\frac{dq_2}{dt}>0$, we plot $(q_1,p_1)$. Each trajectory shown is a (topological) circle
with quasiperiodic motion. The energy $H$ for all the circles shown in the figures, including $B_1$ is the same and $H\approx -2.63$. Part (c) shows in white all the initial points $(q_1,p_1)$ on the Poincar{\'e} surface for which there exists a
$p_2$ so that the Hamiltonian $H$ at $(q_1,q_2=0,p_1,p_2)$ is the same as the one in parts (a) and (b). Part (c) also shows the trajectory which corresponds to the circle $B_1$ in (a) and (b).}
\label{fig:3B_global}
\end{figure}
\begin{figure}
\centering
\subfigure[ ]{\includegraphics[width = .48\textwidth]{threeb-multi1.png}}
\subfigure[ ]{\includegraphics[width = .48\textwidth]{threeb-multi2.png}}
\subfigure[ ]{\includegraphics[width = .48\textwidth]{threeb-multi4.png}}
\subfigure[ ]{\includegraphics[width = .48\textwidth]{threeb-multi3.png}}
\caption{\label{fig:3B3}
{\bf Torus flow for the restricted three-body problem.}
All four views are of the same two-dimensional quasiperiodic torus lying in $\mathbb{R}^4$. Each picture consists of the same trajectory spiralling densely on this torus. This trajectory is the solution of (\ref{eqn:ThreeBody}), shown as curve $B_1$ in Fig. \ref{fig:3B_global}. We require four different
views of this torus because the embedding into three dimensions gives a highly non-intuitive images. The black circle is the set of values of the Poincar{\'e} return map with $q_2=0$ for this flow torus.}
\end{figure}
\begin{figure}[t]
\centering
\subfigure[ ]{\includegraphics[width = .4\textwidth]{ThreeB_angle_convert.png}}
\subfigure[ ]{\includegraphics[height=.2\textwidth]{ThreeB_step=1E-5_conjugacy.png}}
\subfigure[ ]{\includegraphics[height= .4\textwidth]{ThreeB_step=1E-5_2_log-linear.png}}
\subfigure[ ]{\includegraphics[height= .4\textwidth]{ThreeB_step=1E-5_rot_convrg.png}}
\caption{\textbf{Quasiperiodicity for the restricted three-body problem.} For the quasiperiodic circle $B_1$ in Fig. \ref{fig:3B1}, Part(a) shows how the invariant circle is parameterized by coordinates $\phi \in S^1 \equiv [0,1)$. Part (b) depicts the periodic part $g(\theta)$ of the conjugacy between the
quasiperiodic behavior and rigid rotation by $\rho$. See (\ref{eqn:g_conjufacy}) for a description of $g(\theta)$. Part (c) shows the norm of the Fourier coefficients of the conjugacy as a function of index. This exponential decay
indicates that the conjugacy function is analytic, up to numerical precision. Part (d) shows the convergence
rate of the error in the rotation number $\rho_N$ as a function of the number of iterates $N$. The ``error'' is the difference $|\rho_N-\rho_{N^*}|$, where $N^*$ is large enough so that $\rho_N$ appears to have converged.
The step size used for the Runge-Kutta (RK8) scheme is $10^{-5}$.
}
\label{fig:results_3B}
\label{fig:3B2}
\end{figure}
\begin{table}[t]
\begin{center}
{\begin{tabular}{ |c | c| c | c | }
\hline
Example & Equation & Rotation number(s) & Related Figures \\
\hline
Restricted three-body problem & \ref{eqn:ThreeBody} & $0.063961728757453097164077919081024$ & \ref{fig:3B1}, \ref{fig:3B2}, \ref{fig:3B3} \\[1pt]
Standard map & \ref{eqn:StdMap} & $0.12055272197375513300298164369839$ & \ref{fig:StdMap_global}, \ref{fig:StdMap} \\[1pt]
Forced van der Pol oscillator, $F=5$ & \ref{vdPol_forced} & $0.29206126329199589285577578718959$ & \ref{fig:vdP_global} \\[1pt]
Forced van der Pol oscillator, $F=15$ & \ref{vdPol_forced} & $0.37553441113144010884908928083318$ & \ref{fig:vdP_global} \\[1pt]
Forced van der Pol oscillator, $F=25$ & \ref{vdPol_forced} & $0.56235370092685056634419221336154$ & \ref{fig:vdP_global} \\[1pt]
Two-dimensional torus & \ref{eqn:2D_map} & $\rho_1, \rho_2$ in Table~\ref{table} & \ref{fig:2DMap_overview}, \ref{2DTorus_results} \\[1pt]
\hline
\end{tabular}}
\end{center}
\caption{\label{table:summary}
{\bf Summary of our numerical calculations.}
}
\end{table}
\clearpage
\section{Quasiperiodicity}\label{sec:Quasi}
In the introduction, we described quasiperiodic motion as motion that
could be fully understood through a set of angles of rotation. We now
formalize that idea in the following definition for
quasiperiodicity.
\begin{definition}[Quasiperiodicity]
For a dimension $d \ge 1$, let $\rho = (\rho_1,\rho_2,\dots, \rho_d)$ be a rotation vector
such that all $\rho_k$ are irrational and irrationally related (defined below in (\ref{eqn:irratrel})).
Then following map is known as a {\bf rigid irrational rotation}:
\begin{equation}\label{eqn:Pure_rotation}
T_{\vec\rho}(\theta) = \theta + \rho \pmod 1, \mbox{ where } 0 \le \theta_k \le 1, k=1,2,\dots,n.
\end{equation}
A rigid rotation the simplest, albeit least interesting example of a map
with quasiperiodicity. Since $T_{\vec\rho}$ gives the same values on opposite sides
of the unit cube, we identify the sides and refer to the domain of the
rigid rotation as a {\bf circle} in one dimension and an {\bf $n$-torus}
in $n>1$ dimension. From now on we will refer to the circle as
a 1-torus. We define a general map $T$ to be {\bf
quasiperiodic} if either $T$ or some iterate $T^k$ is topologically
conjugate to a rigid rotation. (We will assume $k=1$ in the rest of this
description.) That is, a map $T$ is quasiperiodic if there is a rigid
rotation map $T_{\vec\rho}$ and an invertible {\bf conjugacy map} $h$ such that
\[T(h(\theta)) = h(T_{\vec\rho}(\theta)).\] A flow has quasiperiodic behavior if
its associated first return Poincar{\'e} map has quasiperiodic behavior.
Since we are focused on maps, we restate this definition in
terms of iterates. Let $(x_n)$ be the forward orbit under $T$, and $( \theta_n )$ the
forward orbit under $T_{\vec\rho}$. That is, $x_{n+1} = T(x_n)$ and $\theta_{n+1} = \theta_n + \rho \pmod 1$. Then as long as $\phi_0 = h(\theta_0)=h(0)$,
\begin{equation}\label{eqn:conjugacy1D}
\phi_n = h(\theta_n)~\mbox{ for all } n=0,1,2,3,\ldots .
\end{equation}
The map $h$ maps a torus to the domain of $T$, meaning that the domain of $T$ is
a topological circle or torus. Since $h$ is a homeomorphism, the map
\begin{equation}\label{eqn:g_conjufacy}
g(\theta) := h(\theta)-\theta
\end{equation}
is periodic.
\end{definition}
Thus a map $T$ restricted to an invariant topological circle $C$ is
quasiperiodic if there is an continuous invertible map $h$ from $C$ to the true circle
which maps $T$ to a rigid irrational rotation map.
For an invertible map $F$ to be quasiperiodic on a topological circle
$C$, it is necessary and sufficient that $F$ has a dense trajectory, as
shown in \cite{TransCircleHomeo}. In general, a circle homeomorphism
without periodic points may not be quasiperiodic. However, if we assume
that the map $F$ and the topological circle $C$ are twice continuously
differentiable, then Denjoy~\cite{VanKampen} showed that these
conditions are both necessary and sufficient. Furthermore, clearly any
rigid irrational rotation map is a real analytic map, but even if we
assume that a quasiperiodic function is analytic, Arnold showed that the conjugacy map $h$
may only be continuous for some atypical rotation number. However, Herman (see \cite{HermanSeminal}) proved that for circle
homeomorphisms, most conjugacy maps $h$ are analytic.
\clearpage
\section{Our $\mbox{WB}_N+$ method and its applications}
\subsection{Rotation number}\label{sec:Method_Rot}
Here is our method for calculating the limit of the Birkhoff average $\lim_{N \to
\infty} \Sigma_{n=1}^N f(x_n)/N = \int f \; d\mu$
along an ergodic trajectory (or first return map) $(x_n)$.
Let the weighting function $w$ be defined as
\begin{equation}\label{eqn:weight}
w_{exp}(t) :=w(t) :=\begin{cases}
\exp\left(\frac{1}{t(t-1)}\right), & \mbox{for } t\in(0,1)\\
0, & \mbox{for } t\notin(0,1).
\end{cases}
\end{equation}
Let $T$ be quasiperiodic on some set $X_0$, with $X_0$ a topological torus. Henceforth, $X_0\equiv{\mathbb{T}^d}$ will denote a $d$-dimensional topological torus. For $d=1$, ${\mathbb{T}^d}$ is a topological circle. For a continuous function $f$ and a $C^\infty$ quasiperiodic map $T$
on ${\mathbb{T}^d}$, let $x_n\in{\mathbb{T}^d}$ be such that $x_n = T(x_{n-1})$ for all $n>1$. We define a {\bf Weighted Birkhoff ($\mbox{WB}_N$) average}
of $f$ as
\begin{equation}\label{eqn:QN}
\mbox{WB}_N(f)(x) :=\sum_{n=0}^{N-1} \hat{w}_{n,N}f(x_n),\mbox{ where }\hat{w}_{n,N}=\frac{w(n/N)}{\Sigma_{j=0}^{N-1}w(j/N)}
\end{equation}
In a companion paper~\cite{Das-Yorke}, it is shown that if $f$ and $T$ are $C^\infty$,
then for every positive integer $m$ there is a constant $C_m >0$ such that
\[\left| WB_N(f)(x) - \int f d\mu \right| \le C_m N^{-m} \mbox{ for every }m.
\]
We refer to the above as {\bf super} (polynomial) {\bf convergence} or {\bf exponential convergence}.
Our numerical method converges fast enough to allow us to obtain high precision values for $\int f d\mu$
with relatively low computational cost. In particular, by appropriate choices for $f$,
we obtain rotation numbers and sometimes the Lyapunov exponents,
Fourier coefficients, and conjugacy reconstructions for quasiperiodic
maps and flows potentially in any finite dimension, though here we work mainly in dimension
one, with one example in dimension two. We now show
how to apply this general method to computation of specific quantities. We observe that $N$ must be larger for $\mathbb{T}^2$ than for $\mathbb{T}^1$, to get a 30-digit accuracy.
{\bf Calculation of rotation vectors.}
\begin{figure}[t]
\centering
\subfigure[ ]{\includegraphics[width = .35\textwidth]{standard.jpg}}
\subfigure[ ]{\includegraphics[width = .35\textwidth]{Orbit,StdMap,rho=1_0.png}}
\caption{
{\bf The standard map.} A variety of orbits from different initial
conditions in the standard map $S_1$ defined in~(\ref{eqn:StdMap}) are plotted on the
left. We can see both chaos (in pink) and quasiperiodic orbits under this map. A single
topological circle with quasiperiodic behavior is plotted on the right.
The orbit has initial conditions
$(x, y) \approx (-0.607, 2.01)$
That is, if we restrict the map to this invariant circle, then it is
topologically conjugate to a rigid irrational rotation.
}
\label{fig:StdMap_global}
\end{figure}
\begin{figure}[t]
\centering
\subfigure[ ]{\includegraphics[width = .35\textwidth]{Conjugacy,StdMap,rho=1_0.png}}
\subfigure[ ]{\includegraphics[width = .35\textwidth]{Analyticity,StdMap,rho=1_0.png}}
\caption{\textbf{The Standard map conjugacy.}
Fig. (a) shows the shows the
change of variables which converts the motion on the circle in Fig.~\ref{fig:StdMap_global}b into a pure
rotation.
Fig. (b) shows the decay of the Fourier coefficients. Since
the conjugacy is an odd function, the odd-numbered Fourier sine and
cosine terms are zero and therefore have been omitted from the picture.
The decay of the Fourier terms can be bounded from above be an
exponential decay, which suggests that the conjugacy is analytic.}
\label{fig:StdMap}
\end{figure}
\begin{figure}[t]
\centering
\subfigure[ ]{\includegraphics[width=.3\textwidth]{Orbits,vdPol.png}}
\subfigure[ ]{\includegraphics[width = .3\textwidth]{Conjugacies,vdPol.png}}
\subfigure[ ]{\includegraphics[width = .3\textwidth]{Analiticities,vdPol.png}}
\caption{ {\bf Forced van der Pol oscillator.} \label{fig:vdP_global}
Part (a): Attracting orbits for a number of different forcing values $F$ for the stroboscopic
map of the van der Pol flow given in (\ref{vdPol_forced}). The plot
depicts points
$(X,Y) = (x(t_k),x'(t_k))$, where $t_k = 2 k \pi/0.83, k = 0,1,2,\dots.$
The chaotic orbit lying inside the cycles corresponds to $F=45.0$. There are stable quasiperiodic orbits shown as circles, which from outermost to innermost correspond to $F=5.0$, $15.0$, $25.0$ and $35.0$ respectively. Part (b): the periodic part $g(\theta)$ of
the conjugacy (Eq. \ref{eqn:conjugacy1D}) to a pure rotation, for $F=$ 5, 15 and 25. Part (c): the second figure gives a demonstration of the analyticity of the conjugacy to a pure rotation by studying the decay of the norm of the $k$-th
Fourier coefficients with $k$, up to the resolution of the numerics.
}
\end{figure}
Recall that every quasiperiodic map is conjugate to a pure rotation $T_\rho$ by a vector $\rho = (\rho_1, \dots,\rho_n)$. In our discussion, we assume that the components
of $\rho$ are irrational and irrationally related.
When we
say that $\rho_1,\dots,\rho_d$ are irrationally related, we mean that
there are no integers $k_1,\dots,k_d $
and $k_0$ for which,
\begin{equation}\label{eqn:irratrel}
k_1\rho_1 + k_2 \rho_2 + \dots + k_d \rho_d \ne k_0
\end{equation}
unless all $k_i$ are zero.
Since we are only provided with the
map $T$ and not with the map $R$, we need a way of recovering this
rotation vector using only a forward trajectory of the map $T$. The
following formula gives a way of calculating the rotation vector $\rho$
directly from the original map. The rotation vector given in the formula
is valid well beyond the case of a quasiperiodic function. For example, the rotation vector can
certainly be rational in one or all of its components.
\begin{definition}[Rotation vectors.]
Let $q(x) = x \pmod 1$ and define $q:\mathbb{R}^d \to {\mathbb{T}^d}$ to be the projection map modulo
$\mathbb{Z}$ in each coordinate. For example, in one dimension the circle $[0,1]$ with ends identified is the image of the real
$\mathbb{R}$ under the map $q$.
For any homeomorphism $T:{\mathbb{T}^d} \to {\mathbb{T}^d}$, let $\bar{T}:\mathbb{R}^d\rightarrow\mathbb{R}^d$ be the lift of
$T$. That is,
\begin{displaymath}\label{eqn:q_as_cover}
q\circ\bar{T}=\phi\circ q.
\end{displaymath}
Since $T$ is a homeomorphism, the map $\bar{T}(x)-x$ must be periodic in $x$ with period 1.
The rotation vector of a homeomorphism $T: {\mathbb{T}^d} \to {\mathbb{T}^d}$ starting at lift point $z$ is given by
\begin{equation}\label{eqn:rot_num_defn}
\rho(T,z):=\underset{n\rightarrow\infty}{\lim}\frac{\bar{T}^n(z)-z}{n}.
\end{equation}
\end{definition}
Note that possible rotation numbers $(\rho_1,\rho_2)$ of a quasiperiodic map on a torus ${\mathbb{T}^d}$ depend on the coordinates chosen for $T$ and $z$. For example,
for domain $\mathbb{T}^2$, the set of pairs $(\rho_1,\rho_2)$ possible for a fixed $T$ is dense in
the two-torus. See~\cite{Das-Yorke}.
Based on the above definition, the standard method of computing $\rho$ from an orbit $(y_k)$
of length $n$ is to calculate the average
\begin{equation}\label{eqn:rot_num_alt}
\rho\approx\frac{1}{N} \sum_{n=1}^{N} (\bar{T}(y_n)-y_n).
\end{equation}
This method converges slowly at best, with order of only
$1/N$~\cite{seara:villanueva:06}.
However, since Eq. \ref{eqn:rot_num_alt} can be written as a Birkhoff average by writing $f(y_n)=T(y_n)-y_{n}$, we can apply our method
to this function. That is, let $(y_k)_{k=0}^n$ be a forward orbit for $\bar{T}$.
Our approximation of $\rho$ is given by the weighted average of $f$,
\begin{equation}\label{eqn:weighted_rho}
\mbox{WB}_N (y_{n+1}-y_n) := \sum\limits_{n=0}^{N-1} \hat{w}_{n,N} (y_{n+1}-y_n)\rightarrow \rho.
\end{equation}
{\bf Rotation number for the restricted three-body problem.} In the
introduction, we stated our results, including the rotation number for a
quasiperiodic orbit for the restricted three-body problem.
Fig.~\ref{fig:3B2}(b) shows the convergence rate of the calculation for
this rotation number. .
{\bf Rotation number for the standard map.}
The standard map is an area preserving map on the two-dimensional torus, often studied as a typical
example of analytic twist maps (see \cite{StdMap1}). It is defined as follows\footnote{The
standard map generally depends on a parameter $\alpha$, but we only
consider the case $\alpha=1$.}
\begin{eqnarray}\label{eqn:StdMap}
S_1
\left(
\begin{array}{c}
x \\
y
\end{array}
\right)
=
\left(
\begin{array}{c}
x+y \\
y+ \sin x
\end{array}
\right)
\pmod{2\pi}.
\end{eqnarray}
The left-hand panel of Fig.~\ref{fig:StdMap_global} shows the
trajectories starting at a variety of different initial conditions
plotted in different colors. The shaded set is a large invariant chaotic set
with chaotic behavior, but many other invariant sets consist of
one or more topological circles, on which the system has quasiperiodic
behavior. For example, initial conditions $(\pi,1.65)$ leads to chaos
while $(\pi,1.5)$ leads to a quasiperiodic trajectory. As is clearly
the case here, one-dimensional
quasiperiodic sets often occur in families for non-linear processes,
structured like the rings of an onion. There are typically narrow bands of chaos between quasiperiodic onion rings. Usually these inner rings are differentiable
images of the $n$-torus, Yamaguchi and Tanikawa~\cite{StdMap1} and Chow et. al. in \cite{1halfDegree}
showed that the outermost limit (the onion's skin, to continue the analogy) will
still be a torus, but may not be differentiable. We have computed the rotation number
for the standard map orbit shown in the right panel Fig.~\ref{fig:StdMap_global} using
quadruple precision.
Its 30 digit
value is given in Table~\ref{table:summary}.
{\bf The forced Van der Pol oscillator.} Fig.~\ref{fig:vdP_global} shows orbits for the time-$2\pi/0.83$ map of the
following periodically forced Van der Pol oscillator with nonlinear
damping~\cite{vdPol_forced}
\begin{equation}\label{vdPol_forced}
\frac{d^2 x}{dt^2}-0.2 \left( 1-x^2 \right) \frac{dx}{dt}+20 x^3=F\sin \left(0.83 t \right),
\end{equation}
with a variety of choices of $F$. While the innermost orbit shown is a
chaotic attractor, the outer orbits are topological circles with
quasiperiodic behavior\footnote{As with the standard map, we have
specified all non-essential parameters rather than stating the most
general form of the Van der Pol equation.}. Our computed the rotation number
for the three orbits $F=15,25,35$ are given in Table~\ref{table:summary}.
{\bf A two-dimensional torus map.}
We now introduce an example of a two-dimensional quasiperiodic torus map on $\mathbb{T}^2$. This is a two-dimensional version of Arnold's family of 1D maps (see \cite{Arnold}), originally introduced in the following two papers~\cite{Grebogi:83,Grebogi:85}. The map is given by $(T_1,T_2)$ where
\begin{eqnarray*}\label{eqn:2D_map}
T_1(x,y) = \left[ x+ \omega_1 + \frac{\epsilon}{2 \pi} P_1(x,y) ) \right] \pmod 1\\
T_2(x,y) = \left[ y+ \omega_2 + \frac{\epsilon}{2 \pi} P_2(x,y) \right] \pmod 1,
\end{eqnarray*}
and $P_i(x,y), i=1,2$ are periodic functions with period one in both variables, defined by:
\[
P_i(x,y) = \sum_{j=1}^4 a_{i,j} \sin (2 \pi \alpha_{i,j}), \mbox{ with } \alpha_{i,j}= r_j x + s_j y +b_{i,j}.
\]
The values of all coefficients are given in Table~\ref{table}.
\begin{table}[t]
\begin{center}
{\begin{tabular}{ |c | c| }
\hline
Coefficient & Value \\
\hline
$\epsilon$ & $ 0.4234823$ \\[1pt]
$\omega_1$ & $ 0.71151134457776362264681206697006238 $ \\[2pt]
$\omega_2$ & $ 0.87735009811261456100917086672849971 $ \\[1pt]
$a_{1,j}$ & $(-0.268,-0.9106,0.3,-0.04)$ \\[1pt]
$a_{2,j}$ & $(0.08,-0.56,0.947,-0.4003)$ \\[1pt]
$b_{1,j}$ & $(0.985,0504,0.947,0.2334)$ \\[1pt]
$b_{2,j}$ & $(0.99,0.33,0.29,0.155)$ \\[1pt]
$r_j$ & $(1,0,1,0)$ \\[1pt]
$s_j$ & $(0,1,1,-1)$ \\[1pt]
Computed $\rho_1$ & $0.718053759982066107095244936117$ \\[1pt]
Computed $\rho_2$ & $0.885304666596099792113366824157$ \\[1pt]
\hline
\end{tabular}}
\end{center}
\caption{\label{table}
{\bf Coefficients for the torus map.} All values are used in quadruple precision,
but in this table the repeated zeros on the end of the number are suppressed.}
\end{table}
This choice of this function is based on~\cite{Grebogi:83,
Grebogi:85}. Both papers use the same form of equation, though
the constants are close to but not precisely the same as the ones used previously.
This is fitting with the point of view advocated in by these papers, in that the constants should be randomly chosen. Since we
are using higher precision, we have chosen constants
that are irrational to the level of our precision. The forward orbit is
dense on the torus, and the map is a nonlinear which exhibits two-dimensional quasiperiodic behavior.
Fig.~\ref{fig:2DMap_overview}a depicts iterates of the orbit, indicating that it is
dense in the torus. In order to verify that this map is indeed quasiperiodic,
we used a Birkhoff average to compute the two Lyapunov exponents to verify that both are
zero ({\em cf.} Fig.~\ref{fig:2DMap_overview}b). In terms of method, this is just a matter of changing the function $f$ used in $\mbox{WB}_N$ in Eq. \ref{eqn:QN}.
Likewise, finding rotation numbers in two dimensions uses the same technique as in the one-dimensional case ({\em cf.} Fig.~\ref{fig:2DMap_overview}c).
In all of our calculations, the computation is significantly longer than in one dimension in order
to get the same accuracy, perhaps because in two dimensions, coverage of dense orbit varies like the square
of the side length of the domain.
\begin{figure}[t]
\centering
\subfigure[ ]{\includegraphics[height= .3\textwidth]{orbitontorus.png}}
\subfigure[ ]{\includegraphics[height= .3\textwidth]{lya1compare.png}}
\subfigure[ ]{\includegraphics[height= .3\textwidth]{rot1compare.png}}
\caption{\textbf{Two-dimensional torus map.}
Fig. (a) shows the first $10^{4}$ iterates of an orbit for the two-dimensional quasiperiodic
torus map.
The orbit appears to be dense, indicating quasiperiodicity. This is
confirmed by computing the two Lyapunov exponents of the orbit using
Birkhoff averaging and finding them to be 0. The convergence of this
computation for one of the two Lyapunov exponents is shown in blue in
(b). The highest to lowest curves show the convergence rates resp.
for the first three weighting functions given in~(\ref{eqn:weights}).
Fig. (c) shows the convergence rate for the first rotation number for
the four different weighting functions.
}
\label{fig:2DMap_overview}
\end{figure}
\begin{figure}[t]
\centering
\subfigure[ ]{\includegraphics[height=.4\textwidth] {weighting.pdf}}
\subfigure[ ]{\includegraphics[height=.4\textwidth] {Convergence_rot_num_with_iterates,del=0_8,eps=0_8.png}}
\caption{\textbf{Rate of convergence for different weight functions}. \label{fig:mid_circle}
Left is a plot of the three non-constant weighting functions from~(\ref{eqn:weights})
(quad blue, $\sin^2$ yellow, $\exp$ red).
Since only the shape matters, they have been
rescaled so that each has a peak of approximately one.
For a given $w$ and a given number of iterates $N$, the rotation number
$\hat{\rho}$ approximation is calculated for $B_1$ of the restricted three-body problem, the error of the calculation is the
difference $|\rho-\hat{\rho}|$. The figure shows the convergence rate of
this error as a function of $N$. The exponential weight function is seen to be the
best method.
The error
cannot be reduced below $10^{-32}$ because that is the limit of
quadruple precision. }
\end{figure}
\begin{figure}[t]
\centering
\subfigure[ ]{\includegraphics[width = .46\textwidth]{reconstruction1.png}}
\subfigure[ ]{\includegraphics[width = .46\textwidth]{fouriercomp1.png}}
\caption{\textbf{Conjugacy for the torus.} \label{2DTorus_results}
Fig. (a) depicts the reconstruction of the periodic part $g$ (see \ref{eqn:g_conjufacy}) of the first component of the
conjugacy function for the torus map. The second conjugacy function is similar but not depicted here.
Fig. (b) shows the decay of the Fourier coefficients for this component of the conjugacy function on the
log-linear scale.
}
\end{figure}
{\bf Convergence rate.}
In order to explain why the convergence of our method is so good, we introduce four different
possible values for the weighting function $w$, depicted in Fig.~\ref{fig:mid_circle}a,
and compare the convergence results for computing
the rotation number for each of these choices of $w$.
\begin{eqnarray}\label{eqn:weights}
w_{equal}(t) & = & 1\mbox{ (Birkhoff's choice)}\\
w_{quad}(t) & = & t (1-t)\\
w_{(\sin^2)}(t) & = & \sin^2(\pi t) \\
w_{\exp}(t) & = & \exp\left({-1}/{(t(1-t))}\right).
\end{eqnarray}
If we compute with the first choice of $w$, we recover the a truncated
series in the definition of the Birkhoff average. To estimate the error, we
expect the difference $f(x_{N+1})-f(x_{N})$ to be of order one, implying that
\[\mbox{WB}_{w_{equal},N+1} - \mbox{WB}_{w_{equal},N} \sim 1/N.\]
The choice of a particular starting point also creates a similar uncertainty of order $1/N$.
For all but the first
choice, $w$ is always positive between $0$ and $1$ but vanishes as $t$
approaches $0$ and $1$. In addition, going down the list, increasing
number of derivatives of $w$ vanish for $t \to 0$ and $t\to 1$, with all derivatives of $w_{\exp}$
vanishing at $0$ and $1$. We thus expect the
effect of the starting and endpoints to decay at the same rate as this number of
vanishing derivatives. Indeed, we find that $w_{quad}$ corresponds approximately to order $1/N^2$ convergence,
$w_{(\sin^2)}$ to $1/N^3$ convergence, and $w_{\exp}$ to convergence faster than any polynomial.
Figs.~\ref{fig:mid_circle}b and ~\ref{fig:2DMap_overview}bc show this effect.
{\bf Related methods}. See
\cite{seara:villanueva:06,luque:villanueva:14} for references to earlier
methods for computing rotation numbers. In
\cite{seara:villanueva:06,luque:villanueva:14}, A. Luque and J.
Villanueva develop fast methods for obtaining rotation numbers for
analytic functions on a quasiperiodic torus, sometimes with
quasiperiodic forcing with several rotation numbers. The paper
\cite{luque:villanueva:14} examines a smooth function f on a
quasiperiodic torus. Let $f_n$ denote the value of f at the n-th
trajectory point. From this sequence they can obtain the rotation number
with error satisfying $|error| \le C_p N^{-p}$ for any $p$ where $C_p$
is a constant. The method of computation depends on $p$ and as $p$
increases the computational complexity increases for fixed $N$. If
$T(p,N)$ is their computation time, it appears that
$T(p,N)/N\rightarrow\infty$ as $p\rightarrow\infty$. In comparison, computation time for our weighted Birkhoff average is
simply proportional to $N$ since it requires a sum of $N$ numbers. The paper gives one figure (Fig 6) from which the rate of convergence can be computed, namely a restricted three-body problem. Their rotation-rate error is proportional to $N^{-3.5}$ and is $\approx 10^{-18}$ at $N = 2^{21}$.
Several variants of the Newton's
method have been employed to determine quasiperiodic trajectories in
different settings. In \cite{NumericQuasi3} the monodromy variant of
Newton’s method was applied to locate periodic or quasi-periodic
relative satellite motion.
A PDE-based approach was
taken in \cite{NumericQuasi4}, where the authors defined an invariance
equation which involves partial derivatives. The invariant tori are
then computed using finite element methods. See also Chapter 2,
\cite{NumericQuasi4} for more references on the numerical computation of
invariant tori.
\subsection{Fourier coefficients and conjugacy reconstruction}\label{sec:Fourier}
For a quasiperiodic curve as shown in Fig. \ref{fig:results_3B}a, there are two approaches to representing the curve. Firstly, we can write the coordinates $(X,Y)$ as a function of $\theta\in S^1$, or secondly, we can reduce the dimension and represent the points on the curve by an angle $\phi\in S^1$, that is, $\phi(X(\theta),Y(\theta)$, which is also $h(\theta)=\theta+g(\theta)$. We have shown $g$ in Fig \ref{fig:results_3B}b and the exponential decay of the norm of the Fourier coefficients in Fig. \ref{fig:results_3B}c. To limit the number of graphs in this paper, we have only created the Fourier series for the periodic part $g(\theta)$
Given a continuous periodic map $f:S^1\rightarrow\mathbb{R}$, where $S^1$ is a circle (or
one-torus),
the Fourier sine and cosine representation of $f$ is the following.
\begin{equation}\label{eqn:sine_cosine_expansion}
\mbox{For every }t\in S^1,\ f(t)=\frac{b_0}{2}+ \sum_{k=1}^\infty b_k \cos(2k\pi t)+ \sum_{k=0}^\infty c_k \sin(2k\pi t)
\end{equation}
where the coefficients $b_k$ and $c_k$ are given by the following formulas.
\begin{equation}\label{eqn:a_n}
b_k = 2 \int_{\theta\in S^1} f(\theta) \cos(2 k \pi\theta) \; d\theta,
\end{equation}
\begin{equation}\label{eqn:b_n}
c_k = 2 \int_{\theta\in S^1} f(\theta) \sin(2 k \pi\theta)\; d\theta.
\end{equation}
To be able to use the fast Fourier transform, $2^M$ equally spaced points on the circle are required. If we only have access to an ergodic orbit $(x_n)$ on the circle, then we cannot use the fast Fourier transform as we only have the function values $f(x_n)$ along a quasiperiodic trajectory, and a rotation number $\rho$. So instead, we obtain these coefficients using a weighted Birkhoff average on a trajectory
$(x_n)$ by applying the functional $\mbox{WB}_{w,N}$. For $k=0$, we find $a_0$ by applying $\mbox{WB}$ to the function $1$.
For $k>0$, we find $b_k$ and $c_k$ as follows.
\begin{equation}\label{eqn:a_n_from_Q}
b_k = \mbox{WB}_{w,N} (f(\theta) \cos(2 k \pi\theta)) = \sum_{n=0}^N f(x_n) \cos(2k\pi n\rho) \hat{w}_{n,N}.
\end{equation}
\begin{equation}\label{eqn:b_n_from_Q}
c_k = \mbox{WB}_{w,N} (f(\theta) \sin (2 k \pi\theta))= \sum_{n=0}^N f(x_n) \sin(2k\pi n\rho) \hat{w}_{n,N}.
\end{equation}
By specifying that $\theta_0 = 0$, our computation of rotation number $\rho$ provides all iterates : $\theta_n = n \rho$. Using the Fourier coefficients, we can
thus reconstruct the periodic part of the change of variables function $g$ (see Eq. \ref{eqn:g_conjufacy}).
This is depicted for the restricted three-body problem in Fig.~\ref{fig:3B2},
for the standard map in Fig.~\ref{fig:StdMap}, for the forced van der Pol equation in
Fig.~\ref{fig:vdP_global}.
In all three one-dimensional cases, we depict $\sqrt{b_k^2 + c_k^2}$ as a function of $k$.
Our main observation is that the Fourier
coefficients decay exponentially; that is, for some positive numbers $\alpha$
and $\beta$, in dimension one, the Fourier coefficients $b_k$ and $c_k$ satisfy
\begin{equation}\label{eqn:Fourier_decay}
\sqrt{|b_k|^2+|c_k|^2} \le \alpha e^{- \beta |k|} \mbox{ for all }k\in\mathbb{Z}.
\end{equation}
This is characteristic of analytic functions. We therefore state that all of the conjugacy functions
that we computed in our examples are effectively analytic,
``effectively'' meaning within the precision of our quadruple precision
numerics.
In two dimensions, the computation of Fourier coefficients is similar,
but instead of only having one set of cosine and sine functions, for
each $(j,k)$, we have two linearly independent sets of complex-valued functions,
where $i = \sqrt{-1}$:
\[ e^{i (j x + k y)} \mbox{ and } e^{i (j x - k y)}. \]
We define $a_{j,k}$ and $b_{j,k}$
to be the complex-valued coefficients corresponding to each of these
functions. The reconstructed conjugacy function and decay of
coefficients for the two-dimensional torus is depicted in
Fig.~\ref{2DTorus_results}. The decay of coefficients shows $\sqrt{j^2 +
k^2}$ on the horizontal axis, and $|b_{j,k}|$ and $|c_{j,k}|$ on the
vertical axis, where both of these coefficients are complex, meaning
that $|\cdot|$ represents the modulus. Again here, the coefficients
decay exponentially, though the decay of coefficients is considerable
slower in two dimensions due to the added dimension. The data looks
quite a lot more crowded in this case, since there are many different
values of $(j,k)$ such that the values of $\sqrt{j^2 + k^2}$ are identical or
very close. In addition, the two sets of
coefficients $b_{j,k}$ and $c_{j,k}$ generally converge at different exponential
rates. This is why there is a strange looking set of two different
clouds of data in Fig.~\ref{2DTorus_results}b. While more information on
the difference between these coefficients is gained by interactively
viewing the data in three dimensions, we have not been able to find a
satisfactory static flat projection of this data. We feel that in a still image,
the data cloud shown conveys the maximum information.
\textbf{Accuracy of the calculation of Fourier coefficients}. Our method of calculation of the Fourier coefficients is dependent on the knowledge of the rotation vector $\vec\rho$, or an approximation $\hat{\vec\rho}$. For the rest of the chapter, $m\in\mathbb{N}$ is some fixed integer and $M$ is some integer satisfying $M>d+m(d+\beta)$, where $\beta$ is the Diophantine class of $\vec\rho$. Let $\Delta\vec\rho$ be the error in the approximation of $\rho$, that is,
\[\Delta\vec\rho:=\hat{\vec\rho}-\vec\rho.\]
We are interested in knowing how the error in the calculation of Fourier coefficients $b_k, c_k$ depend on $\Delta\vec\rho$. Let for every $k\in\mathbb{Z}^d$, \boldmath $f_k(\theta)$ \unboldmath $:=e^{i 2\pi k\cdot\theta}$. Then every periodic function $g$ has the complex Fourier series representation
\[g(\theta)=\underset{k\in\mathbb{Z}^d}{\Sigma}a_k f_k,\mbox{ where }a_k=\int_{{\mathbb{T}^d}}g(\theta)f_{-k}(\theta)\]
Therefore, $a_k$ can be approximated as
\[\hat a_k=\mbox{WB}_{N,\hat{\vec\rho}}(g(\theta)f_{-k}(\theta))=\sum\limits_{n=0}^{N-1}g(n\vec\rho)e^{-i 2\pi n k\cdot \hat{\vec\rho}}\hat{w}_{n,N}.\]
\begin{corollary}\label{cor:error_with_rho}
If $N\|k\cdot\Delta\vec\rho\|<<1$, then for each $m\in\mathbb{N}$, there exists a constant $C_{g,w,m}>0$ such that
\[\frac{|\hat a_k(g)-a_k(g)|}{|a_k(g)|}\leq \pi N |k\cdot \Delta\vec\rho|+C_{g,w,m}N^{-m}.\]
\end{corollary}
\begin{proof}
We begin by obtaining bounds on the error in calculating the Fourier coefficients of pure exponentials. Note that $a_k(f_l)$ equals $1$ if $l=k$, equals $0$ if $l\neq k$. We will estimate the error $|\hat a_k(f_l)-a_k(f_l)|$ in each case.
\[|\hat a_k(f_k)=\mbox{WB}_{N,\hat{\vec\rho}}(f_k(\theta)f_{-k}(\theta))| =\sum\limits_{n=0}^{N-1}e^{-i 2\pi n k\cdot \Delta\vec\rho}\hat{w}_{m,N}.\]
Then if $N\|k\cdot\Delta\vec\rho\|<<1$, then using the approximation $e^{i\alpha}-1\approx i\alpha$, one can say that
\[\begin{split}
|\hat a_k(f_k)-a_k(f_k)|&=|\hat a_k(f_k)-1|\\
&=\left|\sum\limits_{n=0}^{N-1}(e^{-i 2\pi n k\cdot \Delta\vec\rho}-1)\hat{w}_{m,N}\right|\\
&\leq\sum\limits_{n=0}^{N-1}|e^{-i 2\pi n k\cdot \Delta\vec\rho}-1|\hat{w}_{m,N}\\
&\leq\sum\limits_{n=0}^{N-1}2\pi n k\cdot |\Delta\vec\rho|\hat{w}_{m,N}\\
&=\pi N |k\cdot \Delta\vec\rho|.\\
\end{split}\]
Since $\vec\rho$ is Diophantine with Diophantine class $\beta\geq 0$, for every $\alpha\in\mathbb{R}$, there exists a constant $C_{\vec\rho}>0$ such that for all $l\in\mathbb{Z}^d$ with $\|l\|$ sufficiently big, one has
\begin{equation}\label{eqn:shifted_Dioph}
|\exp(i2\pi(l\cdot\vec\rho-\alpha)-1)|=|\exp(i2\pi(l\cdot\vec\rho-\alpha)-\exp(i2\pi \alpha))|\geq C_{\vec\rho}\|l\|^{-(d+\beta)}
\end{equation}
By Eq. 14 from \cite{Das-Yorke}, for every $l\neq k$ there exists a constant $C_{w,m}>0$ depending on $w$ and $m$ such that for every $\alpha\in\mathbb{R}$
\begin{equation}\label{eqn:DY_14}
\left|\sum\limits_{n=0}^{N-1}\exp(-i 2\pi n \alpha)\hat{w}_{m,N}\right| \leq C_{w,m}N^{-m}|e^{i 2\pi \alpha}-1|^{-m}
\end{equation}
Combining Eqs. \ref{eqn:shifted_Dioph} and \ref{eqn:DY_14}, we get
\[\begin{split}
|\hat a_k(f_l)-a_k(f_l)|&=|\hat a_k(f_l)|\\
&=\left|\sum\limits_{n=0}^{N-1}\exp(-i 2\pi n (l\cdot \vec\rho-k\cdot \hat{\vec\rho}))\hat{w}_{m,N}\right|\\
&\leq C_{w,m}N^{-m}|\exp(i 2\pi (l\cdot \vec\rho-k\cdot \hat{\vec\rho}))-1|^{-m}\\
&\leq C_{w,m}C_{\vec\rho}N^{-m}\|l\|^{m(d+\beta)}\\
\end{split}\]
Since $f\in C^{M}$, there exists $C_{g,M}>0$ depending on $g$ and $M$ such that for each $l\neq \vec 0$, $|a_l|\leq C_{f,M} \|l\|^{-M}$. Therefore, in a manner similar to the derivation of Eq. 15 from \cite{Das-Yorke}, we can write
\[\begin{split}
|\hat a_k(g)-a_k(g)|&=\left|\underset{l\in\mathbb{Z}^d}{\Sigma}a_l(a_k(f_l)-a_k(f_l))\right|\\
&\leq |a_k(a_k(f_k)-a_k(f_k))|+\left|\underset{l\neq k}{\Sigma}a_l(a_k(f_l)-a_k(f_l))\right|\\
&\leq |a_k|\pi N |k\cdot \Delta\vec\rho|+\underset{l\neq k}{\Sigma}|a_l(a_k(f_l)-a_k(f_l))|\\
&\leq |a_k|\pi N |k\cdot \Delta\vec\rho|+C_{w,m}C_{\vec\rho}N^{-m}\underset{l\neq k}{\Sigma}|a_l|\|l\|^{m(d+\beta)}\\
&\leq |a_k|\pi N |k\cdot \Delta\vec\rho|+C_{w,m}C_{\vec\rho}C_{f,M}N^{-m}\underset{l\neq k}{\Sigma}\|l\|^{-M}\|l\|^{m(d+\beta)}\\
&= |a_k|\pi N |k\cdot \Delta\vec\rho|+C_{w,m}C_{\vec\rho}C_{f,M}N^{-m}\underset{l\neq k}{\Sigma}\|l\|^{-(M-m(d+\beta))}\\
\end{split}\]
Since $M>d+m(d+\beta)$, the sum $\underset{l\neq k}{\Sigma}\|l\|^{-(M-m(d+\beta))}<\infty$. The statement of the corollary now follows. \qed
\end{proof}
\clearpage
\bibliographystyle{unsrt}
|
1,116,691,500,906 | arxiv |
\section{Acknowledgments}
This work was supported in part by National Key R{\&}D Program of China (2020AAA0107700), by National Natural Science Foundation of China (62102353, 62227805), by National Key Laboratory of Science and Technology on Information System Security (6142111210301), by State Key Laboratory of Mathematical Engineering and Advanced Computing, and by Key Laboratory of Cyberspace Situation Awareness of Henan Province (HNTS2022001). We would like to thank Dingkun Wei, Jingjing Wang, Zijing Hu and Yanqing Liu for their implementation of some experiments.
\section{Appendix}
\subsection{Experimental Setup}
\noindent \underline{\textbf{Datasets}}
We use CIFAR10, Purchase100,
FaceScrub530 and UTKFace datasets
which are widely adopted in previous studies on
membership
inference attacks, attribute inference attacks and model inversion attacks.
Additionally, we also evaluate the effectiveness of {\sc Purifier}\xspace against memership inference attacks on CIFAR100, Texas and Location.
\noindent \textbf{CIFAR10.}
It is a machine learning benchmark dataset for evaluating image recognition
algorithms with 10 classes.
It consists of 60,000 color images, each of size 32 x 32.
The dataset has 10
classes, where each class represents an object (e.g., airplane, car, etc.).
\noindent \textbf{Purchase100.}
This dataset is based on Kaggle's ``acquired valued shopper'' challenge.%
\footnote{https://www.kaggle.com/c/acquire-valued-shoppers-challenge/data}
We used the preprocessed and simplified version of this
dataset.
It is composed of 197,324 data records and each data record has 600 binary
features. The dataset is clustered into 100 classes.
\noindent \textbf{FaceScrub530.}
This dataset consists of URLs for 100,000 images of 530 individuals.
We obtained the preprocessed and simplified version of this dataset from
which has 48,579 facial images and each image is
resized to 64 $\times$ 64.
\noindent \textbf{UTKFace.}
This dataset consists of URLs for 22000 images of individuals. We train the classifier to classify the genders and use the race as the sensitive attribute in our experiments.
\noindent \textbf{CIFAR100.}
It is a machine learning benchmark dataset for evaluating image recognition
algorithms with 10 classes.
It consists of 60,000 color images, each of size 32 x 32.
The dataset has 100
classes and each classes has 600 images.
\noindent \textbf{Texas.}
We use the same data as previous studies, which cluster the data with 6169 attributes to 100 classes.
\noindent \textbf{Location.}
This dataset is based on the publicly
available set of mobile users’ location “check-ins” in the
Foursquare social network, restricted to the Bangkok area
and collected from April 2012 to September 2013. The record of Location has 446 attributes and has clustered into 30 classes.
\noindent \underline{\textbf{Target Classifier}}
We use the same model architectures as in previous
work.
For CIFAR10 and CIFAR100 datasets, we use DenseNet121.
We train our classifier with SGD optimizer for
350 epochs with the learning rate 0.1 from epoch 0 to 150, 0.01 from 150 to 250,
and 0.001 from 250 to 350. The classifier is regularized with
$L_2$ regularization (weight decay parameter 5e-4).
For Purchase100 dataset, we use the same model and training strategy as in previous work
to train the target classifier. It is a 4-layer fully
connected neural network.
For FaceScrub530 dataset, we use the same conventional neural network and
the same training strategy as in previous work to train the target
classifier.
For UTKFace dataset, we use the same neural network and
the same training strategy as used in FaceScrub530 dataset, except that the output layer dimension is changed to 2.
For Texas dataset, we use a 4-layer fully
connected neural network with the Tanh as the activation function.
For Location dataset, we use a 3-layer fully
connected neural network with the Tanh as the activation function.
\noindent \underline{\textbf{{\sc Purifier}\xspace}}
We use CVAE to implement the \textit{confidence reformer} $G$. It has the layer size of [20, 32,
64, 128, 2, 128, 64, 32, 20] for CIFAR10, [200, 128, 256, 512, 20, 512, 256,
128, 100] for Purchase100, Texas and location, [1060, 512, 1024, 2048, 100, 2048, 1024, 512,
1060] for FaceScrub530 and CIFAR100. The layer size of [4, 32,
64, 128, 2, 128, 64, 32, 2] for UTKFace, .
We use ReLU and batch normalization in hidden
layers.
We train {\sc Purifier}\xspace on Purchase100 dataset for 150 epochs, CIFAR10, Texas and Location datasets for
100 epochs, FaceScrub530, UTKFace and CIFAR100 datasets for 300 epochs.
We use Adam optimizer with the learning rate 0.01 for CIFAR10, 0.0001 for
Purchase100, Texas and Location, 0.0005 for Facescrub530 and CIFAR10 and 0.0001 for UTKFace.
\noindent \underline{\textbf{Existing Attacks}}
In our experiments, we implement the following data inference attacks.
\noindent \textbf{NSH attack}.
This attack assumes that the attacker knows both the membership labels and ground truth of $D_{aux}$, and thus no shadow is trained.
The membership classifier makes use of both the confidence score vector and the ground truth of a data sample to predict its membership.
\noindent \textbf{Mlleaks attack}.
In this attack, the attacker knows the ground truth of $D_{aux}$ but does not know their membership labels. Therefore, a shadow model is required to replicate the target model.
We use half of $D_{aux}$ to train the shadow model which has the same architecture as the target classifier, and use the whole $D_{aux}$ to train the membership classifier with their labeled membership information in terms of the shadow model.
The membership classifier is a multi-layer perception with a 128-unit hidden layer and a sigmoid output layer.
All weights were initialized with normal distribution with a mean of 0 and standard deviation of 0.01, and all biases are initialized to 0.
We use the Adam optimizer with learning rate 0.001. The number of training epochs is set to 50 for each dataset.
\noindent \textbf{Adaptive attack}.
This is an adaptive version of the Mlleaks attack, where the attacker is
assumed to know all the details of the defender's {\sc Purifier}\xspace and its training
data $D_2$. Hence, the attacker trains the same {\sc Purifier}\xspace and appends it to the
shadow model. The membership classifier is then trained on the purified
confidence score vectors.
\noindent \textbf{BlindMI attack}. We consider BlindMI-DIFF-w/, where the attacker is assumed to know
the soft label and the ground truth of the target dataset. We use sobel to generate non-member samples. The size of the non member dataset $|S_{nonmem}|$ is 20.
\noindent \textbf{Label-only attack}. In Gap Attack, the attacker is assumed to have the ground truth of the data sample and
predicts that it is a member if and only if the target classifier gives the
correct label on the sample.
In the Transfer attack, auxiliary dataset $D_{aux}$ is re-labeled by querying the target model. The shadow models that we adopt share the same architecture with the target model. Three different thresholds are chosen in previous work, and we consider the one that gives the best result on $D_{aux}$. In the Boundary attack, we adopt HopSkipJump noise, with a total evaluation of {15000 per sample}, which ensures the attack performance is stable. Similarly, we report the results on the $L_2$ norm. Due to high query-complexity, the results are reported on a subset that consists of 1000 samples.
\noindent \textbf{Adversarial Model inversion attack}.
The attacker trains an inversion model on $D_{aux}$ to perform the model inversion attack.
\noindent \textbf{Attribute inference attack}.
The attacker trains a classification on $D_{aux}$ to predict the additional sensitive attribute taking the confidence vectors as input. The classifier is a multi-layer perceptron.
All weights were initialized with normal distribution with a mean of 0 and standard deviation of 0.01, and all biases are initialized to 0.
We use the Adam optimizer with learning rate 0.00001. The number of training epochs is set to 1500 for the UTKFace dataset.
\begin{table}[h]
\centering
\tiny
\caption{Data allocation.
A dataset is divided into training set $D_{1}$ of the target classifier, reference set $D_{2}$ and test set $D_{3}$.
In membership inference attack, we assume that the attacker has access to a subset $D^{A}$ of $D_{1}$ and a subset $D^{'A}$ of $D_{3}$.
}
\label{tb:datasplit}
\resizebox{\columnwidth}{!}{
\begin{tabular}{c|l|l|l|c|r}
\hline
Dataset & \multicolumn{1}{c|}{$D_{1}$} & \multicolumn{1}{c|}{$D_{2}$} & \multicolumn{1}{c|}{$D_{3}$} & \multicolumn{1}{c|}{$D^{A}$} & \multicolumn{1}{c}{$D^{'A}$} \\
\hline
CIFAR10 & 50,000 & 5,000 & 5,000 & 25,000 & 2,500\\
Purchase100 & 20,000 & 20,000 & 20,000 & 10,000 & 10,000\\
FaceScrub530 & 30,000 & 10,000 & 8,000 & 15,000 & 4,000\\
UTKFace & 12,000 & 5,000 & 5,000 & 6,000 & 2,500\\
CIFAR100 & 50,000 & 5,000 & 5,000 & 25,000 & 2,500\\
Texas & 10,000 & 10,000 & 10,000 & 5,000 & 5,000\\
Location & 1,600 & 1,600 & 1,600 & 800 & 800\\
\hline
\end{tabular}
}
\end{table}
Table~\ref{tb:datasplit} presents the data allocation in our experiments.
We divide each dataset into the target classifier's training set $D_1$, the reference set $D_2$ and the test set $D_3$.
In membership inference and attribute inference attack,
we assume that the attacker has access to a subset $D^{A}$ of $D_1$ and a subset $D^{'A}$ of $D_3$ to form its auxiliary dataset $D_{aux}$.
We use the remaining data in $D_1$ and $D_3$ to test the membership inference and attribute inference accuracy.
We use $D_2$ as the reference dataset for defenses that require such a dataset.
In the model inversion attack, for the FaceScrub530 classifier, the attacker uses a CelebA\footnote{ICCV 2015 Deep Learning Face Attributes in the Wild}
dataset to train the inversion model. For other classifiers, the attacker samples 80\% from $D_1$, $D_2$ and $D_3$ respectively to train the inversion model, and uses the remaining 20\% data to test the inversion error.
\noindent \underline{\textbf{Metrics}}
We use the following metrics to measure the utility, defense performance and efficiency of a defense method.
\noindent \textbf{Classification Accuracy.} It is measured on the training set and the test set of the target classifier. It reflects how good the
target classifier is on the classification task.
\noindent \textbf{Utility Error.} It is measured on the training set
$D_1$ and the test set $D_3$ of the target classifier. It reflects how good the
target classifier is on the classification task.
\noindent \textbf{Inference Accuracy.} This is the classification accuracy of
the attacker's attack model in predicting the membership and additional sensitive attribute of input samples.
\noindent \textbf{Inversion Error.} We measure the inversion error by computing the mean squared error between the original input sample and the reconstruction.
For the FaceScrub530 classifier, it is measured on $D_1$ and $D_3$. For other classifiers, it is measured on the 20\% of $D_1$ and $D_3$ respectively.
\noindent \textbf{Efficiency.} We measure the efficiency of a defense method by
reporting its training time and testing time relative to the original time required by the target
classifier.
\subsection{Experiment of Adversarial Model Inversion in MNIST Dataset}
We investigate the {\sc Purifier}\xspace against the adversarial model inversion attack on the MNIST dataset and plot the reconstructed images in Figure~\ref{fig:mnist}. We report the inversion error at
the left of Figure~\ref{fig:mnist} to qualify the inversion performance. As
illustrated in Figure~\ref{fig:mnist}, the attacker is able to reconstruct nearly
identical images when no defense is used. However, it is much
more difficult for the attacker to reconstruct the image from the
purifier-defended model. The reconstructed images lose a lot
of details compared with the original one, only representing a
blurred image of their classes.
\begin{figure}[t]
\centering
\subfigure{
\includegraphics[width=1\linewidth]{mnist_defense.pdf}
}
\caption{Model inversion attack against the MNIST classifier
defended by different approaches. Numbers on the left indicate inversion errors}
\label{fig:mnist}
\end{figure}
\subsection{Distribution of the target classifier’s prediction
uncertainty and confidence in correct class}
We examine the indistinguishability of the confidence scores on members and
non-members by plotting the histogram of the target classifier's confidence in
predicting the correct class and the prediction uncertainty in CIFAR10 and FaceScrub530 datasets.
We show the results in Figure~\ref{fig:prediction_acc_other} and Figure~\ref{fig:uncertainty_other}.
As Figure~\ref{fig:prediction_acc_other} and Figure~\ref{fig:uncertainty_other}
show,
{\sc Purifier}\xspace can also reduce the gap between the two curves in CIFAR10 and FaceScrub530 datasets.
This demonstrates that PURIFIER also successfully reduces the
individual differences between members and non-members in CIFAR10 and FaceScrub530 datasets.
\begin{figure}[t]
\centering
\begin{minipage}[b]{1\linewidth}
\centering
\subfigure{
\includegraphics[width=0.5\linewidth]{fig7_cifar10_nodefence.pdf}
\includegraphics[width=0.5\linewidth]{fig7_cifar10_purifier.pdf}
}
\subfigure{
\includegraphics[width=0.5\linewidth]{fig7_facescrub_nodefence.pdf}
\includegraphics[width=0.5\linewidth]{fig7_facescrub_purifier.pdf}
}
\end{minipage}
\caption{Distribution of the target classifier's confidence in predicting the correct class on members and non-members of its training set.}
\label{fig:prediction_acc_other}
\end{figure}
\begin{figure}[h!]
\centering
\begin{minipage}[b]{1\linewidth}
\centering
\subfigure{
\includegraphics[width=0.5\linewidth]{fig8_cifar10_withoutdefence.pdf}
\includegraphics[width=0.5\linewidth]{fig8_cifar10_withdefence.pdf}
}
\subfigure{
\includegraphics[width=0.5\linewidth]{fig8_facescrub_withoutdefence.pdf}
\includegraphics[width=0.5\linewidth]{fig8_facescrub_withdefence.pdf}
}
\end{minipage}
\caption{Distribution of the target classifier's prediction uncertainty on members and non-members of its training set. The uncertainty is measured as the normalized entropy of the confidence score vector.}
\label{fig:uncertainty_other}
\end{figure}
\input{new_dataset_comparison.tex}
\subsection{Experiments on Other Three Datasets}
We evaluate the effectiveness of {\sc Purifier}\xspace against various membership inference attacks on CIFAR100, Texas and Location datasets. Table~\ref{tb:new_dataset} shows that {\sc Purifier}\xspace is able to reduce the performance of the membership inference attacks significantly on the additional three datasets and outperforms SELENA in most of cases. For instance, {\sc Purifier}\xspace can reduce the accuracy of NSH attack on Texas dataset from 66.37\% to 51.29\% while SELENA can only reduce it from 66.37\% to 54.10\%.
\begin{table}[t]
\caption{Results of {\sc Purifier}\xspace against attackers with the more powerful shadow models.}
\label{tb:powerful_shadow}
\begin{tabular}{c|c|c|c}
\toprule
\multirow{2}{*}{Dataset} & \multirow{2}{*}{Defense} & \multirow{2}{*}{NSH Attack} & \multirow{2}{*}{Mlleaks Attack} \\
& & & \\ \hline
\multicolumn{1}{c|}{\multirow{2}{*}{CIFAR10}} & None & 58.46\% & 70.20\% \\ \cline{2-4}
\multicolumn{1}{c|}{} & Purifer & 50.67\% & 50.01\% \\ \hline
\multirow{2}{*}{Purchase100} & None & 88.27\% & 68.50\% \\ \cline{2-4}
& Purifer & 52.74\% & 51.00\% \\ \hline
\multirow{2}{*}{FaceScrub530} & None & 70.30\% & 73.46\% \\ \cline{2-4}
& Purifer & 50.22\% & 50.66\% \\ \bottomrule
\end{tabular}
\end{table}
\subsection{Effectiveness of {\sc Purifier}\xspace against Attackers with More Powerful Shadow Models}
In order to study the effectiveness of {\sc Purifier}\xspace against stronger attackers, we evaluate the performance of {\sc Purifier}\xspace against NSH attack and Mlleaks attack which use the same number of the data as the target model to train a more powerful shadow model. As Table~\ref{tb:powerful_shadow} shown, {\sc Purifier}\xspace is also effective to reduce the attack accuracy of the attackers with more powerful shadow models. For instance, the attack accuracy of the Mlleaks attack drops to 50.01\% from 70.20\%, which means {\sc Purifier}\xspace is able to mitigate stronger attacks which having more information (e.g., an attacker has more data to train shadow models) than that we assumed in the main paper.
\subsection{Inference about the reference data}
Involving an in-distribution reference dataset in the defense mechanism is common in the literature.
For instance, MemGuard uses a reference set to train the defense classifier. Min-Max uses it to train the inference model.
Similarly, our approach uses it to train the {\sc Purifier}\xspace.
Unfortunately, little has been discussed on whether such reference dataset brings vulnerability for data inference attacks.
Assuming the reference data are considered as members, we present the inversion error and the inference accuracy (we consider NSH attack) on the reference set $D_2$ and the test set $D_3$ for each defense in Table~\ref{tb:attack_d2}.
Results show that
the inference accuracy does not increase on the reference set compared with the original training data of the target classifier.
{\sc Purifier}\xspace can still preserve the defense effect against the adversarial model inversion attack and the membership inference attack.
However, there might be opposing views on whether such reference datasets should be considered as members.
\begin{table}[t]
\centering
\small
\caption{Results of model inversion attack and membership inference attack on the reference set for different defenses. The experiments are performed on the FaceScrub530 dataset.}
\label{tb:attack_d2}
\resizebox{\columnwidth}{!}{
\begin{tabularx}{\columnwidth}{l|Y|Y}
\hline
Defense & Inversion error & Inference accuracy\\
\hline
Purifier & 0.0435 & 51.23\%\\
\hline
Min-Max & 0.0222 & 52.07\% \\
\hline
MemGuard & 0.0116 & 52.20\% \\
\hline
\end{tabularx}
}
\end{table}
\subsection{Effect of {\sc Purifier}\xspace's Training Data}
\begin{table}[t]
\centering
\small
\setlength{\tabcolsep}{6pt}
\caption{Effect of the {\sc Purifier}\xspace's in-distribution training data on the
defense performance. The numbers are reported on the Purchase100 dataset.}
\label{tb:effectreference}
\resizebox{\columnwidth}{!}{
\begin{tabularx}{\columnwidth}{c|c|Y|Y|Y}
\hline
Training set & Classification acc & NSH &
Mlleaks & Adaptive \\
\hline
$D_2$ (5,000) & 83.85\% & 52.63\% & 50.09\% &
50.62\% \\
\hline
$D_2$ (10,000) & 83.47\% & 51.72\% & 50.12\% &
50.14\% \\
\hline
$D_2$ (20,000) & 83.23\% & 51.71\% & 50.09\% &
50.13\% \\
\hline
$D_2$ (40,000) & 81.39\% & 51.93\% & 50.09\% &
50.20\% \\
\hline
$D_2$ (60,000) & 81.75\% & 51.84\% & 50.09\% &
50.10\% \\
\hline
$D_1$ (20,000) & 84.23\% & 52.29\% & 50.12\% &
50.07\% \\
\hline
\end{tabularx}
}
\end{table}
We investigate the effect of the {\sc Purifier}\xspace's training data by using different
in/out-distribution data to train {\sc Purifier}\xspace.
Specifically, for in-distribution data, we vary the size of $D_2$ and also
replace $D_2$ with $D_1$. For out-of-distribution data, we use CIFAR10 data to
train the {\sc Purifier}\xspace for the FaceScrub530 classifier, and use randomly generated
data to train the {\sc Purifier}\xspace for the Purchase100 classifier.
We present the effect of the in-distribution training data in
Table~\ref{tb:effectreference}.
{
The results show that {\sc Purifier}\xspace is still effective. The membership inference accuracy (Mlleaks and Adaptive)
is reduced to nearly 50\% regardless of the size of $D_2$. {\sc Purifier}\xspace is also insensitive to the size of the $D_2$. The difference of the defense performance is negligible as the size of $D_2$ changes from 5,000 to 60,000. This is good for the defender, as one can achieve good performance with a small reference set. However, when the size of $D_2$ becomes too large (i.e., 40,000 to 60,000), the classification accuracy drops to a certain extent. The reason could be that {\sc Purifier}\xspace starts to learn the detailed information of the confidence score vectors. As a result, the purified confidence score vector no longer concentrates on general
patterns but becomes an accurate reconstruction, which hinders the classification utility.
When we use the classifier's training data $D_1$ to train the {\sc Purifier}\xspace, the defense performance is comparable to the ones on $D_2$. For example, the attack accuracy of the NSH attack is 52.29\%, which is marginally higher than the result of 51.71\% on $D_2$, but still acceptable.
}
Table~\ref{tb:effectoutof} shows the effect of the out-of-distribution training
data. {\sc Purifier}\xspace can still mitigate the attacks, but at the cost of
sacrificing the utility of the target classifier significantly.
This is not surprising because {\sc Purifier}\xspace cannot extract useful patterns from
the confidence scores on out-of-distribution data, which makes the purified
confidence information meaningless.
\begin{figure}[t]
\centering
\subfigure{
\includegraphics[width=1\linewidth]{near_member.pdf}
}
\caption{
The proportion of noise data that {\sc Purifier}\xspace can detect under the
FGSM attacks on the FaceScrub530 dataset.
}
\label{fig:near_member}
\end{figure}
\begin{table}[h]
\centering
\small
\setlength{\tabcolsep}{1pt}
\caption{Effect of the {\sc Purifier}\xspace's out-of-distribution training data on the
defense performance.}
\label{tb:effectoutof}
\resizebox{\columnwidth}{!}{
\begin{tabularx}{\columnwidth}{l|l|Y|c|c|c}
\hline
Classifier & Purifier & Classification acc & NSH &
Mlleaks & Adaptive \\
\hline
FaceScrub530 & CIFAR10 & 40.05\% & 54.54\% & 50.11\%
& 50.50\% \\
\hline
Purchase100 & Random & 7.76\% & 51.55\% & 50.20\% &
50.71\% \\
\hline
\end{tabularx}
}
\end{table}
\subsection{Effectiveness of {\sc Purifier}\xspace
to detect noisy members}
We investigate the effectiveness of {\sc Purifier}\xspace
to detect noisy members, which means the members added noise by attackers intentionally. We use the adversarial attack methods (FGSM)\footnote{ICLR 2015 Explaining and Harnessing Adversarial Examples}
to create noisy members.
As shown in Figure~\ref{fig:near_member} that {\sc Purifier}\xspace can accurately detect the members
with noise $\left \| \eta \right \|_{\infty}<1e-10$ on FaceScrub530 dataset.
\begin{table}[]
\caption{Ablation study on the Label Swapper.}
\label{tb:Ablation}
\begin{tabular}{c|c|c}
\toprule
\multirow{2}{*}{Dataset} & \multirow{2}{*}{Defense} & \multirow{2}{*}{Transfer Attack} \\
& & \\ \hline
\multicolumn{1}{c|}{\multirow{3}{*}{CIFAR10}} & None & 0.5048 \\ \cline{2-3}
\multicolumn{1}{c|}{} & CVAE & 0.5045 \\ \cline{2-3}
\multicolumn{1}{c|}{} & CVAE + Label Swapper & 0.4974 \\ \hline
\multirow{3}{*}{CIFAR100} & None & 0.6668 \\ \cline{2-3}
& CVAE & 0.6632 \\ \cline{2-3}
& CVAE + Label Swapper & 0.4975 \\ \bottomrule
\end{tabular}
\end{table}
\subsection{Ablation study on the Label Swapper}
We conduct the ablation study on the label swapper to investigate the effectiveness of it in our defense mechanism. Given our design of label swapper to mitigate the attackers using only the hard labels to perform attacks, we verify its efficiency under label only transfer attack. Table~\ref{tb:Ablation} shows a significant AUC drop in label only transfer attack with the help of label swapper compared by those mechanisms without it. Additionally, the result indicates that label only transfer attack can obtain approximately the same AUC among models without any defense and {\sc Purifier}\xspace without the label swapper, which means {\sc Purifier}\xspace without label swapper fails to mitigate the label only attack as we have assumed. In summary, the label swapper plays an indelible role in PURIFIER to mitigate the label only attacks.
\subsection{Cost of {\sc Purifier}\xspace in Real-world Setting}
We discuss the latency and the memory cost introduced by {\sc Purifier}\xspace at this section. We report that the training time of the CIFAR10 model is 3.18h, and the extra cost introduced by {\sc Purifier}\xspace in it is 1.34h. Compared with 71.30h SELENA has introduced, {\sc Purifier}\xspace contributes to less latency time. Besides the training cost, {\sc Purifier}\xspace will spend extra 282.18s to predict 50k images in the inference time. We speculate that the latency of {\sc Purifier}\xspace is mostly caused by $k$NN. Hence we expect to replace $k$NN by a more efficient component in the future work. For the memory cost, {\sc Purifier}\xspace costs extra space for swap set. The sizes of it with different datasets are respectively 78.23K for CIFAR10, 1.14M for Purchase and 13.5M for Facescrub530. We believe the extra memory cost by {\sc Purifier}\xspace is negligible compared to the original training set.
\section{Approach: {\sc Purifier}\xspace}
\begin{figure}[t]
\small
\begin{center}
\vspace{-0.8cm}
\setlength{\abovecaptionskip}{-0.5cm}
\setlength{\belowcaptionskip}{-0.74cm}
\includegraphics[width=1.13\linewidth]{architecture_5.pdf}
\caption{Architecture of {\sc Purifier}\xspace.
}
\label{fig:architecture}
\end{center}
\end{figure}
We propose {\sc Purifier}\xspace as a defense of data inference attacks.
Since membership inference attack is one of the most typical instances of the data inference attacks, we design {\sc Purifier}\xspace against it as the point of penetration and evaluate its defense performance against other data inference attacks.
The main idea of {\sc Purifier}\xspace is to transform the confidence score vectors in such
a way that they appear indistinguishable on members and non-members. We focus on reducing the three underlying distinguishabilities of confidence scores between members and non-members: \textit{individual shape}, \textit{statistical distribution} and \textit{prediction label}.
{\sc Purifier}\xspace consists of a \textit{confidence reformer} $G$ and a \textit{label swapper} $H$, as shown in Figure~\ref{fig:architecture}. $G$ takes as input the original confidence score vectors
and reforms them as if they were predicted on non-members, achieving indistinguishability of {individual shape} and {statistical distribution}.
We design $G$ as a CVAE, with the predicted label being the condition.
In this way, $G$ is able to learn the overall distribution of the confidence scores from all classes by setting the condition to the corresponding class.
The \textit{label swapper} $H$ takes the reformed confidence scores from $G$, and modifies the predicted labels of members to reduce the gap of classification accuracy between members and non-members, achieving indistinguishability of {prediction label}.
\subsection{Achieving Individual Indistinguishability}
In order to achieve the individual indistinguishability of confidence scores between members and non-members, {\sc Purifier}\xspace reforms the confidence scores with the \textit{confidence reformer} $G$, which is a CVAE. $G$ takes the confidence score $F(\vec{x})$ predicted by the target classifier $F$ as input, with the corresponding label $l$ being the condition. $F(\vec{x})$ first goes through the encoder, where it is mapped to the encoded latent space $\vec{r}$. The decoder then maps the confidence score back from the latent space $\vec{r}$,
and the reformed confidence score $G(F(x)|l)$ is obtained. $G$ is trained on the confidence scores predicted by $F$ on the defender’s reference dataset $D_{ref}$, which consists
of non-member samples. As a result, $G$ learns the pattern of \textit{individual shape} on non-member samples. The reforming process of $G$ could remove any difference in the {individual shape} of $F(\vec{x})$, achieving {individual} indistinguishability.
In order to preserve the classification accuracy, we train $G$
to also produce the label predicted by F by adding a label loss. Formally, $G$ is trained to minimize the following objective function.
\vspace{-0.5cm}
\begin{equation}
\small
\vspace{-0.2cm}
\label{eq:loss}
\begin{split}
L(G) = \underset{\vec{x} \sim p_{r}(\vec{x})}{\mathbb{E}}
&[\mathcal{R}((G(F(\vec{x})|l), F(\vec{x}))
+ \lambda \mathcal{L}((G(F(\vec{x})|l), l)]
\end{split}
\end{equation}
where $p_r(\vec{x})$ represents the conditional probability of $\vec{x}$ for
samples in $D_{ref}$, $l$ represents the label of $F(\vec{x})$ (i.e., $l = \argmax(F(\vec{x}))$). $\mathcal{R}$ is a reconstruction loss function ($L_2$
norm) and $\mathcal{L}$ is the cross entropy loss function. The parameter $\lambda$ controls the balance of the two loss functions during training.
\subsection{Achieving Statistical Indistinguishability}
We can observe the {statistical distribution} of $F(\vec{x})$ by plotting $F(\vec{x})$ on the encoded latent space $\vec{r}$. Figure~\ref{fig:latent_space_purifier} shows an example of such {statistical distribution} on CIFAR10 dataset, where different
colors represent different labels. We can observe that confidence score vectors are clustered into several groups according to their labels. However, the members are more densely clustered while non-members are not, which indicates that the distribution of members and non-members are different.
To mitigate the difference in {statistical distribution} between members
and non-members, \textit{confidence reformer} $G$ introduces Gaussian noises in the latent space $r$, where the label $l$ is used as the condition. During the training process, the reconstruction loss $\mathcal{R}$ encourages the decoder of $G$ to generate confidence scores that are similar to the non-member ones on $D_{ref}$ (non-members) with the same label $l$. However, noises introduced in the latent space $\vec{r}$ will increase the reconstruction error. As a result, $G$ learns a robust latent representation that could preserve the {statistical distribution} of the non-members of label $l$ even if noises are added. During the inference process, the added noises breakdown the clustering of confidence scores on members, while the decoder generates the reformed versions that are similar to the ones on $D_{ref}$, mitigating the difference in {statistical distribution}.
\subsection{Achieving Label Indistinguishability}
To cope with the difference in prediction label, we design a \textit{label swapper} $H$, which modifies the prediction labels of members to reduce the gap of classification accuracy between members and non-members. After training the \textit{confidence reformer} $G$, we randomly select training data to replace their predicted labels with the second largest predicted labels at a certain swap rate $p_{swap} = (acc_{train} - acc_{test} )/ acc_{train}$, where $acc_{train}$ and $acc_{test}$ are the training accuracy and the test accuracy of the target classifier respectively. Note that we fix the data at the training stage whose labels will be modified, so when attackers use the same data to query the final model, they will get the same output. Hence {\sc Purifier}\xspace can defend the \textit{replay attack} where attackers exploit the differences between the outputs of multiple same queries to the target model.
Given an input sample $\vec{x}$, $H$ first identifies if $\vec{x}$ is a member whose label needs to be modified.
In order to identify members, the \textit{label swapper} stores information of the original training data. However, it is challenging for the label swapper to efficiently store and index the member information in the run time especially in a large learning task.
To this end,
Label swapper stores $F(\vec{x})$ where $x \in D_{swap}$ as the identifiers to form a prediction indexing set $P_{index}$ whose dimensionality is much smaller than the training data $D_{train}$.
In order to tolerate small perturbations of members added by attackers to indirectly infer membership of a target member sample $x \in D_{swap}$, $H$ uses $k$ nearest neighbor ($k$NN) to identify these suspicious noisy members and also swaps their labels.
\begin{figure}[!t]
\footnotesize
\let\@latex@error\@gobble
\begin{algorithm}[H]
\SetAlgoLined
\SetKwInput{KwInput}{Input}
\SetKwInput{KwOutput}{Output}
\KwInput{The reference dataset $D_{ref}$, the training dataset $D_{train}$, the target classifier $F$, size of mini-batch ${q}$, size of the data need to be modify the labels $t$, number of epochs $P$, learning rate $\eta$, label loss coefficient $\lambda$}
\KwOut{Model parameters $\theta$ of \textit{label reformer} $G_\theta$, The prediction indexing set $P_{index}$}
$\theta \gets \mathbf{initialize}(G_\theta)$ \;
\For{$p=1$ to $P$}{
\For{each mini-batch ${\{(\vec{x}_{{ref}_{j}},y_{{ref}_{j}})\}}^q_{j=1} \subset D_{ref}$}{
$c_{r_j} \gets F(\vec{x}_{ref_j})$\;
$l_{r_j} \gets \mathbf{onehot}(\argmax(c_{r_j})$)\;
$g \gets \nabla_\theta \frac{1}{q} \sum_{j=1}^{q} \mathcal{R}(G_\theta(c_{r_j}|l_{r_j}),c_{r_j}) + \lambda \mathcal{L}(G_\theta(c_{r_j}|l_{r_j}),l_{r_j})$\;
$\theta \gets \mathbf{updateParameters}(\eta,\theta,g)$
}
}
$P_{index} \gets \emptyset$\;
$D_{train} \gets \mathbf{shuffle}(D_{train})$\;
$D_{swap} \gets {\{(\vec{x}_{{train}_{j}},y_{train_j})\}}^t_{j=1} \subset D_{train}$\;
\For{each ${(\vec{x}_{{train}_{j}},y_{train_j})} \in D_{swap}$}{
$c_j \gets F(\vec{x}_{train_j})$\;
$P_{index} \gets P_{index} \cup \{c_j\}$\;
}
\Return $G_{\theta}$, $P_{index}$
\caption{Training process of {\sc Purifier}\xspace.}
\label{alg:train}
\end{algorithm}
\vspace{-0.5cm}
\end{figure}
\noindent \underline{\textbf{Training and inference process of {\sc Purifier}\xspace}}
The training process of {\sc Purifier}\xspace is detailed in Algorithm \ref{alg:train}. For each epoch, we first draw a mini-batch of data points ${\{(\vec{x}_{{ref}_{j}},y_{ref})\}}^q_{j=1}$ from the reference set $D_{ref}$. Then we query the target classifier $F$ to obtain the confidence scores $c_{r_j}$ and the labels $l_{r_j}$ (Line 1-5). After that, the loss is calculated on the objective function~\ref{eq:loss} and gradient descent is used to update the parameters $\theta$ of \textit{confidence reformer} $G$ (Line 6-7).
When the training of $G$ is finished, we select the data from $D_{train}$ at rate $p_{swap}$ randomly to form $D_{swap}$ (Line 10-11).
After that, we query the target classifier $F$ to get the confidence $c_{j}$ of the sample ${(\vec{x}_{{train}_{j}},y_{{train}_{j}})} \in D_{swap}$.
The original confidence score $c_{j}$
is added to the prediction indexing set $P_{index}$ and later used by the \textit{label swapper} to achieve indistinguishability of prediction label (Line 12-15).
In the inference stage,
given an input sample $\vec{x}$, we first query the target classifier $F$ to get the confidence score $c$ and the predicted label $l$. Then, we input $c$ into the \textit{confidence reformer} $G$, with $l$ being the condition, to get the purified confidence vector $p$.
At this stage, $p$ is indistinguishable in individual shape and statistical distribution.
The \textit{label swapper} $H$ checks if $c$ has a match in $P_{index}$ using $k$NN and swaps the label of $p$ if $c$ is matched.
This ensures indistinguishability in terms of prediction label. Finally, {\sc Purifier}\xspace returns the purified confidence scores $p$.
\section{Conclusion}
In this paper, we propose {\sc Purifier}\xspace to defend data inference attacks.
{\sc Purifier}\xspace learns the pattern of non-member confidence score vectors and
purifies confidence score vectors to this pattern. It makes member
confidence score vectors indistinguishable from non-members in terms of
individual shape, statistical distribution and prediction label.
Our experiments show that {\sc Purifier}\xspace is effective and efficient in mitigating existing data inference attacks, outperforming previous defense methods,
while imposing negligible utility loss.
\section{Ethics Statement}
The code for {\sc Purifier}\xspace is available at \url{https://github.com/wljLlla/Purifier_Code}.
\section{Experimental Results}
\input{bigtable}
\subsection{{\sc Purifier}\xspace is Effective in Membership Inference}
\noindent \underline{\textbf{Effectiveness}}
Table~\ref{tb:bigtable} presents the defense
performance of {\sc Purifier}\xspace against different membership inference attacks.
For each classification task, {\sc Purifier}\xspace decreases the attack
accuracy as well as preserves the the classification accuracy.
{\sc Purifier}\xspace reduces the accuracy of NSH attack significantly for
different datasets. For instance, it reduces the accuracy of NSH attack
from 69.34\% to 51.56\% in FaceScrub530 dataset.
As for Mlleaks attack, the model defended with {\sc Purifier}\xspace reduces
the attack accuracy to nearly 50\%.
Comparing with the pure Mlleaks attack, the performance of the adaptive attack does not show a large difference where {\sc Purifier}\xspace reduces the accuracy to nearly 50\%.
{\sc Purifier}\xspace is also effective against BlindMI attack. For example, {\sc Purifier}\xspace reduces the accuracy of BlindMI from 62.61\% to 50.00\% in FaceScrub530
dataset.
\noindent \underline{\textbf{Comparison with other defenses}}
\noindent \textbf{Existing Defenses.}
We compare {\sc Purifier}\xspace with following defenses.
\ding{172}Min-Max~\cite{nasr_machine_2018}.
\ding{173}MemGuard~\cite{jia_memguard_2019}.
\ding{174}Model-Stacking~\cite{salem_ml-leaks_2018}.
\ding{175}MMD Defense~\cite{li_membership_2021}.
\ding{176}SELENA~\cite{selena}.
\ding{177}One-Hot Encoding.
\ding{178}Random Noise.
We attach the details of the above-mentioned defense methods, including their implementations and selections of hyperparameters, to the Appendix.
\input{comparetable}
Table~\ref{tb:comparetable} shows the defense performance of {\sc Purifier}\xspace and
other defense methods against membership inference attacks under different
datasets.
{\sc Purifier}\xspace achieves the best defense
performance against most of the attacks, including the NSH attack, the BlindMI attack and the gap attack compared to other
methods on all of three datasets. For the Mlleaks Attack, {\sc Purifier}\xspace can achieve the second best performance only to One-Hot Encoding and Random Noise.
{\sc Purifier}\xspace also achieves a better security-utility
tradeoff than other defenses. It imposes a reduction in test
accuracy of about 1\%. In comparison, Model-Stacking and SELENA can mitigate membership inference attacks to some
extent, but they incur intolerable reduction in model's test accuracy.
For One-Hot
Encoding and Random Noise,
their transformation on
confidence vectors leads to a large degree of semantic information loss.
MemGuard reaches acceptable defense performance with negligible decline in test
accuracy. However, its defense performance is not as good as that of {\sc Purifier}\xspace.
\subsection{{\sc Purifier}\xspace is Effective in Adversarial Model Inversion}
\noindent \underline{\textbf{Effectiveness}}
We further investigate the defense performance of {\sc Purifier}\xspace against adversarial model
inversion attack.
We train an inversion attack model on top of each classifier
with or without defense on FaceScrub530 dataset.
Although {\sc Purifier}\xspace is designed to protect models from membership
inference attacks, it turns out that the {\sc Purifier}\xspace is also effective in mitigating model inversion attack. Figure~\ref{fig:facescrub}
shows the results of our
experiment on adversarial model inversion attack on FaceScrub530.
We quantify the inversion quality by reporting the average facial similarity scores compared with the ground truth using the Microsoft Azure Face Recognition service~\cite{azure_notitle_2020}, which is shown on left side of Figure~\ref{fig:facescrub}. The less the number is,the less similarity reconstructed samples share with the original samples.
We report all the inversion error under three datasets in Table~\ref{tb:bigtable}. As shown in Table~\ref{tb:bigtable} and Figure~\ref{fig:facescrub}, the inversion loss on the FaceScrub530 dataset is raised 4+ times (i.e. from 0.0114 to 0.0454) after applying {\sc Purifier}\xspace, indicating the performance reduction of the inversion attack is significant.
Note that the effect of defense against the adversarial model inversion attacks on Purchase100 and CIFAR10 seems less significant compared with FaceScrub530
This is because the inversion attack does not perform well on these classifiers even though without any defense.
\noindent \underline{\textbf{Comparison with other defenses}}
{\sc Purifier}\xspace also achieves the best performance in defending model inversion attack on CIFAR10 and Facescrub530.
Table~\ref{tb:comparetable} shows that {\sc Purifier}\xspace has the largest inversion
error(also called reconstruction error) compared with other defenses on these datasets,
quantitatively demonstrating that {\sc Purifier}\xspace achieves better defense performance
against adversarial model inversion attack than other defenses.
Figure~\ref{fig:facescrub} depicts the reconstructed samples from confidence
vectors given by each defense model on FaceScrub530 dataset. With {\sc Purifier}\xspace as
defense, the reconstructed images are much less similar to the ground truth
image and look more blurred. Other defense methods, however, could not protect
the model from adversaries recovering small details of the original image. It
can be quantitatively verified by the similarity scores gathered from the
Microsoft Azure Face Recognition service. For instance, the average similarity scores
of reconstructed images of MemGuard-defended models are 0.17, which are larger
than that of {\sc Purifier}\xspace (i.e., 0.14).
{\sc Purifier}\xspace achieves the smallest similarity scores among other defense
methods, indicating that {\sc Purifier}\xspace can protect the target model against
adversarial model inversion attack effectively.
\subsection{{\sc Purifier}\xspace is Effective in Attribute Inference}
\noindent \underline{\textbf{Effectiveness}}
We deploy {\sc Purifier}\xspace under the attribute inference attack and find that {\sc Purifier}\xspace is also effective in mitigating it. We train an attribute inference classifier on UTKFace dataset to predict the race of the given sample.
Table~\ref{tb:attributeinference} shows the results of our experiment.
The attribute inference accuracy on the UTKFace dataset is reduced to 20.94\% (almost random guessing) after applying {\sc Purifier}\xspace.
\begin{figure}[t]
\centering
\vspace{-0.15cm}
\setlength{\abovecaptionskip}{-0.1cm}
\setlength{\belowcaptionskip}{-0.4cm}
\subfigure{
\includegraphics[width=0.85\linewidth]{facescrub_defense_3.pdf}
}
\caption{Model inversion attack against the FaceScrub530 classifier
defended by different approaches.
}
\label{fig:facescrub}
\vspace{-0.3cm}
\end{figure}
\begin{table}[t]
\centering
\vspace{-0.4cm}
\scriptsize
\vspace{0.2cm}
\setlength{\abovecaptionskip}{-0.01cm}
\setlength{\belowcaptionskip}{0.6cm}
\caption{Attribute inference attack against the UTKFace
classifier with and without {\sc Purifier}\xspace.}
\label{tb:attributeinference}
\resizebox{\columnwidth}{!}{
\begin{tabularx}{\linewidth}{c|Y|Y|Y|c}
\hline
\multirow{2}{*}{Dataset} & \multirow{2}{*}{Defense} & \multicolumn{2}{c|}{Utility} & \multirow{2}{*}{Attack Accuracy} \\
\cline{3-4}
& & \multirow{1}{*}{Train acc} & \multirow{1}{*}{Test acc.} \\
\hline
\multirow{2}{*}{UTKFace} & None &
99.92\% & 83.08\% & 31.06\% \\
\cline{2-5}
& Purifier & 84.20\% & 82.78\% & 20.94\% \\ \hline
\end{tabularx}
}
\vspace{-0.65cm}
\end{table}
\begin{figure}[t]
\vspace{0.3cm}
\setlength{\abovecaptionskip}{-0.2cm}
\setlength{\belowcaptionskip}{-0.5cm}
\centering
\tiny
\begin{minipage}[b]{1\linewidth}
\centering
\subfigure{
\includegraphics[width=0.5\linewidth]{efficiency_training.pdf}
\includegraphics[width=0.5\linewidth]{efficiency_testing.pdf}
}
\end{minipage}
\caption{Efficiency of different defense methods.
}
\label{fig:efficiency}
\vspace{-0.2cm}
\end{figure}
\subsection{Efficiency}
Figure~\ref{fig:efficiency} presents the efficiency of {\sc Purifier}\xspace compared with other defenses.
We perform our experiments on a PC equipped with four Titan XP GPUs with 12GBytes of graphic memory,128 GBytes of memory and an Intel Xeon E5-2678 CPU.
The training time of {\sc Purifier}\xspace is only 0.423 time of the target classifier, which outperforms all the other methods.
The testing time of {\sc Purifier}\xspace is {18.06} times as much as the target
classifier, which is considered acceptable compared to MemGuard whose testing time is 7,000+ times more than the original classifier.
\subsection{Analysis of Purified Confidence Scores}
In this subsection, we analyze how the purified confidence scores affect membership inference attacks by evaluating three indistinguishabilities: individual, statistical and label.
\noindent \underline{\textbf{Individual Indistinguishability of Purified
Confidence}}
\begin{figure}[t]
\centering
\subfigbottomskip=1pt
\setlength{\abovecaptionskip}{0.2cm}
\setlength{\belowcaptionskip}{-0.5cm}
\begin{minipage}[b]{1\linewidth}
\centering
\subfigure{
\includegraphics[width=0.5\linewidth]{fig7_purchase_nodefence.pdf}
\includegraphics[width=0.5\linewidth]{fig7_purchase_purifier.pdf}
}
\vspace{-0.1cm}
\end{minipage}
\vspace{-0.3cm}
\begin{minipage}[b]{1\linewidth}
\centering
\setlength{\abovecaptionskip}{-0.2cm}
\subfigure{
\includegraphics[width=0.5\linewidth]{fig8_purchase_withoutdefence.pdf}
\includegraphics[width=0.5\linewidth]{fig8_purchase_withdefence.pdf}
}
\end{minipage}
\caption{Distribution of the target classifier's confidence in predicting the correct class and the prediction uncertainty on members and non-members of training set.}
\label{fig:prediction_acc}
\vspace{-0.2cm}
\end{figure}
\newcolumntype{P}[1]{>{\Centering\hspace{0pt}}p{#1}}
\newcolumntype{Z}{>{\centering\let\newline\\\arraybackslash\hspace{0pt}}X}
\begin{table}[t]
\centering
\scriptsize
\setlength{\abovecaptionskip}{-0.02cm}
\setlength{\belowcaptionskip}{0.5cm}
\caption{Gap of the classifier's confidence in predicting the correct class(i.e, Confi)
and the prediction uncertainty(i.e, Uncer) between members and non-members.}
\label{tb:measurement}
\resizebox{\columnwidth}{!}{
\begin{tabularx}{\columnwidth}{l|l|Y|Y|Y|Y|Y|Y}
\hline
\multirow{2}{*}{Metric} & \multirow{2}{*}{Defense} &
\multicolumn{2}{Y|}{CIFAR10} & \multicolumn{2}{Y|}{Purchase100} &
\multicolumn{2}{Y}{FaceScrub530} \\
\hhline{~|~|-|-|-|-|-|-}
& & Max & Avg. & Max & Avg. & Max & Avg. \\
\hline
\multirow{2}{*}{Confi} & None & 0.103 & 0.004 & 0.412 & 0.016 &
0.415 & 0.017 \\
\cline{2-8}
& Purifier & 0.009 & 0.000 & 0.019 & 0.001 &0.012 & 0.001 \\
\hline
\multirow{2}{*}{Uncer} & None & 0.114 & 0.005 & 0.201 & 0.015 &
0.418 & 0.017 \\
\cline{2-8}
& Purifier & 0.006 & 0.000 & 0.007 & 0.001 & 0.006 & 0.001 \\
\hline
\end{tabularx}
}
\vspace{-0.5cm}
\end{table}
{\sc Purifier}\xspace reshapes the input confidence score vectors according to the pattern
of the learned non-member samples.
We examine the indistinguishability of the confidence scores on members and
non-members by plotting the histogram of the target classifier's confidence in
predicting the correct class and the prediction uncertainty in Figure~\ref{fig:prediction_acc}.
The prediction uncertainty is measured as the normalized entropy
$\frac{-1}{\log (k)} \sum_i \hat{\vec{y}_i}\log (\hat{\vec{y}_i})$ of the
confidence vector $\vec{y} = F(\vec{x})$, where $k$ is the number of classes.
As Figure~\ref{fig:prediction_acc}
shows,
{\sc Purifier}\xspace can reduce the gap between the two curves.
Similar curves can be obtained on CIFAR10 and FaceScrub530 classifiers. We attach them to Appendix.
We also report the maximum gap and the average gap between the curves
in Table~\ref{tb:measurement}.
The results show that our approach can significantly reduce both the maximum
and average gaps between the target classifier's confidence in predicting the
correct class as well as the prediction uncertainty on its members versus
non-members.
This demonstrates that {\sc Purifier}\xspace successfully reduces the individual
differences between members and non-members.
\noindent \underline{\textbf{Statistical Indistinguishability of Purified
Confidence}}
\begin{figure}[t]
\vspace{-0.2cm}
\setlength{\belowcaptionskip}{-0.65cm}
\setlength{\abovecaptionskip}{0.15cm}
\subfigbottomskip=1pt
\centering
\subfigure[\scriptsize Original member latents.]{
\includegraphics[width=0.4\linewidth]{member_origin_Latent.jpg}
}
\subfigure[\scriptsize Original non-member latents.]{
\includegraphics[width=0.46\linewidth]{nonmember_origin_Latent.jpg}
}
\subfigure[\scriptsize Purified member latents.]{
\includegraphics[width=0.4\linewidth]{member_purified_Latent.jpg}
}
\subfigure[\scriptsize Purified non-member latents.]{
\includegraphics[width=0.46\linewidth]{nonmember_purified_Latent.jpg}
}
\caption{The statistical distribution of latent vectors on the CIFAR10 dataset. Different colors stand for latent vectors with
different labels. (a) and (b) depict latent vectors of the original member
and non-member confidence score vectors;
(c) and (d) depicts latent vectors of member and non-member confidence
score vectors with {\sc Purifier}\xspace defended.}
\label{fig:latent_space_purifier}
\end{figure}
We present the statistical distribution of confidence score vectors in the encoder latent space of the \textit{confidence reformer}. Figure~\ref{fig:latent_space_purifier} visibly displays the differences on the CIFAR10 dataset between
members and non-members in
the latent space.
As illustrated in the first row, latent vectors of the
members tend to cluster together according to their labels, while those of
non-members are more scattered in the map.
The second row of Figure~\ref{fig:latent_space_purifier} also shows the statistical distribution of members and
non-members processed with {\sc Purifier}\xspace in the
latent space. When processed with {\sc Purifier}\xspace, Gaussian noises are added to the confidence score vectors, making the clustered member latent vectors to be more
scattered on the latent space.
This demonstrates that {\sc Purifier}\xspace can reduce the statistical differences
between members and non-members while preserving semantic utility.
\noindent \underline{\textbf{Label Indistinguishability of Purified
Confidence}}
{\sc Purifier}\xspace uses \textit{label swapper} to identify and swap the predicted labels of members.
\textit{label swapper} incurs
negligible reduction of test accuracy. At the same time, swapping the labels of the member samples reduces the training accuracy so that the gap between the accuracy of member and non-members is minimized.
This is shown in
Table~\ref{tb:bigtable}, where the
training accuracy of the model is close to the test accuracy. Many label-only
membership inference attacks are less effective under {\sc Purifier}\xspace with \textit{label swapper}.
This reflects that purified member confidence vectors
are less distinguishable from those of the non-members in terms of label.
\section{Experimental Setup}
\subsection{Datasets \& Models}
\noindent \textbf{Membership Inference Attack.}
We use CIFAR10~\cite{shokri_membership_2017, salem_ml-leaks_2018, li_membership_2020}, Purchase100~\cite{shokri_membership_2017, nasr_machine_2018, salem_ml-leaks_2018, li_membership_2020} and
FaceScrub530~\cite{yang_neural_2019} datasets
which are widely adopted in previous studies on membership
inference attacks.
\noindent \textbf{Model Inversion Attack}.
We use the same datasets as membership inference attacks.
\noindent \textbf{Attribute Inference Attack}.
We use the same dataset UTKFace~\cite{zhang2017age} as in a previous study~\cite{song2020overlearning} where the attacker infers additional attribute (i.e., race of five possible values) beyond the original gender classification task.
We attach the details of the datasets to Appendix, including the introduction, pre-processing and data allocation. We also further elucidate the target classifier and {\sc Purifier}\xspace on different datasets in the Appendix, including their model architectures and hyper-parameters.
\subsection{Existing Attacks}
In our experiments, we implement the following attacks.
\noindent \textbf{Membership Inference Attack.}
We implement the atttacks including \ding{172} {NSH attack}~\cite{nasr_machine_2018}, \ding{173} {Mlleaks attack}~\cite{salem_ml-leaks_2018}, \ding{174} {Adaptive attack}~\cite{salem_ml-leaks_2018} (where the attacker knows all the details about the defense mechanism), \ding{175} {BlindMI attack}~\cite{hui_practical_2021}, \ding{176} {Label-only attack}~\cite{yeom_privacy_2018,
li_membership_2020}. \textbf{Model Inversion Attack.}
The attacker trains an inversion model on $D_{aux}$ to perform the model inversion attack. The inversion model takes $F(\vec{x})$ as input and is able to reconstruct $\vec{x}$~\cite{yang_neural_2019}.
\textbf{Attribute Inference Attack.}
The attacker trains a classification on $D_{aux}$ to infer additional sensitive attribute beyond the original input attributes of the given sample~\cite{song2020overlearning}.
We attach the details of the attack above-mentioned methods, including their implementations, to the Appendix.
\subsection{Metrics}
We use the following 4 metrics to measure the utility, defense performance and efficiency of a defense method.
\noindent \ding{172}\textbf{Classification Accuracy:} It is measured on the training set and the test set of the target classifier.
\ding{173}\textbf{Inference Accuracy:} This is the classification accuracy of
the attacker's attack model in predicting the membership/sensitive attribute of input samples.
\ding{174}\textbf{Inversion Error:} Following~\cite{yang_neural_2019}, We measure the inversion error by computing the mean squared error between the original input sample and the reconstruction.
\ding{175}\textbf{Efficiency:} We measure the efficiency of a defense method by
reporting its training time and test time relative to the original time required by the target
classifier.
\section{Introduction}
Machine learning has been provided as a service by many platforms, transforming various aspects of our daily life such as handling users' sensitive data.
Users access these models through prediction APIs which return a confidence score vector or a label.
Many studies have indicated that the prediction information of a single sample could be exploited
to perform data inference attacks to get useful information about this sample on
which the machine learning model operates
~\cite{shokri_membership_2017, yang_neural_2019,song2020overlearning}.
Data inference attacks could be largely divided into two categories. The first kind of attacks aim at inferring distributional information about a class by observing the prediction changes of different samples~\cite{An.Mirror.NDSS.2022, csmia}, while the second kind of attacks are to infer the individual information of a sample by observing its specific prediction output such as the membership inference attacks~\cite{nasr_machine_2018,salem_ml-leaks_2018,hui_practical_2021,yeom_privacy_2018,li_membership_2020,li_membership_2021}, adversarial model inversion attacks~\cite{yang_neural_2019} and attribute inference attacks~\cite{song2020overlearning}. In this paper, we focus on the second type of data inference attacks.
Among these data inference attacks, \textit{membership inference attack}~\cite{shokri_membership_2017} is one of the most important and exemplary attacks.
In the membership inference attack, the adversary is asked to determine whether a
given data sample is in the target model's training data.
Many studies acknowledge that the confidence score vectors tell more prediction information beyond the label and thus they should be provided in the prediction results. Therefore, a number of approaches have been proposed to defend the membership inference attack while preserving the confidence scores~\cite{shokri_membership_2017, salem_ml-leaks_2018, nasr_machine_2018, abadi_deep_2016, jia_memguard_2019, selena}.
On the other hand, some studies believe that removing the confidence
information in the prediction result is a way of defending the membership
inference attack. However, these defenses are broken by label-only
attacks~\cite{yeom_privacy_2018, choo_label-only_2020a, li_membership_2021},
whereby only the predicted label is exploited to infer the membership.
The major cause of membership inference attack is that the prediction results are distinguishable for members and non-members.
For example, when a model overfits on the training data, it behaves more confidently on predicting the training data (members) than predicting the testing data (non-members).
The prediction differences between
members and non-members exist in their \textit{individual shape},
\textit{statistical distribution} and \textit{prediction label}.
(1) The target classifier often assigns a higher probability to the predicted class when given a member, making the confidence scores distinguishable in individual shape. This is exploited by many attacks~\cite{salem_ml-leaks_2018, nasr_machine_2018}
(2) Confidence scores on members and non-members are also distinguishable in their
statistical distribution. Our experiments show that confidence scores on the
members are more clustered in the encoded latent space, while those on
non-members are more scattered.
BlindMI~\cite{hui_practical_2021} exploits such statistical
difference to infer membership by comparing the
distance variation of the confidence scores of two generated datasets.
(3) In addition, the confidence scores on members and non-members are different in the prediction label. Member samples have a higher probability of being correctly classified than the non-member samples, which leads to the difference in classification accuracy. Various label-only attacks exploit such distinguishability~\cite{yeom_privacy_2018, li_membership_2020, li_membership_2021}.
In this paper, we propose a defense method, namely
{\sc Purifier}\xspace, against the membership inference attack.
The main idea is to directly reduce the distinguishability of confidence scores
and labels on members and non-members by transforming the confidence score
vectors of members as if they were predicted on non-members. It takes as input the
prediction produced by the target model and outputs a transformed
version.
First, we train {\sc Purifier}\xspace on the confidence score vectors predicted by the
target model on non-member data to reconstruct these vectors using a novel
training strategy. This encourages {\sc Purifier}\xspace to learn the individual shape of
these non-member confidence score vectors and eventually to generate confidence
score vectors as if they were drawn from the learned pattern, reducing distinguishability of confidence scores in
\textit{individual shape}.
Second, we use Conditional Variational
Auto-Encoder (CVAE) as a component of {\sc Purifier}\xspace to introduce Gaussian noises to the confidence
scores, such that the statistically clustered confidence scores can be scattered
and become indistinguishable from those on non-members, reducing distinguishability in \textit{statistical distribution}.
Third, to decrease the distinguishability in \textit{prediction labels},
{\sc Purifier}\xspace intentionally modifies the predicted labels of members while preserving those of non-members, which results in a reduction of classification accuracy gap between members and non-members.
Although {\sc Purifier}\xspace is designed to defend the membership inference attacks, it turns out to be also
effective in defending the \textit{adversarial model inversion attack} and the \textit{attribute inference attack}. In the adversarial model inversion attack, the adversary aims at inferring a reconstruction~\cite{yang_neural_2019, fredrikson_model_2015, hitaj_deep_2017} of the data sample.
In the attribute inference attack, the adversary could infer additional sensitive attribute beyond the original input attributes of this data sample~\cite{song2020overlearning}.
We believe that the purification process contributes to the removal of the
redundant information (hidden in the confidence scores) that is useful to
recover the input sample, and preserves only the essential semantic information
for the prediction task. As a result, the adversary can obtain no more useful
information than the prediction itself from the purified prediction results.
We extensively evaluate {\sc Purifier}\xspace on various benchmark datasets and model architectures.
We empirically show that {\sc Purifier}\xspace can defend data inference attacks effectively and efficiently with negligible utility loss.
{\sc Purifier}\xspace can reduce the membership inference accuracy. For example, the NSH attack~\cite{nasr_machine_2018} accuracy drops from 70.36\% to 51.71\% in our experiments, which is significantly more effective than previous defenses. {\sc Purifier}\xspace is also effective against adversarial model inversion attack. For instance,
the inversion loss on the FaceScrub530 dataset is raised 4+ times (i.e. from 0.0114 to 0.0454) after applying {\sc Purifier}\xspace.
Furthermore, {\sc Purifier}\xspace can reduce the attribute inference accuracy from 31.06\% to 20.94\% (almost random guessing) on one of evaluated datasets.
\textbf{Contributions.} In summary, we make the following contributions in this paper.
\begin{itemize}
\setlength\itemsep{0em}
\item To the best of our knowledge, our work is the first to study membership inference attacks comprehensively from the perspectives of \textit{individual shape},
\textit{statistical distribution} and \textit{prediction label}.
\item We design {\sc Purifier}\xspace to
defend against membership inference attacks by reducing the
distinguishability of the confidence scores in terms of the above three aspects with negligible utility loss.
{\sc Purifier}\xspace is shown to be also effective in defending other data inference attacks.
\item We extensively evaluate {\sc Purifier}\xspace and compare it with existing defenses. Our experimental results show that {\sc Purifier}\xspace outperforms existing defenses in both effectiveness and efficiency.
\end{itemize}
\section{Problem Statement}
We focus on classification models of neural
networks, i.e., a machine learning classifier $F$ is trained on its training dataset $D_{train}$ to map a given sample $\textbf{x}$ to a specific class based on the confidence vectors $F(\vec{x})$ which is the classifier output.
We consider the data inference attacks designed to infer useful information about a specific sample $\textbf{x}$ based on the target classifier's output $F(\vec{x})$, for examples, the membership inference attack, adversarial model inversion attack and attribute inference attack. We do not consider the data inference attacks~\cite{An.Mirror.NDSS.2022, csmia} which infer distributional information about a class through observing the output changes of $F$ on different $\vec{x}$ in this paper.
\vspace{-0.15cm}
$$
\vspace{-0.15cm}
\label{define}
\vec{x}, F, D_{aux} \xrightarrow{} \left\{useful\ information\ of\ \vec{x}\right\}
$$
We assume that the attacker has a black-box access to the classifier $F$, where the attacker can only query $F$ with its data sample $\vec{x}$ and obtain the prediction scores $F(\vec{x})$. We also assume that the attacker has an auxiliary dataset $D_{aux}$ to assist its attacks such as a set of data samples drawn from a similar data distribution as the target classifier's training data distribution.
\section{Discussion}
Assuming the reference data are considered as members, we present the inversion error and the inference accuracy
on the reference set for each defense and attach the results to the Appendix.
The Results show that
{\sc Purifier}\xspace can still preserve the defense effect against the adversarial model inversion attack and the membership inference attack.
We also investigate the effect of the {\sc Purifier}\xspace’s training data by using different in-distribution and out-of-distribution data
to train {\sc Purifier}\xspace. The results show that {\sc Purifier}\xspace can still mitigate the attacks, but at the cost of sacrificing the utility significantly when using out-of-distribution data. We attach the results to the Appendix.
Furthermore, we investigate the effectiveness of {\sc Purifier}\xspace to detect noisy members and attach the result to Appendix. It shows that {\sc Purifier}\xspace can accurately detect the members with noise $\left \| \eta \right \|_{\infty}<1e-10$ on FaceScrub530 dataset.
\section{Related Work}
\textbf{Data Inference Attacks.}
In data inference attacks, the attacker aims at inferring information about the data that the target model operates on.
Xiao et al.~\cite{xiao_adversarial_2019} studied the adversarial reconstruction problem.
They studied the prediction model which outputs 40 binary attributes.
Our paper, on the contrary, studies black-box classifiers whose output is constrained by a probability distribution.
Jia and Gong~\cite{jia_attriguard_2018} proposed the adversarial formulation for privacy protection. They aimed at protecting the privacy of users' sensitive attributes from being inferred from their public data. Our work investigates inference attacks that leverage prediction results of machine learning models to infer useful information about the input data.
\textbf{Secure \& Privacy-Preserving Machine Learning.}
A number of studies made use of trusted hardware and cryptographic computing to provide secure and privacy-preserving training and use of machine learning models.
These techniques include homomorphic encryption, garbled circuits and secure multi-party computation on private data~\cite{liu_oblivious_2017, bonawitz_practical_2017, phong_privacy-preserving_2018, dowlin_cryptonets_2016, mohassel_secureml_2017, dwork_privacy-preserving_2018} and secure computing using trusted hardware~\cite{ohrimenko_oblivious_2016, juvekar_gazelle_2018}.
Although these methods protect sensitive data from direct observation by the attacker, they do not prevent information leakage via the model computation. |
1,116,691,500,907 | arxiv | \section{Introduction}
Multi-access edge computing (MEC) is an emerging service network architecture to push powerful services to the proximity of end users, which includes not only the network services as proposed in the formerly mobile cloud computing, but also sensing, computing, storage and intelligence. Thus, service requests from customers could be flexibly provided at edge nodes, e.g., base stations (BSs), access points (APs), and roadside units (RSUs). Imagine that in a smart city envisioned in the last few years, if customized devices with powerful sensing, communications, computing, storage, and intelligence (SCCSI) capability are installed in or co-located with existing infrastructure (e.g., BSs, APs, RSUs, rooftops, lamp posts, etc) or vehicles (e.g., public transits, vehicles, etc), then we will have a powerful MEC system, consisting of mobile infrastructure to take advantage of both spectrum and mobility opportunity, to provide needed SCCSI services \cite{ding2017cognitive2}. Due to the omnipresence of vehicles (public or private) in a city, their mobility could easily reach the proximity of end users, i.e., the edge. Moreover, by shifting processing/computing power from the remote cloud to locally situated, possibly reconfigurable, edge servers (ESs), MEC helps significantly mitigate traffic congestion over backbone networks and reduce end-to-end (e2e) latency for many emerging capability demanding applications, enhancing QoE for end users. This vision can be easily illustrated in Figure \ref{fig:mec_app} where powerful ESs are populated in a smart city to build a robust service network for SCCSI services.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{smart_architecture4mec.eps}
\caption{A typical scenario for MEC.}\label{fig:mec_app}
\vspace*{-0.1in}
\end{figure}
Due to its tremendous potential to revolutionize the telecommunications and computing industries, there is a surge in MEC, particularly in wireless industries \cite{koketsurodrigues2021offloading}. The early concept of MEC was conceived at the early stage of wireless industries, particularly when mobile devices had limitations on resources on their own and there was a need to pull resources from mobile devices in close proximity together for sharing, leading to the cloudlet or the later mobile edge computing (also termed MEC). Following the developments of wireless mesh networks and cloud computing, where specialized powerful routers or computing servers were deployed at strategic locations to beef up the communications and computing at the edge, i.e., push communications and computing services closer to the end users, the true marriage of communications and computing started to emerge, leading to the birth of powerful MEC technologies. In light of integration of AI into future wireless technologies and real-time intelligent control, 5G systems and beyond (simply 5G+) have already taken edge computing capability into their design consideration, positioning it as the enabler to intelligentize the vertical industries.
Although there are extensive works on MEC done in the past few years, many open problems are still unresolved. The challenges lie in the fact that task offloading involves many design factors, not only computing resources, but also transmission capability. A task to be computed has to be successfully delivered to an ES and then computed within the timeline required by the specific applications. Besides, both transmissions and computing systems should have sufficient storage to buffer the task for resource scheduling. Thus, an effective MEC system has to optimally coordinate the distributed spectrum resource, computing resource, and storage resource under the constraints on power consumption and latency.
In this paper, we plan to articulate the problems and challenges in MEC, discuss the possible models and solution approaches, and then identify the future research directions. Different from some existing surveys on MEC which attempt to provide comprehensive summary of research efforts in this area~\cite{mao2017survey,abbas2017mobile, mach2017mobile}, this paper targets at design taxonomy, research problems, typical architectures, and potential solutions by jointly considering communication and computing resources from an end-to-end perspective. Moreover, we will offer enlightening thoughts on several key issues, including the use of queuing model and multi-hop offloading for MEC scenarios.
\begin{comment}
The remainder of this paper is organized as follows. In the next section, we present a few killer applications of MEC to further motivate the research. In Section III, we offer a general framework for MEC, fundamental problems and performance metrics. In Section IV, we discuss the key design issues based on a classification of system settings and modeling approaches, followed by a case study in Section V. In Section VI, we point out a few important research directions in MEC and conclude the paper in the last section.
\end{comment}
\section{Selected Killer Applications}
To demonstrate how important MEC is, in this section, we present an non-exhaustive list of important killer applications to demonstrate how MEC can be leveraged to boost both communications and computing performance for better users' QoE.
\vspace{-0.15in}
\subsection{Connected and Autonomous Driving}
The one on top of attractive applications for MEC should be connected and autonomous driving (CAD). CAD does need near real-time map for collision avoidance and pedestrian safety, thus the simultaneous localization and mapping (SLAM) is crucially important. Yet, SLAM demands timely sensing (e.g., video) and fast computing capability to perform fine-grained video analytics, which can hardly be done by connected and autonomous vehicles (CAVs) on time. Besides, any highly complex computing and processing shifted from CAVs to the edge could significantly reduce the cost of CAVs. One may argue that such tasks can be offloaded to the cloud, but yet the latency may pose serious problem for CAD. Thus, the only alternative is an effective MEC.
\vspace{-0.15in}
\subsection{Industrial Internet of Things}
Many real-time applications such as industrial Internet of Things (IoT) do require low latency for real-time scheduling and control. For example, future smart manufacturing may demand timely sensing to collect much needed data, deliver them to a certain computing facility to carry out necessary computing for intelligence extraction, and then take timely control actions.
Due to the tight latency requirement, MEC seems to be the only choice.
\vspace{-0.15in}
\subsection{Video Surveillance}
To build a smart city, public safety is one of the important design goals. To this end, video surveillance cameras may be deployed. Due to large volume of video data, it is impractical to upload all the raw data to the cloud to carry out video analysis. More often timely video analytics may have to be done within certain time frame in order to fight possible crimes in tough neighborhoods. This is exactly what MEC could come to rescue.
\vspace{-0.15in}
\subsection{Smart and Connected Health}
Information and communications technologies (ICT) and artificial intelligence (AI) have revolutionized what we care for people's health, and smart and connected health (SCH) is a great effort towards this goal. For people who have chronic or life critical diseases, SCH could help collect vital signals under 24/7 clock, perform timely signal analysis to monitor potential changes in signal pattern by applying sophisticated AI algorithms such as machine learning (ML), deep learning (DL), and federated learning (FL), etc. Due to this periodic signal collection, huge volume of data will be generated, resulting in big data to be handled, and it may not be necessary to always interact with the cloud, but yet timely data analytics may have to be done locally. Thus, MEC provides another rich application to address important problems.
\section{General MEC Framework}
With such diverse applications of MEC, the problem is how to design such an MEC system to suit a practical application. In this section, we first present a general architectural design guideline for MEC, and then identify the fundamental design issues under the MEC framework to improve users' QoE. Afterwards we discuss three critical metrics to measure the performance of MEC.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{mec_architecture.eps}
\centering
\caption{The generic architecture of an MEC.}\label{fig:mec}
\vspace*{-0.1in}
\end{figure}
\vspace{-0.15in}
\subsection{Architectural Design for MEC}
An MEC consists of the cloud, edge servers (ESs) and mobile devices (MDs) as shown in Figure \ref{fig:mec}. A set of ESs are populated in the area of service (AoS) while MDs roaming in the AoS could offload their tasks to ESs for computing (for signal processing or machine learning). An ES is typically customized with powerful communications capability (e.g., cognitive radios), high computing power and sufficient storage space, running various AI algorithms, which is typically termed {\em CCSI} capability. If it is also equipped with internal or external sensing features or interconnected with sensors in its proximity, we could enable MEC the SCCSI capability. An MD could access the computing, storage and intelligence resource at its proximity if reliable communication link(s) can be established to an ES (or a subset of ESs). ESs can be connected via high-speed links (solid lines) and/or wireless links (dashed line), depending on the deployment strategy for ESs.
\vspace{-0.15in}
\subsection{Fundamental Problems}
The fundamental problem in MEC is how to facilitate MDs to access the services at ESs with desired quality of service (QoS). This mainly includes the following few design issues.
\textbf{Computation Offloading}. An MD could offload computing task either wholly or partially, depending on both local computing resource in the MD and the channel condition from the MD to an ES(s) it could reach. However, this problem is highly challenging since both the channel condition and computing resource (at both the MD and the chosen ES(s)) should be taken into consideration. Moreover, partial offloading decisions involve additional constraints, e.g., whether the tasks are allowed to be split into variable ratio or not, which is largely ignored in most existing works.
\textbf{Resource Allocation}. After task offloading decision is made, the next design task is to allocate enough communication resource between an MD and selected ESs and sufficient computing power at both the MD and selected ES(s). An efficient resource allocation has to be designed due to the limitation of resources in MEC.
\textbf{Caching and Prefetching}. Although storage can be regarded as a type of resource at ESs, due to its importance, we deal with it separately from resource allocation when caching and prefetching are considered. Caching/prefetching at the edge (or edge caching) is different from the traditional caching in that the goal here is to enable corresponding computing tasks to be executed efficiently at ESs and is often jointly designed with task offloading to maximize MEC performance. Moreover, computing at the edge typically relies on the placement of non-trivial amount of data for training and ML models for edge intelligence while traditional caching works mostly for content distribution. Thus, how to design effective caching or prefetching strategies in MEC system is significantly different from traditional approach and highly challenging.
\textbf{Energy Conservation}. Since MDs tend to be battery powered, energy conservation is always an important design consideration. The energy consumption comes from both transmissions and computing. A computing task can be done locally at an MD or offloaded to an ES or jointly done at both sides in order to conserve energy on the MD.
\textbf{Mobility Management}. Many network entities such as MDs or mobile ESs may be subject to mobility, which will result in degraded communications quality or longer e2e latency. How to manage the mobility to sustain continuous communications services or effective computing is an important design issue. To keep the task computing ESs closer to the end users on the move, mobility management (e.g., user tracking) and task computing migration (e.g., virtual machine migration) must be taken into consideration in practical applications \cite{yuan2020joint}.
\vspace{-0.15in}
\subsection{Performance Metrics}
To assess the performance of MEC, we typically need some key performance indices (KPIs). In this subsection, we present three essential performance metrics for the aforementioned problems from the customers' perspective (i.e., latency and energy consumption at MDs) and service providers' perspective (e.g., task completion rate or simply throughput), respectively.
\textbf{Latency}. MEC targets at providing timely access to edge services, and hence latency is definitely the first metric to guarantee. Latency here means that the time between the instant that an MD submits its computing task to the instant it obtains the computing result of the task. Since this definition is from the perspective of an MD, it is indeed the end-to-end (e2e) latency. Since the MD may make a decision based on the result of the completed task within certain time limit, latency is indeed an important metric.
\textbf{Energy Consumption}. Energy consumption either due to task transmissions or due to task computation locally at an MD is always a major concern for resource-constrained systems. Since many end users tend to use their smart devices for the applications of their interests, it is always good idea to save the battery life of their MDs. Thus, the total energy consumption at an MD for a task completion under a task offloading scheme is another metric to be considered.
\textbf{Throughput (task completion ratio)}. To maximize the revenue, a service provider for an MEC system tends to utilize system resources as efficiently as possible to accommodate customers to maximize the completed tasks, which indicates that the system throughput (or task completion ratio) is a critical metric for MEC. To boost the throughput, load balancing across multiple ESs is an effective way since it improves service availability by spreading the workload based on load status in terms of both communication and computing. To optimize the aggregated throughput and users' QoE, MD's tasks have to be transferred to the edge server with lower computing load via less congested links.
\section{Resource Allocation and Load Balancing: An End-to-End Perspective}
With all basic concepts clarified, based on the current research activities, we are now ready to categorize major research problems on MEC. In this paper, we focus on two major research tasks in MEC, namely, resource allocation and load balancing. Resource allocation tends to address how to allocate resources for end users from one service facility (e.g., an ES) while load balancing focuses on how to balance the workloads over multiple service facilities (e.g., ESs). Typically, depending on specific applications, resource can be different. Although our envisioned MEC consists of SCCSI services, in this paper, we focus mainly on communications and computing. We will articulate the research problems involving with spectrum allocation, computing offloading, and the related load balancing under different system setup settings.
\vspace{-0.15in}
\subsection{Single-Edge and Single-MD Case (SESM)}
We start with the simplest case first to articulate the problem. For an MEC system with SESM, the problem is whether to offload computing from the mobile user device (MD) partially or wholly to the edge server (simply ES). The decision depends on various factors of concern. If we are concerned with the MD's energy conservation, we may compare the energy consumption on data offloading (depending on spectrum availability and channel conditions between MD and ES) and the computing power consumption at MD if computed locally. This problem can also be formulated as partial computing offloading problem by dividing the whole computing task into two parts to be performed at MD and ES separately if a task is divisible. If a computing task is latency sensitive, then the problem can be formulated as MD energy minimization under e2e latency constraint. In \cite{han2020joint2}, we formulated a joint optimization problem to minimize long-term power consumption at the MD under resource constraints on communications and computing and applied Lyapunov optimization to obtain approximate solutions.
If latency is not a deciding factor, e.g., for delay-tolerant computing, a computing task can be queued either at MD waiting for enough spectrum and/or better channel condition for opportunistic offloading or queued at an ES for sufficient computing resource to perform opportunistic computing.
\vspace{-0.15in}
\subsection{Single-Edge and Multi-MD Case (SEMM)}
This is probably the most common scenario investigated in the current literature. Under this scenario, we have only one single server to provide computing services and have a single shared spectrum to provide offloading for all MDs. If too many users attempt to offload, it may not be worthwhile for an MD to offload its task to the ES due to either spectrum shortage or computing overload at the ES, because this may cause excessive energy consumption when uploading the task or intolerable latency due to communication/computing queueing delay. Thus, the problem has become more complicated. Due to the competing nature for both spectrum and computing, energy minimization and latency minimization for task computing should be formulated carefully. The coupling of task offloading and computing may impact the final decision on whether task computing is carried out either locally or at ES.
If queueing is allowed, various queueing models can be developed for conserving energy on MDs or reducing latency for task computing. One MD may choose to opportunistically offload tasks only when channel to the ES is good or choose to offload only when the computing work load at ES is light.
\vspace{-0.15in}
\subsection{Multi-Edge and Single-User Case}
This scenario is not really practical, but may be useful to capture certain essential features of practical problems. It may be used to model the decision process when an MD decide where to offload its computing task, given the MD could offload its task to a subset of ESs. The set of problems for this case can be classified into the following cases. i) Given the channel conditions and computing load on each ES, minimize the MD's energy consumption under latency constraint; ii) Given the channel conditions, computing load on each ES, and residual energy in MD, split computing tasks into local computing and edge computing to maximize the task throughput under latency constraint and MD energy constraint; and iii) Given the channel conditions and computing load on each ES, maximize the task throughput under latency constraint. With multiple edge servers available, the computing tasks from the MD have to consider the load balancing in terms of both communications and computing. In \cite{han2019offloading}, we investigated this problem and identified the bottleneck in the offloading process.
\vspace{-0.15in}
\subsection{Multi-Edge and Multi-User Case}
This is probably the most general scenario that we typically encounter in MEC systems. In the nutshell, the problem at hand is to match the computing tasks with appropriate ESs to optimize the desired design objective, given the channel conditions between MDs and ESs and the computing work loads on ESs. The design objective can be either minimizing MD energy consumption or maximizing the task throughput, both under task latency constraint. Some researchers have also proposed to minimize latency under local power constraints. However, it is noted that minimizing latency may not generate enough incentive for MEC operators as end users will be satisfied and pay the same bills as long as the latency requirement is met. Many papers addressed simplified version of the aforementioned problem: given a set of computing tasks to be offloaded, match the ESs under the latency constraints, which has not taken the task arrival uncertainty into consideration, and hence is not practical at all. How to formulate the right set of optimization problems is of paramount importance, yet highly challenging, particularly under stochasticity in task arrivals, spectrum availability, and computing workload.
Besides, most papers focus on offloading via one-hop transmissions between MDs and ESs. Recently, we have started investigating whether we could leverage multi-hop transmissions to balance the computing workload at ESs and transmission workload from MDs to ESs while meeting the latency constraints. In this subsection, we elaborate research activities based on whether offloading is done via single-hop or multi-hop.
\subsubsection{Single-hop offloading}
Most current research activities focus on single-hop offloading in the sense that an end user is seeking a powerful edge in its proximity and offload its computing tasks directly \cite{poularakis2020service}.
Some works assume that computing resources at ESs are sufficient and formulate the optimization problem either to maximize the sum throughput or to minimize the energy consumption at MDs. Some other works assume that communication resources are sufficient and formulate the optimization problem to balance the computing workload at ESs. Both scenarios, although reducing the problems significantly, are not really practical.
In \cite{poularakis2020service}, Poularakis \textit{et al. } studied probably the most general problem to minimize the workload offloaded to the cloud under constraints on communications, computing and storage at ESs, but did not taking the queueing model into consideration. Recently, we formulate the optimization problem to maximize the task completion rate (throughput) by balancing the workloads for both communications and computing under e2e latency constraints \cite{deng2021throughput2}. The intuitive idea is motivated by the observation that when we offload a task for computation, we should not only offload it to an ES that does have sufficient computing resource to complete the task, but more importantly should have enough communications resource to upload it in the first place in order to complete it. Thus, in \cite{deng2021throughput2}, under certain assumption on the stochasticity of involved processes (task arrivals, random channels and varying computing power), we use the tandem queue to model the joint system of communications and computing, and find the (approximate) solution, which represents significant departure from the main stream research.
\subsubsection{Multi-hop offloading}
Although it is much easier to formulate the optimization problem under one-hop offloading, in reality, due to the locality of end users and their running applications, task offloading may be difficult to achieve due to heterogeneity in the availability in spectrum and computing resource at different location and at different time. Some ESs may have abundant computing resource, but do not have enough spectrum resource at their proximity to receive tasks, while some of them do have sufficient spectrum resource to receive tasks, but do not have enough computing power. Yet, task completion does demand both sufficient spectrum resource and computing power. Thus, if there are ways such as emerging omnipresent vehicular communications systems to relay tasks from congested location to less congested area via multi-hop relays, it could potentially boost the task completion rate (throughput). For example, in~\cite{deng2021leverage}, we have proposed to leverage vehicles to transport tasks from congested areas to less congested areas to significantly increase the task throughput. Data transportation can of course be carried out either through store-carry-forward mechanisms or via V2V relays over (opportunistic) vehicular communications, which leverages both the mobility opportunity and the spectrum opportunity.
One special scenario is that all ESs in a specific area are interconnected via high-speed reliable links like cable or optical fibers so that the workload in the network of ESs can be balanced directly among them. In this case, the aforementioned task offloading problem can be boiled down to the load balance problem over spectrum resource via relaying nodes, that is, tasks can be uploaded to any ES. Thus, the problem to be solved is to design anycast routing schemes over potentially wireless ad hoc networks with latency constraints.
To conserve MDs' energy under multi-hop task offloading strategies, end users could search for agents, e.g., nearby powerful relaying nodes, to help relay their tasks to be computed at appropriate ESs. This would lead to another rich set of optimization problems for MEC task offloading.
\subsubsection{Queueing Consideration}
It is typically observed that many prior works have not really taken specific queuing models into the optimization to leverage the opportunistic communications and computing~\cite{poularakis2020service}. We recently utilize the tandem queue models to capture the stochastic nature of task arrivals and service (transmission time and computing time)~\cite{deng2021throughput2,deng2021leverage}. Queuing scheduling is unavoidable in any service systems which can be utilized to increase the resource efficiency and enhance QoE. This is particularly true for joint distributed communications and computing systems for task offloading which can be better optimized by leveraging both mobility opportunity and spectrum opportunity as in vehicular MEC. Unfortunately, the complexity becomes overwhelming when queuing models are taken into consideration for task offloading in MEC, which demands further investigation.
\begin{comment}
\section{Case Study}
Due to page limit, in this section, we only present one case study to demonstrate the useful applications of MEC to illustrate the benefits.
As in \cite{deng2021throughput2}, we consider a multi-user MEC system with multiple ESs, where a set $\mathcal{U}=\{1,2,\cdot\cdot\cdot,U\}$ mobile devices (MDs) are randomly distributed and potentially moving around while a set $\mathcal{J}=\{1,2,\cdot\cdot\cdot,J\}$ of ESs are fixed in their locations and equipped with multi-core CPUs. Moreover, we develop a discrete-time model to capture the dynamic MEC system, i.e., the whole time $T$ is divided into $t$ time-slots, $t \in \mathcal{T}=\{1,2,\cdot\cdot\cdot,T\}$.
\textit{Communication-computing tandem queue model.} To characterize the joint impact of communication and computing on MEC, we construct a communication-computing tandem queue for resource allocation with the dynamics:
\begin{equation}\label{relationship}
\left\{
\begin{array}{lr}
{Q}_i(t+1)=\max\{{Q}_i(t)-r_i(\boldsymbol p, \boldsymbol \eta, t+1),0\} + a_i(t+1), & \\
{K}_{ij}(t+1)=\max\{{K}_{ij}(t)-c_{ij}(t+1), 0\} + r_i(\boldsymbol p, \boldsymbol \eta, t+1),
\end{array}
\right. \nonumber
\end{equation}
where ${Q}_i(t)$ and ${K}_{ij}(t)$ represent the queue lengths at the $i$-th MD and the ES $j$ at the end of the $t$-th time-slot, respectively; $r_i(\boldsymbol p, \boldsymbol \eta, t)$ and $c_{ij}(t)$ respectively denote the transmission rate and computing rate for MD $i$, where $\boldsymbol p$ and $\boldsymbol \eta$ are the set of transmit powers and the set of user association variables, respectively; and $a_i(t)$ is the number of task arrivals at MD $i$, at the beginning of time-slot $t$.
\textit{Throughput optimization based on Markov decision process.} While MEC has shown its huge potential to enhance application performance, the resource limits at the edge have become the bottleneck as exponentially increasing number of users scramble for MEC service, which is largely ignored in the literature. In \cite{deng2021throughput}, we formulated an optimization problem to maximize the aggregated throughput under the constraints on both spectrum and computing resources while meeting the QoS requirements. We utilized the Markov decision process (MDP) to facilitate the throughput maximization at ESs, where all tandem queue lengths and channel states form the system states while the allocation of transmission power and computing capacities are regarded as the actions.
As illustrated in Fig. \ref{fig:setting}, we evaluate the performance of the proposed MDP-based mechanism under the setting where there are 4 evenly distributed ESs and 40 randomly distributed MDs in an area of $100\times100 \;m^2$. In Fig. \ref{fig:result}, we compare the performance of \textit{MDP-based algorithm} with \textit{computation-based algorithm} (which only considers the computing resource allocation at ESs), \textit{transmission-based algorithm} (which only considers the communication resource allocation at the MDs), and \textit{Lyapunov-based algorithm} (which adopts the Lyapunov optimization in the proposed system model). We observe that, as the task arrival rate increases, the joint resource allocation of both transmission and computing at both MDs and ESs significantly outperform all other algorithms.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{evaluation_setting4case1.eps}
\centering
\caption{Simulation setting with $40$ mobile users and $4$ MEC servers in an area of $100\times100 \;m^2$ \cite{deng2021throughput}}\label{fig:setting}
\vspace*{-0.1in}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{evaluation_result4case1.eps}
\centering
\caption{The average throughput vs. the task arrival rate \cite{deng2021throughput}}\label{fig:result}
\vspace*{-0.1in}
\end{figure}
\end{comment}
\section{Future Research Directions}
Although we have already mentioned some research challenges and possible solutions under various MEC system settings, there still exist many challenging research problems in this area. Here we only list some important research directions we have identified that are urgently needed to be resolved.
\vspace{-0.15in}
\subsection{Place Edge Servers for Effective MEC Design}
Depending on the cost in response to the coverage requirement, ESs have to be appropriately placed in the service area of interest. The closely related research is on placement of ESs aiming at minimizing the access latency from MDs to ESs or energy consumption, where communication-related service quality metrics have been used to address the coverage issue~\cite{zhang2021joint}. However, in terms of computing, how to quantify the service coverage from an end-to-end perspective is still unclear. One possible solution is to consider some use cases such as smart city applications, and formulate the optimization problem to minimize the computing service outage (i.e., the probability that a computing task cannot be completed within the latency requirement).
\vspace{-0.15in}
\subsection{Take Queuing Models into Account}
To simplify the modeling process or reduce the complexity in optimization in MEC systems, queue models for task uploading and computing are typically ignored and even if they are considered, they are just simplified versions, which may not be practical. In reality, the performance highly depends on how a task transmission is done and/or how its computing is scheduled, which is particularly true when leveraging geographical differences of users and ESs as in multi-hop offloading scenarios \cite{deng2021leverage}. Therefore, how to incorporate the appropriate queuing models into optimization is an important yet challenging research task.
\vspace{-0.15in}
\subsection{Leverage In-Network Resources}
Current wireless networking environments are now equipped with rich powerful SCCSI capability which could be leveraged to fulfill needed service provisioning. For example, many today's vehicles, particularly the emerging connected and autonomous vehicles (CAVs), are installed with powerful computing and storage units and sensors, but they are not intended for communications and computing services for external applications. Yet, their omnipresence in smart cities could definitely help provide additional MEC services \cite{ding2019beef2}. How to take advantage of such in-network resources will form a fruitful future research direction.
\vspace{-0.15in}
\subsection{Take Advantage of Mobility and Spectrum Opportunities}
Many powerful communications and computing devices with more rich storage tend to be mobile due to shared mobility, user mobility, or things' mobility in general. For example, public transits carry users with SCCSI capable devices or they themselves are equipped with powerful SCCSI capability, and hence their routine or planned routes would enable them to regularly carry such SCCSI services, a rich set of MEC services that can be leveraged. Moreover, wireless spectrum is location-based, and their utility is shown to be opportunistic spatially and temporally \cite{ding2017cognitive2}. Thus, for an effective MEC system to be designed with specific missions, it is possible to leverage both spectrum and mobility opportunities to boost its service performance without adding extra infrastructure and/or resource. If user owned equipments are used, crowdsourcing and the corresponding incentive mechanism design can be incorporated into this design.
\vspace{-0.15in}
\subsection{Utilize Machine Learning to Improve MEC}
Machine learning (ML) has emerged as the panacea to solve many difficult engineering problems which cannot be resolved easily with traditional mathematical tools. Since computing at edge via an MEC system involves with factors such as computing load, transmission environments and network load, user device capability, and complicated queue scheduling for both communications and computing, traditional mathematical modeling based on first-principles will result in highly complicated optimization problem with high dimension (i.e., the curse of dimensionality). ML may offer certain level of model reduction by capturing only the important features according to the specificity of the involved applications \cite{rodrigues2019machine}. Moreover, for cases that traditional modeling is hard to handle, model-free ML may be evoked to find an approximate model (data-driven approach). For example, the multi-hop offloading problem may be solved via reinforcement learning, which is currently under investigation.
\vspace{-0.15in}
\subsection{Utilize MEC to Improve Machine Learning}
ML typically consumes intensive computing resource, particularly for real-time or delay-sensitive tasks as in connected and autonomous driving, which is why MEC was introduced \cite{rodrigues2019machine}. How to build an effective MEC system for AI at edge (or edge intelligence) to facilitate tremendous IoT applications and smart city initiatives is of paramount importance. Under this design, edge offloading (from MDs to ESs or among ESs) potentially plays a crucial role in providing data input for edge caching, edge training, and edge inference according to the dynamic resources in the MEC systems~\cite{xu2021edge}. By combining MEC with edge intelligence, we could investigate how to leverage MEC to boost the federated learning, which is still in its infancy and demands further research.
\vspace{-0.15in}
\subsection{Develop an Effective Holistic MEC Service Ecosystem}
Although MEC was conceived in response to the edge computing, it turns out MEC as a service system can be leveraged to enable significantly more services beyond computing as we envisioned in \cite{ding2019beef2}. If being built well, MEC can be used to perform sensing and data collection, push delay-tolerant data to the edge to facilitate fast communications, cache popular content of large volume for content distribution, provide temporary buffer/storage for effective transmission scheduling and computing, conserve energy for battery-driven devices, and extract intelligence at edge or at premises where intelligence is needed. This is true for smart city initiatives where omnipresent SCCSI capability is needed for smart city operations, particularly where capability-enabled vehicles, public or private, can be utilized to provide utility-like services \cite{ding2019beef2}. Thus, the future design for MEC should focus on the development of holistic utility-like service ecosystem to enable future intelligentization of life sustaining systems to improve people's quality of life (QoL). This low-cost MEC ecosystem may help ease the digital divide and provide remote education and training services.
\section{Concluding Remarks}
Multi-access edge computing (MEC) provides an effective approach to push computing rich cloud services to the proximity of end users, the so-called edge, in order to facilitate emerging latency-sensitive applications like connected and autonomous driving, smart and connected health, and timely edge intelligence services. Although there exist tremendous research on MEC in the past few years, there are still many design challenges as many interesting applications emerge. In this paper, we attempt to clarify some critical concepts in MEC and realistic design issues, elaborate the important research issues, and identify future research directions. Our focus is much more on the problem identification and optimization formulation from the technical perspective. It is our high hope that this paper could help readers quickly get familiar with the research critical problems and make significant contributions to solve the important problems in this area.
\section*{Acknowledgements}
The work of X. Chen, G. Zhu and Y. Fang was supported in part by US National Science Foundation under IIS-1722791 and CNS-2106589. The work of X. Deng was supported in part by National Natural Science Foundation of China (No. 62172441, 61772553).
\renewcommand\refname{References}
\bibliographystyle{IEEEtran}
|
1,116,691,500,908 | arxiv | \section{Introduction}
The success of the Standard Model (SM) has been confirmed by the LHC and it implies the existence of a mass gap between SM fields and possible new-physics (NP) states. Therefore, effective field theories (EFTs) are very convenient in order to study systematically low-energy data to search for traces of NP scales, the so-called bottom-up approach. Whereas the low-energy constants (LECs) of these EFTs are free-parameters and contain information about heavy scales, the structure of the effective Lagrangian relies on the particle content (SM particles here), the symmetries and the power counting. The symmetries and the power counting depend on the mechanism to introduce the Higgs. Here we follow the most general case, the electroweak (EW) effective theory (EWET): the non-linear realization of the electroweak symmetry breaking is taken and, consequently, no assumptions are made about the relation between the Higgs and the EW Goldstones and a ``chiral'' (generalized momenta) expansion is observed. It is worth noting that the so-called SM effective field theory (SMEFT) is a particular case of the EWET. At higher energies possible heavy resonances are included by following the same scheme, {\it i.e.}, same symmetries and power expansion.
Accordingly, we work with two EFTs: the EWET at long distances, which includes only the SM fields, and an EW resonance theory at shorter distances, which contains in addition heavy resonances. After integrating out the resonances both Lagrangians can be matched and the EWET LECs are determined in terms of resonance parameters. Assuming a good high-energy behavior in the resonance theory allows us to imagine this effective Lagrangian as a bridge between the EWET and the unknown underlying theory and also to reduce the number of unknown resonance parameters. Note that the main focus of this work is to constrain NP scales by using current experimental bounds on EWET LECs and our determinations of these LECs in terms of only a few resonance parameters~\cite{PRD,Pich:2015kwa}.
\section{Theoretical framework}
The EWET Lagrangian is organized following an expansion in generalized momenta~\cite{Weinberg,Buchalla,lagrangian}:
\begin{eqnarray}
\mathcal{L}_{\mathrm{EWET}} &=& \sum_{\hat d\ge 2}\, \mathcal{L}_{\mathrm{EWET}}^{(\hat d)}\,, \label{EWET-Lagrangian0}
\end{eqnarray}
where the chiral dimension $\hat d$ indicates the infrared behavior at low momenta~\cite{Weinberg}. The building blocks and their related power-counting rules can be found in Refs.~\cite{PRD,lagrangian}. In view of the current experimental information, the relevant bosonic part of the LO EWET Lagrangian reads
\begin{eqnarray}
\Delta \mathcal{L}_{\mathrm{EWET}}^{(2)} & =& \frac{v^2}{4}\,\left( 1 +\frac{2\,\kappa_W}{v} h + \frac{c_{2V}}{v^2} \,h^2\right) \langle \, u_\mu u^\mu\,\rangle \, , \label{LOEWET_lagrangian}
\end{eqnarray}
where $h$ is the Higgs field, $u_\mu$ the tensor containing one covariant derivative of the EW Goldstones and $\langle\cdots\rangle$ indicate an $SU(2)$ trace. $\kappa_W$ and $c_{2V}$ parametrize the $hWW$ and $hhWW$ couplings, respectively. Taking into account again the available experimental information, the relevant part of the purely bosonic NLO EWET Lagrangian is given by~\cite{lagrangian}:
\begin{equation}
\Delta \mathcal{L}_{\mathrm{EWET}}^{(4)} =
\Frac{\mathcal{F}_1}{4}\,\langle \, {f}_+^{\mu\nu} {f}_{+\, \mu\nu}- {f}_-^{\mu\nu} {f}_{-\, \mu\nu}\,\rangle +
\Frac{i\, \mathcal{F}_3 }{2} \,\langle \, {f}_+^{\mu\nu} [u_\mu, u_\nu] \,\rangle +
\mathcal{F}_4 \langle \, u_\mu u_\nu\,\rangle \, \langle \, u^\mu u^\nu\,\rangle +
\mathcal{F}_5 \langle \, u_\mu u^\mu\,\rangle\, \langle \, u_\nu u^\nu\,\rangle ,
\label{EWET_lagrangian}
\end{equation}
where $f^{\mu\nu}_\pm$ introduce the field-strength tensors. In the SM, $\kappa_W=c_{2V}=1$ and $\mathcal{F}_{1,3,4,5}=0$. $\mathcal{F}_1$, $\mathcal{F}_{1,3}$ and $\mathcal{F}_{1,3-5}$ contribute, respectively, to the $S$ oblique parameter, the trilinear and the quartic gauge couplings.
Since we are interested only in the resonance contributions to the bosonic ${\cal O}(p^4)$ EWET LECs, only ${\cal O}(p^2)$ operators with up to one bosonic resonance $R$ are required~\cite{lagrangian}. The relevant $CP$-even resonance Lagrangian reads~\cite{PRD,lagrangian}:
\begin{align}
\qquad \Delta \mathcal{L}_{\mathrm{RT}} \,=\,&
\frac{v^2}{4}\!\left(\! 1 \!+\!\frac{2\,\kappa_W}{v} h \!+\! c_{2V} \,h^2\!\right)\! \langle \, u_\mu u^\mu\,\rangle
+\langle \, V_{\mu\nu} \left( \frac{F_V}{2\sqrt{2}} f_+^{\mu\nu} + \frac{i G_V}{2\sqrt{2}} [u^\mu, u^\nu] + \frac{\widetilde{F}_V }{2\sqrt{2}} f_-^{\mu\nu} \right) \,\rangle
\phantom{\bigg( }
\nonumber \\
& \quad
+ \langle \, A_{\,\mu\nu} \left(\frac{F_A}{2\sqrt{2}} f_-^{\mu\nu} + \frac{\widetilde{F}_A}{2\sqrt{2}} f_+^{\mu\nu} + \frac{i \widetilde{G}_A}{2\sqrt{2}} [u^{\mu}, u^{\nu}] \right) \,\rangle
+ \frac{c_{d}}{\sqrt{2}}\, S_1\langle \, u_\mu u^\mu \,\rangle \,.
\label{Lagrangian}
\end{align}
By integrating out the resonance fields in (\ref{Lagrangian}), the EWET Lagrangian of (\ref{EWET_lagrangian}) is recovered and, consequently, the EWET LECs are determined in terms of resonance parameters~\cite{PRD,lagrangian},
\begin{align}
\mathcal{F}_1 &= - \Frac{F_V^2-\widetilde{F}_V^2}{4M_{V}^2} + \Frac{F_A^2-\widetilde{F}_A^2}{4M_{A}^2}\,, &
\mathcal{F}_3 &= - \Frac{F_VG_V}{2M_{V}^2} - \Frac{\widetilde{F}_A\widetilde{G}_A}{2M_{A}^2}\,, \nonumber \\
\mathcal{F}_4 &= \Frac{G_V^2}{4M_{V}^2} + \Frac{{\widetilde{G}_A}^2}{4M_{A}^2} \,,&
\mathcal{F}_5&= \Frac{c_{d}^2}{4M_{S_1}^2}-\Frac{G_V^2}{4M_{V}^2} - \Frac{{\widetilde{G}_A}^2}{4M_{A}^2}\,. \label{integration}
\end{align}
\begin{figure*}[!t]
\begin{center}
\begin{minipage}[c]{7.0cm}
\includegraphics[width=7.0cm]{plotF1_newbis.pdf}
\end{minipage}
\hskip .5cm
\begin{minipage}[c]{7.0cm}
\includegraphics[width=7.0cm]{plotF3_new.pdf}
\end{minipage}
\\[8pt]
\begin{minipage}[c]{7.0cm}
\includegraphics[width=7.0cm]{plotF4_new.pdf}
\end{minipage}
\hskip .2cm
\begin{minipage}[c]{7.0cm}
\includegraphics[width=7.0cm]{plotF45_new.pdf}
\end{minipage}
\end{center}
\caption{{\small $\mathcal{F}_1$, $\mathcal{F}_3$, $\mathcal{F}_4$ and $\mathcal{F}_4+\mathcal{F}_5$ as functions of the corresponding resonance mass ($M_V$, $M_A$ or $M_{S_1^1} \,v/c_d$)~\cite{PRD}.
The 95\% CL experimentally-allowed regions given in Table~\ref{exp} are covered by the green areas, and they are further extended by dashed green bands accounting for our estimation of the one-loop running uncertainties. If the prediction of the LECs depend on both $M_V$ and $M_A$, the gray and/or brown regions cover all possible values for $M_A>M_V$. If the 2nd WSR has been assumed, we indicate it explicitly in the plot, with the corresponding lines for the limit $M_A=M_V$ (orange), $M_A=1.1\,M_V$ (blue), $M_A=1.2\,M_V$ (red) and $M_A\to \infty$ (dark gray). In the case without the 2nd WSR, the prediction for $\mathcal{F}_1$ is given by the gray and brown regions. In case of using only the even-parity operators, it is indicated.
}}
\label{plots1}
\end{figure*}
As stressed previously, high-energy constraints allow us to reduce the number of resonance parameters. For the case at hand, the following short-distance constraints have been used~\cite{PRD,lagrangian}:
\begin{enumerate}
\item Well-behaved two-Goldstone vector form factor (VFF). Assuming that this form factor vanishes at high energies, the following constraint is found:
\begin{equation}
v^2\, -\,F_V\,G_V\, -\,\widetilde{F}_A\,\widetilde{G}_A \,=\, 0 \,. \label{VFF}
\end{equation}
\item Weinberg Sum Rules (WSRs). The $W^3B$ correlator is an order parameter of the EWSB. In asymptotically-free gauge theories it vanishes at short distances as $1/s^3$, implying two superconvergent sum rules~\cite{WSR}: the 1st WSR (vanishing of the $1/s$ term) and the 2nd WSR (vanishing of the $1/s^2$ term). While the 1st WSR is supposed to be also fulfilled in gauge theories with nontrivial ultraviolet (UV) fixed points, the validity of the 2nd WSR depends on the particular type of UV theory considered~\cite{1stWSR}. The 1st and 2nd WSRs read, respectively,
\begin{equation}
F_V^2 +\widetilde{F}_A^2 - F_A^2 - \widetilde{F}_V^2 \,=\, v^2 \,, \qquad \quad
F_V^2 M_V^2 + \widetilde{F}_A^2 M_A^2 - F_A^2 M_A^2 - \widetilde{F}_V^2 M_V^2=0 . \label{WSRs}
\end{equation}
\end{enumerate}
The use of (\ref{VFF}) and (\ref{WSRs}) in (\ref{integration}) lets us get the following determinations or bounds, which are displayed in the plots of Figure~\ref{plots1}~\cite{PRD}:
\begin{enumerate}
\item $\mathcal{F}_1$. Assuming both WSRs, $\mathcal{F}_1$ can be determined in terms of only resonance masses; contrary, discarding the 2nd WSR, we only find a bound in terms of only resonance masses~\cite{PRD}:\footnote{For this bound one needs to assume that $M_A>M_V$ and, in the case of considering $P$-even and also $P$-odd operators, $F_A^2>\widetilde F_A^2$. Both hypothesis seem to be reasonable working assumptions.}
\begin{equation}
\mathcal{F}_1\Big|^{\mathrm{1st\,\&\,2nd\,WSRs}} \,=\, -\displaystyle\frac{v^2}{4} \left( \frac{1}{M_V^2} \!+\! \frac{1}{M_A^2} \right)\,, \qquad \qquad
\mathcal{F}_1\Big|^{\mathrm{1st\,WSR}}\,<\, \displaystyle\frac{-v^2}{4M_V^2}\,. \label{F1}
\end{equation}
These results are shown in the top-left plot in Figure~\ref{plots1}. The dark gray curve shows the upper bound of (\ref{F1}). Therefore, and if only the 1st WSR is obeyed, the whole region below this line (gray and brown areas) would be theoretically allowed. If one accepts moreover the validity of the 2nd WSR, $\mathcal{F}_1$ is determined as a function of $M_V$ and $M_A$, see (\ref{F1}), being the dark gray curve the limit $M_A\to\infty$. The values of $\mathcal{F}_1$ for some representative axial-vector masses are shown in the red ($M_A=1.2\, M_V$), blue ($M_A=1.1\, M_V$) and orange ($M_A=M_V$) curves; being the orange line the lower bound $\mathcal{F}_1=-v^2/(2M_V^2)$. This range of $M_A\sim M_V$ corresponds actually to the most plausible scenario~\cite{ST}. Then, and if both WSRs are followed, only the gray region would be theoretically allowed if one assumes that $M_A>M_V$.\footnote{In the case with only $P$-even operators, following both WSRs implies $M_A>M_V$, but this is only a reasonable assumption if one considers also $P$-odd operators.}
\item $\mathcal{F}_3$. In this case the WSRs are not relevant. Assuming only $P$-even operators, (\ref{VFF}) implies a determination in terms of only resonance masses, whereas the inclusion of $P$-odd operators reduces this determination to a bound~\cite{PRD}:\footnote{For this bound one needs to assume that $M_A>M_V$ and $F_VG_V>0$, reasonable working assumptions.}
\begin{equation}
\mathcal{F}_3\Big|^{\mathrm{P{\text -}even}} \,=\,-\displaystyle\frac{v^2}{2M_V^2}\,, \qquad \qquad
\mathcal{F}_3\Big|^{\mathrm{P{\text -}even\,\&\,P{\text -}odd}} \,<\, -\displaystyle\frac{v^2}{2M_A^2}\,. \label{F3}
\end{equation}
The only $P$-even-operators prediction of (\ref{F3}) is shown by the black curve in the top-right plot of Figure~\ref{plots1}. Adding $P$-odd operators, we have the upper bound of (\ref{F3}), which is represented by the same curve but this time with $M_R=M_A$. Therefore, the whole region below this line (gray area) would be allowed in the most general case.
\item $\mathcal{F}_4$. In order to get a determination in terms of only resonance masses, one needs to consider a well-behaved two-Goldstones VFF, both WSRs and only $P$-even operators~\cite{PRD},
\begin{equation}
\mathcal{F}_4\Big|^{\mathrm{1st\,\&\,2nd\,WSRs;\,P{\text -}even}}\,=\, \displaystyle\frac{v^2}{4} \left( \frac{1}{M_V^2}\!-\!\frac{1}{M_A^2}\right)\,.\label{F4}
\end{equation}
We show it in the bottom-left panel in Figure~\ref{plots1}. The upper bound (dark gray curve) is obtained at $M_A\to\infty$. Thus, the theoretically allowed region is the gray area below that curve. The values of $\mathcal{F}_4$ for some representative axial-vector masses are shown in the red ($M_A=1.2\, M_V$), blue ($M_A=1.1\, M_V$) and orange ($M_A=M_V$) curves again.
\item $\mathcal{F}_4+\mathcal{F}_5$. No constraints are required in this case, since directly from (\ref{integration}) one gets~\cite{PRD}
\begin{equation}
\mathcal{F}_4+\mathcal{F}_5\,=\,\displaystyle\frac{c_d^2}{4M_{S^1_1}^2}\,. \label{F45}
\end{equation}
This clean prediction is shown by the black curve in the bottom-right plot of Figure~\ref{plots1}.
\end{enumerate}
\section{Phenomenology}
\begin{table}[tb]
\begin{center}
\renewcommand{\arraystretch}{1.0}
\begin{tabular}{|r@{$\,<\,$}c@{$\,<\,$}l|c|c||r@{$\,<\,$}c@{$\,<\,$}l|c|c| }
\hline
\multicolumn{3}{|c|}{LEC} & Ref. & Data & \multicolumn{3}{|c|}{LEC} & Ref. & Data \\
\hline \hline
$0.89$ & $\kappa_W$ & $1.13$ & \cite{deBlas:2018tjm}
& LHC &
$-0.06$&$\mathcal{F}_3$ &$0.20$&\cite{Almeida:2018cld} & LEP \& LHC \\ \hline
$-1.02$ & $c_{2V}$ & $2.71$ & \cite{ATLAS:2019dgh} & LHC &
$-0.0006$&$\mathcal{F}_4$&$0.0006$&\cite{Sirunyan:2019der} & LHC \\ \hline
$-0.004$ &$ \mathcal{F}_1$& $0.004$ &
\cite{Tanabashi:2018oca} & LEP &
$-0.0010$&$\mathcal{F}_4+\mathcal{F}_5$&$0.0010$ & \cite{Sirunyan:2019der} & LHC \\ \hline
\end{tabular}
\caption{{\small
Experimental constraints on purely bosonic EWET LECs, at 95\% CL~\cite{PRD}.}} \label{exp}
\end{center}
\end{table}
Experimental constraints on $\mathcal{F}_{1,3,4,5}$ are reported in Table~\ref{exp}~\cite{PRD} and are shown in the plots of Figure~\ref{plots1} as green areas. While these experimental constraints have been obtained at lower scales, the tree-level predictions of (\ref{integration}) are supposed to apply at a scale around the resonance masses. Thus, it is convenient to estimate the one-loop running uncertainties~\cite{Guo:2015isa} due to these different scales and they are indicated in Figure~\ref{plots1} with the dashed green bands that enlarge the experimentally allowed regions. They depend on $\kappa_W$ and $c_{2V}$, which are also given in Table~\ref{exp}.
The principal result of Figure~\ref{plots1} is that the resonance mass scale is pushed to the TeV range, $M_R \stackrel{>}{_\sim} 2\,$TeV, in agreement with Ref.~\cite{ST}, where we estimated theoretically the $S$ and $T$ oblique parameters at NLO. Other minor results are obtained~\cite{PRD}: the $S$ oblique parameter provides the most precise LEC determination ($\mathcal{F}_1$), implying the constraints $M_{V,A}\stackrel{>}{_\sim} 2$~TeV (95\% CL);
the triple gauge couplings give a weaker limit on $\mathcal{F}_3$, translating into the lower bound $M_{V,A}\stackrel{>}{_\sim} 0.5$~TeV (95\% CL);
assuming both WSRs and considering only $P$-even operators, the bounds on $\mathcal{F}_4$ constrain the mass of the vector resonance to $M_V\stackrel{>}{_\sim} 2$~TeV if $M_A/M_V>1.1$ (95\% CL);
the limit on $\mathcal{F}_4+\mathcal{F}_5$ implies that the singlet scalar resonance would have a mass $M_{S_1}\stackrel{>}{_\sim} 2$~TeV (95\% CL), taking a $S_1 WW$ coupling close to the $hWW$ one ($c_d\sim v$). Thus, experimental constraints start already to be competitive and, once new data be available, more precise information will be obtained using this kind of approaches.
\vspace{-0.2cm}
|
1,116,691,500,909 | arxiv | \section{The barycenter method}
I wish to search for a minimizer of a function $f : \mathcal{X} \rightarrow \mathbb{R}$ using the formula
\begin{equation}
\label{eq:barycenter}
\hat{x}_n = \frac{\sum_{i=1}^n x_i \mathrm{e}^{- \nu f(x_i)}}{\sum_{i=1}^n \mathrm{e}^{- \nu f(x_i)}},
\end{equation}
which expresses the center of mass or barycenter of $n$ test points $x_i \in \mathcal{X} \subset \mathbb{R}^{n_x}$ weighted according to the exponential of the value of the function at each point. In this formula $\nu \in \mathbb{R}$ is a positive constant. The point $\hat{x}_n$ will be in the convex hull of the $\{x_i\}$, and we shall assume that the $n_x$--dimensional search space $\mathcal{X}$ is convex. Applications of the method to nonconvex search spaces are conceivable but shall not be entertained here. The rationale behind it is that points where $f$ is large receive low weight in comparison with those for which $f$ is small.
Formula~\eqref{eq:barycenter} is suitable for {direct optimization, once scorned} \cite{once-scorned-wright}, now a respectable research area, both for its scientific challenges and practical applications. Also known as \textsl{derivative--free optimization}, it deals with the search for extrema of a given function, employing only the values of the function and not its mathematical expression. A recent book \cite{derivative-free-book} serves as a good entry point to the literature.
When the 1st and perhaps also 2nd derivatives of the function are available, use of gradients and Hessians leads to steepest--descent and Newton--like search algorithms. Often however derivatives are costly or impossible to compute. The challenge in direct optimization is to obtain algorithms with comparable performance, without knowledge of the derivatives.
The barycenter method has been employed successfully to tune filters in system identification \cite{MOLI-transactions}, see also \cite{RomanoCDC:2014}.
The filter parameters are additional structure parameters that need to be chosen before the model parameters themselves can be optimized, and may be said to perform a role similar to that of hyper--parameters in machine learning. With that experience in mind, the use of the barycenter method to tune hyper--parameters seems worth exploring.
A continuous--time version of the barycenter algorithm was analyzed in \cite{pait2014reading}. Some aspects of the method and its applications were presented at SIAM conferences \cite{SiamSDiego2014,SiamBoston2016}.
A recursive version of the barycenter method follows in
\Cref{sec:recursive}. A randomized search procedure is given a probabilistic analysis in \Cref{sec:gradient}, demonstrating that it exhibits convenient gradient descent--like properties. A complex version is shown, in \Cref{sec:complex}, to put less weight on queries made in regions where the rate of change of the goal function is large, thus discounting tests made away from candidate extrema; it may prove to be particularly suitable for situations in which the goal function cannot be fully evaluated at each optimization step. The method's robustness is discussed in \Cref{sec:noisy}, making a contribution to the study of optimization under noisy measurements. \Cref{sec:paralipomena} contains applications, generalizations, connections with sundry ideas in the literature, and directions for further work. Some technical proofs are put off until \Cref{sec:tech}.
\section{Recursive algorithm}
\label{sec:recursive}
The barycenter or center of mass of a distribution of weights $\mathrm{e}^{-\nu f(x_i)}$ placed at points $x_i$ is the point $x \in \mathbb{R}^{n_\rho}$ that minimizes the weighted sum of distances
\begin{equation*}
\label{eq:sum-of-distances}
\sum_{i=1}^n (x - x_i)^2 \mathrm{e}^{-\nu f(x_i)}.
\end{equation*}
Indeed, the sum is minimized when
\[\frac{\partial}{\partial x} \sum_{i=1}^n \left. (x-x_i)^2 \mathrm{e}^{-\nu f(x_i)}\right|_{\hat{x}_n}
= 2 \sum_{i=1}^n \left. (x-x_i) \mathrm{e}^{-\nu f(x_i)}\right|_{\hat{x}_n} = 0,
\]
so that
$ \hat{x}_n \sum_{i=1}^n \mathrm{e}^{-\nu f(x_i)} = \sum_{i=1}^n x_i \mathrm{e}^{-\nu f(x_i)}$, leading to \eqref{eq:barycenter}. Now writing $m_n = \sum_{i=1}^n \mathrm{e}^{- \nu f(x_i)}$ gives
\[
\hat{x}_n m_n = x_n \mathrm{e}^{-\nu f(x_n)} + \sum_{i=1}^{n-1} x_i \mathrm{e}^{-\nu f(x_i)}
= x_n \mathrm{e}^{-\nu f(x_n)} + m_{n-1} \hat{x}_{n-1},
\]
from which we obtain the recursive formulas
\begin{align}
m_n & = m_{n-1} + \mathrm{e}^{-\nu f(x_n)} \label{eq:mass_update} \\
\hat{x}_n & = \frac{1}{m_n} \left( m_{n-1} \hat{x}_{n-1} + \mathrm{e}^{-\nu f(x_n)} x_n \right). \label{eq:bary_update}
\end{align}
Here $m_0 = 0, \hat{x}_0 $ is arbitrary, and $x_n$ is the
sequence of test values.
From the point of view of recursive search strategies, it can be useful to pick the sequence of test points $x_n$ as the sum of the barycenter $\hat{x}_{n-1}$ of the previous points and a ``curiosity'' or exploration term $z_n$:
\begin{equation}
x_n = \hat{x}_{n-1} + z_n \label{eq:periergia}.
\end{equation}
Then \eqref{eq:bary_update} reads
\begin{equation}
\hat{x}_{n} -\hat{x}_{n-1} =
\frac{\mathrm{e}^{-\nu f(x_n)}}{m_{n-1} + \mathrm{e}^{-\nu f(x_n)}}z_n \label{eq:periergia2}.
\end{equation}
Although the size of the steps $\hat{x}_{n} -\hat{x}_{n-1} \rightarrow 0$ as $n \rightarrow \infty$ if the function $f$ is bounded in the search set, without further assumptions on the function $f$ and on the sequence of oracle queries $x_n$
we cannot be sure that the sequence is Cauchy.\footnote{This analysis was communicated to the author by Guilherme Vicinansa. }
Lack of assured convergence of the barycenter $\hat{x}_{n}$ is a desirable property, because convergence should come only as a consequence of judicious choice of the sequence of tests.
We have not specified how the oracle queries are to be selected, and in fact the barycenter method is not in conflict with derivative--free search algorithms developed in the literature, such as Nelder--Mead's, trust--region methods, or in fact \textsl{any} combination of such strategies.
In \Cref{sec:gradient} a simple randomized strategy is presented and analyzed.
\section{Probabilistic analysis}
\label{sec:gradient}
{A randomized version} of the barycenter algorithm can be studied using formula \eqref{eq:periergia2}.
If we consider $\hat{x}_n$ to be our best guess, on the basis of the information provided by the tests up to $x_n$, of where the minimum of $f(\cdot)$ might be found, then in the absence of any extra knowledge it makes sense to pick the curiosity $z_n$ as a random variable with some judiciously chosen probability distribution.
In light of the central limit theorem, we will analyze the case where $z_n$ is normal.
With the goal of obtaining approximate formulas for the barycenter update rule, in the remainder of this section we will assume that $f(\cdot)$ is twice continuously differentiable with respect to the argument $x$. Differentiability is not required for the barycenter method to be useful, and this assumption will be weakened later on, in \Cref{sec:noisy}.
Define $F_n(z) = \frac{\mathrm{e}^{-\nu f(\hat{x}_{n-1} + z)}}{m_{n-1} + \mathrm{e}^{-\nu f(\hat{x}_{n-1} + z)}}$, and
for subsequent use write
\[\bar{F}_n (z) =
\frac{m_{n-1} \mathrm{e}^{-\nu f(\hat{x}_{n-1} + z)}}{(m_{n-1} + \mathrm{e}^{-\nu f(\hat{x}_{n-1} + z)})^2} =
\frac{m_{n-1} }{m_{n-1} + \mathrm{e}^{-\nu f(\hat{x}_{n-1} + z)}} F_n ,\]
so that $\frac{\partial F}{\partial z} = - \nu \bar{F} \frac{\partial f}{\partial z}.$ Here and in the computations that follow the subscript indicating dependence of the sample ordinality $n$ is omitted if there is no ambiguity.
\begin{theorem} \label{thm:average}
If $z_n$ has a Gaussian distribution, the expected value of $\Delta\hat{x}_n = \hat{x}_n - \hat{x}_{n-1}$ is proportional to the average value of the gradient of $f(\hat{x}_{n-1} + z_n)$ in the support of the distribution of $z$.
\end{theorem}
\begin{proof}
The claim is established by a calculation, which in an appropriate notation is straightforward. Consider the probability density function
\[
p(z)= \frac{1}{\sqrt{(2\pi)^n |\Sigma|}} e^{- \frac{1}{2}(z- \bar{z})^T\Sigma^{-1}(z- \bar{z})}.
\]
Then $\frac{\partial p}{\partial z^\beta} = - \Sigma^{-1}_{\beta \alpha } (z^\alpha- \bar{z}^\alpha) p(z) $ so $z^\alpha p = \bar{z}^\alpha p - \Sigma^{\alpha \beta} \frac{\partial p}{\partial z^\beta}$. Einstein's implicit summation of components with equal upper and lower indices convention is in force, with upper Greek indices for the components of $z$ and of $\Sigma$, and lower indices for the components of $\Sigma$'s inverse.
With $\mathcal X$ the $n$--dimensional set where the curiosity $z$ takes its values, for each component $z^\alpha$ of the vector $z$ we have
\begin{equation*}
E \left [ F(z) z^\alpha \right] = \int_{\mathcal X} F(z) z^\alpha p(z) \, \mathrm{d} z
= \int_{\mathcal X} F(z) \bar{z}^\alpha p(z) \, \mathrm{d} z
- \Sigma^{\alpha \beta} \int_{\mathcal X} F(z) \frac{\partial p}{\partial z^\beta}(z) \, \mathrm{d} z.
\end{equation*}
Using the integration--by--parts formula in Lemma~\ref{lemma:int-by-parts},
\begin{equation*}
\int_{\mathcal X} F(z) \frac{\partial p}{\partial z^\beta} \, \mathrm{d} z
+ \int_{\mathcal X} \frac{\partial F}{\partial z^\beta} p(z) \, \mathrm{d} z
= \int_{\partial \mathcal X} F(z) p(z) \left (\frac{\partial}{\partial z^\beta} \rfloor \mathrm{d} z \right),
\end{equation*}
where $\frac{\partial}{\partial z^\beta} \rfloor \mathrm{d} z$ is an $(n-1)$--form which can be integrated at the boundary $\partial {\mathcal X}$ of the $n$--dimensional set $ {\mathcal X}$. The right--hand side is zero, because $F$ is bounded and $p(z)$ vanishes at the border of $\mathcal X$, which is at infinity, hence
\begin{multline*}
E \left [ F(z) z^\alpha \right]
= \bar{z}^\alpha \int_{\mathcal X} F(z) p(z) \, \mathrm{d} z
+\Sigma^{\alpha \beta} \int_{\mathcal X} \frac{\partial F}{\partial z^\beta} p(z) \, \mathrm{d} z\\
= E \left [ F(z) \right] \bar{z}^\alpha
-\nu \Sigma^{\alpha \beta} \int_{\mathcal X} \bar{F} (z) \frac{\partial }{\partial z^\beta}\left( f(\hat{x}_{n-1} + z)\right) p(z)\, \mathrm{d} z,
\end{multline*}
so
\begin{equation}
\label{eq:approx-grad}
\boxed
{
E \left [ \Delta\hat{x}_n \right] = E \left [ F_n (z) \right] \bar{z} -\nu \Sigma \, E\left[ \bar{F_n}(z) \nabla f(\hat{x}_{n-1} + z) \right],
}
\end{equation}
where the 2nd term is proportional to the negative gradient of $f$ as claimed.
\end{proof}
Formula \eqref{eq:approx-grad} is a key result concerning the barycenter method. It shows that roughly speaking a random search performed in conjunction with the barycenter algorithm follows the direction of the negative average gradient of the function to be minimized, the weighted average being taken over the domain where the search is performed. For a given $\nu$, the step size is essentially given by $\Sigma$. Depending on the shape of the function $f$, large values of the variance of $z$ may compromise the descent property of the search method.
The term in $\bar{z}$ can be employed to incorporate extra knowledge in several manners. For example, if at each step we take $\bar{z} = \xi \; \Delta\hat{x}_{n-1}$, then the gradient term is responsible for the rate of change, or acceleration, of the search process. The factor $0 < \xi < 1$ is chosen to dampen oscillations and prevent instability. The case $\xi = 0$ corresponds to the garden--variety, non--accelerated gradient--like search.
In the context of intended applications to adaptive controller and filter tuning, here the author cannot resist citing his seldom--read paper \cite{acelera-letters} discussing tuners that set the 2nd derivative of the adjusted parameters, rather than the 1st derivative, as is more often done in the literature, as well as an application to filtering theory \cite{max-pait-jojoa}, where the 2nd difference is used in a discrete--time version.
The designer has the freedom to choose the free parameters $\nu$, $\Sigma$, and $\bar{z}$ of the randomized barycenter search procedure in order to achieve the most desirable convergence properties.
{Understanding of the variance of} $\Delta\hat{x}_n $, which depends on the Hessian $\nabla^2 f$ of $f$, is useful in picking these free parameters.
\begin{theorem} \label{thm:variance}
Under the conditions of Theorem~\ref{thm:average} and assuming that the variance of $z$ is small, the variance of $\Delta\hat{x}_n$ for $\bar{z} = 0$ near a critical point of $f(x)$ where $\nabla f= 0$ is approximately
\begin{equation}
\label{eq:approx-variance}
\mathrm{Var} (\Delta \hat{x}) \approx
\Sigma E[F^2 ] - 2\nu \Sigma^T E\left[ F \bar{F} \nabla^2 f \right] \Sigma.
\end{equation}
\end{theorem}
The proof, presented in \Cref{sec:tech}, is similar to the computation of the mean. What Theorem~\ref{thm:variance}
tells us is that, near a minimum of a locally convex function, the variance of the adjustment step grows less than linearly with the variance of the curiosity; the higher the Hessian and the larger $\nu$ is, the smaller the variance. This is a desirable property of the method, because it indicates that the barycenter moves around less than the test points. Formula \eqref{eq:approx-variance} is only approximate, as can be confirmed by considering that the variance of $\Delta \hat{x}$ can only grow with increasing $\Sigma$ and must remain positive--definite.
Note that by construction both functions $F$ and $\bar{F}$ are between 0 and 1. Roughly speaking they tend to decrease together as $n$ becomes larger. If such a decrease is unwelcome, one course of action is to employ a forgetting factor as suggested in \Cref{sec:paralipomena}. To develop an intuition on what happens to the rate of adjustment $\Delta \hat{x}$ over time, it can be useful to consider a situation where the search steps are small and all variables take their mean value. In this case we have
$F_n \approx \frac{\mathrm{e}^{-\nu f(\hat{x}_{n-1})}}{m_{n-1} + \mathrm{e}^{-\nu f(\hat{x}_{n-1})}}$ and
\begin{multline*}
F_{n+1} = \frac{\mathrm{e}^{-\nu f(\hat{x}_{n} + z)}}{m_{n} + \mathrm{e}^{-\nu f(\hat{x}_{n} + z)}}
\approx \frac{\mathrm{e}^{-\nu f(\hat{x}_{n-1} + \Delta \hat{x}_n)}}{m_{n-1} + \mathrm{e}^{-\nu f(\hat{x}_{n-1})} + \mathrm{e}^{-\nu f(\hat{x}_{n})}} \\
\approx \frac{\mathrm{e}^{-\nu f(\hat{x}_{n-1})} \mathrm{e}^{-\nu \nabla f(\hat{x}_{n-1}) \cdot \Delta \hat{x}_n}}{m_{n-1} + \mathrm{e}^{-\nu f(\hat{x}_{n-1})} } \frac{m_n}{m_{n+1}}
\approx F_n \frac{m_n}{m_{n+1}} (1 + \bar{F} \nu^2 \nabla f^T \Sigma \nabla f),
\end{multline*}
where we wrote the 1st order term of the Taylor expansion of the exponential to reveal that step size tends to increase due to the gradient--descent property of the algorithm, counterbalancing the decreases as the previous weights add up. It could be interesting, in further work, to investigate possible connections between the probabilistic barycenter update and quasi--Newton or conjugate descent methods.
\paragraph{We have recovered a type of steepest--descent search using direct methods,\nopunct \!}
without explicitly computing derivatives or even using an expression for the function itself.
The descent properties of the randomized barycenter method suggest that a convergence analysis along the usual lines for gradient--descent like algorithms could be endeavored, taking into account the stochastic nature of the descent. This would of course require assumptions about the properties of the specific function being optimized.
We shall not proceed along this path here.
The analysis presented is of a descriptive nature, showing what one might expect from the method without detailed consideration of a search strategy, rather than a prescription for a particular strategy.
In fact we envision the barycenter method as a useful way of combining different strategies whose qualities are applicable to diverse circumstances. Some points in the sequence $x_i$ could be chosen according to coordinate--search methods, and others using simplicial or line--search ones. Alternatively, the sequence of curiosities $z_i$ could have a multimodal distribution or be heteroscedastic, picking some points near the current barycenter with the goal of improving search precision, while perhaps a smaller fraction would have a high--variance distribution, in order to explore the possibility of distant minima smaller than the local one.
The goals should be balanced taking into account the requirements of each problem. When the main cost of the procedure is computation of the goal function, then perhaps a fair amount of exploration may be justified by the possibility of detecting deeper minima. But in real--time design applications it may be undesirable to test points with high cost, and exploration must be used more sparingly.
\section{A stationary phase algorithm}
\label{sec:complex}
We now develop a version of the barycenter method employing complex weights
\footnote{
We will embark on
a digression that will only become relevant if we try to apply the method to manifolds $\mathcal{X}$ whose geometry is other than Euclidian, and which can be comfortably ignored at 1st reading.
To use complex exponents we need to identify the points $x$ not by coordinates, but by quantities that have an immediate metrical meaning. Define the positive quantity $x^\alpha$ as the distance between point $x$ and the hyperplane where the $\alpha$th coordinate is zero. Although the set of positive quantities $\{x^\alpha \}, \alpha = 1, \ldots, n_x$ is \emph{not} a vector field and does not correspond globally to a coordinate representation on the space $\mathcal X$ in any geometrically meaningful way --- in particular it does not transform correctly under changes of coordinates, each number $x^\alpha$ always remaining positive --- its numerical values coincide with the numerical values of the coordinate components of the vector $x$ in the positive orthant, which may therefore be used as local coordinates. Therefore we will take the liberty of using the habitual coordinate notation in some calculations involving the quantities $x^\alpha$.}
The \emph{complex barycenter} is defined term--by--term by the same formula as the barycenter in \eqref{eq:barycenter}, with a complex exponent $\nu$:
\begin{equation}
\label{eq:barycenter-i}
{\eta}^\alpha_n = \frac{\sum_{i=1}^n x^\alpha_i \mathrm{e}^{-\nu f(x_i)}} {\sum_{i=1}^n \mathrm{e}^{-\nu f(x_i)}},
\end{equation}
but now our estimate of the extremum point is
\begin{equation}
\label{eq:barycenter-modulus}
\hat{x}^\alpha_n = | \eta_n^\alpha |.
\end{equation}
In these formulas all $x_i \geq 0$
\footnote{The operations above are performed term--by--term, because there is no meaningful way of obtaining a coordinate representation of a point from the calculation of the modulus $| \eta | = \sqrt{\eta \bar{\eta}}$ of a complex number $\eta $. Although not a bona fide operation on vector spaces, it could be given the tensor--style expression
\begin{equation*}
\label{eq:barycenter-modulus-symbol}
\hat{x}^\alpha_i = \sqrt{\eta_i^\beta \bar{\eta}_i^\gamma \delta_{\beta \gamma}^\alpha}
\end{equation*}
using the symbol $\delta_{\beta \gamma}^\alpha = 1$ if $\alpha = \beta = \gamma$, and zero otherwise. Having said that, from now on we drop the superscripts $\alpha$ denoting coordinates unless needed.}
In the real case the result of \eqref{eq:barycenter-i} together with \eqref{eq:barycenter-modulus} coincides with that of \eqref{eq:barycenter}.
The algorithm is suggested by Feynman's interpretation of quantum electrodynamics \cite{feynman-qed,feynman2012quantum} and by the stationary phase approximation \cite{bleistein1975asymptotic,hormander2012analysis} used in the asymptotic analysis of integrals.
We are interested in the situation in which $\nu$ has a real as well as an imaginary part, and, in contrast with the asymptotic expansions literature and with quantum mechanics, the imaginary part is bounded and, although perhaps large by design, not overwhelmingly so.
To understand the advantages of using a complex exponential, split the set of queries $x_i$, $i \in \{1,2,\ldots,n\}$ into disjoint subsets $A$ and $B$, whose elements are respectively $x^A_i$, $i \in \{1,2,\ldots,n^A\}$ and $x^B_j$, $j \in \{1,2,\ldots,n^B\}$, with $n^A + n^B = n$. Denote
${m}^A = {\sum_{i=1}^{n^A} \mathrm{e}^{-\nu f(x^A_i)}}$,
${m}^B = {\sum_{j=1}^{n^B} \mathrm{e}^{-\nu f(x^B_j)}}$,
${\eta}^A = {\sum_{i=1}^{n^A} x^A_i \mathrm{e}^{-\nu f(x^A_i)}}/m_A$, and
${\eta}^B = {\sum_{j=1}^{n^B} x^B_j \mathrm{e}^{-\nu f(x^B_j)}}/m_B$. Then
\begin{equation}
\label{eq:barycenter-splitAB}
{\eta}_n = \frac{\sum_{i=1}^{n^A} x^A_i \mathrm{e}^{-\nu f(x^A_i)}+\sum_{j=1}^{n^B} x^B_j \mathrm{e}^{-\nu f(x^B_j)}}
{\sum_{i=1}^{n^A} \mathrm{e}^{-\nu f(xA_i)}+\sum_{j=1}^{n^B} \mathrm{e}^{-\nu f(x^B_j)}}
= \frac{m^A \eta^A + m^B \eta^B}{m_A + m_B}.
\end{equation}
The relative weight of the measurements allocated to subset $A$ is thus determined by the sum $m^A$. The subsets $A$ and $B$ need not be organized into any temporal or other ordering; the formula is valid irrespectively of how the points $\{x_i \}$ are assigned to $A$ and $B$, and can easily be generalized to more subsets as in \Cref{sec:paralipomena}
.
Now suppose that all points $x_i^A$ are concentrated in a small region around a point $x^A$, so that a Taylor expansion up to the linear terms only can be used and
$f(x_i^A) \approx f(x^A) + \nabla f (x^A) \cdot y$, with $y$ in a box such that each of its coordinate components $y^\alpha \in [-\delta , \delta ], \alpha \in {1,2,\ldots,n_x}$.
When $\nu$ is real, by convexity $\eta^A=\hat{x}^A$ will be contained inside the said region, and repeated tests would
have the effect of increasing the combined weight of the points and pulling the barycenter towards the region. If the region does not contain a candidate optimizer, this effect is unwelcome. Perhaps it can be mitigated by judicious choice of the test sequence, but often one does not have complete flexibility in specifying oracle queries, and in any case the derivatives of $f(\cdot)$ being unknown it is not possible to ascertain a priori whether a certain region contains candidate extremizers. The complex version of the barycenter method presents desirable properties in this case.
Perhaps the most transparent way to analyze this situation is to assume that the measurements are equally distributed inside the box with sides $2 \delta $. This is a reasonable practical assumption, as there is not much reason to query the oracle more often at one point of a small region than another.
Under these circumstances the expected value of the combined weight of the measurements $x^A$ can be approximated up to the linear term of a Taylor series
\begin{multline}
\label{eq:interference-product}
E\left [ \sum_{i=1}^{n^A} \mathrm{e}^{- \nu f(x_i^A)} \right ] \approx
n^A E\left [ \mathrm{e}^{- \nu \left ( f(x^A)+ f'_\alpha (x^A) y^\alpha \right )} \right ] \\
= n^A \int_{y^\alpha \in [-\delta_\alpha , \delta_\alpha ]} \mathrm{e}^{- \nu f(x^A)} \; \mathrm{e}^{-\nu f'_1 y^1}\frac{dy^1}{2 \delta_1 } \wedge \cdots \wedge \mathrm{e}^{-\nu f'_{n_x} y^{n_x}}\frac{dy^{n_x}}{2 \delta_{n_x} }\\
= n^A \mathrm{e}^{- \nu f(x^A)} \int_{-\delta_1 }^{\delta_1 } \mathrm{e}^{-\nu f'_1 y^1}\frac{dy^1}{2 \delta_1 } \cdots \int_{-\delta_{n_x} }^{\delta_{n_x} } \mathrm{e}^{-\nu f'_{n_x} y^{n_x}}\frac{dy^{n_x}}{2 \delta_{n_x} }.
\end{multline}
Each of the $n_x$ integrals in the product above is of the form
\begin{equation*}
\int_{-\delta }^{\delta } \mathrm{e}^{-\nu f'_\alpha y^\alpha} \frac{\mathrm{d} y^i\alpha}{2 \delta_\alpha }
= \frac{1}{-\nu f' \delta } \frac{\mathrm{e}^{-\nu f'_i \delta} - \mathrm{e}^{\nu f'_ i \delta} }{2}
= \frac{\sinh (\nu f' \delta )}{\nu f' \delta }.
\end{equation*}
Writing $r= \Re[\nu f' \delta ]$, $q= \Im[\nu f' \delta ]$ compute
\begin{multline*}
\sinh (\nu f' \delta )
= \frac{\mathrm{e}^r (\cos q + j \sin q) - \mathrm{e}^{-r} (\cos q - j \sin q) )}{2} \\
= \frac{(\mathrm{e}^r-\mathrm{e}^{-r}) \cos q + j (e^{r} + \mathrm{e}^{-r}) \sin q}{2}
= \sinh r \cos q + j \cosh r \sin q,
\end{multline*}
so that
\[
\left | \frac{\sinh (\nu f' \delta )}{\nu f' \delta } \right |^2 = \frac{\sinh^2 r + \sin^2 q}{r^2 + q^2 }.
\]
If we consider boxes of sides $\delta$ such that $\Im[\nu f' \delta ] = k \pi$, with $k = 1,2,\ldots$, the term $\sin^2 q = 0$. Then each of the factors in \eqref{eq:interference-product} is a complex number with magnitude $\sinh r / \sqrt{r^2 + q^2}$. Thus the expected value of the total weight of the measurements $x^A$ in a small box such that $\nabla f$ is \textsl{not} close to zero is reduced by a factor $(1 + q^2/r^2)^{n_x/2}$. It seems reasonable to choose the imaginary part of $\nu$ larger than its real part, though not so large as to cause problems with noisy measurements. We summarize these findings as follows.
\begin{theorem} \label{thm:complex-interference}
The expected contribution of measurements made outside of any region where $\nabla f \approx 0$ is discounted by one factor, proportional to $\nabla f$ and to the ratio between the complex magnitude of $\nu$ and its real part, for each dimension of the search space.
\end{theorem}
This destructive interference, so to speak, between repeated measurements near points which are \textsl{not} candidates for minimizers is the justification for employing complex values of $\nu$.
For the purpose of computing $\eta_n$, the sequential ordering of the points $\{x^A_i\}$ and $\{x^B_i\}$ is immaterial. However if we were to compute $\hat{x}_i$ at a certain instant $i$ and use its value to determine the choice of the subsequent oracle queries, along the lines suggested in the recursive algorithm analyzed in \Cref{sec:recursive}, then
property \eqref{eq:barycenter-splitAB} would lose applicability, and the ordering of the query point $x_i$ would need to be taken into consideration.
With some poetic license, we might say that the information responsible for the interference phenomenon, which largely cancels out the results of unproductive tests, is contained in the phase of $\eta$, and would be lost in the moment that the magnitude of $\eta$ is computed.
These considerations do not apply in the real $\nu$ case, in which $\eta_n$ and $\hat{x}_n$ coincide (for positive values of $\hat{x}_n$).
\section{Noisy measurements and non--smooth functions}
\label{sec:noisy}
Oftentimes each measurement of the function $f$ at point $x_i$ is corrupted by noise or experimental errors. In this case we still would like to minimize $f$, but now using oracle answers $f(x_i) + w_i$. For the purpose of analyzing the effect of noise on the results of the barycenter method, we consider the sequence $x_i$ as given and $w_i$ as an ergodic random process. A more elaborate analysis of the effect of noise on the sequence $\{x_i\}$ itself, which would depend on the recursive search algorithm used to generate the oracle queries, will not be attempted in this paper.
With $x_i$ deterministic and $w_i$ random, we wish to compute statistics of
\begin{equation}
\label{eq:barycenter-noise}
{\eta}_n = \frac{\sum_{i=1}^n x_i \mathrm{e}^{-\nu (f(x_i)+ w_i)}} {\sum_{i=1}^n \mathrm{e}^{-\nu (f(x_i)+w_i)}}.
\end{equation}
If $w_i$ are independent, normal, zero--mean variables with standard deviation $\sigma$,
then $E[e^{-\nu w_i}] = \mathrm{e}^{\nu^2 \sigma^2/2} := \bar{w}$ and ${\rm{Var}}[\mathrm{e}^{-\nu w_i}] = \mathrm{e}^{2\nu^2 \sigma^2} - \mathrm{e}^{\nu^2 \sigma^2} = \bar{w}^4 - \bar{w}^2$.
Define the nominal or ``noise--free'' values
$\bar{m} = \sum_{i=1}^n \mathrm{e}^{-\nu f(x_i)}$ and
$\bar{\eta} = { \sum_{i=1}^n x_i \mathrm{e}^{-\nu f(x_i)}}/ \bar{m}$, and also the scalar quantity $\bar{\bar{m}} = \sum_{i=1}^n \mathrm{e}^{-2\nu f(x_i)}$, the vector quantity $\bar{\bar{\eta}} = \sum_{i=1}^n x_i \mathrm{e}^{-2\nu f(x_i)}/ \bar{\bar{m}}$, and the matrix quantity $\breve{\eta} = \sum_{i=1}^n x_i x_i^T \mathrm{e}^{-2\nu f(x_i)} / \bar{\bar{m}}$.
\begin{theorem} \label{thm:under-noise} Assuming that $\sigma$ is small, under the circumstances above the mean and variance of $\eta$ can be expressed approximately as follows:
\begin{align}
\label{eq:mean-eta}
E[\eta] & \approx \bar{\eta} +
\frac{ \bar{\bar{m}} }{\bar{m}^2 } (\bar{\eta} - \bar{\bar{\eta}}) \nu^2 \sigma^2
\text{ and } \\
\label{eq:var-eta}
{\rm{Var}}[\eta] & \approx
\frac{ \bar{\bar{m}} }{\bar{m}^2 }(\bar{\eta} \bar{\eta}^T - \bar{\eta} \bar{\bar{\eta}} - \bar{\bar{\eta}} \bar{\eta} + \breve{\eta})
\nu^2 \sigma^2.
\end{align}
\end{theorem}
The proof is given in \Cref{sec:tech}. What we learn from somewhat involved formulas~\eqref{eq:mean-eta}
and \eqref{eq:var-eta} is that noise or measurement errors generate a bias and a variance in the barycenter, both proportional in 1st approximation to the variance of the noise. These effects conspire to pull the barycenter away from the looked--after minimum of the function. However one may derive a measure of comfort in that the unwelcome errors tend to zero as the noise becomes smaller.
Theorem~\ref{thm:under-noise} indicates that there is little reason to fear that the method breaks down under moderate measurement or computing errors.
The situation where the (unknown) function to be optimized is not smooth can be studied as a particular case of the optimization of $f(x) + w$, where $f(\cdot)$ itself is smooth, and $w(x)$ is the difference between the function under consideration and its smooth approximation, plus a noise or error parcel when applicable. In this case the assumption that $w_i$ is uncorrelated with $x_i$ can be objected to. On the other hand any function can be approximated with arbitrary precision by a smooth function. The analysis presented is reasonable in the practical case when
the approximation error is overshadowed by measurement or numerical errors.
\section{Paralipomena}
\label{sec:paralipomena}
Formula~\eqref{eq:barycenter} is suitable for direct, or derivative--free, optimization: it is applicable when no functional expression for $f(\cdot)$ is known, and the search for an optimizer makes use of a zero--order oracle \cite{nesterov2004introductory} which, when queried, supplies the value of the goal function at a given point, but not of its derivatives. In practical applications the oracle query may take the form of an experiment or simulation. To wrap up this paper we connect to various ideas in the literature: Laplace's method; asymmetry, which is a drawback of the method; forgetting factors; parallelization; and partial evaluation of goal functions. Then we mention applications to feedback control and dynamical system identification and offer some conclusions.
\paragraph{Laplace's method\nopunct \!}
for asymptotic analysis uses an integral equation that can be seen as a limiting case of the barycenter method, in the limit when $\nu \rightarrow \infty$ and is real. In contrast, we are strictly interested in computable algorithms, and $\nu$ takes finite values; in fact we need to keep $\nu$ limited to stay clear of numerical troubles. For the same reason we employ a sum of a finite number of test points, which can be contrasted with Laplace's integral formula, itself more similar to the expectations appearing in the analyses in \Cref{sec:gradient} and \Cref{sec:complex}. Formula \eqref{eq:barycenter} is also related to the expression for the LogSumExp (LSE) function which is used in machine learning, and can perhaps be seen as a way of deploying the Boltzmann distributions of statistical mechanics in an optimization procedure.
\paragraph{If the goal function is not symmetric\nopunct \!}
around the extremum, the barycenter will be pulled towards an average of the test points, which may not coincide with the actual extremum. It would be negligent not to mention this drawback of the barycenter method. In a sufficiently small neighborhood of a twice continuously differentiable function a quadratic approximation can be used, and the issue of asymmetry disappears. Higher values of $\nu$ will also tend to make the resulting bias smaller.
\paragraph{A forgetting factor\nopunct \!}
can be employed
if convergence of the step size is not desired,
and more recent tests are to be given a bigger weight than older ones. Now in place of formulas \eqref{eq:mass_update} and \eqref{eq:bary_update} write
\begin{equation}
\label{eq:forgetting_update}
m_n = \lambda_n m_{n-1} + \mathrm{e}^{-\nu_n f(x_n)}, \qquad
\hat{x}_n = \frac{1}{m_n} \left( \lambda_n m_{n-1} \hat{x}_{n-1} + \mathrm{e}^{-\nu_n f(x_n)} x_n \right).
\end{equation}
The forgetting factor $\lambda_n \leq 1$, whose dependence on $n$ allows for the case when the change happens in some instants but not all, helps the search focus closer and closer, and also to filter out noise in accumulated past measurements.
If $\lambda$ is constant then batch formula \eqref{eq:barycenter}
needs to be replaced by
\begin{equation}
\label{eq:barycenter-forget}
\hat{x}_n = \frac{\sum_{i=1}^n x_i \lambda^{n-i} \mathrm{e}^{- \nu f(x_i)}}{\sum_{i=1}^n \lambda^{n-i}\mathrm{e}^{- \nu f(x_i)}}.
\end{equation}
If in a certain application it becomes necessary to modify step size, then another reasonable course of action is to increase $\nu$ at some, or all, instants $n$. Coherent with this strategy would be to correct $m_{n-1}$ by a factor
$\mathrm{e}^{(\nu_{n}- \nu_{n-1})f(\hat{x}_{n-1})}$, to reflect the sharper weighting on latter points. No recursive formula such as \eqref{eq:forgetting_update} can be used, because the current barycenter with a given $\nu$ does not contain all the information necessary to compute where the barycenter would have been if a different $\nu$ had been employed.
\paragraph{The barycenter method is parallelizable\nopunct \!}
in a natural way. As equation~\eqref{eq:barycenter-splitAB} and its obvious extension
\begin{equation} \label{eq:parallel}
{\eta} = \frac{\sum_{\ell=1}^{n_\ell} m^{A_\ell} \eta^{A_\ell}} {\sum_{\ell=1}^{n_\ell} m^{A_\ell} }
\end{equation}
show, the search procedure may be run as a set of independent simultaneous searches.
The processes responsible for computing the partial barycenters of the sets $A_\ell$ may be several instances of the same algorithm, for example the type of randomized searches discussed in \Cref{sec:gradient} with different initializations or with different statistics. Such a heteroscedastic, multi--modal search can be used to probe distant corners of the search space $\mathcal{X}$, trying to avoid getting stuck at local minima.
While randomized local searches may be among the simplest to implement and analyze, diverse search mechanisms with complementary desirable properties and drawbacks can be incorporated and combined using \eqref{eq:parallel}. The barycenter method is not in conflict with other algorithms previously described in the literature.
Parallelization is an effective way to take advantage of intrinsically parallel computer architectures, because the various sequences need to be combined only at the end of the search process, or perhaps at a limited number of intermediary steps. Parallelization is perhaps a compelling argument for considering the complex version of the barycenter method.
\paragraph{Partial evaluation\nopunct \!}
of the goal function is a situation that can arise if
to each query at $x_i$ the oracle returns a value $f_i(x_i)$. This situation is of practical relevance if, for example, the optimization goal can be expressed as
\begin{equation*}
\label{eq:partial-goal-sum}
f(x) = \sum_{j=1}^k f_k(x),
\end{equation*}
with $k$ a large number (although typically $k \ll n$, the total number of oracle queries performed), which makes it impractical to evaluate the full $f(x)$ at each step. In this case our method is reminiscent of so--called stochastic gradient algorithms, although without the need for computing derivatives.
The complex version of the barycenter method seems particularly appropriate under such circumstances. Because full information about $f(x)$ does not become available with each measurement, several oracle queries at the same or nearby points may become necessary. In hose that happen in regions of high gradient of $f(\cdot) $ are discounted as explained in \Cref{sec:complex}.
Under partial evaluations an occasional measurement of a small value for the goal function could mislead the search procedure.
Although one should be careful not to discount the challenge of adjusting the free parameters of the search procedure, the barycenter method seems more easily applicable in such circumstances then other derivative--free methods. It is not readily apparent how partial--gradient methods might be used if only 0--order information on the goal function may be used.
The method developed might also be applied to situations where the goal function is partially known and can be expressed as $f(x,g(x))$, with either $f(\cdot,\cdot)$ or $g(\cdot)$ of a known form. Partial derivatives can be computed as allowed by functional knowledge, and used to guide the choice of the sequence $x_i$ according to well--established optimization methods. Those in turn can be combined with an exploration term using the barycenter method.
\paragraph{Dynamical system identification and direct adaptive control\nopunct \!}
are applications of interest to the author where partial evaluation comes about. In system identification, one wishes to minimize an error $| \hat\theta - \theta |$ using only available measurements $|\phi(t)( \hat\theta - \theta )|$, with $\phi(t)$ a regressor that depends on the state of the dynamical system to be modeled. Often $\phi(t)$ cannot be chosen directly and the system parameter estimate $\hat\theta$ has to be derived using whatever data becomes available through observation of the process under consideration. For linear model fitting usually $\phi(t)( \hat\theta - \theta )$ is known, and gradient--type or least--squares algorithms are applicable; however there exist problems of interest such as direct adaptive control where only the error's magnitude $|\phi(t)( \hat\theta - \theta )|$ is available for the purpose of tuning $\hat\theta(t)$ \cite{design-direct}; see also \cite{miller-lyapunov}.
When optimization is performed offline, the most important resource to conserve are time--consuming oracle queries themselves. If however optimization is used to make real--time decisions, such as those involved in the tuning of a feedback controller, then the cost of all queries is not equal. The costly ones are those which result in high values of the goal functions, and accordingly a more cautious type of search may be preferable.
\begin{center} * \hspace{2em} * \hspace{2em} * \end{center}
\paragraph{The goal of traditional model--based optimization\nopunct \!}
is to obtain the most precise results using the least amount of computational power. In direct optimization, the most important resource, to be used sparingly, are queries that evaluate the cost to be optimized. The barycenter method appears, from the analysis developed here, to provide a useful compromise between these goals. The computational requirements are relatively modest, as are the data requirements, and the algorithm is flexible enough to incorporate improvements if more data is available, or if there exists previous information concerning the structure of the function to be optimized.
If perhaps not completely obvious, the barycenter method has an a posteriori inevitability. The complex version in particular seems to have useful and elegant properties, which to the best of the author's knowledge have not yet been explored.
{
|
1,116,691,500,910 | arxiv | \section{Introduction}
Understanding the spectrum of hadrons reveals information on the non-perturbative aspects of QCD \cite{Godfrey:1998pd} and on the inner structure of hadrons.
The {Regge trajectory} is one of the effective approaches for studying hadron spectra \cite{Regge:1959mz,Collins:1971ff,Chew:1961ev,Hey:1982aj,Inopin:2001ub,Chen:2016spr,
Chen:2016qju,Inopin:1999nf,Chew:1962eu,Brodsky:2017qno}.
The orbital and radial {Regge trajectories} \cite{note} for pion are often taken as being approximately linear in the $(M^2,\,l)$ plane and in the $(M^2,\,n_r)$ plane, respectively, $M^2={\alpha_1}l+{\alpha_2}n_r+c$ \cite{Anisovich:2000kxa,Afonin:2006vi,Shifman:2007xn,Ebert:2009ub,Masjuan:2012gc,
Kobzarev:1992wt,Lucha:1991ce,Afonin:2021cwo}, where $M$ is the mass, $\alpha_1$ and $\alpha_2$ are the Regge slopes, $l$ is the orbital angular momentum, $n_r$ is the radial quantum number, and $c$ is a constant.
The pion {Regge trajectories} are in fact nonlinear when they are examined more precisely, see Fig. \ref{fig:regm2old}.
Many authors have discussed the nonlinearity of the pion {Regge trajectories}. In Ref. \cite{Tang:2000tb}, the authors note that the orbital pion {Regge trajectory} is nonlinear by the "zone test", $M^2=-0.1077J^2+1.6003J+0.019$, where $J$ is the total angular momentum. In Ref. \cite{Brisudova:1999ut}, the pion {Regge trajectory} is discussed by using the square-root trajectory. In Ref. \cite{Sharov:2013tga}, the author gives the nonlinear pion orbital {Regge trajectory} with corrections based on the string model. In Ref. \cite{Sonnenschein:2014jwa}, the pion {Regge trajectory} is nonlinear as the masses of quarks are considered. In Refs. \cite{Chen:2021kfw,Chen:2018hnx,Chen:2018bbr}, the pion {Regge trajectories} are fitted by a nonlinear formula, $M^2=2.78(0.8+n_r)^{2/3}-2.38$.
It is known that the significant nonlinearity of the {Regge trajectories} for the heavy mesons arises from the nonrelativisity of heavy mesons \cite{Sergeenko:1994ck,Inopin:1999nf,Sonnenschein:2018fph,Sonnenschein:2014jwa,
Cotugno:2009ys,Burns:2010qq,Kruczenski:2004me,MartinContreras:2020cyg,Chen:2021kfw,Chen:2018hnx,
Chen:2018nnr,Chen:2018bbr} due to the heavy masses of quarks. In this work, we revisit the pion {Regge trajectories} and present discussions on the nonlinearity of them.
\begin{figure}[!phtb]
\centering
\subfigure[]{\label{subfigure:fiterr}\includegraphics[scale=0.85]{regorb-pi.eps}}
\subfigure[]{\label{subfigure:fitbeta}\includegraphics[scale=0.85]{regrad-pi.eps}}
\caption{The fitted orbital and radial {Regge trajectories} for pion, which are plotted in the $(M^2,\,l)$ plane and in the $(M^2,\,n_r)$ plane, respectively. These figures are from Ref. \cite{Chen:2021kfw}. The well-established states are given by solid dots and the unwell-established states are given by circles. Open squares are predicted masses.}\label{fig:regm2old}
\end{figure}
This paper is organized as follows: In Sec. \ref{sec:fit}, we fit the orbital and radial pion {Regge trajectories} by using four points and five points on the {Regge trajectories}, respectively. The discussions are presented in Sec. \ref{sec:dis} and the conclusions are in Sec. \ref{sec:conclusions}.
\section{Fit of the pion {Regge trajectories}}\label{sec:fit}
In this section, we fit the orbital and radial {Regge trajectories} for the pion by employing a newly proposed ansatz. The ansatz is model-independent and therefore the fit is model-independent. Then we obtain the fitted parameters without the preset values.
\subsection{Preliminaries}
The {Regge trajectories} can be written in different forms, such as $l=l(M)$, $M^2=f(l,\,n_r)$, $M=M(l,\,n_r)$ \cite{Burns:2010qq}, $(M-m)^2=g(l,\,n_r)$ \cite{Afonin:2020bqc,Chen:2017fcs,Jia:2019bkr} and so on.
We use the following ansatz inspired by Refs. \cite{Chen:2021kfw,MartinContreras:2020cyg,Brau:2000st}
\begin{equation}\label{grform}
M={\beta_x}\left(x+c_0\right)^{\nu}+c_1\;\;(x=l,\,n_r)
\end{equation}
to fit the orbital and radial {Regge trajectories} \cite{note} for the pion in the $(M,\,x)$ plane. $\beta_x$ is the slope. The constants $c_0$ and $c_1$ vary with different {Regge trajectories}. The exponent $\nu$ which relates to the dynamics of mesons is regarded as a free parameter in fit.
As $0<\nu<1$, $\nu=1$ or $\nu>1$, Eq. (\ref{grform}) becomes concave downwards, linear or convex upwards, respectively. $\nu\in(0,1)$ is used to find the appropriate value because the pion orbital and radial {Regge trajectories} are obviously concave in the $(M,\,n_r)$ and $(M,\,l)$ planes, see Fig. \ref{fig:regm}.
The used data are listed in the 3rd column in Table \ref{tab:fitmass}.
The quality of a fit is measured by the quantity $\chi^2$ defined by \cite{Sonnenschein:2014jwa}
\begin{equation}\label{chi2}
\chi^2=\frac{1}{N-1}\sum_{i=1}^N\left(\frac{M_{fi}-M_{ei}}{M_{ei}}\right)^2,
\end{equation}
where $N$ is the number of points on the trajectory, $M_{fi}$ is the fitted value and $M_{ei}$ is the experimental value of the $i$-th particle.
The fit is calculated by using MATHEMATICA program. $c_0$, $c_1$ and $\nu$ are free parameters. $\beta_x$ is calculated by using the FindFit function and Eq. (\ref{grform}). The quantity $\chi^2$ is calculated by using Eq. (\ref{chi2}). By minimizing the $\chi^2$ quantity for a given $\nu$, the parameters $c_0$, $c_1$ and $\beta_x$ are determined for each value of $\nu$.
\begin{table*}[!phtb]
\caption{Mesons' masses (in units of ${\rm MeV}$). The experimental masses are from PDG \cite{ParticleDataGroup:2020ssz}. The fitted results by using the {Regge trajectories} [Eq. (\ref{grform})] are shown in comparison with the theoretical values in the EFG model \cite{Ebert:2009ub} and in the GI model \cite{Godfrey:1985xj}. Fit5 are obtained by using five points on the {Regge trajectories} while Fit4 are fitted by excluding $\pi^0$. A question mark ($?$) indicates an unwell-established state. The fitted {Regge trajectories} are listed in Table \ref{tab:fitreg}.}
\label{tab:fitmass}
\centering
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}ccllllllrll@{}}
\hline\hline
State & Meson & PDG \cite{ParticleDataGroup:2020ssz} & EFG \cite{Ebert:2009ub} & GI \cite{Godfrey:1985xj} & \multicolumn{3}{c}{Fit5} & \multicolumn{3}{c}{Fit4}\\
& & & & & $\nu=0.4$ & $\nu=0.45$ & $\nu=0.5$
& $\nu=0.4$ & $\nu=0.45$ & $\nu=0.5$\\
\hline
$1^1S_0$ & $\pi^0$ & $134.9768\pm0.0005$ &154 &150
& 135 & 136 & 135 & $-$115 & 90 & 255 \\
$2^1S_0$ & $\pi(1300)$ & $1300\pm100$ &1292 & 1300
& 1307 & 1302 & 1269 & 1310 & 1312 & 1315 \\
$3^1S_0$ & $\pi(1800)$ & $1810^{+9}_{-11}$ &1788 &1880
& 1764 & 1756 & 1739 & 1765 & 1759 & 1754 \\
$4^1S_0$ & $\pi(2070)$? & $2070\pm35$ &2073 &
& 2096 & 2095 & 2099 & 2096 & 2093 & 2091 \\
$5^1S_0$ & $\pi(2360)$? & $2360\pm25$ &2385 &
& 2368 & 2376 & 2403 & 2366 & 2370 & 2375 \\
$6^1S_0$ & & & &
& 2601 & 2621 & 2671 & 2598 & 2611 & 2625 \\
$7^1S_0$ & & & &
& 2808 & 2840 & 2913 & 2803 & 2827 & 2851 \\
$8^1S_0$ & & & &
& 2995 & 3039 & 3135 & 2989 & 3023 & 3060 \\
\hline
$1^1S_0$ & $\pi^0$ & $134.9768\pm0.0005$ &154 &150
& 135 & 135 & 135 & $-$175 & 25 & 185 \\
$1^1P_1$ & $b_1(1235)$ & $1229.5\pm3.2$ &1258 &1220
& 1226 & 1235 & 1213 & 1229 & 1231 & 1232 \\
$1^1D_2$ & $\pi_2(1670)$ & $1670.6^{+2.9}_{-1.2}$ &1643 &1680
& 1675 & 1673 & 1660 & 1678 & 1672 & 1666 \\
$1^1F_3$ & $b_3(2030)$? & $2032\pm12$ &1884 &2030
& 2004 & 2000 & 2002 & 2004 & 2002 & 1998 \\
$1^1G_4$ & $\pi_4(2250)$? & $2250\pm15$ & 2092 &2330
& 2272 & 2272 & 2291 & 2270 & 2275 & 2279 \\
$1^1H_5$ & & & &
& 2502 & 2508 & 2545 & 2498 & 2513 & 2526 \\
$1^1I_6$ & & & &
& 2707 & 2719 & 2776 & 2700 & 2726 & 2750 \\
\hline\hline
\end{tabular*}
\end{table*}
\begin{table*}[!phtb]
\caption{The fitted pion {Regge trajectories} (in units of ${\rm GeV}$) by using four points (Fit4) and by using five points (Fit5). }
\centering
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}cllclc@{}}
\hline\hline
& & \multicolumn{2}{c}{Fit4} & \multicolumn{2}{c}{Fit5}\\
& & Traj. & $\chi^2$ & Traj. & $\chi^2$ \\
\hline
& $\nu=0.4$ &$M=1.404\,l^{0.4}-0.175$ & $9.5\times10^{-5}$ &$M=1.424(l+0.03)^{0.4}-0.215$ & $7.7\times10^{-5}$\\
Orbital Traj. & $\nu=0.45$ &$M=1.206\,l^{0.45}+0.025$ & $1.15\times10^{-4}$ & $M=1.198(l+0.004)^{0.45}+0.035$ & $9.0\times10^{-5}$\\
& $\nu=0.5$ & $M=1.047\,l^{0.5}+0.185$ & $1.5\times10^{-4}$ & $M=1.078\,l^{0.5}+0.135$ & $1.9\times10^{-4}$ \\
\hline
& $\nu=0.4$ &$M=1.425\,n_r^{0.4}-0.115$ & $2.8\times10^{-4}$ & $M=1.438(n_r+0.016)^{0.4}-0.14$ & $2.2\times10^{-4}$ \\
Radial Traj. & $\nu=0.45$ & $M=1.222\,n_r^{0.45}+0.09$ & $3.4\times10^{-4}$
& $M=1.241(n_r+0.002)^{0.45}+0.06$ & $2.8\times10^{-4}$ \\
& $\nu=0.5$ & $M=1.060\,n_r^{0.5}+0.255$ & $4.1\times10^{-4}$ & $M=1.134\,n_r^{0.5}+0.135$ & $6.7\times10^{-4}$ \\
\hline\hline
\end{tabular*}\label{tab:fitreg}
\end{table*}
\begin{figure}[!phtb]
\centering
\subfigure[]{\label{subfig:fiterr}\includegraphics[scale=0.74]{fit5err.eps}}
\subfigure[]{\label{subfig:fitc0}\includegraphics[scale=0.7]{fit5c0.eps}}
\subfigure[]{\label{subfig:fitc1}\includegraphics[scale=0.7]{fit5c1.eps}}
\subfigure[]{\label{subfig:fitbeta}\includegraphics[scale=0.7]{fit5beta.eps}}
\caption{The quantities $\chi^2$, $c_1$, $c_0$ and $\beta_x$ ($x=l,\,n_r$) vary with different exponent $\nu$. The orbital {Regge trajectory} (the blue dots) and the radial {Regge trajectory} (the red circles) are fitted by using five points on the {Regge trajectories}, that is, $\pi^0$ is included in fit. The used formulas are Eqs. (\ref{grform}) and (\ref{chi2}).}\label{fig:fit5}
\end{figure}
\begin{figure}[!phtb]
\centering
\subfigure[]{\label{subfigure:fit4err}\includegraphics[scale=0.74]{A4fiterr.eps}}
\subfigure[]{\label{subfigure:fit4c0}\includegraphics[scale=0.7]{A4fitc0.eps}}
\subfigure[]{\label{subfigure:fit4c1}\includegraphics[scale=0.7]{A4fitc1.eps}}
\subfigure[]{\label{subfigure:fit4beta}\includegraphics[scale=0.7]{A4fitbeta.eps}}
\caption{Comparison between the fitted quantities $\chi^2$, $c_1$, $c_0$ and $\beta_x$. The quantities which are obtained from the fitted orbital (the blue dots) and radial (the red circles) {Regge trajectory} as the pion is included (five-point fit) are compared with the quantities obtained from the fitted orbital (the cyan solid squares) and radial (the magenta empty squares) {Regge trajectory} as the $\pi^0$ is excluded (four-point fit). The used formulas are Eqs. (\ref{grform}) and (\ref{chi2}).}\label{fig:rthl}
\end{figure}
\subsection{Fit of the pion {Regge trajectories} by using five points}
In this subsection, the $\pi^0$ is included in fit, that is, five points or five states listed in Table \ref{tab:fitmass} are used to fit the orbital and radial {Regge trajectories} by employing Eq. (\ref{grform}), respectively.
The quantities $\chi^2$, $c_1$, $c_0$ and $\beta_{l}$ ($\beta_{n_r}$) vary with the exponent $\nu$, see Fig. \ref{fig:fit5}. $\nu$ ranges from $0.1$ to $0.9$ because as ${\nu}{\gtrapprox}1$, the fitted values of $c_0$ and $c_1$ become anomalous and omitting $\nu\in(0.9,1)$ does not affect the results.
$\chi^2$ increases with $\nu$. It increases rapidly and becomes large as $\nu{\gtrapprox}0.5$, see Fig. \ref{subfig:fiterr}.
$c_0$ is related with the curvature of the used formula (\ref{grform}). $c_0>0$ implies that the curvature of the formula (\ref{grform}) may be larger than expected while $c_0=0$ suggests that the exponent $\nu$ is appropriate or large. As shown in Fig. \ref{subfig:fitc0}, the fitted results prefer $0<\nu{\lessapprox}0.46$.
$c_1$ is expected to be greater than or equal to $0$, see the appendix \ref{subsec:str2} and Eq. (\ref{mrdef}). The $c_1-\nu$ plot shows that the better value is $\nu{\gtrapprox}0.43$, see \ref{subfig:fitc1}.
As $\nu\in(0.45,\,0.6)$, $\beta_l^2\in(1.436,\,0.942)$ and $\beta_{n_r}^2\in(1.54,\,1.04)$, see Fig. \ref{subfig:fitbeta}.
According to the previous discussions, for the five-point fit $\nu\approx0.45$ is the better fitted value without considering the Regge slopes.
\begin{figure}[!phtb]
\centering
\subfigure[]{\label{sf:fit4o}\includegraphics[scale=0.74]{regmorb4.eps}}
\subfigure[]{\label{sf:fit4r}\includegraphics[scale=0.7]{regmrad4.eps}}
\subfigure[]{\label{sf:fit5o}\includegraphics[scale=0.7]{regmorb5.eps}}
\subfigure[]{\label{sf:fit5r}\includegraphics[scale=0.7]{regmrad5.eps}}
\caption{The fitted orbital and radial {Regge trajectories} for pion by using four points (Fit4) and by using five points (Fit5). They are plotted in the $(M,\,l)$ plane and in the $(M,\,n_r)$ plane, respectively. The red dotted lines are for $\nu=0.4$, the blue dashed lines are for $\nu=0.45$ and the black lines are for $\nu=0.5$. The used formulas are listed in Table \ref{tab:fitreg}.}\label{fig:regm}
\end{figure}
\begin{figure}[!phtb]
\centering
\subfigure[]{\label{sfs:fit4o}\includegraphics[scale=0.74]{regm2orb4.eps}}
\subfigure[]{\label{sfs:fit4r}\includegraphics[scale=0.7]{regm2rad4.eps}}
\subfigure[]{\label{sfs:fit5o}\includegraphics[scale=0.74]{regm2orb5.eps}}
\subfigure[]{\label{sfs:fit5r}\includegraphics[scale=0.7]{regm2rad5.eps}}
\caption{The fitted orbital and radial {Regge trajectories} for pion by using 4 points (Fit4) and by using 5 points (Fit5). They are plotted in the $((M-m_R)^2,\,l)$ plane and in the $((M-m_R)^2,\,n_r)$ plane, respectively. The used formulas for the black lines are from Table \ref{tab:fitreg}. The red dashed lines are the linear fits.}\label{fig:regm2}
\end{figure}
The fitted {Regge trajectories} by using five points are listed in Table \ref{tab:fitreg} with $\nu=0.4,\,0.45,\,0.5$. They are plotted in the $(M,\,l)$ plane and in the $(M,\,n_r)$ plane, respectively, see Fig. \ref{fig:regm}. The fitted masses are listed in Table \ref{tab:fitmass}, which are in agreement with the experimental values.
As mentioned in the introduction, the pion {Regge trajectories} are nonlinear in the $(M^2,\,x)$ ($x=l,\,n_r$) plane, see Fig. \ref{fig:regm2old}. However, both of the orbital pion {Regge trajectory} and the radial pion {Regge trajectory} can be described in a linear form when they are plotted in the $((M-m_R)^2,\,x)$ plane with appropriate $m_R$, see Fig. \ref{fig:regm2}.
\subsection{Fit of the pion {Regge trajectories} by using four points}
The pion can be explained as the pseudo-Nambu-Goldstone boson associated with chiral symmetry breaking \cite{Horn:2016rip,Sonnenschein:2018fph}. The low-mass pion is often excluded from the corresponding {Regge trajectories} formed by its orbitally or radially excited partners. In this subsection, we fit the orbital and radial {Regge trajectories} formed by the four orbitally excited states of $\pi^0$ and by the four radially excited states of $\pi^0$, respectively.
$\chi^2$ for the orbital and radial {Regge trajectories} increases with the exponent $\nu$. $\chi^2$ obtained by using the four points is roughly equal to $\chi^2$ calculated by using the five points as $\nu\in(0,0.5)$, see Fig. \ref{subfigure:fit4err}.
According to Fig. \ref{subfigure:fit4c0}, $\nu\in(0.39,\,0.64)$ for the orbital {Regge trajectory} and $\nu\in(0.15,\,0.6)$ for the radial {Regge trajectory} as $c_0=0$.
$c_1$ and $\beta_x$ calculated by four points have similar behavior as that obtained by using five points, respectively. In the range $\nu\in(0.3,0.5)$, $c_1$ (or $\beta_x$) obtained by using four points and five points are approximately equal.
As shown in Fig. \ref{subfigure:fit4c1}, $\nu{\gtrapprox}0.43$ for the radial {Regge trajectory} and $\nu{\gtrapprox}0.45$ for the orbital {Regge trajectory} as $c_1\ge0$.
As $\nu\in(0.45,\,0.6)$, $\beta_l^2\in(1.454,\,0.659)$ and $\beta_{n_r}^2\in(1.493,\,0.677)$, see Fig. \ref{subfigure:fit4beta}.
We can conclude that the better value of $\nu$ ranges from 0.45 to 0.5 for the four-point fit.
When $\pi^0$ is excluded in fit, the fitted {Regge trajectories} by using 4 points with $\nu=0.4,\,0.45,\,0.5$ agree well with the experimental values, see Table \ref{tab:fitreg}. The extrapolated masses of $\pi^0$ from the fitted radial {Regge trajectories} are $-115$ ${\rm MeV}$, $90$ ${\rm MeV}$ and $255$ ${\rm MeV}$ for $\nu=0.4,\,0.45,\,0.5$, respectively.
For the orbital {Regge trajectory}, the extrapolated masses of $\pi^0$ are $-175$ ${\rm MeV}$, $25$ ${\rm MeV}$ and $185$ ${\rm MeV}$ for $\nu=0.4,\,0.45,\,0.5$, respectively.
The fitted pion {Regge trajectories} are plotted in the $(M,\,l)$ plane and in the $(M,\,n_r)$ plane, respectively, see Fig. \ref{fig:regm}.
Whether the $\pi^0$ is included or not, both the orbital {Regge trajectory} and the radial {Regge trajectory} are linear when they are plotted in the $((M-m_R)^2,\,x)$ plane with nonzero $m_R$, see Fig. \ref{fig:regm2}. For the five-point fit, $m_R=135$ ${\rm MeV}$ for both the orbital {Regge trajectory} and the radial {Regge trajectory}, which is the experimental value of $\pi^0$. But for the four-point fit, $m_R=255$ ${\rm MeV}$ for the radial {Regge trajectory} and $m_R=185$ ${\rm MeV}$ for the orbital {Regge trajectory} in case of $\nu=0.5$.
\section{Discussions}\label{sec:dis}
\subsection{Confining potentials}
The confining potentials taking the power-law form with different power indexes are discussed in many works, such as $V_{conf}{\sim}r$ in the well-known Cornell potential \cite{Eichten:1974af}, $V_{conf}{\sim}r^{0.91}$ \cite{Flamm:1987kx}, $V_{conf}{\sim}r^{3/4}$ \cite{Lichtenberg:1988tn}, $V_{conf}{\sim}r^{2/3}$ \cite{Heikkila:1983wd,song:1991jg}, $V_{conf}{\sim}r^{1/2}$ \cite{Song:1986ix}, $V_{conf}{\sim}r^{0.1}$ \cite{Martin:1980jx,Martin:1980rm} and so on.
In Ref. \cite{Rai:2008sc}, the mass spectrums are extracted and the radial wave functions are reproduced from different models as well as from the {nonrelativistic} phenomenological quark antiquark potential of the type $V(r)=-\alpha_s/r+Ar^{\delta}$ with $\delta$ varying from 0.5 to 2. In Ref. \cite{Patel:2008na}, the power index range of $0.1<\delta<2.0$ has been explored when computing the decay rates and spectroscopy of the $Q\bar{Q}$ mesons in the {nonrelativistic} potential.
The exponent $\nu$ in (\ref{grform}) is related to the confining potential, see Eq. (\ref{rtcom}) and the appendix \ref{subsec:pot}. In case of the {ultrarelativistic} energy region, $\nu=1/2$ indicates the linear confining potential, $V_{conf}{\sim}r$. $\nu=0.45$ implies $V_{conf}{\sim}r^{9/11}$. $\nu=0.4$ arises from the confining potential $V_{conf}{\sim}r^{2/3}$.
In case of the {nonrelativistic} energy region, $\nu=1/2,\,0.45,\,0.4$ give $V_{conf}{\sim}r^a$ with $a=2/3,\,18/31,\,1/2$, respectively.
It is known that the orbitally excited states and the radially excited states of pion are taken as the {ultrarelativistic} systems \cite{Chen:2021kfw}. The suggested $0.45\le\nu\le0.5$ by the fit indicates the confining potential $r^a$ with $9/11{\le}a\le1$.
\subsection{Parameter $c_1$ or $m_R$}
In the {ultrarelativistic} limit, $c_1$ in (\ref{grform}) or $m_R$ in (\ref{reglike}) is usually assumed to be zero, i.e., the {Regge trajectory} takes the form $M^2={\alpha_x}(x+c_0)^{\gamma}$ with $\gamma=1$. According to Eqs. (\ref{sse}), (\ref{massform}), (\ref{reglike}), (\ref{mrdef}), (\ref{lowMass}) and (\ref{highMass}), $c_1$ or $m_R$ is related with the masses of constituents and the constant part of the interaction energy, especially in the {nonrelativistic} energy region. In Ref. \cite{Godfrey:1985xj}, $m_u=m_d=220$ ${\rm MeV}$, $C$ in Eq.
(\ref{potv}) reads $C=-253$ ${\rm MeV}$. Substituting $m_u$, $m_d$ and $\epsilon_c=C$ into Eq. (\ref{mrdef}) gives $c_1=m_R=187$ ${\rm MeV}$. It is in excellent agreement with $185$ ${\rm MeV}$ which is obtained from the fitted orbital {Regge trajectories} ($\nu=0.5$) and is smaller than $255$ ${\rm MeV}$ which is from the fitted radial {Regge trajectories} ($\nu=0.5$) as $\pi^0$ is excluded in fit, see Table \ref{tab:fitreg}.
As shown in the appendix \ref{appsec:b}, nonzero $m_R$, i.e., nonzero $c_1$ in Eq. (\ref{grform}) will shift the slope of the {Regge trajectories} to a lower value. They can give the reasonable slopes. It shows that nonzero $c_1$ or nonzero $m_R$ is appropriate and acceptable.
It is the nonzero $c_1$ or $m_R$ together with $\beta_x(x+c_0)^{1/2}$ that leads to the nonlinearity of the orbital $(M^2,\,l)$ pion {Regge trajectory} and the nonlinearity of the radial $(M^2,\,n_r)$ pion {Regge trajectory}.
As $m_R$ is not equal to zero and is chosen appropriately, the radial pion {Regge trajectory} in the $((M-m_R)^2,\,n_r)$ plane and the orbital {Regge trajectory} in the $((M-m_R)^2,\,l)$ plane are linear whether the $\pi^0$ is included on the {Regge trajectories}, see Fig. \ref{fig:regm2}.
According to Eq. (\ref{grform}) or (\ref{reglike}), $c_1{\ne}0$ [or $m_R{\ne}0$] means that one part of $M$ keeps constant and does not vary with $l$ and $n_r$ while the other portion changes with $l$ or $n_r$. $c_1=0$ [or $m_R=0$] implies that all parts of the bound-state masses are effected by the potentials because $\nu$ is related with the confining potential.
$c_0=0$ indicates that $M$ varies with $l$ or $n_r$ in a simple way. As $c_0{\ne}0$, $c_0$ will be entangled with $l$ ($n_r$) because $\nu{\ne}1$.
\subsection{A note on $\pi^0$}
There are lots of discussions on the intrinsic structure of the pion \cite{Horn:2016rip,Arrington:2021biu,Fariborz:2021gtc}.
In Ref. \cite{Alexandrou:2017itd}, the lattice QCD gives $m_{\pi}=296$ ${\rm MeV}$. In Ref. \cite{Santowsky:2021ugd}, $m_{\pi}=328$ ${\rm MeV}$. In Ref. \cite{Frezzotti:2022dwn}, the pion mass is in the range $250-500$ ${\rm MeV}$. The extrapolated mass are $m_{\pi}=185$ ${\rm MeV}$ for the orbital {Regge trajectory} by using four points and
$m_{\pi}=255$ ${\rm MeV}$ for the radial {Regge trajectory} by using four points in case of the linear confining potential. They are larger than the experimental result and smaller than the results in Refs. \cite{Alexandrou:2017itd, Santowsky:2021ugd}. According to the discussions in section \ref{sec:fit}, it is not foreclosed and reasonable that $\pi^0$ is taken as the first point on the pion {Regge trajectories}, see Figs. \ref{sf:fit4o}, \ref{sf:fit4r}, \ref{sfs:fit4o} and \ref{sfs:fit4r}. It implies that $\pi^0$ can be regarded as the quark-antiquark state like other states on the pion {Regge trajectories}.
\section{Conclusions}\label{sec:conclusions}
The orbital and radial pion {Regge trajectories} are fitted phenomenologically by employing the ansatz $M={\beta_x}\left(x+c_0\right)^{\nu}+c_1$ where $x=l,\,n_r$.
It is shown that nonzero $m_R$ is reasonable and acceptable. Nonzero $m_R$ or $c_1$ gives an explanation that the pion {Regge trajectories} are concave in the usually employed $(M^2,\,x)$ plane as being examined more precisely. As $m_R$ is chosen appropriately, both the orbital and radial pion {Regge trajectories} are linear in the $((M-m_R)^2,\,x)$ plane whether the $\pi^0$ is included or not on the {Regge trajectories}.
It is reasonable and not foreclosed that $\pi^0$ is regarded as the first point on the pion {Regge trajectories}.
The fitted pion {Regge trajectories} suggest $0.45\le\nu\le0.5$. It indicates the confining potential $r^a$ with $9/11{\le}a\le1$.
We present discussions in the appendix \ref{appenda:structa} on the structure of the {Regge trajectories} plotted in the $(M,\,x)$ plane and in the $((M-m_R)^2,\,x)$ plane based on the potential models and the string models.
In the appendix \ref{appsec:b}, the {Regge trajectories} for the light nonstrange mesons with different $m_R$ are shown in the $((M-m_R)^2,\,x)$ plane.
It is illustrated that $m_R$ can be nonzero for the light nonstrange mesons.
\section*{Acknowledgments}
We are very grateful to the anonymous referees for the valuable comments and suggestions. This work is supported by the Natural Science Foundation of Shanxi Province of China under Grant no. 201901D111289, which is sponsored by the Shanxi Science and Technology Department.
|
1,116,691,500,911 | arxiv | \section{Introduction}
Background knowledge plays a very important role in human conversations. For example, to have a meaningful conversation about a movie, one uses their knowledge about the plot, reviews, comments and facts about the movie.
A typical conversation involves recalling important points from this background knowledge and producing them appropriately in the context of the conversation. However, most existing large scale datasets \cite{ubuntu,ritter2010,HRED} simply contain a sequence of utterances and responses without any \textit{explicit} background knowledge associated with them. This has led to the development of models which treat conversation as a simple sequence-to-sequence generation task and often produce output which is both syntactically incorrect and incoherent (off topic). To make conversations more coherent, there is an increasing interest in integrating structured and unstructured knowledge sources with neural conversation models. While there are already some works in this direction \cite{trainable_task_oriented,dst_review,lowe2015,KGNCM} which try to integrate external knowledge sources with existing datasets, we believe that building new datasets where the utterances are \textit{explicitly} linked to external background knowledge will further facilitate the development of such background aware conversation models.
\begin{figure*}
\label{dataset_example}
\centering
\resizebox{12.5cm}{7.7cm}{
\begin{tikzpicture}
\small
\node[draw,text width=3.5cm] at (-7.7,16) {... The lab works on spiders and has even managed to create new species of spiders through genetic manipulation.\textcolor{blue}{\uline{ While Peter is taking photographs of Mary Jane for the school newspaper, one of these new spiders lands on his hand and bites him }} Peter comes home feeling ill and immediately goes to bed. ...};
\node[font=\bfseries] at (-7.7,18.7) {Plot};
\node[draw,text width=3.5 cm] at (-7.7,10.7) {... \textcolor{blue} { \uline{I thoroughly enjoyed ``Spider-Man''}} which I saw in a screening. I thought the movie was very engrossing. Director Sam Raimi kept the action quotient high, but also emphasized the human element of the story. Tobey was brilliant as a gawky teenager...
};
\node[font=\bfseries] at (-7.7,13.2) {Review};
\node[draw,text width=7.1cm] at (-1.75,13.5) {Speaker 1(N): Which is your favourite character?\\
\vspace{2mm}
Speaker 2(C): My favorite character was Tobey Maguire. \\\vspace{2mm}
Speaker 1(N): I thought he did an excellent job as peter parker, I didn't see what it was that turned him into Spider-Man though.\\\vspace{2mm}
Speaker 2(P): Well this happens while Peter is taking photographs of Mary Jane for the school newspaper, one of these new spiders lands on his hand and bites him. \\\vspace{2mm}
Speaker 1 (N): I see. I was very excited to see this film and it did not disappoint! \\\vspace{2mm}
Speaker 2(R): I agree, I thoroughly enjoyed ``Spider-Man''\\\vspace{2mm}
Speaker 1(N): I loved that they stayed true to the comic.\\\vspace{2mm}
Speaker 2(C): Yeah, it was a really great comic book adaptation\\\vspace{2mm}
Speaker 1(N): The movie is a great life lesson on balancing power.\\\vspace{2mm}
Speaker 2(F): That is my most favorite line in the movie, ``With great power comes great responsibility.''\vspace{2mm}
};
\node[font=\bfseries] at (-1.75,18.7){Movie: Spider-Man};
\node[draw,text width=3.5cm] at (4.4,16) {... Crazy attention to detail. \textcolor{blue} {\uline{My favorite character was Tobey Maguire.}} I can't get over the ``I'm gonna kill you dead'' line. It was too heavily reliant on constant light-hearted humor. However the constant joking around kinda bogged it down for me. \textcolor{blue} { \uline{A really great comic book adaptation.}} ....};
\node[font=\bfseries] at (4.25,18.6) {Comments};
\node[text width=0.002cm] at (2.4,10.58) {\begin{tabular}{|l|l|}
\hline
Awards & \begin{tabular}[c]{@{}l@{}}Golden Trailer\\ Awards 2002 \\ \end{tabular} \\ \hline
Taglines & \begin{tabular}[c]{@{}l@{}} \textcolor{blue} {\uline{With great}}\\ \textcolor{blue} {\uline{power comes}}\\ \textcolor{blue} {\uline{great}} \\ \textcolor{blue} {\uline{responsibility.}} \\ Get Ready For\\ Spidey !\end{tabular} \\ \hline
\begin{tabular}[c]{@{}l@{}}Similar\\ Movies\end{tabular} & {\color[HTML]{000000} \begin{tabular}[c]{@{}l@{}}Iron Man\\ Spider-Man 2 \end{tabular}} \\ \hline
\end{tabular}
};
\node[font=\bfseries] at (4.25,13.2) {Fact Table};
\end{tikzpicture}
}
\caption{A sample chat from our dataset which uses background resources. The chosen spans used in the conversation are shown in \textcolor{blue} {\uline{blue}}. The letters in the brackets denote the type of resource that was chosen - P, C ,R, F and N indicate \textbf{P}lot, \textbf{C}omments, \textbf{R}eview, \textbf{F}act Table and \textbf{N}one respectively.}
\end{figure*}
With this motivation, we built a new background aware conversation dataset using crowdsourcing. Specifically, we asked workers to chat about a movie using structured and unstructured resources about the movie such as plots, reviews, comments, fact tables (see Figure \ref{dataset_example}). For every even numbered utterance, we asked the workers to consult the available background knowledge and try to construct a sentence which contains information from this background knowledge and is relevant in the current context of the conversation (akin to how humans recall things from their background knowledge and insert them appropriately in the conversation). For example, in Turn 2, Speaker 2 picked a sentence from the plot which is relevant to the current context of the conversation. Similarly, in Turn 3, Speaker 2 picked a sentence from the movie review. We also asked the workers to suitably modify the content picked from the background knowledge, if needed, so that the conversation remains coherent.
We collected around 9K such conversations containing a total of 90K utterances pertaining to about 921 movies. These conversations along with the background resources will be made publicly available\footnote{\url{https://github.com/nikitacs16/Holl-E}}. For every utterance, we also provide information about the exact span in the resource from which this utterance was created. Lastly note that unlike existing datasets, our \textit{test set} contains multiple reference responses for each test context thereby facilitating better evaluation of conversation models. We believe that this dataset will allow the community to take a fresh look at conversation modeling and will lead to the development of models which can learn to exploit background knowledge to pick appropriate responses instead of generating responses from scratch. Such a conversation strategy which produces responses from background knowledge would be useful in various domains. For example, a troubleshooting bot could exploit the information available in manuals, reviews and previous bug reports about the software. Similarly, an e-commerce bot could exploit the rich information available in product descriptions, reviews, fact tables, \textit{etc.} about the product. While the proposed dataset is domain specific, it serves as a good benchmark for developing creative background-knowledge-aware models which can then be ported to different domains by building similar datasets for other domains.
We establish some initial baselines using three different paradigms to demonstrate the various models that can be developed and evaluated using this dataset. For the sake of completeness, the first paradigm is a hierarchical variant of the sequence to sequence architecture which does not exploit any background knowledge. The second paradigm is the copy-and-generate paradigm wherein the model tries to copy text from the given resources whenever appropriate and generate it otherwise. The third paradigm borrows from the span prediction based models which are predominantly being used for Question Answering (QA). These baseline results along with the dataset would hopefully shape future research in the area of background aware conversation models.
\section{Related Work}
There has been an active interest in building datasets \cite{survey_on_dialog_datasets} for training dialog systems. Some of these datasets contain transcripts of human-bot conversations \cite{DSTC1,DSTC2,DSTC3} while others are created using a fixed set of natural language patterns \cite{weston2017,moviedialog}. The advent of deep learning created interest in the construction of large-scale dialog datasets \cite{ubuntu,ritter2010,sordoninaacl2015} leading to the development of several end-to-end conversation systems \cite{NRM,NCM,personaNCM,HRED} which treat dialog as a sequence generation task.
To make the output of these models more coherent, there is an increasing effort in integrating external background knowledge with these models. This is because human beings rely on background knowledge for conversations as well as other tasks \cite{schallert}. There has been considerable work on incorporating background knowledge in the context of goal-oriented dialog datasets even before the advent of large-scale datasets for deep learning \cite{letsgo,mit_atis} as well as in recent times \cite{trainable_task_oriented,dst_review,incar} where datasets include small sized knowledge graphs as background knowledge.
However, the conversations in these datasets are very templated and nowhere close to open conversations in specific domains such as the ones contained in our dataset.
Even in the case of open domain conversations, there are some works which have integrated external knowledge sources. Most of the entries in 2017 Amazon Alexa Prize \cite{alexa_prize} relied on background knowledge for meaningful response generation. Milabot \cite{milabot} and even the winning entry SoundingBoard \cite{soundingboard} used Reddit pages, Amazon's Evi Service, and large databases like OMDB, Google Knowledge Graph and Wikidata as external knowledge. The submission named Eigen \cite{eigen} used several dialog datasets and corpora belonging to related Natural Language Processing tasks to make their responses more informative. We refer the reader to \cite{alexa_prize} for detailed analysis of these systems. In the space of academic datasets, \cite{lowe2015} report results on the Ubuntu dataset using manpages as external knowledge whereas \cite{KGNCM} use Foursquare tips as external knowledge for social media conversations. However, unlike our work both these works do not create a new dataset where the responses are explicitly linked to a knowledge source. The infusion of external knowledge in both these works is post facto (as opposed to our work where we take a bottom-up approach and explicitly create a dataset which allows exploitation of background knowledge). Additionally, existing large-scale datasets are noisy as they are extracted from online forums which are inherently noisy. In contrast, since we use crowdsourcing, the extent of noise is reduced since there are humans in the loop who were explicitly instructed to use only clean sentences from the external knowledge sources.
We would also like to mention some existing works such as \cite{hehe2017,dealnodeal,edina} which have used crowdsourcing for creating conversation datasets. In fact, our data collection method is inspired by the work of \cite{edina} where the authors use self-dialogs to collect conversation data about movies, music and sports. They are referred to as self-dialogs because the same worker plays the role of both parties in the conversation. However, our work differs from \cite{edina} as we provide explicit background knowledge sources to the workers from where they can copy text with the addition of suitable prefixes and suffixes to generate appropriate coherent responses.
\section{Dataset}
\label{sect:pdf}
In the following sub-sections we describe the various stages involved in collecting our dataset.
\subsection{Curating a list of popular movies}
We created a list of 921 movies containing (i) top 10 popular movies within the past five years, (ii) top 250 movies as per IMDb rankings, (iii) top 10 movies in popular genres, and (iv) other popular movie lists made available elsewhere on the Internet. These movies belonged to 22 different genres such as sci-fi, action, horror, fantasy, adventure, romance, \textit{etc.} thereby ensuring that our dataset is not limited to a specific genre. We considered those movies for which enough background information such as plots, reviews, comments, facts, \textit{etc.} were available on the Internet irrespective of whether they were box-office successes or not. Please find the respective urls in the Appendix.
\subsection{Collecting background knowledge} For each movie, we collected the following background knowledge:
\textbf{1. Review (R):} For each movie, we asked some in-house workers to fetch the top 2 most popular reviews for this movie from IMDb using the \textit{sort by Total Votes} option. We also instructed them to avoid choosing reviews which were less than 50 words but this was typically never the case with popular reviews.
\textbf{2. Plot (P):} For each movie,
we extracted information about the ``Plot'' of the movie from the Wikipedia page of the movie. Wikipedia pages of movies have an explicit section on ``Plot'' making it easy to extract this information using scripts.
\textbf{3. Comments (C):} Websites like \textit{Reddit} have a segment called ``official discussion page about X'' (where X is a movie name) containing small comments about various aspects of movie. We identified such pages and extracted the first comment on every thread on this page. We bundled all these comments into a single text file and refer to it as the resource containing ``Comments''. For a few movies, the official discussion page was not present in which case we used the review titles of all the IMDb reviews of the movie as comments. The difference between Reviews and Comments is that a Review is an opinion piece given by one person thus typically exhibiting one sentiment throughout while Comments include opinions of several people about the same movie ensuring that positive, negative and factual aspects of the movie are captured as well as some banter.
\textbf{4. Meta data or Fact Table (F):} For each movie, we also collected factual details about the movie, \textit{viz.}, box office collection, similar movies (for recommendations), awards and tag-lines from the corresponding IMDb pages and Wikipedia Infoboxes. Such information would be useful for inserting facts in the conversation, for example, \textit{ ``Did you know that the movie won an Oscar?}''. We included only 4 fields in our fact table instead of showing the entire Wikipedia Infobox to reduce the cognitive load on turkers who already had to read the plot, reviews and comments of the movie.
\subsection{Collecting conversation starters} During our initial pilots, we observed that if we asked the workers to converse for at least 8 turns, they used a lot of the initial turns in greetings and general chit-chat before actually chatting about a movie. To avoid this, we collected opening statements using Amazon Mechanical Turk (AMT) where the task for the workers was to answer the following questions ``\textit{What is your favorite scene from the movie X ?}'', ``\textit{What is your favorite character from the movie X ?}'' and ``\textit{What is your opinion about the movie X?}'' (X is the movie name). We paid the workers 0.04\$ per movie and showed the same movie to 3 different workers, thereby collecting 9 different opening statements for every movie. By using these statements as conversation starters in our data collection, the workers could now directly start conversing about the movie.
\subsection{Collecting background knowledge aware conversations via crowdsourcing}
Our aim is to create a conversation dataset wherein every response is explicitly linked to some structured or unstructured background knowledge. Creating such a dataset using dedicated in-house workers would obviously be expensive and time consuming and so we decided to use crowdsourcing. However, unlike other NLP and Vision tasks, where crowdsourcing has been very successful, collecting conversations via crowdsourcing is a bit challenging. The main difficulty arises from the fact that conversation is inherently a task involving two persons but it is hard to get two workers to synchronize and chat on AMT. We did try a few pilot experiments where we setup a server to connect two AMT workers but we found that the probability of two workers simultaneously logging in was very low. Thus, most workers logged in and left in a few seconds because no other worker joined simultaneously. Finally, we took inspiration from the idea of self chats \cite{edina} in which, the same worker plays the role of both Speaker 1 and Speaker 2 to create the chat.
In the above self chat setup, we showed every worker 3 to 4 resources related to the movie, \textit{viz.}, plot (P), review (R), comments (C) and fact table (F). We also showed them a randomly selected opening statement from the 9 opening statements that we had collected for each movie and requested them to continue the conversation from that point. The workers were asked to add at least 8 utterances to this initial chat. While playing the role of Speaker 1, the worker was not restricted to copy/modify sentences from the background resources but was given the freedom to create (write) original sentences. However, when playing the role of Speaker 2, the worker was strictly instructed to copy/modify sentences from the shown resources such that they were relevant in the current context of the conversation. The reason for not imposing any restrictions on Speaker 1 was to ensure that the chats look more natural and coherent. Further, Speaker 2 was allowed to add words at the beginning or end of the span selected from the resources to make the chats more coherent and natural (for example, see the prefix in utterance 2 of Speaker 2 in Figure \ref{dataset_example}).
We paid the workers 40 cents for every chat. Please refer to the Appendix for the instruction screen shots.
\subsection{Verification of the collected chats}
Every chat that was collected by the above process was verified by an in-house evaluator to check if the workers adhered to the instructions and produced coherent chats. Since humans typically tend to paraphrase the background knowledge acquired by reading articles, one could argue that such conversations may not look very natural because of this restriction to copy/modify content from the provided resources. To verify this, we conducted a separate human evaluation wherein we asked 15 in-house evaluators to read conversations (without the background resources) from our dataset and rate them on five different parameters. Specifically, they were asked to check if the conversations were 1) \textbf{intelligible:} \textit{i.e.}, an average reader could understand the conversation 2) \textbf{coherent:} \textit{i.e.}, there were no abrupt context switches 3) \textbf{grammatically correct} 4) \textbf{on-topic:} \textit{i.e.}, the chat revolved around the concerned movie with digression limited to related movies/characters/actors and 5) \textbf{natural two-person chats:} \textit{i.e.}, the role-play setup does not make the chat look unnatural. These evaluators were post-graduate students who were fluent in English and had watched at least 100 Hollywood movies. We did not give them any information about the data creation process.
We used a total of 500 chats for the evaluation and every chat was shown to 3 different evaluators. The evaluators rated the conversations on a scale of 1 (very poor) to 5 (very good). We computed inter-annotator agreement using the mean linearly weighted Cohen's $\kappa$ \cite{cohen_kappa} and mean Krippendorff's $\alpha$ \cite{kripp_alpha}. The average rating for each of the 5 parameters along with the inter annotator agreement are reported in Table \ref{human_evaluation} and are very encouraging.
\subsection{Statistics}
\begin{table}
\centering
\begin{tabular}{llll}
\hline
\textbf{Metric } & \textbf{Rating} & \textbf{$\alpha$} & \textbf{$\kappa$}\\ \hline
\textbf{Intelligible} & 4.47 $\pm$ 0.52 & 0.70 & 0.69\\
\textbf{Coherent} & 4.33 $\pm$ 0.93 & 0.57 &0.71\\
\textbf{Grammar} & 4.41 $\pm$ 0.56 & 0.60 & 0.69\\
\textbf{Two-person-chat} & 4.47 $\pm$ 0.46 & 0.64 &0.70\\
\textbf{On Topic} & 4.57 $\pm$ 0.43 & 0.72 &0.70\\ \hline
\end{tabular}
\caption{Average human evaluation scores with standard deviations for conversations (scale 1-5). We also report mean Krippendorff's $\alpha$ and mean Cohen's $\kappa$}
\label{human_evaluation}
\end{table}
In Table \ref{basic_statistics}, we show different statistics about the dataset collected using the above process. These include average number of utterances per chat, average number of words per utterance, and so on followed by the statistics of the different resources which were used as background knowledge. Please note that the \# unique Plots and \# unique Reviews correspond to unique paragraphs while the \# unique Comments is the count of unique sentences. We observed that 41.2\%, 34.6\%, 16.1\% and 8.1\% of Speaker 2 responses came from Reviews, Comments, Plots and Fact Table respectively.
\begin{table}
\centering
\begin{tabular}{ll}
\hline
\#chats & 9071 \\
\#movies & 921 \\
\#utterances & 90810 \\
Average \# of utterances per chat & 10.01 \\
Average \# of words per utterance & 15.29 \\
Average \# of words per chat & 153.07 \\
Average \# of words in Plot & 186.10 \\
Average \# of words in Review & 384.44 \\
Average \# of words in Comments & 123.81 \\
Average \# of words in Fact Table & 33.47 \\
\# unique Plots & 5157 \\
\# unique Reviews & 1817 \\
\# unique Comments & 12740 \\\hline
\end{tabular}
\caption{Statistics of the dataset}
\label{basic_statistics}
\end{table}
\section{Models}
We evaluate three different types of models as described below. Since these are popular existing models, we describe them very briefly below and refer the reader to the original papers for more details. Note that in this work we merge the comments, reviews, plots and facts into one single document and refer to it as background knowledge. In the rest of the paper, when we refer to a \textit{resource} we mean this single document which is a merger of all the resources unless specified otherwise.
\subsection{Generation based models} We use the standard Hierarchical Recurrent Encoder Decoder model (HRED) \cite{HRED} instead of its variant \cite{vhred} as the standard model performs only slightly poorly than the variant and is much easier to implement. It decomposes the context of the conversation as two level hierarchy using Recurrent Neural Networks (RNN). The lower RNN encodes individual utterances (sequence of words) which is then fed into the higher level RNN as a sequence of utterances. The decoder RNN then generates the output based on this hierarchical context representation.
\subsection{Generate-or-Copy models}
Get To The Point (GTTP) \cite{GTTP} proposed a hybrid pointer generator network for abstractive summarization that learns to copy words from the source document when required and otherwise generates a word like any sequence-to-sequence model. In the summarization task, the input is a \textit{document} and the output is a \textit{summary} whereas in our case the input is a \{\textit{document, context}\} pair and the output is a \textit{response}. Here, the context includes the previous two utterances and the current utterance. We modified the architecture to suit our task. We use an RNN to compute the representation of the document (like the original model) and introduce another RNN to compute a representation of the context by treating it as a single sequence of words. The decoder which is also an RNN then uses the document representation, context representation and its own internal state representation to compute a (i) probability score which indicates whether the next word should be copied or generated (ii) probability distribution over the vocabulary if the next word needs to be generated and (iii) probability distribution over the input words if the next word needs to be copied. These three probability distributions are then combined to produce the next word in the response.
\subsection{Span prediction models}Bi-directional Attention Flow Model (BiDAF) \cite{BIDAF} model is a QA model which was proposed in the context of the SQuAD dataset \cite{SQUAD}. Given a \textit{document} and a \textit{question}, the model uses a six-layered architecture to predict the span in the document which contains the answer. We can use their model as it is for our task without any modifications by simply treating the \textit{context} as the \textit{question} and the \textit{resource} as the \textit{document}.
We chose to evaluate on the modified generate-or-copy model instead of other variants such as \cite{KGNCM,lowe2015} as the modified model already contains the extra encoder for background model which is present in these models. Moreover, the modified model uses a hybrid copy-or-generate decoder which is well-suited to our task.
\section{Experimental Setup}
In this section we describe the train-validation-test splits, the process used for creating training instances, the manner in which the models were trained using our data and the evaluation metrics.
\subsection{Creating train/valid/test splits} On average we have 9.14 chats per movie. We divide the collected chats into train, validation, and test splits such that all the chats corresponding to a given movie are in exactly one of the splits. This ensures that a movie seen in the test or validation set is never seen at training time. We create the splits such that the percentage of chats in the train-validation-test set is roughly 80\%-10\%-10\%.
\subsection{Creating training instances} For each chat in the training data, we construct training instances of the form \{\textit{resource, context, response}\} where the \textit{context} is taken as previous two utterances and current utterance. We consider only the even numbered utterances as training examples as they are generated from the background resources thus emulating a human-bot setup. If a chat has 10 turns, we will have 5 instances. The task then is to train a model which can predict these even numbered responses. At test time the model is shown \{\textit{resource, context}\} and predicts the response. Note that, HRED will ignore the \textit{resource} and only use \{\textit{context, response}\} as input-output pairs. BiDAF and GTTP will use \{\textit{resource, context, response}\} as training data with relevant \textit{span} instead of \textit{response} for BiDAF.
\subsection{Merging resources into a single document} As stated earlier, we simply merge all the background information to create a single document which we collectively refer to as \textit{resource}. For the BiDAF model, we had to restrict the length of the resource to 256 words because we found that even on a K80 GPU with 12GB RAM, this model gives an out of memory error for longer documents. We found this to be a severe limitation of this and other span based models (for example, R-Net \cite{RNET}) .
We experimented with three methods of creating this resource. The first method \textit{oracle} uses the actual resource (plot or comments or reviews) from which the next response was generated as a resource. If that resource itself has more than 256 words then we truncate it from the beginning and the end such that the span containing the actual response is contained within the retained 256 words. The number of words that are discarded from the start or the end is chosen at random so that the correct spans do not end up in similar positions throughout the dataset. The next two methods \textit{mixed-short} and \textit{mixed-long} are created by merging the individual resources. We retain each resource in the merged document proportional to its length. (\textit{i.e,}if there are 400 words in the plot, 200 words in the review and 100 in the comments, the merged resource will contain contiguous sentences from these three resources in the ratio of 4:2:1.) Further, we ensure that the merged resource contains the actual response span. In this way, we create \textit{mixed-short} with 256 words and \textit{mixed-long} with 1200 words (the maximum length of the merged resources). We will henceforth denote \textit{oracle}, \textit{mixed-long} and \textit{mixed-short} using \lq (o) \rq, \lq (ms) \rq and \lq (ml) \rq respectively. We report results for BiDAF(o), BiDAF (ms), GTTP (o) and GTTP (ml).
\begin{table*}
\centering
\begin{tabular}{l|cc|cc|cc|cc|cc}
\hline
\textbf{Model} & \multicolumn{2}{c|}{\textbf{F1}} & \multicolumn{2}{c|}{\textbf{BLEU}} & \multicolumn{2}{c|}{\textbf{Rouge-1}} & \multicolumn{2}{c|}{\textbf{Rouge-2}} & \multicolumn{2}{c}{\textbf{Rouge-L}} \\\hline
\textbf{HRED } & - & - & 5.23 & 5.38 & 24.55 & 25.38 & 7.61 & 8.35 & 18.87 & 19.67 \\\hline
\textbf{GTTP (o) }& - & - & 13.92 & 16.46 & 30.32 & 31.6 & 17.78 & 21.21 & 25.67 & 27.83 \\
\textbf{GTTP (ms)} & - & - & 11.05 & 15.68 & 29.66 & 31.71 & 17.70 & 19.72 & 25.13 & 27.35 \\
\textbf{GTTP (ml)} & - & - & 7.51 & 8.73 & 23.20 & 21.55 & 9.91 & 10.42 & 17.35 & 18.12 \\\hline
\textbf{BiDAF (o)} & 39.69 & 47.18 & 28.85 & 34.98 & 39.68 & 46.49 & 33.72 & 40.58 & 35.91 & 42.64 \\
\textbf{BiDAF (ms)} & 45.73 & 51.35 & 32.95 & 39.39 & 45.69 & 50.73 & 40.18 & 45.01 & 43.46 & 46.95\\\hline
\end{tabular}
\caption{Performance of the proposed models on our dataset. The figures on the left in each column indicate scores on single-reference test dataset while the figures on the right denote scores on multi-reference dataset.}
\label{results}
\end{table*}
\subsection{Evaluation metrics} As HRED and GTTP models are generation based models we use BLEU-4, ROUGE-1, ROUGE-2 and ROUGE-L as the evaluation metrics. For BiDAF we use the above metrics by comparing the predicted span with the reference span. For BiDAF, we also report F1 as stated in \cite{SQUAD}.
In addition to the automatic evaluation, we also collected human judgments using 100 test responses generated for every model for every setup (oracle, mixed-short, mixed-long). These evaluators had the same qualifications as the evaluators who earlier helped us evaluate our dataset. They were asked to rate the response on scale of 1 to 5 (with 1 being the least) on the following four metrics: (1) Fluency(Flu), (2) appropriateness/relevance (apt) of the response in the current context language (3) humanness (Hum) of the response, \textit{i.e.,} whether the responses look as if they were generated by a human (4) and specificity (spec) of the response, \textit{i.e.}, whether the model produced movie-specific responses or generic responses such as ``This movie is amazing''. We report these results in Table \ref{human_evaluation_model}.
\begin{table}[]
\centering
\begin{tabular}{l|llll}
\hline
\textbf{Model} & \textbf{Hum} &\textbf{Apt} & \textbf{Flu} & \textbf{Spec} \\\hline
\textbf{HRED} & 3.08 & 2.49 & 2.64 & 2.06 \\
\textbf{GTTP (o)} & 4.10 & 3.73 & 4.03 & 3.33 \\
\textbf{GTTP (ml)} &2.93 &2.97 &3.42 &2.60 \\
\textbf{BiDAF (o)} & 3.78 & 3.71 & 4.05 & 3.76 \\
\textbf{BiDAF(ms)} & 3.41 & 3.38 & 3.47 & 3.30 \\\hline
\end{tabular}
\caption{Human evaluation results on the model performances.}
\label{human_evaluation_model}
\end{table}
\subsection{Collecting multiple reference responses}
One common issue with evaluating dialog systems is that existing datasets typically contain only one reference response whereas in practice several responses can be correct in a given context.
To solve this to a certain extent, we collected three reference responses for every Speaker 2 utterance in our dataset (note that Speaker 2 is treated as the bot while training/testing our models). We show the previous utterances ending with Speaker 1's response and ask workers to provide three appropriate responses from the given resources. We found that in some cases there was only one appropriate response like factual response and the workers could not provide multiple references . In this way we were able to create a multiple reference test set where 78.04\% of the test instances have multiple responses. In Table \ref{results}, we report two sets of scores based on single-reference test dataset and multi-reference test dataset. While calculating the scores for multi-reference dataset, we take the maximum score over multiple reference responses.
Please refer to the Appendix section for the details of the model, hyperparameters, example of multiple references in our dataset and sample outputs produced by different models.
\section{Results and Discussion}
In this section, we discuss the results of our experiments as summarized in Tables \ref{results} and \ref{human_evaluation_model}.
\textbf{Generation based models v/s Span prediction models:}
We compare the generation based models and span prediction models only based on results in the \textit{oracle} setting. Here, the span based model (BiDAF) outperforms the generation based models (HRED and GTTP). This confirms our belief that the natural language generation (NLG) capabilities of current generation based models are far from being acceptable even in case of generate-or-copy modes. This also emphasizes the importance of this data which allows building models which can exploit well-formed sentences in the background knowledge and reproduce them with minor modifications instead of generating them from scratch. While the results for BiDAF are encouraging, we reiterate that it does not scale to longer documents (we were not able to run it in the \textit{mixed-long} setting). We still need much better models as BiDAF on SQuAD dataset gives an F1 of 81.52 \% which is much higher than the results on our dataset. Further, note that using the predicted span as a response is not natural. This is evident from human likeliness (Hum) score of GTTP (o) being higher than both the BiDAF models. We need models which can suitably alter the span to retain the coherence of the context.
\textbf{Effect of including background knowledge:} We observe that there isn't much difference between the performance of HRED which does not use any background knowledge when compared to GTTP (ml) which actually uses a lot of background knowledge. However, there is a substantial difference between the performance of HRED and GTTP (o) which uses only the relevant background knowledge. Further, without background knowledge, HRED learns to produce very generic responses (Spec score = 2.06). This shows that the background knowledge is important, but the models should learn to focus on the right background knowledge relevant to the current context. Alternately, we can have a two-stage network which first predicts the right resource (plot, review, comments) from which the span should be selected and then selects the span from this chosen resource.
\textbf{\textit{Oracle} v/s \textit{mixed-short} resource:}
We observe that the performance of BiDAF (ms) is actually better than BiDAF (o) even when the resource length for both is 256 words. We would expect a poor performance for BiDAF (ms) as the resource has more noise because of the sentences from irrelevant resources. However, we speculate the model learns to regard irrelevant sentences as noise and learns to focus on sentences corresponding to the correct resource resulting in improved performance (however, this is only a hypothesis and it needs to be verified).
We realize that this is clearly a poor baseline and we need better span prediction based models which can work with longer documents. At the same time, GTTP (o) and GTTP (ms) have comparable (yet poor) performance. There is no co-attention mechanism in this model which can effectively filter out noisy sentences.
\textbf{Observations from the copy-and-gen model:}
We observed that this model produced sentences where on average of 82.18\% (\textit{oracle}) and 71.95\% (\textit{mixed-long}) of the tokens were copied. One interesting observation was that it easily learns to copy longer contiguous sequences one word at a time.
However, as is evident from the automatic evaluation metrics, in many cases, the `copied' spans are not relevant to the current context.
\\\\
\indent \textbf{Evaluating with multiple references}:
When considering multiple references, the performance numbers as reported in Table \ref{results} indeed improve. This shows the importance of having multiple references and the need to develop metrics which account for multiple dissimilar references.
\section{Conclusion}
We introduce a new dataset for building dialog systems which would hopefully allow the community to take a fresh look at this task. Unlike existing datasets which only contain a sequence of utterances, in our dataset each response is explicitly linked to some background knowledge. This mimics how humans converse by recalling information from their background knowledge and use it appropriately in the context of the conversation. Using this dataset, we evaluated models belonging to three different paradigms, \textit{viz.}, generation based models, generate-or-copy models and span prediction models. Our results suggest that the NLG capabilities of existing seq-to-seq models are still far from desirable while span based models which completely bypass the process of NLG show some promise but with clear scope for improvement.
Going forward, we would like to build models which are a hybrid of span prediction models and generation models. Specifically, we would like to build models which can learn to copy a large sequence from the input instead of one word at a time. Another important aspect is to build less complex models which can handle longer documents. For example, the BiDAF model has an expensive outer product between two large matrices which makes it infeasible for long documents (because the size of these matrices grows with the length of the document). Alternately, we would like to build two-stage models which first select the correct resource from which the next response is to be generated and then generate or copy the response from the resource.
\section*{Acknowledgements}
We would like to thank Department of Computer Science and Engineering, and Robert Bosch Center for Data Sciences and Artificial Intelligence, IIT Madras (RBC-DSAI) for providing us with adequate resources. We also thank Gurneet Singh and Sarath Chandar for helping us in the data collection phase two and three respectively. Lastly, we thank all the AMT workers around the world and our in-house evaluators.
\cleardoublepage
\section*{Appendix}
\section*{Model details - GTTP}
Since we modified the existing architecture of Get to the Point \cite{GTTP}, we now provide details of the same.
In the summarization task, the input is a \textit{document} and the output is a \textit{summary} whereas in our case the input is a \{\textit{resource/document, context}\} pair and the output is a \textit{response}. Note that the context includes the previous two utterances (dialog history) and the current utterance. Since, in both the tasks, the output is a sequence (\textit{summary} v/s \textit{response}) we don't need to change the decoder (\textit{i.e.}, we can use the decoder from the original model as it is). However, we need to change the input fed to the decoder. Similar to the original model, we use an RNN to compute the representation of the document. Let $N$ be the length of the document then the RNN computes representations $h_1^r, h_2^r, ..., h_N^r$ for all the words in the resource (we use the superscript $r$ to indicate resource). The final representation of the resource is then the attention weighted sum of these word representations:
\begin{equation}
\label{attention_on_document}
\begin{split}
e_i^t &= v^T tanh(W_r h_i^r + U s_t + b_r)\\
a^t &= softmax(e^t)\\
r_t &= \sum_i a_i^t h_i^r
\end{split}
\end{equation}
where $s_t$ is the state of the decoder at the current time step. In addition, in our case, we also have the context of the conversation apart from the document (resource). Once again, we use an RNN to compute a representation of this context. Specifically, we consider the previous $k$ utterances as a single sequence of words and feed these to an RNN. Let $M$ be the total length of the context (\textit{i.e.}, all the $k$ utterances taken together) then the RNN computes representations $h_1^c, h_2^c, ..., h_M^c$ for all the words in the context (we use the superscript $c$ to indicate context). The final representation of the context is then the attention weighted sum of these word representations:
\begin{equation}
\begin{split}
f_i^t &= v^T tanh(W_c h_i^c + V s_t + b_c)\\
m^t &= softmax(f^t)\\
c_t &= \sum_i m_i^t h_i^c
\end{split}
\end{equation}
where $s_t$ is the state of the decoder at the current time step.
The decoder then uses $r_t$ (document representation), $c_t$ (context representation) and $s_t$ (decoder's internal state) to compute a probability distribution over the vocabulary $P_{vocab}$. In addition the model also computes $p_{gen}$ which indicates that there is a probability $p_{gen}$ that the next word will be \textit{generated} and a probability $(1 - p_{gen})$ that the next word will be \textit{copied}. We use the following modified equation to compute $p_{gen}$
\begin{equation}
p_{gen} = \sigma(w_{r}^T r_t + w_{c}^T c_t + w_s^T s_t + w_x^T x_t + b_{g})
\end{equation}
where $x_t$ is the previous word predicted by the decoder and fed as input to the decoder at the current time step. Similarly, $s_t$ is the current state of the decoder computed using this input $x_t$. The final probability of a word $w$ is then computed using a combination of two distributions, \textit{viz.}, ($P_{vocab}$) as described above and the attention weights assigned to the document words as shown below
\begin{equation}
P(w) = p_{gen} P_{vocab}(w) + (1 - p_{gen}) \sum_{i:w_i=w} a_i^t
\end{equation}
where $a_i^t$ are the attention weights assigned to every word in the document as computed in Equation \ref{attention_on_document}. Thus, effectively, the model could learn to copy a word $i$ if $p_{gen}$ is low and $a_i^t$ is high.
\section*{Example from the multiple reference test set}
As seen from the Table \ref{multi_test}, the given chat on ``Secret Life of Pets'' can have multiple responses for Speaker 2. Notice how Reference 1 talks against low critique scores thus emphasizing that he was totally impressed by the movie while Reference 4 has neutral opinion about the same. At the same time, Reference 3 talks about movie specific details like his favorite character while Reference 4 gives a personal opinion. All these four responses are valid given the current context.
\begin{table*}[]
\centering
\begin{tabular}{|l|l|}
\hline
Movie Name & The Secret Life of Pets \\ \hline
Chat & \begin{tabular}[c]{@{}l@{}}Speaker1 : What do you think about the movie? \\ Speaker2 : I think it was comical and entertaining. \\ Speaker1 : It delivered what was promised.\end{tabular} \\ \hline
Reference 1 & I agree! I'm surprised this film got such a low overall score by users. \\ \hline
Reference 2 & My favorite character was Gidget! She was so much fun and so loyal to her friends! \\ \hline
Reference 3 & \begin{tabular}[c]{@{}l@{}}Yes! As a Great Dane owner, I often wonder what my dogs are thinking. \\ It was fun to see this take on it.\end{tabular} \\ \hline
Reference 4 & It was full of cliches with a predictable story, but with some really funny moments. \\ \hline
\end{tabular}
\caption{Multiple references for the given chat.}
\label{multi_test}
\end{table*}
\section*{Hyper-parameters}
We describe the hyperparameters that we used for each model in this sub section. Following the original paper, we trained the HRED model using Adam \cite{adam} optimizer with an initial learning rate of 0.001 on a minibatch of size 16. We used a dropout \cite{dropout} with a rate of 0.25. For word embeddings we use pre-trained GloVe \cite{glove} embeddings of size 300. For all the encoders and decoders in the model we used Gated Recurrent Unit (GRU) with 300 as the size of the hidden state. We restricted our vocabulary size to 20,000 most frequent words.
We followed the hyperparameters mentioned in the original paper and trained GTTP using Adagrad \cite{adagrad} optimizer with an initial learning rate of 0.15 and an initial accumulator value of 0.1 on a minibatch of size 16. For the encoders and decoders we used LSTMs with 256 as the size of the hidden state. To avoid vanishing and exploding gradient problem we use gradient clipping with a maximum gradient norm of 2. We used early stopping based on the validation loss.
Again following the original paper, we trained BiDAF using AdaDelta \cite{adadelta} optimizer with an initial learning rate of 0.5 on a minibatch of size 32. For all encoders, we use LSTMs with 256 as the size of the hidden state. We used a dropout \cite{dropout} rate of 0.2 across all LSTM layers, and for the linear transformation before the softmax for the answers. For word embeddings we use pre-trained GloVe embeddings of size 100. For both GTTP and BiDAF, we had to restrict context length to 65 tokens for fair comparison. Note that GTTP can scale beyond 65 tokens but BiDAF cannot.
\section*{Sample responses produced by the models}
As seen from Table \ref{example-chat}, HRED isn't able to produce responses that correspond to the given movie or the given context as it lacks any notion of background knowledge associated with it. We will not consider HRED for the following discussion. In Example 1, we can clearly see that only GTTP (oracle) matches with the ground truth. The remaining three models produce varied outputs which are still relevant to the context. In Example 2, we observe that prediction based models produce appropriate recommendation because of better context-document mapping mechanisms. Both the GTTP models produce responses which are copied but irrelevant to the context. At the same time, just producing spans without any structure isn't natural. This explains the need for hybrid models. Example 3 asks for the backstory of a character which requires complex reasoning. The model has to first understand the plot of the movie to locate the sentences which talk about that character's past. As seen from the responses of the given models, all of them except GTTP (ml) pick sentences which are relevant to the character but do not answer the required question. As discussed earlier, these models rely on word-overlap and thus possess limited natural language understanding. Thus, we need models which are capable of going beyond word overlap and producing responses even in such complex scenarios.
\begin{table*}
\centering
\begin{tabular}{|l|p{4cm}|p{4cm}|p{4cm}|}
\hline
Movie & Example 1 (Creed) & Example 2 (Ted) & Example 3 (Jaws) \\ \hline
Context &
S1 : Which was the best scene in the movie according to you? \newline S2 : I liked the one in which Rocky trains and runs.\newline S1 : It was great when the match commences, and although Donnie takes a few hits, he defeats Leo. \newline S2 : Creed goes the distance as an exceptional crowd pleaser. \newline S1 : I agree with grit, style and substance. There's no other way to say it; Creed is a knockout from start to finish. & S2 : The scene where they're yelling at each other out the window was gold \newline S1 : Thats a similar scene from family guy. I couldnt stop laughing. \newline S2 : I thought every scene was gold. I also couldn't believe Liam Neeson was in the movie. \newline S1 : I will love to pick something similar this evening. Do you have a recommendation? & S1 : What do you think about the characters in this movie? \newline S2 : My favorite character was Brody, he was such a fun character.\newline S1 : I didn't get Quint's backstory. \\ \hline
Ground Truth & It wholeheartedly captures what was so brilliant about the first film: the characters. & Yes. You can try 22 Jump Street. & Quint, aboard that ship in World War II when it was sunk by torpedoes, had witnessed the deaths of roughly 800 men that day, many of whom were eaten by sharks as they struggled helplessly in the water. \\ \hline
HRED & yeah , but as the ark is launched , and you are right taxi driver , runs with the beauty , \textless{}UNK\textgreater \textless{}UNK\textgreater and \textless{}UNK\textgreater \textless{}UNK\textgreater . & the plot is a classic in the film , it 's not that it 's an original and cinematic experience . & i 'm not ashamed to admit it , mary poppens is bangin ! ! ! \\ \hline
BiDAF(o) & I can't say enough great things about writer / director Ryan Coogler. & The Hangover & Hooper notices the shark beginning to circle the boat, and Quint rushes out for a look. \\ \hline
BiDAF(ms) & I loved how they introduced each opponent with his record and accomplishments. & 21 Jump Street & The mood suddenly darkens as Quint admits that the former tattoo represented the US Navy cruiser \\ \hline
GTTP(o) & it wholeheartedly captures what was so brilliant about the first film : the characters . & \$ 218,628,680 i think . & brody is stunned and alerts quint . \\ \hline
GTTP(ml) & i can't say enough great things about writer / director ryan coogler . & this movie had way too much product placement . & jaws' is the original summer blockbuster \\ \hline
\end{tabular}
\caption{Examples produced by various models. S1 denotes Speaker1 while S2 denotes Speaker2. \lq o\rq,\lq ml\rq ~and \lq ms\rq ~represent \textit{oracle}, \textit{mixed-long} and \textit{mixed-short} versions of the dataset respectively.}
\label{example-chat}
\end{table*}
\if 0
\section*{Collection of popular movies list}
Following are the urls used to curate the popular movies list:\\
IMDb top 250 :\\ \url{https://www.imdb.com/chart/top}
Popular movies by genre: \\
\url{https://www.imdb.com/feature/genre/}\\
Other popular movies lists: \\
\url{http://www.filmsite.org/guinness.html}\\
\url{https://www.thetoptens.com/best-movie-genres/}\\
\url{http://1001films.wikia.com/wiki/The_List}\\
The imdb ids of the movies will be made available with every example.
\fi
\section*{Data Collection Interfaces used on Amazon Mechanical Turk}
As explained in the paper, we resorted to AMT for three types of data collection \textit{viz.} collection of opening statements as discussed in stage 3 of the dataset collection (Figure \ref{phase_3}), the actual chat data collection explained in stage 4 of the procedure and additional responses collected for the test dataset. We will now show the instruction screens. The html files for the same will be released along with the code.
\begin{figure*}
\centering
\resizebox{7.5cm}{11.2cm}{
\includegraphics{ins2.png}
}
\caption{Instruction screen for collection of multiple-responses for the same chat for the test dataset.}
\label{multi-response.png}
\end{figure*}
\begin{figure*}
\centering
\resizebox{\textwidth}{8cm}{
\includegraphics{instruction_screen.png}
\label{}
}
\caption{Instruction screen for chat data collection from Phase 4 of the dataset collection procedure}
\end{figure*}
\begin{figure*}
\centering
\resizebox{9.5cm}{13cm}{
\includegraphics{dunkirk1.png}
}
\caption{Background resources for the example chat shown to the workers on AMT}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[scale=0.3]{dunkirk2.png}
\caption{Example chat shown to the workers on AMT}
\end{figure*}
\begin{figure*}
\centering
\resizebox{\textwidth}{8cm}{
\includegraphics{ins1.png}
}
\caption{Instruction screen for opening statement collection from the stage of the dataset collection procedure}
\label{phase_3}
\end{figure*}
\newpage
|
1,116,691,500,912 | arxiv | \section{Introduction}\label{sec:intro}
The popularity of cloud computing is clearly demonstrated by its wide adoption: for example, nearly 60\% of the Apache Spark installations are deployed in the Cloud~\cite{cloudera2018}. The benefit from embracing the Cloud paradigms typically lies in the reduced cost of ownership for the infrastructure, that may reach up to 66\% for some cases~\cite{aws2012}.
A critical point for the IaaS Cloud infrastructures is the monitoring and management of the virtual machines (VMs) and physical nodes in a data center. Even just monitoring may present scalability issues, due to the sheer amount of data involved~\cite{NRDC:2014}. In a similar way, the optimization problem for mapping VMs over the infrastructure may exhibit an unmanageable dimensionality forcing the Cloud provider to introduce some over-simplification~\cite{Katz}. A common problem that hinders the scalability of monitoring and management in Cloud data centers is considering each VM as a black-box independent from the others.
Effective proposal to improve the scalability of Cloud monitoring and management using a class based-approach have been recently introduced~\cite{Canali:TGCN, Canali:NCCA15}. These class-based solutions leverage the observation that VMs hosting the same software component of the same application exhibit similar behavior with respect to resource utilization. Hence, by taking into account the similarity in VMs behavior it is possible, for example, to increase by nearly one order of magnitude the number of VMs that can be considered in the of the data-center VMs allocation problem \cite{Canali:NCCA15}.
However, available clustering techniques to identify VMs that exhibit similar behavior, show a trade-off between accuracy of VMs group identification and the amount of observations (and hence the time) required to reach an accurate classification~\cite{Canali:TCC18}. This issue hinders the application of a class-based approach outside a static scenario characterized by long-term commitments~\cite{Durkee:2010}, where cloud customers purchase VMs for extended periods of time. An attempt to address this trade-off has been made through a technique named AGATE (Adaptive Gray Area-based TEchnique)~\cite{Canali:TCC18}, that adapts the amount of observations to the level of certainty of the identification of a VM as belonging to a cluster. However, also this technique does not ensure an upper bound on the time to identify VMs.
In this paper we propose a different approach based on a classifier that uses a deep learning technique to identify the VMs relying on a model obtained from preliminary training. While the need to tune the neural network reduces the flexibility compared to the purely clustering-based approach proposed in~\cite{Canali:TCC18}, the proposed classifier is significantly faster and can identify VMs with a perfect accuracy, observing their behavior for just a few minutes, compared to the hours required in the alternative approaches.
We tested the proposed classifier using traces derived from the work in~\cite{Canali:TCC18}. The results confirm that our proposal significantly outperforms the alternatives in terms of accuracy in the classification and in the time required to reach that accuracy.
The remainder of this paper is organized as follows.
Section~\ref{sec:problem} provides a description of the proposed deep-learning-based classifier including both the model and the implementation.
Section~\ref{sec:results} describes the experimental results. Section~\ref{sec:related} discusses the related work and Section~\ref{sec:conclusions} concludes the paper with some final remarks and outlines future research directions.
\section{Methodology}\label{sec:problem}
Our contribution comprises two learnable models for VM identification, which we call \textit{DeepConv} and \textit{DeepFFT}. While the former employs a convolutional neural network to process time signals, the latter combines the Fast Fourier Transform operator and a convolutional neural network to analyze the VM behaviour in the frequency domain.
In the following, we will first outline the basic elements of a deep learning approach to classification. Next, we will describe the DeepConv network and subsequently we will outline the DeepFFT model by highlighting the differences with the previous approach.
\subsection{Model overview}
We start our analysis with a discussion of the Deep learning approach to classification that is the core of the paper.
To provide a consistent description, we will refer to the symbols and the nomenclature shown in Table \ref{table:tab1}. As further information, we provide the dimensionality/value ranges for the main elements of the model. We do not provide this information for the main layers and components of the deep-learning model (bottom part of the table) because they are described in full detail in the following of the paper.
\begin{table}[ht]
\resizebox{\columnwidth}{!}{
\begin{tabular}{|c|c|c|}
\toprule[1pt]
\textbf{Symbol} & \textbf{Meaning} & \textbf{Dimensionality} \\
\midrule[1pt]
$M$ & Number of dataset metrics& 16\\
$W$ & Sequence length considered & 4 \dots 256 \\
$X$ & Input data & $M \times W$ \\
$C$ & Number of VM classes & 2 \\
$N$ & batch size & 64 \\
$N_b$ & Number of model's blocks & $2 \dots N$ \\
$\mathcal{B}_n$ & n-th Block & $2 \dots N$ \\
$K_s$ & Kernel size in convolutions & 3\\
$s$ & stride (step) in convolutions & 2\\
$out$ & Output of the model & $C$ \\
\midrule
$\mathcal{FC}$ & Fully Connected layer & \\
$\mathcal{BN}$ & Batch-Norm layer \cite{DBLP:journals/corr/IoffeS15} & \\
$\mathcal{C}_{1D}$ & Convolution layer & \\
$\mathcal{A}_{ReLU}$ & ReLU Activation function & \\
$\mathcal{B}_{FFT}$ & Fast Fourier Transform layer & \\
\bottomrule[1pt]
\end{tabular}
}
\caption{Summary of symbols used in the model} \label{table:tab1}
\end{table}
Convolutional Neural Networks (CNNs) are a class of Artificial Neural Networks that have been proven very effective in the last years in solving complex tasks involving multimedia data, such as images and video.
They work well for identifying simple patterns within input data which can then be used to form more complex patterns in subsequent operations and finally be sufficiently informative to be used to perform the specific task at hand (\emph{i.e.\ } classification, regression, etc).
Inspired by deep convolutional neural networks, which are composed of several Convolutional layers followed by a final Fully Connected layer, our models are composed by a variable sequence of blocks and a final Fully Connected layer for the output classification.
VMs behavior is described as a set of time series each describing a specific metric (\emph{e.g.\ } Memory utilization, CPU utilization, Network traffic, etc\dots). For a complete list of the metric used in our experiments, the reader can refer to Table~\ref{tab:metrics} in the next section.
Since each metric is a series of samples taken across time, that are by nature one dimensional signals, we make use of a specific kind of Convolution, \emph{i.e.\ } 1-dimensional Convolutions, as elementary block of our network, and we consider each metric as an input channel on which we calculate the convolution operation independently.
A Convolution 1D is a deep learning linear operation used to extract features from one dimensional data like signals, with the aim to identify local patterns within a certain window, which is called kernel size ($K_s$). The kernel contains the learnable parameters that are used to carry out the operation. Because this kernel is being shifted along the time dimension with a certain step, called stride ($s$), the same calculation is executed on every patch of the data attended by the kernel, so that a pattern learned at one position can also be recognized at a different position, making 1D Convolution translation invariant.
In the simplest case, let's assume a Convolution 1D layer with input size $(N, C_{in}, L_{in})$ and output $(N, C_{out}, L_{out})$, then the values of the output tensor can be computed as follow:
\begin{eqnarray}
out\left[i,j\right]=b\left[j\right] + \sum\limits_{k=0}^{C_{in} - 1} w\left[j, k\right] \star input\left[i,k\right]
\end{eqnarray}
where $\star$ is the convolution operator, $N$ is the batch size, $C$ denotes the number of channels (for the first layer, the channels correspond to the VMs metrics), $L$ is the length of signal sequence (referred to as $input$). Furthermore, $w$ and $b$ are the learnable parameters of the layer, with shape $(C_{out}, C_{in}, K_s)$ and $(C_{out})$, respectively.
The stride ($s$) is an hyper-parameter of the 1-dimensional convolution that controls the step of the kernel: if greater than one, data is scanned with greater steps, hence less times, causing a decrease of the initial dimension length, an effect that can be seen as pooling, a well-known deep learning strategy to reduce data dimensions, useful especially when they are high.
In our models we also make use of Batch Normalization~\cite{DBLP:journals/corr/IoffeS15}, which is a popular operation that normalizes data across the batch dimension considered at a time, where the batch is a subsample of the dataset use in training phase to speed up gradient based optimization. Applying Batch Normalization after each convolutional layer helps deep networks to converge faster; and lastly, we use the non-linear activation function ReLU (Rectified Linear Unit) that is used to stabilize the gradient during training.
\subsection{DeepConv Model}
Each block of our DeepConv network is hence comprised of a Convolution1D layer with kernel size of 3 and stride of 2, followed by a Batch-Normalization layer and a ReLU activation function; this block is repeated a variable number of times depending on the input sequence length ($W$), with a minimum of two times. Following the last block, data is then flattened and applied to a Fully Connected layer which will output the class results.
The number of blocks of the model is given by the following:
\begin{equation}
\label{eq:N_b}
N_b = \max(\log_2(W) - 1 , 2)
\end{equation}
Where $W$ is the input sequence length of the data considered.
Varying the number of blocks is a consequence of a simple consideration: the model needs to have a final layer with neuron’s Receptive Field that can observe, and thus leverage information, over the entire input sequence; hence, given that we do experiments with different input sequence length, we need a flexible model that can adapt its architecture’s depth based on the input at hand.
The Receptive Field of a general neuron is nothing but the portion of input data that the neuron has access to and can influence its activation.
\begin{figure}
\includegraphics[width=\linewidth]{DeepConvimage.eps}
\caption{DeepConv model architecture.}
\label{fig:model}
\end{figure}
Our generic DeepConv model for an input sequence length of 32 timesteps is shown in Figure~\ref{fig:model}, where we outline how the data shape changes passing through each block of the network until the final Fully Connected layer; each metric is exhibited as a column of values, so that putting together $M$ metrics (16 in our case), we obtain an input shape of ($W, M$).
Adding also the Batch-Size dimension $N$, which considers Batch-Size input samples at a time to optimize the network, we obtain the final input shape for training of ($N, W, M$).
The final $out$ class probabilities given by the model is calculated as follows:
\begin{equation}
P(out)=\text{softmax}(out) = \frac{e^{out}}{\sum e^{out}}
\end{equation}
Where $out$ is the output of the model, which is computed as follows:
\begin{equation}
out = (\mathcal{FC} \circ \mathcal{B}_1 \circ \mathcal{B}_2 \circ \dots \circ \mathcal{B}_{N_b})(X)
\label{eq:out}
\end{equation}
with $\circ$ being the concatenation operator for Neural Network blocks, $N_b$ the number of blocks as defined in Eq.~\ref{eq:N_b}, $\mathcal{FC}$ the final Fully Connected layer, and each block defined as:
\begin{equation}
\mathcal{B}_n = (\mathcal{A}_{ReLU} \circ \mathcal{BN} \circ \mathcal{C}_{1D}) (X)
\label{eq:block}
\end{equation}
where $\mathcal{A}_{ReLU}$ indicates the activation function, $\mathcal{BN}$ a Batch Normalization layer, $\mathcal{C}_{1D}$ a Convolution1D layer and $X$ the input tensor.
\subsection{DeepFFT Model}
Given the periodicity of some of the signals involved, we derived a second model, namely DeepFFT, with the same architecture described so far used with time series data, but applied to data frequencies, hence transforming the data to the Fourier Domain before feeding the model.
If we compare DeepFFT with the previously described DeepConv model, the only difference lies in the presence of an additional initial layer which computes the Fast Fourier Transform (FFT) of each metric sequence and returns the magnitude of each frequency. We empirically found that using the magnitude leads to better results than other alternatives, such as using the phase or the raw real and imaginary parts.
Therefore the substantial difference between the two models is that DeepConv works in the time domain whilst DeepFFT works in the Fourier Domain, and we can simply derive its general formulation from Eq.~\ref{eq:out} by adding an initial FFT computation block over the input (\emph{i.e.\ } $\mathcal{B}_{FFT}$). The new layer is placed before every other block, as follows:
\begin{equation}
out = (\mathcal{FC} \circ \mathcal{B}_1 \circ \mathcal{B}_2 \circ \dots \circ \mathcal{B}_{N_b}) (\mathcal{B}_{FFT}(X))
\label{eq:outFFT}
\end{equation}
\subsection{Implementation Details}
We now discuss some details on our implementation of the DeepConv and DeepFFT models.
We implemented the models using the PyTorch framework (code is available at the following URL: \url{https://github.com/MatteoStefanini/DeepVM}).
Our implementation includes a pre-processing of the input data. Specifically, we normalize the data to have zero mean and unit variance in each channel. As the initial stream of data is partitioned into several input sequences with a length defined as the window $W$, we also leverage data augmentation techniques for sequences longer than 64 timesteps; specifically, we apply 75\% overlay between sequences, so that we can obtain more sequences for train and evaluation purpose.
We also balance the data to have the same number of samples within each class and we split the dataset in three parts, \emph{train}, \emph{validation} and \emph{test} sets, with the chosen fractions of 0.7, 0.2 and 0.1, respectively, and then used separately in the training, validation and test phases of the models.
For the training phase we use the Cross Entropy as loss function to evaluate the predictions of the models and back-propagate the error in training phase. In all our experiments we use the Adam optimizer \cite{DBLP:journals/corr/KingmaB14}, with default values, and, after a grid search on learning rate and weight decay hyperparameters, we found that a learning rate of 0.0003 and weight decay of 0.0012 work well in most scenarios. A more detailed sensitivity analysis with multiple scenarios, however, is left as as an open issue to address as a future work. We also reduce the learning rate by a factor of 0.6 when we observe that the validation loss does not decrease for 10 consecutive epochs.
After each training epoch, which is a pass over the entire training set, we evaluate the model performance in a validation phase (using the validation fraction of the dataset). In our experiments, we train each model for 110 epochs, observing that all models converge within this range. The validation phase identifies the best performing model that is used in the final performance evaluation with the test data never used before.
\section{Experimental results}\label{sec:results}
Experiments were carried out using a dataset owned by the University of Modena and Reggio Emilia, consisting of eight real-world cloud virtual machines, monitored for a few years and divided in two classes: Web-server and SQL-server. The list of metric fed into the classifier is provided in Table~\ref{tab:metrics}. The experimental setup used is consistent with the scenario described in~\cite{Canali:TCC18}: in particular, the classes of VMs and the metrics considered are the same.
\begin{table}[ht]
\resizebox{\columnwidth}{!}{
\begin{tabular}{|c|c|}
\toprule[1pt]
\textbf{Metric} & \textbf{Description} \\
\midrule
SysCallRate & Rate of system calls [req/sec] \\
CPU & CPU utilization [\%]\\
IdleCPU & Idle CPU fraction [\%]\\
I/O buffer & Utilization of I/O buffer [\%] \\
DiskAvl & Available disk space [\%]\\
CacheMiss & Cache miss [\%]\\
Memory & Physical memory utilization [\%]\\
UserMem & User-space memory utilization [\%]\\
PgOutRate & Rate of memory pages swap-out [pages/sec]\\
InPktRate & Rate of network incoming packets [pkts/sec]\\
OutPktRate & Rate of network outgoing packets [pkts/sec]\\
InByteRate & Rate of network incoming traffic [KB/sec]\\
OutByteRate & Rate of network outgoing traffic [KB/sec]\\
AliveProc & Number of processes in system\\
ActiveProc & Number of active processes in run queue\\
RunTime & Execution time \\
\bottomrule[1pt]
\end{tabular}
}
\caption{Metrics used for VM classification.} \label{tab:metrics}
\end{table}
In our experiments we aim to validate the ability of the proposed model to provide an accurate identification of VMs based on their behavior. In particular, we consider our proposed models DeepConv and DeepFFT compared with other state-of-the-art solutions such as the AGATE technique~\cite{Canali:TCC18} and a PCA-based clustering solution that exploits the correlation between the time series of the VMs metrics to characterize the VMs behavior. The PCA-based clustering has been used as the best representative of traditional clustering technique in~\cite{Canali:TCC18}.
The main metric in our analyses is the \emph{accuracy}, that is the percentage of samples identified correctly by our classifier. This metric has been used consistently in previous papers on VMs identification based on clustering, such as~\cite{Canali:TCC18}. However, in some cases we refer to the dual metric, that is the classification error percentage.
As our goal is to provide a fast and accurate identification of VMs, we consider important to evaluate how the identification accuracy changes as a function of the window length $W$. Specifically, in our experiments $W$ ranges between 4 and 256 timesteps, that in minutes correspond respectively to 20 and 1280 (slightly more than 21 hours).
As a first result we provide an evaluation of the two proposed models, shown in Figure~\ref{fig:accuracy}. In particular, we show the accuracy achieved by the DeepConv and DeepFFT models as a function of the window length $W$ (we recall that $W$ is measured in 5 minutes time steps). A first significant result of this evaluation is the overall performance of the considered model. Looking at the data, we observe that the accuracy of each model is always higher than 98.5\%, that is a fairly good performance for this type of problems (a quick comparison with the AGATE technique, shows that both proposed models outperforms consistently AGATE). Even more interesting, the DeepConv model achieves a perfect classification of VMs, especially for short time windows, that is the most interesting and challenging scenario because it enables the identification in near real-time of a VM. The DeepConv model worsen its performance as the window increases, likely due to the increased model complexity and the fixed size of convolutional kernels. The DeepFFT model presents an opposite behavior, with performance improving as the window grows. This effect can be explained by considering that a longer time window provides more information for the Fourier-transformed problem and the classifier can work with a more consistent description of the VMs behavior.
\begin{figure}
\includegraphics[width=\linewidth]{accuracy.png}
\caption{Models Accuracy.}
\label{fig:accuracy}
\end{figure}
Having obtained a first assessment of the Deep learning-based models, we compare our proposal with state-of-the-art solutions for VMs classification. In particular, we refer to the AGATE~\cite{Canali:TCC18} technique and to the PCA-based clustering.
For this comparison we refer to the error percentage and, for the AGATE technique, we also show the percentage of unclassified VMs (that are the VMs left in the gray-area for additional data collection).
The results of the study are reported in Table~\ref{table:results} and in Figure~\ref{fig:comparison}. In particular, Table~\ref{table:results} shows the error of the considered alternatives for a window $W$ ranging from 4 to 256 time steps. As we aim to refer as closely as possible to the original results in~\cite{Canali:TCC18}, in the results marked with a star (*), we approximate the results in hours with the closest possible window size. Boldface characters are used to outline, for each value of the window $W$, the best performing solution.
We observe that, as a general rule, the longer the observation window, the better each solution is performing. For the PCA-based clustering, the percentage of errors decreases from nearly 18\% to roughly 15\%. For the AGATE solution the error percentage is not monotone but remains in the range $[1.8\%, 2.4\%]$; however we observe a clear reduction of the un-classified VMs dropping from nearly 50\% to nearly 19\% as the window grows. The AGATE solution outperforms by one order of magnitude the previous solutions. However, the Deep learning-based models are a clear step ahead compared to the AGATE technique. We observe that, for every considered window the best performance are achieved by a deep learning-based model. Furthermore, for small windows (\emph{i.e.\ } $W \le 8$ time steps), the DeepConv model achieves 0\% errors with no need to postpone any classification using the notion of a gray-area.
\begin{table}[ht]
\resizebox{\columnwidth}{!}{
\begin{tabular}{|c c|c|c|c|c|c|c|c|}
\toprule[1pt]
\multicolumn{2}{|c|}{\textbf{Method}} & \textbf{4} & \textbf{8} & \textbf{16} & \textbf{32} & \textbf{64} & \textbf{128} & \textbf{256} \\
\midrule[1pt]
\multicolumn{2}{|c|}{PCA-based} & - & 17.9* & 17.5* & 16.1* & 15.4* & 15.2* & 15.1* \\
\midrule[1pt]
\multirow{2}{*}{AGATE} & error & - & 1.8* & 2.8* & 2.3* & 1.9* & 1.7* & 2.4* \\
& grey-area & - & 47.8* & 41.1* & 27.8* & 22.3* & 19.9* & 18.7* \\
\midrule[1pt]
\multicolumn{2}{|c|}{DeepFFT} & 1.32 & 1.24 & 1.45 & 1.04 & 0.79 & \textbf{0.53} & \textbf{0.43} \\
\midrule[1pt]
\multicolumn{2}{|c|}{DeepConv}& \textbf{0.00} & \textbf{0.00} & \textbf{0.03} & \textbf{0.64} & \textbf{0.76} & 1.27 & 1.17 \\
\bottomrule[1pt]
\end{tabular}
}
\caption{Error comparison with state-of-the-art.} \label{table:results}
\end{table}
The results in Table~\ref{table:results} are more clearly visible if we refer to Fig.~\ref{fig:comparison}. The reduction of the gray area (shown as a gray shadow in the figure) is quite evident and demonstrates how the AGATE technique becomes more effective with time, while the amount of errors (red line) remains in the order of few percentage points. On the other hand, the previously proposed PCA-based clustering (the yellow line) is clearly affected by an unacceptable amount of errors. However, the two deep learning-based models (blue and green lines) are a clear step ahead compared to the existing techniques as they provide even lower error rate than the AGATE alternative, without using a gray area.
\begin{figure}
\includegraphics[width=\linewidth]{comparison.png}
\caption{Comparison with state-of-the-art.}
\label{fig:comparison}
\end{figure}
\section{Related work}\label{sec:related}
The tasks of monitoring and managing Cloud data centers in a scalable way has received lots of attention over the last years~\cite{Aceto:2013,Belogazov:TSE}.
At the level of monitoring scalability, it is common to exploit aggregation and filtering techniques to reduce the amount of data before sending them to the data center controller.
When focusing on the monitoring of a Cloud infrastructure, scalability problems are typically addressed relying on some form of dimensionality reduction (\emph{e.g.\ } filtering or aggregation) that occurs before sending data to the cloud management function. Such dimensionality reduction is performed by ad-hoc software, typically in the form of a library or implemented as an data-collecting software agent. Example of this approach are provided in~\cite{Mehrotra:2011,Shao:2011,Azmandian:2011,Kertesz:2013}.
The actual aggregation policy may vary ranging from extraction of high-level performance indicators~\cite{Shao:2011}; to obtaining parameters that aggregate metrics from different system layers (hardware, OS, application and user) using Kalman filters~\cite{Mehrotra:2011}; to a simple linear combination of OS-layer metrics~\cite{Azmandian:2011}; up to systems that extract data from both the OS and the applications~\cite{Kertesz:2013,Andreolini:CIT11}.
The problem of cloud monitoring is addressed not just by research proposals but also by full-featured frameworks both commercial or open source (just to give a few names, Amazon cloud Watch is a commercial product, while MONASCA, the OpenStack monitor is open source)
However, the common limit of these solution is that each object taken into account in the monitoring process (either VM or physical node) is considered as independent form every other objects. In doing so, these proposals fail to take advantage form the similarities between objects exhibiting a similar behavior.
The management of Cloud systems is another critical topic, where several papers have been published, starting from the early examples based on the principles of autonomic computing applied to the the Cloud~\cite{Buyya2012}. Another interesting example of Cloud managment is represented by the Bobtail library~\cite{Bobtail}, that aims at supporting in each VM the identification of placement problems that result in high communication latency.
All these solutions rely on the assumption that the cloud user is willing to install a specific software layer on each VM, to overcome some limitations of the IaaS vision of the cloud. Our focus is completely different as we place no requirement on the VMs user and we comply completely with the IaaS vision.
Other studies aiming at improving the data center scalability have been proposed, such as~\cite{Canali:TGCN,Canali:NCCA15,Meo:PCV}. Our proposal can be integrated with these solution to improve the scalability of the cloud data center management.
Identifying similarities between VMs in a Cloud infrastructure is the key problem of our research. Several relevant works are discussed in the following. The research in~\cite{Zhang:IOT} aims at identifying similar VMs, but the similarity detection is limited to storage resources and its application scope is that of storage consolidation strategies. Similarly, the study in~\cite{Jayaram:2011} investigates similarities of VMs static images used in public cloud environments to provide insights for de-duplication and image-level cache management. Our approach focuses on a wider range of applications because we do not limit our analysis to a few resources for a limited purpose, but we consider a robust and general-purpose multi-resource similarity identification mechanism. A similar focus on similarity detection in VMs characterizes~\cite{Canali:TCC18}. Such study aims to address the trade-off between a fast identification of VMs and its accuracy using an adaptive approach. Our proposal addresses the same issue relying on a deep learning approach that ensures a very fast and accurate identification of the VMs.
Techniques derived from deep-learning have been recently proposed to address problems in the field of distributed infrastructure such as Cloud data centers and Fog systems. For example, the authors of~\cite{Liu:HFC17} propose a deep reinforced learning technique for the management of VMs allocation in Cloud data center.
Our proposal is completely orthogonal to the proposal in~\cite{Canali:NCCA15} and can be integrated with a class-based approach leveraging the VMs identification proposed in this paper. Another application of Deep-learning in distributed system is related to anomaly or attack detection. For example,~\cite{Diro:DAD18} proposes a deep-learning classifier for attack detection in a Fog system. While the basis of their deep-learning approach are similar to the ones in our proposal, to the best of our knowledge, ours is the first attempt to use deep learning to classify the behavior of VMs to support monitoring or management purposes, rather than aiming to attack or anomaly detection.
\section{Conclusions and future work}\label{sec:conclusions}
In this paper we focused on the scalability problems of a Cloud infrastructure, aiming to enable the adoption of solutions that improve scalability of monitoring and management through a classification of VMs that exhibit a similar behavior.
Existing solution for VMs clustering and classification are characterized by a trade-off between accurate VMs identification and timely response. Previous proposals aiming to address this problem exploited the notion of a gray area. While this approach is viable for the identification of VMs with a long life span, it is hard to apply in cloud infrastructures with on-demand VMs that are typically created and destroyed in just a few hours.
This limitation motivates our proposal of a different approach to the problem that, instead of an unsupervised clustering, exploits classifiers based on deep learning techniques to assign a newly deployed VMs to a cluster of already-known VMs. We propose two deep learning models for the classifier, namely DeepConv and DeepFFT based on convolution neural networks and Fast Fourirer Transform.
We validate our proposal using traces from a real cloud data center and we compare our classifiers with state-of-the-art solutions such as the AGATE technique (that exploits a gray area to adapt the observation time of each VM so that uncertainly classified VMs are not immediately assigned to a group) and a PCA-based clustering solution.
The results confirm that the deep learning models consistently outperforms every other alternative without the need to introduce a gray area to delay the classification. Even more interesting, the proposed classifiers can provide a fast and accurate identification of VMs. In particular, the DeepConv model provides a perfect classification with just a 4 samples of data (corresponding to 20 minutes of observation), making our proposal viable also to classify on-demand VMs that are characterized by a very short life span.
This paper is just a preliminary work in a new line of research that aims to apply deep learning techniques to the problems of cloud monitoring and management. Future works will focus on a more thorough evaluation of the proposed models, with additional sensititiy analyses with respect to the models parameters; on the proposal of additional classification models; and on the application of Generative Adversarial Networks to improve the quality of VMs identification in cases where the quality of data is lower than in the considered example (i.e., due to reduced number of metrics and presence of sampling errors).
\section*{Acknowledgement}
\bibliographystyle{IEEEtran}
|
1,116,691,500,913 | arxiv | \section{Introduction}
\label{sec:intro}
Supervised machine learning generally requires a large amount of labeled training data for an effective training of the underlying prediction model, especially when the prediction model is complex, \textit{e.g.}, Deep Neural Networks (DNN), where the number of parameters is at a scale of thousand millions. However in practice, collecting a sufficient number of manually labeled training samples may prove tedious, time consuming, even impractical, and therefore prohibitive, \textit{e.g.}, object edge detection, medical image segmentation, where a pixel-wise ground truth is needed. This is all the more true when the task is novel or rare. For example, for a fine-grained image classification task, for some rare categories we can only gather very limited number of image samples. Transfer Learning (TL) aims to leverage existing related source domain data for an informed knowledge transfer to a target task and thereby solve or mitigate this kind of "data starvation" problem to help the learning of a target task. As such, TL has received an increasing interest from several research communities \cite{SurveyTL} \cite{shao2015transfer}.
In this paper we consider the Inductive Transfer Learning (ITL)
problem \cite{SurveyTL}, which aims at learning an effective classification model for some target categories with few training samples, by leveraging knowledge from different but related source categories with far more training samples. Given the breakthrough of Deep Neural Networks (DNNs) in an increasing number of applications, a natural yet simple solution to this problem consists of fine-tuning a DNN which is pre-learned on some related source data for a given target classification task \cite{yosinski2014transferable}. However, although this fine-tuning process can inherit or preserve the knowledge learned during pre-training on source data, prior knowledge about the relatedness between source and target tasks is not explicitly explored. As a result, such a fine-tuning process may fall short to achieve an effective adaptation of a pre-trained DNN for a given target task , especially when the latter has very few labeled data.
A recent move is \cite{ge2017borrowing} on selective joint fine-tuning, which tackles this problem by first selecting relevant samples in the source domain, then performing a joint fine-tuning on target training data and the selected source data. Although the selective step ensures the fine-tuning process to use only source samples which are related to a given target domain, in the joint fine-tuning step the source classifier and target classifier are still trained as two different classifiers.
In this paper, we propose to explicitly account for the relatedness between source and target tasks and explore such prior knowledge through the design of a novel loss function, namely Optimal Transport loss (OT loss), which is minimized during joint training of the underlying neural network, in order to bridge the gap between the source and target classifiers. This results in a Joint Transfer Learning Network (JTLN). This JTLN can be built upon common Deep Neural Network structure.
In JTLN, the source data and target data go through same feature extraction layers simultaneously, and then separate into two different classification layers. The Optimal Transport loss is added between the two classification layers' outputs, in order to minimize the distance between two classifiers' predictions.
As the Optimal Transport loss is calculated with a pre-defined cost matrix, this JTLN can therefore incorporate different prior knowledge about the relations between source and target tasks by using different kinds of cost metrics. In this work, we show two examples of using the distance between category distributions as cost metric.
The contributions of this paper are threefold:
\begin{enumerate}
\item We propose a Joint Transfer Learning framework built upon existing Deep Neural Networks for Inductive Transfer Learning.
\item We extend the Wasserstein loss proposed in \cite{frogner2015learning} to a more general Optimal Transport loss for comparing probability measures with different length, and use it as a soft penalty in our JTLN.
\item We show two different ways of using the distance between category distributions as cost metric for OT loss. Experimental results on two ITL image classification datasets show that JTLN with these two cost metrics can achieve better performance than consecutive fine-tuning or simple joint fine-tuning without extra constraint.
\end{enumerate}
\section{Related work}
\label{sec:related_work}
A related problem in transfer learning (TL) is the Domain Adaptation (DA) problem, which is Transductive TL \cite{SurveyTL} and assumes that the source domain and target domain share the same label space, while following different probability distributions. Optimal Transport has already been successfully applied to DA in \cite{courty2016optimal}. Recently several deep joint learning methods have been proposed to solve this problem. For example in \cite{long2015learning} the authors propose to add multiple adaptation layers upon deep convolutional networks. Through these adaptation layers the mean embeddings of source distribution and target distribution are matched, therefore encouraging the network to learn a shared feature space for source domain and target domain. In \cite{long2016unsupervised} the authors extend the previous work by adding additional residual layers to adjust classifier mismatch between source domain and target domain. Although these methods work well for domain adaptation problems, their assumption that the source domain and target domain share a same label space and have a limited distribution discrepancy restrict the possibility of applying these methods for Inductive Transfer Learning.
Until recently most state-of-the-art ITL methods are based on shallow machine learning models \cite{shao2015transfer} \cite{kuzborskij2013from} \cite{kuzborskij2015transfer} \cite{li2017self}. For example in \cite{kuzborskij2015transfer} the authors propose to select relevant source hypotheses and feature dimensions through greedy subset selection. In \cite{li2017self} the authors propose to learn a high quality dictionary for low-rank coding across source domain and target domain for self-taught learning (which is ITL with only unlabeled samples in source domain). To the best of our knowledge, the proposed JTLN is the first work to tackle ITL with Deep Neural Networks and optimal transport theory.
The proposed JTLN has been inspired by a recent work on wasserstein loss \cite{frogner2015learning}. Frogner et al. proposed a wasserstein loss as a soft penalty for multi-label prediction. Although their wasserstein loss is calculated between predicted label vector and ground-truth label vector with the same length, the matrix scaling process used to calculate this wasserstein loss is actually not restricted to square transportation matrix. In this paper, we extend this wassertein loss to a more general Optimal Transport loss for label vectors with different length. As a result, the proposed JTLN enables the exploration of prior knowledge through the initial cost matrix and makes use of the OT loss as a soft penalty for bridging the gap between target and source classifier predictions.
\section{Joint Transfer Learning Network}
\label{sec:JTLN}
\subsection{Problem definition and the JTLN structure}
\label{subsec:problem_definition}
Assume that we have a small target training set $\mathcal{T}= \{ (\mathbf{x}^{t}_{i}, {y}^{t}_{i})\}^{n_t}_{i=1}$ of $n_t$ training samples, with $\mathbf{x}^t_i \in \mathcal{X}_t,~ y^t_i \in \mathcal{L}_t$ and $\mathcal{L}_t = \{ l^t_i \}^{L_t}_{i=1}$ is the target label set.
In Inductive Transfer Learning we are also given a larger source set $\mathcal{S} = \{ (\mathbf{x}^{s}_i, {y}^{s}_i) \}^{n_s}_{i=1}$ of $n_s$ samples, with $ \mathbf{x}^s_i \in \mathcal{X}_s,~ y^s_i \in \mathcal{L}_s$ and $\mathcal{L}_s = \{ l^s_i \}^{L_s}_{i=1}$ is the source label set. (No specific assumption is made for $\mathcal{X}_t$ and $\mathcal{X}_s$, meaning they can either be equal or not equal.)
We assume that $\mathcal{L}_s \neq \mathcal{L}_t$, this means that the target samples and source samples are from different concept categories. We also assume that a cost metric
$c(\cdot,\cdot) : \mathcal{L}_s \times \mathcal{L}_t \to \mathbb{R}$ could be found, which indicates the relationships between each pair of source category and target category (We will show in section \ref{subsec:cost-metric} two examples on defining this cost metric).
We build the Joint Transfer Learning Network upon common Deep Neural Networks (\emph{e.g.} Alexnet for image classification), an illustration of a JTLN built upon Alexnet can be found in Figure \ref{fig:JTLN_structure}. In JTLN the feature extraction layers are shared by source data and target data and give $f(\mathbf{x}_i)$ as the feature vector for input sample $\mathbf{x}_i$. Following are two fully-connected layers with different output dimensions, which are considered as the source classifier and target classifier. The output of the source classifier is noted as: $h_s(\mathbf{x}_i) = a(\mathbf{W}_s \cdot f(\mathbf{x}_i) + \mathbf{b}_s)$, where $a(\cdot)$ is the softmax activation function, $\mathbf{W}_s$ and $\mathbf{b}_s$ are layer weight and bias for source classifier. The output of the target classifier is noted similarly: $h_t(\mathbf{x}_i) = a(\mathbf{W}_t \cdot f(\mathbf{x}_i) + \mathbf{b}_t)$, with $\mathbf{W}_t$ and $\mathbf{b}_t$ the layer weight and bias for target classifier. Two cross-entropy losses are added for joint learning with source data and target data. The source cross-entropy loss term is defined as:
\begin{equation}
\label{eq:loss_src}
\frac{1}{n_s} \sum^{n_s}_{i=1} \ell_{ce}(h_s(\mathbf{x}^{s}_{i}), {y}^{s}_{i})
\end{equation}
where $\ell_{ce}(\cdot,\cdot)$ is the cross-entropy loss function. The target cross-entropy loss term is defined similarly:
\begin{equation}
\label{eq:loss_tar}
\frac{1}{n_t} \sum^{n_t}_{i=1} \ell_{ce}(h_t(\mathbf{x}^{t}_{i}), {y}^{t}_{i})
\end{equation}
To express our prior knowledge about the relatedness between source and target tasks, we propose to add a third Optimal Transport loss term for target data to restrict the distance between source classifier output and target classifier output, the OT loss term is noted as:
\begin{equation}
\label{eq:loss_w}
\frac{1}{n_t} \sum^{n_t}_{i=1} \ell_{ot}(h_s(\mathbf{x}^{t}_{i}),h_t(\mathbf{x}^{t}_{i}))
\end{equation}
where $\ell_{ot}(\cdot,\cdot)$ is the OT loss which will be defined in section \ref{subsec:OTloss}.
Therefore training with JTLN is a problem of minimizing the empirical risk which is a combination of the three loss terms shown above:
\begin{equation}
\label{eq:JTLN_ERM}
\min_{\Theta} \frac{1}{n_t} \sum^{n_t}_{i=1} \ell_{ce}(h_t(\mathbf{x}^{t}_{i}), {y}^{t}_{i}) + \frac{\lambda_s}{n_s} \sum^{n_s}_{i=1} \ell_{ce}(h_s(\mathbf{x}^{s}_{i}), {y}^{s}_{i}) + \frac{\lambda_{ot}}{n_t} \sum^{n_t}_{i=1} \ell_{ot}(h_s(\mathbf{x}^{t}_{i}),h_t(\mathbf{x}^{t}_{i}))
\end{equation}
where $\Theta$ denote the set of all parameters in JTLN
, $\lambda_s$ is the loss weight for source cross-entropy loss and $\lambda_{ot}$ is the loss weight for OT loss.
\begin{figure}[!htb]
\centering
\includegraphics[width=1\linewidth]{JTLN_structure_box.pdf}
\caption{The structure and data flow of a Joint Transfer Learning Network based on Alexnet}
\label{fig:JTLN_structure}
\end{figure}
\subsection{Optimal Transport Loss}
\label{subsec:OTloss}
In this paper we consider the discrete optimal transport problem.
As the output of the source classifier (\emph{i.e.} $h_s(\mathbf{x}_i)$) and that of the target classifier (\emph{i.e.} $h_t(\mathbf{x}_i)$) are outputs of softmax activation, meaning that $\sum^{L_s}_{j=1} (h_s(\mathbf{x}_i))_j = 1$ and $\sum^{L_t}_{j=1} (h_t(\mathbf{x}_i))_j = 1$. We can therefore consider them as two probability measures over their corresponding label space. We define:
\begin{equation}
\label{eq:define_mu_nu}
\begin{split}
\mu = h_s(\mathbf{x}_i) \in \mathcal{Y}_s \\
\nu = h_t(\mathbf{x}_i) \in \mathcal{Y}_t
\end{split}
\end{equation}
where $\mathcal{Y}_s=\mathbb{R}^{L_s}_{+}$ is the space of measures over the source label set $\mathcal{L}_s$ and $\mathcal{Y}_t = \mathbb{R}^{L_t}_{+}$ is the space of measures over the target label set $\mathcal{L}_t$. Note that $\mu$ and $\nu$ defined here are discrete probability measures, \emph{i.e.} histograms in the probability simplex $\Delta^{\mathcal{L}_s}$ and $\Delta^{\mathcal{L}_t}$.
Given a cost metric $c(\cdot,\cdot) : \mathcal{L}_s \times \mathcal{L}_t \to \mathbb{R}$, the optimal transport problem aims at finding the optimal transportation plan $\gamma_0$ which minimizes the cost to transport the mass in probability measure $\mu$ to match that in $\nu$.
The Kantorovich formulation \cite{kantorovich2006translocation} of this discrete optimal transport problem can be defined as follows:
\begin{equation}
\label{eq:ot_exact}
\begin{split}
&\gamma_0 = \argmin_{\gamma \in \Pi(\mu,\nu)} \langle \gamma, \mathbf{C} \rangle_F \\
\Pi(\mu,\nu) = \{ \gamma \in &(\mathbb{R}_+)^{L_s \times L_t} ~\vert~ \gamma \mathbf{1}_{L_t} = \mu,~ \gamma^\top \mathbf{1}_{L_s} = \nu \}
\end{split}
\end{equation}
where $\langle \cdot, \cdot \rangle_F$ is the Frobenius dot product. $\mathbf{C}$ is the cost matrix with $\mathbf{C}_{i,j} = c(l^s_i,l^t_j)$, the cost term $\mathbf{C}_{i,j}$, which can be interpreted as the cost to move a probability mass from $l^s_i$ to $l^t_j$ (In our case we can think it as the cost to transfer the prediction on source category $l^s_i$ to the prediction on target category $l^t_j$). $\Pi(\mu,\nu)$ is the set of all valid transportation plans, \emph{i.e.} the set of joint probability measures on $\mathcal{L}_s \times \mathcal{L}_t$ with $\mu$ and $\nu$ as marginals. $\mathbf{1}_d$ is a $d$-dimensional vector of ones.
If $\mathcal{L}_s = \mathcal{L}_t$, the wasserstein distance can be defined and in \cite{frogner2015learning} the authors use the wasserstein distance formulation as a loss function for their multi-label prediction problem. In our case, we have assumed $\mathcal{L}_s \neq \mathcal{L}_t$, we therefore define our loss function in a similar way directly based on the Optimal Transport formulation:
\begin{defn}
\label{def:otloss}
(Optimal Transport Loss) ~\emph{For any source classifier $h_s: \mathcal{X} \to \Delta^{\mathcal{L}_s}$, and any target classifier $h_t: \mathcal{X} \to \Delta^{\mathcal{L}_t}$, given input $\mathbf{x} \in \mathcal{X}$, and a cost metric $c(\cdot,\cdot) : \mathcal{L}_s \times \mathcal{L}_t \to \mathbb{R}$, the Optimal Transport Loss is defined as:
\begin{equation}
\label{eq:otloss}
\ell_{ot}(h_s(\mathbf{x}),h_t(\mathbf{x})) \triangleq \inf_{\gamma \in \Pi(h_s(\mathbf{x}), h_t(\mathbf{x}))} \langle \gamma, \mathbf{C} \rangle_F
\end{equation}
where $\langle \cdot, \cdot \rangle_F$ is the Frobenius dot product. $\mathbf{C}$ is the cost matrix with $\mathbf{C}_{i,j} = c(l^s_i,l^t_j)$.
$\Pi(h_s(\mathbf{x}),h_t(\mathbf{x}))$ is the set of valid transportation plans defined as:
\begin{equation}
\label{eq:otloss_pi}
\Pi(h_s(\mathbf{x}),h_t(\mathbf{x})) = \{ \gamma \in (\mathbb{R}_+)^{L_s \times L_t} ~\vert~ \gamma \mathbf{1}_{L_t} = h_s(\mathbf{x}),~ \gamma^\top \mathbf{1}_{L_s} = h_t(\mathbf{x}) \}
\end{equation}
where $\mathbf{1}_d$ is a $d$-dimensional vector of ones.}
\end{defn}
This Optimal Transport Loss in Definition \ref{def:otloss} can therefore be calculated by solving the discrete optimal transport problem shown in \eqref{eq:ot_exact}. Problem in \eqref{eq:ot_exact} is a linear programming problem and can be solved with combinatorial algorithms, \emph{e.g.} the simplex methods and its network variants \cite{courty2016optimal}. However, the computational complexity is shown to be $O((L_s+L_t)L_s L_t log(L_s+L_t))$ at best \cite{ahuja1993network}. This limits the usage of this formulation for large scale dataset.
Recently, Cuturi et al. \cite{cuturi2013sinkhorn} \cite{benamou2015iterative} proposed an entropy regularized optimal transport problem, which can be efficiently solved by iterative Bregman Projections. The discrete optimal transport problem with entropy regularization can be defined as:
\begin{equation}
\label{eq:ot_ereg}
\begin{split}
\gamma_0 &= \argmin_{\gamma \in \Pi(\mu,\nu)} \langle \gamma, \mathbf{C} \rangle_F - \frac{1}{\lambda}H(\gamma)\\
\Pi(\mu,\nu) = \{ \gamma &\in (\mathbb{R}_+)^{L_s \times L_t} ~\vert~ \gamma \mathbf{1}_{L_t} = \mu,~ \gamma^\top \mathbf{1}_{L_s} = \nu \} \\
&H(\gamma) = \sum_{i,j} \gamma_{i,j} \log \gamma_{i,j}
\end{split}
\end{equation}
where $H(\gamma)$ is the entropy of $\gamma$ and $-\frac{1}{\lambda}$ is the regularization weight.
This entropy regularization forces the solution of \eqref{eq:ot_ereg} to be smoother as $\frac{1}{\lambda}$ increase, \emph{i.e.}, as $\frac{1}{\lambda}$ increases, the sparsity of $\gamma^{\lambda}_0$ decreases.
This non-sparsity of the solution helps to stabilize the computation by making the problem strongly convex with a unique solution.
The advantage of this entropic regularized OT problem is that its solution is a diagonal scaling of $e^{-\lambda\mathbf{C}-1}$, where $e^{-\lambda\mathbf{C}-1}$ is the element-wise exponential matrix of $-\lambda \mathbf{C}-1$. The solution to this diagonal scaling problem can be found by iterative Bregman projections \cite{benamou2015iterative}.
In this work we calculate the approximation to the Optimal Transport Loss in Definition \ref{def:otloss} by solving the entropic regularized optimal transport problem defined in \eqref{eq:ot_ereg} using iterative Bregman projections as shown in \cite{benamou2015iterative}. The computation of this approximate OT Loss is defined in Algorithm \ref{algo:aproxi_otloss}, where $./$ means element-wise division.
\begin{algorithm}
\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\caption{Computation of the approximate OT Loss}
\label{algo:aproxi_otloss}
\begin{algorithmic}[1]
\REQUIRE
$h_s(\mathbf{x}) \in \Delta^{\mathcal{L}_s}$, $h_t(\mathbf{x}) \in \Delta^{\mathcal{L}_t}$, $\lambda$, $\mathbf{C}$ \\
\STATE
Initialize: $\mathbf{u} = \mathbf{1}_{L_s} / L_s $, ~$\mathbf{v} = \mathbf{1}_{L_t} / L_t$, ~$\mathbf{K} = e^{-\lambda\mathbf{C}-1}$
\WHILE{$\mathbf{u}$ has not converged}
\STATE
$\mathbf{v} = h_t(\mathbf{x}) ./ (\mathbf{K}^{\top} \mathbf{u})$
\STATE
$\mathbf{u} = h_s(\mathbf{x}) ./ (\mathbf{K} \mathbf{v})$
\ENDWHILE
\STATE
$\ell_{ot}(h_s(\mathbf{x}),h_t(\mathbf{x})) = \langle diag(\mathbf{u}) \cdot \mathbf{K} \cdot diag(\mathbf{v}), \mathbf{C} \rangle_F$
\end{algorithmic}
\end{algorithm}
\subsection{Back-propagation with OT Loss}
\label{subsec:otloss-backprop}
The empirical risk minimization problem defined in Equation \ref{eq:JTLN_ERM} is normally solved by a gradient descent algorithm, therefore the gradient of each loss term with respect to their corresponding inputs should be expressed analytically for back-propagation. As in \cite{frogner2015learning} we define the Lagrange dual problem of LP problem \ref{eq:otloss} as :
\begin{equation}
\label{eq:ot_dual}
\begin{split}
\ell_{ot}(h_s(\mathbf{x}), h_t(\mathbf{x})) = \sup_{\alpha,\beta \in \mathcal{C}} {\alpha}^{\top} h_s(\mathbf{x}) + \beta^{\top} h_t(\mathbf{x}) \\
\mathcal{C} = \{ (\alpha,\beta) \in \mathbb{R}^{L_s \times L_t} : \alpha_i + \beta_j \leq \mathbf{C}_{i,j}\}
\end{split}
\end{equation}
The $\mathbf{u}$ and $\mathbf{v}$ defined in Algorithm \ref{algo:aproxi_otloss} can be expressed as: $\mathbf{u} = e^{\lambda \alpha}$ and $\mathbf{v} = e^{\lambda \beta}$. As \ref{eq:otloss} is a linear program, at an optimum the values of the dual and the primal are equal, therefore the dual optimal $\alpha$ is a sub-gradient of the OT loss with respect to $h_s(\mathbf{x})$ and $\beta$ is a sub-gradient of the OT loss with respect to $h_t(\mathbf{x})$.
The gradient of OT loss with respect to its two arguments can therefore be expressed as follows and can be easily computed with the optimal scaling vectors $\mathbf{u}$ and $\mathbf{v}$ after matrix scaling with Algorithm \ref{algo:aproxi_otloss}:
\begin{equation}
\label{eq:otloss_gradients}
\begin{split}
\frac{\partial \ell_{ot}(h_s(\mathbf{x}),h_t(\mathbf{x}))}{\partial h_s(\mathbf{x})} = \alpha = \frac{\log \mathbf{u}}{\lambda} - \frac{\log \mathbf{u}^{\top} \mathbf{1}_{L_s}}{\lambda L_s} \mathbf{1}_{L_s} \\
\frac{\partial \ell_{ot}(h_s(\mathbf{x}),h_t(\mathbf{x}))}{\partial h_t(\mathbf{x})} = \beta = \frac{\log \mathbf{v}}{\lambda} - \frac{\log \mathbf{v}^{\top} \mathbf{1}_{L_t}}{\lambda L_t} \mathbf{1}_{L_t}
\end{split}
\end{equation}
Note that $\alpha$ and $\beta$ are defined up to a constant shift, \emph{i.e.} any upscaling of the vector $\mathbf{u}$ can be paired with a corresponding downscaling of the vector $\mathbf{v}$ (and vice versa) without altering the matrix $\gamma_0$, therefore the second terms in Equation \ref{eq:otloss_gradients} are added to ensure that $\alpha$ and $\beta$ are tangent to their corresponding simplex.
\subsection{Choosing the cost metric}
\label{subsec:cost-metric}
In Definition \ref{def:otloss}, the cost metric $c(\cdot,\cdot)$ can be interpreted as the cost to transfer the prediction on a source category to that of a target category. This cost metric embodies prior knowledge which describes the relatedness between each pair of source and target categories. The choice of this cost metric is crucial in JTLN for having a better joint learning performance.
A reasonable choice is to consider that the sample features in each category follow a probability distribution in a joint feature space, and to define this cost metric as the distance between two distributions. For example given a source category $l^s_i$ and a target category $l^t_j$, and a feature extractor $f(\mathbf{x})$ for sample $\mathbf{x}$. Suppose $\{f(\mathbf{x}^s) ~\vert~ \forall (\mathbf{x}^s,y^s),~ y^s = l^s_i \}$ follows the distribution $\mu_s$, and $\{f(\mathbf{x}^t) ~\vert~ \forall (\mathbf{x}^t,y^t),~ y^t = l^t_j \}$ follows the distribution $\mu_t$, our goal is to define a distance $d(\mu_s,\mu_t)$ between the two distributions as the cost metric for OT loss: $c(l^s_i,l^t_j) = d(\mu_s,\mu_t)$. To simplify the notations, in the following of this section we will use $\mathbf{x}^s$ and $\mathbf{x}^t$ instead of $f(\mathbf{x}^s)$ and $f(\mathbf{x}^t)$ to represent samples from the two distributions.
This definition implies that if the distribution of a target category and that of a source category lie close to each other in the feature space, their corresponding labels are more probably related and therefore cost less effort to transfer the prediction of one to that of the other.
There are various ways to calculate the distance between two distributions. One way is to use the two-sample test with Multi-Kernel Maximum Mean Discrepancy (MK-MMD) as test statistics, which is successfully applied for solving domain adaptation problems \cite{long2015learning}. Another way is to use Optimal Transport and employ a basic distance metric (\emph{e.g.} Euclidean distance) as the cost metric. In the following we show details on how to apply these two methods as cost metrics for evaluating the distance between a given pair of source and target categories.
\subsubsection{MK-MMD as cost metric}
\label{subsubsec:mkmmd}
Given samples from two distributions $\mu_s$ and $\mu_t$, a two-sample test determines whether to reject the null hypothesis $H_0: \mu_s=\mu_t$, based on the value of a test statistics measuring the distance between the samples. One choice of the test statistics is the maximum mean discrepancy (MMD), which is a distance between embeddings of the probability distributions in a reproducing kernel Hilbert space. Here we make use of the multi-kernel variant of MMD (MK-MMD) proposed in \cite{gretton2012optimal}, which maximizes the two-sample test power and minimizes the Type II error (\emph{i.e.}, the probability of wrongly accepting $H_0$ when $\mu_s \neq \mu_t$), given an upper bound on Type I error (\emph{i.e.}, the probability of wrongly rejecting $H_0$ when $\mu_s \neq \mu_t$), by leveraging different kernels for kernel embeddings.
Let $\mathcal{H}_k$ be a reproducing kernel Hilbert space (RKHS) endowed with a characteristic kernel $k$. The mean embedding of distribution $\mu_s$ in $\mathcal{H}_k$ is a unique element $\phi_k(\mu_s) \in \mathcal{H}_k$ such that $\mathbf{E}_{\mathbf{x} \sim \mu_s}g(\mathbf{x}) = \langle g(\mathbf{x}), \phi_k(\mu_s) \rangle_{\mathcal{H}_k}$ for all $g \in \mathcal{H}_k$. The MMD between probability distributions $\mu_s$ and $\mu_t$ is defined as the RKHS distance between the mean embeddings of $\mu_s$ and $\mu_t$. The squared formulation of MMD can be defined as:
\begin{equation}
\label{eq:mk-mmd}
d^2_k(\mu_s,\mu_t) = \parallel \phi_k(\mu_s) - \phi_k(\mu_t) \parallel^2_{\mathcal{H}_k} = \mathbf{E}_{\mathbf{x}^s{\mathbf{x}^s}'}k(\mathbf{x}^s,{\mathbf{x}^s}') + \mathbf{E}_{\mathbf{x}^t{\mathbf{x}^t}'}k(\mathbf{x}^t,{\mathbf{x}^t}') - 2\mathbf{E}_{\mathbf{x}^s \mathbf{x}^t}k(\mathbf{x}^s,\mathbf{x}^t)
\end{equation}
where $\mathbf{x}^s,{\mathbf{x}^s}' ~~\iidsim~~ \mu_s$ and $\mathbf{x}^t,{\mathbf{x}^t}' ~~\iidsim~~ \mu_t$. With $\phi_k$ an injective map, \emph{i.e.} $k$ is a characteristic kernel, the MMD is a metric on the space of Borel probability measures, \emph{i.e.} $d_k(\mu_s,\mu_t) = 0$ if and only if $\mu_s = \mu_t$.
The characteristic kernel $k$ is defined as the convex combination of $m$ PSD (positive semi-definite) kernels $k_u$:
\begin{equation}
\label{eq:mkmmd-kernels}
\mathcal{K} \triangleq \{ k=\sum^m_{u=1}\beta_uk_u \vert \sum^m_{u=1}\beta_u=1, \beta_u \geqslant 0, \forall u \}
\end{equation}
where the constraints on coefficients $\{\beta_u\}$ are imposed to guarantee that the derived multi-kernel $k$ is characteristic. This multi-kernel $k$ can leverage different kernels to enhance the power of two-sample test.
For computation efficiency, we adopt the unbiased estimate of MK-MMD which can be computed with linear complexity:
\begin{equation}
\label{eq:mk-mmd-estim}
d^2_k(\mu_s,\mu_t) = \frac{2}{n}\sum^{n/2}_{i=1}g_k(\mathbf{z}_i)
\end{equation}
where $\mathbf{z}_i \triangleq (\mathbf{x}^s_{2i-1}, \mathbf{x}^s_{2i}, \mathbf{x}^t_{2i-1}, \mathbf{x}^t_{2i})$ is a random quad-tuple sampled from $\mu_s$ and $\mu_t$
, and we evaluate each quad-tuple with $g_k(\mathbf{z}_i) \triangleq k(\mathbf{x}^s_{2i-1}, \mathbf{x}^s_{2i}) + k(\mathbf{x}^t_{2i-1},\mathbf{x}^t_{2i}) - k(\mathbf{x}^s_{2i-1}, \mathbf{x}^t_{2i}) - k(\mathbf{x}^s_{2i},\mathbf{x}^t_{2i-1})$.
\subsubsection{OT as cost metric}
\label{subsubsec:ot-costmetric}
Consider $\mu_s$ and $\mu_t$ as two empirical distributions defined by their corresponding discrete samples:
\begin{equation}
\label{eq:discrete_distributions}
\mu_s = \sum^{n_s}_{i=1}p^s_i \delta_{\mathbf{x}^s_i},~~ \mu_t = \sum^{n_t}_{i=1}p^t_i \delta_{\mathbf{x}^t_i}
\end{equation}
where $\delta_{\mathbf{x}_i}$ is the Dirac function at location $\mathbf{x}_i$. $p^s_i \in \Delta^{n_s}$ and $p^t_i \in \Delta^{n_t}$ are probability masses associated to the $i$-th sample. We can therefore define a discrete optimal transport problem with entropy regularization as in equation \eqref{eq:ot_ereg}:
\begin{equation}
\label{eq:otmetric_def}
\begin{split}
\gamma_0 &= \argmin_{\gamma \in \Pi(\mu_s,\mu_t)} \langle \gamma, \mathbf{C} \rangle_F - \frac{1}{\lambda}H(\gamma)\\
\Pi(\mu_s,\mu_t) = \{ \gamma &\in (\mathbb{R}_+)^{n_s \times n_t} ~\vert~ \gamma \mathbf{1}_{n_t} = \mu_s,~ \gamma^\top \mathbf{1}_{n_s} = \mu_t \} \\
&H(\gamma) = \sum_{i,j} \gamma_{i,j} \log \gamma_{i,j}
\end{split}
\end{equation}
We define the cost metric in Equation \eqref{eq:otmetric_def} as squared Euclidean distance between two samples $\mathbf{C}_{i,j} = \parallel \mathbf{x}^s_i - \mathbf{x}^t_j \parallel^2_2$, and define the distance between the two distributions as $d(\mu_s,\mu_t) = \langle \gamma_0, \mathbf{C} \rangle_F$. This distance can be computed using the same matrix scaling procedure as in Algorithm \ref{algo:aproxi_otloss}.
\section{Experiments}
\label{sec:exps}
In this section we show the experiments of our proposed JTLN built upon Alexnet for Inductive Transfer Learning (ITL) with fine-grained image classification datasets.
We make use of the FGVC-Aircraft Dataset \cite{maji13fine-grained}. The dataset contains 10000 images of aircraft, with 100 images for each of 100 different aircraft model variants. These aircraft variants are from 30 different manufacturers. We build our Inductive Transfer Learning datasets upon this dataset by choosing the model variants from one manufacturer as the target domain categories, and consider the rest of the model variants as source domain categories. The images in source categories are all used for JTLN training, while the images in target categories are split into a subset for training and a subset for testing. We choose the two manufacturers with the most model variants to form two different ITL datasets, the characteristics of these ITL datasets are listed in Table \ref{tab:dataset_aircraft}, where the dataset name is indexed by the target manufacturer name.
Note that all experiments perform fine-tuning based on an Alexnet model pre-trained with the ImageNet database. Further to the recommendations in \cite{yosinski2014transferable}, we fix the first three convolution layers since the features learned by these layers are general for different tasks, fine-tune the 4-th and 5-th convolutional layers with a small learning rate because the features learned in these two layers are less general, and fine-tune the 6-th and 7-th fully connected layers with a larger learning rate because the features learned in these layers are more task specific. The classification layers (fc8 with softmax) are trained from scratch.
We compare our proposed JTLN with three baseline methods: the first one consists of fine-tuning only with target training samples; the second one is the commonly adopted method, which first fine-tunes the pre-trained model with source samples, then continues fine-tuning with target training samples; the third baseline performs fine-tuning jointly with source samples and target training samples without applying the OT loss.
We also show the results of two variants of the proposed JTLN:
(1) JTLN (fc7MKMMD) is JTLN using MK-MMD as cost metric as shown in section \ref{subsubsec:mkmmd}, and using the fc7-layer output of the Alexnet model pre-trained on ImageNet as features for the computation of the MK-MMD distances.
(2) JTLN (fc7OT) is JTLN using OT as cost metric as shown in section \ref{subsubsec:ot-costmetric}, using the same Alexnet fc7-layer output as features.
\begin{table*}[!htb] \footnotesize
\begin{minipage}{0.43\textwidth}
\centering
\caption{ITL Datasets with FGVC-Aircraft images}
\label{tab:dataset_aircraft}
\begin{tabular}{l|cc}
\Xhline{1.2pt}
\Gape[6pt][4pt]{\tabincell{l}{Dataset properties}} & Boeing & Airbus \\
\Xhline{1pt}
\Gape[4pt][4pt]{\tabincell{l}{N\textsuperscript{o} of target \\ categories}} & 22 & 13 \\
\hline
\Gape[4pt][4pt]{\tabincell{l}{N\textsuperscript{o} of target \\ training images}} & 1466 & 867 \\
\hline
\Gape[4pt][4pt]{\tabincell{l}{N\textsuperscript{o} of target \\ testing images}} & 734 & 433 \\
\hline
\Gape[4pt][4pt]{\tabincell{l}{N\textsuperscript{o} of source \\ categories}} & 78 & 87 \\
\hline
\Gape[4pt][4pt]{\tabincell{l}{N\textsuperscript{o} of source \\ images}} & 7800 & 8700 \\
\Xhline{1.2pt}
\end{tabular}
\end{minipage}
\hspace{18pt}
\begin{minipage}{0.5\textwidth}
\centering
\caption{Experimental Results on the ITL Datasets (results are multi-class classification accuracy)}
\label{tab:results_aircraft}
\begin{tabular}{l|cc}
\Xhline{1.2pt}
\Gape[8pt][6pt]{Methods} & Boeing & Airbus \\%& Embraer \\
\Xhline{1pt}
\Gape[4pt][4pt]{\tabincell{l}{Finetuning on target}} & 0.4796 & 0.4965 \\%& 0.7468\\
\hline
\Gape[4pt][4pt]{\tabincell{l}{Consecutive finetuning\\on source+target}} & 0.5286 & 0.545 \\%& \\
\hline
\Gape[4pt][4pt]{\tabincell{l}{Joint finetuning\\on source+target}} & 0.5395 & 0.5497 \\%& \\
\hline
\Gape[4pt][4pt]{\tabincell{l}{JTLN (fc7MKMMD)}} & 0.5422 & \textbf{0.5982} \\%& 0.7682\\
\hline
\Gape[4pt][4pt]{\tabincell{l}{JTLN (fc7OT)}} & \textbf{0.5436} & 0.5704 \\
\Xhline{1pt}
\end{tabular}
\end{minipage}
\end{table*}
The classification accuracies of these methods for the two ITL datasets are shown in table \ref{tab:results_aircraft}. We can see that with fc7MKMMD as cost metric, JTLN for ITL-Airbus successfully improved the performance of joint fine-tuning by 5 points. JTLNs (fc7MKMMD and fc7OT) on ITL-Boeing also improved in comparison with joint fine-tuning. However, the performance increase is not as high as that with the ITL-Airbus dataset. We believe this can be partially explained by the fact that ITL-Boeing has less source categories and less source samples than ITL-Airbus.
\section{Conclusion and Future work}
\label{sec:conlu}
In this paper we have proposed a novel Joint Transfer Learning Network (JTLN) for Inductive Transfer Learning. By adding an Optimal Transport loss (OT loss) between the source and target classifier predictions during the joint fine-tuning process, the proposed JTLN can effectively learn useful knowledge for target tasks from source data. Another advantage of JTLN is the possibility of incorporating prior knowledge about the relatedness between the target and source categories by using different cost metric for OT loss. We show experimental results of JTLN with two different cost metrics in comparison with three baseline methods on two Inductive Transfer Learning datasets. The results verify the effectiveness of the proposed JTLN.
Future work includes further exploration of different cost metrics for OT loss. An interesting
variant of JTLN could be to dynamically learn the cost matrix along the fine-tuning process while using the current fine-tuned model as feature extractor.
{\small
\bibliographystyle{unsrt}
|
1,116,691,500,914 | arxiv | \subsection*{Acknowledgments}
The authors would like to thank CAPES, CNPQ, FUNAPE-GO, and FAPESP for partial financial support.
|
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